[    {        "title": "2303.14426v1.Muon_spin_relaxation_and_emergence_of_disorder_induced_unconventional_dynamic_magnetic_fluctuations_in_Dy___2__Zr___2__O___7__.pdf",        "content": "arXiv:2303.14426v1  [cond-mat.str-el]  25 Mar 2023Muon spin relaxation and emergence of disorder-induced unc onventional dynamic\nmagnetic fluctuations in Dy 2Zr2O7\nSheetal1,†, Pabitra K. Biswas2,$, K. Yokoyama2, D. T. Adroja2and C. S. Yadav1∗\n1School of Physical Sciences, Indian Institute of Technolog y Mandi, Mandi-175075 (H.P.), India\n2ISIS Pulsed Neutron and Muon Source, STFC Rutherford Applet on Laboratory,\nHarwell Campus, Didcot, Oxfordshire OX11 0QX, United Kingd om∗\nThe disordered pyrochlore oxide Dy 2Zr2O7shows the signatures of field-induced spin freezing\nwith remnant zero-point spin-ice entropy at 5 kOe magnetic fi eld. We have performed zero-field and\nlongitudinal field Muon spin relaxation ( µSR) studies on Dy 2Zr2O7. Our zero field studies reveal\nthe absence of both long-range ordering and spin freezing do wn to 62 mK. The µSR relaxation\nrate exhibits a temperature-independent plateau below 4 K, indicating a dynamic ground state of\nfluctuating spins similar to the well-known spin ice system D y2Ti2O7. The low-temperature spin\nfluctuations persist in the longitudinal field of 20 kOe as wel l and show unusual field dependence of\nthe relaxation rate, which is uncommon for a spin-liquid sys tem. Our results, combined with the\nprevious studies do not show any evidence of spin ice or spin g lass ground state, rather point to a\ndisorder-induced dynamic magnetic ground state in the Dy 2Zr2O7material.\nI. INTRODUCTION\nIn geometrically frustrated magnetic systems, the\ncompeting magnetic interactions combined with the\nlattice geometry prevent the system from establishing\nthe long-range order. The corner-sharing tetrahedral\narrangements of the magnetic rare earth ions in the\npyrochlore lattice R 2B2O7(R = rare-earth, B =\ntransition metal) manifest the development of exotic\nmagnetic states, which have been extensively studied\nin the literature [1–3]. Variations at the R site can\nmodulate the coherent magnetic ground state from a\nclassical spin ice (Ho 2Ti2O7, Dy2Ti2O7) to a quantum\nspin ice (Ce 2Zr2O7and Nd 2Zr7O7), which can be\nattributed to rather different crystal-field ground states\nof R sites in the D 3dlocal symmetry. [4–6]. However,\nwhen structural disorder is introduced, the physical\nproperties of the R ions in frustrated magnetic systems\nare strongly influenced. Furthermore, recent theoretical\nand experimental research probing disorder has provided\ninteresting insight on the spin ices, spin liquids and XY\npyrochlores [7–9]. The emergence of disorder-induced\nmagnetic states in frustrated magnets is one of the\nmost important and current aspects of magnetism. The\ndisorder often acts as a significant perturbation force\nthat deviates a spin system far away from its coherent\nquantum ground state. This is particularly true for spin\nice pyrochlore oxides R 2B2O7, in which disorder effects,\nsuch as stuffing and oxygen non-stoichiometry [9–11], or\nsimply structural distortions [7, 12], or disorder due to\nthe mixed B-site [13], are found to have a propounding\nimpact on their magnetic ground state.\n∗ †Current Address: J¨ ulich Centre for Neutron Science JCNS at\nMaier-Leibnitz Zentrum (MLZ), Germany\n;$Deceased author\n;∗Email: shekhar@iitmandi.ac.inFor instance, Dy 2Ti2O7is a clean pyrochlore spin\nice system, whereas Dy 2Zr2O7forms in a chemically\ndisordered structural phase and develops a dynamic\nground state down to 50 mK without zero-point entropy\n[14]. Similar behavior is reported for another spin-ice\nmaterial Ho 2Ti2O7, where the chemical alteration of the\npyrochlore phase precludes the development of spin-ice\ncorrelations in Ho 2Zr2O7[15]. The Tb 2Ti2O7remains\nin a dynamic magnetic state [16, 17] whereas a subtle\ndisorder in the lattice induces spin-glass behavior in\nTb2Hf2O7[18]. Theoretical studies show that varying\nthe degree of disorder can cause a system to transition\nto different magnetic states, providing a better under-\nstanding of the physics of these structurally disordered\nfrustrated systems [19]. While a significant chemical dis-\norder would likely preclude the development of spin ice\ncorrelation [14, 15], a subtly tuned level of disorder may\nlead to a quantum spin liquid (QSL) ground state [3, 19].\nIn a similar manner, it has been found that the disorder\ndue to the site mixture of Mg2+/Ga3+in the recently\ndiscovered triangular antiferromagnet YbMgGaO 4also\nplays an important role in the formation of the QSL\nground state in this compound [20, 21]. These studies\nshow how disorder can lead to a plethora of intriguing\nphenomena in highly frustrated systems. The tuning of\ndisorder offers an intriguing opportunity to investigate\nmuch new physics in frustrated magnets.\nAlong this line, Dy 2Zr2O7has shown the possible re-\nalization of field-induced spin ice state by stabilizing the\nfinite residual entropy of R[ln2-(1/2)ln(3/2)] at 5 kOe\nmagnetic field, equivalent to Dy 2Ti2O7[22]. Unlike the\nspin ice Dy 2Ti2O7, the zero field heat capacity and neu-\ntron studies by Ramon et al.suggests the highly dy-\nnamic spin state without any residual entropy down to\n50 mK [14]. Additionally, the ac susceptibility measure-\nments show the signature of weak spin freezing at ∼1\nK in the highly dynamic ground state [14]. The neu-\ntron diffraction study in the presence of external mag-2\nnetic field by the same group reveals the development of\nmagnetic peak for H≥2 kOe, suggesting the long-range\norder state [23]. This result does not conform with other\nmacroscopic measurements and thus complicates the un-\nderstanding of the origin of observed behavior and true\nground state of Dy 2Zr2O7. Therefore, we have carried\nout zero-field and longitudinal field muon spin relaxation\n(µSR) measurements to understand the impact of the\ndisorder on the low-temperature magnetic state and to\ndetermine whether or not the magnetic field transits the\nsystem to spin ice or the magnetically long-rangeordered\nstate.\nII. EXPERIMENTAL TECHNIQUES\nPolycrystalline Dy 2Zr2O7was prepared by solid-state\nreaction method as described previously [24]. The phase\npurity and crystal structure was characterized by per-\nforming the Rietveld refinement of the powder X-ray\ndiffraction data. The obtained structural parameters\nwere in close agreement with the previously reported pa-\nrameters for the compound and confirmed the formation\nof the disordered pyrochlore phase. The µSR data was\ncollectedattheISISpulsedmuonfacility,RutherfordAp-\npleton Laboratory,United Kingdom [25] using HiFi spec-\ntrometer in the temperature range 62 mK and 4 K using\na dilution refrigerator at various magnetic fields ranging\nbetween 0 - 20 kOe. Additionally, the high-temperature\ndata up to 300 K were collected by transferring the sam-\nple to4He cryostat.\nIII. RESULTS AND DISCUSSSION\nFig. 1 show the selected zero-field µSR asymmetry\nspectra in the temperature range 62 mK and 300 K. The\ninitial asymmetry shows a value of ∼0.25 atT= 300 K\nand relaxes with time owing to the rapid fluctuations of\nthe internal field of Dy3+moments in the paramagnetic\nphase. The relaxation signal decreases continuously on\nlowering the temperatures, and does not show oscilla-\ntions within the analyzed time window of 0.12 to 15 µs.\nAs expected and seen in the bulk magnetization studies,\nthis behavior rules out the presence of long-range mag-\nnetic ordering and the formation of the static uniform\nlocal field. We have fitted the µSR asymmetry data with\nthe stretched exponential function A(t)=Aoexp[−(λt)β]\n+Abg, where A ois the initial asymmetry, λis the spin\nrelaxation rate and βis the stretched exponent. For the\nsystems with a single well-defined spin fluctuation rate,\nthe exponent β= 1 and relaxation fits to the simple\nexponential function [26]. However, for Dy 2Zr2O7, a\nsimple exponential function was inadequate to account\nfor the observed asymmetry spectra, therefore we used\nthe phenomenological stretched exponential function.\nThis indicates the presence of multiple spin fluctuation\nrates or relaxation channels, which is not surprising,/s48 /s50 /s52 /s54/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s51/s48/s46/s48/s52/s48/s46/s48/s53\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s51/s48/s46/s48/s52/s48/s46/s48/s53/s65/s115/s121/s109/s109/s101/s116/s114/s121\n/s116/s32/s40 /s109 /s115/s41/s72/s32/s61/s32/s48/s32/s79/s101/s50/s57/s56/s32/s75\n/s50/s53/s48/s32/s75\n/s50/s48/s48/s32/s75\n/s49/s53/s48/s32/s75\n/s57/s48/s32/s75\n/s52/s48/s32/s75\n/s65/s115/s121/s109/s109/s101/s116/s114/s121\n/s116/s32/s40 /s109 /s115/s41/s32/s54/s50/s32/s109 /s75 \n/s32/s49/s48/s32/s75 \nFIG. 1. High-temperature zero-field µSR spectra for\nDy2Zr2O7. Solid (red) lines are the fit to an stretched ex-\nponential function described in the main text. Inset shows\nthe relaxation spectra at T= 62 mK and 10 K.\ngiven the structurally disordered pyrochlore lattice.\nThis behavior resembles with the Tb 2Hf2O7system\nexhibiting multiple relaxation rates due to the presence\nof anion-disordered pyrochlore lattice [18].\nThe temperature dependence of λandβof stretched\nfunction obtained from the time dependent muon\nasymmetry data are plotted in Fig. 2. As seen in the\nfigure, spin relaxation rate λincreases and the exponent\nβdecreases on cooling. This is due to the cumulative\neffect of the depopulation of Dy3+excited crystal field\nlevels and slowing down of spin fluctuations leading to\na wider field distribution at muon sites. The relaxation\nrate for Dy 2Zr2O7peaks around T∼130 K, showing\nsimilarities with Dy 2Ti2O7system [27, 28]. Such a peak\ninλ(T) is a characteristic feature of the changes in spin\ndynamics, which is often associated with the magnetic\ntransition [29]. While being consistent with the earlier\nµSR studies on powder Dy 2Ti2O7, this feature suggests\na crossover to the slow fluctuating regime in Dy 2Zr2O7\nwithout long-range ordering [27]. The DC susceptibility\nmeasurements (shown in the inset of Fig. 2) also suggest\nthe absence of any long-range magnetic ordering near\nthis temperature and also down to 2 K, indicating the\nparamagnetic ground state [22].\nThe inset of Fig. 2 shows the Arrhenius fit of ex-\ntracted relaxation rate to λ∝exp(Ea/kBT), yielding\nan activation energy of Ea/kB∼533 K for T≥40\nK andEa/kB∼710 K for T≥130 K. Note that the\nArrhenius fits to the relaxation rate in Dy 2Ti2O7yield\nan energy barrier of 468 K for T≥60 K [28] and a\nbarrier of 210 K for high-temperature spin freezing at\n∼16 K [2]. This indicates the thermally activated spin3\n/s48/s46/s48/s48/s52 /s48/s46/s48/s48/s56 /s48/s46/s48/s49/s50/s45 /s49/s48/s49/s50\n/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s50/s52/s54\n/s84/s32/s40/s75/s41/s108 /s32/s40 /s109 /s115/s45/s49\n/s41\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s98/s124 /s69\n/s97 /s124 /s32 /s126 /s32 /s53/s51/s51/s32 /s75 \n/s32 /s70 /s105 /s116 /s108/s110 /s108 /s32/s40 /s109 /s115/s45 /s49\n/s41\n/s49/s47/s84/s32/s40/s75/s45 /s49\n/s41/s124 /s69\n/s97 /s124 /s32 /s126 /s32 /s55/s49/s48/s32 /s75 /s48 /s49/s48/s48 /s50/s48/s48/s48/s51/s54/s32\n/s84/s32/s40/s75 /s41/s99 /s32\n/s40/s101/s109 /s117 /s32/s79/s101/s45 /s49\n/s32/s109 /s111/s108 /s45 /s49\n/s41\nFIG. 2. Temperature dependence of the parameters λandβ\nextracted from the stretched exponential fit of the zero field\nmuon spin relaxation data collected between 40 - 300 K. The\nstandard deviation of the parameters is represented by the\nerror bars. The maximum at ∼130 K suggests the relaxation\nrate going beyond the limit of the instrument as a result of\nlarge moment value ofDy3+ion. Inset (left) DCsusceptibility\nof Dy2Zr2O7taken from Ref[22] does not show any long-range\nmagnetic ordering and (right) shows the Arrhenius fit of the\nrelaxation rate for T≥40 K and T≥130 K.\ndynamics in the high-temperature regime. No activation\nbarrier could be calculated for 10 K ≤T≤40 K region,\nas the polarization signal got completely lost below t\n<<1µs. However, on further cooling below 10 K, a\nrecovery of the resolved asymmetry at t= 0.12µs, (with\ncomparatively less than 1/3 value of initial asymmetry\nvalue as expected for spin glass-like freezing) was\nobserved (see inset of Fig. 1). Generally, for a magnetic\nsystem, the re-polarization of muon signal is a signature\nof the emergence of static internal fields associated with\nthe spin freezing state. However, considering the earlier\nstudies of heat capacity and diffuse neutron scattering,\nre-polarizationof muon signal for Dy 2Zr2O7suggests the\ndevelopment of the fluctuating correlated spin state [14].\nOur preliminary measurements were also limited to 0.12\n≤t≤15µs and it may possible that the information\nabout the low-temperature static spin state (if any)\nis lost in the missing time window of t<0.12µs, as\nexpected for the glassy/freezing state. Therefore, µSR\nmeasurements in the early time windows would be useful\nfor ascertaining the magnetic ground state of Dy 2Zr2O7.\nThe stretched exponent βgradually decreases from ∼\n0.8 at high temperatures to a constant value of ∼0.4\n(see Fig. 2 and 3). In contrast, the spin-glass systems\nshow a dramatic drop in βnearly equaling 1/3 as the\nsystem approaches freezing temperature [30].\nFigure 3 shows that the relaxation rate increases\nbelow 10 K and nearly saturates below 3 K without any\nsignature of the non-relaxing tail. This demonstrates the\nabsence of further slowing-down of the spin dynamics\n/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s98\nFIG. 3. Temperature dependence of the parameters λandβ\nextracted from the stretched exponential fit of the zero field\nmuon spin relaxation data collected between 0.062 - 10 K.\nThe standard deviation of the parameters is represented by\nthe error bars. A peak shape anomaly at ∼1.2 K (in the\nshaded region) is due to the effect of variation of stray field\ndue to field ramp on other nearby instrument for these data\npoints.\nas observed in the spin-ice materials. We observed a\nsubstantial λ∼1µs−1relaxation rate at T≤3 K. This\nbehavior is similar to Dy 2Ti2O7, however, unexpected\nfor spin ice system of Ising spins where the monopole\nexcitations are separated by large energy barriersagainst\nsingle spin-flip process [31]. This suggests that the spin\ncorrelations observed in Ref[14] in diffuse neutron\nscattering at low temperatures fluctuate on a different\ntime scale from the muon Larmor precession frequency\n(MHz). Although a similar persistent spin dynamics\nhave been observed in a variety of frustrated magnets,\nthe origin of the temperature-independent relaxation\nplateau remains unknown [32, 33]. On the other hand,\nquantum spin liquid systems investigated using ZF- µSR\nalso exhibit temperature independent muon relaxation\nrate at the lowest temperature range, and then follows\na power law behavior with increasing temperature\n[34, 35]. It is to mention that Ramon et al.found a\nstrong frequency dependence in Dy 2Zr2O7below 2 K\nin ac susceptibility studies [14], implying the presence\nof unusual spin freezing as seen in spin ice Dy 2Ti2O7,\nHo2Ti2O7and other frustrated magnets. However,\nno such feature is observed in our µSR data, usually\ndistinguished by a peak shape anomaly at the freezing\ntemperature [36]. Upon the recovery of asymmetry, the\nrelaxation rate increases by 60% when cooling from T\n>9 K (Curie-Weiss temperature) down to 0.062 K,\nindicating a weak slowing down of the Dy3+spin fluc-\ntuations. Whereas, below the freezing point, relaxation\nrate in a spin-glass typically increases by several orders\nof magnitude ( λ∼1–20µs−1) [26, 37]. Instead, the\npresence of finite residual spin relaxation rate down to4\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s49/s54/s48/s46/s50/s48/s48/s46/s50/s52/s65/s115/s121/s109/s109/s101/s116/s114/s121\n/s116/s32/s40 /s109 /s115/s41/s32/s48/s32/s79/s101/s32/s32/s32/s32/s32 /s32/s55/s46/s53/s32/s107/s79/s101\n/s32/s50/s46/s53/s32/s107/s79/s101 /s32/s49/s48/s32/s107/s79/s101\n/s32/s53/s32/s107/s79/s101/s32/s32/s32 /s32/s50/s48/s32/s107/s79/s101/s84/s32/s61/s32/s54/s50/s32/s109/s75\nFIG. 4. Muon spin relaxation spectra collected at T= 62\nmK in a longitudinal field of 0 - 20 kOe. Color line represents\nthe fit using stretched exponential function (as described i n\nthe main text).\nthe lowest measuring temperature is consistent with the\nlarge entropy of Rln2 [22] and large susceptibility value\n[14], providing evidences of magnetic excitation in a\nweakly frozen spin state.\nFig. 4 shows the asymmetry versus time spectra\ncollected in the longitudinal fields of H= 0 - 20 kOe.\nSimilar to zero field measurement, µSR spectra in the\nmagnetic field do not exhibit any evidence of magnetic\nordering down to 62 mK even up to the applied field of\n20 kOe. These µSR spectra taken at different fields are\nalso fitted well by the stretched exponential function.\nThe obatined temperature and magnetic field depen-\ndence of the extracted relaxation rate are plotted in Fig.\n5(a) and 5(b) respectively. The relaxation rate has a\ntemperature dependence similar to the zero-field for all\nfields. Although the Dy3+moments undergo a transition\nto the field-induced frozen state on the ac susceptibility\ntime scale below 10 K, fluctuations appear to emerge in\nlongitudinal-field muon experiments down to 62 mK. In\ncomparison to muon technique which probes the spin\ndynamics of the order of 104to 1011Hz, ac susceptibility\nemployed for this study could probe the fluctuations\ndown to the millisecond timescale (or kHz) only [38].\nThe longitudinal relaxation rate as a function of the\napplied field, for different temperatures ( T= 0.062 - 2\nK) shows a peak around 10 kOe, indicating a change in\nspin dynamics. This feature reflects the competition be-\ntween partially frozen (static) and dynamic components\nof the local field at the muon site. Further, the field\ndependence of the relaxation rate and the higher time\ntail observed in the µSR (Fig. 4), are due to the field/s48/s46/s49 /s49 /s49/s48/s50/s52/s54\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48/s51/s54\n/s48/s32/s79/s101/s50/s48/s32/s107/s79/s101\n/s49/s53/s32/s107/s79/s101\n/s49/s48/s32/s107/s79/s101\n/s53/s32/s107/s79/s101/s108 /s32/s40 /s109 /s115/s45/s49\n/s41\n/s84 /s32/s40/s75 /s41/s40/s97/s41/s50/s32/s75 \n/s49/s32/s75 \n/s48/s46/s55/s53/s32/s75 \n/s48/s46/s53/s32/s75 \n/s72/s32/s40/s107/s79/s101/s41/s48/s46/s48/s54/s50/s32/s75 /s51/s54/s51/s54/s108 /s32/s40 /s109 /s115/s45/s49\n/s41/s51/s54/s51/s54\n/s40/s98/s41\nFIG. 5. Temperature (a) and field (b) dependence of longi-\ntudinal relaxation rate of Dy 2Zr2O7. Theλvalues for 15 and\n20 kOe field in (a) are shifted upwards in y-axis (by ∼4µs−1)\nfor better clarity. The stray fields contribution in zero-fie ld\ndata is removed for the sake of comparison.\ndependence of the fast fluctuating electronic relaxation\nand it persists up to the highest field of 20 kOe. This\nconfirms the paramagnetic correlated dynamical ground\nstate without long range ordering uo to 20 kOe at 62\nmK. This behavior is consistent with the field-induced\nrelaxation in ac susceptibility [22]. Though the spin-ice\nentropy is reported to stabilize in Dy 2Zr2O7atT= 5\nkOe below 1 K, the longitudinal field µSR results do\nnot show any supported signature, rather suggest the\ndevelopment of unusual field-induced magnetic state in\na highly dynamic background of fluctuating spins. This\nbehavior is similar to other spin ice systems, where µSR\nreveals a large density of excitation in the two-in-two-out\nfrozen spin state [28].\nIn conclusion, there are no evidences of long-range\nordering and spin freezing down to 62 mK in our\ncomprehensive studies of ZF and LF µSR. In spite of\nthe weak static field as shown by ac susceptibility and\nalso evident from the recovery of asymmetry below\n10 K, Dy 2Zr2O7exhibits the persistence of dynamic\nmagnetism down to 62 mK. Though the absence of\nlong-range ordering, persistent spin fluctuations and\nsaturation of λat low temperatures are the signature of\nquantum spin liquid ground state (QSL), the unusual\ndependence of the relaxation rate on field exclude this\npossibility and imply the development of magnetically\ncorrelated spin state. It is to mention here that, neutron\ndiffraction measurement by Ramon et. al. [23] shows\nthe emergence of magneitc Braggs peaks for H≥2 kOe,\nhowever the broad short-range correlation peak remain\ncentered at 1.2 ˚A−1even at the highest applied magnetic\nfield. This could indicate the coexistence of a dynamic5\nspin state and a partially ordered spin state. In muSR,\nhowever, the homogeneous field cannot be seen in the\npresence of fluctuations. The observed peculiar behavior\nis might be an outcome of the presence of the lattice\ndisorder in comparison to structurally clean pyrochlore\nDy2Ti2O7system. The structural studies Dy 2Zr2O7\nin Ref[23] also suggests the site mixing of Dy/Zr ions\nresults bond randomness in the lattice. Similarly,\nYbMgGaO 4possess QSL gorund state attributed to\nthe presence of Mg/Ga site mixing [20]. Consequently,\nthe dominant effect from A/B site mixing or ran-domness along with magnetic frustration is responsible\nforthecurbingoforderingandthedynamicgroundstate.\nAcknowledgment:\nWe thanks AMRC, Indian Institute ofTechnologyMandi\nfor the experimental facility. Sheetal Acknowledged IIT\nMandi and MHRD India for the HTRA fellowship. Dr.\nD. T. Adroja thank EPSRC UK for funding (Grant No.\nEP/W00562X/1). We would like to thank the ISIS Facil-\nity for beam time, RB2010601 and the data are available\nfrom Ref[39].\n[1] S. Bramwell, M. Harris, B. Den Hertog, M. Gingras,\nJ. Gardner, D. McMorrow, A. 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Telling, Un-\nravelling the magnetic ground state of Dy 2Zr2O7\npyrochlore: A candidate for spin ice state,\nSTFC ISIS Neutron and Muon Source,\nhttps://doi.org/10.5286/ISIS.E.RB2010601 ,\n."    },    {        "title": "1711.07238v1.Dynamical_suppression_of_fluctuations_in_an_interacting_nuclear_spin_bath_of_a_self_assembled_quantum_dot_using_multiple_pulse_nuclear_magnetic_resonance.pdf",        "content": "Dynamical suppression of \ructuations in an interacting nuclear spin bath of a\nself-assembled quantum dot using multiple pulse nuclear magnetic resonance\nA. M. Waeber\u0003\nDepartment of Physics and Astronomy, University of She\u000eeld, She\u000eeld S3 7RH, United Kingdom and\nWalter Schottky Institut and Physik-Department,\nTechnische Universit at M unchen, Am Coulombwall 4, 85748 Garching, Germany\nM. Hopkinson\nDepartment of Electronic and Electrical Engineering,\nUniversity of She\u000eeld, She\u000eeld S1 3JD, United Kingdom\nM. S. Skolnick and E. A. Chekhovichy\nDepartment of Physics and Astronomy, University of She\u000eeld, She\u000eeld S3 7RH, United Kingdom\n(Dated: November 3, 2021)\nElectron spin qubit coherence in quantum dots is ultimately limited by random nuclear\nspin bath \ructuations. Here we aim to eliminate this randomness by making spin bath\nevolution deterministic. We introduce spin bath control sequences, which systematically\ncombine Hahn and solid echoes to suppress inhomogeneous broadening and nuclear-nuclear\ninteractions. Experiments on self-assembled quantum dots show a \fve-fold increase in nuclear\nspin coherence. Numerical simulations show that these sequences can be used to suppress\ndecoherence via qubit-qubit interaction in point defect and dopant spin systems.arXiv:1711.07238v1  [cond-mat.mes-hall]  20 Nov 20172\nThe excellent spin-photon interface of con\fned charges in III-V semiconductor quantum dots\n(QDs) has recently attracted a lot of attention for potential applications in photon-mediated quan-\ntum networks1{3. The large optical dipole moment of QDs makes ultrafast optical spin control\nfeasible and permits unrivalled entanglement generation rates4{6. On the other hand, the coher-\nence properties of the electron or hole spin qubit are strongly a\u000bected by hyper\fne interaction with\nthe \ructuating spin bath of the \u0018105constituent nuclei of the QD7,8.\nRecently, signi\fcant progress has been made in suppressing hyper\fne-induced qubit dephasing in\nstrain-free QDs by applying tailored dynamical decoupling protocols to the qubit spin9,10. However,\nthere are limits to this approach: in a strained QD, static quadrupolar \felds lead to a spread of the\nnuclear spin Larmor frequencies, reducing the e\u000bectiveness of such a spectral \fltering method11{14.\nIn order to improve the properties of the qubit environment directly, we explore the comple-\nmentary pathway of controlling the nuclear spin bath itself with pulsed nuclear magnetic resonance\n(NMR)15,16. Examples for controlling spin-spin interactions are found in NMR spectroscopy where\nsequences such as WAHUHA17and MREV18,19are used to average out dipolar couplings selec-\ntively. However, these `solid echo' cycles do not refocus inhomogeneous broadening caused by\nadditional static or time-dependent \felds. On the other hand, dynamical decoupling sequences\nconsisting of a series of \u0019-pulses can suppress this inhomogeneous dephasing very e\u000bectively20{22,\nbut are unsuitable for controlling dipolar coupling terms. Instead, such sequences even increase\ndipolar dephasing through the parasitic e\u000bect of instantaneous di\u000busion23{25.\nIn this work, we introduce a set of pulse sequences which are designed to combine the features\nof dynamical decoupling with those of solid echoes. Under these combined Hahn and solid echo\n(CHASE) multiple pulse sequences, we engineer the dynamics of the interacting many-body nu-\nclear spin bath in a single InGaAs QD. While it is not possible to eliminate spin bath dynamics\ncompletely, we show that random \ructuations can be transformed into deterministic evolution,\nwhich can in principle be decoupled from the qubit using standard control schemes16. We test\nCHASE sequences experimentally in optically detected nuclear magnetic resonance (ODNMR)\nmeasurements and explore their applicability to systems with and without strong inhomogeneous\nresonance broadening using \frst principle quantum mechanical simulations. Our experimental re-\nsults reveal an up to \fvefold increase in the CHASE nuclear spin coherence time of75As compared\nto the Hahn echo26. Furthermore, our simulations show that cyclic application of CHASE sequences\ncan suppress dephasing arbitrarily well in spin ensembles with weak inhomogeneous broadening\n(e.g. strain-free quantum dots, dilute donor spins or defect centres).\nBefore presenting experimental results, we describe how the control sequences are designed.3\nWe start from an intuitive approach previously used to extend nuclear spin lifetimes in silicon\nand diamond27,28: by combining solid echo cycles with refocusing \u0019-pulses, both inhomogeneous\ndephasing and dipolar coupling can be suppressed. Here we substantiate this approach by applying\na rigorous average Hamiltonian theory (AHT)29, which is a form of perturbation theory based on\nMagnus expansion. The evolution of a spin-1/2 nuclear bath Iiis analysed under a given pulse\ncycle in an external magnetic \feld Bz. We take into account a dipolar coupling term Hzz\ndas well as\na generic resonance o\u000bset Hamiltonian Hz\n0, which describes inhomogeneous resonance broadening\ncaused for example by chemical shifts or by a static quadrupolar interaction:\nH=Hz\n0+Hzz\nd\n=hX\ni\u0001\u0017iIz;i+hX\ni<j\u0017ij(3Iz;iIz;j\u0000Ii\u0001Ij);(1)\nwhere \u0001\u0017idenotes the resonance frequency o\u000bset of the i-th nuclear spin and \u0017ijis the dipolar\ncoupling constant between two spins IiandIj. The free decay of transverse magnetisation under\nthis Hamiltonian is described by a rate \u0000 /1=T\u0003\n2.\n-xy yx x(a)\n0 t=6τevolCHASE-5\nx-y -y-x -x\n+\nt=12τevolCHASE-10\n(d) CHASE-34\n-x-y-yxxx-yx-y-x xx-y\nxxy yx-x-xyx\nt=36τevolyx y-x0x -x-y-x\nxxy -x(b)\nxy yx -x -x-y -y-x x\n+\nt=24τevolCHASE-20(c)ττττττ\nFIG. 1. Pulse sequences for refocusing inhomogeneous and dipolar broadening. The rf carrier phases are\n'x= 0,'\u0000x=\u0019for\u0006x rotations around the ^ exaxis, and'y=\u0019=2,'\u0000y= 3\u0019=2 for\u0006y rotations around\n^ey. Narrow pulses indicate \u0019=2-rotations with pulse time t\u0019=2 and broad pulses correspondingly represent\n\u0019-rotations with duration t\u0019. (a) The combined Hahn and solid echo sequence CHASE-5 with cycle time\ntc= 3t\u0019+\u001cevolconsisting of the total gate time 3 t\u0019and total free evolution time \u001cevol= 6\u001c(where\u001c\u001dt\u0019\nis the free evolution time between two pulses). (b) Extension to CHASE-10, which is less sensitive to \fnite\npulse durations t\u0019>0. (c) Using symmetry considerations, a further optimised sequence CHASE-20 is\nconstructed. (d) The longest sequence CHASE-34 has the best refocusing capability for t\u0019!0.4\nUsing AHT as a benchmark tool (see details in Supplemental Material32) we have analysed\nvarious combinations of \u0019and\u0019=2 pulses to \fnd those that maximise the spin bath coherence\nwhile minimising the pulse sequence length. The shortest e\u000ecient cycle which gives a noticeable\nincrease of bath coherence (CHASE-5) contains only 5 pulses and is illustrated in Fig. 1(a).\nAssuming in\fnitely short pulses ( t\u0019!0), the zeroth order average Hamiltonian /\u0000 vanishes.\nThe leading residual contribution to dephasing is a \frst order term /~tc\u00002mixing contributions\nfrom the inhomogeneous broadening Hamiltonian and the dipolar interaction30,31:\n\u0016HCHASE\u00005=itc\n18~[Hzz\nd\u0000Hxx\nd;Hy\n0] +O(~t2\nc\u00003); (2)\nwheretcis the full cycle time and Hxx\nd,Hy\n0are the dipolar and inhomogeneous broadening Hamil-\ntonians acting along orthogonal equatorial axes ^ exand ^ey(see Supplemental Material32for the full\nde\fnition).\nUnder realistic experimental conditions, the assumption of in\fnitely short pulses is often not\njusti\fed. For \fnite pulse durations t\u0019, the zeroth order average Hamiltonian does not vanish under\nCHASE-5, reducing the capability of the cycle to increase the spin bath coherence. However, we\ncan obtain an average Hamiltonian of the form of Eq. 2 even for \fnite t\u0019by extending our cycle to\nCHASE-10 as illustrated in Fig. 1(b), analogous to the pure solid echo extension from WAHUHA\nto MREV18,19. Furthermore, by adding the pulse block shown in Fig. 1(c) we can symmetrise the\ncycle to CHASE-20 and remove the \frst-order mixing term, condensing the average Hamiltonian\nto\u0016HCHASE\u000020/O(~t2\nc\u00003) independent of the pulse duration t\u0019.\nFinally, we identify a longer sequence CHASE-34 (see Fig. 1(d)) with a total gate time 20 t\u0019\nwhich reduces the average Hamiltonian to a second order mixing term for t\u0019!0 but has non-\nvanishing lower-order terms for \fnite pulse durations. A comprehensive overview of the AHT\ncalculations and residual Hamiltonians for the sequences shown in \fgure 1 can be found in the\nSupplemental Material32.\nWe study the performance of these sequences experimentally on individual charge-free InGaAs\nQDs with\u0018105nuclear spins. Here, we follow the ODNMR pump-probe scheme used in Ref.26:\nthe QD sample is kept at T= 4:2 K and is subjected to a strong magnetic \feld of Bz= 8 T. Using\na confocal microscopy setup in Faraday orientation, we prepare the nuclear spin bath optically\nthrough polarisation-selective pumping of an exciton transition (dynamic nuclear polarisation,\nDNP). In this way, we can achieve hyper\fne-mediated spin bath polarisation degrees of up to\n65%33,34. Radio frequency (rf) \felds are coupled to the QD via a multi-winding copper coil in\nclose proximity to the sample. Changes in the \fnal bath polarisation are probed with a weak5\noptical pulse measuring the splitting of the neutral exciton Zeeman doublet11.\nWe perform resonant pulsed NMR measurements on the inhomogeneously broadened central\nspin transition\u00001=2$+1=2 of the75As (inhomogeneous width of \u0001 \u0017inh\u001840 kHz) and71Ga\n(\u0001\u0017inh\u001810 kHz) nuclear spin ensembles11,26. The phases of the \u0019-pulses of all sequences are chosen\nto produce spin rotations around the ^ exaxis of the rotating frame. In each experiment, a \u0019=2-pulse\nis applied prior to the multipulse cycle to initialise the spin state. We conduct experiments with\ninitial\u0019=2 rotation around the ^ exaxis (Carr-Purcell or CP-like sequences35, denoted as `-X') and\naround the ^ eyaxis (Carr-Purcell-Meiboom-Gill or CPMG-like sequences36, `-Y'): in this way we\ndistinguish between a genuine improvement of the spin coherence and spin locking e\u000bects37{39,\nwhich only stabilize spin magnetization along a certain direction. A \fnal \u0019=2-pulse is an inverse\nof the initialisation pulse and projects the refocused magnetisation along the ^ ezaxis for optical\nreadout26.\nRepresentative data for experiments on75As and71Ga is shown by the solid symbols in Figs\n2(a) and 2(b), respectively. The values of the nuclear spin coherence times T2and echo amplitudes\n\u0001Ehf(\u001cevol= 0) in Figs 2(c-f) are obtained by \ftting the data with a compressed exponential decay\nfunction\n\u0001Ehf(\u001cevol) = \u0001Ehf(\u001cevol= 0)\u0001e\u0000(\u001cevol=T2)\f, (3)\nwhere \u0001Ehfdenotes the change in the measured hyper\fne shift due to rf-induced depolarisation\nof the nuclear spin bath, \u001cevolis the total free evolution time over npulse cycles, \f2[1;2]\nis a compression factor40,T2describes decoherence of the spin bath during free evolution, while\nreduction of the echo amplitude \u0001 Ehf(\u001cevol= 0) compared to the initial magnetization \u0001 Ehf(t= 0)\nquanti\fes the imperfections of pulse spin rotations.\nIn order to analyse the in\ruence of spin locking e\u000bects37{39, we test a series of CP-X and\nCP(MG)-Y sequences with alternating pulse carrier phase (sequence cycle \u0000\u001c=2\u0000\u0019x\u0000\u001c\u0000\u0019\u0000x\u0000\n\u001c=2\u0000). With increasing number of cycles n, expressed in terms of the total rf gate time, we observe\na strong increase of the measured T2under CP-X for both isotopes (black squares in Figs 2(c,d)).\nBy contrast, no signi\fcant increase of T2is observed under CP-Y (blue circles). However, the CP-Y\necho amplitude is rapidly reduced with increasing n(Figs 2(e,f)), owing to the limited available rf\npower resulting in violation of the `hard pulse' condition32.\nThe contrasting behaviour of T2under alternating phase CP-X/Y has been observed in other\nsystems38,39,41and has been attributed variably to spin locking41,42or stimulated echoes43. Here,\nwe ascribe the up-to-fourfold increase of T2under CP-X to a form of pulsed spin locking arising6\n0.11 1 01 000\n123N\nMR Signal ΔEhf (µeV)Free evolution time τevol (ms) \nHahn echo, HE-X \nCP-X-64 \nCHASE-Y-107\n1Ga\n0.11 1 01 000\n1234567Free evolution time τevol (ms)NMR Signal ΔEhf (µeV) Hahn echo, HE-X \nCP-X-48 \nCHASE-Y-10 \nCHASE-X-347\n5As\n1001011025\n101001011021\n23456781\n00101102012345671\n00101102012347\n5AsTotal gate time (in units of tπ) \n   HE-X&CP-X \n   HE-Y&CP(MG)-Y \n   CHASE-X-10/20 \n   CHASE-Y-10/20 \n   CHASE-X-34Nuclear spinc\noherence time T2 (ms)7\n5As7\n1GaN\nuclear spinc\noherence time T2 (ms)Total gate time (in units of tπ)7\n1GaNMR signala\nmpitude ΔEhf(τevol=0) (µeV)T\notal gate time (in units of tπ)N\nMR signala\nmpitude ΔEhf(τevol=0) (µeV)T\notal gate time (in units of tπ)(f)(c)(\ne)(d)(a)( b)\nFIG. 2. Nuclear spin polarisation decay under pulsed control of the75As and71Ga spin ensembles in a\nsingle InGaAs QD. (a,b) Decay of the polarisation \u0001 Ehfas a function of the total free evolution time \u001cevol\nfor di\u000berent control sequences. Symbols mark experimental data and solid lines show best \fts with Eq. 3.\n(c,d) Dependence of the \ftted nuclear spin coherence time T2on the number of sequence cycles nexpressed\nas total rf gate time (in units of the \u0019-pulse time t\u0019). Error bars mark 90% con\fdence intervals. (e,f)\nRespective \ftted echo amplitude \u0001 Ehf(\u001cevol= 0) as a function of the total gate time. The data for one\ncycle of HE and CHASE-10 is combined with the data for integer cycle numbers nof CP and CHASE-20,\nrespectively.7\nfrom dipolar evolution during the \fnite \u0019-pulse duration42: Our interpretation is based on obser-\nvation that the spin lock disappears for small pulse-to-cycle time ratios t\u0019=tc(see Supplemental\nMaterial32).\nWe now examine the spin bath coherence under CHASE-10/20 sequences. In order to account\nfor the spin locking e\u000bects discussed above we conduct all experiments with both -X and -Y\ninitialization. In experiments on75As, a marked increase of TCHASE\u0000Y\u000010\n2 = 10:5\u00060:7 ms compared\nto the Hahn echo decay THE\u0000X\n2 = 4:3\u00060:2 ms is observed even under a single cycle of CHASE-Y-10\n(green triangles in Fig. 2). We \fnd a similar proportional increase from THE\u0000X\n2 = 1:2\u00060:1 ms\ntoTCHASE\u0000Y\u000010\n2 = 2:3\u00060:2 ms for71Ga. In comparison, the additional coherence gain under\nCHASE-Y-20 is only marginal. However, as shown in Figs 2(e,f), the preservation of the echo\namplitude \u0001 Ehf(\u001cevol= 0) under CHASE-Y-20 is more robust compared to CP-Y. The CHASE-X-\n10/20 coherence time (red triangles in Figs. 2(c,d)) exceeds the T2of CHASE-Y-10/20, suggesting\nthe presence of spin-locking under CHASE-X-10/20. Thus we use CHASE-Y-10/20 to examine\nthe spectral properties of the spin bath dynamics in a dynamical decoupling fashion: We observe\nno further coherence gain under up to n= 6 cycles of CHASE-Y-20, suggesting that environment\nnoise (e.g. produced by charge \ructuations) is negligible over a broad frequency domain of up to\n\u0018100 kHz (given by an average inter-pulse delay of \u001810\u0016s), as expected for neutral QDs20,26.\nRecent work has shown, however, that the nuclear spin bath coherence is drastically reduced when\nthe QD is occupied by an electron44. We expect that CHASE sequences will be suited to restore the\nbath coherence, o\u000bering a pathway for electron spin manipulation in a quiescent QD environment.\nFinally, we present an experiment using CHASE-34 (violet stars in Fig. 2). We \fnd that\nonly the CP-like cycle yields a measurable decay curve for75As whereas the71Ga echo amplitude\n\u0001Ehf(\u001cevol= 0) is too small to be resolved. Tantalisingly, we observe TCHASE\u0000X\u000034\n2 = 22:4\u00064:5 ms\n(violet stars in Fig. 2(a)) { nearly a \fvefold increase of the bath coherence time compared to HE.\nNumerical simulations (see Supplemental Material32) indicate that the reduction of \u0001 Ehf(\u001cevol= 0)\nin this 34-pulse cycle is not due to the limited rf pulse bandwidth, but is likely related to small\npulse calibration errors, o\u000bering in principle a route for further improvements.\nThe experimental observations are corroborated by extensive \frst principle quantum mechanical\nsimulations of the nuclear spin bath evolution under the studied pulse sequences. We consider an\nensemble of twelve dipolar coupled75As spins and study the evolution of the resonantly driven\ncentral transition in the limits of large (\u0001 \u0017i\u001d\u0017ij) and vanishing (\u0001 \u0017i\u001c\u0017ij) inhomogeneous\nbroadening. In this way, we can con\frm the experimental results in self-assembled QDs with\nstrong static quadrupolar broadening and explore the applicability of CHASE sequences to more8\n1001011021033\n301\n101001001011021033\n301\n101001\n001011021030.40.60.81.01\n001011021030.40.60.81.0t/s61552=80 µs(a) \nt/s61552→0      t/s61552>0 \n       HE-X&CP-X \n       HE-Y&CP(MG)-Y \n       CHASE-X-10/20 \n       CHASE-Y-10/20 \n       CHASE-X-34 \n       CHASE-Y-34Δ/s61550i << /s61550ijTotal gate time (in units of t/s61552)Nuclear spin coherence t\nime, T2 (ms)t\n/s61552=10 µs(b)Δ\n/s61550i >> /s61550ijN\nuclear spin coherence t\nime, T2 (ms)Total gate time (in units of t/s61552)(\nc)Δ\n/s61550i << /s61550ijt\n/s61552=80 µsFinal magnetization for s\nhort free evolution, 〈Iz(τ=0)〉T\notal gate time (in units of t/s61552)(d)Δ\n/s61550i >> /s61550ijt\n/s61552=10 µsF\ninal magnetization for s\nhort free evolution, 〈Iz(τ=0)〉T\notal gate time (in units of t/s61552)\nFIG. 3. Simulated evolution of a dipolar coupled ensemble of twelve75As spins under control sequences\n(a,c) with small and (b,d) large inhomogeneous (quadrupolar) broadening \u0001 \u0017i. Figs (a) and (b) show\nthe respective \ftted nuclear spin coherence times T2as a function of the total gate time. The gate time\ndependence of the corresponding echo amplitudes for short free evolution hIz(0)iis shown in (c) and (d).\nSimulations are done for both in\fnitely sharp ( t\u0019= 0, open symbols) and \fnite pulses (solid symbols),\nwhere we set t\u0019= 80\u0016s for \u0001\u0017i\u001c\u0017ijand uset\u0019= 10\u0016s in the limit of \u0001 \u0017i\u001d\u0017ij. Further simulation\ndetails and representative decay curves are shown in the Supplemental Material32.\nhomogeneous dilute spin systems as encountered in defect centres in diamond or donors in silicon.\nIn addition, we study the in\ruence of \fnite pulse durations on the bath evolution by testing our\nsequences with both in\fnitely short pulses t\u0019!0 and realistic pulse durations t\u0019\u001810\u0000100\u0016s.\nFigure 3 shows the \ftted coherence times (a,b) and echo amplitudes (c,d) for simulated de-\ncay curves in case of negligible inhomogeneous resonance broadening \u0001 \u0017i\u001c\u0017ij(a,c) and large\nbroadening \u0001 \u0017i\u001d\u0017ij(b,d). Simulation data at in\fnitely short (\fnite) pulses is marked by open9\n(solid) symbols. The simulations with \u0001 \u0017i\u001d\u0017ijand \fnite pulses are in very good agreement\nwith the experiments (cf. Fig. 2(c-f)): namely, the increase in T2with increasing number of cycles\nnunder CP-X (spin locking), the reduction of echo amplitude with growing nunder CP-Y, and\nthe extension of T2under CHASE-10/20 are all well reproduced. Moreover, the CHASE-10/20\necho amplitudes are stable under increasing n, demonstrating the superior performance of CHASE\nunder `soft pulse' conditions compared to previously introduced sequences28,45,46(see additional\nsimulations in Supplemental Material32).\nHaving established the validity of the simulations we apply them to the regimes not accessible\nin experiments on self-assembled QDs. In the case of \u0001 \u0017i\u001c\u0017ijapplicable e.g. to defect spins\nin diamond and SiC, the e\u000bect of the CHASE sequences is analogous to that of a solid echo\nsequence17{19: dipolar broadening can be suppressed arbitrarily well with increasing nresulting in\nincrease of T2(triangles in Fig. 3(a)). Importantly, the CHASE-10/20 echo amplitudes (Fig. 3(c))\nremain close to ideal \u00191 under \fnite ('soft') pulses even at large n: there is a potential in applying\nCHASE-10/20 to electron spin qubits for dynamical decoupling from the nuclear spin bath and for\nsimultaneous suppression of the decoherence that arises from qubit-qubit dipolar interactions15,25\nand can not be handled by the standard \u0019-pulse sequences.\nFinally, we make a note on some peculiarities predicted in the simulations. For large inho-\nmogeneity (\u0001 \u0017i\u001d\u0017ij) and short pulses ( t\u0019!0) the CP-X/Y coherence time T2decreases with\nincreasingn(open squares and circles in Fig. 3(b)) and asymptotically approaches the T2value\nobtained for weak inhomogeneity (\u0001 \u0017i\u001c\u0017ij) (Fig. 3(a)). This is unexpected, since \u0019-pulse trains\ndo not modify the dipolar Hamiltonian. We tentatively ascribe this to fast spin rotations induced\nby the in\fnitely short rf pulses: such rotations e\u000bectively shorten the spin lifetime, broaden the\nhomogeneous NMR linewidths and re-enable the dipolar nuclear spin \rip-\rops which are otherwise\ninhibited by inhomogeneous broadening. Note that similar trends are observed in the simulations\nwith CHASE-10/20 sequences, where the competing e\u000bects of the re-enabled \rip-\rops and conver-\ngence of the average Hamiltonian lead to a non-monotonic dependence of T2onn. We also point\nout that under small inhomogeneous broadening \u0001 \u0017i\u001c\u0017ijand \fnite pulse durations, the increase\n(decrease) of T2under CP-Y (CP-X) observed in Fig. 3(a) is reversed compared to the \u0001 \u0017i\u001d\u0017ij\ncase (Fig. 3(b)) { an unexpected result requiring further investigation.\nIn summary, we have introduced a set of multiple pulse cycles which allow e\u000ecient simultaneous\nrefocusing of inhomogeneous and dipolar broadening. Beyond the current work, these sequences\nmay have potential to restore the spin bath coherence in charged QDs which have recently been\nshown to possess far shorter Hahn echo decay times44. In this way, one would aim to create a10\ndeterministically evolving spin environment for a central electron spin. In addition, CHASE could\nbe used to enhance the coherence of defect centres and dopants beyond the limits of standard\ndynamical decoupling protocols as the parasitic decoherence channel of instantaneous di\u000busion15,25\nis quenched. Further optimisation of spin bath \ructuation freezing can be explored using techniques\nsuch as optimal control47.\nThe authors are grateful to A. I. Tartakovskii for useful discussions. This work was supported by\nthe EPSRC Programme Grant EP/J007544/1, ITN S3NANO. E.A.C. was supported by a Univer-\nsity of She\u000eeld Vice-Chancellor's Fellowship and a Royal Society University Research Fellowship.\nComputational resources were provided in part by the University of She\u000eeld HPC cluster Iceberg.\n\u0003andreas.waeber@wsi.tum.de\nye.chekhovich@she\u000eeld.ac.uk\n1Kimble, H. J. The quantum internet Nature 453, 1023 (2008).\n2Akopian, N. and Wang, L. and Rastelli, A. and Schmidt, O. G. and Zwiller, V. 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Symmetric-cycle pulse sequence for dynamical decoupling of local\n\felds and dipole-dipole interactions Journal of Physics B 48, 135503 (2015).\n47Khaneja, N. and Reiss, T. and Kehlet, C. and Schulte-Herbr uggen, T. and Glaser, S. J. Optimal control\nof coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms Journal of\nMagnetic Resonance 172, 296 (2005).1\nSUPPLEMENTAL MATERIAL\nThe supplemental material contains further details on the theoretical tools used to design the\nCHASE pulse sequences presented in the main text, an analysis of the in\ruence of pulsed spin\nlocking and a discussion of the respective in\ruence of pulse parameters on the performance of\nthese spin control sequences. Finally, the methods used to simulate the spin bath evolution are\ndescribed with additional results for related pulse sequences from literature.\nSupplemental Note 1. AVERAGE HAMILTONIAN THEORY\nAverage Hamiltonian theory (AHT) is an established tool for the theoretical characterisation\nand analysis of pulse sequences for magnetic resonance spin controlS1{S3. Within certain constraints\nit allows the time evolution of a given spin Hamiltonian under interaction with a periodic time-\ndependent external magnetic \feld to be approximated. We use AHT to determine how well a\nfrequency o\u000bset Hamiltonian Hz\n0and a dipolar coupling term Hzz\ndcan be suppressed simultaneously\nby the CHASE sequences introduced in the main text.\nTo this end, we consider a nuclear spin ensemble Iiwith spin 1=2. The evolution of the wavefunc-\ntion (t) describing the state of the nuclear spin bath is determined by the Schr odinger equation:\n@ (t)=@t=\u0000(i=~)H(t) (t); (1)\nH(t) =Hz\nL+Hz\n0+Hzz\nd+Hrf(t); (2)\nwhere the Hamiltonian H(t) is the sum of the Larmor term ^HLdescribing interaction of the spins\nwith a static magnetic \feld Bzalong the ^ezaxis, the o\u000bset term Hz\n0describing static resonance\nfrequency shifts, the dipolar term Hzz\nddescribing nuclear-nuclear spin interaction and the radio-\nfrequency (rf) term Hrf(t) describing the e\u000bect of the oscillating magnetic \feld inducing nuclear\nmagnetic resonance.\nWe use transformation into the frame rotating around the direction of the static magnetic \feld\n(^ezaxis) at the radio-frequency. In this way the e\u000bect of the static magnetic \feld is eliminated\n(Hz\nL= 0) and the oscillating rf \feld becomes static (see Section 5.5 inS4). The explicit time\ndependence inHrf(t) is then only due to the pulsed nature of the rf \feld.2\nThe individual terms are explicitly de\fned as\nHz\n0=hX\ni\u0001\u0017iIz;i; (3)\nHzz\nd=hX\ni<j\u0017ij(3Iz;iIz;j\u0000Ii\u0001Ij); (4)\nHrf(t) =\u0000h\u0017rf(t)X\niI';i: (5)\nIn the studied quantum dots (QDs) the resonance o\u000bset term Hz\n0is dominated by the static\nquadrupolar frequency shifts (\u0001 \u0017ifor thei-th nuclear spin), although in other systems \u0001 \u0017icould\nalso include di\u000berent static frequency o\u000bsets such as chemical shifts or magnetic \feld gradients. In\naddition, we consider a truncated dipolar coupling term Hzz\ndwith coupling strength\n\u0017ij=\u00160\n4\u0019~\n2\u0019\r2\n21\u00003 cos2\u0012\nr3; (6)\nbetween two spins IiandIj. Here,\u00160= 4\u0019\u000110\u00007N/A2is the magnetic constant, \ris the nuclear\ngyromagnetic ratio, and rdenotes the length of the vector connecting the two spins, which forms\nangle\u0012with the ^ezaxis.\nInteraction with resonant rf pulses is described by the time-dependent term Hrf(t) where the \feld\namplitude\u0017rf(t) =\u00170during a pulse and \u0017rf(t) = 0 otherwise. The spin operator I'determines the\nin-plane rotation axis about which the spin bath precesses under the rf \feld with a given phase ':\nI'=Ixcos'+Iysin': (7)\nHere, we want to study the spin bath evolution under the 'internal' static terms Hint=Hz\n0+Hzz\nd\nin the interaction frame of Hrf(t). AHT can give an approximate description of the time evolution\nafter one or more full rf cycles if three conditions are met: (i) The rf Hamiltonian is periodic over\nthe cycle time tc, i.e.Hrf(t+tc) =Hrf(t). (ii) The net spin rotation after a full rf cycle is a multiple\nof 2\u0019. (iii) Since AHT is a perturbation method in terms of tc=T\u0003\n2, the solution only converges\nquickly ifjHz\n0jtc=~\u001c1 andjHzz\ndjtc=~\u001c1.\nWe write the total time-evolution operator as\nU(t) =Texp\u0014\n\u0000i\n~Zt\n0dt0H(t0)\u0015\n=Urf(t)Uint(t); (8)\nwith Dyson time-ordering operator Tand\nUrf(t) =Texp\u0014\n\u0000i\n~Zt\n0dt0Hrf(t0)\u0015\n; (9)\nUint(t) =Texp\u0014\n\u0000i\n~Zt\n0dt0~Hint(t0)\u0015\n; (10)3\nwhere we introduced the toggling frame Hamiltonian\n~Hint(t) =U\u00001\nrf(t)HintUrf(t): (11)\nWhile an exact solution for Eq. (10) is generally challenging to \fnd, an approximate description\nof the spin bath evolution at times t=n\u0001tccan be found if conditions (i)-(iii) are ful\flled. In\nthis case, we can apply a Magnus expansion to replace the expression of Eq. (10) by an e\u000bective\naverage Hamiltonian \u0016Hsuch that\nUint(ntc) = exp\u0014\n\u0000i\n~ntc\u0016H\u0015\n= exp\u0014\n\u0000i\n~ntc\u0010\n\u0016H(0)+\u0016H(1)+:::\u0011\u0015\n;(12)\nwith leading order terms\n\u0016H(0)=1\ntcZtc\n0~H(t)dt; (13)\n\u0016H(1)=\u0000i\n2tc~Ztc\n0dt2Zt2\n0dt1h\n~H(t2);~H(t1)i\n; (14)\n\u0016H(2)=1\n6tc~2Ztc\n0dt3Zt3\n0dt2Zt2\n0dt1\n\u0010h\n~H(t1);h\n~H(t2);~H(t3)ii\n+h\n~H(t3);h\n~H(t2);~H(t1)ii\u0011\n:(15)\nThe contributions of higher order AHT terms to \u0016Hscale astk\nc\u0000k+1for\u0016H(k), with free decay rate\n\u0000/1=T\u0003\n2. We thus see that in the limit of n!1 cycles and cycle time tc!0, only the zeroth\norder term \u0016H(0)remains. However, in practice, higher order contributions are rarely negligible,\nmaking e.g. longer solid echo sequences such as MREVS5,S6and BR-24S3more e\u000ecient than the\nshorter WAHUHA cycleS7in many applications.\nWe calculate the average Hamiltonian for a spin bath interacting with a given pulse sequence\nas described by equations (2)-(5) using Mathematica with the freely-available non-commutative\nalgebra package NCAlgebraS8. For the zeroth order average Hamiltonian, we consider \fnite pulse\ndurations where t\u0019is the time required for a \u0019-rotation. In this case, the cycle time tcis given by\nthe sum of pulse times and pulse spacings \u001c. First and second order terms are only calculated in\nthe limit of in\fnitely sharp rf pulses t\u0019!0.\nThe AHT terms we obtain for the CHASE sequences presented in the main text are listed in\nSupplemental Table 1. For clarity, we split the k-th order average Hamiltonian into contributions\nfrom the resonance o\u000bset ( \u0016H(k)\n0) and dipolar Hamiltonian ( \u0016H(k)\nd). For higher orders k\u00151, we also4\nSupplemental Table 1. Overview of AHT terms up to second order calculated for the CHASE sequences\npresented in the main text. Zeroth order terms are calculated assuming \fnite pulse duration t\u0019whereas\nt\u0019!0 is assumed for higher order terms. Unlisted terms \u0016H(0);(1);(2)\n0 ,\u0016H(0);(1);(2)\nd,\u0016H(1);(2)\nd0are zero.\nAHT term CHASE-5\n\u0016H(0)\n02t\u0019\n\u0019tcHz\n0\n\u0016H(0)\ndit\u0019\n\u0019tcf[Hxx\nd;Iz] + [Hyy\nd;Ix\u0000Iz]\u0000[Hzz\nd;Ix]g\n\u0016H(1)\nd0itc\n18~[Hzz\nd\u0000Hxx\nd;Hy\n0]\n\u0016H(2)\nd1\n2\u0000tc\n18~\u00012[Hzz\nd\u0000Hxx\nd;[Hzz\nd;Hyy\nd]]\n\u0016H(2)\nd03\u0000tc\n18~\u00012f[Hy\n0;[Hy\n0;Hxx\nd\u0000Hzz\nd]] + [Hzz\nd\u0000Hyy\nd;[Hz\n0;Hy\n0]]g\nCHASE-10\n\u0016H(1)\nd0itc\n36~[Hzz\nd\u0000Hxx\nd;Hy\n0]\n\u0016H(2)\nd1\n2\u0000tc\n36~\u00012[Hzz\nd\u0000Hxx\nd;[Hzz\nd;Hyy\nd]]\n\u0016H(2)\nd03\u0000tc\n36~\u00012[Hy\n0;[Hy\n0;Hxx\nd\u0000Hzz\nd]]\nCHASE-20\n\u0016H(2)\nd1\n2\u0000tc\n72~\u00012[Hzz\nd\u0000Hxx\nd;[Hzz\nd;Hyy\nd]]\n\u0016H(2)\nd03\u0000tc\n72~\u00012[Hy\n0;[Hy\n0;Hxx\nd\u0000Hzz\nd]]\nCHASE-34\n\u0016H(0)\ndit\u0019\n\u0019tcf4[Hxx\nd;Iy\u0000Iz] + 2[Hyy\nd;Ix+ 2Iz]\u00002[Hzz\nd;Ix+ 2Iy] +i\u0019Hzz\ndg\n\u0016H(2)\nd03\u0000tc\n108~\u00012f2\n3[Hy\n0;[Hy\n0;Hxx\nd\u0000Hzz\nd]] +2\n3[Hzz\nd\u0000Hyy\nd;[Hz\n0;Hy\n0]]\u00001\n3[Hzz\nd\u0000Hxx\nd;[Hz\n0;Hx\n0]]g\ninclude mixed terms ( \u0016H(k)\nd0). The full k-th order average Hamiltonian is thus given by\n\u0016H(k)=\u0016H(k)\n0+\u0016H(k)\nd+\u0016H(k)\nd0: (16)\nOnly non-vanishing terms are listed, i.e. terms which do not appear in Supplemental Table 1 do\nnot contribute to the total average Hamiltonian.\nThe de\fnition of the additional resonance o\u000bset and dipolar coupling Hamiltonians used in\nSupplemental Table 1 is based on equations (3) and (4), i.e.\nHx\n0=hX\ni\u0001\u0017iIx;i;Hy\n0=hX\ni\u0001\u0017iIy;i; (17)\nHxx\nd=hX\ni<j\u0017ij(3Ix;iIx;j\u0000Ii\u0001Ij);Hyy\nd=hX\ni<j\u0017ij(3Iy;iIy;j\u0000Ii\u0001Ij): (18)\nAs discussed in the main text, we see that CHASE-5 has a non-vanishing zeroth order contribution\nif the pulse duration t\u0019is non-negligible. In order to suppress spin bath dynamics using cycles of\nCHASE-5, it is therefore crucial to minimise the ratio t\u0019=tc. The subsequent longer sequences are5\ninsensitive to \fnite pulse durations in zeroth order and leave progressively fewer higher order AHT\nterms in the t\u0019!0 limit. CHASE-34 forms an exception to this behaviour. While most e\u000ecient\nin suppressing the spin dynamics under ideal conditions, this cycle is also prone to \fnite pulse\ne\u000bects as the zeroth order dipolar average Hamiltonian contributes to dephasing under realistic\nexperimental conditions.\n(a) MS-7\n-xy y-y x\n0 t=6τevolx -x\n(b) MS-12\n-xy y-y x\n0 t=12τevolx -x-yy-yxx\n(c) MKL-68\nx-yy x -x-x-yyx\nt=96τevol0x-yy-x-x-yyx\nx-yy y -x-x-yyxx-yy-x-x-yyx2\nSupplemental Figure 1. Schematic of alternative pulse sequences from literature for refocusing inhomoge-\nneous and dipolar broadening. Labels are de\fned as in Fig. 1 of the main text. (a) MS-7 cycle as introduced\nby Moiseev and Skrebnev in RefS9. (b) MS-12 cycle as introduced by Moiseev and Skrebnev in RefS10. (c)\nMKL-68 cycle as described and used by Maurer, Kucsko et al. in RefS11.\nFor comparison, we also calculate the AHT terms of alternative sequences from literature which\nhave been proposed or used with the aim of suppressing both dipolar coupling and frequency o\u000bset\nterms. These results are listed separately in Supplemental Table 2.\nThe MS-7 pulse cycleS9(Supplemental Fig. 1(b)) yields average Hamiltonians identical to those\nof CHASE-5 in the short pulse limit. Its extension to MS-12S10(Supplemental Fig. 1(c)) removes\nodd-order AHT terms owing to its symmetry properties. However, unlike the longer CHASE\nsequences it is not robust against decoherence during a \fnite pulse duration t\u0019.\nThe intuitive approach of alternating MREV cycles with \u0019-pulses (MKL-68, Supplemental\nFig. 1(d)) was employed by Maurer, Kucsko et al. to extend nuclear spin coherence times in\ndiamondS11. Again, the performance of the cycle is limited under experimental conditions by a\nnon-vanishing zeroth-order term \u0016H(0)\nd.6\nSupplemental Table 2. Overview of AHT terms calculated for other cycles referred to in the main text.\nAHT term MS-7S9\n\u0016H(0)\n02t\u0019\n\u0019tcfHy\n0+Hz\n0g\n\u0016H(0)\ndit\u0019\n\u0019tcf[Hyy\nd\u0000Hzz\nd;Ix] +5\n2i\u0019Hxx\nd+ i\u0019Hyy\nd+ 2i\u0019Hzz\ndg\n\u0016H(1)\nd0itc\n18~[Hzz\nd\u0000Hxx\nd;Hy\n0]\n\u0016H(2)\nd1\n2\u0000tc\n18~\u00012[Hzz\nd\u0000Hxx\nd;[Hzz\nd;Hyy\nd]]\n\u0016H(2)\nd03\u0000tc\n18~\u00012f[Hy\n0;[Hy\n0;Hxx\nd\u0000Hzz\nd]] + [Hzz\nd\u0000Hyy\nd;[Hz\n0;Hy\n0]]g\nMS-12S10\n\u0016H(0)\ndit\u0019\n\u0019tcf[Hyy\nd\u0000Hzz\nd;2Ix] +5\n2i\u0019Hxx\nd+1\n2i\u0019Hyy\nd+5\n2i\u0019Hzz\ndg\n\u0016H(2)\nd1\n2\u0000tc\n36~\u00012[Hzz\nd\u0000Hxx\nd;[Hzz\nd;Hyy\nd]]\n\u0016H(2)\nd03\u0000tc\n36~\u00012f[Hy\n0;[Hy\n0;Hxx\nd\u0000Hzz\nd]] + [Hzz\nd\u0000Hyy\nd;[Hz\n0;Hy\n0]]g\nMKL-68S11\n\u0016H(0)\ndt\u0019\ntcf\u00002Hxx\nd+ 6Hyy\nd\u0000Hzz\ndg\n\u0016H(2)\nd1\n2\u0000tc\n288~\u00012[Hzz\nd\u0000Hxx\nd;[Hzz\nd;Hyy\nd]]\n\u0016H(2)\nd0\u00003\u0000tc\n288~\u00012[Hy\n0;[Hy\n0;Hxx\nd\u0000Hzz\nd]]7\nSupplemental Note 2. PULSED SPIN LOCKING\n0.010 .11 1 01 000.00.20.40.60.81.0Normalised NMR SignalF\nree evolution time τevol (ms)pulse-to-cycle ratio tπ/tc \n37.5% \n25% \n12.5% \n5% \n1.5% \n0.5% \n0.15% \n0.05%\nSupplemental Figure 2. Dependence of the experimentally measured normalised71Ga nuclear spin echo\namplitude on the free evolution time \u001cevolunder the phase-alternating CP-X series tested in the main text.\nEach trace shows data for a \fxed pulse-to-cycle ratio t\u0019=tcand varying \u0019-pulse number. Values are extracted\nfrom exponential decay \fts to experimental data as shown in Fig. 2(b) of the main text. The Hahn echo\ndecay \ft is shown as a solid red line for comparison.\nThe extended nuclear spin polarisation decay times we observe in Figs 2(c,d) of the main text\nunder phase-alternated Carr-Purcell sequences (CP-X) are attributed to a form of pulsed spin\nlocking described theoretically by Li et al.S12This spin locking mechanism arises due to dipolar\nevolution during the non-negligible \u0019-pulse duration t\u0019. As shown by the authors, in this limit\napplication of average Hamiltonian theory yields\n\u0016H(0)=4t\u0019\n\u0019tcHy\n0+1\ntc(4\u001cHzz\nd\u0000t\u0019Hxx\nd); (19)\nfor a cycle ( \u001c\u0000\u0019x\u00002\u001c\u0000\u0019\u0000x\u0000\u001c). The static term /Hy\n0in equation (19) is subsequently removed\nby transformation into a second toggling frame where we time-average over a Rabi cycle in its\ne\u000bective \feld. The twice averaged Hamiltonian is\n\u0016\u0016H(0)=\u00001\ntc\u0012\n2\u001c\u0000t\u0019\n2\u0013\nHyy\nd: (20)\nAs the initial ( \u0019=2)xpulse of the CP-X sequence prepares the spin bath in the state Iywhich\ncommutes withHyy\nd, the magnetisation is preserved or `locked' and no spin echo decay is predicted8\nin zeroth order. Li et al. also examined the cases of \fxed-phase CP-X as well as CP-Y with and\nwithout phase-alternation and found that, in agreement with our experimental results, no such\ne\u000bect is predicted for the CP-Y sequence tested in the current workS12.\nAlternative mechanisms leading to prolonged coherence times under CP-X have been suggested\nby other authorsS13,S14. However, we can con\fdently link our experimental observation to the\npulsed spin locking mechanism outlined above. A key assumption in the transition to the second\ntoggling frame is that the spin bath evolves slowly under the Hamiltonian (4 \u001cHzz\nd\u0000t\u0019Hxx\nd)=tcover\nthe relevant timescale set by the Rabi frequency \n =4t\u0019\u0001\u0017i\n\u0019tc(c.f. condition (iii) for applicability of\nAHT in Supplemental Note 1). Hence we expect a strong dependence of the spin locking e\u000eciency\non the pulse-to-cycle time ratio t\u0019=tc2[0;0:5], wheret\u0019=tc!0 in the limit of in\fnitely short\npulses and t\u0019=tc!0:5 in the limit of continuous rf excitation.\nIn Figs 2(a) and 2(b) of the main text we show the nuclear spin polarisation decay as a function\nof the free evolution time for a \fxed number of pulses and varying pulse spacings. In order to\nverify the dependence of spin locking on the pulse-to-cycle ratio, we now replot this experimental\ndata as a function of the \u0019-pulse number for di\u000berent \fxed pulse-to-cycle ratios. Supplemental\nFig. 2 shows that the spin locking e\u000eciency is noticeably reduced for t\u0019=tc.10% and the Hahn\necho decay (red solid line) is fully restored for t\u0019=tc.0:5%.9\nSupplemental Note 3. PERFORMANCE BEYOND THE `HARD PULSE' LIMIT\nThe central transitions (CTs) of the half-integer quadrupolar nuclear spins in a self-assembled\nQD are typically inhomogeneously broadened to \u0001 \u0017inh\u001810\u000040 kHz by the strain induced electric\n\feld gradientsS15. In order to perform pulse sequences on the full CT experimentally, we need to\napply rf pulses which are su\u000eciently broadband (`hard'), i.e. have a large enough amplitude to\nperform the desired \u0019=2- or\u0019-rotation even for spins Iin the tails of the broadened transition\nspectrum where the rf excitation can have a resonance o\u000bset \u0001 \u0017i&10 kHz. It is often assumed\nimplicitly that this `hard pulse condition' is ful\flled.\nHowever, while this is readily achievable for small pulse numbers (for a single pulse we require\nTRabi.2=\u0001\u0017inh\u001925\u0016s), the hard pulse condition is increasingly di\u000ecult to meet for longer\nsequences. We consider the Bloch equations of motion in the rotating frame of a static magnetic\n\feldBz\n@~M(t)=@t=~\n\u0002~M\u0000~\u0000\u0001(~M\u0000~M0); (21)\ndescribing the evolution of magnetisation ~M=P\ni\rIiunder an angular velocity vector ~\n =\n(\nrfcos';\nrfsin';2\u0019\u0001\u0017i)Tand with relaxation rate ~\u0000 = (T\u00001\n2;T\u00001\n2;T\u00001\n1)Tand equilibrium mag-\nnetisation ~M0. Here, \n rf= 2\u0019=T Rabiis the resonant Rabi frequency. We see that even a small\nresonance o\u000bset \u0001 \u0017iresults in a slight tilt of ~\n towards the ^ ezaxis. As a consequence, an rf pulse\nof duration TRabiand carrier phase 'will no longer result in a perfect 2 \u0019-rotation of the magneti-\nsation~Mabout~\n. Instead, a small rotation angle error is introduced, which can rapidly lead to\nnon-negligible e\u000bects as the errors of subsequent pulses add up. In practice, this is re\rected in a\nloss of NMR signal amplitude after a pulse sequence even in the limit of short free evolution as the\ncontribution of spins with large \u0001 \u0017ito the measured ensemble magnetisation is reduced.\nWe consider this e\u000bect numerically for the71Ga and75As nuclear spin bath ensembles studied\nin this work. To quantify the `hardness' (frequency bandwidth) over which a given pulse sequence\nis stable against resonance o\u000bsets, we evaluate the evolution of a magnetisation vector ~Munder\nthe sequence including initialisation and readout \u0019=2-pulses as a function of \u0001 \u0017. In order to\nkeep our results as general as possible, we rewrite the resonance o\u000bset in terms of the inverse\nresonant Rabi period as \u000e= \u0001\u0017TRabi. We can then describe the rescaled rotation axis as ^ e\n=\n(1+\u000e2)\u00001=2(cos';sin';\u0000\u000e)T. For added simplicity, our model does not consider any spin relaxation\nor dephasing (e.g. through dipolar interaction) and we set T1=T2=1.\nSupplemental Figs 3(a,b) show the simulated dependence of the CP-X/Y sequences with alter-10\n-0.4- 0.20 .00 .20 .40.00.51.0-\n0.4- 0.20 .00 .20 .40.00.51.00\n1 02 03 04 05 00.00.51.075As71Ga(\nc)(b)7\n5As \n Magnetisation Mz/Mz(0)F\nrequency Offset δ (T-1R\nabi)CP-X cycles \nncycle=0.5 \nncycle=8 \nncycle=3271Ga(a)C\nP-Y cycles \nncycle=0.5 \nncycle=8 \n Magnetisation Mz/Mz(0)F\nrequency Offset δ (T-1R\nabi) \n Mz/Mz(0)C\nycle Number ncycleCP-XC P-Y \nGa-71 Ga-71 \nAs-75 As-75\nSupplemental Figure 3. (a,b) Normalised magnetisation along the ^ ezaxis after evolution under a series of\nnon-resonant rf pulses as a function of the resonance frequency o\u000bset \u000efor (a) CP-X and (b) CP-Y sequences\nwith alternating carrier phase. Solid curves show simulated data for di\u000berent cycle numbers ncycle. Dashed\nlines show normalised NMR spectra of the75As (blue) and71Ga (green) central transitions with frequency\naxes rescaled by TRabi. (c) Weighted average of the normalised magnetisation over the75As (circles) and\n71Ga (squares) central transition after a CP-X (solid symbols) or CP-Y (empty symbols) sequence as a\nfunction of refocusing cycle number ncycle.\nnating pulse carrier phase discussed in the main text on the frequency o\u000bset \u000e. Solid lines corre-\nspond to sets of simulations with di\u000berent \u0019-pulse numbers. Thus for example, the orange line in\nSupplemental Fig. 3(a) shows the relative change of the z-component Mz=Mz(0) of a magnetisation\nvector~M(with~M(0) = (0;0;1)T) after evolution under a CP-X sequence ( \u0019x=2\u0000(\u0019x\u0000\u0019\u0000x)8\u0000\u0019x=2)\nfor resonance frequency o\u000bsets \u00001=2TRabi\u0014\u000e\u00141=2TRabi. For comparison, experimental cw NMR\nspectra of the71Ga and75As central spin transitions rescaled by the respective shortest Rabi pe-\nriod we could reach experimentally ( TRabi(71Ga) = 5:6\u0016s andTRabi(75As) = 9:8\u0016s at rf power\nP= 200 W) are shown as dotted lines.\nThe desired performance is characterized by Mz=Mz(0)\u00191 over a wide range of o\u000bsets \u000e. In\nthis respect, we note that the CP-X cycle (Supplemental Fig. 3(a)) is very robust against frequency\no\u000bsets: even at large cycle numbers, the magnetisation vector is largely restored over a large range11\nof o\u000bsets. In practice, this means that we can expect a stable NMR echo amplitude at short evo-\nlution times independent of the number of applied \u0019-pulses (as long as additional pulse calibration\nerrors are negligible). This expected behaviour is shown in Supplemental Fig. 3(c): here, we cal-\nculate the expected experimental echo amplitude from a weighted average of the normalised \fnal\nmagnetisation over the respective71Ga and75As NMR spectra for di\u000berent cycle numbers ncycle.\nIn agreement with the experimental data shown in Figs 2(e,f) of the main text, the CP-X signal\namplitude (solid symbols) is stable. By contrast, we note in Supplemental Figs 3(b,c) that the\n`bandwidth' within which the CP-Y sequence can restore the initial magnetisation rapidly narrows\nwith increasing cycle number ncycle. Again, this is in agreement with the experimental observation\nin the main text, where the CP-Y NMR signal amplitude \u0001 Ehf(\u001cevol= 0) rapidly decreases with\nincreasing\u0019-pulse number. Although periodic revivals of the `restored' magnetisation are seen in\nthe orange line of Supplemental Fig. 3(b), these are experimentally compensated by the contribu-\n-0.4- 0.20 .00 .20 .40.00.51.07\n1Ga75As(d)(b)( a)C\nP-like ('X') \nCHASE-5 \nCHASE-10 \nCHASE-20 \nCHASE-34 \nMS-7 \nMS-12 \nMKL-6875As \n Magnetisation Mz/Mz(0)F\nrequency Offset δ (T-1R\nabi)71Ga(\nc)-0.4- 0.20 .00 .20 .40.00.51.0C\nPMG-like ('Y') \nCHASE-5 \nCHASE-10 \nCHASE-20 \nCHASE-34 \nMS-7 \nMS-12 \nMKL-68 \n Magnetisation Mz/Mz(0)F\nrequency Offset δ (T-1R\nabi)75As7\n1Ga-\n0.4- 0.20 .00 .20 .40.00.51.0C\nHASE-Y-20 \nncycle=1 \nncycle=3 \nncycle=6 \n Magnetisation Mz/Mz(0)F\nrequency Offset δ (T-1R\nabi)75As7\n1Ga-\n0.4- 0.20 .00 .20 .40.00.51.0 \n Magnetisation Mz/Mz(0)F\nrequency Offset δ (T-1R\nabi)CHASE-X-20 \nncycle=1 \nncycle=3 \nncycle=6\nSupplemental Figure 4. Normalised magnetisation along the ^ ezaxis after evolution under a series of non-\nresonant rf pulses as a function of the resonance frequency o\u000bset \u000efor (a) CHASE-X and (b) CHASE-Y\nsequences and additional cycles from literature. Dashed lines show normalised NMR spectra of the75As\n(blue) and71Ga (green) central transitions with frequency axes rescaled by TRabi. (c), (d) Simulated data\nfor CHASE-X/Y-20 with di\u000berent cycle numbers ncycle.12\ntions of spins at intermediate o\u000bsets \u000ewhere the \fnal magnetisation vector is e\u000bectively \ripped.\nThis oscillating behaviour arises as the total pulse rotation error adds up to multiple precessions\nof the magnetisation vector ~Mabout ^e\n.\nWe run the same simulations for the CHASE sequences introduced in the main text and for the\nalternative sequences from literature analysed in Supplemental Note 1. The results for the various\ncycles under `X'- and `Y'-initialisation pulses are depicted in Supplemental Figs 4(a) and 4(b),\nrespectively. We note that most sequences have a similar `hard pulse' bandwidth under both\ninitialisation pulse conditions. Overall, all of the sequences presented are more stable against\nresonance o\u000bsets than the CP-Y sequence studied in the main text. However, the broadband\nperformance of the CP-X sequence (Supplemental Fig. 3(a)) remains higher than that of any\nCHASE cycle. This is in qualitative agreement with our experimental results (compare Figs 2(e,f)\nof the main text).\nAdditionally, we see that multiple cycles of CHASE-X/Y-20 reduce the o\u000bset tolerance to some\nextent (Supplemental Figs 4(c,d)). This is con\frmed experimentally, as the echo amplitude of the\nspectrally broader75As ensemble is noticeably reduced with increasing cycle number in Fig. 2(e)\nof the main text.\nIn summary, the reduced NMR signal amplitudes observed in the experiments on the71Ga and\n75As CTs presented in the main text can be reproduced qualitatively using a simple Bloch model.\nWe conclude that this is not a fundamental limitation of the various pulse cycles we studied, but\ncan be attributed to `soft' rf pulses which for a given spectral broadening of the spin bath can\nin principle be avoided by using higher rf excitation powers. Alternatively, more advanced NMR\ntechniques such as composite pulses could be implemented in future experiments to increase the\n`hardness' of the applied pulsesS16,S17.13\nSupplemental Note 4. NUMERICAL SIMULATION OF THE NUCLEAR SPIN\nEVOLUTION UNDER PULSED RADIOFREQUENCY MANIPULATION\nIn this Note we describe the details of the numerical simulation of the quantum mechanical\nevolution dynamics of the interacting nuclear spin bath.\nA. The model.\nWe consider once more the model introduced in Supplemental Note 1, where the evolution of\nthe wavefunction  (t) describing the state of the nuclear spin bath in the rotating frame of an\nexternal magnetic \feld Bzis determined by:\n@ (t)=@t=\u0000(i=~)H(t) (t);\nH(t) =Hz\n0+Hzz\nd+Hrf(t):(22)\nAs before, the Hamiltonian H(t) is composed of a term Hz\n0describing here quadrupolar interaction\nwith electric \feld gradients, a nuclear dipolar coupling term Hzz\ndand a radio-frequency (rf) term\nHrf(t).\nWe consider half-integer spins Iand simulate evolution only of the Iz=\u00061=2 subspace cor-\nresponding to the NMR experiments on the central transition. The e\u000bect of the quadrupolar\ninteraction (more speci\fcally of its second order term) on the Iz=\u00061=2 manifold is equivalent to\nan additional magnetic \feld that changes the Larmor frequency of the i-th nucleus by \u0001 \u0017i. The\nquadrupolar term can then be written explicitly as:\nHz\n0= 2\u0019~NX\ni=1\u0001\u0017iIz;i; (23)\nwhere the summation goes over all Nnuclei.\nWe consider the case of high magnetic \feld (signi\fcantly larger than the local dipolar \feld), so\nthat the nuclear-nuclear interaction is described by the truncated dipole-dipole Hamiltonian:\nHzz\nd=\u00160\n4\u0019~2\r2X\ni<jx2\ni;j+y2\ni;j\u00002z2\ni;j\n(x2\ni;j+y2\ni;j+z2\ni;j)5=2\u0014\nIz;iIz;j\u0000(I+ 1=2)2\n2(Ix;iIx;j+Iy;iIy;j)\u0015\n; (24)\nwhere\u00160= 4\u0019\u000210\u00007N/A2is the magnetic constant, \ris the nuclear gyromagnetic ratio,\n(xi;j;yi;j;zi;j) is the vector connecting spins iandjand the summation goes over all pairs of\nnon-identical nuclei. We ignore here any possible contributions from pseudo-dipolar or exchange\ninteractions.14\nThe e\u000bect of the rf \feld is described by:\nHrf(t) = 2\u0019~(I+ 1=2)NX\ni=1(\u00171;x(t)Ix;i+\u00171;y(t)Iy;i); (25)\nwhere parameters \u00171;x(t) and\u00171;y(t) characterise the amplitude of the rf magnetic \feld along the\nxandyaxes of the rotating frame respectively. The parameters \u00171;x(t) and\u00171;y(t) are piecewise\nfunctions of time describing the variation of the rf \feld amplitude during the pulse sequence.\nIn the above equations 23, 24 and 25 the operators Ix,IyandIzare spin-1/2 Pauli matrices\nacting on the e\u000bective spin-1/2 subspace of the spin states with spin projections Iz=\u00061=2. The\nfactors (I+ 1=2) in equations 24 and 25 (which di\u000ber them from equations 3, 4, 5) arise from the\nmatrix elements of the IxandIyoperators of the full spin- Inuclei that are projected on to the\nIz=\u00061=2 subspace. All quantities are in SI units, e.g. frequencies \u0017are in hertz and Hamiltonians\nare in joules.\nThe evolution of the nuclear spin magnetisation is calculated by direct propagation of the\nSchr odinger equation (Eq. 22) using the 6-th order Runge-Kutta method implemented in Mathe-\nmatica 10.3 and 11.0.1. The Hamiltonian matrices are stored and handled as `sparse arrays' for\nimproved computation e\u000eciency. For a given system of few spins, direct evaluation of Eq. 22 gives\nan exact evolution of the wavefunction, thus yielding a complete description of the nuclear spin\ndynamics. The maximum number of spins Nthat can be simulated with this direct approach is\nseverely limited by the computation resources, since the memory and the computation time scale\napproximately as \u0018N2and\u0018N4respectively. Thus when performing simulations on a `toy' sys-\ntem with small N, the choice of the parameters and initial states becomes important for obtaining\nresults that are relevant for systems with much larger N(such as quantum dots with N\u001510000).\nThe choice of parameters is discussed in the following section.\nB. Model parameters for simulation of the nuclear spin dynamics in quantum dots.\nFor simulations, nuclear spins are placed at the nodes of the face-centered cubic (fcc) lattice.\nOne nucleus is placed at the origin x=y=z= 0 and the other nuclei are selected from its\nnearest neighbors { this way the nuclear spin cluster is kept as `spherical' and `dense' as possible\nwhich allows approximating the complexity and the magnitude of the dipolar interaction in a 3D\nlattice. Example clusters with N= 6,N= 12 andN= 19 spins are shown in Supplemental\nFig. 5(a). The nuclei are taken to be75As with gyromagnetic ratio \r= 2\u0019\u00027:29\u0002106rad/s.\nThe lattice constant of the fcc lattice is taken to be a0= 0:564786 nm, corresponding to GaAs at15\na temperature of \u00184 K.\nInhomogeneous quadrupolar interaction is introduced by varying the Larmor frequency \u0017Lof\neach spin (Eq. 23). Firstly, the Larmor frequency of the i-th spini2[1;N] in the rotating\nframe is set to be \u0001 \u0017i=\u0017Q(i\u00001=2\u0000N=2). This equidistant set of \u0001 \u0017iis then rescaled as\n\u0001\u0017i!k1\u0002\u0001\u0017i, preserving the mean Larmor frequency \u0001\u0017i= 0. The factor k1is chosen as an\nimplementation of a uniform random distribution in the range [0 :77::1:3]. At the next step, each\nfrequency \u0001 \u0017iis modi\fed further by adding a random o\u000bset k2;i\u0002\u0017Qwithk2;iselected randomly\nfor each nucleus from a uniform distribution in the range [ \u00000:325;0:325]. The purpose of such\na randomisation using parameters k1andk2;iis to eliminate spurious periodic `beatings' in the\nnuclear spin dynamics arising from the small number of the nuclear spins N < 20 used in the\nsimulations. Such `beatings' are not present in experimental decay curves on real III-V quantum\ndots where N > 10000, but similar features are observed for defect spins in dilute nuclear spin baths\n(e.g. NV centres in diamond), where electron-nuclear hyper\fne coupling leads to periodic coherence\ncollapsesS18,S19. The physical meaning of the randomisation procedure can be seen as follows: The\nquantum dot can be viewed as built of a large number of clusters, each containing Nspins with\na di\u000berent random distribution of quadrupolar frequencies. The experimentally measured NMR\nsignal is an average over all such clusters, which is simulated by Monte-Carlo averaging over k1and\nk2;iin the numerical calculations. The step \u0017Qfor the equidistant spacing of Larmor frequencies is\nchosen to be large compared to the dipolar coupling \u0017ijof any two spins, so that the suppression\nof dipolar \rip-\rops arising from quadrupolar interaction (characteristic of self-assembled quantum\ndotsS20) can be e\u000eciently simulated. We typically use \u0017Q= 2000 Hz, which is chosen empirically\nby observing that no change in spin dynamics occurs when \u0017Qis increased further. The ranges for\nk1andk2;iare also chosen from trial simulations to be large enough to suppress spurious `beatings'\nwhile still small enough to ensure that the minimum di\u000berence between any \u0001 \u0017iis large enough to\nemulate strongly inhomogeneous quadrupolar interaction. When simulating spin dynamics of the\nnuclei in the absence of quadrupolar e\u000bects we use the above procedure with \u0017Q,k1andk2;iset to\nzero.\nFor initialisation of the nuclear spin system we use the following procedure. Each of the N\nnuclear spins is randomly initialised in one of the four single-spin eigen states with Iz=\u00003=2::+3=2.\nThe probabilities to \fnd each nucleus in Iz=\u00061=2 states and Iz=\u00063=2 are taken to be 60% and\n40% respectively. The probabilities for the Iz= +1=2 andIz=\u00001=2 states are taken to produce\n75% polarisation degree in the Iz=\u00061=2 subensemble. Such a choice of probabilities corresponds\nclosely to the experimental conditions where optical pumping inducing nuclear spin polarisation16\ndegree of\u001850% is followed by adiabatic radiofrequency sweeps exchanging the populations of Iz=\n\u00061=2 andIz=\u00063=2 statesS20. The nuclei in the Iz=\u00063=2 states are then ignored when simulating\nthe spin dynamics of the Iz=\u00061=2 states. This is justi\fed since the Iz=\u00063=2 states have very\nlong correlation times ( \u001cc\u001810 s, Ref.S21) and act on the Iz=\u00061=2 spins simply as a source of\nquasistatic local magnetic \felds which are already taken into account by the inhomogeneous spread\nof the Larmor frequencies \u0001 \u0017i. This initialisation procedure gives a tensor product random state\nwhich is not an eigenstate but where each nucleus is in a single-spin eigenstate with Iz= +1=2\norIz= +1=2. In each simulation run (Monte-Carlo sample) the initial function is constructed by\nrepeating the above procedure and creating a linear superposition of 1000 basic random states with\nrandom complex weighting coe\u000ecients. Such a highly entangled pure state with \fnite polarisation\nalongzdirection has `self-averaging' properties arising from `quantum parallelism'S22and allows\nfor faster convergence of the Monte-Carlo simulations.\nThe typical number of Monte-Carlo samples is 1000. For each Monte-Carlo sample a set of\nnuclear frequency shifts \u0001 \u0017iis generated (with random parameters k1andk2;i), the wavefunction\nis then initialised into a random superposition state as described above. The time evolution of the\nwavefunction is then calculated numerically from the Schr odinger equation 22 with a Hamiltonian\nwhose time dependence is a piecewise function determined by the rf pulse sequences. The overall\ntime dependence of the nuclear spin polarisation is calculated by averaging over the Monte-Carlo\nsamples. In all simulations the nuclei are \frst initialised in a state polarised along the ^ ezaxis,\nthen a single \u0019=2-pulse is used to rotate the polarisation into the xyplane, then a time sequence\nconsisting of rf pulse rotations and free evolution periods is simulated, \fnally a single \u0000\u0019=2-pulse\nis used to rotate the magnetisation. All of the studied NMR pulse sequences are cyclic, i.e. in the\nlimit of a short free evolution and ideal rf pulses the magnetisation is returned into its original\nstate along the ^ ezaxis (or into a state with inverted z-magnetisation for HE-Y, the Meiboom-Gill\nversion of Hahn echo). In simulations we use both ideal (in\fnitely short, or `hard') and non-ideal\n(\fnite duration, or `soft') rectangular rf pulses. The total free evolution time \u001cevolis varied, and for\neach value of \u001cevolthe \fnal value of the nuclear magnetisation hIzialong the ^ezaxis is computed.\nThe resulting time dependence hIz(\u001cevol)ire\rects the process of nuclear spin decoherence and can\nbe used to derive the coherence time T2.17\n0.11 1 01 000.000.050.100.150.200.252\n46810121416182046810121416182022\n(b)H\nE-Y  CHASE-Y-20   N \n                  6 \n                  12 \n                  19Final spin magnetization 〈Iz(τ)〉//s61518F\nree evolution time, τevol (ms)(a)( c)C\nHASE-Y-20H\nE-YN\nuclear spin coherence time, T2 (ms)N\number of nuclei, Nexperimente\nxperiment\nSupplemental Figure 5. (a) Spatial geometry of the75As nuclear spin cluster used for numerical simulations.\nRed balls show a con\fguration with N=6 spins, red and green with N=12 spins, red, green and blue combined\ntogether form a cluster with N=19 spins. (b) Simulated dependence of the \fnal nuclear spin magnetisation\nhIz(\u001cevol)i(normalised by the number of spins N) on the total free evolution time \u001cevolunder Hahn Echo\n(HE-Y, solid symbols) and CHASE-Y-20 pulse sequences computed for N=6 spins (circles), N=12 spins\n(squares), and N=19 spins (triangles). A Meiboom-Gill version of Hahn echo sequence ( \u0019=2y\u0000\u001cevol=2\u0000\n\u0019x\u0000\u001cevol=2\u0000\u0019=2\u0000y) is used with a \u0019=2-shift between the phases of the \u0019=2- and\u0019-pulses. Lines show\nthe \ftting used to derive nuclear spin decoherence times T2. (c) Nuclear spin decoherence times T2derived\nfrom numerical simulations plotted as a function of the number of nuclear spins for Hahn Echo (circles) and\nCHASE-Y-20 (triangles) pulse sequences. Shaded areas show experimentally measured decoherence times\nof75As spins in a self-assembled quantum dot.\nC. Examples of simulations of the nuclear spin dynamics in quantum dots: dependence on\nthe number of spins N.\nWe now give several examples of the results obtained from the above described numerical\nsimulation procedure. Several simulated hIz(\u001cevol)icurves are shown in Supplemental Fig. 5(b) for\na Meiboom-Gill version Hahn echo (HE-Y, solid symbols) and CHASE-Y-20 (open symbols) pulse\nsequences { these are computed for clusters with di\u000berent numbers of nuclei shown in Supplemental\nFig. 5(a) for the case of large inhomogeneous quadrupolar interaction (\u0001 \u0017i\u001d\u0017ij). Lines show\n\ftting using compressed exponential functions. For the Hahn echo sequence the decay is close to\nGaussian (characterised by compression factor \f\u00192:0\u00002:1), while for CHASE-Y-20 the best \ft\nis for\f\u00191:56\u00001:67, and some deviation from a mono-exponential decay is observed, especially\nat smallN.\nThe nuclear spin decoherence times T2derived from the \fts as in Supplemental Fig. 5(b) are18\nshown in Supplemental Fig. 5(c) by the symbols and are compared to the experimental values\nfor75As spins in self-assembled quantum dots (shaded areas). It can be seen that the number of\nspins a\u000bects the overall timescale of the nuclear spin decoherence { for larger Nthe nuclear spin\ndecoherence is faster as the interaction with a larger number of neighbors is taken into account.\nHowever, the overall trend in variation of T2under di\u000berent pulse sequences is found to be robust\nagainstN. For example, while the decoherence times T2depend on Nas shown in Fig. 5(c),\nthe ratio of the T2values under CHASE-Y-20 and Hahn echo sequences is nearly independent of\nN, ranging between \u00182:74 forN= 19 and\u00183:0 forN= 6 which is in good agreement with\nthe experimental ratio of \u00182:8. These test results justify the use of relatively small N{ for\nmost simulations in this work we employ N= 12 giving a good compromise between accuracy\nand computation time. The fact that the T2simulated for N= 12 di\u000bers from the experimental\nT2on a system with N\u001810000 only by\u001850% indicates the robustness of our approach. Thus\nour simulations (i) give a good quantitative numerical estimate of the absolute T2values, and (ii)\nprovide an excellent tool for examining the e\u000bect of various pulse sequences on T2.\nD. Procedure for derivation of the nuclear spin coherence times and echo amplitudes from\nthe results of numerical simulations.\nWe now present the raw data of the numerical simulations for the CHASE sequences (Supple-\nmental Fig. 6) and discuss the procedure for analysing the raw data and deriving the spin bath\ncoherence times T2and the echo amplitudes hIz(\u001cevol= 0)iin the limit of short free evolution time\n\u001cevol!0. Supplemental Fig. 6(a) shows the simulated spin bath dynamics under CHASE-Y-20.\nThe results are presented for ideal in\fnitely short (`hard') rf control pulses (open symbols) and for\nthe \fnite (`soft') rectangular pulses (solid symbols, \u0019-pulse length of t\u0019= 10\u0016s). The calculations\nwere performed for 1 cycle of the sequence (triangles) and for 4 cycles (pentagons). Lines show\nbest least-squares \fts using compressed exponents. These \fts are used to derive the spin bath\ncoherence times T2. While for 1 cycle the \ft is good, for more complex conditions, e.g. 4 cycles\nandt\u0019= 10\u0016s, there is a considerable deviation between numerical experiment and exponential\n\fts { in such cases the T2times are still derived from \ftting but should be treated as approximate\nvalues.\nSupplemental Fig. 6(b) shows further results for the spin bath dynamics under the CHASE-Y-\n34 pulse sequence. Here deviation from the exponential \ft is observed for 4 cycles even at t\u0019= 0\nwhile att\u0019= 10\u0016s the oscillations and reduction of the echo amplitude at short free evolution time19\n10-410-310-210-11001011020.000.050.100.150.201\n0-410-310-210-11001011020.000.050.100.150.20(a)C\nHASE-Y-20 \n1 cycle, t/s61552=0 µs \n4 cycle, t/s61552=0 µs \n1 cycle, t/s61552=10 µs \n4 cycle, t/s61552=10 µsFinal spin magnetization 〈Iz(τ)〉//s61518F\nree evolution time, τevol (ms)F\ninal spin magnetization 〈Iz(τ)〉//s61518( b)C\nHASE-Y-34 \n1 cycle, t/s61552=0 µs \n4 cycle, t/s61552=0 µs \n1 cycle, t/s61552=10 µs \n        4 cycle, t/s61552=10 µsF\nree evolution time, τevol (ms)\nSupplemental Figure 6. Simulated dependence of the \fnal nuclear spin magnetisation hIz(\u001cevol)i(normalised\nby the number of spins N) on the total free evolution time \u001cevolunder CHASE-Y-20 (a) and CHASE-Y-34\n(b) pulse sequences computed for N=12 spins. The results are presented for in\fnitely short ( t\u0019= 0, open\nsymbols) and \fnite ( t\u0019= 10\u0016s, solid symbols) control pulses. The calculations were performed for 1 cycle\n(triangles) and for 4 cycles (pentagons) of the sequence. Lines show best least-squares \fts using compressed\nexponents. In the case of 4 cycles of CHASE-Y-34 with t\u0019= 10\u0016s the imperfect pulse rotations result in\nsigni\fcant loss of transverse nuclear spin magnetisation even at short \u001cevol{ this prohibits unambiguous\nde\fnition of the coherence time T2, thus no \ftting results are shown.\n\u001cevol!0 are particularly pronounced. This requires care when deriving decoherence parameters.\nFirstly, in our analysis the echo amplitude hIz(\u001cevol= 0)iis derived not from \ftting but rather by\ntaking the average spin magnetisation hIzi(normalised by the number of nuclei N) at short free\nevolution times \u001cevol<5\u0016s { this de\fnition of hIz(\u001cevol= 0)iis not a\u000bected by deviation of the spin\ndecay from the exponential model. Secondly, for any cyclic pulse sequences with ideal `hard' pulses\n(t\u0019= 0) the resulting magnetisation hIz(\u001cevol= 0)iafter the sequence with short free evolution\n\u001cevol!0 is by de\fnition equal to the initial magnetisation hIz(t= 0)ibefore the pulse sequence is\napplied (in the studied example hIz(t= 0)i=N\u00190:217), while for non-ideal pulses ( t\u0019>0) nuclear\nspin magnetisation at \u001cevol!0 may be lost simply due to the imperfect spin rotations (i.e. due\nto the `soft' pulse conditions). Such imperfect rotations mean that the spin bath states during\nfree evolution periods of \fnite duration \u001cevol>0 deviate from the desired sequence. Under such\nconditions (e.g. t\u0019= 10\u0016s in Supplemental Fig. 6) the reduction in hIz(\u001cevol)iis not related to20\ndecoherence as such, prohibiting any unambiguous de\fnition for T2.\nTaking into account the above arguments we establish an approach to the analysis of the nu-\nmerical results which can be summarised as follows: The echo amplitude hIz(\u001cevol= 0)iis derived\nby averaging the Izover short free evolution times \u001cevol<5\u0016s. For echo amplitudes hIz(\u001cevol= 0)i\nbelow 70% of the initial magnetisation hIz(t= 0)ithe coherence time T2is unde\fned, while for\nhIz(\u001cevol= 0)iabove this threshold, the T2is derived from \ftting with compressed exponential\nfunctions. Moreover, in the main text and the subsequent discussion we present echo amplitudes\nat short free evolution times hIz(\u001cevol= 0)inormalised by the initial magnetisation hIz(t= 0)i.\nE. Suppression of the nuclear spin \ructuations under various pulse sequences: results of\nnumerical simulations.\nIn the main text we present the results of numerical simulations for the pulse sequences used\nin the experimental work. Simulations are in good agreement with the experiment and con\frm\nrobust extension of the nuclear spin coherence time T2under CHASE pulse sequences. In this\nsection we present simulated nuclear spin dynamics under alternative pulse sequences reported in\nthe literature and compare their performance to CHASE.\nHaving discussed in the previous section how echo amplitudes hIz(\u001cevol= 0)iand coherence\ntimesT2are derived, we now examine their dependence on the control pulse sequence parameters.\nThe results of the simulations are summarised in Supplemental Fig. 7 for the cases of small inho-\nmogeneous quadrupolar interaction (\u0001 \u0017i\u001c\u0017ij, panels a, c) and large inhomogeneous quadrupolar\ninteraction (\u0001 \u0017i\u001d\u0017ij, panels b, d). Three types of sequences are presented: (i) CHASE-10/20\nas proposed in this work, (ii) the 7 and 12 pulse sequences proposed theoretically by Moiseev and\nSkrebnevS9,S10and labeled MS-7/12 here, (iii) the sequence consisting of 8 MREV-8 pulse trains\ninterwoven with four phase-refocusing \u0019-pulses used by Maurer, Kucsko et al.S11in experiments\non NV centers in diamond and labeled MKL-68. The results for Hahn Echo (HE) and Carr-Parcell\n(CP) sequences are shown as well for a reference. The numbers in the sequence labels stand for\nthe total number of rf control pulses in one cycle. All results in Supplemental Fig. 7 are plotted\nas a function of the total duration of the control rf pulse sequence (total gate time) in the units of\nthe\u0019-pulse duration t\u0019. Similar to the way the results are presented in the main text, we combine\nCHASE-10 with CHASE-20 and MS-7 with MS-12: the points with the shortest total gate time\ncorrespond to one cycle of CHASE-10 and MS-7 sequences, while points with larger gate times\ncorrespond to integer numbers of repeated cycles of CHASE-20 and MS-12. For each sequence we21\n11 01 001 0003\n301\n1010011 01 001 0003\n301\n101001\n1 01 001 0000.00.20.40.60.81.01\n1 01 001 0000.00.20.40.60.81.0t/s61552=80 µs \ntπ→0      tπ>0 \n               HE-X&CP-X \n               HE-Y&CP(MG)-Y \n       CHASE-X-10/20 \n       CHASE-Y-10/20 \n       CHASE-X-34 \n       CHASE-Y-34 \n       MKL-X-68 \n       MKL-Y-68 \n       MS-X-7/12 \n       MS-Y-7/12(a)Δ\nνi/s61574νijTotal gate time (in units of t/s61552)Nuclear spin coherence t\nime, T2 (ms)Improvedefficiencyt\n/s61552=10 µsImprovedefficiency(\nc)Δνi/s61575νijN\nuclear spin coherence t\nime, T2 (ms)Total gate time (in units of t/s61552)(\nb)Δ\nνi/s61574νijt\n/s61552=80 µsFinal magnetization for s\nhort free evolution, 〈Iz(τ=0)〉T\notal gate time (in units of t/s61552)(d)Δ\nνi/s61575νijt\n/s61552=10 µsF\ninal magnetization for s\nhort free evolution, 〈Iz(τ=0)〉T\notal gate time (in units of t/s61552)\nSupplemental Figure 7. Results of the nuclear spin decoherence numerical simulations for N= 12 dipolar\ncoupled75As nuclear spins under rf pulse control sequences for the case of small \u0001 \u0017i\u001c\u0001ij(a,c) and large\n\u0001\u0017i\u001d\u0001ij(b,d) quadrupolar broadening. Symbols in \fgures (a) and (b) show the nuclear spin coherence\ntimesT2for di\u000berent pulse sequences as a function of the total pulse (gate) time in units of t\u0019. The\nplot for each type of sequence is obtained by varying the number of cycle repeats; for Hahn Echo (HE),\nMS-7 and CHASE-10 we consider only one cycle and combine the data with CP, MS-12 and CHASE-20\nrespectively. The dashed lines represent constant e\u000eciencies of the pulse sequences, de\fned as coherence\ntime to gate time ratio. The gate time dependencies of the \fnal magnetisation (echo amplitude) at short\nfree evolutionhIz(\u001cevol= 0)iare shown in (c) and (d), the values are normalised by the magnitude of the\ninitial magnetisation hIz(t= 0)i. Simulations are carried out for both in\fnitely short ( t\u0019= 0, open symbols)\nand \fnite pulses (solid symbols), where we set t\u0019= 80\u0016s for \u0001\u0017i\u001c\u0017ijandt\u0019= 10\u0016s for \u0001\u0017i\u001d\u0017ij. The\nhIz(\u001cevol= 0)ivalues in (c) and (d) are plotted only for t\u0019>0 since att\u0019= 0 one hashIz(\u001cevol= 0)i= 1\nfor any cyclic pulse sequence by de\fnition.\nconsider two cases: with nuclear magnetisation initialised by a \u0019=2-pulse along the same ^ exaxis as22\nthe\u0019-pulses of the sequence (`-X' sequences) and with initialisation along the ^ eyaxis, orthogonal\nto that of the \u0019pulses (Meiboom-Gill version, labeled `-Y').\n1. The case of zero inhomogeneous resonance broadening.\nWe \frst examine the case of zero inhomogeneous resonance broadening \u0001 \u0017i\u001c\u0017ij(negligible\nquadrupolar e\u000bects or chemical shifts) as shown in Supplemental Fig. 7(a),(c). It follows from\nSupplemental Fig. 7(a) that all three types of sequences can be used to achieve arbitrarily long\nnuclear spin coherence time: the T2increases approximately linearly with the increasing number\nof sequence repetitions (increasing total gate time). This is largely expected from AHT { when\nthe number of cycles is increased, the cycle duration tcis reduced, and the average Hamiltonian\nconverges to its remaining zeroth order term \u0016H(0)\nd. As shown in Supplemental Tables 1 and 2,\nthis term vanishes for t\u0019= 0 for all studied sequences. For a given total gate time, the T2\nvalues are very close for all three types of sequences for initial magnetisation along either xor\nyaxes { the di\u000berence is less than a factor of 2. However, the performance of the sequences is\nnotably di\u000berent when non-ideal pulses ( t\u0019>0) are considered. For both MKL and MS sequences\na pronounced loss of magnetisation hIz(\u001cevol= 0)iat short free evolution (echo amplitudes) is\nobserved for the Meiboom-Gill (`Y') versions of the sequences when the number of cycles is increased\n(Supplemental Fig. 7(c)) { this means that strong nuclear spin decoherence is induced by the\n\fnite `soft' control pulses irrespective of decoherence during free evolution between the pulses.\nBy contrast the CHASE sequences show robust performance for an arbitrary direction of the\ninitial nuclear spin magnetisation for the total gate times of up to \u0018200t\u0019studied here, thus\ndemonstrating their capability to dynamically `freeze' arbitrary \ructuation of the transverse nuclear\nmagnetisation.\n2. The case of large inhomogeneous resonance broadening.\nThe case of large inhomogeneous resonance broadening \u0001 \u0017i\u001d\u0017ij(e.g. strong quadrupolar\ne\u000bects) is presented in Supplemental Fig. 7(b),(d). We start by examining the coherence times\nunder ideal `hard' control pulses ( t\u0019= 0, open symbols in Supplemental Fig. 7(b)). The MKL\nsequence exhibits reduced T2times, which are even shorter (for 1 cycle) than in the case of simple \u0019\npulse trains (Carr-Parcell sequences, CP). This is likely due to the fact that the MKL sequence was\nnot designed to be applied to strongly inhomogeneous spin systems in the \frst place. By contrast,23\nall of the CHASE and MS sequences provide enhancement in T2compared to Hahn echo and CP\nand show a similar non-monotonic behaviour on the total gate time which is also presented in Fig.\n3(b) of the main text for CHASE-10/20 sequences. For the total gate times up to \u0018100t\u0019\u0000200t\u0019\nthe nuclear spin coherence time T2is seen to decrease. Such reduction is also observed for the CP\nsequences and is interpreted to arise from fast rotations of the spins by the rf pulses which lead to an\ne\u000bectively shortened spin lifetime and broadened nuclear spin transitions. Such a broadening can\ncompensate for the energy mismatch between the spins induced by the quadrupolar inhomogeneity\nand restores the dipolar exchange spin-spin \rip-\rops. This interpretation is readily con\frmed by\nexamining the CP results: for a large number of pulse cycles (with the total gate time &100t\u0019)\ntheT2of the inhomogeneous (\u0001 \u0017i\u001d\u0017ij, Supplemental Fig. 7(b)) nuclear spin bath reduces to\nexactly the value of T2\u00192:02 ms observed for the homogeneous bath (\u0001 \u0017i\u001c\u0017ij, Supplemental\nFig. 7(a)) where dipolar \rip-\rops are allowed. When the number of CHASE or MS sequence\ncycles is increased further ( &500t\u0019in Supplemental Fig. 7(b)), T2increases steadily, indicating\nsuppression of dipolar interactions and convergence of the average Hamiltonian to zero, similar to\nthe homogeneous case (\u0001 \u0017i\u001c\u0017ij, Supplemental Fig. 7(a)). The interplay between the opposing\ne\u000bects of the reappearance of the \rip-\rops and the convergence of the average Hamiltonian depends\nstrongly on the magnitude of the quadrupolar inhomogeneity and rf pulse duration t\u0019. However, it\nis possible to establish a qualitative agreement between the experiment and the simulations: for a\nwide range of the CHASE-10/20 cycle numbers ( .200t\u0019in Supplemental Fig. 7(b)), T2is nearly\nconstant { this matches the weak dependence of the experimentally measured T2on the number\nof cycles as observed in Figs 2(c,d) of the main text.\nWe now examine the e\u000bect of the \fnite `soft' pulses ( t\u0019>0) under strong inhomogeneous\nbroadening conditions (\u0001 \u0017i\u001d\u0017ij, solid symbols in Supplemental Figs 7(b),(d)). It follows from\nSupplemental Fig. 7(d) that the loss of transverse spin polarisation during the control rf pulses\n(observed as decrease in the initial echo amplitude hIz(\u001cevol= 0)i) is most pronounced for the MS\nsequences { the spin coherence can be maintained well above hIz(\u001cevol= 0)i\u00180:7 only for one\ncycle of MS-7. One cycle of MKL-68 with a total gate time of 36 t\u0019can preserve the echo amplitude\nabove the 70% threshold but the resulting coherence time T2<2 ms is shorter than for Hahn echo.\nBy contrast, the CHASE sequences demonstrate the best performance in terms of both preserving\nthe echo amplitude hIz(\u001cevol= 0)iunder long rf pulse trains and enhancing the coherence time T2.\nWhile CHASE-20 can maintain hIz(\u001cevol= 0)i>0:7 for gate times >100t\u0019, the coherence time T2\ndecreases abruptly above 24 t\u0019in case of the `Y' initialisation pulse. A robust performance in terms\nof `freezing' of the spin bath \ructuation using \fnite pulses is obtained for either CHASE-10/20 or24\nCHASE-34 for the total rf pulse gate times up to 20 \u000024t\u0019with CHASE-34 producing a longer\ncoherence time T2.\n3. Analysis and discussion.\nIn various applications of magnetic resonance it is a common aim to seek for an optimal shape\nof the rf control \feld that produces the desired spin manipulationS23,S24. It is thus useful to\ncompare the di\u000berent pulse sequences discussed here by introducing a quantity that characterises\ntheir e\u000eciency. To this end we take the ratio of the coherence time T2during free evolution\nand the duration of the rf control pulses required to achieve such T2{ in other words, the pulse\nsequence is more e\u000ecient if it yields increased T2at reduced overhead of spin manipulation via\nthe rf control pulses. The dashed lines in Supplemental Figs 7(a),(b) show constant e\u000eciency\nlevels (given by linear functions with di\u000berent slopes). It follows from Supplemental Fig. 7(a)\nthat in case of zero inhomogeneous resonance broadening (\u0001 \u0017i\u001c\u0017ij) the e\u000eciency is nearly\ninvariant, gradually decreasing with the growing number of sequence cycle repeats. In case of\nlarge inhomogeneity (\u0001 \u0017i\u001d\u0017ij) the increase in the number of sequence cycle repeates (total gate\ntime) leads to reduction in e\u000eciency due to the re-appearance of the dipolar \rip-\rops discussed\nabove. Supplemental Fig. 7(b) shows that the best e\u000eciency is achieved for one cycle of either\nMS-7 or CHASE-10, while for one cycle of CHASE-34 the coherence time T2can be extended\nonly with some loss in e\u000eciency. These results indicate that when the dipolar-coupled spin bath\nis inhomogeneously broadened (Supplemental Fig. 7(b)) its coherence can be extended e\u000eciently\nonly by introducing complex pulse sequences that cancel higher order terms of the averaged spin\nHamiltonian { this is di\u000berent from the case of zero inhomogeneous broadening (Supplemental\nFig. 7(a)) where cycles of the basic sequence repeated multiple times e\u000eciently enhance the spin\nbath coherence.\nFurther improvements in simultaneous suppression of spectral broadening and dipolar couplings\nin nuclear spin baths may bene\ft from techniques beyond AHT. One example of such a technique\nare composite pulses. We have conducted preliminary numerical simulations with modi\fed CHASE\nsequences, where each pulse is replaced by a composite broadband BB1 pulseS25. However, these\npulses give no improvement and in fact result in a slight reduction of the nuclear spin coherence\ntimesT2, while requiring signi\fcantly longer gate times (and hence reduced e\u000eciency). Alternative\napproaches may involve more sophisticated tools, including numerical optimization algorithmsS26.\nTo summarise the results of these numerical simulations, we \fnd that optimal control of the spin25\nbath coherence is achieved using one cycle of the CHASE-10, CHASE-20, or CHASE-34 sequences\nas they (i) extend the spin bath coherence time T2both under small and large inhomogeneous\nresonance broadening, (ii) show robust preservation of the spin bath magnetisation even under\nnon-ideal \fnite duration (`soft') control pulses, (iii) e\u000bectively `freeze' nuclear spin \ructuations\nregardless of their direction in the plane perpendicular to the external magnetic \feld. The simula-\ntions for CHASE-34 predict signi\fcant improvement of the coherence compared to CHASE-10/20\nwhen applied to an inhomogeneously broadened system { this is con\frmed in experiment on nuclear\nspins in self-assembled quantum dots, although in the experiment the loss of spin bath magneti-\nsation is found to be larger than expected from simulations, most likely due the accumulation of\namplitude and/or phase errors of the real rf pulses. Overall, the CHASE sequences developed here\nprovide a well balanced performance and can be used to control spin bath \ructuations both in\nsystems with large inhomogeneous resonance broadening (e.g. quantum dots) and systems with\nsmall broadening (e.g. defect spins in diamond) where good tolerance to non-ideal \fnite pulses is\nrequired.\n\u0003andreas.waeber@wsi.tum.de\nye.chekhovich@she\u000eeld.ac.uk\nS1. Haeberlen, U. and Waugh, J. S. Coherent Averaging E\u000bects in Magnetic Resonance Physical Review\n175, 453 (1968).\nS2. Rhim, W. K. and Elleman, D. D. and Vaughan, R. W. Analysis of multiple pulse NMR in solids\nJournal of Chemical Physics 59, 3740 (1973).\nS3. Burum, D. P. and Rhim, W. K. Analysis of multiple pulse NMR in solids. III Journal of Chemical\nPhysics 71, 944 (1979).\nS4. Slichter, C. P. Principles of Magnetic Resonance Springer , corrected third (1996).\nS5. Rhim, W. K. and Elleman, D. D. and Vaughan, R. W. Enhanced resolution for solid state NMR\nJournal of Chemical Physics 58, 1772 (1973).\nS6. Mans\feld, P. and Orchard, M. J. and Stalker, D. C. and Richards, K. H. B. Symmetrized Multipulse\nNuclear-Magnetic-Resonance Experiments in Solids: Measurement of the Chemical-Shift Shielding\nTensor in Some Compounds Physical Review B 7, 90 (1973).\nS7. Waugh, J. S. and Huber, L. M. and Haeberlen, U. Approach to High-Resolution nmr in Solids Physical\nReview Letters 20, 180 (1968).\nS8. Helton, J. W. and de Oliveira, M. C. NCAlgebra http://math.ucsd.edu/ \u0018ncalg/ (2017).\nS9. Moiseev, S. A. and Skrebnev, V. A. Short-cycle pulse sequence for dynamical decoupling of local \felds\nand dipole-dipole interactions Physical Review A 91, 022329 (2015).26\nS10. Moiseev, S. A. and Skrebnev, V. A. Symmetric-cycle pulse sequence for dynamical decoupling of local\n\felds and dipole-dipole interactions Journal of Physics B 48, 135503 (2015).\nS11. Maurer, P. C. and Kucsko, G. and Latta, C. and Jiang, L. and Yao, N. Y. and Bennett, S. D. and\nPastawski, F. and Hunger, D. and Chisholm, N. and Markham, M. and Twitchen, D. J. and Cirac, J.\nI. and Lukin, M. D. Room-Temperature Quantum Bit Memory Exceeding One Second Science 336,\n1283 (2012).\nS12. Li, D. and Dong, Y. and Ramos, R. G. and Murray, J. D. and MacLean, K. and Dementyev, A. E. and\nBarrett, S. E. Intrinsic origin of spin echoes in dipolar solids generated by strong \u0019pulses Physical\nReview B 77, 214306 (2008).\nS13. Franzoni, M. B. and Levstein, P. R. Manifestations of the absence of spin di\u000busion in multipulse NMR\nexperiments on diluted dipolar solids Physical Review B 72, 235410 (2005).\nS14. Ridge, C. D. and O'Donnell, L. F. and Walls, J. D. Long-lived selective spin echoes in dipolar solids\nunder periodic and aperiodic \u0019-pulse trains Physical Review B 89, 024404 (2014).\nS15. Chekhovich, E. A. and Kavokin, K. V. and Puebla, J. and Krysa, A. B. and Hopkinson, M. and\nAndreev, A. D. and Sanchez, A. M. and Beanland, R. and Skolnick, M. S. and Tartakovskii, A. I.\nStructural analysis of strained quantum dots using nuclear magnetic resonance Nature Nanotechnology\n7, 646 (2012).\nS16. Levitt, M. H. and Freeman, R. NMR population inversion using a composite pulse Journal of Magnetic\nResonance 33, 473 (1979).\nS17. Tycko, R. Broadband Population Inversion Physical Review Letters 51, 775 (1983).\nS18. Mims, W. B. Envelope Modulation in Spin-Echo Experiments Physical Review B 5, 2409 (1972).\nS19. Jelezko, F. and Gaebel, T. and Popa, I. and Gruber, A. and Wrachtrup, J. Observation of Coherent\nOscillations in a Single Electron Spin Physical Review Letters 92, 076401 (2004).\nS20. Chekhovich, E. A. and Hopkinson, M. and Skolnick, M. S. and Tartakovskii, A. I. Suppression of\nnuclear spin bath \ructuations in self-assembled quantum dots induced by inhomogeneous strain Nature\nCommunications 6, 6348 (2015).\nS21. Waeber, A. M. and Hopkinson, M. and Farrer, I. and Ritchie, D. A. and Nilsson, J. and Stevenson, R.\nM. and Bennett, A. J. and Shields, A. J. and Burkard, G. and Tartakovskii, A. I. and Skolnick, M. S.\nand Chekhovich, E. A. Few-second-long correlation times in a quantum dot nuclear spin bath probed\nby frequency-comb nuclear magnetic resonance spectroscopy Nature Physics 12, 688 (2016).\nS22. Schliemann, J. and Khaetskii, A. V. and Loss, D. Spin decay and quantum parallelism Physical Review\nB66, 245303 (2002).\nS23. Khaneja, N. and Brockett, R. and Glaser, S. J. Time optimal control in spin systems Physical Review.\nA63, 032308 (2001).\nS24. Souza, A. M. and \u0013Alvarez, G. A. and Suter, D. Robust Dynamical Decoupling for Quantum Computing\nand Quantum Memory Physical Review Letters 106, 240501 (2011).\nS25. Wimperis, S. Broadband, Narrowband, and Passband Composite Pulses for Use in Advanced NMR27\nExperiments Journal of Magnetic Resonance A 109, 221 (1994).\nS26. Khaneja, N. and Reiss, T. and Kehlet, C. and Schulte-Herbr uggen, T. and Glaser, S. J. Optimal control\nof coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms Journal of\nMagnetic Resonance 172, 296 (2005)."    },    {        "title": "1312.7481v1.Effect_of_anisotropic_spin_absorption_on_the_Hanle_effect_in_lateral_spin_valves.pdf",        "content": " 1Effect of anisotropic spin absorption on the Hanle effect in lateral spin valves \nH. Idzuchi1,2*, Y . Fukuma2,3, S. Takahashi4,5, \n S. Maekawa5,6 and Y . Otani1,2* \n1 Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan \n2 Advanced Science Institute, RIKE N, 2-1 Hirosawa, Wako 351-0198, Japan \n3 Frontier Research Academy for Young Researchers, Kyushu Institute of Technology, 680-4 \nKawazu, Iizuka 820-8502, Japan \n4 Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan  \n5 CREST, Japan Science and Technology, Tokyo 102-0075, Japan \n6 Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan \n \nPACS; 72.25.Ba, 72.25.Mk, 72.25.Rb, 75.76.+j,   2Abstract \nWe have succeeded in fully describing dynamic properties of spin cu rrent including the \ndifferent spin absorption mechanism for longitudinal and transverse spins in lateral spin valves, which enables to elucidate intrinsic spin tran sport and relaxation mechanism in the nonmagnet. \nThe deduced spin lifetimes are found independen t of the contact type. From the transit-time \ndistribution of spin current extracted from the Fourier transform in Hanle measurement data, the \nvelocity of the spin current in  Ag with Py/Ag Ohmic contact tu rns out much faster than that \nexpected from the widely used model. \n \n* e-mail address: idzuchi@issp.u-to kyo.ac.jp, yotani@issp.u-tokyo.ac.jp  3Rapid development in spintronics is underpi nned by solid understand ing of fundamental \nproperties of spin transport [ 1,2]. The dynamic transport propertie s of spin current have been \nanalyzed by a response of sp in precession and dephasing since the pioneering wo rk of Johnson \nand Silsbee in 1985 [3] and this so-called Hanle effect analysis has been employed up to the \npresent to extract the spin lifetime, the velocity and the transit-time distribution between the injector and the detector [4-9]. However, recen t experimental progress in  creating spin currents \nrevealed new experimental results  which could not be explained by the previous framework. For \nexample, the Hanle analyses of dynamic spin transport properties of graphene, recently performed by assuming an empirical transit-time distribution, yielded stri kingly different spin \nlifetimes depending on the type of contacts a lthough intrinsic (bulk) pr operties of the spin \ntransport in nonmagnetic materials should be independent of the cont act type [8]. In the case of \nsilicon, the experimental Hanle si gnals could not be fully descri bed by the empirical model based \non a drift-dominated transit-time distribution in spin-transport of se miconductor [10,11]. For \nGaAs, solid analysis of spin relaxation in a two-dimensional electron  gas is hampered by \ncomplexities of charge and spin transports [1 2,13]. Therefore, it is essential to provide a \nframework for understanding the dynamic spin tran sport properties in th e nonmagnetic materials. \nIn this Rapid Communications, we establis h the formalism of Hanle effect to deduce \nintrinsic spin transport propertie s in nonmagnetic materials. The experimental studies are based  4on metallic lateral spin valves, which have comparative advantage in designing the measurement \nscheme owing to clear physics of charge and spin transport and spin relaxation mechanism \n[14,15], good controllability of dimensions where one-dimensional transport model is applicable, \nand comparability of junction property from low re sistive transparent junctions to high resistive \ntunnel junctions [4,7,9,14-21]. As a consequence, we have succeeded in identifying the impact of \nspin absorption effect on the deduced spin lifeti me and obtaining intrinsic spin lifetime which is \ncomparable with other experimental probes su ch as conduction electron spin resonance. \nIn order to establish a model of dynamic spin-transport, the Hanle effect was measured in \nvarious lateral spin valves (LSVs) with Ni 80Fe20 (Permalloy, Py)/Ag Ohmic and with \nPy/MgO/Ag junctions. Samples were prepared on a Si/SiO 2 substrate with a suspended \nresist-mask by using shadow evaporation technique  [21] and fabricated LSVs consist of two \nferromagnetic Py wires (140-nm-wide and 20-nm-thi ck [22]) bridged by a nonmagnetic Ag wire \n(100-140 nm-wide and 100-nm-thick). When the curre nt is applied to the Py/(MgO/)Ag injector \njunction, the diffusive spin current is generated in the nonmagnetic wire. With the perpendicular \nmagnetic field Bz applied, the spins begin to precess, and the transit time for the spin t is deduced \nfrom a change of the angle in the orientation at  the detector, which determines the output signal \nof the device [3,4]. Figure 1 shows Hanle signa l for LSVs with both the Py/Ag and Py/MgO/Ag \njunctions, with the inje ctor-detector separation L varied from 3.00 m to 6.00 m. The spin valve  5signal RS corresponds to the difference in non-loca l resistances between  the parallel and \nantiparallel magnetic configurations of the injector and the detector at BZ = 0. The value of RS \ndecreases with increasing L because the spin accumulation decrea ses due to the spin relaxation in \nAg [15]. Also, the values of RS for Py/Ag junctions are reas onably smaller than those for \nPy/MgO/Ag junctions due to the spin resistance mismatch [20,21]: in the case of Ohmic Py/Ag \njunction, the spin current in the Ag wire is  absorbed into Py, which is  expected from very low \ninterface resistance RI for Py/Ag. In Fig. 1, the first cross-point /2\nzB of the Hanle signal for the \nparallel and antiparallel magnetic configuration of the injector an d detector Py wires corresponds \nto the transit time when the collective /2 rotation of diffusive spins is completed. The /2\nzB \ndecreases with increasing L because of the increased transit time in the Ag wire. Figure 1 also \nshows that the magnitude of /2\nzB alters depending on the type of junctions: for LSV with L = \n6.00 m, the Py/Ag junctions give /2\nzB ~  156 mT whereas the Py/MgO/Ag junctions give  \n120 mT. These values corre spond respectively to /2\nL~2.75  1010 s-1 and 2.11  1010 s-1, \nindicating that faster spin di ffusion for the Py/Ag junctions compared with the Py/MgO/Ag \njunctions. This tendency wa s consistently observed in the LSVs both with L = 4.50 m and 3.00 \nm, the latter of which has the most pronounced difference in /2\nzB between the LSVs with \nPy/Ag and Py/MgO/Ag junctions. \nIn order to understand more explicitly the effect of the spin absorption on the dynamic  6property of spin transport, the transit-time di stribution was examined. Ha nle signal is described \nby integrating the transit-time dist ribution with Larmor precession as \n L00(, ) ( ) c o s ( )y Vd t S x L t d t P t t  ,  (1) \nwhere Sy(x=L,t) is the net spin density along the y direction parallel to the easy axis of \nferromagnet at the detector, t is the transit time and P(t) is the transit-time distribution of the net \nspin density given by its modulus S(L,t) [Sx2(L,t)Sy2(L,t)]1/2 [4, 9]. This means that spins \ninjected at x=0 arrive at the detector position with a probability of P(t) and the detection voltage \nis proportional to the integrated y-component spin density S(x=L,t)cos(Lt) with respect to all the \npossible transit time. After the spin begins to reach the detector, the P(t) increases until the \nspin-flip nature appears, i.e., the transit time beco mes comparable to the spin  lifetime. As a result, \nthe transit-time distribution exhibi ts a typical peak structure as s hown in Fig. 2(a), and is usually \ndescribed by an em pirical distribution \n2\nem N sf\nN1() e x p / 4 (/ ) ,\n4() Pt L D t t\nDt    \n     (2) \nwhere DN is the diffusion constant for spin and sf is the spin lifetime [4]. Considering the fact \nthat the Hanle signal is given by equation (1), P(t) can be directly derived by applying Fourier \ntransform to the experimental Hanle signal [ 10]. Figure 2(a) and 2(b) show the derived P(t) by \nperforming Fourier transform for the 6 m spin transport in LSVs. In the case of LSVs with \nPy/MgO/Ag junctions, experimental data agree excellently with the curve obtained from an  7empirical model equation (2) with the spin lifetime in table I and the diffusion constant derived \nwith Einstein relation, which valid ates this scheme. On the other hand, in the case of LSVs with \nPy/Ag junctions, P(t) from Fourier transform is shifted to th e left-hand side with respect to the \none expected from the empirical equation (2), suggesting the faster spin diffusion. The experimental P(t) is remarkably different from the empirical equation (2); this makes us desire to \nconstruct the model of transit-time distributi on to go beyond the empiri cal one which does not \nconsider the spin absorption. \nIn order to gain the insight of the effect of spin absorption on the dynamic properties of \nspin currents in nonmagnet, we formulate the Ha nle effect for LSVs with low resistive Ohmic \njunctions to tunnel junctions. For this, following tw o issues have to be fully taken into account: \nfirstly, the spin absorption by both injector and detector fe rromagnets, affects a spatial \ndistribution of chemical potential [23,24]. In ad dition to it, a recent experiment of Ghosh et al \nshowed that spin relaxation processes in ferro magnets were different between longitudinal and \ntransverse spin currents [25-27]. Their results suggest th at the spin relaxation is expected to be \nmore pronounced when the diffusive spins are orie nted perpendicular to the magnetization of the \ndetector via precession. The longitudi nal component of spin current \n||\nSiI through i-th junction ( i \n1, 2) is described as ||\nIIF I F | | /( / 2 ) [ ( ) ]Si i i i i i iI PG e G e x     , where IiP is the \ninterfacial-current spin-polarization, GIi is the interface conductance, FF F(+) / 2ii i  , ()\nFi   8is the spin-dependent electrochemical potential of F i, Fi is the spin accumulation of F i at the \ninterface, || F() () / ( )y xS x N  is the longitudinal component of spin accumulation in the Ag \nwire, N(F) is the density of state at Fermi energy, and xi is the contact position ( x1 = 0, x2 L). \n||\nSiI is inversely proportional to  the spin resistance of i-th ferromagnet RFi, as schematically \nshown in Fig. 3(a). In the presen ce of transverse spin accumulationF () () / ( ) ,x xS x N  the \ntransverse spin current SiI is given by (/ )( )Si i iI Ge x \n  , where iG is the real part of spin \nmixing conductance at the i-th interface [28] as schematically shown in Fig. 3(b). The spatial \ndistribution of  and || are illustrated in Figs. 3(c) an d 3(d) with considering different \nmechanism of spin absorption for longitudinal and transverse spin accumulation, based on the \nmodel of Stiles and Zangwill [ 29]. The spin accumulations, ||()x  and ( ) x  in the Ag \nnanowire are given by the complex representation || (,) () ()xtx i x      [22] \nL L|| ||\n11 2 2\nem em00\nFN FN() (,) ( ,)2( ) 2( )it it SS S SIi I Ii Ix dtP x t e dtP x L t eeN A eN A   , (3) \nand the spin current density in the complex representation is N(/ 2 ) ( ) , ()Sj ex x     where \nN is the electrical conductivity of Ag wire. Us ing the boundary conditions that the spin and \ncharge currents are continuous at the interfaces of junctions  1 and 2, we obtain the spin \naccumulation voltage 2VVdetected by Py and the nonlocal resistance V/I of Hanle signal in \nLSV [22], from which parameters can be direc tly deduced without using effective one [23]. \nWhen the junctions are the t unnel junction, the Hanle signal  reduces to the conventional  9expression in LSVs [4,21] in the limit of small spin absorption. \nThe experimental results are well reproduced by  the present theoretical calculations using \nreasonable parameters listed in Table I, as can be seen in Fig. 1. The obtained spin polarizations \nPF and PI agree well with our previous results [21] a nd values reported in [30]. The resistivity of \nPy was 1.75×10-5 cm. The junction resistance of Py/MgO/Ag was 20 , which is enough \nhigher compared with spin resistance RAg = NN/AN = 1 . The interfacial resistance of Ohmic \nPy/Ag junctions and the spin diffusion length of Py are respectively taken as RIAJ = 5  10-4 \nm)2 [30] and Py = 5 nm [31] from the literature. DN = 612  19 cm2/s is derived from \nEinstein relation N  e2DNN(F) where N(F) = 1.55 states/eV/cm3 [32]. While the shape of \nHanle signal is drastically modified  by the junctions as in Fig. 1, the spin lifetimes, 40.8 ± 6.2 ps \nand 40.3 ± 7.3 ps, for Py/Ag and Py/MgO/Ag juncti ons agree well with each other. The spin \nrelaxation mechanism is characteri zed by the spin relaxation ratio a  e/sf with respect to the \nmomentum relaxation time e. For Ag, a = 0.10 ps / 40 ps = 2.5  10-3 obtained in this study is \nconsistent with that (2.50  10-3) derived from the conduction electron spin resonance (CESR) \nexperiment [33], in which the spin relaxation mech anism was identified as Elliott-Yafet type. The \nagreement on the value of a for Al and Cu determined fr om the transport and the CESR \nmeasurements has also been repor ted [3, 14]. Therefore, the spin relaxation time obtained in this \nstudy is an intrinsic property of Ag. The char acterization was only possible by using the device  10structure designed in the present study, where the spin transport channel is much longer than the \njunction size and the surface spin scattering is su ppressed by capping layer [15]. In addition to it, \nthe Fourier transform of the theoretical Hanle signal agr ees with the experimental P(t) not only \nfor LSVs with Py/MgO/Ag junctions but also  for Py/Ag junctions, which complimentary \nsupports the validity of our model.  These results show that equatio n (1) cannot be used with the \nmost widely used P(t) = Pem(t) to analyze Hanle signal in LSVs of which RI is lower than RN due \nto the spin absorption effect. They may provide s purious spin lifetimes with mimicking signals or \nin some cases with different shapes of Hanle signals. In other words, the same spin lifetime \nresults in the different Hanle signal with and without spin ab sorption, the former of which \nexhibits a broader signal as shown in Fig. 1. Th is tendency is consistent with the reported Hanle \nsignals in graphene based LSVs with various type of junctions, where the spin lifetime is \ndeduced as 448-495 ps and 84 ps respectively for tunnel junction and tran sparent junction [8]. \nThe reanalysis of data using our model provid es 448-495 ps and 440 ps fo r tunnel junctions and \ntransparent junctions, respectively [22], which allo ws us to separate the intrinsic and extrinsic \nspin flip mechanisms in graphene. \nSpin absorption effect drastically alters the transit-time distribution. The velocity v is \nestimated as v  L/ttrans where ttrans  0dt[tP(t)] / 0dt P(t). Figures 2(a) and 2(b) show its speed \nas fast as 9.2  104 m/s for Py/Ag junctions and 6.6  104 m/s for Py/MgO/Ag junctions, which  11means the diffusion velocity depends  on not only diffusion constant but also a spatial gradient in \nthe accumulated spins. The velocity for the Py/Ag junctions is accelerated toward the detector \nbecause the spatial distribution of  the electrochemical potential is  strongly modified by the spin \nabsorption while the diffusion coefficient remains constant in consistent with theoretical report \n[34]. Figure 2(c) shows v and the full width at ha lf maximum (FWHM) of P(t) for Py/Ag \njunctions normalized by those for Py/MgO /Ag junctions. For Py/Ag junction not only v is higher \nbut also the FWHM is smaller than those for Py/MgO/Ag junctions, which has the more \npronounced difference for short L. FWHM is the essential parameter to characterize the coherent \nspin precession with respect to the applied fiel d because broad distribution of the dwell time \ngives rise to phase decoherence of the precessi ng spins [9]. The narrower FWHM for the Ohmic \njunction may pave the way for efficient contro l of spins in nonmagnetic material for active \nspintronic devices. \nOur model also enables to derive spin mixing conductance G which is one of the \nprincipal physical quantities ch aracterizing recent novel spintroni c effects such as spin pumping \nand insulating spin Seebeck effect [35, 36]. In the present study experimental G is shown in \nTable I, whereas theoretical G is roughly given by Sharvin conductance GSh\n = e2kF2/4h, \nwhere kF is the Fermi wave number of nonmagne t [37,38]. It provides the value of GPy/Ag\n  GSh\n \n= 3.7  1014 (m2)-1 (kF = 1.20×1010 m-1 is from [39]), which is cons istent with our experimental  12values. The larger theoretical value may be due to a reflection of the spin current at th e interface \n[37]. Similar behavior is reported for GPy/Cu\n of Py/Cu junctions: the experimental value of \nGPy/Cu\n was obtained as 3.9 1014 (m2)-1 from Giant Mageneto Resistance (GMR) study \nanalyzed by circuit theory on ferromagnetic/normal metal hybrid  device developed by Brataas et \nal [35,40], which is also smaller than the theoretical value GSh\n = 4.8  1014 (m2)1 (kF = \n1.36×1010 m-1 is from [39]). The quantitative evaluation of sf on the change of G is shown in \n[22]. We shall note here that G obtained in this study is different from the value obtained from \nspin pumping by a factor of 3-6 [25,41]. For th e spin pumping measurement, the magnetization \ndynamics in ferromagnetic resonance is used for in jecting spins in the no nmagnet, and therefore \nthe spin transport properties at the interface ma y be different from the Hanle effect and GMR \nmeasurements using static spin current [42]. The transport parameters  generally depend on the \nfrequency i.e. G=G(). Therefore, the Hanle measurements provide us an alternative scheme to \ndetermine  G. \nIn summary, we have studied the dynamic trans port properties of spin current in metallic \nlateral spin valves (LSVs) with various juncti ons. The effect of spin  absorption on the Hanle \nsignal was clearly observed in all the devices. The velocity of diffusive spin currents and the \ntransit-time distribution was successfully eval uated by applying Fourier transform to the \nexperimental Hanle signals, resulting in excellent  agreement with the empirical model in the case  13of Py/MgO/Ag junctions. In contrast, we found th at the transit-time dist ribution in LSVs with \nPy/Ag junctions was strongly deviat ed from that expected in the empirical model and that the \nspins diffuse much faster than in LSVs with Py/MgO/Ag junctions, reflecting the spatial distribution of chemical potential affected by the type of junctions. We have successfully \nformulated the Hanle effect for the LSVs with an isotropic spin absorption for the transverse and \nlongitudinal components of the sp in polarization in spin current s relative to the detector \nmagnetization-direction, which enables to elucidate intrinsic spin transport and relaxation mechanisms in the nonmagnet. The model also provides alternative way to determine the spin mixing conductance.  \nThis work was partly supported by Grant-in-Aid for Scientif ic Research (A) (No. 23244071), \n(C) (No. 22540346), Young Scientis t (A) (No. 23681032) from the Ministry of Education, \nCulture, Sports, Science and Technology , Japan, and Hoso Bunka Foundation. \n  14References and footnotes  \n[1] I. Žutić, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004). \n[2] C. Chappert, A. Fert, and F. N. Van Dau, Nature Mater. 6, 813 (2007). \n[3] M. Johnson and R. H. Silsbee, Phys. Rev. Lett. 55, 1790 (1985). \n[4] F. J. Jedema, H. B. Heershe, A. T. Filip, J. J. A. Baselmans and B. J. van Wees, Nature \n416, 713 (2002). \n[5] X. Lou et al. Nat. Phys. 3, 197 (2007). \n[6] T. Sasaki, T. Oikawa, T. Suzuki, M. Shirai shi, Y . Suzuki, and K. Noguchi, Appl. Phys. \nLett. 96, 122101 (2010). \n[7] G. Mihajlovi ć, J. E. Pearson, S. D. Bader, and A. Hoffmann, Phys. Rev. Lett. 104, \n237202 (2010). \n[8] W. Han et al. Phys. Rev. Lett. 105, 167202 (2010). \n[9] H. Idzuchi, Y. Fukuma, and Y. Otani, Scientific Reports 2, 628 (2012) \n[10] H.-J. Jang and I. Appelbaum, Phys. Rev. Lett. 103, 117202 (2009). \n[11] B. Huang, and I. Appelbaum, Phys. Rev. B 82, 241202(R) (2010). \n[12] C. P. Weber, N. Gedik, J. E. Moore, J. Or enstein, J. Stephens and D. D. Awschalom, \nNature 437, 1330 (2005). \n[13] L. Yang, J. D. Koralek, J. Orenstein, D. R. Tibbetts, J. L. Reno and M. P. Lilly, Nat. \nPhys. 8, 153 (2011). \n[14] F. J. Jedema, M. S. Nijboer, A. T. Filip, and B. J. van Wees, Phys. Rev. B 67, 085319 \n(2003). \n[15] H. Idzuchi, Y. Fukuma, L. Wang and Y. Otani, Appl. Phys. Lett. 101, 022415 (2012). \n[16] S. O. Valenzuela and M. Tinkham, Appl. Phys. Lett. 85, 5914 (2004). \n[17] S. Garzon, I. Žuti ć and R. A. Webb, Phys. Rev. Lett. 94, 176601 (2005).  15[18] A. Vogel, J. Wulfhorst and G. Meier, Appl. Phys. Lett. 94, 122510 (2009). \n[19] X. J. Wang, H. Zou, L. E. Ocola and Y. Ji, Appl. Phys. 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B 66, 014407 (2002). \n[30] J. Bass, and W. P. Pratt, J. Magn. Magn. Mater. 200, 274 (1999). \n[31] S. Dubois, L. Piraux, J. M. George, K. Ouna djela, J. L. Duvail and A. Fert, Phys. Rev. \nB 60, 477 (1999). \n[32] D. A. Papaconstantopoulos, Handbook of the Band Structure of Elemental Solids  \n(Plenum, 1986). \n[33] F. Beuneu and P. Monod, Phys. Rev. B 13, 3424 (1976). \n[34] S. Hershfield, and H. L. Zhao, Phys. Rev. B 56, 3296 (1997). \n[35] Y . Tserkovnyak, A. Brataas, and G. E. Bauer, Phys. Rev. B 66, 224403 (2002).  16[36] J. Xiao, G. E. Bauer, K. Uchida, E. Saitoh, and S. Maekawa, Phys. Rev. B 81, 214418 \n(2010). \n[37] J. Barnaś, A. Fert, M. Gmitra, I. Weymann, and V . K. Dugaev, Phys. Rev. B 72, \n024426 (2005). \n[38] A. A. Kovalev, A. Brataas, and G. E. Bauer, Phys. Rev. B 66, 224424 (2002). \n[39] N. W. Ashcroft, and N. D. Mermin, Solid State Physics  Ch. 2 (Brooks/Cole, 1976). \n[40] G. E. Bauer , Y . Tserkovnyak, D. Huerta s-Hernando and A. Brataas, Phys. Rev. B 67, \n094421 (2003). \n[41] F. D. Czeschka et al. Phys. Rev. Lett . 107, 046601 (2011). \n[42] In the case of spin pumping, inje cted spin is in resonant state. \n \n \nFigure captions \n \nFig. 1. Hanle signal in LSVs with Py/Ag junctions  and Py/MgO/Ag junctions with various \nseparations L. Black and red circles show respectively non-local resistance V/I of parallel and \nantiparallel magnetic configurations of th e injector and detect or electrodes at T = 10 K. Curves \nare obtained by the formula of Hanle effect [22]  with adjusting parameters shown in Table I. \nArrows (/2\nzB and /2\nzB) show the first cross-points of th e Hanle signal for the parallel and \nantiparallel configurations corresponding to the collective /2 rotation of diffusive spins. \n  17Fig. 2 (a),(b) Derived transit time distribution of pure spin current  P(t) (red circle) by \nperforming Fourier transform on Ha nle signal shown in Fig. 1(e) and (f). Dashed curves are \nderived by the empirical model, i.e., diffusion distribution with spin-f lip expressed by equation \n(2), with the values of DN and sf listed in Table I. Solid curve shows the distribution including \nthe effect of spin absorption [22]. All P(t) is normalized by P(tmax) where tmax gives the \nmaximum of P(t). (c) Velocity and full width at half  maximum (FWHM) for spin absorption \nmodel normalized by those for empirical m odel. Lines are guides to the eyes. \n \nFig. 3.  (a) Absorbed longitudinal spin current IS|| is proportional to longitudinal spin \naccumulation || and inversely proportional to the spin resistance of ferromagnet RF. (b) \nAbsorbed transverse spin current IS is proportional to transv erse spin accumulation  and the \nreal part of spin mixing conductance G. (c), (d) Schematic of || and  in the vicinity of \nthe detector junction. In ferromagnet,  is decaying with precessing along the magnetization \ndirection, which results in damp ing with oscillation [ 29]. The red and blue curves are calculated \nby the spin diffusion equation with using the equation (48) in [29]. \n  18Tables  \nTable I:  Adjusting parameters for Hanle signals wh ich are shown in Fig. 1. The uncertainties \nof adjusting parameters are determined by the least squares fittings. \n \nJunction  L (m) PF P I(Py/MgO/Ag)  P I(Py/Ag)  sf (ps) G  (m-2-1) \nPy/Ag 3.00  0.57  0.04 N/A 0.80  0.03 40.3  5.3  (3.5  0.9)  1014 \nPy/MgO/Ag 3.00  N/A 0.28  0.02 N/A 38.0  3.9  N/A \nPy/Ag 4.50  0.51  0.14 N/A 0.80  0.10 39.3  5.1  (2.0  0.9)  1014 \nPy/MgO/Ag 4.50  N/A 0.33  0.05 N/A 38.0  6.4  N/A \nPy/Ag 6.00  0.55  0.12 N/A 0.76  0.06 42.9  7.9  (3.6  8.4)  1014 \nPy/MgO/Ag 6.00  N/A 0.26  0.07 N/A  45.0  10.2 N/A \nTable I  Idzuchi et al.  19 \n \nFig.1  Idzuchi et al.  20 \n \n \nFig.2  Idzuchi et al.  21 \n \n \nFig.3  Idzuchi et al.  1SUPPLEMENTARY INFORMATION \nEffect of anisotropic spin absorption on  the Hanle effect in lateral spin valves \nH. Idzuchi1,2*, Y. Fukuma2,3, S. Takahashi4,5, S. Maekawa5,6 and Y. Otani1,2* \n1Institute for Solid State Physics, Unive rsity of Tokyo, Kashiwa 277-8581, Japan \n2Advanced Science Institute, RIKEN, 2-1 Hirosawa, Wako 351-0198, Japan \n3Frontier Research Academy for Young Researchers, Ky ushu Institute of Technology, 680-4 Kawazu, \nIizuka 820-8502, Japan \n4Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan \n5CREST, Japan Science and Technology, Tokyo 102-0075, Japan \n6Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan  \n \nNonlocal resistance in a lateral spin valve \nWhen a magnetic field B  (0, 0, Bz) is applied perpendicular to the plane of a spin \ninjection and dete ction device consisting of a nonmagnetic metal (N) connected to the \nferromagnets of the injector (F1) and the detector (F2) with the magnetizations (white arrows) along the \ny direction, the injected spins in the N electrode precess around the z axis parallel to \nB, as shown in Fig. 1. When the spin-current 1i\nsiIe polarized along the ei direction (, )ix y  \nis injected from F1 into N at x 1(0 )i\nsIthrough the 1st junction and the spin current2i\nsiIe \nis absorbed by F2 at x L 2(0 )i\nsIthrough the 2nd junction, the motion of the spin density S \ndue to the spin accumulation is governed by the diffusion-modified Bloch-Torrey equation \n[21,43] \n2 11\nN\nsf N N\n22\nNN() ()22\n                                               ( ) ( ),22xy\nss\nex y\nxy\nss\nxyIIDx xte A e A\nIIxLx LeA eA \n      \n SSSB S e e\nee\n             (S1)  2 \n \n  \n \nFig. S1. Precession of accumulated spins in N in a lateral spin valve in the presence of perpendicular \nmagnetic field B where the spin accumulation (spin density) S rotates during the travel of distance L \nbetween the injector F1 and the detector F2. The projection of S (S\ny) along the magnetization of F2 is \ndetected by F2 as output voltage V. \n \nwheresfis the spin-life times, DN is the spin diffusion constant, and AN is the cross- sectional \narea of N electrode, ei is the unit vector of the i-th direction, and the spin current is taken to \nbe the same unit as charge cu rrent. In perpendicular magnetic fields smaller than the \ndemagnetization field, the out-of-plane component Sz of the spin density is small and is \ndisregarded for simplicity. In the steady state  (/ 0 ) , tS  Equation (S1) is solved to yield the \nspin density S  (Sx, Sy, 0) in the complex representation [21] \n|| / | | /\n12\nNN() () ()4xx L\nyx s s Sx S x i S x Ie I eeD A                                   ( S 2 )  \nwith the complex representation of spin currents 11 1yx\ns ss I Ii I and 22 2yx\ns ss I Ii I  through \njunction 1 and 2 and \n 3N\nLs f,\n1i\n\n                                            (S3) \nwhereL ezB is the Larmor frequency,B2/eis the gyromagnetic ratio of conduction \nelectrons and Ns f N Dbeing the spin diffusion length in the absence of the \nperpendicular magnetic field. Since  is the complex quantity, exp( | | / ) x exhibits a \ndamped oscillation as a function of x. We note that (S2) is rewritten as [44] \nLL 12\nem em00\nNN() (,) ( ,) ,22it it ssIIS x dtP x t e dtP x L t eeA eA                      ( S 4 )  \nwhere em(,)Px t  is the transit-time distribution function: \n2\nN sf /(4 )\nem\nN/ 1(,)\n4xD t tPx t e e\nDt\n  .                                       ( S 5 )  \nWhen describing the spin transport in the presence of spin precession, it is convenient to \nuse the complex spin accumulationNF() ( 2 / )() / ( )xS x N  NN() ()yxxix   given by \n|| / || /\nN 1 2\nNN, () e exL x\ns sexI IA                                              ( S 6 )  \nwhereNis the electrical conductivity of the N electrode and N(F) is the density of states (per \nspin) at the Fermi energy. The absolute valueN|( ) | x  corresponds to the splitting in the \nelectrochemical potentials (ECP) of the up and down spin electrons. The charge transport is \ndescribed by the average of the ECPs: NN N (/ )  eI A x  for x  0 andN0 (ground level \nof ECP) for x  0. In equation (S6), the first term repr esents the increase of spin accumulation \ndue to spin injection from F1 and the second term  is the decrease due to spin absorption by F2. \nNote that the charge current is absent and th e pure spin current flows in the region of x  0. \nThe spin current density flowing in the x direction is given by the complex representation  4N\nNN () , ()2sjxxe                                         ( S 7 )  \nSince the thicknesses of the F1 and F2 ( nm) are much larger than the spin diffusion \nlength F(~ 5 n m ) , we may take the spin-dependent ECPs in F1 and F2 close to the \ninterfaces in the forms of vertical transport along the z direction [24] \nF\nF/ () F\nF1 F1 F1 1 ()\nF1 I1\n/ () F\nF2 F2 2 ()\nF2 I2() () ,   \n2\n() () ,   \n2z c y\ns\nz c y\nsezI p I e\nA\nezI e\nA\n\n\n\n \n\n \n  \n                                 ( S 8 )  \nwhere F1 F I1 1(/ )ceI A z eV  represents the EPC of charge in F1, AIk is the contact area of \nthe k-th interface,F2 2ceV eV takes a constant potential wit h no charge current in F2, V1 \nand V are the voltage drops across junctions 1 and 2, respectively, Fis the spin-diffusion \nlength of F1 and F2, andFF F F() /kk kp  where FF F()kk is the spin polarization \nof the F k. \nThe interfacial spin-depende nt currents across the k-th junction ( k 1, 2) with the \npolarizations parallel ()kIand antiparallel ()kIto the magnetization direction of F1 are \n[21,24,45,46] \nF NN\nII I\nF NN\nII I(0) ( ) ( )Re Im ,  22\n(0) ( ) ( )Re Im ,22kk k\nkk k k\nkk k\nkk k kxxIG i A Gee e\nxxIG i A Gee e   \n   \n  \n\n         \n       \n                       ( S 9 )  \nwhereIkGis the spin-dependent interface conductance of k-th junction,IkGis the transverse \ninterface conductance per area, so-called th e spin-mixing conductance with the dimension \n1m2, and x1 0 and x2 L. These enable to address the effect of spin absorption not only for  5longitudinal spin accumulation [21] but also for transverse one. We note that the complex \nrepresentations (S9) are equivalent to the vector  representation in [44]. The total charge and \nspin currents across the k-th interface are kkkIII , 12(,  0 )  II I and skkkI II . \nThe above interfacial currents are applicable to junctions from tunneling to transparent \nregime. \nUsing the boundary conditions that the spin and charge currents ar e continuous at the \ninterfaces of junctions 1 and 2, we can derive the matrix equation for the interface spin \ncurrents  \n1\n2 I1 1 F1 F1\n22\nI1 N F1 N 1\n20ˆ 2,0 11\n0y\ns\ny\ns\nx\ns\nx\nsI I\nI PR PRXPR PR I\nI                                       ( S 1 0 )  \nwith the matrix \n//\n1||\n//\n2||\n//\n1\n//\n2Re Re Im Im\nRe Re Im Imˆ ,\nIm Im Re Re\nIm Im Re Re[] [ ] []\n[ ] [] []\n[] [] [ ]\n[] [ ] [][]\n[]\n[]\n[]LL\nLL\nLL\nLLre e\ner eX\ner e\nee r\n\n\n\n  \n\n  \n\n  \n\n    \n \n \n  \n\n\n\n\n \n   \n \n\n\n             ( S 1 1 )  \nwhereN/ and \nIF\n22\nIN FN NI22 1,      ,       ( 1,2) 11kk\nkk\nkk kkRRrr kPR p R RA G   || . \nHere,II1/kkR GII I()kk kGGG is the interface resistance (conductance) of junction k, \nNN N N(/ ) R A  and FF F I(/ )kkR A  are the spin resistances of N and F electrodes, \nNandFare the resistivities, andII II I() / ()kk kk kPG G G G   is the interfacial spin-current \npolarization.  6The boundary conditions also le ad to the nonlocal voltage V due to the spin accumulation \ndetected by F2,  \nF2 F2 I2 I2\nN2 22\nF2 N I2 N,11y\nsPR PRVR IPR P R                                   ( S 1 2 )  \nwhere the minus sign indicates the absorption of spin current by F2. Using the solution of the \nmatrix equation (S10), we obtain the nonlocal resistance \nF1 F1 I1 I1 F2 F2 I2 I2 12\nN 222 2\nF1 N I1 N F2 N I2 N2,ˆ 111 1 det( )PR PR PR P R C VRIP R P R P R P R X                 ( S 1 3 )  \nwhere   ˆ det( )Xis the determinant of the matrix ˆXin (S11) and C12 is the (1, 2) component of \nthe cofactors of ˆX, \n//\n/\n12 1\n//\n2Re Im Im[ ]\ndet Im[ ] Re Re .\nIm Re Re[ ][] []\n[] [ ]\n[] []LL\nL\nLLee\nCr e\nee r\n\n\n \n\n \n\n  \n \n \n\n\n\n \n \n \n\n                      ( S 1 4 )  \nWhen junctions 1 and 2 are tunnel junctionsIN F(, )kkR RR , (S13) reduces to [21,44] \n I1 I2 NN1Re / exp( / ) .2VPPR LI                                    ( S 1 5 )  \nIn the absence of perpendicular magnetic field, (S13) reduces to the previous result of [24].  7Simulated Hanle curves for grap hene based lateral spin valve \nIn order to underscore the validity of our analysis, we fit the reported Hanle signal for \ngraphene based lateral spin valve with transpar ent junctions [8]. As shown in Fig. S2 and \ntable SI, the deduced spin lifetime and diffusi on constant are consistent with those from \ntunnel junctions. wF = 50 nm, wN = 2200 nm, F = 60 nm, F = 6 cm, RI = 285 , N = 0.35 \nmS are taken from [8].  \nTable SI:  Adjusting parameters for Hanle si gnals for graphene based LSV with \nCo/Graphene junction. \nL (m) PF P I(Co/Graphene) sf (ps) DN (cm2/s) G (m-2-1) \n3.00  0.40  0.0088 440  163  1.6  1010 \n \n \nFig. S2. Simulated Hanle curve of graphene base d lateral spin valves with transparent \njunctions. Dots (experimental data) are from [8] and blue lines are calculated from (S13).  8Spin valve measurement \nIn order to make the analysis simple, we used the same widths of Py wires. The switching \nfield of each Py wire was controlled by the do main-wall nucleation, i.e., the injector had a \nlarge domain wall reservoir at the edge, producin g lower switching field than the detector as \nshown in Fig. S3 [47].  \nThe initial state of the Hanle measurement was set as follows. For the parallel state, \nfirstly Py was initialized by the large field (~ 1000 Oe), and then the fi eld was set to zero. For \nthe antiparallel state, firstly Py was initialized by the large field, and secondly the field was \ndecreased to over the first switching field (~ -100 Oe), finally the field was set to zero. \n \n \n  \n \n \nFig. S3. Spin valve measurement (red lines). Blue  line indicates the initialization of Py \nmagnetizations for antiparallel configuration. Magnetic field was applied in parallel with the \neasy axis of Py. Bold arrows show the magnetiz ation states of injector and detector ferro \nmagnet. \n 9The quantitative evaluation of sf on the change of G \nHere we discuss the effect of anisotropy on the evaluation of spin lifetime. In the case of \nisotropic spin absorption, G = 1/ AJ{1/(2RI)+1/(2 RF)}, as shown in (S11). For Ag based LSV \nwith Py/Ag junctions with L = 3 m, the derived sf with isotropic spin absorption is 20 % \nsmaller than the one with anisotropic spin absorption because of underestimation of spin \nabsorption. In contrast, for graphene based LSV with transparent junction ( RI = 285 ) [8], \nthe derived sf with isotropic spin absorption is almost  same, which is consistent with derived  \nG is only 4 % different from 1/ AJ  {1/(2 RI)+1/(2 RF)}. The small effect of isotropy is attributed \nto higher junction resistance compared th e one with Ohmic contact in metallic system. \n \nAdditional Reference \n[43] M. Johnson, and R. H. Silsbee, Phys. Rev. B 37, 5312 (1988). \n[44] F. J. Jedema, M. V. Costac he, H. B. Heersche, J. J. Baselm ans, and B. J. van Wees, Appl. \nPhys. Lett. 81, 5162 (2002). \n[45] T. Valet, and A. Fert, Phys. Rev. B  48, 7099  (1993). \n[46] X. Wang, G. E. Bauer, B. J. van Wees, A.  Brataas, and Y. Tserkovn yak, Phys. Rev. Lett. \n97, 216602 (2006). \n[47] H. Idzuchi, Y. Fukuma, L. Wang and Y. Otani, Appl. Phys. Exp. 3, 063002 (2010). "    },    {        "title": "0903.2186v1.Simulation_of_a_spin_wave_instability_from_atomistic_spin_dynamics.pdf",        "content": "arXiv:0903.2186v1  [cond-mat.mtrl-sci]  12 Mar 2009Simulation of a spin-wave instability from atomistic spin d ynamics\nJ. Hellsvik,∗B. Skubic, L. Nordstr¨ om, and O. Eriksson\nDepartment of Physics and Materials Science, Uppsala Unive rsity, Box 530, SE-751 21 Uppsala, Sweden\n(Dated: February 6, 2020)\nWe study the spin dynamics of a Heisenberg model at finite temp erature in the presence of an\nexternal field or a uniaxial anisotropy. For the case of the un iaxial anisotropy our simulations show\nthat the macro moment picture breaks down. An effect which we r efer to as a spin-wave insta-\nbility (SWI) results in a non-dissipative Bloch-Bloemberg en type relaxation of the macro moment\nwhere the size of the macro moment changes, and can even be mad e to disappear. This relaxation\nmechanism is studied in detail by means of atomistic spin dyn amics simulations.\nPACS numbers: 75.10.-b, 75.20.En, 75.40.Gb\nI. INTRODUCTION\nRelaxation processes for magnetization dynamics are\npoorly understood but play a crucial role for spin dy-\nnamics in general. In this article we address one of\nthe most fundamental processes of magnetization dy-\nnamics, the uniform motion of the magnetization in an\nanisotropy field. The uniform motion of the magneti-\nzation and the relaxation of the uniform motion is of\ncentral importance in applications (magnetic switching\nin storage media etc.).1We study here a ferromagnetic\nsystem which initially is excited by a finite angle rotation\nof the magnetization with respect to the anisotropy axis.\nThis excitation brings the magnetization into a uniform\nmotion where the magnetization eventually relaxes back\nto an alignment with the magnetization direction par-\nallel to the anisotropy axis. Different phenomenological\nmodels, suchasGilbertdampingandBloch-Bloembergen\ndamping,2,3have been used for describing this macro\nlevel relaxation of the magnetization. In this article we\nperform simulations of magnetization dynamics on an\natomicscaleandwestudytheconsequencesforthemacro\nscale behavior of the magnetization dynamics.\nThe initial rotation of the magnetization of a ferro-\nmagnet in an external field can be seen as an excitation\nof a large number of uniform k=0 magnons. During the\nrelaxationprocessthesemagnonsinteract,dissipatingen-\nergy and angular momentum. Relaxation can occur via\ntwo processes, one where both energy and angular mo-\nmentum are transfered out (or in) of the magnetic sys-\ntem and the second where energy is transfered within the\nmagnetic system, to other non-uniform k/negationslash= 0 magnons.\nThe first process, which describes a dissipative damping\nin the equations of motion for magnetization dynamics,\nresults in a Gilbert like relaxation,\n∂M\n∂t=−γM×H+α\nMM×∂M\n∂t, (1)\nwhereMis the macro moment, Hthe effective field, γ\nthegyromagneticratio,and αadampingparameter. The\nsecond process, which is described by the precessional\nterm in the equations of motion, results in a special case(|Mz|constant) of the Bloch-Bloembergen damping,\n∂M\n∂t=−γM×H−Mx\nTˆ ex−My\nTˆ ey,(2)\nwhere the effective field is assumed to lie in the z-\ndirection and Tis a relaxation parameter. This second\nprocess is the focus of this work.\nWe address here a mechanism for the relaxation of\nthe uniform motion of the magnetization within the\nspin-system itself which is seen to result in a Bloch-\nBloembergen like damping of the magnetization of\nthe system. Several such mechanisms exist, such as\nthe Suhl instability,42-magnon scattering,5,64-magnon\nscattering,7etc. All of the mentioned mechanisms rely\non the dipolar interactions resulting in an energy lower-\ning of non-uniform magnons and an energy degeneracy\nbetween the uniform magnons and certain non-uniform\nmagnons.\nIn this article we study a different mechanism, since\ndipolar interaction are not even included in our simula-\ntions. The mechanism that we address is instead due\nto the thermal fluctuations on an atomic scale of the\nmagnetization combined with the nature of the uniax-\nial anisotropy. Such a mechanism, which does not rely\non dipolar interactions, were studied by Safonov et al.8\nand recently by Kashuba9and Garanin et al.10,11where\nit was shown that a spin wave instability (SWI) devel-\nops in a uniaxial anisotropy field. As we show in this\narticle, based on theoretical considerations, the insta-\nbility should develop on the atomic length scale as well\nas on the micrometer length scale which was treated by\nKashuba, provided certain conditions are fulfilled. The\ninstability is shown to be caused by the altering of the\nnon-uniform thermal magnetic excitations of the system\nas it undergoes a uniform rotation.\nII. DETAILS OF THE SPIN-DYNAMICS\nSIMULATIONS\nThere are at least two approaches for studying mag-\nnetization dynamics in simulations. Most common is\nwhat is pursued in micromagnetics, the solution of the2\nphenomenological Landau Lifshitz Gilbert (LLG) equa-\ntion on a micrometer length scale for a continuum\nmagnetization.12An alternative approach, which is uti-\nlized here, is based on solving the equations of motion\nfor magnetization dynamics where the magnetization on\na nanoparticleor atomic scale is represented by a Heisen-\nberg Hamiltonian. The Heisenberg Hamiltonian is often\nsuccessful in describing magnetic systems on an atomic\nscale, especiallywhenusingfirstprinciplescalculationsof\nthe interatomic exchange. The current method is based\non atomistic spin dynamics which has as starting point\na quantum mechanical description from density func-\ntional theory of the evolution of the atomic spins. Other\nworks which have taken this approach can be found in\nRefs. 13,14,15.\nOur simulations are performed using the ASD (Atomic\nSpin Dynamics) package16which is based on an atom-\nistic approach of spin dynamics. Interatomic exchange\nand magneto-crystalline anisotropy (MA) are included\nin the Hamiltonian. We use a parameterization of the\ninteratomic exchange part of the form of a Heisenberg\nHamiltonian, where the exchange parameters are calcu-\nlated from first principles theory. The effect of temper-\nature is modeled by Langevin dynamics. Connection to\nan external thermal bath is modeled with a Gilbert like\ndamping. The simulations are performed on bcc Fe using\nfour coordination shells in the Heisenberg Hamiltonian.\nIn order to ease comparison we used the same exchange\nparameters as in Refs. 15,16.\nIII. DYNAMICS IN AN EXTERNAL FIELD\nDifferent coarse grained levels can be used for describ-\ning magnetization dynamics. Here we will work with two\nlevels: (1) the individual atomic moments, mi, and (2)\na macro moment, M, representing the sum of the indi-\nvidual atomic moments of the total system. Any atomic\nmoment is typically exposed to an interatomic exchange\nfield,Beff,i, of the order of 1000 T. At finite temperature\nthe atomic moments fluctuate around a common direc-\ntion. On average,below the critical temperature, there is\na finite magnetic moment and the interatomic exchange\nfield averagedoverall atoms is directed along the average\nmoment. The size of the average moment or the macro\nmoment depends on the spread of the individual atomic\nmoments, whichisgovernedbythetemperature. The sit-\nuation is illustrated in the top part of Fig. 1, where the\ndistribution of atomic moments is illustrated for T=0 K\n(Fig. 1a) and at finite temperature (Fig. 1b). This de-\nscription of magnetization at finite temperature is the\nstarting point for our discussion.\nIf the system is exposed to an external field, the aver-\nagemomentwillprecessinthisexternalfield. Theatomic\nmoments precess in a uniform way without distortion of\ntheir internal distribution. The torque exerted by the\nexternal field, ∂mi/∂t=−γmi×Beff,ion each atom\ni, results in an equal angular velocity of all the atomicθ\nφ\nAnisotropy ax is External field axisT=0 T>0 a. b.\nd. c.\nFIG. 1: Figures (a) and (b) show the distribution of atomic\nmoments of the spin dynamics simulations. At finite tem-\nperature the orientations of the atomic spins are distribut ed\naround a common axis (b). The angles θandφdiscussed in\nthe text are also defined (a). Figures (c) and (d) show the\nevolution of the spin distribution, as given by the evolutio n of\nthecirculargreydiscrepresentingthedistributionofmag netic\nmoments defined in (b). The system is at finite temperature\nin an external field (c) and in a uniaxial anisotropy (d).\nmoments. This is illustrated in Fig. 2 (top left) where\nthe torque (or ∂mi/∂t) is shown as a function of angle\n(θ) between the magnetic moment and applied field. In\nFig. 2 (top right) we also show the resulting angular ve-\nlocity (∂φi/∂t) of each atomic spin. The angular velocity\nisconstantandseentobe independent of θ, hencethe an-\ngular velocity is the same for all spins and it stands clear\nthat an external field will not influence the relative orien-\ntation of the atomic spins. The evolution of the distribu-\ntion of the atomic moments at finite temperature during\nrelaxation in an external field is schematically shown in\nFig. 1c. The figure illustrates the fact that an external\nfield results in a simple rotationofthe magnetizationand\nthat all individual atomic spins rotate without changing\nthe relative direction to all other atomic spins.\nIV. DYNAMICS IN A UNIAXIAL\nANISOTROPY FIELD\nIf there is a uniaxial anisotropy in the system, such as\nmagneto-crystalline anisotropy or shape anisotropy, an\nexcitation of the macro moment in the anisotropy (by a\nrotation) will in general lead to a precessional motion of3\nthe macro moment in the anisotropy field which appears\nsimilar to the precession in an external field. For the\ncase of a uniaxial magneto-crystalline anisotropy, which\nwe will consider now, there are however important dif-\nferences in the spin dynamics. We define the anisotropy\nenergy for each atomic moment as E=ke2\nzwherekis\nthe anisotropyconstant which determines the strength of\nthe anisotropyand ezis thez-componentof the direction\nof the atomic moment. The torque and angular velocity\non any atomic spin are illustrated in Fig. 2 for both an\neasy-axis anisotropy (middle panels) and an easy-plane\nanisotropy (lower panels). The torque is clearly differ-\nent than in the case of an applied field (top panels), and\nmore importantly the angular velocity of each spin is no\nlonger independent of θ. Hence, spin dynamics in a uni-\naxial anisotropy field does not lead to a uniform rotation\nof the atomic spins. Instead the internal distribution\nof the atomic moments is distorted during the rotation,\nas illustrated in Fig. 1d. For the case of the external\nfield the distribution of atomic spins remained constant\nwith the result that the size of the macro moment re-\nmained constant during the precession. Hence, the pro-\ncess could conveniently be described within a macro mo-\nment picture. This is not true for the case of the uniax-\nial anisotropy and the macro moment description breaks\ndown. Since the internal distribution of the atomic mo-\nments is changed duringthe precession, and the direction\nof any atomic moment in general changes relative to all\nother moments in the system, the size of the macro mo-\nment changes which leads to a considerably more com-\nplex macro level behavior. In the rotating frame of the\naveragemoment, theeasy-axisanisotropyisseentocoun-\nteract the precession of atomic moments in the effective\nexchange field, while the hard-axis anisotropy is seen to\nenhance the precession in the effective exchange field. As\nthe average moment precesses in the uniaxial anisotropy\nthe atomic moments will have a tendency to spread re-\nducingthenetmomentofthesystem,asshowninFig.2d.\nWe will refer to this behavior as a spin-wave instability\n(SWI), according to the discussion by Kashuba (Ref. 9).\nAs we will show in our simulations, the SWI results\nin an apparent damping of the uniform motion of the\nmacro moment. We define the anisotropy axis as the\nz-axis. What is significant for this damping is the van-\nishing of the macro moment components perpendicular\n(x,y) to the anisotropy axis and the constant value of\nthe parallel macro moment component ( z). Hence, dur-\ning the SWI, the average magnetization of the system\ndrops and only the z-component of the average magneti-\nzation remains finite as the xandy-components vanish.\nThis givesthe relaxationofthe macromoment due to the\nSWI a Bloch-Bloembergen form where |Mz|is constant.\nThus, due to the SWI there is an alignment of the macro\nmoment with the anisotropy axis where the alignment\noccurs maintaining a constant value of the z-component\nof the macro moment. This is illustrated in Fig. 3. This\nshowsthataredistributionofangularmomentumanden-\nergy within the magnetic system is taking place. Hence,0 0.5 100.51dm/dt (γ|m||Hext|)\n0 0.5 1-0.500.5dm/dt (2γK|m|2)\n0 0.5 1\nθ (π)-0.500.5dm/dt (2γK|m|2)0 0.5 1-1.5-1-0.500.511.5dφ/dt (γ|Hext|/2π)\n0 0.5 1-1.5-1-0.500.511.5dφ/dt (γK|m|/π)\n0 0.5 1\nθ (π)-1-0.500.51dφ/dt (γK|m|/π)a. External field\nc.          Easy-axis \n             anisotropy\ne. Easy-plane \n    anisotropyf. Easy-plane \n    anisotropyd.          Easy-axis \n             anisotropyb. External field\nFIG. 2: The plots illustrate the change of the magnetic mo-\nment due to an external field, easy-axis anisotropy and an\neasy-plane anisotropy. The graphs on the left hand side give\nthe magnitude |∂m/∂t|while the graphs on the right hand\nside give the angular velocity of the atomic spins with respe ct\nto angle θbetween spin and applied field or anisotropy axis.\nNote that in the case of a uniaxial anisotropy field θis defined\nas the angle between moment and a fixed crystallographic di-\nrection of the anisotropy field (e.g. 100). H is the strength o f\nthe external field and K the strength of the anisotropy field.\nthere is a relaxation taking place even though the dissi-\npative damping, α, is set to zero. In reality there is also a\nfinite dissipativedamping, α, and thereforealsoa Gilbert\ncontribution to the relaxation of the macro moment. In\nsome of our simulations, in order to clearly observe the\nBloch-Bloembergen damping with Mz=constant, we set\nα= 0. The fact that the value of the z-component of the\nmacro moment is constant during the SWI is expected\nsince with zero damping the precessional torque of the\nuniaxial anisotropy is the only source or drain of angular\nmomentum within the spin system and this torque lacks\nz-component.\nA. Simulating bcc Fe with different strengths of\nuniaxial anisotropy\nIn order to study the SWI of bcc Fe we choose a\n20×20×20 cell with periodic boundary conditions, en-\ncompassing16000atomicspins, andthreedifferentvalues\nof the strength of an uniaxial anisotropy: -2 mRy/atom,\n-0.2 mRy/atom and -0.02 mRy/atom, with an easy axis\ndirected along the z-axis. Materials with Fe atoms in a\nbcc environment and enhanced anisotropy may be found\nexperimentally in magnetic multilayers, e.g. with Pt.\nThe anisotropy can here be significantly stronger than\nin the bulk case. The magnetic anisotropy of a tetrago-4\na. Bloch-Bloembergen \n             da mping \n          (| M |=const. )b. Gilbert damping \n        (| M|=const. )\nz\nFIG. 3: The figures illustrate the Bloch-Bloembergen (a)\ndamping and the Gilbert (b) damping for a macro moment.\n0 0.2 0.4 0.6 0.8 1\nTime (ps)00.20.40.60.81Normalized magnetic moment, M/M0\n0 0.2 0.4 0.6 0.8 1\nTime (ps)00.20.40.60.81Normalized magnetic moment, M/M0\n0 0.2 0.4 0.6 0.8 1\nTime (ps)00.20.40.60.81Normalized magnetic moment, M/M0\nα=0.1, θ=45o\nα=0.01, θ=45o\nα=0.001, θ=45o\nα=0.1, θ=90o\nα=0.01, θ=90o\nα=0.001, θ=90o\nFIG. 4: Calculated evolution ofthe total magnetization of b cc\nFe as a function of time for different angles between the initi al\nmagnetization and the applied field and for different values o f\nthe damping parameter. The temperature was 100 K.\nnal FeCo/Pt(001) superlattice was measured17toKu=\n2.28 MJm−3, corresponding to Ku≈0.012 mRy/atom.\nThe perpendicular magnetic anisotropy of (Co, Fe)/Pt\nmultilayers was measured by Sato et al.18toKu=\n0.25erg/cm−2, correspondingto Ku≈0.027mRy/atom.\nThe strongest magnetic anisotropy found in experiments\nis for SmCo 519,20with values of Ku= 7.7 MJm−3,\ncorresponding to Ku≈0.31 mRy/formula unit. As\nwill be presented below we see SWI phenomena in our\nsimulations for the anisotropy values -2 mRy/atom, -\n0.2 mRy/atom but not for -0.02 mRy/atom. In order\nto simplify, we have not considered non-magnetic (e.g.\nPt) atoms in the simulations, but only the effect they\nhave on the uniaxial anisotropy field. As we will show,\nthese systems can display an instability on a time scaleof\npicoseconds(showninFig.4). Wenowinvestigatethede-\npendence of the SWI on thermal fluctuations and damp-\ning and we investigate the redistribution of the atomic\nmoments which takes place.\nIn Fig. 4, we show a series of simulations for three dif-\nferent damping parameters, α, and two different initial\nangles,θ= 45◦andθ= 90◦. In these simulations weused a uniaxial energy of -2 mRy/atom. For the macro\nmoment there is now a Bloch-Bloembergen like damp-\ning due to the SWI and a Gilbert like damping due to\nthe inclusion of a dissipative damping in the microscopic\nequations of motion. For the case of θ= 45◦we see\nthe presence of both these damping terms (see Fig. 4).\nForα=0.1 the Gilbert term is seen to dominate. After a\nshort dip in the magnitude of the magnetization due to\nthe SWI the magnitude of the magnetization is seen to\nrecover. For α=0.01 and 0.001 the Bloch-Bloembergen\ndamping is seen to dominate, and the size of the mag-\nnetic moment reaches a value of M/M 0≈0.6-0.7. For\nθ= 90◦the situation is slightly different. At this specific\nangle only the SWI contributes to the relaxation of the\nsystem. For this reason the behavior in Fig. 4 is fairly in-\ndependent of the magnitude of α, and the magnetization\nevolves with time to a value where M/M 0≈0-0.1.\nThe cause of the SWI is an internal redistribution of\nthe atomic moments. In Fig. 5 we show a histogram of\nthe angles of the atomic moments with respect to the\naverage atomic moment. The distribution is shown for\ndifferent points in time for two damping parameters, α=\n0.0andα= 0.1(withauniaxialenergyof-2mRy/atom).\nAs a first observation, in contrast to what one might\nexpect, thesizeof αdoesnotchangetherateatwhichthe\ndirections of the atomic moments are redistributed. This\nis illustrated in both the upper and lower panel of Fig. 5.\nOne would, simplemindedly, expect a large damping of\nthe atomic moments in the interatomic exchange field to\nreduce the spread of the atomic moments, which would\ncounteract the SWI. However, this does not happen. A\nsecond observation (see upper panel of Fig. 5) is that the\ndistribution of θis smeared out during the SWI. This\nis consistent with the fact that the net moment of the\nsystem is reduced. A third observation (see lower panel\nof Fig. 5) is that the distribution of φis heavily distorted\nduring the SWI. At t= 0 the distribution is constant,\nwhichalsoisillustratedbythe circulardisc inFig. 1d. At\nt= 0.2psthedistributionisdistorted,whichisconsistent\nwith the development of an elliptically shaped disc in\nFig. 1d.\nIn order to explain the observations in Fig. 5 we show\nin Fig. 6 a histogram of the energy distribution of the\nmagnetic moments at different points in time during the\nsimulation. Thehistogramsfortheenergydistributionat\ndifferent points in time fall on top of each other and co-\nincide with the Boltzmann distribution at 300 K, which\ndemonstrates that the simulations are done at thermal\nequilibrium, throughout the SWI. This explains the first\nobservation of Fig. 5. The effect of the dissipative damp-\ning in Langevin dynamics is to bring the system to ther-\nmal equilibrium. But since the SWI conserves the ther-\nmal distribution of the system, damping has no net ef-\nfect on the distribution of the directions of the atomic\nmoments. The second and third observation from Fig. 5,\nconcern the change in angular distribution of the atomic\nmoments and explain how the fact that the system re-\nmains in thermal equilibrium can be consistent with a5\nFIG. 5: Distribution of the angles between the average macro\nmoment and each atomic moment of the bcc Fe simulation\ncell. In the simulation the initial angle between the averag e\nmagnetization and the anisotropy axis is θ= 90◦. The top\npanel shows the distribution of θfor the different atomic spins\nand the bottom panel shows the distribution of φ, defined in\nFig. 1.\nreduction in the average magnetization. The angular\ndistribution of the atomic moments is heavily distorted\nwhereas the energy distribution remains constant. For\nfinite damping, the situation changes slightly. We show\nin Fig. 7 how the magnetic energy, which here is the sum\nof exchange and anisotropy energy, evolves in time. For\nthe zero damping case in of Fig. 7 the lowering of the\nanisotropy energy is compensated by an increase in the\nexchange energy, leaving the total energy constant. This\nis contrasted by the finite damping cases with α= 0.001\nrespective 0.1, were the initial increase in exchange en-\nergy decays towards its equilibrium value at the given\ntemperature. In both cases the time-evolution of the\natomic moments lowers the total energy. The reason\nfor the different behaviors can be explained as follows.\nFor the zero damping case, α= 0.000, the sum of the ex-\nchangeandtheanisotropyenergyisaconstantofmotion.\nAt the start ofthe simulations the magnetic moments are\nin thermal equilibrium at T= 300 K and the total ex-\nchange energy is constant. When the anisotropy field is\n’turned on’ at t= 0, the system is not in an anisotropy\nenergy minima as the average magnetization is at an an-\ngleθ= 90◦to the easy axis. The evolving magnetic mo-\nments lower their anisotropy energy with an amount of\nenergy that is in its entity transfered to exchange energy,0 5 10 15 20 25\nEnergy (mRy)00.20.40.60.81Number of states0 ps\n0.1 ps\n0.2 ps\n0.3 ps\nBoltzman distribution 300 K\nFIG. 6: Histogram of the energies of the atomic spins for a\nsimulation of bcc Fe with α= 0.0 andθ= 90◦. Although\nthere is a large drop of the average moment of the system\nthe energy distribution does not change significantly durin g\nthe development of the SWI. Data for different times of the\nsimulation are shown.\n0 1 2 3 4 5−12−10−8−6−4−20\nTime t (ps)Energy/Atom (mRy)\n  \nEtot α=0.0000\nEExc α=0.0000\nEAni α=0.0000\nEtot α=0.0010\nEExc α=0.0010\nEAni α=0.0010\nEtot α=0.1000\nEExc α=0.1000\nEAni α=0.1000\nFIG. 7: (color online) The evolution in time of the total en-\nergy, the exchange energy and the anisotropy energy, for var -\nious damping parameters.\nsince energy can not dissipate in or out of the system.\nWith a small but finite damping, α= 0.001, the mag-\nnetic excitations can dissipate and lower the exchange\nenergy. With a large damping of α= 0.100 the exchange\nenergy dissipates to within 5 ps to reach its equilibrium\nvalue at temperature T= 300 K.\nThermal fluctuations play an important role for the\ndevelopment of the SWI. Naturally there is therefore a\ndependenceofthetime-scaleoftheinstabilityonthetem-\nperature. We found however that in the range 10-300 K\nthe time-scale is fairly independent on the temperature\nas shown in Fig. 8 (again we used a uniaxial energy of\n-2 mRy/atom for these simulated data). The thermal\nfluctuations also have another role. For systems where\nthe macro moment is unable to relax along an anisotropy\naxis (i.e. when θ= 90◦) thermal fluctuations turn out\nas the only mechanism for the system to come out of\nthe chaotic SWI state when the anisotropy field is re-6\n0 0.2 0.4 0.6 0.8 1\nTime (ps)00.20.40.60.81Normalized magnetic moment, M/M0\nT=10 K\nT=100 K\nT=300 K\nFIG. 8: Simulations at different temperatures of bcc Fe with\nα= 0.0 andθ= 90◦. The SWI develops on the same time\nscale for different temperatures.\n0 2 4 6 8 10\nTime (ps)00.10.20.30.40.50.6Normalized magnetic moment, M/M0\nλ=0.001\nλ=0.01\nλ=0.1\nFIG. 9: Starting from bcc Fe in a SWI state the anisotropy\nfield is removed (at T=300 K). The system is seen to evolve\nback slowly toward a ferromagnetic state.\nmoved. Starting from a chaotic state where the SWI has\nbeen allowed to bring the system to a zero total moment\nstate we suddenly remove the anisotropy field and ob-\nserve the evolution of the system (see Fig. 9). It is now\nonly the complete randomness of the thermal fluctua-\ntions that eventually is able to evolve the system back\nto a ferromagnetic state. The thermal fluctuations will\neventually bring the spin distribution which has a total\nmoment close to zero, to a spin distribution with a to-\ntal moment approaching a finite value. The process is\nhowever very time consuming, as shown in Fig. 5, and\nonly observed for the largest damping parameter in the\npresent simulations.\nWe now compare simulated results using different\nstrengths of the uniaxial anisotropy as well as different\nvalues of the damping parameter. The interatomic ex-\nchange interactions and the size of the simulation cell\nwere kept the same as in previous simulations. The00.20.40.60.8\n  \nα=0.0000\nα=0.0001\nα=0.0010\nα=0.0100\nα=0.1000\n00.20.40.60.8Average, normalized magnetic moment, M/M0\n  \nα=0.0000\nα=0.0001\nα=0.0010\nα=0.0100\nα=0.1000\n0 1 2 3 4 500.20.40.60.8\nTime t (ps)  \nα=0.0000\nα=0.0001\nα=0.0010\nα=0.0100\nα=0.1000a)\nKx=−2.0 mRy,  θ=90°\nb)\nKx=−0.20 mRy,  θ=90°\nc)\nKx=−0.02 mRy,  θ=90°\nFIG. 10: (color online) Calculated evolution of the magneti -\nzation of bcc Fe as a function of time with different values of\nthe uniaxial anisotropy and for different values of the damp-\ningparameter. The magnetization is initially at angle θ= 90◦\nto the anisotropy axis.\ntemperature was 300 K. In Fig. 10 we show the case\nwhereθ=90, for three values of uniaxial anisotropy,\n−2 mRy/atom, −0.2 mRy/atom and −0.02 mRy/atom.\nNote that the case with an anisotropy of −2 mRy/atom\nwas also considered in Fig. 4, although in Fig. 10 we\nshow the dynamical response over a larger time interval,\n5 ps. For the strongest value of the uniaxial anisotropy\nthe SWI develops rather easily, whereas for the lowest\nvalue of the uniaxial anisotropy the SWI does not de-\nvelop at all, at least not in the time interval considered.\nThe intermediate value of the uniaxial anisotropy results\nin an intermediate situation where the macro moment\noscillates in time (at least in this time-interval, we will\nreturn to this situation below). The reason behind the\ndifferent behaviors shown in Fig. 10, is a competition be-\ntween the strength of the uniaxial anisotropy, which in\nline with the discussion around Fig. 1 tends to spread the\ndistribution of all atomic moments, and the importance\nofthe otherrelevant interactionsin the system, primarily\nthe strength of the interatomic exchange interaction.\nIn Fig. 11 we show very similar simulations as in\nFig. 10, with the only difference being that we show re-\nsults for the case when θ= 45◦. Here the intermediate\nand lowest value of the uniaxial anisotropy does not have\nsufficient strength to drive a SWI, whereas the largest\nvalue of the anisotropy the SWI develops and a Bloch-\nBloembergen damping occurs. This was also illustrated\nin Fig. 4, but over a shorter time-interval.\nThe case when θ= 45◦and with a uniaxial anisotropy\nof -0.2 mRy/atom is, as Fig. 10 suggests, a particularly\ninteresting case, since here the anisotropy and exchange\ninteractions seems to be tuned into a situation where\nboth are very influential for the evolution of the macro\nspin. In fact, Fig. 10 suggests that in this case the mag-\nnetization oscillates between a Gilbert like damping and\nBloch-Bloembergen like damping. For this reason we7\n00.20.40.60.8\n  \nα=0.0000\nα=0.0001\nα=0.0010\nα=0.0100\nα=0.1000\n00.20.40.60.8Average, normalized magnetic moment, M/M0\n  \nα=0.0000\nα=0.0001\nα=0.0010\nα=0.0100\nα=0.1000\n0 1 2 3 4 500.20.40.60.8\nTime t (ps)  \nα=0.0000\nα=0.0001\nα=0.0010\nα=0.0100\nα=0.1000a)\nKx=−2.0 mRy,  θ=45°\nb)\nKx=−0.20 mRy,  θ=45°\nc)\nKx=−0.02 mRy,  θ=45°\nFIG. 11: (color online) Same as Fig. 10 but with the magne-\ntization initially at angle θ= 45◦to the anisotropy axis.\nhave extended the simulations over a larger time interval\n(50 ps), and the results are shown in Fig. 12. It is to\nbe noted from this figure that for this borderline case,\nthe evolution of the macro spin depends not only on the\ncompetition between interatomic exchange and uniaxial\nanisotropy, but also on the value of the damping param-\neter. For large values of the damping a regular Gilbert\ndamping behaviors is found. For small and intermediate\nvalues of the damping the macro spin is found to oscil-\nlate in time, but otherwise following a dynamic response\nwhichresemblesBloch-Bloembergendamping. Hencethe\ndata in Fig. 12 show that by careful tuning ofthe relative\nimportance of the uniaxial anisotropy, exchange interac-\ntion and damping, one may obtain a behavior which is\nmore complex than that given by pure Gilbert or Bloch-\nBloembergen damping.\nThe finite size of the simulation cell restricts the pos-\nsible spin wave excitations. The simulations described so\nfar were all for L= 20 corresponding to 16000 magnetic\nmoments. With a smaller cell only the modes with short\nwave lengths can occur. This means that the weaker\nuniaxial anisotropy cannot drive an SWI unless the sim-\nulation cell is large enough. The trend for if a SWI can\noccur or not for the different simulation cells, with cell\nsizeL= 10, 15, 20 and 25, are presented in Tables I, II,\nfor the 90 degree and 45 degree case, respectively. In the\ntable for the 90 degreecase we havedefined a strong SWI\nasthecasewherethemagnetizationdropsbelow0 .2M0, a\nmedium SWI as when it drops below 0 .6M0, a weak SWI\nas when the magnetization drops with 0 .05−0.20M0and\nno SWI when the magnetization drops less than 0 .05M0.\nIn the table for the 45 degree case the same notation is\nusedapartfromthatwehereredefinestrongSWIaswhen\nthemagnetizationdropsbelow0 .65M0(whichherecorre-\nspondsto aBloch-Bloembergendamping). Theresultsof\nTables. I,II correspond well to the results of Ref. 10 (see\ne.g. Eqn. 27). For the anisotropy values −2 mRy/atom,\n−0.2 mRy/atom and −0.02 mRy/atom and with the ex-\nchange energy summed up over all coordination shells toTABLE I: Simulations for varying cell size, 90 degree case.\nThe entries describe the possible occurrence of a SWI during\nthe simulation time t= 5 ps.\nKu(mRy/atom) αL=10L=15 L=20 L=25\n-2.0 0-0.1strongstrong strong strong\n-0.2 0-0.1 noweakmedium medium\n-0.02 0-0.1 nono no no\nTABLE II: Simulations for varying cell size, 45 degree case.\nThe entries describe the possible occurrence of a SWI during\nthe simulation time t= 5 ps.\nKu(mRy/atom) αL=10 L=15 L=20 L=25\n-2.0 0-0.01 strong strong strong strong\n-2.0 0.1weak weak weak weak\n-0.2 0-0.01 no no noweak\n-0.2 0.1 no no no no\n-0.02 0-0.1 no no no no\n≈10 mRy/atom we get Nmax= 5,16,51 where Nmaxis\nthe largest cell size that suppress SWI effects.\nV. DISCUSSION AND SUMMARY\nIn this paper we have investigated the conditions when\na spin-wave instability (SWI) may occur. In order for\nthis to happen, a number of requirements must be met.\nFirst, theremustbeaninitialperturbationtothesystem,\ne.g. thermal fluctuations, such that the atomic moments\nstart to deviate from the direction of the macro moment.\nSecondly, theremust be amagneticanisotropyin the sys-\n0 10 20 30 40 5000.10.20.30.40.50.60.70.80.91\nTime t (ps)Average, normalized magnetic moment, M/M0\n  \nα=0.0000\nα=0.0001\nα=0.0010\nα=0.0100\nα=0.1000θ=90%\nFIG. 12: (color online) Same as the middle panel of Fig. 10\nbut showing the evolution of the magnetization up to 50 ps.\nForα= 0...0.01 the magnetization oscillates in the interval\n0.55−0.70M0. Forα= 0.1 the magnetization recovers after\n∼20 ps to the value M= 0.88M0which is the thermally\nequilibrated value at temperature T= 300 K.8\ntem. The presence of a SWI is found to a large degree\nbe determined by a competition between the magnetic\nanisotropy and the strength of the exchange interaction.\nIn some special cases, where these two contributions are\nvery delicately balanced, the value of the damping pa-\nrameter can finally determine whether or not a SWI oc-\ncurs. We have also found that the size of the simulation\ncell is influential for if a SWI occurs, a conclusion which\nis in agreement with the results of Ref.10.\nAnother conclusion we reach from our simulations is\nthat due to thermal fluctuations the simple model of\na macro moment precessing in a uniaxial anisotropy is\nfound to be inaccurate. The uniaxial anisotropy leads\nto a non-uniform rotation of the composing atomic mo-\nments. On a short time scale the effect is small. On\na longer time scale or for larger anisotropies there are\nsevere consequences. An instability appears which ef-\nfectively leads to a Bloch-Bloembergen damping of the\nmagnetization.\nOur simulations point to a technical avenue for de-\nsigning media for data-storage and magnetic memories,\nwhere e.g. the grain size of the storage media would be a\nmaterials property which one could compare to the vari-ous sizes of our simulation cell. Media with a small grain\nsize could possibly then exhibit a weaker tendency for\na SWI to be observed. If experimental evidence for the\nspin wave instability could be demonstrated, it would\nimply that there is an increased importance to a fine\ngrain description of the magnetization dynamics in sim-\nulations and it would show that macro moment mod-\nels lose accuracy when anisotropies are involved in the\ndynamics. Further experimental studies addressing this\nissue are highly desired.\nAcknowledgments\nFinancial support from the Swedish Foundation for\nStrategic Research (SSF), the Swedish Research Council\n(VR), the Royal Swedish Academy of Sciences (KVA),\nLiljewalchs resestipendium and Wallenbergstiftelsen is\nacknowledged. Calculations have been performed at the\nSwedish national computer centers UPPMAX, HPC2N\nand NSC.\n∗johan.hellsvik@fysik.uu.se\n1J. St¨ ohr and H.-C. Siegmann, Magnetism: From funda-\nmentals to nanoscale dynamics (Springer Verlag, Berlin,\n2006).\n2F. Bloch, Physical Review 70, 460 (1946).\n3N. Bloembergen, Physical Review 78, 572 (1950).\n4H. Suhl, Journal of Physics and Chemistry of solids 1, 209\n(1957).\n5R. Arias and D. L. Mills, Physical Review B 60, 7395\n(1999).\n6M. J. HurbenandC. E. Patton, Journal of AppliedPhysics\n83, 4344 (2008).\n7A. Y. Dobin and R. H. Victoria, Physical Review Letters\n90, 167203 (2003).\n8V. L. Safonov and H. Neal Bertram, Physical Review B\n63, 094419 (2001).\n9A. Kashuba, Physical Review Letters 96, 047601 (2006).\n10D. A. Garanin, H. Kachkachi, and L. Reynaud, EPL (Eu-\nrophysics Letters) 82, 17007 (2008).\n11D. G. Garanin and H. Kachkachi, Magnetization reversal\nvia internal spin waves in magnetic nanoparticles (2009),\nhttp://arxiv.org/abs/0902.1492v1.\n12A. Aharoni, Introduction to the theory of ferromagnetism(Oxford university press, Oxford, 2000).\n13V. P. Antropov, M. I. Katsnelson, B. N. Harmon, M. v.\nSchilfgaarde, and D. Kusnezov, Physical Review B 54,\n1019 (1996).\n14B. Ujfalussy, B. Lazarovits, L. Szunyogh, G. M. Stocks,\nand P. Weinberger, Physical Review B 70, 100404 (2004).\n15X. Tao, D. P. Landau, T. C. Schulthess, and G. M. Stocks,\nPhysical Review Letters 95, 087207 (2005).\n16B. Skubic, J. Hellsvik, L. Nordstr¨ om, and O. Eriksson,\nJournal of Physics Condensed Matter 20, 315203 (2008),\nURLhttp://www.fysik.uu.se/cmt/asd/ .\n17P. Warnicke, G. Andersson, M. Bj¨ orck, J. Ferr´ e, and\nP. Nordblad, Journal of Physics Condensed Matter 19,\n226218 (2007).\n18T. Sato, T. Goto, H. Ogata, K. Yamaguchi, and\nH. Yoshida, Journal of Magnetism and Magnetic Materials\n272-276 , E951 (2004).\n19K. Strnat, G. Hoffer, J. Olson, W. Ostertag, and J. J.\nBecker, Journal of Applied Physics 38, 1001 (1967).\n20E. A. Nesbitt, R. H. Willens, R. C. Sherwood, E. Buehler,\nand J. H. Wernick, Applied Physics Letters 12, 361 (2009)."    },    {        "title": "1101.5789v1.Spin_transport_in_magnetically_ordered_systems__effect_of_the_lattice_relaxation_time.pdf",        "content": "arXiv:1101.5789v1  [cond-mat.stat-mech]  30 Jan 2011SPIN TRANSPORT IN MAGNETICALLY ORDERED\nSYSTEMS: EFFECT OF THE LATTICE RELAXATION TIME\nY. Magnin, Danh-Tai Hoang, and H. T. Diep∗\nLaboratoire de Physique Th´ eorique et Mod´ elisation,\nUniversit´ e de Cergy-Pontoise,\nCNRS, UMR 8089\n2, Avenue Adolphe Chauvin,\n95302 Cergy-Pontoise Cedex, France\nSpin resistivity Rhas been shown to result mainly from the scattering of itiner ant\nspins with magnetic impurities and lattice spins. Ris proportional to the spin-spin\ncorrelation so that its behavior is very complicated near an d at the magnetic phase\ntransition of the lattice spins. For the time being there are many new experimental\ndata on the spin resistivity going from semiconductors to su perconductors. Depend-\ning on materials, various behaviors have been observed. The re is however no theory\nso far which gives a unified mechanism for spin resistivity in magnetic materials.\nRecently, we have showed Monte Carlo results for different sys tems. We found that\nthe spin resistivity is very different from one material to ano ther. In this paper, we\nshow for the first time how the dynamic relaxation time of the l attice spins affects\nthe resistivity of itinerant spins observed in Monte Carlo s imulation.\nPACS numbers: 05.60.Cd Classical transport ; 75.47.-m Magnetotra nsport phenomena; ma-\nterials for magnetotransport ; 75.10.Hk Classical spin models ; 05.10 .Ln Monte Carlo meth-\nods\nI. INTRODUCTION\nThe resistivity is an important subject in condensed-matter physic s. It has been studied\nexperimentally and theoretically already with old classical physics. Ho wever, only from\n∗Corresponding author, E-mail: diep@u-cergy.fr2\nthe fifties, with notions borrowed from microscopic modern physics that the resistivity has\nbeen viewed as a consequence of microscopic mechanisms which gove rn physical behaviors\nof materials through which conduction electrons travel. In this pap er, we are interested in\nthe resistivity caused by magnetic scattering of itinerant electron ic spins by localized lattice\nspins in magnetic materials (ferromagnets and anfiferromagnets) . The resulting resistivity\nis called hereafter ”spin resistivity” which is to be distinguished from t he resistivity due to\nspin-independent scattering, for example by phonons and non mag netic impurities.\nThe spin resistivity, R, was shown to depend onthe spin-spin correlationin ferromagnetic\ncrystals by de Gennes and Friedel1, Fisher and Langer2among others. At low temperatures\n(T), the spin-waves are shown to be responsible for the T2behavior of the spin resistivity in\nferromagnets3,4. Note however that in these calculations the itinerant electrons ha ve been\nconsidered as free electrons interacting with the lattice spins, but there is no interaction\nbetween them. We have showed5,6that when an interaction between itinerant electrons is\nintroduced, the itinerant electrons can be crystallized at low Tgiving rise to an increase of\nRasT→0. Experimental data in various materials show this behavior7–11, but we would\nwarnthat there maybeother mechanisms involved aswell. At the mag neticphase transition\ntemperature TC, the spin-spin correlationdiverges inmagnetic materialswith asecon d-order\nphase transition. The theory of de Gennes-Friedel predicts that Rshould show a divergent\npeak. However, experiments in various magnetic materials ranging f rom semiconductors\nto superconductors7–17show indeed an anomaly at the transition temperature TC, but the\npeak is more or less rounded, not as sharp as expected from the div ergence of the correlation\nlength. It has been shown in fact that2,18the form of the peak depends on the length of\nthe correlation included in the calculation of R: if only short-range correlations are taken\ninto account, then the peak is very rounded. A justification for th e use of only short-range\ncorrelations comes from the fact that the mean free path of itiner ant spins is finite at TC.\nWhen scattering is due to impurities, the peak has been shown to dep end on the localization\nlength19. In the case of antiferromagnets, Haas has shown the absence o f a resistivity peak20.\nOur recent works using Monte Carlo (MC) simulations have shown tha t there is indeed an\nanomaly at TCin various magnetic models from ferromagnets5,21,22, antiferromagnets5,23to\nfrustrated spin systems6,24. The shape of the anomaly depends on many ingredients such as\ncrystal structures, spin models, and interaction parameters.\nIn this paper, we will show new results obtained by MC simulation when w e take into3\naccount the temperature dependence of the relaxation time of loc alized lattice spins in the\nsimulation. We will show that this temperature dependence affects t he shape of the peak in\nthe phase transition region.\nSection II is devoted to a description of the general model and the MC method. We\nintroduce in this section the temperature dependence of the relax ation time. Results are\nshown and discussed in section III for both ferro- and antiferrom agnets in terms of critical\nslowing-down. Concluding remarks are given in section IV.\nII. MODEL AND METHOD\nThe model we use in our MC simulation is very general. The itinerant spin s move in a\ncrystal whose lattice sites are occupied by localized spins. The itiner ant spins are assumed\nto be of Ising type, but the method of simulation can be used for oth er spin models23. The\nlocalizedspinsmaybeofIsing, XYorHeisenberg models. Theirinterac tionisusuallylimited\nto nearest neighbors (NN) but this assumption is not necessary. I t can be ferromagnetic or\nantiferromagnetic.\nA. Interactions\nWe consider a thin film of a given lattice structure where each lattice s ite is occupied\nby a spin. The interaction between the lattice spins is limited to NN with t he following\nHamiltonian :\nHl=−/summationdisplay\n(i,j)Ji,j/vectorSi./vectorSj (1)\nwhere/vectorSiis an Ising spin whose values are ±1,Ji,jthe exchange integral between the NN\nspin pair /vectorSiand/vectorSj. Hereafter we take Ji,j=Jfor all NN spin pairs, for simplicity. As a\nconvention, ferromagnetic (antiferromagnetic) interaction has positive (negative) sign. The\nsystem size is Nx×Ny×NzwhereNi(i=x,y,z) is the number of lattice cells in the i\ndirection. Periodic boundary conditions (PBC) are used in the xandydirections while the\nsurfaces perpendicular to the zaxis are free. The film thickness is Nz.\nWe define the interaction between the itinerant spins and the localize d lattice spins as4\nfollows\nHr=−/summationdisplay\ni,jIi,j/vector σi./vectorSj (2)\nwhereσiis the Ising spin of the i−thitinerant electron and Ii,jdenotes the interaction\nthat depends on the distance between an electron iand the spin /vectorSjat the lattice site j. For\nsimplicity, we use the following interaction expression\nIi,j=I0e−αrij(3)\nwhererij=|/vector ri−/vector rj|,I0andαare constants. In the same way, interaction between itinerant\nelectrons is defined by\nHm=−/summationdisplay\ni,jKi,j/vector σi./vector σj (4)\nKi,j=K0e−βrij(5)\nwithKi,jbeing the interaction that depends on the distance between electr onsiandj. The\nchoice of the constants K0andβwill be discussed below.\nDynamics of itinerant electrons is ensured by an electric field applied a long the xaxis.\nElectrons travel in the xdirection, leave the system at the end. The PBC on the xyplanes\nensure that the electrons who leave the system at one end are to b e reinserted at the other\nend. For the zdirection, we use the mirror reflection at the two surfaces. These boundary\nconditions are used in order to conserve the average density of itin erant electrons. One has\nHE=−e/vector ǫ./vectorℓ (6)\nwhereeis the charge of electron, /vector ǫthe applied electric field and /vectorℓthe displacement vector\nof an electron.\nSince the interaction between itinerant electron spins is attractive , we need to add a\nchemical potential in order to avoid a possible agglomeration of elect rons into some points\nin the crystal and to ensure a homogeneous spatial distribution of electrons during the\nsimulation. The chemical potential term is given by\nHc=D/vector∇rn(/vector r) (7)\nwheren(/vector r) is the concentration of itinerant spins in the sphere of D2radius, centered at /vector r.\nDis a constant parameter appropriately chosen.5\nB. Simulation Method\nThe procedure of our simulation can be split into two steps. The first step consists\nin equilibrating the lattice at a given temperature Twithout itinerant electrons. When\nequilibrium is reached, in the second step, we randomly add N0polarized itinerant spins\ninto the lattice. Each itinerant electron interacts with lattice spins in a sphere of radius D1\ncentered at its position, and with other itinerant electrons in a sphe re of radius D2.\nThe procedure of spin dynamics is described as follows. After inject ingN0itinerant\nelectrons in the equilibrated lattice, we equilibrate the itinerant spins using the following\nupdating. We calculate the energy Eoldof an itinerant electron taking into account all\ninteractions described above. Then we perform a trial move of leng thℓtaken in an arbitrary\ndirection with random modulus in the interval [ R1,R2] whereR1= 0 and R2=a(NN\ndistance), abeing the lattice constant. Note that the move is rejected if the ele ctron falls in\na sphere of radius r0centered at a lattice spin or at another itinerant electron. This exc luded\nspaceemulatesthePauliexclusion. Wecalculatethenewenergy EnewandusetheMetropolis\nalgorithmtoaccept orreject theelectrondisplacement. Wechoos eanother itinerantelectron\nand begin again this procedure. When all itinerant electrons are con sidered, we say that\nwe have made a MC sweeping, or one MC step/spin. We have to repeat a large number of\nMC steps/spin to reach a stationary transport regime. We then pe rform the averaging to\ndetermine physical properties such as magnetic resistivity, electr on velocity, energy etc. as\nfunctions of temperature.\nWe emphasize here that in order to have sufficient statistical avera ges on microscopic\nstates of both the lattice spins and the itinerant spins, we use the f ollowing procedure: after\naveraging the resistivity over N1steps for ”each” lattice spin configuration, we thermalize\nagain the lattice with N2steps in order to take another disconnected lattice configuration .\nThen we take back the averaging of the resistivity for N1steps for the new lattice config-\nuration. . We repeat this cycle for N3times, usually several hundreds of thousands times.\nThe total MC steps for averaging is about 4 ×105steps per spin in our simulations. This\nprocedure reduces strongly thermal fluctuations observed in ou r previous work22.\nOf course, the larger N1andN3are the better the statistics becomes. The question is\nwhat is the correct value of N1for averaging with each lattice spin configuration at a given\nT? This question is important because this is related to the relaxation t imeτLof the lattice6\nspins compared to that of the itinerant spins, τI. The two extreme cases are i) τL≃τI, one\nshould take N1= 1, namely the lattice spin configuration should change with each mov e\nof itinerant spins ii) τL≫τI, in this case, itinerant spins can travel in the same lattice\nconfiguration for many times during the averaging.\nIn order to choose a right value of N1, we consider the following temperature dependence\nofτLin non frustrated spin systems. The relaxation time is expressed in t his case as25\nτL=A\n|1−T/TC|zν(8)\nwhereAisaconstant, νthecorrelationcriticalexponent, and zthedynamicexponent. From\nthisexpression, weseethatas TtendstoTC,τLdiverges. Inthecriticalregionaround TCthe\nsystemencountersthustheso-called”criticalslowingdown”: the spinrelaxationisextremely\nlong due to the divergence of the spin-spin correlation. In our prev ious papers5,6,21,22,24, we\ndid not take into account the temperature dependence of τL. We propose to study here the\nspin resistivity using Eq. (8).\nWe define spin resistivity ρas :\nρ=1\nne(9)\nwhereneis the number of itinerant electron spins crossing a unit slice perpend icular to the\nxdirection per unit of time.\nC. Choice of parameters and units\nThe spin resistivity is dominated by the two interactions Eqs. (2) and (5). As said\nearlier, our model is very general. Several kinds of materials such a s metals, semiconductors,\ninsulating magnetic materials etc. can be studied with our model, prov ided an appropriate\nchoice of the parameters. For example, non magnetic metals corre spond to Ii,j=Ki,j= 0\n(free conduction electrons). The case of magnetic semiconducto rs corresponds to the choice\nof parameters K0andI0so as the energy of an itinerant electron due to the interaction\nHrshould be much lower than that due to Hm, namely itinerant electrons are more tightly\nbound to localized atoms. Note that Hmdepends on the concentration of itinerant spins: for\nexample the dilute case yields a small Hm. We will show below results obtained for typical\nvalues of parameters which correspond more or less to semiconduc tors. The choice of the7\nparameters has been made after numerous test runs. We describ e the principal requirements\nwhich guide the choice: i) We choose the interaction between lattice s pins as unity, i. e.\n|J|= 1, ii) We choose interaction between an itinerant and its surroundin g lattice spins so\nas its energy Eiin the low Tregion is the same order of magnitude with that between lattice\nspins. To simplify, we take α= 1. This case corresponds more or less to a semiconductor,\nas said earlier, iii) Interaction between itinerant spins is chosen so th at this contribution\nto the itinerant spin energy is smaller than Eiin order to highlight the effect of the lattice\nordering on the spin current. To simplify, we take β= 1, iv) The choice of Dis made in\nsuch a way to avoid the formation of clusters of itinerant spins (agg lomeration) due to their\nattractive interaction [Eq. (5)], v) The electric field is chosen not so strong in order to avoid\nits dominant effect that would mask the effects of thermal fluctuat ions and of the magnetic\nordering, vi) The density of the itinerant spins is chosen in a way that the contribution\nof interactions between themselves is much weaker than Ei, as said above in the case of\nsemiconductors.\nA variation of each parameter respecting the above requirements does not change qual-\nitatively the results shown below. Only the variation of D1in some antiferromagnets does\nchange the results (see Ref.6).\nThe energy is measured in the unit of |J|. The temperature is expressed in the unit of\n|J|/kB. The distance ( D1andD2) is in the unit of a.\nIII. RESULTS AND DISCUSSION\nInthissection, weshowforcomparisontheresultsobtainedwithan dwithouttemperature\ndependence of the lattice relaxation time for both ferromagnets a nd antiferromagnets. In\neach case we use the same set of interaction parameters in order t o outline the effect of the\ntemperature-dependent relaxation time.\nIn this paper we use the lattice size Nx=Ny= 20 and Nz= 8 and we consider the body-\ncentered cubic (BCC) lattice for illustration. The lattice constant is a. The spin resistivity\nis calculated with N0= (Nx×Ny×Nz)/2 itinerant spins (one electron per two lattice cells).\nExcept otherwise stated, we choose interactions I0= 2,K0= 0.5,D1=a,D2=a,D= 0.5,\nǫ= 1,N0= 1600, and r0= 0.05a. A discussion on the effect of a variation of each of these\nparameters is given above.8\nNote that, due to the form of the interaction given by Eq. (5), the itinerant spins have\na tendency to form compact clusters to gain energy. This tendenc y is neutralized by the\nconcentration gradient term, i. e. a chemical potential, given by Eq . (7). The value of Dhas\ntobechosen so astoavoid anagglomerationofitinerant spins. Thisc hoice depends ofcourse\non the values of D1andD2. Examples have been shown elsewhere.6,24For the temperature\ndependence of the lattice relaxation time τL, we take ν= 0.638 (3D Ising universality) and\nz= 2.02.26By choosing A= 1, we fix τL= 1 atT= 2TCdeep inside the paramagnetic\nphase far above TC. This value is what we expect for thermal fluctuations in the disorde red\nphase.\nFigure 1 shows the spin resistivity Rin a BCC ferromagnet. Note that the transition\ntemperature for this thin film of size 20 ×20×8 with Ising spins interacting via the NN\ncoupling is TC≃6.35. Several remarks are in order:\ni) The results obtained with and without temperature-dependent r elaxation time for\nT < T Ccoincide with each other\nii) AtTC, for the set of parameters used here, the results using the temp erature-\nindependent relaxation shows a broad maximum above TCwhile those using the\ntemperature-dependent relaxation strongly decreases at TCgiving rise to a sharp peak.\nWe show now in Fig. 2 the spin resistivity Rin a BCC antiferromagnet. Note that the\ntransition temperature is the same as that of the ferromagnet co unterpart shown above.\nHere we observe that Rin the case of temperature-dependent relaxation is lower than tha t\nin the case of temperature-independent one in the whole temperat ure range. Note that the\nvalue of the peak is much smaller here than in the ferromagnet case.\nWe show in Fig. 3 the two curves of ferromagnet and antiferromagn et withT-dependent\nrelaxation time. We observe here that below TC, the resistivity of antiferromagnet is higher\nthan that of ferromagnet, while for T > T Cthe reverse is true.\nIt is interesting to calculate the relaxation time τIof the itinerant spins. We define τIin\nthe simulations as the MC time (in unit of one MC step/spin) between tw o ”MC collisions”,\nnamely the lapse of time between two ”rejections” of a spin to advan ce. Of course this\nquantity is averaged over all itinerant spins and over the simulation t ime. Figure 4 shows\nτ−1\nIobtained by simulation using τL. As forRseen above, the temperature dependence and\nindependence are markedly different only for T > T C. The same thing is observed for the\ncase of antiferromagnet shown in Fig. 59\nFIG. 1: BCC ferromagnetic thin film: Resistivity Rwith temperature-independent relaxation\n(white circles) and temperature-dependent relaxation (bl ack circles) in arbitrary unit versus tem-\nperature T, in zero magnetic field, with electric field ǫ= 1,I0= 2,K0= 0.5.\nFIG. 2: BCC antiferromagnetic thin film: Resistivity Rwith temperature-independent relax-\nation (white circles) and temperature-dependent relaxati on (black circles) in arbitrary unit versus\ntemperature T, in zero magnetic field, with electric field ǫ= 1,I0= 2,K0= 0.5.\nFigure 6 shows τ−1\nIfor both ferro- and antiferromagnetic cases, for comparison. T he\nantiferromagnet has τ−1\nIlarger at T < T C. Note that the resistivity is proportional to τ−1\nI.\nLet us discuss the reason why the temperature dependence of th e lattice relaxation time\naffects so strongly the shape of the spin resistivity for T≥TC. First, we emphasize on two\n”empirical” rules that we observed and verified in a number of cases:\n•a) itinerant spins move easily when they are energetically unstable. T he electric field\nthen drives them easily forward. On the other hand, when they are ”at ease” with\nsurrounding spins, namely their energy is low, they will not move easily . We have\nchecked this rule by calculating their velocity as a function of their en ergy610\nFIG. 3: BCC ferromagnetic and antiferromagnetic films: Resi stivityRwith temperature-\ndependent relaxation for ferro- (black circles) and antife rromagnet (white circles) in arbitrary\nunit versus temperature T, in zero magnetic field, with electric field ǫ= 1,I0= 2,K0= 0.5.\nFIG. 4: BCC ferromagnetic film: Inverse of relaxation time of itinerant spins τ−1\nIcalculated with\ntemperature-independent (white circles) and temperature -dependent (black circles) of the lattice\nspins versus temperature T, in zero magnetic field, with electric field ǫ= 1,I0= 2,K0= 0.5.\n•b) in the case where the energy of an itinerant is low, the move of itine rant spins\ndepends on the energy difference between its initial and final positio ns. Consider\nthe ordered phase of the lattice: the energy at any point is very low and the energy\ndifference between any two points is close to zero (ordered state) . So, by the MC\nupdating criterion, the electric field dominates again the move of itine rant spins. This\nexplains the very small resistivity at low Twith respect to that at high T(except\nwhenT→0 where other mechanisms come to play).\nFor the effect of τL, several important points are in order:\ni) ForT < T Cthe lattice is ordered, therefore itinerant spins do not see the diffe rence11\nFIG. 5: BCC antiferromagnetic film: Inverse of relaxation ti me of itinerant spins τ−1\nIcalculated\nwith temperature-independent(whitecircles) and tempera ture-dependent(black circles) relaxation\ntime of the lattice spins versus temperature T, in zero magnetic field, with electric field ǫ= 1,\nI0= 2,K0= 0.5.\nFIG. 6: BCC ferromagnet and antiferromagnet cases: Inverse of relaxation time of itinerant spins\nτ−1\nIcalculated with temperature-dependent relaxation time of the lattice spins for ferromagnet\n(black circles) and antiferromagnet (white circles) versu s temperature T, in zero magnetic field,\nwith electric field ǫ= 1,I0= 2,K0= 0.5.\nwhen the lattice changes its microstates more frequently or less fr equently. This explains the\nsame values obtained for Rwith and without temperature dependence of τLin ferromagnets.\nIn antiferromagnets, one observes a small difference due to the p resence of lattice down spins\nwhich act differently on up-polarized itinerant spins.\nii) ForT > T C, the lattice is disordered: the lattice spins are frequently flipped. I tinerant\nspins have to move constantly to accommodate themselves to the fl uctuating environment.\nThus,τIis long by definition because there are very few rejections to move. Consequently,12\nRis small in the paramagnetic phase.\niii) Finally, it is striking to observe a strong correlation between τLandτI: SinceτLis\nvery large in the transition region where the lattice is in the regime of c ritical slowing down,\nitinerant spins have time to find themselves in energetically favorable positions. Once they\nare there they refuse to move (first rule mentioned above). As a c onsequence τIis very small\n(for example τI= 1 if they refuse to move at every update trial). Ris thus very high. We\nhave showed the inverse of τIin Figs. 4, 5 and 6 because Ris inversely proportional to τI.\nThe correlation between τLandτIis thus ”high τLcorresponds to low τI” and vice-versa.\nNow, let us show the effect of the choice of Aof Eq. (8) in Fig. 7. The higher A(i.\ne. higher τL) induces an increase of RnearTCbut gives the same value as Tis far away\nfrom the critical point. Thus, the width of Rat the transition temperature can serve as a\nmeasure of the relaxation time of the lattice spins.\nFIG. 7: BCC ferromagnet. Effect of the choice of the constant A: i)A= 1 (black circles), ii)\nA= 2 (white circles) versus temperature T, in zero magnetic field, with electric field ǫ= 1,I0= 2,\nK0= 0.5. See text for comments.\nIV. CONCLUSION\nIn this paper, we have shown the effect of the temperature depen dence of the relaxation\ntime on the spin resistivity for both ferromagnetic and antiferroma gnetic films with BCC\nlattice structure.\nIn the ferromagnetic case, the long relaxation time in the critical re gion compared to\nthat of the paramagnetic phase gives rise to a sharp peak of the sp in resistivity at TC.\nThe resistivity in the low- Tregion is insensitive to the relaxation time while in high- T13\nregion, the resistivity is much smaller than that obtained with the tem perature-independent\nrelaxation time. The same tendency is observed for the antiferrom agnetic case: while the\nspin resistivity in the case of temperature-independent relaxation time does not show a\npeak atTC, the extremely long relaxation in the critical region with respect to t hat of the\nparamagnetic phase gives rise to a pronounced rounded peak at TC. It is very interesting to\nstudy other systems such as spin glasses where the relaxation time is extremely long even\nat temperatures far below TC.\n1P.-G. de Gennes and J. Friedel, J. Phys. Chem. Solids 4(1958) 71.\n2M. E. Fisher and J.S. Langer, Phys. Rev. Lett. 20(1968) 665.\n3T. Kasuya , Prog. Theor. Phys. 16(1956) 58.\n4E. A. Turov, Iza. Akad. Nauk. SSSR. Serb. Fiz. 19(1955) 426.\n5Y. Magnin, K. Akabli, H. T. Diep and Isao Harada, Comp. Mat. Sci. Computational Materials\nScience49(2010) S204.\n6Y. Magnin, K. Akabli and H. T. Diep, Phys. Rev. B, to appear.\n7S. Chandra, L. K. Malhotra, S. Dhara and A. C. Rastogi, Phys. Rev. B54(1996) 13694.\n8J. Du, D. Li, Y. B. Li, N. K. Sun, J. Li and Z. D. Zhang, Phys. Rev. B76(2007) 094401.\n9M. A. McGuire, A. D. Christianson, A. S. Sefat, B. C. Sales, M. D. Lumsden, R. Jin, E. A.\nPayzant, D. Mandrus, Y. Luan, V. Keppens, V. Varadarajan, J. W. Brill, R. P. Hermann, M.\nT. Sougrati, F. Grandjean and G. J. Long, Phys. Rev. B78(2008) 094517.\n10C. L. Lu, X. Chen, S. Dong, K. F. Wang, H. L. Cai, J.-M. Liu, D. Li and Z. D. Zhang, Phys.\nRev.B79(2009) 245105.\n11Tiffany S. Santos, Steven J. May, J. L. Robertson and Anand Bhat tacharya, Phys. Rev. B80\n(2009) 155114.\n12Alla E. Petrova, E. D. Bauer, Vladimir Krasnorussky, and Ser gei M. Stishov, Phys. Rev. B74\n(2006) 092401.\n13S. M. Stishov, A.E. Petrova, S. Khasanov, G. Kh. Panova, A.A. Shikov, J. C. Lashley, D. Wu,\nand T. A. Lograsso, Phys. Rev. B76(2007) 052405.\n14Jing Xia, W. Siemons, G. Koster, M. R. Beasley and A. Kapituln ik,Phys. Rev. B79(2009)\n140407(R).14\n15X. F. Wang, T. Wu, G. Wu, Y. L. Xie, J. J. Ying, Y. J. Yan, R. H. Liu and X. H. Chen, Phys.\nRev. Lett. 102(2009) 117005.\n16Y. B. Li, Y. Q. Zhang, N. K. Sun, Q. Zhang, D. Li, J. Li and Z. D. Zh ang,Phys. Rev. B72\n(2005) 193308.\n17Y. Q. Zhang, Z. D. Zhang and J. Aarts, Phys. Rev. B79(2009) 224422.\n18Mitsuo Kataoka, Phys. Rev. B63(2001) 134435-1.\n19G. Zarand, C. P. Moca and B. Janko, Phys. Rev. Lett. 94(2005) 247202.\n20C. Haas, Phys. Rev. 168(1968) 531.\n21K. Akabli, H. T. Diep and S. Reynal, J. Phys.: Condens. Matter 19(2007) 356204.\n22K. Akabli and H. T. Diep, Phys. Rev. B77(2008) 165433.\n23K. Akabli, Y. Magnin, M. Oko, Isao Harada and H. T. Diep, submi tted toPhys. Rev. B.\n24Danh-Tai Hoang, Y. Magnin and H. T. Diep, submitted to Modern Phys. Lett. B\n25P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49(1977) 435.\n26Vladimir V. Prudnikov, Pavel V. Prudnikov, Aleksandr S.Kri nitsyn, Andrei N. Vakilov, Evgenii\nA. Pospelov, and Mikhail V. Rychkov, Phys. Rev. E81(2010) 011130, and references therein."    },    {        "title": "1710.09623v1.Dynamical_spin_accumulation_in_large_spin_magnetic_molecules.pdf",        "content": "Dynamical spin accumulation in large-spin magnetic molecules\nAnna P lomi\u0013 nska,1,\u0003Ireneusz Weymann,1,yand Maciej Misiorny2, 1,z\n1Faculty of Physics, Adam Mickiewicz University, 61-614 Pozna\u0013 n, Poland\n2Department of Microtechnology and Nanoscience MC2,\nChalmers University of Technology, SE-412 96 G oteborg, Sweden\n(Dated: November 10, 2021)\nThe frequency-dependent transport through a nano-device containing a large-spin magnetic\nmolecule is studied theoretically in the Kondo regime. Speci\fcally, the e\u000bect of magnetic anisotropy\non dynamical spin accumulation is of primary interest. Such accumulation arises due to \fnite o\u000b-\ndiagonal in spin components of the dynamical conductance. Here, employing the Kubo formalism\nand the numerical renormalization group (NRG) method, we demonstrate that the dynamical trans-\nport properties strongly depend on magnetic con\fguration of the device and intrinsic parameters\nof the molecule. Speci\fcally, the e\u000bect of dynamical spin accumulation is found to be greatly af-\nfected by the type of magnetic anisotropy exhibited by the molecule, and it develops for frequencies\ncorresponding to the Kondo temperature. For the parallel magnetic con\fguration of the device,\nthe presence of dynamical spin accumulation is conditioned by the interplay of ferromagnetic-lead-\ninduced exchange \feld and the Kondo correlations.\nI. INTRODUCTION\nOver the past two decades, nano-devices involving\nindividual spin impurities strongly tunnel-coupled to\nleads have proven to be an excellent test-bed for study-\ning quantum many-body e\u000bects in electronic transport,\namong which the Kondo e\u000bect is one of the most promi-\nnent ones [1{3]. In principle, the role of such a spin\nimpurity can be played by any system that either in-\nherently exhibits spin or is capable to accommodate\na single conduction electron, which has been experi-\nmentally demonstrated for various nanoscopic structures,\nsuch as, quantum dots [4{7], magnetic adatoms [8{12]\nor molecules [13{22]. A proper understanding of the ef-\nfect of charge and spin correlations on electronic trans-\nport is especially sought for devices based on large-\nspin (S >1=2) impurities. Importantly, such systems\nare a suitable platform for applications in emerging tech-\nnologies for storage and processing information [23, 24],\nwhose aim is to utilize magnetic properties of single\natoms [25{27] or molecules [28{31]. The key property\nof a large spin to serve as a base for a memory de-\nvice is the uniaxial magnetic anisotropy that introduces\nan energy barrier for spin reversal [32]. The uniaxial\ncomponent of magnetic anisotropy is, however, often ac-\ncompanied by the transverse one [33] that, allowing for\nthe under-barrier transitions [34], has the parasitic ef-\nfect on the spin stability. In the stationary transport\nregime, the interplay between the Kondo correlations and\nthe magnetic anisotropy of a spin impurity has been pre-\ndicted to signi\fcantly a\u000bect transport characteristics of\na device. Speci\fcally, this interplay leads to a number\nof spectroscopic features ranging from the current sup-\npression due to the spin reversal barrier [35{38] to some\n\u0003anna.plominska@amu.edu.pl\nyweymann@amu.edu.pl\nzmisiorny@amu.edu.plmore intricate, Berry's phase-related e\u000bects originating\nfrom the quantum tunneling of spin [39{45].\nIn the present paper, on the other hand, we address\nthe dynamical aspect of spin-dependent transport\nthrough magnetic molecules in the Kondo regime.\nWhereas this problem has been studied for spin one-half\nimpurities [46{55], it has only recently attracted some\nattention in the context of large-spin impurities [56].\nIn general, by analyzing the dynamical response of a sys-\ntem to an external time-dependent bias one obtains a di-\nrect access to the \ructuations in the system, which is en-\nsured by the \ructuation-dissipation theorem [57] linking\nthe dynamical conductance of the system with its noise\npower spectral density.\nHere, we speci\fcally focus on the in\ruence of mag-\nnetic anisotropy on the e\u000bect of dynamical spin accu-\nmulation , which can be attributed to the non-zero o\u000b-\ndiagonal in spin component of frequency-dependent con-\nductance [48, 50]. The physical meaning of such accu-\nmulation can be better understood if one imagines that\nit actually corresponds to the situation when, for in-\nstance, one injects electrons of given spin orientation\nbut detects the current of opposite spin direction. For\nthis purpose, we consider a magnetic molecule as an ex-\nemplar of a large-spin impurity. Formed by a single\nconducting orbital exchange-coupled to an anisotropic\nmagnetic core, such a model of a molecule captures\ne\u000bects both due to charging and magnetic anisotropy.\nThe dynamical linear-response transport characteristics\nof the system are obtained using a combination of the\nKubo approach [56, 57] and the numerical renormaliza-\ntion group (NRG) method [58, 59].\nWe show that the e\u000bect of dynamical spin accumu-\nlation strongly depends on the intrinsic parameters of\nthe molecule and the magnetic con\fguration of the de-\nvice. For the antiparallel magnetic con\fguration and\nin the case of an easy-axis type of uniaxial magnetic\nanisotropy, the spin accumulation becomes generally sup-\npressed, however, it can be restored if a transversearXiv:1710.09623v1  [cond-mat.mes-hall]  26 Oct 20172\nelectrodeLeft\nelectrodeRight\nGate electrodee\nmagnetic\ncoreorbital\nlevelMagnetic molecule\nxyz\nFigure 1. Schematic depiction of a large-spin magnetic mole-\ncule embedded in a magnetic tunnel junction. For a detailed\ndescription of the model system see Sec. II.\nanisotropy component is also present. This is contrary to\nthe case of an easy-plane type of anisotropy, where a pro-\nnounced dynamical spin accumulation can develop both\nin the absence and presence of transverse anisotropy.\nFor all considered cases, we \fnd that a local maximum\nin the dynamical spin accumulation emerges for energy\nscale corresponding to the Kondo temperature, with the\nheight dependent on the strength of Kondo correlations.\nFurthermore, for the parallel magnetic con\fguration of\nthe device, we demonstrate that the presence and mag-\nnitude of dynamical spin accumulation is conditioned\nby the interplay of ferromagnetic-lead proximity-induced\nquadrupolar exchange \feld and the correlations respon-\nsible for the formation of the Kondo e\u000bect.\nThe paper is organized as follows: In Sec. II an ac-\ncount of basic premises and assumptions of the model\nis provided, while in Sec. III the theoretical framework\nused in calculations of the dynamical conductance is out-\nlined. Numerical results are discussed in Sec. IV, which\nbegins with a detailed review of energy reference scales\nand model parameters (Sec. IV A). The analysis of the re-\nsults we start for an anisotropic molecule, and next, we\ninclude stepwise the uniaxial (Sec. IV B 1) and trans-\nverse (Sec. IV B 2) component of magnetic anisotropy\ninto the picture. The e\u000bects of the spin polarization\nand magnetic con\fguration of electrodes are discussed\nin Sec. IV C and Sec. IV D, respectively. In Sec. IV E\nthe transport behavior in the case of the antiferromag-\nnetic coupling between the molecule's core spin and\nthe orbital level is discussed. Finally, the main conclu-\nsions are presented in Sec. V\nII. MODEL SYSTEM\nIn order to study the e\u000bect of dynamical spin accu-\nmulation in the case of a large-spin impurity, we employ\nhere the model system that consists of a large-spin mag-\nnetic molecule embedded in the magnetic tunnel junc-\ntion, see Fig. 1. Speci\fcally, the magnetic molecule is\nrepresented as a large internal spin ^S(referred to alsoas magnetic core), with S > 1=2, coupled viaexchange\ninteraction Jto a single orbital level (OL). It is as-\nsumed that the molecule is tunnel-coupled to ferromag-\nnetic electrodes of the junctions only through the OL,\nwhich essentially means that transport of electrons across\nthe junction takes place exclusively through this orbital\n[60, 61]. Moreover, spin-dependent electron tunneling\nprocesses between the OL and electrodes lead to broad-\nening of the former, and this broadening is described\nby the spin-dependent hybridization function \u0000q\n\u001bwith\nq=L(eft);R(ight).\nThe full Hamiltonian ^Htotalcharacterizing the system\nunder consideration has the following form\n^Htotal=^HOL+^Hcore+^HOL-core +^Hel+^Htun:(1)\nHere, the \frst three terms are related to the magnetic\nmolecule. In particular, ^HOLaccounts for the key prop-\nerties of the conducting OL and it reads as\n^HOL=\"X\n\u001b^n\u001b+U^n\"^n#; (2)\nwith the \frst term representing the contribution due to\noccupation of the OL by an electron of spin \u001band en-\nergy\", and the second term including the Coulomb inter-\nactionUthat arises in the situation when two electrons\nof opposite spins reside in the OL. The relevant occupa-\ntion operator ^ n\u001b= ^cy\n\u001b^c\u001bis de\fned in terms of electron\ncreation (^cy\n\u001b) and annihilation (^ c\u001b) operators for the OL.\nWe note that application of a voltage to the gate elec-\ntrode allows for tuning the OL energy \". Furthermore,\nthe second term of Hamiltonian (1) describes magnetic\nanisotropy of the molecule's magnetic core within the\ngiant-spin approach [32],\n^Hcore=\u0000D^S2\nz+E\u0000^S2\nx\u0000^S2\ny\u0001\n; (3)\nwithDandEdenoting the uniaxial and transverse mag-\nnetic anisotropy constants, respectively. Finally, the ex-\nchange interaction between the magnetic core e\u000bective\nspin ^Sand the spin of a single electron occupying the or-\nbital ^s= (1=2)P\n\u001b\u001b0^\u001b\u001b\u001b0^cy\n\u001b^c\u001b0, where ^\u001b\u0011(^\u001bx;^\u001by;^\u001bz)\nstands for the Pauli spin operator, is given by\n^HOL-core =\u0000J^s\u0001^S; (4)\nwith theJ-coupling being ferromagnetic (FM) forJ >0\nand antiferromagnetic (AFM) for J <0.\nFerromagnetic electrodes of the junction are approxi-\nmated as reservoirs of non-interacting and spin-polarized\nelectrons and described by the Hamiltonian\n^Hel=X\nq\u001bZW\n\u0000Wd\u000f\u000f^aqy\n\u001b(\u000f)^aq\n\u001b(\u000f); (5)\nwhere ^aqy\n\u001b(\u000f)\u0002\n^aq\n\u001b(\u000f)\u0003\nis the operator responsible for cre-\nation [annihilation] of a spin- \u001belectron in the qth elec-\ntrode, and Wdenotes the conduction band half-width.3\nMoreover, only the case of a collinear relative orientation\nof the spin moments of electrodes, that is, the parallel (P)\nand antiparallel (AP) magnetic con\fguration, is consid-\nered. It is also assumed that the orientation of these\nspin moments is collinear with the principal axis of the\nmolecule corresponding to the uniaxial component of its\nmagnetic anisotropy.\nUltimately, the single electron tunneling processes be-\ntween the OL and electrodes are captured by the last\nterm of Hamiltonian (1),\n^Htun=X\nq\u001br\n\u0000q\n\u001b\n\u0019ZW\n\u0000Wd\u000fh\n^aqy\n\u001b(\u000f)^c\u001b+ ^cy\n\u001b^aq\n\u001b(\u000f)i\n:(6)\nLet us introduce the total broadening \u0000q= \u0000q\n\"+ \u0000q\n#of\nthe OL due to its tunnel-coupling to the qth electrode,\nand de\fne the spin polarization coe\u000ecient for theqth\nelectrode as pq=\u0000\n\u0000q\n\"\u0000\u0000q\n#\u0001\n=\u0000\n\u0000q\n\"+ \u0000q\n#\u0001\n. Next, assuming\nthat both electrodes are made of the same material ( pL=\npR\u0011p) and that the OL is tunnel-coupled symmetrically\nto both electrodes (\u0000L= \u0000R\u0011\u0000), one can parametrize\nthe hybridization functions as follows: \u0000L\n\"(#)= \u0000R\n\"(#)=\n(\u0000=2)(1\u0006p) for the parallel magnetic con\fguration, and\n\u0000L\n\"(#)= \u0000R\n#(\")= (\u0000=2)(1\u0006p) for the antiparallel one.\nIII. DYNAMICAL SYSTEM RESPONSE\nSince the main goal is to analyze the e\u000bect of dy-\nnamical spin accumulation, below we outline a deriva-\ntion of the frequency-dependent conductance (admit-\ntance) in terms of relevant spectral functions. In the\nnext step, these functions will be calculated with the help\nof the Wilson's numerical renormalization group (NRG)\nmethod [58, 59, 62, 63].\nTo begin with, let us assume that an external bias volt-\nageVL(R)(t) modulated periodically in time is applied\nto the ferromagnetic electrodes. To take into account\nthe e\u000bect of such a time-dependent bias, the full Hamil-\ntonian (1) of the system becomes modi\fed by adding a\nnew term [47, 48, 51, 56],\n^Hbias=X\nq\u001b^Qq\n\u001bVq(t); (7)\nwith the operator ^Qq\n\u001bdescribing the spin- \u001bcomponent\nof charge induced in the qth electrode de\fned as\n^Qq\n\u001b=\u0000jejZW\n\u0000Wd\u000f^aqy\n\u001b(\u000f)^aq\n\u001b(\u000f): (8)\nTo calculate the current \rowing through the system of\na large-spin magnetic molecule, we use the Kubo formula\nIq(t)\u0011h^Iq(t)i=X\nq0\u001b\u001b0Z\ndt0Gqq0\n\u001b\u001b0(t\u0000t0)Vq0(t0);(9)whereGqq0\n\u001b\u001b0(t\u0000t0) stands for the time-dependent conduc-\ntance and takes the following form\nGqq0\n\u001b\u001b0(t\u0000t0) =\u0000i\n~\u0012(t\u0000t0)\n[^Iq\n\u001b(t);^Qq0\n\u001b0(t0)]\u000b\n; (10)\nwith the current, ^Iq\n\u001b(t) = d ^Qq\n\u001b(t)=dt, and charge, ^Qq0\n\u001b0(t0),\noperators given in the interaction picture, and h:::i\ndenoting the quantum-statistical average. Next, after\nFourier-transforming Eq. (10) and performing laborious,\nalbeit straightforward calculations, one can \fnd a gen-\neral expression for the frequency-dependent (dynami-\ncal) conductance Gqq0\n\u001b\u001b0(!) |for a detailed derivation see,\ne.g., Ref. [56]. At this point, let us focus on the cur-\nrent response IR(!) in the right electrode, and use that\nthis current is invariant under an overall potential shift\nby\u0000VR(!), which yields\nIR(!) =X\n\u001b\u001b0GRL\n\u001b\u001b0(!)\u0002\nVL(!)\u0000VR(!)\u0003\n: (11)\nThus, taking into consideration only the real part of\ntheright-left component of the dynamical conductance,\nGc(!)\u0011P\n\u001b\u001b0\u0002\nReGRL\n\u001b\u001b0(!)\u0003c;for the parallel ( c= P)\nand antiparallel ( c= AP) magnetic con\fguration of the\njunction one obtains\nGc(!) =X\n\u001b\u001b0Gc\n\u001b\u001b0(!); (12)\nwith the spin-resolved components Gc\n\u001b\u001b0(!) of the form\nGc\n\u001b\u001b0(!) =G0\n2\u000bc\n\u001b\u001a\n\u000e\u001b\u001b0\u0002\ngOL\n\u001b(!)]c\n+1\n2\fc\n\u001b\u001b0\u0002\ngI\n\u001b\u001b0(!)\u0003c\u001b\n:(13)\nThe factors \u000bc\n\u001band\fc\n\u001b\u001b0in the equation above depend\nonly on the magnetic con\fguration and the spin polar-\nization coe\u000ecient pof electrodes,\n\u000bP\n\u001b= 1 +\u0011\u001bpand\u000bAP\n\u001b= 1\u0000p2; (14)\nwith\u0011\"(#)=\u00061,\n\fP\n\u001b\u001b0=r1 +\u0011\u001b0p\n1 +\u0011\u001bpand\fAP\n\u001b\u001b0=1 +\u0011\u001bp\n1 +\u0011\u001b0p: (15)\nFurthermore, in Eq. (13), G0\u00112e2=his the conduc-\ntance quantum, while gOL\n\u001b(!) and gI\n\u001b\u001b0(!) represent two\ndi\u000berent (dimensionless) contributions to the conduc-\ntance [48],\ngOL\n\u001b(!) =1\n2!Z\nd!0AOL\n\u001b(!0)\nA0\u0002\nf(!0\u0000!)\u0000f(!0+!)\u0003\n;(16)\nand\ngI\n\u001b\u001b0(!) =\u00001\n!\u0001AI\n\u001b\u001b0(!)\n~\u001aA0: (17)4\nImportantly, the physical origin of each of these contri-\nbutions can be deduced from analysis of the two spectral\nfunctionsAOL\n\u001b(!) andAI\n\u001b\u001b0(!) occurring in Eqs. (16)\nand (17), respectively. In particular, the former spec-\ntral function describes the orbital level (OL), and it is\nde\fned as\nAOL\n\u001b(!)\u0011\u00001\n\u0019Imhhc\u001bjcy\n\u001biir\n!: (18)\nThe latter, on the other hand, is given by\nAI\n\u001b\u001b0(!)\u0011\u00001\n\u0019Imhh^I\u001bj^Iy\n\u001b0iir\n!; (19)\nand this spectral function is associated with the dimen-\nsionless current operator\n^I\u001b\u0011^cy\n\u001b^\t\u001b\u0000^\ty\n\u001b^c\u001b: (20)\nHere, the \feld operator ^\t\u001bcorresponds essentially to the\neven linear combination of electrode operators,\n^\t\u001b=p\u001aZ\nd\u000fh\n\u0003L\n\u001b^aL\n\u001b(\u000f) + \u0003R\n\u001b^aR\n\u001b(\u000f)i\n(21)\nwith \u0003q\n\u001b=p\n\u0000q\n\u001b=(\u0000L\u001b+ \u0000R\u001b) and\u001a= 1=(2W) being the\ndensity of states of a conduction band. Finally, the scal-\ning factorA0= 1=(\u0019\u0000) in Eqs. (16)-(17) denotes the\nspectral function of a single-level quantum dot (or in\nother words, that of the OL disconnected from the inter-\nnal spin,J= 0) at!= 0 and for nonmagnetic electrodes.\nThe function f(!) in Eq. (16) is the Fermi-Dirac distri-\nbution,f(!) =\b\n1+exp[ ~!=k BT]\t\u00001, withTbeing tem-\nperature and kBstanding for the Boltzmann constant.\nAt this point, we would like to emphasize that one\nof the main quantities of interest in this paper is associ-\nated with the o\u000b-diagonal in spin components of the spin-\nresolved dynamical conductance. In particular, Gc\n\"#(!)\nandGc\n#\"(!) take into account the spin correlations be-\ntween the spin-up and spin-down channels and their \f-\nnite values can be associated with the e\u000bect of dynamical\nspin accumulation that can build up in the molecule at\n\fnite driving frequencies ![48, 50].\nAs one can see from Eqs. (13) and (16)-(17), in or-\nder to calculate the spin-resolved components Gc\n\u001b\u001b0(!)\nof the dynamical conductance, one needs to know \frst\nthe spectral functions: AOL\n\u001b(!), Eq. (18), and AI\n\u001b\u001b0(!),\nEq. (19). In the present work, these functions are de-\nrived using the NRG method [58, 59, 62, 63]. The idea\nof NRG is based on the logarithmic discretization of the\nconduction band with the discretization parameter \u0003. In\nthe next step, such a discretized model is mapped onto\na semi-in\fnite chain with exponentially decaying hop-\npings. Importantly, the \frst site of semi-in\fnite chain\nis coupled to a spin impurity. To obtain the follow-\ning results, we used the discretization parameter \u0003 = 2,\nand we kept Nk= 2560 states during calculations. The\nhigh accuracy of calculations was achieved by averag-\ning the spectral data over Nz= 4 di\u000berent discretizationmeshes [64]. Moreover, when discussing the behavior\nof the conductance in the zero-frequency limit, we will\npresent the data obtained from the full density-matrix\nNRG approach [62, 65] by assuming T=W = 10\u000012, which\nis much smaller than the other energy scales considered\nthroughout this paper.\nIV. NUMERICAL RESULTS AND DISCUSSION\nThe main goal of this paper is to investigate the e\u000bect\nof dynamical spin accumulation in large-spin magnetic\nmolecules, and in particular, to discuss how this e\u000bect is\na\u000bected when the molecular spin is subject to magnetic\nanisotropy. For this purpose, we consider the model of\na magnetic molecule introduced in Sec. II with the mag-\nnetic core characterized by spin S= 2. To conduct a\nsystematic analysis of the problem, and to understand\nhow the dynamical spin accumulation manifests itself in\nfrequency-dependent transport characteristics, \frst we\nwill address the simplest example of a spin-isotropic\nmolecule (D= 0 andE= 0). Next, we will discuss in\nSec. IV B 1 how dynamical spin accumulation changes\nwhen the uniaxial component of magnetic anisotropy be-\ncomes gradually involved ( D6= 0 andE= 0). Finally,\nalso the transverse component ( D6= 0 andE6= 0) will\nbe included in Sec. IV B 2 to establish the complete pic-\nture of the problem.\nA. Energy reference scales and model parameters\nIn the regime of strong tunnel coupling of a molecule\nto electrodes, which is of key interest here, transport\nproperties of the system are determined by strong charge\nand spin correlations. As a result, one can generally ex-\npect the Kondo e\u000bect to play a dominant role as soon as\nthe OL is occupied by a single electron and temperature\nis lower than some characteristic energy scale, referred to\nas the Kondo temperature TK[2]. In general, TKcan de-\npend on di\u000berent parameters of a system under consider-\nation. For instance, in the case of a single-level quantum\ndot (the Anderson impurity) attached to ferromagnetic\nelectrodesTK, in the vicinity of the particle-hole symme-\ntry point (\"=U\u0019\u00000:5), is determined by the Coulomb\ninteraction U, the broadening of the level \u0000 and the spin\npolarization of electrodes p[66, 67]. On the other hand,\nin large-spin systems the Kondo temperature can be ad-\nditionally a\u000bected by other parameters of the model, such\nas, the spin length Sor magnetic anisotropy constants D\nandE[35, 39, 40, 44]. Thus, in the following calculations\nwe use the Kondo temperature TKfor a bare OL ( J= 0)\ntunnel-coupled to nonmagnetic electrodes ( p= 0), given\nin energy units ( kB\u00111), as a consistent energy reference\nscale insensitive to magnetic properties of the molecule.\nSuch a choice of TKcorresponds in fact to the Kondo\ntemperature of a single-level quantum dot, and hence-\nforth we will refer to this temperature as T0\nK. Speci\fcally,5\nthe temperature dependence of zero-frequency normal-\nized linear conductance G(!= 0;T)=G(!= 0;T= 0)\nat the particle-hole symmetry point ( \"=U=\u00000:5) is\nused for estimation of T0\nKfrom the following condi-\ntionG(!= 0;T=T0\nK)=G(!= 0;T= 0) = 1=2.\nIn this paper, all the results are obtained for T= 0.\nWith regard to the magnetic con\fguration of the junc-\ntion, the main discussion is carried out for the case of the\nantiparallel orientation of spin moments in electrodes. In\nsuch a con\fguration the e\u000bective spintronic dipolar [67]\nand quadrupolar [68] exchange \felds are generally absent,\nwhich allows us to analyze how the dynamical spin accu-\nmulation is a\u000bected exclusively by the intrinsic molecular\nmagnetic anisotropy. Later on, we will also include the\nspintronic contribution to magnetic anisotropy by switch-\ning the junction into the parallel magnetic con\fguration.\nNote that throughout the paper a molecule is assumed\nto be electrostatically tuned viaa gate electrode to the\nparticle-hole symmetric point \"=U=\u00000:5, so that in the\nparallel magnetic con\fguration the dipolar exchange \feld\ndoes not arise [37]. In this way, we can consistently ex-\nclude any e\u000bects stemming from the presence of a mag-\nnetic \feld, either real or e\u000bective, which are not the sub-\nject of the present analysis.\nMoreover, the Coulomb energy is chosen U=W = 0:4,\nwith the half-width of conduction band Wserving here\nas the energy unit (that is, W\u00111), whereas the mag-\nnitude of the constant Jdescribing the exchange cou-\npling between the OL and the electrodes is taken to be\njJj=W= 0:0045. Depending on whether Jispositive or\nnegative , one can expect either the underscreened [69{72]\nortwo-stage [73{75] Kondo e\u000bect, respectively, to arise\nin the system [76]. Here, we focus on the case of J >0,\nthat is, on the ferromagnetic type of the J-coupling, and\nthe key di\u000berences occurring for the antiferromagnetic\ncoupling (J < 0) will be addressed only at the end, in\nSec. IV E. Finally, the broadening of the OL due to tun-\nneling of electrons to/from external electrodes is assumed\nto be \u0000=U= 0:1, while the spin polarization of electrodes\nisp= 0:5, unless stated otherwise. Consequently, for the\nparameters assumed above one \fnds the reference Kondo\ntemperature to be T0\nK=W= 0:002.\nB. The e\u000bect of magnetic anisotropy on the\ndynamical spin accumulation\nBefore we proceed to a discussion of how the dy-\nnamical spin accumulation is a\u000bected by the presence\nof magnetic anisotropy, it may be instructive to focus\n\frst brie\ry on frequency-dependence transport features\nof a spin-isotropic molecule1. This case is illustrated\n1Note that in order to enable qualitative comparison of the present\nresults with the previous studies for a single-level quantum dot\n(QD) [47, 50, 51], in Fig. 2 dotted lines representing the latterwith the solid line in Fig. 2, where the dynamical con-\nductanceGAP(!) for the antiparallel magnetic con\fg-\nuration of the junction is shown as a function of fre-\nquency!. SinceGAP(!) can be in general resolved into\nfour spin components GAP\n\u001b\u001b0(!), see Eq. (12), in Fig. 2\napart from the total conductance GAP(!) [plotted in\n(a,e)] we also present the spin-diagonal GAP\n\u001b\u001b(!) [in (b,f)]\nand o\u000b-diagonal GAP\n\u001b\u001b(!) [in (c,g)] contributions, with the\nnotation\u001bto be read as\"\u0011# and#\u0011\" . In particular,\nhere only the components for \u001b=\"are shown, because\nin the antiparallel magnetic con\fguration the following\nsymmetries hold:2\nGAP\n\"\"(!)=GAP\n##(!) andGAP\n#\"(!)=(1\u0000p)2\n(1+p)2GAP\n\"#(!):\nMoreover, since the dynamical spin accumulation essen-\ntially leads to enhancement of imbalance in the number\nof electrons with opposite spin orientations transferred\nacross the junction viaa molecule, we introduce here the\nfrequency-dependent parameter P(!) characterizing the\nspin polarization of the current injected into a drain elec-\ntrode de\fned as:\nP(!)\u0011I\"(!)\u0000I#(!)\nI\"(!) +I#(!): (22)\nIn the situation under consideration, the role of a drain\nis played by the right electrode, I\u001b(!)\u0011IR\n\u001b(!), Eq. (11),\nandI\u001b(!)/G\u001b\u001b(!) +G\u001b\u001b(!), so that\nP(!) =P0(!) +Pdsa(!): (23)\nImportantly, the current spin polarization parameter\nP(!) consists of two terms: the \frst representing the\ndiagonal in spin contribution to the conductance,\nP0(!) =G\"\"(!)\u0000G##(!)P\n\u001b\u001b0G\u001b\u001b0(!); (24)\nand the second arising exclusively due to the dynamical\nspin accumulation,\nPdsa(!) =G\"#(!)\u0000G#\"(!)P\n\u001b\u001b0G\u001b\u001b0(!); (25)\ncase have been added. However, in the main text we do not\ndiscuss this case.\n2The origin of the two symmetries can be explained by noting that\nthe tunnel-coupling of a molecule to two electrodes, Eq. (6), can\nbe e\u000bectively reduced to a single channel problem by applying\nan appropriate unitary transformation [77]. Interestingly, in the\ncase of antiparallel magnetic con\fguration of the junction the\nnew e\u000bective spin-dependent tunnel coupling becomes actually\nindependent of the spin polarization pof electrodes. As a re-\nsult, calculations of the spectral functions AOL\n\u001b(!), Eq. (18), and\nAI\n\u001b\u001b0(!), Eq. (19), proceed as if electrodes were nonmagnetic , so\nthat eventually the values of these functions do not depend on\nspin indices.6\n00.20.40.60.8\n00.10.20.30.4\n00.10.20.3\n00.10.20.30.40.50.6\n10\u0000610\u0000410\u0000210010\u0000610\u0000410\u00002100\nConductance GAP(!)=G0\nQD\njDj=T0\nK= 0\njDj=T0\nK= 10\u00004\njDj=T0\nK= 10\u00002\njDj=T0\nK= 0.5 00.20.40.60.8 (a)Uniaxial magnetic anisotropy:\neasy-axis type ( D>0)\n(e)easy-plane type ( D<0)GAP\n\"\"(!)=G0\n00.10.20.30.4\n(b) (f)GAP\n\"#(!)=G0\n00.10.20.3(c) (g)Spin polarization PAP(!)\nFrequency !=T0\nK00.10.20.30.40.50.6\n10\u0000610\u0000410\u00002100(d)\nFrequency !=T0\nK10\u0000610\u0000410\u00002100(h)\nFigure 2. The e\u000bect of the uniaxial component of\nmagnetic anisotropy ( D6= 0 and E= 0) on the frequency-\ndependent conductance of a large-spin magnetic molecule\nshown for the FM J-coupling and the junction in the antipar-\nallel (AP) magnetic con\fguration. Left [right ]column corre-\nsponds to the molecule exhibiting the easy-axis (D > 0) [easy-\nplane (D < 0)] type of magnetic anisotropy. Bottom pan-\nels(d,h) present the current spin polarization PAP(!) which\narises here solely due to the dynamical spin accumulation,\ni.e.,PAP(!) =PAP\ndsa(!), Eq. (25). The solid lines represent\nthe case of a spin-isotropic molecule, while the thin dotted\nlines are for a single-level quantum dot (QD), i.e., for J= 0.\nThe vertical dashed lines indicate the corresponding excita-\ntion energies, for details see the main text. The scaling fac-\ntorG0stands for the conductance quantum. For a discussion\nof parameters assumed in calculations see Sec. IV A.\nwhich vanishes in the limit of !!0 [50]. Recall\nthat the spin-resolved components G\u001b\u001b0(!) of conduc-\ntance are given by Eq. (13). One can, thus, imme-\ndiately conclude that in the antiparallel magnetic con-\n\fgurationPAP\n0(!) = 0, and consequently, no spin po-\nlarization of the current occurs in the stationary case,PAP(!= 0) = 0, whereas for \fnite-frequency transport\nPAP(!) =PAP\ndsa(!) |namely, the spin polarization of\nthe current is here a purely dynamical e\u000bect.\nFirst of all, we recall that the conductance of a\nspin-isotropic system in the zero-frequency limit ap-\nproaches lim !!0GAP(!) = (1\u0000p2)G0, andGAP(!= 0)\nconsists solely of the diagonal-in-spin components,\nthat is,GAP(!= 0) =P\n\u001bGAP\n\u001b\u001b(!= 0), with the o\u000b-\ndiagonal components being identically equal to zero,\nGAP\n\u001b\u001b(!= 0) = 0. As the driving frequency !gets\nlarger, one observes a monotonic decrease in GAP(!), see\nFig. 2(a). However, a closer examination of spin compo-\nnents of the conductance reveals that whereas GAP\n\"\"(!)\ndecreases in value as well, GAP\n\"#(!) actually follows the\nopposite trend and exhibit a maximum. The broad max-\nimum inGAP\n\"#(!) appears approximately at the frequency\ncorresponding to the Kondo temperature TK[56]. As one\nmay notice, here TK\u001cT0\nK, which stems from the fact\nthatTKbecomes suppressed with the increase of J[76].\nSuch a \fnite-frequency feature in the o\u000b-diagonal in spin\ncomponents of GAP(!) is a hallmark of the dynamical\nspin accumulation occurring in the system. Moreover,\nunlikeGAP\n\"#(!), the spin polarization PAP(!) of the cur-\nrent increases monotonically with !until!\u0019T0\nK, where\na local maximum occurs, see Fig. 2(d).\nOn the other hand, in the limit of large frequen-\ncies,!&T0\nK, one can already point out that the ef-\nfect of magnetic anisotropy is expected to be negligi-\nble (ifjDj<T0\nK). This stems from the fact that by\nfurther raising frequency !, one in fact increases the\namount of energy pumped into the system. As a re-\nsult, excitations between states belonging to the two\nspin multiplets S+ 1=2 andS\u00001=2, arising due to\ntheJ-coupling, take place, and they are observed as\na small resonance in GAP(!) at frequency !\u0019J, indi-\ncated by the arrow in Fig. 2(a). Interestingly, such res-\nonant transitions enhance the diagonal-in-spin conduc-\ntance, Fig. 2(b), whereas they have a detrimental e\u000bect\non the dynamical spin accumulation, leading to a local\nminimum in GAP\n\"#(!), Fig. 2(c), and consequently, also\nin the current spin polarization PAP(!), Fig. 2(d). Fi-\nnally, with the further increase of !one expects that\nexcitations related to the Coulomb interaction should be-\ncome the key factor determining the behavior of conduc-\ntance [51, 56]. Since the purpose of the present work is to\nstudy the e\u000bects due to magnetic anisotropy, which take\nplace at frequencies corresponding to energy scales set by\nmagnetic anisotropy constants DandE, Eq. (3), such a\nlarge-frequency regime ( !\u0019U) is not relevant and, thus,\nit will not be considered here.\n1. Uniaxial magnetic anisotropy\nThe situation changes when the molecular spin starts\nenergetically preferring some speci\fc spatial orienta-\ntion(s), which essentially means that the spin is sub-7\nject to magnetic anisotropy described generally by the\nHamiltonian (3). Let us \frst analyze the case when such\na preferable orientation of the spin is related to a spe-\nci\fc axis, customarily associated with the zquantiza-\ntion axis, see Fig. 1, and represented by the term \u0000D^S2\nz\nin Eq. (3). One can immediately see that depending\non the sign of the uniaxial magnetic anisotropy con-\nstantD, the energy of the system becomes minimized if\nthe molecular spin is oriented either along (for D> 0)\nor perpendicular to (for D< 0) thezaxis. The for-\nmer case is often referred to as the `easy-axis' type of\nmagnetic anisotropy, while the latter one as the `easy-\nplane' type. For systems characterized by a half-integer\nlarge spin ( S >1=2), as the one considered here, it is a\nknown, experimentally observed fact that their Kondo-\ndominated zero-frequency linear current response re-\nmains una\u000bected by the uniaxial magnetic anisotropy\nifD< 0 [10, 11, 78, 79], whereas for D> 0 the trans-\nport becomes suppressed [18, 19, 22, 80].\nThe dynamical conductance for the case of uniaxial\nmagnetic anisotropy of the easy-axis type ( D> 0) is pre-\nsented in the left column of Fig. 2. Several distinctive\nfeatures in GAP(!) that evolve with the increase of D\ncan be immediately spotted in Fig. 2(a). To begin with,\nthe zero-frequency conductance GAP(!= 0) becomes re-\nduced and GAP(!) remains frequency-independent as !\ngrows. Once the frequency reaches !\u0019D, a small\npeak forms, and for even larger values of !the conduc-\ntance gets diminished. Comparing the diagonal-in-spin\ncomponent GAP\n\"\"(!) in Fig. 2(b) with the o\u000b-diagonal\noneGAP\n\"#(!) in Fig. 2(c), one can conclude that the en-\nhancement of the conductance at !\u0019Dcan be fully at-\ntributed to the e\u000bect of the dynamical spin accumulation.\nImportantly, it can be noticed that, unlike in the spin-\nisotropic case where the spin accumulation, PAP(!)6= 0,\npersists over a wide range of frequencies, now the e\u000bect\narises only above some threshold frequency !\u0003. More-\nover, for!&!\u0003the diagonal component GAP\n\"\"(!) de-\ncreases monotonically (until !\u0019T0\nK), whereas GAP\n\"#(!)\n\frst builds up rapidly and then changes only insigni\f-\ncantly up to the limit of !&T0\nK. As a result, the cur-\nrent spin polarization PAP(!), similarly as in the spin-\nisotropic case, increases steadily for larger and lager !,\nthough the achievable values of PAP(!) are appreciably\nsmaller than for a spin-isotropic molecule.\nThe presence of the threshold frequency !\u0003above\nwhich the dynamical spin accumulation takes place\ncan be understood by considering the mechanism un-\nderlying the spin-exchange (Kondo) processes respon-\nsible for \ripping the spin orientation of an electron\nin the molecular OL. To gain an intuitive picture\nof such processes, it is instructive to analyze the\neigenstates of a free-standing molecule, described by\nthe Hamiltonian ^Hmol=^HOL+^Hcore+^HOL-core;see\nEqs. (2)-(4), which participate in transport. Since\nthe results are obtained at T= 0, it su\u000eces to con-\nsider only the states of lowest energy. For J >0,\nthe ground state doublet of the S+1=2 spin multiplet hasthe formjStot\nz=\u00065=2i\u0011j\u0006 1=2iOL\nj\u0006 2icore;with\njsz(Sz)iOL(core) denoting the spin state of the OL (mag-\nnetic core). Because the spin-exchange processes, oc-\ncurring in the OL due to its strong hybridization to\nelectrodes, can just lead to \ripping of the OL spin,\nj\u00001=2iOL$j1=2iOL, without a\u000becting the state of the\ninternal spinjSzicore, no direct transitions between the\ndoublet ground states are possible. In fact, any pair\nof molecular states jStot\nz;1iandjStot\nz;2ican support the\nspin-exchange processes due to tunneling of electrons\nonly ifjStot\nz;1\u0000Stot\nz;2j= 1, which basically stems from\nthe fact that angular momentum exchanged between\nthe tunneling current and the molecule has to be con-\nserved. It means that the only allowed transitions\nfrom the statesjStot\nz=\u00005=2iandjStot\nz= 5=2ican be\nthose to the \frst excited doublet states3jStot\nz=\u00003=2i\nandjStot\nz= 3=2i, respectively. Importantly, these states\nare separated from the ground state doublet by an en-\nergy gap \u0001. As a result, the spin exchange processes,\nwhich underlie the dynamical spin accumulation, be-\ncome active if the energy pumped into the molecule\nby means of periodic driving external potential sat-\nis\fes the condition !&!\u0003\nD>0\u0019\u0001. In the limit of\nD\u001cJand at the particle-hole symmetry point ( \"=U=\n\u00000:5), one can thus estimate [38]: !\u0003\nD>0\u0019KSDwith\nKS= 2S(2S\u00001)=(2S+ 1). Those excitation energies\nare marked in left column of Fig. 2 with vertical dashed\nlines. As can be seen, the agreement between this esti-\nmate and numerical data is quite satisfactory. It proves\nthat the dynamical spin accumulation builds up in the\nmolecule for frequencies !&!\u0003\nD>0.\nThe picture developed above changes signi\fcantly if\na molecule is characterized by the uniaxial magnetic\nanisotropy of the easy-plane type ( D< 0), see the right\ncolumn of Fig. 2. In particular, the major di\u000berence is\nthat in such a situation at frequencies !.Dthe conduc-\ntanceGAP(!) [see Fig. 2(e)] always reaches the limit of\nunitary transport, that is, GAP(!) = (1\u0000p2)G0, which\nis a signature of the Kondo e\u000bect. It is clear from\nFigs. 2(f)-(g) that this e\u000bect is not related to the dynam-\nical spin accumulation, as at low frequencies ( !\u001cD)\ntransport is fully determined only by the conductance\ncomponents diagonal in spin, GAP(!)\u0019P\n\u001bGAP\n\u001b\u001b(!).\nFurthermore, it can be noticed that the components o\u000b-\ndiagonal in spin are now characterized by smaller thresh-\nold frequencies !\u0003, and that their magnitudes are larger,\ncompare Fig. 2(g) with Fig. 2(c). This, in turn, af-\nfects values of the current spin polarization PAP(!),\nwhich in the present case exceed those for a spin-isotropic\nmolecule, see Fig. 2(h). However, in other respects,\nthe behavior of the relevant quantities under discussion,\nshown in Figs. 2(e)-(h), qualitatively resembles that ob-\n3Note that the states jStot\nz=\u00063=2iconstituting the \frst excited\ndoublet of the S+ 1=2 spin multiplet have the from of a linear\ncombination of the following states:\b\nj\u00061=2iOL\nj\u0006 1icore;\nj\u00071=2iOL\nj\u0006 2icore\t\n:8\nserved forD> 0.\nTo understand the origin of the Kondo e\u000bect re-\nvival we again invoke the spectrum of a free-standing\nmolecule. Since the molecular spin is character-\nized by the uniaxial magnetic anisotropy of the easy-\nplane type, it means that the ground state dou-\nblet of the S+ 1=2 spin multiplet is formed by\nthe states with the lowest Stot\nzcomponent, namely,\njStot\nz=\u00061=2i. These states arise as superpositions\nof states\b\nj\u00061=2iOL\nj0icore;j\u00071=2iOL\nj\u0006 1icore\t\n,\nfrom which it is clear that the ground state doublet\ncan now support the electron spin exchange processes\nin the OL |the mechanism underlying the Kondo e\u000bect.\nThe e\u000bective exchange interaction between the molecule\nand the leads is conditioned by the excitation energies be-\ntween the ground state doublet and the empty and fully-\noccupied orbital-level molecular states, which basically\ndepends on all model parameters in a nontrivial fashion.\nConsequently, it is a tedious task to provide a simple an-\nalytical formula for the energy scale !\u0003. Instead, let us\njust conclude from the inspection of frequency-dependent\ntransport characteristics shown in the right column of\nFig. 2 that the Kondo temperature is of the order of the\nmagnetic anisotropy constant, TK\u0018jDj, while the en-\nergy scale!\u0003\nD<0is slightly smaller than TKand grows\nlinearly withjDj. This behavior can be clearly seen in\nthe dynamical spin accumulation shown in Fig. 2(g). The\nonset ofGAP\n\"#(!) moves to larger frequencies with increas-\ningjDjand in the limit of very large magnetic anisotropy\nthe system's dynamical behavior approaches the quan-\ntum dot case, indicated by the thin dotted line.\nFrom the above discussion one can already formulate\nsome more universal statements concerning the behav-\nior of the dynamical spin accumulation. It is clear that\nthis e\u000bect is most e\u000bective when the spin-exchange pro-\ncesses are relevant. This happens for frequencies corre-\nsponding to the energy scale responsible for the forma-\ntion of the Kondo state. Thus, one can observe that\nthe maximum in GAP\n\u001b\u001b(!) develops for some resonant fre-\nquency!r, which is of the order of the Kondo temper-\nature,!r\u0019TK. On the other hand, the width of this\nmaximum depends strongly on the frequency range of\nthe slope when the conductance as a function of !in-\ncreases due to the Kondo e\u000bect |note that we discuss\nthe behavior on logarithmic scale. This is why a broad\nmaximum can be observed for spin-isotropic molecules,\nwhile for \fnite magnetic anisotropy the frequency range\nof enhanced spin accumulation is much reduced. When\nthe conductance GAP(!) reaches a plateau with lower-\ning!, the spin-\rip processes become quenched and a\nmany-body delocalized screened-spin state is formed be-\ntween the molecule's spin and the spins of conduction\nelectrons. The electrons, when tunneling through the\njunction, experience then only a phase shift and spin-\n\rip processes are suppressed [2]. As a consequence, the\no\u000b-diagonal components of frequency-dependent conduc-\ntance get suppressed and the e\u000bect of dynamical spin ac-\ncumulation disappears. The energy scale when this hap-pens is, in turn, described by !\u0003, which corresponds to\nthe onset of dynamical spin accumulation with increasing\nthe driving frequency !.\nAs far as the height of the maximum in GAP\n\u001b\u001b(!) is\nconcerned, one can see that if the value of zero-frequency\nconductance is smaller than its maximum value, which\ne\u000bectively means that the Kondo e\u000bect cannot fully de-\nvelop in the system, the magnitude of GAP\n\u001b\u001b(!) gets re-\nduced. This can be especially seen for the easy-axis type\nof magnetic anisotropy presented in the left column of\nFig. 2. On the other hand, for magnetic molecules with\nanisotropy of the easy-plane type, the ground state is al-\nways a spin doublet, so that at low frequencies the Kondo\ne\u000bect can fully develop and, consequently, while the po-\nsition of maximum in GAP\n\u001b\u001b(!) depends on D, its maxi-\nmum value does not. In fact, the maximum value of the\ndynamical spin accumulation is then comparable to the\nquantum dot case, see the right column of Fig. 2.\n2. Uniaxial and transverse magnetic anisotropy\nThe uniaxial component of magnetic anisotropy along\nthezaxis is often accompanied by the transverse one, de-\nscribed by the term E\u0000^S2\nx\u0000^S2\ny\u0001\nin Eq. (3). In essence, it\ncaptures the e\u000bect of breaking the rotational symmetry\naround the zaxis, which, in other words, means that\nthe internal spin has a tendency to align along some\ndirections with respect to the plane perpendicular to\nthezaxis. In particular, for E > 0 the energy of the spin\nis minimized if it is oriented along the yaxis. The signif-\nicance of this transverse term of magnetic anisotropy lies\nin the fact that such a term leads to mixing of the axial\nspin statesjSzicore, which can be easily seen if one intro-\nduces in Eq. (3) the ladder operators ^S\u0006\u0011^Sx\u0006i^Sy:\nGenerally, this mixing is at the foundation of many im-\nportant e\u000bects in\ruencing transport, such as, the quan-\ntum tunneling of spin [39, 44], or the Berry-phase block-\nade [41{43].\nFigure 3 illustrates how inclusion of the transverse\nmagnetic anisotropy ( D6= 0 andE6= 0) a\u000bects the \fnite-\nfrequency conductance and the dynamical spin accumula-\ntion in the case of the easy-axis ( D> 0, left column) and\neasy-plane ( D< 0, right column) type of uniaxial mag-\nnetic anisotropy. Let us \frst focus on the case of D> 0.\nThe \frst noticeable di\u000berence, as compared with the case\nofD> 0 andE= 0 [see Figs. 2(a)-(d)], is that one ob-\nserves the revival of the Kondo e\u000bect for su\u000eciently low\nfrequencies. Such a restoration of transport occurs as a\nconsequence of the mixing caused by the second term of\nHamiltonian (3), because now each molecular spin state\ne\u000bectively becomes a superposition of all possible OL\nelectronic spin and internal spin states [44]. This, in turn,\nmeans that the spin exchange processes leading to tran-\nsitions between the states of the ground state doublet are\npermitted. Furthermore, for !\u0019!ra pronounced max-\nimum in the dynamical conductance GAP(!) is visible,\nFig. 3(a), and it stems from the dynamical spin accu-9\n00.20.40.60.8\n00.10.20.30.4\n00.10.20.3\n00.10.20.30.40.50.6\n10\u0000610\u0000410\u0000210010\u0000610\u0000410\u00002100\nConductance GAP(!)/G0\nE=jDj= 0\nE=jDj= 1=10\nE=jDj= 1=5\nE=jDj= 1=3 00.20.40.60.8\n(a)Uniaxial andtransverse magnetic anisotropy:\neasy-axis type ( D>0)\n(e)easy-plane type ( D<0)GAP\n\"\"(!)/G0\n00.10.20.30.4 (b) (f)GAP\n\"#(!)/G0\n00.10.20.3(c) (g)Spin polarization PAP(!)\nFrequency !=T0\nK00.10.20.30.40.50.6\n10\u0000610\u0000410\u00002100(d)\nFrequency !=T0\nK10\u0000610\u0000410\u00002100(h)\nFigure 3. Analogous to Fig. 2 except that now also the ef-\nfect of the transverse component of magnetic anisotropy Eis\nincluded for a selected value of uniaxial magnetic anisotropy\nconstantjDj=T0\nK= 10\u00002. Here, the transverse constant Eis\nalways assumed positive. Note that to enable an easy com-\nparison with results in Fig. 2, the scales are kept identical\nas in Fig. 2, and the \fnely dashed line representing the case\nofE= 0 is added to serve as the reference line.\nmulationGAP\n\"#(!), which at this particular frequency !r\nexhibits a sharp resonance, as one can see in Fig. 3(c).\nWe also note that at !ra kink develops in the conduc-\ntance component diagonal in spin GAP\n\"\"(!), Fig. 3(b),\nwhile spin polarization PAP(!) exhibits a local maxi-\nmum, Fig. 3(d).\nThe behavior of dynamical transport properties in the\npresence of transverse component of magnetic anisotropy\ncan be understood by invoking the discussion in the pre-\nvious section. Generally, one can again notice that the\n10\u0000810\u0000710\u0000610\u0000510\u0000410\u00003\n10\u0000210\u0000110\u0000610\u0000510\u0000410\u0000310\u0000210\u00001100\n10\u0000210\u00001\nResonance frequency !r=T0\nK\nTransverse anisotropy E=Dantiparallel (AP)\n10\u0000810\u0000710\u0000610\u0000510\u0000410\u00003\n10\u0000210\u00001(a)\nindepend. of p\nE=D=1=3\n!r=!r(E=D=1=3)\nTransverse anisotropy E=Dp= 0.25\np= 0.5\np= 0.7510\u0000610\u0000510\u0000410\u0000310\u0000210\u00001100\n10\u0000210\u00001(b)\nparallel (P)Figure 4. (a) Dependence of the position !rof the resonance\nin the conductance component o\u000b-diagonal in spin, G\u001b\u001b(!),\non the transverse magnetic anisotropy constant E. For the\nantiparallel (AP) magnetic con\fguration (squares) the data\npoint for E=D = 1=3 (marked by a \fnely dashed vertical line)\ncorresponds to the resonance indicated in Fig. 3(c) by the ar-\nrow. Note that for this magnetic con\fguration !ris indepen-\ndent of the spin polarization p, see Fig. 5(g), and that lines\nconnecting the data points serve as a guide for eyes. The data\npoints at E=D = 1=3 for the parallel (P) magnetic con\fgura-\ntion represent the position of relevant peaks in Fig. 7(g). (b)\nData points shown in (a) rescaled by !r(E=D = 1=3) to high-\nlight the change of the slope occurring in the parallel magnetic\ncon\fguration. Here, D=T0\nK= 10\u00002and other parameters as\nin Fig. 3.\nvalue of!rcoincides with the Kondo temperature, that\nis, the dynamical spin accumulation exhibits a maximum\nat!=!r\u0019TK. Such a feature arises for all nonzero\nvalues of anisotropy Econsidered in the \fgure, and im-\nportantly, the position of this feature depends strongly\non the transverse anisotropy component |small changes\ninElead to a large shift of the resonance frequency !r.\nThis is directly related to a strong dependence of the\nKondo temperature on the model parameters and, in\nparticular, \fnite transverse anisotropy [44]. An explicit,\nnumerically determined dependence of TKonEcan be\nseen in Fig. 4, which illustrates how the position of the\nresonance in the dynamical spin accumulation GAP\n\u001b\u001b(!)\nevolves when the transverse anisotropy parameter Eis\nmodi\fed. Clearly, small changes in Eresult in large\nmodi\fcation of !rand, thus, TK. From the slope of\nthe calculated curve, see squares in Fig. 4, we estimate\nthat!r/(E=D )\u00003. We also note that the position of\nthis curve is insensitive to the spin polarization pof elec-\ntrodes. Moreover, it can be observed in Fig. 3(c) that\nthe width of the peak in GAP\n\"#(!) andPAP(!) |plotted\non a logarithmic scale| hardly depends on E. One can\nconclude, thus, that the width of the resonance in dy-\nnamical spin accumulation, which occurs at !r\u0019TK, is\nalso approximately given by the Kondo temperature.\nOn the contrary, in the case of the uniaxial magnetic\nanisotropy of the easy-plane type ( D< 0), presented\nin the right column of Fig. 3, the dynamical conduc-\ntanceGAP(!) is modi\fed more subtly. Now, one ob-\nserves the well-developed Kondo e\u000bect [Figs. 3(e)-(f)] and\na pronounced maximum both in the dynamical spin ac-10\ncumulation GAP\n\"#(!) [Fig. 3(g)] and in the current spin\npolarizationPAP(!) [Fig. 3(h)], already visible in the\nabsence of transverse component of magnetic anisotropy.\nThe increase of Eresults only in a small reduction of the\nthreshold frequency !\u0003at whichGAP\n\"#(!) starts building\nup, and at which also the suppression of the Kondo ef-\nfect takes place |in other words, the raise of Eleads to\na slight decrease of the Kondo temperature. Moreover,\nfor largerEthe maximum in GAP\n\"#(!) gets broader and\neventually a small dip on the top of it develops. Inter-\nestingly, at the frequency where this dip is observed, one\ncan also notice a local maximum in GAP\n\"\"(!), see Fig. 3(f).\nC. In\ruence of the spin polarization of electrodes\nIn order to explore further the subtle interplay between\nthe Kondo e\u000bect and the dynamical spin accumulation,\nhere we consider the e\u000bect of spin polarization of elec-\ntrodes. For this purpose, in Fig. 5 and Fig. 6 we show\nhow the dynamical transport response of the system un-\nder investigation changes for di\u000berent values of the spin\npolarization parameter pin the case of D> 0 (Fig. 5)\nandD< 0 (Fig. 6), respectively. The left (right) column\nin those \fgures corresponds to the case of zero (\fnite)\ntransverse magnetic anisotropy constant E.\nLet us \frst consider the case of D> 0 shown in Fig. 5.\nGenerally, as expected for the Kondo e\u000bect, with the in-\ncrease ofpthe low-frequency conductance GAP(!.!\u0003),\nthat is, below the threshold frequency !\u0003for the dynam-\nical spin accumulation to kick in, becomes suppressed,\nsee Figs. 5(a,e). This behavior stems from the fact that\nin electrodes characterized by a large degree of spin po-\nlarization, there is a great imbalance between the num-\nbers of spin-majority and spin-minority electrons to be\ninvolved in the spin exchange processes leading to the\nKondo e\u000bect. In consequence, the larger the imbalance\nis, the less e\u000bective these processes become, and the more\nthe Kondo e\u000bect becomes suppressed. The dependence\nof the zero-frequency conductance GAP(!= 0) in the an-\ntiparallel magnetic con\fguration on the spin polarization\npof electrodes is presented in the inset to Fig. 5(c) and\nit can be described by a simple formula, GAP(!= 0)\u0011\nGAP(!= 0;p) = (1\u0000p2)GAP(!= 0;p= 0).\nThe situation changes qualitatively for frequen-\ncies!&!\u0003, where the o\u000b-diagonal-in-spin component\nof conductance GAP\n\"#(!), Figs. 5(c,g), starts contributing\nsigni\fcantly, so that the dynamical spin accumulation\nemerges as the dominant e\u000bect. Importantly, although\none can notice that GAP\n\"#(!) does not vanish even when\nthe electrodes are non-magnetic ( p= 0), no spin polar-\nization of the current is observed in such a case, that is,\nPAP(!) = 0, as one can see in Figs. 5(d,h). Moreover,\nthe behavior of the dynamical conductance GAP(!) is\nthen primarily governed by its diagonal-in-spin compo-\nnentGAP\n\"\"(!) |compare in Figs. 5 panels (a,e) with (b,f).\nOn the other hand, in the opposite limit of strongly spin-\npolarized electrodes ( p>0:5), whereGAP\n\"\"(!) gets pro-\n00.20.40.60.81\n00.10.20.30.40.5\n00.10.20.30.40.5\n00.20.40.60.81\n00.25 0.50.75 1\n00.20.40.60.81\n10\u0000410\u0000210010\u0000410\u00002100\nConductance GAP(!)/G0\np= 0\np= 0.25\np= 0.5\np= 0.75\np= 0.95\n00.20.40.60.81(a)Easy axis ( D>0)\nWithout transverse\nanisotropy ( E=D= 0)\n(e)With transverse\nanisotropy ( E=D= 1=3)GAP\n\"\"(!)/G0\n00.10.20.30.40.5(b)\n(f)GAP\n\"#(!)/G0\n00.10.20.30.40.5(c)GAP(!=0)=G0\nSpin polarization p00.20.40.60.81\n00.25 0.50.75 1E=D= 0E=D= 1=3 (g)Spin polarization PAP(!)\nFrequency !=T0\nK00.20.40.60.81\n10\u0000410\u00002100(d)\nFrequency !=T0\nK10\u0000410\u00002100(h)Figure 5. Evolution of the frequency-dependent conduc-\ntance GAP(!) in (a,e), its spin components GAP\n\"\"(!) in (b,f)\nandGAP\n\"#(!) in (c,g), as well as the current spin polariza-\ntionPAP(!) in (d,h) presented as a function of the spin-\npolarization coe\u000ecient pof electrodes for D=T0\nK= 10\u00002(the\nuniaxial magnetic anisotropy of the easy-axis type). Left\n(right )column corresponds to the case without (with) the\ntransverse component of the magnetic anisotropy included,\nthat is, for E= 0 ( E=D=3). The inset in (c) presents the\ndependence of the zero-frequency conductance GAP(!= 0) on\nthe spin polarization pin the case of E= 0 and E=D = 1=3.\nOther parameters are the same as in Fig. 2.\ngressively suppressed for large p, the features associated\nwith the dynamical spin accumulation GAP\n\"#(!) become in\nfact increasingly visible in the total conductance GAP(!)\nwithin the entire range of frequencies !|compare in\nFigs. 5 panels (a,e) with (c,g). This observation illus-\ntrates the key generic di\u000berence between the response of11\n00.20.40.60.81\n00.10.20.30.40.5\n00.10.20.30.40.5\n00.20.40.60.81\n10\u0000410\u0000210010\u0000410\u00002100\nConductance GAP(!)/G0\np= 0\np= 0.25\np= 0.5\np= 0.75\np= 0.95\n00.20.40.60.81\n(a)Easy plane ( D<0)\nWithout transverse\nanisotropy ( E=jDj= 0)\n(e)With transverse\nanisotropy ( E=jDj= 1=3)GAP\n\"\"(!)/G0\n00.10.20.30.40.5\n(b) (f)GAP\n\"#(!)/G0\n00.10.20.30.40.5(c) (g)Spin polarization PAP(!)\nFrequency !=T0\nK00.20.40.60.81\n10\u0000410\u00002100(d)\nFrequency !=T0\nK10\u0000410\u00002100(h)\nFigure 6. Analogous to Fig. 5 except that here the case of the\nuniaxial magnetic anisotropy of the easy-plane type ( D=T0\nK=\n\u000010\u00002) is considered. Note that the dependence of the zero-\nfrequency dynamical conductance GAP(!= 0) on pis the\nsame regardless of whether the transverse component of mag-\nnetic anisotropy is present or not, and it is given by the curve\nforE=D = 1=3 shown in the inset to Fig. 5(c).\nthe Kondo e\u000bect and the dynamical spin accumulation\nto a large spin polarization of electrodes. Speci\fcally,\nunlike the Kondo e\u000bect, the dynamical spin accumula-\ntion is augmented with the increase of p. This is a direct\nconsequence of the fact that GAP\n\"#(!) is associated with\nmajority spin bands of both leads, while the diagonal-in-\nspin components GAP\n\u001b\u001b(!) depend on both majority and\nminority spin bands.\nIn addition, one can note that while the low ( !\u001c!\u0003)\nand high-frequency ( !\u001d!\u0003) behavior of the dynamicalconductance and its spin-resolved contributions is qual-\nitatively similar in the case of E= 0 and \fnite E, huge\ndi\u000berences occur when !\u0019!\u0003. ForE= 0, the dynamical\nspin accumulation starts growing for !&!\u0003to reach a\nplateau, whereas for \fnite E,GAP\n\"#(!) increases to form a\nstrong maximum, the height of which grows with increas-\ningp. To understand this di\u000berence let us recall that in\nthe absence of transverse magnetic anisotropy the Kondo\ne\u000bect develops only partially. This implies that dynam-\nical spin accumulation has a moderate, relatively wide\nin frequencies, maximum. On the other hand, for \fnite\ntransverse magnetic anisotropy the Kondo resonance can\nfully develop with a clear sharp maximum in dynamical\nspin accumulation at !r\u0019TK, which becomes greatly en-\nhanced with increasing spin polarization p. In fact, in the\nlimit of half-metallic leads ( p= 1), the dynamical con-\nductance would be exclusively due to the e\u000bect of dynam-\nical spin accumulation, that is, GAP(!) =P\n\u001bGAP\n\u001b\u001b(!).\nWe also note that while the spin-resolved components\nof the frequency-dependent conductance strongly depend\nonp, the characteristic energy scales, !\u0003,TKand conse-\nquently!r, hardly do so, see Fig. 5.\nThe above discussion is also relevant to the case of\neasy-plane type of magnetic anisotropy ( D< 0), which\nis shown in Fig. 6. With raising the spin polarization,\nthe diagonal-in-spin components of the dynamical con-\nductance become suppressed [Figs. 6(b,f)], while the o\u000b-\ndiagonal component GAP\n\"#(!) increases [Figs. 6(c,g)], and\nfor su\u000eciently large pgives a dominant contribution to\nthe total conductance. In such a situation, the interplay\nof these two contributions results in a non-monotonic fre-\nquency dependence of GAP(!), see Figs. 6(a,e), which\nexhibits a local maximum due to dynamical spin ac-\ncumulation. Similar to the case of easy-axis magnetic\nanisotropy presented in Fig. 5, large spin polarization p\nof the leads induces large spin polarization of the cur-\nrent, see Figs. 6(d,h). As far as the e\u000bects related to the\ntransverse component of magnetic anisotropy are con-\ncerned, \fnite Eresults mainly in quantitative changes in\nthe dynamical response of the system, cf. the left and\nright column of Fig. 6, manifesting as a small reduction\nof the Kondo temperature and, consequently, the energy\nscale!\u0003. However, a qualitative di\u000berence can be still\nobserved in the dynamical spin accumulation, which in\nthe case ofE=jDj=3 exhibits a small dip at intermediate\nfrequencies, see Fig. 6(g) for !=T0\nK\u001910\u00002.\nD. Parallel magnetic con\fguration of the junction\nUp to this point, the discussion has been concentrated\non the situation of the junction in the antiparallel mag-\nnetic con\fguration, which allowed us to exclude from the\npicture some subtle e\u000bects due to the spintronic e\u000bective\nexchange \felds. To acquire a complete understanding of\nhow the magnetic con\fguration of the junction a\u000bects\nthe process of dynamical spin accumulation, we now also\nconsider the parallel magnetic con\fguration of the junc-12\n00.20.40.60.81\n00.10.20.30.40.5\n00.050.10.15\n00.20.40.60.81\n00.25 0.50.75 1\n00.20.40.60.81\n10\u0000810\u0000610\u0000410\u0000210010\u0000810\u0000610\u0000410\u00002100\nConductance GP(!)=G0\np= 0\np= 0.25\np= 0.5\np= 0.75\np= 0.95\n00.20.40.60.81(a)Easy axis ( D>0)\nWithout transverse\nanisotropy ( E=D= 0)\n(e)With transverse\nanisotropy ( E=D= 1=3)GP\n\"\"(!)=G0\n00.10.20.30.40.5\n(b) (f)GP\n\"#(!)=G0\n00.050.10.15(c)\nGP(!=0)=G0\nSpin polarization p00.20.40.60.81\n00.25 0.50.75 1E=D= 0E=D= 1=3 (g)Spin polarization PP(!)\nFrequency !=T0\nK00.20.40.60.81\n10\u0000810\u0000610\u0000410\u00002100(d)\nFrequency !=T0\nK10\u0000810\u0000610\u0000410\u00002100(h)\nFigure 7. Analogous to Fig. 5 but now the parallel (P)\nmagnetic con\fguration of the junction is shown. Note that at\npresent GP\n\"#(!) =GP\n#\"(!), and thus,Pdsa(!) = 0, so that the\ncurrent spin polarization occurs only due to the diagonal-in-\nspin terms GP\n\"\"(!)6=GP\n##(!), that is,PP(!)\u0011PP\n0(!). Note\nthat although, for the sake of consistency, we plot here the\nsame set of pvalues as in previous \fgures, the frequency range\nhas been extended here to include lower values of !. More-\nover, the long-dashed line is identical to that in Fig. 5 and\nit serves as the reference line. Recall that D=T0\nK= 10\u00002and\nthe other parameters are the same as in Fig. 2.\ntion.\nFor this purpose, we present in Figs. 7 and 8 how the\ndynamical transport characteristics of the system depend\non the spin polarization pof electrodes for the parallel\nmagnetic con\fguration. To begin with, let us \frst focus\non the case of the easy-axis type of uniaxial magnetic\n00.20.40.60.81\n00.10.20.30.40.5\n00.050.10.15\n00.20.40.60.81\n10\u0000810\u0000610\u0000410\u0000210010\u0000810\u0000610\u0000410\u00002100\nConductance GP(!)=G0\np= 0\np= 0.25\np= 0.5\np= 0.75\np= 0.9500.20.40.60.81\n(a)Easy plane ( D<0)\nWithout transverse\nanisotropy ( E=jDj= 0)\n(e)With transverse\nanisotropy ( E=jDj= 1=3)GP\n\"\"(!)=G0\n00.10.20.30.40.5\n(b) (f)GP\n\"#(!)=G0\n00.050.10.15(c) (g)Spin polarization PP(!)\nFrequency !=T0\nK00.20.40.60.81\n10\u0000810\u0000610\u0000410\u00002100(d)\nFrequency !=T0\nK10\u0000810\u0000610\u0000410\u00002100(h)Figure 8. Analogous to Fig. 7 except that here the case of the\nuniaxial magnetic anisotropy of the easy-plane type ( D=T0\nK=\n\u000010\u00002) is considered. The other parameters are the same as\nin Fig. 2.\nanisotropy ( D> 0) shown in Fig. 7. One can see that\nif only the uniaxial component of magnetic anisotropy\nis present the dynamical conductance GP(!) does not\ndi\u000ber qualitatively from the antiparallel case, compare\nFig. 7(a) with Fig. 5(a). Nevertheless, two key quan-\ntitative di\u000berences can be spotted immediately: First,\nthe di\u000berent values of conductance in the zero-frequency\nlimit,GAP(!= 0)>GP(!= 0) for 0 <p.0:85 [com-\npare the insets in Fig. 5(c) and Fig. 7(c)], and second,\nthe behavior of the resonance around the threshold fre-\nquency!\u0003which marks the onset of the dynamical spin\naccumulation, as discussed in Sec. IV B 1. Speci\fcally,\nwith the increase of the spin polarization pof electrodes13\nthis resonance becomes shifted towards larger frequencies\nand its magnitude gets attenuated.\nThe origin of the suppression of GP(!= 0) and the\nshift can be explained by taking into account the fact\nthat in the present con\fguration the spintronic ef-\nfective exchange \felds can arise. Since we consider\nthat the molecule is tuned to the particle-hole symme-\ntry point ( \"=U=\u00000:5), one expects actually only the\nquadrupolar \feld [68], which essentially provides an addi-\ntional uniaxial contribution \u0000Ds\u0000^Sz+^sz\u00012(withDs>0)\nto the magnetic anisotropy Hamiltonian (3). As a result,\nthe energy separation \u0001 /D+Dsbetween the states\nparticipating in transport, that is, the ground state dou-\nblet and \frst excited doublet in the S+ 1=2 spin multi-\nplet, increases. This, in turn, translates into the larger\nthreshold frequency !\u0003\nD>0and also means that the spin\nexchange processes leading to the Kondo e\u000bect at low\nfrequencies are more subdued, as compared with the an-\ntiparallel case where Ds= 0.\nOn the other hand, the suppression of the features oc-\ncurring for !&!\u0003has its roots in the response of the\ndynamical spin accumulation GP\n\"#(!) to increasing p, see\nFig. 7(c). Importantly, this response is strikingly dif-\nferent from that for the antiparallel magnetic con\fgu-\nration in Fig. 5(c). First of all, it should be noted\nthat now one \fnds GP\n\"#(!) =GP\n#\"(!), which straightfor-\nwardly leads to the conclusion that the dynamical spin\naccumulation does not contribute to the spin polariza-\ntion of the current, PP\ndsa(!) = 0. In fact, the current\nspin polarization PP(!) shown in Fig. 7(d) is exclusively\ndue to the di\u000berence between the diagonal-in-spin com-\nponentsGP\n\"\"(!) andGP\n##(!), that is,PP(!)\u0011PP\n0(!).\nMoreover, the intensity of GP\n\"#(!) is signi\fcantly reduced\nwith respect to GAP\n\"#(!), and it exhibits the opposite be-\nhavior with the increase of p, namely, in the parallel mag-\nnetic con\fguration the dynamical spin accumulation is\ndiminished for large p. This behavior can be understood\nby realizing that in the antiparallel con\fguration the\no\u000b-diagonal conductance GAP\n\"#(!) is associated with the\nmajority-spin subbands of both leads. Consequently, in-\ncreasing the spin polarization results in an enhancement\nof dynamical spin accumulation. On the other hand, in\nthe parallel con\fguration, the o\u000b-diagonal components\ndepend on both majority and minority spin subbands of\nboth leads, such that the minority spin channel provides\na bottleneck for GP\n\"#(!). Consequently, in the parallel\nmagnetic con\fguration the e\u000bect of dynamical spin accu-\nmulation becomes suppressed with increasing p, contrary\nto the case of antiparallel con\fguration. Note also that in\nthe limit of half-metallic leads ( p!1),GP\n\"#(!) would be\nfully suppressed and the total conductance would be ex-\nclusively given by the majority spin diagonal component\nof the conductance, while in the antiparallel con\fgura-\ntion all components would disappear, except for GAP\n\"#(!),\nnamely, the total conductance would be only due to the\ndynamical spin accumulation.\nLet us now take the transverse component of mag-netic anisotropy into consideration, see the right column\nof Fig. 7. Comparing with the case of the antiparallel\nmagnetic con\fguration shown in Fig. 5(e), one observes\nin Fig. 7(e) that the suppression of the Kondo e\u000bect oc-\ncurring for large pproceeds now in a qualitatively di\u000ber-\nent manner. With the increase of the spin polarization,\nthe Kondo temperature TKinitially decreases whereas\nthe zero-frequency conductance GP(!= 0) =G0remains\nuna\u000bected, and only above some threshold value of p\nalsoGP(!= 0) becomes diminished |for a detailed evo-\nlution ofGP(!= 0) as a function of psee the inset in\nFig. 7(c). In consequence, as long as the Kondo e\u000bect\ndominates transport, the current injected into the right\nelectrode does not display the spin polarization, that is,\nPP(!) = 0 for!.TK, see Fig. 7(h). Furthermore, we\nnotice that in the large frequency regime !&!\u0003, the\ntransport features due to the dynamical spin accumu-\nlation behave identically to those discussed in the sit-\nuation without the transverse component of magnetic\nanisotropy. On the other hand, in the opposite limit it\ncan be seen that, unlike for the antiparallel con\fguration,\nthe resonance arising in GP\n\"#(!) shifts its position !rto-\nwards smaller frequencies when pbecomes larger. This\nresults from the dependence of the Kondo temperature\nonp, which decreases as praises.\nThe explicit dependence of !ron the transverse mag-\nnetic anisotropy is presented in Fig. 4 for a couple of\nselected values of spin polarization. It can be clearly\nseen that the resonance frequency, and, thus, also the\nKondo temperature, strongly depends now both on the\nvalue ofEand the spin polarization p. Interestingly, the\nslope of the dependence of !ronE=D di\u000bers slightly\nfrom the case of antiparallel con\fguration, and it is also\nsensitive to a change of p, see Fig. 4(b). Note that the\ndata points (squares) in Fig. 4 corresponding to the an-\ntiparallel magnetic con\fguration are actually valid also\nfor the nonmagnetic junction ( p= 0). One can see that\nwith increasing p, the dependence of the Kondo tem-\nperature on Ebecomes sharper. Moreover, because the\nKondo temperature greatly depends on spin polarization,\nforp&0:75, the Kondo e\u000bect actually develops at ex-\ntremely small energy scales, which are completely not\nrelevant from the experimental point of view.\nThe behavior of dynamical system's response changes\nif one considers the uniaxial magnetic anisotropy of the\neasy-plane type ( D< 0), see Fig. 8. It can be seen that\nin such a case, already without the transverse compo-\nnent of magnetic anisotropy, a signi\fcant qualitative dif-\nference in evolution of the dynamical conductance as a\nfunction of parises between the antiparallel (left column\nin Fig. 6) and parallel (left column in Fig. 8) magnetic\ncon\fguration of the junction. One can notice that, unlike\nfor the antiparallel case, the transition between the highly\nconducting state for small pand the weakly conducting\nstate for large poccurring for !.!\u0003is rather abrupt.\nFurthermore, also when including the transverse com-\nponent of magnetic anisotropy (right column in Fig. 8),\nthis transition proceeds in a qualitatively di\u000berent man-14\n00.20.40.60.81\n0 0.2 0.4 0.6 0.8 100.040.080.12\n0 0.2 0.4 0.6 0.8 100.20.40.60.81\n0 0.2 0.4 0.6 0.8 100.040.080.12\n0 0.2 0.4 0.6 0.8 100.20.40.60.81\n10\u0000810\u0000610\u0000410\u0000210000.20.40.60.81\n10\u0000810\u0000610\u0000410\u0000210000.20.40.60.81\n10\u0000810\u0000610\u0000410\u0000210000.20.40.60.81\n10\u0000810\u0000610\u0000410\u00002100\nConductance GP(!)=G0\nSpin polarization p!= 000.20.40.60.81\n0 0.2 0.4 0.6 0.8 1(c)\nGP\n\"#(!)=G0\nSpin polarization p00.040.080.12\n0 0.2 0.4 0.6 0.8 1(d)\nConductance GP(!)=G0\nSpin polarization p00.20.40.60.81\n0 0.2 0.4 0.6 0.8 1(g)\nGP\n\"#(!)=G0\nSpin polarization p00.040.080.12\n0 0.2 0.4 0.6 0.8 1(h)0 0.2 0.4 0.6 0.8 1Conductance GP(!)=G0Spin polarization p\nFrequency !=T0\nK00.20.40.60.81\n10\u0000810\u0000610\u0000410\u00002100(a)Uniaxial magnetic anisotropy of the easy-plane type ( D<0)\nWithout transverse anisotropy ( E=jDj= 0)!=T0\nK= 10\u00006\n!=T0\nK= 10\u00004\n!=T0\nK= 10\u00003\n!=T0\nK= 10\u000010 0.04 0.08 0.12GP\n\"#(!)=G0Spin polarization p\nFrequency !=T0\nK00.20.40.60.81\n10\u0000810\u0000610\u0000410\u00002100(b)0 0.2 0.4 0.6 0.8 1Conductance GP(!)=G0Spin polarization p\nFrequency !=T0\nK00.20.40.60.81\n10\u0000810\u0000610\u0000410\u00002100(e)With transverse anisotropy ( E=jDj= 1=3)\n0 0.04 0.08 0.12GP\n\"#(!)=G0Spin polarization p\nFrequency !=T0\nK00.20.40.60.81\n10\u0000810\u0000610\u0000410\u00002100(f)\nFigure 9. Evolution of the dynamical conductance GP(!) and its o\u000b-diagonal-in-spin component GP\n\"#(!) as functions of the\nspin polarization pof electrodes for the uniaxial magnetic anisotropy of the easy-plane type ( D=T0\nK=\u000010\u00002).Left panels [(a)-\n(d)] correspond to the case without the transverse component of magnetic anisotropy ( E= 0), whereas right panels [(e)-(f)] to\nthe case when the transverse component is included ( E=jDj= 1=3).Bottom row presents cross-sections of the respective map\nplots in the top row for chosen values of frequencies !marked in (a), with the solid line standing for the zero-frequency limit.\nThe other parameters are the same as in Fig. 2.\nner as compared to the antiparallel case (right column\nin Fig. 6). Importantly, it can be observed that above\nsome threshold value of pthe dynamical spin accumula-\ntionGP\n\"#(!) shown in Fig. 8(g) changes its character and\nit develops a resonance typical to the easy-axis type of\nuniaxial magnetic anisotropy, see Fig. 7(g).\nIt is worth noting that the di\u000berence in the behavior\nof the dynamical conductance observed in the parallel\nand antiparallel magnetic con\fgurations on the value of\nthe spin polarization of electrodes has its origin in com-\npletely di\u000berent mechanisms governing the suppression\nof the Kondo e\u000bect. In the case of antiparallel con\fg-\nuration, the system behaves e\u000bectively as if coupled to\nnonmagnetic leads and the e\u000bect of spin polarization en-\nters only through the prefactors in appropriate formu-\nlas. In the parallel con\fguration, on the other hand,\nthe e\u000bective couplings do depend on spin, which results\nin nontrivial spin-resolved molecule-bath renormalization\ne\u000bects that give rise to \fnite local exchange \felds. As a\nconsequence, the interplay between the degree of spin\npolarization, which conditions the strength of exchange\n\felds, and the electronic correlations driving the Kondo\ne\u000bect, gives rise to large qualitative di\u000berences, as com-\npared to the case of antiparallel con\fguration.In order to gain a better insight into how the dynam-\nical conductance evolves with increasing the spin po-\nlarizationp, in Fig. 9 we analyze in detail the depen-\ndence of the total dynamical conductance GP(!) and\nits o\u000b-diagonal-in-spin component GP\n\"#(!) onp. Indeed,\nwe \fnd for E= 0 [Figs. 9(a)-(d)] that in the paral-\nlel magnetic con\fguration the transport response of the\nsystem is very sensitive to the change of p, with the\nsuppression of the Kondo e\u000bect taking place suddenly\natp0\u00190:5 [Fig. 9(a)] and the value of GP(!) altering\nforp > p 0only slightly, see the relevant cross-sections\ngiven by the dashed lines in Fig. 9(c). As discussed above,\nthe dynamical spin accumulation [Fig. 9(b)] arises only\nfor!&!\u0003, but at present the threshold energy !\u0003de-\npends onpnon-monotonically. Namely, it \frst decreases\naspgrows, while above p0the opposite trend is visible,\nandGP\n\"#(!) becomes quickly attenuated. In the left col-\numn of Fig. 10 we present how the total dynamical con-\nductanceGP(!) [Fig. 10(a)] and its o\u000b-diagonal-in-spin\ncomponent GP\n\"#(!) [Fig. 10(b)] vary in paroundp0. One\ncan see that, in fact, GP(!) does not change smoothly\nduring the transition between p.p0andp&p0and two\ncharacteristic features arise |also two pronounced peaks\nare visible in GP\n\"#(!). Again, such a behavior can be at-15\n00.20.40.60.81\n00.040.080.12\n0.485 0.49 0.495 0.5 0.38 0.4 0.42 0.44\nConductance GP(!)=G0\n!=T0\nK= 10\u00008\n!=T0\nK= 10\u00007\n!=T0\nK= 10\u00006\n!=T0\nK= 10\u0000500.20.40.60.81\n(a)Easy plane ( D<0)\nWithout transverse\nanisotropy ( E=jDj= 0)\n(c)With transverse\nanisotropy ( E=jDj= 1=3)GP\n\"#(!)=G0\nSpin polarization p00.040.080.12\n0.485 0.49 0.495 0.5(b)\nSpin polarization p0.38 0.4 0.42 0.44(d)\nFigure 10. (a)-(b) [(c)-(d)] Cross-sections of the map plots\nshown in the top row of Fig. 9 for selected values of frequen-\ncies!and resolved around the features marked by the arrow\nin Fig. 9(a) [(e)].\ntributed to the presence of the spintronic component Ds\nof magnetic anisotropy. Speci\fcally, recall that unlike the\nintrinsic uniaxial component D, which only a\u000bects the\nspin ^Szof the molecular magnetic core [see Eq. (3)], the\nspintronic component Dshas the e\u000bect on the total spin\nof the molecule, ^Sz+ ^sz. Importantly, it leads to such a\nsituation that di\u000berent doublets within the S+ 1=2 spin\nmultiplet e\u000bectively respond in a somewhat dissimilar\nway to varying of p. SinceDs>0, with the increase of p\nthe e\u000bect of intrinsic uniaxial anisotropy ( D< 0) gets re-\nduced, the system undergoes a transition from the ground\nstate doubletjStot\nz=\u00061=2ifor smallptojStot\nz=\u00065=2i\nfor largep. This transition, however, is not direct\nand it proceeds viathe doubletjStot\nz=\u00063=2i, namely:\nThe \frst (left one) of two additional features visible in\nFigs. 10(a)-(b) appears when the doublets jStot\nz=\u00061=2i\nandjStot\nz=\u00063=2iare degenerate, and the latter doublet\nbecomes the new ground state. With the further increase\nofp, at some other critical value pthis doublet becomes\neventually degenerate with jStot\nz=\u00065=2i, which results\nin formation of the right feature in Figs. 10(a)-(b). For\neven larger p, the spintronic component Dsstarts dom-\ninating over the intrinsic one Dand the dynamical spin\naccumulation GP\n\"#(!) displays characteristics of the uni-\naxial magnetic anisotropy of the easy-axis type, compare\nalso Fig. 8(c) with Fig. 7(c). Moreover, since in this limit\nthe ground state is jStot\nz=\u00065=2i, a signi\fcant suppres-\nsion of the dynamical spin accumulation occurs.\nWe note that the range of spin polarization values \u0001 p,\nfor which this transition occurs, is conditioned by the in-terplay of the quadrupolar exchange \feld and the Kondo\ntemperature. More speci\fcally, the Kondo e\u000bect be-\ncomes approximately suppressed once D+Ds&TK. As\ncan be seen in Fig. 9(a) for p<p 0, the Kondo temper-\nature is of the order of TK=T0\nK\u001910\u00003. On the other\nhand, the dependence of Dson the spin polarization\ncan be approximated by Ds/\u00002p2=U[68]. This implies\nthat \u0001p\u0019pTKU=\u0000\u00190:02, which agrees reasonably well\nwith the numerical data shown in Fig. 10(a).\nThe e\u000bect of the spintronic component of magnetic\nanisotropy is even more pronounced if the molecule ex-\nhibits also the intrinsic transverse component of mag-\nnetic anisotropy ( E6= 0). The corresponding dynamical\ntransport quantities for this case are shown in Figs. 9(e)-\n(h). First of all, the presence of non-zero Eresults in\npersistence of the Kondo e\u000bect to large values of p, as\ncompared to the case of E= 0 discussed above. One can\nsee that larger values of spin polarization are needed to\nsuppress the Kondo peak in GP(!). One can also observe\nthat forp>p 0the dynamical spin accumulation GP\n\"#(!)\n[shown in Fig. 9(f)] displays a pronounced maximum\nfor frequencies corresponding to the Kondo temperature.\nThis resonance is a clear signature of the e\u000bective uni-\naxial magnetic anisotropy of the easy-axis type, as dis-\ncussed in Sec. IV B 2 [see especially Fig. 3(c)]. More-\nover, analogously to the case of E= 0 [see Figs. 10(a)-\n(b)],GP\n\"#(!) is characterized by two maxima at p\u00190:39\nandp\u00190:44, which translate into a non-monotonic de-\npendence of GP(!) on the spin polarization at low fre-\nquencies!, see Figs. 9(e,g) and also the magni\fcation\nof the relevant range of pin Figs. 10(c)-(d). Noticeably,\nthese features are a pure dynamical e\u000bect and they dis-\nappear in the zero-frequency limit, as illustrated by the\nsolid line in Figs. 9(g,h). The occurrence of these two\nmaxima can be understood by invoking exactly the same\narguments as those used above to explain the origin of\nthe two resonances in GP\n\"#(!) forE= 0. The only dif-\nference is associated with the increased separation of the\ntwo present resonances. It occurs as a result of mod-\ni\fcation of states and energies of the molecule due to\nthe presence of the transverse (second) term in Hamilto-\nnian (3), which are further renormalized non-trivially by\nspin-dependent electron tunneling processes.\nE. Antiferromagnetic coupling between the OL and\nthe magnetic core\nFinally, in this section we discuss the main results in\nthe case when the coupling between the molecule's mag-\nnetic core and the orbital level is of the antiferromag-\nnetic (AFM) type ( J <0). For this purpose, in Fig. 11\nwe show the frequency-dependent transport coe\u000ecients\nin the antiparallel con\fguration calculated for uniaxial\nmagnetic anisotropy of the easy-axis type ( D> 0) when\nthe transverse component of magnetic anisotropy is ab-\nsent (E= 0, left column) and \fnite ( E6= 0, right col-\numn), respectively.16\n00.20.40.60.8\n00.10.20.30.4\n00.10.20.3\n00.20.40.60.8\n10\u0000810\u0000610\u0000410\u0000210010\u0000810\u0000610\u0000410\u00002100\nConductance GAP(!)=G0\nQD\njDj=T0\nK= 0\njDj=T0\nK= 10\u00004\njDj=T0\nK= 10\u00002\njDj=T0\nK= 0.5\n00.20.40.60.8\n(a)Uniaxial magnetic anisotropy of easy-axis type ( D>0)\nWithout transverse\nanisotropy ( E=D= 0)\nE=D= 0\nE=D= 1=5\nE=D= 1=3(e)With transverse\nanisotropy ( E=D= 1=3)GAP\n\"\"(!)=G0\n00.10.20.30.4\n(b) (f)GAP\n\"#(!)=G0\n00.10.20.3(c) (g)Spin polarization PAP(!)\nFrequency !=T0\nK00.20.40.60.8\n10\u0000810\u0000610\u0000410\u00002100(d)\nFrequency !=T0\nK10\u0000810\u0000610\u0000410\u00002100(h)\nFigure 11. The e\u000bect of magnetic anisotropy on the fre-\nquency-dependent conductance of a large-spin magnetic\nmolecule shown for the AFM J-coupling ( J=T0\nK=\u00002:25) and\nthe junction in the antiparallel (AP) magnetic con\fgura-\ntion. Note that only the uniaxial magnetic anisotropy of\nthe easy-axis plane ( D > 0) is presented. Left column [(a)-\n(d)] corresponds to the molecule exhibiting exclusively the\nuniaxial component of magnetic anisotropy ( E= 0), whereas\nright column [(e)-(h)] shows the e\u000bect of the transverse com-\nponent for D=T0\nK= 10\u00002. All other parameters are the same\nas in Fig. 2.\nWe recall that a spin-isotropic molecule ( D= 0) with\nthe AFMJ-coupling generically exhibits the two-stage\nKondo e\u000bect, so that for frequencies !smaller than the\nenergy scale characteristic of the second stage of screen-\ningT\u0003\nK,!.TK, the value of conductance is expected\nto drop signi\fcantly, see the solid line in the left columnof Fig. 11. Such a suppression of conductance GAP(!)\n[Fig. 11(a)], and also of the dynamical spin accumula-\ntionGAP\n\u001b\u001b(!) [Fig. 11(c)], is a direct consequence of the\nAFM coupling between the spin of an electron in the or-\nbital level and the molecule's core spin. This coupling\nsurpasses the AFM interaction between the OL's spin\nand spins of conduction electrons that gives rise to the\nKondo resonance. Note that for the assumed value of\nexchange coupling J, the temperature T\u0003\nKis relatively\nhigh, so that the conductance only slightly increases with\nlowering!due to the \frst-stage Kondo e\u000bect and then\nimmediately drops down, which results in relatively low\nmaximum around !\u0019TK.\nAdding the uniaxial component of magnetic anisotropy\ndoes not a\u000bect signi\fcantly the low-frequency results, as\nthe dynamical conductance and its spin-resolved compo-\nnents are suppressed already for !.TK, sinceT\u0003\nKandTK\nare in fact of a similar order. Moreover, similarly as\nin the case of the FM J-coupling, see Figs. 2(a)-(d), a\nthreshold frequency !\u0003arises above which the dynami-\ncal spin accumulation GAP\n\u001b\u001b(!) is observed, as shown in\nFig. 11(c). Even though the low-frequency transport is\nat present substantially reduced, the values of the spin\npolarization of the current injected into a drain electrode\nachieved for !>!\u0003can be still quite large and become\nsuppressed with increasing D, see Fig. 11(d).\nWhen the molecule also possesses transverse compo-\nnent of magnetic anisotropy, the second stage of Kondo\nscreening can become suppressed. This is seen in\nFigs. 11(e)-(f) where \fnite Erestores the Kondo peak\nat low frequencies. As expected, since now a pronounced\nKondo resonance is established for !.TK, the dynami-\ncal spin accumulation exhibits a maximum with its height\nbeing of the same order as in the quantum dot case, see\nFig. 11(g). A maximum at the same frequency is also\nobserved in the spin polarization [Fig. 11(h)].\nFinally, we note that for the uniaxial magnetic\nanisotropy of the easy-plane type ( D< 0), and also when\nthe magnetic con\fguration of the device is parallel, one\n\fnds a qualitatively similar behavior to the case of ferro-\nmagnetic exchange interaction discussed in previous sec-\ntions. However, due to the dominance of the second-stage\nKondo e\u000bect the transport is generally suppressed.\nV. CONCLUSIONS\nIn this paper we have analyzed the dynamical trans-\nport properties of magnetic molecules coupled to ferro-\nmagnetic electrodes in the Kondo regime. The molecule\nwas modeled by a LUMO level, directly coupled to ex-\nternal leads and additionally coupled through a ferro-\nmagnetic exchange interaction to the core spin of the\nmolecule. We have focused on the e\u000bect of dynami-\ncal spin accumulation, which can be associated with the\no\u000b-diagonal-in-spin component of the dynamical conduc-\ntance. We have in particular addressed the question of\nhow the dynamical spin accumulation becomes a\u000bected17\nby the presence of uniaxial, either of easy plane or easy\naxis type, and transverse anisotropy of the molecule. Our\nconsiderations have been performed in the linear response\nregime by using the Kubo formula, while all the dynam-\nical response functions were determined by using the nu-\nmerical renormalization group method.\nWe have generally shown that, in the case of antipar-\nallel magnetic con\fguration of the device, the dynam-\nical spin accumulation can develop for frequencies cor-\nresponding to the energy scale responsible for the for-\nmation of the Kondo e\u000bect, since then the spin-exchange\nprocesses are most e\u000bective. A local maximum in G\u001b\u001b(!)\nthus develops for resonant frequency !r, which is of the\norder ofTK. The width of this local maximum depends\nin turn on the energy scale where the Kondo state is\nformed. Consequently, while for spin isotropic molecules\ndynamical spin accumulation exhibits a broad maximum,\nin the presence of magnetic anisotropy the width of this\nmaximum becomes reduced. Another important energy\nscale describing the behavior of the dynamical spin accu-\nmulation denoted by !\u0003corresponds to the frequency be-\nlow which the conductance reaches a plateau. Then, the\nspin-\rip processes become quenched, such that G\u001b\u001b(!)\nvanishes. On the other hand, the height of the maxi-\nmum in dynamical spin accumulation turned out to de-\npend strongly on the magnitude of the Kondo e\u000bect. We\nhave shown that if the Kondo resonance develops fully\nfor!<T K, the spin accumulation exhibits a local max-\nimum, the height of which is of the same order as in\nthe case of quantum dots. However, when due to the\npresence of magnetic anisotropy the Kondo e\u000bect cannot\nfully emerge, the e\u000bect of dynamical spin accumulation\ncorrespondingly becomes reduced.\nWhen the junction is in the parallel magnetic con\fg-\nuration, the situation drastically changes, since then the\nspin-resolved renormalization of the molecule comes into\nplay. As a result, dynamical transport properties, and\nalso the dynamical spin accumulation, are conditioned\nby the interplay of the quadrupolar exchange \feld and\nKondo correlations. For the out-of-plane type of mag-\nnetic anisotropy we have predicted a large dynamical\nspin accumulation if the transverse anisotropy is present.\nOn the other hand, for the easy plane type of magnetic\nanisotropy, we have shown that, with increasing the spin\npolarization of the leads, there is a transition from a spin\ndoublet to higher-spin state, which results in a suppres-\nsion of the Kondo resonance. Interestingly, the suppres-\nsion of the low-frequency conductance occurs in a non-monotonic fashion, which is an indication of a highly non-\ntrivial competition between the intrinsic and spintronic\ncontributions to the magnetic anisotropy of the molecule.\nIn addition, we have also performed the analysis as-\nsuming that the exchange coupling between the orbital\nlevel and magnetic core is of antiferromagnetic type.\nThen, in the absence of magnetic anisotropy and for\nnonmagnetic leads, the system exhibits two-stage Kondo\ne\u000bect. We have shown that \fnite magnetic anisotropy\nof the molecule leads to the suppression of both the\nfrequency-dependent conductance and the e\u000bect of dy-\nnamical spin accumulation.\nFinally, we recall that all the considerations presented\nin this paper were performed for the particle-hole symme-\ntry point of the model. However, it is worth emphasizing\nthat the results discussed in the case of the antiparal-\nlel magnetic con\fguration are qualitatively valid also out\nof the particle-hole symmetry point, provided the sys-\ntem is in the Kondo regime (the orbital level is singly\noccupied). On the contrary, in the case of the paral-\nlel magnetic con\fguration, detuning from the symmetry\npoint results in a development of the e\u000bective dipolar ex-\nchange \feld. Because this \feld acts in a similar fashion to\nexternal magnetic \feld and splits the orbital level, most\nof the presented e\u000bects will become suppressed once the\nmagnitude of exchange \feld surpasses the relevant energy\nscales for considered phenomena. Last but not least, we\nwant to notice that, since the position of the orbital level\ncan be tuned by a gate voltage, the e\u000bects predicted in\nthis paper can be tested in near future with present-day\nexperimental techniques.\nACKNOWLEDGMENTS\nWork supported by the Polish Ministry of Science and\nEducation as Iuventus Plus project (IP2014 030973) in\nyears 2015-2017 (A.P., M.M.) and the Polish National\nScience Centre from funds awarded through the decision\nNo. DEC-2013/10/E/ST3/00213 (I.W.). M.M. also ac-\nknowledges \fnancial support from the Polish Ministry\nof Science and Education through a young scientist fel-\nlowship (0066/E- 336/9/2014) and from the Knut and\nAlice Wallenberg Foundation. 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Sherman\nDepartment of Physical Chemistry, Universidad del Pa´ ıs Va sco UPV/EHU, 48080 Bilbao, Spain and\nIKERBASQUE Basque Foundation for Science, 48011 Bibao, Biz kaia, Spain\n(Dated: July 13, 2018)\nOne-dimensional quantum wires are considered as prospecti ve elements for spin transport and\nmanipulation in spintronics. We study spin dynamics in semi conductor GaAs-like nanowires with\ndisorder and spin-orbit interaction by using a rotation in t he spin subspace gauging away the spin-\norbit field. At a strong enough disorder spin density, after a relatively fast relaxation stage, reaches\na plateau, which remains a constant for long time. This effect is a manifestation of the Anderson\nlocalization and depends in a universal way on the disorder a nd the spin-orbit coupling strength.\nAs a result, at a given disorder, semiconductor nanowires ca n permit a long-term spin polarization\ntunable with the spin-orbit interactions.\nPACS numbers: 72.25.Rb,72.70.+m,78.47.-p\nI. INTRODUCTION\nThe main idea of spintronics - the design and applica-\ntion of devices controlling not only the charge dynam-\nics but also the electron spin evolution - can be use-\nful for information storage, transfer, and manipulation\ntechnologies.1–3Possible realizations of spintronics de-\nvices can be based on semiconductor nanowires4–10for\nquasi-ballistic electron transport, coherent transmission\nof information, and spin control. These systems attract\na great deal of attention due to a clear interplay of trans-\nport and spin-orbit (SO) coupling physics.11–16\nThis control faces the problem of inevitable spin relax-\nation due to the coupling of electron spin to environment\nthrough SO coupling. As a result, the factors determin-\ningthespinrelaxationratebecomeofcrucialimportance.\nTwolimiting casesofspin relaxationarewellunderstood.\nFor the itinerant electrons spin relaxation in mainly de-\ntermined by the Dyakonov-Perel’ mechanism, that is by\nrandom precession of electron spin due to the random in\ntime electron momentum.\nA different approach should be applied for elec-\ntrons localized in a regular external potential form-\ning quantum dots promising for quantum information\napplications.17Here momentum is not a well-defined\nquantity, and the momentum-dependent splitting re-\nquired for the Dyakonov-Perel mechanism vanishes. As\na result, spin relaxation through SO interaction requires\nphonon-induced coupling of different orbital states of the\nlocalized electron and nonzero external magnetic field.18\nIn the absence of magnetic field and spin-orbit coupling,\nspin relaxationcanoccurdue to the hyperfinecouplingof\nelectronspintospinsoflatticenuclei.19Inbothcases,the\ninitial spin polarization goes asymptotically to zero. The\ncharacteristic timescale of spin relaxation of electrons lo-\ncalized in quantum dots is expected to be several orders\nof magnitude longer than that of itinerant electrons.\nWhile these two limits of free and strongly localizedFIG. 1: (Color online) Semiconductor nanowire with random\nimpurities shown as filled circles. Although we consider a\none-dimensional electron motion, impurities can be random ly\ndistributed over the cross-section of the wire.\nelectrons are well understood, the interplay of disorder-\ninduced localization and spin relaxation of itinerant elec-\ntrons remains an open question although some aspects\nof the problem have been addressed.20–23The questions\nhere are (i) how the localization forms the spin relax-\nation, and (ii) whether the initially prepared spin density\nrelaxes to zero. As a nontrivial example of this interplay\nwe mention that weak localization of two-dimensional\nelectrons leads to a long power-like rather than exponen-\ntial spin relaxation.24,25Here we analyze this problem\nfor the one-dimensional system, providing, on one hand,\nthe basic example of localization physics in a random\npotential,26,27and, on the other hand, an example of a\nsystem, where spin-orbit coupling can be gauged away\nby a SU(2) transformation.\nThis paper is organized as follows. In Sec. II, we show\nhow to treat spin relaxation in one-dimensional systems\nwith the gauge transformation and introduce the tight-\nbinding Hamiltonian for the model. The spin dynamics\nwill be analyzed by a numerically exact calculation in\nSec. III, where we show that spin density does not re-\nlax to zero, in contrast to what expected. In addition, in\nthisSec. III westudyhowasymptoticvalueofspinpolar-\nization depends on the disorder and spin-orbit coupling.\nConclusions summarize the results in Sec. IV.2\nII. MODEL\nA. Hamiltonian and gauge transformation\nThe investigated structure is a quantum wire extended\nalong thexaxis, as shown in Fig. 1. The total Hamilto-\nnian has the form\nˆH=¯h2\n2m(kx−Ax)2+U(x)−mα2\n2¯h2,(1)\nwhereAx=−mασy/¯h2standsfortheRashbacoupling28\nwith the strength α,σyis the Pauli matrix, kxis the elec-\ntron wavevector, and mis the effective mass. The Dres-\nselhaus coupling29is obtained with Ax=−mβσx/¯h2,\nwhereβis the coupling constant. Without loss of gener-\nality, we concentrate here on the Rashba coupling, which\ncan be changed on demand by applying external electric\nfield across the structure.30\nThe SO interaction can be removed from ˆHin Eq.(1)\nthrough a gauge transformation31,32with a SU(2) spin\nrotation: ˆS= exp(−ixσy/2ξ),whereξ= ¯h2/2mαis\nthe spin-precession length. After this transformation the\nsystem Hamiltonian has the form:ˆ/tildewideH= ¯h2k2\nx/2m+U(x).\nSince for the Hamiltonian (1), σyis the integral of mo-\ntion, the spin density component along the y-axis is time\nindependent. A nontrivial dynamics of the transformed\nspin occurs for the γ= (x,z) spin components /an}bracketle{t/tildewidesγ(x,t)/an}bracketri}ht\nand can be expressed in terms of the spin diffusion\n/an}bracketle{t/tildewidesγ(x,t)/an}bracketri}ht=/integraldisplay\nDγβ(x−x′,t)/an}bracketle{t/tildewidesβ(x′,0)/an}bracketri}htdx′,(2)\nwhereDγβ(x,t) is the exact disorder-dependent one-\ndimensional spin diffusion Green’s function. In a non-\nmagnetic system without SO coupling Dγβ(x,t) =\nδγβD(x,t) is diagonalin the spin subspace. As a result of\nthe gauge transformation, the uniform density dynamics\nis determined by only the Fourier component25\nD(q,t) =/integraldisplay∞\n−∞dxe−iqxD(x,t) (3)\nwithq= 1/2ξandEq. (2) simplifies forthe physicalmea-\nsurable spins as /an}bracketle{tsγ(t)/an}bracketri}ht=/an}bracketle{tsγ(0)/an}bracketri}htD(1/2ξ,t). Here we\nwill use a similar, however, somewhat different approach\nbased on numerically exact analysis of the direct time\nevolution of the initial spin-polarized states. It will be\nshown that the resulting spin dynamics has unexpected\nfeatures, including a long-time plateau in the spin polar-\nization.\nThe eigenfunctions ofˆ/tildewideHcan be taken in the form\nψ(x) =ψ(x)|1/an}bracketri}htandψ(x) =ψ(x)|−1/an}bracketri}ht, where|±1/an}bracketri}htare\nthe eigenstates of σzwith the corresponding eigenvalues.\nThe eigenstates of ˆH,φ(x) can be obtained by spin ro-\ntation of the ψ(x)|σ/an}bracketri}htstates. For example, with spin-up\ninitial state ψ(x)|1/an}bracketri}htone obtains:\nφ(x) =ψ(x)/bracketleftbigg\ncos/parenleftbiggx\n2ξ/parenrightbigg\n|1/an}bracketri}ht+sin/parenleftbiggx\n2ξ/parenrightbigg\n|−1/an}bracketri}ht/bracketrightbigg\n.(4)\nFIG. 2: (Color online) Site dependent components of\nφN/4(xn) for a qualitative description of entanglement in-\nduced by the gauge transformation for (a) α= 0, (b)α=\n0.125×10−6meVcm, and (c) α= 10−6meVcm (U0= 55\nmeV). The solid and dashed lines represent |1/angbracketrightand|−1/angbracketrightcom-\nponents, respectively.\nThe spin dynamics and spin relaxation in the system, as\nit will be shown below, is solely due to the entanglement\nof spin and coordinate in Eq.(4).\nB. Tight-binding model and disorder\nWe perform numerical analysis using the tight-binding\nmodel, employing the approach similar to Refs.[16,33].\nThe one-dimensional electron gas is sampled with N=\n213(8192) grid points xn=nl, where 1 ≤n≤Nand\nlis the effective lattice constant with periodic boundary\nconditions.34The effective hopping matrix element be-\ntween two nearest neighbors is chosen as t= 50 meV,\nand the kinetic energy is E(kx) = 2t(1−cos(kxl)). As a3\nresult, the eigenenergies span the range of [0 ,200] meVs.\nThe distance between two neighbor grid points becomes\nl= ¯h/√\n2mt= 3.37 nm to satisfy the electron effective\nmassm= 0.067m0in GaAs semiconductor with m0\nbeing the free electron mass.\nThe random potential Un=U(xn) uniformly spans\nthe range [ −U0/2,U0/2] with the white noise correla-\ntor/an}bracketle{tU(xn1)U(xn2)/an}bracketri}ht=/an}bracketle{tU2/an}bracketri}htδn1,n2, where/an}bracketle{tU2/an}bracketri}ht=U2\n0/12.\nThe effects of disorder can be approximately charac-\nterized through the energy-dependent momentum relax-\nation timeτE, which wedefine as¯ h/τE=/an}bracketle{tU2/an}bracketri}htlνE, where\nνE=√m/π¯h√\n2Eis the density of states per spin com-\nponent. The resulting mean free path ℓE=vEτE, where\nvE=/radicalbig\n2E/mis the electron speed and the correspond-\ning diffusion coefficient DE=v2\nEτE.\nIn this representation the eigenstates ofˆ/tildewideHandˆHform\nbasis sets, {ψi}and{φi}respectively, where 1 ≤i≤2N.\nForthesame i, these twosetsarerelatedbythe localspin\nrotation ˆS.Weassumethat ψi=ψi(xn)|1/an}bracketri}htfor1≤i≤N\nandψi=ψi−N(xn)|−1/an}bracketri}htforN <i≤2N.\nIII. SPIN DYNAMICS\nWe study dynamics of initial ψistates with 1 ≤i≤N,\ncorresponding to the evolution upon instant switching of\nthe SO coupling. The time dependence can be expressed\nwith the spectral decomposition as:\nψso\nj(t) =/summationdisplay\ni=1,2Naijφie−itεi/¯h, (5)\nwhereaij=/an}bracketle{tφi|ψj/an}bracketri}ht, andεiare the corresponding\neigenenergies. The spin component expectation value\n/an}bracketle{tσz(t)/an}bracketri}htj=/angbracketleftBig\nψso\nj(t)/vextendsingle/vextendsingle/vextendsingleσz/vextendsingle/vextendsingle/vextendsingleψso\nj(t)/angbracketrightBig\nisdetermined by the spec-\ntrum and eigenstates of the system.\nIn order to give an idea of the entanglement induced\nby SO coupling, we present in Fig. 2 the evolution of\nφN/4(xn) state with the increase in the spin-orbit cou-\npling. Atα= 0, we obtain a product state φN/4(xn) =\nψN/4(xn)|1/an}bracketri}ht, and with the increase in αentangled states\nareformed. Theoverlapof φi(xn) andψj(xn)eigenstates\nis characterized by two sets of matrix elements aij; for\nexample, for 1 ≤j≤N:\naij=/summationdisplay\nncos/parenleftbiggxn\n2ξ/parenrightbigg\nψi(xn)ψj(xn),1≤i≤N(6)\naij=/summationdisplay\nnsin/parenleftbiggxn\n2ξ/parenrightbigg\nψi(xn)ψj(xn), N <i ≤2N.\nThe behavior of aijpresented Fig. 3 demonstrates that\nfor givenjit has nonnegligible values only in a certain,\nrather narrow, range of i.\nTo illustrate the role of the random potential, we con-\nsider as examples weak ( U0= 15 meV, U0≪t) and\nFIG. 3: (Color online) Absolute values of aijaround the ini-\ntial spin-up state ψN/4; hereα= 10−6meVcm (strong SO\ncoupling) and U0=55 meV (strong disorder).\nFIG. 4: (Color online) Inverseparticipation ratio ζfor the low\npart of the energy spectrum; gray (red) circles denote stron g\ndisorder (U0=55 meV) and black circles denote weak disorder\n(U0=15 meV). Since even for U0=55 meV we obtain ζ≪1,\nthe localized states are distributed over many lattice site s,\nconfirming applicability of the tight-binding Hamiltonian for\nthe localization problem.\nstrong (U0= 55 meV, U0> t) disorder. For free elec-\ntrons in state j=N/4 andE= 31 meV, the result-\ning ¯h/τEis about 0.1 and 1 meV, respectively. For a\nfree electron with the energy E≈20 meV the velocity\nvE≈3.5×107cm/s, the mean free path ℓE∼2.5×10−5\ncm (¯h/τE= 1 meV), and the corresponding diffusion co-\nefficientDE∼103cm2/s. These parameters provide an\neffective integral characteristic of the disorder and corre-\nspond to realistic parameters of the wires, which, how-\never, can strongly vary from sample to sample and from\nexperiment to experiment.\nThe effect of localization by disorder is seen in the in-\nverse participation ratio35(IPR)ζi=/summationtext\nn/vextendsingle/vextendsingleψ4\ni(xn)/vextendsingle/vextendsingle. The4\nFIG. 5: (Color online) Time-dependent polarization in the\nweak-disorder regime ( U0=15 meV). The initial bins are cen-\ntered at the states (a) N/4 (bin width 6.8 meV), (b) N/8\n(bin width 3.7 meV), and (c) and N/16 (bin width 2.1 meV)\nwith energies decreasing in the same order. The curves\nfor SO couplings 0 .125×10−6meVcm, 0.5×10−6meVcm,\nand 2×10−6meVcm are drawn with circles, triangles, and\nsquares, respectively. Note that after the relaxation stag e the\nspin density remains a finite constant.\nIPR calculated for the low-energy spectrum is presented\ninFig.4. Asexpected,thedegreeoflocalizationincreases\nwithU0and this effect is more pronounced for the elec-\ntrons with lowest energies. In contrast to the results of\nRef.[23], the IPR in this system does not depend on the\nSO coupling. We now study the effects of disorder and\nspin-orbit coupling on the averagespin dynamics of a bin\nof 256 initial spin-up states and 8 realizations of the ran-\ndom potential. The statistical error of this approach is,\ntherefore1/√\n2048=2.2%, makingthe resultsstatistically\nrepresentative.\nWe take three example bins with three different de-\ngrees of localization. The bins are centered around the\nspin-up states ψN/4,ψN/8, andψN/16, whose IPR val-\nues increase in the same order (energies decreasing, see\nFig. 4). The calculated bin- and potential realization-\naveraged spin dynamics is shown in Figs. 5 and 6, re-\nvealing strong influence of the disorder-induced spatial\nlocalization of states. Physically, collisions of electrons\nwith impurities force electron spin to frequently reverse\nthe precession direction. In the classical picture, this\nleads to a long Dyakonov-Perel’ spin relaxation. If the\nFIG. 6: (Color online) Time-dependent spin polarization in\nthe strong-disorder regime ( U0=55 meV) with the same no-\ntations as in Fig. 5. (a) N/4 (bin width 7.2 meV), (b) N/8\n(bin width 4.5 meV), and (c) N/16 (bin width 3.7 meV). Note\nthat forα= 0.125×10−6meVcm the spin is almost constant\nin time, thus suitable for spin-based operations.\nquantum effects of localization are important, the result-\ning effect is the “freezing” of the electronic spin. As one\ncan see in Figs. 5 and 6, the electron spin density re-\nlaxes for ≃5 ps and then remains constant in time for\ninfinitely long (beyond 0.2 ns in our computation). As\nexpected, the spin polarization plateau is higher (i) for\nlocalized states and (ii) for weak SO interaction. Al-\nmost time-independent spin states are achieved e.g., at\nU0= 55 meV and α= 0.125×10−6meVcm.\nTo gain insight into the problem, we study the depen-\ndence of asymptotic spin density on SO coupling and the\nlocalization of electrons in more detail. The long-term\ndensities are plotted in Fig. 7 against parameter ξ/an}bracketle{tζ/an}bracketri}ht.\nThis parameter combines the two factors determining\nthe spin dynamics, SO coupling and spatial localization,\nwhere/an}bracketle{tζ/an}bracketri}htis averaged over 256 bin states and 8 realiza-\ntion of the random potential. The given values follow a\nuniversal dependence indicating a unique trend for long-\nterm spins against SO coupling and localization through\ndisorder. This trend corresponds to a fast increase in the\nasymptoticsteadypolarizationfor ξ/an}bracketle{tζ/an}bracketri}ht<1andasmooth\nincreaseandsaturationfor ξ/an}bracketle{tζ/an}bracketri}ht>1. Theseresultscanbe\nunderstood as follows. To show an efficient spin dynam-\nics, the electron should move the distance of the order\nofπξ. Therefore, the spatial spread of the correspond-5\nFIG. 7: (Color online) Long-term relative polarization as a\nfunction of ξ/angbracketleftζ/angbracketrightfor three different degrees of localization. Pa-\nrameterξis modified by changing the coupling constant α.\ning states should be larger than πξ. With a stronger\nlocalization, the spread and the overlap decrease leading\nto the universal behavior shown in Fig.7. Qualitatively,\nin the “clean” ξ/an}bracketle{tζ/an}bracketri}ht ≪1 regime the spin relaxation has\nthe Dyakonov-Perel’ mechanism either purely exponen-\ntial for Ω EτE≪1 or a combination of oscillations and\nexponential decay if Ω EτE≥1, where the spin preces-\nsion rate Ω E= 2α√\n2mE/¯hcorresponds to the electron\nmomentum at given energy E.\nIV. CONCLUSION\nTo summarize, localization effects of disorder and SO\ncoupling in semiconductor nanowires determine the dy-\nnamics of electronic spins. Our tight-binding model cal-culations show that a prepared spin density relaxes un-\ntil reaching a plateau, directly related to the disorder\nand strength of SO interaction. In contrast to the ex-\npected decay to zero, a long-time constant polarization\nplateau survives to infinite time. The asymptotic spin\ndensity has a universal dependence on the product of\nthe inverse participation ratio and the spin precession\nlength. In the absence of magnetic field, the hyperfine\ncoupling to the spins of nuclei will lead to spin relaxation\non timescales at least two orders of magnitude longer\nthan the timescale of the plateau formation of the or-\nder of 10 ps.19As the experiments on spin transport did\nnot reveal electron-electron interaction effects,8here we\nhave neglected them. Furthermore, whether there exists\na range of parameters where the Coulomb forces can be\nstrong enough to modify our results for localized states,\nremains to be investigated.\nAn immediate consequence of this result is the ability,\nby choosing the desired Rashba SO parameter for a given\nwire, to produce and destroy steady spin states, which\nare of interest for spin-based operations. These results\nsuggest that semiconductor nanowires can be used for\ncoherenttransmissionandstorageofinformation, manip-\nulated by spatially and temporally modulated spin-orbit\ncoupling.\nV. ACKNOWLEDGMENTS\nWe thank G. Japaridze, J. Siewert, and L.A. Wu for\nhelpful discussions. This workwassupportedbythe MCI\nof Spain grant FIS2009-12773-C02-01and ”Grupos Con-\nsolidados UPV/EHU del Gobierno Vasco” grant IT-472-\n10.\n1I. Zuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004); J. Fabian, A. Matos-Abiague, C. Ertler, P.\nStano, and I. Zutic, Acta Physica Slovaca 57, 565 (2007).\n2A. 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Liu, and J. Sinova, Phys.\nRev. B84, 035318 (2011)."    },    {        "title": "1106.3687v1.Electrical_control_of_spin_dynamics_in_finite_one_dimensional_systems.pdf",        "content": "Electrical control of spin dynamics in \fnite one-dimensional systems\nA. Pertsova, M. Stamenova and S. Sanvito\nSchool of Physics and CRANN, Trinity College Dublin, Dublin 2, Ireland\n(Dated: May 22, 2022)\nWe investigate the possibility of the electrical control of spin transfer in monoatomic chains\nincorporating spin-impurities. Our theoretical framework is the mixed quantum-classical (Ehrenfest)\ndescription of the spin dynamics, in the spirit of the s-d-model, where the itinerant electrons are\ndescribed by a tight-binding model while localized spins are treated classically. Our main focus is\non the dynamical exchange interaction between two well-separated spins. This can be quanti\fed by\nthe transfer of excitations in the form of transverse spin oscillations. We systematically study the\ne\u000bect of an electrostatic gate bias Vgon the interconnecting channel and we map out the long-range\ndynamical spin transfer as a function of Vg. We identify regions of Vggiving rise to signi\fcant\nampli\fcation of the spin transmission at low frequencies and relate this to the electronic structure\nof the channel.\nPACS numbers: 75.78.-n,75.30.Hx,73.63.-b,85.75.-d\nI. INTRODUCTION\nThe rapid development of the \feld of spintronics1over\nthe past two decades has uncovered exciting and novel\nphenomena related to the dynamics of the electronic spin\nin a wide variety of systems, ranging from bulk materi-\nals to spatially-con\fned structures2. Fueled by the ever-\ngrowing needs for speed, capacity and energy-e\u000eciency in\ncomputing, the central objective in understanding and ul-\ntimately controlling the spin properties in the solid state\nhas been constantly shifting towards the nano-scale. At\nthese lengths and times the conventional methods for spin\ncontrol, based on magnetic \felds alone, are limited by\nscalability issues. Alternative approaches are thus sought\nand they typically involve electric \felds of some form3.\nOne way to manipulate spins by purely electrical\nmeans relies on the spin-transfer torque mechanism4.\nThis approach uses spin-polarized currents to control the\ndirection of the magnetization and has been realized in\nvarious nano-structured materials ranging from magnetic\nmultilayers5to, more recently, single atoms in STM-type\ngeometries6. Alternative to electric current control is the\noptical control, such as in the laser-driven ultrafast mag-\nnetization switching7,8. In a somewhat di\u000berent context,\nthe optical manipulation of single spins in bulk media9\nis at the heart of the most promising candidates for the\nquantum information processing technology10,11.\nAnother alternative is based on the idea of an electro-\nstatic control of the spin-density, i.e. of the construction\nof spin-transistor type devices12. Recently, the concept\nof gate-modulated spin-pumping13transistors has been\nstudied theoretically in in\fnite graphene strips with pat-\nterned magnetic implantations14. Importantly, such de-\nvices rely on the e\u000ecient transport of spin information\nbetween two points in space and time, and require the\npossibility to actively tune the propagating spin-signal\nduring its transport. In this work we explore this possi-\nbility for atomistic spin-conductors. We consider a \fnite\nmono-atomic wire linking two localized spin-carrying im-\npurities. When one of the localized spins is set into pre-cession, it generates a perturbation in the spin-density.\nThis perturbation is carried through the wire by con-\nducting electrons and can be detected in the dynamical\nresponse of the second spin. We show that the propaga-\ntion of the spin-signal and consequentially the dynami-\ncal communication between the two spin centers, can be\ntuned by means of an electrostatic gate applied to the in-\nterconnecting wire. Our main \fnding is an enhancement\nof the communication for a certain range of gate volt-\nages. This is linked to the modi\fcation of the electronic\nstructure of the wire induced by the applied electrostatic\ngate.\nBecause of their considerable size the systems investi-\ngated here are still beyond the present numerical capabil-\nities of \frst-principles dynamics15and as such they are\ndescribed by model Hamiltonians. Usually the dynam-\nics is approached within the linear response approxima-\ntion14. In this paper we go beyond the linear response\nlimit and propose a fully microscopic description based\non the time-dependent Schr odinger equation, which al-\nlows us to describe arbitrary excitations. In other words\nour simulations are not limited to small gate voltages or\nsmall-angle spin precession. We use a single-band tight-\nbinding Hamiltonian to model the itinerant s-electrons\nin our metallic wires and include local Heisenberg inter-\nactions to a number of magnetic ions (typically one or\ntwo). The latter are modeled as classical spin degrees of\nfreedom and enter on equal footing in the common mixed\nquantum-classical dynamic portrait of the system.\nThis paper is organized as follows. In Section II we\nintroduce the model system and our theoretical frame-\nwork. Section III contains the results of our investiga-\ntions. Firstly we investigate the electron response of an\natomic wire, not including magnetic impurities, to lo-\ncal spin excitations. Secondly, we study the dynamical\ninterplay between two spin-impurities electronically con-\nnected by the wire and show how such a dynamics is\na\u000bected by the gate voltage. Finally we draw some con-\nclusions.arXiv:1106.3687v1  [cond-mat.mes-hall]  18 Jun 20112\nII. THE MODEL\nA cartoon of the device considered is presented in Fig.\n1. This consists of an N-sites long atomic wire intercon-\nnecting two magnetic centers. The latter, represented in\nthe \fgure by the spin-vectors, enter our model as substi-\ntutional magnetic impurities positioned at the two ends\nof the wire (the local spin originates from the deeply lo-\ncalizedd-electrons). One of them, say the left-hand side\nspin-centerS1, is labeled as \\driven\", as its precession is\ninduced and sustained by a local (at that site) magnetic\n\feld. The other localized spin, S2, located at the other\nend of the chain is the \\probe\" spin and it is not directly\ncoupled to any external magnetic \feld but only to the\nelectron gas. In this way S2probes the magnetic excita-\ntions produced by the driven spin S1as these propagate\nthrough the interconnecting wire. The device is described\nas a mixed quantum-classical system in the spirit of the\n(s-d)-model16, where the time-dependent Hamiltonians\nof the two exchange-coupled spin sub-systems read\n^Hel(t) =X\ni;j= 1;N\n\u000b= 1;2HTB\nijc\u000by\nic\u000b\nj (1)\n\u0000J2X\n\u000b;\f=1h\nS1(t)c\u000by\n1c\f\n1+S2(t)c\u000by\nNc\f\nNi\n\u0001^\u001b\u000b\f;\nHS(t) =\u0000S1(t)\u0001[Js1(t) +g\u0016BB]\u0000JS2(t)\u0001sN(t):(2)\nThe top expression is for the quantum electrons. Here\nHTB\nij=\"i\u000eij+\r\u000ei;i\u00061is a single-orbital tight-binding\nHamiltonian with on-site energies \"iand hopping inte-\ngral\r(\rsets the relevant energy scale for the entire sys-\ntem);c\u000by\ni(c\u000b\ni) is the creation (annihilation) operator for\nan electron with spin-up ( \u000b= 1) or spin-down ( \u000b= 2) at\nthe atomic site i;^\u001b=1\n2(\u001bx;\u001by;\u001bz) is the electron spin\noperator,f\u001blgl=x;y;z being the set of Pauli matrices; S1;2\nis the unit vector in the direction of the localized spin;\nJ > 0 is the exchange coupling strength.\nThe classical dynamics of the local spins is governed\nbyHS(t), which describes the interaction of S1;2with the\nmean-\feld local electron spin-density si\u0011h^\u001bii, taken as\nthe instantaneous expectation value of the conduction-\nelectron spin at site i(see further in text for the exact\nde\fnition). The classical Hamiltonian also includes in-\nteraction with the external magnetic \feld B= (0;0;Bz),\nwhich is applied locally to S1only, in order to drive its\nprecession. We further assume g= 2 for the localized\nspins and\u0016B= 5:788 10\u00005eV/T is the Bohr magneton.\nIn order to study the dynamics of this system in the\ntime-domain we integrate the coupled quantum and clas-\nsical equations of motion (EOM) of the two spin sub-\nsystems17,18. In our spin-analogue to Ehrenfest molecu-\nlar dynamics19, the full set of coupled Liouville equations\nFIG. 1: (Color online) Model system investigated in this work:\ntwo localized spins are electronically connected by a mono-\natomic wire, where electrons can \row. An electrostatic gate\nis applied to some of the interconnecting sites.\nread\nd^\u001a\ndt=i\n~h\n^\u001a;^Heli\n; (3)\ndSn\ndt=fSn;HSg=Sn\nS\u0002\u001a\nJs1+g\u0016BBforn= 1\nJsN forn= 2:\nHeref;grepresents the classical Poisson bracket, [ ;]\nstands for the quantum-mechanical commutator and S=\n~is the magnitude of the two classical spins. The \frst\nEOM is for the electron density matrix ^ \u001a. At the initial\ntime,t0, ^\u001ais constructed from the eigenstates fj'\u0017ig2N\n\u0017=1\nof the spin-polarized electron Hamiltonian ^Hel(t0) [see\nEq. (2)], with the composite index \u0017=fi;\u001bglabel-\ning the set of 2 Nspin-polarized eigenstates. We de\fne\n\u001a(t0) =P\n\u0017f\u0017j'\u0017ih'\u0017jwithf\u0017=\u0011F(\u000f\u0017\u0000EF) being\nthe occupation numbers distributed according to Fermi-\nDirac statistics. The instantaneous onsite spin-density is\nthus generated as\nsi(t)\u0011h\u001bii(t) = Tr [\u001a(t)\u001b]i=X\n\u000b\f\u001a\u000b\f\nii(t)\u001b\f\u000b:(4)\nThe coupled EOMs are integrated numerically by using\nthe fourth-order Runge-Kutta (RK4) algorithm20. As a\nresult the set of trajectories for the local electronic spin-\ndensitiessi(t) are obtained as well as those of the driven\nand the probe classical spins S1;2(t). The typical length\nof the chain that we consider is N= 100.\nThe e\u000bect of an electrostatic gate is incorporated in our\nmodel as a rigid shift of the onsite energies \"i!\"i+Vg,\nwherei2[i1;i2] is a certain range of sites in the middle\nof the chain and Vgis the gate voltage25.3\nIII. DYNAMICS OF THE ITINERANT SPINS\nWITH FROZEN IMPURITIES\nA. No external gate\nAs a preliminary step towards the combined quantum-\nclassical dynamics we \frst address the dynamics of the\nspin-density of the itinerant electrons in the presence\nof the two local spins, whose directions are \fxed, e.g.\nS1;2jj^z. A spin excitation is produced by a small but \f-\nnite spatially-localized perturbation in the spin-density.\nAs initial density matrix at t0we use the one that corre-\nsponds to a perturbed Hamiltonian ^Hel(t0\u0000\u000et) in which\none of the localized spins is slightly tilted in the x\u0000z\nplane, such that S1(t0\u0000\u000et)\u0001^z\u0019d\u0012.5o. However at\nt0we bringS1back to its original direction ( jj^z) where\nit stays throughout the simulation. In other words we\nstudy the time evolution of the system with the two lo-\ncal spins parallel to each other starting from the ground\nstate electronic charge density of the system where one\nof the two spins is tilted by a small angle.\nThe evolution of the density matrix for t>t 0is then\ngiven by ^\u001a(t) =e\u0000i^Helt=~^\u001a(t0)ei^Helt=~, which translates\ninto the following expression for the density matrix ele-\nments\n\u001a\u001b\u001b0\nkl(t) =X\nmne\u0000i!mntc\u001b\nkmc\u0003\u001b0\nlnX\ni\u000bX\nj\fc\u0003\u000b\nimc\f\njn\u001a\u000b\f\nij(t0):\n(5)\nIn Eq. (5) we have de\fned the frequencies !mn\u0011em\u0000\nen)=~corresponding to di\u000berences between the eigen-\nvaluesemof^Hel(t0). The coe\u000ecients c\u001b\nkm=hk\u001bj'mi\nare the projections of the eigenvectors j'mion the spin-\nresolved atomic orbital basis jk\u001bi\u0011jki\nj\u001bi, wherek\nrepresents the atomic site and \u001b=\";#is the spin compo-\nnent.\nThe typical evolution of the spin density of the itiner-\nant electrons resulting from excitation described above is\npresented in Fig. 2. The trajectories of the individual on-\nsite spin polarizations \u0001 sz\ni(t) =sz\ni(t)\u0000sz\ni(t0) stemming\nfrom from Eq. (5) are perfectly identical (on the scale\nof the graph) to those obtained by the numerical inte-\ngration of the EOM [i.e. from Eq. (2)], con\frming the\nreliability of our time-integrator for the typical duration\nof the simulations.\nThe spatial distribution of the initial spin polarization,\nsz\ni(t0), is shown in Fig. 2(b) and its subsequent evolu-\ntion, presented in Fig. 2(c) can be qualitatively charac-\nterized as the propagation of a spatially-localized spin\nwave-packet. This travels along the wire with a practi-\ncally uniform velocity very close to the Fermi level group\nvelocity, as expected in view of the rather small over-\nall local spin-polarization of the chain. The packet then\ngradually loses its sharpness as it disperses. However,\nthe backbone of the packet is detectable even after a few\ntens of re\rections at the ends of the wire. Such a feature\ndemonstrates that this \fnite 1D model atomic system is\na relatively e\u000ecient waveguide for spin wave-packets (ofsub-femtosecond duration) at least over the investigated\ntime-scales of a few picoseconds.\nFIG. 2: (Color online) Spin excitation of one dimensional\nwire. (a) Time dependence of the spin-polarization on the\n\frst site (\u0001 sz\n1) as obtained from the numerical integration\nof the EOMs and directly from Eq. (5). (b) Initial spin-\npolarization sz(t0) as a function of site index. (c) Time and\nspace evolution of \u0001 sz\niwith the color shade representing the\nmagnitude of absolute value of \u0001 sz\ni(t). Note that 2 \u001cis the\nwave-packet round-trip time as also seen in the bottom panel.\nEq. (5) can also be used to calculate directly the spec-\ntra of particular spin observables. These match perfectly\nwith those calculated by performing the discrete Fourier\nTransform21(dFT) of the time-dependent spin-densities\nobtained over the \fnite duration of the dynamic simu-\nlation. Clearly, in the absence of the localized impuri-\nties (i.e. for a \fnite homogeneous tight-binding chain)\n!mncan be calculated exactly. The spectrum of \u0001 sz\ni\nhas monotonously decreasing amplitudes from the lowest\npossible frequency !min\u00193\u00192j\rj\n(N+1)2~to the maximum one\n!max\u00194j\rj\n~(expressed in the limit of N!1 ). This is\nalso true for the case of interaction with the two frozen\nlocal impurities (e.g. for J=\r) as these have a minor\ne\u000bect on the electron Hamiltonian. For the parameters\ntypically used in our simulations the corresponding max-\nimum period Tmax=2\u0019\n!min.1 ps is within the total time\nof the simulation while the corresponding minimum pe-\nriodTmin=2\u0019\n!max\u00191fsis much larger than the typical\ntime step \u0001 t= 0:01 fs.\nA more detailed spectral analysis is obtained by the\ntwo-dimensional dFT power portraits21of \u0001sz\ni(see Fig.\n3) denoted as dFT[\u0001 sz\ni(t)](k;!). In Fig. 3 we com-\npare such spectra for the itinerant spin-dynamics to its\nexact counterpart in the case of a \fnite homogeneous\ntight-binding chain. These portraits reveal key features\nof the one-dimensional fermionic system that vary sys-\ntematically with the band-\flling \u001a0, namely (i) a near-\ncontinuum of allowed modes in a certain ( k;!)-space\nregion, de\fned by a low- and a high-energy dispersion\nfunctions, (ii) a linear dispersion for small k(!/kfor\nk!0). An analogous mode-occupation patterns have\nbeen rigorously analyzed in relation to the dynamical\nproperties of one-dimensional quantum Heisenberg spin\nchains which too have a tight-binding type dispersion re-4\nlation23. The low-energy mode-occupation limit is due\nto the fact that the energy of the electron-hole excita-\ntion can approach zero only for \u0001 k!0 and \u0001k!2kF,\nwherekFis the Fermi vector ( kF=\u0019\n2afor\u001a0= 0:5, cor-\nresponding to one electron per site). The variation of the\nband \flling away from the half-\flling results into a fold-\ning of the low-energy limit as shown in Figs. 3(b), 3(c),\n3(e) and 3(f). Note that due to electron-hole symmetry\nwe only show the spectra for \u001a0\u00150:5.\nFIG. 3: (Color online) dFT[\u0001 sz\ni(t)](k;!) for di\u000berent band-\n\flling: (a, d) \u001a0= 0:5 , (b, e)\u001a0= 0:6 and (c, f) \u001a0= 0:8.\nThe top panels show the exact analytical excitation spectrum\nfor a homogeneous tight-binding chain without local spins.\nThe bottom panels show the results of our numerical sim-\nulations. The inter-site distance ais arbitrary. The color\nshade in the bottom panel represents the absolute value of\ndFT[\u0001sz\ni(t)](k;!) and is logarithmically scaled for better con-\ntrast.\nB. Electrostatic gate applied\nThe dynamics of the itinerant spins, as described by\nEq. (5), in the case of an electrostatic gate applied to a\nsection of the wire (typically in the middle of the wire)\ncannot be expressed in a closed form as a function of\nVgfor arbitrarily big systems. As such we resort to our\nnumerical integration scheme. Before addressing the dy-\nnamics, however, we \frst analyze the ground-state elec-\ntronic structure of the gated wires for di\u000berent values of\nthe gate potential Vg. Displayed in Fig. 4 is the adiabatic\nvariation of the eigenvalues for a non spin-polarized chain\nwithN= 30 sites (without localized spins) as a function\nof the gate potential Vg(the gate is applied to 10 sites\nin the middle of the chain, i.e. at the sites with indexes\ngoing from i1= 11 toi2= 20). Note that energy-related\nquantities on all the \fgures are in units of the hopping\nintegral\r. ForVgclose to 0, the discrete energy spec-\ntrum spans in the range [ \u00002\r;2\r]. With the increase of\nVgthe levels spacing distorts as the eigenstates are af-\nfected di\u000berently by the gate. Certain eigenvalues grownearly linearly with increasing Vg(these are shown as red\nsquares in Fig. 4). The spatial distribution of these eigen-\nstates is predominantly concentrated in the gated region,\ni.e. they corresponds to the region where the local onsite\nenergy has been modi\fed.\nFIG. 4: (Color online) Schematic of the gate-dependence\nof the ground state energy spectrum of ^Hel. According to\ntheir weights at the three subsections of the chain \n L;M;R=P\ni2L;M;Rjcinj2we distinguish three types of states depend-\ning on which of the three partial weights is the greatest. We\nuse black circles for the case of \n L, red squares for \n Mand\ngreen diamonds for \n R. Note that the eigenstates correspond-\ning to the two ungated regions (\n Land \n R) are degenerate\nby symmetry, as the gate is applied in the exact center of the\nwire. The inset shows a magni\fcation of an area of intermedi-\nateVg, illustrating the situation of avoided crossings. A short\nchain with N= 30 sites is used for simplicity.\nFor extremely large values of Vg(Vg>4\r) the chain is\ne\u000bectively split into three energetically-decoupled parts,\nthe gated middle and the two identical un-biased ends on\nthe left-hand side and on the right-hand side. More inter-\nesting for us is the range of intermediate gate voltages,\nfor which the three parts of the wire are substantially\na\u000bected but not yet decoupled by the gate voltage. For\nsuchVgavoided crossings occur between states localized\nin the gated and non-gated regions. This gives rise to\nadditional low-frequency lines in the dynamical spectrum\nas described by Eq. (5). As we will demonstrate in the\nAppendix the presence of these avoided crossings around\nthe Fermi level for certain intermediate Vgyields an en-\nhanced transmission through the gated wire (waveguide)\nat low frequencies.\nThe two-dimensional dFT images (Fig. 5) of the spin\ndynamics in the presence of the gate bring additional di-\nmension to the electron spectroscopy analysis. The dif-\nference here with respect to the case depicted in Fig. 3(d)\nis that a gate has been applied to the middle of the chain\nin the ground state, i.e. VgP\n\u000b;i2[i1;i2]c\u000by\nicihas been\nadded to ^Helatt=t0. Again we use the half-\flling case,\n\u001a0= 0:5 (one electron per atom). From the \fgure it is5\nimmediately noticeable that the e\u000bect of the variation of\nVgon the excitation spectra is quite similar to the e\u000bect\nof the band-\flling in the non-gated case. Due to the gate\npotential, the relative electron populations of the gated\nand gate-free parts of the chain change (the gated re-\ngion is depopulated). For intermediate values of Vg[see\nFig. 5(b)] the excitation spectrum is rather a superpo-\nsition of two spectra with band \fllings below and above\n0:5 (approximately 0 :2 and 0:6). Furthermore we \fnd a\nsubstantially increased population of the low-frequency\nmodes for all k-vectors, i.e. of states were forbidden by\nsymmetry for Vg= 0. For large Vg[see Fig. 5(c)] the\nmiddle part of the wire becomes almost completely de-\npleted and the ( k;!)-portrait corresponds e\u000bectively to\nthe excitation spectrum of a single chain with a band-\n\flling\u001a0\u00190:75, which is similar to the non-gated case\npresented in Fig. 3(f).\nFIG. 5: (Color online) dFT[\u0001 sz\ni(t)](k;!) for di\u000berent values\nof the gate potential: (a) Vg= 0:2\r, (b)Vg= 2:2\rand (c)\nVg= 4:6\r. The color shade is identical to that on Fig. 3.\nIV. COMBINED QUANTUM-CLASSICAL SPIN\nDYNAMICS IN THE PRESENCE OF A GATE\nThe inclusion of dynamic local spin-impurities in the\nmodel requires the numerical integration of the set of cou-\npled non-linear EOMs of Eq. (3). For a small number of\nclassical spins the dynamics of the itinerant spin-density\nis qualitatively very similar to what is described by Eq.\n(5). The frequency-domain analysis of the classical spin\ntrajectories by means of dFT reveals the characteristic\nsignature of the discrete electronic spectrum with only\nadditional modulations in the amplitudes. We focus now\non the case in which S1is driven by a local magnetic \feld\ninto a precession, hence it acts as a spin-pump (see car-\ntoon in Fig. 1). It is important to note that what we refer\nto as a local magnetic \feld B= (0;0;Bz) is only instru-\nmental to trigger and sustain a uniform Larmor preces-\nsion ofS1, i.e. it does not produce any Zeeman splitting\nin the itinerant electrons spectrum. We consider now the\nfollowing situation: the quantum-classical spin-system is\nin its ground state until t=t0, when the \frst local spin\nS1starts \ructuating to form a small misalignment with\nB. This sets the entire quantum-classical spin system\ninto motion. The typical trajectories of the transverse\ncomponents Sx\n1(t) andSx\n2(t) and their frequency-domaindFT images for di\u000berent values of the gate voltage are\npresented in Fig. 6.\nFIG. 6: (Color online) Time evolution of the x-components of\nthe localized spins, Sx\n1(t) (blue) and Sx\n2(t) (red) [left] and the\ncorresponding spectra, dFT[ Sx\ni(t)](!), [right] for three di\u000ber-\nent gate voltages (a, d) Vg= 0:1\r, (b, e)Vg= 2:2\rand (c,\nf)Vg= 4:6\r. The frequency-domain is represented by the di-\nmensionless quantity ~!=\rand \u0016!L=~!L=\r= 1:16 for\r= 1\neV. Note that in the case of Vg= 2:2\rdi\u000berent scales are\nused for the magnitudes of Sx\n1(t) andSx\n2(t) and their spectra\n(axes corresponding to Sx\n2(t) are marked in red).\nIn this case the excitation of a transverse spin-density\natt= 0 by the initiation of the Larmor precession is very\nsimilar in nature to the excitation induced by tilting one\nlocal spin investigated before (section III). In fact it in-\ndeed develops in a very similar way in the early stages of\nthe time evolution. Just like in Fig. 2 a non-equilibrium\ntransverse spin-density spin packet is sent along the wire\nand both the localized spins respond each time the packet\nreaches them. Figure 6(a) displays the case of a very\nsmall gate voltage Vg\u001c\r. The frequency-domain image\nof the precessional motion of Sx\n1(t) has the signature of\nthe discrete electronic spectrum with an amplitude enve-\nlope that peaks at the Larmor frequency !L= 2\u0016BB1=~.\nThe dFT-portrait of the probe spin Sx\n2(t) is very simi-\nlar to that of Sx\n1(t). Although signi\fcantly down-scaled\nin amplitude, it has the electronic-structure modes im-\nprinted in its classical motion in the same way as it is\nfor the driven spin. This qualitative behavior is observed\nforVg= 0 and a range of small voltages Vg\u0014\rbut\nit changes dramatically for gate voltages Vg> \r [see\nFig. 6(b) and (c)]. The case of extremely big voltages\nVg>4\r(depicted in the bottom panel) is a trivial one for\nwhich the electrostatic barrier Vgis completely opaque to\nthe electrons and the probe spin S2does not respond to\nthe Larmor precession of S1(note also the rarefaction of\nmodes in the Sx\n1spectrum due to the e\u000bective shortening\nof the chain by 2/3).\nSpecial attention must be devoted to the intermediate6\nVgregime in which the dynamics of the probe spin is qual-\nitatively di\u000berent and uncorrelated to that of the driven\none [see Fig. 6(b)]. The typical outcome of the real time\ndynamics at such intermediate voltages \r < Vg<4\ris\nthat the probe spin starts accumulating very big trans-\nverse de\rections. This is then fed back into the pumping\nspin which also de\rects more but still preserves, to a\ngreat extent, its precession about the magnetic \feld.\nIt is worth noting that this self-ampli\fcation of the\nspin-dynamics does not come at a total energy cost, as\nthe model system described by Eg. (2) is completely\nconservative. As such we simply observe a conversion\nof electrostatic energy into \\spin-energy\" in the form\nof a spin amplitude transverse to the driving magnetic\n\feld. This accumulation is related to the increasing en-\nergy transfer from the itinerant electrons to the localized\nmoments, as illustrated in Fig. 7(a). Such an energy\ntransfer is de\fned as \u0001 E(t) =j\u0001Eel(t)\u0000\u0001ES(t)j, where\n\u0001Eel=S(t) =Eel=S(t)\u0000Eel=S(t0),Eel(t) = Tr[\u001a(t)\u0001Hel(t)]\nandES(t) =\u0000g\u0016BS1(t)\u0001B. The total energy conser-\nvation \u0001Eel(t) =\u0000\u0001ES(t) has been veri\fed within a\nrelative error of 10\u00009%. The ampli\fcation of the energy\ntransfer26is characteristic of this Vgrange (\r <Vg<4\r)\n[see the inset of Fig. 7(a)].\nThe frequency-domain image of the localized spin tra-\njectories [see Fig. 6(e)] for \r <Vg<4\rshows a substan-\ntial qualitative di\u000berence for the driven and the probe\nspins. While around !Lthe shape of the spectrum of\nS1is similar to the two extreme Vgcases, a signi\fcant\namplitude is accumulated now at low frequencies. The\nmain di\u000berence from the small Vgcase, however, is in the\nspectrum ofS2which shows practically no response at\nthe Larmor frequency but has very large amplitudes in\nthe low frequency range.\nThis behaviour in the classical spin dynamics charac-\nteristic of certain gate voltages stems from the dominance\nof the low frequency modes in the electron dynamics at\nthoseVg. The e\u000bect can be seen also in the much simpler\ncase of a non-spin-polarized uniform atomic wire sub-\njected to a sudden switch of an electrostatic gate in the\nmiddle. In the Appendix we consider the charge dynam-\nics associated to this situation and analyze the lowest\nfrequency components entering in Eq. (5). The initial\ncharge excitation, similarly to the transverse spin polar-\nization in the model above, is triggered by a local elec-\ntrostatic perturbation at the \frst site. It follows from\nEq. (5) that, for certain values of Vgfor which two adja-\ncent adiabatic eigenenergies ( ei;ei+1) around the Fermi\nlevel come closer to each other in an avoided crossing, the\ndynamical charge amplitude at the opposite end of the\nwire exhibits a peak at that particular very low frequency\n(ei+1\u0000ei)=~. In other words, a certain gate voltage con-\ndition is created for resonant charge transfer between the\nthree parts of the chain at low frequencies. This, in the\ncase of itinerant transverse spin-polarization, promotes\nthe enhanced reaction of the probe spin to the spin pump-\ning produced on the other side of the gate.\nThe electronic part of full quantum-classical spin dy-\nFIG. 7: (Color online) (a) Time evolution of the energy\ntransfer, \u0001 E(t), between electronic and localized spins for\nVg= 2:2\r. The inset shows the maximum value of the energy\ntransfer, \u0001Emax, for a \fxed duration of simulation (2 ps) as\na function of Vg. A red arrow indicates the particular voltage\nfor which \u0001 E(t) is plotted. (b) Evolution of the electron pop-\nulations of di\u000berent parts of the chain for Vg= 2:2\r. The two\nblack dashed lines show the equilibrium populations of the\ngate-free parts (upper line) and of the gated middle section\n(lower line), i.e. the populations corresponding to the case\nwhen the gate is applied in the ground state.\nnamics in the presence of the gate (Fig. 6) is, in many\nrespects, similar to the charge-only case, described in Sec-\ntion III B and in the Appendix. As before we construct\nthe two-dimensional dFT portraits of the spin-density\nevolution. Providing additional k-resolved information\nthese give another perspective for interpreting the dra-\nmatic increase of the low-frequency oscillations of the\nprobe spin at certain gate voltages, when compared to\nthe case of frozen Siand electrons at t=t0\u0000\u000etrelaxed\nto the external gate potential (Fig. 5). In the latter case,\nthe increased low-frequency occupations (with respect to\nthe non-gated case) were attributed to the split of the\none-dimensional system into subsystems with di\u000berent\naverage electron densities. Here we \fnd a similar dFT\npattern, now in the presence of dynamically-coupled lo-\ncal spins and a gate abruptly introduced at t=t0. Those\ntwo additional time-dependent factors result in a signif-\nicant variation in time of the electron distributions in\nthe gated and non-gated segments of the chain [see Fig.\n7(b)]. The dFT image in this case can be interpreted\nas a superposition of spectra corresponding to di\u000berent\n\u001a0, resulting in a smearing of the low-frequency structure\npreviously present in the ( k;!)-space image. A compar-\nison of the results for Vg= 0 andVg= 2:2\ris presented\nin Fig. 8. We \fnd that the full dynamical Vg= 0 por-\ntrait is rather similar to the results obtained for frozen-\nimpurities (see Fig. 5), the only additional feature being\nthe adiabatic excitation at the Larmor frequency of the\ndriven spin. The di\u000berence (point-by-point subtraction of\nthe contour plots) with the corresponding gated case, de-\npicted Fig. 8(b) shows that it is indeed mainly the lower\nfrequency band of the spectrum (for any wave-vector)7\nthat gains occupation in the gated case.\nFIG. 8: (a) dFT[\u0001 sz\ni(t)](k;!) forVg= 0. (b) The\ndi\u000berence between dFT[\u0001 sz\ni(t)](k;!) forVg= 2:2\rand\nforVg= 0 de\fned as \u0001 = jdFT[sx\ni(t)](k;!)Vg=2:2\r\u0000\ndFT[sx\ni(t)](k;!)Vg=0j. The color shade represents the abso-\nlute value of dFT[\u0001 sz\ni(t)](k;!) and \u0001, respectively, and is\nlogarithmically scaled for better contrast.\nV. CONCLUSIONS\nIn summary, we have employed atomistic dynamical\nsimulations to investigate the e\u000bect of an electrostatic\ngate as means of controlling the indirect coupling be-\ntween two distant localized spin impurities in a \fnite\nmetallic wire comprising one hundred atoms. One of the\nimpurities, precessing in external magnetic \feld, plays\nthe role of a spin-pump and the response of the second\n(probe) spin was analyzed as a function of the gate po-\ntential applied to the interconnecting wire. We identi\fed\na range of external gate potentials for which the spin-\npumping is extremely e\u000ecient and leads to a substantial\nexcitation of transversely-polarized itinerant spin density\nin the non-gated parts of the chain. This, in turn, pro-\nduces a massive low-frequency swing of the probe spin.\nSuch a resonant e\u000bect has been related to gate-induced\navoided crossings in the electronic structure of the in-\nterconnect. For certain gate voltages these occur in the\nvicinity of the Fermi level and assist the enhanced trans-\nfer of charge/spin across the barrier. Evidences for this\ne\u000bect are also identi\fed in the Fourier portraits of the\ncalculated time-dependent spin-distribution.\nAcknowledgments\nThis work is sponsored by Science Foundation of Ire-\nland (Grant No. 07/IN.1/I945). Computational re-\nsources have been provided by the Trinity Center for\nHigh Performance Computing and by the Irish Center\nfor High-End Computing.Appendix\nWe consider the eigenstate-resolved dynamical ampli-\ntudeAmncorresponding to a mode with frequency !mn,\nassociated to the transitions between the eigenstates em\nanden. The system is the simpli\fed wire used for the\nspectroscopic analysis of Fig. 4, i.e. it is a non-spin-\npolarized 30-atom long chain. Analogously to the spin-\npolarized case, the t=t0excitation is applied simply as\na potential shift at the \frst atomic site. From Eq. (5)\nthe dynamically-driven charge-accumulation amplitude\nat siteiis de\fned as\nAi\nmn=cimc\u0003\ninX\nklc\u0003\nkmcln\u001akl(t0); (A.1)\nsuch that\n\u001aii(t) =X\nmnAi\nmne\u0000i!mnt(A.2)\nis the dynamic electron occupation at site i. This quan-\ntity is calculated at the last site, i=N, i.e. at the oppo-\nsite side of the chain with respect to the perturbation. In\nEq. (A.1)cim=hij'miare the onsite components of the\neigenvectorsj'miof the non-spin-polarized Hamiltonian\n^Hel(Vg) for which the gate voltage Vgis applied (between\nthe sitesi= 11 andi= 21).\nFrom all the pairs of adjacent eigenstates ( ei;ei+1),\nwhich give rise to the lowest frequency modes ( !i;i+1=\n(ei+1\u0000ei)=~) in the dynamics, we identify in Fig. 9 those\nwhich have the highest (in absolute value) amplitude\nAN\ni;i+1. For small gate voltages these are the eigenstates\naround the Fermi level but as Vgincreases the modes\nwith the largest amplitudes tend to arise where avoided\ncrossings of eigenstates occur. These are also the modes\nwith frequencies close to the lowest possible frequency for\nthat particular gate voltage.\nThe actual highest amplitude at the last site, corre-\nsponding to the highlighted modes in Fig. 9, are de-\npicted as a function of Vgin Fig. 10(a), together with\nthe quantity AN\nsum\u0011qP\ni(AN\ni;i+1)2. This represents\nthe pessimistic estimate (considering all modes orthogo-\nnal in phase) of the total low-frequency amplitude corre-\nsponding to all pairs of adjacent eigenvalues. Such curves\ndemonstrate the resonant nature of the dynamical site-\noccupation as a function of the gate voltage. Fig. 10(a)\nalso shows that the low frequency charge excitation peaks\nfor intermediate gate voltages (for this case Vg\u00182:5\r)\ncan be related to the observed enhanced low-frequency\ncoupling between the itinerant and localized spins for\nthe intermediate Vgrange. Interestingly, the two A(Vg)\ncurves coincide for very low and very high voltages, show-\ning that in this case the entire dynamics is practically due\nto just one mode (marked in black in Fig. 9 ). In the\nintermediate regime it is clear that there are more than\none low-frequency modes which have a signi\fcant ampli-\ntude, although the one marked in Fig. 9 gives the leading\ncontribution.8\nFIG. 9: (Color online) Adiabatic variation with the gate volt-\nage of a few eigenvalues around the Fermi level (we have used\n\"0= 0 andj\rj= 1 eV, hence EF= 0 atVg= 0). Marked with\nblack lines are the pairs of eigenstates ( i;i+1) which give rise\nto the highest (in absolute value) dynamical amplitude AN\ni;i+1\nat the last site N.\nFIG. 10: (Color online) (a) The maximum single-mode am-\nplitudeAN\nmaxand an estimate for the total low-frequency am-\nplitudeAN\nsum(see text for details) as a function of the applied\ngate voltage. (b) The frequency of the mode related to the\npair of eigenstates ( i= 15) around the Fermi level, adiabati-\ncally evolved with the gate voltage and (c) its corresponding\namplitude as de\fned in Eq. (A.1).For a particular pair of adjacent eigenstates there are a\nfew instances where the adiabatic eigenvalues come closer\ntogether as a function of the gate voltage. In Fig. 10(b,c)\nwe analyze the dependence of the frequency and the am-\nplitude corresponding to the pair of eigenvalues just be-\nlow and above the Fermi level in the ground state ( i= 15\nfor the 30-atom chain at half-\flling). We \fnd that the\ndynamical charge amplitude on the last site peaks (as\nabsolute value) every time the frequency passes through\na minimum.\n1T. Dietl, D. D. Awschalom, M. Kaminska and H. Ohno,\nSpintronics (Elsevier, Amsterdam, 2008), Vol. 82.\n2B. Hildebrand and K. Ounadjela, Spin dynamics in con-\n\fned magnetic structures (Springer-Verlag, Berlin, 2002).3N. Baadji, M. Piacenza, T. Tugsuz, F. D. Sala, G. Maruc-\ncio and S. Sanvito, Nat. Mat. 8, 813 (2009).\n4J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n5E. B. Mayers, D. C. Ralph, J. A. Katine, R. N. Louie, R.9\nA. Buhrman, Science 285, 867 (1999).\n6S. Loth, K. von Bergmann, M. Ternes, A. F. Otte, C. P.\nLutz and A. J. Heinrich, Nature Physics 6, 340 (2010).\n7A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. Pisarev, A.\nM. Balbashov and Th. Rasing, Nature 435, 655 (2005).\n8A. Kirilyuk, A.V. Kimel and Th. Rasing, Rev. Mod. Phys.\n82, 2731 (2010).\n9R. Hanson and D. D. Awschalom, Nature 453, 1043\n(2008).\n10D. D. 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Turnbull, and H.\nEhrenreich (Academic, New York, 1969), Vol. 23, p. 184.\n17M. Stamenova and S. Sanvito, in The Oxford Handbook\non Nanoscience and Technology (Oxford University Press,\nOxford, 2010), Vol. 1, p. 169.\n18M. Stamenova, T.N. Todorov and S. Sanvito, Phys. Rev.\nB77, 054439 (2008).19A. P. Hors\feld, D. R. Bowler and A. J. Fisher, J. Phys.:\nCondens. Matter 16, L65 (2004).\n20J. Thijssen, Computational physics (Cambridge University\nPress, 2007), p. 473.\n21W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P.\nFlannery, Numerical recipes: the art of scienti\fc comput-\ning(Cambridge University Press, 2007), p. 600.\n22T. Giamarchi, Quantum Physics in One Dimension\n(Clarendon Press, Oxford, 2004), p. 12.\n23J. des Cloizeaux and J. J. Pearson, Phys. Rev. 128, 2131\n(1962); G. Muller and H. Thomas, Phys. Rev. B 24, 1429\n(1981); O. Derzhko, in Condensed Matter Physics in the\nPrime of the 21st Century: Phenomena, Materials, Ideas,\nMethods , proceedings of the 43rd Karpacz Winter School\nof Theoretical Physics, Ladek Zdroj, Poland, edited by J.\nJedrzejewski (World Scienti\fc, 2008), p. 35.\n24A. P. Hors\feld, D. R. Bowler, A. J. Fisher, T. N. Todorov\nand M. J. Montgomery, J. Phys.: Cond. Mat. 16, 3609\n(2004)\n25This region can also be re-interpreted as part of a di\u000berent\nmaterial so that the same model system also describes a\none-dimensional tri-layered heterostructure in which the\ntwo peripheral layers are identical.\n26This e\u000bect can appear as a particular form of \\heating\" for\nthe local spins, at the expense of electron gas, as within the\ntime covered by the simulation ( \u001910 ps) the energy trans-\nfer seems to be irreversible. It is well-known, however, that\nthe typical Ehrenfest molecular dynamics (EMD) su\u000bers\nfrom the inability to account for Joule heating processes24\ndue to the lack of dynamical correlations between the elec-\ntrons and the ions. Our simulations are limited to picosec-\nond times and by no means imply a possible qualitative\ndi\u000berence of the EMD and its spin couterpart."    },    {        "title": "2107.12075v2.Universal_scaling_of_spin_mixing_dynamics_in_a_strongly_interacting_one_dimensional_Fermi_gas.pdf",        "content": "Universal spin mixing oscillations in a strongly interacting one-dimensional Fermi gas\nGiovanni Pecci,1Patrizia Vignolo,2and Anna Minguzzi1\n1Univ. Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France\n2Université Côte d’Azur, CNRS, Institut de Physique de Nice, 06560 Valbonne, France\nWe study the spin-mixing dynamics of a one-dimensional strongly repulsive Fermi gas under har-\nmonic confinement. By employing a mapping onto an inhomogeneous isotropic Heisenberg model\nand the symmetries under particle exchange, we follow the dynamics till very long times. Starting\nfrom an initial spin-separated state, we observe superdiffusion, spin-dipolar large amplitude oscil-\nlations and thermalization. We report a universal scaling of the oscillations with particle number\nN1=4. Our study puts forward one-dimensional correlated fermions as a new system to observe the\nemergence of non-equilibrium universal features.\nElucidating the dynamics of interacting Fermi gases\nis important for understanding a large variety of physi-\ncal phenomena, from condensed matter to plasmas and\nastrophysical objects as neutron stars. The strongly out-\nof-equilibrium dynamics of interacting quantum systems\nis currently one of the most challenging open problems.\nIn this context, the spin dynamics deserves a specific\nfocus. Spin currents can be easily damped by inter-\nparticle collisions [1] and the continuity equation for the\nspindensityincludesbothorbitalcurrentandspintorque\ncontributions [2]. Spin drag is another manifestation\nof interactions among the spin species, inducing spin-\ndiffusive or non-dissipative dynamics depending on the\ninteraction regimes [1, 3–7]. Ultracold atomic gases pro-\nvide an ideal platform for exploring in isolated condi-\ntions the out-of-equilibrium spin dynamics [8–10]. In a\nthree dimensional geometry, the oscillatory dynamics of\na strongly interacting Fermi gas with initially spatially\nseparated spin components was studied in [11]. The spin\ndrag, spin diffusivity and spin susceptibility were ob-\ntained, and a universal limit for spin diffusivity at low\ntemperature was reported for the unitary Fermi gas.\nArelevantquestioniswhathappenstotheabovequan-\ntities when reducing the dimensionality of the system\nto quasi one-dimensional, and what type of universality\nemerges. One-dimensional (1D) systems display specific\nfeatures, as the enhancement of quantum fluctuations\nand correlations and they can be described by a wealth\nof theoretical and numerical methods [12–15]. The quan-\ntum dynamics may be strongly affected by the geomet-\nrical constraints, as well as by the presence of a large\nnumber of conserved quantities, as demonstrated e.g. in\nthe quantum Newton’s cradle experiment [16].\nWe address this question by following the dynamics of\nstrongly repulsive fermions subjected to a longitudinal\nharmonic confinement in a tight waveguide. As in the\nthree-dimensional case of Ref.[11], we start from an ini-\ntially imbalanced state with all spin up on the left and all\nspin down on the right of the harmonic trap, and we fol-\nlow the damped oscillations of the magnetization. While\nthe fully quantum dynamics at arbitrary interactions can\nbe followed only at short times with a classical simula-\ntor, we focus here on the strongly correlated regime of\nvery large interactions, close to the integrable point atinfinite repulsions [17–23]. In this regime, the dynamics\nof the charge and spin decouple, and the spin dynamics\ncan be followed exactly till very long times by means of\na mapping onto the one of an inhomogeneous, isotropic\nHeisenberg model [24, 25] with site-dependent couplings\n[26, 27].\nFIG. 1. Left panel: spin up \u001a\"in orange (light grey) and\ndown\u001a#in violet (dark grey) spatial densities (in units of the\ninverse harmonic oscillator length `\u00001, with`=p\n\u0016h=m! 0)\nas a function of position in the trap (in units of `) at times\n!0t= 0;33;200from top to bottom. The two initially sepa-\nrated clouds start oscillating in the trap and eventually fully\nmix, approaching to a zero-magnetization state. Right panel:\nmagnetization as a function of x(in units of `) andt(in units\nof!\u00001\n0) forN= 12fermions. The green line corresponds to\ncenter of mass d(t)of the magnetization.\nAn overview of the full spin dynamics is provided in\nFig. 1, where three main dynamical regimes arise. At\nshort times, we predict the emergence of a superdif-\nfusive behaviour, compatible with Kardar-Parisi-Zhang\n(KPZ) universality, in striking difference from the diffu-\nsive one found in the three-dimensional counterpart[11].\nWe thus identify 1D correlated fermions as a new system\nto observe the emergence of non-equilibrium universal-\nity, largely explored in homogeneous Heisenberg models\n[28–35] and experimentally evidenced in quantum mag-\nnets and in ultracold atoms on a lattice [36–39]. At in-\ntermediate times, we observe large-amplitude spin-dipole\noscillations and we obtain the spin drag decay rate. We\nunveil aN1=4scaling in the oscillation frequency, imply-\ning a slow-down of the motion and the decrease of thearXiv:2107.12075v2  [cond-mat.quant-gas]  30 May 20222\nzero-temperature spin drag rate as the particle number\ngrows.\nAt long times, the oscillations are damped out and the\nsystem thermalizes to the diagonal ensemble [40]. From\nthe analysis of the energy levels distribution we find that\nthe system is weakly non-integrable. The proposed setup\nallows to explore the conditions for emergence of non-\nequilibrium universal behaviour in relation to the break-\ning of its integrability in one dimension.\nModel and dynamics We consider a one-dimensional\nSU(2) interacting Fermi gas confined in a harmonic trap.\nThe Hamiltonian for such system reads\nH=NX\ni=1\u0010p2\ni\n2m+m!2\n0x2\ni\n2\u0011\n+gX\ni6=j\u000e(xi\u0000xj);(1)\nwhereN=N\"+N#is the total number of particles and\nwe takeN\"=N#,!0is the frequency of the harmonic\ntrap and we model the interspecies interaction using a\ndelta potential of strength g. Hamiltonian (1) is char-\nacterized by the symmetry under exchange of particles\nhaving the same spin. For SU(2) fermions, the eigen-\nstates can be classified by the irreducible representations\nof the permutation group (see eg. [20]).\nWe focus on the strongly repulsive limit g!1: in\nthis regime the model is exactly solvable [18] and the\nwavefunction is given by:\n\t =X\nP\u0012(xP(1)<:::<xP(N))aP\tA(x1;:::xN);(2)\nwhere\u0012(x1<x2<x3<:::xN)is the characteristic func-\ntion of the coordinate sector fx1<x2<x3<:::<xNg,\naPare phases depending on the spin ordering of the cor-\nresponding coordinate sector and the summation is per-\nformed over all the possible permutations PofNele-\nments. The function \tAis the wavefunction of a N-\nparticle non-interacting Fermi gas in the same external\npotential, i.e. the anti-symmetric product of Neigen-\nfunctions of the harmonic oscillator. Remarkably, in the\ng!1limit the spin and spatial (’charge’) degrees of\nfreedom are decoupled in the wavefunction.\nWe determine the phases apin Eq.(2) to first order\nin1=gby mapping the Hamiltonian (1) into an effective\nspin chain:\nHs= (EF\u0000N\u00001X\niJi)1+N\u00001X\ni=1JiPi;i+1;(3)\nwherePi;i+1is the transposition operator on the chain of\nNsites andEF=N2\u0016h!0=2is the Fermi energy. The co-\nefficientsJiare site-dependent hopping parameters ofthe\nchain, carrying information on the external potential and\non the atom-atom interaction of the original fermionic\nproblem [41]. The explicit expression reads [42]:\nJi=1\ngZ1\n\u00001dx1:::dxN\u000e(xi\u0000xi+1)\u0012(x1<:::<xN)\f\f\f@\tA\n@xi\f\f\f2\n:\n(4)We classify the basis vectors of the Hilbert space as-\nsociated to (3) according to the spin ordering on the\nchain (the so-called snippet basis [43]). For example,\nforN\"=N#= 2the vectorj\"\"##iindicates that all the\nspins\"are placed in the left half of the chain. There-\nfore, the dimension of the Hilbert space is s=N!\nN\"!N#!.\nThe diagonalization of Eq. (3) allows us to calculate the\naPand thus several observables such as the spin densi-\nties\u001a\";#(x;t). This allows to study the dynamics of the\ntrapped system with an arbitrary initial state.\nIn this work we follow the fermion dynamics starting\nfrom the initially strongly out-of-equilibrium state j\u001f(t=\n0)i=j\"\"\":::###i, as in Ref.[11], where the spins up and\ndown are separated in the two opposite sides of the trap.\nSince the harmonic trap is unchanged, the spatial part of\nthewavefunction(2)isconstantduringthemotion, hence\nJiare constant in time. The time evolution involves only\nthe spin degrees of freedom and can be obtained using\nthe effective spin chain Hamiltonian (3). Recalling that\nthe spin operators are related to the permutation opera-\ntor by the relation Pk;k+1=1\n2(1+\u001bk\u0001\u001bk+1), Hamilto-\nnian (1) can be mapped to the one of an inhomogeneous\nisotropic Heisenberg model HH=PN\u00001\nj=1Jj~ \u001bj\u0001~ \u001bj+1, but\nin particle space , ie each lattice site is associated to a par-\nticle index. The equation of motion for the spin operator\n~Sj=1\n2(\u001bx\nj;\u001by\nj;\u001bz\nj)for thej-th particle reads\ndS\u0016\nj\ndt=i[Hs;S\u0016\nj] = (~ \u001cj\u0002~Sj)\u0016; (5)\nwhere\u0016=x;y;z. and~ \u001cj=Jj\u00001~ \u001bj\u00001+Jj~ \u001bj+1is the\ntorque acting on a fixed particle due to the coupling with\nthe neighbouring ones [41]. As a result of Eq. (5) we\nconclude that in our case the spin dynamics is entirely\ndue to spin torque [41].\nThe experimentally accessible component of such spin\nvector isSz\nj, associated to the local magnetization\nm(x;t) =NX\nj=1mj(t)\u001aj(x); (6)\nwhere\u001aj(x)is the spatial density of the j-th parti-\ncle in the trap [26, 41], mj(t) =h\u001f(t)jSz\njj\u001f(t)iand\nj\u001f(t)i=e\u0000iHstj\u001f(0)iis the time-evolved spin state, ob-\ntained from the diagonalization of Hsby exploiting all\nits symmetries. The magnetization is experimentally ac-\ncessible by recording the population imbalance among \"\nand#fermions,m(x;t) =\u001a\"(x;t)\u0000\u001a#(x;t).\nAnother important observable for the dynamics is the\nspin current density\nj(x;t) =1\n2N\u00001X\nj=1jj(t)(\u001aj(x) +\u001aj+1(x));(7)\nwherejj(t)areobtainedfromthe zcomponentofEq.(5),\njj(t) =Jj(\u001bx\nj\u001by\nj+1\u0000\u001by\nj\u001bx\nj+1):We detail in the following\nthe spin-mixing dynamics in the various relevant time3\nregimes. Systems up to N= 12particles have been an-\nalyzed using exact diagonalization, while we used trun-\ncated Taylor series approximation of the time evolution\noperator [44] and tDMRG [45] to study larger systems.\nFIG. 2. (a) Early-time magnetization dynamics m(x;t)from\nexact solution for N= 12as a function of space ( xin units\nof`) and time t(in units of !\u00001\n0). (b) Integrated current\n\u000ej(t)in units of`\u00001from DMRG, as a function of time t(in\nunits of!\u00001\n0) evaluated in the center of the trap x= 0(solid\nlines), together with ballistic \u000ej\u0018t, KPZ\u000ej\u0018t2=3and\ndiffusive behaviour \u000ej\u0018t1=2(dashed straight lines). Larger\nnumber of particles are associated with increasingly darker\ncolors. (c) KPZ scaling of the magnetization mn(t), shown as\na function of y=n=(!0t)2=3. We show the comparison with\nthe derivative of the suitably renormalized 1d KPZ scaling\nfunctiong(y)[46].\nShort times The short time dynamics, before the first\nspin oscillation, is shown in Fig. 2. We show the mag-\nnetization as a function of space and time, showing that\nthe initially sharp magnetization interface spreads with\ntime, till it starts feeling the effect of the confining poten-\ntial. To identify the nature of the magnetization spread-\ning, it is useful to study the time-integrated spin current\ndensity as in [28], \u000ej(t) =Rt\n0dt0j(0;t0), wherej(x;t)is\ndefined in Eq.(7) and it is calculated at the center of the\ntrap. We see that the integrated spin current displays a\nsuperdiffusive behaviour \u000ej(t)\u0018t\u0011, with power law ex-\nponent\u0011\u00180:638(1). This is compatible with the value\n\u0011= 2=3predicted for the homogeneous spin chain and\nclearly not ballistic nor diffusive.Remarkably, low-energy\ndynamics described by Luttinger liquid predicts ballistic\nbehaviour[1]: thedeviationfromthispredictiondiscloses\nthe marked out-of-equilibrium features of the physical\nsystem that are described in an exact way by our model.\nUsing DMRG calculations we have checked that the KPZ\nregion persists for larger numbers of particles (see Fig.2)\nThe deviation at later times from KPZ behaviour is due\nto onset of the oscillatory dynamics associated to the\npresence of the external trap. Within the KPZ region,\nwe find that the magnetization profiles collapse onto each\nother if plotted as a function of xn=(!0t)1=z,z= 3=2be-\ning the KPZ value for the dynamical critical exponent.\nIntermediate times We next focus on the interme-\ndiate time regime, when the particles undergo large-\namplitude spin-dipole oscillations in the trap. We followthe center of mass oscillations of the magnetization,\nd(t) =1\nNZ1\n\u00001dx x m (x;t): (8)\nThe time evolution of d(t)is shown in Fig. 3a for various\nvalues of the number of particles. We observe damped\noscillations tending to a plateau corresponding to zero\nmagnetization. At later times (not shown in the figure),\nthe dynamics undergoes several partial revivals, as ex-\npected since we describe a closed quantum system. Quite\nremarkably, we find that the various curves for different\nparticle numbers collapse one to another if we scale the\ntime axis by a factor N\u000bwith\u000b= 0:25. Using DMRG\nsimulations up to N= 60particles [41], we have tested\nthat the scaling is robust at increasing particle numbers.\nThe magnetization oscillations are well approximated by\na damped harmonic oscillator F(t) =f0e\u0000\rtcos(\nt)(see\nFig. 3b). This allows us to obtain the oscillation fre-\nquency \nfor the various Nvalues. We find that the\ndamping rate \rdoesn’t depend on the number of parti-\ncles. Combining the two values, we obtain the spin drag\nrate as \u0000sd= \n2=\r[3, 4, 11, 47].\nWe also perform a spectral analysis of d(t)by intro-\nducing the spectral function A(!) =R1\n0dt d(t)ei!t. In\nFig. 3c we show jA(!)jand the Fourier transform of\nthe fitted signal ~F(!) =R1\n0dt F (t)ei!tas a function\nof the rescaled frequencies. The spectral function shows\ntwo main peaks centered around \u0006\nuniv. Several excita-\ntion frequencies contribute to the overall shape and the\nlinewidth of ~F(!)[41]. To estimate the scaling expo-\nnent\u000bwe evaluate the position PN(\u000b) =!PN\u000bof the\nmaximum of ~F(!N\u000b)at positive frequencies, such that\n~F(!PN\u000b) = max!>0~F(!N\u000b), as a function of a scaling\nexponent\u000b. As we show in Fig. 3d, the universal scaling\nis reached for \u000b= 0:25.\nThe universal scaling observed in Fig. 3 allows us to\nestimate the spin-dipole oscillation frequency at larger\nNas\nN'\nuniv=N1=4, with \nuniv'0:19!0. Corre-\nspondingly, we find that the spin drag scales as \u0000sd=\n\n2\nuniv=(\rN1=2), hence vanishing at large particle num-\nbers, as also predicted in [1] for low-energy excitations of\nthe spectrum.\nLong times Finally, we study the long-time regime at\nwhich the damped dynamics becomes dominant and the\nsystem approaches to a zero-magnetization state. Since\nthe Hamiltonian (1) is not integrable at finite interaction\nstrength, we expect some traces of chaoticity to emerge\nduring the dynamics [40, 48, 49]. In this case the sys-\ntem thermalizes to a state described by the diagonal\nensemble, coinciding in our case with the microcanoni-\ncal ensemble [50]. We verify this by calculating the dis-\ntanceR(t) =R\ndxj\u001a\"(x;t)\u0000\u001a\";MC(x)j2between the spin\nup density and its value in the microcanonical ensem-\nble\u001a\";MC(x). The results are presented in Fig. 4a. At\ntimes corresponding to the zero-magnetization plateau in\nFig. 4,R(t)vanishes and the spin density approaches to\nthe steady state value. At later times, revivals occur and\nthe system deviates from this configuration.4\nFIG. 3. (a) Center of mass of the magnetization d(t), in units of `, as a function of time, in units of !\u00001\n0, and scaled by a factor\nN1=4to evidence the universal behaviour of the oscillations. (b) d(t)forN= 12fitted with a damped harmonic oscillator\nF(t) =f0e\u0000\rtcos(\nt). (c) Modulus of A(!), in units of `=! 0, for different number of particles as a function of the universal\nfrequencies !N1=4=!0, compared to the modulus of the Fourier transform ~F(!)of the fitF(t)(dotted violet line). Colors codes\nare the same as in panel (a). (d) Position PN(\u000b)of the peaks of ~F(!N\u000b), in units of !0, as a function of the scaling parameter\n\u000b.\n  \n 0 20 40 60 80 100 120\n 0 200 400 600 800 1000R(t) l\nω0 t(a)\n 0 2 4 6 8 10\n 50 150 250\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8\n 0 1 2 3 4 5 6W(∆ε)\n∆ε(b)\n 0 0.2 0.4 0.6 0.8 1\n 0 1 2 3 4 5 6\n∆ε\nFIG. 4. (a) Distance R(t)(in units of `\u00001) as a function\nof time (in units of !\u00001\n0). The inset shows a zoom of the\narea indicated by the rectangle. (b) Level spacing distribu-\ntionW(\u0001\u000f)for the unfolded spectrum in a sector at fixed\nsymmetry. The orange and the blue curves show respec-\ntively the Wigner-Dyson WWD(\u0001\u000f)and the Brody distribu-\ntionWB(\u0001\u000f)with\f= 0:22. Theinsetshowsthelevel-spacing\ndistribution of the whole unfolded spectrum and the Poisson\ndistribution WP(\u0001\u000f)(green line). In all the panels, N= 14.\nTo further provide evidence for chaotic behaviour,\nwe analyze the level-spacing distribution W(\u0001\u000f)[51–53],\nconstructed using the unfolded dimensionless energy lev-\nels [49, 54–56]. The spectrum of an integrable system\nfollows a Poissonian distribution WP(\u0001\u000f)=e\u0000\u0001\u000f, while\na chaotic system is described by a Wigner-Dyson one\nWWD(\u0001\u000f)=\u0019\n2\u0001\u000fe\u0000\u0019\n4\u0001\u000f2. Weinterpolatebetweenthetwo\nregimes, thus quantifying the level of chaoticity encoded\nin the spectrum, through the Brody distribution [57]:\nWB(\u0001\u000f) = (\f+ 1)b\u0001\u000f\fe\u0000b\u0001\u000f\f+1; (9)\nwhereb=f\u0000[(\f+ 2)=(\f+ 1)]g\f+1and\u0000is the Euler\nGamma function. The Brody distribution reduces to the\nPoissonorWigner-Dysononesfor \f= 0or1respectively.\nTo obtain the level-spacing distribution it is importantto take into account the symmetries of the system [58],\nwhich in our case are the spatial parity and the symme-\ntry under particle exchange. Our choice of basis vectors\nallows ustoreadilycheck theparity ofthe eigenstates. In\norder to identify the symmetry under particle exchange\nassociated to a given Young tableau, we diagonalize the\nHeisenbergHamiltonianinthebasisofthepermutational\nsymmetry [20, 59]. We then partition the energy levels\naccording to the quantum numbers of the corresponding\neigenstates. In the inset in Fig.4b we show the distribu-\ntionofalltheunfoldedlevelspacings,irrespectivelyofthe\nsymmetry constraints. In this case the chaoticity is hid-\nden and the distribution is Poissonian. The level-spacing\ndistribution of the largest subspace at fixed symmetry\nis shown in the main panel of Fig.4b. We find that the\nlevel-spacing distribution is well described by the Brody\ndistribution with parameter \f= 0:22. This shows that\nlarge interactions destroy only partially the integrability\nof the infinite-repulsion model. A moderately chaotic be-\nhaviour also emerges from the study of the localization\nproperties of the eigenstates of the Hamiltonian (1) [41].\nSuch intermediate regime is typical of integrable systems\nsubjected to small perturbations [49, 52, 60].\nConclusions We have studied the strongly out-\nof-equilibrium spin-mixing dynamics of repulsive 1D\nfermions under harmonic confinement, starting from an\ninitial spatially separated spin configuration. Thanks to\nthe mapping to an inhomogeneous Heisenberg model on\nan effective lattice in particle space, we have followed the\nreal-space magnetization dynamics till very long times.\nAt short times, as specific of one-dimensional systems\nand different from the three-dimensional strongly inter-\nacting Fermi gas, we observe superdiffusive behaviour of\nthe magnetization profile in time. The system here con-\nsidered is weakly not integrable,hence equivalent to the\ncase where KPZ universality was reported in the short\ntimes dynamics [35, 61, 62]. Our observations call for the5\nexploration of the universal properties of the correspond-\ning spin model. At intermediate times, we have obtained\ndamped spin-dipole oscillations characterized by a uni-\nversal scaling of the oscillation time with N1=4, thus pre-\ndictingaslow-downoftheoscillationanddecreaseofspin\ndrag at large particle numbers. 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Strongly repulsive regime\nWe show here an intrinsic property of the strong-coupling expansion we are using to study the system: a variation\nof the coupling constant gto a new value g0induces a scaling of all many-body energy levels Ento new values E0\nnsuch\nthatgEn=g0E0\nn, without affecting the eigenvectors. This follows from the expression of Hsand of the coefficient Ji\ngiven in the main text and may be used as to identify the strongly interacting regime.\nThe above property implies that the frequencies !driving the dynamics, i.e. the energy spacings, scale accordingly.\nAs a consequence, by rescaling the time scale likewise, the dynamics doesn’t depend on the actual value of the coupling\nconstantg.\nIn Fig. 5 we show the center of mass of the magnetization d(t)as a function of the time by rescaling the time axis\nof a factor of 1=~g, being ~g=g=(\u0016h!0l). We observe that, for various values of the coupling constant, the spin dynamics\nhas exactly the same features.\n  \n-10-8-6-4-2 0 2 4 6\n 0  1  2  3  4  5  6d(t)/l\nω0 t/ g~g~=10\ng~=15\ng~=20\ng~=30\nFIG. 5. Center of mass of the magnetization d(t), in units of `, forN= 8particles, as a function of the rescaled time !0t=~g,\nfor various values of the dimensionless coupling constant ~g. The curves corresponding to different ~gare not resolved since they\ncollapse one onto the other.\nB. Continuity equation\nWe demonstrate here Eq. (5), showing that the dynamics we study in the main text is purely due to a torque\nmoment among the spins. Consider indeed\n[Hs;S\u0016\nj] =1\n2X\nlX\n\u0015=x;y;zJl[\u001b\u0015\nl\u001b\u0015\nl+1;\u001b\u0016\nj]: (10)\nDecomposing the commutator and using the relation [\u001b\u0016\ni;\u001b\u0015\nj] = 2iP\n\r\u000eij\u000f\u0016\u0015\r\u001b\r\niwe get\ndS\u0016\nj\ndt=\u0000X\n\r;\u0015\u000f\u0016\u0015\r(Jj\u00001\u001b\u0015\nj\u00001+Jj\u001b\u0015\nj+1)\u001b\r\nj; (11)\nfrom which we readily obtain Eq. (5).\nIn order to have a better understanding of the dynamics, we write the three components of the last equation on the\nsnippet basis which is our computational basis. To do so, we recall the action of the Pauli matrices on the single-spin\nHilbert space. From their explicit expression one sees that both \u001bx\njand\u001by\njinduce a spin flip on the j-th site, the latter8\nalso imprinting a spin-dependent phase. On the other hand, \u001bz\njdoesn’t invert the spin on the site jand its action it’s\nequivalent to the identity if the j-th spin is up and induces a phase shift of \u0019if this spin is down. Consequently, as\nthe components xandyof Eq. (11) involve respectively products of \u001by\nj\u001bz\nj+1and\u001bx\nj\u001bz\nj+1, they modify the total spin\nof the state.\nFrom the above considerations and since we are working at fixed total spin, we can assess that the expectation\nvalue ofdSx\nj\ndtanddSy\nj\ndtare vanishing on the basis we are considering. This is not the case fordSz\nj\ndt, whose expectation\nvalue provides access to the expression of the spin current in terms of the permutation operator,\nhepjjljeqi= 2Jl(\u00001)\u000eq(l);\"hepjPk;k+1jeqi(1\u0000\u000epq): (12)\nC. Orbital current\nWe show here that the particle orbital current is zero and consequently the dynamics described in the main text in\nthe strongly interacting regime is only due to spin torque.\nThe equation of motion for the density of spin \"particles reads (an analogous definition hold for spin #)\n@n\"(x;t)\n@t=Z1\n\u00001dx1:::dxN@\n@t\u0010\n\t\u0003N\"X\nj=1\u000e(x\u0000xj)\t\u0011\n: (13)\nUsing the wavefunction reported in Eq.(2) of the main text, and recalling that there is no time-dependence of the\nsingle-particle orbitals in the quench protocol here considered, we obtain\n@n\"(x;t)\n@t=X\nP;Q\u0010@a\u0003\nP\n@taQ+a\u0003\nP@aQ\n@t\u0011Z1\n\u00001dx1:::dxNN\"X\nj=1\u0012P\u0012Q\u000e(x\u0000xj)j\tAj2; (14)\nwhereweset \u0012P=\u0012(xP(1)<xP(2)<:::xP(N)). Remarkably, theproduct \u0012P\u0012Qisnon-zeroonlywhenthepermutations\nPandQcoincide. Consequently, the expression above can be simplified as in the following:\n@n\"(x;t)\n@t=\nX\nP\u0010@a\u0003\nP\n@taP+a\u0003\nP@aP\n@t\u0011N\"X\nj=1Z1\n\u00001dx1:::dxN\u0012P\u000e(x\u0000xj)j\tAj2=\nX\nP\u0010@a\u0003\nP\n@taP+a\u0003\nP@aP\n@t\u0011N\"X\nj=1\u001aP(j)(x);(15)\nwhere in the last line we used the definition of the spatial density of the j-th particle in the coordinate sector\nindividuated by the permutation P. From Eq. (15) we see that the only contribution to the equation of motion for\nthe spin up density comes from the spin sector, whose time evolution is uniquely fixed by the Hamiltonian Hs. The\ntotal density (’charge’) sector, that in the dynamical protocol we consider is constant in time, only contributes with\na spatial envelope given by the densities of the particles in the harmonic trap.\nD. Parity of oscillation frequencies\nWe derive here the analytical expression for the magnetization, that illustrates the energy levels involved in the\ntime evolution. Starting from the definition given in the main text, the magnetization can be written as m(x;t) =Ps\nj=1mj(t)\u001aj(x). The probability of the i-th particle to have magnetization \u00061is:\nmj(t) =h\u001f(t)jSz\njj\u001f(t)i=sX\np=1jap(t)j2(\u000ep(j);\"\u0000\u000ep(j);#); (16)\nwherej\u001f(t)i=Ps\np=1apjepiis the spin component of the wavefunction, being jepithe snippet basis. We also\nintroduced \u000ep(j);\"(#)that is equal to one if the j-th component of the p-th element of the basis is a spin up (down)9\n  \n-1.2-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6\n 0  50  100  150  200dD(t)\nω0t(a)\nd(t)/A(12)\n 0 20  40  60  80\nω0t/N0.25(b)\nN=10\nN=8\nN=12\nN=14\nN=16\nN=20\nN=2234567\n 4 6 8 10 12A(N)\nNN-1/2\nFIG. 6. (a) Discrete and continuum center of mass of magnetization for N= 12as a function of time. The two curves are\nproportional at any time. In the inset, we show the proportionality constant A(N)as a function of N. (b) Discrete center of\nmass of magnetization dD(t)as a function of time for different number of particles. The time axis has been rescaled by a factor\nofN1=4to stress the universal behaviour of the frequency of the spin oscillations.\nand zero otherwise. We want to study the symmetry of Eq. (16), starting from the explicit expression for jap(t)j2,\njap(t)j2=sX\nl;ke\u0000i(Ek\u0000El)tfp;kfp;l l0 k0; (17)\nwherefp;k=hepj kiand k0=h kj\u001f(0)i, withj kieigenstates of Hs. WhenN\"=N#the Hamiltonian is symmetric\nunder spin inversion: one can check that this implies fpk=\u0006fs\u0000p+1;k;;8k. The choice of the sign depends on the\nindexk. Consequently, we split the sums in (17) according to the parity of the eigenstates. To do so, we divide\nall the possible indexes k;lin four setsf\u0003++;\u0003\u0000+;\u0003+\u0000;\u0003\u0000\u0000gsuch that if k;l2\u0003abthenfp;k=afs\u0000p+1;kand\nfp;l=bfs\u0000p+1;l; a;b = +;\u0000.\nOnecanalsoshowthatthespininversionsymmetryinducestherelation \u000ep(j);\"=\u000e(s\u0000p+1)(j);#8p. Asaconsequence,\nwe can write Eq.(16) as in the following:\nmj(t) = 4s=2X\np=1X\nl;k2\u0003+\u0000cos(!k;lt)fp;kfp;l l0 k0(\u000ep(j);\"\u0000\u000ep(j);#): (18)\nFrom Eq. (18) we see that the frequencies !k;l= (Ek\u0000El)=\u0016h, that have a non-vanishing contribution in the\nmagnetization dynamics, are the ones connecting eigenstates with different parity. As immediate remark, we see that\nthe frequencies !k;k= 0don’t affect the time evolution consistently with the Fourier analysis presented in Fig. 3c.\nE. Larger systems\nThe results presented in the main text are obtained using exact techniques. Such approach yields a deep under-\nstanding of the physical phenomena involved, however, the exact solution gets more and more complex as the number\nof degrees of freedom increases.10\n  \n 0 200 400 600 800 1000 1200 1400 (a)I\n0x1001x10-32x10-33x10-3\n(b)O\n012345678\n-5 0 510(c)SH\nE/(¯ hω0)\n 0 200 400 600 800 1000 1200 1400 (a)I\n0x1001x10-32x10-33x10-3\n(b)O\n012345678\n-5 0 510(c)SH\nE/(¯ hω0)\n 0 200 400 600 800 1000 1200 1400 (a)I\n0x1001x10-32x10-33x10-3\n(b)O\n012345678\n-5 0 510(c)SH\nE/(¯ hω0)\n 0 200 400 600 800 1000 1200 1400 (a)I\n0x1001x10-32x10-33x10-3\n(b)O\n012345678\n-5 0 510(c)SH\nE/(¯ hω0)\n 0 200 400 600 800 1000 1200 1400 (a)I\n0x1001x10-32x10-33x10-3\n(b)O\n012345678\n-5 0 510(c)SH\nE/(¯ hω0)\n 0 200 400 600 800 1000 1200 1400 (a)I\n0x1001x10-32x10-33x10-3\n(b)O\n012345678\n-5 0 510(c)SH\nE/(¯ hω0)\n 0 200 400 600 800 1000 1200 1400 (a)I\n0x1001x10-32x10-33x10-3\n(b)O\n012345678\n-5 0 510(c)SH\nE/(¯ hω0)\n 0 200 400 600 800 1000 1200 1400 (a)I\n0x1001x10-32x10-33x10-3\n(b)O\n012345678\n-5 0 510(c)SH\nE/(¯ hω0)\n 0 200 400 600 800 1000 1200 1400 (a)I\n0x1001x10-32x10-33x10-3\n(b)O\n012345678\n-5 0 510(c)SH\nE/(¯ hω0)\nFIG. 7. From left to right respectively: (a) inverse partecipation ratio Ir(b) overlap between the probability distributions of\nneighboring eigenstates Oand (c) Shannon entropy SHas functions of the energy, in units of \u0016h!0, for a system of N= 14\nparticles. The green lines indicate the corresponding value predicted by Random Matrix Theory.\nIn order to overstep such limitation and to corroborate the results presented in the main text, we have performed\nnumerical simulations using the time-dependent Density-Matrix-Renormalization Group (DMRG) method [63, 64] .\nIn particular, in order to describe the long-time dynamics, we implemented an approximation of the time evolution\nbased on the Taylor expansion of the evolution operator [44].\nFor the short time dynamics, the DMRG results for particle numbers N= 20toN= 60are presented in Fig. 2\nof the main text. In Fig. 6 we show the results for intermediate times. We first of all compare the time evolution\nof the discrete center of mass of the magnetization dD(t) =P\njj mj(t)with the corresponding quantity calculated\nexactly in the continuum case d(t)(panel a). We find that the two curves are proportional one to the other, i.e.\nd(t) =A(N)dD(t). The constant A(N)depends on the details of the harmonic trap and scales as /N\u00001=2(see inset\nin the panel a) of Fig. 6). In panel (b) of the same figure we show dD(t)for different number of particles, as a function\nof the rescaled time !0t=N1=4. Apart from a small variation of the details of the oscillations, due to the higher number\nof modes involved in the dynamics, we find that curves collapse onto each other, thus confirming that the universal\nscaling reported in the main text holds also for larger values of N.\nF. Further probes of chaoticity\nWe provide here further arguments on the breaking of the integrability of the model induced by the strong coupling\nexpansion to first order in 1=g. In the spirit of Ref. [49], we calculated other quantities to probe the presence of chaotic-\nity in the system. In particular, we focused on the eigenstates of the Hamiltonian and on their localization properties.\nIn Fig. 7 we show from left to right respectively the inverse participation ratio I=Ps\np=1jfp;jj\u00004, the overlap between\nconsecutive probability distribution, O=Ps\u00001\np=1jfp;jj2jfp;j+1j2and the Shannon entropy SH=\u0000Ps\np=1jfp;jj2lnjfp;jj2.\nWe compare the three quantities with the corresponding value predicted by the Random Matrix Theory, indicated by\nthe green lines. By comparing the results with the ones of Ref. [49], we see that the eigenstates shows some similarities\nin terms of statistical properties, but are not completely sparse as they would be in the integrable system."    },    {        "title": "1406.1192v2.SU_3__semiclassical_representation_of_quantum_dynamics_of_interacting_spins.pdf",        "content": "SU(3)semiclassical representation of quantum dynamics of interacting spins\nShainen M Davidson\u0003and Anatoli Polkovnikov\nDepartment of Physics, Boston University, Boston, MA 02215, USA\n(Dated: January 23, 2015)\nWe present a formalism for simulating quantum dynamics of lattice spin-one systems by \frst\nintroducing local hidden variables and then doing semiclassical (truncated Wigner) approximation\nin the extended phase space. In this way we exactly take into account the local on-site Hamiltonian\nand approximately treat spin-spin interactions. In particular, we represent each spin with eight\nclassicalSU(3) variables. Three of them represent the usual spin components and \fve others are\nhidden variables representing local spin-spin correlations. We compare our formalism with exact\nquantum dynamics of fully connected spin systems and \fnd very good agreement. As an application\nwe discuss quench dynamics of a Bose-Hubbard model near the super\ruid-insulator transition for a\n3D lattice system consisting of 1000 sites.\nRecent experiments in such areas as ultra-cold gases,\ncoupled atom-photon systems, ultrafast pump-probe\nspectroscopy in solids, and others have stimulated active\ntheoretical research in quantum dynamics of interacting\nsystems. While there has been signi\fcant progress in var-\nious directions, our understanding is still quite limited.\nPartly this is due to a lack of reliable and controllable\nnumerical methods. Perhaps the most powerful methods\nin equilibrium based on quantum Monte-Carlo are not\nreadily adopted to non-equilibrium systems. The other\navailable methods include: simulations based on exact di-\nagonalization; dynamical renormalization group and re-\nlated matrix product states methods [1]; dynamical mean\n\feld theory based methods [2, 3] mostly developed for\nfermions and only recently applied to bosons [4]; quan-\ntum kinetic equations and Keldysh diagrammatic tech-\nnique [5]; and phase space methods. The latter has re-\ncently become a major tool for studying dynamics of var-\nious systems, from interacting atomic clocks to the early\nUniverse [6{12]. These methods are based on mapping\nthe density matrix and operators into classical functions,\nwhich depend on phase space variables like coordinates\nand momenta, complex wave amplitudes, or classical spin\ndegrees of freedom.\nPhase space methods are very e\u000ecient for systems near\nthe classical or non-interacting limit, where the quan-\ntum evolution is well described by classical trajectories.\nThe main idea of the present work is that we can signi\f-\ncantly improve the accuracy of these methods by extend-\ning classical phase space by introducing new (hidden)\nphase space variables. In this way we can take into ac-\ncount local quantum \ructuations exactly and treat other\ndegrees of freedom approximately. We illustrate this idea\nby focusing on an example of spin-one coupled systems\nwhere, in addition to three variables representing the\nx;y;z components of the spin, we introduce \fve addi-\ntional hidden variables representing local spin-spin cor-\nrelations. We believe these ideas can be further extended\nand applied to study a large class of interacting systems\nboth in and out of equilibrium.\nBefore proceeding with our ideas we brie\ry review thephase space representation of quantum dynamics of in-\nteracting bosonic systems. For concreteness we focus on\nthe Wigner-Weyl quantization (see Supplementary Ma-\nterial for a brief overview). Any operator of a quan-\ntum system that we would normally represent through\na function of boson operators ^ aand ^aycan be mapped\nto a function over the classical phase space of (complex)\ncanonical variables \u000band\u000b\u0003. This function is called\nthe Weyl symbol of the operator and it is uniquely de-\n\fned. The Weyl symbol of the density matrix is known\nas the Wigner function [10, 15]. It plays the role of the\n(quasi)-probability distribution. In general, time evolu-\ntion of the Wigner function is given by a Fokker-Planck\nequation with high derivatives, which is hard to handle.\nHowever, near the classical limit or for non-interacting\nsystems one can use the so-called truncated Wigner ap-\nproximation (TWA) [6, 7, 10], where the Wigner func-\ntion, like a classical probability distribution, is conserved\non classical trajectories. Then the expectation value of\nany operator can be straightforwardly computed at any\nmoment in time:\nh^\n(t)i\u0019Z\nd~ \u000b0d~ \u000b\u0003\n0W(~ \u000b0;~ \u000b\u0003\n0)\nW(~ \u000bcl(t);~ \u000b\u0003\ncl(t));(1)\nwhereW(~ \u000b0;~ \u000b\u0003\n0) is the Wigner function representing the\ninitial state of the system, \n W(~ \u000b;~ \u000b\u0003) is the Weyl symbol\nof the operator ^\n, and the classical paths are found by\nsolving Hamilton's equations,\ni_\u000bcl=@HW\n@\u000b\u0003\ncl: (2)\nTWA becomes more accurate as the number of particles\nincreases. It is always exact when the Hamiltonian is\nnon-interacting, i.e. quadratic in boson operators.\nOne way to implement TWA with spin operators is to\nuse the Schwinger boson representation for spin, where\nthe spin operators are built from 2 creation/annihilation\npairs of boson operators contracted with the Pauli ma-\ntrices:\n^S\u000b= ^ay\ni1\n2\u001bij\n\u000b^aj: (3)arXiv:1406.1192v2  [quant-ph]  21 Jan 20152\nThe boson pairs \\inherit\" the commutation relations of\nthe matrices they are contracted with, in this case the\nspin algebra of SU(2). Representing the spin Hamilto-\nnian in this way, we can use the machinery of coherent\nstate TWA to approximate the dynamics (see [10] Sec-\ntion 3.3). Since TWA dynamics is exact for a Hamilto-\nnian that is quadratic in boson operators, it is also exact\nfor a system linear in spin operators.\nWe can similarly represent any SU(N) group of op-\nerators with commutation relations [ ^X\u000b;^X\f] =if\u000b\f\r^X\r\nusingNboson operator pairs and the Ndimensional ma-\ntricesTij\n\u000bwith the corresponding algebra:\n^X\u000b= ^ay\niTij\n\u000b^aj: (4)\nNote that any Hamiltonian that acts on a Hilbert space\nof dimension N(or indeed any Hermitian operator of di-\nmensionN) can be represented as a linear superposition\nof the generators of the SU(N) group and an identity\nmatrix. Again, the TWA dynamics of any Hamiltonian\nlinear inSU(N) variables will be exact. This statement\ncan be alternatively understood based on the linearity\nof the Heisenberg equations of motion as was recently\nexplored in Ref. [16].\nTaking the Wigner-Weyl transform of (22), we can de-\n\fne real number valued variables\nX\u000b=\u000b\u0003\niTij\n\u000b\u000bj; (5)\nand the classical equations of motion for the canonical\nvariables (2) will lead to dynamics for these variables as\n(see Supplementary Material for details)\n_Xcl\n\u000b=f\u000b\f\r@HW\n@Xcl\n\fXcl\n\r: (6)\nIn the case of the SU(2) group representing e.g. spin\nin a magnetic \feld the structure constants are given by\nf\u000b\f\r=\u000f\u000b\f\r, where\u000fis the fully antisymmetric tensor.\nThen it is easy to see that the equation above represents\nthe standard Bloch equations_~X=~X\u0002~B, whereH=\n\u0000~B~Xand~Bis a (possibly time-dependent) magnetic\n\feld.\nIf we have a single spin 1/2 degree of freedom then its\nHamiltonian can always be represented as a linear su-\nperposition of Pauli matrices (generators of the SU(2)\ngroup) and thus semiclassical dynamics are exact. Con-\nsider now the slightly more complicated situation of an\nisolated spin-one degree of freedom. A generic Hamil-\ntonian for spin-one can include interactions, i.e. terms\nnon-linear in spin operators. Just to be speci\fc consider\nthe interactions of the type ^S2\nzsuch that\n^HI=\u0000~B~^S+ (U=2)^S2\nz: (7)\nNote that both ~BandUcan explicitly depend on time.\nIf the interaction coupling Uis zero we are back to thesituation discussed before, where the Hamiltonian is a lin-\near superposition of SU(2) generators and semiclassical\ndynamics are exact. However, for \fnite interactions the\nsemiclassical approximation breaks down at long times.\nHowever, since spin-one operators exist in a 3 dimen-\nsional Hilbert space, we can make the semiclassical dy-\nnamics exact by enlarging the group to SU(3) by using\nthe 8 3-dimensional matrix generators of SU(3) as the\nmatricesT\u000bin (22). Those can be chosen to be Gell-\nMann matrices, but it is more convenient to use linear\ncombination such that the \frst three generators are the\nspin-one spin matrices [17], e.g.\nT1=0\nB@01p\n20\n1p\n201p\n2\n01p\n201\nCA (8)\nand thus ^X1=\u0010\n^ay\n1^a2+ ^ay\n3^a2+h:c:\u0011\n=p\n2 and so on (see\nSupplementary Material for details).\nBecause any Hermitian 3 \u00023 matrix can be expressed\nthrough a linear combination of SU(3) matrices (and the\nidentity) the interaction term also becomes linear in the\nSU(3) representation. For example, for our choice of\nSU(3) generators we have ^S2\nz= (2\u0000p\n3^X8)=3.\nThus the whole spin-1 Hamiltonian becomes linear\nin the generators of SU(3) and the equations of mo-\ntion (6) become exact. These equations can be inter-\npreted as an exact classical representation of quantum\ndynamics of a spin-one system in the eight-dimensional\nphase space spanned by the classical phase space vari-\nablesX1;:::;X 8[18].X1;X2;X3represent the three spin\ncomponents, while the remaining \fve variables represent\nnonlinear spin terms. These are e\u000bectively hidden vari-\nables in our approach. The advantage of our approach\nbecomes apparent when we start coupling spin-one sys-\ntems together and start doing approximations.\nNote that the eight equations (6) are not completely\nindependent: they satisfy constraints set by conservation\nof the Casimir operators:\nC1=X\n\u000bX2\n\u000b; C2=X\n\u000b\f\rd\u000b\f\rX\u000bX\fX\r; (9)\nwhered\u000b\f\rare the symmetric structure constants of the\nSU(3) group. Interestingly X2\n1+X2\n2+X2\n3, representing\nthe sum of squares of the classical spin components, is not\nconserved under SU(3) dynamics. There is no paradox\nhere: recall all operators including ^S2\nx;y;z are represented\nthrough linear combination of operators ^X\u000bso e.g.X2\n1\ndoes not have the simple physical interpretation of S2\nx\neven though X1does represent Sx.\nTo illustrate the di\u000berence between the \\naive\" SU(2)\nTWA and the new SU(3) TWA we consider the Hamilto-\nnian (7) with Bx=By= 0 andBz= 1. TheSU(2) and\nSU(3) Weyl symbols corresponding to this Hamiltonian3\ntime<Sx>\n0 10-11\nFIG. 1. Comparison of the dynamics of h^Sxifor a spin-one\nparticle initially pointed in the x-direction subjected to the\nHamiltonian1\n2^S2\nz\u0000^Sz. The dynamics are calculated with\nexact diagonalization (solid, blue), SU(3) TWA (dashed, red),\nandSU(2) TWA (dotted, yellow). (Color online.)\nare\n(HI)SU(2)\nW = (U=2)X2\n3\u0000X3\u00001=2;\n(HI)SU(3)\nW = (U=6)(2\u0000p\n3X8)\u0000X3: (10)\nNote that in the SU(2) case we chose the spin-one repre-\nsentation of the spin operators given by the \frst three op-\nerators of the SU(3) representation. The additional con-\nstant term\u00001=2 in theSU(2) Hamiltonian comes from\n(^X2\nk)W=X2\nk\u0000tr(T2\nk)=4. For concreteness we choose\nU= 1 and start with the spin pointing along the x-\ndirection and observe the expectation value of ^Sxas a\nfunction of time. In Fig. 1 we show comparison of the\nresulting exact dynamics with SU(2) andSU(3) TWA\napproximations. As expected the SU(3) TWA is exact\nwhile theSU(2) semiclassical dynamics are only accurate\nat short times. The di\u000berence comes from the fact that\nany interaction terms in the SU(2) case are represented\nby non-linearity while in the SU(3) case they are rep-\nresented by additional (hidden) variables, which in turn\nhave their own quite complex dynamics.\nNext let us consider a more complicated setup, where\nwe deal with a system of interacting spin-one degrees of\nfreedom such that the Hamiltonian becomes\n^H=X\nn^H(n)\nI+^HC (11)\nwhereH(n)\nIis the local spin-one Hamiltonian (7) describ-\ningn-th spin and\n^HC=\u0000JX\nn6=m(^Sn\nx^Sm\nx+^Sn\ny^Sm\ny): (12)\nWe have chosen a fully connected Hamiltonian to allow\nfor comparison of TWA and exact dynamics for larger\nsystem sizes.\nThe Weyl symbol of the coupling term is the same for\ntheSU(2) and the SU(3) representations because it doesnot involve local nonlinear spin-operators,\n(HC)W=\u0000JX\nn6=m(Xn\n1Xm\n1+Xn\n2Xm\n2): (13)\nFor multi-spin systems we do not use the exact Wigner\nfunction, which is de\fned and integrated over the coher-\nent state variables. Instead, we use a multivariate Gaus-\nsian distribution fn(Xn\n1;:::;Xn\nN2\u00001) for each site nand\nintegrate over the N2\u00001SU(N) variables:\nh^\n(t)i\u0019ZY\nndXnfn(~Xn)\nW(~Xcln(t)); (14)\nwhere the mean and covariance matrix for each fis\n\fxed by the quantum expectation values of the initial\nstate of the system, hXn\n\u000bifn=h^Xn\n\u000biandhXn\n\u000bXn\n\fifn=\nh(^Xn\n\u000b^Xn\n\f+^Xn\n\f^Xn\n\u000b)=2i(see Supplementary Material for\ndetails). We use this best Gaussian approximation for\ntwo reasons. First, the exact Wigner function will in\ngeneral have negative values, so the integration depends\non the cancellation of positive and negative contribu-\ntions, which numerically requires more sample points to\nconverge. Secondly, we numerically found that the best\nGaussian TWA results are consistently more accurate.\nFormally this Gaussian scheme is justi\fed if we increase\nthe spin size (proportional to the conserved value of the\nCasimir operator). For an initial correlated (not prod-\nuct) state one should use the multivariate Gaussian which\ncorrectly reproduces both local and non-local correlation\nfunctions likehXm\n\u000bXn\n\fi. For observables, instead of Weyl\nordering one can use direct quantum classical substitu-\ntion ^Xn\nm!Xn\nmbecause any onsite observable is linear in\n^Xand for linear operators this substitution is exact [10].\nIn Fig. 2 we show the dynamics of the spin \ructua-\ntionsh^S2\nziper site obtained by exact diagonalization, and\nSU(2) andSU(3) TWA. The system is initially prepared\nwith all spins pointing in the x-direction. We compare\nthe dynamics for a fully connected system for di\u000berent\nvalues of the coupling Jand for di\u000berent system sizes.\nAs the coupling is lowered and the on-site term in the\nHamiltonian becomes more dominant, the SU(3) TWA\nbecomes a better approximation, while the SU(2) be-\ncomes worse. When the on-site term is 5 times as dom-\ninant as the coupling term, the SU(3) TWA is indistin-\nguishable from exact quantum dynamics. As the system\nsize increases, and hence each site is connected to more\nsites, theSU(3) TWA dynamics approach exact quantum\ndynamics. Similarly to the SU(2) case,SU(3) TWA fails\nto describe quantum revivals, which occur later and later\nin time as the system size increases.\nAs a more practical example, we model the Bose-\nHubbard model using the e\u000bective Hamiltonian [19]\nHeff=U\n2X\ni(^Si\nz)2\u0000J\u0016nX\nhiji(^Si\nx^Sj\nx+^Si\ny^Sj\ny)\u0000\u0016X\ni^Si\nz;\n(15)4\nM=50,Jz=0.2\ntime<Sz2>\n0 500.30.5\nM=3,Jz=0.2\ntime<Sz2>\n0 500.30.5\nM=50,Jz=1\ntime<Sz2>\n0 500.20.5\nM=5,Jz=0.2\ntime<Sz2>\n0 500.30.5\nM=7,Jz=0.2\ntime<Sz2>\n0 500.30.5\nFIG. 2. Comparison of the dynamics of h^S2\nzifor a fully con-\nnected system with Mspins due to Hamiltonian (11) calcu-\nlated with exact diagonalization (solid, blue), SU(3) TWA\n(dashed, red), and SU(2) TWA (dotted, yellow). Initially\nall spins are pointed in the x-direction. On the left are the\nresults for various coupling strengths Jz, wherez=M\u00001\nis the connectedness (the inset in the bottom plot shows the\nsame plot for times 0 to 50). On the right are the results for\ndi\u000berent system sizes. (Color online.)\nwhere \u0016nis the mean particle density. This truncation of\nthe Hilbert space to three dimensions per site is accept-\nable in the vicinity of the Mott insulating state [20]. We\nuseSU(3) TWA to determine the dynamics of the order\nparameter \u001as=P\ni6=jh^S+\ni^S\u0000\nji=M2(representing super-\n\ruid density) for a 3D system with M= 103sites in\na cubic lattice with periodic boundary conditions. The\nresults are plotted in Fig. 3.\nFirst we quench from the Mott insulator phase, i.e. a\nFock state on each site (the ground state for J= 0).\nIn terms of the e\u000bective spin Hamiltonian, this corre-\nsponds to a product state of j^Sz= 0i. The dynamics\narise from an instantaneous quench to a \fnite coupling,\neitherJ\u0016nz=U = 0:2 orJ\u0016nz=U = 1. In each case, the\nsystem moves away from a pure Mott insulator state; for\na smaller coupling, there is some oscillation which is ab-\nsent for a larger coupling. The super\ruid density remains\nsmall, as a sudden quench leads to a high temperature\nstate which does not exhibit long range order [21].\nWe also show a quench from the super\ruid phase (the\nground state for U= 0), which in terms of the e\u000bec-\ntive spin Hamiltonian corresponds to a product state of\nj^Sx= +1i. When the system is quenched to J= 0, each\nsite precesses independently. Thus we can calculate the\ndynamics using exact diagonalization, SU(3) TWA, and\nSU(2) TWA. Since the on-site Hamiltonian can be lin-\nM=103,Jn-\nzU=0.2\ntUrs\n0 5000.007\nM=103,Jn-\nzU=1\ntUrs\n0 5000.05\ntUrs\n0 5001\ntUrs\n0 5001FIG. 3. The dynamics of the order parameter \u001asfor the e\u000bec-\ntive Bose-Hubbard model (15), calculated with SU(3) TWA.\nThe top row shows the results starting in the Mott insulator\nphase. The bottom row begins in the super\ruid phase: in the\nleft-hand plot, the system is quenched to J= 0, while in the\nright hand-hand plot the system is quenched to J\u0016nz=U = 0:1.\n(Color online.)\nearized in terms of SU(3) variables, the SU(3) TWA re-\nproduces the exact quantum dynamics, including quan-\ntum recurrences, while the SU(2) TWA decays. When\nwe instantaneously quench to J\u0016nz=U = 0:1, theSU(3)\nTWA still reproduces the oscillations of quantum recur-\nrences, damped by the coupling to the larger system.\nIn summary, we have introduced a semiclassical for-\nmalism for simulating the quantum dynamics of strongly\ninteracting coupled-spin systems. We have shown that\nby increasing the phase space and introducing new (hid-\nden) degrees of freedom one can partially account for\nlocal quantum \ructuations and signi\fcantly improve the\naccuracy of the semiclassical description of the dynam-\nics. We have argued and shown numerically that the\naccuracy of this method increases as we increase connec-\ntivity of the system. We have demonstrated numerically\nthat this method accurately reproduced results of quench\ndynamics of coupled spin-one systems in a broad range\nof parameters including the strong coupling regime. As\nanother illustration we analyzed quench dynamics across\nthe super\ruid-insulator transition in a three-dimensional\nBose-Hubbard model.\nWhile here we only presented results for SU(3) vari-\nables, this formalism can be straightforwardly applied\nto anySU(N) group (albeit with a larger phase space),\nwhereNis the dimension of a local Hamiltonian. Thus\nwe can use classical dynamics to exactly do local quan-\ntum dynamics which are linear in any SU(N) represen-\ntation. This should allow one to take into account quan-\ntum \ructuations within larger clusters and then use the\nTWA approximation to treat inter-cluster coupling. We\nare planning to analyze this possibility in a future work.\nAn important and open question is \fnding the optimal\nway of introducing hidden variables keeping their number5\nlarger than in the naive classical limit, yet much smaller\nthan the Hilbert space size.\nWe thank V. Oganesyan for the initial idea of in-\ncreasing phase space and A.M. Rey for sharing details\nof unpublished work. This work was supported by\nAFOSR FA9550-313 13-1-0039, NSF DMR-1206410, and\nthe Natural Sciences and Engineering Research Council\nof Canada.\n\u0003Corresponding author: shainen@bu.edu\n[1] U. Schollwock, Rev. Mod. Phys. 77, 259 (2005).\n[2] H. Aoki, N. Tsuji, M. Eckstein, and M. Kollar,\narXiv:1310.5329 (2013).\n[3] C. Trefzger and K. Sengupta, Physical Review Letters\n106, 095702 (2011).\n[4] H. U. R. Strand, M. Eckstein, and P. Werner,\narXiv:1405.6941 (2014).\n[5] A. Kamenev and A. Levchenko, Advances in Physics 58,\n197 (2009).\n[6] M. J. Steel, M. K. Olsen, L. I. Plimak, P. D. Drummond,\nS. M. Tan, M. J. Collett, D. F. Walls, and R.Graham,\nPhys. Rev. A 58, 4824 (1998).\n[7] P. Blakie, A. S. Bradley, M. Davis, R. Ballagh, and\nC. Gardiner, Advances in Physics 57, 363 (2008).\n[8] A. Altland, V. Gurarie, T. Kriecherbauer, and\nA. Polkovnikov, Phys. Rev. A 79, 042703 (2009).\n[9] J. Lancaster, E. Gull, and A. Mitra, Phys. Rev. B 82,\n235124 (2010).\n[10] A. Polkovnikov, Annals of Physics 325, 1790 (2010).\n[11] B. Opanchuk, R. Polkinghorne, O. Fialko, J. Brand, and\nP. D. Drummond, Annalen der Physik 525, 866 (2013).\n[12] A. M. Rey, A. Gorshkov, C. Kraus, M. J. Martin,\nM. Bishof, M. D. Swallows, X. Zhang, C. Benko, J. Ye,\nN. Lemke, and A. Ludlow, Annals of Physics 340, 311\n(2014).\n[13] C. Gardiner and P. Zoller, Quantum Noise , 3rd ed.\n(Springer-Verlag, Berlin Heidelberg, 2004).\n[14] D. Walls and G. Milburn, Quantum Optics (Springer-\nVerlag, Berlin, 1994).\n[15] M. Hillery, R. O'Connell, M. Scully, and E. Wigner,\nPhysics Reports 106, 121 (1984).\n[16] V. Galitski, Phys. Rev. A 84, 012118 (2011).\n[17] M. N. Kiselev, K. Kikoin, and M. B. Kenmoe, Europhys.\nLett.104, 57004 (2013).\n[18] That the dynamics of a spin-one system can be rep-\nresented by eight linear equations in classical variables\nshould not be surprising: the von Neumann equation is\nalso of this form, where the eight variables are the eight\nindependent elements of the 3 \u00023 density matrix.\n[19] E. Altman and A. Auerbach, Phys. Rev. Lett. 89, 250404\n(2002).\n[20] S. D. Huber, E. Altman, H. P. B uchler, and G. Blatter,\nPhysical Review B 75, 085106 (2007).\n[21] A. Polkovnikov, S. Sachdev, and S. M. Girvin, Physical\nReview A 66, 053607 (2002).6\nSupplementary Material\nTRUNCATED WIGNER APPROXIMATION\nPerhaps the best known phase space representation is\nbased on the Wigner-Weyl quantization [10, 15], where\nthe Weyl symbol of an operator is represented through\nthe partial Fourier transform. In the coherent state rep-\nresentation corresponding to second-quantized language\none de\fnes\n\nW(~ \u000b) =Zd~ \u0011d~ \u0011\u0003\n2h~ \u000b\u0000~ \u0011=2j^\nj~ \u000b+~ \u0011=2ie(~ \u0011\u0003~ \u000b\u0000~ \u0011~ \u000b\u0003)=2:\n(16)\nThe Wigner function is simply the Weyl transform of\nthe density matrix. There are other equivalent rep-\nresentations of the operators [13, 14], for example, P-\nrepresentation or Q-representation (also known as the\nHusimi representation). For our purposes it is not es-\nsential which of these representations is used but we will\nrefer to Wigner-Weyl quantization to be speci\fc. Within\nphase space methods one can describe expectation val-\nues of arbitrary operators as a standard average, where\nthe Wigner function W(\u000b;\u000b\u0003) plays the role analogous\nto the classical probability distribution:\nh\ni=Z\nd~ \u000bd~ \u000b\u0003W(~ \u000b;~ \u000b\u0003)\nW(~ \u000b;~ \u000b\u0003): (17)\nThe equation of motion for the Wigner function\nreads [10]\ni_W= 2Wsin\u0012\u0003c\n2\u0013\nHW; (18)\nwhere\n\u0003c=X\nj \u0000@\n@\u000bj\u0000 !@\n@\u000b\u0003\nj\u0000 \u0000@\n@\u000b\u0003\nj\u0000 !@\n@\u000bj(19)\nis the so-called symplectic operator and HWis the Weyl\nsymbol of the Hamiltonian. In general this partial-\ndi\u000berential equation is hard to solve. However, it be-\ncomes simple if one can expand the sin function to lead-\ning order in \u0003 c:\ni_W\u0019W\u0003cHW: (20)\nIn this limit the right-hand side becomes just the Pois-\nson bracket of the Wigner function and the Hamiltonian.\nTherefore the resulting equation is identical to the clas-\nsical Liouville equation for the probability distribution.\nWe note that the factor of icomes from using the coher-\nent state Poisson brackets (see Ref. [10] for details). In\nturn this Liouville equation locally conserves the prob-\nability along the classical trajectories described by thecorresponding equations of motion\ni_\u000bclj=@HW\n@\u000b\u0003\nclj; (21)\nin this case the Gross-Pitaevskii equations. The Wigner\nfunction simply describes the distribution of initial con-\nditions. This linearization of the equations of mo-\ntion (known as the truncated Wigner approximation\n(TWA) [6]) can be justi\fed near the classical limit, where\n\u0003cis small. For coherent states it is proportional to 1 =N,\nwhereNis the occupation of modes; in the coordinate-\nmomentum representation \u0003 is explicitly proportional to\n~; and, for spin systems \u0003 is inversely proportional to\nthe spinS. TWA also becomes exact for harmonic sys-\ntems, where the Hamiltonian HWdoes not contain terms\nhigher than quadratic in phase space variables. Indeed,\nin this case all terms containing \u00033\ncand higher will vanish\nbecause they contain at least three derivatives.\nSU(N)TWA\nFor a more detailed account of using speci\fcally SU(2)\nSchwinger bosons in TWA, see [10] Section 3.3.\nWe can write operators with SU(N) algebra using N\nboson operator pairs and the Ndimension matrix repre-\nsentation of the operators Tij\n\u000b:\n^X\u000b= ^ay\niTij\n\u000b^aj: (22)\nWe de\fne the classical SU(N) variables as the Wigner-\nWeyl transform of the SU(N) operators (recall the ma-\ntrices are traceless):\nX\u000b=\u000b\u0003\niTij\n\u000b\u000bj: (23)\nUsing this as a de\fnition, we can rewrite the symplectic\noperator (19) in terms of the classical SU(N) variables:\n\u0003c= \u0000@\n@X\u000bif\u000b\f\rX\r\u0000 !@\n@X\f: (24)\nIn TWA, the dynamics are approximated to follow clas-\nsical trajectories, so the classical SU(N) variables, as\nfunctions of the classical canonical variables, simply fol-\nlow the path determined by Hamilton's equation:\n@\n@tXcl\n\u000b=iHW\u0003cXcl\n\u000b (25)\n=f\u000b\f\r@HW\n@Xcl\n\fXcl\n\r; (26)\nwhere we have also rewritten HWin terms of classical\nSU(N) variables. We can see how this follows directly7\nfrom the classical equations of motion for the canonical\nvariables; if we multiply equation (21) by \u000b\u0003\niTij\n\u000b, and\nthe complex conjugate equation by Tji\n\u000b\u000bi(where the \u000b\nrepresent classical paths), their addition yields the time\ndependency of the SU(N) variables:\n@\n@tXcl\n\u000b=\u000b\u0003\niTij\n\u000b@\u000bj\n@t+@\u000b\u0003\ni\n@tTij\n\u000b\u000bj (27)\n=\u0000i\u000b\u0003\niTij\n\u000b@HW\n@\u000b\u0003\nj+i@HW\n@\u000biTij\n\u000b\u000bj: (28)\nTaking the derivative of equation (23) to get\n@\n@\u000bj=\u000b\u0003\niTij\n\u000b@\n@Xcl\u000b; (29)\nwe get\n@\n@tXcl\n\u000b=\u0000i\u000b\u0003\niTij\n\u000bTjk\n\f\u000bk@HW\n@X\f+i@HW\n@X\f\u000b\u0003\nkTki\n\fTij\n\u000b\u000bj\n(30)\n=\u0000i\u000b\u0003\ni[T\u000b;T\f]ij\u000bj@HW\n@Xcl\n\f(31)\n=f\u000b\f\rXcl\n\r@HW\n@Xcl\n\f: (32)\nSU(3)SCHWINGER BOSONS\nWe use a fundamental representation of SU(3) with\nthe following basis matrices (similar to [17]):\nT1=0\nB@01p\n20\n1p\n201p\n2\n01p\n201\nCA;T2=0\nB@0\u0000ip\n20\nip\n20\u0000ip\n2\n0ip\n201\nCA;\n(33)\nT3=0\n@1 0 0\n0 0 0\n0 0\u000011\nA;T4=0\n@0 0 1\n0 0 0\n1 0 01\nA; (34)\nT5=0\n@0 0\u0000i\n0 0 0\ni0 01\nA;T6=0\nB@0\u00001p\n20\n\u00001p\n201p\n2\n01p\n201\nCA;(35)\nT7=0\nB@0ip\n20\n\u0000ip\n20\u0000ip\n2\n0ip\n201\nCA;T8=0\nB@\u00001p\n30 0\n02p\n30\n0 0\u00001p\n31\nCA:\n(36)\nNote that these are normalized for spin-one, such that\ntr(TiTj) = 2\u000eij: (37)\nThe non-zero structure constants are\nf123=f147=f165=f246=f257=f367= 1; (38)\nf178=f286=p\n3; (39)\nf345= 2; (40)where the rest can be determined by permuting the in-\ndices (they are ant-symmetric under permutations).\nThe non-zero symmetric structure constants are\nd114=d125=d477= 1; (41)\nd136=d224=d237=d466=d567=\u00001; (42)\nd118=d228=d668=d778=1p\n3; (43)\nd338=d448=d558=d888=\u00002p\n3; (44)\nwhere the rest can be determined by permuting the in-\ndices (they are symmetric under permutations).\nSince the \frst three matrices have the algebra of\nSU(2), they can be used to construct Schwinger bosons\nthat can represent any magnitude of spin. We have the\nmapping\nT1=^Sx (45)\nT2=^Sy (46)\nT3=^Sz: (47)\nFor the special case of spin-one, we can construct any\noperator with all the appropriate algebra as a linear com-\nbination of the eight SU(3) matrices. The generators\nthemselves are related to spin-one operators as follows:\nT4= (Sx)2\u0000(Sy)2(48)\nT5= [Sx;Sy]+ (49)\nT6= [Sx;Sz]+ (50)\nT7= [Sy;Sz]+ (51)\nT8=1p\n3\u0000\n(Sx)2+ (Sy)2\u00002(Sz)2\u0001\n; (52)\nwhere the hats have been left o\u000b the spins to emphasize\nthat these equalities are only true for the spin-one spin\nmatrices, not the spin operators in general.\nFor example, we can represent the squares of the spins\nas\n(Sx)2=1\n3(21+p\n3\n2T8) +1\n2T4 (53)\n(Sy)2=1\n3(21+p\n3\n2T8)\u00001\n2T4 (54)\n(Sz)2=1\n3(21\u0000p\n3T8): (55)\nGAUSSIAN APPROXIMATE WIGNER\nFUNCTIONS\nUsing Gaussian distributions instead of the exact\nWigner functions does not necessarily worsen the approx-\nimation, since the di\u000berence between the exact Wigner\nfunction and the appropriately chosen Gaussian distri-\nbution approximation is of the same order in 1 =Sas the8\ncorrections to the TWA dynamics. Formally one can un-\nderstand this by requiring that expectation values of the\nspin variables ^X1;:::^X8and their variances agree with\nthose given by the Gaussian up to order 1 =S2, whereS\nis the spin size (proportional to the conserved Casimir\noperator as we discuss next). The Gaussian distribu-\ntion has a clear advantage in that it is always positive\nand thus easy to handle. Numerically we found that the\naccuracy of TWA with the Gaussian distribution is con-\nsistently better than that of the TWA with the exact\nWigner function.\nNote that the initial states we choose are all product\nstates, so the Wigner functions are simply a product of\nsingle site Wigner functions. We do this for simplicity;\nthe initial Wigner function can also represent an entan-\ngled initial state.\nIn the Gaussian approximation, then, the expectation\nvalue for an operator ^\n will be\nh^\ni\u0019ZY\nmY\nkdXm\nke\u0000(RklXm\nl\u0000\u0016k)2=2\u001b2\nk\np\n2\u0019\u001bk\nW;(56)\nwheremruns over the number of sites and kover the\neightSU(3) variables.\nThe expectation values of the means and symmetrized\ncorrelation functions, which we require to exactly match\nthose in the initial state, are given on each site by\ntr(^\u001a0^Ti) =ZY\nkdXke\u0000(RklXl\u0000\u0016k)2=2\u001b2\nk\np\n2\u0019\u001bkXi (57)\nand\ntr(^\u001a01\n2(^Ti^Tj+^Tj^Ti)) =\nZY\nkdXke\u0000(RklXl\u0000\u0016k)2=2\u001b2\nk\np\n2\u0019\u001bkXiXj;(58)\nwhereR;\u0016 and\u001bare free parameters of the Gaussian,\nwhich we determine from the initial conditions.\nFor example, for \u001a0=j^Sx= 1ih^Sx= 1j, we rotate with\nR=0\nBBBBBBBBBBBB@0 0 0\u00001\n20 0 0p\n3\n2\n0 0 0 0 0 0 1 0\n0 0\u00001p\n20 01p\n20 0\n01p\n20 01p\n20 0 0\n0 0 0p\n3\n20 0 01\n2\n0 01p\n20 01p\n20 0\n0\u00001p\n20 01p\n20 0 0\n1 0 0 0 0 0 0 01\nCCCCCCCCCCCCA(59)and have Gaussians determined by\n\u0016=0\nBBBBBBBBBB@0\n0\n0\n0\n1p\n3\n0\n0\n11\nCCCCCCCCCCA\u001b=0\nBBBBBBBBBB@1\n1\n1\n1\n0\n0\n0\n01\nCCCCCCCCCCA: (60)\nNote that alternatively one can write the multivariate\nGaussian distribution in a more conventional form\nf(X1;:::;X 8) =1p\n(2\u0019)8j\u0006je\u0000(Xi\u0000\u0016i)\u0006\u00001\nij(Xj\u0000\u0016j):(61)\nHere the vector \u0016jis the vector of means and \u0006 is the\ncovariance matrix.\nCASIMIR OPERATORS\nThe conservation of the classical form of the two\nCasimir operators is made plain when they are written\nin terms of the conjugate variables:\nC1=bX\n\u000b=1X2\n\u000b=4\n3\u0000\nj\u000b1j2+j\u000b2j2+j\u000b3j2\u00012; (62)\nC2=X\n\u000b\f\rd\u000b\f\rX\u000bX\fX\r=16\n9\u0000\nj\u000b1j2+j\u000b2j2+j\u000b3j2\u00013:\n(63)\nThe conservation of the Casimir operators is assured by\nconservation of the total number of Schwinger bosons;\nconservation of total number of Schwinger bosons is guar-\nanteed by the global U(1) symmetry of the Hamiltonian\n\u000bi!\u000biei\u0012which is imposed by only including conjugate\nvariables as they are combined in SU(3) operators. Note\nthat the Casimir operators become larger as we approach\nthe classical limit so the inverse of say C1determines the\nstrength of quantum \ructuations in the system."    },    {        "title": "1512.05673v3.Multi__Q__hexagonal_spin_density_waves_and_dynamically_generated_spin_orbit_coupling__time_reversal_invariant_analog_of_the_chiral_spin_density_wave.pdf",        "content": "Multi- Qhexagonal spin density waves and dynamically generated spin-orbit coupling:\ntime-reversal invariant analog of the chiral spin density wave\nJ. W. F. Venderbos\nDepartment of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA\n(Dated: August 16, 2021)\nWe study hexagonal spin-channel (\\triplet\") density waves with commensurate M-point propa-\ngation vectors. We \frst show that the three Q=Mcomponents of the singlet charge density and\ncharge-current density waves can be mapped to multi-component Q= 0 nonzero angular momen-\ntum order in three dimensions (3D) with cubic crystal symmetry. This one-to-one correspondence\nis exploited to de\fne a symmetry classi\fcation for triplet M-point density waves using the stan-\ndard classi\fcation of spin-orbit coupled electronic liquid crystal phases of a cubic crystal. Through\nthis classi\fcation we naturally identify a set of noncoplanar spin density and spin-current density\nwaves: the chiral spin density wave and its time-reversal invariant analog. These can be thought of\nas 3DL= 2 and 4 spin-orbit coupled isotropic \f-phase orders. In contrast, uniaxial spin density\nwaves are shown to correspond to \u000bphases. The noncoplanar triple- Mspin-current density wave\nrealizes a novel 2D semimetal state with three \ravors of four-component spin-momentum locked\nDirac cones, protected by a crystal symmetry akin to nonsymmorphic symmetry, and sits at the\nboundary between a trivial and topological insulator. In addition, we point out that a special class\nof classical spin states, de\fned as classical spin states respecting all lattice symmetries up to global\nspin rotation, are naturally obtained from the symmetry classi\fcation of electronic triplet density\nwaves. These symmetric classical spin states are the classical long-range ordered limits of chiral spin\nliquids.\nI. INTRODUCTION\nSpin-orbit coupling, an intrinsic property of electrons\nin solids, is at the root of many phenomena currently\nattracting a great deal of attention in condensed matter\nphysics. The topological insulators are a prime example\nof a novel material class of which spin-orbit coupling is\na key characteristic [1, 2]. The importance of spin-orbit\ncoupling is most clearly re\rected in the spin-momentum\nlocking of the celebrated Dirac surface states mandated\nby nontrivial bulk topology. In two dimensions graphene\nis a topological insulator in the presence of spin-orbit cou-\npling [3], however, in practice the spin-orbit interaction\nturns out to be too weak.\nWhereas the topological insulators have been success-\nfully described in terms of non-interacting electron band\ntheory, spin-orbit coupled Mott insulators [4] are mate-\nrials for which strong correlation e\u000bects are important.\nThe iridium-based oxides provide hallmark examples of\nspin-orbit coupled materials where electron interactions\nhave been proposed to give rise to intriguing phases of\nmatter such as Kitaev spin liquids [4, 5], magnetic-order-\ninduced Weyl semimetals [6], and quantum critical nodal\nnon-Fermi liquids [7]. Spin-orbit coupling is key, since it\nbreaks spin-rotation symmetry and thus allows, for in-\nstance, for Kitaev-type terms in the spin Hamiltonians\nof e\u000bective local moments describing these materials.\nIn both these cases, topological insulators and spin-\norbit coupled Mott insulators, the spin-orbit interaction\nis intrinsic and of relativistic origin. Spin-orbit coupling\ncan, however, also be dynamically generated in nonrela-\ntivistic systems by interactions. Instabilities of the Fermi\nliquid can lead to condensation of particle-hole pairs into\nnematic phases with anisotropic distortions of the Fermisurface [8, 9], possibly in combination with time-reversal\nsymmetry breaking [10], or into isotropic spin-orbit cou-\npled phases [11]. Such electronic orders belong to the\nclass of phases called electronic liquid crystals [8], which\nare quantum analogs of classical liquid crystals, as they\nexhibit symmetry breaking but remain metallic.\nIn this work, we show how aspects of these three phe-\nnomena involving spin-orbit coupling (i.e., dynamic gen-\neration of spin-orbit coupling, electron correlation, and\nnontrivial topology) come together in two-dimensional\nsystems with hexagonal symmetry. Speci\fcally, we study\ndensity wave formation, i.e., condensation of particle-\nhole pairs, at \fnite commensurate wave vectors associ-\nated with nested van Hove singularities. Our study com-\nprises a classi\fcation of density waves in the spin channel,\nreferred to as triplet density waves [12], building on pre-\nvious work which has focused on the singlet channel [13].\nOne of our main results is to show that multi-component\ndensity waves with nonzero (commensurate) wave vector\nin 2D can be mapped to and classi\fed as condensates\nof particle-hole pairs with nonzero angular momentum\nin three dimensions (3D). For the triplet case this estab-\nlishes a connection between the phenomenology of multi-\ncomponent spin density wave states and spin-orbit cou-\npled\u000band\fliquid crystal phases [11]. The latter are\nparticle-hole analogs of the AandBphases of super\ruid\n3He [14].\nVan Hove singularities connected by inequivalent com-\nmensurate wave vectors generically occur for electrons\nin simple hexagonal lattices such as the triangular and\nhoneycomb lattices when doped to band structure sad-\ndle points. Doped graphene is a notable example [15].\nThe van Hove singularities are located at the three cen-\nters of the Brillouin zone edges, i.e., the Mpoints (asarXiv:1512.05673v3  [cond-mat.str-el]  5 Mar 20162\nshown in Fig. 1), and the wave vectors connecting them\nare theM-point vectors themselves.\nA number of studies have addressed the e\u000bect of\ninteractions between the three \ravors of saddle point\nelectrons, predicting exciting unconventional correlated\nphases such as topological chiral superconductivity, as\nwell as Chern-insulating chiral spin density waves [16{\n30]. These works highlight the rich physics expected\nin a broad class of (doped) hexagonal materials, with\ndoped graphene as a concrete example. The purpose of\nthis paper is to propose a comprehensive classi\fcation\nfor density wave states that can arise when multi-\ravor\nsaddle-point electrons condense. Known phases such as\nthe chiral and uniaxial spin density wave are shown to\nbe natural products of such a classi\fcation. In addition,\nwe \fnd novel density wave states with topological quasi-\nparticle spectra and hidden order.\nWe now give a brief overview of the content of this\nwork and summarize the main results.\nOverview and main results\nIn this work, we present a symmetry classi\fcation of\nhexagonal triplet M-point order. At the heart of such\nclassi\fcation is the notion of extended point group sym-\nmetry. Extended point groups are crystal point groups\nsupplemented with those lattice translations that do not\nmap the enlarged unit cell, i.e., the unit cell de\fned by\nthe ordering vector of the density wave, to itself. Con-\nsequently, extended point groups provide a natural and\nsystematic way to study particle-hole condensation at \f-\nnite commensurate wave vector, as density waves can be\nclassi\fed in terms of extended point group representa-\ntions in the same way as angular momentum channels\nare labeled by point group representations [31, 32].\nIn previous work we have analyzed hexagonal lattice\nM-point order in the singlet channel using extended point\ngroup symmetry [13] and found a set of charge density ( s)\nwaves, in addition to a set of time-reversal odd charge-\ncurrent density ( d) waves. Both sets of orders correspond\nto nesting instabilities. On the basis of that analysis, here\nwe address the triplet variants of these orders. The tri-\nangular and honeycomb lattices will serve as examples of\nsystems with hexagonal symmetry to which our analy-\nsis and results apply, with an emphasis on the triangular\nlattice.\nA central theme of our study is dynamically gener-\nated spin-orbit coupling. By spin-orbit coupling here we\nmean the coupling of the spin and angular momentum\nof the particle-hole pairs in the condensed phase. The\nconcept of dynamically generated spin-orbit coupling is\nintroduced in more detail in the next section, where we\nconsiderQ= 0 condensation. Q= 0 condensation\nis caused by Pomeranchuk-type Fermi liquid instabili-\nties and gives rise to phases that exhibit (anisotropic)\nFermi surface distortions as a result of (rotational) sym-\nmetry breaking [8{11, 33]. In systems with hexagonalsymmetry, spin-orbit coupled phases of the general form\nh^ y\n\u001b(~k)^ \u001b0(~k)i=~\u0001(~k)\u0001~ \u001b\u001b\u001b0with~\u0001(~k) a linear com-\nbination of degenerate d-wave form factors (relevant at\nvan Hove doping), can be distinguished by total angu-\nlar momentum quantum numbers, as we explain in more\ndetail in the following. One of such orders, the d+id\n\f-phase [11, 33], is favored when nesting is weak [34].\nThe discussion of Q= 0d-wave orders, highlighting the\ncoupling of degenerate orbitals to spin, will set the stage\nfor the classi\fcation and study of triplet M-point order\nof the general form h^ y\n\u001b(~k+~M\u0016)^ \u001b0(~k)i=~\u0001\u0016(~k)\u0001~ \u001b\u001b\u001b0\n(\u0016= 1;2;3, see Fig. 1). As a \frst step, we will consider\nthe nesting instabilities at the Mpoints in the spin chan-\nnel, i.e., the triplet M-point instabilities. Based on the\nnesting instabilities and their symmetry properties, we\nwill derive and discuss three main results.\n(i)We show that the s-wave and d-wave nesting in-\nstabilities map to sets of L= 2 and 4 angular momenta,\nrespectively, transforming as partners of representations\nof the cubic group Oh. This result is established by con-\nstructing a mapping between elements of the extended\nhexagonal point group C000\n6vand elements of the cubic\ngroup, realizing an isomorphism between the two groups.\nUsing this mapping, we will demonstrate that coupling\ntheL > 0 orbitals to spin ( S), in the same spirit as\ntheQ= 0d-waves, de\fnes a symmetry classi\fcation of\nhexagonal triplet M-point order: total angular momen-\ntumJ=L+Sbecomes a symmetry label for density\nwaves. We argue that, from the perspective of symmetry\nand phenomenology, we can interpret distinct triplet M-\npoint orders as electronic liquid crystal ( \u000band\f) phases\nof a 3D Fermi liquid with cubic symmetry. In addition,\nwe present a dual interpretation in which electrons in\na cubic crystal with intrinsic orbital degrees of freedom\nspontaneously develop orbital order.\n(ii)Within the framework of the symmetry classi\f-\ncation, we identify two distinct spin-orbit coupled cu-\nbic crystal \f-phases, which are in correspondence with\nwhat we call scalar M-point density wave orders. They\nare scalar orders in the sense that they can be viewed\nas total angular momentum J= 0 terms. The \frst\noriginates from the set of charge density or swaves\nand corresponds to a full spin-rotation symmetry bro-\nken spin density wave state, the so-called chiral spin\ndensity wave [26, 27, 35, 36], associated with a gapped\nmean-\feld spectrum. The mean-\feld ground state is a\nChern insulator. The second is a time-reversal invariant\ntripletd-wave or spin-current state, breaking no symme-\ntries other than spin-rotation symmetry. We show that\nthe mean-\feld ground state is a symmetry-protected 2 D\nDirac semimetal [37], protected by symmetries closely re-\nlated to non-symmorphic crystal symmetry, and sits at\nthe boundary between a trivial and a topological insula-\ntor. Using simple symmetry arguments, we discuss how\nsymmetry breaking perturbations can drive the Dirac\nsemimetal into either a trivial or topological electronic\nstate. As such, the scalar (i.e., J= 0) triplet d-wave\nstate realizes a novel electronic phase, the dynamically3\ngenerated 2D Dirac semimetal.\n(iii)The symmetry classi\fcation of triplet M-point or-\nder can be used to obtain the symmetric spin states of a\ngiven (hexagonal) lattice. Symmetric spin states are clas-\nsical spin states that respect all symmetries of the crys-\ntal lattice up to a global rotation of all spins [38]. Such\nstates are relevant in the context of magnetic materials,\ni.e., systems described by pure spin model Hamiltonians,\nas well as materials in which itinerant carriers are coupled\nto (large) localized spins. We will demonstrate that the\nsymmetry classi\fcation provides a straightforward and\nconstructive derivation of symmetric spin states.\nTo summarize the organization of the paper, Sec. II\nwill introduce dynamically generated spin-orbit coupling.\nIn Sec. III, triplet M-point order is considered, by \frst\ndiscussing nesting instabilities and then proceeding to in-\ntroduce the mapping from hexagonal to cubic symmetry.\nUsing the mapping to de\fne the symmetry classi\fcation,\nwe identify and study two types of scalar triplet density\nwave states. Sec. IV aims at understanding the proper-\nties of the scalar triplet orders by focusing on low-energy\nelectronic degrees of freedom. In Sec. V, we point out the\nconnection to classical spin liquid states, and \fnally, in\nSec. VI we summarize and discuss the results presented.\nII.Q= 0TRIPLET d-WAVES\nIn case of hexagonal C6vsymmetry, the d-wave\nchannel is two-fold degenerate, with the d-wave or-\nbitals (dx2\u0000y2;dxy) transforming as partners of a two-\ndimensional representation. Therefore, if the leading in-\nstability is in the Q= 0d-wave channel, a general linear\ncombination of the two d-wave orbitals is allowed. This\nsituation has been shown to occur for the triangular and\nhoneycomb lattices electrons at van Hove doping, when\nnesting is weak [34, 39]. In the singlet channel only real\nsuperpositions of the two degenerate orbitals are allowed.\nIn the triplet channel, however, both real and complex\nor \\chiral\" linear combinations are possible.\nIn the triplet channel, real and chiral linear combina-\ntions of the dwaves are lattice analogs of the \u000band\felec-\ntronic liquid crystal phases with dynamically generated\n\\spin-orbit coupling\" in continuum Fermi liquids [11, 33].\nBoth in the \u000band\fphases spin-rotation symmetry is\nspontaneously broken. In the \u000bphase the spin order is\nuniaxial and spin rotation is only partially broken. Spa-\ntial rotations are broken due to d-wave nature of the or-\nbital angular momentum. In contrast, spin-rotation sym-\nmetry is fully broken in the \fphase, yet the \fphase is\nisotropic due to the coupling of spin and orbital angular\nmomentum: combined spin and spatial rotations leave\nthe state invariant. The isotropy can be thought of as a\nconsequence of two angular momenta adding to form a\nrotationally invariant singlet state.\nLet us show this more explicitly. We collect the two\nd-wave orbitals in a vector ~\u0015(~k) = (\u0015d1;\u0015d2) (explicit\nexpressions of lattice form factors can be found in Table Vof Appendix D) and then write the \u000bphase as\nh^ y\n\u001b(~k)^ \u001b0(~k)i=~\u0001\u000b\u0001~\u0015(~k)\u001b3\n\u001b\u001b0: (1)\nThe two-component order parameter ~\u0001\u000bis real, which\nfollows from the requirement of Hermiticity. As a result,\n~\u0001\u000bis a nematic order parameter breaking lattice rota-\ntional symmetry [33]. Due to the uniaxial spin polar-\nization along the zaxis, spin-rotation symmetry is only\npartially broken.\nInstead, the \fphase takes the form of a dot product\nofdwaves and spin, ~\u0015\u0001~ \u001b, given by\nh^ y\n\u001b(~k)^ \u001b0(~k)i= \u0001\f[\u0015d1(~k)\u001b1\n\u001b\u001b0+\u0015d2(~k)\u001b2\n\u001b\u001b0]:(2)\nThis is a chiral superposition of d-waves, as is easily seen\nby considering the o\u000b-diagonal elements h^ y\n#(~k)^ \"(~k)i=\n\u0001\f(\u0015d1+i\u0015d2)\u0018d+id. As a result of the spin-orbit\ncoupling~\u0015\u0001~ \u001bmomentum and spin are locked together in\nsuch a way that (properly chosen) simultaneous rotations\nmake the\fphase isotropic.\nAs a spoiler for the next section, the coupling of d-\nwaves and spin in the \f-phase can be obtained more for-\nmally by the standard recipe for addition of angular mo-\nmenta in a crystal. In hexagonal symmetry the d-waves\ntransform as the nematic doublet E2, and the spin com-\nponents (\u001b1;\u001b2) as the vector doublet E1. Addition of\nangular momenta is then given by E2\u0002E1=B1+B2+E1.\nThe \frst two terms are scalars, ~\u0015\u0001~ \u001band ^z\u0001~\u0015\u0002~ \u001b, which\nboth correspond to isotropic \f-phases.\nIt should be noted that even though the \u000bphase is\nnon-chiral whereas the \fphase is chiral, both break time-\nreversal symmetry since both are triplet d-wave states. In\naddition, it should be noted that unitary rotations in spin\nspace will yield equivalent states in case of both types of\nphases, since the normal state is spin-rotation invariant.\nWhen the dominant instability is in the Q= 0d-wave\nparticle-hole channel, it was shown that the chiral d+id\n\f-phase is favored [34]. This result mirrors the result\nfound in the Cooper channel, in which case chiral d+id\nsuperconductivity is favored [17, 18]. These \fndings are\nrooted in hexagonal symmetry and therefore apply to\nboth the triangular and honeycomb lattices. An expres-\nsion similar to Eq. (2), taking the sublattice structure\ninto account, can be written for the honeycomb lattice.\nWe summarize this section by noting that hexagonal\nQ= 0 triplet states are examples of dynamically gener-\nated spin-orbit coupling. In particular, the \f-phase of\nEq. (2) is a spin-orbit coupled scalar, analogous to total\nangular momentum J=L+S= 0 states arising from\naddition of two angular momenta LandS.\nIII. HEXAGONAL TRIPLET Q=MSTATES\nParticle-hole condensation at \fnite wave vector is ex-\npected when the Fermi surface is strongly or perfectly\nnested by that wave vector. For hexagonal lattices such4\nas the triangular and honeycomb lattices, Fermi surface\nnesting can occur at three inequivalent wave vectors ~M\u0016.\nIn this section, we study triplet density waves of hexago-\nnal lattices with commensurate ordering vectors ~M\u0016hav-\ning the general form\nh^ y\n\u001b(~k+~M\u0016)^ \u001b0(~k)i=~\u0001\u0016(~k)\u0001~ \u001b\u001b\u001b0; (3)\nwhere~\u0001\u0016(~k)\u0001~ \u001b\u001b\u001b0is the triplet version of the singlet\nterm \u0001\u0016(~k)\u000e\u001b\u001b0(suppressing sublattice indices). As a\n\frst step towards this goal, we derive all distinct nest-\ning instabilities using a simple algebraic approach. We\nestablish the symmetry of the nesting instabilities using\nextended point group representations. Based on that we\nde\fne a symmetry classi\fcation of triplet nesting insta-\nbilities in terms of spin-orbit coupling, in close analogy\nwith the spin-orbit coupled Q= 0 phases. To this end,\nwe introduce a mapping from hexagonal symmetry to cu-\nbic symmetry. We then analyze a speci\fc class of spin-\norbit coupled orders, the total angular momentum J= 0\norders, in more detail.\nA. Nesting instabilities\nThe analysis of hexagonal lattice nesting instabilities\nin the presence of spin is a straightforward extension of\nthe analysis without spin degree of freedom [13]. The\ngoal is to construct an algebra of the van Hove electrons\nand use that algebra to determine the nesting instabil-\nities. For lattices with hexagonal symmetry, the three\nvan Hove singularities are located at the inequivalent M-\npoint momenta ~M\u0016, shown in Fig. 1. The algebra of the\nM-point electrons can be expressed in terms of the van\nHove electron operators ^\b,\n^\b =0\nB@^ \u001b(~M1)\n^ \u001b(~M2)\n^ \u001b(~M3)1\nCA\u00110\nB@^ 1\u001b\n^ 2\u001b\n^ 3\u001b1\nCA; (4)\nwhere\u001blabels the spin. The full hexagonal van Hove\nalgebra is then de\fned by the bilinears ^\by\u0003i\u001bj^\b, where\n\u0003iis the set of Gell-Mann matrices, i.e., the generators\nof SU(3) (we also include the identity matrix), and \u001bj\nare Pauli matrices [generators of SU(2)] which act on the\nelectron spin. The matrices \u0003iact on the species index\n\u0016corresponding to ~M\u0016. The explicit form of the set of\neight \u0003iis given in Appendix A. We \fnd it convenient to\ngroup them into three subsets, denoted by ~\u0003a,~\u0003b, and\n~\u0003c.\nFrom the set of SU(3) generators we can form three\nSU(2) sub-algebras, each of which acts in the subspace\nof a pair of van Hove electrons. In this way the sub-\nalgebras are associated with the three ways of connecting\ntwoM-points, i.e., coupling the van Hove electrons. For\ninstance, the pair of van Hove electrons ^ 1and ^ 2is con-\nnected by wave vector ~M3. The SU(2) algebra speci\fed\nkxky~M1~M2~M3kxkyM01M02M03FIG. 1. (Left) Brillouin zone (BZ) of hexagonal lattices with\nspecialM-point momenta ~M\u0016. Red hexagon represents the\nFermi surface at Van Hove \flling; small black hexagon is the\nfolded BZ of the quadrupled unit cell. (Right) Folded BZ and\nhigh symmetry M0-points; red line segments are the folded\nFermi surface. Dashed lines are the unfolded BZ (black) and\nFermi surface (red).\nby (\u00031\na;\u00031\nb;\u00031\nc) acts in the subspace ( ^ 1;^ 2) and governs\nthe nesting instabilities at wave vector ~M3. The matrix\n\u00031\ncis diagonal and describes the population imbalance of\nvan Hove electrons ^ 1and ^ 2. All bilinears that do not\ncommute with ^\by\u00031\nc^\b give rise to nesting instabilities.\nThe non-commuting set of matrices is given by \u00031\naand\n\u00031\na\u001bj, corresponding to charge and spin density waves,\nrespectively, in addition to \u00031\nband \u00031\nb\u001bj, which corre-\nspond to charge currents and spin currents.\nNesting instabilities at the wave vectors ~M1and~M2\nare obtained by either explicitly constructing the van\nHove sub-algebras of the doublets ( ^ 2;^ 3) and ( ^ 3;^ 1),\nor more directly by using rotational symmetry. One \fnds\nthat the bilinears speci\fed by the matrices ~\u0003acorrespond\nto three degenerate charge density wave instabilities at\nthe three di\u000berent wave vectors, whereas the set ~\u0003bcorre-\nsponds to three degenerate charge-current density waves.\nIn the spin channel, the set ~\u0003a\u001bjdescribes spin density\nwaves and the set ~\u0003b\u001bjdescribes spin-current density\nwaves.\nLet us consider the symmetry properties of the nest-\ning instabilities. Charge density wave order given by ~\u0003a\nrespects time-reversal symmetry while orbital current or-\nder~\u0003bis odd under time reversal. Since spin is odd under\ntimereversal, the triplet orders ~\u0003a\u001bj(spin density waves)\nand~\u0003b\u001bj(spin-current density waves) are odd and even\nunder time reversal, respectively. As far as spatial sym-\nmetries are concerned, the components of ~\u0003atransform\nas the representation F1of the extended point group C000\n6v,\nand the components of ~\u0003basF2(see also Appendix D).\nTheF1representation describes s-wave form factors at\neach wave vector and the F2representation describes d-\nwave form factors, so we may use these symmetry labels\nto refer to charge and orbital current order (see also Ta-\nble I). In particular, we refer to ~\u0003a\u001bjas triplets-wave or\ntripletF1order, and similarly for ~\u0003b\u001bj. We assume ab-5\nsence of spin-orbit coupling in the normal state leading to\na full SU(2) rotation symmetry in spin space. Therefore,\nwe simply distinguish singlet and triplet instabilities. We\nthus obtain\nF1!~\u0003a(+);~\u0003a\u001bj(\u0000);\nF2!~\u0003b(\u0000);~\u0003b\u001bj(+); (5)\nwhere (\u0006) indicates even or odd under time-reversal.\nB. Global spin rotation and mapping to cubic L>0\nangular momentum phases\nThe purpose of this section is to establish the corre-\nspondence between symmetries of the density waves in\n2D and symmetries of angular momenta in 3D. This re-\nquires studying the action of symmetries of the hexagonal\nlattice on the density waves in more detail. To proceed,\nwe therefore explore the properties of the M-point rep-\nresentation of hexagonal symmetry. The M-point rep-\nresentation is simply given by the action of symmetries\non the three Mpoints (or, equivalently, the three den-\nsity wave components). It can be de\fned in terms of a\nvector~ v, the components of which are the linearly inde-\npendent functions describing modulations with M-point\npropagation vectors. It is given by\n~ v(~ x) =0\nB@cos~M1\u0001~ x\ncos~M2\u0001~ x\ncos~M3\u0001~ x1\nCA; (6)\nwhere~ xlabels the sites of the triangular Bravais lattice.\nThe e\u000bect of elementary translations T(~ ai) (i= 1;2;3) on\n~ vis given by~ v(~ x+~ ai) =Gi~ v(~ x). Here,~ a1;2= (1;\u0006p\n2)=2\nand~ a3= (\u00001;0). We de\fne the action of the group gen-\neratorsC6and\u001bvon~ vasX~ v(~ x) andY~ v(~ x), respectively,\nwhich are given by\nX=0\nB@0 1 0\n0 0 1\n1 0 01\nCA; Y =0\nB@0 0 1\n0 1 0\n1 0 01\nCA; (7)\n(more details in Appendix A). All matrices Giare diag-\nonal, an example is G1= diag(\u00001;\u00001;1). Consequently,\ntheO(3) matrices Gi,X, andYgenerate a representa-\ntion of the extended hexagonal point group C000\n6v.\nA general modulation of electron density with M-point\npropagation vectors can be written as a linear combina-\ntion of the three components of ~ v,~ w\u0001~ v(~ x), where~ w\nis an arbitrary vector. Except for certain special cases,\nsuch linear combinations will not respect lattice symme-\ntries. This is certainly true for the translations: Gi~ w6=~ w\nfor at least one Giand general ~ w. When spin is taken\ninto account, however, a more careful consideration of\nthe symmetries of M-point order is required. We now\ndemonstrate this based on the simplest case of M-pointmodulations described by single ~ w[i.e.,~ w\u0001~ v(~ x)], hence\navoiding unessential details such as sublattice structure.\nGeneral modulations of spin density, with spin repre-\nsented by the Pauli matrices ~ \u001b, are written in terms of a\nmatrixWas\n~ \u001b\u0001W\u0001~ v(~ x): (8)\nThis can be thought of as a vector ~ wfor each spin di-\nrection. The e\u000bect of a translation is then expressed as\n~ \u001b\u0001W\u0001Gi~ v(~ x), where the Giact onWfrom the right.\nNote that matrices acting from the right act on the M-\npoint indices \u0016, whereas matrices acting from the left of\nWact on the spin indices i(i.e.,~ \u001b\u0001W\u0001~ v=\u001biWi\u0016v\u0016).\nDue to the spin degree of freedom, it can occur that in\nsome cases multiplication by Gifrom the right is equal\nto multiplication by some O(3) matrixRifrom the left,\n~ \u001b\u0001W\u0001Gi~ v=~ \u001b\u0001RiW\u0001~ v: (9)\nIn this way, the action of translations is carried over to\nspin space, since Riis a global spin-rotation matrix. A\ntrivial example of this is the identity matrix, i.e. W=I,\nin which caseRi=Gi[40].\nThe key implication of the relation WGi=RiWis the\nequivalence of translations and spin rotations. Speci\f-\ncally, it implies that the term ~ \u001b\u0001W\u0001~ v=~ \u001b\u0001~W(with\n~W\u0011W\u0001~ v) can be made invariant under translations\nwith the help of a unitary matrix U2SU(2) associated\nwithR, expressed as\n~ \u001b\u0001~W=Uy~ \u001b\u0001(R~W)U: (10)\nThe unitary matrix Uacts on the matrix components\nof the\u001bj, and through the correspondence with Rim-\nplements the global spin rotation. Hence, even though\nM-point modulations would seem to break translational\nsymmetry, in certain cases (for certain W) they are in-\nvariant up to global spin rotation [26].\nThe signi\fcance of these global spin rotations is the\nresulting invariance of the Hamiltonian H(~k). A trans-\nlation combined with a global spin rotation leaves the\nHamiltonian invariant and this e\u000bective symmetry will\nlead to degeneracies of the spectrum.\nThe same argument applies to the point group gen-\neratorsC6and\u001bv, in which case the global spin rota-\ntion matrices are denoted as RXandRY. It is impor-\ntant, however, to distinguish proper and improper R,\nsince one can only associate an SU(2) matrix to a proper\nR. Improper rotations acquire an extra minus sign, i.e.,\n~ \u001b\u0001(R~W) =\u0000U~ \u001b\u0001~WUy, as a consequence of inversion.\nAgain taking the simplest case W=Ias an example, it\nis easy to see that all re\rections, which are composed of\nY, are improper. As a result, M-point spin density mod-\nulations of the form ~ \u001b\u0001~ v(~ x) are odd under re\rections.\nThese considerations demonstrate that in order to\nproperly analyze the symmetry properties of triplet M-\npoint density waves, it is important to take the global\nspin rotations into account.6\nThe expression ~ \u001b\u0001W\u0001~ v=~ \u001b\u0001~Win (8) bears a sugges-\ntive resemblance to the familiar spin-orbit coupling term\n~S\u0001~L, where~Sis spin and~Lis orbital angular momentum.\nIndeed, as is clear from the preceding discussion, ~ \u001b\u0001~W\nimplies an entangling of spatial symmetries and spin. We\nnow show how this resemblance can be formalized by ex-\nploiting the mapping between the extended hexagonal\npoint group C000\n6vand the cubic group Oh. We then ex-\nplain how such mapping provides a way to classify and\ndetermine the symmetry of triplet M-point density wave\norders.\nWe start by explicitly constructing the mapping from\nthe groupC000\n6vto the cubic group Oh. It is easy to see\nthat the matrices Gi,X, andY, associated with the gen-\nerators ofC000\n6v, areO(3) rotation matrices. The key ob-\nservation is that they correspond to rotations that leave a\ncube invariant. For instance, the matrices Giare two-fold\nrotation about the principal axes, and Xis a three-fold\nrotation about the body diagonal. As a result, the M-\npoint representation also realizes a representation of Oh.\nThis is, however, not a faithful representation, since the\ntwo-fold rotation C2=C3\n6inC000\n6vis represented by the\nidentity, i.e., X3= 1. A faithful representation is ob-\ntained by rede\fning the generator matrix of C6as\u0000X.\nIn this way, the twofold rotation in C000\n6vis mapped to\nthe inversion in Oh: (\u0000X)3=\u00001. This establishes a\none-to-one correspondence between elements of C000\n6vand\nOh. In particular, the representations of the two groups\nand their basis functions are in one-to-one correspon-\ndence. For instance, the cubic representation generated\nbyfGi;\u0000X;Ygis given byT1usymmetry, and it is simple\nto check that the representation generated by fGi;X;Yg\nisT2g. The three-component nature of representations\nofC000\n6vis rooted in nonzero ( M-point) linear momenta,\nwhereas the three-component nature of Ohrepresenta-\ntions is due to nonzero angular momenta. Angular mo-\nmentum orbitals with T1uandT2gsymmetry are given\nby (kx;ky;kz) and (kykz;kzkx;kxky).\nFrom this follows one of the main results of this pa-\nper: hexagonal density waves with M-point wave vectors\ncan be mapped to Q= 0 nonzero angular momentum\n(L6= 0) order in a three-dimensional cubic crystal. An-\nother way of stating it is that particle-hole pairs with\n\fnite momentum in 2D can be uniquely associated with\nparticle-hole pairs with nonzero angular momentum in\n3D. The latter belong to the class of liquid crystal phases.\nWhat is the implication of this correspondence? To an-\nswer that question, we \frst relate the C000\n6vrepresentations\nF1andF2, which describe the symmetry of the charge\nand charge-current density waves, to representations of\nthe cubic group. The former corresponds to T2g, whereas\nthe latter is generated by fGi;X;\u0000Ygand corresponds\ntoT1g. The cubic representations T2gandT1gdescribe\nthree-fold degenerate orbitals coming from L= 2 and\n4 angular momenta, respectively. Evidently, the charge\ndensityswaves, which have F1symmetry, are mapped\nto theT2gorbitals (kykz;kzkx;kxky). The charge-current\nd-density waves with F2symmetry are mapped to T1gor-Order type Extended group C000\n6v Cubic group Oh\nCharge\n(s-wave)F10\nB@1\n1\n11\nCAT2g0\nB@kykz\nkzkx\nkxky1\nCA\nCharge-current\n(d-wave)F2i0\nB@k2\n3\u0000k2\n1\nk2\n1\u0000k2\n2\nk2\n2\u0000k2\n31\nCAT1g0\nB@kykz(k2\ny\u0000k2\nz)\nkzkx(k2\nz\u0000k2\nx)\nkxky(k2\nx\u0000k2\ny1\nCA\nTABLE I. Table showing the correspondence of representa-\ntions of the extended hexagonal point group C000\n6vand the cu-\nbic groupOh. In addition, we list the angular momentum\nfunctions transforming as these representations. Hexagonal\nF1andF2order is shown to be of s- andd-wave type, re-\nspectively. Their cubic equivalents are composed of L= 2 (d\nwave) and L= 4 orbitals. Note that hexagonal d-waveM-\npoint order (i.e., F2) is imaginary and breaks time-reversal\nsymmetry. Here, k1;2;3=~k\u0001~ a1;2;3, where~ a1;2;3are the lattice\nvectors of the triangular Bravais lattice.\nbitals, which are listed in Table I.\nThe usefulness of the mapping between hexagonal and\ncubic symmetry becomes apparent once spin is consid-\nered. It can be stated as follows: spin-orbit coupled cu-\nbic liquid crystals, i.e., spin-rotation symmetry broken \u000b\nand\fphases formed from T1gandT2gorbitals, can be\nused to de\fne a classi\fcation of hexagonal triplet den-\nsity wave states. Since each spin-orbit coupled \u000band\f\nphase uniquely corresponds to a density wave state, a\nclassi\fcation of the former implies a classi\fcation of the\nlatter. This is a second key result of this paper and we\nnow demonstrate this in detail.\nThe spin-orbit coupled cubic liquid crystal phases with\nT1gandT2gangular momenta are direct analogs of the\n\u000b- and\f-phases discussed in the previous section. Con-\nsider \frst the \fphase case. Spin angular momentum ~ \u001b\ntransforms as T1gunder cubic symmetry. The \fphase\nis a spin-orbit coupled state for which only total angular\nmomentum is a good quantum number. Taking the T2g\norbitals as a \frst example, the good quantum numbers in\na cubic crystal are obtained from the product representa-\ntionT1g\u0002T2gwhich is decomposed as A2g+Eg+T1g+T2g.\nThis is the lattice analog of the addition of a pair of an-\ngular momenta L= 1 (orbital) and S= 1 (spin) in the\npresence of full rotational symmetry, giving total angular\nmomentum J=L+S= 0;1;2. The term A2gshould\nbe interpreted as the J= 0 case and corresponds to an\nisotropic\f-phase. Collecting the T2gorbitals in a vector\n~d(~k) = (kykz;kzkx;kxky), and denoting electron opera-\ntors of a 3D Fermi liquid (with cubic crystal anisotropy)\nby ^\u001f(~k), the spin-orbit coupled \fphase takes the form\nh^\u001fy\n\u001b(~k)^\u001f\u001b0(~k)i= \u0001\f~d(~k)\u0001~ \u001b\u001b\u001b0; (11)\nwhich is a 3D analog of Eq. (2). Other terms in the de-\ncomposition of T1g\u0002T2gcorrespond to multi-component\nanisotropic spin-orbit coupled liquid crystals.7\nThe terms in the decomposition are symmetry labels\n(quantum numbers) for spin-orbit coupled liquid crystals\nand provide a way to classify these phases, in the same\nway asJ= 0;1;2 classi\fes total angular momentum.\nThen, as a consequence of the one-to-one correspondence\nbetween the cubic orbitals and hexagonal M-point den-\nsity waves, this implies a symmetry classi\fcation of the\ntripletM-point density waves. Each spin-orbit coupled\nliquid crystal state can be mapped back to a unique den-\nsity wave state, and its symmetry follows directly from\nthe one-to-one correspondence. A key property of the\nclassi\fcation obtained in this way is that it manifestly\ntakes global spin-rotation invariance into account.\nTo illustrate this, let us consider the T2g\f-phase with\nA2gsymmetry. Comparing the character tables of Oh\nandC000\n6vwe \fnd that A2gsymmetry in the former corre-\nsponds toA2symmetry in the latter. Importantly, the\nA2representation is a translationally invariant represen-\ntation. This shows that global spin-rotation equivalence\nis manifest, since the only way to preserve translations\nforM-point order is to combine them with global spin ro-\ntations. Note that a spin density wave with A2symmetry\nbreaks re\rection symmetry.\nSimilar to the case of T2gorbitals, we can obtain the\n\f-phase in case of the T1gorbitals. Spin-orbit coupling\nleads to the decomposition\nT1g\u0002T1g=A1g+Eg+T1g+T2g: (12)\nHere theA1gterm corresponds to a J= 0\f-phase.A1g\nsymmetry implies full invariance under cubic symmetry\nand therefore full invariance under hexagonal symmetry.\nAs a result, the triplet spin-current d-density wave which\nuniquely corresponds to the A1g\f-phase breaks no spa-\ntial symmetries.\nIn addition to the \fphase we can also consider the ana-\nlog of the\u000bphase. For instance, the cubic \u000bphase with\nconstituent dwave orbitals ~d(~k) = (kykz;kzkx;kxky) is\nwritten as\nh^\u001fy\n\u001b(~k)^\u001f\u001b0(~k)i=~\u0001\u000b\u0001~d(~k)\u001bj\n\u001b\u001b0; (13)\nwhich should be compared to Eq. (1). Here, jis a given\ndirection in spin space, for instance, the global zaxis,\nmaking it a uniaxial spin ordered state with only par-\ntially broken spin-rotational symmetry. We will see in\nthe following that the \u000bphase corresponds to a uniaxial\nspin density wave.\nLet us summarize the main result of this section. By\nestablishing a one-to-one correspondence between the cu-\nbic groupOhand the hexagonal group C000\n6v, we have refor-\nmulated hexagonal M-point order associated with nest-\ning instabilities in terms of electronic liquid crystal states\nwith nonzero angular momentum in a cubic crystal. Cou-\npling spin and orbital angular momentum allowed us to\nassign a symmetry label to hexagonal triplet ordering in\na way that gives the correct symmetry of the orders.\nSpeci\fcally, we obtain two special triplet density wave\nstates (\\\fphases\") labeled by nondegenerate representa-\ntions and transforming as scalars. These can be viewedas isotropic total angular momentum J= 0 states. Due\nto this, in what follows we will refer to these orders as\nscalar triplet states, or alternatively a scalar spin-orbit\ncoupled states. In the remainder of this section we focus\non these two scalar triplet states with high symmetry and\nstudy them in more detail based on their realization on\nthe triangular and honeycomb lattices.\nC.s-wave triplet states: Spin density waves\nTriplets-wave states are the spin density waves de-\nrived from the F1representation and correspond to the\nnesting instabilities given by ~\u0003a\u001bjin Eq. (5). The sym-\nmetry classi\fcation of triplet swaves gives rise to a scalar\ntriplet state with A2symmetry, which is the T2g\fphase\nmapped back to a density wave state. Here, we con-\nstruct this scalar order explicitly for both the triangu-\nlar and honeycomb lattices and study its properties. In\nboth cases, the scalar order is obtained by pairing each\nM-point component ~M\u0016with a spin component \u001bj(i.e.,\nPauli matrix). This is schematically shown in Fig. 2. For\nthe triangular lattice, this directly leads to\nh^ y\n\u001b(~k+~M\u0016)^ \u001b0(~k)i= \u0001A2\u001b\u0016\n\u001b\u001b0: (14)\nIn case of the honeycomb lattice the components of F1or-\nder are speci\fed by two vectors ~ wAand~ wBwhich collect\nthe order-parameter components for each of the two sub-\nlattices (see Ref. 13 for details). To construct the scalar\ntriplet state, we pair each component with a distinct spin\ncomponent, leading to\nh^ y\ni\u001b(~k+~M\u0016)^ j\u001b0(~k)i= \u0001A2w\u0016\ni\u000eij\u001b\u0016\n\u001b\u001b0: (15)\nThe vectors ~ wAand~ wBare given by ~ wA= (\u00001;\u00001;1)\nand~ wB= (1;\u00001;\u00001).\nThese two triangular and honeycomb triple- Mspin\ndensity wave states are examples of non-coplanar spin\norder. In fact, these spin density waves, which we have\nderived from symmetry principles here, are nothing but\nthe chiral spin density waves found both in itinerant clas-\nsical magnets and mean-\feld Hubbard model calculations\non the triangular [26, 41{43], honeycomb [27, 29, 44]\nand kagome lattices [30, 35, 36]. The chiral spin den-\nsity wave was also found in a honeycomb Hubbard\nmodel study using advanced quantum many-body tech-\nniques [18, 19, 23].\nChiral spin density waves owe their name to the prop-\nerty of having nonzero chirality \u0014, de\fned as \u0014=~\u00011\u0001\n~\u00012\u0002~\u000136= 0 [where ~\u0001\u0016\u0018P\nkh^ y(~k+~M\u0016)~ \u001b^ (~k)i=N].\nThe chiral spin density waves of (14) and (15) were shown\nto induce a full spectral gap and the mean-\feld ground\nstate is a Chern insulator with spontaneous quantum Hall\n(QH) e\u000bect [26, 27]. This is consistent with A2symme-\ntry, i.e., the breaking of all re\rections, and broken time-\nreversal symmetry caused by non-coplanar spin order.\nThe mean-\feld spectra of these scalar s-wave states\nare presented in Fig. 3. They show the spectral gap at8\n~M1~M2~M3~M1~M2~M3\u00003\u00002\u00001\u00003\u00003\u00003↵\u0000phase(J= 0)phase\nFIG. 2. Schematic representation of F1s-wave triple- M\norder. (Left) Each order parameter component (i.e., M-point)\nis associated with an s-wave orbital and the same spin matrix\n\u001b3giving a uniaxial state. This state is related to the cubic\n\u000bphase of Eq. (13). (Right) Same as on the left, except\nthat eachM-point is associated with a di\u000berent spin matrix,\nresulting in the chiral state of (14) and (15). The chiral states\nare related to the cubic \f-phase of Eq. (11).\nvan Hove \flling. Note that the spectra exhibit a manifest\ntwo-fold degeneracy, i.e., each band is fully two-fold de-\ngenerate. This can be traced back to the e\u000bective trans-\nlation invariance of the chiral spin density wave: trans-\nlations are preserved when followed by a global rotation.\nAs a consequence of A2symmetry there can be no fur-\nther degeneracies. To obtain the mean-\feld spectra we\nused a tight-binding mean-\feld Hamiltonian H0+H\u0001,\nwhereH0contains a nearest-neighbor hopping t= 1 and\nH\u0001contains the mean-\felds de\fned above.\nThe scalar triplet s-wave states are the spin density\nwaves obtained from mapping the T2g\f-phase back to a\ndensity wave. A natural question then is: What is the\ndensity wave state corresponding to the T2g\u000b-phase of\nEq. (13)? Clearly, this must be a uniaxial spin density\nwave, so each ( M-point) component is associated with\nthe same spin matrix (see Fig. 2). Let us consider a\nparticular \\ \u000bphase\", where each component has equal\namplitude ~\u0001\u000b\u0018\u0001\u000b(1;1;1). In the spin density wave\nlanguage this is a triple- Muniaxial spin density wave\ngiven by\nh^ y\n\u001b(~k+~M\u0016)^ \u001b0(~k)i= \u0001 uniaxial\u001b3\n\u001b\u001b0; (16)\nwith uniform order parameter magnitude \u0001 uniaxial but\n\u0014= 0. This state is the uniaxial spin density wave of\nRef. 29.\nThe connection between the uniaxial triplet states and\nthe chiral triplet states has been demonstrated from the\nperspective of their mean-\feld spectra [30]. The former\nare semi-metallic with a (spin-\fltered) quadratic band\ncrossing at the reduced zone center [29]. Smoothly de-\nforming the uniaxial spin state so as to give it nonzero\nspin chirality \u0014gaps out this quadratic band crossing\npoint and results in a state adiabatically connected to the\ngapped scalar spin-orbit coupled state of Eq. (14) [30].\nWe thus \fnd that by mapping the cubic spin-orbit cou-\npled liquid crystal phases back to density wave states, we\nare naturally led to the chiral and uniaxial spin density\n-3.0-2.5-2.0-1.5-1.0-0.5 0\n-6-5-4-3-2-1 0 1 2 3\n\u0000K0+K0\u0000M0\u0000M01M02K0\u0000K0+\u0000K0+K0\u0000M0\u0000Energy (t)FIG. 3. . Mean-\feld spectra of the scalar triplet s-wave states\nof the triangular lattice (left) given in Eq. (14), and of the\nhoneycomb lattice (right) given in Eq. (15). In both cases\nwe show spectra for \u0001 A2= 0:25. The inset on left shows\nthe reduced BZ with the path along which bands are plotted.\nNote that all bands are doubly degenerate, as explained in\nthe text. In case of the honeycomb lattice (right) we only\nshow the lower half of the spectrum, i.e., up to E= 0. At\n\fllingsn= 3=4 (triangular) and n= 3=8 (honeycomb) the\nmean \feld ground state is insulating and has nonzero Chern\nnumber.\nwave states. The upshot of this mapping is that their\nsymmetries are made transparent.\nD.d-wave triplet states: Spin-current density\nwaves\nThe triplet d-wave states have d-wave form factors as-\nsociated with each of the three ordering components ~M\u0016.\nAs a consequence, they are imaginary, preserve time-\nreversal symmetry, and are described by the nesting in-\nstability matrices ~\u0003b\u001bjin Eq. (5). We found that the\nscalar triplet order constructed from d-wave components\nhasA1symmetry: it is the T1g\fphase mapped back to\na density wave state.\nIt is obtained by pairing each d-wave component at\nwave vector ~M\u0016with a di\u000berent spin component \u001bj. In\ncase of the triangular lattice, writing the condensate in\nterms of \u0001 \u0016(~k) ash^ y\n\u001b(~k+~M\u0016)^ \u001b0(~k)i= [\u0001\u0016(~k)]\u001b\u001b0, we\n\fnd\n\u00011(~k) =i\u0001A1(cosk3\u0000cosk1)\u001b1;\n\u00012(~k) =i\u0001A1(cosk1\u0000cosk2)\u001b2;\n\u00013(~k) =i\u0001A1(cosk2\u0000cosk3)\u001b3: (17)\nHere, \u0001A1is a real order parameter and ki=~k\u0001~ aiwith\n~ aithree (triangular) lattice vectors related by three-fold\nrotations. These triplet density waves preserve time-\nreversal symmetry, since the combination of complex con-\njugation and spin \rip leaves the state invariant. In that\nsense, it represents genuine dynamically generated spin-\norbit coupling and is therefore the time-reversal invariant\nanalog of the chiral spin density wave.\nWe note that this is di\u000berent for the cubic equivalent\nof the scalar hexagonal d-wave state, i.e., the \fphase of\nL= 4T1gorbitals. Collecting the T1gorbitals listed in9\n\u0000K0+K0\u0000M0\u0000M01M02K0\u0000K0+\u0000K0+K0\u0000M0\u0000Energy (t)-6-5-4-3-2-1 0 1 2 3\n-3.0-2.5-2.0-1.5-1.0-0.5 0\nFIG. 4. . Mean-\feld band structure of the scalar triplet d-\nwave state of Eq. (17) (left), and the equivalent state on the\nhoneycomb lattice (right). Inset on the left shows the path\ntaken in the reduced BZ. On the left we plot \u0001 A1= 0:25\n(black) and \u0001 A1=\u00000:25 (red), whereas on the right we plot\n\u0001A1=\u00060:15 (black/red). All bands are two-fold degenerate.\nThe key feature of the d-wave band structures shown here is\nthe symmetry protected degeneracy at the M0points, and the\nresulting the Dirac semimetal mean-\feld state.\nTable I in the vector ~ g(~k), the cubic L= 4\fphase takes\nthe form\nh^\u001fy\n\u001b(~k)^\u001f\u001b0(~k)i= \u0001\f~ g(~k)\u0001~ \u001b\u001b\u001b0: (18)\nHermiticity requires \u0001 \fto be real, implying that the\ncubic spin-orbit coupled \fphase and the scalar M-point\nd-wave state are equivalent with respect to all spatial\nsymmetries, yet di\u000ber with respect to time reversal.\nLet us study the mean-\feld spectrum of the density\nwave state of Eq. (17), which is shown in Fig. 4. We \frst\nnote a full two-fold degeneracy of each band, as was the\ncase for the scalar s-wave state. The reason for the de-\ngeneracy is the same: translations (in combination with\nspin rotations) are good symmetries. In the present case,\nhowever, the presence of both time-reversal symmetry\nand inversion also mandates a two-fold degeneracy, which\nwill therefore be preserved even if translations are bro-\nken. We further observe that the spectrum depends on\nthe sign of \u0001 A1, as the black (red) curves correspond\nto positive (negative) sign. For \u0001 A1>0 (black bands),\nwe \fnd that the mean-\feld ground state is a semimetal,\nwith linearly dispersing nodal points located at the M0\npoints of the reduced BZ. Due to the two-fold degener-\nacy of each band, the Dirac nodes come in pairs at each\nM0point, giving rise to three \ravors of four-component\nDirac nodes.\nThese nodal points are protected by crystal symme-\ntries, and as a result the semi-metallic mean-\feld state\nis a symmetry-protected Dirac semimetal in two dimen-\nsions [37, 45]. We show this explicitly in the next section.\nAs such, the scalar triplet d-wave state with A1symmetry\nshould be contrasted with graphene [15]. Graphene has\nfully spin-degenerate Dirac nodes which can be gapped\nby symmetry-preserving spin-orbit coupling [3]. In con-\ntrast, in the present case the mean-\feld Dirac quasipar-\nticles can only become massive by breaking symmetries,\nleading to either a trivial insulating state or a topological\ninsulator. Therefore, the Dirac semimetal state originat-\ning from scalar triplet d-wave state sits at the boundary\n~M1~M2~M3\ndx23\u0000x21dx21\u0000x22dx22\u0000x23~M1~M2~M3\ndx23\u0000x21dx21\u0000x22dx22\u0000x23\u00003\u00003\u00003\u00002\u00001\u00003\n↵\u0000phase(J= 0)phaseFIG. 5. Schematic representation of F2d-wave triple- Mor-\nder. TheM-points are associated with d-wave orbitals rotated\nby\u00062\u0019=3 with respect to each other. (Left) Each M-point is\nassociated with the same spin matrix giving a uniaxial QSH\nstate. (Right) Each M-point is associated with a di\u000berent spin\nmatrix, leading to the scalar triplet d-wave state with Dirac\nsemimetallic spectrum. The pairings on the left and right\nmay be interpreted as cubic \u000band\fphases, respectively.\nbetween a trivial and topological insulator [37].\nThe symmetry protected 2D Dirac semimetal is a\ngeneric feature of triplet M-point order, since it is rooted\nin hexagonal symmetry. This is con\frmed by Fig. 4,\nwhich shows the mean-\feld spectrum of the honeycomb\nlattice scalar triplet d-wave state. The key characteris-\ntics of the honeycomb lattice spectrum are identical to\nthe triangular lattice spectrum, notably the Dirac nodes\nat theM0points.\nHexagonal lattice triplet d-wave order is schematically\nshown and summarized in Fig. 5. On the right side, each\nwave vector ~M\u0016is paired with its corresponding d-wave\nform factor and a di\u000berent spin matrix. This depicts spin-\norbit coupling. On the left each wave vector is paired\nwith itsd-wave form factor, however, each wave vector\ncarries the same spin. We refer to this as uniaxial order,\nin analogy with s-wave order in Fig. 2. The uniaxial\norder corresponds to an L= 4 cubic\u000bphase, similar to\ntheL= 2\u000bphase of Eq. (13). The \u000b-phase is interesting\nin its own right, since it corresponds to a quantum spin\nHall (QSH) phase [3, 46]. It can be viewed as two copies\nof ad-wave Chern insulator, one for each spin species\nwith opposite sign.\nIV. LOW-ENERGY PROPERTIES OF\nHEXAGONAL TRIPLET ORDERS\nThe aim of this section is to give a more elaborate\nanalysis of the spectral properties of states discussed in\nthe previous section. In particular, we examine to what\nextent the characteristics of the mean-\feld ground state\ncan be understood from a low-energy description. Such\na description is independent of a given lattice model and\ntherefore helps to put the result on a more general foot-\ning. First, we study the lifting of energy level degenera-\ncies at the Mpoints where the van Hove electrons are\nlocated. We then derive the low-energy Dirac theory of10\nthe nodal degeneracies that arise in the triplet d-wave\nstate of Eq. (17).\nA. Electrons at the Mpoints\nWe start from the electron operator ^\b of Eq. (4) and\nconsider the action of symmetries on ^\b. The action of\nthe symmetry group C000\n6von the three \ravors of M-point\nelectrons is given by the M-point representation matri-\ncesfGi;X;Ygintroduced in the previous section (see\nalso Appendix A). Speci\fcally, for the symmetry group\ngenerators we have\nT(~ a1) :^\b!G1^\b;\nC6:^\b!X^\b\n\u001bv:^\b!Y^\b: (19)\nDensity wave ordering is expressed in terms of fermion\nbilinears, ^\by\u0003i\u001bj^\b, as discussed in the beginning of the\nprevious section.\nIn case of singlet (spin-rotation invariant) order, i.e.\n^\by\u0003i^\b, the transformation properties of the M-point\nelectrons can be used to show that the set of Gell-Mann\nmatrices~\u0003ahasF1symmetry and the set ~\u0003bhasF2sym-\nmetry [13]. Based on that, we found that the symmetry\nof triple-Morder, ordering of each of the M-point com-\nponents simultaneously with equal amplitude, is A1for\nthe former and A2for the latter. The energy levels of\ntriple-Morder are given by the two Gell-Mann matrices\n\u00032\ncand \u00031\nc, respectively, in the corresponding eigenbasis.\nThe e\u000bect of spin structure is best accounted for by\nexplicitly distinguishing the two types of triplet order,\nuniaxial and spin-orbit coupled, and analyzing them sep-\narately. In the \frst case, that of uniaxial triplet order,\nthe analysis is a straightforward extension of singlet or-\nder [13]. In the second case, that of scalar spin-orbit\ncoupled order with full SU(2) symmetry breaking, the\nanalysis of energy level splittings at the Mpoints is more\nsubtle and requires the notion of double groups. It turns\nout that, as a consequence of the intimate connection to\ncubic symmetry, energy levels are governed by the double\ncubic group, as we will explain in what follows.\nLet us \frst focus on the uniaxial triplet orders. To\nbe speci\fc, we take the spin polarization axis to be the\nzaxis.M-point electron bilinears can then be written\nas a simple product of Gell-Mann matrices and \u001b3. In\nparticular, the two bilinears describing uniaxial triple- M\norder are given by \u00032\nc\u001b3(uniaxial spin density wave) and\n\u00031\nc\u001b3(uniaxial orbital spin currents) [47]. As a result,\nthe analysis of the spin-rotation invariant case e\u000bectively\napplies to each spin sector separately. For the uniaxial\nspin density waves of Eq. (16), this implies a two-fold\ndegeneracy in each spin sector. However, due to the rel-\native sign di\u000berence ( \u0018\u001b3), these two-fold degenerate\nlevels in each spin sector are split with respect to each\nother. In addition, there are two non-degenerate levels,one for each spin species. Importantly, both spin-\fltered\ntwo-fold degeneracies constitute a spin-\fltered quadratic\nband touching (QBT) protected by rotational symme-\ntry and an e\u000bective time-reversal symmetry [29, 30, 48].\nThis directly follows from the symmetry of triple- Mor-\nder [13, 49]. Furthermore, since such argument only re-\nlies on symmetry, it proves that the \\half-metal\" state\nof Ref. 29, i.e., a metal with fully spin-polarized Fermi\nsurface electrons, is a generic feature of uniaxial M-point\nspin density waves.\nThe gap matrix \u00031\nc\u001b3, which corresponds to orbital\nspin currents, leads to a double degeneracy of all three\nenergy levels of \u00031\nc: each level occurs once for each spin\nprojection. The matrix \u00031\ncimplies time-reversal symme-\ntry breaking, but in combination with \u001b3time-reversal\nsymmetry is preserved. Moreover, since the gap matrix\n\u00031\nccorresponds to a spontaneous QH phase, \u001b3promotes\nthe ground to a QSH phase [3, 46].\nNext, we consider the case of spin-orbit coupled scalar\norder. Contrary to uniaxial order, spin-orbit coupled or-\nder does not decouple into two separate spin sectors. For\ninstance, scalar s-wave order of Eqs. (14) and (15) is writ-\nten in terms of van Hove electron bilinears as ~\u0003a\u0001~ \u001b. As a\nresult, one needs to consider the combined e\u000bect of sym-\nmetries on spin and M-point degrees of freedom. We now\nshow that degeneracies in the subspace given by ^\b can\nbe derived by exploiting the mapping between hexagonal\ntripletM-point order and cubic orbital order.\nTo demonstrate that the energy levels of van Hove\nelectrons ^\b are e\u000bectively governed by the cubic dou-\nble group, we construct an operator ^\u0007 so that its orbital\ndegree of freedom transforms in the same way under Oh\nas^\b underC000\n6v. The operator ^\u0007 is given by the T2g\norbitals, ^\u0007 = ( ^ yz\u001b;^ zx\u001b;^ xy\u001b) (see also Appendix A).\nFor instance, under a two-fold rotation about the z-axis\n(equivalent to the translation T(~ a1) inC000\n6v)^\u0007 transforms\nasG1^\u0007. Similarly, other elements of Ohact on ^\u0007 as\nproducts offGi;X;Yg, the generators of the M-point\nrepresentation.\nWith spin-orbit coupling symmetries act on ^\u0007 as dou-\nble group elements. The action of symmetries such as\nthe rotation G1is thenUG1G1^\u0007, where the SU(2) ma-\ntrixUG1implements the rotation G1in spin space. In\ngeneral, the matrix Ugimplements the symmetry g2Oh.\nSymmetry-mandated degeneracies of ^\u0007 follow from rep-\nresentations of the double group. The cubic double group\nadmits 2D and 4D spin-orbit coupled representations,\ncorresponding to total angular momenta j= 1=2 and\nj= 3=2. It is known that under cubic symmetry the\nT2gorbitals split into a two-fold degenerate j= 1=2 dou-\nblet and a four-fold degenerate j= 3=2 quadruplet. This\nsituation applies to the scalar order with A1gsymmetry\n(i.e., symmetric under all elements of the cubic group) of\nEq. (12). Instead, the scalar order with A2gsymmetry\nis symmetric under all elements of Th. With spin-orbit\ncoupling the double group of Thonly admits 2 Drepre-\nsentations, and as a result degeneracies will be two-fold.11\nRep. Type triple- M GS triple- M GS triple- M GS\nsinglet uniaxial SOC scalar\nF1s-waveA1(+;\u0000) insulator/QBT A1(\u0000;\u0000) spin-\fltered QBT A2(\u0000;+) Chern insulator\nF2d-waveA2(\u0000;\u0000) Chern insulator A1(+;\u0000) QSH insulator A1(+;+) spin-locked Dirac SM\nTABLE II. Summary of hexagonal lattice s- andd-density wave states at the M-points. Table lists the symmetry and nature\nof the mean-\feld ground state (GS) of singlet triple- M, uniaxial triple- M, and spin-orbit coupled (SOC) scalar triple- Morder.\nThe symmetry label A1;2refers to point group symmetry, and ( \u0006;\u0006) refers to preserved/broken time-reversal (\frst entry) and\ntranslational symmetry (second entry). Note that in case of uniaxial order we give the label A1since one can consider each\nspin species separately, and invert the spin if necessary.\nThe key result is that degeneracies of ^\u0007 carry over to\n^\b. The splitting of M-point electrons is identical to the\nsplitting of the T2gorbitals. This is a direct consequence\nof the mapping between cubic and hexagonal C000\n6vsym-\nmetry: symmetries acting on ^\u0007 must act in the same\nway on ^\b and therefore degeneracies are preserved. The\nmean-\feld spectra of scalar triplet M-point order con-\n\frm this. The spectra of the scalar triplet d-wave states\nshown in Fig. 5 exhibit a two-fold and four-fold degener-\nacy at \u0000, the order of which depends on the sign of \u0001 A1\n(i.e., black and red bands). Instead, all levels at \u0000 are\ntwo-fold degenerate in case of scalar s-wave triplet order\nshown Fig. 2 (recall that all bands are two-fold degener-\nate).\nIn addition to explaining degeneracies, the electron op-\nerator ^\u0007 gives rise to a dual description of the map-\nping from hexagonal density waves to nonzero angular\nmomentum condensation in 3 D. Instead of associating\nthe angular momentum with the condensed particle-hole\npairs, as we have done so far, it can be associated with an\ninternal electronic orbital degree of freedom given by the\nT2gorbitals of ^\u0007. Condensation in the s-wave (L= 0)\nchannel then corresponds to spontaneous orbital order,\nwhich is expressed by the bilinears\nh^\u0007y~\u0003a\u001bj^\u0007i;h^\u0007y~\u0003b\u001bj^\u0007i: (20)\nSince ^\u0007 and ^\b transform equivalently under cubic and\nhexagonal symmetry, respectively, these orbital order pa-\nrameters are symmetry-equivalent to the spin density and\nspin-current density waves.\nThe considerations based on a description in terms\nof low-energy M-point electrons are summarized in Ta-\nble II. The two sets of M-point order components, swave\n(F1symmetry) and dwave (F2symmetry), can condense\nin singlet or triplet channel. In the latter case, assuming\ntriple-Mordering, there is a uniaxial phase and a spin-\norbit coupled scalar phase. In the uniaxial state trans-\nlational symmetry is broken and the mean-\feld ground\nstate is a spin-\fltered QBT semimetal or a QSH insula-\ntor. In the scalar spin-orbit coupled state translational\nsymmetry is preserved and the mean-\feld ground state\nis a Chern insulator or a symmetry protected Dirac semi-\nmetal.B. Low-energy theory at the M0point: 2D Dirac\nsemimetal\nThe main characteristic of the scalar triplet d-wave\nstate is the nodal Dirac degeneracy at the M0points of\nthe reduced BZ, as shown in Fig. 4. We argued that\nthese Dirac points are symmetry protected and we will\nnow prove this. To this end we choose the ~M0\n2point\nand write the electron operator at ~M0\n2(in case of the\ntriangular lattice) as\n^\bM0\u00110\nBBBB@^ \u001b(~M0\n2)\n^ \u001b(~M0\n2+~M1)\n^ \u001b(~M0\n2+~M2)\n^ \u001b(~M0\n2+~M3)1\nCCCCA: (21)\nTo facilitate expressing the action of lattice symmetries,\nwe de\fne a set of Pauli matrices \u001ciacting within the\nblocks (~M0\n2;~M0\n2+~M1) and (~M0\n2+~M2;~M0\n2+~M3), in addi-\ntion to a set of matrices \u0017iacting on the block degree of\nfreedom (e.g., exchanging blocks). The ~M0\n2point is left\ninvariant by the inversion C2, the re\rection \u001bv, and no-\ntably the translations T(~ ai), all of which are symmetries\nof the scalar triplet d-wave state (in combination with\nglobal spin rotations).\nWe \fnd that the inversion acts as \u00171^\bM0, whereas\nthe translations T(~ a2) andT(~ a3) act as\u001b1\u00173^\bM0and\n\u001b2\u001c3^\bM0(see Appendix B). The Hamiltonian matrix at\n~M0\n2is linear combination of matrices \u001bi\u0017j\u001ck(i;j;k =\n0;1;2;3) and must be invariant under the symmetry op-\nerations. Taking into account the constraints coming\nfromC2\u001bvand time-reversal Twe \fnd only two allowed\nterms at~M0\n2, which are \u001c3and\u001c2(\u001b1\u0000\u001b3\u00171). These two\nterms anti-commute, implying two eigenvalues at ~M0\n2,\u000f,\nand\u0000\u000f, each four-fold degenerate. We conclude that the\ndouble Dirac node at ~M0\n2, and thus at all M0points, is\nmandated by symmetry.\nBased on this conclusion, we ask whether symmetry\nbreaking perturbations can gap out the Dirac nodes in\ninteresting ways. First, we expand the mean-\feld band\nstructure corresponding to the Hamiltonian H0+H[\u0001]\n[with \u0001\u0011\u0001A1of Eq. (17)] around ~M0\n2, assuming we are\nin the semi-metallic state shown in Fig. 4 (black bands).12\nWe take this node as an example, the theory around ~M0\n1;3\nmay be developed in a similar way. Details of the deriva-\ntion are presented in Appendix B, and we simply quote\nthe result here:\nH(~ q) = (v1q\u0000+v0q+)~\u00171~\u001c2+ (v2q+\u0000v0q\u0000)~\u00173~\u001c2:(22)\nThe Pauli matrices ~ \u0017iand ~\u001ciact on an e\u000bective valley and\npseudospin degree of freedom of the double Dirac node,\nspeci\fed in Appendix B together with Fermi velocity co-\ne\u000ecientsv1;2;v0. We have de\fned q\u0006=q1\u0006q2, where\nqi=~ q\u0001~ ai. As shown in Fig. 6, q+\u0018qxandq\u0000\u0018qy,\nimplying that q\u0000is along the direction of the undistorted\nFermi surface (see Fig. 1), whereas q+is orthogonal to\nit. We expect that when \u0001 A1!0 the Hamiltonian H(~ q)\nonly depends on q+. This is veri\fed by checking the\nbehavior of v1;2andv0, which become v1;v0!0, and\nv2!2t. As a result, the Hamiltonian (22) describes the\ndouble Dirac node with linear dispersion in q\u0006as func-\ntion of the order parameter \u0001 A1.\nThe symmetry protection of the double Dirac node (at\n~M0\n2) critically relies on the invariant translations T(~ ai).\nTo study the fate of the Dirac node, we therefore consider\na perturbation that breaks translational symmetry. Such\na perturbation is given by the charge modulation\n\u000eH=mX\n\u0016;\u001b;~k^ y\n\u001b(~k)^ \u001b(~k+~M\u0016) + H.c.; (23)\nand we \fnd that \u000eHgaps out the Dirac nodes at the M0\npoints. The gapped state respects time-reversal and in-\nversion symmetries, and we calculate the Fu-Kane invari-\nant\u00170[50] to determine the nature of the ground state.\nQuite remarkably, we \fnd that the topological invariant\n\u00170depends on the sign of m, i.e., (\u00001)\u00170= sgn(m). This\nresult may be understood as follows, starting from dou-\nble Dirac node at ~M0\n2. The double Dirac node consists of\ntwo Kramer's doublets with opposite inversion eigenval-\nues. The time-reversal invariant perturbation \u000eHsplits\nthe double node, leaving only one of the Kramer's dou-\nblets occupied. The sign of mcontrols which Kramer's\ndoublet, and consequently which inversion eigenvalue, is\noccupied. If the even eigenvalue is occupied at ~M0\n2, the\nsame must be true for the other M0points, implying that\nthe product of inversion eigenvalues of occupied bands\nat time-reversal invariant momenta, and thus \u00170, isodd\n(since the product of the \u0000\u000fsubspace is odd). This shows\nthat sgn(m) determines whether the gapped state is a\ntopological or trivial insulator. In Appendix B, we o\u000ber\nan alternative interpretation of this result.\nThe main features of the mean-\feld Dirac semi-metal\nstate are summarized in Fig. 6. In particular, Fig. 6 high-\nlights that the Dirac semimetal sits at the boundary be-\ntween a trivial and topological insulator. Both phases are\naccessible by a single perturbation parameter m, given by\nEq. (23). We stress that this applies in general to lattices\nwith hexagonal symmetry. We leave a comprehensive in-\nvestigation of the double Dirac node theory, including a\nclassi\fcation of all possible mass terms, for future study.\nM01M02M03\n0!mITIq1\u0000q2q1+q2FIG. 6. (Left) Schematic representation of the double Dirac\nnodes of the scalar triplet d-wave state of Eq. (17) and Fig. 4,\nlocated at the three inequivalent M0points. (Right) The\nDirac points can be gapped out by a charge density mod-\nulation [Eq. (23)] controlled by mass parameter m, the sign\nof which determines whether gapped state is a topological\ninsulator (TI) or a trivial insulator (I).\nV. GENERAL SYMMETRIC SPIN STATES\nThe symmetry classi\fcation of spin density waves has\na connection to a special class of classical spin states, the\nsymmetric classical spin states. Symmetric classical spin\nstates are con\fgurations of classical spins that respect\nall symmetries of the crystal lattice, up to a global O(3)\nrotation. The concept of symmetric classical spin states\nwas introduced in Ref. 38, where they were referred to\nasregular magnetic orders . The signi\fcance of the global\nspin rotations lies in the fact that most spin Hamiltonians\nare functions of O(3) invariant bilinears such as ~Si\u0001~Sj. In\nRef. 38 it was shown that symmetric classical spin states\nare good variational classical ground states of those spin\nHamiltonians.\nThe scalar triplet s-wave states of (14) and (15) are\nexamples of such symmetric spin states, if we interpret\nthem as classical spin states. Formally, we can identify\nthese triplet density waves with classical spin con\fgura-\ntions~S(~ x) =P\n~ q~S(~ q)e\u0000i~ q\u0001~ xby taking the Fourier com-\nponents~S(~M\u0016) equal to\nSi(~M\u0016) =1\nNX\n\u001b\u001b0~k\u001bi\n\u001b\u001b0h^ y\n\u001b(~k+~M\u0016)^ \u001b0(~k)i: (24)\n(For simplicity we have suppressed the sublattice index.)\nThe resulting spin con\fguration respects all lattice sym-\nmetries up to global spin rotation. To see this, we recall\nthat the scalar triplet states transform as scalars under\nC000\n6v, including the translations, since global spin rota-\ntion invariance is a manifest feature of the classi\fcation\nof spin density waves. The scalar triplet states are in-\nvariant up to sign, which implies that the classical spin\nstates derived from them are indeed symmetric.\nWe can ask the following question: Can we obtain\nall symmetric spin states with M-point modulation on\na given hexagonal lattice using the symmetry classi\fca-\ntion? The answer is yes. To demonstrate this, it is help-\nful to review how the scalar triplet orders of Eqs. (14)\nand (15) are constructed. The starting point is three-13\ncomponent charge order with F1symmetry, which is then\nidenti\fed with T2gorbital angular momenta in a cubic\ncrystal. Coupling the ( \u0018L= 1) angular momenta to\nspin (\u0018S= 1) and decomposing them into total an-\ngular momentum states yields an e\u000bective J= 0 state\ntransforming as a scalar. This scalar non-coplanar spin\ndensity wave corresponds to a symmetric con\fguration\nof classical spins.\nApart from charge order with F1symmetry, hexago-\nnal lattices can support other charge ordered states with\nM-point modulations (the honeycomb lattice is an ex-\nample), with di\u000berent symmetry [13] (we brie\ry review\nthis in Appendix C). Since representations of C000\n6vmap\nto representations of the cubic group, distinct charge or-\nders map to distinct angular momenta, such as porf\norbitals, which are three-fold degenerate in cubic sym-\nmetry. Coupling these angular momenta to spin, as dis-\ncussed in Section III B, and decomposing into total angu-\nlar momentum states will result in a scalar J= 0 state.\nThat state, when transformed back to spin density wave\nmust correspond to a symmetric classical spin state, as\nit is invariant under all lattice symmetries up to a global\nspin rotation.\nWe illustrate this using the honeycomb and kagome\nlattices as examples. In addition to charge order with F1\nsymmetry, the honeycomb lattice supports charge order\nwithF4symmetry. The equivalent of F4symmetry in\nthe cubic group is T2usymmetry (i.e., forbitals), and\nspin-orbit coupling yields T2u\u0002T1g=A2u+:::(we are\nonly interested in the scalar representation). The A2u\nterm implies the existence of a triplet density wave state\nwithB1symmetry, which corresponds to a symmetric\nspin state using Eq. (24). As a result, the honeycomb\nlattice admits two symmetric classical spin states with\nM-point wave vectors. Applying the same method to the\nkagome lattice we \fnd three sets of M-point charge or-\nder:F1,F3, andF4symmetries (see Appendix C). These\nare mapped to d-,p- andf-wave angular momenta, re-\nspectively. Spin-orbit coupling yields three distinct scalar\nJ= 0 states, from which we conclude that the kagome\nlattice admits three symmetric spin states with M-point\nmodulations. This is summarized in Table III. Explicit\ncomparison of the spin con\fgurations shows that our re-\nsult is in agreement with Ref. 38.\nThe constructive derivation presented here provides\na straightforward route to obtain the set of symmetric\nspin states of a given lattice. It is limited only in the\nsense that the modulation wave vectors are \fxed ahead\nof time. Here we explicitly constructed M-point sym-\nmetric spin states. Consequently, symmetric spin states\nwithK-point modulation, for instance, are inevitably\nmissed. This may be remedied, however, by simply de-\nriving allK-point charge order representations and as-\nsociating them with spin similar in spirit to spin-orbit\ncoupling. Indeed, in case of the kagome lattice, symmet-\nric spin states with K-point wave vector can be extracted\nfromK-point charge order.\nSymmetric classical spin states are good variationalTriangular Honeycomb Kagome\nCharge order F1F1+F4F1+F3+F4\nCubic symmetry T2gT2g+T2uT2g+T1u+T2u\nOrbitals fdg fdg+ffg fdg+fpg+ffg\nScalar (J= 0)A2A2+B1A2+B2+B1\nTABLE III. List of the M-point modulated symmetric clas-\nsical spin states on the triangular, honeycomb and kagome\nlattices. Top row shows the symmetry of the charge ordered\nstates they are derived from, and the second row lists the\ncubic representations corresponding to charge order with Fi\nsymmetry. The angular momenta (i.e., p;d;f -waves) trans-\nforming as those representations of the cubic group are listed\nin the third row. The bottom row lists the symmetry of the\nelectronic scalar spin density wave state (i.e., the total angular\nmomentum J= 0 state) that follows from coupling orbitals\nand spin.\nground states of a large class of spin Hamiltonians, in\nsome cases saturating rigorous lower bounds on ground\nstate energies [38]. Apart from lattice magnets described\nby spin Hamiltonians, symmetric spin states are also rel-\nevant for materials described by itinerant carriers cou-\npled to localized (classical) spins. For instance, mag-\nnetic states on the kagome lattice, stabilized by itinerant\nelectron-mediated interactions at speci\fc electron densi-\nties [35, 36], were found to be the symmetric spin states.\nIn general, the itinerant carrier density, tuned to com-\nmensurate doping, sets the magnetic modulation wave\nvectors. Therefore, the present approach is particularly\nsuited for deriving variational magnetic states of coupled\nspin-electron models, as the ordering vectors are prede-\ntermined. An added bene\ft is that the symmetry of the\nelectronic Hamiltonian directly follows from the deriva-\ntion.\nIn this section we have shown that the symmetric spin\nstates correspond to the cubic J= 0 singlets obtained\nfrom spin-orbit coupling. We conclude by noting that the\nfull set of multiplets (i.e., all representations, including\nthe multicomponent representations) may be viewed as\nan exhaustive symmetry classi\fcation of all (classical)\nspin states on a given hexagonal lattice.\nVI. DISCUSSION AND CONCLUSION\nIn this work we introduced a classi\fcation of hexago-\nnal triplet density waves on the basis of a mapping from\n2D order at \fnite wave vector to 3D Q= 0 order with\nnonzero angular momentum. The mapping follows from\nthe isomorphism between the extended hexagonal point\ngroupC000\n6vand the cubic point group Oh. Let us sum-\nmarize a number of key consequences of the mapping\nto cubic symmetry. Two important results directly fol-\nlow. First, in order to correctly determine the symme-\ntry of hexagonal triplet M-point order, it is necessary14\nto consider composites of spatial symmetries and global\nspin rotations. These composites are naturally obtained\nby mapping to cubic symmetry. Global spin rotation\nequivalence, which, for instance, mandates the double\ndegeneracy of electronic bands, is manifestly built into\nthe classi\fcation in terms of cubic L>0 orders. There-\nfore, the correct symmetry of the hexagonal triplet den-\nsity waves follows naturally from the mapping to cubic\nsymmetry. This is particularly important since the 2D\nDirac semimetal is protected by composite symmetries.\nSecond, in order to understand the splitting of energy\nlevels in the ^\b subspace [see Eq. (4)], at \u0000, one needs to\ninvoke the double group of Oh, as we demonstrated in\nSec. IV A. The fact that ^\b splits into a j= 1=2 doublet\nandj= 3=2 quadruplet, or three j= 1=2 doublets, is\ninextricably linked to the equivalence of ^\b and ^\u0007.\nA third consequence deserving a comment, is that the\nsymmetry equivalence of hexagonal triplet orders and cu-\nbic liquid crystal phases implies a common phenomeno-\nlogical Ginzburg-Landau description. As a result, from\nthe perspective of e\u000bective theories based on the sym-\nmetry of the order parameter, the two seemingly di\u000ber-\nent classes of orders have shared properties. A thorough\nsurvey of the features of such a Landau theory is left\nfor future study, but we expect that the connection be-\ntween 2D and 3D orders gives rise to additional insight.\nAn interesting aspect which is worth mentioning is that\nmulti-component orders, such as the M-point spin and\nspin-current density waves, can in general give rise to dis-\ntinct types of composite orders. These composites may\norder at temperatures above the transition temperature\nof the primary order, leading to, for instance, nematic\nor charge density wave order. The latter possibility has\nbeen addressed for the case of the uniaxial spin density\nwave in Ref. 51. Condensation of composite order pa-\nrameters has been studied to great extent in the context\nof the iron-pnictide materials, with a focus on nematic-\nity [52{57] and more recently charge density wave and\nvector chiral order [58], the latter constituting a triplet\nd-wave order.\nFourth, the classi\fcation introduced in this work leads\nto one of our key results: the identi\fcation of a new set of\nphases, the time-reversal invariant scalar triplet d-wave\nstates. In addition, it follows naturally from classi\fcation\nthat the scalar triplet dwave is the time-reversal invari-\nant analog of the chiral spin density wave. The scalar\ntripletdwave has remarkable symmetry properties: it\ndoes not break any spatial symmetries, is time-reversal\ninvariant, but does break spin-rotation symmetry. As\nsuch, it bears similarity to spin nematic order, and due\nto its peculiar symmetry may be called a \\hidden order\"\nstate.\nIt is important to mention that, even though these\ntwo types of spin-orbit coupled scalar orders can be in-\nterpreted as 3D electronic liquid crystal phases, there is\na signi\fcant di\u000berence between the two. Whereas elec-\ntronic liquid crystal phases exhibit Fermi surface distor-\ntions or reconstructions but remain metallic, the elec-tronic (mean-\feld) states corresponding to the scalar\ntriplet orders are insulating ( swave) and semi-metallic ( d\nwave). Indeed, the two systems (i.e., a 2D nested Fermi\nsurface and a 3D Fermi liquid) are di\u000berent. Therefore,\nin spite of the correspondence, the nature of both the\nnormal state and the condensate is di\u000berent for the two\ncases. There is, however, a similarity in the following\nsense. We have seen that the scalar s-wave state, i.e., the\nchiral spin density wave, gives rise to a fully gapped spec-\ntrum, whereas the spectrum of the scalar d-wave state has\nnodal degeneracies due to higher symmetry. The cubic\nT2g\fphase, in contrast to a p-wave (orT1u)\fphase, ex-\nhibits nodes. The cubic T1g\fphase has the same nodes,\nbut as a result of full cubic symmetry has an extra set of\nnodes: higher symmetry mandates an extra set of nodes.\nThe scalar spin density wave and scalar spin-current\ndensity waves, i.e., the \f-phase density waves, both con-\nstitute topological phases, in the sense that their mean-\n\feld band structures are topological. The chiral spin den-\nsity wave has nonzero Chern number in the ground state.\nIt is an example of so-called topological Mott (Chern) in-\nsulators [59]. The scalar triplet d-wave state is a novel\ntype of semimetal. It realizes a dynamically generated\nDirac semi-metal in two dimensions, protected by crys-\ntal symmetry. Perturbations can gap out the Dirac nodes\nand drive the system into either the trivial electronic in-\nsulator, or the topological insulator. This is achieved by\na very simple perturbation: charge density modulations\nthat only break translational symmetry. Interestingly,\ntheZ2topological index depends on the sign of the charge\ndensity perturbation. This transition, controlled by the\nsign of the mass perturbation, may be understood as a\nband inversion at an odd number of time-reversal invari-\nant momenta, i.e., all the M0points. In this respect, the\nDirac semimetal state signi\fcantly di\u000bers from graphene,\nwhich has double Dirac nodes (counting spin) at each of\nthe twoKpoints. The present spin-orbit coupled Dirac\nsemimetal has a double node at the threeM0points, i.e.,\nthree \ravors of double nodes. In this respect it bears\nsome similarity to the (111) surface states of topological\ncrystalline insulators in the SnTe material class [60{62],\nwhich are located at the M-points of the surface BZ. It\nwill be interesting to further develop the Dirac theory\nof the three M0points and consider the e\u000bect of various\nsymmetry breaking perturbations.\nWe have shown that the classical spin state analogs\nof non-coplanar \fphase spin density waves are symmet-\nric spin states. Recently, the latter were shown to be\nthe classical long-range ordered limits of time-reversal\nsymmetry broken or chiral spin liquid phases [63]. Very\nrecently, it was shown that quantum disordering the non-\ncoplanar chiral spin state realized in a chiral spin model\non the honeycomb lattice can result in a chiral spin liq-\nuid [64]. Furthermore, recent theoretical work has con-\nsidered quantum disordering the electronic chiral spin\ndensity wave state of a quarter doped honeycomb lat-\ntice model, and found that the resulting spin-charge liq-\nuid state is topologically ordered [23]. This establishes15\nan exciting connection between the electronic \f-phase\nspin density waves and spin liquid physics. In particu-\nlar, it will be interesting to explore to connection of the\nscalar spin-current d-wave state, the time-reversal invari-\nant analog of the chiral spin density wave, to the physics\nof spin(-charge) liquids. In this respect, we note that,\nsince thed-wave state is time-reversal invariant, it can be\nconverted into a triplet spin correlation function as [12]\nh~S(~k+~Q\u0016)\u0002~S(~k)i= \u0001~d(~k); (25)\nwhere the~d(~k) vector lives in spin space and the com-\nponents are given by Eq. (17). Therefore, the triplet\ndensity waves studied in this work give rise to intriguing\nquestions to be addressed in future work.\nACKNOWLEDGMENTS\nI have bene\fted greatly from helpful and stimulating\nconversations and with L. Fu, C. Ortix, J. van Wezel, J.\nvan den Brink, M. Daghofer, J. Hoon Han, Y. Ran, T.\nIadecola and C. Chamon. I gratefully acknowledge K.\nBarros, G.W. Chern and C.D. Batista for collaborations\non related subjects. I am particularly indebted to C.\nOrtix and J. van den Brink for a careful reading of the\nmanuscript and thoughtful comments. This work was\nsupported by the Netherlands Organization for Scienti\fc\nResearch (NWO).\nAppendix A: M-point representation of hexagonal\nsymmetry\nTheM-point representation of hexagonal symmetry is\nspeci\fed by the action of elements of the symmetry group\non the vector ~ v=~ v(~ x) de\fned as\n~ v(~ x) =0\nB@cos~M1\u0001~ x\ncos~M2\u0001~ x\ncos~M3\u0001~ x1\nCA: (A1)\nThe translations T(~ ai), where~ ai(i= 1;2;3) are the ele-\nmentary lattice vectors, are represented by the matrices\nGide\fned through the equation\n~ v(~ x+~ xi)\u0011Gi~ v(~ x); i= 1;2;3: (A2)\nExplicitly,G1andG2are given by\nG1=0\nB@\u00001\n\u00001\n11\nCA; G2=0\nB@1\n\u00001\n\u000011\nCA: (A3)\nThe matrices Giinherit the algebraic properties of the\ntranslations. They satisfy G2\ni= 1, they mutually com-\nmute and multiplication of two of them gives the third,\ni.e.G1G2=G3.The rotations and re\rections can be expressed in terms\nof the generators C6and\u001bv. The action of C6is de\fned\nas~ v0(~ x) =~ v(C\u00001\n6~ x) and is given by the matrix X,\n~ v(C\u00001\n6~ x) =X~ v(~ x); X =0\nB@0 1 0\n0 0 1\n1 0 01\nCA (A4)\nNote thatXhas the property X3= 1 and thus X\u00001=\nX2. In addition, one has X\u00001=XT, whereXTis the\ntranspose. It thus follows that ~ v(C\u00001\n3~ x) =X2~ v(~ x) =\nXT~ v(~ x). For the re\rection \u001bvwe de\fne the matrix Yas\n~ v(\u001b\u00001\nv~ x) =Y~ v(~ x); Y =0\nB@0 0 1\n0 1 0\n1 0 01\nCA: (A5)\nAll rotations and re\rections can be represented by a\nproduct of powers of XandY, i.e.XmYn. An arbitrary\nstring ofXandYmatrices can be brought into this form\nusing (XY)2= 1, which is equivalent to XY=YXT.\nThe set of generator matrices fG1;X;Yg(translations\nG2andG3can be written as products of the generators)\nde\fnes an embedding of the group C000\n6vinO(3), the group\nof orthogonal matrices in three dimensions. Clearly, this\nmapping is not invertible, as the inversion C2=C3\n6is\nmapped to identity through X3= 1. An invertible em-\nbedding is obtained by rede\fning the set of generators as\nfG1;\u0000X;Yg, i.e., associating the O(3) matrix\u0000Xwith\nC6. In this way, two-fold rotation C22C000\n6vis mapped\nto the inversion P2O(3).\nAs we explained in the main text, the key property of\nsuch mapping is that it establishes an exact isomorphism\nbetweenC000\n6vand the cubic group Oh, a subgroup of O(3).\nIndeed, direct inspection shows that the character tables\nof both groups are identical, which is a manifestation of\nthe isomorphism. When mapped onto elements of the\ncubic group, the translations T(~ ai) (represented by Gi)\nare given by two-fold rotations around the x,yandz\naxes. The six-fold rotations C6are interpreted as three-\nfold rotations about body diagonals of the cube combined\nwith inversion (i.e., \u0000X). The re\rection Yis mapped to\na two-fold rotation about the axis ^ x\u0000^zcombined with\nrotation.\nThe mapping to the cubic group implies that the rep-\nresentations de\fned by fG1;\u0000X;YgandfG1;X;Ygcan\nbe labeled using the Ohcharacter table. The faithful\nrepresentation generated by fG1;\u0000X;Ygis equal to the\nmatrix representation of ( x;y;z ) under cubic symme-\ntry and hence given by T1u. The representation gener-\nated byfG1;X;Ygcorresponds to the cubic representa-\ntionT2g, which describes the symmetry of the d-orbitals\n(yz;zx;xy ). In general, the 3D representations of the\ncubic group,fT1g;T2g;T1u;T2ug, are in correspondence\nwith the 3D representations of C000\n6v,fF2;F1;F3;F4g(in\nthat order). The latter describe translational symmetry\nbrokenM-point modulations. The mapping between cu-\nbic and hexagonal symmetries is summarized in Table IV.16\nHexagonalC000\n6v CubicOh\nReps. F1,F2,F3,F4T2g,T1g,T1u,T2u\nG1,G2,G3 Translations T(~ ai) Twofold rotations C2i\n\u0000I3 Twofold Rotation C2 InversionP\nX,XTTwofold rotations Threefold rotations\n(PrincipalC3) (Body diagonal C3)\nY Re\rection\u001bv Roto-re\rection PC0\n2\nTABLE IV. Table summarizing the mapping between the\nhexagonal group C000\n6vand the cubic group Oh. HereI3is\nthe identity matrix.\nAn equivalent de\fnition of the M-point representation\nfollows from considering the action of symmetry opera-\ntions on the van Hove electron operator ^\b introduced in\nthe main text [see Eq. (4)] and given by\n^\b =0\nB@^ \u001b(~M1)\n^ \u001b(~M2)\n^ \u001b(~M3)1\nCA: (A6)\nEvaluating the action of the generators of C000\n6von the\nM-point index \u0016, i.e.~M\u0016one \fnds\nT(~ x1) :^\b!G1^\b;\nC6:^\b!X^\b\n\u001bv:^\b!Y^\b; (A7)\nwhereG1,XandYare the matrices given in (A2){(A5).\nWe stress that here we only consider the action of symme-\ntries on the M-point index, and for the moment disregard\nthe action on the internal spin degree of freedom.\nStarting from ^\b, the mapping to cubic symmetry can\nbe formulated in terms of a d-orbital electron operator ^\u0007\nde\fned as,\n^\u0007 =0\nB@^ yz\u001b\n^ zx\u001b\n^ xy\u001b1\nCA; (A8)\nTheM-point representation generated by fG1;X;Yg\narises from considering the action of symmetries on ^\b,\nand as a result of the isomorphism between C000\n6vandOh,\nit arises equivalently from considering the action of Oh\nelements on ^\u0007. Speci\fcally, one \fnds that\nC2:^\u0007!G1^\u0007;\nPC3:^\u0007!X^\u0007;\nPC0\n2:^\u0007!Y^\u0007; (A9)\nwhereC2is a two-fold rotation about the principal z-axis,\nPis the inversion, C3is a three-fold rotation about a\nbody diagonal, and C0\n2is another (inequivalent) two-foldrotation. We conclude that ^\u0007 transforms in exactly the\nsame way under Ohsymmetry as ^\b underC000\n6vsymmetry.\nNote that the representation matrices act on the orbital\ndegree of freedom and not on spin, and the equivalence\nof^\u0007 and ^Phipertains to the spatial (i.e., orbital) degree\nof freedom.\nInsofar as spin is concerned, ^\u0007 transforms according to\nthe cubic double group. Speci\fcally, each element g2Oh\nis associated with Ug2SU(2) such that ^\u0007!UgOg^\u0007,\nwhereOgthe matrix representation of gobtained from\nfG1;X;Yg. For instance, the three two-fold rotations\nabout the principal axis have fUC(x)\n2;UC(y)\n2;UC(z)\n2g=\nf\u0000i\u001b1;\u0000i\u001b2;\u0000i\u001b3g. In addition, the rotation Xis ac-\ncompanied with UX=ei\u0019\u001b3=4ei\u0019\u001b2=4, and the rotation\n\u0000YwithUY=ei\u0019\u001b2=4e\u0000i\u0019\u001b3=2=\u0000iei\u0019\u001b2=4\u001b3.\nWe conclude this appendix by providing explicit ex-\npression for the matrices of fermion bilinears ^\u0003 given by\n^\u0003 = ^\by\n\u0016\u0003\u0016\u0017^\b\u0017. Here \u0003 is an Hermitian matrix and the\nspace of these M-point Hermitian matrices is spanned by\nthe Gell-Mann matrices, the generators of SU(3). In this\nwork we choose to group them in three sets de\fned by\n~\u0003a,~\u0003band~\u0003c. They are given by\n\u00031\na=0\nB@0 1 0\n1 0 0\n0 0 01\nCA;\u00032\na=0\nB@0 0 0\n0 0 1\n0 1 01\nCA;\u00033\na=0\nB@0 0 1\n0 0 0\n1 0 01\nCA\n\u00031\nb=0\nB@0\u0000i0\ni0 0\n0 0 01\nCA;\u00032\nb=0\nB@0 0 0\n0 0\u0000i\n0i01\nCA;\u00033\nb=0\nB@0 0i\n0 0 0\n\u0000i0 01\nCA\n\u00031\nc=0\nB@1 0 0\n0\u00001 0\n0 0 01\nCA;\u00032\nc=1p\n30\nB@1 0 0\n0 1 0\n0 0\u000021\nCA:(A10)\nWith the help of the symmetry transformation properties\nof Eq. (A7), it is straightforward to establish that ~\u0003a\ntransforms as F1and~\u0003basF2.\nAppendix B: Low-energy Dirac theory at the\nM0-points\n1. Proof of degeneracy\nThe proof of the symmetry protected denegeracy at\nthe~M0\n2point of the folded BZ requires evaluating the\ne\u000bect of symmetries leaving ~M0\n2invariant on the electron\noperator of Eq. (21). These symmetries are inversion C2,\nre\rection\u001bvand translations T(~ ai). The action of C2is\ngiven by\nC2!0\nBBBB@^ \u001b(\u0000~M0\n2)\n^ \u001b(\u0000~M0\n2+~M1)\n^ \u001b(\u0000~M0\n2+~M2)\n^ \u001b(\u0000~M0\n3+~M3)1\nCCCCA=\u00171^\bM0: (B1)17\nFrom this we conclude that the Hamiltonian at ~M0\n2can\nonly have terms \u001bi\u001cjor\u001bi\u001cj\u00171, where it is understood\nthati;j= 0;1;2;3. From Appendix A we know that the\ntranslation T(~ a2) is associated with G2, i.e., a rotation\naround the xaxis by\u0019, and as a result the symmetry\nT(~ a2) acts as [disregarding U(1) phases]\nT(~ a2)!\u001b1\u00173^\bM0: (B2)\nThis leaves us with the allowed terms \u001cj,\u001b1\u001cj,\u001b2\u001cj\u00171\nand\u001b3\u001cj\u00171. Similarly, the translation T(~ a3) is associated\nwithG3and therefore e\u0000i\u0019\u001b2=2. Hence, the action of the\ntranslation symmetry is\nT(~ a3)!\u001b2\u001c3^\bM0; (B3)\nwhich leaves us with the following allowed terms\n\u001c3; \u001b1\u001c1; \u001b1\u001c2; \u001b2\u00171; \u001b2\u001c3\u00171;\n\u001b3\u001c1\u00171; \u001b3\u001c2\u00171:\nWe are left with two re\rections leaving ~M0\n2invariant. We\nconsiderC2\u001bv, the action of which on ^\bM0is\nC2\u001bv!ei\u0019\u001b2=4\u001b30\nBBBB@1 0 0 0\n0 0 0 1\n0 0 1 0\n0 1 0 01\nCCCCA^\bM0: (B4)\nThe spin rotation UY=\u0000iei\u0019\u001b2=4\u001b3is the global spin\nrotation associated with Y(see also Appendix A). This\ntransformation property immediately leads to the exclu-\nsion of\u001b2\u001c3\u00171and\u001b2\u00171. The term \u001c3is clearly left\ninvariant. The remaining four terms must be combined\nin order to represent invariant terms, and in the end we\n\fnd the following three terms allowed by symmetry\n\u001c3; \u001c1(\u001b1\u0000\u001b3\u00171); \u001c2(\u001b1\u0000\u001b3\u00171):\nApplying a basis transformation e\u0000i\u0019\u001b1\u00171=4ei\u0019\u001b3=8brings\nthem into the form \u001b1\u001c1,\u001b1\u001c2and\u001c3. Clearly, these\nthree matrices mutually anti-commute and as result any\nlinear combination of these terms, i.e., the most general\nHamiltonian allowed by spatial symmetry, can only have\ntwo eigenvalues \u000fand\u0000\u000f. Each eigenvalue must be four-\nfold degenerate, proving the symmetry protection of the\ndouble Dirac nodes at ~M2. Clearly, the same is true for\nthe otherM0points.\nWe can exclude one more term using time-reversal\nsymmetryT. Time-reversal acts as\nT ! \u0000 i\u001b2\u00171^\bM0; (B5)\nfrom which we conclude that the only allowed terms are\n\u001c3and\u001c2(\u001b1\u0000\u001b3\u00171).2. Low-energy Dirac theory\nThe low-energy theory of the mean-\feld Dirac nodes\nat~M0\n2is constructed by expanding the mean-\feld band\nstructure to linear order around ~M0\n2. The \frst step is\nto diagonalize the Hamiltonian H(~M0\n2), which is given\nbyH(~M0\n2) =\u00002t\u001c3+ \u0001(\u0000\u001b1\u001c2+\u001b3\u001c2\u00171). We per-\nform a basis transformation UyH(~M0\n2)UwithU=\nei\u0019\u001b1\u00171=4ei\u0019\u001b3=8e\u0000i\u0019\u001b1=4. This yields the Hamiltonian\nH(~M0\n2) =\u00002t\u001c3+p\n2\u0001\u001c2\u001b3\u0011\u0000\u0018\u001c3+\u0011\u001c2\u001b3(B6)\nAs proven earlier, the Hamiltonian has two eigenvalues,\n\u000f\u0006=\u0006\u000f\u0011\u0006p\n\u00182+\u00112. The eigenvectors corresponding\nto\u000f+, i.e. the subspace of the relevant Dirac nodes, are\ngiven byj'mni,\nj'11i= (u;0;\u0000iv;0;0;0;0;0);\nj'12i= (0;0;0;0;u;0;\u0000iv;0);\nj'21i= (0;u;0;iv;0;0;0;0);\nj'22i= (0;0;0;0;0;u;0;iv); (B7)\nwhereuandvare de\fned as\nu=1p\n2r\n1\u0000\u0018\n\u000f; v =1p\n2r\n1 +\u0018\n\u000f: (B8)\nNote that if \u0001!0 (i.e.,\u0011!0) one hasv= 1 andu= 0,\nas expected.\nThe next step is to expand the mean-\feld Hamilto-\nnianH(~k) in small~ qwith respect to ~M0\n2, retaining only\nthe linear terms. We then perform the same basis trans-\nformationUand project the expanded Hamiltonian into\nthe subspace given by j'mni. Theq-linear part of the\nHamiltonian at ~M0\n2is\nHq=\u0018(q2\u00173\u0000q1\u00173\u001c3) +\u0011(q1\u00173\u001c2\u001b1\u0000q1\u00172\u001c3\u001b2)=p\n2\n+\u0011(q2\u00172\u001c1\u001b3\u0000q2\u00172\u001b2)=p\n2: (B9)\nTo express the resulting Dirac Hamiltonian in terms of\ne\u000bective valley and pseudospin degrees of freedom, we\nde\fne two sets of Pauli matrices, ~ \u0017and ~\u001c, which act on\nmandnofj'mni, respectively. In addition, we take\nq\u0006=q1\u0006q2, whereqi=~ q\u0001~ ai. We \fnd the Hamiltonian\nH(~ q) = (v1q\u0000+v0q+)~\u00171~\u001c2+ (v2q+\u0000v0q\u0000)~\u00173~\u001c2(B10)\nwithv1=\u0018u2+\u0011v2=p\n2,v2=\u0018v2+\u0011u2=p\n2, andv0=\n2\u0011uv=p\n2. As a result, in the limit \u0001 !0, i.e., the\nabsence of any order, v2=\u0018andv1;v0= 0. This is as\nexpected, since q\u0000is in the direction of the undistorted\nFermi surface, implying there is no dispersion in that\ndirection.\nThe inversion C2is given by\u00171^\bM0. Projecting \u00171into\nthe Dirac spinor subspace de\fned by j'mniwe \fnd ~\u001c1.\nProjecting the perturbation \u000eHgiven in Eq. (23) into\nthe same subspace yields m~\u001c1. This is consistent with18\nthe fact that \u000eHis symmetric under inversion. More\nimportantly, this demonstrates that mcontrols what the\ninversion eigenvalue of the occupied Kramer's doublet is.\nAn alternative way to understand the dependence of\nthe topological invariant (in this case given by the Fu-\nKane formula [50]) on sgn( m) is to start from the charge\ndensity modulations \u000eHand consider the M-point elec-\ntrons at \u0000:\n^\b\u0000=0\nB@^ \u001b(~M1)\n^ \u001b(~M2)\n^ \u001b(~M3)1\nCA\u00110\nB@^ 1\u001b\n^ 2\u001b\n^ 3\u001b1\nCA: (B11)\nIn the presence of the perturbation \u000eH(assuming \u0001 =\n0), the ^\b\u0000states split into non-degenerate level and a\ndegenerate doublet ( ^\t1\u001b;^\t2\u001b) [13], the latter given by\n^\t1\u001b= (^ 1\u001b+^ 2\u001b\u00002^ 3\u001b)=p\n6;\n^\t2\u001b= (\u0000^ 1\u001b+^ 2\u001b)=p\n2: (B12)\nThe sign of mdetermines the relative energetic ordering\nof the doublet and the non-degenerate level.\nIfm< 0, the doublet is higher in energy, and the Fermi\nlevel is at the semimetallic quadratic band crossing point\nde\fned by ( ^\t1;^\t2) and governed by the quadratic band\ncrossing Hamiltonian H(~ q)\u0018(q2\nx\u0000q2\ny)~\u001c3+ 2qxqy~\u001c1(see\nRef. 13). Here, ~ \u001c3=\u00061 labels the states ^\t1;2.\nWe can now consider \fnite \u0001, i.e., \fnite \u0001 A1in\nEq. (17). Projecting the \\perturbation\" coming from \u0001\ninto the subspace given by ( ^\t1\u001b;^\t2\u001b) we \fnd \u0001 ~ n\u0001~ \u001b~\u001c2,\nwith~ n= (1;1;1). This is recognized as a QSH gap of\na quadratic band crossing: ~ \u001c2constitutes the quantum\nanomalous Hall gap, and ^ n\u0001~ \u001bgives it opposite sign for\nthe two spin projections. This proves that the result-\ning state, which is adiabatically connected to the Dirac\nsemimetal with gap m < 0 at theM0points, is a topo-\nlogical insulator state.\nIn contrast, had we assumed m > 0, the quadratic\nband crossing point de\fned by ( ^\t1;^\t2) would be fully\noccupied (i.e., the Fermi level would notbe precisely at\nthe quadratic band crossing point) and the insulating\nstate with \fnite \u0001 would be adiabatically connected to\nthe trivial insulator de\fned by \u000eHwithm> 0.\nAppendix C: Derivation of symmetric classical spin\nstates\nWe brie\ry review the derivation of charge order repre-\nsentations, introduced in Ref. 13, based on the example\nof the honeycomb and kagome lattices discussed in Sec-\ntion V.\nModulations with M-point propagation vectors lead to\na quadrupling of the unit cell. In case of the honeycomb\nlattice the enlarged unit cell contains ns= 4\u00022 = 8\nsites, whereas in case of the kagome lattice the enlarged\nunit cell contains ns= 4\u00023 = 12. We label all sites andcollect them in a vector ~ sgiven byfsigns\ni=1. Extended\npoint group operations will permute the sites and the\npermutation matrices de\fne a representation of extended\npoint group. We write the representation as PM\nsand\nit has dimension ns. The superscript Mis meant to\nindicateM-point modulation (unit cell quadrupling).\nThe representation PM\nsis reducible and can be decom-\nposed into irreducible representations of the extended\npoint group. In case of the honeycomb one \fnds\nPM\ns=A1+B2+F1+F4; (C1)\nwhereas the kagome lattice yields\nPM\ns=A1+E2+F1+F3+F4: (C2)\nTheFisignal translational symmetry breaking and con-\nstitute the M-point modulated content of the decompo-\nsition. These representations are listed in Table III. We\nobserve that the honeycomb lattice has two independent\nrepresentations F1andF4, and the kagome lattice admits\nthree,F1,F3, andF4. From this we conclude that the\nformer admits two classical spin liquid states, whereas\nthe latter admits three.\nAppendix D: Extended point groups and character\ntables\nHere, we provide a basic review of the essentials of\nextended point group symmetry used in the main text.\nThe crystal point group of 2D hexagonal materials is C6v,\nwhich is identical to the dihedral group D6for spinless\nelectrons. For spinful electrons the point groups D6and\nC6vare technically distinct, however, for the purpose of\nthis work we consider them equivalent, as our results are\nindependent of technical di\u000berences, and focus on the\npoint group C6v. Note that time reversal acts as T=\ni\u001b2K(Kis complex conjugation).\nThe group C6vis generated by a six-fold rotation C6\nand a re\rection \u001bv, where the re\rection is de\fned as\n(x;y)!(x;\u0000y). The space group Sis given by all point\ngroup elements and all translations over lattice vectors\n~ x, i.e.,T(~ x), where the lattice vectors are generated by\n~ a1and~ a2. As a result, the translation subgroup Tis\ngenerated by T(~ a1) andT(~ a2).\nIn general, a point group Gcan be obtained as the\nfactor group of the space group Sand the translation\nsubgroupT. The extended point groups are obtained as\nthe factor group of the space group and a modi\fed trans-\nlation subgroup ~T, de\fned as the group of translations\ncompatible with ordering vectors, or equivalently, with\nthe enlarged unit cell. All translation in ~Tmap the en-\nlarged unit cell to itself. The enlargement is \fxed by the\nordering vectors, in our case the M-point vectors. Hence,\nthe extended point group is de\fned as ~G=S=~T. Clearly,\n~Gis larger than G, as the translations in Tbut no longer\nin~Tare now in ~G.19\nRep. Type Label Expression\nA1s \u0015 s(~k) (cosk1+ cosk2+ cosk3)=p\n3\nE2dx2\u0000y2\u0015d1(~k) (cosk1+ cosk2\u00002 cosk3)=p\n6\ndxy\u0015d2(~k) (cos k1\u0000cosk2)=p\n2\nTABLE V. Lattice angular momentum form factors trans-\nforming as representations of C6v, corresponding to near-\nest neighbor hopping on the triangular lattice. We de\fned\nki=~k\u0001~ aiwith~ aigiven in Sec. III. Note that ( \u0015d1;\u0015d2)\u0018\n(k2\nx\u0000k2\ny;2kxky) when expanded in ~k.\nIn this work, we consider hexagonal symmetry, C6v,\nand ordering at the Mpoints. The latter implies trans-\nlational symmetry breaking such that eTis generated\nbyT(2~ a1) andT(2~ a2). The translations T(~ a1)\u0011t1,\nT(~ a2)\u0011t2andT(~ a1+~ a2)\u0011t3are added to the point\ngroup. Since three translations are added to the point\ngroup, we denote the extended point group of C6vas\nC000\n6v. The character table of C000\n6vis given in Table VI.In the main text, we use an isomorphism between the\nhexagonal extended point group C000\n6vand the cubic point\ngroupOh. This connection is discussed in more detail in\nAppendix A. The character table of the cubic group is\ngiven in Table D 2.\n1. Lattice angular momentum basis functions\nThe triangular lattice angular momentum form factor\nfunctions, used to express condensate functions, are given\nin Table V. Table V lists the functions that transform\nas representations of C6vand correspond to form factors\noriginating from triangular lattice nearest-neighbor (e.g.,\n~ ai) hopping.\n2. Character tables\nFor completeness and convenience, here we reproduce\nthe character tables of the extended point groups C000\n6v\n(see Table VI) and Oh(see Table D 2).\n[1] Z. Hasan, and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[2] X.-L. Qi., and S.-C. Zhang, Rev. Mod. Phys. 83, 1057\n(2011).\n[3] C. L. Kane, and E. J. Mele, Phys. Rev. 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B\n83, 184401 (2011).20\nConjugacy class C000\n1C000\n2C000\n3C000\n4C000\n5C000\n6C000\n7C000\n8C000\n9C000\n10\nPoint group t1,t2t1C2,t2C2tiC3,tiC\u00001\n3tiC6,tiC\u00001\n6 3\u001bv,t1\u001bv2t1\u001bv,t2\u001bv3\u001bd,t2\u001bd1t1\u001bd1,t3\u001bd1\nC000\n6vI t 3C2t3C2C3,C\u00001\n3C6,C\u00001\n6t2\u001bv3,t3\u001bv1t2\u001bv2,t3\u001bv2t3\u001bd2,t1\u001bv3t1\u001bd2,t2\u001bd2\nt1\u001bv3,t3\u001bv3 t2\u001bd3,t3\u001bd3\nA1 1 1 1 1 1 1 1 1 1 1\nA2 1 1 1 1 1 1 \u00001\u00001\u00001\u00001\nB1 1 1\u00001\u00001 1 \u00001 1 1 \u00001\u00001\nB2 1 1\u00001\u00001 1 \u00001\u00001\u00001 1 1\nE1 2 2\u00002\u00002\u00001 1 0 0 0 0\nE2 2 2 2 2 \u00001\u00001 0 0 0 0\nF1 3\u00001 3\u00001 0 0 1 \u00001 1 \u00001\nF2 3\u00001 3\u00001 0 0 \u00001 1 \u00001 1\nF3 3\u00001\u00003 1 0 0 1 \u00001\u00001 1\nF4 3\u00001\u00003 1 0 0 \u00001 1 1 \u00001\nTABLE VI. Character table of the point group C000\n6v[65]. Translations t1andt2correspond to T(~ a1) andT(~ a2), respectively.\nt3=T(~ a1+~ a2). The irreducible representations that arise as a consequence of the added translations are F1,F2,F3andF4,\nall three-dimensional.\nPoint group OhI 3C2\n4 6C4 6C0\n2 8C3P 3PC2\n4 6PC 4 6PC0\n2 8PC 3\nKoster Mulliken\n\u0000+\n1 A1g 1 1 1 1 1 1 1 1 1 1\n\u0000+\n2 A2g 1 1 \u00001\u00001 1 1 1 \u00001\u00001 1\n\u0000+\n3 Eg 2 2 0 0 \u00001 2 2 0 0 \u00001\n\u0000+\n4 T1g 3\u00001 1 \u00001 0 3 \u00001 1 \u00001 0\n\u0000+\n5 T2g 3\u00001\u00001 1 0 3 \u00001\u00001 1 0\n\u0000\u0000\n1 A1u 1 1 1 1 1 \u00001\u00001\u00001\u00001\u00001\n\u0000\u0000\n2 A2u 1 1 \u00001\u00001 1 \u00001\u00001 1 1 \u00001\n\u0000\u0000\n3 Eu 2 2 0 0 \u00001\u00002\u00002 0 0 1\n\u0000\u0000\n4 T1u 3\u00001 1 \u00001 0 \u00003 1 \u00001 1 0\n\u0000\u0000\n5 T2u 3\u00001\u00001 1 0 \u00003 1 1 \u00001 0\nTABLE VII. Character table of the point group Oh.\n[39] B. Valenzuela and M. A. H. Vozmediano, New J. 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B 77, 224413 (2008)."    },    {        "title": "2012.08711v3.Ab_initio_Ultrafast_Spin_Dynamics_in_Solids.pdf",        "content": "Ab initio Ultrafast Spin Dynamics in Solids\nJunqing Xu,1Adela Habib,2Ravishankar Sundararaman,3and Yuan Ping1\n1Department of Chemistry and Biochemistry, University of California, Santa Cruz, CA 95064, USA\n2Department of Physics, Applied Physics and Astronomy,\nRensselaer Polytechnic Institute, 110 8th Street, Troy, New York 12180, USA\n3Department of Materials Science and Engineering,\nRensselaer Polytechnic Institute, 110 8th Street, Troy, New York 12180, USA\n(Dated: October 1, 2021)\nSpin relaxation and decoherence is at the heart of spintronics and spin-based quantum infor-\nmation science. Currently, theoretical approaches that can accurately predict spin relaxation of\ngeneral solids including necessary scattering pathways and capable for ns to ms simulation time are\nurgently needed. We present a \frst-principles real-time density-matrix approach based on Lind-\nblad dynamics to simulate ultrafast spin dynamics for general solid-state systems. Through the\ncomplete \frst-principles descriptions of pump, probe and scattering processes including electron-\nphonon, electron-impurity and electron-electron scatterings with self-consistent electronic spin-orbit\ncouplings, our method can directly simulate the ultrafast pump-probe measurements for coupled spin\nand electron dynamics over ns at any temperatures and doping levels. We \frst apply this method\nto a prototypical system GaAs and obtain excellent agreement with experiments. We found that\nthe relative contributions of di\u000berent scattering mechanisms and phonon modes di\u000ber considerably\nbetween spin and carrier relaxation processes. In sharp contrast to previous work based on model\nHamiltonians, we point out that the electron-electron scattering is negligible at room temperature\nbut becomes dominant at low temperatures for spin relaxation in n-type GaAs. We further examine\nultrafast dynamics in novel spin-valleytronic materials - monolayer and bilayer WSe 2with realistic\ndefects. We \fnd that spin relaxation is highly sensitive to local symmetry and chemical bonds\naround defects. For the bilayer WSe 2, we identify the scattering pathways in ultrafast dynamics\nand determine relevant dynamical properties, essential to its utilization of unique spin-valley-layer\nlocking e\u000bects. Our work provides a predictive computational platform for spin dynamics in solids,\nwhich has unprecedented potentials for designing new materials ideal for spintronics and quantum\ninformation technology.\nI. INTRODUCTION\nSpin is a fundamental quantum mechanical property of\nelectrons and other particles. The spin states can be used\nas the basis of quantum bits in quantum information sci-\nence (QIS)1, in addition to being used in spintronics anal-\nogous to electrical charge in conventional electronics2.\nThe key property for spintronics and spin-based QIS\nis the lifetime of spin states. Stable manipulations of\nspin states in practical applications require lifetimes on\nthe order of hundreds of nanoseconds or even millisec-\nonds. Determining the underlying mechanisms and con-\ntrolling spin relaxation are vital to reach long spin life-\ntimes at room temperature. Experimentally spin relax-\nation can be studied through ultrafast magneto-optical\npump-probe3,4and spin transport measurements5, allow-\ning the direct observations of dynamical processes and\nquantitative determination of spin relaxation time, \u001cs.\nDespite signi\fcant experimental progresses and sev-\neral proposed systems in the past decades2,6, materials\nwith properties required for practical QIS and spintron-\nics applications such as long \u001csat room temperature re-\nmain to be found1,5,7. Theoretical predictions of mate-\nrials properties have been mostly focused on electronic\nexcitations8{10and electron-hole recombinations11{13of\npotential spin defects for QIS applications. Reliable pre-\ndiction of spin lifetime and dominant relaxation mecha-\nnism will allow rational design of materials in order toaccelerate the identi\fcation of ideal materials for quan-\ntum technologies, while forgoing the need of experimental\nsearch over a large number of materials.\nUntil recently, most state-of-the-art theoretical meth-\nods to study spin dynamics of solid-state materials are\nlimited to simpli\fed and system-speci\fc models that re-\nquire prior input parameters2,14{16. These methods laid\nimportant theoretical foundation for spin dynamics, such\nas the spin-Bloch kinetic equations developed from Non-\nequilibrium Green's Function theory (NEGFT)17. Ref.\n18 derived a closed equation of motion for the electronic\nsingle-particle density matrix, including di\u000berent scatter-\ning matrices, which may be applicable to spin dynamics.\nHowever, because of the simpli\fed electronic structure\nand electron-phonon coupling matrices, quantitative pre-\ndiction of spin relaxation remains out of reach. Occasion-\nally, even trends in \u001cspredicted by such models may be\nincorrect as shown recently for graphene.19Furthermore,\nthese models are unable to provide predictive values for\nnew materials where prior inputs are not available.\nPrior to our work, the existing \frst-principles method-\nology for spin lifetime has been mostly based on spin-\rip\nmatrix elements in a specialized Fermi's Golden rule20{22,\nwhich is only applicable to systems with Kramers' degen-\neracy or spatial inversion symmetry, not suitable to lots\nof materials promising for quantum computing and spin-\ntronics applications2,5. Other \frst-principles techniques\nlike real-time Time-Dependent Density Functional The-arXiv:2012.08711v3  [cond-mat.mtrl-sci]  30 Sep 20212\nory (TDDFT)23are challenging for crystalline systems\ndue to high computational cost for describing phonon re-\nlaxations that require large supercells. More importantly,\nlong simulation time over nanoseconds often required by\nspin relaxation is a major di\u000eculty for TDDFT, which\nis only practical for tens to a few hundred femtoseconds.\nWhile spin dynamics based on TDDFT has been recently\nperformed for ultrafast demagnetization of magnetic sys-\ntems within tens of fs24{26, the intrinsic time scale and\nsupercell limitations mentioned above remain.\nWe recently derived a generalized rate equation27\nbased on \frst-principles density matrix (DM) Lindblad\ndynamics framework, which provides accurate spin re-\nlaxation time due to spin-orbit and electron-phonon cou-\nplings for a broad range of materials, with arbitrary sym-\nmetry. However, our previous work requires the system\nis already at a quasi-equilibrium state when the dynam-\nics can be described by a single-exponential decay, which\ncan not describe coupled spin and carrier dynamics at\nan out-of-equilibrium state in ultrafast pump-probe ex-\nperiments. In this work, we develop a real-time ab initio\nDM dynamics method based on this theoretical frame-\nwork, with complete descriptions of scattering processes\nincluding electron-phonon (e-ph), electron-impurity (e-i),\nand electron-electron (e-e), being adequate for over ns to\nms simulation time. Speci\fcally, compared to the gener-\nalized rate equation in our previous work, DM dynamics\nwith explicit real-time evolutions allow coupled carrier\nand spin relaxation away from quasi-equilibrium with all\ndecoherence pathways simultaneously. This will facili-\ntate direct prediction of experimental signatures in ul-\ntrafast magneto-optical spectroscopy to unambiguously\ninterpret experimental probes of spin and electron dy-\nnamics.\nWe will demonstrate the generality of our approach\nby considering two prototypical and disparate systems -\nGaAs and few-layer WSe 2, which have very di\u000berent spin\nrelaxation mechanisms.\nWe will \frst apply our DM dynamics methodology\nto investigate ultrafast spin dynamics of GaAs, which\nhas broad interest in spintronics over past decades2,28{31\nand more recently32{34, partly due to its long spin life-\ntime especially in the n-doped material at relatively low\ntemperature28. Despite various experimental28,29,35{37\nand theoretical2,30,31,38{40(mostly using parameterized\nmodel Hamiltonian) studies previously, the dominant\nspin relaxation mechanism of bulk GaAs under various\ntemperatures and doping levels remains unclear. For ex-\nample, Refs. 30 and 39 claimed e-i and e-ph scatterings\ndominate spin relaxation at low and room temperatures,\nrespectively; however, Refs. 31 and 40 conclude that e-e\nmay be more important at room temperature and even\nmore at lower temperatures. Moreover, electron-phonon\nscattering matrices which can be accurately obtained\nfrom \frst-principles, are very di\u000ecult to be precisely de-\nscribed in parameterized models used previously. Most\nimportantly, the applicability of empirical D'yakonov-\nPerel' (DP) relation, which is widely used for describinginversion-asymmetric systems including GaAs, needs to\nbe carefully examined. Throughout this work, we pro-\nvide complete and unambiguous insights on the underly-\ning mechanism of spin relaxation and applicability of the\nDP relation for GaAs from \frst-principles DM dynamics.\nDue to broken inversion symmetry and strong SOC,\nmonolayer transition metal dichalcogenides (TMDs) ex-\nhibit exciting physical properties including valley-speci\fc\noptical excitation and spin-valley locking e\u000bects. In Ref.\n41,42, it has been shown that by introducing doping in\nmonolayer TMDs, ultraslow decays of Kerr rotations,\nwhich correspond to ultralong spin/valley lifetimes of res-\nident carriers especially resident holes can be observed\nat low temperatures. Those features make monolayer\nTMDs advantageous for spin-valleytronics and (quan-\ntum) information processing.\nBesides monolayers, bilayer TMDs recovering inversion\nsymmetry have also attracted signi\fcant interests be-\ncause of the new \\layer\" degree of freedom or layer pseu-\ndospin in addition to spin and valley pseudospin43{45.\nPrevious studies already concluded that electronic states\natK/K0valleys of a bilayer TMD are approximately a\nsuperposition of those of two monolayers. This allows\nus to tune which layer carriers/spins are localized by a\nperpendicular electric \feld Ez, and make use of the spin-\nvalley-layer locking e\u000bects for spin-valleytronic applica-\ntions.\nAlthough spin/valley relaxation of resident carri-\ners in monolayer TMDs, which is most relevant to\nspin-valleytronic applications, have been extensively\nexamined27,41,42,46, the underlying dynamics especially\nthe e\u000bects of di\u000berent types of impurities have not\nbeen investigated through predictive ab initio simula-\ntions. Furthermore, for bilayer TMDs, the study on\nspin/valley dynamics is still in infancy with few ultra-\nfast measurements which however do not exhibit long\nspin relaxation time and are lack of spin-valley-layer\nlocking property47{49. There is a lack of knowledge of\nthe role of scattering processes and the scattering path-\nways for spin/valley dynamics of free carriers in bilay-\ners, which prevents researchers to realize and manipu-\nlate spin-valley-layer locking e\u000bects for designing spin-\nvalleytronic devices.\nIn this work, we will answer the above questions by\nperforming ab initio real-time dynamics simulations with\na circularly polarized pump pulse and relevant scattering\nmechanisms. We focus on WSe 2due to its larger valence\nband SOC splitting and focus on dynamics of holes since\n\u001csof holes seem longer than electrons.\nIn the following, we \frst introduce our theoretical for-\nmalism of real-time density-matrix approach with var-\nious scattering processes and pump-probe spectroscopy.\nIn particular, we focus on spin-orbit mediated spin relax-\nation and decoherence processes under the existences of\nelectron scatterings, which are rather common in semi-\nconductors and metals2,17. We use this method to sim-\nulate pump-probe Kerr rotation and real-time spin dy-\nnamics, by using GaAs as a prototypical example and3\ncomparing with experiments. Next, we study spin life-\ntime dependence on the temperature and doping level,\nwhere the dominant mechanisms can vary signi\fcantly.\nWe then discuss the roles of di\u000berent scattering mecha-\nnisms and phonon modes in carrier and spin relaxations,\nrespectively, in order to resolve related long-standing con-\ntroversies. We further simulate ultrafast dynamics in\nmonolayer and bilayer WSe 2and extract useful dynami-\ncal properties. Our work provides predictive theory and\ncomputational platform for open quantum dynamics, and\no\u000bers new and critical insights for spin relaxation and de-\ncoherence in general solid-state systems.\nII. THEORY\nA. Real-time density-matrix dynamics and spin\nrelaxation time\nTo provide a general formulation of quantum dynamics\nin solid-state materials, we start from the Liouville-von\nNeumann equation in the interaction picture,\nd\u001a(t)\ndt=\u0000i[H0(t);\u001a(t)]; (1)\nH0(t) =H(t)\u0000H0(t); (2)\nwhereH,H0andH0are total, unperturbed and per-\nturbed Hamiltonian, respectively. In this work, the total\nHamiltonian is\nH=H0+Hpump +He\u0000i+He\u0000ph+He\u0000e;(3)\nH0=He;0+He\feld +Hz+Hph; (4)\nwhereHe;0is electronic Hamiltonian under zero exter-\nnal \feld. In this work, He\feld is Hamiltonian induced\nby a perpendicular electric \feld Ezalong the vacuum\ndirection. Hzis the Zeeman Hamiltonian correspond-\ning to an external magnetic \feld B,Hz=gs\u0016BB\u0001s,\nwhere s= (sx;sy;sz) andsiis spin matrix in Bloch ba-\nsis under zero \feld. gsisgfactor and \u0016Bis the Bohr\nmagneton. Hpump is the Hamiltonian of the pump pulse\nand will be described below. Hphis the phonon Hamilto-\nnian, while He\u0000i,He\u0000phandHe\u0000edescribe the electron-\nimpurity, electron-phonon and electron-electron interac-\ntions respectively. The detailed forms of the interaction\nHamiltonians are given in Appendix A.\nIn practice, the many-body density matrix master\nequation in Eq. 1 is reduced to a single-particle one and\nthe environmental degrees of freedom are traced out50.\nThe total rate of change of the density matrix is separated\ninto terms related to di\u000berent parts of Hamiltonian,\nd\u001a\ndt=d\u001a\ndtjcoh+d\u001a\ndtjscatt; (5)\nwhere\u001ais the density matrix of electrons. Above,d\u001a\ndtjcoh\ndescribes the coherent dynamics of electrons under po-\ntentials or \felds, e.g. the applied pump pulse, whiled\u001a\ndtjscatt captures the scattering between electrons and\nother particles.\nTo obtain Eq. 5 which involves only the dynamics of\nelectrons or the electronic subsystem, we have assumed\nthe environmental subsystem is not perturbed by the\nchange of the electronic subsystem, which in this work\nmeans there is no dynamics of phonons. This assump-\ntion is valid when the system is not far from equilibrium,\ne.g., when excitation is weak. In most spin dynamics ex-\nperiments, it is desirable to work in the low excitation\ndensity limit to avoid additional complexities and focus\non the physics of spin dynamics. Indeed in many exper-\niments, e.g., in Refs. 28 and 51, pump \ruence and exci-\ntation density are controlled to be low, e.g., excitation\ndensity 2\u00021014cm\u00002for GaAs. Therefore, phonon dy-\nnamics can be safely excluded in the current stage. The\ninclusion of phonon degrees of freedom in the density-\nmatrix dynamics has been discussed in detail in Refs.\n50,52 with model Hamiltonian, which can be our future\nwork to implement from \frst-principles.\nTo de\fne spin lifetime, we follow the time evolution of\nthe observable\nSi= Tr (si\u001a); (6)\nwheresiis the spin operator ( i=x;y;z ). This time\nevolution must start at an initial state (at t=t0) with\na net spin i.e. \u000e\u001a(t0) =\u001a(t0)\u0000\u001aeq6= 0 such that\n\u000eSi(t0) =Si(t0)\u0000Seq\ni6= 0, where \\eq\" corresponds to\nthe \fnal equilibrium state. We evolve the density matrix\nthrough Eq. 5 using an adaptive Runge-Kutta fourth-\norder method for a long enough simulation time, typi-\ncally from tens of ps to several ns, until the evolution of\nSi(t) can be reliably \ftted by\nSi(t)\u0000Seq\ni= [Si(t0)\u0000Seq\ni]exp\u0014\n\u0000t\u0000t0\n\u001cs;i\u0015\n\u0002cos [!B(t\u0000t0) +\u001e]: (7)\nto extract the relaxation time, \u001cs;i. Above,!Bis oscilla-\ntion frequency due to energy splitting in general, which\nunder an applied magnetic \feld Bwould include a con-\ntribution\u00190:5gs\u0016B\u0010\nB\u0002^Si\u0011\n.\nIn order to examine whether the spin relaxation time\ndepends on how the spin imbalance is generated, we im-\nplement two general ways to initialize \u000e\u001a(t0). First, for\nsimulating pump-probe experiments, we choose \u000e\u001a(t0)\ncorresponding to interaction with a pump pulse. Sec-\nond, we use the technique proposed previously in Ref. 27\nby applying a test magnetic \feld at t=\u00001, allowing the\nsystem to equilibrate with a net spin and then turning it\no\u000b suddenly at t0.4\nB. Scattering terms\nThe scattering part of the master equation can be sepa-\nrated into contributions from several scattering channels,\nd\u001a\ndtjscatt=X\ncd\u001a\ndtjc; (8)\nwhereclabels a scattering channel. Under Born-Markov\napproximation, in general we have18\nd\u001a12\ndtjc=1\n2X\n3452\n4(I\u0000\u001a)13Pc\n32;45\u001a45\n\u0000(I\u0000\u001a)45Pc;\u0003\n45;13\u001a323\n5+H:C:; (9)\nwherePcis the generalized scattering-rate matrix and\nH.C. is Hermitian conjugate. The subindex, e.g., \\1\", is\nthe combined index of k-point and band. The weights\nof k points must be considered when doing sum over k\npoints. Note that Pcin the interaction picture is related\nto its value PS;cin the Schrodinger picture as\nPc\n1234(t) =PS;c\n1234exp [i(\u000f1\u0000\u000f2\u0000\u000f3+\u000f4)t]; (10)\nwhere\u000fiare single-particle eigenvalues of H0. Be-\nlow, we consider three separate scattering mechanisms\n- electron-impurity (e-i), electron-phonon (e-ph) and\nelectron-electron (e-e), and describe the matrix elements\nfor each.\nFor electron-phonon scattering, the scattering matrix\nis given by18\nPS;e-ph\n1234 =X\nq\u0015\u0006Aq\u0015\u0006\n13Aq\u0015\u0006;\u0003\n24; (11)\nAq\u0015\u0006\n13=r\n2\u0019\n~gq\u0015\u0006\n12q\n\u000eG\u001b(\u000f1\u0000\u000f2\u0006!q\u0015)q\nn\u0006\nq\u0015;(12)\nwhereqand\u0015are phonon wavevector and mode, gq\u0015\u0006\nis the electron-phonon matrix element, resulting from\nthe absorption (\u0000) or emission (+) of a phonon, com-\nputed with self-consistent spin-orbit coupling from \frst-\nprinciples,53n\u0006\nq\u0015=nq\u0015+ 0:5\u00060:5 in terms of phonon\nBose factors nq\u0015, and\u000eG\n\u001brepresents an energy conserv-\ning\u000e-function broadened to a Gaussian of width \u001b.\nNext, for electron-impurity scattering, the scattering\nmatrix is given by\nPS;e-i\n1234=Ai\n13Ai;\u0003\n24; (13)\nAi\n13=r\n2\u0019\n~gi\n13q\n\u000eG\u001b(\u000f1\u0000\u000f3)p\nniVcell; (14)\ngi\n13=h1jVij3i; (15)\nwhereniandVcellare impurity density and unit cell vol-\nume, respectively, and Viis the impurity potential. In\nthis work, we deal with ionized and neutral impurities\ndi\u000berently. For ionized impurities, Viis proportional\nto screened Coulomb potential54; for neutral impurities,we compute impurity potentials with supercell methods\nfrom DFT. (See Appendix A for further details).\nFinally, for electron-electron scattering, the scattering\nmatrix is given by18\nPS;e-e\n12;34=2X\n56;78(I\u0000\u001a)65A15;37A\u0003\n26;48\u001a78; (16)\nA1234=1\n2(A1234\u0000A1243); (17)\nA1234=1\n2r\n2\u0019\n~h\nge\u0000e\n1234(\u000eG\n\u001b;1234)1=2+ge\u0000e\n2143(\u000eG\n\u001b;2143)1=2i\n;\n(18)\nge-e\n1234=h1 (r)jh2 (r0)jV(r\u0000r0)j3 (r)ij4 (r0)i; (19)\nwhereV(r\u0000r0) is the screened Coulomb potential and\n\u000eG\n\u001b;1234=\u000eG\n\u001b(\u000f1+\u000f2\u0000\u000f3\u0000\u000f4) is a Gaussian-broadened\nenergy conservation function. The screening is described\nby Random-Phase-Approximation (RPA) dielectric func-\ntion (details in Appendix A). Although the above equa-\ntions describe all possible scattering processes between\nelectrons and holes, we only consider those between\nconduction electrons here, which are appropriate for n-\ntype Group III-V semiconductors30,39. The electron-\nhole scattering can be important for intrinsic and p-type\nmaterial.30,39We note that unlike the e-ph and e-i chan-\nnels,PS;e-e(as well as the dielectric screening in V) is a\nfunction of \u001aand needs to be updated during time evolu-\ntion of\u001a. This is a clear consequence of the two-particle\nnature of e-e scattering. PS;e-ecan be written as the\ndi\u000berence between a direct term and an exchange term,\nPS;e-e=PS;e-e;d\u0000PS;e-e;x; (20)\nPS;e-e;d=X\n56;78(I\u0000\u001a)65A15;37A\u0003\n26;48\u001a78; (21)\nPS;e-e;x=X\n56;78(I\u0000\u001a)65A15;37A\u0003\n26;84\u001a78: (22)\nAccording to Ref. 50, the direct term is expected to dom-\ninate the dynamical scattering processes between con-\nduction or valence electrons, allowing us to neglect the\nexchange term here.\nC. Pump-probe simulation\nIn nonrelativistic limit, the light-matter interaction\nHamiltonian operator ( bHe\u0000p) reads55\nbHe\u0000p=e\nmeA(t)\u0001bp\n+e\n2meA(t)\u0001A(t) +ge\u0016Bbs\u0001(5\u0002A(t));\nwhere A(t) is the vector potential and A(t) =\nA0(t)e\u0000i!t+A\u0003\n0(t)ei!twithA0(t) being the complex\namplitude and !being photon frequency. bpis mo-\nmentum operator. ge\u00192:0023192 is anomalous gy-\nromagnetic ratio. The second quadratic term plays a5\nrole only when pump \ruence is higher by several orders\nof magnitude than that in usual spin dynamics experi-\nments and can be safely neglected. Since 5\u0002A(t) =\n\u0000iqphoton\u0002A(t)55and the photon wavevector qphoton is\nquite small (the photon wavelength is much longer than\nthe scale of unit cells), the third term is also negligible.\nTherefore, we will only keep the \frst A(t)\u0001bpterm.\nThe interaction with a pump pulse of frequency !pump\nin the interaction picture is given by\nHpump;k;mn (!pump;t) =e\nmeA0(t)\u0001pk;mneit(\u000fm\u0000\u000fn\u0000!pump )\n+H:C:; (23)\nwherem;n represent the band indices and krepresents\nthe k point sampling in the \frst Brillouin zone. For a\nGaussian pulse centered at time tcenter with width \u001cpump,\nA0(t) =A01pp\u0019\u001cpumpexph\n\u0000(t\u0000tcenter )2=\u0000\n2\u001c2\npump\u0001i\n:\n(24)\nNote that the corresponding pump \ruence is Ipump =\n!2\npumpjA0j2=(8\u0019\u000b), where\u000bis \fne structure constant.\nAs a part of the coherent portion of the time evolution,\nthe dynamics due to this term are captured directly in\nthe Liouville form,56,57\nd\u001a\ndtjpump =\u0000i[Hpump;\u001a]: (25)\nThe probe pulse interacts with the material similarly\nto the pump pulse, and could be described in exactly\nthe same way in principle. However, this would require\nrepeating the simulation for several values of the pump-\nprobe delay. Instead, since the probe is typically chosen\nto be of su\u000eciently low intensity, we use second-order\ntime-dependent perturbation theory to capture its inter-\naction with the system,\n\u0001\u001aprobe=1\n2X\n3458\n<\n:[I\u0000\u001a(t)]13Pprobe\n32;45\u001a(t)45\n\u0000[I\u0000\u001a(t)]45Pprobe;\u0003\n45;13\u001a(t)329\n=\n;+H:C:;\n(26)\nwherePprobeis the generalized scattering-rate matrix for\nthe probe in the interaction picture. Its corresponding\nSchrodinger-picture quantity is\nPS;probe\n1234 =X\n\u0006Aprobe;\u0006\n13Aprobe;\u0006;\u0003\n24; (27)\nAprobe;\u0006\n13 =r\n2\u0019\n~e\nme\u0010\nAprobe\n0\u0001p\u0011q\n\u000eG\u001b(\u000f1\u0000\u000f3\u0006!probe):\n(28)\nThe dielectric function change \u0001 \u000fbetween the excited\nstate and ground state absorption detected by the probe\nis then\nIm\u0001\u000f=2\u0019\n(!probe)3jAprobe\n0j2Tr\u0000\nH0\u0001\u001aprobe\u0001\n: (29)Note that \u0001 \u001aprobecontainsjAprobe\n0j2so that Im\u0001 \u000fis in-\ndependent of Aprobe\n0. The above Im\u0001 \u000fis a functional\nof the density matrix according to Eq. 26 and is an ex-\ntension of the usual independent-particle Im \u000fdepending\non just occupation numbers.58After computing Im\u0001 \u000f\nabove, the real part Re\u0001 \u000fcan be obtained from the\nKrames-Kronig relation.\nBy summing up the dielectric function change \u0001 \u000fcom-\nputed above with the dielectric function for ground state\nabsorption, we can obtain the excited-state \u000fas inputs\nfor Kerr and Faraday rotation calculations.59These cor-\nrespond to the rotations of the polarization plane of a\nlinearly polarized light, re\rected by (Kerr) and trans-\nmitted through (Faraday) the material, after a pump ex-\ncitation with a circularly-polarized light. Time-Resolved\nKerr/Faraday Rotation (TRKR/TRFR) has been widely\nused to study spin dynamics of materials28,36. In a\nTRKR experiment, a circularly-polarized pump pulse is\nused to excite valence electrons of the sample to conduc-\ntion bands. The transitions approximately satisfy the\nselection rule of \u0001 mj=\u00061 for left and right circularly-\npolarized pulses, respectively, where mjis secondary to-\ntal angular momentum. TRKR works by measuring the\nchanges of polarization of re\rected light, which qualita-\ntively is proportional to the small population imbalance\nof electronic states with di\u000berent mj.\nSpeci\fcally, the Kerr rotation angle \u0012Kis computed\nwith dielectric functions by\n\u0012K=Imp\u000f+\u0000p\u000f\u0000\n1\u0000p\u000f+p\u000f\u0000; (30)\nwhere\u0006denotes the left and right circular polarization,\nrespectively.\nIII. COMPUTATIONAL DETAILS\nThe ground-state electronic structure, phonon, and e-\nph matrix element calculations of GaAs and few-layer\nWSe 2are \frst calculated using Density Functional The-\nory (DFT) with relatively coarse kandqmeshes in the\nJDFTx plane-wave DFT code.60For GaAs, we use the\nexperimental lattice constant of 5.653 \u0017A,61and select the\nSCAN exchange-correlation functional62for an accurate\ndescription of the electron e\u000bective mass (see section II\nin Supplemental Materials63). We also apply a scissor\noperator to the DFT values to reach experimental band\ngap 1.43 eV64. For WSe 2, we used PBE exchange cor-\nrelation functional along with the DFT-D2 pair poten-\ntial dispersion corrections65. The resulting lattice con-\nstant is 3.32 \u0017A and distance between two W-atom planes\nis 6.419 \u0017A close to experimental values of bulk WSe 2,\n3.297 and 6.491 \u0017A66. The phonon calculations of GaAs\nand WSe 2employ a 4\u00024\u00024 and 6\u00026 supercell, re-\nspectively. We use Optimized Norm-Conserving Van-\nderbilt (ONCV) pseudopotentials67with self-consistent\nspin-orbit coupling throughout, which we \fnd converged6\nat a plane-wave kinetic energy cuto\u000b of 34 and 62 Ry for\nGaAs and WSe 2, respectively. With these computational\nparameters, we \fnd the e\u000bective mass of conduction elec-\ntrons of GaAs to be 0.054 me, close to the experimental\nvalue of 0.067 me64. (More convergence tests can be found\nin Supporting Information (SI)63).\nWe then transform all quantities from plane wave ba-\nsis to maximally localized Wannier function basis68, and\ninterpolate them53,69{73to substantially \fner k and q\nmeshes. The Wannier interpolation approach fully ac-\ncounts for polar terms in the e-ph matrix elements and\nphonon dispersion relations, using the approach devel-\noped by Verdi and Giustino74for 3D and using the meth-\nods in Ref. 75 and Ref. 76 for 2D systems. The Born\ne\u000bective charges and dielectric constants are calculated\nfrom open-source code QuantumESPRESSO77.\nFor GaAs, the \fne kandqmeshes are 288\u0002288\u0002288\nfor simulations at 300 K and are \fner at lower tempera-\nture, e.g., 792\u0002792\u0002792 for simulations at 30 K. This\nis necessary to sample enough electronic states around\nband edges and for spin lifetime convergence within 20%.\nThekandqconvergence are easier for WSe 2due to\nmuch larger e\u000bective masses and we used 168 \u0002168 and\n600\u0002600 meshes at 50 and 10 K, respectively. The com-\nputation of e-i and e-e matrix elements and the real-time\ndynamics simulations are done with a new custom code\ninterfaced to JDFTx. The energy-conservation smearing\nparameter\u001bis chosen to be comparable or smaller than\nkBTfor each calculation. Detailed convergence tests of\nnumber of k points and energy window for electronic\nstates at various smearing parameters can be found in\nSupplemental Materials63.\nIV. RESULTS AND DISCUSSIONS\nA. Applications to n-doped GaAs\n1. Spin dynamics and its relation to TRKR\nIn general, time evolution of Kerr rotation angle \u0012K\n(see Eq. 30) is not equivalent to that of spin along the\ndirection of re\rected light, and in fact, they can be quite\ndi\u000berent in some cases41. There are few \frst-principles\nstudies of TRKR considering scattering processes in a\nform of semiclassical Boltzmann equation58. A full quan-\ntum description of scatterings with non-diagonal density\nmatrix in TRKR has not been presented in previous \frst-\nprinciples studies, to the best of our knowledge. And the\nrelation between dynamics of \u0012Kand spin observable for\ngeneral systems including GaAs has not yet been well\nexamined.\nUsing our density-matrix approach, we are able to di-\nrectly simulate the nonequilibrium ultrafast dynamics of\noptically excited systems during which the dynamics of\ndi\u000berent electronic quantities such as spin and carriers\ncan be strongly coupled. We include all scattering terms\nin a full quantum description as shown in the theory sec-\nFIG. 1. The energy-resolved dynamics of carriers (a)\n\u0001n(\u000f;t) =n(\u000f;t)\u0000n(\u000f;0) and (b) spins Sz(\u000f;t) of con-\nduction electrons with a circularly polarized pump pulse cen-\ntered at 0.5 ps. The insets on the top right of both panels\nshow \u0001n(\u000f;t) andSz(\u000f;t) at\u000f=1.45 eV. The pump energy\n!pump=1.47 eV is chosen to be higher than band gap 1.43\neV64. The width of the pump pulse \u001cpump is 100 fs. The\npump \ruence Ipump is low at 0.01 \u0016J cm\u00002. The dynamics\ncan be approximately divided into three regions - Region I,\nII and III labeled in this \fgure. In Region I, the system is\nexcited by a pump pulse. In Region II, pump processes are\nalready \fnished, then both carriers and spins relax simulta-\nneously. In Region III, carrier distribution stays unchanged\nwhile spins keep decaying.\ntion II.B and Appendix A. We perform the real-time dy-\nnamics simulations of n-type GaAs for tens of ps at room\ntemperature and several ns at low temperature until the\n\ftted spin lifetime does not change any more. Having\ntemporal density matrix, we can further analyze the dy-\nnamics of various observables, including occupation, spin\nand Kerr rotation angle easily. We then examine the re-\nlation between \u0012Kand spin in the dynamics.\nFigure 1 shows the energy-resolved dynamics of car-7\nFIG. 2. (a) Compare the dynamics of Kerr rotation angle \u0012K\nat di\u000berent probe energies !probe excited by a circularly po-\nlarized pump pulse in \frst few ps. Fast oscillations in \frst 2\nps are due to the pump pulse and coupled spin and carrier re-\nlaxation. (b) Compare relaxation of di\u000berent observables - \u0012K\nwith di\u000berent !probe (denoted by black and red lines) and Sz\nwith initial spin imbalance generated by a pump pule (Pump)\nor a test magnetic \feld along z direction \u0015s;z\u00180:001\u00000:1\nTesla (blue and green lines). !pump=1.47 eV. The pump pulse\nis centered at 0.5 ps. The longer time-scale dynamics (over 10\nps) for\u0012KandSzhas similar relaxation time independent on\nthe generation method of spin imbalance and speci\fc probe\nenergies.\nriers \u0001n(\u000f;t) =n(\u000f;t)\u0000n(\u000f;0) and spins Sz(\u000f;t).\nThe energy-resolved observable O(\u000f) is de\fned as\nRehP\nk;mnok;mn\u001ak;nm\u000e(\u000f\u0000\u000fkm)i\n, whereois operator\nmatrix. We can see that during the \frst ps (region I in\nFig. 1), both observables vary quickly due to the exis-\ntence of the pump processes and both have their maxi-\nmum at an energy slightly lower than the pump energy,\n1.47 eV (slightly larger than the band gap 1.43 eV), at\na time shortly after the time center of the pump pulse -\n0.5 ps. Interestingly, after pump being not active or af-\nter 0.8-1 ps, carriers and spins simultaneously relax until\n2-3 ps (region II in Fig. 1a and 1b). Afterward (region\nIII in Fig. 1a and 1b), carriers stay unchanged but spins\nSz(\u000f;t) decay exponentially as shown in the insets of Fig.\n1a and 1b.\nWe have further analyzed the dynamics of Kerr ro-\ntation angle \u0012Kand compared it with spin dynamics.\nFrom Fig. 2a, we can see that during pump processes and\nshortly after them (from 0 to 2 ps), \u0012K(t) has strong os-\ncillations and sensitive to the probe energy !probe. The\n!probe-sensitivity may be partly attributed to the energy\ndependence of carrier and spin dynamics. From Fig. 2a\nand 2b, it can be seen that after 3 ps (or in time region\nIII de\fned in Fig. 1), \u0012Kwith di\u000berent !probe decay ex-\nactly the same. We can also \fnd that with a pump pulse,\nrelaxation time of the Kerr rotation is the same as that of\nSz, i.e.\u001cs;z. Moreover, it turns out that \u001cs;zdoes not de-\npend on how spin imbalance is generated - by a circularly\npolarized pump pulse or by turning o\u000b a test magnetic\n\feld alongzdirection (see Sec. II A). This may indicate\nthat if the system is not extremely far from equilibrium,\nFIG. 3. Theoretical spin lifetime with (black solid square) and\nwithout (black empty square) the electron-electron scattering\ncompared with experimental data. Exp. A, B, C and D are\nexperimental data from Refs. 36,37,78 and 28, respectively.\nspin relaxation along direction iis not sensitive to the\nway of generating spin imbalance, as long as the degrees\nof freedom other than Siare not relevant or disappear\nin a short time. According to these observations, here-\ninafter, we will do real-time dynamics starting from a \u000e\u001a\ngenerated by turning o\u000b a test magnetic \feld and \ft \u001cs;z\nfrom time evolution of Sz.\nWe have also studied the e\u000bects of !pump and pump\n\ruenceIpump on spin relaxation of n-GaAs at 300 K. We\n\fnd that!pump has very weak e\u000bects on spin relaxation\nbut\u001cs;zdecreases with pump \ruence. See more details\nin Appendix C.\n2. Temperature-dependence of spin lifetime and its\ndominant relaxation mechanism\nAs discussed earlier, long-standing controversies re-\nmain for the dominant spin relaxation mechanism of\nGaAs at di\u000berent temperature and doping level30,31,39,40,\nwhich will be resolved in the following sections. We start\nfrom study \u001cs;zofn-GaAs as a function of temperature\nat a moderate doping level (2 \u00021016cm\u00003). For sim-\nplicity, we assume all impurities are fully ionized, so that\nthe impurity density niis equal to the free carrier den-\nsitynfree. We \frst compared our calculated spin lifetime\nwith experimental results in Fig. 3. Our results of \u001cs;z\nofn-GaAs give good agreement with experiments at var-\nious temperatures28,36,37,78. Di\u000berent experiments have\nslight variations between each other due to sample prepa-\nration conditions and speci\fc measurement techniques.\nThe spin lifetime increases from tens of ps at room tem-\nperature to tens of ns at low temperature. Note that e-e\nscattering plays an essential role at low temperatures,\ni.e. by comparing with (black solid square) and without\n(black empty square) in Fig. 3. The correct temperature8\nFIG. 4. Spin and carrier lifetimes of n-GaAs with ni=\n2\u00021016cm\u00003with di\u000berent scattering mechanisms and dif-\nferent phonon modes. In (a) and (b), \\All\" represents all the\ne-ph, e-i and e-e scattering mechanisms being considered. TA,\nLA and LO represent transverse acoustic, longitudinal acous-\ntic and longitudinal optical modes, respectively. The carrier\nlifetimes\u001cppresent are the inverse of averaged carrier scatter-\ning rates\n\u001c\u00001\np\u000b\n. The method of carrier lifetime calculations is\ngiven in Appendix B. himeans taking average around chem-\nical potential \u0016. For a state-resolved quantity Akn, its aver-\nage is de\fned ashAi=P\nknAkn[feq]0(\u000fkn)=P\nkn[feq]0(\u000fkn),\nwhere [feq]0is the derivative of Fermi-Dirac function.\ndependence of \u001cs;zcan be reproduced only if e-e scatter-\ning is included.\nWe further examine the contributions of di\u000berent scat-\ntering mechanisms to carrier and spin lifetime respec-\ntively, as a function of temperature. Di\u000berent from\nspin lifetime obtained from real-time DM dynamics in-\ncluding all scattering processes simultaneously, the car-\nrier lifetime ( \u001cp) is de\fned through the inverse of the\naveraged carrier scattering rate( \u001c\u00001\np):\u001cp= 1=\n\u001c\u00001\np\u000b\n.\nVarious scattering processes (e-e, e-i and e-ph) con-\ntribute to the total carrier scattering rates through\n\u001c\u00001\np= (\u001ce\u0000e\np)\u00001+ (\u001ce\u0000i\np)\u00001+ (\u001ce\u0000ph\np)\u00001.himeans tak-\ning average around chemical potential \u0016. For a state-\nresolved quantity Akn, its average is de\fned as hAi=P\nknAkn[feq]0(\u000fkn)=P\nkn[feq]0(\u000fkn), where [feq]0is the\nderivative of Fermi-Dirac function. For both carrier and\nspin lifetime, the lifetime due to the most dominant scat-\ntering channel is the closest to the one including all pro-\ncesses (black squares in both Fig. 4a and b). For spin\nrelaxation in Fig. 4a, at low temperature below 50 K,\ne-e scattering is the most dominant process as discussed\nabove. However, the e-ph process becomes more domi-\nnant above 100K. On the other hand, for carrier relax-\nation in Fig. 4b, the e-i process is dominant over a wide\ntemperature range from low to right below room tem-\nperature. At room temperature, for both spin and car-\nrier lifetimes, the e-ph scattering is the most important\nprocess (closest to the total lifetime with all scatteringprocesses).\nOur observations di\u000ber from those in Refs. 31 and 40,\nwhere the authors also found that e-e scattering domi-\nnates spin relaxation at lower temperatures, e.g. 77 K,\nbut their results showed that at room temperature e-\ne scattering can be more important than other scatter-\nings and enhances \u001cs;zofn-GaAs by about 100% with\nmoderate doping concentrations. The overestimate of\nthe e\u000bects of e-e scattering at room temperature is most\nlikely a limitation of the semiclassical method employed\ntherein.\nSimilarly, we also \fnd that di\u000berent phonon modes\ncan play di\u000berent roles in carrier and spin relaxations as\nshown in Fig. 4c and 4d. For example, at room temper-\nature, LO (longitudinal optical) mode is most important\nfor carrier relaxation but seems less important than TA\n(transverse acoustic) modes for spin relaxation. The sit-\nuation is the opposite at 100 K where TA/LO is most\nimportant for carrier/spin relaxation. Our \fnding that\nTA modes are slightly more important than LO mode\nin spin relaxation at room temperature is di\u000berent from\nwhat have been believed in previous model studies30,40,\nwhere they declared that the electron-LO-phonon scat-\ntering dominates spin relaxation at high temperatures\nespecially at room temperature. This disparity is most\nlikely due to di\u000berences in the e-ph matrix elements and\nelectronic quantities, where we used fully \frst-principles\napproaches instead of parameterized models in previous\nwork.\nIn addition, we \fnd the total spin lifetime is the longest\nwhen considering all scattering processes in Fig. 4a; in\ncontrast, the carrier lifetime is the shortest including all\nscattering mechanism in Fig. 4b. This follows the inverse\nrelation between spin and carrier lifetime in the empir-\nical D'yakonov{Perel' (DP) mechanism2,14for systems\nwithout inversion symmetry, as will be discussed in more\ndetails in next section.\n3. Doping-level-dependence of spin lifetime and its\ndominant relaxation mechanism\nFigure 5 shows the carrier and spin lifetimes with dif-\nferent doping density niat 30 K with individual and\ntotal scattering pathways, respectively. Similar to tem-\nperature dependence and phonon contributions, it is also\nfound that the roles of di\u000berent scattering mechanism dif-\nfer considerably between spin and carrier relaxation pro-\ncesses. Speci\fcally, for the carrier relaxation in Fig. 5b,\nexcept when niis very low (e.g. at 1014cm\u00003), the\nelectron-impurity scattering (e-i) dominates, similar to\nthe case of carrier lifetime over a large range of temper-\nature at a moderate doping in Fig. 4b. On the other\nhand, for the spin relaxation in Fig. 5a, the e-e scatter-\ning dominates except at very high concentration (above\n1017cm\u00003), while e-i scattering is only important in the\nvery high doping region (close to or above 1017cm\u00003).\nFigure 5 shows the calculated \u001cshas a maximum at9\nFIG. 5. (a) Spin and (b) carrier lifetimes of n-GaAs with di\u000berent doping concentrations at 30 K with di\u000berent scattering\nmechanisms. \\All\" represents all the e-ph, e-i and e-e scattering mechanisms being considered. (c)\n\n2\u000b\n\u0000\n\n2\ni\u000b\nas a function\nof carrier density, where \nis the Larmor frequency due to the \\internal\" magnetic \feld computed from \frst-principles, which\ndescribes the SOC term induced by inversion asymmetry.\nni= 1-2\u00021016cm\u00003, and\u001csdecreases fast with ni\ngoing away from its peak position. This is in good agree-\nment with the experimental \fnding in Ref. 28, which\nalso reported \u001csatni= 1016cm\u00003is longer than \u001cs\nat other lower and higher niat a low temperature (a\nfew Kelvin). The nidependence of \u001csmay be qualita-\ntively interpreted from the commonly used empirical DP\nrelation2for inversion-asymmetric systems, \u001cs;i\u0018\u001cDP\ns;i=\n1=\u0002\n\u001cp\u0001\u0000\n\n2\u000b\n\u0000\n\n2\ni\u000b\u0001\u0003\n, where\u001cpis the carrier lifetime,\nand\nis the Larmor frequency due to the \\internal\"\nmagnetic \feld, which describes the SOC term induced\nby inversion asymmetry. For spin 1/2 systems, the inter-\nnal magnetic \feld at k(\nk) will induce an energy split-\nting \u0001 kand polarize the spin along the direction of \nk.\nPreviously, \nkwas mostly obtained with model Hamilto-\nnian with Dresselhaus SOC \feld79, which is rather qual-\nitative. Instead, we obtained k-dependent internal mag-\nnetic \feld \nkfrom \frst-principles calculations, by using\n\nk;i= 2\u0001 k\u0001sexp\nk;i=~, wheresexp\nk;iis the spin expectation\nvalue.\nFrom Fig. 5, we \fnd that with nifrom 1014cm\u00003\nto 5\u00021015cm\u00003, carrier lifetime \u001cpdecreases rapidly\n(black curve in Fig. 5b) and\n\n2\u000b\n\u0000\n\n2\ni\u000b\nremains \rat\nin Fig. 5c, which may explain why spin lifetime ( \u001cs) in-\ncreases in Fig. 5a based on the DP relation; however,\nwhenni>1016cm\u00003,\u001cpdecreases with a similar speed\nbut\n\n2\u000b\n\u0000\n\n2\ni\u000b\nexperiences a sharp increase, which may\nexplain why spin lifetime decreases in Fig. 5b and owns\na maximum at 1016cm\u00003.\nNote that although the above empirical DP relation is\nintuitive to understand the cause of doping-level depen-\ndence of spin lifetime, it may break down when we eval-\nuate individual scattering processes. For example, when\nniincreases from 1014cm\u00003to 1015cm\u00003, both carrier\nlifetime\u001cpand spin lifetime \u001cs;zdue to e-i scattering\ndecrease while the internal magnetic \feld remains un-\nchanged. Moreover, the simple empirical relation cannot\npossibly explain our \frst-principles results that the e-e\nand e-i scatterings have largely di\u000berent contributions in\ncarrier and spin relaxation. First-principles calculationsare critical to provide unbiased mechanistic insights to\nspin and carrier relaxation of general systems.\nB. Applications to few-layer WSe 2\n1. Spin/valley relaxation of resident holes of monolayer\nWSe 2\nFor holes of monolayer WSe 2, spin/valley relaxation\nis mostly determined by intervalley spin-\rip scattering\nprocesses between KandK0valleys because of the spin-\nvalley locking. Previously, we reported spin/valley life-\ntimes of resident holes of monolayer TMDs at T \u001550 K\nwith e-ph scattering27. At very low temperatures, e.g., 10\nK, intervalley e-ph scattering is however not activated as\nthe corresponding phonon occupation is negligible; there-\nfore, other scattering mechanisms are necessary to be in-\ncluded. Note that e-e scattering should not play an im-\nportant role in spin relaxation of holes of TMDs. The rea-\nson is: The e-e scattering is a two-particle process where\na transition is accompanied by another transition with\nenergy and momentum being conserved. Considering the\nfact that only the highest occupied band is involved (see\nband structure in Fig. S5) in dynamics of TMD holes,\nfor an e-e process, a K!K0(K0!K) spin-\rip transition\nmust be accompanied by an opposite K0!K(K!K0)\nspin-\rip transition. Overall, e-e scattering processes have\nnegligible contributions to spin relaxation of TMD semi-\nconductors. As a result, we will include only e-ph and\ne-i scatterings for WSe 2. We use the supercell method\nto compute e-i scattering matrix elements for neutral de-\nfects with self-consistent SOC and more details can be\nfound in Appendix A.\nExperimentally several types of impurities/defects ex-\nist in TMD samples. Here we pick four types of impu-\nrities with di\u000berent symmetries and chemical bonds (see\nFig. 6(a)) - Se vacancy (V Se), two neighboring Se vacan-\ncies (V 2Se\u0000N), W vacancy (V W) and two Se vacancies\nwith the same in-plane position (V 2Se\u0000S). As most point10\nFIG. 6. (a) The schematics of four types of impurities in\nWSe 2. (b) Spin lifetimes of holes of monolayer WSe 2with\na relatively low hole density 1011cm\u00002with impurities com-\npared with experimental data. Exp. A, B, C and D are exper-\nimental data from Refs. 42, 80, 46 and 81, respectively. The\nchoices of impurity concentration niof di\u000berent impurities are\ngiven in the main text.\ndefects are relatively deep with large ionization energies\nin semiconducting TMDs82, we mostly consider neutral\ndefects here. According to Refs. 83{85, the impurity con-\ncentrationniranges from 8\u00021010to 1014cm\u00002depend-\ning on samples. Considering that V Seis often regarded\nas the most abundant impurity, we choose a reasonable\nimpurity density niof V Se- 7\u00021011cm\u00002, within the\nexperimental range and for better comparison with ex-\nperimental \u001csat T\u001520 K shown in Fig. 6(b). niof\nV2Se\u0000Nis chosen as 8\u0002109cm\u00002, two order of magnitude\nlower than V Sebecause of its larger formation energy85\nand better comparison with experimental \u001cs.niof V W\nand V 2Se\u0000Sare chosen arbitrarily as we \fnd they have\nrather weak e\u000bects on spin relaxation and are 7 \u00021011and\n3.5\u00021011cm\u00002, respectively.\nFrom Fig. 6, we \frst \fnd that at T >20 K, spin relax-\nation is almost driven by e-ph scattering and impurities\ncan only a\u000bect spin relaxation at T \u001420 K. For the ef-\nfects of di\u000berent impurities on spin relaxation, we have\nV2Se\u0000N\u001dVSe\u001dVW\u0018V2Se\u0000S. Such di\u000berences are\ndirectly related to the large di\u000berences among various\nimpurities in electron-impurity matrix elements for the\nintervalley processes (scattering between KandK0val-\nley), i.e. much larger matrix elements jgijfor interval-\nFIG. 7. (a) Schematic diagram of scattering pathways of\nexcited holes in 1% compressed bilayer WSe 2with a low equi-\nlibrium hole density and 1.4 \u00021012cm\u00002VSe(double of the\nmonolayer value in Fig. 6) under \fnite Ezduring and after\na circularly polarized pump, which initially excite holes (la-\nbelled as half circles) at both KandK0valleys and two top\nvalence bands (\\A +\" atK0and \\B+\" atK). The lower-energy\nexcited holes will decay to the band edge (most to \\A +\" atK0\nand fewer to \\A*\" atK) through di\u000berent scattering path-\nways. A state being labeled by \\A\"(\\B\") means the wave-\nfunction of this state is mostly localized in layer A(B) of the\nbilayer.*and+represent spin-up and spin-down, respec-\ntively. The color of electronic state represents spin polariza-\ntion - red and blue mean spin-up and spin-down, respectively.\n(b) Time evolutions of valley- and band-resolved (excited or\nexcess) hole densities at 50 K under two Ezwith a circularly\npolarized pump. The pump energy is selected to excite elec-\ntrons at two top valence bands. The pump center is at 0.5\nps.nVrepresents excess hole density at valley V. The insets\nof panel (b) are the schematics of energies of two top valence\nbands atK,K0and \u0000 valleys under two Ez(see calculated\nband structures in SI Fig. S7).\nley scattering at V 2Se\u0000Nand V Secompared to the ones\nat V Wand V 2Se\u0000Sas shown in SI Fig. S9. Moreover,\nthe temperature dependence of \u001cswith V 2Se\u0000Nis much\nweaker and in better agreement with experiments than\nthat with V Se. Therefore, the observed weak tempera-\nture dependence in some experiments is probably related\nto the existence of larger size impurities with lower sym-\nmetries (e.g. V 2Se\u0000N). Our observations suggest that\nthe local symmetry and chemical bonds surrounding an\nimpurity have large impact on spin relaxation.11\nAdditionally, we have simulated spin lifetimes of mono-\nlayer WSe 2at 15 K with di\u000berent hole densities and \fnd\na strong hole density dependence. The related results are\nshown in SI Fig. S12.\n2. Ultrafast dynamics of holes of bilayer WSe 2\nUnderstanding detailed dynamical processes and re-\nlated scattering mechanism can help develop strategy\nof controlling and manipulating spin/valley relaxation\nthrough tuning external \felds, materials composition,\nand strain. In the following, we will \frst identify the\nscattering pathways of excited holes and spins in bilayer\nWSe 2(AB stacking as shown in Fig. 7a). Next through\nreal-time simulations, we determine dynamical quanti-\nties like spin lifetimes and valley polarization at di\u000berent\nexternal \felds and strain. Finally, we show the carrier\noccupation on each layer is fully spin polarized and can\nbe switched by an external electric \feld.\nAccording to previous studies86,87and our calcula-\ntions, for valence bands of unstrained bilayer WSe 2, \u0000\nvalley is slightly higher than K/K0valley, which is usu-\nally undesirable. The K/K0valley can be pushed higher\nthan \u0000 valley by applying slight in-plane compressive\nstrain (corresponding band structures can be found in\nSI Fig. S6). Moreover, under zero Ez, the bilayer WSe 2\nhas inversion symmetry, leading to carriers being equally\npopulated at KandK0valleys and at two layers, with\nspin up and down degeneracy. A \fnite Ezcan break in-\nversion symmetry (thus break Kramers degeneracy) and\ninduce non-zero layer polarization or layer population dif-\nference. For example, two top valence bands from layer\nA with*(up spin) and layer B with +(down spin) are\ndegenerate at Kwithout electric \feld but split under\nelectric \feld. Thus each band is associated with a partic-\nular spin channel, valley and layer, i.e. spin-valley-layer\nlocking e\u000bect. Also by tuning the sign and magnitude of\nEz, we are able to control various physical quantities like\nlayer pseudospin, band splitting energy, etc. Therefore,\nto ensure spin-valley-layer locking e\u000bects being observed,\nwe will study slightly compressed inversion-symmetric bi-\nlayer WSe 2under \fnite Ez.\nFigure 7(a) shows the scattering pathway schematics\nfor hole bands (half circles) during the \frst few ps of\nslightly compressed (1%) bilayer WSe 2excited by a cir-\ncularly polarized pump pulse under \fnite Ezat 50 K,\nbefore exciton recombination processes happen typically\nat tens of ps timescale at this temperature.88,89Similar\nto the GaAs case (see Sec. IV A 1), the hole spins will un-\ndergo the following processes: optical generation, decay\nto band edges and at the end slow relaxation. Initially,\nholes with the same spin polarization are excited at both\nKandK0valleys and two layers equally (e.g. down spin\nholes generated at Kvalley and layer B and K0valley\nand layer A as shown in Fig. 7(a)). During and after\nthe excitations, lower-energy holes at K(K0) valley will\ndecay to the band edge through two possible scattering\nFIG. 8. Spin lifetime and maximum valley polarization\nPV=jnK\u0000nK0j=ntotof bilayer WSe 2at 50 K, where nK(K0)\nis excess hole density at K(K0) valley and ntotis total ex-\ncess hole density. Negative strain means compressive strain.\nPVwill be 90% of its maximum shortly after the pump and\nreach the maximum at a time from 1 to 20 ps depending on\nEzand the strain. After reaching its maximum, since most\ncarriers/spins have already decayed to the band edge, PVcan\nrelax only very slowly through intervalley spin-\rip scattering.\nThis implies that having a high maximum of PVis important\nto ensure a high PVduring a long time.\npathways: (i) Direct pathway through interlayer spin-\nconserving scattering (solid blue line); (ii) Indirect path-\nway through \u0000-valley-related scattering (dashed lines for\nboth spin conserving and \rip processes). After all holes\ndecay to the band edge, most of their carried spins are\n\\locked\" at a certain layer and valley (e.g., \\A +\" atK0in\nFig. 7) due to weak intervalley spin-\rip scattering (green\narrow in Fig. 7(a)), which is the so-called \\spin-valley-\nlayer locking\".\nIn Fig. 7(b), we show time evolutions of valley- and\nband-resolved (excited or excess) hole densities under two\nEz. It can be seen that under a low Ez(0.5 V/nm, Fig.\n7(b) left panel), the main scattering pathway is the direct\none mentioned above, i.e. the spin down holes scattered\nfrom B+atKin dashed blue line with decreasing pop-\nulation to A+atK0in solid blue line with increasing\npopulation. Under a higher Ez(2 V/nm, Fig. 7(b) right\npanel), although the direct scattering pathway still exists,\nthe indirect one through the \u0000 valley also becomes impor-\ntant because the band energy at \u0000 is pushed higher than\nthe second valence band at KandK0under this electric\n\feld (see inset of Fig. 7(b) right panel). Here occupation\nat B+atKin dashed blue line rapidly decreased while\noccupation at \u0000 with both up and down spins in solid\nblack temporarily increased through indirect scattering,\nand most importantly the A *atKin solid red also in-\ncreased due to the scattering through \u0000 valley. Increased\npopulation at A*atKrepresents weakening the spin-\nvalley-locking e\u000bect. Therefore, the indirect scattering\npathway will lead to the reduction of spin density and\nvalley polarization.\nWe then show two key dynamical quantities - spin life-\ntime\u001csin Fig. 8(a) and maximum valley polarization\nof bilayer WSe 2in Fig. 8(b) as a function of Ezwith12\nFIG. 9. Time evolution of top layer hole occupation ftop\nhand\nspinstop\nz(see their de\fnitions in Appendix D) normalized by\ncorresponding total quantities of compressed bilayer WSe 2at\n50 K after the sign of Ezis switched at t= 0. Att=\u000010 ps,\na pump pulse centered at t=\u00009:5 ps with\u001cpump = 100 fs is\nstarted to be applied and real-time density matrix dynamics\nis run under Ez= 0:5 V/nm until t= 0 to allow holes/spins\nto decay to the band edge where states are localized in bottom\nlayer. When the sign of Ezis suddenly switched, holes/spins\nare still localized in the same bottom layer but the new eigen-\nstates around the band edge are localized in top layer, thus\nholes/spins will transfer from bottom to top layer.\ndi\u000berent percentages of strain. Valley polarization is de-\n\fned asPV=jnK\u0000nK0j=ntot, wherenK(K0)is excess\nhole density at K(K0) valley and ntotis total excess\nhole density. A high maximum of PVis necessary to\nensure a high PVfor extended time. This is because:\nwhenPVreaches its maximum, since most carriers/spins\nhave already decayed to the band edge at the same time,\nPVwill decay very slowly through intervalley spin-\rip\nscattering. Obviously, unstrained system is not suitable\nto utilize spin-valley-layer locking considering its short\nspin lifetimes and low maximum PVunder a relatively\nlow electric \feld. As we discussed above, a slight strain\nis helpful to ensure K=K0valleys are su\u000eciently higher\nthan \u0000 valley. Moreover, from Fig. 8, an optimized range\nofEzis from\u00180.2 to\u00182 V/nm, where the system shows\nlong\u001csand high maximum PV, consistent with the fact\nmentioned above: indirect scattering pathway, which be-\ncomes important under high Ez, will cause loss of spin\nand valley polarization.\nFinally, in a bilayer, the percentage of carriers/spins\nlocalized in top/bottom layer at equilibrium can be con-\ntrolled by a static Ez. For device applications, it may\nbe desirable to dynamically and spatially tune the loca-\ntions of holes/spins by controlling Ez. For this purpose,\na knowledge of the speed of carriers/spins transferring\nbetween two layers can be useful.\nTo extract the time scale of such transfer, we \frst\ngenerate holes/spins by applying a circularly polarizedpump pulse centered at \u00009:5 ps and let the system\nevolves until t= 0 under Ez= 0:5 V/nm to ensure\nalmost all holes/spins being localized in bottom layer,\nthen by switching Ezsuddenly to\u00000:5 V/nm at t= 0,\nholes/spins will start to transfer to top layer.\nIn Fig. 9, we show time evolution of top layer hole\noccupation ftop\nhand spinstop\nz(see their de\fnitions in\nAppendix D) normalized by corresponding total quan-\ntities of 2% compressed bilayer after the sign of Ezis\nswitched. From Fig. 9, at 50 K, 90% switching of ftop\nh\nandstop\nztakes\u00186 ps. This time constant is much shorter\nthan\u001cs, which means such tuning is fast enough to use\nan electric \feld as a \"switch\" in spintronic devices.\nV. CONCLUSIONS\nIn this article, we present a \frst-principles real-time\ndensity-matrix approach to simulate ultrafast spin-orbit-\nmediated spin dynamics in solids with arbitrary crystal\nsymmetry. The complete ab initio descriptions of pump,\nprobe and three scattering processes - the electron-\nphonon, electron-impurity and electron-electron scatter-\ning in the density-matrix master equation, allows us to di-\nrectly simulate the nonequilibrium ultrafast pump-probe\nmeasurements and makes our method applicable to any\ntemperatures and doping levels. This method has been\napplied to simulate spin relaxation of n-GaAs. We con-\n\frm that relaxation time of Kerr rotation and that of spin\nobservables are almost identical and \fnd that relaxation\ntime of spin polarization is relatively robust, i.e. insen-\nsitive to how spin imbalance is initialized. Furthermore,\nwe have studied the temperature and doping-level depen-\ndencies of spin lifetime and examined the roles of various\nscattering mechanisms. Overall our theoretical results\nare in good agreement with experiments. Importantly,\nour \frst-principles simulations provide rich mechanistic\ninsights of spin relaxation of n-GaAs: we point out that\nalthough at low temperatures and moderate doping con-\ncentrations e-i scattering dominates carrier relaxation,\ne-e scattering is the most dominant process in spin re-\nlaxation. The relative contributions of phonon modes\nalso vary considerably between spin and carrier relax-\nation. We have further examined ultrafast dynamics in\nfew-layer WSe 2with realistic impurities. We \fnd that\nspin relaxation can highly depend on local symmetry and\nchemical bonds surrounding impurities. For the bilayer,\nwe identify the scattering pathways of holes in ultrafast\ndynamics and determine relevant dynamical properties,\nincluding\u001cs, maximum valley polarization and layer pop-\nulation/spin switch time, which are essential to utilize\nits unique spin-valley-layer locking e\u000bects. Our method\nopens up the pathway to predict spin relaxation and de-\ncoherence for general materials and provide unbiased in-\nsights and guidelines to experimental materials design,\nwhich have the potential to revolutionize the \feld of spin-\ntronics and quantum information technologies.13\nACKNOWLEDGEMENTS\nWe thank Hiroyuki Takenaka for helpful discussions.\nThis work is supported by National Science Founda-\ntion under grant No. DMR-1956015 and the Air Force\nO\u000ece of Scienti\fc Research under AFOSR Award No.\nFA9550-YR-1-XYZQ. A. H. acknowledges support from\nthe American Association of University Women(AAUW)\nfellowship program. This research used resources of the\nCenter for Functional Nanomaterials, which is a US DOE\nO\u000ece of Science Facility, and the Scienti\fc Data and\nComputing center, a component of the Computational\nScience Initiative, at Brookhaven National Laboratory\nunder Contract No. DE-SC0012704, the lux supercom-\nputer at UC Santa Cruz, funded by NSF MRI grant\nAST 1828315, the National Energy Research Scienti\fc\nComputing Center (NERSC) a U.S. Department of En-\nergy O\u000ece of Science User Facility operated under Con-\ntract No. DE-AC02-05CH11231, the Extreme Science\nand Engineering Discovery Environment (XSEDE) which\nis supported by National Science Foundation Grant No.\nACI-154856290, and resources at the Center for Compu-\ntational Innovations at Rensselaer Polytechnic Institute.\nAPPENDIX A: INTERACTION HAMILTONIAN\nTERMS AND MATRIX ELEMENTS\nThree interaction Hamiltonian terms in Eq. 3 read\nHe-ph=X\n12q\u0015cy\n1c2\u0010\ngq\u0015\u0000\n12bq\u0015+gq\u0015+\n12by\nq\u0015\u0011\n; (31)\nHe-i=niVcellX\n12cy\n1c2gi\n12; (32)\nHe-e=X\n1234cy\n1cy\n2c3c4ge-e\n1234: (33)\nThe e-ph matrix gq\u0015\u0006are computed with self-\nconsistent spin-orbit coupling and Wannier interpolation\nby the supercell method.\nHere we assume impurity density is su\u000eciently low and\nthe average distance between neighboring impurities is\nsu\u000eciently long so that the interactions between impuri-\nties are negligible. The e-i matrix giis\ngi\n13=h1j\u0001Vij3i; (34)\n\u0001Vi=Vi\u0000V0; (35)\nwhereViis the potential of the impurity system and V0\nis the potential of the pristine system. In this work, gifor\nneutral and ionized impurities are computed di\u000berently\nas follows.\nFor neutral impurities, Viis computed using a large\nsupercell including an impurity with self-consistent SOC\nat DFT. This is important for including the detailed po-\ntential pro\fle and chemical bonding environment for dif-\nferent impurities. To speed up the supercell convergence,we used the potential alignment method developed in\nRef. 91. We checked the supercell size convergence of\ngifor neutral V Sein monolayer WSe 2and found that\nthe corresponding spin lifetime with 6 \u00026 and 8\u00028 super-\ncells di\u000bers by only a few percent. Thus 6 \u00026 supercell is\nenough for giof neutral impurities in monolayer WSe 2.\nFor ionized impurities, we use approximate impurity\npotentials as detailed below. In general, \u0001 Vimay\nbe separated into two terms \u0001 Vi= \u0001Vi\nns+ \u0001Vi\nsoc;\nwhere \u0001Vi\nnsis the spin-independent part and \u0001 Vi\nsoc=\u0002\n~=\u0000\n4m2c2\u0001\u0003\nr\u0000\n\u0001Vi\nns\u0001\n\u0002p\u0001\u001bis the SOC correction. For\nionized impurities, we approximate \u0001 Vi\nnsas the poten-\ntial of point charge and is simply the product of the im-\npurity charge Zand the screened Coulomb potential54,\ni.e., \u0001Vi\nns=ZVscr. Such approximate describes the\nlong-range part of spin-independent di\u000berential impurity\npotential \u0001 Vi\nnsaccurately, which is often the most im-\nportant contribution from ionized impurities. Consider-\ning that \u0001Vi\nsocis commonly neglected in previous theo-\nretical studies on spin relaxation30,92whenever screened\nCoulomb potential Vscris used, we will not include \u0001 Vi\nsoc\nfor ionized impurities either. Note that such potential for\nionized impurities still relaxes spin through spin-mixing\nand spin-precession. The e-e matrix ge-eis\nge-e\n1234=h1 (r)jh2 (r0)jV(r\u0000r0)j3 (r)ij4 (r0)i;(36)\nwhereV(r\u0000r0) is the screened electron-electron inter-\naction. The SOC corrections on V(r\u0000r0)93,94will not\nbe included similar to the ionized impurity case. Thus,\nV(r\u0000r0) is simply the screened Coulomb potential Vscr.\nTherefore, in both calculations of giandge-e, the compu-\ntation of the screened Coulomb potential Vscris of key\nimportance.\nCurrently, we use the static RPA (Random Phase Ap-\nproximation) dielectric function for the screening and\nneglect local-\feld e\u000bects. We then show the e-e self-\nenergy (Im\u0006) obtained with such dielectric function well\nreproduces the one obtained with dynamically screened\nCoulomb interaction with full RPA dielectric matrix in\nthe relevant energy range as shown in Fig. 10. The di-\nelectric function has the form\n\u000f(q) =\u000fs\u000fintra(q); (37)\nwhere\u000fsis the static background dielectric constant\nand can be calculated by Density Functional Perturba-\ntion Theory (DFPT)95.\u000fintra(q) is the intraband contri-\nbution which involves only states with free carriers and is\ncritical for doped semiconductors. It is computed using\nRandom Phase Approximation (RPA),\n\u000fintra(q) =1\u0000Vbare(q)X\nkmn0\n@fk\u0000q;m\u0000fkn\n\u000fk\u0000q;m\u0000\u000fk;n\u0002\njhuk\u0000q;mjuknij21\nA;\n(38)\nwhere the sum runs over only states having free car-\nriers, e.g., for a n-doped semiconductor, mandnare14\nconduction band indices. In the above formula, fis\ntime-dependent non-equilibrium occupation instead of\nthe equilibrium one feq. Therefore, if pump is activated\nor optical \feld A0(t) of the pump pulse is not negligible,\n\u000fintra(q) will be updated in every time step, as fwill dif-\nfer fromfeqand the magnitude of di\u000berence depends on\nthe excitation density. Vbare(q) =e2=\u0000\nVcell\"0jqj2\u0001\nis the\nbare Coulomb potential with Vcellthe unit cell volume\nand\"0vacuum permittivity. uknis the periodic part of\nthe Bloch wave function.\nWe then have the matrix elements in reciprocal space,\ngi\n13=ZVscr(q13)hu1ju3i; (39)\nge-e\n1234=Vscr(q13)\u000ek1+k2;k3+k4hu1ju3ihu2ju4i;\n(40)\nVscr(q13) =Vbare(q13)=\u000f(q13); (41)\nwhereVscr(q) is the screened Coulomb potential and\nq13=k1\u0000k3.\u000ek1+k2;k3+k4is Kronecker delta function\nand means k1+k2=k3+k4.hu1ju3iis the overlap\nmatrix element between two periodic parts of the Bloch\nwave functions.\nAPPENDIX B: CARRIER SCATTERING RATE\nAND IM \u0006FROM THE DENSITY-MATRIX\nAPPROACH\nAt the semiclassical limit, density matrix \u001ais replaced\nby (non-equilibrium) occupation f, then the scattering\nterm originally with a full quantum description in Eq. 9\nrequired by DM dynamics becomes:\ndf1\ndtjc=X\n26=1\u0002\n(1\u0000f1)Pc\n11;22f2\u0000(1\u0000f2)Pc\n22;11f1\u0003\n;(42)\nusing the facts that P11;22is real and \\2=1\" term is\nzero. \\c\" represent a scattering channel. Note that the\nweights of k points must be considered when doing sum\nover k points.\nSupposefis perturbed from its equilibrium value by\n\u000ef, i.e.,f=feq+\u000ef, then insert fafter perturbation\ninto Eq. 42 and linearize it,\ndf1\ndtjc=\u0000X\n26=1\u0002\nPc\n11;22feq\n2+ (1\u0000feq\n2)Pc\n22;11\u0003\n\u000ef1;(43)\nusing the fact that \u000eP11;22is always zero, even for the\ne-e scattering.\nDe\fne carrier relaxation time of state \\1\" \u001cc\np;1by\ndf1\ndtjc=\u0000\u000ef1\n\u001cc\np;1, we have\n1\n\u001cc\np;1=X\n26=1\u0002\nPc\n11;22feq\n2+ (1\u0000feq\n2)Pc\n22;11\u0003\n: (44)\nThe linewidth or the imaginary part of the self-energy\nfor the scattering channel cis related to the carrier re-\nlaxation time by Im\u0006c\n1=~=\u0000\n2\u001cc\np;1\u0001\n.\nFIG. 10. Im\u0006 due to e-e scattering of valence electrons of p-\ntype silicon computed by Eq. 46 (Density-Matrix) compared\nwith those calculated by the \fnite-temperature GW method\n(FT-GW)97.\u0016is set to 0.05 eV lower than Valence Band\nMaximum (VBM). For simplicity, SOC is not considered in\nthis test.\nUsing Eq. 44, we have calculated the e-ph scattering\nrates and they are in good agreement with previous the-\noretical results96. For e-ph scattering, Eq. 44 will repro-\nduce the imaginary part of the well-known Fan-Migdal\nself-energy53.\nFor e-i scattering, we have\n1\n\u001ce-i\np;1=2\u0019\n~niVcellX\n2jgi\n12j2\u000eG\n\u001b(\u000f1\u0000\u000f2): (45)\nThe above equation (Eq. 45) is consistent with Ref. 54.\nFor e-e scattering, neglecting the exchange contribu-\ntion, which is a commonly-used approximation50,97,\n1\n\u001ce-e\np;1=2\u0019\n~X\n26=1;34jA1324j22\n4feq\n2feq\n4(1\u0000feq\n3) +\n(1\u0000feq\n2)feq\n3(1\u0000feq\n4)3\n5:\n(46)\nTo verify our implementation of e-e scattering term, we\nhave calculated Im\u0006 due to e-e scattering of valence\nelectrons of p-type silicon based on the above equa-\ntion and compare it with those calculated by the \fnite-\ntemperature GW method from \frst-principles,97as im-\nplemented in JDFTx.60The JDFTx implementation,\nin turn, has been benchmarked to reproduce the ex-\npected dependence with temperature and carrier energy,\nIm\u0006e-e/(\"\u0000\"F)2+ (\u0019kBT)2, as expected for metals.73\nFrom Fig. 10, we can see the results by two methods\nagree well for the energy range close to the Fermi level\nwhich is relevant to e-e scatterings due to energy conser-\nvation. This veri\fes our implementation of e-e scattering\npart.15\nFIG. 11.Sz(t) ofn-GaAs with ni= 2\u00021016cm\u00003at 300\nK with di\u000berent pump pulse energies ( !pump) varying with\nseveralkBT.\nFIG. 12. (a) The excitation density as a function of the\npump \ruence (left panel) and (b) the spin lifetime as a func-\ntion of the excitation density generated by a circularly polar-\nized pump pulse for n-GaAs with ni= 1014cm\u00003at 300 K.\n!pump=1.47 eV.\nAPPENDIX C: THE EFFECTS OF !pump AND\nPUMP FLUENCE ON SPIN RELAXATION OF\nGAAS AT 300 K\nIn Fig. 11, we study the Szrelaxation dependence on\npump-pulse energy changes with several kBT. We can\nsee that variation of !pump has very weak e\u000bects on spin\ndynamics of n-GaAs at 300 K.\nIn Fig. 12a, we study the e\u000bects of the pump \ruence\nIpump on spin relaxation. Firstly, we can see that in the\nlow pump \ruence region or when Ipump<1\u0016J/cm\u00002, the\nexcitation density increases linearly with Ipump but when\nIpump>1\u0016J/cm\u00002, the excitation density increases\nslower. This is because in high \ruence cases, during the\nexcitation by a pump pulse, a signi\fcant amount of con-\nduction states have been already \flled, which reduce the\nprobability of the transitions from valence bands to con-duction bands. From Fig. 12b, we \fnd that spin lifetime\nofn-GaAs decreases with the excitation density. This\ndependence may be explained based on the empirical DP\nrelation2\u001cs;i\u0018\u001cDP\ns;i= 1=\u0002\n\u001cp\u0001\u0000\n\n2\u000b\n\u0000\n\n2\ni\u000b\u0001\u0003\nas we dis-\ncussed in Sec. IV A 2. At 300 K, generally\n\n2\u000b\n\u0000\n\n2\ni\u000b\nwill increase with increasing free carrier density (through\nan increase of excitation density here), similar to what\nwe \fnd at 30 K shown in Fig. 5c. On the other hand, \u001cp\ndue to the electron-phonon scattering, which dominates\ncarrier relaxation at 300 K, is less sensitive to the varia-\ntion of excitation density. Therefore, it is the increase of\n\n2\u000b\n\u0000\n\n2\ni\u000b\ncausing the decrease of spin lifetime when\nincreasing excitation density.\nAPPENDIX D: LAYER-RESOLVED QUANTITIES\nOF BILAYER WSE 2\nFor a bilayer with one layer above z= 0 and another\nbelow, the operator ^ltopprojecting any local quantity\nA(r) to the top layer can be de\fned through the relation\n^ltopA(r) =H(z)A(r); (47)\nwherezis the third component of randH(z) is Heav-\niside step function\nH(z) =f1;z> 0\n0;z\u00140: (48)\nTherefore, top layer hole occupation ftop\nhis\nftop\nh=Tr\u0010\n^ltop(1\u0000f)\u0011\n(49)\n=X\nknltop\nk;nn(1\u0000fkn); (50)\nwherefis occupation and ltop\nk;mn=hkmj^ltopjkni=\nhkmjH(z)jkni. 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Although robust spin and\nvalley degrees of freedom are inferred from po-\nlarized photoluminescence (PL) experiments [4–\n8], PL timescales are necessarily constrained by\nshort-lived (3-100ps) electron-hole recombination\n[9, 10]. Direct probes of spin/valley polariza-\ntion dynamics of resident carriers in electron (or\nhole) doped TMDCs, which may persist long af-\nter recombination ceases, are at an early stage\n[11–13]. Here we directly measure the coupled\nspin-valley dynamics in electron-doped MoS 2and\nWS2monolayers using optical Kerr spectroscopy,\nand unambiguously reveal very long electron spin\nlifetimes exceeding 3ns at 5K (2-3 orders of mag-\nnitude longer than typical exciton recombina-\ntion times). In contrast with conventional III-\nV or II-VI semiconductors, spin relaxation ac-\ncelerates rapidly in small transverse magnetic\nfields. Supported by a model of coupled spin-\nvalley dynamics, these results indicate a novel\nmechanism of itinerant electron spin dephasing\nin the rapidly-fluctuating internal spin-orbit field\nin TMDCs, driven by fast intervalley scattering.\nAdditionally, a long-lived spin coherence is ob-\nserved at lower energies, commensurate with lo-\ncalized states. These studies provide crucial in-\nsight into the physics underpinning spin and val-\nley dynamics of resident electrons in atomically-\nthin TMDCs.\nStudies of optical spin orientation and spin relaxation\nusing polarized light have a long and exciting history\nin conventional III-V and II-VI semiconductors [14, 15].\nEarly seminal works focused on magneto-optical stud-\nies of polarized PL from recombining excitons [14], from\nwhich spin lifetimes could be indirectly inferred. How-\never, it was the direct observation of very long-lived\nspin coherence of resident electrons in materials like\nGaAs and ZnSe [16, 17] – revealed unambiguously by\ntime-resolved Faraday and Kerr rotation studies – that\ncaptured widespread interest and helped to launch the\nburgeoning field of “semiconductor spintronics” in the\nlate 1990s [15]. With a view towards exploring cou-\npled spin/valley physics of resident electrons in the new\natomically-thin and direct-bandgap TMDC semiconduc-\ntors, here we apply related experimental methods and\ndirectly reveal surprisingly long-lived and coherent spindynamics in monolayer MoS 2and WS 2.\nFigure 1a depicts the experimental setup. High-\nquality monolayer crystals of n-type MoS 2and WS 2,\ngrown by chemical vapor deposition on SiO 2/Si sub-\nstrates [18], were selected based on low-temperature re-\nflectance and PL studies (see Methods). Transverse\nmagnetic fields ( By) were applied using external coils.\nA weak pump laser illuminates individual crystals with\nright- or left-circularly polarized light (RCP or LCP) us-\ning wavelengthsnear the lowest-energy“A” exciton tran-\nsition, which primarily photoexcites spin-polarized elec-\ntrons and holes into the KorK′valley, respectively [1–\n10]. Any induced spin and valley polarization is then\ndetected via the optical Kerr rotation (KR) or Kerr el-\nlipticity(KE)that isimpartedtoalinearly-polarizedand\nwavelength-tunable probe laser that is normally incident\nand focused on to the crystal. Either continuous-wave\n(cw) or pulsed pump/probe lasers can be used; both\ntypes of experiments will be discussed.\nThe diagrams in Fig. 1a depict the conduction and va-\nlence bands in the KandK′valleys in a typical mono-\nlayer TMDC, along with the relevant spin/valley optical\nselection rules and scattering processes. In n-type ma-\nterial, the resident spin-up and spin-down electrons in\nthe conduction band of the K(K′) valley have densi-\ntiesn↑andn↓(n′\n↑andn′\n↓), which all share a common\nchemical potential in thermal equilibrium. However, fol-\nlowing pulsed photoexcitation and fast ( ∼10 ps) recom-\nbination with photogenerated holes [9, 10], these resi-\ndent electron densities may be unequalandout of equi-\nlibrium, as depicted. This can arise, e.g., from many-\nbody correlations while holes are present [2, 20], or pref-\nerential non-radiative recombination of holes with par-\nticular electron states (nonradiative processes account\nfor the majority of recombination in TMDCs [5]). In\nthis way, photoexcitation can impart a net spin polariza-\ntion (Sz=n↑−n↓+n′\n↑−n′\n↓) and/or valley polarization\n(Nv=n↑+n↓−n′\n↑−n′\n↓) onto the resident electrons,\nthat may remain even after all holes have recombined.\n(Analogously, a weak steady-state photoexcitation can\nestablish a nonequilibrium steady-state polarization of\nresident electrons). The intrinsic relaxation dynamics of\nthis polarization, which proceeds without the perturb-\ning influence of the holes, is the principal focus of these\nstudies.\nCrucially, any long-lived polarization of the resident\nelectrons can be directly monitored using Kerr spec-\ntroscopy. This stands in marked contrast with polar-\nizedPLstudies, whichexplicitly requiretheparticipation2\ny\nzlinearsample\ntunable probe \n(cw or pulsed)in cryostata\npol.By\nKerr rotation  pump RCP/LCP \n(cw or pulsed) \n  bc By(mT)=0.0\n3.4\n6.8\n16.9\n33.7\n67.5\n135\n270\ncw pump: 632.8 nm\ncw probe: 675.0 nmhwhm ~ 7mT\n05101520\n0.51\n-200 -100 0 100 200Pump-induced Kerr rotation ( µrad)\nPL polarization\nBy (mT)pulsed pump,probe: 660 nm T=5K0\n0 1 2Kerr rotation (arb. units)\nTime (ns)f\ndK\n LCPRCPn↑ ↓n\nK’n’ ↑↓n’ \nγvγs\n↑↑\n↑\n↑n↑↓n’ -∝\ne236 mT304 mT\n270 mT\n0.2 0.4 0.6\nTime (ns)\n0 1 201Calculated spin polarization S\nz(t) \nTime (ns)ΩL(x10 9) \n0\n4\n8\n12\n24\n-40 -20 0 20 40Calculated Hanle signal                  S \nz(t) dt \nLarmor Frequency ΩL(x10 9)∫\n0\nFIG. 1. Long-lived electron spin dynamics in n-type MoS 2at 5K. a , Experimental schematic. A weak pump laser\nilluminates the TMDC crystals with right- or left-circular ly polarized (RCP or LCP) light. The induced spin/valley pol arization\nis detected via the optical Kerr rotation (KR) or Kerr ellipt icity (KE) that is imparted on a linearly-polarized, wavele ngth-\ntunable probe laser. Pump and probe lasers can be either cont inuous-wave (cw) or pulsed. The diagrams depict a simple\nsingle-electron picture of the conduction and valence band s at the KandK′valleys of electron-doped monolayer MoS 2, along\nwith the relevant optical selection rules and scattering pr ocesses. In each valley, the conduction bands are separatel y drawn\nas spin-up (left) and spin-down (right) components (with sl ightly different curvature and with small splitting ∆ c; these effects\ngenerate the effective spin-orbit field ±ˆzBso). The densities of the resident electrons are n↑,n↓,n′\n↑, andn′\n↓. Near the lowest-\nenergy “A” exciton, RCP light couples only to spin-up electr on states in the Kvalley (n↑), while LCP light couples only to\nspin-down electron states in the K′valley (n′\n↓). Densities n↓andn′\n↑do not couple directly to light at the A exciton energy.\nFollowing recombination with photogenerated holes, KR and KE therefore depend on the difference n↑−n′\n↓. Electron spin\nrelaxation within a given valley ( γs) couples n↑withn↓(andn′\n↑withn′\n↓), while spin-conserving intervalley scattering γvcouples\nn↑withn′\n↑(andn↓withn′\n↓).b, Using weak cw pump and probe lasers, the red curve shows the i nduced KR versus applied\nmagnetic field Byfrom monolayer MoS 2at 5 K. This “Hanle-Kerr” experiment reveals rapid reductio n of optically-induced\nspin polarization by very small By, suggesting long spin relaxation times of resident electro ns. The PL polarization shows\nno change over this field range. c, Time-resolved KR data (using pulsed pump and probe lasers) directly reveals very long\nelectron relaxation dynamics in MoS 2, at different By.d,e, Calculated Hanle-Kerr and time-resolved dynamics of itin erant\nresident electrons, based on the model developed in the main text. Fast intervalley scattering γvand associated fluctuating\n±ˆzBsodrives the rapid dephasing of electron spin polarization in smallBy.f, Expanded view of residual long-lived electron\nspin coherence, likely due to additional contributions fro m localized states (curves offset for clarity).\nof (and recombination with) a photo-excited hole. Kerr\neffects depend only on the difference between a mate-\nrial’s RCP and LCP absorption and indices of refraction.\nPer the selection rules in monolayer MoS 2and related\nTMDCs [1, 2], RCP light near the lowest-energy “A”\nexciton in TMDCs couples to the resident electron den-\nsityn↑in theKvalley. Similarly, LCP light couples to\nn′\n↓in theK′valley. Nonequilibrium perturbations to\nthe densities n↑andn′\n↓shift their chemical potentials,\nwhich change the absorption and refraction of RCP andLCP probe light, particularly at wavelengths near opti-\ncal transitions. Thus, to leadingorder, Kerrsignals using\nlight near the A exciton are proportional to n↑−n′\n↓, to\nwhich both spin and valley polarization can contribute,\nviz.(Sz+Nv)/2.\nFigure 1b demonstrates optically-induced polarization\nin monolayerMoS 2using cwpump and probelasers. The\nwavelengths λpump(632.8 nm) and λprobe(675 nm) were\nchosen to address only the “A” exciton. At zero applied\nfield, a steady-state KR of ∼15µrad is induced by the3\npump onto the probe laser. Surprisingly, however, this\npolarizationsignalissharplyreducedbyverysmalltrans-\nverse fields By. The very narrow and nearly Lorentzian-\nshaped dependence of the measured KR on By(the half-\nwidth is only ∼7 mT) strongly suggests a long-lived spin\npolarization of resident electrons. (In contrast, the PL\npolarization, which probes primarilyshort-lived excitons,\nis unchanged overthis field range, in agreement with ear-\nlier studies [7]). These Kerr data are very reminiscent of\ntraditional “Hanle-effect” studies in conventional semi-\nconductors like n-type GaAs [14, 21, 22], wherein By\ndephases an optically injected steady-state electron spin\npolarization ( Sz) due to spin precession about By. This\nleadstoaLorentziandependenceof SzonBy, fromwhich\nthe spin lifetime τscan be directly inferred via the half-\nwidthBy0of the Hanle peak if the g-factor geis known;\nnamely,τ−1\ns=geµBBy0/¯h.\nIt is therefore tempting to associate the data in Fig.\n1b with spin dephasing in MoS 2due solely to precession\nof resident electrons about By, and to infer a spin life-\ntime. However, in contrast to conventional III-V or II-VI\nsemiconductors, electrons in the high-momentum Kand\nK′valleysofmonolayerMoS 2arepredicted toexperience\nstrong spin-orbit coupling, due to the different curvature\nof the spin-up and spin-down conduction bands and to\nany intrinsic splitting ∆ cof these bands [1, 2, 23, 24].\nThis coupling can be viewed as a large (of order 10 T)\nout-of-plane effective magnetic field Bso. Importantly,\nBsois oriented parallel or antiparallel to ˆ zdepending\non whether the electron resides in the KorK′valley.\nSpin-conserving intervalley electron scattering, which is\nnot forbidden in the conduction band and is expected\nto be fast ( γ−1\nv∼0.1-1ps), therefore leads to a rapidly\nfluctuating effective magnetic field ‘seen’ by electrons.\nThis fluctuating field alone will not affect (dephase)\nelectron spins that are also oriented along ±ˆz. However,\nin the additional presence of By, electron spins willpre-\ncess about the total fluctuating field ˆ yBy±ˆzBso, which\nis no longer oriented along ˆ z. This leads to a valley-\ndependent spin precession and associated spin dephasing\nthat is analogous to momentum-dependent spin preces-\nsion and dephasing common in conventional semiconduc-\ntors (e.g., the Dyakonov-Perel mechanism [14]) or to the\nelectron spin depolarization in germanium that is also\ndrivenbyintervalleyscattering[25]. A direct and testable\nconsequence is that electron spin relaxation in MoS 2is\nexpected to depend strongly on By– even for small val-\nues ofBy– in marked contrast to ordinary III-V and\nII-VI bulk semiconductors.\nThe narrow Hanle-Kerr data in Fig. 1b are consistent\nwith this scenario. However, to directlymeasure elec-\ntron spin relaxation and test this hypothesis, we turn\nto time-resolved KR studies using ultrafast pump and\nprobe lasers. Figure 1c shows the measured decay of the\noptically-induced electron polarization in the same MoS 2\ncrystal, using photons with energy just below the “A”\nexciton resonance. A key finding of this work is that the\npolarization decay time is extremely long ( ∼3 ns) at zero0.40.60.8 R/R 0\n0\n600 620 640 660 680Induced signal (arb. units)Probe wavelength (nm)KR KE \n632.8 nm \n543.5 nm pump “A” “B” \nFIG. 2. Spectral dependence of the optically-induced\nKerr rotation/ellipticity signals in monolayer MoS 2.\nThe upper (black) trace shows the normalized reflectance\nspectrum R/R0from a MoS 2crystal at 5 K. “A” and “B” ex-\nciton features are clear. The lower traces show the opticall y-\ninduced KR and KE signals as a function of the probe laser\nwavelengthat By=0. Astrongresonance atthe“A”excitonis\nobserved. Results using two different pump laser wavelength s\nare shown (632.8 nm and 543.5 nm)\nfield. This timescale exceeds typical PL recombination\ntimes by two to three orders of magnitude [9, 10], further\nimplicating resident electrons as the source of long-lived\npolarization. Very recent studies of monolayer MoS 2and\nWSe2using related optical techniques did not reveal any\nlonglivedpolarizationimpartedtoresidentelectrons,per-\nhaps due to elevated temperatures [13] or to the use of\nprobe photons with higher energy [11, 12]. The nanosec-\nond electron spin relaxation times observed here at zero\nfield mayultimately be limited by intra-valleyDyakonov-\nPerel or Elliot-Yafet processes [26, 27], or spin-flip scat-\ntering with magnetic impurities or long-wavelength flex-\nural phonons [28].\nCrucially, the polarization decay time rapidly de-\ncreases with small increasing By, and no prominent spin\nprecession is observed. These two observations directly\nsupport the scenario described above, of depolarization\ndue to a rapidly-fluctuating Bsodriven by fast inter-\nvalley electron scattering, for the following reasons: i)\nIfBsowere small or did not exist, then pronounced spin\nprecession about Bywould be observed (as is the case\nfor resident electrons in GaAs, ZnSe, GaN, or CdTe [15–\n17, 29, 30]), and ii) if Bsowere static and not fluctu-\nating, then spins would be effectively pinned along Bso\nand the small By(≪ |Bso|) would have little influence\non the long spin decay. Neither of these phenomena were\nobserved. Rather, these data are consistent with a novel\nspin relaxation mechanism in monolayer TMDCs that is\ndriven by fast inter-valley scattering (rapidly fluctuating\nBso), and activated by small By.\nA model of coupled spin-valley dynamics captures the\nunderlying physics and reproduces essential features of\nthe data. The phenomenological equation of motion for4\nthe electron spin polarizations SKandSK′in the two\nvalleys reads:\ndSτ\ndt=ΩL×Sτ+τΩsoˆz×Sτ−γsSτ−γv(Sτ−S−τ),(1)\nwhereτ=τ(t) =±1 is the index for KandK′val-\nleys respectively. The first two terms describe spin pre-\ncession about an applied field ΩL=gµBB/¯hand the\ninternal spin-orbit field Ωso=±ˆzgµBBso/¯h. The 3rd\nterm describes intrinsic spin relaxation within a given\nvalley with rate γs, and the 4th term is spin-conserving\ninter-valley scattering with rate γvthat is expected to\nbe fast and which leads to fast relaxation of any val-\nley polarization. We compute the total spin polariza-\ntionS=SK+SK′. For large Bso(Ωso≫ΩL)\nand fast intervalley scattering ( γv>Ωso), and as-\nsuming initial polarization S0ˆz, the solution is Sz(t) =\nS0/summationtext\nj=1,2Ajeiωjt. Here,A1,2= (1±Γv//radicalbig\nΓ2v−Ω2\nL)/2\nand the two eigenmodes iω1,2=−γs−Γv±/radicalbig\nΓ2v−Ω2\nL,\nwhere Γ v≡Ω2so/4γv. There is a critical applied field\nΩc\nL= Γv, below which the modes are purely decaying\nand above which the modes oscillate. The Hanle curve\nis calculated by integrating Sz(t) for each Ω L; namely/integraltext∞\n0Sz(t)dt=S0(γs+ 2Γv)/[Ω2\nL+(γ2\ns+ 2γsΓv)]. Thus,\nHanle-Kerr data are predicted to be Lorentzian (as ob-\nserved) with half-width/radicalbig\nγ2s+2γsΓv.\nFigures 1d and 1e show the calculated Hanle curves\nand electron spin dynamics. The model reproduces the\nrapid drop in electron spin polarization with increas-\ning (small) By, and captures the shallow dip and subse-\nquent recovery of the electron spin polarization at short\ntimescales. The model does not, of course, capture the\noffset of the measured Hanle-Kerr data (see Fig. 1b),\nthe origin of which suggests an additional long-lived\nand field-independent polarization, perhaps from local-\nized states in MoS 2, which have been studied recently.\nThis offset is also manifested in the time-resolved data\nas the very slowly decaying and largely field-independent\nsignal that persists at long delays.\nA surprising additional observation is the appearance,\nat larger magnetic fields, of a small but long-lived os-\ncillatory signal visible between ∼200-800 ps (red trace\nin Fig. 1c). Figure 1f shows an expanded view of this\nsignal at By=236, 270, and 304 mT. The oscillation fre-\nquency scales linearly with By, indicating that this sig-\nnal arises from coherently-precessing electrons with g-\nfactor|ge| ≃1.8. Such long-lived coherence signals are\nnot expected from itinerant resident electrons for rea-\nsons described above. However, they may arise from\ncontributions from an additional population of localized\nstates that do not undergo rapid intervalley scattering\nand which precess about the bare applied field By. Time\nresolved measurements at lower photon energies below\nthe A exciton are consistent with this scenario (Fig. S1).\nWe confirm that these Kerr signals originate from\nMoS2by showing their spectral dependence. Figure 2\nshows the reflectivity of the MoS 2crystal (top black\ntrace), below which are the peak Hanle-Kerr signals at-200 0 200Kerr rotation (arb. units)\nBy (mT)0 1 2Kerr rotation (arb. units)\nTime (ns)T=5K\n10K\n15K\n20K\n30K\n40K\n01\n0a b\nc\n0123\n050100\n0 10 20 30 40Decay time (ns)\nInverse linewidth (1/T)\nT (K)\nFIG. 3. Temperature dependence of electron spin re-\nlaxation in monolayer MoS 2at zero magnetic field.\na, Hanle-Kerr data (using cw lasers; λpump=632.8 nm,\nλprobe=675 nm). With increasing temperature, the curve\nwidth increases while the amplitude drops, indicating fast er\nspin relaxation of resident electrons. b, Corresponding time-\nresolved KR directly reveals faster spin relaxation with in -\ncreasing temperature. Here, λpump=635 nm with ∼100 ps\npulse duration; λprobe=672 nm with ∼200 fs pulse duration.\nc, The measured relaxation time, and inverse Hanle-Kerr\nlinewidth, versus temperature.\nBy=0. Both KR and KE show a strong resonance at\nthe “A” exciton. Data using 632.8 nm and 543.5 nm\ncw pump lasers are shown; the latter allows to track the\ninduced Kerr signals out to shorter probe wavelengths\nthat overlap with the higher-energy “B” exciton which\nelicits a smaller response. This behavior was confirmed\non many different MoS 2crystals. We note that the res-\nonances are redshifted ∼25 meV from the exciton peak,\nsuggesting that the resident electrons’ polarization is re-\nvealed preferentially through trion-related (rather than\nneutral exciton-related) optical transitions [2], consistent\nwith analogous studies in III-V and II-VI semiconductor\nquantum wells [29, 30]. The dependence of Hanle-Kerr\ndata on probe wavelength is shown in Fig. S2.\nAn essential consideration for spin-coherent phenom-\nena in semiconductors is its dependence on tempera-\nture. Systematic studies on MoS 2(see Fig. 3) reveal\nvery long spin lifetimes at 5K ( ∼3 ns), which decrease\nrapidly to <200 ps by 40 K. This behavior is in quali-\ntative agreement with recent theoretical predictions [28]\nthat account for Elliot-Yafet spin-flip processes due to\nelectron-phonon scattering with long-wavelengthflexural\nphonons in TMDCs. A comparison of supported versus\nunsupported monolayercrystalscouldconfirm this mech-\nanism.5\n12\n-100 -50 0 50 100 Induced KR (abs. value) \nBy (mT)cw probe: 607.0 nm cw pump: 543.5 nm \n01\n0 1 2 3Induced KR (abs. value) \nTime (ns)pulsed pump/probe: 608 nm \n0a b\nc “A”\n01\n580 600 620 640 660R/R0\nInduced KR\nProbe wavelength (nm)cw pump: 543.5 nm \nFIG. 4. Long-lived spin polarization dynamics in\nmonolayer WS 2at 5 K.a, Hanle-Kerr measurement of\noptically-induced spin polarization at the “A” exciton, us ing\ncw lasers. (At this probe wavelength the induced KR has\nnegative sign, thus the absolute value is shown for clarity) .\nb, Time-resolved KR at zero field (using ultrafast pump and\nprobe pulses). c, The normalized reflectivity of monolayer\nWS2(left axis), showing the “A” exciton feature. Right axis:\nwavelength dependence of the optically-induced KR signal a t\nzero applied field (using cw lasers).\nFinally, we show that these long-lived electron spin\npolarizations are not unique to MoS 2, but also appear in\nother TMDCs. Figure 4 shows Hanle-Kerr and time-\nresolved KR measurements in monolayer WS 2, along\nwith its spectral dependence. Like MoS 2, WS2also ex-\nhibits very narrow Hanle-Kerr signals and nanosecond\npolarization relaxation times. The peak Hanle-Kerr sig-\nnal also occurs on the low-energyside of the “A” exciton,\nagain suggesting that coupling to resident electrons may\nproceed preferentially through trion transitions. The\nfield-independent background on the Hanle data is sig-\nnificantly larger (about 80% of the peak, rather than\n30-40% for MoS 2), which may indicate lesser material\nquality.\nIn summary, steady-state and time-resolved Kerr-\nrotation spectroscopy directly and unambiguously reveal\nthe very long spin relaxation and spin coherence of res-\nident electrons in n-type MoS 2and WS 2monolayers.\nThese nanosecond timescales exceed by 2-3 orders of\nmagnitude typical PL recombination times. In contrast\nwith conventional III-V and II-VI semiconductors, rapid\nspin depolarization of itinerant electrons in MoS 2occurs\nin very small applied By. Supported by a model of cou-\npled spin-valley relaxation, these data point to a novel\ndephasing mechanism generated by precession about the\nrapid-fluctuating internal spin-orbit field that is a hall-markofTMDCmaterials,drivenbyfastinter-valleyscat-\ntering. These measurements open the door for studies\nof the spin- and valley- dynamics of intrinsic and resi-\ndentcarriersinelectron-andhole-dopedtwo-dimensional\nTMDC semiconductors.\nMETHODS\nHigh-quality triangular crystals of n-type (electron doped)\nmonolayer MoS 2and WS 2were grown by chemical vapor de-\nposition on SiO 2/Si substrates following [18]. Typical lateral\ndimensions ranged from 10-20 µm. The background electron\ndoping level in these crystals is estimated to be ne∼5×1012\ncm−2based on transport studies of field-effect transistor de-\nvices fabricated from similarly-grown crystals. The sampl es\nwere mounted on the vacuum cold finger of a small variable-\ntemperature liquid helium optical cryostat (3-300 K). Appl ied\nmagnetic fields up to 300 mT were generated using an exter-\nnal electromagnet. Individual crystals were screened for a\nhigh degree of circularly-polarized exciton PL ( >80%) and\nsharp reflectivity spectra.\nAll Kerr-effect measurements were performed in a reflec-\ntion geometry, as depicted in Fig. 1a. For Hanle-Kerr mea-\nsurements of steady-state polarization, continuous-wave (cw)\npumpandprobelasers were used. The632.8 nmand543.5nm\nlines of HeNe lasers were used for the pump, while the probe\nlaser was typically a tunable narrowband dye laser or fixed-\nwavelength diode lasers. Time-resolved measurements used\nultrafast 250 fs pulses from a wavelength-tunable 76 MHz op-\ntical parametric oscillator (OPO). Time-resolved studies typ-\nically used wavelength-degenerate pump and probe pulses (a n\nexception is the temperature-dependent data of Fig. 3, wher e\na 635 nm pulsed diode laser with ∼100 ps pulse duration was\nsynchronized to the OPO and used as a non-degenerate pump\nlaser. In all experiments, the pump laser was weakly focused\nso as to illuminate the entire TMDC crystal, thereby miti-\ngating any possible influence of density gradients or carrie r\ndiffusion, while the linearly-polarized and normally-inci dent\nprobe laser was more tightly focused to a ∼4µm spot in the\ncenter of the crystal. It was verified that the small off-axis a n-\ngle of the pump laser (about 10 degrees from sample normal)\ndid not influence the results – similar data were obtained wit h\nco-propagating normally-incident pump and probe lasers.\nExcept where otherwise noted, the pump laser polariza-\ntion was modulated between RCP and LCP by a photoelastic\nmodulator to facilitate lock-in detection. Detection of th e in-\nduced optical polarization rotation (KR) and ellipticity ( KE)\nimparted on to the probe beam was achieved with a standard\noptical bridge arrangement usingbalanced photodiodes. 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Electron spin relaxation due to\nD’yakonov-Perel’ and Elliot-Yafet mechanisms in mono-\nlayer MoS 2: Role of intravalley and intervalley processes.\nPhys. Rev. B 89, 115302 (2014).\n[28] Song, Y. & Dery, H. Transport theory of monolayer\ntransition-metal dichalcogenides through symmetry. Phys.\nRev. Lett. 111, 026601 (2013).\n[29] Zhukov, E.A., Yakovlev, D.R., Bayer, M., Glazov, M.M.,\nIvchenko, E.L., Karczewski, G., Wojtowicz, T. & Kos-\nsut, J. Spin coherence of a two-dimensional electron gas\ninduced by resonant excitation of trions and excitons\nin CdTe/(Cd,Mg)Te quantum wells. Phys. Rev. B 76,\n205310 (2007).\n[30] Chen, Z., Carter, S. G., Bratschitsch, R. & Cundiff, S. T.\nOptical excitationandcontrol ofelectron spinsin semicon -\nductor quantum wells. Physica E 42, 1803-1819 (2010).7\nI. SUPPLEMENTAL INFORMATION\nA. Wavelength dependence of time-resolved Kerr\nand Hanle-Kerr measurements\nFigure S1 shows time-resolved Kerr-rotation measurements\nin monolayer MoS 2, atBy= 0 and 270 mT, using degenerate\npump and probe lasers for different photon energies span-\nning the “A” exciton. At short wavelengths (high photon\nenergies), the induced KR signals are small. At intermedi-\nate wavelengths λjust on the low-energy side of the exciton\nresonance ( e.g.λ=654-660 nm), the KR signals are large and\ndecay slowly at zero field, consistent with the long-lived de cay\nof polarized itinerant and resident electrons. At By=270 mT,\nthe large KR signals decay quickly due to dephasing caused\nby precession about the combined applied field Byand the\nlarge effective internal spin-orbit field ±ˆzBsothat is rapidly\nfluctuating due to fast intervalley scattering of itinerant elec-\ntrons. At these wavelengths, the additional long-lived osc illa-\ntory (coherent) spin precession signal is barely visible.\nHowever at even longer wavelengths further on the low-\nenergy side of the exciton resonance, the KR signals reduce\nuntil only this small oscillatory signal remains. These re-\n8\n6\n4\n2\n0 x10 -1  \n0.8 0.6 0.4 0.2 0.0x103 660 nm 8\n6\n4\n2\n0 x10 -1  \n0.8 0.6 0.4 0.2 0.0x103 657 nm 8\n6\n4\n2\n0 x10 -1  \n0.8 0.6 0.4 0.2 0.0x103 654 nm \n8\n6\n4\n2\n0 x10 -1  \n0.8 0.6 0.4 0.2 0.0x103 663 nm \n0.6\n0.4\n0.2\nx10 -1  \n0.8 0.6 0.4 0.2 0.0x103 666 nm 0.3\n0.2\n0.1\nx10 -1  \n0.8 0.6 0.4 0.2 0.0x103 670 nm \nTime (ns)Induced Kerr rotation (arb. units) 0.8\n0.6\n0.4\n0.2R/R 0\n680 660 640 620 600\nWavelength (nm)\"A\" \"B\" a\nb8\n6\n4\n2\n0\nx10 -1  \n0.8 0.6 0.4 0.2 0.0x103 646 nm \n By (mT)\n 0 \n 2708\n6\n4\n2\n0\nx10 -1  \n0.8 0.6 0.4 0.2 0.0x103 650 nm \nFIG. 5. Supplemental Figure S1: Wavelength depen-\ndence of time-resolved Kerr rotation studies in MoS 2.\na, Normalized reflectance spectrum R/R0from a monlayer\nMoS2crystal at 5 K. The arrows indicate the wavelengths λ\nat which the degenerate pump-probe KR measurements were\nperformed. b, Time-resolved KR in monolayer MoS 2for vari-\nousλatBy= 0 and 270 mT. Note the changing y-axis scales.sults are consistent with the laser pulses pumping (and prob -\ning) predominantly itinerant electrons in MoS 2at intermedi-\nate wavelengths (654-660 nm), but then pumping and prob-\ning more localized states at longer wavelengths (663-670 nm).\nBeing localized, these electrons may not undergo rapid inte r-\nvalley scattering and may not ‘see’ the large spin orbit field\n±ˆzBso, andtherefore precessaboutthebareappliedmagnetic\nfieldBy. Localized states in monolayer TMDC materials have\nbeen observed in recent experimental studies [1].\nFigure S2 shows Hanle-Kerr data from monolayer MoS 2at\n5K. A fixed cw pump laser was used (632.8 nm), and the\ndifferent Hanle curves show data obtained with different cw\nprobe laser wavelengths. The inversion of the induced KR\nsignal is clearly observed (as in Fig. 2 of the main text).\nThese curves have somewhat greater width than those shown\nin the main text ( e.g., Figure 1), due to the use of higher\npump and probe laser intensity. Minimum Hanle widths (im-\nplying long spin lifetimes of resident electrons) are obtai ned\nat long wavelengths on the low-energy side of the “A” exciton\nresonance.\n10 \n5\n0\n-5 Kerr rotation (arb. units) \n-300 -200 -100 0 100 200 300\nBy (mT) λprobe (nm)\n 664.9\n 662.6\n 669.8\n 674.7\n 680.0\n 660.0\n 643.8\n 648.3\n 657.7\n 653.0\n λpump =632.8 nm\nFIG. 6. Supplemental Figure S2: Probe wavelength\ndependence of Hanle-Kerr measurements in MoS 2.\nUsing cw pump and probe lasers, the plot shows the Hanle-\nKerr curves measured in MoS 2at 5K using different probe\nwavelengths spanning the “A” exciton resonance.8\npump/probe\nRCP/RCP \nRCP/LCP \n0 1 2\nTime (ns)0 1 2∆R\nTime (ns)By=0 mT By=270 mT a b\n0\nFIG. 7. Supplemental Figure S3: Selective right- and\nleft-circularly polarized (RCP/LCP) pump-probe\nstudies of induced reflectivity . Using RCP pump pulses,\ntheKvalley of monolayer MoS 2is weakly photoexcited at\nthe “A” exciton, while the induced change in reflectivity ∆ R\nof both RCP and LCP probe pulses is detected as a function\nof time. BothRCP and LCP probes (probing KandK′val-\nleys, respectively) show an induced longlived change. λpump\n=λprobe= 660 nm; T=5K.a, Measurement at By=0.b,\nMeasurement at By=270 mT.B. Selectively probing polarization dynamics in K\nandK′valleys\nBecause Kerr signals scale as the difference between RCP\nand LCP optical constants, it is natural to ask which compo-\nnent contributes primarily. That is, does resonant pumping\ntheKvalley with RCP pump light affect only RCP probe\nlight, or does italso affectLCP light (whichprobes the K′val-\nley)? Figure S3 shows time-resolved measurements of pump-\ninducedreflectivityinMoS 2usingco-andcross-circular pump\nand probe pulses. Interestingly, the reflectivity of bothRCP\nand LCP probe light increases immediately following pho-\ntoexcitation (albeit by different amounts), consistent wit h re-\ncent studies on very short ( <10 ps) timescales [2], suggesting\nthatstrongexcitonCoulombcorrelation effectswhileholes are\npresentgenerate theinitial non-equilibriumelectron den sities,\nwhich then, following recombination, relax on long timesca les.\n[1] He, Y.-M., Clark, G. Schaibley, J.R., He, Y., Chen, M.-\nC., Wei, Y.-J., Ding, X., Zhang, Q., Yao, W., Xu, X., Lu,\nC.-Y. & Pan, J.W., Single quantum emitters in monolayer\nsemiconductors. arXiv:1411.2449.[2] Mai, C., Barrette, A., Yu, Y., Semenov, Y.G., Kim, K.W.,\nCao, L. & Gundogdu, K. Many-body effects in valleytron-\nics: Direct measurement of valley lifetimes in single-laye r\nMoS2.Nano Lett. 14, 202-206 (2014)."    },    {        "title": "1209.4567v1.Quench_dynamics_in_spin_crossover_induced_by_high_pressure.pdf",        "content": "arXiv:1209.4567v1  [cond-mat.str-el]  20 Sep 2012Quench dynamics in spin crossover induced by high pressure\nA. I. Nesterov∗\nDepartamento de F´ ısica, CUCEI, Universidad de Guadalajar a,\nAv. Revoluci´ on 1500, Guadalajara, CP 44420, Jalisco, M´ ex ico\nS. G. Ovchinnikov†\nL. V. Kirensky Institute of Physics, SB RAS, 660036, Krasnoy arsk , Russia\nInstitute of Engeniering Physics and Radioelectronics,\nSiberian Federal University, Krasnoyarsk 660041, Russia\n(Dated: November 22, 2018)\nIn this paper we have studied analytically and numerically d ynamics of spin crossover induced\nby time-dependent pressure. We show that quasi static press ure, with a slow dependence on time,\nyields a spin crossover leading to transition from the state of quantum system with high spin (HS)\nto the low spin (LS). However, a quench dynamics under shock- wave load is more complicated. The\nfinal state of the system depends on the amplitude and pulse ve locity, resulting in the mixture of\nthe HS and LS states.\nPACS numbers: 03.65.Vf, 14.80.Hv, 03.65.-w, 03.67.-a, 11. 15.-q\nKeywords: Energy level crossing, crossover, quantum phase transition\nI. INTRODUCTION\nSpin crossover in condensed matter physics is a trans-\nformation of a system with one spin S1at each lattice\nsite into another state with spin S2induced by some ex-\nternal parameter like strong magnetic field, high pres-\nsure etc. It accompanies by the energy level E1andE2\ncrossing, where Eais the local energy of the magnetic\nion with spin Sa(a= 1,2). Recently spin crossovers\nin magnetic oxides have been found under high pressure\ninFeBO 31,CdFe3(BO3)42,BiFeO 33,Fe3O44. Below\nthe Curie temperature ofmagneticorderspin crossoveris\naccompanied by the sharp change of the magnetization,\nneverthelessitmaybeobservedintheparamagneticstate\nlike inCdFe3(BO3)42as the sharp change of the XES\nsatellite/mainpeakintensityratiowithpressureincrease.\nEnergy levels crossing results in the lost of analyticity\nin the energy spectrum at the critical point (in the ther-\nmodynamic limit)5. Near the critical point adiabatic-\nity breaks down and non-equilibrium phenomena asso-\nciated with the drastically grown quantum fluctuations\ncan drive the system away from the ground state. The\nfinal result depends on how fast the transition occurs. If\nthe quench process is sufficiently fast, large numbers of\ntopological defects are created and the final state, being\ncharacterizedbymixtureofhighandlowspinphases, can\nbe essentially different from that been obtained as result\nof slow evolution. Qualitatively, quench dynamics can be\ndescribed by the Kibble-Zurek theory of nonequilibrium\nphase transitions6–8.\nIn this paper we consider quench dynamics in spin\ncrossover induced by time-dependent pressure. The pa-\nper is organized as follows. In Sec. II, a general model of\nspin crossover under high pressure is introduced. In Sec.\nIII, we study quench dynamics. We consider two cases:\na) Pressure is a linear function of time; b) Pressure is\ndefined by a pulse of a given shape. We conclude in Sec.IV with a discussion of our results.\nII. MODEL\nThe multielectron ion in a crystal field has the energies\nof terms for dnconfigurations determined numerically by\nthe Tanabe-Sugano diagrams9as a solution of the eigen-\nvalue problem. Simple analytical calculations of the low\nenergy terms with different spin value that is sufficient\nto study spin crossover has been done recently10. The\ncrystal field parameter increase linearly with pressure P.\nThus the multielectron energies for spin S1andS2(E1\nandE2)arealsolinearfunctionsof P. Todistinguishtwo\ndifferent spin states in the lattice we introduce the Ising\npseudospin states |i∝angbracketrightand|−i∝angbracketrightfor|dn\ni,Si\n1∝angbracketrightand|dn\ni,Si\n2∝angbracketright,\nwhereiruns over all sites in the lattice. Thus we neglect\nthe spin degeneracy of the dn\niterms but capture the pos-\nsibility of energy level crossing that is the essential part\nof the spin crossover. Then, in the basis |+i∝angbracketright,|−i∝angbracketright, the\nHamiltonian of the system can be written as follows\nH=/summationdisplay\ni/parenleftbig\nλi\n01 1+εiˆσz\ni)+/summationdisplay\nijHij, (1)\nwhereλi\n0= (Ei\n1+Ei\n2)/2,εi= (Ei\n1−Ei\n2)/2, and 1 1, ˆσzare\nthe identity and Pauli matrices, respectively; the Hamil-\ntonian of interaction between the spins being Hij.\nTheHijincludes the isotropic Heisenberg term with\nthe exchange interaction Iijbetween nearest spins and\nthe anisotropic term HA. The interatomic interaction\nIijis negligibly small in comparison with the interatomic\nHund’s coupling (ratio 10−2). Thus its contribution to\nthe localized spin energy E1andE2due to the effective\nmolecular field can be neglected. Nevertheless the ex-\nchange interaction plays very important role: it results\nin the long range order and synchronize each spin in the\nsame quantum state providing a cooperative behavior of2\nthe spin system. If it were the ferromagnetic interaction,\neach spin at T=0 would have the maximal projection S\nwith integer magnetic moment 2 S.\nIn all examples given above there is the antiferro-\nmagnetic interaction. The ground state of the isotropic\nHeisenberg antiferromagnet has non integer local mag-\nnetic moment due to the quantum spin fluctuations. It\nis known that for large spin Seffect of quantum fluctua-\ntionsislessimportantthenforsmallspin, andfor FeBO 3\nspin is 5/2. Moreover the magnetic anisotropy addition-\nally suppresses quantum fluctuations. For example, in\nFeBO 3the anisotropy field is 0 .3T11and the measured\nvalue of the effective moment 2/radicalbig\nS(S+1) = 5.9 is very\nclose to the calculated for S= 5/2 value 5.916.\nThus we conclude that due to anisotropy the magnetic\nmoment at T= 0 has integer value ( of course it is a\nproperty of the magnetic insulator that does not hold for\nitinerant magnets), and due to exchange interaction all\nspins are in the same quantum state. So spin crossover\natT= 0 is the transition of the whole crystal from one\nmagnetically ordered state to another. Nevertheless the\ncriterium of the transition can be found from consider-\nation of the single ion energies crossover due to space\nuniform cooperative magnetic order. Anisotropic rela-\ntivistic interactions, for example a spin-orbital interac-\ntion, are also important because can mix different spin\nstates inside single ion.\nWe consider spin crossover far from the thermody-\nnamic phase transition in the paramagnetic phase, it al-\nlows us to simplify this interaction and substitute the ef-\nfect of exchange with the effective mean field. This mean\nfield is spatially uniform for the ferromagnetic insulator\nor two-sublattice for the antiferromagnet one. Examples\ngiven above1–4correspond to the anti- or ferrimagnetics.\nIn any case this mean field just renormalizes the interi-\nonic multielectron energies E1andE2, and is irrelevant\nto the crossover phenomenon. Another interaction that\nis smaller then the exchange one is given by relativistic\nanisotropy contribution to the Hij. For example a spin-\norbital interaction can mix different spin states inside\nsingle ion, and it occurs to be important in our problem.\nIn what follows we will consider the simplified spa-\ntially uniform model12. Motivation for this simplifica-\ntion is as follows. Despite that spin crossover is related\nto many body system, the essential features of its dy-\nnamics can be described by effective, Landau-Zenertype,\nHamiltonian13.\nThe Hamiltonian of our model is given by H=/summationtextN\ni=1Hi, where\nHi=/parenleftbigg\nλ00\n0λ0/parenrightbigg\n+/parenleftbigg\nε ρe−iϕ\nρeiϕ−ε/parenrightbigg\n.(2)\nThe energy spectrum is given by ε±=λ0±/radicalbig\nε2+ρ2.\nBothλ0andεare pressure dependent. Further we as-\nsume that the spin excitation gap is given by\nε(P) =ε0/parenleftbigg\n1−P\nPc/parenrightbigg\n. (3)Thecrossovertakesplaceatthe point Pcwhenε(Pc) = 0.\nThe spin-orbit coupling λ=ρeiϕ(withρ≪ε0) mixes\nthe different spin states, and it plays the role of quan-\ntum fluctuations in our Ising pseudospin basis. The di-\nmensionless spin gap, P/Pc−1, plays a role of relative\ntemperature ( T/Tc−1) near the critical point Tc14.\nIII. QUENCH DYNAMICS INDUCED BY\nHIGH PRESSURE\nWe consider time dependent Schr¨ odinger equation for\nthe Hamiltonian (2) assuming for simplicity that the\npressure is the linear function of time, P=Pc(1+t/τQ).\nInserting this expression into Eq. (3), we obtain ε(t) =\n−ε0t/τQ. The parameter τQdepends on ˙Pand can be\nwritten asτQ=Pc/˙P.\nLet|1∝angbracketrightand|0∝angbracketrightare eigenstates of the operator ˆ σz, so\nthat ˆσz|1∝angbracketright=|1∝angbracketrightand ˆσz|0∝angbracketright=−|0∝angbracketright. Expressing a generic\nstate vector as\n|ψ(t)∝angbracketright=e−i/integraltext\nλ0(t)dt(C1(t)e−iϕ/2|1∝angbracketright+C0(t)eiϕ/2|0∝angbracketright),\n(4)\nwe find that the coefficients C1(t) andC0(t) satisfy the\nSchr¨ odinger equation with the time-dependent Hamilto-\nnian in the Landau-Zener (LZ) form (in units /planckover2pi1= 1)\nid\ndt/parenleftbigg\nC1(t)\nC0(t)/parenrightbigg\n=/parenleftbigg\n−∆t ρ\nρ∆t/parenrightbigg/parenleftbigg\nC1(t)\nC0(t)/parenrightbigg\n,(5)\nwhere ∆ = ε0/τQ.\nIn terms of dimensionless scaled time τ=√\n∆t=\nτ0(P/Pc−1) withτ0=√ε0τQ=/radicalBig\nε0Pc/˙P, the Landau-\nZener model is described by the Hamiltonian\nH=/parenleftbigg\n−τ ω\nω τ/parenrightbigg\n, (6)\nwhereω=ρ/√\n∆ =τ0ρ/ε0is the dimensionless coupling\nconstant. Writing |u(τ)∝angbracketright=C1(τ)|1∝angbracketright+C0(τ)|0∝angbracketright, one can\nrecast the Schr¨ odinger equation (5) as\nid\ndτ|u(τ)∝angbracketright=H(τ)|u(τ)∝angbracketright. (7)\nHere the time τruns from the initial time τi=−√ε0τQ,\ncorresponding to the initial pressure Pi= 0, to final τf=\n(Pf/Pc−1)√ε0τQ, corresponding to Pfat the end of\nquench (Pf> Pc). Further we assume that ε0τQ≫1,\nthen timeτcan be extended to ±∞, and the problem\nbecomes fully equivalent to the LZ problem.\nTheenergyspectrumoftheHamiltonian(6)isgivenby\nε±(τ) =±√\nτ2+ω2, and its instantaneous eigenvectors\ncan be written as\n|u−(τ)∝angbracketright=/parenleftBigg\n−sinθ(τ)\n2\ncosθ(τ)\n2/parenrightBigg\n,|u+(τ)∝angbracketright=/parenleftBigg\ncosθ(τ)\n2\nsinθ(τ)\n2/parenrightBigg\n(8)3\n1 1\n0.5 0.51 1−1−1−0.5−0.5\n0 0\nx x0.5 0.5z z0 0\n0 0\n□ □0.5 0.5\n−0.5−0.51 1\n−0.5−0.5−1−1\nFIG. 1: (Color online) Bloch vector’s evolution ( ω= 3). The\nmotion starts at the south pole of the sphere and asymptoti-\ncally ends at the north pole.\nwhere cosθ(τ) =−τ/√\nτ2+ω2. The energygapbetween\nthe ground and excited states equals 2√\nτ2+ω2.\nFrom Eq. (8) it follows that while the ground state be-\nhaves atτ=±∞as|u−(−∞)∝angbracketright → |0∝angbracketrightand|u−(+∞)∝angbracketright →\n|1∝angbracketright, theexcitedstatebehavesasfollows: |u+(−∞)∝angbracketright → |1∝angbracketright\nand|u+(+∞)∝angbracketright → |0∝angbracketright. The state |u−(−∞)∝angbracketrightcorresponds\nto the high spin (HS) of the system and |u+(−∞)∝angbracketrightcor-\nresponds to the low spin (LS). Thus, if the system ini-\ntially was in the HS state, at the end of the evolution its\nground state corresponds to the LS. Tunnelling between\nthe positive and negative energy eigenstates, leading to\nthe mixture ofHS and LS, happens in the neighbourhood\nof the critical point τc= 0 (P=Pc) whenτ∈(−ω,ω)15.\nWe assume further that the evolution of the system\nstarts at the moment of time τi=−√ε0τQ(P(τi) = 0)\nfrom the ground state |u−(τi)∝angbracketright. Sinceρ≪ε0, we have\n|u−(τi)∝angbracketright ∝ |0∝angbracketright. This yields the following initial condi-\ntions:C0(τi) = 1 andC1(τi) = 0. At the end ofevolution\nwe obtain, |u−(τ)→ |1∝angbracketright, whileτ→+∞.\nIn Figs. 1 – 3 we present the results of numerical solu-\ntion of the Schr¨ odinger equation. In Fig. 1, 2 time evo-\nlution of the Bloch vector, n=∝angbracketleftu|σ|u∝angbracketright, is shown. The\nmotion begins at the south pole of the two-dimensional\nsphereS2and forω= 3 ends at the north pole. However\nfor the choice of the parameter ω<1, numerical simula-\ntion shows that the Bloch vector never reaches the north\npole, which corresponds to the LS state. This implies\nthat at the end of evolution the quantum system does\nnot remains in the ground state and its final state is the\nmixture of the LH and LS states.\nIn Fig. 3 the probability Pτ=|C1(τ)|2of transition\n|0∝angbracketright → |1∝angbracketrightis depicted for various values of ω. As can\nbe seen, with decreasing of adiabadicity parameter ωthe−1−1 1 1−1−1−0.5−0.5\n−0.5−0.5 0.5 0.5z z0 00.5 0.5\n0 01 1\n0 0\n□ □ x x0.5 0.5 −0.5−0.5\n−1−1\nFIG. 2: (Color online) Bloch vector’s dynamics ( ω= 0.75).\nThe evolution starts at the south pole of the sphere.\nFIG. 3: (Color online) Probability of transition Pτobtained\nfrom exact solution of LZ problem as function of τ. From up\nto down: ω= 3,0.5,0.25.\ntransition probability decreases as well. Its asymptotic\nbehaviour is described by the LZ formula (13).\nA. Exact solution of the Landau-Zener problem\nTheexactsolution ofthe Eq.(7) isgivenin termsofthe\nparabolic cylinder functions16–18,D−1−iω2/2(z), where\nz=√\n2τe−iπ/4. Assuming that initially the system was\nin the ground state, |u−(−∞)∝angbracketright → |0∝angbracketright, we obtain the ini-\ntialconditionsasfollows: C0(−∞) = 1andC1(−∞) = 0.\nThe probability of transition to the state |1∝angbracketrightat timeτis4\ngiven by18–20\nPτ=ω2\n2e−πω2/4/vextendsingle/vextendsingleD−1−iω2/2(τ√\n2e3iπ/4)|2.(9)\nAs can be shown, the condition ε0≫ρmay be recast\nto|τi| ≫ω, whereτi=−√ε0τQis the initial time. Since\n|τi| ≫1, forτ >|τi|the so-called weak-coupling asymp-\ntoticapproximation may be applied18. The asymptotic\nof the transition probability is\nPτ∼1−e−πω2−2ω\nτe−πω2/2/radicalbig\n1−e−πω2cosξw(τ),\n(10)\nwhere\ncosξw(τ) =π\n4+τ2+ω2\n2τ2ln2+argΓ/parenleftbigg\n1−iω2\n2/parenrightbigg\n.\n(11)\n1. Adiabatic approximation\nIn the adiabatic approximation the probability of the\nsystem to remain in the ground state may be described\nby\nPad(τ) =|∝angbracketleftu−(τ)|0∝angbracketright|2=1\n2/parenleftbigg\n1+τ√\nτ2+ω2/parenrightbigg\n.(12)\nThe condition for adiabatic evolution, required by the\nadiabatic theorem, is ω2≫119. The probability of\nremaining in the ground state at the end of evolution\n(τ→ ∞) is given by Pad=|C1(+∞)|2. For slow evo-\nlution we can use the LZ formula21,22to describe the\nprobability of adiabatic evolution:\nPad= 1−e−πω2. (13)\nIn Figs. 4, 5 the probability of the adiabatic transition\n(red line) and the results of the exact solutions (blue\nline) are depcted. As can be seen for ω= 3 there is a\ngoodagreementbetweenexactsolutionandthe adiabatic\nformula (12). For ω= 0.5 the disagreement takes place.\nUsing the so-called adiabatic-impulse (AI)\napproximation13,15, qualitatively, the dynamics of the\nLandau-Zener model can be described by the Kibble-\nZurek theory of nonequilibrium phase transitions6–8.\nThe AI-approximation assumes that the whole evolution\ncan be divided in three parts and up to the phase factor\nthe wave |u(t)∝angbracketrightfunction approximately can be described\nas\nτ∈[−∞,−ˆτ] : |u(τ)∝angbracketright ≈ |u−(τ)∝angbracketright\nτ∈[−ˆτ,ˆτ] : |u(τ)∝angbracketright ≈ |u−(−ˆτ)∝angbracketright\nτ∈[ˆτ,+∞] :|∝angbracketleftu(τ)|u−(τ)∝angbracketright|2= const\nwhere the time ˆ τ, introduced by Zurek7, is called the\nfreeze-out time and define the instant when behaviourFIG. 4: (Color online) Probability of transition |0/angbracketright → |1/angbracketrightas\nfunction of dimensionless time τ(ω= 3). Red line: proba-\nbility of adiabatic transition, Pad(τ). Blue line: probability\nof transition Pτobtained from the exact solution of the LZ\nproblem.\nFIG. 5: (Color online) Probability of transition |0/angbracketright → |1/angbracketrightas\nfunction of τ(ω= 0.5). Red (upper) line: probability of\nadiabatic transition. Blue (lower) line: transition proba bility\nobtained from exact solution of the LZ problem.\nof the system changes from the adiabatic regime to an\nimpulse one where its state is effectively frozen and then\nback from the impulse regime to the adiabatic one.\nIf the evolution starts at moment τi≪ −ˆτfrom\nthe ground state, the equation for determining ˆ τreads\nπˆτ/2 = 1/gap(ˆτ) (for details of calculation see Ref.13),\nand its solution is given by\nˆτ=ω√\n2/radicalBigg/radicalbigg\n1+4\nπ2ω4−1. (14)\nUsing the relation τ=τ0(P/Pc−1), we find that the\nchange of the adiabatic regime to a non-adiabatic one5\nFIG. 6: (Color online) Probability, Pe, of finding the system\nin the excited state ( x=πω2): blue line – Pe≈PAI(16),\ndashed red line – the LZ expression, Pe=e−πω2.\noccurs when the pressure is P1=Pc/parenleftbig\n1−ˆτ/τ0/parenrightbig\n, and the\nnon-adiabatic evolution becomes the adiabatic evolution\nagain when the pressure increases up to P2=Pc/parenleftbig\n1 +\nˆτ/τ0/parenrightbig\n. From here we find that within the interval of\npressure, ∆ ˆP=P2−P1= 2Pcˆτ/τ0, the behaviour of the\nsystem is described by the impulse regime.\nEmploying Eq. (14), we approximate ∆ ˆPfor fast\n(ω2≪1) and slow ( ω2≫1) transitions as\n∆ˆP\nPc=\n\n1√πτ0, ω2≪1\n1\nπωτ0, ω2≫1(15)\nIn the AI approximation the probability, Pe, of find-\ning the system in the excited state at τf≫ˆτcan be\ncalculated as follows13,15:\nPe≈PAI=|∝angbracketleftu+(ˆτ)|u−(−ˆτ)∝angbracketright|2=ˆτ2\nω2+ ˆτ2(16)\nSubstituting ˆ τfrom (14), we obtain\nPAI=2\nx2+x√\nx2+4+2, (17)\nwherex=πω2.\nForω2≪1, from Eq. (17) it follows PAI≈1−πω2.\nIn the first order this coincides with the result predicted\nby exact LZ formula: Pe=e−πω2. For the adiabatic\nevolution,ω2≫1, we obtain PAI≈1/π2ω4(See Fig.\n6.). As canbeseen, theAI approximationisgoodenough\nforω2≤1 and in the limit ω2≫1.\nComparison with experimental data. – To com-\npare our theoretical finding with the experiments on spin\ncrossover under the high pressure we use the data from\nRefs.1–4. The typical value of the critical pressure is\nPc= 50GPa, and the rest of parameters are taken as\nfollows:ε0=eV,ρ= 0.01eV and τQ≈104s. The\ncomputation yields: τ0≈109andω≈107. Thus the\nspin crossover under slowly changed pressure realizedin the cited experiments is a highly adiabatic process\n(ω2≫1). Using (15), we find the that the domain of\nnon-adiabaticity is defined by ∆ ˆP≈10−7Pa. The cor-\nresponding interval of time ˆt≈1/(πρτ0) isˆt≈10−21s.\nB. Quench dynamics under shock-wave load\nIn this section we study the quench dynamics in the\nspin system under time-dependent pressure, P(t). We\nassume that at the initial moment of time P(ti) = 0, and\nat the end of evolution P(tf) =P0. Further it is conve-\nnient to present P(t) asP(t) =P0s(t), wheres(ti) = 0\nands(tf) = 1.\nOne can observe that the first term in the Hamiltonian\n(2), yielding contribution to the total phase factor of the\nthe wave function, does not affects the dynamics of the\nsystem and may be omitted. Setting for simplicity ϕ= 0\nand using Eq. (3), one can recast the time-dependent\ndriving Hamiltonian as follows:\nHτ(t) =/parenleftbigg\nε0(1−as(t))ρ\nρ−ε0(1−as(t))/parenrightbigg\n,(18)\nwherea=P0/Pc. For a given a, the crossover occurs at\nthe critical point sc=s(tc) defined as sc= 1/a.\nIn what follows we specify the pressure as a pulse with\nthe shape determined by\nP=/braceleftbigg\n0, t<0\nP0tanh(αt), t≥0,(19)\nwhereP0is the pulse height. For P0=Pc, expanding\nP(t) near of the critical point up to the first order, we\nobtain the related LZ problem: P(t) =Pc(1+αt). From\nhere we obtain the correspondingLZ adiabaticity param-\neter asωLZ=ρ/√αε0.\nApplying the adiabatic theorem we find that condition\nfor adiabatic evolution can be written as follows:\n1\nρ2max\n0<P<P 0/vextendsingle/vextendsingle/vextendsingle/vextendsingle∝angbracketleftu+|dHτ\ndt|u−∝angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪1. (20)\nNext using Eq. (19), we obtain\n1\nω2\nLZmax\n0<s<1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleaβ(1−s2)/radicalbig\n(1−as)2+β2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪1,(21)\nwhereβ=ρ/ε0. In terms of the function ω(s) defined as\n1\nω2=1\nω2\nLZaβ(1−s2)/radicalbig\n(1−as)2+β2, (22)\nthe condition of adiabaticity (21) can be recast to the\nfollowing:\nmax\n0<s<1/parenleftbigg1\nω2(s)/parenrightbigg\n≪1 (23)6\nFIG. 7: (Color online) Parameter of adiabaticity as functio n\nofs. From the right to the left: a= 10,5,2.5,1.25 (ωLZ= 1).\nFIG. 8: (Color online) Dependence of f(a) on the amplitude\na=P0/Pc(β= 0.01).\nIn Fig. 7 the function 1 /ω2(s) is depicted ( ωLZ=1).\nWe observe that 1 /ω2has a maximum in the critical\npoint. This is in agreement with the general observation\non breaking-down of adiabaticity in the neighbourhood\nof the critical point.\nForβ≪1 we obtain the following estimate\n1\nω2≈f(a)\nω2\nLZ, (24)\nwhere\nf(a) =\n\n2aβ√\n1−a2\n(1+√\n1−a2)/radicalbig\n1−a2+β2, a≤1\na−1/a, a≥1(25)\nIn Fig. 8 the function f(a) is depicted. We observe that\nf(a)≪1 fora≤1 and it is increasing function, f(a)>1\nfora>1.\nIn Figs. 9 - 11 we present the results of numerical\ncalculations for different choice of parameter a. In all\ncalculations the parameters were chosen as follows: ε0=FIG. 9: (Color online) Dependence of transition probabilit y\nPτon time t, for different initial conditions ( P0=Pc). From\nup to down: |C0(0)|2= 0,0.25,0.5,0.75,1 (α= 10ns−1).\nFIG. 10: (Color online) Transition probability Pτfor different\nvalues of the pulse height as function of time t. From up to\ndown:a=P0/Pc= 1.01,1.001,0.999,0.99 (α= 10ns−1).\n1eV,ρ= 0.01eV andα= 10ns−1. This yields ωLZ= 1,\nand forβ=ρ/ε0, we obtain β= 0.01.\nFig. 9 shows that for P0=Pc, independently of the\ninitial conditions, the final state of the system is defined\nby the equal mixture of LH and LS states. The reason\nfor this phenomena is in growing quantum fluctuations\nwhich lead to the mixture of LH and LS states at the\ncritical pressure Pc. In Figs. 10 and 11 the dependence\nof transition probability on time and pressure, respec-\ntively, is depicted for different values of the dimension-\nless amplitude of the pulse, a=P0/Pc. In the interval\nof pressure, 0 <P/lessorsimilar1.25Pc, the evolution is highly adi-\nabatic, and for P0≈1.25Pcthe system comes to the LS\nstate with the minimal density of defects. However with\nthe increasing of pressure amplitude, P0, the probability\nto pass to the LS state is decreasing, and the system re-\nmains in the mixture of the HS ans LS states. Further\nincreasing of the amplitude leads to the domination of\nthe HS population over LS population.7\nFIG. 11: (Color online) Transition probability Pτas func-\ntion of scaled pressure sfor different values of the pulse\nheight. From up to down: a=P0/Pc= 1.25,2.5,5,10,20\n(α= 10ns−1).\nIV. SUMMARY AND CONCLUSIONS\nWe have analytically studied the properties of spin\ncrossover under the high pressure. We showed that at\nstatic loading and P=Pcoccupation numbers of both\nHS and LS states are equal at zero temperature ( T= 0),\nnHS=nLS= 0.5. ForP < P cwe obtainnHS= 1 and\nnLS= 0, whilefor P >P conehasnHS= 0andnLS= 1.\nStatic transition at T= 0 is a sharpquantum phasetran-\nsition with geometric Berry-like phase being the order\nparameter12. Finite temperature removes singularity inthenHS(P) dependence. Thermal fluctuations between\ntwo states |0∝angbracketrightand|1∝angbracketrightresults in a smooth crossover in-\nstead of the quantum phase transition at T= 0.\nToverifyourtheoreticalpredictionswehaveperformed\nnumerical simulations. Fig. 10 shows that the tempo-\nral quantum fluctuations have an effect similar to the\nthermal fluctuations. Small deviations of the shock wave\namplitudeP0/Pcfrom the unity results not in a sharp\nchange of the probability Pτeither to zero or to unity\nbut to a continuous deviation of the Pτfrom the 0.5\nvalue. At the same time for P0/Pc= 1 any initial dis-\ntribution of the HS and LS states will end its evolution\nin the equilibrium state nHS=nLS= 0.5 (Fig.9). This\nconclusion is valid for any choice of parameter α.\nAnotherdynamicaleffectwewouldliketodiscussisthe\nadiabaticity violations near the critical pressure (Fig.11).\nThis is a manifestation of the general Kibble-Zurek the-\nory. The results shown in Fig.11 are contra intuitive at\nthe first glimpse. The shock wave with larger amplitude\nhas smaller final probability for spin crossover and larger\nprobability to stay in the initial HS state. To understand\nthis effect one should note that the characteristic scale of\nthe pressure increase is given by the factor aαfor the\nwave with amplitude a. Thus larger amplitude wave also\nis faster.\nAcknowledgments\nThis research was supported by the President of Rus-\nsia Grant NSh-1044.2012.2, Presidium of the Russian\nAcademy of Science Project 2.16, RFBR Grant 12–02-\n90410a, A.I. Nesterov acknowledges the support from\nthe CONACyT, Grant No. 118930.\n∗Electronic address: nesterov@cencar.udg.mx\n†Electronic address: sgo@iph.krasn.ru\n1V.A. Sarkisyan, I.A. Trojan, I.S. Lyubutin et al, JETP\nLett.76, 664 (2002).\n2A.G. Gavriliuk, S.A. Kharlamova, I.S. Lyubutin et al,\nJETP Lett. 80, 426 (2004).\n3A.G. Gavriliuk, V.V. Struzhkin, I.S. Lyubutin et al, Phys.\nRev. B77, 155112 (2008).\n4Y. Ding, D. Haskel, and S. G. Ovchinnikov et al, Phys.\nRev. Lett. 100, 045508 (2008).\n5S. Sachdev, Quantum Phase Transitions (Cambridge Uni-\nversity Press, Cambridge, 2001).\n6T. W. B. Kibble, Journal of Physics A: Mathematical and\nGeneral 9, 1387 (1976).\n7W. H. Zurek, Nature 317, 505 (1985).\n8W. H. Zurek, Phys. Rep. 276, 177 (1996).\n9Y. Tanabe and S. Sugano, J. Phys. Soc. Jap. 9, 753 (1954).\n10S. G. Ovchinnikov, JETP 140, 107 (2008).\n11L. V. Velikov and et al, JETP39, 909 (1974).\n12A. I. Nesterov and S. G. Ovchinnikov, JETP Lett. 90, 580\n(2009).13B. Damski and W. H. Zurek, Phys. Rev. A 73, 063405\n(2006).\n14W. H. Zurek, U. Dorner, and P. Zoller, Phys. Rev. Lett.\n95, 105701 (2005).\n15B. Damski, Phys. Rev. Lett. 95, 035701 (2005).\n16M. Abramowitz and I. A. Stegun, eds., Handbook of Math-\nematical Functions (Dover, New York, 1965).\n17M. N. Lebedev, Special Functions &Their Applications\n(Dover, New York, 1972).\n18N. V. Vitanov and B. M. Garraway, Phys. Rev. A 53, 4288\n(1996).\n19N. V. Vitanov, Phys. Rev. A 59, 988 (1999).\n20S. Suzuki and M. Okada, in Quantum Annealing and Re-\nlated Optimization Methods , edited by A. Das and B. K.\nChakrabarti (Springer, 2005), vol. 679 of Lecture Notes in\nPhysics, pp. 207 – 238.\n21L. Landau and E. M. Lifshitz, ”Quantum Mechanics”\n(Pergamon, New York, 1958).\n22C. Zener, Proc. R. Soc. A 137, 696 (1932)."    },    {        "title": "2009.10628v1.From_Chaotic_Spin_Dynamics_to_Non_collinear_Spin_Textures_in_YIG_Nano_films_by_Spin_Current_Injection.pdf",        "content": "From Chaotic Spin Dynamics to Non-collinear Spin Textures in\nYIG Nano-\flms by Spin Current Injection\nHenning Ulrichs\u0003\nI. Physical Institute, Georg-August University of G ottingen,\nFriedrich-Hund-Platz 1, 37077 G ottingen, Germany\n(Dated: September 23, 2020)\nAbstract\nIn this article I report about a numerical investigation of nonlinear spin dynamics in a magnetic\nthin-\flm, made of Yttrium-Iron-Garnet (YIG). This \flm is exposed to a small in-plane oriented\nmagnetic \feld, and strong spin currents. The rich variety of \fndings encompass dynamic regimes\nhosting localized, non-propagating solitons, a turbulent chaotic regime, which condenses into a\nquasi-static phase featuring a non-collinear spin texture. Eventually, at largest spin current, a\nhomogeneously switched state is established.\n1arXiv:2009.10628v1  [cond-mat.mes-hall]  22 Sep 2020INTRODUCTION\nRecent advancements[1{5] in the art of thin-\flm growth allows nowadays to prepare YIG\n\flms with nanometer thickness. These \flms in particular feature magnetic losses comparable\nor lower than metallic ferromagnets like the widely used Permalloy or amorphous CoFeB\nalloys. Such YIG nano-\flms are of great interest to implement functionalities based on\nwave interference in magnon spintronic applications [6{9]. Being electrically insulating, YIG\nallows to completely disentangle spin and charge current related physics, which makes this\nmaterial in particular attractive for studies on spin-related transport phenomena [10{16]. In\naddition to its appearance in these topical research \felds, YIG is since its discovery a great\nmedium to study highly nonlinear spin dynamics. Turbulence [17], parametric instabilities\n[18{21], and even Bose-Einstein-condensation (BEC) [21{23] have been studied in YIG since\nquite a few decades. But so far many of these intriguing phenomena could only be realized\non rather macroscopic scales, rendering them less attractive for practical application. To\naddress these e\u000bects, YIG samples are usually exposed to strong, monochromatic microwave\nradiation, whose magnetic part can directly drive magnetization dynamics.\nOn the other hand, if single-frequency excitation is not a prerequisite, spin currents can\nbe considered as a convenient method, realizing a broad-band excitation. Spin currents can\nbe generated by a charge current when being lead through a spin-Hall material [24{27],\nwhich are for example the very common heavy metals platinum, or tungsten. In a simple\npicture, one can relate the appearance of spin currents in patterned \flms consisting of these\nmaterials to spin-orbital coupling (SOC): If a lateral charge current carried by a priori\nnot-spin-polarized electrons, experiences scattering with SOC, this scattering gives rise to\na vertical spin imbalance, building up between top and bottom surfaces of the conducting\n\flm. When deposited on top of a YIG nano-\flm, the spin accumulation at the interface can\ninteract with the magnetic moments in the YIG. This in particular can result in an e\u000bective\nreduction of magnetic losses of magnons. A critical current can in this context be de\fned\nas the magnitude at which the mode with lowest losses reaches the point of full damping\ncompensation. The spin current-induced instability of a particular mode is the essential\nmechanism behind spin-Hall oscillators, which have been realized with metallic Permalloy\n[28, 29], as well as with insulating YIG [30, 31] as active magnetic media.\nThe \fndings presented in the following in particular shed light on the question what\n2happens if one exceeds the instability threshold, in a situation when the injection of spin\ncurrents is only con\fned in one lateral dimension, or even not at all. Thereby the inves-\ntigations presented here complement recent \fndings [12, 14, 15] about magnon transport\nphenomena in YIG nano-\flms, and theoretical investigations, predicting BEC in such an\nexperimental situation [32, 33]. The here pursued micromagnetic approach provides a view\ninside the \flm, circumventing spatial and temporal resolution limitations encountered in\ncommon experimental approaches like Brillouin Light scattering [34], used to image mag-\nnetization dynamics. When con\fning the spin current, I have found \frst the nucleation of\nso-called spin wave bullets, whose density quickly increases, leading into a chaotic regime.\nAt even larger spin current a novel quasi-static phase condenses out of these turbulent \ruc-\ntuations. This phase is characterized by a stripe-like, non-collinear magnetization texture.\nAt higher current, this texture gradually disappears, and a fully switched, homogeneous\nmagnetic state is established.\nThis paper is organized as follows. First, I provide details about the numerical method.\nIn particular, I will explain how I take temperature-related e\u000bects into account. Then,\nI present results obtained for the case of con\fned spin-current injection. Subsequently, I\npresent the \fndings for the case of unrestricted injection. In the \fnal discussion, I \frst\nexplain the magnitude of the numerically found threshold current density, and explain why\nsubsequently turbulence arises. Finally, a tentative interpretation for the emerging quasi-\nstatic texture is presented.\nEXPERIMENTAL DETAILS\nTo simulate the spin-current injection into a YIG nano-\flm with a thickness of tYIG=\n20 nm, the micromagnetic simulation code MuMax3 [35] was used. In this \fnite-di\u000berences\nnumerical code, the magnetic \flm is divided into cubic cells of size 5 nm by 5 nm by\n20 nm. Each cell hosts a magnetic moment with a \fxed vectorial length, interacting via\nmicromagnetic exchange and dipolar \felds with its surroundings. A total lateral area of\n2560 nm by 2560 nm was considered. For the YIG \flm at 285 K a saturation magnetiza-\ntion ofM0= 0:11 MA/m, an exchange constant of A= 3:7 pJ/m2, a gyromagnetic ratio of\n\r= 1:7588\u000110111/Ts, and a Gilbert damping constant of \u000b= 0:001 were assumed. The\nspin torque generated via the spin-Hall-e\u000bect in a tPt= 3:5 nm thick Pt layer was taken into\n3Figure 1. In\ruence of Joule heating on static and dynamic magnetization. (a) Temperature\ndependence of magnetization according to [38], solid line is a power law \ft. Dashed vertical line\nmarks the Curie temperature TC. (b) Current density dependence of the temperature underneath\nthe Pt stripe according to [15], solid line is a quadratic \ft. Dashed horizontal line marks the Curie\ntemperature TC. (c) Derived current density dependence of magnetization M0. (d) Derived current\ndensity dependence of exchange constant A.\naccount by adding the Slonczewski torque term [36, 37] to the equation of motion of the\nmagnetization. As a conversion factor between charge and spin current, a spin-Hall angle\nof\u0012SHE= 0:11, and an interface transparency of about \u001ci= 0:47 were employed. These\nmaterial parameters resemble typical experimental values, as used in reference [15]. The\nMuMax3 script \fle in the supplement to this article provides all information to reproduce\nthe simulations.\nNote that, I did not consider the Oersted \feld created by the charge current. In the\nSupplementary Information I show that, due to its small magnitude, the Oersted \feld does\nnot in\ruence the dynamics. In contrast, the in\ruence of sample temperature on the mag-\nnetization and exchange, enhanced by Joule heating is taken into account. For simplicity, I\nhave assumed homogeneous heating. Laterally inhomogeneous temperature pro\fles do not\nimpact the dynamics, as discussed in the Supplementary Information. In the simulation,\n4a static reduction of the magnetization as well as temperature driven \ructuations, imple-\nmented by means of a \ructuating thermal \feld [35], are taken into account by the method\ndescribed in the following. I assume the temperature dependence of the magnetization\nshown in Figure 1(a), which was published in [38]. Note that, these data are well described\nby a phenomenological power law with exponent 0 :511(5) (red curve). Figure 1(b) shows\nexperimental temperature calibration data from [15], which extrapolates quadratically (red\ncurve) toTC= 560 K at about j= 8\u00011011A/m2. Combining both data sets and \ftting\ncurves, I have constructed the current dependence of the magnetization shown in Figure\n1(c). This curve is taken for rescaling of the e\u000bective magnetization at a given current and\ntemperature in the simulation: this means that in practice the length of the magnetization\nvector in each simulated cell is adjusted accordingly. Note that, in the simulation also long\nrange / low frequency \ructuation are included, stochastically excited by the thermal \feld.\nSuch \ructuations further reduce the e\u000bective magnetization. Across the whole temperature\nrange (285 K to 560 K) valid here, I have found the necessity to increase the magnetization\nby about 1 percent in order to take the additional reduction of the e\u000bective magnetization\nby such \ructuations into account. For the exchange constant I have assumed the classical\nmicromagnetic expectation A(T)/M0(T)2[39, 40]. The resulting current dependence is\nshown in 1(d).\nFigure 2 shows the experimental sample designs considered in this work. In Fig. 2(a)\nthe case of a spatially con\fned spin current injection is depicted, realized by patterning the\ncharge current carrying Pt layer to a stripe with a width of w= 500 nm. In the simulation\nthe Pt stripe is considered only implicitly, by enabling the Slonczewski torque only in the\ninjection region beneath the conductor. Absorbing boundary conditions (ABC) were applied\nto the edges parallel to the wire, and periodic boundary condition (PBC) were applied to\nthe perpendicular edges. Thereby an in\fnitely long wire was simulated. The external \feld\nhas a magnitude of \u00160H= 50 mT, and is oriented in the \flm plane, perpendicular to the\nwire. The detection stripe included in Figure 2(a) is depicted in order to graphically de\fne\nthe region underneath in the YIG \flm. This region is used for probing dynamics outside\nthe actively excited region.\nIn Fig. 2(g) the case of homogeneous spin current injection is depicted. Here, the PBCs\nare applied to all edges.\n5Figure 2. Sketch of the experimental situations and snapshots of simulated magnetization dynamics\nin terms of the normalized magnetic vector \feld m(x;y). (a) Sketch for con\fned spin current\ngeneration and injection. (g) Sketch for homogeneous spin current generation and injection. Edges\nmarked by PBC and ABC refer to periodic and absorbing boundary conditions. Note that both\nsketches include color-coded maps of m, referring to a current density below the onset of bullet\nformation, at \u0000 = \u00000:27 as indicated. (b) to (f) show snapshots of mfor increasing \u0000 as indicated\nfor the case of con\fned spin current injection. Dashed lines mark the boundaries of the Pt stripe.\n(h) to (l) depict analogously snapshots of mfor uncon\fned spin current injection.\nRESULTS\nFigure 2 depicts snapshots of the dynamics obtained for the two cases of con\fned, and\nrestricted spin current injection, at current densities above and below a certain critical\nthresholdjth. Note that I quantify this threshold later from the data, and use it to de\fne\nthe overcriticality \u0000 by\n\u0000 =j\njth\u00001: (1)\n6Figure 3. Spectral characterization of dynamics in the injection and detection area. (a) Part\nof typical transient dynamics in terms of the magnetic component mi\ny(md\ny), spatially averaged\nover the injection (detection) area, obtained at \u0000 = 7. (b) Corresponding Fourier power spectra\ncalculated from a in total 50 ns long transient, featuring a dominant peak at frequency fb, marked\nby the vertical blue dashed line (close to the frequency of Ferromagnetic Resonance f0, marked\nby the vertical green dashed line). (c) and (d) Dependency of power spectra in the injection and\ndetection area on the current density. The green dashed line marks the calculated f0(j). The blue\ndashed line serves as a guide to the eye for fb(j). Blue and orange arrows indicate the spectra\nshown in (b).\nCon\fned spin current injection\nLet us begin the inspection of the results by analyzing the case of spin current injection\ncon\fned to a stripe. Figure 2(a) to (f) shows snapshots of the magnetization after dynamic\nequilibrium has been established. For a current density below a certain threshold j < j th\n(\u0000<0, see Fig. 2(a)), no dynamic response can be seen. When increasing the current to\nj > j th(\u0000>0), this situation changes. Now, the simulation features localized hot spots,\nwhere the \flm is strongly excited (see Fig. 2(b)).\n7The normalized magnetization component mi\ny=hMyiinjection\nM0averaged across the injection\narea provides quantitative access to these dynamics. A representative time series obtained\nat \u0000 = 7 is shown in Figure 3(b). The Fourier transform power spectrum shown in Fig.\n3(c) is dominated by a strong peak at the frequency fb= 2:4 GHz. Note that this value\nis smaller than the bottom of the linear spin wave spectrum at about f0= 2:7 GHz (green\ndashed line in Fig. 3(c)). Both spectral and spatial features are typical for so-called spin-\nwave bullet modes [41]. Such a bullet is a nonlinear, non-propagating solitonic solution of\nthe gyromagnetic equation of motion. On the other hand, the dynamics in the detection\narea captured by md\ny=hMyidetection\nM0(see Fig. 3(b) and (c)), shows oscillations at a frequency\nclose to the frequency of Ferromagnetic Resonance (FMR)\n!0= 2\u0019f0=p\n!H(!H+!M(j)); (2)\nwhere!H=\r\u00160H, and!M(j) =\r\u00160M0(j).[42] When increasing the current density,\nthe number of simultaneously existing bullets in the injection area increases, as Fig. 2(c)\nillustrates. Simultaneously, their frequency fbdecreases, as shown in Fig. 3(c). This down-\nshift in frequency is well-known for bullets in in-plane magnetized magnetic \flms. In the\ndetection area, the frequency f0of the dominating FMR mode follows the thermally driven\ndecrease of the magnetization due to Joule heating (see for a plot of Eq. (2) the green dashed\nline in Fig. 3(d)). When reversing the current polarity, the dynamics in the injection as well\nas in the detection area are progressively suppressed and dominated by the FMR mode, as\ndemonstrated by the good agreement of the spectral maxima with the calculated dependence\nof the FMR frequency f0onjshown in Fig. 3(c) and (d).\nThe emergence of the bullets in the injection area can be characterized by an order\nparameter\n\t =1\u0000mi\nx\n2; (3)\nwheremi\nx=hMxii\nM0. The order parameter \t in essence captures how strong the magnetiza-\ntion deviates from the equilibrium orientation in the absence of a spin current, when MkH.\nFigure 4 shows a plot of the dependence of \t on j. One can see a quick initial growth,\nfollowed by an intermediate slowing down of the growth, which then speeds up again to\nreach values \t >0:5. Let us take a closer look at the initial growth. For a continuous phase\n8Figure 4. Dependence of the order parameter \t (blue circles) and of the magnon emission \u0006\n(orange rectangles) on the current density j. The red dashed line is a \ft of Eq. (3). The inset\nmagni\fes the behavior close the threshold current density, marked by the horizontal green dashed\nline. The horizontal green and blue dashed lines mark the onset of spin wave bullet formation, and\nthe emergence of the quasi-static texture, respectively. Orange dashed line is a guide to the eye.\ntransition one can expect according to Landau [43] a generic dependence\n\t =\u0012j\njth\u00001\u0013\"\n= \u0000\": (4)\nIndeed, \ftting Eq.(4) to the data yields a critical exponent of \"= 0:72(3), and a threshold\ncurrent density of jth= 0:17(1)\u00011011A/m2(see also inset in Fig. 4). Fig. 4 clearly shows that,\nat around \u0000 = 32, further evolution of the order parameter deviates from Eq. (4). Indeed,\nthe order paramter soon exceeds \t = 0 :5, which implies that on average, the magnetization is\nthen aligned antiparallel to the external \feld. Before this switching is completely achieved,\na quasi-static magnetic texture emerges (see Fig.2(e)). The spin-torque induced magnon\nemission from the injection area can be captured by\n\u0006(j) =hMx(js= 0)i2\nd\u0000hMx(j)i2\nd; (5)\nwhere the spatial average across detection area hMx(js= 0)idrefers to a simulation\n9conducted at \fnite temperature T(j), as caused by Joule heating, but without taking into\naccount the spin current js\rowing from the Pt stripe into the YIG \flm. In contrast\nhMx(j)idrefers to a simulation including the action of the spin current. By construction, \u0006\nis proportional to the number of magnons emitted from the injection area, which are caused\nonly by the spin injection, without compromising the thermal background. The current\ndependence \u0006( j) is included in Fig. 4. It displays a quick initial growth, followed by a\nsaturation around \u0000 = 30. Thereafter, \u0006 quickly decreases down to zero emission.\nUnrestricted spin current injection\nIn this section the situation sketched in Figure 2(g) is analyzed, where no spatial restric-\ntions are imposed on the spin current injection (see Fig. 2(g) to (l) for typical snapshots).\nAlso here, spin wave bullets appear, albeit chaos sets in earlier. The motivation for this\nexperiment is to analyze and better understand the transition from bullets to the emer-\ngence of the quasi-static stripe-like texture. The evolution of this transition is elucidated in\nFigure 5 in terms of 2d spatial, and spatio-temporal Fast Fourier transform (FFT) power\nmapsPFFT(kx;ky) andPFFT(kx;f)ky=0ofmz. In the left panel of Figure 5(a) one can see\nPFFT(kx;ky) of an already chaotic state, obtained at \u0000 = 1. The magnetization displays\nno clear structure, as the quite isotropic Fourier spectrum demonstrates. The agreement\nbetween the computed dispersion of plane spin waves [44] with the maxima of the spatio-\ntemporal Fourier spectrum PFFT(kx;f)ky=0depicted in the right panel shows that, the \ruc-\ntuations here still mainly correspond to linear spin waves. At \u0000 = 6 :4 (Fig. 5(b)), short\nwave length \ructuations strongly increase. Secondly, one can see a signature of the bullets\nappearing in the spatio-temporal Fourier spectrum. Namely, the largest spectral weight ap-\npears around kx= 0 at frequencies below the computed spin wave dispersion (dashed line).\nThis deviation is even more pronounced at \u0000 = 21 (see Fig.5(c)). At \u0000 = 43, the short wave\nlength \ructations are suppressed, and the spectrum displays a peculiar anisotropy, corre-\nsponding to the stripe-like magnetic texture shown in Figure 2(k). The static behaviour of\nthis state is re\rected by the spatio-temporal Fourier spectrum, which shows two maximima\nat frequency f= 0. Now, these maxima can not be related to linear spin waves at all\n(dashed white curve).\n10Figure 5. Spatio-temporal spectral characterization of spin dynamics in case of uncon\fned spin\ninjection. The left panels show 2d spatial FFT power maps PFFTfmz(x; y)g(kx;ky) of snapshots\nof the magnetization component mz. The right panels shows spatio-temporal FFT power maps\nPFFT(kx;f) alongkxforky= 0. The di\u000berent sub\fgures refer to speci\fc overcriticalites \u0000 as\nindicated.\n11DISCUSSION\nBullet dynamics\nIn the simulations considering a spatially restricted spin current injection, one sees the\nappearance of localized modes above a current density of jth= 0:17\u00011011A/m2. This number\ncan be compared with a simple expectation. In case of YIG nano-\flms, the mode with lowest\nlosses is the FMR mode. Without spin currents, its relaxation rate reads [45]\n!R=\u000b(!H+ 0:5!M): (6)\nThe spin torque pumps energy into the magnetic oscillations with a rate [36, 37]\n\f=j\u0001\r~\n2eM0tYIG\u0002SHE\u001ci (7)\nExact compensation, that is !R=\f, leads to a theoretical critical current density of 0 :16\u0001\n1011A/m2in the Pt stripe. Only when exceeding this value, the magnetization can become\nunstable. Indeed, the observed threshold almost exactly coincides with this theoretical\nexpectation. All properties derived from inspecting the current dependency of the dynamics\ncomply with the interpretation that, the unstable mode is a spin-wave bullet [41].\nTurbulence\nAs more and more bullets appear with increasing current, the dynamics quickly becomes\nchaotic. Note that, this chaos is deterministically driven by the spin current injection. As\nsignature of deterministic chaos, I \fnd that, in all spectra discussed in this article, the phases\nare random, and react sensitively on small perturbation of the initial state. This sensitivity\nis maintained, when excluding the thermal \ructuation \feld.\nIn the Supplementary Information accompanying this article, an analysis of spectral\nproperties of this chaotic state is shown. Chaos appears, because with increasing current,\nfor a larger and larger part of the spin-wave spectrum losses are compensated. Therefore,\ndissipation can only occur when three-magnon or higher-order scattering pushes energy into\nhigher-frequency modes, whose losses are not yet overcompensated by the injected spin\ncurrent. These nonlinear processes inevitably set in when the unstable modes have achieved\n12large enough amplitudes. Such an energy cascade is indeed prototypical for turbulence\n[46, 47]: energy is injected into the low wave number, low frequency part of the spectrum,\nand energy is dissipated as it reaches the large wave number, high frequency part.\nFurthermore, there is an interesting connection to classical pipe \row experiments. There,\nso-called pu\u000bs appear as precursors to turbulence.[48, 49] At a \frst glance, pu\u000bs and bullets\nseem to have a lot in common, as both appear prior to the onset of turbulence, and both\ndynamics are nonlinear and localized. Similar to the pu\u000bs in pipes, the bullets have a \fnite\nlifetime. How far does the analogy hold? I would like to emphasize that, in contrast to\npu\u000bs, the bullets do not move. They remain stationary inside the injection region. Note\nthat, this re\rects a quite di\u000berent experimental situation: in pipe \row experiments, one\ninduces turbulence locally, by placing objects in the \row, or by a nozzle. Here, my focus is\non a spatially extended injection region for the spin current injection, giving rise to chaotic\ndynamics in this region. The data presented in Figure 3 shows that, outside this region,\nthe magnetic \flms behaves mainly like a normal, thermally excited system. Secondly, with\nincreasing spin current injection, turbulence evolves, and the lifetime of the bullets decreases.\nAt even larger current density, the turbulence disappears again, in favor of a quasi-static\ntexture. In contrast, pu\u000bs moving down-stream have an increasing lifetime as a function\nof the Reynolds number. As I explain in the Supplementary Information, the latter can\nbe regarded as e\u000bectively controlled by the spin current injection. To further investigate\nsimilarities and di\u000berences, one could envisage a di\u000berent sample design, in which the Pt\ninjection stripe consists of two adjacent sections with large and small width, with a metallic\nferromagnetic \flm below. Then, one can locally induce bullets (=pu\u000bs) below the small\nwidth part (=reservoir under pressure). In addition, one may be able to push the bullets\ninto the large width part (=pipe) by means of the spin torque due to the current \row inside\nthe ferromagnet, similar to a moving domain wall.\nNon-collinear spin texture\nAt larger overcriticality, the progressive softening of the bullet mode culminates in a quasi-\nstatic pattern. Note that, besides softening, also the local switching of the magnetization\ndrives the condensation into the stripe pattern: wherever Mk\u0000H, the injected spin exerts\na damping-like torque. Only at small overcriticality, MkHstill holds on average, and the\n13torque is anti-damping like.\nRegarding the quasi-static texture, one may recall that Bender et al. [32] proposed in\n2014 that Bose-Einstein condensation of magnons should set in under spin current injection.\nIn their theory, a phase diagram is derived under the assumption of small angle dynamics.\nI here emphasize that, in case of a strongly excited YIG nano-\flm, the nonlinear spin-\nwave bullets have to be considered as dynamic modes undergoing condensation. Their local\noscillation angle is large. Therefore the theory of reference [32] cannot be applied directly. To\nfurther understand the classical condensation phenomenon observed in this micromagnetic\nsimulation work, I suggest to \frst-of-all consider the dispersion of bullets, which I here\napproximate by\n!b(k) =p\n(!H\u0000a!Mk2) (!H\u0000a!Mk2+!M); (8)\nwherea=2A\n\u00160M2\n0. Note that, in this expression the wave number k/1\ndbcharacterizes\nthe diameter of the non-propagating bullet [41]. Comparing the maxima of the Fourier\npower in Figure 5(d) with the overlaid dispersion curve Eq.(8) (dashed green line), I \fnd\nan intersection approximately at the point of vanishing frequency. To rule out that this\nis a mere coincidence, I have repeated the simulations for di\u000berent external \felds between\n\u00160H= 25 mT and 400 mT. At all \felds I have found at a current density of j= 7:5\u0001\n1011A/m2(corresponding to \u0000 = 43) the stripe texture, and determined the corresponding\ncharacteristic wave number k0. The \feld dependence of k0is plotted in Figure 6, on top of a\ncoloured map encoding the \feld and wave number dependence of !b(H;k). For all selected\n\feldsH, the characteristic wave numbers k0lie approximately on the isocontour \u0010!=0of\nvanishing frequency. This re\rects the \fnding that the emerging texture is a quasi-static\nfeature.\nWhy should this particular mode be chosen? Recall that the conventional dissipation\nargument for spin-wave instabilities implies that, the mode with smallest losses is selected\n[17]. For a magnon BEC, this is also the mode with the lowest frequency. Here, this\nargument fails, because the spin torque anyway compensates the direct dissipative losses.\nBut by pushing the bullets as far away from the linear spin-wave spectrum as possible, the\nsystem minimizes nonlinear losses, which occur due to multiple-magnon scattering. Such\nprocesses redistribute energy from the bullets into high frequency magnons, whose losses\nare not compensated by the spin current. This indirect route remains as active dissipation\n14Figure 6. Field dependence of quasi-static texture. The colored map in the background shows the\n\feld and wave number dependence of Equation (8). The characteristic wave numbers k0(open\ncircles) lie on the isocontour \u0010!=0(red line).\nchannel as long as the bullet frequency does not vanish.\nSUMMARY\nTo summarize, the overall picture for spin current induced magnetization dynamics in\nYIG nano-\flms obtained from micromagnetic simulation is like this: when exceeding the\nthreshold current density jth= 0:17\u00011011A/m2, \frst-of-all single spin-wave bullets ap-\npear, whose number quickly increases with increasing current. The bullets then give rise to\ndeterministic chaos. This turbulent state eventually freezes out, in favor of a quasi-static,\nnon-collinear magnetic texture, which \fnally gradually turns over into a completely switched\nstate. Note that, combining materials with large spin-Hall angles like \f-tungsten [50], with\noptimally grown YIG nano-\flms, displaying Gilbert damping constants as small as only\n7\u000210\u00005[2, 4], opens up a realistic, and fruitful perspective for studying samples with large\nactive areas ( w\u001dk\u00001\nb). Then, one might be able to observe turbulent dynamics, as well as\nthe novel, quasi-static texture. While so far, no experimental reports about the emergence\nof such a texture exist, the possibility to establish a connection to experimental work (see\nSupplementary Information of this article) further supports this chance. In addition to such\nexperimental opportunities, the \fndings presented in this paper also open in interesting\nperspective for the application of spin hydrodynamic theory [51{54]. 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Lett.\n123, 117203 (2019).\n19"    },    {        "title": "1310.4840v1.Electrical_Detection_of_Direct_and_Alternating_Spin_Current_Injected_from_a_Ferromagnetic_Insulator_into_a_Ferromagnetic_Metal.pdf",        "content": "arXiv:1310.4840v1  [cond-mat.mtrl-sci]  17 Oct 2013Electrical Detection of Direct and Alternating Spin Curren t Injected from a\nFerromagnetic Insulator into a Ferromagnetic Metal\nP. Hyde, Lihui Bai, D.M.J. Kumar, B.W. Southern, and C.-M. Hu∗\nDepartment of Physics and Astronomy, University of Manitob a, Winnipeg, Canada R3T 2N2\nS. Y. Huang, B. F. Miao, and C. L. Chien\nDepartment of Physics and Astronomy, Johns Hopkins Univers ity, Baltimore, MD 21218, USA\n(Dated: June 8, 2021)\nWe report room temperature electrical detection of spin inj ection from a ferromagnetic insulator\n(YIG) into a ferromagnetic metal (Permalloy, Py). Non-equi librium spins with both static and\nprecessional spin polarizations are dynamically generate d by the ferromagnetic resonance of YIG\nmagnetization, and electrically detected by Py as dc and ac s pin currents, respectively. The dc spin\ncurrent is electrically detected via the inverse spin Hall e ffect of Py, while the ac spin current is\nconverted to a dc voltage via the spin rectification effect of P y which is resonantly enhanced by\ndynamic exchange interaction between the ac spin current an d the Py magnetization. Our results\nreveal a new path for developing insulator spintronics, whi ch is distinct from the prevalent but\ncontroversial approach of using Pt as the spin current detec tor.\nDeveloping new methods for generating and detecting\nspin currents has been the central task of spintronics.\nIn the pioneering work of Johnson and Silsbee [1], the\ngeneration and detection of spin-polarized currents were\nboth achieved through the use of ferromagnetic metals\n(FM). Recent breakthroughs reveal ferromagnetic insu-\nlators (FI) to be promising spin current sources, in which\nspin currents can be generated without the presence of\nany charge current [2, 3]. In the ground-breaking exper-\niment performed 3 years ago by Kajiwara et al.[2], elec-\ntrical detection of the spin current generated by yttrium\niron garnet (Y 3Fe5O12, YIG) was achieved by utilizing\nthe heavynormalmetal platinum (Pt), in whichspin cur-\nrent was detected via the inverse spin Hall effect (ISHE).\nSince then, nearly the entire insulator-spintronics com-\nmunityhasfollowedsuitandusedPtasthe standardspin\ndetector. But so far, consensus has not yet been achieved\non a few critical spin-dependent material issues of Pt [4–\n7]. Given the fact that ferromagnetic metals are broadly\nused as spin detectors in both semiconductor [8, 9] and\nmetallic spintronics devices [1, 10, 11], it is noteworthy\nthat the appealing topic of how a FM material may de-\ntect the spin current generated by a FI has barely been\ninvestigated. Elucidating this issue is of broad interest\nfor making insulator-spintronics device compatible with\nboth semiconductor and metallic spintronics devices.\nIn this letter, we report room temperature detection\nof spin current generated in YIG by feromagnetic res-\nonance (FMR). Distinct from the popular approach of\nusing Pt as the spin detector, we use the ferromagnetic\nmetal Permalloy (Py) instead, and demonstrate that Py\nnot only detects the dc spin current from YIG, but most\nstrikingly, it also detects the recently predicted ac spin\ncurrent [12] by directly converting it into a dc voltage,\nwhich makes Py a superior spin detector compared to\nPt. Two very recent experiments make this work possi-\nble: (i) the discovery of the ISHE in Py [13], and (ii) theestablishment of a universal method for clearly separat-\ning spin rectification from spin pumping [14].\nWe begin by highlighting the basic ideas. As shown in\nFig. 1, let us consideraPy/YIGbilayerunder microwave\nirradiation in an external magnetic field H. Choosing the\nxaxis as the longitudinal direction for measuring the dc\nvoltages, and the zaxis as perpendicular to the interface,\nthe direction of His described by the polar (with respect\nto thezaxis) and azimuth (with respect to the xaxis)\nangles of θandφ, respectively, as shown in Fig. 1(c).\nAt the FMR frequency ωYIGof YIG, the magnetization\nof YIG precesses about its saturation magnetization M,\nwhich pumps non-equilibrium spins diffusing across the\nPy/YIG interface. Hence, a dc spin current jscarries\nstatic non-equilibrium spin angular momentum which is\nantiparallel to M, while an ac spin current js(ωYIG) car-\nries dynamic non-equilibrium spin angular momentum\nwhich is precessing about M[12]. Both spin currents\nflow along the zdirection, as shown in Fig. 1(c).\nBased on the recently discovered ISHE in Py [13], the\nidea of using Py to detect jsis straightforward as shown\nin Fig. 1(a). It can be detected by the dc voltage VSP\nin Py produced through spin pumping and the ISHE,\ni.e.,VSPis proportional to js. In contrast, detecting the\nhigh-frequency ac spin current js(ωYIG) is nontrivial and\nis currently of great interest. Two groups have very re-\ncently developed very smart methods to solve this prob-\nlem [15, 16]. Both use a microwave detector for measur-\ning the ac current in Pt induced by js(ωYIG). Different\nfrom the two methods [15, 16], our idea is inspired by\nthe pioneering work of the forgotten masters Silsbee et\nal., who performed 35 years ago the first spin pumping\nexperiment via the enhanced spin resonance [17]. And\nwe utilize the spin rectification effect in Py which we\nhave systematically studied [18–21]. At the Py FMR fre-\nquencyωPy, the precessing magnetization leads to the\nspin rectification which induces a dc voltage VSRpropor-2\n(a)\n(b)\n(c)\nFIG. 1: (Colour online) (a) At φ=90◦, the dc spin current\npumped by YIG FMR can be detected in Py via the ISHE\ninduced dc voltage VSP. (b) At φ=0◦, the Py FMR can be\ndetected by VSRvia the spin rectification effect. (c) At the\nequal-resonance condition, the ac spin current pumped from\nthe YIG enhances the FMR of Py, which can be detected by\nthe increased VSR. Correspondingly, enhanced YIG FMR can\nbe detected via the increased VSP.\ntional to the precession angle, as shown in Fig. 1(b). At\nthe equal-resonance condition where ωYIG=ωPy[shown\nin Fig. 1(c)] the ac spin current precessing at ωYIGmay\nenhance the FMR of Py via dynamic exchange interac-\ntion, in a process similar to the enhanced electron spin\nresonances discovered by Silsbee et al.[17]. Thus, mea-\nsuring the enhanced VSRin Py may permit direct elec-\ntrical detection of the ac spin current without the use of\nany microwave detectors.\nSuch a method needs two prerequisites: (i) a clear pro-\ncedure for distinguishing VSPfromVSR, and (ii) a prac-\ntical way for setting the equal-resonance condition where\nωYIG=ωPyatthesamemagneticfield H, orequivalently,\nsettingthe FMR resonancefield HYIG=HPyatthe same\nmicrowave frequency ω. The required procedure has re-\ncently been established [14] so that we may use the fol-\nlowing angular condition and symmetries to clearly sep-\narate and identify the dc voltages induced by pure spinpumping ( VSP) and pure spin rectification ( VSR):\nAt φ= 90◦,VSP(θ,H) =−VSP(θ,−H) =−VSP(−θ,H);\nAt φ= 0◦,VSR(θ,H) =VSR(θ,−H) =−VSR(−θ,H).\n(1)\nTheequal-resonancecondition, aswedemonstratebelow,\ncan be set by adjusting the Hfield direction, making use\nof the different magnetic anisotropies of Py and YIG.\nSamples were prepared by magnetron sputtering and\npatternedusingaphoto-lithographyandliftofftechnique.\nA 10-nm thick Py thin film was deposited on a YIG sub-\nstrate (10 mm ×4 mm in area) and patterned into Hall\nbar structure with lateral dimensions of 5 mm ×0.2 mm.\nA 100-mW microwave was applied to excite FMR in the\nbilayer through a rectangular waveguide. By sweeping\ntheHfieldatafixedmicrowavefrequency, dcvoltagesin-\nduced by FMR were detected along the xaxis of the Hall\nbar using a lock-in amplification. Here, the microwave\npower was modulated at a frequency of 8.33 kHz.\nFigure 2 shows typical voltage signals measured at\nω/2π= 11 GHz. While sweeping the Hfield applied\natφ= 90◦, we observe a background signal of ±0.3µV\nand sharp resonances at µ0HR=±0.484 T with a line\nwidth of 10.0 mT as shown in Fig. 2(a). At the lower\n(inner) field side ofthe sharp resonance, there is a weaker\nresonance together with a series of resonances too weak\nto be accuratelydistinguished. Both the backgroundand\nresonance signals have an odd symmetry with respect to\ntheHfield direction, i.e,V(H) =−V(−H). The data\nplotted in Fig. 2(a) was taken at θ= 25◦, but data with\nan odd symmetry was measured at other angles of θ(not\nshown), provided φ= 90◦. In contrast, by setting φ=\n0◦, both the background and the two sharp resonances\nnearlydisappear, asshownin Fig. 2(b). Instead, broader\nresonances at ±1.137 T with a line width of 17.5 mT are\nobserved, which have an asymmetric line shape but even\nfield symmetry of V(H) =V(−H). Again, as long as φ\n= 0◦, the broad resonances with even field symmetry are\nobserved at arbitrary angle of θ, but note that they do\nnot appear in the spectrum measured at φ= 90◦.\nSimilarbackgroundvoltage Vbghasbeenfoundinother\nbilayer devices such as Pt/YIG under microwave excita-\ntion [22]. In general, for devices with a thin metallic\nlayer deposited on a thick substrate, microwave heating\nis known to cause a temperature gradient perpendicular\nto the interface [23]. Hence, a simple interpretation is\nthat such a vertical temperature gradient may drive a dc\nspin current via the spin Seebeck effect [7], which may\nbe detected via the ISHE as Vbg. Indeed, we find that\nVbg∝sin(φ) as expected from the spin Seebeck effect.\nHowever, we note that such an angular dependence can\nnot irrefutably rule out the possibility that Vbgis caused\nby the anomalous Nernst effect [5] which leads to the\nsame relation of Vbg∝sin(φ). Hence, we leave the in-\ntriguing origin of Vbgto a future study, and focus in this3\n-1.00.01.0VSR (µV)\n-1.4 -0.7 0.0 0.7 1.4\nµ0H (T)-0.50.00.5VSP (µV)\nθ = 25o   φ =  90o \nθ =  2o  φ = 0oω/2π =   11 GHz (a)\n(b)\n12\n8\n4ωr/2π  (GHz)\n1.5 1.2 0.9 0.6 0.3 0.0\nµ0HR (T) PyFMR\n YIGFMRθ = 90o φ = 45o\nθ = 1o φ = 45o(c)\nFIG. 2: (Colour online) (a) The YIG and (b) the Py FMR\nelectrically detected via VSPatφ= 90◦andVSRatφ= 0◦,\nrespectively. (c) ωr−HRdispersions ofthePyandYIGFMRs\nmeasured at in-plane ( θ= 90◦) and out-of-plane ( θ≈0◦) field\nconfigurations. Curves are calculated theoretically.\npaper on the detection of spin currents via FMR, which\ncan be conclusively verified.\nWhenφ∝negationslash=n×π/2 where nis an integer, we find\nthat both the sharp and broad resonances appear in the\nsame voltage trace. Although their relative strength de-\npends on φ, as we have discussed, neither of their res-\nonance fields is sensitive to this angle; both depend on\nthe polar angle, θ. Setting φ= 45◦, the dispersions for\nboth resonances were measured at θ= 1◦and 90◦, cor-\nresponding to perpendicular and in-plane Hfield direc-\ntions, respectively. They are plotted in Fig. 2(c) for\ncomparison. To identify these resonances, we have cal-\nculated the FMR conditions for the Py/YIG bilayer by\nlinearizing the Landau-Lifshitz-Gilbert equations about\nthe equilibrium determined by the Hfield strength and\ndirection. Because of the macroscopic lateral size of the\ndevice, wemakethe simplestapproximationtomodel the\nmagnetic anisotropy by using a perpendicular demagne-\ntization field µ0Mdas the fitting parameter. From the\nbest fits we find µ0Md= 0.147 and 0.910 T for YIG and\nPy, respectively. The gyromagnetic factor is found to be\nγ= 27.0 and 26.2 GHz/T for YIG and Py, respectively.\nNote that the thin Py film has a much larger perpendic-\nular anisotropy than YIG, as expected.\nThe calculated dispersions are plotted in Fig. 2(c) assolid curves. The good agreement allows us to identify\nthe sharp and broad resonances in Fig. 2(a) and (b) as\nthe FMR of YIG and Py, respectively. Their different\nline widths are consistent with the fact that the damping\nconstant of YIG is much smaller than that of Py. To\nkeep the focus, our simple model includes neither the\nexchange coupling nor the high-order anisotropy of YIG,\nhence it does not explicitly explain the origin of the weak\nresonance in Fig. 2(a), which could be the spin wave\nobserved previously [2]. Following Eq. 1, at φ= 90◦, the\nmeasured field symmetry of V(HYIG)≃ −V(−HYIG) as\nshown in Fig. 2(a) allows us to identify the dc voltage of\nthe YIG FMR as VSP[14]. Hence, the dc spin current js\ninjected from the YIG into the Pyis electricallydetected.\nSimilarly, at φ= 0◦, the measured field symmetry of\nV(HPy)≃V(−HPy) as shown in Fig. 2(b) confirms\nthat the Py FMR is electrically detected via pure spin\nrectification [14, 18], which we now use to detect the ac\nspin current js(ωYIG).\nAs shown in Fig. 2(c), at the same microwave fre-\nquency, the Py FMR measured in the in-plane configu-\nration with θ= 90◦appears on the low field side of the\nYIG FMR. Due to the larger perpendicular anisotropy\nof Py, in the perpendicular configuration with θ= 1◦,\nthe Py FMR moves to the high field side. Hence, the\nequal-resonance condition of Py and YIG can be set by\ntuning the polar angle θ. With the obtained parameters\nwe have calculated and found that it occurs at θ= 12◦.\nWe thus proceeded to study the ac spin current en-\nhanced FMR signal near θ= 12◦. Following Eq. 1 by\nsettingφ= 0◦, we can trace the electrically detected\nFMR of Py when θis tuned through 12◦, as shown in\nFig. 3(a). The shaded areas are the approximate calcu-\nlated FMR fields of Py. When θis tuned from 9◦to 12◦,\nthe peak-to-peak amplitude of the FMR signal is seen\nto increase by more than a factor of 4 (from below 0.5\nµV to above 2 µV). When θis further tuned from 12◦to\n19◦, the FMR signal amplitude drops back below 0.5 µV.\nNote that the detailed line shape of the FMR signal de-\npends sensitively on the external field direction [21], but\natφ= 0◦the amplitude of the FMR signal, electrically\ndetected via spin rectification, provides a good measure\nof the cone angle of the magnetization precession [18].\nIn order to rule out the possibility that the dramati-\ncally enhanced Py FMR signal is just due to the static\ninterlayer exchange coupling [24], we monitor the θde-\npendence ofthe YIG FMR signaldetected by spin pump-\ning atφ= 90◦. For the static coupling of Py and YIG\nmagnetizations, one would only observe an anti-crossing\nof their FMRs, with the enhancement of one mode ac-\ncompanied by the suppression of the other [24]. In con-\ntrast, as shown in Fig. 3(b), the FMR signal of YIG is\nalso found to be enhanced dramatically when θis tuned\nthrough 12◦.\nSuchasimultaneousenhancementofbothFMRsignals\nis more clearly seen from the systematic data measured4V\n0.6 0.4 0.2\nµ0H (T)0.2 µVθ  = 9o\nθ  = 12o\nθ  = 19oφ = 90oV\n0.6 0.4 0.2\nµ0H (T)2.0 µVθ = 9o\nθ = 12o\nθ = 19oφ = 0o(a) (b)\nFIG. 3: (Colour online) At θ= 12◦, both amplitudes of (a)\nthe Py FMR measured by VSR(φ= 0◦) and (b) the YIG\nFMR measured by VSP(φ= 90◦) are greatly enhanced. In\nboth (a) and (b) ω/2π= 7GHz.\natω/2π= 7 GHz. As shown in Fig. 4(a), going from\nthe perpendicular down to the in-plane configuration by\nincreasing θ, the FMR field of Py deceases much faster\nthan that of YIG due to their different perpendicular\nanisotropies. It crosses first at θ= 12◦with the YIG\nFMR (as calculated), then it crosses at about 14◦with\nthe weak resonance mode YIG WR. Fig. 4(b) shows the\namplitude of Py FMR signal measured at φ= 0◦via\nspin rectification, which is normalized by the maximum\namplitude of 2.67 µV atθ= 12◦. For comparison, the\namplitude of the YIG FMR measured at φ= 90◦via\nspin pumping is plotted in Fig. 4(c), which is normal-\nized by the maximum amplitude of 0.35 µV, also at θ=\n12◦. Clearly, at the equal-resonance condition, the am-\nplitudes of both the Py and YIG FMR voltages increase\ndramatically and simultaneously.\nIt is intriguing to compare the simultaneously en-\nhanced FMRs electrically detected in Py/YIG bilayer\nwith the simultaneously narrowing of the FMRs mea-\nsured by absorption spectroscopy on Fe/Au/Fe layers\n[25]. The absorption experiment performed by Heinrich\net al.is enlightening since it reveals the exact cancel-\nlation of the spin currents flowing in opposite directions\nat equal-resonancecondition, which reduces the damping\nof spin pumping. In our experiment, the dc voltage de-\ntected via the spin rectification effect measures the cone\nangle of Py FMR. At the equal-resonance condition, the\nac spin current pumped by YIG FMR injects into Py,\nwhich reduces the damping and therefore enhances the\ncone angle of the Py FMR. In the phenomenological the-\norydevelopedbySilsbee et al.[17], suchan enhancement0.6\n0.4\n0.2µ0HR (T)\n20 16 12 8\nθ  (degree) PyFMR\n YIGFMR\n YIGWRω/2π =   7 GHz\n1.0\n0.5\n0.0\n20 16 12 8\nθ  (degree) PyFMR\n1.0\n0.5\n0.0\n20 16 12 8\nθ  (degree) YIGFMR\n YIGWR(a)\n(b)\n(c) Normalized Amplitude\nFIG. 4: (Colour online) (a) The polar angular dependence\nof the resonance fields measured at ω/2π= 7GHz, showing\nthe Py FMR crosses the YIG resonances at θ= 12◦and 14◦.\nThe normalized amplitudes of (b) the Py FMR and (c) the\nYIG resonances showing the simultaneous enhancement at\nequal-resonance conditions. Solid curves in (a) are calcul ated\ntheoretically, dashed curves in (b) and (c) are guide to eyes .\nof spin resonance is caused by the dynamic exchange in-\nteraction between the ac spin current and the spin an-\ngular momentum. Either of these two pictures allow us\nto conclude that, by using the spin rectification of Py,\nthe ac spin current of YIG can be electrically detected\nat the equal-resonance condition, as demonstrated in our\nexperiment.\nIn summary, we have demonstrated new methods for\nthe electrical detection ofdc and ac spin currents in YIG.\nBothareachievedby usingPyas the spin detector. Since\nthe magnetization in Py is very easy to control by either\ntuning anexternalmagneticfield orbytailoringitsshape\nanisotropy, we expect that our straightforward methods\npermit the advancement of insulator spintronics in a dis-\ntinct new path, setting it free from relying on Pt as the\nspin detector, in which the pivotal spin Hall effect is still\ncontroversial and is very difficult to tune.\nWork in Manitoba has been funded by NSERC, CFI,\nURGP, and UMGF grants (C.-M.H. and B.W.S.). Work\nat JHU has been funded by NSF (DMR-1262253).\nD.M.J.K was supported by Mitacs Globalink Program.\nS.Y.H was partially supported by STARnet sponsored\nby MARCO and DARPA.5\n∗Electronic address: hu@physics.umanitoba.ca\n[1] M. Johnson, R. H. Silsbee, Phys. Rev. Lett. 55, 1790\n(1985).\n[2] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K.\nTakanashi, S. Maekawa, and E. Saitoh, Nature 464, 262\n(2010).\n[3] Ken-ichi Uchida, Hiroto Adachi, Takeru Ota, Hiroyasu\nNakayama, Sadamichi Maekawa, and Eiji Saitoh, Appl.\nPhys. Lett. 97, 172505 (2010).\n[4] S.Y. Huang, W. G. Wang, S. F. Lee, J. Kwo, and C. L.\nChien, Phys. Rev. Lett. 107, 216604 (2011).\n[5] S.Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J.\nWu, T.Y. Chen, J. Q. Xiao, and C. L. Chien, Phys. Rev.\nLett.109, 107204 (2012).\n[6] D. Qu, S.Y. Huang, Jun Hu, Ruqian Wu, and C. L.\nChien, Phys. Rev. Lett. 110, 067206 (2013).\n[7] T. Kikkawa, K. Uchida, Y. Shiomi, Z. Qiu, D. Hou, D.\nTian, H. Nakayama, X.-F. Jin, and E. Saitoh, Phys. Rev.\nLett.110, 067207 (2013).\n[8] C.-M. Hu, J. Nitta, A. Jensen, J. B. Hansen, and H.\nTakayanagi, Phys. Rev. B 63, 125333 (2001).\n[9] Xiaohua Lou, et al., Nature Physics 3, 197 (2007).\n[10] F. J. Jedema, A. T. Filip, and B. J. van Wees, Nature.\n410, 345, (2001).\n[11] F. J. Jedema, H. B. Heersche, A. T. Filip, J. J. A. Basel-\nmans and B. J. van Wees, Nature 416, 713, (2002).\n[12] H.J. Jiao and G. E. W. Bauer, Phys. Rev. Lett. 110,\n217602 (2013).\n[13] B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, Phys.Rev. Lett. 111, 066602, (2013).\n[14] Lihui Bai, P. Hyde, Y. S. Gui, C.-M. Hu, V. Vlam-\ninck, J. E. Pearson, S. D. Bader, and A. Hoffmann,\narXiv:1308.4041.\n[15] Dahai Wei, Martin Obstbaum, Christian Back, and\nGeorg Woltersdorf, arXiv:1307.2961.\n[16] C. Hahn, G. de Loubens, M. Viret, and O. Klein,\narXiv:1308.3433.\n[17] R. H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B.\n19, 4382, (1979).\n[18] Y. S. Gui, N. Mecking, X. Zhou, Gwyn Williams, and\nC.-M. Hu, Phys. Rev. Lett. 98, 107602 (2007).\n[19] N. Mecking, Y. S. Gui, and C.-M. Hu, Phys. Rev. B 76,\n224430 (2007).\n[20] A. Wirthmann, X. Fan, Y. S. Gui, K. Martens, G.\nWilliams, J. Dietrich, G.E. Bridges, andC.-M. Hu, Phys.\nRev. Lett. 105, 017202 (2010).\n[21] M. Harder, Z. X. Cao, Y. S. Gui, X. L. Fan, and C.-M.\nHu, Phys. Rev. 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Lett. 90,\n187601 (2003)."    },    {        "title": "0912.3517v2.An_improved_effective_one_body_Hamiltonian_for_spinning_black_hole_binaries.pdf",        "content": "arXiv:0912.3517v2  [gr-qc]  26 Feb 2010An improved effective-one-body Hamiltonian for spinning bl ack-hole binaries\nEnrico Barausse1and Alessandra Buonanno1\n1Maryland Center for Fundamental Physics, Department of Phy sics, University of Maryland, College Park, MD 20742\n(Dated: September 3, 2018)\nBuilding on a recent paper in which we computed the canonical Hamiltonian of a spinning test\nparticle in curved spacetime, at linear order in the particl e’s spin, we work out an improved effective-\none-body (EOB) Hamiltonian for spinning black-hole binari es. As in previous descriptions, we\nendow the effective particle not only with a mass µ, but also with a spin S∗. Thus, the effective\nparticle interacts with the effective Kerr background (havi ng spinSKerr) through a geodesic-type\ninteraction and an additional spin-dependent interaction proportional to S∗. When expanded in\npost-Newtonian (PN) orders, the EOB Hamiltonian reproduce s the leading order spin-spin coupling\nand the spin-orbit coupling through 2.5PN order, for any mas s-ratio. Also, it reproduces allspin-\norbitcouplings inthe test-particle limit. Similarly toth etest-particle limit case, when we restrict the\nEOBdynamicstospins aligned or antialigned with theorbita l angular momentum, for whichcircular\norbits exist, the EOBdynamics has several interesting feat ures, suchas the existence of an innermost\nstable circular orbit, a photon circular orbit, and a maximu m in the orbital frequency during the\nplunge subsequent to the inspiral. These properties are cru cial for reproducing the dynamics and\ngravitational-wave emission of spinning black-hole binar ies, as calculated in numerical relativity\nsimulations.\nPACS numbers: 04.25.D-, 04.25.dg, 04.25.Nx, 04.30.-w\nI. INTRODUCTION\nCoalescing black-hole binaries are among the most\npromising sources for the current and future laser-\ninterferometer gravitational-wave detectors, such as the\nground-based detectors LIGO and Virgo [1, 2] and the\nspace-based detector LISA [3].\nThe search for gravitational waves from coalescing bi-\nnaries and the extraction of the binary’s physical param-\neters are based on the matched filtering technique, which\nrequires accurate knowledge of the waveform of the in-\ncoming signal. Because black holes in general relativity\nare uniquely defined by their masses and spins, the wave-\nforms for black-hole binaries on a quasi-circular orbits\ndepend on eight parameters, namely the masses m1and\nm2and the spin vectors S1andS2. Due to the large pa-\nrameter space, eventually tens of thousands of waveform\ntemplates may be needed to extract the gravitational-\nwave signal from the noise, an impossible demand for\nnumerical-relativity alone. Fortunately, recent work at\nthe interface between analytical and numerical relativity\nhas demonstratedthe possibility ofmodeling analytically\nthe dynamics and the gravitational-waveemission of coa-\nlescingnon-spinning black holes, thus providing data an-\nalysts with analytical template families [4–7] to be used\nfor the searches (see also Ref. [8], which considers the\ncases of extreme mass-ratio inspirals). The next impor-\ntant step is to extend those studies to spinning precessing\nblack holes.\nSofar, theanalyticalmodelingoftheinspiral,plunge1,\n1We refer to plunge as the dynamical phase starting soon after the\ntwo-body system passes the last stable orbit. During the plu ngemerger2, and ringdown has been obtained within either\nthe effective-one-body (EOB) formalism [4, 6, 7, 9–17] or\nin Taylor-expanded PN models [13], both calibrated to\nnumerical-relativity simulations, or in phenomenological\napproaches [5, 18] where the numerical-relativity wave-\nforms are fitted to templates which resemble the PN ex-\npansion, but in which the coefficients predicted by PN\ntheory are replaced by many arbitrary coefficients. Con-\nsidering the success of the EOB formalism in under-\nstanding the physics of the coalescence of non-spinning\nblack holes and modeling their gravitational-wave emis-\nsion with a small number of adjustable parameters, in\nthis paper we will use that technique, adapting it to the\ncase of spinning black-hole binaries.\nThe first EOB Hamiltonian which included spin effects\nwas computed in Ref. [19]. In Ref. [20], the authors used\nthe non-spinning EOB Hamiltonian augmented with PN\nspin terms to carry out the first exploratory study of the\ndynamics and gravitational radiation of spinning black-\nholebinariesduringinspiral, mergerandringdown. More\nrecently, Ref. [21] extended the model of Ref. [19] to in-\nclude the next-to-leading-orderspin-orbitcouplings. The\nEOBformalismdeveloped in Refs. [19, 21] highlightssev-\neral features of the spinning two-body dynamics and was\nrecently compared to numerical-relativity simulations of\nspinning non-precessing black holes in Ref. [22]. In this\npaper we build on Refs. [19, 21] and also on Ref. [23], in\nwhich we (in collaboration with Etienne Racine) derived\nthe canonical Hamiltonian for a spinning test-particle in\ncurved spacetime, at linear order in the particle’s spin,\nthe motion is driven mostly by the conservative dynamics.\n2We refer to merger as the dynamical phase in which the two-\nbody system is described by a single black hole.2\nandworkoutanimprovedEOBHamiltonianforspinning\nblack-hole binaries. In particular, our EOB Hamiltonian\nreproduces the leading order spin-spin coupling and the\nspin-orbit coupling through 2.5PN order, for any mass-\nratio. Also, it resums allthe test-particle limit spin-orbit\nterms. Moreover, when restricted to the case of spins\naligned or antialigned with the orbital angular momen-\ntum, it presents several important features, such as the\nexistence of an innermost stable circular orbit, a photon\ncircular orbit, and a maximum in the orbital frequency\nduring the plunge subsequent to the inspiral. All of these\nfeatures are crucial for reproducing the dynamics and\ngravitational-wave emission of spinning coalescing black\nholes, as calculated in numerical relativity simulations.\nThis paper is organized as follows. After presenting\nour notation (Sec. II), in Sec. III we build on Ref. [23]\nand derive the Hamiltonian for a spinning test particle in\naxisymmetric stationary spacetimes. In Sec. IV, we spe-\ncializetheaxisymmetricstationaryspacetimetothe Kerr\nspacetime in Boyer-Lindquist coordinates. In Sec. V we\nwork out the EOB Hamiltonian of two spinning precess-\ning black holes. In Sec. VI we restrict the dynamics to\nspins aligned or antialigned with the orbital angular mo-\nmentum and determine several properties of the circular-\norbit dynamics. Section VII summarizes our main con-\nclusions. More details on how the spin-spin sector of the\nEOB Hamiltonian is constructed are eventually given in\nAppendix A.\nII. NOTATION\nThroughout this paper, we use the signature\n(−,+,+,+) for the metric. Spacetime tensor indices\n(ranging from 0 to 3) are denoted with Greek letters,\nwhile spatial tensor indices (ranging from 1 to 3) are\ndenoted with lowercase Latin letters. Unless stated oth-\nerwise, we use geometric units ( G=c= 1), although we\nrestore the factors of cwhen expanding in PN orders.\nWe define a tetrad field as a set consisting of a timelike\nfuture-oriented vector ˜ eµ\nTand three spacelike vectors ˜ eµ\nI\n(I= 1,...,3) — collectively denoted as ˜ eµ\nA(A= 0,...,3)\n— satisfying\n˜eµ\nA˜eν\nBgµν=ηAB, (2.1)\nwhereηTT=−1,ηTI= 0,ηIJ=δIJ(δIJbeing the\nKronecker symbol).\nInternal tetrad indices denoted with the uppercase\nLatin letters A,B,CandDalwaysrunfrom 0to3, while\ninternal tetrad indices with the uppercase Latin letters\nI,J,KandL, associated with the spacelike tetrad vec-\ntors, run from 1 to 3 only. The timelike tetrad index is\ndenoted by T.\nTetrad indices are raised and lowered with the metric\nηAB[e.g., ˜eµ\nA=ηAB(˜eB)µ]. We denote the projections\nof a vector Vonto the tetrad with VA≡Vµ˜eA\nµ, and\nsimilarly for tensors of higher rank. Partial derivativeswill be denoted with a comma or with ∂, and covariant\nderivatives with a semicolon.\nIII. HAMILTONIAN FOR A SPINNING\nTEST-PARTICLE IN AXISYMMETRIC\nSTATIONARY SPACETIMES\nFollowing Ref. [24], we write a generic axisymmetric\nstationary metric in quasi-isotropic coordinates as\nds2=−e2νdt2+R2sin2θB2e−2ν(dφ−ωdt)2\n+e2µ/parenleftbig\ndR2+R2dθ2/parenrightbig\n, (3.1)\nwhereν,µ,Bandωare functions of the coordinates R\nandθ. Introducing the cartesian quasi-isotropic coordi-\nnates\nX=Rsinθcosφ, (3.2a)\nY=Rsinθsinφ, (3.2b)\nZ=Rcosθ, (3.2c)\nwe can write Eq. (3.1) as\nds2=e−2ν/bracketleftbig\nB2ω2/parenleftbig\nX2+Y2/parenrightbig\n−e4ν/bracketrightbig\ndt2\n+2B2e−2νω(Y dX−XdY)dt\n−2/parenleftbig\nB2e−2ν−e2µ/parenrightbig\nXY\nX2+Y2dXdY\n+e2µX2+B2e−2νY2\nX2+Y2dX2\n+B2e−2νX2+e2µY2\nX2+Y2dY2+e2µdZ2.\n(3.3)\nIt is straightforward to see that in the flat-spacetime\nlimit (ω=ν=µ= 0,B= 1) Eq. (3.3) reduces to\nthe Minkowski metric.\nReference[23] computed the Hamiltonian ofaspinning\ntest-particle in curved spacetime at linear order in the\nparticle’s spin, and showed that it can be written as\nH=HNS+HS, (3.4)\nwhereHNSis the Hamiltonian for a non-spinning test\nparticle of mass m, given by\nHNS=βiPi+α/radicalBig\nm2+γijPiPj,(3.5)\nwith\nα=1/radicalbig\n−gtt, (3.6)\nβi=gti\ngtt, (3.7)\nγij=gij−gtigtj\ngtt, (3.8)3\nand\nHS=−/parenleftBigg\nβiFK\ni+FK\nt+αγijPiFK\nj/radicalbig\nm2+γijPiPj/parenrightBigg\nSK,\nwhere the coefficients FI\nµcan be expressed in terms of a\nreference tetrad field ˜ eAas\nFK\nµ=/parenleftbigg\n2EµTI¯ωJ\n¯ωT+EµIJ/parenrightbigg\nǫIJK,(3.9)\nEλµν≡1\n2ηAB˜eA\nµ˜eB\nν;λ, (3.10)\nwith\n¯ωµ=¯Pµ−m˜eT\nµ, (3.11)\n¯Pi=Pi, (3.12)\n¯Pt=−βiPi−α/radicalBig\nm2+γijPiPj,(3.13)\n¯ωT= ¯ωµ˜eµ\nT=¯Pµ˜eµ\nT−m, (3.14)\n¯ωI= ¯ωµ˜eµ\nI=¯Pµ˜eµ\nI. (3.15)\nReference [23] also showed that in order to obtain a\nHamiltonian giving the usual leading-order spin-orbitcoupling without gauge effects (or, equivalently, HS= 0\nin flat spacetime), the reference tetrad field must become\ncartesian in the flat-spacetime limit. We find that the\nfollowing choice for the reference tetrad\n˜eT\nα=δt\nα(−gtt)−1/2=eνδt\nα, (3.16a)\n˜eα\n1=Be−µX2+eνY2\nB(X2+Y2)δα\nX+(Be−µ−eν)XY\nB(X2+Y2)δα\nY,\n(3.16b)\n˜eα\n2=(Be−µ−eν)XY\nB(X2+Y2)δα\nX+eνX2+Be−µY2\nB(X2+Y2)δα\nY,\n(3.16c)\n˜eα\n3=e−µδα\nZ, (3.16d)\nindeed reduces to the cartesian tetrad ˜ eT\nα= 1, ˜eα\nI=δα\nI\nin the flat-spacetime limit.\nWe can then use the tetrad defined by Eqs. (3.16a)–\n(3.16d) to calculate the coefficients FK\nµin Eq. (3.9), and\nobtain\nHS=HSO+HSS, (3.17)\nwith\nHSO=e2ν−µ(eµ+ν−B)(ˆP·ξR)SZ\nB2√QR2ξ2+eν−2µ\nB2/parenleftbig√Q+1/parenrightbig√QR2ξ2/braceleftBigg\nBcosθeµ+ν(ˆP·ξR)/parenleftBig/radicalbig\nQ+1/parenrightBig\n(S·N)ξ2\n+R(S·ξ)/bracketleftBig\nµR(ˆP·VR)/parenleftBig/radicalbig\nQ+1/parenrightBig\n−µcosθ(ˆP·N)ξ2−/radicalbig\nQ(νR(ˆP·VR)+(µcosθ−νcosθ)(ˆP·N)ξ2)/bracketrightBig\nB2\n+eµ+ν(ˆP·ξR)/parenleftBig\n2/radicalbig\nQ+1/parenrightBig/bracketleftBig\nνRR(S·V)−νcosθ(S·N)ξ2/bracketrightBig\nB−BReµ+ν(ˆP·ξR)/parenleftBig/radicalbig\nQ+1/parenrightBig\nR(S·V)/bracerightBigg\n,\n(3.18)\nHSS=ωSZ+e−3µ−νωR\n2B/parenleftbig√Q+1/parenrightbig√QRξ2/braceleftBigg\n−eµ+ν(ˆP·VR)(ˆP·ξR)(S·ξ)B+e2(µ+ν)(ˆP·ξR)2(S·V)\n+e2µ/parenleftBig\n1+/radicalbig\nQ/parenrightBig/radicalbig\nQR2(S·V)ξ2B2+(ˆP·N)R/bracketleftBig\n(ˆP·VR)(S·N)−(ˆP·N)R(S·V)/bracketrightBig\nξ2B2/bracerightBigg\n+e−3µ−νωcosθ\n2B/parenleftbig√Q+1/parenrightbig√QR2/braceleftBigg\neµ+ν(ˆP·N)(ˆP·ξR)R(S·ξ)B−e2(µ+ν)(ˆP·ξR)2(S·N)\n+/bracketleftBig\n(S·N)(ˆP·VR)2−(ˆP·N)R(S·V)(ˆP·VR)−e2µ/parenleftBig\n1+/radicalbig\nQ/parenrightBig/radicalbig\nQR2(S·N)ξ2/bracketrightBig\nB2/bracerightBigg\n,(3.19)\nQ= 1+γijˆPiˆPj= 1+e−2µ(ˆP·N)2+e−2µ(ˆP·VR)2\nR2ξ2+e2ν(ˆP·ξR)2\nB2R2ξ2, (3.20)\nwhere we denote\nˆP=P\nm, (3.21)\nN=X\nR, (3.22)\nξ=eZ×N=−YeX+XeY\nR,(3.23)\nV=N×ξ, (3.24)and\nfR≡∂f(R,cosθ)\n∂R, (3.25)\nfcosθ≡∂f(R,cosθ)\n∂(cosθ). (3.26)4\nHere, the generic function fcan stand for B,ω,µorν.\nNote that because ωis proportional to gtφ[see Eq. (3.1)]\nand thus to the spin of the spacetime, HSS(which is pro-\nportional to ωand its derivatives) gives the leading-order\ncoupling between the particle’s spin and the spin of the\nbackground spacetime (together with other higher order\nterms). Also, because ˆP·ξR=ˆPφin spherical coordi-\nnates,HSOis the part ofthe Hamiltonian which givesthe\nleading-order spin orbit coupling (again, together with\nother higher order terms). Moreover, note that HS= 0\nin a flat spacetime, thus confirming the absence of gauge\neffects in the leading order spin-orbit coupling.\nAs a consistency test, we specialize to the case of a\nspherically symmetric spacetime in quasi-isotropic coor-\ndinates, which was considered in Ref. [23] (see Sec. V\nA therein). Because the metric for such a spacetime is\ngiven by\nds2=−f(R)dt2+h(R)(dX2+dY2+dZ2),(3.27)\na comparison with Eq. (3.1) immediately reveals that\nB=/radicalbig\nf(R)h(R), (3.28)\nω= 0, (3.29)\nν=1\n2log[f(R)], (3.30)\nµ=1\n2log[h(R)]. (3.31)\nInserting Eqs.(3.28)–(3.31) in Eqs.(3.17)–(3.20), we find\nHS=L·S\n2mR/radicalbig\nf(R)h(R)2√Q(1+√Q)×\n/braceleftBig/radicalbig\nQ[f′(R)h(R)−f(R)h′(R)]−f(R)h′(R)/bracerightBig\n,\n(3.32)\nwhere\nQ= 1+1\nhˆP2, (3.33)\nL=X×P, (3.34)\nin agreement with Eq. (5.7) in Ref. [23].Let us now investigate how the Hamiltonian (3.17) is\naffectedby achangeofthe radialcoordinate R. Denoting\nthe new radial coordinate by r=|x|and defining\nJ−1≡dR\ndr, (3.35)\nthe radial derivatives of the metric potentials can be re-\nexpressed as\nfR=frJ, (3.36)\nwhere again f=B,ω, ν,µ . The spin S, the derivatives\nof the metric potentials with respect to cos θ, and the\nquantities\nN=n=x\nr, (3.37)\nξ=eZ×N=ez×n, (3.38)\nV=v=n×ξ (3.39)\nare not affected by the coordinate change. The same\napplies to the quantities ˆP·VRandˆP·ξRappearing\nin Eqs. (3.18), (3.19) and (3.20). In fact, in spherical\ncoordinates,wehave ˆP·VR=−ˆPθsinθandˆP·ξR=ˆPφ,\nhence\nˆP·VR=ˆp·vr, (3.40)\nˆP·ξR=ˆp·ξr, (3.41)\nwhereˆp=p/mandpis the conjugate momentum in the\nnew coordinate system, i.e., pi=∂Xj/∂xiPj. On the\ncontrary, since ˆP·N=ˆPR, we have\nˆP·N= (ˆp·n)J. (3.42)\nIt is therefore straightforward to compute HSin a coor-\ndinate system related to quasi-isotropic coordinates by a\nrescaling of the radius. We have\nHS=HSO+HSS, (3.43)\nwhere\nHSO=e2ν−µ(eµ+ν−B) (ˆp·ξr)Sz\nB2√QR2ξ2+eν−2µ\nB2/parenleftbig√Q+1/parenrightbig√QR2ξ2/braceleftBigg\nBcosθeµ+ν(ˆp·ξr)/parenleftBig/radicalbig\nQ+1/parenrightBig\n(S·n)ξ2\n+R(S·ξ)J/bracketleftBig\nµr(ˆp·vr)/parenleftBig/radicalbig\nQ+1/parenrightBig\n−µcosθ(ˆp·n)ξ2−/radicalbig\nQ(νr(ˆp·vr)+(µcosθ−νcosθ)(ˆp·n)ξ2)/bracketrightBig\nB2\n+eµ+ν(ˆp·ξr)/parenleftBig\n2/radicalbig\nQ+1/parenrightBig/bracketleftBig\nJνrR(S·v)−νcosθ(S·n)ξ2/bracketrightBig\nB−JBreµ+ν(ˆp·ξr)/parenleftBig/radicalbig\nQ+1/parenrightBig\nR(S·v)/bracerightBigg\n,\n(3.44)5\nHSS=ωSz+e−3µ−νJωr\n2B/parenleftbig√Q+1/parenrightbig√QRξ2/braceleftBigg\n−eµ+ν(ˆp·vr)(ˆp·ξr)(S·ξ)B+e2(µ+ν)(ˆp·ξr)2(S·v)\n+e2µ/parenleftBig\n1+/radicalbig\nQ/parenrightBig/radicalbig\nQR2(S·v)ξ2B2+J(ˆp·n)R[(ˆp·vr)(S·n)−J(ˆp·n)R(S·v)]ξ2B2/bracerightBigg\n+e−3µ−νωcosθ\n2B/parenleftbig√Q+1/parenrightbig√QR2/braceleftBigg\n−e2(µ+ν)(ˆp·ξr)2(S·n) +eµ+νJ(ˆp·n)(ˆp·ξr)R(S·ξ)B\n+/bracketleftBig\n(S·n)(ˆp·vr)2−J(ˆp·n)R(S·v)(ˆp·vr)−e2µ/parenleftBig\n1+/radicalbig\nQ/parenrightBig/radicalbig\nQR2(S·n)ξ2/bracketrightBig\nB2/bracerightBigg\n,(3.45)\nQ= 1+γijˆpiˆpj= 1+e−2µ(ˆp·n)2J2+e−2µ(ˆp·vr)2\nR2ξ2+e2ν(ˆp·ξr)2\nB2R2ξ2, (3.46)\nand whereRmust of course be expressed in terms of the\nnew radial coordinate r.\nIV. HAMILTONIAN FOR A SPINNING\nTEST-PARTICLE IN KERR SPACETIME IN\nBOYER-LINDQUIST COORDINATES\nIn this section, we will specialize the Hamiltonian de-\nrivedintheprevioussectiontothe caseofKerrspacetime\nin Boyer-Lindquist coordinates.\nWe start from the metric potentials appearing in\nEq. (3.1), which in the case of a Kerr spacetime take\nthe form [25]\nB=√\n∆\nR, (4.1)\nω=2aMr\nΛ, (4.2)\ne2ν=∆Σ\nΛ, (4.3)\ne2µ=Σ\nR2, (4.4)\nwith\nΣ =r2+a2cos2θ, (4.5)\n∆ =r2+a2−2Mr, (4.6)\n̟2=r2+a2, (4.7)\nΛ =̟4−a2∆sin2θ, (4.8)\nwhere the parameter a, which has the dimensions of a\nlength, is related to the spin vector SKerrof the Kerr\nblack hole by\na=|SKerr|\nM. (4.9)\nThe Boyer-Lindquist coordinate ris related to the quasi-\nisotropic coordinate Rby\nr=R+M+R2\nH\nR, (4.10)whereRH=√\nM2−a2/2isthehorizon’sradiusinquasi-\nisotropic coordinates. Note that the inverse of this trans-\nformation is given, outside the horizon, by\nR=1\n2/parenleftbig\nr−M+√\n∆/parenrightbig\n. (4.11)\nWe then obtain that the derivatives of the metric poten-\ntials take the form\nBr=r−M−√\n∆\nR√\n∆, (4.12a)\nωr=2aM[Σ̟2−2r2(Σ+̟2)]\nΛ2,(4.12b)\nνr=r−M\n∆+r\nΣ\n−2r̟2−a2(r−M) sin2θ\nΛ,(4.12c)\nµr=r\nΣ−1√\n∆, (4.12d)\nBcosθ= 0, (4.12e)\nωcosθ=−4a3Mr∆ cosθ\n(∆Σ+2Mr̟2)2, (4.12f)\nνcosθ=2a2Mr̟2cosθ\n(∆Σ+2Mr̟2)Σ, (4.12g)\nµcosθ=a2cosθ\nΣ, (4.12h)\nand we also have\nJ−1=dR\ndr=R√\n∆. (4.13)\nInserting Eqs. (4.12a)–(4.12h) and Eq. (4.13) into\nEqs. (3.43)–(3.46), we find that Rcancels out both in\nQ, that is\nQ= 1+∆(ˆp·n)2\nΣ+(ˆp·ξr)2Σ\nΛ sin2θ+(ˆp·vr)2\nΣ sin2θ,(4.14)6\nand in the Hamiltonian HS. In conclusion, the Hamil-\ntonian of a spinning test-particle in Kerr spacetime in\nBoyer-Lindquist coordinates is\nH=HNS+HS, (4.15)\nwith\nHNS=βipi+α/radicalBig\nm2+γijpipj, (4.16)whereα,βiandγijare given in Eqs. (3.6)–(3.8) and\nneed to be computed using the Kerr metric coefficients\n(4.1)–(4.8), and with\nHS=HSO+HSS, (4.17)\nwhere\nHSO=e2ν−˜µ/parenleftBig\ne˜µ+ν−˜B/parenrightBig\n(ˆp·ξr)(S·ˆSKerr)\n˜B2√Qξ2+eν−2˜µ\n˜B2/parenleftbig√Q+1/parenrightbig√Qξ2/braceleftBigg\n(S·ξ)˜J/bracketleftBig\nµr(ˆp·vr)/parenleftBig/radicalbig\nQ+1/parenrightBig\n−µcosθ(ˆp·n)ξ2\n−/radicalbig\nQ(νr(ˆp·vr)+(µcosθ−νcosθ)(ˆp·n)ξ2)/bracketrightBig\n˜B2+e˜µ+ν(ˆp·ξr)/parenleftBig\n2/radicalbig\nQ+1/parenrightBig/bracketleftBig\n˜Jνr(S·v)−νcosθ(S·n)ξ2/bracketrightBig\n˜B\n−˜J˜Bre˜µ+ν(ˆp·ξr)/parenleftBig/radicalbig\nQ+1/parenrightBig\n(S·v)/bracerightBigg\n, (4.18)\nand\nHSS=ω(S·ˆSKerr)+e−3˜µ−ν˜Jωr\n2˜B/parenleftbig√Q+1/parenrightbig√Qξ2/braceleftBigg\n−e˜µ+ν(ˆp·vr)(ˆp·ξr)(S·ξ)˜B+e2(˜µ+ν)(ˆp·ξr)2(S·v)\n+e2˜µ/parenleftBig\n1+/radicalbig\nQ/parenrightBig/radicalbig\nQ(S·v)ξ2˜B2+˜J(ˆp·n)/bracketleftBig\n(ˆp·vr)(S·n)−˜J(ˆp·n)(S·v)/bracketrightBig\nξ2˜B2/bracerightBigg\n+e−3˜µ−νωcosθ\n2˜B/parenleftbig√Q+1/parenrightbig√Q/braceleftBigg\n−e2(˜µ+ν)(ˆp·ξr)2(S·n)+e˜µ+ν˜J(ˆp·n)(ˆp·ξr)(S·ξ)˜B\n+/bracketleftBig\n(S·n)(ˆp·vr)2−˜J(ˆp·n)(S·v)(ˆp·vr)−e2˜µ/parenleftBig\n1+/radicalbig\nQ/parenrightBig/radicalbig\nQ(S·n)ξ2/bracketrightBig\n˜B2/bracerightBigg\n, (4.19)\nwhere we define\n˜B=BR=√\n∆, (4.20)\n˜Br=BrR=r−M−√\n∆√\n∆,(4.21)\ne2˜µ=e2µR2= Σ, (4.22)\n˜J=JR=√\n∆, (4.23)\nˆSKerr=SKerr\n|SKerr|(4.24)\nand we recall that ξ2= sin2θ. We stress that because\nthisHamiltonianisexpressedintermsofquantitieswhich\nare scalar under spatial rotations, we can express it in\na cartesian coordinate system in which the spin of the\nKerr black hole is not directed along the z-axis. For\nthat purpose, it is sufficient to replace rwith (x2+y2+\nz2)1/2, cosθwithˆSKerr·n,ezwithˆSKerrin Eq. (3.38),\nand express the vectors appearing in Eqs. (4.16)–(4.19)\nin terms of their cartesian components.\nAs a consistency check, we can compute the Hamilto-nian for a spinning test-particle in a Schwarzschildspace-\ntime in Schwarzschild spherical coordinates by setting\na= 0, and compare the result to the expression com-\nputed in Ref. [23] [see Eq. (5.12) therein]. We find\nHS=ψ6\nR3√Q(1+√Q)×\n/bracketleftbigg\n1−M\n2R+2/parenleftbigg\n1−M\n4R/parenrightbigg/radicalbig\nQ/bracketrightbigg\n(L·S∗),(4.25)\nwhere\nS∗=M\nmS, (4.26)\nψ=/parenleftbigg\n1+M\n2R/parenrightbigg−1\n, (4.27)\nR=1\n2/parenleftBig\nr−M+/radicalbig\nr2−2Mr/parenrightBig\n,(4.28)\nand\nQ= 1+(ˆp·n)2/parenleftbigg\n1−2M\nr/parenrightbigg\n+(ˆp·v)2+(ˆp·ξ)2\nsin2θ,(4.29)7\nin agreement with Ref. [23]. Also, it is worth noting\nthat the Hamiltonian (4.25) is the same as the quasi-\nisotropic Schwarzschild Hamiltonian (3.32), expressed in\nterms of the Schwarzschild coordinate r. This is because\nthe scalar product L·Sis unaffected by a change of the\nradial coordinate.\nV. EFFECTIVE-ONE-BODY HAMILTONIAN\nFOR TWO SPINNING BLACK HOLES\nThe EOB approach was originally introduced in\nRefs. [9–11, 19] to provide us with an improved (re-\nsummed) Hamiltonian that could be used to evolve a\nbinary system not only during the long inspiral, but also\nduring the plunge, and that could supply a natural mo-\nmentat which to switch from the two body description\nto the one-body description, in which the system is rep-\nresented by a superposition of quasi-normal modes of the\nremnant black hole.\nA crucial ingredient of the EOB approach is the real\nPN-expanded Arnowitt-Deser-Misner (ADM) Hamilto-\nnian (or realHamiltonian) describing two black holes of\nmassesm1,m2and spins S1,S2. The real Hamiltonian\nisthencanonicallytransformedandsubsequently mapped\nto aneffective Hamiltonian Heffdescribing a test-particle\nof massµ=m1m2/(m1+m2) and suitable spin S∗,\nmoving in a deformed Kerr metric of mass M=m1+m2\nand suitable spin SKerr. The parameter regulating the\ndeformation is the symmetric mass ratio of the binary,\nη=µ/M, which ensuresthat the deformation disappears\nin the case of extreme mass-ratio binaries. The resulting\nimproved EOB Hamiltonian then takes the form\nHimproved\nreal=M/radicalBigg\n1+2η/parenleftbiggHeff\nµ−1/parenrightbigg\n.(5.1)\nThe computation of the improved EOB Hamiltonian\nconsists of several stages. For this reason, we briefly re-\nview here the main steps and the underpinning logic that\nwe will follow in the rest of this section:\n(i) We apply a canonical transformation to the PN-\nexpanded ADM Hamiltonian using a generating\nfunction which is compatible with the one used in\nprevious EOB work, obtaining the PN-expanded\nHamiltonian in EOB canonical coordinates (see\nSec. VA);\n(ii) We compute the effective Hamiltonian correspond-\ning to the canonically transformed PN-expanded\nADM Hamiltonian (see Sec. VB);\n(iii) We deform the Hamiltonian of a spinning test-\nparticle in Kerr derived in Sec. IV by deforming\nthe Kerr metric (see Sec. VC) , and expand this\ndeformed Hamiltonian in PN orders (see Sec. VD);\n(iv) Comparing (iii) and (iii), we work out the mapping\nbetween the spin variables in the real and effectivedescriptions, and write the improved EOB Hamil-\ntonian (see Sec. VE).\nA. The ADM Hamiltonian canonically transformed\nto EOB coordinates\nWedenotetheADM canonicalvariablesinthebinary’s\ncenter-of-mass frame with r′andp′. It is convenient to\nintroduce the following spin combinations:\nσ=S1+S2, (5.2)\nσ∗=S1m2\nm1+S2m1\nm2, (5.3)\nσ0=σ+σ∗. (5.4)\nMoreover, in order to consistently keep track of the PN\norders, we will restore the speed of light cand rescale the\nspins variables as σ∗→σ∗candσ→σc.3The canoni-\ncal ADM Hamiltonian is known through 3PN order [26–\n30] and partially at higher PN orders [31, 32]. In par-\nticular, the spin-orbit and spin-spin coupling terms agree\nwith those computed via effective-field-theory techniques\nat 1.5PN, 2PN and 3PN order [33–36]. In this paper, we\nuse the spin-independent part of the ADM Hamiltonian\nthrough 3PN order, but we only use its spin-dependent\npart through 2.5 PN order, i.e., we consider the leading-\norder (1.5 PN) and the next-to-leading order (2.5PN)\nspin-orbit couplings, but only the leading order (2PN)\nspin-spin coupling. The expressions for these couplings\nare [19, 27]\nHADM\nSO(r′,p′,σ∗,σ) =1\nc3L′\nr′3·(gADM\nσσ+gADM\nσ∗σ∗),\n(5.5)\nHADM\nSS(r′,p′,σ∗,σ) =1\nc4η\n2r′3/bracketleftbig\n3(n′·σ0)2−σ2\n0/bracketrightbig\n,\n(5.6)\nwithL′=r′×p′,n′=r′/r′, and\ngADM\nσ= 2+1\nc2/bracketleftbigg19\n8ηˆp′2+3\n2η(n′·ˆp′)2\n−(6+2η)M\nr′/bracketrightbigg\n, (5.7a)\ngADM\nσ∗=3\n2+1\nc2/bracketleftbigg/parenleftbigg\n−5\n8+2η/parenrightbigg\nˆp′2+3\n4η(n′·ˆp′)2\n−(5+2η)M\nr′/bracketrightbigg\n, (5.7b)\n3This is appropriate for black holes or a rapidly rotating com -\npact stars. In the black-hole case, S=χM2/c, withχrang-\ning from 0 to 1. In the rapidly spinning star case one has\nS=Mvrotr∼Mcrs∼M2/c(where we have assumed that the\nrotational velocity vrotis comparable to cand that the stellar\nradiusris of the order of the Schwarzschild radius rs∼M/c2).8\nwherewehaveintroducedthe rescaledconjugatemomen-\ntumˆp′=p′/µ.\nWe now perform a canonical transformation from the\nADM canonicalvariables r′andp′to the EOB canonical\nvariables randp. Let us first consider the purely orbital\ngenerating function\nG(r′,p) =r′·p+GNS(r′,p), (5.8)\nGNS(r′,p) =GNS1PN(r′,p)\n+GNS2PN(r′,p)+GNS3PN(r′,p),(5.9)\nwherethe1PN-accurategeneratingfunction GNS1PNwas\nderived in Ref. [10],\nGNS1PN(r′,p) =1\nc2r′·p/bracketleftbigg\n−1\n2ηˆp2+M\nr′/parenleftbigg\n1+1\n2η/parenrightbigg/bracketrightbigg\n,\n(5.10)\nwhile the 2PN and 3PN accurate generating functions,\nGNS2PNandGNS3PN, were derived in Refs. [10] and [11],\nrespectively. From the definition of generating function,\nit followsthat the transformationofthe phase-spacevari-\nables is implicitly given by\nxi=x′i+∂GNS(x′,p)\n∂pi, (5.11)\npi=p′\ni−∂GNS(x′,p)\n∂x′i, (5.12)\nwhile the Hamiltonian transforms as H(r,p) =\nHADM(r′,p′). At linear order, which is enough for\nour purposes, Eqs. (5.11) and (5.12) can be written as\ny=y′−{GNS,y′}, where{...}are the Poisson brackets\nand where ystands for either xorp. The transforma-\ntion of the Hamiltonian, again at linear order, is then\nH(y) =HADM(y) +{GNS,HADM}(y) [21]. Similarly, if\none considers a generating function which depends not\nonly on the orbital variables, but also on the spins,\nG(r′,p,σ∗,σ) =r′·p\n+GNS(r′,p)+GS(r′,p,σ∗,σ) (5.13)\nthe Hamiltonian will again transform as H(y) =\nHADM(y) +{GNS,HADM}(y) +{GS,HADM}(y), where\nnow the Poisson brackets in the term {GS,HADM}will\ninvolve also the spin variables [21]. In particular, let us\nconsider a spin-dependent generating function\nGS(r′,p,σ∗,σ) =GS2PN(r′,p,σ)\n+GS2.5PN(r′,p,σ∗,σ) (5.14)\n+GSSS2.5PN(r′,p,σ∗,σ).\nwhere the 2PN-accuratespin-dependent generating func-tionGS2PNwas implicitly4used in Ref. [19],\nGS2PN(r′,p,σ) =−1\n2c4M2r′2/braceleftBig\n[σ2−(σ·n′)2](r′·p)\n+(σ·n′)(r′×p)·(σ×n′)/bracerightBig\n;\n(5.15)\nthe 2.5PN-accurategeneratingfunction GS2.5PNlinearin\nthe spin variables was introduced in Ref. [21],\nGS2.5PN(r′,p,σ∗,σ) =1\nµr′3c5(r′·p)(r′×p)·\n[a(η)σ+b(η)σ∗],(5.16)\na(η) andb(η) being arbitrary gauge functions; also, for\nreasons which will become clear in Sec. VD, we include\nthe following 2.5PN-accurate generating function, cubic\nin the spins,\nGSSS2.5PN(r′,p,σ∗,σ) =µ\n2M3r′4c5(σ·r′)[σ∗·(σ×r′)].\n(5.17)\nWhen applying the generating function (5.13) to the\nADM 2PN spin-spin Hamiltonian (5.6), we obtain\nHSS2PN(r,p,σ∗,σ) =HADM\nSS2PN(r,p,σ∗,σ)\n+{GS2PN,HNewt}(r,p,σ),\n(5.18)\nwith\nHNewt=−Mµ\nr+p2\n2µ, (5.19)\n{GS2PN,HNewt}(r,p,σ) =−1\nc4η\n2r3/bracketleftbig\n(n·σ)2−σ2/bracketrightbig\n+1\n2µM2r2c4/braceleftBig\n−[p2−2(p·n)2]σ2\n+[(p−2(p·n)n)·σ]p·σ/bracerightBig\n.(5.20)\nSimilarly, if we apply the same generating function to\nthe ADM spin-orbit Hamiltonian (5.5), the 1.5PN order\nterm remains unaltered [21], while the 2.5PN order term\ntransforms as [21]\nHSO2.5PN(r,p,σ∗,σ) =HADM\nSO2.5PN(r,p,σ∗,σ)\n+{G2.5PN,HNewt}(r,p,σ∗,σ),\n+{GNS1PN,HADM\nSO1.5PN}(r,p,σ∗,σ) (5.21)\nwhere\nG2.5PN=GSS2.5PN+GSSS2.5PN, (5.22)\n4See discussion in Sec II D of Ref. [19]. The need for this gener -\nating function will become apparent with Eq. (5.55) in Sec. V D.9\n{G2.5PN,HNewt}(r,p,σ∗,σ) =\n1\nr3c5L·[b(η)σ∗+a(η)σ]/bracketleftbigg\n−M\nr+ˆp2−3(ˆp·n)2/bracketrightbigg\n+[σ∗·(σ×n)][σ·(p−2(p·n)n)]\nM3r3c5\n+(L·σ∗)σ2−(L·σ)(σ∗·σ)\n2M3r4c5, (5.23)\nand\n{GNS1PN,HADM\nSO1.5PN}(r,p,σ∗,σ) =\n−3L\n2r3c5·/parenleftbigg3\n2σ∗+2σ/parenrightbigg/braceleftbigg\n−M\nr(2+η)+η/bracketleftbig\nˆp2+2(ˆp·n)2/bracketrightbig/bracerightbigg\n.\n(5.24)\nTherefore, the complete real Hamiltonian in the EOB\ncanonical coordinates is\nH(r,p,σ∗,σ) =Hnospin(r,p,σ∗,σ)\n+HADM\nSO(r,p,σ∗,σ)\n+HADM\nSS(r,p,σ∗,σ)\n+{G2.5PN,HNewt}(r,p,σ∗,σ)\n+{GNS1PN,HADM\nSO1.5PN}(r,p,σ∗,σ)\n+{GS2PN,HNewt}(r,p,σ),\n(5.25)\nwhereHnospinis the 3PN ADM Hamiltonian for non-\nspinning black holes, canonically transformed to EOB\ncoordinates, which can be obtained from Ref. [11].\nB. Spin couplings in the effective Hamiltonian\nFollowing Refs. [9, 11, 19], we map the effective and\nreal two-body Hamiltonians as\nHeff\nµc2=H2\nreal−m2\n1c4−m2\n2c4\n2m1m2c4,(5.26)\nwhereHrealis the real two-body Hamiltonian containing\nalsotherest-masscontribution Mc2. Wedenotethenon-\nrelativistic part of the real Hamiltonian by HNR, i.e.,\nHNR≡Hreal−Mc2. Identifying HNRwithHasgivenin\nEq. (5.25), and expanding Eq. (5.26) in powers of 1 /c, we\nfind that the 1.5PNand 2.5PNorderspin-orbit couplings\nof the effective Hamiltonian are\nHeff\nSO(r,p,σ∗,σ) =1\nc3L\nr3·/parenleftbig\ngeff\nσσ+geff\nσ∗σ∗/parenrightbig\n+[σ∗·(σ×n)][σ·(p−2(p·n)n)]\nM3r3c5\n+(L·σ∗)σ2−(L·σ)(σ∗·σ)\n2M3r4c5,\n(5.27)where [21]\ngeff\nσ= 2+1\nc2/braceleftbigg/bracketleftbigg3\n8η+a(η)/bracketrightbigg\nˆp2\n−/bracketleftbigg9\n2η+3a(η)/bracketrightbigg\n(ˆp·n)2\n−M\nr[η+a(η)]/bracerightbigg\n, (5.28a)\ngeff\nσ∗=3\n2+1\nc2/braceleftbigg/bracketleftbigg\n−5\n8+1\n2η+b(η)/bracketrightbigg\nˆp2\n−/bracketleftbigg15\n4η+3b(η)/bracketrightbigg\n(ˆp·n)2\n−M\nr/bracketleftbigg1\n2+5\n4η+b(η)/bracketrightbigg/bracerightbigg\n,(5.28b)\nand the 2PN order spin-spin coupling is\nHeff\nSS(r,p,σ∗,σ) =1\nc4η\n2r3(3ninj−δij)σi\n0σj\n0\n−1\nc4η\n2r3/bracketleftbig\n(n·σ)2−σ2/bracketrightbig\n+1\n2µM2r2c4/braceleftBig\n−[p2−2(p·n)2]σ2\n+[(p−2(p·n)n)·σ]p·σ/bracerightBig\n.(5.29)\nC. The Hamiltonian of a spinning test-particle in a\ndeformed Kerr spacetime\nWe now deform the Hamiltonian of a spinning test-\nparticle in a Kerr spacetime computed in Sec. IV [see\nEqs. (4.16), (4.17), (4.18) and (4.19)] by deforming the\nKerr metric. The deformation that we introduce is reg-\nulated by the parameter η=µ/M, and therefore dis-\nappears in the test-particle limit. Also, the deformed\nHamiltonianwillbesuchastoreproduce,whenexpanded\nin PN orders, the spin couplings of the effective Hamil-\ntonian given in Sec. VB.\nWhenthespinoftheKerrblackholeiszero,thatis a=\n0, we require the metric to coincide with the deformed-\nSchwarzschildmetricused in the EOBformalismfor non-\nspinning black-hole binaries [10, 11]. That deformation\nsimply amounts to changing the components gttandgrr\nof the metric. In the spinning case, following Ref. [21],\nwe seek an extension of this deformation by changing the\npotential ∆ appearing in the Kerr potentials (4.1)–(4.4).\nIt is worth noting, however, that we are not allowed\nto deform the Kerr metric in an arbitrary way. We re-\ncall indeed that the Hamiltonian that we have derived\nin Sec. IV is only valid for a stationary axisymmetric\nmetric, and in coordinates which are related to quasi-\nisotropic coordinates by a redefinition of the radius. In\nother words, it must be possible for our deformed metric\nto be put in the form (3.1) by a coordinate change of the\ntypeR=R(r). For this reason we cannot deform the\nmetric exactly in the same way as in Ref. [21]. Here we10\npropose to deform the metric potentials in the following\nmanner\nB=√∆t\nR, (5.30)\nω=/tildewideωfd\nΛt, (5.31)\ne2ν=∆tΣ\nΛt, (5.32)\ne2µ=Σ\nR2, (5.33)\nand\nJ−1=dR\ndr=R√∆r, (5.34)\nwhere the relation between randRcan be found by\nintegrating Eq. (5.34):\nR= exp/parenleftbigg/integraldisplaydr√∆r/parenrightbigg\n. (5.35)\nThe deformed metric therefore takes the form\ngtt=−Λt\n∆tΣ, (5.36a)\ngrr=∆r\nΣ, (5.36b)\ngθθ=1\nΣ, (5.36c)\ngφφ=1\nΛt/parenleftbigg\n−/tildewideω2\nfd\n∆tΣ+Σ\nsin2θ/parenrightbigg\n,(5.36d)\ngtφ=−/tildewideωfd\n∆tΣ, (5.36e)\nwhich does not depend on R. Therefore, as we will show\nexplicitly later in this section, we do notneed to com-\npute the integral (5.35) to write the Hamiltonian. The\nquantities ∆ t, ∆r, Λtand/tildewideωfdin Eqs. (5.36a)–(5.36e) are\ngiven by\n∆t=r2/bracketleftbigg\nA(u)+a2\nM2u2/bracketrightbigg\n, (5.37)\n∆r= ∆tD−1(u), (5.38)\nΛt=̟4−a2∆tsin2θ, (5.39)\n/tildewideωfd= 2aMr+ωfd\n1ηaM3\nr+ωfd\n2ηMa3\nr,(5.40)\nwhereu=M/r,ωfd\n1andωfd\n2are adjustable parameters\nwhich regulate the strength of the frame-dragging, and\nthrough 3PN order [9, 11]\nA(u) = 1−2u+2ηu3+η/parenleftbigg94\n3−41\n32π2/parenrightbigg\nu4,(5.41)\nD−1(u) = 1+6ηu2+2(26−3η)ηu3. (5.42)We find that our deformed metric is the same as the de-\nformed metric of Ref. [21], except for gφφandgtφ.5As\nwe prove below, the differences between our deformation\nand the deformation of Ref. [21] appear in the Hamilto-\nnian at PN orders higher than 3PN.\nTo obtain the total Hamiltonian (4.15), that is H=\nHNS+HS, we first compute the Hamiltonian HNSfor a\nnon-spinningparticleinthedeformed-Kerrmetric. Using\nEq. (4.16) and Ref. [11], we have\nHNS=βipi+α/radicalBig\nm2+γijpipj+Q4(p),(5.43)\nwhereQ4(p) is a term which is quartic in the space mo-\nmentapiand which was introduced in Ref. [11], and\nα=1/radicalbig\n−gtt, (5.44)\nβi=gti\ngtt, (5.45)\nγij=gij−gtigtj\ngtt. (5.46)\nIn Eqs. (5.44)–(5.46) the metric components have to be\nreplaced with those of the deformed-Kerrmetric (5.36a)–\n(5.36e). When expanded in PN orders, Eq. (5.43) co-\nincides, through 3PN order, with the Hamiltonian of a\nnon-spinning test particle in the deformed-Kerr metric\ngiven by Ref. [21].\nSecond, to calculate HSgiven by Eqs. (3.43), (3.44)\nand (3.45), we need to compute the derivatives of the\nmetric potentials. We obtain\nBr=√∆r∆t′−2∆t\n2√∆r∆tR, (5.47a)\nωr=−Λ′\nt/tildewideωfd+Λt/tildewideω′\nfd\nΛ2\nt, (5.47b)\nνr=r\nΣ+̟2/parenleftbig\n̟2∆t′−4r∆t/parenrightbig\n2Λt∆t,(5.47c)\nµr=r\nΣ−1√∆r, (5.47d)\nBcosθ= 0, (5.47e)\nωcosθ=−2a2cosθ∆t/tildewideωfd\nΛ2\nt, (5.47f)\nνcosθ=a2̟2cosθ(̟2−∆t)\nΛtΣ,(5.47g)\nµcosθ=a2cosθ\nΣ, (5.47h)\nwhere the prime denotes derivatives with respect to r.\nAs already stressed, although the metric potentials B,\n5Ref. [21] chooses gφφ= (−a2sin2θ+ ∆t)/(∆tΣsin2θ) and\ngtφ=a(∆t−̟2)/(∆tΣ), which are different from our expres-\nsions (5.36d) and (5.36e) even for ωfd\n1=ωfd\n2= 0.11\nω,νandµdepend on R, the factors Rcancel out in\nthe deformed-Kerr metric. Therefore, those factors must\ncancel out also in HS. This happens because the ref-\nerence tetrad field ˜ eAwhich, together with the metric,\ncompletely determines the Hamiltonian [see Eq. (3.9)],\ncan be defined independently of R. Indeed, this turns\nout to be the case, and if we introduce the rescaled po-\ntentials\n˜B=BR=/radicalbig\n∆t, (5.48)\n˜Br=BrR=√∆r∆t′−2∆t\n2√∆r∆t,(5.49)\ne2˜µ=e2µR2= Σ, (5.50)\n˜J=JR=/radicalbig\n∆r (5.51)\nand define\nQ= 1+∆r(ˆp·n)2\nΣ+(ˆp·ξr)2Σ\nΛtsin2θ+(ˆp·vr)2\nΣ sin2θ,(5.52)\nthe Hamiltonian HSfor the deformed-Kerr metric takes\nexactlythe sameform asin the Kerrcase[seeEqs.(4.17),\n(4.18) and (4.19)], where we recall that ξ2= sin2θand\nwherenowωanditsderivatives, νanditsderivatives,and\nthe derivatives of µare given by Eqs. (5.31), (5.32), and\nEqs. (5.47a)–(5.47h). Also, as we have already stressed,\nin order to express the Hamiltonian HSin a cartesian\ncoordinatesysteminwhichthespinofthe deformed-Kerr\nblack hole is not directed along the z-axis, it is sufficient\nto replacerwith (x2+y2+z2)1/2, cosθwithˆSKerr·n,\nezwithˆSKerrin Eq. (3.38), and to express the vectors\nappearing in the Hamiltonian in terms of their cartesian\ncomponents.\nD. PN expansion of the deformed Hamiltonian\nWe nowexpandthe deformed Hamiltonian H=HNS+\nHSderived in the previous section into PN orders. We\nwill denote the spin of the deformed-Kerr metric with\nSKerr, while for the test particle’s spin we introduce the\nrescaled spin vector S∗=SM/m,Sbeing the physical,\nunrescaled spin. Also, we rescale the spins as SKerr→\nSKerrcandS∗→S∗c, so as to keep track of the PN\norders correctly. Moreover, we set SKerr=χKerrM2,\nχKerrbeing the dimensionless spin of the deformed-Kerr\nblack hole, with norm |χKerr|ranging from 0 to 1.\nAs already mentioned, the part of the Hamiltonian\nwhich does not depend on the test particle’s spin, HNS,\nagrees through 3PN order with the corresponding HNS\ncomputed in Ref. [21]. Moreover, although the metric\n(5.36a)–(5.36e) only coincides with the Kerr metric for\nη= 0, the dependence on ηappears neither in the 2PN\norder coupling of the deformed-Kerr black hole’s spin\nwith itself, nor in its 1.5PN and 2.5PN order spin-orbit\ncouplings. Those couplings are therefore the same as inthe case of the Kerr metric, and they are given by\nHNS\nSO1.5PN=1\nc32\nr3L·SKerr, (5.53)\nHNS\nSO2.5PN= 0, (5.54)\nHNS\nSS2PN=1\nc4m\n2Mr3(3ninj−δij)Si\nKerrSj\nKerr\n−1\nc4m\n2Mr3/bracketleftbig\n(n·SKerr)2−S2\nKerr/bracketrightbig\n+1\n2m(Mr)2c4/braceleftBig\n−[p2−2(p·n)2]S2\nKerr\n+[(p−2(p·n)n)·SKerr]p·SKerr/bracerightBig\n.\n(5.55)\nExpandingtheninPNordersthe partoftheHamiltonian\nthat depends on the test particle’s spin, that is HS, we\nfind\nHS\nSO1.5PN=3\n2r3c3L·S∗, (5.56)\nHS\nSO2.5PN=1\nr3c5/bracketleftbigg\n−M\nr/parenleftbigg1\n2+3η/parenrightbigg\n−5\n8ˆp2/bracketrightbigg\nL·S∗\n+[S∗·(SKerr×n)][SKerr·(p−2(p·n)n)]\nM3r3c5\n+(L·S∗)S2\nKerr−(L·SKerr)(S∗·SKerr)\n2M3r4c5.\n(5.57)\nHS\nSS2PN=m\nMr3c4(3ninj−δij)Si\nKerrSj\n∗.(5.58)\nWe recall that the Hamiltonian for a spinning test parti-\ncle in curved spacetimefrom which we started the deriva-\ntion of our novel EOB model [see Eq. (3.9)] is only valid\nat linear order in the particle’s spin. Therefore, the\nsame restriction applies to the Hamiltonian derived in\nSec. VC. In particular, that Hamiltonian does not in-\nclude the couplings of the particle’s spin with itself. We\nintroduce those couplings by hand, at least at the leading\norder (2PN), by adding a quadrupole deformation [19]\nhµν, quadratic in the particle’s spin, to the deformed-\nKerr metric in Sec. VC [see Eqs. (5.36a)–(5.36e)]. The\nexpression for hµνand the details of the above proce-\ndure — together with a way in which it can in principle\nbe extended to reproduce also the next-to-leading order\ncoupling of the particle’s spin with itself — are given\nin Appendix A. For the purpose of the present discus-\nsion, however, it is sufficient to mention that the addition\nof this quadrupole deformation to the metric (5.36a)–\n(5.36e) augments Eq. (5.55) by the term\nm\n2Mr3c4(3ninj−δij)Si\n∗Sj\n∗.(5.59)\nTherefore, the total leading order spin-spin Hamiltonian12\nis\nHSS2PN=HS\nSS2PN+HNS\nSS2PN\n+m\n2Mr3c4(3ninj−δij)Si\n∗Sj\n∗\n=m\n2Mr3c4(3ninj−δij)Si\n0Sj\n0\n−1\nc4m\n2Mr3/bracketleftbig\n(n·SKerr)2−S2\nKerr/bracketrightbig\n+1\n2mM2r2c4/braceleftBig\n−[p2−2(p·n)2]S2\nKerr\n+[(p−2(p·n)n)·SKerr]p·SKerr/bracerightBig\n,(5.60)\nwithSi\n0=Si\nKerr+Si\n∗.\nAs we will show in Sec. VE, a proper choice of the\nvectorsSKerrandS∗in terms of the vectors σandσ∗,\ndefined in Eqs. (5.2) and (5.3), allows us to reproduce\nthe PN-expanded effective Hamiltonian [see Eqs. (5.27)–\n(5.29)] using the PN-expanded deformed-Kerr Hamilto-\nnian that we have just derived.\nFinally, it is worth noting that the presence of terms\nquadratic in the deformed-Kerr black hole’s spin in\nEq. (5.57) explains why we introduced the 2.5PN-\naccurate canonical transformation (5.17). Indeed, the\nlatter produces exactly the same terms in the PN-\nexpanded effective Hamiltonian (5.27) at 2.5PN order.\nQuite interestingly, the terms quadratic in SKerrappear-\ning in Eq. (5.57) could also be eliminated with a suitable\nchoice of the reference tetrad ˜ eA. In fact, as stressed in\nSec. III and in Ref. [23], a choice of the reference tetrad\nfield corresponds to choosing a particular gauge for the\nparticle’s spin. In agreement with this interpretation,\nwe find that the terms of Eq. (5.57) which are quadratic\ninSKerrdisappear if the initial tetrad (3.16a)–(3.16d) is\nchanged to a different tetrad ˜e′\nArelated to the original\none by the following purely-spatial rotation:\n˜e′T=˜eT,˜e′\nI=RIJ˜eJ, (5.61)\nwhere the rotation matrix RIJis given by\nR=RY/bracketleftbigg\n−a2XZ\n2R4/bracketrightbigg\nRX/bracketleftbigg\n−a2Y Z\n2R4/bracketrightbigg\n,(5.62)\nRX[ψ] andRY[φ] being rotations of angles ψandφ\naround the axis XandY, respectively.\nAs a consistency check, we have verified that this new\ntetrad is the same as that used in Ref. [23] when com-\nputingtheHamiltonianinADMcoordinates,wherethose\ntermsquadraticin SKerrdo notappear. Wehavechecked\nthis by transforming the new tetrad (5.61) from quasi-\nisotropic to ADM coordinates [which are related by the\ncoordinate transformation(49) in Ref. [31]], and compar-\ningittothetetradgiveninEqs.(6.9a)–(6.9b)ofRef.[23],\nand find that the two tetrads agree through order 1 /c8.E. The effective-one-body Hamiltonian\nIn this section we first find the mapping between\nthe masses µ,Mand the spins σandσ∗of the ef-\nfective Hamiltonian derived in Sec VB, and those of\nthe deformed-Kerr Hamiltonian derived in Secs. VC\nand VD, that is m,M,SKerrandS∗. Then, we derive\nthe improved (resummed) EOB Hamiltonian.\nAs shown in Ref. [9], matching the non-spinning parts\nHNSof these Hamiltonians forces us to identify the total\nmassMofthe twoblackholes in the PNdescription with\nthe deformed-Kerr mass Mof the test-particle descrip-\ntion, thus justifying our choice of using the same symbol\nfor these two a priori distinct quantities. Similarly, we\nfind thatm=µ[9]. Assuming this mapping between the\nmasses and imposing that the PN-expanded deformed-\nKerr Hamiltonian given by Eqs. (5.53)–(5.60) coincides\nwith the effective Hamiltonian given by Eqs. (5.27)–\n(5.29), we obtain the following mapping between the\nspins\nS∗=σ∗+1\nc2∆σ∗, (5.63)\nSKerr=σ+1\nc2∆σ, (5.64)\nwhere we have set for simplicity a(η) = 0 andb(η) = 0\nand where\n∆σ=−1\n16/braceleftBigg\n12∆σ∗+η/bracketleftbigg2M\nr(4σ−7σ∗)\n+6(ˆp·n)2(6σ+5σ∗)−ˆp2(3σ+4σ∗)/bracketrightbigg/bracerightBigg\n.\n(5.65)\nHere,∆σ∗is an arbitrary function going to zero at least\nlinearly in ηwhenη→0, so as to get the correct\ntest-particle limit. In fact, if ∆σ∗satisfies this condi-\ntion and if we assume, as appropriate for black holes,\nS1,2=χ1,2m2\n1,2(with|χ1,2| ≤1 and constant)6, when\nm2∼0 we have SKerr=S1+O(m2). Similarly, for\nm2∼0 the physical unrescaled spin of the effective par-\nticle isS=S∗m/M=S2+O(m2)2. The equations of\nmotion of our initial Hamiltonian (3.9) coincide with the\nPapapetrou equations [23], which describe the motion of\na spinning test-particle in a curved spacetime [38, 39].\nAssuming the canonical commutation relations between\nxi,pj,S1andS2, we obtain that the Hamilton equa-\ntions for the effective deformed-Kerr Hamiltonian are\n˙y= ˙yP+O(m2). Here, the dot denotes a time deriva-\ntive,yis a generic phase-space variable ( xi,pj,S1or\n6As noted by Ref. [21], a spin mapping such as ours also gives\nthe correct test particle limit if |S1,2|/m1,2= const., but this\nscaling of the spins with the masses is not appropriate for bl ack\nholes [37].13\nS2), and ˙y= ˙yPare the Papapetrou equations expressed\nin Hamiltonian form. Therefore, our mapping repro-\nduces the correct test-particle limit, and the remainders\nSKerr−S1=O(m2)andS−S2=O(m2)2produceextra-\naccelerations of order O(m2) or higher. This is compa-\nrable to the self-force acceleration [40], which appears at\nthe next order in the mass ratio beyond the test-particle\nlimit.\nAlthough different choices for the function ∆σ∗are in\nprinciple possible, we choose here\n∆σ∗=η\n12/bracketleftBig2M\nr(7σ∗−4σ)+ˆp2(3σ+4σ∗)\n−6(ˆp·n)2(6σ+5σ∗)/bracketrightBig\n, (5.66)\nwhich gives, when inserted into Eq. (5.65), ∆σ= 0. Be-\ncause this form for ∆σ∗is clearly not covariant under\ngeneric coordinate transformations, we choose instead\nthe following form for the mapping of the spins, which is\ncovariant at least as far as the square of the momentum\nis concerned:\n∆σ= 0, (5.67)\n∆σ∗=η\n12/bracketleftBig2M\nr(7σ∗−4σ)+(Q−1)(3σ+4σ∗)\n−6∆r\nΣ(ˆp·n)2(6σ+5σ∗)/bracketrightBig\n, (5.68)\nwhere we have replaced ˆp2withγijˆpiˆpj=Q−1 [where\nQis given in Eq. (5.52)] and ( ˆp·n)2= ˆp2\nrwith\n∆r(ˆp·n)2/Σ =grrˆp2\nr. This form agrees with the pre-\nvious mapping through order 1 /c2, but differs from it at\nhigher orders. Although neither this form is completely\ncovariant, not even under a rescaling of the radial coor-\ndinate (as it still features a dependence on the radius r),\nit proved slightly better as far as the dynamics of the\nEOB model, analyzed in the next section, is concerned.\nIn particular, the factor grr, which becomes zero at the\nhorizon, quenches the increase of ˆ prat small radii, thus\ngiving a more stable behavior during the plunge subse-\nquent to the inspiral. (A similar effect was observed in\nRef. [22], where the radial momentum was expressed in\ntortoise coordinates to prevent it from diverging close to\nthe horizon.)\nHaving determined the mass and spin mappings, we\ncan write down the improved (resummed) Hamiltonian\n(or EOB Hamiltonian) for spinning black holes. To this\npurpose, it is sufficient to invert the mapping between\nthe real and effective Hamiltonians [Eq. (5.26)]. In units\nin whichc= 1, we obtain\nHimproved\nreal=M/radicalBigg\n1+2η/parenleftbiggHeff\nµ−1/parenrightbigg\n,(5.69)\nwith\nHeff=HS+βipi+α/radicalBig\nµ2+γijpipj+Q4(p)\n−µ\n2Mr3(δij−3ninj)S∗\niS∗\nj.\n(5.70)Here, the −µ/(2Mr3)(δij−3ninj)S∗\niS∗\njterm is the\nquadrupole deformation introduced in the previous sec-\ntion to account for the leading order coupling of the par-\nticle’s spin with itself (see also Appendix A); βi,αand\nγijare computed using the deformed-Kerr metric, that\nis inserting Eqs. (5.36a)–(5.36e) into Eqs. (5.44)–(5.46);\nHSis obtained by inserting Eqs. (5.31), (5.32), and\nEqs. (5.47a)–(5.52) into Eqs. (4.17), (4.18) and (4.19).\nLastly, the spin SKerrenters this Hamiltonian through\nthe parameter a=|SKerr|/Mappearingin the deformed-\nKerr metric.\nBefore completing this section, we want to discuss the\ndeformation of the Kerr potentials ∆ tand ∆ rgiven in\nEqs. (5.37) and (5.38), which play an important role in\nthe EOB Hamiltonian (5.69). It is convenient to re-write\nthe function ∆ tas\n∆t=r2∆u(u), (5.71)\n∆u(u) =A(u)+a2\nM2u2. (5.72)\nIn previous EOB investigations the Pad´ e summation was\napplied to the function ∆ uto enforce the presence of a\nzero, correspondingto the EOBhorizon, both in the non-\nspinning [11] and spinning case [19, 21]. Reference [22]\npointed out that when including the 4PN and 5PNterms\nin the function A(u), the Pad´ e summation generates\npolesifspinsarepresent. Also, the Pad´ esummationdoes\nnot always ensure the existence of an innermost stable\ncircular orbit (ISCO) for spins aligned and antialigned\nwith the orbital angular momentum and, even when it\ndoes, the position of the ISCO does not vary monotoni-\ncally with the magnitude of the spins. For these reasons,\nwe propose here an alternative way of enforcing the exis-\ntence of the EOB horizons. Working through 3PN order,\nwe write\n∆u(u) =¯∆u(u)/bracketleftbig\n1+η∆0+ηlog/parenleftbig\n1+∆1u+∆2u2\n+∆3u3+∆4u4/parenrightbig/bracketrightbig\n, (5.73)\nwhere\n¯∆u(u) =a2\nM2/parenleftBigg\nu−M\nrEOB\nH,+/parenrightBigg/parenleftBigg\nu−M\nrEOB\nH,−/parenrightBigg\n(5.74)\n=a2u2\nM2+2u\nηK−1+1\n(ηK−1)2,(5.75)\nrEOB\nH,±=/parenleftBig\nM±/radicalbig\nM2−a2/parenrightBig\n(1−Kη).(5.76)\nHere,rEOB\nH,±are the EOB horizons, which differ from the\nKerr horizons when the adjustable parameter Kis differ-\nent from zero, and where the log is introduced to quench\nthe divergence of the powers of uat small radii. We\ncould in principle replace the logarithm with any other\nanalytical function with no zeros (e.g., an exponential).\nHowever, when studying the dynamics ofthe EOB model\n(see Sec. VI) the results are more sensible if we choose a\nfunction, such as the logarithm, which softens the diver-\ngence of the truncated PN series.14\nThe coefficients ∆ 0, ∆1, ∆2, ∆3and ∆ 4can be de-\nrived by inserting Eq. (5.73) into Eq. (5.71), expanding\nthrough3PNorder,andequatingtheresulttoEqs.(5.71)and (5.72), with A(u) given by its PN expansion (5.41).\nDoing so, we obtain\n∆0=K(ηK−2), (5.77)\n∆1=−2(ηK−1)(K+∆0), (5.78)\n∆2=1\n2∆1(−4ηK+∆1+4)−a2\nM2(ηK−1)2∆0, (5.79)\n∆3=1\n3/bracketleftBig\n−∆3\n1+3(ηK−1)∆2\n1+3∆2∆1−6(ηK−1)(−ηK+∆2+1)−3a2\nM2(ηK−1)2∆1/bracketrightBig\n,(5.80)\n∆4=1\n12/braceleftBig\n6a2\nM2/parenleftbig\n∆2\n1−2∆2/parenrightbig\n(ηK−1)2+3∆4\n1−8(ηK−1)∆3\n1−12∆2∆2\n1+12[2(ηK−1)∆2+∆3] ∆1\n+12/parenleftbigg94\n3−41\n32π2/parenrightbigg\n(ηK−1)2+6/bracketleftbig\n∆2\n2−4∆3(ηK−1)/bracketrightbig/bracerightBig\n. (5.81)\nBy construction, if we expand Eq. (5.73) in PN orders,\nKcan only appear at 4PN and higher orders, because\nwe must recover the PN expansion (5.37)–(5.41) through\n3PN order. In this sense, Kparameterizes our ignorance\nof the PN expansion at orders equal or higher than 4PN\n(i.e.,Kwould not play any role if the PN series were\nknown in its entirety). Similarly, we re-write the poten-\ntial ∆r[Eq. (5.38)] as\n∆r= ∆tD−1(u), (5.82)\nD−1(u) = 1+log[1+6 ηu2+2(26−3η)ηu3].\n(5.83)\nThe coefficients in the above function D−1(u) are such\nthat, when PN expanded, it gives the PN result (5.42),\nand the logarithmic dependence is once again chosen to\nquench the divergence of the truncated PN series.\nFinally, let us stress that if we included PN orders\nhigher than 3PN in the functions A(u) andD(u), we\nwould need to add higher order coefficients ∆ iwithi>4\nin Eq. (5.73).\nVI. EFFECTIVE-ONE-BODY DYNAMICS FOR\nCIRCULAR, EQUATORIAL ORBITS\nInthissectionwestudy thedynamicsofthe novelEOB\nmodel that we developed in Sec. VE. We will show that\n(i) Our EOB model has the correct test-particle limit,\nfor both non-spinning and spinning black holes, for\ngenericorbits and arbitrary spin orientations;\n(ii) There exist an ISCO when the spins are aligned or\nantialigned with the orbital angular momentum L;\n(iii) The radius, energy, total angular momentum, or-\nbital angular momentum and frequency at theISCO exhibit a smooth dependence on the binary\nmass-ratio and spins. Also, this dependence looks\nreasonable based on what we expect from the test-\nparticle limit and from numerical-relativity simula-\ntions;\n(iv) The frequency at the ISCO for an extreme mass-\nratio non-spinning black-hole binary agrees with\nthe exact result computed by Ref. [41];\n(v) During the plunge subsequent to the ISCO, the\norbital frequency of black-hole binaries with spins\naligned or antialigned with Lgrows and reaches\na maximum, after which it decreases. The ra-\ndius at which the frequency peaks is very close to\nthe radius of the equatorial, circular light ring (or\nphoton orbit). This feature generalizes the non-\nspinning behavior [9], and it has a clear physical\ninterpretation in terms of frame-dragging. As in\nthe non-spinning case[9], it providesa natural time\nat which to match the two-body description of the\ninspiral and plunge to the one-body description of\nthe merger and ringdown.\nWe stress that only (i) applies to generic orbits and spin\norientations, while (ii), (iii), (iv) and (v) are true for\nblack-hole binaries with spins aligned or antialigned with\nL. (It should be noted that circular or spherical orbits,\nand therefore the ISCO, are not even present for generic\norbitsandspin orientations,becausethe systemis notin-\ntegrable, not even in the test-particle limit [37]). While\nwe will tackle the study of generic orbits and arbitrary\nspin orientations in a follow-up paper, we argue that the\npreliminary study presented here is already sufficient to\nillustrate the potential of the novel EOB model. We re-\ncall [22] that the only existing EOB model for spinning\nblack-hole binaries, proposed in Refs. [19, 21], (i) repro-\nducesonlyapproximatelythetestparticlelimit; (ii)when15\nincluding non-spinning terms at 4PN and 5PN order, it\ndoes not always present an ISCO for binaries with spins\nparallel to L, and when it does the spin dependence of\nquantities evaluated at the ISCO is unusual; (iii) gen-\nerally, the orbital frequency does not peak during the\nplunge, making the prediction of the matching time from\nthe two-body to the one-body description quite problem-\natic.\nLet us now go through the points of the list that we\npresented at the beginning of this section. In order to\nprove point (i) we first need to observe that the de-\nformed metric (5.36a)–(5.36e) [with the potentials ∆ r\nand ∆ tgiven by Eqs. (5.82) and (5.73)] reduces to the\nKerr metric as η→0, and the deformation is linear in\nηwhenη∼0. Therefore, the acceleration produced by\nthis deformation on the test-particle is comparableto the\nself-force acceleration, which appears at the next order\nin the mass ratio beyond the test-particle limit. Sec-\nond, as already proved in Sec. VE, the mapping (5.63)–(5.64) of the spins reduces to SKerr=S1+O(m2) and\nS=S∗m/M=S2+O(m2)2whenm2∼0, where the\nremainders produce accelerations which are again com-\nparable to the self-force acceleration.\nTo prove points (ii), (iii), (iv) and (iv), we need to\nwrite the effective EOB Hamiltonian (5.70) for equato-\nrial orbits and for spins parallel to the orbital angular\nmomentum (chosen to be along the z-axis). We obtain\nHeff=HS+βipi+α/radicalBig\nµ2+γijpipj+Q4(p)\n−µ\n2Mr3S2\n∗,\n(6.1)\nHS=geff\nSOL·S∗+geff\nSSS∗,\n(6.2)\nwhere\ngeff\nSO=e2ν−˜µ/bracketleftBig\n−√Q∆r(˜Br−2˜Bνr)+/parenleftBig\ne˜µ+ν−˜B/parenrightBig/parenleftbig√Q+1/parenrightbig\n+(˜Bνr−˜Br)√∆r/bracketrightBig\n˜B2M/parenleftbig√Q+1/parenrightbig√Q, (6.3)\ngeff\nSS=µ\nM/bracketleftBigg\nω+1\n2˜Be−˜µ−νωr/radicalbig\n∆r+/parenleftbiggL2\nz\nµ2−˜B2e−2(˜µ+ν)∆rp2\nr\nµ2/parenrightbiggeν−˜µωr√∆r\n2˜B/parenleftbig√Q+1/parenrightbig√Q/bracketrightBigg\n, (6.4)\nwith\nQ= 1+∆rp2\nr\nµ2r2+L2\nzr2\nµ2(̟4−a2∆t).(6.5)\nThe above equations can be evaluated explicitly by us-\ning Eqs. (5.31), (5.32), (5.47a)–(5.47d), (5.48)–(5.50),\n(5.71)– (5.82)7. To calculate the radius and the or-\nbital angular momentum at the ISCO for the EOB\nmodel, we insert Eq. (6.1) into the real EOB Hamilto-\nnian (5.69), and solve numerically the following system\nof equations [9]\n∂Himproved\nreal(r,pr= 0,Lz)\n∂r= 0,(6.6)\n∂2Himproved\nreal(r,pr= 0,Lz)\n∂r2= 0,(6.7)\nwith respect to randLz. Moreover, the frequency for\ncircular orbits is given by\nΩ =∂Himproved\nreal(r,pr= 0,Lz)\n∂Lz, (6.8)\n7A Mathematica notebook implementing the Hamiltonian (6.1) –\n(6.5) is available from the authors upon request.which follows immediately from the Hamilton equations\nbecauseLz=pφ. Finally the binding energy is Ebind=\nHimproved\nreal−M.\nHenceforth, we set the adjustable frame-dragging pa-\nrametersωfd\n1=ωfd\n2= 0 [see Eq. (5.40)] and write Kin\nEq. (5.76) as a polynomial of second order in η,\nK(η) =K0+K1η+K2η2. (6.9)\nK0,K1andK2being constants. We find that if we\nimpose\nK(1/4) =1\n2,dK\ndη(1/4) = 0 (6.10)\nthe functional dependence on ηandχof several physical\nquantities evaluated at the ISCO is quite smooth and\nregular. Therefore, imposing these constraints we obtain\nK(η) =K0(1−4η)2+4(1−2η)η. (6.11)\nIt is worth noting that the values of KanddK/dηatη=\n1/4 have a more direct meaning than the coefficients K1\nandK2. In fact, current numerical-relativity simulations\ncan evolve binary black holes with η≈0.25 (with only\nfew runs having η∼0.1). Thus, they can determine\nK(1/4)−1/2, while the value of ( dK/dη)(1/4) can be16\nFIG. 1: The frequency at the EOB ISCO for binaries having\nspins parallel to L, with mass ratio q=m2/m1and with\nspin-parameter projections onto the direction of Lgiven by\nχ1=χ2=χ. As expected, the frequency increases with χ\nfor a given mass ratio, while for fixed χit increases with qif\nχ/lessorsimilar0.9, while it decreases with qifχis almost extremal (see\ntext for details).\nhopefully determined when more numerical simulations\nwithη= 0.1–0.25 become available.\nHereafter, we will use Eq. (6.11) and set K0= 1.4467.\nThe latter is determined by requiring that the ISCO fre-\nquency for extreme mass-ratio non-spinning black-hole\nbinaries agrees with the exact result of Ref. [41], which\ncomputed the shift ofthe ISCOfrequency due to the con-\nservative part of self-force (see also Ref. [42] where the\nresultofRef.[41]wascomparedtothenon-spinningEOB\nprediction which resums the function (5.41) ` a laPad´ e.).8\nIn Fig. 1 we plot the orbital frequency at the ISCO for\nbinaries with mass ratio q=m2/m1ranging from 10−6\nto 1 and spins aligned with L. In particular, denoting\nbyS1,2=χ1,2m2\n1,2the projections of the spins along the\ndirection of L, we consider binaries with χ1=χ2=χ.\nWe see that the ISCO frequency increases with the mag-\nnitude of the spins χif the mass ratio is fixed, as ex-\npected from the test-particle case. Also, if the spins\nare kept fixed and small, the ISCO frequency increases\nwith the mass-ratio, as it should be to reproduce the\n8We stress that the most general form of K(η) can include terms\ndepending on a2, witha=|SKerr|/M. In particular, a term not\ndepending on ηand proportional to a2could be determined by a\ncalculation similar to that in Ref. [41], that is by computin g the\nshiftof the ISCO frequency caused by the conservative parto fthe\nself-force, for a non-spinning test-particle in a Kerr spac etime.FIG. 2: The final spin parameter χfinas inferred at the EOB\nISCO, for binaries having spins parallel to L, with mass ratio\nq=m2/m1and with spin-parameter projections onto the\ndirection of Lgiven by χ1=χ2=χ. As expected, χfin\nflattens for large χin the comparable mass case (see text for\ndetails).\nresults of numerical-relativity simulations (see, e.g., the\nnon-spinning EOB models of Ref. [6, 7]). However, if the\nspins are close to χ= 1, the ISCO frequency decreases\nwhen the mass ratio increases. This crossover is mir-\nrored by a similar behavior of other quantities evaluated\nat the ISCO — such as the energy, the orbital angular\nmomentum, andthecoordinateradius—anditsphysical\nmeaning can be explained as follows. When comparable-\nmass almost-extremal black holes merge, the resulting\nblack-hole remnant has a spin parameter that is slightly\nsmaller than the spin parameters of the parent black\nholes. This is a consequence of the cosmic censorship\nconjecture (see Ref. [46] and references therein) which\nprevents black holes with spin χ >1 to be formed [47].\nTherefore, because in the EOB model the position, and\ntherefore the frequency, of the ISCO (together with the\nlossofenergyand angularmomentum duringthe plunge)\nregulate the final spin of the remnant, and because for\nan isolated black hole the ISCO frequency increases with\nthe spin, any EOB model that satisfies the cosmic cen-\nsorship conjecture must have an ISCO frequency that\nslightly decreases with the mass ratio when χ∼1. This\ninterpretation can be confirmed by computing the final\nspin of the remnant black hole as estimated at the ISCO.\nWe have\nχfin=S1+S2+LISCO\n(M+Ebind\nISCO)2, (6.12)\nwhich is plotted in Fig. 2. Although the final spin gets\nslightlylargerthan1forhighinitialspins(becauseweare17\nFIG. 3: The final spin parameter χfinas inferred at the EOB\nISCO for binaries having spins parallel to L, with mass ratio\nq=m2/m1and with spin-parameter projections onto the di-\nrection of Lgiven by χ1=χ2=χ, comparedtotheremnant’s\nfinal spin parameter predicted by the formula presented in\nRef. [43] (“BR09”), which accurately reproduces numerical -\nrelativity results. The EOB model and the BR09 formula\nagree when the mass ratio is small ( q= 0.1), because the\nemission during the plunge, merger and ringdown is negligib le\nin this case. For q= 0.5 andq= 1, there is an offset, because\nthe EOB result, at this stage, neglects the gravitational-w ave\nemission during the plunge, merger and ringdown (see text\nfor details).\nneglecting here the energy and angular momentum emit-\nted during the plunge, merger and ringdown), the curves\nare remarkably smooth and monotonic (see the corre-\nsponding Fig. 5 of Ref. [21]) and they flatten at high ini-\ntialspins, asexpected. Inparticular,inFig.3wefocuson\nmass ratios q= 1,q= 0.5 andq= 0.1, and plot the final\nspinχfinas inferred from the ISCO energy and angular\nmomentum, together with the final spin of the remnant\npredicted by the formula presented in Ref. [43] which ac-\ncurately reproduce the numerical-relativity results (see\nalso Refs. [45, 49–54] for other formulas for the final spin\nof the remnant). It is remarkable that in spite of the off-\nsetbetweenthepredictionsoftheformulaofRef.[43]and\ntheEOBresult, whichisduetoneglectingtheenergyand\nangular momentum emitted during plunge, merger and\nringdown, the qualitative behavior of the curves in Fig. 3\nis the same. Also, we observe that the difference between\ncorresponding curves decreases with the mass ratio, with\nthe EOB and the numerical-relativity–based results be-\ning in very good agreement for q= 0.1. This happens\nbecause the energy and angular momentum emitted dur-\ning plunge, merger and ringdown become negligible forFIG. 4: The mass loss inferred at the EOB ISCO for binaries\nhaving spins parallel to L, with mass ratio q=m2/m1and\nwith spin-parameter projections onto the direction of Lgiven\nbyχ1=χ2=χ, compared to the total mass lost during the\ninspiral, merger and ringdown, as predicted by the formulas\npresented in Ref. [44] (“AEI09”) and in Ref. [45] (“RIT09”),\nwhich reproduce numerical-relativity results, although w ith\ndifferent accuracies because of the different parameter regi ons\nthey cover (see the text for details). The EOB model and the\nAEI09 and RIT09 fits agree when the mass ratio is small\n(q= 0.1), while there is an offset for q= 0.5 andq= 1.\nThe reason is that the ringdown emission, which is negligibl e\nfor small mass-ratios, is not taken into account by our EOB\nmodel at this stage.\nsmall mass-ratios.9. Similarly, in Fig. 4 we plot the bind-\ning energy at the ISCO for mass ratios q= 1,q= 0.5 and\nq= 0.1 and compare it with fits to numerical-relativity\ndata for the total mass radiated in gravitational waves\nduring the inspiral, merger and ringdown. In particular,\nfor theq= 1 case we use the fit in Ref. [44] (“AEI09”),\nwhile forq= 0.5 andq= 0.1 we use the fit recently pro-\nposed by Ref. [45] (“RIT09”). While the AEI fit is more\naccurate than the RIT one for the particular configura-\ntionconsideredhere(seeFig. 11andrelateddiscussionin\nRef. [44]), the AEI fit is only applicable for comparable-\nmass binaries10, and for this reasonwe resortto the more\n9This can be seen by noting that, for a test-particle with mass\nmaround a black holes with mass M, the final plunge lasts a\ndynamical time ∼M[9], while the inspiral from large radii to\nthe ISCO lasts ∼M2/m.\n10The limited applicability of the AEI fit (which is only valid f or\nequal-mass binaries with spins aligned or anti-aligned) is indeed\none reason why it turns out to be more accurate, for the config-\nuration under consideration, than the RIT fit, which is inste ad\napplicable to more generic binaries.18\nFIG. 5: The frequency at the EOB ISCO for binaries having\nspins parallel to L, with mass ratio q=m2/m1and with\nspin-parameter projections onto the direction of Lgiven by\nχ1=−χ2=χ. As expected, the frequency is constant in\nthe equal-mass case, because the spins of the two black holes\ncancel out (see text for details).\nFIG. 6: The final spin parameter χfinas inferred at the EOB\nISCO, for binaries having spins parallel to L, with mass ratio\nq=m2/m1and with spin-parameter projections onto the\ndirection of Lgiven by χ1=−χ2=χ. The results are the\nsame for all equal-mass binaries, for which the spins of the\ntwo black holes cancel out (see text for details).FIG. 7: The maximum of the EOB orbital frequency during\nthe plunge, for binaries having spins parallel to L, with mass\nratioq=m2/m1and with spin-parameter projections onto\nthe direction of Lgiven by χ1=χ2=χ. As expected, the\nfrequency increases with χfor a given mass ratio, while for\nfixedχit increases with qifχ/lessorsimilar0.9, while it decreases with\nqisχis almost extremal (see text for details).\ngeneralRIT fitin the q= 0.5andq= 0.1cases. [In Fig.4\nwe show the predictions of both the AEI and the RIT fit\nin theq= 1 case. Being the AEI fit more accurate, its\ndifference from the RIT fit gives an idea of the error bars\nwhich should be applied to the predictions of the RIT fit\nforq= 0.5 andq= 0.1.]\nInFigs.5and6wepresentsimilarresults, fortheISCO\nfrequency and for the final spin estimated at the ISCO,\nin the case of spins antialigned with the orbital angular\nmomentum. The most apparent feature of these figures\nis that, in the equal-mass case, the quantities under con-\nsideration are independent of χ. This happens because\nin this case the spins S1andS2are equal and opposite,\nwhich results in a zero value for the spins SKerrandS∗\nentering the EOB Hamiltonian (5.69). As such, in the\nEOB model, equal-mass binaries with equal and oppo-\nsite spins behave as non-spinning binaries. This feature,\nwhich is also shared by the PN-expanded Hamiltonian,\nuntil the PN order which is currently known, is also in\nagreement with the results of numerical simulations. In\nfact, equal-mass binaries with equal and opposite spins\nwould be indistinguishable with LISA, Virgo and LIGO\nobservations [44, 55]. Except for this feature, and simi-\nlarly to the aligned case discussed above, the behavior of\nthe curves in Figs. 5 and 6 is quite smooth and regular\nwhen going from the equal-mass case to the test-particle\ncase.\nIn Fig. 7 we plot the maximum value of the orbital\nfrequency during the plunge subsequent to the inspi-19\nFIG. 8: For binaries having spins parallel to L, with mass ratio q=m2/m1= 1 (left panel) and q=m2/m1= 0.1 (right panel)\nand with spin-parameter projections onto the direction of Lgiven by χ1=χ2=χ, we plot twice the maximum of the EOB\norbital frequency during the plunge against the frequencie s of the first 8 overtones of the ℓ= 2,m= 2 quasi-normal mode of a\nKerr black hole. The quasi-normal mode frequency is compute d using the final spin and final mass of the remnant. The final\nspin is estimated by applying the formula of Ref. [43], while for the final mass we use the formula of Ref. [44] (“AEI09”) in\ntheq= 1 case and that of Ref. [45] (“RIT09”) in the q= 0.1 case. As can be seen, in the q= 0.1 case the peak frequencies\nlie among the high overtones of the ℓ= 2,m= 2 mode, while in the q= 1 case they are generally lower than them. In the\nq= 1 case we also mark with a square the numerical gravitationa l-wave frequency at the peak of the h22mode when χ= 0.\nThis gravitational-wave frequency coincides with (twice) the maximum of the EOB orbital frequency at the time when the\nmatching of the quasi-normal modes is performed in the non-s pinning EOB model of Ref. [7]. The numerical gravitational- wave\nfrequency is computed from the numerical simulation of Refs . [48] (“Caltech-Cornell”).\nral, for binaries with mass ratio q=m2/m1and with\nχ1=χ2=χ. Moreprecisely, weassumethatthe particle\nstarts off with no radial velocity at the ISCO (thus hav-\ning angular momentum LISCOand energy EISCO), and\nwe compute prassuming that the energy and angular\nmomentum are conserved during the plunge. We find\nthat the orbital frequency presents a peak for any value\nof the spins and any mass ratio, and we denote the value\nof the frequency at the peak with MΩmax. We note that\nthe behavior of MΩmaxas a function of the mass ratio is\nsimilar to that of MΩISCO. In particular, its dependence\nonηchanges sign when going from small to large spins.\nThe physical interpretation of the peak of the orbital\nfrequency is that the frequency increases as the effec-\ntive particle spirals in, but when the effective particle\ngets close to the black hole, the orbital frequency has\nto decrease because the particle’s motion gets locked to\nthe horizon (this is a well-known phenomenon, see for\ninstance Ref. [56, 57] for some of its effect on the test-\nparticle dynamics). Said in another way, the orbital fre-\nquency of the effective particle for an observer at infinity\ngoestoaconstant(ortozerointhenon-spinningcase[9])\non the EOB horizon. As a consequence, the peak in the\nfrequency can be used to signal the transition betweentwo regimes [9]: one in which the deformed black hole\nand the effective particle have different frequencies and\noneinwhichthe twobodies basicallymoveandradiateas\na single perturbed black hole. For this reason the peak of\nthe frequency provides the EOB approach with the nat-\nural point where to switch to the one-body description,\ni.e., the point where to start describing the gravitational\nwaveforms as a superposition of quasi-normal modes.\nWe find that the values of MΩmaxare roughly those\nneeded to attach the quasi-normal modes used in EOB\nmodelstodescribethemergerandtheringdown[6,7,22].\nThisisshowninFig.8, whereweplottwicethemaximum\nof the orbital frequency, MΩ22for binaries with q= 1\n(left panel) and q= 0.1 (right panel) and with spins\nχ1=χ2=χ. We compare MΩ22with the frequency of\nthe first 8 overtones of the ℓ= 2,m= 2 quasi-normal\nmode of a Kerr black hole, computed using the final spin\nand the final mass of the black-hole remnant [58]. [The\nfinal spin parameter is estimated by applying the for-\nmula of Ref. [43], while for the final mass we use the\nformula of Ref. [44] (“AEI09”) in the q= 1 case and\nthe one of Ref. [45] (“RIT09”) in the q= 0.1 case.] In\ntheq= 1 case we also mark with a square the numerical\ngravitational-wavefrequency at the peak ofthe h22mode20\nFIG. 9: The maximum of the EOB orbital frequency during\nthe plunge, for binaries having spins parallel to L, with mass\nratioq=m2/m1and with spin-parameter projections onto\nthe direction of Lgiven by χ1=−χ2=χ. The results are\nthe same for all equal-mass binaries, for which the spins of\nthe two black holes cancel out (see text for details).\nwhenχ= 0. This gravitational-wavefrequency coincides\nwith (twice) the maximum of the EOB orbital frequency\natthetimewhenthematchingofthequasi-normalmodes\nis performed in the non-spinning EOB model of Ref. [7].\nThe numerical gravitational-wavefrequency is computed\nfrom the numerical simulation of Refs. [48] (“Caltech-\nCornell”). As can be seen, while in the q= 0.1 case\nthe peak frequencies lie among the high overtones of the\nℓ= 2,m= 2 quasi-normal mode, in the q= 1 case\nthey are generally lower than them. Quite interestingly,\nwe find that the values of MΩ22forχ/greaterorsimilar0.4 can be in-\ncreased up to the frequencies of the quasi-normal modes\nby assuming ωfd\n2∼30–70 in Eq. (5.40). Nevertheless,\nthe frequencies that we obtain are comparable to those\nused for the matching with the quasi-normal modes in\nRef. [7, 22], and we therefore expect such a matching to\nbe possible also in our EOB model.\nAlso, it is interesting to note that the position rmaxof\nthe frequency peak is quite close (to within 8%) to the\nposition of the light ring (or circular photon orbit). This\nfact, which holds exactly in the non-spinning case [9],\nfurther confirms that the potential barrier for massless\nparticles (such as gravitational waves) lies at r∼rmax.\nFinally, inFig.9weshowthemaximumvalueoftheor-\nbital frequency during the plunge for binaries with mass\nratioq=m2/m1and withχ1=−χ2=χ. As for the\nISCO quantities, the dependence on the spins and the\nmassratiosis muchsimplerthan in the alignedcase, with\nthe black-holespins cancelling out in the equal mass-case\nand thus giving results which are independent of χ. Also,weseeasmooth transitionfrom theequal-massto the ex-\ntrememass-ratiocase,thatourmodelreproducesexactly.\nAlso in this antialigned case, the radius rmaxagrees with\nthe light-ring position to within 4%.\nVII. CONCLUSIONS\nIn this paper, building on Ref. [23], we computed the\nHamiltonian of a spinning test particle, at linear order in\nthe particle’s spin, in an axisymmetric stationary metric\nand in quasi-isotropic coordinates. Then, by applying a\ncoordinate transformation, we derived the Hamiltonian\nof a spinning test particle in Kerr spacetime in Boyer-\nLindquist coordinates\nWe used those results to construct an improved EOB\nHamiltonian for spinning black holes. To achieve this\ngoal, we followed previous studies [19, 21] and mapped\nthe real two-body dynamics into the dynamics of an ef-\nfective particle with mass µand spin S∗moving in a\ndeformed-Kerr spacetime with spin SKerr, the symmet-\nric mass-ratio of the binary, η, acting as the deformation\nparameter.\nTo derive the improved EOB Hamiltonian, we pro-\nceeded as follows. First, we applied a suitable canonical\ntransformationtotherealADM Hamiltonianandworked\nout the PN-expanded effective Hamiltonian through the\nrelation\nHeff\nµ=H2\nreal−m2\n1−m2\n2\n2m1m2. (7.1)\nThen, we found an appropriate deformed-Kerr met-\nric such that the corresponding Hamiltonian, when ex-\npanded in PN orders, coincided with the PN-expanded\neffective Hamiltonian through 3PN order in the non-\nspinning terms, and 2.5PN order in the spinning terms.\nThe (resummed) improved EOB Hamiltonian is then\nfound by inverting Eq. (7.1), which gives\nHimproved\nreal=M/radicalBigg\n1+2η/parenleftbiggHeff\nµ−1/parenrightbigg\n,(7.2)\nwithHeffgiven by Eq. (5.70), where α,βiandγijare ob-\ntained by inserting Eqs. (5.36a)–(5.36e) into Eqs. (5.44)–\n(5.46); where HSis obtained by inserting Eqs. (5.31),\n(5.32), and Eqs. (5.47a)–(5.52) into Eqs. (4.17), (4.18)\nand (4.19); and where the effective particle’s spin S∗\nand the deformed-Kerr spin SKerr(witha=|SKerr|/M)\nare expressed in terms of the real spins by means of\nEqs. (5.63), (5.64), (5.2) and (5.3).\nThe crucial EOB metric potential for quasi-circular\nmotion is the potential ∆ t(r) (which reduces in the non-\nspinningcaseto the radialpotential A(r) ofRefs. [9, 10]).\nTo guarantee the presence of an inner and outer horizons\nin the EOB metric, we proposed to re-write the poten-\ntial ∆ t(r) in a suitable way [see Eqs. (5.71) and (5.73)],21\nintroducing the adjustable EOB parameter K(η) regu-\nlating the higher-order, unknown PN terms. The rea-\nson why we did not re-write the potential ∆ t(r) using\nthe Pad´ e summation [21] is because Ref. [22] found that\nwhen including non-spinning terms at 4PN and 5PN or-\nder, the Pad´ e summation produces spurious poles, does\nnot always ensure the presence of an ISCO for binaries\nwith spins parallel to Land, even when it does, the spin\ndependence of physical quantities evaluated at the ISCO\nis quite unusual.\nRestricting the study to circular orbits in the equato-\nrial plane and assuming spins aligned or antialigned with\nthe orbital angular momentum, we investigated several\nfeatures of our improved EOB Hamiltonian. Using an\nexpression of the EOB adjustable parameter K(η) which\nreproduces the self-force results in the non-spinning ex-\ntreme mass-ratio limit [41, 42], we computed the orbital\nfrequency at the EOB ISCO, we estimated the final spin\nfromtheEOBISCO,andthemaximumorbitalfrequency\nduring the plunge. We found that these predictions are\nquite smooth and regular under a variation of ηand of\ntheblack-holespins. Quiteinterestingly,themaximumof\nthe orbital frequency during the plunge alwaysexists and\nis close to the light-ring position, as in the non-spinning\ncase [9]. For this reason, as in the non-spinning case [9],\nthe orbital-frequency peak can be used within the EOB\nto markthe matching time at which the mergerand ring-\ndown start, i.e, the time when, in the EOB formalism,\nthe gravitational waveforms start being described by a\nsuperposition of quasi-normal modes. This will be useful\nin future comparisons of the EOB model with numerical-\nrelativity simulations.\nThe results of Sec. VI are an example of the per-\nformances that our improved EOB Hamiltonian can\nachieve. We expect several refinements to be possibly\nneeded when comparing our EOB model with accurate\nnumerical-relativity simulations of binary black holes.\nWe may, forexample, extend ourmodel toreproducealso\nthe next-to-leading order spin-spin couplings, which are\nknown and appear at 3PN order [28–30, 33–36]. Also, we\nmight introduce a different mapping between the black-\nhole spins S1,S2andS∗,SKerr, a different form of the\nadjustable parameter K(η), and re-write differently the\nEOB metric potential ∆ t(r). We could also introduce\nin it the adjustable parameters a5anda6at 4PN and\n5PN order, respectively. Moreover, other choices of the\nreference tetrad used to work out the Hamiltonian for\na spinning test-particle in an axisymmetric stationary\nspacetime could be in principle used, leading to a differ-\nent (canonically related) EOB Hamiltonian. Lastly, the\nmapping (7.1)-(7.2) could me modified by introducing a\ndependence on the spin variables.\nIn conclusion, the most remarkable feature of our im-\nproved EOB Hamiltonian is that in the extreme mass-\nratio limit, it exactly reproduces the Hamiltonian of a\nspinning test particle in a Kerr spacetime, at linear order\nin the particle’s spin and at all PN orders.Acknowledgments\nWe thank Yi Pan for several useful discussions. E.B.\nand A.B. acknowledge support from NSF Grants No.\nPHYS-0603762 and PHY-0903631. A.B. also acknowl-\nedges support from NASA grant NNX09AI81G.\nAppendix A: Incorporating spin-spin couplings in\nthe effective-one-body Hamiltonian\nThe Hamiltonian for a spinning particle in a Kerr\nspacetime that we derived in Sec. IV, and the Hamilto-\nnian for a spinning particlein a deformed-Kerrspacetime\nthat we derived in Sec. VC are only valid at linear order\nin the particle’s spin. However, as suggested in Ref. [19],\nwe can introduce the terms that are quadratic in the par-\nticle’s spin by modifying the quadrupole moment of the\nKerr metric.\nIn particular, we can add a quadrupole which is\nquadratic in the particle’s spin to the quadrupole of the\nKerr metric (which is quadratic in SKerr). The expres-\nsion for the metric perturbation correspondingto a slight\nchange of the Kerr quadrupole can be extracted from\nthe Hartle-Thorne metric [59, 60], which describes the\nspacetime of a slowly rotating star. Ref. [61] gives this\nexpression in quasi-Boyer-Lindquist coordinates (i.e., in\ncoordinates that reduce to Boyer-Lindquist coordinates\nif the quadrupole perturbation is zero, thus reducing the\nspacetime to pure Kerr). This is exactly what is needed\nfor our purposes, since we work in quasi-Boyer-Lindquist\ncoordinates too.\nIn particular, our procedure for introducing the terms\nwhich are quadratic in the particle’s spin into our Hamil-\ntonian consists of modifying the effective metric (5.36a)–\n(5.36e) by adding the quadrupole metric\nhµν=1\nM4QijS∗\niS∗\nj¯hµν, (A1)\nwhere the quadrupole tensor Qijis given by\nQij=δij−3ninj, (A2)\nand¯hµνis given by [61]\n¯htt=1\n1−2M/rF1(r),¯hti= 0, (A3)\n¯hij=−F2(r)/braceleftbigg\nδij−ninj/bracketleftbigg\n1+/parenleftbigg\n1−2M\nr/parenrightbiggF1(r)\nF2(r)/bracketrightbigg/bracerightbigg\n.\n(A4)\nThe functions F1,2(r) in the above equation are derived\nin Ref. 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Babak, Classical and Quantum\nGravity 23, 4167 (2006)."    },    {        "title": "1207.5231v1.Manipulating_the_Voltage_Dependence_of_Tunneling_Spin_Torques.pdf",        "content": "Manipulating the Voltage Dependence of Tunneling Spin\nTorques\nA. Manchon\nPhysical Science and Engineering, King Abdullah University of Science and Technology\n(KAUST), Thuwal 23955, Saudi Arabia\nABSTRACT\nVoltage-driven spin transfer torques in magnetic tunnel junctions provide an outstanding tool to design advanced\nspin-based devices for memory and reprogrammable logic applications. The non-linear voltage dependence of the\ntorque has a direct impact on current-driven magnetization dynamics and on devices performances. After a brief\noverview of the progress made to date in the theoretical description of the spin torque in tunnel junctions, I present\ndi\u000berent ways to alter and control the bias dependence of both components of the spin torque. Engineering the\njunction (barrier and electrodes) structural asymmetries or controlling the spin accumulation pro\fle in the free\nlayer o\u000ber promising tools to design e\u000ecient spin devices.\nKeywords: Spin Transfer Torque, Magnetic Tunnel Junction, Magnetic Random Access Memory\n1. INTRODUCTION\nWhile most of the commercial microelectronic devices are based on the charge of the carriers (electrons and holes),\nSpintronics relies on the spin angular momentum of the carrier to generate low energy consumption functional\ndevices. The most impressive demonstration of the technological relevance of Spintronics is the implementation\nof Giant Magnetoresistive read heads in magnetic data storage.1As illustrated by the current state of the art\nof data storage, the \frst virtue of magnetic devices is their non-volatility: in the absence of external input\nand thermal activation, the magnetic state remains frozen in its equilibrium direction. Therefore, non-volatile\nMagnetic Random Access Memories (MRAM) composed of magnetic tunnel junctions are expected to present\noutstanding performances down to 30nm and below.\nThere are currently various types of MRAM. The original MRAM concept, commercially available, is based\non \feld-induced switching but has limited performances when scaling down the device area.2Another candidate\nis the thermally-assisted MRAM, or TAS-RAM, where \feld-induced switching is assisted by thermal activation.3\nBeyond these di\u000berent alternatives, spin transfer-MRAMs (STT-MRAMs) present a particular interest for both\napplied and fundamental condensed matter physicists2through the electrical manipulation of the magnetic\ncon\fguration of the device. This has been made possible by the prediction of the so-called spin transfer torque\n(STT) by Slonczewski and Berger:4A spin-polarized current impinging on a ferromagnet can transfer its spin\ndegree of freedom to the local magnetization exerting a torque on it. This e\u000bect has been observed in metallic\nspin-valves (two ferromagnetic layers separated by a spacer), tunneling junctions and even magnetic domain\nwalls and is shown to induce current-control magnetic excitations, GHz self-sustained magnetic precessions and\nmagnetization switching.5{7This area has known tremendous developments in the past ten years with the\nrealization of e\u000ecient devices such as spin torque MRAMs, radio-frequency oscillators and reprogrammable\nlogics.\nBeyond technological challenges, spin transfer torque in magnetic tunnel junctions (MTJs) presents a number\nof puzzles. Although STT in MTJs is considered to have reached maturity, the bias dependence extracted from\nroutine measurements is still not well understood nor controlled. In this article, I propose an overview of the\nmeans to control the bias dependence of STT in MTJs. I present \frst the nature of the spin torque expected in\nan \"ideal\" ballistic tunnel junction. Then, I brie\ry review the in\ruence of asymmetries and interfacial scattering\nand \fnally I discuss the in\ruence of spin di\u000busion in the free layer.\nFurther author information: E-mail: aurelien.manchon@kaust.edu.sa, Telephone: (+966)544700061, Website:\nhttp://spintronics.kaust.edu.saarXiv:1207.5231v1  [cond-mat.mes-hall]  22 Jul 20122. STATE OF THE ART: EXPERIMENTS\nSpin transfer torque in tunnel junctions has been \frst demonstrated in MgO-based junctions by Sun and in AlOx-\nbased junctions by Huai et al. and Fuchs et al.8However, Fe/MgO/Fe-type junctions very rapidly supplanted the\nAlOx junctions due to their outstanding transport properties. Since then, a number of exciting realizations have\nbeen achieved including switching, excitations, precessions, spin diode etc. We refer the reader to Ref. [6] for\nmore details about the experimental achievements. Among the most important results, it has been demonstrated\nexperimentally9and theoretically10{16that the spin torque has the following form\nT=Tjjm\u0002(p\u0002m) +T?m\u0002p; (1)\nTjj=a1V+a2V2; T?=b0+b2V2: (2)\nHere,mandpare the magnetization direction of the free and pinned layers, respectively and Vis the bias\nvoltage applied across the junction. The \frst term Tkin Eq. (1), referred to as the in-plane torque, competes\nwith the magnetic damping allowing for self-sustained magnetic precessions and switching, whereas the second\ntermT?, referred to as the out-of-plane torque, acts like an e\u000bective \feld applied along p. As we discuss in the\nnext section, this dependence is theoretically predicted in ballistic junction when the transverse spin density is\nabsorbed at the interface.\nHowever, recent experiments have reported important discrepancies between the actual bias dependence\nof the spin torque and the one proposed in Eqs. (1)-(2).17{20In particular, Oh et al.17showed that in an\nasymmetrically designed MTJ, the bias dependence of the out-of-plane torque acquires a linear contribution.\nThis linear contribution has also been observed by Petit et al. and Heinonen et al.19as small biases. Deac et\nal.18and Li et al.20have also uncovered complex bias dependences at large bias voltages. The aim of the present\narticle is to present a coherent description of the spin transfer torque in tunnel junctions in order to account for\nthese di\u000berent observations.\n3. BALLISTIC INTERFACIAL SPIN TORQUE\n3.1 Ballistic Tunneling\nAt this early stage of the theory, we directly connect the spin torque acting on the volume of the free magnetic\nlayer to the interfacial spin current transverse to the local magnetization, Tjj(?)=Jx(y)\nsjinterface . By de\fnition,\nthe spin transfer torque is due to the amount of spin angular momentum transferred to the magnetic layer,\nminus the amount of spin lost through spin-\rip mechanisms. In the present case of ballistic transport (no spin\nrelaxation), incoming spins gradually align on the local magnetization by transferring its transverse momentum\nto it, as schematically depicted in Fig.1(a). This alignment takes place on a very small distance near the interface,\nnamed the spin dephasing length \u0015\u001e. Assuming that the spin torque is related to the interfacial spin current\nmeans that the thickness of the free layer is much larger than the spin dephasing length ( d>>\u0015\u001e). As we will\ndiscuss in the following, this de\fnition is not appropriate in the general case and quantum resonances as well\nas spin di\u000busion e\u000bects can signi\fcantly a\u000bect the torque in the case of ultra thin free layers. Nonetheless, we\nassume for now that the spin transfer is interfacial and given by the interfacial tunneling spin current.\nThe available theoretical descriptions of the bias dependence of spin torque in tunnel junctions have been ob-\ntained through transfer matrix formalism,11free electron,12{14,21tight binding10,16and ab-initio calculations.15\nAlthough these methods capture some or most of the band structure details of the system, they do not provide\nconceptual pictures about the symmetries of the bias dependence of the spin torque. In the present paragraph,\nwe wish to provide such a qualitative description, disregarding any band structure details. A convenient method\nis the Bardeen Transfer Matrix (BTM) approach,22that was widely used by Slonczewski in his descriptions of\nspin torque in tunnel junctions.11The tunneling transport is expressed as the product of the interfacial densities\nof states through a transfer matrix. The simplest version of this model proposed by Julliere23had a reasonable\nsuccess in explaining the tunneling magnetoresistance. Formally, BTM approach can actually be recovered as\nan asymptotic limit of the Keldysh tight-binding model in the case of thick and large barriers:24As long as the\ninterface states are decoupled (tunneling is treated perturbatively), BTM is applicable. Therefore, the present\ndiscussion does not reproduce quantitatively the physics of ultra thin barrier MTJs, but we believe it does providep\"m\"x\"y\"z\"p\"m\"(a)\"(b)\"\nλφ#\nθ\"ϕ\"Figure 1. (Color online) (a) Schematics of the spin transfer process in a magnetic tunnel junction. The black arrows refer\nto the magnetizations direction, and the red arrows represent the itinerant electron spins. Note that the itinerant spins\nget aligned on the local magnetization of the free layer mover a distance \u0015\u001e, called spin dephasing length; (b) Schematics\nof the alignment of the impinging spin in the rigid spin picture (red arrow) and in the realistic transport description (blue\narrow). This supplementary angle gives rise to the out-of-plane torque.\nthe relevant symmetries and tendencies. In the spin-dependent BTM formalism, the charge and spin current\ndensities are given in the spinor form in the 2 \u00022 spin space\n^J= 2\u0019e\n\u0016hZ\nd\u000f[^\u001aL^TL!R^\u001aR^Ty\nL!RfL(1\u0000fR)\u0000^\u001aR^TR!L^\u001aL^Ty\nR!LfR(1\u0000fL)]; (3)\nwherefL;Rand ^\u001aL;Rare the Fermi distribution function and electronic density of states at the left (right)\ninterfaces, and ^TL!R(^TR!L) is the spin-dependent transfer matrix accounting for both elastic and inelastic\ntunneling. In the spinor formalism, the charge current and spin current are expressed je=Tr[^J] andJs=\nTr[^\u001b^J], where ^\u001bis the vector of Pauli spin matrices. The hat^denotes 2 \u00022 matrices in spin space. This form\naccounts for contributions of rightward and leftward electrons originating from the left (L) or right (R) reservoirs.\nUsing the above de\fnition assumes that tunneling is only due to interfacial densities of states and transmission\nthrough the barrier.\nThe important point here is the de\fnition of the transmission matrices ^TL!Rand ^TR!L. In the case of spin\ntorque, we have to extend the BTM formalism to non-collinear magnetization directions. That is to say, the\ninterfacial densities of states must be rewritten in the absolute quantization axis that we chose aligned on the\nmagnetization of the right layer. Slonczewski11and Levy and Fert25used this approach to derive explicit low\nbias expressions of the in-plane torque. They assumed that the ballistic tunneling through the barrier is spin\nindependent. In other words, the spin state does not rotate during the tunneling. The spin impinging on the free\nlayer is then simply aligned on the polarizer orientation p, as depicted in Fig. 1(b). This picture, that we refer\nto as the rigid spin assumption, allows to get an expression for the in-plane torque, but does not provide any\nindications about the nature of the out-of-plane torque. However, numerical calculations10,12{15and experimental\nevidence9,17{20have demonstrated the existence of a large out-of-plane component of the spin torque. This\nindicates that a fundamental ingredient is missing in the original BTM approach and corrections to the rigid\nspin assumption must be considered. Actually, when a spin-polarized electron impinges on a ferromagnet, its\ntransmission through the interface is accompanied by a spin rotation . This can be easily understood using a toy\nmodel. Let's consider a rightward electron in the rotating frame of the right electrode. Its original wave function\nj\ti0is transformed into j\titunder transmission through the interface\nj\ti0= cos\u0012\n2j\"i\u0000 sin\u0012\n2j#i)j \tit=^Tj\ti0= (T\"cos\u0012\n2j\"i\u0000T#sin\u0012\n2j#i): (4)\nSince the interface between the barrier and the free magnetic layer is spin-dependent, it \flters the electron spin\nand its majority and minority projections are not transmitted the same way: the interaction with the interface\ninduces a phase shift between the majority and minority projections of the spin. In other words, since the\nimpinging spin feels a magnetic interaction at the interface, it rotates upon transmission/re\rection. This e\u000bect\nhas been identi\fed by Stiles and Zangwill26in metallic spin-valves and by Manchon et al.12in magnetic tunnel\njunctions. These references demonstrate that the supplementary angle gained over transmission can be quitelarge and depends on the direction of incidence. In metallic systems, averaging over the Fermi surface quenches\nthis contribution,26whereas in magnetic tunnel junctions, the wave vector \fltering yields an e\u000bective angle that\nis quite large and has a component out of the ( m;p) plane12[see Fig. 1(b)]. Consequently, the e\u000bective spin\ndensity in the free layer is rotated out of this plane and produces a so-called out-of-plane torque.\nThis implies that the tunneling matrix is no more spin independent and that the incident spin \"sees\" a\nfree layer magnetization with a virtual orientation ( \u0012,'),'being the phase induced by the spin-dependent\ntransmission. The expression for the tunnel current spinor in the quantization axis of the right (free) layer is\nthen\n^J= 2\u0019e\n\u0016hZ\nd\u000f[R^\u001aL^TL!RR'L^\u001aRRy\n'L^Ty\nL!RRyfL(1\u0000fR)\u0000^\u001aR^TR!LR'R^\u001aLRy\n'R^Ty\nR!LfR(1\u0000fL)]; (5)\nwhere we de\fned the interfacial density of state of the ith electrode and the rotations matrices RandR'\n^\u001ai=1\n2\u0012\n\u001ai+ \u0001\u001ai 0\n0\u001ai\u0000\u0001\u001ai\u0013\n;R=\u0012\ncos\u0012\n2\u0000sin\u0012\n2\nsin\u0012\n2cos\u0012\n2\u0013\n;R'=\u0012\ncos\u0012\n2\u0000ei'sin\u0012\n2\ne\u0000i'sin\u0012\n2cos\u0012\n2\u0013\n:(6)\nAs mentioned above, note that the matrices R'iaccount for the rotation of the electron spin when tunneling\nthrough the barrier whereas Rensures that the spinor current is de\fned in the quantization frame of the right\nlayer. A straightforward calculation leads to\nTjj(\u000f) = 2\u0019\n\u0016hZ\nd\u000fjTj2\u001aR\u0001\u001aL(cos'LfL\u0000cos'RfR); (7)\nT?(\u000f) = 2\u0019\n\u0016hZ\nd\u000fjTj2(\u001aR\u0001\u001aLsin'LfL+\u001aL\u0001\u001aRsin'RfR); (8)\nAt this stage we have made no assumption about the symmetry of the junction. Note that in general the\ncorrection from the interfacial precession is small ' << 1. In this case, it clearly appears that the in-plane\ntorque is antisymmetric (in the limit of small ') when exchanging L and R indices, whereas the out-of-plane\ntorque remains symmetric . We considered ballistic tunneling in a thick barrier, but it is in principle possible\nto have asymmetric electrodes or asymmetric barrier pro\fle. Let's now assume a symmetric tunnel junction\nsubmitted to a bias voltage. In this case, we can rewrite the explicit energy dependence of the di\u000berent terms\ninvolved in the spin transport\n\u001ai(\u000f) =\u001a(\u000f\u0006eV=2);\u0001\u001ai(\u000f) = \u0001\u001a(\u000f\u0006eV=2); 'i(\u000f) ='i(\u000f\u0006eV=2): (9)\nwhere the sign\u0006is associated with the left (right) electrode respectively, and the Fermi-Dirac distribution is\ngiven byfL;R(\u000f) = (e(\u000f\u0000\u000fF\u0006eV=2)=kBT+ 1)\u00001. Then, the in-plane and out-of-plane torques become\nTjj=Z\nd\u000f[^\u001cjj(\u000f+eV=2)\u0000^\u001cjj(\u000f\u0000eV=2)]; (10)\nT?=Z\nd\u000f[^\u001c?(\u000f+eV=2) + ^\u001c?(\u000f\u0000eV=2)]: (11)\nTherefore, independently on the detail of the band structure and in the limit of the model, if the junction is\nsymmetric, it implies that\nTjj=a1V; (12)\nT?=b0+b2V2; (13)\nin qualitative agreement with the numerical estimates.10,12{15Note that the actual energy dependence of the\ninterfacial spin-dependent densities of states in\ruences the bias dependence of the spin torque at large voltages.\nFirst, as experimentally demonstrated by Valenzuela,27the polarization of the impinging electrons depends on\nthe bias polarity which induces quadratic corrections to the simple linear voltage dependence of the in-plane\ntorque presented in Eq. 12. Second, at large bias dependence, Fowler-Nordheim processes start dominating the\ntransport which induces oscillatory voltage dependences of the tunneling magnetoresistance and spin torque, as\ncalculated by Tang et al.16,283.2 Materials Consideration\nWhen a spin-polarized electron enters into the free layer, its spin precesses around the local magnetization with\na spatial period of 2 \u0019=(k\"\u0000k#).12,26Since the angle between the electron spin and the local magnetization\ndepends on the incident angle of the electron, the precession period also depends on the angle of incidence. After\naveraging over the Fermi surface, as mentioned above, the e\u000bective magnetization displays a oscillatory damped\nspatial pro\fle as depicted schematically in Fig. 2.\nThe most common system investigated to date is the MgO-based magnetic tunnel junctions with CoFeB\nelectrodes. For thick enough barriers, the electron wave function is \fltered so that the transport is dominated\nby \u0001 1symmetries.29Since \u0001 1bands are half metallic in CoFeB, the junctions display huge tunneling magne-\ntoresistance ratios.30In this limit, the majority projection of the electron spin propagates regularly in the free\nlayer whereas the minority spin wave function is evanescent. As a consequence, the spin density pro\fle is heavily\ndamped, as shown in Fig. 2, and the spin torque is very strongly localized at the interface. Thus, the model of\ninterfacial spin torque proposed above is valid. Note however that in thin MgO barriers, resonant states arise31\nthat alter the tunnel magnetoresistance by allowing for minority electrons from other band symmetries to tunnel\nthrough the barrier. In this case, the minority waves are no more evanescent and are allowed to propagate in\nthe free layer, yielding a torque that can extend through the volume of this layer.\nFigure 2. (Color online) Schematics of the spatial pro\fle of the spin density transverse to the local magnetization, extending\nin the free layer. We represent typical pro\fles in the cases of a weak ferromagnet (long precession length - red), strong\nferromagnet (short precession length - black) and half-metallic free layer (exponential decay at the interface - blue).\n3.3 Structural Asymmetries\nThe previous simple model provides a qualitative description of the bias dependence expected in the case of\nasymmetric junctions. When the work function of the left and right electrodes are di\u000berent or when the materials\nof the electrodes are di\u000berent from each other, they induce asymmetries in the transport: the spin torque is\nmore e\u000ecient at one bias polarity than at the other. In other words, when inserting structural asymmetries,\n^\u001cL!R6= ^\u001cR!L. The \frst implication is that the bias dependence of the out-of-plane torque deviates from the\n\"conventional\" quadratic dependence. This is schematically represented in Fig. 3, left panel. In this \fgure,\nwe represented the bias dependence obtained using a free electron model in the case of symmetric (solid lines)\nand asymmetric tunnel barrier (dashed lines). Note the presence of a quadratic component V2in the in-plane\ntorque, that arises from the energy dependence of the interfacial densities of states. Details can be found\nelsewhere.12,14,16\nThis feature has been recently exploited by Oh et al.17to reduce the back-hopping process observed at\nlarge biases.32As discussed above, in a symmetric junction, whereas the sign of the in-plane torque depends\non the polarity of current injection ( /V), the out-of-plane torque is always in the same direction ( /V2).\nConsequently, at positive polarity both torques favor the antiparallel con\fguration, whereas at negative polarity,\nthe in-plane torque favors the parallel con\fguration while the out-of-plane torque favors the antiparallel one.\nThis competition leads to back-hopping of the magnetization state which is detrimental for applications such as/Minus0.4/Minus0.2 0.2 0.4Bias Voltage/LParen1V/RParen1\n/Minus0.4/Minus0.20.20.40.60.8Spin Torque/LParen1arb.units/RParen1\n2 4 6 8 10Thickness/LBracket1nm/RBracket10.51.01.52.0Spin Torque/LParen1arb.units/RParen1Figure 3. (Color online) (a) Bias dependence of the in-plane (black) and out-of-plane (red) torques in the case of symmetric\n(solid) and asymmetric barrier (dashed). See Refs. *** for details; (b) Spin torque magnitude as a function of the free\nlayer thickness in the case of coherent ballistic transport (quantum interferences are visible - black), spin di\u000busion in the\nshort spin dephasing limit (the thickness dependence in /1=d- red) and long spin dephasing (deviation from 1 =d- blue).\nMRAMs.17,32Introducing arti\fcial asymmetries in the system, by using either di\u000berent electrode materials or\nby engineering the barrier potential pro\fle, Oh et al.17showed that it is possible to quench this back-hopping\nprocess by adding a linear part to the bias dependence of the out-of-plane torque.\n3.4 Interfacial Scattering\nWe now brie\ry review previous works on the in\ruence of interfacial spin scattering in magnetic tunnel junc-\ntions.21,25It is well known that electron interactions with thermal excitations can a\u000bect magnetic tunnel junc-\ntions properties such as the magnetoresistance.33Hot electrons-induced magnon emission and absorption a\u000bect\nthe polarization of the tunneling electrons and electron-phonon interactions result in thermally-assisted tunnel-\ning. In the presence of electron-magnon and electron-phonon interactions, the transfer matrix becomes both spin\nand energy dependent\n^Te\u0000m\nL!R=^TL!R \n^I+r\nQm\nN(^\u001b:SR\ntr+^\u001b:SL\ntr)!\n; (14)\n^Te\u0000ph\nL!R=^TL!R0\n@1 +s\nQph\nq\nN(bq+b+\nq)1\nA^I; (15)\nwhereQm(Qph\nq) is the phenomenological electron-magnon (electron-phonon) e\u000eciency, Nis the number of atoms\nper cell,\u001bis the vector of Pauli spin matrices and SL(R)\ntr are the transverse part of the magnetizations of the\nleft and right electrodes. Details about the derivation of Eqs. (14)-(15) can be found in Ref. [25]. Assuming\nthat the electron spin-dependent densities of state do not vary much over the range eV, and considering acoustic\nphonons (!/q,Qq/q) with a density of states of the form \u001aph(!)/!\u0017, we obtain, at T=0 K and low bias\nvoltage\nTjj=G0PLsin\u0012(1 +\u0010phjVj\u0017+2)V; (16)\nT?\u0000T?0=G0PR\u001eLsin\u0012\u0010phjVj\u0017+3; (17)\n\u0010phbeing a coe\u000ecient that depends on the electron-phonon coupling, Fermi energy, Debye temperature \u0002 Detc...\nNote that the symmetry of the out-of-plane torque against the bias is conserved, whereas the in-plane torque\nacquires an antisymmetric component.\nIn the case of electron-magnon interaction, the transfer matrix [Eq. (14)] possesses non-diagonal elements\nthat are responsible for spin-\rip scattering. We then expect a much more complex in\ruence on the torque.\nAssuming a magnon density of states of the form \u001am(!) =!\u0017, symmetric electrodes ( Pi=P,'i='R) andT=0 K, we \fnd:\nTjj\u0000Tjj0/sin\u0012[P(1 +P)\u0000(1\u0000P)(1 +Pcos\u0012)]V\u0017+2; (18)\nT?\u0000T?0/P\u001esin\u0012(1\u0000cos\u0012)jVj\u0017+2: (19)\nThe detail of these expressions can be found in Ref. [25]. Interestingly the out-of-plane torque and the conduc-\ntance (not shown) both acquire a component that is symmetric against the bias because neither electron-magnon\nnor electron-phonon scattering break the junction symmetry. Furthermore, since the electron-magnon interac-\ntion mixes the majority and minority channels, the angular dependence is also a\u000bected, contrary to the case of\nelectron-phonon coupling.\n3.5 Finite thicknesses in ballistic regime\nIn the case where the electrodes' thickness is much larger than the spin dephasing length, the incident spin\ncurrent is totally absorbed in the free layer. Therefore, the torque is determined by the interfacial spin current\nonly. For example, in the case of half-metallic transport of \u0001 1electrons in FeCo/MgO/FeCo junctions, since\n\u00011minority electrons cannot propagate into the free layer, the length over which spin precesses (or in other\nwords, the length over which spin transfer takes place) is reduced to one to two monolayers.15In this case, the\ntotal torque exerted on the layer does not depend on the layer thickness since all the injected transverse spin\ncurrent has been absorbed at the interface. Therefore, the average torque acting on the free layer is inversely\nproportional to the layer thickness ( /1=d).\nIn the case of normal metallic behavior, both majority and minority spin propagate in the free layer, giving\nrise to a non-vanishing coherence length (see Fig. 2). In the case of free layers of thicknesses comparable to the\nspin precession length, the transverse spin current is not fully absorbed in the free layer and is re\rected by the\nsecond interface. This induces quantum interferences between the injected spin current and the re\rected (counter\npropagating) one. The precession pattern of the spin density is therefore a combination between exp[ i(k\"\u0000k#)x]\nand exp[i(k\"+k#)x]. This interference pattern is therefore a\u000bected by the thickness of the free layer and\nproduces a complex thickness dependence, as schematically depicted in Fig. 3(b). This dependence has been\nstudied numerically in several references.14,15Note that quantum coherent is seminal to obtain this pattern and\ninterfacial roughness or impurity scattering can easily destroy these interferences.\n4. DIFFUSION OF TUNNELING SPIN TORQUE\nIn the present section, we brie\ry present recent results obtained in the opposite limit of di\u000busive transport in\nthe free layer.34As seen in the previous section, in the coherent ballistic regime, quantum interferences occur\nwhen the free layer thickness is \fnite. However, in the limit of di\u000busive transport, quantum interferences are\nquenched and spin relaxation and di\u000busion dominate the transport. The tunneling process through the insulating\nbarrier imposes a ballistic injection of carriers at the interface between the tunnel barrier and the free layer which\nconstitutes a boundary condition to the coupled di\u000busive spin transport equations for the transverse component\nof the spin accumulation vector s. Along these guidelines, the spin dynamics of the transverse spin accumulation\nin the free layer is governed by the following coupled equations in steady state\nr\u0001J =\u00001\n\u001cJs\u0002m\u00001\n\u001c\u001em\u0002(s\u0002m)\u00001\n\u001csfs; (20)\nJ=\u0000Dr\ns; (21)\nwheresis the spin accumulation, mis the direction of the local magnetization, and Jis the spin current\ntensor. The di\u000busion is characterized by the di\u000busion constant Dand the spin dynamics is controlled by the\nspin precession time \u001cJ, spin decoherence time \u001c\u001eand spin relaxation time \u001csf. Whereas the spin relaxation\na\u000bects the three spin components, the spin precession and spin decoherence terms only a\u000bect the two transverse\ncomponents of the spin accumulation vector. The spin torque is de\fned as the spatial change of spin current,\ncompensated by the spin relaxation term\nT=1\n\nZ\n\nd\n\u0012\n\u0000r\u0001J\u00001\n\u001csfs\u0013\n: (22)Here, \n is the volume of the magnetic layer. Note the seminal di\u000berence with the de\fnition exploited in the\nprevious section: the spatial variation of the spin current is now partially balanced by the loss of the spin angular\nmomentum through spin \rip scattering and the electron is now propagating di\u000busively in the free layer yielding\na speci\fc spatial distribution of the spin current that needs to be evaluated.\nWe consider a \fnite free layer embedded between the tunnel barrier and a normal metallic capping layer.\nAssuming that a spin current J0=Jk+iJ?is imposed by the ballistic tunnel at the interface with the barrier,\nand considering a standard continuity condition at the interface with the capping layer,34the total spin torque\nexerted on the ferromagnet is\nTk+iT?=J0\ndL2\nL2\n0coshd\nL+\u0011sinhd\nL\u00001\ncoshd\nL+\u0011sinhd\nL; (23)\nwhere1\nL2\n0=\u0000i\n\u00152\nJ+1\n\u00152\n\u001e,1\nL2=\u0000i\n\u00152\nJ+1\n\u00152\n\u001e+1\n\u00152\nsf,dis the free layer thickness and \u0011is a parameter that only\ndepends on the bulk characteristics of the free and capping layers. We clearly see that in general the in-plane\nand out-of-plane torques are a mixture between in-plane and out-of-plane interfacial spin currents\nTk=\u000bJk+\fJ?; (24)\nT?=\u0000\fJk+\u000bJ?; (25)\nThese expressions reduce to the ballistic case studied in the previous section in the case of in\fnitely free layer\nthickness (d!1 ) and in\fnite spin di\u000busion length ( \u0015sf!1 ). However, in the most general case, it appears\nthat the spin torque can not be simply identi\fed to the interfacial spin current. This property arises from the\nfact that (i) spin relaxation is always present, alters the propagation of the spin density in the metallic layers\nand redistributes the spin degree of freedom on the two torque components; (ii) when the free layer thickness\nis comparable to the spin dephasing length, the transverse spin density di\u000buses towards the interface with the\ncapping layer which in turns in\ruences the spin dynamics in the free layer. This last e\u000bect virtually enhances\nthe impact of the spin di\u000busion length on the spin current mixing. These two situations are illustrated in Fig. 4.\n1234Position @nmD0.20.40.60.8Spin Density Harb.unitsL\nFigure 4. (Color online) Spatial distribution of the out-of-plane (black) and in-plane (red) components of the spin accu-\nmulation in the free layer/capping layer bilayer structure for short spin dephasing length ( \u0015\u001e= 0:5nm - solid lines), long\nspin dephasing length ( \u0015\u001e= 1:5nm - dashed lines) for short spin relaxation length in the capping layer ( \u0015N\nsf= 1nm). The\ndotted lines display the spin accumulation pro\fle for long spin relaxation length in the capping layer ( \u0015\u001e= 1:5nm and\n\u0015N\nsf= 10nm). The interface with the tunnel barrier is located at x= 0 and the interface between the free layer and the\ncapping layer is at x= 2nm. These curves were calculating following the theory developed in Ref. 34.\nThe di\u000busion of spin current in the free layer has two major implications: the redistribution of the spin angular\nmomentum is thickness-dependent and as soon as the thickness of the free layer exceeds the spin dephasing length,\nthe interfacial spin current limit is recovered [see Fig. 3(b), red line]. Note that the deviation we obtain hereis not related to quantum interferences (which are destroyed by impurity scattering and roughness), but rather\nto the incomplete absorption of the spin current: when the thickness of the free layer is smaller than the spin\ndephasing length \u0015\u001e, the transverse spin current responsible for the spin torque is not fully absorbed in the free\nlayer and di\u000buses towards the capping layer. This induces a deviation from the usual 1/d-thickness dependence\n[see Fig. 3(b), blue line]. Increasing the thickness of the free layer improves the absorption of the spin current\nand for thicknesses much larger than the spin dephasing length, the thickness dependence of the torque recovers\nthe 1/d limit. Note that in the case of half-metallic behavior, as in Fe/MgO/Fe tunnel junctions, the minority\nband of \u0001 1symmetry does not propagate in the ferromagnet which induces a quenching of the spin dephasing\nlength and reducing the spin torque to a 1/d behavior.\nSecond, this new dynamics mixes the two transverse components of the spin current and is therefore respon-\nsible for the deviation from the ballistic bias dependence shown in Eq. (2). If one assumes a bias dependence\nof the spin current on the form Jk=a1VandJ?=b2V2, as expected and observed for systems such as\nFe/MgO/Fe,9,15then both in-plane and out-of plane spin torque components will be a mixture of linear and\nquadratic bias dependences\nTk=\u000ba1V+\fa2V2; (26)\nT?=\u0000\fa1V+\u000ba2V2; (27)\nThis spin mixing may explain the recently obtained experimental results that show a strong linear voltage\ndependence of the out-of-plane torque, in contrast with the predictions.10,12{14\n5. CONCLUSION\nIn conclusion, the role of spin di\u000busion in the metallic layers of MTJs has been addressed theoretically. Assuming\nan interfacial bias-driven spin current at the interface between the insulator and the ferromagnet, the spin\ndi\u000busion equation is solved and describes a complex spin dynamics in the metallic layers. It is found that\nthis dynamics mixes the components of the spin current tranverse to the local magnetization which results in a\nsuperposition between linear and quadratic bias dependence of the out-of-plane torque. The thickness dependence\nof the spin transfer torque is also altered for small thicknesses.\nACKNOWLEDGMENTS\nThe author acknowledge S. Zhang, K.-J. Lee, J. Grollier, H. Ja\u000br\u0012 es, R. Matsumoto and A. Fert for stimulating\ndiscussions and fruitful collaborations.\nREFERENCES\n1. C. Chappert, A. Fert and F. Nguyen Van Dau, Nature Materials 6, 813 (2007).\n2. S. Ikeda, J. Hayakawa, Y. M. Lee, F. Matsukura, Y. Ohno, T. Hanyu, and H. Ohno, IEEE Trans. Elec. Dev.\n54, 991 (2007).\n3. I.L. Prejbeanu, M. Kerekes, R. C. Sousa, H. Sibuet, O. Redon, B. Dieny, and J. P. Nozieres, J. Phys.\nCondensed Matter 19, 165 (2007).\n4. J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996); L. Berger, Phys. Rev. B 549353, (1996).\n5. D. C. Ralph and M. D. Stiles, J. Magn. Magn. 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Min et al. , J. Appl. Phys. 105, 07D126 (2009).\n33. S. Zhang, P. M. Levy, A. C. Marley, and S. S. P. Parkin, Phys. Rev. Lett. 79, 3744 (1997); J. S. Moodera,\nJ. Nowak, and R. J. M. van de Veerdonk, Phys. Rev. Lett. 80, 2941 (1998).\n34. A. Manchon, R. Matsumoto, H. Ja\u000br\u0012 es and J. Grollier, arXiv:1204.5000 (2012)."    },    {        "title": "2002.01151v1.Isotropic_All_electric_Spin_analyzer_based_on_a_quantum_ring_with_spin_orbit_coupling.pdf",        "content": "arXiv:2002.01151v1  [cond-mat.mes-hall]  4 Feb 2020Isotropic All-electric Spin analyzer based on a quantum rin g with spin-orbit couplings\nShenglin Peng,1,2Wenchen Luo,2,∗Jian Sun,2Ai-Min Guo,2Fangping Ouyang,1,2,3,†and Tapash Chakraborty4,‡\n1State Key Laboratory of Powder Metallurgy, and Powder Metal lurgy Research Institute,\nCentral South University, Changsha, P. R. China 410083\n2School of Physics and Electronics, Central South Universit y, Changsha, P. R. China 410083\n3School of Physics and Technology, Xinjiang University, Uru mqi, P. R. China 830046\n4Department of Physics and Astronomy, University of Manitob a, Winnipeg, Canada R3T 2N2\n(Dated: February 5, 2020)\nHere we propose an isotropic all electrical spin analyzer in a quantum ring with spin-orbit cou-\npling by analytically and numerically modeling how the char ge transmission rates depend on the\npolarization of the incident spin. The formalism of spin tra nsmission and polarization rates in an\narbitrary direction is also developed by analyzing the Ahar onov-Bohm and the Aharonov-Casher\neffects. The topological spin texture induced by the spin-or bit couplings essentially contributes to\nthe dynamic phase and plays an important role in spin transpo rt. The spin transport features de-\nrived analytically has been confirmed numerically. This int eresting two-dimensional electron system\ncan be designed as a spin filter, spin polarizer and general an alyzer by simply tuning the spin-orbit\ncouplings, which paves the way for realizing the tunable and integrable spintronics device.\nI. INTRODUCTION\nManipulation of the spin degrees of freedom and the\nconduction charges in low-dimensional quantum struc-\ntures has been attracting considerable interest, due to\nwide range of potential applications in semiconductor\nspintronics and quantum computation. How to control,\nmodulate, or detect the spin degree of freedom at the\nmesoscopic scale is a key step for the application of the\nspin coherence in electronic devices. The quantum ring\n[1, 2] is an ideal platform to take into consideration the\nAharonov-Bohm (AB) and the Aharonov-Casher (AC)\neffects to show the nature of the quantum interference in\nconductance. The transport properties of similar nano-\ndevices have received considerable attention, especially\nin the spin transport device subject to the Rashba spin-\norbit coupling (SOC) [3–8], but the presence of Dressel-\nhaus SOC or combination of both SOCs[9, 10] have not\nbeen investigated sufficiently as yet.\nThe interplay of the Rashba SOC and the quantum\ninterference has been widely reported in the literature.\nNo spin is being polarized [11–13] in the transmission in\nthe two-lead rings with equal arm length and without a\nmagnetic flux or an impurity. This is because in this case\nthe interference phase of the two different eigentransport\nchannels is entirely due to the AC effect. The signs of\nphases are opposite but the absolute values are equal\nresulting in equal transmission rate for opposite spins.\nTopolarizethe spin, weneed to introducemagneticfields\n[14–17], use unequal length arms [11, 18, 19], doping [20–\n22], or contact three or more leads [23, 24].\nQuantum interference between the two arms of the\nring provides suitable means for controlling the spin in\n∗Electronic address: luo.wenchen@csu.edu.cn\n†Electronic address: ouyangfp@csu.edu.cn\n‡Electronic address: Tapash.Chakraborty@umanitoba.cathe nano-scale, which has been proven by the Green’s\nfunction method [18, 25] or Griffith’s boundary condi-\ntions [15, 19–21, 23, 24, 26]. The first order linear ap-\nproximation with full transparent contacts was also re-\nported [11, 14, 16, 27], albeit without the backscattering\neffect. The S-matrix method [28, 29] presents a rough\nassessment of the backscattering by fixing the energy-\ndependentcouplingparameterbetweentheleadsandring\nas constant. We note that previous works on spin trans-\nport properties in the quantum ring were not compre-\nhensive. For spin-unpolarized input current these works\noften only focused on spin polarization in the zdirection\nor the direction of the eigenstates of the ring. The total\npolarizability, polarization direction, and spin polariza-\ntion in arbitrary directions were rarely discussed. Work\nin the case of the arbitrarily spin-polarized incident are\ndifficult to find in the literature.\nIn this work, we present an analytical model for one-\ndimensional (1D) rings and numerical studies of real-\nistic two-dimensional (2D) quantum rings in the non-\nequilibrium Green’s function (NEGF) method [6] where\nboth the Rashba and Dresselhaus SOCs are present. We\nderivethe formulaforthe transmissionratesforarbitrary\nspin polarization and generalize them to the cases of the\npolarized incident spin. A density matrix describing the\nspin-polarized (in arbitrary direction) input current is\nalsointroducedintothe Green’sfunction equation, which\nresults in the same results obtained by the analytical 1D\nmodel. The transmission rate Tcan be up to unity with\nthe fully polarized output in a proper magnetic field and\nwith a proper Rashba SOC.\nWhen the input current is spin-polarized the transmis-\nsion rate depends on the direction of the input polariza-\ntion and the output current is still spin polarized. So\nthe quantum ring is also acting as a spin torque which\nmay be useful in spintronics. This property also guides\nus finding the way to design an omnidirectional spin an-\nalyzer. In contrast, the optical polarization analyzer is\nsimpler since the polarization is perpendicular to the di-2\nrection of the light. However, the spin polarization can\nbe along an arbitrary direction on the Bloch sphere. The\nspin analyzer in a particular direction can be achieved\nin the ferromagnetism systems [30]. The arbitrary spin\nanalyzer needs the light involved [31, 32], which is diffi-\ncult to be integrated. Here, we just need to measure the\nconductances in different strengthes of the SOC to ob-\ntain the polarization of the incident spin, which is easier\nto integrate on the chip. It is interesting that in such a\nsimple system, the spin filter, spin polarizer and spin an-\nalyzer can be achieved by just tuning the magnetic field\nor the Rashba SOC via the gate [33–36].\nII. THE TRANSPORT PROPERTIES IN THE\nONE-DIMENSIONAL MODEL\nTo understand the transport properties in a quantum\nring, the one-dimensional (1D) model is usually applied.\nThe ring is contacted with the left and the right leads at\nϕ=πand 0, respectively. In this work, we suppose that\nthe electron is injected from the left lead, then it travels\nthrough the ring in two different paths, one from ϕ=π\nto 0 clockwise (the upper arm) and the other from ϕ=π\nto 2πcounterclockwise (the lower arm), as shown in Fig.\n1(a).\nAs discussed in the previous work [37], the 1D model\nworks very well when the radius is not too large. The\n1D model here, at least, is a good approximation which\nresults in the correct physical pictures. Another approx-\nimation of neglecting the Zeeman effect is also adopted.\nIn the relatively low magnetic field ( B <3T), the Zee-\nman coupling is weak and could be neglected. We can\nalso numerically verify that this approximation is appro-\npriate in low magnetic fields.\nFor simplicity, we first consider only the Rashba SOC\nbeing present. If the Zeeman coupling is neglected the\nenergyspectrum of the 1D ring is given by [12, 15, 16, 38,\n39]Eµ\nn=τ/parenleftBig\nnµ\nj−ΦAB\n2π−Φµ\nAC\n2π/parenrightBig2\nwherenµ\njis the orbital\nquantum number, and the index µ= 1,2 represents the\nspin eigenstates |↑∝an}bracketri}htand|↓∝an}bracketri}ht, andj=±represents the\nclockwise and counterclockwise electron motions, respec-\ntively. Also, τ=/planckover2pi12\n2m∗r2\n0is the energy unit, Φ AB= 2πNis\nthe AB phase with the relative magnetic flux N=eBr2\n0\n2/planckover2pi1,\nand Φµ\nAC= (−1)µ/parenleftBig/radicalbig\n1+4β2\n1−1/parenrightBig\nπis the AC phase\nwithβ1=g1m∗r0//planckover2pi1.\nThe corresponding eigenstates are given by Ψµ\nj(ϕ) =\n1√\n2πe−inµ\njϕχµ(ϕ), whereχ1(ϕ) =/parenleftbig\ncosθ1\n2,−eiϕsinθ1\n2/parenrightbigT\nandχ2(ϕ) =/parenleftbig\nsinθ1\n2,eiϕcosθ1\n2/parenrightbigT, with tanθ1= 2β1[39].\nIt is clear that the directions of the spin polarization are\nalong (θ1,ϕ) and (π−θ1,π+ϕ) for the two eigenstates\nrespectively.\nThe schematic diagram of the total transport is explic-\nitly drawn in Fig. 1(a). The incident current can be de-\ncomposed into the two eigenstates χ1,2, and the electronis transported by these two channels. The transmission\nrate is given by (see the Method),\nTµ=KΦµ\nKK′+/bracketleftbig\n4k2\n0(Φµ−K′)+k2sin2(πk0r0)/bracketrightbig2,(1)\nwhereK= 16k2k2\n0sin2(πk0r0), Φµ= cos2ΦAB+Φµ\nAC\n2and\nK′= cos2(πk0r0). For vanishing magnetic field the AB\nphase vanishes and the SOC induced energy shift U0is\nneglected, then Eq. ( 1) agrees with the results obtained\nin Ref. [12, 15]. If there is a constant potential Uadded\nat the contact then the magnetic field for Tµ= 0 is not\nchanged while the magnetic field for Tµ= 1 is slightly\nshifted. Hence, for the spin filter the contact defect is\nnot very important.\nThe numerator of Eq. ( 1) indicates that the trans-\nmission rate oscillates with the incident energy E, and\ncos2ΦAB+Φµ\nAC\n2means that Tµoscillates with the increase\nof the magnetic field Bor the coupling strength of the\nSOCg1. When the magnetic field vanishes, Φ AB= 0 and\nT1=T2, resulting in a completely unpolarized transport\nif the incident spin is unpolarized. However, if both of\nthe AB and the AC phases are taken into consideration\nin a proper magnetic field the spin can be fully polarized\nafter traversing the ring.\nIf we want an 100% polarized spin current output\nthen the phases must satisfy Φ AB+ Φµ\nAC=πso that\nthe eigenstate χµis completed filtered out, and only\nthe other eigenstate is left. For a given SOC differ-\nent AB phases (different magnetic flux) lead to differ-\nent eigenstate filtering. The magnetic flux difference\nof the two nearest eigenstates filtering is then given by\n∆N=1\n2(/radicalbig\n1+4β2\n1−1). This result of constructing a\nperfect spin filter is consistent with the results of the S-\nmatrix method [28].\nIf the incident spin is unpolarized, then the spin\ncan be composed of an arbitrary direction ( θ′,ϕ′) and\nits opposite direction ( π−θ′,π+ϕ′) independently.\nThe two transport channels do not interfere with each\nother, and the transmission rates can be obtained by\nprojecting the two eigen transmission rates onto the\ntwo directions, T(θ′,ϕ′)+=/summationtext\nµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftBig\nχ(θ′,ϕ′)/bracketrightBig†\nχµ(0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nTµ,\nandT(θ′,ϕ′)−=/summationtext\nµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftBig\nχ(π−θ′,π+ϕ′)/bracketrightBig†\nχµ(0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nTµ,where\nχ(θ′,ϕ′)≡/parenleftBig\ncosθ′\n2,eiϕ′sinθ′\n2/parenrightBigT\n. The upper index of χ\nstands for the direction of the spin of the state. The spin\npolarization of the outcoming current in an arbitrary di-\nrection can be found to be\nP(θ′,ϕ′)=/bracketleftbig\nχ1(0)/bracketrightbig†σ(θ′,ϕ′)χ1(0)Pχ,(2)\nwhere the spin matrix along the direction of ( θ′,ϕ′) is\nσ(θ′,ϕ′)= (σxcosϕ′+σysinϕ′)sinθ′+σzcosθ′,andPχ=\n(T1−T2)/(T1+T2)isthespinpolarizationinthedirection\nof the two eigenstates at the contact, ( θ1/2,0). Since3\n|P(θ′,ϕ′)| ≤ |Pχ|, the outcoming polarization is always\nalong the direction of the eigenstate χ1orχ2.\nThe transmission rates when the incident spin is un-\npolarized are well studied. Next we consider the case\nwhere the incident spin is polarized in an arbitrary di-\nrection along ( θ,ϕ). Irrespective of the incident electron\nis a pure or a mixed state, the transmission rate is always\nobtained by\nT(θ,ϕ)=/summationdisplay\nµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBig\nχ(θ,ϕ)/parenrightBig†\nχµ(π)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nTµ,\n=T1cos2/parenleftbiggθin\n∆\n2/parenrightbigg\n+T2sin2/parenleftbiggθin\n∆\n2/parenrightbigg\n,(3)\nwhereθin\n∆is the angle between the direction ( θ,ϕ) and\nthe direction of the spin polarization of χ1(π) which is\n(θ1,0). It means that the arbitrarily polarized spin is\nprojected to the two conjugate eigensatesofthe ring, and\nthe transmission rate of the spin is the sum of the two\neigen channels. Moreover, for the unpolarized incident\ncurrent, we can decompose it into two conjugate parts,\nand we get T(θ,ϕ)+T(π−θ,π+ϕ)=T1+T2.\nIn fact, we can define the transmission rate T(θ,ϕ)\n(θ′,ϕ′)±\nwhere the upper index is the polarization of the inci-\ndent spin and the lower index represents the transmis-\nsion rate along the direction ( θ′,ϕ′) (for (θ′,ϕ′)+) or\n(π−θ′,π+ϕ′) (for (θ′,ϕ′)−). If the incident electrons\narebeinginjectedonebyoneandissupposedtobeapure\nstate, then the outcoming wave function at the right lead\ncanbefoundas χ(θ,ϕ)\nout=/summationtext\nµ[χµ(π)]†χ(θ,ϕ)tµχµ(0).The\ntransmission and the polarization rates are then given by\nT(θ,ϕ)\n(θ′,ϕ′)±=/bracketleftBig\nχ(θ,ϕ)\nout/bracketrightBig†\nσ(θ,ϕ)±χ(θ,ϕ)\nout (4)\nP(θ,ϕ)\n(θ′,ϕ′)=/bracketleftBig\nχ(θ,ϕ)\nout/bracketrightBig†\nσ(θ,ϕ)χ(θ,ϕ)\nout, (5)\nwhereσ(θ,ϕ)±=|(θ,ϕ)±∝an}bracketri}ht∝an}bracketle{t(θ,ϕ)±|is the density matrix\nof the eigenstate of the matrix σ(θ,ϕ).\nBy analyzing Eq. ( 3), it is easy to obtain\nthat max(T(θ,ϕ)) = max( T1,T2) and min( T(θ,ϕ)) =\nmin(T1,T2). Therefore, the incident spin having maxi-\nmum and the minimum transmission rates must be par-\nallel to the polarization directions of the two eigenstates,\nrespectively.\nThe generic spin torquing is given by Eq. ( 5), but the\npresence of a magnetic field makes the analytical result a\nbit complicated. Forsimplicity, weconsiderthe magnetic\nfield approaching zero, so that Φ AB→0 andT1=T2.\nThe spin polarizations for an arbitrarily polarized inci-\ndent current are\n\n\nP(θ,ϕ)\nx=−sin2θ1cosθ+cos2θ1sinθcosϕ,\nP(θ,ϕ)\ny= sinϕsinθ,\nP(θ,ϕ)\nz= cos2θ1cosθ+sin2θ1sinθcosϕ.(6)\nWhenϕ= 0, then P(θ,ϕ)\nx=−sin(2θ1−θ),P(θ,ϕ)\ny=\n0,P(θ,ϕ)\nz= cos(2θ1−θ). It means that the incident andoutcoming spins are all in the xOzplane, the spin passes\nthe ring and is torqued a fixed angle in the xOzplane,\n(θout,ϕout) = (θ−2θ1,0). It can be intuitively under-\nstood by the spin textures of the eigenstates that there\nis noycomponent spin at ϕ= 0,πin the ring [37].\nThe torqued angle is only related to the strength of the\nSOC. This special case goes back to the result obtained\nin Ref. [40], and the more special case, P(0,0)\nz= cos(2θ1)\nwas obtained in the path-integral approach [17]. A se-\nries of the ring may be able to tune the spin polarization\narbitrarily.\nIf only the Dresselhaus SOC is present, the anal-\nysis above is still valid, but some terms need to be\nchanged. The AC phase needs to be replaced by Φµ\nAC=\n−(−1)µ/parenleftBig/radicalbig\n1+4β2\n2/parenrightBig\nπwhereβ2=g2m∗r0//planckover2pi1. Theeigen-\nstates of the ring also need to be changed to χ1(ϕ) =/parenleftbig\ncosθ2\n2,ie−iϕsinθ2\n2/parenrightbigT, χ2(ϕ) =/parenleftbig\nsinθ2\n2,−e−iϕcosθ2\n2/parenrightbigT,\nwith tanθ2= 2β2, and the additional potential is U0=\n−β2\n2. All other calculations remain unchanged.\nWhen both the SOCs are present then it would be\ndifficult to have analytical results for the transport prob-\nlem. WethenseekthesolutionsnumericallyintheNEGF\nmethod.\nIII. NUMERICAL RESULTS OF THE SPIN\nAND CHARGE TRANSPORT PROPERTIES IN\nTWO-DIMENSIONAL MODELS\nThe spin transmission rates Tαand the spin polar-\nization rates Pαare important variables characterizing\nthe transport properties. The spin polarization rate Pα\nis the probability of the spin of the outcoming electron\nprojected to the αaxis.P0is the total polarizationof the\noutcoming spin. If P0= 1, then the spin of the current\nat the drain is fully polarized along a certain direction,\notherwise the outcoming current contains different com-\nponents of the spin at the same time and it is not fully\npolarized. We here numerically calculate the two rates to\nexplore the transport properties of a more realistic two-\ndimensional quantum ring contacted by the source and\ndrain on the two ends of a diameter. We then discuss\nhow the spin of the current is polarized and filtered by\nthe quantum ring with the SOCs when the incident elec-\ntron is spin unpolarized, and compare the 2D numerical\nresults with the analysis in the 1D model.\nFor simplicity and without loss of generality we con-\nsider the ring on the surface of the InAs semiconductor.\nWe adopt the tight-binding Hamiltonian (details shown\nin the appendix) to perform the numerical calculations\nby applying the Green’s functions [6, 41]. The device\nis indicated in Fig. 1(b), in which the lattice constant\nis 1 nm [42]. The energy spectrum of the ring without\nthe source and the drain in such a tight-binding model is\nsimilar to that of the ring in the parabolic potential cal-\nculated in Fock-Darwin basis [43], as shown in Fig. 1(c).\nSo the tight-binding model itself is reliable and is a very4\n/uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013/uni0000001a/uni00000018/uni00000014/uni00000013/uni00000013/uni00000014/uni00000015/uni00000018/uni00000014/uni00000018/uni00000013/uni00000014/uni0000001a/uni00000018/uni00000015/uni00000013/uni00000013/uni00000015/uni00000015/uni00000018/uni00000015/uni00000018/uni00000013\n/uni000000ed/uni00000014/uni00000011/uni00000018 /uni000000ed/uni00000014/uni00000011/uni00000013 /uni000000ed/uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000011/uni00000018/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013\nLwr0rw\nrw\nΨµ\nin Ψµ\noutΨµ\n1\nΨµ\n2\n/uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000018 /uni00000013/uni00000011/uni00000014/uni00000013 /uni00000013/uni00000011/uni00000014/uni00000018 /uni00000013/uni00000011/uni00000015/uni00000013/uni0000001a/uni00000018/uni00000014/uni00000013/uni00000013/uni00000014/uni00000015/uni00000018/uni00000014/uni00000018/uni00000013/uni00000014/uni0000001a/uni00000018/uni00000015/uni00000013/uni00000013/uni00000015/uni00000015/uni00000018/uni00000015/uni00000018/uni00000013\n/uni00000015/uni00000011/uni0000001b /uni00000016/uni00000011/uni00000013 /uni00000016/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000015/uni00000018 /uni00000013/uni00000011/uni00000018/uni00000013 /uni00000013/uni00000011/uni0000001a/uni00000018 /uni00000014/uni00000011/uni00000013/uni00000013¯hg2= 0nm·meV¯hg1= 20nm ·meVB= 3.0TTz↓Tz↑\n/uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000015/uni00000018 /uni00000013/uni00000011/uni00000018/uni00000013 /uni00000013/uni00000011/uni0000001a/uni00000018 /uni00000014/uni00000011/uni00000013/uni00000013/uni00000014/uni00000013/uni00000013/uni00000013/uni00000015/uni00000013/uni00000013/uni00000013/uni00000016/uni00000013/uni00000013/uni00000013/uni00000017/uni00000013/uni00000013/uni00000013/uni00000018/uni00000013/uni00000013/uni00000013/uni00000019/uni00000013/uni00000013/uni00000013/uni0000001a/uni00000013/uni00000013/uni00000013(a) ( b) ( c)\n(d) ( e)B[T] x/r0\nT B[T] k[π/a] k[π/a]\nenergy[meV]y/r0energy[meV]energy[meV]\nFIG. 1: (color online). (a) The schematic transport in a 1D ri ng. (b) The device with r0=15 nm, rw=5 nm, lead (red) width\nLw=10 nm and lattice constant a=1 nm. (c) The energy spectrum of tight-biding ring (black) w ith/planckover2pi1g1= 20.0 nm ·meV and\ng2= 0. (d) The energy spectrum of the lead without the Zeeman cou pling or SOCs. (e) Three figures from left to right: The\nenergy of the input electrons; Around B= 3T, the energy spectrum of the ring within the energy of the i nput electrons, i.e.\nE <250 meV; The transmission versus the energy of the input elec trons.\naccurate approximation for the real physical system.\nFig.1shows the two-lead transport device and an ex-\nample of the transport property of the ring. The lead is\n10 nm wide in the ydirection and is semi-infinite along\nthexaxis. We consider low-energy transport only, and\nthe input electronsare on the lowestenergyband allowed\nin the lead as shown in Fig. 1(d). In Fig. 1(e), we find\nthat the transmissions are only allowed when the energy\nof the incident electron is close to the energy levels of the\nring.\nPrevious studies have offered the possibilities that the\nquantumringwith SOCscanactasthespinfilterandpo-\nlarizer. We apply the NEGF method in a full 2D model\nquantum ring with SOCs, as shown in Fig. 1. More-\nover, we take all the realistic conditions, Zeeman cou-\npling, finite width of the ring and the rotational symme-\ntry breaking, into consideration. In fact, the 1D model\nstill works qualitatively. The spin filtering can be basi-\ncally explained by the transmission rates Tµin Eq. (1)\nvarying with the competition of the AB and the AC\nphases in different magnetic fields. In a proper magnetic\nfield and with a proper SOC, one channel can be shielded\nand the other one is fully survived, so that both T0and\nP0can be up to 1. Details can be found in the appendix.\nAs discussed in the 1D model, if only the Rashba SOC\nis present then TyandPywill be suppressed. The direc-\ntion of the spin polarizer can be tuned by the strength of\nthe Rashba SOC in the plane xOz. If only the Dressel-haus SOC is present, then TxandPxwill be suppressed.\nThe direction of the spin polarizer is then in the plane\nyOz. If both of the SOCs are present then the situation\nbecomes complex and the spin polarizer can be controled\nmorewidely. However, we find that if the outcoming spin\nneeds to be polarized well, then it is better to keep one\nSOC dominating the system. The competition of the two\nSOCs makes the spin more difficult to be polarized, as\nshown in the appendix.\nWe note that in such a simple device the spin filter-\ning and spin polarizer can be realized. The unpolarized\nspinistransportedthroughthesimplequantumringwith\nRashba SOC, and then the outcoming spin is polarized.\nThe directions other than the outcoming polarization are\nfiltered, and the current is spin polarized. Moreover, the\nRashbaSOCcan be easilytuned, sothat the polarization\nof the outcoming spin can be easily tuned by a gate.\nIV. ISOTROPIC ALL-ELECTRIC SPIN\nANALYZER\nNow we would like to consider the case when the in-\ncident electrons are already fully polarized. Similar to\nthe light polarizer, the ring with the SOCs in fact can\nbe acted as a spin analyzer. If the incident electron is\nalready spin polarized in the direction of ( θin,ϕin) in the\nspherical coordinate of the spin space, then the transmis-5\nsion rate is given by\nTα(E) = Tr/bracketleftbig\nσαΓ(E)G(E)σ(θin,ϕin)+Γ(E)G†(E)/bracketrightbig\n,(7)\nwhereσ(θin,ϕin)+is the density matrix of the polarized\nstate,the Green’sfunction Gandthe broadeningfunction\nΓ can be found in Ref. [41]. We note that the outcoming\nspin is still spin polarized, but is torqued by an angle\ngiven by Eq. ( 6).\nUsing Eq. ( 7), we can clearly decompose the unpo-\nlarized incident ψinin the basis of σz. In the density\nmatrix form, |ψin∝an}bracketri}ht∝an}bracketle{tψin|=/parenleftBig\n|ψin\nz↑∝an}bracketri}ht∝an}bracketle{tψin\nz↑|+|ψin\nz↓∝an}bracketri}ht∝an}bracketle{tψin\nz↓|/parenrightBig\n/2.\nThe incident wave function can be divided into two parts\nwith opposite spin polarization, and each part provides\na transport channel. The total transmission rate is the\nsum of the transmission rates of the two channels, since\nthere is no coherence between the two channels. In the\nappendix, we can clearly see how the spin textures and\nthe current evolve in the ring for different channels in\nwhich the spin is decomposed along + zor−z.\nA. Transmission rates for the polarized incident\nspin current\nWesupposethattheringisonlycoupledbytheRashba\nspin-orbitinteraction /planckover2pi1g1= 20nm·meVandtheincident\ncurrent is already spin polarized. The polarization direc-\ntion of the incident spin ( θin,ϕin) varies and the charge\ntransmission rate is indicated in Fig. 2. Fig.2(a) shows\nthe case when T1= 1,T2= 0 andP0= 1. The one-\ndimensional analytical model predicts that the outcom-\ningpolarizationisalongthe eigenstate χ1(0), (0.161π,π),\nand theχ2channel is closed T2= 0. In the two-\ndimensional model, it indicates that the outcoming po-\nlarization is along ( θout,ϕout) = (0.214π,0.984π) always,\nwhere the channel of χ1is free to transport and the other\nchannel (χ2) is completely closed. It means that the po-\nlarization angle of the eigenstate of the 2D ring at ϕ= 0\nis (0.214π,0.984π). This difference comes from the Zee-\nman effect and the finite width. It implies that these\neffects can also generate a finite Pyin the ring with the\nRashba SOC only, which is significantly differenti from\nthe 1D model.\nMoreover,the incident polarizationwith the maximum\ntransmission rate among all the directions in the spin\nspaceisalsoalongthe eigenstate χ1\n2D(π), (θmax\nin,ϕmax\nin) =\n(0.214π,0.016π). We note that ( θmax\nin,ϕmax\nin) and\n(θout,ϕout) are mirror symmetry to the zaxis.\nThe transmission of the spin-polarized input current\nin arbitrary direction is determined by the projection of\n(θin,ϕin) to (θmax\nin,ϕmax\nin), since the channel of χ2\n2Dis\nclosed. For a more general case, both transport channels\nof the eigensates allow electrons to pass ( Tmax\n0,Tmin\n0>\n0), as shown in Figs. 2(b) and (c), the maximum trans-\nmission rate Tmax\n0= 0.962 corresponds to the incident\npolarization ( θmax\nin,ϕmax\nin) = (0.792π,0.963π), and its\noutput polarization is ( θmax\nout,ϕmax\nout) = (0.792π,0.037π).For the minimum transmission rate, we have Tmin\n0=\n0.591, (θmin\nin,ϕmin\nin) = (0.208π,1.963π), (θmin\nout,ϕmin\nout) =\n(0.792π,1.037π). In this case, ( θout,ϕout) is no longer\na fixed angle along χ1\n2D, but changes with the angle of\nincidence (θin,ϕin).\nInterestingly, we also find numerically that the gen-\neral relation between the incident angle and the charge\ntransmission rates T0is given by\nT0=Tmax\n0cos2/parenleftbiggθ∆\n2/parenrightbigg\n+Tmin\n0sin2/parenleftbiggθ∆\n2/parenrightbigg\n,(8)\nwhereθ∆is the angle between the incident spin po-\nlarization ( θin,ϕin) and the special angle ( θmax\nin,ϕmax\nin),\nwhether the outcoming spin is polarized or not. This\nequation is exactly the same as Eq. ( 3) that we found for\nthe 1D model. The only difference is that in Eq. ( 3),T1,2\ncorrespond to the transmission rate of the eigenstates of\ntheringχ1,2. However,inthe2Dringthe Tmax\n0andTmin\n0\ncorrespond to the eigenstates of the 2D ring which are a\nlittle different from those of the 1D ring. The arbitrary\nspin is projected to the angles of the eigenstates of the\nring, (θmax\nin,ϕmax\nin) and(π−θmax\nin,ϕmax\nin+π). This projec-\ntion then gives directly the transmission rate in Eqs. ( 3)\nand (8). The Zeeman coupling, circle symmetry break-\ning, and finite width only change the spin-polarization\ndirection of eigenstates χµ, the properties predicted by\nthe 1D analytical model are retained, which implies that\nwe could use the quantum ring to design the integrable\nspin devices.\nB. Design of a spin analyzer\nThe ring acts as a spin torque: it allows the electron\nto pass but the spin polarization must be torqued. If the\nring is coupled by the Dresselhaus spin-orbit interaction\nonly, the similar spin torque occurs. The only difference\nis the outcoming angle of the spin, which is the mirror\nsymmetry of the incident angle (for the maximum trans-\nmission rate only) to the plane xOz. The direction de-\npendent transmission rate is also given by Eqs. ( 3) and\n(8).\nAccording to the property of the angle dependent\ntransmission rate in Eq. ( 8), we can realize a spin ana-\nlyzerintheringdevice. Beforethemeasurement,weneed\nto knowTmax\n0andTmin\n0in a given magnetic field. They\ncan be determined by the measurement of the transmis-\nsion rates of the known spin polarized incidents. We\nuse three spin polarized incident with Px,y,z= 1, re-\nspectively, and one spin unpolarized incident to identify\nthe following parameters: Tmax\n0,Tmin\n0,θmax\nin,ϕmax\nin. The\ntransmission rate for the unpolarized incident is marked\nasT, and we have already known T=Tmax\n0+Tmin\n0. The\nthree transmission ratesfor different spin polarizationin-\ncident are marked as T(x),T(y) andT(z). Applying Eq.\n(8) toT(x),T(y) andT(z), we find another three equa-\ntions. So four equations in all can be solved and the four\nvariablesTmax\n0,Tmin\n0,θmax\nin,ϕmax\ninare found.6\n/uni00000013/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000028c /uni00000014/uni00000011/uni00000013/uni0000028c\n/uni00000010/uni00000014/uni00000011/uni00000013/uni00000013/uni0000028c/uni00000010/uni00000013/uni00000011/uni00000019/uni0000001a/uni0000028c/uni00000010/uni00000013/uni00000011/uni00000016/uni00000016/uni0000028c/uni00000013/uni00000011/uni00000013/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000016/uni00000016/uni0000028c/uni00000013/uni00000011/uni00000019/uni0000001a/uni0000028c/uni00000014/uni00000011/uni00000013/uni00000013/uni0000028c\n/uni00000013/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000028c /uni00000014/uni00000011/uni00000013/uni0000028c\n/uni00000013/uni00000011/uni00000013/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000014/uni0000001a/uni0000028c/uni00000013/uni00000011/uni00000016/uni00000016/uni0000028c/uni00000013/uni00000011/uni00000018/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000019/uni0000001a/uni0000028c/uni00000013/uni00000011/uni0000001b/uni00000016/uni0000028c/uni00000014/uni00000011/uni00000013/uni00000013/uni0000028c\n/uni00000013/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000028c /uni00000014/uni00000011/uni00000013/uni0000028c\n/uni00000013/uni00000011/uni00000018/uni0000001c/uni00000013/uni00000011/uni00000019/uni00000018/uni00000013/uni00000011/uni0000001a/uni00000014/uni00000013/uni00000011/uni0000001a/uni0000001b/uni00000013/uni00000011/uni0000001b/uni00000017/uni00000013/uni00000011/uni0000001c/uni00000013/uni00000013/uni00000011/uni0000001c/uni00000019\nθin θin θinϕout θout T0\n(d) (c) (b)B= 1.500T\n¯hg1= 33.5nm·meV\n¯hg2= 0nm ·meV\n/uni00000013/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000028c /uni00000014/uni00000011/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000018/uni0000028c/uni00000014/uni00000011/uni00000013/uni0000028c/uni00000014/uni00000011/uni00000018/uni0000028c/uni00000015/uni00000011/uni00000013/uni0000028c\n/uni00000013/uni00000011/uni00000013/uni00000013/uni00000013/uni00000011/uni00000014/uni0000001a/uni00000013/uni00000011/uni00000016/uni00000016/uni00000013/uni00000011/uni00000018/uni00000013/uni00000013/uni00000011/uni00000019/uni0000001a/uni00000013/uni00000011/uni0000001b/uni00000016/uni00000014/uni00000011/uni00000013/uni00000013\nθinT0ϕin(a)B= 2.408T\n¯hg1= 33.5nm·meV\n¯hg2= 0nm ·meV\nFIG. 2: (color online). The energy of the incident electron i sEin= 198.5 meV. (a) The total transmission rate T0for different\nangles of the polarization of the incident electron ( θin,ϕin) when/planckover2pi1g1= 33.5 nm·meV and g2= 0 atB= 2.408 T. (b) The total\ntransmission rate T0and (c)-(d) the angles of the polarization of the outgoing el ectron (θout,ϕout) for different polarization of\nthe incident electron ( θin,ϕin) atB= 1.5 T.\nThe scenario to analyze the spin polarization by de-\ntecting the charge transmission rates for different SOCs\nthen can be established. The scheme is described as fol-\nlows:\nFirst, the ring is coupled by the two spin-orbit interac-\ntions (g1,g2). The transmission rates are shown in colors\nin Fig.3(a).Tmax\n0and the corresponding incident po-\nlarization ( θmax\nin,ϕmax\nin)1is already known, as the blue\nvector in Fig. 3(d). Once we measure the transmission\nrateT0, we can find the angle θ∆1between the incident\npolarization angle ( θin,ϕin) and (θmax\nin,ϕmax\nin)1by apply-\ning Eq. ( 8). However, the possible polarization direction\nin the three-dimensional space of the spin can be along\nany element of the cone, shown as the blue circle in Fig.\n3(d). we project the angles of the elements of the cone\nonto the (θ,ϕ) plane to obtain the solid line in Fig. 3(a).\nSecond, we tune the Rashba SOC and the transmis-\nsion rates are shown in colors in Figs. 3(b). The angle of\nthe maximum transmission rate, ( θmax\nin,ϕmax\nin)2, is repre-\nsented by the green vector in Fig. 3(d). Then measure\nthe transmission rate to obtain the angle θ∆2to find the\nsecond cone. The spin polarization is possibly located in\nthe solid green line in Fig. 3(b), where the dashed line\nrepresents the first measurement. So the incident polar-\nization must be at one of the intersection points of the\ntwo lines.\nThirdly, we tune the Rashba SOC again to find the\nthird line which is shown in Fig. 3(c). The three lines\nmust intersect at the same point which is the unique di-\nrection of the polarization of the incident spin. The in-\ntersection point can also be seen in the spin space in Fig.\n3(d).\nHere the external magnetic field is fixed and can be\nintegrated on the chip. In fact, the three curves in the\n(θ,ϕ) plane always intersect at the same point for anymagnetic field. A proper magnetic field results in better\ndiscrimination.\nWe note that the spin analyzer could be also achieved\nby a single SOC. In the 1D model, a single SOC only\ntwists the spin in one direction ( xory). The incident\nangle can not be uniquely determined, there are always\ntwo intersection points, no matter how many times we\ntune the strength of the SOC. However, in the real 2D\nring, the spin can be twisted more widely. The unique\nintersection can appear. We show the numerical results\nin Fig.3(e) where only the Rashba SOC exists and the\nDresselhaus SOC is absent. It is clear that after three\nmeasurements with different strengthes of the Rashba\nSOC, all the cones intersect at the unique intersection\nand the other intersection has been lifted. So the spin\npolarization can also be identified more easily.\nV. CONCLUSION\nIn summary, we present a detailed study of the trans-\nport properties of the device in which a quantum ring is\nin contact with two leads at the ends of one diameter.\nWhen the SOC is introduced, different phases are added\nin the matter wave of the electron with different spins.\nSo that the transmission rates for different spins are no\nlonger degenerate. By detailed analytical and numerical\nstudies, we find that in a simple quantum ring device, the\nspin unpolarized current can be spin polarized parallel to\nthe eigenstates of the ring for appropriate SOC and the\nmagnetic field. The direction of the polarization can be\ntuned easily by the SOC and the magnetic field as well.\nThis simple device is therefore proposed to be a spin po-\nlarizer. Moreover, similar to the light polarizer/analyzer,\nit can also be designed as an omnidirectional all-electric7\n(d)\nxyz\n/uni00000013/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000028c /uni00000014/uni00000011/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000013/uni0000028c/uni00000014/uni00000011/uni00000013/uni0000028c/uni00000015/uni00000011/uni00000013/uni0000028c\n/uni00000013/uni00000011/uni0000001a/uni00000015/uni00000013/uni00000011/uni0000001a/uni00000018/uni00000013/uni00000011/uni0000001a/uni0000001c/uni00000013/uni00000011/uni0000001b/uni00000016/uni00000013/uni00000011/uni0000001b/uni00000019/uni00000013/uni00000011/uni0000001c/uni00000013/uni00000013/uni00000011/uni0000001c/uni00000017\n¯hg1= 40nm ·meVT0\nθinϕin(c)B= 1.8T ¯hg2= 20nm ·meV\n/uni00000013/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000028c /uni00000014/uni00000011/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000013/uni0000028c/uni00000014/uni00000011/uni00000013/uni0000028c/uni00000015/uni00000011/uni00000013/uni0000028c\n/uni00000013/uni00000011/uni0000001b/uni00000016/uni00000013/uni00000011/uni0000001b/uni00000019/uni00000013/uni00000011/uni0000001b/uni0000001b/uni00000013/uni00000011/uni0000001c/uni00000014/uni00000013/uni00000011/uni0000001c/uni00000016/uni00000013/uni00000011/uni0000001c/uni00000019/uni00000013/uni00000011/uni0000001c/uni0000001b\n¯hg1= 10nm ·meVT0\nθinϕin(b)B= 1.8T ¯hg2= 20nm ·meV\n/uni00000013/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000028c /uni00000014/uni00000011/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000013/uni0000028c/uni00000014/uni00000011/uni00000013/uni0000028c/uni00000015/uni00000011/uni00000013/uni0000028c\n/uni00000013/uni00000011/uni0000001b/uni00000014/uni00000013/uni00000011/uni0000001b/uni00000017/uni00000013/uni00000011/uni0000001b/uni0000001a/uni00000013/uni00000011/uni0000001c/uni00000013/uni00000013/uni00000011/uni0000001c/uni00000016/uni00000013/uni00000011/uni0000001c/uni00000019/uni00000013/uni00000011/uni0000001c/uni0000001c\n¯hg1= 0nm·meVT0\nθinϕin(a)B= 1.8T ¯hg2= 20nm ·meV\n(f)\nxyz\n/uni00000013/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000028c /uni00000014/uni00000011/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000013/uni0000028c/uni00000014/uni00000011/uni00000013/uni0000028c/uni00000015/uni00000011/uni00000013/uni0000028c\n¯hg1= 10nm ·meV\n¯hg1= 25nm ·meV\n¯hg1= 35nm ·meV\nθinϕin(e)B= 1.8T ¯hg2= 0nm·meV\nFIG. 3: (color online). The energy of the incident electron i sEin= 198.5 meV, and the background magnetic field B= 1.8\nT. (a)-(d) The colors represent the transmission rates for t he SOCs /planckover2pi1(g1,g2) = (0,20),(10,20),(40,20) nm·meV, respectively.\nThe solid curve in (a) - (d) represents the possible angles of the incident polarization after the first, second and the thi rd\nmeasurements, respectively. The dash lines in (b) and (c) re presents the previous measurements. (d) The possible polar ization\nis a cone for each measurement in the spin space. The blue, gre en and red arrows stand for the vector ( θmax\nin,ϕmax\nin)1,2,3,\nrespectively. The intersection of the three cones is the pur ple vector representing the incident polarization. (e)-(f ) The\nintersections of the possible angles in three Rashba SOCs /planckover2pi1g1= 10,25,35 nm·meV, respectively, when the Dresselhaus is\nabsent.\nspin analyzer by simply measuring the transmission rate\nof the polarized incident via Eq. ( 8). These findings pave\nthe way to control the system in spintronics and may be\nuseful in quantum computation. It also contributes an\neasy and controllable proposal to the design of the high-\nperformance all-electric transport device.\nVI. ACKNOWLEDGEMENT\nThis workis supported by the NSF-China under Grant\nNo. 11804396. J.S. is supported by the NSF-China un-\nder Grant No. 11804397. A.G. acknowledges financial\nsupport by the NSF-China under Grant No. 11874428,\n11504066, and the Innovation-Driven Project of Central\nSouth University (CSU) (Grant No. 2018CX044). F.O.\nacknowledges financial support by the NSF-China under\nGrant No. 51272291, the Distinguished Young Scholar\nFoundation of Hunan Province (Grant No. 2015JJ1020),\nand the CSU Research Fund for Sheng-hua scholars\n(Grant No. 502033019). T.C. would like to thank Jun-\nsaku Nitta for helpful discussion, in particular, for point-\ning out Ref. [28].Appendix A: Method and Formalism\nHere we explicitly derive the transmission rate in\nEq. (1). We consider electrons as plane waves in the\ntwo leads with momentum /planckover2pi1kand energy E=/planckover2pi12k2\n2m∗.\nNote that there is an additional potential U0=−β2\n1in-\nduced by the Rashba SOC [38], so that in the two arms,\nE=/planckover2pi12k2\n0\n2m∗+U0. The wave vector in the two arms are [12]\nkµ\nj=k0+j/parenleftBig\nΦAB\n2πr0+Φµ\nAC\n2πr0/parenrightBig\n.The incident current can be\ndecomposed into the two eigenstates χ1,2, and the elec-\ntron is transported by this two channels. We note that\nin this case the two channels are independent and there\nis no interference between the two eigenstates. If the\nincident spin is polarized along χµthen the outcoming\npolarization is still along χµ. If the incident current is\ndecomposed into other two orthogonal states, then the\ninterference is difficult to deal with.\nThe wave function at the left lead contains the inci-\ndent and the reflection, Ψµ\nin. The wave function of the\nupper arm also contains two parts, the clockwise and the\nanticlockwise movements, Ψµ\n1. In the same manner, the\nwave function of the lower arm is Ψµ\n2. The output wave\nfunction is marked Ψµ\nout. All these wave functions are8\ngiven by\nΨµ\nin=/parenleftbig\neikx+rµe−ikx/parenrightbig\nχµ(π),(S1)\nΨµ\n1=/summationdisplay\njCjejikµ\njxχµ(φ), (S2)\nΨµ\n2=/summationdisplay\njDjejikµ\n−jxχµ(φ), (S3)\nΨµ\nout=tµeikxχµ(0), (S4)\nwhererµis the reflection rate, C,Dare the parameters\nwhich can be determined by the continuous condition,\nandtµis the variable characterizing the transport prop-\nerties of the device. The transmission rate is thus given\nbyTµ=|tµ|2. By applying the Griffith boundary condi-\ntions [15, 19–21, 23, 24, 26], the wave functions and the\ncurrents must be continuous at the two leads ( x=±r0\norθ= 0,π), we obtain six equations to solve the six vari-\nables. Among them, the most wanted transmission rate\ncan be solved,\nTµ=KΦµ\nKK′+/bracketleftbig\n4k2\n0(Φµ−K′)+k2sin2(πk0r0)/bracketrightbig2,(S5)\nwhich is shown as Eq. (1).\nIn order to calculate the transport properties numeri-\ncally, it is convenient to discretize the continuous Hamil-\ntonian. We discretize Hon the sites of a square lattice\nwith the lattice constant ato obtain the tight binding\nHamiltonian. It is obtained by calculating the matrix el-\nements in the basis of position. The tight binding Hamil-\ntonian is given by\nH=/summationdisplay\ni/parenleftbigg\nVi+4t+∆\n2σz/parenrightbigg\nc†\nici−/summationdisplay\n/angbracketlefti,j/angbracketright(t+sij)c†\nicjeiθij,\n(S6)\nt=/planckover2pi12\n2m∗a2, (S7)\nsij=−i/planckover2pi1g1\n2a2/parenleftbig\nσx∆y−σy∆x/parenrightbig\n−i/planckover2pi1g2\n2a2/parenleftbig\nσy∆y−σx∆x/parenrightbig\n,\n(S8)\nθij=e\n/planckover2pi1(Axi∆x+Ayi∆y), (S9)\nwhereiruns over all sites, ∝an}bracketle{ti,j∝an}bracketri}htrepresents the nearest\nneighbouring hopping only, xiandyiare thexandy\ncoordinates of site i, and ∆xij=xj−xi, ∆yij=yj−yi.\nFor convenience, we apply a hard-wall potential instead\nof the parabolic potential,\nVi=/braceleftBigg\n0|ri−r0|/lessorequalslantrw\n∞ |ri−r0|>rw, (S10)\nwhereri=/radicalbig\nx2\ni+y2\niand the width of the ring is rw. We\nconnect two parallel leads to the ring, then the transmis-\nsion properties can be obtained by using the nonequilib-\nrium Green’s function (NEGF).It is worthwhile to note that the continuous model and\nthe tight binding model are compatible and all the ob-\nservable quantities in these two models are almost equal\n(the small errors vanish when a→0). Moreover, the en-\nergy spectrum has no essential difference in a parabolic\npotentialfromthatinahard-wallpotential, if rwmatches\nthe confinement /planckover2pi1ω. Then we consider the transport\nproperties in such a lattice model with tight-binding\nHamiltonian. The spin transmission rate Tof the elec-\ntron transporting from the left lead to the right lead is\ndefined by using the NEGF method [6, 41],\nTα(E) = Tr{σα[ΓR(E)GRL(E)ΓL(E)G†\nLR(E)]},(S11)\nwhereα∈ {x,y,z},σx,y,zare the Pauli matrices and σ0\nis the unit matrix. The Green’s function is defined by\nthe projection of the full Green’s function [41],\nGRL=PRGPL, (S12)\nG(E) = (E−H−ΣR−ΣL)−1,(S13)\nwherePR,PLare the projection operators to the right\nand the left leads, Σ R,ΣLare the self-energy of the right\nand the left leads, respectively. The broadening function\nis defined by Γ j=i/bracketleftBig\nΣj−Σ†\nj/bracketrightBig\n.\nTα(E) is the transmission rate of the ( α∈ {x,y,z})\ncomponent of the spin or the total charge transmission\n(α= 0) while the energy of the incident electron is E.\nThen the spin polarization rate Pis defined as:\nPα=Tα\nT0×100%,α∈ {x,y,z} (S14)\nandP0=/radicalBig\nT2x+T2y+T2z/T0=/radicalBig\nP2x+P2y+P2z.P0\nrepresents the spin polarization of the outcoming elec-\ntron. IfP0= 1, then the spin is fully polarized. If\nP0= 0, the spin is fully unpolarized.\nBy diagonalizing the tight-binding Hamiltonian in Eq.\n(S6), we can have the value of the wave functions at each\nsite,ψ(ri), which is a two component spinor. The phys-\nical quantities can then be obtained. The spin fields are\ncalculated by\nσα(ri) =ψ†(ri)σαψ(ri), (S15)\nand the density is given by n(ri) =ψ†(ri)ψ(ri). The\naverage value of the observable quantity is thus given by\n∝an}bracketle{tA∝an}bracketri}ht=/summationtext\niψ†(ri)Aψ†(ri)∆x∆y. The in-plane field can\nbe described by the vector field σ(r) = (σx(r),σy(r)).\nThe current operators can be derived by jµ=−δH\nδA, so\nthat the on-site current densities are given by\njx(ri) =e\n2m∗/bracketleftBig\nψ†(ri)Pxψ(ri)+(Pxψ(ri))†ψ(ri)/bracketrightBig\n−eψ†(ri)(g1σy+g2σx)ψ(ri), (S16)\njy(ri) =e\n2m∗/bracketleftBig\nψ†(ri)Pyψ(ri)+(Pyψ(ri))†ψ(ri)/bracketrightBig\n+eψ†(ri)(g1σx+g2σy)ψ(ri). (S17)9\n/uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017/uni00000013/uni00000011/uni00000019/uni00000013/uni00000011/uni0000001b/uni00000014/uni00000011/uni00000013(b)\n¯hg1= 20nm ·meV\n¯hg2= 0nm·meV\nE= 198.5meVBmaxT\nB′\nmaxT\nBminT\nB′\nminT\n/uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017/uni00000013/uni00000011/uni00000019/uni00000013/uni00000011/uni0000001b/uni00000014/uni00000011/uni00000013(a)\n¯hg1= 0nm·meV\n¯hg2= 0nm·meV\nE= 198.5meV\nTz↑Tz↓BmaxT\nBminT\n/uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017/uni00000013/uni00000011/uni00000019/uni00000013/uni00000011/uni0000001b/uni00000014/uni00000011/uni00000013(c)\n¯hg1= 0nm·meV\n¯hg2= 20nm ·meV\nE= 198.5meV\n/uni00000014/uni00000013/uni00000013 /uni00000014/uni00000018/uni00000013 /uni00000015/uni00000013/uni00000013 /uni00000015/uni00000018/uni00000013/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013(e)\n¯hg1= 0nm·meV\n¯hg2= 20nm ·meV\nB= 2.67T\n/uni00000014/uni00000013/uni00000013 /uni00000014/uni00000018/uni00000013 /uni00000015/uni00000013/uni00000013 /uni00000015/uni00000018/uni00000013/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013TxTyTzT0\n(d)\n¯hg1= 20nm ·meV\n¯hg2= 0nm·meV\nB= 2.67T∆BmaxT=B′\nmaxT−BmaxT\n∆BminT=BminT−B′\nminT\n∆BT=B′\nminT−B′\nmaxTB[T] B[T]\nB[T]\nE[meV]TT\nE[meV]T\nFIG. S1: (color online). Tz↑andTz↓(a) without SOC, (b)\nwith the Rashba SOC only, and (c) with the Dresselhauss\nSOC only. Tαα∈(x,y,z) (d) with the Rashba SOC only\nand (e) with the Dresselhauss SOC only.\nThe current is contributed by three parts,\njα(ri)≡jz↑,α(r)+jz↓,α(r)+jSOC,α(r),(S18)\nwhere\njz↑,α(ri) =e\n2m∗/bracketleftbig\nψ∗\n↑Pαψ↑+(Pαψ↑)∗ψ↑/bracketrightbig\n,(S19)\njz↓,α(ri) =e\n2m∗/bracketleftbig\nψ∗\n↓Pαψ↓+(Pαψ↓)∗ψ↓/bracketrightbig\n,(S20)\njSOC,x(ri) =−eψ†(g1σy+g2σx)ψ, (S21)\njSOC,y(ri) =eψ†(g1σx+g2σy)ψ. (S22)\nThe on-site wave function spinor is ψ=/parenleftbigψ↑ψ↓/parenrightbigT, and\n↑,↓are related to the eigenstates of the spin operator σz.\nAppendix B: Spin transmissions in different SOCs –\nSpin filtering and spin polarizer\nWestudyindetailshowthetransmissionrateisrelated\nto the magnetic field and the SOCs. We suppose that\nthe input electrons are spin unpolarized. If there is no\nSOC, the transported electrons are spin unpolarized as\nwell, i.e.,Tz↑=Tz↓forg= 0. However, the Zeeman\ncoupling makes the transmission rate different, especially\nin a strong magnetic field, as shown in Fig. S1(a). Since\nthe minimum transmission rates for spin up and down\nare all located at the same magnetic field, it would bedifficult to suppress one spin to zero and keep the other\nspin finite.\nIf the SOCs are introduced into the system, we find\nthat the transmission rate curves for spin down and spin\nup arewell separated. Ifonly the RashbaSOC is present,\nthe curve of Tz↓is shifted left and the curve of Tz↑is\nshifted right as shown in Fig. S1(b). If only the Dressel-\nhaus SOC exists, the shift of the curves is just opposite\nto that in a Rashba ring, as shown in Fig. S1(c).\nIf there is no magnetic field, the electron transports\nthrough the upper arm and the lower arm with the same\nphase added ( T1=T2), so that the transmission rates for\ndifferent spins depend on the ring itself and are equal, as\nshowninFigs. S1(b)and(c). However,infinitemagnetic\nfields the time-reversal symmetry is broken, the spin de-\ngeneracy in the ring will be lifted more in the presence of\neither Rashba or Dresselhas SOC strongly. Such a com-\nbination of the magnetic field and the SOCs can lead to\nsignificant spin filtering effect.\nIn the numerical curves (Figs. S1(b) and (c)), the spin\nfiltering appears periodically in a magnetic field, since\nthe term Φ AB+ΦACin transmission rate Eq. (1) only\ndepends on the magnetic field. For instance, if only the\nRashba SOC is present and the energy of the incident\nelectron is Ein= 198.5 meV in Fig. S1(b), the lowest\nmagnetic field where the spin down is suppressed is at\nB= 2.67T, which can be further lowered by increasing\nthe radius of the ring (at the same magnetic flux). In\nthis case,Tz↓→0 andTz↑is finite, so that the output\nelectrons are almost polarized to spin up, Pz→1. For\ndifferent energies of the input electrons, the transmission\nrates are shown in Figs. S1(d) and (e). The negative Ti\nrepresentsthe icomponentofthe outputspinispolarized\nin the negative direction of the iaxis. In fact, according\ntoFig.S1(d)andtheanalysisofthe1Dmodel, theoutput\nspin is polarized along χ1,2between the zand−xaxis,\nsinceTy≈0 andTx,zare finite.\nWe now compare the transmission curves of the ring\nwithout SOC(Fig. S1(a)) andthe ringwith RashbaSOC\n(Fig.S1(b)). Thefirstmaximumratefor Tz↑isatBmaxT\nin the ring without SOC. After the Rashba SOC is set in,\nboth of the Tz↑andTz↓transmission rates are shifted.\nWesupposethatthe firstmaximumvalueof Tz↑isshifted\ntoB′\nmaxT. In the same manner, the first minimum rate\nin the ring without SOC is at BminT= 2.83T, while the\nfirst minimum rate Tz↓is shifted to B′\nminT. We define\nthe parameters ∆ BmaxT=B′\nmaxT−BmaxT, ∆BminT=\nBminT−B′\nminT, and ∆BT=B′\nminT−B′\nmaxTto study\nhow the Rashba SOC shifts the transmission rate curve\nand changes the polarization of the spin.\nWe show that in Fig. S2(a) ∆BTdecreases with the\nincrease of the Rashba SOC g1, due to the change of the\nAC phase Φµ\nAC, just as predicted in the 1D model. When\n/planckover2pi1g1= 33.5 nm·meV, ∆BT= 0, which means that the\ntransmission rate of spin down is suppressed to minimum\nand the transmission rate of spin up is maximum. Inter-\nestingly, at this point the total charge transmission rate\nis exactly 1 as shown in Fig. S2(b). Hence, χ1is com-10\n/uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000019/uni00000013/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013(b)\nTx\nTy\nTz\nT0\n/uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000019/uni00000013/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013(a)\n∆BminT\n∆BmaxT\n∆BT\n/uni00000014/uni0000001a/uni00000018 /uni00000015/uni00000013/uni00000013 /uni00000015/uni00000015/uni00000018/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013(e)\nPx\nPy\nPz\nP0\n/uni00000014/uni0000001a/uni00000018 /uni00000015/uni00000013/uni00000013 /uni00000015/uni00000015/uni00000018/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013(d)\nTx\nTyTz\nT0\n/uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000019/uni00000013/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013\nPx\nPy\nPz\nP0(c)¯hg1[nm·meV] ¯hg1[nm·meV]\nE[meV] E[meV]∆B[T]\n¯hg1[nm·meV]\nT TorPP\nFIG. S2: (color online). (a) Magnetic field shifts ∆ Binduced\nby the Rashba SOC. (b) Spin transmission rates Tand (c)\nspin polarization Pin different g1with incident electron en-\nergy 198.5 meV, in the magnetic field where the minimum Tz↓\nis, so that P0= 1 always. (d) The spin transmission rate T\nand (e) the spin polarization Pat/planckover2pi1g1= 33.5 nm ·meV and\nB=2.41 T (corresponding to ∆ BT= 0) with energy.\npletely suppressed, and χ2passes the ring freely. Mean-\nwhile, theycomponent of the spin is almost suppressed,\nTy≈0, and both |Tx|and|Px|increase, as shown in\nFigs.S2(b) and (c). We are then able to control the\ndirection of the spin polarization by tuning the Rashba\nSOC. The tunable spin polarizer is thereby established.\nIn Fig.S2(d), we show how the spin polarization and the\ntransmission rate vary with the energy of the incident\nelectron. Basically, Tyis close to 0 and Pyis also always\nvery small. It is nonzero comparing with the 1D model,\ndue to the Zeeman coupling and the width of the ring.\nIn Figs.S2(d) and (e) we show that if the energies of the\nincident electrons are in the region [185 ,205] meV, the\nspin transmission and polarization are stable. So that\nthe outcoming current which is obtained by integral of\nthe transmission rate Tover this region is almost fully\nspin polarized. In order to exclude the unwanted trans-\nmission below 185meV, we can apply a gate to lift the\nwhole energy band of the lead.\nInourringdevice, theRashbaSOCtilts the spintothe\nxaxis, while the Dresselhaus SOC flip the spin towards\ntheydirection. It can also be understood simply as fol-\nlows. When the magnetic field is absent, the effective\nvector potential induced by the SOCs is\nASOC\nx=−m\ne/planckover2pi1(g1σy+g2σx), (S1)\nASOC\ny=m\ne/planckover2pi1(g1σx+g2σy). (S2)\nSuppose the incident wave function is spin polarized,\nψin\n+= (1 0)T, then the outcoming wave function influ-\nenced by the SOC is given by ψout∝e−iAx·2(r0+rw)ψin,since the coordinate difference in the ydirection is zero.\nIf there is only Rashba existing,\nψout\nR∝(1+iγg1σy)/parenleftbigg\n1\n0/parenrightbigg\n=/parenleftbigg\n1\n−γ/parenrightbigg\n,(S3)\nwhereγ= 2(r0+rw)m\ne/planckover2pi1g1>0. So we have ∝an}bracketle{tσx∝an}bracketri}ht=\n−2γ <0 and∝an}bracketle{tσy∝an}bracketri}ht= 0. The spin is torqued from\nthe +zdirection to −x. If the incident electron is spin\ndown,ψin\n−= (0 1)T, thenψout\nR= (γ1)T, and then\n∝an}bracketle{tσx∝an}bracketri}ht= 2γ >0 and∝an}bracketle{tσy∝an}bracketri}ht= 0. In Fig. S2(d), however\nspin down is suppressed in the transport, and the spin\nupψin\n+is flipped to the −xaxis. On the other hand, if\nonly the Dresselhaus SOC is present, we can do the same\ncalculation. For ψin\n+,ψout\nD= (1iγ)T, so that ∝an}bracketle{tσx∝an}bracketri}ht= 0\nand∝an}bracketle{tσy∝an}bracketri}ht= 2γ >0. Forψin\n−,ψout\nD= (iγ1)T, so that\n∝an}bracketle{tσx∝an}bracketri}ht= 0 and ∝an}bracketle{tσy∝an}bracketri}ht=−2γ <0. In Fig. S2(e), the spin up\nis suppressed, while the spin down ψin\n−is flipped to the\n−yaxis in the transport. The analysis agrees with the\nnumerical results perfectly.\nAs drawn in Fig. S3, we show the relation between\nthe spin transmission rates and the spin polarizations\nfor different SOCs. If only the Rashba SOC is existing,\nthenTyandPywill be suppressed shown in Fig. S3(a).\nThe direction of the spin polarizer can be tuned by the\nstrength of the Rashba SOC in the plane xOz. If only\nthe Dresselhaus SOC is present, then TxandPxwill be\nsuppressedshowninFig. S3(b). Thedirectionofthespin\npolarizer is then in the plane yOz. If both of the SOCs\nare present, then the situation becomes complicated and\nspin polarizer can be controled more widely, as shown in\nFig.S3(c). However, we find that if the outcoming spin\nneeds to be polarized well, then it is better to keep one\nSOC dominating the system. The competition of the two\nSOCs makes the spin more difficult to be polarized.\nAppendix C: Spin textures and current in the\ntransport\nThe incident spin is supposed to be unpolarized, so\nthat the wave function of the incident electrons ψin\ncan be decomposed to two parts in any direction of\nthe spin polarization. Without the loss of generality,\nwe decompose the incident electron in the basis of σz,\n|ψin\nz↑|2=|ψin\nz↓|2. The spins of the two parts are indepen-\ndently polarized along zor−zdirection, respectively.\nFor each part of the incident electron, it contributes one\ntransmission channel in the transport. Then we can fig-\nure out which channel plays more important role in the\ntransport. The wave function of the incident electron is\nsupposed to be the wave function of the lowest band of\nthe lead. By employing Eq. (7) we can obtain the wave\nfunction in the ring by the Green’s function method,\nψring=GτLψin\nz↑(↓), (S1)\nwhereτLis the coupling matrix between the incident\n(left) lead and the ring[41]. We againemploy the current11\n/uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019/uni0000001a/uni00000018/uni00000014/uni00000013/uni00000013/uni00000014/uni00000015/uni00000018/uni00000014/uni00000018/uni00000013/uni00000014/uni0000001a/uni00000018/uni00000015/uni00000013/uni00000013/uni00000015/uni00000015/uni00000018/uni00000015/uni00000018/uni00000013\n/uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019\n/uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019\n/uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019\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\n/uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019/uni0000001a/uni00000018/uni00000014/uni00000013/uni00000013/uni00000014/uni00000015/uni00000018/uni00000014/uni00000018/uni00000013/uni00000014/uni0000001a/uni00000018/uni00000015/uni00000013/uni00000013/uni00000015/uni00000015/uni00000018/uni00000015/uni00000018/uni00000013\n/uni00000013/uni00000011/uni00000013/uni00000013/uni00000013/uni00000011/uni00000016/uni00000016/uni00000013/uni00000011/uni00000019/uni0000001a/uni00000014/uni00000011/uni00000013/uni00000013/uni00000014/uni00000011/uni00000016/uni00000016/uni00000014/uni00000011/uni00000019/uni0000001a/uni00000015/uni00000011/uni00000013/uni00000013\n/uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019/uni0000001a/uni00000018/uni00000014/uni00000013/uni00000013/uni00000014/uni00000015/uni00000018/uni00000014/uni00000018/uni00000013/uni00000014/uni0000001a/uni00000018/uni00000015/uni00000013/uni00000013/uni00000015/uni00000015/uni00000018/uni00000015/uni00000018/uni00000013\n/uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019\n/uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019\n/uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019\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\n/uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019/uni0000001a/uni00000018/uni00000014/uni00000013/uni00000013/uni00000014/uni00000015/uni00000018/uni00000014/uni00000018/uni00000013/uni00000014/uni0000001a/uni00000018/uni00000015/uni00000013/uni00000013/uni00000015/uni00000015/uni00000018/uni00000015/uni00000018/uni00000013\n/uni00000013/uni00000011/uni00000013/uni00000013/uni00000013/uni00000011/uni00000016/uni00000016/uni00000013/uni00000011/uni00000019/uni0000001a/uni00000014/uni00000011/uni00000013/uni00000013/uni00000014/uni00000011/uni00000016/uni00000016/uni00000014/uni00000011/uni00000019/uni0000001a/uni00000015/uni00000011/uni00000013/uni00000013\n/uni00000013/uni00000017/uni0000001b/uni00000014/uni00000015/uni00000014/uni00000019/uni00000015/uni00000013/uni0000001a/uni00000018/uni00000014/uni00000013/uni00000013/uni00000014/uni00000015/uni00000018/uni00000014/uni00000018/uni00000013/uni00000014/uni0000001a/uni00000018/uni00000015/uni00000013/uni00000013/uni00000015/uni00000015/uni00000018/uni00000015/uni00000018/uni00000013\n/uni00000013/uni00000017/uni0000001b/uni00000014/uni00000015/uni00000014/uni00000019/uni00000015/uni00000013\n/uni00000013/uni00000017/uni0000001b/uni00000014/uni00000015/uni00000014/uni00000019/uni00000015/uni00000013\n/uni00000013/uni00000017/uni0000001b/uni00000014/uni00000015/uni00000014/uni00000019/uni00000015/uni00000013\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\n/uni00000013 /uni00000017 /uni0000001b/uni00000014/uni00000015 /uni00000014/uni00000019 /uni00000015/uni00000013/uni0000001a/uni00000018/uni00000014/uni00000013/uni00000013/uni00000014/uni00000015/uni00000018/uni00000014/uni00000018/uni00000013/uni00000014/uni0000001a/uni00000018/uni00000015/uni00000013/uni00000013/uni00000015/uni00000015/uni00000018/uni00000015/uni00000018/uni00000013\n/uni00000013/uni00000011/uni00000013/uni00000013/uni00000013/uni00000011/uni00000014/uni00000019/uni00000013/uni00000011/uni00000016/uni00000015/uni00000013/uni00000011/uni00000017/uni0000001a/uni00000013/uni00000011/uni00000019/uni00000016/uni00000013/uni00000011/uni0000001a/uni0000001c/uni00000013/uni00000011/uni0000001c/uni00000018¯hg1= 20nm ·meV,¯hg2= 0nm·meV\n¯hg1= 0nm·meV,¯hg2= 20nm ·meV\n¯hg2= 5nm·meV,B= 3.0T(a)\n(b)\n(c)P0 Pz Py Px T0\nP0 Pz Py Px T0B[T] B[T] B[T] B[T] B[T0]\n¯hg1[nm·meV] ¯hg1[nm·meV] ¯hg1[nm·meV] ¯hg1[nm·meV] ¯hg1[nm·meV]energy[meV] energy[meV] energy[meV]\nFIG. S3: (color online). The transmission Tand spin polarizability Pwith (a) Rashba SOC only, (b) Dresselhauss SOC only,\nand (d) fixed Dresselhauss SOC but tunable Rashba SOC.\ndensities jz↑,jz↓andjSOCdefined in Eqs. ( S19) to (S22),\nwhereψneeds to be replacedby ψring. We can define the\ntransmission density tα(r) proportional to the current,\ntα(r) =−/planckover2pi1\n2e·meVjα(r). (S2)\nThe transmission rate is thus obtained by Tα=/summationtext\nitα(ri), whereiincludes all the sites between the lead\nand the ring.\nFor simplicity, we consider only the Rashba SOC in\ntwo different cases: (i) /planckover2pi1g1= 20 nm ·meV atB= 2.76T,\nthe electron of the incident energy Ein= 198.5 meV has\nthe transmission rate T0= 0.372; (ii) /planckover2pi1g1= 20 nm ·meV\natB= 0.1T, the transmission rate of the electron with\nEin= 172 meV is T0= 1.998. In Figs. S4(a) and (b),\nwe show how the incident wave functions ψin\nz↑= (1 0)T\nandψin\nz↓= (0 1)Tare transported through the ring,\nrespectively, where the outcoming spin is polarized and\nthe transport rate is relatively low. In Figs. S4(c) and\n(d), we show how the incident wave functions transport\nin the ring when the magnetic field is B= 0.1T, where\nthe electrons pass through the ring freely but the spin is\nnot polarized at all.\nWhen the transportreachesthe equilibriumstatus, the\nspin and charge densities and the current densities areplotted in Fig. S4. Both the charge densities and the\nspin textures shown in the first two columns (from left to\nright) of Fig. S4are periodically distributed in the ring\nas a stationary wave, due to the interference of the mat-\nter wave of the electron. The spin textures also support\nthe analysis of the outcoming spins derived in Eq. ( S3),\ni.e. at the right lead ∝an}bracketle{tσy∝an}bracketri}ht= 0, thexcomponent spin is\ngenerated in the transport by the SOC and the direction\nofσx(r) depends on the polarization of the incident spin.\nComparing with the case without the magnetic field\n[3, 4], here the vector potential of the external magnetic\nfield and the effective vector potential induced by the\nSOC give different phases to the upper and the lower\narms, respectively. This phase difference leads to differ-\nent transmission for different spins and can be observed\nby the transport experiment.\nIn the spin up channel in case (i), electron is mostly\ntransported by the current jz↑, which means the SOC\ndoes not contribute a lot in the transmission. In the spin\ndown channel, the SOC flips spin and induces stronger\ntransmission. However, the transmission of this channel\nisstillweak, onlycontributes1 /20ofthe spinup channel.\nIn this case, the spin is thus strongly polarized. In the\ncase (ii), both of the two channels have high transmission12\n/uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014\n/uni000000ed/uni00000014 /uni00000013 /uni00000014\n/uni000000ed/uni00000014 /uni00000013 /uni00000014\n/uni000000ed/uni00000014 /uni00000013 /uni00000014/uni00000013/uni00000014/uni00000015×/uni00000014/uni00000013/uni000000ed/uni00000015\n/uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014\n/uni000000ed/uni00000014 /uni00000013 /uni00000014\n/uni000000ed/uni00000014 /uni00000013 /uni00000014\n/uni000000ed/uni00000014 /uni00000013 /uni00000014/uni00000013/uni00000014/uni00000015/uni00000016×/uni00000014/uni00000013/uni000000ed/uni00000016\n/uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014\n/uni000000ed/uni00000014 /uni00000013 /uni00000014\n /uni000000ed/uni00000014 /uni00000013 /uni00000014\n /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018×/uni00000014/uni00000013/uni000000ed/uni00000015\n/uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014\n/uni000000ed/uni00000014 /uni00000013 /uni00000014\n /uni000000ed/uni00000014 /uni00000013 /uni00000014\n /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018×/uni00000014/uni00000013/uni000000ed/uni00000015\n/uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014\n/uni000000ed/uni00000014/uni00000013/uni00000014×/uni00000014/uni00000013/uni000000ed/uni00000018tz↑ tz↓ tSOC ttotal˚A−2 ˚A−10.354\n/uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014\n/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018×/uni00000014/uni00000013/uni000000ed/uni00000018¯hg1= 20nm ·meV,¯hg2= 0nm·meV,B= 2.67T,E= 198.5meVy/r0˚A−2(a)\n/uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014\n/uni000000ed/uni00000014/uni00000013/uni00000014×/uni00000014/uni00000013/uni000000ed/uni00000018˚A−2 ˚A−10.017\n/uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014\n/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018×/uni00000014/uni00000013/uni000000ed/uni00000018y/r0˚A−2(b)\n/uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014\n/uni000000ed/uni00000015/uni00000013/uni00000015×/uni00000014/uni00000013/uni000000ed/uni00000019˚A−2 ˚A−11.000\n/uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014\n/uni00000013/uni00000014/uni00000015/uni00000016×/uni00000014/uni00000013/uni000000ed/uni00000019¯hg1= 20nm ·meV,¯hg2= 0nm·meV,B= 0.10T,E= 172.0meVy/r0˚A−2(c)\n/uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014\n/uni000000ed/uni00000015/uni00000013/uni00000015×/uni00000014/uni00000013/uni000000ed/uni00000019\nx/r0 x/r0 x/r0 x/r0 x/r0˚A−2 ˚A−1\nσz\n0.998\n/uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014\n/uni00000013/uni00000015/uni00000017×/uni00000014/uni00000013/uni000000ed/uni00000019\nx/r0y/r0˚A−2\nρ(d)\nFIG. S4: (color online). Transport status and transmission flow density for electrons with fixed incident energy and magn etic\nfields in a Rashba ring. The first two columns (from left to righ t) show the charge and spin fields when the transport reaches\nthe equilibrium status, the colors represent the density an dσz(r), respectively. In (a) and (b), strong spin filtering is foun d, i.e.\nthe transmission rate of spin up is much higher than that of sp in down. In (c) and (d) the outcoming spin is fully unpolarize d\nalthough the total transmission rate is almost 1. For (a) and (b),/planckover2pi1g1= 20 nm ·meV,B= 2.67 T and the incident energy is\n198.5 meV. For (c) and (d), /planckover2pi1g1= 20 nm ·meV,B= 0.1 T and the incident energy is 172 meV.\nrate, close to 1. 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For simplicity and saving\ncomputing resources we utlized a larger lattice constant.\n[43]T. Chakraborty, Quantum Dots (Elsevier, 1999)."    },    {        "title": "1806.01978v1.Control_of_magnetization_dynamics_by_spin_Nernst_torque.pdf",        "content": "1 \n Control of magnetization dynamics by spin Nernst torque  \nArnab Bose, Ambika Shankar Shukla, Sutapa Dutta , Swapnil Bhuktare, Hanuman Singh, Ashwin A . \nTulapurkar  \nDepartment of Electrical Engineering, Indian Institute of Technology -Bombay, Powai, Mumbai, India 400076  \nRelativistic interaction between electron’s spin and orbital angular momentum has provided efficient \nmechanism to control magnetization of nano -magnets. Extensive research has been done to \nunderstand and improve spin -orbit interaction driven torques generated by non -magnets while \napplying electric current. In this work , we show that heat current  in non -magnet can also couple to \nits spin -orbit interaction to produce torque on adj acent ferromagnet. Hence, this work provides a \nplatform to study spin -orbito -caloritronic effect s in heavy metal/ferromagnet bi -layers.  \nSince last few years considerable attention has been drawn to control the magnetization \ndynamics by  pure spin current ge nerated by spin Hall effect  (SHE)1,2,3,4 and interfacial magnetic \nfields5,6 by Rashba effect . SHE and Rashba effects  are relativistic phenomena which couple electron ’s \nspin and orbital motion and can be used  to exert  spin-orbit torques7,8. On the other hand , thermal \ngradient in ferromagnet  can also create pure spin current9,10,11,12,13 which can further produce thermal \nspin torques14,15,16,17,18 and domain wall motion19,20. Conversion of heat current into spin current in a \nnon-magnet has been shown recently via the  spin Nernst effect (SNE)21,22,23,24. But an important \nquestion remains unanswered whether thermal gradient in non -magnet can generate spin toque owing \nto its spin -orbit coupling, which in turn could be used for manipulating magnetization. In this letter \nwe demonstrate that , interplay of heat current and spin -orbit coupling in non -magnetic P latinum (Pt) \ncan generate thermally driven spin -orbit torque (equivalent to  spin Nernst torque (SNT )). SNT has \nbeen  predicted  recently25,26 but it is lacking the experimental evidence.  Here,  we show that effective \nmagnetic damping can be controlled by SNT while creating thermal gradient in  Pt/Ni 81Fe19 bilayer . \nThis can open a new avenue to manipulate  spins in magnetic nano structures  for technological \napplications27,28,29. \n \nFig. 1. Schematic representation of the experiment.  (a) Schematic diagram of Pt/Py \nbilayer with direction of spin separation due to SHE or SNE. (b,c) Change of linewidth (or \neffective damping) on application of dc spin current  (generated by SHE or SNE) . (d) \nDirection of anti -damp ing and damping like torques. (e ) Coloured SEM image of the \nfabricated device with current and voltage terminals.  \n2 \n  \nWhile thermal gradient is established in heavy metal  (Pt in this work) , spins of opposite \npolarity separate out in a direction orthogonal to the direction of heat flow due to SNE as shown in \nFig. 1(a) . This situation is thermal analogous to SHE. Thus Pt converts heat current into pure spin \ncurrent which is then injected into neighbouring ferromagnet  (FM) . If the injected spin current \ndensity is enough,  spin torque is expected  on the FM causing enhanced or reduced damping \ndepending upon the direction of spin vectors absorbed by FM ( Fig 1(b -c)). We compare  the change \nin resonance linewidth of FM due to the spin torque generated by SNE and SHE  by performing  spin-\ntorque ferromagnetic resonance (ST -FMR) experiment2,4,6,30,31,32. Basic working principle of ST -\nFMR is the following.  Radio frequency (rf) current is applied along X axis and dc voltage is measured \nin the same terminals using a bias -T (AMR based detection of ST -FMR2,32) while external magnetic \nfield is swept  at angle θ with respect to X-axis ( Fig. 1(e) ). Pt converts rf charge current into rf spin \ncurrent which is injected to ferromagnet (Ni 81Fe19: Py here after).  Radio frequency  spin cu rrent and \ncurrent induced rf fields excite  the magnet to undergo small osc illation  around its equilibrium \nposition . Due to the AMR effect, resistance of the magnet (hence FM/HM stack) also oscillates.  \nHomodyne mixture of RF applied current and RF resistance of the sample produce s dc voltage. \nAdvantage of ST -FMR is that at resona nce large dc voltage can be obtained . This dc voltage is \ntypically combination of symmetric Lore ntzian ( VS) and anti -symmetric Lore ntzian (VA).  \n()2\n1 2 2\n0 4SVC\nHH=\n− +\n and \n()\n()0\n2 2 2\n04\n4AHHVC\nHH−=\n− + where C1 and C2 are the amplitude  of VS \nand VA respectively,  H is external ly applied  field, H0 is resonant field position  and ∆ is the resonance \nlinewidth (FWHM).  VS indicate s the contribution of spin current induced torque and VA indicates in-\nplane field induced torques. In Pt/Py bi -layers , the Oerst ed magnetic field is the dominant source of \nin-plane field.2,32. So charge to spin current conversion efficiency (or effective spin Hall angle) in \nHM can be quantified  from C1/C2 ratio as following: \n( )1/2 0 ' 1\n21/2S Pt Py\nSH Oe M t tCHHC⊥ =+\n  where \nMS is saturation magnetization, tPt, tPy are thickness of Pt and Py film respectively, \nH⊥  is \nperpendicular magnetic anisotropy  field. Now , if dc current is superimposed on the RF current then \nnon-zero dc spin current is injected which  can change the resonance linewidth of Py2,3,32. It also \nprovides direct quantification of effective spin Hall angle as following:  \n0 ' 21\ncos 4 2S Py\nSH\ncSlopeMt edHHf dJ ⊥      =+         \nwhere JC is charge current density through Pt, f \nis frequency of applied rf cur rent, γ is gyromagnetic ratio . In this work, we s have superimposed  a dc \nheat current on rf charge current. We could modulate the resonance line width of Py which  provide s \na direct evidence of control of magnetization dynamics by spin Nernst torque  (SNT) . \n We fabricated the device as shown in Fig. 1(e) . The crossbar is made of  Pt (15 nm) and a \nrectangular shaped dot of  Py (2  nm) is deposited at the centre of Pt  crossbar . Top of Py was  capped \nwith Ta (1.5 nm).  On the top and bottom lead of the Pt crossbar , two heater lines are fabricated which \nare electrically isolated by SiOx (30 nm) from Pt. Heater lines are prepared with Ta/Pt (6 0 nm). \nNumbers in bracket indicate thickness of metals. Entire fabrication is done by standard electron beam \nlithography, sputtering and lift -off technique.  Before deposition of Py , surface of Pt was cleaned by \nAr ion without breaking the vacuum. We have earlier shown that linewidth change can be sensitively \nmeasured in planar Hall structure  doing ST -FMR  [Ref.  32]. We follow the same approach in this \nwork  to compare the strength of spin Hall torque (SHT) and spin Nernst torque (SNT). Our detection 3 \n method is the following. Radio frequency current is applied along X-axis;  dc voltage is measured \nalong same direction  (hence AMR based detection) but dc heat current (or charge current) is passed \nalong Y-axis to modulate  the line width by SNT (or SHT) as shown in Fig. 1(e) . Application of dc \ncurrent  (heat current or charge current)  perpendicular to the direction of volta ge measurement reduces \nnoise and hence measurement sensitiv ity significantly increases as shown in Ref 32. \n \nFig. 2. Characterization of spin Hall torque.  (a) DC voltage generated by ST -FMR for \ndifferent frequencies of applied rf current. (b) Fit of dc vol tage by VS and VA. Inset of (b) \nshows Kittel’s fit. (c) ST -FMR spectrum when dc charge current is applied perpendicular to \napplied rf current. (d) Modulation of linewidth on application of dc charge current.   \n \n Fig. 2  shows the characterization of spin-Hall torque by measuring ST-FMR.  Fig. 2(a)  shows \nthe typical dc voltage spectrum as external magnetic field is swept for different frequencies of applied \ncurrent.  This dc voltage can be fit ted to the  sum of VS and VA (Fig. 2(b) ). Red squares in Fig. 2(b) \nshow the experimental data  and black curve shows fitting  which is sum of VS (pink curve) and VA \n(blue curve). The symmetric component confirms the spin -orbit torque generated by spin Hall effect.  \nInset of Fig. 2(b) shows the Kittel’s fit for resonant magnetic fields and frequencies. From this we \nobtain \nH⊥ =8.05 kOe  (which corresponds to Ms = 6.5×105 A/m.) Fig. 2(c)  shows the voltage spectrum \nwhen dc charge current is applied orthogonal to the d irection of rf current flow. We can clearly see \nthe dominant change in the shape of the voltage signal  as the linewidth  significantly  changes.  For the \npositive current linewidth is more in positive field value s and it is less in negative field values (vice \nversa for negative applied current).  Our detection method is so sensitive that linewidth change is \nclearly visible in Fig. 2(c)  itself.  Linewidth (∆) as a function of applied dc current is shown in top \npanel of Fig. 2(d) . It shows expected linear dependence as function of applied current.  Bottom panel \nof Fig. 2(d)  shows the difference in linewidth for θ=350 and θ=2150 (∆’=∆(350)-∆(2150)) as a function \nof applied current which also shows linear  dependence . Bottom panel of Fig. 2(d) represents  only the \ncontribution of spin current induced damping change eliminating the overall heating effect  if any . \nFrom this measurement  of linewidth modulation,  we can extract the effective spin Hall angle of Pt to \nbe 0.12±0.06 . Extracted  value o f spin Hall angle from C1/C2 ratio is somewhat lower. We have further \n4 \n noticed that in this planar Hall geometry when dc current is applied perpendicular to rf current \nmodulation in VS is higher  compared to the line -width modulation  as a function of applied dc current. \nSame behaviour  was also observed in our previous work ( Fig. 3(c) of ref 32 and Fig. 2(c) in this \nwork).  However, we have verified that the line width modulation is the same  irrespective of the \ndetection methods (AMR or  PHE based detection or  hybrid planar Hall detection  method) . So in this \nwork we quantify the strength of SHT and SNT in Pt by measuring linewidth change by injecting dc \nspin current by SHE and SN E respectively.  \n \nFig. 3. Characterization of SNT.  (a) dc voltage spectrum on application of heat current \nalong + Y axis and –Y axis. (b)  dc voltage spectrum when positive and negative field data \npoints are superimposed for dc thermal gradient along Y axis. ( c,d) Difference in linewidth  \nbetween positive an d negative field values  (∆’) for different applied heater powers and \ndifferent angles ( θ) (respectively).  \n \nNow we show the evidence of spin Nernst torque by passing  dc heat current instead of \napplying dc charge current while doing ST -FMR as discussed above.  Two different heater lines are \nfabricated on Hall bar ( Fig. 1(e) ) to create thermal gradient along ± Y-axis. When current flows in \nheaterline -1 (HL1)  it becomes hot due to Joule’s  heating and most of the heat is carried by Pt below \nthe heat line.  This creates thermal gr adient in Pt/Py bilayers along + Y axis. Similarly , when current \nis applied in heaterline -2 (HL2) the rmal gradient is created along -Y axis. Fig. 3(a)  shows the dc \nvoltage spectrum generated by ST -FMR while thermal gradient is create d along ± Y axis. We can  see \nthe difference in dc voltage spectrum for positive and negative thermal gradients.  This cannot be \nexplained by overall heating effect since overall heating would be same for both the direction of \nthermal gradients.  All these mea surements are performed when θ is 350. In Fig. 3(b)  we further show \nthat when voltage signal of positive field and negative field is superimposed , there is distinct change \nin the shape of voltage spectrum (hence the linewidth) which furthe r confirms the existence of SNT. \nDifference in the shape of voltage  signal  in Fig. 3(b) cannot be explained by overall heating effect as \nit would be same for both 350 and 2150. As shown  earlier ( Fig. 2(d) ), line-width difference for positive \nand negative f ield (∆’=∆(θ) - ∆(θ+1800)) is measured for different applied heater powers ( Fig. 3(c) ) \nand for  different angles ( Fig. 3(d) ). ∆’ is proportional to the heater power ( Fig. 3(c) ) and it closely \nfollows cosθ dependence  (Fig. 3(d) ). In our geometry ( Fig. 1 (a)) the line width modulation due to \nSNT is expected to be maximum  at θ=00, but ST -FMR signal becomes zero at θ=00. We further \nconfirm ed that polarity of heater current has negligible  effect on ∆’ as heater line is electrically \ninsulated from Pt hall bar and Py dot is quite far away (1 .5 μm) from the heater line  to get affected \nby magnetic (Oersted) field produced by the heater current.  In our control experiment we  have  \napplied heater current in both HL1 and HL2 which would  cause the same overall heating but fail to \nset up  well directed thermal gradient . We found negligible effect on the line width in this case.  Our \nobserved results shown in Fig. 3 strongly supports the evidence of spin Nernst torque in Pt.  \n5 \n   \nFig. 4.  Temperature profile . (a) Surface plot of 2D  thermal gradient. Estimated temperature \n(b) and temperature gradient (c) in Pt/Py interface.  \n \n We have followed the same approach to find thermal gradient in Pt as reported in Ref  24. \nFrom the resistance value of the heater line, its overall temperature is known (on chip temperature \ncalibration).  Once the temperature of heater line is known temperature profile of Pt/Py interface can \nbe calculated from  COMSOL simulation  (Fig. 4(b)  and 4(c)). Overall temperature rise is around 25 \nK which is fairly small and  hence we observe negligible contribution from overall heating effect.  We \nconsider the thermal gradient at the interface to be 15 K/μ m while estimating SNT.  Our estimated \ntemperature gradient in this kind of geometry is in good agreement with previous resu lts20. \nComparing the line -width modulation by SHT and SNT we can quantify that 15 K/  μm horizontal \nthermal gradient in Pt/Py interface is equivalent to the application of 4.9×109 A/m2 amount of charge \ncurrent density in Pt.  This reported value of heat current to spin current conversion efficiency by SNE \nis consistent with our previous report of SNE24 and comparable to reports by other groups21,23. \n In conclusion, we have demonstrated the control of magnetization dynamics by thermally \ndriven spin Hall torque (or spin Nernst torque). We report  approximately 0.9 % line -width change \ndue to spin Nernst torque effect.  It indicates that about 100 times more thermal gradient needs  be \ncreated to achieve switching by spin Nernst torque  which could be achieved in material having large \nspin Nernst angle and implementing efficient mechanism (such as Laser heating) to create large \nthermal  gradient  at nano scale.  Further by using ferromagnets with less out -of-plane anisotropy, the \nthermal gr adients required for switching can be reduced. These results will beneficial in energy \nharvesting sector where heat current  in non -magnet s can be utilized to write magnetic memories for \nnon-volatile memory applications.  \nReferences  \n1 J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jung wirth,  “Spin Hall effects ”. Rev. Mod. \nPhys.  87, 1213 (2015)  \n2 L. Q. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, “Spin-Torque Ferromagneti c Resonance \nInduced by the Spi n Hall Effect ”, Phys. Rev. Lett. 106, 036601 (2011)  \n3 K. Ando, S. Takahashi, K. Harii, K. Sasa ge, J. Ieda, S. 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Watanabe, and S. Yuasa, “Spin Torq ue Diode effect in Magnetic Tunnel Junctions ”, Nature (London)  \n438, 339 (2005)  7 \n                                                                                                                                                                                \n31 A. Bose, S . Dutta, S. Bhuktare, H. Singh , and A. A.  Tulapurkar, “Sensitive measurement of spin orbit \ntorque  driven ferromagnetic resonance detected by planar Hall  geometry ”, Appl. Phys. Lett. 111, 162405 \n(2017)  \n32 A. Bose, D. D. Lam, S. Bhuktare, H. Singh , S. Miwa  and A. Tulapurkar,  “Observation of anomalous spin -\ntorque generated by a ferromagnet ”, arXiv:1706.07245 (2017)  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 8 \n                                                                                                                                                                                \nSupplementary information  \nArnab Bose, Ambika Shukla, Sutapa Dutta, Swapnil Bhuktare, Hanuman Singh, Ashwin A \nTulapurkar  \nDepartment of Electrical Engineering, Indian Institute of Technology -Bombay, Powai, Mumbai, \nIndia 400076  \n \nS1. Temperature calibration  \nThermal gradient is obtained by doing on -chip calibration.  As current is passed through heater line it becomes \nhot due to Joule’s heating and its resistance increases. Hence resistance of heater line  provides information of \naverage temperature in heater line.  Same approach w as adopted in our previous work [S1]  and by other \nresearchers  [S2,S3,S4] . Heater line was made of Ta/Pt (~50 nm).  Its resistance changes from 50 Ω to 65 Ω on \napplication of maximum  heater power (0.3 W). This corresponds to overall temperature of heater line to be \napproximately 420 K which is obtained from the temperature dependent resistance measurement of the heater \nline. We observed minor change in resistance in Pt Hall bar which contains rectangular Py dot a t the centre.  It \nindicates that the temperature of horizontal Pt line does not increase much and the heat is locally centred near \nthe heater line.  To get exact temperature profile , COMSOL simulation is done with heat transfer m odule with \nfollowing boundary conditions: (1) temperature of heater line is 420 K (experimentally obtained) and (2) \ntemperature of the bottom of Si is 293 (lab temperature).  \n \nFig. S1. (a) Temperature profile at the interface of Pt/Py.  (b) Isothermal temperature contour. (c) \nSchematics of device structure.  Arrow indicates direction of heat flow.  (d) Temperature along Y-axis. \n(e) Temperature gradient along Y-axis. \n \nFig. S1(a) shows the surface temperature. Bright yellow colour indicates  hot region and temperature of heater \nis set 420 K.  Deep red indicates the colder region.  Fig. S2(b) shows the isothermal contour which shows that \n9 \n                                                                                                                                                                                \nheat is also localized in very small region (approx. 10×10×10 μm3) as expected.  Fig. S1(d) shows the \ntemperat ure as along Y axis which shows exponential decay of thermal gradient  (Fig. S1(e)) . Temperature \nbecomes around 300 K when we go 10 μm away from the heater line.  It shows that overall temperature of Py \nis slightly more than room temperature (~325 K).  Maximu m thermal gradient along Y axis in Pt/Py interface \nis nearly 20 K/μm.  Estimated temperature profile is in good agreement with previous works  [S1-S6]. We have \nconsidered below parameters in simulation. We have also checked that slight variation of these par ameter does \nnot influence the simulated result much. Finally estimated thermal gradient will be in order tens of K/μm in \nthis kind of geometry.  \nMaterial  Pt Si SiO2  \nThermal conductivity (SI unit)  85 120 1.4 \n \nReferences  \n[S1] A. Bose, S. Bhuktare , H. Singh, S. Dutta, V. Achanta, A. A. Tulapurkar. Appl. Phys. Lett. (accepted)  \n[S2] P. Krzysteczko,  X. Hu, N. Liebing, S. Sievers, and H. W. Schumacher . Domain wall magneto -Seebeck \neffect . Phys. Rev. B 92, 140405(R) (2015)  \n[S3] S. Meyer, Y. -T. Chen, S. W immer, M. Althammer, T. Wimmer, R. Schlitz, S. Geprägs, H. Huebl, D. \nKödderitzsch, H. Ebert, G.E.W. Bauer, R. Gross, and S.T.B. Goennenwein, Nat. Mater. 16, 977 (2017)  \n[S4] A. Pushp, T. Phung, C. Rettner, B. P. Hughes, S. H. Yang, and S. S. P. Parkin, “Giant thermal spin torque -\nassisted magnetic tunnel junction switching,” Proc. Natl. Acad. Sci. U.S.A. 112, 6585 –6590 (2015).  \n[S5] A. Slachter, L. F. Bekker, J. -P. Adam, and J. B. van Wees, “Thermally driven spin injection from a \nferromagnet into a non -magnetic metal,” Nat. Phys. 6, 879 –882 (2010)  \n[S6] M. Walter, J. Walowski, V. Zbarsky, M. M€ unzenberg, M. Sch€afers, D. Ebke, G. Reiss, A. Thomas, P. \nPeretzki, M. Seibt et al., “Seebeck effect in magnetic tunnel junctions,” Nat. Mater. 10, 742 –746 (2011)  \n \n "    },    {        "title": "0805.1320v2.Spin_dynamics_in__III_Mn_V_ferromagnetic_semiconductors__the_role_of_correlations.pdf",        "content": "arXiv:0805.1320v2  [cond-mat.str-el]  25 Aug 2008Spin dynamics in (III,Mn)V ferromagnetic semiconductors: the role of correlations\nM. D. Kapetanakis and I. E. Perakis\nDepartment of Physics, University of Crete, and Institute o f Electronic Structure & Laser,\nFoundation for Research and Technology-Hellas, Heraklion , Crete, Greece\n(Dated: November 6, 2018)\nWe address the role of correlations between spin and charge d egrees of freedom on the dynamical\nproperties of ferromagnetic systems governed by the magnet ic exchange interaction between itiner-\nant and localized spins. For this we introduce a general theo ry that treats quantum fluctuations\nbeyond the Random Phase Approximation based on a correlatio n expansion of the Green’s function\nequations of motion. We calculate the spin susceptibility, spin–wave excitation spectrum, and mag-\nnetization precession damping. We find that correlations st rongly affect the magnitude and carrier\nconcentration dependence of the spin stiffness and magnetiz ation Gilbert damping.\nPACS numbers: 75.30.Ds, 75.50.Pp, 78.47.J-\nIntroduction— Semiconductors displaying carrier–\ninduced ferromagnetic order, such as Mn–doped III-V\nsemiconductors, manganites, chalcogenides, etc, have re-\nceived a lot of attention due to their combined magnetic\nand semiconducting properties [1, 2]. A strong response\nof their magnetic properties to carrier density tuning via\nlight, electrical gates, or current[3, 4, 5] canlead to novel\nspintronics applications [6] and multifunctional magnetic\ndevices combining information processing and storage on\na single chip. One of the challenges facing such magnetic\ndevices concerns the speed of the basic processing unit,\ndetermined by the dynamics of the collective spin.\nTwo key parameters characterize the spin dynam-\nics in ferromagnets: the spin stiffness, D, and the\nGilbert damping coefficient, α.Ddetermines the long–\nwavelength spin–wave excitation energies, ωQ∼DQ2,\nwhereQis the momentum, and other magnetic prop-\nerties.Dalso sets an upper limit to the ferromagnetic\ntransition temperature: Tc∝D[1]. So far, the Tcof\n(Ga,Mn)As has increased from ∼110 K [2] to ∼173 K\n[1, 7]. It is important for potential room temperature\nferromagnetism to consider the theoretical limits of Tc.\nTheGilbertcoefficient, α, characterizesthedampingof\nthe magnetization precession described by the Landau–\nLifshitz–Gilbert (LLG) equation [1, 8]. A microscopic\nexpression can be obtained by relating the spin suscepti-\nbility of the LLG equation to the Green’s function [9]\n≪A≫=−iθ(t)<[A(t),S−\nQ(0)]> (1)\nwithA=S+\n−Q,S+=Sx+iSy.∝angbracketleft···∝angbracketrightdenotes the\naverage over a grand canonical ensemble and SQ=\n1/√\nN/summationtext\njSje−iQRj, whereSjare spins localized at N\nrandomly distributed positions Rj. The microscopic ori-\ngin ofαisstill notfully understood[9]. Amean–fieldcal-\nculation of the magnetization damping due to the inter-\nplay between spin–spin interactions and carrier spin de-\nphasingwasdevelopedin Refs.[9, 10]. Themagnetization\ndynamics can be probed with, e.g., ferromagnetic res-\nonance [11] and ultrafast magneto–optical pump–probe\nspectroscopy experiments [5, 12, 13, 14]. The interpre-tation of such experiments requires a better theoretical\nunderstanding of dynamical magnetic properties.\nIn this Letter we discuss the effects of spin–charge cor-\nrelations, due to the p–d exchange coupling of local and\nitinerant spins, on the spin stiffness and Gilbert damp-\ningcoefficient. Wedescribequantumfluctuationsbeyond\nthe Random Phase Approximation (RPA) [15, 16] with a\ncorrelationexpansion[17]ofhigherGreen’sfunctionsand\na 1/S expansion of the spin self–energy. To O(1/S2), we\nobtain a strong enhancement, as compared to the RPA,\nof the spin stiffness and the magnetization damping and\na different dependence on carrier concentration.\nEquations of motion— The magnetic propertiescan be\ndescribedby the Hamiltonian [1] H=HMF+Hcorr, where\nthe mean field Hamiltonian HMF=/summationtext\nknεkna†\nknaknde-\nscribes valence holes created by a†\nkn, wherekis the mo-\nmentum, nis the band index, and εknthe band disper-\nsion in the presenceof the mean field created by the mag-\nnetic exchangeinteraction[16]. The Mn impurities act as\nacceptors, creating a hole Fermi sea with concentration\nch, and also provide S= 5/2 local spins.\nHcorr=βc/summationdisplay\nq∆Sz\nq∆sz\n−q+βc\n2/summationdisplay\nq(∆S+\nq∆s−\n−q+h.c.),(2)\nwhereβ∼50–150meV nm3in (III,Mn)V semiconductors\n[1] is the magnetic exchane interaction. cis the Mn spin\nconcentration and sq= 1/√\nN/summationtext\nnn′kσnn′a†\nk+qnakn′the\nhole spin operator. ∆ A=A− ∝angbracketleftA∝angbracketrightdescribes the quan-\ntum fluctuations of A. The ground state and thermo-\ndynamic properties of (III,Mn)V semiconductors in the\nmetallic regime ( ch∼1020cm−3) are described to first\napproximation by the mean field virtual crystal approxi-\nmation,HMF, justified for S→ ∞[1]. Most sensitive to\nthe quantum fluctuations induced by Hcorrare the dy-\nnamical properties. Refs.[9, 15] treated quantum effects\ntoO(1/S) (RPA). Here we study correlations that first\narise atO(1/S2). By choosing the z–axis parallel to the\nground state local spin S, we have S±= 0 and Sz=S.\nThe mean hole spin, s, is antiparallel to S,s±= 0 [1].2\nThe spin Green’s function is given by the equation\n∂t≪S+\n−Q≫=−2iSδ(t)+βc≪(s×S−Q)+≫\n−i∆≪s+\n−Q≫+βc\nN×\n/summationdisplay\nkpnn′≪(σnn′×∆Sp−k−Q)+∆[a†\nknapn′]≫,(3)\nwhere ∆ = βcSis the mean field spin–flip energy gap\nands= 1/N/summationtext\nknσnnfknis the ground state hole spin.\nfkn=∝angbracketlefta†\nknakn∝angbracketrightis the hole population. The first line on\nthe right hand side (rhs) describes the mean field pre-\ncession of the Mn spin around the mean hole spin. The\nsecond line on the rhs describes the RPA coupling to the\nitinerant hole spin [10], while the last line is due to the\ncorrelations. The hole spin dynamics is described by\n(i∂t−εkn′+εk−Qn)≪a†\nk−Q↑ak↓≫\n=βc\n2√\nN/bracketleftbigg\n(fk−Qn−fkn′)≪S+\n−Q≫\n+/summationdisplay\nqm≪(σn′m·∆Sq)∆[a†\nk−Qnak+qm]≫\n−/summationdisplay\nqm≪(σmn·∆Sq)∆[a†\nk−Q−qmakn′]≫/bracketrightbigg\n.(4)\nThe firstterm on the rhsgivesthe RPAcontribution[10],\nwhile the last two terms describe correlations.\nThe correlation contributions to Eqs.(3) and (4) are\ndetermined by the dynamics of the interactions be-\ntween a carrier excitation and a local spin fluctuation.\nThis dynamics is described by the Green’s functions\n≪∆Sp−k−Q∆[a†\nknapn′]≫, whose equations of motion\ncouple to higher Green’s functions, ≪Sa†aa†a≫and\n≪SSa†a≫, describingdynamicsof threeelementaryex-\ncitations. To truncate the infinite hierarchy, we apply a\ncorrelation expansion [17] and decompose ≪Sa†aa†a≫\ninto all possible products of the form ∝angbracketlefta†aa†a∝angbracketright ≪S≫,\n∝angbracketleftS∝angbracketright∝angbracketlefta†a∝angbracketright ≪a†a≫,∝angbracketlefta†a∝angbracketright ≪∆S∆[a†a]≫, and∝angbracketleftS∝angbracketright ≪\na†aa†a≫c, where≪a†aa†a≫cis obtained after sub-\ntracting all uncorrelated contributions, ∝angbracketlefta†a∝angbracketright ≪a†a≫,\nfrom≪a†aa†a≫(we include all permutations of mo-\nmentum and band indices) [18]. Similarly, we decompose\n≪SSa†a≫into products of the form ∝angbracketleftSS∝angbracketright ≪a†a≫,\n∝angbracketleftS∝angbracketright∝angbracketlefta†a∝angbracketright ≪S≫,∝angbracketleftS∝angbracketright ≪∆S∆[a†a]≫, and∝angbracketlefta†a∝angbracketright ≪\n∆S∆S≫. This corresponds to decomposing all opera-\ntorsAinto average and quantum fluctuation parts and\nneglecting products of three fluctuations. We thus de-\nscribe all correlations between any twospin and charge\nexcitations and neglect correlations among threeor more\nelementary excitations (which contribute to O(1/S3))\n[18]. In the case of ferromagnetic β, as in the mangan-\nites, we recover the variational results of Ref.[19] and\nthus obtain very good agreement with exact diagonaliza-\ntionresultswhilereproducingexactlysolvablelimits (one\nelectron, half filling, and atomic limits, see Refs.[18, 19]).When treating correlations in the realistic (III,Mn)V\nsystem, the numerical solution of the above closed sys-\ntem of equations of motion is complicated by the cou-\npling of many momenta and bands and by unsettled is-\nsues regarding the role on the dynamical and magnetic\nanisotropy properties of impurity bands, strain, localized\nstates, and sp–d hybridization [1, 20, 21, 22, 23]. In the\nsimpler RPA case, which neglects inelastic effects, a six–\nband effective mass approximation [16] revealed an order\nof magnitude enhancement of D. The single–band RPA\nmodel [15] also predicts maximum Dat very small hole\nconcentrations, while in the six–band model Dincreases\nand then saturates with hole doping. Here we illustrate\nthe main qualitative features due to ubiquitous corre-\nlations important in different ferromagnets [19, 24] by\nadopting the single–band Hamiltonian [15]. We then dis-\ncuss the role of the multi–band structure of (III,Mn)V\nsemiconductors by using a heavy and light hole band\nmodel.\nIn the case of two bands of spin– ↑and spin– ↓states\n[15], we obtain by Fourier transformation\n≪S+\n−Q≫ω=−2S\nω+δ+ΣRPA(Q,ω)+Σcorr(Q,ω),(5)\nwhereδ=βcsgives the energy splitting of the local spin\nlevels. Σ RPAis the RPA self energy [15, 16].\nΣcorr=βc\n2N/summationdisplay\nkp/bracketleftBigg\n(Gpk↑+Fpk)ω+εk−εk+Q\nω+εk−εk+Q+∆+iΓ\n−(Gpk↓−Fpk)ω+εp−Q−εp\nω+εp−Q−εp+∆+iΓ/bracketrightBigg\n(6)\nis the correlated contribution, where\nGσ=≪S+∆[a†\nσaσ]≫\n≪S+≫, F=≪∆Sza†\n↑a↓≫\n≪S+≫.(7)\nΓ∼10-100meV is the hole spin dephasing rate [25]. Sim-\nilar to Ref.[10] and the Lindblad method calculation of\nRef.[14], we describe such elastic effects by substituting\nthe spin–flip excitation energy∆ by ∆+ iΓ. We obtained\nGandFbysolvingthecorrespondingequationstolowest\norder in 1/S, with βSkept constant, which gives Σcorrto\nO(1/S2). More details will be presented elsewhere [18].\nResults— Firstwestudythe spinstiffness D=DRPA+\nDcorr\n++Dcorr\n−. The RPA contribution DRPAreproduces\nRef.[15]. The correlated cotributions Dcorr\n+>0 and3\n0 0.1 0.2 0.3 0.4 0.5\np00.020.040.060.08D/D0 D\nDRPA\nDRPA+D(-)\n0 0.2 0.4\np00.020.040.06\n50 100 150\nβc  (meV)00.010.02D/D0\n50 100 150\nβc  (meV)00.020.040.06a) βc =70meV b) βc =150meV\nc) p =0.1 d) p =0.5\nFIG. 1: (Color online) Spin stiffness Das function of hole\ndoping and interaction strength for the single–band model.\nc= 1nm−3, Γ=0,D0=/planckover2pi12/2mhh,mhh= 0.5me.\nDcorr\n−<0 were obtained to O(1/S2) from Eq.(6) [18]:\nDcorr\n−=−/planckover2pi12\n2mhS2N2/summationdisplay\nkp/bracketleftBigg\nfk↓(1−fp↓)εp(ˆp·ˆQ)2\nεp−εk\n+fk↑(1−fp↑)εk(ˆk·ˆQ)2\nεp−εk/bracketrightBigg\n, (8)\nDcorr\n+=/planckover2pi12\n2mhS2N2/summationdisplay\nkpfk↓(1−fp↑)×\n/bracketleftBig\nεk(ˆk·ˆQ)2+εp(ˆp·ˆQ)2/bracketrightBig\n×\n/bracketleftbigg2\nεp−εk+1\nεp−εk+∆−∆\n(εp−εk)2/bracketrightbigg\n,(9)\nwhereˆQ,ˆk, andˆ pdenote the unit vectors.\nFor ferromagnetic interaction, as in the manganites\n[19, 24], the Mn and carrier spins align in parallel. The\nHartree–Fock is then the state of maximum spin and\nan exact eigenstate of the many–body Hamiltonian (Na-\ngaoka state). For anti–ferromagnetic β, as in (III,Mn)V\nsemiconductors, the ground state carrier spin is anti–\nparallel to the Mn spin and can increase via the scat-\ntering of a spin– ↓hole to an empty spin– ↑state (which\ndecreases Szby 1). Such quantum fluctuations give rise\ntoDcorr\n+, Eq.(9), which vanishes for fk↓= 0.Dcorr\n−comes\nfrom magnon scattering accompanied by the creation of\naFermi seapair. In the caseofaspin– ↑Fermi sea, Eq.(8)\nrecovers the results of Refs.[19, 24].\nWe evaluated Eqs.(8) and (9) for zero temperature\nafter introducing an upper energy cutoff corresponding\nto the Debye momentum, k3\nD= 6π2c, that ensures the\ncorrect number of magnetic ion degrees of freedom [15].0 0.1 0.2 0.3 0.4 0.5\np00.20.4D/D0\n0 0.1 0.2 0.3 0.4 0.5\np00.20.4\n0 0.1 0.2 0.3\nεF  (eV)00.010.02D/D0\n0 0.1 0.2 0.3 0.4 0.5\nεF (eV)00.020.04a) βc =70meV b) βc =150meV\nc) βc =70meV d) βc =150meV\nFIG. 2: (Color online) Spin stiffness Dfor the parameters of\nFig. 1. (a)and(b): two–bandmodel, (c)and(d): dependence\non the Fermi energy within the single–band model.\nFigs. 1(a) and (b) show the dependence of Don hole\ndoping, characterized by p=ch/c, for two couplings β,\nwhile Figs. 1(c) and (d) show its dependence on βfor\ntwo dopings p. Figure 1 also compares our full result, D,\nwithDRPAandDRPA+Dcorr\n−. It is clear that the cor-\nrelations beyond RPA have a pronounced effect on the\nspin stiffness, and therefore on Tc∝D[1, 7] and other\nmagnetic properties. Similar to the manganites [19, 24],\nDcorr\n−<0 destabilizes the ferromagneticphase. However,\nDcorr\n+stronglyenhances Das comparedto DRPA[15] and\nalso changes its p–dependence.\nThe ferromagnetic order and Tcvalues observed in\n(III,Mn)V semiconductors cannot be explained with the\nsingle–band RPA approximation [15], which predicts a\nsmallDthat decreases with increasing p. Figure 1\nshows that the correlations change these RPA results in\na profound way. Even within the single–band model,\nthe correlations strongly enhance Dand change its p–\ndependence: Dnow increases with p. Within the RPA,\nsuch behavior can be obtained only by including multiple\nvalence bands [16]. As discussed e.g. in Refs.[1, 7], the\nmain bandstructure effects can be understood by con-\nsidering two bands of heavy ( mhh=0.5me) and light (\nmlh=0.086me) holes. Dis dominated and enhanced by\nthe more dispersive light hole band. On the other hand,\nthe heavily populated heavy hole states dominate the\nstatic properties and EF. By adopting such a two–band\nmodel, we obtain the results of Figs. 2(a) and (b). The\nmain difference from Fig. 1 is the order of magnitude en-\nhancement of all contributions, due to mlh/mhh= 0.17.\nImportantly,thedifferencesbetween DandDRPAremain\nstrong. Regarding the upper limit of Tcdue to collective\neffects, we note from Ref.[7] that is is proportional to D\nand the mean field Mn spin. We thus expect an enhance-\nment, as compared to the RPA result, comparable to the4\n0 0.5 100.020.04αα\nαRPA\n0 0.5 100.020.04\n0 0.5 1\np00.020.04α\n0 0.5 1\np00.020.04a) βc =70meV b) βc =100meV\nc) βc =120meV d) βc =150meV\nFIG. 3: (Color online) Gilbert damping as function of hole\ndoping for different interactions β.c= 1nm−3,Γ = 20meV.\ndifference between DandDRPA.\nThe dopingdependence of Dmainlycomesfromits de-\npendence on EF, shown in Figs. 2(c) and (d), which dif-\nfers strongly from the RPA result. Even though the two\nband model captures these differences, it fails to describe\naccuratelythe dependence of EFonp, determined by the\nsuccessive population of multiple anisotropicbands. Fur-\nthermore, thespin–orbitinteractionreducesthe holespin\nmatrix elements [22]. For example, |σ+\nnn′|2is maximum\nwhen the bandstates arealsospin eigenstates. The spin–\norbit interaction mixes the spin– ↑and spin– ↓states and\nreduces|σ+\nnn′|2. From Eq.(3) we see that the deviations\nfromthe meanfield resultaredetermined bythe coupling\nto the Green’s functions ≪σ+\nnn′∆[a†\nnan′]≫(RPA),≪\n∆Szσ+\nnn′∆[a†\nnan′]≫(correctiontoRPAdueto Szfluctu-\nationsleadingto Dcorr\n+>0), and≪∆S+σz\nnn′∆[a†\nnan′]≫\n(magnon–Fermi sea pair scattering leading to Dcorr\n−<0).\nBoth the RPA and the correlation contribution arising\nfrom ∆Szare proportional to σ+\nnn′. Our main result, i.e.\ntherelativeimportance of the correlation as compared to\nthe RPA contribution, should thus also hold in the real-\nistic system. The full solution will be pursued elsewhere.\nWe now turn to the Gilbert damping coefficient, α=\n2S/ω×Im≪S+\n0≫−1atω→0 [9]. We obtain to\nO(1/S2) thatα=αRPA+αcorr, where αRPArecovers\nthe mean–field result of Refs [9, 10] and predicts a linear\ndependence on the hole doping p, while\nαcorr=∆2\n2N2S2/summationdisplay\nkpIm/bracketleftBigg\nfk↓(1−fp↑)\n∆+iΓ×\n/parenleftbigg1\nεp−εk−δ+1\nεp−εk+∆+iΓ/parenrightbigg/bracketrightBigg\n(10)\narises from the carrier spin–flip quantum fluctuations.Fig.(3) compares αwith the RPA result as function of\np. The correlations enhance αand lead to a nonlinear\ndependence on p, which suggests the possibility of con-\ntrolling the magnetization relaxation by tuning the hole\ndensity. A nonlinear dependence of αon photoexcitation\nintensity was reported in Ref.[13] (see also Refs.[12, 21]).\nWe conclude that spin–charge correlations play an im-\nportant role on the dynamical properties of ferromag-\nnetic semiconductors. For quantitative statements, they\nmust be addressed together with the bandstructure ef-\nfects particular to the individual systems. The correla-\ntions studied here should play an important role in the\nultrafast magnetization dynamics observed with pump–\nprobe magneto–optical spectroscopy [12, 13, 14, 21, 22].\nThis work was supported by the EU STREP program\nHYSWITCH.\n[1] T. Jungwirth et al., Rev. Mod. Phys. 78, 2006.\n[2] H. Ohno, Science 281, 951 (1998).\n[3] S. Koshihara et al., Phys. Rev. Lett. 78, 4617 (1997).\n[4] H. Ohno et al., Nature 408, 944 (2000).\n[5] J. Wang et al., Phys. Rev. Lett. 98, 217401 (2007).\n[6] S. A. Wolf et al., Science 294, 1488 (2001).\n[7] T. K. Jungwirth et al., Phys. Rev. B 72, 165204 (2005).\n[8] L. D. Landau, E. M. Lifshitz, and L. P. Pitaeviski, Sta-\ntistical Physics, Part 2 (Pergamon, Oxford, 1980).\n[9] J. Sinova et. al., Phys. Rev. B69, 085209 (2004); Y.\nTserkovnyak, G.A.Fiete, andB. I.Halperin, Appl.Phys.\nLett.84, 25 (2004).\n[10] B. Heinrich, D. Fraitov´ a, and V. Kambersk´ y, Phys. Sta t.\nSol.23, 501 (1967).\n[11] S. T. B. Goennenwein et al., Appl. Phys. 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B 76, 045205\n(2007).\n[23] X. Liu et. al., Phys. Rev. B 71, 035307 (2005); K.\nHamaya et. al., Phys. Rev. B 74, 045201 (2006).\n[24] D. I. Golosov, Phys. Rev. Lett. 84, 3974 (2000); N.\nShannon and A. V. Chubukov, Phys. Rev. B 65, 104418\n(2002).5\n[25] T. Jungwirth et. al., Appl. Phys. Lett. 81, 4029 (2002)."    },    {        "title": "1901.00714v3.Observing_zero_field_spin_dynamics_with_spin_noise_in_a_pi_pulse_modulated_field.pdf",        "content": "arXiv:1901.00714v3  [physics.atom-ph]  17 Jun 2019Observing zero-field spin dynamics with spin noise in a pi-pu lse modulated field\nGuiying Zhang,1,2Ya Wen,1Jian Qiu,3and Kaifeng Zhao1,∗\n1Institute of Modern Physics, Department of Nuclear Science and Technology and Applied Ion Beam Physics Laboratory,\nKey Laboratory of the Ministry of Education, Fudan Universi ty, Shanghai 200433, China\n2College of Science, Zhejiang University of Technology, Hang zhou 310023, China\n3Insitute for Electric Light Sources, School of information Science and Engineering, Fudan University, Shanghai 20043 3, China\n(Dated: June 18, 2019)\nSpin noise spectroscopic study of spin dynamics in a zero mag netic field is commonly masked by\nthe dominating 1 /fnoise. We show that in a pi-pulse modulated magnetic field, sp in noise spectrum\ncentered at one-half of the modulation frequency reveals sp in dynamics in a zero-field free of any\nlow-frequency noise.\nA sample of Nparamagnetic spins at thermal equi-\nlibrium generates spin fluctuations of the order of√\nN.\nAccording to the fluctuation-dissipation theorem [1], the\npower spectrum of fluctuations, whether they are clas-\nsical or quantum in nature, is proportional to the fre-\nquency response of the system to a small driving force,\nand vice versa. The measurement of such fluctuations,\nspin noise spectroscopy, was pioneered in the early 1980s\n[2, 3]. With the modern instrumental advancement such\nas real-time spectrum analyzer and ultra-fast digitizer, it\nhas become a powerful non-perturbative way to obtain\ninformation about the spin dynamics of various systems\nincluding atomic vapors [4], semiconductors [5–7], and\nquantum dots [8, 9]. The most efficient non-perturbative\nspin noise detection method is by measuring the Faraday\nrotation (FR) of an off-resonant linearly polarized beam\npassing through a strictly unpolarized sample [10, 11].\nFR is also widely used for calibrating the spin noise for\nquantummetrology,whereanunpolarizedsystemismore\nfavorable than a polarized one which is prone to convert-\ning classical noises into spin fluctuations or to introduce\nback-action noises, both of which scale as Nand over-\nwhelm the projection noise [12, 13].\nIn conventional FR spin noise measurements, a DC\nmagnetic field transverse to the probing direction is ap-\npliedtoshift spinnoisespectra(SNS) toahigh-frequency\nregion free from any technical noises, especially the dom-\ninating 1 /fnoise [2]. But spin dynamics in (near) zero-\nfieldcanbe verydifferentfromthatinlargefields[14–17].\nCross-correlation SNS has been used to study heteroge-\nnous interacting spin system in zero-field. [18]. Although\nSNS in zero-field can be obtained by subtracting SNS in\nzeroand high transversefield fromeach other[16], due to\nthe wandering 1 /fnoise, such approachonly workswhen\nthelinewidth orthepowerofthespinnoiseismuchlarger\nthose of the 1 /fnoise. Here we show that by using a π\npulse modulated (PM) field, one can observe the spin\ndynamics in zero-field with SNS signals shifted to one-\nhalf of the field modulation frequency. We demonstrate\nthis technique by studying the spin-exchange relaxation\n(SER) in a87Rb atomic vapor, which consists of two spin\nspecies with equal but opposite g-factors. We also pro-\nFaFb\nP/g84 /g83(a) (b)\nFa\nFb2( - )P/g83 /g84 ()zB t ()zB t\nFIG. 1. (color online) Spin precessions on the two hyperfine\nsublets in a π-PM (a) or an arbitrary θp-PM field (b).\npose a generalized π-PM field for studying interactions\nbetween spins with an arbitrary ratio of g-factors. Sup-\npressing SER by pulsed modulated field was first demon-\nstrated with 2 π-PM [19], which has no time-averaged\nspin precession. π-PM with synchronous σ+ andσ−\npumping has been used to achieve SERF magnetometry\nand to suppress the back-polarization field of 87Rb in a\nHelium-Neon comagnetometer [20]. It should be noted\nthat a very similar technique was independently devel-\noped by Zhang and Zhao to achieve SERF Bell-Bloom\nmagnetometry in large fields with phase sensitive detec-\ntion [21] following their proposal of using π-PM field to\nevadelight-shift-back-action-noise[22]. SNSinsinusoidal\nbias fields has been studied theoretically [23] on the 2nd-\norder spin correlations and experimentally on the 4th-\norder spin correlations [24]. Interestingly, ultra-high fre-\nquency SNS can be shifted to the low-frequency region\nwith a pulsed probe laser [25, 26].\nThe basicideaofourmethod is shownin Fig.1(a). The\nground state of alkali metal atoms has two hyperfine sub-\nletswithcorrespondingatomicspinnumbers Fa=I+1/2\nandFb=I−1/2, where Iis the nuclear spin number.\nThe two sublets have equal but opposite g-factors, thus\nFaandFbprecess in the opposite direction under a con-\nstant magnetic field. Without loss of generality, assum-\ning both spins are in the same direction initially, their\nrelative angle, as well as the magnitude of the total spin\nchanges coherently with time. If a random spin exchange\ncollision (SEC) takes place between a pair of atoms dur-\ning the precession, while the total spin is conserved, each\natom flips between the two hyperfine sublets, reversing\nits spin precession direction. Thus their coherence is de-\nstroyed and the magnitude of the total spin relaxes. If2\nwe replace the constant field by a π-PM one and assume\nthat the pulse-width is so narrow that SECs can only\ntake place between the pulses, then each pulse rotates\nFaby an angel of πandFbby−π, making them point-\ningin the same directionatthe end. The internalstateof\nthe atomic system is the same before and after the pulse,\nexcept for an overall spatial rotation which has no effect\non the microscopic SECs. Therefore, such a PM field is\nequivalent to a zero-field for SEC even though each spin\nreverses its direction pulse after pulse.\nWe first give a simple theory for the SNS in a PM\nfield. According to the Wiener-Khintchine theorem, the\npower spectral density (PSD) of a random spin fluctua-\ntion,F(t), is equal to the Fourier transform of the F(t)’s\nautocorrelation function, C(τ)≡ ∝ang⌊ra⌋ketleftF(t)F(t+τ)∝ang⌊ra⌋ketright,\nS(ν) = 2/integraldisplay∞\n0cos(2πντ)C(τ)dτ. (1)\nFor spins with a constant relaxation rate in a bias mag-\nnetic field, C(τ) is given by\nC(τ) =∝ang⌊ra⌋ketleftF2\n0∝ang⌊ra⌋ketrighte−Γ|τ|cos[θ(τ)], (2)\nwhere Γ is the transverse relaxation rate, θ(τ) is the spin\nprecession angle during time τand∝ang⌊ra⌋ketleftF2\n0∝ang⌊ra⌋ketright=F(F+1)/3 is\nthe variance of spin noise. Under a DC field of strength\nB,θ(τ) = 2πνLτ, where 2 πνL=gµBBis the Larmor\nfrequency with µBbeing the Bohr magneton. Assuming\nΓ<<2πνL, we have\nS(ν) =∝ang⌊ra⌋ketleftF2\n0∝ang⌊ra⌋ketright\n2πδ/2\n(ν−νL)2+δ2/4. (3)\nwhereδ= Γ/πisthefull-width-half-maximum(FWHM).\nIf the bias field is modulated with sharp pulses at fre-\nquencyνp,θ(τ) can be approximated by a staircase func-\ntion,θ(τ) =⌊(τ+τ′)/Tp⌋θp, whereTp= 1/νpis the time\ninterval between successive pulses, θpis the pulse area\nwhich is the spin precession angle caused by each pulse,\nτ′is the time delay between the occurrence of a random\nspin fluctuation and the last pulse, and ⌊x⌋is the floor\nfunction which gives the largest integer ≤x. Since spin\nfluctuation is a stationary random process, τ′should be\ndistributed between (0 ,Tp) uniformly [23]. Thus\nC(τ) =1\nTp/integraldisplayTp\n0dτ′∝ang⌊ra⌋ketleftF2\n0∝ang⌊ra⌋ketrighte−Γ|τ|cos[⌊(τ+τ′)/Tp⌋θp].(4)\nWhen Γ ≪¯νL, where 2 π¯νL=θp/Tpis the average Lar-\nmor frequency of the pulsed field, we find\nS(ν)∝∞/summationdisplay\nn=1,3,5+/summationdisplay\ns=−1−cos(θp)\nT2pν2n,sδ/2\n(ν−νn,s)2+δ2/4,(5)\nwhereνn,±= [n±(θp−π)/π]νp/2 represent the center\nfrequency of each resonance. Note, nsums over all the\nprobe BPD \nWollaston SA \nlinear polarizer y\nx z/g11/g12ZB t\ntTp\ndT p2/g79\nFIG. 2. (color online) Experiment setup and parameters of\nthePM field. BPD:balanced photodetector, Wollaston: Wol-\nlaston prism, λ/2: half-wave plate, d: duty cycle.\npositive odd numbers. When θp∝negationslash=π, each index ncorre-\nsponds to a doublet centered at nνp/2 with a frequency\nsplitting of νn,+−νn,−=νp(θp−π)/π. Whenθp=π,\nthe doublet merges into a single peak at nνp/2, and\nS(ν)∝∞/summationdisplay\nn=1,3,58\nn2δ/2\n(ν−nνp/2)2+δ2/4.(6)\nOur experimental set up is shown in Fig.2. A 2 .4cm\ncubic Pyrex cell containing87Rb with no buffer gas is\nplaced in a ceramic oven. The inner wall of the cell is\ncoated with octadecyltrichlorosilane [27] to preserve the\nspin coherence over several hundred wall collisions. The\noven is heated by a nonmagnetic wire and is well insu-\nlated so that after the heating current is turned off, the\ntemperatureofthecelldropslessthan0 .2Kwithin80sof\nsignal averaging time. A 30cm diameter Helmholtz coil\ncontrolled by a low noise current source (ADC6156) pro-\nvides the constant field. The pulsed-field is providedby a\n18cm diameter coil system driven by a homemade pulse\ncurrent supply controlled by a function generator. The\ninductance of pulse coil is 16 µH. The whole setup is en-\nclosedbyafour-layer µ-metalshield toreducethe residue\nfield below 1nT after degaussing. A 1MHz linewidth ex-\nternal cavity diode laser (Toptica DLpro) red-detuned\n1.8GHz from 87Rb D1 line F= 2 toF′= 1 transi-\ntion is used as the probe beam. The detuning is much\nlarger than the Doppler broadening (300MHz) and ho-\nmogeneous broadening ( ∼10MHz) of the optical reso-\nnance, ensuring negligible photon absorption. However,\nit is much smaller than the ground state hyperfine split-\nting so that the FR signal is mostly contributed by the\nspin orientation from the atoms on the F= 2 hyperfine\nsublet. The probe beam passes through a single mode\noptical fiber and is collimated into a beam of 4mm di-\nameter. It is then linearly polarized before entering the\nvapor cell with a power of 0 .5mW. The FR of the probe3\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48/s53/s46/s48/s69/s45/s49/s54/s49/s46/s48/s69/s45/s49/s53/s49/s46/s53/s69/s45/s49/s53/s50/s46/s48/s69/s45/s49/s53/s50/s46/s53/s69/s45/s49/s53\n/s49/s54/s46/s48 /s49/s54/s46/s53 /s49/s55/s46/s48 /s49/s55/s46/s53 /s49/s56/s46/s48/s48/s46/s48/s69/s43/s48/s48/s50/s46/s48/s69/s45/s49/s53/s52/s46/s48/s69/s45/s49/s53/s54/s46/s48/s69/s45/s49/s53/s80/s83/s68/s32/s40/s114/s97/s100/s50\n/s47/s72/s122/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s107/s72/s122/s41/s100/s32/s61/s32/s51/s50/s51 /s32/s72/s122/s32/s100/s32/s61/s32/s52/s54 /s32/s72/s122\n/s32/s32\nFIG. 3. (color online) SNSs under a DC field ( νL= 17kHz)\n(filled black dots) and a π-PM field ( νp= 34kHz ,d= 2.5%)\n(empty red circles). The cell temperature is 106◦C. The\nresolution of the spectrum analyzer for the main plot and\nthe inset is 250Hz and 8Hz, respectively. For the DC field,\nthe SNS peaks at 17kHz with a wide transit pedestal and\na narrow Ramsey peak. For the PM field, the two peaks\nat 34 and 68kHz, corresponding to integer multiples of νp,\nare parasitic signals stimulated by the magnetic pulses. Th e\nthreepeaksat17, 51and85kHz, correspondingtohalf-integ er\nmultiples of νp, are the SNS predicted by Eq.(6).\nis measured with a polarizing beam splitter and a bal-\nanced photodetector (Thorlabs PDB210A) whose output\nis fed into a spectrum analyzer (SRS760) for PSD mea-\nsurement. All spectra are linearly rms averaged 5000\ntimes. The time of averaging is about 80s for a 1 .6kHz\nspan and decreases with increasing span range.\nFig.3 compares the measured SNS in a DC and a π-\nPM field. The main plot shows the whole spectrum of\na 100kHz span while The inset shows a zoomed scan\naround the first Larmor resonance. Under the DC field,\nthe SNS centers at the Larmor frequency νLof the field,\nandcontainsawidepedestalduetothetransittimeeffect\nandanarrowpeakduetothewallinduced Ramseyeffect.\nFor the transit pedestal, the observed spin autocorrela-\ntion decays quickly as the atoms fly out of the probe\nregion, while for the Ramsey peak, the observed spin\nautocorrelation lasts over a much longer time as atoms\ntransverse the probe region many times due to coherence\npreserving collisions with the anti-relaxation cell wall.\nThe linewidth of the Ramsey peak is mostly contributed\nby the SER broadening which is qσnv/π∼290Hz, where\nσis the Rb SE cross-section ∼2×10−14cm2,nthe Rb\nnumber density ∼0.85×1012cm−3,vthe relative atomic\nspeed∼430m/s andq= 1/8 the nuclear slowing-down-\nfactor for the F= 2 sublevel [28]. The rest of about\n33Hz is contributed by wall relaxations. Under the π-\nPM field, the SNS is composed of multiple resonances\ncentered at the odd harmonics of the average Larmor\nfrequency νp/2 adjusted in our experiment to be equal\ntoνLof the DC field. The hight and width of the Ram-/s49/s54 /s49/s55 /s49/s56 /s53/s48 /s53/s49 /s53/s50 /s56/s52 /s56/s53 /s56/s54/s48/s46/s48/s69/s43/s48/s48/s49/s46/s48/s69/s45/s49/s53/s50/s46/s48/s69/s45/s49/s53/s51/s46/s48/s69/s45/s49/s53/s52/s46/s48/s69/s45/s49/s53/s53/s46/s48/s69/s45/s49/s53/s82/s97/s109/s115/s101/s121/s32/s72/s97/s114/s109/s111/s110/s105/s99/s115/s32/s40/s114/s97/s100/s50\n/s47/s72/s122/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s107/s72/s122/s41/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s110 /s32/s32/s49 /s32/s51 /s32/s53\n/s32/s32 /s32/s110\n/s110/s32/s40/s107/s72/s122/s41 /s32/s49/s55 /s53/s49 /s56/s53\n/s32/s32/s32/s32/s32/s100 /s32/s40/s72/s122/s41 /s32/s52/s55 /s52/s53 /s52/s54\n/s32/s32/s114/s101/s108/s46/s32/s104/s101/s105/s103/s104/s116/s32/s32/s32 /s32/s32/s49/s32/s32/s32/s32/s32/s32/s32/s48/s46/s49/s49/s32/s32/s32/s32/s48/s46/s48/s51/s56\n/s32/s114/s101/s108/s46/s32/s104/s101/s105/s103/s104/s116/s32/s40/s116/s41 /s32/s32/s49/s32/s32/s32/s32/s32/s32/s48/s46/s49/s49/s49/s32/s32/s32/s32/s48/s46/s48/s52\nFIG. 4. (color online) Joint plot of the SNS of all Ramsey\nharmonics in a π-PM field. Each peak is obtained by sub-\ntracting the transit pedestal from the spectrum scanned wit h\na 1.6kHz frequency span. The inset table lists all the parame-\nters for each harmonic with the last row being the theoretica l\nrelative peak heights, which are 1 ,1/32,1/52.\nsey peaks are inaccurate in the main plot due to its low\nfrequency-resolution. Fromthehighfrequency-resolution\ninset, it is clear that the Ramsey peak of the π-PM field\nis much narrower than that of the DC field. We will only\nfocus onthe line shape ofthe Ramseypeaksbecause they\ncorrespond to the single exponential relaxation assumed\nin our theory and contain the information of SECs.\nFig.4 plots all the Ramseyharmonics in the π-PM field\nwithin the 100kHz bandwidth of the spectrum analyzer.\nThe parameters of each peak are listed in the inset table.\nThe measured values agree very well with the theoretical\nresults given by Eq.(6). Because/summationtext∞\nn=1,3,51/n2=π2/8,\nthe 1st harmonic contains ∼81% of total noise power.\nThus it alone is strong enough for studying spin dynam-\nics, while all the other harmonics are just its small repli-\ncas.\nThe equivalence of a PM field to a zero-field depends\non two critical parameters, duty cycle dand pulse area\nθp. Whenθp=π, the PM field becomes more and more\nequivalent to a zero-field as d→0, since the probabil-\nity of SECs taking place during the pulse is equal to the\nduty cycle. As a result, the FWHM of the Ramsey peak\ndecreases linearly with the das is shown in Fig.5. On the\nother hand, as is shown in Fig.6, for d≪1, spin dynam-\nics and the line shape of the Ramsey peaks are governed\nbyθp, which can be changed in our experiment by vary-\ning the voltage of the pulse driver. The dependence of δ\nonθpat fixedνpanddis plotted in Fig.7. As shown in\nFig.1(b), when θpis close but not equal to π, each pulse\nchanges the relative angle between the atomic spins on\nthetwohyperfinesubletsby2( θp−π)foreverytimeinter-\nvalTp, equivalent to a DC field with Larmor frequency,\nωef=|θp−π|/Tpasfarastheevolutionofthisrelativean-\ngle is concerned. When ωefis nonzero, the orientation of\nspins gets more and more diffused after each pulse when4\n/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53 /s48/s46/s51/s48 /s48/s46/s51/s53 /s48/s46/s52/s48 /s48/s46/s52/s53/s50/s53/s53/s48/s55/s53/s49/s48/s48/s49/s50/s53/s49/s53/s48/s49/s55/s53/s50/s48/s48/s100 /s32/s40/s72/s122/s41\n/s68/s117/s116/s121/s32/s99/s121/s99/s108/s101/s121/s45/s105/s110/s116/s101/s114/s99/s101/s112/s116/s58/s32/s51/s54/s32/s72/s122\nFIG. 5. (color online) Duty cycle dependence of the δof the\n1st Ramsey peak. The extrapolated δat 0 dutycycle is 36Hz,\nwhich should be free of any SER and solely contributed by\nthe wall relaxation.\n/s49/s53 /s49/s54 /s49/s55 /s49/s56 /s49/s57/s113\n/s112/s32/s61/s32/s48/s46/s57/s57/s55 /s112/s113\n/s112/s32/s61/s32 /s112 /s32\n/s113\n/s112/s32/s61/s32/s48/s46/s57/s57/s50 /s112\n/s113\n/s112/s32/s61/s32/s48/s46/s57/s52/s51 /s112/s78/s111/s105/s115/s101/s32/s80/s83/s68/s32/s40/s97/s46/s117/s46/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s107/s72/s122/s41\nFIG. 6. (color online) SNSs with different pulse areas. The\nvalueθpis calculated from the frequency splitting of the dou-\nblet,νn,+−νn,−=νp(θp−π)/π. The experimental data\n(black circles) are fitted by a double Lorentzian of equal\nlinewidth (red solid line). The relative height of the two pe aks\nfor each θpdoes not agree with that of Eq.5 very well, espe-\ncially for θp> π. It might be caused by the asymmetric\n(shark-fin) shape of the actual field pulses, but the exactly\nreason is not understood yet.\nthe atoms jumps back and forth between the two hyper-\nfine states of different g-factors, causing SER broadening\nwhich saturates until ωefis much larger than the SEC\nrate. The exact dependence of SER on the strength of a\nDC field has been solved for polarized spins [28]. A more\nrigourous treatment may need the stochastic fluctuation\ntheory outlined in Ref.[18]. The solid line in Fig.7 is\nthe calculated total broadening using [28] with the SER\nrate as the only adjustable fitting parameter. The fitted\nSER rate is found to be corresponding to the Rb vapor\ndensity at 104◦C, which is in good agreement with our\nmeasured cell stem temperature of 105 .4◦C, since alkali\nvapor densities are usually lower in coated cells.\nThis method can also be extended to study the spin\ncorrelations between spin species of arbitrary g-factors./s45/s48/s46/s53 /s45/s48/s46/s52 /s45/s48/s46/s51 /s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48/s100 /s32/s40/s72/s122/s41\n/s40/s113\n/s112/s45/s112 /s41/s32/s47 /s112\nFIG. 7. (color online) Pulse area dependence of the δof the\n1st Ramsey peak with d= 5%. The Black squares are ex-\nperimental values and the blue solid line is the theoretical\ncalculation. The minimum δis about 50Hz, contributed by\nthe 36Hz wall relaxation and the 14Hz residue SER due to\nthe finite pulse width.\nπ\nx x \nx xy\nyy\nyz z\nz zπ/2 \nπ/2 s1\ns2\n s1s2\ns2s1\ns1s2\nFIG. 8. (color online) A π/2pulse for S1aboutz-axis followed\nby aπpulse for S2about the y-axis followed by a π/2 pulse\nforS1about z-axis effectively rotate both spins about the z-\naxis byπregardless of the ratio of their g-factors. The probe\nbeam is along the x direction.\nFor two spin species whose ratio of g-factors can be re-\nduced to a ratio of two odd numbers, e.g. n/m, annπ-\npulse for one spin is simultaneously an mπ-pulse for the\nother, hence an effective π-pulse for both spins. Even for\ntwo spins ofan arbitraryratio of g-factors, an effective π-\npulse can be formed by a sequence of pulses as illustrated\nin Fig.8. Assume both spins point in the x-direction ini-\ntially, the first pulse rotates S1about the z-axis by π/2\nand points it to the y-axis; the second pulse rotates S2\nabout the y-axis by π, causing it to lead/lag S1on the\nx-y plane by the same angle it lags/leads S1after the\nfirst pulse; the last pulse rotates S1about the z-axis by\nπ/2 again, brings both spins to the −x-direction at the\nend. The net result is that both spins rotated a πradian\nregardless of the ratio of their g-factors.\nAπ-PM field with duty cycle dand average Larmor\nfrequency ¯ νLrequires the pulse height and width to be5\n2π¯νL/dandd/2¯νLrespectively, which are limited by the\ncoil inductance in practice, since the coil has to be large\nenough to guarantee the field uniformity within the sam-\nple volume and its inductance is proportional to its size.\nFor our 2cm cubic cell, the preliminary current supply\ncan create field pulses up to 40KHz repetition rate at\nd= 2.5%. Such a limit can be easily increased by 3 to 4\norders of magnitude for micrometer-scale samples.\nIn summary, we show that spin dynamics of a strictly\nunpolarized system in zero-field can be revealed, free\nfrom the 1 /fnoise, with SNS in a π-pulse-modulated\nmagnetic field. In fact, any magnetic field pulse which\nendsupgeneratinganoverallrotationofthesystemwhile\nmaintaining the relative orientations of the internal spins\nis capable of creating a macroscopically spin oscillation\nsignal while keeping the system invariant for microscopic\nisotropic spin interactions. Therefore, this method is not\nonly limited to spin-exchange-collisions, and also the π-\npulse presented here is just a special case which gives the\nstrongest macroscopic signal. The same technique also\ncan be applied to polarized systems to suppress SER as\nwellasanyspinrelaxationswhoselongitudinalrelaxation\ntime is longer than the transverse one, which will ben-\nefit precision measurements such as magnetometry and\ngyroscope.\nWe thanks Kangjia Liao for helpful discussions. This\nwork is supported by National Key Research Program\nof China under Grant No. 2016YFA0302000, NNSFC\nunder Grant No. 91636102, and SNSF under Grant No.\n16ZR1402700. ZhangGYacknowledgesthesupportfrom\nNNSFC under Grant No. 11704335.\n∗zhaokf@fudan.edu.cn\n[1] R. Kubo, Rep. Prog. Phys. 29, 255 (1966).\n[2] E. B. Aleksandrov and V. S. Zapassky,\nZhurnal Eksperimentalnoi Teor. Fiz. 81, 132 (1981).\n[3] T. Sleator, E. L. Hahn, C. Hilbert, and J. Clarke,\nPhys. Rev. Lett. 55, 1742 (1985).\n[4] S. A. Crooker, D. G. Rickel, A. V. Balatsky, and D. L.\nSmith, Nature 431, 49 (2004).\n[5] M. Oestreich, M. Romer, R. J. Haug, and D. Hagele,\nPhys. Rev. Lett. 95, 216603 (2005).[6] G. M. Muller, M. Oestreich, M. Romer, and J. Hubner,\nPhysica E 43, 569 (2010).\n[7] J. Hubner, F. Berski, R. Dahbashi, and M. Oestreich,\nPhysica Status Solidi B 251, 1824 (2014).\n[8] S. A. Crooker, J. Brandt, C. Sandfort, A. Greilich, D. R.\nYakovlev, D. Reuter, A. D. Wieck, and M. Bayer,\nPhys. Rev. Lett. 104, 036601 (2010).\n[9] Y. Li, N. Sinitsyn, D. L. Smith, D. Reuter, A. D.\nWieck, D. R. Yakovlev, M. Bayer, and S. A. Crooker,\nPhys. Rev. Lett. 108, 186603 (2012).\n[10] V. S. Zapasskii, Adv. Opt. Photonics 5, 131 (2013).\n[11] N. A. Sinitsyn and Y. V. Pershin,\nRep. Prog. Phys. 79, 106501 (2016).\n[12] J. L. Sorensen, J. Hald, and E. S. Polzik,\nPhys. Rev. Lett. 80, 3487 (1998).\n[13] G. Vasilakis, V. Shah, and M. V. Romalis,\nPhys. Rev. Lett. 106, 143601 (2011).\n[14] W. Happer and H. Tang,\nPhys. Rev. Lett. 31, 273 (1973).\n[15] A. T. Dellis, M. Loulakis, and I. K. Kominis,\nPhys. Rev. A 90, 032705 (2014).\n[16] R. Dahbashi, J. Hubner, F. Berski, K. Pierz, and\nM. Oestreich, Phys. Rev. Lett. 112, 156601 (2014).\n[17] F. Berski, J. Huebner, M. Oestreich, A. Lud-\nwig, A. D. Wieck, and M. Glazov,\nPhys. Rev. Lett. 115, 176601 (2015).\n[18] D. Roy, L. Y. Yang, S. A. Crooker, and N. A. Sinitsyn,\nSci Rep 5, 9573 (2015).\n[19] A. Korver, R. Wyllie, B. Lancor, and T. G. Walker,\nPhys. Rev. Lett. 111, 043002 (2013).\n[20] M. Limes, D. Sheng, and M. Romalis,\nPhys. Rev. Lett. 120, 033401 (2018), 00000.\n[21] G. Zhang, Atomic magnetometry in vapor cells with anti-\nrelaxation coatings , Ph.D., Fudan University, Shanghai\n(2016).\n[22] G.-Y. Zhang and K.-F. Zhao,\nPhys. Rev. A 93, 033841 (2016).\n[23] M. BraunandJ. Konig,Phys. Rev. B 75, 085310 (2007).\n[24] F. X. Li, S. A. Crooker, and N. A. Sinitsyn,\nPhys. Rev. A 93, 033814 (2016).\n[25] G. M. Muller, M. Romer, J. Hubner, and M. Oestreich,\nPhys. Rev. B 81, 121202(R) (2010).\n[26] S. Starosielec and D. Hagele,\nAppl. Phys. Lett. 93, 051116 (2008).\n[27] G. Zhang, L. Wei, M. Wang, and K. Zhao,\nJ. Appl. Phys. 117, 043106 (2015).\n[28] W. Happer and A. C. Tam,\nPhys. Rev. A 16, 1877 (1977)."    },    {        "title": "2206.00784v2.Substrate_Effects_on_Spin_Relaxation_in_Two_Dimensional_Dirac_Materials_with_Strong_Spin_Orbit_Coupling.pdf",        "content": "Substrate E\u000bects on Spin Relaxation in Two-Dimensional Dirac Materials with Strong\nSpin-Orbit Coupling\nJunqing Xu1,\u0003and Yuan Ping1,y\n1Department of Chemistry and Biochemistry, University of California, Santa Cruz, CA 95064, USA\n(Dated: December 6, 2022)\nUnderstanding substrate e\u000bects on spin dynamics and relaxation in two-dimensional (2D) mate-\nrials is of key importance for spintronics and quantum information applications. However, the key\nfactors that determine the substrate e\u000bect on spin relaxation, in particular for materials with strong\nspin-orbit coupling, have not been well understood. Here we performed \frst-principles real-time\ndensity-matrix dynamics simulations with spin-orbit coupling (SOC) and quantum descriptions of\nelectron-phonon and electron-impurity scattering for the spin lifetimes of supported/free-standing\ngermanene, a prototypical strong SOC 2D Dirac material. We show that the e\u000bects of di\u000ber-\nent substrates on spin lifetime ( \u001cs) can surprisingly di\u000ber by two orders of magnitude. We \fnd\nthat substrate e\u000bects on \u001csare closely related to substrate-induced modi\fcations of the SOC-\feld\nanisotropy, which changes the spin-\rip scattering matrix elements. We propose a new electronic\nquantity, named spin-\rip angle \u0012\"#, to characterize spin relaxation caused by intervalley spin-\rip\nscattering. We \fnd that the spin relaxation rate is approximately proportional to the averaged value\nof sin2\u0000\n\u0012\"#=2\u0001\n, which can be used as a guiding parameter of controlling spin relaxation.\nINTRODUCTION\nSince the long spin di\u000busion length ( ls) in large-area\ngraphene was \frst reported by Tombros et al.[1], sig-\nni\fcant advances have been made in the \feld of spin-\ntronics, which has the potential to realize low-power\nelectronics by utilizing spin as the information car-\nrier. Various 2D materials have shown promising spin-\ntronic properties[2], e.g., long lsat room temperatures\nin graphene[3] and ultrathin black phosphorus[4], spin-\nvalley locking (SVL) and ultralong spin lifetime \u001csat\nlow temperatures in transition metal dichalcogenides\n(TMDs)[5] and germanene[6], and persistent spin helix\nin 2D hybrid perovskites[7].\nUnderstanding spin relaxation and transport mecha-\nnism in materials is of key importance for spintronics\nand spin-based quantum information technologies. One\ncritical metric for ideal materials in such applications is\nspin lifetime ( \u001cs), often required to be su\u000eciently long\nfor stable detection and manipulation of spin. In 2D-\nmaterial-based spintronic devices, the materials are usu-\nally supported on a substrate. Therefore, for the design\nof those devices, it is crucial to understand substrate ef-\nfects on spin relaxation. In past work, the substrate ef-\nfects were mostly studied for weak SOC Dirac materials\nlike graphene[8{12]. How substrates a\u000bect strong SOC\nDirac materials like germanene is unknown. In partic-\nular, the spin relaxation mechanism between weak and\nstrong SOC Dirac materials was shown to be drastically\ndi\u000berent. [6] Therefore, careful investigations are required\nto unveil the distinct substrate e\u000bects on these two types\nof materials.\nHere we focus on the dangling-bond-free insulating\n\u0003jxu153@ucsc.edu\nyyuanping@ucsc.edusubstrates, which interact weakly with the material thus\npreserve its main physical properties. Insulating sub-\nstrates can a\u000bect spin dynamics and relaxation in sev-\neral aspects: (i) They may induce strong SOC \felds, so\ncalled internal magnetic \felds Binby breaking inversion\nsymmetry[9] or through proximity e\u000bects[10]. For ex-\nample, the hexagonal boron nitride substrate can induce\nRashba-like \felds on graphene and dramatically accel-\nerate its spin relaxation and enhance the anisotropy of\n\u001csbetween in-plane and out-of-plane directions[8]. (ii)\nSubstrates may introduce additional impurities [11, 12]\nor reduce impurities/defects in material layers, e.g.,\nby encapsulation[13]. In consequence, substrates may\nchange the electron-impurity (e-i) scattering strength,\nwhich a\u000bects spin relaxation through SOC. (iii) Ther-\nmal vibrations of substrate atoms can introduce addi-\ntional spin-phonon scattering by interacting with spins\nof materials[9].\nPreviously most theoretical studies of substrate e\u000bects\non spin relaxation were done based on model Hamil-\ntonian and simpli\fed spin relaxation models[9, 11, 12].\nWhile those models provide rich mechanistic insights,\nthey are lack of predictive power and quantitative ac-\ncuracy, compared to \frst-principles theory. On the other\nhand, most \frst-principles studies only simulated the\nband structures and spin polarizations/textures of the\nheterostructures[14{16], which are not adequate for un-\nderstanding spin relaxation. Recently, with our newly-\ndeveloped \frst-principles density-matrix (FPDM) dy-\nnamics approach, we studied the hBN substrate e\u000bect on\nspin relaxation of graphene, a weak SOC Dirac material.\nWe found a dominant D'yakonov-Perel' (DP) mechanism\nand nontrivial modi\fcation of SOC \felds and electron-\nphonon coupling by substrates[8]. However, strong SOC\nDirac materials can have a di\u000berent spin relaxation mech-\nanism - Elliott-Yafet (EY) mechanism[17], with only\nspin-\rip transition and no spin precession, unlike the DParXiv:2206.00784v2  [cond-mat.mes-hall]  4 Dec 20222\nmechanism. How substrates a\u000bect spin relaxation of ma-\nterials dominated by EY mechanism is the key question\nhere. Furthermore, how such e\u000bects vary among di\u000ber-\nent substrates is another outstanding question for guiding\nexperimental design of interfaces.\nIn our recent study, we have predicted that mono-\nlayer germanene (ML-Ge) is a promising material for\nspin-valleytronic applications, due to its excellent prop-\nerties including spin-valley locking, long \u001csandls, and\nhighly tunable spin properties by varying gates and ex-\nternal \felds[6]. As discussed in Ref. 6, ML-Ge has strong\nintrinsic SOC unlike graphene and silicene. Under an\nout-of-plane electric \feld (in consequence broken inver-\nsion symmetry), a strong out-of-plane internal magnetic\n\feld forms, which may lead to mostly EY spin relax-\nation [6]. Therefore, predicting \u001csof supported ML-Ge\nis important for future applications and our understand-\ning of substrate e\u000bects on strong SOC materials. Here,\nwe examine the substrate e\u000bects on spin relaxation in\nML-Ge through FPDM simulations, with self-consistent\nSOC and quantum descriptions of e-ph and e-i scatter-\ning processes[6, 8, 18{20]. We study free-standing ML-\nGe and ML-Ge supported by four di\u000berent insulating\nsubstrates - germanane (GeH), silicane (SiH), GaTe and\nInSe. The choice of substrates is based on similar lat-\ntice constants to ML-Ge, preservation of Dirac Cones,\nand experimental synthesis accessibility[21, 22]. We will\n\frst show how electronic structures and \u001csof ML-Ge\nare changed by di\u000berent substrates - while \u001csof ML-\nGe on GeH and SiH are similar to free-standing ML-\nGe, the GaTe and InSe substrates strongly reduce \u001csof\nML-Ge due to stronger interlayer interactions. We then\ndiscuss what quantities are responsible for the disparate\nsubstrate e\u000bects on spin relaxation, which eventually an-\nswered the outstanding questions we raised earlier.\nRESULTS AND DISCUSSIONS\nSubstrate e\u000bects on electronic structure and spin\ntexture\nWe begin with comparing band structures and spin\ntextures of free-standing and supported ML-Ge in Fig. 1,\nwhich are essential for understanding spin relaxation\nmechanisms. Since one of the most important e\u000bects of\na substrate is to induce an out-of-plane electric \feld Ez\non the material layer, we also study ML-Ge under a con-\nstantEzas a reference. The choice of the Ezis based on\nreproducing a similar band splitting to the one in ML-Ge\nwith substrates. The band structure of ML-Ge is similar\nto graphene with two Dirac cones at KandK0\u0011\u0000K,\nbut a larger band gap of 23 meV. At Ez= 0, due to\ntime-reversal and inversion symmetries of ML-Ge, every\ntwo bands form a Kramers degenerate pair[17]. A \fnite\nEzor a substrate breaks the inversion symmetry and in-\nduces a strong out-of-plane internal B \feld Bin(Eq. 21),\nwhich splits the Kramers pairs into spin-up and spin-down bands[6]. Interestingly, we \fnd that band struc-\ntures of ML-Ge-SiH (Fig. 1c) and ML-Ge-GeH (Fig. S4)\nare quite similar to free-standing ML-Ge under Ez=-7\nV/nm (ML-Ge@-7V/nm, Fig. 1b), which indicates that\nthe impact of the SiH/GeH substrate on band structure\nandBinmay be similar to a \fnite Ez(see Fig. S4). This\nsimilarity is frequently assumed in model Hamiltonian\nstudies[9, 11]. On the other hand, the band structures of\nML-Ge-InSe (Fig. 1d) and ML-Ge-GaTe (Fig. S4) have\nmore di\u000berences from the free-standing one under Ez,\nwith larger band gaps, smaller band curvatures at Dirac\nCones, and larger electron-hole asymmetry of band split-\ntings. This implies that the impact of the InSe/GaTe\nsubstrates can not be approximated by applying an Ez\nto the free-standing ML-Ge, unlike SiH/GeH substrates.\nWe further examine the spin expectation value vectors\nSexpof substrate-supported ML-Ge. Sexpis parallel to\nBinby de\fnition (Eq. 21). Sexp\u0011\u0000\nSexp\nx;Sexp\ny;Sexp\nz\u0001\nwithSexp\nibeing spin expectation value along direction i\nand is the diagonal element of spin matrix siin Bloch\nbasis. Importantly, from Fig. 1e and 1f, although Sexpof\nML-Ge on substrates are highly polarized along z(out-of-\nplane) direction, the in-plane components of Sexpof ML-\nGe-InSe (and ML-Ge-GaTe) are much more pronounced\nthan ML-Ge-SiH (and ML-Ge-GeH). Such di\u000berences are\ncrucial to the out-of-plane spin relaxation as discussed in\na later subsection.\nSpin lifetimes of germanene on substrates and spin\nrelaxation mechanism\nWe then perform our \frst-principles density-matrix\ncalculation [6, 18{20] at proposed interfaces, and examine\nthe role of electron-phonon coupling in spin relaxation of\nML-Ge at di\u000berent substrates. Throughout this paper,\nwe focus on out-of-plane \u001csof ML-Ge systems, since their\nin-plane\u001csis too short and less interesting. We com-\npare out-of-plane \u001csdue to e-ph scattering between the\nfree-standing ML-Ge (with/without an electric \feld) and\nML-Ge on di\u000berent substrates in Fig. 2a. Here we show\nelectron\u001csfor most ML-Ge/substrate systems as intrin-\nsic semiconductors, except hole \u001csfor the ML-Ge-InSe\ninterface. This choice is because electron \u001csare mostly\nlonger than hole \u001csat lowTexcept for the one at the\nML-Ge-InSe interface; longer lifetime is often more ad-\nvantageous for spintronics applications. From Fig. 2, we\n\fnd that\u001csof ML-Ge under Ez= 0 and -7 V/nm are\nat the same order of magnitude for a wide range of tem-\nperatures. The di\u000berences are only considerable at low\nT, e.g, by 3-4 times at 20 K. On the other hand, \u001csof\nsupported ML-Ge are very sensitive to the speci\fc sub-\nstrates. While \u001csof ML-Ge-GeH and ML-Ge-SiH have\nthe same order of magnitude as the free-standing ML-\nGe, in particular very close between ML-Ge-GeH and\nML-Ge@-7 V/nm, \u001csof ML-Ge-GaTe and ML-Ge-InSe\nare shorter by at least 1-2 orders of magnitude in the\nwhole temperature range. This separates the substrates3\nFIG. 1. Band structures and spin textures around the Dirac cones of ML-Ge systems with and without substrates. (a)-(d) show\nband structures of ML-Ge under Ez= 0 and under -7 V/nm and ML-Ge on silicane (SiH) and on InSe substrates respectively.\n(e) and (f) show spin textures in the kx-kyplane and 3D plots of the spin vectors Sexp\nk1on the circlej\u0000 !kj= 0:005 bohr\u00001of\nthe band at the band edge around Kof ML-Ge on SiH and InSe substrates respectively. Sexp\u0011\u0000\nSexp\nx;Sexp\ny;Sexp\nz\u0001\nwithSexp\ni\nbeing spin expectation value along direction iand is the diagonal element of spin matrix siin Bloch basis. The red and blue\nbands correspond to spin-up and spin-down states. Due to time-reversal symmetry, band structures around another Dirac cone\natK0=\u0000Kare the same except that the spin-up and spin-down bands are reversed. The grey, white, blue, pink and green\nballs correspond to Ge, H, Si, In and Se atoms, respectively. Band structures of ML-Ge on germanane (GeH) and GaTe are\nshown in Fig. S4 in the Supporting Information, and are similar to those of ML-Ge on SiH and InSe substrates, respectively.\nIn subplots (e) and (f), the color scales Sexp\nzand the arrow length scales the vector length of in-plane spin expectation value.\ninto two categories, i.e. with a weak e\u000bect (ML-Ge-GeH\nand ML-Ge-SiH) and a strong e\u000bect (ML-Ge-GaTe and\nML-Ge-InSe).\nWe further investigate the role of electron-impurity (e-\ni) scattering in spin relaxation under di\u000berent substrates,\nby introducing defects in the material layer. We consider\na common type of impurity - single neutral Ge vacancy,\nwhose formation energy was found relatively low in previ-\nous theoretical studies[23, 24]. From Fig. 2b, we can see\nthat\u001csof all \fve systems decrease with impurity density\nni. Since carrier scattering rates \u001c\u00001\np(carrier lifetime \u001cp)\nincreases (decrease) with ni, we then obtain \u001csdecreases\nwith\u001cp's decrease, an evidence of EY spin relaxation\nmechanism. Moreover, we \fnd that \u001csis sensitive to the\ntype of the substrate with all values of ni, and for each of\nfour substrates, \u001csis reduced by a similar amount with\ndi\u000berentni, from low density limit (109cm\u00002, where e-\nph scattering dominates) to relatively high density (1012\ncm\u00002, where e-i scattering becomes more important).\nSince the bands near the Fermi energy are composed\nof the Dirac cone electrons around KandK0valleys in\nML-Ge, spin relaxation process arises from intervalleyand intravalley e-ph scatterings. We then examine rel-\native intervalley spin relaxation contribution \u0011(see its\nde\fnition in the Fig. 2 caption) in Fig. 2c. \u0011being close\nto 1 or 0 corresponds to intervalley or intravalley scatter-\ning being dominant in spin relaxation. \u0011becomes close\nto 1 below 70 K for electrons of ML-Ge-SiH, and below\n120 K for holes of ML-Ge-InSe. This indicates that at\nlowTonly intervalley scattering processes are relevant\nto spin relaxation in ML-Ge on substrates. This is a re-\nsult of spin-valley locking (SVL), i.e. large SOC-induced\nband splittings lock up or down spin with a particular K\nor K' valley [6]. According to Fig. 1 and 2c, the SVL\ntransition temperature ( TSVL; below which the propor-\ntion of intervalley spin relaxation rate \u0011is close to 1)\nseems approximately proportional to SOC splitting en-\nergy \u0001SOC, e.g. for electrons (CBM) of ML-Ge-GaTe and\nML-Ge-SiH, and for holes (VBM) of ML-Ge-InSe, \u0001SOC\nare\u001815,\u001824 and 40 meV respectively, while TSVLare\n50, 70 and 120 K respectively. As \u0001SOCcan be tuned by\nEzand the substrate, TSVLcan be tuned simultaneously.\nUnder SVL condition, spin or valley lifetime tends to be\nexceptionally long, which is ideal for spin-/valley-tronic4\nFIG. 2. The out-of-plane spin lifetime \u001csof intrinsic free-standing and substrate-supported ML-Ge. (a) \u001csof ML-Ge under\nEz= 0, -7 V/nm and substrate-supported ML-Ge as a function of temperature without impurities. Here we show electron \u001cs\nfor intrinsic ML-Ge systems except that hole \u001csis shown for ML-Ge-InSe, since electron \u001csare longer than hole \u001csat lowT\nexcept ML-Ge-InSe. (b) \u001csas a function of impurity density niat 50 K. The impurities are neutral ML-Ge vacancy with 50% at\nhigher positions and 50% at lower ones of a Ge layer. The dashed vertical line corresponds to the impurity density where e-ph\nand e-i scatterings contribute equally to spin relaxation ( ni;s). And e-ph (e-i) scattering is more dominant if ni<(>)ni;s. (c)\nThe proportion of intervalley spin relaxation contribution \u0011of (electrons of) ML-Ge-SiH and (holes of) ML-Ge-InSe without\nimpurities. \u0011is de\fned as \u0011=(\u001cinter\ns;z)\u00001\n(\u001cinters;z)\u00001+w(\u001cintras;z)\u00001, where\u001cinter\ns;z and\u001cintra\ns;z are intervalley and intravalley spin lifetimes,\ncorresponding to scattering processes between KandK0valleys and within a single KorK0valley, respectively. \u0011being close\nto 1 or 0 corresponds to dominant intervalley or intravalley spin relaxation, respectively. wis a weight factor related to what\npercentage of total Szcan be relaxed out by intravalley scattering itself. wbeing close to 0 and 1 correspond to the cases that\nintravalley scattering can only relax a small part (0) and most of excess spin (1) respectively. In Supporting Information Sec.\nSII, we give more details about de\fnition of w. (d) Electron and hole \u001csat 20 K of ML-Ge without impurities on hydrogen-\nterminated multilayer Si, labeled as Si nH withnbeing number of Si layers. Si nH is silicane if n= 1, and hydrogen-terminated\nSilicon (111) surface if n=1.\napplications.\nAdditionally, the studied substrates here are mono-\nlayer, while practically multilayers or bulk are more com-\nmon, thus it is necessary to understand how \u001cschanges\nwith the number of substrate layers. In Fig. 2d, we\nshow\u001csat 20 K of ML-Ge on hydrogen-terminated mul-tilayer Si, ML-Ge-Si nH, withnbeing number of Si layer.\nSinH becomes hydrogen-terminated Silicon (111) surface\nifn=1. We \fnd that \u001csare changed by only 30%-40%\nby increasing nfrom 1 to 3 and kept unchanged after\nn\u00153. For generality of our conclusion, we also test\nthe layer dependence of a di\u000berent substrate. We found5\nthe\u001csof ML-Ge on bilayer InSe ( n= 2) is changed by\n\u00188% compared to monolayer InSe at 20 K, even smaller\nchange than the one at Si nH substrates. Given the dis-\nparate properties of these two substrates, we conclude\nusing a monolayer is a reasonable choice for simulating\nthe substrate e\u000bects on \u001csin this work.\nThe correlation of electronic structure and phonon\nproperties to spin relaxation at di\u000berent substrates\nWe next analyze in detail the relevant physical quan-\ntities, and determine the key factors responsible for sub-\nstrate e\u000bects on spin relaxation. We focus on results\nunder lowTas spin relaxation properties are superior at\nlowerT(the realization of SVL and longer \u001cs).\nFirst, to have a qualitative understanding of the\nmaterial-substrate interaction strength, we show charge\ndensity distribution at the cross-section of interfaces in\nFig. 3a-d. It seems that four substrates can be catego-\nrized into two groups: group A contains GeH and SiH\nwith lower charge density distribution in the bonding re-\ngions (pointed by the arrows); group B contains GaTe\nand InSe with higher charge density distribution in the\nbonding regions. In Fig. S5, we investigate the charge\ndensity change \u0001 \u001ae(de\fned by the charge density dif-\nference between interfaces and individual components).\nConsistent with Fig. 3, we \fnd that \u0001 \u001aefor GaTe and\nInSe substrates overall has larger magnitude than the\none for GeH and SiH substrates. Therefore the material-\nsubstrate interactions of group B seem stronger than\nthose of group A. Intuitively, we may expect that the\nstronger the interaction, the stronger the substrate e\u000bect\nis. The FPDM simulations in Fig. 2a-b indeed show that\nthe substrate e\u000bects of group B being stronger than those\nof group A on \u001cs, consistent with the above intuition.\nNext we examine electronic quantities closely related\nto spin-\rip scattering responsible to EY spin relaxation.\nQualitatively, for a state k1, its spin-\rip scattering rate\n\u001c\u00001\ns(k1) is proportional to the number of its pair states\nk2allowing spin-\rip transitions between them. The num-\nber of pair states is approximately proportional to den-\nsity of states (DOS) around the energy of k1. Moreover,\nfor EY mechanism, it is commonly assumed that spin\nrelaxation rate is proportional to the degree of mixture\nof spin-up and spin-down states (along the zdirection\nhere), so called \\spin-mixing\" parameter[17] b2\nz(see its\nde\fnition in Sec. SII), i.e., \u001c\u00001\ns/\nb2\nz\u000b\n, where\nb2\nz\u000b\nis the\nstatistically averaged spin mixing parameter as de\fned in\nRef. 6. Therefore, we show DOS, energy-resolved spin-\nmixingb2\nz(\") and\nb2\nz\u000b\nas a function of temperature in\nFig. 3e-g.\nWe \fnd that in Fig. 3e DOS of ML-Ge-GeH and ML-\nGe-SiH are quite close to that of ML-Ge@-7V/nm, while\nDOS of ML-Ge-GaTe and ML-Ge-InSe are 50%-100%\nhigher around the band edge. Such DOS di\u000berences\nare qualitatively explained by the staggered potentials of\nML-Ge-GaTe and ML-Ge-InSe being greater than thoseof ML-Ge-GeH and ML-Ge-SiH according to the model\nHamiltonian proposed in Ref. 25. In Fig. 3f-g, b2\nzof ML-\nGe-GeH and ML-Ge-SiH are found similar to ML-Ge@-\n7 V/nm, and not sensitive to energy and temperature.\nOn the contrast, for ML-Ge-GaTe and ML-Ge-InSe, their\nb2\nz(\") and\nb2\nz\u000b\nincrease rapidly with energy and temper-\nature. Speci\fcally, we can see at 300 K,\nb2\nz\u000b\nof ML-Ge-\nGaTe and ML-Ge-InSe are about 4-20 times of the one\nof ML-Ge-GeH and ML-Ge-SiH in Fig. 3g. Thus the one\norder of magnitude di\u000berence of \u001csbetween group A (ML-\nGe-GeH and ML-Ge-SiH) and group B (ML-Ge-GaTe\nand ML-Ge-InSe) substrates at 300 K can be largely ex-\nplained by the substrate-induced changes of DOS and\nb2\nz\u000b\n. On the other hand, at low T, e.g., at 50 K,\nb2\nz\u000b\nof ML-Ge-GaTe and ML-Ge-InSe are only about 1.5 and\n2.5 times of the ones of ML-Ge-GeH and ML-Ge-SiH,\nand DOS are only tens of percent higher. However, there\nis still 1-2 order of magnitude di\u000berence of \u001csbetween\ndi\u000berent substrates. Therefore, the substrate e\u000bects on\n\u001cscan not be fully explained by the changes of\nb2\nz\u000b\nand\nDOS, in particular at relatively low temperature.\nWe then examine if substrate-induced modi\fcations of\nphonon can explain the changes of spin relaxation at dif-\nferent substrates, especially at low T. We emphasize that\nat lowT, since spin relaxation is fully determined by\nintervalley processes (Fig. 2c), the related phonons are\nmostly close to wavevector K. From Fig. 4, we \fnd that\nthe most important phonon mode for spin relaxation at\nlowThas several similar features: (i) It contributes to\nmore than 60% of spin relaxation (see Fig 4a). (ii) Its\nenergy is around 7 meV in the table of Fig. 4a. (iii)\nIts vibration is \rexural-like, i.e., atoms mostly vibrate\nalong the out-of-plane direction as shown in Fig. 4b-\nd. Moreover, for this mode, the substrate atoms have\nnegligible thermal vibration amplitude compared to the\none of the materials atoms. This is also con\frmed in\nthe layer-projected phonon dispersion of ML-Ge-InSe in\nFig. 4e. The purple box highlights the critical phonon\nmode around K, with most contribution from the mate-\nrial layer. (iv) The critical phonon mode does not couple\nwith the substrate strongly, since its vibration frequency\ndoes not change much when substrate atoms are \fxed\n(by comparing Fig. 4e with f). We thus conclude that\nthe substrate-induced modi\fcations of phonons and ther-\nmal vibrations of substrate atoms seem not important for\nspin relaxation at low T(e.g. below 20 K).\nTherefore, neither the simple electronic quantities\nb2\u000b\nand DOS nor the phonon properties can explain the sub-\nstrate e\u000bects on spin relaxation at low T.\nThe determining factors of spin relaxation derived\nfrom spin-\rip matrix elements\nOn the other hand, with a simpli\fed picture of spin-\n\rip transition by the Fermi's Golden Rule, the scattering\nrate is proportional to the modulus square of the scat-\ntering matrix elements. For a further mechanistic un-6\nFIG. 3. Charge density, density of states (DOS), and spin mixing parameters of free-standing and substrate-supported ML-Ge.\nCross-section views of charge density at interfaces of ML-Ge on (a) GeH, (b) SiH, (c) GaTe, and (d) InSe. The Ge layers\nare above the substrate layers. The unit of charge density is e=bohr3. Charge densities in the regions pointed out by black\narrows show signi\fcant di\u000berences among di\u000berent systems. (e) DOS and (f) energy-solved spin-mixing parameter along zaxis\nb2\nz(\") of ML-Ge under Ez=-7 V/nm and on di\u000berent substrates. \"edgeis the band edge energy at the valence band maximum\nor conduction band minimum. The step or sudden jump in the DOS curve corresponds to the edge energy of the second\nconduction/valence band or the SOC-induced splitting energy at K. (g) The temperature-dependent e\u000bective spin-mixing\nparameter\nb2\nz\u000b\nof various ML-Ge systems.\nderstanding, we turn to examine the modulus square of\nthe spin-\rip matrix elements, and compare their qual-\nitative trend with our FPDM simulations. Note that\nmost matrix elements are irrelevant to spin relaxation\nand we need to pick the \\more relevant\" ones, by de\fning\na statistically-averaged function. Therefore, we propose\nan e\u000bective band-edge-averaged spin-\rip matrix element\njeg\"#j2(Eq. 8). Here the spin-\rip matrix element can be\nfor general scattering processes; in the following we focus\non e-ph process for simplicity. We also propose a so-called\nscattering density of states DSin Eq. 9, which measures\nthe density of spin-\rip transitions and can be roughly re-\ngarded as a weighted-averaged value of the usual DOS.\nBased on the generalized Fermi's golden rule, we approx-\nimately have \u001c\u00001\ns/jeg\"#j2DSfor EY spin relaxation (see\nthe discussions above Eq. 11 in \\Methods\" section).\nAs shown in Fig. 5a, \u001c\u00001\nsis almost linearly propor-\ntional tojeg\"#j2DSat 20 K. As the variation of DSamong\nML-Ge on di\u000berent substrates is at most three times (see\nFig. 3e and Fig. S6), which is much weaker than the\nlarge variation of \u001c\u00001\ns, this indicates that the substrate-\ninduced change of \u001csis mostly due to the substrate-\ninduced change of spin-\rip matrix elements. Although\njeg\"#j2was often considered approximately proportionalto\nb2\u000b\n, resulting in \u001c\u00001\ns/\nb2\u000b\n, our results in Fig. 3\nin the earlier section indicate that such simple approx-\nimation is not applicable here, especially inadequate of\nexplaining substrate dependence of \u001csat lowT.\nTo \fnd out the reason why jeg\"#j2for di\u000berent sub-\nstrates are so di\u000berent, we \frst examine the averaged\nspin-\rip wavefunction overlap jo\"#j2(with the reciprocal\nlattice vector G= 0), closely related to jeg\"#j2(Eq. 18\nand Eq. 17). From Fig. 5b, \u001c\u00001\nsandjo\"#j2have the same\ntrend, which implies jeg\"#j2andjo\"#j2may have the same\ntrend. However, in general, the G6=0elements ofjo\"#j2\nmay be important as well, which can not be unambigu-\nously evaluated here. (See detailed discussions in the\nsubsection \\Spin-\rip e-ph and overlap matrix element\"\nin the \\Methods\" section).\nTo have deeper intuitive understanding, we then pro-\npose an important electronic quantity for intervalley\nspin-\rip scattering - the spin-\rip angle \u0012\"#between two\nelectronic states. For two states ( k1;n1) and (k2;n2) with\nopposite spin directions, \u0012\"#is the angle between \u0000Sexp\nk1n1\nandSexp\nk2n2or equivalently the angle between \u0000Bin\nk1and\nBin\nk2.\nThe motivation of examining \u0012\"#is that: Suppose two\nwavevectors k1andk2=\u0000k1are in two opposite valleys7\n(a)\nSubstrate !K(meV) Contribution\nGe@-7V/nm 7.7 78%\nGe-GeH 6.9 70%\nGe-SiH 7.1 64%\nGe-GaTe 6.4 90%\nGe-InSe 7.2 99%\nFIG. 4. (a) The phonon energy at wavevector Kof the mode\nthat contributes the most to spin relaxation, and the per-\ncentage of its contribution for various systems at 20 K. We\nconsider momentum transfer K, as spin relaxation is fully de-\ntermined by intervalley processes between KandK0valleys.\n(b), (c) and (d) Typical vibrations of atoms in 3 \u00023 supercells\nof (b) ML-Ge@-7 V/nm, (c) ML-Ge-SiH, and (d) ML-Ge-\nInSe of the most important phonon mode at Karound 7 meV\n(shown in (a)). The red arrows represent displacement. The\natomic displacements smaller than 10% of the strongest are\nnot shown. (e) The layer-projected phonon dispersion of ML-\nGe-InSe within 12 meV. The red and blue colors correspond\nto the phonon displacements mostly contributed from the ma-\nterial (red) and substrate layer (blue) respectively. The green\ncolor means the contribution to the phonon displacements\nfrom the material and substrate layers are similar. The pur-\nple boxes highlight the two most important phonon modes\naroundKfor spin relaxation.(f) Phonon dispersion of ML-\nGe-InSe within 12 meV with substrate atoms (InSe) being\n\fxed at equilibrium structure and only Ge atoms are allowed\nto vibrate.\nQand -Qrespectively and there is a pair of bands, which\nare originally Kramers degenerate but splitted by Bin.\nDue to time-reversal symmetry, we have Bin\nk1=\u0000Bin\nk2,\nwhich means the two states at the same band natk1\nandk2have opposite spins and \u0012\"#between them is\nzero. Therefore, the matrix element of operator bAbe-\ntween states ( k1;n) and (k2;n) -Ak1n;k2nis a spin-\ripone and we name it as A\"#\nk1k2. According to Ref. 26,\nwith time-reversal symmetry, A\"#\nk1k2is exactly zero. In\ngeneral, for another wavevector k3within valley - Qbut\nnot\u0000k1,A\"#\nk1k3is usually non-zero. One critical quan-\ntity that determines the intervalley spin-\rip matrix ele-\nmentA\"#\nk1k3for a band within the pair introduced above\nis\u0012\"#\nk1k3. Based on time-independent perturbation theory,\nwe can prove that\f\fA\"#\f\fbetween two states is approxi-\nmately proportional to\f\fsin\u0000\n\u0012\"#=2\u0001\f\f. The derivation is\ngiven in subsection \\Spin-\rip angle \u0012\"#for intervalley\nspin relaxation\" in \\Methods\" section.\nAs shown in Fig. 5c, \u001c\u00001\nsof ML-Ge on di\u000berent\nsubstrates at 20 K is almost linearly proportional to\nsin2(\u0012\"#=2)DS, where sin2(\u0012\"#=2) is the statistically-\naveraged modulus square of sin\u0000\n\u0012\"#=2\u0001\n. This indicates\nthat the relation jeg\"#j2/sin2(\u0012\"#=2) is nearly perfectly\nsatis\fed at low T, where intervalley processes dominate\nspin relaxation. We additionally show the relations be-\ntween\u001c\u00001\nsandjeg\"#j2DS,jo\"#j2DSandsin2(\u0012\"#=2)DSat\n300 K in Fig. S7. Here the trend of \u001c\u00001\nsis still approxi-\nmately captured by the trends of jeg\"#j2DS,jo\"#j2DSand\nsin2(\u0012\"#=2)DS, although not perfectly linear as at low T.\nSince\u0012\"#is de\fned by Sexpat di\u000berent states, \u001csis\nhighly correlated with Sexpand more speci\fcally with\nthe anisotropy of Sexp(equivalent to the anisotropy of\nBin). Qualitatively, the larger anisotropy of Sexpleads to\nsmaller\u0012\"#and longer\u001csalong the high-spin-polarization\ndirection. This \fnding may be applicable to spin re-\nlaxation in other materials whenever intervalley spin-\rip\nscattering dominates or spin-valley locking exists, e.g., in\nTMDs[5], Stanene[27], 2D hybrid perovskites with persis-\ntent spin helix[7], etc.\nAt the end, we brie\ry discuss the substrate e\u000bects\non in-plane spin relaxation ( \u001cs;x), whereas only out-of-\nplane spin relaxation was discussed earlier. From Table\nSI, we \fnd that \u001cs;xof ML-Ge@-7V/nm and supported\nML-Ge are signi\fcantly (e.g., two orders of magnitude)\nshorter than free-standing ML-Ge, but the di\u000berences\nbetween\u001cs;xof ML-Ge on di\u000berent substrates are rela-\ntively small (within 50%). This is because: With a non-\nzeroEzor a substrate, the inversion symmetry broken\ninduces strong out-of-plane internal magnetic \feld Bin\nz\n(>100 Tesla), so that the excited in-plane spins will pre-\ncess rapidly about Bin\nz. The spin precession signi\fcantly\na\u000bects spin decay and the main spin decay mechanism\nbecomes DP or free induction decay mechanism[28] in-\nstead of EY mechanism. For both DP and free induc-\ntion decay mechanisms[20, 28], \u001cs;xdecreases with the\n\ructuation amplitude (among di\u000berent k-points) of the\nBincomponents perpendicular to the xdirection. As\nthe \ructuation amplitude of Bin\nzof ML-Ge@-7V/nm and\nsupported ML-Ge is large (Table SI; much greater than\nthe one ofBin\ny), their\u001cs;xcan be much shorter than the\nvalue of ML-Ge at zero electric \feld when EY mechanism\ndominates. Moreover, since the \ructuation amplitude of\nBin\nzof ML-Ge on di\u000berent substrates has the same or-8\nFIG. 5. The relation between \u001c\u00001\nsand the averaged modulus square of spin-\rip e-ph matrix elements jeg\"#j2, of spin-\rip overlap\nmatrix elementsjo\"#j2andsin2(\u0012\"#=2) multiplied by the scattering density of states DSat 20 K. See the de\fnition of jeg\"#j2,\njo\"#j2andDSin Eq. 8, 19 and 9 respectively. \u0012\"#is the spin-\rip angle between two electronic states. For two states ( k;n) and\n(k0;n0) with opposite spin directions, \u0012\"#is the angle between \u0000Sexp\nknandSexp\nk0n0.sin2(\u0012\"#=2) is de\fned in Eq. 24. The variation\nofDSamong di\u000berent substrates is at most three times, much weaker than the variations of \u001c\u00001\nsand other quantities shown\nhere.\nder of magnitude (Table SI), \u001cs;xof ML-Ge on di\u000berent\nsubstrates are similar.\nCONCLUSIONS\nIn this paper, we systematically investigate how spin\nrelaxation of strong SOC Dirac materials is a\u000bected by\ndi\u000berent insulating substrates, using germanene as a pro-\ntotypical example. Through FPDM simulations of \u001csof\nfree-standing and substrate supported ML-Ge, we show\nthat substrate e\u000bects on \u001cscan di\u000ber orders of magni-\ntude among di\u000berent substrates. Speci\fcally, \u001csof ML-\nGe-GeH and ML-Ge-SiH have the same order of mag-\nnitude as free-standing ML-Ge, but \u001csof ML-Ge-GaTe\nand ML-Ge-InSe are signi\fcantly shortened by 1-2 orders\nwith temperature increasing from 20 K to 300 K.\nAlthough simple electronic quantities including charge\ndensities, DOS and spin mixing\nb2\nz\u000b\nqualitatively ex-\nplain the much shorter lifetime of ML-Ge-GaTe/InSe\ncompared to ML-Ge-GeH/SiH in the relatively high T\nrange, we \fnd they cannot explain the large variations\nof\u001csamong substrates at low T(i.e. tens of K). We\npoint out that spin relaxation in ML-Ge and its inter-\nfaces at low Tis dominated by intervalley scattering pro-\ncesses. However, the substrate-induced modi\fcations of\nphonons and thermal vibrations of substrates seem to be\nnot important. Instead, the substrate-induced changes\nof the anisotropy of Sexpor the spin-\rip angles \u0012\"#which\nchanges the spin-\rip matrix elements, are much more cru-\ncial.\u0012\"#is at the \frst time proposed in this article to the\nbest of our knowledge, and is found to be a useful elec-\ntronic quantity for predicting trends of spin relaxation\nwhen intervalley spin-\rip scattering dominates.Our theoretical study showcases the systematic inves-\ntigations of the critical factors determining the spin re-\nlaxation in 2D Dirac materials. More importantly we\npointed out the sharp distinction of substrate e\u000bects on\nstrong SOC materials to the e\u000bects on weak SOC ones,\nproviding valuable insights and guidelines for optimizing\nspin relaxation in materials synthesis and control.\nMETHODS\nFirst-Principles Density-Matrix Dynamics for Spin\nRelaxation\nWe solve the quantum master equation of density ma-\ntrix\u001a(t) as the following:[19]\nd\u001a12(t)\ndt= [He;\u001a(t)]12+\n0\nBBB@1\n2P\n3458\n<\n:[I\u0000\u001a(t)]13P32;45\u001a45(t)\n\u0000[I\u0000\u001a(t)]45P\u0003\n45;13\u001a32(t)9\n=\n;\n+H:C:1\nCCCA;\n(1)\nEq. 1 is expressed in the Schr odinger picture, where the\n\frst and second terms on the right side of the equa-\ntion relate to the coherent dynamics, which can lead\nto Larmor precession, and scattering processes respec-\ntively. The \frst term is unimportant for out-of-plane\nspin relaxation in ML-Ge systems, since Larmor preces-\nsion is highly suppressed for the excited spins along the\nout-of-plane or zdirection due to high spin polarization\nalongzdirection. The scattering processes induce spin9\nrelaxation via the SOC. Heis the electronic Hamiltonian.\n[H;\u001a] =H\u001a\u0000\u001aH. H.C. is Hermitian conjugate. The\nsubindex, e.g., \\1\" is the combined index of k-point and\nband.P=Pe\u0000ph+Pe\u0000iis the generalized scattering-\nrate matrix considering e-ph and e-i scattering processes.\nFor the e-ph scattering[19],\nPe\u0000ph\n1234 =X\nq\u0015\u0006Aq\u0015\u0006\n13Aq\u0015\u0006;\u0003\n24; (2)\nAq\u0015\u0006\n13=r\n2\u0019\n~gq\u0015\u0006\n12q\n\u000eG\u001b(\u000f1\u0000\u000f2\u0006!q\u0015)q\nn\u0006\nq\u0015;(3)\nwhereqand\u0015are phonon wavevector and mode, gq\u0015\u0006\nis the e-ph matrix element, resulting from the absorp-\ntion (\u0000) or emission (+) of a phonon, computed with\nself-consistent SOC from \frst-principles,[29] n\u0006\nq\u0015=nq\u0015+\n0:5\u00060:5 in terms of phonon Bose factors nq\u0015, and\u000eG\n\u001b\nrepresents an energy conserving \u000e-function broadened to\na Gaussian of width \u001b.\nFor electron-impurity scattering[19],\nPe\u0000i\n1234=Ai\n13Ai;\u0003\n24; (4)\nAi\n13=r\n2\u0019\n~gi\n13q\n\u000eG\u001b(\u000f1\u0000\u000f3)p\nniVcell; (5)\nwhereniandVcellare impurity density and unit cell vol-\nume, respectively. giis the e-i matrix element computed\nby the supercell method and is discussed in the next sub-\nsection.\nStarting from an initial density matrix \u001a(t0) prepared\nwith a net spin, we evolve \u001a(t) through Eq. 1 for a long\nenough time, typically from hundreds of ps to a few \u0016s.\nWe then obtain spin observable S(t) from\u001a(t) (Eq. S1)\nand extract spin lifetime \u001csfromS(t) using Eq. S2.\nComputational details\nThe ground-state electronic structure, phonons, as well\nas electron-phonon and electron-impurity (e-i) matrix\nelements are \frstly calculated using density functional\ntheory (DFT) with relatively coarse kandqmeshes in\nthe DFT plane-wave code JDFTx[30]. Since all sub-\nstrates have hexagonal structures and their lattice con-\nstants are close to germanene's, the heterostructures\nare built simply from unit cells of two systems. The\nlattice mismatch values are within 1% for GeH, GaTe\nand InSe substrates but about 3.5% for the SiH sub-\nstrate. All heterostructures use the lattice constant 4.025\n\u0017A of free-standing ML-Ge relaxed with Perdew-Burke-\nErnzerhof exchange-correlation functional[31]. The in-\nternal geometries are fully relaxed using the DFT+D3\nmethod for van der Waals dispersion corrections[32].\nWe use Optimized Norm-Conserving Vanderbilt (ONCV)\npseudopotentials[33] with self-consistent spin-orbit cou-\npling throughout, which we \fnd converged at a ki-\nnetic energy cuto\u000b of 44, 64, 64, 72 and 66 Ry forfree-standing ML-Ge, ML-Ge-GeH, ML-Ge-SiH, ML-Ge-\nGaTe and ML-Ge-InSe respectively. The DFT calcula-\ntions use 24\u000224kmeshes. The phonon calculations em-\nploy 3\u00023 supercells through \fnite di\u000berence calculations.\nWe have checked the supercell size convergence and found\nthat using 6\u00026 supercells lead to very similar results of\nphonon dispersions and spin lifetimes. For all systems,\nthe Coulomb truncation technique[34] is employed to ac-\ncelerate convergence with vacuum sizes. The vacuum\nsizes are 20 bohr (additional to the thickness of the het-\nerostructures) for all heterostructures and are found large\nenough to converge the \fnal results of spin lifetimes. The\nelectric \feld along the non-periodic direction is applied\nas a ramp potential.\nFor the e-i scattering, we assume impurity density is\nsu\u000eciently low and the average distance between neigh-\nboring impurities is su\u000eciently long so that the interac-\ntions between impurities are negligible, i.e. at the dilute\nlimit. The e-i matrix gibetween state ( k;n) and (k0;n0)\nisgi\nkn;k0n0=hknjVi\u0000V0jk0n0i, whereViis the poten-\ntial of the impurity system and V0is the potential of the\npristine system. Viis computed with SOC using a large\nsupercell including a neutral impurity that simulates the\ndilute limit where impurity and its periodic replica do\nnot interact. To speed up the supercell convergence, we\nused the potential alignment method developed in Ref.\n35. We use 5\u00025 supercells, which have shown reasonable\nconvergence (a few percent error of the spin lifetime).\nWe then transform all quantities from plane wave basis\nto maximally localized Wannier function basis[36], and\ninterpolate them[29, 37{41] to substantially \fner k and\nq meshes. The \fne kandqmeshes are 384\u0002384 and\n576\u0002576 for simulations at 300 K and 100 K respectively\nand are \fner at lower temperature, e.g., 1440 \u00021440 and\n2400\u00022400 for simulations at 50 K and 20 K respectively.\nThe real-time dynamics simulations are done with our\nown developed DMD code interfaced to JDFTx. The\nenergy-conservation smearing parameter \u001bis chosen to\nbe comparable or smaller than kBTfor each calculation,\ne.g., 10 meV, 5 meV, 3.3 meV and 1.3 meV at 300 K,\n100 K, 50 K and 20 K respectively.\nAnalysis of Elliot-Yafet spin lifetime\nIn order to analyze the results from real-time \frst-\nprinciples density-matrix dynamics (FPDM), we com-\npare them with simpli\fed mechanistic models as dis-\ncussed below. According to Ref. [18], if a solid-state\nsystem is close to equilibrium (but not at equilibrium)\nand its spin relaxation is dominated by EY mechanism,\nits spin lifetime \u001csdue to the e-ph scattering satis\fes (for\nsimplicity the band indices are dropped)10\n\u001c\u00001\ns/N\u00002\nk\n\u001fX\nkq\u00158\n<\n:jg\"#;q\u0015\nk;k\u0000qj2nq\u0015fk\u0000q(1\u0000fk)\n\u000e(\u000fk\u0000\u000fk\u0000q\u0000!q\u0015)9\n=\n;;(6)\n\u001f=N\u00001\nkX\nkfk(1\u0000fk); (7)\nwherefis Fermi-Dirac function. !q\u0015andnq\u0015are\nphonon energy and occupation of phonon mode \u0015at\nwavevector q.g\"#is the spin-\rip e-ph matrix element\nbetween two electronic states of opposite spins. We will\nfurther discuss g\"#in the next subsection.\nAccording to Eq. 6 and 7, \u001c\u00001\nsis proportional to jg\"#\nqj2\nand also the density of the spin-\rip transitions. Therefore\nwe propose a temperature ( T) and chemical potential\n(\u0016F;c) dependent e\u000bective modulus square of the spin-\n\rip e-ph matrix element jeg\"#j2and a scattering density\nof statesDSas\njeg\"#j2=P\nkqwk;k\u0000qP\n\u0015jg\"#;q\u0015\nk;k\u0000qj2nq\u0015P\nkqwk;k\u0000q; (8)\nDS=N\u00002\nkP\nkqwk;k\u0000q\nN\u00001\nkP\nkfk(1\u0000fk); (9)\nwk;k\u0000q=fk\u0000q(1\u0000fk)\u000e(\u000fk\u0000\u000fk\u0000q\u0000!c); (10)\nwhere!cis the characteristic phonon energy speci\fed\nbelow, and w k;k\u0000qis the weight function. The matrix\nelement modulus square is weighted by nq\u0015according to\nEq. 6 and 7. This rules out high-frequency phonons at\nlowTwhich are not excited. !cis chosen as 7 meV\nat 20 K based on our analysis of phonon-mode-resolved\ncontribution to spin relaxation. w k;k\u0000qselects transitions\nbetween states separated by !cand around the band edge\nor\u0016F;c, which are \\more relevant\" transitions to spin\nrelaxation.\nDScan be regarded as an e\u000bective density of spin-\n\rip e-ph transitions satisfying energy conservation be-\ntween one state and its pairs. When !c= 0, we\nhaveDS=R\nd\u000f\u0010\n\u0000df\nd\u000f\u0011\nD2(\u000f)=R\nd\u000f\u0010\n\u0000df\nd\u000f\u0011\nD(\u000f) with\nD(\u000f) density of electronic states (DOS). So DScan\nbe roughly regarded as a weighted-averaged DOS with\nweight\u0010\n\u0000df\nd\u000f\u0011\nD(\u000f).\nWithjeg\"#j2andDS, we have the approximate relation\nfor spin relaxation rate,\n\u001c\u00001\ns/jeg\"#j2DS: (11)\nSpin-\rip e-ph and overlap matrix element\nIn the mechanistic model of Eq. 6 in the last section,\nthe spin-\rip e-ph matrix element between two electronic\nstates of opposite spins at wavevectors kandk\u0000qof\nphonon mode \u0015reads[29]g\"#;q\u0015\nkk\u0000q=D\nu\"(#)\nk\f\f\f\u0001q\u0015vKS\f\f\fu#(\")\nk\u0000qE\n; (12)\n\u0001q\u0015vKS=s\n~\n2!q\u0015X\n\u0014\u000be\u0014\u000b;q\u0015@\u0014\u000bqvKS\npm\u0014; (13)\n@\u0014\u000bqvKS=X\nleiq\u0001Rl@VKS\n@\u001c\u0014\u000bjr\u0000Rl; (14)\nVKS=V+~\n4m2c2rrV\u0002p\u0001\u001b; (15)\nwhereu\"(#)\nkis the periodic part of the Bloch wavefunc-\ntion of a spin-up (spin-down) state at wavevector k.\u0014is\nthe index of ion in the unit cell. \u000bis the index of a di-\nrection. Rlis a lattice vector. Vis the spin-independent\npart of the potential. pis the momentum operator. \u001bis\nthe Pauli operator.\nFrom Eqs. 12-15, g\"#can be separated into two parts,\ng\"#=gE+gY; (16)\nwheregEandgYcorrespond to the spin-independent\nand spin-dependent parts of VKSrespectively, called El-\nliot and Yafet terms of the spin-\rip scattering matrix\nelements respectively.[28]\nGenerally speaking, both the Elliot and Yafet terms\nare important; for the current systems \u001cswith and with-\nout Yafet term have the same order of magnitude. For\nexample,\u001csof ML-Ge-GeH and ML-Ge-SiH without the\nYafet term are about 100% and 70% of \u001cswith the Yafet\nterm at 20 K. Therefore, for qualitative discussion of \u001cs\nof ML-Ge on di\u000berent substrates (the quantitative calcu-\nlations of\u001csare performed by FPDM introduced earlier),\nit is reasonable to focus on the Elliot term gEand avoid\nthe more complicated Yafet term gY.\nDe\fneVE\nq\u0015as the spin-independent part of \u0001 q\u0015vKS,\nso thatgE=D\nu\"(#)\nk\f\f\fVE\nq\u0015\f\f\fu#(\")\nk\u0000qE\n. Expanding VE\nq\u0015as\nP\nGeVE\nq\u0015(G)eiG\u0001r, we have\ngE=X\nGeVE\nq\u0015(G)o\"#\nkk\u0000q(G); (17)\no\"#\nkk\u0000q(G) =D\nu\"(#)\nk\f\f\feiG\u0001r\f\f\fu#(\")\nk\u0000qE\n; (18)\nwhereo\"#\nkk\u0000q(G) isG-dependent spin-\rip overlap func-\ntion. Without loss of generality, we suppose the \frst\nBrillouin zone is centered at \u0000.\nTherefore,gEis not only determined by the long-range\ncomponent of o\"#\nkk\u0000q(G), i.e.,o\"#\nkk\u0000q(G= 0) but also the\nG6= 0 components. But nevertheless, it is helpful to\ninvestigate o\"#\nkk\u0000q(G= 0) and similar to Eq. 8, we pro-\npose an e\u000bective modulus square of the spin-\rip overlap\nmatrix elementjo\"#j2,11\njo\"#j2=P\nkqwk;k\u0000qP\n\u0015jo\"#\nk;k\u0000q(G= 0)j2\nP\nkqwk;k\u0000q: (19)\nInternal magnetic \feld\nSuppose originally a system has time-reversal and in-\nversion symmetries, so that every two bands form a\nKramers degenerate pair. Suppose the k-dependent spin\nmatrix vectors in Bloch basis of the Kramers degenerate\npairs are s0\nkwiths\u0011(sx;sy;sz). The inversion symme-\ntry broken, possibly due to applying an electric \feld or\na substrate, induces k-dependent Hamiltonian terms\nHISB\nk=\u0016BgeBin\nk\u0001s0\nk; (20)\nwhere\u0016Bgeis the electron spin gyromagnetic ratio.\nBin\nkis the SOC \feld and called internal magnetic \felds.\nBinsplits the degenerate pair and polarizes the spin along\nits direction. The de\fnition of Bin\nkis\nBin\nk\u00112\u0001SOC\nkSexp\nk=(\u0016Bge); (21)\nwhere Sexp\u0011\u0000\nSexp\nx;Sexp\ny;Sexp\nz\u0001\nwithSexp\nibeing spin\nexpectation value along direction iand is the diagonal\nelement ofsi. \u0001SOCis the band splitting energy by SOC.\nSpin-\rip angle \u0012\"#for intervalley spin relaxation\nSuppose (i) the inversion symmetry broken induces Bin\nk\n(Eq. 21) for a Kramers degenerate pair; (i) there are two\nvalleys centered at wavevectors Qand\u0000Qand (iii) there\nare two wavevectors k1andk2nearQand\u0000Qrespec-\ntively. Due to time-reversal symmetry, the directions of\nBin\nk1andBin\nk2are almost opposite.\nDe\fne the spin-\rip angle \u0012\"#\nk1k2as the angle between\n\u0000Bin\nk1andBin\nk2, which is also the angle between \u0000Sexp\nk1\nandSexp\nk2. We will prove that for a general operator bA,\n\f\f\fA\"#\nk1k2\f\f\f2\n\u0019sin2\u0010\n\u0012\"#\nk1k2=2\u0011\f\f\fA##\nk1k2\f\f\f2\n; (22)\nwhereA\"#\nk1k2andA##\nk1k2are the spin-\rip and spin-\nconserving matrix elements between k1andk2respec-\ntively.\nThe derivation uses the \frst-order perturbation theory\nand has three steps:\nStep 1: The 2\u00022 matrix of operator bAbetween k1and\nk2of two Kramers degenerate bands is A0\nk1k2. According\nto Ref. 26, with time-reversal symmetry, the spin-\rip\nmatrix element of the same band between kand\u0000kis\nexactly zero, therefore, the spin-\rip matrix elements ofA0\nk1k2are zero at lowest order as k1+k2\u00190, i.e.,A0;\"#\nk1k2\u0019\nA0;#\"\nk1k2\u00190.\nStep 2: The inversion symmetry broken induces Bin\nk\nand the perturbed Hamiltonian HISB\nk(Eq. 20). The\nnew eigenvectors Ukare obtained based on the \frst-order\nperturbation theory.\nStep 3: The new matrix is Ak1k2=Uy\nk1A0\nk1k2Uk2. Thus\nthe spin-\rip matrix elements A\"#\nk1k2with the inversion\nsymmetry broken are obtained.\nWe present the detailed derivation in SI Sec. III.\nFrom Eq. 22, for the intervalley e-ph matrix elements\nof ML-Ge systems, we have\n\f\f\fg\"#\nk1k2\f\f\f2\n\u0019sin2\u0010\n\u0012\"#\nk1k2=2\u0011\f\f\fg##\nk1k2\f\f\f2\n: (23)\nAs\f\f\fg\"#\nk1k2\f\f\f2\nlargely determines \u001csof ML-Ge systems,\nthe di\u000berences of \u001csof ML-Ge on di\u000berent substrates\nshould be mainly due to the di\u000berence of sin2\u0010\n\u0012\"#\nk1k2=2\u0011\n.\nFor the intervalley overlap matrix elements, we should\nhave\f\f\fo\"#\nk1k2\f\f\f2\n\u0019sin2\u0010\n\u0012\"#\nk1k2=2\u0011\f\f\fo##\nk1k2\f\f\f2\n. Since\f\f\fo##\nk1k2\f\f\f2\nis of order 1,\f\f\fo\"#\nk1k2\f\f\f2\nis expected proportional to\nsin2\u0010\n\u0012\"#\nk1k2=2\u0011\nand have the same order of magnitude as\nsin2\u0010\n\u0012\"#\nk1k2=2\u0011\n.\nFinally, similar to Eq. 8, we propose an e\u000bective mod-\nulus square of sin2\u0010\n\u0012\"#\nk1k2=2\u0011\n,\nsin2(\u0012\"#=2) =P\nkqwk;k\u0000qsin2\u0010\n\u0012\"#\nk;k\u0000q=2\u0011\nP\nkqwk;k\u0000q: (24)\nDATA AVAILABILITY\nThe data that support the \fndings of this study are\navailable upon request to the corresponding author.\nCODE AVAILABILITY\nThe codes that were used in this study are available\nupon request to the corresponding author.\nACKNOWLEDGEMENTS\nWe thank Ravishankar Sundararaman for helpful dis-\ncussions. This work is supported by the Air Force Of-\n\fce of Scienti\fc Research under AFOSR Award No.\nFA9550-YR-1-XYZQ and National Science Foundation\nunder grant No. DMR-1956015. This research used\nresources of the Center for Functional Nanomaterials,12\nwhich is a US DOE O\u000ece of Science Facility, and the\nScienti\fc Data and Computing center, a component of\nthe Computational Science Initiative, at Brookhaven Na-\ntional Laboratory under Contract No. DE-SC0012704,\nthe lux supercomputer at UC Santa Cruz, funded by NSF\nMRI grant AST 1828315, the National Energy Research\nScienti\fc Computing Center (NERSC) a U.S. Depart-\nment of Energy O\u000ece of Science User Facility operated\nunder Contract No. DE-AC02-05CH11231, and the Ex-\ntreme Science and Engineering Discovery Environment\n(XSEDE) which is supported by National Science Foun-\ndation Grant No. ACI-1548562 [42].AUTHOR CONTRIBUTIONS\nJ.X. performed the \frst-principles calculations. J.X.\nand Y.P. analyzed the results. J.X. and Y.P. designed all\naspects of the study. 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Neumann,1J¨urgen Henk,1and Ingrid Mertig1, 2\n1Institut f¨ ur Physik, Martin-Luther-Universit¨ at Halle-Wittenberg, D-06099 Halle (Saale), Germany\n2Max-Planck-Institut f¨ ur Mikrostrukturphysik, D-06120 Halle (Saale), Germany\n(Dated: March 29, 2019)\nOur joint theoretical and computer experimental study of heat-to-spin conversion reveals that noncollinear\nantiferromagnetic insulators are promising materials for generating magnon spin currents upon application\nof a temperature gradient: they exhibit spin Seebeck and spin Nernst e \u000bects. Using Kubo theory and spin\ndynamics simulations, we explicitly evaluate these e \u000bects in a single kagome sheet of potassium iron jarosite,\nKFe 3(OH) 6(SO 4)2, and predict a spin Seebeck conversion factor of 0 :2\u0016V=K at a temperature of 20 K.\nIntroduction. Interconversion phenomena between physi-\ncal quantities like sound, charge, spin, or heat [ 1] are corner-\nstones in the solid-state research for next-generation alterna-\ntives to today’s CMOS technology. Two particularly active\nfields are those of spinelectronics (charge to spin and vice\nversa ) [2] and spincaloritronics (heat to spin and vice versa )\n[3]. While the former relies on electrons, the latter may dis-\nregard electrons as fundamental carriers in favor of collective\nmagnetic excitations, i. e., magnons, thereby circumventing\nJoule heating.\nA prominent magnonic heat-to-spin conversion phe-\nnomenon, which promises temperature control as well as waste-\nheat recovery, is the spin Seebeck e \u000bect (SSE) [ 4], comprising\na spin current in a magnetic insulator as response to an applied\ntemperature gradient. Magnons that “flow down” the gradient\ncarry spin from the hot to the cold side of the sample. Accu-\nmulated at these ends, the spin di \u000buses into an adjacent heavy\nmetal layer and gets converted into a transverse charge current\nby the inverse spin Hall e \u000bect [5].\nWhile the SSE is natural to ferromagnets, it does not show\nup in uniaxial collinear antiferromagnets, because of their\nspin-degenerate magnon bands. Only an external magnetic\nfield, which causes a Zeeman splitting of the magnon bands,\nintroduces nonzero spin Seebeck signals [ 6–11]. Thus, the\nstatus quo is that the heat-to-spin conversion by means of the\nSSE is possible in either ferromagnets (e. g., LaY 2Fe5O12[4])\nor uniaxial collinear antiferromagnets or paramagnets (e. g.,\nMnF 2[10] and GGG [ 12], respectively) in magnetic fields,\nor biaxial collinear antiferromagnets (e. g., NiO [ 13]) with\nnondegenerate magnon bands in zero field.\nAn alternative to the SSE is o \u000bered by the spin Nernst e \u000bect\n(SNE), which describes a transverse spin current as a response\nto a temperature gradient in magnetic insulators. It is found\nboth in ferromagnets [ 14–17], collinear antiferromagnets [ 17–\n20], and paramagnets [ 21]. However, its proportionality to the\nstrength of spin-orbit coupling (SOC) renders the heat-to-spin\nconversion rather ine \u000ecient. Therefore, it is about time to\nconsider spin transport in a di \u000berent material class, namely in\nnoncollinear antiferromagnetic insulators (NAIs).\nHerein, we show that NAIs are, in principle, materials for the\ngeneration of bulk magnon spin currents in zero magnetic field\nandwithout SOC . Taking a single kagome sheet of the NAI\npotassium iron jarosite KFe 3(OH) 6(SO 4)2as an example, weidentify spin Seebeck and planar spin Nernst signals due to in-\nplane polarized bulk spin currents both within Kubo transport\ntheory as well as atomistic spin dynamics simulations. Using\nsuperordinate symmetry arguments, these SSEs and planar\nSNEs are established as the magnonic version of spin-polarized\nelectron currents in noncollinear antiferromagnetic metals [ 22,\n23].\nHeat-To-Spin Conversion in Potassium Iron Jarosite. We\nconsider a single kagome sheet of potassium iron jarosite,\nKFe 3(OH) 6(SO 4)2, which is an electrically insulating mineral\nbuilt from Fe kagome planes [cf. Fig. 1(a)] stacked along the c\ndirection in ABC sequence [ 24]. The almost classical S=5=2\nspins order below 65 K in the positive vector chiral (PVC)\nphase [ 25,26] depicted in Fig. 1(b). This phase is characterized\nby a positive zcomponent of the vector spin chirality \u0014=\nS1\u0002S2+S2\u0002S3+S3\u0002S1, where Si(i=1;2;3) are the three\nspins in the magnetic unit cell as indicated in Figs. 1(a) and\n(b).\nThe two-dimensional (2D) spin Hamiltonian [27, 28]\nH=1\n2~2X\ni j\u0010\nJi jSi\u0001Sj+Di j\u0001Si\u0002Sj\u0011\n+g\u0016B\n~BzX\niSz\ni(1)\nincludes antiferromagnetic exchange Ji jbetween nearest\n(3:18meV [28,29]) and next-nearest neighbors (0 :11meV\n[28,29]), which—due to geometric frustration—favors any\nclassical 120\u000eground state. The Dzyaloshinskii-Moriya inter-\naction (DMI) [ 30,31] between nearest neighbors is described\nby the vector Di j=\u0006(Dkˆni j+Dzˆz) [positive (negative) sign for\ncounterclockwise (clockwise) circulation], possessing out-of-\nplane ( Dz=\u00000:062J[28,29]) as well as in-plane components\n(Dk=0:062J[28,29]). The latter arise because the kagome\nplanes are no mirror planes [ 27].Di jis orthogonal to the\ni jbond and ˆni j=ˆnjiis an in-plane unit vector as shown in\nFig. 1(a). A sign convention opposite to that of Ref. 27is\nused: Dz<0 stabilizes the PVC phase [ 27] and Dkcauses a\ntiny out-of-plane canting [ 27] (canting angle 1 :98\u000e[28,29]).\nFinally, we consider a magnetic field Balong zdirection (with\ng-factor g=2:13 [32] and Bohr’s magneton \u0016B).\nAt zero magnetic field the spin textures of adjacent kagome\nlayers in bulk KFe 3(OH) 6(SO 4)2are exactly opposite [ 25,26]\ndue to weak interlayer coupling ( \u0019\u00000:03meV [33]) that is\nneglected here. Upon application of a su \u000eciently large mag-\nnetic field of Bz\u0019\u000017 T [ 33,34] the spin orientations in everyarXiv:1903.11896v1  [cond-mat.mes-hall]  28 Mar 20192\n1 23(a) M/prime\nx(b)\nΓ K\nM xy\nz(c)\nK \u0000 M K051015Energy \"(meV)\n(d)\n0.000.010.020.030.040.05 (e)\nFIG. 1. Single kagome layer of KFe 3(OH) 6(SO 4)2. (a) Structural\nlattice with three atoms per unit cell. DMI vectors are indicated as\narrows for counterclockwise circulation. (b) In-plane components of\nthe magnetic PVC order with M0\nxtime-reversal mirror plane. (c) First\nBrioullin zone of the kagome lattice with indicated high symmetry\npoints. (d) Magnon dispersion relation (at Bz=\u000017 T) along the path\nmarked in (c). (e) Wave vector resolved spin expectation values of\nthe lowest band in the vicinity of the \u0000point [see dashed circle in\n(c)]; color indicates the absolute value (in units of ~) and arrows the\ndirection.\nsecond layer flip and the kagome sheets exhibit identical mag-\nnetic orders [ 32–34]. As we show in the following, identical\ntextures are important to ensure a finite spin current generation\nupon application of a temperature gradient. Thus, the results\nobtained for the 2D model at hand apply to the actual bulk\nmaterial for Bz.\u000017 T.\nTaking this magnetic field into account, we determine the\nresulting canting angle numerically ( \u00192:85\u000e) and carry out\nlinear spin-wave calculations (cf. Sec. I A of the Supplemental\nMaterial (SM) [ 35]). We obtain the magnon energies \"nk(with\nn=1;2;3) shown in Fig. 1(d) along high symmetry lines\ndepicted in Fig. 1(c). Following Ref. 36, we calculate the spin\nexpectation values of magnons in the lowest band close to the\nBrillouin zone center [Fig. 1(e)]. We find the double winding\nof the magnon spin direction about the \u0000point known from\nRef. 36. This spin-momentum locking suggests the possibility\nof net spin currents in nonequilibrium.\nWe are interested in the magnetothermal transport tensor\n\u0007\r\n\u0016\u0017, that mediates between the temperature gradient r\u0017Tin\u0017\ndirection and the nonequilibrium spin current density hj\r\n\u0016iof\r\nspin in\u0016direction:hj\r\n\u0016i=\u0007\r\n\u0016\u0017(\u0000r\u0017T). This tensor comprises\nthe SSE (diagonal elements, \u0016=\u0017), the SNE (o \u000b-diagonal\nelements,\u0016,\u0017; antisymmetric part of \u0007\r), and the planar SNE\n(symmetric part of \u0007\r). Applying Kubo’s theory [ 19,37–39]\nand considering only the intraband contributions proportional\nto a phenomenological magnon relaxation time \u001c, we find\n(cf. Sec. I B of SM [ 35] for the derivation and a discussion ofapproximations)\n\u0007\r\n\u0016\u0017=\u001c\n2VTX\nk2NX\nn=1Reh\u0010\nJ\r\nk;\u0016\u0011\nnni\u0000Qk;\u0017\u0001\nnn \n\u0000@\u001a\n@\"!\n:(2)\n(J\r\nk;\u0016)nnand ( Qk;\u0016)nndenote diagonal matrix elements of\nthe spin and heat current operators, respectively. \u001a=\n[exp(\f(GEk)nn)\u00001]\u00001is the Bose-Einstein distribution func-\ntion,\f=(kBT)\u00001with kBdenoting Boltzmann’s constant,\nandVis the sample’s volume. Gis the bosonic metric and\nEkthe paraunitarily diagonalized Hamiltonian, containing the\nmagnon energies (cf. Sec. I A of SM [35]).\nEq.(2)describes the time-odd part of the full magnetother-\nmal transport tensor (cf. Sec. I B of SM [ 35]). To see so, recall\nthat J\r\nk;\u0016is even but Qk;\u0016odd under time reversal. Thus, re-\nversal of the magnetic texture reverses the sign of \u0007\r\n\u0016\u0017, which\nis well-known for the SSE in ferromagnets [ 4]. This also ex-\nplains the absence of the SSE in those antiferromagnets for\nwhich time reversal can be “repaired” by a sublattice swap:\nsuch antiferromagnets are e \u000bectively time-even and as such\nincompatible with a time-odd transport response. In zero field,\nbulk KFe 3(OH) 6(SO 4)2exhibits such a symmetry due to the\nopposite spin textures of adjacent kagome sheets and we expect\nzero\u0007. This is why we consider the spin-flopped phase with\nBz<\u000017 T modeled by a single layer.\nApplying Eq. (2)to the 2D model of KFe 3(OH) 6(SO 4)2, we\ncalculate the elements \u0007\r\n\u0016\u0017for\u0016;\u0017;\r =x;y(in-plane transport\nof in-plane polarized spins); results are shown in Fig. 2 [ 40].\nThere are several nonzero elements, which are identical in\nmodulus (red line); the transport tensor assumes the form\n\u0007x= 0\u0007x\nxy\n\u0007x\nxy0!\n; \u0007y= \u0007x\nxy 0\n0\u0000\u0007x\nxy!\n: (3)\nConsequently, when applying rTinxorydirection, there is a\nlongitudinal magnon particle current density, consisting of y\nspin-polarized magnons (diagonal elements of \u0007y): there is a\nSSE. Moreover, there is a transverse x-polarized spin current\n(o\u000b-diagonal elements of \u0007x), i. e., a planar SNE. At 20 K\nwe find a spin Seebeck coe \u000ecient of about 60 keV=(Km) (left\nordinate in Fig. 2). Assuming an inverse-spin-Hall spin-to-\ncharge current conversion factor of 1 :3\u000210\u00004Vs=(Am) [41] in\nan adjacent platinum layer, this corresponds to a spin Seebeck\nconversion factor (SSCF) of 0 :2\u0016V=K(right ordinate in Fig. 2).\nThis is to be compared with values of ferrimagnetic YIG ( .\n5\u0016V=K[42]) or of collinear antiferromagnets in magnetic\nfields like Cr 2O3(0:015\u0016V=Kat 14 T and 35 K [ 9]), MnF 2\n(41:2\u0016V=Kat 14 T and 15 K [ 10]), or\u000b-Cu 2V2O7(0:1\u0016V=K\nat 5 T and 2 K [ 43]). Hence, the SSE in KFe 3(OH) 6(SO 4)2is\nwell within experimental range; the same arguments apply to\nthe planar SNE.\nIn contrast to the aforementioned collinear antiferromagnets\nthat exhibit the SSE only in magnetic fields, we also obtain\na SSE in zero magnetic field (cf. Sec. II of SM [ 35]), which\nwould be experimentally accessible in a single kagome sheet.3\n0 10 20 30050100\n00.20.4\nTemperature T(K)Υ(keV\nKm)\nSSCF (µV\nK)Υx\nxy=Υx\nyx=Υy\nxx=−Υy\nyy\nΥx\nxx=Υx\nyy=Υy\nxy=Υy\nyx\nFIG. 2. Temperature dependence of \u0007\r\n\u0016\u0017(\u0016;\u0017;\r =x;y) in\nKFe 3(OH) 6(SO 4)2forBz=\u000017 T. Left ordinate: natural units of \u0007in\nthree dimensions. Right ordinate: corresponding spin Seebeck conver-\nsion factor (SSCF; inverse spin Hall voltage divided by temperature\ngradient) in an adjacent platinum layer.\nComputer experiment. Since spin is not conserved, spin\ncurrents are detected indirectly by measurement of the observ-\nable spin accumulation they bring about in samples with finite\ndimensions; standard means include the inverse spin Hall ef-\nfect in an adjacent normal metal layer [ 5], spin torque in an\nadjacent ferromagnet [ 44], or magnetooptical Kerr microscopy\n[45]. We now demonstrate that the SSE and planar SNE in\nNAIs cause finite nonequilibrium spin accumulations.\nThe actual experimental situation, in which a temperature\ngradient is applied to the magnet, is simulated by relying on\nclassical atomistic spin dynamics simulations based on the\nstochastic Landau-Lifshitz-Gilbert equation [ 46]. We set up a\nrectangular stripe of a single KFe 3(OH) 6(SO 4)2kagome sheet\n(built from about 50 000 spins) with finite width in ydirection\nand periodic boundary conditions along the longer xdirection.\nThe orientation of the kagome triangles and the spin ordering\nis as indicated in Fig. 1(b). Each spin is coupled to its own heat\nbath at a spatially varying temperature as shown in Fig. 3(a);\nthe temperature profile exhibits two opposite gradients in x\ndirection. Then, inspired by Ref. 47, we measure a position-\nresolved steady-state nonequilibrium spin accumulation h\u0001Sii\n(cf. Sec. III of SM [35] for technical details).\nThe position-resolved x,y, and zcomponents ofh\u0001Siiare\nshown in panels (b), (c) and (d) of Fig. 3, respectively. In\npanel (b), an accumulation of xspin is observed at the edges\nof the sample in those regions with a finite rxT(cf. red and\nblue horizontal stripes), indicating a transverse spin current as\nexpected from the o \u000b-diagonal elements of \u0007x[cf. Eq. (3)]. In\ncontrast, there is zero xspin accumulation at the ends of the\ngradients, which is in agreement with zero diagonal elements\nof\u0007x. Overall, Fig. 3(b) proves the existence of the planar\nSNE with x-polarized transverse spin currents.\nIn Fig. 3(c), a finite yspin accumulation is observed at\nthe ends of the thermal gradients (cf. red and blue vertical\nstripes), which is in accord with the nonzero diagonal elements\nof\u0007y[cf. Eq. (3)]. However, no yspin accumulates at the\nedges of the sample (zero o \u000b-diagonal elements of \u0007y). These\n0.511.5\n(a)\n4\n2\n0\n−2\n−4\n10−3¯hT(K)\n020406080\n(b)\nxy/angbracketleft∆S x/angbracketrightPosition y\n020406080\n(c) /angbracketleft∆S y/angbracketrightPosition y\n0 30 60 90 120 150 180020406080\n(d) /angbracketleft∆S z/angbracketright\nPosition x(lattice constants)Position yFIG. 3. Nonequilibrium spin accumulation in a single kagome sheet of\nKFe 3(OH) 6(SO 4)2as obtained by spin dynamics simulations. (a) Tem-\nperature profile with two opposite gradients. (b)–(d) Position-resolved\nx,y, and zcomponents of the nonequilibrium spin accumulation h\u0001Si.\nWhile xspin accumulates at the sample’s edges [SNE geometry, (b)],\nyandzspin accumulates in longitudinal direction [SSE geometry,\n(c) and (d)], i. e., at the ends of the temperature gradient. Opposite\ntemperature gradients cause opposite accumulations. Red /white /blue\ncolor indicates positive /zero/negative accumulation.\nresults demonstrate the existence of the SSE with y-polarized\nlongitudinal spin currents.\nWe also observe a longitudinal accumulation of zspin in\nFig. 3(d). It is caused by the small out-of-plane canting induced\nby the in-plane DMI, due to which all magnons acquire a spin\ncomponent in zdirection (cf. Sec. IV A of SM [ 35]). This\ne\u000bect is just the usual SSE associated with ferromagnetism.\nIn Sec. III D of the SM [ 35] we show that reversal of the\nmagnetic texture leads to a sign reversal of spin accumulations,\nunambiguously relating the accumulations with the time-odd\npart of\u0007, which is in accordance with theory. This finding\ncorroborates further that spin current responses of kagome\nlattices with opposite textures cancel out.\nOrigin of the SSE and SNE. The qualitative results we\nhave obtained here, namely the existence of both a SSE and\nSNE, are not limited to the material under consideration. They\nequally apply to any kagome antiferromagnet in the PVC phase\nas is evident from superordinate symmetry considerations.\nThe slightly out-of-plane tilted PVC phase has a three-fold\nrotational axis pointing out of the plane ( C3z) and a M0\nxtime-\nreversal mirror plane whose normal points along x[Fig. 1(b)].\nApplying Neumann’s principle [ 48], i. e., requiring the magne-\ntothermal transport tensor to be invariant under the magnetic\ncrystal symmetries, the shape of \u0007in Eq. (3)can be derived\nrigorously (cf. Sec. IV A of SM [ 35]). In essence, the SSE and4\nplanar SNE are allowed to exist, because the symmetry of the\nnoncollinear PVC texture is too low to forbid spin currents.\nThis finding is in accordance with the symmetry-restricted\nspin transport tensor shapes studied in Ref. 49and the elec-\ntronic spin-polarized currents in noncollinear antiferromag-\nnetic metals [22, 23] [50].\nIn the Introduction we promised spin currents in zero field\nandwithout SOC . Armed with the above symmetry arguments,\nwe construct a gedanken magnet that keeps these promises.\nStarting from a KFe 3(OH) 6(SO 4)2sheet, the limit of zero SOC\nis obtained by setting the DMI zero. This results in a perfectly\ncoplanar PVC phase (zero magnetization) stabilized by the\nsecond-nearest neighbor exchange [51–53]. In addition to the\nC3zandM0\nxsymmetries, there are now a Myand a M0\nzsymmetry.\nThe latter are still insu \u000ecient to forbid spin-polarized currents\n(Sec. IV B of SM [ 35]) and the form of \u0007remains as in Eq. (3).\nThus, assuming a single kagome sheet with a single magnetic\ndomain, we obtain both an SSE and SNE in zero field [54].\nWe have numerically verified the SSE and planar SNE in\nthegedanken magnet by calculating \u0007by Eq. (2)(Sec. IV B\nof SM [ 35]). There are three Goldstone modes (cf. Fig. S3\nof SM [ 35]) [51,52], associated with a global rotation of\nthe texture; all three magnon branches contribute to trans-\nport (cf. Fig. S4 of SM [ 35]). Unfortunately, we are not\naware of a material which strictly realizes this model, ow-\ning to the inevitibility of nearest-neighbor DMI on the kagome\nlattice. Since a single KFe 3(OH) 6(SO 4)2kagome sheet devi-\nates only slightly from the gedanken magnet, single-crystalline\nbulk KFe 3(OH) 6(SO 4)2[32] serves as a candidate material\non which a proof-of-principle experiment can be performed.\nOther iron jarosites [ 55] are also candidates, e. g., silver iron\njarosite AgFe 3(OH) 6(SO 4)2, which orders below 59 K [ 33] and\ntakes Bz<\u000010 T [33] to ensure identical layer spin textures.\nOther antiferromagnetic textures. By extending the sym-\nmetry considerations to other noncollinear antiferromagnetic\ntextures, it becomes evident that the SSE and planar SNE are\ninherent phenomena in NAIs.\nIn contrast to the PVC phase, the negative vector chiral\n(NVC) phase (cf. Fig. S5 of SM [ 35]), which is stabilized for\nDz>0 [27] (recall opposite sign convention) and characterized\nby\u0014z<0, does not have a C3axis. Thus, symmetry consid-\nerations are restricted to an M0\nxtime-reversal mirror plane,\nyielding (Sec. IV C of SM [35])\n\u0007x= 0\u0007x\nxy\n\u0007x\nyx0!\n; \u0007y= \u0007y\nxx0\n0\u0007y\nyy!\n: (4)\nSince in general \u0007y\nxx,\u0007y\nyyand\u0007x\nxy,\u0007x\nyx, there is on top of the\nplanar SNE of x-polarized spin (symmetric part of \u0007x), also a\n“magnetic” [ 56] SNE (antisymmetric part of \u0007x). A 90\u000erotated\nversion of the NVC phase appears in cadmium kapellasite due\nto local anisotropies [57] (Sec. IV D of SM [35]).\nConcerning three-dimensional NAIs, we note that py-\nrochlores with the all-in–all-out texture as, e. g., Sm 2Ir2O7\n[58], are expected to exhibit a planar SNE with the property\nthat the force, the current, and the transported spin are mutually\northogonal to each other (Sec. IV E of SM [35]).Conclusion. We showed that NAIs can exhibit bulk\nmagnon spin currents, thus complementing the recent pro-\nposal of an interfacial SSE [ 59]. As results, NAIs o \u000ber the\ncombined advantages of nonelectronic spin transport and an-\ntiferromagnetism. They may replace ferromagnets as spin-\nactive components of next-generation spin(calori)tronic de-\nvices and introduce a paradigm of an “antiferromagnetic insu-\nlator spincaloritronics”, as the magnonic pendant to the thriving\nfield of “antiferromagnetic spinelectronics” [60–67].\nBesides an experimental proof of principle of our quantita-\ntive predictions for KFe 3(OH) 6(SO 4)2our work calls for the\ndevelopment of a theory of magnon spin and heat di \u000busion [ 68]\nin NAIs, an investigation of the influence of noncollinearity-\ninduced magnon-magnon interactions [ 53] on spin transport, of\nthe dynamics of noncollinear antiferromagnetic domain walls\n[69] in temperature gradients, and a material search for NAIs\nwith ordering temperatures above room temperature.\nBy virtue of Onsager’s reciprocity relation [ 70], the exis-\ntence of SSEs and SNEs in noncollinear antiferromagnets,\nimmediately implies that of the spin Peltier and spin Ettings-\nhausen e \u000bects. 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Pure-\nspin dynamics is obtained by tracing out the orbital degrees of freedom, whereas pseudo-spin dynamics retains\n(as is conventional) an implicit coordinate dependence. We show that magnetic field inhomogeneity and spin-\norbit interaction result in a non-unitary evolution in pure-spin dynamics, whereas these interactions contribute\nto the effective pseudo-spin Hamiltonian via terms that are asymmetric in spin perm utations, in particular, the\nDzyaloshinskii-Moriya (DM) spin-orbit interaction. We nu merically investigate the non-unitary effects in the\ndynamics of the triplet states population, purity, and Lamb energy shift, as a function of interdot distance and\nmagnetic field difference ∆/vectorB. The spin-orbit interaction is found to produce effects of r oughly four orders of\nmagnitude smaller than those due to ∆/vectorBin thepure-spin model. We estimate the spin-orbit interaction mag-\nnitude in the DM-interaction term. Our estimate gives a smal ler value than that recently obtained by Kavokin\n[Phys. Rev. B 64,075305(2001)],whodidnotincludedoubleoccupancyeffec ts. Weshowthatanecessaryand\nsufficientcondition forobtaining a universal setof quantu m logicgates, involvingonlytwospins, inboth pure-\nandpseudo-spin models is that the magnetic field inhomogeneity ∆/vectorBand the Heisenberg interaction are both\nnon-vanishing. We also briefly analyze pure-spin dynamics in the electron on liquid helium system recen tly\nproposed by Lyon[Phys. Rev. A 74, 052338 (2006)].\nPACS numbers:\nI. INTRODUCTION\nThe spin degree of freedom of a localized particle, e.g., an\nelectron or nucleus, is a popular carrier of quantum informa -\ntion. Itservesasaqubitwhichcanbemanipulatedinorderto\naccomplisha computationaltask. Thespin ofelectronsloca l-\nized in quantum dots (QDs) or by donor atoms has been the\nsubjectofextensiverecentstudies1,2,3,4,5,6,7,8,9,10,11,12,13,14,15.\nConsider two electrons trapped in two sites AandB, e.g.,\ntwoQDs eachcontainingoneelectron. Thetwo-electronsys-\ntemisfullydescribedbythetotalwavefunction |Ψtot/an}b∇acket∇i}ht,which\ndependsontheelectrons’coordinates /vector randspin variables σ.\nThetwo-electronspin-densitymatrix,obtainedbytracing out\nthe orbital degrees of freedom, ρ= Tr/vector r|Ψtot/an}b∇acket∇i}ht/an}b∇acketle{tΨtot|, fully\ndescribesthespindynamicsaslongasonecannotordoesnot\nwishtoapplymeasurementsthatcanseparateorlocalizeele c-\ntrons spatially; the only observable is then the electron sp in,\nsα=1\n2σα,whereσαarethePaulispinone-halfmatriceswith\nα=x,y,z. Since the spin system is not closed – there is a\ncouplingtotheelectrons’spatialdegreesoffreedom–weob -\nserve open system effects, i.e., the spin dynamicsbecomesi n\ngeneral non-unitary. We refer to this dynamics as pure-spin\ndynamics.\nIn contrast, pseudo-spin dynamics is the standard case\nwhere the electron spin observable is not free from coordi-\nnatedependencebutincludesinformationabouttheelectro n’s\nlocalization orbital. In the pseudo-spin case one defines the\nelectron spin operator as a bilinear combination of electro n\nannihilation and creation Fermi operators, cAs,c†\nAs, in a lo-\ncalized orbital φA(sis a spin index, Ais the QD index):sα\nA=1\n2/summationtext2\nss′=1c†\nAs(σα)ss′cAs′,α=x,y,z. Thenthe oper-\nators{sα\nA}αobeytheusualsu (2)commutationrules.\nThis paper is the sequel to our work Ref. 16 (henceforth\n“part I”), where we derived an operator-sum representation\n(OSR) as well as a master equationin the Lindbladand time-\nconvolutionless(TCL) forms for the spin-density matrix of a\ntwo-electron system. In this sequel we focus on a detailed\ncomparison of pureandpseudo-spin dynamics. We are in-\nterested in particular in how non-unitary effects in pure-spin\ndynamicsaretranslatedintothecorrespondingunitaryone sin\npseudo-spin dynamics and vice versa. We show that as long\nasthereis nomagneticfield inhomogeneitythe pure-spindy-\nnamics is unitary, but in the presence of magnetic field inho-\nmogeneitythisdynamicsisnon-unitary\nThe paper is organized as follows. We begin, in Section\nII by highlighting the differences and relationship betwee n\npseudo-andpure-spinmodels. SectionIIIprovidesaconcrete\nillustrationintermsofasystemoftwoQDstrappingoneelec -\ntron each. In it, we examine the role of different interactio ns\nin bothpseudo-andpure-spin dynamics. We first derive the\ncoordinate part of the Hamiltonian (subsection IIIA) and th e\nform of the dipolar interaction (subsection IIIB). In subse c-\ntionsIIICandIIID,respectively,wethenpresentcalculat ions\nillustratingeffectsduetobothexternalmagneticfieldinh omo-\ngeneity and the spin-orbit interaction in the pure-spin model.\nIn subsections IIIE and IIIF, we discuss universal quantum\ngatesin both pseudo-andpure-spinmodels. SubsectionIIIG\npresents our estimates for spin-orbit interaction effects in the\npseudo-spin model, and compares these estimates to the re-\nsultsofRef.17. We concludein SectionV.2\nAtomic units, /planckover2pi1=e=me= 1,1/c≃1/137, are used\nthroughoutthepaperunlessstatedotherwise.\nII.PSEUDO- VSPURE-SPINAPPROACHES\nIn this section we discuss the relation between the present\napproach based on the spin-density matrix and the pseudo-\nspin effectiveHamiltonianapproach. Thelatter is usually de-\nveloped as a low-energy mapping within the Hubbard model\nHamiltonian of interacting electrons9,11,18,19,20,21,22,23. We do\nnotfollowtheHubbardmodelsinceitishighlysimplifiedand\nneglectsmanyinteractionswhichwe wouldlike tokeephere.\nForHubbardmodelanalysesinthequantumcomputationcon-\ntext,see,e.g.,Ref.9.\nA.Pseudo-spineffective Hamiltonian\nIn order to keep the present treatment as simple as possi-\nble we restrict ourselves to the two orbitals approximation\nused in part I; inclusion of excited-state orbitals is strai ght-\nforward. Consider the four single-occupancy basis states\n{Φs1,Φti,i= 1,2,3}, whereΦs1is a singlet wavefunction\nwithtwoelectronslocalizedondifferentQDs, AandB,while\nΦtiare the corresponding triplet wavefunctions. The two\ndouble-occupancystates {Φs2,Φs3}describetwoelectronsin\na singlet state, localized on the same QD, AorB. The total\nwavefunction Ψtot(t)inthisbasisset takesthe form\nΨtot(t) =3/summationdisplay\ni=1(asi(t)Φsi+ati(t)Φti),\nwherethecomplexamplitudes {asi(t),ati(t)}define,respec-\ntively, the singlet and triplet states population. In the to tal\nHilbert space, the state is defined by 11 real parameters [12\nreal parameters defining {asi(t),ati(t)}minus a normaliza-\ntion condition]. The unitary evolution in the total Hilbert\nspaceisdescribedby\n/parenleftbigg\n|as(t)/an}b∇acket∇i}ht\n|at(t)/an}b∇acket∇i}ht/parenrightbigg\n= exp(−iHt)/parenleftbigg\n|as(0)/an}b∇acket∇i}ht\n|at(0)/an}b∇acket∇i}ht/parenrightbigg\n,\nwhereHisthetotal two-electronsystem Hamiltonian.\nSince these basis states are orthonormal, projection opera -\ntorsintothecorrespondingsubspacescanbewrittenas\nP=|Φs1/an}b∇acket∇i}ht/an}b∇acketle{tΦs1|+3/summationdisplay\ni=1|Φti/an}b∇acket∇i}ht/an}b∇acketle{tΦti|,\nQ=|Φs2/an}b∇acket∇i}ht/an}b∇acketle{tΦs2|+|Φs3/an}b∇acket∇i}ht/an}b∇acketle{tΦs3|, (1)\nwhereQprojects onto the double occupancy states. Then,\nusing the method of projection operators, one obtains the\nSchr¨ odinger (eigenvalue) equation projected into the P-\nsubspace\n(Heff(E)−E)PΨ = 0, (2)where\nHeff(E) =PHP+PHQ1\nE−QHQQHP. (3)\nObserve that Eq. (2) is exactbut non-linear, and has 6 solu-\ntions.\nDue to interelectron repulsion the double occupancystates\nare usually much more energetic than the singly-occupied\nones if the electrons are well localized in QDs. We consider\nthe low-energy physics described by Eq. (2) where the total\nenergyEis near the energies of singly-occupied states. In\ngeneralHeffis not a Hamiltonian since it is a function of the\nenergyE. However, if the energy gap between the P- and\nQ-statesislargeenough,onecanexpandandapproximate\nHeff(E) =Heff(¯E)+∞/summationdisplay\nn=1PHQ/parenleftbig¯E−E/parenrightbign\n/parenleftbig¯E−QHQ/parenrightbign+1QHP\n=Heff(¯E)+H(1)\neff(¯E)/parenleftbig¯E−E/parenrightbig\n+O[/parenleftbig¯E−E/parenrightbig2]\n≈ Heff(¯E)+H(1)\neff(¯E)/parenleftbig¯E−E/parenrightbig\n, (4)\nwhere¯Eis an average energy in the P-subspace and\nH(1)\neff(¯E) =PHQ/parenleftbig¯E−QHQ/parenrightbig−2QHP. Keeping terms up\nto the first order in Eq. (4), the non-linear Eq. (2) can be\nreducedto ageneralizedlinearequationproblem\n/parenleftig\nHeff(¯E)+H(1)\neff(¯E)¯E−(1+H(1)\neff(¯E))E/parenrightig\nPΨ = 0.(5)\nSolvingEq. (5),we obtainfourlowenergysolutions;thetwo\nhigh energy, double-occupancy solutions are lost in this ap -\nproximation. Therefore, in the low energy, pseudo-spin ap-\nproximationthe state isdescribedby7 realparameters.\nIn the following, we assume H(1)\neff(¯E)≡0for simplicity.\nTheeffectiveHamiltonianEq. (4)canberecastintoa pseudo-\nspinform. UsingEq. (1)we have\nHeff=Hss|Φs1/an}b∇acket∇i}ht/an}b∇acketle{tΦs1|+3/summationdisplay\ni,j=1Htt\nij|Φti/an}b∇acket∇i}ht/an}b∇acketle{tΦtj|\n+3/summationdisplay\ni=1/parenleftbig\nHst\ni|Φs1/an}b∇acket∇i}ht/an}b∇acketle{tΦti|+Hts\ni|Φti/an}b∇acket∇i}ht/an}b∇acketle{tΦs1|/parenrightbig\n,(6)\nwhere\nHss=/an}b∇acketle{tΦs1|Heff(¯E)|Φs1/an}b∇acket∇i}ht,Htt\nij=/an}b∇acketle{tΦti|Heff(¯E)|Φtj/an}b∇acket∇i}ht,\nHst\ni=/an}b∇acketle{tΦs1|Heff(¯E)|Φti/an}b∇acket∇i}ht,Hts\ni=/parenleftbig\nHst\ni/parenrightbig∗. (7)\nInthesecondquantizationrepresentation,the P-subspaceba-3\nsisvectorstaketheform\n|Φs1/an}b∇acket∇i}ht=1√\n2/parenleftig\nc†\nA↑c†\nB↓−c†\nA↓c†\nB↑/parenrightig\n|0/an}b∇acket∇i}ht=\n1√\n2(|↑/an}b∇acket∇i}htA⊗|↓/an}b∇acket∇i}htB−|↓/an}b∇acket∇i}htA⊗|↑/an}b∇acket∇i}htB),\n|Φt1/an}b∇acket∇i}ht=c†\nA↑c†\nB↑|0/an}b∇acket∇i}ht=|↑/an}b∇acket∇i}htA⊗|↑/an}b∇acket∇i}htB,\n|Φt2/an}b∇acket∇i}ht=1√\n2/parenleftig\nc†\nA↑c†\nB↓+c†\nA↓c†\nB↑/parenrightig\n|0/an}b∇acket∇i}ht=\n1√\n2(|↑/an}b∇acket∇i}htA⊗|↓/an}b∇acket∇i}htB+|↓/an}b∇acket∇i}htA⊗|↑/an}b∇acket∇i}htB),\n|Φt3/an}b∇acket∇i}ht=c†\nA↓c†\nB↓|0/an}b∇acket∇i}ht=|↓/an}b∇acket∇i}htA⊗|↓/an}b∇acket∇i}htB, (8)\nwhere we introduced pseudo-spin states |s/an}b∇acket∇i}htα,s=↑,↓,α=\nA,B, localized near the AandBsites [the term pseudoem-\nphasizes the fact that these are not really spin states since\nthey depend on the electron orbital degrees of freedom].\nEqs. (8)establishaone-to-onecorrespondencebetween4ba -\nsis states {Φs1,Φti,i= 1,2,3}and4 tensor-product pseudo-\nspinstates |s/an}b∇acket∇i}htα⊗|s′/an}b∇acket∇i}htβ,wheres,s′=↑,↓,α,β=A,B. Then,\nrelabeling the pseudo-spin states as |0,1/an}b∇acket∇i}ht=|↑,↓/an}b∇acket∇i}htand intro-\nducingthe pseudo-spinPauliandidentityoperators\nσx=|0/an}b∇acket∇i}ht/an}b∇acketle{t1|+|1/an}b∇acket∇i}ht/an}b∇acketle{t0|,\nσy=−i(|0/an}b∇acket∇i}ht/an}b∇acketle{t1|−|1/an}b∇acket∇i}ht/an}b∇acketle{t0|),\nσz=|0/an}b∇acket∇i}ht/an}b∇acketle{t0|−|1/an}b∇acket∇i}ht/an}b∇acketle{t1|,\nI=|0/an}b∇acket∇i}ht/an}b∇acketle{t0|+|1/an}b∇acket∇i}ht/an}b∇acketle{t1| (9)\nwhere we temporarily dropped the subscripts AandB, one\neasilyfindsthat\n|Φs1/an}b∇acket∇i}ht/an}b∇acketle{tΦs1|=S,|Φti/an}b∇acket∇i}ht/an}b∇acketle{tΦtj|=Tij,\n|Φs1/an}b∇acket∇i}ht/an}b∇acketle{tΦti|=Ki. (10)\nHerethepseudo-spinoperators S,Tijaredefinedas\nS=1\n4I−/vector sA·/vector sB,T11=1\n4I+1\n2Sz+sAzsBz,\nT22=1\n4I+sAxsBx+sAysBy−sAzsBz,\nT33=1\n4I−1\n2Sz+sAzsBz,\nT12=1√\n2/bracketleftbig1\n2S++Js/bracketrightbig\n,T23=1√\n2/bracketleftbig1\n2S+−Js/bracketrightbig\n,\nT13=sAxsBx−sAysBy+i(sAxsBy+sBxsAy),\nT=3\n4I+/vector sA·/vector sB,T21=T†\n12,\nT31=T†\n13,T32=T†\n23\n(11)\nwhere\nJs=sAzsBx+sAxsBz+i(sAzsBy+sAysBz),\nS±=Sx±iSy,/vectorS=/vector sA+/vector sB,\nandKisdefinedas\nK1=−i\n2√\n2/braceleftbigg/parenleftig\n/vectorJas/parenrightig\nx−i/parenleftig\n/vectorJas/parenrightig\ny/bracerightbigg\n,K2=i\n2/parenleftig\n/vectorJas/parenrightig\nz,\nK3=i\n2√\n2/braceleftbigg/parenleftig\n/vectorJas/parenrightig\nx+i/parenleftig\n/vectorJas/parenrightig\ny/bracerightbigg\n(12)where/vectorJas=/bracketleftig\n/vector sB−/vector sA×/vectorS/bracketrightig\n. In fact, Eqs. (11) and (12) can\nbe obtainedfromthe correspondingonesin part I if the pure-\nspin operators /vector s1,2are replaced respectively by the pseudo-\nones,/vector sA,B. We reproduce these formulas here in order to\nmakethepresentationasself-containedaspossible.\nAsisseenfromEqs. (11)and(12),thefirstlineofEq. (6)is\nsymmetricwith respect to spin permutations[ A↔B], while\nthe second one is asymmetric representing, in particular, t he\nDzyaloshinskii-Moriya (DM-type) interaction term.24,25No-\nticethattheseasymmetric(inspinpermutations)termscan cel\nout of unitary spin dynamics after averaging over orbital de -\ngreesoffreedom,asdemonstratedinpartI.However,theydo\nnotdisappearcompletely,butratherareconvertedintothe cor-\nrespondingnon-unitarytermsplusthe Lambshift termaswill\nbe seen in next subsection. From the symmetric part of the\nHamiltonian (6), using Eqs. (11) one can derive the isotropi c\nHeisenbergexchangeinteractionterm\nHH=JH/vector sA·/vector sB, (13)\nwhere the Heisenberg exchange interaction constant JH=\n1\n3/summationtext\niHtt\nii−Hss;incontrast,aswasdemonstratedinpartI,the\nHeisenberginteractiontermdoesnotaffectthe unitaryevo lu-\ntion of the spin density matrix, apart from the Lamb-energy\nshift. In subsection IIIC, we demonstratenumericallythe e f-\nfects of the Heisenberg interaction on both the Lamb-energy\nshift and the non-unitary part of the spin density matrix evo -\nlution.\nObserve that the asymmetric part of the Hamiltonian Eq.\n(6) is proportional to the singlet-triplet subspace intera ction\nmatrixHst\ni,whichisresponsibleforthecouplingbetweensin-\nglet and triplet states. As will be demonstratedin Section I II,\nthe non-zero coupling between these states is due to /vectorB-field\nspatial inhomogeneity (i.e., it cannot arise due to the homo -\ngeneouscomponentoftheexternalmagneticfield),aswellas\ndueto thespin-orbitinteraction.\nB. Spindensitymatrix\nIn part I we derived the Lindblad-typemaster equation for\nthespindensitymatrix\n∂ρ(t)\n∂t=−i/bracketleftig\n˜Htt\nα,ρ(t)/bracketrightig\n+Lα[ρ(t)], (14)\n˜Htt\nα=/summationdisplay\nij/parenleftbigg\nHtt+1\n2Pα/parenrightbigg\nijTij=Htt+1\n2Pα,\nLα[ρ(t)] =1\n2/summationdisplay\nij(χα)ij/parenleftig/bracketleftig\nKi,ρ(t)K†\nj/bracketrightig\n+/bracketleftig\nKiρ(t),K†\nj/bracketrightig/parenrightig\n,\nwhere the first and second terms describe, respectively, uni -\ntary and non-unitary contributions to the evolution. ˜Htt\nαis\naneffective pure-spinHamiltonianwhichincludesthe Lamb-\nshift term,1\n2Pα; thepure-spin operators TijandKiare de-\nfined by Eqs. (11) and (12) where /vector sA,B→/vector s1,2. The index\nα={s,t,m}specifies which initial state ρ(0), singlet, s,\ntriplet,t,ora mixedone, m, istaken.4\nAs mentionedin part I, all the matrixfunctionsin Eq. (14)\naswellasthe pseudo-spinHamiltonianEq. (6)areexpressible\nintermsof Hmatrixelements\nH=/parenleftbigg\nHssHst\nHtsHtt/parenrightbigg\n, (15)\nHαβ\nij=/an}b∇acketle{tΦαi|H|Φβj/an}b∇acket∇i}ht, α,β=s,t, i,j = 1,2,3.\nIn the following example, we consider the triplet case for\nwhichwehave\nχT\nα=i/parenleftbig\nQα−Q†\nα/parenrightbig\n, (16)\nPα=Qα+Q†\nα\nwhere\nQα=/summationtext\nkexp(−iεkt)Hts[|esk/an}b∇acket∇i}ht/an}b∇acketle{tesk|Rα(0)+|esk/an}b∇acket∇i}ht/an}b∇acketle{tetk|]×\n/parenleftbigg/summationtext\nkexp(−iεkt)[|etk/an}b∇acket∇i}ht/an}b∇acketle{tesk|Rα(0)+|etk/an}b∇acket∇i}ht/an}b∇acketle{tetk|]/parenrightbigg−1\n.\n(17)\nHere,Rα(0)isacorrelationmatrix,whichestablishesanini-\ntial correlationbetweenthesingletandtripletamplitude s\nas(0) =Rα(0)at(0) (18)\n[in the triplet case, we have Rα=t(0)≡0; in the mixed case,\nwhere both as(0)/ne}ationslash= 0andat(0)/ne}ationslash= 0,Rα=m(0)/ne}ationslash= 0; for the\nsinglet case, see part I] and εk,|esk/an}b∇acket∇i}ht, and|etk/an}b∇acket∇i}htare solutions\ntothe eigenvalueproblem\n/parenleftbigg\nHssHst\nHtsHtt/parenrightbigg/parenleftbigg\n|esk/an}b∇acket∇i}ht\n|etk/an}b∇acket∇i}ht/parenrightbigg\n=εk/parenleftbigg\n|esk/an}b∇acket∇i}ht\n|etk/an}b∇acket∇i}ht/parenrightbigg\n, k= 1,···,6.\n(19)\nIII. EXAMPLE:SYSTEMOF TWOQUANTUMDOTS\nIn this section, we investigate the role of different inter-\nactions in the calculation of the Hmatrix. Let us consider\na system of two electrons trapped at sites /vector rAand/vector rB(/vector rA,B\nare radius-vectors of the centers of QDs in the z= 0plane)\ncreated by a system of charged electrodes in a semiconduc-\ntor heterostructure so that the electrons are confined in the\nz= 0plane or a system of localized conduction-band elec-\ntrons inn-doped GaAs as in our calculation example. The\nheterostructuretrappingpotential\nVtr(z,/vector r) =V⊥(z)+VA(/vector r)+VB(/vector r)(20)\nisseparableinthein-planeandout-of-planedirections; V⊥(z)\nandVA,B(/vector r)arethetrappingpotentialsin the z-directionand\nin thez= 0plane around /vector rA,Brespectively. If the elec-\ntronsystemisplacedina constantmagneticfield /vectorB0directed\nalong the z-axis (with vector potential /vectorA0=1\n2/bracketleftig\n/vector r×/vectorB0/bracketrightig\n),\nthen the in-plane motion, in a superposition of the in-planeconfining oscillatory potential and a perpendicular magnet ic\nfield,isdescribedbytheFock-Darwin(FD)states.26Approx-\nimatingtheconfiningpotentialbya quadraticone\nVA,B(/vector r)≈1\n2ω2\nA,B(/vector r−/vector rA,B)2, (21)\nwe can take as basis “atomic” orbitals the ground-state func -\ntions\nφA,B(z,/vector r) =ϕ0(z)RFD\nA,B(|/vector r−/vector rA,B|),(22)\nwheretheout-of-planemotioninthe z-directionis“frozen”in\nthegroundstate ϕ0(z)in thepotential V⊥(z), andthe ground\nFD state is\nRFD\nA,B=1√\n2πlA,Bexp/parenleftigg\n−r2\n4l2\nA,B/parenrightigg\n, (23)\nlA,B=lc\n4/radicalig\n1+4ω2\nA,B/ω2c, lc=/radicalbiggc\nB0.\nHerelA,Bis the effective length scale, equal to the magnetic\nlengthlcin the absence of the confiningpotential, ωA,B≡0;\nωc=B0/cis thecyclotronfrequency.\nThe orbitals Eq. (22) must be orthogonalized. One way\nto dothisis a simpleGram-Schmidtorthogonalizationproce -\ndure:\n˜φA=φA,\n˜φB=1√\n1−S2(φB−SφA),(24)\nwhere the overlap matrix element SAB=SBA=S=\n/an}b∇acketle{tφA|φB/an}b∇acket∇i}htcanbecalculatedanalytically\nS=2lAlB\nl2\nA+l2\nBexp/parenleftbigg\n−r2\nAB\n4(l2\nA+l2\nB)/parenrightbigg\n.(25)\nFor appropriate values of system parameters such as the in-\nterdot distance rABand the external magnetic field B0, the\noverlapbecomesexponentiallysmall.\nThe other, more symmetric way is to make a transition to\nthe“molecular”ortwo-centeredorbitalsbypre-diagonali zing\nthecoordinatepartofPauli’s non-relativisticHamiltoni anˆhc,\nwhichdescribestheelectron’smotioninasuperpositionof the\ntrappingpotentialandmagneticfields:\n˜φA=cAAφA+cABφB,\n˜φB=cBAφA+cBBφB,/angbracketleftig\n˜φi/vextendsingle/vextendsingle/vextendsingle˜φj/angbracketrightig\n=δij, i,j =A,B,/angbracketleftig\n˜φi/vextendsingle/vextendsingle/vextendsingleˆhc/vextendsingle/vextendsingle/vextendsingle˜φj/angbracketrightig\n=εiδij, i,j =A,B.(26)\nThe two-state eigenvalue problem Eq. (26) is solved analyt-\nically in terms of “atomic” orbitals matrix elements: hij=\n/an}b∇acketle{tφi|ˆhc|φj/an}b∇acket∇i}ht.\nIn general, given the “molecular” Eq. (26) or “half-\nmolecular” Eq. (24) basis choices, one cannot ascribe a spin\nto a particular QD, since an electron in a molecular orbital\nbelongstobothQDs.\nThe total Hamiltonian contains both coordinate and spin-\ndependentterms. First we consider the coordinatepart of th e\nHamiltonianinthe ˜φA,Bbasisset.5\nA. Coordinate part of theHamiltonian\nIn view of the orthogonality of the singlet and triplet spin\nwave functions,the spin-independentpart of the Hamiltoni an\ndoes not contribute to the singlet-triplet coupling, Hst\nc=\nHts\nc= 0, whereas for the singlet-singlet and triplet-triplet\nHamiltonianswe get\nHss\nc={Hss\ncij}3\ni,j=1, Hss\ncij=Hss∗\ncji\nHss\nc11=˜hAA+˜hBB+ ˜vee(AB;AB)+ ˜vee(AB;BA),\nHss\nc12=√\n2/parenleftig\n˜hBA+ ˜vee(AB;AA)/parenrightig\n, (27)\nHss\nc13=√\n2/parenleftig\n˜hAB+ ˜vee(AB;BB)/parenrightig\n,\nHss\nc22= 2˜hAA+ ˜vee(AA;AA), Hss\nc23= ˜vee(AA;BB),\nHss\nc33= 2˜hBB+ ˜vee(BB;BB),\nHtt\nc=εtI, (28)\nεt=˜hAA+˜hBB+ ˜vee(AB;AB)−˜vee(AB;BA),\nwhere\n˜hij=/angbracketleftig\n˜φi/vextendsingle/vextendsingle/vextendsingleˆhc/vextendsingle/vextendsingle/vextendsingle˜φj/angbracketrightig\n, i,j=A,B\n˜vee(ij;kl) =/angbracketleftig\n˜φi(1)˜φj(2)/vextendsingle/vextendsingle/vextendsingle1\nεr12/vextendsingle/vextendsingle/vextendsingle˜φk(1)˜φl(2)/angbracketrightig\n,(29)\ni,j,k,l=A,B\nwith˜hij=εiδijfor “molecular” orbitals and ˜veebeing the\ninterelectron electrostatic interaction matrix elements . The\nmatrixHss\ncis diagonally dominated if the overlap S≪1;\nHss\nc11isthesingletenergyofthesinglyoccupiedstatewhereas\nHss\nc22andHss\nc33are energies of doubly occupied states if one\nneglects the coupling between single- and double-occupanc y\nstates. Observe that the Heisenberg constant JH=εt−εs\nwhereεsisthe lowesteigenvalueofthematrix Hss\nc. Thema-\ntrix elements ˜hijand˜vee(ij;kl)can trivially be expressed in\ntermsofthecorrespondingmatrixelements hijandvee(ij;kl)\nwheretheorthonormalizedstates ˜φi’sarereplacedby φi’sus-\ningtherelationsEq. (24) or(26).\nB. Dipolespin-spininteraction\nIn the total spin representation, the dipole spin-spin inte r-\nactioncanberewrittenas27\nVdip=1.45\n2meV/parenleftigg\nS2r2\n12−3(/vectorS·/vector r12)2\nr5\n12−\n8π\n3/parenleftbigg\nS2−3\n2/parenrightbigg\nδ(/vector r12)/parenrightbigg\n. (30)\nSince/vectorS|χs/an}b∇acket∇i}ht= 0andft(/vector r1=/vector r2) = 0,where|χs/an}b∇acket∇i}htandftare\nsinglet-state spin and triplet-state coordinate wavefunc tions,we haveHst\ndip=Hts\ndip= 0and a non-zero contribution to\nHss\ndipcomesonlyfromthecontactterm:\n(Hss\ndip)ij= (1.45·2π) meV/an}b∇acketle{tfsi|δ(/vector r12)|fsj/an}b∇acket∇i}ht\n= 1.09ΛmeV\nd11d12d13\nd12d221\n2d11\nd131\n2d11d33\n(31)\nwhereΛisaneffectiveconstantoftheinteractionthatconfines\nelectronsin the z-planeand\nd11= 2/angbracketleftig\n˜φ2\nA/vextendsingle/vextendsingle/vextendsingle˜φ2\nB/angbracketrightig\n=1\nl2\nA+l2\nBexp/parenleftbigg\n−r2\nAB\n2(l2\nA+l2\nB)/parenrightbigg\n,\nd12=√\n2/angbracketleftig\n˜φ3\nA/vextendsingle/vextendsingle/vextendsingle˜φB/angbracketrightig\n=√\n2lB/lA\n3l2\nB+l2\nAexp/parenleftbigg\n−3\n4r2\nAB\n3l2\nB+l2\nA/parenrightbigg\n,\nd13=√\n2/angbracketleftig\n˜φA/vextendsingle/vextendsingle/vextendsingle˜φ3\nB/angbracketrightig\n=√\n2lA/lB\n3l2\nA+l2\nBexp/parenleftbigg\n−3\n4r2\nAB\n3l2\nA+l2\nB/parenrightbigg\n,\nd22=/angbracketleftig\n˜φ2\nA/vextendsingle/vextendsingle/vextendsingle˜φ2\nA/angbracketrightig\n=1\n4l2\nA, d33=/angbracketleftig\n˜φ2\nB/vextendsingle/vextendsingle/vextendsingle˜φ2\nB/angbracketrightig\n=1\n4l2\nB.(32)\nThe magneticdipolecontributionto the triplet-triplet in terac-\ntionHamiltoniancanbewrittenas\nHtt\ndip= 0.36meV\n¯t0−3√\n2¯t∗\n1−3¯t∗\n2\n−3√\n2¯t1−2¯t03√\n2¯t∗\n1\n−3¯t23√\n2¯t1¯t0\n(33)\nwhereti, i= 0,1,2aredipoletensoroperators\nt0=1−3cos2θ12\nr3\n12,\nt1=sin2θ12exp(iϕ12)\nr3\n12, (34)\nt2=sin2θ12exp(2iϕ12)\nr3\n12\nwith(r12,θ12,ϕ12)being spherical coordinates of the inter-\nelectron radius-vector /vector r12=/vector r1−/vector r2; the bar over tidenotes\naveragingoverthetripletcoordinatewavefunction:\n¯ti=/integraldisplay /integraldisplay\nd3/vector r1d3/vector r2|ft(/vector r1,/vector r2)|2ti(/vector r1,/vector r2)(35)\nTakingintoaccountthefactthattheelectronsareexponen-\ntially localized at sites /vector rAand/vector rBin theftstate, a good ap-\nproximation to ¯tiis to approximate the function tiby a con-\nstant value at those points where ft(/vector r1,/vector r2)is localized, thus\nobtainingtheestimate\nHtt\ndip=0.36\nr3\nABmeV\n1 0−3\n0−2 0\n−3 0 1\n.(36)\nInordertofurtherimprovetheestimate,thefunction ti(/vector r1,/vector r2)\ncan be expanded in a Taylor’s series around the localization\npoints and the remaining integrals in the expansion terms be\ncalculated analytically. From Eq. (36) we find the estimate\nHtt\ndip∼(0.36/r3\nAB)meV≈5.0·10−8meVatrAB= 100˚A.6\nC. The /vectorB-fieldinteractioninthe pure-spinmodel\nForthemagneticfield onegets\nHtt(/vectorB) =\nBavzB−\nav0\nB+\nav0B−\nav\n0B+\nav−Bavz\n,(37)\nwhere\n/vectorBav=1\n2/parenleftig/angbracketleftig\n˜φA/vextendsingle/vextendsingle/vextendsingle/vectorB/vextendsingle/vextendsingle/vextendsingle˜φA/angbracketrightig\n+/angbracketleftig\n˜φB/vextendsingle/vextendsingle/vextendsingle/vectorB/vextendsingle/vextendsingle/vextendsingle˜φB/angbracketrightig/parenrightig\n,\nB±\nav=1√\n2(Bavx±iBavy). (38)\nUsing Eqs. (11) and (37), one derivesthe Zeeman interac-\ntion Hamiltonian of the total spin /vectorSwith the magnetic field\n/vectorBav:\nHtt(/vectorB) =3/summationdisplay\nij=1Htt\nij(/vectorB)Tij=/vectorBav·/vectorS.(39)\nSimilarly,forthesinglet-tripletmatrixwehave\nHst(/vectorB) =\n1\n2√\n2∆B+−1\n2∆Bz−1\n2√\n2∆B−\n1\n2δB+−1√\n2δBz−1\n2δB−\n−1\n2δB−∗1√\n2δB∗\nz1\n2δB+∗\n,(40)\nwhere\n∆/vectorB=/angbracketleftig\n˜φB/vextendsingle/vextendsingle/vextendsingle/vectorB/vextendsingle/vextendsingle/vextendsingle˜φB/angbracketrightig\n−/angbracketleftig\n˜φA/vextendsingle/vextendsingle/vextendsingle/vectorB/vextendsingle/vextendsingle/vextendsingle˜φA/angbracketrightig\n,\n∆B±= ∆Bx±i∆By,\nδ/vectorB=/angbracketleftig\n˜φA/vextendsingle/vextendsingle/vextendsingle/vectorB/vextendsingle/vextendsingle/vextendsingle˜φB/angbracketrightig\n, δB±=δBx±iδBy.(41)\nIf the/vectorB-field is homogeneous, from Eq. (41) we obtain\n∆/vectorB=δ/vectorB= 0andHst(/vectorB) = 0. In this case, the spin\ndynamicsisunitaryandis describedbytheZeemanHamilto-\nnianHtt(/vectorB)Eq. (39);thespin-spindipoleinteraction Htt\ndipis\ntoosmall andcanusuallybesafelyneglected.\nLet us consider modifications due the to /vectorB-field inhomo-\ngeneityinthe pure-spinmodel. Neglectingcontributionsfrom\nthedouble-occupancystateswithinthefirst-orderperturb ation\napproximation in the singlet-triplet interaction Hst, we find\nforthenon-unitaryterminEq. (14)\nLt=1\n2/summationdisplay\nij(χt)ij/parenleftig/bracketleftig\nKi,ρK†\nj/bracketrightig\n+/bracketleftig\nKiρ,K†\nj/bracketrightig/parenrightig\n=sin(JHt)\nJH/parenleftbig/bracketleftbig\nK,ρK†/bracketrightbig\n+/bracketleftbig\nKρ,K†/bracketrightbig/parenrightbig\n,(42)\nwhere\nK=/summationdisplay\niHst\n1i(/vectorB)Ki=−i\n4∆/vectorB·/vectorJas (43)\nand\n/vectorJas=/bracketleftig\n/vector s2−/vector s1×/vectorS/bracketrightig\n= 2[/vector s2×/vector s1]+i(/vector s2−/vector s1)(44)/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48/s49/s46/s49/s84/s114/s105/s112/s108/s101/s116/s32/s115/s116/s97/s116/s101/s115/s32/s112/s111/s112/s117/s108/s97/s116/s105/s111/s110/s44/s32/s124/s97\n/s116/s40/s116/s41/s124/s50\n/s116/s105/s109/s101/s44/s32/s50 /s116/s47/s74\n/s72/s32/s124 /s66\n/s122/s47/s74\n/s72/s124/s61/s48/s46/s49/s50\n/s32/s48/s46/s51/s48\n/s32/s48/s46/s53/s57\n/s32/s48/s46/s56/s57\n/s32/s49/s46/s49/s56\n/s32/s50/s46/s51/s55/s114\n/s65/s66/s61/s52/s48/s48/s32/s65/s110/s103/s115/s116/s114/s111/s109\nFIG. 1: (color online) The triplet states population of elec trons in\nshallow QD centers in GaAs, as a function of time at different mag-\nnetic field differences ∆Bz= 0.01,0.025,0.05,0.075,0.1,0.2T,\nnormalized to the Heisenberg exchange JHconstant. Initially the\nsystemisinthetripletstate |S= 1,MS= 0/angbracketright. Thedistancebetween\nQDsis 400 ˚A.\nis an asymmetric spin operator containing both linear and\nbilinear parts. Observe that Lt= 0at the “swap” times\ntn=πn/JH,n= 0,1,....\nSimilarly,forthe Lambshift inEq. (14) wehave\nLt=1\n2/summationdisplay\nij(Pt)ijK†\niKj=1−cos(JHt)\nJHK†K.(45)\nObserve that LtandLtare quadratic in the difference field\n∆/vectorB. Besides, notice that the magnetic field due to spin-orbit\ncouplingdoesnotcontributetothedifferencefield ∆/vectorBso= 0\nbutcontributestothe δ/vectorBso-fieldthatispresentinthecoupling\nbetween the triplet states and the double occupancy, single t\nstates,inEq. (40). Iftheexternalmagneticfield /vectorBexishomo-\ngeneous, then the singlet-triplet states coupling comes on ly\nfrom the spin-orbit interaction. Since the double-occupan cy\nstates should be involved in the dynamics in order to obtain\nnon-zero spin-orbit interaction effects, these effects ar e ex-\npected to be especially small, proportional to δB2\nso,in the\npure-spin model. An estimate of these spin-orbit effects will\nbegivenina numericalexamplein thenextSection.\nClearly,thereisanimportantqualitativedifferencebetw een\npure-andpseudo-spin models. In the former, the singlet-\ntriplet states coupling is a second order effect, while in th e\nlatter this coupling is of first order in Hst[cf., Eq. (42) and\n(6)]. Thus, in pure-spin models effects due to /vectorB-field inho-\nmogeneityshouldbe especially (quadratically)small as co m-\nparedtothecorresponding pseudo-spinmodeleffects. Incase\nofnegligible /vectorB-fieldinhomogeneity,asfollowsfromEq. (40),\nthepure-spindynamicsis unitaryandis governedbythe spin\nHamiltonian Htt.\nLet us now consider a simple numerical example for the\nnon-unitaryeffectsduetothedifferencefield ∆/vectorBforanelec-\ntronlocalizedona donorimpurityin an n-dopedGaAs semi-7\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s52/s53/s48/s46/s53/s48/s48/s46/s53/s53/s48/s46/s54/s48/s48/s46/s54/s53/s48/s46/s55/s48/s48/s46/s55/s53/s48/s46/s56/s48/s48/s46/s56/s53/s48/s46/s57/s48/s48/s46/s57/s53/s49/s46/s48/s48/s49/s46/s48/s53/s80/s117/s114/s105/s116/s121/s44/s32/s112/s40/s116/s41\n/s116/s105/s109/s101/s44/s32/s50 /s116/s47/s74\n/s72/s32/s124 /s66\n/s122/s47/s74\n/s72/s124/s61/s48/s46/s49/s50\n/s32/s48/s46/s51/s48\n/s32/s48/s46/s53/s57\n/s32/s48/s46/s56/s57\n/s32/s49/s46/s49/s56\n/s32/s50/s46/s51/s55/s114\n/s65/s66/s61/s52/s48/s48/s32/s65/s110/s103/s115/s116/s114/s111/s109\nFIG. 2: (color online) The purity p(t) = Trρ2(t)for the same\nparameters as inFig. 1.\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s76/s97/s109/s98/s32/s101/s110/s101/s114/s103/s121/s32/s115/s104/s105/s102/s116/s44/s32/s69\n/s76/s97/s109/s98/s47/s74\n/s72\n/s116/s105/s109/s101/s32/s50 /s116/s47/s74\n/s72/s32/s124 /s66\n/s122/s47/s74\n/s72/s124/s61/s48/s46/s49/s50\n/s32/s48/s46/s51/s48\n/s32/s48/s46/s53/s57\n/s32/s48/s46/s56/s57\n/s32/s49/s46/s49/s56\n/s32/s50/s46/s51/s55/s114\n/s65/s66/s61/s52/s48/s48/s32/s65/s110/s103/s115/s116/s114/s111/s109\nFIG. 3: (color online) The Lamb shift energy as a function of t ime\nfor the same parameters as inFig. 1.\nconductor. To simplify numerics, we assume that Hss=\ndiag(εs1,εs2,εs3)is diagonalandthe singlet-tripletcoupling\nfield,∆/vectorB, has only a non-zero z-component, ∆Bz. Then,\nthe corresponding eigenvalue problem, Eq. (19), can be re-\nduced to a biquadratic polynomial equation which could, in\nprinciple, be solved exactly. If we neglect the exponential ly\nsmall coupling field δBz, proportional to the overlap S, the\nbiquadratic equation reduces to a quadratic one and we find\nforthenon-unitaryterm\nLt=ωsinωt\nJ2\nH+1\n2∆B2z(1+cosωt)×\n/parenleftbig/bracketleftbig\nK,ρK†/bracketrightbig\n+/bracketleftbig\nKρ,K†/bracketrightbig/parenrightbig\n(46)\nK=−i\n4∆Bz·Jasz/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s84/s114/s105/s112/s108/s101/s116/s32/s115/s116/s97/s116/s101/s115/s32/s112/s111/s112/s117/s108/s97/s116/s105/s111/s110/s44/s32/s124/s97\n/s116/s40/s116/s41/s124/s50\n/s116/s105/s109/s101/s44/s32/s50 /s116/s47/s124 /s66\n/s122/s124/s32/s114\n/s65/s66/s61/s51/s48/s48/s32/s65/s110/s103/s46/s32\n/s32/s51/s53/s48\n/s32/s52/s48/s48\n/s32/s52/s53/s48\n/s32/s53/s48/s48\n/s32/s54/s48/s48/s66\n/s122/s61/s48/s46/s48/s53/s32/s84\nFIG. 4: (color online) Triplet states population dependenc e on in-\nterdot separation rAB= 300,350,400,450,500, and600˚A, at a\nfixed∆Bz= 0.05T.\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s49/s48/s49/s50/s51/s52/s53/s54/s55/s56/s76/s97/s109/s98/s32/s101/s110/s101/s114/s103/s121/s32/s115/s104/s105/s102/s116/s44/s32/s69\n/s76/s97/s109/s98/s47/s124 /s66\n/s122/s124\n/s116/s105/s109/s101/s44/s32/s50 /s116/s47/s124 /s66\n/s122/s124/s32/s114\n/s65/s66/s61/s51/s48/s48/s32/s65/s110/s103/s46\n/s32/s51/s53/s48\n/s32/s52/s48/s48\n/s32/s52/s53/s48\n/s32/s53/s48/s48\n/s32/s54/s48/s48/s66\n/s122/s61/s48/s46/s48/s53/s32/s84\nFIG. 5: (color online) Lamb energy shift dependence on inter dot\ndistancerAB= 300,350,400,450,500, and600˚A, at a fixed\n∆Bz= 0.05T.\nandtheLambshift\nLt=JH(1−cosωt)\nJ2\nH+1\n2∆B2z(1+cosωt)K†K,(47)\nwhereω=/radicalbig\nJ2\nH+∆B2z. In the limit of small mag-\nnetic field inhomogeneity, |∆Bz/JH| ≪1,ω→ |JH|\nand Eqs. (46) and (47) go over into (42) and (45), respec-\ntively. Eq. (47) describes the Lamb energy shift of the\ntriplet state |S= 1,MS= 0/an}b∇acket∇i}htdue to the coupling between\nsinglet and triplet states induced by the magnetic field inho -\nmogeneity ∆Bz. At the magnetic field geometry we have\nchosen there is no coupling between |S= 1,MS=±1/an}b∇acket∇i}htand\n|S= 1,MS= 0/an}b∇acket∇i}htstates.\nIn Figs. 1-3, we show the results of calculations for the\ntriplet states population, purity, and the Lamb shift energ y,\nrespectively, as a function of time at a fixed interdot separa -8\ntion (rAB= 400˚A) and different ∆Bz. For the Heisenberg\ninteraction constant JHwe used an asymptotically correct\nexpression28,29,30obtained for hydrogenlike centers in GaAs\n[note that our JH=εt−εsis related to the exchange in-\ntegralJin Ref. 28 via JH=−2J]. Initially, the system\nis assumed to be in the |S= 1,MS= 0/an}b∇acket∇i}htstate. As can be\nseen from Fig. 1, there is a re-distribution between singlet\nand triplet states population due to the singlet-triplet su b-\nspace coupling. At |∆Bz/JH|/lessorsimilar0.1, the probability of re-\ndistribution is negligible and the time-evolution is basic ally\nunitary. With increasing ∆Bz, this probabilityre-distribution\nis seen to be more pronounced,time-evolution becomesnon-\nunitary (Fig. 2), and |at(t)|2can drop to the value J2\nH/ω2at\nt=πn/ω, n = 1,3,.... Observe that the non-unitary dy-\nnamicsrevealsrepetitionsintimeandatmomentsofmaximal\n(minimal) singlet-triplet states probability re-distrib ution we\nfind maximal (minimal) Lamb energy shifts (Fig. 3). Thus,\nthe non-unitary effects observed are not irreversible and t hey\ndo not result in a real decoherence process. We do not have\nin our two-electronmodela real, externaland infinite “bath ”,\ncouplingtowhichwouldresultinirreversibledecoherence ef-\nfects in the spin system. In Figs. 4,5 we demonstrate the de-\npendence of triplet states population and Lamb energy shift s\nontheinterdotdistance rABat afixed ∆Bz= 0.05T.\nD. Spin-orbitinteractionin pure-spinmodel\nIn this subsection we estimate the non-unitary effects in\nthepure-spin model due to spin-orbit interaction. For sim-\nplicity we assume that the external magnetic field is homo-\ngeneous and directed along the z-axis, with Bozbeing its z-\ncomponent. Since δ/vectorBsois a pure imaginary field (its compo-\nnents are matrix elements between the real states ˜φAand˜φB\nof an odd vector function of the momentumoperator, both in\nvacuumandin the bulkof semiconductorsthat lack inversion\nsymmetry, Dresselhaus fields,31as well as in heterostructure\nzinc-blendes,Rashbafields,32) wehave\n(δB±\nso)∗=−δB∓\nso, δB∗\nsoz=−δBsoz.(48)\nUsingthese relationships,the singlet-tripletspin-orbi tcou-\nplingcanbe writtenas\nHst(/vectorBso) =\n0 0 0\n1\n2δB+\nso−1√\n2δBsoz−1\n2δB−\nso\n1\n2δB+\nso−1√\n2δBsoz−1\n2δB−\nso\n.(49)\nThe couplings between double-occupancy, singlet and\ntriplet states are seen to be the same. We assume that\nHss= diag(εs,εdo,εdo),whereεsandεdoare the sin-\nglet and double-occupancy states energies, and Htt=\ndiag(εt+,εt,εt−),whereεtisatripletstateenergyand εt±=\nεt±B0z. Withintheseapproximations,the6-by-6eigenvalue\nproblem Eq. (19) is then reduced to computing the roots of\nthebiquadraticequation33\nE4+a3E3+a2E2+a1E+a0= 0 (50)where\na3=−4/summationdisplay\ni=1εi, a 2=/summationdisplay\ni/negationslash=jεiεj−/summationdisplay\nα=x,y,z|δBsoα|2,\na1=−/summationdisplay\ni/negationslash=j/negationslash=kεiεjεk+\n/parenleftbigg\n|δBsoz|2+1\n2[|δBsox|2+|δBsoy|2]/parenrightbigg\n(ε2+ε4)+\n(|δBsox|2+|δBsoy|2)ε3,\na0=4/productdisplay\ni=1εi−|δBsoz|2ε2ε4−\n1\n2[|δBsox|2+|δBsoy|2]ε3(ε2+ε4),\nε1=εs, ε 2=εt+, ε 3=εt, ε 4=εt−.\nFor hydrogenlikecentersone can estimate the energies εdo\nandεtas follows. The ground energy of two well separated\nhydrogen atoms is E2H≈ −27.2eV. Using for GaAs the\nscalingfactor KGaAs=m∗/ε2≈4.6·10−4onecanestimate\nεt≈KGaAsE2H=−12.6meV.εdois located higher than\nεtdue to mainly interelectronrepulsion ˜vee(AA;AA)so that\nεdo−εt= ˜vee(AA;AA)≈12.6meV.\nIfδBsoα≡0, α=x,y,z, the roots of Eq. (50) Eiare\nequal toεi,i= 1,...,4. The two other roots are E5=εdo\nandE6=εs. Thecorrespondingeigenvectorsare\n|ek/an}b∇acket∇i}ht=dk/parenleftigg\n0,1,1,−δB−\nso\nEk−ε2,√\n2δBsoz\nEk−ε3,δB+\nso\nEk−ε4/parenrightiggT\n,\nk= 1,...,4,\n|e5/an}b∇acket∇i}ht=1√\n2(0,1,−1,0,0,0)T,\n|e6/an}b∇acket∇i}ht= (1,0,0,0,0,0)T, (51)\nwhere\ndk=/parenleftbig\n2+[|δBsox|2+|δBsoy|2]×\n/parenleftbigg1\n(Ek−ε2)2+1\n(Ek−ε4)2/parenrightbigg\n+2|δBsoz|2\n(Ek−ε3)2/parenrightbigg−1/2\n.\nNoticethattheaboveformulasarenotvalidinthedegener-\nate case: B0z= 0andε2=ε3=ε4=εt. In this case the\nbiquadraticEq. (50) reducesto two quadraticones, two root s\nof which are degenerate, E1=E2=εt. Formally, one gets\nsingularities in Eq. (51) at E1=E2=εt. Therefore, the\nsimpler, degenerate case should be analyzed separately and\nthecorrespondingformulas[notshownhere]canbederived.\nLetusnowfindthe spin-orbitfield\nδ/vectorBso=/angbracketleftig\n˜φA/vextendsingle/vextendsingle/vextendsingle/vectorBso(/vector p)/vextendsingle/vextendsingle/vextendsingle˜φB/angbracketrightig\n≈ /an}b∇acketle{tφA|/vectorBso(/vector p)|φB/an}b∇acket∇i}ht\n=/integraltext\nd/vector rφA(|/vector r−/vectorR|)/vectorBso(−i∇/vector r)φB(r)\n=/vectorBso(−i∇/vectorR)/integraltext\nd/vector rφA(|/vector r−/vectorR|)φB(r)\n=/vectorBso(−i∇/vectorR)S(R)(52)9\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s57/s57/s57/s57/s55/s48/s48/s46/s57/s57/s57/s57/s55/s53/s48/s46/s57/s57/s57/s57/s56/s48/s48/s46/s57/s57/s57/s57/s56/s53/s48/s46/s57/s57/s57/s57/s57/s48/s48/s46/s57/s57/s57/s57/s57/s53/s49/s46/s48/s48/s48/s48/s48/s48/s84/s114/s105/s112/s108/s101/s116/s32/s115/s116/s97/s116/s101/s115/s32/s112/s111/s112/s117/s108/s97/s116/s105/s111/s110/s44/s32/s124/s97\n/s116/s40/s116/s41/s124/s50\n/s116/s105/s109/s101/s44/s32/s50 /s116/s47/s124\n/s100/s111/s45\n/s116/s124/s32/s114\n/s65/s66/s61/s50/s48/s48/s32/s65/s110/s103/s115/s116/s114/s111/s109\n/s32/s50/s53/s48\n/s32/s51/s48/s48\n/s32/s51/s53/s48\n/s32/s52/s48/s48\n/s32/s53/s48/s48\n/s61 /s47/s52/s44/s32 /s61 /s47/s51\nFIG. 6: (color online) Triplet states population dependenc e on spin-\norbitinteractionasafunctionoftime,atdifferentinterd otseparations\nrABand fixed orientation of the interdot radius-vector, /vector rAB, (θ=\nπ/4,ϕ=π/3) [see text]. The initial triplet states population is\ntaken tobe equal. The external magnetic field B0z= 0.5T.\nwhere/vectorBso(/vector p)is an odd function of the momentum operator\n/vector p=−i∇/vector rand/vectorR=/vector rAB. In particular, in zinc-blende semi-\nconductors such as GaAs, /vectorBsois cubic in the componentsof\n/vector p:31,34\nBsoα=Asopα(p2\nβ−p2\nγ),\nα,β,γ={cyclicpermutationsof x,y,z}\nAso=αso/parenleftig\nm∗/radicalbig\n2m∗Eg/parenrightig−1\n, (53)\nwherem∗istheeffectivemassoftheelectron, Egistheband\ngap [m∗≈0.072, Eg≈1.43eVfor GaAs], px,py,pzare\ncomponents of the momentum along the cubic axes [100],\n[010], and[001]respectively. The dimensionless coefficient\nαso= 0.07forGaAs. FromEqs. (52)and(53)we obtain\nBsoα=iAsoRα(R2\nβ−R2\nγ)\nR3/braceleftig\nS′′′(R)−\n3\nRS′′(R)+3\nR2S′(R)/bracerightbigg\n(54)\nThe overlap integral35S(r=R/aB) = (1 + r+\nr2/3)exp(−r)for hydrogenlike centers [ aB≈92˚Afor\nGaAs]andEq. (54) reducesto\nBsox=i(0.83meV)sin θcosϕ(sin2θsin2ϕ−cos2θ)×/parenleftbigg\n−1\n3/parenrightbigg\nr2exp(−r) (55)\nwhere(R,θ,ϕ)aresphericalcoordinatesofthevector /vectorR,with\nothercomponentsbeingobtainedbycyclic interchangeofin -\ndices.\nIn Figs. 6 and 7 we display the time-dependence of the\ntriplet states population and the purity, which is induced b y\nthe spin-orbit interaction, Eq. (55), at a fixed orientation ,/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s57/s57/s57/s57/s52/s48/s46/s57/s57/s57/s57/s53/s48/s46/s57/s57/s57/s57/s54/s48/s46/s57/s57/s57/s57/s55/s48/s46/s57/s57/s57/s57/s56/s48/s46/s57/s57/s57/s57/s57/s49/s46/s48/s48/s48/s48/s48/s80/s117/s114/s105/s116/s121/s44/s32/s112/s40/s116/s41\n/s116/s105/s109/s101/s44/s32/s50 /s116/s47/s124\n/s100/s111/s45\n/s116/s124/s32/s114\n/s65/s66/s61/s50/s48/s48/s32/s65/s110/s103/s115/s116/s114/s111/s109\n/s32/s50/s53/s48\n/s32/s51/s48/s48\n/s32/s51/s53/s48\n/s32/s52/s48/s48\n/s32/s53/s48/s48\n/s61 /s47/s52/s44/s32 /s61 /s47/s51\nFIG. 7: (color online) The purity in the presence of spin-orb it cou-\npling. Allparameters are the same as inFig. 6.\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s57/s57/s57/s57/s50/s53/s48/s46/s57/s57/s57/s57/s51/s48/s48/s46/s57/s57/s57/s57/s51/s53/s48/s46/s57/s57/s57/s57/s52/s48/s48/s46/s57/s57/s57/s57/s52/s53/s48/s46/s57/s57/s57/s57/s53/s48/s48/s46/s57/s57/s57/s57/s53/s53/s48/s46/s57/s57/s57/s57/s54/s48/s48/s46/s57/s57/s57/s57/s54/s53/s48/s46/s57/s57/s57/s57/s55/s48/s48/s46/s57/s57/s57/s57/s55/s53/s48/s46/s57/s57/s57/s57/s56/s48/s48/s46/s57/s57/s57/s57/s56/s53/s48/s46/s57/s57/s57/s57/s57/s48/s48/s46/s57/s57/s57/s57/s57/s53/s49/s46/s48/s48/s48/s48/s48/s48/s49/s46/s48/s48/s48/s48/s48/s53/s84/s114/s105/s112/s108/s101/s116/s32/s115/s116/s97/s116/s101/s115/s32/s112/s111/s112/s117/s108/s97/s116/s105/s111/s110/s44/s32/s124/s97\n/s116/s40/s116/s41/s124/s50\n/s116/s105/s109/s101/s44/s32/s50 /s116/s47/s124\n/s100/s111/s45\n/s116/s124/s32/s40 /s61 /s47/s52/s44 /s61/s48/s41\n/s32/s40 /s61 /s47/s52/s44 /s61 /s47/s50/s41\n/s32/s40 /s61 /s47/s52/s44 /s61 /s41\n/s32/s40 /s61 /s47/s51/s44 /s61/s48/s41\n/s32/s40 /s61 /s47/s51/s44 /s61 /s47/s50/s41\n/s32/s40 /s61 /s47/s51/s44 /s61 /s41\n/s114\n/s65/s66/s61/s50/s48/s48/s32/s65/s110/s103/s115/s116/s114/s111/s109\nFIG. 8: (color online) Effect of spin-orbit angular anisotr opy on\ntripletstates population. Allparameters are the same as in Fig. 6.\n(θ=π/4,ϕ=π/3), and different rABin the range 200-\n500˚A. Observe that the maximal re-distribution of singlet-\ntripletprobabilityoccursat 2πt/|εdo−εt|= 0.5andthespin-\norbitinteractioneffectdiminishesas rABincreases. Themax-\nimalsinglet-stateprobabilityachievedat rAB= 200˚Aisseen\ntobequitesmall, ∼10−5. Ascomparedtothenon-unitaryef-\nfectsinducedby /vectorB-fieldinhomogeneity,thespin-orbiteffects\nare onaveragefourordersof magnitudesmaller. Theangular\ndependence of the population of triplet states on the interd ot\nradius-vectororientationatafixed rAB= 200˚Aisillustrated\ninFig.8.10\nE. The /vectorB-fieldinteractioninthe pseudo-spin model\nUsing Eqs. (37) and (40), the effectiveHamiltonian matrix\nEq.(6)in thebasis {Φs1,Φti,i= 1,2,3}canberewrittenas\nHeff=\nεs1\n2√\n2∆B+−1\n2∆Bz−1\n2√\n2∆B−\n1\n2√\n2∆B−εt+BavzB−\nav 0\n−1\n2∆BzB+\nav εt B−\nav\n−1\n2√\n2∆B+0 B+\navεt−Bavz\n\n(56)\nwhere for simplicity we neglected contributions from the\ndouble-occupancystates [the resolvent term in Eq. (3)]. Al -\nternatively,inthe pseudo-spinrepresentationwe get\nHeff=/parenleftbigg1\n4εs+3\n4εs/parenrightbigg\nI+JH/vector sA·/vector sB+\n/vectorBav·(/vector sA+/vector sB)+∆/vectorB\n2·(/vector sA−/vector sB)\n=/parenleftbigg1\n4εs+3\n4εt/parenrightbigg\nI+JH/vector sA·/vector sB+\n/vectorBA·/vector sA+/vectorBB·/vector sA, (57)\nwhere/vectorBA=/vectorBav+∆/vectorB/2and/vectorBB=/vectorBav−∆/vectorB/2are the\nlocal magneticfields at sites AandB, respectively. The term\nJH/vector sA·/vector sBis the familiar Heisenberg interaction. In matrix\nform,Eq. (57) canberewrittenas\nHeff=\nε11\n2B−\nA0 0\nB+\nAε21\n2JH0\n01\n2JHε31\n2B−\nB\n0 01\n2B+\nBε4\n(58)\nε1=εt+1\n2BAz, ε2=εt−1\n2(JH+BAz),\nε3=εt−1\n2(JH−BBz), ε4=εt−1\n2BBz\nwhereB±\nA,B=BA,Bx±iBA,By.\nTheHamiltonianEq. (56) generatesaunitaryevolution\nUeff(t) = exp(−itHeff) (59)\ninC4. At a fixed set of parameters εs,εt,/vectorBA,/vectorBBthe propa-\ngatorUeff(t)does not providea universal set of unitary gates\ninC4. Any unitary transformation U∈U(4)can be repre-\nsented as a product of a phase factor exp(iα), whereαis a\nreal parameter, and a unitary transformation US∈SU(4).\nAny transformation USis determined by M= 42−1 = 15\nindependentrealparameters (θ1,...,θ 15)so that\nUS(θ1,...,θ M) = exp/parenleftigg\n−iM/summationdisplay\ni=1θiFi/parenrightigg\n,(60)\nwhere the set ofgenerators {Fi}is an orthonormalizedtrace-\nless,HermitianmatrixsetthatformsaLiealgebra su(4)[Fi’s\nform a complete basis in a real M-dimensionalvector space;\ntheyareanalogsofPauli matrices, σα,α=x,y,z,insu(2)-see,e.g.,Ref.36]. FromtherepresentationEq. (60)itfoll ows\nimmediately that Ueff,S(t) = exp/parenleftbig\n−it/parenleftbig1\n4εs+3\n4εt/parenrightbig/parenrightbig\nUeff(t)\ncannot match an arbitrary US, because the number of inde-\npendent parameters in Eq. (56) is at most 8– fewer than the\nnumberof θi’s. Thiscanalsobeunderstoodfromthefactthat\nthe form of the Hamiltonian matrix, Eq. (56), is not generic.\nIn particular, the matrix is sparse, i.e., the entries (2,4)and\n(4,2)arezeros.\nHowever,compositionsofunitarytransformationsEq. (59)\ntaken at different sets of parameters can provide a universa l\nsetofunitarygatesin C4. Awell-knownexampleofuniversal\ngates is provided by the Heisenberg interaction (at JH/ne}ationslash= 0)\nwithsingle-spinaddressing(at ∆/vectorB/ne}ationslash= 0).1\nFrom Eq. (56) it follows that a necessary and sufficient\ncondition for obtaining a universal set of gates on two spins\nis to have an inhomogeneity in the magnetic field ∆/vectorB/ne}ationslash= 0\n[the source of inhomogeneity can be different, it can be ei-\nther strongly localized magnetic fields or g-factor engineer-\ning] and the Heisenberg interaction, JH/ne}ationslash= 0. The reason\nis that when ∆/vectorB= 0, the Hamiltonian Eq. (56) and the\ncorresponding unitary transformations take a block-diago nal\nform, with singlet-triplet entries being zeros, while when\nJH= 0, the Hamiltonian form (58) will have zero off-\ndiagonalblock-matrices. Clearly, evena compositionof su ch\nunitary transformations taken at different sets of paramet ers,\neitherεs,εt,/vectorBA=/vectorBBorεs=εt,/vectorBA,/vectorBB, will be in a\nblock-diagonalformandit cannotreproduceanarbitraryun i-\ntary transformation. Note that when one allows for encoding\na qubit into three or more spins, the Heisenberg interaction\nalone is universal in the pseudo-spin model,37,38and Heisen-\nberg along with an inhomogneousmagnetic field is universal\nforanencodingofa singlequbitintopairofspins.39\nMoreover,itshouldbenotedthatinthehomogeneousmag-\nneticfieldcaseunitarytransformationsrestrictedtothet riplet\nsubspace will not provide a universal set of gates. To prove\nthis statement, let us consider a composition of two unitary\ntransformationsinthetripletsubspace:\nexp(−it1Htt(/vectorB1))exp(−it2Htt(/vectorB2)) =\nexp(−i(t1+t2)Htt\neff) =\nexp/parenleftig\n−i(t1+t2)εt−it1Htt(/vectorB1)−it2Htt(/vectorB2)−\nt1t2\n2/bracketleftig\nHtt(/vectorB1),Htt(/vectorB2)/bracketrightig\n+···/parenrightig\n,(61)\nwhereontheright-hand-sideweusedtheCampbell-Hausdorf f\nformula.40FromEq. (37) oneobtains\n/bracketleftig\nHtt(/vectorB1),Htt(/vectorB2)/bracketrightig\n=i\n2BzB−0\nB+0B−\n0B+−2Bz\n,(62)\nwhere/vectorB=[/vectorB1×/vectorB2]andB±=/vectorBx±i/vectorBy. Since the\nhigher-order terms in the Campbell-Hausdorff formula con-\nsist of nested commutators between Htt(/vectorB1)andHtt(/vectorB2),\nwe find that the effective Hamiltonian Htt\neffcorresponding to\nthe product of two unitary transformations will still have a\nsparseform,withthe (1,3)and(3,1)entriesbeingzeros.11\nF. Isit possibletoobtainauniversal set ofgates inthe\npure-spinmodel?\nSimultaneously, we have just proved that in the pure-spin\nmodel, in the case of a homogeneous magnetic field, unitary\ntransformationsin the triplet subspace will not providea u ni-\nversal set of gates. On the other hand, at ∆/vectorB/ne}ationslash= 0, we have\nalready shown that the evolution of the spin-density matrix\nis non-unitary. Let us assume that we have a non-unitary\ngateL1so thatρ(t1) =L1(t1)ρ(0). How could one define\na composition of two non-unitary gates, L2L1? In order to\ndo this unambiguously, L2should obey a compatibility con-\ndition with the initial state [because a non-unitary L-gate is\nnot totally independent of the initial state, it includes so me\nsort of correlation information encoded in the initial stat e],\nthat is a correlation Rm(t1)established between as(t1)and\nat(t1)amplitudes at t=t1should be included in in the def-\ninition of the corresponding dynamics generator operators in\nL2. Eq. (17), where the left-hand-side and Htsshould be\nreplacedby Rm(t1)andtheidentitymatrix,respectively,pro-\nvidesarelationshipbetween Rm(t1)andRm(0). Ifthecorre-\nlationbetweentheamplitudesat t= 0andt=t1isthesame,\nRm(t1) =Rm(0),thenweobviouslyhave L1=L2=Land\nL2(t2)L1(t1) =L(t1+t2).\nInthetotal Hilbertspace,thestate isdefinedby11realpa-\nrameters. While in the reduced description, the spin-densi ty\nmatrix is defined by 5 real parameters (for more on the spin\ndensitymatrixparametrizationintermsof a’s,seeSectionV).\nFixing a correlation in the initial state, as(0) =Rm(0)at(0),\nwe have 3 complex equations between the amplitudes as(0)\nandat(0), which define a 5D real manifold embedded into\nthetotalHilbertspace. Usingtheseequationswecansepara te\n6 extra real degreesof freedomthat we have in the total state\ndescription from those in the spin-densitymatrix descript ion.\nHowever, these extra degrees of freedom are not eliminated\nin the spin-density description, they are included in the fo rm\nof correlation matrix Rα(0),α={s,t,m}. It was shown in\npart I that Eq. (14) providesan exactdescription of quantum\nevolution,in the spin-densityspace. Therefore,as long as we\nhaveauniversalset ofunitarygatesinthe totalHilbertspa ce,\nthis set of gateswill be translated into the correspondingu ni-\nversal set of non-unitarygatesgeneratedby Eq. (14) becaus e\nno information is lost in our “reduced” spin-density matrix\ndescription.\nG. Spin-orbitinteraction inthe pseudo-spinmodel\nLetusconsiderspin-orbiteffects,whichareproportional to\nδBso, inthepseudo-spinmodel. FromEq. (6)we obtain\nHso(δ/vectorBso) =/summationdisplay\nkk′=2,3Hss\n1k/parenleftbigg1\n¯EI−Hss/parenrightbigg−1\nkk′Fk′(δ/vectorBso),\n(63)where\nFk(δ/vectorBso) =3/summationdisplay\ni=1/parenleftig\nHst\nki(δ/vectorBso)Ki+(Hst\nki(δ/vectorBso))∗K†\ni/parenrightig\n(64)\nIt follows from Eq. (49) that Hst\n2i=Hst\n3i,F2=F3and\nEq.(64) canbereducedto\nF2,3(δ/vectorBso) =F(δ/vectorBso) =i√\n2δ/vectorBso[/vector sA×/vector sB](65)\nThen,Eq. (63) canberewrittenas\nHso(δ/vectorBso) =AdoF(δ/vectorBso) (66)\nwhere the coefficient Adois proportional to the ratios of the\namplitudes of double occupancy transitions ( Hss\n12andHss\n13)\nand the energies of interelectron interaction ( Hss\n22−¯Eand\nHss\n33−¯E)in doublyoccupiedQDs:\nAdo=/summationdisplay\nkk′=2,3Hss\n1k/parenleftbigg1\n¯EI−Hss/parenrightbigg\nkk′\n=1\n∆/bracketleftbig\nHss\n12(¯E−Hss\n33−Hss\n32)+\nHss\n12(¯E−Hss\n22−Hss\n23)/bracketrightbig\n≈Hss\n12\n¯E−Hss\n22+Hss\n13\n¯E−Hss\n33. (67)\nHere,thedeterminantis\n∆ = (¯E−Hss\n22)(¯E−Hss\n33)−|Hss\n23|2.(68)\nTheeffectivespin-orbitinteractionHamiltonianEq. (66) is\ndifferent from the correspondingone obtained by Kavokin.17\nIn our derivation the double-occupancy states are essentia l,\nwhereas in Ref. 17 these states are totally neglected. We\nshowedabovethatneglectingdouble-occupancystatesresu lts\ninzerospin-orbitcoupling. Theveryphysicalpictureputf or-\nwardinRef.17tosupportthederivationwasbasedontheas-\nsumptionthat whenoneof thetwo electronslocalizedat cen-\ntersAorBtunnelstotheadjacentcenter(say,from AtoB),it\nexperiencestheinfluenceofthespin-orbitfieldresultingf rom\nthe under-barrier motion of the electron. Neglecting doubl e\noccupancy states means that the second electron should si-\nmultaneouslytunnelfrom BtoAsothatthetwoelectronscan\nneverbefoundinthesameQD.Indeed,inRef.17thissimul-\ntaneoustwo-electrontransition is describedby the produc tof\ntwo matrix elements: the overlap Sand(δ/vectorBso)α,α=x,y,z\n[Hso∼S(δ/vectorBso·[/vector sA×/vector sB]),ourδ/vectorBsoisrelatedtoKavokin’s\nb-field via δ/vectorBso=−ib, and the overlap via S= Ω]. With\nthe orthogonalized molecular-type, two-center orbitals s uch\na one step two-electron transition gives a zero contributio n\nsince the spin-orbit interaction is a one-electronoperato r and\nthe overlap ˜S=/angbracketleftig\n˜φA/vextendsingle/vextendsingle/vextendsingle˜φB/angbracketrightig\n= 0. Eq. (63) describes the\ntwo-step mechanism: in the first step, the two-electron sys-\ntem makes a transition from the singly-occupiedstate Φs1to\ntheintermediate,double-occupancystates Φs2,Φs3duetothe\ninter-electron interaction ( Hss\n1k, k= 1,2terms). Then in the12\n/s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48 /s49/s50/s48/s48 /s49/s52/s48/s48 /s49/s54/s48/s48 /s49/s56/s48/s48 /s50/s48/s48/s48 /s50/s50/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48/s75\n/s115/s111\n/s114\n/s65/s66/s32/s40/s65/s110/s103/s115/s116/s114/s111/s109/s41\nFIG. 9: Spin-orbit interaction reduction coefficient Eq. (7 3) as a\nfunction of interdot distance for GaAs.\nsecond step, as a result of the spin-orbit interaction, the s ys-\ntem makes transitions from Φs2,Φs3toΦtitriplet states (the\nHst\nkiterms).\nLetusfindanestimate for\nAdo(R)≈ −2√\n2\n|εdo−εt|/integraldisplay/integraldisplay\nd/vector r1d/vector r2φ2(r1)×\nφ(|/vector r2−/vectorR|)1\nε|/vector r2−/vector r1|φ(r2),(69)\nwhere the hydrogenlike orbital φ(r) =\n(πa3\nB)−1/2exp(−r/aB). Since electron 1 in Eq. (69)\nis localized around the effective Bohr radius aB, one can\napproximate\n1\n|/vector r2−/vector r1|≈1\n|/vector r2−aB/vector r1/r1|. (70)\nThen, the remaining integrals can be calculated exactly and\nwe obtain\nAdo(r=R/aB)≈ −4√\n2\ner{4eF1(r)−\n2[F1(r+1)+F1(r−1)]−\n[F2(r+1)−F2(r−1)]+\n2[F0(r+1)−F0(r−1)]}(71)\nwhere\nF0(x) =sign(x)\n48/bracketleftbig\n15(1+|x|)+6|x|2+|x|3/bracketrightbig\nexp(−|x|),\nF1(x) =1\n48/bracketleftbig\n3|x|(1+|x|)+|x|3/bracketrightbig\nexp(−|x|),\nF2(x) =sign(x)\n2(1+|x|)exp(−|x|).\n(72)\nHere, we used the same estimate for εdo−εtas in Section\nIIID.\nInordertocompareourcalculationstoKavokin’sresultfor\nGaAs, we have plotted in Fig. 9 the spin-orbit interaction re -\nductioncoefficient\nKso=|Ado|√\n2S, (73)which is exactly the ratio of our and Kavokin’s estimates,\nas a function of interdot distance. As one can see, Ksode-\ncreases from 0.98 to 0.46 in the range rAB∼300−700˚A.\nInterestingly, our results qualitatively agree with the re sults\nof Ref. 28, which obtained in the region of interest [ rAB∼\n(3−7)aB] a reduction of about one-half relative to that of\nRefs. 17,41. According to Ref. 28, Hso∼2JγGK, whereJ\nis the exchange integral calculated using the medium hyper-\nplane method29,30andγGKis an angle of spin rotation due\nto spin-orbit interaction introduced by Gor’kov and Krotko v\n[γGK≈1\n2γK,γKbeingthe correspondingangle of spin rota-\ntion introduced by Kavokin]. Note that Kavokin’s γKandJ\nare not independent parameters. In Ref. 17, γKwas defined\nasΩb/Jso that their product 2JγK= 2Ωbdoes not depend\nexplicitlyon J.\nIV. ELECTRONSON LIQUID HELIUM\nRecently,Lyonsuggestedthatthespinofelectronsfloating\non the surface of liquid helium (LHe) will make an excellent\nqubit.42Lyon’s proposal, instead of using the spatial part of\nthe electron wavefunction as a qubit as in the charge-based\nproposal,43,44,45,46takesadvantageofthesmallervulnerability\nof the electron’s spin to external magnetic perturbationsa nd,\nasaconsequence,alongerspin-coherencetime. Italsohast he\nimportant advantage over semiconductor spin-based propos -\nals (asfirst pointedout in the charge-basedproposal43,44,45,46)\nthat, with the electrons residing in the vacuum, several im-\nportantsourcesofspindecoherenceareeliminatedso thatt he\nenvironmenteffectsarehighlysuppressed(thespin-coher ence\ntimeisestimated42tobeT2>100s).\nThe geometry of the system with the electrons trapped at\nthe LHe-vacuum interface (see Fig. 2 in Ref. 42), is concep-\ntually similar to that of semiconductor heterostructures. The\ntwo electronsare trappedat sites /vector rAand/vector rB(/vector rA,Bare radius-\nvectors of the centers of quantum dots in the z= 0plane)\nbythe twoattractivecenterscreatedbytwochargedspheric al\nelectrodes located below the LHe surface at distance hand\nseparatedbytheinterdotdistance rAB=|/vector rA−/vector rB|:\nVtr(z,/vector r) =−Λ\nz+VA(z,/vector r)+VB(z,/vector r)(74)\nVA,B(z,/vector r) =−QA,B/radicalig\n(z+h)2+(/vector r−/vector rA,B)2,(75)\nwherethe first potential, −Λ/z,is dueto attractionto the im-\nage charge induced by an electron in the LHe [ Λ = (ε−\n1)/(4(ε+1))≈7·10−3,withε≈1.057beingthedielectric\nconstant of helium]. For purposes of interaction control, t he\nQA,Bchargesontheelectrodescanbemadevariableintime.\nTheelectronsarepreventedfrompenetratingintoheliumby a\nhigh potential barrier ( ∼1eV) at the helium surface, so that\nformally one can put Vtr=∞atz <0. The in-plane and\nout-of-planemotionofelectronsinthepotentialofEq.(74 )is\nin generalnon-separable. However,near the electrode’spo si-\ntion/vector rA,BthepotentialofEq.(75) isapproximatelyseparable13\ninthezand/vector rcoordinates\nVA,B(z,/vector r)≈ −QA,B\nh+E⊥A,Bz+1\n2ω2\nA,B(/vector r−/vector rA,B)2,\n(76)\nwhere it is assumed that zand|/vector r−/vector rA,B| ≪handE⊥A,B=\nQA,B/h2,ωA,B=/parenleftbig\nQA,B/h3/parenrightbig1/2. In the separable approx-\nimation of Eq. (76), the electron’s motion in the z-direction\n[z >0] is described by a 1D-Coulomb potential perturbed\nbya small Starkinteractionandthe in-planemotionbya 2D-\noscillatory potential. We assume that the out-of-plane mo-\ntionin the z-directionis “frozen”inthe groundstate ofa 1D-\nCoulombpotential\nϕC\n0(z) = 2√\nΛ(Λz)exp(−Λz), (77)\nand the in-plane motion, in a superposition of the in-plane\nconfining oscillatory potential and, possibly, a perpendic ular\nmagnetic field, is described by the Fock-Darwin (FD) states\nof Eq. (23). Then, the calculation of hijandvee(ij;kl)in\nEqs. (27)-(29) in the chosen basis set is reduced to the one-\ndimensionalintegrals\nhAB=/bracketleftbigg\n−Λ\n2+1\n4l2\nA−2ΛQAgc(α,β,λ A)−\n2ΛQBgc(α,β,λ B)]SAB,\nhAA=−Λ\n2+1\n4l2\nA−2ΛQAgc(α,βA,0)−\n2ΛQBgc(α,βA,2ΛrAB),\nhBB=−Λ\n2+1\n4l2\nB−2ΛQAgc(α,βB,2ΛrAB)−\n2ΛQBgc(α,βB,0),\ngc(α,β,λ) =/integraldisplay∞\n0dxJ0(λx)×\nexp/parenleftbig\n−αx−βx2/parenrightbig\n/(x+1)3, (78)\nα= 2Λh, β=(2ΛlAlB)2\nl2\nA+l2\nB,\nβA,B= 2(ΛlA,B)2, λA,B=2Λl2\nB,A\nl2\nA+l2\nBrAB,\nvee(AB;CD) =Neegee(a,b),\nNee=ΛlAlBlClD\n(l2\nA+l2\nC)(l2\nB+l2\nD)×\nexp/parenleftbigg\n−r2\nAC\n4(l2\nA+l2\nC)−r2\nBD\n4(l2\nB+l2\nD)/parenrightbigg\n,\ngee(a,b) =/integraldisplay∞\n0dxJ0(bx)×\n(3x2+9x+8)(x+1)−3, (79)\na= 4Λ2/parenleftbiggl2\nAl2\nC\nl2\nA+l2\nC+l2\nBl2\nD\nl2\nB+l2\nD/parenrightbigg\n,\nb= 2Λ/vextendsingle/vextendsingle/vextendsingle/vextendsinglel2\nC/vector rA+l2\nA/vector rC\nl2\nA+l2\nC−l2\nD/vector rB+l2\nB/vector rD\nl2\nB+l2\nD/vextendsingle/vextendsingle/vextendsingle/vextendsingle,/s53/s48/s48 /s54/s48/s48 /s55/s48/s48 /s56/s48/s48 /s57/s48/s48 /s49/s48/s48/s48/s49/s69/s45/s49/s49/s49/s69/s45/s49/s48/s49/s69/s45/s57/s49/s69/s45/s56/s49/s69/s45/s55/s49/s69/s45/s54/s49/s69/s45/s53/s49/s69/s45/s52/s49/s69/s45/s51/s48/s46/s48/s49/s48/s46/s49/s73/s110/s116/s101/s110/s115/s105/s116/s121/s44/s32/s109/s101/s86\n/s73/s110/s116/s101/s114/s100/s111/s116/s32/s68/s105/s115/s116/s97/s110/s99/s101/s32/s114\n/s65/s66/s44/s32/s65/s110/s103/s115/s116/s114/s111/s109/s32/s72/s101/s105/s115/s101/s110/s98/s101/s114/s103/s32/s73/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110\n/s32/s68/s105/s112/s111/s108/s101/s32/s73/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110\nFIG. 10: The Heisenberg and dipole-dipole spin interaction mag-\nnitudes as a function of interdot distance. The distance fro m the\nelectrodes to the LHe surface h= 800˚Aand the charges on the\nelectrodes QA=QB= 1a.u.The magnetic field /vectorB0= 0.\nwherei=A,B,C,D in the two-electron matrix elements\ndenotes orbitals with the effective lengths lilocalized at /vector ri,\nandJ0(x)is the zeroth order Bessel function. From Eq. (79)\none can obtain the following expression for the Heisenberg\ninteractionconstant\nJH=−1.36·104Λgee(a,0)S2meV (80)\na=8Λ2l2\nAl2\nB\nl2\nA+l2\nB.\nNote that JHis proportionalto the square of the overlapma-\ntrix element S, Eq. (25), and the integral gee(a,0)does not\ndependontheinterdotdistance rAB. Asaroughestimate,one\ncanapproximatethe rationalfunctionin theintegral gee(a,0)\nbyaconstant8,asaresultobtaining gee(a,0)≈4/radicalbig\nπ/a.\nFigure 10 shows the magnitudes of the Heisenberg and\ndipole-dipole spin interaction, |JH|andJdipfrom Eqs. (80)\nand (36) respectively, as a function of interdot distance. A s\nestimated, the magnitude of the Heisenberg interaction is\ncomparable to the weak dipole-dipole interaction at rAB≃\n1000˚A. However, we remark that the strong dependence of\nS∼exp(−αr2\nAB)(quadratic in the interdot distance) is due\ntothequadraticdependenceoncoordinatesintheexponento f\nthecorrespondingoscillatorywavefunctions. Asymptotic ally,\nthe confining potential Eq. (75) behaves as a 2D Coulomb\npotential, so that one should expect a milder coordinate de-\npendence, S∼exp(−αrAB), at large distances (assuming\nB0= 0) and the rough estimate of Eq. (80) providesa lower\nboundfortheHeisenberginteractionstrength.\nLyon suggested42using, instead of the exchange interac-\ntion,themagneticdipole-dipoleinteractionbetweenthes pins\nin order to implement two-qubit gates, motivating this by th e\nstrong sensitivity of the exchange coupling to the paramete rs\nof the system and, hence, the corresponding difficulties wit h\nattempting to control this interaction. Our analysis confir ms\nthis, though, of course it is not easy to control the dipole-14\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s84/s114/s105/s112/s108/s101/s116/s32/s115/s116/s97/s116/s101/s115/s32/s112/s111/s112/s117/s108/s97/s116/s105/s111/s110/s44/s32/s124/s97\n/s116/s40/s116/s41/s124/s50\n/s116/s105/s109/s101/s44/s32/s50 /s116/s47/s124/s74\n/s72/s124/s32 /s66\n/s122/s61/s48/s46/s48/s53/s32/s84\n/s32/s48/s46/s49/s32/s84/s114\n/s65/s66/s61/s57/s48/s48/s32/s65/s110/s103/s115/s116/s114/s111/s109\nFIG. 11: (color online) The triplet states population of ele ctrons\ntrapped in QDs on a LHe surface, as a function of time at differ -\nent magnetic field differences ∆Bz= 0.05and 0.1 T. Initially the\nsystemisinthetripletstate |S= 1,MS= 0/angbracketright. Thedistancebetween\nQDs is900 ˚A.Allother parameters are the same as inFig.10.\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48/s80/s117/s114/s105/s116/s121/s44/s32/s112/s40/s116/s41\n/s116/s105/s109/s101/s44/s32/s50 /s116/s47/s124/s74\n/s72/s124/s32 /s66\n/s122/s61/s48/s46/s48/s53/s32/s84\n/s32/s48/s46/s49/s32/s84/s114\n/s65/s66/s61/s57/s48/s48/s32/s65/s110/s103/s115/s116/s114/s111/s109\nFIG.12: (coloronline)Thepurity p(t)forthesameparametersasin\nFig.11.\ndipole interaction either: Eq. (36) shows that this interac tion\ndepends on only one controllable parameter, the interdot di s-\ntance(the g-factorisaconstantinvacuum).\nSimilarly to Figs. 1 and 2, Figs. 11 and 12 demonstrate\nnon-unitary effects in the pure-spin model, due to magnetic\nfield inhomogeneityin the electrons-on-LHesystem. The in-\nterdot distance shown is rAB= 900˚A. At this distance the\nHeisenberg interaction still prevails over the dipole inte rac-\ntionbyatleastanorderofmagnitude. Again,thepatternsee n\nin the singlet-triplet states populationredistribution i s clearly\noscillatory.V. DISCUSSIONAND CONCLUSION\nWe have performed a comparative study of pure- and\npseudo-spin dynamics for a system of two interacting elec-\ntrons trapped in two QDs. We have shown that when there\nis negligible coupling between the spin and orbital degrees\nof freedom, which is the case of near /vectorB-field homogeneity\nand negligible spin-orbit interaction, the system spin dyn am-\nics is unitary in both pure- andpseudo-spin models and is\ngoverned by the Zeeman interaction Hamiltonian of the to-\ntal spin/vectorS(S= 1) with the magnetic field /vectorBav. The sin-\nglet and triplet states are totally decoupled; the total spi n is\nconserved. The spin system Hilbert space can be decom-\nposed into two independent, singlet and triplet subspaces,\nthe singlet spin states being magnetically inactive ( S= 0).\nThus, the two-electron spin system restricted to the triple t\nsubspace physically embodies a qutrit. The Heisenberg in-\nteraction operates differently in pure- andpseudo-spin mod-\nels. If for simplicity we neglect double-occupancystates, the\npseudo-spin state is totally defined by four complex ampli-\ntudes:{as1(0),ati(0), i= 1,2,3}in the basis {Φs1,Φti};\nso that the Heisenberg interaction results in a phase uni-\ntary transformation: {as1(t) = exp( −iεst)as1(0), ati(t) =\nexp(−iεtt)ati(0)}. Since the spin-density matrix ρ(t)is a\nbilinear combination of a’s:ρtij(t) =ati(t)a∗\ntj(t),ρs(t) =\n1−/summationtext\niρtii(t), theρ-state will not by affected by the unitary\ntransformationinducedbytheHeisenberginteraction.\nWe have also shown that unitaryquantumgatesrealized in\nboth spin models do not provide a universal set of gates un-\nder the condition ∆/vectorB= 0. In order to obtain a universal\nset of gates, there should be both non-zero coupling between\nsinglet and triplet states ( ∆/vectorB/ne}ationslash= 0) and non-zero Heisenberg\ninteraction ( JH/ne}ationslash= 0). Although at ∆/vectorB/ne}ationslash= 0pure-spin dy-\nnamics becomesnon-unitary,one can establish a relationsh ip\nbetween unitary gates in pseudo-spin and the corresponding\nnon-unitarygatesin pure-spindynamicssothatauniversalset\nof quantum gates constructed within the pseudo-spin model\nwill generatea universalset of non-unitarygatesin pure-spin\ndynamics.\nTo demonstrate the non-unitary effects, which are pro-\nportional to the square of the magnetic field inhomogene-\nity, inpure-spin dynamics, we have calculated how singlet\nand triplet states populations, as well as the purity and the\nLamb energy shift, are affected by ∆/vectorB/ne}ationslash= 0and spin-\norbit interactionin n-dopedGaAs semiconductors. These ef-\nfects are found to be strongly dependent on the ratio of /vectorB-\nfield inhomogeneityand the Heisenberg interaction constan t,\n|∆Bz/JH|. For example, the singlet-triplet states popula-\ntion re-distribution is maximal at t=πn/ω, whereω=/radicalbig\nJ2\nH+∆B2zandn= 1,3,..., and the singlet state pop-\nulation can achieve the value ∆B2\nz/ω2. Thus, we can con-\nclude that the Heisenberg interaction, characterized by th e\ninteraction constant JH,plays an essential role in producing\nnon-unitaryeffectsin pure-spindynamics. Spin-orbitinterac-\ntion effects are found to be roughly four ordersof magnitude\nsmaller ascomparedto those caused by /vectorB-field inhomogene-\nity.15\nAs shown in Figs. 1-8, there are clear oscillations in the\npure-spindynamicsand the non-unitarybehaviorof the spin-\ndensity matrix does not show the decaying pattern character -\nistic of a real decoherence process. This should be expected\nsince the “bath” – the electron orbitals – in our spin model\nis not a real stochastic or infinite external bath, an interac tion\nwithwhichmayresultinirreversibledecoherence. Inessen ce,\nthe spin dynamics is embedded in space and our bath is too\nsmall. ThecoordinateHilbertspacein thetwo-orbitalgrou nd\nstate approximation adopted in the present paper is repre-\nsented by 4 two-electron coordinate basis wavefunctions. I n\nprinciple, the coordinate bath can be large in a system where\ncouplings between excited and ground state orbitals are not\nnegligible. This is an interesting question for future inve sti-\ngation: how will couplings to excited orbitals affect the no n-\nunitaryspindynamics? Theotherinterestinggeneralizati onof\nthepresentmodelisinclusionofrealenvironmenteffects, i.e.,\ntherealstochasticbathrepresentingtheinteractionofel ectron\nspinswiththesemiconductormedium. Wewillconsiderthese\nandothergeneralizationsin afuturepublications.\nIn thepseudo-spin model, where /vectorB-field inhomogeneity\nresults in first-order effects, we have estimated the contri -\nbution of the spin-orbit interaction to the effective pseudo-\nspin Hamiltonian, namely the Dzyaloshinskii-Moriya spin-\norbit interaction term, and have suggested a two-step mecha -\nnism: couplingbetween the singly-occupiedsinglet state a nd\ntripletstatesoccursviaintermediate,double-occupancy states\n(direct couplingbetween these states turnsout to be zero du e\ntoorthogonalityoftheorbitalsinvolvedinthetransition ). Ourcalculations predict a smaller magnitude of the spin-orbit in-\nteractionascomparedtotheestimatesofRef.17,butarecon -\nsistent withtheresultsofRef.28.\nInoursecondapplicationwedemonstrated,inFigs.11and\n12, non-unitary effects due to ∆/vectorB/ne}ationslash= 0in a system of elec-\ntronstrappedabovea liquidhelium surface, namely the spin -\nbased quantum computingproposal by Lyon42. A more thor-\noughinvestigationofspindynamicsinthissystemisleftfo ra\nfuturepublication.\nIn conclusion, we note that the two-electron spin-density\nmatrix description advocated in this paper is expected to be\nuseful when electrons trapped in QDs are not spatially re-\nsolved or resolvable. The spin-dynamics is then completely\ndescribed by the spin-density matrix ρ. Although the ρ-\ndynamics becomes non-unitary in general (at ∆/vectorB/ne}ationslash= 0), it\nis controllableby modulatingthe interactionparameters, JH,\n/vectorBav,∆/vectorB. Since the non-unitarity comes from the mag-\nnetic field inhomogeneities, ∆/vectorBand/orδ/vectorBso, and since the\nρ-dynamics is quadratically protected from these fields, thi s\nmight prove to be important in practical quantum computing\nas minimizing coupling between spin and orbital degrees of\nfreedomquadratically improves the fidelity of unitary gates\nintheρ-state space.\nAcknowledgments\nThisworkwassupportedbyNSFGrantNo.CCF-0523675.\n1D.Loss and D.P.DiVincenzo, Phys.Rev. A 57, 120 (1998).\n2B.E.Kane, Nature 393, 133 (1998).\n3R. Vrijen, E. Yablonovitch, K. Wang, H.W. Jiang, A. Balandin ,\nV. Roychowdhury, T. Mor, and D. DiVincenzo, Phys. Rev. A 62,\n012306 (2000).\n4X.Huand S.Das Sarma,Phys.Rev. A 61, 062301 (2000).\n5J. Schliemann, D. Loss, and A.H. MacDonald, Phys. Rev. B 63,\n085311 (2001).\n6X.Huand S.Das Sarma,Phys.Rev. A 66, 012312 (2002).\n7B. Koiller, X. Hu, and S. Das Sarma, Phys. Rev. B 66, 115201\n(2002).\n8T.A. Kaplan and C. Piermarocchi, Phys. Rev. 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B 67,\n155402 (2003).\n46M.I.Dykman, P.M.Platzman,and Seddighrad, Physica E 22, 767\n(2004)."    },    {        "title": "0807.3654v1.Dynamics_of_coupled_spins_in_quantum_dots_with_strong_spin_orbit_interaction.pdf",        "content": "arXiv:0807.3654v1  [cond-mat.mes-hall]  23 Jul 2008Dynamics of coupled spins in quantum dots with strong spin-o rbit interaction\nA. Pfund1, I. Shorubalko1, K. Ensslin1and R. Leturcq1,2\n1Solid State Physics Laboratory, ETH Z¨ urich, 8093 Z¨ urich, Switzerland, E-mail: pfund@phys.ethz.ch\n2Institut d’ ´Electronique de Micro´ electronique et de Nanotechnologie ,\nCNRS-UMR 8520, Dpt ISEN, Avenue Poincar´ e,\nBP 60069, 59652 Villeneuve d’Ascq Cedex, France\nWe investigated the time dependence of two-electron spin st ates in a double quantum dot fabri-\ncated in an InAs nanowire. In this system, spin-orbit intera ction has substantial influence on the\nspin states of confined electrons. Pumping single electrons through a Pauli spin-blockade configu-\nration allowed to probe the dynamics of the two coupled spins via their influence on the pumped\ncurrent. We observed spin-relaxation with a magnetic field d ependence different from GaAs dots,\nwhich can be explained by spin-orbit interaction. Oscillat ions were detected for times shorter than\nthe relaxation time, which we attribute to coherent evoluti on of the spin states.\nDouble quantum dots (DQDs) are considered as model\nsystems for quantum bits (qubits) in spin-based solid\nstate quantum computation schemes [1]. The combi-\nnation of single qubit rotations and so-called two-qubit√\nSWAP gates would facilitate universal quantum op-\nerations. Fast control of the exchange coupling allows\nto coherently manipulate coupled spin qubits [2] and to\nquantify the relevantspin relaxationand coherencetimes\n[3, 4] in GaAs based quantum dots. Beyond the two-\nqubit operations, controlled rotation of a single spin has\nbeen demonstrated [5]. Especially appealing for a scal-\nable technology is the possibility to perform these single\nqubit operations with electric gate signals mediated by\nthe spin-orbit interaction (SOI) [6]. This has stimulated\nthe interest in alternative systems with strong spin-orbit\ninteraction, as recently detected in InAs nanowires [7, 8]\nand carbon nanotubes [9].\nComplementary to being a tool for single spin rota-\ntion, SOI can have substantial influence on two-qubit\noperations via exchange gates [10, 11] or direct spin-spin\ncoupling [12]. Here we investigate the dynamics of two\ncoupled, spatially separated spins in a DQD fabricated\nin an InAs nanowire, where SOI is orders of magnitudes\nstronger than in GaAs [7, 8].\nWe employ a charge pumping scheme [13, 14] to mea-\nsure the time dependence of two-electron spin states\nby transport through the DQD. When the system con-\ntains two (excess) electrons, the Pauli exclusion principle\nsuppresses certain transitions [15]. This spin-blockade\n(SB) can be used to electrically determine the spin state\n[2, 3, 5, 16]. The pumped current is strongly reduced in\nthe blockaded direction compared to cycling in the oppo-\nsite way, which reflects the spin transition rules leading\nto the SB. We concentrate on the evolution of those two-\nelectron spin states, where the electrons are distributed\nbetweenthecoupleddots(the (1 ,1)occupancy). Adecay\nof the SB is observed on a timescale of ∼300ns, which\nwe relate to relaxation towards a state with (1 ,1)-triplet\ncharacter. In contrast, no decay is observed up to sev-\neralµs when both electrons occupy the same dot (the\n(0,2) occupancy). The observed time-dependence dif-(a) \n1mG1 \nG2 GC V\nt\n(b) (1,1) \n(0,2) (0,1) (1,2) VG1 \nVG2 \nV\nt(c) \n(d) NW S\nD0\n-10 \nB (mT) \n-20 \nI (pA) \n0 0.5 (meV) ε\n-20 0 +20 0100 \nB (mT) I (pA) \nFIG. 1: (color online). (a)Scanning electron microscope im-\nage of the measured device. Top-gates G1,G2 andGCdefine\na double quantum dot in the InAs nanowire (NW) between\nsource (S) and drain ( D). Fast voltage pulses can be applied\ntoG1 andG2. The external magnetic field is parallel to the\nNW.(b)Sketch of a charge stability diagram section of the\ndouble dot. Numbers ( n,m) label the ground state electron\nconfiguration. The axis εdefines the detuning of the elec-\ntrochemical potentials in the two dots for 2 electrons in the\nsystem. The dotted line indicates the pumping cycle used for\nthe time-dependent measurements. (c)Current ISDfor finite\nbiasVSD= +0.7mV as a function of magnetic field Band\ndetuning εat the (1 ,1)-(0,2) transition. Spin-blockade sup-\npresses the current around B= 0.(d)Cross section along\nε≈0 as indicated by the dashed line in Fig.1(c).\nfers significantly from measurements in GaAs DQDs and\ncannotbeexplainedbymodelsaccountingonlyforhyper-\nfine interaction. Instead, the magnetic field dependence\nis consistent with SOI mediated relaxation [17, 18, 19].\nOn a shorter timescale ( ∼100ns), we detect oscillations\nbetween the spin-states. These findings suggest, that co-\nherence times are similar to GaAs DQDs.\nWe investigate a DQD formed by lithographically de-\nfined top-gates on an epitaxially grown InAs nanowire\n[8, 16], see Fig.1(a). Transport measurements were per-\nformed in a dilution refrigerator at an electronic temper-\nature of 130mK. A magnetic field can be applied parallel2\nto the nanowire. Thermalized coaxial cables allow to ap-\nply voltage pulses with a typical rise time of 2ns to the\ntop-gates. Bias tees at low temperature are used to ad-\nmix AC and DC signals.\nThe gates G1,G2 define tunnel barriers and tune en-\nergy levels in dot 1 and 2. The center gate GCseparates\nthe two quantum dots. In the presented measurements,\nthe center gate voltage is fixed and defines a tunnel cou-\npling≈3µeV. Due to Coulomb blockade, the number\nof electrons in each dot is fixed for specific regions in\ntheVG1-VG2-plane [20]. A part of the charge stability\ndiagram is sketched in Fig.1(b) and the electronic con-\nfiguration ( n,m) is labeled by the number of electrons n\nin dot 1 (m in dot 2). These labels refer to the number\nof excess electrons in addition to spin-less filled shells of\nelectrons [8, 16]. Variation of the gate voltages along the\ngreen arrow in Fig.1(b) detunes the levels in the dots by\nan energy ε.\nIn the case without spin dependent interactions, two\nelectrons form either a singlet Sor triplet states Tσ\n(σ= 0,±denotes the z-component of the spin state). If\nthe detuning εis positive, both electrons are in the same\ndot and the ground state is the singlet S(0,2). Triplets\nin (0,2) have higher energies because they involve occu-\npation of an excited orbital state. For ε <0, the singlet\nS(1,1) and the triplets T0,±(1,1) are close in energy at\nzero magnetic field [21]. Since tunneling preserves spin,\na transition from a (1 ,1)-triplet to S(0,2) is forbidden.\nVarious experiments show that the singlet-triplet picture\ndescribes well the SB in GaAs DQDs [2, 3, 15, 21]. In the\nfollowing, this model is used for a qualitative description.\nIn Fig.1(c) the current through the device is shown\nas a function of detuning εand magnetic field B, when\nno pulses are applied to the gates. A finite source-drain\nbiasVSD= 0.7mV is applied. Sequential transport from\n(1,1) to (0,2) is in principle allowed, if the relevant lev-\nels are within the bias window: 0 ≤ε≤ |eVSD|. Around\nzerofieldhowever,thecurrentinFig.1(c)isstronglysup-\npressed. In the basic picture described above, blockade\narises once a (1 ,1)-triplet is loaded: the state can neither\ntunnel to S(0,2) nor unload again to the source, if it is\nwithin the bias window. Not explained by this model is\nthe strong current which sets in for small magnetic fields\nas shown in Fig.1(d). This behavior is not reported in\nGaAs DQD tuned to the same coupling, but also occurs\nin other DQDs with strong SOI, as recently found in car-\nbon nanotubes [9, 22]. In the following, we identify SOI\nmediated relaxation to T+(1,1) as the origin of this dif-\nference to GaAs.\nTo probe the time evolution of the spin-states, we\nuse pumping cycles where single electrons are shuttled\nthrough the DQD [13, 14]. Fast (ns) pulses are applied\nto the gates in a loop around the (0 ,1)-(1,1)-(0,2)-triple\npoint in the charge stability diagram. The voltages are\nswitched rapidly along the dotted line in Fig.1(b) and\nwaiting times t(0,1),t(1,1),t(0,2) are spent in each\nI=+ ef\nI=+ef/4 \nI=-efI/e (MHz) +5 \n0\n-5 (0,2) (1,1) \n(0,1) spin-\nblockade\ne- e-\n(0,2) (1,1) \n(0,1) e-e-(1,1) (0,2) ST T\nS\n(1,1) (0,2) ST T\nSB=1T \nB=0T \nf (MHz) 5 15 \nFIG. 2: (color online). Pumped electrons per time I/eat\nzero bias as a function of pumping frequency ffor cycles as\nindicated in Fig.1(b). The lowest curve (red) shows I/efor\nanti-clockwise cycling as indicated in the lower round inse t.\nFor clockwise cycling (see upper round inset), the pumped\ncurrentis significantly reducedby Pauli spin-blockadefor zero\nmagnetic fields (blue curve) compared to large fields (green\ncurve, 1T). Insets sketch the level energies for the transit ion\n(1,1)-(0,2) in the clockwise cycle. For B=0T (blue), spin-\nblockade suppresses the transition from triplets T(1,1) to the\n(0,2)-singlet. For B=1T (green), (0 ,2)-singlet and triplet\nbecome degenerate and are mixed by spin-orbit interaction.\nThen no spin-blockade occurs.\nstate. The pumped current is measured with zero bias\nacross the device and each point is averaged over 2s.\nIn Fig.2 the pumped current is shown as a function of\ncycling frequency for cycles with t(0,1)=t(1,1)=t(0,2).\nThe behavior is different for the two possible pumping\ndirections. The lowest (red) curve shows the result for\nanti-clockwise cycling (lower round inset). The current\nis negative and equal to the elementary charge times the\ncycle frequency up to several MHz as expected. When\ncycling in the opposite direction (upper round inset), the\ncurrentis reversedand the pumping efficiency is sensitive\nto magnetic field. For B= 0T (middle curve, blue), we\nfind a significantly reduced current compared to the anti-\nclockwise direction. If a high magnetic field B= 1T is\napplied, charge is again pumped with the full efficiency\nof one electron per cycle (upper curve, green).\nWe never observed pumping currents higher than one\nelectron per cycle. The tunnel rates in our device cor-\nrespond to timescales <1ns (estimated from measure-\nments as in Fig.1(c) [8]). The pulses are slow with re-\nspect to the tunnel rate. Therefore the charge configura-\ntion (n,m) during the cycle follows the ground state in3\n(a) \n100mT 28mT \n0200 1000 0.4 0.6 1\nt(1,1) (ns)   <N> \n8mT \n0mT (b) \n-10 +10 B (mT) t(1,1)=1µs \n(c) mixed states \n[1] [2] [3] \n[4] \ncycle evolution (time) energy \n[5] B>0 \n(0,1) (1,1) (0,2) (0,1) S(0,2) S(1,1) \n(0,1) T (1,1) \nT (1,1) +-\nT (1,1) 0\n(0,1) \nFIG. 3: (color online). (a)Average number of pumped elec-\ntrons per cycle /angbracketleftN/angbracketrightas a function of the time t(1,1) for dif-\nferent magnetic fields. (b)Dependence of the long time limit\nof/angbracketleftN/angbracketright(t(1,1) = 1µs) on the external magnetic field B.(c)\nScheme of the energy levels along the pumping cycle for a\nmagnetic field B >0. The system evolves along the thick\nlines (labels [1]-[5], gray areas represent waiting times) : [1]\nstart in (0 ,1); [2] tunneling of an electron into one of the\n(1,1) states (arrows); [3] evolution and relaxation in the ( 1,1)\nsubspace; [4] transition along the detuning axis ε, [5] tunnel-\ning out. Only electrons coming from S(0,2) give rise to a\npumped current (lowest arrow at [5]). Electrons coming from\n(1,1)-states are only shuttled back and forth (empty arrows).\nAt the transition [4], hyperfine interaction or SOI hybridiz e\ndifferent spin states (avoided crossings). During step [3], evo-\nlution between the mixed states and relaxation to T+(1,1)\ncan occur.\nthe charge stability diagram - provided the transition is\nnot forbidden by spin selection rules. Beyond that, the\npumping efficiency depends on the size of the pulse loop\nin Fig.1(b). For example, if the (0 ,2)-corner is chosen at\na too high detuning, the transition from (1 ,1) to (0,2)\noccurs by electron escape via (0 ,1) [3]. We adjusted the\npulsing parameters so that these processes are minimal.\nThe pumping scheme allows to study the time evo-\nlution of the quantum states involved in the SB. For a\ntolerable signal-to-noise ratio of the pumped current, the\ntotal cycle times should be shorter than ≈2µs. Within\nthis limit, we observe no dependence of the pumping ef-\nficiency when varying separately the times t(0,1) and\nt(0,2) (not shown). However, t(1,1) has a strong influ-\nence on the pumped current.\nIn Fig.3(a), the average number of pumped electrons\nper cycle /angbracketleftN/angbracketrightis plotted as a function of t(1,1). To main-\ntain a well detectable signal, the total cycle period is\nfixed to 1 .2µs. Since /angbracketleftN/angbracketrightonly depends on t(1,1) for\nthese timescales, we fix t(0,2) = 100ns and compensatethe time spent in (1 ,1) by shortening the time in (0 ,1)\ncorrespondingly. A monotonic long-time increase of /angbracketleftN/angbracketright\nis found for times >200ns [24]. At finite field, this effect\nis much more pronounced than at B= 0T.\nThe long-time limit of /angbracketleftN/angbracketrightis studied as a function of\nB-field in Fig.3(b). For t(1,1) = 1µs,/angbracketleftN/angbracketrightis sensitive to\nmagnetic fields of a few mT. This behavior is in line with\nthe field dependence of the current through SB at finite\nbias (Fig.1(d)).\nIn order to analyze the behavior of the pumped cur-\nrent, we use the singlet-triplet model for SB [25]. The\nvalues of the pumped currents in Fig.2 are related to the\nspin-transition rules between the corners of the pumping\nloop. For the anti-clockwise cycle (lower round inset),\nthe transition from (0 ,2) to (1,1) is always allowed and\none electron is transfered from right to left during each\nroundtrip. In the opposite direction (upper round inset),\nthe transition from (1 ,1) to (0,2) is spin selective. The\ntripletsT0,±(1,1) are blocked and only the singlet can\npass, which reduces the pumped current. At B= 1T,\nthe excited triplet T+(0,2) comes close in energy to the\nground state S(0,2) and both are mixed by SOI [8]. This\nway SB is lifted and the full pumping current is recov-\nered.\nTo understand the decay in Fig.3(a), we analyze the\nspin-selective transition (1 ,1)-(0,2) for different mag-\nnetic fields. A contribution to the pumped current is\ngenerated only by those (1 ,1)-states, which are trans-\nfered into a singlet during the pulse. In other states, the\nelectron is blocked.\nAtB= 0T, all (1 ,1)-states are close in energy [2, 21]\nand becomemixed bydifferent spin coupling mechanisms\nduringthe time t(1,1). The pumped currentthen reflects\nthe overlap with the singlet. In Fig.3(a), the curve for\nB= 0T shows only a weak time dependence. This sup-\nports that there is no preferential evolution towards a\ncertain state, but mixing between all states.\nFor finite field, the level evolution along the triangular\npumping cycle is sketched in Fig.3(c). Between the (1 ,1)\nand (0,2) corners, triplets and singlet levels would cross\nat two points (label [4] in Fig.3(c)). In the presence of\nSOIorhyperfineinteraction, hybridizationofstatesleads\nto avoided crossings at these points [21, 23].\nZeeman splitting lowers the energy of the state with\nT+(1,1)-character. Relaxation to this new ground state\noccursduringthetime t(1,1). Thisincreasesthepumped\ncurrent, because T+(1,1)is admixedto the singlet during\nthe charge transition (label [4] in Fig.3(c)). We estimate\na relaxation time T1(1,1)≈300ns by fitting with an ex-\nponential curve. A comparable relaxation process is not\nreported in GaAs DQDs, where SB is generally restored\nwith finite magnetic fields [2, 3, 21].\nThe B-dependence of /angbracketleftN/angbracketrightfor long t(1,1) (Fig.3(b))\nsuggests a SOI mediated relaxation. The relaxation rate\nfor these processes generally increases with the splitting\nof the involved states [17, 18, 19]. In contrast, spin state4\n100mT (a) \n<N> \n0mT \n0.4 0.6 \n  \n(b) \n0 20 80 0.4 0.6 \n  \ninit in (0,1) \ninit in (1,2) •<N> \nt(1,1) (ns) \nFIG. 4: (color online). Number of pumped electrons per cycle\nas a function of t(1,1).(a)Dependent on the external field,\n/angbracketleftN/angbracketrightshows oscillations with a period 9 .4ns and characteristic\ndecay time of 25ns (for 0mT) and 45ns (100mT). (b)When\nchanging the initialization state of the cycle, the phase of the\noscillations changes by π. Cycles are (0 ,1)-(1,1)-(0,2)-(0,1)\n(blue dots) and (1 ,2)-(1,1)-(0,2)-(1,2) (black rhombs).\ndecay due to hyperfine interaction with the nuclei is sup-\npressed in a field which splits T±(1,1) [21].\nFor times shorter than the relaxation time, the curves\ninFig.3(a)showup-turnswhicharenotfullyunderstood.\nHowever, high resolution measurements in this region re-\nveal striking oscillations of /angbracketleftN/angbracketrightas a function of the time\nt(1,1), as shown in Fig.4(a). As above, the total cycle\ntime is constant (140ns) in a regime, where the signal\nonly depends on t(1,1) (t(0,2) = 20ns fixed). The os-\ncillation period does not vary with magnetic field, but\nthe decay is changed. A purely exponentially decaying\nfunction cannot be fitted to the amplitude. Neverthe-\nless it allows to estimate a decay time, which increases\nmonotonically from 25ns at 0T to 45ns at 100mT.\nThe oscillations as a function of t(1,1) are robust\nagainst variation of the two other waiting times and the\ntotal cycle period. The period corresponds to an energy\nsplitting of h/9.4ns = 0.44µeV, which is consistent with\nthe energy scales for exchange coupling, hyperfine inter-\naction and spin-orbit interaction (at small fields) in the\nsystem[8]. Theseenergyscales,themagneticfielddepen-\ndence of the decay and the selective time dependence on\nt(1,1) suggest coherent evolution in the (1 ,1) subspace\nas the origin of the oscillations.\nA detection of coherent oscillations in the pumping\nscheme would imply a selective state preparation. In\nFig.4(b) we observe a striking dependence of the phase\nof the oscillations on the way the two-electron state is\nloaded. Moving the initial state from (0 ,1) to (1,2) in\nthe chargestability diagram(Fig.1(b)) results in a phase\nshift ofπ(in both cases, the charge is pumped in the\ndirection of SB). These observations suggest that the na-\nture and the coupling of spin-states in DQDs are signifi-\ncantly changed by the SOI compared to the well under-stood situation in GaAs dots.\nBy pumping single electrons through a spin-blockaded\nInAs DQD, westudied the dynamics oftwocoupled spins\nin the presence of strong SOI. Beyond the spin-selection\nrulesleadingtoSB,SOImediatedrelaxationtothe(1 ,1)-\ntriplet ground state was observed at finite magnetic field.\nFor times shorter than the relaxation time, oscillations\nwere detected in the pumped current. These processes\ncan influence the operation of two-qubit gates in systems\nwith strong SOI.\nWe thank B. Altshuler, A. Imamoglu, D. Klauser,\nD. Loss, C. Marcus, Y. Meir, L. Vandersypen and\nA. Yacoby for stimulating discussions, M. Borgstr¨ omand\nE. Gini for advice in nanowire growth. We acknowledge\nfinancial support from the ETH Zurich.\n[1] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120\n(1998).\n[2] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird,\nA. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson,\nand A. C. Gossard, Science 309, 2180 (2005).\n[3] A. C. Johnson, J. R. Petta, J. M. Taylor, A. Yacoby,\nM. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C.\nGossard, Nature 435, 925 (2005).\n[4] E. A. Laird, J. R. Petta, A. C. Johnson, C. M. Marcus,\nA. Yacoby, M. P. Hanson, and A. C. Gossard, Phys. Rev.\nLett.97, 056801 (2006).\n[5] F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink,\nK. C. Nowack, T. Meunier, L. P. Kouwenhoven, and\nL. M. K. Vandersypen, Nature 442, 766 (2006).\n[6] K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, and\nL. M. K. Vandersypen, Science 318, 1430 (2007).\n[7] C. Fasth, A. Fuhrer, L. Samuelson, V. N. Golovach, and\nD. Loss, Phys. Rev. Lett. 98, 266801 (2007).\n[8] A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq,\nPhys. Rev. B 76, 161308 (2007).\n[9] F. Kuemmeth, S. Ilani, D. C. Ralph, and P. L. McEuen,\nNature452, 448 (2008).\n[10] G. Burkard and D. Loss, Phys. Rev. Lett. 88, 047903\n(2002).\n[11] D. Stepanenko, N. E. Bonesteel, D. P. DiVincenzo,\nG. Burkard, and D. Loss, Phys. Rev. B 68, 115306\n(2003).\n[12] M. Trif, V. N. Golovach, and D. Loss, Phys. Rev. B 75,\n085307 (2007).\n[13] H. Pothier, P. Lafarge, C. Urbina, D. Esteve, and M. H.\nDevoret, Europhys. Lett. 17, 249 (1992).\n[14] A. Fuhrer, C. Fasth, and L. Samuelson, Appl. Phys. Lett.\n91, 052109 (2007).\n[15] K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha, Sci-\nence297, 1313 (2002).\n[16] A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq,\nPhys. Rev. Lett. 99, 036801 (2007).\n[17] S. Amasha, K. MacLean, I. P. Radu, D. M. Zumbuhl,\nM. A. Kastner, M. P. Hanson, and A. C. Gossard, Phys.\nRev. Lett. 100, 046803 (2008).\n[18] T. Meunier, I. T. Vink, L. H. W. van Beveren, K.-J.\nTielrooij, R. Hanson, F. H. L. Koppens, H. P. Tranitz,5\nW. Wegscheider, L. P. Kouwenhoven, and L. M. K. Van-\ndersypen, Phys. Rev. Lett. 98, 126601 (2007).\n[19] R. Hanson, L. H. W. van Beveren, I. T. Vink, J. M. Elz-\nerman, W. J. M. Naber, F. H. L. Koppens, L. P. Kouwen-\nhoven, and L. M. K. Vandersypen, Phys. Rev. Lett. 94,\n196802 (2005).\n[20] W. G. van der Wiel, S. D. Franceschi, J. M. Elzerman,\nT. Fujisawa, S. Tarucha, and L. P. Kouwenhoven, Rev.\nMod. Phys. 75, 1 (2002).\n[21] F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Han-\nson, L. H. W. van Beveren, I. T. Vink, H. P. Tranitz,\nW. Wegscheider, L. P. Kouwenhoven, and L. M. K. Van-\ndersypen, Science 309, 1346 (2005).\n[22] H. O. H. Churchill, D. Marcos, F. Kuemmeth, S. K. Wat-\nson, and C. M. Marcus, Contribution to INTNAN8 con-ference (2008).\n[23] D. J. Reilly, J. M. Taylor, J. R. Petta, C. M. Marcus,\nM. P. Hanson, and A. C. Gossard (2008).\n[24] The minimum at t(1,1)≈160ns is also observed for\ndifferent total cycle periods and in schemes, where only\nt(1,1) is varied. The origin is not fully understood, but\nit does not affect the analysis of the relaxation process\nabove 200ns.\n[25] Since the (0 ,2) singlet-triplet splitting is much larger\nthen the energy scale of the SOI [8], the (0 ,2)-states\nare reasonably described as triplets T0,±(0,2) and sin-\ngletS(0,2). The nature of the (1 ,1)-levels could however\nbe strongly modified by SOI."    },    {        "title": "2401.00174v1.Spin_current_generation_due_to_differential_rotation.pdf",        "content": "Spin current generation due to differential rotation\nTakumi Funato1,2, Shunichiro Kinoshita3,4, Norihiro Tanahashi4, Shin Nakamura4, and Mamoru Matsuo2,5,6,7\n1Center for Spintronics Research Network, Keio University, Yokohama 223-8522, Japan\n2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n3Department of Physics, College of Humanities and Sciences, Nihon University, Tokyo 156-8550, Japan\n4Department of Physics, Chuo University, Tokyo 112-8551, Japan\n5CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan and\n7Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan\n(Dated: January 2, 2024)\nWe study nonequilibrium spin dynamics in differentially rotating systems, deriving an effective\nHamiltonian for conduction electrons in the comoving frame. In contrast to conventional spin current\ngeneration mechanisms that require vorticity, our theory describes spins and spin currents arising\nfrom differentially rotating systems regardless of vorticity. We demonstrate the generation of spin\ncurrents in differentially rotating systems, such as liquid metals with Taylor-Couette flow. Our\nalternative mechanism will be important in the development of nanomechanical spin devices.\nIntroduction .— Generating and controlling spin cur-\nrents is a key challenge in spintronics [1]. Recent ad-\nvancements of nanofabrication have enabled us to utilize\nmechanical motions in materials for spin transport [2–\n18]. In particular, the gyromagnetic effect [19–21], the\nconversion of mechanical angular momentum to electron\nspin, is increasingly significant in this context. Since Bar-\nnett, Einstein, and de Haas discovered [19–21], the gy-\nromagnetic effect has been observed across a vast range\nof rotational speeds from a few Hz to 1022Hz in both\nmagnetic and non-magnetic materials [22–34]. More-\nover, this effect defies the constraints of spin-orbit in-\nteraction strength to generate spin current [35–40]. A\nprime example of the spin-orbit free mechanism relying\non the gyromagnetic effect is the generation of spin cur-\nrents in Cu thin films [36], conventionally considered un-\nsuitable due to weak spin-orbit interactions, using non-\nuniform rotations in surface acoustic waves. This rev-\nelation opens new avenues in material selection for mi-\ncro/nanomechanical spin devices.\nTraditionally, spin density in steady rigid body ro-\ntation is generated through the spin-rotation coupling\nHsr=−s·Ω, where sis spin and Ωis angular veloc-\nity [41], known as the Barnett effect [19]. If this inter-\naction can be localized, spin density gradient and spin\ncurrent via diffusion may result. Previously, such ‘lo-\ncalization’ was embodied by the spin-vorticity coupling\nHsv=−s·ω, in which the rigid angular velocity Ωis pro-\nmoted to the vorticity ω= (1/2)∇×vof the velocity field\nvof the lattice. Here, a coupling with differential rotation\nΩ(r) =r×v/r2provides another way to localize the in-\nteraction, though it has been overlooked in conventional\ntheories. This new coupling enables spin current genera-\ntion in systems with non-uniform rotational motion but\nwithout vorticity, and it may expand our understanding\nof spin transport driven by non-uniform rotation.\nIn this study, we investigate the non-equilibrium spindynamics in differentially rotating systems within a mi-\ncroscopic theory. By mapping into a comoving frame, we\nconstruct an effective Hamiltonian for conduction elec-\ntrons in these systems, demonstrating the emergence of\neffective gauge fields. Furthermore, we derive micro-\nscopic expressions for the spin density and spin current\nof conduction electrons driven by these emergent gauge\nfields. By applying this to a liquid metal and a non-\nmagnetic metallic cantilever as examples of differentially\nrotating systems, we estimate the concrete amount of the\nspin current. In particular, we show that even in cases\nsuch as Taylor-Couette flow where the vorticity-gradient\nis zero, spin currents can be generated due to the differ-\nential rotation. Consequently, we uncover mechanisms\nof angular momentum transfer that have not been cap-\ntured by traditional frameworks, specifically those involv-\ning temporally and spatially modulated lattices transfer-\nring momentum to conduction electron spins. Our re-\nsults will contribute to the rapidly expanding field of\nnon-equilibrium spin physics in nanomechanical systems.\nEmergent Gauge Fields in Comoving Frame .—We con-\nsider the free electron system subject to momentum scat-\ntering and spin-orbit scattering due to the impurities. In\nthe inertial laboratory frame the Hamiltonian is given by\nˆH′=Z\nd3xˆψ†(x)\u001a\n−ℏ2\n2m∇2−ϵF+V′\nimp(x, t)\n+λsoσ·[∇V′\nimp(x, t)×(−iℏ∇)]\u001b\nˆψ(x),(1)\nwhere ˆψ(x) is the electron field operator, ϵFis the Fermi\nenergy, σ= (σx, σy, σz) are the Pauli matrices, and\nλsois the strength of the spin-orbit interaction. The\nthird term represents the impurity scattering and the\nfourth term represents the spin-orbit scattering. Here,\nV′\nimp(x, t) =P\nju(x−r′\nj(t)) is the total impurity po-\ntential, where u(x−r′\nj(t)) is a single impurity potential\ndue to the j-th impurity located at the position r′\nj(t).arXiv:2401.00174v1  [cond-mat.mes-hall]  30 Dec 20232\nIt is worth noting that the electrons are subject to the\nmoving impurities because we suppose the total system\nis differentially rotating. To characterize the differen-\ntial rotation of the system, we introduce a rotation an-\ngle Φ( x, t) around the z-axis, which is chosen as a rota-\ntion axis. When we take a cylindrical coordinate system,\nthe coordinate transformation from the laboratory frame\nr′= (r′, φ′, z′) to the rotating frame r= (r, φ, z ) can\nbe written as r=r′,z=z′, and φ=φ′−Φ(r′, t).\nNote that Φ is independent of φ, i.e., ∂φΦ = 0 because\nof axisymmetry. Supposing Φ( x, t) = 0 at an initial time\nt= 0, the position of the j-th impurity at tis given\nbyr′\nj(t) =Rz[Φ(rj, t)]rj, where Rzdenotes rotation\naround the z-axis and rjis the position at the initial\ntime.\nNow, we define a generator of the differential rotation\nwith angle Φ( x, t) as\nˆQΦ(t) =Z\nd3xΦ(x, t)ˆψ†(x)Jzˆψ(x), (2)\nwhere Jzis the total angular momentum operator acting\non coordinates and spin space as Jz=−iℏ∂φ+ℏσz/2.\nNote that Jzand Φ( x, t) are commutative. For an ar-\nbitrary state vector in the laboratory frame |Ψ′(t)⟩, the\nstate vector in the rotating frame is given by\n|Ψ(t)⟩= exp\u0014i\nℏˆQΦ(t)\u0015\n|Ψ′(t)⟩. (3)\nThe Schr¨ odinger equation in the laboratory frame,\niℏ∂t|Ψ′(t)⟩=ˆH′|Ψ′(t)⟩, yields\niℏ∂\n∂t|Ψ(t)⟩= (eiˆQΦ/ℏˆH′e−iˆQΦ/ℏ−ˆQ∂tΦ)|Ψ(t)⟩\n=ˆHT|Ψ(t)⟩,(4)\nwhere ˆQ∂tΦ=R\nd3x∂tΦ(x, t)ˆψ†(x)Jzˆψ(x) and ˆQΦcom-\nmute because of ∂φΦ = 0. The Hamiltonian HTwhich\ngoverns dynamics in the rotating frame. The density\noperator in the rotating frame, ˆ ρ(t), is given by ˆ ρ(t) =\neiˆQΦ/ℏˆρ′(t)e−iˆQΦ/ℏ, where ˆ ρ′(t) is the density operator in\nthe laboratory frame. The time evolution of ˆ ρ(t) is deter-\nmined by iℏ∂tˆρ(t) = [ ˆHT,ˆρ(t)]. Assuming that the single\nimpurity potential u(x) is isotropic and its typical range\na, such that u(x)≃0 for|x| ≫a, is much smaller than\na typical scale of the gradient of the differential rotation,\ni.e.,a|∇Φ| ≪1, the Hamiltonian in the rotating frame\ncan be rewritten as\nˆHT=Z\nd3xˆψ†(x)\u001a1\n2m(−iℏ∇−AsJz)2−As,0Jz−ϵF\n+Vimp(x) +λsoσ·[∇Vimp(x)×(−iℏ∇−AsJz)]\u001b\nˆψ(x),\n(5)\nwhere the time and spatial derivatives of the rotation\nangle are denoted by\nAs,µ(x, t) =\u0010\n∂tΦ(x, t),∇Φ(x, t)\u0011\n(µ= 0, x, y, z ).(6)We call As,µ(x, t) “emergent gauge field” in this pa-\nper. In the rotating frame, the effects of the differ-\nential rotation are represented by the emergent gauge\nfields, whereas the impurity potential given by Vimp(x) =P\nju(x−rj) does not depend on time under the assump-\ntiona|∇Φ| ≪1.\nSetup .—We present the Fourier representation of the\ntotal Hamiltonian in the rotating frame to facilitate cal-\nculations: ˆHT=ˆH0+ˆHimp+ˆHso+ˆH′(t), where ˆH′(t) is\nthe contribution of the emergent gauge field, and we treat\nit as a perturbation. The first term ˆH0=P\nkϵkˆψ†\nkˆψk\nrepresents the kinetic term, where ϵk=ℏ2k2/2m−ϵF\nis the kinetic energy, and ˆψkis the Fourier component\nof the electron annihilation operator. The second and\nthird terms describe the momentum scattering and the\nspin-orbit scattering due to the impurities, respectively.\nThese are expressed as ˆHimp=P\nkk′Vk−k′ˆψ†\nkˆψk′and\nˆHso=iℏλsoP\nkk′Vk−k′(k×k′)·ˆψ†\nkσˆψk′, where Vkde-\nnotes the Fourier component of the impurity potential\nVimp(x). We assume a short-range impurity potential,\ni.e.,u(x−rj) =uiδ(x−rj), where uiis the strength\nof the impurity potential defined by ui=R\nd3xu(x) in\ngeneral. While, the perturbed part, denoted by ˆH′(t) =\nˆHs+O(Lz) with Lz=−iℏ∂φbeing the orbital angular\nmomentum, represents the effect of the emergent gauge\nfields. The Hamiltonian ˆHsincorporates the electron\nspin, given by\nˆHs=−ℏ2\n2mX\nkk′qˆψ†\nk+\u0012\nkσzδkk′−1\n2As,k−k′\u0013\nˆψk′\n−·As(q)\n−ℏ\n2ˆs(q)As,0(q), (7)\nwhere As,qis the Fourier component of the emergent\ngauge fields, and k±=k±q/2 are defined.\nTo define the spin-current operator, we consider the\ntemporal modulation of the z-polarized spin density,\n∂tˆs(q) =−iq·ˆjs(q) +ˆTq, where ˆ s(q) =P\nkˆψ†\nk−σzˆψk+\nis the spin-density operator, and ˆTqdescribes the spin\ntorque due to the spin-orbit interaction of the impuri-\nties. The spin-current density operator polarized in the\nz-direction is defined by ˆjs(q) =P\nkk′ˆψ†\nk′\n−js,k′kˆψk+,\nwhere the matrix elements js,k′kis given by\njs,k′k=δk′kvkσz+λsoVk′−k[ez×(k′−k)]−ℏAs,k′−k\n2m,\n(8)\nwhere vk=ℏk/mis the velocity and ezis the unit vector\ninz-direction.\nCalculation of Spin Current.— We now compute the\nspin current induced by the emergent gauge fields. The\nstatistical average of the spin density and spin current is3\ngiven by\n⟨ˆjµ(q, ω)⟩=Z∞\n−∞dϵ\n2πiX\nkk′Trh\njsµ,k′kG<\nk+,k′\n−(ϵ+, ϵ−)i\n,\n(9)\nwhere ϵ±=ϵ±ω/2,js0,k′k=σzδk′k, and the\ntrace is taken for the spin space. Here, the four-\nvector ˆjµ= (ˆs,ˆjs) represents the spin density and\nspin current operators. The function G<\nk+,k′\n−(ϵ+, ϵ−)\nis the lesser component of the nonequilibrium path-\nordered Green function, defined by Gk,k′(t, t′) =\n−i⟨TKˆψk+(t)ˆψ†\nk′\n−(t′)⟩, where TKis a path-ordering op-\nerator, ˆψ(t) = ˆU†(t)ˆψ(t)ˆU(t) is the Heisenberg repre-\nsentation with ˆU(t) =Texp[−(i/ℏ)Rt\n−∞ˆHT(τ)dτ] and T\nbeing time-ordering operator, and ⟨···⟩ = tr(ˆ ρ···) rep-\nresents the expectation value with the density operator\nˆρ.\nAssuming that the characteristic energy scales of the\nmomentum scattering and the spin-orbit scattering due\nto the impurities are much smaller than the Fermi en-\nergy, i.e., niui≪ϵFand ℏ2λ2\nsok4\nF≪1, we treat them\nin the Born approximation. With the uniformly ran-\ndom distribution of impurities, we perform the aver-\nage of their positions to obtain the retarded/advanced\nGreen function: gr/a\nk(ϵ) = 1 /(ϵ−ϵk±iℏγ), where\nℏγ=πniu2\niν0(1 + 2 ℏ2λ2\nsok4\nF/3) is the damping constant\ncalculated with the density of state per spin at Fermi\nlevel ν0=mkF/2π2ℏ2. We assume that ℏγ≪ϵF. This\ncondition is well-satisfied when uiν0≲1.\nThe spin-current density in linear response to the\nemergent gauge fields is expressed as ⟨ˆjµ(q, ω)⟩=\nKµν(ω)As,ν(q, ω), where Kµν(ω) is the response func-\ntion. It is presumed that the time and spatial varia-\ntion of the differential rotation are much slower than the\nelectron mean-free path l=vFτand momentum relax-\nation time τ= 1/2γ, respectively, i.e., l|∇Φ| ≪1 and\nτ|∂tΦ| ≪1, where vF=ℏkF/mis the Fermi velocity and\nkF=p\n2mϵF/ℏ2is the Fermi wavenumber. In terms of\nFourier space, conditions lq≪1 and τω≪1 hold. By\nincluding the ladder vertex corrections due to the impu-\nrities, and using the relations ℏvFq/2≪ℏγ≪ϵFand\nℏω/2≪ℏγ≪ϵF, the response function is calculated as\nKµν(ω) =δµ0δν0ℏσc\n2e2D\n+iωℏ\n4πX\nkvk,µTr\u0002\nσzgr\n+σz(vk,ν+ Λs\nν)ga\n−\u0003\n,(10)\nwhere σc=nee2τ/m is the Drude conductivity with ne=\n4ϵFν0/3 being the number density of the electrons and\ne(>0) being the elementary charge, and D=v2\nFτ/3 is\nthe diffusion constant. We set vk,0= 1 and vk,i=ℏki/m.\nHere, Λs\nνdescribes the three-point vertex corrections, and\ngr/a\n±=gr/a\nk±(±ω/2) are specified. The first term of theresponse function represents the spin susceptibility for\nthe rigid rotation [42], known as the Barnett effect.\nPerforming straightforward calculation, we derive the\nrotation-induced spin density and spin current:\n⟨ˆs(q, ω)⟩=−iωℏσc\n2e2Dτ−1\ns\nDq2−iω+τ−1sΦ(q, ω),(11)\n⟨ˆjs(q, ω)⟩=iωℏσc\n2e2τ−1\ns\nDq2−iω+τ−1siqΦ(q, ω),(12)\nwhere τs= 9τ/8ℏ2λ2\nsok4\nFis the spin-relaxation time.\nCombining these results in the real space, we obtain\nFick’s law, js=−D∇s. This implies that our spin cur-\nrent is a diffusive flow produced by the gradient of the\nspin density, in which the impurity scattering governs the\ndiffusion.\nNow, we focus on long-term dynamics such that time\nscales are longer than the period of the rotation, ω≲Ω.\nIf the spin relaxation is much faster than typical scales\nof the angular velocity and the spatial variation of the\ndifferential rotation, i.e., Dq2, ω≲Ω≪τ−1\ns, which is\nwell-satisfied in metals, the rotation-induced spin current\nreduces to the following form in the real space:\njs(x, t) =−ℏσc\n2e2∇∂tΦ(x, t). (13)\nIn addition, the rotation-induced spin density reduces to\ns(x, t) =ℏσc\n2e2D∂tΦ(x, t), (14)\nwhich is the Barnett effect generalized to differential rota-\ntions. The susceptibility given by ℏσc/2e2Dis identical\nto that of the Barnett effect for rigid rotations. These\nresults suggest that the spin density and current are po-\nlarized along the rotation axis and the spin current is\ndriven in the direction of the spatial gradient of the an-\ngular velocity. By contrast, if the spin relaxation is so\nslow that τ−1\ns≪ω≲Ω, the spin density (11) as well as\nthe spin current (12) vanish, which implies that the spin\nrelaxation is necessary to generate the spin current and\nspin density. Despite this fact, the magnitude of the spin\ncurrent (13) is independent of the spin relaxation time.\nThe absence of τsfrom the long-term dynamics of the\nspin density and the spin current is explained as follows.\nIn the response function (10), the first term that orig-\ninates from the spin-rotation coupling As,0Jzin (5) is\nprincipal, while the other terms including the spin-orbit\ncoupling are suppressed by τsω≪1. This means that\nthe spin density is determined only by the susceptibility\nof the Barnett effect and the angular velocity. The gradi-\nent of this spin density produces the spin current due to\nthe diffusion caused by the impurity scattering, as shown.\nThus, the spin density and current are independent of τs.\nThe spin-orbit interaction contributes only to the tran-\nsient process that is necessary to drive the system to the\nfinal steady state, but it does not contribute to long-term4\ndynamics. Indeed, for ω≳Ω, (11) and (12) provide the\nfollowing spin transport equation:\n∂s\n∂t+∇·js=−s\nτs+ℏσc\n2e2Dτs∂tΦ, (15)\nwhich describes the transient process with a time-scale\nω≃τ−1\ns. We expect to obtain similar diffusive spin cur-\nrents as long as there are not only the spin-orbit interac-\ntions as presented here but also other interactions that\ncan produce transient processes satisfying Ω ≪τ−1\ns≪\nτ−1.\nTaylor-Couette Flow .—As an explicit example, let us\nconsider a two-dimensional steady flow with concentric\ncircular streamlines. In this case, the flow velocity is\nparallel to the φ-direction, v= (0, vφ,0), satisfying the\nfollowing Navier-Stokes equation: ∂2\nrvφ+ (∂rvφ)/r−\nvφ/r2= 0. The general solution is vφ=c1/r+c2rwith\nintegration constants c1andc2determined by boundary\nconditions. The first term represents irrotational flow,\nwhile the second term represents rigid-rotation flow. We\nconsider the two infinitely long coaxial cylinders of radii\nr1andr2(r2> r1), and the inner and outer cylinders are\nrotating at constant angular velocities Ω 1and Ω 2, respec-\ntively. Under these boundary conditions, vφ(r1) =r1Ω1\nandvφ(r2) =r2Ω2, the constants are obtained as c1=\n(Ω1−Ω2)r2\n1r2\n2/(r2\n2−r2\n1) and c2= (Ω 2r2\n2−Ω1r2\n1)/(r2\n2−r2\n1).\nThis concentric steady flow, known as the Taylor-Couette\nflow [43], induces the steady differential rotation with an-\ngular velocity Ω( r) =c1/r2+c2, leading to the generation\nof spin current (see Fig. 1(a)):\njs(r) =erℏσc\ne2r2\n1r2\n2\nr2\n2−r2\n1Ω1−Ω2\nr3, (16)\nwhere erbeing the unit vector in the r-direction. No-\ntably, since the vorticity in this system is constant,\n∇×v= 2c2ez, the conventional spin currents owing\nto the spin-vorticity coupling, which require the vorticity\ngradient [35, 44] or time-dependent vorticity [45], do not\nappear. On the other hand, our theory predicts the gen-\neration of spin current even in vorticity-free cases c2= 0.\nTo estimate the magnitude of the spin current, we as-\nsume that the radii of the two cylinders are much larger\nthan the gap between them d=r2−r1, i.e., r1, r2≫d,\nand only the outer cylinder is rotating, Ω 1= 0 and\nΩ2̸= 0, for simplicity. Under this assumption, the spin\ncurrent is approximated as js=ℏσcΩ2/2e2d. We con-\nsider (Ga,In)Sn as the fluid with the electric conductiv-\nityσc= 3.26×106(Ω·m)−1[38]. Set d∼1µm and\nΩ2∼102kHz, the magnitude of the spin current in charge\ncurrent units is estimated as ejs∼1.07×102A·m−2.\nTorsional Oscillation of Cantilever .—As another ex-\nample, we focus on the torsional oscillation of a can-\ntilever, wherein one of the ends is securely fixed while\nexternal forces are exerted on the opposite end. These\nforces induce only a twisting motion in the cantilever, not\n(a) (b)\nFIG. 1. Schematic illustration showing the generation of spin\ncurrent due to (a) the Taylor-Couette flow in a liquid metal\nand (b) torsional motion of a cantilever.\nbending or other deformations. In this case, the angu-\nlar velocity of the system varies along the rotation axis\nrather than the radial direction. The distortion angle\nφ(z, t) of the cantilever dictates the subsequent equation\nof motion: C∂2\nzφ=ρmI∂2\ntφ, where Cis the torsional\nrigidity, ρmis the mass density, and Iis the moment of\ninertia of the cross-section about its center of mass. By\nsolving the equation of motion under the boundary con-\nditions φ(0, t) = 0 and ∂zφ(l, t) = 0 and considering the\ninitial conditions φ(l,0) = φ0and∂tφ(z,0) = 0, we de-\nrive the solution as φn(z, t) =φ0sinknzcosωnt, where\nkn= (2n−1)π/2landωn=vknwith the integer n≥1\nand the velocity v=p\nC/ρ mI. The spin current, driven\nby the n-th torsional oscillation of cantilever, flows along\nthez-direction as given by (see Fig. 1(b))\njs,n(z, t) =ezℏσcφ0v\n2e2k2\nncosknzsinωnt. (17)\nThe mechanism under investigation in this study rep-\nresents a universal phenomenon, irrespective of material\nchoice, and fundamentally distinct from the previous the-\nory [46] that focus solely on magnetic materials.\nFinally, we estimate the magnitude of the spin cur-\nrent driven by the torsional oscillation. For a plate-\nshaped cantilever with width a, thickness band length\nl, the quantities CandIare calculated as C≃µab3/3\nandI≃a3b/12 (a≫b) with Lam´ e constant µ. The\nmagnitude of the total spin current in charge current\nunits is denoted by Jn=eabj s,n(0,0), while that at-\ntributed to the first torsional oscillation mode is given\nbyJ1= (π2ℏσcφ0/4e)(b/l)2p\nµ/ρm. We consider that\nthe cantilever is composed of copper with weak spin-\norbit interaction. By using the charge conductivity σc=\n6.45×107Ω−1m−1, the Lam´ e constant µ= 48.3 GHz and\nthe mass density ρm= 8.96×103kg/m3, the total spin\ncurrent is estimated as J1∼0.15µA for φ0∼0.01 and\nb/l= 1/4.\nConclusion and discussion .— We studied non-\nequilibrium spin dynamics in differentially rotating sys-\ntems within a microscopic theory. We obtained a Hamil-\ntonian with emergent gauge fields by using a mapping to\nthe comoving frame of the differential rotation. We esti-\nmated the spin currents generated by differential rotation\nin a liquid metal and a non-magnetic metallic nanome-5\nchanical system for experimental reference. Our mech-\nanism produces spin currents even in vorticity-free sys-\ntems. Hence, this mechanism of spin current generation\nis novel and distinct from the known mechanisms based\non the spin-vorticity coupling.\nDistinctions between our mechanism and those pro-\nposed in other literature [35, 44, 45] can be summarized\nas follows. In terms of spin transport equations, the\nsource terms that violate the conservation of the spin\ndensity differ in the mechanisms. The spin current is\nproportional to the spatial gradient of these source terms.\nFor our case, the spin transport equation is given in (15)\nand the source term is proportional to Ω, the angular\nvelocity of the orbital rotation around a fixed axis. On\nthe other hand, the source term given in [45] and those in\n[35, 44] are proportional to ∂t˜ωand ˜ω, respectively, where\n˜ωis the vorticity. These distinctions can be detected ex-\nperimentally. In an irrotational flow, the vorticity is zero,\npreventing the spin current generation due to the spin-\nvorticity coupling. Creating an irrotational flow akin to\na differentially rotating system allows us to detect spin\ncurrents specific to our mechanism. One may achieve\nsuch flow in the Taylor-Couette flow by taking appropri-\nate boundary conditions that realize c2= 0 with c1̸= 0.\nFurthermore, we can distinguish the mechanisms even in\nthe case of c2̸= 0. Since the vorticity in this system is\nuniform and time-independent, neither the source term\nin [45] nor that in [35, 44] can contribute to the spin\ncurrent. As a result, the Taylor-Couette flow can gen-\nerate the spin current in our mechanism but not in the\nother ones. Exploring these experiments would be highly\nvaluable.\nWe have comments on possible pictures that come from\n(13). If we interpret As,0=∂tΦ as a “chemical potential”\nfor the spin density, (13) suggests that the diffusive spin\ncurrent is produced as a result of the position-dependent\n“chemical potential” for the spin density. Also, rewriting\n(13) as js(x, t)/ne= [−(ℏ/2)∇∂tΦ]τ/m, we may inter-\npret that a “thermodynamic force” −(ℏ/2)∇∂tΦ is act-\ning on the electrons depending on their spin, since τ/m\nis the mobility of the electron and js(x, t)/necan be un-\nderstood as the mean velocity of the electrons. These\ninterpretations should be further refined by future stud-\nies.\nAcknowledgements .—We would like to thank D. Oue\nand Y. Nozaki for the valuable and informative discus-\nsion. This work was partially supported by JST CREST\nGrant No. JPMJCR19J4, Japan. We acknowledge JSPS\nKAKENHI for Grants (Nos. JP21H01800, JP21H04565,\nJP23H01839, JP21H05186, JP19K03659, JP19H05821,\nJP18K03623 and JP21H05189). The work was sup-\nported in part by the Chuo University Personal Research\nGrant. The authors thank RIKEN iTHEMS NEW work-\ning group for providing the genesis of this collaboration.[1] S. Maekawa, S. O. Valenzuela, E. Saitoh, and T. 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Taylor, Philos. Trans. R. Soc. London, Ser. A 223,\n289 (1923).\n[44] M. Matsuo, Y. Ohnuma, and S. Maekawa, Phys. Rev. B\n96, 020401 (2017).\n[45] M. Matsuo, J. Ieda, K. Harii, E. Saitoh, and S. Maekawa,\nPhys. Rev. B 87, 180402 (2013).\n[46] J. Fujimoto and M. Matsuo, Magnon current genera-\ntion by dynamical distortion, Phys. Rev. B 102, 020406\n(2020).7\nTRANSFORMATION TO DIFFERENTIALLY ROTATING FRAME\nBy using the total angular momentum operator acting on coordinates and spin space,\nJz≡ −iℏ(x×∇)z⊗I+ 1⊗ℏ\n2σz, , (18)\nwe define the following operator:\nˆQΦ(t) =Z\nd3xΦ(x, t)ˆψ†(x)Jzˆψ(x), (19)\nwhich is a generator of the differential rotation with angle Φ( x, t) around the z-axis. Note that we assume the rotation\nangle is axisymmetric, i.e., ∂Φ/∂φ= 0. The canonical commutation relation {ˆψ(x),ˆψ†(y)}=δ(x−y) yields\n[ˆQΦ(t),ˆψ(x)] =−Φ(x, t)Jzˆψ(x),[ˆQΦ(t),ˆψ†(x)] = Φ( x, t)(Jzˆψ(x))†. (20)\nWe have\nexp\u0014i\nℏˆQΦ(t)\u0015\nˆψ(x) exp\u0014\n−i\nℏˆQΦ(t)\u0015\n=e−iΦ(x,t)Jz/ℏˆψ(x)\n=e−iΦ(x,t)σz/2ˆψ(x′) ( x′≡e−Φ(x,t)(x×∇)zx).(21)\nNote that it can be explicitly written in matrix form as\ne−Φ(x,t)(x×∇)zx=R−1\nz[Φ(x, t)]x,Rz(φ)≡\ncosφ−sinφ0\nsinφcosφ0\n0 0 1\n. (22)\nThe free part of the Hamiltonian (namely, the kinetic term) transforms as\neiˆQΦ/ℏˆH0e−iˆQΦ/ℏ=ˆH0+i\nℏZ1\n0dλeiλˆQΦ/ℏ[ˆQΦ,ˆH0]e−iλˆQΦ/ℏ\n=ˆH0+i\nℏ[ˆQΦ,ˆH0]−1\n2ℏ2[ˆQΦ,[ˆQΦ,ˆH0]]−i\n3!ℏ3[ˆQΦ,[ˆQΦ,[ˆQΦ,ˆH0]]] +···\n=1\n2mZ\nd3x[−iℏ∇ˆψ(x)−∇Φ(x, t)Jzˆψ(x)]†·[−iℏ∇ˆψ(x)−∇Φ(x, t)Jzˆψ(x)],(23)\nwhere we have used the following equations:\n[ˆQΦ,ˆH0] =iℏ2\n2mZ\nd3x∇Φ(x, t)·h\n(−i∇ˆψ(x))†Jzˆψ(x) + (Jzˆψ(x))†(−i∇ˆψ(x))i\n, (24)\n[ˆQΦ,[ˆQΦ,ˆH0]] =−ℏ2\nmZ\nd3x|∇Φ(x, t)|2(Jzˆψ(x))†Jzˆψ(x), (25)\n[ˆQΦ,[ˆQΦ,[ˆQΦ,ˆH0]]] = 0 . (26)\nThe impurity potential part is\neiˆQΦ/ℏ\u0014Z\nd3xˆψ†(x)V′\nimp(x, t)ˆψ(x)\u0015\ne−iˆQΦ/ℏ=Z\nd3xˆψ†(x′)V′\nimp(x, t)ˆψ(x′)\n=X\njZ\nd3xˆψ†(x′)u(x−r′\nj(t))ˆψ(x′) (r′\nj(t) =Rz[Φ(rj, t)]rj)\n=X\njZ\nd3xˆψ†(x)u(Rz[Φ(x, t)]x− R z[Φ(rj, t)]rj)ˆψ(x)\n≃X\njZ\nd3xˆψ†(x)u(x−rj)ˆψ(x) =Z\nd3xˆψ†(x)Vimp(x)ˆψ(x),(27)8\n(a)\n(b)\nFIG. 2. The diagrams representing the (a) response functions and the (b) three-point vertices. The black circles represent the\nvertices and the shaded region represents the ladder vertex corrections. The dotted lines represent the impurity potential and\nthe crosses represent the impurities.\nwhere we have changed a variable of integration as x→ R z[Φ(x, t)]xin the third line. Because we assume that a\nsingle impurity potential u(x) is isotropic and has compact support in |x|< a, we have\nu(Rz[Φ(x, t)]x− R z[Φ(rj, t)]rj) =u(x− R z[Φ(rj, t)−Φ(x, t)]rj)≃u(x−rj) (28)\nfor|x−rj|< a≪1/|∇Φ|. In a similar manner, the spin-orbit interaction part is\neiˆQΦ/ℏ\u0014\nλsoZ\nd3xˆψ†(x)σ·[∇V′\nimp(x, t)×(−iℏ∇)]ˆψ(x)\u0015\ne−iˆQΦ/ℏ\n=λsoZ\nd3xˆψ†(x)σ·[∇Vimp(x)×(−iℏ∇−∇Φ(x, t)Jz)]ˆψ(x).(29)\nDEFINITION OF SPIN CURRENT OPERATOR\nIn this section, we define the spin-current density operator through the continuum equation with respect to the\nspin density. The z-polarized spin density operator is defined by\nˆs(x)≡ˆψ†(x)σzˆψ(x). (30)\nThe time derivative of the z-polarized spin density operator in the Heisenberg representation is given by\n∂\n∂tˆs(x, t) =−∇·ˆjs(x, t) +ˆT(x, t). (31)\nThe first term is the divergence of the spin-current density:\nˆjs(x) =ˆψ†(x)\u0014ℏσz\n2im← →∇+λsoez×∇V−ℏ\n2mAs\u0015\nˆψ(x). (32)\nThe second term represents the spin torque due to the impurity spin-orbit interaction:\nˆT(x) =−i\nℏλsoˆψ†(x)\u0014\nσzez×\u0012\n∇V×ℏ\ni∇\u0013\u0015\nˆψ(x). (33)\nCALCULATION OF SPIN DENSITY AND SPIN CURRENT DENSITY\nIn this section, we demonstrate the calculation of the spin density and spin-current density driven by the differential\nrotation in linear response to the emergent gauge fields. The response functions of spin density and spin-current density\ncorresponding to the diagrams shown in Fig. 2(a) are expressed as\nKµν(ω) =δµ0δν0ℏσc\n2e2D+iωℏ\n4πX\nkvk,µtr[σzgr\n+σz(vk,ν+ Λs\nν)ga\n−], (34)9\nwhere the three-point vertices Λs\nνare shown in Fig. 2(b). Up to the second order in the wavenumber qand frequency\nω, the response functions are calculated as\nK00(ω) =ℏσc\n2e2D+iωℏ\n2πh\n1 + Λs\n0i\nI0=ℏσc\n2e2DDq2+τ−1\ns\nDq2−iω+τ−1s, (35)\nK0j(ω) =iωℏ\n2πh\n1 + Λs\n0i\nIj=−iωℏσc\n2e2iqj\nDq2−iω+τ−1s, (36)\nKi0(ω) =iωℏ\n2πIih\n1 + Λs\n0i\n=−iωℏσc\n2e2iqi\nDq2−iω+τ−1s, (37)\nKij(ω) =iωℏ\n2πh\nIij+IiΛs\nji\n=iωℏσc\n2e2\u0012\nδij−Dqiqj\nDq2−iω+τ−1s\u0013\n, (38)\nwhere the Latin indices represent the spacial directions, i.e., i=x, y, z . Here, the integrations Iµνare defined by\nIµν=X\nkvk,µvk,νgr\n+ga\n−, (39)\nand given by\nI0=I00=πν0\nℏγ[1−τ(Dq2−iω)], (40)\nIi=Ii0=I0i=−Diqiπν0\nℏγ, (41)\nIij=v2\nF\n3πν0\nℏγ\u0014\nδij+δijτ\u0012\niω−3\n5Dq2\u0013\n−6\n5τDq iqj\u0015\n. (42)\nNote that the response functions satisfy the following identities:\n−iωK 00+iqjK0j=−iωℏσc\n2e2Dτ−1\ns\nDq2−iω+τ−1s,−iωK i0+iqjKij=iωℏσc\n2e2τ−1\ns\nDq2−iω+τ−1siqi. (43)\nThe spin density is given by\n⟨ˆs(q, ω)⟩=K00As,0+K0jAs,j\n=ℏσc\n2e2Dτ−1\ns\nDq2−iω+τ−1sAs,0+ℏσc\n2e2iqj\nDq2−iω+τ−1s(−iωAs,j−iqjAs,0)\n=ℏσc\n2e2Dτ−1\ns\nDq2−iω+τ−1s(−iωΦ). (44)\nThe spin-current density is given by\n⟨ˆjs,i(q, ω)⟩=Ki0As,0+KijAs,j\n=iωℏσc\n2e2τ−1\ns\nDq2−iω+τ−1sAs,i\n+iωℏσc\n2e21\nDq2−iω+τ−1s(−iωAs,i−iqiAs,0) +iωℏσc\n2e2iDqj\nDq2−iω+τ−1s(iqiAs,j−iqjAs,i)\n=iωℏσc\n2e2τ−1\ns\nDq2−iω+τ−1siqiΦ. (45)\nWe note that ( As,0, As,i) = (−iωΦ, iqiΦ).\nLadder vertex corrections\nIn this section, we calculate ladder vertex corrections due to the impurity scattering and spin-orbit scattering. First,\nwe define the elementary vertex fabshown in Fig. 3(a) corresponding to the coupling to the single impurity:\nfab=δab+iℏλso(k×k′)·σab, (46)10\n(a) (b)\nFIG. 3. (a) The elementary vertex due to the single impurity. (b) The four-point vertex in the ladder approximation.\nwhere the Latin indices a, bdescribe the spin space. The proper four-point vertex Γ0shown in the first term on the\nright-hand side of Fig. 3(b) is calculated as\nΓ0\nab,cd=niu2\ni⟨fadfcb⟩FS=ℏ\nπν0\u0012\nγ0δadδcb+1\n3γsoσad·σcb\u0013\n, (47)\nwhere ⟨···⟩ FSmeans averaging over at the Fermi surface, ℏγ0=πniu2\niν0is the damping due to the impurity scattering,\nandγso=ℏ2λ2\nsok4\nFγ0is the damping due to the spin-orbit scattering. The four-point vertex shown in Fig. 3(b) is\ndetermined by the following Dyson equation:\nΓab,cd(q) = Γ0\nab,cd+ Γ0\nab,efI0(q)Γfe,cd(q)\n= Γc(q)δabδcd+ Γs(q)σab·σcd, (48)\nwhere a, . . . , f are the spin indices, and\nΓc(q) =ℏ\n4πν0τ21\nDq2−iω, (49)\nΓs(q) =ℏ\n4πν0τ21−τ\nτs\nDq2−iω+τ−1s. (50)\nTherefore, the three-point vertices Λs\nνare calculated by\nσα\nabΛs\nν(q) =σα\ndcΓab,cd(q)Iν, (51)\nand given by\nΛs\n0(q) =1\nτ(Dq2−iω+τ−1s)−1, (52)\nΛs\nj(q) =−Diqj\nτ(Dq2−iω+τ−1s). (53)"    },    {        "title": "1602.00404v2.Equations_of_motion_of_test_particles_for_solving_the_spin_dependent_Boltzmann_Vlasov_equation.pdf",        "content": "arXiv:1602.00404v2  [nucl-th]  12 May 2016Equations of motion of test particles for solving the spin-d ependent Boltzmann-Vlasov\nequation\nYin Xia,1,2Jun Xu∗,1Bao-An Li,3,4and Wen-Qing Shen1\n1Shanghai Institute of Applied Physics, Chinese Academy of S ciences, Shanghai 201800, China\n2University of Chinese Academy of Science, Beijing 100049, C hina\n3Department of Physics and Astronomy, Texas A &M University-Commerce, Commerce, TX 75429-3011, USA\n4Department of Applied Physics, Xi’an Jiao Tong University, Xi’an 710049, China\n(Dated: May 13, 2016)\nA consistent derivation of the equations of motion (EOMs) of test particles for solving the spin-\ndependent Boltzmann-Vlasov equation is presented. The res ulting EOMs in phase space are similar\nto the canonical equations in Hamiltonian dynamics, and the EOM of spin is the same as that in\nthe Heisenburg picture of quantum mechanics. Considering f urther the quantum nature of spin and\nchoosing the direction of total angular momentum in heavy-i on reactions as a reference of measuring\nnucleon spin, the EOMs of spin-up and spin-down nucleons are given separately. The key elements\naffecting the spin dynamics in heavy-ion collisions are iden tified. The resulting EOMs provide asolid\nfoundation for using the test-particle approach in studyin g spin dynamics in heavy-ion collisions\nat intermediate energies. Future comparisons of model simu lations with experimental data will\nhelp constrain the poorly known in-medium nucleon spin-orb it coupling relevant for understanding\nproperties of rare isotopes and their astrophysical impact s.\nPACS numbers: 25.70.-z, 24.10.Lx, 13.88.+e, 21.30.Fe, 21. 10.Hw\nIntroduction: The importance of nucleon spin degreeof\nfreedomwasfirstrecognizedmorethan50yearsagowhen\nMayer and Jensen introduced the spin-orbit interaction\nandusedittoexplainsuccessfullythemagicnumbersand\nshell structure of nuclei [1, 2]. Subsequently, the nuclear\nspin-orbit interaction was found responsible for many in-\nterestingphenomenainnuclearstructure[3–8]. Italsoaf-\nfectssomefeaturesofnuclearreactions,suchasthefusion\nthreshold [9], the polarization measured in terms of the\nanalyzing power in pick-up or removal reactions [10–13],\nandthespindependenceofnucleoncollectiveflow[14,15]\nin heavy-ion collisions (HICs). However, the role of nu-\ncleon spin is much less known in nuclear reactions than\nstructures. In HICs at intermediate energies, a central\nissue is the density and isospin dependence of the spin-\norbit coupling in neutron-rich medium, see, e.g., Ref. [16]\nfor a recent review. It is also interesting to mention that\nthe study of spin-dependent structure functions of nu-\ncleons and nuclei has been at the forefronts of nuclear\nand particle physics [17]. This study will be boosted by\nfuture experiments at the proposed electron-ion collider\nusing polarized beams [18]. In this work, we derive for\nthe first time equations of motion (EOMs) of nucleon\ntest particles [19] for solving the spin-dependent Boltz-\nmann (Vlasov)-Uehling-Uhlenbeck (BUU or VUU) equa-\ntion. These EOMs provide the physics foundation for\nsimulating spin transport for not only nucleons in heavy-\nion reactions but also electrons for understanding many\ninterestingphenomena, suchasthespinwave[20–23], the\nspin-Hall effect [24–26], etc.\nConsidering the spin degree of freedom, the Wigner\n∗Corresponding author: xujun@sinap.ac.cnfunction in phase space becomes a 2 ×2 matrix [27].\nIts time evolution is governed by the Boltzmann-Vlasov\n(BV) equation obtained by a Wigner transformation of\nthe Liouville equation for the density matrix [20, 28, 29]\n∂ˆf\n∂t+i\n¯h/bracketleftBig\nˆε,ˆf/bracketrightBig\n+1\n2/parenleftBigg\n∂ˆε\n∂/vector p·∂ˆf\n∂/vector r+∂ˆf\n∂/vector r·∂ˆε\n∂/vector p/parenrightBigg\n−1\n2/parenleftBigg\n∂ˆε\n∂/vector r·∂ˆf\n∂/vector p+∂ˆf\n∂/vector p·∂ˆε\n∂/vector r/parenrightBigg\n= 0, (1)\nwhere ˆεandˆfare from the Wigner transformation of\nthe energy and phase-space density matrix, respectively,\nand they can be decomposed into their scalar and vector\nparts, i.e.,\nˆε(/vector r,/vector p) =ε(/vector r,/vector p)ˆI+/vectorh(/vector r,/vector p)·/vector σ, (2)\nˆf(/vector r,/vector p) =f0(/vector r,/vector p)ˆI+/vector g(/vector r,/vector p)·/vector σ, (3)\nwhere/vector σ= (σx,σy,σz) andˆIare respectively the Pauli\nmatrices and the 2 ×2 unit matrix, εandf0are\nthe scalar part of the effective single-particle energy\nˆεand phase-space density ˆf, respectively, and /vectorhand\n/vector gare the corresponding vector distributions. Adding\nthe Uehling-Uhlenbeck collision term, the resulting spin-\ndependent BUU equation can be used to describe the\nspin-dependent dynamics in various systems.\nWhile the spin-dependent BUU equation can be taken\nas the starting point of investigating spin dynamics in\nHICs, a consistent derivation of the EOMs of nucleon\ntest particles for simulations is still lacking. The sit-\nuation is quite different for electron spin transport in\nsolid state physics. To our best knowledge, the treat-\nment of electron spin transport relevant to the present\nstudy mostly follows two approaches. One way (Method2\nI) is to start from a model Hamiltonian and use the\ncanonical EOMs for the time evolution of the electron’s\ncoordinate and momentum, while the time evolution of\nthe electron’s spin is given by its commutation relation\nwith the model Hamiltonian as in the Heisenberg pic-\nture of quantum mechanics [24, 30, 31]. Moreover, an\nadiabatic approximation is often used so that the time\nevolution of spin can be solved first. Inserting the solu-\ntion for spin evolution into the EOMs for coordinate and\nmomentum then leads to the Berry curvature terms [32–\n34]. Another method (Method II) frequently used is to\nlinearize the spin-dependent BUU equation for spin-up\nand spin-down particles separately through the relax-\nation time approximation [25, 35–37]. In nuclear physics,\nEOMs of nucleon test particles should be derived consis-\ntently from the BUU transport equation used to model\nHICs. It is well known that the spin-independent BV\nequation can be solved numerically by using the test-\nparticle method [38, 39]. In particular, it was shown that\nthe EOMs of test particles are identical to the canonical\nEOMs if only the lowest order term in expanding the\nWigner function is considered (see Ref. [19] and com-\nments in Ref. [40]). Applying the test-particle approach\nto solving the spin-dependent BUU equation for the first\ntime, we found that the EOMs of nucleon test particles\nare similar to those for electrons obtained within Method\nI described above, albeit with different forms of interac-\ntions.\nDecomposition of the spin-dependent phase-space dis-\ntribution function and its evolution: To avoid confusion,\nwe begin by first commenting on the two approaches of\nderiving the BV equation with different definitions of the\nspinor Wigner distribution function often used in the lit-\nerature. Equation (1) was derived by means of the den-\nsity matrix method and taking the semiclassical limit as\noutlined by Smith and Jensen [20]. It can be separated\ninto two equations governing the scalar and vector dis-\ntributions, respectively,\n∂f0\n∂t+∂ε\n∂/vector p·∂f0\n∂/vector r−∂ε\n∂/vector r·∂f0\n∂/vector p+∂/vectorh\n∂/vector p·∂/vector g\n∂/vector r−∂/vectorh\n∂/vector r·∂/vector g\n∂/vector p= 0,(4)\n∂/vector g\n∂t+∂ε\n∂/vector p·∂/vector g\n∂/vector r−∂ε\n∂/vector r·∂/vector g\n∂/vector p+∂f0\n∂/vector r·∂/vectorh\n∂/vector p\n−∂f0\n∂/vector p·∂/vectorh\n∂/vector r+2/vector g×/vectorh\n¯h= 0. (5)\nAnother way to get the spin-dependent BV equation\nis to start from the time-dependent Hartree-Fock equa-\ntions for the one-body density matrix with spin degree of\nfreedom and rewrite the equations with the help of the\nWigner transformation, see, e.g., Refs. [28, 29]. In this\nway, one will get four coupled equations which describe\nthe time evolution of the four-component Wigner phase-\nspace densities from the 2 ×2 density matrix with spin.\nThe definition of the Wigner function of particles withspin-1/2 was suggested in Refs. [27, 41] as\nfσ,σ′(/vector r/vector p,t) =/integraldisplay\nd3se−i/vector p·/vector s/¯hψ∗\nσ′(/vector r−/vector s\n2,t)ψσ(/vector r+/vector s\n2,t),(6)\nf(/vector r/vector p,t,0) =f1,1(/vector r/vector p,t)+f−1,−1(/vector r/vector p,t), (7)\nτ(/vector r/vector p,t,x) =f−1,1(/vector r/vector p,t)+f1,−1(/vector r/vector p,t), (8)\nτ(/vector r/vector p,t,y) =−i[f−1,1(/vector r/vector p,t)−f1,−1(/vector r/vector p,t)], (9)\nτ(/vector r/vector p,t,z) =f1,1(/vector r/vector p,t)−f−1,−1(/vector r/vector p,t), (10)\nwithσ(σ′) = 1 for spin up and −1 for spin down. The\nabove definitions are convenient in treating the expec-\ntation values of the spin components. Equation (6)\ngives the matrix components of the Wigner function\nwith spin degree of freedom. f(/vector r/vector p,t,0) is the ordinary\nWigner phase-space density irrespective of the particle\nspin, while τ(/vector r/vector p,t,x),τ(/vector r/vector p,t,y), andτ(/vector r/vector p,t,z), repre-\nsenting the three components of the spin Wigner density\n/vector τ(/vector r,/vector p,t), are the probabilities of the spin projection on\nthex,y, andzdirections, respectively. With the above\ndefinitions, the Wigner density f(/vector r,/vector p,t) in Eq. (7) and\nthe spin Wigner density /vector τ(/vector r,/vector p,t) (Eqs. (8 - 10)) can be\nexpressedintermsofthe f0(/vector r,/vector p,t)and/vector g(/vector r,/vector p,t)inEq.(3)\nas [41]\nf(/vector r,/vector p,t) = 2f0(/vector r,/vector p,t), (11)\n/vector τ(/vector r,/vector p,t) = 2/vector g(/vector r,/vector p,t). (12)\nIn this way, the two approaches using two different defi-\nnitions of the spinor Wigner function lead to exactly the\nsame spin-dependent BV equation.\nSingle-particle energy with spin-orbit interaction:\nWhile our derivation is general, to be specific, for the\nspin-dependent part of the single-particle Hamiltonian\nin Eq. (2) we take the Skyrme-type effective two-body\ninteraction including the spin-orbit coupling [42, 43]\nˆhso\nq=−1\n2W0∇·(/vectorJ+/vectorJq)+/vector σ·[−1\n2W0∇×(/vectorj+/vectorjq)]\n+1\n4iW0[(∇×/vector σ)·∇(ρ+ρq)+∇(ρ+ρq)·(∇×/vector σ)]\n−1\n4iW0[∇·(∇×(/vector s+/vector sq))+(∇×(/vector s+/vector sq))·∇],(13)\nwhereq=norpis the isospin index, and ρ,/vector s,/vectorj, and\n/vectorJare the number, spin, momentum, and spin-current\ndensities, respectively. According to the definition of the\nWignerfunctioninEq.(6), thesedensitiescanbedirectly\nexpressed as [42, 43]\nρ(/vector r) =/integraldisplay\nd3pf(/vector r,/vector p), (14)\n/vector s(/vector r) =/integraldisplay\nd3p/vector τ(/vector r,/vector p), (15)\n/vectorj(/vector r) =/integraldisplay\nd3p/vector p\n¯hf(/vector r,/vector p), (16)\n/vectorJ(/vector r) =/integraldisplay\nd3p/vector p\n¯h×/vector τ(/vector r,/vector p). (17)3\nAfter a Wigner transformation, the Eq. (13) can be read-\nily expressed in terms of the above densities as\nhso\nq(/vector r,/vector p) =h1+h4+(/vectorh2+/vectorh3)·/vector σ(18)\nwithh1,/vectorh2,/vectorh3, andh4given by\nh1=−W0\n2∇/vector r·[/vectorJ(/vector r)+/vectorJq(/vector r)], (19)\n/vectorh2=−W0\n2∇/vector r×[/vectorj(/vector r)+/vectorjq(/vector r)], (20)\n/vectorh3=W0\n2∇/vector r[ρ(/vector r)+ρq(/vector r)]×/vector p, (21)\nh4=−W0\n2∇/vector r×[/vector s(/vector r)+/vector sq(/vector r)]·/vector p. (22)\nComparing with Eq. (2), the effective single-particle en-\nergy ˆεcan be written as\nεq(/vector r,/vector p) =p2\n2m+Uq+h1+h4, (23)\n/vectorhq(/vector r,/vector p) =/vectorh2+/vectorh3, (24)\nwhereUqis the spin-independent mean-field potential.\nThe nuclear tensor force can be implemented in a similar\nwayif needed, once the scalarand the vectorcomponents\nare decomposed from the corresponding energy-density\nfunctional.\nSpin-dependent EOMs of test particles: We now de-\nrive the EOMs from the decoupled spin-dependent BV\nequation [Eqs. (4) and (5)] by using the test-particle\nmethod [19, 38]. The vector part /vector g(/vector r,/vector p) of the spinor\nWigner function distribution in Eq. (3) can be repre-\nsented by a real unit vector /vector ntimes a scalar function\nf1(/vector r,/vector p), i.e.,\n/vector g(/vector r,/vector p) =/vector nf1(/vector r,/vector p). (25)\nHere we assume that /vector nis independent of /vector rand/vector p, which\nis valid if/vector nevolves much faster than the phase-space\ncoordinates /vector rand/vector por if/vector nis a global constant. Under\nthisassumptionandbysubstitutingEq.(25)intoEqs.(4)\nand (5), we obtain\n∂f0\n∂t+∂ε\n∂/vector p·∂f0\n∂/vector r−∂ε\n∂/vector r·∂f0\n∂/vector p+(∂/vectorh\n∂/vector p·/vector n)·∂f1\n∂/vector r\n−(∂/vectorh\n∂/vector r·/vector n)·∂f1\n∂/vector p≈0, (26)\n∂f1\n∂t/vector n+ (∂ε\n∂/vector p·∂f1\n∂/vector r)/vector n−(∂ε\n∂/vector r·∂f1\n∂/vector p)/vector n+∂f0\n∂/vector r·∂/vectorh\n∂/vector p\n−∂f0\n∂/vector p·∂/vectorh\n∂/vector r+(2/vector n×/vectorh\n¯h+∂/vector n\n∂t)f1≈0.(27)\nGenerally, the magnitude of the Poisson bracket {f0,/vectorh},\ni.e., (∂f0/∂/vector r)·(∂/vectorh/∂/vector p)−(∂f0/∂/vector p)·(∂/vectorh/∂/vector r), is much\nsmaller than that of /vectorhor{f0,ǫ}. In this approximationandbyseparatingcomponentsparallelandperpendicular\nto/vector n, Eq. (27) can be divided into two parts, i.e.,\n∂f1\n∂t/vector n+ (∂ε\n∂/vector p·∂f1\n∂/vector r)/vector n−(∂ε\n∂/vector r·∂f1\n∂/vector p)/vector n+∂f0\n∂/vector r·∂/vectorh\n∂/vector p\n−∂f0\n∂/vector p·∂/vectorh\n∂/vector r= 0, (28)\n∂/vector n\n∂t≈2/vectorh×/vector n\n¯h. (29)\nAs/vector nis a unit vector, we have /vector n·/vector n= 1. By taking the\ninner product of /vector nwith Eq. (28) (or Eq. (27)) on both\nsides, one obtains\n∂f1\n∂t+∂ε\n∂/vector p·∂f1\n∂/vector r−∂ε\n∂/vector r·∂f1\n∂/vector p+∂f0\n∂/vector r·(∂/vectorh\n∂/vector p·/vector n)\n−∂f0\n∂/vector p·(∂/vectorh\n∂/vector r·/vector n) = 0. (30)\nAdding and subtracting Eqs. (26) and (30), we get two\nequations for two types of particles with phase-space dis-\ntribution functions f±=f0±f1, i.e.,\n∂f+\n∂t+/parenleftbigg∂ǫ\n∂/vector p+∂Vhn\n∂/vector p/parenrightbigg\n·∂f+\n∂/vector r−/parenleftbigg∂ǫ\n∂/vector r+∂Vhn\n∂/vector r/parenrightbigg\n·∂f+\n∂/vector p= 0,(31)\n∂f−\n∂t+/parenleftbigg∂ǫ\n∂/vector p−∂Vhn\n∂/vector p/parenrightbigg\n·∂f−\n∂/vector r−/parenleftbigg∂ǫ\n∂/vector r−∂Vhn\n∂/vector r/parenrightbigg\n·∂f−\n∂/vector p= 0,(32)\nwithVhn≡/vectorh·/vector n. It can be understood from Eqs. (3) and\n(25) thatf+andf−are the eigenfunctions of ˆf, rep-\nresenting the phase-space distributions of particles with\ntheir spin in + /vector nand−/vector ndirections, respectively, i.e.,\nspin-up and spin-down particles.\nFollowing the test-particle method and using an auxil-\niary variable /vector s, the time evolution of the Wigner function\nf±(/vector r,/vector p) can be expressed as [19]\nf±(/vector r,/vector p,t) =/integraldisplayd3r0d3p0d3s\n(2π¯h)3exp{i/vector s·[/vector p−/vectorP(/vector r0/vector p0/vector s,t)]/¯h}\n×δ[/vector r−/vectorR(/vector r0/vector p0/vector s,t)]f±(/vector r0,/vector p0,t0), (33)\nwheref±(/vector r0,/vector p0,t0) is the Wigner functions at time\nt0with the initial conditions /vectorR(/vector r0/vector p0/vector s,t0) =/vector r0and\n/vectorP(/vector r0/vector p0/vector s,t0) =/vector p0. Our main task is now to find the\nnew phase-space coordinates /vectorR(/vector r0/vector p0/vector s,t) and/vectorP(/vector r0/vector p0/vector s,t)\nand to obtain the Wigner function at the next time step\nt=t0+∆twith a small increment ∆ t. By substituting4\nEq. (33) into Eqs. (31) and (32), we obtain\n[−∂/vectorR(/vector r0/vector p0/vector s,t)\n∂t+∂ε\n∂/vector p]·∂f±(/vector r,/vector p,t)\n∂/vector r±∂Vhn\n∂/vector p·∂f±(/vector r,/vector p,t)\n∂/vector r\n+/integraldisplayd3r0d3p0d3s\n(2π¯h)3{f±(/vector r0,/vector p0,t0)[−i/vector s\n¯h·∂/vectorP(/vector r0/vector p0/vector s,t)\n∂t\n−[ε(/vector r−/vector s\n2,t)−ε(/vector r+/vector s\n2,t)]\ni¯h]\n∓f±(/vector r0,/vector p0,t0)[[Vhn(/vector r−/vector s\n2,t)−Vhn(/vector r+/vector s\n2,t)]\ni¯h]}\n×exp{i/vector s·[/vector p−/vectorP(/vector r0/vector p0/vector s,t)]/¯h}\n×δ[/vector r−/vectorR(/vector r0/vector p0/vector s,t)] = 0. (34)\nComparing similar terms in the above equation, we get\nthe following equations for /vectorRand/vectorP, respectively, i.e.,\n/bracketleftBigg\n−∂/vectorR(/vector r0/vector p0/vector s,t)\n∂t+∂ε\n∂/vector p/bracketrightBigg\n·∂f±(/vector r,/vector p,t)\n∂/vector r±∂Vhn\n∂/vector p·∂f±(/vector r,/vector p,t)\n∂/vector r\n= 0, (35)\nand\nf±(/vector r0,/vector p0,t0){−i/vector s\n¯h·∂/vectorP(/vector r0/vector p0/vector s,t)\n∂t\n−[ε(/vector r−/vector s\n2,t)−ε(/vector r+/vector s\n2,t)]\ni¯h}∓f±(/vector r0,/vector p0,t0)\n×{[Vhn(/vector r−/vector s\n2,t)−Vhn(/vector r+/vector s\n2,t)]\ni¯h}= 0.(36)\nIn order to satisfy the above two equations with arbi-\ntraryf±(/vector r,/vector p,t) andf±(/vector r0,/vector p0,t0), we get the equations\nof motion for /vectorR\n∂/vectorR(/vector r0/vector p0/vector s,t)\n∂t=∂ε\n∂/vector p±∂Vhn\n∂/vector p, (37)\nand for/vectorP\n/vector s·∂/vectorP(/vector r0/vector p0/vector s,t)\n∂t= [ε(/vector r−/vector s\n2,t)−ε(/vector r+/vector s\n2,t)]\n±[Vhn(/vector r−/vector s\n2,t)−Vhn(/vector r+/vector s\n2,t)].(38)\nExpandingEq.(38)in /vector sandkeepingonlythelowestorder\nterm, we then obtain\n∂/vectorP(/vector r0/vector p0/vector s,t)\n∂t=−∂ε\n∂/vector r∓∂Vhn\n∂/vector r. (39)\nConsidering Eqs. (18-24) and combining Eqs. (37), (39),\nand (29), the EOMs of test particles for solving the spin-\ndependent BV equation with the spin-orbit interaction\nare thus\n∂/vectorR\n∂t=/vector p\nm+∇/vector p(h1+h4)±∇/vector p(/vectorh2·/vector n+/vectorh3·/vector n),(40)\n∂/vectorP\n∂t=−∇/vector rUq−∇/vector r(h1+h4)∓∇/vector r(/vectorh2·/vector n+/vectorh3·/vector n),(41)\n∂/vector n\n∂t=2(/vectorh2+/vectorh3)×/vector n\n¯h, (42)with the upper sign for f+and lower sign for f−, re-\nspectively, and h1,/vectorh2,/vectorh3, andh4given by Eqs. (19-22).\nAccording to the definition of the spinor Wigner phase-\nspace density distribution in Eqs. (3) and (25), /vector nis the\ndirection (unit vector) of the local spin polarization in\n3-dimensional coordinate space, which can be expressed\nby the test-particle method [38, 39] as\n/vector n=/summationtext\ni/vector niδ(/vector r−/vector ri)δ(/vector p−/vector pi)\n|/summationtext\ni/vector niδ(/vector r−/vector ri)δ(/vector p−/vector pi)|, (43)\nwith/vector nibeing the spin expectation direction of the ith\nnucleon. Based on Eqs. (7-10), the scalar Wigner den-\nsity distribution f(/vector r,/vector p) and the vector Wigner density\ndistribution /vector τ(/vector r,/vector p) can be expressed by the test-particle\nmethod as\nf(/vector r,/vector p) =1\nNTP/summationdisplay\niδ(/vector r−/vector ri)δ(/vector p−/vector pi),(44)\n/vector τ(/vector r,/vector p) =1\nNTP/summationdisplay\ni/vector niδ(/vector r−/vector ri)δ(/vector p−/vector pi),(45)\nwithNTPbeing the number of test particles per nu-\ncleon. In this way, the number, spin, momentum, and\nspin-current densities can also be calculated via\nρ(/vector r) =1\nNTP/summationdisplay\niδ(/vector r−/vector ri), (46)\n/vector s(/vector r) =1\nNTP/summationdisplay\ni/vector niδ(/vector r−/vector ri), (47)\n/vectorj(/vector r) =1\nNTP/summationdisplay\ni/vector pi\n¯hδ(/vector r−/vector ri), (48)\n/vectorJ(/vector r) =1\nNTP/summationdisplay\ni(/vector pi\n¯h×/vector ni)δ(/vector r−/vector ri).(49)\nQuantum nature of spin: The above EOMs for the\nphase-space coordinates are the same as the canonical\nequationsinHamiltoniandynamics, andtheEOMofspin\nis the same as that in the Heisenburg picture of quantum\nmechanics. These EOMs have been applied in our previ-\nous studies [14–16]. As first demonstrated by the Stern-\nGerlach experiment [44], the projection of spin onto any\nreferencedirectionusedinmeasurementsisquantized. In\nnon-central HICs, the angular momentum is in the ydi-\nrection perpendicular to the reaction plane ( x-o-zplane).\nItisthusnaturaltoset ydirectionasthethird(magnetic)\nspin direction. We then fix /vector n= ˆyin Eqs. (40) and (41)\nand set/vector ni=±ˆyin Eqs. (45), (47), and (49) depending\non whether the ith particle is spin-up or spin-down with\nrespect to the yaxis. In this way the time evolution of /vector n\n(Eq. (42)) is not needed. Thus, as describing the isospin\ndynamics with separate EOMs for neutrons and protons\n[45], wenowhaveseparateEOMsforspin-up(uppersign)\nand spin-down (lower sign) particles\n∂/vectorR\n∂t=/vector p\nm+∇/vector p(h1+h4)±∇/vector p(h2y+h3y),(50)\n∂/vectorP\n∂t=−∇/vector rUq−∇/vector r(h1+h4)∓∇/vector r(h2y+h3y).(51)5\nIt is seen that the h2y≡/vectorh2·/vector nandh3y≡/vectorh3·/vector nlead to\nthe spin-dependent motion while the h1andh4affect the\nglobal motion in phase space.\nTo summarize, the spin-dependent Boltzmann-Vlasov\nequation can be solved by extending the test-particle\nmethod. Considering the quantum nature of spin and\nchoosing the direction of total angular momentum in\nheavy-ion reactions as a reference of measuring nucleon\nspin, the EOMs of spin-up and spin-down nucleons are\nderived. The key elements affecting the spin dynamics in\nheavy-ion collisions are identified. The derived EOMs of\ntest particles provide the theoretical foundation of sim-\nulating spin-dependent dynamics in intermediate-energy\nheavy-ion collisions. Future comparisons of model sim-\nulations with experimental data will help constrain the\npoorly known in-medium nucleon spin-orbit coupling.\nThis work was supported by the Major State BasicResearch Development Program (973 Program) of China\nunder Contract Nos. 2015CB856904and 2014CB845401,\nthe National Natural Science Foundation of China un-\nder Grant Nos. 11320101004, 11475243, and 11421505,\nthe ”100-talent plan” of Shanghai Institute of Applied\nPhysics under Grant Nos. Y290061011 and Y526011011\nfrom the Chinese Academy of Sciences, the Shanghai\nKeyLaboratoryofParticlePhysicsandCosmologyunder\nGrant No. 15DZ2272100, the ”Shanghai Pujiang Pro-\ngram” under Grant No. 13PJ1410600, the US National\nScience Foundation under Grant No. PHY-1068022,\nthe U.S. Department of Energy, Office of Science, un-\nder Award Number de-sc0013702, and the CUSTIPEN\n(China-U.S. Theory Institute for Physics with Exotic\nNuclei) under the US Department of Energy Grant No.\nDEFG02- 13ER42025.\n[1] M. G. Mayer, Phys. Rev. 75, 1969 (1949).\n[2] M. G. Mayer and J. H. D. Jenson, Elementary Theory of\nNuclear Shell Structure (Wiley, New York, 1955).\n[3] G. A. Lalazissis et al., Phys. Lett. B 418, 7 (1998).\n[4] M. Morjean et al., Phys. Rev. Lett. 101, 072701 (2008).\n[5] J. P. Schiffer et al., Phys. Rev. Lett. 92, 162501 (2004).\n[6] M. Bender et al., Phys. Rev. C 80, 064302 (2009).\n[7] B. Chen et al., Phys. Lett. B 355, 37 (1995).\n[8] K.-L. Kratz, K. Farouqi, and B. 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Rep. 464,\n113 (2008)."    },    {        "title": "2306.07864v1.Nonequilibrium_spin_transport_in_integrable_and_non_integrable_classical_spin_chains.pdf",        "content": "Nonequilibrium spin transport in integrable and non-integrable classical spin chains\nDipankar Roy,1,∗Abhishek Dhar,1,†Herbert Spohn,2,‡and Manas Kulkarni1,§\n1International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India\n2Zentrum Mathematik and Physik Department, Technische Universit¨ at M¨ unchen, Garching 85748, Germany\n(Dated: June 14, 2023)\nAnomalous transport in low dimensional spin chains is an intriguing topic that can offer key\ninsights into the interplay of integrability and symmetry in many-body dynamics. Recent studies\nhave shown that spin-spin correlations in spin chains, where integrability is either perfectly preserved\nor broken by symmetry-preserving interactions, fall in the Kardar-Parisi-Zhang (KPZ) universality\nclass. Similarly, energy transport can show ballistic or diffusive-like behaviour. Although such be-\nhaviour has been studied under equilibrium conditions, no results on nonequilibrium spin transport\nin classical spin chains has been reported so far. In this work, we investigate both spin and energy\ntransport in classical spin chains (integrable and non-integrable) when coupled to two reservoirs at\ntwo different temperatures/magnetization. In both the integrable case and broken-integrability (but\nspin-symmetry preserving), we report anomalous scaling of spin current with system size ( Js∝L−µ)\nwith an exponent, µ≈2/3, falling under the KPZ universality class. On the other hand, it is note-\nworthy that energy current remains ballistic ( Je∝L−ηwith η≈0) in the purely integrable case\nand there is departure from ballistic behaviour ( η >0) when integrability is broken regardless of\nspin-symmetry. Under nonequilibrium conditions, we have thoroughly investigated spatial profiles\nof local magnetization and energy. We find interesting nonlinear spatial profiles which are hallmarks\nof anomalous transport. We also unravel subtle striking differences between the equilibrium and\nnonequilibrium steady state through the lens of spatial spin-spin correlations.\nAnomalous behaviour in low dimensional systems is a\nfascinating phenomenon with great significance in theory\nand applications. Recent discoveries on such phenom-\nena that deviates from Fourier law in one-dimensional,\nintegrable spin chains has generated a lot of interest\n[1–3]. It has been observed that the well-known quan-\ntum Heisenberg spin-1\n2chain (also called quantum XXX\nmodel) exhibits anomalous behaviour for nonequilibrium\nspin transport [4, 5]. In particular, this anomalous be-\nhaviour involves deep connections to the Kardar-Parisi-\nZhang (KPZ) universality class [6, 7], wherein spin cor-\nrelation is characterized by the KPZ scaling function\n[8, 9]. Such anomalous behaviour, often referred to as the\nKPZ superdiffusion , also exists in other integrable quan-\ntum models with higher symmetry [10, 11]. Moreover,\nthere is numerical evidence that the KPZ superdiffusion\nis present in all integrable models having non-Abelian\nsymmetry [12]. There has been significant progress in\nthe experimental side as well, with contemporary experi-\nments detecting the 1D KPZ physics in real systems mod-\nelled by the quantum Heisenberg spin-1\n2model [13, 14].\nTo understand such behaviour in the quantum models, a\nnumber of analytical studies have focused on the relation\nbetween integrability and the KPZ superdiffusion [1, 15–\n17]. In addition to these results for integrable models,\nmore recent studies have considered the impact of addi-\ntional integrability-breaking terms. Weak integrability-\nbreaking, spin-symmetry preserving terms leave the KPZ\nsuperdiffusion intact in the quantum Heisenberg chain\n[18]. However, normal diffusion with enhanced diffusion\ncoefficient is observed for spin in the quantum Heisen-\nberg model at an infinite temperature in the presence of\nstrong noisy couplings [19]. The question of integrabilitybreaking in quantum systems is still open because various\nparameter regimes are yet to be understood [2, 3, 20, 21].\nRemarkably, even in classical spin chains KPZ su-\nperdiffusion has been reported. Numerical investigations\nin Refs. 22–24 show that the KPZ superdiffusion de-\nscribes spin correlations in the integrable lattice Landau-\nLifshitz (ILLL) also known as the Ishimori-Haldane (IH)\nmodel [25, 26]. Studying classical analogues are specially\nbeneficial because of severe numerical challenges in quan-\ntum models.\nIn a recent study motivated by the quantum-classical\ncorrespondence [23, 27–30], robustness of the KPZ su-\nperdiffusion is predicted in the classical integrable spin\nchain in the presence of integrability-breaking but spin-\nsymmetry preserving terms [31]. When spin-symmetry\nis not preserved, either diffusive or anomalous (but non-\nKPZ) behaviour arises. Interestingly, a special situation\noccurs at low temperatures even in the classical Heisen-\nberg (CH) model which is non-integrable: KPZ superdif-\nfusion is observed due to near-integrability [32]. At inter-\nmediate temperatures, the possibility of anomalous be-\nhaviour is debatable [32–34].\nMost of these aforementioned studies rely on the equi-\nlibrium correlations to glean information about transport\nproperties. The number of studies on spin or energy\ntransport under nonequilibrium conditions are limited.\nOn the quantum side, there has been some work on dif-\nfusive energy and/or spin transport [4, 35, 36]. On the\nclassical side, thermal tranpsort has been investigated in\nthe CH model [37, 38]. It is worth noting that there\nhas been no work reported on nonequilibrium spin trans-\nport in classical spin chains even for the well-known CH\nmodel.arXiv:2306.07864v1  [cond-mat.stat-mech]  13 Jun 20232\nBefore proceeding further, we recall some important\naspects of nonequilibrium energy (or heat) transport\nwhich has been a topic of intense research [39–46]. Early\nstudies on heat conduction and temperature profile in\nmicroscopic models include the theoretical works on\nharmonic chains, both regular and disordered, in the\nnonequilibrium steady state (NESS) [47–50]. These stud-\nies observe diverging conductivity (with system size) in\nharmonic chains. Such divergence in the thermodynamic\nlimit is also observed in the Fermi-Pasta-Ulam-Tsinghou\n(FPUT) model (see [51] for a review) where anharmonic\ninteractions are present [52–54]. There are also one-\ndimensional systems of particles with alternating masses\nwhich exhibit anomalous transport [54–59]. Moreover,\nthere has been investigations in other low-dimensional\nsystems, such as, the discrete nonlinear Schr¨ odinger\nequation [60–62], the 1D Toda chains [55, 63, 64], rotor\nmodels [65], nanosystems [46], to name a few. To obtain\nan understanding of anomalous heat transport starting\nfrom a microscopic model, there has been analytical stud-\nies on simple models as well, for example, the stochastic\nmodel in Refs. [66, 67], the L´ evy walk model [68], and\nthe harmonic chain momentum exchange model [69].\nGiven this enormous interest about anomalous trans-\nport in low-dimensional systems, we investigate the\nspin and energy current in classical integrable and non-\nintegrable spin chains. It is to be noted that even in the\nnon-integrable model there are four conservation laws,\nnamely, the energy and three components of spin (due to\nisotropy). The key findings of our work are as follows:\n(i)Integrable case (IH): We observe that spin current\nscales with system size as L−µwith µ≈2/3 in the\nintegrable IH model [see Fig. 2(a)]. This is a hall-\nmark of KPZ superdiffusion. The energy current\nhowever is observed to be ballistic [see Fig. 3(a)]\nthereby bringing out the signatures of integrabil-\nity.\n(ii)Non-integrable case (CH): In the non-integrable\nCH model, for the spin current, we find hints\nof KPZ superdiffusion at low temperature and\nconventional diffusion at high temperature [see\nFig. 2(a)]. On the other hand, for energy, we\nfind the energy current approaches conventional\ndiffusive behaviour [see Fig. 3(a)] thereby con-\nsistent with the expectation from generic non-\nintegrable/chaotic systems.\n(iii)Broken-integrability case (IH-CH): This case per-\ntains to a case where the IH chain is perturbed with\na non-integrable (CH) Hamiltonian. Remarkably,\nwe find that the spin current scaling with system\nsize falls into the KPZ universality class even in the\nnon-perturbative regime [see Fig. 2(b)].\nThe model we consider (Fig. 1) is a classical spin chain\nofL+2 three-component spins, ⃗Sn, n= 0,1,2, . . . , L +1,\nFIG. 1: (Color online) A schematic diagram showing\nthe 1D classical spin chain ( ⃗S1,⃗S2, . . . , ⃗SL) in contact\nwith two reservoirs represented by ⃗S0and⃗SL+1. We in-\ntroduce two auxiliary fields ⃗ gland⃗ grto ensure that the\nboundary spins ⃗S0and⃗SL+1are maintained at a desired\nmagnetization.\nof unit length. The Hamiltonian of the system is\nH=L/summationdisplay\nn=0en,\nen=−Jln/parenleftig\n1 +⃗Sn·⃗Sn+1/parenrightig\n−λ⃗Sn·⃗Sn+1(1)\nwith J, λ⩾0 describing the relative strength of the inte-\ngrable and non-integrable parts. We refer to this model\nas the Ishimori-Haldane-classical-Heisenberg or IH-CH\nspin chain. Note that enin Eq. (1) is local energy. It is\nalso to be noted that λ= 0 corresponds to the IH spin\nchain or the ILLL model [23, 25, 26, 70, 71] and J= 0\ncorresponds to the CH spin chain [72].\nIn order to study nonequilibrium spin and energy\ntransport, in addition to the Hamiltonian dynamics, we\nincorporate local heat bath dynamics at the boundary\nspins ⃗S0and⃗SL+1(see Supplementary Material [73] for\ndetails). The local heat bath dynamics of the two bound-\nary spins is specified by their temperatures TlandTr\nand by two auxiliary boundary magnetic fields ⃗ gland⃗ gr.\nThe two boundary spins thus play the role of reservoirs,\nwhich means that the bond energy of the pair of spins\n⃗S0and⃗S1, and the magnetization of ⃗S0are set by the\ntemperature Tland field ⃗ gl. Likewise, the bond energy of\nthe pair of spins ⃗SLand⃗SL+1, and the magnetization of\n⃗SL+1are set by the temperature Trand field ⃗ gr. Our full\ndynamics consists of alternating periods of Hamiltonian\nevolution of the L+ 2 spins and Monte-Carlo evolution\nof the two boundary spins [73].\nThe dynamics of the bulk spins in the IH-CH model is\ngiven by\nd⃗Sn\ndt=−/parenleftig\n⃗Js\nn−⃗Js\nn−1/parenrightig\n, (2)\nwhere we define local spin current as ⃗Js\nn=−cn(⃗Sn×\n⃗Sn+1) with cn=λ+J/(1 +⃗Sn·⃗Sn+1).Similarly, we can\nwrite an equation for local energy en\nden\ndt=−/parenleftbig\nJe\nn−Je\nn−1/parenrightbig\n, (3)3\nwhere local energy current Je\nn=−cn⃗Sn·⃗Js\nn+1. We con-\nsider the lattice-averaged currents Jsand Jedefined as\nJs=1\nL−2L−2/summationdisplay\nn=1/angbracketleftig\n⃗Js\nn·ˆz/angbracketrightig\n, Je=1\nL−2L−2/summationdisplay\nn=1/angbracketleftig\nJe\nn/angbracketrightig\n,(4)\nwhere the average ⟨·⟩is over time (in NESS) as well as\ndifferent realizations and ˆ zis the unit vector correspond-\ning to the third component. Note that the lower and\nupper limit of the summation in Eq. (4) is chosen so as\nto ensure that ⃗S0and ⃗SN+1do not contribute to the\ncurrents. We expect that spin and energy currents scale\nwith system size as\nJs∝L−µ,Je∝L−η, µ, η ⩾0. (5)\nWe focus on determining the exponents µandηin our\nnumerical simulations (see supplementary material [73]).\nIt is worth pointing out that for some models, the the-\noretically best understood cases are ballistic, KPZ, and\ndiffusive for which the exponent equals 0 ,2/3, and 1 re-\nspectively.\nModel Case J λ T µ η\nCHI 0 10.2 0.61 0.63\nII 0 1 1 0.91 0.89\nIH/ILLL 1 0 1 0.75−0.02\nIH-CHI 10.11 0.72 0.26\nII 10.51 0.64 0.72\nIII 11.51 0.58 0.70\nIV 0.21 1 0.65 1.10\nTABLE I: The models and corresponding values of\nthe parameters Jandλconsidered in this study. The\ncorresponding exponents µandηfor spin and energy\ntransport respectively from our simulations are listed. T\nhere is average temperature of the two reservoirs, i.e.\nT= (Tl+Tr)/2. The column “Case” labels the parame-\nter set.\nWe consider seven cases (see Table I), i.e. points in the\n(J, λ, T ) space, for our IH-CH model to investigate spin\nand energy transport. One can distinguish two broad\ncategories for these seven cases on the basis of the pa-\nrameters Jandλ: (a) when J= 1, λ= 0 or J= 0, λ= 1\nwhich correspond to purely integrable (IH) or purely non-\nintegrable (CH) respectively, and (b) J, λ > 0 which rep-\nresent the mixed case where integrable Hamiltonians are\ncoupled to terms resulting in breaking of integrability,\nbut preserving spin-symmetry. The figures presented in\nthis paper are divided in these two categories as well.\n103\nL100101103Js\n∝L−2/3(a)\nCH I\nCH II\nIH1030.51.0µ\n103\nL100101103Js\n∝L−2/3(b)\nIH-CH I\nIH-CH II\nIH-CH III\nIH-CH IV1030.51.0µ\nFIG. 2: Plots of spin current Js[Eq. (4)] versus sys-\ntem size Lfor (a) the IH and CH models, and (b) the\nIH-CH spin chain. The error bars (computed using the\naverage bond currents) are smaller than the symbols for\nall data-points in both the plots and hence not visible.\nWe average over 48 −50 independent simulations over 4\ndifferent time intervals of length Tav= 105in the NESS.\nThe two insets show the exponent µcomputed from two\nneighbouring data points. The horizontal dashed line in\nthe insets correspond to µ= 2/3 (KPZ superdiffusion).\n103\nL10−210−1100101102102Je\n∝L−1(a)CH I\nCH II\nIH\n10301η\n103\nL10−210−1100101102102Je\n∝L−1(b)IH-CH I\nIH-CH II\nIH-CH III\nIH-CH IV\n10301η\nFIG. 3: Plots of the energy current Je[Eq. (4)] versus\nsystem size Lfor (a) the CH and IH models, and (b) the\nIH-CH spin chain. The error bars (computed using the\naverage bond currents) are smaller than the symbols for\nall data-points in both the plots. We average over 48 −50\nindependent simulations over 4 −5 different time intervals\nof length Tav= 105in the NESS. The two insets show the\nexponent ηcomputed from two neighbouring data points.\nThe horizontal dashed line in the insets correspond to\nη= 1 (conventional diffusion).\nOther parameters [ Tl, Tr,⃗ gl= (0,0, g),⃗ gr= (0,0,−g)]\nthat are used in our simulations are described in detail\nin the Supplementary Material [73]. Below we discuss\nour findings.\nCH model (CH I and CH II). – The nature of spin\ntransport depends on the temperature in the CH model.\nIn particular, we observe KPZ superdiffusion at low tem-\nperatures where the scaling of spin current is Js∼L−µ\nwith µ≈0.61 at T= 0.2 [see Fig. 2(a) and inset]. This\nfinding of KPZ nonequilibrium transport is rooted to the\npossible deep connection between low temperature limit\nof CH model and continuum Landau-Lifshitz or IH model\n[32]. On the other hand, in high temperature regime, we4\nfind close to diffusive behaviour. At T= 1, we obtain an\nexponent µ≈0.91. For the energy current, at low tem-\nperature ( T= 0.2), we find that Je∼L−ηwith η≈0.63.\nHowever, we see a trend that suggests that the exponent\nis likely to increase as Lis increased [see Fig. 3(a) and\ninset]. At high temperature ( T= 1), we find close to con-\nventional diffusion for the energy current with the scaling\nexponent η≈0.89.\nIH model. – Departure from Fourier law are expected\nfor both spin and energy transport in this model because\nof its integrability. We find that the spin current scales\nasJ∼L−µwith µ≈0.72 [see Fig. 2(a) and inset]. This\nKPZ scaling of spin current with system size is indeed re-\nmarkable and consistent with scaling exponents obtained\nfrom equilibrium correlations [23]. On the other hand,\nthe energy transport is observed to be ballistic with the\nenergy current scaling as ∼L−ηwith η≈ −0.02 [see\nFig. 3(a) and inset]. It is worth noting that having spin\ncurrent scaling with system size to be superdiffusive and\nenergy current to be ballistic in the same model is rather\nrare and remarkable.\nHaving discussed the purely integrable ( J= 1, λ= 0)\nand the purely non-integrable ( J= 0, λ= 1) cases so far,\nnext we discuss the mixed cases ( J, λ > 0).\nWeakly perturbative regime (IH-CH I). – First, we set\nλ= 0.1 and J= 1 such that integrability is broken with\na weak perturbation. In this weakly perturbative regime,\nwe observe an exponent µ≈0.72 [see Fig. 2(b) and inset]\nwhich indicates a behaviour close to KPZ superdiffusion.\nOn the other hand, for energy current, we find that the\nexponent is η≈0.26 as shown in Fig. 3(b). This be-\nhaviour is due to finite system size and upon increasing\nthe system size we expect that ηwould tend to 1 (con-\nventional diffusion).\nIntermediate regime (IH-CH II). – We increase the\nstrength of the perturbation by setting λ= 0.5 keeping\nJ= 1. In this intermediate regime, where energy contri-\nbution from the integrable and nonintegrable terms are\ncomparable, we also observe the KPZ behaviour for spin\ntransport with µ≈0.64 [see Fig. 2(b) and inset]. The\ncloseness to KPZ exponent even when the perturbation\nis considerable is remarkable. On the other hand, the\nenergy current scales with system size with an exponent\nη≈0.73 for energy [see Fig. 3(b) and inset]. This again\nis a finite size effect and upon increasing system size we\nexpect ηto increase to 1 (conventional diffusion).\nNonperturbative regime (IH-CH III). – To access the\nnonperturbative regime, we increase the perturbation\nstrength to λ= 1.5. Here also, we find evidence of close\nto KPZ behaviour for spin transport with the exponent\nµ≈0.58 [see Fig. 2(b) and inset]. Just like in the pre-\nvious case, we obtain an exponent η≈0.73 for energy\ncurrent [see Fig. 3(b) and inset]. The exponent is ex-\npected to increase to 1 (conventional diffusion) when the\nsystem size is increased.Other perturbative regime (IH-CH IV). – In addition to\nthe three cases mentioned above (IH-CH I, IH-CH II,\nIH-CH III), we also check a case where the integrable\npart plays the role of perturbation. We achieve this by\nsetting J= 0.2 and λ= 1. We find µ≈0.65 for the\nspin current [see Fig. 2(b) and inset]. This suggests the\nKPZ superdiffusion for spin transport. We also compute\nthe energy currents for different system sizes and find\nan exponent η≈1.10 [see Fig. 3(b) and inset]. This\nindicates diffusive energy transport.\n0.00 0.25 0.50 0.75 1.00\nx−6−4−20246102m(x)(a)\nL= 28\nL= 29\nL= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx0.51.01.52.0103Js(x)(b)L= 28\nL= 29L= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx−6−4−20246102m(x)(c)\nL= 28\nL= 29\nL= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx0.51.01.52.02.53.0103Js(x)(d)L= 28\nL= 29L= 210\nL= 211\nFIG. 4: (a) Plot of average magnetization m(x) =\n⟨Sz\n[xL]⟩for the CH model (CH II) where we have cho-\nsenT= 1. We see that the magnetization profile is\nlinear which is a fingerprint of conventional diffusive be-\nhaviour. The horizontal dashed-dotted lines show ex-\npected values of magnetizations at the boundaries. The\ntop horizontal line shows the expected magnetization at\nthe left boundary ( ⃗S0) while the bottom line shows the\nexpected magnetization at the right boundary ( ⃗SL+1).\n(b) Plot of average bond current Js(x) versus normal-\nized site position xfor the CH model. This plot shows\nthat we have reached NESS. (c) Average magnetization\nm(x) =⟨Sz\n[xL]⟩for the IH model where we have chosen\nT= 1. Nonlinear magnetization profile is a hallmark of\nanomalous transport. As in (a), the horizontal dashed-\ndotted lines show expected values of magnetizations at\nthe boundaries. (d) Average bond current Js(x) versus\nnormalized site position xfor IH model which confirms\nthat we have reached NESS.\nHaving discussed results on nonequilibrium spin and\nenergy transport in detail, we now discuss an important\nquantity namely spatial profiles of average magnetiza-\ntion when the spin chain is placed out of equilibrium. In\nFig. 4, we present results for the average magnetization\nm(x) =⟨Sz\n[xL]⟩where [ xL] equals the integer closest to5\n0 20 40 60 80 100 120\nn0.00.10.20.3Cneq(n)\nCH II,L= 27\nCH II,L= 28\nIH,L= 27\nIH,L= 28\n0.0 0.2 0.4\nn/L10−310−210−1100Cneq\nFIG. 5: Plots of correlations Cneq(n) [Eq. (6)] for the\nCH II and IH spin chains at T= 1 with no field [ g= 0,\ntherefore equilibrium]. Red cross and blue plus denote\nequilibrium data for CH (II) and IH models respectively.\nThe nonequilibrium case with nonzero field ( g= 0.5) is\nshown for two system sizes as indicated in the legend.\nWe set Tl=Tr= 1. We see a stark difference between\nequilibrium ( ⃗ gl=⃗ gr) and nonequilibrium ( ⃗ gl̸=⃗ gr) situa-\ntion. We choose system size L= 256 for the equilibrium\ncase ( g= 0). In the nonequilibrium case, we present two\ndifferent system sizes ( L= 128 and L= 256) to show\nthe finite size effects, that are absent in the equilibrium\ncase. In the inset, we plot Cneqversus n/L which shows\na good data-collapse.\nxLand recall that ⟨·⟩denote average over time (in NESS)\nas well as different realizations. We see nonlinear spatial\nprofiles in the IH case [Fig. 4(c)] which is the hallmark\nof anomalous behaviour [74, 75] and linear profiles in the\nCH case [Fig. 4(a)] which is expected in conventional dif-\nfusive systems. It is to be kept in mind that nonlinear\nprofiles can also occur in conventional diffusion equation\nwith a density dependent diffusion constant but the dif-\nference in the anomalous case is that the nonlinearity\npersists for arbitrarily small temperature (or field) dif-\nference applied at the two ends.\nIn addition to local spatial profiles discussed above, it\nis interesting to investigate spatial correlations in NESS.\nTo do so, we investigate a quantity Cneq(n) given by\nCneq(n) =⟨⃗SL\n2·⃗SL\n2+n⟩ − ⟨⃗SL\n2⟩ · ⟨⃗SL\n2+n⟩. (6)\nIn Fig. 5, we plot this correlation for the IH and CH mod-\nels and demonstrate stark difference between equilibrium\nand nonequilibrium situation. In particular, in the equi-\nlibrium case, we have exponentially decaying correlation\n(short ranged), but in the nonequilibrium case we see\nlong-range correlations. The long-range correlations and\ntheir scaling form (as seen in the inset of Fig. 5) is one\nof the hallmarks of the NESS [76] and has also been ob-\nserved in systems with anomalous transport [69]. Wealso benchmark our results with known equilibrium re-\nsults (see Supplemental Material [73]).\nIn conclusion, we have considered purely integrable\n(IH) and purely non-integrable (CH) spin chains along\nwith perturbed integrable cases while maintaining spin-\nsymmetry (IH-CH). We primarily focus on system size\nscaling of spin and energy currents and this is summa-\nrized in Table I. Notably, we find strong evidence of KPZ\nsuperdiffusion in nonequilibrium spin transport in (i) IH\ncase, (ii) CH case at low temperature, and (iii) IH-CH\ncases. To the best of our knowledge, this is the first\nwork reporting spin transport in classical spin chains. We\nalso demonstrate the existence of nonlinear spatial pro-\nfiles of magnetization which is fingerprint of anomalous\nbehaviour (Fig. 4). We compute spatial correlations and\nremark on the striking difference between the equilibrium\nand nonequilibrium cases (Fig. 5).\nIn future, it is interesting to develop an analytical\nframework presumably based on effective fractional dif-\nfusion equation [75] for these various class of spin chains\nconsidered in our work. Exploring the role of disorder\n[77] and additional spin-symmetry preserving terms [10]\nis interesting both from theoretical and experimental per-\nspective. The presence of anisotropy parameter that vi-\nolate spin-symmetry preservation [4, 31] is expected to\nlead to superdiffusion-diffusion crossover and this is an\ninteresting direction to explore.\nH.S., A.D. and M.K. acknowledge the support from the\nScience and Engineering Research Board (SERB, gov-\nernment of India), under the VAJRA faculty scheme\n(No. VJR/2019/000079). M.K. would like to ac-\nknowledge support from the project 6004-1 of the\nIndo-French Centre for the Promotion of Advanced\nResearch (IFCPAR), SERB Early Career Research\nAward (ECR/2018/002085) and SERB Matrics Grant\n(MTR/2019/001101) from the Science and Engineering\nResearch Board (SERB), Department of Science and\nTechnology (DST), Government of India. A.D. and\nM.K., and acknowledge support of the Department of\nAtomic Energy, Government of India, under Project No.\n19P1112RD. M. K. thanks the hospitality of LPENS\n(Paris), LPTHE (Paris) and LPTMS (Paris-Saclay).\n∗dipankar.roy@icts.res.in\n†abhishek.dhar@icts.res.in\n‡spohn@ma.tum.de\n§manas.kulkarni@icts.res.in\n[1] V. B. Bulchandani, S. Gopalakrishnan, and E. Ilievski,\nSuperdiffusion in spin chains, Journal of Statistical Me-\nchanics: Theory and Experiment 2021 , 084001 (2021).\n[2] S. Gopalakrishnan and R. 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B 80,\n115104 (2009).Supplemental material\nNonequilibrium spin transport in integrable and non-integrable classical spin chains\nDipankar Roy,1,∗Abhishek Dhar,1,†Herbert Spohn,2,‡and Manas Kulkarni1,§\n1International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India\n2Zentrum Mathematik and Physik Department, Technische Universit¨ at M¨ unchen, Garching 85748, Germany\n(Dated: June 14, 2023)\nCONTENTS\nI. Simulation details 1\nII. Equilibrium correlations 3\nA. Analytical expressions for the two-point\ncorrelation function in the the\nIshimori-Haldane model 4\nB. Exact formulas for equilibrium two-point\ncorrelations in the CH model in absence of field4\nIII. Profiles in NESS 5\nReferences 6\nI. SIMULATION DETAILS\nIn this section, we discuss the direct numerical simula-\ntion (DNS) of the nonequilibrium steady state (NESS) for\nthe open spin chain. We recall the schematic diagram in\nFig. S1. It is necessary to choose an appropriate protocol\nfor the dynamics of the boundary spins ⃗S0and⃗SL+1such\nthat these act as effective reservoirs. Our prescription\ninvolves a protocol which is a combination of (i) purely\ndeterministic dynamics of the bulk spins, ⃗S1, . . . , ⃗SLand\n(ii) a mixed dynamics (deterministic and Monte Carlo)\nof the boundary spins, ⃗S0and⃗SL+1. We will discuss the\ndetails below. Recall that the Hamiltonian is given by\nH=L/summationdisplay\nn=0en,\nen=−Jln/parenleftig\n1 +⃗Sn·⃗Sn+1/parenrightig\n−λ⃗Sn·⃗Sn+1(S1)\nwith J, λ⩾0 describing the relative strength of the inte-\ngrable and non-integrable parts.\nInitial configuration preparation. – First, we generate a\nthermal configuration for the bulk spins ( ⃗S1,⃗S2, . . . , ⃗SL)\nat temperature Tusing Monte Carlo (MC) algorithm. In\ndoing so, we consider the entire Hamiltonian in Eq. (S1),\n∗dipankar.roy@icts.res.in\n†abhishek.dhar@icts.res.in\n‡spohn@ma.tum.de\n§manas.kulkarni@icts.res.in\nFIG. S1: (Color online) We recall the schematic dia-\ngram showing the 1D classical spin chain ( ⃗S1,⃗S2, . . . , ⃗SL)\nin contact with two reservoirs represented by ⃗S0and\n⃗SL+1. We introduce two auxiliary fields ⃗ gland⃗ grto en-\nsure that the boundary spins ⃗S0and⃗SL+1are maintained\nat a desired magnetization. The detailed simulation pro-\ncedure is described in Section I.\n0 2 4 6 8 10\nx10−510−410−310−210−1100C(x)\nCH II,g= 0\nCH II analytical\nIH,g= 0\nIH analytical\nFIG. S2: Plots of correlations Cpb(x) [Eq. (S4)] for the\nclassical Heisenberg (CH II) and the Ishimori-Haldane\n(IH) spin chains at T= 1 with no field ( g= 0). The\ndashed blue and the solid red line represent analytical\nresults for the CH II and IH models respectively given in\nEqs. (S18) and (S15) respectively. Here the system size\nused for numerics is L= 32.\nbut update only spins ⃗S1,⃗S2, . . . , ⃗SL. Recall that the\nbulk is defined solely as ⃗S1,⃗S2, . . . , ⃗SL. We now discuss\nthe boundary spins ( ⃗S0and⃗SL+1). To study the energy\ntransport, we fix the temperature of the left ( ⃗S0) and\nright ( ⃗SL+1) boundary spins as Tl=T+ ∆TandTr=arXiv:2306.07864v1  [cond-mat.stat-mech]  13 Jun 20232\n0.00 0.25 0.50 0.75 1.00\nx−4−2024102m(x)(a)\nL= 28\nL= 29\nL= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx0.51.01.52.02.5103Js(x)(b)L= 28\nL= 29L= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx−4−2024102m(x)(c)\nL= 28\nL= 29\nL= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx0.51.01.52.02.5103Js(x)(d)L= 28\nL= 29L= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx−5.0−2.50.02.55.0102m(x)(e)\nL= 28\nL= 29\nL= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx1.01.52.02.53.0103Js(x)(f)L= 28\nL= 29L= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx−6−4−20246102m(x)(g)\nL= 28\nL= 29\nL= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx0.51.01.52.02.5103Js(x)(h)L= 28\nL= 29L= 210\nL= 211\nFIG. S3: Plots of magnetization profiles in (a), (c),\n(e), and (g) for the integrability-broken cases IH-CH I,\nII, III, and IV respectively. The parameters for all these\ncases are provided in Table I. The temperature is fixed\natT= 1. In all these plots, we observe nonlinear-shaped\nprofiles which is a hallmark of anomalous transport. The\nhorizontal dashed-dotted lines show expected values of\nmagnetizations at the boundaries. In other words, the\ntop horizontal line shows the expected magnetization at\nthe left boundary ( ⃗S0) while the bottom line shows the\nexpected magnetization at the right boundary ( ⃗SL+1).\nIn order to benchmark the numerical procedure and to\nensure we have truly reached NESS in the right column\n[(b), (d), (f), and (h)], we have shown the local average\nbond spin current. The flat profiles confirm that we have\nreached NESS.\n0.00 0.25 0.50 0.75 1.00\nx−8.2−8.1−8.0−7.9−7.810e(x)(a)\nL= 28\nL= 29\nL= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx1.01.52.02.53.03.5103Je(x)(b)L= 28\nL= 29L= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx−3.4−3.3−3.2−3.1−3.0−2.910e(x)(c)\nL= 28\nL= 29\nL= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx0.20.40.6103Je(x)(d)L= 28\nL= 29L= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx−2.2−2.1−2.0−1.9−1.8−1.710e(x)(e)\nL= 28\nL= 29\nL= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx1.151.201.25102Je(x)(f)L= 28\nL= 29L= 210\nL= 211FIG. S4: Plots of bond energy profiles in (a), (c) and (e)\nfor the CH I, CH II, and IH models. The parameters for\nall these cases are provided in Table I. The temperature\nisT= 0.2 for the CH I model whereas the temperature\nis fixed at T= 1 for CH II and IH models. For CH I\nand CH II, we observe linear profiles. This indicates dif-\nfusive transport for CH I and CH II. On the other hand\nwe observe flat profile for the IH model which suggests\nballistic transport. As in Fig. S3, the horizontal dashed-\ndotted lines show expected values of bond energy at the\nboundaries. In other words, the top horizontal line shows\nthe expected bond energy at the left boundary while the\nbottom line shows the expected bond energy at the right\nboundary. As before, in order to benchmark the numeri-\ncal procedure and to ensure we have truly reached NESS\nin the right column [(b), (d), and (f)], we have shown\nthe local average bond current. The flat profiles indeed\nconfirm that we have reached NESS.\nT−∆Trespectively. Keeping ⃗S1and⃗SLfixed at the\nvalues obtained by Monte Carlo (MC), we separately do\nan MC simulation to update ⃗S0and⃗SL+1by considering\nterms in Hamiltonian in Eq. (S1) but involving only the\nbond term ⃗S0·⃗S1and⃗SL·⃗SL+1respectively. Until now,\nwe have discussed the preparation of an initial state, i.e.\na configuration ⃗S0,⃗S1, . . . , ⃗SL+1. The next step is the\ntime evolution.\nTime evolution. – Starting from the initial configura-\ntion prepared in the manner discussed above at t= 0,3\n0.00 0.25 0.50 0.75 1.00\nx−2.7−2.6−2.5−2.4−2.310e(x)(a)\nL= 28\nL= 29\nL= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx89101112103Je(x)(b)L= 28\nL= 29L= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx−5.3−5.2−5.1−5.0−4.910e(x)(c)\nL= 28\nL= 29\nL= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx2345678103Je(x)(d)L= 28\nL= 29L= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx−13.5−13.4−13.3−13.2−13.1−13.010e(x)(e)\nL= 28\nL= 29\nL= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx23456103Je(x)(f)L= 28\nL= 29L= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx−4.4−4.3−4.2−4.1−4.0−3.910e(x)(g)\nL= 28\nL= 29\nL= 210\nL= 211\n0.00 0.25 0.50 0.75 1.00\nx0.250.500.751.001.251.50103Je(x)(h)L= 28\nL= 29L= 210\nL= 211\nFIG. S5: Plots of bond energy profiles in (a), (c), (e)\nand (g) for the integrability-broken cases IH-CH I, II, III,\nand IV respectively. The parameters for all these cases\nare provided in Table I. For all these cases, we observe\nlinear profiles. This indicates diffusive transport. As in\nFig. S4, the horizontal dashed-dotted lines show expected\nvalues of bond energy at the boundaries. In other words,\nthe top horizontal line shows the expected bond energy\nat the left boundary while the bottom line shows the\nexpected bond energy at the right boundary. As before,\nin order to benchmark the numerical procedure and to\nensure we have truly reached NESS in the right column\n[(b), (d), (f) and (h)], we have shown the local average\nbond current. The flat profiles indeed confirm that we\nhave reached NESS.we evolve allspins ( ⃗S0,⃗S1, . . . , ⃗SL+1) via determinis-\ntic Hamiltonian dynamics using a fourth order adaptive\nRunge-Kutta method (ARK4) until a time (say denoted\nbyt1) in the simulation becomes just above a chosen\nthreshold time denoted by tup. At t1, single MC step is\nperformed at each of the boundary spins ( ⃗S0and⃗SL+1\nusing only the bond terms ⃗S0·⃗S1and⃗SL·⃗SL+1repec-\ntively). If the steps of either of the spins are success-\nful, we update the boundary spins accordingly. On the\nother hand, however, we do not change the bulk spins.\nWith this updated spin configuration (boundary spins)\natt=t1, we repeat the deterministic dynamics of the\nwhole chain ⃗S0,⃗S1, . . . , ⃗SL+1, and subsequently do the\nMC update of the boundary spins. This entire process is\nrepeated for a long time so that the system reaches the\nNESS. We denote this time as Tss. At this time the bond\ncurrents are independent of bond index (see, for exam-\nple, Fig. S4 to be discussed later) which is an indication\nof successfully reaching NESS.\nIt is important to note that, Runge-Kutta methods\ndo not preserve conserved quantities. In particular, the\nlengths of the spins may differ from unity especially if\nTssis too large (for example, say >5×105). Evolv-\ning the spins furthermore is expected to incur consider-\nable errors especially in quantities, such as the averaged\ncurrents which are generally small ( ≪1) for the chosen\nparameters of interest. To circumvent this issue, we nor-\nmalize all the spins at t=Tss. Even though this is an\napproximation, the system will still be very close to the\ndesired NESS.\nHaving reached NESS, we still need to repeat the above\nprescription in order to compute time-averaged currents.\nWe start by carrying out the dynamics for a short interval\nof time (say Tsh) just to ensure that we are in the NESS.\nWe then run the simulation for a time Tavafter which\nwe record the data. This process is itself performed a\nnumber of times (say 4 −10) to get average value of the\ndata.\nUntil now, for simplicity, we have restricted our discus-\nsions to the case where only the boundary temperatures\nare specified. We now briefly outline the extension of the\nprocedure to the case with specified magnetic fields at the\ntwo ends. In that case we set (i) ∆ T= 0 and (ii) consider\ntwo boundary magnetic fields ⃗ gland⃗ grat the boundary\nspins ⃗S0and⃗SL+1respectively. In order to update the\nboundary spins ⃗S0and⃗SL+1using MC, we consider the\nadditional energy contribution of these fields −⃗ gl·⃗S0and\n−⃗ gr·⃗SL+1respectively. This ensures that any desired\nmagnetization is incorporated at the boundaries.\nII. EQUILIBRIUM CORRELATIONS\nRecall that we computed the correlation function\nCneq(n) defined as\nCneq(n) =⟨⃗SL\n2·⃗SL\n2+n⟩ − ⟨⃗SL\n2⟩ · ⟨⃗SL\n2+n⟩. (S2)4\nIn this section we discuss some analytical forms of equi-\nlibrium correlation function and compare our numerics\nwith it. For this purpose it is useful to define the fol-\nlowing correlation for the open ( Cob) and periodic ( Cpb)\nchains\nCob(n) =⟨⃗SL/2·⃗SL/2+n⟩eq (S3)\nCpb(n) =1\nLL/summationdisplay\nl=1/angbracketleftbig⃗Sl·⃗Sl+n/angbracketrightbig\neq, (S4)\nwhere ⟨·⟩eqis average in equilibrium. In the case of open\nboundaries, we employ the procedure in Section I with\n∆T= 0 and g= 0. This enables us to reach equilibrium\nand we then average over time to compute Cob(n). On\nthe other hand, we compute Cpb(n) by averaging over\nequilibrium configurations of the periodic chain gener-\nated by MC. Next, we discuss some analytical forms\nfor the equilibrium correlation functions in the Ishimori-\nHaldane model (Section II A) and the Classical Heisen-\nberg model (Section II B).\nA. Analytical expressions for the two-point\ncorrelation function in the the Ishimori-Haldane\nmodel\nWe show here that\nCpb(n) =⟨⃗Sa·⃗Sa+n⟩eq=1\n3|n|, n∈Z, (S5)\nand this holds true in the infinite system-size limit L→\n∞. Recall that the Hamiltonian for the IH model with\nLthree-component spins is given by\nH=−L/summationdisplay\nk=1ln/parenleftig\n1 +⃗Sk·⃗Sk+1/parenrightig\n, (S6)\nwhere we assume periodic boundary condition ⃗SL+1=⃗S1\nand set inverse temperature β= 1. To prove (S5), we\nwrite the spins ⃗Skas\nSx\nk= sin θkcosϕk,\nSy\nk= sin θksinϕk,\nSz\nk= cos θk,(S7)\nwhere ϕk∈[0,2π) and θk∈[0, π]. We introduce the\nnotations:\n/integraldisplay\nDS≡/integraldisplay\n···/integraldisplayL/productdisplay\nk=1d⃗Sk, (S8a)\n/integraldisplay\nd⃗Sk≡/integraldisplayπ\n0sinθkdθk/integraldisplay2π\n0dϕk, (S8b)\npkl≡⃗Sk·⃗Sl. (S8c)\nThus we have\npkl= sin θksinθlcos (ϕk−ϕl) + cos θkcosθl.(S9)Note that\n/integraldisplay\nd⃗Slpkl= 0 (S10a)\n/integraldisplay\nd⃗Slp2\nkl=16π2\n3(S10b)\n/integraldisplay\nd⃗Slp3\nkl= 0 (S10c)\n/integraldisplay\nd⃗Slpklplm=4π\n3pkm. (S10d)\nUsing these relations, we find that the partition function\nZLatβ= 1 is given by\nZL=/integraldisplay\nDSL/productdisplay\nk=1/bracketleftbig\n1 +pk(k+1)/bracketrightbig\n= (4π)L+ 3/parenleftbigg4π\n3/parenrightbiggL\n.\n(S11)\nNote that in Eq. (S11), the notation pk(k+1)stands for\n⃗Sk·⃗Sk+1and Eq. (S11) holds for any L. In addition to\nthis normalization factor ZLgiven in Eq. (S11), we also\nneed to evaluate the following integral\nNL(n) =/integraldisplay\nDS pa(a+n)L/productdisplay\nk=1/bracketleftbig\n1 +pk(k+1)/bracketrightbig\n. (S12)\nWhen n= 0, we have NL(0) = ZL. For |n|⩾1, the\nterms which survive on expanding the product are\n/integraldisplay\nDS pa(a+n)a+n/productdisplay\nk=apk(k+1)= (4π)L1\n3|n|,\n/integraldisplay\nDS pa(a+n)L+a/productdisplay\nk=a+npk(k+1)= (4π)L1\n3L−|n|.(S13)\nThus the correlation ⟨⃗Sa·⃗Sa+n⟩eqis given by\n⟨⃗Sa·⃗Sa+n⟩eq=NL(n)\nZL\n=1\n3|n|+1\n3L−|n|\n1 + 3/parenleftbig1\n3/parenrightbigL(S14)\nAssuming |n| ≪Land taking the limit L→ ∞ , we\nfind\n⟨⃗Sa·⃗Sa+n⟩eq=NL(n)\nZL=1\n3|n|. (S15)\nIn Fig. S2, we compare the analytical results provided\nin Eq. (S15) with the direct numerics of the IH model\nand observe that the numerical results agree well with\nthe analytical results.\nB. Exact formulas for equilibrium two-point\ncorrelations in the CH model in absence of field\nIn this subsection, we recall the exact formulas for the\nequilibrium correlations in the CH model [1, 2]. Recall5\nthat the CH model with Lthree-component spins is given\nby\nH=−L/summationdisplay\nk=1⃗Sk·⃗Sk+1 (S16)\nIt turns out that the two-point correlation is\n⟨⃗Sa·⃗Sa+n⟩eq= [L(1)]n(S17)\nwhere the Langevin function L(x) is given by\nL(x) = coth( x)−1\nx. (S18)\nIn Fig. S2, we compare the analytical results provided in\nEq. (S18) with the direct numerics of the CH model and\nfind that the agreement is excellent.\nModel Case J λ T ∆T g\nCHI 0 10.20.025 0.01\nII 0 1 1 0.10.1\nIH/ILLL 1 0 1 0.10.1\nIH-CHI 10.11 0.090.08\nII 10.51 0.060.05\nIII 11.51 0.040.05\nIV 0.21 1 0.070.08\nTABLE I: The values of the parameters T, ∆Tand\ngused in DNS. Recall that the auxiliary fields ⃗ gland\n⃗ grare set in terms of the parameter gas⃗ gl= (0,0, g)\nand⃗ gr= (0,0,−g) as mentioned in the main text. The\nvalues of ∆ Twere chosen to keep the energy difference\nof the left and the right reservoirs approximately at 0 .05.\nThe values of gare chosen such that the absolute value\nof magnetization at the boundaries is close to 0 .06. The\ncolumn “Case” labels the parameter set.\nIII. PROFILES IN NESS\nIn this section, we discuss the the spatial profiles of the\nlocal magnetization and energy. The spatial profiles can\nindicate the nature of nonequilibrium transport. For e.g.,\na linear spatial profile is indicative of conventional diffu-\nsive transport and a nonlinear profile is often a finger-\nprint of anomalous transport. A horizontally flat profileis a sign of ballistic transport behaviour. Below we dis-\ncuss these spatial profiles. For all cases discussed in this\nsection, the values of parameters, such as the bulk tem-\nperature T, boundary temperatures ( T±∆T), and the\nstrength of boundary fields ( g), are provided in Table I.\nFirst, we describe the local magnetization m(x) (de-\nfined in the main text) which we recall to be\nm(x) =⟨Sz\n[xL]⟩, (S19)\nwhere [ xL] equals the integer closest to xLand recall\nthat⟨·⟩denote average over time (in NESS) as well as\ndifferent realizations. In Fig. S3 (a), (c), (e), and (g), we\nshow the magnetization profiles for the four integrability-\nbroken cases. For all cases, the magnetization profile is\nnonlinear. This nonlinearity is a hallmark of anomalous\n(KPZ) transport. In the case of IH-CH III [Fig. S3 (e)]\nwhere J= 1, λ= 1.5, we are in the non-perturbative\nregime and the nonlinear nature is not yet clearly seen.\nThe apparent linear profile for the magnetization profile\nin this case is potentially a finite-size effect. We expect\nto obtain a nonlinear profile when the system size Lis\nhigher. In order to benchmark the numerical procedure\nand to ensure we have truly reached NESS in the right\ncolumn of Fig. S3 [i.e., (b), (d), (f), and (h)], we have\nshown the local average bond spin current. The flat pro-\nfiles confirm that we have reached NESS.\nWe next discuss the spatial energy profiles. The local\nenergy enis defined in the main text which we recall to\nbe\nen=−Jln/parenleftig\n1 +⃗Sn·⃗Sn+1/parenrightig\n−λ⃗Sn·⃗Sn+1 (S20)\nwith J, λ⩾0 describing the relative strength of the in-\ntegrable and non-integrable parts. We show the spatial\nenergy profiles in Fig. S4 for the CH and IH models. The\nenergy profile is linear for the CH model at both low and\nhigh temperatures [see Figs. S4 (a) and (c)]. This linear\nprofiles indicate diffusive behaviour for the CH model.\nIn the IH model, the energy profile is flat for all L[see\nFig. S4 (e)]. This is a signature of ballistic transport. We\nshow the energy profile for the integrability-broken cases\nin Fig. S5 (a), (c), (e), and (g). The profiles are linear.\nHowever, the profiles are observed to converge with the\nsystem size clearly only for the case IH-CH IV [see Fig. S5\n(g)]. This hints at diffusive behaviour for the case IH-CH\nIV. For other cases, we expect that the profiles would\nconverge with system size as we increase system sizes to\nlarger values. As before, in order to benchmark the nu-\nmerical procedure and to ensure we have truly reached\nNESS in the right column of both Fig. S4 and Fig. S5, we\nhave shown the local average bond energy current. The\nflat profiles indeed confirm that we have reached NESS.6\n[1] G. S. Joyce, Classical Heisenberg Model, Phys. Rev. 155,\n478 (1967).\n[2] D. Bagchi and P. K. Mohanty, Thermally driven classi-cal Heisenberg model in one dimension, Phys. Rev. B 86,\n214302 (2012)."    },    {        "title": "2311.10110v2.Control_of_individual_electron_spin_pairs_in_an_electron_spin_bath.pdf",        "content": "Control of individual electron-spin pairs in an electron-spin bath\nH. P. Bartling1,2,∗N. Demetriou1,2,∗N. C. F. Zutt1,2, D. Kwiatkowski1,2, M. J. Degen1,2,\nS. J. H. Loenen1,2, C. E. Bradley1,2, M. Markham3, D. J. Twitchen3, and T. H. Taminiau1,2†\n1QuTech, Delft University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands\n2Kavli Institute of Nanoscience Delft, Delft University of Technology,\nPO Box 5046, 2600 GA Delft, The Netherlands and\n3Element Six, Fermi Avenue, Harwell Oxford, Didcot, Oxfordshire, OX11 0QR, United Kingdom\n(Dated: November 29, 2023)\nThe decoherence of a central electron spin due to the dynamics of a coupled electron-spin bath\nis a core problem in solid-state spin physics. Ensemble experiments have studied the central spin\ncoherence in detail, but such experiments average out the underlying quantum dynamics of the bath.\nHere, we show the coherent back-action of an individual NV center on an electron-spin bath and\nuse it to detect, prepare and control the dynamics of a pair of bath spins. We image the NV-pair\nsystem with sub-nanometer resolution and reveal a long dephasing time ( T∗\n2= 44(9) ms) for a qubit\nencoded in the electron-spin pair. Our experiment reveals the microscopic quantum dynamics that\nunderlie the central spin decoherence and provides new opportunities for controlling and sensing\ninteracting spin systems.\nSolid-state spins provide a versatile platform for quan-\ntum science and technology, as well as for studying the\nfundamentals of spin coherence. A canonical case is the\ncentral spin problem: a single, central spin coupled to\na surrounding bath of interacting spins [1–10]. A com-\nmon approach to protect the central spin from decoher-\nence due to the bath spins is to use echo or decoupling\nsequences. Under such echo sequences, the system un-\ndergoes complex quantum dynamics that depend on the\nmicroscopic bath configuration and on the back-action of\nthe central spin on the interacting spin bath [1–10].\nFor an electron spin in a nuclear-spin bath, the large\nmagnetic moment of the electron spin strongly affects the\nnuclear-spin bath evolution. The resulting back-action\ncreates rich dynamics under echo sequences on the cen-\ntral spin [11–13] and has enabled the control of tens of\nindividual nuclear spins [14, 15], pairs of coupled nuclear\nspins [12, 13, 16, 17], and collective excitations [18, 19]\nin the spin bath.\nFor an electron spin in an electron-spin bath, the ef-\nfect of back-action is more subtle. All couplings are of\nsimilar strength and they are typically weak compared\nto the energy splittings. The resulting central spin deco-\nherence has been investigated in detail in ensemble ex-\nperiments [3, 5, 9, 10, 20], which have been described\nby semi-classical models, as well as by fully quantum\nmodels using approximate numerical methods [3, 5, 8–\n10, 21]. In such ensemble experiments, the underlying\nmicroscopic quantum dynamics are averaged out. Single-\nspin experiments have been performed with NV centers\nin diamond [2, 4, 22–28]. However, the coherence under\necho sequences [29] could be satisfactorily described by\nan effective magnetic field noise (an Ornstein-Uhlenbeck\nprocess) [1, 4, 7, 9]. In this classical model of the spin\nbath, the back-action of the central spin is neglected and\nthe central limit theorem is used to approximate the bath\nas Gaussian, forgoing the microscopic quantum dynam-\nA\nBCD\n14N12C\nNV electron spinP1 electron spin\nX\n(i, mI )\nD1,(i, mI )\nD2,(i, mI )FIG. 1. Schematic of the spin system. We detect and\ncontrol the dynamics of a pair of P1 electron spins in a P1\nbath through the back-action from an NV center. The inset\nshows the lattice structure of a P1 center in diamond with\nthe14N nuclear spin (spin states mI∈ {− 1,0,+1}) and four\nJahn-Teller axes ( i∈ {A, B, C, D }).X(i,mI)is the effective\ncoupling between the P1 electron spins and D(i,mI)are the\neffective couplings with the NV center electron spin.\nics.\nHere, we show that the microscopic flip-flop dynamics\nin an electron-spin bath can be experimentally accessed\nand controlled using echo sequences on a central spin.\nCompared to previous work, we observe a single central\nelectron spin, rather than an ensemble average, and use\ntime-resolved correlations and real-time logic to prepare\nand observe specific configurations of the bath, rather\nthan averaging over all states. We demonstrate strong\nback-action of a central electron spin on the dynamics of\nan individual spin pair in the bath, and use this coherent\ninteraction to detect, image and control the spin pair. We\nshow that the spin pair can be used to encode a control-\nlable qubit with long coherence times ( T∗\n2= 44(9) ms),\ndue to a combination of a decoherence-free subspace and\na clock transition. Our results directly access the micro-\nscopic quantum dynamics that underlie the central spin\ndecoherence and provide new opportunities for control-\nling interacting spin systems.arXiv:2311.10110v2  [quant-ph]  28 Nov 20232\nWe investigate a single nitrogen-vacancy (NV) center\nin diamond surrounded by a bath of P1 centers (nitro-\ngen defects) at a temperature of 3.3 K (Fig. 1). The P1\nconcentration is ∼75 ppb, and the estimated13C con-\ncentration is 0 .01% [24, 30]. The NV electron spin acts as\nthe central spin and is initialized optically and read out\nusing spin-selective optical excitation (637 nm) [16, 24].\nThe P1 centers have multiple internal, dynamic degrees\nof freedom: the electron spin-1/2, four different Jahn-\nTeller (JT) axes and a spin-114N nuclear spin (Fig. 1).\nThe P1 Hamiltonian for JT axis i∈ {A, B, C, D }is [31]\nHP1,i=γeB·J+γnB·I+J·Ai·I+I·Pi·I.(1)\nwhere γe(γn) is the electron (nitrogen) gyromagnetic\nratio and J(I) is the electron spin-1/2 (14N nuclear\nspin-1) operator vector. Ai(Pi) is the hyperfine\n(quadrupole) tensor where the subscript iindicates the\nJahn-Teller axis [24, 32]. We apply a few-degree mis-\naligned magnetic field with respect to the NV axis B=\n[2.43(2) ,1.42(3) ,45.552(3)] G to lift the degeneracy for\nthe different JT axes (Supplementary Sec. VI).\nSince the NV-P1 dipolar coupling can be approximated\nasˆSzˆJz[33], echo sequences on the NV electron spin pri-\nmarily probe the energy-conserving flip-flop dynamics of\nthe P1 bath due to the dipolar P1-P1 couplings. Whether\nflip-flops between two P1 centers are allowed depends on\ntheir electron and14N spin states, on their JT axes, and\non the local magnetic field due to the NV,13C spins,\nand other nearby P1 centers. Therefore, the dynamics\nare complex, depend strongly on the specific microscopic\nconfiguration, and change over time.\nWe probe and prepare specific bath configurations by\nperforming time-resolved experiments through repeated\nNV measurement sequences. This is made possible by\nthe long lifetime of the JT axis and14N spin at cryogenic\ntemperatures and by a low-intensity resonant readout of\nthe NV spin that only weakly perturbs the P1 center\nstates [24]. Previous room-temperature experiments with\nhigh-power off-resonant lasers rapidly average over all P1\nstates [2, 4, 22, 34, 35].\nWe apply dynamical decoupling sequences consisting of\nπ-pulses with variable spacing 2 τ(Fig. 2a), which sense\nthe bath dynamics around a frequency of 1 /(4τ). We\nrepeatedly apply the sequence, bin moutcomes together\nand analyze the signal and correlations over time. Figure\n2b shows a time trace, revealing discrete jumps in the\nNV coherence. A longer-time histogram (Fig. 2c) reveals\nthat the signal is a rare occurrence, which would be easily\nlost in the noise in a time-averaged measurement.\nWe create a map of the bath dynamics by collecting\nhistograms as a function of the interpulse delay τ(Fig.\n2d). The result shows distinct resonances for various val-\nues of τ, which we attribute to two coupled P1 centers\nin the bath switching to different electron-spin, JT and\n14N configurations. For each configuration, the P1 spin\nOutcomes ms = 0\nτ (μs)Occurrence(b)\n(d)\nNormalized\noccurrenceτ (μs)Outcomes ms = 0(e)\nOutcomes ms = 0\nBin number\nOccurrence(a)\nparitym\nτ τ 2τπ π\nreadout\n reset  8my y\n(c)\nFIG. 2. Repetitive dynamical decoupling spectroscopy\nof a P1 center bath. (a) Experimental sequence. (b)Time\ntrace for τ= 14.2μs and bin-size m= 200. (c)Histogram of\na 3-minute-long time trace for τ= 14.2μs and m= 200. We\nrarely observe a high number of ms= 0 occurrences ( ∼1.3%).\nDue to limited observations, the fraction of ∼1.3% is likely\nnot a precise measure of the probability of occurrence. (d)\nRepetitive dynamical decoupling spectroscopy of a P1 bath\nsurrounding an NV center. We apply the sequence shown in\n(a) for m= 200. (e)Simulation of the repetitive dynamical\ndecoupling spectroscopy in (d) for a system of one NV center\nand two P1 centers with the positions as obtained in Fig. 4.\npair flip-flops with a characteristic frequency, which is\nresonant with the sensing sequence for a particular τ.\nTo analyze the results, we consider a single pair of\nP1 centers. For a large magnetic field, the electron-\nand nuclear-spin basis states are proper P1 eigenstates.\nEnergy-conserving electron-spin flip-flops are then al-\nlowed when the two P1 centers have identical JT and14N\nstates. As exploited extensively for nuclear-spin pairs\n[12, 13, 16, 17], the dynamics can then be described by a\npseudo-spin in the anti-parallel spin subspace ( |⇑⟩=|↑↓⟩\nand|⇓⟩=|↓↑⟩).\nThe pseudo-spin Hamiltonian [12, 13, 16, 17], including\nthe effect of the NV center, is:\nH(i,m I)=X(i,m I)ˆSx+msZ(i,m I)ˆSz, (2)\nwhere ˆSx,ˆSzare spin-1/2 operators, X(i,m I)is the effec-\ntive spin-pair coupling, Z(i,m I)is a detuning due to the\ndifferent couplings to the central NV spin, and msis the\nNV spin projection. For large magnetic fields, there are\n12 such Hamiltonians (four JT axes and three14N states),\nall with equal values for XandZ(Supplementary Sec.3\nII). For the field applied here, which is of the order of the\nhyperfine interaction ( γeB∼A∥, A⊥), the electron- and\nnuclear-spin states mix. Therefore, flip-flop interactions\ninvolving the nuclear spin are possible, and X(i,m I)and\nZ(i,m I)depend on the JT and spin states involved. We\nuse the high-field spin labels for simplicity, but take the\nmodified eigenstates and additional flip-flop interactions\ninto account in our analysis.\nNext, we demonstrate the initialization, control and\nmeasurement of the P1-pair state. From the mathe-\nmatical equivalence with previous work [6, 11, 16], it\nfollows that the Hamiltonian in Equation 2 yields an\neffective ˆSNV\nzˆSzinteraction under a resonant dynami-\ncal decoupling sequence with 2 τ=π/ωrwith ωr=/radicalbig\nX2+ (Z/2)2. The NV electron spin thus picks up a\npositive or negative phase depending on the state of the\nP1-pair pseudo-spin [16]. No phase is picked up when\nthe pair is in the parallel subspace ( |↑↑⟩,|↓↓⟩), nor for\nany combination of JT and14N states that do not cause\nflip-flop dynamics at the resonant frequency ωr.\nTo initialize the P1 pair in a particular JT and14N\nstate and in the anti-parallel subspace, we apply repeated\n‘parity’ readouts (Fig. 2a), and put a threshold on the\nobtained counts. We implement real-time logic to speed\nup the initialization procedure: during the 50 parity read-\nouts we keep track of the obtained counts and we restart\nthe procedure if heralding successful preparation becomes\nunlikely (Supplementary Sec. XI). This yields a ∼10x\nspeed-up of the experiments and is essential for enabling\nthe presented measurements.\nTo initialize the spin pair, we apply repeated ‘spin’\nreadouts (Fig. 3a). Subsequent spin measurements are\ntime-matched to account for the evolution of the spin pair\nduring one spin readout, similar to previous experiments\nwith repeated measurements on precessing nuclear spins\n[16, 36] (Supplementary Sec. XII).\nBy choosing a different interpulse delay τ, we can ad-\ndress different JT and14N states. We can thus measure\nthe dependence of the electron-electron couplings ( Xand\nZ) on the JT and14N states by performing Ramsey ex-\nperiments using different values of τfor preparation and\nmeasurement (Fig. 3b).\nTo investigate the spin-pair coherence, we measure the\ndephasing time for τ= 14 .0μs (Fig. 3c). We find\nT∗\n2= 44(9) ms, among the longest reported for solid-\nstate electron-spin qubits [37]. Compared to the single P1\nelectron-spin coherence T∗\n2= 50(3) μs [24], this is a three-\norder-of-magnitude improvement in the same nuclear-\nand electron-spin bath. Two mechanisms contribute to\nthis long dephasing time. First, the anti-parallel spin-\npair states from a decoherence-free subspace: they are\ninsensitive to the partially correlated noise from the P1\nbath [16]. And second, the spin pair forms a clock transi-\ntion due to the P1-P1 coupling [16] (Supplementary Sec.\nX).\nNext, we determine what JT and14N states are as-\nFidelity\nFree evolution time t (ms)τ = 11.2 μs\nτ = 18.6 μsτ = 14.0 μs τ = 16.4 μs\nτ = 29.0 μsspinn\nparity50\nspin5t spin6(a)\n(b)\n(c)τ τ 2τπ π\nreadout\n reset  4nx yFidelity\nFree evolution time t (ms)\nFidelity\nFree evolution time t (s)FIG. 3. Ramsey measurements and coherence of the\nP1 spin pair. (a) Experimental sequence to measure the\npseudo-spin state ( {|↑↓⟩} vs.{|↓↑⟩} ) (Supplementary Sec.\nIII).(b)Ramsey measurements for five interpulse delays τ\nchosen at signal dips in Fig. 2d. We apply 50 parity readouts\nand herald initialization for ≥15 counts. We then apply\n5 spin readouts, which herald initialization in |↑↓⟩(|↓↑⟩) for\n≥1 (= 0) counts. We use 6 spin readouts to measure the final\nspin-pair state, assigning ≥2 (≤1) counts to |↑↓⟩(|↓↑⟩). The\ncontrast is limited by the pseudo-spin dephasing during the\nspin initialization and readout (Supplementary Sec. XII). (c)\nBloch vector length measurement of the spin pair at τ= 14.0\nμs, obtaining T∗\n2= 44(9) ms. The data is not corrected for\nthe readout infidelity.\nsociated to the signals for the different interpulse de-\nlaysτ. Due to the electron-nuclear hyperfine interac-\ntion and misaligned, finite magnetic field, the electron-\nspin transition frequency is different for each JT and\n14N state. After initializing the electron-spin pair in\n1\n2|↑↓⟩⟨↑↓| +1\n2|↓↑⟩⟨↓↑| for an unknown JT and14N state,\nwe apply a radio-frequency (RF) pulse that - when reso-\nnant - can flip the spin pair to the parallel subspace re-\nsulting in a change in signal on the NV center (Fig. 4a).\nThe RF frequencies at which electron-spin flips occur,\ngive information about the JT and14N state associated\nto that τ(Supplementary Sec. IV).\nFigure 4a shows the data for both τ= 11 .2μs and\nτ= 14.0μs. In order to map the obtained frequencies\nto the JT and14N state, we simulate the experiment for\neach JT and14N configuration (Supplementary Sec. IV).\nFrom this, we obtain a set of possible Jahn-Teller axis and\n14N spin state assignments (Supplementary Sec. IV). We\nperform the same analysis for τ= 16.8μs,τ= 18.6μs\nandτ= 29.0μs (Supplementary Sec. V).\nWhile the JT axis can be directly assigned, the14N4\n(b) (c)\nP1-P1 NV-P1 fit error (nm)\nxyz1 nm \nc-c \nbondCounts\nRF frequency (MHz)τ = 11.2 μs\nτ = 14.0 μs\nRF frequency (MHz)Countsτ = 11.2 μs\nτ = 14.0 μsparity50\nparity100RF (a)\nNVP1P1\nxyz\nxz[-20.3(5), -7.7(8), 9.8(2)] nm[6.7(2), -2.5(2), 7.3(2)] nm\nFIG. 4. RF driving and imaging of the P1 electron-\nspin pair. (a) (Top) Experimental sequence. The spin pair\nis initialized in1\n2|↑↓⟩⟨↑↓| +1\n2|↓↑⟩⟨↓↑| , after which an RF\npulse of varying frequency can flip the spin pair to the parallel\nsubspace (Supplementary Sec. IV). (Bottom) Spectra for τ=\n11.2μs (top) and τ= 14.0μs (bottom). A reduction in counts\ngives information about the JT and14N state probed for that\nτ.(b)Fitted positions (up to inversion symmetry) of the two\nP1 centers (blue) with respect to the NV center (purple). (c)\nFit errors for the P1-P1 and NV-P1 positions.\nspin-state assignment is more complicated. Due to the\nelectron-nitrogen spin mixing, flip-flops can occur that in-\nvolve both the electron and nitrogen spin, creating addi-\ntional possible transitions (Supplementary Sec. II). Next,\nwe resolve this ambiguity by determining for which of the\npossible state assignments a consistent spatial structure\nof the system can be found.\nThe spin-pair couplings obtained in Fig. 3b combined\nwith the associated JT and14N states allow us to image\nthe P1 electron-spin pair [38]. We fit the five measured\nP1-P1 couplings to different spatial configurations of the\nP1 pair. Multiple14N spin-state assignments are possi-\nble for three out of five measured couplings (Supplemen-\ntary Sec. V). We resolve this by considering all possible\nassignments in the fit. We benchmark the fitting algo-\nrithm on 10 randomly generated P1 pairs (Supplemen-\ntary Sec. VII), after which we apply it to the measured\ncouplings. The result yields the relative position of the\ntwo P1 centers with an uncertainty close to the diamond\nbond length (Fig. 4b,c), as well as the JT and14N states\nassociated to the τresonances (Supplementary Sec. V)\nWe find the position of the NV center, using a similarfitting algorithm (Supplementary Sec. VII). In Degen et\nal. [24] double-resonance sequences were used to measure\nthe dipolar couplings of the NV center to the two P1\ncenters. We fix the obtained P1-P1 relative position and\nfit to the NV-P1 couplings (up to inversion symmetry).\nThe result is shown in Fig. 4b,c.\nA coherent interaction between the NV and the P1-\npair requires the interaction time (set by 1 /Z) to be small\ncompared to the spin-pair dephasing time ( T∗\n2) and the\nNV electron-spin coherence time under decoupling ( T2).\nA key element in our experiment is that the large num-\nber of different internal P1 states cause energy differences\nthat prevent flip-flop dynamics in the bath. This extends\nthe NV spin coherence and facilitates selective address-\ning of individual spin pairs, at the cost of a lower success\nprobability for finding a given pair in the desired config-\nuration. The full microscopic dynamics including the P1\nand13C nuclear-spin bath are complex. Predicting the\ntypical number of P1 spin pairs that would be observed\nfor randomly drawn NV centers (i.e. bath instances),\nlikely requires detailed numerical simulation, which we\ndo not pursue here.\nIn conclusion, we experimentally demonstrated the\ndetection, imaging and control of an electron-spin pair\nin a spin bath through the back-action from a central\nspin. These results experimentally access the underlying\nmicroscopic quantum dynamics which are central to\ntheoretical methods, such as correlated cluster expansion\n(CCE), that have been widely used to understand time-\nand ensemble-averaged measurements [3, 5, 6, 8, 10, 21].\nThe long dephasing times indicate electron-spin pairs\nbased on P1 centers or other defects [25, 26, 39, 40]\nmight be interesting qubits. While the added complexity\nfrom the P1 internal states limits its use as a qubit, this\ncould be partly overcome by applying a large, aligned\nmagnetic field (Supplementary Sec. II). Lastly, the\npresented methods could contribute to efforts towards\natomic-scale magnetic resonance imaging of complex\nspin samples outside of the diamond by directly detect-\ning and imaging spin pairs [15, 41].\nWe thank V.V. Dobrovitski and M.H. Abobeih for\ndiscussions. This work was supported by the Nether-\nlands Organisation for Scientific Research (NWO/OCW)\nthrough a Vidi grant, as part of the Frontiers of\nNanoscience (NanoFront) programme and through the\nproject QuTech Phase II funding: “Quantum Technol-\nogy for Computing and Communication” (Project No.\n601.QT.001). This project has received funding from\nthe European Research Council (ERC) under the Euro-\npean Union’s Horizon 2020 research and innovation pro-\ngramme (grant agreement No. 852410). This project\n(QIA) has received funding from the European Union’s\nHorizon 2020 research and innovation programme under\ngrant agreement No. 820445. The fitting algorithms were\nperformed on the DelftBlue supercomputer [42].5\n∗These authors contributed equally to this work.\n†T.H.Taminiau@TUDelft.nl\n[1] J. R. Klauder and P. W. Anderson, Spectral diffusion\ndecay in spin resonance experiments, Phys. Rev. 125,\n912 (1962).\n[2] R. Hanson, V. V. Dobrovitski, A. E. Feiguin, O. Gywat,\nand D. D. 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De-\ngen, and J. M. Abendroth, Diamond surface engineering\nfor molecular sensing with nitrogen—vacancy centers, J.\nMater. Chem. C 10, 13533 (2022).\n[42] Delft High Performance Computing Centre (DHPC),\nDelftBlue Supercomputer (Phase 1), https:\n//www.tudelft.nl/dhpc/ark:/44463/DelftBluePhase1\n(2023).1\nSupplemental Material\nCONTENTS\nI. System Hamiltonian 2\nA. NV-P1-P1 Hamiltonian 2\nB. Effective coupling 3\nC. P1 center orientations 3\nII. Effects of P1 electron-nitrogen spin mixing 4\nIII. Spin and parity readout 7\nIV. RF simulations 8\nV. RF Rabi oscillations 12\nVI. Magnetic field fluctuations 14\nVII. Imaging the system 15\nA. Benchmarking 15\nB. Permutations 16\nVIII. Simulation of dynamical decoupling spectrum 18\nIX. Ramsey data with the NV electron spin in ms=−1 18\nX. Dephasing mechanisms 20\nA. Correlated and uncorrelated noise 20\nB. Nuclear-spin bath 20\nC. Electron-spin bath 21\nD. External magnetic field 21\nXI. Optimization of parity initialization 25\nXII. Spin readout calibration 27\nReferences 29arXiv:2311.10110v2  [quant-ph]  28 Nov 20232\nI. SYSTEM HAMILTONIAN\nIn this section, we discuss the Hamiltonian that describes the dynamics of the NV-P1-P1 system. Then, we explain\nhow we calculate the effective coupling between the defects’ electron spins. Lastly, we examine the possible orientations\nof the P1 centers in the diamond lattice.\nA. NV-P1-P1 Hamiltonian\nThe Hamiltonian of the NV-P1-P1 system consists of the individual Hamiltonians of the NV and P1 centers and\nthe Hamiltonians for the dipolar interactions between their respective electron spins [1, 2]. Note that we omit the\nHamiltonian term that describes the NV nitrogen spin since the zero-field splitting suppresses spin mixing between\nthe NV nitrogen and electron spin and the external magnetic field is (nearly) aligned along the NV symmetry axis.\nWe also omit any Hamiltonian terms that describe the dipolar coupling between the14N nuclear spin of one defect and\nthe electron or14N nuclear spin from other defects, due to the large difference in gyromagnetic ratios ( γe/γn≈9000)\n[2].\nH=HNV+2/summationdisplay\nj=1HP1,j+3/summationdisplay\nj<kHDjk (S1)\nwhere\nHNV= ∆ˆS2\nz+γeB·S\nHP1,j=γeB·Jj+γnB·Ij+Jj·Ai·Ij+Ij·Pi·Ij\nHDP1,j−NV=D·(3(Jj·ˆr)(S·ˆr)−Jj·S)\nHDP1,j−P1,k=D·(3(Jj·ˆr)(Jk·ˆr)−Jj·Jk)(S2)\nIj=/bracketleftbigˆIx,ˆIy,ˆIz/bracketrightbig\njSpin-1 operators for the jth P114N nuclear spin\nJj=/bracketleftbigˆJx,ˆJy,ˆJz/bracketrightbig\njSpin-1/2 operators for the jth P1 electron spin\nS=/bracketleftbigˆSx,ˆSy,ˆSz/bracketrightbig\nSpin-1 operators for the NV electron spin\nB=/bracketleftbigBx, By, Bz/bracketrightbig\nB-field vector\nAi=RTAdiagR Hyperfine coupling tensor\nPi=RTPdiagR Quadrupolar coupling tensor\nwith\nD=−µ0γ2\neℏ\n4πr3.\nµ0is the vacuum magnetic permeability, γe≈2.8024 MHz/G and γn≈0.3078 kHz/G are the gyromagnetic ratios of\nthe electron and14N spin respectively. ris the physical separation of the electron spins and ˆ r=r/|r|the unit vector\nbetween them. Rare the rotation matrices of the SO(3) group, with Euler angles α, β, γ [1, 2]:\nR(α, β, γ ) =\ncos(γ) cos( β) cos( α)−sin(γ) sin(α) cos( γ) cos( β) sin(α) + sin( γ) cos( α)−cos(γ) sin(β)\n−sin(γ) cos( β) cos( α)−cos(γ) sin(α)−sin(γ) cos( β) sin(α) + cos( γ) cos( α) sin( γ) sin(β)\nsin(β) sin(α) sin( β) sin(α) cos( β)\n (S3)\nThe rotation matrices Rrotate the hyperfine and quadrupolar tensors depending on the Jahn-Teller axis, i.e Ai=\nR(α, β)TAdiagR(α, β). The Euler angles α, β define the Jahn-Teller principal axis (Fig. S1). Due to the axial\nsymmetry of the P1 center in its principal axis ( Ax=AyandPx=Py), we can set γ= 0 without loss of generality\n[2]. The rotation matrix Rsimplifies to\nR(α, β) =\ncos(β) cos( α) cos( β) sin(α)−sin(β)\n−sin(α) cos( α) 0\nsin(β) cos( α) sin( β) sin(α) cos( β)\n (S4)\nThe hyperfine and quadrupolar tensors are Adiag= diag[114 .03,81.31,81.31] MHz and Pdiag= diag[2 .65,1.32,1.32]\nMHz [3]. Note that the effect of the NV-P1 dipolar coupling can be approximated as pure dephasing of the form\nˆSzˆJzdue to the large NV-P1 energy difference and the large NV and P1 Zeeman energies compared to the NV-P1\ncoupling.3\nB. Effective coupling\nNow, we describe how we calculate the effective couplings between the different defect spins (P1-P1 and NV-P1).\nThese effective couplings provide the parameters X(i,mI)andZ(i,mI)of the pseudo-spin Hamiltonian.\nWe start by constructing the Hamiltonian of the system of interest, i.e HNV-P1 orHP1-P1 . To calculate the effective\ncoupling between the electron spins, we consider the energy differences between their eigenstates. For example, for\nthe NV-P1 system, with the P1 center in a particular Jahn-Teller axis iand its14N nuclear spin in |+⟩(mI= +1),\nthe effective coupling between the electron spins is given by [2]\nD+,i= (λ|−1,+,↓⟩,i−λ|−1,+,↑⟩,i) + (λ|0,+,↑⟩,i−λ|0,+,↓⟩,i), (S5)\nwhere λare the eigenvalues corresponding to the eigenvectors denoted in their subscripts. The indicated eigenstates\ncorrespond to the NV electron spin, the P1 nuclear spin, and the P1 electron spin respectively.\nFor two P1 centers undergoing flip-flop dynamics, the effective coupling is given by\nX=λ1√\n2(|+,↓,+,↑⟩+|+,↑,+,↓⟩)−λ1√\n2(|+,↓,+,↑⟩−|+,↑,+,↓⟩), (S6)\nwhere for this example the nitrogen spin for both P1 centers is in the |+⟩(mI= +1) state. The indicated eigenstates\ncorrespond to the nitrogen and electron spin of the first P1 center respectively, followed by the nitrogen and electron\nspin of the second P1 center.\nC. P1 center orientations\nP1 centers appear in two different orientations in the diamond lattice, as depicted in Fig. S1. In this section, we\nshow that these two different orientations do not result in an observable difference in our experiments. The hyperfine\nand quadrupolar tensors are transformed differently by the two Euler angles αandβdepending on the Jahn-Teller\naxis. In the table in Fig. S1, the first (second) set of α,β(α′,β′) corresponds to the left (right) picture of the P1\ncenter orientations.\nWe now show that the transformation of a diagonal matrix Mi=RT(αi, βi)MdiagR(αi, βi) is equivalent for both\norientations ( αi, βi) and ( α′\ni, β′\ni). Let R=R(α, β) and R′=R(α′, β′). We can then write\nR′MiR′T=R′RTMdiagRR′T\n=MdiagR′RTRR′T\n=Mdiag\n=⇒Mi=R′TMdiagR′(S7)\nHere we made use of the fact that R′RT= diag(1 ,−1,−1) is a diagonal matrix for α′=α+πandβ′=π−β. It\nthus commutes with the diagonal matrix Mdiag. Since the dipole term in the Hamiltonian is also invariant under the\ndifferent Jahn-Teller distortions, the couplings describing the dynamics remain unchanged for the two orientations,\nand we do not expect an observable difference in our measurements.\nA\nBCD\n14N12C\nAB\nC\nD14N12C\nJTα β α′β′\nA0 109.5 180 70.5\nB120 109.5 300 70.5\nC240 109.5 60 70.5\nD0 0 180 180\nFIG. S1. Schematic of the two possible orientations of a P1 center with corresponding rotation angles. On the\nleft, the two possible orientations of a P1 center in the diamond lattice are shown. The Jahn-Teller axes are indicated by\nA, B, C andD. The Euler angles corresponding to these Jahn-Teller axes are given in degrees on the right, where the left\n(right) column corresponds to the left (right) orientation.4\nII. EFFECTS OF P1 ELECTRON-NITROGEN SPIN MIXING\nIn the following, we will denote the single P1 eigenbasis as |− ↓⟩ ,|− ↑⟩ ,|0↓⟩,|0↑⟩,|+↓⟩,|+↑⟩, where the first\nentry refers to the nitrogen spin ( mI=−1,0,+1) and the second entry refers to the electron spin ( ↑or↓). In a\nsimple picture, one would expect to observe flip-flop dynamics of two spins if they are degenerate and have nonzero\ndipolar coupling. For example, for a pair of P1 centers the states |− ↓ − ↑⟩ and|− ↑ − ↓⟩ are degenerate. The\ndipolar coupling between the two electron spins can induce flip-flops between them. Then, the eigenstates become\n1√\n2(|− ↓ − ↑⟩ ± |− ↑ − ↓⟩ ). Consider two other states of the pair of P1 centers: |− ↓0↑⟩and|0↑ − ↓⟩ . These two\nstates are degenerate as well. However, in principle, we do not get any flip-flop dynamics, because the dipolar coupling\nbetween the two electron spins cannot induce nitrogen spin flips.\nThis simple picture holds when the magnetic field is much larger than the hyperfine interaction with the P1 nitrogen\nspin-1 ( γeB≫A∥, A⊥). Due to the finite magnetic field we work at ( γeB∼A∥, A⊥), the single P1 eigenstates are\nnot separable into a nitrogen spin-1 and an electron spin-1/2 part. Instead, the nitrogen and electron spin become\nmixed. The finite magnetic field mostly leads to mixing of {|− ↑⟩ ,|0↓⟩}and{|0↑⟩,|+↓⟩}. This implies that the\ndipolar coupling between the two electron spins can induce flip-flop dynamics between states such as |0↓ − ↑⟩ and\n|− ↑0↓⟩.\nTherefore, we expect to observe signal on the NV center due to two types of flip-flop states. First, there is the simple\ncase in which the nitrogen spin states are approximately equal and fixed, and the electron spins form superpositions\nof|↑↓⟩and|↓↑⟩. Second, the mixing between the nitrogen spin and electron spin of the P1 centers allows for more\ncomplicated flip-flop dynamics.\nIn Fig. S2 we show the simulated dynamical decoupling signal on the NV center per Jahn-Teller state for the\nNV-P1-P1 positions obtained in this work. For each Jahn-Teller state, we calculate the expected NV electron signal\nper P1-P1 eigenvector. Only eigenvectors generating significant signal (fidelity loss of >0.1) are shown. The labels\nare obtained by calculating the overlap of the P1-P1 eigenvector with the spin basis vectors and indicating the largest\noverlap.\nFrom Fig. S2 it can be seen that most NV electron coherence loss due to P1 electron-spin pairs is due to the simple\nflip-flop states: |+↑+↓⟩,|0↑0↓⟩,|− ↑ − ↓⟩ . However, there are more complicated flip-flop states due to the mixing\nbetween the nitrogen spin and electron spin of the P1 centers, amongst which are |0↓ − ↑⟩ and|+↓0↑⟩.\nTo further illustrate the origin of the more complicated flip-flop states, we simulate the same system at a high\nmagnetic field ( B= [1,1,100] G). It is expected that only the simple flip-flop states remain, because in this regime\nit holds that γeB > A ∥, A⊥, resulting in negligible P1 electron-nitrogen spin mixing. The result of the simulation is\nshown in Fig. S3, where we do indeed find that only the simple flip-flop states remain.\nAt the magnetic field used in the experiments in this paper ( B∼45 G), the P1 electron-nitrogen spin mixing\nis not negligible. Hence, we have to consider the possibility of measuring dipole-dipole couplings resulting from\nelectron-nitrogen spin mixing. To address this possibility, we also fit the measurements to these flip-flop states, see\nSupplementary Section VII.\nFor completeness, we simulate the same system at a very high, but misaligned magnetic field of B= [100 ,100,10000]\nG. The result is shown in Fig. S4. We observe that all Jahn-Teller axes and nitrogen-spin states give approximately\nthe same signal. For these values of the magnetic field, the effect of the electron-nitrogen spin mixing is close to\nnegligible and the P1-P1 electron-electron coupling is therefore almost the same for all configurations.\nImportantly, the two P1 centers still need to be in the same Jahn-Teller axis and nitrogen spin state to be degenerate\nand to flip-flop. However, the amplitude of a dip in the dynamical decoupling signal goes up from 1/288 to 1/24.\nThe ratio of 1/288 comes from 4 Jahn-Teller axes for each P1 center, 3 nitrogen spin states for each P1 center and 2\nelectron spin states for each P1 center, two of which ( |↑↓⟩,|↓↑⟩) generate signal. Since all 12 possibilities in Fig. S4\ngive the same signal, the amplitude goes up from 1/288 to 1/24.\nWhen the external magnetic field is very high and aligned, the Jahn-Teller axes A,BandCalso become degener-\nate. This increases the fraction 1 /24 further to 5 /48, since six additional Jahn-Teller configurations exhibit flip-flop\ndynamics at the flip-flop rate.5\nA B\nC D\nτ (μs) τ (μs)Fidelity Fidelity\nFIG. S2. Simulation of the effect of a pair of P1 centers on dynamical decoupling of the NV electron spin\ngrouped per Jahn-Teller state. We simulate the system discussed in this paper: a single NV center coupled to two P1\ncenters. We use the same magnetic field as in the experiments: B= [2.43,1.42,45.552] G. The position vectors for the NV\ncenter and two P1 centers are the ones extracted in the main text. Then, we consider the two P1 centers to be in the same\nJahn-Teller state ( A,B,CorD) and calculate the expected signal on the NV center for each P1-P1 eigenvector and for each\nJahn-Teller state. Each of the four figures corresponds to a different Jahn-Teller state, indicated by the title. We only show\nstates that lead to a significant coherence loss on the NV electron spin: less than a fidelity of 0.9. The legends on the right\nof each graph show which P1-P1 states cause the NV electron coherence loss. We observe coherence loss from both types of\nflip-flop states: when the nitrogen is fixed as well as more complicated flip-flop states that involve nitrogen spin flips. The\nlatter states can give significant signal, comparable to the flip-flop states that do not involve the nitrogen spin.\nA B\nC D\nτ (μs)τ (μs)Fidelity Fidelity\nFIG. S3. Simulation of the effect of a pair of P1 centers on dynamical decoupling of the NV electron spin\ngrouped per Jahn-Teller state at a high magnetic field. We simulate the same system as in Fig. S2 at a magnetic field\nof [Bx, By, Bz] = [1 ,1,100] G. At magnetic fields significantly greater than the P1 electron-nitrogen hyperfine coupling, only\nsimple flip-flop states generate signal on the NV center.6\nA B\nC D\nτ (μs)τ (μs)Fidelity Fidelity\nFIG. S4. Simulation of the effect of a pair of P1 centers on dynamical decoupling of the NV electron spin\ngrouped per Jahn-Teller state at a very high magnetic field. We simulate the same system as in Fig. S2 at a magnetic\nfield of [ Bx, By, Bz] = [100 ,100,10000] G. At this magnetic field, the external magnetic field is significantly greater than the\nelectron-nuclear hyperfine interaction and the nuclear quadrupole interaction. Hence, we observe close to the same signal for\nall Jahn-Teller axes and all nitrogen-spin states.7\nIII. SPIN AND PARITY READOUT\nIn Fig. S5, we show the spin and parity readout together with Bloch spheres indicating the phase picked up by the\nNV center electron spin. When the P1 electron-spin pair is in the parallel state ( {|↑↑⟩ ,|↓↓⟩} ), there are no flip-flop\ndynamics and the NV electron spin does not pick up any phase. However, when the P1 electron-spin pair is in the\nanti-parallel subspace ( {|↑↓⟩ ,|↓↑⟩} ), the NV picks up a positive or negative phase depending on the pseudo-spin state.\nNote that the NV electron spin only picks up phase when the interpulse delay τis resonant with the P1 pair flip-flop\ndynamics (and when the P1 pair is thus in that particular JT and14N configuration).\nFor the spin readout, we tune the number of dynamical decoupling units such that the NV electron spin picks up\na phase of ±π/2 for the two anti-parallel spin-pair states |↑↓⟩and|↓↑⟩. If we then read out along the y-axis, we can\ndistinguish between |↑↓⟩and|↓↑⟩. For the parity readout, we use double the number of dynamical decoupling units\nsuch that the NV electron spin picks up a phase of ±π. If we read out along the x-axis, we can distinguish between\nthe spin pair being in the parallel and anti-parallel subspace. By combining parity and spin readouts (Fig. 3), we can\ninitialize the P1 spin pair in a specific anti-parallel state. In the final readout, we use spin readouts to distinguish\nbetween the two anti-parallel states (Fig. 3).\nx\nyRO axisRO axisx\nyparity spin(b) (a)\nτ τ2τππ\nreadout\n reset  8y y\nτ τ2τππ\nreadout\n reset  4x y\nFIG. S5. Evolution of the NV center during spin and parity readout. (a) Spin readout sequence and the corresponding\nphase pick-up of the NV center for the four different spin-pair states shown on a 2D Bloch sphere. We calibrate the number of\ndynamical decoupling units such that the NV electron spin picks up a ±π/2 phase for |↑↓⟩and|↓↑⟩. Then, we read out along\nthey-axis. (b)Parity readout sequence and the corresponding phase pick-up of the NV center for the four different spin-pair\nstates shown on a 2D Bloch sphere. We calibrate the number of dynamical decoupling units such that the NV electron spin\npicks up a ±πphase for |↑↓⟩and|↓↑⟩. Then, we read out along the x-axis.8\nIV. RF SIMULATIONS\nIn this section, we simulate the application of a radio-frequency (RF) pulse on a single P1 center. To take into\naccount the electron-nitrogen interaction in a single P1 center, we simulate the full time-dependent Hamiltonian under\napplication of a single RF pulse by adding the Hamiltonian term\nHRF= Ω cos(2 πft+ϕ)ˆJx+γn\nγeΩ cos(2 πft+ϕ)ˆIx (S8)\nwhere γe(γn) is the electron (nitrogen) spin gyromagnetic ratio. For the simulations, we set Ω = 250 kHz,\ncomparable to the Rabi frequency in the experiment (Supplementary Sec. V). Then, Ω ≫Xwhich means we can\nneglect the effect of the P1-P1 dipole-dipole coupling under the application of an RF pulse. Thus, it suffices to\nsimulate the application of an RF pulse on a single P1 center, which makes the simulation of the time-dependent\nHamiltonian significantly faster.\nIn Figures S6, S7 the results are shown for each of the four Jahn-Teller axes separately. For each relevant RF\nfrequency, we simulate the evolution of the system for the six different eigenstates the P1 center can be in. For a\nparticular frequency, certain eigenstates will give signal but others do not. Hence, observing a Rabi oscillation gives\ninformation about the Jahn-Teller axis as well as the nitrogen-spin state of the P1 center.\nNext, we convert Figs. S6, S7 to truth tables in order to make it straightforward to compare against experiment. If\nthe Rabi oscillation of a particular eigenstate dips below 0.95, we consider that to be an observable signal and indicate\nit with a “1” in Tables I, II, III, IV. If the Rabi oscillation does not dip below 0.95, we do not consider that to be an\nobservable signal and indicate it with a “0” in Tables I, II, III, IV. The frequencies between the different tables are\ndifferent, since each Jahn-Teller axis has different eigenfrequencies due to the misaligned magnetic field. This makes it\nrelatively straightforward to determine the Jahn-Teller axis. To determine the combination of eigenstates that cause\nflip-flop dynamics, we use a combination of the measurements and the fitting procedure (Supplementary Sec. VII).9\nFidelity\nPulse duration (μs)\nFidelity\nPulse duration (μs)\nFIG. S6. Simulation of Rabi oscillations of a P1 center for various RF frequencies and initial states. Rabi\noscillations are simulated for two different Jahn-Teller axes A(left) and B(right). The frequency at which the RF pulse is\napplied is indicated on top of each plot. For each plot, we simulate the application of the RF pulse for each eigenstate. We\ndenote the eigenstates with −/0/+ indicating the (approximate) nitrogen-spin state and ↑/↓indicating the (approximate)\nelectron-spin state. The signals originating from particular eigenstates can be the same, which makes their Rabi oscillations\noverlap. We clarify this by converting the simulated Rabi oscillations to Tables I, II.10\nFidelity\nPulse duration (μs)\nFidelity\nPulse duration (μs)\nFIG. S7. Simulation of Rabi oscillations of a P1 center for various RF frequencies and initial states. Rabi\noscillations are simulated for two different Jahn-Teller axes C(left) and D(right). The frequency at which the RF pulse is\napplied is indicated on top of each plot. For each plot, we simulate the application of the RF pulse for each eigenstate. We\ndenote the eigenstates with −/0/+ indicating the (approximate) nitrogen-spin state and ↑/↓indicating the (approximate)\nelectron-spin state. The signals originating from particular eigenstates can be the same, which makes their Rabi oscillations\noverlap. We clarify this by converting the simulated Rabi oscillations to Tables III, IV.11\nf(MHz) |+↓⟩|0↓⟩|− ↓⟩|− ↑⟩|0↑⟩|+↑⟩\n27.645/27.715 1 1 0 1 1 0\n238.079 1 0 0 0 0 1\n80.127 0 1 1 0 0 0\n189.902 1 1 0 1 1 0\n189.831 1 1 0 1 1 0\n82.06 0 0 1 1 0 0\n20.532 0 0 0 0 1 1\nTABLE I. Truth table for Jahn-Teller state A. For each frequency, it is indicated per (approximate) P1 eigenstate whether\na Rabi oscillation is observed (“1”) or not (“0”).\nf(MHz) |+↓⟩|0↓⟩|− ↓⟩|− ↑⟩|0↑⟩|+↑⟩\n28.441 1 1 0 0 0 0\n239.035 1 0 0 0 0 1\n80.119 0 1 1 0 0 0\n189.114 0 1 0 0 1 0\n81.106 0 0 1 1 0 0\n27.89 0 0 0 1 1 0\n21.48 0 0 0 0 1 1\nTABLE II. Truth table for Jahn-Teller state B. For each frequency, it is indicated per (approximate) P1 eigenstate whether\na Rabi oscillation is observed (“1”) or not (“0”).\nf(MHz) |+↓⟩|0↓⟩|− ↓⟩|− ↑⟩|0↑⟩|+↑⟩\n29.281 1 1 0 0 0 0\n240.127 1 0 0 0 0 1\n80.128 0 1 1 1 0 0\n79.952 0 1 1 1 0 0\n188.31 0 1 0 0 1 0\n28.23 0 0 0 1 1 0\n22.535 0 0 0 0 1 1\nTABLE III. Truth table for Jahn-Teller state C. For each frequency, it is indicated per (approximate) P1 eigenstate\nwhether a Rabi oscillation is observed (“1”) or not (“0”).\nf(MHz) |+↓⟩|0↓⟩|− ↓⟩|− ↑⟩|0↑⟩|+↑⟩\n257.994 1 0 0 0 0 1\n86.055 0 1 1 0 0 0\n177.2 1 1 0 1 1 0\n177.125 0 1 0 0 1 0\n47.132 0 0 1 1 0 0\n43.938/44.013 1 1 0 1 1 0\n36.856 0 0 0 0 1 1\nTABLE IV. Truth table for Jahn-Teller state D. For each frequency, it is indicated per (approximate) P1 eigenstate\nwhether a Rabi oscillation is observed (“1”) or not (“0”).12\nV. RF RABI OSCILLATIONS\nTo assign Jahn-Teller axes and nitrogen-spin states to signals observed at different values of the interpulse delay τ,\nwe measure Rabi oscillations and compare the frequencies at which signal was observed against Tables I, II, III, IV.\nIn Figure S8 we show the observed Rabi oscillations for each value of τ. The contrast for τ= 18.6μs is relatively\npoor, since there are other resonances close by (Fig. 2).\nDue to mixing of the electron spin and nitrogen spin (Supplementary Sec. II), the combinations of P1 eigenstates\nthat can generate flip-flop dynamics can also include nitrogen spin flips. In particular, |0↓⟩and|− ↑⟩ as well as\n|0↑⟩and|+↓⟩are mixed at our magnetic field and can therefore exhibit flip-flop dynamics. In Table V we show\nthe Jahn-Teller axes and potential flip-flop states for each value of τthat we obtain from the combination of the\nRabi oscillation experiments in Fig. S8 and Tables I, II, III, IV. To evaluate which of the potential flip-flop states\ncorrespond to our observed dynamics, we enter all this information into the fit (Supplementary Sec. VII). Then, we\nobtain the fitted flip-flop states as shown in Table V.\nτ(us) Jahn-Teller axis potential flip-flop states fitted flip-flop states\n11.2 B |+↑⟩,|+↓⟩or|+↓⟩,|0↑⟩ |+↑⟩,|+↓⟩\n14.0 A |0↓⟩,|− ↑⟩ or|− ↑⟩ ,|− ↓⟩ |− ↑⟩ ,|− ↓⟩\n16.4 A |0↑⟩,|0↓⟩ |0↑⟩,|0↓⟩\n18.6 D |0↑⟩,|+↓⟩or|+↑⟩,|+↓⟩ |+↑⟩,|+↓⟩\n29.0 B |0↑⟩,|0↓⟩ |0↑⟩,|0↓⟩\nTABLE V. Potential and fitted flip-flop states for each measured τ.Each row indicates the Jahn-Teller axis, potential\nand fitted flip-flop states for the indicated value of τ. The first index of the ket refers to the nitrogen spin, the second to the\nelectron spin. When for example |+↑⟩,|+↓⟩are the indicated basis states resulting in flip-flop dynamics, the corresponding\nP1-P1 eigenstates are1√\n2(|+↑+↓⟩ ± | +↓+↑⟩).13\nCountsτ = 11.2 μs\nτ = 14.0 μs\nPulse duration (μs)τ = 29.0 μsτ = 16.8 μs\nτ = 18.6 μs82.1 MHz 80.118 MHz239.02 MHz 21.49 MHz\n80.12 MHz 189.18 MHz20.52 MHz 80.11 MHz 189.83 MHz\n44.02 MHz 258.0 MHzCounts Counts Counts Counts\nPulse duration (μs)(a)\n(b)\n(c)\n(d)\n(e)\n(f)parity50\nparity100RF \nFIG. S8. Rabi oscillations for different interpulse delays τ. (a) Experimental sequence. We use 50 parity readouts to\ninitialize the spin pair in the antiparallel subspace by selecting on ≥15/50 counts. The value of the interpulse delay τin the\nparity readout determines which combination of Jahn-Teller and nitrogen-spin state we are initializing. Then, a radio-frequency\n(RF) pulse is applied. If resonant, the spin pair can flip to the parallel subspace and a Rabi oscillation is observed. (b)Results\nforτ= 11.2μs.(c)Results for τ= 14.0μs.(d)Results for τ= 16.8μs.(e)Results for τ= 18.6μs.(f)Results for τ= 29.0\nμs.(b-f) The corresponding RF frequencies are given in the inset of each plot.14\nVI. MAGNETIC FIELD FLUCTUATIONS\nThe external magnetic field (orientation) can change the effective P1-P1 electron-electron dipole interaction. The\nmagnetic field fluctuations result from temperature fluctuations of the permanent magnets and from the presence of\n6-9 T magnetic field systems in nearby laboratories. The effect of these fluctuations on the measured dipole-dipole\ninteraction (Fig. S10) could ultimately limit the accuracy of the fitted P1-P1 position. To that end, we quantify the\nexternal magnetic field fluctuations by monitoring four single P1 frequencies using double electron-electron resonance\n(DEER). See Ref. [2] for more details.\nThe result is shown in Fig. S9. These are all the magnetic field measurements taken during the experimental\nperiod in which the Ramsey measurements in Fig. 3 of the main text were measured. Typical Bx, Byfluctuations\nare on the order of 30 mG and the Bzfluctuation is 3 mG. The relative stability of Bzis explained by the periodic\nrecalibration of the magnetic field using the NV electron ms= 0 to ms=−1 frequency. However, during periods\nof the measurements, larger drifts are observed of about 100 mG peak-to-peak in Bx, Byand 20 mG in Bz. In both\nthese regimes, we quantify the effect of such fluctuations on the measured dipolar coupling (Fig. S10). On average\nwe find fluctuations with a standard deviation of σ∼30 Hz, which amounts to 0 .2% relative to the dipolar coupling.\nBz (G)Occurrence\nBx (G) By (G)Occurrence\nOccurrenceσ = 23 mG σ = 32 mG σ = 3 mG\nFIG. S9. Magnetic field fluctuations during Ramsey experiments. We plot the magnetic fields in the x,yandz\ndirection during the Ramsey measurements in Fig. 3 of the main text. The magnetic fields are Bx= 2.43(2) G, By= 1.42(3)\nG and Bz= 45.552(3) G. The standard deviation of the distribution σis given in the graphs. The components are obtained\nby measuring four single P1 frequencies using DEER. See ref. [2] for more details.\nX (kHz)σ = 30 Hz σ = 94 HzOccurrence\nOccurrence\nX (kHz)\nFIG. S10. Effect of magnetic field fluctuations on P1-P1 electron-electron coupling. We calculate the effective P1-P1\nelectron-electron dipolar coupling with the NV in ms= 0 for various magnetic fields. Specifically, we take the obtained P1-P1\nposition and monitor the dipolar coupling when both P1 centers are in the Jahn-Teller state Aand the nitrogen-spin state\nmI= 0. The magnetic field is B= [2.43(2) ,1.42(3) ,45.552(3)] G. On top of that, we add a random fluctuation on each value\ndrawn from a Gaussian distribution with set standard deviations. (left) The fluctuations on BxandByare 30 mG and the\nfluctuation on Bzis 3 mG, consistent with σin Fig. S9. (right) The fluctuations on BxandByare 100 mG and the fluctuation\nonBzis 20 mG, consistent with the approximate peak-to-peak values in Fig. S9. In this worst-case scenario we obtain an error\non the dipolar coupling of <1%.15\nVII. IMAGING THE SYSTEM\nCombining the P1-P1 couplings obtained in this work with the NV-P1 couplings reported in previous work on this\nsample [2], we aim to resolve the spatial configuration of the NV-P1-P1 system. In our simulations, we construct a\nfunction that, given a magnetic field Band spatial configuration of the P1 centers, returns a set of couplings C:\nf(B,r12,r23) =C (S9)\nwhere\nr12= [r12, θ12, ϕ12] vector from the NV center to the first P1 center\nr23= [r23, θ23, ϕ23] vector from the first P1 center to the second P1 center\nTo resolve the physical position of the three defects, we use the least-squares fitting method scipy.optimize.leastq\nwhich uses the Levenberg-Marquadt algorithm. Using the function in Eq. S9, we provide a set of measured couplings\nC′and request a position that minimizes the residual sum of squares (RSS) between the calculated and measured\ncoupling sets: C′andC. The B-field is a known and therefore fixed parameter. Finally, the errors on the obtained\nvariables are the standard deviations. They are calculated from the covariance matrix of the variables returned from\nthe fitting procedure.\nWe first find the relative position between the two P1 centers, which we label S1andS2. We obtain two possible\nsolutions. These are the two mirrored vectors corresponding to a permutation of the two P1 centers: S1− →S2or\nS2− →S1. This symmetry is expected, due to the symmetry of the dipolar coupling. We then lock the P1-P1 position\nto one of the above vectors and find the position of the NV center with respect to the P1 pair. In total, this allows\nfor two symmetrically inverted solutions. Figure S11 shows a simplified schematic of the two possible solutions.\nA. Benchmarking\nTo quantitatively analyse the performance of our imaging algorithm, we benchmark the fitting method in this\nsection, for both the P1-P1 and NV-P1 pair, on numerically generated data for which the NV-P1-P1 positions\nare known. We generate 10 random positions, calculate the exact couplings between the defects and introduce\nrandom errors on the couplings that reflect our measurement uncertainties. We sample these coupling errors from\na normal distribution with a standard deviation of 0 .2%, reflecting the standard deviation of the magnetic field\nfluctuations during our experiments in Supplementary Section VI. We then apply our imaging algorithm and examine\nNV electron spinP1 electron spin\nNV electron spinB\nz\nx\nFIG. S11. Inversion symmetric solutions of the spatial configuration of the NV-P1-P1 system. We find the relative\nposition of one P1 center to the other. Due to the system’s symmetry there are two possible solutions. For each, there is one\nunique solution for the relative position of both P1 centers to the NV.16\nP1 - P1 rms (nm)\nxyz1 nm \nc-c bond\nNV - P1 pairrms (nm)\nxyz1 nm \nc-c bond\nFIG. S12. Benchmarking of the fitting algorithm for the P1-P1 position and for the position of the NV center.\nFor the P1-P1 imaging benchmarking (top) we generate 10 random positions and calculate the exact couplings between them.\nWe then create 200 different noisy coupling sets and execute our imaging process, with 300 initial guesses for each set. We\naccept the fit result with the lowest RSS value to the exact couplings and calculate the average deviation from the true position,\nin Cartesian coordinates. We repeat the same procedure to image the pair with respect to the NV center (bottom), where the\nP1 centers are explicitly set to their relative position, as obtained from the experimental couplings. We use 400 initial guesses.\nNote that although for the P1-P1 position, we achieve a resolution better than the diamond bond length, for the NV positions\nsome errors exceed the nanometer mark.\nits performance by comparing the generated positions with the positions obtained through our fitting method. Due\nto multiple local minima of the function in Eq. S9, the fitting results are sensitive to the initial guess. To tackle\nthis, we fit each case with randomly generated initial guesses; 300 and 400 for the P1-P1 and NV-P1 pair systems\nrespectively. Finally, we accept the outcome with the lowest RSS as the final position.\nB. Permutations\nAs discussed in Supplementary Section IV, the RF measurements provide insight into the Jahn-Teller axis and\nnitrogen-spin state corresponding to a particular interpulse delay τ. However, as discussed in Supplementary Section\nV, we cannot uniquely identify the Jahn-Teller and nitrogen-spin state of the P1 pair directly from those measurements.\nUsing the fitting algorithm described above, we consider all the possible states of the system, as indicated in Table\nV. Figure S13 shows the RSS values of the least-squares optimization method, for the 8 possible permutations.\nThe assignment with the lowest RSS value corresponds to the states where the nitrogen spin is fixed. For a more\ndetailed discussion on the effect of electron-nitrogen spin mixing on the observed dynamical decoupling spectrum, see\nSupplementary Section II.17\nPermutation indexrms (Hz)\nFIG. S13. Comparison of the RSS value for all possible nitrogen-spin state assignments. We fit the experimental\ncouplings for all the possible nitrogen-spin state combinations allowed by the RF measurements in Supplementary Section\nV. The permutations are indexed according to Table VI. The assignment with the lowest residual is the nitrogen-spin state\nassignment where the nitrogen spin is always the same for both P1 centers.\nPerm. Index τ= 11.2μs (B)τ= 14.0μs (A)τ= 16.4μs (A)τ= 18.6μs (D)τ= 29.0μs (B)\n1 |+↑⟩,|+↓⟩ |− ↑⟩ ,|− ↓⟩ |0↑⟩,|0↓⟩ |+↑⟩,|+↓⟩ |0↑⟩,|0↓⟩\n2 |+↑⟩,|+↓⟩ |− ↑⟩ ,|− ↓⟩ |0↑⟩,|0↓⟩ |0↑⟩,|+↓⟩ |0↑⟩,|0↓⟩\n3 |+↑⟩,|+↓⟩ |0↓⟩,|− ↑⟩ |0↑⟩,|0↓⟩ |+↑⟩,|+↓⟩ |0↑⟩,|0↓⟩\n4 |+↑⟩,|+↓⟩ |0↓⟩,|− ↑⟩ |0↑⟩,|0↓⟩ |0↑⟩,|+↓⟩ |0↑⟩,|0↓⟩\n5 |+↓⟩,|0↑⟩ |− ↑⟩ ,|− ↓⟩ |0↑⟩,|0↓⟩ |+↑⟩,|+↓⟩ |0↑⟩,|0↓⟩\n6 |+↓⟩,|0↑⟩ |− ↑⟩ ,|− ↓⟩ |0↑⟩,|0↓⟩ |0↑⟩,|+↓⟩ |0↑⟩,|0↓⟩\n7 |+↓⟩,|0↑⟩ |0↓⟩,|− ↑⟩ |0↑⟩,|0↓⟩ |+↑⟩,|+↓⟩ |0↑⟩,|0↓⟩\n8 |+↓⟩,|0↑⟩ |0↓⟩,|− ↑⟩ |0↑⟩,|0↓⟩ |0↑⟩,|+↓⟩ |0↑⟩,|0↓⟩\nTABLE VI. The possible nitrogen-spin state assignments for all values of τ.We fit the measured couplings to all\nthe possible combinations allowed by Tables I-IV. The permutation index respects the order of Fig. S13. The first entry of\nthe denoted states refers to the nitrogen spin of the P1 center and the second to the electron spin. Each column indicates\nthe possible flip-flop states for that particular interpulse delay τ. The top of each column also indicates the corresponding\nJahn-Teller axis in brackets.18\nVIII. SIMULATION OF DYNAMICAL DECOUPLING SPECTRUM\nThe NV-P1-P1 positions obtained in the main text allow us to simulate the dynamical decoupling spectrum mea-\nsured in Fig. 2d. The result is shown in Fig. 2e. We observe good qualitative agreement between the simulated\ndynamical decoupling spectrum and the measured spectrum.\nTo simulate the spectrum, we have to consider all possible Jahn-Teller axes and P1-P1 eigenstates. Therefore,\nwe simulate the dynamical decoupling signal of the NV center electron spin for each Jahn-Teller axis and for each\ncorresponding eigenstate of the P1-P1 Hamiltonian. We then convert the simulated fidelity to an observable number\nof counts. When the NV center electron spin is in ms= 0, the probability of measuring a photon count is 70%.\nAnd when the electron spin is in ms=−1, the probability of not measuring a photon count is 99%. Given the\n200 repetitive dynamical decoupling repetitions, we can thus convert the simulated NV electron spin fidelity to the\nexperimental number of counts. Finally, we add Poissonian noise on the photon counts to simulate shot noise.\nGiven the expected signal per Jahn-Teller axis and corresponding eigenstate, we now assume each Jahn-Teller axis\nand eigenstate has equal probability of occurrence. Then, we sum all the individual signals and normalise the result.\nThis procedure results in the simulated dynamical decoupling signal in Fig. 2e.\nIn the simulated spectrum, we observe signal originating from flip-flop states that only involve the electron spin,\nbut we also observe signal from the more complex flip-flop states that involve both the nitrogen and the electron spin.\nThe latter signals mainly occur at τ= 20−25μs (Supplementary Sec. II). In the experimental data (Fig. 2d) we also\nsee clear signal at these values of τ, indicating that we observe flip-flop dynamics involving the two nitrogen spins in\nexperiment.\nIX. RAMSEY DATA WITH THE NV ELECTRON SPIN IN ms=−1\nIn the main text, we perform Ramsey experiments using five values of the interpulse delay τ(Fig. 3). For these\nmeasurements, the NV electron spin is in ms= 0 during free evolution. This is done to isolate the P1-P1 dipolar\ncoupling from the NV-P1 dipolar couplings. In that way, we can fit to the P1-P1 position first before bringing in the\nNV electron spin.\nIn Fig. S14 we show the Ramsey experiments for each of the five values of τwith the NV electron spin in ms=−1.\nNext to the major contribution of the P1-P1 dipolar coupling, we also get a contribution of the NV-P1 dipolar\ncoupling, which adds a detuning to the P1-P1 coupling and thereby alters the measured flip-flop frequency. In Table\nVII we compare the measured evolution frequencies of the Ramsey experiment to the expected values from the NV-\nP1-P1 position found in the main text. Overall we find good agreement, but we do observe deviations of up to a few\nhundred Hz. In particular, the deviations for τ= 16.4μs and τ= 18.6μs with the NV electron spin in ms=−1 are\nlarger than expected. Currently we cannot explain these deviations.\nAn additional observation from Fig. S14 is that the inhomogeneous dephasing times with the NV electron spin in\nms=−1 are about an order of magnitude smaller than those with the NV electron spin in ms= 0. This is expected,\nsince the additional field gradient due to the presence of the NV electron spin detunes the P1 electron-spin pair away\nfrom the anticrossing making it more susceptible to magnetic field noise (i.e. a less effective clock transition).\nτ(us) measured fms=−1(kHz) calculated fms=−1(kHz) measured fms=0(kHz) calculated fms=0(kHz)\n11.2 22.52(1) 22.378 22.152(2) 22.106\n14.0 18.160(7) 18.323 17.943(1) 18.114\n16.4 15.55(9) 15.027 14.87(8) 14.837\n18.6 13.21(6) 13.856 13.7(1) 13.414\n29.0 8.80(4) 8.892 8.2(3) 8.591\nTABLE VII. Measured and calculated frequencies of the P1 electron-spin pair. For each τ, we indicate the measured\nfrequency when the NV electron spin is in ms=−1 as well as the calculated frequency based on the obtained position in the\nmain text. For completeness, we also give the frequencies when the NV electron spin is in ms= 0.19\nFidelity\nFree evolution time t (ms)τ = 11.2 μs\nτ = 18.6 μsτ = 14.0 μs\nτ = 16.4 μs\nτ = 29.0 μsf = 22.52(1) kHz\nf = 18.160(7) kHz\nf = 15.55(9) kHz\nf = 13.21(6) kHz\nf = 8.80(4) kHz\nparitym\nspinn\nparity50\nspin5 t spin6(a)\n(b)\n(c)τ τ 2τπ π\nreadout\n reset  8my y\nτ τ 2τπ π\nreadout\n reset  4mx y\nFIG. S14. Ramsey measurements of the P1 electron spin pair when the NV electron spin is in ms=−1. (a,b)\nPulse sequences to measure the parity ( {|↑↑⟩ ,|↓↓⟩} vs.{|↑↓⟩ ,|↓↑⟩} ) and spin ( {|↑↓⟩} vs.{|↓↑⟩} ) of the electron-spin pair.\n(c)Ramsey measurements for five different interpulse delays τ. Each τcorresponds to an identifiable dip in the dynamical\ndecoupling spectrum of Fig. 2d. The obtained frequency fas well as the interpulse delay τused are indicated on the right.20\nX. DEPHASING MECHANISMS\nIn this section, we discuss the mechanisms involved in the P1-P1 electron-spin pair T∗\n2. The spin-pair dephasing\noriginates from magnetic field fluctuations, either from the permanent magnets used to apply our magnetic field or\nfrom the local spin baths surrounding the spin pair. The electron-spin pairs in our diamond are surrounded by two\ndifferent spin baths:\n•13C spin bath with a concentration of ∼0.01% [4]\n•P1 electron-spin bath with an estimated concentration of ∼75 ppb [2]\nThe combined noise of the external magnetic field and the two local spin baths can have a different effect on the\ntwo P1 centers forming the pair. We will denote the magnetic field noise on the first (second) P1 center electron spin\nasδB1(δB2). These noise terms affect the evolution frequency of the spin pair, which results in dephasing. To solve\nthis generally, we have to consider the full effect of δB1,2on the P1-P1 Hamiltonian, including the time dependence\nofδB1,2. Here, we will assume that δB1,2is quasi-static, which is not necessarily true for the bath of P1 centers.\nFurthermore, we will consider the effects of global, correlated noise and local, uncorrelated noise separately.\nA. Correlated and uncorrelated noise\nThe quantisation axis of the P1 center is generally not along the direction of the external magnetic field due to\nthe strong electron-nuclear hyperfine interaction ( γeB∼A∥, A⊥). Correlated noise, such as fluctuations of the global\nmagnetic field strength, therefore has a non-negligible effect on the effective coupling Xbetween the two electron\nspins. In contrast,13C nuclear spin pairs form a more ideal decoherence-free subspace, since fluctuations of the\nglobal magnetic field strength do not influence the nuclear-nuclear effective coupling at all [5]. On the other hand,\nuncorrelated, local noise from the spin bath affects the detuning of the two P1 center electron spins. In the pseudo-spin\npicture, we can write this uncorrelated noise ∆ Zas [5]:\nH=XˆSx+ [msZ+ ∆Z]ˆSz. (S10)\nTo examine the difference between the effect of correlated (global) noise and uncorrelated (local) noise on a P1-P1\nelectron-spin pair, we simulate the P1-P1 system obtained in the main text for the resonance at τ= 14.0μs. We\nconsider correlated noise to be the exact same noise on both P1 centers, also in magnitude, and uncorrelated noise\nto be completely independent noise on each P1 center. To simulate an example of correlated noise, we vary the\nz-component of the external magnetic field with a standard deviation of σ= 0.3 mG (Fig. S15a). To simulate\nuncorrelated noise, we vary the z-component of the external magnetic field for only one of the two P1 centers (Fig.\nS15b). From the results in Fig. S15, we conclude that the effect of correlated noise on the spin-pair frequency is\nnegligible compared to the effect of uncorrelated noise.\nAdditionally, the two types of noise result in different distributions, which can be understood from the different\norigins of noise. Correlated noise does not affect the detuning Zbetween the two spins, but it does affect the coupling\nX. The noise therefore adds linearly to X, leading to a relatively symmetric distribution. On the other hand,\nuncorrelated noise adds quadratically to the frequency, as is shown in Equation S10. This results in a one-sided\ndistribution, as the effect of negative and positive noise ∆ Zis the same. Figure S15b also highlights that the spin\npair is only second-order sensitive to uncorrelated noise for ms= 0 and therefore forms a clock transition.\nFor similar magnitudes of correlated and uncorrelated noise, the uncorrelated noise dominates in limiting the T∗\n2\nof a spin pair. In other words, it is the difference between δB1andδB2that determines ∆ Z, which is the main\ncontributor to the inhomogeneous dephasing of the spin pair.\nB. Nuclear-spin bath\nTo estimate the typical noise strength from both the nuclear-spin bath and the electron-spin bath, we consider the\nNV electron spin T∗\n2, which is measured to be 94(2) μs [4]. Assuming quasi-static, Gaussian noise this gives a standard\ndeviation of the frequency of σf=1√\n2πT∗\n2= 2.39(5) kHz.\nTo estimate what part of this noise is typically due to the13C bath, we follow Ref. [4] where the T∗\n2of a13C spin\ndue to the13C bath in the same device was measured to be 0 .66(3) s. This gives σf= 0.34(2) Hz. To convert this to\nthe noise on the electron spin, we multiply byγe\nγcwhere γe(γc) is the electron (13C) gyromagnetic ratio. We obtain21\nNoise source Magnitude Type Pseudo-spin effect Expected T∗\n2\nNuclear-spin bath 0.89(4) kHz Mostly uncorrelated X, Z ∼10−40 ms\nP1 spin bath 2.22(5) kHz Uncorrelated\n& correlatedX, Z Uncorrelated: ∼5 ms\nCorrelated: ≫1 s\nExternal B-field [σx, σy, σz] = [23 ,32,3] mG\n[σx, σy, σz] = [3 ,3,3] mGCorrelated X ∼7.5 ms\n∼83 ms\nTABLE VIII. Summary of noise sources and their effects. For each noise source, we summarise their magnitude, type\n(correlated or uncorrelated), expected T∗\n2and whether that noise source mainly affects the coupling between the spins Xor\nwhether it affects both the coupling Xand the detuning between the two spins Z.\nσf= 0.89(4) kHz. As an alternative approach, we simulate 104configurations of13C spins surrounding an electron\nspin with a concentration of 0 .01%. The result is shown in Fig. S16. The average noise an electron spin experiences\ndue to a13C bath is 1.2 kHz, similar to the value of 0 .89(4) kHz obtained from the measurement in Ref. [4]. Note\nthat the exact values for each spin depend strongly on the local environment, and therefore these numbers should\nonly be interpreted as estimates for typical values of the standard deviation of the noise and its distribution.\nImportantly, the13C spins are relatively close to the P1 centers. Therefore, the noise due to the13C spins on both\nP1 centers of the pair is likely uncorrelated (local), although we cannot quantify how uncorrelated it is exactly. Since\nthe effect of uncorrelated noise relative to correlated noise is large, we can follow the analyses of Dobrovitski et al.\n[6] and Bartling et al. [5] to analyse Equation S10. We plot the analytical solution in Fig. S18a for a quasi-static\nbath with a noise magnitude of 0.3 mG and a coupling X= 18.114 kHz. We roughly reproduce the timescales of the\nobserved decay, suggesting that the uncorrelated noise from the13C spin bath plays an important role in limiting the\nP1 pair inhomogeneous dephasing time. Note that we have assumed that all estimated13C bath noise is uncorrelated,\nwhile it is conceivable it is partially correlated. That would increase the inhomogeneous dephasing time in Fig. S18a.\nC. Electron-spin bath\nWe estimate the noise from the electron spin bath on a single electron spin considering the noise on the NV center\n(2.39(5) kHz) and the nuclear-spin bath noise (0 .89(4) kHz): belectron =/radicalbig\nb2\ntotal−b2\nnuclear= 2.22(5) kHz. This noise\nfigure consists of a correlated and an uncorrelated part on the pair of P1 centers. We do not know exactly what part\nof the 2 .22(5) kHz noise is correlated and what part is uncorrelated. In Fig. S17 we show the effect of the P1 bath\nnoise either being fully correlated or fully uncorrelated. When we assume the noise to be fully correlated, we obtain\na modulation of the coupling Xwith a standard deviation of 6 mHz, resulting in T∗\n2values exceeding a second.\nWhen the noise is uncorrelated, we can perform the same analysis as for the nuclear spins. We follow the analyses\nof Dobrovitski et al. [6] and Bartling et al. [5] to analyse Equation S10. This results in the dephasing as shown in\nFig. S18b. We observe a decay time of a few milliseconds. Note that in this analysis we assume that the P1 bath is\nquasi-static, which is likely not a valid assumption for longer times.\nD. External magnetic field\nThe long-term magnetic field fluctuations are σx= 23 mG, σy= 32 mG and σz= 3 mG. In Figure S10 we simulate\nthe effect of external magnetic field fluctuations on the electron-electron coupling. Due to the relative magnitude of\ntheσxandσyfluctuations, a significant standard deviation of σ= 30 Hz on the electron-electron coupling is obtained.\nThis translates to T∗\n2≈7.5 ms, smaller than the observed value of T∗\n2= 44(9) ms.\nFor the Bloch vector measurements in Fig. 3d, we measure ⟨Y⟩and⟨Z⟩for each point after which we obtain the\nBloch vector length as/radicalbig\n⟨Y⟩2+⟨Z⟩2. Slow magnetic field fluctuations over the course of the measurement do not\naffect the Bloch vector measurement as much as a Ramsey measurement. Only the relative phase between ⟨Y⟩and\n⟨Z⟩within a single data point is prone to fluctuations, but the Bloch vector measurement is not sensitive to different\nexternal magnetic fields between data points, contrary to a Ramsey measurement.\nIn our experiment, the external magnetic field fluctuations are typically much slower than the duration of a single\nmeasurement point. If we then assume more conservative fluctuations during the Bloch vector measurement of\nσx=σy=σz= 3 mG, we observe an effect on the coupling Xas shown in Fig. S19. The standard deviation on the\ncoupling Xis 2.7 Hz, which translates to T∗\n2= 83 ms.\nWe conclude that all three noise sources can have a significant effect on the observed inhomogeneous dephasing\ntime of T∗\n2= 44(9) ms. In Table VIII we summarise the effects of the three noise sources.22\nFor the nuclear- and electron-spin baths, the correlated noise is negligible compared to the uncorrelated noise. The\nnuclear-spin bath noise is likely more uncorrelated due to their closeness to the P1 centers. However, it is unclear\nexactly what part of the noise is correlated and what part is uncorrelated. To obtain a more thorough description\nof the spin-bath noise, the complex dynamics of the P1 bath would have to be taken into account using for example\ncorrelated cluster expansion (CCE).\nThe magnetic field fluctuations are larger in magnitude, but only introduce correlated noise. The Bloch vector\nlength measurement is crucial to mitigate the longer-time fluctuations of the external magnetic field.\nf (Hz)Occurrence\nOccurrence\nf (Hz)(a) (b)\nFIG. S15. Simulation of the effect of correlated and uncorrelated noise of the nuclear-spin bath on the two P1\ncenters at τ= 14.0μs.We simulate the P1-P1 system discussed in this paper for Jahn-Teller axis Aand nitrogen-spin state\nmI=−1, which corresponds to the resonance at τ= 14.0μs. Then, we calculate the effective electron-electron coupling for\ntwo different situations. (a)We add noise with a standard deviation of σ= 0.3 mG to the z-direction of the external magnetic\nfield. The value of 0.3 mG corresponds to the estimated noise due to the13C bath. In this simulation, we assume the noise is\ncommon to both P1 centers and therefore correlated. Therefore, its main effect is to modulate the effective electron-electron\ncoupling. (b)We vary the magnetic field in the z-direction of only one of the P1 centers. The standard deviation is also\nσ= 0.3 mG. This simulation emulates the noise from nearby nuclear spins that have a different effect on each of the P1 centers\nforming the pair. The noise is therefore uncorrelated.23\nFIG. S16. Simulated electron-spin noise for varying13C spin configurations. We simulate 104configurations of13C\nspins surrounding an electron spin in a diamond lattice for a concentration of 0.01 %. The standard deviation of the noise\ngenerated by one such13C spin bath is b. The average of the distribution is 1.2 kHz. The right tail is due to more strongly\ncoupled spins (we excluded spins with a coupling larger than 10 kHz).\nf (Hz)Occurrence\nOccurrence\nf (Hz)(a) (b)\nFIG. S17. Simulation of the effect of correlated and uncorrelated noise of the P1 bath on the two P1 centers at\nτ= 14.0μs.We simulate the P1-P1 system discussed in this paper for Jahn-Teller axis Aand nitrogen-spin state mI=−1,\nwhich corresponds to the resonance at τ= 14.0μs. Then, we calculate the effective electron-electron coupling for two different\nsituations. (a)We add noise with a standard deviation of σ= 0.8 mG to the z-direction of the external magnetic field. The\nvalue of 0.8 mG corresponds to the estimated noise due to the P1 bath. This noise is common to both P1 centers and therefore\ncorrelated. Therefore, its main effect is to modulate the effective electron-electron coupling. (b)We vary the magnetic field\nin the z-direction of only one of the P1 centers. The standard deviation is also σ= 0.8 mG. This simulation emulates the\nnoise from nearby electron spins that have a different effect on each of the P1 centers forming the pair. The noise is therefore\nuncorrelated.24\nt (s)Fidelity\n(a) (b)\nFidelity\nt (s)\nFIG. S18. The expected P1 pair inhomogeneous dephasing under the assumption of uncorrelated noise. We plot\nthe analytical solutions to Equation S10 obtained from Refs. [5, 6]. (a)The spin-pair coupling is X= 18.114 kHz and the\nstandard deviation of the noise is 0.3 mG, consistent with the expected noise from the nuclear-spin bath. We find a timescale\ncomparable to the experimentally observed T∗\n2= 44(9) ms. (b)With a noise of 0.8 mG, consistent with the noise from the\nelectron-spin bath under the assumption that all P1 bath noise is uncorrelated and quasi-static.\nf (Hz)Occurrence\nFIG. S19. Simulation of the effect of correlated and uncorrelated noise of the external magnetic field on the two\nP1 centers at τ= 14.0μs.We simulate the P1-P1 system discussed in this paper for Jahn-Teller axis Aand nitrogen spin\nstate mI=−1, which corresponds to the resonance at τ= 14.0μs. Then, we calculate the effective electron-electron coupling\nfor a standard deviation of σ= 3.0 mG on the x-,y- and z-direction of the external magnetic field. This noise is common to\nboth P1 centers and therefore correlated. Therefore, its main effect is to modulate the effective electron-electron coupling.25\nXI. OPTIMIZATION OF PARITY INITIALIZATION\nWe perform measurement-based initialization: measurements are performed to confirm that the system is currently\nin the desired state (JT,14N spin and spin state for both P1 centers). Due to the many possible states the P1\npair can take, the probability to start in the subspace is approximately 1/288, making initialization time-consuming.\nIn this section, we describe how we optimize the initialization procedure for speed and fidelity, by using repeated\nmeasurements and intermediate dynamic decision making combined with the capability to scramble the states. The\nfactor ∼10x speed-up realized is essential for enabling the experiments in the main text.\nTo initialize the P1 electron spin pair, we execute mparity measurements, followed by nspin measurements. The\nparity measurement initializes the two electron spins of the P1 centers into the antiparallel subspace and the desired\n14N and JT configuration. The spin measurement initializes in one of the two antiparallel spin states. If the P1 centers\nare in a particular configuration, such that their coupling is resonant with the dynamical decoupling interpulse delay,\nthe NV electron is projected into the bright state and we detect photons (Fig. 2). To find a robust initalization\nscheme, we implement repetitive parity measurements, at some specific interpulse delay τ, and obtain a time-trace\nfor the pair, as shown in Fig. 2b. By collapsing the time-trace in bins of 200 parity measurements we make two\nobservations (Fig. 2c). We note a well-separated peak between 120 and 150 counts. This allows us to introduce\na threshold check for initialization, which we implement with 50 parity measurements. If we observe more than 15\nphotons in 50 parity measurements, the pair is initialized in the Jahn-Teller and nitrogen spin configuration resonant\nto the used interpulse delay τ, and in the anti-parallel electron spin state.\nTo speed up the initialization time, we introduce intermediate photon count thresholds. We check after a number\nof parity measurements whether we already have observed photon counts, and decide whether we want to abort and\nrestart the initialization. For example, consider a total of 50 parity measurements with a threshold of 15 photon\ncounts. If we perform an intermediate check at the 30th iteration and we have not detected any photons, it is highly\nunlikely to detect 15 photons at the 50th parity measurement. In this scenario, we abort the sequence early. We\nrandomize the Jahn-Teller axes, and nitrogen-spin states of the P1 centers by applying a green laser pulse and restart\nthe initialization procedure [2].\nWe optimize for minimum initialization time by analyzing data sets such as in Fig. 2b with a Monte Carlo sampling\nmethod. By starting at a random point along the time trace, we emulate parity measurements of P1 pairs that are\nin a random Jahn-Teller and nitrogen configuration. To emulate restarting the initialization procedure, we jump to\nanother random point along the trace, simulating the scrambling of the P1 pair Jahn-Teller axis and nitrogen spin\nstate with the green laser.\nWe examine an initialization scheme with a single intermediate threshold check by sampling the data set for varying\nbin sizes Θ = 3 ,5,7,9 and thresholds Λ = [0 ,8]. We find the minimum initialization time by sweeping over the possible\nthresholds Λ for each size Θ. As a figure of merit, we calculate the total time needed to achieve 5000 successful P1\npair initializations. This is defined as both the intermediate and final check of 50 parity outcomes surpassing their\nrespective thresholds. The results are plotted in Fig. S20. We find an optimum set of parameters for a single check\nof Θ = 3 and Λ = 2. In this setting, the average time for each successful P1 pair initialization is 1 .38(4) s. By\nintroducing another intermediate check with bin size Θ = 10 and threshold Λ = 3 the initialization time slightly\nimproves to 1 .37(4) s.\nThe final result provides a factor 10 improvement over the basic initialization without intermediate thresholds,\nessential to make the experiments in the main text feasible. We note that this is a crude optimization and that more\nadvanced methods, like Bayesian inference or techniques based on machine learning, are likely to provide additional\nspeed-ups.26\nFIG. S20. Optimization of the initialization with a single threshold check. We optimize the initialization time by\nsampling at different bin sizes Θ and sweep the allowed thresholds Λ. We find that the optimal combination is Θ = 3 and\nΛ = 2, for which the average initialization time is 1 .38(4)s.\nFIG. S21. Initialization schemes with single and double intermediate threshold checks. For a successful initialization,\nwe check for 15 observed photons out of a total of 50 parity measurements. However, we abort the procedure at one (top) or\ntwo (bottom) intermediate stages, depending on the photon counts thus far. With intermediate checks at 2 out of 3 photons,\nfollowed by 3 out of 10, we achieve an average initialization time of 1 .37(4) s.27\nXII. SPIN READOUT CALIBRATION\nIn this section, we describe the optimization of the spin readout. In particular, we show results for τ= 11.2μs,\nbut note that the procedure and results are similar for other values of τ. The experiments typically consist of a two-\nstep initialization process, containing first parity readouts and then spin readouts, and a one-step readout process,\ncontaining only spin readouts (Fig. 3). The optimization of the parity initialization has been described in detail in\nsection XI. In the following, we will therefore assume that we aim to distinguish between the two anti-parallel states\n|↑↓⟩and|↓↑⟩.\nFor spin initialization, we do not perform any real-time thresholding. Passing the parity readout is a rare event\n(section XI) and it is therefore beneficial to collect all data. In the analysis, we then distinguish between the two\npseudo-spin states by thresholding on the number of counts obtained in the spin readouts ( ≥1/5 or 0 /5).\nDuring a spin readout, the electron-spin pair evolves with frequency X, which is the dipolar coupling between the\ntwo spins of the spin pair. To make sure that each spin readout repetition measures along the same basis, we calibrate\nthe time between subsequent spin readouts (Fig. S22a). If the timing is correct, |↑↓⟩and|↓↑⟩will show a difference\nin obtained counts during the spin readout. The optimal timing is found around 48 .5μs. Note that the spin readout\nwait time has to be calibrated for each τseparately since the frequency Xis different for each τ.\nWe perform the readout optimization as described in the supplementary material of Ref. [5]. We also discuss the\nprocess here for completeness. The two states that we want to optimally distinguish are |↑↓⟩and|↓↑⟩, which we will\nwrite as |a⟩and|b⟩for simplicity. In the initialization step, we use krepetitions and record N(k) counts. We set strict\nthresholds to make the initialization as good as possible: N(k)> N aandN(k)< N bwhere Na(Nb) is the threshold\nto initialize in |a⟩(|b⟩). In the readout step, we use nrepetitions and obtain N(n) counts. To optimally distinguish\nstates |a⟩and|b⟩, we sweep a threshold Tand obtain the combined initialization and readout fidelity as:\nF=F|a⟩+F|b⟩\n2=1\n2P(N(n)≥T|N(k)> N a) +1\n2P(N(n)< T|N(k)< N b). (S11)\nWe then optimize Ffor the number of readouts nand the threshold T. In Fig. S22b, we show a histogram of\nthe obtained counts for the two different spin states |↑↓⟩and|↓↑⟩forn= 6 spin readouts, obtained using strict\ninitialization thresholds of >8/10 and <1/10. In Fig. S22c, we sweep the number of spin readouts nas well as the\nthreshold T. We find an optimal number of readouts n= 6 with a threshold T= 2 obtaining a combined initialization\nand readout fidelity of F= 94.8(6)%. In Fig. S22d, we show a sweep of the threshold Tforn= 6 spin readouts,\ngiving the optimal threshold of 2.\nFor each value of τ, we performed this characterization separately. However, the optimal parameters are very similar\nfor other values of τ. Hence, the number of spin readouts n= 6 and threshold T= 2 was used for all measured values\nofτ.28\n(b) (a)\nparity50\nspin5\nspin20 ≥15/50 >0/5\n<1/5parity50\nspin10\nspin6 ≥15/50 >8/10\n<1/10\nparity50\nspin10\nspinn ≥15/50 >8/10\n<1/10parity50\nspin10\nspin6 ≥15/50 >8/10\n<1/10Spin readout wait time (μs) Counts\nThresholdFidelity\nThresholdFidelity\nNumber of readouts n\nFractionAverage counts\n(c) (d)\nFIG. S22. Optimization of spin readout for τ= 11.2μs. (a) Sweep of the wait time between subsequent spin readouts.\nWe find a maximum contrast between the two spin states around 48 .5μs.(b)Histogram for the two different spin states |↑↓⟩\nand|↓↑⟩forn= 6 spin readouts. (c)Sweep of the number of readouts n, showing the optimal fidelity Fand optimal threshold\nTfor each number of readouts. (d)Sweep of the threshold Tforn= 6 spin readouts. The optimal spin readout parameters\naren= 6 readouts with a threshold of T= 2, giving a combined initialization and readout fidelity of F= 94.8(6)%.29\n[1] W. V. Smith, P. P. Sorokin, I. L. Gelles, and G. J. Lasher, Electron-spin resonance of nitrogen donors in diamond, Phys.\nRev.115, 1546 (1959).\n[2] M. J. Degen, S. J. H. Loenen, H. P. Bartling, C. E. Bradley, A. L. Meinsma, M. Markham, D. J. Twitchen, and T. H.\nTaminiau, Entanglement of dark electron-nuclear spin defects in diamond, Nature Communications 12, 3470 (2021).\n[3] H. S. Knowles, D. M. Kara, and M. Atat¨ ure, Observing bulk diamond spin coherence in high-purity nanodiamonds, Nature\nMaterials 13, 21 (2014).\n[4] C. E. Bradley, S. W. de Bone, P. F. W. M¨ oller, S. Baier, M. J. Degen, S. J. H. Loenen, H. P. Bartling, M. Markham,\nD. J. Twitchen, R. Hanson, D. Elkouss, and T. H. Taminiau, Robust quantum-network memory based on spin qubits in\nisotopically engineered diamond, npj Quantum Information 8, 122 (2022).\n[5] H. P. Bartling, M. H. Abobeih, B. Pingault, M. J. Degen, S. J. H. Loenen, C. E. Bradley, J. Randall, M. Markham, D. J.\nTwitchen, and T. H. Taminiau, Entanglement of spin-pair qubits with intrinsic dephasing times exceeding a minute, Phys.\nRev. X 12, 011048 (2022).\n[6] V. V. Dobrovitski, A. E. Feiguin, R. Hanson, and D. D. Awschalom, Decay of Rabi oscillations by dipolar-coupled dynamical\nspin environments, Phys. Rev. Lett. 102, 237601 (2009)."    },    {        "title": "2101.08868v1.Effects_of_the_dynamical_magnetization_state_on_spin_transfer.pdf",        "content": "E\u000bects of the dynamical magnetization state on spin transfer\nNeil Tramsen,\u0003Alexander Mitrofanov,\u0003and Sergei Urazhdin\nDepartment of Physics, Emory University, Atlanta, GA, USA\nWe utilize simulations of electron scattering by a chain of dynamical quantum spins, to analyze\nthe interplay between the spin transfer e\u000bect and the magnetization dynamics. We show that the\ncomplex interactions between the spin-polarized electrons and the dynamical states of the local spins\ncan be decomposed into separate processes involving electron re\rection and transmission, as well\nas absorption and emission of magnons - the quanta of magnetization dynamics. Analysis shows\nthat these processes are substantially constrained by the energy and momentum conversation laws,\nresulting in a signi\fcant dependence of spin transfer on the electron's energy and the dynamical state\nof the local spins. Our results suggest that exquisite control of spin transfer e\u000eciency and of the\nresulting dynamical magnetization states may be achievable by tailoring the spectral characteristics\nof the conduction electrons and of the magnetic systems.\nI. I. INTRODUCTION\nSpin transfer e\u000bect (ST) - the transfer of spin angu-\nlar momentum from spin-polarized conduction electrons\nto magnetic systems [1{4] - is one of the most exten-\nsively studied e\u000bects in modern nanomagnetism, thanks\nto the unique fundamental insights it provides into elec-\ntron spin physics, a plethora of related magnetoelectronic\nand dynamical e\u000bects, as well as viable applications in\ninformation technology [5{9]. ST can result in magne-\ntization reversal [10, 11], precession [12{14] and other\ndynamical e\u000bects [15{17]. Magnetic switching driven by\nST is \fnding applications in memories and biomimet-\nics, while its ability to generate magnetization dynamics\nprovides unique opportunities for magnonics - the infor-\nmation and telecommunication technology utilizing the\nquanta of magnetization dynamics (magnons) as infor-\nmation carriers [18, 19].\nST is a consequence of spin angular momentum con-\nservation in the process of scattering of spin-polarized\nconduction electrons by magnetic systems. This pro-\ncess has been extensively analyzed in the semiclassical\napproximation for the magnetic systems, in which the\nmagnetic order is approximated as a continuous classical\nvector \feld with \fxed magnitude, or as an array of lo-\ncalized classical magnetic moments. The latter approx-\nimation is commonly utilized in micromagnetic simula-\ntions. The magnetization dynamics of ferromagnets is\nusually described by the semiclassical Landau-Lifshitz-\nGilbert (LLG) equation with an additional Slonczewski's\nterm arising from ST [1]. The possibility to include ST in\nthe LLG equation, instead of jointly solving the dynam-\nical equations for the conduction electrons and the mag-\nnetization coupled by the exchange interaction, requires\nan adiabatic approximation, i.e., it is assumed that the\nrelevant magnetization dynamics are signi\fcantly slower\nthan the spin dynamics of conduction electrons involved\nin ST.\nWe now discuss recent developments in the studies of\n\u0003N.T. and A.M. contributed equally to this work.ST, relevant to the present work, that transcend these\napproximations. In the semiclassical approximation, the\nmagnitude of magnetization in ferromagnets is \fxed, so\nST is forbidden by angular momentum conservation if\nthe electron is spin-polarized collinearly with the mag-\nnetic order. However, recent experimental measure-\nments [20, 21] and theoretical studies [22{26] revealed\na contribution to ST, termed the quantum ST, which\npersists in the collinear geometry. In ferromagnets, this\ne\u000bect becomes noticeable only at cryogenic temperatures.\nHowever, it may be dominant in antiferromagnets, and\nis the only expected contribution to ST in spin liquids,\nwhose magnetic state cannot be described semiclassi-\ncally [26, 27].\nTheoretical studies have shown that ST leads to quan-\ntum entanglement between conduction electrons and\nmagnetization [23, 25], and can also mediate entangle-\nment within the magnetic system [26], with possible ap-\nplications in quantum information technologies [28, 29].\nFurthermore, it was shown that the conservation of the\ntotal energy and linear momentum of the quasiparticles\ninvolved in ST, the electrons and the magnons, can im-\npose substantial constraints on the magnetization dy-\nnamics induced by ST, and on scattering of electrons\nby magnetic systems [30]. While the roles of energy and\nlinear momentum in ST were analyzed already in the\nsemiclassical models [31{33], the constraints imposed by\nthe magnon energy and momentum, in the de Broglie\nsense, were not captured by these models.\nIn this work, we utilize quantum simulations of elec-\ntron scattering by a quantum spin chain initially popu-\nlated with one magnon, to analyze the e\u000bects of magne-\ntization dynamics on ST. In principle, such e\u000bects can\nbe introduced already in the semiclassical approach, by\njointly solving the coupled dynamical equations for the\nconduction electrons and the magnetization [34]. How-\never, our quantum simulations show that the conserva-\ntion of the total energy and linear momentum (in the de\nBroglie sense) of the quasiparticles involved in ST, the\nelectrons and the magnons, plays a central role in the\nstudied phenomena. Thus, our results provide further\nevidence for the signi\fcance of non-classical aspects of\nST, and suggest a new route for controlling its e\u000eciency.arXiv:2101.08868v1  [cond-mat.mtrl-sci]  21 Jan 20212\nThe rest of this paper is organized as follows. In Sec-\ntion II, we describe the model and provide the compu-\ntational details. In Section III, we classify and analyze\ndi\u000berent scattering processes involved in ST, for a spin\nchain populated with one magnon. In Section IV, we con-\nsider ST for the electron spin-polarized orthogonally to\nthe equilibrium magnetization, and show that this case,\ncommon in the ST studies, includes all the contributions\nanalyzed in Section III. We summarize our observations\nin Section V.\nII. II. MODEL AND SIMULATION DETAILS\nWe consider an electron initially propagating in a non-\nmagnetic medium, and subsequently scattered by a fer-\nromagnet (FM) modeled as a chain of n= 10 localized\nspins-1/2. In the tight-binding approximation, this sys-\ntem can be described by the Hamiltonian [23, 30]\n^H=\u0000X\nibjiihi+ 1j\n\u0000X\nj(J^Sj\u0001^Sj+1+Jsdjjihjj\n^Sj\u0001^s);(1)\nwhere the \frst term on the right describes hopping of the\nitinerant electron, the second - its exchange interaction\nwith the local spins, and the last term - the exchange\nbetween the local spins. Indices i= 1:::180 in Eq. (1)\nenumerate the tight-binding sites of the entire considered\nsystem, including the FM and the non-magnetic medium\nsurrounding it, while indices j= 70\u000080 enumerate the\nsites occupied by the localized spin-1/2 chain represent-\ning the FM. ^Sj,^sare the spin operators of the electron\nand of the local spins, bis the electron hopping parame-\nter,Jis the exchange sti\u000bness of the local spins, and Jsd\nis their exchange interaction with the electron. The cal-\nculations below use b= 1 eV,J=Jsd= 0:1 eV, unless\nspeci\fed otherwise.\nWe use periodic boundary conditions for both the elec-\ntron and the spin chain, to avoid artifacts associated with\nre\rections at the boundaries. Spin-orbit interactions are\nneglected in our model. Nevertheless, we expect our re-\nsults to be broadly relevant to spin-orbit torques, thanks\nto the general validity of conservation laws governing\nelectron-magnon scattering. For typical experimental\nmagnetic \felds, the e\u000bects of the Zeeman interaction are\nnegligible on the considered time scales, aside from de\fn-\ning the quantization axis for the local spin dynamics. In\nthe following, we assume that the local spins are aligned\nwith the z-axis in their ground state, with hSzi= 5\u0016h.\nThe same value aof the lattice constant, which de\fnes\nthe possible values of quasiparticle wavevectors, is used\nthroughout the entire system.\nTo analyze ST, the system is initialized with the elec-\ntron forming a Gaussian wavepacket centered around the\nwavevector k(i)\neand polarized in the + z,\u0000z, orxdirec-\ntion. The local spins are populated with one magnonwith wavevector k(i)\nm. In this state, hSzi= 4\u0016h. The\nsystem is then evolved using the Schrodinger equation\nwith the Hamiltonian Eq. (1). We choose the width of\nthe electron wavepacket so that it remains well-de\fned\nthroughout the scattering process, allowing us to clearly\nidentify the time intervals corresponding to its propaga-\ntion in the non-magnetic medium before and after ST,\nand enabling us to unambiguously determine the associ-\nated changes of physical quantities.\nTo analyze the evolution of the two subsystems, we in-\ntroduce the density matrices ^ \u001ae= Tr m^\u001aand ^\u001am= Tr e^\u001a\nfor the electron and the local spins, respectively, by trac-\ning out the full density matrix ^ \u001awith respect to the other\nsubsystem [23]. The expectation value of a physical quan-\ntity ^Aassociated with the electron isD\n^AE\n= Tr( ^A^\u001ae),\nwhile the probability of its value aisPa=h aj^\u001aej ai,\nwhere ais the corresponding eigenstate. Similar re-\nlations are used to analyze the observables associated\nwith the local spins. For instance, the expectation val-\nues of di\u000berent contributions to the system's energy are\nobtained by using the corresponding terms in the Hamil-\ntonian Eq. (1) as ^A. The distribution of electron mo-\nmentumpeis obtained by projecting onto the plane-\nwave eigenstates j ki\u0018eikex. Here,ke=pe=\u0016his the\nwavevector describing the corresponding Fourier compo-\nnent of the electron wave. For brevity, we interchange-\nably use the terms \"wavevector\" (or \"wavenumber\") and\n\"momentum\" (in the de Broglie sense), since the two\nquantities are simply related by the Planck's constant\n\u0016h. For magnons, we utilize the Bethe ansatz to classify\nthe eigenstates, as described below, and project ^ \u001amonto\nthese states to determine the distribution of magnon pop-\nulations and their energies/momenta.\nIII. III. CLASSIFICATION AND ANALYSIS OF\nTHE SCATTERING PROCESSES INVOLVED IN\nST\nIn this Section, we identify and characterize several\nscattering processes involved in ST, for a spin chain ini-\ntially populated with one magnon, and demonstrate that\nthese processes lead to distinct outcomes. For clarity,\nFig. 1 illustrates these processes separately for the spin-\nup and spin-down polarization (i.e., polarization in the\n+zand\u0000zdirections) of the incident electron. In the\nnext Section, we show that scattering of the incident\nelectron polarized in the x-direction can be interpreted\nas a superposition of all these processes, demonstrating\ntheir relevance for the generic ST geometries involving\nspin currents non-collinear with the magnetization.\nThe incident electron can be either transmitted or re-\n\rected, and its polarization can either change or remain\nthe same. Spin-up is parallel to the local spins in their\nground state. Since each magnon carries spin 1, angular\nmomentum conservation requires that electron scattering\neither results in the absorption of the initially present\nmagnon (panel a), or one magnon remains in the system3\n(a)Before scattering After scattering\n(b)\n(c)0 magnons\n2 magnons1 magnon\n1 magnon\n1 magnon\n1 magnon\nFigure 1. (Color online) Processes involved in ST, for the\nlocal spins populated with one magnon. (a) For a spin-up\nincident electron, the magnon can be absorbed, \ripping elec-\ntron spin to down. (b) For a spin-down incident electron, an\nadditional magnon can be emitted, \ripping electron spin to\nup. (c) The electron can be scattered without spin-\ripping\nor changing the number of magnons in the system, but nev-\nertheless exchange energy and momentum with the existing\nmagnon. In all cases, the electron can be either re\rected or\ntransmitted.\n(panel c). In the former case, the electron must spin-\rip\nto spin-down, while in the latter its spin must remain\nthe same. We note that even if the magnon population\ndoes not change, energy and linear momentum can be ex-\nchanged between the electron and the magnon as a result\nof scattering.\nBy the same spin angular momentum conservation ar-\ngument for the incident spin-down electron, ST can result\nin the generation of a second magnon accompanied by\nelectron spin-\ripping into the spin-up state [Fig. 1(b)].\nThe electron can be also scattered without ST, but\nnevertheless exchange energy and momentum with the\nmagnon, similarly to the spin-up electron.\nWe con\frmed the scenarios identi\fed in Fig. 1 by pro-\njecting the results of the simulations onto the eigenstates\nof the system, as described in Section II. The dependen-\ncies of the probabilities of di\u000berent scattering outcomes\non the initial magnon wavevector are shown for spin-up\nand spin-down incident electrons in Figs. 2 (a) and (b),\nrespectively.\nWe note several important features of scattering. First,\ndi\u000berent scattering processes contribute di\u000berently to\nST. For instance, absorption by a spin-up electron of a\nmagnon with wavenumber k(i)\nm= 0 is accompanied by\nelectron transmission (labeled \\t. m. abs.\"). Second,\nthe probabilities of di\u000berent processes are strongly de-\npendent on the magnitude of the magnon wavevector.\nFor instance, for the spin-up polarization of the incident\nelectron, the probability of electron transmission with-\nout ST (labeled \\t. no ST\" in Fig. 2(a)) is close to 1\nr. no STr . m. em.r\n. no STr . m. abs.t. m. abs.t. no ST (b)( a)0\nπ - πPk\nma10\n.50\nt. m. em.t. no ST 0\n.50\n1Pk\nma-π0 π Figure 2. (Color online) Probabilities of di\u000berent scattering\noutcomes, as classi\fed in Fig. 1, for spin-up (a) and spin-down\n(b) incident electron vs k(i)\nma, fork(i)\nea= 0:6. In the abbre-\nviated labels, electron transmission and re\rection is denoted\nas \\t.\" and \\r.\", respectively, while \\m. abs.\", \\m. em.\",\nand \\no ST\" denote the cases of magnon absorption, magnon\nemission, and the absence of ST, respectively.\nfor most values of km>0, except for a pronounced dip\naroundkm= 0. Meanwhile, the probability of re\rection\nwithout ST, labeled \\r. no ST\", remains negligible at all\nk(i)\nm. As a consequence, the total probability of scatter-\ning without ST, and conversely the magnitude of ST, is\ndependent on k(i)\nm. A clear asymmetry of these results\nwith respect to the sign of k(i)\nmindicates that both the\nmagnitude and the direction of the magnon momentum\nplay important roles in the scattering processes.\nThe dominance of transmission without ST among the\nscattering processes for spin-up electrons can be quali-\ntatively understood as a consequence of weak e\u000bects of\nthes-dinteraction between the conduction electron's spin\nand the local spins that are almost parallel to each other.\nThese e\u000bects are signi\fcantly stronger for the spin-down\nincident electron. In particular, while electron transmis-\nsion without ST is still dominant, the probability of ST\naccompanied by both electron transmission and re\rection\nis signi\fcantly larger [Fig. 2(b)]. We also note that the\ndependencies on k(i)\nmfor the spin-down electron are sub-\nstantially di\u000berent from those for the spin-up electron,\nsuggesting that ST involves a complex interplay between\nthe spin and the orbital degrees of freedom of the mag-\nnetic system and of the electron.\nIII.1. A. Classi\fcation of the dynamical states of\nthe magnetic system and of the electron\nThe results of Fig. 2 demonstrate that electron scatter-\ning and ST are strongly a\u000bected by the dynamical state\nof the magnetic system. We now quantitatively analyze\nthese e\u000bects, and show that they are governed by the\nconservation of energy and linear momentum, in the de\nBroglie sense, of quasiparticles involved in ST - the elec-\ntrons and the magnons [30].\nTo determine the characteristics of magnons generated\nby ST, we classify the eigenstates of the magnetic system\nusing the Bethe ansatz [35, 36]. The one-magnon states4\nare plane waves with wave numbers km:\nj i=nX\nx=11pnexp (ikmax)jxi; (2)\nwherenis the number of the local spins, and ais the lat-\ntice constant. Inserting this ansatz into the Hamiltonian,\nwe obtain the dispersion relation:\nEm= 4J(1\u0000coskma)\u0000E0: (3)\nwhereE0=\u0000Jnis the ground state energy of FM.\nThe two-magnon states are characterized by two\nwavenumbers k1,k2that are either real or complex-\nconjugates, and satisfy k1+k2\u0011k2m= 2\u0019=n(\u00151+\u00152),\nwhere\u0015iare integer Bethe numbers [36]. The values of\nk1,k2are obtained numerically by plugging the Bethe\nansatz for the two-magnon states into the Hamiltonian.\nThe eigenenergies have the form\nE2m(k) = 4JX\ni(1\u0000coskia)\u0000E0; (4)\nwherei= 1;2. Forn= 10 spins in the chain, there are\n45 two-magnon states.\nFor the electron wavepacket centered at the wavevector\nkeand localized outside the magnetic system before or\nafter scattering, the dispersion is\nEe= 2b(1\u0000coskea); (5)\nas determined from the hopping term in the Hamiltonian.\nWe will now utilize this classi\fcation of the dynam-\nical states to separately analyze the distinct scattering\nprocesses involved in ST.\nIII.2. B. Magnon absorption\nAn incident spin-up electron can absorb a magnon,\n\ripping its spin, and bringing the magnetic system to\nits ground state [Fig. 1(a)]. As Fig. 2(a) illustrates, for\nthe transmitted electron, the probability of this process\nis maximized for the initial magnon momentum k(i)\nm= 0.\nMeanwhile, for the re\rected electron, the probability is\nmaximized at large negative k(i)\nm. These observations can\nbe explained by the constraints imposed by the conser-\nvation of energy and linear momentum, in the de Broglie\nsense, as follows.\nSince the Hamiltonian Eq. (1) is time-independent, the\ntotal energy of the system comprising the electron and\nthe spin chain must be conserved. This conservation law\ntakes the simplest form when the electron is localized in\nthe non-magnetic medium before or after scattering and\nthe two subsystems do not interact,\nE(i)\ne+E(i)\nm=E(f)\ne; (6)\nwhereEe(Em) is the electron (magnon) energy, in the de\nBrogile sense, and superscripts ( i), (f) denote the charac-\nteristics of the quasiparticle before and after scattering,\nrespectively.\n02-\n2k(i)e\nak(i)e\na−\nπ0\nJ=0.2J\n=0.1ππ\n0k(i)m\nak\n(i)e\na(b)( a)0\nπ - πk\naE, eVk\n(f)m\nak(i)m\nak(f)e\naFigure 3. (Color online) Analysis of the magnon absorption\nprocess. (a) Electron dispersion (solid curve) and magnon dis-\npersion (dashed curve). The states of the subsystems before\nand after scattering are shown with open and \flled symbols,\nrespectively, for k(i)\nm=\u00002:3=a,k(i)\ne= 1=aallowing magnon\nabsorption. (b) Relationship between the electron momen-\ntum and the magnon momentum allowing magnon absorp-\ntion, for the labeled values of J. Curves: numerical solution\nof Eqs. (6), (7), symbols: results of simulations. Dashed lines\nshowk(i)\nm=\u00002k(i)\neandk(i)\nm=\u00002k(i)\ne+ 2\u0019=a.\nFor the momentum (or more precisely, quasi-\nmomentum for the discrete lattice), the situation is more\ncomplicated, because the spin chain breaks the transla-\ntion symmetry of the electron's spatial domain, so the\nmomentum needs not be conserved. Nevertheless, the\nconservation of linear momentum can be interpreted as a\nconsequence of the constructive wave interference among\nthe quasiparticles involved in the scattering process. This\ncondition can be expressed by [30]\nk(i)\ne+k(i)\nm=k(f)\ne+ 2\u0019l=a; (7)\nwhereke(km) is the electron (magnon) wavenumber, and\nthe last term with integer laccounts for the umklapp\nprocesses.\nWe note that interference takes place inside the spin\nchain, while k(i)\neandk(f)\neare de\fned outside the chain.\nThe s-d exchange may be expected to result in the shift\nof the electron's momentum as it crosses the boundary of\nthe spin chain. It is common to account for such e\u000bects\nby using the approximation of s-d exchange-induced con-\nduction band splitting. However, the s-d exchange term\nin the Hamiltonian Eq. (1) is itself the mechanism of ST\ndiscussed in this work, i.e., at small Jsdit can be consid-\nered as a perturbation for an electron whose dispersion\nis not modi\fed by s-d exchange. In quantum-mechanical\nterms, the complexity of this problem stems from the\nquantum entanglement between the conduction electron\nand the local spins, such that single-quasiparticle disper-\nsion relations become insu\u000ecient to describe the dynam-\nics of the system [23, 25]. However, these e\u000bects are small\nif the electron energy is substantially larger than the s-d\nexchange energy, and will be neglected here.\nEquations (6) and (7), with the quasiparticle energies\nrelated to their momenta by dispersions Eqs. (3), (5), give\ntwo possible values for k(i)\nmfor a givenk(i)\ne(or vice versa)5\nthat permit magnon absorption. Since magnon disper-\nsion is gapless [37], the magnon with k(i)\nm= 0,E(i)\nm= 0\ncan be absorbed, while the electron is transmitted with-\nout changing its energy or momentum.\nTo analyze the second possibility, we consider the\nelectron and the magnon dispersions [Fig. 3(a)]. Elec-\ntrons are signi\fcantly more dispersive than magnons. If\nmagnon dispersion were negligible, the electron could\nbe elastically re\rected while absorbing a magnon with\nk(i)\nm=\u00002k(i)\ne+ 2\u0019l=a, where integer laccounts for the\numklapp processes. Non-negligible magnon dispersion\nresults in an increase of the re\rected electron's energy,\nwhile the absorbed magnon's momentum shifts in the\nnegative direction. Symbols in Fig. 3(a) show the mo-\nmenta and energies of quaisparticles before and after\nscattering, obtained from the simulations for k(i)\ne= 1=a,\nk(i)\nm=\u00002:3=aallowing magnon absorption. The rela-\ntions among the momenta and energies are consistent\nwith our analysis.\nFigure 3(b) shows the dependence of the wavenum-\nber of the absorbed magnon on the wavenumber of the\nincident electron, both for the transmitted and for the\nre\rected electrons. Calculations based on the conserva-\ntion laws [curves], are in agreement with the results of\nquantum simulations [symbols]. The transmitted elec-\ntron absorbs a zero-momentum magnon, regardless of k(i)\ne\nor the exchange sti\u000bness J, as expected from energy and\nmomentum conservation [horizontal line in Fig. 3(b)].\nFor the re\rected electron, the dependence k(i)\nm(k(i)\ne) is\nslightly shifted below the linear form k(i)\nm=\u00002k(i)\ne+\n2\u0019l=a expected for negligible magnon dispersion [dashed\nlines in Fig. 3(b)], with an abrupt jump close to k(i)\ne<\u0018\n\u0019=2adue to umklapp. The downward shift increases with\nincreasingJthat controls magnon dispersion, consistent\nwith the discussed mechanisms.\nThe two main takeaways from our analysis of magnon\nabsorption are i) this process is selective with respect to\nthe magnon characteristics. The selectivity is controlled\nby the electron's energy and momentum, and ii) re\rected\nelectrons contribute to this process di\u000berently from the\ntransmitted electrons.\nIII.3. C. Scattering without spin transfer\nStudies of the e\u000bects of spin-polarized currents on mag-\nnetic systems usually focus on the transfer of spin. From\nthis perspective, no e\u000bects on the state of the magnetic\nsystem would be expected in the absence of ST. We show\nbelow that this is not the case. While the magnon pop-\nulation does not change in the absence of ST, magnon\nenergy can be modi\fed by electron scattering. The re-\nsults discussed below were obtained for scattering of the\nspin-up electron. They were similar for the spin-down\nincident electron.\nIn the absence of ST, the energy and momentum con-\nk(i)m\na= -1.25k(i)m\na= -1.25(b)( a)−\nπ0π0\nπ Δkmak\n(i)e\na−π0π0\nπ k(f)e\nak\n(f)e\n= -k(i)e\nk (i)m\na=1.25k(i)m\na=1.25k\n(i)e\naFigure 4. (Color online) E\u000bects of electron scattering with-\nout ST. (a) The change of magnon's wavenumber vs the\nwavenumber of the incident electron, (b) The wavenumber of\nthe electron after scattering vs its initial wavenumber. Curves\nare calculations based on the conservation laws Eqs. (8), sym-\nbols are the results of simulations. Dashed dotted curves are\nfork(i)\nm=\u00001:25=aandk(i)\nm= 1:25=a, respectively, the solid\nline isk(f)\ne=\u0000k(i)\ne. The trivial forward-scattering solution\nk(f)\ne=k(i)\neis not shown.\nservation relations are\nE(i)\ne+E(i)\nm=E(f)\ne+E(f)\nm;\nk(i)\ne+k(i)\nm=k(f)\ne+k(f)\nm+ 2\u0019l=a;(8)\nwith integer laccounting for umklapp. For given k(i)\nmand\nk(i)\ne, Eqs. (8) give two solutions for the \fnal state.\nOne of the two solutions is trivial - the wavenumbers of\nboth the electron and the magnon remain the same, i.e.,\nit is an elastic forward-scattering process. Naively, one\nmay expect that the second solution must correspond to a\nsimilar elastic electron re\rection. However, the reversal\nof the sign of electron's wavevector must be associated\nwith the exchange of momentum, and consequently of\nenergy, between the electron and the magnetic system,\nresulting in the modi\fcation of the magnon properties\neven without ST.\nFigure 4(a) shows the change of the magnon's\nwavenumber as a function of the incident electron's\nwavenumber, for two opposite values of the initial\nmagnon wavenumber. These results demonstrate that\nnon-ST electron scattering results in a large magnon drag\ne\u000bect - a shift of the magnon momentum determined by\nthe initial momentum of the electron. At small k(i)\ne, the\nmagnon's wavenumber is shifted in the direction of in-\ncident electron's momentum. At large k(i)\ne, the shift\nswitches to the opposite direction, due to the onset of\numklapp process at k(i)\neclose to\u0019=2a.\nThe reciprocal e\u000bect on the scattered electron is il-\nlustrated in Fig. 4(b), which shows the dependence of\nthe electron's wavenumber after scattering on its initial\nwavenumber. At k(i)\ne< \u0019= 2a, the wavenumber of the\nscattered electron is shifted in the direction opposite to\nthe wavenumber of the magnon, relative to the depen-\ndencek(f)\ne=\u0000k(i)\neexpected for the elastic electron back-6\n0.00.1k(i)e\nak(i)e\na, k(f)2\nmak\n(i)m\nak(f)2\nma(b)π0\n−\nπ(a)−\nπ0 π reflectedtransmittedPk\n(f)2\nma−π0 π \nFigure 5. (Color online) Excitation of the two-magnon\nstate by the spin-down electron. (a) Probability distribu-\ntion of the two-magnon wavenumber k2mfor the re\rected\n(squares connected by solid lines) and transmitted electron\ncomponents (circles connected by dashed lines), at k(i)\ne= 1=a,\nk(i)\nm= 1:88=a. (b) The resonant value of k(i)\nevsk(i)\nm, and the\nresulting values of the two-magnon wavevector k(f)\n2m.\nscattering. At larger k(i)\ne, the sign of the shift is reversed\ndue to the onset of magnon umklapp.\nFor most values of k(i)\ne, the electron is backscattered,\nk(f)\ne<0. However, for ki\neclose to the center or the\nboundary of the Brillouin zone and ki\nm<0, the electron\nbecomes forward-scattered. This e\u000bect can be described\nas suppression of electron backscattering by the magnetic\nmaterial due to the constraints imposed by the conserva-\ntion laws.\nFor parameters corresponding to the transition be-\ntween forward- and back-scattering, the velocity of the\nscattered electron vanishes. This outcome may be partic-\nularly useful for current-driven phenomena, for two rea-\nsons. First, all the initial kinetic energy of the electron\nis transferred to the magnetic system. Second, the scat-\ntered electron remains in the magnetic system for a long\nperiod of time, increasing the probability of magnon gen-\neration due to the s-d exchange.\nIII.4. D. Excitation of two-magnon states\nAn incident electron polarized in the \u0000zdirection can\nexcite two-magnon states [Fig. 1(b)]. In this case, one can\nexpect the conservation laws to be less restrictive than in\nmagnon absorption or magnon number-conserving pro-\ncesses, because of the additional degrees of freedom of\ntwo magnons in the \fnal state. Indeed, the probability\nof two-magnon excitation remains \fnite for all k(i)\nmat a\ngivenk(i)\ne, both for the re\rected and for the transmitted\nelectron component, as illustrated by the curves labeled\n\"r. m. em.\" and \"t. m. em.\" in Fig. 2(b). Neverthe-\nless, these curves exhibit sharp peaks resulting from the\nconservation laws, as follows.\nThe energy conservation relation for the two-magnon\nstate excitation is\nE(i)\ne+E(i)\nm=E(f)\ne+E(f)\n2m; (9)whereE(f)\n2mis the energy of the two-magnon state given\nby the Bethe ansatz Eq. (4).The wavefunction of the two-\nmagnon state can be characterized by the two-magnon\nwavenumber k2m=k1+k2= 2\u0019=n(\u00151+\u00152), where\nn= 10 is the size of the spin chain, and \u0015iare the in-\nteger Bethe numbers [36]. The values of k1,k2can be\ncomplex due to the magnon-magnon interaction, so for\nsome two-magnon states they cannot be interpreted as\nsingle-magnon wavevectors. The momentum conserva-\ntion relation is\nk(i)\ne+k(i)\nm=k(f)\ne+k(f)\n2m+ 2\u0019l=a: (10)\nThe conservation relations Eqs. (9), (10) do not pre-\nvent excitation of the two-magnon state for any given\nk(i)\nm,k(i)\ne. For instance, for the transmitted electron, there\nis always a solution k(f)\n2m=k(i)\nmwithk1= 0 andk2=k(i)\nm,\nwithE(k(f)\n2m) =E(k(i)\nm). Nevertheless, these relations re-\nsult in well-de\fned characteristics of scattered electron\nand of the excited two-magnon state, as illustrated in\nFig. 5(a) for the two-magnon momentum, for a generic\nset of initial conditions. As expected, for the transmitted\nelectron component, the two-magnon momentum is the\nsame as the initial magnon momentum. For the re\rected\nelectron, the \fnal-state momentum is determined by the\ndispersions of the involved quasiparticles.\nThe probability of two-magnon excitation is strongly\ndependent on the initial state. For the transmitted elec-\ntron, it is maximized for k(i)\nm= 0, i.e. when k1=k2=\nk(i)\nm= 0 - the additional excited magnon has the same\nwavevector as the initial magnon wavevector, which re-\nmains unchanged [see the curve labeled \"t. m. em.\" in\nFig. 2(b)]. This result can be interpreted as stimulated\nmagnon emission, i.e., electron spin \rip-driven emission\nof an additional magnon with the same characteristics\nas the magnon(s) initially in the system. Stimulated\nmagnon emission is the quantum-mechanical picture for\nthe semi-classical spin torque [2, 20, 22].\nThe situation is more complicated for the re\rected\nelectron. Figure 5(b) shows the dependence of k(i)\neon\nk(i)\nmmaximizing the probability of two-magnon excita-\ntion by the re\rected electron. The \"resonant\" condition\nisk(i)\ne\u0019k(i)\nm=2 +\u0019l=a, such that the linear momentum\nconservation relation Eq. (10) gives k(f)\n2m= 2k(i)\nm+2\u0019l0=a,\nwherel0is an integer accounting for umklapp [circles and\ndashed lines in Fig. 5(b)].\nThe relation k(f)\n2m= 2k(i)\nmseems to suggest that the\nresonant condition is associated with stimulated magnon\nemission by the re\rected electron, i.e., k1=k2=k(i)\nm.\nHowever, because of magnon interactions, bound two-\nmagnon states with complex k2=k\u0003\n1become formed\ninstead of real k1=k2[36]. Analysis of the simulation\nresults reveals that the resonantly excited two-magnon\nstates are not such bound states, but rather unbound\ntwo-magnon states with k1,k2shifted with respect to k(i)\nm\nin the opposite directions. For instance, for the resonant\nvaluesk(i)\ne= 1:1=a,k(i)\nm= 1:88=a, we obtain k1= 1:51=a,7\n-0.060.000.060\n.10.20.30.40\n10\n.00.10.2ΔE, eVk\nma−ΔSzΔSx(\nd)( c)(b)( a)−\nΔSz, ΔSx(hbar)k\nma−π0 π −\nπ0 π − π0 π −π0 π m\nagnon emissionmagnon absorptionno z-STPk\nmaReflectedT ransmitted P\nk\nmamagnon absorptionmagnon emission\nFigure 6. (Color online) Scattering of the electron polarized\nin thexdirection vs k(i)\nm, forJ= 0:2 eV,kea= 0:8. (a)\nEnergy transfer \u0001 E=E(i)\ne\u0000E(f)\nefrom the electron to the\nlocal spins. (b) Transfer of the z(open symbols) and x(solid\nsymbols) electron spin components to the local spins. Lines\nconnecting symbols are guides for the eye. (c),(d) Probabil-\nities of di\u000berent scattering scenarios for the transmitted (c)\nand re\rected (d) electron. Re\rection without ST is negligible\nand not shown.\nk2= 2:26=afor the re\rected electron. This \fnding pro-\nvides insight into the problem of stimulated scattering in\nnonlinear systems, warranting further studies beyond the\nscope of this work.\nIV. IV. SCATTERING OF ELECTRON\nPOLARIZED IN THE X DIRECTION\nIn this Section, we analyze the scattering of an electron\npolarized in the xdirection, perpendicular to the local\nspins in their ground state. Absorption of the spin cur-\nrent component perpendicular to the magnetization plays\na central role in the semiclassical theories of ST. Naively,\nthis process is unrelated to magnon absorption or emis-\nsion, which requires transfer of the z-component. Never-\ntheless, we show that scattering of the x-polarized elec-\ntron can be interpreted as a superposition of scattering\nof spin-up and spin-down electrons, as may be expected\nfrom the spin decomposition jsxi= (j\"i+j#i)=p\n2. As a\nconsequence, the outcomes of this process are determined\nby the conservation laws discussed above.\nEnergy transfer between the electron and the local\nspins exhibits complex variations with k(i)\nm, Fig. 6(a).\nOverall, at small k(i)\nmenergy \rows from the electron to\nthe local spins, while at large k(i)\nmit \rows from the lo-\ncal spins to the electron. The transfer of the x- and\nz-components of spin also exhibit a complex dependenceonk(i)\nm, Fig. 6(b). The transferred x-component is always\npositive, i.e., the initial electron spin is always partially\nabsorbed by the local spins, consistent with the usual\napproximations of semiclassical ST theories. In contrast,\nthe transferred zcomponent of spin is always negative,\nindicating that the magnon emission process is dominant\nover magnon absorption. We now demonstrate that the\ndependencies in Fig. 6(a),(b) result from the constraints\nimposed by the conservation laws.\nFigures 6(c),(d) show the probabilities of di\u000berent scat-\ntering scenarios identi\fed in Fig. 1, determined from the\nquantum simulations by projecting the \fnal state of the\nsystem onto the corresponding eigenstates. The magnon\nabsorption probability becomes \fnite at k(i)\nm= 0 for the\ntransmitted electron component, and at large negative\nk(i)\nmfor the re\rected electron component, due to the en-\nergy and momentum constraints discussed in Section III.\nThe magnon emission probability remains \fnite for all\nk(i)\nm, and exhibits peaks at k(i)\nm= 0 for the transmit-\nted electron component, and at k(i)\nm= 1:3=afor the re-\n\rected component, in accordance with the magnon emis-\nsion mechanisms discussed in Section III.\nThe variations of the energy and spin transfer are ex-\nplained in terms of these contributions, as follows. The\nminimum in the transferred energy [Fig. 6(a)], coincid-\ning with the maximum of the transferred z-component\nof spin [Fig. 6(b)], is explained by the resonant absorp-\ntion of the initial magnon as well as a broad minimum\nin magnon emission for the re\rected electron component\n[Fig. 6(d)]. On the other hand, a peak in the energy\ntransfer coinciding with a minimum in the transfer of\nthe z-component of spin, is explained by the resonant\nmagnon emission. The negative energy transfer for large\ninitial magnon momentum can be understood as a con-\nsequence of the availability of many two-magnon states\nwith energies smaller than that of the initial one-magnon\nstate, which is not the case for small k(i)\nmdue to the con-\nstraints imposed by momentum conservation.\nThe transfer of the xspin component [solid symbols\nin Fig.6(b)] does not follow the trends discussed above.\nTo analyze this e\u000bect in terms of the conservation laws,\nthe states of the magnetic system must be expanded in\nthexbasis. In the absence of anisotropy or a Zeeman\n\feld, a magnon in the xbasis is a superposition of two\nmagnons polarized in zand\u0000zdirections. Addition-\nally, the ground state of the z-basis is transformed into\na highly excited state in the x-basis. Analysis of ST in\nthese highly excited states is beyond the scope of the\npresent work.\nV. V. CONCLUSIONS\nIn this work, we utilized simulations of electron scatter-\ning by a chain of quantum spins, to reveal the quantum\nconstrains on ST that may not be captured by the semi-\nclassical approximation for the magnetic system. Our8\nmain results are:\n•The generally complex process of ST can be de-\ncomposed into a superposition of several distinct\nprocesses with qualitatively di\u000berent characteris-\ntics. These processes are magnon absorption, emis-\nsion, and the magnon number-conserving process,\naccompanied by electron transmission or re\rection,\n•In addition to angular momentum conservation\nusually considered in the analysis of ST, energy\nand momentum conservation laws, in the de Broglie\nsense of energy and momentum of quasiparticles in-\nvolved in ST, the electrons and the magnons, sub-\nstantially constrain the energy and the momentum\nof magnons that can be absorbed or emitted in\nthe ST process. These constraints may enable the\nrealization of selective cooling of speci\fc magnon\nmodes, or laser-like emission of speci\fc modes not\nlimited to the lowest-frequency dynamical states\nexcited in typical ST experiments,\n•Even in the absence of ST, spin-polarized electri-\ncal current can change the dynamical state of the\nmagnetic system. In particular, we demonstrated\nthe magnon drag e\u000bect - the change of the magnon\nmomentum due to the electron current. Generally,\nST processes are highly asymmetric with respect to\nthe direction of magnon momentum relative to the\nelectron current,\n•The magnon emission process, which is central to\nthe ST-induced generation of coherent dynamical\nmagnetization states, involves a complex interac-\ntion with the existing magnon in the system, which\nbears some signatures of the usual stimulated emis-\nsion of bosons, but is more complex due to the non-\nlinear magnon interactions, which warrants further\ntheoretical and experimental studies.\nThe presented analysis was based on numerical simu-\nlations of the simplest Heisenberg Hamiltonian and the\ns-dexchange approximation. Because of the exponential\nscaling of the dimensionality of the Hilbert space with the\nsize of the quantum systems, our simulations were lim-\nited to a chain of 10 local spins representing the magneticsystem. We expect these results to be directly relevant\nto quasi-1D systems. However, some of our conclusions\nwill likely become signi\fcantly modi\fed in higher dimen-\nsions. For instance, in 2D and 3D, the momentum con-\nservation will not limit the possible scattering processes\nonly to re\rection or transmission, expanding the range\nof accessible dynamical states. Interface roughness and\ndefects braking the translational invariance should fur-\nther relax the constraints imposed by momentum con-\nservation. Nevertheless, based on our analysis, we can\nconclude that the spatial characteristics of the magnetic\nsystems, such as the quality of their interfaces and the\nspatial homogeneity, which do not play any role in the\nsemiclassical limit, must signi\fcantly a\u000bect the e\u000eciency\nof ST and the spectral characteristics of the dynamical\nstates induced by this e\u000bect.\nAnother demonstrated e\u000bect, expected to be relevant\nto ST regardless of the system dimensionality, is the\nmagnon drag e\u000bect - the directionality of the magnon \row\ninduced by ST and controlled by the direction of electron\n\row. This e\u000bect is highly attractive both for magnon-\nics and for spin-caloritronics. The demonstrated e\u000bects\nare also relevant for the reciprocal phenomena associated\nwith electron transport. For instance, using conserva-\ntion laws as selection rules, the electron's dynamics can\nbe controlled, enabling almost re\rectionless \row of elec-\ntrons through magnetic systems. Furthermore, certain\ndynamical magnetization states can serve as electron ac-\ncelerators, increasing the transmitted electron's velocity\ndue to energy and momentum transfer from the magnetic\nsystem. Yet another possibility highlighted by our sim-\nulations is electron stopping due to the interaction with\nmagnetic systems, enabling the formation of entangled\nelectron-magnon state that can be useful for the quantum\ninformation technologies [28, 29]. 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Beyond\nLinear Response of Spin Baths\nChang-Yu Hsieh1, 2and Jianshu Cao1, 3\n1)Department of Chemistry, Massachusetts Institute of Technology,\n77 Massachusetts Avenue, Cambridge, MA 02139\n2)Singapore University of Technology and Design, Engineering Product Development,\n8 Somapah Road, Singapore 487372\n3)Singapore-MIT Alliance for Research and Technology (SMART) Centre,\nSingapore 138602\nWe use the \\generalized hierarchical equation of motion\" proposed in Paper I to study\ndecoherence in a system coupled to a spin bath. The present methodology allows a\nsystematic incorporation of higher order anharmonic e\u000bects of the bath in dynamical\ncalculations. We investigate the leading order corrections to the linear response\napproximations for spin bath models. Two types of spin-based environments are\nconsidered: (1) a bath of spins discretized from a continuous spectral density and (2)\na bath of physical spins such as nuclear or electron spins. The main di\u000berence resides\nwith how the bath frequency and the system-bath coupling parameters are chosen\nto represent an environment. When discretized from a continuous spectral density,\nthe system-bath coupling typically scales as \u00181=pNBwhereNBis the number of\nbath spins. This scaling suppresses the non-Gaussian characteristics of the spin bath\nand justify the linear response approximations in the thermodynamic limit. For the\nphysical spin bath models, system-bath couplings are directly deduced from spin-\nspin interactions and do not necessarily obey the 1 =pNBscaling. It is not always\npossible to justify the linear response approximations in this case. Furthermore,\nif the spin-spin Hamiltonian and/or the bath parameters are highly symmetrical,\nthese additional constraints generate non-Markovian and persistent dynamics that is\nbeyond the linear response treatments.\n1arXiv:1701.05713v1  [physics.chem-ph]  20 Jan 2017I. INTRODUCTION\nUnderstanding the dissipative quantum dynamics of a system embedded in a complex\nenvironment is an important topic across various sub-disciplines of physics and chemistry.\nSigni\fcant progress in the understanding of condensed phase dynamics has been achieved\nwithin the context of a few prototypical models1{3such as Caldeira-Leggett model and\nspin-boson model. The environment is often modeled as a bosonic bath characterized with\na spectral density, from which bath-induced decoherence can be deduced. This prevalent\nadoption of bosonic baths is based on following considerations: (1) the simple mathematical\ntractability of a Gaussian bath model, (2) the linear response of an environment is often\nsu\u000ecient to account for quantum dissipations and (3) the spectral density can be extracted\nfrom the classical molecular dynamics simulations\nDespite the above-mentioned merits, there exists scenarios in which the \\bosonization\"\nof an environment is inadequate. For instance, the electron-transfer reaction in condensed\nphase is often approximated with the spin boson model. The abstract model treats the\ngeneric quantum environment as a set of harmonic oscillators, which corresponds to taking\nonly linear response of solvent e\u000bects outside a solvation shell while imposing a \\harmonic\napproximation\" on the vibrations modes inside the shell. The anharmonicity and higher-\norder nonlinear response can be substantial when the donor-acceptor complex is strongly\ncoupled to some low-frequency vibrational modes or present in a nonpolar liquids. To better\nunderstand the anharmonic e\u000bects of the environment, several groups including ours4,5have\nstudied correlation functions of anharmonic oscillators and a generalized spin boson model\nwith a bath of independent Morse or quartic oscillators. Similarly, a spin bath can be viewed\nas an extreme limit of anharmonic oscillators and provides additional insights into condensed\nphase dynamics.\nOn the other hand, physical spin bath models, corresponding to localized electron and/or\nnuclear spins, have received increased attentions due to ongoing interests in developing\nvarious solid-state quantum technologies6{8under the ultralow temperature regime when\ninteractions with the phonons or vibrational modes are strongly suppressed. For these spin-\nbased environments, the spectral density is no longer a convenient characterization of the\nbath. Instead, each bath spin is explicitly speci\fed with parameters f!k;gkg, the intrinsic\nenergy scale and the system-bath coupling coe\u000ecients, respectively.\n2In this work, we investigate corrections to the standard linear response treatment of\nquantum dissipations induced by a spin bath9. To quantitatively capture these higher or-\nder responses, we utilize our recently proposed generalized hierarchical equation of motion\n(gHEOM) method10to incorporate higher order cumulants of an in\ruence functional into\nan extended HEOM framework11through a stochastic formulation12?{14. Even though it\nis possible to derive the gHEOM directly through the path integral in\ruence functional\napproach15,16, we emphasize that the stochastic approach provides an extension to addi-\ntional methodology developments such as hybrid deterministic-stochastic algorithms17. This\nis a direction we are pursuing. Due to the enormous numerical e\u000borts needed to generate\naccurate long-time results, we restrict the numerical illustrations in the short-time limit. Re-\ncently, our group18and others have proposed methods to construct the memory kernel of a\ngeneric bath from numerically exact short-time dynamical results. Hence, the gHEOM pro-\nvides an invaluable tool to capture the anharmonicity and non-Gaussian e\u000bects of a generic\nquantum environment when used along with these other methods to correctly reproduce\nlong-time results. Furthermore, starting from the stochastic formalism or the gHEOM, it\nalso serves as a starting point to derive master equations19incorporating these higher order\nnonlinear e\u000bects and can be more e\u000eciently solved to extract long-time results.\nAs alluded earlier, we cover both scenarios: the spin bath as a speci\fc realization of\nan anharmonic condensed-phase environment and the physical spin bath. In this work, we\nalways explicitly take a spin bath as a collection of \fnite number of spins. For physical spin\nbath models, this restriction re\rects the reality that there could only be a \fnite number of\nspins in the surrounding environment. For an anharmonic condensed phase environment,\nif we simply take the spin bath as a realization of an in\fnitely large heat bath then it\nhas been rigorously shown16,20that all higher order response function must vanish in the\nthermodynamic limit. On the other hand, if we perform atomic simulation of solvents in a\ncondensed phase, the anharmonicity can probably be attributed to a few prominent degrees\nof freedom. Therefore, we restrict the number of bath spins in order to probe the e\u000bects of\nhigher order response functions. Many earlier studies21{26on the spin bath focused on the\nthermodynamic limit and restricted to analyze the linear response only. Some interesting\nphenomena include more coherent population dynamics23in the nonadiabatic regime at ele-\nvated temperature and the onset of negative thermal conductivity22in a molecular junction\ncoupled to two large spin baths held at di\u000berent temperatures. In App. B, we report our own\n3investigation on di\u000berences between a spin bath and a bosonic bath in the linear response\nlimit when the spin bath can be accurately mapped onto an e\u000bective bosonic bath.\nIn this work, the main di\u000berence between the two types of environment comes down to\nhow the parameters f!k;gkgare assigned to each bath spin as explained later. In general,\nwe \fnd the anharmonicity is more pronounced at the low-frequency end when the spectral\ndensity for condensed phase environment is the commonly used Ohmic form. Therefore,\na slow spin bath at low temperature could potentially pose the most di\u000eculty for a linear\nresponse treatment of the bath. On the other hand, for a physical spin bath model, highly\nnon-Markovian and persistent dynamics27{31emerge under a combination of narrowly dis-\ntributed bath parameters and highly symmetrical system-bath Hamiltonians. To accurately\nreproduce these results might require taking higher order response of the spin bath into\naccount.\nThe paper is organized as follows. In Sec.II, we introduce the spin bath models of interest.\nIn Sec.III, we provide a brief account of the stochastic formalism10and how to use it to\nderive the gHEOM with a systematic inclusion of higher order cumulants of an in\ruence\nfunctional. In Sec.IV, we \frst discuss an exactly solvable dephasing model to stress the\nimportance of higher order cumulant corrections and present a benchmark to validate our\nnumerical method then move on to study both \fnite size representation of the condensed-\nphase environment and an Ising spin bath. A brief summary is given in Sec.V. In App.A,\nwe provide additional materials on the stochastic derivation of the generalized HEOM. In\nApp.B, we investigate the linear response e\u000bects of the spin bath and physical signatures\nthat one can use to distinguish a spin based condensed-phase environment from a bosonic\none.\nII. SPIN BATH MODELS\nWe consider the following Hamiltonian in this work,\n^H=^Hs+^HB+^Hint\n=\u000f\n2^\u001bz\n0+\u0001\n2^\u001bx\n0+^HB+^Hint; (1)\nwhere ^HB=P\nk>0(!k=2)^\u001bz\nkand the standard partition of the system (subscript s), bath\n(subscript B) and the mutual interaction (subscript int) is implied. The general spin-spin\n4interaction takes the form ^ \u001b\u000b\n0^\u001b\f\n1, where\u000b=\f denotes the Cartesian components of Pauli\nmatrices. Among the choices, most common system-bath interactions read,\n^Hint=8\n>>>>><\n>>>>>:P\nk>0gk^\u001bz\n0^\u001bx\nk;\nP\nk>0gk^\u001bz\n0^\u001bz\nk;\nP\nk>0gk(^\u001b+\n0^\u001b\u0000\nk+ ^\u001b\u0000\n0^\u001b+\nk);\nP\nk>0gk(^\u001bx\n0^\u001bx\nk+ ^\u001by\n0^\u001by\nk+ ^\u001bz\n0^\u001bz\nk):(2)\nIn this work, we should explicitly consider \frst two interaction forms. The \frst form is\nappropriate for modelling condensed-phase environment, while the second form is useful in\nthe decoherence study in quantum computing and related contexts.\nA. Anharmonic condensed-phase environment\nOne simple-model approach to investigate anharmonicity of a condensed-phase environ-\nment is to generalize the typical bosonic bath by substituting harmonic oscillators with the\nquartic oscillators or Morse oscillators32. We brie\ry illustrate the steps to obtain a spin bath\nmodel corresponding to the low-energy spaces of a bath of Morse oscillators4.\nWe still begin with Eq. (1) but having a di\u000berent bath part,\n^HB=X\nk>0\u0012P2\nk\n2+Dk\u0000\n1\u0000e\u0000\u000bkXk\u00012\u0013\n;\n^Hint= ^\u001bz\n0X\nk>0ckXk: (3)\nThen-th eigen-energy of the k-th Morse oscillator is given by\nEn;k=!k\u0012\nn+1\n2\u0013\n\u0000\u001fk\u0012\nn+1\n2\u00132\n; (4)\nwhere the fundamental frequency !k=\u000bkp2Dkand the anharmonicity factor \u001fk=\u000b2\nk=2.\nFollowing Ref. 32, one can characterize the anharmonicity of each mode by imposing a\nparameter, \u0003 (the number of bound states in a Morse oscillator). Under the condition of\n\fxed \u0003, one gets\nDk=!k\u0003\n2; \u000bk=r!k\n\u0003;and\u001fk=!k\n2\u0003; (5)\nfor each Morse oscillator with a free parameter !k. As clearly implied in Eqs. (3) and (4), the\nMorse potential and the energy level spacing smoothly reduce back to those of a harmonic\n5oscillator in the limit of \u0003 !1 . The recovered harmonic bath is characterized by\n^HB=X\nk>0\u0012P2\nk\n2+1\n2!2\nkX2\nk\u0013\n;\n^Hint= ^\u001bz\n0X\nk>0ckXk: (6)\nOn the other hand, by setting \u0003 = 2, an e\u000bective spin bath emerges. The Equation (3) can\nnow be cast as\n^HB=X\nk>0!k\n2^\u001bz\nk;\n^Hint= ^\u001bz\n0X\nk>0ckp2!k^\u001bx\nk:; (7)\nwhich correspond to the \frst interaction form in Eq. (2).\nThe mapping of a generic anharmonic environment onto a spin bath is more universal\nthan the speci\fc example of Morse oscillators. A general approach to achieve the mapping is\nto formulate an in\ruence functional of the bath then perform a cumulant expansion, which\ncharacterizes the bath-induced decoherence through multi-time correlation functions. One\nthen has a clear set of criteria to construct a spin bath to reproduce the bath's response\nup to a speci\fc cumulant expansion order. This is achievable as a set of spins (or qubits)\nconstitute a versatile quantum simulator33that can simulate other simple quantum systems.\nB. Physical Spin Bath\nIn the present context, the spin bath is not just a conceptual model but represent the ac-\ntual spin-based environment composed of nuclear / electron spins in the surrounding medium\nof a physical system. Depending on details regarding a system, spin-spin interactions could\ntake on a number of di\u000berent forms such as the Ising, \rip-\rop (or XX) and Heisenberg\ninteractions in Eq. (2). For simplicity, we investigate the Ising interaction34,35in addition to\nthe \frst form of interaction in Eq. (2).\nThe physical spin bath must contain a \fnite number of bath spins. In certain systems,\nsuch as electrically-gated quantum dots6in GaAs, there could be as many as 105\u0000106\nnuclear spins within the quantum-dot con\fning potential. While only a small fraction of\nbath spins are strongly coupled to the system; it is often possible to make semi-classical\napproximations to simplify the calculations. On the other hand, for NV centers8and related\n6system, the relevant spin bath contains only 101\u0000102spins. The bath could be potentially\ntoo small for semi-classical approximation and too large for a full dynamical simulations.\nAlthough we have seen impressive advances in simulation methods23,24,36{38for large spin\nsystems in the last decade, there still exists rooms for improvement.\nC. Spin Bath Parameters\nA \fnite-size bath model is fully characterized by pairs of parameters, f!k;gkg. In mod-\nelling physical spin system, these parameters are often randomly drawn from narrow prob-\nability distributions as done and justi\fed in earlier works39{41. In particular, we will sample\nboth!kandgkfrom the uniform distributions.\nWhen addressing spin bath as a representation for anharmonic condensed phase envi-\nronment, we consider an alternative assignment of parameters, f!k;gkg, by discretizing an\nOhmic spectral density, J(!)/!exp(\u0000!=!c) according to the scheme given in Ref. 16. In\nthe thermodynamic limit, it has already been shown16that the spin bath can be exactly\nmapped onto a bosonic one with a temperature-dependent spectral density\nJe\u000b(!) = tanh\u0012\f!\n2\u0013\nJ(!): (8)\nOur present focus is to investigate the nonlinear e\u000bects beyond the e\u000bective spectral density\nprescription.\nIII. METHODOLOGY\nA. Stochastic decoupling of many-body quantum dynamics\nWe now present an approach to systematically incorporate the non-linear bath e\u000bects\ninto the HEOM framework through a stochastic calculus based derivation. Given the model\ndescribed earlier, the exact quantum dynamics of the composite system (central spin and\n7the bath) can be cast into a set of coupled stochastic di\u000berential equations,\nd~\u001as=\u0000idth\n^Hs;~\u001asi\n\u0000idtB(t) [A;~\u001as] (9)\n\u0000ip\n2dW\u0003A~\u001as+ip\n2dV\u0003~\u001asA;\nd~\u001aB=\u0000idth\n^HB;~\u001aBi\n+1p\n2dW(B\u0000B(t)) ~\u001aB\n+1p\n2dV~\u001aB(B\u0000B(t)): (10)\nwhere ~\u001as=B(t) refers to the stochastically evolved density matrices in the presence of the\nwhite noises, implied by the Wiener di\u000berential increments dWanddV, and the bath-\ninduced stochastic \feld acting on the system,\nB(t) =X\nkgkTrBf~\u001aB(t)\u001bx\nkg: (11)\nThe reduced quantum dynamics of the central spin is recovered after averaging ~ \u001as(t) over\ndi\u000berent noise realizations in the above equations,\n\u001as(t) =E(~\u001as(t)): (12)\nAs implied in Eq. (9), the dissipative e\u000bects of the bath are completely captured by the\ninterplay of the stochastic \feld B(t) and the white noises. To determined the stochastic\n\feld, one can use Eq. (10) to derive a closed form expression for B(t) in the case of bosonic\nbath,\nB(t) =1p\n2\u0012Zt\n0dW(s)\u000b(s) +Zt\n0dV(s)\u000b\u0003(s)\u0013\n; (13)\nwhere\u000b(t) =R1\n0d!J(!) (coth(\f!=2) cos(!t)\u0000isin(!t)) is the two-time correlation func-\ntions. For non-Gaussian bath models, such as the spin bath, the stochastic \feld is determined\nby multi-time correlation functions as follows,\nB(t) =1p\n2Zt\n0dW(s)\b2;1(t;s) +1p\n2Zt\n0dV(s)\b2;2(t;2) +\n\u00121p\n2\u00133Zt\n0Zs1\n0Zs2\n0dW(s1)dW(s2)dW(s3)\b4;1(t;s1;s2;s3) +\u0001\u0001\u0001+\n\u00121p\n2\u00133Zt\n0Zs1\n0Zs2\n0dV(s1)dV(s2)dV(s3)\b4;8(t;s1;s2;s3) +:::; (14)\nwhere the de\fnition on correlation functions \b n;m(t;t1;:::;tn\u00001) are delegated to the App. A.\nThe equation (14) assumes the odd-time correlation functions vanish with respect to the\n8initial thermal equilibrium state. For now, we simply note that there are 2n\u00001possiblen-time\ncorrelation functions. The second subscript m= 1;:::2n\u00001labels these functions. Not all of\nthen-time correlation functions are independent; every correlation function and its complex\nconjugate version are counted separately in this case. The motivation to distinguish the\ncorrelation functions is actually to di\u000berentiate all possible sequence of multiple integrations\nof noise variables associated with the n-time correlation functions as shown in Eq. (14). In\nthis equation, we have explicitly expanded this formal expression up to the fourth-order\ncorrelation functions.\nSubstituting Eq. (14) into Eq. (9), one then derives a closed stochastic di\u000berential equa-\ntion for the central spin. Direct stochastic simulation schemes so far have only been proposed\nfor the Gaussian baths when the stochastic \feld is exactly de\fned by the second cumulant\nas in Eq. (13). In this case, the stochastic \feld can be combined with the white noises to\nde\fne color noises with statistical properties speci\fed by the bath's two-time correlation\nfunction. When higher order cumulant terms are needed to properly characterize B(t), a di-\nrect stochastic simulation becomes signi\fcantly more complicated. In this study, we choose\nto convert Eqs. (9) and (14) to a hierarchy of deterministic equations involving auxiliary\ndensity matrices.\nB. Generalized hierarchical equations of motion\nTo derive the hierarchical equation, we begin with Eq. (9) and take a formal ensemble\naverage of the noises to get\nd\u001as\ndt=\u0000ih\n^Hs;\u001asi\n\u0000i[A;E(B(t)~\u001as)]: (15)\nTo arrive at the equation above, we invoke the relation of Eq. (12) and the fact E(dW) =\nE(dV) = 0. This deterministic equation now involves an auxiliary density matrix E(B(t)~\u001as(t))\nthat needs to be solved too. Working out the equation of motion for the auxiliary density\nmatrix (ADM), one is then required to de\fne additional ADMs and a hierarchy forms.\nFollowing a recently proposed scheme, we introduce a complete set of orthonormal func-\ntionsf\u001ej(t)gand express all the multi-time correlation functions as\n\bn+1;m(t;t1;:::;tn) =X\njjj\u001fn+1;m\njjj\u001ej1(t\u0000t1)\u0001\u0001\u0001\u001ejn(tn\u0000t1); (16)\n9wherejjj= (j1;:::jn). Due to the completeness, one can also cast the derivatives of the basis\nfunctions in the form,\nd\ndt\u001ej(t) =X\nj0\u0011jj0\u001ej0(t): (17)\nNext we de\fne the cumulant block matrices\nAn=2\n6664an\n1j1\u0001\u0001\u0001an\n1jk;0;:::\n...\nan\n2nk1\u0001\u0001\u0001an\n2njk;0;:::3\n7775; (18)\nwhere each composed of 2n+1row vectors with inde\fnite size and the 0 in each row im-\nplies all zeros beyond this point. For instance, A1has two row vectors while A2has four\nrow vectors etc. The m-th row vector of matrix Ancontains matrix elements denoted by\n(an\nmj1;an\nmj2;:::an\nmjk). Each of this matrix element can be further interpreted by\nan\nmj\u0011\u00121p\n2\u0013nZt\n0Zs1\n0:::Zsn\u00001\n0dU(s1):::dU (sn)\u001ej1(t\u0000s1)\u001ej2(s2\u0000s1)\u0001\u0001\u0001\u001ejn(sn\u0000s1);\n(19)\nwheredU(sj) can be either a dW(sj) ordV(sj) stochastic variable depending on index m.\nWith these new notations, the multi-time correlation functions in Eq. (14) are now concisely\nencoded by\nB(t) =X\nn;m;jjj\u001fn+1;m\njjjan\nmjjj: (20)\nNow we introduce a set of ADM's\n\u001a[A1][A2][A3]\u0001\u0001\u0001\u0011E Y\nn;m;kan\nmjk~\u001as(t)!\n; (21)\nwhich implies the noise average over a product of all non-zero elements of each matrix Ai\nwith the stochastically evolved reduced density matrix of the central spin. The desired\nreduced density matrix would correspond to the ADM in which all matrices are empty.\nFurthermore, the very \frst ADM we discuss in Eq. (15) can be cast as\nE(B(t)~\u001as) =X\nn;m;jjj\u001fn+1;m\njjj\u001a:::[An]:::(t); (22)\n10where each ADM, \u001a:::[An]:::, on the RHS of the equation carries only one non-trivial matrix\nelementan\nm;jjjinAn. Finally, The hierarchical equations of motion for all ADMs can now be\nput in the following form,\n@t\u001a[A1][A2][A3]=\u0000i\u0002\nHs;\u001a[A1][A2][A3]\u0003\n\u0000iX\nn;m;jjj\u001fn;m\njjj\u0002\nA;\u001a\u0001\u0001\u0001[An+(m;jjj)]\u0001\u0001\u0001\u0003\n\u0000iX\nn;m;jjj\u001ej1(0)A\u001a\u0001\u0001\u0001[An\u00001+(m0;jjj1)][An\u0000(m;jjj)]\u0001\u0001\u0001\u0000iX\nn;m;jjj\u001ej1(0)\u001a\u0001\u0001\u0001[An\u00001+(m0;jjj1)][An\u0000(m;jjj)]\u0001\u0001\u0001A\n+X\nn;m;j\u0011jj0\u001a\u0001\u0001\u0001h\nan\nmj!an\nmj0i\n\u0001\u0001\u0001(23)\nIn this equation, we introduce a few compact notations that we now explain. We use\n[An\u0006(m;jjj)] to mean adding or removing an element an\nmjjjto them-th row. We also use\n\u0002\nan\nmj!an\nmj0\u0003\nto denote a replacement of an element in the m-th row of An. On the second\nline, we specify an element in a lower matrix given by ( m0;j1). The variable j1implies\nremoving the \frst element of the jarray and the associated index m0is determined by\nremoving the \frst stochastic integral in Eq. (19). After the \frst term on the RHS of Eq. (23),\nwe only explicitly show the matrices Ana\u000bected in each term of the equation.\nIV. RESULTS AND DISCUSSIONS\nIn the following numerical examples, we investigate the dynamics of a central spin cou-\npled to a spin bath, Eq. (1). First two forms of system-bath interactions in Eq. (2) will be\naddressed. We use the gHEOM to simulate dynamics and adopt the Chebyshev polynomials\nas the functional basis to interpolate high-dimensional multi-time correlation functions. We\nnote that there exists simpler choices11of functional basis when only the two-time correla-\ntion functions are needed as in the bosonic bath models. We consider the standard initial\nconditions,\n^\u001a(0) = ^\u001as(0)\n^\u001aeq\nB; (24)\nwhere the bath density matrix is simply a tensor product of the thermal equilibrium state\nfor each individual mode.\nThrough the examples in this section, we investigate whether it is generally a valid idea\nto map a spin bath model onto an e\u000bective bosonic one, such as obtained through a second\norder cumulant expansion of the spin bath's in\ruence functional. The gHEOM presented in\n11Sec. III B will be used to quantify the contributions of higher order cumulant corrections to\nthe quantum dynamics of the central spin.\nA. Pure dephasing\nWe \frst analyze a pure dephasing model42, i.e. \u0001 = 0 in Eq. (1) and adopt the \frst\ninteraction ^Hint= ^\u001bz\n0P\nkgk^\u001bx\nkin Eq. (2). This case can be analytically solved and provides\ninsights into the higher order response functions of the bath. The coherence of the central\nspin can be expressed as\nh\"j\u001as(t)j#i=h\"j\u001as(0)j#ie\u0000i\u000fte\u0000(t); (25)\nwhere the decoherence factor \u0000( t) reads,\n\u0000(t) =X\nklnD\neiH(k)\n+te\u0000iH(k)\n\u0000tE\n=X\nkln\u0014\n1\u00004g2\nk\n\n2\nk(1\u0000cos \nkt)\u0015\n;\n\u0019X\nk\"\n4g2\nk\n!2\nk(1\u0000cos(!kt))\u0000\u00124g2\nk\n!2\nk\u00132\nsin(!kt) (sin(!kt)\u0000!kt)#\n; (26)\nwith ^H(k)\n\u0006= (!=2)^\u001bz\nk\u0006gk^\u001bx\nkand \nk=!kp\n1 + (4g2\nk=!2\nk). On the last line, we expand \u0000( t) to\nget the two leading contributions with respect to \u0015k= 4g2\nk=!2\nk. These two terms correspond\nto the second order and fourth order cumulant expansion of the in\ruence functional for this\nparticular model, respectively.\nEven with this simple case, one can draw important remarks regarding spin bath mediated\ndecoherence. First of all, the exact result in Eq. (26) implies the perturbations coming from\na speci\fc spin bath mode are modulated with an interaction renormalized frequency \n kas\nopposed to the bare frequency !k, which is the way bosonic modes perturb a system through\na linear coupling. The origin of this interaction dressed \n kcan be understood by inspecting\nthe time evolution of Pauli matrices associated with individual spin modes,\n^\u001b\u0006\nk(t) =e\u0006i!t^\u001b\u0006\nk(0)\u0007igkZt\n0d\u001ce\u0006i!k(t\u0000\u001c)^\u001bz\nk(\u001c);\n^\u001bz\nk(t) = ^\u001bz\nk(0)\u0000i2gkZt\n0d\u001c\u0000\n^\u001b+\nk(\u001c)\u0000^\u001b\u0000\nk(\u001c)\u0001\n: (27)\nBy converting the raising and lowering Pauli matrices into the xandyPauli matrices, it is\nobvious that di\u000berent components of the bath spin get coupled together via the system-bath\n12interaction in a non-trivial way and renormalize the frequency at which a spin precesses.\nThere is no such coupling of the internal structure for harmonic oscillators due to the funda-\nmental di\u000berences in the commutation properties of bosonic creation/annihilation operators\nand Pauli matrices for spins.\nSecondly, when \u0015k\u001d1 then \n kis signi\fcantly shifted from the bare frequency !k. The\ndensity of states of the bath will be dramatically re-organized with respect to \n k. Any\nmethods expanding around !kwill be di\u000ecult to provide accruate results when system-bath\ncoupling is strong and/or when !kis small. A striking example would be a single-frequency\nbath, in which all modes possess the identical energy scale !k=!and coupled non-uniformly\nto the system. For a bosonic bath as well as the second-order cumulant expansion for a spin\nbath, there is essentially no dephasing. According to the second-order result in Eq. (26), the\ncoherence of the central spin is periodically recovered at time points !t=n2\u0019withnan\ninteger. However, for an exact treatment of the spin bath, non-trivial dephasing happens as\nlong as not all coupling coe\u000ecients gkare identical. Due to the system-bath interaction, the\nsingle-frequency distribution of !can be broadened to a \fnite bandwidth corresponding to\nthe dressed \n k. This analysis is con\frmed in Fig. 1, where the exact result (green) undergo\nan irrversible decay but not for the second-order result (red). In the \fgure, we also consider a\nmodi\fed second order cumulant result (blue dotted) which expands the decoherence function\n\u0000(t) with respect to \n kinstead of!k. As shown, this modi\fed expansion works extremely\nwell. In general, this dressing of \n kimplies a faster dephasing rate is expected from a spin\nbath when compared to a similar bosonic bath, i.e. a bath of harmonic oscillators sharing\nthe same set off!k;gkg.\nFinally, the temperature independence of \u0000( t) in Eq. (26) is another distinguishing prop-\nerty to set apart spin bath from what has been known for the bosonic bath models. This\ntemperature-independent dephasing can already be inferred from the expression of the e\u000bec-\ntive spectral density in Eq. (8), and is further con\frmed to hold beyond the linear response\nregime in this pure dephasing model as the temperature factors is missing in the exact\nexpression in Eq. (26).\nTo estimate the contributions of higher order cumulant corrections (HOCCs) to the cen-\ntral spin deocherence, we note the fourth order cumulants in the last line of Eq. (26) can\nbecome dominant under two conditions: (1) the perturbation parameters satisfy \u0015k>1\nand/or (2) when t>1=(\u00152\nk!k) such that the terms linearly proportional to !ktdominates the\n13FIG. 1. The magnitude of quantum coherence, jh\"j\u001as(t)j#ij for a central spin coupled to 200\nbath spins with same frequency, !k=!. The coupling coe\u000ecients gkare sampled from a uniform\ndistribution. The green, red, and blue-dashed curves correspond to the exact, second-order and\nmodi\fed (see text) second-order expansion of \u0000( t).\nsecond order cumulant. The second condition implies a potential linear time tdivergence.\nThis instability is an artifact of cumulant expansion and can be removed by introducing\nhigher order cumulants. When the bath parameters f!k;gkgare obtained by discretizing a\ncontinuous spectral density as discussed in Sec. II C, all \u0015k\u00181=NBcan be made arbitrarily\nsmall when su\u000eciently large number of spin modes are used. Hence, the arti\fcial diver-\ngence due to unstable part of the fourth-order cumulants can often be suppressed within\nthe typical time domain for simulating condensed-phase dynamics. However, even if just a\nfew modes, satisfying \u0015k\u00151, could potentially contribute immensely to the overall HOCCs\nbecause of the logarithmic form for each spin's contribution to the exact expression for \u0000( t)\nin Eq. (26).\nAs mentioned in the case of physical spin models9, there is no reason that \u0015kshould be\nrelated to the number of bath spins. Hence, the e\u000bects of HOCCs can become noticeable if\nnot all\u0015kare su\u000eciently small within the simulation time window to suppress the divergence\nassociated with the unstable part of higher order cumulants. In Fig. 2, we look at the\ndephasing rate, \u0000( t)=tfor two di\u000berent cases to analyze HOCCs. The initial condition of\nthe central spin is taken to be the pure state j i=1p\n2(j\"i+j#i). This calculation also\nserves as a benchmark to validate that the gHEOM correctly resolves the second and fourth\n14FIG. 2. The decoherence rate \u0000( t)=tas function of time. The red, black and blue curves are\nthe exact rate, the fourth-order and and the second-order rate according to Eq. (26). The open\ncircles on the curves are generated numerically from the gHEOM method. The bath is composed\nof 50 spins with parameters !kandgksampled uniformly from the following ranges. (Panel a):\n!k2[0:4;0:6] andgk2[0:08;0:12]. (Panel b): !k2[0:4;0:6] andgk2[0:18;0:22].\ncumulant contributions when compared to the exact expansions in Eq. (26). The parameters\n\u000f= 2 and \u0001 = 0 are used for the system Hamiltonian. We assign random samples of f!k;gkg\nfrom two uniform distributions centered on !oandgo, respectively, with details given in the\n\fgure caption. In panel (a), the fourth order results can su\u000eciently reproduce the exact rate\nand it starts to deviate from the linear response rate around t\u00183. In panel (b), by further\nincreasing the average coupling g0, even the fourth order corrections starts to fall short of\nreproducing the exact rate around t\u00183. For both cases considered in Fig. 2, \u0015k<1 hold for\nall bath modes, which ensure all perturbative expansion parameters \u0015kare well-behaved in\nthe short-time limit. It is clear that the HOCCs becomes important to simulate the system\nrelaxation when the spin bath parameters do not satisfy the scaling relation \u0015k\u00181=NB.\nB. Anharmonic Condensed-Phase evironment\nWe consider the same model as in Sec. IV A but with \u0001 6= 0 in Eq. (1). The central spin\ncould su\u000ber relaxation due to interaction with the bath in this case. The parameters, f!k;gkg,\nare assigned by discretizing an Ohmic spectral density. This \fnite-size restriction is a neces-\nsity to observe any deviations from linear response results as explained in the introduction.\n15Furthermore, the discretization allows us to numerically compute the multi-time correlation\nfunctions, needed for the gHEOM calculations, by summing over contributions from each\nbath spins.\nOne can estimate the leading order corrections beyond the linear response approxima-\ntion by a perturbative expansion with respect to \u0015kas done in the previous section. We\n\frst analyze the population dynamics in the Markovian limit. We adapt the NIBA (Non-\nInteracting Blip Approximation) equation to the spin bath model22with a symmetric system\nHamiltonian, Hs=\u0001\n2\u001bx\n0. We \fnd\nd\ndt<\u001bz\n0(t)>=\u0000\u00012Zt\n0dse\u0000QR(t\u0000s)cos (QI(t\u0000s))<\u001bz\n0(s)>; (28)\nwhere theQRandQIfunctions read\nQR(t)\u0019X\nk\u001a\n\u0015k(1\u0000cos (!kt)) +\u00152\nk\n2\u0014\nsin(!kt)!kt+ sin2(!kt) sech2\u0012\f!k\n2\u0013\u0015\u001b\n;\nQI(t)\u0019X\nk\u001a\u0014\n\u0015ksin(!kt) +\u00152\nk\n2[sin(2!kt)\u0000cos(!kt)!kt]\u0015\ntanh\u0012\f!k\n2\u0013\u001b\n; (29)\nwith\u0015k\u0011(4gk=!k)2. The equation (28) is a second-order expansion (with respect to the\no\u000b-diagonal element, \u0001) of the memory kernel. The functions, QR(t) andQI(t), in Eq. (29)\nare further expanded up to \u00152\nk. If one only retains the \frst term, proportional to \u0015k, then\nQR(t) andQI(t) reduce to the standard NIBA expressions for an e\u000bective bosonic bath with\na temperature dependent spectral density, Eq. (8).\nTo \fnish the Markovian approximation, we replace < \u001bz\n0(s)>with< \u001bz\n0(t)>on the\nRHS of Eq. (28), extend the integration limit to in\fnity in both directions, and perform a\nshort-time expansions of QRandQIto keep terms up to \u0018t2. One then integrates out the\nmemory kernel in Eq. (28) and obtains a simple rate equation with the Fermi golden rule\nrate given by\nk= \u00012r\u0019\na1(1 +\u0018)\u00001=2exp\u0012\n\u0000b2\n1\n4a1(1 +\u0010)2\n1 +\u0018\u0013\n;\n\u0019\u00012r\u0019\na1exp\u0012\n\u0000b2\n1\n4a1\u0013\nexp\u0012\n\u00001\n2\u0018\u0013\n;\n=klinexp\u0012\n\u00001\n2\u0018\u0013\n(30)\nwhere\u0018=a2=a1,\u0010=b2=b1,a1=P\nk\u0015k!2\nk=2,a2=P\nk\u00152\nktanh(\f!k=2)!2\nk=2,b1=\nP\nk\u0015k!ktanh(\f!k=2) andb2=P\nk\u00152\nk!ktanh(\f!k=2)=2. As\u0015k!0, the rate approaches\n16to the linear response / e\u000bective bosonic bath results: k!klin. The expression on the\nsecond and third line constitutes a good approximation when both \u0018and\u0010are small. In\nparticular, the last exponential factor isolates the leading-order correction to the rate con-\nstant,\u0011= exp\u0000\n\u00001\n2\u0018\u0001\n. Whenever the scaling \u0015k\u00181=NBis imposed, \u0018will scale as 1 =NB\ntoo and\u0011will be exponenially suppressed. Furthermore, we note that the leading-order cor-\nrection reduces the relaxation rate at low temperature and gradually converge to the linear\nresponse result klinwith increasing temperature due to the factor tanh( \f!k=2) contained in\na2variable.\nNext, we investigate numerically the convergence of a spin bath to the linear response\nresults within the gHEOM calculations. We use the following parameters \u0001 = 1 and \u000f= 0 for\nthe system Hamiltonian and the spectral density parameters: !c= \u0001,\f\u0001 = 2,\u000b= 2:3. For\nOhmic spectral density, coupling coe\u000ecients satisfy g2\nk/!kand the perturbative parameter\n\u0015k/1=!k. Therefore, the low frequency bath modes are more severely in\ruenced by the\nsystem-bath interactions with \n k=!kp1 +\u0015kshifted further from !k. To investigate\ndeviations from the linear response results, the highest frequency we consider is !max= 2!c\nin the discretization. The convergence of dynamics is demonstrated in Figs. 3a and 3b, the\npopulation and the coherences of the central spin are plotted up to \u0001 t= 3:5, respectively, for\na total number of 35, 70, 105, 210 and 500 spins including up to the fourth-order cumulant\nexpansions. As the number of bath spins increase, the results converge smoothly to those\nof the condensed phase environment in the thermodynamic limit. At NB= 500, the fourth\norder results already converge with the second order results.\nNext, we investigate the temperature dependence of HOCCs. The same set of Hamilton-\nain and bath's spectral density parameters as above is used except the temperature will be\nvaried for analyses. We use a set of NB= 35 spins for illustrations. In many earlier studies,\ndistinctive properties of a spin bath (in comparison to a bosonic counterpart) are found to\nbe temperature-related and are attributed to the temperature-dependent spectral density,\nEq. (8). However, restricting the discussions to the e\u000bective spectral density imply the\ncomparisons focused on the linear response limit. We would like to further analyze the con-\ntributions of HOCCs to these temperature-dependent e\u000bects. In Fig. IV B, we compare the\nsecond-order (dashed curves) and fourth-order (solid curves) results to inspect the contribu-\ntion of the HOCCs. In short, the HOCCs become more pronounced when the temperature\nis lowered and the divergence between second-order and four-order results increase. The nu-\n17FIG. 3. Convergence study of the population and coherence dynamics with respect to the number\nof spins discretized from an Ohmic spectral density. The population (a) and coherence (b) dy-\nnamics are obtained with the fourth-order gHEOM. For the 500-spin case, the fourth-order results\nessentially converge to the second-order results for both population and coherence dynamics.\nmerical results is also consistent with the earlier conclusion drawn from the rate expression,\nEq. (30). Similar observations16,23have been reported in the literature where it was found\nthat more number of discretized bath spins are needed to reach the linear response limit\nat low temperatures. Despite the simple argument that the linear response of a spin and a\nharmonic oscillator converge in the zero temperature limit, the two bath models actually do\nnot converge except in the cases of large bath size. This is because the higher order response\nfunctions for spins become more prominent in the low temperature regime. The primary\nreason is due to the way temperature enters the correlation functions as tanh( \f!k=2).\nC. Ising Spin Bath\nFinally, we consider another system-bath interaction, ^Hint= ^\u001bz\n0P\nkgk^\u001bz\nk. The Ising spin-\nspin interaction prohibits bath spins to be \ripped but still entangles system and bath. While\nthis spin bath model is appropriate in certain quantum computing contexts34,35, it does not\nrelate to a condensed phase environment. Hence, we will follow the physical spin bath\napproach to sample !kandgkfrom uniform distributions.\nWe \frst consider a pure dephasing case with \u0001 = 0. The coherence of the central spin\n18FIG. 4. Temperature dependence of the population (a) and coherence (b) dynamics. In both\npanels, four temperature cases \f\u0001 = 0:25 (red),\f\u0001 = 0:5 (blue,+), \f\u0001 = 1:0 (green,\u0002) and\n\f\u0001 = 1:5 (black,o) are considered. The solid and dashed curves correspond to the second-order\nand fourth-order gHEOM calculations, respectively.\ncan be cast in the general form of Eq. (25) but with a di\u000berent decoherence function,\n\u0000(t) =X\nkln (cos(2gkt)\u0000i\rksin(2gkt)); (31)\nwhere\rk= tanh(\f!k=2). Unlike the previouse pure-dephasing model in Sec. IV A, \u0000( t)\nacquires a temperature dependence through \rk. Since ^HBand ^Hintcommutes, the bath\nHamiltonian can be removed from the dynamical equation in a rotated frame and no dressed\nbath frequencies \n kappear as in Sec. IV A. For the Ising spin bath, the bath frequencies\n!kenter the decoherence function through \rk, re\recting the thermal equilibrium initial\ncondition\u001aeq\nB.\nAccording to Eq. (31), \rkmodulate the magnitude of \u0000( t) and the Ising spin bath becomes\nless e\u000ecient at interacting with the system in the high temperature limit and / or the low-\nfrequency limit when \rk\u001c1. The system-bath coupling coe\u000ecients gkdetermine the\noscillatory behaviors of \u0000( t) in time domain. When gkare narrowly distributed around a\nmean value go, one expects a partially periodic recurrence of \u0000( t).\nNow we investigate e\u000bects of HOCCs. Two points make the Ising spin model an interest-\ning case to analyze. First, an expansion of Eq. (31) with respect to gkreveals that the even\n19cumulants do not contribute to the imaginary component of \u0000( t), which drives a rotation of\nthe central spin on the Bloch sphere. A second order cumulant expansion will completely\nmiss this rotation. This point is illustrated in Fig. 5 where the central spin's initial condition\nis taken to bej s(0)i= (j\"i+j#i)=p\n2. The second-order result (red curve) reproduces the\nreal part of the coherence, h\"j\u001as(t)j#i, but completely misses the growth of the imaginary\ncomponent. Once we incorporate the third and fourth cumulants in the calculation, the re-\nsult (green curve) converges better to the exact one in Fig. 5b. Secondly, take B=P\nkgk^\u001bz\nk\nas the bath part of ^Hintand it is easy to verify the multi-time correlation functions, such\nas Trf\u001aeq\nBB(t1)B(t2):::B (tn)g, are time invariant (i.e. the memory kernel of the bath do\nnot decay in time). The highly non-Markovian nature of the Ising spin bath can give rise to\nnon-trivial steady states in the long time limit. In Fig. 6, we consider another case and \fnd\nthe second-order result suggests a fully decayed coherence while the exact result shows a\npersistent oscillation. Although the higher-order result (green curve) can better capture the\non-going oscillating behavior in the transient regime, the arti\fcial divergence of cumulant\nexpansion (explained in Sec. IV A) suggests more cumulant terms should be included within\nthe simulated time domain.\nGoing beyond the pure dephasing case, we restore the o\u000b-diagonal coupling \u0001. We exam-\nine both the coherence and population dynamics in Fig. 7. Similar to pure dephasing case,\none expects a slow decoherence when gkare narrowly distributed. For both coherence and\npopulation dynamics, we identify the clear insu\u000eciency of second-order results and higher-\norder cumulants again helps to restore the coherence and the beatings of the population\ndynamics in the transient regime.\nV. CONCLUSIONS\nIn summary, we use our recently proposed gHEOM to investigate the e\u000bects of HOCCs\non the quantum dissipations induced by \fnite-size spin bath models. The gHEOM can\nsystematically incorporate the higher order cumulants of the bath's in\ruence functional into\ncalculations. The controlled access to non-Gaussian e\u000bects of the bath allows us to assess\nthe su\u000eciency of a linear response approximation. Besides the spin baths, the methodology\ncan be similarly applied towards other types of anharmonic environments. However, due to\nthe prohibitive numerical resources required to accurately characterize the high-dimensional\n20FIG. 5. The real (a) and imaginary (b) part of coherence, h\"j\u001as(t)j#i, for a central spin coupled\nto 50 bath spins. The system Hamiltonian is absent, i.e. \u000f= 0 and \u0001 = 0. The bath parameters\nare randomly drawn from the range: !k2[14;15] andgk2[0:006;0:0018] and\f= 0:2.\nmulti-time correlations functions, the best usage of this method is to combine it with transfer\ntensor method (TTM) proposed by one of us. One can use the gHEOM to quantitatively\ncapture the exact short-time dissipative dynamics embedded in a non-Gaussian bath. These\nshort-time results are then fed to the TTM method to reproduce the correct memory kernel\nof the environment and allow an e\u000ecient and stable long-time simulation of dissipative\ndynamics.\nThrough the analyses done in Sec. IV B, we \fnd the linear response approximation pro-\nvides a highly e\u000ecient and accurate result for a \fnite spin bath over a wide range of pa-\nrameters. This is mainly because the next leading order correction scale as 1 =NBin the\ncumulant expansion. We present one \\extreme\" result for a relatively slow Ohmic bath in\norder to observe appreciable corrections coming from the higher order cumulant terms in\nthe short time limit. Even in this case, the higher order e\u000bects still vanish when NB= 500.\nAlthough, the low-temperature condition should exacerbate the discrepancy between exact\nand linear-response results, we \fnd the actual e\u000bects rather minimal in the short-time limit.\nConsidering the signi\fcant numerical costs to access higher order cumulants, it certainly\nmake linear response approximation a highly appealing option in dealing with a spin-based\n21FIG. 6. The real part of coherence, h\"j\u001as(t)j#i, for a central spin coupled to 30 bath spins. The\nsystem Hamiltonian parameters are \u000f= 1 and \u0001 = 0. The bath parameters are randomly drawn\nfrom the range: !k2[7:8;8:1] andgk2[0:005;0:007] and\f= 0:5.\ncondensed phase environment. In App. B, we further investigate the di\u000berences between a\nspin and bosonic bath in the linear response limit. We con\frm the lack of appreciable tem-\nperature dependence on the dissipative dynamics and the emergence of negative di\u000berential\nthermal conductance are two robust physical signatures to distinguish a spin bath from a\ncorresponding bosonic one as explained in the appendix.\nIt is much simpler to devise numerical examples in which the higher order cumulants\nplay critical roles in a physical spin bath model. The most critical factor is the probability\ndistributions forf!k;gkg. Narrow distributions will make the spin bath model more di\u000ecult\nfor linear-response approximations, and it is likely to have such narrow distributions in real\nspin-based environments. In such cases, linear-response results could deviate extremely from\n22FIG. 7. The coherence (a) and population (b) dynamics for a central spin coupled to 45 bath spins.\nThe system Hamiltonian parameters are \u000f= 1 and \u0001 = 1. The bath parameters are randomly\ndrawn from the range: !k2[6:8;7:2] andgk2[0:04;0:06] and\f= 0:5.\nthe exact results such as shown in Fig. 1 and Fig. 6. Secondly, for highly symmetric spin-\nspin interaction such as the Ising Hamiltonian considered in Sec. IV C, the second order\ncumulants fail to generate a rotation of the spin state, which could only be accounted by\nthe odd-order cumulants. Finally, the physical spin bath could be di\u000ecult to handle due to\nthe possibility of extreme non-Markovianity. In the Ising Hamiltonian example, we see the\nextreme case of having all bath's multi-time correlation functions to be time-invariant and\nwe need to expand deep down the hierarchy to obtain converged results. The highly non-\nMarkovian nature of spin bath is not a rare exception. In addition to Ising Hamiltonian, the\n\rip-\rop and Heisenberg Hamiltonian in combination with narrow distribution of f!k;gkg\ncan also result in highly symmetric systems (central spin plus the bath) with a rich set of\nnon-Markovian and persistent dynamics. With the gHEOM method, we can systematically\nincorporate higher order cumulants to improve the simulation results for physical spin bath\nmodels.\n23ACKNOWLEDGMENTS\nC.H. acknolwedges support from the SUTD-MIT program. J.C. is supported by NSF\n(grant no. CHE-1112825) and SMART.\nAppendix A: Stochastic mean \feld and multi-time correlation functions\nFrom Eq. (11), it is clear that equation of motions for B(t) can be obtained from the\nstochastic dynamical equations for the bath density matrix in Eq. (10). More speci\fcally,\nthe time evolution of the stochastic \feld B=P\nkgk(hby\nki+hbki) is jointly determined by\ndhby\nki=i!khby\nkidt+1p\n2gkdW\u0003G+\u0000\nk+1p\n2gkdV\u0003G\u0000+\nk; (A1)\ndhbki=\u0000i!khbkidt+1p\n2gkdW\u0003G\u0000+\nk+1p\n2gkdV\u0003G+\u0000\nk: (A2)\nThe expectation values in Eqs. (A1)-(A2) are taken with respect to the stochastically evolved\n~\u001ak(t). The generalized cumulants above are de\fned as\nG\u000b1\u000b2\nk=hb\u000b1\nkb\u000b2\nki\u0000hb\u000b1\nkihb\u000b2\nki: (A3)\nIn this case of bosonic bath models, it is straightforward to show that the time derivatives\nofG\u000b1\u000b2\nk vanish exactly. Hence, the second order cumulants are determined by the thermal\nequilibrium conditions of the initial states. Immediately, one can identify the relevant quan-\ntityG+\u0000\nk=nB(!k), the Bose-Einstein distribution for the thermal state of the bath. The\ntime invariance of the second order cumulants make Eq. (A1)-(A2) amenable to deriving a\nclosed form solution. On the other hand, for non-Gaussian bath such as the spin bath, the\nsecond cumulants are not time invariant. One way to determine their time evolution is to\nwork out their equations of motion by iteratively applying Eq. (10). It is straightforward to\nshow these equations couples di\u000berent orders of generalized cumulants,\ndG\u000b\u000b\u000b\nk=idtj\u000b\u000b\u000bj!kG\u000b\u000b\u000b\nk+1p\n2dW\u0003\ns\u0010\nG[\u000b\u000b\u000b;+]\nk+G[\u000b\u000b\u000b;\u0000]\nk\u0011\n+1p\n2dV\u0003\ns\u0010\nG[+;\u000b\u000b\u000b]\nk+G[\u0000;\u000b\u000b\u000b]\nk\u0011\n; (A4)\nwhere\u000b\u000b\u000b= (\u000b1;\u000b2:::\u000bn) speci\fes a sequence of raising and lowering spin operators that\nconstitute this particular n-th order cumulant and j\u000b\u000b\u000bj=P\ni\u000biwith\u000bi=\u00061 depending on\nwhether it refers to a raising (+) or lowering (-) operator, respectively. We use [ \u000b\u000b\u000b;\u0006]\u0011\n24(\u000b1;:::\u000bn;\u0006) to denote an n+ 1-th cumulant obtained by appending a spin operator to \u000b\u000b\u000b.\nA similar de\fnition is implied for [ \u0006;\u000b\u000b\u000b]. More speci\fcally, these cumulants are de\fned via\nan inductive relation that we explicitly demonstrate with an example to obtain a third-order\ncumulant starting from a second-order one given in Eq. (A3),\nG[(\u000b1;\u000b2);\u0006]\nk =hb\u000b1b\u000b2(b\u0006\u0000hb\u0006i)i+hb\u000b1(b\u0006\u0000hb\u0006i)ihb\u000b2i+hb\u000b1ihb\u000b2(b\u0006\u0000hb\u0006i)i:(A5)\nThe key step in this inductive procedure is to insert an operator identity b\u0006\u0000hb\u0006iat the\nend of each expectation bracket de\fning the n-th cumulant. If a term is composed of m\nexpectation brackets, then this insertion should apply to one bracket at a time and generate\nm terms for the n+ 1-th cumulant. Similarly, we get G[\u0006;\u000b\u000b\u000b]by inserting the same operator\nidentity to the beginning of each expectation bracket of G\u000b\u000b\u000b\nk.\nFor the spin bath, these higher order cumulants persists up to all orders. In any calcu-\nlations, one should certainly truncate the cumulants at a speci\fc k-th order by imposing\nthe time invariance, G\u000b\u000b\u000b\nk(t) =G\u000b\u000b\u000b\nk(0) and evaluate the lower order cumulants by recursively\nintegrating Eq. (A4). Through this simple prescription, one derives Eq. (14).\nAppendix B: Physical signatures of the spin bath models in the\nthermodynamic limit\nIn the main text, we focus predominantly on the higher order corrections to the quantum\ndissipations induced by a spin bath. Now we turn attention to the linear response limit,\nand we look for physical signatures that can distinguish the spin and bosonic bath models\n. The condensed-phase spin bath (with practically in\fnite number of modes) can be rigor-\nously mapped onto an e\u000bective bosonic bath with a temperature-dependent spectral density,\nEq. (8). To facilitate the calculations in this appendix, we should take the spin bath as an\ne\u000bective bosonic model and adopt the superohmic spectral density for convenience. The\nresults below compliments those of earlier studies22{26on the same subject.\n251. Electronic Coherence of a Two-Level System\nWe \frst compare the dynamics of a dimer coupled to (1) a bosonic bath and (2) a spin\nbath. The spectral densities for the two models read\nJb(!) = 2K!3\n!2\nce\u0000!=!c;\nJs(!) = 2K!3\n!2\nce\u0000!=!ctanh (\f!=2); (B1)\nwhere the subscript b=sdenotes the bosonic bath and the spin (e\u000bective bosonic) bath model,\nrespectively. In the study of FMO-like molecular systems, surprisingly long coherence times\nwere discovered and successfully explained through an enhanced NIBA formalism3which\ntakes into account the \frst order blip interactions. By considering the same dimer system\nand a similar set of experimentally relevant parameters as in Ref. 43, we investigate how much\nthe coherent population dynamics of an e\u000bective dimer will change when the environment is\nreplace by a spin bath with an identical set of parameters as the original harmonics-oscillator\nbased condensed phase. The enhanced NIBA formalism is also adopted here as it provides\nsu\u000eciently accurate results over a long time span in the parameter regime considered here.\nThe parameters are K= 0:16,\u000f=\u0001 = 0:6,!c=\u0001 = 2:0 and \u0001\u0019106:2cm\u00001.\nFrom Fig. 8, the dimer's population dynamics in the presence of the bosonic (panel a) and\nthe spin (panel b) bath at two di\u000berent temperatures: T= 290Kand 75Kare presented.\nThe signi\fcant temperature-dependent relaxation is observed in the standard bosonic bath;\nwhile the almost temperature independent relaxation is found in the spin bath model. These\nresults con\frm that the atypical temperature dependence of a spin bath induced relaxation\ncan be quite pronounced and easily detected in the experimentally accessible regime.\n2. Energy transport through a non-equilibrium junction\nWe next explore additional features of a spin-based environment in the non-equilibrium\nsituations. We study the energy transport across a molecular junction connected to two heat\nbaths composed of non-interacting spins. The extended double-bath model should read,\n^H=^Hs+X\n\u0016=L;R^HB;\u0016+X\nk;\u0016=L;R^\u001bz\n0(gk\u0016\u001by\nk;\u0016+gk\u0016\u001bk;\u0016); (B2)\nwhere ^Hsand ^HB;\u0016are the standard system and bath Hamiltonian with \u0016=L;R to denote\nthe two bath at left or right end. Recent study22tried to model anharmonic junctions with\n26 0 0.5 1\n 0  100  200  300  400  500  600\nt [fs]T=75K, state 0\nT=75K, state 1\nT=290K, state 0\nT=290K, state 1\n 0 0.5 1\n 0  100  200  300  400  500  600\nt [fs]T=75K, state 0\nT=75K, state 1\nT=290K, state 0\nT=290K, state 1FIG. 8. Relaxation in the presence of a bosonic bath (left) and a spin bath (right) at high and low\ntemperatures.\nthe spin bath and hinted several qualitative di\u000berences in the transport phenomena. In\nthis work, we adopt a non-equilibrium Polaron-transformed Red\feld equation (NE-PTRE)\nin conjunction with the full counting statistics to compute the steady-state energy transfer\nthrough the junction. In Ref.44, it was demonstrated that NE-PTRE can be reliably used to\ncalculate energy currents (through a molecular junction coupled to bosonic baths) from the\nweak to the strong system-bath coupling regimes as the corresponding analytical expressions\nfor the energy current reduced elegantly to either the standard Red\feld (weak coupling) or\nNIBA (strong coupling) results in the appropriate limits. In this study we apply the NE-\nPTRE formalism to calculate and compare the energy current through the junction while\ncontacted by (a) two spin baths and (b) two bosonic baths held at di\u000berent temperatures.\nSimilar to App. B 1, the actual calculations below will treat the spin bath as an e\u000bective\nbosonic bath. The same superohmic spectral density, Eq. (B1), is adopted here. The\nparameters are \u0001 = 1, \u000f= 10,!c= 10 andK= 3:5. The left and right bath share an\nidentical set of parameters except the temperature, and the fast bath (or scaling) limit with\n!c\u001d1 is imposed in both bath models.\nIn Fig. 9, the heat transfer exhibits a negative di\u000berential thermal conductance (NDTC)\nfor spin baths but not for bosonic baths. The temperature, kBT\u000bare measured in terms of\n\u0001 withkB= 1 in the present case. We remark that the NDTC (for bosonic bath models)\nreported in some earlier studies are found to be an artifact of the Marcus approximation (over\na wide range of parameter regimes) as clari\fed by the NE-PTRE method reported in Ref. 44.\nIn Fig. 9, the NE-PTRE predicts the NDTC for the spin bath model but not for the bosonic\n27 0 0.0015 0.003 0.0045 0.006 0.0075\n 0  7  14  21  28J\n∆TFIG. 9. The steady-state energy current through a two-level junction as a function of the tempera-\nture di\u000berence between the left and right baths. The energy current for the spin bath (blue,dotted)\nand bosonic bath (red,solid) in the strong coupling regimes, respectively. The temperature incre-\nment \u0001T=TL\u0000TRis applied symmetrically to both baths such that TL+TRis \fxed.\nbath model. In a separate work (not shown here), we also investigate a single-frequency spin\nbath and compute the energy current by using the NIBA method and \fnd the same NDTC\nappearing in the strong coupling regime. This helps to validate that the present result is\nnot speci\fc to a particular spectral density. This qualitative di\u000berence between the bosonic\nand spin baths results make the NDTC another strong physical signature to distinguish the\ntwo bath models.\nREFERENCES\n1A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger,\nRev. of Mod. 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A 83, 032105 (2011).\n43L. A. Pach\u0013 on and P. Brumer, J. Phys. Chem. Lett. 2, 2728 (2011).\n44C. Wang, J. Ren, and J. S. Cao, Sci. Rep. p. 1187 (2015).\n30"    },    {        "title": "1408.6700v1.Interplay_of_spin_orbit_and_hyperfine_interactions_in_dynamical_nuclear_polarization_in_semiconductor_quantum_dots.pdf",        "content": "Interplay of spin-orbit and hyper\fne interactions in dynamical nuclear polarization in\nsemiconductor quantum dots\nMarko J. Ran\u0014 ci\u0013 c and Guido Burkard\nDepartment of Physics, University of Konstanz, D-78457 Konstanz, Germany\n(Dated: August 28, 2021)\nWe theoretically study the interplay of spin-orbit and hyper\fne interactions in dynamical nuclear\npolarization in two-electron semiconductor double quantum dots near the singlet ( S) - triplet (T+)\nanticrossing. The goal of the scheme under study is to extend the singlet ( S) - triplet ( T0) qubit\ndecoherence time T\u0003\n2by dynamically transferring the polarization from the electron spins to the\nnuclear spins. This polarization transfer is achieved by cycling the electron spins over the S\u0000\nT+anticrossing. Here, we investigate, both quantitatively and qualitatively, how this hyper\fne\nmediated dynamical polarization transfer is in\ruenced by the Rashba and Dresselhaus spin-orbit\ninteraction. In addition to T\u0003\n2, we determine the singlet return probability Ps, a quantity that can be\nmeasured in experiments. Our results suggest that the spin-orbit interaction establishes a mechanism\nthat can polarize the nuclear spins in the opposite direction compared to hyper\fne mediated nuclear\nspin polarization. In materials with relatively strong spin-orbit coupling, this interplay of spin-orbit\nand hyper\fne mediated nuclear spin polarizations prevents any notable increase of the S\u0000T0qubit\ndecoherence time T\u0003\n2.\nI. INTRODUCTION\nElectron spins in semiconductor quantum dots are con-\nsidered to be excellent candidates for qubits [1]. In or-\nder for a full scale quantum computer to be produced,\na successful ful\fllment of the DiVincenzo criteria [2] is\nnecessary. Accurate qubit manipulation [3, 4] and reli-\nable state preparation [5] are some of the requirements\nthat have been satis\fed in the past years. Techniques\nfor qubit identi\fcation and fast readout are also known,\ne.g., the spin readout for a two-electron double quan-\ntum dot is most commonly done in the regime of Pauli\nspin blockade [6] using spin to charge conversion mea-\nsurements [7]. Still, one challenge remains - su\u000eciently\nisolating the qubit from the corruptive e\u000bects of its sur-\nroundings.\nDue to the in\ruence of its surroundings, a qubit will\nirreversibly lose information. Di\u000berent types of informa-\ntion losses happen on di\u000berent time scales. The time in\nwhich a qubit relaxes to a state of thermal equilibrium\nis the relaxation time T1, whereas the time in which a\nqubit loses coherence due to the collective e\u000bects of its\nsurroundings is the decoherence time T\u0003\n2. Although ex-\nperimental and theoretical solutions for overcoming these\ninformation losses have been steadily developed for years,\n[8-13] overcoming qubit decoherence caused by a \ructu-\nating nuclear spin bath is still an ongoing task.\nSilicon [14] and graphene [15] have stable isotopes with\na zero nuclear spin. Therefore, they can be isotopi-\ncally puri\fed leaving only spin zero nuclei which do not\ncontribute to the electron spin qubit decoherence. On\nthe other hand, III-IV semiconductors, and particularly\nInxGa1\u0000xAs structures only have stable isotopes with a\nnon-zero nuclear spin. An electron con\fned in a typi-\ncal InxGa1\u0000xAs quantum dot interacts with 104\u0000106\nnuclear spins, which contribute strongly to electron spin\nqubit decoherence. Optically [16-18] or electrically polar-\nizing the nuclear spins can prolong the coherence timesof electron spins. Such a polarization of nuclear spins is\nachieved by transferring spin from the electron spins to\nthe nuclear spins in a procedure called dynamical nuclear\npolarization (DNP) [19].\nA suitable system for conducting DNP is a gate de\fned\ndouble quantum dot loaded with two electrons. There\nhas been a variety of proposals [20, 3] to use DQDs as\nqubits, e.g., by focusing on the singlet jSi= 1=p\n2(j\"ij#i\n\u0000j#ij\"i ) and tripletjT0i= 1=p\n2(j\"ij#i +j#ij\"i ) logi-\ncal subspace [21], where the generated nuclear di\u000berence\n\feld and the exchange interaction are used to perform\nuniversal control of the qubit on the Bloch sphere. Other\nthan the already mentioned DNP, the e\u000bects of dephas-\ning caused by a nuclear spin bath, can be canceled by\napplying a Hahn echo sequence [22], or the more elabo-\nrate CPMG sequences [21].\nThe generation of a nuclear gradient \feld, required to\ncontrol the S\u0000T0qubit [21], can be achieved by cy-\ncling the electron spins over the anticrossing between\nthe singletjSi= 1=p\n2(j\"ij#i\u0000j#ij\"i ) and triplet\njT+i=j\"ij\"i states. During such a S\u0000T+cycle, the\nelectron spins transfer polarization to the nuclear spins\n[23], and a nuclear di\u000berence \feld is generated. Further-\nmore, a higher degree of nuclear spin polarization causes\na longer spin coherence time of the S\u0000T0qubit. In mate-\nrials with sizable spin-orbit interaction, the spin-orbit in-\nteraction induces electron spin \rips, and this mechanism\ncompetes with the hyper\fne mediated electron spin \rips\nrequired for DNP. In such materials, we theoretically ex-\nplore the interplay of spin-orbit and hyper\fne e\u000bects on\nnuclear spin preparation schemes, in the vicinity of the\nS\u0000T+anticrossing.\nWe assume that the dots are embedded in the semicon-\nductor material In xGa1\u0000xAs with 0\u0014x\u00141. We model\n150 nuclear spins per dot fully quantum mechanically,\nkeeping track of how the probabilities and coherences of\nall nuclear states change in time. As compared to our\nmodel, recent models treating more [23] or fewer [24] nu-arXiv:1408.6700v1  [cond-mat.mes-hall]  28 Aug 20142\nclear spins fully quantum mechanically, do not take into\naccount the spin-orbit interaction. Although there has\nbeen some work on the interplay of spin-orbit and nu-\nclear e\u000bects in GaAs double quantum dots [25-28], to\nour best knowledge none of these theoretical frameworks\ntreat the nuclear spin dynamics fully quantum mechan-\nically, nor investigate the nuclear spin dynamics when\nsubjected to a large number ( \u0019300) of DNP cycles. On\nthe other hand, again to our best knowledge, there has\nbeen no theoretical work to describe the S\u0000T+DNP in\nmaterials having strong spin-orbit interaction, e.g., InAs.\nExperiments in InAs have been carried out with a single\nelectron spin in a single quantum dot [29], or in a double\nquantum dot, by using a di\u000berent, more elaborate puls-\ning sequence [30]. As a consequence of our fully quantum\ntreatment we can give precise estimations of T\u0003\n2, compare\nthem to known experiments in GaAs [31], and calculate a\nvalue forT\u0003\n2in InxGa1\u0000xAs. Our results can also be be\nextrapolated to materials with even stronger spin-orbit\ncoupling as compared to InAs such as, e.g., InSb.\nThis paper is organized as follows. In Section II we\ndescribe our model, in Section III we discuss the total\nnuclear spin angular momentum basis which signi\fcantly\nreduces the dimension of our Hilbert space. In Section\nIV we study the time evolution during the DNP cycle, in\nSection V we present results on In 0:2Ga0:8As, a material\nwith an intermediate strength of spin-orbit interaction,\nand in Section VI we compare results for di\u000berent abun-\ndances of indium in In xGa1\u0000xAs. We conclude in Section\nVII.\nII. MODEL\nThe con\fnement in a quantum dot is modeled with a\nquadratic potential and the electronic wave functions are\ncalculated according to the Hund-Mulliken theory [32].\nOur approach is a good approximation in the regime\nwhere half of the interdot separation ais larger thanthe e\u000bective Bohr radius, a>\u0018aB=p\n\u0016h=m\u0003!0. Here,\n!0is the circular frequency of the con\fning potential,\nwhich we later assume to be \u0016 h!0= 3:0 meV, and m\u0003\nis the e\u000bective electron mass ( m\u0003= 0:067m0for GaAs\nandm\u0003= 0:023m0for InAs). The interdot separation\n2aneeds to be chosen su\u000eciently large, due to the fact\nthat the Hund-Mulliken theory is valid in the regime of\nweakly interacting quantum dots. On the other hand,\nthe extended tunneling matrix element tHneeds to be\nnonvanishing, so that our DNP sequence is still possi-\nble. Therefore, for In 0:2Ga0:8As, which is the material\nwe study in Section V, we want tH\u00190:01U, whereU\nis the Coulomb energy of the electrons. This is why we\nseta= 46:3 nm. A magnetic \feld of B= 110 mT is\napplied perpendicular to the plane spanned by the [110]\nand [ \u0016110] crystallographic axes, see Fig. 1. The speci\fc\nvalue of the magnetic \feld is chosen so that the S\u0000T+\nanticrossing is located at \"\u00193U=2, where\"is the energy\ndi\u000berence between the quantum dots, Fig. 2.\nAll stable isotopes of gallium and arsenide have a nu-\nclear spinj= 3=2, while stable isotopes of indium have\na nuclear spin j= 9=2. Here we discuss a simpli\fed\nmodel in which all of the nuclear spins are assumed to be\nj= 1=2 [33]. Also, spin-orbit e\u000bects depend strongly on\nthe homogeneity of the distribution of In and Ga atoms in\nInxGa1\u0000xAs. Here, we assume a completely homogenous\ndistribution of In and Ga. For numerical convenience\nwe model a geometry in which the [110], [ \u0016110] crystal-\nlographic axes and the interdot connection axis p\u0018lie in\nplane (Fig. 1). We develop a numerical method for mod-\neling up to N= 150 nuclear spins per dot, a constraint\nimposed by our current computational resources.\nThe total Hamiltonian describing the electronic and\nnuclear degrees of freedom is\nH=H0(\") +HHF+HSO: (1)\nHereH0(\") is the non-relativistic Hamiltonian of two\nelectrons in a QD [32],\nH0(\") =0\nBBBBBB@U\u0000\" X\u0000p\n2tH 0 0 0\nX U +\"\u0000p\n2tH 0 0 0\n\u0000p\n2tH\u0000p\n2tHV+ 0 0 0\n0 0 0 V\u0000+g\u0016BBz\n0 0 0 0 V\u0000 0\n0 0 0 0 0 V\u0000\u0000g\u0016BBz1\nCCCCCCA; (2)\nin the basis offS(2;0);S(0;2);S(1;1);T+(1;1);T0(1;1);\nT\u0000(1;1)g. The letter Sdenotes the singlet state, and T+,\nT\u0000,T0are triplet states with the total spin projections\nms= +1,ms=\u00001,ms= 0. The numbers in the paren-\ntheses indicate the charge state. More speci\fcally, (2 ;0)\ndenotes a state where the left dot is occupied with two\nelectrons and the right dot is empty, (0 ;2) denotes a statewhere the right dot is being occupied with two electrons\nand the left dot is empty, and (1 ;1) stands for each dot\nbeing occupied with one electron. The Hamiltonian [Eq.\n(2)] acquires time dependence through the bias energy \".\nTo describe the DNP process, the bias energy \"will be\nassumed to be a linear function of time \"=rt;where we\nsetr= 2U=\u001c, and where \u001c= 50 ns is the duration of the3\nFigure 1. (Color online) Geometry of the problem. The\nstrength of spin-orbit interaction is tuned by varying the an-\ngle\u0012between the [110] crystallographic axis and the interdot\nconnection axis p\u0018. Spin-orbit interaction generates an e\u000bec-\ntive magnetic \feld \nalong theyaxis. The external magnetic\n\feld is perpendicular to the [110] - p\u0018plane.\nbias sweep. The value of ris chosen so that \"= 2Uat\nthe beginning of the sweep ( t= 0),\"= 0 at the and of\nthe sweep ( t=\u001c), as in the experiment by Petta et al.\n[5].\nThe quantities in H0are the on-site Coulomb energy\nU\u00181 meV, the coordinated hopping from one dot to\nthe otherX\u00180:1\u0016eV, the doubly occupied singlet and\ntriplet matrix elements, V+; V\u0000\u001810\u0016eV, and the ex-\ntended hopping parameter, tH\u00180:01U[32]. The Zee-\nman energy is given as g\u0016BBz, wheregis the electron\ngfactor (g=\u00000:44 for GaAs, g=\u000014:7 for InAs),\nthe Bohr magneton is \u0016B= 5:79\u000210\u00005eV/T and\nBz= 110 mT is the magnetic \feld. For an electron con-\n\fned in an GaAs QD the Zeeman energy at this \feld is\nEz= 2:8\u000210\u00006eV. Due to the fact that we are inter-\nested in the S\u0000T+transition, we focus our attention\non the energy subspace spanned by the states fS(2;0),\nS(1;1),T+(1;1)g. The singlet S(0;2) is high in energy\nwith respect to the other two singlets [cf. Fig. 2] (for\npositive values of the detuning \") whereas the remain-\ning two singlets S(2;0) andS(1;1) are close in energy.\nThe triplet states T0(1;1), andT\u0000(1;1) are split o\u000b from\ntheT+(1;1) by the Zeeman energy. It should be men-\ntioned that we treat the Hamiltonian [Eq. (2)] using the\nadiabatic approximation, meaning that the system will\nremain in its instantaneous eigenstates. This allows us\nto obtain the eigenenergies by diagonalizing the Hamilto-\nnianH0in the subspace of fS(2;0);S(1;1)g. As a result\nof the diagonalization we obtain the two hybridized sin-\ngletsjS+i,jS\u0000i[32, 34] with energies\nES\u0006=U\u0000\"+V+\n2\u0006r\n(U\u0000\"+V+)2\n4+ 2t2\nH;(3)\n-10123\n0 0.5 1 1.5 2E/U\nε/UT−(1,1)\nT0(1,1)\nT+(1,1)\nS−S(0,2)\nS+\nFigure 2. (Color online) Two-electron spectrum of a DQD\nin InAs as a function of the interdot bias \", obtained by di-\nagonalizing the Hamiltonian H0[Eq. (2)]. The energy E\nand the detuning \"are expressed in units of the Coulomb en-\nergyU. The parameters of the plot are the magnetic \feld\nB= 1 T, the Coulomb energy U= 4:86 meV, the extended\ntunneling hopping tH= 0:11 meV, the triplet matrix element\nV+= 2:16\u0016eV, the doubly occupied singlet matrix element\nV\u0000= 0:42\u0016eV, half of the interdot separation a= 73:6 nm.\nIncluding hyper\fne interaction and/or spin-orbit interaction\nopens up an avoided crossing \u0001 [34] (upper inset). The mag-\nnetic \feld is chosen large, as compared to the value in the re-\nmainder of the paper, for visualization purposes. The S(2;0)\nandS(0;2) are singly occupied singlets, S(1;1) is the doubly\noccupied singlet. T+,T0andT\u0000are triplet states correspond-\ning toms= 1,ms= 0 andms=\u00001. TheS\u0000andS+are\nthe lower and the upper hybridized singlet [see Eq. (4) and\nEq. (5)].\nand eigenvectors\njS\u0000i=c(\")jS(1;1)i+p\n1\u0000c(\")2jS(2;0)i; (4)\njS+i=p\n1\u0000c(\")2jS(1;1)i\u0000c(\")jS(2;0)i: (5)\nWithc(\") = cos we denote the charge admixture coef-\n\fcient which can be expressed with the charge admixture\nangle , where\ncos 2 =U\u0000V+\u0000\"p\n(U\u0000V+\u0000\")2+ 8t2\nH: (6)\nWe only take into account the transitions between the\nlower hybridized singlet jS\u0000iand tripletjT+ibecause\nthe upper hybridized singlet jS+iis higher in energy, and\ntherefore can be neglected, as shown in Fig. 2 .\nThe spin-orbit term HSOin the Hamiltonian is a func-\ntion of the angle \u0012[cf. Fig 1] between the [110] crystal-\nlographic axis and the interdot connection axis p\u0018[34],\nHSO=i\n2\n(\u0012)\u0001X\ns;t=\";#(cy\nLs\u001bstcRt\u0000h:c:); (7)4\nwhere \n(\u0012) is the spin-orbit e\u000bective magnetic \feld de-\n\fned by\ni\n(\u0012) =h\bLj^p\u0018j\bRi((\f\u0000\u000b) cos\u0012e[\u0016110]+(\f+\u000b) sin\u0012e[110]):\n(8)\nHere\u000band\fare the Rashba [35] and Dresselhaus [36]\ncoe\u000ecients, the cy\nr;soperator creates an electron with spin\ns=\";#, in the right or left dot, r=R;L. Further, \u001bs;t\nis the vector of Pauli matrices and \b L;Rare the spatial\nparts of the wavefunctions corresponding to the left and\nthe right dot respectively [34] and ^ p\u0018is the component\nof the momentum operator along the interdot connection\naxis.\nFor computational simplicity, we choose our coordinate\nsystem such that the matrix elements of the spin-orbit\npart of the Hamiltonian [Eq. (7)] are always real. This is\nachieved by setting the eyaxis of our coordinate system\nparallel with \n[34], as shown in Fig. 1. When the spin-\norbit interaction is excluded, our xandyaxes are parallel\nto the crystallographic axes.\nFinally, the hyper\fne part of the Hamiltonian is given\nby [23]\nHHF=S1\u0001h1+S2\u0001h2=1\n22X\ni=1(2Sz\nihz\ni+S+\nih\u0000\ni+S\u0000\nih+\ni);\n(9)\nwhereS(\u0006)\niare theith electron spin ladder operators,\nSz\niandhz\niare thezcomponents of the ith electron spin\noperator and Overhauser \feld operator. Furthermore,\nh\u0006\ni=hx\ni\u0006ihy\niare the ladder operators of the Overhauser\n\feld,\nhi=n(i)X\nk=1Ak\niIk\ni; (10)\nwhere Ik\niare the nuclear spin operators for the kth nu-\nclear spin in contact with the ith electron spin. The\nstrength of the hyper\fne coupling between the ith elec-\ntron and the kth nuclear spin is labeled Ak\ni. In general Ak\ni\ncan have a di\u000berent value for every nuclear spin, but we\nsimplify this by assuming a constant hyper\fne coupling\nAk\ni=Atot=N[24].\nPerforming a diagonalization in the singlet subspace\nspanned byfS(2;0),S(1;1)g, we \fnd that the singlet\neigenfunctions are bias dependent and therefore time de-\npendent [Eq. (4) and Eq. (5)]. This implies that the cou-\npling between the lower hybridized singlet jS\u0000iand the\njT+itriplet state is time dependent as compared to time\nindependent coupling between the jS(1;1)iandjS(2;0)i\nsinglets and the jT+itriplet. The time dependence of\nthe coupling originates on the fact that the coupling de-\npends on the charge state of the hybridized singlet [Eq.\n(4) and Eq. (5)]. The state S(2;0) couples to T+only\nvia the spin-orbit interaction and S(1;1) couples to T+\nonly by means of the hyper\fne interaction. By using thewavefunctions of the lower hybridized singlet (see Eq. (4)\nwe can calculate the matrix element of the Hamiltonian\nbetween the lower hybridized singlet jS\u0000iand the triplet\njT+i\nhS\u0000jHjT+i=c(\")hS(1;1)jHHFjT+i\n+p\n1\u0000c(\")2hS(2;0)jHSOjT+i:(11)\nIt should be mentioned that due to time dependent\ninteractions, the model discussed here must go beyond\nthe Landau-Zener model [37-39].\nIII. THE BASIS OF TOTAL ANGULAR\nMOMENTUM\nIn our model, all nuclear spins are treated as having\nspinj= 1=2. This means that the total number of nu-\nclear spin states is dim( H) = 2N, whereNis the num-\nber of nuclear spins in a quantum dot. Because the total\nnumber of nuclear spin states scales exponentially with N\nit would be impossible to treat a large number ( N= 150)\nof nuclear spins with the computational power at our dis-\nposal. In order to make the problem treatable we \frst\nmake a basis change from the product basis f\";#g, to\nthe basis of total angular momentum fjj;mig. Herejis\nthe total nuclear spin quantum number, 0 \u0014j\u0014N=2,\nandmis the total nuclear spin projection along the z\naxis,\u0000j\u0014m\u0014j. Now the total number of states can\nbe written as\ndim(H) =N=2X\nj=0X\nperm(2j+ 1) = 2N: (12)\nThe inner sum runs over all permutation symmetries for\na given value of j. The basis of total angular momentum\nstill scales as dim( H) = 2N, but now certain states in\nthe inner sum in Eq. (12) do not need to be taken into\naccount, and states with higher jin the outer sum in\nEq. (12) can be neglected due to the low probability\nof their occurrence. In the remainder of this section we\nwill describe in more detail how we reduce the number of\nnuclear spin states from dim( H) = 2Nto dim(H0)\u001c2N.\nNeither the hyper\fne nor the spin-orbit interaction\nmix states with di\u000berent j, and thus the matrix represent-\ning our Hamiltonian is block diagonal with every block\ncorresponding to a value of j=j0; j0+ 1; :::N= 2. The\nvalue ofj0depends on the parity of N, for an even N,\nj0= 0 and for an odd N,j0= 1=2. The probability\ndistribution of nuclear spin states, with respect to the\nquantum number jis a Gaussian (in the limit N!1 )\nwith its maximum located at \u0019p\nN=2, Fig. 3. From\nnow on we will refer to this value of jas its most likely\nvalue,jml\u0019p\nN=2. The nuclear spin probability distri-\nbution, with respect to the number of nuclear spins per\ndotNand quantum number jis given by the following\nformula [40]5\n00.020.040.060.080.1\n0 10 20 30 40 50 60 70p(j, N )\nj\nGaussian fit\nincluded states\nneglected states\nFigure 3. (Color online) Initial nuclear spin probability distri-\nbution with respect to the quantum number jforN= 150 nu-\nclear spins 1 =2, wherejml=p\nN=2 andjmax= 18. Through-\nout our calculations we only consider the states 0 \u0014j\u0014jmax\n(blue diamonds) and do not consider the states j > j max\n(black circles).\np(N;j) =(2j+ 1)2N!\n(N=2 +j+ 1)!(N=2\u0000j)!2N: (13)\nThejandmquantum numbers are generally not suf-\n\fcient to describe all possible nuclear spin states. Other\nthanjandm, the nuclear spin states are described by\ntheir permutation symmetries. For example, for three\nnuclear spins de\fned by quantum numbers j= 1=2 and\nm= 1=2, there are two states j1=2;1=2iandj1=2;1=2i0\nwith distinct permutation symmetries. These two states\nare not mixed by homogenous hyper\fne or by spin-orbit\ninteractions. Furthermore, they remain equally probable\nas the matrix elements of the Hamiltonian only depend\nonjandmand not on the symmetry properties. There-\nfore, by evaluating our system for a certain symmetry\nj1=2;1=2iwe would also know the behavior of the state\nwith a di\u000berent permutation symmetry j1=2;1=2i0. By\ngeneralizing this simple example to N-spin systems we\ncan signi\fcantly reduce the number of the states we con-\nsider. For every value of jwe need to evaluate only one\nstate of symmetry in Eq. (12), and therefore for each\nvalue ofjthe inner sum in Eq. (12) can be replaced by\none representing term.\nWe can reduce the number of states further by choosing\nthe maximum value of jwe take into consideration, jmax\nin a manner thatp\nN=2\u001cjmax\u001cN=2. The omission of\nall states with j > j maxis justi\fed because these states\noccur with a very low probability (see Fig. 3 and Eq.\n(13)). Now the total number of the states we consider\nscales with jmaxas\ndim(H0) =jmaxX\nj=0(2j+ 1)\u0018=(jmax+ 1)2\u001c2N:(14)Due to the fact that the states with di\u000berent jdo not\nmix by any interaction we consider, we can analyze our\nsystem for one value of jat a time and \fnally average\nover all included values of j. By doing so, we average over\nclose to (but not exactly) 100% of all possible states. In\nour case,N= 150 nuclear spins per dot and 0 \u0014j\u001475.\nConstraining ourselves to 0 \u0014j\u0014jmax= 18, we average\nover 97:8% of all possible nuclear spin con\fgurations, as\nshown in Fig. 3. The e\u000eciency of our approach can be\nillustrated best if we calculate the number of states in the\nf\",#gbasis and in the fjj;migbasis after we consider\nonly one symmetry state for every jand consider only\n0\u0014j\u0014jmax. ForN= 150, Eq. (12) yields dim( H)\u0019\n1:4\u00021045and forjmax= 18, Eq. (14) yields dim( H0) =\n361.\nIV. TIME EVOLUTION DURING DNP\nWe now describe a single step in the DNP proce-\ndure. The system is initialized in a singlet state S(2;0),\nwhere both electrons are occupying the same dot. Af-\nterwards, the electronic system is driven with a \fnite\nvelocity through the S\u0000T+anticrossing (see Fig. 2) by\nvarying the voltage bias \". The electronic state is then\nmeasured, and \fnally the system is reset quickly to the\ninitial state S(2;0) [23]. Accordingly, we propagate the\ndensity matrix of the system \u001aaccording to the update\nrule\n\u001a(i+1)=MSU\u001a(i)UyMS+MTU\u001a(i)UyMT: (15)\nHere\u001a(i)and\u001a(i+1)are the total density matrices be-\nfore and after the i-th DNP step, Uis the unitary time\nevolution operator and MSandMTare the singlet and\ntriplet projection operators [41]. They satisfy the rela-\ntionsMS+MT=I;andMSMT= 0:\nAfter the evolution of the system, a measurement of\nthe electronic state takes place. This measurement pro-\ncedure has two outcomes: either a singlet Sor a triplet\nT+is detected. The nuclear density matrix is updated\naccordingly,\n\u001an=PS\u001aS\nn+PT\u001aT\nn; (16)\nwhere\u001anis the nuclear density matrix and\nPS= Tr[MSU\u001a(i)UyMS] andPT= Tr[MTU\u001a(i)UyMT]\nare the singlet and the triplet outcome probabilities.\nThe superscripts SandTstand for a nuclear density\nmatrix related to the singlet and the triplet measurement\noutcome. For a certain value of jwe calculate the singlet\nreturn probability PS, and the standard deviation of the\nnuclear di\u000berence \feld, \u001b(z)=p\nh(\u000ehz)2i\u0000h\u000ehzi2[13].\nAfter averaging over all included j, we use the stan-\ndard deviation of the nuclear di\u000berence \feld to evaluate\ntheS\u0000T0spin qubit decoherence time, T\u0003\n2= \u0016h=\u001b(z)[13].\nWe compute the propagator Uby discretizing the time\ninterval (0;\u001c). Our model describes the passage through6\nthe anticrossing with q= 100 equally spaced, step-like\ntime increments. The procedure of computing the prop-\nagator is the following: For every discrete point in time\ntiwe compute the Hamiltonian H(ti). We approximate\nthe propagator for the \fxed time point ti,\nUti=e\u0000iH(ti)\u0001t=\u0016h; (17)\nwith \u0001t=\u001c=q. By repeating the procedure for every\ndiscrete step we obtain the total time evolution operator\nU=UtqUtq\u00001:::Ut1: (18)\nTuning the system across the S\u0000T+point and measuring\nthe electronic state after every forward sweep changes the\nprobabilities and coherences of the electronic and the nu-\nclear states. The qualitative picture is simpler if we \frst\ndisregard the spin-orbit interaction. When the spin-orbit\ninteraction is excluded, both the electronic spin singlet\nand the triplet outcomes increase the probability for nu-\nclear spins to be in the spin down state [23], correspond-\ning to generating negative values of nuclear spin polariza-\ntionP= (n\"\u0000n#)=(n\"+n#), wherePis the nuclear spin\npolarization, n\"is the number of nuclear spins pointing\nup andn#is the number of nuclear spins pointing down\n[cf. Figs. 4(a-d)].\nThere is one more possible process, involving spin-orbit\ninteraction, which is not shown in Fig. 4. After cycling\nthe electronic system across the S\u0000T+anticrossing the\nsystem can end up in a virtual T+state due to spin-orbit\ninteraction, but is instantaneously transferred to a singlet\nstate due to hyper\fne interaction, accompanied by a \rip\nof the nuclear spin from down to up, thus changing the\nnuclear spin polarization closer to positive values. This\nis a process that, along with the process visualized on\nFig. 4(d), competes with the hyper\fne-mediated gener-\nation of negative polarization of the nuclear spins (down\npumping). These two processes combined compensate\nthe down pumping in systems with strong spin-orbit in-\nteraction.\nTo make an e\u000bective comparison between In xGa1\u0000xAs\nsystems with di\u000berent indium content xwe keep the same\nvalues forBzandd=a=aB= 2:186. This implies that\nthe single particle tunneling and the overlap between the\nquantum dots would remain the same for every value of x\n(see Ref. [32]). For a comparison between di\u000berent ma-\nterials, the relative strength of the spin-orbit interaction\ncan be quanti\fed by the ratio of \u0004 = 4 a=\u0003SO, where \u0003 SO\nis the spin-orbit length de\fned by\n1\n\u0003SO=m\u0003\n\u0016hq\ncos2\u0012(\u000b\u0000\f)2+ sin2\u0012(\u000b+\f)2:(19)\nHere,m\u0003is the e\u000bective electron mass, \u000band\fare the\nRashba and Dresselhaus constants and \u0012is the angle be-\ntween the [110] crystallographic axis and the interdot\nconnection axis p\u0018[cf. Fig.4].\nThe spin-orbit length is the distance which an electron\nneeds to travel in order to have its spin \ripped due to\nspin-orbit interaction. If the electrons are initialized in\n(a)\n(b)\n(c)\n(d)\nFigure 4. (Color online) System initialization and measure-\nment outcomes. (a) Initially, the quantum dots have an en-\nergy bias\"and the two electrons rest in a singlet (2 ;0) state\non the left dot. (b) After slowly tuning \"to zero, and measur-\ning a singlet outcome, due to the weak measurement the spin\nof the nuclear bath decreases. (c) In the case of a spin triplet\noutcome an electron spin \rips and the spin of the nuclear\nbath is changed accordingly. (d) The electronic spin can also\nbe \ripped due to spin-orbit, and the spin of the nuclear bath\nis pumped in the opposing direction (up) due to the weak\nmeasurement. With \"we denote the voltage bias, \u0012is the\nangle between the [110] crystallographic axis and the interdot\nconnection axis p\u0018,\nis the spin-orbit e\u000bective magnetic \feld.\na singlet state the probability for \ripping the tunneling\nelectron due to spin-orbit interaction is P\rip= 1=2 at7\n00.20.4\n-15 -10 -5 0 5 10 15p(m)\nm(a) left dot\n-8 -6 -4 -2 0 2 4 6 8\nm(b) right dot\nFigure 5. (Color online) (a) Probability distribution in the\nleft quantum dot with respect to the nuclear spin projection\nquantum number mforjL= 14. Blue circles represent the\ninitial probability distribution, black triangles represent the\nprobability distribution after 300 cycles with spin-orbit in-\nteraction excluded, and red squares represent the probability\ndistribution after 300 cycles with spin-orbit interaction in-\ncluded. (b) Probability distribution in the right quantum dot\nwith respect to the nuclear spin projection quantum number\nmforjR= 7. Red pentagons present the initial probability\ndistribution, green triangles represent the probability distri-\nbution after 300 cycles with spin-orbit interaction excluded\nand black diamonds represent the probability distribution af-\nter 300 cycles when spin-orbit interaction is included corre-\nsponding to \u0012=\u0019=2. Here,\u0012is the angle between the [110]\ncrystallographic axis and the interdot connection axis p\u0018. The\nnumber of nuclear spins per quantum dot is N= 150.\n2a= \u0003 SO=2. This further implies that if \u0004 <1, the\nsystem is more probable to remain in a singlet state. If\n\u0004 = 1 the SandT+outcomes due to spin-orbit cou-\npling are equally probable and \fnally if 1 <\u0004<2 a\nT+outcome due to spin-orbit is more probable, because\nthe probability that the tunneling electron has \ripped its\nspin is greater than P\rip>0:5. In our study \u0003 SO=2\u001d2a\nwhich implies \u0004 \u001c1, thus singlet outcomes due to\nspin-orbit interaction are always more probable even in\npure InAs with the strongest possible value of spin-orbit\n(\u0012=\u0019=2). In pure InAs, with \u0012=\u0019=2, \u0004\u00190:63 for\nd=a=aB= 2:186.\nV. RESULTS FOR In 0:2Ga0:8As\nOur attention is now focused on In 0:2Ga0:8As, a ma-\nterial with an intermediate strength of spin-orbit cou-\npling, as compared to the relatively weak spin-orbit cou-\npling in GaAs and relatively strong spin-orbit coupling in\nInAs. We have evaluated the system of N= 150 nuclear\nspins per dot, for di\u000berent values of the angle \u0012and with\njmax= 18. States with j >j maxwould further lower the\nT\u0003\n2andPsand increase \u001b(z). Therefore, we point out\nthat our results provide an upper bound for T\u0003\n2(includ-\ning states with j >jmax= 18 could lower T\u0003\n2for at most\n2:2%, see Fig. 3 and Eq. (13)) and Psand a lower bound\nfor\u001b(z). We study the e\u000bect of 300 DNP cycles on the\nnuclear spin state. We \fnd that the spin-orbit interac-\n0.50.60.70.80.91\n0 50 100 150 200 250 300Ps\nNumber of cycles\nspin-orbit excluded\nθ= 0\nθ=π/12\nθ=π/6\nθ=π/4\nθ=π/3\nθ=π/2Figure 6. (Color online) The singlet return probability PS\nas a function of the number of cycles across the S\u0000T+an-\nticrossing in In 0:2Ga0:8As. Here,\u0012is the angle between the\n[110] crystallographic axis and the interdot connection axis\np\u0018.\ntion has a notable e\u000bect on nuclear state preparation. In\nFig. 5, we plot the probabilities of nuclear spin states for\na case with a given value of jL;Rin the left and the right\ndot.\nForjL= 14 andjR= 7 the pumping procedure has al-\ntered the nuclear probability distribution from a uniform\ndistribution (with respect to the quantum number m) to\na probability distribution where states with negative m\nare more likely. In the case without spin-orbit interac-\ntion, two processes contribute to this negative pumping\nof the nuclear spin [23] - the singlet detection accompa-\nnied by a weak measurement of the nuclear spin state\nand theT+detection, which \rips the nuclear spin down\nto conserve the total spin of the system [cf. Fig. 4(b)\nand Fig. 4(c)]. Although including spin-orbit interaction\n[cf. Fig. 5(a), Fig. 5(b)], changes the \fnal distribution\nof nuclear spin states only slightly, spin-orbit e\u000bects still\nhave a notable e\u000bect on the singlet return probability\nPS= Tr[MSU\u001a(i)UyMS]. In Fig. 6, we plot PSas a\nfunction of the number of cycles across the S\u0000T+anti-\ncrossing for In 0:2Ga0:8As. Here, we tune the strength of\nthe spin-orbit interaction by varying the angle \u0012between\nthe [110] crystallographic axis and the interdot connec-\ntion axisp\u0018. As shown in Fig. 6 (solid red line), re-\npeatedly cycling the system across the anticrossing point\npolarizes the nuclear spins, which leads to Ps= 1 af-\nter 300 cycles [23]. The situation changes dramatically\nwhen we include the spin-orbit interaction, which com-\npetes with the hyper\fne mediated down pumping of the\nnuclear spin.\nBy theoretically varying the strength of the spin-orbit\ninteraction, we \fnd that when the spin-orbit interaction\nhas the largest possible value for \u0012=\u0019=2, it signi\fcantly\na\u000bects the singlet return probability Ps\u00190:72 (Fig. 6).8\n405060708090100\n0 50 100 150 200 250 300σ(z)[neV]\nNumber of cycles\nspin-orbit excluded\nθ= 0\nθ=π/6\nθ=π/3\nθ=π/2\nFigure 7. (Color online) Standard deviation of the nuclear\ndi\u000berence \feld \u001b(z)with respect to the number of DNP cycles\nacross theS\u0000T+anticrossing and di\u000berent values of angle\n\u0012in In 0:2Ga0:8As. Here, \u0012is the angle between the [110]\ncrystallographic axis and the interdot connection axis p\u0018.\nIncluding spin-orbit interaction generates a mechanism\nwhich polarizes nuclear spins in the up direction (see Sec-\ntion IV and Fig. 5). As a consequence of this behavior,\nthe nuclear preparation mechanism is not e\u000ecient when\nspin-orbit e\u000bects are strong. The interplay of the hyper-\n\fne and spin-orbit interactions on nuclear state prepa-\nration can be observed better if we plot the standard\ndeviation of the nuclear di\u000berence \feld \u001b(z)(Fig. 7).\nWe notice that the spin-orbit interaction has prevented\nthe reduction of the standard deviation of the nuclear\ndi\u000berence \feld (0 \u0014\u0012\u0014\u0019=2, see Fig. 7). Spin-orbit\ninteractions a\u000bect the e\u000borts to increase the spin S\u0000T0\nqubit decoherence time T\u0003\n2, see Fig. 8. The strongest\nspin-orbit coupling, corresponding to \u0012=\u0019=2, slightly\nlowers the resulting decoherence time from T\u0003\n2\u001915 ns\n(red line) to T\u0003\n2\u001913 ns (black dashed line with black x\nsymbols).\nWithout the spin-orbit interaction our theory predicts\nthat the ratio of the \fnal decoherence time (after the cy-\ncling is complete) T\u0003\n2;fand initial decoherence time (be-\nfore the cycling starts) T\u0003\n2;iisT\u0003\n2;f=T\u0003\n2;i\u00192:28 [cf. Fig.\n9]. The situation changes when we include spin-orbit in-\nteraction. For \u0012= 0 we \fnd a value of T\u0003\n2;f=T\u0003\n2;i\u00192:20,\nwhile for\u0012=\u0019=2 the ratio is T\u0003\n2;f=T\u0003\n2;i\u00192:04.\nAfter the inclusion of the spin-orbit interaction the ra-\ntioT\u0003\n2;f=T\u0003\n2;idecreases with \u0012. Our results suggest that\ntheS\u0000T+dynamical nuclear polarization is not as e\u000bec-\ntive in materials with intermediate strength of spin-orbit\ninteraction, as compared to those without spin-orbit cou-\npling. Nevertheless, the DNP still provides a notable en-\nhancement of the S\u0000T0qubit decoherence time T\u0003\n2. We\nwork in the so called \"giant spin model\" and we model\nthe behavior of 104\u0000106nuclear spins with signi\fcantly\nfewer spins,\u0018102\u0000103. In general \u001b(z)\ni/Ak\ni, which\n68101214\n0 50 100 150 200 250 300T∗\n2[ns]\nNumber of cycles\nspin-orbit excluded\nθ= 0\nθ=π/6\nθ=π/3\nθ=π/2Figure 8. (Color online) S\u0000T0qubit decoherence time\nT\u0003\n2as a function of the number of DNP cycles across the\nS\u0000T+anticrossing and strength of spin-orbit interaction in\nIn0:2Ga0:8As. Here,\u0012is the angle between the [110] crystal-\nlographic axis and the interdot connection axis p\u0018.\n22.052.12.152.22.252.3\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6T∗\n2,f/T∗\n2,i\nθ[rad]\nspin-orbit excluded\nspin-orbit included\nFigure 9. (Color online) The ratio of the \fnal T\u0003\n2;fand initial\nT\u0003\n2;idecoherence times in In 0:2Ga0:8As, for di\u000berent values of\nthe angle\u0012between the [110] crystallographic axis and the\ninterdot connection axis p\u0018.\nwould give rise to a much higher standard deviation of\nthe nuclear di\u000berence \feld than expected. Therefore, we\nrescale the hyper\fne constant, such that \u001b(z)\nihas the\nsame value for N\u0019106, andN= 150,jml=p\nN=2.\nThe predicted decoherence time before the start of the\nDNP isT\u0003\n2\u00196:2 ns while measurements yield T\u0003\n2\u001910 ns\nfor pure GaAs [5] (where excluding spin-orbit e\u000bects is a\ngood approximation). Since \u001b(z)\ni/p\nN, and\u001b(z)\nfdoes\nnot depend on Nbut on di\u000berent parameters, we can\nestimate that T\u0003\n2;f=T\u0003\n2;i\u0018p\nNfor our case of N= 150\nand the realistic case N= 106(for an electrically de-\n\fned quantum dot in In xGa1\u0000xAs). Therefore, we can9\n68101214\n0 50 100 150 200 250 300T∗\n2[ns]\nNumber of cycles\nx= 0%\nx= 20%\nx= 40%\nx= 60%\nx= 80%\nx= 100%\nFigure 10. (Color online) S\u0000T0electron spin coherence time\nT\u0003\n2as a function of the number of DNP cycles across the\nS\u0000T+anticrossing, for di\u000berent abundances of indium xin\nInxGa1\u0000xAs and for \u0012=\u0019=2. Here,\u0012is the angle between\nthe [110] crystallographic axis and the interdot connection\naxisp\u0018.\nestimate the maximum possible ratio of initial and \f-\nnal decoherence times for the realistic case of N= 106\nspins and spin-orbit interaction excluded and included\nto beT\u0003\n2;f=T\u0003\n2;i\u0019175 without spin-orbit interaction,\ncompared to T\u0003\n2;f=T\u0003\n2;i\u001994 for GaAs in reference [23],\nT\u0003\n2;f=T\u0003\n2;i\u0019174 for\u0012= 0,T\u0003\n2;f=T\u0003\n2;i\u0019163 for\u0012=\u0019=2.\nVI. RESULTS FOR In xGa1\u0000xAs\nIn this section we will compare the T\u0003\n2results for\nInxGa1\u0000xAs with varying In content x. We vary the\nconcentration of indium xin the sample between 0 and\n1 with a 0:2 increment. For the sake of computational\ne\u000eciency, and the fact that we are interested in a mere\ncomparison between materials with di\u000berent percentages\nof indium, our computational method is slightly simpli-\n\fed now. Instead of averaging over all possible states\nranging from jmin\u0000jmaxwe setjL=jR=jml=p\nN=2\nfor the left and the right quantum dot. This e\u000bectively\nmeans that we are simulating a situation where an exper-\niment is performed only once with the most likely nuclear\nspin con\fguration.\nFrom Fig. 10 we conclude that raising the concentra-\ntion of indium in a In xGa1\u0000xAs sample has a detrimental\ne\u000bect on the e\u000eciency of the S\u0000T+DNP scheme. By\ndoping the system with indium, the Rashba spin-orbit\ncoupling is strengthened, thus reducing the overall \u0003 SO\n[Eq. (19)], which as a consequence has more virtual and\nrealT+outcomes due to the spin-orbit interaction. The\nvirtualT+will relax to S, quickly \ripping a nuclear spin\nfrom down to up in the process. The real spin-orbit me-\ndiatedT+outcomes will also pump the nuclear spin to-\n68101214\n0 50 100 150 200 250 300T∗\n2[ns]\nNumber of cycles\nGaAs\nInAsFigure 11. (Color online) S\u0000T0electron spin coherence time\nT\u0003\n2for GaAs and InAs as a function of the number of DNP\ncycles across the S\u0000T+anticrossing, for \u0012= 0, i.e. the\ncase where the [110] crystallographic axis and the interdot\nconnection axis p\u0018are aligned.\nwards the positive values of the polarization (up). This\nprocess can completely vain e\u000borts to increase T\u0003\n2, even\nat intermediate concentrations of 40% In (Fig. 10). At\nhigher indium concentrations, DNP is totally suppressed\nfor all values of \u0012[cf. Fig. 11].\nVII. CONCLUSIONS AND FINAL REMARKS\nOur results show that pure InAs is a not a suitable can-\ndidate forS\u0000T+DNP, due to the fact that the enhance-\nment ofT\u0003\n2is strongly suppressed even for the smallest\npossible strength of the spin-orbit interaction correspond-\ning to\u0012= 0. Dynamical nuclear polarization in InAs\ncould still be achieved by using single spin single quan-\ntum dot systems [29] or by using a more elaborate pulsing\nsequence [30]. A similar behavior could be expected in\nmaterials with even stronger spin-orbit as compared to\nInAs and that is, e.g., InSb.\nTo conclude, we have discussed a nuclear polarization\nscheme in In xGa1\u0000xAs double quantum dots with spin-\norbit interaction included. In the presence of spin-orbit\ninteraction a suppression of the enhancement of T\u0003\n2is\npredicted. Our conclusions are also valid for materials\nwith fewer nuclear spins. We underline that the S\u0000T+\nDNP sequence is highly sensitive to the strength of the\nspin-orbit coupling, and therefore the e\u000eciency of the\nS\u0000T+DNP sequence will depend on the angle \u0012and\nthe In content xin InxGa1\u0000xAs. A stronger spin-orbit\ninteraction will establish a process that will quickly neu-\ntralize any e\u000borts to prolong T\u0003\n2. 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Lett. 96, 100501\n(2006)."    },    {        "title": "0710.5193v2.Electron_Transport_Driven_by_Nonequilibrium_Magnetic_Textures.pdf",        "content": "arXiv:0710.5193v2  [cond-mat.mes-hall]  2 Apr 2008Electron Transport Driven by Nonequilibrium Magnetic Text ures\nYaroslav Tserkovnyak and Matthew Mecklenburg\nDepartment of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA\n(Dated: November 30, 2018)\nSpin-polarized electron transport driven by inhomogeneou s magnetic dynamics is discussed in the\nlimit of a large exchange coupling. Electron spins rigidly f ollowing the time-dependent magnetic\nprofile experience spin-dependent fictitious electric and m agnetic fields. We show that the electric\nfield acquires important corrections due to spin dephasing, when one relaxes the spin-projection\napproximation. Furthermore, spin-flip scattering between the spin bands needs to be taken into\naccount in order to calculate voltages and spin accumulatio ns induced by the magnetic dynamics.\nA phenomenological approach based on the Onsager reciproci ty principle is developed, which allows\nus to capture the effect of spin dephasing and make a connectio n to the well studied problem of\ncurrent-driven magnetic dynamics. A number of results that recently appeared in the literature are\nrelated and generalized.\nPACS numbers: 72.15.Gd,72.25.Ba,75.47.-m,75.75.+a\nInterest in magnetic heterostructures,1which was ini-\ntially fueled by the discovery of the giant magnetoresis-\ntance and, a decade later, by the current-induced switch-\ning in spin valves and related systems, has more recently\nspilled over into current-driven phenomena in magnetic\nbulk, individual magnetic films, and nanowires.2Par-\nticular attention was given to the problems of current-\ndriven Doppler shift of spin waves, magnetic instabilities,\nand domain-wall motion. The latter has also enjoyed a\nvery vibrant experimental activity, which is in part mo-\ntivated by a promising application potential in spintron-\nics. The past year3,4,5,6saw a revival of interest in the\ninverseeffect of electromotiveforcesinduced by the time-\ndependent magnetization, which were previously studied\nin variousphysicalcontexts(see, e.g., Refs.7,8,9). Inthis\npaper, we will exploit the reciprocal relation between the\ntwo phenomena, which will allow us to understand im-\nportant spin-dephasing corrections to the electromotive\nforce. Such correctionswerefirst mentioned in Ref. 4 and\nthe Onsager principle in the present context was invoked\nin Ref. 5. Reference 3 reported the magnetically-induced\nelectromotive forces as a manifestation of the position-\ndependent Berry phase accumulation and Ref. 6 consid-\nered these forces acting on semiclassical wave packet mo-\ntion, mainly reproducing results from Ref. 9. For com-\npleteness, it shouldalsobe mentioned that amuch earlier\npaper10alreadycontains some seminal phenomenological\ninsights related to the problem of the electric response to\nthe magnetic domain-wall dynamics.\nIn the following, we start by recalling how most of the\nresults recently discussed in the literature can be cap-\ntured by an SU(2) gauge transformation together with\nthe projection of spins on the magnetic direction.9The\ncorrections due to the remaining transverse spin dynam-\nics are governed by spin dephasing, which have already\nbeen studied for the reciprocal process of current-driven\nmagnetic dynamics,2and can be translated to the cur-\nrent problem by the Onsagerprinciple. We will develop a\ngeneral framework, that will allow us to relate and gener-\nalizethemorespecializedcases,whichwererecentlystud-ied using different methods.3,4,5,6Finally, we will derive\nspin-charge diffusion equations, accounting for spin-flip\nscattering and respecting local charge neutrality, which\nis necessary in order to relate the microscopic electromo-\ntive forces to measurable quantities, such as an induced\nvoltage and spin accumulation.\nMost of our analysis will pertain to the following time-\ndependent Hamiltonian:\nH(t) =p2\n2m+∆xc\n2ˆσ·m(r,t)+Vc(r,t)+Hσ.(1)\nHere,Hσis the contribution due to spin-relaxation pro-\ncesses, which will be characterized by a Bloch-type T1\nspin flipping and T2spin dephasing, and Vc(r,t) stands\nfor a Hartree charging potential, which will be taken into\naccount only insofar as enforcing local charge neutral-\nity. ∆ xcis the ferromagnetic exchange band splitting, ˆσ\nis the vector of Pauli spin matrices, and mstands for\nthe local magnetization direction unit vector, so that the\nmagnetization is given by M=Mm. The exchange field\n∆xcmmay in practice be provided by localized magnetic\ndorbitals (as in the so-called s−dmodel) or it may\nself-consistently be governed by the itinerant electron\nspin density (as in the Stoner model or local spin-density\napproximation).2We will first perform a microscopic cal-\nculation for the idealized Hamiltonian (1) neglecting Hσ\nand subsequently utilize the Onsager theorem to capture\nthe spin-dephasing corrections. The spin-flip scattering\nwill be included phenomenologically in the final diffusion\nequation.\nBy disregarding Hσ, we can perform an SU(2) gauge\ntransformation by rotating mto point along the zaxis\nfor allrandt.9,11This is conveniently achieved by the\nHermitian spin-rotation matrix ˆU=ˆσ·n(such that ˆU=\nˆU†=ˆU−1), where nis the unit vector n∝m+z. It is\neasy to see that ˆU(ˆσ·m)ˆU= ˆσz(sinceˆUcorresponds\nto aπ-angle spin rotation around n). By applying this\ngauge transformation to the spinor wave function, we get2\nfor the transformed Hamiltonian\nH′(t) =1\n2m/parenleftBig\np−ˆA/parenrightBig2\n+ˆV+∆xc\n2ˆσz+Vc,(2)\nwhere the SU(2) vector potential is given by ˆAi=\ni/planckover2pi1ˆU∇iˆU=−/planckover2pi1ˆσ·(n×∇in) and the SU(2) ordinary po-\ntential is ˆV=−i/planckover2pi1ˆU∂tˆU=/planckover2pi1ˆσ·(n×∂tn) (setting the par-\nticle chargeandspeed oflighttounity). p=−i/planckover2pi1∇isthe\ncanonical momentum. If the exchange field ∆ xcis large\nand the magnetic texture is sufficiently smooth and slow,\nwe can project the fictitious potentials on the zaxis as\nˆV→Vˆσz, whereV=/planckover2pi1z·(n×∂tn) =/planckover2pi1sin2(θ/2)∂tφ, and\nsimilarly for the vector potential, Ai=−/planckover2pi1z·(n×∇in) =\n−/planckover2pi1sin2(θ/2)∇iφ, where ( θ,φ) are the spherical angles\nparametrizing m. We thus get for the effective electric\nfield4,6,9\nE=−∂tA−∇V=/planckover2pi1\n2sinθ[(∂tθ)(∇φ)−(∂tφ)(∇θ)],\n(3)\nor, written in the explicitly spin-rotationally-invariant\nform,\nEi=/planckover2pi1\n2m·(∂tm×∇im). (4)\nThe effective magnetic field is6,9,12\nB=∇×A=/planckover2pi1\n2sinθ(∇φ)×(∇θ)\n=/planckover2pi1\n4ǫijkmi(∇mk×∇mj),(5)\nwhereǫijkis the antisymmetric Levi-Civita tensor, and\na summation over repeated indices is implied. The total\nforce on a spin- ↑(↓) electron moving with velocity vis\nthus given by\nF↑(↓)=±(E+v×B). (6)\nEqs.(3)and(5)werefirstderivedinRef.9andrecently\nrederived within a semiclassical wave-packet analysis.6\nThe gauge-transformation based approach9puts the re-\nsult into a broader perspective, allowing us, for exam-\nple, to consider the effect of the magnetic field (5) on\nthe quantum transport corrections, such as a weak local-\nization, as well as to include spin-independent electron-\nelectron interactions, which would not modify fictitious\nfields (3) and (5). Note, in particular, that the magnetic\nfield (5) can in practice be quite large: For example, for\na static magnetic variation on the scale of 10 nm, the\ncorresponding fictitious field is of the order of 10 T. It\nis most convenient to estimate the strength of the elec-\ntric field (3) by the characteristic voltage it induces over\na region where the magnetization direction flips its di-\nrection: /planckover2pi1ω/e, whereωis the frequency of the magnetic\ndynamics. In the following, we will concentrate on the\nsemiclassical spin and charge diffusion generated by the\neffective electric field E. In order to make a closer con-\nnection to the experimentally relevant quantities, we willneed to take into accountspin relaxationand alsoenforce\nlocal charge neutrality for electron diffusion.\nThe role of spin relaxation can be twofold. First of all,\nspin accumulation, which will generally be generated by\nthe spin-dependent force (6) will relax, characterized by\nthe longitudinal spin-flip time T1. There is also another\nmore subtle effect, which is due to the dephasing of elec-\ntronspins followinga dynamic magneticprofile, since the\nexchange field ∆ xcis not infinite and spins do not per-\nfectly align with the local magnetization. Hence, there is\ngenerallyafinite spinmisalignment, which dephaseswith\na characteristic time T2. This gives corrections to the re-\nsults obtained by a rigid projection of spins on the local\nmagnetization direction. We will see that such correc-\ntions turn out to be important for the currents generated\nby magnetic dynamics, in the same sense that analogous\ncorrectionsare crucial for understanding current-induced\nmagnetic motion.2\nLet us now take a step aside, by recalling the general\nexpression for the dynamics of an isotropic ferromagnet\nwell below the Curie temperature:2\n∂tM=−γM×Heff+α\nMM×∂tM\n+/planckover2pi1(σ↑−σ↓)\n2S(∇iµ)/parenleftbigg\n1−β\nMM×/parenrightbigg\n∇iM,(7)\nwhich is valid for spatially smooth magnetic profiles (the\nso-called adiabatic approximation) and weak currents.\nHere,αis the Gilbert damping constant, βis another di-\nmensionless phenomenological parameter whose physical\nmeaning will be discussed later, µis the electrochemical\npotential, σsis the spin- sconductivity (along the local\nmagnetization direction m) relating particlecurrents to\n∇µ,Sis the equilibrium spin density of the ferromag-\nnet along m, andγ=M/Sis the gyromagnetic ratio.\nRecall that the effective field Heffis the quantity defined\nto bethermodynamically conjugate to the magnetization:\nHeff=∂MF(note the sign difference from the standard\ndefinition), where Fis the free energy and ∂Mstands\nfor the functional derivative. The other thermodynamic\nvariable we will consider is the electron density ρ(r,t),\nwhose thermodynamically conjugate counterpart is the\nelectrochemical potential µ=∂ρF.\nSuppose we perturb the electron density with respect\nto an equilibrium with some static magnetic texture\nand uniform chemical potential, and consider the en-\nsuing magnetic response. Eq. (7) then describes the\nnonequilibrium coupling of the magnetization dynam-\nics to the electron density’s thermodynamic conjugate,\nwhich is slightly out of equilibrium. The Onsager reci-\nprocity principle13allows us to immediately write down\ntheresponseoftheelectrondensitytoasmallmodulation\nof the effective field Heff, with respect to an equilibrium\nconfiguration. To simplify things, let us for a moment\ndisregard Gilbert damping αin Eq. (7) and return to\ninclude it later on. An electric response to a magnetic3\nperturbation then becomes15\n∂tρ=−γ/planckover2pi1(σ↑−σ↓)\n2∇i{Heff·[(1+βm×)∇im]}.(8)\nBy comparingEq. (8) with the continuity equation ∂tρ=\n−∇iji, we can identify the particle current as\nji=γ/planckover2pi1(σ↑−σ↓)\n2Heff·[(1+βm×)∇im].(9)\nSince for each spin species, js=σsFs, whereFsis the\neffective force, we finally get for the latter F↑,↓=±F16,\nwhere\nFi=/planckover2pi1\n2(m×∂tm)·[(1+βm×)∇im]\n=/planckover2pi1\n2[m·(∂tm×∇im)+β(∂tm·∇im)],(10)\nafter inverting the magnetic equation of motion (7) in\norder to express the effective field Heffin terms of the\nmagnetization dynamics m(r,t). (Note that since the\ncurrents themselves are now generated by the magneti-\nzation dynamics, we can neglect their backaction on the\nmagnetic response, when inverting the equation of mo-\ntion to express Heffin terms of m, since it would give\nrise to higher-order terms.) Equation (10) is a key result\nof this paper. It is also easy to show that taking into\naccount Gilbert damping αhas no consequences for the\nfinal result (10) [after rewriting Eq. (7) in the Landau-\nLifshitz form, in order to eliminate the ∂tterm on the\nright-hand side and thus make the equation suitable for\nthe Onsager theorem]. This is not surprising, since the\nphysics of the Gilbert damping αdoes not have to be\nrelated to the magnetization—particle-density coupling\nthat determines the force (10).2\nPhysically, the βcorrection in Eq. (10) is related\nto a slight spin misalignment of electrons propagating\nthrough an inhomogeneous magnetic texture with the lo-\ncal direction of the magnetization m. In the limit of\n∆xc→ ∞, this misalignment vanishes and so should β,\nreducing the result (10) to Eq. (4). Indeed, a microscopic\nderivation of Eq. (7) shows β∼/planckover2pi1/(T2∆xc), where T2is\nthe characteristic transverse spin relaxation time.2The\nβterm in Eq. (10) can thus be viewed as a correction to\nthe topological structure of the electron transport rigidly\nprojected on the magnetic texture, due to the remaining\ntransverse spin dynamics and dephasing. Such a βcor-\nrection was first reported in Ref. 4, which used a very\ndifferent and more technical language and did not bene-\nfit from the reciprocity relation with the current-driven\nmagnetic dynamics (7). Our phenomenological deriva-\ntion of Eq. (10) based on the Onsager theorem provides\na much simpler frameworkfor studying these subtle spin-\ndephasing effects.\nLet us now discuss the measurable consequences of\nEq. (10) in two simple scenarios sketched in Fig. 1. Con-\nsider a nontrivial one-dimensional magnetic profile along\nthexaxis, such as a magnetic domain wall in a nar-\nrowwire, withnegligibletransversespininhomogeneities.\nFIG. 1: Two simple scenarios for voltage generation by the\nmagnetic dynamics: (1) Magnetic texture m(x,t), such as a\ndomain wall along the xaxis, is steadily rotating around the\nxaxis and (2) the same texture rigidly sliding along the x\naxis. In the former case, the force Fxacting on electrons is\nproportional to the frequency of rotation ω, with the domi-\nnant term having a purely geometric meaning in terms of the\nposition-dependent Berry-phase accumulation rate. (An al -\nternative physical explanation can also be provided by tran s-\nforming to the rotating frame of reference and applying the\nLarmor theorem.) In the case of the sliding dynamics, the\nleading contribution to the magnetically-induced force is pro-\nportional to the spin-dephasing rate (parametrized by β) and\nthe “curvature” of the texture profile ( ∂xm)2.\nFirst, let us look at a steady rotation of the entire one-\ndimensional texture around the xaxis, with a constant\nfrequency ω. Then,∂tm=ωx×mand\n∆V=−/integraldisplay\ndxFx=−/planckover2pi1ω\n2x·/integraldisplay\n(dm+βm×dm).(11)\nIn the absence ofspin dephasing β, this result can be eas-\nily understood by transforming into the rotating frame\nof reference: By the Larmor theorem, this corresponds\nto a fictitious field along the xaxis:H′=−(/planckover2pi1ω/2)ˆσx.\nFor spins up (down) projected on the local magnetiza-\ntion direction, this corresponds to the potential V=\n∓(/planckover2pi1ω/2)x·m. It is equally straightforward to interpret\nthis result in terms of the rate of the Berry phase accu-\nmulation by spins adiabatically following the steady ex-\nchange field precession,3,14which is proportional to the\nposition-dependent solid angle enclosed by spin preces-\nsion. The βterm in Eq. (11) gives a correction to these\nidealistic considerations, which depends on the geometry\nof the magnetic texture. Next, we consider the voltage\ninduced by a rigid translation of a one-dimensional mag-\nnetictexture m(x−vt) alongthe xdirectionwithvelocity\nv. The corresponding force\nFx=−/planckover2pi1\n2βv(∂xm)2(12)\nis then entirely determined by the βterm, which drags\nspins down along the direction of the magnetic texture\nmotion and spins up in the opposite direction. This is\nanalogous to the current-driven domain wall velocity in\none dimension, which, for smooth walls and low currents,\nis proportional to β.24\nFinally, we need to include spin-flip relaxation time\nT1and derive spin-charge diffusion equations, enforcing\nlocalchargeneutrality. Assumingdiffusive transport, the\nforce (10) can now be added as a contribution to the\ngradient of the effective electrochemical potential. The\ndiffusion equation for spin- sparticles is then given by\n(∂t−Ds∇2)ρs+σs/parenleftbig\ns∇·F−∇2Vc/parenrightbig\n=ρ−s\nτ−s−ρs\nτs,(13)\nwhereρsis the nonequilibrium (spin- s) particle density,\nDsis the diffusion coefficient, and τsis the spin-flip time.\nRecall that the conductivity is related to the density of\nstatesNsbytheEinstein’srelation: σs=NsDs.Vcisthe\nelectric potential, which has to be found self-consistently\nby enforcing local charge neutrality. Note that the equi-\nlibrium considerations require that τs/τ−s=Ns/N−s.\nWe shouldalsostressthatthe force(10) mayhaveafinite\ncurl, so that we cannot generally describe it by a ficti-\ntious potential. After straightforward manipulations, we\ncan decouple the diffusion equation for the spin accumu-\nlationµσ(defined as the difference between the spin-up\nand spin-down electrochemical potentials, divided by 2)\nfrom the average electrochemical potential µas follows:\n/parenleftbig\n∂t+τ−1−D∇2/parenrightbig\nµσ=−D∇·F,\n−∇2µ=P/parenleftbig\n∇2µσ−∇·F/parenrightbig\n.(14)\nHere,P= (σ↑−σ↓)/(σ↑+σ↓) is the conductivity polar-ization,D= (D↑+D↓)/2−P(D↑−D↓)/2 is the effective\nspin-diffusion constant, and τ−1=τ−1\n↑+τ−1\n↓is the char-\nacteristic T−1\n1rate for spin flipping. If ∇×F= 0, we\ncan integrate the second equation to express the electro-\nchemical potential gradient in terms of the force Fand\nthe spin accumulation gradient as follows:\n∇µ=P(F−∇µσ), (15)\nassuming the appropriate boundary conditions. Accord-\ning to Eqs. (14), the spin accumulation decays in the\nabsence of the force Fon the scale of the spin-diffusion\nlengthλsd=√\nDτ. Away from the dynamic magnetic\ntexture (on the scale of λsd), the generated electrochemi-\ncal potential (15) will then be determined simply by inte-\ngrating the force F. In general, however, especially when\n∇×F∝ne}ationslash= 0, one has to revert to Eqs. (14).\nIn summary, we theoretically studied electron trans-\nport generated by a dynamic magnetization texture. We\nreproduced and generalized the results that recently ap-\npeared in literature,3,4,6revealing an intricate connec-\ntion with the theory of the current-induced magnetiza-\ntion dynamics.2We expect that in practice it is consider-\nably simpler to solve this reciprocal problem, especially\nfor including subtle corrections to the topological Berry\nphase structure of spins assumed to rigidly follow the\ntime-dependent magnetic profile.\n1D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320,\n1190 (2007), and references therein.\n2Y. Tserkovnyak, A. Brataas, and G. E. Bauer, J. Magn.\nMagn. Mater. 320, 1282 (2008), and references therein.\n3S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98, 246601\n(2007).\n4R. A. Duine, Phys. Rev. B 77, 014409 (2008).\n5W. M. Saslow, Phys. Rev. B 76, 184434 (2007).\n6S. A. Yang, D. Xiao, and Q. Niu, cond-mat/0709.1117.\n7A. Stern, Phys. Rev. Lett. 68, 1022 (1992).\n8M. Stone, Phys. Rev. B 53, 16573 (1996).\n9G. E. Volovik, J. Phys. C: Sol. State Phys. 20, L83 (1987).\n10L. Berger, Phys. Rev. B 33, 1572 (1986).\n11G. Tatara, H. Kohno, J. Shibata, Y. Lemaho, and K.-J.\nLee, J. Phys. Soc. Jpn. 76, 054707 (2007).\n12J. Ye, Y. B. Kim, A. J. Millis, B. I. Shraiman, P. Majum-\ndar, and Z. Teˇ sanovi´ c, Phys. Rev. Lett. 83, 3737 (1999).\n13L. D. Landau and E. M. Lifshitz, Statistical Physics, Part\n1, vol. 5 of Course of Theoretical Physics (Pergamon, Ox-\nford, 1980), 3rd ed.\n14M. V. Berry, Proc. R. Soc. London A 392, 45 (1984).\n15Note that the Onsager reciprocity relations are somewhat\nspecial in the present case: since one of the thermody-namic quantities (namely, the magnetization) is odd un-\nder time reversal, the matrix that couples the rates of the\nrelaxation of the involved quantities to their thermody-\nnamic conjugates is antisymmetric, which is subject also\nto changing the sign of the magnetic field and the equi-\nlibrium magnetization.13Hence the overall minus sign in\nEq. (8) and the sign change in front of β.\n16Strictly speaking, the analysis based on Eq. (7) can only\ncapture thetotal particle current(9). Inorder torigorous ly\ncalculate the spin-resolved forces, we would have to explic -\nitly take into account one more thermodynamic variable,\nnamely, the nonequilibrium spin accumulation (which is\nproportional to the chemical potential mismatch between\nthe up and down electrons along the local magnetization)\nand consider its action on the magnetic dynamics. The\nlatter program would push us too much off track, and we\nchoose not to pursue it here. We only wish to note that\ntheβterm describing the spin-dephasing correction to the\nfictitious field (4) may in general become different for the\ntwo spin species when the ferromagnetic exchange energy\nis comparable to the Fermi energy."    },    {        "title": "1611.03378v3.Spin_charge_coupled_dynamics_driven_by_a_time_dependent_magnetization.pdf",        "content": "Spin-charge coupled dynamics driven by a time-dependent magnetization\nSebastian T olle,1Ulrich Eckern,1and Cosimo Gorini2\n1Universit at Augsburg, Institut f ur Physik, 86135 Augsburg, Germany\n2Universit at Regensburg, Institut f ur Theoretische Physik, 93040 Regensburg, Germany\nThe spin-charge coupled dynamics in a thin, magnetized metallic system are investigated. The\ne\u000bective driving force acting on the charge carriers is generated by a dynamical magnetic tex-\nture, which can be induced, e.g., by a magnetic material in contact with a normal-metal system.\nWe consider a general inversion-asymmetric substrate/normal-metal/magnet structure, which, by\nspecifying the precise nature of each layer, can mimick various experimentally employed setups. In-\nversion symmetry breaking gives rise to an e\u000bective Rashba spin-orbit interaction. We derive general\nspin-charge kinetic equations which show that such spin-orbit interaction, together with anisotropic\nElliott-Yafet spin relaxation, yields signi\fcant corrections to the magnetization-induced dynamics.\nIn particular, we present a consistent treatment of the spin density and spin current contributions\nto the equations of motion, inter alia identifying a novel term in the e\u000bective force which appears\ndue to a spin current polarized parallel to the magnetization. This `inverse spin \flter' contribution\ndepends markedly on the parameter which describes the anisotropy in spin relaxation. To further\nhighlight the physical meaning of the di\u000berent contributions, the spin pumping con\fguration of\ntypical experimental setups is analyzed in detail. In the two-dimensional limit the build-up of a DC\nvoltage is dominated by the spin galvanic (inverse Edelstein) e\u000bect. A measuring scheme that could\nisolate this contribution is discussed.\nKeywords: charge and spin transport, spin-orbit coupling, spin pumping, metallic \flm, Boltzmann theory\nI. INTRODUCTION\nThe active control of the spin degrees of freedom in a\nsolid state system is the central concern of spintronics.1\nThe exchange coupling between the magnetization and\nthe spin of charge carriers is routinely exploited two ways:\nto generate spin currents and non-equilibrium spin polar-\nizations, and to employ such currents and polarizations|\ngenerated by other means|to exert a torque on the mag-\nnetization. In this work we are concerned with the \frst\nscenario only, though all setups that will be discussed\ncan, and typically are, used for both purposes.\nIn this context, spin pumping2{5and the inverse spin\nHall e\u000bect (ISHE)5{9are the tools of choice for gener-\nation and detection of electronic spin currents, respec-\ntively. The typical spin pumping setup consists of a\nmagnet10/normal-metal bilayer. The magnetization of\nthe magnetic material is driven such that it performs\na conical precession, and a spin current perpendicular\nto the interface (here, along the z-direction) builds up,\njz\u0018g\"#\nrn\u0002_n+g\"#\ni_n. The vector components of jz, i.e.,\nja\nz, witha=x;y;z; represent the spin polarization, nis\nthe instantaneous magnetization direction, and g\"#\nr(g\"#\ni)\nis the real (imaginary) part of the spin-mixing conduc-\ntanceg\"#.3Due to the ISHE in the bulk of the normal\nmetal, this spin current can be detected by measuring the\ninverse spin Hall voltage appearing therein. The inverse\nspin Hall voltage is associated with the spin Hall angle\n\u0012sH, which is de\fned as the ratio of the spin Hall and\ncharge conductivities. Large spin Hall angles are typi-\ncally found in transition metals such as Au,9,11Pt7,9,12{14\nor Ta.14,15The same class of setups is also used to study\nthe reciprocal e\u000bect, when spin currents generated in the\nnormal layer enter the magnetic material and exert atorque on its magnetization.13,15{17The spin-galvanic ef-\nfect (SGE),18,19which can be related to the ISHE,20,21\nrepresents another channel for spin-to-charge conversion.\nIt is also referred to as the inverse Edelstein e\u000bect,22,23\nand consists in the generation of a charge current per-\npendicular to the polarization of a nonequilibrium spin\ndensity. Of course, its inverse can as well be used to\ninduce a torque on magnetizations.24,25\nBesides the magnet/normal-metal system just dis-\ncussed, di\u000berent spin pumping setups are possible. For\nexample, spin-charge coupled transport in a Fe/GaAs\nbilayer can be understood as taking place in an e\u000bec-\ntive two-dimensional (2D) magnetized electron gas at\nthe Fe/GaAs interface,26,27which can be regarded as a\nmagnet/normal-metal system with the normal metal in\nthe 2D-limit. Indeed, experimentally realized thin \flms\nspan the range of thicknesses from a few monolayers28,29\nup to tens of nanometers,11,14,30so that the full three-\ndimensional (3D) to 2D range is available. Clearly, the\nanalysis of spin pumping is di\u000berent in the 3D or 2D\nscenario. In the latter case no spin current can \row\nperpendicular to the 2D metal, while in-plane spin cur-\nrents will be generated by the driving magnetization as\nsoon as in-plane spin-orbit coupling is taken into account,\nthus leading to in-plane ISHE physics. We will concen-\ntrate on the 2D to quasi-2D regime, in a sense to be\nmade more precise later, and connect our analysis to\nthe one usually performed for 3D systems in the clos-\ning. Note that this kind of 2D analysis is also relevant\nfor magnet/topological-insulator structures, which have\nbeen recently employed for both spin pumping and recip-\nrocal torque-inducing purposes,31,32due to the intrinsic\n2D nature of the topological surface states.\nTypically, spin pumping is most e\u000bective as long as\nthe thickness of the \flm does not exceed the spin re-arXiv:1611.03378v3  [cond-mat.mes-hall]  2 Mar 20172\nlaxation length of the normal metal.3,4In thin \flms, on\nthe other hand, Elliott-Yafet scattering, an important\nspin-relaxation mechanisms in various metals,1should\nbe more e\u000bective for in-plane spins than for out-of-plane\nones|in the 2D limit it does not lead to any relaxation of\nthe out-of-plane spins at all.33Furthermore, corrections\narise in the presence of magnetic textures and intrinsic\nspin-orbit \felds, and indeed such corrections turn out to\nbe necessary for the physical consistency of the spin dy-\nnamics. These corrections are taken into account, as well\nas anisotropic spin relaxation.\nWhile the magnetization of the spin pumping se-\ntups mentioned above is homogeneous, the situation is\neven more interesting in the presence of magnetic tex-\ntures/spin waves, when complex spin-charge and magne-\ntization dynamics takes place.34{37Hence, we will con-\nsider the general situation where the driving is due to\na time-dependent magnetic texture, whose spatial and\ntemporal pro\fle can have any form, and only needs to\nbe smooth on the Fermi wavelength and energy scales.\nWe will model the thin metallic system as a nearly\nfree electron gas, and employ an SU(2)-covariant kinetic\nformulation38to compute the e\u000bective forces which act\non the conduction electrons. The latter are generated by\nthe interplay of the magnetization dynamics and spin-\norbit coupling. We remark that our kinetic treatment\ncan include \fner details of the spin-orbit \feld, such as\nthose described in Ref. 26, where the latter is shown to\ndepend on the direction of the main (static) component\nof the magnetization. However, in order to focus on the\nessentials and avoid overburdening the equations, we as-\nsume a Rashba-like spin-orbit \feld only. Such an e\u000bective\n\feld is taken to be homogeneous across the whole|not\nnecessarily strictly 2D|sample, similar to Refs. 17 and\n39. The opposite limit of a bulk metal with a sharp \u000e-\nlike Rashba spin-orbit coupling at the interface has also\nbeen considered,21,40{42and recently discussed in great\ndetails.43,44\nThe outline of the article is as follows. We \frst (Sec. II)\nintroduce the system, and connect its model form to real-\nworld structures. In so doing we also clarify the meaning\nof the important parameters related to the physical en-\nergy scales of the problem. In Sec. III we introduce the\nmodel in detail, as well as the transport equations for the\ncharge and spin distribution functions. The general the-\noretical results, in particular, the derivation of the gener-\nalized e\u000bective force acting on the conduction electrons\nin the presence of spin-orbit coupling, are presented in\nSec. IV. Secs. III and IV are technically more involved,\nand can be skipped by the reader mostly interested in\ntheir speci\fc physical consequences. An experimentally\nrelevant example is dealt with in Sec. V, which analyzes\nthe typical spin pumping con\fguration. More precisely,\nwe show that the build-up of a DC electric \feld in a\nnarrow metallic \flm is mainly due to the SGE and sub-\nstantially modi\fed by spin relaxation, and suggest that\nthis can be probed by comparing a longitudinal and an\northogonal measurement on the same sample. Here wealso comment on the connection between the 2D analysis\nof spin pumping, and the established 3D one. A brief\nconclusion is given in Sec. VI. Finally, the appendices\nshow the detailed derivation of the collision integrals and\nof the generalized spin di\u000busion equations.\nII. THE SYSTEM AND ITS ENERGY SCALES\nFerromagnet\nNormal metal\nSubstraten(r, t)\nStructure\ninversion\nasymmetry\n(a)\nFe\nGaAs2DEG\n(b)\nFIG. 1. A sketch of the considered structure is shown in (a);\nn(r;t) is the magnetization direction in the magnetic mate-\nrial, here illustrated as a N\u0013 eel domain wall. The precise nature\nof the magnetic layer, e.g., ferro- or ferrimagnetic insulator, is\ninconsequential for our treatment, although it will determine\nthe value of the physical parameters entering the e\u000bective\nHamiltonian. The same holds for the normal metal, whose\nthickness can be anything from a few monolayers (2D) up to\ntens of nanometres (3D). Electrons therein feel an e\u000bective\nRashba spin-orbit \feld due to inversion symmetry breaking,\nas well as (random) spin-orbit scattering from impurities. The\nsubstrate is a generic structureless insulator, possibly the vac-\nuum. Panel (b) shows a possible experimental realization of\na spin pumping setup, where precession of the (here homoge-\nneous) Fe magnetization drives the spin-charge dynamics of\na 2D electron gas formed at the interface with GaAs.\nThe system consists of a substrate/normal-\nmetal/magnetic material structure, as sketched in\nFig. 1(a), and is characterized by various energy scales,\nwhich will now be introduced. First of all, the Fermi\nenergy\u000fFis assumed to be much larger than any other\nrelevant energy, i.e., we are dealing with a `good metal'.\nWe then assume a proximity induced magnetization in\nthe metallic \flm. The coupling between the itinerant s-\nand the localized d-electrons, i.e., the induced magnetic\ntexture, is described within the s-dmodel:\nHsd= \u0001 xcn(r;t)\u0001\u001b\n2; (1)3\nwhere\u001b= (\u001bx;\u001by;\u001bz) denotes the vector of Pauli matri-\nces, \u0001 xcthe ferromagnetic exchange band splitting, and\nnthe magnetization direction. At \frst sight this model\nmight appear questionable, since we wish to study the\ndynamics in the non-magnetic metallic \flm. However, it\nis known that metals like Pt or Pd can be magnetized\ndue to the magnetic proximity e\u000bect,45{47and that the\nexchange energy may be large, i.e., much larger than the\ndisorder broadening, \u0001 xc\u001c=~\u001d1, with\u001cthe momentum\nrelaxation time. The precise value of \u0001 xcdepends on the\nmaterial properties of the magnetic and non-magnetic\nlayers, and of the interface.\nFurthermore, we assume two types of spin-orbit cou-\npling, a Rashba-like spin-orbit term due to structure in-\nversion asymmetry, see Fig. 1(a), and extrinsic spin-orbit\ncoupling due to impurities. The Rashba spin-orbit cou-\npling Hamiltonian reads\nHR=\u0000\u000b\n~\u001b\u0002^z\u0001p; (2)\nwith the Rashba parameter \u000b, estimated to be between\n0:03 and 3 eV \u0017A, depending on the structure and mate-\nrial properties of the system.48The associated spin-orbit\nsplitting \u0001 so= 2\u000bpF=~is taken as small, in the sense\nthat \u0001 so\u001c=~\u001c1. This condition is often appropriate\nand, in addition, useful for obtaining physically trans-\nparent equations for the spin-charge coupled dynamics.\nHowever, since typical values for the spin-orbit splitting\nare in the range 10\u00003:::10\u00001eV,26,28,29this condition\nis not universally realistic. Extrinsic spin-orbit coupling\nwith impurities is described by\nHext=\u0000\u00152\n4~\u001b\u0002rV(r)\u0001p; (3)\nwhere\u0015is the e\u000bective Compton wavelength, whose\nstrength is material and impurity-type dependent, and\nV(r) is the disorder potential. Due to the two\ntypes of spin-orbit coupling, both Dyakonov-Perel (DP)\nand Elliott-Yafet (EY) spin relaxation mechanisms are\npresent. The corresponding energies are given by\n~\n\u001cDP=~\u00122m\u000b\n~2\u00132\nD; (4)\n~\n\u001cs=~\n\u001c\u0012\u0015pF\n2~\u00134\n; (5)\nrespectively, with the e\u000bective mass m, the Fermi mo-\nmentumpF, and the di\u000busion constant D=v2\nF\u001c=d,\nwhered= 2;3 represents the dimensionality. Note\nthat the expression for ~=\u001cDPfollows from the condi-\ntion \u0001 so\u001c=~\u001c1. Dyakonov-Perel relaxation is intrin-\nsically non-isotropic, since the Rashba term (2) contains\nonly in-plane momenta, and while its strength depends\non the dimensionality of electronic motion through D,\nits anisotropy does not. Elliot-Yafet relaxation is, on the\nother hand, strongly anisotropic only in the 2D limit.\nHere, however, `2D' does not refer to the electronic mo-\ntion, being rather determined by the ratio of the metalthickness,tm, to the spin relaxation length: Elliott-Yafet\nis 2D (3D) roughly for tmsmall (large) in this sense. The\ntransition is modelled by introducing a phenomenologi-\ncal parameter 0 <\u0010 < 1, with\u0010= 0$2D; \u0010 = 1$3D\n(see Sec. III). We focus on the experimentally relevant\nregime 1=\u001cDP\u001d1=\u001cs.22Together with the strong ex-\nchange assumption, \u0001 xc\u001c=~\u001d1,47this leads to the fol-\nlowing hierarchy of energy scales:\n~\n\u0001xc\u001cs|{z}\n\fs\u001c~\n\u0001xc\u001cDP|{z}\n\fDP\u001c~\n\u0001xc\u001c\u001c1\u001c~\n\u0001so\u001c(6)\nEquation (6) de\fnes the spin torque parameters \fsand\n\fDP, which will appear repeatedly below.\nThe magnetization is assumed to be smooth on the\nFermi wavelength ( \u0015F) scale, and the frequency of its\ntime-dependence is taken as small compared to the spin-\n\rip rate,\n!\u001cs=~\u001c1; (7)\napplicable for the typical adiabatic pumping regime. In\nSec. IV, we will in addition consider the di\u000busive regime,\n!\u001c;ql\u001c1; (8)\nwithqthe typical wavevector of the system inhomo-\ngeneities, and l=vF\u001cthe mean free path.\nIII. THE KINETIC EQUATIONS\nIn order to describe the transport phenomena in the\npresence of spin-orbit coupling and a magnetic texture,\nwe use the SU(2) formulation of the Boltzmann-like\nequation.38The Hamiltonian of the system reads\nH=1\n2m\u0012\np+Aa\u001ba\n2\u00132\n+e\b+\ta(r;t)\u001ba\n2+V(r)+Hext;\n(9)\nwhere Aais an SU(2) vector potential which describes\nthe intrinsic spin-orbit coupling, \b is the electric poten-\ntial withe=jej, and \tais an SU(2) scalar potential.\nHere and throughout the paper, upper (lower) indices\nwill indicate spin (real space) components. A summa-\ntion over repeated indices is implied.\nFor the system as discussed in Sec. II, we have\nHsd ! \ta(r;t) = \u0001 xcna(r;t); (10)\nHR ! Ax\ny=\u0000Ay\nx=2m\u000b\n~; (11)\ncompare Eqs. (1) and (2).\nAccording to Ref. 38 and for \u000e-correlated (short-range)\ndisorder, the Boltzmann equation for the distribution\nfunctionf=f0+f\u0001\u001b, withf0(f) the particle (spin)\ndistribution function, reads\n~@tf+p\nm\u0001~rrf+1\n2fF;rpfg=\u00001\n\u001c(f\u0000hfi) +IEY[f];\n(12)4\nwhereh:::idenotes the angular average w.r.t. the mo-\nmentum. The covariant time (spatial) derivative ~@t(~rr)\nand the generalized force Fare given by\n~@t=@t\u0000i\n~\u0014\n\ta\u001ba\n2;\u0001\u0015\n; (13)\n~rr=rr+i\n~\u0014\nAa\u001ba\n2;\u0001\u0015\n; (14)\nF=\u0000eE\u0000\u0010\nE+p\nm\u0002B\u0011a\u001ba\n2; (15)\nand\nEa\ni=\u0000ri\ta\u00001\n~\"abc\tbAc\ni; (16)\nBa\ni=\u00001\n2~\"ijk\"abcAb\njAc\nk; (17)\nridenoting the i-th component of rr. The dot within\nthe commutator in Eqs. (13) and (14) is a placeholder for\nthe object on which the covariant derivative acts. Using\nEqs. (10) and (11) the i-th component of the generalized\nforce reads\nFi=\u0000eEi+ \u0001 xc\u0010\n~rin\u0011\n\u0001\u001b\n2\u00002m\u000b2\n~3\"ijzpj\u001bz:(18)Thei-th component of the (3D) covariant derivative ~ri\nis de\fned as\n~ri=ri+ [ai]\u0002; (19)\nwhere we have introduced the notation for an anti-\nsymmetric matrix [ v]\u0002with its components de\fned by\n([v]\u0002)ab=\u0000\"abcvc. This de\fnition corresponds to a\ncross product in the sense that [ v]\u0002b=v\u0002bfor an\narbitrary vector b. The vector aiis de\fned as ai=\n2m\u000b=~2(\u0000\u000eiy;\u000eix;0). Analogously, from now on we use\nthe `3D covariant time derivative' de\fned as\n~@t=@t+\u0001xc\n~[n]\u0002: (20)\nThe quantity IEYon the r.h.s. of Eq. (12) is the Elliott-\nYafet collision operator, representing spin-\rip processes.\nIt follows from the impurity averaged self-energy as de-\npicted in Fig. 2, and is substantially modi\fed in the\npresence of intrinsic spin-orbit coupling and magnetic\ntextures. The corrections are obtained via a \frst-order\nSU(2) shift (see App. A), which yields the following gen-\neralized collision integral:\nIEY=I0\nEY+\u000eI\t\nEY+\u000eIA\nEY; (21)\nwhere\nI0\nEY=\u00001\n2N0\u001c\u0012\u0015p\n2~\u00134Z\ndp0\u000e(\u000fp\u0000\u000fp0) [\u0000 (fp+fp0)]\u0001\u001b; (22)\n\u000eI\t\nEY=mp2\nN0\u001c\u0012\u0015\n2~\u00134Z\ndp0\u000e(\u000fp\u0000\u000fp0)(\u0000\t)\u0001h\nf0\np\u001b+fp\u0000\u0010\n1 +\u000fp0@\u000fp0\u0011\u0010\nf0\np0\u001b\u0000fp0\u0011i\n; (23)\n\u000eIA\nEY=1\nN0\u001c\u0012\u0015\n2~\u00134Z\ndp0\u000e(\u000fp\u0000\u000fp0)Aa\u0001Lp;p0\u0002\n\u001ba\u0000\nf0\np\u0000f0\np0\u0001\n+\u0000\nfa\np+fa\np0\u0001\u0003\n; (24)\nwith\t= \u0001 xcn,dp0\u0011ddp0=(2\u0019~)d,Lp;p0= (p02+p\u0001\np0)p\u0000(p2+p\u0001p0)p0, the density of states per volume\nand spinN0, and \u0000 = diag(1 ;1;\u0010). The latter takes into\naccount the anisotropy of spin-\rip processes, as discussed\nabove, hence depends on the thickness tmof the normal\nmetal. Clearly 0 \u0014\u0010\u00141, with\u0010= 0 representing the\nlimit that the normal metal is a 2D gas, i.e., for small\ntm, whereas one may assume \u0010= 1 whentm>`s, where\n`sis the spin relaxation length of the normal metal. We\nemphasize that Eqs. (23) and (24) are valid for arbitrary\nspin-orbit \felds, not only Rashba coupling, and magnetic\ntextures.\nThe above expressions, Eqs. (23) and (24), are `\frst-\norder' in the SU(2) \felds, provided we take the spin dis-\ntribution function fto be `zero-order'. However, as dis-\ncussed in the next section in connection with Eq. (30), f\ncontains a local-equilibrium part, feq, which formally is\nalso `\frst-order'. Thus, in order to treat relaxation dueto the EY collision operator consistently, it becomes nec-\nessary to include also a speci\fc second-order correction\nin the SU(2) shift, which is given by\n\u000eIA;\t\nEY=\u00001\n4\u001c\u0012\u0015\n2~\u00134\n\t\u0001Aipip2@\u000fp\u0000\n\u000fp@\u000fpf0\neq\u0001\n;(25)\nwheref0\neqis the Fermi function. As a consequence, only\nthe non-equilibrium part of the spin density, \u000es, will enter\nthe e\u000bective force, Eq. (48). An even more detailed inves-\ntigation of the EY collision operator is well underway.49\nIV. SPIN-CHARGE COUPLED DYNAMICS\nHere we present the coupled equations for the electron\ndensity, the electron current, the spin density, and the5\nFIG. 2. Impurity averaged self-energy which determines\nthe Elliott-Yafet collision operator. The boxed crosses, the\ndashed line, and the arrowed double line represent spin-orbit\ncoupling, impurity correlations, and the Green's function in\nKeldysh space, respectively.\nspin current, respectively de\fned as follows:50\nn= 2Z\ndpf0; (26)\nji= 2Z\ndppi\nmf0; (27)\ns=Z\ndp f; (28)\nja\ni=Z\ndppi\nmfa: (29)\nWe focus \frst on the spin sector (IV A), and second on\nthe charge sector (IV B), and third discuss the interpre-\ntation of the di\u000berent contributions to the e\u000bective force\n(IV C).\nA. Spin sector\nIn order to study the spin sector, we multiply the\nBoltzmann equation with the Pauli vector \u001band per-\nform the trace. Before doing so, it is convenient to split\nthe spin distribution function fas\nf=feq+\u000ef (30)\nwithfeq=\u0000\n\u0000@\u000fpf0\neq\u0001\n(\u0001xc=2)n, wheref0\neqis the Fermi\nfunction. This is motivated by the form of the spin den-\nsity\ns=seq+\u000es; (31)\nwhere seq= (N0\u0001xc=2)nis the equilibrium part of s\nwhich adiabatically follows the magnetization. The dy-\nnamics of the itinerant electrons, which is typically much\nfaster than the magnetization dynamics, leads to the\nnonequilibrium contribution \u000es.\nWe trace over the spin sector and obtain the following\n3\u00023 matrix equation:\nM\u000ef=Nh\u000efi+S; (32)\nwith\nM= 1 +\u001c\n2\u001cs\u0000 +\u001c~@t+\u001cpi\nm~ri; (33)\nN= 1\u0000\u001c\n2\u001cs\u0000; (34)\nS=\u0000\n@\u000fpf0\neq\u0001\u001c\u0001xc\n2_n: (35)Note that we have neglected small deviations of f0from\nits angular average, f0'hf0i, since these are at least\n\frst order in the electric \feld Eor the magnetic texture,\ni.e.,rinor_n. Furthermore we assume hf0i'f0\neq.\nBy an integration of the spin sector over the momen-\ntum we obtain\n~@t\u000es+~riji=\u00001\n\u001cs\u0000\u000es\u0000N0\u0001xc\n2_n: (36)\nNext, we consider the quasiadiabatic limit, \u001cs@t\u000es\u001c\u000es,\nas well as\u001cs@t\u000es\u001c\u0010\u000es. We are then able to solve for\nthe nonequilibrium spin density:\n\u000es= (\u000es)n+ (\u000es)js; (37)\nwith\n(\u000es)n=\u0000N0\u0001xc\u001cs\n2\u0000\n\u0000 +\f\u00001\ns[n]\u0002\u0001\u00001_n (38)\nthe part of \u000eswhich is associated directly with the mag-\nnetization, and with\n(\u000es)js=\u0000\u001cs\u0000\n\u0000 +\f\u00001\ns[n]\u0002\u0001\u00001~riji (39)\nthe part which is associated with the spin current. In\ngeneral the spin current itself depends on the spin density,\nwhich has to be kept in mind when solving for the spin\ndensity from Eq. (36). The split in Eq. (37) is found to\nbe technically convenient.\nIn the following, we shall calculate the spin current\nin the di\u000busive regime. Our approach is to rewrite the\nmatrix Min Eq. (32) as follows:\nM= (1 +\u0018)M; (40)\nwhere\nM=\u0012\n1 +\u0001xc\u001c\n~[n]\u0002\u0013\n; (41)\nand\n\u0018=\u0012\u001c\n2\u001cs\u0000 +\u001c@t+\u001cpi\nm~ri\u0013\nM\u00001: (42)\nIn the di\u000busive regime we can approximate (1 + \u0018)\u00001'\n1\u0000\u0018, and rewrite Eq. (32) as\n\u000ef=M\u00001(1\u0000\u0018)\u0010\nNh\u000efi+S\u0011\n: (43)\nBy multiplying the latter equation with pi=mand inte-\ngrating over the momentum, we obtain the spin current:\nji= (ji)n+ (ji)s; (44)\nwith\n(ji)n=\u001c\n2DN0\u0001xcM\u00001~riM\u00001_n (45)\nthe part arising directly from the magnetization, and\n(ji)s=\u0000DM\u00001~riM\u00001\u000es (46)\nthe part of the spin current which has its source in the\nspin density.6\nEffective force\nFi= (Fi)s+ (Fi)jsSpin density\nδs= (δs)n+ (δs)jsSpin current\nji= (ji)n+ (ji)sMagnetization\nFIG. 3. Scheme of the various contributions to the e\u000bective\nforce.\nB. Charge sector\nFor the charge sector, we trace over the Boltzmann\nequation (12), multiply with pi, and integrate over the\nmomentum, with the following result:\n(1 +\u001c@t)ji+Drin=\u0000n\u0016Ei+\u001cN0\u0001xc\nmFi;(47)\nwhere\u0016=e\u001c=m is the electron mobility, and n\u0016=\u001bD=e\nwith\u001bDthe Drude conductivity. The e\u000bective force Fi\ncombines the contributions of the nonequilibrium part of\nthe spin density and the spin current.\nA scheme of the various contributions to the e\u000bective\nforce is depicted in Fig. 3. We split Fi= (Fi)s+ (Fi)js\ninto two terms which represent the contribution of the\nspin density, ( Fi)s, and a term representing the direct\ncontribution of the spin current, ( Fi)js. According to\nthis split it is clear that ( Fi)sis associated with the SGE,\nand (Fi)jswith the ISHE. The two contributions to the\ne\u000bective force explicitly read\n(Fi)s=1\nN0\u0014\u0010\n~rin\u0011\n\u0001\u000es+2m\u000b\n~2\fs(^z\u0002\u000es)i\u0015\n;(48)\n(Fi)js=1\nDN0\u0014\n\fDP(jz\u0002^z)i+\u001c\n\u001csn\u0010\u0001ji\u0015\n; (49)\nwithn\u0010= \u0000n. With respect to the second term on the\nr.h.s. of Eq. (48), compare the discussion in connection\nwith Eq. (25).\nWe further divide ( Fi)sinto contributions arising from\n(\u000es)nand (\u000es)js:\n(Fi)s= (Fi)s;n+ (Fi)s;js; (50)\nwhere (Fi)s;nand (Fi)s;jshave the same form as ( Fi)sin\nEq. (48), but with \u000esbeing replaced by ( \u000es)nand (\u000es)js,\nrespectively. Analogously, we de\fne\n(Fi)js= (Fi)js;n+ (Fi)js;s (51)\nwith (Fi)js;nand (Fi)js;sof the same form as ( Fi)jsin\nEq. (49), but with jibeing replaced by ( ji)nand ( ji)s,\nrespectively. The idea behind this separation is to express\nthe e\u000bective force in terms of the drive, _n, and the spin\ncurrent, the subscripts indicating the respective origins.1. The contribution (Fi)s;n\nThe spin density related directly to the dynamical\nmagnetization, see Eq. (38), is explicitly given by\n(\u000es)n=~N0\n21\nn2\n\u0010\u0014\nn\u0010\u0002_n\u0000\fs\u0010\u0000\u00001_n\u0015\n: (52)\nWe insert Eq. (52) into Eq. (48) and \fnd\n(Fi)s;n=~\n21\nn2\n\u0010(\n(rin)\u0001\u0000\nn\u0010\u0002_n\u0000\fs\u0010\u0000\u00001_n\u0001\n+2m\u000b\n~2\u0002\n(n\u0001n\u0010)^z\u0002_n\n+\fs^z\u0002(n\u0010\u0002_n+\u0010n\u0002\u0000\u00001_n)\u0003\ni)\n:(53)\nAssuming\u0010= 1 (i.e.,tm\u001d`s) the last equation reduces\nto Eq. (11) in Ref. 51; see also Refs. 35, 52{56. Recall\nthat\u0010describes the anisotropy of spin-\rip relaxation and\nthat\u0010= 1 corresponds to the isotropic case. As far as we\nknow, this is the \frst time that such anisotropy ( \u0010 <1)\nis explicitly taken into account. However, experiments\non thin \flms typically deal with samples on the scale of\na few nanometres, hence it is appropriate to include this\ne\u000bect.\n2. The contributions (Fi)s;jsand(Fi)js;n\n(homogeneous case)\nFor the sake of simplicity and since we are mostly in-\nterested in the competition between the SGE and the\n(in-plane) ISHE, we shall focus here on the Rashba con-\ntribution, i.e., we consider ~ri\u0019[ai]\u0002in Eqs. (39) and\n(48), which corresponds to a spatially homogeneous sit-\nuation. Neglecting terms \u0018\f2\nsand smaller for the spin\ndensity dependent contribution, we \fnd\n(Fi)s;js=1\nDN01\nn2\n\u0010\fDP\u001a\n(n\u0001n\u0010)\u0002^z\u0002(jz)n\u0003\ni\n+ 2X\na=x;y\u0002\nnz(ja\na)n\u0000na(jz\na)n\u0003\n(^z\u0002n)i\u001b\n:(54)\nAdding the contribution by ( ji)n[Eq. (49) with ji!\n(ji)n] we obtain\n(Fi)s;js+(Fi)js;n\n=1\nDN0(\n\fDP\u0010(1\u0000\u0010)n2\nz\nn2\n\u0010[^z\u0002(jz)n]i\n+ 2\fDPX\na=x;y\u0002\nnz(ja\na)n\u0000na(jz\na)n\u0003\n(^z\u0002n)i)\n+\u001c\n\u001csn\u0010\u0001(ji)n: (55)7\nNote that ( Fi)js;nfeatures a direct contribution due to\nthe driving source, i.e., _n, whereas ( Fi)js;scontributes\nmore indirectly through \u000es. Apparently Eq. (55) yields a\nnon-trivial interplay between the two `origins' (spin den-\nsity versus spin current, cf. Eqs. (50) and (51)) since the\n\frst term on the r.h.s. of Eq. (49) is cancelled to some\nextent by the \frst term on the r.h.s. of Eq. (54). This\ndemonstrates once more that the interplay between SGE\nand ISHE is non-trivial.\nC. Discussion: e\u000bective forces\nWe emphasize that the above expressions, Eqs. (48)\nand (49), obtained by properly integrating the kinetic\nequation, are of general validity. Before going into the\ndetails of the spin pumping con\fguration (Sec. V), we\ncomment on the relation of these equations to previous\nresults. First, neglecting the Rashba contribution in the\n\frst term on the r.h.s. of Eq. (48), i.e., ~rin! rin,\nconsidering the isotropic case ( \u0010= 1), and taking into\naccount that\n\u000es=~N0\n2[n\u0002_n\u0000\fs_n] (56)\nin this limit, this force term agrees with the result given\nin Ref. 35. Second, including the Rashba contribution\nbut neglecting the spin current contribution to \u000es, cf.\nEq. (39), and again for \u0010= 1, Eq. (48) reduces to Eq. (11)\nin Ref. 51. Third, the \frst term on the r.h.s. of Eq. (49),\nwhich is due to Rashba spin-orbit coupling, is related to\nthe ISHE: a charge current in the x-y-plane is generated\nby a spin current (in that plane) polarized in z-direction,\nwith the charge current direction being perpendicular to\nthe spin current direction, \u0018jz\u0002^z, cf. Ref. 6.\nNote, however, that the various terms are not inde-\npendent from each other. In particular, spin density and\nspin current are closely related, as already pointed out\nin Ref. 57, due to the interplay of Rashba coupling and\nEY relaxation. This is apparent from Eq. (36), which for\ntime-independent and spatially homogeneous situations\nreads:\n\u0001xc\n~n\u0002\u000es+ai\u0002ji=\u00001\n\u001cs\u0000\u000es: (57)\nTaking this equation into account, it is possible to relate\nthe total e\u000bective force derived here to the one discussed\nin Ref. 23 (see Eq. (12) therein). First, consider the limit\n\u0001xc!0, which leaves only the second term in (48) and\nthe \frst term in (49). The latter is readily identi\fed\nwith the `Hall-like' force;23however, using Eq. (57) we\n\fnd that the new term, Eq. (48), which results from the\nEY collision operator, Eq. (24), gives an identical contri-\nbution, thus our result appears to be larger by a factor of\ntwo. Further di\u000berences become apparent for \fnite \u0001 xc.\nLast but not least, the second term on the r.h.s. of\nEq. (49) is denoted `inverse spin \flter' term, as it de-\nscribes a force arising from a spin current which is po-\nlarized parallel the magnetization (roughly speaking), itsstrength being\u0018\u001c\u00001\ns. To the best of our knowledge,\nsuch a term has not been explicitly considered before.\nHowever, it can be related to the anomalous Hall e\u000bect:\nimagine that an electric \feld in x-direction creates a spin\ncurrent via the spin Hall e\u000bect (in y-direction, polarized\ninz-direction). This spin current leads via the inverse\nspin \flter term to a charge current in y-direction. Note\nthat a non-zero \u0010is required for this argumentation to\nbe valid. In this context, see also the discussions given\nin Refs. 58 and 47.\nV. SPIN PUMPING CONFIGURATION\nn(t)z\nxyjy\nEx\n(a)\n (b)\nFIG. 4. (a) The studied con\fguration, i.e., a metallic \flm on\ntop of a magnetic material (shown in blue). (b) Sketch of the\nconical precession of the magnetization, de\fning the relevant\nangles\u0012and\u001e.\nIn this section we consider the magnetization dynamics\nto be \fxed, namely as a precession with a cone angle\n\u0012and angular frequency !about an axis \fxed by an\nexternal static and homogeneous magnetic \feld.59The\nmagnetization direction is parametrized as follows:\nn(t) =R\u001e0\n@n0\n\u000eny(t)\n\u000enz(t)1\nA=R\u001e0\n@cos\u0012\nsin\u0012cos!t\nsin\u0012sin!t1\nA; (58)\nwithR\u001ea rotation matrix around the z-axis,\nR\u001e=0\n@cos\u001e\u0000sin\u001e0\nsin\u001ecos\u001e0\n0 0 11\nA; (59)\nwhere\u001eis the angle between the x-axis and the cone\naxis, see Fig. 4.\nFurthermore, we assume open circuit conditions in x-\nandy-direction, in order to determine the electric \feld\nalong these directions. Since the magnetization is ho-\nmogeneous we expect the particle current to be homoge-\nneous as well, and due to the open circuit condition we\nhavejx;y= 0 in the whole sample. From Eq. (47) we \fnd\n\u001bDEx;y=e\u001cN 0\u0001xc\nmFx;y: (60)8\nAccording to Ref. 57 the spin Hall conductivity can be\nexpressed as60\u001bsH=e~\u001cN0=2m\u001cs(for\u001cs\u001d\u001cDP), hence\nwe may rewrite Eq. (60) as\neEx;y=2\u0012sH\n\fsFx;y; (61)\nwhere\u0012sH=e\u001bsH=\u001bDis the spin Hall angle. In order of\nmagnitude, \u0012sH\u0018~=\u000fF\u001cs. Our focus in the following is\non the DC contribution to the electric \feld, thus we will\naverage Eq. (61) with respect to time.\nLet us now explicitly consider the x-component of the\ne\u000bective force. According to Eqs. (53) and (55), as well\nas Eq. (49) with ji!(ji)s, and it is the sum of the\nfollowing three contributions:\n(Fx)s;n=\u0000m\u000b\n~\u0014\n_ny+\fs(1 +\u0010) (n\u0002_n)y\u0015\n;(62)\n(Fx)s;js+ (Fx)js;n\n=\u0000\fDP\nDN0\u001a\n\u0010(1\u0000\u0010)n2\nz\u0000\njz\ny\u0001\nn\n+2nyX\na=x;y[nz(ja\na)n\u0000na(jz\na)n]\u001b\n+1\nDN0\u001c\n\u001csn\u0010\u0001(jx)n; (63)\n(Fx)js;s=1\nDN0\u0014\n\fDP\u0000\njz\ny\u0001\ns+\u001c\n\u001csn\u0010\u0001(jx)s\u0015\n;(64)\nwhere, in order to derive Eqs. (62) and (63), we approx-\nimated n2\n\u0010'1 and n\u0010\u0001n'1 since the cone angle \u0012is\nusually small.8For this reason, we shall also allow only\nterms up to sin2\u0012when performing the time average,\nhence we neglect terms of order n2\nzsince the time aver-\nage would lead to \u0018sin4\u0012terms. We realize that the\n\frst term on the r.h.s. of Eq. (63), which has its origin\nin the interplay of the spin density and the spin current,\nis negligible.\nWe rewrite Eqs. (62){(64) in order to elucidate the\ndi\u000berent e\u000bects:\nF(A)\nx=\fDP\nDN0\u0000\njz\ny\u0001\ns; (65)\nF(B)\nx=\u0000m\u000b\n~\u0014\n_ny+\fs(1 +\u0010) (n\u0002_n)y\u0015\n\u00002\fDP\nDN0nyX\na=x;y[nz(ja\na)n\u0000na(jz\na)n];(66)\nF(C)\nx=1\nDN0\u001c\n\u001csn\u0010\u0001jx; (67)\nwhereF(A)\nxcan be related to the ISHE, and F(B)\nxto the\nSGE. The last term, F(C)\nx, describes the build-up of an\ne\u000bective force (or electric \feld) due to the spin current\npolarized parallel to the magnetization,61and is denoted\ninverse spin \flter term, as discussed in Sec. IV B.Analogously, we obtain for the y-component:\nF(A)\ny=\u0000\fDP\nDN0(jz\nx)s; (68)\nF(B)\ny=m\u000b\n~\u0014\n_nx+\fs(1 +\u0010) (n\u0002_n)x\u0015\n+2\fDP\nDN0nxX\na=x;y[nz(ja\na)n\u0000na(jz\na)n];(69)\nF(C)\ny=1\nDN0\u001c\n\u001csn\u0010\u0001jy: (70)\nIn the following subsections, we consider a narrow wire\n(see Fig. 5) and the electric \feld that will be measured in\na longitudinal and an orthogonal measurement. We as-\nsume that the wire is `narrow' such that the width of the\nwire is smaller than the spin di\u000busion length `s=pD\u001cs.\nFor such a con\fguration, the spin current contribution\npolarized parallel to the magnetization and \rowing par-\nallel to the narrow edge vanishes, see App. B.\nA. Longitudinal measurement\nLet us consider a narrow wire as depicted in Fig. 5. For\nsuch a sample we \fnd a homogeneous spin current \rowing\ninx-direction which, in particular, has a contribution\npolarized along n, giving rise to the force in Eq. (67) when\nperforming a longitudinal measurement. We perform the\ntime average of Eqs. (65){(67), insert the results into\nEq. (61), and obtain the following DC electric \felds:\nD\nE(A)\nxE\nt\u0018\fDP\u0012sHF\u000b\ne\u001c\u0012sHF\u000b\ne; (71)\nD\nE(B)\nxE\nt=\u00002\u0012sHF\u000b\ne(1 +\u0010) sin\u001esin2\u0012; (72)\nD\nE(C)\nxE\nt=\u0000\u0012sHF\u000b\ne(1\u0000\u0010) sin\u001esin2\u0012; (73)\nwithF\u000b\u0011\u000b!m=~. We realize that the ISHE ( A) term\nplays only a minor role for the total electric \feld, hExit=\nhE(A)\nx+E(B)\nx+E(C)\nxit. Note that for \u0010'0 the inverse\nspin \flter contribution is of the same order of magnitude\nas the SGE term, whereas it vanishes for \u0010= 1.\nB. Orthogonal measurement\nIn the case of an orthogonal measurement, see Fig. 5,\nr.h.s., the contribution given in Eq. (70) vanishes since\nthe spin current jylacks a contribution parallel to the\nmagnetization ( ly\u001c`s). For the DC electric \feld along\ny-direction, we \fnd\nD\nE(A)\nyE\nt\u0018\fDP\u0012sHF\u000b\ne\u001c\u0012sHF\u000b\ne; (74)\nD\nE(B)\nyE\nt=\u00002\u0012sHF\u000b\ne(1 +\u0010) cos\u001esin2\u0012; (75)\nD\nE(C)\nyE\nt= 0; (76)9\nleaving only the SGE term to contribute to the total DC\nelectric \feld.\nC. Discussion\nFIG. 5. Bottom part of \fgure: top view of the setup; the\nlength is denoted lx, the widthly(ly\u001clx). Top part of \fgure:\nqualitative plot of the DC electric \felds, heExitandheEyit,\nfor a longitudinal (left) and orthogonal (right) measurement.\nIn both cases we set \u0010= 0.\nComparing the results for the longitudinal and the or-\nthogonal measurement, we see that the signal can be up\nto 1:5 times larger in the longitudinal measurement (for\n\u0010= 0), see Fig. 5. For a 2D electron gas one should thus\nbe able to probe a Rashba-induced SGE, and by com-\nparing samples of di\u000berent thicknesses, to additionally\nobtain estimates of \u000band\u0010.\nRecent articles3,8,9,62discussed spin pumping and the\ninduced ISHE on the basis of a spin current jzwhich \rows\nperpendicular to the interface, i.e., in z-direction into the\nnormal-metal \flm. This is signi\fcantly di\u000berent from the\nsituation we are discussing here ( jz= 0). Nevertheless,\nthe electric \feld estimated in such a way8shows the same\nangular-dependence of the magnetization as our result:\nRef. 8)Ex\u0018Fg\"#sin\u001esin2\u0012; (77)\nEq. (72))Ex\u0018F\u000bsin\u001esin2\u0012: (78)\nComparing the relevant forces, Fg\"#=e2!g\"#=(4\u001bD)\nandF\u000b, for reasonable parameter values, g\"#\u0019\n2:1\u00021019m\u00002and\u001bD\u00192:4\u0002106\n\u00001m\u00001for a Pt\n\flm,8we \fnd the forces to be of the same order of mag-\nnitude for \u000b\u00190:3 eV\u0017A. Thus we conclude that the\nSGE contribution and the inverse spin \flter e\u000bects due\nto Rashba-induced spin currents and spin density, as dis-\ncussed here, should both be taken into account when in-\nterpreting experiments.\nVI. CONCLUSIONS\nWe have studied the spin-charge coupled dynamics in\na magnetized thin metallic \flm with Rashba spin-orbit\ncoupling. In particular, we have considered a generalized\nElliott-Yafet collision integral, valid for arbitrary spin-\norbit \felds and magnetic textures, and taken into account\nanisotropic spin-\rip processes. Signi\fcant modi\fcationsof the kinetic equations describing spin and charge trans-\nport have been found. The e\u000bective force acting on the\ncharge carriers in the presence of spin-orbit coupling has\nbeen derived in a very general form, and evaluated in\ndetail for the case of a time-dependent texture. For a\nnarrow wire in the typical spin pumping con\fguration an\nin-plane electric \feld is generated, for which the spin gal-\nvanic e\u000bect is crucial, while the (in-plane) spin Hall e\u000bect\nturns out to be negligible. However, an additional con-\ntribution of similar strength from an `inverse spin \flter'\ne\u000bect is found to be relevant for a longitudinal measure-\nment, while it vanishes for an orthogonal measurement.\nThis suggests the possibility of determining the strength\nof the spin galvanic e\u000bect and the spin-orbit coupling\nparameter|Rashba in our speci\fc scenario|by perform-\ning both measurements on the same sample.\nACKNOWLEDGMENTS\nWe acknowledge stimulating discussions with C. Back,\nL. Chen, M. Decker, and R. Raimondi. We especially\nare indebted to L. Chen for helpful comments on the\nmanuscript. We acknowledge \fnancial support from the\nGerman Research Foundation (DFG) through TRR 80\n(UE, ST) and SFB 689 (CG).\nAppendix A: The Elliott-Yafet collision operator\nIn this appendix we follow the procedure outlined in\nRef. 38. We start by deriving the Elliott-Yafet colli-\nsion operator within \frst order in the SU(2) \felds from\nthe microscopic Green's function Gand the Elliott-Yafet\nself-energy \u0006 EY(diagrammatically depicted in Fig. 2) in\nthe two-dimensional case, and comment on the three-\ndimensional case at the end.\nThe Elliott-Yafet collision operator is given by\nIEY=\u00001\n4\u0019~Z\nd\u000f~LK; (A1)\nwithL= [\u0006 EY;G]. The superscript Krepresents the\nKeldysh component, and the tilde the SU(2) shift; clearly\n~L= [~\u0006EY;~G]. In \frst order, we have\n~L\u0019L\u00001\n2\b\nA\u0016;@\u0016\npL\t\n; (A2)\nwith the four-potential A\u0016= (\u0000\t;A) and@\u0016\np=\n(@\u000f;rp). We recall that the components of the four-\npotential are SU(2) gauge \felds, i.e., \t = \ta\u001ba=2 and\nA=Aa\u001ba=2. In order to derive the explicit expression\nfor the collision operator, we need in the \frst step the im-\npurity averaged and SU(2) shifted self-energy, compare\nFig. 2, which reads\n~\u0006EY=~\u00060\nEY+\u000e~\u0006\t\nEY+\u000e~\u0006A\nEY; (A3)\nwith10\n~\u00060\nEY=CZd2p0\n(2\u0019~)2\u001bz~G(p0)\u001bz(p\u0002p0)2\nz; (A4)\n\u000e~\u0006\t\nEY=C\n2@\u000fZd2p0\n(2\u0019~)2\u0012\n\u001bz\u001a\n\ta\u001ba\n2;~G(p0)\u001b\n\u001bz\u0000\u001a\n\ta\u001ba\n2;\u001bz~G(p0)\u001bz\u001b\u0013\n(p\u0002p0)2\nz; (A5)\n\u000e~\u0006A\nEY=C\n2Zd2p0\n(2\u0019~)2\u0012\n\u001bz\u001a\nAa\u001ba\n2;[rp0~G(p0)]\u001b\n\u001bz\u0000\u001a\nAa\u001ba\n2;\u001bz~G(p0)\u001bz\u001b\nrp\u0013\n(p\u0002p0)2\nz; (A6)\nwhereC= (2\u0019\u001c~3N0)\u00001(\u0015=2)4. For the Green's func-\ntions we have\n~GK=\u0010\n~GR\u0000~GA\u0011\n(1\u00002f); (A7)\nwheref=f0+f\u0001\u001bdenotes the distribution function\n2\u00022 matrix; furthermore,\n~GR\u0000~GA=\u00002\u0019i\u000e(\u000f\u0000\u000fp): (A8)\nNote that the SU(2) shifted retarded and advanced\nGreen's functions are diagonal in spin. Inserting ~Linto\nEq. (A1) and using Eqs. (A4){(A8) leads to the collision\noperators as given in Eqs. (22){(24) for \u0010= 0, corre-\nsponding to the 2D case.\nWhen we consider the bulk 3D case we have the fol-\nlowing replacement:\n\u001bz:::\u001bz(p\u0002p0)2\nz!\u001ba:::\u001bb(p\u0002p0)a(p\u0002p0)b(A9)\nwithin the integrals in Eqs. (A4){(A6), respectively.\nThen we obtain, by the same procedure as outlined in\nSec. I, Eqs. (22){(24), for \u0010= 1, except for a small nu-\nmerical di\u000berence related to the angular average in 2D\nversus 3D. In order to describe the anisotropy in the in-\ntermediate regime, we insert the matrix \u0000 = diag(1 ;1;\u0010)\nwith 0\u0014\u0010\u00141; cf. Eqs. (22) and (23).\nAppendix B: Narrow wires\nHere we show that the contribution which is polarized\nparallel to the magnetization of the spin current \row-\ning in the narrow direction vanishes in the homogeneous\ncase. We consider a narrow wire along x-direction, i.e.,\nly\u001c`sandlx\u001d`s, and an open circuit condition,\njy(y= 0) = jy(y=ly) = 0. For the sake of simplic-\nity we put \u0010= 1. Since the system is homogeneous,\nwe assume that the spin current \rowing in x-direction is\nhomogeneous as well, and given by Eqs. (44){(46) with\n~rx![ax]\u0002. According to Eqs. (37){(39) the spin den-\nsity can be expressed as\n\u000es(y) =\u000es0\u0000\u001csM\u00001\ns~ryjy(y) (B1)\nwith\n\u000es0= (\u000es)n\u0000\u001csM\u00001\ns[ax]\u0002jx: (B2)In addition, Ms=M(\u001c!\u001cs), whereM, cf. Eq. (41), is\nexplicitly given by\nM= 1 +\u0001xc\u001c\n~[n]\u0002=\f\u00001\n\u001c0\n@\f\u001c\u0000nzny\nnz\f\u001c\u0000nx\n\u0000nynx\f\u001c1\nA;(B3)\nwhere\f\u001c=~=\u0001xc\u001c. The spin current \rowing in y-\ndirection is given by Eqs. (44){(46):\njy(y) =DM\u00001~ryM\u00001\u0000\nN0\f\u00001\n\u001c_n\u0000\u000es(y)\u0001\n: (B4)\nWe insert Eq. (B1) into Eq. (B4) and obtain approxi-\nmately ( ~ryjy'ryjy) the following di\u000berential equation:\n\u0000\n1\u0000`2\nsM\u00001\nsM\u00002r2\ny\u0001\njy(y) =jy;0; (B5)\nwhere the r.h.s., which is spatially constant, is given by\njy;0=DM\u00001[ay]\u0002M\u00001\u0000\nN0\f\u00001\n\u001c_n\u0000\u000es0\u0001\n: (B6)\nIt is apparent that jy;0is a particular solution of Eq. (B5).\nIn order to determine the complete solution jy=jy;h+\njy;0, we have to add the solution of the homogeneous\ndi\u000berential equation, which can be written as follows:\n\u0000\nMsM2\u0000`2\nsr2\ny\u0001\njy;h(y) = 0: (B7)\nIt is convenient to change the basis by the following trans-\nformation:\nR=0\n@nx_nx=j_nj(n\u0002_n)x=j_nj\nny_ny=j_nj(n\u0002_n)y=j_nj\nnz_nz=j_nj(n\u0002_n)z=j_nj1\nA; (B8)\nwhich replaces nbyexin Eq. (B7):\nh\u0000\n1 +\f\u00001\ns[ex]\u0002\u0001\u0000\n1 +\f\u00001\n\u001c[ex]\u0002\u00012\u0000`2\nsr2\nyi\n~|y;h(y) = 0;\n(B9)\nwith~|y;h=RTjy;h. Thex-component of ~|y;hrepresents\nthe contribution of the spin current which is parallel to\nthe magnetization. The matrix in Eq. 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B 67,\n104430 (2003)."    },    {        "title": "1106.3124v2.Synchronous_Spatial_Oscillation_of_Electron__and_Mn_Spin_Polarizations_in_Dilute_Magnetic_Semiconductor_Quantum_Wells_under_Spin_Orbit_Effective_Magnetic_Fields.pdf",        "content": "Synchronous Spatial Oscillation of Elect ron- and Mn-Spin Polarizations in \nDilute-Magnetic-Semiconducto r Quantum Wells under Spin-Orbit Effective Magnetic \nFields \n \nTakuma TSUCHIYA \n \nDivision of Applied Physics, Faculty of Engineering, Hokkaido University (Hokudai) \nKita 13 Nishi 8, Kita -ku, Sapporo 060-8628, Japan \n \n (Received                                 ) \n \nIn semiconductors, spin-orbit effective magnetic fields, i.e., the \nRashba and Dresselhaus fields, are used to control electron-spin polarization. This operation, howev er, destroys the electron-spin \ncoherence, and the spin polarization is limited to the vicinity of a ferromagnetic source electrode. In this paper, we propose the use of dilute magnetic semiconductors to improve the coherence of spatially \noscillating electron-spin polar ization. In dilute magnetic \nsemiconductors, the electron-spin polarization near the source \nelectrode dynamically induces the local spin polarization of magnetic impurities through s-d spin-flip scattering. This impurity-spin polarization improves, in turn, th e coherence of the electron-spin \npolarization, and this improved el ectron-spin polarization induces \nimpurity-spin polarization farther in  the adjacent region. Because of \nthis positive feedback, the coherent and synchronized spatial oscillations of electron- and impurity-spin polarizations grow \ncooperatively. A numerical calcula tion for a CdMnTe quantum well \ndemonstrates the validity of this mechanism.  Keywords: Rashba field, electron- spin precession, electron-spin \npolarization, dilute magnetic semiconductors, s-d interaction, \nCdMnTe, impurity Overhauser effect, Mn-spin polarization, \nquantum wells, Monte Carlo method  \n \n2 \n 1. Introduction \nControlling the spatial distri bution of electron-spin polariz ation, i.e., the local average \nof the spin vectors of many electrons, wit hout deteriorating electron-spin coherence is \nquite important for spintroni cs device applications.1) However, it is not an easy task. In \nsome devices, such as spin field-effect transistors (spin-FETs),2) electron spins, which are \ninjected into the two-dimensional (2D) ch annel from a ferromagnetic source electrode, \nare controlled through electron-spin precession caused by the Rashba3,4) and \nDresselhaus5) effective magnetic fields, both orig inating from the atomic spin-orbit \ninteraction. However, because these spin-orbit fields and the resulting electron-spin \nprecession depend on the electron wave vector, the spin coherence length is limited through the D’yakonov-Perel’ (DP) relaxation mechanism,\n6) and the electrons-spin \npolarization is localized  near the source electrode. Therefore, it is worthwhile to search \nfor methods to overcome the DP spin relaxation. \nDilute magnetic semiconducto rs (DMSs), investigated ac tively for over thirty years,7) \nhave been considered inappr opriate for coherent electron -spin transport. In these \nmaterials, various fascinating phenomena, su ch as carrier-induced ferromagnetism, giant \nZeeman effect, and spin polaron effect,8) are caused by the s-d exchange interaction \nbetween the spins of conduction s-electron s and localized d-electrons of magnetic \nimpurities. Regrettably, this s-d interaction destroys the electron-sp in coherence because \nof the s-d spin-flip scattering caused by the s- d interaction. Turning our attention to the \npolarization of impurity spins, however, we note a possibility to improve the electron-spin coherence. It was pointed out in  ref. 9 that the impurity spins are polarized \ndynamically by s-d spin-flip scattering unde r electron-spin polarization, and this \ndynamical impurity-spin polarization could induce ferromagnetic ordering even under  \n \n3 \n the small splitting of the up-sp in and down-spin quasi-Fermi le vels for carriers. This is a \nresult of the giant Zeeman splitting for electrons under the induced impurity-spin polarization. If the electr on-spin polarization oscillates spatially, the induced \nimpurity-spin polarization also os cillates. In this case, the electron spins are not always \nparallel to the impurity-spin polarization, and the electron-spin precession is expected to \nbe changed by the effective magnetic field originating from the impurity-spin polarization. It is quite desirable if this change in electron-spi n precession improves the \nelectron-spin coherence. \nIn this paper, we propose a possible mechan ism to improve the spatial electron-spin \ncoherence by DMSs and demonstrate its vali dity through a numerical  calculation for the \nspin transport of conduction electrons injected  from a ferromagnetic source electrode into \na 5 nm Cd\n0.99Mn 0.01Te quantum well (QW). This paper is  organized as follows. In §2, we \npropose a possible mechanism to overcome the DP spin relaxation for conduction \nelectrons. In §3, a method of nu merical calculation is explained. Numerical results are \nshown in §4, and §5 is devoted to a summary. In  Appendix, we give a brief review for the \nelectron-spin polarization in  nonmagnetic quantum wells. \n \n2. Dynamical Mn-spin polarization and im provement of electron-spin coherence \nWe consider the transport of spin-polarized conduction electrons in dilute-magnetic \nQWs. As is shown schematically in Fig. 1, electrons are injected from a ferromagnetic \nsource electrode into the QW at ()( ) , 0,arbitraryxy = . Their spins are assumed to be \npolarized along the z-direction. When the electrons are injected, the direction kθ of the \n2D wave vector () ( ) ,c o s , s i nx yk kkk k θ θ == ⋅k&&  of each electron is random, and the 2D  \n \n4 \n wave number k& is distributed in accordance with th e Fermi distribution function. In the \nQW, the electric field xE between a source electrode and a drain electrode is applied \nalong the x-axis. In the present paper, we ignore the Dresselhaus field for simplicity and \ntake into account only the Rashba field fo r the spin-orbit effective magnetic field. \nBecause the system has a translational symmetry along the y-axis, the electron-spin \npolarization is uniform along the y-axis and depends only on x. We do not consider the \ndetails of the ferromagnetic electrode, because the details of the spin injection are beyond \nthe scope of this study. \nFor electron transport, we assume that th e state of an electron along the QW plane is \ngiven by well-defined position a nd momentum. For this assumption, it is necessary that \nthe uncertainties of the position and momentum are much smaller than the electron mean free path and momentum, respectively. This condition is known to be satisfied usually in semiconductors, and this assumption is widely  employed in the Monte Carlo simulations \nof electron transport in semiconductors.\n10,11) Thus, the state of an electron is specified by \nconfinement wave function ()zϕ , 2D position (),xy=r& , 2D wave vector k&, and spin \nvector ( ) ,,x yzsss=s . Under the effective-mass approximation, the 2D electron velocity \nis given by *m =vk&&= , where *m is the effective mass of the conduction band. The \nelectrons suffer momentum scatterings by phono ns and impurities, as is schematically \nshown in Fig. 1. \nThe precession of an electron spin is determin ed, in general, by the precession equation \n pr,d\ndt= ×sΩ s (1) \nwhere prΩ is the precession vector.12) For the Rashba effective magnetic field induced by  \n \n5 \n the applied electric field zE along the growth direction, the precession vector is given by \n () () ()Rashba\nRashba Rashba2,, 0 ,yxkkαγ=− = − Ω kB k&&= (2) \nwhere RashbaB  is the Rashba field, γ the gyromagnetic ratio of a conduction electron, and \n()Rashba zE α ∝  the Rashba coefficient.3,4) A schematic illustration of electron-spin \nprecession is shown in Fig. 2(a) . Because the direction of RashbaΩ  is along the QW and \nperpendicular to k&, the electron spin rotates in the plane including k& and the z-axis. \nThis k& dependence of the precession and the mome ntum scatterings due to acoustic and \noptical phonons, and nonmagnetic impurities cause the DP spin relaxation. As a result, the \nspin polarization ()xs , or the local average of many electron spins in a small spatial  \nregion around x, is localized near the source electrode as will be shown in Fig. 5(c) in \nAppendix.  \nLet us consider the effects of the s-d exch ange interaction on elect ron-spin polarization. \nThis interaction causes two phenomena important for the present research: the dynamical impurity-spin polarization originating from the s-d spin-flip scattering\n9) and the \nelectron-spin precession due to this im purity-spin polarization. The s-d Hamiltonian7) is \ngiven by \n ( )Mn Mn\ns-d s-dˆ ˆ ,ii\niHa δ =− ⋅ −∑Ss r R  (3) \nwhere ˆsand Mnˆ\niS are the spin operators of a conduction s-electron and the i-th Mn \nimpurity, and ( ) ,,xyz =r  and ( )Mn,,ii i i XYZ = R  are their coordinates, respectively. s-da \nis the s-d coupling constant. Considering the pz-axis as a principal axis for spins and \nusing the rising and lowering operators \np pˆ ˆˆxy ssi s±= ±  and \np pMn Mn Mn\n,,ˆˆ ˆ\nii x i ySS i S±=± , we obtain   \n \n6 \n  () ()ppMn Mn Mn Mn\ns-d s-d1ˆ ˆ ˆ ˆˆ ˆ\n2ii i z z i\niH a Ss Ss Ss δ+− −+⎡⎤=− + + −⎢⎥⎣⎦∑ rR . (4) \nIn this equation, the term Mn Mnˆ ˆ ˆˆiiSs Ss+− −++   gives the s-d spin-flip scattering. Because this \nscattering transfers electron spins into  Mn spins, the Mn-spin polarization ()Mn,xt S , or \nthe local average of MnS in a small spatial region around x, is induced dynamically \nunder the electron-spin polarization. This dynamical Mn-spin polarization9) is similar to \nthe Overhauser effect between sp in-polarized electrons and nuclei.12,13)  \nThe other term \np pMnˆˆiz zSs  in eq. (4) causes the energy sp litting between the spin-up and \nspin-down states of electrons under a finite ()Mn,xt S , resulting in electron-spin \nprecession for the mixed spin state. Because each electron is assumed to form a wave \npacket, it is reasonable to cons ider the electron-spin precession vector due to the Mn-spin \npolarization to be determined by ()Mn,xt S . The precession vector is given by  \n () () ()Mn s-d\nMn Mn Mn ,, ,xtaxtn x t γ−=− S B Ω \u0011=, (5) \nwhere Mnn is the Mn density. The total electron-pr ecession vector is given simply by the \nsum of eqs. (2) and (5) as \n () () () () ()pr Rashba Mn Rashba Mn ,, , ,xtx t x t γγ =+ = − − ΩkΩ kΩ Bk B&& & . (6) \nIf the condition Mn RashbaBB\u0015  is satisfied, MnB  dominates the electron-spin \nprecession, and the spatial electron-spin cohere nce is expected to be improved. Let us \nconsider the behavior of an  electron spin vector under ()Mnx B . For example, we consider \n()Mnx B  to be proportional to ()Mnx S  shown in Fig. 3(b2), which will be obtained \nnumerically in §4. This ()Mnx S  is induced by electron-spin  polarization similarly to  \n \n7 \n ()xs  shown in Fig. 3(a1) and is almost parallel  to it. At the source edge, the injected \nelectron spins are parallel to ( )Mn 0x= B , and the direction of ()Mnx B  changes gradually \nas the electrons move in the channel. When MnB is strong enough, the electron spin \nalmost follows ()Mnx B  adiabatically regardless of () RashbaBk&, as is schematically \nshown in Fig. 2(b). This mechanism is r obust against the momentum scatterings of \nelectrons, because the change in () RashbaBk& is negligible for Mn RashbaBB\u0015 . This is in \ncontrast to electron-spin precession only by RashbaB , as schematically shown in Fig. 2(a), \nwhere the precession depends on kθ. Therefore, all electron spins at x are almost \nparallel to ()Mnx B , and the spatial electron-spin cohere nce is expected to be improved. \nThis improved spatial electron-spin cohe rence elongates the region of the induced \nMn-spin polarization in turn. This extende d Mn-spin polarization further improves the \nspatial electron-spin coherence. This positive feedback between the electron- and Mn-spin polarizations elongates their coherence lengths successively.  \nThe condition \nMn RashbaBB\u0015  necessary for the present mechanism is easily satisfied \nin typical II-VI DMS QWs on a time scale of 10 ns. To perform an order estimation, we \nconsider a 10 nm Cd 0.99Mn 0.01Te QW with the electron sheet density 12\nS10 N=  cm-2. The \nelectron-spin splitting Mn\nMn Mn 0 s-d Mn Na x Δ= =Ω S = under fully polarized Mn spins, \nor Mn52= S , is estimated to be 5.5 meV , where 3\n04 Na=  is the density of cation \nsites with a being the lattice constant, ( )Mn 0.01 x =  the Mn mole fraction, and \n0s - d 220 Na =  meV for \nMn Mn1Cd Mn Texx− .7) This exceeds the spin splitting by the Rashba  \n \n8 \n field Rashba Rashba F 20 . 6 k α Δ= =  meV for an electron at  the 2D Fermi surface for 1210SN=  \ncm-2 or F0.03 k\u0011 Å-1, and Rashba 10 α = meVÅ for example. This Rashba field corresponds \nto the applied gate electric field 18zE\u0011 mV/nm estimated from \n 2\nSO g SO\nRashba\ngg S O g S O(2 )\n2( ) ( 3 2 )zEeEmE E Eα∗Δ +Δ=× ,+Δ + Δ= (7) \nwhere g1.606E=  eV is the band-gap energy, SO0.8 Δ=  eV the spin-orbit splitting in the \nvalence band,14) and e the elementary charge. \nThe time scale of the growth of the Mn-spin polarization is obtaine d to be 10 ns from \nsf\neτ, or the s-d spin-flip scattering time of  electrons, which is estimated to be sf\ne100 τ∼  ps \nfor the present QW.15)  Since the density of Mn, 18\nMn 0 Mn 147 10 nN x=×\u0011  cm-3, is about \n100 times larger than the electron volume density 18\neS 10 nN d==  cm-3 for the well \nthickness 10d=nm, the order of Mn spin-flip scattering time is Mn Mn e e 10 nn τ τ = ∼ ns. \nMn-spin scattering mechanisms  other than the s-d scatte ring, which originate, for \nexample, from phonon scatterings,8) can be ignored for low temperatures, because their \ntime scale, 0.1 ms at 4.2 K,16-23)  is much longer than the s-d scattering time. \nIn addition to the s-d spin-flip scattering, the spatial Mn-spin fl uctuation causes the \nelectron-spin relaxation. Although the experimental relaxation time24) of fluct\ne 10 τ\u0011  ps \nwas observed for 8 nm Cd 0.955Mn 0.045Te QWs with 10\nS31 0 N=× and 1071 0×  cm-2, we \nexpect that this type of relaxation can be  ignored under the in-plane electric field xE, \nparticularly because of the following three reasons: First, fluct\neτ  in the present \nCd0.99Mn 0.01Te QW is expected to be about 45 ps, because fluct\neM n 1x τ ∝ .8) Second, it  \n \n9 \n takes only 5 ps, or one order shorter than th e relaxation time, for electrons to drift 1 μm \nunder 1xE= kV/cm for example, when  the electron mobility 421 0μ=×  cm2/Vs, which \nis obtained by the Monte Carlo calculatio n in §4. Actually, the higher mobility \n52.6 10μ=×  cm2/Vs was observed experimentally for the 20 nm CdTe QW at 0.6 K.25) \nThird, the above spin relaxations are expect ed to be relieved w ith increasing Mn-spin \npolarization, because the Mn-spi n fluctuation is suppressed. \n \n3. Numerical Method \nIn order to demonstrate the validity of th e above-proposed mechanism, we perform a \nnumerical calculation. In this calculation, processes essential for the present mechanism, \ni.e., electron transport with momentum s catterings, dynamical Mn-spin polarization, and \nelectron-spin precession due to MnB and RashbaB , are considered. For the wave function of \nelectrons confined in the QW, we take  into account only the ground subband for \nsimplicity and employ the in finite-barrier approximati on. Then, the electron wave \nfunction along the z-direction is given by () ( ) 2s i n zd z dϕπ = , in which we have \nignored the modification by zE for simplicity. \nTo simulate electron transport, we employ the Monte Carlo method.10,11,26-28) Virtual \nelectrons, spin-polarized along the z-axis, are injected into the channel from the \nferromagnetic source electrode. The energy ε of each electron is given by the Monte \nCarlo method in accordance with the Fermi distribution. The 2D momentum of this \nelectron is given by ( ) 2* c o s , s i nkk mε θθ =×k& = , where kθ is given by a uniform \npseudo-random number between 2π−  and 2π. These electrons are accelerated by an \nelectric field xE along the channel and scattered sometimes by acoustic and optical  \n \n10 \n phonons, nonmagnetic impurities, and magnetic Mn  impurities. The probabilities of these \nscatterings are estimated through Fermi’s golden rule in the usual manner, and the timing of each scattering event is determined also by the Monte Carlo method.\n11)  Because the \ndetails of nonmagnetic-impurity scattering are not important for the present calculation, \nwe assume the impurity scattering time to be 15 ps. Since the anomalous and spin Hall effects are beyond the scope of this study, processes related to them, such as skew scattering and side jump,\n29) are not taken into account. The source a nd drain electrodes \nare assumed to absorb all incoming virtual el ectrons. When a virtual electron is absorbed \nby these electrodes, a new virtual electron is emitted from the source electrode. The drain \nelectrode is at 6x= μm for the present calculation. \nWe estimate the rate of the s-d spin-flip scattering between electrons and Mn spins \nunder the assumption that  the Mn spins in Cd 0.99Mn 0.01Te are isolated. For this Mn \nconcentration, the strong anti-ferromagnetic  interaction between nearest-neighbor Mn \nimpurities can be ignored,7) because the probability for a Mn impurity to have at least one \nnearest-neighbor Mn impurity is ()41 0.99 0.04− \u0011 . The probability of spin-flip \nscattering from spin-up to spin-down electron states is given by Fermi’s golden rule for \nthe s-d Hamiltonian, eq. (4), as15) \n () ()()\n() ( ) ( ) ()22 22 2\n2 s-d s-d\nsf s-d 34*,22 *\n5511 11 ,22ii\ni\ni\nii i i i i iZZ k mLWa F m E ELm\nF m j j mm mmϕϕ π\nπ+−⎛⎞ ⎛⎞=Θ + − ⎜⎟ ⎜⎟ ⎜⎟⎝⎠ ⎝⎠\n⎛⎞= + −+ = + −+ ⎜⎟⎝⎠∑&=\n= (8) \nwhere 2L is the area of the QW, Θ Heaviside’s step function, s-dE+ and s-dE− are the spin \nenergies originating from the s-d interac tion for the spin-up and spin-down states, \n( )52ij=  is the length of the Mn spin, and im is the Mn-spin component along the  \n \n11 \n direction of the electron spin. We ignore the Pauli exclusion principle for the final-state \noccupation of electrons for simplicity. This is  reasonable for sufficiently spin-polarized \nelectrons, which are important for the presen t mechanism, because the final spin-down \nstates are almost unoccupied. Although the ra te of this spin-flip  scattering due to \nindividual Mn impurities depends in reality on iZ and ()i Fm , we consider, for \nsimplicity, an average over Mn impurities in  the present calculation. Assuming that the \nMn concentration Mnn is uniform in the QW, we obtain15) \n () () ()224\nMn0\n22 2\n4 Mn Mn Mn\n3201*\n44 3 6sin .16 2d\nii\ni\ndZZ n d L z d zd\nnd L nL nL dzdd ddϕϕ ϕ\nπ⋅\n⎛⎞== = ⎜⎟⎝⎠∑ ∫\n∫\u0011\n (9) \nFor the factor F in eq. (8), we assume the phenomenological form \n ()()()()\n()av35cos for 065,\n35for 06j\nj\nj\njmx\nmx\nFm x\nmxπ ⎧ ⎛⎞\n⎪ ⎜⎟ ≥⎪ ⎜⎟= ⎝⎠ ⎨\n⎪\n< ⎪⎩ (10) \nwhere ()jmx  is an average of im in a small grid between xjx=Δ and  ( )1j x +Δ , \nwhere 0,  1,  2,j=\" and xΔ is the width of grids. This form reproduces \n()()av 03 5 6j Fm x ==  for a random distribution of im, and ()av52 0F = and \n( )av52 5 F−= for fully polarized Mn spins.15)  Hence, we obtain \n () ()()22 2\ns-d s-d s-d Mn\nsf av 33.82 *jjk am nWx F m x E Edm+−⎛⎞Θ+ −⎜⎟⎜⎟⎝⎠&=\u0011= (11) \nIn order to estimate the time evolution of ()Mn,xt S , we use the total change of \nelectron spin (),jkxt Δs , obtained by the Monte Carlo proc ess, in the space grid and a  \n \n12 \n time grid between tk t=Δ  and ()1kt+Δ with the integer k. Because the total angular \nmomentum is conserved in th e s-d spin flip, we obtain \n ( ) ( ) eM n M n ,, 0 ,jk jk nx t n x tΔ+ Δ =sS  (12) \nwhere ()Mn ,jkxt ΔS  is the change in () Mn ,jxt S . Taking into account the spin-lattice \nrelaxation time lattice\nMnτ  caused by phonon scatterings,8) we obtain the time evolution of the \nMn-spin polarization by \n () () ()\n() ()()e\nMn 1 Mn\nMn\n0\nMn Mn\nlattice\nMn,, ,\n,, ,\n,jk jk jk\njk jknxt xt xtn\nxtx t T t\nτ+=− Δ\n⎡⎤ − Δ⎣⎦−SSs\nSS s (13) \nwhere ()()0\nMn ,,jkxtT Ss  is the Brillouin function of Mn spins for the effective \nmagnetic field originating from ( ),jkxt s  at the temperature T. Actually, 0\nMnS is \nnegligible, because the Mn-spin splitting is one order of magnitude smaller than the \nthermal energy even at 4.2 K. We employ lattice\nMn 0.1 τ =  ms at 4.2T=  K.16-23) Although the \nactual s-d spin-flip scatteri ng rate of Mn depends on iZ, we have ignored this \ndependence, in accordance with the averagi ng over Mn impurities employed above. The \nspin diffusion of Mn is also ignored, because the experimental diffusion constant is small \nfor similar II-VI DMSs;  8\ndiff 71 0 K−=×  cm2/s at 1.8 K for Zn 0.99Mn 0.01Se.30) The \nprecession of Mn spins due to the effective magnetic field caused by (),xts  is \nneglected, because ()Mn ,xt S  is almost parallel to (),xts  and the precession \nfrequency is low.  \n \n13 \n It should be noted that ()Mn ,xt S  is overestimated in some degree because of the \naveraging. However, this overestimation is ex pected not to change the numerical results \nmuch. In reality, the average of the Mn sp ins should be given in equilibrium by \n ()\n()Mn\nMn\nMn\nMn52\n12 12\nMn e e\n52\nMn 52'12 12\nee\n'5 2z\nz\nz\nzSz\nS z\nS\nSSp p\nS\npp+−\n=−\n+−\n=−⋅\n=∑\n∑, (14) \nbecause the Mn-spin population Mn\nMnzSp satisfies Mn Mn 1 12 12\neM n eM nzzSSpp pp+ +−= , where 12\nep+ and \n12 12\nee 1 pp−+=−  are the electron-sp in populations for 12zs=+  and 12− , respectively. \nThis equation gives Mn 1.5zS\u0011 and 2, which correspond to Mn 3.3 Δ\u0011  and 4.4 meV , for \n( )12 12\nee 20 . 1 5zsp p+−=− \u0011  and 0.25, respectively, for example. Thus, the \nelectron-spin splitting MnΔ due to Mn-spin polarization is much larger than that by the \nRashba field, Rashba 0.6 Δ=  meV , even under a weak electr on-spin polarizat ion. Therefore, \nthe overestimation changes the numerical results much only in regions where the \nelectron-spin polarization is quite weak. The above estimation also indicates that the \npresent mechanism is valid even for the in sufficient spin polarization of injected \nelectrons. \nThe spin precession of individual electrons is calculated numerically from eqs. (1), (2), \n(5), and (6). MnΩ is assumed to be constant within a time and space grid. The widths of \nthe time and space grids used in the present calculation are 0.03xΔ=  μm and 1tΔ=  ps, \nrespectively. We have assumed ( )Mn,0 0xt< = S , and the s-d interaction is switched on \nat 0t=. The number of virtual electrons used in  the calculation is 50,000. The parameters \nused in the present calculation are shown in Table I.   \n \n14 \n  \n4. Numerical Results \nIn Fig. 3, we show the electron- and Mn-spin polarizations, ()xs  and ()Mnx S , in a \n5 nm Cd 0.99Mn 0.01Te QW with the electron sheet density 12\nS10 N=  cm-2, the Rashba \ncoefficient Rashba 10 α =  meVÅ, the in-plane electric field 1xE=− kV/cm, and the \ntemperature 4.2T=  K as a function of the distance x from the source electrode for the \nelapsed times 0,  5,  and 40t=  ns. At 0t=, Mn spins are not polarized or ()Mn 0 x= S  , \nas shown in Fig. 3(a2), and the electron- spin precession is caused only by the Rashba \nfield. In Figs. 4(a1)-4(a5), we show the co mponents of individual electron spins as a \nfunction of xfor the constant ( ) cos ,sinkk k θ θ =⋅k&&  with 0kθ=, 6π± , and 3π± , \nand 6\nF21 . 7 7 1 0 kk== ×&  cm-1, where Fk is the Fermi wave number for spin-polarized \nelectrons with the sheet density 12\nS10 N=  cm-2. Although it is found that these profiles \ndepend on the direction kθ, they are independent of the wave number k&. This is because \nboth v& and () RashbaΩ k& are proportional to k&. Because of these kθ dependence and \nmomentum scatterings, ()xs  is damped within a half os cillation period and vanishes \nfor 2x≥ μm. This profile is essentially the same  as the result for the nonmagnetic CdTe \nQW shown in Fig. 5(c) in Appendix, because th e effect of the s-d spin-flip scattering on \nelectron-spin polarization is weak. \nFor 5t= ns, ()Mnx S  is partially polarized for 1.5x<  μm, as is shown in Fig. 3(b2). \nThis profile is almost proportional to ()xs  at 0t= in Fig. 3(a1). At the same time, the \nspatial coherence of ()xs  shown in Fig. 3(b1) is improve d in this region. The profiles  \n \n15 \n of individual electr on spins under this ()Mnx S  are shown in Figs. 4(b1) -4(b5). As has \nbeen expected, the electron spins are almost parallel to ()Mnx S  regardless of k& for \n1.5x<  μm, where Mn spins are polarized. On the contrary, the spin profile depends on \nk& for 1.5x>  μm. For 1.5x<  μm, a small oscillation of ys is found. This oscillation is \ndue to a small tilt of  pr Mn Rashba=+ΩΩΩ  with respect to the axis of s. The period of this \noscillation is much shorter than that by the Rashba field in Figs. 4(a1)-4(a5), because \nprΩ  at 5t= ns for  1.5x<  μm is much larger than RashbaΩ . \nAs is shown in Fig. 3(c2), ()Mnx S  is polarized almost fully at 40t=  ns. At the same \ntime, ()xs  in Fig. 3(c1) shows a clear oscillation synchronous with ()Mnx S  in the \nentire region in the figure. This is explai ned by the behavior of individual electron spins \nshown in Figs. 4(c1)-4(c5), which is almost independent of k& and almost parallel to \n()Mnx S . These results clearly demonstrate that  the present mechanism is valid for \novercoming the DP spin relaxation. It should be noted that that both ()xs  and  \n()Mnx S  have small y-components for larger x values and the individual electron \nspins show an additional small and rapid oscillation pronounced for larger kθ values. The \norigin of the latter oscillation is the same as that of the oscillation at 5t= ns for 1.5x<  \nμm. The small Mn,yS -component is induced by ()ysx in the past. Actually, we find in \nFigs. 4(b1)-4(b5) that ()ysx  tends to have a positive y-component for 2x> μm at \n5t= ns. Because MnΩ  is comparable to RashbaΩ  around 2x≈ μm, the precession  \n \n16 \n vector prR a s h b a M n=+ΩΩ Ω  tilts toward the y-direction. As a result, electron spins have \nan ys-component. \n \n5. Summary \nIn this paper, we have proposed a possible mechanism to overcome the \nD’yakonov-Perel’ spin relaxation for conductio n electrons and to improve the spatial \ncoherence of spatially oscillating spin polariz ation as a result. In this mechanism, the \npolarization of magnetic impurities in di lute-magnetic semiconductors, induced \ndynamically by the s-d interaction between c onduction electrons and the impurities, plays \nan important role. The effective magnetic fi eld due to the impurity-spin polarization can \nbe stronger than the Rashba and Dresselhau s fields, and dominate the electron-spin \nprecession. Under this condition, a ll electron spins follow the impurity field, regardless of \ntheir wave vectors and trajec tories, and the spatial electr on-spin coherence is improved. \nNumerical calculations, in which the Rashba effective magnetic field, the s-d interaction, and electron transport have been considered, have demonstrated that the synchronized \nand spatially coherent oscillations of elec tron- and magnetic-impur ity-spin polarizations \ngrow cooperatively owing to the positive feedback between them.  \n \n17 \n  \nAppendix: Electron-Spin Transport in Non-Magnetic Quantum Wells \nIn this Appendix, we discuss the eff ects of the in-plane electric field xE and \nmomentum scatterings on the electron-spin polarization ()xs  under the spin-orbit \neffective magnetic fields in nonmagnetic QWs. Th is is a starting point of the present study. \nAlthough we ignore the Dresse lhaus effective field for simplicity, the results are \nqualitatively the same even under the Dre sselhaus field. We c onsider the system \nschematically shown in Fig. 1. The electron-spin polariz ation, which is along the \nz-direction at the source edge, rotates spatially because of the spin precession of \nindividual electrons due to  the Rashba field, and it is relaxed through the DP \nspin-relaxation mechanism.6) The method of numerical calculation is the same as that \nexplained in § 3, except that the s-d interaction is not included. \nFor the present discussion, the kθ dependence of ()xs, or the x dependence of spin \nprecession for each electron, discussed already in §4 and shown in Figs. 4(a1)-4(a5) is \nimportant. Although these figures are for the 5 nm CdMnTe QW, the results are \nessentially the same as those for CdTe QWs, because ()Mn 0 x= S  in both cases. In Fig. \n5(a), a numerical result of ()xs  for the 5 nm CdTe QW is shown. In this calculation, we \nassume that  k& for each electron is given randomly and is time-independent. This means \nthat we assume 0xE= and ignore momentum scatterings for electrons. The electron-spin \npolarization rotates in the xs-zs plane, in spite that ys is finite for individual electrons \nwith 0kθ≠. This is due to the cancellation of ys between electrons with opposite kθ \nvalues. The amplitude ()xs  decreases rapidly for 1x< μm and gradually for  \n \n18 \n 1x> μm because of the kθ dependence of electron-spin precession. \nIn the case of a finite xE, electrons are accelerated along the x-axis, and their 2D wave \nvector direction converges to 0kθ→. As a result, spin relaxation is expected to be \nreduced. The result for 1xE=−  kV/cm is shown in Fig. 5(b). As is expected, the spatial \nelectron-spin coherence is improved considerab ly. In reality, however, it is necessary to \ntake into account momentum scatteri ngs, which change the direction of k& of individual \nelectrons frequently. As a result, the spin  precession of each electron is randomized, and \nthe spin-coherence length is strongly reduced , as is shown in Fig. 5(c). This result is \nessentially the same as that for the magnetic CdMnTe QW with ()Mn 0 x= S  shown in \nFig. 3(a). Thus, the momentum scatterings for electrons accelerate the DP spin relaxation \nstrongly. \nUnder a finite xE, xk, or xk averaged over all conduction electrons, and the \nresulting spin splitting along the y-axis due to the Rashba field are expected to be finite. \nThus, we might anticipate a finite ys. However, this is not the case. To discuss spin \npolarization, it is necessary to consider  the entire electron distribution in the k&-space. As \nis shown in Fig. 6(a), the 2D Fermi su rfaces for electrons with spins along the \ny±-direction for 0xE= at absolute zero are circular. This is because the dispersion \nrelations for both spin states are parabolic and isotropic with  respect to their bottoms at \nuO and dO , even under the Rashba field. Since the areas of these circles are the same, the \nnumbers of spin-up and down electrons are e qual, and no spin polarization arises. Under a \nfinite xE, these circles are shifted along the xk-axis, but their shapes are not changed \nwithin the Ohmic conduction regime, as is schematically shown in Fig. 6(b).31) Although  \n \n19 \n the circles may be warped in the non-Oh mic regime, the numbers of spin-up and \nspin-down electrons are expected to be the same. As a result, spin polarization is not \ninduced by the Rashba field even under a finite xE. 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Saikin, M. Shen, M.-C. Cheng, and V . Privman: J. Appl. Phys. 94 (2003) 1769. \n28) T. Tsuchiya: J. Phys.: Conf. Ser. 61 (2007) 1191. \n29) See for example, P. Nozières a nd C. Lewiner: J. Phys. (Paris) 34 (1973) 901.  \n \n22 \n 30) A. Maksimov, D. R. Yakovlev, J. Debus, I. I.  Tartakovskii, A. Waag, G. Karczewski, T. \nWojtowicz, J. Kossut, and M. Bayer: Phys. Rev. B 82 (2010) 035211. \n31) C. Kittel: Introduction to Solid State Physics  (Wiley, Hoboken, NJ, 2005) 8th ed. \n32) D. L. Rode: Phys. Rev. B 2 (1970) 4036.  \n \n23 \n  \n \n \nFig. 1. (Color online) Schematic illustrati on of electron transport with momentum \nscatterings and electron-spin precession under spin-orbit effective magnetic fields. \n  \n \n24 \n  \n \n \nFig. 2. (Color online) Schematic illustration of  spin precession of an electron (a) without \nand (b) with strong Mn-spin polarization. (a) Without Mn-spi n polarization, the axis of \nthe electron-spin precession is parallel to  the Rashba field. (b) Under the Mn-spin \npolarization, the electron spin follows the Mn-field adiabatically. Because of the small difference between the direction of the electron spin and the total effective magnetic field, \na fine oscillation around the total field emer ges. For visibility, the directions of the \nRashba and Mn fields are made opposite to  the system of the present numerical \ncalculation.  \n \n25 \n  \n \n \nFig. 3. (Color online) Electron- and Mn-spin polarizations in a 5nm Cd 0.99Mn 0.01Te \nquantum well with electron sheet density 12\nS10 N=  cm-2, Rashba coefficient Rashba 10 α = \nmeVÅ, and in-plane electric field 1xE=−kV/cm as a function of distance x from the \nsource electrode at temperature 4.2T=  K. For electron spins, the effective magnetic \nfield due to the dynamically induced Mn-spin polarization and the Ra shba field are taken \ninto account. The small y-component is caused by the small tilt of the total effective \nmagnetic field.  \n \n26 \n  \n \nFig. 4. (Color online) Spin components of an  electron with consta nt 2D wave vector \n( )( )F2c o s , s i nkk k θ θ =⋅k&  in 5 nm Cd 0.99Mn 0.01Te quantum well with electron sheet \ndensity 12\nS10 N=  cm-2 under the Mn-spin polarization show n in Figs. 3(a2)-3(c2) and the \nRashba field of Rashba 10 α = meVÅ at 4.2T=  K. Without Mn-spin polarization or for \n0t=, ()xs  depends on kθ. Under the polarized  Mn spins, at 40t=  ns and 1.5x<  μm  \n \n27 \n at 5t= ns, electron spins are synchronized with Mn polarizat ion regardless of kθ. The \nfine oscillation prominent for 40t=  ns originates from the small difference in direction \nbetween the total effective magne tic field and the electron spin.  \n \n28 \n  \n \nFig. 5. (Color online) Electr on-spin polarization in a 5n m CdTe quantum well as a \nfunction of distance x from the ferromagnetic source electrode at (a) 0xE= without \nmomentum scatterings, (b) 1xE=−  kV/cm without momentum  scatterings, and (c) \n1xE=−  kV/cm with momentum scatterings . The electron sheet density is 12\nS10 N=  cm-2, \nthe Rashba coefficient Rashba 10 α =  meVÅ, and temperature 4.2T=  K.  \n \n29 \n  \n \n \nFig. 6. (Color online) Schematic illustration of electron distribution for spin-up and down \nstates along the y-axis under the Rashba effective ma gnetic field: (a) circular Fermi \nsurfaces for 0xE= and (b) shifted Fermi circles for 0xE< in the Ohmic conduction \nregime. The centers of the Fermi circles for spin-up and spin-down electrons are denoted \nby  uO and dO for 0xE= and by u'O and 'dO for 0xE≠, respectively. \n  \n \n30 \n  \nTable I. Parameters used in the numerical calc ulations. Some paramete rs used to estimate \nelectron scattering rates are not referred in the text. \nMaterial Cd 0.99Mn 0.01Te \nLattice constant a 0.6482 nm (ref. 14) \nWell thickness d 5 nm \nTemperature T 4.2 K \nElectron sheet density SN 1012 cm-2 \nIn-plane electric field xE -1 kV/cm \nEffective mass *m 0.09 0m (ref. 14) \nRashba constant Rashbaα  10 meVÅ \ns-d coupling constant 0s - dNa  220 meV (ref. 7) \nDensity of cation sites 3\n04/ Na=  1.468 x 1022 cm-3 \nSpin lattice relaxation time for Mn lattice\nMnτ  0.1 ms (4.2 K) \nStatic dielectric constant 0ε 10.2 (ref. 14) \nHigh frequency dielectric constant ε∞ 7.1 (ref. 14) \nLO phonon energy LOω=  21.01 meV (ref. 14) \nAcoustic deformation potential ad 9.5 eV (ref. 32) \nLongitudinal elastic constant LC 6.97x1010 N/m2 (ref. 32) \nImpurity scattering ti me for electrons imp\neτ 15 ps \n0m: electron rest mass\n \n  "    },    {        "title": "2304.14957v1.Competing_signatures_of_intersite_and_interlayer_spin_transfer_in_the_ultrafast_magnetization_dynamics.pdf",        "content": "Competing signatures of intersite and interlayer spin transfer in the ultrafast\nmagnetization dynamics\nSimon Häuser,1,∗Sebastian T.Weber,1Christopher Seibel,1Marius Weber,1Laura Scheuer,1\nMartin Anstett,1Gregor Zinke,1Philipp Pirro,1Burkard Hillebrands,1Hans\nChristian Schneider,1Bärbel Rethfeld,1Benjamin Stadtmüller,1, 2,†and Martin Aeschlimann1\n1Department of Physics and Research Center OPTIMAS,\nRheinland-Pfälzische Technische Universität Kaiserslautern-Landau, 67663 Kaiserslautern, Germany\n2Institute of Physics, Johannes Gutenberg University Mainz, 55128 Mainz, Germany\n(Dated: May 1, 2023)\nOptically driven intersite and interlayer spin transfer are individually known as the fastest pro-\ncesses for manipulating the spin order of magnetic materials on the sub 100fs time scale. However,\ntheir competing influence on the ultrafast magnetization dynamics remains unexplored. In our\nwork, we show that optically induced intersite spin transfer (also known as OISTR) dominates the\nultrafast magnetization dynamics of ferromagnetic alloys such as Permalloy (Ni 80Fe20) only in the\nabsence of interlayer spin transfer into a substrate. Once interlayer spin transfer is possible, the\ninfluence of OISTR is significantly reduced and interlayer spin transfer dominates the ultrafast mag-\nnetization dynamics. This provides a new approach to control the magnetization dynamics of alloys\non extremely short time scales by fine-tuning the interlayer spin transfer.\nIncreasing the operating speed of modern spintronic\ndevices requires new strategies to manipulate, transport,\nand store digital information encoded in the spin angu-\nlar momentum of electrons on shorter time scales. One\npromising way to achieve this goal is to use ultrashort\nfemtosecond (fs) light pulses as external stimuli to mod-\nify the material properties of spintronic relevant mate-\nrials, such as ferromagnets and antiferromagnets. The\nfeasibility of this approach for the realization of ultrafast\nspintronics has been demonstrated by pioneering studies\ninthelasttwodecades, whichrevealedthe(sub-)picosec-\nond loss of magnetic order in ferro- and antiferromagnets\nafter excitation with ultrashort optical and THz pulses\n[1–11] and even reported the all optical magnetization re-\nversal by fs light pulses [12–14]. In most cases, however,\nthe time scale of the material response and the corre-\nsponding change in the spin order of the material is not\nrelatedtothedurationoftheopticalexcitationitself(i.e.,\nthe length of the fs-light pulse). Instead, the magnetiza-\ntion dynamics evolves on a significantly longer intrinsic\ntime scale that is governed by secondary angular mo-\nmentum dissipation processes such as electron-electron\nscattering [15, 16], Elliot-Yafet electron-phonon spin flip\nscattering [16–19], generation of ultrafast non-coherent\nmagnons [9, 20]. These processes limit the optical mate-\nrial response in metallic 3d ferromagnets, such as nickel,\nto≈100fs [1, 17, 18].\nFaster material responses have only been reported for\nmagnetic materials where the ultrafast magnetization\ndynamics are dominated, or at least significantly influ-\nenced, by optically induced spin transport and transfer\nprocesses. One of these processes is the superdiffusive\nspin transport [21–24] in magnetic bilayer and multilayer\nstructures. In this case, the different velocities of the op-\ntically excited minority and majority carriers in the fer-\nromagnet lead to an effective ultrafast transport of spinangular momentum out of the magnetic material into an\nadjacent layer and thus to an ultrafast demagnetization\nthat can be faster than 50fs [22]. The second spin trans-\nfer process relevant on these ultrashort time scales is the\nso-called optical intersite spin transfer (OISTR) [25]. It\nrefers to an optically induced spin transfer between dif-\nferent magnetic subsystems of a magnetic alloy or mul-\ntilayer system, which is purely mediated by the optical\ntransitions between different electronic states of the ma-\nterials [26–32]. As a result, the influence of OISTR on\nthe magnetization dynamics of materials is determined\nsolely by the pulse length of the optical excitation.\nIn reality, however, OISTR and superdiffusive spin\ntransport influence the demagnetization dynamics on\nnearly identical time scales. This is mainly due to the\nfact that the fs light pulses used to manipulate magnetic\nmaterials are typically generated with pulse durations in\nthe range between 20fs and 50fs. This could potentially\nlead to a competing influence of OISTR and superdif-\nfusive spin transport on the ultrafast magnetization dy-\nnamicsofmagneticmultilayerstructures. Understanding\nand exploiting this competition thus offers an intriguing\npathwaytoopticallyengineertheultrafastmagnetization\ndynamics on the sub 100fs time scale.\nIn this work we investigate the competing influence\nof OISTR and interlayer spin transfer via superdiffusive\nspin transport on the ultrafast magnetization dynam-\nics of the ferromagnetic alloy Permalloy (Py, Ni 80Fe20).\nTuningthesubstratematerialofthePyfilmsfromthein-\nsulator MgO to gold films of different thicknesses ( 10 nm\nand100 nm) allows us to gradually increase the role of\ninterlayer spin transfer for the ultrafast demagnetization\ndynamics. Our joint experimental and theoretical work\nprovides substantial evidence that the OISTR signature\nand thus the relevance of OISTR for the demagnetization\ndynamics of Py decreases with increasing importance ofarXiv:2304.14957v1  [cond-mat.mtrl-sci]  28 Apr 20232\nFIG. 1. (Left) Sketch of the time-resolved Kerr spectroscopy\nexperiment. An optical pump pulse ( 1.55eV, 30fs) excites\nthe sample while the element specific magnetization dynam-\nics are monitored by changes in the magnetic asymmetry at\nthe Fe M 2,3and Ni M 3absorption edges. (Right) Schematic\nrepresentation of the density of states of Fe and Ni in Py and\nthe population changes due to the OISTR. Optical excitation\nby the IR pump leads to an effective spin transfer from the\noccupied Ni minority channel to the Fe minority channel.\ninterlayerspintransferfromPyintotheadjacentmetallic\nlayer. Our results underscore the competing roles of in-\ntralayerandinterlayerspintransferfortheultrafastmag-\nnetization dynamics of magnetic multilayer structures on\nthe ultrafast, sub 100fs time scale.\nInourstudy, weinvestigatethreePermalloy(Ni 80Fe20)\nthin films ( 10 nm) deposited on different substrates: (i)\ndirectly on the insulating substrate MgO, (ii) on a 10 nm\nAu film on MgO, and (iii) on a 100 nmAu film. All\nsamples were protected from oxidation by a 2 nmAl2O3\ncapping layer.\nThe time- and element-resolved magnetization dynam-\nics of these samples were investigated by time-resolved\nKerr spectroscopy with fs- extreme UV (XUV) radiation\nin transverse geometry (T-MOKE). Using neon for the\ngeneration of the fs XUV radiation by high harmonic\ngeneration (HHG) (similar to Ref. [33]), we can cover a\nspectral range of 40−72eV that coincides with the char-\nacteristic M 2,3absorption edges of Fe and Ni at ~ 52.7 eV\nand ~ 66.2 eV[34], respectively.\nThe magnetic contrast is obtained from the absorption\nspectra by recording the reflectivity of the whole fs-XUV\nspectrum of the sample for two alternating directions of\nthe magnetic B-fieldI+andI−as shown in Figure 1.\nThen, wecalculatethemagneticasymmetry Aby[33,35]\nA=I+−I−\nI++I−∝Spin Polarization . (1)\nThe energy resolved asymmetry Ais proportional to the\nspin polarization (SP) [36, 37] and gives clear signatures\nof OISTR as shown by Hofherr et al. [26].\nFortheopticalexcitationofoursample, weusedalaser\nfluence of 28.1mJ/cm2, which resulted in a different loss\nof magnetization for each sample. Therefore, we applieda dedicated normalization procedure to all magnetization\ntraces using the data for Py/Au(10nm) as a reference.\nWe start our discussion with the magnetization dy-\nnamics of Py/MgO where the initial magnetization dy-\nnamics are dominated by OISTR and interlayer spin\ntransfer into the substrate is absent. As reported for\nanother FeNi alloy[26], the optical excitation by the IR\npump pulse leads mainly to a transfer from the minority\nelectrons of Ni into the unoccupied minority states of Fe\nclose to the Fermi energy EF, as shown in the inset of\nFigure 1. This intralayer spin transfer can be recorded\nand visualized in the T-MOKE experiment by extracting\nthetime-dependentspinpolarizationforselectedspectral\nregionsinclosevicinitytotheM 2,3absorptionedgesofNi\n(~66.2eV) and Fe (~ 52.7eV). In particular, we follow our\nprevious work [26] and evaluate the spin polarization for\nan energy range corresponding to the occupied Ni states\nbelow the Fermi energy and the unoccupied Fe states lo-\ncated slightly above EF. The corresponding traces are\nshown as blue (Ni states) and red (Fe states) curves in\nFigure 2.\n-4000 4 008 000.70.80.91.01.1d\nelay (fs)norm. asymmetry (arb. u.)N\niF\ne-\n0.4-0.3-0.2-0.10.00.1O\nISTR trace (arb. u.)\nFIG. 2. Temporal evolution of the spin polarization of the\ncorresponding Ni states (blue line) below and Fe states (red\nline) above EFfor Py/MgO. The difference between the spin\npolarization of the Ni and Fe states is shown as a green curve.\nThe SP of the Ni states increases instantaneously upon\nlaser excitation, coinciding with a decrease in the SP of\nthe Fe states. This opposite behavior has been reported\nas the spectroscopic OISTR signature and characterizes\nthe initial influence of OISTR on the ultrafast magneti-\nzation dynamics of Py. It can be further quantified by\nthe so-called OISTR trace (OT), i.e. the difference be-\ntween the SP of the Ni and Fe states, shown as a green\ncurve in Figure 2. The OT reaches its maximum after\nabout 200 fsand disappears again within the next 250 fs.\nThe rise of the OT can be directly related to the OISTR\nand the corresponding initial changes in the magnetiza-\ntion of the Fe and Ni sublattices. The decay of the OT3\nis attributed to exchange scattering [38, 39] and spin flip\nscattering processes occurring locally within the Py film.\nTo explore the competing influence of OISTR and in-\nterlayerspintransportontheultrafastmagnetizationdy-\nnamics of Py, we now turn to the OT for similar Py\nfilms on metallic Au films of 10nm and 100nm thick-\nness. We recorded similar time-resolved T-MOKE data\nsets for both samples (see Supplementary Material) and\ndetermined the OT using the identical procedure as de-\nscribed above for the Py film on MgO. The resulting OTs\nare summarized for all three sample systems in Figure 3.\nWe find a clear decrease in the magnitude of the OT for\nPy/Au( 10 nm) compared to Py/MgO. More importantly,\nthe OT disappears completely for Py on the 100 nmAu\nfilm.\n-4000 4 008 00-0.050.000.050.100.15 \nPy/MgO \nPy/Au10nm \nPy/Au100nmOISTR trace (arb. u.)d\nelay (fs)\nFIG. 3. Temporal evolution of the OT for Py on three differ-\nent substrates: insulating MgO (green curve), metallic 10 nm\nAu film, and 10 nmAu film.\nOur observation clearly demonstrates that the ini-\ntial changes in spin polarization and the magnetization\ndynamics due to OISTR are greatly reduced or even\nsuppressed by replacing the insulating substrate with\na metallic thin film. Therefore, we propose that these\nchanges in OT are due to interlayer spin transfer and\ntransport, ratherthanamodificationofthespinflipscat-\ntering rate in Py due to the hybridization of Py and Au\nat the interface. In particular, no local interfacial ef-\nfect could account for the observed thickness dependent\nchanges in the OT.\nFigure 4 outlines a possible scenario explaining the re-\nduction of OT in the Py film by interlayer spin trans-\nfer: In the absence of OISTR, interlayer spin transfer\nleads to a flow of spin-polarized charges from the ferro-\nmagnetic material into the non-magnetic substrate and a\ncorresponding backflow of unpolarized charges to main-\ntain charge neutrality. This is shown in Figure 4(a). The\nFIG. 4. Schematic illustration of the different optically in-\nduced interlayer spin transfer pathways of majority and mi-\nnority carriers across the Py/Au interface. The hypotheti-\ncal flow of spin-polarized carriers in the absence of OISTR\nis shown in panel (a), the additional interlayer spin transfer\npathways due to OISTR are highlighted in panel (b).\nmagnitude of the interlayer spin transfer for both major-\nity and minority carriers depends (apart from the inter-\nfacial transmission efficiency), on the number of carriers\nin the ferromagnetic material, i.e. in Py, and the number\nof available states in the substrate material, i.e. in Au.\nThis results in an overall majority carrier dominated spin\ntransfer from Py to Au. In the case of OISTR, the op-\ntical excitation additionally redistributes minority elec-\ntrons from Ni states below EFto Fe states above EF, as\nshown on the right side of Figure 1. Thus, OISTR leads\nto a significant change in the number of carriers and final\nstates available for interlayer spin transfer. This popu-\nlation change leads to additional interlayer spin transfer\nchannels as shown in Figure 4(b). The optically excited\nFe minority electrons can now travel through the inter-\nface, reducing the excited state spin population initially\ncreated by the OISTR. On the other hand, the backflow\nof electrons from Au to Py can repopulate the minor-\nity states in Ni that were initially depopulated by the\nOISTR. In this way, both additional transport channels\nwould substantially counteract the changes in the spin-\ndependent population created by the OISTR and thus\nreduce the OT observed in our experiment. However, our\nmodel does not yet account for the increasing reduction\nin OT magnitude with increasing film thickness.\nTo adress this final question theoretically, we model\nthe ultrafast population of states around the Fermi en-\nergy in Au with the help of a kinetic approach consider-\ning the optical excitation as well as the electron-electron\nand electron-phonon interactions, each with a full Boltz-4\nmann collision integral in dependence of time and energy\n[40]. This allows us to determine the non-equilibrium\npopulation and depopulation of states on the same time\nscales as the measured OISTR signal in the Py layer.\nThe excitation strength is determined from the experi-\nmental fluence and absorption calculations for the exper-\nimental multilayer including capping and substrate with\nrefractive indices from Ref. [41]. The resulting absorp-\ntion profiles for a 10 nmAu-film and for a 100 nmAu-\nfilm are shown in Figure 5 a). The dashed line marks\nthe backside of the 10 nmAu-film, which is responsible\nfor the back reflections, leading to a much larger total\nabsorption in the thinner Au film as compared to the\nthicker film, note the logarithmic scale in Figure 5 a).\nFurthermore, we assume a homogeneous distribution of\nthe absorbed energy for the thin film of 10nm thickness.\nThis is already roughly fulfilled by the spatial absorp-\ntion profile depicted in Figure 5a), but also because the\nrange of ultrafast homogeneous energy distribution by\nballistic electrons has been estimated to be about 100 nm\nfor Au [4], which is much larger than the thin Au sub-\nstrate. In contrast, for the 100nm Au-film, we determine\ntwo different limits of absorption strength based on the\nprofile shown in Figure 5a), as will be described later.\nWith the given amount of absorbed energy we model the\nlaser-excitationofelectronsinAu. Typicalresultingnon-\nequilibrium electron distributions in noble metals can be\nfound, e.g., in Refs. [40, 42, 43].\nWe evaluate the change of occupation in the states at\nE=EF−1.8 eVandE=EF+ 1.35 eVaccording to ex-\nperimentally measured energy regions. The width of the\nevaluated interval, ∆E= 0.7 eV, is chosen correspond-\ning to the width of the HHG peaks of the experiment.\nAs indicated in Figure 4, the energy window above the\nFermi energy EFcorresponds to Fe states providing mi-\nnority carriers for the transport into the Au substrate,\nwhile the energy below the Fermi energy is relevant for\nreceiving minority carriers from Au in Ni.\nThe time evolution of the occupation is given as\nn(t) =E+∆E/2/integraldisplay\nE−∆E/2f(/epsilon1,t)D(/epsilon1)d/epsilon1, (2)\nwherefdenotes the electronic distribution function and\nDits density of states (DOS).\nThe change of the occupation n(t)as compared to its\ninitial value at room temperature is shown in Figure 5b).\nIn the case of the 10 nmAu layer, the excitation leads to\nafastincreaseoftheoccupationabove EFandadecrease\nbelowEFwithin the time scale of the laser pulse. After\nthe laser pulse, relaxation processes lead to a decrease of\nboth signals.\nThis shows that the laser excitation is blocking the\nstates relevant for non-local transport across the inter-\nface effectively on the time scale of the pump pulse dura-\ntion and the subsequent several 100 fs, see solid lines in\n10−210−11Absorption (%\nnm)\n0 20 40 60 80 100\nPosition (nm)Au 10 nm\nAu 100 nm\nPy Au a)\n-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.5Occupation change (10271\nm3)\n0 100 200 300 400 500\nTime (fs)Fe-Au receiving states\nNi-Au providing statesb)\n10 nm Au\n100 nm AuFIG. 5. a) Structure of the samples and absorption pro-\nfiles. b) Change of the occupation due to excitation for states\nslightly above Fermi energy (red lines) and slightly below\nFermi energy (blue lines) in Au, respectively. The solid lines\nshow the occupation difference for the 10 nmAu layer. The\nshaded areas mark the occupation difference for the case of\nthe100 nmAu layer for different assumptions on laser-energy\ntransport (see text). The temporal shape of the laser pulse is\nshown in gray.\nFigure5b). Thus, thetransportscenariodepictedinFig-\nure 4 is partially blocked and while the measured OISTR\nsignal in Py is reduced as compared to an insulating sub-\nstrate, but it does not vanish completely.\nIn the case of the thicker 100 nmAu layer, the spin\ntransport from Py to Au is not blocked, leading to a\ncomplete extinction of the resolvable OISTR signal. We\nshowthisfortwolimitingcasesofAuexcitation. Inafirst\ncalculation, we assume that the fraction of pump energy\nabsorbed in Au, see Figure 5a), is homogeneously dis-\ntributed by ballistic transport over the 100 nmAu layer.\nThe occupation change in the relevant Au states is not\nresolved in Figure 5 b), see dotted lines. Thus, in this\ncase, Aucanactasaspinacceptororsource, respectively,\nand thus eliminate the OT. To evaluate the possibility of\nstate-blocking in the 100 nmAu substrate as we found\nfor the thin 10 nmAu film, we consider no energy trans-\nport within the Au layer and average just over the first\n10 nmclosest to the Py interface. The excitation of this\nvirtual layer is lower than in the case of the 10 nmAu\nfilm, because no back reflection effects are active in the\nbulk substrate. The resulting occupation changes within5\nthe energy intervals relevant for spin transport are shown\nby the dashed lines in Figure 5 b), which represent the\nmaximum occupation change caused by the given exci-\ntation parameters. While these occupation changes are\nqualitatively similar to the corresponding changes we de-\ntermined for the thin film (solid lines), their magnitude\nis much weaker. Thus, the spin current from Py into Au\nis not blocked by the excited non-equilibrium electron\ndistribution and efficiently extincts the OISTR signal.\nOur theoretical model thus explains the existence of\nan OISTR signal in Py on a metallic substrate by a\nblocked spin transport across the interface due to a tran-\nsientnon-equilibriumelectrondistributioninthemetallic\nsubstrate. This state blocking is more effective at higher\nabsorption strength in the metal. For a thin, 10 nmAu\nfilm, we find a higher absorption due to multireflection in\nthe thin film. Moreover, the absorbed energy cannot be\ndissipated to the depth by ballistic energy transport as it\nis known for gold films [4], but remains confined in this\nthin layer close to the experimentally studied Py film.\nIn conclusion, we uncover the competing influence of\nOISTR and interlayer spin transfer on the initial ultra-\nfast magnetization dynamics of the ferromagnetic alloy\nPermalloy. We show that OISTR dominates the ultra-\nfast magnetization dynamics of the alloy only in the\nabsence of interlayer spin transfer into a metallic sub-\nstrate. However, once interlayer spin transfer is possible,\nthe optically redistributed spins due to OISTR are, at\nleastpartially, transferredfromthealloyintothemetallic\nsubstrate and thus no longer dominate the initial ultra-\nfast demagnetization dynamics. Thus, our study demon-\nstrates a clear way to tune the competing influence of\nOISTR and interlayer spin transfer on the magnetization\ndynamics by controlling the efficiency of interlayer spin\ntransfer across interfaces.\nThe experimental work was funded by the Deutsche\nForschungsgemeinschaft (DFG, German Research Foun-\ndation) - TRR 173 - 268565370 Spin + X: spin in its\ncollective environment (Project A08 and B11). 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B 99, 174314\n(2019).Competing signatures of intersite and interlayer spin transfer in the ultrafast\nmagnetization dynamics - Supplemental Material\nSimon H¨ auser,1,∗Sebastian T. Weber,1Christopher Seibel,1Marius Weber,1\nLaura Scheuer,1Martin Anstett,1Gregor Zinke,1Philipp Pirro,1Burkard Hillebrands,1Hans\nChristian Schneider,1B¨ arbel Rethfeld,1Benjamin Stadtm¨ uller,1, 2,†and Martin Aeschlimann1\n1Department of Physics and Research Center OPTIMAS,\nRheinland-Pf¨ alzische Technische Universit¨ at Kaiserslautern-Landau, 67663 Kaiserslautern, Germany\n2Institute of Physics, Johannes Gutenberg University Mainz, 55128 Mainz, Germany\n(Dated: May 1, 2023)\nSAMPLE PREPARATION\nPy thin films and Py/Au bilayers were grown on 0 .5 mm-thick MgO (100) by molecular beam epitaxy (MBE)\ntechnique in an ultrahigh vacuum (UHV) chamber at a base pressure of 5 ×10−9mbar. The cleaning protocol of the\nMgO substrates included ex-situ chemical cleaning with acetone and isopropanol and in-situ heating at 600 °C for 1 h.\nPy deposition was performed at room temperature with a growth rate in the range of R= 0.15˚A/s ( R= 0.15 nm/s)\ncontrolled by a quartz crystal oscillator. In the case of the bilayer structures, a 10 nm or 100 nm Au film was first\ngrown directly on the MgO substrate using similar parameters.\nEXPERIMENTAL DETAILS\nOur time-resolved experiments were performed with a high harmonic generation transversal magneto optical Kerr\neffect (HHG-T-MOKE) pump-probe setup [1, 2]. An amplified Ti:sapphire laser system (1.57 eV, 30 fs, 6 kHz, 1.7 mJ\nper pulse) was used to generate the pump and probe beams. To generate the XUV probe beam, we used the high\nharmonic generation process with a fiber based setup and neon as the excited noble gas [1, 2]. We determined the\ntemporal pulse length at the sample position to 80 fs for the 1.57 eV pump pulse using an autocorrelation method.\nBoth p-polarized pulses were focused on the sample at an angle of incidence of 45 degrees. The reflected XUV light\nis separated from the residual fundamental light of 1.57 eV by two 100 nm aluminum filters and then detected by a\nspectrometer to ensure energy and therefore element resolution. The estimated energy resolution is ≈0.8 eV.\nSPECTRAL RANGES FOR EVALUATION\n506 07 0/s8722/s48/s46/s500.0asymmetry (arb. u.)e\nnergy (eV)(a)i\nntensity\n506 07 0/s8722/s48/s46/s500.0asymmetry (arb. u.)e\nnergy (eV)(b)i\nntensity\n506 07 0/s8722/s48/s46/s500.0asymmetry (arb. u.)e\nnergy (eV)(c)i\nntensity\nFIG. S1: Detected XUV-spectra (cyan and magenta) and the resulting asymmetries of Py on (a) MgO, (b) 10 nm Au\nand (c) 100 nm Au. The red and blue shaded areas correspond to the evaluated spectral regions for Fe and Ni,\nrespectively.\nSCALING OF MAGNETIZATION DYNAMICS\nAs mentioned in the experimental details of the paper, we applied the same fluence for each Py sample with a\ndifferent substrate, resulting in different quenching. These results are shown in Fig. S2 (a). Here, the Py/Au(10 nm)arXiv:2304.14957v1  [cond-mat.mtrl-sci]  28 Apr 20232\nand Py/Au(100 nm) sample systems lost almost the same amount of spin polarization (about 30% and 28%) after\noptical irradiation, although the remagnetization is much faster in the case of Py/Au(100 nm). For Py/MgO a\nquenching of about 15% was found. The corresponding OISTR-trace (OT), as the difference of the Ni and Fe signal,\nis shown in Fig. S2 (b). While the OT for Py/Au(100 nm) is close to zero, the OT for Py/MgO and Py/Au(10 nm) is\nsimilar, but slightly higher in the case of Py/MgO. One conclusion out of this is, that although the absorbed energy\nin the spin system is almost 50% lower in Py/MgO than in Py/Au(10 nm), the OT is still higher in Py/MgO.\n01 2 0.70.80.91.01.1norm. asymmetry (arb. u.)d\nelay (ps) Fe (MgO) \nNi (MgO) \nFe (Au10nm) \nNi (Au10nm) \nFe (Au100nm) \nNi (Au100nm)(a)\n/s8722/s52/s48/s480 4 008 00/s8722/s48/s46/s48/s530.000.050.100.15OISTR trace (arb. u.)d\nelay (fs) Py/MgO \nPy/Au10nm \nPy/Au100nm(b)\n01 2 0.70.80.91.01.1norm. asymmetry (arb. u.)d\nelay (ps) Fe (MgO) \nNi (MgO) \nFe (Au10nm) \nNi (Au10nm) \nFe (Au100nm) \nNi (Au100nm)(c)\n/s8722/s52/s48/s480 4 008 00/s8722/s48/s46/s48/s530.000.050.100.15OISTR trace (arb. u.)d\nelay (fs) Py/MgO \nPy/Au10nm \nPy/Au100nm(d)\nFIG. S2: (a) Raw dynamic of the spin polarization of Py on MgO, Au(10 nm) and Au(100 nm). (b) Resulting\nOISTR trace of (a). (c) Dynamic of the spin polarization of Py on MgO, Au(10 nm) and Au(100 nm) normalized to\n≈30% quenching. (d) Resulting OISTR trace of (c).\nFor better comparison we have scaled the quenching of each sample to the value of 30%, using Py/Au(10 nm) as a\nreference (Fig. S2 (c)) to linearly extrapolate the OT of Py/MgO with ≈30% of quenching. This was done by first\nselecting a time window between 644 fs and 991 fs for Py/MgO, 340 fs and 810 fs for Py/Au(100 nm). The averaged\nspin polarization in this window was then calculated to be 0 .828 for Py/MgO and 0 .728 for Py/Au(100 nm). This\nvalue was then subtracted from the whole time-resolved curve, resulting in different normalized levels before time zero\nlT0, 0.165 for Py/MgO and 0 .265 for Py/Au(100 nm). Now the whole time resolved curve was multiplied by 0 .3/lT0\nand summed to 0.7. The result is shown in Fig. S2 (c), where all time resolved curves show similar quenching of\n≈30%. We then extract the OT of each sample, as shown in Fig. S2 (d). Now the OT of Py/MgO is about a factor\nof two larger than Py/Au(10 nm), showing the trend of decreasing OT with increasing Au-thickness.3\nTHEORETICAL MODEL\nWe used a model based on full Boltzmann collision integrals to calculate the dynamics of the electronic distribution,\nwhich we used to determine the time-dependent occupation of states as described in the main text. The full details\nof the model are described in Ref. [3]. To determine the absorption profiles for the considered scenarios, we solved\nthe Helmholtz equation for Al 2O3(2 nm)/Py(10 nm)/Au(d)/MgO(500 nm) heterostructures with d= 10 nm and d=\n100 nm, respectively. Details of the calculations can be found in the supplementary information of Ref. [4]. The\nrefractive index for Al 2O3was taken from Ref. [5], for Py from Ref. [6], for Au from Ref. [7] and for MgO from\nRef. [8]. For all values, a wavelength of 800 nm was considered in accordance with our experiments. We determined\nthe absorbed energy density by integrating the absorption profiles in the respective materials and considering the\nexperimental fluence of 28 .1 mJ cm−2. The intensity of the laser excitation was then adjusted in the model to reproduce\nthe energy absorbed in the experiment. For the two limiting cases of no transport and full ballistic transport in the\ncase of the 100 nm Au layer, we took into account only the first 10 nm of the absorption profile for the former case\nand the full 100 nm for the latter case. The resulting averaged fractions of absorbed energy are 0 .36 % nm−1for the\n10 nm Au layer, 0 .03 % nm−1for the 100 nm Au layer and 0 .15 % nm−1for the limiting case of no transport in the\n100 nm Au layer.\n∗shaeuser@rptu.de\n†b.stadtmueller@rptu.de\n[1] C. La-O-Vorakiat, E. Turgut, C. A. Teale, H. C. Kapteyn, M. M. Murnane, S. Mathias, M. Aeschlimann, C. M. Schneider,\nJ. M. Shaw, H. T. Nembach, and T. J. Silva, Phys. Rev. X 2, 011005 (2012).\n[2] S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens,\nS. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, Proceedings of the National\nAcademy of Sciences 109, 4792 (2012).\n[3] B. Y. Mueller and B. Rethfeld, Phys. Rev. B 87, 035139 (2013).\n[4] C. Seibel, M. Weber, M. Stiehl, S. T. Weber, M. Aeschlimann, H. C. Schneider, B. Stadtm¨ uller, and B. Rethfeld, Phys.\nRev. B 106, L140405 (2022).\n[5] R. Boidin, T. Halenkoviˇ c, V. Nazabal, L. Beneˇ s, and P. Nˇ emec, Ceramics International 42, 1177 (2016).\n[6] K. K. Tikuiˇ sis, L. Beran, P. Cejpek, K. Uhl´ ıˇ rov´ a, J. Hamrle, M. Vaˇ natka, M. Urb´ anek, and M. Veis, Materials & Design\n114, 31 (2017).\n[7] S. M. Werner, K. Glantschnig, and C. Ambrosch-Draxl, J. Phys. Chem. Ref. Data 38, 1013 (2009).\n[8] R. Stephens and I. Malitson, J. Res. Natl. Bur. Stand. 49, 249 (1952)."    },    {        "title": "1005.0259v3.Dynamical_exchange_interaction_between_localized_spins_out_of_equilibrium.pdf",        "content": "Dynamical exchange interaction between localized spins out of equilibrium\nJ. Fransson\u0003\nDepartment of Physics and Astronomy, Uppsala University, Box 530, SE-751 21 Uppsala\n(Dated: March 22, 2021)\nThe electron mediated exchange interaction between local spins adsorbed on two-dimensional\nsurface is studied under non-equilibrium conditions. The e\u000bective spin-spin interaction is found to\ndepend both on the spin-polarization of the substrate and the excitation spectrum of the local spins.\nFor spatially anisotropic spin-polarization of the substrate, the spatial dependence of the interaction\ncomprise components decaying as sin(2 kFR)=(2kFR) and sin(2 kFR)=(2kFR)2.\nPACS numbers: 75.30.Et, 71.70.Gm, 75.30.Hx\nThe excitation spectra of spin systems strongly de-\npends of the type of interactions that are involved. The\nmagnetic moment of e.g. single Co [1{8], Fe [9, 10],\nCr [10], and Mn [11] atoms become strongly anisotropic\ndue to symmetry reduction in the interaction with elec-\ntron medium. Studies of inelastic scattering processes\nof layered materials [8] and single atoms [3{7, 9{13]\nhave given deepened insight to the excitation spectra\nof various elements, which then provide further detail\nto the understanding of the involved interactions. phys\nFor magnetic systems, the interactions between the lo-\ncal spins can be of di\u000berent character, which is often\nmodeled using e.g. the Ising and Heisenberg Hamilto-\nnians, but also anisotropic models such as e.g. XY-\nor anisotropic Heisenberg Hamiltonians. Regardless\nof model, the interaction parameters describe a phys-\nical interaction between the spins, which result from\ndi\u000berent mechanisms. The spin-spin interaction may\nbe direct in the sense that the exchange Coulomb in-\ntegral (R\n y\n\u001b(r) y\n\u001b0(r0)V(r;r0) \u001b(r0) \u001b0(r)drdr0) is non-\nnegligible, or of indirect nature, e.g. super-exchange\nor double-exchange. Of particular interest is the\nRuderman-Kittel-Kasuya-Yosida (RKKY) interaction\n[14{16], which is generated by a coupling between the\nlocal spins and the surrounding electron medium, such\nthat the spin-spin exchange interaction is mediated by\nthe electronic environment.\nIn this paper, we address the electron mediated ex-\nchange interaction between localized spin under non-\nequilibrium conditions in two-dimensional systems. The\nquestion is pertinent to recent measurements using e.g.\nscanning tunneling microscopy (STM) where local non-\nequilibrium conditions are created by the tunneling cur-\nrent. It is demonstrated that the resulting spin-spin ex-\nchange interaction depends on the spin-polarization of\nthe electron medium and on the excitation spectrum\nof the localized spins. For spatially anisotropic spin-\npolarized surface electrons the spin-spin interaction com-\nprise the Ising, Heisenberg, and Dzyaloshinski-Moriya in-\nteractions, where the latter is shown to asymptotically\ndecay as sin(2 kFR)=(2kFR). In contrast to previous\nstudies of the non-equilibrium RKKY interaction [17, 18],\nwe here also include the proper time-dependence of thelocal spins.\nThe presence of the local spin (or magnetic) moments,\nresults in a spatially inhomogenous surface electron spin-\npolarization which can be transformed into a spatially\nnon-uniform spin bias distribution between the spin-\nprojections of the surface electrons. Under the spin bi-\nased conditions, electrons \row between the local spin mo-\nments, however, di\u000berent spin projections travel in di\u000ber-\nent directions, thus, establishing a net equilibrium. The\nsetup is, hence, reminiscent of the electron spin resonance\nsituation discussed in Refs. 19, 20, and is ideal for inves-\ntigating the electron mediated exchange interaction in\nterms of non-equilibrium formalism.\nWe begin by considering localized spins Srat the posi-\ntionsrinteracting with a continuum, treated in the closed\ntime-path Green function formalism [21] which was re-\ncently applied to spin dynamics in a Josephson junction\n[22] and three-dimensional metallic systems [23]. We cal-\nculate the partition function (in units: ~=c= 1)\nZ[Sn(t)] =trTCeiS; (1a)\nS=SWZWN +Sext+I\nC[HK+HT]dt; (1b)\nI\nC(\u0001)dt=Z1\n\u00001(\u0001)dt+\u0000Z1\n\u00001(\u0001)dt\u0000; (1c)\nwhere we have omitted unimportant contributions from\nthe electron gas with quadratic dispersion and isotropic\ne\u000bective mass m, as we are considering conduction elec-\ntrons in the continuum approximation. The STM tip\nis assumed to have negligible e\u000bect on the spin cluster.\nSWZWN =P\nrR\nSr(t)\u0001[Sr(t)\u0002_Sr(t)]dt=S2\nr,Sr=jSrj,\nis the Wess-Zumino-Witten-Novikov (WZWN) term de-\nscribing the Berry phase accumulated by the local spins.\nThe trace runs over the degrees of freedom for the elec-\ntrons in the tip and substrate in order to provide an ef-\nfective spin action, which in the present situation repre-\nsents the interaction of the magnetic spins with a non-\nequilibrium environment. Sextrepresents the coupling\nbetween the system with the external electromagnetic\n\feld. The Hamiltonians inside the contour integral de-\n\fne the (Kondo) coupling between the local spins and the\nsurface electrons, HK=\u0000vuJKP\nrSr(t)\u0001s(r;t), and thearXiv:1005.0259v3  [cond-mat.str-el]  24 Nov 20102\ncoupling to external electrodes HTwhich generates a tun-\nneling current in and/or out from the two-dimensional\nsurface. For example, recent STM measurements mo-\ntivates to model the tunneling current between the tip\nand surface [13] using HT=P\nrP\npk\u001b\u001b0cy\np\u001b(\u000e\u001b\u001b0T0+\nT1\u001b\u001b\u001b0\u0001Sr)ck\u001b0eik\u0001r+i\u001e(t)+H:c:, where p(k) denotes the\nmomentum for electrons in the tip (substrate), whereas\nT0,T1are the ( pandkdependent) rates for the di-\nrect and exchange coupled tunneling. Here, s(r;t) is the\nelectron spin density, whereas vuandJKde\fnes a unit\nsurface element and the Kondo coupling to the electrons.\n\u001e(t) =eRt\n\u00001Vsd(t0)dt0gives the energy shift due to the\nbias voltage Vsd(t) applied between the tip and surface\nand\u001bis the vector of Pauli spin matrices.\nThe procedure in [22{24] yields the e\u000bective action\nS=SWZWN +ZX\nr[g\u0016BB(r;t) +j(1)(r;t)=e]\u0001S2\nr(t)dt\n\u0000(vuJK)2ZX\nrr0S2\nr(t)\u0001Fr(r;r0;t;t0)S1\nr0(t0)dtdt0\n+1\neZX\nrr0S2\nr(t)\u0001j(2)(r;r0;t;t0)S1\nr0(t0)dtdt0; (2)\nwhere S1(t) = [ S(t+) +S(t\u0000)]=2 and S2(t) =\nS(t+)\u0000S(t\u0000), whereasFr\nij(r;r0;t;t0) = (\u0000i)\u0012(t\u0000t0)\u0002\nh[si(r;t);sj(r0;t0)]iis the retarded spin GF of the surface\nelectrons. j(1)(r;t) =j(1)(r;t)^zandj(2)(r;r0;t;t0) are the\nspin-polarized current density and spin current density,\nrespectively, between the tip and the substrate generated\nby the spin-imbalance and non-equilibrium conditions in\nthe electrodes [24, 25]. This, general, formulation of the\naction is motivated from the perspective of recent tun-\nneling experiments. In this paper the focus, however, is\non the third term to the right in Eq. (2), which rep-\nresents the RKKY interactions as it emerges from the\n(Kondo) coupling between the localized spin moments\nand the surface electrons.\nOwing to the general non-equilibrium conditions, the\nretarded spin GF is expressed in terms of the lesser and\ngreater surface electron GFs G<>(r;r0;t;t0), that is,\nFr\nij(r;r0;t;t0) =(\u0000i)\u0012(t\u0000t0)trS[\u001biG>(r;r0;t;t0)\n\u0002\u001bjG<(r0;r;t0;t)\u0000\u001biG<(r;r0;t;t0)\n\u0002\u001bjG>(r0;r;t0;t)]; (3)\nwhere the trace tr Sis taken over spin space of the sur-\nface electrons. For non-interacting but spin-polarized\nsurface electrons we write the real space GFs according to\nG<>(R;\u001c) =R\nG<>(k;\u001c)eik\u0001Rdk=(2\u0019)2, where R=r\u0000r0\nand\u001c=t\u0000t0. The lesser and greater forms of the GF\ncan be written\nG<>(k;\u001c) =X\n\u001b(\u001b0+\u001b\u0001\u0001\u001bz\n\u001b\u001b)G<>\n\u001b(k;\u001c)=2;(4)whereG<>\n\u001b(k;\u001c) = (\u0006i)f(\u0006\"k\u001b) exp(\u0000i\"k\u001b\u001c) with\"k\u001b=\n\"k+\u001bj\u0001j=2, whereas\u001b0is the identity matrix, and f(x)\nis the Fermi function. The e\u000bective spin-splitting \u0001is\ngenerated by the surface electrons due to coupling to in-\nternal and/or external spin degrees of freedom. \u0001can\npartially be due to e.g. the spatially inhomogeneous\nmean \feld\u0000vuJKP\nrhSriwhich is generated by the ad-\nsorbed spins, and partially due to e.g. spin-orbit inter-\nactions in the surface, pertinent to recent STM measure-\nments of Co/Pt(111) [3]. Using Eq. (4) and the identity\n(A\u0001\u001b)(B\u0001\u001b) = (A\u0001B)\u001b0+i(A\u0002B)\u0001\u001b[26] we \fnd,\nafter some algebra,\nS2\nr(t)\u0001Fr(R;t;t0)S1\nr0(t0) =\n=1\n2X\n\u001b\u001b0F\u001b\u001b0(R;\u001c)n\n2\u001bz\n\u001b\u001b\u001bz\n\u001b0\u001b0[S2\nr(t)\u0001\u0001][S1\nr0(t0)\u0001\u0001]\n+ [1\u0000j\u0001j2\u001bz\n\u001b\u001b\u001bz\n\u001b0\u001b0]S2\nr(t)\u0001S1\nr0(t0)\n\u0000i[\u001bz\n\u001b\u001b\u0000\u001bz\n\u001b0\u001b0]\u0001\u0001[S2\nr(t)\u0002S1\nr0(t0)]o\n; (5)\nwhere the dynamical range functions F\u001b\u001b0are given by\nF\u001b\u001b0(R;\u001c) =(\u0000i)\u0012(\u001c)Z\n[f(\"k0\u001b0)\u0000f(\"k\u001b)]ei(k\u0000k0)\u0001R\n\u0002e\u0000i(\"k\u001b\u0000\"k0\u001b0)\u001cdk\n(2\u0019)2dk0\n(2\u0019)2: (6)\nBelow, we shall calculate those functions explicitly. Be-\nfore we proceed, however, we note that the electron medi-\nated spin-spin interaction described in Eq. (5) comprise\nthree di\u000berent kinds of interactions; Ising, Heisenberg,\nand Dzyaloshinski-Moriya (DM) types of interactions, re-\nspectively, in agreement with Ref. [27].\nThe \frst term (\u0018[Sq\nr(t)\u0001\u0001][Sc\nr0(t0)\u0001\u0001]) is a general-\nized Ising-type of interaction. That it is of Ising-type can\nbe understood since it provides the interaction between\nthe spins projected onto the direction of the \feld \u0001. By\nrotating the reference frame such that e.g. \u0001= \u0001 ^z,\none \fnds that [ Sq\nr(t)\u0001\u0001][Sc\nr0(t0)\u0001\u0001] = \u00012Sq;z\nr(t)Sc;z\nr0(t0).\nThe Ising interaction vanishes for non-spin polarized con-\nduction electrons, since the range functions are spin-\nindependent, i.e. F\u001b\u001b0=F, under such conditions.\nThe \frst two contributions are expected to be present\nbetween spins interacting via metallic or semi-conducting\nmedium. The last contribution is, on the other hand,\nexpected to arise in anisotropic systems [28, 29]. Here,\nthis anisotropy is generated by the spin polarized surface\nelectrons. For non-chiral spin-polarization of the surface\nelectrons, the range functions F\"#=F#\", which leads to\nthat the DM interaction vanishes, as expected.\nIt is interesting to note, that the DM interaction is\nnon-vanishing whenever the local spins create a spatially\ninhomogeneous spin-polarization of the surface electrons.\nIn this sense, the induced spin-polarization can be viewed\nas an e\u000bective defect induced spin-orbit interaction.\nNext, we calculate the dynamical range functions. The\nangular integrals in Eq. (6) results in the factors J0(kR),3\nwhereJ0(x) is the Bessel function. The Fourier transform\nof the exponential e\u0000i(\"k\u001b\u0000\"k0\u001b0)\u001cis given by ( !\u0000\"k\u001b+\ni\u000e)\u00001(!0\u0000\"k0\u001b0\u0000i\u000e)\u00001. Thus, for quadratic dispersion\n\"k=k2=2N0,N0=m, changing momentum to energy\nintegrations, and carrying out those energy integrals, give\nF\u001b\u001b0(R;\u001c) =iN2\n0\n4\u0012(\u001c)e\u0000i\n\u001b\u001b 0\u001cZ1+i\u000e\n\u00001+i\u000e[f\u001b0(!0)\u0000f\u001b(!)]\n\u0002H(1)\n0(~Rp!)H(1)\n0(~Rp\n!0)e\u0000i(!\u0000!0)\u001cd!\n2\u0019d!0\n2\u0019;\n(7)\nwhereH(1)\nn(x) is the Hankel function, \n \u001b\u001b0=j\u0001j(\u001b\u0000\n\u001b0)=2,~R=Rp2N0, andf\u001b(!) =f(!+\u001bj\u0001j=2). Using\nthat (\u0000i)\u0012(\u001c)e\u0000ix\u001c=R\n(\n\u0000x+i\u000e)\u00001e\u0000i\n\u001cd\n=2\u0019, we can\n\fnally write F\u001b\u001b0(R;\u001c) =R\nF\u001b\u001b0(R;\n)e\u0000i\n\u001cd\n=(2\u0019),\nwhere\nF\u001b\u001b0(R;\n) =(\u0000i)N2\n0\n4Z\n[f\u001b(!\u0000\n) +f\u001b(!)]H(1)\n0(~Rp!)\n\u0002H(1)\n0(~Rp\n!\u0000\n + \n\u001b\u001b0)d!\n2\u0019(8)\nshowing that the dynamical range function depends on\ntime\u001c, the spin bias \n \u001b\u001b0, and on the spin-chemical po-\ntential\u0016\u001b=\"F\u0000\u001bj\u0001j=2. Eq. (8) is the central result of\nthis paper and below we analyze a few of its consequences\nto the electron mediated spin-spin exchange interaction.\nThe static regime, which corresponds to assuming\nfrozen spin moments, is de\fned for \n = 0. For small\nspin biases \n \u001b\u001b0=\"F\u001c1, such that H(1)\n0(~Rp!+ \n\u001b\u001b0)\u0019\nH(1)\n0(~Rp!), the integral in Eq. (8) can be analytically\ncalculated for low temperatures ( T!0), for which\nF\u001b\u001b0(R) =N0\n2\u0019k2\n\u001bX\nn=0;1\u0014\nJn(Rk\u001b)Yn(Rk\u001b)\n\u0000i\n2[J2\nn(Rk\u001b)\u0000Y2\nn(Rk\u001b)]\u0015\n; (9)\nwherek\u001b=p\nk2\nF\u0000\u001bj\u0001jN0with the Fermi vector kF=p2N0\"F, whereasYn(x) is the Neumann function. Here,\nthe real part captures previous results [30{32], in the\nspin-degenerate limit k\u001b!kF, whereas the imaginary\npart accounts for the retardation and damping e\u000bects of\nthe interaction due to the electron medium.\nFor 2kFR\u001d1, the asymptotic expansion of Eq. (9)\n[30] leads to that Re F\u001b\u001b0(R)\u0018sin(2k\u001bR)=(2k\u001bR)2, in\nagreement with previous results [30{32], which provides\nthe usual spatial decay for the isotropic electron medi-\nated Heisenberg-like exchange, see Eq. (5). Analogously,\nthe asymptotic expansion of the imaginary part of Eq.\n(9) gives Im F\u001b\u001b0\u0018cos(2k\u001bR)=(2k\u001bR)2. In contrast, the\nelectron mediated DM interaction asymptotically decays\nas sin(2kFR)=(2kFR), which can be seen from the fol-\nlowing observation. The DM interaction depends on the\nrange functions as F\"#\u0000F#\", see Eq. (5). By Taylor ex-\npanding the imaginary part of this di\u000berence and usingthatk\u001b\u0019kF\u0000\u001bj\u0001jN0=(2kF), one \fnds that it asymp-\ntotically reduces to\nImX\n\u001b\u001bz\n\u001b\u001bF\"#(R)\u0018j\u0001jN0\n2k2\nF\u0001sin 2kFR\n2kFR; (10)\nwhich to the best of our knowledge has not been re-\nported previously. Hence, despite the DM interaction\ndepends quadratically on the anisotropy \feld \u0001, it tends\nto dominate over the Heisenberg, and Ising, exchange for\nsu\u000eciently large distances between the spins. Thus, in\nabsence of e\u000bective magnetic \felds acting on the local\nspins, two spins con\fgure themselves perpendicular to\none another when being separated by a su\u000eciently large\ndistance since the DM interaction dominates their cou-\npling. A collinear alignment of the spins is typically fa-\nvorable whenever the spins are close to one another, since\nthe Heisenberg interaction provides the strongest contri-\nbution to the coupling. The cross over distance at which\nthe DM interaction begins to dominate over the Heisen-\nberg interaction is roughly given by RC\u0019kF=(2j\u0001jN0).\nThis cross over distance is obtained under the assumption\nthatN0j\u0001j=k2\nF\u001c1, which is reasonable from the point\nof view of Ref. [33], where kF'0:17\u0017A\u00001whereas the\nspin-splitting due to the Rashba e\u000bectp\nN0j\u0001j'0:012\n\u0017A\u00001(j\u0001j\u00181 meV). In terms of these values, the cross\nover distance RC\u00191;180\u0017A. For larger spin-splitting,\nthe above approximations are not valid, however, we ex-\npect that the e\u000bect of the DM interaction will be even\nstronger and that the cross over distance is signi\fcantly\nshorter.\nFor the remainder of this paper, we consider the lo-\nFIG. 1: Energy (\n) and spatial ( R) dependence of the real,\n(a), (c), and imaginary, (b), (d), parts of F\"#(R;\n). The\ndotted lines in panels (a) and (b) indicate the energies at\nwhich the line scans in panels (c) and (d) are extracted. In\npanels (c), (d), the plots are vertically shifted for clarity. Here,\nkF= 0:17\u0017A\u00001, \n\"#= 10 meV at T= 10K.4\ncal spin moments to be time-dependent, i.e. \n 6=\n0, as they would be under the in\ruence of e.g. an\ne\u000bective magnetic \feld. In Eq. (5), the dynam-\nical range function F\u001b\u001b0(R;\u001c) mediates the interac-\ntion between the spins signi\fed by S(1)(t0) and S(2)(t),\nwhich have di\u000berent time-arguments. Using that e.g.\nS(1)(t) =R\nS(1)(x)eixtdx=(2\u0019) and integrating over t\nandt0[c.f. Eq. (2)], we can write, for instance,\nthe Heisenberg type of exchange interaction between\ntwo spins located at randr0as\u0000(vuJK)2P\n\u001b\u001b0[1\u0000\nj\u0001j\u001bz\n\u001b\u001b\u001bz\n\u001b0\u001b0]R\nF\u001b\u001b0(R;\n)S(2)\nr(\n)\u0001S(1)\nr0(\u0000\n)d\n=(4\u0019), and\nanalogously for the other two types of interactions. The\nexchange interaction between the localized spins, hence,\nstrongly depends on the excitation spectra of the spins.\nIn Fig. 1, we plot the real, panels (a), (c), and the imag-\ninary, panels (b), (d), parts of F\u001b\u001b0(R;\n) for a given\nspin bias, \n\"#\u001810 meV, and temperature, T= 10 K.\nAs expected from previous results, and from the above\ndiscussion, in the regime near \n = 0, F\u001b\u001b0acquires an\noscillatory decaying behavior as function of R.\nThe expression given in Eq. (8) suggests to inter-\npret the e\u000bective exchange interaction as the interfer-\nence between (spin-dependent) charge density waves with\ntheir frequencies set by the spin-chemical potential \u0016\u001b,\nthe spin bias \n \u001b\u001b0, and the excitation spectra of the\nlocalized spins. Particularly, the waves are described\nin terms of the wave vectors k=p2N0!andk\u001b\u001b0=p\n2N0(!\u0000\n + \n\u001b\u001b0), respectively. Fig. 1 (a), (b), dis-\nplay how the oscillatory character of F\u001b\u001b0(R;\n), as func-\ntion ofR, changes as the energy \n varies along the verti-\ncal axis. The period of the oscillations is shorter for en-\nergies \n6= 0 compared to the period in the static regime.\nIn the static regime, the frequency of the charge density\nwave di\u000ber only by the spin bias \n \u001b\u001b0, which typically is\nsmall compared to the Fermi energy. Hence, the charge\ndensity waves are almost in phase with one another. For\n\fnite energies, \n 6= 0, the di\u000berence in frequency be-\ntween the two waves increases, such that the waves go\nout of phase. Hence, due to the incommensurability of\nthe waves, the resulting period of F\u001b\u001b0(R;\n), as function\nofR, changes for increasing j\nj.\nIn conclusion, we have studied the electron mediated\nspin-spin exchange interaction under non-equilibrium\nconditions for localized spins embedded in a two-\ndimensional system. It was demonstrated that the range\nfunction depends dynamically on time, the spin excita-\ntion spectrum, and the spin-bias between the spin chan-\nnels in the electron medium. This leads to that the elec-\ntron mediated exchange interaction between the local-\nized spins is determined by the spin-polarization of the\nelectron medium as well as of the excitation spectra of\nthe local spins. In the case of spatially anisotropic spin-\npolarized surface electrons, the electron mediated spin-\nspin exchange interaction comprise the Ising, Heisenberg,\nand Dzyaloshinski-Moriya type of interactions, capturingthe static case previously reported [27]. It was, moreover,\nshown that earlier results for the range function, which\nwere derived for systems in the static regime [27, 30{\n32], can be straightforwardly extended to slowly \ructu-\nating spins. Particularly, the Dzyaloshinski-Moriya in-\nteraction was shown to decay as sin(2 kFR)=(2kFR) for\nweakly spin-polarized electrons.\nThe author thanks A. V. Balatsky, A. Bergman, O.\nEriksson, and L. Nordstr om for valuable discussions, and\nfor support from the Swedish Research Council.\n\u0003Electronic address: Jonas.Fransson@fysik.uu.se\n[1] P. Gambardella, S. Rusponi, M. Veronese, S. S. Dhesi,\nC. Grazioli, A. Dallmeyer, I. Cabria, R. Zeller, P. H.\nDederichs, K. Kern, C. Carbone, and H. 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Matter, 15, S693 (2003)."    },    {        "title": "2103.10711v1.Domain_wall_dynamics_of_ferrimagnets_induced_by_spin_current_near_the_angular_momentum_compensation_temperature.pdf",        "content": "Domain wall dynamics of ferrimagnets induced by spin-current near the angular\nmomentum compensation temperature\nV.V. Yurlov,1K.A. Zvezdin,2, 3,\u0003P.N. Skirdkov,2, 3and A.K. Zvezdin2\n1Moscow Institute of Physics and Technology, Institutskiy per. 9, 141700 Dolgoprudny, Russia\n2Prokhorov General Physics Institute of the Russian Academy of Sciences, Vavilova 38, 119991 Moscow, Russia\n3New Spintronic Technologies, Russian Quantum Center,\nBolshoy Bulvar 30, bld. 1, 121205 Moscow, Russia\n(Dated: March 22, 2021)\nWe report on a theoretical study of the spin-current excited dynamics of domain walls (DWs) in\nferrimagnets in the vicinity of the angular momentum compensation point. E\u000bective Lagrangian\nand nonlinear dynamic equations are derived for a two-sublattice ferrimagnet taking into account\nboth spin-torques and external magnetic \feld. The dynamics of the DW before and after the Walker\nbreakdown is calculated for any direction of the spin current polarization. It is shown that for the\nin-plane polarization of the spin current, the DW mobility reaches a maximum near the temperature\nof the angular momentum compensation. For the out-of-plane spin polarization, in contrast, a spin\ncurrent with the densities below the Walker breakdown does not excite the dynamics of the DW.\nAfter overcoming the Walker breakdown, the domain wall velocity increases linearly with increasing\nthe current density. In this spin-current polarization con\fguration the possibility of a gigahertz\noscillation dynamics of the quasi-antiferromagnetic vector under the action of a damping-like torque\nin the angular momentum compensation point is demonstrated. Possible structures for experimental\ndemonstration of the considered e\u000bects are discussed.\nI. INTRODUCTION\nSpintronics, which is a rapidly developing branch of\nnanoelectronics, is based on the concept that the prin-\ncipal role in information processing belongs to spins of\nelectrons instead of charges[1, 2]. The mainstream of\nspintronics is attaining the winning combination of the\nspin transport e\u000eciency and nanoscale size of spintronic\ndevices. In this regard, magnetic DWs attract increas-\ning attention[3{5]: they can be used to store and trans-\nmit information in race track or magnetic random access\nmemories (MRAM)[6{10].\nConventional spintronic devices use ferromagnetic\n(FM) materials owing to their property to create and\nsubsequently to use spin-polarization of the conducting\nelectrons. The nanural restrictions of ferromagnetic spin-\ntronic devices relate to the limited operation frequen-\ncies and general energy e\u000eciency. Recent advances in\nspin current injection into insulating antiferromagnets\n(AFMs) have revealed the prospects of AFM spintron-\nics, whose advantages is an extremely high frequency in\ncomparison to the operation frequency of ferromagetic\ndevices[11, 12]. At the same time the AFM spintron-\nics has its own shortcomings associated with the di\u000ecul-\nties to detect the magnetization states and magnetiza-\ntion dynamics. These motivate a boosting development\nof ferrimagnetic (FiM) spintronics, which combines the\nultra-high operation frequences close to those of AFM de-\nvices, with much more reliable ways to detect its magne-\ntization states. Very rich and interesting magnetization\ndynamics[13, 14], in terms of fundamental and applied\n\u0003zvezdin.ka@phystech.eduphysics, is observed in these materials near the points of\ncompensation of magnetization and angular momentum.\nMoreover, by manipulating the temperature of the ferri-\nmagnet near the compensation points, outstanding mag-\nnetization switching characteristics can be obtained[15{\n17]. It has been shown that the electrical current can\nbe an e\u000ecient approach to magnetization switching[18{\n21]. GdFeCo FiM layer demonstrates ultrafast magneti-\nzation reversal in\ruenced by femtosecond laser pulses in\nvarious experiments[22]. These results suggests that the\nFiMs based structures can form a promising technologi-\ncal platform for ultrafast spintronic memory devices.\nAngular momentum compensation point TA, where\nM1=\r1=M2=\r2,\riis the gyromagnetic ratio of the i-\nsublattice (i = 1, 2), represents a very promising line\nof research of FiMs magnetization dynamics[13, 23, 24].\nRecent \feld-driven experiments demonstrated high ve-\nlocity and great mobility of a domain wall in FiM near\ntheTA[25]. The next natural step in this direction is to\nuse the spin-currents to manipulate DWs position and\ndynamics[26]. While the spin-current induced phenom-\nena in FMs seems to be well comprehensible, the mech-\nanisms of spin transfer in AFMs and compensated FiMs\nare still not \fgured out properly.\nIn the present research we develop a model to de-\nscribe DW motion in ferrimagnets near the angular mo-\nmentum compensation point in case of arbitrary spin\ncurrent polarization and torque type. DW dynamics\nin\ruenced by a spin-current in FiMs is studied gener-\nally by using collective coordinates model and Landau-\nLifshitz-Gilbert equation with addition of spin-transfer\ntorque components[27{29]. Here instead we employ\nthe Lagrangian formalism for two subblatice ferrimag-\nnet. This approach allows us to strictly de\fne ferrimag-\nnetic parameters such as width of the DW, velocity ofarXiv:2103.10711v1  [physics.app-ph]  19 Mar 20212\nthe magnons, transverse magnetic susceptibility, e\u000bec-\ntive Gilbert damping parameter and gyromagnetic ratio\nby using perturbation theory.\nWe derive non-linear dynamic equations based on the\nLagrangian formalism, which is similar to Slonszewski\nequations[30]. Using this model, we calculate the dynam-\nics of the domain wall in FiM, depending on the direction\nof the polarizer, electric current density and temperature,\nbefore and after the Walker breakdown. In our model-\ning we observe that for the in-plane polarization of the\nspin current, the DW mobility reaches a maximum near\nthe temperature of the angular momentum compensa-\ntion, and vanishes after bypassing the Walker breakdown.\nFor the out-of-plane spin polarization, in contrast, a spin-\ncurrent with the densities below the Walker breakdown\ndoes not excite the stationary dynamics of the DW. After\novercoming the Walker breakdown, the domain wall ve-\nlocity increases linearly with increasing the electric cur-\nrent density. In this con\fguration of the spin current,\nnear the compensation point TAwe observe gigahertz os-\ncillations of the quasi-antiferromagnetic vector.\nII. MODEL AND BASIC EQUATIONS\nz\nxyL\nθ\nφ\nFiMs layerDW\nspin-currentσVℓ\nFIG. 1. Schematic of considered FiMs with single domain\nwall,\u001bis polarization vector of the spin-current, l- is thick-\nness of the sample; \u0012and'are the polar and azimuthal angels\nof quasi-antiferromagnetic vector L.\nWe develop a model based on the Lagrange formalism\nfor describing DW dynamics due to spin-current. For two\nsublatticed FiMs ordering parameters related to magne-\ntizations M1,M2of these sublattices can be introduced\nin the vicinity of compensation temperatures as quasi-\nantiferromagnetic vector L=M1\u0000M2and ferromag-\nnetic M=M1+M2order parameters. We consider a\nspin-current with a polarization \u001b\rowing through theFiM \flm (see Fig. 1). By analogy with the approach\nused for ferromagnets, we use the adiabatic approxima-\ntion, assuming that the AFM order parameter Lchanges\nslowly in comparison with the spins of the injected elec-\ntrons. The spin-current excites a spin transfer torque\nacting on local magnetization of the i-sublatice respec-\ntively and in general case composed to the in{plane and\nthe out{of plane components Ti\nST=Ti\nFL+Ti\nDL. The\nTi\nFLcomponent due to its symmetry is usually refer-\need as a \feld{like torque and has the following form:\nTi\nDL\u0018[Mi\u0002\u001b]. The Ti\nDLtorque component has a sym-\nmetry similar to the damping torque and usually refereed\nas an damping-like torque (or anti-damping-like torque):\nTi\nDL\u0018[Mi\u0002[Mi\u0002\u001b]]. Typically the magnitude of\nthe anti-damping-like torque component is signi\fcantly\nlager than the \feld-like one for magnetic tunnel junc-\ntions, however in case of spin-orbit torques can be of a\nsimilar magnitude.\nThe magnetization dynamics is described by a system\nof Euler-Lagrange equations:\n8\n>><\n>>:d\ndt\u0010@L\n@_\u0012i\u0011\n\u0000\u000eL\n\u000e\u0012i=\u0000@Ri\n@_\u0012i\u0000@W\n@_\u0012i\nd\ndt\u0010@L\n@_'i\u0011\n\u0000\u000eL\n\u000e'i=\u0000@Ri\n@_'i\u0000@W\n@_'i; (1)\nwhereLandRiare the Lagrangian and Rayleigh func-\ntions,Wis spin transfer torque power density; \u0012iand\n'iare the polar and azimuthal angles characterizing the\norientation of the i-th sublattice magnetization (i = 1,\n2). Note that \u000eWrepresents external spin-current ef-\nfect on magnetic structure and consists of the damping-\nlike and the \feld-like components. Due to its symme-\ntry the \feld-like component can be included in the La-\ngrangian by using the quasi-antiferromagnetic approxi-\nmation. Thus, \u000eWin the Euler-Langrange equations con-\nsist of only damping-like spin-current component. Here-\ninafter we turn to the e\u000bective Lagrangian Le\u000b, e\u000bective\nRayleigh function Re\u000band power density of a spin cur-\nrent\u000eWin quasi-antiferromagnetic approximation appli-\ncable in the vicinity of compensation temperatures (see\nin Supplementary)[31]\nLe\u000b=\u001f?\n2\u0010_\u0012\n\re\u000b\u00112\n+m\u0010\nH\u0000_'\n\re\u000b\u0011\ncos\u0012+\n\u001f?\n2\u0010\nH\u0000_'\n\re\u000b\u00112\nsin2\u0012\u0000Kusin2\u0012\u0000\n\u0000K?sin2\u0012sin2'\u0000A\u0010\u0010d\u0012\ndx\u00112\n+ sin2\u0012\u0010d'\ndx\u00112\u0011\n\u0000\n\u0000\u001f?\n2\u0010B\nM\u00112\u0010\nsin2\u0012n?+\n+ cos2\u0012cos2('\u0000 )nk+ sin2('\u0000 )nk\u0011\n;\nRe\u000b=\u000be\u000bM\n\re\u000b\u0010\n_\u00122+ sin2\u0012\u0001_'2\u0011\n;\n\u000eW=\u0000Asin('\u0000 )nk\u0001\u000e_\u0012+\n+(\u0000An?sin2\u0012+Ankcos\u0012cos('\u0000\u0012))\u0001\u000e_';(2)3\nwherem=M2\u0000M1,M=M1+M2;\u001f?=M=Hex\nis transverse magnetic susceptibility, Hexis an exchange\nmagnetic \feld acting between sublattices; KuandK?are\nconstants of uniaxial and in-plane magnetic anisotropies\nrespectively; A is an exchange sti\u000bness constant, \u0012and\n'are the polar and azimuthal angles of an quasi-\nantiferromagnetic vector L,H= (0;0;Hz) is a mag-\nnetic \feld applied along the \\easy magnetization axis\";\n\u000be\u000b=\u000bm=(m\u0000m0),\re\u000b=\rm=(m\u0000m0),\re\u000b=\n\r\u0001(1\u0000m\u0001m0=M2)\u00001,\u000b= (\u000b1\r2+\u000b2\r1)=2(\r1+\r2),\n1=\r= (1=\r1+ 1=\r2)=2, where\u000biand\riare a damping\nconstant and a gyromagnetic ratio for the i-sublatice re-\nspectively,m0=M(\r1\u0000\r2)=(\r1+\r2);A=~JPDL=(2el)\nandB=~JPFL=(2el) are the \feld-like and the damp-\ning (or anti-damping) spin transfer torque coe\u000ecients,\nwhereJis electrical current density, lis the thickness of\nthe magnetic \flm, e>0 is the electron charge; n?andnk\nare the out-of- and the in- plane components of unit vec-\ntorn= (nx;ny;nz) along the polarization of spin-current\n\u001b, is an angle between the projection of polarization\nvector of the spin-current \u001bon the x-y plane and the\nx-axis;PDLandPFLare the \feld-like and the damp-\ning (or anti-damping) polarizations of the spin current,\nrespectively.\nImplementing the procedure which is described in the\nSupplementary, we derive the system of dynamic equa-\ntions for the 180\u000eDW without external magnetic \feld:\n8\n><\n>:2\u000bM\n\r\u00010_q+m_'\n\re\u000b=fT\u0012\n\u0000\u001f?\n\r2\ne\u000b'+m\n\re\u000b_q\n\u00010\u0000K?sin 2'\u00002\u000bM\n\r_'=fT';\n(3)\nwhere q is a coordinate of the DW centre, \u0001 0=p\nA=Ku\nis a width of the DW. The spin transfer torque compo-\nnents are written as\nfT\u0012=\u0000\u0019\n2Asin('\u0000 )nk\nfT'=\u0000An?+\u001f?\n2\u0010B\nM\u00112\nsin 2('\u0000 )nk: (4)\nNote that in general case the width of the DW is deter-\nmined as \u0001 = \u0001 0p\n1\u0000( _q=c)2, wherec=\re\u000bp\n2A=\u001f?is\na magnons velocity (see in Supplementary). For our set\nof parameters it can be estimated as c\u00188 km/s. As a\nresult, the variation of the DW width for the considered\nvelocities is of the order of one percent (\u0001 =\u00010\u00180:01).\nThus we can assume that _ q\u001ccand DW width \u0001 \u0019\u00010.\nIII. DYNAMIC EQUATION ANALYSIS\nTo understand peculiar features of the current induced\nDW dynamics in compensated FiMs following from eqs.\n(3) and (4) we analyze several particular cases. To calcu-\nlate the DW dynamics we use typical GdFeCo param-\neters: [25]: Ku\u00181\u0001105erg/cc,M \u0019 900 emu/cc,\n\u000b\u00180:02,\r\u00182\u0001107,A\u00181\u000110\u00006erg/cm,gd= 2:2,gf= 2,TM= 220 K,TA= 310 K,l= 10 nm, where\ngdandgfare Lande g-factors for d- and f-sublatices re-\nspectively. The constant of in-plane magnetic anisotropy\nin case of in\fnite \flm is K?= 2\u0019m2, however in case\nof a narrow FiMs nanowire it has a di\u000berent form due\nto magnetostatic interaction. Note, that all dynamic pa-\nrameters (such as velocity, DW displacement and others)\nare functions of \u0017=m=M, which can be rewritten in\nterm of temperature Tby using the following expression\n:\n\u0017=m\nM=T\u0000TM\nT\u0003; (5)\nwhereT\u0003= 1891 K is obtained from the GdFeCo\nparameters[25]. For all further mentioned modelling re-\nsultsPDL= 0:3 andPFL= 0:03.\nT=TA\nT=290 K\nT=280 K\nT=270 KT=350 K\nJT=TA\nT=290 K\nT=280 K\nT=270 KT=350 K\nJIn-plane spin-current polarization (n =1)║\n(a) (b)\nFIG. 2. a) Average DW velocity in Walker and post Walker\nregimes as a function of the electrical current density J; b)\nAbsolute value of the azimuthal angle 'in Walker and post\nWalker regimes as a function of the electrical current density\nJ; blue, green, yellow, red and black curves correspond to the\ntemperatures T= 270 K,T= 280 K,T= 290 K,T=TA\nandT= 350 K respectively; the black arrow indicate the\ntransition in the post Walker regime. All curves are plotted\nfor the in-plane spin current polarization.\nFirst, let us analyze DW dynamics for the in-plane\nspin-current when K?6= 0 and the spin polarization\nalong the y axis n= (0;1;0) ( =\u0019=2 andnk= 1). In\nthis geometry stationary DW motion ( _ '= 0 and a con-\nstant DW) is observed below Walker breakdown. In the\nTAthe azimuthal angle 'tends to zero (see red curve in\nFig. 2(b)) and the stationary DW motion is observed with\nthe velocity _ q=\u00010=\u0019\rA=4\u000bM, which follows from the\n\frst equation in (3). As follows from (3) and (4) in this\ncase the damping-like (or anti-damping-like) spin trans-\nfer torque component with magnitude Ainitiates DW\ndynamics, while the \feld-like one only modi\fes the mag-\nnetostatic term. The magnitude of the azimuthal angle '\nincreases with increasing in the electric current density\nand tends to \u0019=2 which is demonstrated in Fig. 2(b).\nNote that in the case of the in-plane polarizer after\nreaching the critical current density corrisponding to the\nWalker breakthrough J\u0003= 16elj\u000be\u000bjK?=\u0019\u0017~PAD, there\nis no domain wall motion observed - it is indicated by the4\nblack arrows in the Fig. 2(a) and Fig. 2(b). This range\ncorresponds to the constant azimuthal angle '\u0019\u0019=2.\nSpin-current cannot push the domain-wall when angle '\nexceeds\u0019=2 (see Fig. 2(b)) for the case of in-plane polar-\nization. This means that the steady precessional motion\nof DW is impossible for in-plane polarized spin current\nand DW velocity eventually drops to zero for all temper-\natures except for TA.\nNow let us discuss a more di\u000ecult situation when the\n\u001bis parallel to the z-axis n= (0;0;1) andn?= 1. Actu-\nally, if we assume that K?= 0,\u001f?\u001c1 and consider the\ntemperatures in the vicinity of angular momentum com-\npensation point TA, the system (4) describes the steady\nmotion of the DW with velocity _ qand precession rate _ ':\n_q\n\u00010=\u0000\r\n2M\u000b\u0001A\u0017=2\u000be\u000b\n1 + (\u0017=2\u000be\u000b)2; (6)\n_'=\r\n2M\u000b\u0001A\n1 + (\u0017=2\u000be\u000b)2: (7)\nBy using the equation (5) we can rewrite the (6) and\n(7) in term of temperature T and study the dependence\nof the DW velocity and precession rate on temperature\nand current density. Fig. 3(a) demonstrates that the DW\nvelocity has two maximum values near the angular mo-\nmentum compensation point and these values increase\nwith growth in current density. These curves are asym-\nmetric with respect to TA. Thus, the velocity of the DW\nchanges its sign passing through the angular momentum\ncompensation point. This situation is also realized in\nFig. 3(b), where the dependence of the DW velocity on\nelectric current density at di\u000berent temperatures is given.\nAs it is seen from the equation (6) DW velosity linearly\ndepends on the electrical current density. The blue and\ngreen line (see Fig. 3(b)) lie below the TAand the slope\nof this curves (DW mobility _ q=J) decreases. The DW ve-\nlocity changes its direction above the angular momentum\ncompensation point (red curve in Fig. 3(b)).\nNote, that the DW velocity reaches 260 m/s at cur-\nrent densities by about 3 \u0002107A/cm2. Precession rate is\nnot zero _'6= 0 in the vicinity of the angular momentum\ncompensation temperature compared with \feld-driving\nDW motion[32] (where magnetic \feld is applied along\nthe easy magnetization axis). The equation (7) shows\nthat _'reaches its maximum by about 17 GHz at low\ncurrent density (\u00183\u0002107A/cm2) near theTA(see in\nFig. 3(c)). As it follows from the equations (6) and (7)\nin case of considered polarization direction both oscilla-\ntion of the 'angel and DW motion is triggered by the\ndamping (or anti-damping) spin transfer torque compo-\nnent with magnitude A; hence DW dynamic and oscilla-\ntion freezes without spin-current. Field-like spin transfer\ntorque is neglected at the out-of-plane spin-current po-\nlarization case due to decomposition of Lagrangian of the\ntwo-sublattice ferrimagnet as a next order small param-\neter (see in Supplementary).\n(a) (c)\n(b)Out-of-plane spin-current polarization (n⟂=1) \nT=290 K\nT=300 K\nT=320 KTA TMJ=1×107 A/cm2\nJ=2×107 A/cm2\nJ=3×107 A/cm2J=1×107 A/cm2\nJ=2×107 A/cm2\nJ=3×107 A/cm2\nTA TM\nJFIG. 3. a) Dependence of the DW velocity on temperature\nat the di\u000berent current densities; b) Dependence of the DW\nvelocity on electrical current density at the di\u000berent temper-\natures; c) Precession rate as a function of temperature at the\ndi\u000berent current densities. All curves are plotted for the out-\nof-plane spin current polarization.\nA\nBStationary mode\n         J<J*Non-stationary mode\n            J>J*Out-of-plane spin-current polarization (n⟂=1) J\nFIG. 4.J\u0000Tdiagram which describes the ranges of a steady\nand non-steady motion of the DW, green curve shows the\ntemperature dependence of the critical current J\u0003; the ranges\nabove (J > J\u0003) and below ( J < J\u0003) of the green curve\ncorrespond to non-stationary (post Walker) and stationary\n(Walker) mode of the DW, respectively. Insets show the time\ndependence of DW displacement in non-stationary range for\npoint A (T= 320 K and J= 2\u0001107A/cm2) and stationary\nrange for point B ( T= 280 K and J= 0:3\u0001107A/cm2) of\nthe diagram. All curves are plotted for the out-of-plane spin\ncurrent polarization.\nLet us analyse the DW dynamic in the presence of in-\nplane magnetic anisotropy K?6= 0 and n= (0;0;1). We\n\fnd out that there are two di\u000berent regimes of the DW\nmotion: steady ( _ '= 0) and non-steady ( _ '6= 0). Let us\ndiscuss the non-steady one. An analytical solution to the5\nsystem of di\u000berential equations (4) can be written as:\ntan'=J\u0003\nJ+r\n1\u0000\u0010J\u0003\nJ\u00112\ntan(!0t\u0000'0);(8)\nwhereJ\u0003=4\u0019M2\u00172el\n~PDLis the critical current density,\n!0=\r\n\u000b\u0019\u00172Mp\n1\u0000(J\u0003=J)2\n1+(\u0017=2\u000beff)2,'0= arctanJ\u0003=Jp\n1\u0000(J\u0003=J)2. The\nnon-stationary regime realises when the current density J\nhigher than critical current J\u0003(J >J\u0003). This situation\nis represented on the J\u0000Tdiagram in Fig. 4, where the\ngreen curve is the temperature dependence of the critical\ncurrent density J\u0003. The equation (8) describes the oscil-\nlation of the angle 'in the non-stationary range of the\nJ\u0000Tdiagram (J >J\u0003) and inset for the point A in Fig. 4\nshows the time dependence of the DW displacement in\nthis range at \fxed temperature T= 320 K and current\ndencityJ= 2\u0001107A/cm2. Stationary regime of the DW\nrealises when the current density Jis lower than critical\ncurrentJ\u0003. Inset in Fig. 4 for the point B ( T= 280 K\nandJ= 0:3\u0001107A/cm2) shows that after a small period\nof time\u00180:15 ns the DW displacement stops changing\nin time. Therefore, the precession rate _ '= 0, velocity\nof the DW tends to zero and the magnetization freezes\nin the stable state, which corresponds to the equation\nsin 2'=J=J\u0003as follows from the (3). Hence stationary\n(Walker) mode in considered case corresponds to absence\nof DW motion, while non-stationary mode is responsible\nfor DW motion.\nJT=290 K\nT=TA\nT=330 K\nJ*\n1 J*\n2Out-of-plane spin-current polarization (n⟂=1) \nFIG. 5. Average DW velocity in stationary and non-\nstationary mode as function of the electrical current den-\nsity J; blue, green and red curves correspond to temperatures\nT= 290 K,T=TAandT= 330 K, respectively; J\u0003\n1;2are\ncritical current densities which corresponds to T= 290 K\nandT= 330 K, respectively. All curves are plotted for the\nout-of-plane spin current polarization.\nThe dependence of the average DW velocity on theelectrical current density in the stationary and non-\nstationary regimes is shown in Fig. 5. In the stationary\nmode the DW velocity is zero. Near the critical current\nJ\u0003an increase in the value of velocity occurs. However,\nin the non-stationary mode ( J > J\u0003) the average DW\nvelocity linearly increases. Note that in the angular mo-\nmentum compensation point the average velocity is equal\nto zero (green curve in Fig. 5). Besides Fig. 5 shows that\nvelocity changes its sign passing through the T A, which\nis demonstrated by blue ( T= 280 K< TA) and red\n(T= 330 K< TA) curves in Fig. 5. These results are\nconsistent to the J\u0000Tdiagram in Fig. 4. It's impor-\ntant to note that in the non-stationary mode nonlinear\nspin waves can be excited and a\u000bect the dynamics of\nthe DW. However, frequencies of the spin-wave in ferri-\nmagnetic or antiferromagnetic materials lies in terahertz\nrenge[27, 33, 34]. In contrast precession rate of quasi-\nantiferromagnetic vector lies in gigahertz range and we\nsuppose that nonlinear spin waves have weak e\u000bect on\nthe DW dynamics. Moreover, our model itself has a lim-\nitation (see Supplementary) in precession rate, which co-\nincides with the frequencies of spin waves.\nNow, let us discuss the directions of the spin-current\npolarisation \u001band type of torques, which can lead to\ne\u000bects mentioned above, and possibility of their experi-\nmental realization. As follows from the reported results,\nthe damping (or anti-damping) spin transfer torque is\nresponsible for considered motion and oscillation regimes\nfor both planar and perpendicular spin-current polarisa-\ntion\u001b.\nThe \frst possible way to create damping (or anti-\ndamping) torque is to use magnetic tunnel junction\n(MTJ) structure. It is consist of free magnetic layer\nand polariser, which are separated by thin insulating ma-\nterial (usually MgO). In such a structure electric cur-\nrent \rows perpendicularly to the plane and creates Slon-\nczewski torque in the free layer, while spin-current polar-\nisation \u001bdirection is determined by magnetization direc-\ntion of the polariser. Example of MTJ structure is pre-\nsented in Fig. 6(a). The typical polarization value PDL\nin MTJ with ferromagnets is about 0.2-0.4. Hence one\ncan add thin FM layer between MgO and FIMs, which is\nusually done even in classic MTJ to improve TMR and\npolarization values [10], to achieve the level of PDL= 0:3\nused in our simulations.\nAnother way to create damping (or anti-damping)\ntorque is to use heavy metal / FIMs heterostructure. In\nsuch structure electric current \rows through heavy metal\n(like Ta, W, Pt, Au etc.) in plane of the \flm and due\nto the spin Hall e\u000bect creates perpendicular spin current\nwith polarization \u001b, which is perpendicular to the both\nelectric and spin currents. This spin current can cre-\nate anti-damping torque in FIMs. The examples of spin\nHall based geometry in case of perpendicular and pla-\nnar polarization \u001bis presented in Fig. 6(b) and Fig. 6(c)\nrespectively. The value of polarisation in these cases is\nequal to spin Hall angle. This angle can be up to 0.3\n[35, 36], which again makes our PDL= 0:3 is reasonable.6\nz\nx y\nFiMs layerDW\nJV\nmrefMgOFM layer(a)\n(b)\nV\nFiMs layer\nJHMσ\nFiMs layerV\nHM\nJ(c)\nσ\nFIG. 6. Examples of a) MTJ base, b)-c) spin Hall based struc-\ntures, which can be used to observe reported DW motion and\noscillation regimes in FIMs \flm or nanostripe. b) corresponds\nto perpendicular and c) - to planar direction of spin-current\npolarisation \u001b.\nMoreover, it is possible to use topological insulator in-\nstead of heavy metal to achieve spin Hall angles more\nthan 1 [37], which signi\fcantly decrease required current\ndensities.IV. DISCUSSION AND CONCLUSION\nThe theoretical study of the DW dynamics caused by\nthe spin-current near the angular momentum compensa-\ntion point is performed by using the Lagragian formal-\nism. The non-linear dynamic equations describing the\nDW motion are derived from the e\u000bective Lagrangian\nof the two sublattice ferrimagnets. We analyse the DW\nmotion at di\u000berent directions of the spin-current polar-\nizations and show the di\u000berent types of magnetic het-\nerostructures where this spin-current polarization can\nbe realised. In the case of the out-of-plane polarizer\n(n= (0;0;1)) we analyse dependence of DW velocity\nand precession rate on temperature and current den-\nsity. We foresee the possibility to generate oscillations of\nthe quasi-antiferromagnetic vector Lwith the frequen-\ncies by about 17 GHz at low current densities in the\nvicinity of the angular momentum compensation tem-\nperature. This oscillations are initiated by the damping\n(or anti-damping) component of the spin-transfer torque.\nThis precession movement can be associated with a re-\ncent micromagnetic modelling of THz oscillation caused\nby a spin current in antiferromagnetic materials[38] at\nhigh current densities. Furthermore, the DW velocity\nchanges the direction passing trough this temperature\nand this e\u000bect is observed experimentally in GdFeCo fer-\nrimagnet due to spin-current[39]. We explore the DW\nmotion in the stationary (Walker) and non-stationary\n(post Walker) modes and construct the diagram that pro-\nvides the values of current densities and temperatures for\nwhich these modes are realised. The model shows that\nin the Walker regime no DW motion occurs, while in the\npost Walker range DW velocity linearly increases with\nthe current. Note, that the similar dependence of the\nDW velocity was observed due to the spin Hall e\u000bect[26]\nin the TbCo ferrimagnet sample in presence of external\nmagnetic \feld. We also analyze the DW dynamics for\nthe in-plane spin-current polarization and obtain the de-\npendence of the DW velocity as a function of current in\nthe Walker and post Walker regimes. Finally, we deter-\nmine the directions of the spin-current polarisation \u001band\ntype of torques, which lead to e\u000bects mentioned above,\nand possibility of their experimental realization. These\nresults may be useful for experimental studying of do-\nmain wall dynamics in ferrimagnets.\nThis research has been supported by RSF grant No.\n19-12-00432.\n[1] M. Tsoi, R. E. Fontana, and S. S. P. Parkin,\nApplied Physics Letters 83, 2617 (2003),\nhttps://doi.org/10.1063/1.1578165.\n[2] J. Grollier, P. Boulenc, V. Cros, A. Hamzi\u0013 c, A. Vaur\u0012 es,\nA. Fert, and G. 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Ono, Nature Electronics 2,\n389393 (2019)."    },    {        "title": "1003.4232v1.Dynamical_central_peak_and_spinon_deconfinement_in_frustrated_spin_chains.pdf",        "content": "arXiv:1003.4232v1  [cond-mat.str-el]  22 Mar 2010Dynamical central peak and spinondeconfinement infrustrat ed spinchains\nJ. Kokalj1and P. Prelovˇ sek1,2\n1J. Stefan Institute, SI-1000 Ljubljana, Slovenia and\n2Faculty of Mathematics and Physics, University of Ljubljan a, SI-1000 Ljubljana, Slovenia\n(Dated: June 12, 2018)\nStudying the dynamical spin structure factor in frustrated spin chains with spontaneously dimerized ground\nstate we show that besides the gapped spin-wave excitations there appears at finite temperatures also a sharp\ncentralpeak. Thelattercanbeattributedtodeconfined spin ons, accountedwellwithinthe variationalapproach.\nThe central peak remains well pronounced withinthe local sp in dynamics and may be relevant for experiments\nonmaterials with1D frustratedspinchains.\nPACS numbers: 71.27.+a, 75.10.Pq\nFrustrated spin systems have been intensively investigate d\nboth theoretically an experimentally in last decades, offe ring\nnovelphenomenaandchallengesaswellasabroaderviewon\nstrongly correlated electron systems. Among 1D models the\nspin-1/2antiferromagnetic(AFM)Heisenbergchain(nearest-\nneighborinteraction J >0)frustratedwith the secondneigh-\nbor AFM interaction J′>0[1] has attracted wide attention\nalso due to its relevance to the quasi-1D material CuGeO 3\nexhibitingthespin-Peierlstransitionat TSP= 14K. Forpar-\nticular parameters J′/J=α= 0.5the exact ground state\n(g.s.) foundby Majumdarand Ghosh (MG) [2] is doublyde-\ngenerateanddimerizedwithaspingaptoexcitedstates. Suc h\na spin-liquid state without a long range magnetic order has\nbeen shown to extend in a wider range around this point, i.e.\nin the regime α > α c∼0.241.[3–5] Excited states at the\nMGpointhavebeendeterminedanalytically[6,7]andcanbe\nrepresented to a good approximation as a pairs of S= 1/2\nsolitonsorspinonswithagappeddispersion,theconceptco n-\nfirmedby detailed numericalstudiesusing the density-matr ix\nrenormalization-group(DMRG)method. [8]\nThe dynamical propertiesof the frustrated J-J′spin chain\nhave been so far mostly studied via the dynamicalspin struc-\nture factorS(q,ω)motivated again by the inelastic neutron\nscattering (INS) results on CuGeO 3.[9] AtT= 0the con-\ntinuum ofS= 1excitation in S(q,ω)in a wide range of α\ncan be well representedwith a pair of spinons, [8, 10] in par-\nticular if the phenomenologicallyintroducedmatrix eleme nts\nare taken into account [11] in analogy to the basic α= 0\nAFM Heisenberg model.[12] So far, there are very few the-\noretical results on finite- Tproperties of frustrated system. It\nhasbeenshownthattheupperboundaryofspinoncontinuum\ninS(q,ω)persists even at T >0,[13] while the maximumin\nthe static structurefactor S(q)exhibita shiftto incommensu-\nrateq<πat largerαandT. [14,15]\nIn the following we present evidence that T >0dy-\nnamicsoffrustratedspin-chainmodelwiththespontaneous ly\ndimerizedg.s. exhibitsseveral strikingandrather unexpe cted\nfeatures. Most evident, numerically calculated S(q,ω)re-\nveals at low but finite T >0a sharp central ω∼0peak\nwell pronounced in the region q∼π, coexisting with the\ngapped two-spinon continuum known already from T= 0\nstudies.[8, 10] The central peak has its manifestation in aunusualT-dependence of static susceptibility χq∼π(T)with\na maximum at T >0. It shows up also in the local ( q-\nintegrated)SL(ω)as relevant, e.g., for the NMR spin-lattice\nrelaxation. Using a variational presentation of excited tw o-\nspinon states and relevant matrix elements we show that the\nphenomenoncan be directly traced back to spinons and their\ndeconfinednature.\nInthefollowingwestudythefrustratedspinmodelona1D\nchain\nH=J/summationdisplay\ni[Si·Si+1+αSi·Si+2], (1)\nwhereSiare localS= 1/2operators and the only rele-\nvant parameter is α=J′/J(we choose furtheron J= 1).\nThe model has been invoked as the microscopic model for\nCuGeO 3withα∼0.36(realized above T < T SPwhere\nlattice-deformation-induced dimerization is zero). But i t as\nwell represents the 1D zig-zag spin system, example being\ndouble-chaincompound SrCuO 2within the opposite limit of\nlarge|α| ∼5−10. [16]\nAs the central quantity we calculate dynamical S(q,ω)at\nT >0. We employ two numerical approaches. Finite- T\nLanczos method (FTLM) [17] based on the Lanczos diago-\nnalization of small systems covers the whole Trange but is\nrestricted to system sizes N≤28whereby we use periodic\nboundary conditions (b.c.). Finite- Tdynamical extension of\nthe DMRG (FTD-DMRG) method recently developedby the\npresentauthors[18]combinestheDMRGoptimizationofba-\nsis states with the FTLM method for dynamical correlations\natT >0and offers more powerful method for low T. The\nmodel, Eq.(1), is here studied with open b.c. The reachable\nsystemsizesdependon Tandthemethodshowsgoodconver-\ngence,atleastforlow ω,forsystems N <60forT≤0.5pre-\nsented here, with the typical subblock dimension m≤256.\nConcentrating on the low- ωdynamical window the method\nis used as presented in Ref.[18], while high- ωresults are im-\nproved by the application of correction vectors increasing at\nthe same time computation demand. The advantage of both\nmethodsisverygoodspectralresolution,sothattypically only\na minor additional ωdependent broadening of δ∼0.02at\nω∼0andδ∼0.06athigherωisemployedinpresentations.\nLetusfirst presentresultsfortheMG modelwith α= 0.5.2\nWhileS(q,ω)atT= 0is rather well understood and inves-\ntigated numerically,[10] we concentrate in Figs. 1 and 2 on\nT >0FTD-DMRG results for different T/J≤0.5. The\nhigh-ωcontinuum appearing at T= 0above the two-spinon\ngap∆0∼0.25J[5] is qualitativelynot changedfrom T= 0\nspectra. The evident new feature is the central peak at ω∼0\nmostpronouncedat q∼π. Itswidthwisverynarrowbutstill\nofintrinsicnature(beinglargerthantheadditionalbroad ening\nδ= 0.02). It is evident that at fixed Tthe widthwincreases\naway fromq=πwhereby the peak also looses the intensity.\nStill it remains well pronounced in wide region q >0.7π.\nThecomparisonofFigs. 1and2 also revealsthat wincreases\nas well with Tand finally mergesin a broadercontinuumfor\nhighT.\nq\nω 0 0.5 1\nS(q,ω)\n 0 1 2 3\n-0.5 0 0.5 1 1.5 2 2.5S(q,ω)\nFigure 1: (Color online) Dynamical spin structure factor S(q,ω)for\nthe MG model at T/J= 0.1within the whole range of q≤π,\ncalculatedbytheFTD-DMRGmethodonasystem of N= 60sites.\nCorrection vector improvement of high ωpart was preformed only\nforq≥13π/15.\n 0 0.2 0.4 0.6 0.8 1\n-1  0  1  2  3S(q=π,ω)\nωT=0.0\nT=0.1\nT=0.3\nT=0.5\nFigure 2: (Color online) S(q=π,ω)for different T/J=\n0.0,0.1,0.3,0.5where the broadening of central peak with increas-\ningTis wellpronounced.\nIn the following we present the analysis showing that the\nemergence of the central peak in S(q,ω)at lowT < Jcanbe described well in terms of spinons as relevant excitation s\nof the system and their deconfinement. At the MG point\nα= 0.5theg.s. (foreven Nandperiodicb.c.) hastheenergy\nE0=−3NJ/8and the wavefunction which can be written\nasthe productoflocal singlets Ψ0= [1,2][3,4]·[N−1,N].\nIt is doubly degenerate with the correspondingeigenstate ˜Ψ0\nhaving for one site shifted singlets. It has been already rea l-\nized [1, 6, 7] that lowest excitations can be well represente d\nin terms of spinon states. In particular, the lowest branch o f\napproximate triplet ( S= 1,Sz= 1) eigenstates can be con-\nstructedfromthelocaltriplettwo-spinonstates\nψt(p,m) = [1,2]...[2p−3,2p−2]↑2p−1[2p,2p+1]...\n[2m−2,2m−1]↑2m[2m+1,2m+2].... (2)\nwherethe first spinonis onsite 2p−1and the secondon site\n2m. Sincethetotalmomentum Qisconservedduetoperiodic\nb.c.,therelevanttwo-spinonfunctionsare\nψt\nQ(k) =1\nMM/summationdisplay\np,mei(Q+k)p+i(Q−k)mψt(p,m),(3)\nwhere sums run over M=N/2double cells. In the further\nanalysis difficulties arise since ψt(p,m)are not orthogonal\nevenfor distant |p−m| ≫1and furthermorefor p∼m. To\nfindpropereigenfunctionswe follow the procedureand nota-\ntion of Ref.[6] which for each momentum subspace Qyields\nnontrivialmatrixelements\n∝angbracketleftψt\nQ(k′)|ψt\nQ(k)∝angbracketright=9J2\n64ω−ω+δk,k′+1\nMχQ(k,k′),(4)\n∝angbracketleftψt\nQ(k′)|˜H|ψt\nQ(k)∝angbracketright=9ǫQ(k)J2\n64ω−ω+δk,k′+1\nMhQ(k,k′),\nwhere˜H=H−E0,ω±=ω((Q±k)/2),ω(p) = (5/4+\ncos2p)J/2are(approximate)single-spinonenergiesand\nǫQ(k) =ω++ω−= (5\n4+cosQcosk)J.(5)\nThe off-diagonal terms χQ(k,k′),hQ(k,k′)(not presented\nhere)emergingfromnonorthogonalityof ψt\nQ(k)arethesame\nas given in Ref.[6]. Within the triplet two-spinon basis,\nEqs.(2)and(3),propereigenstates(neverthelessnotyete xact\neigenstatesof Eq. (1)) are obtainedvia the diagonalizatio nof\nEqs.(4)andcanbedenoted Ψt\nQ(k)(wherebykremainsonlya\nlabelandnotawelldefinedwavevector). Incontrastto ψt\nQ(k),\nΨt\nQ(k)are ortho-normalized. The correspondingtwo-spinon\nexcitation energies et\nQ(k)are well approximated as the sum\noftwo(deconfined)freespinons,i.e. et\nQ(k)∼ǫQ(k), Eq.(5).\nOur goal is, however, to understand low- Tproperties of\nS(q,ω). Before discussing T >0results, we first have to re-\nconsiderthe T= 0spectrumS0(q,ω)whichhasbeenalready\ninterpretedintermsoftwo-spinonexcitations.[8, 10, 19] Still\nthe corresponding matrix element has been postulated so far\nonly phenomenologically [11, 20] in analogy with previous3\nworksontheunfrustratedHeisenbergmodel.[12]Wenotetha t\natT= 0withinthechosensubspace,Eq.(2),wecanexpress\nS0(q,ω) =1\n2/summationdisplay\nk|∝angbracketleftΨt\nq(k)|S+\nq|Ψ0∝angbracketright|2δ(ω−eq(k)),(6)\nand\nS+\nq|Ψ0∝angbracketright=1−e−iq\n2M/summationdisplay\nkψt\nq(k). (7)\nSince Eq.(7) is an exact representation of the S+\nqoperator,\nwe canevaluate matrixelements ζq(k) =∝angbracketleftΨt\nq(k)|S+\nq|Ψ0∝angbracketrightby\ndiagonalizingnumericallyequations,Eqs.(4). Takingint oac-\ncount thatet\nq(k)∼ǫq(k)we can then evaluate S0(q,ω)in\nthe two-spinon approximation. Results within such a frame-\nwork are presented for q=πin Fig. 3 along with the full\nnumerical results obtained via the T= 0FTD-DMRG (for\nT= 0identical to the more standard dynamical DMRG)\nevaluated within a system of N= 100sites. The agree-\nmentisverysatisfactoryexceptatthehigher- ωendwherethe\nobtained intensity is too low as well as Eq.(4) seem to gen-\nerate a high two-spinon anti-bound state (peak in S0(π,ω))\nbesides the free two-spinon dispersion, Eq.(5). It should b e\nnoted that obtained ζq(k)is quite far from the oversimplified\nspinon picture with ζq(k)∼1.[20] Still it is hard to find for\nit an appropriate analytical expression.[6, 20] One possib il-\nity is to neglect non-orthogonalities in Eqs.(4) which yiel ds\n˜ζq(k)∝1/(ω+ω−)1/2. Corresponding“free”spinonsresults\nforS0(q=π,ω)also presented in Fig. 3 show qualitatively\nreasonable trend (fall-off for higher ω). Still they give an in-\ncorrect behavior at lower and higher cut-off due to divergen t\ntwo-spinondensityofstates.\n 0 0.5 1 1.5\n 0  1  2  3S0(q=π,ω)\nω2 spinons: free  \n2 spinons: exact\nDMRG\nFigure 3: (Color online) T= 0dynamical spin structure factor\nS0(q=π,ω)as calculated via the DMRG for N= 100sites (full\nline),numericallywithinthetwo-spinonapproximation(d ashedline)\nand usingsimplified ˜ζq(k)(dotted line).\nThe above agreement of numerical T= 0results with the\ndescriptionintermsofthetwo-spinonbasis,Eq.(2),gives firm\nsupport also to the interpretation of T >0dynamics. In the\nlow-Tregime we are in S(q,ω)predominantly dealing withexcitations increasing the number of spinons, ns→ns+ 2,\nanalogousto thosein S0(q,ω), Eqs.(6) and(7). Theircontri-\nbution analogousto T= 0Fig. 3 is evident also at T >0in\nFigs.1 and2.\nHowever,inadditiontherearepossibletransitionsbetwee n\nexcitedstatesconserving ns. Inparticular,thematrixelement\nγqQ(k,k′) =∝angbracketleftΨt\nq+Q(k)|S+\nq|Ψs\nQ(k′)∝angbracketrightas introduced already\nin Ref.[20] is finite and nontrivial. Here Ψs\nQ(k′)are singlet\ntwo-spinonseigenstates. We evaluate γqQ(k,k′)numerically\nassuming two-spinon approximation, Eq.(4). Results show\nthat elements are nearly diagonal, i.e., γqQ(k,k′)∼δk′,k+Q\nleading inS(q,ω)to the contributionat ω∼et\nq+Q(k+q)−\nes\nQ(k). Since at low- Tfavored are lowest excited states, i.e.,\nfrom Eq.(5) Q∼0,k∼πandQ∼π,k∼0with a\nBoltzmann weight p∝exp(−∆0/T)(where∆0∼ǫπ(0) =\nJ/4). Numerical solution of two-spinon problem shows that\nes\nQ(k)∼et\nQ(k)∼ǫQ(k)consistent with the picture of un-\nbound (deconfined) spinons. Hence, the strongest transitio ns\nareatω∼ǫq+Q(k+q)−ǫQ(k). Thisevidentlyleadsat q∼π\nto a sharp central peak at ω∼0with the strength increasing\nas∝exp(−∆0/T).\nFromaboveperspectivewethereforeconcludethatthepro-\nnounced central peak in Figs. 1,2 at q∼πconfirm the pre-\nsentedanalysisofnearly-freeordeconfinedspinonsasexci ted\nstates at the MG point. On the other hand, even without the\nextensive calculations it is evident that the central peak c an\nonly appear if triplet and singlet spinon states are nearly d e-\ngenerateagainonlypossiblefordeconfinedspinons.\nThe emergence of the central peak is, however, not re-\nstricted to the MG point α= 0.5but appears to be related\nclosely to the existence of the spontaneous dimerization at\nα > α cand the spin gap ∆0>0. We tested numerically\nalso the case α= 0.7(only partly presented here) where the\nspin gapis larger ∆0∼0.4J. [5] Consequentlyalso the cen-\ntralpeakfeatureisevenmorepronouncedandextendedinthe\nqspaceaswellaspersiststohigher T.\nItis evidentthatthe centralpeakhasasubstantialeffecto n\nthestatic susceptibility χq(T),\nχq(T) =/integraldisplay∞\n−∞dω\nω[1−e−ω/T]S(q,ω),(8)\nbeing sensitive to low- ωdynamics. In Fig. 3 we show the\nFTLM results obtained on systems with N= 28sites for\nχq(T)with various qand againα= 0.5. Most pronounced\nis the variation at q=πwhere the g.s. value χπ(0)is deter-\nminedwith the dimerizationgap,i.e. χπ(0)∝1/∆0. Instead\nof naively expected monotonouslydecreasing χq(T), we ob-\nservein Fig.4simultaneouslywiththeemergenceofthecen-\ntral peak an increasing χπ(T)in the regime 0< T < T∗\nwherebyT∗∼∆0/2. On the other hand, for T > T∗the\nfall-off is uniform with χπ(T)∝1/Twhich is close to the\ncharacteristic critical q=πbehavior for the simple AFM\nHeisenbergmodel.[18] Results for q<πare quiteanalogous\ntakingintoaccountthattherelevantspin gapis ∆q>∆0.\nThe effect of the central peak is visible also in local spin\ncorrelations SL(ω) = (1/N)/summationtext\nqS(q,ω)as presented in4\nq\nT 0.5 1.5 2.5\nχq(T)\n4π/75π/76π/7π\n 0 0.2 0.4 0.6 0.8 1χq(T)\nFigure 4: (Color online) Staticsusceptibility χq(T)vs.Tfor differ-\nentq≤πforα= 0.5as obtained withthe FTLM.\nFig. 5. They are accessible directly via FTD-DMFT by cal-\nculating local spin correlations ∝angbracketleftSz\ni,Sz\ni∝angbracketrightωlocating the site\ni∼N/2to avoid effects of open b.c. As well we can eval-\nuate them within the FTLM and periodic b.c. summing all\nS(q∝negationslash= 0,ω)(q= 0contribution is delta function due to\nthe conserved Sz\ntot). Clearly,in bothapproachesthe diffusion\ncontribution q∼0is not represented correctly but it is ex-\npected to be subdominant.[21] FTD-DMRG results in Fig. 5\npresented for α= 0.5,0.7reveal a central peak at ω∼0\nwell separated from the higher- ωtwo-spinons continuum as\nfar asT/lessorsimilar∆0. The peakgainsthe weight at T∼T∗and for\nT > T∗steadily becomes broader, finally merging with the\ncontinuumfor T >∆0. Moreoverweobserveforboth αthat\nSL(ω= 0)isnearlyconstantinabroadrange T∗<T <2J.\n 0 0.2 0.4 0.6\n 0  1  2SL(ω)\nωα=0.5 T=0.1\nα=0.5 T=0.3\nα=0.7 T=0.1\nα=0.7 T=0.3\nFigure 5: (Color online) Local spin correlations SL(ω)forα=\n0.5,0.7atT= 0.1,0.3as obtained withthe FTD-DMRGmethod.\nIt should be noted that within the simplest approximation\n(withq-independent form factor) the NMR or NQR spin-\nlattice relaxation should be a closely related to SL(ω), i.e.\nthe relaxation rate is given by 1/T1∝SL(ω= 0). Fol-\nlowingaboveresultswe wouldobtainforconsideredsystems\n1/T1∼const in a broad range T > T∗similar to theoretical\npredictionsfor the 1D (unfrustrated)AFM Heisenbergmodel\nand CuGeO 3[21]. While the agreement for higher T >∆0\nwith the Heisenberg model is not surprising the novel contri -butionofthecentralpeakisthatthe validityofthisuniver sal-\nity is extended to lower T > T∗. It should be also reminded\nthat such relaxation is far from the usual Korringa relaxati on\nwith1/(TT1)∼const.\nInconclusion,wehaveshownthatthefrustratedspinchain\nasmanifestedwithin the 1D J-J′modelwithα>α creveals\nbesides the gap in spin excitations at T= 0also very un-\nusual spin dynamics at finite but low T <∆0. The central\npeak which appears in S(q,ω)atq∼πas well in the q-\nintegratedlocal SL(ω)isverysharpanddominatesthelow- ω\nresponse at low T. It is a direct consequence and the signa-\nture of deconfinement of spinon excitations in such systems.\nIt remains to be investigated whether such a behavior is re-\nstricted to the particular case of investigated model or the re\nare other gapped spin systems with similar phenomena. As\nfar as experimental relevance is concerned extensively inv es-\ntigatedCuGeO 3abovethe spin-Peierlstransition T >T SPis\ninterpretedwith a frustrated spin-chain model with α∼0.36\nandcouldpartlyexhibitmentionedphenomenainspiteofpre -\nsumablyverysmall scale ∆0<0.02J[8].\nWe authors acknowledge helpful discussions with T. To-\nhyama as well as the support of the Slovenia-JapanResearch\nCooperative grant and the Slovenian Agency grant No. P1-\n0044.\n[1] for a review see P. Lecheminant, Frustrated Spin Systems (ed.\nH. T.Diep, WorldScientific,p.307, 2004).\n[2] C.K. Majumdar, D.K. Ghosh, J. Math. Phys. 10, 1388, 1399\n(1969).\n[3] K. Okamoto, K. Nomura, Phys.Lett.A 169, 433 (1992).\n[4] S.Eggert, Phys.Rev. B 54, R9612 (1996).\n[5] S.R.White,I.Affleck, Phys.Rev. B 54, 9862 (1996).\n[6] B.S.Shastry, B.Sutherland, Phys.Rev. Lett. 47, 964 (1981).\n[7] W.J. Caspers, K.M. Emmett, W. Magnus, J. Phys. A 17, 2687\n(1984).\n[8] E. Sørensen, I. Affleck, D. Augier, D. Poilblanc, Phys. Re v. B\n58, R14701 (1998).\n[9] M. Arai, M. Fujita, M. Motokawa, J. Akimitsu, S.M. Bennin g-\nton, Phys. Rev. Lett. 77, 3649 (1996).\n[10] H. Yokoyama, Y. Saiga,J.Phys. Soc.Jpn. 66, 3617 (1997).\n[11] R.R.P. Singh, P. Prelovˇ sek, B.S. Shastry, Phys. Rev. L ett.77,\n4086 (1996).\n[12] G.M¨ uller,H.Beck,J.C.Bonner,Phys.Rev.Lett. 43,75(1979).\n[13] K. Fabricius,U.L¨ ow, Phys.Rev. B 57, 13371 (1998).\n[14] S.Watanabe, H. Yokoyama, J. Phys.Soc.Jpn 68, 2073 (1999).\n[15] I.Harada,Y.Nishiyama,Y.Aoyama,S.Mori,J.Phys.Soc .Jpn,\nSuppl. A 69, 339 (2000).\n[16] M. Matsuda, K. Katsumata, K.M. Kojima, M. Larkin, G.M.\nLuke, J. Merrin, B. Nachumi, Y.J. Uemura, H. Eisaki, N. Mo-\ntoyama et al.,Phys.Rev. B 55, R11953 (1997).\n[17] J. Jakliˇ c, P.Prelovˇ sek, Phys.Rev. B 49, 5065 (1994).\n[18] J. Kokalj, P.Prelovˇ sek, Phys.Rev. B 80, 205117 (2009).\n[19] D.Poilblanc,J.Riera,C.A.Hayward,C.Berthier,M.Ho rvati´ c,\nPhys. Rev. B 55, R11941 (1997).\n[20] B.S.Shastry, D.Sen,Phys. Rev. B 55, 2988 (1997).\n[21] M. Itoh, M. Sugahara, T. Yamauchi, Y. Ueda, Phys. Rev. B 54,\nR9631 (1996)."    },    {        "title": "0807.2524v3.Spin_currents_and_spin_superfluidity.pdf",        "content": "March 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics\nVol. 00, No. 00, June 2008, 1{68\nREVIEW\nSpin currents and spin super\ruidity\nE. B. Sonin\u0003\nRacah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel\n(received )\nThe present review analyzes and compares various types of dissipationless spin transport: (1)\nSuper\ruid transport, when the spin-current state is a metastable state (a local but not the\nabsolute minimum in the parameter space). (2) Ballistic spin transport, when spin is trans-\nported without losses simply because sources of dissipation are very weak. (3) Equilibrium\nspin currents, i.e., genuine persistent currents. (4) Spin currents in the spin Hall e\u000bect. Since\nsuper\ruidity is frequently connected with Bose condensation, recent debates about magnon\nBose condensation are also reviewed. For any type of spin currents simplest models were\nchosen for discussion in order to concentrate on concepts rather than details of numerous\nmodels. The various hurdles on the way of using the concept of spin current (absence of the\nspin-conservation law, ambiguity of spin current de\fnition, etc.) were analyzed. The \fnal\nconclusion is that the spin-current concept can be developed in a fully consistent manner,\nand is a useful language for description of various phenomena in spin dynamics.\nKeywords: spin current; spin super\ruidity; easy-plane (anti)ferromagnet; Landau criterion;\nspin-orbit coupling; spin Hall e\u000bect\n1. Introduction\nThe problem of spin transport occupies minds of condensed matter physicists for\ndecades. A simple example of spin transport is spin di\u000busion, which is a process\naccompanied with dissipation. Conceptually more complicated is \\dissipationless\"\nspin transport, which was also discussed long time but was in the past and remains\nnow to be a matter of controversy. The main source of controversy is that spin is not\na conserved quantity. This leads to many complications and ambiguities in de\fning\nsuch concepts as spin \row, current, or transport . Sometimes these complications\nare purely semantic. However, this does not make them simpler for discussion.\n\\Semantic traps\" very often are a serious obstacle for understanding physics and\nfor deriving proper conclusions concerning observation and practical application of\nthe phenomenon. The best strategy in these cases is to focus not on names but\non concepts hidden under these names. Only after this one may \\take sides\" in\nsemantic disputes not forgetting, however, that choosing names is to considerable\nextent a matter of convention and taste.\nDuring long history of studying the problem of \\dissipationless\" spin transport\none can notice three periods, when studies in this \feld were especially intensive.\nThe \frst period started from theoretical suggestions on possible \\super\ruidity of\nelectron-hole pairs\" [1], which were later extended on possible spin super\ruidity\n[2, 3]. At the same period the concept of spin super\ruidity was exploited [4{6] for\n\u0003Email: sonin@cc.huji.ac.il\nISSN: 0001-8732 print/ISSN 1460-6976 online\nc\r2008 Taylor & Francis\nDOI: 10.1080/0001873YYxxxxxxxx\nhttp://www.informaworld.comarXiv:0807.2524v3  [cond-mat.other]  3 Mar 2010March 3, 2010 17:4 Advances in Physics SpinRev\n2 E. B. Sonin\ninterpretation of experiments demonstrating unusually fast spin relaxation in3He-\nA [7]. The second period was marked by intensive theoretical and experimental\nwork on spin super\ruidity in3He-B starting from interpretation of experiments\non the so-called Homogeneously Precessing Domain (HPD) [8] in terms of spin\nsupercurrents [9]. Finally in these days (the third period) we observe a growing\ninterest to dissipationless spin currents in connection with work on spintronics\n[10]. The \fnal goal of spintronics is to create devices based on spin manipulation,\nand transport of spin with minimal losses is crucial for this goal. Now one can \fnd\nreviews summarizing the investigations done during the \frst [11] and the second\n[12, 13] periods of works on dissipationless spin transport. On the other hand,\nthe work on spin transport in spintronics is a developing story, and probably it is\nstill premature to write summarizing reviews. Nevertheless, some reviews mostly\naddressing the spin Hall e\u000bect have already appeared [14, 15]. It looks also useful to\nhave a glance on the current status of the \feld from a broader viewpoint and to \fnd\nbridges between current investigations and those done in the \\last millennium\".\nThe present review aims at this goal. The intention is to discuss mostly concepts\nwithout unnecessary deepening in details, and simplest models were chosen for this.\nThe term \\super\ruidity\" is used in the literature to cover a broad range of phe-\nnomena, which have been observed in super\ruid4He and3He, Bose-Einstein con-\ndensates of cold atoms, and, in the broader sense of this term, in superconductors.\nIn the present review super\ruidity means only a possibility to transport a physi-\ncal quantity (mass, charge, spin, ...) without dissipation. Exactly this phenomenon\ngave a rise to the terms \\superconductivity\" and \\super\ruidity\", discovered nearly\n100 years and 70 years ago respectively. It is worthwhile to stress that one should\nnot understand the adjective \\dissipationless\" too literally. In reality we deal with\nan essential suppression of dissipation due to the presence of energetic barriers of\nthe topological origin. How essential suppression could be, is a matter of a special\nanalysis. In the present review we restrict discussion with the question whether\nactivation barriers, which suppresses dissipation, can appear.\nBut super\ruidity is not the only reason for suppression of dissipation in the\ntransport process, and it is important to understand the di\u000berence between various\ntypes of dissipationless transport. In the present review we shall discuss four types\nof them:\n\u000fSuper\ruid transport: The spin-current state is a metastable state (a local but\nnot the absolute minimum in the parameter space).\n\u000fBallistic transport. Here spin is transported without losses simply because\nsources of dissipation are very weak.\n\u000fEquilibrium currents. Sometimes symmetry allows currents even at the equilib-\nrium. A superconductor in a magnetic \feld is a simple example. Equilibrium spin\ncurrents are also possible, though there is a dispute on whether they have some-\nthing to do with spin transport. Equilibrium spin currents are genuine persistent\ncurrents, since no dissipation is possible at the equilibrium by de\fnition.\n\u000fSpin currents in the spin Hall e\u000bect. These currents are also called dissipationless\nsince they are normal to the driving force (electric \feld) and therefore do not\nproduce any work. However, in the spin Hall e\u000bect there is dissipation connected\nwith a longitudinal charge current through a conducting medium. On the other\nhand, it was recently revealed that the spin Hall e\u000bect is possible also in insulators\nwhere a charge current is absent. Then spin currents are not accompanied by\nany dissipation becoming similar to equilibrium spin currents.\nThe second type (ballistic) looks mostly trivial: dissipation is absent because\nsources of dissipation are absent. Still it is worth of short discussion since some-March 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 3\ntimes they confuse ballistic transport with super\ruid transport (an example of it\nis discussed in section 7.2). The super\ruid transport does not require the absence\nof dissipation mechanisms. One may expect that in an ideally clean metal at zero\ntemperature resistance would be absent. But this would not be superconductivity.\nSuperconductivity is the absence of resistance in a dirty metal at T >0.\nThe \frst two types of spin currents are discussed in Part I of the review, which is\ndevoted to magnetically ordered systems. The third and the fourth types are dis-\ncussed mostly in Part II, which addresses time-reversal-invariant systems without\nmagnetic order, though in magnetically ordered media equilibrium spin currents\nare also possible (section 9). Since from the very beginning of the theory of super-\n\ruidity the relation between super\ruidity and Bose condensation was permanently\nin the focus of attention, discussing spin super\ruidity one cannot avoid to consider\nthe concept of magnon Bose condensation, which is vividly debated nowadays.\nSection 10 addresses this issue.\nPart I: Spin currents in magnetically ordered systems\n2. Mass supercurrents\nSince the idea of spin super\ruidity originated from the analogy with the more\ncommon concept of mass super\ruidity let us shortly summarize the latter. The\nessence of the transition to the super\ruid or superconducting state is that below\nthe critical temperature the complex order parameter  =j jei', which has a\nmeaning of the wave function of the bosons or the fermion Cooper pairs, emerges\nas an additional macroscopical variable of the liquid. For the sake of simplicity, we\nrestrict ourselves to the case of a neutral super\ruid at zero temperature putting\naside the two-\ruid theory for \fnite temperatures. Then the theory of super\ruidity\ntells that the order parameter  determines the particle density n=j j2and the\nvelocity of the liquid is given by the standard quantum-mechanical expression\nv=\u0000i~\n2mj j2( \u0003r \u0000 r \u0003) =~\nmr': (1)\nThus the velocity is a gradient of a scalar, and any \row is potential. Since the\nphase and the particle number are a pair of canonically conjugate variables, one\ncan write down the Hamilton equations for the pair of the canonically conjugated\nvariables \\phase { density\":\n~d'\ndt=\u0000\u000eE\n\u000en;dn\ndt=\u000eE\n~\u000e': (2)\nHereE=R\nd3REis the total liquid energy, whereas Eis the energy density, and\n\u000eE=\u000enand\u000eE=\u000e'are functional derivatives of the total energy:\n\u000eE\n\u000en=@E\n@n\u0000r\u0001@E\n@rn\u0019@E\n@n=\u0016; (3)\n\u000eE\n\u000e'=@E\n@'\u0000r\u0001@E\n@r'=\u0000r\u0001@E\n@r'=\u0000~r\u0001g: (4)March 3, 2010 17:4 Advances in Physics SpinRev\n4 E. B. Sonin\na)\nb)\nFigure 1. Phase (inplane rotation angle) variation at the presence of mass\n(spin) supercurrents. a) Oscillations in a sound (spin) wave). b) Stationary\nmass (spin) supercurrent.\nIn these expressions \u0016is the chemical potential,\ng=nv=@E\n~@r'(5)\nis the particle current, and the dependence of the energy on the density gradient\nwas ignored. Eventually the Hamilton equations are reduced to the equations of\nhydrodynamics for an ideal liquid:\nmdv\ndt=\u0000r\u0016; (6)\ndn\ndt=\u0000r\u0001g: (7)\nA crucial property of the system is the gauge invariance: the energy does not\ndepend on the phase directly ( @E=@' = 0) but only on its gradient. According to\nNoether's theorem this must lead to the conservation law for a conjugate variable,\nthe total number of particles. The conservation law manifests itself in the conti-\nnuity equation (7), which contains the particle supercurrent. The pre\fx \\super\"\nstresses that this current is not connected with dissipation. It is derived from the\nHamiltonian or the Lagrangian but not from the dissipation function. In contrast to\nthe di\u000busion current proportional to the density gradient, the supercurrent is pro-\nportional to the phase gradient. Therefore it appears only in a coherent state with\nbroken gauge invariance. The equations of super\ruid hydrodynamics can be derived\nfrom the Gross{Pitaevskii equation for a weakly non-ideal Bose-gas. However, they\nare much more general than this model. They can be formulated from the most gen-\neral principles of symmetry and conservation laws. Indeed, deriving the two-\ruid\ntheory of super\ruidity Landau did not use the concept of Bose-condensation.\nAn elementary collective mode of the ideal liquid is a sound wave. In a sound\nwave the phase varies in space, i.e., the wave is accompanied by mass supercur-\nrents (\fgure 1a). An amplitude of the time and space dependent phase variationMarch 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 5\nis small, and currents transport mass on distances of the order of the wavelength.\nA really super\ruid transport on macroscopic distances is related with stationary\nsolutions of the hydrodynamic equations corresponding to \fnite constant currents\nwith constant nonzero phase gradients (current states). In the current state the\nphase rotates through a large number of full 2 \u0019-rotations along streamlines of the\ncurrent (\fgure 1b).\nThe crucial point of the super\ruidity concept is why the supercurrent is a persis-\ntent current, which does not decay despite it is not the ground state of the system\nand has a larger energy. The \frst explanation of the supercurrent stability was\ngiven on the basis of the well known Landau criterion [16]. According to this cri-\nterion, the current state is stable as far as any quasiparticle of the Bose-liquid in\nthe laboratory frame has a positive energy and therefore its creation requires an\nenergy input. Let us suppose that elementary quasiparticles of the Bose-liquid at\nrest have an energy spectrum \"(p) wherepis the quasiparticle momentum. If the\nBose-liquid moves with the velocity vthe quasiparticle energy in the laboratory\nframe is\"(p)+p\u0001v. The energy cannot be negative (which would mean instability)\nif\nv<vL= min\"(p)\np: (8)\nIn super\ruid4He the Landau critical velocity vLis determined by the roton part\nof the spectrum. But in this review we focus on the long-wavelength collective\nexcitations, which are phonons with the spectrum \"=usp. Then according to\nequation (8) the supercurrent cannot be stable if the velocity vexceeds the sound\nvelocityus.\nAs far as one wants to check the Landau criterion for long-wavelength collective\nmodes like sound waves, it is not necessary to solve a dynamical problem looking\nfor the spectrum of phonons. It is enough to estimate the energy of possible static\n\ructuations in the stationary current state with particle density n0and velocityv0.\nLet us write down the energy of the current state taking into account possible local\n\ructuations of the particle density, n0=n\u0000n0, and of the velocity, v0=v\u0000v0,\nup to the terms of the second order:\nE=Z\nd3R\u0014\n\u00160(n0)n0+@\u00160(n0)\n@nn02\n2+m(n0+n0)(v0+v0)2\n2\u0015\n:\nHere\u00160=@E0=@n andE0are the chemical potential and the energy density of\nthe liquid at rest. One may neglect terms of the \frst order with respect to the\ndensity \ructuation n0and the velocity v0since we look for an energy extremum\nat \fxed averaged density and velocity and the \frst-order term must vanish after\nintegration. Using the thermodynamic relation mu2\ns=n=@\u00160=@n=@2E0=@n2and\nomitting the subscript 0 in n0andv0, one obtains\nE\u0019Z\nd3R\u0014mu2\nsn02\n2n+mn0v\u0001v0+mnv02\n2\u0015\n: (9)\nThe quadratic form under the integral is positive de\fnite, i.e., the current state\ncorresponds to the energy minimum, if the condition u2\ns> v2is satis\fed. This\ncondition is identical to the Landau criterion equation (8) for the phonon spectrum\n\"=usp.\nThe theory of super\ruidity tells that the Landau criterion is a necessary but\nnot su\u000ecient condition for current metastability. The Landau criterion checks onlyMarch 3, 2010 17:4 Advances in Physics SpinRev\n6 E. B. Sonin\nπ−π/2 5π/2π/2 3π/22π 0π3π/4 5π/4π/2 3π/2π/4 7π/42π0\na)\nb)rmrm\nFigure 2. Mass and spin vortices. a) Mass vortex or spin vortex in an easy-\nplane ferromagnet without inplane anisotropy. b) Spin vortex at small spin\ncurrents (hr'i\u001c 1=l) for four-fold inplane symmetry. The vortex line is a\ncon\ruence of four 90\u000edomain walls (solid lines).\nsmall deviations from the current state. Meanwhile the current state can be de-\nstroyed via large perturbations of the current state. In super\ruids these large per-\nturbations are vortices. In the current state the phase rotates along the current\ndirection. The current can relax if one can remove one 2 \u0019-turn of the phase. This\nrequires that a singular vortex line crossed or \\cut\" the channel cross-section. The\nprocess is called \\phase slip\".\nIf the vortex axis (vortex line) coincides with the zaxis, the phase gradient\naround the vortex line is given by\nr'v=[^z\u0002r]\nr2; (10)\nwhereris the position vector in the xyplane. The phase changes by 2 \u0019around\nthe vortex line (\fgure 2a). Creation of the vortex requires some energy. The vortex\nenergy per unit length (line tension) is determined by the kinetic (gradient) energy:\n\u000f=Z\nd2r~2n(r'v)2\n2m=\u0019~2n\nmlnrm\nrc; (11)\nwhere the upper cut-o\u000b rmis determined by geometry. For example, for the vortex\nshown in \fgure 2a it is the distance of the vortex line from a sample border. The\nlower cut-o\u000b rcis the vortex-core radius. It determines the distance rat which the\nphase gradient is so high that the hydrodynamic expression for the energy becomesMarch 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 7\nRe ψ\nIm ψ ψ\nRe ψ\nIm ψ ψa)\nb)\nFigure 3. Topology of the uniform mass current and the vortex states. a) The\ncurrent state in a torus maps onto the circumference j j=j 0j=const in the\ncomplex - plane, where  0is the equilibrium order parameter wave function\nof the uniform state. b) The vortex state maps onto the circle j j\u0014j 0j.\ninvalid. A good estimation for rcisrc\u0018\u0014=us, where\u0014=h=m is the circulation\nquantum of the velocity. Inside the core the modulus of the order parameter goes\ndown to zero eliminating the singularity in the kinetic energy at the vortex axis.\nFor the weakly non-ideal Bose-gas this estimation yields the coherence length.\nNow suppose that a vortex appears in the current state with the constant gradient\nr'0: The phase gradients induced by the vortex are superimposed on the constant\nphase gradient related to the current: r'=r'0+r'v. The total gradient\nenergy includes that of the current, the vortex energy given by equation (11),\nand the energy from the cross terms of the two gradient \felds. Only the last two\ncontributions are connected with the vortex, and their sum determines the energy\nof the vortex in the current state:\n~\u000f=\u0019~2n\nmLlnrm\nrc\u00002\u0019~2n\nmSr'0; (12)\nwhereLis the length of the vortex line and Sis the area of the cut, at which\nthe phase jumps by 2 \u0019. For the 2D case shown in \fgure 2a (a straight vortex in\na slab of thickness Lnormal to the picture plane) S=Lrm. One can see that\nvortex motion across the channel (growth of rm) is impeded by the barrier, which\nis determined by variation of the energy ~ \u000fwith respect to rm. The peak of the\nbarrier corresponds to rm= 1=2r'0. The height of the barrier is\n\u000fm\u0019\u0019~2n\nmLln1\nrcr'0: (13)\nThus the barrier disappears at gradients r'0\u00181=rc, which are of the same order\nas the critical gradient determined from the Landau criterion. In the 3D geometry\nthe phase slip is realized with expansion of vortex rings. For the ring of radius R\nthe vortex-length and the area of the cut are L= 2\u0019RandS=\u0019R2respectively,\nand the barrier disappears at the same critical gradient \u00181=rcas in the 2D case.\nThe barriers stabilizing metastable current states are connected with topology\nof the order parameter space. In a super\ruid the order parameter is a complex\nwave function  (r). At the equilibrium  =j 0jei', where the modulus j 0jis a\nconstant determined by minimization of the energy and the phase 'is a degen-\neration parameter since the energy does not depend on '. Any current state in\na closed annular channel (torus) with the phase change 2 \u0019naround the channelMarch 3, 2010 17:4 Advances in Physics SpinRev\n8 E. B. Sonin\nmaps onto a circumference j j=j 0jin the complex plane (\fgure 3a) winding the\ncircumference ntimes. It is evident that it is impossible to change nkeeping the\npath on the circumference j j=j 0jall the time. Thus nis atopological charge .\nOne can change it (removing, e.g., one winding around the circumference) only\nby leaving the circumference (the equilibrium order parameter space in the case).\nThis should cost energy, which is spent on creation of a vortex. Figure 3b shows\nmapping of the vortex state onto the circle j j\u0014j 0j.\nWithout such topological barriers super\ruidity is ruled out. However, barriers\ndo not automatically provide the life-time of currents long enough. In practice,\ndissipation via phase slips is possible even in the presence of barriers due to thermal\n\ructuations or quantum tunneling. Here and later on we address only \\ideal\"\ncritical currents (the upper bound for critical currents) at which barriers disappear\nleaving \\practical\" critical currents beyond the scope of the present review.\n3. Phenomenology of magnetically ordered systems and spin currents\nThe main interaction responsible for magnetic order is exchange interaction, which\nis invariant with respect to rotations of the whole spin system. Then according\nto Noether's theorem the total spin must be conserved. For ferromagnets where\nthe order parameter is the spontaneous magnetization M, this means that the\nexchange energy can depend on the absolute value of Mbut not on its direction.\nOther contributions to the free energy (anisotropy energy or dipole-dipole inter-\naction) are related with spin-orbit interaction, which does not conserve the total\nspin. But these interactions are relativistically small, i.e., governed by the small\nrelativistic parameter v=c, wherevis a typical electron velocity and cis the speed\nof light. The spin-orbit interaction does depend on Mdirection, but because of its\nweakness cannot a\u000bect the absolute value Min slow dynamics. This is a crucial\npoint in the phenomenological theory of magnetism of Landau and Lifshitz [17],\nwhich determines the form of the equation of motion for ferromagnet magnetization\nknown as the Landau-Lifshitz equation [18]:\n@M\n@t=\r[Heff\u0002M]; (14)\nwhere\ris the gyromagnetic ratio between the magnetic and mechanical moment\n(M=\u0000\rS). The e\u000bective magnetic \feld is determined by the functional derivative\nof the total free energy F=R\nd3RFwith density F:\nHeff=\u0000\u000eF\n\u000eM: (15)\nAccording to the Landau-Lifshitz equation, the absolute value Mof the magnetiza-\ntion cannot vary. The evolution of Mis a precession around the e\u000bective magnetic\n\feldHeff.\nAt \frst let us discuss exchange approximation , in which relativistic e\u000bects are\nignored and the conservation of total spin is not violated. In this approximation\nthe free energy density is\nF(M) =F0(M) +\u000b\n2riM\u0001riM: (16)\nThe \frst exchange-energy term F0(M), being the largest term, is crucial for deter-\nmination of the equilibrium value of M. But after determination of Mit can beMarch 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 9\nignored as an inessential large constant. Indeed, its contribution to the e\u000bective \feld\nin Landau-Lifshitz equation (14) does not produce any e\u000bect: the contribution is\nparallel toMand vanishes in the vector product. In the absence of external \felds,\nwhich break invariance with respect to rotations in the spin space, the Landau-\nLifshitz equation reduces to the continuity equations for components of the spin\ndensityS=\u0000M=\r:\n@Si\n@t=\u00001\n\r@Mi\n@t=\u0000rjJi\nj; (17)\nwhere\nJi\nj=\u0000\u0014\nM\u0002@F\n@rjM\u0015\ni=\u0000\u000b[M\u0002rjM]i=\u0000\u000b\"iklMkrjMl (18)\nis thejth component of the spin current transporting the ith component of spin.\nThus in an isotropic ferromagnet all three components of spin are conserved.\nThe Landau-Lifshitz equation has plane-wave solutions describing spatially\nnonuniform precession of the magnetization M=M0+maround the ground-\nstate magnetization M0:m/eikr\u0000i!t. Here the magnetization deviation mis\nsmall and normal to M0. Linearizing with respect to m, one obtains spin waves\nwith the spectrum\n!=\r\u000bM 0k2: (19)\nIn an isotropic ferromagnet spin waves at k6= 0 are accompanied by spin currents,\nbut super\ruid spin transport is impossible as will be clear from section 4.\nNext we shall consider the case when spin-rotational invariance is partially bro-\nken, and there is uniaxial crystal magnetic anisotropy given by the third term in\nthe phenomenological free energy:\nF=F0(M) +\u000b\n2riM\u0001riM+EAM2\nz\n2M2: (20)\nIf the anisotropy energy EAis positive, it is the \\easy plane\" anisotropy, which\nkeeps the spontaneous magnetization M0in thexyplane (the continuous limit\nof theXY model). In this model the zcomponent of spin is conserved, because\ninvariance with respect to rotations in the easy plane remains unbroken. Since the\nabsolute value of magnetization is \fxed, the vector Mof the magnetization is\nfully determined by the angle 'showing the direction of Min the easy plane xy\n(Mx=Mcos',My=Msin') and by the zcomponent of the magnetization mz.\nWe use the notation mzinstead ofMzin order to emphasize that mzis a small\ndynamic correction to the magnetization, which is absent at the equilibrium. In\nthe new variables the free energy is\nF=Z\nd3RF=Z\nd3R\u0014m2\nz\n2\u001f+A(r')2\n2\u0015\n: (21)\nThe constant A=\u000bM2is sti\u000bness of the spin system determined by exchange\ninteraction, and the magnetic susceptibility \u001f=M2=EAalong thezaxis is de-\ntermined by the uniaxial anisotropy energy EAkeeping the magnetization in the\nplane. The Landau-Lifshitz equation reduces to the Hamilton equations for a pair of\ncanonically conjugate continuous variables \\angle{angular momentum\" (analogousMarch 3, 2010 17:4 Advances in Physics SpinRev\n10 E. B. Sonin\nto the canonically conjugate pair \\coordinate{momentum\"):\nd'\ndt=\u0000\r\u000eF\n\u000emz=\u0000\r@F\n@mz; (22)\n1\n\rdmz\ndt=\u000eF\n\u000e'=@F\n@'\u0000r\u0001@F\n@r'; (23)\nwhere functional derivatives on the right-hand sides are taken from the free energy\nFgiven by equation (21). Using the expressions for functional derivatives one can\nwrite the Hamilton equations as\nd'\ndt=\u0000\rmz\n\u001f; (24)\n\u00001\n\rdmz\ndt+r\u0001Jz= 0; (25)\nwhere\nJz=\u0000@F\n@r'=\u0000Ar' (26)\nis the spin current.\nThere is an evident analogy of equations (24) and (25) with the hydrodynamic\nequations (6) and (7) for an ideal liquid, equation (25) being the continuity equa-\ntion for spin. This analogy was exploited by Halperin and Hohenberg [19] in their\nhydrodynamic theory of spin waves. In contrast to the isotropic ferromagnet with\nthe quadratic spin-wave spectrum, the spin wave in the easy-plane ferromagnet has\na sound-like spectrum as in a super\ruid: !=csk, where the spin-wave velocity\niscs=\rp\nA=\u001f. Halperin and Hohenberg introduced the concept of spin current,\nwhich appears in a propagating spin wave like a mass supercurrent appears in a\nsound wave (\fgure 1a). This current transports the zcomponent of spin on dis-\ntances of the order of the wavelength. But as well as the mass supercurrent in\na sound wave, this small oscillating spin current does not lead to super\ruid spin\ntransport, which this review addresses. Spin super\ruid transport on long distances\nis realized in current states with magnetization rotating in the plane through a\nlarge number of full 2 \u0019-rotations as shown in \fgure 1b.\nLet us consider now the case of antiferromagnetic order. The simplest model of\nan antiferromagnet is two sublattices with magnetizations M1andM2. In the\nabsence of weak ferromagnetism and external magnetic \felds two magnetizations\nM1=\u0000M2completely compensate each other without producing any total mag-\nnetizationm=M1+M2. However, a small magnetization mdoes appear due to\nexternal magnetic \felds or dynamical e\u000bects. The amplitudes of M1andM2and\ntheir mutual orientation are mostly determined by strong exchange interaction, but\nthe latter does not \fx the direction of the staggered magnetization L=M1\u0000M2,\nwhich is the order parameter of a two-sublattice antiferromagnet.The equations\nof motion for two vectors Landmcan be derived from the two Landau-Lifshitz\nequations for M1andM2taking into account the exchange interaction between\ntwo sublattices. But it would be useful to present a more general version of the\nmacroscopic phenomenological theory, which is able to describe an antiferromag-March 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 11\nnetic structure of any complexity [20, 21]. The theory of spin dynamics in su-\nper\ruid phases of3He developed by Leggett and Takagi [22] also belongs to this\nclass. Following the same principle \\exchange is the strongest interaction\" as in\nthe Landau-Lifshitz theory, macroscopic theories of this type deal with phenomena\nat scales essentially exceeding microscopic scales (the coherence length in the case\nof3He), at which the exchange energy establishes the tensor structure of the order\nparameter. This permits to assume that the entire dynamic evolution of the order\nparameter reduces to rotations in the 3D spin state, which cannot change the ex-\nchange energy. Then the dynamics of the system is described by three independent\npairs of canonically conjugated variables \\angle{moment\" 'i{mi=\r(i= 1;2;3):\n@'i\n@t=\u0000\r\u000eF\n\u000emi;\n1\n\r@mi\n@t=\u000eF\n\u000e'i: (27)\nHere'iare the angles of spin rotations around three Cartesian axes ( i=x;y;z ).\nApart from spatial dependence of the variables, these equations are similar to the\nequations of motion of a 3D rigid top. In our case the top is an antiferromagnetic\nspin order parameter rigidly \fxed by exchange interaction. As in the case of a\ntwo-sublattice antiferromagnet, magnetization mresults from deformation of the\nequilibrium spin structure. The approach is valid as far as this deformation is weak,\ni.e.,mis smaller than the characteristic moments of the antiferromagnetic structure\n(staggered magnetization Lin the case of a two-sublattice antiferromagnet). Since\nrotation around the vector Lhas no e\u000bect on the state of the system the latter has\nonly two degrees of freedom corresponding to two pairs \\angle{moment\". Then\nthe equations become the equations of motion of a rotator. In contrast to the\nspontaneous magnetization Min the Landau-Lifshitz equation (14), the small\nabsolute value of the magnetization mis not kept constant.\nBecause the group of 3D rotations is non-commutative, the state of the system\ndepends on the order, in which rotations around di\u000berent axes are performed.\nIn practice they frequently use the Euler angles (they are introduced in section\n7). For the most content of Part I (except for section 7), one can choose one\ndegree of freedom connected with the conjugate pair 'z{mz, and the problem of\nnon-commutativity is absent (further we shall omit the subscript zof the angle\n'z). If the energy of the ground state does not depend (or depends weakly as\ndiscussed in section 5) on the angle ', the equations of motions for 'andmzare\nthe same Hamilton equations (22) and(23), which were formulated for an easy{\nplane ferromagnet. In the case of a two-sublattice antiferromagnet the angle 'is\nthe angle of the staggered magnetization Lin the easy plane.\nThe discussion of this section has not made any reference to a concrete micro-\nscopic model of magnetism. Indeed, the approach is general enough and is valid\nfor models of magnetism based on the concepts of either localized or itinerant\nelectrons. In particular, ferromagnetism of localized electrons is described by the\nHeisenberg model with the Hamiltonian:\nH=\u0000JX\ni;jsi\u0001si+1; (28)\nwhereJ >0,siare spins at the sites i, and the summation over jincludes only the\nnearest neighbors to the site i. In the continuum limit, when the spin rotates very\nslowly at scales of the intersite distance a, the Hamiltonian (28) reduces to the freeMarch 3, 2010 17:4 Advances in Physics SpinRev\n12 E. B. Sonin\nenergy (16) in the Landau-Lifshitz theory with the magnetization M=\u0000\rhsii=a3\nand the sti\u000bness constant \u000b=Ja5=\r2.\nThe debates on reliability of the general phenomenological approach to mag-\nnetism are as old as the approach itself. Nearly sixty years ago Herring and Kit-\ntel [23] argued with their opponents that their phenomenological theory of spin\nwaves \\is not contingent upon the choice of any particular approximate model\nfor the ferromagnetic electrons\". Interestingly these discussions are still continu-\ning in connection with the concept of the spin current, which originates from the\ngeneral phenomenological approach. Originally they connected spin supercurrents\nwith counter\rows of particles with opposite spins, for example, of He atoms in\nthe A-phase of3He [4, 5]. Bunkov [13] insisted that only a counter\row of particles\nwith opposite spins would lead to super\ruid spin transport, thus ruling out spin\nsuper\ruidity in materials with magnetic order resulting from exchange interaction\nbetween localized spins (see p. 93 in his review). However, the spin current does not\nrequire itinerant electrons for its existence [2]. The presumption that spin transport\nin insulators is impossible is still alive nowadays. According to Shi et al. [24], it is\na critical \raw of spin-current de\fnition if it predicts spin currents in insulators.\n4. Stability of spin-current states\nFor the sake of simplicity further we focus on current states in an easy-plane fer-\nromagnet, though the analysis can be easily generalized to other magnetically or-\ndered systems discussed in the previous section. In the current state the sponta-\nneous magnetization M(r) rotates in the easy plane through a large number of\nfull 2\u0019-rotations when the position vector ris varying along the direction of spin\ncurrent (\fgure 1b). The spin-current state is metastable if it corresponds to a local\nminimum of the free energy, i.e., any transition to nearby states would require an\nincrease of energy. This condition is an analog of the Landau criterion for mass\nsupercurrents discussed in section 2. In order to check current metastability, one\nshould estimate the energy of possible small static \ructuations around the sta-\ntionary current state. For this estimation, one should take into account that the\nsti\u000bness constant Ais proportional to the squared inplane component of the spon-\ntaneous magnetization M2\n?=M2\n0\u0000m2\nz, and in the presence of large angle gradients\nAmust be replaced with A(1\u0000m2\nz=M2\n0). So the free energy is\nF=Z\nd3R\u0014m2\nz\n2\u001f+A(1\u0000m2\nz=M2\n0)(r')2\n2\u0015\n=Z\nd3R\u0014m2\nz\n2EA\u0000A(r')2\nM2\n0+A(r')2\n2\u0015\n: (29)\nOne can see that if r'exceedsp\nM2\n0=\u001fA =p\nEA=Athe current state is unstable\nwith respect to the exit of M0from the easy plane. This is the Landau criterion\nfor the stability of the spin current.\nLike in super\ruids, stability of current states is connected with topology of the\norder parameter space. For ferromagnets the order parameter is the magnetization\nvectorM. For isotropic ferromagnets the space of degenerated equilibrium states\nis a spherejMj=jM0j, whereas for an easy-plane ferromagnet this space reduced\nto an equatorial circumference on this sphere (\fgure 4a). Thus the order parameter\nspace for an easy-plane ferromagnet is topologically equivalent to that space for su-\nper\ruids (the circumference on the complex plane shown in \fgure 3). Spin-current\nstates are stable because they belong to the topological classes di\u000berent from theMarch 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 13\na)\nc)b)\nFigure 4. Topology of the uniform spin current and the spin vortex states.\na) The current state in a torus maps onto the equatorial circumference of\nthe order parameter sphere jMj=const . b) Isotropic ferromagnet: continu-\nous deformation reduces a circumference (current state) to a point (uniform\ncurrent-free state). c) The spin vortex state maps onto either an upper or a\nlower half of the sphere jMj=const .\nclass of the uniform ground state and cannot be reduced to the latter by continuous\ndeformation of the path. In contrast, for an isotropic ferromagnet the path around\nthe equatorial circumference can be continuously transformed to a point on the\nsphere as shown in \fgure 4b. In this process the energy monotonously decreases,\nand topological barriers are absent. Topology of an easy-axis (anti)ferromagnet\nalso does not allow stable spin-current states.\nThe connection of super\ruidity-like phenomena with topology of the order pa-\nrameter space is universal and not restricted with the examples of mass and spin\nsuper\ruidity considered here. The same arguments support possibility of exciton\nsuper\ruidity, which was discussed even earlier than spin super\ruidity (see the in-\ntroductory section 1). Though the whole problem of super\ruid exciton transport\nis far from its resolution, in some special case experimental evidences of this trans-\nport has already been reported. Kellogg et al. [25] observed vanishing resistance\nin double quantum Hall layers, which was interpreted as a consequence of Bose\ncondensation of interlayer excitons (or pseudospin ferromagnetism).\nAs well as in the theory of mass super\ruidity, the Landau criterion is a nec-\nessary but not su\u000ecient condition for current metastability. One should also to\ncheck stability with respect to large perturbations, which are magnetic vortices .\nThe magnetic vortices were well known in magnetism. Bloch lines in ferromagnetic\ndomain walls are an example of them [26]. In the spin-current state the magneti-\nzationMtraces a spiral at moving along the current direction. The spin current\ncan relax if one can remove one turn of the spiral. This requires that a singular\nline (magnetic vortex) crossed or \\cut\" the channel cross-section [2, 3] as shown in\n\fgure 2b.\nThe structure of the magnetic vortex outside the vortex core is the same as ofMarch 3, 2010 17:4 Advances in Physics SpinRev\n14 E. B. Sonin\nthe mass super\ruid vortex given by equation (10). Correspondingly, the magnetic\nvortex energy is determined by the expression similar to equation (11):\n\u000f=Z\nd2rA(r'v)2\n2=\u0019Alnrm\nrc; (30)\nwhere the upper cut-o\u000b rmdepends on geometry. However, the radius rcand the\nstructure of the magnetic vortex core are determined di\u000berently from the mass\nvortex. In a magnetic system the order parameter must not vanish at the vortex\naxis since there is a more e\u000bective way to eliminate the singularity in the gradient\nenergy: an excursion of the spontaneous magnetization out of the easy plane xy.\nThis would require an increase of the uniaxial anisotropy energy, which keeps M\nin the plane, but normally this energy is much less than the exchange energy,\nwhich keeps the order-parameter amplitude Mconstant. Finally the core size rc\nis determined as a distance at which the uniaxial anisotropy energy density EAis\nin balance with the gradient energy A(r')2\u0018A=r2\nc. This yields rc\u0018p\nA=EA.\nFigure 4c shows mapping of the spin vortex state onto the order parameter space.\nIn contrast to super\ruid vortices mapping onto a plane circle, the spin vortex state\ncan map onto one of two halves of the sphere jMj=const . Thus a magnetic (spin)\nvortex has an additional topological charge having two values \u00061 [27].\nThe energy of the spin-current state with a vortex and the energy of the barrier,\nwhich blocks the phase slip, i.e., the decay of the current, are determined similarly\nto the case of mass super\ruidity [see equations (12) and (13)]:\n~\u000f=\u0019LA lnrm\nrc\u00002\u0019ASr'0; (31)\n\u000fm\u0019\u0019LA ln1\nrcr'0; (32)\nwhereLis the length of the vortex line and Sis the area of the cut, at which the\nangle jumps by 2 \u0019. Thus the barrier disappears at gradients r'0\u00181=rc, which are\nof the same order as the critical gradient determined from the Landau criterion.\nThis is a typical situation in the super\ruidity theory. But sometimes the situation\nis more complicated as we shall see in section 7.2.\n5. Spin currents without spin conservation law\nThough processes violating the conservation law for the total spin are relativisti-\ncally weak, their e\u000bect is of principal importance and in no case can be ignored. The\nattention to super\ruid transport in the absence of conservation law was attracted\n\frst in connection with discussions of super\ruidity of electron-hole pairs. The num-\nber of electron-hole pairs can vary due to interband transitions. As was shown by\nGuseinov and Keldysh [28], interband transitions lift the degeneracy with respect\nto the phase of the \\pair Bose-condensate\" and make the existence of spatially\nhomogeneous stationary current states impossible. On the basis of it Guseinov and\nKeldysh concluded that there is no analogy with super\ruidity. This phenomenon\nwas called \\\fxation of phase\". However some time later it was demonstrated [29]\nthat phase \fxation does not rule out existence inhomogeneous stationary currentMarch 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 15\nstates, which admit some analogy with super\ruid current states1. This analysis\nwas extended on spin currents [2, 3].\nIn the spin system the role of the phase is played by the angle of the magnetization\nMin the easy plane, and the degeneracy with respect to the angle is lifted by\nmagnetic anisotropy in the plane. Adding the n-fold inplane anisotropy energy to\nthe total free energy (21) the latter can be written as\nF=Z\nd3R\u001am2\nz\n2\u001f+A(r')2\n2+K[1\u0000cos(n')]\u001b\n: (33)\nThen the spin continuity equation (25) becomes\n1\n\rdmz\ndt=r\u0001Jz+nKsin(n') =\u0000A\u0014\nr2'\u0000sin(n')\nl2\u0015\n; (34)\nwhere\nl2=A\nnK: (35)\nExcludingmzfrom equations (24) and (34) one obtains the sine Gordon equation\nfor the angle ':\n@2'\n@t2\u0000c2\ns\u0014\nr2'\u0000sin(n')\nl2\u0015\n= 0; (36)\nwherecs=p\n\rA=\u001f is the spin-wave velocity. According to this equation, the\ninplane anisotropy leads to a gap in the spin-wave spectrum:\n!2=nc2\ns\nl2+c2\nsk2: (37)\nThere are one-dimensional solutions '(x\u0000vt) of the sine Gordon equation with\nnon-zero average hr'i, which correspond to a periodic lattice of solitons (domain\nwalls) of the width \u0018~l=lp\n1\u0000v2=c2swith the period x0= 2\u0019=nhr'imoving\nwith the velocity v. The function inverse to '(x\u0000vt) is\nx\u0000vt=rn\n2~lZ'd'0\np\u0014\u0000cosn'0; (38)\nwhere the constant \u0014>1 is determined by the equation\nx0=2\u0019\nnhr'i=rn\n2~lZ2\u0019=n\n0d'0\np\u0014\u0000cosn'0: (39)\nThe free energy of the soliton lattice is given by\nF=A\"\n1\u0000\u0014\nnl2+hr'i\n\u0019~lrn\n2Z2\u0019=n\n0p\u0014\u0000cosn'd'#\n: (40)\n1Similar conclusions have been done with respect to possibility of supercurrents in systems with spatially\nseparated electrons and holes [30, 31].March 3, 2010 17:4 Advances in Physics SpinRev\n16 E. B. Sonin\n/angb∇acketleft∇ϕϕ\nϕ/angb∇acket∇ight≪1\nl\n/angb∇acketleft∇ϕ/angb∇acket∇ight ≫1\nlx\nx\nFigure 5. The nonuniform spin-current states with hr'i\u001c1=landhr'i\u001d\n1=l.\nIt is possible to develop the hydrodynamic theory of the soliton lattice in the\nterms of local density and velocity of solitons [32], which is able to describe defor-\nmations of the lattice slow in space and time. Here we focus on stationary current\nstates when dmz=dt= 0 (v= 0). At small average twisting of the spontaneous\nmagnetizationhr'i\u001c 1=lthe structure constitutes domains that correspond to\nthenequivalent easiest directions in the easy plane. In this limit ( \u0014!1) the free\nenergy density is the product of the energy of an isolated domain wall and the\ndensity of domain walls nhr'i=2\u0019:\nF=A4hr'i\n\u0019pnl: (41)\nSpin currents (gradients) inside domains are negligible but there are essential spin\ncurrents inside domain walls where r'\u00181=l. This hardly reminds genuine su-\nper\ruid transport on macroscopical scales: spin is transported over distances on\nthe order of the domain-wall width l. With increasing hr'ithe density of domain\nwalls grows, and at hr'i\u001d1=lthey coalesce while for a displacement along the\ndirection of the gradient hr'i, the end point of the vector Mdescribes, a line close\nto a helix. The nonuniform states with hr'i\u001c1=landhr'i\u001d1=lare shown in\n\fgure 5. Thus the processes violating spin conservation law are not important for\nlarge deformations (gradients) of the spin structure. This means that the analogy\nof these deformed states with current states in super\ruids makes sense.\nStudying stability of nonuniform current states it is possible to ignore the inplane\nanisotropy only for large spin currents when r'\u001d1=l. Let us consider the opposite\nlimit ofhr'i\u001c1=lwhen the spin structure reduces to a chain of domain walls. The\nrelaxation of the spin current, which is proportional to the wall density, requires\nthat some domain walls vanish from the channel. This process is illustrated in \fgure\n2b for the four-fold inplane symmetry ( n= 4). When a magnetic vortex appears, n\ndomain walls \fnish not at the wall but at the vortex line, around which the angleMarch 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 17\n'changes by 2 \u0019. The 2\u0019angle jump occurs at the cut restricted by the vortex\nline. Thendomain walls disappear via motion of the vortex line across the channel\ncross section. In the course of this process, the change of the energy consists of the\nvortex-line energy, which is proportional to the line length L, and of a decrease\nof the surface energy of the nwalls themselves proportional to the cut area S.\nThe latter contribution is determined by the product of the free energy density\n(41) and the volume 2 \u0019S=hr'i. Taking these two contributions into account, the\nenergy during the process of annihilation of nwalls is\n~\u000f=\u0019LA lnrm\nrc\u00008ApnS\nl: (42)\nComparing it with the energy given by equation (31) one sees that the gradient\nr'0is replaced by the maximum gradient \u00181=linside the domain wall. Corre-\nspondingly for the 2D case shown in \fgure 2b the expression (32) for the barrier\nenergy must be replaced by\n\u000fm\u0019\u0019LA lnl\nrc\u0019\u0019\n2LAlnEA\nK; (43)\nwhere the two lengths rc\u0018p\nA=EAandl\u0018p\nA=K are determined by the uniaxial\nand the inplane anisotropy energies EAandK. Thus large barriers stabilizing spin-\ncurrent states are possible only if the condition EA\u001dKis satis\fed. This conclusion\n[2, 3, 11] was recently con\frmed by the analysis of K onig et al. [33].\nAn important di\u000berence with conventional mass super\ruidity is that in conven-\ntional super\ruidity the barrier, which suppresses supercurrent relaxation, grows\nunrestrictedly when the gradient r'decreases. In contrast, in spin super\ruidity\nthe barrier growth stops when the gradient reaches the values of the order 1 =l(in-\nverse width of the domain wall). Since the current relaxation time exponentially\ndepends on the barrier (whether the barrier is overcome due to thermal \ructuations\nor via quantum tunneling) the life time of the current state in conventional super-\n\ruidity diverges when the velocity (phase gradient) decreases. In contrast, the life\ntime of the spin current can be exponentially large but always \fnite . This provides\nan ammunition for rigorists, who are not ready to accept the concept \\spin super-\n\ruidity\" (or super\ruidity of any non-conserved quantity) in principle. In principle ,\none could agree with them. But in practice , whatever we call it, \\non-ideal su-\nper\ruidity\" or \\quasi-super\ruidity\", some consequences should outcome from the\nfact of the existence of topological barriers suppressing relaxation of spin-current\nstates. A key point is whether these consequences are observable. This is the topic\nof the next section.\n6. Is super\ruid spin transport \\real\"?\nFrom early days of discussions on spin supercurrents and up to now there are ar-\nguments on whether the spin supercurrent can result in \\real\" transport of spin.\nPartially this is a semantic problem: One must carefully de\fne what \\real\" trans-\nport really means. Let us suppose that one has a usual super\ruid mass persistent\ncurrent in a ring geometry. Nobody doubts that real mass transport occurs in this\ncase, but how can one notice it in the experiment? In any part of the ring channel\nthere is no accumulation (increase or decrease) of the mass. Of course, one can\ndetect gyroscopic e\u000bects related with persistent currents, but it is an indirect evi-\ndence. What may be a direct evidence? One could suggest a Gedanken ExperimentMarch 3, 2010 17:4 Advances in Physics SpinRev\n18 E. B. Sonin\nSuperflow of angular momen tumTorque\nFigure 6. Mechanical analogue of a persistent current: A twisted elastic rod\nbent into a closed ring. There is a persistent angular-momentum \rux around\nthe ring.\nin which the ring channel is suddenly closed in some place. In the wake of it one\ncan observe that the mass increases on one side from the closure and decreases on\nthe other side. This would be a real transport if one required a demonstration of\nmass accumulation as a proof of it. Accepting this de\fnition of transport reality\none can notice real transport only in a non-equilibrium process, when the trans-\nported quantity decreases in some place and increases in another . Naturally one\ncan discard these semantic exercises as irrelevant for practice, but only as far as\nthey refer to mass currents. In the case of spin currents in the past and nowadays\nspin accumulation sometimes is considered as a necessary proof of real spin trans-\nport. Therefore, in old publications on spin super\ruidity [2, 3, 11] much attention\nwas paid to possible experimental demonstration of spin transport from one place\nto another.\nBefore starting discussion of possible spin-transport demonstration it is useful to\nconsider a mechanical analogue of super\ruid mass or spin supercurrent [11]. Let us\ntwist a long elastic rod so that a twisting angle at one end of the rod with respect\nto an opposite end reaches values many times 2 \u0019. Bending the rod into a ring\nand connecting the ends rigidly, one obtains a ring with a circulating persistent\nangular-momentum \rux (\fgure 6). The intensity of the \rux is proportional to\nthe gradient of twisting angle, which plays the role of the phase gradient in the\nmass supercurrent or the spin-rotation-angle gradient in the spin supercurrent.\nThe analogy with spin current is especially close because spin is also a part of the\nangular momentum. The deformed state of the ring is not the ground state of the\nring, but it cannot relax to the ground state via any elastic process, because it\nis topologically stable. The only way to relieve the strain inside the rod is plastic\ndisplacements . This means that dislocations must move across rod cross-sections.\nThe role of dislocations in the twisted rod is the same as the role of vortices in\nthe mass or spin current states: In both of the cases some critical deformation\n(gradient) is required to switch the process on. There are various ways to detect\ndeformations or strains in an elastically deformed body. Similarly, it is certainly\npossible, at least in principle, to notice deformation (angle gradient) of the spin\nstructure in the spin-current state. It would be a legitimate evidence of the spin\ncurrent, not less legitimate than a magnetic \feld measured around the ring as an\nevidence of the persistent charge current in the ring.\nOf course, it is not obligatory to discuss the twisted rod in terms of angular-\nmomentum \rux. One can describe it only in terms of deformations, stresses, andMarch 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 19\nJz\nx\nJ\nLz\nxSpin injection \nSpin injection Spin injection Medium without \nspin super/f_luidity \nMedium with \nspin super/f_luidity \nmxzmz\n0\nFigure 7. Spin injection to a spin-nonsuper\ruid and a spin-super\ruid\nmedium.\nelastic sti\u000bness. So we must have in mind that there are two languages, or descrip-\ntions of the same physical phenomenon. A choice of one of them is a matter of taste\nand tradition. For example, in order to describe the transfer of momentum they\nuse the momentum-\rux tensor (\\\rux\", or \\current\" language) in hydrodynamics,\nwhile in the elasticity theory they prefer to call the same tensor as stress tensor . In\nprinciple one can avoid the term \\super\ruidity\" and speak only about the \\phase\nsti\u000bness\" even in the case of mass supercurrents.\nLet us return to possible demonstration of \\real\" spin transport. Suppose that\nspin is injected into a sample at the sample boundary x= 0 (\fgure 7). The injection\ncan be realized practically either with an injection of a spin-polarized current (for\nthe sake of simplicity we put aside the problem what happens with charge in this\ncase), or with pumping the spin with a circularly polarized microwave irradiation.\nIf the medium at x>0 cannot support super\ruid spin transport, the only way of\nspin propagation is spin di\u000busion described by the equations\n@mz\n@t\u0000\rr\u0001Jz\nd+mz\nT1= 0;Jz\nd=Dsrmz; (44)\nwhereDsis the spin-di\u000busion coe\u000ecient and T1is the time characterizing the Bloch\nlongitudinal relaxation, which violates the spin-conservation law. In the stationary\ncase@mz=@t= 0, and both the spin current and the nonequilibrium magnetization\nmzexponentially decay inside the sample: Jz\nd/mz/e\u0000x=Ls, whereLs=pDsT1\nis the spin-di\u000busion length. So no spin can reach the other boundary x=Lof the\nsample provided L\u001dLs.\nNow let us suppose that the medium at 0 <x<L is magnetically ordered and\ncan support super\ruid spin transport. If the injection is so weak that the angle gra-\ndientr'is much less than 1 =l, the perturbation of the medium by the injection can\npenetrate at the length not longer than the domain-wall width l. So the injection\npushes a piece of a domain wall into the sample. The spin current exponentially\ndecays like in the medium without super\ruidity. If the injection is so strong that\nthe angle gradient r'exceeds its maximum value 1 =pnlin a center of an isolated\ndomain wall, continuity of the spin current on the boundary requires appearance\nof a soliton lattice with a period of the order or smaller than l. This means that\nthe spin current can reach the other boundary x=L. In the thermodynamic limit\nL!1 one may expect a stationary soliton lattice with mz= 0. However, at the\nboundaryx=Lthe spin current must be injected into the medium without spin\nsuper\ruidity. This is impossible without a \fnite mz, and the \fnite mzmeans thatMarch 3, 2010 17:4 Advances in Physics SpinRev\n20 E. B. Sonin\nthe soliton lattice is moving . In the limitr'\u001d1=lthe soliton lattice is rather\ndense and one may neglect periodical spin-current modulation caused by inplane\nanisotropy. Then spin transport is described by the equations\nd'\ndt=\u0000\rmz\n\u001f; (45)\ndmz\ndt\u0000\rr\u0001Jz+mz\nT1= 0; (46)\nwith the boundary conditions for the supercurrent Jz(0) =Jz\n0atx= 0 and\nJz(L) =\u0000fmz(L) atx=L. The current Jz\n0in the \frst condition is the spin-\ninjection current, while the second boundary condition takes into account that\nthe medium at x > L is not spin-super\ruid and spin injection there is possible\nonly if some non-equilibrium magnetization mz(L) is present. The coe\u000ecient fcan\nbe found by solving the spin-di\u000busion equations in the medium at x > L [3]. It\nalso depends on properties of the contact at x=L. While the inplane anisotropy\nviolating the spin conservation (phase \fxation) was neglected, one cannot neglect\nirreversible dissipative processes, which also violate the spin-conservation law. The\nsimplest example of such a process is the longitudinal spin relaxation characterized\nby timeT1.\nThe stationary solution of equations (45) and (46) is\nmz=\u0000\rT1\nL+f\rT 1Jz\n0\u0019\u0000\rT1\nLJz\n0; Jz(x) =Jz\n0\u0012\n1\u0000x\nL+f\rT 1\u0013\n\u0019Jz\n0L\u0000x\nL:(47)\nThough the solution is stationary in the sense that @mz=@t= 0, but@'=@t6= 0.\nWe consider a non-equilibrium process (otherwise spin accumulation is impossible),\nwhich is accompanied by the precession of Min the easy plane. But the process\nis stationary only if the precession angular velocity is constant in space. The con-\nditionmz=const, which results from it, is similar to the condition of constant\nchemical potential in super\ruids or electrochemical potential in superconductors\nin stationary processes. If this condition were not satis\fed, there would be steady\ngrowth of the angle twist as is evident from equation (45). As already mentioned\nabove, the nonzero mzmeans that the soliton lattice is moving. In our case the\nsoliton velocity is rather slow since it is inversely proportional to L:v=c2\nsT1=L.\nOne can see that irreversible loss of spin is a more serious obstacle for super-\n\ruid spin transport than coherent phase \fxation, to which most of attention was\nattracted in the literature. Because of spin relaxation, the spin current inevitably\ndecreases while moving away from the injection point, in contrast to constant su-\nper\ruid mass currents. However, in a spin-super\ruid medium this decrease is linear\nand therefore less destructive than exponential decay of currents in non-super\ruid\nmedia. So one have a good chance to notice spin accumulation in the medium at\nx>L rather distant from the place of original spin injection. This justi\fes using\nthe term \\super\ruid\".\nIn the presented analysis we assumed that at the boundary the entire spin-\ninjection current is immediately transformed into a supercurrent. Actually spin\ninjection can also generate the di\u000busion current close to the boundary. However, at\nsome distance from the boundary the di\u000busion current inevitably transforms into\na supercurrent [3]. If this distance (healing length) is much shorter than the size L\nof the sample our boundary condition at x= 0 is fully justi\fed. Similar e\u000bects take\nplace at contacts \\normal metal - superconductor\": The current from the normalMarch 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 21\nmetal to the superconductor is completely transformed into the supercurrent at\nsome \fnite distance from the contact.\nSpin injection is not the only method of generation of spin currents. One can\ngenerate spin currents by a rotating inplane magnetic \feld, which is applied to\none end of the sample and is strong enough to orient the magnetization parallel\nto it. Because of the sti\u000bness of the spin system, the spin rotation at one end is\ntransmitted to the other end of the sample, which is not subject to the direct e\u000bect\nof the rotating magnetic \feld. Transmission of the torque through the sample is\nspin current . Since the rotating \feld is acting on the phase (angle) of the order\nparameter, it can be called coherent method, in contrast to the incoherent method\nof spin injection. The coherent method of spin-current generation has no analog in\nsuper\ruids and superconductors, since in the latter cases there is no \feld linked\nto the phase of the order parameter. Referring to the set up shown in \fgure 7\nwith spin injection replaced by rotating magnetic \feld, in the coherent method the\nmagnetization mzis \fxed by the frequency of the rotating \feld. In this case there\nis no threshold for spin-current generation, and the spin current appears whatever\nsmall the frequency could be. But if the frequency (and mzproportional to it) is low,\nthe spin transmission is realized via generation of a chain of well separated solitons\n(domain walls), which propagate to the other end of the sample. Thus, a \\moving\nsoliton lattice\" is another synonym for spin super\ruid transport. Long-distance\npropagation of solitons through a slab of the Aphase of super\ruid3He generated\nby a pulse of a radio-frequency magnetic \feld has already experimentally realized\nby Bartolac et al. [34]. This experiment was discussed in terms of spin transport\nin reference [32].\n7. Spin-precession super\ruidity in super\ruid3He-B\n7.1. Stationary uniform precession in3He-B\nNow we focus on the experimental and theoretical investigations of super\ruid spin\ntransport in the Bphase of super\ruid3He. The spin super\ruidity in the Bphase\nhas several important features, which distinguish it from the spin super\ruidity dis-\ncussed previously in this review. First, in contrast to what was considered earlier,\nobserved spin-current states in the Bphase are dynamical nonlinear states very\nfar from the equilibrium, which require for their support permanent pumping of\nenergy. Thus dissipation is always present, and speaking about \\super\ruidity\", i.e.,\n\\dissipationless\" spin transport, we have in mind the absence of additional dissipa-\ntion connected with the spin current itself. Second, while the previous discussion\ndealt with the transport of a single spin component ( z-component), in the Bphase\nspin vector performs a more complicated 3D rotation and the spin current refers\nto the transport of some combination of spin components. This combination may\nbe called \\precession moment\" because it is a canonical conjugate of the preces-\nsion rotation angle (precession phase) rather the rotation angle of genuine spin\nin the spin space. So one should discern two types of spin super\ruid transport:\ntransport of spin precession and transport of spin [35]. In the experiment [8] they\nused slightly nonuniform magnetic \felds, and precession took place only inside\nthe homogeneously precessing domain (HPD). But for discussion of super\ruid spin\ntransport it is not so important, and in the following we consider only processes\ninside the HPD ignoring gradients of the magnetic \feld.\nThe spin dynamics of super\ruid phases of3He is described by the theory of\nLeggett and Takagi [22], which is an example of the general phenomenological\ntheory of magnetically ordered systems in terms of conjugate canonical variablesMarch 3, 2010 17:4 Advances in Physics SpinRev\n22 E. B. Sonin\nFigure 8. Euler angles for spin precession in3He-B. The rigid top presents\nthe rigid spin order parameter structure controlled by large exchange energy.\nThe rotation of the coordinate frame transforming the axis zto the axis \u0018is\nperformed around the axis N(line of nodes).\n\\angle{moment\", which was shortly discussed in section 3. As well as in studies\nof rotating solid tops, sometimes it is more convenient to describe spin rotations\nvia the Euler angles \u000b,\f, and \u0000 (\fgure 8). In3He super\ruid spin dynamics these\nangles were used by Fomin [9, 12], who, however, replaced the angle \u0000 by the angle\n\b =\u000b+ \u0000. The angle \fis the precession tipping angle, and \u000bis the precession\nphase determining the direction of the line of nodes N. The angle \b characterizes\nthe resultant rotation of the order parameter in the laboratory frame, and in the\nlimit\f!0 (no precession) becomes the angle of rotation around the zaxis. The\nmagnetic moments canonically conjugate to the angles \u000b,\f, and \b are P=mz\u0000\nm\u0018,m\f, andm\u0018respectively, where mzis thezcomponent of the magnetization\nmin the laboratory coordinate frame, m\u0018is the projection of mon the\u0018axis\nof the rotating coordinate frame (see \fgure 8), and m\fis the projection of mon\nthe line of nodes N, which is perpendicular to the axes zand\u0018. The free energy\ndensity consists of three terms, F=F0+Fr+V, where\nF0=m2\n2\u001f\u0000m\u0001H (48)\nincludes the magnetization and the Zeeman energies, and the gradient energy Fr\nand the order-parameter dependent energy V(dipole energy in the case of3He)\nwill be determined later on. Here \u001fis the magnetic susceptibility.\nFor the phenomena observed experimentally only one degree of freedom is essen-\ntial, which is connected with precession, i.e., with the conjugate pair \\precession\nphase\u000b{precession moment P\". In contrast to the mode connected to the longitu-\ndinal magnetic resonance (oscillations of the longitudinal spin component), which\nwas discussed in previous sections, the precession mode is connected with the trans-\nverse magnetic resonance (nuclear magnetic resonance in the case of3He), in whichMarch 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 23\nmzdoes not oscillate essentially.\nNeglecting nutation, the directions of the axis \u0018and of the moment mcoincide.\nThen\fis constant, m\u0018=m,m\f= 0,P=m(cos\f\u00001), and the free energy\ndensityF0becomes\nF0=m2\n2\u001f\u0000(m+P)H; (49)\nwhereHis a strong constant magnetic \feld parallel to the zaxis. The Hamilton\nequations for the precession mode are:\n@\u000b\n@t=\r\u000eF\n\u000eP=\u0000!L+\r@(Fr+V)\n@P;\n1\n\r@P\n@t=\u0000\u000eF\n\u000e\u000b=\u0000@(Fr+V)\n@\u000b+ri@Fr\n@ri\u000b; (50)\nwhere!L=\rHis the Larmor frequency and F=R\nd3RFis the total free energy.\nSince this section deals with nuclear spins, here \ris the nuclear gyromagnetic ratio.\nIn the experiment the magnetization amplitude is determined by the magnetic \feld:\nm=\u001fH.\nEquation (50) describes free precession without dissipation. One of the most\nimportant mechanisms of dissipation is the Leggett{Takagi mechanism [22] related\nwith the process of equilibration of the magnetization of the normal component\nwith the precessing magnetization of the super\ruid component (for details see the\nreviews by Bunkov [13] and Fomin [12] and references therein). This mechanism\nbecomes ine\u000bective at low temperatures, and this leads to the Suhl instability of\nthe uniform precession, which is discussed below. So one cannot observe uniform\nprecession in3He-B at very low temperatures. The dissipation leads to a precession\ndecay, and in order to support the state of uniform precession in the experiment\nthe energy dissipation must be compensated by the energy pumped by the rotating\ntransverse magnetic \feld. Assuming that the balance of the pumped energy and\nthe dissipated energy eventually leads to stationary precession, we may further\nignore the both. The stationary precession state corresponds to the extremum of\nthe Gibbs thermodynamic potential, which is obtained from the free energy with\nthe Legendre transformation: G=F+!PP=\r, where the precession frequency\n!P=\u0000@\u000b=@t plays the role of the \\chemical potential\" conjugate to the precession\nmoment density P.\nUp to now the theory was rather general and valid for precession in any magnet-\nically ordered system. Referring to3He-B particularly, the dipole energy in3He-B\nis [12, 13]\nV=2\u001f\n2\n15\r2\u0014\n(1 + cos \b)u+ cos \b\u00001\n2\u00152\n; (51)\nwhere \n is the longitudinal NMR frequency and u= cos\f. At the stationary\nprecession the angle \b does not vary in time and can be found by minimization of\nthe Gibbs potential. In the state of uniform precession without spatial gradients\nonly the dipole energy depends on \b, and the equation for \b is\n@V\n@\b=4\u001f\n2\n15\r2\u0014\n(1 + cos \b)u+ cos \b\u00001\n2\u00152\n(1 +u) sin \b = 0: (52)March 3, 2010 17:4 Advances in Physics SpinRev\n24 E. B. Sonin\nSolution of this equation yields\ncos \b =1=2\u0000u\n1 +u; V(u) = 0 for\f <104\u000e(u>\u00001=4);\ncos \b = 1; V(u) =8\u001f\n2\n15\r2\u00121\n4+u\u00132\nfor\f >104\u000e(u<\u00001=4): (53)\nThus atu >\u00001=4 (\f <104\u000e) and atu <\u00001=4 (\f >104\u000e) one must choose two\ndi\u000berent branches of the solution of equation (52). The critical angle \fc= 104\u000eis\ncalled the Leggett magic angle.\nIt has already been known about 50 years from studies of nonlinear ferromagnetic\n[36] and antiferromagnetic [37] resonance that the state of uniform spin precession\nwith \fnite precession angle can be unstable with respect to excitation of spin\nwaves (Suhl parametric instability). Though Suhl instability is a phenomenon of\nthe nonlinear classical wave theory [38] it is easier to qualitatively explain it in\nterms of spin-wave quanta (magnons). The processes leading to instability are\ntransformations of nquanta of uniform precession into two spin-wave quanta with\nwave vectors\u0006k:\nn!P=!(k) +!(\u0000k): (54)\nThe precession is unstable if at least one of these processes is allowed by the laws\nof energy and momentum conservation. The three-magnon process ( n= 1) cor-\nresponds to the \frst order Suhl instability. The process is possible if a quantum\nof uniform precession of frequency !Pcan dissociate into two quanta of the lower\nspectral branch with frequency !P=2. Another possibility to destabilize uniform\nprecession is a four-magnon process of transformation of two quanta of uniform\nprecession into two quanta of the same spectral branch with \fnite wave vectors \u0006k\n(the second-order Suhl instability, n= 2). The process becomes possible if a non-\nlinear correction to the frequency of uniform precession has an opposite sign with\nrespect to the frequency dispersion d!2=dk2. In the theory of nonlinear waves the\nlatter condition is called Lighthill's condition, which is necessary for modulation\ninstability [39]. The second-order Suhl instability is an example of it. In all known\nexamples of uniform precessions in magnetically ordered systems there are condi-\ntions for at least one type of Suhl instability. In super\ruid3He Suhl instability is\npossible in the A [40, 41] and B [42, 43] phases. But as any parametric instabil-\nity, the Suhl instability can be suppressed by dissipation, which leads to a critical\nprecession angle below which the state of uniform precession remains stable. In\nthe B phase stable uniform precession is possible at temperatures T > 0:4Tc. At\nlower temperatures dissipation is weak and cannot block the Suhl instability. This\nexplains a sudden transition to the regime of \\catastrophic relaxation\" observed\nby Bunkov et al. [44].1\n7.2. Stability of spin-precession supercurrents (Landau criterion)\nLet us consider the state with uniform spin-precession current proportional to the\ngradientr\u000b. The total free energy should now include the gradient energy of3He-B\n1In contrast to Surovtsev and Fomin [42, 43], who explained catastrophic relaxation with the bulk Suhl\ninstability, Bunkov et al. [45] suggested another mechanism of Suhl instability, which exists near the surface\n(see arguments over the two mechanisms by Fomin [46] and Bunkov et al. [47])March 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 25\n[48]:\nFr=A(u)r\u000b2\n2+\u001f\n\r2\u0014\nc2\nkr\b2\n2+c2\n?ru2\n2(1\u0000u2)\u0015\n; (55)\nwhere\nA(u) =\u001f\n\r2[c2\nk(1\u0000u)2+c2\n?(1\u0000u2)]; (56)\nu= cos\f, andckandc?are longitudinal and transverse spin wave velocities. The\nexpression for Frassumes that all gradients are normal to the axis zparallel to\nthe dc magnetic \feld.\nAn important feature of the dipole energy in3He-B is its independence of the\nprecession angle \u000b. In accordance with Noether's theorem this means that the\nprecession moment is strictly conserved. Thus the equations describing stationary\nprecession with frequency !Pare\n\u0000!P=\u0000!L+\r\u000e(Fr+V)\n\u000eP; (57)\nr\u0001J= 0; (58)\nwhere\nJ=\u0000@F\n@r\u000b=\u0000A(u)r\u000b (59)\nis the spin-precession current.\nApart fromr\u000b, other gradients are absent: ru=r\b = 0. Then equations (53)\nand (55){(57) yield the following equation for u:\n(!P\u0000!L)!L\u0000[c2\nk+ (c2\n?\u0000c2\nk)u]r\u000b2= 0 at\f <104\u000e(u>\u00001=4);\n(!P\u0000!L)!L\u0000[c2\nk+ (c2\n?\u0000c2\nk)u]r\u000b2+4\n2(1 + 4u)\n15= 0 at\f >104\u000e(u<\u00001=4):\n(60)\nThe proper way to check stability of the current state given by equation (60) is to\nuse the Landau criterion [35]. Since we check stability of the relative minimum at\n\fxed averaged gradient r\u000b, we must do a new Legendre transformation choosing\na new Gibbs thermodynamic potential\n~G=G+J\u0001r\u000b=F+!PP\n\r+J\u0001r\u000b; (61)\nwhich has a minimum at the speci\fed values of the precession P=M(1\u0000u0) and\nthe gradientr\u000b0. Now we must \fnd the energy increase due to \ructuations. It is\neasy to check that \ructuations of \b always increase ~G, so it su\u000eces to retain in\nthe \ructuation energy only terms quadratic in small deviations u0=u\u0000u0and\nr\u000b0=r\u000b\u0000r\u000b0from the stationary values u0andr\u000b0(we omit hereafter theMarch 3, 2010 17:4 Advances in Physics SpinRev\n26 E. B. Sonin\nsubscript 0):\n\u000e~G=A(u)(r\u000b0)2\n2+A0(u)u0r\u000b0+1\n2\u0014\nV00(u) +A00(u)r\u000b2\n2\u0015u02\n2: (62)\nThe \ructuation energy is positive de\fnite, i.e., the current state is stable, so long\nasr\u000bdoes not exceed the critical value\nr\u000bc=\u0014A(u)V00(u)\nA02(u)\u0000A00(u)A(u)=2\u00151=2\n=4\n5p\n3(c2\nk(1\u0000u)2+c2\n?(1\u0000u2)\n[c2\nk+ (c2\n?\u0000c2\nk)u]2+c4\n?=3)1=2\n:(63)\nThis expression yields the critical gradient on the order of the inverse dipole length\n1=\u0018d= \n=c?, but it is valid only at u<\u00001=4. Atu>\u00001=4 (\f <104\u000e) the dipole\nenergy vanishes,r\u000bc, and the super\ruid precession transport is impossible.\nAnother de\fnition of the critical gradient was suggested by Fomin [49]: He be-\nlieved that the spin-current state can be stable as far as the gradient does not\nexceed the value r\u000bc=p\n(!P\u0000!L)!L=c?, which is the maximum gradient at\nwhich equation (60) for uhas a solution. Fomin's theory allows stable supercur-\nrents for\f <104\u000ewhen the dipole energy and the Landau critical gradient vanish.\nArguing in favor of his critical gradient, Fomin [50] stated that the Landau crite-\nrion is not necessary for the super\ruid spin transport since emission of spin waves,\nwhich comes into play after exceeding the Landau critical gradient, is not essen-\ntial in the experimental conditions (see also the similar conclusion after equation\n(2.39) in the review by Bunkov [13]). This argument is conceptually inconsistent.\nIf the experimentalists observed \\dissipationless\" spin transport simply because\ndissipation were weak, they would deal with ballistic rather than super\ruid trans-\nport. An example of ballistic spin transport will be considered in section 8. As was\nstressed in section 1, the essence of the phenomenon of super\ruidity is not the ab-\nsence of sources of dissipation, but ine\u000bectiveness of these sources due to energetic\nand topological reasons. The Landau criterion is an absolutely necessary condition\nfor super\ruidity. Fortunately for the super\ruidity scenario in the3He-B, Fomin's\nestimation of the role of dissipation by spin-wave emission triggered by violation\nof the Landau criterion is not conclusive. He found that this dissipation is weak\ncompared to dissipation by spin di\u000busion. But this is an argument in favor of im-\nportance rather than unimportance of the Landau criterion. Indeed, spin-di\u000busion,\nwhatever high the di\u000busion coe\u000ecient could be, is ine\u000bective in the subcritical\nregime, in which the gradient of the \\chemical potential\" is absent. On the other\nhand, in the supercritical regime the \\chemical potential\" is not constant anymore\nand this triggers the strong spin-di\u000busion mechanism of dissipation.\n7.3. Experimental evidence of the super\ruid spin-precession transport\nAs was discussed above, the appearance of spin current itself is not yet a manifes-\ntation of spin super\ruid transport. Supercurrents appear in spin waves or domain\nwalls where they transport spin on distances of the order of wavelength or width\nof domain walls, but hardly it would be reasonable to call it super\ruid transport.\nSimilarly spin currents in the domain wall separating HPD from the bulk without\nprecession cannot be a manifestation of super\ruid transport. A convincing evidence\nof spin super\ruidity would be spin transport on long distances. This evidence was\npresented by Borovik-Romanov et al. [51] studying spin current through a long\nchannel connecting two cells \flled by HPD. The schematic set up of their exper-\niment is shown in \fgure 9. There is a dc magnetic \feld parallel to the verticalMarch 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 27\nzz\nβz\nββ\nFigure 9. Spin-precession transport through a channel connecting two cells\n\flled by HPD. The horizontal arrow shows the direction of the spin-precession\ncurrent in the channel. The precession angle \fin the channel is less than in\nthe cells (the analogue of the the Bernoulli e\u000bect, see the text).\naxisz. The HPDs in the two cells were supported with independently rotating rf\nmagnetic \felds and with di\u000berent precession phases as a result of it. A small dif-\nference in the frequencies of the two rf \felds leads to a linear growth of di\u000berence\nof the precession phases \u000bin the cells. This creates a phase gradient r\u000bin the\nchannel accompanied by a spin-precession supercurrent. The rf coils can monitor\nprecession phases in di\u000berent parts of the set up. Due to a linear growth of r\u000b\nin time eventually it reaches the critical value at which a 2 \u0019phase slip occurs. It\nis possible to register this event via its e\u000bect on NMR absorption [13]. Thus the\ncritical gradient can be measured as a function of the precession frequency.\nDespite aforementioned conceptual \raw of Fomin's theory, Borovik-Romanov et\nal. [51, 52] found an agreement of this theory with the experiment. Let us compare\nnow the experiment with the theory based on the Landau criterion. Equation (63)\nfor the Landau critical argument contains the value u= cos\finside the channel,\nwhich is di\u000berent from uin the cells where there is no precession phase gradients\nr\u000b(see \fgure 9). This is an analogy with the Bernoulli law in hydrodynamics\n(liquid density is less in areas with higher currents). The value of uin the channel\ngrows withr\u000baccording to equation (60) at \fxed precession frequency !P. The\nlatter is controlled in the experiment and in stationary states does not vary in\nspace, exactly like the chemical potential in stationary states of super\ruids. If\n(!P\u0000!L)!L<A0(\u00001=4)\n2\r2\n\u001f(r\u000bc)2; (64)\nwherer\u000bcis the critical gradient from equation (63) at u=\u00001=4,ureaches the\nvalue -1/4 earlier than r\u000breaches the Landau critical gradient. Since no stable\nsupercurrent is possible at u >\u00001=4 the critical argument is determined as a\nsolution of equation (60) at u=\u00001=4:\nr\u000bc=\"\n4!L(!P\u0000!L)\n5c2\nk\u0000c2\n?#1=2\n: (65)\nThe experiment was done at small !P\u0000!L, and its results must be compared with\nequation (65). For the ratio c2\nk=c2\n?= 4=3 [50] the latter gives the value of r\u000bc\nby the numerical factorp\n12=17\u00190:84 smaller than Fomin's result, which was in\nabout 1.5 times larger than the critical gradient in the experiment. Thus the theory\nbased on the Landau criterion even better agrees with the experiment [51, 52], andMarch 3, 2010 17:4 Advances in Physics SpinRev\n28 E. B. Sonin\nan approximate agreement of Fomin's result with the experiment cannot be used\nas an argument in its favor. At larger values of !L\u0000!P, when the condition (64)\nis not satis\fed, the critical argument is determined by (63) and not proportional\ntop\n(!P\u0000!L)!L. So the di\u000berence with Fomin's theory becomes more essential\n(for more details see reference [53]).\nFurther important development in experimental studies of spin-precession super-\n\ruidity was observation of a spin-current analog of the Josephson e\u000bect [54]. The\nweak link was formed by making a constriction of the channel (ori\fce) connecting\nthe two cells.\n7.4. Spin-precession vortex and its nucleation\nAs was already discussed in section 4, at gradients less than the Landau critical\ngradient, the barrier, which impedes the current decay, is related to vortex motion\nacross the \row streamlines (phase slips). Similarly, vortices called spin-precession\nvortices appear in the spin-precession \row, and the vortex core radius was estimated\nto be on the order of the dipole length \u0018d[35]. The barrier for vortex growth in\nthe phase-slip process vanishes at phase gradients of the order of the inverse core\nradius. So the threshold for vortex instability agrees with the critical gradient from\nthe Landau criterion [equation (63)]. This is usual in the conventional super\ruidity\ntheory [11].\nLater Fomin [50] showed that the vortex core must be determined by another\nscale\u0018F=c?=p\n(!P\u0000!L)!L, where!Pand!Lare the precession and the Larmor\nfrequencies. This was supported by Misirpashaev and Volovik [55] on the basis of\nthe topological analysis. According to equation (60) in the ground state without\nspin currents ( !P\u0000!L)!L= 16\n2ju+ 1=4j=15. So ifuis not too close to -1/4\n\u0018d=c?=\n and\u0018Fare of the same order of magnitude. But if u!\u0000 1=4, i.e. the\nprecession angle \fapproaches to the critical value \fc= 1:82 rad (or 104\u000e) the core\nradius becomes rc\u0018\u0018F\u0018\u0018d=(\f\u0000\fc), i.e., by the large factor 1 =(\f\u0000\fc) di\u000bers\nfrom the earlier estimation rc\u0018\u0018d[35]. So the latter is valid only far from the\ncritical angle, where \f\u0000\fc\u00181. Since no barrier impedes vortex expansion across\na channel if the gradient is on the order of 1 =rc, the large core rc\u0018\u0018d=(\f\u0000\fc) at\n\f!\fcleads to the strange (from the point of view of the conventional super\ruidity\ntheory) conclusion: The instability with respect to vortex expansion occurs at the\nphase gradients\u00181=rcessentially less than the Landau critical gradient \u00181=\u0018d,\nobtained for any\f >\fc. Recently a resolution of this paradox was suggested [56]:\nAt precession angles close to 104\u000eat phase gradients less than the Landau critical\ngradient but larger than the inverse core radius, no barrier impedes phase slips at\nthe stage of vortex motion across streamlines, but there is a barrier, which blocks\nphase slips on the very early stage of nucleation of the vortex core. So for these\ngradients stability of current states is determined not by vortices but by vortex-core\nnuclei.\nIt should be stressed that, in contrast to the previous subsection, where the\ngrowth ofr\u000bwas accompanied by the growth of uat \fxed (!P\u0000!L)!L, the\npresent analysis is performed at \f= arccosu\fxed in the channel excepting an\narea a vortex core or its nucleus. It never exactly equal to \fcthough\f\u0000\fccould\nbe whatever small. Thus ( !P\u0000!L)!Lgrows withr\u000b. Vortex nucleation starts from\na \"protonucleus\", which is a slight localized depression of the super\ruid density\n[determined by A(u) in our case]. The nucleus, which is related with a peak of a\nbarrier, corresponds to an extremum (saddle point) of the Gibbs potential given\nby equation (61). Therefore, the nucleus structure should be found from solution\nof the Euler-Lagrange equations for this Gibbs potential. The \frst step is to varyMarch 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 29\nthe Gibbs potential with respect to \u000b. Let us restrict ourselves with a 1D problem,\nwhen the distribution in the nucleus depends only on one coordinate x. Then the\ndistribution ofr\u000bis given byr\u000b=\u0000J=A(u), whereJis equal to the spin-\nprecession current J=\u0000A(u1)r\u000b0, which is determined by the gradient r\u000b0far\nfrom the nucleus center. Expanding with respect to small deviation g=u\u0000u1\nfrom the \fxed value u1at in\fnity one obtains\n~G=\u0000J2\n2A(u)+\u001fc2\n?\n\r2\u0014(ru)2\n2(1\u0000u2)+u\n\u00182\nF\u0015\n+V(u)\n\u0019\u001fc2\n?\n\r2(rg)2\n2(1\u0000u21)+g\u001ad\ndu\u0014\n\u0000J2\n2A(u1)+V(u1)\u0015\n+\u001fc2\n?\n\r21\n\u00182\nF\u001b\n+g2\n2d2\ndu2\u0014\n\u0000J2\n2A(u1)+V(u1)\u0015\n+g3\n6J2\n2d3A(u1)\u00001\ndu3; (66)\nwhere we took into account that d3V(u)=du3= 0. The linear in gterms must\nvanish at the stationary current state. The term quadratic in gdetermines the\nstability of the current state: it vanishes at the Landau critical current\nJ2\nc= 2d2V(u1)\ndu2\u001ad2[A(u1)\u00001]\ndu2\u001b\u00001\n: (67)\nThis is exactly the Landau critical argument r\u000bc, which was determined in sec-\ntion 7.2 [equation (63)]. Considering the case of the current close to the critical\nvalue and using the Taylor expansion of A\u00001(u) aroundu=\u00001=4 one obtains\n~G=16\u001fc2\n?\n15\r2\u0014(rg)2\n2+ag2\n2\u0000bg3\n6\u0015\n; (68)\nwhere\na= 0:239\u0012\r2\n\u001fc2\n?\u00132\n(J2\nc\u0000J2); b= 0:577\u0012\r2\n\u001fc2\n?\u00132\nJ2\nc: (69)\nThe Euler-Lagrange equation for this Gibbs potential, \u0000\u0001g=ag\u0000bg2=2 = 0,\ndetermines the distribution of g:\ng=g0\u0012\n1\u0000tanh2x\nrp\u0013\n; (70)\nwhereg0= 3a=b= 1:24(J2\nc\u0000J2)=J2\ncis the value of gin the nucleus center and\nrp= 2=pa= 4:1\u001fc2\n?=\r2p\nJ2c\u0000J2is the nucleus size. The energy of the nucleus,\n\u000f=16\u001fc2\n?S\n15\r2Z3a=b\n0r\nag2\u0000bg3\n3dg=64\u001fc2\n?\n25\r2a5=2\nb2S= 0:214S(J2\nc\u0000J2)5=2\nJ4c;(71)\ndetermines the barrier for the process of the vortex core nucleation. Here Sis\nthe cross-section area of the channel. Since in the limit J!Jcthe nucleus size\nrpdiverges, our 1D description is always valid close enough to the critical point,\nwhererp\u001dp\nS. Whenrpbecomes smaller than the transverse size of the channel,\none should consider the 3D or 2D (in the case of a thin layer) nucleus. The \frstMarch 3, 2010 17:4 Advances in Physics SpinRev\n30 E. B. Sonin\nstage of this problem is to \fnd the distribution of r\u000bfrom the continuity equation\nr[gr\u000b] = 0. Its solution demonstrates that outside the nucleus the distribution\nofr\u000bis the same as around the vortex ring (3D case) or the vortex dipole (2D\ncase). In particular, in the 2D case\nr\u000b=r\u000b0\u0000Z1\n0g(r1)r2\n1dr1\u0014r\u000b0\nr2\u00002r(r\u0001r\u000b0)\nr4\u0015\n: (72)\nIn contrast to the 1D case, the relation between r\u000bandgis not local, so the\nfollowing variation of the Gibbs potential with respect to gleads to an integro-\ndi\u000berential equation. However using the scaling arguments one may conclude that\nthe nucleus energy can be roughly estimated from the expression (71) for the 1D\ncase with replacing Sbyr2\npor byrpLfor the 3D and the 2D cases respectively ( L\nis the thickness of the 2D layer).\nThis analysis demonstrates an unusual feature of the super\ruid spin-precession\ntransport at the precession angle close to the critical angle 104\u000e: A bottleneck of\nthe phase slip process is not connected with expansion of already formed (i.e., with\nsizes exceeding the core size) vortices but with the early stage of vortex nucleation.\nThe spin-precession vortex in3He-B was detected experimentally [57, 58].\n8. Ballistic spin transport by magnons\nAnother interesting case of spin transport in magnetically ordered system is con-\nnected with magnons [59, 60]. This is an analogue of the normal mass current in\nsuper\ruids, which arises due to transport of mass by quasiparticles (e.g., phonons\nat low temperatures). Let us \fnd \frst what contribution to the spin current comes\nfrom one magnon. The simplest case is a magnon in an isotropic ferromagnet.\nMetastable spin-current states are impossible in such a ferromagnet but it is not\nessential for now: We look for a \\normal\" spin current.\nWe consider an isotropic ferromagnet subject to a magnetic \feld Hwith the free\nenergy density\nF=\u0000M\u0001H+\u000b\n2riM\u0001riM: (73)\nThe free energy is invariant with respect to rotations around the magnetic \feld H\n(thezaxis), so in accordance with Noether's theorem the zcomponent of spin is\nconserved.\nIn the homogeneous ground state the spontaneous magnetization M0is par-\nallel toH. Linearizing the Landau-Lifshitz equation (14) with respect to small\nperturbation m=M\u0000M0(m?M0) one obtains\n1\n\rdm\ndt=\u0000\u000bri[M0\u0002rim] + [H\u0002m]: (74)\nThis equation yields spin waves /eik\u0001r\u0000i!twith the spectrum\n!(k;H) =\r(H+\u000bM0k2); (75)\nwhich in contrast to the spectrum (19) without magnetic \feld, has a gap \rHequal\nto the frequency of the ferromagnetic resonance.March 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 31\nIn the linear approximation the spin current Jz, which is determined by equa-\ntion (18), vanishes after averaging over the period of the wave. So we need the\nterms of the second order in m, which yield the current\nJz\nj=\u000bh^z\u0001[m\u0002rjm]i=\u000bkjhm2i: (76)\nThe energy density in a single spin-wave mode is \u000f= (!=2\rM0)hm2iV, whereV\nis the sample volume. Contributions of a magnon with the energy \u000f=~!to the\nsquared transverse magnetization and to the spin current are \u000ehm2i= 4\u0016BM0=V\nand\u000eJz=~v(k)=Vrespectively. Here \u0016B=\r~=2 is the Bohr magneton and\nv(k) =d!=dkis the magnon group velocity. If there is an ensemble of magnons\nwith the distribution function n(k) the total spin current will be\nJz=~\nVX\nkn(k)v(k) =~\n(2\u0019)3Z\ndkn(k)v(k): (77)\nFor the axisymmetric equilibrium Planck distribution the spin currents vanishes.\nBut the spin current appears if the magnon distribution is the Planck distribution\nwith a drift velocity vn:\nn0(k;H) =1\ne~[!(k;H)\u0000k\u0001vn]=T\u00001: (78)\nThe magnon drift velocity vnis an analogue of the normal velocity in a super\ruid\nliquid. Expanding the right-hand side of equation (77) in small vnone obtains the\nlinear relation between the spin current and the drift velocity. The Plank distribu-\ntion with a drift is valid only if interaction between magnons is more e\u000bective than\ninteraction of magnons with other quasiparticles (electron, phonons), or lattice de-\nfects. Otherwise the magnon distribution must be determined from Boltzmann's\nequation, and the spin current appears only if there is a gradient of the magnetic\n\feldH, which plays a role of the chemical potential for spin since H=\u0000@F=@m.\nThe spin current proportional to the gradient of His accompanied by dissipation\nand is determined by \\spin-conductivity\" Jz=rH.\nAnother way to obtain the spin current in the magnon system is to connect a\nquasi-one-dimensional ferromagnetic channel of \fnite length with two bulk ferro-\nmagnets (they act as reservoirs for spin) via ideal contacts [60]. Magnons cross\nthe channel without scattering (ballistic regime). A spin current appears if there\nis a di\u000berence \u0001 Hof the magnetic \felds in the two leads. The physical picture\nis similar to that for electron ballistic transport analyzed in the framework of the\nLandauer-B uttiker approach [61]: The \\right-movers\" (magnons moving to right)\nand \\left-movers\" (magnons moving to left) are described by the equilibrium dis-\ntribution inside the leads, which magnons come from. Bearing in mind that the\nwave vector has the only component along the channel the spin current is given by\nJz=~\n2\u0019\u0014Z1\n0dkd!\ndkn0(k;H + \u0001H) +Z0\n\u00001dkd!\ndkn0(k;H)\u0015\n=\u0016B\n\u0019\u0001H\ne\u0016BH=2T\u00001:\n(79)\nHeren0(k;H) is the Planck distribution given by equation (78) at vn= 0, taking\ninto account the dependence of !onH. This yields the spin conductance Jz=\u0001H\nobtained by Meier and Loss [60]. The origin of dissipation is similar to that for\nballistic charge transport along a 1D channel [61]: There is no dissipation in theMarch 3, 2010 17:4 Advances in Physics SpinRev\n32 E. B. Sonin\nchannel itself, but magnons arriving to the leads have the distribution di\u000berent\nfrom the equilibrium distribution inside the lead. Its relaxation to the equilibrium\nleads to dissipation.\nIt is worthwhile to stress again (see also section 7.2) that though dissipation in\nthe channel is absent in the ballistic regime, this is not super\ruidity: In the ballistic\nregime dissipation is absent because there is no sources of dissipation.\n9. Equilibrium spin currents in helimagnets\nThough the previous analysis addressed the case of ferromagnets it can be ex-\ntended on anti- and ferrimagnets. The analysis is also relevant for spin transport\nin spinor Bose condensates of cold atoms, for which the stability of spin-current\nstates (spiral structures) was also analyzed in the spirit of the Landau criterion\n[62]. The condition for super\ruid spin transport is a proper topology of the mag-\nnetic order parameter: the magnetic order parameter space can be mapped onto\na circumference with only weakly broken symmetry (or no broken symmetry at\nall) with respect to rotation around the circumference. However, spin currents in\nhelimagnets still require a special discussion.\nThe magnetic structure in helimagnets is a spatial rotation of the magnetization\n(spin) in the easy plane. This structure appears due to Dzyaloshinskii-Moria in-\nteraction linear in gradients of magnetization, which break invariance with respect\nto space inversion. Its energy is given by D\u0001[M\u0002[r\u0002M]] [17]. This energy\nis relativistically small and can a\u000bect only the direction of the magnetization M,\nbut not its absolute value M. In easy-plane magnets Mis fully determined by its\nangle in the easy-plane, and the free energy is\nF=Z\nd3RF=Z\nd3R\u001am2\nz\n2\u001f+A(r')2\n2+K[1\u0000cos(n')]\u0000DM2rz'\u001b\n;(80)\nwhere it is supposed that the magnetic spiral is oriented along the axis z. This\nenergy di\u000bers from the free energy in equation (33) with the term linear on the\nangle gradientrz'.\nThe term linear in the angle gradient does not a\u000bect the equations of motion\nbut it is important for de\fnition of the equilibrium magnetic structure: the phase\ngradient is present in the ground state. It also changes the expression for the spin\ncurrent replacing equation (26) with\nJz=\u0000@F\n@r'=\u0000A(r'\u0000k); (81)\nwherek= (DM2=A)^z. The ground state is determined by minimization of the free\nenergy with respect to the average gradient hr'i, which determines the density\nnhr'i=2\u0019of domain walls. Focusing on the limit of low density of domain wall\n[see equation (41)] the free energy density is\nF=Ahr'i\u00124\n\u0019pnl\u0000k\u0013\n: (82)\nThus atk < 4=\u0019pnl(strong anisotropy) there is a complete \\phase \fxation\",\nspin being directed along some of the in-plane easy axes, and the ground state is\nuniform (r'= 0). This means that the spin current Jz=Akis present in the\nground state. At k= 4=\u0019pnthere is the phase transition to the spiral structureMarch 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 33\nwithhr'i6= 0, which is a chain of solitons of the sine Gordon equation obtained by\nvariation of the free energy. Near the phase transition the equilibrium spin current\nis still close to Jz=Ak. However in the limit k\u001d1=lthe anisotropy energy is\nnot essential and r'\u0019k. Then the equilibrium spin current is absent despite the\npresence of the spiral structure with r'6= 0.\nThe equilibrium spin currents in helimagnets were calculated by Heurich et al.\n[63] and by Bostrem et al. [64]. In their calculations rotational invariance in the easy\nplane was broken by an inplane magnetic \feld H?. This corresponds to the free\nenergy (80) with K=M0H?andn= 1. Our analysis shows that an equilibrium\nspin current is a generic feature of any helimagnet even without an in-plane \feld\nsince one cannot imagine an easy-plane magnetic material without at least some\n\fnite in-plane anisotropy. The phase transition between the phase-\fxed state and\nthe the state with a helical structure is a typical example of the commensurate{\nincommensurate phase transition, which is common in condensed matter physics\n[65]. In particular, such a transition driven by a in-plane magnetic \feld was studied\nexperimentally [66] and theoretically [67] in the quantum Hall bilayers.\nIn addition to equilibrium spin currents there are also possible metastable spin\ncurrents. Their stability is determined like it was done in section 4, but in the\nLandau criterion, r' <p\nEA=A, and in the expression, which determines the\nbarrier for vortex expansion [equation (32)], one should replace the phase gradient\nr'withjr'\u0000kj.\nThe helical structure at the equilibrium (and possibly in the metastable state) is\nstationary and does not move. However, as discussed in section 6, spin injection to\na medium, which does not support super\ruid spin transport, is impossible without\nsome nonequilibrium magnetization. Then the helical structure must move, but this\nmotion can be blocked by pinning. The e\u000bect of pinning on moving incommensurate\nstructures is well known [65]. It is similar to the e\u000bect of coercivity on motion\nof domain walls. However, in the limit of high spin currents most e\u000bective for\ndissipationless spin transport pinning is suppressed due to strong overlapping of\ndomain walls. If they overlap so strongly that the phase (angle) gradient does not\nvary in space, the pinning force must vanish.\nEquilibrium spin currents are possible not only in helimagnets. They were re-\nvealed also in spin-Heisenberg rings subject to inhomogeneous magnetic \felds\n[68, 69]. These currents are connected with the geometric phase quantization in\nring geometry and, being inversely proportional to the length of the ring, are pos-\nsible only in mesoscopic rings. Their origin and properties are similar to mesoscopic\ncurrents arising in systems without magnetic order, which will be considered in sec-\ntion 12.2.\nEquilibrium spin currents were considered by a number of researchers as a bizarre\nand unphysical outcome. They believed that they are background currents, which\ndo not correspond to real spin transport [70{72]. This stimulated attempts to\nrede\fne the de\fnition of spin current in order to avoid spin currents in the ther-\nmodynamic equilibrium. For Heisenberg magnets it was done by Sch utz et al. [72].\nIn many aspects confusion about a proper spin-current de\fnition in magnetically\norder systems re\rects the similar problem in systems without magnetic order, and\nwe postpone its discussion for section 11.\n10. Magnon Bose-Einstein condensation vs spin super\ruidity\nConnection of Bose-Einstein condensation (BEC) with super\ruidity was debated\nfrom the very beginning of the theory of super\ruidity. This connection is not\nstraightforward. BEC not necessarily leads to super\ruidity as the simple case ofMarch 3, 2010 17:4 Advances in Physics SpinRev\n34 E. B. Sonin\nthe ideal Bose-gas demonstrates. On the other hand, one may explain super\ruidity\nwithout referring to the concept of BEC as Landau did creating his two-\ruid theory.\nThere are debates also what the term BEC really means and how widely one may\nuse it. Nowadays there is a strong tendency to see manifestation of BEC in various\ncases of formation of coherent states in magnetically ordered systems. The BEC\nscenario was suggested for the phase transition to the antiferromagnetic state from\nthe saturated ferromagnetic state in a strong magnetic \feld in magnetic insulators\nTlCuClO 3[73, 74], and Cs 2CuCl 4[75]. In yttrium-iron-garnet \flms Demokritov\net al. [76] [see also [77]] observed condensation of magnons in a single state in-\nterpreting it in terms of BEC. Finally, Bunkov and Volovik [78{80]) reinterpreted\nspin-precession super\ruidity in3He-B (see section 7) in terms of BEC. Dealing\nwith semantics it is di\u000ecult to decree strict rules and to reach a broad consensus.\nWithout trying to do it, we shall simply follow recent debates on the question when\none may and when one may not use the label \\magnon BEC\".\nI start from a general argument why the term \\magnon BEC\" is at least problem-\natic (but not forbidden!). Let us remind what did BEC mean at the time of Bose,\nEinstein, and London when the concept was coined. The quantum theory showed\nthat if identical particles are described by the Bose-statistics, at low temperatures\nthey can condense at the state of the lowest energy. Since all Bose-condensed par-\nticles are described by the same wave-function, the BEC yields a new coherent\nclassical \feld, which cannot be imagined in the classical theory of particles. So\nfrom this point of view the non-trivial essence of the BEC is appearance of a co-\nherent classical \feld, which does not exist in the classical description of particles.\nReturning now to the magnon BEC, a magnon can be a result of quantization of\na classical \feld (spin waves). If one claims magnon BEC, this in fact means that\n\\particles\" magnons can be described by a classical \felds. However, this is a logical\ncircle: First we state that we need to quantize a classic \feld and to introduce quanta\nof this \feld, then we tell that we should describe condensed \\particles\" (quanta) by\na classical \feld. So we return exactly to the same classical \feld, which we started\nfrom! In other words, in this special case quantization of spin waves was not neces-\nsary. Calling coherent spin waves by a Bose-condensate of magnons, one must allow\nalso to call laser modes \\Bose-condensates of photons\", or deformed states of solids\n\\Bose-condensates of phonons\". Analogy between Bose-condensates and coherent\nstates was well known long ago [81]. But analogy does not mean identity. Some\nreasonable restriction should be imposed on using the term BEC. It seems that\nmore careful is to use this term only referring to objects, which can be particles\nin the classical limit. This means that coherent sound (phonons), electromagnetic\n(photons) or spin (magnons) waves are not Bose-condensates, but excitons in semi-\nconductors are. However, no clever classi\fcation can avoid \\gray areas\", where it\nfails to provide de\fnite conclusions. Can one call coherent polariton states as Bose-\ncondensates keeping in mind that the polariton is a superposition of an exciton and\na photon? I leave this question without an answer.\nPutting aside this argument as too rigorist let us discuss various claims of\n\\magnon BEC\". They can be classi\fed as equilibrium phenomena, or non-\nequilibrium ones, which require energy pumping for their observation. Let us start\nfrom the equilibrium case of the phase transition to the antiferromagnetic state\nfrom the saturated ferromagnetic state in a strong magnetic \feld in magnetic insu-\nlators TlCuClO 3[73, 74], BaCuSi 2O6[82], and Cs 2CuCl 4[75]. The theoretical basis\nfor it was known from 80s [83] (see also [84]): The transition from the ferromagnet\n(all spin parallel to the magnetic \feld) to the two-sublattice antiferromagnet with\nspins of sublattices in the plane can be described in terms of magnon Bose-operators\nin the ferromagnetic state. This transition is one from numerous examples of ori-March 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 35\nentational phase transitions in magnetism. If the BEC label is used for this case\nit could be used for any other. More generally, any transition described in terms\nof soft-mode instability (e.g., structure transitions in solids) may be considered as\nBEC. Then the phenomenon of BEC loses its uniqueness. Reservations concerning\nusing the term \\magnon BEC\" for equilibrium phenomena were expressed in a\nnumber of publications [79, 85].\nAnother class of magnon BEC is non-equilibrium states, which require en-\nergy pumping for their support. There are two ways to support a coherent non-\nequilibrium state. The \frst one is to pump energy at the same frequency with\nwhich the phase of the coherent state varies in time. An example of such non-\nequilibrium coherent states is the coherent precession in3He-B discussed above in\nsection 7. In these cases there is a magnetic \feld rotating with the same frequency\nas the precession angular velocity. Snoke [86] called these examples of coherent\nstates \\driven condensates\" suggesting that in such cases the word \\condensate\"\nwas not appropriate and contradicted to the spirit of the term \\condensate\" as a\nspontaneous thermodynamic phenomenon. Putting aside this argument for a while\nand accepting the view of Bunkov and Volovik [78{80], who considered the co-\nherent spin precession as a BEC phenomenon, the question arises why they do it\nonly with respect to the coherent precession in3He-B. The coherent precession\ntakes place in the nonlinear ferromagnetic or antiferromagnetic resonances, which\nwere studied more than 50 years. Already at that time it was well realized that\nprecessing magnetization may be treated purely classically as a gigantic magnetic\ndipole, which emits coherent radiation similar to Dicke superradiance [87]. Later\nexperimental evidence of this radiation was reported [88]. Following the semantic\ncriteria of Bunkov and Volovik one may call it \\discovery of magnon BEC\".\nReally spontaneous magnon condensation was observed by Demokritov et al. [76]\nin yttrium-iron-garnet \flms: they pumped microwave energy at higher magnon\nmodes and observed accumulation of magnons in the lowest-energy magnon state,\nwhich corresponded to nonzero wave vector. It is important that pumped magnons\nwere incoherent [89], so ensuing accumulation in a single state was spontaneous in-\ndeed. Earlier the phenomenon was discussed theoretically by Kalafati and Safonov\n[90]. Condensation requires some threshold pumping level. Accepting undisputed\nsimilarity of this phenomenon with BEC, it is worthwhile to draw attention to\nanother analogy, which is not less \\intriguing\" than the BEC analogy: the analogy\nwith lasers. Keldysh [91] de\fned \\lasing\" in his review on the exciton BEC (see\np. 255 there): \\The accumulation of a macroscopic number of initially incoher-\nent excitation quanta in a single-photon mode is lasing\". Replacing \\single-photon\nmode\" by \\single-magnon mode\" one can apply this de\fnition to the experiment\nof Demokritov et al. [76]. On the basis of this analogy and since magnons are cou-\npled with electromagnetic waves, one should expect coherent microwave radiation\nsimilar to laser radiation. This radiation was really observed recently by Dzyapko\net al. [92]. Radiation results from con\ruence of two magnons with opposite wave\nvectors (one-magnon radiation is forbidden by the momentum conservation law). It\nwas demonstrated that the frequency of radiation is determined by the condensed\nmagnon frequency but not the frequency of the pumping signal. Description of\nmagnon accumulation in a single state in terms of coherent magnon states was\nsuggested many years ago [93] by analogy with coherent photon states in lasers.\nOn the basis of this idea recently Rezende [94] developed the theory of coherent\nmicrowave radiation detected by Dzyapko et al. [92]. If it were possible to check\ncoherence of this radiation, it would be a demonstration of magnon laser .\nIn order to distinguish between BEC and laser, it was suggested to use BEC\ntrademark only if there is a quasi-equilibrium distribution of non-condensedMarch 3, 2010 17:4 Advances in Physics SpinRev\n36 E. B. Sonin\nmagnons with a well de\fned chemical potential that requires an e\u000bective magnon\nthermalization [79] (in lasers non-condensed photons are not in equilibrium). As one\nof possible semantic restrictions on using the term BEC, this criterion of BEC is for-\nmally legitimate. Demokritov et al. [95] experimentally demonstrated that magnon\ncondensation in the momentum space in yttrium-iron-garnet \flms was accompanied\nwith the quasi-equilibrium Bose-Einstein distribution of magnons. On the other\nhand, since the property of coherence is related only with condensed magnons it is\ndi\u000ecult to understand why the distribution function of non-condensed magnons is\nso crucial.\nUp to now we discussed the property of coherence but not of super\ruidity. It is\nnecessary to stress that coherence does not lead automatically to super\ruidity as\nsometimes assumed [80]. An example of the coherent state without super\ruidity\nwas presented in section 4: an (anti)ferromagnet fully isotropic, or with uniaxial\neasy-axis anisotropy. There is long-range correlation of spins in these cases, but\nno metastable spin-current states. The property of super\ruidity is determined by\ntopology of the order parameter for a concrete type of a condensed state. Let us\ncheck it for various BEC cases considered in the present section. The equilibrium\nphase transition in strong magnetic \felds from ferromagnetic to antiferromagnetic\nordering certainly leads to potential possibility of super\ruid spin transport: In\nthe antiferromagnetic phase the strong magnetic \feld keeps the magnetizations\nof the sublattices in the plane normal to the \feld. So spiral structures with the\nantiferromagnetic vector rotating in the plane can be stable and provide dissipa-\ntionless spin currents. The precession states in3He-B also can be accompanied by\nstable spin-precession currents as was demonstrated theoretically and experimen-\ntally (section 7). As for magnon condensation in yttrium-iron-garnet \flms [76], one\ncannot expect super\ruid spin transport in this case since there is no easy plane\nfor order-parameter rotation in the experimental geometry (the magnetic \feld was\nparallel to the \flm). However, if the modi\fcation of the experimental geometry (the\nmagnetic \feld normal to the \flm) recently suggested by Tupitsyn et al. [96] were\nrealized, the question on possible super\ruid spin transport would become relevant.\nPart II:Spin currents without magnetic order\n11. De\fnition of spin current\nUp to now we considered spin currents, which are possible only in magnetically\nordered media. Meanwhile, a lot of attention was devoted to other types of dissi-\npationless spin currents, which can appear without magnetic ordering. They are\nanalogues of persistent charge currents in normal metals [97] and can appear even\nin equilibrium. Certainly in this case they cannot relax and are genuinely persistent,\nin contrast to super\ruid spin currents, which are only metastable.\nWe shall analyze the spin transport in normal systems using the simplest model:\nnoninteracting electrons. Then one can start from a single electron in magnetic\n(H) and electric ( E) \felds, which are not uniform in general. The electron wave\nfunction is a two-component spinor\n\t=\u0012\n \"\n #\u0013\n: (83)March 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 37\nWe need the Hamiltonian with relativistic corrections [98]:\n^H=~2\n2m\u0012\n\u0000ir\u00002\u0019A\n\b0\u0000\u0015[\u001b\u0002E]\u00132\n\u0000\u0016BH\u0001\u001b; (84)\nwhere \b 0=hc=e is the single-electron magnetic-\rux quantum and \u001bis the vector of\nPauli matrices. In vacuum the constant \u0015is equal toe=4mc2, but in crystals it can\nbe much larger [14]. The electric \feld Emay include not only an external \feld but\nalso crystal \felds. The crystal electric \feld is possible in 3D systems with broken\nspace-inversion symmetry. In 2D electron systems on semiconductor surfaces the\ne\u000bective electric \feld normal to the surface can result from asymmetry of the\nelectrostatic potential con\fning the 2D system to the surface (structure inversion\nasymmetry). This leads to Rashba spin-orbit coupling (section 13.1). Since the\nPauli matrices do not commute, the spin-orbit term proportional to Ecan be\nconsidered as a non-Abelian gauge \feld similar to the Yang-Mills \felds [99].\nWriting down the Schr odinger equation for the Hamiltonian (84),\ni~@\t\n@t=^H\t; (85)\none can derive the continuity equations for charge:\ne@(\ty\t)\n@t=\u0000r\u0001j; (86)\nand for spin:\n@S\f\n@t=~\n2@(\ty\u001b\f\t)\n@t=\u0000r\u0001j\f+G\f; (87)\nwith the following expressions for the currents:\nj=e~\nm\u001a\n\u0000i\n2(\tyr\t\u0000r\ty\t)\u00002\u0019A\n\b0(\ty\t)\u0000\u0015[(\ty\u001b\t)\u0002E]\u001b\n;(88)\nj\f=~2\n2m\u001a\n\u0000i\n2(\ty\u001b\fr\t\u0000r\ty\u001b\f\t)\u00002\u0019A\n\b0(\ty\u001b\f\t)\n\u0000\u0015\n2[(\ty(\u001b\f\u001b+\u001b\u001b\f)\t)\u0002E]\u001b\n: (89)\nSince spin is not conserved, there are source terms (torques) in the spin-balance\nequations:\nG\f=\u0000\r~\n2[H\u0002(\ty\u001b\t)]\u000b\u0000i~\u0015\n2f\ty[\u001b\u0002[E\u0002(r\u00002\u0019iA=\b0)]]\f\t\n+[[E\u0002(r+ 2\u0019iA=\b0)\ty]\u0002\u001b]\f\tg:(90)\nThe spin current, equation (89), agrees with the current de\fnition used by Rashba\n[100] and many others:\nj\f=~\n4(\tyf\u001b\fv+v\u001b\fg\t); (91)March 3, 2010 17:4 Advances in Physics SpinRev\n38 E. B. Sonin\nwhere\nv=~\nm\u0012\n\u0000ir\u00002\u0019A\n\b0\u0000\u0015[\u001b\u0002E]\u0013\n(92)\nis the operator of the electron group velocity.\nOne may write down the spinor wave function \tas\n\t=pn\u0012\ncos\r\n2ei\u001e\"\nsin\r\n2ei\u001e#\u0013\n; (93)\nwheren= (\ty\t) is the electron density, \ris the tipping angle of the average spin\nwith respect to the z-axis,\u001e\"and\u001e#are the phases of the two spinor components.\nNote that here we consider a single-electron wave function, and the electron density\nis normalized to unity:R\nn(R)dR= 1. Introducing the global phase \u001e=1\n2(\u001e\"+\u001e#)\nand the relative phase \u001ez=\u001e#\u0000\u001e\"the averaged spin depends on the latter, which\nis the angle of rotation around the zaxis in the spin space:\nh\u001bxi=(\ty\u001bx\t)\nn= sin\rcos\u001ez;h\u001byi=(\ty\u001by\t)\nn= sin\rsin\u001ez;\nh\u001bzi=(\ty\u001bz\t)\nn= cos\r: (94)\nIn the variables n,\r,\u001e, and\u001ezthe electron (free) energy is given by\nF=Z\nd3RF=Z\ndRn(\n~2\n2m\"\ncos2\r\n2\u0012\nr\u001e\"\u00002\u0019A\n\b0\u00132\n+ sin2\r\n2\u0012\nr\u001e#\u00002\u0019A\n\b0\u00132#\n\u0000\u0016BBcos\r+~2\u0015\n2m[^z\u0002E]r\u001ez\n\u0000~2\u0015\n2m[h\u001bi\u0002E]\u0001(r\u001e\"+r\u001e#) +~2\u00152\n2mE2+\u000fr(rn;r\r)\u001b\n; (95)\nwhere the two last terms are not essential for the further analysis. The charge and\nthezspin component currents are:\nj=e\n~@F\n@r\u001e=ne~\nm\u001a\nr\u001e\u00002\u0019A\n\b0\u0000cos\r\n2r\u001ez\u0000\u0015[h\u001bi\u0002E]\u001b\n; (96)\njz=\u0000@F\n@r\u001ez=n~2\n2m\u001a\ncos\r\u0012\nr\u001e\u00002\u0019A\n\b0\u0013\n\u0000r\u001ez\n2\u0000\u0015[^z\u0002E]\u001b\n=~h\u001bzi\n2ej+Jz:\n(97)\nHere we divided the spin current on two parts. The \frst one is a convection: the\ncharge current is transferring also the average spin. The second one may be called\napure spin current, which is determined in the coordinate frame moving together\nwith electrons, i.e., with the velocity v=j=en:\nJz=~2\n2mn\u001a\n\u0000sin2\r\n2r\u001ez\u0000\u0015[(^z\u0000h\u001bih\u001bzi)\u0002E]\u001b\n: (98)March 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 39\nNeglecting the term proportional to the electric \feld this spin current is propor-\ntional to the gradient of the angle \u001ezof the spin rotation around the axis zin\naccordance with Noether's theorem. The coe\u000ecient before the gradient (sti\u000bness)\nis proportional to the squared inplane component of the spin ( /sin2\r) similarly\nto the spin current in magnetically ordered media (see the \frst paragraph of sec-\ntion 4).\nLet us discuss various terms in the expressions for charge and spin currents. The\nterms related to the global-phase gradient r\u001eand the electromagnetic vector Aare\ncommon and do not require any special comment. The terms proportional to the\nspin-angle gradient r\u001ezare related with the geometrical (Berry) phase [101{103],\nwhich results from the transport of spin over a closed path in a parametric space,\nwhich is the con\fgurational space in our case. For spin1\n2the Berry phase is \u00001\n2\n,\nwhere \n is the solid angle subtended by the varying spin during the cyclic process.\nSo the Berry-phase \\gradient\" is r\u0000 =\u00001\n2r\n =\u00001\n2(1\u0000cos\r)r\u001ez. The gradient\nis in quotation marks because it is a well de\fned gradient of a scalar function only\nat constant tipping angle \r. In general the Berry phase depends on the path used\nfor its de\fnition. In order to separate the Berry-phase current from that connected\nwith the global phase the latter must be rede\fned. Introducing ~\u001e=\u001e\"instead of\n\u001e=1\n2(\u001e\"+\u001e#) the charge current is\nj=e~\nmn\u001a\nr~\u001e\u0000r\u0000\u0000A\n2\u0019\b0\u0000\u0015[h\u001bi\u0002E]\u001b\n: (99)\nIt is worthwhile to stress that the Berry phase not only path-dependent but is not\nsingle-valued even for a given path. Indeed, one may rede\fne the solid angle \n in the\nBerry phase \u0000 =\u00001\n2\n replacing d\n = 2\u0019(1\u0000cos\r)d\u001ezbyd\n =\u00002\u0019(1+cos\r)d\u001ez.\nSo the Berry phase depends on what pole of the sphere was included into the solid\nangle. The di\u000berence between two values can be accounted for in the phase ~\u001e. Thus\nthe Berry phase is a part of the global geometric phase, which the electron obtains\nafter moving around the closed trajectory, and separation of the Berry phase from\nthe rest part of the global phase is not unique.\nThe charge and the spin currents also contain a spin-orbit term proportional to\nan electric \feld. These terms are a manifestation of the Aharonov{Casher e\u000bect\n[104] related with the non-relativistic interaction of the magnetic moment (spin in\nour case) with the electric \feld. The Rashba spin-orbit interaction (section 13) and\nthe spin Hall e\u000bect (section 14) originate from this term. It can also be attributed\nto the Berry phase for the parallel transport in the momentum space [103].\nWe have discussed spin currents in the free-electron model, but the presence\nof various components of the current is determined by symmetry and topology\nwithout being restricted with a particular microscopic model. For example, the\nAharonov-Casher e\u000bect was considered in the tight-binding model [105] and in the\npresence of random spin-orbit interaction [106]. As already stressed in the end of\nsection 3, presumption that spin transport requires mobile carriers of spin [13, 24]\nis not justi\fed: For phenomenology it does not matter whether spin is transported\nby itinerant carriers or via exchange interaction between localized electrons.\nThere were worries in the literature about ambiguity of the spin-current de\fni-\ntion [70]. Indeed, the de\fnition of the spin current given above [equation (91)] is\nnot the only possible choice. One can rede\fne the spin current by adding to it any\ncurrent\u000ej\f\ni(j\f\ni!j\f\ni+\u000ej\f\ni), if it is accompanied by rede\fnition of the spin torque:\nG\f!G\f+ri\u000ej\f\ni. This is a purely formal ambiguity of current de\fnition (like\nfreedom to choose various de\fnitions of potentials in electrodynamics), which must\nnot lead to any ambiguity in physical predictions. This only means that any de\f-March 3, 2010 17:4 Advances in Physics SpinRev\n40 E. B. Sonin\nnition of current is not complete without accompanying de\fnition of spin torque.\nIn principle, it is possible even to avoid the spin-current term at all, treating the\nwhole current divergence rij\f\nias a part of the spin torque. This would exclude\nthe word spin current from the scienti\fc lexicon. As discussed in section 6, one can\navoid using such terms as \roworcurrent , and describe the same phenomena using\nonly concepts of deformation, spin sti\u000bness, or torque [11, 63, 107]. However, it is\nnot necessary: In many cases the \\current language\" provides a very transparent\nphysical picture of processes in various spin systems.\nThere were also attempts to rede\fne the spin-current so that to eliminate the\ntorque term G\ffrom the spin continuity equation making it to look as a continuity\nequation for a conserved quantity. For example, Shi et al. [24] (see also the discus-\nsion of their work by Bray-Ali and Nussinov [108]) used the de\fnition of a torque\ndipole term by Culcer et al. [107] rewriting the torque density as a divergence of a\ntorque dipole density P\f:\nG\f=\u0000r\u0001P\f: (100)\nThe torque dipole density was included into the rede\fned spin current T\f=j\f+\nP\f. Then the spin continuity equation (87) becomes\n@S\f\n@t+r\u0001T\f= 0: (101)\nHowever, there is a payo\u000b for this formally legitimate operation: The spin current\nbecomes non-local since it is determined by an integral over the torque distribution\nin the whole bulk. Eventually any choice of the spin-current de\fnition must yield\nthe same predictions for observations if the whole procedure is correct. But using\nnonlocally de\fned spin currents it is more di\u000ecult to study e\u000bects related to local\nprocesses, which do not conserve the total spin. We shall return back to this is-\nsue discussing the observation of spin currents in the equilibrium Rashba medium\n(section 13.4).\n12. Equilibrium spin currents in one-dimensional rings\n12.1. A 1D ring in a homogeneous magnetic \feld\nThe simplest example of the equilibrium spin and charge currents is persistent\ncurrents in a one-dimensional ring [101, 109]. Let us consider \frst a ring of radius\nRin a constant magnetic \feld H=H^znormal to the xyplane of the ring,\nwhich provides the magnetic \rux \b = \u0019R2Hthrough the ring. We neglect the\nZeeman energy since its e\u000bect (splitting of the Fermi energies for spins up and\ndown) becomes important for the magnetic \felds \u0018\b0n=R much higher than the\n\felds\u0018\b0=R2when the Aharonov-Bohm phase shift /\b becomes essential. Here\nn=N=2\u0019Ris the electron density for Nelectrons in the ring. We consider zero\ntemperature. Introducing the azimuthal angle \u001efor the position of the electron in\nthe ring, the Hamiltonian is\n^H=~2\n2mR2\u0012\n\u0000i@\n@\u001e\u0000\u001a\u00132\n; (102)\nwhere\u001a= \b=\b0. Taking into account the periodic boundary conditions, the elec-\ntron spinor components are eigenfunctions of the angular momentum:  \";#(\u001e)/March 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 41\neis\u001e, wheresis an integer. Let us consider electrons with the same spin (up or\ndown). The total energy of the Fermi sea and the ground-state persistent currents\ndepend on whether the number of electrons s even or odd. If the number 2 p+ 1 of\nelectrons is odd their energy is:\nE=~2\n2mR2pX\ns=\u0000p(s\u0000\u001a)2: (103)\nThis energy corresponds to the ground state only if \u00001=2<\u001a< 1=2. In other cases\nthe electron Fermi see should be shifted. For example if \u001abecomes larger than 1/2,\none must sum sfrom\u0000p+ 1 top+ 1. The particle current at \u00001=2<\u001a< 1=2 is\ngiven by\nj=\u0000e\n2\u0019~@E\n@s=\u0000(2p+ 1)\u001ae~\n2\u0019mR2\u0019\u0000\u001aen~\n2mR: (104)\nThis function must be periodically extended on any \u001abeyond the interval\n(\u00001=2;1=2). This yields the periodic sawtooth dependence of the current on \u001awith\nthe period 1. If the number 2 pof electrons is even one should shift this dependence\nwith the half-period 1/2.\nTaking into account the both directions of spin, the result depends on whether\nthe electron number Nis even or odd. If Nis even the numbers of electrons with\nspins up and down are even or odd together. Then the charge current obtained for\none value of spin is doubled, whereas the spin current, which is proportional to the\ndi\u000berence of charge currents for two values of spin, vanishes. On the other hand,\nifNis odd the charge-current dependence for one spin direction is shifted with\nrespect to another. This lead to the persistent spin current jz=~(j+\u0000j\u0000)=2e\nwith the period 1, where j\u0006are charge currents for two values of spin. The spin\ncurrent jumps between values \u0006~2n=2mRevery half-period.\n12.2. A 1D ring in an inhomogeneous magnetic \feld: Berry-phase currents\nLet us consider now a ring in an inhomogeneous crown-shaped magnetic \feld dis-\ntribution. This problem was considered in a number of papers starting from Loss\net al. [101]. There is a constant z-component Hcos\u001fand a rotating transverse\ncomponent: Hx=Hsin\u001fcos\u001eandHy=Hsin\u001fsin\u001e(crown-shaped \feld distri-\nbution). Here \u001fis the tipping angle of the magnetic \feld on the ring with respect\nto thezaxis (\fgure 10). The Hamiltonian for a single electron with a position\ncharacterized by the azimuthal angle \u001eis\n^H=~2\n2mR2\u0012\n\u0000i@\n@\u001e\u0000\u001a\u00132\n\u0000\u0016BHfsin\u001f[cos\u001e^\u001bx+ sin\u001e^\u001by] + cos\u001f^\u001bzg:(105)\nThis Hamiltonian can be exactly diagonalized [102, 110]. The eigenstates are\nspinors\nj\t+(l)i=\u0012cos\r\n2eil\u001e\nsin\r\n2ei(l+1)\u001e\u0013\n;\nj\t\u0000(l)i=\u0012cos\r+\u0019\n2eil\u001e\nsin\r+\u0019\n2ei(l+1)\u001e\u0013\n=\u0012\u0000sin\r\n2eil\u001e\ncos\r\n2ei(l+1)\u001e\u0013\n(106)March 3, 2010 17:4 Advances in Physics SpinRev\n42 E. B. Sonin\n/vectorH\nφ\nχ\nΩz\nFigure 10. A 1D ring in a crown-shape magnetic \feld. The solid angle \nsubtended by the spin parallel to Hdetermines the Berry phase \u0000 = \u00001\n2\n.\nwith the eigenvalues of the energy\n\u000f\u0006=E\u0006\u0007r\n\u0001E2\n4+ (\u0016BHsin\u001f)2(1\u0000cos\r): (107)\nHerelis the quantum number for the orbital moment, which should include also\nthe Aharonov-Bohm phase shift, and \ris the tipping angles for the average spin\nwith respect to the z-axis determined by\ncos\r=\u0001Ep\n\u0001E2+ 4(\u0016BBsin\u001f)2: (108)March 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 43\nTwo states correspond to two opposite directions of spin with angles \rand\r+\u0019.\nThe energies\nE+=~2l2\n2mR2\u0000\u0016BHcos\u001f;\nE\u0000=~2(l+ 1)2\n2mR2+\u0016BHcos\u001f; (109)\nare obtained taking into account only the Zeeman energy in the longitudinal mag-\nnetic \feldHz=Hcos\u001f, and\n\u0001E=E\u0000\u0000E+=~2(2l+ 1)\n2mR2+ 2\u0016BHcos\u001f: (110)\nThe signs + and - label the states with spin up and down in the limit of the\nlongitudinal magnetic \feld ( \u001f!0). In the eigenstates the average spin rotates\nin space around the zaxis with the same speed as the magnetic \feld. Because of\ninhomogeneity of the magnetic \feld the states are not eigenstates of the operator\nof the orbital moment but are mixtures of the states with two di\u000berent moments.\nHowever, they are eigenstates of the total moment (orbital moment + spin) with\nthe quantum number l+1\n2since cylindrical symmetry is not broken.\nThe charge and spin currents are obtained by summation over all land two\npossible spin states:\nj=e~\n2\u0019mR2(X\nl\u001a\nl+1\n2[1\u0000cos\r(l)]\u001b\n+X\nl\u001a\nl+ 1\u00001\n2[1\u0000cos\r(l)]\u001b)\n;(111)\njz=~2\n4\u0019mR2(X\nl\u001a\nlcos\r(l)\u00001\n2[1\u0000cos\r(l)]\u001b\n\u0000X\nl\u001a\n(l+ 1) cos\r(l) +1\n2[1\u0000cos\r(l)]\u001b)\n: (112)\nThe second terms /(1\u0000cos\r) in the sums originate from the geometrical (Berry)\nphase [101]. The simplest case is the limit of strong magnetic \feld, when the spins\nin two states are parallel or antiparallel to the magnetic \feld, and \r=\u001fdoes not\ndepend onl. Then the Berry contribution to the spin current jzis proportional to\nthe total electron density, while the Berry contribution to the charge current jis\nproportional to the di\u000berences of the densities in the two spin states, i.e., to the\nspin polarization.\nSimilar e\u000bects exist in the presence of a spatially rotating electric \feld (external\nor crystal), which generates an e\u000bective magnetic \feld via spin-orbit interaction\n[103, 110]. The equilibrium spin currents can appear without electron mobility\nwhen no charge current is possible. An example of it is the persistent spin current\nin a spin-1/2 Heisenberg ring [68, 69]: Similarly to the case considered above, the\ncrown-shape e\u000bective magnetic \feld makes the spin to subtend a solid angle, which\nleads to the Berry phase contribution to the spin current.\nAll e\u000bects discussed in the present section are purely mesoscopic: The derived\ncharge and spin currents decrease with increasing size of the ring. The case of\nmacroscopic equilibrium spin currents, which are \fnite in the thermodynamic limit,March 3, 2010 17:4 Advances in Physics SpinRev\n44 E. B. Sonin\nis discussed in the next section.\n13. Equilibrium spin currents in the 2D electron gas with spin-orbit\ninteraction\n13.1. Currents in a uniform 2D gas with spin-orbit interaction\nSpin-orbit interaction allows to govern the spin transport by electric \feld, which is\nexpected to be useful for applications [10]. This explains a great interest to systems\nwith spin-orbit interaction. A classical example of such a system is a 2D electron\ngas with the Rashba spin-orbit term (let us call it Rashba medium ). Rashba [100]\nrevealed that in the Rashba medium spin currents appear even at equilibrium.\nHowever, he quali\fed them as \\not real\", which cannot lead to transport and\naccumulation of spin. Their presence in the ground state was considered as an\ninherent problem in the spin current concept [70, 71]. In the present section I shall\naddress equilibrium spin currents in the 2D electron gas with spin-orbit coupling\nand analyze whether these currents have something to do with spin transport.\nThe Hamiltonian for a 2D electron gas with spin-orbit interaction is\nH=~2\n2mn\nr\tyr\t+i\u000b(r)(\ty[\u001b\u0002^z]iri\t\u0000ri\ty[\u001b\u0002^z]i\t)\n+i\f(r)[\ty(\u001bxrx\t\u0000\u001byry\t)\u0000(\u001bxrx\ty\u0000\u001byry\ty)\t)o\n; (113)\nwhere\u000b(r) and\f(r) are parameters of the Rashba and Dresselhaus spin-orbit\ninteraction respectively. In general they may depend on the 2D position vector r.\nFor simplicity, we shall concentrate the further analysis on the Rashba interaction\n(\f= 0) postponing its extension on Dresselhaus spin-orbit interaction for the end\nof this subsection. The Rashba term /\u000bis a particular case of the spin-orbit-\ninteraction term in the Hamiltonian equation (84), assuming that \u0015E=\u000b^z. The\nSchr odinger equations for components of the spinor \tare:\ni~_ \"=~2\nm\u0012\n\u00001\n2r2 \"+\u000b@ #\n@x\u0000i\u000b@ #\n@y+1\n2@\u000b\n@x #\u0000i\n2@\u000b\n@y #\u0013\n;\ni~_ #=~2\nm\u0012\n\u00001\n2r2 #\u0000\u000b@ \"\n@x\u0000i\u000b@ \"\n@y\u00001\n2@\u000b\n@x \"\u0000i\n2@\u000b\n@y \"\u0013\n: (114)\nThe spin current and the torque in the spin-continuity equation (87) are\nj\f\ni=\u0000i~2\n4m(\ty\u001b\fri\t\u0000ri\ty\u001b\f\t)\u0000\u000b~2\n4m(\tyf\u001b\f[\u001b\u0002^z]i+ [\u001b\u0002^z]i\u001b\fg\t);(115)\nG\f=\u0000i\u000b~2\n2mn\u0010\n\tyf[\u001b\u0002[^z\u0002r]]\f\tg\u0011\n\u0000\u0010\nf[[r\u0002^z]\u0002\u001b]\f\tyg\t\u0011o\n:(116)\nThe Greek super(sub)script \frefers to three components x;y;z in the 3D spin\nspace, whereas the Latin super(sub)script iis related with the two coordinates x;y\nin the 2D electron layer.March 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 45\na) b)\nΕ Ε\nk k+-+-\nFigure 11. The ground state of the Rashba medium. a) The case km> \u000b,\nthe Fermi sea (shaded blue) \flls the upper (+) and the lower (-) band. b)\nThe casekm<\u000b, the Fermi sea \flls only the lower band.\nIn the uniform Rashba medium eigenstates are plane waves given by spinors\n1p\n2\u0012\n1\n\u0014\u0013\neikr; (117)\nwhere\u0014=\u0007iei','is the angle between the wave vector kand the axis x(kx=\nkcos',ky=ksin'), and the upper (lower) sign corresponds to the upper (lower)\nbranch of the spectrum (band) with the energies (see \fgure 11)\n\u000f=~2\nm\u0012k2\n2\u0006\u000bk\u0013\n=~2(k2\n0\u0000\u000b2)\n2m: (118)\nThe energy is parametrized by the wave number k0, which is connected with ab-\nsolute values of wave vectors in two bands as k=jk0\u0007\u000bj. The eigenstates are\nspin-polarized in the plane with spins\ns=\u0006~\n2[k\u0002^z]\nk(119)\nparallel or antiparallel to the e\u000bective spin-orbit magnetic \feld. There is no spin\ncomponent normal to the plane ( zaxis). The group velocities in two bands are\ngiven by\nv(k) =~k\nm+2\u000b\nm[^z\u0002s] =~k0\nmk\nk: (120)\nSpin torque in the eigenstates is absent, but there are inplane spin currents:\nji\nj\u0006(k) =~2\n2m\u0012\n\u0006\"isks\nkkj+\u000b\"ij\u0013\n; (121)\nwhere\"ijis a 2D antisymmetric tensor with components \"xy= 1 and\"yx=\u00001.\nThe spin current does not vary in space since there is no precession of spin oriented\nalong the e\u000bective spin-orbit magnetic \feld. The latter is constant for a plane wave,\nand there is no torque on the spin violating its conservation.March 3, 2010 17:4 Advances in Physics SpinRev\n46 E. B. Sonin\nThough any eigenstate is spin-polarized, after averaging over the equilibrium\nFermi sea (we consider the T= 0 case) the total spin vanishes. But the total spin\ncurrents do remain. The Fermi energy is \u000fF=~2(k2\nm\u0000\u000b2)=2, wherekmis the\nmaximum value of k0. In the case km>\u000b, when the both electron bands are \flled\n(\fgure 11a), the inplane spin currents are sums of contributions from two bands\n[112]:\nji\nj=~2\n4\u0019m\"ij\u0014Zkm\u0000\u000b\n0\u0012k\n2+\u000b\u0013\nkdk+Zkm+\u000b\n0\u0012\n\u0000k\n2+\u000b\u0013\nkdk\u0015\n=\u000b3~2\n6\u0019m\"ij:(122)\nAtkm<\u000bonly the lower band is \flled (\fgure 11b), and\nji\nj=~2\n4\u0019m\"ijZkm+\u000b\n\u0000km+\u000b\u0012\n\u0000k\n2+\u000b\u0013\nkdk=~2\n4\u0019m\"ij\u0012\n\u0000k3\nm\n3+km\u000b2\u0013\n:(123)\nHere we presented the calculation of equilibrium spin currents in the 2D gas\nwith Rashba spin-orbit interaction at zero temperature. Recently Bencheikh and\nVignale [113] performed a more general calculation taking into account temperature\ne\u000bects and Dresselhaus spin-orbit coupling. In the presence of Dresselhaus spin-\norbit coupling equation (122) is generalized to\nji\nj=~2\n6\u0019m[\u000b(\u000b2\u0000\f2)\"ij+\f(\u000b2\u0000\f2)(\u001bz)ij]: (124)\nThe existence of equilibrium spin currents is related with broken space inversion\nsymmetry and is a generic phenomenon in systems with spin-orbit interaction re-\nlated to the non-Abelian gauge invariance [114]. In order to demonstrate that an\nequilibrium spin current is able to transport spin, we should consider nonuniform\nmedia.\n13.2. Spin currents in a nonuniform Rashba medium\nLet us consider a slightly modulated Rashba medium with the Rashba parameter\nvarying in space as [112]:\n\u000b(r) =\u000b0+\u000b1cos(p\u0001r): (125)\nThe eigenstates found above must be corrected using the perturbation theory with\nrespect to\u000b1:\t=\t0+\t0. Here \t0is the spinor for a uniform medium with\n\u000b=\u000b0given by equation (117). The equations for the \frst order correction \t0are\n(in components):\nm\n~2\u0001\u000f 0\n\"\u0000\u000b0[k\u0014\u0003(k) +p\u0014\u0003(p)] 0\n#=\u000b1\u0014(k)p\n2\u0014\nk\u0014\u0003(k) +p\u0014\u0003(p)\n2\u0015\neik\u0001rcos(p\u0001r);\n\u0000\u000b0[k\u0014(k) +p\u0014(p)] 0\n\"+m\n~2\u0001\u000f 0\n#=\u000b1p\n2\u0014\nk\u0014\u0003(k) +p\u0014\u0003(p)\n2\u0015\neik\u0001rcos(p\u0001r);\n(126)\nwhere\n\u0001\u000f=\u0000~2\nm\u0012p2\n2+p\u0001k\u0007\u000b0k\u0013\n: (127)March 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 47\nThe solution of linear equations for \t0should be used for derivation of all relevant\nphysical quantities (densities, torques, and currents). The general expressions are\nrather cumbersome. Moreover, the perturbation theory fails in the limit p!0.\nTherefore, we restrict ourselves with the limit p\u001dk;\u000b 0. The torques and currents\nfor inplane spin components are (only linear in \u000b1terms are kept)\nGi\u0006(k) =\u00062\u000b1\u000b0~2\nm\"ijpjk\np2\u0014\n1\u0000(p\u0001k)2\np2k2\u0015\nsin(p\u0001r); (128)\nji\nj\u0006(k) =\u000b1~2\n2m\u001a\n\u0000pj\"isps\np2+\"ij\u00074\"ij\u000b0k\np2\u0014\n1\u0000(p\u0001k)2\np2k2\u0015\u001b\ncos(p\u0001r):(129)\nThe torque and the current for the z-component of spin are given by terms of higher\norder in 1=pand vanish after integration over the Fermi sea. The integration of\nthe torque and the current for inplane spin over the Fermi sea yields for the case\nkm>\u000b 0:\nGi=1\n4\u00192Z\nGi+(k)dk+1\n4\u00192Z\nGi\u0000(k)dk\n=\u0000\u000b1\u000b0~2\n\u0019m\"ijpj\np2\u0012\nk2\nm\u000b0+\u000b3\n0\n3\u0013\nsin(p\u0001r); (130)\nji\nj=1\n4\u00192Z\nji\nj+(k)dk+1\n4\u00192Z\nji\nj\u0000(k)dk\n=\u000b1~2\n8\u00192m\u0014\u0012\n\"ij\u0000pj\"isps\np2\u0013\nn+8\u0019\"ij\u000b0\np2\u0012\nk2\nm\u000b0+\u000b3\n0\n3\u0013\u0015\ncos(p\u0001r):(131)\nIfkm<\u000b 0:\nGi=1\n4\u00192Z\nGi\u0000(k)dk=\u0000\u000b1\u000b0~2\n\u0019m\"ijpj\np2\u0012\nkm\u000b2\n0+k3\nm\n3\u0013\nsin(p\u0001r);(132)\nji\nj=1\n4\u00192Z\nji\nj\u0000(k)dk\n=\u000b1~2\n8\u0019m\u0014\u0012\n\"ij\u0000pj\"isps\np2\u0013\nn+8\u0019\"ij\u000b0\np2\u0012\nkm\u000b2\n0+k3\nm\n3\u0013\u0015\ncos(p\u0001r):(133)\nThe \frst term in the spin current, which is proportional to electron density n, is\ndivergence-free, whereas the divergence of the second term does not vanish and\ncompensates the spin torque in the spin balance. Thus the second term is respon-\nsible for spin transport from areas, where spin is produced ( Gi>0) to areas where\nspin is absorbed ( Gi<0). One may consider this as a manifestation of spin trans-\nport even though it does not result in spin accumulation. Thus equilibrium spin\ncurrents can transport spin, and an attempt to distinguish equilibrium (persistent)\ncurrents from transport spin currents [115] hardly would be reasonable.\nIt is worthwhile to note that the spin current in a modulated Rashba medium is\nlinear in the spin-orbit coupling constant \u000b0, whereas the dependence of the current\non\u000bin the uniform Rashba medium is cubic. Apparently the linear dependence isMarch 3, 2010 17:4 Advances in Physics SpinRev\n48 E. B. Sonin\nelectric current \nFigure 12. Spin-dependent re\rection of electrons from an ideal impenetrable wall. The electron from the\nupper band ( ki\n+) is re\rected either as an electron from the same band ( kr\n+), or as an electron from the\nlower band ( kr\n\u0000).\na general property of nonuniform media. In particular, the same dependence was\npredicted by Sablikov et al. [116], who considered equilibrium spin currents along\nthe interface between the media with and without spin-orbit coupling.\n13.3. Interference and torque at edges of the Rashba medium\nOne cannot understand the physical meaning of spin currents without a clear pic-\nture of what is going on at borders of a system with a bulk spin current. Suppose\nthat the Rashba medium occupies the semispace x <0 while atx >0 spin-orbit\ninteraction is absent. The two semispaces have also di\u000berent potentials, so the\nHamiltonian is\nH=~2\n2mn\nr\tyr\t+i\u000b(r)(\ty[\u001b\u0002^z]iri\t\u0000ri\ty[\u001b\u0002^z]i\t)o\n+V(r)\ty\t;(134)\nwhere\nV(r) =\u001a\n0 atx<0\nUatx>0; \u000b(r) =\u001a\n\u000batx<0\n0 atx>0: (135)\nThe electron states near the interface between two regions with di\u000berent spin-orbit\nconstants have already been analyzed earlier [117, 118]. The spin currents in a\nhybrid ring consisting of parts with and without Rashba spin-orbit coupling were\nalso analyzed by Sun et al. [119].\nAtx<0 one should look for a superposition of plane waves: one incident wave,\nwhich is coming from x=\u00001, and two re\rected waves (\fgure 12). For high-\nenergy electrons with k0> \u000b and the incident electron in the upper band, the\nsuperposition is\n\t=eikyy\np\n2\u0014\u0012\n1\n\u0014+\u0013\neik+xx+r1\u0012\n1\n\u0014\u0003\n+\u0013\ne\u0000ik+xx+r2\u0012\n1\n\u0014\u0003\n\u0000\u0013\ne\u0000ik\u0000xx\u0015\n;(136)March 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 49\nwhere\u0014\u0006=\u0007iei'\u0006,'\u0006= arctan(ky=k\u0006x) are the azimuthal angles of the wave\nvectors in the plane xy, andk\u0006x=q\n(k0\u0007\u000b)2\u0000k2yare thexcomponents of the\nwave vectors corresponding to states of the same energy in the upper (+) and the\nlower (-) band. At x > 0 the wave function is evanescent: \t=\u0012\nt\"\nt#\u0013\neikyye\u0000kbx,\nwherekb=p\n2mU= ~2\u0000k2\n0+\u000b2.\nThe wave superposition should satisfy the boundary conditions, which include\ncontinuity of the both components of the spinor and jumps of \frst derivatives of\nthese components [117, 118] related with derivatives of \u000bin equation (114):\n@\t\n@x\f\f\f\f\n+0\u0000@\t\n@x\f\f\f\f\n\u00000=\u0000i\u000b\u001by\t: (137)\nThe expressions for the re\rection coe\u000ecients are rather cumbersome in general,\nand we restrict ourselves with the case of an in\fnite potential step at the x= 0\n(U;kb!1 ). Then the spinor wave function at x= 0 vanishes, and expressions for\nthe re\rection coe\u000ecients become simple:\nr1=\u0014+\u0000\u0014\u0003\n\u0000\n\u0014\u0003\n\u0000\u0000\u0016\u0014+=ei('++'\u0000)\u00001\nei('\u0000\u0000'+)+ 1; r2=\u0014\u0003\n+\u0000\u0014+\n\u0014\u0003\n\u0000\u0000\u0014\u0003\n+=\u00002ei'\u0000cos'+\nei('\u0000\u0000'+)+ 1:(138)\nThe relation between the angles '+and'\u0000is determined from the condition that\nscattering does not change the component ky= (k0\u0000\u000b) sin'+= (k0+\u000b) sin'\u0000.\nFor low-energy electrons k0< \u000b all three waves in the superposition belong to\nthe two parts of the lower band, either to the right ( k >\u000b ) or to the left ( k <\u000b )\nfrom the energy minimum (\fgure 11). If the incident wave corresponds to the state\nwithk>\u000b , the superposition is\n\t=eikyy\u0014\u0012\n1\n\u0014+\u0013\neik+x+r1\u0012\n1\n\u0014\u0003\n+\u0013\ne\u0000ik+x+r2\u0012\n1\n\u0014\u0000\u0013\neik\u0000x\u0015\n; (139)\nwhere\u0014\u0006=iei'\u0006,'\u0006= arctan(ky=k\u0006x), andk\u0006x=q\n(\u000b\u0006k0)2\u0000k2y. The positive\nsign before k\u0000in the exponent of the second re\rected wave was chosen because the\nnegative group velocity of this wave. Since the electron transport is determined by\nthe group velocity, the latter should be directed from the boundary into the bulk\neven though the wave vector is directed to the boundary. The re\rection coe\u000ecient\nare\nr1=\u0014+\u0000\u0014\u0000\n\u0014\u0000\u0000\u0014\u0003\n+=ei('+\u0000'\u0000)\u00001\ne\u0000i('\u0000+'+)+ 1; r 2=\u0014\u0003\n+\u0000\u0014+\n\u0014\u0000\u0000\u0014\u0003\n+=\u00002e\u0000i'\u0000cos'+\ne\u0000i('\u0000+'+)+ 1:(140)\nWhereas in plane-wave eigenstates of the Rashba Hamiltonian the spin has no\nzcomponent, the interference between the waves in the superposition leads to\npartial spin polarization along the zaxis. Fork0> \u000b the oscillating spin densityMarch 3, 2010 17:4 Advances in Physics SpinRev\n50 E. B. Sonin\nsz= (~=2)\ty^\u001bz\t(Friedel-like oscillation) is given by\ns+z(k) =~\n4n\nr1(e\u00002i'++ 1)e\u00002ik1x+r2[e\u0000i('++'\u0000)\n+1]e\u0000i(k+x+k\u0000x)x+r\u0003\n1r2[ei('+\u0000'\u0000)+ 1]ei(k+x\u0000k\u0000x)xo\n+ c.c.\n=~(sin'++ sin'\u0000) cos'+\n1 + cos('+\u0000'\u0000)[sin(2k+xx)\n\u0000sin(k+xx+k\u0000xx)\u0000sin(k+xx\u0000k\u0000xx)]: (141)\nExchanging + and \u0000one obtains the spin density s\u0000z(k) for the incident electron\nfrom the lower band. Similar expressions can be derived for the low-energy case\nk0<\u000b, when all waves belong to the lower band.\nThe expressions given above are valid only if ky<k+, or sin'\u0000<jk0\u0000\u000bj=(k0+\n\u000b). Atk\u0000> ky> k +the re\rection of the incident electron from the lower band\nto the upper one is forbidden by the conservation law. But the contribution of the\nupper band into the wave superposition is still present in the form of the evanescent\nmode. The wave superposition in this case is\n\t=eikyy\np\n2\u0014\u0012\n1\n\u0014\u0000\u0013\neik\u0000xx+r1\u0012\n1\n\u0014\u0003\n\u0000\u0013\ne\u0000ik\u0000xx+g\u0012\n1\ns\u0013\nepx\u0015\n; (142)\nwhere\u0014\u0000=iei'\u0000and\np=q\n(k0+\u000b)2sin2'\u0000\u0000(k0\u0000\u000b)2; s=ky\u0000p\nk0\u0000\u000b\nr1=\u0000s\u0000iei'\u0000\ns+ie\u0000i'\u0000; g=\u00002icos'\u0000\ns+ie\u0000i'\u0000: (143)\nThezspin density for this wave superposition contains not only the interference\ncontributions but also the contribution from the evanescent component /epx:\ns\u0000z(k) =pcos2'\u0000\nk0sin'\u0000[e2px+ cos(2k\u0000xx)\u00002epxcos(k\u0000xx)]\n+cos'\u0000\nk0sin'\u0000[2k0\u0000(k0+\u000b) cos2'\u0000][sin(2k\u0000xx)\u00002epxsin(\u0000xkx)]: (144)\nThis expression is valid independently of whether the electron energy is high ( k0>\n\u000b) or low (k0<\u000b).\nAll contributions to the zspin density are odd with respect to the sign of kyand\nvanish in the equilibrium state. But in the presence of the voltage bias along the\nyaxis the distribution function also has an odd component, and spin polarization\nbecomes possible. This leads to the edge spin accumulation (polarization), which\nis important for investigation of the intrinsic spin Hall e\u000bect (section 14.3).\nThe wave interference near the edge leads not only to zspin polarization but\nalso to the spin torque. The existence of this torque is required by the spin balance\n(87): If there is no current in the vacuum and there is a bulk current normal\nto the boundary, the presence of an edge torque is inevitable and its total value\n(integral over the whole edge area) must be equal to the spin current from the bulk\nindependently of particular properties of the edge (an ideally re\recting wall in our\ncase). Moreover, the integral edge torque should compensate the bulk spin currentMarch 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 51\nnot only at the border with the vacuum but also at the interface between the\nRashba medium and a medium, which does not allow dissipationless spin currents.\nIndeed, in the latter case spin di\u000busion accompanied by spin accumulation is the\nonly mechanism of spin transport, but in equilibrium no dissipative process is\npossible and the spin current must vanish at the interface.\nBut the type of the edge does in\ruence the spatial distribution of the torque.\nWe shall derive this distribution for a simpler case km\u001c\u000b(\fgure 11b) when\nall expressions can be expanded in k0. Herekmis the maximum value of k0cor-\nresponding to the Fermi level. In this limit the main contribution to the torque\noriginates from interference of the incident wave with the second re\rected wave in\nthe superposition (139):\nGy+(k) =\u0000\u000b~2ky\nmRen\ne\u0000ik+x+ik\u0000x\u0000\n1\u0000\u0014\u0003\n+\u0014\u0000\u0001\nr2o\n: (145)\nA similar contribution Gy\u0000comes from the conjugate superposition, in which the\nincident plane wave /e\u0000ik\u0000xcorresponds to the wave number k < \u000b with the\nnegative group velocity. The subsequent integration over the whole Fermi sea in\nthe lower band yields\nGy(x) =\u0000\u000b2~2k2\nm\n\u0019m\u0014\n1F2\u0012\n\u00001\n2; 1;3\n2;\u0000k2\nmx2\u0013\n\u00001\n21F2\u0012\n\u00001\n2;3\n2;3;\u0000k2\nmx2\u0013\n+2\n3kmx\u0015\n; (146)\nwherepFq(a1;:::;ap;b1;:::;bq;z) is the generalized hypergeometric function [120].\nThe total torque over the whole bulkR0\n\u00001Gy(x)dx=~2\u000b2km=4\u0019mexactly com-\npensates the bulk spin current [see equation (123) in the limit km\u001c\u000b].\nAt large distances from the border the torques for single modes oscillate fast, so\nthe asymptotic behavior of the torque can be analyzed using the steepest-descent\nmethod. This yields the asymptotic torque at x!\u00001 :\nGy=\u0000r\u0019\nkm\u000b2~2\n4\u00192mjxj5=2sin\u0010\n2kmx\u0000\u0019\n4\u0011\n: (147)\nThis Friedel-like oscillation may be suppressed by disorder or electron-electron\ninteraction, which were neglected in our analysis.\nIn summary, we have obtained the following picture of spin currents and torques\nin the restricted Rashba medium. There is no spin torque inside the medium far\nfrom medium edges, but there is a constant spin current there. On the other hand,\ninterference of incident and re\rected plane waves leads to spin torques of opposite\nsigns (source and drain of spin) near the two edges. The role of the bulk equilibrium\nspin current is to transport spin from the spin source near one edge to the spin\ndrain near the opposite edge.\n13.4. Experimental detection of equilibrium spin currents\nThe central question for understanding the physical sense of the spin current is\nhow is it possible, if possible at all, to detect the existence of spin currents ex-\nperimentally . This question is especially acute for equilibrium spin currents since\nthey do not lead to any spin accumulation. However, spin current leads to electric\npolarization, which might be detected via electric \felds produced by it. Indeed,March 3, 2010 17:4 Advances in Physics SpinRev\n52 E. B. Sonin\nτ -τ\nτ\nh\nlzy\nxa)\nb)\nFigure 13. The cantilever with the integrated Rashba medium (thick solid\nline) on it. (a) A rigid substrate. The red arrow (above the Rashba medium)\nshows the direction of the spin current. The green arrow (inside the substrate)\nshow the direction of the orbital-moment current. The currents result in me-\nchanical torques\u0006\u001cat the edges of the substrate (b) The substrate is now a\n\rexible cantilever. The edge torque at its free end leads to its displacement\nh.\nthe spin currents is a \row of magnetic moments related with spin. According\nto classical electrodynamics [121] a magnetic moment mmoving with velocity\nvcreates an electric dipole p= [v\u0002m]=2c. The terms mivjin this expression\nare proportional to the spin currents ji\nj:mivj=\rji\nj. So the dipole moment is\npi=\"ijk\rjj\nk=2c=\"ijkejj\nk=2mc2. But this relation does not take into account crys-\ntal e\u000bect, which can strongly amplify spin-orbit interaction comparing with vacuum\nelectrodynamics. A more reliable de\fnition of spin-orbit dipole moment is using\nthe thermodynamic relation p=@h^Hi=@Eapplied to the Hamiltonian (84) [70].\nThis yields pi=\"ijk\u0015jj\nk. For the Rashba medium with the spin-orbit constant \u000b\nthe spin current ji\njis determined by equations (122) and (123), \u0015=@\u000b=@Ez, and\nthe electric dipole is parallel to the zaxis.\nThe electric \felds induced by stationary spin currents in conducting media were\ndiscussed by Hirsch [122] and Sun et al. [123]. These \felds were also discussed\nfor magnetically order systems [60, 69, 124, 125], where there are no itinerant\ncarriers. This phenomenon is an e\u000bect inverse to the spin Hall e\u000bect (generation\nof spin current by an electric \feld), which will be discussed in the next section\n14. The inverse spin Hall e\u000bect was observed experimentally by Valenzuela and\nTinkham [126]. Though the experiment was realized for spin-di\u000busion currents but\nnot equilibrium spin current discussed here, there is no evident reason why the\norigin of the spin current would be essential for the existence of the e\u000bect. In any\ncase, this suggests at least a Gedanken experiment for detection of equilibrium spin\ncurrents: as well as the charge currents in currents loops, which do not lead to any\ncharge accumulation, are detected by magnetic-\feld measurement, one may detect\nequilibrium spin currents by electric-\feld measurement even in the absence of any\nspin accumulation.\nRecently it was suggested [127] to detect an equilibrium spin current in the\nRashba medium by measuring a mechanical torque on a substrate at edges of the\nRashba medium caused by the spin current. If the substrate is \rexible, the edge\ntorques should deform it (\fgure 13), and measurement of this deformation would\nprovide a method to detect equilibrium spin currents experimentally. An appropri-March 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 53\nate experimental technique for such a measurement is already known: a mechanical\ncantilever magnetometer with an integrated 2D electron system [128]. Earlier me-\nchanical detectors were suggested for detection of non-equilibrium di\u000busion spin\ncurrents [129]. A mechanical stress produced by spin currents in mesoscopic struc-\ntures with collinear magnetic order was also studied by Dugaev and Bruno [130].\nThey called it the magneto mechanical e\u000bect .\nDerivation of the mechanical torque produced by bulk spin currents is based\non the conservation law for the total angular momentum (the spin + the orbital\nmoment) in the system \\2D electron gas + substrate\". The continuity equation\n(87) for spin must be supplemented by the continuity equation for orbital moment:\n@L\f\n@t+r\r~J\f\n\r=\u0000G\f: (148)\nHereL\fare\fcomponents of the orbital-moment densities and ~J\f\n\ris the orbital-\nmoment \rux-tensor. The torque G\fin this equation is the same as in the con-\ntinuity equation (87) for spin, but appears with an opposite sign. This provides\nthe conservation of the total angular momentum S+L. We consider currents of\nycomponents along the axis x, so\f=yand\r=x. At the equilibrium the time\nderivatives of momenta are absent. In section 13.3 we saw that inside the Rashba\nmedium there is a spin current but no torque, while at the edge there is an edge\nspin torque, which compensates the bulk current. According to the total angular\nmomentum conservation law this leads to an edge orbital torque and to a \rux of\nthe orbital moment with a sign opposite to that of the spin current, as shown in\n\fgure 13. Since the 2D electron gas has no ycomponent of the orbital moment,\nthe whole orbital torque must be applied to an edge of the substrate. Now if the\nsubstrate is a cantilever rigidly \fxed at one end (\fgure 13), the mechanical torque\n\u001c=~Jy\nx=\u0000jy\nx=R\nGy(x)dxwill deform the cantilever, and the displacement of its\nfree end can be measured.\nThe fact that the spin current must be accompanied by an opposite orbital-\nmoment current was already noticed by Sheng and Chang [131] and Zhang and\nYang [132]. Transformation of spin to angular momentum at an edge of the Rashba\nmedium was recently discussed by Teodorescu and Winkler [133]. Since the coun-\nter\row of the spin and the orbital moment does not lead to any \row of the total\nmoment, Zhang and Yang [132] have concluded that the spin current is not observ-\nable and cannot induce electric \felds discussed in the beginning of this subsection.\nHowever, this conclusion ignores the fact that the spin and the orbital moments\nhave di\u000berent gyromagnetic ratios. Therefore though the \row of the total mechan-\nicalmoment really vanishes, the \row of the total magnetic moment does not. In\nfact, the orbital moment current, which compensates the spin current, is not an\nobstacle but an instrument for spin-current detection as is shown here.\nFor numerical estimation of the e\u000bect we shall use the maximal value of the\nspin-orbit coupling \u000b~2=m= 6\u000210\u00009eV cm = 10\u000020erg cm = 10\u000011J m\nquoted by Rashba [111] for InAs based quantum wells. When calculating the spin\ntorque, it was simpler to consider the case km\u001c\u000b. But the mechanical torque\nreaches its maximum at km> \u000b [see equation (122)] when the spin current is\njy\nx=\u0000\u000b3~2=6\u0019m\u001810\u00008erg/cm=10\u00003J/m. In order to estimate the displace-\nmenthof the cantilever end (see \fgure 13), we use the cantilever parameters from\nreference [134]: the length l= 120\u0016m and the spring constant k=F=h= 86\u0016N/m,\nwhereFis the force on the cantilever end. Using the theory of elastic plates [135],\none obtains that the torque \u001c=\u0000jy\nxproduces the displacement h= 3\u001c=2kl.\nThis yields h\u00180:45\u0016m. Less optimistic estimations of the spin-orbit couplingMarch 3, 2010 17:4 Advances in Physics SpinRev\n54 E. B. Sonin\n(\u000b~2=m= 3\u000210\u00009eV cm for InAs, or 1 :4\u000210\u00009eV cm for GaSb [136]) predict\nan order or more smaller displacements, but certainly measurable with the modern\nmicromechanical technique. The torque can be enhanced and tuned by an external\nmagnetic \feld.\nIn the presence of the external magnetic \feld one should add the Zeeman energy\n\u0000\u0016B\u001b\u0001Hto the Hamilton equation (113). The spin current in a single-electron\nstate is determined by the spin, which is parallel or antiparallel to the \\e\u000bective\"\nmagnetic \feld H\u0000\u000b\b0[k\u0002^z]=\u0019acting on the electron. Here \b 0=hc=e is the single-\nelectron \rux quantum. Integrating the single-state spin current over the whole k\nspace and assuming that the Zeeman energy is much larger than the spin-orbit\nenergy, one obtains the bulk spin current\njy\nx=\u0007\u000b\n8\u0019\u000f2\nF\u0000\u00162\nBH2\n\u0016BH3(H2\nz+H2\nx); (149)\nwhere\u000fFis the Fermi energy. In the interval \u0000\u0016BH <\u000fF<\u0016BHelectrons \fll only\nthe lower band [the lower sign in equation (149)]. Then in terms of the electron\ndensityn=m(\u000fF+\u0016BH)=2\u0019~2\njy\nx=\u0000\u000b~2\b0\n2mHn\u0012H\n\b0\u0000n\u0013H2\nz+H2\nx\nH3: (150)\nIf\u000fF> \u0016BH, electrons \fll the both bands, the contributions from two bands to\nthe spin current cancel each other, and spin current vanishes in our approximation.\nThe present analysis ignores the e\u000bect of the electromagnetic vector potential on\nthe electron momentum, but this e\u000bect (which deserves a special analysis) is absent\nfor the inplane magnetic \feld Hx.\nIt is worthwhile to comment that ambiguity of spin-current de\fnition, which\nwas intensively discussed in the literature, has no impact on the e\u000bect considered\nhere. As discussed in the end of section 11, one may rede\fne the spin current\nji\njby adding to it an arbitrary term ( ji\nj!ji\nj+\u000eji\nj) but at the same time it is\nnecessary to compensate it by rede\fnition of the spin torque ( Gi!Gi+rj\u000eji\nj).\nIf the balance of the orbital part of the angular momentum is also considered, the\nde\fnitions of the orbital torque and \rux must be compatible with those of the spin\npart, in order not to violate the conservation law of the total angular momentum.\nEventually whatever de\fnition was used any correct calculation must predict the\nsame observable e\u000bect (displacement of the cantilever). After we de\fned the spin\ncurrent by equation (115), we are not free anymore in the choice of the de\fnition of\nthe torque and the current of the orbital angular momentum: The \rux of the orbital\nangular momentum in the elastic cantilever should be de\fned as ~Ji\nj=\"imnxmTn\nj\nwhereTn\njis the elastic stress tensor. This choice looks most natural since it de\fnes\nthe mutual torque between the spin and the orbital moment as a derivative of the\nspin-orbit energy with respect to the rotation angle.\nLet us look at an alternative de\fnition of spin current [24], which was discussed\nin the end of section 11. The \\spin-conserving\" current T\f(x) =jz(x) +P\f(x) =\njz(x)+R0\nxG\f(x0)dx0, which includes the dipole torque term P\f(x), is constant, and\nit is di\u000ecult to notice in this picture that there is a process of angular-momentum\ntransfer between spin and orbital degrees of freedom. This is due to a nonlocal\ncharacter of the current T\f: it controls only the global balance of spin. Globally\nno change of spin occurs in the sample, spin being generated at one edge and ab-\nsorbed at another. Meanwhile, exactly local torques, which do not violate the global\nspin balance, are responsible for the angular-momentum transfer to the orbital sub-March 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 55\nsystem, which leads to the mechanical deformation discussed in this section.\n14. Spin Hall e\u000bect\n14.1. Phenomenology of spin Hall e\u000bect\nIf an electrical current \rows through a conductor with spin-orbit coupling, this\ncan give rise to a spin current \rowing normally to the direction of the electrical\ncurrent. This e\u000bect predicted by Dyakonov and Perel [137] was later called spin\nHall e\u000bect [138]. In contrast to the usual Hall e\u000bect, the spin Hall e\u000bect originates\nfrom the e\u000bective magnetic \feld produced by spin-orbit interaction and does not\nrequire an external magnetic \feld for its existence. Experimental evidences of the\ne\u000bect in bulk semiconductors [139], in a hole [140] and an electron [141] 2D gas\nhave already been reported. Observation of the inverse spin Hall e\u000bect [126] also\nprovides evidence for the spin Hall e\u000bect since the direct and the reverse e\u000bects\nare connected with the Onsager relations.\nThere are two possible origins for the spin Hall e\u000bect. The \frst one, which was\ndiscussed by Dyakonov and Perel [137], is due to spin-dependent scattering on\nimpurities and is called extrinsic spin Hall e\u000bect. But broken space inversion sym-\nmetry also can allow an intrinsic e\u000bect, which is not connected with impurities\ndirectly, though can be strongly a\u000bected by them (see below).\nIn general spin currents in the spin Hall e\u000bect are not equilibrium currents con-\nsidered in the previous section. They are accompanied by dissipation and require\nan energy input for their existence, despite that dissipation is related not with the\nspin current itself but with the longitudinal charge current. What does unite them\nwith the spin currents discussed through the present review, is common controver-\nsies about their de\fnitions and the impact of the absence of spin conservation. We\nconsider the 2D electron gas assuming the presence of spin-orbit interaction and a\nweak electric \feld parallel to the axis y. Independently of the microscopic origin\nof the spin Hall e\u000bect, the broken space inversion symmetry allows the current of\nthezspin component along the axis xgiven by\njz\nx=\u001bSHE; (151)\nwhere\u001bSHis the Hall spin conductivity. In a uniform system with constant \u001bSHthis\ncurrent is constant by de\fnition. On the other hand, at the sample border the spin\ncurrent must vanish. Thus one should look for a spin current of another nature or an\nedge torque, which would compensate the spin current of electric origin. According\nto Dyakonov and Perel [137] (see also the recent phenomenological analysis by\nDyakonov [142]) another current is a dissipative spin di\u000busion current, i.e. the\ntotal spin current is jz\nx+jz\nD, where\njz\nD=\u0000DsrxSz: (152)\nThis current is not constant because of inevitable longitudinal spin relaxation de-\ntermined by the time T1. Thus the continuity equation for the zspin component\nis\n@Sz\n@t=\u0000rxjz\nD\u0000Sz\nT1=Dsr2\nxSz\u0000Sz\nT1: (153)\nFor the stationary process with @Sz=@t= 0 and under the condition that the totalMarch 3, 2010 17:4 Advances in Physics SpinRev\n56 E. B. Sonin\ncurrent vanishes at the sample border x= 0 one obtains\njz\nD=\u0000jz\nxe\u0000x=Ls; Sz=\u0000jz\nxT1\nLse\u0000x=Ls; (154)\nwhereLs=pDsT1is the spin-di\u000busion length, which has already been introduced\nin section 6. Thus the spin Hall e\u000bect leads to accumulation of the zspin component\nat the sample edge (called also spin orientation). The degree of spin accumulation\nis governed by dissipation parameters T1andDs. A reader can \fnd a detailed\ndiscussion of physical mechanisms responsible for dissipation in the review by \u0014Zuti\u0013 c\net al. [10].\n14.2. Extrinsic spin Hall e\u000bect\nThe extrinsic spin Hall e\u000bect originates from the spin-orbit interaction related to\nthe local electric \felds induced by impurities or defects. A well accepted model for\nthis interaction [14] is described by the Hamiltonian term\nHextr=\u0015\u001b\u0001[p\u0002rVi(r)]; (155)\nwherepis the electron momentum and Vi(r) is the impurity axisymmetric poten-\ntial, which depends on the distance rfrom the impurity. The e\u000bect of the spin-orbit\ninteraction on elastic scattering was known long time ago [144]. The interaction\nleads to asymmetry of scattering (skew scattering), and the di\u000berential cross section\ndepends on spin of an incident electron:\n\u001b\u0006(\u0012) =\u001b0(\u0012)[1\u0006S(\u0012)]; (156)\nwhere\u001b0(\u0012) is an even function of the scattering angle \u0012and determines scattering\nof unpolarized electron beams, while S(\u0012) is an odd function of \u0012called Sherman\nfunction. The Sherman function determines the skew scattering. The upper and\nthe lower signs correspond to spins +1 =2 and\u00001=2.\nVarious types of impurities and of 2D quantum wells were discussed in the litera-\nture (see, e.g., [145] and references therein). Here we present the simple but general\ndiscussion in terms of e\u000bective cross sections similar to that by Engel et. al. [146]. A\nstraightforward and physically transparent method to calculate a bulk spin current\nis solution of the Boltzmann equation. Studying the spin Hall e\u000bect they usually\nused the quantum Boltzmann equation, in which the distribution function was a\nmatrix 2\u00022 in spin indices [15, 147{149]. However, for the extrinsic e\u000bect under\nconsideration the zspin component is a good quantum number, which is a\u000bected\nneither by external electric \feld, nor by spin-orbit interaction determined by equa-\ntion (155). Indeed, the e\u000bective spin-orbit magnetic \feld is normal to the 2D layer\nplane and scattering cannot lead to any spin-\rop. This means that o\u000b-diagonal\nelements of the spin-density matrix vanish. Two diagonal elements correspond to\ntwo distribution functions f\u0006(k) =f0(k) +f0\n\u0006(k) for electrons with positive (+)\nand negative (-) zspins. Here f0(k) is the Fermi equilibrium distribution function,\nwhich does not depend on spin and direction of the electron wave vector k. The\nBoltzmann equation for the non-equilibrium distribution function f0\n\u0006(k) generated\nby a weak inplane electric \feld Eis\ne\n~E@f0(k)\n@k=nivFZ\n[f\u0006(\u001e)\u0000f\u0006(\u001e0)]\u001b\u0006(\u0012)d\u0012; (157)March 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 57\nwherevFis the Fermi velocity, niis the impurity density, and \u001eand\u001e0=\u001e+\u0012are\nangles between kandEbefore and after a collision. The solution of this equation,\nas one may check by substitution, is\nf0\n\u0006(k) =e~\u001c\nm@f0(\u000f)\n@\u000f\u0012\nE\u0001k\u0006\u001bs\n\u001btE\u0001[^z\u0002k]\u0013\n: (158)\nHere\u000f=~2k2=2mis the electron energy, and the relaxation time\n\u001c=\u001bt\nnivF(\u001b2\nt+\u001b2s)(159)\nis determined by the transport cross section,\n\u001bt=Z\nI(\u0012)(1\u0000cos\u0012)d\u0012; (160)\nand the e\u000bective cross section related to scattering asymmetry,\n\u001bs=Z\nI(\u0012)S(\u0012) sin\u0012d\u0012: (161)\nThe parameter \u001bs=\u001btis called transport skewness [146].\nKnowing the non-equilibrium distribution function one can calculate the charge\ncurrent parallel to Eand thezspin current transverse to Eby integration over the\nmomentum space and summation over two spin values. Let us consider the zero\ntemperature limit when @f0(\u000f)@\u000f=\u000e(\u000f\u0000\u000fF). The charge current is given by the\nusual Drude formula,\nj=\u001bE=e2\n\u0019E\u001c\"F\n~2; (162)\nwhile the spin current jz=\u001bSHEis determined by spin Hall conductivity [146]\n\u001bSH=e\n2\u0019\u001c\"F\n~\u001bs\n\u001bt=~\n2e\u001bs\n\u001bt\u001b: (163)\nHere\u001bis the ohmic electron conductivity.\nIt is believed that the extrinsic spin Hall e\u000bect was detected by Kato et al. [139]\nwho observed spin accumulation on edges of n-GaAs layers. Engel et al. [146] have\nfound a rough quantitative agreement of the experiment with the theory presented\nin this subsection, though the signs of the e\u000bects were opposite. This disagreement\nremains unresolved (see discussion in [146]).\n14.3. Intrinsic spin Hall e\u000bect and edge spin accumulation\nThe intrinsic spin Hall e\u000bect does not rely on spin-dependent e\u000bects in scattering\nbut is related entirely to the uniform bulk spin-orbit interaction, which leads to\nspin-orbit-split band structure. The e\u000bect was proposed by Murakami et al. [150] in\nbulkp-type semiconductors using the e\u000bective Luttinger Hamiltonian for holes and\nby Sinova et al. [151] in 2D electron systems with the Rashba spin-orbit interaction.\nIn the literature there were debates on the strength of the intrinsic spin Hall e\u000bect inMarch 3, 2010 17:4 Advances in Physics SpinRev\n58 E. B. Sonin\nthe 2D electron gas. Relating the spin current to the acceleration of electrons by the\nelectric \feld in a pure material, Sinova et al. [151] concluded that the intrinsic spin\nHall e\u000bect is characterized by the \\universal\" spin Hall conductivity \u001bSH=e=8\u0019.\nOriginally it was believed that this result is insensitive to a small concentration of\nimpurities, despite the fact that a steady state of a conductor in an electric \feld\nis impossible without collisions, which compensate the monotonous acceleration of\nelectrons by the electric \feld. Later on it turned out that in the Rashba medium\neven rare collisions should not be ignored and eventually the intrinsic spin-Hall\nconductivity at zero frequency vanishes in an in\fnite system (see discussion and\nreferences in the review by Engel et al. [14]). But this conclusion is valid only for\nRashba spin-orbit coupling linear in the electron wave vector k, and the intrinsic\nspin Hall e\u000bect for Rashba coupling nonlinear in k, or in the Luttinger model\nfor spin-orbit coupled systems [152] is not ruled out. The intrinsic e\u000bect in the\nmedium with nonlinear in kRashba coupling follows from the theory using the\nquantum Boltzmann equation for the spin density matrix [149, 153]. In contrast to\nthe extrinsic spin Hall e\u000bect, the intrinsic spin Hall e\u000bect is not possible without\no\u000b-diagonal matrix elements since the zspin current is proportional to them. This\nis because the bulk spin-orbit interaction keeps the spin inside the xyplane, and\nonly correlation between eigenstates of the Rashba Hamiltonian (like interference\nbetween plane-wave states near edges) leads to zspin polarization and zspin\ncurrents.\nHowever, the experimental detection of the intrinsic spin Hall is rather problem-\natic. The experimental evidences of the spin Hall e\u000bect reported in the literature\nwere based on optical measurement of the spin accumulated on the sample edges\npresumably resulting from the spin Hall e\u000bect. Meanwhile, spin accumulation (po-\nlarization) near boundaries might not be so simply related to spin currents in the\nbulk as repeatedly pointed out [15, 143, 149]. As was shown above (section 13.1),\nthe bulk spin current is not necessarily accompanied by spin accumulation. More-\nover, spin accumulation at sample edges is possible even without bulk spin current.\nIt was demonstrated for the ballistic spin Hall e\u000bect [154{156], when the electron\nmean-free path exceeds the sample sizes. Edge spin accumulation without bulk\nspin currents takes place also in the standard collisional regime when the electron\nmean-free path is much shorter than the sizes of the sample but still longer than\nthe distance where interference of incident and re\rected waves responsible for edge\naccumulation takes place [157]. Let us discuss edge accumulation without bulk spin\ncurrents in more details.\nSince our goal is to analyze the case without bulk spin current, we may use\nthe standard linear-in-momentum Rashba Hamiltonian. Moreover, we do not need\nto deal with the quantum Boltzmann equation and restrict ourselves with the\nclassic Boltzmann equation for two scalar distribution functions corresponding to\ntwo diagonal elements of the spin density matrix. Indeed, according to references\n[148, 149], for the linear-in-momentum Rashba interaction bulk spin currents vanish\ntogether with o\u000b-diagonal matrix elements, to which they are proportional.\nIn section 13.3 it was shown that the eigenstates of the Rashba Hamiltonian are\npartiallyzspin polarized near the boundaries, though in the equilibrium there is\nno total spin polarization after integration over the Fermi sea. But in the spin Hall\ne\u000bect there is a charge current (along the yaxis at our choice of the coordinate\nframe, see \fgure 12), and the non-equilibrium distribution function has a compo-\nnent odd with respect to the wave vector component ky. First we shall consider the\nballistic regime when the voltage drop Voccurs at the contacts, and there is no\nelectric \feld inside the sample. In the narrow interval of energies \u000fF+eV >\u000f>\u000f F\naround the Fermi surface only left-moving electrons with ky>0 are present. TheyMarch 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 59\nare responsible for the edge accumulation of the zspin. Bearing in mind that\neV=d\u000f= (~2=m)k\u0006Fdk, the two band contributions to the total spin density\nsz(x) =s+z(x) +s\u0000z(x) are determined by integrals over the Fermi circumferences\nof the two bands with the Fermi wave vectors k\u0006F=jkm\u0007\u000bj:\ns\u0006z(x) =meVk\u0006F\n4\u00192~2kmZ\u0019=2\n0s\u0006z(k)d'\u0006: (164)\nHerekmis the value of k0at the Fermi circumferences. The spin densities s\u0006z(k)\nfor the eigenstates, which correspond to the wave vectors kof incident waves, were\nfound in section 13.3. The asymptotic behavior of the spin density is determined\nby the evanescent-mode contribution [see equation (144)] and at x!\u00001 is given\nby\nsz(x) =meV\n8\u00192~r\u000b\nkm1\nk2\n+Fk\u0000F1\njxj3: (165)\nThe total accumulated spin is given by [156]\nSz=Z0\n\u00001sz(x)dx=meV\n8\u00192~\u000b\u0012\nlnkm+\u000b\njkm\u0000\u000bj\u00002\u000b\nkm\u0013\n: (166)\nFor further comparison with the collisional regime it is convenient to connect the\ntotal spin not with the voltage Vbut with the electric current,\nj=e2nV\n\u0019~km\u0002(\n2k2\nm\nk2\nm+\u000b2atkm>\u000b\n1 atkm<\u000b; (167)\nwhere the 2D electron density is n= (k2\nm+\u000b2)=2\u0019atkm>\u000bandn=\u000bkm=\u0019at\nkm<\u000b. Then\nSz=mj\n8\u0019en\u000b\u0012\nlnkm+\u000b\njkm\u0000\u000bj\u00002\u000b\nkm\u0013\n\u0002(\nk2\nm+\u000b2\n2kmatkm>\u000b\nkmatkm<\u000b: (168)\nIn the limits of weak ( \u000b!0) and strong ( \u000b!1 ) spin-orbit interaction this yields\nSz= (mj=24\u0019en)(\u000b2=k2\nm) andSz=\u0000mj=4\u0019enrespectively. At km=\u000bthere is a\nlogarithmic divergence, which can be cut either by the sample size or by nonlinear\ne\u000bects.\nLet us switch now to the collisional regime. As was explained above, since the\nbulk spin current is absent, one may use the standard Boltzmann equation for two\nscalar distribution functions f\u0006(k) =f0(\u000f) +f0\n\u0006(k) for the two bands, where f0(\u000f)\nis the equilibrium Fermi distribution function, which depends only on energy \u000f. The\nstationary solution of the Boltzmann equation for the non-equilibrium distribution\nfunctionf0in a weak electric \feld along the yaxis is [159]\nf0\n\u0006=e\u001cE\n~@f0(k)\n@k=eE\u001c~km\nmsin'\u0006\u000e(\u000f\u0000\u000fF): (169)\nHere the relaxation time \u001cfor elastic scattering on defects is determined by the\ntransport cross section and in general is a function of k. In principle \u001cshould\ndi\u000ber for two bands. But the di\u000berence vanishes for weak spin-orbit coupling andMarch 3, 2010 17:4 Advances in Physics SpinRev\n60 E. B. Sonin\nfurther will be neglected. The functions f0\n\u0006determine the electric current equal to\nj=e2E\u001ck2\nm=2\u0019mforkm>\u000b and toj=e2E\u001c\u000bkm=2\u0019mforkm<\u000b. Thezspin\ndensities for the two bands instead of (164) are given by\ns\u0006z(x) =eE\u001ck\u0006F\n4\u00192~Z\u0019=2\n\u0000\u0019=2sin'\u0006s\u0006z(k)d'\u0006: (170)\nTedious but straightforward integrations similar to those for the ballistic regime\nyield the total edge spin:\nSz=\u0000mj\n32\u00192enk2\nm+\u000b2\nk4m\u0014\n3(k2\nm\u0000\u000b2) arctan2p\u000bkm\nkm\u0000\u000b\n\u00002p\u000bkm(3k2\nm+ 2km\u000b+ 3\u000b2)\nkm+\u000b+\u0019(k2\nm+ 3\u000b2)\u0015\n(171)\nfor the high-energy case km>\u000b, and\nSz=\u0000mj\n16\u00192en\u000bkm\u0014\n3(\u000b2\u0000k2\nm) arctan2p\u000bkm\n\u000b\u0000km\n\u0000p\u000bkm(6\u000b2+ 6k2\nm+ 4\u000bkm)\n\u000b+km+ 4\u0019\u000bkm\u0015\n(172)\nfor the low-energy case \u000b>km.\nWhen\u000b!1 the di\u000berence between the ballistic and collisional regime vanishes.\nOn the other hand, in contrast to the ballistic regime, in the collisional regime the\naccumulated spin remains \fnite even in the limit of zero spin-orbit coupling \u000b!0\nbeing equal to\nSz=\u0000mj\n32\u0019en: (173)\nThis paradoxical result is explained by the divergence of the width \u00181=(k\u0000x\u0000k+x)\nof the spin accumulation area in this limit. In the ballistic regime this divergence\nis canceled after summation over the two bands. However, our analysis is valid\nonly if all relevant scales including 1 =(k\u0000x\u0000k+x) are less than the electron mean-\nfree path. When this condition is violated the spin accumulation should go down.\nFigure 14 shows the reduced total accumulated spin ~Sz= 4\u0019enSz=mj for the\nballistic (curve 1) and the collisional (curve 2) regimes as functions of the density-\ndependent parameter \u000b=p\u0019n. Edge spin accumulation without bulk spin currents is\npossible for other types of spin-orbit interaction. In particular, Bokes and Horv\u0013 ath\n[160] considered spin-orbit interaction related to the edge potential V(x) con\fning\nthe electron gas: HSO=\u000bE\u001b\u0001[k\u0002rV(x)]. They obtained the edge accumulation\nlinear in the spin-orbit constant \u000bE, which is insensitive to the details of the edge\npotential. This allows to expect that the assumption of the hard-wall potential\nmade in our calculation for the Rashba spin-orbit interaction also is not so crucial\nfor the \fnal outcome of the calculation.\nOriginally the spin Hall e\u000bect was de\fned as an e\u000bect related to bulk spin cur-\nrents. The present discussion shows that spin accumulation is not really a probe of\nthe bulk spin current: the former can be absent in the presence of the bulk current\nand can appear in the absence of the latter. Therefore the question arises whether\nedge accumulation without bulk currents may be called the spin Hall e\u000bect. AMarch 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 61\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 1.0 0.5 0.0 0.5 1.0 \n-\n-1\n2\nFigure 14. The plot of the reduced total spin ~Sz= 4\u0019enS z=mj as functions of \u000b=p\u0019n.1{ the ballistic\nregime. 2{ the collisional regime.\nchoice of terminology usually is a matter of convention, taste, or tradition. Edge\nspin accumulation and spin currents require the same symmetry, and one may call\nthe edge spin accumulation without bulk currents the edge spin Hall e\u000bect.\nIn order to compare the edge and the bulk spin Hall e\u000bects, we scale the latter\nusing the \\universal\" spin conductivity \u001bSH=jz=E=e=8\u0019, though in reality this\nis far from being universal [14]. Here jzis the bulk current of the zspin. Assuming\nthat at the edge the bulk spin current is fully compensated with spin di\u000busion\ncurrent (section 14.1), the total accumulated spin is jzT1=eET 1=8\u0019, whereT1is\nthe longitudinal spin relaxation time. So ratio of the edge to the bulk spin Hall\ne\u000bect is\u0018\u001c=T 1.\nFor comparison with the spin Hall e\u000bect observed in the 2D hole gas [140, 158] on\nmay use\u001c\u001810~=EF= 20=kmvF,n= 2\u00021012cm\u00001, and the accumulation area\nwidth 10 nm given by Nomura et al. [158]. Then the total spin accumulated due to\nthe edge spin Hall e\u000bect at \u000b!0 is about 70 % of the experimental value. So the\ninterpretation of this experiment in the terms of the bulk spin currents probably\nmust be reconsidered even if the spin-orbit interaction for these materials should\nbe described by a model more general than used for the present analysis.\nThere are other cases of electrically generated edge spin accumulation without\nbulk spin currents inside the sample. Adagideli and Bauer [161] discussed edge\nspin accumulation governed by spin di\u000busion, which occurred within the distance\nof the order of the spin-di\u000busion length Lsfrom the interface between the media\nwith and without spin-orbit coupling. In contrast, interference spin accumulation\ndiscussed above, is not related to any dissipative process and occurs at the distance\non the order spin-orbit length, which was assumed to be much shorter than the\nmean-free path. It is interesting that the di\u000busion governed accumulation provides\nthe total accumulated spin \u0018eE\u001c [71] of the same order of magnitude as the\ninterference mechanism. Since Lsmust essentially exceed the mean-free path, the\ninterference mechanism provides much higher spin density but in a much narrower\nlayer. Another example of edge spin accumulation without bulk spin currents will\nbe discussed in section 14.5 addressing the quantum spin Hall e\u000bect .March 3, 2010 17:4 Advances in Physics SpinRev\n62 E. B. Sonin\n14.4. Spin currents in spin Hall insulators\nAs was already mentioned, though transverse spin currents induced by the spin\nHall e\u000bect are dissipationless themselves, the whole process is accompanied by\ndissipation in the longitudinal channel and therefore requires an energy source.\nMurakami et al. [162] (see also reference [163]) proposed the totally dissipation-\nless spin-Hall e\u000bect in insulators with spin-orbit interaction. In a band insulator\nat zero temperature and the Fermi level within the gap between conductance and\nthe valence bands a weak electric \feld cannot generate an electric current, but\nmay result in a transverse spin current, i.e., the spin conductivity \u001bSHin the spin-\nHall relation (151) is \fnite being determined by topological invariants of the band\nstructure. Materials with this property are called spin Hall insulators . The dissi-\npationless spin currents in insulators, when the Fermi level is inside the forbidden\ngap, were revealed by numerical calculations using realistic parameters of semi-\nconductor band structures [164]. The spin Hall e\u000bect is possible not only in band\ninsulators. Meier and Loss [60] considered the spin Hall e\u000bect in a two-dimensional\nHeisenberg model consisting of localized spin, in contrast to itinerant electrons in\nband insulators.\nThe spin Hall e\u000bect in insulators has a very important feature discerning it\nfrom the spin Hall e\u000bect in conductors. Since there is no charge current in the\nformer case, there is no Joule heating and no energy input [163]. As a result, no\ndissipation process is possible, and the problem reduces to the equilibrium problem\nof an insulator in an electric \feld. The bulk spin current cannot be compensated\nby the dissipative mechanism of Dyakonov and Perel [137], and one should look for\na Hamiltonian mechanism of absorption (generation) of spin near sample edges.\nSo there is a close analogy between spin currents in the spin Hall insulator and\nequilibrium currents in the Rashba medium analyzed in section 13. Indeed, the\nRashba spin-orbit interaction is connected with an internal or external electric \feld\nnormal to the electron-gas plane. The electric \feld is not able produce a current\nsince electrons are con\fned in a 2D layer. So the 2D gas is an insulator in the z\ndirection, but this does not rule out a Hall spin current along the layer.\nThe equilibrium character of spin currents in spin Hall insulator impose a serious\nrestriction on methods of spin-current detection. In particular, spin injection into\na non-magnetic material without spin-orbit interaction is impossible, since in this\nmaterial spin can be transported only by a dissipative (di\u000busion) current, which\nmust be supported by energy pumping. On the other hand, it is possible to extract\nspin from a spin Hall insulator putting it into a contact with a magnetically ordered\nmedium supporting dissipationless spin transport.\n14.5. Spin accumulation and the quantum spin Hall e\u000bect in topological\ninsulators\nRecently great attention was attracted to remarkable properties of topological insu-\nlators , in which the quantum spin Hall e\u000bect was predicted [165, 166]. The hallmark\nof a topological insulator is a forbidden gap originated from spin-orbit coupling and\nthe presence of topologically stable helical edge states. Topological criteria and\nclassi\fcation for these materials have already been carefully analyzed [167{169].\nThe outcome of this analysis most important for the goals of the present review\nis illustrated in \fgure 15. At two edges of the sample parallel to the electric \feld\n(we address the 2D case) each spin is able to move only in one direction, which\nis opposite for two spin directions. The edge states cross the whole forbidden gap\nconnecting the valence-band and the conduction-band bulk continua. The system\nis time-reversal invariant, and two states with opposite directions of the spin andMarch 3, 2010 17:4 Advances in Physics SpinRev\nAdvances in Physics 63\nelectrode electrode \nFigure 15. Edge states in a topological insulator. Wide blue arrows show spin direction (spin quantization\naxis is not necessarily in the plane as in the \fgure). At the upper edge spins up are rightmovers while\nspins down are leftmovers. At the lower edge directions of motion are opposite. The ballistic edge channels\nshort-circuit the bulk insulator with in\fnite resistance. Therefore the whole sheet is globally a ballistic\nconductor without electric \feld inside. The external voltage bias Vdrops inside electrodes (see voltage\ndistribution in the lower part of the \fgure).\nthe wave vectors form a Kramers degenerate pair. Because of helicity (one-way mo-\ntion) the edge states are robust against elastic backscattering and therefore may\nbe treated as ballistic. This means that the ballistic edge channels short-circuit the\nbulk insulator with in\fnite resistance, and the whole sample is globally a ballistic\nconductor without electric \feld inside.\nThus, whatever the bulk spin conductivity \u001bSH=jz=Ecould be, bulk spin\ncurrents are absent simply because the bulk electric \feld is absent. Nevertheless,\nthe voltage drop between two leads (see the voltage distribution along the sample\nin \fgure 15) leads to edge spin accumulation similar to that considered in section\n14.3 for bulk conductors. However, mechanisms of the edge accumulation in two\ncase are di\u000berent. Whereas in bulk conductors the spin accumulation arose from\ninterference of electron plane waves re\rected from the boundary, in topological\ninsulators spin polarization appears because the leftmoving and rightmoving edge\nstates with oppositely directed spins have di\u000berent densities proportional to the\nvoltage bias. The latter directly follows from the Landauer-B uttiker approach. The\nleftmovers occupy states below the Fermi level in the right electrode, whereas the\nrightmovers occupy states below the Fermi level in the left electrode (\fgure 15).\nThe di\u000berence between the two Fermi energies is eV, and the edge spin density\n(spin per unit length along the edge) is determined by the number of states in this\nenergy interval:\nSz=ese\nhvFeV; (174)\nwherevFe=d\"(ky)=~dkyis the Fermi velocity of the edge mode with the spectrum\n\"(ky) andseis the e\u000bective \\spin\" of the edge states, which may be di\u000berent from\n\u0006~=2 in general. The parameters of edge modes were calculated numerically and\nanalytically using simple but reliable models [167, 168]. The word \\spin\" is in\nquotation marks since it is actually a combination of the electron spin (which isMarch 3, 2010 17:4 Advances in Physics SpinRev\n64 E. B. Sonin\n\u0006~=2 of course) and the orbital moment of the electron in the band. Though it is\ndangerous to jump to general conclusions on which angular momentum eventually\nis relevant for various experiments, one may de\fnitely expect that electromagnetic\nexperiments (Kerr or Faraday e\u000bect, induced electric \felds) address the magnetic\nmoment in the edge mode.\nFor estimation of the e\u000bective spin related to the magnetic moment one can\nconsider the model usually used for the topological insulator [167, 168]: the edge\nstate crosses the forbidden gap separating the conduction and the valence bands,\nwhich originate from s-type (l= 0) andp-type (l= 1) atomic orbitals. The p-\ntype orbital corresponds to quantum numbers j= 3=2 for the total moment and\nmj=\u00061=2 for its projection on the quantization axis. The Lande factor,\ngL= 1 +j(j+ 1)\u0000l(l+ 1) +s(s+ 1)\n2j(+1);\nis equal to the electron gfactorge= 2 for the s-type orbital and to the factor\ngv= 4=3 for thep-type orbital. The edge mode is a superposition of the two states\nwith the weights teandtvrespectively ( te+tv= 1). Then the e\u000bective spin in\nequation (174) is\nse=~\n2\u0012\ntc+gv\ngetv\u0013\n=~\n2\u0012\ntc+2tv\n3\u0013\n: (175)\nUsing this value of the e\u000bective spin the accumulated magnetization is obtained\nfrom expression (174) by multiplying with the electron gyromagnetic relation.\nIt is interesting to compare the edge spin accumulation due to the quantum spin\nHall e\u000bect with the accumulation due to the intrinsic spin Hall e\u000bect in conductors\n(section 14.3). Introducing the average electric \feld E=V=L, whereLis the length\nof the sample, equation (174) at se\u0018hyieldsSz\u0018eEL=vFe. Comparing this with\nSz\u0018eE\u001c for the accumulation in conductors one sees that this transforms to the\naccumulated spin in topological insulators after replacing the relaxation time \u001cby\nthe time of \right L=vFethrough ballistic edge channels.\nSoon after the theoretical prediction the topological insulators were experimen-\ntally detected in the HgTe quantum well [170] by studying charge transport. It\nwas demonstrated that at the quantum well thickness exceeding the critical value\n6.3 nm there was an interval of gate voltages where the conductance reaches the\nvalue 2e2=hindependently of the sample width W(see \fgure 15). This is a clear\nevidence of the ballistic transport through edge states while the main bulk is not\nconducting. The topological insulators states were also detected in BiSb [171], BiSe\n[172], and BiTe [173] compounds by the methods of angle-resolved photoemission\nspectroscopy (APRES). In the literature the topological insulator state is called\nalso the quantum spin Hall state. It is worthwhile of mentioning that despite im-\npressive experimental demonstration of the quantum spin Hall state the quantum\nspin Hall e\u000bect itself is still wants its experimental con\frmation: A \\smoking gun\"\nof edge spin accumulation have not yet been reported.\n15. Conclusions\nThe present review focused on four types of dissipationless spin transport: (1) Su-\nper\ruid transport, when the spin-current state is a metastable state (a local but\nnot the absolute minimum in the parameter space). (2) Ballistic spin transport,\nwhen spin is transported without losses simply because sources of dissipation areMarch 3, 2010 17:4 Advances in Physics SpinRev\nREFERENCES 65\nvery weak. (3) Equilibrium spin currents, i.e., genuine persistent currents. (4) Spin\ncurrents in the spin Hall e\u000bect. The dissipationless spin transport was a matter\nof debates during decades, though sometimes they were to some extent semantic.\nTherefore it was important to analyze what physical phenomenon was hidden un-\nder this or that name remembering that any choice of terminology is inevitably\nsubjective and is a matter of taste and convention. The various hurdles on the way\nof using the concept of spin current (absence of the spin-conservation law, ambi-\nguity of spin current de\fnition, etc.) were analyzed. The \fnal conclusion is that\nthe spin-current concept can be developed in a fully consistent manner, though\nthis is not an obligatory language of description: Spin currents are equivalent to\ndeformations of the spin structure, and one may describe the spin transport also\nin terms of deformations and spin sti\u000bness.\nThe recent revival of interest to spin transport is motivated by emerging of spin-\ntronics and high expectations of new applications based on spin manipulation. 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Chen at al., Science 325 (2009), p. 178."    },    {        "title": "0709.4172v1.Spin_polarized_Current_induced_Instability_in_Spin_Valve_with_Antiferromagnetic_Layer.pdf",        "content": "arXiv:0709.4172v1  [cond-mat.mtrl-sci]  26 Sep 2007Spin-polarized Current-induced Instability in Spin-Valv e with\nAntiferromagnetic Layer\nHelen V. Gomonay and Vadim M. Loktev\nNational Technical University of Ukraine “KPI”\n37, ave Peremogy, Kyiv, 03056, Ukraine\nAbstract\nIn the framework of phenomenological model we consider dyna mics of a compensated collinear\nantiferromagnet (AFM) in the presence of spin-polarised cu rrent. The model is based on the\nassumption that AFM spins are localised and spin torque is tr ansferred to each magnetic sublattice\nindependently. It is shown that in AFM spin current i) can be a source of the ”negative friction”;\nandii) modifiesspin-wavefrequencies. Equilibriumstate o fAFM can bedestabilized bythecurrent\npolarized in parallel to AFM vector. Threshold current at wh ich the loss of stability takes place\ndepends upon the magnetic anisotropy of AFM.\nPACS numbers:\nKeywords: antiferromagnet, spin-torque effect, spin-valve, sp in-polarised current\n1I. INTRODUCTION\nThe phenomenon of spin transfer from conductivity electrons to m agnetization of fer-\nromagnetic (FM) layer is widely used in engineering of the magnetic mem ory devices.\nWhile flowing from nonmagnetic to ferromagnetic layer spin-polarised electrons transfer\nspin torque1and additional magnetization2thus inducing reorientation or even dynamically\nstable rotation of localised magnetic moments. Physical interpreta tion of these phenomena\nis based on the law of spin conservation and s−dexchange interaction between free carriers\nand localized moments.1,3\nRecent experiments with nanopillars4,5point out that spin-polarised current also can\nchange the stateof antiferromagnetic (AFM) layer andcharacte ristic value of critical current\nat which reorientation of spins takes place could be much smaller than in FM. From general\npoint of view, study of spin transport effects in AFMmay open either moreefficient methods\nfor spintronics or much more reach fundamental phenomena. In p articular, in the AFM\nmetal with spin-density waves (SDW) s−dexchange couples spins of free electrons with\norientation of AFM vector and influence of spin torque is substantia lly enhanced.6\nIn the present paper we address another question: “Is it possible to control the state\nof an AFM metal without SDW with spin-polarized current?” As a starting point we\nconsider the “toy” model of the compensated collinear AFM in which t he magnetic order\nis mainly caused by localised spins. Fe 50Mn50alloy widely used in spin-valve structures can\nbe considered as an example of such a material.\nII. MODEL\nWeconsider a spin-valve structure (analogousto that studied inRe f. 4,5 consisting ofFM\nand AFM layers separated by a nonmagnetic metallic spacer (Fig.1) r ather thin in order to\ncondition a ballistic regime for conductivity electrons. FM with an easy axis inpdirection\nacts as a spin polarizer for the electron current Iflowing through the whole structure.\nThe magnetic state of AFM is unambiguously defined by sublattice mag netizations Mj\n(j=1,2). We assume that due to a local character of s−dexchange, spin conservation\nlaw is fulfilled independently for each act of conductivity-to-localized spin interaction7. So,\naccording to Slonczewskii mechanism1, each sublattice magnetization experiences a spin\n2torqueTj=σI[Mj×[Mj×p]]/M0, where coefficient σ=εηµ0g/(2M0Ve) depends upon\nthe geometry of contact (volume V), and spin-polarization efficiency ε,M0=|Mj|. Hereη\nis the Plank constant, gis gyromagnetic ratio and eis electron charge. Depending on the\ndirection of the electron current the sign of Ican be either positive (incoming spin flux) or\nnegative (outcoming spin flux). \nFM  AFM NM  X \nj pM1 \nM2 Z \nFigure 1: (Color online) Spin-valve structure consisting o f FM and AFM layers and nonmagnetic\n(NM) spacer. In the standard spin-valve structure5NM spacer is absent.\nIn the present paper we restrict ourselves with the case of small e xternal exposure, i.e.,\nthe work of the polarized current over the localized spins is suppose d to be much smaller\nthan the exchange energy that keeps magnetizations M1,M2antiparallel. To this end,\nmacroscopic magnetization M=M1+M2is much smaller than AFM vector L=M1−M2,\n|M| ≪ |L|, and one can reduce the description of AFM dynamics to a Lagrange form with\nLas a generalized variable and the Lagrange function in a form8\nL=χ⊥\n8g2M2\n0˙L2−A\n8M2\n0(∇L)2−wan(L). (1)\nHereχ⊥,A, andwan(l) arethemagnetic susceptibility, inhomogeneous exchange consta nt\nand magnetic energy of AFM layer, respectively.\nWithin the framework of the Lagrange formalism, all the dissipative p rocesses (Gilbert\ndamping and spin-torque induced rotation of magnetic moments) co uld be adequately de-\n3scribed with the Relay function\nR=χ⊥αG\n8g2M2\n0˙L2−σI\n4gM0(p,L×˙L), (2)\nwhere the damping parameter αGis equal to the linewidth of AFM resonance (see, e.g.,\nRef.9).\nIII. GENERAL CONSIDERATIONS\nSome peculiarities of AFM dynamics in the presence of spin-polarized c urrent can be\ndeduced from the analysis of the Relay function (2) that describes the rate of energy losses\nin the system.\n•Likein FM, spin-polarized current may work as a source of the external energy pump-\ning (“negative” friction) and suppress the Gilbert damping. This tak es place for a\ncertain value of current, I≥Ic1, and noncollinear orientation of FM and AFM easy\naxes, e.g., p⊥L(0). Critical current Icrat which the effective damping changes sign\nis calculated from the condition of negative dissipation ˙L(∂R/∂˙L)≤0. In the case of\nsteady precession of AFM vector with a frequency ωaround the equilibrium direction\nL(0), the critical current is given by the expression Icr∝χ⊥αGω/σ, analogous to that\nin FM material.10In contrast to FM, the value of critical current in AFM is sub-\nstantially reduced due tostrong exchange interaction between th emagnetic sublattices\n(AFM susceptibility χ⊥is small).\n•Anefficientenergypumpingtakesplacefortheprecessional motion only, i.e., when L⊥\n˙L. Linear oscillations of AFM vector (with L∝ba∇dbl˙L) are always dissipative, ˙L(∂R/∂˙L)≥\n0.\n•UnlikeFM, the presence of spin-polarized current may change spin-wave spectra of\nAFMandthusgiverisetoinstabilityandakindofspin-floptransitionint hecasewhen\nFM and AFM easy axes are parallel, p∝ba∇dblL(0). As can be seen from (2), small deviations\nδL⊥⊥L(0)oriented perpendicular toequilibrium vector L(0)induce ageneralized force\nF=−∂R/∂˙L=p×δL⊥(Iσ/4gM0). This force is a linear function of δL⊥and thus\nmay compete with the restoring force produced by the magnetic an isotropy field.\nIn the next section we consider the last case in more details.\n4IV. CURRENT-INDUCED INSTABILITY\nThetypical AFMmetalused inspin-valves canbethought ofasan“e asy-plane” AFMbe-\ncause ofi) a very small anisotropy of bulk materials and ii) possible out-of-plane anisotropy\nproduced by the shape and interfacial interactions. Let equilibrium orientation of AFM vec-\ntorL(0)be parallel to FM magnetization p∝ba∇dblZin the film plane. In this case the linearized\nLagrange equations for small excitations Lx,Lyare obtained from (1), (2) as follows:\n¨Lx+αG˙Lx−c2∇2Lx+ω2\nx(0)Lx−σIgM 0χ−1\n⊥Ly= 0,\n¨Ly+αG˙Ly−c2∇2Ly+ω2\ny(0)Ly+σIgM 0χ−1\n⊥Lx= 0, (3)\nwhere the gaps ωj(0) =g/radicalbig\nKj/χ⊥, (j=x,y) in spin-wave spectra are expressed through\nthe effective anisotropy constants Kx,Ky,c=g/radicalbig\nA/χ⊥is a spin-wave velocity, Xaxis is\ndirected perpendicular to the film plane.\nThe analysis of shows that depending onthe current value Iequations (3), have two types\nof solutions. Below the threshold I≤Ith1≡g|Kx−Ky|/(2M0σ) AFM vector oscillates\naround equilibrium direction with eigenfrequencies\nΩ2\n1,2(k) =1\n2/parenleftbig\nω2\nx+ω2\ny/parenrightbig\n+c2k2±1\n2/parenleftbig\nω2\nx−ω2\ny/parenrightbig/radicalbig\n1−(I/Ith1)2, (4)\nwherekis wave-vector. Both modes are linearly polarized. The greater the current, the\ngreater istheout-of-planecomponent Lx. Energy dissipation is dueto internal frictionsolely\nand thus, equilibrium state with L(0)∝ba∇dblpis stable.\nWith increase of Ithe difference between the frequencies Ω 1and Ω 2decreases until at\nI=Ith1the spectrum became degenerate, Ω 1= Ω2. Polarization of eigen modes can be\neither linear or circular and energy dissipation is governed by two mec hanisms: damping and\npumping. In the interval Ith1≤I≤Ith2≡/radicalbig\nI2\nth1+I2crdamping is stronger that pumping\nand the state with L(0)∝ba∇dblpis stable. The value of critical current\nIcr≡αG/radicalbig\nKyχ⊥\n2M0σ=αG\nωy(0)Ky\nKx−KyIth1 (5)\nis calculated from the condition of accurate compensation of two dis sipation mechanisms.\nFor definiteness we assume that in-plane anisotropy Kyis weaker than out-of-plane Kx.\nAtI≥Ith2an amplitude of at least one of the modes grows exponentially with the\ncurrent-dependent increment α=αG(I−Ith2)/Icr. This means that the state with L(0)∝ba∇dblp\n5becomes unstable and the system evolves to a new state, e.g. to an other (nonparallel to\np) equilibrium orientation of AFM vector in the film plane. Such a behaviou r is somehow\nanalogoustospin-floptransitionobserved inAFMsofthe“easy-pla ne” typeunder theaction\nof external magnetic field applied in parallel to AFM vector.\nV. DISCUSSION\nThe described dynamics of AFM in the presence of spin-polarised cur rent differs sub-\nstantially from that in FM materials. The difference can be intuitively un derstood from the\ngeometry of spin rotation (see Figs.2 and 3). In the FM characteris ed by a single magnetic\nvectorMmagnetization has only two degrees of freedom. In the absence of any dissipative\nprocesses magnetic excitations take a form of precessional motio n ofMaround its equi-\nlibrium direction M0(double-line ellipse in Fig.2). The motive force of the precession is\nan effective internal field that keeps magnetization direction along a n easy axis M0(in the\nparticular case, parallel to spin polarisation axis p). Spin torque Tacts in such a way as to\nchange an angle between MandM0, and, correspondingly, energy of excitation. Thus, in\nFM spin torque always acts as an energy source (or drain) and thus its effect is equivalent\nto positive/negative friction. Nondissipative dynamics in FM is possible only in the case of\nprecise balance between the torque-induced pumping and internal damping.\nM0, p \nM \nT \nFigure 2: (Color online) Rotation of magnetization Munder the action of spin torque.\nAFM with two magnetic sublattices has more degrees of freedom. In the absence of\ndissipation the low-energy excitations correspond to coherent pr ecession of both sublattice\nmagnetizations M1andM2(double-line ellipses in Fig.3). The effective internal fields rotate\n6magnetizations in opposite directions so that an AFM Lvector can oscillate within a plane.\nSpin torques T1andT2turn both magnetizations M1andM2in the same direction\n(“up” in Fig.3).\nT1 \nT2 M10 , p \nM20  L \nM2 M1 \nFigure 3: (Color online) Rotation of sublattice magnetizat ionsM1,2and AFM vector Lunder the\naction of spin torques T1,2.\nSo, for one sublattice an angle between the excited and equilibrium or ientations increases\nand for another sublattice decreases. This means that for a cert ain relation between spin\ncurrent and internal field (defined by the magnetic anisotropy con stantsKx,y) corresponding\nchanges inenergy could betotally compensated andtorque-induce d motion is nondissipative\neven in the absence of the internal damping.\nThe described (“nondissipative”) influence of spin-current on AFM below the threshold\ncurrentI < I th1is to a certain extent analogous to the affect of an external magne tic\nfield applied in parallel to L(0). Both the magnetic field and spin-current may give rise\nto the softening of one of the spin-wave modes and cause spin-flop transition or transition\nto dynamically stable stationary state. Each effect is insensitive to t he reversal of field\ndirection, spin polarization and current direction. On the contrary , due to the difference of\nsymmetry properties and depending on mutual orientation of field, spin and current flow,\n7combined application of the magnetic field and spin-polarized current , may give rise to an\nenhancement or to reduction of threshold current and spin-flop fi eld. Detailed analysis of\nthis situation is beyond the scope of the paper.\nThe dynamics obtained in the framework of a very simple “toy” model is nevertheless\nqualitatively consistent with the observed5direct effect of electron current on the magnetic\nstate of FeMn, namely, irreversible switching of spin-valve structu re at a threshold current\nI∝5÷7.5 mA. The effect was observed in the presence of the external mag netic field\nH∝0.1T.Theauthorsattributedthisbehaviorto“reorientationsofma gneticconfiguration\nof FeMn among a few metastable states”. We think that the reason of reorientation can be\nthe current-induced instability described above. External field ap plied in-parallel to L(0)is\na source of additional magnetic anisotropy.\nThe value of threshold current can be roughly estimated using the v alue of FeMn bulk\nsusceptibility11χ⊥= 10−5(SI units), magnetization 2 µ0M0=0.1 T and typical AFM layer\ndimensions5120x60x1.5 nm3. We assume that out-of-plane anisotropy Kxcan be as large\nas 105J/m3due to the interface effects (e.g., coupling strength between Co/F eMn layers is\nestimated12as 10−4J/m2and monolayer thickness is ∝3·10−10m). Altimately, in the case\nof 100% spin-polarization efficiency, we get Ith1∝KxVe/η∝10 mA. Threshold current Ith2,\nwhich separates reversible/irreversible rotation of vector Lis of the same order of value, at\nleast in the case of the pronounced anisotropy ( Kx−Ky∝Ky) and quality factor of AFM\nresonance ωy/αG≥10. Really, in this case, as follows from (5), critical current Icr≤0.1Ith1\nandIth2≈Ith1∝10 mA that agrees in order of value with experimental results.\nVI. CONCLUSIONS\nIn summary, we have considered the dynamics of a compensated AF M with localized\nspins in the presence of spin-polarized current. In contrast to FM , spin current is not only\na source of “negative friction” but it also acts as an “effective field” that modifies spin-\nwave modes and gives rise to the loss of stability of the state with par allel orientation of\nspin polarization and AFM vector. Predictions of the above “toy” mo del are in qualitative\nagreement with experimentally observed influence of spin current o nAFMFeMn. Estimated\nvalues of the threshold current are of the same order of value as c haracteristic currents in\nthe experiment.5\n8Acknowledgments\nH.G. is grateful to Prof. A. N. Slavin for discussions and drawing her attention to the\nproblem considered in the paper.\n1J. Slonczewski, JMMM 159, L1 (1996).\n2P. E. Z. Yu. V. Gulyaev and E. M. Epshtein, JETP Letters 84, 344 (2006).\n3M. D. Stiles, J. Xiao, and A. Zangwill, Phys. Rev. B 69, 054408 (2004).\n4Z. Wei, A. Sharma, A. S. Nunez, P. M. Haney, R. A. Duine, J. Bass , A. H. MacDonald, and\nM. Tsoi, Physical Review Letters 98, 116603 (2007).\n5S. Urazhdin and N. Anthony, Phys. Rev. Lett. 99, 046602 (2007).\n6A. Nunez, R. Duine, P. Haney, and A. MacDonald, Phys. Rev. B 73, 214426 (2006).\n7P. M. Haney and A. H. MacDonald, cond-mat/0708.3231v1 (2007 ).\n8V. G. Bar’yakhtar, B. A. Ivanov, and M. V. Chetkin, Uspekhi Fi zicheskikh Nauk 28, 1425\n(1986).\n9V. G. Bar’yakhtar, JETP 87, 1501 (1984).\n10A. N. Slavin and V. S. Tiberkevich, Phys. Rev. B 72, 094428 (2005).\n11Y. Endoh and Y. Ishikawa, J. Phys. Soc. Japan 30, 1614 (1971).\n12W. Kuch, F. Offi, L. I. Chelaru, M. Kotsugi, K. Fukumoto, and J. K irschner, Phys. Rev. B 65,\n140408 (2002).\n9"    },    {        "title": "1612.09019v1.Theory_of_electron_transport_and_magnetization_dynamics_in_metallic_ferromagnets.pdf",        "content": "May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 1\n1\nTheory of electron transport and magnetization dynamics in\nmetallic ferromagnets\nGen Tatara\nRIKEN Center for Emergent Matter Science (CEMS)\n2-1 Hirosawa, Wako, Saitama 351-0198, Japan\n\u0003E-mail: gen.tatara@riken.jp\nMagnetic electric e\u000bects in ferromagnetic metals are discussed from the view-\npoint of e\u000bective spin electromagnetic \feld that couples to conduction electron\nspin. The e\u000bective \feld in the adiabatic limit is the spin Berry's phase in space\nand time, and it leads to spin motive force (voltage generated by magnetization\ndynamics) and topological Hall e\u000bect due to spin chirality. Its gauge coupling\nto spin current describes the spin transfer e\u000bect, where magnetization structure\nis driven by an applied spin current. The idea of e\u000bective gauge \feld can be\nextended to include spin relaxation and Rashba spin-orbit interaction. Voltage\ngeneration by the inverse Edelstein e\u000bect in junctions is interpreted as due to\nthe electric component of Rashba-induced spin gauge \feld. The spin gauge\n\feld arising from the Rashba interaction turns out to coincides with troidal\nmoment, and causes asymmetric light propagation (directional dichroism) as a\nresult of the Doppler shift. Rashba conductor without magnetization is shown\nto be natural metamaterial exhibiting negative refraction.\nKeywords : Spintronics, Spin-charge conversion, Gauge \feld, Rashba spin-orbit\ninteraction\n1. Introduction\nOur technology is based on various electromagnetic phenomena. For de-\nsigning electronics devices, the Maxwell's equation is therefore of essential\nimportance. The mathematical structure of the electromagnetic \feld is\ngoverned by a U(1) gauge symmetry, i.e., an invariance of physical laws\nunder phase transformations. The gauge symmetry is equivalent to the\nconservation of the electric charge, and was established when a symme-\ntry breaking of uni\fed force occured immediately after the big bang. The\nbeautiful mathematical structure of charge electromagnetism was therefore\ndetermined when our universe started, and there is no way to modify its\nlaws.\nInterestingly, charge electromagnetism is not the only electromagnetismarXiv:1612.09019v1  [cond-mat.mes-hall]  29 Dec 2016May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 2\n2\nFig. 1. The spin of a conduction electron is rotated by a strong sdinteraction with\nmagnetization as it moves in the presence of a magnetization texture, resulting in a spin\ngauge \feld. Magnetization texture is therefore equivalent to an e\u000bective electromagnetic\n\feld for conduction electron spin.\nallowed in the nature. In fact, electromagnetism arises whenever there is a\nU(1) gauge symmetry associated with conservation of some e\u000bective charge.\nIn solids, there are several systems which have the U(1) gauge symmetry as\na good approximation. Solids could thus display several types of e\u000bective\nelectromagnetic \felds. A typical example is a ferromagnetic metal. In fer-\nromagnetic metals, conduction electron spin (mostly selectron) is coupled\nto the magnetization (or localized spins of delectrons) by an interaction\ncalled thesdinteraction, which tends to align the electron spin parallel (or\nanti-parallel) to the localized spin. This interaction is strong in most 3 d\nferromagnetic metals, and as a result, conduction electron's spin originally\nconsisting of three components, reduces to a single component along the lo-\ncalized spin direction. The remaining component is invariant under a phase\ntransformation, i.e., has a U(1) gauge symmetry just like the electric charge\ndoes. A spin electromagnetic \feld thus emerges that couples to conduction\nelectron's spin.\nThe subject of the present paper is this spin electromagnetic \feld. Spin\nelectromagnetic \feld drives electron's spin, and thus plays essential roles\nin spintronics. There is a gauge \feld for the spin electromagnetic \feld, a\nspin gauge \feld, which couples to spin current of the conduction electron.\nThe gauge coupling describes the e\u000bects of spin current on the localized spin\ndynamics. As we shall see, when a spin-polarized electric current is applied,\nthe adiabatic spin gauge \feld leads to spin-transfer torque and moves the\nmagnetization structure (Sec. 6). The world of spin electromagnetic \feld\nis richer than that of electric charge, since the electron's spin in solids is\nunder in\ruence of various interactions such as spin-orbit interaction. We\nshall show that even magnetic monopoles can emerge (Sec. 4)May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 3\n3\nFig. 2. Ferromagnetic metals have magnetization and conduct electricity, indicating\nexistence of localized spins and conduction electrons.\nA spin electromagnetic \feld was \frst discussed in the context of a volt-\nage generated by a canting of a driven domain wall by L. Berger1, and\nmathematically rigorous formulation was given by G. Volovik2. The idea\nof e\u000bective gauge \feld was shown to be extended to the cases with spin\nrelaxation3, and Rashba interaction4{7.\nSome of the phenomena discussed in this paper overlaps those in the\npaper by R. Raimondi in this lecture series, studied base on the Boltzmann\nequation approach8.\n2. Ferromagnetic metal\nLet us start with a brief introduction of ferromagnetic metals (Fig. 2).\nFerromagnets have magnetization, namely, an ensemble of localized spins.\nDenoting the localized spin as S, the magnetization is M=\u0000~\r\na3S, where\n\r(>0) andaare gyromagnetic ratio and lattice constant, respectively.\nAs the electron has negative charge, the localized spin and magnetization\npoints opposite direction. In 3 dtransition metals, localized spins are aligned\nspins of 3delectrons. Ferromagnetic metals have \fnite conductivity, indi-\ncating that there are conduction electrons, mainly 4 selectrons. The con-\nduction electrons and delectrons are coupled via sdmixing. As a result,\nthere arises an exchange interaction between conduction electron spin, s,\nand localized spin, which reads\nHsd=\u0000JsdS\u0001s; (1)\nwhereJsdrepresents the strength. In this article, the localized spin is\ntreated as classical variable, neglecting the conduction of delectrons.\nThe dynamics of localized spin is described by the Landau-Lifshiz-May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 4\n4\nGilbert (LLG) equation,\n_n=\rB\u0002n+\u000bn\u0002_n; (2)\nwheren\u0011S=Sis a unit vector representing the direction of localized spin,\nBis the total magnetic \feld acting on the spin. The last term of the right\nhand side represents the relaxation (damping) of localized spin, called the\nGilbert damping e\u000bect and \u000bis the Gilbert damping constant. The Gilbert\ndamping constant in most metallic ferromagnets are of the order of 10\u00002.\nWe shall now start studying phenomena arising from the exchange in-\nteraction, Eq. (1), between localized spin and conduction electron.\nFig. 3. Schematic \fgures showing conduction electron injected to a domain wall. (a):\nIn the adiabatic limit, i.e., for a large domain wall width, the electron goes through the\nwall with a spin \rip (left). (b): Non adiabaticity due to \fnite domain wall width leads\nto re\rection and electric resistance (right).\n3. Electron transport through magnetic domain wall :\nphenomenology\nWe consider a ferromagnetic domain wall, which is a structure where lo-\ncalized spins (or magnetization) rotate spatially (Fig. 3). Its thickness,\n\u0015, in typical ferromagnets is \u0015= 10\u0000100nm. Let us consider here what\nhappens when a conduction electron goes through a domain wall. The wall\nis a macroscopic object for electrons, since thickness is much larger than\nthe typical length scale of electron, the Fermi wavelength, 1 =kF, which is\natomic scale in metals. The electron is interacting with localized spin via\nthesdexchange coupling, Eq. (1). We consider the case of positive Jsd,\nbut the sign does not change the scenario. The sdinteraction tends to align\nparallel the localized spin and conduction electron spin. If localized spin\nis spatially uniform, therefore, the conduction electron is also uniformly\npolarized, and electron transport and magnetism are somewhat decoupled.\nInteresting e\u000bects arise if the localized spins are spatially varying like theMay 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 5\n5\nλzV\nFig. 4. Potential energy V(z) for conduction electron with spin !and as a result of\nsdexchange interaction. Dotted lines are the cases neglecting spin \rip inside the wall,\nwhile solid lines are with spin \rip.\ncase of a domain wall. We choose the zaxis along the direction localized\nspins change. The lowest energy direction (magnetic easy axis) for local-\nized spins is chosen as along zaxis. (The mutual direction between the\nlocalized spin and direction of spin change is irrelevant in the case without\nspin-orbit interaction.) The wall in this case is with localized spins inside\nthe wall changing within the plane of localized spin, and such wall is called\nthe N\u0012 eel wall. At z=1the localized spin is Sz=S, and isSz=\u0000Sat\nz=\u00001, and those states are represented a !and , respectively. For\n electron, the potential in the left regime is low because of sdexchange\ninteraction, while that in the right region is high (dotted lines in Fig. 4).\nThat is, the localized spin structure due to a domain wall acts as a spatially\nvarying magnetic \feld, resulting in potential barriers, V!(z) =\u0000JsdSz(z)\nandV (z) =JsdSz(z). Considering the domain wall centered at x= 0\nhaving pro\fle of\nSz(z) =Stanhz\n\u0015; Sx(z) =S\ncoshz\n\u0015; Sy= 0; (3)\nconduction electron's Schr odinger equation with energy Ereads\n\u0014\n\u0000~2\n2md2\ndz2\u0000JsdS\u0012\n\u001bztanhz\n\u0015+\u001bx1\ncoshz\n\u0015\u0013\u0015\n\t =E\t; (4)\n\t(z) = (\t!(z);\t (z)) begin the two-component wave function. If the\nspin direction of the conduction electron is \fxed along the zaxis, the po-\ntential barrier represetned by the term proportional to \u001bzleads to re\rection\nof electron, but in reality, the electron spin can rotate inside the wall as a\nresult of the term proportional to \u001bxin Eq. (4). The mixing of  and!\nelectron leads to the smooth potential barrier plotted as solid lines in Fig.\n4.May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 6\n6\nLet us consider an incident  electron from the left. If the electron\nis slow, the electron spin can keep the lowest energy state by gradually\nrotating its direction inside the wall. This is the adiabatic limit. As there\nis no potential barrier for the electron in this limit, no re\rection arises from\nthe domain wall, resulting in a vanishing resistance (Fig. 3(a)) In contrast,\nif the electron is fast, the electron spin cannot follow the rotation of the\nlocalized spin, resulting in a re\rection and \fnite resistance (Fig. 3(b)). The\ncondition for slow and fast is determined by the relation between the time\nfor the electron to pass the wall and the time for electron spin rotation.\nThe former is \u0015=vFfor electron with Fermi velocity vF(=~kF=m) (spin-\ndependence of the Fermi wave vector is neglected and mis the electron\nmass). The latter time is ~=JsdS, as the electron spin is rotated by the sd\nexchange interaction in the wall. Therefore, if\n\u0015\nvF\u001d~\nJsdS; (5)\nis satis\fed, the electron is in the adiabatic limit9. The condition of adia-\nbatic limit here is the case of clean metal (long mean free path); In dirty\nmetals, it is modi\fed10,11.\nThe transmission of electron through a domain wall was calculated by G.\nG. Cabrera and L. M. Falicov12, and its physical aspects were discussed by\nL. Berger1,13. Linear response formulation and scattering approach were\npresented in Refs.14{16. The adiabaticity condition was discussed by X.\nWaintal and M. Viret9.\n3.1. Spin-transfer e\u000bect\nAs we discussed above, in the adiabatic limit, the electron spin gets rotated\nafter passing through the wall (Fig. 3(a)). The change of spin angular\nmomentum, 2\u0002~\n2=~, must be absorbed by the localized spins. (Angular\nmomentum dissipation as a result of spin relaxation is slow compared to\nthe exchange of the angular momentum via the sdexchange interaction.)\nTo absorb the spin change of ~, the domain wall must shift to the right,\nresulting in an increase of the spins  . We consider for simplicity the\ncase of cubic lattice with lattice constant a. The distance of the wall shift\n\u0001Xnecessary to absorb the electron's spin angular momentum of ~is then\n[~=(2~S)]a(Fig. 5)). If we apply a spin-polarized current through the wall\nwith the density js(spin current density is de\fned to have the same unit of\nA/m2as the electric current density.) The rate of the angular momentum\nchange of the conduction electron per unit time and area is ~js=e. As theMay 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 7\n7\n∆X\n∆XSa∆XSa−\nFig. 5. The shift of the domain wall by a distance \u0001 Xresults in a change of the spin\nof the localized spins\u0001X\naS\u0000\u0010\n\u0000\u0001X\naS\u0011\n= 2S\u0001X\na. The angular momentum change is\ntherefore ~if \u0001X=a\n2S.\nnumber of the localized spins in the unit area is 1 =a2, the wall must keep\nmoving a distance of ( js=e)(a3=2S) per unit time. Namely, when a spin\ncurrent density is applied, the wall moves with the speed of\nvs\u0011a3\n2eSjs: (6)\nThis e\u000bect was pointed out by L. Berger1in 1986, and is now called the\nspin-transfer e\u000bect after the papers by J. Slonczewski17.\nFrom the above considerations in the adiabatic limit, we have found that\na domain wall is driven by spin-polarized current, while the electrons do not\nget re\rected and no resistance arises from the wall. These two facts naively\nseem inconsistent, but are direct consequence of the fact that a domain\nwall is a composite structure having both linear momentum and angular\nmomentum. The adiabatic limit is the limit where angular momentum is\ntransfered between the electron and the wall, while no linear momentum is\ntransfered.\n4. Adiabatic phase of electron spin\nTransport of conduction electrons in the adiabatic (strong sd) limit is theo-\nretically studied by calculating the quantum mechanical phase attached to\nthe wave function of electron spin. We here consider a conduction electron\nhopping from a site rto a neighboring site at r0\u0011r+a(ais a vector\nconnecting neighboring sites)(Fig. 6). The localized spin direction at those\nsites aren(r)\u0011nandn(r+a)\u0011n0, respectively, and the electron's waveMay 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 8\n8\nFig. 6. Left: A Unitary transformation U(\u0012;\u001e) relates the two spin con\fgurations j\"i\nandjniasjni=Uj\"i. Right: The overlap of the wave functions at sites randr0is\nhn(r)jn(r0)i=h\"jU(r0)\u00001U(r)j\"i.\nfunction at the two sites are\njni= cos\u0012\n2j\"i+ sin\u0012\n2ei\u001ej#i\njn0i= cos\u00120\n2j\"i+ sin\u00120\n2ei\u001e0j#i; (7)\nwhere\u0012,\u001eand\u00120,\u001e0are the polar angle of n(r) andn(r0), respectively (Fig.\n6). The wave functions are concisely written by use of matrices, U(r) and\nU(r0), which rotates the spin state j\"itojni(Fig. 6), asjni=U(r)j\"iand\njn0i=U(r0)j\"i. The rotation matrix is given by18(neglecting irrelevant\nphase factors)\nU(r) =ei\n2(\u001e\u0000\u0019)\u001bzei\n2\u0012\u001bye\u0000i\n2(\u001e\u0000\u0019)\u001bz=\u0012cos\u0012\n2sin\u0012\n2ei\u001e\n\u0000sin\u0012\n2e\u0000i\u001ecos\u0012\n2\u0013\n:(8)\nThe overlap of the electron wave functions at the two sites is thus hn0jni=\nh\"jU(r0)\u00001U(r)j\"i. When localized spin texture is slowly varying, we\ncan expand the matrix product with respect to aasU(r0)\u00001U(r) = 1\u0000\nU(r)\u00001(a\u0001r)U(r) +O(a2) to obtain\nhn0jni'1\u0000h\"jU(r)\u00001(a\u0001r)U(r)j\"i'ei'; (9)\nwhere\n'\u0011ia\u0001h\"jU(r)\u00001rU(r)j\"i\u0011a\u0001As: (10)\nSince (U\u00001rU)y=\u0000U\u00001rU,'is real. A vector Ashere plays a role of\na gauge \feld, similarly to that of the electromagnetism, and it is called\n(adiabatic) spin gauge \feld. By use of Eq. (8), this gauge \feld reads (the\nfactor of1\n2represents the magnitude of electron spin)\nAs=~\n2e(1\u0000cos\u0012)r\u001e: (11)May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 9\n9\nFor a general path C, the phase is written as an integral along Cas\n'=e\n~Z\nCdr\u0001As: (12)\nExistence of path-dependent phase means that there is an e\u000bective magnetic\n\feld,Bs, as seen by rewriting the integral over a closed path by use of the\nStokes theorem as\n'=e\n~Z\nSdS\u0001Bs; (13)\nwhere\nBs\u0011r\u0002As; (14)\nrepresents the curvature or e\u000bective magnetic \feld. This phase ', arising\nfrom strong sdinteraction, couples to electron spin, and is called the spin\nBerry's phase. Time-derivative of phase is equivalent to a voltage, and thus\nwe have e\u000bective electric \feld de\fned by\n_'=\u0000e\n~Z\nCdr\u0001Es; (15)\nwhere\nEs\u0011\u0000 _As; (16)\n(For a gauge invariant expression of Es, we need to include the time com-\nponent of the gauge \feld, As;019.) In terms of vector nthe e\u000bective \felds\nread\nEs;i=\u0000~\n2en\u0001(_n\u0002rin)\nBs;i=~\n4eX\njk\u000fijkn\u0001(rjn\u0002rkn): (17)\nThese two \felds couple to the electron spin and are called spin electromag-\nnetic \felds (Asis spin gauge \feld). They satisfy the Faraday's law,\nr\u0002Es+_Bs= 0; (18)\nas a trivial result of their de\fnitions. De\fning the spin magnetic charge as\nr\u0001Bs\u0011\u001am; (19)\nwe see that \u001am= 0 as a local identity, since spin vector with \fxed length\nhas only two independent variables, and thereforeP\nijk\u000fijk(rin)\u0001(rjn\u0002\nrkn) = 0. However, there is a possibility that the volume integral,May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 10\n10\nFig. 7. Magnetization structures, n(r), of a hedgehog monopole having a monopole\ncharge ofQm= 1 and the one with Qm= 2 . At the center, n(r) has a singularity and\nthis gives rise to a \fnite monopole charge.\nEs\nBs\nFig. 8. Spin electric \feld Esand spin magnetic \feld Bsact oppositely for electrons\nwith opposite spin, and thus are useful for generation of spin current.\nQm\u0011R\nd3r\u001am, is \fnite; In fact, using the Gauss's law we can write (R\ndS\nrepresents a surface integral)\nQm=h\n4\u0019eZ\ndS\u0001\n; (20)\nand it follows that Qm=h\ne\u0002integer since1\n4\u0019R\ndS\u0001\nis a winding number,\nan integer, of a mapping from a sphere in the coordinate space to a sphere\nin spin space. If the mapping is topologically non-trivial as a result of a\nsingularity, the monopole charge is \fnite. Typical nontrivial structures of n\nare shown in Fig. 7. The singular structure with a single monopole charge\nis called the hedgehog monopole.\nThe Faraday's law similarly reads ( r\u0002Es)i+_Bsi=~\n4eP\nijk\u000fijk_n\u0001\n(rjn\u0002rkn)\u0011jm, which vanishes locally but is \fnite when integrated,\nindicating that topological monopole current jmexists.\nThe other two Maxwell's equations describing r\u0001Esandr\u0002Bsare\nderived by evaluating the induced spin density and spin current based on\nlinear response theory4,19.\n5. Detection of spin electromagnetic \felds\nThe spin electromagnetic \felds are real \felds detectable in transport mea-\nsurements. They couples to the spin polarization of the electrons (Fig.May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 11\n11\n8), and because spin density and spin current in ferromagnetic metals is\nalways accompanied with electric charge and current, respectively, the ef-\nfects of the spin magnetic \felds are observable in electric measurements.\nThe electric component Esis directly observable as a voltage generation\nfrom magnetization dynamics, and the voltage signals of \u0016V order have\nbeen observed for the motion of domain walls and vortices20,21. The spin\nmagnetic \feld causes an anomalous Hall e\u000bect of spin, i.e., the spin Hall\ne\u000bect called the topological Hall e\u000bect. The spin electric \feld arises if mag-\nnetization structure carrying spin magnetic \feld becomes dynamical due to\nthe Lorentz force from Bsaccording to Es=v\u0002Bs, wherevdenotes the\nelectron spin's velocity. The topological Hall e\u000bect due to skyrmion lattice\nturned out to induce Hall resistivity of 4n\ncm22,23. Although those signals\nare not large, existence of spin electromagnetic \felds is thus con\frmed ex-\nperimentally. It was recently shown theoretically that spin magnetic \feld\ncouples to helicity of circularly polarized light (topological inverse Faraday\ne\u000bect)24, and an optical detection is thus possible.\n6. E\u000bects of spin gauge \feld on magnetization dynamics\nAs discussed in the previous section, the spin gauge \feld are measured by\ntransport experiments. Here we study the opposite e\u000bect, the e\u000bects of spin\ngauge \feld on magnetization dynamics when spin current is applied. The\nspin gauge \feld is expected to couple to the spin current of the electron,\njs, via the minimal coupling,\nHAs=Z\nd3r\u0014\n\u0000~\neAs\u0001js+n~2\n2m(As)2\u00002~As;0\u001as\u0015\n; (21)\nwherenis the electron density, and \u001as=1\n2(n+\u0000n\u0000) is the electron spin\ndensity,n\u001b(\u001b=\u0006) representing the density of electron with spin \u001b. The\n\feldAs;0is the time component of spin gauge \feld (Eq. (11) with spa-\ntial derivative replaced by time derivative). (For rigorous derivation of the\ncoupling, see Eqs. (38)(39).) As the spin gauge \feld is written in terms of\nlocalized spin variables, \u0012and\u001e, as a result of Eq. (11), this interaction\ndescribes how the spin current and electron density a\u000bects the magnetiza-\ntion dynamics. Here we study the adiabatic limit, where the contribution\nsecond order in As(the second term of the right hand side of Eq. (21) is\nneglected. Including the gauge interaction, the Lagrangian for the localized\nspin reads\nLS=Z\nd3r\u00142\na3As;0\u0000\nS+\u001asa3\u0001\n\u0000~\neAs\u0001js\u0015\n\u0000HS; (22)May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 12\n12\nwhereHSis the Hamiltonian. We see that the magnitude of localized spin is\nmodi\fed to be the e\u000bective one S\u0011S+\u001asa3including the spin polarization\nof the conduction electron. Writing the gauge \feld terms explicitly, we have\nLS=Z\nd3r\u0014\nS(1\u0000cos\u0012)\u0012@\n@t\u0000vs\u0001r\u0013\n\u001e\u0015\n\u0000HS; (23)\nwherevs\u0011a3\n2eSjs. The velocity vshere agrees with the phenomenological\none, Eq. (6), if electron spin polarization is neglected (i.e., if S=S).\nIn the adiabatic limit, therefore, the time-derivative of the localized spin\nin the equation of motion is replaced by the Galilean invariant form with\na moving velocity of vswhen a spin current is present. The equation of\nmotion derived from the Lagrangian (23) reads\n\u0012@\n@t\u0000vs\u0001r\u0013\nS=\u0000\rBS\u0002S; (24)\nwhereBSis the e\u000bective magnetic \feld due to HS. From Eq. (24), it is ob-\nvious that the magnetization structure \rows with velocity vs, and this e\u000bect\nis in fact the spin-transfer e\u000bect discussed phenomenologically in Sec. 3.1.\nIt should be noted that the e\u000bect is mathematically represented by a simple\ngauge interaction of Eq. (22). The equation of motion (24) is the Landau-\nLifshiz-Gilbert (LLG) equation including adiabatic spin-transfer e\u000bect. It\nwas theoretically demonstrated that the spin-transfer torque induces a red\nshift of spin wave, resulting in instability of uniform ferromagnetic state\nunder spin-polarized current25.\nIn reality, there is nonadiabatic contribution described by spin-\rip inter-\nactions. Such contribution leads to a mixing of the electron spins resulting\nin a scattering of the conduction electron and a \fnite resistance due to the\nmagnetization structure15,16. This scattering gives rise to a force on the\nmagnetization structure as a counter action26.\nAs we have seen, the concept of adiabatic spin gauge \feld is useful to give\na uni\fed description of both electron transport properties in the presence\nof magnetization structure and the magnetization dynamics in the presence\nof spin-polarized current.\n7. Field-theoretic description\nSo far we discussed that an e\u000bective spin gauge \feld emerges by looking\ninto the quantum mechanical phase factor attached to conduction electron\nin the presence of magnetization structures. Existence of e\u000bective gauge\n\feld is straightforwardly seen in \feld-theoretic description.May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 13\n13\nA \feld-theoretical description is based on the Lagrangian of the system,\n^L=i~Z\nd3rX\n\u001b^cy\n\u001b_^c\u001b\u0000^H; (25)\nwhere ^H=^K+^Hsdis the \feld Hamiltonian. Here\n^K=Z\nd3rX\n\u001b^cy\n\u001b\u0012\n\u0000~2\n2mr2\u0013\n^c\u001b=~2\n2mX\n\u001bZ\nd3r(r^cy\n\u001b)(r^c\u001b) (26)\ndescribes the free electron part in terms of \feld operators for conduction\nelectron, ^c\u001band ^cy\n\u001b, where\u001b=\u0006denotes spin. The sdexchange interaction\nis represented by\n^Hsd=\u0000JsdS\n2Z\nd3r^cy(n\u0001\u001b)^c: (27)\nWe are interested in the case where n(r;t) changes in space and time slowly\ncompared to the electron's momentum and energy scales. How the electron\n'feels' when \rowing through such slowly varying structure is described by\nintroducing a rotating frame where the sdexchange interaction is locally\ndiagonalized. In Sec. 4, we introduced a unitary matrix U(r;t), and this\nmatrix is used here to introduce a new electron operator as\n^a(r;t) =U(r;t)^c(r;t): (28)\nThe new operator ^ adescribes the low energy dynamics for the case of\nstrongsdexchange interaction. In fact, the sdexchange interaction for this\nelectron is diagonalized to be\n^Hsd=\u0000MZ\nd3r^ay\u001bz^a; (29)\nwhereM\u0011JsdS\n2. Instead, the kinetic term for the new electron is modi\fed,\nbecause derivative of the electron \feld is modi\fed as\nr^c=U(r+iAs)^a; (30)\nwhere\nAs\u0011\u0000iUyrU: (31)\nHereAsis a 2\u00022 matrix, whose componets are represented by using Pauli\nmatrices as\nAs;i=X\n\u000b=x;y;zA\u000b\ns;i\u001b\u000b: (32)May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 14\n14\nEquation (30) indicates that the new electron \feld ^ ais interacting with an\ne\u000bective gauge \feld, As. This gauge \feld has three components, is non-\ncommutative and is called the SU(2) gauge \feld. The three components\nexplicitly read\n0\n@Ax\ns;\u0016\nAy\ns;\u0016\nAz\ns;\u00161\nA=1\n20\n@\u0000@\u0016\u0012sin\u001e\u0000sin\u0012cos\u001e@\u0016\u001e\n@\u0016\u0012cos\u001e\u0000sin\u0012sin\u001e@\u0016\u001e\n(1\u0000cos\u0012)@\u0016\u001e1\nA: (33)\nDue to Eq. (30), the kinetic term ^Kis written in terms of ^ aelectron as\n^K=~2\n2mZ\nd3r[(r\u0000iAs)^ay][(r+iAs)^a]: (34)\nSimilarly, time-component of the gauge \feld\nAs;0\u0011\u0000iUy@tU; (35)\narises from the time-derivative term ( i~^cy\n\u001b_^c\u001b) of the Lagrangian (25). The\nLagrangian in terms of ^ aelectron therefore reads\n^L=Z\nd3r\u0014\ni~^ay_^a\u0000~2\n2mjr^aj2+\u000fF^ay^a+M^ay\u001bz^a\n+i~2\n2mX\ni(^ayAs;iri^a\u0000(ri^ay)As;i^a)\u0000~2\n2mA2\ns^ay^a\u0000~^ayAs;0^a#\n:(36)\nIf we introduce electron density operator, ^ n\u0011^ay^a, and operators for spin\ndnesity and spin current density as\n^\u001as\u000b\u00111\n2^ay\u001b\u000b^a;^j\u000b\ns;i\u0011\u0000i\n2m^ay$\nri\u001b\u000b^a\u0011\u0000i\n2m\u0002\n^ay\u001b\u000b(ri^a)\u0000(ri^ay)\u001b\u000b^a\u0003\n;\n(37)\nit reads\n^L=Z\nd3r\u0014\ni~^ay_^a\u0000~2\n2mjr^aj2+\u000fF^ay^a+M^ay\u001bz^a\u0000^j\u000b\ns;iA\u000b\ns;i\u0000~2\n2mA2\ns^n\u0000^\u001as\u000bA\u000b\ns;0\u0015\n:\n(38)\nIn the case of M=\u000fF\u001d1 (largeJsd), the electron with spin #has high\nenergy because of strong spin splitting, Eq. (29), and is neglected. In\nthis case, only the zcomponent of the gauge \feld, Az\ns;i, survives. This\ncomponent is thus essentially a U(1) gauge \feld, which coincides with the\nU(1) gauge \feld we have obtained from the argument of electron's phase\nfactor, namely, As=Az\ns. The total Hamiltonian in the limit of large JsdMay 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 15\n15\ntherefore reduces to the one for a charged particle in the presence of a U(1)\ngauge \feldAs;\n^H=Z\nd3r\u0014~2\n2m[(r\u0000iAs)^ay\n\"][(r+iAs)^a\"]\u0000JsdS\n2^ay\n\"^a\"\u0015\n:(39)\nThe \feld-theoretic method present here is highly useful, as it leads to a\nconclusion of the existence of an e\u000bective gauge \feld for spin simply by\ncarrying out a unitary transformation to diagonalize strong sdexchange\ninteraction.\n8. Non-adiabaticity and spin relaxation\nIn reality, there is a deviation from the adiabatic limit we have considered\nso far. One origin is the fact that the magnetization structure is not in the\nslowly varying limit, but has a \fnite length scale of spatial modulation. This\ne\u000bect, we call the non adiabaticity, leads to re\rection of conduction electron\nby magnetization structures as in Fig. 3(b), resulting in a force on the\nmagnetization structure when an electric current is applied13,26. In terms of\ntorque, the e\u000bect of the force due to re\rection is represented by a non-local\ntorque, as it arises from \fnite momentum transfer27. Another e\u000bect we need\nto take into account is the relaxation (damping) of spin schematically shown\nin the Fig. 9. In metallic ferromagnets, the damping mostly arises from\nthe spin-orbit interaction, as seen from the fact that the Gilbert damping\nparameter\u000band thegvalue has a correlation of \u000b/(g\u00002)2as shown\nin Ref.28. Spin relaxation generates a torque perpendicular to the motion\nof the spin, resulting in a canting of the precession axis. Similarly, when a\nspin currentjsis applied, the spin relaxation thus was argued to induce a\ntorque perpendicular to the spin-transfer torque, i.e.,\n\u001c\f\u0011\fa3\n2en\u0002(js\u0001r)n; (40)\nwhere\fis a coe\u000ecient representing the e\u000bect of spin relaxation29,30.\nThose e\u000bects of non adiabaticity and spin relaxation can be calculated\nfrom a microscopic viewpoint27,31. Let us go back to the LLG equation for\nlocalized spin interacting with conduction electron spin via the sdexchange\ninteraction. The total Hamiltonian is HS\u0000MP\nrn(r)\u0001\u001b\u0000He, whereHS\nandHeare the Hamiltonian for localized spin and conduction electron,\nrespectively. The equation of motion for localized spin is given by\n_n=\rBS\u0002n+\rBe\u0002n; (41)May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 16\n16\nFig. 9. Spin relaxation induces a torque perpendicular to the spin motion and let the\nspin relax to the stable direction along the external magnetic \feld.\nwhere\rBS\u00111\n~\u000eHS\n\u000enand\n\rBe\u00111\n~\u000eHe\n\u000en=\u0000M\n~h\u001bi; (42)\nare the e\u000bective magnetic \feld arising from the localized spin and conduc-\ntion electron, respectively. The \feld Beis represented by the expectation\nvalue of electron spin density, h\u001bi, and all the e\u000bects from the conduction\nelectron is included in this \feld; Equation (41) is exact if h\u001biis evaluated\nexactly. Field theoretic approach is suitable for a systematic evaluation of\nthe electron spin density. We move to a rotated frame where the electron\nspin is described choosing the local zaxis along the localized spin. In the\ncase we are interested, namely, when the e\u000bect of non adiabaticity and\ndamping are weak, these e\u000bects are treated perturbatively.\nThe spin density in the laboratory frame is written in terms of the spin\nin the rotated frame ~sassi=Rij~sj, where\nRij\u00112mimj\u0000\u000eij; (43)\nis a rotation matrix, m\u0011(sin\u0012\n2cos\u001e;sin\u0012\n2sin\u001e;cos\u0012\n2) being the vector\nwhich de\fne the unitary rotation. The perpendicular components (denoted\nby?) of electron spin density in the rotated frame are calculated as11\n~s?=\u00002\u001as\nMA?\ns;0\u0000a3\neMjs\u0001A?\ns\u0000\u000bsr\nM(^z\u0002A?\ns;0)\u0000\fsr\neM(^z\u0002(js\u0001A?\ns)):(44)\nThe e\u000bect of spin relaxation is included in \u000bsrand\fsr=~=(2M\u001cs), both\nproportional to the spin relaxation time, \u001cs31. The \frst term of Eq. (44)\nrepresents the renormalization of the localized spin as a result of electron\nspin polarization and the second term, induced in the presence of applied\nspin current, describes the adiabatic spin-transfer torque. Using the iden-\ntity\nRij(As;\u0016)?\nj=\u00001\n2(n\u0002@\u0016n)i; Rij(^z\u0002A?\ns;\u0016)j=1\n2@\u0016ni; (45)May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 17\n17\nwe see that Eq. (44) leads to\n(1 +\u001asa3)_n=\u000bn\u0002_n\u0000a3\n2e(js\u0001r)n\u0000\fa3\n2e[n\u0002(js\u0001r)n] +\rBS\u0002n;\n(46)\nwhich is the LLG equation taking into account the torque due to electrons.\nHere\u000b\u0011\u000bsrand\f\u0011\fsr, neglecting other origins for Gilbert damping and\nnonadiabatic torque.\n9. Current-driven domain wall motion\nLet us brie\ry discuss dynamics of a domain wall based on the LLG equation\n(46) including the current-induced torques. We consider an one-dimensional\nand rigid wall, neglecting deformation. For a domain wall to be created,\nthe system must have an easy axis magnetic anisotropy energy. We also\ninclude the hard-axis anisotropy energy, which turns out to govern the\ndomain wall motion. Choosing the easy and the hard axises along the z\nand theydirections, respectively, the anisotropy energy is represented by\nthe Hamiltonian\nHK\u0011Zd3r\na3\u0014\n\u0000KS2\n2cos2\u0012+K?S2\n2sin2\u0012sin2\u001e\u0015\n; (47)\nwhereKandK?are the easy- and hard-axis anisotropy energies (both are\npositive). We need to take into account of course the exchange coupling,\nwhich is essential for ferromagnetism, which in the continuum expression\nreads\nHJ\u0011Z\nd3rJS2a2\n2(rn)2: (48)\nThe domain wall solution obtained by minimizing HKandHJis Eq. (3)\nwith\u0015=p\nJ=K. Considering a rigid wall, we assume that K\u001dK?.\nThe low energy dynamics of the wall is then described by two variables\n(called the collective coordinates), the center coordinate of the wall, X(t),\nand the angle \u001e(t) out-of the easy plane11,32. The wall pro\fle including the\ncollective coordinates is\nnz(z;t) = tanhz\u0000X(t)\n\u0015; n\u0006(z;t)\u0011nx\u0006iny=e\u0006i\u001e(t)\ncoshz\u0000X(t)\n\u0015:(49)May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 18\n18\nThe equation of motion for domain wall is obtained by putting the wall\npro\fle (49) in Eq. (46) and integrating over spatial coordinate as\n_\u001e+\u000b_X\n\u0015=P\f\n\u0015~j\n_X\u0000\u000b\u0015_\u001e=\u0000vcsin 2\u001e+P~j; (50)\nwhereP\u0011js=jis spin polarization of the current, and both vc\u0011K?\u0015S\n2~\nand~j\u0011a3\n2eSjhave dimension of velocity.\nWhen\f= 0, the wall velocity when a constant ~jis applied is easily\nobtained as26\n_X=(0 ( ~j <~ji\nc)\njPj\n1+\u000b2q\n~j2\u0000(~jic)2 (~j\u0015~ji\nc)(51)\nand~ji\nc\u0011vc\nPis the intrinsic threshold current density26. Namely, the wall\ncannot move if the applied current is lower than the threshold value. This\nis because the torque supplied by the current is totally absorbed by the wall\nby tilting the out of plane angle to be sin 2 \u001e=P~j=vcwhen the current is\nweak (jP~j=vcj\u00141) and thus the wall cannot move. This e\u000bect is called\nthe intrinsic pinning e\u000bect11. For larger current density, the torque carried\nby the current induces an oscillation of the angle similar to the Walker's\nbreakdown in an applied magnetic \feld, and the wall speed also becomes\nan oscillating function of time.\nWhen nonadiabaticity parameter \fis \fnite, the behavior changes\ngreatly and intrinsic pinning e\u000bect is removed and the wall can move with\nin\fnitesimal applied current as long as there is no extrinsic pinning. In\nfact, when the applied current density is ~j >~ja, where\n~ja\u0011vc\nP\u0000\f\n\u000b; (52)\nthe solution of Eq. (50) is an oscillating function given by11\n_X=\f\n\u000b~j+vc\n1 +\u000b2\u0010~j\n~ja\u00112\n\u00001\n~j\n~ja\u0000sin(2!t\u0000#); (53)\nwhere\n!\u0011vc\n\u0015\u000b\n1 +\u000b2s\u0012~j\n~ja\u00132\n\u00001; sin#\u0011vc\n(\f\n\u000b\u0000P)~j: (54)May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 19\n19\nj jcivw\nβ=0\nβ=0.002\nβ=0.005\nβ=0.015\nβ=0.02\nFig. 10. Time averaged wall velocity vwas function of applied spin-polarized current j\nfor\u000b= 0:01. Intrinsic pinning threshold ji\ncexists only for \f= 0. The current density\nwhere derivative of vwis discontinuous corresponds to ~ja.\nThe time-average of the wall speed is\n_X=\f\n\u000b~j+vc\n1 +\u000b21\n~jaq\n~j2\u0000~j2a: (55)\nFor current density satisfying ~j <~ja, the oscillation in Eq. (53) is replaced\nby an exponential decay in time, and the wall velocity reaches a terminal\nvalue of\n_X!\f\n\u000b~j: (56)\nThe angle of the wall also reaches a terminal value determined by\nsin 2\u001e!\u0012\f\n\u000b\u0000P\u0013~j\nvc: (57)\nThe averaged wall speed (Eq. (55)) is plotted in Fig. 10.\nThe intrinsic pinning is a unique feature of current-driven domain wall,\nas the wall cannot move even in the absence of pinning center. In the unit\nof A/m2, the intrinsic pinning threshold is\nji\nc=eS2\nPa3~K?\u0015: (58)\nFor device applications, this threshold needs to be lowered by reducing the\nhard-axis anisotropy and wall width33. At the same time, the intrinsic\npinning is promising for stable device operations. In fact, in the intrin-\nsic pinning regime, the threshold current and dynamics is insensitive to\nextrinsic pinning and external magnetic \feld26, as was con\frmed experi-\nmentally34. This is due to the fact that the wall dynamics in the intrinsicMay 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 20\n20\npinning regime is governed by a torque (right hand side of the second equa-\ntion of Eq. (50)), which governs the wall velocity _X, while pinning and\nmagnetic \feld induce force, which governs _\u001e; The forces due to sample ir-\nregularity therefore does not modify the motion induced by a torque in the\nintrinsic pinning regime.\nExperimentally, intrinsic pinning is observed in perpendicularly magne-\ntized materials34, perhaps due to relatively low intrinsic pinning threshold,\nwhile materials with in-plane magnetization mostly are in the extrinsic\npinning regime governed by the nonadiabatic parameter \fand extrinsic\npinning. In this regime, the threshold current of the wall motion is given\nby35\nje\nc/Ve\n\f; (59)\nwhereVerepresents strength of extrinsic pinning potential like those gen-\nerated by geometrical notches and defects. Control of nonadiabaticity pa-\nrameter is therefore expected to be useful for driving domain walls at low\ncurrent density.\nOf recent interest from the viewpoint of low current operation is to use\nmultilayer structures. For instance, heavy metal layers turned out to lower\nthe threshold current by exerting a torque as a result of spin Hall e\u000bect36,\nand synthetic antiferromagnets turned out to be suitable for fast domain\nwall motion at low current37,38.\n10. Interface spin-orbit e\u000bects\nPhysics tends to focus on in\fnite systems or bulk system approximated as\nin\fnite, as one of the most important objective of physics is to search for\nbeautiful general law supported by symmetries. In the condensed matter\nphysics today, studying such 'beautiful' systems seems to be insu\u000ecient\nanymore. This is because demands to understand physics of interfaces and\nsurfaces has been increasing rapidly as present devices are in nanoscales\nto meet the needs for fast processing of huge data. Systems with lower\nsymmetry are therefore important subjects of material science today.\nSurfaces and interfaces have no inversion symmetry, and this leads\nto emergence of an antisymmetric exchange interaction (Dzyaloshinskii-\nMoriya interaction)39,40in magnetism . As for electrons, broken inversion\nsymmetry leads to a peculiar spin-orbit interaction, called the Rashba in-\nteraction41, whose Hamiltonian is\nHR=i\u000bR\u0001(r\u0002\u001b); (60)May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 21\n21\nwhere\u001bis the vector of Pauli matrices and \u000bRis a vector representing the\nstrength and direction of the interaction. The form of the interaction is the\none derived directly from the Dirac equation as a relativistic interaction, but\nthe magnitude can be strongly enhanced in solids having heavy elements\ncompared to the vacuum case.\nAs is obvious from the form of the Hamiltonian, the Rashba interaction\ninduces electromagnetic cross correlation e\u000bects where a magnetization and\nan electric current are induced by external electric and magnetic \feld, E\nandB, respectively, like represented as\nM=\rME(\u000bR\u0002E);j=\rjB(\u000bR\u0002B); (61)\nwhere\rMEand\rjBare coe\u000ecients, which generally depend on frequency.\nThe emergence of spin accumulation from the applied electric \feld, men-\ntioned in Ref.41, was studied by Edelstein42in detail, and the e\u000bect is\nsometimes called Edelstein e\u000bect. The generation of electric current by\nmagnetic \feld or magnetization, called the inverse Edelstein e\u000bect43, was\nrecently observed in multilayer of Ag, Bi and a ferromagnet44.\n10.1. E\u000bective magnetic \feld\nEquation (60) indicates that when a current density jis applied, the con-\nduction electron has average momentum of p=m\nenj(nis electron density),\nand thus an e\u000bective magnetic \feld of Be=ma3\n\u0000e~2\r\u000bR\u0002j;acts on the\nconduction electron spin ( \r(=jej\nm) is the gyromagnetic ratio). When the sd\nexchange interaction between the conduction electron and localized spin is\nstrong, this \feld multiplied by the the spin polarization, P, is the \feld act-\ning on the localized spin. Namely, the localized spin feels a current-induced\ne\u000bective magnetic \feld of\nBR=Pma3\n\u0000e~2\r\u000bR\u0002j: (62)\nOne may argue more rigorously using \feld theoretic description. Con-\nsidering the case of sdexchange interaction stronger than the Rashba in-\nteraction, we use a unitary transformation to diagonalize the sdexchange\ninteraction (Eq. (28)). The Rashba interaction in the \feld representation\nthen becomes\nHR=\u0000Z\nd3rm\n~e\u000fijk\u000bR;iRkl~jl\ns;j; (63)\nwhere ~jl\ns;j\u0011\u0000i~e\n2may$\nrj\u001blais the spin current in the rotated frame, Rij\nis given in Eq. (43). Terms containing spatial derivatives of magnetizationMay 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 22\n22\nstructure is neglected, considering the slowly-varying structure. In this\nadiabatic limit, spin current is polarized along the zdirection, i.e., ~jl\ns;j=\n\u000el;zjs. We therefore obtain using Rkz=nk,\nHR=Z\nd3rm\n~ejs\u0001(\u000bR\u0002n); (64)\nwhich results in the same expression as Eq. (62).\nThe strength of the Rashba-induced magnetic \feld is estimated (choos-\ninga= 2\u0017A) asBR= 2\u00021016\u0002\u000bR(Jm)js(A/m2); For a strong Rashba\ninteraction \u000bR= 1 eV \u0017A like at surfaces45,BR= 4\u000210\u00002T atjs= 1011\nA/m2. This \feld appears not very strong, but is su\u000ecient at modify the\nmagnetization dynamics. In fact, for the domain wall motion, when the\nRashba-induced magnetic \feld is along the magnetic easy axis, the \feld is\nequivalent to that of an e\u000bective \fparameter of\n\fR=2m\u0015\n~2\u000bR; (65)\nwhere\u0015is the wall thickness. If \u000bR= 1 eV \u0017A,\fRbecomes extremely large\nlike\fR'250 for\u0015= 50 nm. Note that \farising from spin relaxation is\nthe same order as Gilbert damping constant, namely of the order of 10\u00002.\nSuch a large e\u000bective \fis expected to leads to an extremely fast domain\nwall motion under current46,47.\nExperimentally, it was argued that fast domain wall motion observed\nin Pt/Co/AlO was due to the Rashba interaction48, but the result is later\nassociated with the torque generated by spin Hall e\u000bect in Pt layer36. It\nwas recently shown theoretically that strong Rashba-induced magnetic \feld\nworks as a strong pinning center when introduced locally, and that this\nRashba pinning e\u000bect is useful for highly reliable control of domain walls\nin racetrack memories49.\n10.2. Rashba-induced spin gauge \feld\nSince the interaction (63) is the one coupling to the spin current, the Rashba\ninteraction is regarded as a gauge \feld acting on electron spin as far as the\nlinear order concerns. The gauge \feld de\fned by Eq. (63) is\nAR\u0011\u0000m\ne~(\u000bR\u0002n): (66)May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 23\n23\nα\nnERn n.\njR\n’\nFig. 11. Schematic \fgure depicting spin relaxation contribution of Rashba-induced spin\nelectric \feld E0\nRgenerated by magnetization precession. Electric current jis induced\nas a result of motive force E0\nRin the direction perpendicular to both n\u0002_nand Rashba\n\feld\u000bR.\nExistence of a gauge \feld naturally leads to an e\u000bective electric and mag-\nnetic \feld5,7\nER=\u0000_AR=m\ne~(\u000bR\u0002_n)\nBR=r\u0002AR=\u0000m\ne~r\u0002(\u000bR\u0002n): (67)\nIn the presence of electron spin relaxation, the electric \feld has a perpen-\ndicular component6\nE0\nR=m\ne~\fR[\u000bR\u0002(n\u0002_n)]; (68)\nwhere\fRis a coe\u000ecient representing the strength of spin relaxation. For\nthe case of strong Rashba interaction of \u000bR= 3 eV \u0017A, as realized in Bi/Ag,\nthe magnitude of the electric \feld is jERj=m\ne~\u000bR!= 26kV/m if the\nangular frequency !of magnetization dynamics is 10 GHz. The magnitude\nof relaxation contribution is jE0\nRj\u0018260V/m if \fR= 0:01. The e\u000bective\nmagnetic \feld in the case of spatial length scale of 10 nm is high as well;\nBR\u0018260T.\nThe Rashba-induced electric \felds, ERandE0\nR, are important from the\nviewpoint of spin-charge conversion. In fact, results (67)(68) indicates that\na voltage is generated by a dynamics magnetization if the Rashba interac-\ntion is present, even in the case of spatially uniform magnetization, in sharp\ncontrast to the conventional adiabatic e\u000bective electric \feld from the spin\nBerry's phase of Eq. (17). In the case of a think \flm with Rashba interac-\ntion perpendicular to the plane and with a precessing magnetization, the\ncomponentER/_nhas no DC component, while the relaxation contribu-\ntionE0\nRhas a DC component perpendicular to n\u0002_nkn. The geometry of\nthis (spin-polarized) current pumping e\u000bect, j/E0\nR/\u000bR\u0002n, is thereforeMay 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 24\n24\nthe same as the one expected in the case of inverse Edelstein e\u000bect (Fig.\n11). In the present form, there is a di\u000berence between the Rashba-induced\nelectric \feld e\u000bect and the system in Ref.44, that is, the former assumes\na direct contact between the Rashba interaction and magnetization while\nthey are separated by a Ag spacer in Ref.44. It is expected, however, that\nthe Rashba-induced electric \feld becomes long-ranged and survives in the\npresence of a spacer if we include the electron di\u000busion processes. The spin-\ncharge conversion observed in junctions will then be interpreted as due to\nthe Rashba-induced electromagnetic \feld. For this scenario to be justi\fed,\nit is crucial to con\frm the existence of magnetic component, BR, which can\nbe of the order of 100T. In the setup of Fig. 11, BRis alongn. The \feld\ncan therefore be detected by measuring \\giant\" in-plane spin Hall e\u000bect\nwhen a current is injected perpendicular to the plane.\n11. Application of e\u000bective vector potential theory\n11.1. Anomalous optical properties of Rashba conductor\nThe idea of e\u000bective gauge \feld is useful for extending the discussion to\ninclude other degrees of freedom, like optical properties. In fact, the fact\nthat the Rashba interaction coupled with magnetization leads to an e\u000bec-\ntive vector potential AR(Eq. (66)) for electron spin indicates that the\nexistence of intrinsic spin \row. Such intrinsic \row a\u000bects the optical prop-\nerties, as incident electromagnetic waves get Doppler shift when interacting\nwith \rowing electrons, resulting in a transmission depending on the direc-\ntion (directional dichroism), as was theoretically demonstrated in Refs.50,51.\nThe magnitude of the directional dichroism for the case of wave vector q\nis given byq\u0001(\u000bR\u0002n). The vector ( \u000bR\u0002n) is called in the context of\nmultiferroics the toroidal moment, and it was argued to acts as an e\u000bective\nvector potential for light52.\nIt was shown also that Rashba conductor, even without magnetization,\nshows peculiar optical properties such as negative refraction as a result of\nspin-charge mixing e\u000bects50. In fact, spin-charge mixing e\u000bects of Eq. (61)\nleads to a current generated by applied electric \feld, E, given by\njIE\u0001E=\u0000~\r\u0014EIE[\u000bR\u0002(\u000bR\u0002E)]; (69)\nwhere\u0014EIEis a coe\u000ecient (Fig. 12). As it is opposite to the applied \feld,\nthe mixing e\u000bect results in a softening of the plasma frequency as for the\nEhaving components perpendicular to \u000bR. The electric permitivity of theMay 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 25\n25\nFig. 12. Schematic \fgure showing the cross-correlation e\u000bects in the plane perpen-\ndicular to the Rashba \feld \u000bR. Edelstein e\u000bect (E) generates spin density, sE, from\nthe applied electric \feld, and inverse Edelstein e\u000bect (IE) generates current jIE\u0001Efrom\nmagnetization ME.\nsystem is therefore anisotropic; Choosing \u000bRalong thezaxis, we have\n\"z= 1\u0000!2\np\n!(!+i\u0011); \" x=\"y= 1\u0000!2\nR\n!(!+i\u0011); (70)\nwhere!p=p\ne2ne=\"0mis the bare plasma frequency ( neis the electron\ndensity), and !R\u0011!pp\n1 + ReC(!R)< ! pis the plasma frequency re-\nduced by the spin mixing e\u000bect.50(C(!) represents the correlation function\nrepresenting the Rashba-Edelstein e\u000bect, and its real part is negative.) The\nfrequency region !R< ! < ! pis of interest, as the system is insulating\n(\"z>0) in the direction of the Rashba \feld but metallic in the perpen-\ndicular direction ( \"x<0). The dispersion in this case becomes hyperbolic,\nand the group velocity and phase velocity along qcan have opposite direc-\ntion, resulting in negative refraction. Rashba system is, therefore a natural\nhyperbolic metamaterial53. A great advantage of Rashba conductors are\nthat the metamaterial behavior arises in the infrared or visible light region,\nwhich is not easily accessible in fabricated systems. For instance, in the\ncase of BiTeI with Rashba splitting of \u000b= 3:85 eV \u0017A54, the plasma fre-\nquency is!p= 2:5\u00021014Hz (corresponding to a wavelength of 7 :5\u0016m)\nforne= 8\u00021025m\u00003and\u000fF= 0:2 eV55. We then have !R=!p= 0:77\n(!R= 1:9\u00021014Hz, corresponding to the wavelength of 9 :8\u0016m), and hy-\nperbolic behavior arises in the infrared regime. The directional dichroism\narises in the infrared-red light regime50.\n11.2. Dzyaloshinskii-Moriya interaction\nAnother interesting e\u000bect of spin gauge \feld pointed out recently is to in-\nduce the Dzyaloshinskii-Moriya interaction. Dzyaloshinskii-Moriya (DM)\ninteraction is an antisymmetric exchange interaction between magnetic\natoms that can arise when inversion symmetry is broken. In the continuumMay 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 26\n26\nlimit, it is represented as\nHDM\u0011Z\nd3rD\u000b\ni(rin\u0002n)\u000b; (71)\nwhereDa\niis the strength, \u000bandidenotes the spin and spatial direction,\nrespectively. It was recently discussed theoretically that the interaction is a\nresult of Doppler shift due to an intrinsic spin current generated by broken\ninversion symmetry56. In fact, spin current density, js, which is odd and\neven under spatial inversion and time-reversal, respectively, is induced by\nspin-orbit interaction in systems with broken inversion symmetry. Spatial\nvariation of localized spins observed by the \rowing electron spin is then\ndescribed by a covariant derivative,\nDin=rin+\u0011(js;i\u0002n); (72)\nwhere\u0011is a coe\u000ecient. This covariant derivative leads to the magnetic\nenergy generated by the electron of ( Din)2= (rn)2+ 2\u0011P\nijs;i\u0001(n\u0002\nrin) +O(\u00112). We see that the second term proportional to jsis the DM\ninteraction, and thus the coe\u000ecient is Da\ni/j\u000b\ns;i.\nMore rigorous derivation is performed by deriving an e\u000bective Hamilto-\nnian. The electrons interacting strongly with localized spin is described by\na Lagrangian (38), where A\u000b\ns;iis an SU(2) gauge \feld describing the spatial\nand temporal variation of localized spin. To discuss DM interaction, we\ninclude a spin-orbit interaction with broken inversion symmetry,\nHso=Z\nd3ri\n2cy\u0014\n\u0015i\u0001\u001b !ri\u0015\nc; (73)\nwhere\u0015is a vector representing the broken inversion symmetry. (Multior-\nbital cases are treated similarly56.) As is obvious from this form linear in\nspatial derivative and Pauli matrix, the spin-orbit interaction generates a\nspin current proportional to \u0015. From Eq. (38), the e\u000bective Lagrangian for\nlocalized spin to the linear order in derivative is\nHe\u000b=Z\nd3rX\nia~ja\ns;iAa\ns;i; (74)\nwhere ~ja\ns;i\u0011D^~ja\ns;iE\nis the expectation value of the spin current density in\nthe rotated frame. In terms of the spin current in the laboratory frame,\nja\ns;i, the e\u000bective Hamiltonian reads\nHe\u000b=Z\nd3rDa\ni(rin\u0002n)a; (75)May 25, 2022 20:24 WSPC Proceedings - 9in x 6in tatara page 27\n27\nwhere\nDa\ni\u0011j?;a\ns;i; (76)\nandj?;a\ns;iis a component of ja\ns;iperpendicular to the local magnetization\ndirection,n. We therefore see that the DM coe\u000ecient is indeed given by\nthe expectation value of the spin current density of the conduction elec-\ntrons. 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B 93, p. 235131 (Jun 2016)."    },    {        "title": "1912.10599v1.Majorana_mediated_spin_transport_without_spin_polarization_in_Kitaev_quantum_spin_liquids.pdf",        "content": "Majorana-mediated spin transport without spin polarization in Kitaev quantum spin liquids\nTetsuya Minakawa,1Yuta Murakami,1Akihisa Koga,1and Joji Nasu2\n1Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152- 8551, Japan\n2Department of Physics, Yokohama National University, Hodogaya, Yokohama 240-8501, Japan\n(Dated: December 24, 2019)\nWe study the spin transport through the quantum spin liquid (QSL) by investigating the real-time and real-\nspace dynamics of the Kitaev spin system with a zigzag structure in terms of the time-dependent Majorana\nmean-field theory. After the magnetic field pulse is introduced to one of the edges, the spin moments are excited\nin the opposite edge region although no spin moments are induced in the Kitaev QSL region. This unusual\nspin transport originates from the fact that the S=1=2 spins are fractionalized into the itinerant and localized\nMajorana fermions in the Kitaev system. Although both Majorana fermions are excited by the magnetic pulse,\nonly the itinerant Majorana fermions flow through the bulk regime without the spin excitation, resulting in the\nspin transport in the Kitaev system. We also demonstrate that this phenomenon can be observed even in the\nsystem with the Heisenberg interactions using the exact diagonalization.\nSpin transport without an electric current has attracted not\nonly practical interest in spintronics but also considerable at-\ntention in modern condensed matter physics. In insulating\nmagnets, the carriers of the spin current are conventionally\nconsidered to be magnons, which are elementary excitations\nin a magnetically ordered state [1–4]. By contrast, the possi-\nbility of the spin transport in the quantum spin liquid (QSL)\nhas been discussed recently. One of the typical examples is an\nantiferromagnetic Heisenberg spin-1 /2 chain, where elemen-\ntary excitations are described by the spinon with an S=1=2\nspin. The spin Seebeck experiments for the cuprate Sr 2CuO 3\nhave clarified that the spin current arises even in the QSL [5].\nTherefore, the spinons, instead of the magnons, can be respon-\nsible for the spin transport in the nonmagnetic system.\nAnother interesting playground of the QSL is the Ki-\ntaev model [6], which has been studied intensively in this\ndecade [7–34]. The Kitaev model consists of bond-dependent\nIsing interactions between spin-1 /2 moments on a honeycomb\nlattice, and its ground state is exactly shown to be a QSL.\nOne of the interesting features is the spin fractionalization.\nNamely, the spins are fractionalized into itinerant and local-\nized Majorana fermions. Since both quasiparticles are charge\nneutral, the thermal transport is one of the most promising\nphenomena to grasp the presence of the Majorana fermions.\nParticularly, a half quantized plateau in the thermal quantum\nHall e \u000bects has been successfully observed in the Kitaev can-\ndidate material \u000b-RuCl 3[35–38], which is a direct evidence\nof a topologically protected chiral Majorana edge mode [39].\nOn the other hand, less is known about the Majorana-mediated\nspin transport in the Kitaev QSL although it has recently been\ndiscussed in the related systems [40–42].\nIn the Kitaev model, spin correlations are extremely short-\nranged due to the existence of the local Z2symmetry, in con-\ntrast to the Heisenberg chain with power-low spin correla-\ntions. However, it does not necessarily mean the absence\nof the spin transport in the Kitaev model. When small lo-\ncal perturbations are present in the system, eg.the magnetic\nfield, edges, defects, etc. the Z2symmetry is lost in certain re-\ngions [43]. Therefore, intriguing phenomena are expected to\nbe induced in these regions. For example, the spin excitationcould flow through the Kitaev QSL region without spin polar-\nization. Thus, it is highly desired to examine the spin transport\nin the nonequilibrium dynamics, which should be important\nto observe the itinerant nature of the Majorana fermions in the\nbulk.\nIn this Letter, to address the spin transport through the Ki-\ntaev QSL, we investigate the real-time dynamics triggered by\nan impulse magnetic field on one of the edges. Using the\ntime-dependent mean-field (MF) theory, we examine the time\nevolution of the magnetization and dynamics of the fraction-\nalized Majorana quasiparticles. We demonstrate that a spin-\npolarized wavepacket created at the edge propagates to the\nother edge even when the two edges are separated by the QSL\nregion without spin polarization. We also address how robust\nthis anomalous phenomenon is against the Heisenberg interac-\ntions by means of the exact diagonalization (ED). Finally, we\npropose the ways to extract the results intrinsic to the Kitaev\nQSL with the fractionalized quasiparticles in experiments.\nWe consider the Kitaev model in the La\u0002Lbcluster of the\nhoneycomb lattice with zigzag edges, which is schematically\nshown in Fig. 1. The norms of the primitive translational vec-\ntorsaandbare assumed to be unity. The periodic boundary\ncondition is imposed along the b-direction. The system we\nconsider here is composed of three regions. In the middle (M)\nregion, no magnetic field is applied and the Kitaev QSL is re-\nalized without spin polarization. In the right (R) region, the\nstatic magnetic field hRis applied. We introduce LR, which is\ndefined as the number of zbonds included in this region with\nrespect to the adirection (see Fig. 1). Moreover, we term the\nL region composed of the left-edge sites. In this region, we\nintroduce the time-dependent magnetic field hL(t). The corre-\nsponding Hamiltonian is\nH(t)=\u0000JKX\n\r=x;y;zX\nhi;ji\rS\r\niS\r\nj\u0000hRX\ni2RSz\ni\u0000hL(t)X\ni2LSz\ni;(1)\nwhere S\r\niis the\r(=x;y;z) component of an S=1=2 spin op-\nerator at the ith site. The ferromagnetic exchange JK(>0)\nis defined on three di \u000berent types of the nearest-neighbor\nbonds, x(red), y(blue), and z(green) bonds (see Fig. 1). It\nis known that, in the uniform lattice, the magnetic field in-arXiv:1912.10599v1  [cond-mat.str-el]  23 Dec 20192\nabM L R\nx\nFIG. 1. Kitaev model on a honeycomb lattice with zigzag edges.\nRed, blue, and green lines represent x;y, and zbonds, respectively.\nSolid (open) circles represent spin-1 /2 in the A(B) sublattice. In\nthis figure, four z(green) bonds exist along the adirection in the R\nregion, namely, LR=4.\nduces the phase transition to the spin-polarized state around\nhc=JK\u00180:042 [30] within the MF theory. Therefore, we re-\nstrict ourselves to the case with hR<hcto discuss the spin\ntransport inherent in the Kitaev QSL.\nWe study the time evolution of the system upon stimuli of\nthe magnetic pulse in the L region (see Fig. 1) [44, 45]. To\nthis end, we introduce the time-dependent Majorana MF the-\nory. The details of the formulations are shown in Ref. [46].\nThe Hamiltonian Eq. (1) is obtained as a fermion model by\napplying the Jordan-Wigner transformation to the spin opera-\ntors [47–49]. Furthermore, by introducing two kinds of Majo-\nrana fermions, \rand ¯\r, from a complex fermion at each site,\nEq. (1) is rewritten as\nH(t)=\u0000JK\n4X\nr\u0010\ni\rA\nr\u0000a+b\rB\nr+i\rA\nr+b\rB\nr\u0011\n\u0000JK\n4X\nri\rA\nr\rB\nri¯\rA\nr¯\rB\nr\n\u0000hR\n2X\nr2R\u0010\ni\rA\nr¯\rA\nr\u0000i\rB\nr¯\rB\nr\u0011\n+hL(t)\n2X\nr2Li\rB\nr¯\rB\nr; (2)\nwhere rindicates the position of the zbond (the center of the\nzbond; see Fig. 1). \rA\nrand ¯\rA\nr(\rB\nrand ¯\rB\nr) are the Majorana\nfermion operators connected with the zbond in the sublattice\nA(B), as shown in Fig. 1. When hR=hL(t)=0, [H;\u0011r]=0\nat each r, and\u0011r(=i¯\rA\nr¯\rB\nr) is the Z2local conserved quan-\ntity. In the case, the model is solvable as the Hamiltonian\nis bilinear in terms of \r, and its low-energy dispersion is\ngiven as\"k'vjk\u0000kKjaround the K point with the veloc-\nityv=p\n3JK=4. This indicates that \rand ¯\rare regarded as\nthe itinerant and localized Majorana fermions, respectively.\nSince the magnetic field hybridizes two kinds of the\nMajorana fermions, the Hamiltonian is no longer exactly\nsolvable. Here, we apply the Hartree-Fock type decou-\npling to the interaction on the zbond as i\rA\nr\rB\nri¯\rA\nr¯\rB\nr\u0018\ni\rA\nr\rB\nr\u00021(x;t)+\u0002 2(x;t)i¯\rA\nr¯\rB\nr\u0000\u00021(x;t)\u00022(x;t)\u0000i\rA\nr¯\rA\nr\u00023(x;t)\u0000\n\u00024(x;t)i\rB\nr¯\rB\nr+\u0002 3(x;t)\u00024(x;t)\u0000i\rA\nr¯\rB\nr\u00025(x;t)\u0000\u00026(x;t)i¯\rA\nr\rB\nr+\n\u00025(x;t)\u00026(x;t), where we have introduced the six kinds of\nx- and t-dependent MFs as \u00021(x;t)=hi¯\rA\nr¯\rB\nri \u0011 h\u0011i(x;t),\n\u00022(x;t)=hi\rA\nr\rB\nri \u0011 h\u0018i(x;t),\u00023(x;t)=hi\rB\nr¯\rB\nri=\n\u00002hSz\nBi(x;t)=\u00002hSzi(x\u0000xd=2;t),\u00024(x;t)=hi\rA\nr¯\rA\nri=\n050100150t/J−1\nK(a)\n∆SzR\n−2−1012×10−3\n0 10 20 30 40\nx050100150t/J−1\nK(b)\n∆SzR M ←L\n−2−1012×10−3FIG. 2. Real-time dynamics of the Kitaev spin system with La=50\nandLb=300 after the magnetic-field pulse at t=0. The contour\nplot of \u0001Sz(x;t) on the plane of the time tand the space xin the\nsystem (a) without and (b) with the M (white) region where the QSL\nis realized with no spin polarization between the L (left gray) and\nR (right gray) regions (see the top of the panels). The dashed lines\nrepresent x=vtwith the Majorana velocity v(see text).\n2hSz\nAi(x;t)=2hSzi(x+xd=2;t),\u00025(x;t)=hi¯\rA\nr\rB\nri,\u00026(x;t)=\nhi\rA\nr¯\rB\nri, where xis the horizontal coordinate of randxd=\n1=(2p\n3). This MF theory is exact when hL=hR=0.\nTherefore, we believe that our MF results are reliable as far\nas the small fields are applied. In the MF theory, the many-\nbody wave function is expressed as a direct product of one-\nbody states, whose time-evolution is described by the MF\nHamiltonian. Determining the MFs at each time from the\nmany-body wave function, we compute the time-evolution\nof the one-body states with the extended Euler method [50–\n55]. The time-dependent magnetic field is explicitly given as,\nhL(t)=Ap\n2\u0019\u001bexph\n\u0000t2\n2\u001b2i\n, where Aand\u001bare constants for the\nGaussian pulse. In the following, we fix the system size as\nLa=50 and Lb=300, the static field as hR=0:01JK, and\npulse parameters as A=1 and\u001b=2J\u00001\nK.\nBefore discussing the spin transport through the M region,\nwe examine the system without this region, namely, the static\nmagnetic field hRis applied to the sites in the R region with\nLR=La\u00001. In this case, there are no local conserved quan-\ntities, leading to nonzero local spin moments. Figure 2(a)\nshows the contour plot of the changes in the spin moments\n\u0001Sz(x;t), where \u0001O(x;t)=hO(x;t)i\u0000h O(x;\u00001)i. As ex-\npected, we find that the wavepacket created by the magnetic-\nfield pulse flows to the right edge. Note that its velocity al-\nmost coincides with the Majorana velocity of the genuine Ki-\ntaev model, v, which is shown as the dashed line in Fig. 2(a).\nThis indicates that the propagation is attributed to the gapless\nMajorana excitation in the bulk within a small static field.3\n050100150t/J−1\nK(a)\n∆ξR M ←L\n−4−2024×10−3\n0 10 20 30 40\nx050100150t/J−1\nK(b)\n∆ηR M ←L\n−1.0−0.50.00.51.0×10−3\nFIG. 3. Real-time evolution of (a) \u0001\u0018, and (b) \u0001\u0011of the Kitaev spin\nsystem with La=50 and Lb=300 after the magnetic-field pulse at\nt=0. The setup and parameters are the same as in Fig. 2(b).\nNow, we consider the real-time dynamics of the non-\nmagnetic Kitaev spin system triggered by the magnetic-field\npulse at the left edge to discuss how the wavepacket flows\nthrough the M region (the Kitaev QSL without spin polariza-\ntion). In this region, the local conserved quantity is present\nin each hexagon, and spin correlations are extremely short-\nranged [56]. Figure 2(b) shows the time evolution of \u0001Sz(x;t)\nin the system with LR=10. We find that the magnetic moment\nis always zero in the M region and no proximity e \u000bect is found\naround the interface between L and M regions. Nevertheless,\nin the R region, \u0001Sz(x;t) is induced and the wavepacket flows\nwith the Majorana velocity v. This result indicates that the\nspin excitations propagate in the nonmagnetic region via the\nitinerant Majorana fermions, which cannot be explained by\nclassical pictures such as the spin wave theory.\nTo discuss the propagation of the spin excitation through\nthe QSL region in more detail, we examine the time evolutions\nof\u0001\u0018and\u0001\u0011, which correspond to the dynamics of itinerant\nand localized Majorana fermions, as shown in Figs. 3(a) and\n3(b). We find in Fig. 3(a) that the excitation created in the left\nedge at t=0 propagates in the whole region, which results\nfrom the motion of the itinerant Majorana fermions. By con-\ntrast, Fig. 3(b) shows that \u0001\u0011vanishes in the M region owing\nto the existence of the local Z2symmetry, while it appears in\nthe R region. This is the similar behavior as in \u0001Sz. These\nsuggest that, after the excitation at the left edge, only the itin-\nerant Majorana fermions propagate in the bulk, where no os-\ncillation appears in the magnetization, and finally reach the R\nregion. The weak magnetic field in the R region yields the hy-\nbridization between the itinerant Majorana fermions and the\nlocalized fermions, resulting in time-dependent nonzero spin\nmoments there. Thus, in the Kitaev QSL, the spin transport\n01020t/J−1\nK(a)JH/JK=0\n∆Sz|A=1M R L\n−2−1012×10−2\n0 1 2 3\nx01020t/J−1\nK(b)JH/JK=0.03\n∆Sz|A=1M R L\n−2−1012×10−2\n01020t/J−1\nK(c)JH/JK=0.03\n(∆Sz|A=1+ ∆Sz|A=−1)/2M R L\n−2−1012×10−2\n0 1 2 3\nx01020t/J−1\nK(d)JH/JK=0.03\n∆Sz|A=5M R L\n−6−3036×10−2FIG. 4. Contour plots of time-dependent spin moment \u0001Sz(x;t) in\nthe 24-site cluster with hR=0:01JKwhen (a) ( JH=JK;A)=(0;1),\n(b) (0:03;1) and (d) (0 :03;5). (c) Contour plot for the average of\n\u0001Sz(x;t) with ( JH=JK;A)=(0:03;1) and (0:03;\u00001). The dashed\nlines represent x=vtwith the Majorana velocity v(see text).\nis mediated by the Majorana fermions although the spin mo-\nments never appear. Moreover, we have confirmed that the\nmagnitude of the spin moment induced in the R region ex-\nhibits a power-low decay as a function of the length of the M\nregion. This is ascribed to the gapless dispersion of the Ma-\njorana fermions, in contrast to the existence of the gap in the\nspin excitation in the Kitaev model.\nThe pulse-amplitude dependence in this phenomenon is\nalso remarkable. In the R region, \u0001Szturns out to be pro-\nportional to A2. This can be explained by considering the lo-\ncal symmetry at the left edge [57]. This non-linear feature is\nintrinsic in the Kitaev model, in contrast to the conventional\nsystems with \u0001Sz/A. To study how visible anomalous be-\nhavior is in the system with the Heisenberg interaction, we ap-\nply the ED method to the Hamiltonian H(t)+JHP\nhi;jiSi\u0001Sj\nwith the antiferromagnetic Heisenberg coupling JH(>0). It is\nknown that, when hR=hL(t)=0, the Kitaev QSL is stable\nagainst small JH[9, 12, 19, 25]. In our calculations, the initial\nground state is obtained with the Lanczos method and the time\nevolution is simply evaluated by the Runge-Kutta method.\nThe obtained results for the 24-site cluster with La=4,\nLb=3, and LR=1 are shown in Fig. 3. In the calcu-\nlations, we have confirmed that the induced moment is al-\nways parallel to the zdirection. First, we show the results\nfor the genuine Kitaev model with JH=0 in Fig. 4(a). One\ncan find the propagation of the magnetic excitation from one\nedge to the other through the QSL region without spin po-\nlarization, which is consistent with the Majorana MF result\ndiscussed above. In the presence of the Heisenberg term\n(JH=JK=0:03),\u0001Sz(x;t) takes nonzero values in the M re-\ngion, as shown in Fig. 4(b), suggesting that the Heisenberg\ninteraction a \u000bects the flow of the spin excitation. In partic-\nular, the spin modulation in the M region is more prominent\ncompared to that in the R region. This di \u000berence from the gen-\nuine Kitaev model originates from the fact that the Heisenberg\ninteraction yields the interaction between itinerant and local-4\nized Majorana fermions. Therefore, the spin moments appear\nin the M region near the interface to the L region as a proxim-\nity e\u000bect, as shown in Fig. 4(b).\nWe note that \u0001Szin the R region is similar to the case with-\nout the Heisenberg interactions. This implies that the spin\ntransport inherent in the Kitaev model still survives. It is\nnaively expected that in the M and R region, the Heisenberg\nand Kitaev interactions mainly give AandA2contributions in\nthe spin oscillation, as discussed above. Therefore, the unique\nfeature for the Kitaev system is extracted by examining the\naverage of the magnetic responses after the magnetic pulses\nwith Aand\u0000A. In Fig. 4(c), this quantity is hardly seen in the\nM region but clearly observed in the R region, which is a con-\nsequence of the Kitaev QSL with itinerant Majorana fermions.\nWhen the pulse amplitude Ais relatively large, the Kitaev\ninteraction plays a dominant role for the spin propagation and\nthe spin transport without spin polarization becomes practi-\ncally prominent. Figure 4(d) presents the results with the large\nA. The spin moments induced in the M region are relatively\nsmall, but the spin excitation propagates to the right edge, at\nwhich the spin moments induced are much larger than those in\nthe M region. This phenomenon is essentially the same as that\nin the genuine Kitaev case shown in Fig. 4(a). The above two\nresults suggest that the spin transport mediated by the frac-\ntionalized itinerant quasiparticles without spin excitations can\nbe observed even in the presence of additional interactions.\nFinally, we discuss the relevance of the present results to\nreal materials. The setup of our study could be implemented\nby considering a Kitaev candidate material sandwiched by fer-\nromagnetic insulators. The candidate materials have been pro-\nposed as AIrO 3(A=Na, Li) [58–63] and \u000b-RuCl 3[35–38].\nThe stimuli of the magnetic field pulse can be injected from\na ferromagnetic insulator by the spin pumping [64–67] or cir-\ncular polarized light irradiation [68, 69]. Our results suggest\nthat the spin-excited flow propagates to the other edge even if\nthe magnetic polarization is absent in the Kitaev magnet, and\ntherefore, we expect that the time-dependent magnetic mo-\nment is observed in the ferromagnetic insulator connected to\nthe other side of the Kitaev magnet with a small overlapping.\nThis time evolution can be experimentally measured by the\nKerr or Faraday rotations [68, 70], which will provide con-\nvincing evidence of the fractionalized itinerant quasiparticles\nin the bulk of the Kitaev magnet.\nNote that in the real system, a magnetic order hinders\nthe appearance of the Kitaev QSL [71–76]. This e \u000bect can\nbe avoided by the finite temperature measurement above the\nNéel temperature, where the itinerant quasiparticles are active,\nand/or the recent progress of the thin film [13, 77–85], which\nsuppresses the magnetic ordering due to the suppression of\nthe interlayer coupling. Moreover, by changing the intensity\nof the injection of the spin excitation, one could estimate the\nmagnitude of the additional interactions such the Heisenberg\none. The e \u000bect of the o \u000b-diagonal interactions, so called \u0000\nterm, is not addressed in the present study but we expect that\nthis gives a similar e \u000bect to the Heisenberg one [86–88].\nIn summary, we have demonstrated that, after the magneticexcitation at one of the edges in the Kitaev spin system, the\nspin moments never appear in the bulk, but are fluctuated in\nthe opposite edge. We have revealed that this unusual spin\ntransport is governed by the fractionalized itinerant Majorana\nfermions. The spin transport without spin polarization should\nbe visible even in the system with the Heisenberg coupling by\nusing the pulse field dependence in \u0001Sz.\nWe also note that it might be possible to control the mo-\ntion of the localized Majorana fermions (vison) in the bulk, by\nswitching on /o\u000bthe magnetic field. This should be important\nfor realizing the vison transport in the experiments. It is also\ninteresting to study the spin transport in the generalized Kitaev\nmodels [89–91], where the existence of spin fractionalization\nhas also been suggested [90, 92, 93]. The real-time spin dy-\nnamics should be one of the possible candidates to clarify the\npresence of the quasiparticles.\nThe authors thank T. Mizoguchi and M. Udagawa for\nfruitful discussions. Parts of the numerical calculations\nwere performed in the supercomputing systems in ISSP,\nthe University of Tokyo. This work was supported by\nGrant-in-Aid for Scientific Research from JSPS, KAK-\nENHI Grant Nos. 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Zarand3,4\n1Department of Physics, Ben Gurion university, Beer Sheva 84 105 Israel\n2CNRS-Laboratoire de Physique Th´ eorique de l’Ecole Normal e Sup´ erieure, 24 rue Lhomond,75231 Cedex 05, Paris\nFrance\n3Freie Universi¨ at Berlin, Fachbereich Physik, Arnimallee 14, D-14195 Berlin, Germany\n4Theoretical Physics Department, Budapest University of Te chnology and Economics, Budafoki ut 8, H-1521, Hungary\nPACS72.25.Rb – Spin relaxation and scattering\nPACS71.70.Ej – Spin-orbit coupling, Zeeman and Stark splitting, Jahn-Te ller effect\nPACS03.65.Vf – Phases: geometric; dynamic or topological\nAbstract. - We study zero temperature spin dynamics of a particle confin ed to a ring in presence\nof spin orbit coupling and Ohmic electromagnetic fluctuatio ns. We show that the dynamics of\nthe angular position θ(t) are decoupled from the spin dynamics and that the latter is m apped to\ncertain correlations of a spinless particle. We find that the spin correlations in the zdirection\n(perpendicular to the ring) are finite at long times, i.e. do n ot dephase. The parallel (in plane)\ncomponents for spin1\n2do not dephase at weak dissipation but they probably decay as a power\nlaw with time at strong dissipation.\nIntroduction. – Due to recent advances in semicon-\nductor technology, it became possible to isolate and ma-\nnipulate spins of individual electrons [1,2]. For efficient\nspin manipulation, however, slow spin decay is needed.\nSpin decayin mesoscpopicdevicesis generatedbytwoma-\njor sources: Hyperfine interaction with nuclear spins [3] is\nresponsible for spin decay in most materials. However,\nspin-orbit (SO) coupling can also induce spin relaxation,\nand under certain conditions, phonon- [4] or electromag-\nnetic field-induced SO relaxation [7] can dominate the de-\ncay[5,6]. AsshowninRef.[7]two-photon(ortwo-phonon)\nprocesses lead to geometrical spin relaxation even in the\nabsence of external field and, as pointed out recently, this\nmechanism can become even dominant in hole-doped sys-\ntems [8,9].\nHere we make an attempt to understand, whether the\nabove-mentioned geometrical spin relaxation can survive\neven atT= 0 temperature. Although Ohmic electromag-\nnetic fluctuations were found to lead to a vanishing spin\nrelaxation rate at T= 0 [7], the results of Ref. [7] are\nnot conclusive, since they allow for non-exponential relax-\nation, common in Ohmic systems. To address this issue\nmore rigorously, we consider a ring geometry. Studying\na ring is, however, not of pure theoretical interest; high\nquality semiconductor rings [10,11] can in fact also beused as quantum spin qubits [11], and usefulness of these\ndevices depends on spin dephasing, a topic under active\nexperimental study [5,6,8,12,13].\nThere are two types of spin-orbit coupling in two-\ndimensional electron systems: the Rashba interaction in-\nduced, e.g., by an electric field perpendicular to a two-\ndimensional (2D) layer [14], and the Dresselhaus coupling\ninduced by bulk inversion asymmetry [15]. Our aim is to\nstudy, how these couplings influence spin coherence for\nan electron confined to a ring, in the presence of Ohmic\nfluctuations. We shall first derive the appropriate Hamil-\ntonian for a confined electron, and show that the presence\nof the spin does not influence the orbital motion of the\nconfined electron, which is governed exclusively by fluctu-\nations of the external electric field. The dynamics of the\nspin, on the other hand, is determined by the orbital mo-\ntion of the electron, and has a topological character. We\nfind that for weak dissipation the spin does not dephase,\nbut certain spin components are reduced by fluctuations.\nFor strong dissipation, however, we find that certain com-\nponents of the spin probably relax even at T= 0 tem-\nperature, due to the disordering of the orbital degrees of\nfreedom [16]. The relaxation we find is, however, not ex-\nponential but of a power law, typical of confined particles\nat temperature T= 0 [17,18].\np-1B. Horovitz1 P. Le Doussal2 G. Zarand3,4\nHamiltonian. – Let us start by projecting a 2D spin-\norbit Hamiltonian on a ring, a procedure which is not en-\ntirely trivial [19,20]. In addition to the kinetic terms, the\n2D Hamiltonian consists of a potential V0(r) that confines\nthe particle to a ring of radius R±δR, withδR≪R.\nWe write the total Hamiltonian in polar coordinates as\nH0+H′where\nH0=−¯h2\n2me/bracketleftbigg∂2\n∂r2+1\nr∂\n∂r/bracketrightbigg\n+V0(r), (1)\nH′=p2\nθ\n2mer2+α0(Sxpy−Sypx)+β0(Sxpx−Sypy).\nHerepθ=−i¯h∂/∂θ,Sare spin operators, meis the elec-\ntron mass and pxandpydenote the xandycomponents\nof the momentum. The α0term is the Rashba coupling\nwhileβ0denotes the Dresselhaus coupling. Labeling the\nradial eigenstates of H0by|n/an}bracketri}htand their energies by En,\nouraimistoproject H′onthesubspace, |0/an}bracketri}ht, whilekeeping\nterms up to order O(δR), a procedure that involves some\nsubtleties. First we rewrite H′=H1+H2by introducing\nS±\nr≡cosθSx±sinθSyandS±\nθ≡cosθSy∓sinθSx,\nH1=p2\nθ\n2mr2+α0\n2r{S+\nr,pθ}−β0\n2r{S−\nθ,pθ},\nH2=iα0¯hS+\nθ(∂r+1\n2r)−iβ0¯hS−\nr(∂r+1\n2r).(2)\nAs noticed by Meijer et al. [19], for any state ψ(r) that\nis radially localized near R, one has /an}bracketle{tψ|2∂r+1\nr|ψ/an}bracketri}ht= 0.\nTherefore, to first order in the SO coupling, H2does not\ngive a contribution to the projected Hamiltonian. Never-\ntheless, as previously overlooked, H2cannot be ignored:\nlocalization on a scale δRimplies∂r∼1/δRand hence\n2nd order perturbations in H2do give a contribution,\n∼ H2\n2/En=O(1), sinceEn∼1/(δR)2. The next or-\nder contributions scale as H3\n2/E2\nn=O(δR), and vanish in\nthe limitδR→0, similar to all higher order terms in the\nperturbation series.\nPerturbation theory to 2nd order yields therefore the\nprojected spin and angle dependent effective Hamiltonian\nHring=/an}bracketle{t0|H1|0/an}bracketri}ht−/summationdisplay\nn/negationslash=0/an}bracketle{t0|H2|n/an}bracketri}ht/an}bracketle{tn|H2|0/an}bracketri}ht\nEn−E0+O(δR).(3)\nThe sum in Eq. (3) can be evaluated analytically by mak-\ninguseofasumrule[22,23], andthesecondtermofEq.(3)\nsimply becomes1\n2me(α0Sθ−β0S′\nr)2.\nIntroducing the vector h(θ) viah≡(αcosθ−\nβsinθ,αsinθ−βcosθ), and with the dimensionless\nRashba and Dresselhaus couplings defined as α≡mRα0\nandβ≡mRβ0, we can finally rewrite our effective ring\nHamiltonian in the δR→0 limit as\nHring=¯h2\n2meR2[pθ+h(θ)·S]2. (4)\nWe remark, in particular, that the term ∼αβsin2θin the\neffective Hamiltonian of Ref. [20] is exactly canceled by\nthe 2nd order terms. As a consequence, Eq. (4) possesses\na conserved ”momentum”, ˆQ≡pθ+h(θ)·S.N(   )θ0\nθ0θ + 2π0Ω(2π)\nΩ(0)\nFig. 1: Evolution of the spin (coherent state) while the elec tron\nmakes a circle, θ0→θ0+2π. The initial state is rotated around\nan axisN(θ0) by an angle Γ θ0.\nSpectrum. – The eigenstatesandeigenenergiesof(4)\ncan be analytically computed for β= 0, when the system\nis rotationally invariant and therefore Jz=pθ+Szis also\nconserved. The Hamiltonian can then be written as\nHring=¯h2\n2meR2[Jz−n(θ)·S√\n1+α2]2,(5)\nwithn(θ) = (−hx(θ),−hy(θ),1)/√\n1+α2a unit vector.\nThe energy spectrum and the eigenvalues can then easily\nbe found by constructing common eigenstates of the two\ncommutingoperators, Jzandn(θ)·S. ForS= 1/2,n(θ)·S\nandJzhave eigenvalues n(θ)·S=σ/2 andJz=m+\nσ/2, respectively, with σ=±andman integer. The\nspectrum is ǫm,σ=1\n2meR2/bracketleftbig\nm+σ(1\n2−1\n2√\n1+α2)/bracketrightbig2, and\nthe eigenstates are of the form/integraltext\nθeimθ\n√\n2π|θ/an}bracketri}ht⊗|±n(θ)/an}bracketri}ht, with\n|±n/an}bracketri}htdenoting spin coherent states, defined through the\nusual relation, Ω·S|Ω/an}bracketri}ht=S|Ω/an}bracketri}ht[24]. The wave functions\ncan be explicitly expressed as\nψm,+(θ) = eimθ/parenleftbig\ncos¯α\n2,−eiθsin¯α\n2/parenrightbig\n,\nψm,−(θ) = eimθ/parenleftbig\ne−iθsin¯α\n2,cos¯α\n2/parenrightbig\n,(6)\nwith ¯αdefined as ¯α≡arctan(α). The states ψ±m,±are\nrelated by time reversal, and their energies equal, Em,+=\nE−m,−. Forα<√\n3 the ground state has m= 0.1\nDissipation. – Having understood the properties of\nan isolated ring, we now couple the motion of the particle\nto the coordinate ξof a dissipative environment, i.e., we\nconsider the total Hamiltonian as H=Hring+V(θ,ξ).\nThroughout most of this paper we shall assume that\nV(θ,ξ) describes the coupling to a Caldeira-Leggett (CL)\nenvironment, appropriate for small rings in an Ohmic\n(metallic) environment.2Thenξrepresents the random\nforce generated by the environment, V=ξ−eiθ+ξ+e−iθ,\nand theT= 0 Fourier transform of the environment cor-\nrelations is /an}bracketle{tξ−ξ++ξ+ξ−/an}bracketri}htω= ¯h2η|ω|, withηthe dimen-\nsionless friction coefficient.\nThe corresponding equations of motion for θ(t) are\n˙θ=pθ+h·S\nmeR2,¨θ=−1\nmeR2∂θV(θ,ξ).(7)\n1Here we used the phase convention of Ref. [24]. This construc -\ntion can be generalized for larger spins.\n2In a dirty metal environment, e.g., one needs the ring’s radi us\nto be smaller than the mean free path.\np-2Geometric spin dephasing on a ring\nHence, as a consequence of the simple form of Hring,\nEq. (4), the dynamics of θin the dissipative environment\nare not affected by the spin-orbit couplings. This decou-\npling allows us to describe the θ(t) evolution by a path\nintegral, where for each trajectory the spin dynamics fol-\nlow from Eq. (4)\ndS\ndt=˙θh(θ)×S⇒dS\ndθ=h(θ)×S.(8)\nViewingθasa”time”variable,these dynamicscorrespond\nto a spin precession around a ”time” dependent magnetic\nfieldh(θ). Note that switching to ”Schr¨ odinger” picture,\nthe spin coherent states have a simple θevolution, too.\nApart from a phase, they evolve as |Ω(θ)/an}bracketri}ht, where Ω(θ)\nis the vector solution of (8), i.e.dΩ\ndθ=h(θ)×Ω. In\nparticular, the vector n(θ) can also be shown to satisfy\nthis equation.\nIn termsofthe spin operators,Eq. (8) issolvedasa sim-\nple linear mapping Si(θ) =Rij(θ,θ0)Sj, withRij(θ,θ0) a\nrotation matrix. The rotation matrix R2π(θ0) =R(θ0+\n2π,θ0), corresponding to the particle going once around\nthe ring by 2 π, is of special interest. We denote by the\nunit vector N(θ0) its axis of rotation and by Γ θ0the cor-\nresponding rotation angle (see Fig. 1). In particular, for\nβ= 0 we find that the angle Γ θ0is independent of the\ninitial value, Γ = 2 π(1−√\n1+α2), and is typically incom-\nmensurate with 2 π.\nMapping to a spinless system. – For a given evo-\nlution,θ0→θ, we can obtain the evolution of the spin\npart of the wave function from Eq. (8), which is described\nby a unitary operator, Uspin(θ,θ0). Here the Hamilto-\nnian to describe the θ(”time”) evolution of the spin is\nHs=h(θ)·S. We proceed to study the case β= 0. Then,\nas in the standard NMR rotating field problem, the spinor\ntransformation ψ′≡ei(θ−θ0)Szψto the ”rotating frame”\ncancels the ”time” ( θ) dependence, and amounts in re-\nplacingHs→h(θ0)·S−Sz=−√\n1+α2n(θ0)·S. For\nS= 1/2this leadstotheevolutionoperator, Uspin(θ,θ0) =\ne−iθ−θ0\n2σzei√\n1+α2θ−θ0\n2n(θ0)·σ. Using now the expression of\nˆQwe find that the θevolution of the spin states has a\nparticularly simple form\nUspin(θ,θ0)ψm,±(θ0) = eiqm,±(θ0−θ)ψm,±(θ),(9)\nwhereqm±=m±1\n2∓1\n2√\n1+α2denote the eigenvalues\nof the momentum ˆQ. After a 2πrotation the state ψm,±\npicksup anincommensuratephase, 2 πqm,±. Note thatthe\nsemiclassical evolution involves a similar incommensurate\nangle, Γ, as discussed below Eq. (8) (see also Fig. 1).\nMaking use of the decoupling of orbital and spin de-\ngrees of freedom, we can construct a mixed path integral\nformalism (to be detailed in Ref. [23]), where the spin is\ntreated in an evolution operator formalism, while the or-\nbital motion of the particle is developed in a path integral\nformalism. The full evolution for a given environment his-tory is then obtained as:\nψm,±(θt,t) =/summationdisplay\nn/integraldisplay2π\n0dθ0/integraldisplayθt+2πn\nθ0Dθ\neiSP(θ,ξ)Uspin(θt+2πn,θ0)ψm,±(θ0).(10)\nNote thatθin this equation is a non-compact variable,\nand an additional integration over the environment con-\nfigurations has to be carried out in the end. Importantly,\nthe action SP(θ,ξ) =/integraltextt\n0[1\n2mer2˙θ2−V(θ,ξ)] describes a\nparticle on the ring in the presence of dissipation for a\ngiven environment history, and is independent of the spin\nevolution.\nForβ= 0 we can make use of Eq. (9) and obtain a par-\nticularly simple path integral representation for the spin\nevolution. Consider spin correlations with an initial den-\nsity matrix |σ/an}bracketri}ht/an}bracketle{tσ|built from one of the two Kramers de-\ngenerate ground states of m= 0 andσ=±, having mo-\nmentaˆQ=q0,±=±GwithG=1\n2−1\n2√\n1+α2. Using\nEqs. (10) and (9) for the forward and backward spin evo-\nlutions, we find an exact mapping of the spin correlations\nonto a superposition of equilibrium correlations of spin-\nless particles on a ring with a flux Φ = ±G(in units of\nquantum flux):\nPa,Φ(t21) =/an}bracketle{te−iaθ(t2)eiaθ(t1)/an}bracketri}htΦ. (11)\nHereagain, θ(t)isanon-compactvariablewithin( −∞,∞)\nto be used within the path integral representation of the\nspinless problem. Note that the bath still couples to e±iθ\nhence we expect that (11) depends only on the noninte-\nger part of Φ. For /an}bracketle{tSx(t)Sx(0)/an}bracketri}htwe obtain the following\nidentity for an initial density matrix, |+/an}bracketri}ht/an}bracketle{t+|,\nCx\n++(t) =1\n4sin2¯α(P1,G(t)+P−1,G(t)) (12)\n+ cos4¯α\n2P−2G,G(t)+sin4¯α\n2P2−2G,G(t).\nForC−−(t) the same result holds with all subscripts of\nPa,Qreversing sign. For the Szcorrelations, on the other\nhand, we obtain\nCz\n++(t) = cos2¯α+P−1−2G,G(t)sin2¯α, (13)\nandforCz\n−−(t)thesameholdswith P1+2G,−G. Noticethat\nthe degeneracy point, α=√\n3, corresponding to fluxes\nΦ =±1\n2represents a special case, and is not studied here.\nWhilethecorrelationfunctionofthe z-componentofthe\nspin,Cz(t), obviously contains a constant non-decaying\npiece, the correlation function Cx(t) contains only phase\ncorrelation functions Pa,Φwitha/ne}ationslash= 0. There is some ev-\nidence that these correlations decay in time. In particu-\nlarP1,0∼1/t2from the XY lattice model [16] and from\nsmallηexpansion[25]. Correlationswith incommensurate\nawerestudied in a related system ofdissipative Josephson\njunctions [27], and found to decay algebraically. Further\nevidence is for large η, as discussed below. To further\np-3B. Horovitz1 P. Le Doussal2 G. Zarand3,4\nappreciate these correlations we have evaluated the path\nintegrals in (11) analytically for η= 0, and surprisingly,\nwe findPη=0\n∓2G,±G(t) = 1. As a consequence, for η= 0 the\ncorrelation function Cxcontains a piece which does not\noscillate. As discussed below, though reduced, this part\nseems to survive for very weak dissipation, η≪1, while\nit apparently decays algebraically for strong dissipation,\nη≫1.\nStrong dissipation limit. – In the strong dissipa-\ntion limit, η≫1, we can describe the evolution of the\nphase through a Langevin equation, and an expansion in\n1/ηis possible [17,18]. In this limit, all correlation func-\ntionsPa,qwitha/ne}ationslash= 0 are found to decay algebraically.\nLargeηperturbation theory yields that Pa,Φ∼t−a2/πη\nand thex-component of the spin also decays algebraically,\nwhile thez-component remains finite and does not de-\ncay. This result, however, holds only for up to times\nlnt < O(η) beyond which effects of renormalization of η\ncannot be neglected. In a recent work [18] we have shown\nthat in presence of a weak DC electric field there is a crit-\nicalηc= 1/2πsuch thatη > ηcflows toηcwhich would\nindicatePa,Φ∼t−2a2. In some sense the fluctuating spin\ncorresponds to a time dependent flux, i.e. an electric field,\nthough the correspondence is not precise.\nWeak dissipation. – The rather different behavior\nofSx,yandSzshould already be manifest in the weak\ndissipation limit, where we can perform perturbation the-\nory in the strength of the dissipation, η. To do pertur-\nbation theory, we restrict ourselves to the case S= 1/2\nandβ= 0, and use Abrikosov’s pseudofermion method\nto represent each spinor ψmσof (6) by a pseudofermion\noperator,fmσ. In this language the ring Hamiltonian be-\ncomesHring=/summationtext\nm,σǫm σf†\nm σfm σ, while the interaction\nis expressed as\nV=/summationdisplay\nm,σ/parenleftbig\nξ−f†\nm σfm−1σ+ξ+f†\nm σfm+1σ/parenrightbig\n,(14)\nand standard field-theoretical methods can be used to\nevaluate physical quantities. A renormalization group\nanalysis of the vertex function and the pseudofermions’\nself-energy reveals that, although ultraviolet logarithmic\ndivergencies appear in both quantities, they cancel and\nthe dissipation parameter ηis in leading order, neverthe-\nless,exactly marginal , and themassofthe particleremains\nalso unrenormalized [23].\nIn this perturbative regime, fingerprints of a non-\nexponential spin decay should appear in the susceptibility,\nχ, which, in the absence of spin decay, should contain a\nCurie part. To compute χ, we first express the impurity\nspin operator in terms of pseudofermions as\nSi=/summationdisplay\nm,σ,m′,σ′Si\nm,σ,m′,σ′f†\nm σfm′σ′, (15)\nwiththematrixelementssimplydeterminedfromthewave\nfunctions (6), as Si\nm,σ,m′,σ′=/an}bracketle{tΨm,σ|Si|Ψm,σ/an}bracketri}ht. The lead-\ning corrections to χare shown in Fig. 2. Although allFig. 2: Zero and leading order ( ∼η) corrections to the spin\nsusceptibility. Continuous lines and wavy lines represent the\npseudofermion and bosonic propagators, respectively, whi le the\ndots represent spin vertices. Logarithmic divergencies in the\nthree diagrams above exactly cancel.\ncorrections shown contain logarithmic ultraviolet singu-\nlarities, remarkably, all these singularities exactly cancel,\nand one finally obtains just a finite renormalization of the\nperpendicular Curie susceptibility\nχx,y=cos4(¯α/2)\n4T/bracketleftBig\n1−η\n2π[1\nGln/parenleftBig1+2G\n1−2G/parenrightBig\n−4]+O(η2)/bracketrightBig\n.\nThe prefactor cos4(¯α/2) originates from the coefficient of\nP−2G,G(t) in Eq. (12), and accounts for g-factor renor-\nmalization in the isolated ring. The correction ∼η, on the\notherhand, representstheenvironment-inducedrenormal-\nization of the xandycomponents of the spin (g-factor).\nTheaboveperturbativeresultandthesurvivaloftheCurie\nsusceptibility indicates that the term P−2G,G(t) decays to\na reduced but non-zero value for small η.\nIn contrastto the xandycomponents, the zcomponent\nofthesusceptibility, χz, isfoundtoremainunrenormalized\nbyηto leading order in the dissipation. These results\nimply that, for weak Ohmic dissipation, the only effect of\ndissipation is to slightly and anisotropically renormalize\ntheg-factor, but apart from that the spin behaves as a\nfree spin, and does not decay.\nCase ofβ/ne}ationslash= 0. –So far we discussed only the case\nβ= 0. We show now that the system with both α,βfinite\nis equivalent to the Hamiltonian (14). Assume a state |q/an}bracketri}ht\nthat is an eigenstate of ˆQ|q/an}bracketri}ht=q|q/an}bracketri}ht. This state generates\na ladder of states, |m+q/an}bracketri}ht ≡eimθ|q/an}bracketri}ht, with integer mby\nˆQeimθ|q/an}bracketri}ht= (m+q)eimθ|q/an}bracketri}ht. (16)\nSinceˆT−1ˆQˆT=−ˆQ, a sequence of time reversed states is\nalso generated by the time reversal operator, ˆT:ˆQˆT|m+\nq/an}bracketri}ht=−(m+q)ˆT|m+q/an}bracketri}ht.All these states are orthogonal\nsince they correspond to different energy eigenvalues, and\nthe environment couples the mandm±1 states, exactly\nas forβ= 0. The only difference is that E0↑(α,β), which\nis not known analytically, changesthe factor1\n2−1\n2√\n1+α2\ninEm↑, Em↓. Hence Eq. (14) is a correct representation\nalso of theβ/ne}ationslash= 0 case.3\n3The matrix elements of the spin operators are nevertheless d if-\nferent, changing e.g. the overall coefficient in Eq. (15).\np-4Geometric spin dephasing on a ring\nConclusions. – We derived the effective Hamiltonian\nof an electron confined to a ring within a 2-dimensional\nelectron gas, in the presence of SO coupling, and subject\nto a dissipative environment. We have shown that the\norbital motion of the particle decouples from the spin evo-\nlution, and correspondingly, spin decay has a geometric\ncharacter [7]. For an Ohmic environment, we mapped the\nspin relaxation problem to that of a spinless particle on a\nringpiercedbyamagneticflux[Eqs.(12,13)]. Wefindthat\nthezcomponent of the ground state spin is not affected\nby dissipation. The xandyin-plane spin components are,\non the other hand, reduced by dissipation, but we find no\ndephasingfor spin1\n2and weak dissipation. However, these\ncomponents seem to dephase at large dissipation.\nWe should remark that these latter results are based on\nthe assumption of Ohmic dissipation. The situation may,\nhowever, change for subohmic dissipation or 1 /ωγnoise,\npresent in many systems. In this case, the decoupling of\nthe spin and orbital motion and thus Eqs. (12,13) remain\nvalid, however, for subohmic dissipation ηis arelevant\nperturbation, and even a small dissipation could possibly\nlead to the decay of the xandyspin components. This\npossibility, however, needs to be further explored.\n∗∗∗\nWe acknowledge useful discussions with B. Dou¸ cot, W.\nZwerger and A. Zaikin. This research has been supported\nby the Hungarian Research Funds OTKA and NKTH un-\nder Grant Nos. K73361, T ´AMOP-4.2.1/B-09/1/KMR-\n2010-0002, and the EU-NKTH GEOMDISS project and\nby THE ISRAEL SCIENCE FOUNDATION (grant No.\n1078/07).\nREFERENCES\n[1]J. M. Elzerman, et al. ,Nature,430(2004) 431\n[2]K. C. Nowack, F.H.L. Koppens, Yu.V. Nazarov, and\nL.M.K. Vandersypen ,Science,318(2007) 1430\n[3]Khaetskii, A.V., Loss D. andGlazman L. ,Phys. Rev.\nLett.,88(2002) 186802.\n[4]A. V. 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B ,100(1955) 580.\n[16]H. Spohn and W. Zwerger ,J. Stat. Phys. ,94(1999)\n1037; In this work only the decay of imaginary time corre-\nlations is considered. However, the 1 /τ2found in imaginary\ntime is strongly suggestive of a corresponding 1 /t2decay\nin real time.\n[17]D. S. Golubev, G. Sch ¨on and A. D. Zaikin ,\ncond-mat/0208548; G. Schn and A. D. Zaikin ,Phys.\nRep.,198(1990) 237.\n[18]Y. Etzioni, B. Horovitz and P. Le Doussal , Phys.\nRev. Lett. (in press) [arXiv:1101.4160].\n[19]F. E. Meijer, A. F. Morpungo and T. M. Klapwijk ,\nPhys. Rev. B ,66(2002) 033107.\n[20]J. S. Sheng and K. Chang ,Phys. Rev. B ,74(2006)\n235315.\n[21]L.D. Landau and E.M. Lifshitz ,Quantum Mechan-\nics: Non-Relativistic Theory , Vol.3(3dedition, Pergamon)\n1991.\n[22]Kohn W. ,Phys. Rev. ,133(1964) A171.\n[23]B. Horovitz, P. Le Doussal, and G. Zar ´and, unpub-\nlished.\n[24]Auerbach A. ,Interacting Electrons and Quantum Mag-\nnetism(Springer-Verlag, New York) 1994.\n[25]B. Horovitz and P. Le Doussal ,Phys. Rev. B ,82\n(2010) 155127; The small ηexpansion at imaginary times\ncan be extended to finite flux Φ /negationslash=1\n2withP1,Φ∼1/t2. The\nsame result is obtained through a pseudofermion calcula-\ntion [23].\n[26]A.A. Abrikosov ,Physics,2(1965) 5; A.A. Abrikosov ,\nPhysics,2(1965) 21.\n[27]S. L. Lukyanov and P. Werner ,J. Stat. Mech. , (2007)\nP06002\np-5"    },    {        "title": "2203.14068v2.Nonequilibrium_dynamics_in_a_spin_valve_with_noncollinear_magnetization.pdf",        "content": "Nonequilibrium dynamics in a spin valve with noncollinear magnetization\nRudolf Smorka,1Pavel Bal\u0013 a\u0014 z,2Michael Thoss,1, 3and Martin \u0014Zonda4, 1\n1Institute of Physics, Albert-Ludwigs-Universit at Freiburg,\nHermann-Herder-Stra\u0019e 3, 79104 Freiburg i. Br., Germany\n2FZU { Institute of Physics of the Czech Academy of Sciences,\nNa Slovance 1999/2, 182 21 Prague 8, Czech Republic\n3EUCOR Centre for Quantum Science and Quantum Computing,\nAlbert-Ludwig University Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany\n4Department of Condensed Matter Physics, Faculty of Mathematics and Physics,\nCharles University, Ke Karlovu 5, Praha 2 CZ-121 16, Czech Republic\n(Dated: November 28, 2022)\nWe utilize a hybrid quantum-classical equation of motion approach to investigate the spin dy-\nnamics and spin-transfer torque in a spin valve under bias voltage. We show that the interplay\nbetween localized classical magnetic moments and conduction electrons induces a complex e\u000bective\nexchange coupling between the magnetic layers. This leads to a declination of magnetizations from\nlayers anisotropy axes even in equilibrium. Introducing a \fnite bias voltage triggers spin currents\nand related spin-transfer torques which further tilt the magnetizations and govern the relaxation\nprocesses of the spin dynamics. Analyzing di\u000berent scenarios of the applied bias voltage, we show\nthat symmetric and asymmetric voltage drops can lead to relaxation times of the spin dynamics\nthat di\u000ber by several orders of magnitude at comparable charge currents. In both cases we observe\nresonant features, where the relaxation is boosted whenever the chemical potential of the leads\nmatches the maxima in the density of the states of the spin-valve electrons.\nI. INTRODUCTION\nMagnetic multilayer devices, where the exchange cou-\npling between magnetic layers is suppressed by a non-\nmagnetic interlayer, e.g., spin valves or magnetic tunnel\njunctions [1, 2], have attracted a lot of attention from\nengineers and scientists working in di\u000berent \felds. Be-\nsides their direct applicability as, e.g., various types of\nsensors [3], in magnetic recording systems [3{5] or in the\nbroader context of spintronics [6, 7], they also provide\na rich and accessible theoretical as well as experimental\nplatform for the investigation of important physical phe-\nnomena. For example, in recent years, multilayer devices\nplayed a crucial role in the study of spin-hall e\u000bect [8{\n10], ultra-fast demagnetization [11{14], domain-wall dy-\nnamics [15, 16], various types of spin-transfer torques\n(STT) [3, 17{21] and the interplay between electronic\ntransport and dynamics of localized magnetic moments\nin general [22{24]. In addition, spin-torque oscillators\nbased on magnetic vortices in spin valves or tunnel junc-\ntions became a promising candidate for neuromorphic\ncomputing systems [25{29].\nBecause of the diversity of these devices, which spread\nfrom molecular valve systems up to bulk [1, 30{32], a\nmultitude of di\u000berent theoretical methods are used to ra-\ntionalize their properties and predict new features. Ar-\nguably, the most popular ones are classical micromag-\nnetic simulations [33] and atomistic spin dynamics [34]\nbased on the Landau{Lifshitz{Gilbert (LLG) equation.\nA clear advantage of these methods is the large number\nof highly optimized and versatile computer codes avail-\nable [34, 35], which allow to address large systems and\nto incorporate experimentally measured parameters [33].\nHowever, to model phenomena where transport playsa crucial role, the LLG equation must be extended by\nphenomenological or approximate torque and damping\nterms, which describe the e\u000bective in\ruence of spin cur-\nrents on the magnetization [36]. Because transport is\ninherently a quantum phenomenon and because these\nterms are in general in\ruenced by the changing state\nof the spin valve, such a simplistic treatment can miss\nimportant physics, especially in the case of systems far\naway from equilibrium.\nOn the other hand, fully quantum-mechanical ap-\nproaches that are able to capture the quantum nature of\nthese devices are usually constrained to small systems,\nstatic magnetic con\fgurations, short times, or rely on\nsevere approximations [30, 31, 37, 38].\nA natural compromise between completely classical\nand fully quantum-mechanical approaches present hybrid\nmethods which combine both classical and quantum de-\ngrees of freedom [15, 39{54]. In the case of magnetic sys-\ntems, these methods consider classical localized magnetic\nmoments interacting with quantum conduction electrons.\nIn their simplest form, a strict separation of time scales\nis assumed; that is, the dynamics of the classical spins\nis considered to be much slower than the one of elec-\ntrons. Under this assumption, electrons respond instan-\ntaneously to the slow time-dependent potential of the\nclassical degrees of freedom and, therefore, can be de-\nscribed by steady-state approaches, e.g., via nonequilib-\nrium Green functions (NEGF) [40, 41, 46{48, 55]. How-\never, several recent studies have shown that this approach\nis often invalid [15, 43{45, 49{53] because the two time-\nscales can not be strictly separated in general.\nTo take account of this issue, one has to resort to non-\nMarkovian approaches, in which electrons react in a \fnite\ntime to the changes of the classical spins [15, 43{45, 49{arXiv:2203.14068v2  [cond-mat.mes-hall]  24 Nov 20222\n53, 56]. These approaches reveal a time-dependent mis-\nalignment between the localized magnetic moments and\nthe local nonequilibrium spin density of conduction elec-\ntrons. Its most important consequences materialize in\nthe form of additional torques and time-retarded damp-\ning e\u000bects [15, 43{45, 49{53].\nNevertheless, there are quantum e\u000bects not fully cap-\ntured even by these methods. For example, they do not\naccount for so-called quantum spin-transfer torque re-\nsulting due to the quantum many-body states [57] and\nKondo e\u000bect [43, 58], as neither is included in the e\u000bec-\ntive single-particle picture of the hybrid methods. It is\nalso questionable if they can describe the relaxation of\nlarge spins into an excited state due to the coherent cou-\npling to reservoirs observed in quantum systems [59{61],\nalthough there are some examples of nonthermal steady\nstates in quantum-classical systems [62]. In the context of\nour study, it is also important to note that hybrid meth-\nods tend to underestimate the damping of the magnetic\nnutations [45, 63].\nDespite these di\u000berences, the methods that combine\nclassical localized spins with quantum conduction elec-\ntrons are in a rather good qualitative agreement with\nthe full quantum mechanical treatments [44, 45, 51, 57]\nand capture most of its details. As such, these hybrid\ntechniques proved to be extremely useful in the inves-\ntigation of various phenomena not described by classi-\ncal or adiabatic LLG based approaches, e.g., geometrical\ntorque [51], magnetic inertia [15], chiral spin and charge\npumping [49], formation of some nontrivial magnetic tex-\ntures [53, 64] or resonant dependence of the spin damping\non voltage [56, 65]. In addition, they are generalizable to\nrealistic band structures [50].\nIn this paper, we use a quantum-classical equations\nof motion (QC-EOM) approach for open quantum sys-\ntems [15, 56], to study the dynamics of a spin valve\nsystem sandwiched between two metallic leads with \f-\nnite voltage di\u000berence. QC-EOM is an Ehrenfest-type\nmethod [39, 66, 67] used, e.g., to study nuclear dynamics\nin quantum transport [68, 69] or current-induced bond\nrupture in single-molecule junctions [70]. Its advantage\nis that in the case of noninteracting conduction electrons\nthe hierarchy is terminated exactly at the second tier for\na general metallic band of the leads or at the \frst tier\nfor the wide band limit (WBL) approximation [71{76].\nThe method is therefore numerically exact even far away\nfrom equilibrium, allows to reach long simulation times,\nand avoids approximate or phenomenological terms not\nresulting directly from the Hamiltonian of the model.\nUsing the QC-EOM method, we argue that the quan-\ntum character of the conduction electrons is crucial in\nunderstanding of the nonequilibrium dynamics of a spin\nvalve. We show that this is true even in a seemingly sim-\nple case where the spin dynamics of the entire magnetic\nlayer can be truthfully represented by a single aggregated\nmacrospin. Moreover, the character of the voltage drop\ncrucially a\u000bects its magnetic dynamics. The two com-\nmonly used types, namely, a \fnite voltage introducedby shifting the chemical potential of one lead and by an\nequal opposite shift of chemical potentials in both leads,\nshow spin relaxation times that di\u000ber by several orders\nof magnitude at comparable charge currents.\nThe rest of the paper is organized as follows. In\nsection II, we introduce the model and the QC-EOM\nmethod. In the results section III, we \frst discuss the\nisolated spin valve (Sec. III A). Here we introduce the\nmacrospin approximation and show that the e\u000bective ex-\nchange coupling between magnetic layers is a complicated\nfunction of model parameters and system geometry re-\n\recting the density of states of the conduction electrons.\nWe then move to the driven system in Sec. III B where we\ninitially discuss the transient dynamics, showing the cru-\ncial di\u000berence between various types of driving. Next, the\nrelaxation of the magnetizations is analyzed and we show\na staggering di\u000berence between symmetric and asymmet-\nric voltage drop cases. Finally, we address current-driven\ntorques and their e\u000bect on the steady state magnetization\nof the spin valve. Some technical aspects and derivations\nare provided in the Appendices.\nII. MODEL AND METHODS\nThe spin valve heterostructure under consideration\nis illustrated in Fig. 1. It consists of two ferromag-\nnetic (FM) layers known as pinned (PL) and free layer\n(FL) separated by a nonmagnetic (NM) spacer layer\n(SL) [1, 77]. We employ a hybrid quantum-classical de-\nscription where localized spins are treated within a clas-\nsical approximation but movable electrons are quantum\nparticles.\nThe tight-binding Hamiltonian describing the inter-\naction of quantum electrons with local time-dependent\n\felds resulting from the interactions with localized spins\non a lattice reads\nH(t) =\u0000\rX\nhj;j0i\u0010\ncy\njcj0+ h:c:\u0011\n+\u0016X\njcy\njcj\n+Jsd\n2X\nj2PL;FLcy\nj\u001b\u0001Sj(t)cj; (1)\nFIG. 1. Schematic of a spin valve modeled on a two-\ndimensional square lattice which consists of ferromagnetic\n(FM) pinned layer (PL or l, red spheres), nonmagnetic (NM)\nspacer layer (SL) and magnetic free layer (FL or r, wine\nspheres). Arrows depict localized classical spins fSjgj2PL;FL.\nThe spin valve is coupled to two non-interacting metallic leads\n(gray area).3\nwhere spinors cj= (cj\";cj#)Tandcy\nj= (cy\nj\";cy\nj#) consist\nof annihilation or creation operators of the conduction\nelectrons with spins \";#at sitej. Their kinetic energy\nis described by the \frst term of the Hamiltonian. For\nsimplicity, we set the electron nearest neighbor hopping\nparameter\rconstant in the whole spin valve. We use \r\nas the energy scale, i.e., all energies presented in the text\nor \fgures are in the units of \rand time is in units of \r\u00001\n(the typical range of \ris 0:1\u00002 eV [15, 49, 52]). The\nsecond term of the Hamiltonian describes the in\ruence of\na constant electrochemical potential \u0016which governs the\nelectron occupation of the isolated spin valve. Here we as-\nsume that the system is small enough that its equilibrium\nelectrochemical potential can be set and manipulated ex-\nternally, e.g., by an auxiliary gate lead, which does not\ncontribute to the charge and spin transport. If not stated\notherwise, the electrochemical potential \u0016is taken zero\nwhich in equilibrium or for isolated valve sets the half-\n\flling condition. The last term describes a local sd-like\ninteraction between the electrons and classical spins with\nexchange coupling Jsd. Here\u001bis the Pauli vector and\nSj(t) is the magnetic moment vector localized at site j.\nThe localized magnetic moments in the particular mag-\nnetic layer`= PL;FL\u0011l;rare described by the classical\nHamiltonian\nH`\nC(t) =JexX\nhj;j0iSj(t)\u0001Sj0(t)\u0000X\nj2`B\u0001Sj\n\u0000K`X\nj2`(Sj(t)\u0001e`)2+JsdX\nj2`sj(t)\u0001Sj(t);\n(2)\nwhereJexis the intralayer exchange coupling between the\nneighboring localized magnetic moments, Bis the vec-\ntor of external magnetic \feld, K`is the layer-dependent\nmagnetic anisotropy constant, while e`is a unit vector\naligned with the local anisotropy easy axis. The last\nterm couples the classical magnetic system to the quan-\ntum electrons. Here, vector sj(t) =1\n2Tr\u001aj(t)\u001bis the\ntime-dependent electron spin-density, where \u001aj(t) is the\nreduced nonequilibrium single-particle density matrix of\nelectrons on site j. All relevant physical constants have\nbeen absorbed into the parameters of the model and the\nmagnitude of the localized magnetic moments (spins) is\n\fxed to one. For simplicity, we address the structure de-\nscribed by the coupled Hamiltonians (1) and (2) as the\nspin valve.\nWhen considering an isolated system, that is, in the\nabsence of fermionic reservoirs, the time evolution of the\nquantum part of the spin valve described by Hamilto-\nnian (1) is governed by the Liouville-von Neumann equa-\ntion for the single-particle electron density matrix \u001a(t)\n@\u001a(t)\n@t=\u0000i[H(t);\u001a(t)]: (3)\nIn the presence of fermionic reservoirs, a system of\nequations of motion for the reduced density matrix \u001a(t) is\nobtained by tracing out the reservoir degrees of freedomfrom the whole density matrix. In particular, to describe\nthe dynamics of the magnetic junction, we use a hierar-\nchical equations of motion approach [78, 79]. For the case\nof noninteracting fermions, studied in the present paper,\nthe hierarchy of equations of motion for the auxiliary\ndensity matrices terminates at the second tier exactly\n[71{76].\nThe equation of motion for the reduced single-particle\ndensity matrix reads\n@\n@t\u001a(t) =\u0000i[H(t);\u001a(t)] +X\n`\u0010\n\u0005y\n`(t) +\u0005`(t)\u0011\n;(4)\nwhere the second term on the right hand side of Eq. (4)\ngenerates dissipation, a nonunitary time evolution due to\nthe coupling of the central system to the fermionic reser-\nvoirs. The current matrices \u0005`(t) are expressed using the\nnonequilibrium single-particle greater/lesser Green func-\ntions\n\u0005`(t) =Zt\n\u00001d\u001c\u0002\nG>(t;\u001c)\u0006<\n`(\u001c;t)\u0000G<(t;\u001c)\u0006>\n`(\u001c;t)\u0003\n:\n(5)\nHere, \u00067\n`is the lesser/greater self-energy matrix due to\nthe coupling between the reservoir `and the spin valve.\nAssuming a constant density of states (WBL) in the\nreservoirs (leads) with constant broadening function \u0000 `\n(the matrix \u0000`has components \u0000 `at the interface `and\nis zero otherwise), chemical potential \u0016`and temperature\nT= 1=\fwe get\n\u0006<\n`(t;\u001c) =iZ1\n\u00001d\"\n2\u0019f`(\";\u0016`;\f)e\u0000i\"(t\u0000\u001c)\u0000`; (6)\n\u0006>\n`(t;\u001c) =\u0000iZ1\n\u00001d\"\n2\u0019[1\u0000f`(\";\u0016`;\f)]e\u0000i\"(t\u0000\u001c)\u0000`:\nHere,f`(\";\u0016`;\f) is the Fermi function of reservoir `which\ncan be approximated by a sum over Nppoles using the\nPad\u0013 e representation [80]\nf(\")\u00191\n2\u00001\n\fNpX\np=1\u0011p \n1\n\"\u0000\u001f\u0000\np`+1\n\"\u0000\u001f+\np`!\n;(7)\nwhere\u001f\u0006\np`=\u0016`\u0006i\u0018p\f\u00001and\u0011pare Pad\u0013 e coe\u000ecients.\nEmploying the residue theorem, the above expansion al-\nlows to write the current matrices in an explicit form\n\u0005`(t) =1\n4(1\u00002\u001a)\u0000`+NpX\np=1\u0005`;p(t); (8)\nwhere the Pad\u0013 e-resolved auxiliary matrices \u0005`;pfollow\nthe equations of motion\n@\n@t\u0005`;p(t) =\u0000i\u0011p\n\f`\u0000`\u0000i\u0012\nH\u0000i\n2\u0000\u0000\u001f+\np1\u0013\n\u0005`;p(t);(9)\nwith\u0000=P\n`\u0000`. Hence, within the wide band approxi-\nmation used here, we get a closed (exact) system of EOM4\nalready at Eq. (9). Note that the formalism is gauge in-\nvariant in the sense that the results for transport do not\nchange if all three of the chemical potentials ( \u0016l,\u0016rand\n\u0016) are shifted by the same value.\nFinally, using the extension of classical Poisson-\nbrackets to spin systems [81, 82], the classical spin equa-\ntion of motion for the magnetic moment at position j\nreads\n@Sj(t)\n@t=fSj(t);HC(t)g=\u0000Sj(t)\u0002rSj(t)HC:(10)\nTo obtain the overall time-dependence, we evolve the\nset of Eqs. (10) together with Eq. (3) for an isolated\nspin valve or, in the case of heterostructure, together\nwith Eq. (4) and Eq. (9) supplied by Eq. (8). In both\ncases, we evolve the system using the fourth-order (3/8-\nrule) Runge-Kutta method with equal time steps for the\nquantum and classical subsystem.\nThe current matrices \u0005`(t) can be used to calculate\nthe charge and spin currents between the spin valve and\nthe leads`=l;r\nI`(t) =\u0006Re Tr ( \u0005`(t)); (11)\nJ\u000b\n`(t) =\u0006Re Tr ([ 1N\n\u001b\u000b]\u0005`(t)); (12)\nwhere the plus sign is for currents from the left ( l) reser-\nvoir into the spin valve and minus for currents from the\nspin valve into the right ( r) reservoir,\u001b\u000bis the Pauli\nmatrix and Nis the total system size (number of lat-\ntice points). Similarly, the nonequilibrium single particle\ndensity matrix can be used to calculate the local charge\nand spin currents between particular monolayers [50, 83].\nWe pay special attention to the charge and spin resolved\ncurrent at the SL-FL interface\nIF(t) =i\n2X\nhj;j0i2IFTr [Hj;j0\u001aj0;j\u0000\u001aj;j0Hj0;j];(13)\nJ\u000b\nF(t) =i\n2X\nhj;j0i2IFTr [\u001b\u000b(Hj;j0\u001aj0;j\u0000\u001aj;j0Hj0;j)];\n(14)\nwhere the sum runs over coupled pairs of nearest neigh-\nborshj;j0iwithjtaken from the last monolayer of SL\nandj0from of the \frst monolayer of the FL. Hj;j0and\n\u001aj;j0are the respective 2 \u00022 submatrices of the quan-\ntum Hamiltonian and the nonequilibrium density ma-\ntrix. The di\u000berence between the SL-FL interface spin\ncurrentsJ\u000b\nF(t) and FL-lead interface spin current J\u000b\n`(t)\ncan be used to enumerate the aggregated current-driven\nSTT [84]. However, because there can be a \fnite torque\nacting on the localized spins even in equilibrium, one has\nto subtract from the net torque equilibrium contributions\nto obtain the current-driven part of the STT\nTcd(t) =JF(t)\u0000Jr(t)\u0000Jeq\nF: (15)\nTo analyze nonequilibrium results, we also make use\nof the Landauer-B uttiker approach for the transmissionfunction \u0002, its spin-resolved polarization P, and the den-\nsity of states of the heterostructure DOSh [55, 85] for\na \fxed con\fguration of classical spins S(typically the\nequilibrium one)\n\u0002(\";S) = Trf\u0000lGR(\")\u0000rGA(\")g; (16)\nP(\";S) = Trf\u0000lGR(\")\u0000\n\u0000\"\nr\u0000\u0000#\nr\u0001\nGA(\")g;(17)\nDOSh(\";S) = TrifGR(\")\u0000GA(\")g=2\u0019N; (18)\nwhereGR(A)is the retarded (advanced) Green function\nof the coupled system and \u0000\u001b\nl;rare the spin-resolved cou-\npling matrix between the system and the left ( l) or right\nlead (r).\nIII. RESULTS\nA. Isolated spin valve\nBefore investigating the spin dynamics in an exter-\nnally driven spin valve, it is instructive to \frst discuss\nthe dynamics of an isolated spin valve. We use a one-\ndimensional case to discuss the e\u000bect of the electronic\nspectrum on the spin dynamics and the role of the non-\nmagnetic layer in the relaxation process, which both play\nan important role in the driven system.\n1. Electronic spectrum\nWe \frst show that the details of the electronic spec-\ntrum signi\fcantly in\ruence the magnetization dynamics\nof the spin valve. In general, the spectrum is sensitive\nto the orientation of the classical spins and acquires time\ndependence through their dynamics [56, 65]. Already a\nsimple system of just two localized spins coupled through\nspin-dependent currents can have very complicated dy-\nnamics, including some chaotic regimes [42, 82]. There-\nfore, to make our argument more comprehensible, we ad-\ndress here a case in which the electronic spectrum can be\nassumed to be mostly static and the dynamics of a par-\nticular spin in a spin valve is not too complicated with\nrespect to its neighbors.\nWe investigate a one-dimensional chain where the mag-\nnetic layers consist of one to ten sites each. We set the fer-\nromagnetic Heisenberg exchange coupling Jex=\u00001 and\nswitch o\u000b the anisotropies Kl=Kr= 0. This stabilizes\na nearly ideal ferromagnetic ordering in both magnetic\nlayers and signi\fcantly simpli\fes the dynamics.\nBefore addressing the time evolution, it is useful to\nbrie\ry discuss the electronic spectrum for some relevant\nstatic con\fgurations of localized spins. Fig. 2 shows the\ndependence of the spectrum on Jsdfor a linear system of\ntotal length N= 20 (NNM= 10) and two static con\fg-\nurations of the localized spins ( Nl(PL)=Nr(FL)= 5), a\nparallel one (a) and perpendicular one (b). Both spectra\ndisplay a similar band splitting from one mixed band at5\nFIG. 2. Electronic spectrum for a static one-dimensional layer\nwithNl(PL) = 5,NNM= 10,Nr(FL) = 5 as a function of\nelectron-spin coupling Jsd. The two cases represent settings\nwith ferromagnetic con\fguration within the layer and total\nnormalized magnetization components within the layers be-\ning (a) parallel: Mz\nl= 1 andMz\nr= 1 where the two colors\nrepresent states with up (red) and down (blue) magnetic po-\nlarization, (b) perpendicular Mx\nl= 1 andMz\nr= 1. Panels on\nright show the details of the spectrum in the vicinity of the\nFermi level.\nsmall coupling ( Jsd<3) to three distinct bands in the\nstrong coupling ( Jsd>7) regime. Here, the top and bot-\ntom bands re\rect the states predominately localized in\nthe ferromagnetic layers (note their spin polarizations in\nFig. 2(a1) and the discussion in Appendix B). Therefore,\neven when the presence of the central band can lead to\nseemingly \fnite DOS (under suitable broadening) at the\nFermi-level for arbitrary Jsd, the transport characteris-\ntics in the strong coupling regime can be still insulating-\nlike. The reason is that the local DOS calculated for\nmagnetic layers typically shows a large gap around the\nFermi level. However, even in that case, the central band\nhas an important in\ruence on the spin dynamics, be-\ncause the conduction electrons mediate an e\u000bective ex-\nchange interaction ( Je\u000b) between the magnetic layers. As\nwe discussed below, Je\u000bis governed by the states in the\nvicinity of the Fermi-level. The real challenge is that even\nsuch a simple case as presented here shows a complicated\nJsddependence, including a rather complex avoided level\ncrossing for weak coupling [Fig. 2(b 2)].\n2. Macrospin approximation\nThe magnetic layers are coupled by an e\u000bective ex-\nchange interaction Je\u000bdue to the presence of spin-\npolarized conduction electrons in the valve. Because of\nthe strong exchange coupling Jexwhich stabilizes the rel-\native dynamics of spins within one layer, we can extract\nJe\u000bfrom the spin evolution by analyzing the dynamics\nof the net layer magnetizations. To this goal, we intro-duce a simple macrospin approximation with an e\u000bective\nHamiltonian described by a bilinear form\nHMS=\u0000Je\u000bMl(t)\u0001Mr(t); (19)\nwhere each magnetic layer is characterized by a local\nmagnetization Ml;r=PNS\njSj, withNS=Nl=Nr\nbeing the number of spins in a layer. The validity of\nthis approximate model is discussed in Appendix C. The\ntime evolution of one macrospin described by this form\nis given by the equations of motion\n@M`\n@t=Je\u000bM`\u0002M`; (20)\nwhere`;`2fl;rgand`6=`. Under some simple assump-\ntions (e.g., thatjM`j=NS= 1), these nonlinear coupled\nordinary di\u000berential equations can be solved analytically\nby rotating the system to the plane of the limit cycle\nand then back. In accordance with the later investigated\ncase of an open system we set the initial condition to\na parallel formation of classical spins within a layer but\nperpendicular between the magnetic layers (as illustrated\nin Fig. 1). In particular, initially Ml(t= 0) points to\nxdirection and Mr(t= 0) tozdirection. The initial\ncondition of the electrons is set by exact diagonalization\nunder the half-\flling condition ( \u0016= 0). The solution of\nEq. (20) with the above initial state reads (for details see\nAppendix C):\nM`(t) =0\n@cos\u00120 sin\u0012\n0 1 0\n\u0000sin\u00120 cos\u00121\nA0\nB@NSp\n2cos(!t+\u001e`)\nNSp\n2sin(!t+\u001e`)\nNS=p\n21\nCA:(21)\nHere, the right vector represents the solution in the frame\nof the limit cycle and the left matrix is the reverse rota-\ntion around the y-axis to the original frame of the spin\nvalve, where \u0012=\u0019=4,\u001el= 0 and\u001er=\u0019. The charac-\nteristic frequency is given by !=p\n2NSJe\u000b. To quantify\nthe in\ruence of the electronic spectrum on spin dynam-\nics, we use a least-squares \ftting of this analytical solu-\ntion to the numerical data (obtained within the QC-EOM\napproach). This also allows us to test the validity of the\nmacrospin approximation and, in some limiting cases, the\nprecision of our numerical integration. Note that there\nare parameter regimes where we also need a second \ft-\nting parameter 'which shifts the phases to \u001el='and\n\u001er=\u0019+'in cases where the limit cycle is reached only\nafter some signi\fcant time.\n3. Spin valve dynamics\nWe \frst demonstrate the validity of the macrospin ap-\nproximation by comparing it with the numerical simu-\nlations. Fig. 3 depicts the dynamics of the magnetiza-\ntion in the \frst layer (solid lines) and its macrospin \ft\n(dashed lines) for system size N= 20 (NS= 5) and\nthreeJsdvalues. The macrospin approximation \fts the6\nexact dynamics almost perfectly for Jsd= 1 because here\nthe single-spin \ructuations are e\u000bectively suppressed al-\nready at small times. If we neglect the small \ructuations\nand oscillations imposed on top of the main dynamics,\nwhich are not visible on the scale presented in Figs. 3(c),\na similar conclusion can be drawn also for Jsd= 5. Inter-\nestingly, it is the case of intermediate coupling Jsd= 3:3\nwhere the full dynamics becomes rather complicated, for\nexample, it takes some transient time ( t\u0019500) until the\nlimit cycle is reached. Although the dominant preces-\nsion frequency can be still extracted for this case, there\nis some modulation and the \ft is far from perfect.\n-0.8-0.40.00.40.8\n 0  2000  4000  6000(c)Ml /NS\ntime tJsd = 5-0.8-0.40.00.40.8(a)Ml /NS\nMx\nl\nMy\nlfit Mx\nl\nfit My\nl Jsd = 1\n-0.8-0.40.00.40.8(b)Ml /NS\nJsd = 3.3\nFIG. 3. Dynamics of the normalized layer magnetizations\n(solid lines) and their approximate analytical macrospin so-\nlution (dashed lines) with \ftted !. The plotted data are for\nthe \frst ferromagnetic layer calculated for a one dimensional\nspin valve of total length N= 20 (NS\u0011NFM1=NFM2= 5)\nand couplings Jsd= 1 (a), 3:3 (b) and 5 (c).\nTo understand how the coupling Jsda\u000bects the magne-\ntization dynamics, we analyze a system with the spacer\nlayer of length NNM= 10 and three di\u000berent sizes of mag-\nnetic layers. The Jsddependence of the simplest NS= 1\ncase, plotted with the red dashed line in Fig. 4(a), shows\na single broad maximum at Jsd\u00192:5. With increasing\nnumber of spins, the dependence becomes rather compli-\ncated. It exhibits several local maxima and minima for\nNS= 5 (black bullets) and 10 (blue circles) and becomes\nmonotonous only for Jsd&4:5 where all Je\u000b\u0002N2\nScurves\napproach each other.\nThis complicated behavior re\rects the (static) elec-\ntronic spectrum shown for the NS= 5 case in Fig. 4(c).\nBoth the weak coupling Jsd.3 and strong coupling\nJsd&4:5 cases can be qualitatively understood by fol-\nlowing the energy di\u000berence \u0001 \"[Fig. 4(b)] between the\ntwo highest occupied energy states in the static spec-\ntrum marked by the dashed line in Fig. 4(c). Here, the\nenergy di\u000berence \u0001 \"signalizes the magnitude of mag-netic splitting (see the blue and red lines in Fig. 2(a) for\nillustration). The e\u000bective coupling Je\u000btakes local min-\nima when \u0001 \"approaches zero. The reason is that the\nnonmagnetic states do not couple to the classical spin\nand can not mediate the e\u000bective exchange coupling [56].\nConsequently, the largest Je\u000bre\rects the maximum in\n\u0001\"andvice versa . In the case of large spin-electron cou-\nplingJsdthe spectrum is divided into three bands and\nthe states that are mostly localized to the ferromagnetic\nlayers are far away from the Fermi level. Because we do\nnot change the size of the spacer layer, the splittings \u0001 \"\nfor variousNSapproach each other and the same pattern\nis followed by Je\u000b.\nThe only regime where the \ftted Je\u000bdeparts quali-\ntatively from \u0001 \"(gray area in Fig. 4(a) for NS= 5)\ncoincides with the splitting of the three bands illustrated\nin Fig. 2. Here we observe the transition from metallic\nto insulating character of the valve (see also discussion\nin Appendix B). This is accompanied by strong electron-\ninduced spin \ructuations on a time scale much shorter\nthan the main precession. These spin \ructuations lead\nto a deviation from the initial FM ordering which sig-\nni\fcantly modi\fes the electronic spectrum, and there-\nfore also \u0001\"(t), which can not be considered static any-\nmore. This is clearly re\rected in the Je\u000bin this regime\nwhich does not follow the \u0001 \"(t= 0) (for details see Ap-\npendix D). Nevertheless, we can conclude that the sensi-\ntivity ofJe\u000bon the details of the electronic spectrum, in\nall above discussed regimes, underlines the importance of\ntreating electrons as quantum particles instead of using\ne\u000bective classical approximations.\nOutside the regime 3 .Jsd.4:5, the macrospin ap-\nproximation works well also in the case of varying width\nof the spacer layer. Figs. 5(a,b) show the dependen-\ncies ofJe\u000bonNNMfor weakJsd= 1 and strong spin-\nelectron coupling Jsd= 5. The alternation of Je\u000bbe-\ntween ferromagnetic and antiferromagnetic character (a)\nas well as the algebraic decay with increasing NNM(b)\nare in qualitative compliance with previous results [86{\n88]. These features are captured already by perturbation\napproaches, e.g., the theory of Ruderman-Kittel-Kasuya-\nYosida (RKKY) interaction [89{91] which for a one di-\nmensional electron gas predicts Je\u000b/J2\nsd[Si(\u0019NNM)\u0000\n\u0019=2] where Si( x) is the sine integral function [91]. The\nlarge di\u000berence in magnitude between the odd and even\nNNMshown in Figs. 5(b), results from the di\u000berence be-\ntween the polarizations of states at the Fermi level for\nchains of odd and even point numbers.\nAlthough useful, the above \ftting to the macrospin\ndynamics is bound to fail in a more realistic setup. The\nreason is that the above mean-\feld theory cannot cap-\nture some important features of the whole dynamics. For\nexample, it actually takes a \fnite time for electrons to re-\nact to a new position of the classical spins [43{45] and\ncarry the excitation from one magnetic layer to the other.\nFor a long spacer layer, this can lead to a signi\fcant de-\nlay between the dynamics of the two magnetic layers. In\naddition, the microscopic dynamics of electrons gener-7\n0 2 4 6 8 10 Jsd−1.0−0.50.00.51.0DOS(ε)(c)0.10.20.3∆ε(b)\n0.010.020.030.040.050.06Jeff×N2\nS(a)NS= 1\nNS= 5\nNS= 10\nmin max\nFIG. 4. (a) E\u000bective coupling Je\u000b(multiplied by N2\nS) ex-\ntracted from least-square \ft analysis of the numerical spin dy-\nnamics. The shadowed area indicates a regime with high \ft-\nting uncertainty, i.e., where the analytical macrospin solution\nsigni\fcantly departs from the full numerical one. (b) Energy\ndi\u000berence \u0001 \"between the two highest occupied energy levels\n[marked by dashed line in (c)] for NS= 1;5 and 10. (c) Detail\nof the static density of states DOS( \") forNS= 5. All pre-\nsentedJsddependencies were calculated for a one-dimensional\nchain with spacer layer size NNM= 12 and exchange coupling\nJex=\u00001.\nates a time-retarded damping in the dynamics of classi-\ncal spins [43, 49]. We illustrate this in Fig. 5(c) using a\nlong nonmagnetic layer NNM= 400 with the same ini-\ntial condition as before, however, we also introduce an\nexternal magnetic \feld in the zdirection,Bz= 1, which\ntriggers Larmor oscillations in the \frst magnetic layer.\nIt is clear that it takes a \fnite time ( t\u0019NNM=2\r) before\nthe excitation from the \frst magnetic layer (red curve)\nreaches the second one (blue curve). What is even more\nimportant in the context of our work is that the presence\nof a long spacer layer leads to a relaxation of the spin\noscillations.\nA similar relaxation e\u000bect can be obtained even for\na short spacer layer by coupling the spin valve to semi-\nin\fnite metallic leads [56, 92]. In addition, coupling to\nthe leads allows us to address a system in\ruenced by an\nexternal voltage drop [56].\nB. Voltage driven spin valve\nTo investigate the in\ruence of nonequlibrium charge\nand spin currents, resulting as a consequence of an ex-\nternal voltage drop, on the magnetization dynamics, we\nnow turn our attention to a two-dimensional spin valve\n(N= 12\u00024,NS= 4\u00024) with non-collinear magne-\n-1.0-0.50.00.51.0\n 0  100  200  300  400  500  600Nl = 5, NNM = 400, Nr = 5\nJsd = 3.3, Bz = 1(c)\nM\ntime tMx\nl\nMz\nl\nMx\nr-0.04-0.020.000.020.04\n 5  10  15  20  25  30(a)Jeff NS\nNNMJsd = 1\nJsd = 5\n10-410-310-2\n 1  10  100(b)\n|Jeff |\nNNMJsd = 1\nJsd = 5FIG. 5. (a,b) Fitted e\u000bective coupling Je\u000b, respective its mag-\nnitudejJe\u000bj, as a function of the size of the non-magnetic\nspacer layer. (c) Normalized layer magnetization for left ( Mx\nl,\nMz\nl) and right magnetic layer ( Mx\nr) in spin valve with long\nnon-magnetic layer NNM= 400,Jsd= 3:3 in homogeneous\nmagnetic \feld Bz= 1.\ntization sandwiched between two semi-in\fnite metallic\nleads (Fig. 1). As before, we set the model parameters\nwith the aim to make the analysis tractable by simplify-\ning the spin dynamics. We set an intermediate coupling\n\u0000\u0011\u0000l= \u0000r= 1 and use a \fxed temperature in the\nleadsT= 0:025. The intermediate \u0000 reduces re\rection\nof the conducting electrons at the valve-lead interface [56]\n(typical for weak \u0000), provides su\u000ecient broadening [55]\nbut does not dominate over other energy scales of the\nmodel. In addition, because \u0000 \u001dT, \fnite temperature\ne\u000bects are suppressed and, therefore, not discussed in de-\ntail here. The exchange coupling is set to Jex=\u00001 which\nis strong enough to allow us to represent the dynamics\nof a magnetic layer by its normalized net magnetization\nM`=NS(macrospins) [93]. The anisotropy in the left\nlayer is set to be large, Kl= 0:4 (therefore pinned layer\nPL) and points to x-direction. The anisotropy in the\nsecond ferromagnetic layer is set to Kr= 0:02 (therefore\nfree layer FL) and points to z-direction.\nWe work within a partition-free approach [74, 94]\nwhere the system-reservoir coupling \u0000 is assumed to be\n\fnite at all times. To investigate the e\u000bect of a \fnite\nbias voltage on the magnetization dynamics, we employ\na three-stage switching protocol which leads to the net\nlayer magnetization dynamics illustrated in Fig. 6.\nStage 0: We assume that at t<t0, the classical spins\nin each layer are perfectly parallel to the direction of the\nlayer anisotropy: Mx\nPL=NSandMz\nFL=NS. The spin\nvalve is in equilibrium with the electronic reservoirs with\nV= 0. This is ensured by solving Eq. (4) and Eq. (9)\nford\ndt\u001a= 0 andd\ndt\u0005`;p(t) = 0. Stage 1: Att=t0\u001c0\nwe ease the condition of perfect alignment of the spins\nwith the anisotropy \felds. Therefore, the e\u000bective cou-\npling between the magnetic layers, alike the one discussed8\n-0.25-0.20-0.15-0.10-0.050.000.050.100.150.20\n 2000  4000(a)\n∆Mx\nt0 ≡ 0 t’V = 0 V = 1\nJsd = 2, Γ = 1Mr /NS\ntime tMx\nr / NS\nMy\nr / NS\n1 - Mz\nr / NS\n-0.20-0.100.000.100.200.300.40\n 1000  3000  5000  7000  9000(b)\n∆Mx\nt0 ≡ 0 t’V = 0 V = 1\nJsd = 6, Γ = 1Mr /NS\ntime t\nFIG. 6. Examples of dynamics of normalized magnetization in\nthe free layer calculated for two-dimensional spin valve with\nsizeN= 12\u00024,NS=NPL=NFL= 4\u00024 and model\nparameters: Jsd= 1 (a),Jsd= 5 (b), and \u0000 = 1, Kl= 0:4,\nKr= 0:02. Yellow background marks stage one with V= 0.\nAtt=t0f\u00110ga quench to \fnite symmetric voltage drop\nV= 1 is introduced.\nfor the isolated spin valve, triggers spin dynamics. Be-\ncause of the damping driven mainly by the dissipation\nof polarized electrons, the subsystem of localized spins\nrelaxes towards a new (static) con\fguration (this stage\nis marked by the yellow background in Fig. 6). Stage 2:\nAtt=t0= 0, when the system has already relaxed into\nthe equilibrium state, we induce again a nonequilibrium\nsituation by suddenly switching on a \fnite bias voltage\nV6= 0. However, as we argue below, the way how the\ndrop is introduced plays a crucial role in the transient\ndynamics as well as in the steady-state. Therefore, we\nintroduce the voltage in two distinct ways. Either by\nshifting the chemical potential of both leads around the\nequilibrium state ( \u0016l=\u0016r=\u0016= 0):V=\u0016l\u0000\u0016rwith\n\u0016l=\u0000\u0016rwhich we call the symmetric case (in the sense\nofj\u0016lj=j\u0016rj), or by moving only the chemical potential\nof the left reservoir coupled to the pinned layer: V0=\u0016l\nwith\u0016r=\u0016= 0 addressed in the text as the asymmetric\ncase. Note that these two ways mimic di\u000berent physical\nrealizations. For example in the case of small system the\nsymmetric voltage drop can model a gate-tunable junc-\ntion or a bridge and the asymmetric one a scanning tun-\nneling microscope (STM) like geometry where the chem-\nical potential of the surface electrode is aligned with the\ngate induced electrochemical potential.\nIn our model the di\u000berence between these two scenar-\nios lies in the position of the chemical potential of the\nright lead with respect to the equilibrium electrochemi-\ncal potential \u0016\fxed by an auxiliary gate. This di\u000berence\nis important for understanding the results. Basically,\nthe drop of the chemical potential in the leads changesthe electron occupation in the valve, however, the non-\nequilibrium distribution of the charge does not modify \u0016\n\fxed by the auxiliary gate and, therefore, does not shift\nthe spectrum of the valve. A self-consistent calculation\nadjusting\u0016after introducing the leads, which might be\nnecessary for systems without a gate, is not part of the\npresented model calculations. A detailed recipe on how\nto address the problem of gauge-invariant density matrix\nin steady state nonequilibrium linear response calcula-\ntions for systems without a gate can be found in Ref. [95].\nWe show in Appendix G that for the method used here\nthe two discussed voltage-drop scenarios, although dif-\nferent in general, lead to the same charge current in the\nlinear response regime.\nIn practice, we calculate the stages zero and one only\nonce for all required voltage drops for the same system\nparameters. At t=t0\u00110 we then store the state of the\nsystem, meaning the orientation of localized spins, the\nequilibrium single particle density matrix \u001aeq(t= 0) and\nall auxiliary current matrices, and use it as the initial con-\ndition for the second stage. This gives us a well-de\fned\ninitial equilibrium state for the coupled system at V= 0\nwhich di\u000bers from the stage zero result. This is crucial,\nnot only because it enables us to calculate the actual\nchange of measured quantities due to the \fnite voltage,\ne.g., the magnetization change \u0001 M(t) =M(t)\u0000Meqor\ncurrent-driven torques, but it also allows us to investigate\na well de\fned relaxation.\n1. Transient dynamics\nThere are various relevant time scales associated with\nthe dynamics of the coupled system. Some are related to\ntwo distinct anisotropies and lead to di\u000berent spin oscil-\nlations in the pinned and free layer. Others are demon-\nstrated already by the examples of time evolution of the\nmagnetization in the free layer shown in Fig. 6. The dif-\nference between the time at which the system reaches\nthe steady-state magnetization and the relaxation time\nof the magnetization oscillations depends on the param-\neters of the model and the voltage drop. For example,\nforJsd= 2 andV= 1 in Fig. 6(a) the oscillations are\ncentered around the steady state values of Mx;z\nrquite\nearly on (t\u0018500), but the relaxation time of the oscil-\nlations themself is much longer. On the other hand, in\nthe caseJsd= 6 andV= 1 magnetization Mx\nrdecreases\nto its steady state value only very slowly and it is here\nthe longer of the two mentioned timescales. Neverthe-\nless, these are not the only relevant time scales of the\ntransient dynamics.\nFig. 7 shows examples of the time evolution of charge\nand spin currents in the second stage for the same model\nparameters as in Fig. 6. The voltage in Fig. 7(a) and (b)\nwas introduced by a symmetric voltage drop ( V= 1) and\nin panels (c) and (d) by an asymmetric one ( V0= 0:5).\nThe solid lines represent currents at the FL-lead inter-\nface, namely, green for the total charge current [Eq. (11)],9\n-0.20-0.100.000.10\n10-1100101102103104Jsd = 6, V = 1(b)\nI, J\ntI\nJ x\n2\nJ y\n2\nJ z\n2IF\nJ x\nF\nJ y\nF - J y\nF(0)\nJ z\nF\n-0.08-0.040.000.04\n10-1100101102103104Jsd = 6, V’ = 0.5(d)\nI, J\nt-0.040.000.040.080.12\n      Jsd = 2, V’ = 0.5(c)\nI, J-0.100.000.100.20\n      Jsd = 2, V = 1(a)\nI, J\nFIG. 7. Second stage of the time evolution of charge ( I) and\nspin currents (J) calculated for symmetric voltage drop V=\n1 withJsd= 2 (a) and Jsd= 6 (b) and asymmetric voltage\ndropV0= 0:5 withJsd= 2 (c) and Jsd= 6 (d). The solid\nlines show currents measured at the FL-lead interface and\ndashed lines represent respective currents at SL-FL interface.\nThe color key is the same for all panels.\nred, blue, and black for the x;y;z -component of the spin\ncurrent, respectively [Eq. (12)]. The dashed lines of the\nsame colors are local currents measured at the interface\nof the spacer and free layer [Eqs. (13), (14)].\nAlmost all currents are zero or negligible at t= 0 (not\nshown because of the logarithmic scale). The only excep-\ntion is the equilibrium local spin current Jy\nF(t= 0)6= 0.\nThis is related to the e\u000bective exchange interaction be-\ntween PL and FL and torque resulting from it that tilts\nthe spins of the magnetic layers away from the direc-\ntion of the anisotropy even in equilibrium (see yellowstage in Fig. 6). In Fig. 7 we subtracted this equilib-\nrium component from the nonequilibrium value Jy\nF(t) =\nJy\nF(t)\u0000Jy\nF(0) because here we want to investigate the\nin\ruence of the \fnite bias voltage. In accordance with\nother components, we address this di\u000berence simply as\nthe local current for brevity.\nApplication of a symmetric bias voltage induces tran-\nsient currents simultaneously through both system-lead\ninterfaces. Therefore, as shown in Fig. 7 (a) and (b),\nthere are signi\fcant currents \rowing through the right\nsystem-lead interface already after a very short time. Ap-\nproximately at t\u00192 the excitation from the right spin\nvalve interface arrives to the SL-FL interface leading to\nthe formation of local currents there. Next, at t\u00194 the\nexcitation from the left system edge reaches the SL-FL in-\nterface, which marks sudden changes in the pro\fle of the\ncurrents. Considering longer times, because the voltage\ndrop is symmetric, the spin currents at the system-lead\ninterface vanish in the steady state. Qualitatively this\ncan be understood following the pro\fle of the equilib-\nrium transmission function polarization (see the discus-\nsion in Appendix E) which is antisymmetric around the\nFermi level. Therefore, the relevant Fermi window con-\ntains compensating spin-resolved transmission channels\nand the steady-state spin current vanishes.\nThe only non-negligible long-time nonequilibrium spin\ncurrent isJy\nF, which is therefore the sole component\nof the current-driven torque in Eq. (15). This means\nthat the spin currents are fully absorbed by the mag-\nnetic layers. As we will discuss in the next section, this\nhas important consequences for the relaxation and for\nthe steady state magnetization. It also means that the\ndi\u000berence in magnetization (\u0001 Ml;r) is driven primarily\nby the e\u000bective exchange interaction between the mag-\nnetic layers. However, contrary to the isolated system\ndiscussed in Section III A, this e\u000bective coupling might\nbe strongly a\u000bected by the electronic states which are far\naway from the Fermi level of the spin valve. In addition,\nthe nonequilibrium system density of states depends on\nthe voltage drop and the orientation of the spins and is\ntherefore time dependent. The slow change of Jy\nF(t) in\nFig. 7(b) for Jsd= 6 can be attributed to the time evolu-\ntion of DOSh, which is expected to change more for the\nstrongly interacting case ( Jsd= 6) than weakly coupled\nelectrons and localized spins ( Jsd= 2).\nThe asymmetric voltage-drop examples shown in\nFigs. 7 (c) and (d) di\u000ber from the symmetric case. Be-\ncause the voltage drop is introduced only at the side of\nthe pinned layer, the free layer stays for a while in equi-\nlibrium with the right lead. Finite local currents ap-\npear around t\u00194 when the excitation from the left edge\nof the spin valve reaches the SL-FL interface and only\nlater we observe \fnite currents at the right system inter-\nface. Because the voltage drop probes the spin-dependent\ntransmission function asymmetrically, the spin currents\nat negative and positive energies do not compensate each\nother and saturate to \fnite values. Consequently, there\nare \fnite current-driven torques in both xandydirec-10\ntions even at long times.\n2. Relaxation\nThe quantity that clearly demonstrates the qualita-\ntive di\u000berence between the symmetric and asymmetric\nvoltage drop is the relaxation time tRof the magnetiza-\ntion oscillations (Fig. 8). We estimate tRby \ftting the\nenvelope of Mx\nroscillations in the second stage of the\nevolution by the exponential formula\nEM(t) =Aexp[\u0000t=tR] +Mx\nr(t!1 ); (22)\nwhere the amplitude Aand the relaxation time tRare\nthe \ftting parameters and Mx\nr(t!1 ) is the extrapo-\nlated steady-state magnetization component. We disre-\ngard in the \ftting procedure the initial evolution in the\nsecond stage ( typically up to t\u0019100\u0000300) to avoid\nthe distortions from the complicated short-time dynam-\nics discussed above. We focus on the weakly coupled\ncasesJsd= 2 and 3 to avoid the long time scales typical\nfor strongsdcoupling.\n 100 200 300 400 500 600 700\n 0  1  2  3  4  5(b)tR\nV’Jsd = 2\nJsd = 3102103104105\n 0  2  4  6  8  10(a)\ntR\nVJsd = 2\nJsd = 3\nFIG. 8. Relaxation time estimated from the decay of the os-\ncillations of the x-component in the FL magnetization. Panel\n(a) shows the symmetric voltage case (note the logarithmic\ny-scale), panel (b) the asymmetric one.\nThe \ftted relaxation time tRshows a qualitatively\ndi\u000berent dependence on voltage for the symmetric\n[Fig. 8(a)] and asymmetric [Fig. 8(b)] cases. The relax-\nation time calculated for the symmetric case is changing\nby several orders of magnitude with increasing voltage\nV. It grows to very large values tR\u0019105at high volt-\nages (see also the discussion on the numerical precision\nin Appendix A). On the other hand, the relaxation time\nin Fig. 8(b) is relatively stable. It changes within \fve\nhundred time units and saturates for high voltage V0,\nwheretRis several orders of magnitude smaller than for\nthe symmetric case.\n-0.10-0.050.000.050.100.15\n 0  200  400  600  800  1000Jsd = 2, V’ = 4(b)J, Tcd\nt-0.10-0.050.000.050.10\n      Jsd = 2, V = 8(a)J, TcdJ x\nr\nJ y\nr\nJ z\nrT x\ncd\nT z\ncdFIG. 9. Detail of the time evolution of spin currents ( J) and\ncurrent-driven torques ( Tcd) calculated for symmetric voltage\ndropV= 8 withJsd= 2 (a) and asymmetric voltage drop\nV0= 4 withJsd= 2 (b).\n-0.5-0.4-0.3-0.2-0.10.00.10.2\n 0  200  400  600  800  1000Jsd = 2, V’ = 4(b)\n∆M/NS\nt∆Mx\nr\n∆My\nr\n∆Mz\nr-0.4-0.3-0.2-0.10.00.10.20.3\n      Jsd = 2,  V = 8(a)\n∆M/NS\nFIG. 10. Detail of the time evolution of non-equilibrium\nmagnetization di\u000berence (\u0001 M=NS) calculated for symmet-\nric voltage drop V= 8 withJsd= 2 (a) and asymmetric\nvoltage drop V0= 4 withJsd= 2 (b).\nThis signi\fcant discrepancy in relaxation time can be\nattributed to the di\u000berences in the spin currents at the\nsystem-lead interface and related torques. The damping\nof the localized spin dynamics comes from the interac-\ntion with the leads, which act as reservoirs that carry\naway spin excitations from the system. However, this11\nis possible only when they couple to the spin-resolved\nelectronic states, i.e., when there are signi\fcant spin cur-\nrents \rowing between the system and the leads. This\nis not the case for the symmetric voltage drop as it is\nillustrated in Fig. 9(a) where we show the charge cur-\nrents and current-driven torques for V= 8 andJsd= 2.\nBoth currents and relevant torques are quickly dimin-\nishing, and hence do not exert any signi\fcant torque on\nthe localized spins. Therefore, the magnetization shows\na Larmor-like precession due to the e\u000bective \felds as il-\nlustrated in Fig. 10(a). All this is in clear contrast with\nthe respective asymmetric case ( V0= 4) illustrated in\nFig. 9(b) and Fig. 10(b).\nHowever, the dependence of the relaxation time on the\nvoltage drop is far from monotonous. The origin of the\ncomplicated pro\fle can be traced to the density of states.\nIn Fig. 11(a) we show the relaxation rates R= 1=tRat\nJsd= 2 calculated for symmetric (red) and asymmetric\n(blue) voltage drop. The sharp maxima in both Rcurves\nfollow the pro\fle of DOSh in Fig. 11(b) calculated for\nthe equilibrium spin con\fguration at V= 0. This can be\nattributed to the boost of relaxation whenever the chem-\nical potential of the leads is aligned with the (polarized)\nstates in the system. Such a boost is in compliance with\nsingle spin studies [56, 65, 96, 97] and shows the impor-\ntance of the correct treatment of the electronic spectrum.\nInterestingly, for the symmetric voltage drop, the boost\nhappens even for states with energies close to the edge\nof the spectrum. These are localized predominately on\nthe ferromagnetic layers and as such practically do not\ncontribute to the steady state transport (see discussion\nin Appendix E). However, these states can contribute to\nthe relaxation if they are aligned with the chemical po-\ntential of the neighboring lead. In the asymmetric case,\nthis contribution is overshadowed by the damping regu-\nlated by the \fnite spin currents.\n3. Steady state\nFig. 12 shows (up to a constant factor) the steady\nstate magnetization di\u000berence \u0001 Mas a function of the\nsymmetric voltage drop (circles and crosses in the \fg-\nures). The dependence is rather complicated and does\nnot straightforwardly follow the equilibrium DOSh even\nfor smallJsd[compare Fig. 12(a) and Fig. 11(b)]. Nev-\nertheless, it can be fully explained by considering the\ncurrent-driven torques acting on the magnetic layers.\nIn the steady state, the net sum of all torques in the\nsystem is zero. In our case, the dominant contribution to\nthe e\u000bective local \felds acting on the localized spin, and\ncounteracting the current-driven torques, should come\nfrom the misalignment of the spin with the anisotropy\n\feld and from the interaction with neighboring localized\nspins through Jex[see Eq. (10)]. However, assuming that\nthe classical spins are aligned with each other, we can\napproximate the net e\u000bective torque acting on the right\nmagnetic layer by Te\u000b\u00192(Krer)\u0002\u0001Mr.\n 0 0.2 0.4 0.6 0.8\n 0  1  2  3  4  5(b)\nDOSh\nε 0 0.001 0.002 0.003 0.004 0.005 0.006\n 0  1  2  3  4  5(a)R = 1 / tR\nV/2, V’Symmetric V\nAsymmetric V’FIG. 11. (a) Relaxation rates calculated for symmetric (red)\nand asymmetric (blue) voltage drop at Jsd= 2. Thex-scale\nfor the symmetric voltage is scaled by factor 0 :5 to align volt-\nages with the same chemical potential \u0016lprobing energies \".\n(b) Equilibrium density of states of the heterostructure.\n-0.20-0.15-0.10-0.050.00\n 0  2  4  6  8  10Jsd = 6(c)\nTcd y   , Teff y\nV2Krz ∆Mrx \n2Klx ∆Mlz \nTcd y   -0.14-0.12-0.10-0.08-0.06-0.04-0.020.000.02\n      (a)\nKlx = 0.4, Krz = 0.02,   Jsd = 2Tcd y   , Teff y2Krz ∆Mrx \n2Klx ∆Mlz \nTcd y   \n-0.20-0.15-0.10-0.050.00\n      (b)\nJsd = 3Tcd y   , Teff y2Krz ∆Mrx \n2Klx ∆Mlz \nTcd y   \nFIG. 12. Comparison of the steady state current-driven\ntorquesTcdand e\u000bective local torques Te\u000b= 2K`e`\u0002\u0001M\ncalculated as function of Vfor symmetric voltage drop.12\n-0.10-0.08-0.06-0.04-0.020.00\n 0  1  2  3  4  5(b)\nTcd y   ,  Teff y \nV2Krz ∆Mrx -0.020.000.020.040.060.080.100.12\n      (a)\nTcd x   ,  Teff x Jsd = 2\nJsd = 3\nJsd = 6\n-2Krz ∆Mry \nFIG. 13. Comparison of the steady state current-driven\ntorquesTcdand e\u000bective local torques Te\u000b= 2K`e`\u0002\u0001M\ncalculated as function of V0for asymmetric voltage drop.\nBecause in the symmetric case the only signi\fcant tilt\nof the spins due to the \fnite voltage is in the x-direction\nand there are no steady state spin currents at the system-\nlead interfaces, the e\u000bective local torques can be further\napproximated by Ty\ne\u000b= 2Kr\u0001Mx\nr(blue circles in Fig. 12)\nfor free layer and 2 Kl\u0001Mz\nl(black crosses in Fig. 12) for\npinned layer and compared directly to the Tcd(red lines\nin Fig. 12). There is almost a perfect agreement between\nthese three quantities plotted in Fig. 12 for various Jsd.\nConsidering the asymmetric case, the magnetization\n\u0001Mx\nrshown in Fig. 13(b) di\u000bers from the symmetric one\nin Fig. 12 mostly in its magnitude. However, the asym-\nmetric case also shows a signi\fcant steady state decli-\nnation for the ycomponent of the magnetization \u0001 My\nr.\nThis di\u000berence can be again explained by the current-\ndriven torques. There are \fnite spin currents \rowing\nbetween the system and leads. Using the same assump-\ntions as for the symmetric case, we can estimate the ef-\nfective local torques in the FL to be Tx\ne\u000b= 2Kr\u0001My\nr\nandTy\ne\u000b= 2Kr\u0001Mx\nr. The comparison with the current-\ndriven torque for various Jsd(green line in Fig. 12) shows\nagain a very good agreement. We can therefore conclude\nthat the \fnite spin-polarized currents tilt the spins of the\nfree layer not only to the direction of the magnetization\nin the pinned layer (respective opposite to it), but also\nperpendicular to both anisotropy \felds.\nThis, however, opens an interesting question consider-\ning the steady state of the symmetric case. There are no\nspin-polarized steady-state currents \rowing between the\nsystem and the leads. Yet, the orientation of the clas-\nsical spins can not be a\u000bected by the non-polarized cur-\nrents. Therefore, the steady state magnetization seems to\nbe fully dictated by the intravalve spin-polarized currentJF. Following the analysis of the transient dynamics,\none can conclude that JFre\rects the polarization of the\ndensity of states probed by the chemical potential of the\nleads. Considering the symmetry of the spectrum as well\nas the system symmetry and the fact that the charge cur-\nrent plays no role in current-driven torque, there arises a\nquestion, if a similar e\u000bect can be achieved also in equi-\nlibrium.\n-0.15-0.10-0.050.000.050.100.150.20\n 0  2  4  6  8  10Jsd = 3Mrx\nV, 2 µ-Mrx(V)\nMrx(µ-  )\nFIG. 14. Dependence of steady state orientation of the mag-\nnetizationMx\nron symmetric voltage drop V(red line) and\nthe dependence of equilibrium magnetization Mx\nron the spin-\nvalve electrochemical potential \u0016(circles) for Jsd= 3. The\nx-axis for the later case is scaled by factor of 2 to account for\nV= 2\u0016l= 2j\u0016rj.\nThis is indeed the case as shown in Fig. 14. Here we\ncompare the steady state FL magnetization plotted as a\nfunction of voltage (line) with its equilibrium counterpart\ncalculated for a \fxed \u0016l=\u0016r= 0 as a function of the\nelectrochemical potential \u0016forJsd= 3 (circle). They are\nin perfect agreement. For the symmetric voltage drop the\ne\u000bective exchange coupling between the magnetic leads\nis de\fned by the polarization of the steady state DOSh.\nHowever, it is not dictated by the Fermi-level of the iso-\nlated valve, but by the states probed by the chemical\npotential of the leads.\nIV. SUMMARY\nMagnetic multilayer devices are, besides being impor-\ntant components in a multitude of industrial applica-\ntions, an ideal tool to investigate various physical con-\ncepts. In this paper, we examined the spin-transfer\ntorque and related relaxation processes in a spin-valve\nsystem under external voltage bias. To this goal we have\nadapted a QC-EOM method, which bridges the classical\nand quantum mechanical approaches by treating the lo-\ncalized spins as classical degrees of freedom that interact\nwith conduction electrons treated as quantum particles.\nWe intentionally focused on regimes where the dynamics\nof the localized spins can be, to a high degree, repre-\nsented by the net magnetization in the magnetic layer\n(macrospin). This allowed us to analyze the numerical\nresults using intuitive approximations. We have shown\nthat even in such idealized cases the dynamics is rather13\ncomplicated because it re\rects a complex relation be-\ntween the localized spins and conduction electrons.\nIn the case of an isolated spin valve, the interplay of\nclassical and quantum degrees of freedom induces a com-\nplicated e\u000bective exchange interaction Je\u000bbetween the\nferromagnetic layers. Although Je\u000bis a\u000bected mostly by\nelectron states near the Fermi-level, it shows a complex\nnonmonotonous dependence on the spin-electron cou-\nplingJsd.\nCoupling the spin valve to metallic leads and introduc-\ning a \fnite bias voltage by shifting their chemical poten-\ntials triggers nonequilibrium spin currents, and therefrom\nspin-transfer torques in the system. Besides in\ruencing\nthe magnetizations in the spin valve, the spin currents\nalso control the relaxation processes of the spin dynam-\nics. We have observed a resonant character of the relax-\nation, which is boosted whenever the chemical potential\nof at least one of the leads matches the maxima in the\nelectronic density of the states of the spin-valve electrons.\nHowever, there is a qualitative di\u000berence in the tran-\nsient dynamics, spin relaxation, and even steady state\ncharacteristics between the system under symmetric or\nasymmetric voltage drop with respect to the electrochem-\nical potential of the valve. For example, the relaxation\ntime at high voltages can be of several orders of magni-\ntude longer for the symmetric case than for the asymmet-\nric one. This is a consequence of the fact that there is no\nlong-time spin-polarized current \rowing between the sys-\ntem and the leads in the symmetric case. Interestingly,\nthe steady state magnetization governed by the symmet-\nric voltage drop Vcan be mapped to a magnetization of\nthe spin valve in equilibrium ( V= 0) with the electro-\nchemical potential adjusted to \u0016=V=2 (whereVre\rects\nthe voltage drop in the nonequilibrium case).\nV. ACKNOWLEDGEMENTS\nThe authors acknowledge support by the state of\nBaden-W urttemberg through bwHPC and the German\nResearch Foundation (DFG) through grant no INST\n40/467-1 FUGG (JUSTUS cluster). M. \u0014Z. acknowledges\nsupport by the Czech Science Foundation via Project\nNo. 22-22419S. This work was supported by the Ministry\nof Education, Youth and Sports of the Czech Republic\nthrough the e-INFRA CZ (ID:90140). The authors thank\nRichard Koryt\u0013 ar and Tom\u0013 a\u0014 s Novotn\u0013 y for helpful discus-\nsions.\nAppendix A: Pad\u0013 e representation\nBesides the integration step the only convergence pa-\nrameter of the here used QC-EOM is the number of Pad\u0013 e\npoles in the representation of the Fermi function.\nA rather low temperature of T= 0:025 (\f= 40) used\nin our study requires a relatively high number of Pad\u0013 e\npoles. The comparison of exact Fermi function and theapproximated one, Eq. (7), with various number of Pad\u0013 e\npoles is shown in Fig. 15. We have found, that a su\u000ecient\nprecision was obtained in our calculations for NP= 30\nwhich we use throughout the paper.\n 0 0.2 0.4 0.6 0.8 1\n-50 -25  0  25  50β = 40f(ε)\nεFermi fun.\nNP = 10\nNP = 20\nNP = 30\nNP = 50\nFIG. 15. Comparison of exact Fermi function with its approx-\nimation Eq. (7) calculated for NP= 10;20;30;50.\nAppendix B: Localization in one-dimensional valve\nIt is important to note that the regimes of weak\n(Jsd.3) and strong coupling (4 :5.Jsd) di\u000ber signi\f-\ncantly. In the transient regime 3 .Jsd.4:5 we observe\nthe opening of the gap in the local DOS of the mag-\nnetic layers and with it related transition from metallic\nto insulating like character. In addition, here, the lo-\ncalization of electronic eigenstates starts to change sig-\nni\fcantly. In Fig. 16 we show the spatial distribution\nPn\n`=P\nj2`j\u001en(rj)j2, where energy eigenstate \u001en(rj) is\nthe space resolved eigenstate and the sum is restricted to\npositions within the layer `= FL/PL,SL. We focus on\nthe two eigenstates ( n= 0;1) with the lowest eigenen-\nergies, which exhibit the most pronounced localization\ne\u000bects upon increasing Jsd. ForJsd<3, it is not pos-\nsible to infer the typical localization properties of eigen-\nstates from the local probabilities Pn\nL. However, there is\na signi\fcant changes for Jsd>4, where a clear localiza-\ntion of eigenstates can be observed. The point at which\nthis change occurs, coincides precisely with the point at\nwhich spin \ructuations discussed in Section III A 3 are\nthe strongest.\nAppendix C: Rationalization of the macrospin model\nThe macrospin approximation, used in the analysis of\nthe dynamics of the closed spin valve system, can be\njusti\fed by the scheme illustrated in Fig. 17.\nIn the \frst step of this approximation we split Sj(t) =\nSj(0)+\u0001Sj(t) and diagonalize the single particle Hamil-\ntonian from Eq. (1) at t= 0 resulting in a transformed14\n0 2 4 6 8 10 Jsd0.00.30.60.9Pn\n/lscriptN= 20 Pn\nSL\nPn\nPL/FLn= 0\nn= 1\nFIG. 16. Spatial distribution (local probability) Pn\n`of energy-\neigenstate\u001enin layer`, withPn\nPL=FLsummed over all sites\nj2PL=FL (black for n= 0 and gray for n= 1), andPn\nSL\nover allj2SL (red for n= 0 and orange for n= 1). Red\nvertical lines denote Jsdfor which \u0001 \"has a local maximum,\nblack vertical lines those Jsdwhere \u0001\"has a local minimum,\nand gray shaded area denotes the parameter regime in which\nside-bands split of the main band, as in Fig. 4.\nFIG. 17. Schematic illustration of the macro-spin approxi-\nmation: Spin-electron coupled hybrid multi-spin system (left\npanel) is reduced to two-spin system coupled to the electronic\nspectrum (middle panel). After integrating out the electronic\ndegrees of freedom, the system is further reduced to a two-\nspin problem coupled via an e\u000bective exchange interaction JR\n(right panel).\nsystem\nH=X\n\u000b\"\u000bcy\n\u000bc\u000b+JsdX\n\u000b;\u000b02PL;FLcy\n\u000bS\u000b;\u000b0(t)c\u000b0;\n(C1)\nS\u000b;\u000b0(t) =X\nj2PL;FLX\n\u001b\u001b0Uj;\u001b;\u000b(\u001b\u0001\u0001Sj(t) )\u001b\u001b0Uj;\u001b0;\u000b0;\nwhereUj;\u001b0;\u000b0are components of the eigenvectors of\nH(t= 0). Next, a coarse-grained description of the\nlocalized spins is applied, where each magnetic layer is\ncharacterized by a local magnetization Ml;r=PNS\njSj,\nwhereNSis the number of spins in a layer ( Nl(PL) =\nNr(FL)=NS). We can interpret the simpli\fed system as\ntwo macrospins coupled through a spectrum of single par-\nticle energies \"\u000bvia complex time-dependent couplings.\nNote that using this interpretation one can argue that\nfor low enough temperatures only a few states near the\nFermi level will play an important role in the dynamics.\nAs a last step in the macro-spin approximation, the\ncentral part is approximated by an e\u000bective direct ex-\nchange coupling Je\u000bbetween the spins, which we assume\nto be time-independent. Under these assumptions, the\nproblem is reduced to two spins coupled by Je\u000b.The exact solution of Eq. (20) used in the extraction of\nthe e\u000bective exchange interaction Je\u000bbetween the mag-\nnetic layers of the closed spin valve system can be derived\nby the following steps. By recognizing that the cross-\nproductMl(r)\u0002Mr(l)can be rewritten as a matrix vec-\ntor multiplication A\u0002B= [A]\u0002B. Here,A;B2R3\nand [A]\u0002=P3\n\u000b=1A\u000bL\u000b, withL\u000bbeing the basis of the\nLie-algebra SO(3). These elements generate in\fnitesi-\nmally rotations in R3. and the cross-product in R3can\nbe expressed using in\fnitesimal rotations around axis A\nd\nd\u0012\f\f\f\f\n\u0012=0R(\u0012;A)B=A\u0002B: (C2)\nThus, information about the trajectories of the\nmacrospins can be obtained from in\fnitesimal rotations.\nEach of the macrospins is tracing out a trajec-\ntory around the instantaneous position of the other\nmacrospin. Due to the antisymmetry of the cross prod-\nuct, the center of spins is conserved Ml+Mr\u0011M=\nconst. Without loss of generality, we assume M=\r^ez\nwith\r=Mz\nl(t0)+Mz\nr(t0) determined by the initial con-\ndition of both macro-spins because @tM= 0. This as-\nsumption is equivalent to a change of the basis into a\nframe of reference by a rotation of \u0012=\u0000\u0019=4 around the\nCartesiany-axis in the original frame. The trajectory of\nMl;Mris an intersection between the unit sphere S2and\na straight plane at z=\r. These constraints are ful\flled\nby a circleCr\u0018=S1with radius r`=p\nM2\n`\u0000\r2, where\nM`=jM`j2. Under these considerations, the general\nsolution is of the form in the rotated frame is\nM`(t) =0\n@r`cos(!t+\u001e`)\nr`sin(!t+\u001e`)\n\r1\nA; ! = 2\rJe\u000b:(C3)\nand thus Eq. (21) is obtained by rotating Eq. (C3) into\nthe original frame of reference by applying the rotation\nmatrixRy(\u0012) as given in Eq. (21). Due to the symme-\ntry of the system of equations !l=!r=!, where!\noriginates directly from solving Eq. (20), and the phase\ndi\u000berence is exactly \u001el\u0000\u001er=\u0019.\nAppendix D: Spin Fluctuations in the intermediate\ncoupling regime\nIn the main text, we argue that the Jsddependence of\nthe e\u000bective coupling Je\u000bfollows the equilibrium energy\ndi\u000berence \u0001 \"between the two highest occupied energy\nstates (Figure 4). However, this correspondence is invalid\nin the regime 3 .Jsd.4:5. The purpose of this section\nis to elucidate the origin of this discrepancy.\nIn Figure 18 (a) we show the details of the dynamics of\na single classical spin in a FM spin valve with the same\nparameters as in Section III A 3 for Jsd= 0:45, 3:5 and\n5. We diagonalize the Hamiltonian H(t)\u0011H(fSi(t)g)\nfor each time tto obtain the respective spectrum f\"ng15\nfor the spin con\fguration fSi(t)g, and compute there-\nfrom the magnetic splitting \u0001 \"(t) shown in Figure 18\n(b) and the gap (c). It is obvious that in contrast to the\nweak and strong coupling regime, both \u0001 \"(t) and the\ngap show strong \ructuations. In addition, the gap sig-\nni\fcantly departures from its initial value [dotted black\nline in (c)]. Note that neither \u0001 \"(t) nor the gap shown in\nFig. 18 present the actual nonequilibrium values. Never-\ntheless, they both imply that in contrast to the two other\nregimes, the static spectrum is insu\u000ecient for the analy-\nsis of the intermediate regime and so is the macroscopic\napproximation which assumes jMl;rj=NS= 1.\n0.00.51.0Sx\n1Jsd=0.45\nJsd=3.5Jsd=5.0(a)\n0.00.10.2∆e(b)\n0 200 400 600 800 1000\nt0.10.20.3band gapt=0(c)\nFIG. 18. Dynamics of representative classical spin Sx\n1at site 1\nin the PL (a) and time-dependence of energy-di\u000berence \u0001 \"for\nthe same parameter (b) for Jsd= 0:45 (blue) and Jsd= 3:5\n(black) in a spin valve with the same system parameters as\ndiscussed in Section III A 3.\nAppendix E: Equilibrium spectral properties\nThe equilibrium spin con\fguration gives access to the\nequilibrium DOSh, charge and spin-resolved transmission\nfunctions (16)-(18) calculated using the equilibrium ori-\nentations of the localized spins. In cases where the in-\ntroduction of the \fnite voltage leads only to a relatively\nsmall reorientation of the classical spins, and therefore a\nsmall change of DOSh (e.g., weak interaction Jsd), these\nequilibrium functions of energy are helpful in the inter-\npretation of some nonequilibrium results. Fig. 19 illus-\ntrates the equilibrium DOSh and transmission functions\ncalculated for the two cases shown in Fig. 6. The sharp\nstates in DOSh for J= 6 in Fig. 6(a) re\rect the fast\nvanishing broadening of the states, originating from the\ncoupling to the leads, in the central part of the system for\nstrong interaction Jsd[55, 98{100]. Related to the strong\nJsdis also the signi\fcant drop of transmission when com-\npared with Jsd= 2 in Fig. 6(b). This drop re\rects the\nopening of the gap in the magnetic layers and the related\n0.00.20.40.60.81.0\n-6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6(a)\nDOS\nεJsd = 2\nJsd = 6\n0.01.02.03.04.05.0\n-6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6(b)\nΘ2\nεJsd = 2\nJsd = 6\n-2.0-1.5-1.0-0.50.00.51.01.52.0\n-6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6(c)\nP\nεJsd = 2\nJsd = 6FIG. 19. Examples of equilibrium density of states (a), trans-\nmission function (b) and spin polarization (c) calculated for\nthe same cases as shown in Fig. 6.\nchange of the character of the spin valve from metallic-\nlike to insulating-like. Note that the states far away from\nthe Fermi level, belonging mostly to the magnetic layers\n(see Fig. 2), are less relevant for the charge transport\nfrom left to right lead than the central ones. Neverthe-\nless, as we discuss later on, they play a role in the relax-\nation. The spin-polarization of the transmission function\nmeasured at the right system interface in Fig. 6(c) shows\na rather complicated energy dependence, however, what\nis important for our analysis is that it is antisymmetric\naround the Fermi-level.\nAppendix F: System with valve-lead interfaces\nIn the main text we focus on a simple model where\nthe valve was coupled directly to the semi-in\fnite leads\nwhose in\ruence on the system is modeled by the cur-\nrent matrices. However, in real systems the surface of\nthe leads can get spin polarized due to the proximity\nof the magnetic layer and related spin-currents. There-\nfore, there arises a question if the di\u000berences between the\nsymmetric and asymmetric voltage drop survive such an\ne\u000bect. To partially address this problem we introduce\nmetallic interfaces between the valve and the leads. Ba-\nsically, the valve is prolonged by six monolayers before\nthe pinned layer and by six metallic monolayers after the\nfree magnetic layer. That way we investigate a valve\nwith 24\u00024 points where PL starts at Nl= 7 and FL16\n 0 1 2 3 4\n 2  4  6  8  10  12  14  16  18  20  22  24(b)\nVSymmetric voltage drop\n-0.4-0.3-0.2-0.1 0 0.1\nPzjl 0 1 2 3 4(a)\nVAsymmetric voltage drop\n-0.4-0.3-0.2-0.1 0 0.1\nPzjl\n-0.4-0.2 0\n 2  4  6  8  10  12  14  16  18  20  22  24(c)V = V’ = 4PzjlSymmetric, metallic interface\nAsymmetric, metallic interface\nSymmetric\nAsymmetric\n-0.4-0.2 0\n 2  4  6  8  10  12  14  16  18  20  22  24(d)V = V’ = 2Pzjl\nNjlSymmetric, metallic interface\nAsymmetric, metallic interface\nSymmetric\nAsymmetric\nFIG. 20. Local electron spin polarization Pz\njlin each mono-\nlayer for a valve extended by \fnite metallic interface with\nJsd= 2. The vertical lines show the edges of the bare\nvalve. All other parameters are identical to the system stud-\nied in Section III B. Panels (a) and (b) show the evolution\nofPz\njlwith Asymmetric (a) and symmetric (b) voltage drop.\nPanles (c) and (d) compare Pz\njlof bare and extended valve at\nV=V0= 4 anV=V0= 2.\natNl= 15 where Nlcounts the monolayers from the\nleft edge of the system. In Fig. 20 we show the local\nsteady state electron spin polarization in z-directionPz\njl,\ni.e., the normalized electron spin-density calculated by\nsumming and normalizing all steady-state contributions\nin a vertical monolayer\nPz\njl=X\njvTr\u001bz\u001afjl;jvg=X\njvTr\u001afjl;jvg; (F1)wherefjl;jvgare longitudinal and vertical coordinates\nof lattice point j. The top two panels show the evolution\nofPz\njlwith voltage for asymmetric (a) and symmetric\n(b) voltage drop. The bottom two panels present a com-\nparison of Pz\njlfor systems with and without the \fnite\nmetallic interface at V= 4 andV= 2. The dashed ver-\ntical lines mark the edges of the original valve without the\n\fnite metallic interface. The tendency towards the po-\nlarization of the metallic interface is most visible for the\nasymmetric voltage drop case at high V0[note the yellow\narea in panel (a) and the elevation of the black curve at\nNl>18 in panel (c)]. The e\u000bect is most pronounced at\nthe edge of the FL, where it opposes the strong polariza-\ntion observed within the FL, and vanishes with increasing\ndistance from the FL edge. This e\u000bect is, naturally, not\ncaptured by the simple model without the interface.\n0.000.250.500.75\n 0  0.5  1  1.5  2  2.5  3  3.5  4Jsd = 2(c)\nI\nV, V’0.000.050.100.15(b)\n|Jr z|Symmetric with metallic interface\nAsymmetric with metallic interface\nSymmetric v.d.\nAsymmetric v.d.0.000.010.02(a)\n|Jr x|\nFIG. 21. Comparison of the spin (a),(b) and charge (c)\ncurrents calculated for a valve without metallic interfaces\n(N= 12\u00024,NS= 4\u00024) and with \fnite metallic interface\n(N= 24\u00024,NS= 4\u00024) for both symmetric and asymmetric\nvoltage drops and Jsd= 2. All other parameters are identical\nto the system discussed in Section III B.\nHowever, when comparing the spin and charge currents\nmeasured at the right system-lead interface (Fig. 21) we\nsee the same qualitative behavior for the system with-\nout \fnite metallic interface (red and blue lines) and with\nit (orange and black lines). Note that di\u000berences in the\ncourse of the current functions are expected. As dis-\ncussed in the main text, the system is small enough for\ncurrents to be sensitive to the energy spectrum of the\nvalve. This is signi\fcantly modi\fed by adding the in-17\nterface which doubles the number of sites of the lattice.\nNevertheless, in both cases (with and without the \fnite\ninterface) there are \fnite steady-state spin currents for\nthe asymmetric voltage drop and none for the symmet-\nric one. As discussed in the main text this di\u000berence is\nthe main reason for the dramatic di\u000berence in the spin\nrelaxation of these two cases.\nOn the other hand, the enlargement of the valve by\nmetallic interfaces seems to broaden the range of volt-\nages at which is the I\u0000Vcharacteristic approximately\nlinear. As a consequence, for the extended valve there\nis a better agreement between the symmetric and asym-\nmetric charge currents for 0 :1<V < 2 than for the bare\nvalve. We discuss the linear regime in more detail in the\nnext appendix.\nAppendix G: Linear response regime\nIn this appendix we focus on the regime of small volt-\nage. In Fig. 22 we show the details of I\u0000Vcharacteristics\ncalculated for the case of a bare valve with di\u000berent Jsd\nas well as for a valve extended by the metallic interface\ndiscussed in Appendix F. In all presented cases (and also\nfor various one-dimensional geometries not shown here)\nthe asymmetric and symmetric voltage drops give the\nsame charge current for small voltages if Idepends ap-\nproximately linearly on Vas expected [95]. For larger\nvoltages, a clear di\u000berence appears between symmetric\nand asymmetric voltage drop. Here we are entering a\nnon-linear regime where the system is more similar to a\nresonant level model. The transport is sensitive to the\ncomplex density of states of the heterostructure. For ex-\nample, the current is signi\fcantly enhanced whenever the\nchemical potential of a lead is aligned with a maximum\nin DOSh.\n0.000.010.020.030.040.05\n 0  0.1  0.2  0.3  0.4  0.5Jsd = 6(c)\nI\nV, V’0.000.100.20\n      (a)\nKlx = 0.4, Krz = 0.02,   Jsd = 2ISymmetric, metallic interface\nAsymmetric, metallic interface\nSymmetric\nAsymmetric\n0.000.050.10\n      (b)\nJsd = 3IFIG. 22.I\u0000Vcharacteristics at small voltages illustrating\nthat both symmetric and asymmetric voltage drops lead to\nthe same results in the linear response regime observed for\nV\u001c1. The orange and black points in panel (a) show the\nresult for a valve extended by an \fnite metallic interface dis-\ncussed in Appendix F. The red and blue data points were\ncalculated for the same parameters as discussed in Sec. III B.\n[1] B. Dieny, Journal of Magnetism and Magnetic Materials\n136, 335 (1994).\n[2] D. L. Mills and J. A. C. 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Institute of Theoretical Physics, Department of Physics,\nUniversity of Hamburg, Jungiusstra\u0019e 9, 20355 Hamburg, Germany\n2The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany\nWe discuss the emergence of an e\u000bective low-energy theory for the real-time dynamics of two clas-\nsical impurity spins within the framework of a prototypical and purely classical model of indirect\nmagnetic exchange: Two classical impurity spins are embedded in a host system which consists of\na \fnite number of classical spins localized on the sites of a lattice and interacting via a nearest-\nneighbor Heisenberg exchange. An e\u000bective low-energy theory for the slow impurity-spin dynamics\nis derived for the regime, where the local exchange coupling between impurity and host spins is\nweak. To this end we apply the recently developed adiabatic spin dynamics (ASD) theory. Besides\nthe Hamiltonian-like classical spin torques, the ASD additionally accounts for a novel topological\nspin torque that originates as a holonomy e\u000bect in the close-to-adiabatic-dynamics regime. It is\nshown that the e\u000bective low-energy precession dynamics cannot be derived from an e\u000bective Hamil-\nton function and is characterized by a non-vanishing precession frequency even if the initial state\ndeviates only slightly from a ground state. The e\u000bective theory is compared to the fully numerical\nsolution of the equations of motion for the whole system of impurity and host spins to identify\nthe parameter regime where the adiabatic e\u000bective theory applies. E\u000bective theories beyond the\nadiabatic approximation must necessarily include dynamic host degrees of freedom and go beyond\nthe idea of a simple indirect magnetic exchange. We discuss an example of a generalized constrained\nspin dynamics which does improve the description but also fails for certain geometrical setups.\nI. INTRODUCTION\nThe coupling between two magnetic moments can be\na so-called direct coupling, such as the usually short-\nranged quantum Heisenberg exchange interaction or the\nlong-ranged classical dipole interaction, or an indirect\ncoupling [1{3]. All indirect coupling mechanisms, e.g.,\nAnderson's superexchange [4], double exchange [5, 6],\nthe Ruderman-Kittel-Kasuya-Yosida (RKKY) interac-\ntion [7{9], or more exotic mechanisms [10, 11], have in\ncommon that they are derived perturbatively. They rep-\nresent e\u000bective interactions between the magnetic mo-\nments or spins, generically of the form Je\u000bS1S2, where\nthe e\u000bective interaction strength Je\u000bis typically more\nthan an order of magnitude smaller than the typical en-\nergy scales of the host, in which the spins S1andS2are\nembedded.\nIn the RKKY case, for example, two impurity spins are\nembedded in an electronic host system, typically a metal-\nlic Fermi liquid. To avoid complications due to Kondo\ne\u000bect [12] and its intertwining with the RKKY coupling\n[13{17], the impurity spins are represented by classical\nspin vectorsS1andS2. If the local exchange coupling\nKof the impurity spins to the local magnetic moments\nof the electron system is weak as compared to the en-\nergy scales of the host, e.g., the Fermi energy, one may\nuse standard perturbation theory to derive the e\u000bective\nRKKY Hamilton function He\u000b(S1;S2) =JRKKYS1S2.\nThe RKKY interaction JRKKY =K2\u001f12(!= 0) is given\nin terms of the nonlocal retarded static (zero-frequency)\nmagnetic susceptibility \u001f12(!= 0), which is an oscilla-\ntory and decaying function of the inter-impurity distance.\nThe condition He\u000b(S1;S2) = min then provides us with\nthe impurity-spin ground state con\fguration. Obviously,the derivation of simple e\u000bective models is only possible\nif there is a clear separation of energy scales.\nE\u000bective low-energy exchange couplings, like the\nRKKY interaction, are also employed to predict the real-\ntime spin dynamics in atomistic spin-dynamics theories\n[18]. This is justi\fed, for instance, if only the impurity\nspin degrees of freedom are driven out of equilibrium so\nthat one stays in the low-energy sector. In other words,\ne\u000bective low-energy magnetic couplings also govern the\nreal-time dynamics, if the dynamics of the impurity spins\nis slow compared to the fast electron dynamics and if\nonly the former are excited initially. Generally, this argu-\nment exploits that a separation of energy scales obviously\ntranslates into a separation of time scales.\nA su\u000eciently weak coupling Knot only leads to a sep-\naration of energy or time scales but also implies that\nlinear-response theory applies, i.e., in \frst-order-in- K\ntime-dependent perturbation theory [19{21]. Apart from\nsetups which intrinsically prepare non-equilibrium states\n[22, 23], such as transport through nano-structures cou-\npled to leads, linear-response theory predicts that e\u000bec-\ntive impurity-spin couplings are ground-state properties\nof the host. In the RKKY case, it is the ground-state\nmagnetic susceptibility \u001f12(!= 0) that determines the\nRKKY e\u000bective interaction. Besides this, full linear-\nresponse theory and ground-state response functions also\ndescribe other e\u000bects, such as Gilbert damping or inertia\ne\u000bects [20, 21, 24{27], but those come at higher order in\nan expansion in the typical memory time scale and can\nthus be classi\fed as being of secondary importance.\nClosely related to linear-response theory is the idea\nthat the state of the host system, at any instant of time t,\nis the ground state for the given con\fguration of the clas-\nsical impurity spins ( S1(t);S2(t)) at this time. This isarXiv:2009.06296v2  [cond-mat.mes-hall]  28 Jan 20212\nS1S2?host spinseffective impurity-spin dynamicsfull dynamicsimpurity spinssi\nFIG. 1. Left: Two classical impurity spins S1andS2weakly\ncoupled to a system of classical host spins silocalized at the\nsitesiof some lattice. Right: Which type of coupling governs\nthe resulting e\u000bective impurity-spin dynamics?\nreminiscent of the Born-Oppenheimer approach in molec-\nular dynamics with the nuclei treated classically [28].\nAdiabatic dynamics represents another consequence of\nthe weakness of Kand the resulting separation of time\nscales. In case of a host system with a gapped electronic\nstructure, the adiabatic theorem rigorously enforces per-\nfect adiabatic dynamics. In other cases, it is expected\nto represent an excellent approximation, which is moti-\nvated physically by the idea that the host state should\nbe close to the momentary ground state, if the typical\nrelaxation times of the host dynamics are much shorter\nthen the time scale on which the impurity-spin dynamics\ntakes place.\nWith the present paper we reconsider this paradigm by\nstudying an even simpler problem: We still focus on two\nclassical impurity spins but replace the electronic host\nsystem by a system that also consists of classical spins.\nThis setup is illustrated in Fig. 1. The host-spin system\nis given by a classical Heisenberg model with nearest-\nneighbor interaction Jbetween host spins sithat are\nlocalized on the sites iof some lattice. In addition, there\nare two impurity spins coupled to two host spins at sites\ni1andi2via a local Heisenberg interaction K. We as-\nsume a bipartite lattice, such that the K= 0 host-system\nground states are easily found, and we assume a separa-\ntion of energy scales in the form jKj\u001cjJj.\nFormally, the static problem is then treated easily: The\ntotal energy for a given impurity-spin con\fguration is\nE0(S1;S2) = minH(fsig;S1;S2), where the minimiza-\ntion over all host-spin con\fgurations fsigbecomes trivial\nin the limitjKj \u001c jJj. Therewith we have an e\u000bec-\ntive Hamiltonian He\u000b(S1;S2) =E0(S1;S2) which, for\nan SO(3) spin-symmetric situation must have the form\nHe\u000b(S1;S2) =Kf(S1\u0001S2). Here, one should note\nthat, opposed to the RKKY setup, our model system is\nequipped with essentially a single model parameter K=J\nonly, such that in the weak-coupling limit jKj\u001cjJjthe\nstrength of the e\u000bective exchange must scale with K,\nwhile the scalar (smooth) function f:R!Rdepends\non the system geometry only. We will formally derive\nHe\u000b(S1;S2) in the body of the paper.\nOur main interest, however, is focussed on the emer-\ngent real-time adiabatic dynamics in case of time-scale\nseparation 1 =jKj\u001d 1=jJj. Assuming that this is fullydetermined by He\u000b(S1;S2), as the main paradigm sug-\ngests, we get the following equations of motion: _Sr=\n@He\u000b=@Sr\u0002Srforr= 1;2, i.e.,\n_S1=Kf0(S1S2)S2\u0002S1;_S2=Kf0(S1S2)S1\u0002S2;\n(1)\nwheref0is the derivative of f. One easily sees that the\nscalar product S1S2is a constant of motion. Hence, the\nimpurity-spin dynamics could equivalently be deduced\nfrom an e\u000bective Hamiltonian of the form He\u000b(S1;S2) =\nJe\u000bS1S2with e\u000bective interaction Je\u000b=Kf0(S1S2).\nWe call this the naive approach.\nEven in the limit K\u001cJ, the naive approach is shown\nto fail in many cases, depending on the system geome-\ntry. This is worth mentioning since the approach is very\ntempting and, furthermore, the reason for its failure is\nvery interesting and instructive: The pitfall is that the\nconsequences of the assumption that the motion can be\ndescribed as adiabatic are not taken seriously. Assum-\ning that the host system is at time tin its momentary\nground state corresponding to the impurity-spin con\fg-\nuration (S1(t);S2(t)) means that the state of the whole\nsystem (impurity andhost spins) lives in a very small ac-\ncessible con\fguration space, parameterized by a product\nof two Bloch spheres ( S1;S2)2S2\u0002S2. This may lead\nto holonomy e\u000bects [29, 30], i.e., to e\u000bects arising from\nthe holonomic constraints responsible for restricting the\ncon\fguration space. Varying the impurity-spin con\fgu-\nration, the ground state of the host-spin system evolves\nin a geometrically nontrivial way which mathematically\nwould be expressed in terms of the holonomy of a con-\nnection on the manifold of impurity-spin con\fgurations.\nIn particular, as we have shown in a recent paper [31],\nthis leads to the appearance of an additional topological\nspin torque. The topological spin torque is given in terms\nof a topological charge density which, when integrated,\ntakes quantized values only. In Ref. [31] we have worked\nout the general adiabatic spin dynamics (ASD) theory\nfor classical spin systems. The application of ASD to the\ncase of a single impurity spin ( R= 1), coupled to a host-\nspin environment and subjected to a local magnetic \feld\nhas shown that the novel topological spin torque leads\nto an anomalous precession frequency. In the present\npaper we work out the ASD for the case of R= 2 impu-\nrity spins and analyze the impact of the topological spin\ntorque on the time-dependent indirect exchange. For the\ntwo-spin case one may expect a simple precessional dy-\nnamics, similar to Eq. (1), but possibly with a renormal-\nized frequency. The goal of the present paper is to answer\nthis question, to derive, if possible, the correct e\u000bective\nHamilton function, and to check the applicability of ASD\ntheory.\nThe rest of the paper is organized as follows: The next\nsection introduces the model, some basic notations and\nthe fundamental equations of motion. Sec. III brie\ry re-\nviews the general ASD theory for Rimpurity spins cou-\npled to a classical host spin system. The ASD is spin\ndynamics subject to a formal constraint that enforces3\nadiabaticity. We have to carefully specify this constraint.\nThis is done in Sec. IV and used in Sec. V to set up the\ne\u000bective Hamiltonian and to discuss the resulting naive\nimpurity-spin dynamics. In Sec. VI we then compute the\ntopological spin torque, and we work out the implications\nin Sec. VII. The predictions of the ASD, and of the naive\napproach, can be compared with the numerical solution\nof the full set of equations of motions. This is done in\nSec. VIII. We discuss the parameter regimes, where the\nimpurity-spin dynamics is close to adiabatic. In Sec. IX\nwe \fnally discuss an approach which goes beyond the adi-\nabatic approximation and beyond an e\u000bective two-spin\ndynamics. Conclusions are given in Sec. X.\nII. CLASSICAL SPIN MODEL\nWe consider a set of Rimpurity spins embedded in\na lattice of Lhost spins. The host system consists of\nclassical spins si=sniof lengthsand directions given\nby unit vectors ni=si=s. They are localized at the\nsitesi= 1;:::;L of aD-dimensional lattice, and spins\nsiandsi0interact via an antiferromagnetic Heisenberg\nexchangeJii0. The characteristic time scale of the host\nspin system is set by 1 =J(we choose units with ~\u00111).\nImpurity spins are given by classical vectors Srfor\nr= 1;:::;R , and each impurity spin is written in the\nformSr=Smrwith lengths S=jSrjand unit vectors\nmr=Sr=S. We will focus on the case of R= 2 impurity\nspins but develop the theory for the general case of an\narbitrary number R. The impurity spins are locally ex-\nchange coupled to the host spins at sites i1;:::;iRof the\nlattice, and the strength of the local antiferromagnetic\n(\\Kondo\") coupling is denoted as K.\nA particular state of this classical spin model\nis speci\fed by a con\fguration ( s(t);S(t))\u0011\n(s1(t);:::;sL(t);S1(t);:::;SR(t)) of host and impu-\nrity spins at a time t. Its time evolution is governed by\nthe Hamilton function H(s;S):\nH=1\n2LX\ni;i0=1Jii0sisi0+KRX\nr=1sirSr\u0000RX\nr=1SrBr:(2)\nGenerically, we have Jii0=Ji0i=Jbetween nearest\nneighborsiandi0only. We have also added a term de-\nscribing a local magnetic \feld Brcoupling to the impu-\nrity spinSr. In most cases, however, we set Br= 0.\nThe Hamilton function leads to the following coupled set\nof non-linear ordinary di\u000berential equations of motion,\n_si=@H\n@si\u0002si;_Sr=@H\n@Sr\u0002Sr; (3)\nwhich determine the time evolution of an arbitrary given\ninitial spin con\fguration. Note that the lengths of siand\nofSrare conserved. This allows us to absorb constants,\nlike gyromagnetic ratios, in siandSr. The model (2) can\nbe seen as the classical isotropic multi-impurity Kondo-\nnecklace model [13] or simply as a classical Heisenbergmodel with a special multi-impurity geometry. There is\nno direct coupling of the impurity spins but an indirect\ncoupling is mediated via the host.\nWe will study the model in the limit of weak local cou-\nplingK\u001cJ. In this parameter regime the system ex-\nhibits two very di\u000berent time scales, K\u00001andJ\u00001, such\nthat the fast host spins almost instantaneously follow the\nmotion of the slow impurity spins. In this adiabatic limit,\none can expect a strong conceptual simpli\fcation, pro-\nviding us with an e\u000bective theory for the slow degrees of\nfreedom only.\nIII. ADIABATIC SPIN DYNAMICS THEORY\nUsing the notation n(t)\u0011(:::;ni(t);:::) andm(t)\u0011\n(:::;mr(t);:::) to characterize the con\fgurations of the\nfast and of the slow spins at a time tby the respective\nunit vectors, one can state that the time evolution is\nadiabatic, if, at any instant of time t, the con\fguration\nof the fast spins n(t) is the ground-state con\fguration\nfor the present con\fguration m(t) of the slow spins at\ntimet:\nn(t) =n0(m(t)): (4)\nWe expect adiabatic spin dynamics to be realized for\nK\u001cJ(andBr\u001cJ). The condition n=n0(m)\nspeci\fes a hyper surface fn=n0(m)jmarbitrarygin\nn-space, see Eq. (4), i.e., in the product of Bloch spheres,\nn2QL\ni=1S2. Upon approaching the weak-coupling limit\nin parameter space, the fast-spin con\fguration will be\nmore and more constrained to this hyper surface. This\nmeans that a strongly simpli\fed description, i.e., adia-\nbatic spin dynamics (ASD), should be possible in this\nlimit.\nUsing the constraint (4), one can de\fne an e\u000bective\nHamilton function,\nH(s;S)7!H(sn0(m);Sm)\u0011He\u000b(m); (5)\nwhich depends on the slow-spin degrees of freedom only.\nIt is therefore tempting to derive the slow-spin dynamics\nsolely from this e\u000bective Hamiltonian via the Rremaining\ndi\u000berential equations\nS_mr=@He\u000b(m)\n@mr\u0002mr (6)\nform(t), whilen(t) can be obtained from Eq. (4). This\nconstitutes an approach which we will refer to as the\nnaive theory of adiabatic dynamics.\nWe have recently shown that the naive approach may\nlead to incorrect results [31]. To eliminate the fast spin\ndegrees of freedom correctly, one must rather switch\nto a Lagrangian formulation and employ the general\naction principle, \u000eR\ndtL(n;_n;m;_m) = 0. In this\nframework one may safely make use the holonomic con-\nstraints (4) to eliminate the host degrees of freedom4\nand to set up an e\u000bective Lagrangian, Le\u000b(m;_m)\u0011\nL(n0(m);(d=dt)n0(m);m;_m), for the slow-spin degrees\nof freedom only. In terms of the e\u000bective Lagrangian, the\naction principle reads \u000eR\ndtL e\u000b= 0, where \u000eis variation\nof the slow-spin con\fguration monly. This provides us\nwith the ASD equations of motion for the slow spins mj:\nS_mr=@He\u000b(m)\n@mr\u0002mr+Tr\u0002mr: (7)\nAs compared to the \\naive\" adiabatic theory, Eq. (6),\nthere is an additional term due to a \feld\nTr\u0016=Tr\u0016(m;_m) =X\ns\u0017\nr\u0016;s\u0017(m) _ms\u0017; (8)\nwheres= 1;:::;R and\u0016;\u00172fx;y;zg. Here,\n\nr\u0016;s\u0017(m) = 4\u0019X\nise(i)\nr\u0016;s\u0017(m); (9)\nwith\ne(i)\nr\u0016;s\u0017(m) =1\n4\u0019@n0;i(m)\n@mr\u0016\u0002@n0;i(m)\n@ms\u0017\u0001n0;i(m) (10)\nis a rank-2 tensor for each pair of impurities r;s. It de-\nscribes certain topological properties of the ground state\nof the fast-spin subsystem on the hyper surface speci\fed\nby Eq. (4). In fact, each tensor element for \fxed r;sde-\n\fnes a topological charge density, which becomes a quan-\ntized homotopy invariant, namely a topological winding\nnumbere(i)\nrswith a quantized value e(i)\nrs2Zwhen inte-\ngrated. There is a close analogy to the concept of the\nskyrmion density [32{34]. Here, however, the skyrmions\nlive on a product of Bloch spheres rather than in Eu-\nclidean space. Note that the resulting topological spin\ntorqueTr\u0002mrin Eq. (7) involves the time derivative\n_ms. It nevertheless respects total-energy conservation,\nunlike a Gilbert damping term [35, 36]. Details of the\nderivation of Eq. (7) and its interpretation are given in\nRef. 31.\nIV. HOLONOMIC CONSTRAINT\nFor the computation of the topological spin torque,\nthe topological charge density (10) is required. This is a\nground-state property of the host system. Similar to a\ntwo-point response function, it depends on two \fxed posi-\ntionsirandis, i.e., on the two sites at which the r-th and\nthes-th impurity spin couple to the host. For a concrete\ncalculation, we need the explicit form of the constraint\n(4). This means to \fnd the ground-state con\fguration\nof the host spins for an arbitrarily given con\fguration\nm= (m1;:::;mR) of all impurity spins.\nWe start by considering the ground state at K= 0. In\nthis case, the host subsystem is described by a classical\nHeisenberg Hamiltonian Hhost(n) =1\n2s2P\ni;i0Jii0nini0,see Eq. (2). For any choice of the matrix of coupling con-\nstantsJii0, the Hamiltonian is invariant under SO(3) spin\nrotations such that the ground-state manifold is highly\ndegenerate. We pick an arbitrary ground-state con\fg-\nurationn0= (n0;1;:::;n0;L) which minimizes Hhost(n),\ni.e.,Hhost(n0) =E0= min., where E0is the ground-\nstate energy. The SO(3) symmetry implies that Rn0\u0011\n(Rn0;1;:::;Rn0;L) for any rotation matrix R2SO(3) is\na ground state as well: Hhost(Rn0) =E0.\nAt \fniteK, the degeneracy of the ground-state energy\nis lifted. Depending on the strength of K, on the given\nimpurity spin con\fguration m, and on the coupling con-\nstantsJii0, the host ground state can strongly di\u000ber from\nRn0and must be determined numerically in general. For\nweak Kondo coupling K\u001cJ, however, the impurity\nspins basically act as in\fnitesimally weak external local\nmagnetic \felds, which merely break the host SO(3) in-\nvariance and typically favor exactly one state out of the\nK= 0 ground-state manifold, without further disturbing\nthe spin con\fguration of that state. At the same time,\nK\u001cJjust speci\fes a limit where the ASD is expected\nto apply { as will be discussed later by comparing with\nresults from the numerical solution of the full set of equa-\ntions of motion given by Eq (3). Hence, we will focus on\nthe weak-coupling limit.\nThe host ground state for K6= 0 andK\u001cJ\nis obtained by minimization of the Hamilton function\nH(n;m)\u0011H(sn;Sm), whereni=Rn0;i, for given\n\fxedm, and with respect to all R2SO(3). Note that\nthis minimization problem is much simpler as compared\nto a high-dimensional minimization with respect to arbi-\ntrary host-spin con\fgurations, which would be necessary\nbeyond the weak-coupling limit. For the minimization,\nwe can also disregard the magnetic \feld term in Eq. (2)\nas this is independent of nand thus of R. Furthermore,\ndue to the invariance of the inner product si\u0001si0under\nspin rotations, also the Heisenberg term in His constant.\nHence, it is su\u000ecient to focus on the Kondo term only:\nKRX\nr=1(Rn0;ir)\u0001mr!= min.: (11)\nWe make use of the fact that R2SO(3) has the form\nR=Ra(') = exp('La), whereLwith components\nL\u000b2so(3) are the real and antisymmetric generators of\nSO(3), and where the unit vector aspeci\fes the rotation\naxis and'the rotation angle. Let us now assume that\nn0is the desired ground-state con\fguration for given m.\nThis implies that the Hamilton function reaches its min-\nimum at'= 0 for any rotation axis a. With the general\nrelation\n@\n@'exp('La)\f\f\f\n'=0(\u0001) =a\u0002(\u0001); (12)\nwe thus \fnd the following necessary condition for the\nminimum:\nRX\nr=1a\u0002n0;ir\u0001mr= 0: (13)5\nSince this must be satis\fed for rotations around an arbi-\ntrary axisa, we get\nRX\nr=1n0;ir\u0002mr= 0: (14)\nThis means that the total torque on the impurity spins\nmust vanish for the ground-state con\fguration n0.\nWe now specialize to the case of host spins on a bipar-\ntite lattice with nearest-neighbor interactions Jii0, where\nthe ground-state spin structure is collinear, i.e., we have\nn0;i=zi\u0011 (15)\nwithzi2f+1;\u00001gand with a unit vector \u0011to be de-\ntermined from Eq. (14). The host spin structure is in-\nvariant under the simultaneous transformation zi7!\u0000zi\nand\u00117! \u0000\u0011. Hence, we can choose the overall sign\nof (z1;:::;zL) as is convenient, e.g., such that zi1= +1.\nPhysical properties do not depend on this choice. To \fx\n\u0011, we insert Eq. (15) in the equilibrium condition (14).\nThis yields\n\u0011\u0002RX\nr=1zirmr= 0: (16)\nWe de\fne the (staggered) sum of the impurity-spin unit\nvectorsmr\nm0=X\nrzirmr (17)\nandm0=jm0j. With this we have \u0011\u0002m0= 0 and thus\n\u0011=zKm0\nm0: (18)\nThe total energy is minimized for the sign zK=\n\u0000signK=\u0000K=jKjas argued below, see Eq. (21). With\nEq. (15) the explicit constraint Eq. (4) \fnally reads as\nn0;i(m) =zizKP\nrzirmr\njP\nrzirmrj: (19)\nNote that the function n0;i(m) is singular for m0=P\nrzirmr= 0. The condition m0= 0 speci\fes a\nsubmanifold embedded in the full con\fguration space.\nThough this has zero measure, we have to keep this in\nmind and must exclude trajectories m(t) crossing the\nsubmanifold. More importantly, one cannot choose ini-\ntial conditions with m0(t= 0) = 0 within the ASD the-\nory. Consider the case of two impurity spins R= 2\nas an example. Since zi1= +1 is already \fxed (see\nabove), there are two cases to be taken into account:\nzi2=\u00061. Forzi2=\u0000zi1=\u00001, the parallel con\fgura-\ntionm1=m2is singular, while for zi2=zi1= +1 the\nantiparallel con\fguration m1=\u0000m2is singular must\nbe excluded. The physical meaning of the singularity\nbecomes obvious at a later stage (see Sec. VIII).V. EFFECTIVE HAMILTONIAN AND NAIVE\nADIABATIC THEORY\nHaving the explicit form of the constraint, Eq. (19),\nat hand, we proceed with derivation of the e\u000bective\nHamiltonian He\u000b(m) for arbitrary R. This is obtained\nfromH(n;m) by using the constraint to eliminate the\nhost-spin degrees of freedom, which yields He\u000b(m) =\nH(n0(m);m). Concretely, for the considered case of a\ncollinear host-spin con\fguration, Eq. (15), we have\nHe\u000b(m) =E0+KsSRX\nr=1zir\u0011mr\u0000SRX\nr=1mrBr;(20)\nwhereE0= (s2=2)PL\ni;i0=1Jii0zizi0is them-independent\nground-state energy of the host-spin Hamiltonian\nHhost(n). With\u0011=zKm0=m0, see Eq. (18), we \fnd\nHe\u000b(m) =E0+zKKsSm 0\u0000SRX\nr=1mrBr: (21)\nNote that the total energy is at a minimum for zKK=\n\u0000jKj<0. This justi\fes the above choice zK=\u0000signK.\nAs mentioned before, it is very tempting to derive the\nslow-spin dynamics solely from this e\u000bective Hamilto-\nnian, see Eq. (6) and the related discussion. This con-\nstitutes the naive adiabatic theory. The corresponding\nequations of motion of the naive theory are easily de-\nrived. We note that @m0=@mr=zirm0=m0and \fnd\n@He\u000b(m)\n@mr=\u0000jKjsSzirm0\nm0\u0000SBr: (22)\nThis yields\n_mr=1\nS@He\u000b(m)\n@mr\u0002mr=\u0012\n\u0000jKjszirm0\nm0\u0000Br\u0013\n\u0002mr\n(23)\nor\n_mr=\u0000jKjs\nm0zirX\nr0zir0mr0\u0002mr\u0000Br\u0002mr:(24)\nThis is a comparatively simple nonlinear system of R\ndi\u000berential equations of motion for the Rimpurity spins.\nThe naive adiabatic theory is in fact conceptually in-\ncorrect, as the constraint is directly used to simplify the\nHamiltonian, which is generally not justi\fed. Before we\nproceed with the (conceptually correct) adiabatic spin\ndynamics (ASD), however, let us discuss some special\ncases and consequences of the naive theory.\nForBr= 0, the e\u000bective slow impurity-spin dynamics\ntakes place on the time scale set by 1 =K, as is obvious\nfrom Eq. (24). One also easily veri\fes that the total en-\nergyHe\u000band the total impurity spin Smtot\u0011SP\nrmr,\nand alsom0are conserved quantities. Multiplying Eq.\n(24) withzirand summing over r, we see that m0pre-\ncesses around mtot:\n_m0=jKjs\nm0mtot\u0002m0 (25)6\nwith frequency\n!p=jKjsmtot=m0: (26)\nFor two impurity spins, R= 2, we have mtot=m1+\nm2, and withmtotthe angle#that is enclosed by m1\nandm2is conserved as well. In this case, as is directly\nseen from Eq. (24), m1andm2precess with the same\nfrequency!paroundmtot.\nThe precession frequency decisively depends on the ge-\nometry. Let us assume that zi1=\u0000zi2(note that we\n\fxedzi1= +1). This happens to be the case, e.g., for an\nantiferromagnetic ground-state con\fguration of the host\nspins,n0;i=\u0006(\u00001)i\u0011, if the distance i1\u0000i2between the\ntwo impurity spins is odd. Then m0=m1\u0000m2and\nthusm0=p\n2p\n1\u0000cos#. Withmtot=p\n2p\n1 + cos#\nwe \fnd\n!p=jKjsr\n1 + cos#\n1\u0000cos#=jKjscot(#=2): (27)\nForzi1= +zi2= +1, on the other hand, we have m0=\nmtotby de\fnition, and Eq. (25) is useless. The same\nholds for arbitrary Randzir= +1 for allr, such that\nagainm0=mtot. Going back to Eq. (24), we have\n_mr=\u0000(jKjs=m tot)mtot\u0002mrin this case, i.e., each\nimpurity spin precesses with frequency\n!p=jKjs (28)\naround the total impurity spin.\nLet us \fnally emphasize that the naive theory violates\ntotal spin conservation (for Br= 0). The total impu-\nrity spinStot=Smtotis a constant of motion as stated\nabove. The total host spin stot=P\nisi=sP\nizi\u0011,\nhowever, has a nontrivial precession dynamics in the case\nzi1=\u0000zi2and for \u0001\u0011P\nizi6= 0 or, equivalently, for\nstot=s\u0001\u00116= 0. Hence,Stot+stot6= const. This short-\ncoming is cured by the adiabatic spin dynamics theory,\nsee below.\nVI. TOPOLOGICAL CHARGE DENSITY AND\nSPIN TORQUE\nIn addition to the term on the right-hand side of Eq.\n(24), there is an additional contribution to the equations\nof motion of the ASD resulting from the topological spin\ntorque, see Eq. (7). To derive this contribution, we must\ncalculate the topological charge density Eq. (10) from the\nexplicit form of the constraint, Eq. (19).\nThis is done straighforwardly. First, we have:\n@n0;i\u0016(m)\n@mr0\u00160=zizKzir01\nm0\u0012\n\u000e\u0016\u00160\u0000m0\u0016m0\u00160\nm2\n0\u0013\n;(29)\nwhere themdependence is due to m0=P\nrzirmr.Inserting this expression in Eq. (10),\ne(i)\nr\u0016;s\u0017(m) =1\n4\u0019X\n\u001a\u001b\u001c\"\u001a\u001b\u001czizKzir1\nm0\u0012\n\u000e\u001a\u0016\u0000m0\u001am0\u0016\nm2\n0\u0013\n\u0002zizKzis1\nm0\u0012\n\u000e\u001b\u0017\u0000m0\u001bm0\u0017\nm2\n0\u0013\nzizKm0\u001c\nm0;\n(30)\nexpanding, exploiting the condition (14), and carrying\nout the sums over \u001a;\u001b, we \fnd\ne(i)\nr\u0016;s\u0017(m) =1\n4\u0019zizKzirzisX\n\u001c\"\u0016\u0017\u001cm0\u001c1\nm3\n0: (31)\nSummation over iyields the tensor \feld de\fned in Eq.\n(9):\n\nr\u0016;s\u0017(m) =zirziszKs\u0001X\n\u001c\"\u0016\u0017\u001cm0\u001c1\nm3\n0: (32)\nHere, we have de\fned\n\u0001\u0011X\nizi: (33)\nWe see that, in case of a collinear host-spin ground\nstate, Eq. (15), a nonzero \feld \n r\u0016;s\u0017(m) requires a\nground state with a \fnite total host-spin magnetization\nntot=P\nin0;i=P\nizi\u0011= \u0001\u0011. Its modulus ntot= \u0001 is\nnonzero, for instance, in case of ferromagnetic exchange\ncouplingsJ <0, or in case of antiferromagnetic couplings\nJ >0 whenLis odd.\nInserting the result for the tensor \feld into Eq. (8)\nyields:\nTr=X\n\u0016Tr\u0016(m;_m)e\u0016=X\ns\u0016\u0017\nr\u0016;s\u0017(m) _ms\u0017e\u0016\n=zirs\u0001\nm3\n0X\nszisX\n\u0016\u0017\u001c\"\u0016\u0017\u001cm0\u001c_ms\u0017e\u0016\n=zirzKs\u0001\nm3\n0_m0\u0002m0; (34)\nso that the additional topological spin torque on the\nright-hand side of Eq. (7) reads as:\nTr\u0002mr=zirzKs\u0001\nm3\n0(_m0\u0002m0)\u0002mr:(35)\nNote that the torque is independent of the coupling\nparameters and depends on the system geometry only.\nCombining this with Eq. (7) and Eq. (24), we arrive at\nthe ASD equations of motion:\n_mr=\u0012\n\u0000zirjKjs\nm0m0+zirzKs\u0001\nSm3\n0_m0\u0002m0\u0000Br\u0013\n\u0002mr:\n(36)\nThe \frst term is the same as in the naive approach, the\nsecond one is due to the topological spin torque.7\nVII. ADIABATIC SPIN DYNAMICS\nFor the discussion of the equations of motion of adia-\nbatic spin dynamics, see Eq. (36), we will consider sev-\neral cases. Let us \frst check the case R= 1,K > 0,\nJ > 0,i1= 1, and odd L, i.e., there is a single impu-\nrity spin only, m\u0011m1, which couples antiferromagnet-\nically at the \frst site of an antiferromagnetic host with\nan odd number of sites. We have m0=mtot=mand\n\u0001 = +1. The naive equation of motion, Eq. (24), reads\n_m=m\u0002B, and thus predicts precession around the ex-\nternal magnetic \feld with Larmor frequendy !p=B. In-\ncluding the additional topological spin torque, however,\nwe \fndS_m=Sm\u0002B\u0000s(_m\u0002m)\u0002m=Sm\u0002B+s_m.\nThis can be written in the form of the Landau-Lifschitz\nequation but leads to an anomalous precession frequency\n!p=B\n1\u0000s=S: (37)\nThis is precisely the result derived in Ref. [31].\nNext we discuss constants of motion for the general\ncase (but we assume Br= 0). We start by checking\nthat the equation of motion respects the conservation of\njmrj= 1. This is the case (also for \fnite Br) since\nmr_mr= 0, see Eqs. (7) or (36).\nTotal energy energy conservation is ensured on gen-\neral grounds as the equation of motion is derived within\nthe standard Lagrange formalism and employing a scle-\nronomic holonomic constraint. This has also been proven\nexplicitly and discussed in detail in Ref. [31].\nConservation of the total spin, i.e., the sum of the total\nimpurity spin Stot=SP\nrmrand the total host spin\nstot=sP\nin0;i(m), can be shown for the case of an\nSO(3) symmetric e\u000bective Hamiltonian He\u000b(m). This is\ndetailed in Appendix A.\nConservation of the total impurity spin mtot=P\nrzirmris not expected in general. Summing both\nsides of Eq. (36) over r= 1;:::;R we get (forBr= 0):\n_mtot=zKs\u0001\nSm3\n0(_m0\u0002m0)\u0002m0; (38)\nwhich is nonzero for a \fnite topological spin torque, i.e.,\nfor \u00016= 0. Note that this immediately implies _mtotm0=\n0.\nSumming both sides of Eq. (36) over rafter multiplying\nwithzir, we can derive an equation for the \\staggered\"\nsum of the impurity spins m0=P\nrzirmr. ForBr= 0\nwe get:\n_m0=\u0000jKjs\nm0m0\u0002mtot+zKs\u0001\nSm3\n0(_m0\u0002m0)\u0002mtot:\n(39)\nWe immediately have _m0mtot= 0. Together with the\nabove relation _mtotm0= 0, this implies that the in-\nner productm0mtotis conserved. Furthermore, we have\n(_m0\u0002m0)\u0002mtot=\u0000_m0(mtotm0) and therewith wecan convert Eq. (39) into an explicit di\u000berential equation:\n_m0=1\n1 +zKs\u0001\nSm3\n0m0mtotjKjs\nm0mtot\u0002m0: (40)\nMultiplying both sides of the equation with m0, yields\nm0_m0= 0 and hence m0= const (ifBr= 0). Hence,\nthe prefactor of m0\u0002mtotin Eq. (40) is a constant of\nmotion.\nWithm0_m0= 0 we can also simplify the double cross\nproduct in Eq. (38):\n_mtot=\u0000zKs\u0001\nSm0_m0: (41)\nMultiplying with mtotand using _m0mtot= 0, we see\nthat the norm of mtotis conserved. Furthermore, _m0\non the right-hand side can be eliminated using Eq. (40),\nsuch that we are \fnally left with:\n_mtot=zKs\u0001\nSm01\n1 +zKs\u0001\nSm3\n0m0mtotjKjs\nm0m0\u0002mtot:\n(42)\nSumming up, for Br= 0, we have, besides energy and\ntotal spin conservation, the following conserved quanti-\nties:\nm0= const.; m tot= const.;m0mtot= const.\n(43)\nFurthermore, there are two coupled nonlinear ordinary\ndi\u000berential equations, Eq. (40) and Eq. (42), for m0and\nmtot.\nThere is a link to the naive adiabatic theory, namely\nin the topologically trivial case where \u0001 = 0, i.e., where\nthe topological spin torque vanishes. Here, the total\nimpurity spin is conserved, _mtot= 0, and according\nto Eq. (40), m0precesses around mtotwith frequency\n!p=jKjsmtot=m0. This precisely recovers the results of\nthe naive theory, see Eq. (26).\nIn the nontrivial case for \u0001 6= 0, an analytical solution\nof Eq. (40) and Eq. (42) is obtained easily. To this end,\nwe rewrite the equations as\n_m0=c0mtot\u0002m0;_mtot=ctotm0\u0002mtot;(44)\nwith constants c0andctot, and employ a scaling trans-\nformation to new variables fm0=\u000b0m0andfmtot=\n\u000btotmtotsuch that the prefactors in the transformed\nequations of motion are equal (this is the case for zK\u0001>\n0) or di\u000ber by a sign only ( zK\u0001<0). Details are given\nin Appendix B. We \fnd that both, m0andmtot, are\nprecessing around the conserved total spin stot+Stot,\nand the corresponding precession frequency is:\n!p=q\nc2\ntotm2\n0+c2\n0m2\ntot\u00062jc0ctotjm0mtot:(45)\nWith the solutions m0,mtotat hand, the dynamics of\nthe individual impurity moments mrcan be obtained8\nfrom a numerical solution of Eq. (36) for each mrsep-\narately . This situation is di\u000berent from but reminiscent\nof gyroscope theory, since mrprecesses around a mo-\nmentary axis speci\fed by m0and _m0, whilem0itself is\nprecessing around an axis \fxed in space.\nFinally, we consider the special case of two impurity\nspinsR= 2 and vanishing external \felds Br= 0. Here,\nit is su\u000ecient to analyze the two coupled equations Eq.\n(40) and Eq. (42), rather than reverting to Eq. (36) again,\nsince the dynamics of m1andm2is fully determined via\nmtot=m1+m2andm0=zi1m1+zi2m2.\nFor the case zi1=zi2= 1, we have m0=mtot, such\nthat Eq. (40) reduces to Eq. (42). Eq. (42) implies that\nthe total impurity spin is conserved, _mtot= 0, and thus\nthe topological spin torque vanishes. This is the topolog-\nically trivial case. Both impurity spins precess around\nmtotwith frequency !p=jKjs, as discussed above, see\nEq. (28).\nFor the nontrivial case zi1=\u0000zi2= 1, we have m0=\nm1\u0000m2. This implies m0mtot= (m1\u0000m2)(m1+\nm2) = 0, sincem1andm2are unit vectors, such that\nthe last statement of Eq. (43) becomes trivial. Further,\nthe \frst and second one imply m1m2= cos#= const.\nTherewith, the equations of motion for m0andmtot\nsimplify and read:\n_m0=jKjs\nm0mtot\u0002m0 (46)\nand\n_mtot=zKs\u0001\nSm0jKjs\nm0m0\u0002mtot: (47)\nThe precession frequency is:\n!p=jKjs1\nm0r\ns2j\u0001j2\nS2+m2\ntot: (48)\nAssuming that s=Sand thatj\u0001j= 1 (antiferromag-\nnetic host-spin con\fguration and odd L), we have (see\nAppendix B):\n!p=jKjsp\n1 +m2\ntot\nm0=jKjs1\n2 sin#=2r\n1 + 4 cos2#\n2:\n(49)\nThis also applies to the individual impurity spins: For\nR= 2, the dynamics of the impurity spins, m1andm2,\nis simple: They precess around the axis speci\fed by the\ntotal spin with the same frequency as m0andmtot.\nNote that the precession frequency approaches !p!\njKjs=2 for#!\u0019, i.e., when one approaches the global\nground state. This must be compared with the result\n!p!0 that is obtained by the naive adiabatic theory.\nFor#!0, on the other hand, i.e., for m0!0, the\nprecession frequency diverges as !p!p\n5jKjs=#. This\ndivergence originates from the above-mentioned singular-\nity, cf. the discussion following Eq. (19).\nWe would like to emphasize that already the simple\nR= 2 case demonstrates that an e\u000bective impurity-\nspin dynamics is not Hamiltonian. There is no e\u000bective\nsiS1S2host spinsimpurity spinsKJS1S2m0⌘<latexit sha1_base64=\"q783FQz9T9wnYdA2lv+U3CVDQDg=\">AAAB+HicdVDLSgMxFM3UV62Pjrp0EyyCq2HaTl+7ghuXLdha6Awlk962oZkHSUaoQ8H/cONCEbd+ijv/xvQhqOiBkMM595KT48ecSWXbH0ZmY3Nreye7m9vbPzjMm0fHXRklgkKHRjwSPZ9I4CyEjmKKQy8WQAKfw40/vVz4N7cgJIvCazWLwQvIOGQjRonS0sDMu37Eh3IW6MsFRQZmwbbqlWq5VMe2VapWnXpDk3LVqdTKuGjZSxTQGq2B+e4OI5oEECrKiZT9oh0rLyVCMcphnnMTCTGhUzKGvqYhCUB66TL4HJ9rZYhHkdAnVHipft9ISSAX2fRkQNRE/vYW4l9eP1GjupeyME4UhHT10CjhWEV40QIeMgFU8ZkmhAqms2I6IYJQpbvK6RK+for/J92SVXSsRtspNNv3qzqy6BSdoQtURDXURFeohTqIogQ9oCf0bNwZj8aL8boazRjrCk/QDxhvn+cSlGQ=</latexit>\n#<latexit sha1_base64=\"arADQHr4CIsIQf+sjU+jeonNDwo=\">AAAB8HicdVDLSgMxFM3UV62vqks3wSK4GjKd4rS7ghuXLdiHtEPJpJk2NJMZkkyhDAX/wY0LRdz6Oe78G9OHoKIHLhzOuTe59wQJZ0oj9GHlNja3tnfyu4W9/YPDo+LxSVvFqSS0RWIey26AFeVM0JZmmtNuIimOAk47weR64XemVCoWi1s9S6gf4ZFgISNYG+muP8VSj6nGg2IJ2ahWdlAVItt1rmpexRDPdT3kQsdGS5TAGo1B8b0/jEkaUaEJx0r1HJRoPzPPMcLpvNBPFU0wmeAR7RkqcESVny0XnsMLowxhGEtTQsOl+n0iw5FSsygwnRHWY/XbW4h/eb1Uh1U/YyJJNRVk9VGYcqhjuLgeDpmkRPOZIZhIZnaFZIwlJtpkVDAhfF0K/yftsu1U7FqzUqo371dx5MEZOAeXwAEeqIMb0AAtQEAEHsATeLak9Wi9WK+r1py1jvAU/ID19gmmTpFz</latexit>FIG. 2. System geometry: L= 5 host spins (length s= 1),\nR= 2 impurity spins ( S= 1) coupled to the host spins at sites\ni1= 1,i2= 4, antiferromagnetic exchange J;K > 0, hence:\nzi1= +1,zi2=\u00001, and \u0001 = 1. Initial angle enclosed by\nthe two impurity spins: #. Initially the host-spin system is in\nits antiferromagnetic ground state for the given impurity-spin\ncon\fguration, i.e., \u0011=\u0000m0=m0.\nHamilton function He\u000b(m1;m2) with which the equa-\ntions of motion (46) and (47) can be reproduced. Any\nnontrivial two-impurity-spin Hamiltonian model would\nbe of the form He\u000b(m1;m2) =Kf(m1m2) with some\nsmooth real function f, which would immediately, and\nincorrectly, imply that m1+m2is conserved.\nVIII. NUMERICAL RESULTS\nIt remains to check the validity of the adiabatic spin-\ndynamics theory, i.e., to \fnd out in which parameter\nregime the adiabatic approximation is justi\fed. We\ntherefore compare the predictions of the ASD with the\nnumerical solution of the full set of equations of motion\n(3). For the sake of simplicity, we \frst pick a geometry\nwithL= 5 host spins and R= 2 impurity spins with\nantiferromagnetic couplings J;K > 0, see Fig. 2. The\nimpurity spins are coupled to the host at sites i1= 1 and\ni2= 4. Since Lis odd and zi1=\u0000zi2= 1, this is a\nrealization of the topologically nontrivial case.\nWith Fig. 3 we give an example result which is charac-\nteristic of the real-time spin dynamics if KandJare of\nthe same order of magnitude, and if the initial con\fgura-\ntions of impurity spins is far from the (antiferromagnetic)\nground state con\fguration. The initial host-spin con\fg-\nuration is taken to be the ground-state con\fguration for\nthe given impurity-spin directions. As is demonstrated\nwith Fig. 3, we \fnd an extremely complex dynamics as it\nis characteristic for a nonlinear classical Hamiltonian sys-\ntem with several degrees of freedom. For long times, the\ntrajectories cover the entire phase space that is accessible\nunder total energy and total spin conservation.\nClearly, adiabatic spin dynamics is only expected to\nbe realized in the weak-coupling limit K\u001cJ. Fig. 4\nprovides an example for the same setup and parame-\nters as in Fig. 3 but for K=J = 10\u00005. The motion is\nmainly precessional but there is an additional nutation\nvisible. This nutation e\u000bect is not captured by the ASD\nbut gets weaker and \fnally almost disappears when the\ninitial impurity-spin con\fguration is chosen closer and\ncloser to a ground-state con\fguration.\nIf the initially enclosed angle #\u0019\u0019, the dynamics is9\ntmax = 20tmax = 100tmax = 1000z\nyxz\nyxz\nyx\nFIG. 3. Trajectories of the two impurity spins on the Bloch sphere as obtained by numerical solution of the full set of equations\nof motion (3) for K=J,#=\u0019=2 for di\u000berent maximal propagation times tmaxas indicated (in units of K\u00001). At timet= 0\nthe system is prepared as indicated by the arrows. m1: blue,m2: red. The host spins are in their ground-state con\fguration\nfor given impurity spins. The total spin is parallel to the z-axis.\neven more regular. Fig. 5 displays an example, where\n#= 0:95\u0019and where the host is in the corresponding\nground state initially. Still, for K=J = 1, the individual\nimpurity spins mrforr= 1;2 show a rather compli-\ncated time evolution (not shown). The staggered sum\nm0=P\nrzirmr=m1\u0000m2, on the other hand, is al-\nready much closer to a purely precessional motion. The\n\fgure shows the time dependence of the azimuthal angle\n', modulo 2\u0019, ofm0with respect to the total conserved\nspinstot+Stot. This azimuthal angle more or less grows\nlinearly in time but with some weak additional struc-\nture superimposed. With decreasing ratio K=J, the ad-\nditional superimposed oscillations get weaker and weaker\nand are only hardly visible when K=J = 0:01.\nThe green line in Fig. 5 shows the result of the ASD\ntheory, which predicts a purely precessional motion and\ncorrespondingly a linear increase of 'as a function of t.\ntmax = 100z\nyx\nFIG. 4. The same as Fig. 3 but for K=J = 10\u00005. Maximal\npropagation time tmax= 100K\u00001.We see that with decreasing ratio K=J, the trajectory of\nm0, obtained from the full theory, appears to converge\nto the ASD result. Not only the additional structure di-\nminishes further and further but also the ASD prediction\nof the angular velocity d'=dt seems to be approached in\ntheK=J!0 limit.\nOn the contrary, the prediction of the naive adiabatic\ntheory, see Eq. (25), which is directly obtained from the\ne\u000bective Hamiltonian, Eq. (21), is completely o\u000b. The\napproach does yield a purely precessional motion but\nmistakenly around the total impurity spin mtot, which\nis a constant of motion within the naive theory but not\nwithin the ASD and the full theory. Furthermore, the\nangular velocity is by far too small or, as can be seen in\nASDK/J=10-210-1K/J=100naive theory\nFIG. 5. Time dependence (in units of K\u00001) of the azimuthal\nangle (mod 2 \u0019) in the precession dynamics of m0around the\nconserved total spin for the geometry displayed in Fig. 2 and\nfor#= 0:95\u0019. Numerical solution of the full set of equations\nof motion (3) for various ratios K=J as indicated: blue. Naive\ntheory: orange. Adiabatic spin dynamics (ASD): green.10\nASDfull theorynaive theory\nFIG. 6. Precession frequency !pof the staggered sum m0\nas a function of the coupling strength K=J at#= 0:95\u0019.\nResults obtained from the numerical solution of the full set\nof equations of motion, Eq. (3), compared to the predictions\nof the ASD and of the naive adiabatic spin dynamics in the\nlimitK=J!0.\nthe \fgure, the period 2 \u0019=! pis by far too large.\nWith decreasing K=J also other quantities appear to\nconverge to the predictions of the ASD (not shown). We\n\fnd, for example, that the modulus of the staggered and\nof the total impurity spin, m0andmtot, and the scalar\nproductm0mtotor, equivalently, the enclosed angle #\napproach constants when K=J!0, as is stated in Eq.\n(43). Furthermore, also the dynamics of individual im-\npurity spins seem to more and more approach a purely\nprecessional motion with the same precession frequency.\nThere is, however, a \fnite residual di\u000berence between\nthe full theory, Eq. (3), and the ASD persisting in the\nlimitK=J!0. This is demonstrated in Fig. 6 where\nthe precession frequency in the real-time dynamics of\nm0is plotted as a function of K=J. Since#= 0:95\u0019,\nthe impurity-spin con\fguration is close to a ground-state\ncon\fguration, such that there is a frequency with dom-\ninant weight in the Fourier analysis of the data. This\nfrequency smoothly depends on K=J and approaches the\nfrequency of the almost pure precessional dynamics that\nremains in the limit K=J!0. Already at K=J\u001910\u00003\nit approaches saturation, although at a level that di\u000bers\nfrom the ASD result (green arrow) by about 15%. This\nimplies that even in the weak-coupling limit, the host\nspins do not completely adiabatically follow the impurity-\nspin dynamics. The observation of a close-to-adiabaticity\ndynamics has already been made earlier for the single-\nspin (R= 1) case [21]. Turning to the naive adiabatic\ntheory, see orange arrow in Fig. 6, the predicted preces-\nsion frequency is again by far too small, aside from the\nfact that the precession axis is predicted incorrectly.\nWhile the ASD theory is at least qualitatively correct\nat smallK=J and#!\u0019, it must break down for initial\nimpurity-spin con\fgurations that are far from a ground-\nstate con\fguration. This is demonstrated with Fig. 7,\nwhere the analytical result (49) for the ASD precession\nASDfull theorynaive theoryFIG. 7. #dependence of the precession frequency !pas\npredicted by the ASD (green), see Eq. (49), and compared to\nthe numerical data obtained at K=J = 10\u00005by solving the\nfull set of equations of motion (blue points), Eq. (3), and to\nthe naive theory (orange), Eq. (27).\nfrequency is plotted against #and compared to the nu-\nmerical data for a coupling strength K=J = 10\u00005deep in\nthe weak-coupling limit. For #!\u0019the ASD is close to\nthe numerical data and correctly predicts a \fnite nonzero\nfrequency!p=jKjs=2 in the limit, while there is a re-\nmaining discrepancy visible, as discussed above.\nWith decreasing #and increasing parametric distance\nto the ground state, however, the ASD is less reliable.\nThis is understood easily and eventually results from the\nsingular constraint, see Eq. (19), on the m0= 0 man-\nifold. For#!0, i.e., form1=m2initially, we have\nm0= 0 initially, and thus m0= 0 at all times within\nthe ASD. This singularity leads to the divergence of the\nfrequency!p!p\n5jKjs=#for#!0, see Eq. (49). It\nresults from the fact that the ground-state host-spin con-\n\fguration cannot be determined unambiguously for the\nmaximally excited impurity-spin con\fguration, and that\nthere is two-dimensional manifold of degenerate host-spin\ncon\fgurations in this case.\nVice versa, a divergent precession frequency implies a\nfastimpurity-spin dynamics, i.e., a violation of the cen-\ntral assumption of a slow, adiabatic or close-to-adiabatic\nmotion. We note that this kind of inherently built-\nin breakdown of the theory is already known from the\nsingle-spin ( R= 1) case with a \fnite external magnetic\n\feld [31], where it shows up, however, at a di\u000berent point\nin parameter space, namely for s!S, see Eq. (37).\nFinally, the naive adiabatic spin-dynamics theory is\nneither correct in the #!\u0019nor in the #!0 limit.\nIn the latter case, the precession frequency diverges as\n!p!2jKjs=#.\nThe discussion of the results and the conclusions also\napply to larger system sizes. This is demonstrated with\nFig. 8, where the normalized di\u000berence between the ASD\nprecession frequency and the precession frequency of the11\n0.0 0.2 0.4 0.6 0.8 1.0\n/\n100101(A F)/F\nL=5\nL=15\nL=105\nFIG. 8. Normalized di\u000berence of the ASD precession fre-\nquency!Aand the frequency !Fobtained numerically from\nthe full solution of Eq. (3) as function of #atK=J = 10\u00003.\nResults for various system sizes L= 5;15;105. The impurity\nspins couple to positions i1= 1 andi2=L\u00001.\nnumerical solution of the full set of Hamilton equations\nof motion is plotted against #for di\u000berent L. SinceLis\nodd in all cases and since the impurity spins are coupled\nto the host at i1= 1 andi2=L\u00001, we have the topolog-\nically non-trivial case at hand. We see that the di\u000berence\ndiverges for #!0, as discussed above. For #!\u0019, on the\nother hand, the residual di\u000berence becomes larger with\nincreasingL. This means that it becomes more and more\ndi\u000ecult to enforce close-to-adiabatic dynamics. Obvi-\nously, this is due to the necessity to communicate the\nrelative impurity-spin con\fguration over large distances.\nIX. BEYOND THE ADIABATIC\nAPPROXIMATION\nOne way to improve the theory and to go beyond\nthe adiabatic approximation is to relax the constraint\nEq. (19) de\fning the ASD. For the weak-coupling limit\nK\u001cJ, it is tempting to keep the host spins tightly cou-\npled together but to relax the demand that the host-spin\ncon\fguration should be given, at any instant of time, by\nthe ground-state con\fguration for the currently present\ncon\fguration of the impurity spins. This idea can be\nformalized by substituting Eq. (19) by the constraint\nni!=n0;i(\u0011) =zi\u0011; (50)\nwhere\u0011is a (three-component) dynamical degree of free-\ndom normalized to unity, \u00112= 1. If we consider host\nspins on a bipartite lattice or, for the sake of simplic-\nity, on a one-dimensional chain of sites i= 1;:::;L that\nare tightly bound together via a strong antiferromagnetic\ncouplingJ >0, we have zi= (\u00001)i+1, with the conven-\ntionz1= +1.A conceptual disadvantage of an e\u000bective spin-\ndynamics theory under these tight-binding constraints is\nthat it necessarily involves (with \u0011) dynamical host de-\ngrees of freedom, such that one will not end up with an\ne\u000bective theory of the impurity-spin degrees of freedom\nonly. Clearly, this is the price to be paid when aiming at\nan improved theory beyond the ASD. On the other hand,\na formal advantage is that there is no singularity and that\nno submanifold of spin con\fgurations must be excluded,\nas compared to the ASD, cf. the discussion following Eq.\n(19).\nAgain, one must be very careful when imposing the\nconstraint Eq. (50). In Appendix C, it is demonstrated\nthat one runs into unacceptable inconsistencies, if one\nattempts to use the constraint (50) directly for a simpli-\n\fcation of the full set of equations of motion (3). The\nproper way is rather to start from the action principle\nagain, to set up the Lagrangian of the full theory yield-\ning the equations of motion (3), and to treat Eq. (50)\nas a holonomic constraint to simplify the Lagrangian to\nan e\u000bective Lagrangian Le\u000b(\u0011;_\u0011;m;_m) with a strongly\nreduced number of degrees of freedom.\nIn Appendix D this program is carried out for an arbi-\ntrary function n0;i(\u0011) without further speci\fcation. The\nform of the resulting equation of motion, Eq. (D13),\n0 =@He\u000b(\u0011;m)\n@\u0011\u0002\u0011+T\u0002\u0011; (51)\nturns out as quite unusual as it lacks an explicit _\u0011term.\nHowever, similar to the ASD, see Eq. (7), there is an ad-\nditional topological spin-torque term resulting from the\nconstraint. This has the form T=_\u0011\u0002\n, where \nis the pseudo-vector corresponding to an antisymmetric\ntensor \n\u0016\u0017that derives from the topological charge den-\nsity [31] or magnetic vorticity [37], see Eqs. (D16) and\n(D17). This topological spin torque brings the _\u0011depen-\ndency back into the theory.\nEq. (51) holds generally for constraints of the form\nni=n0;i(\u0011). In Appendix E, we evaluate the topo-\nlogical charge density and the resulting topological spin\ntorque for the constraint Eq. (50) explicitly. This leads\nto equations of motion, which, for \u0001 =P\nizi6= 0, have\nthe familiar Hamiltonian form, cf. Eq. (E10):\n\u0001_\u0011=SKX\nrzirmr\u0002\u0011;\n_mr=sKzir\u0011\u0002mr\u0000SBr\u0002mr: (52)\nSome general properties and conservation laws related to\nthese equations are discussed in Appendix E as well.\nHere, we consider the setup discussed in the previ-\nous section, see Fig. 2, and compare the numerical so-\nlution of Eqs. (52) with that of the full set of equations\nof motion (3) and with the predictions of the ASD. For\nR= 2 impurity spins, the constrained spin dynamics is\nin fact more complicated. In particular, there is no sim-\nple precessional motion at moderate K=J. For a com-\nparison with the full spin-dynamics theory, Eq. (3), we12\n105\n104\n103\n102\n101\n100\nK/J0.20.40.60.8/\n102\n101\n100\n(C F)/F\nFIG. 9. Main precession frequency in the Fourier spectrum\nof the real-time dynamics of m0obtained from constrained\nspin-dynamics theory !Cas function of #andK=J. Color\ncode: normalized di\u000berence with the result of the full spin-\ndynamics theory !F.\nnevertheless concentrate on the dominant peak in the\nFourier spectrum and the corresponding precession fre-\nquency!p. Fig. 9 demonstrates that spin dynamics un-\nder the tight-binding constraint in fact substantially im-\nproves the description and is reliable in the weak-coupling\nlimitK=J\u001c1for all angles#specifying the initial\nimpurity-spin con\fguration at time t= 0. This is op-\nposed to the ASD, which requires weaker couplings K=J\nand which captures the full spin dynamics for angles close\nto#=\u0019only, as is shown in Fig. 10.\nThe situation is completely di\u000berent, however, for the\ncase \u0001 = 0, i.e., if the chain of host spins consists of\nan even number of sites, such that for antiferromagnetic\ncouplingJthe total host spin vanishes. Solving the equa-\ntions of motion (3) of the full theory for R= 2 impurity\nspins forK=J\u001c1, one \fnds a nontrivial spin dynamics\n105\n104\n103\n102\n101\n100\nK/J0.20.40.60.8/\n101\n100101\n(A F)/F\nFIG. 10. The same as Fig. 9 but comparing the ASD and\nthe full spin-dynamics theory.with a precessional motion of \u0011whilemtot= const. For\nR= 1, there is no spin dynamics at all, since for \u0001 = 0\nthe total spin is solely given by the single impurity spin,\nand, therefore, the impurity spin is \fxed to its initial di-\nrection due to total spin conservation. Hence, we note\nthat there are no special features here.\nTurning to the constrained spin dynamics and special-\nizing Eq. (52) to the case \u0001 = 0, R= 1 andB= 0,\nprovides us with the two equations 0 = m\u0002\u0011and\n_m=sK\u0011\u0002m, which correctly imply m= const. How-\never, the \frst equation is dubious, since it may con\rict\nwith an initial condition where m\u0002\u00116= 0. On the other\nhand, such an initial state is perfectly allowed by our con-\nstraint Eq. (50). This clearly implies that the constrained\nspin dynamics is inherently inconsistent.\nWe have analyzed the origin of this inconsistency in\nAppendix F. In fact, in the case \u0001 = 0, the e\u000bective\nLagrangian of the constrained spin-dynamics theory is\nsingular. This can be made explicit with a proper gauge\ntransformation after which Le\u000b(\u0011;_\u0011;m;_m) becomes in-\ndependent of _\u0011. Hence, it cannot describe situations\nwhere the\u0011degrees of freedom are dynamic. Actually,\nthis represents a clear example of a non-admissible e\u000bec-\ntive Lagrangian theory.\nLet us \fnally turn to the case \u0001 6= 0 once more. It is\nworth mentioning that the ASD can be newly derived by\nstarting from the e\u000bective Lagrangian for spin dynamics\nunder the tight-binding constraint ni=n0;i(\u0011) and by\nimposing the additional constraint\u0011=zKm0\nm0expressing\nadiabaticity, see Eq. (18). A heuristic argument is given\nin Appendix G for the single-impurity-spin case R= 1.\nThe formal derivation for the general case is worked out\nin Appendix H. This must be seen as a successful consis-\ntency check of the formal theory.\nX. CONCLUSIONS\nClassical Heisenberg spin models are frequently used in\natomistic spin-dynamics studies of condensed-matter sys-\ntems, nanostructures or molecular systems. From a prag-\nmatic point of view, they are quite attractive since the\ncorresponding classical Hamiltonian equations of motion\nform a nonlinear set of ordinary di\u000berential equations,\nwhich can be integrated by numerical means, such that,\nas compared to quantum-spin models, long propagation\ntimes for a large number of spins are easily accessible.\nUsually, for generic model parameters, the resulting mi-\ncroscopic spin trajectories are chaotic and cover the entire\naccessible phase space, as it is expected for a nonlinear\nclassical ergodic system.\nMore regular dynamics is obtained for cases with\nstrongly varying exchange-coupling parameters or, equiv-\nalently, for systems with a clear separation of intrinsic\ntime scales. Such situations are often quite realistic, and\na typical setup has been considered here. We have per-\nformed a comprehensive study of a prototypical model\nconsisting of two impurity spins that are weakly cou-13\npled to an antiferromagnetically coupled host-spin sys-\ntem, i.e., slow impurity spins are interacting with fast\nhost spins. For initial states with energy close to the\nground-state energy, a very regular, mainly precessional\ndynamics emerges, which calls for an e\u000bective low-energy\ntheory.\nThe purely classical system studied here must be seen\nas a simple model system and would have to be re\fned to\ndescribe a realistic material that is accessible experimen-\ntally. Several issues must be considered, such as longer-\nranged interactions between the host spins, nonlocal cou-\npling between impurity and host spins, anisotropies and\nmore. Actually, the model studied here is probably the\nsimplest one that serves our theoretical purposes. The\nmain conclusions, however, will all carry over qualita-\ntively to more realistic setups.\nConceptually, the most interesting \fnding is that the\nspin dynamics is unexpectedly non-Hamiltonian in many\ncases, i.e., there is no e\u000bective RKKY-like Hamiltonian\nthat merely consists of the impurity-spin degrees of free-\ndom and is able to reproduce the impurity-spin dynam-\nics. The reason is that, quite generally, the time-scale\nseparation leads to the emergence of a topological spin\ntorque, which profoundly a\u000bects the spin dynamics. This\nis reminiscent of the Berry phase that emerges in a quan-\ntum (host) system upon slow variation of classical model\nparameters (the impurity spins). An important di\u000ber-\nence, however, is that the (purely classical) topological\nspin torque actually represents a back-reaction of the lo-\ncal topological charge density of the host system on the\nslow impurity spins.\nOur main ansatz for constructing an e\u000bective low-\nenergy impurity-spin dynamics has been the adiabatic\napproximation, which is formulated as a constraint for\nthe host-spin con\fguration. This constraint has to be in-\ncorporated carefully: Making use of the constraint on the\nlevel of the equations of motion runs into unacceptable\ninconsistencies. A consistent e\u000bective theory is obtained\nwhen using the constraint to simplify the original Hamil-\ntonian. This naive e\u000bective theory, however, runs the\nrisk of not respecting certain conservation laws, e.g., to-\ntal spin conservation and has been explicitly shown to fail\nin cases, where the host-spin system has a \fnite total spin\nmoment. A satisfactory e\u000bective theory rather requires\nto work in the Lagrange formalism which allows us to\ninclude arbitrary constraints in a consistent way. Using\nthe adiabatic constraint de\fnes adiabatic spin dynam-\nics (ASD). For the relevant weak-coupling limit, we were\nable to work out the non-Hamiltonian e\u000bective equations\nof motion analytically. The big impact of the topological\nspin torque appearing in the ASD equations becomes ev-\nident when comparing the ASD results with those of the\nnaive theory.\nFrom a theoretical perspective, the ASD appears as a\nvery attractive approach: It follows a clear construction\nprinciple, it maintains conservation laws resulting from\nthe symmetries of the original Hamiltonian, it provides\na true e\u000bective theory formulated in terms of the slowimpurity-spin degrees of freedom only, and it brings a\nhidden topological structure to light that substantially\nmodi\fes the slow spin dynamics. On the other hand, the\napplicability of the ASD stands and falls with the validity\nof the constraint imposed, and unfortunately, contrary to\nquantum systems, there is no direct classical equivalent\nof the adiabatic theorem which ensures adiabaticity in\ncertain limits. Comparison of the predictions of the ASD\nwith those of the full theory treating all, slow and fast\ndegrees of freedom, is thus necessary. In fact, this has\nuncovered some de\fciencies: While the ASD applies to\nthe weak-coupling limit only, as it was anticipated, it\nalso requires that the initial impurity-spin con\fguration\nis not too far from the ground-state con\fguration, and\neven in this case there is a good but not fully convincing\nagreement with the full theory.\nWe have therefore studied another version of a con-\nstrained spin dynamics assuming that the host-spin sys-\ntem is tightly bound but not necessarily in the ground\nstate for the present impurity-spin con\fguration at any\ninstant of time. Also this constrained spin dynamics\nmust be worked out carefully within the Lagrange for-\nmalism, and again there is a topological spin torque in-\nvolved. Spin dynamics under the tight-binding constraint\nsomewhat relaxes the ASD constraint. In fact, the ASD\ncould be newly derived by enforcing the missing piece\nagain. Comparing with the full theory, we found that\nthe relaxation of the constraint indeed results in an im-\nproved e\u000bective theory, which now covers the entire weak-\ncoupling limit. This advantage, however, also comes at a\ncost: Spin dynamics under the tight-binding constraint\nnecessarily involves host degrees of freedom, i.e., it fails\nto provide an e\u000bective impurity-spin dynamics theory.\nMore severely, however, the e\u000bective Lagrangian is sin-\ngular in the case of a nonmagnetic host with a vanishing\ntotal spin.\nOur present study can be seen as a \frst step towards an\ne\u000bective theory of RKKY real-time dynamics, i.e., where\nimpurity spins are coupled to a conduction-electron sys-\ntem, and work on this quantum-classical problem in al-\nready in progress. Clearly, this problem is more involved\nsince with the Fermi energy of the electronic system there\nis an additional energy scale to be considered. This also\nimplies the emergence of a length scale, resulting, e.g., in\nthe nontrivial distance dependence of the e\u000bective RKKY\nexchange. Furthermore, we expect the resulting e\u000bective\ntheory to be of non-Hamiltonian character as well. We\nalso expect to make contact with the Berry curvature\nof the electronic system and a corresponding topolog-\nical spin torque, replacing the topological charge den-\nsity of the purely classical host-spin system studied here.\nClearly, the quantum-classical problem is more relevant\nfor interpreting experimental \fndings. We believe that\nthe insights gained from our present classical study will\nbe very helpful for this next step.14\nACKNOWLEDGMENTS\nThis work was supported by the Deutsche Forschungs-\ngemeinschaft (DFG) through the Cluster of Excellence\\Advanced Imaging of Matter\" - EXC 2056 - project ID\n390715994, and by the DFG Sonderforschungsbereich 925\n\\Light-induced dynamics and control of correlated quan-\ntum systems\" (project B5).\nAppendix A: Total spin conservation within the ASD\nWithin adiabatic spin dynamics the total spin, i.e., the sum of the total impurity spin Stotand the total host\nspinstot, is conserved, if He\u000bis SO(3) symmetric. Here, we prove total-spin conservation for a collinear host-spin\nstructure. We start by computing the time derivative of the total spin:\nd\ndt(Stot+stot) =SX\nr_mr+sX\nid\ndtn0;i(m) =X\nr@He\u000b\n@mr\u0002mr+X\nrTr\u0002mr+sX\nir\u0016_mr\u0016@n0;i(m)\n@mr\u0016; (A1)\nwhere, in the \frst step, we have inserted the equation of motion Eq. (7) to eliminate _mr. Consider the second term\nin the last expression. Making use of Eq. (35) we \fnd:\nX\nrTr\u0002mr=X\nrzirzKs\u0001\nm3\n0(_m0\u0002m0)\u0002mr=zKs\u0001\nm3\n0(_m0\u0002m0)\u0002m0: (A2)\nTo treat the third term, we employ Eq. (29):\nsX\nir\u0016_mr\u0016@n0;i(m)\n@mr\u0016=sX\nir\u0016zizK_mr\u0016zir1\nm0\u0012\ne\u0016\u0000m0\u0016\nm2\n0m0\u0013\n=szKX\nizi1\nm0\u0012\n_m0\u0000(_m0m0)m0\nm2\n0\u0013\n=zKs\u00011\nm3\n0m0\u0002(_m0\u0002m0): (A3)\nThis cancels the second term. SO(3) symmetry implies that He\u000b(m) has the general form\nHe\u000b(m) =f((crr0)r;r0=1;:::;R); (A4)\nwherefis an arbitrary smooth function of all inner products crr0\u0011mrmr0. Sincem2\nr= 1, we have\nX\nr@He\u000b\n@mr\u0002mr=X\nrX\nr0r00@f\n@cr0r00@cr0r00\n@mr\u0002mr=X\nrX\nr0r00@f\n@cr0r00(\u000err0mr00+\u000err00mr0)\u0002mr= 0: (A5)\nThis proves that Stot+stot= const.\nAppendix B: Solution of coupled ODE's\nWe consider the following system of two (three-\ncomponent) ordinary di\u000berential equations\n_x1=c1x2\u0002x1;\n_x2=c2x1\u0002x2; (B1)\nwherec1;c2are constants. With the scaling transforma-\ntion\ny1=\u000b1x1;y2=\u000b2x2; (B2)\nwhere\u000b1;\u000b2are constants to be determined, we have\n_y1=c1\n\u000b2y2\u0002y1;\n_y2=c2\n\u000b1y1\u0002y2: (B3)Choosing\n\u000b1=s\f\f\f\fc2\nc1\f\f\f\f; \u000b 2=s\f\f\f\fc1\nc2\f\f\f\f=1\n\u000b1; (B4)\nwe get\n_y1=s1p\njc1c2jy2\u0002y1;\n_y2=s2p\njc1c2jy1\u0002y2; (B5)\nwheres1=c1=jc1jands2=c2=jc2jare sign factors. We\ndistinguish the two cases s1=\u0006s2and conclude that\nthere is a conserved vector\ny\u0011y1\u0006y2=s\f\f\f\fc2\nc1\f\f\f\fx1\u0006s\f\f\f\fc1\nc2\f\f\f\fx2= const. (B6)15\nSince Eqs. (B5) and (B6) imply\n_y1=\u0006s1p\njc1c2jy\u0002y1;\n_y2=s2p\njc1c2jy\u0002y2; (B7)\nwe see thaty1andy2and, thus,x1andx2precess with\nequal orientation around y. To compute the precession\nfrequency!p=p\njc1c2jjyjwe need the length of y:\njyj2=jy1j2+jy2j2\u00062y1y2\n=\f\f\f\fc2\nc1\f\f\f\fx2\n1+\f\f\f\fc1\nc2\f\f\f\fx2\n2\u00062x1x2: (B8)\nHere, we have used Eq. (B4) and \u000b1\u000b2= 1 in particular.\nNote that Eq. (B1) immediately implies that x1;x2and\nx1x2are constant. The precession frequency\n!p=q\nc2\n2x2\n1+c2\n1x2\n2\u00062jc1c2jx1x2 (B9)\ndepends on the lengths of x1andx2and on the angle\nenclosed initially.\nLet us now consider the di\u000berential equations Eq. (40)\nand Eq. (42) for x1=m0andx2=mtot. The coe\u000e-\ncients corresponding to m0andmtotread\nc0=1\n1 +zKs\u0001\nSm3\n0m0mtotjKjs\nm0;\nctot=zKs\u0001\nSm01\n1 +zKs\u0001\nSm3\n0m0mtotjKjs\nm0;(B10)\nrespectively. With ctot=c0=zKs\u0001=Sm 0, this means pre-\ncession around the axis\ny=s\ns\nSj\u0001j\nm0m0\u0006s\nS\nsm0\nj\u0001jmtot= const. (B11)\nwhere the \\+\"-sign applies for zK\u0001>0 and the \\\u0000\"-\nsign forzK\u0001<0. After rescaling, we note that yis\ncollinear to\n\u0006sj\u0001j\nm0m0+Smtot= const. (B12)\nSincestot=sP\nizi\u0011=s\u0001zKm0=m0=\u0006sj\u0001jm0=m0,\nthis means that the precession axis is just de\fned by the\ntotal spinstot+Stot, as expected on physical grounds.\nA simple result for the precession frequency is obtained\nin the case of two impurity spins ( R= 2) with zi1=\n\u0000zi2, where we can exploit the relation m0mtot= (m1\u0000\nm2)(m1+m2) = 0:\n!p=jKjs1\nm0r\ns2j\u0001j2\nS2+m2\ntot: (B13)\nAssuming that s=Sand thatj\u0001j= 1 (antiferromag-\nnetic host-spin con\fguration and odd L), the precessionfrequency is\n!p=jKjsp\n1 +m2\ntot\nm0=jKjs1\n2 sin#=2r\n1 + 4 cos2#\n2;\n(B14)\nwhere#2]0;\u0019] is the conserved angle enclosed by m1\nandm2.\nAppendix C: Using tight-binding constraints to\nsimplify the equations of motion\nIn an attempt to construct an alternative e\u000bective the-\nory, let us start from the fundamental equations of mo-\ntion (3) for the impurity spins Sr=Smr. Using the\nnotations of Sec. II, we have\n_mr=Ksnir\u0002mr; (C1)\nwhere we have assumed Br= 0, for simplicity. Further,\nthe equations of motion for sir=snirread\n_nir=KSmr\u0002nir+sX\ni0Jiri0ni0\u0002nir: (C2)\nWe want to exploit the constraint\nni=zi\u0011 (C3)\n(zi=\u00061). This tight-binding constraint expresses that\nforK\u001cJall host spins are tightly bound together such\nthat, irrespective of the impurity-spin con\fguration, all\nniare collinear to a unit vector \u0011at all times t.\nIt is tempting, but incorrect, to use the constraint to\nsimplify the equations of motion as will be shown here.\nEq. (C3) implies that the second term in Eq. (C2) van-\nishes. Using the constraint once more, we can eliminate\nthe host spins and are left with\n_mr=Kszir\u0011\u0002mr; (C4)\nand\n_\u0011=KSmr\u0002\u0011 (C5)\nfor allr= 1;:::;R . This yields\n(mr\u0000mr0)\u0002\u0011= 0; (C6)\nor\n\u0011=\u0006mr\u0000mr0\njmr\u0000mr0j; (C7)\nfor allr;r0. For arbitrary directions mrand forR\u00153,\nhowever, this obviously leads to contradictions.16\nAppendix D: Lagrange formalism using tight-binding constraints\nThe correct dynamics under a constraint of the form n=n0(\u0011) (ni=n0;i(\u0011)) can be derived from the e\u000bective\nLagrangian\nLe\u000b(\u0011;_\u0011;m;_m)\u0011L(n0(\u0011);(d=dt)n0(\u0011);m;_m); (D1)\nwhereL(n;_n;m;_m) =SP\njA(mj)_mj+sP\niA(ni)_ni\u0000H(n;m) is the full Lagrangian. Here, we use the short-\nhand notation n0= (n0;1;:::;n0;L),n= (n1;:::;nL) andm= (m1;:::;mR). Furthermore, A(r) is a vector \feld\nsatisfying r\u0002A(r) =\u0000r=r3, and which can thus be interpreted as the vector potential of a unit magnetic (Dirac)\nmonopole located at r= 0. In the standard gauge [38], this is given by A(r) =\u0000(1=r2)(ez\u0002r)=(1 +ezr=r). The\nequations of motion deriving from the full Lagrangian are equivalent with the Hamilton equations (3), see Ref. [31]\nfor further details.\nWith\nd\ndtn0;i(\u0011) = ( _\u0011r)n0;i(\u0011) (D2)\nwe \fnd:\nLe\u000b(\u0011;_\u0011;m;_m) =SX\njA(mj)_mj+sX\niA(n0;i(\u0011))\u0010\n(_\u0011r)n0;i(\u0011)\u0011\n\u0000He\u000b(\u0011;m); (D3)\nwherei= 1;:::;L andj= 1;:::;R , and where He\u000b(\u0011;m) =H(sn0(\u0011);Sm). To get the Lagrange equations of motion,\nwe \frst compute\n@Le\u000b(\u0011;_\u0011;m;_m)\n@mr=SX\n\frrA\f(mr) _mr\f\u0000@He\u000b(\u0011;m)\n@mr: (D4)\nand\n@Le\u000b(\u0011;_\u0011;m;_m)\n@\u0011=sX\ni\fA\f(n0;i(\u0011)) (_\u0011r)rn0;i\f(\u0011) +sX\ni\u000b\f@A\f(n0;i(\u0011))\n@n0;i\u000brn0;i\u000b(\u0011)(_\u0011r)n0;i\f(\u0011)\u0000@He\u000b(\u0011;m)\n@\u0011:\n(D5)\nHere,rr=@=@mr, and Greek indices \u000b;\f;:::2fx;y;zg. Furthermore,\n@Le\u000b(\u0011;_\u0011;m;_m)\n@_mr=SA(mr);@Le\u000b(\u0011;_\u0011;m;_m)\n@_\u0011=sX\ni\u000bA\u000b(n0;i(\u0011))rn0;i\u000b(\u0011); (D6)\nwhich yields\nd\ndt@Le\u000b(\u0011;_\u0011;m;_m)\n@_mr=S(_mrrr)A(mr) (D7)\nand\nd\ndt@Le\u000b(\u0011;_\u0011;m;_m)\n@_\u0011=sX\ni\u000b\f@A\u000b(n0;i(\u0011))\n@n0;i\f(_\u0011rn0;i\f(\u0011))rn0;i\u000b(\u0011) +sX\ni\u000bA\u000b(n0;i(\u0011)r(_\u0011rn0;i\u000b(\u0011)):(D8)\nThe last term equals the \frst term on the right-hand side of Eq. (D5) in the Lagrange equations, since rand _\u0011r\ncommute, such that we are left with:\n0 =d\ndt@Le\u000b\n@_mr\u0000@Le\u000b\n@mr=S(_mrrr)A(mr)\u0000SX\n\frrA\f(mr) _mr\f+@He\u000b(\u0011;m)\n@mr\n=S(rr\u0002A(mr))\u0002_mr+@He\u000b(\u0011;m)\n@mr; (D9)\nand\n0 =d\ndt@Le\u000b\n@_\u0011\u0000@Le\u000b\n@\u0011=@\n@\u0011He\u000b(\u0011;m)\u0000sX\ni\u000b\f@A\f(n0;i(\u0011))\n@n0;i\u000brn0;i\u000b(\u0011)(_\u0011r)n0;i\f(\u0011)\n+sX\ni\u000b\f@A\u000b(n0;i(\u0011))\n@n0;i\f(_\u0011rn0;i\f(\u0011))rn0;i\u000b(\u0011)\n=@He\u000b(\u0011;m)\n@\u0011+T; (D10)17\nwhereTstands for the last two terms. Taking in Eq. (D9) the cross product from the right, ( :::)\u0002mr, we \fnd\nS((rr\u0002A(mr))\u0002_mr)\u0002mr+@He\u000b(\u0011;m)\n@mr\u0002mr= 0: (D11)\nUsingrr\u0002A(mr) =\u0000mr=m3\nr, expanding the remaining double cross product and exploiting that mris a unit\nvector, yields:\nS_mr=@He\u000b(\u0011;m)\n@mr\u0002mr; (D12)\nwhich just recovers the standard form of the equation of motion for mr. On the contrary, the equation of motion for\n\u0011, which is obtained from Eq. (D10) by taking the cross product with \u0011, is unconventional:\n0 =@He\u000b(\u0011;m)\n@\u0011\u0002\u0011+T\u0002\u0011: (D13)\nNote that actually we should have added Lagrange-\nmultiplier terms, Le\u000b(\u0011;_\u0011;m;_m)7!Le\u000b(\u0011;_\u0011;m;_m)\u0000P\nr\u0015r(m2\nr\u00001)\u0000\u0015(\u00112\u00001), to account for the normaliza-\ntion conditions m2\nr= 1 and\u00112= 1. However, this would\nmerely have resulted in additional summands 2 \u0015rmrand\n2\u0015\u0011on the right-hand sides of Eqs. (D9) and (D10), re-\nspectively, which do not contribute after taking the re-\nspective cross products ( :::)\u0002mrand (:::)\u0002\u0011. On the\nother hand, taking the dot products, ( :::)\u0001mrand (:::)\u0001\u0011,\nin Eqs. (D9) and (D10), respectively, just yields the nec-\nessary conditional equations for \u0015rand\u0015, if these were\nrequired.\nTgives rise to a geometrical spin torque T\u0002\u0011and\ncan be read o\u000b from Eq. (D10):\nT=sX\ni\u000b\f\u0012@A\u000b(n0;i(\u0011))\n@n0;i\f\u0000@A\f(n0;i(\u0011))\n@n0;i\u000b\u0013\n\u0001(_\u0011r)n0;i\f(\u0011)rn0;i\u000b(\u0011): (D14)\nExploiting once more the de\fning property of the vec-\ntor potential, ri\u0002A(n0;i) =\u0000n0;i=n3\n0;i, and using the\nnormalization n0;i= 1 in the end, we \fnd:\nT=sX\ni\u000b\f\r\u000f\u000b\f\rrn0;i\u000b(\u0011) (_\u0011r)n0;i\f(\u0011)n0;i\r(\u0011)\n=sX\niX\n\u0016\u0017r\u0016n0;i(\u0011)\u0002r\u0017n0;i(\u0011)\u0001n0;i(\u0011) _\u0011\u0017e\u0016:\n(D15)\nThe scalar triple product de\fnes an antisymmetric tensor\nof rank two:\n\n\u0016\u0017=sX\ni@n0;i(\u0011)\n@\u0011\u0016\u0002@n0;i(\u0011)\n@\u0011\u0017\u0001n0;i(\u0011)\n=\u0000\n\u0017\u0016=X\n\u001a\u000f\u0016\u0017\u001a\n\u001a; (D16)\nwhere the last equation de\fnes the pseudovector \nwith\ncomponents \n \u001a=1\n2P\n\u0016\u0017\u000f\u0016\u0017\u001a\n\u0016\u0017:\n\n=s\n2X\niX\n\u000b\f\r\u000f\u000b\f\rrn0;i\u000b\u0002rn0;i\fn0;i\r; (D17)which has precisely the form of the \\magnetic vorticity\"\n[37]. Hence:\nT=X\n\u0016\u0017\n\u0016\u0017_\u0011\u0017e\u0016=_\u0011\u0002\n: (D18)\nNote thatT_\u0011= 0. Inserting the result for Tin the\nequation of motion, we obtain\n0 =@He\u000b(\u0011;m)\n@\u0011\u0002\u0011+ (_\u0011\u0002\n)\u0002\u0011:(D19)\nIf the pseudovector \nis interpreted as a magnetic \feld\nin\u0011-space,T=_\u0011\u0002\nis the Lorentz force (per unit\ncharge) and T\u0002\u0011the corresponding torque. On the\nother hand, the analogy cannot be made complete, as\nthe curl of the vector potential of a \\magnetic monopole\",\nri\u0002A(n0;i) =\u0000n0;i=n3\n0;i, is a \feld in n0;i-space.\nAppendix E: Spin dynamics under tight-binding\nconstraints\nStarting from the constraint, ni=n0;i(\u0011) =zi\u0011, the\ncomputation of the topological spin torque is straightfor-\nward. We have:\n@n0;i(\u0011)\n@\u0011\u0016=zie\u0016 (E1)\nand thus\n\n\u0016\u0017=sX\ni(zie\u0016)\u0002(zie\u0017)\u0001(zi\u0011) =s\u0011X\nizie\u0016\u0002e\u0017;\n(E2)\nor in terms of the psuedovector\n\n=s\u0001\u0011; (E3)\nwith\n\u0001\u0011LX\ni=1zi: (E4)18\nFor an antiferromagnetic host, e.g., we have \u0001 = \u00061 ifL\nis odd, and \u0001 = 0 if Lis even. Generally, \nis just the\ntotal host spin:\nstot=X\nisi=X\niszi\u0011=s\u0001\u0011=\n: (E5)\nNow, the topological spin torque reads as\nT\u0002\u0011= (_\u0011\u0002\n)\u0002\u0011=s\u0001 (_\u0011\u0002\u0011)\u0002\u0011\n=\u0000s\u0001_\u0011=\u0000s_ntot=\u0000_stot; (E6)\nand therefore the set of equations of motion is given by\ns\u0001_\u0011=@He\u000b(\u0011;m)\n@\u0011\u0002\u0011;\nS_mr=@He\u000b(\u0011;m)\n@mr\u0002mr: (E7)\nWe see that, for \u0001 6= 0 and opposed to the ASD, one\narrives at a standard Hamiltonian dynamics for the re-\nmaining degrees of freedom \u0011andmgoverned by an\ne\u000bective Hamiltonian which is obtained by making use of\nthe constraint in the original Hamiltonian.\nExplicitly, the e\u000bective Hamiltonian, He\u000b(\u0011;m) =\nH(si=szi\u0011;Sr=Smr), reads\nHe\u000b(\u0011;m) =E0+KsS\u0011RX\nr=1zirmr\u0000SRX\nr=1mrBr:(E8)\nThis describes a central spin model: The impurity spins\nSr=Smrcouple with strengths zirKto the central spin\nstot=sntot=s\u0001\u0011. With\n@He\u000b(\u0011;m)\n@\u0011=sSKX\nrzirmr;\n@He\u000b(\u0011;m)\n@mr=sSKzir\u0011\u0000SBr; (E9)\nthe Hamiltonian equations are given by:\ns\u0001_\u0011=sSKX\nrzirmr\u0002\u0011;\nS_mr=sSKzir\u0011\u0002mr\u0000SBr\u0002mr:(E10)\nLet us start the discussion with the case \u0001 6= 0 and de-\nrive some consequences of the equations of motion. First,\nwe note that the total spin is conserved, if Br= 0:\nd\ndt(sntot+Smtot) =s_ntot+X\nrS_mr\n=@He\u000b(\u0011;m)\n@\u0011\u0002\u0011+X\nr@He\u000b(\u0011;m)\n@mr\u0002mr= 0;\n(E11)\nexploiting Eq. (E9). Hence, sntot+Smtot= const. En-\nergy conservation ( d=dt)He\u000b(\u0011;m) = 0 follows by con-\nstruction and can also be veri\fed explicitly. There arefurther conserved quantities. From Eq. (E10) we imme-\ndiately \fndjmrj= const. = 1,j\u0011j= const. = 1, and for\nBr= 0 we can derive\n\u0001_\u0011=SKm0\u0002\u0011;\n_m0=sK\u0011\u0002mtot;\n_mtot=sK\u0011\u0002m0: (E12)\nFurther, Eq. (E10) yields m0mtot= const., and for\nR= 2, in particular, we trivially have m0mtot= 0. Gen-\nerally, one cannot inferm0= const. or mtot= const.,\nopposed to the ASD and Eq. (43). We rather have\nm2\n0+m2\ntot= const. and ( m0\u0006mtot)2= const. only. We\nconclude that the e\u000bective spin dynamics under tight-\nbinding constraints di\u000bers from the naive adiabatic the-\nory as well as from ASD.\nAppendix F: Non-admissible e\u000bective Lagrangian in\nthe case \u0001 = 0\nWe proceed with the discussion of the topologically\ntrivial case \u0001 = 0. Here, Eq. (E10) implies that m0\u0002\u0011=\n0, and this yields \u0011=\u0006m0=m0, and\u0011=zKm0=m0\nif, initially, the dynamics starts with the ground-state\ncon\fguration of the host spins for given m. We would\nthus exactly recover the naive adiabatic theory, see Sec.\nV, and Eq. (24) in particular. However, the constrained\nLagrangian theory is inconsistent in general, as m0\u0002\u0011=\n0 may con\rict with an initial state of the system where\nm0\u0002\u00116= 0.\nIn the case \u0001 = 0, one can in fact show that con-\nstraining the spin system by imposing Eq. (50) leads to\na singular e\u000bective Lagrangian. This singularity is sub-\ntle. For a discussion, we \frst start with a short note on\nLagrange mechanics for a system of point particles de-\nscribed by Ncoordinates q= (q1;:::;qN). Consider the\nEuler-Lagrange equations\n0 =@L(q;_q)\n@qj\u0000d\ndt@L(q;_q)\n@_qj\n=@L\n@qj\u0000X\ni@2L\n@qi@_qj_qi\u0000X\ni@2L\n@_qi@_qjqi:(F1)\nThe Lagrangian is called singular, if the Hesse matrix\nHij=@2L=@_qi@_qjcannot be inverted. This is the typi-\ncal case in classical spin dynamics and explains why there\nis no simple connection between Hamiltonian and La-\ngrangian formalism mediated by a Legendre transforma-\ntion (see, e.g., the discussion in the supplemental ma-\nterial of Ref. [31], section B) and why it is convenient\nto stay with in Lagrangian framework when discussing\nconstrained classical spin systems. In principle, how-\never, a Hamiltonian formulation can be derived directly\nfrom a singular Lagrangian with the Dirac-Bergmann for-\nmalism [39{41]. However, the problem is more severe if\nL(q;_q) =L(q). In such a case not only the Hesse matrix\nis singular (in fact, H= 0), but also the coe\u000ecient matrix19\nKij=@2L=@qi@_qjvanishes. This may lead to inconsis-\ntencies and, hence, such Lagrangians are not admissible.\nThis means that imposing the constraint is unphysical\nand does not lead to a valid e\u000bective theory.\nThe latter exactly applies to our spin system when\nimposing the constraint (50) in the case \u0001 = 0. This\ncan be seen by a gauge transformation of the e\u000bective\nLagrangian. We use the constraint Eq. (50) explicitly to\nrewrite the e\u000bective Lagrangian (D1). With\nd\ndtn0;i(\u0011) = ( _\u0011r)n0;i(\u0011) =zi_\u0011 (F2)\nwe \fnd\nLe\u000b(\u0011;_\u0011;m;_m) =SX\njA(mj)_mj+sX\niA(zi\u0011)zi_\u0011\n\u0000He\u000b(\u0011;m): (F3)\nThe curl of the vector potential A(r)\u0011Aez(r) =\n\u0000(1=r2)(ez\u0002r)=(1 +ezr=r) is invariant under a gauge\ntransformation that replaces ezby an arbitrary unit vec-\ntore. The Euler-Lagrange equations are in fact invariant\nunder a local,i-dependent gauge transformation, speci-\n\fed byez7!ziez, of the second term on the right-hand\nside:\nAez(zi\u0011)7!Azie(zi\u0011) =\u00001\n\u00112ez\u0002\u0011\n1 +ez\u0011=\u0011: (F4)\nNote that this does no longer depend on the site index\ni. The result of the transformation is that the second\nterm on the right-hand side of Eq. (F3) vanishes if \u0001 =P\nizi= 0 and, hence, the transformed but equivalent\ne\u000bective Lagrangian,\nLe\u000b(\u0011;m;_m) =SX\njA(mj)_mj\u0000He\u000b(\u0011;m);(F5)\nlacks the dependence on _\u0011. Therefore, it is not admissi-\nble.\nAppendix G: Single impurity spin coupled to a\nmagnetic \feld\nForR= 1, i.e., for a single impurity spin S=Sm,\nand for \u00016= 0, Eq. (E10) reads:\n\u0001_\u0011=SKm\u0002\u0011;_m=sK\u0011\u0002m\u0000B\u0002m;(G1)\nwhere we have set, without loss of generality, zi1= +1.\nThis implies\n_m=\u0000s\nS\u0001_\u0011\u0000B\u0002m: (G2)\nFor givenm, the ground state of the tightly bound host-\nspin subsystem is given by ni=zi\u0011with\u0011=zKm=\n\u0000signKm. Taking the time derivative of this additional\ncondition yields:\n_\u0011=zK_m: (G3)Inserting this relation into Eq. (G2), we \fnd in case of\nantiferromagnetic Kondo coupling ( zK<0), antiferro-\nmagnetic host-spin structure and odd L(\u0001 = +1)\n_m=1\n1\u0000s=Sm\u0002B: (G4)\nThis describees precession with a renormalized frequency\nas predicted by the ASD, see Ref. [31] and Eq. (37). It\nseems that the ASD can be re-derived by imposing the\nabove additional constraint. For the general case of arbi-\ntraryR, however, we must carefully base the considera-\ntions on the action principle, as shown below, since using\nEq. (G3) to simplify the equation of motion lacks formal\njusti\fcation.\nAppendix H: Alternative derivation of the ASD\nInterestingly, one can indeed give an alternative deriva-\ntion of the ASD by starting from the e\u000bective spin dy-\nnamics under tight-binding constraints discussed above\nand by imposing the additional constraint\n\u0011=\u00110(m)\u0011zKm0=m0; (H1)\nwithzK=\u0000K=jKj, i.e., assuming that the mutually\nbound host spins are, at any instant of time, in the\nground-state con\fguration corresponding to the respec-\ntive impurity-spin con\fguration.\nTo prove our claim, we start from the e\u000bective Hamil-\ntonian Eq. (E8). Inserting the constraint, Eq. (H1), in\nthe e\u000bective Hamiltonian He\u000b(\u0011;m), yields an e\u000bective\nHamiltonian depending on the impurity-spin degrees of\nfreedom only,\nHe\u000b(m) =E0\u0000jKjsSm 0\u0000SRX\nr=1mrBr; (H2)\nwhich, of course, equals the one derived earlier, see Eq.\n(21). For the derivation of the e\u000bective equation of mo-\ntion formunder the additional constraint, we use the\naction principle and follow the steps outlined in Ref.\n[31]. The unconstrained dynamics is governed by the\nLagrangian\nLe\u000b(\u0011;_\u0011;m;_m) =A(\u0011)s\u0001_\u0011+X\nrA(mr)S_mr\n\u0000He\u000b(\u0011;m): (H3)\nUsing the constraint Eq. (H1), we get an e\u000bective La-\ngrangian depending on m;_monly:\nLe\u000b(m;_m) =A(\u00110(m))s\u0001X\nr(_mrrr)\u00110(m)\n+X\nrA(mr)S_mr\u0000He\u000b(m): (H4)20\nThis form of Le\u000bdi\u000bers only slightly from the one dis-\ncussed in Ref. [31] such that the resulting Lagrange equa-\ntions have exactly the same form as Eq. (7):\nS_mr=@He\u000b(m)\n@mr\u0002mr+Tr\u0002mr: (H5)\nHereTris given via\nTr\u0016=Tr\u0016(m;_m) =X\ns\u0017\nr\u0016;s\u0017(m) _ms\u0017; (H6)\nin terms of\n\nr\u0016;s\u0017(m) = 4\u0019s\u0001er\u0016;s\u0017(m); (H7)\nwhich di\u000bers from Eq. (9) by the missing sum over iand\nthe additional factor \u0001. We have:\ner\u0016;s\u0017(m) =1\n4\u0019@\u00110(m)\n@mr\u0016\u0002@\u00110(m)\n@ms\u0017\u0001\u00110(m):(H8)This topological charge density can be computed as in\nSec. VI, and we \fnd\ner\u0016;s\u0017(m) =1\n4\u0019zKzirzisX\n\u001c\"\u0016\u0017\u001cm0\u001c1\nm3\n0: (H9)\nTherewith, we have\n\nr\u0016;s\u0017(m) =zirziszKs\u0001X\n\u001c\"\u0016\u0017\u001cm0\u001c1\nm3\n0: (H10)\nThis is exactly the result found earlier, see Eq. (32), and\nthus yields the same expression for the topological spin\ntorque, Eq. (35), and the same equations of motion, Eq.\n(36), formr.\n[1] D. C. Mattis, The Theory of Magnetism (Springer,\nBerlin, 1981).\n[2] A. Auerbach, Interacting electrons and quantum mag-\nnetism (Springer, New York, 1994).\n[3] W. Nolting and A. Ramakanth, Quantum Theory of Mag-\nnetism (Springer, Berlin, 2009).\n[4] P. W. Anderson, Phys. Rev. 79, 350 (1950).\n[5] C. Zener, Phys. 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This e\u000bect is related to the competition between the spin blockade\ne\u000bect and the hyper\fne interaction. The spin noise spectrum of the system universally diverges as\nln(1=!) at low frequencies.\nI. INTRODUCTION\nThe most fascinating discoveries in the solid state\nphysics in the XXIstcentury are related to the spin de-\ngree of freedom of electrons. Intense studies of the spin-\nrelated phenomena led to the formation of a new branch\nin the solid state physics { spintronics [1]. The spin re-\nlated phenomena are most pronounced in the low dimen-\nsional structures due to the enhanced role of the spin-\norbit and hyper\fne interactions [2{4]. From practical\npoint of view, the most promising for quantum informa-\ntion processing are zero-dimensional nanosystems, such\nas shallow impurities, color centers and quantum dots\n(QDs).\nThere are two complementary approaches to study\nspin-related phenomena in the QDs. The \frst one is\nbased on the optical spin orientation, manipulation and\ndetection, and is usually applied to the self-organized\nquantum dots [5{7]. The second one is based on the\nelectrical spin injection and detection in the gate-de\fned\nquantum dots [8], which makes use of the external mag-\nnetic \feld. An interesting and promising system for the\nlatter approach is a double quantum dot [9, 10], which\ndemonstrates the Pauli or spin blockade e\u000bect [11, 12].\nThis e\u000bect was studied in detail theoretically [13{18], but\nthe spin dynamics was investigated mainly in the pres-\nence of electric current and external magnetic \feld.\nIn this work we study manifestations of the spin block-\nade e\u000bect in the spin dynamics of double quantum dot\nisolated from the environment in the absence of external\nmagnetic \feld. Our theory can be also applied for an iso-\nlated pair of donors, which are close to each other, but\nfar enough from the other donors.\nThe spin dynamics in quantum dots in zero mag-\nnetic \feld is largely driven by the hyper\fne interaction\nwith the host lattice nuclear spins [19]. In a double\n\u0003vmantsev@gmail.com\nysmirnov@mail.io\u000be.ruquantum dot the exchange interaction [20, 21] and elec-\ntron hopping [22, 23] are also important and a\u000bect the\nspin dynamics. We stress, that we consider only hop-\nping between the QDs, but not to the contacts or sub-\nstrate [24, 25]. The interplay between exchange inter-\naction, hopping and hyper\fne interaction can be hardly\ninvestigated for the large spin ensembles. But the dou-\nble quantum dot system considered here allows for the\nexact solution and gives some hints about spin dynamics\nin larger spin systems.\nIn our study we focus on two e\u000bects: the spin relax-\nation and spin noise. The \frst one assumes the spin ori-\nentation and measurement of the spin polarization decay.\nThe second one is based on the continuous measurement\nof the dynamics of spin \ructuations in the thermal equi-\nlibrium [26, 27]. We demonstrate, that in both cases the\nspin dynamics essentially consists of the spin precession\nin a random nuclear \feld and slow power law relaxation.\nThe latter e\u000bect is a consequence of the interplay between\nthe spin blockade and the hyper\fne interaction.\nThe paper is organized as follows. In the next sec-\ntion we present the model of the system under study. In\nSec. III we present our approach to calculate the spin\ndynamics and spin noise spectrum. We show numeri-\ncal results for the arbitrary relation between the system\nparameters and stress the universality of the power law\n/1=tspin relaxation. Then in Sec. IV we derive the\nanalytical results in the limiting cases, which explain the\nnumerical results. Further, in Sec. V we discuss the lim-\nits of applicability of our model, and \fnally summarize\nour \fndings in Sec. VI.arXiv:1905.07305v1  [cond-mat.mes-hall]  17 May 20192\nFIG. 1. Sketch of the double QD system. The two electrons\nare represented by the blue balls, and their spins | by the\nred arrows. Orange arrows show the Overhauser \feld acting\nin each dot, magenta spring denotes the exchange interaction.\nThe navy arrows show the possible hops of electrons between\nthe QDs (green transparent balls).\nII. MODEL\nWe consider a double QD with two electrons, as shown\nin Fig. 1. We assume that the two electrons can be local-\nized either in di\u000berent or in the same QD, and can hop\nbetween the QDs. We take into account the exchange\ninteraction between electrons and their hyper\fne inter-\naction with the host lattice nuclear spins. The Hilbert\nspace of the system under study consists of six states: the\ntwo singlet states, when the two electrons are localized in\nthe same QD, plus another singlet state and three triplet\nstates, when the two electrons are localized in di\u000berent\nQDs.\nThe Hamiltonian of the system has the form:\nH=X\ni;\u001bEin\u001b\ni+X\niUin+\nin\u0000\ni+Js1s2+~X\ni\nisi:(1)\nHere in the \frst term Eiare the localization energies of\nelectrons in the ith QD (i= 1;2) andn\u001b\ni(\u001b=\u0006) are the\noccupancies of the states, characterized by the spin index\n\u001b. The corresponding operators can be written using\nthe Fermi creation (annihilation) operators cy\ni;\u001b(c\u001b\ni) as\nn\u001b\ni=cy\ni;\u001bci;\u001b. The second term in Eq. (1) describes the\non-site electron repulsion with the Hubbard energy Ui.\nThe third term is the exchange interaction, characterized\nby the constant J. The spin operators can be expressed\nas\nsi=1\n2\u001b\u001b\u001b0cy\ni\u001bci\u001b0; (2)\nwhere\u001b= (\u001bx;\u001by;\u001bz) is the vector composed of the\nPauli matrices. Finally, the last term in Eq. (1) is the\nhyper\fne interaction, where \niis the spin precession fre-\nquency in the \ructuation of host lattice nuclear spin po-\nlarization. In this study we assume the number of host\nlattice nuclear spins to be large, so that the Overhauser\n\feld can be considered as static (\\frozen\") [28].\nThe electron hopping being inelastic process, can not\nbe described be electron Hamiltonian solely. Thereforeone has to consider the total Hamiltonian\nHtot=H+Hph+V; (3)\nwhich takes into account a phonon Hamiltonian Hphand\nan electron phonon interaction V. The phonon energy is\ngiven by\nHph=~X\nq\nqby\nqbq; (4)\nwhere \n qis the phonon frequency, corresponding to the\nwavevectorq, andby\nq(bq) is the phonons creation (anni-\nhilation) operator. We assume the phonon polarization\nindex to be included in q. The electron-phonon interac-\ntion after the canonical (polaron) transformation [29, 30]\nreads\nV=tX\n\u001bexp(\n\u0000X\nq\rq\u0002\u0000\neiqR1\u0000eiqR2\u0001\nbq\u0000h:c\u0003)\n\u0002cy\n1\u001bc2\u001b+ h:c:;(5)\nwheretis the hopping constant, \rq=vq=(~\nq) withvq\nbeing the electron-phonon interaction constant, and R1;2\nbeing the coordinates of the QDs.\nThe spin dynamics in the system can be described us-\ning the density matrix formalism. In the description of\nelectron hopping we assume that the \frst two terms in\nthe Hamiltonian (1) and the temperature exceed by far\nthe two latter terms, so the states, where two electrons\nare in the di\u000berent QDs, have nearly the same energy.\nIn this case the o\u000b diagonal matrix elements between\nthe states with the essentially di\u000berent energy can be ne-\nglected. As a result the system is described by the 4 \u00024\ndensity matrix \u001ain the basis of the four states of two\nelectrons in di\u000berent QDs, and the two probabilities Pi\nto \fnd the two electrons in the QD i.\nIn the two lowest orders in the hopping amplitude ( t)\nthe total density matrix of the electron system \u001atotsat-\nis\fes the equation\n_\u001atot=\u0000i\n~[H;\u001atot]\n+\u0019\n~\n\u0002\n2V\u001atotV\u0000\u001atotV2\u0000V2\u001atot\u0003\n\u000e(Ei\u0000Ef)\u000b\nph;(6)\nwhere the \frst line describes the coherent spin dynamics\nand the second one | the electron hopping. The angular\nbrackets denote averaging of the phonon creation and an-\nnihilation operators over the phonon states. The energies\nEiandEfare the total energies of the system before and\nafter the hop, respectively, including the phonon energy.\nNote that we do not include VinH, assuming it to be\nnegligible in comparison with the other terms.\nFrom Eq. (6) we \fnd, that the electron density matrix\n\u001aobeys the master equation\n_\u001a=\u0000i\n~[H;\u001a] +1\n2X\niX\n\u001b;\u001b0h\n2\u0000i\u0016{cy\ni\u001bc\u0016{\u001bP\u0016{cy\n\u0016{\u001b0ci\u001b0\n\u0000\ri\u0016{\u0010\n\u001acy\n\u0016{\u001b0ci\u001b0cy\ni\u001bc\u0016{\u001b+cy\n\u0016{\u001b0ci\u001b0cy\ni\u001bc\u0016{\u001b\u001a\u0011i\n;(7)3\nwhere the symbol \u0016 {denotes the other quantum dot than\ni(\u0016{= 2 ifi= 1 and \u0016{= 1 ifi= 2) and we introduce the\nrates\ri\u0016{and \u0000i\u0016{describing the hopping from QD \u0016 {toi\nwhen the QD iis occupied or empty, respectively. In the\nsecond order in the electron-phonon interaction ( \rq) the\nhopping rate with the change of energy by \u0001 Eis [4, 31]\n\r(\u0001E) =2\u0019\n~t22\r2\nq\u0001ED(\u0001E) [N\u0001E+\u0012(\u0000\u0001E)];(8)\nwhereq\u0001Eis the phonon wave vector corresponding to\na phonon with the energy \u0001 E,D(\u0001E) stands for the\ndensity of phonon states, N\u0001E= 1=[exp(\u0001E=kBT)\u00001]\nis the occupancy of the corresponding states with Tbeing\nthe temperature, and \u0012(x) is the Heaviside step function.\nIn the higher orders in the electron-phonon interaction\nthe expression for the hopping rate is di\u000berent, but its\nexplicit form is unimportant for our study. The speci\fc\nhopping rates between the QDs are given by\n\ri\u0016{=\r(Ei\u0000E\u0016{+Ui); (9a)\n\u0000i\u0016{=\r(Ei\u0000E\u0016{\u0000Ui): (9b)\nOne can see, that in the general case the relation \u0000 i\u0016{\u0015\ri\u0016{\nholds, because of the electron Coulomb repulsion in the\nsame QD.\nSimilarly to Eq. (7) the probabilities Piobey\n_Pi=1\n2X\n\u001b;\u001b0h\n2\ri\u0016{cy\ni\u001bc\u001b\u001acy\n\u0016{\u001b0ci\u001b0\n\u0000\u0000\u0016{i\u0010\nPicy\ni\u001b0c\u0016{\u001b0cy\n\u0016{\u001bci\u001b+cy\ni\u001b0c\u0016{\u001b0cy\n\u0016{\u001bci\u001bPi\u0011i\n:(10)\nThe electron conservation rule for this system can be\nwritten as\nP1+P2+P12= 1 (11)\nwhereP12= (n+\n1+n\u0000\n1)(n+\n2+n\u0000\n2) = Tr\u001ais the probability\nto \fnd the two electrons in the di\u000berent QDs.\nWe recall that we assume the nuclear \felds \nito be\nfrozen. They are created by the nuclear spin \ructuations\nand are described by the Gaussian distribution function\nF(\ni) =1\n(p\u0019\u000e)3e\u0000\n2\ni=\u000e2; (12)with the parameter \u000echaracterizing the dispersion. In\norder to obtain experimentally observable spin dynam-\nics, the solution of the spin dynamics equations should\nbe averaged over this distribution function. In the next\nsection we demonstrate, that this procedure ultimately\nleads to the power low spin decay /1=tat long time\nscales.\nIII. SPIN RELAXATION AND SPIN NOISE\nThe master Eq. (7) can be rewritten in the form of\nequations for the spin operators siand their correlation\nfunctionss\u000b\nis\f\nj, where\u000band\fare the Cartesian indices.\nTheir average values can be expressed through the den-\nsity matrix ashsii= Tr(si\u001a) andhs\u000b\nis\f\nji= Tr(s\u000b\nis\f\nj\u001a).\nThe electron spins obey\ndsi\ndt=\ni\u0002si+J\n~s\u0016{\u0002si\u0000\r\n2(si\u0000s\u0016{); (13a)\nwhere\r=\r12+\r21is the total hopping rate for the\nsinglet state of the two electrons in di\u000berent QDs. The\n\frst term on the right hand side in this equation de-\nscribes the spin precession with the frequency \ni. Simi-\nlarly the second term describes the electron spin preces-\nsion in the e\u000bective exchange magnetic \feld of another\nelectron. Finally, the last term describes the electron\nhopping. One can see, that this term vanishes in the\ncase ofs1=s2due to the spin blockade. By contrast,\nthe hopping rate equals to \rwhen the two electrons are\nin the singlet state, i.e. s1=\u0000s2. We recall, that the\nrelationkBT;U 1;2\u001dJ;~\u000eis assumed, so there is no spin\npolarization in the thermal equilibrium.\nWe stress, that the term s1\u0002s2in Eq. (13a) does\nnot simply reduce to the product of the two average val-\nues, but should be treated as a vector composed of the\nspin correlators. The spin correlation functions in general\nobey the equations\nd\ndt\u0010\ns\u000b\n1s\f\n2\u0011\n=\"\u000b\r\u000e\n\r\n1s\u000e\n1s\f\n2+\"\f\r\u000e\n\r\n2s\u000b\n1s\u000e\n2+J\n4~\"\u000b\f\r(s\r\n1\u0000s\r\n2)\u0000\r\n2\u0010\ns\u000b\n1s\f\n2\u0000s\f\n1s\u000b\n2\u0011\n+\u000e\u000b\f\n2(\rPs\u0000\u000021P1\u0000\u000012P2);(13b)\nwherePs=P12=4\u0000s1s2is the occupancy of the singlet\nstate in the two di\u000berent QDs. Explicit form of these\nequations is given in Appendix A. The \frst two terms in\nthe right hand side of this expression describe the spin\nprecession in the nuclear \feld. The third term is relatedto the exchange interaction and reduces to the \frst power\nof spin operators for the spin one half particles. The rest\nof the terms describe the hopping of electrons and deserve\na longer discussion.\nThe hopping of two electrons to the same QD brings4\nthe system to the singlet state with the zero total an-\ngular momentum. In the same time, the hopping does\nnot change the total angular momentum, so it is allowed\nonly for the singlet spin state in agreement with the Pauli\nexclusion principle. The correlators s\u000b\n1s\f\n2can be com-\nbined in the groups, which transform according to the\nrepresentations D2,D1andD0of SU(3) group. The \fve\ncorrelators s\u000b\n1s\f\n2+s\f\n1s\u000b\n2with\u000b6=\fbelong toD2rep-\nresentation and do not decay, because they require the\ntwo electron spins to be parallel. The three combinations\ns\u000b\n1s\f\n2\u0000s\f\n1s\u000b\n2belong toD1representation and decay with\nthe rate\r, which is described by the penultimate term\nin Eq. (13b). Finally, the correlator s1s2belongs toD0\nrepresentation, and it couples to the scalar occupancies\nP1,P2andP12, which is described by the last term with\nthe square brackets. This term consists of two contribu-\ntions: hopping to the states, where the two electrons are\nlocalized in the same QD with the rate \r, and hopping\nfrom these states with the rates \u0000 12and \u0000 21.\nThe above equations describe the spin dynamics and\nshould be accompanied by kinetic equations for the oc-\ncupancies of the states. Taking into account Eq. (11) it\nis enough to write the two equations\n_Pi=\u00002\u0000\u0016{iPi+ 2\ri\u0016{Ps; (14)\nThe set of 18 Eqs. (11), (13), and (14) is equivalent to\nEqs. (7) and (10) and completely describes the spin and\ncharge dynamics.\nTo simplify the following analysis we set all the hop-\nping rates, \u0000 12, \u000021,\r12and\r21, equal to\r=2. Physically\nthis corresponds to the high temperature limit (the ther-\nmal energy much larger, than the Hubbard energies Ui).\nThen it is convenient to introduce the parameter\nX=P1+P2\u00002Ps; (15)\nwhich describes the deviation of the occupancies from\ntheir steady state values. This parameter simply obeys\ndX\ndt= 2(\n1\u0000\n2)(s1\u0002s2)\u00004\rX: (16)\nNote that the same parameter describes also the dynam-\nics of the spin correlators in Eq. (13b). Thus in this case\none can consider only 16 equations: Eqs. (13) and (16).\nThe spin dynamics can be calculated for the given initial\nconditions, and the double QD system is characterized in\ntotal by the three parameters: J,\u000eand\r.\nTo describe the spin relaxation we consider the ini-\ntial conditions s1(0) =s2(0) =ez=2, whereezis a unit\nvector along some zaxis. These initial conditions corre-\nspond to the optical spin orientation, and they are oppo-\nsite to what is realized in electrically controlled double\nQD system [16]. Fig. 2(a) shows the evolution of the z\ncomponent of the total spin S=s1+s2in the most com-\nplicated case, when all the parameters are of the same\norderJ=~\u0018\u000e\u0018\r(their values are given in the \fgure\ncaption). The spin dynamics can be separated into two\ncontributions, below we describe them separately:\nFIG. 2. (a) The spin relaxation for the initial conditions\nsi(0) = ez=2. The red dashed lines show the asymptote /1=t.\n(b) The spin noise spectrum. The red dashed curve in the in-\nset shows the asymptote /ln(1=!). The parameters of the\ncalculation are J= 0:9~\u000eand\r= 1:1\u000e.\n(i) The total spin quickly decays from 1 to less than\n0:1 and then increases again at t\u000e\u00185. This time depen-\ndence is typical for the spin dephasing in random Over-\nhauser \feld [28, 32, 33]. Notably, the spin polarization\ndoes not decay to zero due to the conservation of the spin\ncomponent parallel to the Overhauser \feld in each QD.\nThe exchange interaction \\exchanges\" the electrons in\nthe two QDs, so the direction of precession of the given\nelectron spin changes. This, however, also does not lead\nto the complete spin relaxation [21]. Indeed, in the limit\nof very strong exchange interaction the hyper\fne \feld\ndoes not mix the singlet and triplet states, so the com-\nponent of the total spin Salong the average Overhauser\n\feld\n= (\n1+\n2)=2 is conserved.\n(ii) At the long timescales the total spin slowly decays5\nto zero due to the hopping of electrons between the QDs.\nIn fact, this is a power law decay Sz(t)/1=t, as shown\nby the red dashed line in Fig. 2(a). This asymptotic is\nmore clearly shown in the inset, where the longer time\nscales are shown in the bilogarithmic scale. We checked\nnumerically, that this law of spin relaxation is valid for\narbitrary relations between the parameters J,\rand\u000e.\nMoreover this law will be derived analytically in a num-\nber of limiting cases in the next section. To understand\nthe e\u000bect qualitatively we note, that in the exceptional\ncase\n1kezand\n2kezthe total spin does not change,\nand the hopping is also forbidden for the initial condition\ns1=s2[see Eqs. (13)]. So in this case the spin polariza-\ntion (in our model) does not decay at all. In the more\nprobable situation, when \n1k\n2the component of the\ntotal spin along this direction does not decay either be-\ncause of the spin blockade. Finally, in the general case of\narbitrary angle between \n1and\n2the spin polarization\ndecays the longer the smaller is the angle. Averaging over\nthe Gaussian distribution of the Overhauser \felds results\nin the power law decay Sz(t)/1=tat long time scales.\nThe slow spin decay can be conveniently revealed in\nthe frequency domain. Experimentally the spin dynam-\nics at low frequencies can be studied by means of the\nspin noise spectroscopy [26]. This method is based on\nthe measurement of the correlation functions of the spin\n\ructuations in the thermal equilibrium. The spin noise\nspectrum (\u000eS2\nz)!is de\fned as a Fourier transform of the\nautocorrelation function\n(\u000eS2\nz)!=1Z\n\u00001h\u000eSz(t)\u000eSz(t+\u001c)iei!\u001cd\u001c; (17)\nwhere the angular brackets denote averaging over t.\nIn the equilibrium the spin polarization is absent, so\nhSz(t)i= 0 and\u000eS=Sin the system under study.\nTo calculate the correlation functions we note, that\nthe correlators at \u001c= 0 can be simply found from the\nsteady state solution of the equations of motion. One\n\fnds, that the correlation functions of Szwith all the\nother operators in Eqs. (13) and (16) are zero except for\nhSzsz\nii=hP12i\n4: (18)\nIn the thermal equilibrium hs1s2i= 0, so from Eqs. (14)\nand Eq. (11) we \fnd that\nhP12i=\u0012\n1 +\r12\n4\u000021+\r21\n4\u000012\u0013\u00001\n: (19)\nIn the high temperature limit ( \ri\u0016{= \u0000i\u0016{=\r=2) one has\nhP12i= 2=3 in agreement with Eq. (16). So for the total\nspin we obtain\nhS2\nzi=1\n3: (20)\nThe correlators de\fne the initial conditions for the time\ncorrelation functions.Then the set of the correlators of \u000eSz(t) with the other\noperators taken at time t+\u001cobey the same equations\nof motion, Eqs. (13) and (16), for \u001c >0 [34]. Moreover,\nthe spin autocorrelation function is an even function of\n\u001c, which allows us to \fnd h\u000eSz(t)\u000eSz(t+\u001c)iand the spin\nnoise spectrum ( \u000eS2\nz)!after Eq. (17). We note, that the\nspin noise spectrum can be also calculated directly in\nthe frequency domain replacing the time derivatives in\nequations of motion with the multipliers \u0000i![35].\nThe spin noise spectrum is shown in Fig. 2(b) for the\nsame system parameters as in the panel (a). It again\nconsists of two contributions. (i) A peak at frequency\n!\u0018\u000e, which corresponds to the spin precession in the\nOverhauser \feld [36, 37]. Its shape reproduces the distri-\nbution function of the absolute values of the Overhauser\n\feld [36, 38]. (ii) A peak at zero frequency, which corre-\nsponds to the slow spin decay at long times. This peak\nshows the divergence ( \u000eS2\nz)!/ln(1=!) at!!0 in\nagreement with the asymptotic h\u000eSz(t)\u000eSz(t+\u001c)i/1=\u001c\nin the time domain. The logarithmic asymptote for the\nspin noise spectrum is shown in the inset in Fig. 2(b).\nThus the spin relaxation and the spin noise spectrum\nessentially describe the same spin dynamics in the time\nand frequency domains, respectively.\nIV. LIMITING CASES\nThe main result of the previous section is the very slow\npower law decay /1=tof the spin polarization despite\nall the necessary ingredients for the spin relaxation in\nthe model. This result corresponds to the divergence of\nthe spin noise spectrum at zero frequency /ln(1=!). In\nthis section we derive these asymptotes in limiting cases,\nwhen one of the system parameters \u000e,J=~, or\ris much\nlarger than the two others.\nA. Strong hyper\fne interaction\nThe limit\u000e\u001dJ=~;\rcorresponds to the two nearly\nindependent QDs, where the spins s1;2precess around\nthe corresponding nuclear \felds \n1;2. As a result of this\nprecession the initial spin polarization on average decays\nthree times on the timescale \u00181=\u000e[32]. The one third\nof spin polarization on average is parallel to the static\n\ructuation of the Overhauser \feld and does not decay\nat this timescale. The exchange interaction only slightly\nchanges the eigenfunctions and does not lead to the com-\nplete spin relaxation. By contrast, the hopping, being\nincoherent process, leads to the complete decay of the\nspin polarization. As a result the exchange interaction in\nthis limit can be neglected, while the hopping can not.\nThe spin dynamics in this limit can be described by\nEqs. (13) with J= 0:\n_si=\ni\u0002si\u0000\r\n2\u0001(si\u0000s\u0016{): (21)6\nFIG. 3. Relaxation of the spin polarization Sz(t) calculated numerically (black solid curves) and analytically (red dashed\ncurves) for the three limiting cases (a) J= 0:04~\u000eand\r= 0:2\u000e[Eq. (25)], (b) J= 0:2~\u000eand\r= 5\u000e[Eq. (34)], and (c)\nJ= 5~\u000eand\r= 0:2\u000e[Eq. (41)]. The initial conditions are s1;2=ez=2. The insets show the power law decay /1=tfor the\nsame parameters.\nOne can separate the spin components parallel and per-\npendicular to the nuclear \feld as\nsik=nisi;si?=si\u0000nisik; (22)\nwhereni=\ni=\ni. These components approximately\nobey [39]\n_si?=\ni\u0002si;?\u0000\r\n2si;?; (23a)\n_sik=\u0000\r\n2\u0000\nsik\u0000s\u0016{kcos\u0012\u0001\n; (23b)\nwhere cos\u0012=n1n2. The solution of these equations\ngives\ns1(t)+s2(t) =X\ni[si?(0) cos(\nit) +ni\u0002si(0) sin(\nit)]\n+n1+n2\n2\u0002\ns1k(0) +s2k(0)\u0003\ne\u0000t\r(1\u0000cos\u0012)=2\n+n1\u0000n2\n2\u0002\ns1k(0)\u0000s2k(0)\u0003\ne\u0000t\r(1+cos\u0012)=2:(24)\nThis expression should be averaged over the distribution\nof\ni[see Eq. (12)]:\nhs1(t) +s2(t)i= [s1(0) +s2(0)]\n\u00022\n3(\"\n1\u0000(\u000et)2\n2#\ne\u0000(\u000et)2=4+e\u0000\rt+\rt\u00001\n(\rt)2)\n:(25)\nThis expression is shown by the red dashed curve in\nFig. 3(a) and agrees with the numerical calculations,\nshown by the black solid curve. At t\u001d1=\rthis ex-\npression yields the power law decay [40]\nhs1(t) +s2(t)i=2\n3\rt: (26)\nThis expression is shown by the red dashed line in the\ninset in Fig. 3(a).Since the spin correlation functions also obey the equa-\ntion like Eqs. (21), the spin noise spectrum can be\nfound simply as a Fourier transform of Eq. (25) with\ns1(0) +s2(0) =ez=3 [see Eq. (20)]:\n\u0000\nS2\nz\u0001\n!=1\n\u000ef\u0010!\n\u000e\u0011\n+1\n\rg\u0012!\n\r\u0013\n; (27)\nwhere we introduce the functions\nf(x) =8\n9p\u0019x2e\u0000x2; (28)\ng(x) =2\n9\u0002\n\u0019jxj+ ln\u0000\n1 + 1=x2\u0001\n\u00002 (1 +xarctgx)\u0003\n:\n(29)\nThe analytical expression for the spin noise spectrum in\nthis limit is shown in Fig. 4 by the blue dashed curve and\nagrees with the numerical calculations (blue solid curve).\nAt low frequencies the spin noise spectrum diverges as\n\u0000\nS2\nz\u0001\n!=4\n9\rln\u0010\r\n!\u0011\n; (30)\nas expected.\nB. Fast hopping between QDs\nIn the limit \r\u001d\u000e;J=~one could expect, that the\nspin polarization quickly decays to zero because of the\nfast hops of electrons into one QD, where the total spin\nis zero. This, however does not happen, because of the\nspin blockade: when the two spins are parallel to each\nother, the electrons do not hop.\nIt is convenient to rewrite Eqs. (13a) as\n_S=\n\u0002S+ \u0001\n\u0002\u0001S; (31a)\n\u0001_S= \u0001\n\u0002S\u0000\r\u0001S\u00002J\n~s1\u0002s2; (31b)7\nFIG. 4. Spin noise spectra calculated numerically (solid\ncurves) and analytically (dashed curves) for the same param-\neters as in Fig. 3(a) (blue curves), (b) (red curves), and (c)\n(black curves), see Eqs. (27), (36), and (43), respectively.\nwhere \u0001S=s1\u0000s2,\n= (\n1+\n2)=2 and \u0001 \n=\n(\n1\u0000\n2)=2. In the lowest order in J=(~\r) the term\nwiths1\u0002s2can be neglected, while \u0001 Sin the second\nequation quickly relaxes to the value\n\u0001S=\u0001\n\u0002S\n\r: (32)\nFrom Eq. (31a) one can see, that Sprecesses around \n,\nwhile its projection on \ndecays due to the second term.\nTherefore one can solve separately equation for these two\ncomponents and \fnd\nhS(t)i=S(0)\n\u0002\u001c\nsin2(\u0012) cos (\nt) + cos2(\u0012) exp\u0012\n\u0000\u0001\n2sin2(\u00120)\n\rt\u0013\u001d\n;\n(33)\nwhere the angular brackets denote averaging over \n1;2,\n\u0012is the angle between \nandS(0), and\u00120is the angle\nbetween \u0001 \nand\n.\nThe frequencies \nand \u0001 \nare normally distributed,\nsimilarly to Eq. (12), but with \u000ebeingp\n2 times smaller.\nThis allows us to \fnd ultimately\nhS(t)i=S(0)2\n3\u0014\u0012\n1\u0000(\u000et)2\n4\u0013\ne\u0000(\u000et)2=8+\r\n2\r+\u000e2t\u0015\n:\n(34)\nThis expression is plotted in Fig. 3(b). One can see, that\nit is very similar to the panel (a) despite the opposite\nrelation between the parameters. At long time scales the\nspin polarization decays as\nhS(t)i=S(0)2\r\n3\u000e2t(35)in agreement with the general result of the previous sec-\ntion.\nSimilarly to the previous subsection the spin noise\nspectrum can be derived simply performing the Fourier\ntransform of Eq. (34):\n\u0000\nS2\nz\u0001\n!=p\n2\n\u000ef p\n2!\n\u000e!\n+\r\n\u000e2h\u00122\r!\n\u000e2\u0013\n; (36)\nwhere we introduced\nh(x) =2\n9fsin(jxj) [\u0019\u00002 Si(jxj)]\u00002 cos(x) Ci(x)g;(37)\nwith Si(x) and Ci(x) being the sine and cosine integral\nfunctions, respectively. This expression is shown by the\nred dashed curve in Fig. 4 and agrees with numerical\ncalculations. At low frequencies one \fnds\n\u0000\nS2\nz\u0001\n!=4\r\n9\u000e2ln\u0012\u000e2\n\r!\u0013\n; (38)\nso the spectrum again diverges logarithmically.\nC. Strong exchange interaction\nIn the limit J=~\u001d\u000e;\rthe spins strongly couple into\nthe triplet and singlet. The hyper\fne interaction weakly\nmixes these states, while the electron hopping is possi-\nble only in the singlet state because of the Pauli spin\nblockade.\nThe equations for spin dynamics in this limit can be\nobtained from Eqs. (13) in the lowest order in ~\n1;2=J.\nNote, that the terms containing \ragain can not be ne-\nglected, because they lead to the complete spin decay at\nlong time scales. As a result we obtain\n_S=\n\u0002S+ \u0001\n\u0002\u0001S; (39a)\n\u0001_S= \u0001\n\u0002S\u0000\r\u0001S\u00002J\n~s1\u0002s2; (39b)\n(s1\u0002s2) _ =J\n2~\u0001S\u0000\rs1\u0002s2: (39c)\nOne can see, that \u0001 Sands1\u0002s2decay much faster,\nthanS, so the time derivatives in the second and third\nequations can be set to zero. This gives\ns1\u0002s2=J\n2~\r\u0001S; (40a)\n\u0001S=~2\r\nJ2\u0001\n\u0002S: (40b)\nSubstituting the last expression in Eq. (39a) and averag-\ning the solution over the nuclear \felds we \fnd\nhS(t)i=S(0)2\n3\u0014\u0012\n1\u0000(\u000et)2\n4\u0013\ne\u0000(\u000et)2=8+J2\n2J2+~2\r\u000e2t\u0015\n:\n(41)8\nThis expression is shown in Fig. 3(c), and one can see,\nthat the spin polarization in this case decays particularly\nslow. Indeed at long time scales one \fnds\nhS(t)i=S(0)2J2\n3~2\r\u000e2t; (42)\nso the prefactor of 1 =tis parametrically large in the limit\nunder study, J=~\u001d\u000e;\r.\nThe spin noise spectrum in this limit can be calculated\nsimilarly to the previous subsections and the result reads\n\u0000\nS2\nz\u0001\n!=p\n2\n\u000ef p\n2!\n\u000e!\n+J2\n~2\r\u000e2h\u00122J2!\n~2\r\u000e2\u0013\n:(43)\nThis expression is shown by the black dashed curve in\nFig. 4 and again agrees with the numerical calculations.\nAt low frequencies one \fnds\n\u0000\nS2\nz\u0001\n!=4J2\n9~2\r\u000e2ln\u0012~2\r\u000e2\nJ2!\u0013\n; (44)\nwhich shows once again, that the spin correlations decay\nparticularly slow in this limit.\nV. DISCUSSION\nWe demonstrated, that the spin polarization decays as\n/1=tfor any relation between the system parameters.\nNow let us discuss the applicability limits of our model.\nIn typical GaAs based self assembled QDs the charac-\nteristic time scales is de\fned by 1 =\u000e\u00181 ns [41, 42]. At\nlonger time scales a few mechanisms of the spin relaxation\ncan come into play, which can limit the applicability of\nour model.\nA zero magnetic \feld the on-site electron spin \rip-\rops\ndue to the electron-phonon and spin-orbit interactions\nhave very low rates because of, e.g., the zero phonon\ndensity of states with zero energy. Indeed, according\nto Refs. 43{48 the spin relaxation caused by the direct\nspin-phonon coupling [49, 50] or spin admixture mecha-\nnisms [49, 51] should exceed 1 s at magnetic \feld smaller\nthan 1 T.\nThe spin-orbit interaction during the hops leads to the\nspin rotations [52, 53], which is not taken into account\nby our model. However, this e\u000bect can be simply ac-\ncounted for by the rotation of the coordinate frames for\nthe two QDs in the spin space [20]. Still the small ran-\ndom deviations of the electron hopping trajectory from\nthe semiclassical one [54] can not be compensated in the\nsame way.\nThe most probable limitations of our model are the ex-\nternal excitation of the system, e.g. by optical pulses [55,\n56], and the nuclear spin dynamics. The nuclear spin pre-\ncession can be caused either by the strain in the QDs and\nthe quadrupole interaction, or by the Knight \feld created\nby electrons. These e\u000bects take place at the microsecondtime scales [57, 58] and quench /1=tasymptotic be-\nhaviour. Nevertheless, we assume that our theory will\ncorrectly describe the spin dynamics in double QD on\nthe sub microsecond time scales. In particular the spin\nrelaxation should be described by the power law from a\nfew nanoseconds to a few microseconds, e.g. three orders\nof magnitude in time and frequency domains.\nVI. CONCLUSION\nTo summarize, we studied the spin dynamics in a dou-\nble QD taking into account the interplay between the\nhyper\fne interaction, exchange interaction and electrons\nhopping. We demonstrated numerically, that for arbi-\ntrary relation between the system parameters the spin\nrelaxation consists of the partial spin dephasing in a ran-\ndom nuclear \feld and a universal power law decay /1=t\nat large time scales. The spin noise spectrum of the sys-\ntem similarly consists of the two contributions and di-\nverges as/ln(1=!) at low frequencies. We proved our\nresults analytically in the limits, when one of the system\nparameters exceeds by far the others. We believe, that\nour result will stimulate further experimental investiga-\ntions of spin relaxation and spin noise in double QD.\nVII. ACKNOWLEDGMENTS\nWe gratefully acknowledge the fruitful discussions with\nM. M. Glazov. All numerical calculations were performed\nunder the Russian Science Foundation \fnancial support\n(RSF No. 18-72-10002). D.S.S. was partially supported\nby the RF President Grant No. MK-1576.2019.2, Russian\nScience Foundation grant No. 19-12-00051 and the Basis\nFoundation.\nAppendix A: Explicit form of spin dynamics\nequations\nWe \fnd it useful to rewrite Eqs. (13) in an explicit\nform:\ndsx\n1\ndt= \ny\n1sz\n1\u0000\nz\n1sy\n1+J\n~(sz\n1sy\n2\u0000sy\n1sz\n2)\u0000\r\n2(sx\n1\u0000sx\n2);(A1a)\ndsy\n1\ndt= \nz\n1sx\n1\u0000\nx\n1sz\n1+J\n~(sx\n1sz\n2\u0000sz\n1sx\n2)\u0000\r\n2(sy\n1\u0000sy\n2);(A1b)\ndsz\n1\ndt= \nx\n1sy\n1\u0000\ny\n1sx\n1+J\n~(sy\n1sx\n2\u0000sx\n1sy\n2)\u0000\r\n2(sz\n1\u0000sz\n2);(A1c)\ndsx\n2\ndt= \ny\n2sz\n2\u0000\nz\n2sy\n2+J\n~(sy\n1sz\n2\u0000sz\n1sy\n2)\u0000\r\n2(sx\n2\u0000sx\n1);(A1d)9\ndsy\n2\ndt= \nz\n2sx\n2\u0000\nx\n2sz\n2+J\n~(sz\n1sx\n2\u0000sx\n1sz\n2)\u0000\r\n2(sy\n2\u0000sy\n1);(A1e)\ndsz\n2\ndt= \nx\n2sy\n2\u0000\ny\n2sx\n2+J\n~(sx\n1sy\n2\u0000sy\n1sx\n2)\u0000\r\n2(sz\n2\u0000sz\n1):(A1f)\nd\ndt(sx\n1sy\n2) = \ny\n1sz\n1sy\n2\u0000\nz\n1sy\n1sy\n2+ \nz\n2sx\n1sx\n2\u0000\nx\n2sx\n1sz\n2\n+J\n4~(sz\n1\u0000sz\n2)\u0000\r\n2(sx\n1sy\n2\u0000sy\n1sx\n2);(A2a)\nd\ndt(sx\n1sz\n2) = \ny\n1sz\n1sz\n2\u0000\nz\n1sy\n1sz\n2+ \nx\n2sx\n1sy\n2\u0000\ny\n2sx\n1sx\n2\n+J\n4~(sy\n2\u0000sy\n1)\u0000\r\n2(sx\n1sz\n2\u0000sz\n1sx\n2);(A2b)\nd\ndt(sy\n1sx\n2) = \nz\n1sx\n1sx\n2\u0000\nx\n1sz\n1sx\n2+ \ny\n2sy\n1sz\n2\u0000\nz\n2sy\n1sy\n2\n+J\n4~(sz\n2\u0000sz\n1)\u0000\r\n2(sy\n1sx\n2\u0000sx\n1sy\n2);(A2c)d\ndt(sy\n1sz\n2) = \nz\n1sx\n1sz\n2\u0000\nx\n1sz\n1sz\n2+ \nx\n2sy\n1sy\n2\u0000\ny\n2sy\n1sx\n2\n+J\n4~(sx\n1\u0000sx\n2)\u0000\r\n2(sy\n1sz\n2\u0000sz\n1sy\n2);(A2d)\nd\ndt(sz\n1sx\n2) = \nx\n1sy\n1sx\n2\u0000\ny\n1sy\n1sx\n2+ \ny\n2sz\n1sz\n2\u0000\nz\n2sz\n1sy\n2\n+J\n4~(sy\n1\u0000sy\n2)\u0000\r\n2(sz\n1sx\n2\u0000sx\n1sz\n2);(A2e)\nd\ndt(sz\n1sy\n2) = \nx\n1sy\n1sy\n2\u0000\ny\n1sx\n1sy\n2+ \nz\n2sz\n1sx\n2\u0000\nx\n2sz\n1sz\n2\n+J\n4~(sx\n2\u0000sx\n1)\u0000\r\n2(sz\n1sy\n2\u0000sy\n1sz\n2):(A2f)\nd\ndt(sx\n1sx\n2) = \ny\n1sz\n1sx\n2\u0000\nz\n1sy\n1sx\n2+ \ny\n2sx\n1sz\n2\u0000\nz\n2sx\n1sy\n2\u0000\r\n4X;\n(A2g)\nd\ndt(sy\n1sy\n2) = \nz\n1sx\n1sy\n2\u0000\nx\n1sz\n1sy\n2+ \nz\n2sy\n1sx\n2\u0000\nx\n2sy\n1sz\n2\u0000\r\n4X;\n(A2h)\nd\ndt(sz\n1sz\n2) = \nx\n1sy\n1sz\n2\u0000\ny\n1sx\n1sz\n2+ \nx\n2sz\n1sy\n2\u0000\ny\n2sz\n1sx\n2\u0000\r\n4X;\n(A2i)\nWe recall, that in order to calculate the spin noise spec-\ntrum the time derivatives can be replaces with \u0000i!, and\n1=6 should be added in the right hand side of Eqs. 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Ames Laboratory and Iowa State University, Ames, Iowa 50011, USA \n†e-mail: g.delange@tudelft.nl  \n*e-mail: r.hanson@tudelft.nl  Abstract \nUnderstanding and mitigating decoherence is  a key challenge for quantum science \nand technology. The main source of decoheren ce for solid-state spin systems is the \nuncontrolled spin bath environment. Here , we demonstrate quantum control of a \nmesoscopic spin bath in diamond at room temperature that is co mposed of electron \nspins of substitutional nitrogen impurities.  The resulting spin bath dynamics are \nprobed using a single nitrogen-vacancy (NV)  centre electron spin as a magnetic field \nsensor. We exploit the spin bath control to dynamically suppress dephasing of the \nNV spin by the spin bath. Furthermore, by combining spin bath control with \ndynamical decoupling, we directly measure th e coherence and temporal correlations \nof different groups of bath spins. These results uncover a new arena for \nfundamental studies on decoherence and enable novel avenues for spin-based \nmagnetometry and quantum information processing.  \nIn the past few years, new advances in quantum science and technology have \nunderscored the importance of understanding an d controlling decoherence of single solid-\nstate spins1-3. Decoherence of a single central sp in in contact with a spin bath \nenvironment has been intensively studied in various systems  such as quantum dots4-7, \ndonors in silicon8 and defects in diamond9-12 through control and readout of the central \nspin. Here, we implement quantum control of  both the central spin and its spin bath \nenvironment, thereby enabling a range of  new experiments on the fundamentals of \ndecoherence. Moreover, spin bath control is  a crucial ingredient of recent proposals for \nenvironment-assisted magnetometry13, room-temperature quantum computing using spins \nin diamond14,15 and spin squeezing16. \n 2Our study focuses on the electronic spin bath environment formed by nitrogen \nimpurities surrounding a single NV centre in diam ond (Fig 1a). The electron spin of the \nNV centre can be initialized and read out opti cally, and coherently controlled with high \nfidelity at room temperature using microwave magnetic pulses17-19. For controlling the \nquantum state of the bath spins, we appl y short (tens-of-nanoseconds) radiofrequency \n(RF) pulses to the sample. The high intensity control fields that allow fast control of both \nthe central NV spin and the bath spins ar e delivered through a broadband coplanar \nwaveguide (CPW) fabricated  on the diamond substrate. \nThe state of the (optically inactive) sp in bath can in principle be monitored \ndirectly via the emitted RF radiation as in conventional electron spin resonance. However, this method requires many orders of  magnitude more spins than contained in \nour mesoscopic region of interest, and is lim ited to high magnetic fields. Instead, we \nexploit the coupling of the bath  spins to the single NV centre. The near-atomic size of the \nNV centre, combined with the strong (~1/r\n3) distance dependence of the dipolar coupling \nto the surrounding bath spins, renders the NV spin mainly sensitive to a small number N \n(a few tens) of bath spins. This local spin bath exhibits a large statistical polarization \n(~1/√N) that is felt by the NV centre as a magnetic dipolar field δb. The spin bath \npolarization and the corresponding value of the bath field δb change in time due to flip-\nflop processes within the bath, leading to dephasing of the NV centre spin on a timescale \n of about 300 ns19. This quasi-static dephasing is  compensated in a spin echo \nsequence with a refocusing π-pulse (Fig.1b), yielding decay on a much longer timescale \nT2,NV = (2.6 ±0.1) μ s19. However, if we induce changes in the state of the bath spins (thus \nchanging the value of δb) by applying a high intensity RF  pulse halfway the NV spin *\n2,NVT\n 3echo sequence, the refocusing is ineffective and the NV spin echo amplitude is reduced \n(Fig.1b). Therefore, by incorporating the spin bath control within a spin echo sequence of \nthe NV centre the resulting spin bath dynamics can be probed.  \nResults \nMagnetic resonance spectroscopy of bath spins.  To identify the environmental \nspins we perform magnetic resonance spectro scopy by sweeping the frequency of applied \nRF pulses while monitoring the NV spin echo amplitude (upper panel of Fig 1c). Several \nsharp dips are observed, demonstrating that sp ins in the environment are being rotated at \nthese specific frequencies. The obtained spect rum matches that of single electron spins ( S \n= 1/2) belonging to substitutional nitrogen (N) impurities20. We find excellent agreement \nwith a theoretical spectrum (lower panel Fig. 1c) calculated using known values for the \nZeeman energy and hyperfine interaction with  the N nuclear spin, which is anisotropic \ndue to a static Jahn-Teller distortion (Fig. 1a).  Since the resonance frequencies are spaced \nby several line widths, only spins that belong to the same spectral group can exchange \nenergy via flip-flop processes. The spin e nvironment of the NV centre can therefore be \ndecomposed into different spectral groups of  electron spins (labelled I to V) that are \ndistinguished by their hype rfine interaction with the host N nuclear spin.  \nWith the resonance frequencies known, we can coherently control the spin \nenvironment. Fig. 2a shows the effect of s hort RF pulses at the resonance frequency of \ngroup II spins. Periodic revivals in the NV sp in echo amplitude are observed as a function \nof RF pulse length, with a frequency that increases with RF pulse amplitude. This \nbehaviour is the key signature of coherently driven (“Rabi”) oscillations, demonstrating \nthat we have achieved quantum control of the spin environment. We note that the NV \n 4spin echo revives almost completely whenever the bath spins are rotated by a multiple of \n2π, indicating that the environm ent has returned to the stat e it had before the RF pulse. \nWe can control all other spin bath groups in  a similar manner (Fig. 2b). In addition, our \nsetup allows us to rotate several or all of the groups simultaneously with multi-frequency control pulses.   \nSpin echo double resonance (SEDOR). The ability to control both the NV \ncentre spin and its spin bath environment opens up a range of new possible experiments \naimed at studying and manipulating the coupli ng between a central sp in and a spin bath \nas well as investigating the internal bath dynamics. We firs t apply the bath control to \nmeasure the coupling of each of the bath spin  groups to the NV centre spin using a spin \necho double resonance (SEDOR) scheme\n21 (Fig. 3a). With this scheme the dephasing of \nthe central spin induced by one particular group of bath sp ins can be probed, while the \neffect of all other dephasing channels (including other spin bath groups) is refocused. We \nfind that, whereas the NV spin echo amplitude decays as , the SEDOR \nscheme yields a faster, Gaussian-shaped decay  (see Fig.3a). The Gaussian shape indicates \nthat the decay observed with SEDOR is domin ated by the quasi-static static dephasing \nchannel that we have selectively turn ed on. Therefore, the SEDOR decay time  \ndirectly yields the r.m. s. interaction strength bi between the NV spin and the ith spin bath \ngroup via3\n2,NV exp[ ( / ) ] tT−\nSEDOR, i T\nSEDOR,1/ / 2ii Tb= . We find bI = (0.83±0.02) μ s-1, bII = (1.59±0.03) μ s-1, bIII = \n(1.58±0.04) μs-1 , bIV = (1.63±0.04) μs-1 and bV  = (0.80±0.02) μs-1 (see Supplementary \nFig. 1). These values are close to the ratio of II II I II VV: : : : 1 : 3:2: 3: 1 bb b b b =  \nexpected from the abundance of each spectral group20, except for the slightly lower value \n 5for group III. This group is actually composed  of two subgroups wh ich spectrally do not \ncoincide perfectly. The control fidelity is therefore lower for this group which results in a \nlower measured coupling in the SEDOR experiment \nThe r.m.s. field fluctuations generated by the full electron spin  bath are given by \n2\nspin bath i\nib=∑b= (3.01 ± 0.04) μs-1. This value falls short of the measured total \ndephasing rate of btotal = (3.6 ± 0.1) μs-1 (see Supplementary Fig. 2a), suggesting the \npresence of additional dephasing channels , such as the carbon-13 nuclear spins9,10 and \nmagnetic field drifts, with a strength  of 22\nexcess total spin bathbb b=−   = (1.97 ± 0.04) μs-1. This \ninterpretation is supported by independent  measurements on an NV centre in a pure \ndiamond sample with low nitrogen content unde r the same experimental conditions that \nyield bexcess = (2.06 ± 0.04) μs-1 (see Supplementary Fig. 2b) .  \nFor applications in quantum information processing1 and spin-based dc-\nmagnetometry2,3, suppressing dephasing is crucial. We now demonstrate that quantum \ncontrol of the spin bath can be used to elim inate the effect of the spin bath on the free \nevolution dynamics of the NV centre spin (see  Fig. 3b). By flipping all bath spins, the \ninteraction between the NV centre and bath spins can be time-averaged to zero. This procedure is akin to dynamical decoupling as recently demonstrated on single NV centre \nspins\n19,22,23, but has the advantage that no cont rol pulses on the NV centre itself are \nrequired. We find that a refocusing π-pulse applied simultaneously to all bath spins (Fig. \n3b) increases  up to the limit set by bexcess, indicating that dephasing by the electron *\n2,NVT\n 6spin bath is suppressed. Similar enhancemen t is achieved by continuous driving of all \nbath spins (see Supplementary Fig. 2a). \n \nCoherence and temporal correlations of bath spins. By combining the spin \nbath control with the ability to freeze the evolution of the NV spin by dynamical \ndecoupling19,22,23, coherence and temporal  correlations during free evolution within the \nspin bath can be directly probed. We repla ce the single refocusing pulse of the spin echo \nsequence on the NV spin by a dynamical decoupling (DD) sequence with a net π-rotation \n(see Supplementary information). The DD sequence provides a means to temporarily turn \noff our sensor (the NV centre) as it is made insensitive to the magnetic environment for the duration of the DD sequence; the net π-rotation ensures that the refocusing action of \nthe sequence is preserved. The two periods of free evolution τ\ns of the NV spin now serve \nas sensing stages which each sample the dipo lar field generated by the bath spins. The \nNV echo amplitude is therefore a measure of the correlation between the dipolar fields \nmeasured during the two sensing stages. While the sensor is switched off, we can apply \nmulti-pulse RF sequences to individual spec tral groups of bath spins to study their \ncoherence during free evolution. An RF Ramsey sequence (Fig. 4a) and Hahn-echo \nsequence (Fig. 4b) is ap plied to spectral group i to measure its spin dephasing time  \nand coherence time  respectively. Data is shown for spectral groups I and II.  *\n2,iT\n2,iT\nThe values we find for  are similar for the two groups as expected, since all \nbath spins suffer from the same dephasing channels formed by spins from all groups. *T2,i\n 7From the value of  we estimate the local density of bath spins to be about 100 parts \nper million (see Supplementary information). *\n2,iT\nThe bath spin-echo sequence yi elds different decay times, T2,I = (1.9 ± 0.6) μs  \nand T2,II = (0.89 ± 0.13) μs, for spectral groups I and II. Th e difference in coherence times \nbetween different spectral groups may arise due to dephasing cause d spins within the \nsame group, in a process which is  known as instantaneous diffusion24,25. The RF π-pulse \ndoes not refocus the dipolar interactions be tween spins of the sa me spectral group since \nthese spins are themselves rotated by the RF π-pulse. The resulti ng intra-group dephasing \nis much stronger in group II because it cont ains three times more spins than group I. \nTo characterize the temporal correlations  resulting from the dynamics in the \nenvironment we perform a direct measuremen t of the auto-correla tion function and its 1/ e \ndecay time τC (see Supplementary information). The field generated by the complete \nmagnetic environment is sampled during two sensing stages separated by a variable \nwaiting time during which we turn off the NV centr e sensor (Fig. 4c) and let the spin bath \nevolve freely26. As the sensor off-time is increased, the initial correlation between the two \nfields is gradually lost resulting in decrea sing NV echo amplitude. We observe a decay of \nthe auto-correlation function on a timescale of about 20 µs. \nWe can also find the correlation time of an individual spin bath group by inserting \na SEDOR sequence in the sensing stage, as  demonstrated for group II (Fig. 4c). The \nmeasured correlation time of group II is comp arable to that of the complete magnetic \nenvironment, indicating that the coherence time of the NV centre is indeed limited by the \n 8dynamics of the electron spin bath. The measur ed correlation time is comparable to the \nvalue  ≈ 13 μs expected from mean-field theory19,24. 23\nspin bath 2, /12NV cbTτ=\nDiscussion \n In conclusion we have demonstrated full quantum control of a spin bath \nsurrounding a single NV centre. These result s pave the way for a new class of \nexperiments on spin bath decoherence, such  as manipulating the correlation time of \ndifferent spin bath groups and generating squeezed spin bath states16. Furthermore, the \nsuppression of spin dephasing by spin-bath control may be exploited for protecting \ncoherence in spin-based quantum technologies1,2,3,14,27,28. Finally, quantum control of \nnitrogen electron spins close to NV centres as demonstrated here enables  implementation \nof quantum registers of individual N electron spins29,30, scalable coupling of NV centre \nquantum bits via spin chains15 and ultra-sensitive environment-assisted magnetometry13.  \n \nMethods \nDevice and setup.  The diamond sample is a single  crystal type Ib plate from \nElement Six, with a Nitrogen concentration specified to be below 200 parts-per-million. \nA high-bandwidth golden coplanar  waveguide (CPW) structure for magnetic resonance is \nfabricated by electron beam lithography dire ctly on top of the bulk diamond sample. A \nstatic magnetic field of 132 G aligned along the symmetry axis of the NV centre is \ngenerated by a custom-built four-coil vector magnet setup (Alpha Magnetics). Spin-\ndependent fluorescence of the NV-centre is detected using a home-built confocal microscope. To determine that the detect ed fluorescence originated from a single NV \ncentre, we performed a measurement of the second-order intensity autocorrelation \n 9function ()2()gτ to verify that the antibunching dip reached a value well below 0.5 \n. Spin initialization and readout is achieved by 600 ns laser (532 \nnm) pulses. The timings of all microwave, ra diofrequency and laser pulses are controlled \nby a 4-channel arbitrary waveform generato r (Tektronix AWG5014). Each experiment is \nrepeated for ~2M times to achieve statistical  noise levels of order 1%. All experiments \nare performed at room temperature. ()2(0) 0.1g ± ( 0.15=)\nPhase and frequency control of MW and RF pulses is provided by IQ modulation \nusing two vector sources (R&S SMBV 100A) with carrier set to 2.5 GHz and 400 MHz \nto control the NV spin and bath spins respec tively. I and Q inputs of each vector source \nare controlled by two analogue channels of the AWG. Single and multi-frequency pulses \non the RF channel are generated by proper IQ mixing to achieve image-frequency \nrejection. The modulated MW and RF signals are fed to two high power amplifiers \n(Amplifier Research 25S1G4 and 30W1000B respectively). After amplification the MW \nand RF signals are combined and deli vered to the on-chip CPW. RF field \ninhomogeneities are <1 % over the relevant le ngth scales (10 nm) and therefore do not \nlimit the control fidelity of the bath pulses. \n \nSpin bath description . Substitutional nitrogen impur ities in diamond (also called \nP1 centres in literature) exhibit trigonal symm etry due to a static Jahn-Teller distortion in \nwhich one of the four N-C bonds is elongated. Most of the electron density is concentrated at the antibonding orbital belonging to this elongated N-C bond\n31. The close \nproximity of and partial overlap between the electron wavefunction and the N nucleus, \ncombined with the trigonal symmetry, give s rise to a strong anisotropic hyperfine \n 10interaction between the defect’s electronic spin ( S = 1/2) and the host N nuclear spin ( I = \n1). The internal Hamiltonian of a nitr ogen impurity in diamond is given by  \n()2\nint //ˆˆ ˆ ˆˆˆ ˆ\nzz xx yy z HA S I A S I S IP I⊥ =+ + − ˆ\n⊥    (1) \nwith A// = 114 MHz, = 86 MHz and P = 4.2 MHz and Ŝ and Î are the operators \nfor the electron and nuclear spin respectively of the nitrogen impurity20. The \ndirection of the z-axis is set by the JT distortion axis and is therefore oriented \nalong one of the four N-C bonds20. The outer two satelli tes (belonging to group I \nand V) in the spectrum of Fig. 1c result from nitrogen impurities which have a JT \naxis oriented along the exte rnally applied field.  A\nBoth the reorientation of the JT axis32 and the nuclear spin-f lips of the nitrogen \nimpurities occur on timescales ranging from milliseconds to seconds. This is much shorter than the time required to build up stat istics (tens of minutes per data point) but \nmuch longer than the time it takes to perf orm a single experimental run (several \nmicroseconds). These processes can therefore be  treated as being qua si-static within one \nrun, while at the same time we are averag ing in the complete experiment over the full \nthermal distribution of all nuclear spin pr ojections and JT orientations. Also, the \ndifferences in local configurations of  N impurities around different NV centres will \nchange the  and  parameters of the NV spin, but not the relevant ensemble \nproperties of the spin bath\n33. *\n2T2T\n 11References  \n1. Ladd, T. D. et al.  Quantum computers. Nature 464, 45-53 (2010). \n2. Degen, C. Scanning magnetic field mi croscope with a diamond single-spin \nsensor. Appl. Phys. Lett.  92, 243111 (2008). \n3. Taylor, J. M. et al . High-sensitivity diamond ma gnetometer with nanoscale \nresolution. Nature Phys.  4, 810 (2008). \n4. Bluhm, H. et al. Dephasing time of GaAs electr on-spin qubits coupled to a \nnuclear bath exceeding 200 μs. Nature Phys. 7, 109–113 (2011). \n5. Churchill, H. O. H. et al. Electron-nuclear interac tion in 13C nanotube double \nquantum dots. Nature Phys. 5, 321-326 (2009). \n6. Vink, I. T. et al . Locking electron spins into magnetic resonance by electron–\nnuclear feedback. Nature Phys.  5, 764-768 (2009). \n7. Latta, C. et al . Confluence of resonant laser exci tation and bidirectional quantum-\ndot nuclear-spin polarization. Nature Phys.  5, 758-763 (2009). \n8. Morello, A. et al . Single-shot readout of an  electron spin in silicon. Nature  467, \n687-691 (2010). \n9. Childress, L. et al. Coherent dynamics of coupled electron and nuclear spin qubits \nin diamond. Science 314, 281–285 (2006). \n10. Mizuochi, N. et al. Coherence of single spins coupl ed to a nuclear spin bath of \nvarying density. Phys. Rev. B  80, 041201 (2008). \n11. Hanson, R., Dobrovitski, V. V., Feguin, A. E., Gywat, O. & Awschalom, D. D. \nCoherent dynamics of a single spin interacting with an adjustable spin bath. \nScience  320, 352-355 (2008). \n 1212. Balasubramanian, G. et al. Ultralong spin coherence time in isotopically \nengineered diamond. Nature Mat.  8, 383-387 (2009).  \n13. Goldstein , G. et al . Environment assisted precision measurement. Phys. Rev. Lett. \n106, 140502 (2011). \n14. Yao, N. Y. et al. Scalable architecture for a room temperature solid-state quantum \nprocessor. arXiv :1012.2864v1 (2011). \n15. Cappellaro, P., Viola, L. & Ramanathan, C. Coherent state transfer via highly \nmixed quantum spin chains . Phys. Rev. A  83, 032304 (2011). \n16. Rudner, M. S., Vandersypen, L. M .K., Vuletic, V. & Levitov, L. S. Generating \nentanglement and squeezed states of  nuclear spins in quantum dots. \narXiv:1101.3370v2 (2011). \n17. Fuchs, G. D., Dobrovitski, V. V., Toyli, D. M., Heremans, F. J. & Awschalom, D. D. Gigahertz dynamics of a strongly driven single quantum spin. Science \n326, \n1520-1522 (2009). \n18. Jelezko, F. & Wrachtrup, J. Single de fect centers in diamond: A review. Phys. \nStat. Sol. 203, 3207-3225 (2006). \n19. de Lange, G., Wang, Z. H., Ristè, D., Dobrovitski, V. V.  & Hanson, R. Universal \ndynamical decoupling of a single solid state spins from a spin bath. Science  330, \n60-63 (2010).  \n20. Smith, W. V., Sorokin, P. P., Gelles, I. L.  & Lasher G. J. El ectron-spin resonance \nof nitrogen donors in diamond. Phys. Rev.  115, 1546-1553 (1959). \n21. Slichter, C. P.  Principles of Magnetic Resonance  (Springer-Verlag, New York, \n1990).  \n 1322. Ryan, C. A., Hodges, J. S. & Cory, D. G. Robust decoupling techniques to extend \nquantum coherence in diamond. Phys. Rev. Lett.  105, 200402 (2010). \n23. Naydenov, B. et al. Dynamical decoupling of a si ngle-electron spin at room \ntemperature. Phys. Rev. B 83, 081201 (2011). \n24. Klauder, J. & Anderson, P. Spectral diffusion decay in spin resonance experiments . Phys. Rev.  \n125, 912-932 (1962). \n25. Tyryshkin, A. M., Lyon, S. A., Astashki n, A. V. & Raitsimring, A. M. Electron \nspin relaxation times of phosphorus donors in silicon. Phys. Rev. B  68, 193207 \n(2003). \n26. Laraoui, A., Hodges, J. S., Ryan , C. A., Meriles, C. A. The diamond Nitrogen-Vacancy center as a probe of random fluc tuations in a nuclear spin ensemble. \narXiv:1104.2546v1 (2011). \n27.\n Takahashi, R., Kono, K., Tarucha, S. & Ono,  K. Bias-voltage c ontrol of reversible \nnuclear spin polarizati on in double quantum dots. arXiv:1012.4545v2 (2011).  \n28. Makhonin, M. N. et al. Full coherent control of nuclear spins in an optically \npumped single quantum dot. arXiv :1102.3636v1 (2011). \n29. Gaebel, T. et al . Room-temperature coherent coup ling of single spins in diamond. \nNature Phys. 2, 408-413 (2006). \n30. Hanson, R., Mendoza, F. M., Epstein, R. J. & Awschalom, D. D. Polarization and \nreadout of coupled single spins in diamond. Phys. Rev. Lett.  97, 087601 (2006). \n31. Loubser, J. H. N. & van Wyk, J. A. Electron spin resonance in the study of \ndiamond. Rep. Prog. Phys. 41, 1201 (1978). \n 1432. Davies, G. The Jahn-Teller effect and vibronic coupling at deep levels in \ndiamond. Rep. Prog. Phys. 44, 787 (1981). \n33.  Dobrovitski, V. V., Feguin, A. E., Awschalom, D. D. & Hanson, R. Decoherence dynamics of a single spin versus spin ensemble. Phys. Rev.  \nB 77, 245212 (2008). \nAcknowledgements \nWe sincerely thank D. D. Awschalom, S. Frol ov, G. D. Fuchs, K. Nowack, D. Ristè and \nL. M. K. Vandersypen for useful discussi ons. We gratefully acknowledge support from \nthe Defense Advanced Research Project s Agency, the Dutch Organization for \nFundamental Research on Matter (FOM), th e Netherlands Organiza tion for Scientific \nResearch (NWO), and the European Un ion SOLID programme. Work at Ames \nLaboratory was supported by the Department of Energy—Basic Energy Sciences under \nContract No. DE-AC02-07CH11358.   \nAuthor contributions \nG.d.L., T.v.d.S. and M.S.B. conducted the e xperiments, V.V.D. and Z.H.W. developed \nthe theory. G.d.L. and R.H. wrote the paper. V.V.D. and R.H. supervised the project. All \nauthors discussed the results, analysed th e data and commented on the manuscript.  \n \nAdditional information \n \nSupplementary Information  is available online\n 15Competing financial interests \nThe authors declare no compe ting financial interests. \n \nCorresponding authors \nCorrespondence to: Gijs de Lange or Ronald Hanson. \n   \nFigure captions \nFigure 1 | Magnetic resonance spectroscopy of a spin bath using a single spin sensor. \na, Schematic of the system: a single NV centre electronic spin ( S = 1) is surrounded by a \nbath of electron spins (S = 1/2) belonging to substitutional N impurities. The applied \nexternal magnetic field B is aligned with the symmetry axis of the NV centre, which is \noriented along the [111] crystallographic di rection. Nitrogen impur ities exhibit a static \nJahn-Teller distortion, which results in an elongation of one of the four N-C bonds. As a \nresult, the defect has a symmetry axis, also called the Jahn-Teller axis (indicated red), \nwhich is oriented along randomly along one of the crystallographic axes. There are \ntherefore two geometric types of bath spins which are distinguished by their orientation \nof the Jahn-Teller axis relative to the external field B: those that have their JT axis at an \nangle α = 0o, and those with α = 109.5o. b, Measurement sequence for spin bath \nspectroscopy. A spin echo sequence is applied to the NV spin using MW pulses; the bath \nspins are controlled by RF pulses. The evol ution of the NV spin during the sequence is \nsketched in the Bloch spheres at the bottom,  both for the case of no spin bath control \n 16(solid line), and for the case of spin bath control applied (dashed line) (see \nSupplementary information for details). c, Upper panel: Magnetic resonance \nspectroscopy of the spin bath. A magnetic fi eld of 132 G is applied along the NV centre \nsymmetry axis. Roman numbers label the diff erent groups of N electron bath spins, \naccording to their nuclear spin projections mI and angle α  between their Jahn-Teller axis \nand the external magnetic field:  I,V: mI = ±1 and α  = 0o, II,IV: mI = ±1 and α  = 109.5o\n, \nIII: mI = 0 and α  = 0o or 109.5o. Lower panel: Calculation of the spectrum (see \nSupplementary information).  \n \nFigure 2 | Coherent control of the spin bath. a,  Coherent driven oscillations of group II \nbath spins, using RF pulses of 298 MHz. Revivals in NV echo amplitude are observed \nwhenever the bath spins from group II perform a 2 π rotation. The maximum Rabi \nfrequency extracted from fitting the upper trace at 20 dBm source power is f1 = (20.5 ± \n0.1) MHz. b, Independent coherent contro l of all groups of bath sp ins. Solid lines are fits \nto /\n1 ec o s ( 2DtT)ftπ−∝  with t  the length of the RF pulse (s ee Supplementary information).  \nFigure 3 | Control of NV spin coherence by spin bath manipulation. a,  Spin echo \ndouble resonance (SEDOR) experiment. The RF pulses have been calibrated to rotate a \npreselected group of bath spins over a π angle. The NV spin echo curve is fit to \n, SEDOR curves are fit to (3\n2,NV exp 2 / Tτ⎡∝−⎢⎣)⎤⎥⎦()2\nSEDOR, exp 2 /i Tτ⎡ ⎤ ∝−⎢ ⎥ ⎣ ⎦ (see \nSupplementary information). b, Dynamical suppression of NV centre spin dephasing \nthrough spin bath control. Free evolution of th e NV spin is shown with (blue) and without \n 17(red) RF π-pulse applied to the bath spins. Solid lines are fits that include the detuning \n( MHz) of the MW driving field compared  to the NV spin splitting, and local \nhyperfine interaction of 2.2 MH z with the host nuclear N spin . This hyperfine interaction \nis responsible for the observe d beating pattern (see Suppl ementary information). The \noverall signal decays with a Gaussi an envelope, with decay constant  = (278 ± 5) ns \n(btotal = (3.60 ± 0.06) µs-1) in the absence of the spin ba th control (red) and with decay \nconstant  = (450 ± 9) ns ( bexcess = (2.11 ± 0.07) µs-1) in the case where the bath \ncontrol pulses are applied (blue).  30fΔ=\nT*\n2,NVT\n*\n2,NV\nFigure 4 | Coherent dynamics and tempor al correlations of the spin bath. a, \nMeasurement of decay during free evolution of spin bath groups I and II. The two sensing \nstages marked by τs serve to sample the magnetic environment before and after the \nRamsey sequence is applied to the bath spins.  The NV spin sensor is turned off while the \nRF Ramsey sequence is applied to the bath sp ins (see main text for details). Instead of \ndetuning the RF pulse field with  respect to the transition to observe fringes, an artificial \ndetuning fa is introduced by changing th e phase of the final RF π/2-pulse linearly with \npulse separation 2  afϕπτ=\n(*\n2,/ec o s 2iT\ni. Solid lines are fits to \n)a i fτ\n0\nI,IIii yA\n=∝+∑ πτφ−⎡+ Δ+⎣⎤⎦ (see Supplementary information for details). \nAdditional contributions resulting from the off-resonant dr iving of spins from the nearest \nneighbouring group is taken into account by an  extra oscillating term with frequency iΔ, \nwhich is given by the detuning of this group w ith respect to the pulse carrier. The fast \nmodulation of Ramsey fringes for group I (upper panel) result from off-resonant driving \n 18of the more abundant spins from group II ( () 2 34.4 0.4IIπΔ= ±i\n*\n2,IIT s-1). From the fits we \nextract the decay constants  = (97 ± 11) ns and = (91 ± 7) ns.  b,  Spin echo on \nspin bath groups I and II. The phase of the final RF π/2-pulse is changed as a function of \ntotal free evolution time τ as *\n2,IT\n2  afϕπτ=  with fa = 10 MHz, resulting in oscillations in \nthe NV spin echo amplitude with free evolution time τ. Solid lines are fits to \n(2,/ec o s 2i\naiTf)τπτφ−∝+  from which the decay times T2,I = (1.9 ± 0.6) µs and T2,II = \n(0.89 ± 0.13) µs are extracted (see Supplementary information). c, Measurement of \ntemporal correlations on the full environmen t and on spin bath group II alone. During the \nsensing stages marked bysτ, MW and RF π-pulses can be applied simultaneously to the \nNV spin and to the bath spins to selectively measure the correlation time of a particular \ngroup of bath spins. Solid lines are fits to ()/ 22exp[ 1Ct\nsbeττ−∝− − ] (see Supplementary \ninformation). \n   \n 19ca\n0.8\n300 400 500\nFrequency (MHz)NV spin echo amplitudeTheory\nB =132GI\nII III IVVExperiment\n0.4\n0.8\n0.4z\nx\ny\nz\nxy\nz\nxyττ\nWithout bath pulse\nWith bath pulse\nδb\nπNVπ 2 π 2NV\nspin:\nR(θ)Bath \nspins:b\nR(θ)\nπNVBath spins\nNC CC\nCCCVB\nC\nC C\nN\nC\nC\nC C\nN\nCNV spin\nα = 109.5oα = 0oFig. 1\n1\n0 100 300820\n412\n20016\nBath pulse length (ns) Source power (dBm)0NV echo amplitude\n02 0 0 4 0 0123\nPulse length (ns)NV echo amplitudeV\nIV\nIII\nII\nIabFig. 200 . 51Without bath π-pulse\nWith bath π-pulse\n00.51P(m  = 0)s\nNV free evolution time τ (μs)π 2 π 2τ\nΣR     (π)iab\n0123400.51NV echo amplitude\nNV free evolution time 2 τ (μs)π π 2 π 2\nSpin echo\nGroup I \nGroup II SEDOR:τ τ\nR     (π)iNV\nspin:\nBath \nspins:NV\nspin:\nBath \nspins:Fig. 3Free evolution time τ (μs)NV echo amplitude\nFree evolution time τ (μs)NV echo amplitudeπ\n0 1 02 03 04 05 00.40.8Norm. NV echo amplitude  Sensor off-time ( μs)Group IIComplete magnetic \n          environmentabc\nτ  = 24± 8  μsCτ  = 30± 8  μsC0.50.6\n00 . 3 0 . 6 0 . 9 1 . 20.20.40.60.8\n00 . 1 0 . 2 0 . 30.20.40.40.8τSensor off\n(1.5 μs)\nπ\n2[π\n2ϕ[Bath \nspins:NV\nspin:\nBath \nspins:R(π)+ DDπ\n2π\n2\nτSensor off\n(400 ns)\n[π\n2π\n2ϕ[NVspin:\nBath \nspins:R(π)\n+ DDπ\n2π\n2R(π)+ DDoff-time\nπ\n2π\n2Sensor\nNV\nspin:π() π()\nπ() π()τs τs τs τs τs τs\nFree evolution Spin-echo Correlation timeFig. 4"    },    {        "title": "1704.06717v2.Rattleback_dynamics_and_its_reversal_time_of_rotation.pdf",        "content": "Rattleback dynamics and its reversal time of rotation\nYoichiro Kondo1,\u0003and Hiizu Nakanishi1\n1Department of Physics, Kyushu University, 33, Fukuoka 819-0395, Japan\n(Dated: June 13, 2017)\nA rattleback is a rigid, semi-elliptic toy which exhibits unintuitive behavior; when it is spun in one\ndirection, it soon begins pitching and stops spinning, then it starts to spin in the opposite direction,\nbut in the other direction, it seems to spin just steadily. This puzzling behavior results from the\nslight misalignment between the principal axes for the inertia and those for the curvature; the\nmisalignment couples the spinning with the pitching and the rolling oscillations. It has been shown\nthat under the no-slip condition and without dissipation the spin can reverse in both directions,\nand Garcia and Hubbard obtained the formula for the time required for the spin reversal tr[Proc.\nR. Soc. Lond. A 418, 165 (1988)]. In this work, we reformulate the rattleback dynamics in a\nphysically transparent way and reduce it to a three-variable dynamics for spinning, pitching, and\nrolling. We obtain an expression of the Garcia-Hubbard formula for trby a simple product of four\nfactors: (1) the misalignment angle, (2) the di\u000berence in the inverses of inertia moment for the two\noscillations, (3) that in the radii for the two principal curvatures, and (4) the squared frequency of\nthe oscillation. We perform extensive numerical simulations to examine validity and limitation of\nthe formula, and \fnd that (1) the Garcia-Hubbard formula is good for both spinning directions in\nthe small spin and small oscillation regime, but (2) in the fast spin regime especially for the steady\ndirection, the rattleback may not reverse and shows a rich variety of dynamics including steady\nspinning, spin wobbling, and chaotic behavior reminiscent of chaos in a dissipative system.\nPACS numbers: 45.40.-f, 05.10-a, 05.45.-a\nI. INTRODUCTION\nSpinning motions of rigid bodies have been studied for\ncenturies and still are drawing interest in recent years, in-\ncluding the motions of Euler's disks [1], spinning eggs [2],\nand rolling rings [3], to mention just a few. Also, macro-\nscopic systems which convert vibrations to rotations have\nbeen studied in various context such as a circular gran-\nular ratchet [4], and bouncing dumbbells, which show a\ncascade of bifurcations [5]. Another interesting example\nof rigid body dynamics which involves such oscillation-\nrotation coupling is a rattleback, also called as a celt\nor wobble stone, which is a semi-elliptic spinning toy\n[Fig. 1(a)]. It spins smoothly when spun in one direc-\ntion; however, when spun in the other direction, it soon\nstarts wobbling or rattling about its short axis and stops\nspinning, then it starts to rotate in the opposite direction.\nOne who has studied classical mechanics must be amazed\nby this reversal in spinning, because it apparently seems\nto violate the angular momentum conservation, and the\nchirality emerges from a seemingly symmetrical object.\nThere are three requirements for a rattleback to show\nthis reversal of rotation: (1) the two principal curvatures\nof the lower surface should be di\u000berent, (2) the two hori-\nzontal principal moments of inertia should also be di\u000ber-\nent, and (3) the principal axes of inertia should be mis-\naligned to the principal directions of curvature. These\ncharacteristics induce the coupling between the spinning\nmotion and the two oscillations: the pitching about the\nshort horizontal axis and the rolling about the long hori-\n\u0003ykondo@stat.phys.kyushu-u.ac.jpzontal axis. The coupling is asymmetric, i.e., the oscilla-\ntions cause torque around the spin axis and the signs of\nthe torque are opposite to each other. This also means\nthat either the pitching or the rolling is excited depend-\ning on the direction of the spinning. We will see that the\nspinning couples with the pitching much stronger than\nthat with the rolling; therefore, it takes much longer time\nfor spin reversal in one direction than in the other direc-\ntion, and that is why most rattlebacks reverse only for\none way before they stop by dissipation.\nIn the 1890s, a meteorologist, Walker, performed the\n\frst quantitative analysis of the rattleback motion [6].\nUnder the assumptions that the rattleback does not slip\nat the contact point and that the rate of spinning speed\nchanges much slower than other time scales, he linearized\nthe equations of motion and showed that either the pitch-\ning or the rolling becomes unstable depending on the\ndirection of the spin. More detailed analyses were per-\nformed by Bondi [7], and recently by Wakasugi [8]. Case\nand Jalal [9] derived the growth rate of instability at slow\nspinning. Markeev [10], Pascal [11], and Blackowiak et\nal. [12] obtained the equations of the spin motion by\nextracting the slowly varying amplitudes of the fast os-\ncillations of the pitching and the rolling. Mo\u000batt and\nTokieda [13] derived similar equations to those of Mar-\nkeev [10] and Pascal [11], and pointed out the analogy to\nthe dynamo theory. Garcia and Hubbard [14] obtained\nthe expressions of the averaged torques generated by the\npure pitching and the rolling, and derived the formula for\nspin reversal time.\nAs the \frst numerical study, Kane and Levinson [15]\nsimulated the energy-conserving equations and showed\nthat the rattleback changes its spinning direction inde\f-arXiv:1704.06717v2  [physics.class-ph]  11 Jun 20172\nnitely for certain parameter values and initial conditions.\nThey also demonstrated the coupling between the oscilla-\ntions and the spinning by showing that it starts to rotate\nwhen it begins with pure pitching or rolling, but the di-\nrection of the rotation is di\u000berent between pitching and\nrolling. Similar simulations were performed by Lindberg\nand Longman independently [16]. Nanda et al. simu-\nlated the spin resonance of the rattleback on a vibrating\nbase [17].\nEnergy-conserving dynamical systems usually conserve\nthe phase volume, but the present rattleback dynamics\ndoes not explore the whole phase volume with a given\nenergy because of a non-holonomic constraint due to the\nno-slip condition. Therefore, the Liouville theorem does\nnot hold, and such a system has been shown to behave\nmuch like dissipative systems. Borisov and Mamaev in\nfact reported the existence of \\strange attractor\" for cer-\ntain parameter values in the present system [18]. The\nno-slip rattleback system has been actively studied in\nthe context of chaotic dynamics during the last decade\n[19, 20].\nE\u000bects of dissipation at the contact point have been\ninvestigated in several works. Magnus [21] and Kara-\npetyan [22] incorporated a viscous type of friction force\nproportional to the velocity. Takano [23] determined the\nconditions under which the reversal of rotation occurs\nwith the viscous dissipation. Garcia and Hubbard [14]\nsimulated equations with aerodynamic force, Coulomb\nfriction in the spinning, and dissipation due to slippage,\nthen they compared the results with a real rattleback.\nThe dissipative rattleback models based on the contact\nmechanics with Coulomb friction have been developed by\nZhuravlev and Klimov [24] and Kudra and Awrejcewicz\n[25{27].\nThis paper is organized as follows. In the next section,\nwe reformulate the rattleback dynamics under the no-\nslip and no dissipation condition in a physically transpar-\nent way. In the small-spin and small-oscillation approx-\nimation, the dynamics is reduced to a simpli\fed three-\nvariable dynamics. We then focus on the time required\nfor reversal, or what we call the time for reversal , which is\nthe most evident quantity that characterizes rattlebacks,\nand obtain a concise expression for the Garcia-Hubbard\nformula for the time for reversal [14]. In Sec. III, the re-\nsults of the extensive numerical simulations are presented\nfor various model parameters and initial conditions in or-\nder to examine the validity and the limitation of the the-\nory. Discussions and conclusion are given in Sec. IV and\nSec. V, respectively.\nII. THEORY\nA. Equations of motion\nWe consider a rattleback as a rigid body, whose con\fg-\nuration can be represented by the position of the center\nof mass G and the Euler angles; both of them are ob-\nFIG. 1. (a) A commercially available rattleback made of plas-\ntic. (b) Notations of the rattleback. (c) A schematic illustra-\ntion of the shell-dumbbell model.\ntained by integrating the velocity of the center of mass v\nand the angular velocity !around it [28].\nWe investigate the rattleback motion on a horizontal\nplane, assuming that it is always in contact with the\nplane at a single point C without slipping. We ignore\ndissipation, then all the forces that act on the rattleback\nare the contact force Fexerted by the plane at C and the\ngravitational force \u0000Mgu, where urepresents the unit\nvertical vector pointing upward [Fig. 1(b)]. Therefore,\nthe equations of motion are given by\nd(Mv)\ndt=F\u0000Mgu; (1)\nd(^I!)\ndt=r\u0002F; (2)\nwhereMand ^Iare the mass and the inertia tensor\naround G, respectively, and ris the vector from G to\nthe contact point C.\nThe contact force Fis determined by the conditions\nof the contact point; our assumptions are that (1) the\nrattleback is always in contact at a point with the plane,\nand (2) there is no slip at the contact point. The second\nconstraint is represented by the relation\nv=r\u0002!: (3)\nBefore formulating the constraint (1), we specify the co-\nordinate system. We employ the body-\fxed co-ordinate\nwith the origin being the center of mass G, and the axes\nbeing the principal axes of inertia; the zaxis is the one\nclose to the spinning axis pointing downward, and the x\nandyaxes are taken to be Ixx>Iyy(Fig. 2).\nIn this co-ordinate, the lower surface function of the\nrattleback is assumed to be given by\nf(x;y;z ) = 0; (4)\nwhere\nf(x;y;z )\u0011z\na\u00001 +1\n2a2(x; y)^R(\u0018)^\u0002^R\u00001(\u0018)\u0012\nx\ny\u0013\n;(5)3\nwith\n^R(\u0018)\u0011\u0012\ncos\u0018;\u0000sin\u0018\nsin\u0018;cos\u0018\u0013\n;^\u0002\u0011\u0012\n\u0012;0\n0\u001e\u0013\n: (6)\nHereais the distance between G and the surface at x=\ny= 0, and\u0018is the skew angle by which the principal\ndirections of curvature are rotated from the x-yaxes,\nwhich we choose as the principal axes of inertia (Fig. 2).\n\u0012=aand\u001e=aare the principal curvatures at the bottom,\nnamely at (0 ;0;a)t.\nNow, we can formulate the contact point condition (1);\nthe components of the contact point vector rshould sat-\nisfy Eq. (4), and the normal vector of the surface at C\nshould be parallel to the vertical vector u. Thus we have\nukrf; (7)\nwhich gives the relation\nr?\na=1\nuz^R(\u0018)^\u0002\u00001^R\u00001(\u0018)u?; (8)\nwhere a?represents the xandycomponents of a vector\nain the body-\fxed co-ordinate.\nBefore we proceed, we introduce a dotted derivative\nof a vector ade\fned as the time derivative of the vector\ncomponents in the body-\fxed co-ordinate. This is related\nto the time derivative by\nda\ndt=_a+!\u0002a: (9)\nNote that the vertical vector udoes not depend on time,\nthus we have\ndu\ndt=_u+!\u0002u=0: (10)\nThese conditions, i.e., the no-slip condition (3), the con-\nditions of the contact point (4) and (8), and the vertical\nvector condition (10), close the equations of motion (1)\nand (2).\nFollowing Garcia and Hubbard [14], we describe the\nrattleback dynamics by uand!. The evolution of !is\nobtained as\n^I_!\u0000Mr\u0002(r\u0002_!) =\u0000!\u0002(^I!)\n+Mr\u0002(_r\u0002!+!\u0002(r\u0002!)) +Mgr\u0002u(11)\nby eliminating the contact force Ffrom the equations\nof motion (1) and (2), and using the no-slip condition\n(3). The state variables uand!can be determined by\nEqs. (10) and (11) with the contact point conditions (4)\nand (8).\nThe rattleback is characterized by the inertial parame-\ntersM,Ixx,Iyy,Izz, the geometrical parameters \u0012,\u001e,a,\nand the skew angle \u0018. For the stability of the rattleback,\nboth of the dimensionless curvatures \u0012and\u001eshould be\nsmaller than 1; without loss of generality, we assume\n0<\u001e<\u0012< 1; (12)\nFIG. 2. (color online) A body-\fxed co-ordinate viewed from\nbelow. The dashed lines indicate the principal directions of\ncurvature, rotated by \u0018from the principal axes of inertia (the\nx-yaxes).\nthen, it is enough to consider\n\u0000\u0019\n2<\u0018< 0; (13)\nfor the range of the skew angle \u0018. The positive \u0018case\ncan be obtained by the re\rection with respect to the x-z\nplane.\nAt this stage, we introduce the dimensionless inertial\nparameters \u000b,\f, and\rfor later use after Bondi [7] as\n\u000b\u0011Ixx\nMa2+ 1; \f\u0011Iyy\nMa2+ 1; \r\u0011Izz\nMa2; (14)\nwhich are dimensionless inertial moments around the\ncontact point C. Note that\n\u000b>\f > 1; (15)\nbecause we have assumed Ixx>Iyy.\nB. Small amplitude approximation of oscillations\nunder!z= 0\nWe consider the oscillation modes in the case of no\nspinning!z= 0 in the small amplitude approxima-\ntion, namely, in the linear approximation in j!xj;j!yj\u001cp\ng=a, which leads to jxj;jyj\u001ca,juxj;juyj\u001c 1, and\nuz\u0019\u00001. In this regime, the xandycomponents of\nEq. (10) can be linearized as\n_u?\u0019^\"!?;^\"\u0011\u0012\n0;1\n\u00001;0\u0013\n=^R(\u0000\u0019=2): (16)4\nBy using Eq. (8) with uz\u0019\u00001, Eq. (11) can be linearized\nas\n^J_!?\u0019g\na2(r\u0002u)?\n=\u0000g\na^\"[\u0000^R(\u0018)^\u0002\u00001^R\u00001(\u0018) + 1]u?; (17)\nwith the inertial matrix\n^J\u0011\u0012\n\u000b;0\n0; \f\u0013\n: (18)\nFrom the linearized equations (16) and (17), we obtain\n^J!?=\u0000g\na(^\u0000\u00001)!?; (19)\nwhere\n^\u0000\u0011^R(\u0018+\u0019=2)^\u0002\u00001^R\u00001(\u0018+\u0019=2): (20)\nAt this point, it is convenient to introduce the bra-ket\nnotation for the row and column vector of !?ash!?j\nandj!?i, respectively. With this notation, Eq. (19) can\nbe put in the form of\nj~!?i=\u0000^Hj~!?i; (21)\nwith\nj~!?i\u0011^J1=2j!?i;^H\u0011g\na^J\u00001=2(^\u0000\u00001)^J\u00001=2;(22)\nwhere ^His symmetric. The eigenvalue equation\n^Hj~!ji=!2\njj~!ji (23)\ndetermines the two oscillation modes with j=porr,\nwhose frequencies are given by\n!2\np;r=1\n2\u0014\n(H11+H22)\u0006q\n(H11\u0000H22)2+ 4H2\n12\u0015\n(24)\nwith\n!p\u0015!r: (25)\nHere,Hijdenotes the ijcomponent of ^H. The orthog-\nonal condition for the eigenvectors j~!piandj~!rican be\nwritten using ^ \"as\nj~!pi= ^\"j~!ri;j~!ri=\u0000^\"j~!pi; (26)\nh~!rj=h~!pj^\";h~!pj=\u0000h~!rj^\": (27)\nIn the case of zero skew angle, \u0018= 0, we have\n!2\np=\u0010g\na\u00111=\u001e\u00001\n\u000b\u0011!2\np0; (28)\n!2\nr=\u0010g\na\u00111=\u0012\u00001\n\f\u0011!2\nr0; (29)\nand the eigenvectors j!piandj!riare parallel to the\nxand theyaxes, thus these modes correspond to the\npitching and the rolling oscillations, respectively. This\ncorrespondence holds for j\u0018j\u001c 1 and!p0> !r0as for\na typical rattleback parameter, the case we will discuss\nmainly in the following [29].C. Garcia and Hubbard's theory for the time for\nreversal\nBased on our formalism, it is quite straightforward\nto derive Garcia and Hubbard's formula for the rever-\nsal time of rotation.\n1. Asymmetric torque coe\u000ecients\nDue to the skewness, the pitching and the rolling are\ncoupled with the spinning motion. We examine this cou-\npling in the case of !z= 0 by estimating the averaged\ntorques around the vertical axis caused by the pitching\nand the rolling oscillations. From Eqs. (1) and (2) and\nthe no-slip condition Eq. (3), the torque around uis given\nby\nT\u0011u\u0001(r\u0002F)\u0019\u0000Ma2[_!?\u0001^\"(^\u0000\u00001)^\"u?];(30)\nwithin the linear approximation in !?,u?, and r?dis-\ncussed in Sec. II B.\nWe de\fne the asymmetric torque coe\u000ecients Kpand\nKrfor each mode by\n\u0000Kp\u0011Tp\nEp; K r\u0011Tr\nEr; (31)\nwhereTj(j=porr) is the averaged torque over the\noscillation period generated by each mode, and Ejis the\ncorresponding averaged oscillation energy which can be\nestimated within the linear approximation as\nE\u0019Ma2(\u000b!2x+\f!2y): (32)\nThe minus sign for the de\fnition of Kpis inserted in\norder that both KpandKrshould be positive for typical\nrattleback parameters as can be seen below. Note that\nthe asymmetric torque coe\u000ecients are dimensionless.\nFrom Eqs. (30) and (32), \u0000Kpis given by\n\u0000Kp=h!pj^\"(^\u0000\u00001)^\"^\"j!pi\nh!pj^Jj!pi\n=\u0000(a=g)h~!pj^J\u00001=2^\"^J1=2^Hj~!pi\nh~!pj~!pi(33)\n=\u0000!2\np(a=g)h~!pj^J\u00001=2^\"^J1=2j~!pi\nh~!pj~!pi: (34)\nIn the same way, Kris given by\nKr=\u0000(a=g)h~!rj^J\u00001=2^\"^J1=2^Hj~!ri\nh~!rj~!ri(35)\n=!2\nr(a=g)h~!pj(^J\u00001=2^\"^J1=2)yj~!pi\nh~!pj~!pi: (36)\nEquations (33){(36) yield simple relations for KpandKr\nas\nKp\nKr=!2\np\n!2r(37)5\nand\nKp\u0000Kr=(a=g)\nh~!pj~!piTrh\n^J\u00001=2^\"^J\u00001=2^Hi\n=\u00001\n2sin(2\u0018)\u00121\n\f\u00001\n\u000b\u0013\u00121\n\u001e\u00001\n\u0012\u0013\n:(38)\nEquations (37) and (38) are enough to determine\nKp=\u00001\n2sin(2\u0018)\u00121\n\f\u00001\n\u000b\u0013\u00121\n\u001e\u00001\n\u0012\u0013!2\np\n!2p\u0000!2r;(39)\nKr=\u00001\n2sin(2\u0018)\u00121\n\f\u00001\n\u000b\u0013\u00121\n\u001e\u00001\n\u0012\u0013!2\nr\n!2p\u0000!2r:(40)\nNote that Eqs. (39) and (40) are consistent with the three\nrequirements of rattlebacks: \u00186= 0,\u000b6=\f, and\u00126=\n\u001e. Equations (39) and (40) are shown to be equivalent\nto the corresponding expressions Eq. (42a,b) in Garcia\nand Hubbard [14] although their expressions look quite\ninvolved. These results also show that\nKpKr>0 and hence TpTr<0; (41)\nnamely, the torques generated by the pitching and the\nrolling always have opposite signs to each other.\n2. Typical rattleback parameters\nTypical rattleback parameters fall in the region that\nsatis\fes the following two conditions: (1) the skew angle\nis small,\nj\u0018j\u001c1; (42)\nand (2) the pitch frequency is higher than the roll fre-\nquency. Under these conditions, the modes pandrof\nEq. (23) correspond to the pitching and the rolling oscil-\nlations respectively, and\n!2\np\u0019!2\np0; !2\nr\u0019!2\nr0 (43)\nin accord with the inequality (25) [29]. From Eqs. (31),\n(39), and (40), the signs of the asymmetric torque coef-\n\fcients and the averaged torques for typical rattlebacks\nare given by\nKp>0 andKr>0; (44)\nand\nTp<0 andTr>0; (45)\nby noting\u0018<0,\u000b>\f ,\u0012>\u001e .\nThe fact that !p0> ! r0for a typical rattleback\nmeans that the shape factor, 1 =\u001e\u00001 or 1=\u0012\u00001, con-\ntributes much more than the inertial factor, 1 =\u000bor 1=\f,\nin Eqs. (28) and (29) although these two factors com-\npete, i.e. 1 =\u001e\u00001>1=\u0012\u00001 and 1=\u000b < 1=\f. This is\na typical situation because the two curvatures of usualrattlebacks are markedly di\u000berent, i.e., \u001e\u001c\u0012 < 1\nas can be seen in Fig. 1(c). Moreover, we can show\nthat the pitch frequency is always higher for an ellip-\nsoid with a uniform mass density whose surface is given\nbyx2=c2+y2=b2+z2=a2= 1 (b2> c2> a2). This also\nholds for a semi-ellipsoid for b2> c2>(5=8)a2, where\nthe co-ordinate system is the same as the ellipsoid.\n3. Time for reversal\nNow we study the time evolution of the spinnde\fned\nas the vertical component of the angular velocity\nn\u0011u\u0001!; (46)\nassuming that the expressions for the asymmetric torque\ncoe\u000ecients, KpandKr, obtained above are valid even\nwhen!z6= 0. We consider the quantities n,Ep, and\nEr, averaged over the time scale much longer than the\noscillation periods, yet much shorter than the time scale\nfor spin change. Then, these averaged quantities should\nfollow the following evolution equations:\nIe\u000bdn(t)\ndt=\u0000KpEp(t) +KrEr(t); (47)\ndEp(t)\ndt=Kpn(t)Ep(t); (48)\ndEr(t)\ndt=\u0000Krn(t)Er(t): (49)\nHere,Ie\u000bis the e\u000bective moment of inertia around u\nunder the existence of the oscillations, and is assumed to\nbe constant; it should be close to Izz. As can be seen\neasily, the total energy Etotde\fned by\nEtot\u00111\n2Ie\u000bn(t)2+Ep(t) +Er(t) (50)\nis conserved. It can be seen that there is another invari-\nant,\nCI\u00111\nKplnEp+1\nKrlnEr; (51)\nwhich has been discussed in connection with a Casimir\ninvariant [13, 30]. With these two conservatives, general\nsolutions of the three-variable system (47){(49) should\nbe periodic.\nLet us consider the case where the spin is positive at\nt= 0 and the sum of the oscillation energies are small\ncompared to the spinning energy:\nn(0)\u0011ni>0;Ep(0) +Er(0)\u001c1\n2Ie\u000bn2\ni: (52)\nFor a typical rattleback, the pitching develops and the\nrolling decays as long as n > 0 as can be seen from\nEqs. (44), (48) and (49). Thus the rolling is irrelevant6\nand can be ignored, i.e., Er(t) = 0, to estimate the time\nfor reversal. Then we can derive the equation\ndn(t)\ndt=\u0000Kp\n2\u0000\nn2\n0\u0000n(t)2\u0001\n; (53)\nwhere the constant n0>0 is de\fned by\n1\n2Ie\u000bn2\n0\u0011Etot: (54)\nThis can be easily solved as\nn(t) =n0(n0+ni) exp(\u0000n0Kpt)\u0000(n0\u0000ni)\n(n0+ni) exp(\u0000n0Kpt) + (n0\u0000ni);(55)\nand we obtain the time for reversal trGH +for theni>0\ncase as\ntrGH +=1\nn0Kpln\u0012n0+ni\nn0\u0000ni\u0013\n; (56)\nby just setting n= 0 in Eq. (55).\nSimilarly, in the case of ni<0, only the rolling de-\nvelops and the pitching is irrelevant, thus we obtain n(t)\nand the time for reversal trGH\u0000as\nn(t) =\u0000n0(n0+jnij) exp(\u0000n0Krt)\u0000(n0\u0000jnij)\n(n0+jnij) exp(\u0000n0Krt) + (n0\u0000jnij)(57)\nand\ntrGH\u0000=1\nn0Krln\u0012n0+jnij\nn0\u0000jnij\u0013\n: (58)\nEquations (56) and (58) are Garcia-Hubbard formulas for\nthe times for reversal [14].\nFrom the expressions of KpandKrgiven by Eqs. (39)\nand (40), we immediately notice that (1) the time for\nreversal is inversely proportional to the skew angle \u0018in\nthe small skewness regime, and (2) the ratio of the time\nfor reversal trGH\u0000=trGH +is simply given by the squared\nratio of the pitch frequency to the roll frequency !2\np=!2\nr,\nprovided initial values n0andniare the same except\ntheir signs.\nFor a typical rattleback, !2\np\u001d!2\nr, thustrGH +\u001c\ntrGH\u0000, i.e., the time for reversal is much shorter in the\ncase ofni>0 than in the case of ni<0. Thus we call\nthe spin direction of ni>0 the unsteady direction [14],\nand that of ni<0 the steady direction .\nIn the small skewness regime, this ratio of the squared\nfrequencies is estimated as\n!2\np\n!2r\u0019!2\np0\n!2\nr0=\f\n\u000b1=\u001e\u00001\n1=\u0012\u00001: (59)\nThis becomes especially large as \u0012approaches 1 or as \u001e\napproaches 0, namely, as the smaller radius of principal\ncurvature approaches a, or as the larger radius of princi-\npal curvature becomes much larger than a. We remark\nthat both of the inertial parameters \u000band\fare largerthan 1 by de\fnition Eq. (14), and cannot be arbitrarily\nlarge for a typical rattleback.\nLet us consider these two limiting cases: \u001e!0 and\n\u0012!1 withj\u0018j\u001c1. In the case of \u001e!0,\nKp!1; K r!(\u0000\u0018)\u00121\n\f\u00001\n\u000b\u0013\u000b\n\f\u00121\n\u0012\u00001\u0013\n;(60)\nthus the time for reversal trGH\u0000remains constant while\ntrGH +approaches 0. In the case of \u0012!1,\nKp!(\u0000\u0018)\u00121\n\f\u00001\n\u000b\u0013\u00121\n\u001e\u00001\u0013\n; K r!0; (61)\nand thustrGH +remains constant while trGH\u0000diverges\nto in\fnity, i.e., the negative spin rotation never reverses.\nIII. SIMULATION\nWe perform numerical simulations for the times for\nthe \frst spin reversal and compare them with Garcia-\nHubbard formulas (56) and (58).\nA. Shell-dumbbell model\nTo consider a rattleback whose inertial and geometri-\ncal parameters can be set separately, we construct a sim-\nple model of the rattleback, or the shell-dumbbell model ,\nwhich consists of a light shell and two dumbbells: the\nlight shell de\fnes the shape of the lower part of the rat-\ntleback and the dumbbells represent the masses and the\nmoments of inertia. The shell is a paraboloid given by\nEq. (4). The dumbbells consist of couples of weights,\nmx=2 andmy=2, \fxed at (\u0006rx;0;0) and (0;\u0006ry;0) in the\nbody-\fxed co-ordinate, respectively [Fig. 1(c)]. Then the\ntotal mass is\nM=mx+my (62)\nand the inertia tensor is diagonal with its principal mo-\nments\nIxx=myr2\ny; I yy=mxr2\nx; (63)\nIzz=myr2\ny+mxr2\nx: (64)\nNote that the simple relation\nIzz=Ixx+Iyy (65)\nholds for the shell-dumbbell model. We de\fne\nfsd\u0011Iyy=Izz; (66)\nthen the dimensionless parameters \u000b; \f; and\rde\fned\nby Eq. (14) are given by,\n\r=Izz=Ma2; \u000b= (1\u0000fsd)\r+ 1; \f=fsd\r+ 1:(67)7\n 0 0.2 0.4 0.6 0.8 1\n 0  1  2  3  4  5relative cumulative frequency\neigenfrequency / ω~ωp0\nωp\nωr0\nωr 0 0.2 0.4 0.6 0.8 1\n 0 1 2 3 4 5 6 7(a)\n(b)relative cumulative frequency\n ωp / ωr \nFIG. 3. (color online) (a) Cumulative distributions of the\npitch and the roll frequencies for the parameter set SD in\nTable I;!pand!rof Eq. (23) and their zeroth order approx-\nimation!p0and!r0by Eqs. (28) and (29)\nare shown. The inset shows the cumulative distribution of\n!p=!r. The number of samples is 106.\n 0 0.2 0.4 0.6 0.8 1\n10-410-310-210-1100101(a)relative cumulative frequency\nKp , KrKp\nKr(b)\n01234567\nKp 0 0.2 0.4Kr\n-8-6-4-2 0\nFIG. 4. (color online) (a) Cumulative distributions of the\nasymmetric torque coe\u000ecients KpandKrfor SD (Table I).\nThe number of samples is 105. (b) A 2D color plot for the\ndistribution of ( Kp,Kr). The color code shown is in the\nlogarithmic scale for the relative frequency P(Kp; Kr), i.e.,\n\u00009\u0014log10P(Kp; Kr)\u00140. The number of samples is 108.\nThe parameter fsdsatis\fes 0<fsd<0:5, since we have\nassumed\u000b>\f .\nThe shell-dumbbell model makes it easier to visualize\nan actual object represented by the model with a set of\nparameters, and is used in the following simulations for\ndetermining the parameter ranges.\nB. Methods\nThe equations of motion (10) and (11) with the contact\npoint conditions (4) and (8) are numerically integrated\nby the fourth-order Runge-Kutta method with an initial\ncondition !(0) and u(0). In the simulations, we take\nu(0) = (0;0;\u00001)t(68)and specify !(0) as\n!(0) = (j!xy0jcos ;j!xy0jsin ;\u0000ni) (69)\nin terms ofj!xy0j, , andni. According to the simpli-\n\fed dynamics (47){(49), the irrelevant mode of oscilla-\ntion does not a\u000bect the dynamics sensitively as long as\nthe relevant mode exists and the initial spin energy is\nmuch larger than the initial oscillation energy. Thus we\nchoosej!(0)i= (!x0;!y0)tin the direction of the rele-\nvant eigenmode,\n = pforni>0;and = rforni<0;(70)\nwhere pand rare the angles of the eigenvectors j!pi\nandj!rifrom thex-axis, respectively.\nNumerical results are presented in the unit system\nwhereM,a, and\n~t\u00111=~!\u0011p\na=g (71)\nas units of mass, length, and time. The size of the time\nstep for the numerical integration is taken to be 0 :002~t.\nIn numerics, we determine the time for reversal trby the\ntime at which n=!\u0001ubecomes zero for the \frst time,\nand they are compared with Garcia-Hubbard formulas\n(56) and (58); n0is determined as\n\rn2\n0\n2=1\n2(\u000b!2\nx0+\f!2\ny0+\r!2\nz0); (72)\nassumingIe\u000b=Izzatt= 0. Here the potential energy\nU(u) is set to zero where u(0) = (0;0;\u00001)t.\nThe parameters used in the simulations are listed in\nTable I. For the parameter set SD, the ranges are shown.\nWhen numerical results are plotted against KporKr,\ngiven by Eqs. (39) or (40), respectively, sets of parame-\nters are chosen randomly from the ranges until resulting\nKporKrfalls within the range of \u00060:1% of a target\nvalue. The ranges of SD are chosen to meet the following\ntwo conditions: (1) 0 < \u001e\u001c\u0012 <1,\f < \u000b , andj\u0018j\u001c1\nand (2) the pitch frequency should be higher than the roll\nfrequency. As argued in Sec. II C, usual rattlebacks such\nas one in Fig. 1(a) satisfy these two conditions. Figure 3\nshows the cumulative distributions for the eigenfrequen-\ncies!pand!r, and their approximate expressions !p0\nand!r0for the parameter set SD; it shows ( !p=!r)>1:3\nin accordance with the condition (2).\nThe parameter set GH gives Kp= 0:553 andKr=\n0:0967, and the distributions of KpandKrfor SD are\nshown in Fig. 4, where one can see Kp\u001dKr. From\nEq. (37), this corresponds to !2\np\u001d!2\nr, i.e., the pitch\nfrequency is signi\fcantly faster than the roll frequency.\nConsequently, the time for reversal is much shorter for\nthe unsteady direction ni>0, where the pitching is in-\nduced, than for the steady direction ni<0, where the\nrolling is induced. We denote the time for reversal for the\nunsteady direction as truand that for the steady direc-\ntion astrswhen we consider a speci\fc spinning direction.8\nTABLE I. Two sets of parameters used in the simulations: GH used by Garcia and Hubbard [14] and SD for the present shell-\ndumbbell model. For SD, the parameter values are chosen randomly from the ranges shown, and averages and/or distributions\nof simulation results are presented.\n\r f sd \u000b; \f \u0012 \u001e \u0000\u0018(deg)\nGH 12 :28 | 13.04, 1.522 0.6429 0.0360 1.72\nSD [5 ;15] [0 :05;0:15] | [0 :6;0:95] [0.01,0.1] (0,6]\n-0.1 0 0.1(a-1)\ntruinitially unsteady directionn / ω~\n-0.1 0 0.1(a-1)\ntruinitially unsteady directionn / ω~\n-0.1 0 0.1(a-1)\ntruinitially unsteady directionn / ω~\n-0.2-0.1 0 0.1 0.2(a-2)ωx / ω~\n-0.4-0.2 0 0.2 0.4\n0 200 400(a-3)ωy / ω~ \nt / t~(b-1)\ntrsinitially steady direction (b-1)\ntrsinitially steady direction (b-1)\ntrsinitially steady direction\n(b-2)\n0 400 800 1200(b-3)\nt / t~  \nFIG. 5. A typical spin evolution and the corresponding !x\nand!yfor GH (Table I). (a) The case of the initial spin in\nthe unsteady direction. The initial condition is speci\fed by\nEqs. (68){(70) with ni= 0:1 ~!andj!xy0j= 0:01 ~!. (b) The\ncase of the initial spin in the steady direction with ni=\u00000:1 ~!\nandj!xy0j= 0:01 ~!. The dashed lines in (a-1) and (b-1) show\nGarcia and Hubbard's solution n(t) given by Eqs. (55) and\n(57), respectively.\nC. Results\n1. General behavior for the parameter set GH\nIn Fig. 5 we show a typical simulation result of the\ntime evolution of the spin n(t) along with the angular\nvelocities!x(t) and!y(t) for the parameter set GH (Ta-\nble I) in the case of the unsteady direction ni>0 (a),\nand the steady direction ni<0 (b).\nFigure 5(a-1) shows that the spin nchanges its sign\nfrom positive to negative at tru\u0019112~t, and Fig. 5(b-\n1) shows the spin nchanges its sign from negative to\npositive at trs\u0019810~t. Garcia and Hubbard's solutions\nn(t) of Eqs. (55) and (57) are shown by the dashed lines\nin Figs. 5(a-1) and (b-1), respectively; they are in good\nagreement with the numerical simulations.\nThe angular velocities !xand!yoscillate in much\nshorter time scale, and their amplitudes evolve di\u000ber-\nently depending on the spin direction. In the case of\nFig. 5(a), where the positive initial spin reverses to neg-\native, the amplitude of !xbecomes large and reaches\nits maximum around tru; the amplitude of !yalso be-\ncomes large around both sides of trubut shows the local\nminimum at tru. Both!xand!yoscillate at the pitchfrequency!p\u00191:44 ~!. In the case of Fig. 5(b) where\nthe negative spin reverses to positive, the situation is\nsimilar but the amplitude of !yreaches its maximum\naroundtrs, and!xand!yoscillate at the roll frequency\n!r\u00190:602 ~!.\nThese features can be understood based on the analy-\nsis in the previous section as follows. The positive spin\ninduces the pitching, which is mainly represented by !x\nbecause the eigenvector of the pitching j!piis nearly par-\nallel to the xaxis, i.e., p\u0019\u000017\u000e. Likewise, the nega-\ntive spin induces the rolling, mainly represented by !y,\nbecause r\u001988\u000e. The local minima of the amplitude\nfor!yin Fig. 5(a-3), or !xin Fig. 5(b-2), at the times for\nreversal are tricky; it might mean that the eigenvector of\nthe pitching (rolling) deviates more from the xaxis (y\naxis) for!z6= 0 than that for !z= 0; as a result, the\npitching (rolling) mode has a larger projection on the y\naxis (xaxis) for!z6= 0.\nNote that for given jnij, the maximum value of !yin\nFig. 5(b-3) is larger than that of !xin (a-2). This is due\nto\u000b\u001d\f; the oscillation energy around zero spin for the\nboth cases should be the same, which gives \u000b!2x\u0019\f!2y\nthusq\n!2x<q\n!2y.\n2. Simulations with the parameter set SD\nWe present detailed results of the simulations for the\nranges of the parameters given by SD in Table I.\na. Unsteady initial spin direction (ni>0).In this\ncase, the system behaves basically as we expect from the\nGarcia-Hubbard formula unless the initial spin or oscil-\nlation is too large. Figure 6 shows the time for reversal\ntruas a function of Kpwhen spun in the unsteady direc-\ntion. The results are plotted against Kpby the procedure\ndescribed in Sec. III B.\nWhen the initial spin niisni.0:2 ~!withj!xy0j=\n0:001~!;0:01~!,truis in good agreement with the Garcia-\nHubbard formula trGH +of Eq. (56), i.e., almost inversely\nproportional to Kpwith small scatter around the av-\nerage. For a given ni, as the initial oscillation ampli-\ntudej!xy0jbecomes large, the standard deviations of\ntrubecome large, and the average of trudeviates up-\nward from the Garcia-Hubbard formula trGH +, which is\nderived with the small amplitude approximation of !x\nand!y. For larger ni,trGH +also underestimates tru,\nas already noted by Garcia and Hubbard [14] for the pa-\nrameter set GH. The underestimation can be also seen in9\n101102103\nni =0.1 ω~slope = -1tru / t~\nni = 0.2 ω~\n0.1 1ni = 0.3 ω~\n|ωxy0| =0.001 ω~\n |ωxy0| =0.01 ω~\n |ωxy0| =0.1 ω~\n101102103\n0.1 1ni =0.4 ω~tru / t~\n0.1 1ni = 0.5 ω~\nKp 00.20.40.6\n0 100(b)\nn / ω~\nt  /  t~\nFIG. 6. (color online) (a) Time for reversal of the unsteady\ndirectiontrufor the parameter set SD (Table I) as a function\nof the asymmetric torque coe\u000ecient Kpin the logarithmic\nscale. The error bars indicate one standard deviation of 1000\nsamples for each data point. The solid lines are trGH +given\nby Eq. (56), calculated using the mean values of n0. (b) A\ntypical spin evolution with ni= 0:5 ~!;j!xy0j= 0:01 ~!. The\nparameter set GH is used.\nFig. 5(a-1), where one can see that Garcia and Hubbard's\nsolutionn(t) of Eq. (55) changes its sign earlier than the\nsimulation.\nForni&0:4 ~!,trudeviates notably upward from the\nGarcia-Hubbard formula trGH +. Asniincreases, the av-\nerage oftruincreases and the standard deviations become\nlarge. Figure 6(b) shows a typical spin evolution with\nni= 0:5 ~!. The spin oscillates widely at the pitch fre-\nquency, which is qualitatively di\u000berent from typical spin\nbehaviors at small niand from Garcia and Hubbard's\nsolutionn(t) of Eq. (55) as in Fig. 5(a-1). In this region,\nthe Garcia-Hubbard formula is no longer valid.\nb. Steady initial spin direction (ni<0).Much more\ncomplicated phenomena are observed when spun in the\nsteady direction. When the initial spin jnijis small\nenough, the spin simply reverses as shown in Fig. 5(b-\n1). We call this simple reversal behavior Type R. For\nlargerjnij, however, there appear some cases where the\nspin never reverses; in such cases there are two types\nof behaviors: steady spinning at nss(Type SS), and spin\nwobbling around nw(nss<nw<0, Type SW). For Type\nSS samples, nssis slightly less than ni, i.e.,nss<ni<0,\nbecause small initial rolling decays and its energy is con-\nverted to the spin energy. Typical spin evolutions of a\nType SS sample and a Type SW sample are shown in\nFigs. 7(b-1) and (b-2).\nFigure 7(a) shows the Krdependence of the fractions\nof Types R, SS, and SW for various initial conditions\ngiven byniandj!xy0j. For each sample, we wait up to\nt= 5trGH\u0000; the spin evolution is classi\fed as Type R\nif it reverses. If it does not, the spin evolution is clas-\nsi\fed as Type SS if the initial rolling amplitude decaysmonotonously, and classi\fed as Type SW if the spin n\nstarts wobbling by the time 5 trGH\u0000. The other samples,\nin which the rolling grows slowly yet shows no visible\nspin change by the time 5 trGH\u0000, are labeled \\unclassi-\n\fed\" in Fig. 7. Such samples may show spin reversal or\nspin wobbling if we take a much longer simulation time.\nType SS appears for jnij&0:3 ~!and its fraction increases\nasjnijincreases. The fraction is larger for smaller Krand\nsmallerj!xy0j, i.e.,j!xy0j= 0:001 ~!. Type SW appears\nforjnij&0:1 ~!and its fraction is also larger for smaller\nKr, but stays around 0 :2 forjnij&0:4 ~!.\nFigure 8 shows the Krdependence of trsonly for the\nsamples of Type R, which shows a spin reversal behav-\nior. For smalljnij.0:2 ~!withj!xy0j= 0:01 ~!;0:001 ~!,\ntrsis in good agreement with Garcia-Hubbard formula\ntrGH\u0000of Eq. (58), and the average of truis almost in-\nversely proportional to Kr. As in the case of the unsteady\ndirection, the standard deviations of trsbecome large,\nand the average trsdeviates downward from trGH\u0000as\ninitial oscillation amplitude j!xy0jbecomes large. Note\nthattrGH\u0000tends to overestimate trs, in contrast to the\ncase of the unsteady direction, where trGH +underesti-\nmatestru. This has also been noted by Garcia and Hub-\nbard [14] for the parameter set GH, and can be seen by\nGarcia and Hubbard's solution n(t) in Fig. 5(b-1). For\njnij&0:3 ~!, one may notice the standard deviations are\nlarge forKr\u001c0:1. In these cases, we \fnd that some\nsamples appear to spin stably for quite a long time, i.e.,\nseveral times of trGH\u0000, and then abruptly starts to re-\nverse its sign. During the time period t<t rs, the rolling\ngrows much more slowly than it should as predicted by\nthe theory in Sec. II. Such samples make both the average\nand standard deviation large as Fig. 8.\nNext we consider the Type SS samples. There always\nexists a steady solution, !(0) = (0;0;const:)tandu(0) =\n(0;0;\u00001)t, and Bondi [7] has shown that for the steady\ndirection, this solution is linearly stable for n<n c1<0,\nwherenc1(<0) is given by\nn2\nc1\u0011g\na\u0000(1\u0000\u0012)(1\u0000\u001e)\n2\u0000(\u0012+\u001e)\u0000(\u000b+\f\u0000\r)(\u0012+\u001e\u00002\u0012\u001e):(73)\nWhen the denominator of Eq. (73) is positive, such a\nthreshold does not actually exist, and the steady solution\nis always unstable. Note that nc1does not depend on \u0018.\nIn Fig. 7, the \flled triangles show the fraction of sam-\nples whosejnc1jis smaller thanjnij, which should corre-\nspond with the ratio of Type SS. For j!xy0j= 0:001 ~!,\nall samples whose jnc1jis smaller thanjnijactually show\nType SS behaviors and vice versa. On the other hand,\nforj!xy0j= 0:1 ~!, there are some samples whose jnc1j\nis smaller thanjnijyet do not show Type SS behav-\nior; forni=\u00000:3 ~!, there are only several Type SS\nsamples out of 8000 samples, which cannot be seen in\nFig. 7(a), and for jnij&0:4 ~!, the fractions of Type SS for\nj!xy0j= 0:1 ~!are smaller than those for j!xy0j= 0:001 ~!.\nThis may be because j!xy0j= 0:1 ~!is not small pertur-\nbation, and the spin might have escaped from the basin\nof attractor of Type SS behavior.10\n 0 0.2 0.4 0.6 0.8 1\n0.01 0.1 Krni = -0.1 ω~\n|ωxy0|=0.1 ω~\n 0 0.2 0.4 0.6 0.8 1\n0.01 0.1 Krni = -0.1 ω~\n|ωxy0|=0.1 ω~\n0.01 0.1 Krni = -0.2 ω~\n|ωxy0|=0.1 ω~\n0.01 0.1 Krni = -0.2 ω~\n|ωxy0|=0.1 ω~\n0.01 0.1 Krni = -0.3 ω~\n|ωxy0|=0.1 ω~\n0.01 0.1 Krni = -0.3 ω~\n|ωxy0|=0.1 ω~\n0.01 0.1 Krni = -0.3 ω~\n|ωxy0|=0.1 ω~00.20.40.60.81(a)\nfraction of each typeType R\nType SWType SS\nunclassified\nni = -0.1 ω~\n|ωxy0|=0.001 ω~\n00.20.40.60.81(a)\nfraction of each typeType R\nType SWType SS\nunclassified\nni = -0.1 ω~\n|ωxy0|=0.001 ω~ni = -0.2 ω~\n|ωxy0|=0.001 ω~ni = -0.2 ω~\n|ωxy0|=0.001 ω~ni = -0.3 ω~\n|ωxy0|=0.001 ω~ni = -0.3 ω~\n|ωxy0|=0.001 ω~ni = -0.3 ω~\n|ωxy0|=0.001 ω~\n0.01 0.1 Krni = -0.4 ω~\n|ωxy0|=0.1 ω~\n0.01 0.1 Krni = -0.4 ω~\n|ωxy0|=0.1 ω~\n0.01 0.1 Krni = -0.4 ω~\n|ωxy0|=0.1 ω~\n0.01 0.1 Krni = -0.5 ω~\n|ωxy0|=0.1 ω~\n0.01 0.1 Krni = -0.5 ω~\n|ωxy0|=0.1 ω~\n0.01 0.1 Krni = -0.5 ω~\n|ωxy0|=0.1 ω~ni = -0.4 ω~\n|ωxy0|=0.001 ω~ni = -0.4 ω~\n|ωxy0|=0.001 ω~ni = -0.4 ω~\n|ωxy0|=0.001 ω~ni = -0.5 ω~\n|ωxy0|=0.001 ω~ni = -0.5 ω~\n|ωxy0|=0.001 ω~ni = -0.5 ω~\n|ωxy0|=0.001 ω~\n-0.50\n0 3000(b-1) Type SSn  / ω~ \nt / t~0 3000(b-2) Type SW\nt / t~0 4000(b-3) Type SW\nt / t~ (chaotic)\nFIG. 7. (color online) (a) Fractions of Types R, SS, and SW for the steady direction for eight values of Krwith various initial\nconditionsj!xy0jandni. Parameters are randomly chosen from SD (Table I). The number of the samples is 1000 for each\nKr. Filled triangles show the fractions of samples whose jnc1jis smaller thanjnij. (b) Typical spin evolutions of a Type SS\nsample (b-1) and a Type SW sample (b-2), along with an example of \\chaotic\" oscillation (b-3) found for Kr= 0:0041 with\nni=\u00000:5 ~!.\n102103104\nni = -0.1 ω~slope = -1trs / t~\nni = -0.2 ω~\n0.01 0.1ni = -0.3 ω~\n|ωxy0|=0.001 ω~\n|ωxy0|=0.01 ω~\n|ωxy0| =0.1 ω~\n102103104\n0.01 0.1ni = -0.4 ω~trs / t~\n0.01 0.1ni = -0.5 ω~\nKr \nFIG. 8. (color online) Time for reversal trsfor the steady\ndirection as a function of Krin the logarithmic scale. Each\ndata point represents the average with the standard deviation\nof Type R samples out of 1000 simulations from the parameter\nset SD (Table I).\nLast we consider the Type SW samples. The time\nwhen the spin starts to wobble roughly corresponds with\ntrsof Type R in Fig. 8; the center of wobbling nwand\nits amplitude vary from sample to sample. As in the\ncase of Type R, there are some samples which start towobble after several times of trGH\u0000whereKr\u001c0:1.\nWobbling behaviors of such samples are similar to those\nwhich start wobbling around trGH\u0000. We remark that\nthere are two qualitatively di\u000berent Type SW behaviors.\nWhenjnij.0:4 ~!, the spin of Type SW sample oscillates\nalmost periodically. However, when ni=\u00000:5 ~!and\nKr\u001c0:1, we \fnd some samples that show \\chaotic\"\noscillations as an example shown in Fig 7(b-3).\nIV. DISCUSSION\nIn the present work, we study the minimal model for\nthe rattleback dynamics, i.e., a spinning rigid body with\na no-slip contact ignoring any form of dissipation. We\nhave reduced the original dynamics to the simpli\fed dy-\nnamics (47){(49) with the three variables. The assump-\ntions and/or approximations used in the derivation are\n(1) the amplitudes of the oscillations are small, (2) the\ncoupling between the spin and the oscillations does not\ndepend on the spin, and (3) the time scale for the spin\nchange is much longer than the oscillation periods. It is\ninteresting to note that the last assumption is apparently\nanalogous to that used in the derivation of an adiabatic\ninvariant for some systems under slow change of an exter-\nnal parameter if the spin variable is regarded as a slow\nparameter. In the present case with this separation of\ntime scales, the dynamics conserves the \\Casimir invari-11\n-0.1 0 0.1(a) θ = 0.6429n / ω~\n-0.1 0 0.1(b) θ = 0.82n / ω~\n-0.1 0 0.1\n0 2000 4000 6000 8000 10000 12000(c) θ = 0.9n / ω~\nt / t~\nFIG. 9. Three types of spin behaviors after the \frst\nreversal period in the small spin regime with ni= 0:1~!,\nj!xy0j= 0:01~!. (a) A quasi-periodic behavior with the pa-\nrameter set GH ( \u0012= 0:6429), (b) a chaotic behavior with\n\u0012= 0:82, (c) a quasi-periodic behavior with a period shorter\nthan the \frst one with \u0012= 0:9. All the other parameters\nfor (b) and (c) are the same as GH. The dashed lines show\nthe spin evolutions for the corresponding simpli\fed dynamics,\nwhereEp(0) =Ma2[\u000b(j!xy0jcos p)2+\f(j!xy0jsin p)2]=2,\nEr(0) = 3\u000210\u00005Ma2~!2, andn(0) =ni.\nant\"CIof Eq. (51).\nOur simpli\fed dynamics can be compared with some\nprevious works. Based on Bondi's formulation [7], Case\nand Jalal obtained the growth rates \u000exand\u000eyof the\npitching and the rolling amplitudes around the xand\nyaxes, respectively, at a small constant spin and small\nskewness [9]. Their results can be expressed as\n\u000ex=n\n2Kp; \u000e y=\u0000n\n2Kr; (74)\nusing our notations. The factor 1 =2 comes from the\nchoice of the variables; they chose the contact point co-\nordinates, while we choose the oscillation energies, which\nare second order quantities of their variables.\nMo\u000batt and Tokieda [13] obtained equations for the\noscillation amplitudes of pitching and rolling, PandR,\nand the spinning Sfor small spin and skewness as\nd\nd\u001c0\nB@P\nR\nS1\nCA=0\nB@R\n\u0015P\n01\nCA\u00020\nB@P\nR\nS1\nCA=0\nB@\u0015PS\n\u0000RS\nR2\u0000\u0015P21\nCA;(75)\nwhere\u001cis rescaled time, and \u0015is the squared ratio of\nthe pitch frequency to the roll frequency. Equation (75)\nis equivalent with Eqs. (47){(49); again the di\u000berence\ncomes from choice of the variables. The mathematical\nstructures of Eq. (75) have been investigated recently in\nmore detail by Yoshida et al. [30] in connection with the\nCasimir invariant and chaotic behavior of the original\ndynamics.\nAfter the \frst round of spin reversals, our simpli\fed\ndynamics (47){(49) repeats itself and shows periodic be-\nhavior as well as the dynamics studied by Mo\u000batt andTokieda Eq. (75) because the system with only three vari-\nables has two conservatives, i.e., the total energy and the\nCasimir invariant. However, the Casimir invariant is an\napproximate one in the original dynamics, and invariant\nonly under the approximations given at the beginning of\nthis section. The Casimir \\invariant\" actually varies and\nthe original system shows aperiodic behaviors.\nA few examples for longer time evolutions of spin n(t)\nare given in Fig. 9 for the system with the parameter\nset GH except for the curvature in the rolling direction\n\u0012= 0:6429 (a) for GH, 0 :82 (b), and 0 :9 (c) along with\nthose by the corresponding simpli\fed dynamics. The\n\frst example (a) almost shows a periodic spin reversal\nbehavior as is expected by the simpli\fed dynamics. It\nis, however, only quasi-periodic with \ructuating period-\nicity. The second example (b) does not show a periodic\nbehavior; the initial spin reversal till t=~t\u0019100 is nearly\nthe same with (a), but after the time of the second spin\nreversal around t=~t\u00193000, it turns into chaotic, devi-\nating from the simpli\fed dynamics. The third example\n(c) may look similar to (a) but is peculiar; it shows a\nquasi-periodic behavior after the initial round of spin re-\nversals, and its periodicity is much shorter than that by\nthe simpli\fed dynamics.\nThe simpli\fed dynamics seems to work reasonably well\nfor the case of smaller \u0012in (a) but fails for larger \u0012close to\n1 in (b) and (c). This indicates that the approximations\nor assumptions used to derive the simpli\fed dynamics are\nnot valid for the larger curvature in the rolling direction\n\u0012; as the radius of curvature 1 =\u0012becomes small and close\nto 1, i.e., the height of the center of mass, the restora-\ntion force for the rolling oscillation becomes weak. This\nshould result in the rolling oscillation with larger ampli-\ntude and the slower frequency, thus the assumptions (1)\nand (3) given at the beginning of this section may not be\ngood enough.\nThe fact that the system shows a di\u000berent behavior\nafter the \frst round of spin reversals is reminiscent of\nthe existence of attractors, which is normally prohib-\nited in a conserving system by Liouville theorem. In the\npresent system, however, the theorem is invalidated by\nthe non-holonomic constraint due to the no-slip condi-\ntion Eq. (3) [31]. As mentioned already, the existence of\nstrange attractors in an energy conserving system with\na non-holonomic constraint has been studied by Borizov\net al. [20], and chaotic behavior in the rattleback sys-\ntem has been discussed in connection with the Casimir\ninvariant by Yoshida et al. [30].\nV. SUMMARY AND CONCLUSION\nWe have performed the theoretical analysis and numer-\nical simulations on the minimal model of rattleback. By\nreformulating Garcia and Hubbard's theory [14], we ob-\ntained the concise expressions for the asymmetric torque\ncoe\u000ecients, Eqs. (39) and (40), gave the compact proof\nto the fact that the pitching and the rolling generate the12\ntorques with the opposite sign, and reduced the original\ndynamics to the three-variable dynamics by a physically\ntransparent procedure.\nOur expressions for the asymmetric torque coe\u000ecients\nare equivalent to those by Garcia and Hubbard, but we\nexplicitly elucidate that the ratio of the two coe\u000ecient for\nthe pitching and the rolling oscillation is proportional to\nthe squared ratio of those frequencies. Since the pitching\nfrequency is signi\fcantly higher than that of the rolling\nfor a typical rattleback, the time for reversal to one spin\ndirection (or unsteady direction) is much shorter than\nthat to the other direction (or steady direction); the spin\nreversal for the latter direction is not usually observed in\na real rattleback due to dissipation.The simulations on the original dynamics for various\nparameter sets demonstrate that Garcia-Hubbard formu-\nlas for the \frst spin reversal time (56) and (58) are good\nin the case of small initial spin and small oscillation for\nboth the unsteady and the steady directions. The devi-\nation from the formula is especially large for the steady\ndirection in the fast initial spin and small Krregime,\nwhere the rattleback may not reverse and shows a vari-\nety of dynamics, that includes steady spinning, periodic\nand chaotic wobbling.\nIn conclusion, the rattleback is simple but shows fasci-\nnatingly rich dynamics, and keeps attracting physicists'\nattention.\n[1] H. K. Mo\u000batt, Nature (London) 404, 833 (2000).\n[2] H. K. Mo\u000batt and Y. Shimomura, Nature (London) 416,\n385 (2002).\n[3] M. A. Jalali, M. S. Sarebangholi, and M. R. Alam, Phys.\nRev. E 92, 032913 (2015).\n[4] M. Heckel, P. M uller, T. P oschel, and J. A. C. Gallas,\nPhys. Rev. E 86, 061310 (2012).\n[5] Y. Kubo, S. Inagaki, M. Ichikawa, and K. Yoshikawa,\nPhys. Rev. E 91, 052905 (2015).\n[6] G. T. Walker, Q. J. Pure Appl. Math. 28, 175 (1896).\n[7] H. Bondi, Proc. R. Soc. Lond. A 405, 265 (1986).\n[8] M. Wakasugi, Master's thesis, Tokyo University, (2011).\n[9] W. Case and S. Jalal, Am. J. Phys. 82, 654 (2014).\n[10] A. P. Markeev, Prikl. Mat. Mekh. 47, 575 (1983).\n[11] M. Pascal, Prikl. Mat. Mekh. 47, 321 (1983).\n[12] A. D. Blackowiak, R. H. Rand, and H. Kaplan, in Pro-\nceedings of the ASME Design Engineering Technical Con-\nferences, Sacramento, California , paper DETC97/VIB-\n4103 (ASME, 1997).\n[13] H. K. Mo\u000batt and T. Tokieda, P. Roy. Soc. Edinb. A\n138, 361 (2008).\n[14] A. Garcia and M. Hubbard, Proc. R. Soc. Lond. A 418,\n165 (1988).\n[15] T. R. Kane and D. A. Levinson, Int. J. Nonlin. Mech.\n17, 175 (1982).\n[16] R. E. Lindberg, Jr. and R. W. Longman, Acta Mech. 49,\n81 (1983).\n[17] A. Nanda, P. Singla, and M. A. Karami, J. Sound. Vib.\n369, 195 (2016).\n[18] A. V. Borisov and I. S. Mamaev, Phys. Usp. 46, 393(2003).\n[19] A. V. Borisov, A. A. Kilin, and I. S. Mamaev, Dokl. Phys.\n51, 272 (2006).\n[20] A. V. Borisov, A. O. Kazakov, and S. P. Kuznetsov, Phys.\nUsp.57, 453 (2014).\n[21] K. Magnus, Z. Angew. Math. Mech. 54, 54 (1974).\n[22] A. V. Karapetyan, Prikl. Mat. Mekh. 45, 42 (1981).\n[23] H. Takano, Regul. Chaotic Dyn. 19, 81 (2014).\n[24] V. P. Zhuravlev and D. M. Klimov, Mech. Solids 43, 320\n(2008).\n[25] J. Awrejcewicz and G. Kudra, Shock Vib. 19, 1115\n(2012).\n[26] G. Kudra and J. Awrejcewicz, Eur. J. Mech. A-Solid 42,\n358 (2013).\n[27] G. Kudra and J. Awrejcewicz, Acta Mech. 226, 2831\n(2015).\n[28] H. Goldstein, C. Poole, and J. Safko, Classical Mechan-\nics, 3rd ed. (Addison Wesley, New York, 2002).\n[29] Note1, Note that in the atypical case of !p0< !r0, i.e.\nthe pitching is slower than the rolling, we have !p\u0019!r0\nand!r\u0019!p0forj\u0018j\u001c1 because!p>!rby Eq. (25).\n[30] Z. Yoshida, T. Tokieda, and P. J. Morrison,\narXiv:1609.09223v1 (2016).\n[31] Note2, The no-slip condition should be violated in the\nsituations where the ratio of the vertical and the inplane\ncomponents of the contact force, i.e., Fk\u0011F\u0001uand\nF?\u0011jF\u0000(F\u0001u)uj, exceeds the friction coe\u000ecient. The\nratioF?=Fkbecomes large when the angular momentum\naround uchanges. In the cases given in Fig. 9, its largest\nvalue is around 0.2."    },    {        "title": "1806.07092v2.Dynamical_exchange_and_phase_induced_switching_of_a_localized_molecular_spin.pdf",        "content": "Dynamical exchange and phase induced switching of a localized molecular spin\nH. Hammar and J. Fransson\nDepartment of Physics and Astronomy, Uppsala University, Box 530, SE-751 21 Uppsala\n(Dated: April 21, 2022)\nWe address the dynamics of a localized molecular spin under the in\ruence of external voltage\npulses using a generalized spin equation of motion which incorporates anisotropic \felds, nonequi-\nlibrium conditions, and non-adiabatic dynamics. We predict a recurring 4 \u0019-periodic switching of\nthe localized spin by application of a voltage pulse of temporal length \u001c. The switching phenomena\ncan be explained by dynamical exchange interactions, internal transient \felds, and self-interactions\nacting on the localized spin moment.\nI. INTRODUCTION\nDynamics of open systems is an active area of research\n[1, 2]. Recent theoretical predictions have suggested that\nperiodical out-of-equilibrium driving can induce tempo-\nral phases of matter [3], which subsequently have been ex-\nperimentally corroborated [4, 5]. Light induced ultra-fast\ndemagnetization has shown that fast responses to exter-\nnal forces can change the long-term magnetic properties,\napproaching stationary regimes not accessible through\nadiabatic processes [6]. These examples vividly illustrate\nthat the equilibrium paradigm is insu\u000ecient when at-\ntempting to treat rapid dynamics and nonequilibrium\nsystems. Thus, when approaching the quantum limit\nin both spatial and temporal dimensions, models based\non instantaneous or local interactions with no record of\nthe past evolution or spatial surrounding can always be\nquestioned. Non-linearities and feedback between inter-\nnal components require a higher level of sophistication\nin the theoretical modeling, allowing to go beyond the\nequilibrium narrative, especially when con\fnement plays\nan important role as in single molecules.\nNonequilibrium open systems such as nanojunctions,\nquantum dots, and single molecules have been studied\nextensively, both experimentally and theoretically. Stud-\nies include electron dynamics [7, 8], vibrating quantum\ndots [9], pulse-enhanced thermoelectric e\u000eciency [10, 11],\nnonequilibrium thermodynamics [12, 13], and optoelec-\ntronics and spectroscopy [14, 15]. Due to size con\fne-\nment, the systems exhibit intrinsic out-of-equilibrium na-\nture and can be controlled by pulses and external forces,\nthus well suited for studying non-adiabatic quantum dy-\nnamics.\nIn this article we predict a novel type of phase in-\nduced switching phenomenon of localized spin embed-\nded in a tunnel junction between metallic leads, across\nwhich a time-dependent voltage, V(t), is applied. By\napplication of a voltage pulse of temporal length \u001c, we\nobserve a recurring switching property of the localized\nspin, essentially whenever the total accumulated phase\n'(V;\u001c)\u0011eV\u001c=~2(2\u0019;4\u0019) mod 4\u0019. The build up of\nthe accumulated phase generates highly anisotropic in-\nternal transient \felds which act on the local spin, ex-\nerting a torque which counteracts the externally applied\nmagnetic \feld. The altered spin con\fguration is stabi-lized by an intrinsic uniaxial anisotropy \feld of the lo-\ncalized spin and the internal \felds crucially governs the\ndynamics long after the voltage pulse is turned o\u000b. This\nnovel switching phenomenon can be explained in terms\nof induced internal transient \felds emerging during the\nvoltage pulse. These can be partitioned into four com-\nponents: i) internal magnetic \feld, ii) Heisenberg, iii)\nDzyaloshinskii-Moriya (DM), and iv) Ising type of self-\ninteractions between the spin at di\u000berent times. While\nall four components are essential for the switching, we\nnotice in particular that the intrinsic uniaxial anisotropy\nand the dynamic Ising interaction creates an energy bar-\nrier between degenerate solutions for the spin, see Fig.\n1(a), which is crucial to stabilize the steady state after\nswitching, whereas the DM interaction provides a torque\nthat is required to drive the spin out of its initial state\ninto the a new \fnal state, see Fig. 1(b). The switch-\ning depends heavily on the sign of the DM interaction\nwhich can be controlled by tuning the intrinsic uniaxial\nanisotropy, exchange coupling, temperature and external\nmagnetic \feld.\nOur results are obtained from a generalized spin equa-\ntion of motion (SEOM) developed for nonequilibrium\nconditions [16{20] and which allows for calculations of\ndynamic exchange interactions [16, 21{24]. Similar ap-\nproaches have previously been used in the stationary\nlimit [20, 25{28]. In comparison to previous studies us-\ning, e.g., quantum master equation [29{31] and stochas-\ntic Landau-Lifshitz-Gilbert equation [32, 33], our ap-\nproach makes a full account of the non-adiabatic dynam-\nics, including temporal non-local properties of the inter-\nnal \felds. This has shown to be of great importance in\nstudies of, e.g., ultra-fast spin dynamics [34{38].\nFIG. 1: Illustration of the contribution of (a) the Ising interac-\ntion/uniaxial anisotropy and (b) the DM interaction. (a) The\nHeisenberg interaction supports two degenerate solutions for\nthe local spin for which there is no energy barrier in between.\nThe Ising interaction and intrinsic uniaxial anisotropy intro-\nduces a potential barrier, and by that creating two separate\nminima. (b) The DM interaction provides the mechanism of\nthe spin to switch and fall into the potential wells.arXiv:1806.07092v2  [cond-mat.mes-hall]  29 Oct 20182\nOur test bench model represents a single-\nmolecule magnet, for instance M-porphyrins and\nM-phthalocyanines where Mdenotes, e.g., a transi-\ntion metal element, which serve as good models for\nfundamental studies [39{41] comprising an inherent\nnonequilibrium nature. Experiments have revealed\ndistance dependent e\u000bects in the exchange interactions\n[42{45], large anisotropy of individual molecules [46{49],\nas well as collective spin excitations and Kondo e\u000bect\n[50{52]. Experiments have also shown the control and\nread-out of spin states of individual single-molecule\nmagnets [53{61].\nII. MODEL\nWe consider a magnetic molecule, embedded in a tun-\nnel junction between metallic leads, comprising a local-\nized magnetic moment Scoupled via exchange to the\nhighest occupied or lowest unoccupied molecular orbital\nhenceforth referred to as the QD level. We de\fne our\nsystem Hamiltonian as\nH=H\u001f+HT+HQD+HS: (1)\nHere,H\u001f=P\nk\u001b2\u001f(\"k\u001f\u0000\u0016\u001f)cy\nk\u001f\u001bck\u001f\u001b;is the Hamil-\ntonian for the left ( \u001f=L) or right ( \u001f=R) lead,\nwherecy\nk\u001f\u001b(ck\u001f\u001b) creates (annihilates) an electron in\nthe lead\u001fwith energy \"k\u001f, momentum k, and spin\n\u001b=\";#, while\u0016\u001fdenote the chemical potential such\nthat the voltage Vacross the junction is de\fned by\neV=\u0016L\u0000\u0016R. Tunneling between the leads and the\nQD level is described by HT=HTL+HTR, where\nHT\u001f=T\u001fP\nk\u001b2\u001fcy\nk\u001f\u001bd\u001b+H:c:. The single-level QD\nis represented by HQD=P\n\u001b\"\u001bdy\n\u001bd\u001b, wheredy\n\u001b(d\u001b) cre-\nates (annihilates) an electron in the QD with energy \"\u001b=\n\"0+g\u0016BBext\u001bz\n\u001b\u001b=2 and spin\u001b, depending on the external\nmagnetic \feld Bext=Bext^z, where g is the gyromagnetic\nratio and\u0016Bthe Bohr magneton. The energy of the local\nspin is described by HS=\u0000g\u0016BS\u0001Bext\u0000vs\u0001S+DS2\nz\nwherevis the exchange integral between the localized\nand delocalized electrons, the electron spin is denoted\ns= y\u001b =2 in terms of the spinor  = (d\"d#),\u001bis the\nvector of Pauli matrices and D is an intrinsic uniaxial\nanisotropy \feld in the magnetic molecule.\nThe local spin dynamics is calculated using our previ-\nously developed generalized SEOM [62], that is,\n_S(t) =S(t)\u0002\u0000\n\u0000g\u0016BBe\u000b\n0(t)\n+1\neZ\n(J(t;t0) +D)\u0001S(t0)dt0\u0013\n: (2)\nHere, Be\u000b\n0(t) is the e\u000bective magnetic \feld acting on\nthe spin, de\fned by Be\u000b\n0(t) = Bext+v\ng\u0016Bm(t)\u0000R\nj(t;t0)dt0=eg\u0016B, where the second contribution is the\nlocal electronic magnetic moment, de\fned as m(t) =\nhs(t)i=1\n2\n (t)y\u001b (t)\u000b\n=1\n2Imsp\u001bG<(t;t), where sp de-\nnotes the trace over spin-1/2 space. The third term is theinternal magnetic \feld due to the electron \row. The \feld\nJ(t;t0) is the dynamical exchange coupling between spins\nat di\u000berent times and D=D^ z^ zis due to the intrinsic\nuniaxial anisotropy.\nThe generalized SEOM makes use of the Born-\nOppenheimer approximation which is motivated as the\nenergy scales of single molecule magnets are in meV\nwhich results in spin dynamics of picoseconds. This is or-\nders of magnitudes smaller than the recombination time-\nscales of the electrons in the junction in the orders of\nfemtoseconds. We also remark that despite the semi-\nclassical nature of the generalized SEOM, it incorporates\nthe underlying quantum nature of the junction through\nthe dynamical \felds jandJ. This is especially impor-\ntant in the transient regime, where the classical Landau-\nLifshitz-Gilbert equation is incapable to provide an ad-\nequate description of the dynamics [63]. The treatment\ngoes beyond the adiabatic limit considered in previous\nworks, e.g., Ref. [33], while still containing important\nattributes as dissipative \felds and spin-transfer torques.\nThe internal magnetic \feld due to the electron \row\nis de\fned as j(t;t0) =iev\u0012(t\u0000t0)h[s(0)(t);s(t0)]i, where\nthe on-site energy distribution is represented by s(0)=P\n\u001b\"\u001bdy\n\u001bd\u001b. This two-electron propagator j(t;t0) is ap-\nproximated by decoupling into single electron nonequi-\nlibrium Green functions (GFs), G<=>, according to\nj(t;t0)\u0019iev\u0012(t\u0000t0)sp\u000f\u0010\nG<(t0;t)\u001bG>(t;t0)\n\u0000G>(t0;t)\u001bG<(t;t0)\u0011\n; (3)\nwhere \u000f= diagf\"\"\"#g. This internal \feld mediates both\nthe magnetic \feld generated by the charge \row as well\nas the e\u000bect of the external magnetic \feld causing the\nZeeman split in the QD.\nThe spin susceptibility tensor J(t;t0) =i2ev2\u0012(t\u0000\nt0)h[s(t);s(t0)]imediates the interactions between the lo-\ncalized magnetic moment at the times tandt0. Decou-\npling into single electron GFs, yields\nJ(t;t0)\u0019ie\n2v2\u0012(t\u0000t0)sp\u001b\u0010\nG<(t0;t)\u001bG>(t;t0)\n\u0000G>(t0;t)\u001bG<(t;t0)\u0011\n: (4)\nThis current mediated interaction can be decomposed\ninto an isotropic Heisenberg interaction JH, and the\nanisotropic Dzyaloshinski-Moriya (DM) Dand Ising I\ninteractions [22, 62].\nThe dynamical QD electronic structure is calculated by\nusing nonequilibrium GFs taking into account the back\naction from the local spin dynamics by perturbation the-\nory. Expanding the contour ordered single electron GF\nG(t;t0) to \frst order in the time-dependent expectation\nvalue of the spin, we obtain\nG(t;t0) =g(t;t0)\u0000vI\nCg(t;\u001c)hS(\u001c)i\u0001\u001bg(\u001c;t0)d\u001c: (5)3\nHere, g(t;t0) is the bare (spin-dependent) QD GF given\nby the equation of motion\n(i@t\u0000\u000f)g(t;t0) =\u000e(t\u0000t0)\u001b0+Z\n\u0006(t;\u001c)g(\u001c;t0)d\u001c; (6)\nwhere the self-energy is \u0006(t;t0) = \u0006(t;t0)\u001b0with\n\u0006(t;t0) =P\n\u001fP\nk2\u001fjT\u001fj2gk(t;t0),\u001b0is the 2\u00022 iden-\ntity matrix and gk(t;t0) is the lead GF. Using the wide-\nband limit we can de\fne the tunneling coupling \u0000\u001f=\n2\u0019jT\u001fj2P\nk2\u001f\u000e(!\u0000\"k) between the lead and the QD\nand the lesser self-energy becomes\n\u0006<(t;t0) =iX\n\u001f\u0000\u001fZ\nf\u001f(!)e\u0000i!(t\u0000t0)+iRt\nt0\u0016\u001f(\u001c)d\u001cd!\n2\u0019:\n(7)\nThe self-energy carries the information of the pulse due\nto the time integration of the chemical potential for each\nlead, i.e.,iRt\nt0\u0016\u001f(\u001c)d\u001c. We refer to Ref. 62 for more\ndetails.\nIII. RESULTS\nIn absence of a voltage across the junction, there is\nno current and the local spin remains in its initial state.\nTaking this as the initial condition for our simulations,\nat timet0we apply a constant voltage of amplitude V,\nwhich is subsequently terminated at t1, and let the sys-\ntem evolve towards its stationary state. The plot in Fig.\n2(a) shows the time-evolution of the local spin orien-\ntation for increasing phase '\u0011eV(t1\u0000t0)=~, where\nbright (dark) corresponds to a spin orientation parallel\n(anti-parallel) to the external \feld. The plot clearly illus-\ntrates that the spin either remains in its initial state or\nis switched to the parallel state, depending on the phase.\nIn particular for phases when '2(0;2\u0019) mod 4\u0019, the\ngeneral orientation of the spin remains unchanged by\nthe temporary nonequilibrium conditions while the spin\naligns anti-parallel to the external \feld whenever '2\n(2\u0019;4\u0019) mod 4\u0019. However, due to non-linearities in Eq.\n(2), the two solutions are not perfectly con\fned to phases\nin the intervals '2(0;2\u0019) mod 4\u0019and'2(2\u0019;4\u0019)\nmod 4\u0019. We shall, nonetheless, henceforth refer to the\nformer regime as spin-conserving and the latter as spin-\n\ripping .\nThe spin current IS=P\n\u001b\u001bz\n\u001b\u001bI\u001b, whereI\u001bis the spin\nresolved electron current through the junction, is plotted\nin Fig. 2(b). The signatures in the spin current orig-\ninates from the variations in the local spin orientation\nas function of the phase '. This is expected since the\nspin-dependent current is sensitive to the local magnetic\nenvironment which strongly depends on whether the local\nspin is parallel or anti-parallel to the external magnetic\n\feld.\nThe origin of the phase induced switching phenomenon\ncan be understood by analyzing the change of the spinsusceptibility tensor, Eq. (4), and the internal magnetic\n\feld, Eq. (3), due to the voltage pulse. The periodic-\nity shown in Fig. 2 originates from the self-energy, Eq.\n(7), where an applied pulse generates the phase factor\nexpfi'gafter the pulse is turned o\u000b. In Fig. 3(a) {\n(d) we plot the integrated underlying \felds for pulses\nof di\u000berent temporal length, i.e., j(t) =R\nj(t;t0)dt0and\nJ(t) =R\nJ(t;t0)dt0. It represents the \frst case of switch-\ning in Fig. 2(a) where the spin switches for '=2\u0019= 1:59\nand 3:34.\nThe internal magnetic \feld in the z-direction, jz(t),\nis shown in Fig. 3(a). It illustrates rapid change im-\nmediately after the pulse is turned o\u000b and approaches\na \fnite value in the long time limit. The internal \feld\ngives mixed contributions depending on the voltage ap-\nplied. In the spin-\ripping regime, '2(2\u0019;4\u0019) mod 4\u0019,\nexempli\fed by '=2\u0019= 1:59 (blue) and 3 :18 (black) in the\n\fgure, the \feld exhibits a drastic varying behavior and\nthen reaches a constant value. The drastic behavior oc-\ncurs during the spin \rip where the peak at '(t)='= 1:5\nis when _Szreaches its peak value. In the spin-conserving\nregime,'2(0;2\u0019) mod 4\u0019,'=2\u0019= 2:39 (red) in the\n\fgure, the changes in the \feld is less drastic and reaches\nabout half the strength in the long time limit. The sig-\nni\fcant change in the long time limit can be attributed\nto the change of the direction of the local spin moment\nas it is encoded in the GFs of the QD, cf., Eq. 5.\nConsidered as a self-interaction in the time-domain\nthe Heisenberg interaction, JHis of anti-ferromagnetic\ncharacter (positive) for all pulse lengths, see Fig. 3(b).\nHere, the change is not that signi\fcant for di\u000berent pulse\nlengths although the time-evolution and the terminal\nvalue is clearly di\u000berent in the two regimes. The DM\ninteraction Dzchanges sign in the spin-\ripping regime,\nwhereas it is strictly positive in the spin-conserving,\nsee Fig. 3(c). The Ising interaction includes both\nthe dynamic contribution Izzand the intrinsic uniaxial\nanisotropy D. The dynamic contribution is small but \f-\nnite and it can easily be seen that the intrinsic contri-\nbution is dominating, see Fig. 3(d). We also observe\nthat the characteristics for '=2\u0019= 2:39 is smaller by\namplitude in comparison to the other pulses. All \felds\ndepend strongly on the pulse length, bias voltage, tem-\nperature, magnetic \feld, exchange coupling and tunnel-\ning coupling.\nA conclusion that can be drawn from the plots in Fig.\n3 is that within the spin-\ripping regime, the induced in-\nteractions have a tendency to grow larger with increasing\npulse length. The analogous behavior cannot, however,\nbe observed by increasing the voltage bias and simulta-\nneously decreasing the pulse length while preserving the\nphase'. Although the non-linearity of the dynamical\nspin equation prevents us from determine the exact ori-\ngin of this property, we conjecture that the di\u000berent con-\nditions leading to either conservation or \ripping of the\nlocalized spin are not governed solely by the phase. It is\nrather a combination of the appropriate phase and that\nthe time-evolution of the surrounding electronic structure4\nFIG. 2: Resulting evolution of (a) Sz, showing the spin \rip, and (b) the spin current, for di\u000berent pulse lengths, plot against\n'=2\u0019. Here,eV= 2\u0000,v= \u0000=2,D= 0:3\u0000, T = 0.0862 \u0000 =kBand B = 0.1158 \u0000 =g\u0016B. The dotted line indicates when the pulse\nends.\nFIG. 3: (a) The internal magnetic \feld, (b) the Heisenberg\ninteraction, (c) the DM interaction and (d) the Ising inter-\naction for di\u000berent values of '=2\u0019. The \fgures plot against\n'(t)='after the pulse is turned o\u000b where '(t) =eV(t\u0000t0)~\nand the inset in (b) show the same \feld against time in ~=\u0000\nfor the full process. The pulse length is 5 (blue), 7.5 (red)\nand 10 (black) in units of ~=\u0000.'(t) and'is in terms of mod\n2\u0019. Other parameters as in Fig. 2.\naccumulates density di\u000berently in the two cases.\nAlthough the dominant \felds in the transient dynamics\nare the Heisenberg interaction and the internal \feld, the\nanisotropic \felds are crucial for the switching to occur.\nDue to the isotropic nature of the Heisenberg interaction,\nits corresponding potential landscape supports a degener-\nate set of stationary solutions for the spin, see left panel\nin Fig. 1(a). Hence, the stationary solution is always\ngoverned by the external \feld. While the degeneracy of\nthe potential landscape is not broken by the Ising inter-\naction and the intrinsic uniaxial anisotropy, it creates anenergy barrier between the degenerate solutions, see right\npanel of Fig. 1(a). The height of this barrier e\u000bectively\ndetermines an upper boundary for the temperature in\norder to prevent thermal random drift between the two\nsolutions. The DM interaction generates a spin transfer\ntorque which, when su\u000eciently strong, can push the spin\nover the energy barrier, see Fig. 1(b). As retardation is\ninherent in the generalized SEOM by construction, both\nspin orientations, parallel and anti-parallel to the exter-\nnal \feld, constitute stable \fxed points in the phase space\nof the dynamical system. Hence, the torque generated by\nthe DM interaction merely has to be su\u000eciently large to\npush the system into the realms of the opposite solution\nfor the switching to occur. This is similar to the case\nwhere anisotropy is introduced in the system by mag-\nnetic leads of di\u000berent polarization [62].\nTuning the DM interaction and the resulting spin\ntransfer torque is of fundamental importance in order\nfor the switching to occur. It is tuned by several com-\npeting parameters, e.g., intrinsic uniaxial anisotropy, lo-\ncal exchange, temperature, external magnetic \feld, and\ntunneling coupling to the leads. The intrinsic uniaxial\nanisotropyDof the localized spin is required in order to\ncreate two separate ground states in the long time limit\nafter the dynamic \felds are switched o\u000b, cf., Fig. 1. This\ncan be seen in Fig. 4(a), which shows the time evolution\nof the spin orientations for increasing anisotropy Daf-\nter a given pulse. The required anisotropy \feld needs to\nsatisfyD&\u0000=5 in order to give a large enough barrier\nto overcome the thermal \ructuations. Fig. 4(b) shows\nthe corresponding DM \feld in the z-direction for di\u000ber-\nent uniaxial anisotropy and it can readily be shown that\natD\u0019\u0000=5 the interaction changes sign, thus causing\na switching by spin transfer torque. Variations between\nthe two stationary spin orientations are governed by the\nlocal exchange coupling vbetween the spin and the elec-\ntrons in the QD level. A local exchange integral satisfying\nv.\u0000=3, does not sustain su\u000eciently strong transient in-\nternal \felds to enable the switching. This can be seen5\nFIG. 4: Resulting evolution of Szand corresponding change in the DM z\feld for varying (a, b) uniaxial anisotropy D, (c, d)\ndi\u000berent exchange coupling strength v, (e, f) temperature T and external magnetic \feld Bext. Here a pulse t1\u0000t0= 4:5~=\u0000 is\napplied and other parameters are hold constant with the same values as in Fig. 2. The vertical dotted line indicates when the\npulse ends.\nin Fig. 4(c), which shows the time evolution of the spin\norientations for increasing coupling vafter a given pulse.\nAs the exchange integral satis\fes v&\u0000=3, the spin un-\ndergoes a reorientation. This is also clearly illustrated by\nthe DM \feld in Fig. 4(d) where there is \frst signi\fcant\ncontributions above v&\u0000=3.\nThe switching is limited by the temperature and exter-\nnal magnetic \feld. From our simulations we can see that\nthe limit on temperature Tand an e\u000bective spin switch-\ning requires that TkB.\u0000=2, wherekBis the Boltzmann\nconstant, see Fig. 4(e). This happens as the temper-\nature introduces thermal \ructuations to counteract the\nbarrier between the two stable solutions, cf., Fig 1, and\nerases the dynamic features of the \felds. It can be il-\nlustrated by the DM \feld in the z-direction for di\u000berent\ntemperatures where the negative features vanish, see Fig.\n4(f). Moreover, magnetic \feld strengths g\u0016BBext.\u0000=3\nis necessary for the spin switching since the induced \felds\ncannot overcome too strong external magnetic \felds, see\nFig. 4(g). It is clearly shown in the DM \feld that it\nchanges sign when the spin no longer switches, see Fig.\n4(h).\nRegarding limitations in our approach we have not con-\nsidered quantum spins or strongly correlated spins. How-\never, our model is essentially applicable for strongly lo-\ncalized spins, pertinent to, e.g., atomic transition metal\nand rare earth elements in molecular compounds such as\nphthalocyanines and porphyrins [50, 64{66]. Therefore,\nour model is restricted to large spin moments, for which\na classical description is viable, while quantum spins are\nbeyond our approach. We, moreover, assume the QD\nlevel to be resonant with the equilibrium chemical poten-\ntial, hence, avoiding possible Kondo e\u000bect that otherwise\nmay occur. While neglecting the local Coulomb repulsion\nis a severe simpli\fcation of the QD description, it is justi-\fed since it is typically negligible for the sp-orbitals that\nconstitute the conducting levels in the molecular ligands\nstructure.\nFurthermore, we have not considered the e\u000bect of a\nthermal and random noise in the generalized SEOM. As\nmotivated in Ref. [62] this requires that the energies\nof the interactions in the problem considered are larger\nthan the energies of these thermal noise \felds. Including\nsuch e\u000bects would add to the limitation of temperature\nalready stated in the results.\nIV. CONCLUSION\nIn conclusion, we have demonstrated that phase in-\nduced switching of a localized magnetic moment embed-\nded in a tunnel junction can be obtained for short voltage\npulses\u001c, satisfying '2(2\u0019;4\u0019) mod 4\u0019. 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Thomson Avenue, Cambridge CB3 0HE, United Kingdom\n3PRESTO, Japan Science and Technology Agency, Kawaguchi, Sa itama 332-0012, Japan\nCoupling between conduction electrons and localized magne tization is responsible for a variety\nof phenomena in spintronic devices. This coupling enables t o generate spin currents from dynam-\nical magnetization. Due to the nonlinearity of magnetizati on dynamics, the spin-current emission\nthrough the dynamical spin-exchange coupling offers a route for nonlinear generation of spin cur-\nrents. Here, we demonstrate spin-current emission governe d by nonlinear magnetization dynamics\nin a metal/magnetic insulator bilayer. The spin-current em ission from the magnetic insulator is\nprobed by the inverse spin Hall effect, which demonstrates no ntrivial temperature and excitation\npower dependences of the voltage generation. The experimen tal results reveal that nonlinear mag-\nnetization dynamics and enhanced spin-current emission du e to magnon scatterings are triggered by\ndecreasing temperature. This result illustrates the cruci al role of the nonlinear magnon interactions\nin the spin-current emission driven by dynamical magnetiza tion, or nonequilibrium magnons, from\nmagnetic insulators.\nDynamical magnetization in a ferromagnet emits a\nspin current,1,2enabling to explore the physics of spin\ntransport in metals and semiconductors.3–22The dy-\nnamical spin-current emission has been achieved utiliz-\ning ferromagnetic metals, semiconductors, and insula-\ntors.23–26In particular, the discovery of the spin-current\nemission from a magnetic insulator yttrium iron gar-\nnet, Y 3Fe5O12, has drawn intense experimental and the-\noretical interests, opening new possibilities to spintronics\nbasedonmetal/insulatorhybrids, whereangularmomen-\ntum can be carried by both electrons and magnons.\nA ferrimagnetic insulator yttrium iron garnet,\nY3Fe5O12, is characterized by the exceptionally small\nmagnetic damping, making it a key material for the de-\nvelopment of the physics of nonlinear magnetization dy-\nnamics.27–29The nonlinear magnetization dynamics in\nY3Fe5O12has been extensively studied both experimen-\ntally and theoretically in the past half a century, bene-\nfited by the exceptional purity, high Curie temperature,\nand simplicity of the low-energy magnon spectrum.28–31\nRecently, thenonlinearmagnetizationdynamicshasbeen\nfound to affect the spin-current emission from the mag-\nnetic insulator; the spin-current emission is enhanced by\nmagnon scattering processes [see Fig. 1(a)], triggered by\nchangingthe excitationfrequencyorpowerofthe magne-\ntization dynamics.14,15,32These findings shed new light\nonthelong-standingresearchonnonlinearmagnetization\ndynamics, promising further development of spintronics\nand magnetics based on the magnetic insulator.\nIn this work, we demonstrate that the spin-current\nemission from Y 3Fe5O12is strongly affected by nonlinear\nmagnetization dynamics at low temperatures. The spin-\n∗Correspondence and requests for materials should be addres sed to\nando@appi.keio.ac.jpcurrent emission is probed by the inverse spin Hall effect\n(ISHE) in aPt film attachedto the Y 3Fe5O12film,11,33,34\nwhich enables to measure temperature dependence of the\nspin-current emission from the magnetic insulator under\nvarious conditions. In spite of the simple structure of the\nmetal/insulator bilayer, we found nontrivial variation of\nthe spin-current emission; the temperature dependence\nof the spin-current emission strongly depends on the mi-\ncrowave frequency and excitation power. This result re-\nveals that nonlinear spin-current emission due to three\nand four magnon scatterings emerges by decreasing tem-\nperature, even at constant magnon excitation frequency\nand power. This finding provides a crucial piece of infor-\nmation for understanding the spin-current emission from\nferromagneticmaterialsandinvestigatingthe magnonin-\nteractions in the metal/insulator hybrid.\nA single-crystal Y 3Fe5O12(111) film (3 ×5 mm2) with\na thickness of 5 µm was grown on a Gd 3Ga5O12(111)\nsubstrate by liquid phase epitaxy (purchased from In-\nnovent e.V., Jena). After the substrates were cleaned by\nsonication in deionized water, acetone and isopropanol, a\npiranha etching process, a mixture of H 2SO4and H 2O2\n(with the ratio of 7 : 3), was applied, then to be able\nto remove any residuals an oxygen plasma cleaning was\nperformed outside a sputtering chamber. On the top of\nthe film, a 10-nm-thick Pt layer was sputtered in an Ar\natmosphere. Prior to sputtering 10-nm-thick Pt layer,\nan argon plasma cleaning was also performed in-situ.\nThe Pt/Y 3Fe5O12bilayer film was placed on a copla-\nnar waveguide, where a microwave was applied to the\ninput of the signal line as show in Fig. 1(b). Two elec-\ntrodes were attached to the edges of the Pt layer. The\nsignal line is 500 µm wide and the gaps between the sig-\nnal line and the ground lines are designed to match to\nthe characteristic impedance of 50 Ω. An in-plane exter-\nnal magnetic field Hwas applied parallel to the signal\nline, or perpendicular to the direction across the elec-2\n(a) \nf (GHz) (b) \n0 10 -5 05\n  \n-10 \n(H - H R) (mT) µ0dV/dH  (  V/mT) µ(e) Pt/YIG \ncoplanar waveguide \n10 210 410 5\nk (cm -1 )10 310 210 410 510 3f = f 0/2 \nf = f min f = f 0(d) \nVISHE +H\n-H02\n-2V (  V) µ\n(c) \nPabs \n02P (mW) \n-10 0\n(H - H R) (mT) 10 \nµ0uniform magnon \nFIG. 1: Detection of spin-current emission. (a) The magnon dispersion in Y 3Fe5O12, wherefandkare the frequency\nand wavenumber of magnons, respectively. The dispersion of the first 40 thickness modes propagating along and opposite t o\nthe magnetic field is shown. The blue and red arrows represent the four and three magnon scatterings. The magnon dispersio n\nshows that both the three and four magnon scatterings create secondary magnons with small group velocity. The lowest\nfrequency is f=fmin. (b) The experimental setup. The Pt/Y 3Fe5O12film placed on the coplanar waveguide was cooled using\na Gifford-McMahon cooler. (c) Magnetic field ( H) dependence of the microwave absorption Pfor the Pt/Y 3Fe5O12film at\nf0= 7.6 GHz and Pin= 10 mW. µ0HR= 183 mT is the resonance field. Pabsis the definition of the magnitude of the microwave\nabsorption intensity. The absorption peak structure compr ises multiple signals due to spin-wave modes. (d) Hdependence of\nthe electric voltage V.VISHEis the magnitude of the electric voltage. The blue and red dat a were measured with the in-plane\nmagnetic field Hand−H, respectively. (e) Hdependence of dV(H)/dHfor the Pt/Y 3Fe5O12film. The damping constant of\nthe Pt/Y 3Fe5O12film was roughly estimated to be 5 ×10−4fromf0dependence of the linewidth at 5 mW.\ntrodes.11Figure 1(c) shows the in-plane magnetic field\nHdependence of the microwave absorption Pmeasured\nby applying a 10 mW microwave with the frequency of\nf0= 7.6 GHz at T= 300 K. Under the ferromagnetic\nresonance condition H=HR, dynamical magnetization\nin the Y 3Fe5O12layer emit a spin current jsinto the\nPt layer, resulting in the voltage generation through the\nISHE as shown in Fig. 1(d).1,2The sign of the voltage is\nchanged by reversing H, consistent with the prediction\nof the spin-current emission from the magnetic insula-\ntor.35Here, the absorption spectrum comprises multiple\nresonancesignalsduetospin-wavemodes, includingmag-\nnetostatic surface waves and backward-volume magneto-\nstatic waves in addition to the ferromagnetic resonance.\nTo extract the damping constant for the Pt/Y 3Fe5O12\nfilm, we have plotted dV/dHin Fig. 1(e), which allows\nrough estimation of the damping constant, α∼5×10−4.\nFigure 2(a) shows temperature dependence of\nVISHE/Pabs, whereVISHEandPabsare the magnitude of\nthe microwave absorption and electric voltage, respec-\ntively;VISHE/Pabscharacterizes the angular-momentum\nconversion efficiency from the microwaves into spin cur-\nrents. Notably, VISHE/Pabsincreases drastically below\nT= 150 K by decreasing Tatf0= 4.0 GHz. This\ndrastic change is irrelevant to the temperature depen-\ndence of the spin pumping and spin-charge conversion\nefficiency in the Pt/Y 3Fe5O12bilayer, such as the spin\nHall angle θSHE, the spin pumping conductance geff, the\nspin diffusion length λ, and the electrical conductivity\nσ. Figure 2(b) shows the temperature dependence of\nthe electrical conductivity σand the spin Hall conduc-\ntivityσs. The spin Hall conductivity was obtained from\nthe temperature dependence of VISHE/Pabsat 10 mW for\nf0= 7.6 GHz shown in Fig. 2(a); the value of VISHE/Pabs\nis insensitive to the excitation power from 5 to 15 mW,indicating that the spin-current emission is reproduced\nwith a liner spin-pumping model:36\nVISHE\nPabs=2ewFσsf0λgefftanh(d/2λ)\nµ0σ2dvFMs∆H/radicalBig\n(γµ0Ms)2+(4πf0)2,(1)\nwherewF= 3.0 mm and vF= 7.5×10−11m3are the\nwidth and volume of the Y 3Fe5O12film.d= 10 nm is\nthe thickness ofthe Pt layer. µ0∆His the half-maximum\nfull-width of the ferromagnetic resonance linewidth. For\nthe calculation of σs, we used the measured parameters\nof the electrical conductivity σand saturation magneti-\nzationMs. The spin-diffusion length37λ= 7.7 nm and\nspin pumping conductance38geff= 4.0×1018m−2were\nassumed to be independent of temperature, as demon-\nstratedpreviously.39The spin Hall conductivity ofthe Pt\nlayer shown in Fig. 2(b) increases with decreasing tem-\nperature above 100 K. Below 100 K, the spin Hall con-\nductivity decreases with decreasing temperature. This\nfeature is qualitatively consistent with the previous re-\nport.39Although the spin Hall conductivity varies with\ntemperature, the variation of the spin Hall conductiv-\nity alone is not sufficient to explain the drastic increase\nofVISHE/Pabsforf0= 4 GHz shown in Fig. 2(a). Thus,\nthe drasticchangein VISHE/Pabsacross150K at f0= 4.0\nGHzcanbeattributedtothechangeinthemagnetization\ndynamicsintheY 3Fe5O12layer. Infact, bydecreasing T,\nthemicrowaveabsorptionintensity Pabsdecreasedclearly\nacrossT= 150 K as shown in Fig. 2(c), suggesting the\nchange of the magnetization dynamics in the Y 3Fe5O12\nlayer across T= 150 K.\nThe origin of the temperature-induced drastic change\nof the spin-conversion efficiency VISHE/Pabsshown in\nFig. 2(a) is enhanced spin-current emission triggered by\nthe three magnon splitting. The three-magnon splitting3\n 7.6 GHz \n 4.0 GHz \nPin  = 10 mW\n0VISHE /Pabs  ( µV/mW) \n369\n100 200\nT (K)300 100 200\nT (K) 300 (10 6 Ω-1 m-1 )\n1.0 1.5 2.0 (b) (a)\n(c)σPabs /Pin \n50 100 150 200\nT (K) 250 3000.20\n0.15\n0.10 f0 = 4.0 GHz6\n4\n2\n0 (10 5 Ω-1 m-1 )\nσS\nFIG. 2: Temperature evolution of spin-current emis-\nsion.(a) Temperature ( T) dependence of VISHE/Pabsfor the\nPt/Y3Fe5O12film atf0= 7.6 (the black circles) and 4.0 GHz\n(the red circles). The data were measured with Pin= 10\nmW microwave excitation. (b) Tdependence of the electri-\ncal conductivity σand the spin Hall conductivity σsfor the\nPt/Y3Fe5O12film. (c) Tdependence of Pabs/Pin, wherePabs\nis the microwave absorption intensity, for Pin= 10 mW and\nf0= 4.0 GHz.\ncreates a pair of magnons with the opposite wavevec-\ntors and the frequency f0/2 from the uniform magnon\nwithf0[see also Fig. 1(a)]. The splitting process redis-\ntributes the magnons and changes the relaxation rate of\nthe spin system, increasing the steady-state angular mo-\nmentum stored in the spin system, or resulting in the\nstabilized enhancement of the spin-current emission.14,32\nThe splitting is allowed only when f0/2> fmin, where\nfminis the minimum frequency of the magnon disper-\nsion, because of the energy and momentum conservation\nlaws. This condition can readily be found by finding fmin\nfor the thin Y 3Fe5O12film from the lowest branch of the\ndipole-exchangemagnondispersion forthe unpinned sur-\nface spin condition:40\nf=/radicalbig\nΩ(Ω+ωM−ωMQ), (2)\nwhere Ω = ωH+ωM(D/µ0Ms)k2,ωH=γµ0H,ωM=\nγµ0Ms, andQ= 1−[1−exp(−kL)]/(kL).D=\n5.2×10−13Tcm2is the exchange interaction constant,\nL= 5µm is the thickness of the Y 3Fe5O12layer, and kis\nthe wavenumber of the magnons. γ= 1.84×1011Ts−1is\nthe gyromagnetic ratio. In Figs. 3(a) and 3(b), we show\nthe lowest branch of the magnon dispersion at different\ntemperatures for the Pt/Y 3Fe5O12film, calculated us-\ning Eq. (2). For the calculation, we used the saturation\nmagnetization Msat each temperature [see Fig. 3(c)],\nestimated from the resonance field data with Kittel’s for-\nmula. We assumed that Dis independent of tempera-(b)(a)\n(c)\nMs (mT) \nµ0300\n240\n180\n120\n300 200 100\nT (K) 300 K\n270 K\n240 K\n210 K\n180 K150 K\n120 K\n105 K90 K\n75 K\n50 K\n2.0 2.5 3.0 3.5 4.0 4.5 f (GHz) \n1.5 \n10 210 4\nk (cm -1 )10 310 5\nf = f0/2 \nk (cm -1 )1×1041×1052.0 2.2 2.4 f (GHz) \nFIG. 3:Magnon dispersion. (a) The lowest-energy branch\nof the magnon spectra for the Pt/Y 3Fe5O12film calculated\nfor the resonance condition at f0= 4.0 GHz. The dispersions\nwere calculated using γ= 1.84×1011Ts−1. The dotted\nred line denotes f=f0/2 = 2.0 GHz. (b) The magnified\nview of the lowest-energy branch of the magnon spectra. (c)\nTemperature dependence of the saturation magnetization Ms\nestimated from the resonance field data.\nture, as demonstrated in literature.32,41,42Although D\ncan slightly depend on temperature,43the shape of the\nmagnon dispersion is not sensitive to the small varia-\ntion ofD. Figures 3(a) and 3(b) demonstrate that the\nminimum frequency fmindecreases with decreasing tem-\nperature and the splitting condition f0/2> fminis sat-\nisfied below T= 150 K; the magnon redistribution is\nresponsible for the enhancement of VISHE/Pabs. Thus,\nthis result demonstrates that the enhanced spin-current\nemission can be induced not only by changing the excita-\ntion frequency or power of the magnetization dynamics,\nbut also by changing temperature.\nFigures 4(a) and 4(b) show temperature dependence\nof the spin-conversion efficiency VISHE/Pabsat different\nmicrowave excitation powers Pinforf0= 7.6 and 4.0\nGHz, respectively. At f0= 4.0 GHz, the enhancement\nofVISHE/Pabsdue to the three-magnon splitting below\n150 K is observed for all the excitation powers as shown\nin Fig. 4(b). The drop in VISHE/PabsatT= 50 K for\nf0= 4.0 GHz is induced by the decrease of the spin\nHall conductivity shown in Fig. 2(b); below 100K, the\nspin Hall conductivity, or the spin Hall angle, decreases\nwith decreasing temperature, whereas the spin-current\nenhancement through the magnon splitting increases by\ndecreasing temperature. The competition gives rise to\nthe peak structure in VISHE/Pabsaround 70 K for 4.04\n50 100 150 200\nT (K)250 300f0 = 7.6 GHzVISHE /Pabs  ( µV/mW) \n0.51.01.52.0\n0VISHE /Pabs  ( µV/mW) -0.5\n-1.0\n-1.5\n-2.0\nVISHE /Pabs  ( µV/mW) -3 \n-6 \n-9 \n-120 100 mW\n 80 mW\n 60 mW\n 40 mW\n 20 mW 15 mW\n 12.5 mW\n 10 mW\n 7.5 mW\n 5 mWf0 = 4.0 GHzVISHE /Pabs  ( µV/mW) \n36912 \n50 100 150 200\nT (K) 250 300(a) (b)\n2.5\n2.0\n1.5\n1.0\n150 300\nT (K) [VISHE /Pabs ] 100 mW  / [VISHE /Pabs ] 5 mW (c)\n+H +H\n-H -H\nFIG. 4: Temperature evolution of spin-current emission for differe nt microwave powers. (a) Temperature T\ndependence of VISHE/Pabsatf0= 7.6 GHz for the in-plane magnetic field H(the upper panel) and reversed in-plane magnetic\nfield−H(the lower panel). (b) Tdependence of VISHE/Pabsatf0= 4.0 GHz for the in-plane magnetic field H(the upper panel)\nand−H(the lower panel). (c) Tdependence of [ VISHE/Pabs]100 mW/[VISHE/Pabs]5 mWatf0= 7.6 GHz. [ VISHE/Pabs]100 mW\nand [VISHE/Pabs]5 mWareVISHE/Pabsmeasured at Pin= 100 mW and 5 mW, respectively.\nGHz. This result also shows that the enhancement factor\nis almost independent of the excitation power. In con-\ntrast, notably, the variation of VISHE/Pabsdepends on\nthe excitation power, especially below 150 K, at f0= 7.6\nGHz as shown in Fig. 4(a). These features for f0= 7.6\nand 4.0 GHz were confirmed in VISHE/Pabsmeasured\nwith the reversed external magnetic field [see the exper-\nimental data for −Hin Figs. 4(a) and 4(b)], indicating\nthat the change of the spin-current emission from the\nmagnetic insulator is responsible for the nontrivial be-\nhavior of VISHE/Pabsat low temperatures.\nTo understand the temperature and power depen-\ndences of VISHE/Pabsatf0= 7.6 GHz in details, we\nplot [VISHE/Pabs]100 mW/[VISHE/Pabs]5 mWin Fig. 4(c).\nFor the spin-current emission in the linear magnetiza-\ntion dynamics regime, VISHE/Pabsis constant with Pin,\nor[VISHE/Pabs]100 mW/[VISHE/Pabs]5 mW= 1becausethe\nemitted spin current is proportional to Pin.35Since the\nthree-magnon splitting is prohibited at f0= 7.6 GHz,\n[VISHE/Pabs]100 mW/[VISHE/Pabs]5 mW≈1.2, atT= 300\nK, demonstrates enhanced spin-current emission without\nthe splitting of a pumped magnon.\nThe observed enhancement of the spin-current emis-\nsion atT= 300 K is induced by the four magnon scat-\ntering, where two magnons are created with the annihi-\nlation of two other magnons [see also Fig. 1(a)].44,45The\nfour-magnon scattering emerges at high microwave exci-\ntation powers Pin> Pth, known as the second order Suhl\ninstability,46wherePthisthe thresholdpowerofthescat-\ntering. Although this process conserves the number of\nmagnons, the magnon redistribution can decrease the re-\nlaxation rate of the spin system through the annihilation\nof the uniform magnons with large damping η0and cre-\nationofdipole-exchangemagnonswithsmalldamping ηq.Thisresultsin the steady-stateenhancement ofthe angu-\nlarmomentum storedinthe spinsystem, ortheenhanced\nspin-current emission.32In the Pt/Y 3Fe5O12film, the\ndamping η0oftheuniformmagnonatlowexcitationpow-\ners is mainly dominated by the two-magnon scattering;\nthe temperature dependence of the ferromagnetic reso-\nnance linewidth is almost independent of temperature as\nshown in the inset to Fig. 5, indicating that the damping\nη0isnotdominatedbythetemperaturepeakprocessesor\ntheKasuya-LeCrawmechanism.47Incontrast,thedamp-\ningηqof the secondary magnons created by the four-\nmagnon scattering is dominated by the Kasuya-LeCraw\nmechanism, since the two-magnon scattering events are\nsuppressed due to the small group velocity; the group\nvelocity of the secondary dipole-exchange magnons cre-\natedatthesamefrequencyastheuniformmagnoncanbe\nclosetozerobecauseofthe exchange-dominatedstanding\nspin-wave branches [see Fig. 1(a)].44,48–50The exchange-\ndominated branches, i.e. the thickness modes, show the\nenergyminimum notonly atthe bottom ofthe dispersion\nbut also at the excitation frequency. Therefore, in the\npresent system, the damping η0of the uniform magnonis\ndominated by the temperature-independent two-magnon\nscattering, whereas the damping ηqof the secondary\nmagnon is dominated by temperature-dependent three-\nparticle confluences, such as the Kasuya-LeCraw pro-\ncess.47In the presence of the four magnon scattering,\nthe total number of the nonequilibrium magnons Ntis\nexpressed as32\nNt\nPabs=1\n2πηq/planckover2pi1f0/bracketleftbigg\n1−χ′′\n2γMs(η0−ηq)/bracketrightbigg\n,(3)\nwhereηqis defined as the average decay rate to the\nthermodynamic equilibrium of the degenerate secondary5\n 120 K\n 105 K\n 90 K\n 75 K\n 50 K 300 K\n 270 K\n 240 K\n 210 K\n 180 K\n 150 K[VISHE /Pabs ] / [VISHE /Pabs ] 5 mW 2.4\n2.0\n1.6\n1.2\n-2.4-2.0-1.6-1.2\n56789\n10 2 3 456789\n100[VISHE /Pabs ] / [VISHE /Pabs ] 5 mW \nPin  (mW)+H\n-H\n00.20.4\n  \n100 200 \nT (K)300H (mT) \nµ0∆\nFIG. 5: Microwave power dependence of spin-current\nemission at different temperatures. Microwave excita-\ntion power Pindependence of [ VISHE/Pabs]/[VISHE/Pabs]5 mW\natf0= 7.6 GHz for different temperatures. The in-plane\nmagnetic field is Hfor the upper panel and −Hfor the lower\npanel, respectively. The inset shows Tdependence of the\nhalf-maximum full-width µ0∆Hof ferromagnetic resonance\nfor the Pt/Y 3Fe5O12film.\nmagnons for simplicity. The imaginary part of the sus-\nceptibility is expressed as\nχ′′=2γMs\nη0+ηspf(Pin), (4)\nwhere\nf(Pin) =1/radicalbig\n1−[χ′′(η0+ηsp)/(2γMs)]4(Pin/Pth)2.(5)\nHere,ηspis the decay constant of the uniform precession\nto degenerate magnons at f0due to scattering on sample\ninhomogeneities. Under the assumption that the spin-\npumping efficiency is insensitive to the wavenumber kof\nthe nonequilibrium magnons, that is VISHE∝js∝Nt,\nEq. (3) is directly related to the spin-conversion effi-\nciency:VISHE/Pabs∝Nt/Pabs.\nThe above model reveals that the spin-current en-\nhancement due to the four-magnon scattering is re-\nsponsible for the nontrivial behavior of the volt-\nage generation shown in Fig. 4(a). As shown in\nFig. 4(c), the nonlinearity of the spin-current emis-\nsion is enhanced by decreasing temperature, from\n[VISHE/Pabs]100 mW/[VISHE/Pabs]5 mW≈1.2 atT= 300\nK to [VISHE/Pabs]100 mW/[VISHE/Pabs]5 mW≈2.4 at 50\nK.Figure5showsmicrowaveexcitationpower Pindepen-\ndenceofVISHE/Pabsforf0= 7.6GHzatdifferenttemper-\natures. Thisresultclearlyshowsthatthethresholdpower 75 K[VISHE /Pabs ] / [VISHE /Pabs ] 5 mW \n 300 K2.0\n1.5\n1.01.2\n1.1\n1.0\nPin  (mW)0 20 40 60 80 100[VISHE /Pabs ] / [VISHE /Pabs ] 5 mW \nFIG. 6: Threshold power of spin-current enhance-\nment. Microwave excitation power Pindependence of\n[VISHE/Pabs]/[VISHE/Pabs]5 mWatf0= 7.6 GHz for T= 300\nK andT= 75 K.\nPthof the spin-current enhancement decreases with de-\ncreasingtemperature, whichisthe originofthenontrivial\nbehavior of the temperature dependence of VISHE/Pabs\nshown in Figs. 4(a) and 4(c). The threshold power of the\nspin-current enhancement through the four-magnon pro-\ncess is very low at low temperatures, making it difficult\nto observe the threshold behavior. In fact, VISHE/Pabs\ndeviates from the prediction of the linear model even\nat the lowest microwave excitation power that is nec-\nessary to detect the ISHE voltage in the Pt/Y 3Fe5O12\nfilm atT= 75 K [see the orange circles in Fig. 6]. At\nT= 300K, a clear threshold is observedaround Pin= 40\nmW. The threshold power of the four-magnon scattering\nis given by47Pth∝h2\nth= (η0/γ)2(2ηq/σq), where hth\nis the threshold microwave field and σqis the coupling\nstrength between the uniform and secondary magnons.\nForsimplicity, weneglectthesurfacedipolarinteractions,\norL→ ∞. Under this approximation, the ferromag-\nnetic resonance condition is given by f0=γµ0Hand the\ncoupling strength can be approximated as σq=γµ0Ms.\nThus, thethresholdpowerforthefour-magnonscattering\nis proportional to\nh2\nth=/parenleftbiggη0\nγ/parenrightbigg2/parenleftbigg2ηq\nγµ0Ms/parenrightbigg\n. (6)\nEquation (6) predicts that the threshold power of the\nspin-currentenhancementdecreaseswithdecreasingtem-\nperature, since Msincreases by decreasing temperature\nas shown in Fig. 3(c). Although the damping η0of the\nuniform magnon is almost independent of temperature\nas shown in the inset to Fig. 5, the damping ηqof the\ndipole-exchange magnon tends to decrease the thresh-\nold power, since ηq, dominated by the Kasuya-LeCraw\nprocess is approximately proportional to temperature.47\nAt high power excitations, the competition between the\nincreaseofthe spin-current enhancement due to the four-\nmagnonscatteringandthedecreaseofthespinHalleffect\nby decreasing temperature gives rise to the peak struc-\nture inVISHE/Pabsaround 100 K for f0= 7.6 GHz [see\nFig. 4(a)].\nIn summary, we have demonstrated that the spin-6\ncurrent emission from a Y 3Fe5O12film is strongly af-\nfected by nonlinear magnetization dynamics at low tem-\nperatures. The spin-current emission has been demon-\nstrated to be enhanced even in the absence of the three-\nmagnon splitting.15The experimental results presented\nin this paper are consistent with this result and further\nextend the physics of the nonlinear spin-current emis-\nsion from the magnetic insulator. Our study reveals that\nthe spin-current enhancement arises from both the three\nand four magnon scatterings depending on the excitation\nfrequency and temperature. We show that the enhanced\nspin-currentemissioncanbetriggeredbydecreasingtem-\nperature, which is evidenced by our systematic measure-\nments for the Pt/Y 3Fe5O12film; the spin-current emis-sion can be enhanced not only by changing the magnon\nexcitation frequency or power, but also by changing tem-\nperature. 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A. Serga, and B. Hillebrands, Nature\nCommun. 5, 4700 (2014)."    },    {        "title": "2003.02226v1.Dynamics_of_the_relativistic_electron_spin_in_an_electromagnetic_field.pdf",        "content": "Dynamics of the relativistic electron spin in an\nelectromagnetic field\nRitwik Mondal1, Peter M. Oppeneer2\n1Fachbereich Physik and Zukunftskolleg, Universität Konstanz, DE-78457 Konstanz,\nGermany\n2Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120\nUppsala, Sweden\nE-mail: ritwik.mondal@uni-konstanz.de\nAbstract. A relativistic spin operator cannot be uniquely defined within relativistic\nquantum mechanics. Previously, different proper relativistic spin operators have\nbeen proposed, such as spin operators of the Foldy-Wouthuysen and Pryce type,\nthat both commute with the free-particle Dirac Hamiltonian and represent constants\nof motion. Here we consider the dynamics of a relativistic electron spin in an\nexternal electromagnetic field. We use two different Hamiltonians to derive the\ncorresponding spin dynamics. These two are: (a) the Dirac Hamiltonian in presence\nof an external field, (b) the semirelativistic expansion of the same. Considering the\nFoldy-Wouthuysen and Pryce spin operators we show that these lead to different spin\ndynamics in an external electromagnetic field, which offers possibilities to distinguish\ntheir action. We find that the dynamics of both spin operators involve spin-\ndependentandspin-independentterms, however, theFoldy-Wouthuysenspindynamics\nadditionally accounts for the relativistic particle-antiparticle coupling. We conclude\nthat the Pryce spin operator provides a suitable description of the relativistic spin\ndynamics in a weak-to-intermediate external field, whereas the Foldy-Wouthuysen spin\noperator is more suitable in the strong field regime.\n1. Introduction\nSpin, in quantum mechanics, is an intrinsic property of an elemental particle e.g., of the\nelectron. However, in contrast to nonrelativistic quantum mechanics, the definition of\nthespinoperatorisnotuniqueinrelativisticquantummechanics[1–5]. Innonrelativistic\nquantum mechanics, the spin is expressed by the Pauli spin matrices as \u001band the\ncorresponding spin angular momentum by S=\u001b\n2(assuming units such that ~= 1). The\nlatter definition is valid for the two component Schrödinger or Pauli Hamiltonian that\nrelates directly the spin operator to the Pauli spin matrices. However, in a relativistic\nformulation the spin angular momentum cannot be defined separately because the total\nangular momentum has to be conserved. Therefore, the definition of spin angular\nmomentum depends on the definition of the orbital angular momentum. Generally, the\norbitalangularmomentumisdefinedas L=r\u0002psuchthatthetotalangularmomentumarXiv:2003.02226v1  [quant-ph]  28 Feb 2020Dynamics of the relativistic electron spin in an electromagnetic field 2\nis calculated as J=L+Sin a nonrelativistic framework. Even so, in relativistic\nquantum mechanics, the position operator is not uniquely defined and, consequently,\nthe spin angular momentum does not have a unique definition [1, 2, 6]. In fact, for both\nthe orbital and spin angular momentum several definitions have been proposed [6].\nWhile the definition of the relativistic spin operator might seem a semantic issue,\nits formulation does in fact matter when relativistic spin dynamics is considered. Spin\ndynamics has previously been computed starting from the nonrelativistic spin operator,\ni.e.the Pauli spin matrices \u001b[7, 8]. The resulting equation of motion is found to be\ncomposed of spin precession, spin relaxation and even spin nutation (inertial dynamics),\nterms that are consistent with the well-known Landau-Lifshitz-Gilbert (LLG) equation\nof spin dynamics. Even though the LLG equation has been used for the spin dynamics\nat ultrashort timescales, its applicability at these timescales has been questioned [9].\nHowever, the relativistic spin dynamics has not yet been derived from a relativistic spin\noperator. With this objective, we derive in this article the spin dynamics for relativistic\nspin operators, in particular, we treat the previously proposed proper relativistic spin\noperators due to Foldy-Wouthuysen [10, 11] and Pryce [12, 13]. We consider three\ndifferent Hamiltonians to derive the relativistic spin dynamics: (1) the free-particle\nDirac Hamiltonian, (2) the Dirac Hamiltonian in an electromagnetic (EM) environment,\n(3) the Foldy-Wouthuysen (FW) transformation of the Dirac Hamiltonian in an EM\nenvironment. The results show that the corresponding spin dynamics leads to the\nLLG equation of motion, however with additional contributions due to relativistic spin\noperator formulations. Comparing the relativistic dynamics for an electron spin in an\nEM field, we draw the conclusion that the Pryce spin operator provides a suitable\nformulation of the relativistic electron spin dynamics in the weak to intermediate field\nregime, however, the FW spin operator is more applicable for describing spin in the\nrelativistic strong field regime.\nIn the following we first introduce the relativistic spin operators, especially the\nFW and Pryce spin operators. Thereafter, in Sec. 3 we formulate the three different\nHamiltonians that will be used to evaluate the relativistic spin dynamics. Then, in Sec.\n4 we derive the spin dynamics corresponding to the FW and Pryce spin operators and\ndiscuss the obtained results. Conclusions are drawn in Sec. 5.\n2. Relativistic spin operators\nThe free-particle Dirac Hamiltonian reads [14–16]\nH0\nD=c\u000b\u0001p+\fm 0c2; (1)\nwiththerestmass m0, and\u000band\farethe 4\u00024Diracmatriceswhichobeythefollowing\nrelations [17]\n\u000b2\ni=1; \f2=1; \u000b i\u000bj+\u000bj\u000bi= 2\u000eij; \u000b i\f+\f\u000bi= 0; (2)Dynamics of the relativistic electron spin in an electromagnetic field 3\nwhere 1isthe 4\u00024identitymatrix. Thecorrespondingspinoperatorisafour-component\noperator that describes both the particle spin up (down) and antiparticle spin up (down)\nstates. The Dirac spin operator has hence the definition SD=\u0006\n2, with the components\nof the operator \u0006(=1\n\u001b) as\n\u0006j=\u0000i\u000bk\u000bl: (3)\nIn addition, the orbital angular momentum is given as LD=r\u0002p, which has to be\nmultiplied by a 2-units block diagonal matrix of 2\u00022. The total angular momentum is\nthengivenby JD=LD+SDanditrecoverstheangularmomentuminthenonrelativistic\nframework when taken in two-component form.\nThe spin does not couple to the orbital angular momentum in nonrelativistic\nquantum mechanics, in fact, the spin operator Sis a constant of motion when the\nSchrödinger Hamiltonian is considered. However, for a free-particle Dirac Hamiltonian,\nthe calculation of the spin dynamics with the Dirac spin operator reveals\ndSD\ndt=\u0000c\u000b\u0002p; (4)\nmeaning that the Dirac spin operator is not a constant of motion. As one expects, the\ncorresponding dynamics contains the particle-antiparticle coupling strength, following\nthe feature of the Dirac Hamiltonian. Moreover, the dynamics suggests that the Dirac\nmatrices\u000biare coupled to the orbital angular momentum via p=\u0000ir. Furthermore,\nit has been shown that the eigenvalues of the Dirac spin operator deviate from \u00061=2\nfor materials having higher atomic numbers [18]. The latter is understandable because,\nfor higher atomic numbers, the spin cannot be considered as an independent quantity,\nrather the spin is coupled to the orbital degrees of freedom due to larger spin-orbit\ncoupling. Thus, the two major drawbacks of the Dirac spin operator is that (a) it\ndoes not commute with the free-particle Dirac Hamiltonian, (b) the eigenvalue does\nnot correspond to \u00061=2for systems with higher atomic number, which implies that the\nDirac spin operator cannot be considered as a proper relativistic spin operator.\nA proper relativistic spin operator should have the following properties [4, 18]:\n(i) It has to commute with the relativistic free-particle Dirac equation. This implies\nthat the spin operator is a constant of motion for a Dirac free-particle.\n(ii) It has to obey the SU(2) algebra of spin operators. The commutator of two spin\noperators should follow the relation\n[Si;Sj] =i\u000fijkSk; (5)\nwhere\u000fijkis the anti-symmetric Levi-Civita tensor.\n(iii) The spin operator must have two eigenvalues of \u00061\n2.\n(iv) The total angular momentum has to be conserved.Dynamics of the relativistic electron spin in an electromagnetic field 4\nThere have been a number of relativistic spin operators reported in the literature\n[4, 5, 18, 19]. However, all of these existing spin operators do not obey all the\naforementioned conditions. It is known that onlythe relativistic Foldy-Wouthuysen\nand Pryce spin operators [10–13] satisfy all the above-mentioned properties. Therefore,\none can infer that they can be considered as proper spin operators [18].\nIn the following, we calculate the spin dynamics corresponding to both of these\noperators for a system excited by an electromagnetic field (e.g., a laser pulse).\n2.1. FW spin operator\nThe FW spin operator has the following definition [10, 11, 17, 20, 21]\nSFW=1\n2\u0006+i\fp\u0002\u000b\n2Ep\u0000p\u0002(\u0006\u0002p)\n2Ep(Ep+m0c2); (6)\nwith the energy Ep=p\np2c2+m2\n0c4. Correspondingly, the position operator is also\ndefined as\nrFW=r\u0000i\f\u000b\n2Ep+i\f(\u000b\u0001p)p\u0000(\u0006\u0002p)jpj\n2Ep(Ep+m0c2)jpj; (7)\nsuch that the total angular momentum is exactly the same as that of the nonrelativistic\ncase i.e.,JFW=LFW+SFW=rFW\u0002p+SFW=r\u0002p+\u0006\n2. This construction ‘made\nby hand’ reflects that the total angular momentum has to be conserved, and has to be\nequal to the total angular momentum for the Pauli representation when we consider the\ntwo-component form.\n2.2. Pryce spin operator\nThe Pryce spin operator has the following definition [12, 13]:\nSPy=1\n2\f\u0006+1\n2(1\u0000\f)(\u0006\u0001p)p\np2; (8)\nand the corresponding position operator has the form\nrPy=r\u00001\n2(1\u0000\f)\u0006\u0002p\np2; (9)\nsuch that the total angular momentum is written as JPy=LPy+SPy=rPy\u0002p+SPy=\nr\u0002p+\u0006\n2. The derived total angular momentum for the Pryce spin and orbital\nmomentum operator is equal to the total angular momentum in the Pauli representation\nas argued earlier.\nAstrikingdifferencebetweenFWandPrycespinoperatorsisthatFWspinoperator\ncontains a coupling term, i.e., the second term of Eq. (6), however, such coupling terms\ndo not appear in the Pryce spin operator in Eq. (8). One immediately notices that\nthe spin operators contain not only the spin angular momentum, but also, the orbitalDynamics of the relativistic electron spin in an electromagnetic field 5\nangular momentum in the form of p. The same is valid for the position operators as\nwell, because of the following reasons. For the FW and Pryce position operators, we\nobtain (neglecting higher-order terms)\nr2\nFW=r2+1\n4E2\np+i\f(r\u0001p)(\u000b\u0001p)\nEp(Ep+m0c2)jpj+\u0006\u0001L\nEp(Ep+m0c2); (10)\nr2\nPy=r2+ (1\u0000\f)\u0006\u0001L\np2; (11)\nrespectively. Here, thelastcorrectiontermsrepresentthewell-knownspin-orbitcoupling\nthat is missing in a nonrelativistic description. Note that there is another relativistic\ncorrection term that appears in the FW position operator which is notably off-diagonal\nintheparticle-antiparticleHilbertspace. Havingtheseproperrelativisticspinoperators,\nwe derive their spin dynamics, particularly, in an applied EM field. While both\noperators are proper spin operators, their formulation is evidently different, and it is\nunknown which spin operator provides a more suitable description of the dynamics. In\nparticular, we are keen to understand the effects of relativistic coupling terms within\nthe corresponding spin dynamics.\n3. Relativistic Hamiltonians\nFor deriving the spin dynamics, we consider three different Hamiltonians. The first one\nis the Dirac free-particle Hamiltonian that has already been introduced in Eq. (1). The\nsecond one is the Dirac equation in the presence of an external EM field that is described\nby the magnetic vector and scalar potentials as A(r;t)and\u001e(r;t). This modified Dirac\nequation can be expressed by the minimal coupling as [17]\nHEM\nD=c\u000b\u0001(p\u0000eA) +\fm 0c2+e\u001e: (12)\nNotethatwehavenotincludedmagneticexchangeinteractioninthefollowingderivation\nbecause of its additional complexity. A rigorous calculation of spin dynamics with\nmagnetic exchange for the nonrelativistic spin operator can be found in Refs. [8, 22].\nNow, we perform the FW transformation of the above Hamiltonian and transform\nthe Hamiltonian as an even Hamiltonian [10, 17, 21, 23]. The FW transformation can be\nsummarized asHFW=eiU\u0000\nHEM\nD\u0000i@\n@t\u0001\ne\u0000iU+i@\n@t, whereUdefines a unitary operator\nobtained from the odd terms (i.e., off-diagonal in the particle-antiparticle space) of the\nHamiltonianHEM\nD. The FW transformed Hamiltonian of Eq. (12) takes the form [24]\nHFW=\fm 0c2+\f\u0012O2\n2m0c2\u0000O4\n8m3\n0c6\u0013\n+E\u00001\n8m2\n0c4[O;[O;F]]\n+\f\n16m3\n0c6fO;[[O;F];F]g; (13)\nwith the following definitions of odd and even terms O=c\u000b\u0001(p\u0000eA)andE=e\u001e,\nrespectively. [A;B]defines the commutator, while fA;Bgdefines the anti-commutatorDynamics of the relativistic electron spin in an electromagnetic field 6\nfor any two given operators AandB. Within the FW transformation, the even terms\nandi@\n@ttransform in a similar way, therefore, we introduce a combined term F=E\u0000i@\n@t\n[25–28]. We calculate the corresponding four-component diagonalized Hamiltonian in\nthe particle-antiparticle space that has the form\nHFW=\fm 0c2+\f(p\u0000eA)2\n2m0\u0000e\f\n2m0\u0006\u0001B\u0000\f(p\u0000eA)4\n8m3\n0c2+e\f\n8m3\n0c2\b\n(p\u0000eA)2;\u0006\u0001B\t\n\u0000\fe2B2\n8m3\n0c2\u0000e\n8m2\n0c2r\u0001E+e\n8m2\n0c2\u0006\u0001[(p\u0000eA)\u0002E\u0000E\u0002(p\u0000eA)]\n\u0000ie\f\n16m3\n0c4\u0006\u0001\u0014\n(p\u0000eA)\u0002@E\n@t+@E\n@t\u0002(p\u0000eA)\u0015\n: (14)\nWe have used the following definitions for the Maxwell fields: B=r\u0002A,E=\n\u0000@A\n@t\u0000r\u001e. The above-derived Hamiltonian is very crucial for understanding the light-\nparticle (antiparticle) interaction at low energy excitation. Eq. (14) can be understood\nascomprisingofnonrelativistictermsandrelativisticterms[29]. Thefirsttermdescribes\nthe rest mass energy which has to be subtracted from the total energy in order to obtain\nthe Pauli Hamiltonian for quantum particles. The second term describes the kinetic\nenergy term in the Schrödinger equation. The third term is the direct Zeeman coupling\nof spins with the external magnetic field. The fourth term is the representation of\nrelativisticmasscorrectionterms. Thefifthtermisanindirectcouplingofspinswiththe\nexternal fields. The sixth term is the relativistic correction to the Zeeman coupling. The\nseventh term explains the Darwin term. The last two terms represent the generalized\nform of spin-orbit coupling. We note that the direct coupling terms of the spin and\nthe external field are the important ones to describe the corresponding interactions\nand dynamics [30, 31], however, the indirect coupling terms could also be interesting\nas well [31, 32]. We also mention that a full Hamiltonian together with the exchange\ninteraction has also been derived in earlier works where the relativistic corrections to\nthe exchange interactions are obtained [8, 22, 29, 33, 34]. The Hamiltonian in Eq. (14)\nhas been used in calculating the general spin dynamics with the Pauli spin operator\n[7, 8, 22, 24, 29, 30, 35–40]. We comment that the calculated spin dynamics could\nexplain the precession, spin relaxation of Gilbert type and even nutation dynamics of a\nsingle spin [37]. However, the derivation of the spin dynamics has been calculated using\na two component extended Pauli Hamiltonian and the nonrelativistic spin operator.\nHere, our goal is to calculate the spin dynamics from relativistic spin operators.\nThe spin-orbit coupling terms can be recast in a more simplified form by using\nthe well-known Maxwell’s equations. Moreover, we can ignore the rest mass energy\nand constant energy terms in the Hamiltonian of Eq. (14), because we work out the\ndynamical equation of motion. The rest of the terms can be simplified to a\nH0\nFW=\f(p\u0000eA)2\n2m0\u0000e\f\n2m0\u0006\u0001B\u0000\f(p\u0000eA)4\n8m3\n0c2+\f\n8m3\n0c2\b\n(p\u0000eA)2;\u0006\u0001B\t\n\u0000e\n8m2\n0c2r\u0001E\u0000~e\n8m2\n0c2\u0006\u0001\u0014\n2E\u0002(p\u0000eA)\u0000i~@B\n@t\u0015\n+e\f\n16m3\n0c4\u0006\u0001@2B\n@t2:(15)Dynamics of the relativistic electron spin in an electromagnetic field 7\nFurther, we can also ignore the Darwin term in our calculation because the Darwin\nterm involves the density of charges according to the Maxwell theory. As we mentioned\nearlier, the direct coupling terms provide an opportunity to directly manipulate the\nspins. Therefore, we evaluate in the following the spin dynamics with those terms.\nTraditionally one is interested in the direct spin-EM field coupling terms to derive\nthe spin dynamics [30, 31]. However, the definition of FW or Pryce spin operator\nsuggests that one also needs to consider the terms that do not explicitly depend on\nthe spins. The reason is that the orbital angular momentum enters in the relativistic\nspin operators in the form of p. Now, the derivation of spin dynamics follows the\ntime evolution of spin operators that involves the commutators of spin operators with\nthe considered Hamiltonian terms. The commutators of the nonrelativistic Pauli spin\noperator with spin-independent terms do not contribute to the dynamics. However, for\nthe relativistic spin operator the spin-independent terms have to be considered as well.\nTherefore, we restrict our derivations to the following FW transformed Hamiltonian\nHspin\ndirect =\f(p\u0000eA)2\n2m0\u0000e\f\n2m0\u0006\u0001B\u0000e\n8m2\n0c2\u0006\u0001\u0014\n2E\u0002(p\u0000eA)\u0000i@B\n@t\u0015\n+e\f\n16m3\n0c4\u0006\u0001@2B\n@t2: (16)\nNote that the other relativistic terms will contribute to the dynamical equation of\nmotion as well, however, for simplicity of the calculations, we consider only the above-\nmentioned direct spin-field interaction terms, which are expected to constitute the main\ncontribution. Moreover, Eq. (16) contains the linear and quadratic interactions in\nthe field,A(r;t). The quadratic terms will become important for the strong field\nregime [41, 42]. In fact, it has been shown that without these quadratic terms, one\ncannot describe the spin dynamics qualitatively and quantitatively at the strong field\nregime [35, 42]. In the below, we calculate the spin dynamics with the linear-order\ninteraction terms with the gauge choice, A=B\u0002r\n2which holds for uniform ( slowly-\nvarying) magnetic field such that r\u0002A=Bandr\u0001A= 0.\n4. Derivation of spin dynamics\n4.1. FW spin operator\nTo derive the spin dynamics we calculate the Heisenberg operator dynamics [23].\n4.1.1. Free-particle Dirac Hamiltonian. The spin dynamics with the free-particle Dirac\nHamiltonian is calculated as\ndSFW\ndt=1\ni\u0002\nSFW;H0\nD\u0003\n=1\n2i\u0002\n\u0006;H0\nD\u0003\n+1\n2Ep\u0002\n\fp\u0002\u000b;H0\nD\u0003\n\u00001\ni\u0014p\u0002(\u0006\u0002p)\n2Ep(Ep+m0c2);H0\nD\u0015\n= 0: (17)Dynamics of the relativistic electron spin in an electromagnetic field 8\nThe meaning of Eq. (17) is that the FW spin operator is constant of motion when a\nfree-particle Dirac Hamiltonian is considered. This result is expected according to the\nfirst condition of a proper spin operator [18]. Therefore, the FW spin operator can be\ntaken as a proper relativistic spin operator.\n4.1.2. Dirac Hamiltonian with EM field. The spin dynamics for the FW spin operator\nfor the Dirac equation with an external EM field can be calculated as follows,\ndSFW\ndt=1\ni\u0002\nSFW;HEM\nD\u0003\n=1\n2i\u0002\n\u0006;HEM\nD\u0003\n+1\n2Ep\u0002\n\fp\u0002\u000b;HEM\nD\u0003\n\u00001\ni\u0014p\u0002(\u0006\u0002p)\n2Ep(Ep+m0c2);HEM\nD\u0015\n=\u0000c\u000b\u0002(p\u0000eA) +c\f\nEpp\u0002(p\u0000eA) +cp2\nEp(Ep+m0c2)\u000b\u0002(p\u0000eA)\n+ce\n2Ep(Ep+m0c2)[(\u000b\u0001r)(B\u0001p)p\u0000(\u000b\u0001B)(r\u0001p)p]\n+ce\n4Ep(Ep+m0c2)h\n\u0006[\u000b\u0001(B\u0002p)] + (\u0006\u0001\u000b)(B\u0002p)\n\u0000[\u0006\u0001(p\u0002\u000b)]B\u0000(\u0006\u0001B)(p\u0002\u000b)i\n: (18)\nThe derivation followed from the three fundamental commutation relations: [\u001bi;\u001bj]\u0000=\n2i\u000fijk\u001bk;f\u001bi;\u001bjg+= 2\u000eijI2\u00022and[ri;pj] =i\u000eij, withI2\u00022the2\u00022identity matrix. It is\nevident that when A=B= 0in Eq. (18), the spin dynamics in Eq. (17) is recovered.\nThe meaning of the dynamical terms are explained in the following way. The first term\nalready explains the coupling dynamics for particles and antiparticles. The second term\ndetermines the individual dynamics without coupling, however, if A= 0, this dynamics\nvanishes because the curl of a gradient is always zero. More importantly, this term does\nnot involve spins because of the fact that the Dirac matrices \u000band\fanti-commute\nwith each other. The rest of the dynamical terms in Eq. (18) are due to the relativistic\npart of the FW spin operator i.e., the last term of Eq. (6). We note that these terms\ninvolve, not only, the spins, but also, the product of spins in the dynamics. One of\nsuch terms constitutes as \u0006\u0001\u000b, which can be recast as \u001b2\ni= 3I2\u00022(assuming Einstein\nsummation convention). Therefore, this dynamical term does actually not depend on\nthe spins. Similarly, the other terms containing products of spins can be recast as\n\u001bi\u001bj=\u000eijI2\u00022+i\u000fijk\u001bk, where the first part is again spin-independent, while the second\npart explicitly depends on spins. We conclude for the FW spin-operator dynamics that,\nalong with the spin-dependent dynamics, there are also the spin-independent parts that\ncontribute to the relativistic spin-operator dynamics.\n4.1.3. FW transformation of the Dirac Hamiltonian with EM field. Next, we evaluate\nthe FW spin-operator dynamics with the FW transformed Hamiltonian in Eq. (16). TheDynamics of the relativistic electron spin in an electromagnetic field 9\ncalculated spin dynamics is\ndSFW\ndt=1\nih\nSFW;Hspin\ndirecti\n=1\n2ih\n\u0006;Hspin\ndirecti\n+1\n2Eph\n\fp\u0002\u000b;Hspin\ndirecti\n\u00001\ni\u0014p\u0002(\u0006\u0002p)\n2Ep(Ep+m0c2);Hspin\ndirect\u0015\n=e\f\n2m0\u0006\u0002B+1\nEp\u0014p\u0002\u000b(p2\u0000eB\u0001L)\n2m0+e(\u0006\u0001\u000b)B\u0002p\n6m0\u0015\n\u0000e\f\n4m0p\u0002[(\u0006\u0002B)\u0002p]\nEp(Ep+m0c2)\n+e\n4m2\n0c2\u0014\n\u0006\u0002(E\u0002p) +i(p\u0002\u000b)\u0002(E\u0002p)\nEp\u0000p\u0002[(\u0006\u0002[E\u0002p])\u0002p]\nEp(Ep+m0c2)\u0015\n\u0000ie\n8m2\n0c22\n4\u0006\u0002_B+i(p\u0002\u000b)\u0002_B\nEp\u0000\u000b\u0002[_B\u0002(\u0006\u0002p)]\n2Ep+p\u0002h\u0010\n\u0006\u0002_B\u0011\n\u0002pi\nEp(Ep+m0c2)3\n5\n\u0000e\n16m3\n0c42\n4\f\u0006\u0002B+(\u0006\u0001\u000b)B\u0002p\n3Ep+\fp\u0002h\u0010\n\u0006\u0002B\u0011\n\u0002pi\nEp(Ep+m0c2)3\n5: (19)\nAs we have started from a semi-relativistic expansion of the Dirac Hamiltonian, it is\nevidently diagonal in the spin space. However, the calculated spin dynamics suggests\nthat the particle-antiparticle coupling terms (off-diagonal) are nonetheless important,\nwhen one considers the relativistic FW spin operator. Furthermore, the spin dynamics\nshows the importance of spin-independent terms in the Hamiltonian in Eq. (16).\nThe kinetic energy term in Eq. (16) is explicit spin independent, however, this term\ncontributestothespindynamicsduetotheformoftherelativisticspinoperator. Infact,\nthecommutator [\fp\u0002\u000b;\fp\u0001p]leadstoananti-commutator fp\u0002\u000b;p\u0001pgbecausethe\nDirac matrices \u000band\fanti-commute with each other and contribute to the dynamical\nequation of motion. The diagonal terms in Eq. (19) have useful meanings as discussed\nin the context of magnetization dynamics [22, 24, 37]. The first term \u0006\u0002Bsignifies the\nprecession of a single spin around a field, the terms \u0006\u0002(E\u0002p)and\u0006\u0002_Bexplains\nthe energy dissipation in terms of damping processes [7, 8]. The higher-order energy\ndissipation terms stem from the relativistic parts of the spin operator. These terms can\nbe identified as the last terms in the second and third lines of Eq. (19). The other terms\nin the second and third lines of Eq. (19) are evidently off-diagonal, thus, they pertain\nto the particle-antiparticle interactions. Higher order relativistic spin dynamical terms\ncan be noticed from the last line of Eq. (19). Such terms have been associated with\nspin dynamics in the inertial regime [43–46], which is a higher-order relativistic spin-\norbit coupling effect [30, 37]. Note that the dynamical term with \u0006\u0001\u000bcan be seen\nas a spin-independent term as described previously. Overall, the spin dynamics with\nthe relativistic FW spin operator exhibits a dynamics that has two contributions: (1)\nspin-dependent and (2) spin-independent terms.\n4.2. Pryce spin operator\nAnother proper relativistic spin operator has been proposed by Pryce [13].Dynamics of the relativistic electron spin in an electromagnetic field 10\n4.2.1. Free-particle Dirac Hamiltonian. ThePrycespindynamicswiththefree-particle\nDirac Hamiltonian is calculated as\ndSPy\ndt=1\ni\u0002\nSPy;H0\nD\u0003\n=1\n2i\u0002\n\f\u0006;H0\nD\u0003\n+1\n2i\u0014\n(1\u0000\f)(\u0006\u0001p)p\np2;H0\nD\u0015\n= 0: (20)\nAgain, this result is expected and the Pryce spin operator can be considered as a proper\nspin operator, similar to the case of the FW spin operator.\n4.2.2. Dirac Hamiltonian with an EM field. However, thespindynamicswiththeDirac\nHamiltonian in the presence of an EM field is rather different and calculated as\ndSPy\ndt=1\ni\u0002\nSPy;HEM\nD\u0003\n=1\n2i\u0002\n\f\u0006;HEM\nD\u0003\n+1\n2i\u0014\n(1\u0000\f)(\u0006\u0001p)p\np2;HEM\nD\u0015\n=ec\n4p2(\u0006\u0002B)\u000b\u0001p+ec\n2p2[(\u000b\u0001r)(B\u0001p)p\u0000(r\u0001p)(\u000b\u0001B)p]:(21)\nIn the above derivation, the first commutator [\f\u0006;c\u000b\u0001p]exactly cancels the last\ncommutator [\f(\u0006\u0001p)p\np2;c\u000b\u0001p]. Therefore, only the remaining commutator [(\u0006\u0001p)p\np2;c\u000b\u0001p]\ncontributes to the spin dynamics. Note that in the absence of the EM field i.e., B= 0,\nthe dynamics in Eq. (21) recovers the spin dynamics for a free Dirac particle in Eq.\n(20). It is interesting to point out that the dynamics in Eq. (21) contains only the\noff-diagonal elements in the matrix formalism. The latter means that this dynamics\nis governed by the coupling between the particles and antiparticles which comes from\nthe Dirac Hamiltonian itself, the term \u000b\u0001p. This feature of the Pryce spin dynamics\nstands in contrast to the FW spin dynamics in Eq. (18), where both diagonal and off-\ndiagonal terms contribute. In fact, the FW spin dynamics contain terms with only\ndiagonal contributions. The first term in Eq. (21) is notably off-diagonal and can\nbe represented by \u001bi\u001bj. Following the similar argument, this term can be split into\na spin-independent part and a spin-dependent part. Thus, the Pryce spin dynamics\ncontains also spin dependent and independent contributions, like the FW spin dynamics\nas discussed earlier. .\n4.2.3. FW transformation of the Dirac Hamiltonian with an EM field. Next, we\ncalculate the spin dynamics from the transformed Hamiltonian in Eq. (16). The derivedDynamics of the relativistic electron spin in an electromagnetic field 11\ndynamical equation is\ndSPy\ndt=1\nih\nSPy;Hspin\ndirecti\n=1\n2ih\n\f\u0006;Hspin\ndirecti\n+1\n2i\u0014\n(1\u0000\f)(\u0006\u0001p)p\np2;Hspin\ndirect\u0015\n=e\n2m0\u0006\u0002B+e\f(1\u0000\f)\n4m0p2\u0006\u0002[p\u0002(B\u0002p)] +e\f\n4m2\n0c2\u0006\u0002(E\u0002p)\n+e(1\u0000\f)\n8m2\n0c2p2\u0014\n(\u0006\u0001p)(\u0006\u0001_B)p\u0000(\u0006\u0001p)(L\u0001_B)p\u0000\u00062p+ (\u0006\u0001p)\u0006\n2(_B\u0001p)\u0015\n\u0000ie\f\n8m2\n0c2\"\n\u0006\u0002_B+\f(1\u0000\f)[(\u0006\u0002_B)\u0001p]p\np2#\n\u0000e\n16m3\n0c4\"\n\u0006\u0002B+\f(1\u0000\f)[(\u0006\u0002B)\u0001p]p\np2#\n: (22)\nAs we have started from a diagonalized Hamiltonian and the Pryce spin operator which\nis diagonal, too, all the derived dynamical terms are diagonal as well. This means\nthat the corresponding dynamics only describes the particles and antiparticles, not the\ncoupling between them. To derive such dynamics, one has to note that the kinetic\nenergy does commute with the first term of the Pryce spin operator in Eq. (8), however,\nit does not commute with the second term because the latter contains the momentum\noperator as well. Such non-commutator implies that not only the spin, but also the\norbital momentum contributes to the relativistic spin dynamics through the spin-orbit\ncoupling-like mechanisms that is considered in the relativistic spin operator of Pryce\ntype. The dynamical terms in Eq. (22) can be related to the similar terms as was\nderived in Eq. (19). For example, the first term in Eq. (22) describes the spin precession\naround a field, the third term and the first terms of third line in Eq. (22) explain the\nenergy dissipation from spin to other degrees of freedom. The first term of the last line\nin Eq. (22) accounts for the spin dynamics in the inertial regime. The other remaining\nterms in Eq. (22) do not directly correspond to the FW dynamics in Eq. (19). However,\nthey derive from the relativistic part of the Pryce spin operator. Thus, they contain\neither (1\u0000\f)or\f(1\u0000\f)as appear in Eq. (22).\n5. Summary and Discussions\nTraditionally, the spin dynamics is derived for the nonrelativistic spin operator (see,\ne.g., [22]). Here, we have derived the spin dynamics with relativistic spin operators.\nWe have used three different Hamiltonians to derive the corresponding spin dynamics:\n(1) free-particle Dirac Hamiltonian, (2) Dirac Hamiltonian in an EM environment,\n(3) diagonalized Dirac Hamiltonian in the presence of an EM field. The relativistic\nspin dynamics is a constant of motion when the free-particle Dirac Hamiltonian is\nconsidered. This result however only holds for relativistic spin operators of FW andDynamics of the relativistic electron spin in an electromagnetic field 12\nPryce type, making them ideal candidates for proper relativistic spin operators. These\ntwo relativistic spin operators are however very different: the FW spin operator has\ndiagonal and off-diagonal elements in spin space, whereas, the Pryce operator has only\ndiagonal elements. These spin operators involve not only spin angular momentum\nin terms of Pauli spin matrices, but also, the orbital angular momentum in terms of\nmomentum operator p. The derived dynamics of these operators in an EM field provides\ntwo important informations: (1) the particle-antiparticle coupling terms contribute to\nthe spin dynamics, even if one starts with a diagonalised Hamiltonian, (2) there exist\ntwo separate parts (spin-dependent and spin-independent terms) of the derived spin\ndynamics. We note that some dynamical terms appear in both the FW and Pryce spin\ndynamics in similar way, however, due of the relativistic spin operators’ construction,\nadditionaltermsexist. Thederiveddynamicsrevealsthatcouplingoftheorbitalangular\nmomentumwithspincontributestothespindynamics, moreover, afewdynamicalterms\nonlydepend on the orbital angular momentum.\nElectromagnetic Field strengthRelativity\nPauli spin operatorPryce spin operatorFW spin operator\nNon-relativisticRelativisticRelativistic\nNo spin-orbitSpin-orbitSpin-orbit\nNo particle-antiparticleNo particle-antiparticleParticle-antiparticle\ncouplingcouplingcoupling\nFigure 1. (Color Online): A schematic for operational spin dynamics of three\ndifferent spin operators at varying EM field strengths. For the weak field strength and\nnonrelativistic regime, the Pauli spin operator is enough to describe the corresponding\nspin dynamics [8], however, in the relativistic regime and for intermediate to strong\nfield strengths the Pryce and FW spin operators are respectively suitable.\nSeveral terms in the derived spin dynamics equations of the two considered proper\nspin operators are rather distinct. The FW spin operator has diagonal and off-diagonal\ncomponents which means it accounts for the coupling terms in the particle-antiparticle\nHilbert space. When we compare the two equations for spin motion, Eq. (19) for FW\ndynamics and Eq. (22) for Pryce dynamics, which have been derived from the same\nHamiltonian, we observe that the Pryce dynamics in Eq. (22) is diagonal and does notREFERENCES 13\ninvolve such coupling terms. In fact, the Pryce dynamics involves terms which have\n(1\u0000\f) that translates to zero contribution for the upper component in 2\u00022formalism.\nTherefore, the 2\u00022Pryce dynamics recovers exactly the same dynamical terms as the\nPauli spin dynamics [37]. As already mentioned, the FW dynamics in Eq. (19) contains\ndiagonal as well as off-diagonal terms. To achieve a 2\u00022electron spin dynamics, one has\nto diagonalize. Even then, the additional terms appear apart from the standard Pauli\nspin dynamics. Moreover, the additional terms account for spin angular momentum and\norbital contributions as well. Therefore, one can conclude that for the spin dynamics\nin an applied EM field, the two spin operators have their own validity regime. We\nthus consider three operational field regimes: weak, intermediate, and strong. In the\nweak field regime, the Pauli spin operator can describe the spin dynamics, while, for\nan intermediate field regime where the spin-orbit coupling is important, the Pryce spin\noperator seems to describe the proper spin dynamics. However, in the stronger field\nregime, where the spin-orbit and relativistic particle-antiparticle couplings are present,\nthe FW spin operator suits the best for describing the spin dynamics. The derived\noperational spin dynamics regimes of the Pauli, Pryce and FW spin operators are\nschematically summarized in Fig. 1.\n6. 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Department of Physics, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong, \nChina \n3. Synergetic Innovation Center of Quantum Information & Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China \n4. London Centre for Nanotechnology, University College London, London WC1H 0AH, United Kingdom  \n5. Department of Materials, Oxford University, Oxford OX1 3PH, United Kingdom  \n6. Beijing Computational Science Research Center, Beijing, China  \n7. Department of Electronic & Electrical Engineering, University College London, London WC1E 7JE, United Kingdom  \n8. Center for Quantum Coherence, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China  \n9. Institute of The oretical Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong \nKong, China  \n† These authors contributed equally to this work.  \n* Corresponding authors . ABSTRACT:  \nMany- body correlations can yield key insights into the nature of interacting systems ; \nhowever, detecting them is often very challenging in  many -particle physics , especially in \nnanoscale systems . Here, tak ing a phosphorus  donor electron spin in a natural -abundance  \n29Si nuclear spin bath as our model system,  we discover both theoretically and \nexperimentally  that many -body correlations in nanoscale  nuclear spin baths produce \nidentifiable  signatures in the  decoherence of the central spin under multiple -pulse \ndynamical decoupling control. We find that when the number of decoupling π-pulses is \nodd, central spin decoherence is primarily  driven  by second -order nuclear spin correlations \n(pair wise flip-flop processes). In contrast,  when the number of π-pulses is even, fourth -\norder nucl ear spin  correlations  (diagonal interaction renormalized pairwise flip -flop \nprocesses ) are principally responsible for the central spin decoherence. Many- body \ncorrelations of different  orders can  thus  be selectively detected  by central spin decoherence \nunder different  dynamical decoupling controls , providing  a useful  approach to prob ing \nmany -body proc esses in nanoscale nuclear spin  baths.  FULL TEXT:  \nDecoherence of a central spin in a solid -state environment is not only an ideal model \nproblem for understanding the foundation of  quantum physics  [1-3] but also a critical issue in a \nnumber of quantum technologies including spin- based quantum information processing  [4, 5] \nand ultrasensitive magnetometry  [6-10]. For example, decoherence from the environmental spin \nbath is often a limiting factor when using systems such as phosphorous  donors in silicon [ 11-16], \nsemiconductor quantum dots  [17, 18 ] and nitrogen- vacancy centers in diamond [ 19, 20 ], as \nquantum bits or sensors. Studying  central spin decoherence caused by environmental fluctuations \nor elementary excitations may  yield key insights into the nature of many -body interactions in the \nenvironment. Furthermore, dynamical control over t he central spin can affect  the dynamics of the \nenvironment  in a detectable manner [ 8, 18].  In the light of these ideas, exploiting central spin \ndecoherence for sensing single nuclear spins or nuclea r spin clusters in spin baths has been \ntheoretically proposed [ 6-8] and experimentally demonstrated  [9,10]. Recently, this idea has \nbeen pus hed to new depths: theoretical studies show that the central spin decoherence can be a \nnovel probe to many -body physics, in particular, phase transitions in spin baths  [21-24]. \nMultiple -spin correla tions are o ne of the  essential characteristics  in spin baths  [11-20], but \ndetection of such correlations is a long -standing challenge in many -body physics. Here we \naddress  this problem with the firs t experimental demonstration of detection of many -body \ncorrelations via central spin decoherence, laying a foundation for studying many -body physics \nand phase transitions in spin baths  [21-24]. \nPrevious approaches to study ing multiple -particle  correlations include the use of nonlinear \noptical spectroscopy of  excit ons in semiconductors  [25-28], nuclear magnetic r esonance (NMR) \nspectroscopy  of nuclear spins in molecules [ 29], and the generalisation of multi -dimensional NMR to  optical spectroscopy [ 30, 31] . Nevertheless,  the detection and characterization  of many -\nbody correlations in nanoscale systems  [32, 33]  remain  highly challenging due to the weak  \nsignals in such small systems.  In this article, we find  that many -body  correlations in nanoscale \nnuclear spin baths have identifiable effects  on the decoherence of a central spin . This enables us \nto propose  and implement a scheme to detect many -body correlations of different orders i n the \nnuclear  spin bath through monitoring  the central spin decoherence. We can distinguish  the \nsecond -order nuclear spin correlations from  the fourth -order nuclear spin correlations by \napplying different  numbers  of pulses  in dynamical  decoupling control  of the central spin. Our \nproposal is  particularly suited for the detection of many -body correlations  in nanoscale systems.  \nResults  \nSystem and  model.  We consider the electron spin ( S= 1/2) of a phosphorus donor localized in \nsilicon as the central spin  (Fig. 1a ). This donor  electron  spin is coupled with a  29Si nuclear spin \nbath ( I = 1/2 and natural abundance  of 4.7%  throughout the host lattice ) by the contact hyperfine \ninteractions and dipolar interactions  [14]. In a strong external magnetic field the Zeeman  \nenergies of the donor spin and nuclear spins are conserved, so the total Hamiltonian can be \nwritten in the  secular form [12, 13 ] \n( 4 ),z z zz\ne z z i i n i ij i j i j i j\ni i ijH S S AI I D I I I I I Iωω+− −+\n<=+ − + +− ∑ ∑∑   (1) \nwhere //en en B ωγ=  is the Larmor frequency of the  donor  electron  spin /bath  nuclear spins, /enγ is \nthe gyromagnetic  ratio of the donor electron spin /bath  nuclear spins, and B is the  external \nmagnetic field  applied along the z-axis. The coupling coeffic ient between  the donor spin and the \ni-th nuclear spin is2 23\n0 [8 / 3 | ( ) | (| | )(3cos 1)/ | | ]i en i i i i Arγγ π ψ θ θ = +− − RR R , where ()i ψR is \nthe donor electron wave function at the  position of the i -th nuclear spin, ()rθ  is the Heaviside  step function and iθ is the angle between the nuclear  spin position vector iR and the magnetic \nfield vector B . In this expression of iA, the first part represents the  contact  hyperfine  interaction \nwhile the second part represents the dipolar  interaction which  starts contributing  for 0 || 2ir>=R  \nnm. The dipolar interaction between the nuclear spins is  22 3(3cos 1)/4| |ij n ij ij D γθ= − R, where ijθ\nis the angle  between =ij i j − RR R  and B. \nWe assume that the donor  electron  spin is initially prepared in  the coherent state \n( )/2 ++− by a /2π -rotation  (with +/− being spin -up/down along the magnetic field \ndirection ). In the subsequent evolution, the  central spin suf fers decoherence as a result of  its \ncoupling to the  nuclear  spin bath. However, by apply ing dynamical decoupling  (DD) control [ 34, \n35] to the central  spin (consisting  of a sequence of π-flips at time s\n ), we can reduce \nits sensitivity to the bath  in general while selectively enhancing  the effect of certain multiple -spin \ndynamics  [8]. With DD,  the restored central spin coherence following  a total evolution time  T is  \n() ( )()†\n,() ,n nn\nJ\nJL T P J UTUTJ+− + − =∑     (2) \nwith \n0 0 2 1 01 [ ( 1) ]( ) ( )( ) ( )()n\nn iV H T t iV H t t iV H t nUT e e e− +− − − − − ±\n± = ,    (3) \nwhere 01\n2z\niiiH AI=∑  and ( 4)zz\nij i j i j i j ijV D II II II+− −+\n<= +−∑ . Here, t he nuclear Zeeman term  \nz\nni iI ω∑ is dropped  since it has no  contribution to the spin decoherence. The nuclear spin bath is \nassumed  to be in an  infinite -temperature (fully mixed) state  with density matrix\n0 /2M\nJJJ ρ=∑ where J is an eigenstate of  z\niiI∑ and M being  the number of nuclear \nspins in the bath.\n We consider two families of DD sequences:  Carr -Purcell -Meiboom -Gill (CPMG) [ 36-38] \nand Uhrig dynamic al decoupling (UDD) [ 39, 40]  (Fig. 2a) . An n-pulse  CPMG sequence  \nperiodically flip s the central spin  at time (2 1) / 2ct c Tn= − , while n- pulse UDD f lips the central \nspin at time 2sin [c / (2 2)]ctT n π = + , where T  is the total evolution time and  1cn=. It should \nbe not ed that CPMG and UDD  are equivalent for 2n≤, and for 1n= simply correspond to the  \nHahn echo.  \nMany -body correlation effects  on central spin decoherence. According to the linked- cluster \nexpansion (LCE) theorem in many -body physics  [41], the quantum evolution of a nuclear spin \nbath can be factorized into contributions of different orders of irre ducible many -body \ncorrelations , namely,  \n( ) , 1234 ( ) expnLT V V V V+− = ++++ ,    (4) \nwith the l -th order many -body correlation  \n() () { } 1 C11ˆT,!lk lCCV dt dt J V t V t Jl=∫∫    (5) \nwhere CˆT is the time -ordering operator along the contour (0 0)CT →→ , and \n() ( ) ( )00 exp exp V t iH t V iH t = − is the intra -bath coupling in the  interaction picture . We show \nsome  examples  of the expan sion terms diagrammatically in Fig. 1b (see  Fig. S1 in \nSupplementary Information for more diagrams) . Here we assume the nuclear spin bath starts \nfrom a pure product state  J. The thermal ensemble results can be obtained by sampling over  \ndifferent initial states and then tak ing a statistical average.  \nFor each LCE term, the real part contribute s to the spin decoherence while the imaginary \npart just produces  a coherent phase  shift (corresponding to self -energy renormalization of the \nprobe spin).  Under CPMG -n or UDD- n control, the first -order LCE term ( 1l=) vanishes due to the contour integral. The second -order LCE term  ( 2l=) corresponds to the  pairwise flip-flop \nprocesses in the nuclear spin bath , in which the bath dynamics is approximated as a product of \nevolutions  of nuclear spin  pairs [15, 17, 18] . Previous studies identified this term as the main \ncause of spin decoherence for the free -induction decay and Hahn echo in the strong magnetic \nfield regime [ 15, 17, 18] . The pairwise flip-flop processes of nuclear spins i , j can be mapped to \nthe precession of a pseudospin ijσ about a pseudofield ( , 0, / 2)ij ij ij D ω±=h  conditioned on the \ncentral spin state  ± [17] (see Supplementary Information) , where ( )/2ij i j AA ω= −  is the \nenergy cost of the flip -flop process . If the central spin is under  CPMG -n control, we have \n() ()odd 2 2\n2 Re 4 4cos cos 2 3ij ij ij ij ijVD tt ωω ω− = −− ∑ when n is odd, but  even\n2 Re =0V when  n \nis even  (see the schem atics in Fig. 3a) , where /2 tT n= . For UDD- n control, the real part of \nsecond -order LCE term also vanishes when n is even and is nonzero when n is odd (see \nSupplementary Information for detail ed derivations) . \nFor higher -order LCE terms, there are three groups of diagrams : ring diagrams, diagonal -\ninteraction renormalized diagram s, and locked diagrams  [41]. Generally , the leading terms  of the \nl-th order diagrams are proportional to ( )/l\nij ijDω. Due to the random distribution of nuclear spins, \nthe contributions from different nuclear spin clusters add destructively when lis odd but add \nconstructively when l is even. Hence,  the odd- order LCE terms contribute negligibly to the spin \ndecoherence.  \nThe central spin decoherence problem can be exactly solved by the cluster -correlation \nexpansion (CCE) method [ 42]. To identify the  contributions of  different many -body correlations \nto the central spin decoherence, we compare the approximate results  obta ined by the LCE  to the \nexact  numerical results obtained by  the CCE (Fig. 2 b). We  see that the second- order  pairwise  flip-flop LCE term  (2V) almost fully reproduces  the CCE results  for DD  controls of odd pulse \nnumber, while the contribution of  the fourth -order  diagonal -interaction renormalized  LCE term  \n(4zV) coincide s with the CCE results  for DD controls of even pulse number.This  indicates  that \nwe can  selectively  detect  either  the second- order or fourth -order many -body correlations by \nchoosing  an appropriate  number of DD control  pulses . Similar pulse -number parity effects were \ntheoretically noticed before [38], however, without analyzing the underlying microscopic \nprocesses . \nThe different  correlations actually present different central sp in decoherence features.  In \nparticular,  the 2V correlation causes  decoherence with a faster initial decay but  a longer decay \ntail (odd 2\n, ln | ( ) |LT T+− −  ); while the decoherence induced by the 4zV correlat ion is better  preserved \nin the short time regime but decays faster in the long time  regime  (even 4\n, ln | ( ) |LT T+− −  ). \nIt should be pointed out that the LCE -4zV term contain s two-body, three- body and four -\nbody nuclear spin correlat ions ( Fig. 1a ). The two -body fourth -order correlations  have no \ncontribution to decoherence, because the pair wise flip-flop of two nuclear spins is independent of  \nthe diagonal interaction between them . The nuclear spin clusters contributing the most to cent ral \nspin decoherence are those four -spin or three -spin clusters  with small inter -nuclei distances  ( <1 \nnm), so that the energy cost of the pairwise flip -flop processes of two nuclear spins is  \nsignificantly changed by the other nuclear spins in the cluster (see Supplementary Information). In the  calculations , we consider a bath volume with radius 8 nm  from the central spin, \ncorresponding to 5000 nuclear spins. Statistical studies (Fig. 3b) show that there are about  \n41.8 10×  such four -spin clusters and 42.6 10×  three- spin clusters in the bath. In Fig. 3c  we \ncompare the contributions of  different many -body correlations and find that the four -body correlations are the main contribution to the central spin decoherence under DD  control of even \nnumber of pulses. T he three- body correlations  are non- zero but relatively sm all. \nExperimental results . We have  observed the pulse -number parity effect in DD experiments on \nP-donors in natural Si  (Fig. 4). The measured de coherence decays fi t well in stretched \nexponential functions  ( )ID SD exp / / TTλττ −− (see Fig. S 3 in Supplementary Information) . \nHere the first term /IDTeτ− represents the instantaneous diffusion caused by dipolar coupling to \nother P -donor  electron spin s in the sample  ([P] = 3x1014/cm3), and the second term (/ )SDTeλτ −\nrepresents the central spin decoherence (spectral diffusion) caused by the 29Si nuclear spin bath.  \nIn Fig s. 4a-b, we show the measured decays, corrected to  exclud e the instan taneous \ndiffusion ( with IDτ=10 ms  determined by the initial exponential decay of the raw experimental \ndata in  Fig. S3 of  Supplementary Information) . The measured and calculated results  agree well \nfor both CPMG -n and UDD -n controls , without any adjustable parameters in the calculation s. In \nFigs. 4c-d, we compare the  central spin coherence  decay time SDτ and exponent  stretching factor  \nλ of the measured and numerical data as functions  of the pulse number n. The  quantitative and \nqualitative agreement  is remarkable, the only exception being  that the  measured decay time  SDτ \noscillates with  n somewhat less strongly than expected . As predicted, the s tretching factor  λ \noscillates between about 2 and 4 as  n increases, mean ing that either the second -order \ncorrelations or fourth- order correlations contribute  dominantly to central spin decoherence. The \nslight decrease of the stretched exponent λwith n can be ascribed to the emergence of the \n“Markovian ” decoherence when the coherence time is prolonged to exceed the pairwise flip -flop \ntime and the higher -order many -body correlations become more important  [42]. Discussion  \nThe different signatures of the many -body correlations  under DD control  of the central spin , \nin particula r the pulse -number parity effect in  the number of DD control pulses, provide a  useful \napproach  to study ing many -body physics in the nuclear spin bath. Note that the parity effect is \nnot affected by the type of DD sequences adopted  in this paper - it exists in both CPMG and \nUDD control s. It is remarka ble that the many -body correlations between nuclear spins have \nsizable  effects even at temperatures (a few Kelvin in our experiments) much higher than the \ncoupling strengths between the nuclear spins (a few nano- Kelvin). \nThe pulse -number parity effect  should  be observable  in a broad range of  central spin \nsystems  as long as the following conditions are satisfied: (i)  pure dephasing condition-  the \nexternal magnetic field should be large so that the energy -non-conserving processes (such as \nsingle nucle ar spin  rotations) are highly suppressed (i.e., the total Hamiltonian can be written in \nthe secular form); (ii) slow/non- Markovian bath condition -  the couplings between nuclear spins \nshould be much weaker than the inverse decoherence time (under this condition th at the LCE \nterms converge rapidly  with increasing orders and the central spin decoherence is mainly  \ninduce d by the lowest -order non- zero LCE terms ). \nThe detection of many -body correlations  may find applications  in id entifying the structures \nof mole cules. In particular, the pulse -number parity effect can be adopted to tell whether the \nmolecules that form the nuclear spin bath have two -body or higher -order interactions among the \nnuclei.  It should be noted that the current scheme can only detect up to the four th-order (four-\nbody ) correlations. Generalization to detection of higher order correlations is in principle \npossible by using more complicated dynamical control (in timing, composition, etc) and/or different types of probes (e.g., higher spins). Exploration along this line will be interesting topics \nfor future studies.  \nMethod  \nNumerical simulation method . The P -donor  electron  spin decoherence in a natural  abundance  \n29Si  nuclear spin bath was  numerically solved by the well -established cluster -correlation \nexpansion (CCE) method [ 42]. The central spin coherence time depends on the random  \nconfiguration of  29Si nuclear spin  positions in the lattice. To compare with the experimental \nresul ts, we ran simulations for  100 random nucl ear spin  configurations and took the ensemble \naverage of  the corresponding time -domain spin coherence. Since the central spin decoherence is \nalmost independent of the initial state of the nuclear spin bath, we just  took a random single -\nsample state  J (an eigenstate of  {}z\niI) as the in itial state of the nuclear spin bath . \nExperimental setup . Experimental results were measured on a natural silicon Czochralski wafer \ndoped with 3x1014/cm3 phosphorus, using a Bruker Elexsys58 0 X band (9.6 GHz) spectrometer . \nAll decay times were obtained on the high -field ESR line ( 1/2Im= − ) at 3452 G at 6 K [ where \nthe electron  spin relaxation processes  (1T≈1 s) did not contribute to decoherence  over  the \ntimescale s considered in this paper ]. The multiple pulses required for the DD  sequence s can \nresult in “stimulated echoes” , and other unwanted echoes, in the experiment due to pulse \ninfidelities. When such echoes overlap with the desired one ( from spin packets which have been \nflipped by all the  π pulses), the  experimentally observed  decay curves gain unwanted \ncontributions. W e therefore cycled the phase s of the applied π pulses  in such a way as to remove \nthe contribution of all undesired echoes . For UDD , the timings between each pulse are different \nand most stimulated echoes fall outside the desired one which  can then be isolated. For example, the phase cycling sequence for UDD -4 requires simply subtracting the echo from two \nexperiments where the first t wo pulses are  changed from +π  to –π and the last two are +π. For \nCMPG, this is more challenging  as the intervals are eq ual and we did not  suppress all possible \nstimulated echoes for CPMG -5 and CPMG -6. \nReferences  \n1. Zurek, W. H. 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Tchernov for his contributions to the experimental design . This work was \nsupported by Hong  Kong RGC/GRF Project 401413 , The Chinese University of Hong Kong Focused Investments \nScheme, National Basic Research Program of Chin a (973 Program) Grant No. G2009CB929300,  and National \nNatural Science Foundation of China Grant No. 61121491.  Work at UCL was supported by the European Research  \nCouncil  under the European Community’s  Seventh Framework  Programme (FP7/2007- 2013)/ERC (Grant N o. \n279781), and by the Engineering and Physical Sciences Research Council (EPSRC) grants  EP/K025945/1 and \nEP/I035536/2.  J. J. L. M. is supported  by the Royal Society.  \nAuthor Contributions  R.B.L. conceived the idea. W.L.M.  and N.Z . perfo rmed the theoretical  study, G.W. and \nJ.J.L.M. carried out the experimental  study . W.L.M., G.W. and R.B.L . wrote the paper . S.S.L . discussed the scheme \nand the results . All authors analyzed the results and commented on the manuscrip t. \nCompeting financial interests The authors declare no competing financial interests.  \nCorrespondence and requests for materials should be addressed to  R.B.L.(email: rbliu@phy.cuhk.edu.hk) or \nJ.J.L.M. (email: jjl.morton@ucl.ac.uk ). \n   \nFigure 1 | Many -body correlations in the 29Si nuclear spin bath p robed  by a phosphorus \ndonor electron spin . (a) Due to the extended donor wavefunction, the  P-donor electron spin \n(blue  arrow ) interacts with a bath of  29Si nuclear spin s (red arrows ) possessing various  many -\nbody c orrelations.  (b) Topologically inequivalent  connected diagrams  (LCE diagrams)  \ncorresponding to different many -body correlations in the nuclear spin bath: (I) 2V-second -order \npairwise flip-flop diagram, (II -V) 4zV-fourth -order diagonal interaction renor malized  pairwise \nflip-flop diagrams . Here the nuclear spin operators iI+, iI−, z\niI are represented in turn by filled \ncircles, empty circles or empty squares . The off -diagonal (diagonal) i nteraction terms are \nrepresented by wavy (dashed) lines. The solid arrows represent  nuclear spin correlation functions \nbetween  iI±\n and iI or z\niI with the arrows indicating the direction of propagation time . \n  \n \nFigure 2 | Effects of different orders of many -body correlations on c entral spin decoherence \nunder  dynamical decoupling . (a) Schematics  of various CPMG and UDD pulse sequences. (b) \nComparisons of the P -donor electron spin de coheren ce in a  natural -abundance  29S inuclear spin \nbath calculated  by the numerically exact  CCE method (lines) and those by the LCE \napproximation (symbols)  to determine the many -body correlations that contribut e significantly  to \nthe spin decoherence under various  CPMG and UDD control s. Here , LCE -2V (crosses)  \nrepresents the pairwise flip -flop processes in the nuclear spin bath which dominate for sequences \nwith an odd number of  π pulses, while LCE-4zV (squares) represen ts the diagonal interaction \nrenormalized pairwise flip -flop processes  which dominate for the even -numbered sequences \nwhere LCE -V2 is zero ( see Fig.  1b). The magnetic field was set as 0.3B=  T applied along the \n[110] lattice direction.  \n \n \nFigure 3 | Contributions  of three- body  and four -body  correlations to  the central spin  \ndecoherence under  CPMG -2 control . (a) Schematic s of bifurcated pseudo- spin evolutions  \nconditioned on the c entral spin state under  CPMG -2 (or UDD -2) control. The conjugat e pseudo -\nspins ()ijt±σ (corresponding to the central spin in the state ±) describe the dynamics of two -spin \ncorrelations. The more the trajectories are separated, the greater the central spin decoherence. \nThe conj ugate pseudo -spins exchange their pseudo -fields ij±h at time ,3tττ=  when the central \nspin is flipped by a π-pulse.Without the diagonal interaction renormalization the conjugate \ntrajectories are symmetric and coincide at time T in the leading order of the evolution time, \nleading to cancellation of decoherence . (b) Histogram of the numbe r of nuclear spin clusters \n(with inter -nuclei distances <1 nm) in 200 different bath configurations. (c ) Decomposition of \nthe LCE -4zV term into three -body and four -body correlations (see Fig 1a)  for CPMG -2 (or UDD-\n2) control of the central spin. The magnetic field was 0.3B=  T applied along the [ 110] lattice \ndirection. \n   \nFigure 4 | Comparison between theoretical and experimental results of natSi:P electron spin  \ndecoherence under dynamical d ecoupling . (a,b) Measured (solid lines) and calculated (dashed \nlines) coherence of the P -donor electron spin in the natural 29Si nuclear spin bath under (a)  \nCPMG or  (b) UDD control . We attribute the deviation seen at ~1 ms for CPMG -6 to an overlap \nwith uncorrected stimulated/unwanted echoes . (c,d)  Comparisons of the experimental (solid lines) \nand theoreti cal (dashed line) decay time s SDτ (blue) and stretched exponent s λ (green) of the \ncentral spin decoherence u nder (c) CPMG or (d) UDD control . The magnetic field was 0.3B=  T \napplied along the [110] lattice direction.  \n  \nSUPPLEMENTARY INFORMATION \nfor \nUncover ing many -body correlations in nanoscale nuclear spin baths by \ncentral spin decoherence  \nWen-Long Ma ,Gary Wolfowicz , Nan Zhao, Shu- Shen Li, John J. L. Morton &  Ren-Bao Liu \nI. Analytical Derivation  of LCE terms  \nA. Interaction picture  \nThe propagators of the nuclear spin bath can be written as  [41] \n{ }\n{ }000\n00ˆ exp[ ( ) ] exp( )T exp ( )\nˆT exp ( ) exp( ),t\nti V H t iH t i V t dt\ni V t t dt iH t ′′ −+ = − −\n ′′ = −− −∫\n∫  (S1a)  \n{ }\n{ }000\n00ˆ exp[ ( ) ] exp( )T exp ( )\nˆT exp ( ) exp( ),t\nti V - H t iH t i V t dt\ni V t dt iH t ′′ − = −−\n ′′ = −−∫\n∫   (S1b)  \nwhere ˆT is the time -ordering operator and \n00 () e x p ( ) e x p ( ) () () () () 4 ,zz\nij i j i j i j\nijV t iH t V iH t D I t I t I t I t I I+− −+\n< = −= + −∑  (S2) \nwith ()iit\niiI t Ieω± ±±=  and / 2.iiA ω=  By the relations above, the operator  ()nUT±  can be rewritten \nin the intera ction picture as the product of several evolution operators. For example, f or the \nCPMG -1 (UDD- 1) and CPMG -2 (UDD- 2) controls  \n{ }{ }1\n00ˆˆ (2 ) T exp ( ) T exp ( ) ,ttU t i b t t dt i b t dt±   ′ ′ ′′ = − − −−    ∫∫   (S3a)\n { }{ }\n{ }{ }2\n00\n00ˆˆ (4 ) T exp ( ) T exp ( )\nˆˆT exp ( ) T exp ( ) ,tt\nttU t i b t t dt i b t dt\ni b t t dt i b t dt±   ′′ ′′ = −− − ×    \n   ′ ′ ′′ − − −−    ∫∫\n∫∫\n   (S3b)  \nwith ()2 tT n= . \nB. Generalized Wick ’s theorem for spin  1/2 operators  \nWick ’s theorem  for bosons or fermions cannot be directly used for the  nuclear spins, \nbecau se the commutation brackets of spin operators do not yield c -number s. Previous studies \ngeneralized Wick ’s theor em to spin 1/2 operators [ 41, S1 ]. First we define t he contraction of two \nspin operators  as \n{ }{ }ˆˆ () ( ) T () ( ) N () ( ) ,ii ii iiI tI t I tI t I tI tαβ αβ αβ′′ ′= −\n    (S4) \nwhere {}ˆNis the normal -ordered operator  depending on the state of the nuclear spin iψ such \nthat {}ˆN0iψ= . For example,  \n{ }ˆN () ( ) ( ) ( ) () ( ) 0zz\nii i i iiiiI tI t I t I t I tI t+− − +′ ′′ ′ ′′ ↑= ↑= ,    (S5a)  \n{ }ˆN () ( ) ( ) () ( ) ( ) 0zz\nii i ii iiiI tI t I t I tI t I t+− +−′ ′′ ′ ′′ ↓= ↓= .    (S5b)  \nIf the nuclear spin i  is in the spin -down state (iψ= ↓ ), we have  the following contraction \nrelations [ S1] \n11\n11( ) ( ; ) 2 ( ; ) ( ) ( ) ( )exp( ),\n(; ) ( ; ) (; ) ( ) ,\n() ( ; )z\ni j m ij i m ij\nz\ni m j ij i m\nz\ni j ijI t I t t t I tt t t t t t t i t\nIt t tIt t It t t t t t\nI tI tt Iδ θθ θ ω\nδθ\nδ+−\n−−\n+′ ′′ = − −− −\n′ ′′ ′ ′′ =−−\n′ ′′=\n\n( ) ( ).it ttθ+′− (S6) \nwhere ()tθ is the Heavi side step function. If iψ= ↑ , we can get the new contraction relations \nfrom (S6) by the  transformation  iiII±→−. Now we can state the generalized Wick ’s theor em for spin 1/2 operators: the time -ordered \nproduct of a set of time -dependent spin operators is equal to the sum of all possible fully \ncontracted products which contains only z\niIoperators [ 41, S1 ]. \nC. Derivation of LCE terms  \nNow we can derive the analytical forms of the LCE terms. First we calculate t he LCE -1V \nterm [see Fig. S1 (a)], \n() { } 1 C1 1 1ˆT ( 4 ) 0.zz\nij i j\nij CCV J V t J dt D ij I I ij dt= = −=∑ ∫∫\n  (S7) \nwhere Jj= ⊗  and ij i j= ⊗ . We see that t his term vanishes due to the contour integral. \nThe LCE -2V term  [see Fig. S1(b)]  is \n()() { }\n{ }2 12 C 1 2\n12 C 1 1 2 2\n12 1 1 2 21ˆT2!\nˆT () () () ()\n() () () ()  .C\ni jji\nij C\ni j ji\nij CV dt dt J V t V t J\ndt dt ij I t I t I t I t ij\ndt dt ij I t I t I t I t ij+−+ −\n>\n+−+−\n>=\n==∫\n∑∫\n∑∫\n    (S8)\n \nFor the CPMG -n control, we have () ()22\n2 Re 4 4cos cos 2 3ij ij ij ij ijVD t t ωω ω− = −− ∑  when n is \nodd, and  2 Re =0V when n is even. For the UDD -n control, we also have 2 Re =0V  when n \nis even, but 2 ReV cannot be written in a simple compactform as in the CPMG case when n is \nodd ( 2n>). \nThe LCE -4zVterm i ncludes four diagrams [Fig. S1(g -j)]. However, the last two diagrams \n[Fig. S1(i -j)] have little contribution to central sp in decoherence, because the pairwise flip -flop \nprocesses of nuclear spins ( i, j) are independent of the diagonal interactions between them (zz\nij i jDII ) [so the 4- th order terms in Fig. S1( i)-(j) approximately reduce to the same form as  in \nFig. S1 (c)-(d), respectively, but are higher -order small quantities] . For the diagrams in Fi g. S1(g -\nh), we can get analytical results of the three -body and four -body correlations  for the CPMG and \nUDD control of even pulse number as follows  \n( )\n( )( )2\n2 2\n4\n2\n4ln ,\nln ,ij z\nijk k ik jk\nij\nij zz\nijkl k l ik jk il jl\nijDL I DD\nDL I I DD DDω\nω−−\n− −−\n\n    (S9) \nwhere ijkL and ijklL denote the central spin decoherence caused by the  diagonal interaction \nrenormalized pairwise flip -flop processes  (ij↔ ) in the three -spin cluster { },,i jk  [Fig. S2(b) ] \nand four -spin clusters { } ,,,i jkl  [Fig. S2 (c)], respectively,  and zz\nkkI JI J≡ . These a nalytical \nexpressions imply that to have significant contributions to the central spin decoherence the \nnuclear spin clusters should s atisfy the following conditions : (i) the inter -nuclei distances in four -\nspin clusters or three -spin clusters should be rather small ( <1 nm); (ii) the renorm alization to the \nenergy cost of the pair flip -flop ( i, j) should be substantial  as compared with the bare energy cost, \ni.e., ( )1 z\nij k ik jkIDD ω−−  should be large for three -spin clusters { },,i jk  while\n( )( )2 zz\nij k l ik jk il jlI I DD DD ω−−−  should be positive and large for four spin clusters { } ,,,i jkl . \nII. Pseudo -spin Model  \nTo get an intuitive understanding of the pulse -number parity  effect, we use the pseudo -spin \nmodel  [17] to describe the dynamics of two nuc lear spins. In the strong f ield regime, the \nHamiltonian of  the i -th and j -th nuclear spins conditioned on the central spin state  \n/2 ,ij\nij z ij x HD ωσ σ±= ±+      (S10)  where the basis set is defined as { } ,↑↓ ↓↑ . Note that the two pseudo- fields corresponding to \nthe two opposite central spin states lie in the xz -plane and are symmetric with respect to the x-\naxis. The time evolution operator is  \n( ) cos ( )sin ,xx zz U t in n φ σσφ±= −±      (S11)  \nwhere t φκ= , 22/4ij ijD κω= + , /x ijnD κ = , /z ijnωκ= . If the central spin is under CPMG -n \nor UDD-  n control, the  time evolution operator nU± can be obtained by the above formula . For \nCPMG -1 (UDD- 1) and CPMG -2 (UDD- 2) controls, we have  \n( )1 22 2\n2 22 22 2( ) 1 2 sin (2 sin sin 2 ),\n( ) 1 2 sin 2 2 sin 2 1 2 sin 2 sin .x xz y x\nx x x xx zU T n in n\nU T n in n nφ φσ φσ\nφ φ φ σ φσ±\n±= −−\n = −− −\n (S12) \nFor t he donor spin in silicon, we ha ve ij ijD ω , so 2/x ij ijnD ω ≈  is a small quantity.  The \ndifference between ()nUT+  and ()nUT−  causes the central spin decoherence ,()nLT+− . When \n21nk= + , we have 21 21kk\nx UU n++\n+−− and 21 2\n, 21 () 1 ()k\nxk L T nf T+\n+− + ≈− . However, when 2nk= , \ndue to the symmetry between the two pseudo- fields corresponding to the two opposite central \nspin states, the two  conjugate trajectories of the pseudo- spin under the two pseudo- fields cross \ninto each other (in the leading order of evolution time) at the end of the DD control. Therefore\n222kk\nx UU n+−− and 24\n,2() 1 ()k\nxk L T nf T+− ≈− . Here ()nfT  is a function of the total evolution time \nT and the pulse number of DD control n. \nIf we consider all the nuclear spins in the bath, then the central spin decoherence can be \nexpressed as the product of the decoherence contributed by each pair of nuclear spins. Then we \nhave  () ()22\n21\n, 21 21 2244( ) 1 exp ,ij ij k ij ij\nkk\nij ij ij ijDDL T fT fTωω+\n+− + +  \n≈− ≈ −       ∏∏\n  (S13a)  \n() ()44\n2\n, 21 21 4416 16( ) 1 exp .ij ij k ij ij\nkk\nij ij ij ijDDL T fT fTωω+− + +  \n≈− ≈ −       ∏∏\n  (S13b)  \nThese results are consistent with results obtained by the LCE method. Recall that  the LCE -lV \nterms are proportional to (/)l\nij ijDω. Therefore, for CPMG or UDD control of odd pulse number s, \nthe second -order correlations contribute the most to the central spin decoherence. But for the \nCPMG or UDD control of even pulse number s, the second -order correlations are cancel led and \nthe fourth- order correlations corresponding  to the ring diagrams 4rV and l ocked diagrams4lV (see \nFig S1) would contribute the most to the central spin decoherence. It should be pointed out that \nin the discussion above we have not  consider ed the diagonal interactions between the nuclear \nspins i , j and other nuclear spins in this pse udo-spin model. Actually such  diagonal interactions \nwill renormalize the pseu do-spin Hamiltonian  and break the symmetry between the two \nconjugate pseudo- fields for  the pseudo- spin. Therefore, the diagonal interaction renormalized \npairwise flip -flop (instead of 4lV and 4rV) would be the dominant  contribution to the central spin \ndecoherence when the number of pulses is  even.  \n \nSupplementary References  \nS1. Giovannini, B. & Koide, S. Perturbation theory for magnetic impurities in metals. Prog. \nTheor. Phys.34, 705- 725 (1965).   \nFigure S1| Topologically inequivalent connected diagrams corresponding to different \nmany -body correlations in the nuclear spin bath up to the fourth order . (a) 1V-first-order \ndiagram, (b) 2V-second -order pair wise flip-flop diagram, (c) -(d) 3zV-third -order diagonal \ninteraction renormalized pair wise flip-flop diagrams, ( e) 3rV-third -order ring diagram, (f) 4rV-\nfourth -order ring diagram, ( g)-(j) 4zV-fourth -order diagonal interaction renormalized pair wise \nflip-flop diagrams, (k)- (l) 4lV-fourth -order locked diagrams.(m) -(n) 4rzV-fourth -order diagonal \ninteraction renormalized  ring diagrams.  \n  \n \nFigure S2| Decomposition of  many -body correlations into  LCE diagrams. We only consider \nthe 2V and 4zV terms contributing most to central spin decoherence. The fourth -order diagonal -\ninteracti on renormalized pair flip -flop processes (4zV) can be two -body, three -body or  four-body \ncorrelations. The two -body correlations describe the pairwise flip -flop processes of  nuclear spins \ni, j renormalized by the diagonal couplings be tween i  and j  while th e three -body (four -body) \ncorrelations describe the pair wise flip-flop processes of nuclear spins i , j renormalized by the \ndiagonal couplings of i , j to nuclear spin k  (k, l) in the nuclear spin bath. Note that in this figure \nthe verti ces along  the same  horizontal line are of the same spin. \n \n \nFigure S3  | Numerical fit s of experimental  and theoretical results of natSi:P electron spin  \ndecoherence by exponential functions ( ) λ\nID SD exp -T /τ- T/τ . (a,c) Experimental or (b,d)  \ntheoretical   (solid lines)  and fitted (dashed lines) coherence of the P -donor electron spin in the \nnatural 29Si nuclear spin bath under (a ,b) CPMG or (c,d) UDD control . Here the same value of \nIDτ=10 ms was used in all the fits. We attribute the  deviation seen at ~1 ms for CPMG -6 to an \noverlap with uncorrected stimulated/unwanted echoes . The magnetic field was 0.3B=  T applied \nalong the [110] lattice direction. \n"    },    {        "title": "0905.3826v1.Multiple_Quantum_Coherence_and_Entanglement_Dynamics_in_Spin_Clusters.pdf",        "content": "Multiple Quantum Coherence and Entang lement Dynamics in Spin Clusters \n \nG. B. Furman1,2, V.M.Meerovich1, and V.L.Sokolovsky1\n1Department of Physics, Ben Gurion University, Beer Sheva 84105, Israel \n2Ohalo College, Qazrin, 12900, Israel \n25 May 2009 \n \nAbstract \nWith the purpose to reveal consistency between multiple quantum (MQ) coherences and \nentanglement, we investigate numerically the dynamics of these phenomena in one-\ndimensional linear chains and ring of nuclear spins 1/2 coupled by dipole–dipole \ninteractions. As opposed to the calculation of  the MQ coherence intensity based on the \ndensity matrix describing the spin system as  a whole, we consider  the \"differentiated\" \nintensity related only to the chosen spin pa ir based on the reduced density matrix. It is \nshown that the entanglement and the MQ co herence have similar dynamics only for \nnearest neighbors while we did not obtained  any consistency for remote spins.   \n \n \n \n \n \n \n \n \n \n \n \n \n \nKeywords: multiple quantum coherences, spin correlations, spin dynamics, entanglement, concurrence  PACS numbers: 03.67.Mn, 03.65.-w,  03.67.Bg,  76.60.-k \n \n \nI. Introduction \n       Entanglement is a central phenom enon of quantum mechanics [1-4] and plays a \npredominant role in quantum information pr ocessing applications  such as quantum \ncomputing [5], quantum communication [6], a nd quantum metrology [7, 8].  At the last \ndecade, entanglement appears as a major goal of many studies to be aimed for creation \nvarious quantum states with photons, trapped ions, cold atoms or spins. While being \nintensive study of the entanglement propertie s, including both quali tative and quantitative \naspects [3, 4, 9], to decide whether a quantum state is entangled or not is still an unsolved \nproblem in general.  \n         For a spin system, which we consider in this paper, entanglement is regarded as a \nresult of a quantum correlation between remo te particles of the spin system [3, 4]. \n    On the other hand, correlations between spins lead to the appearance of multiple-quantum (MQ) coherences [10]. Whereas MQ coherences is a collective effect of the \ncorrelation of all spins in the system [10, 11], widely used m easures of entanglement, the \nvon Neumann entropy and concurrence, descri be entanglement between two spins only \n[3, 4].  Therefore, we will try to extract the intensity of the MQ coherence related to two \nselected spins in a spin cluster and to comp are it with the concurrence of these spins.  \nFor this purpose we will calculate the se cond and zero order coherences using the \nreduced density matrix as it is done for entangle ment measure for a sele cted spin pair [12, \n13]. Using the reduced density matrix we num erically simulate the MQ coherence and \nentanglement dynamics in one-dimensional linea r chains and ring of nuclear spins 1/2 \ncoupled by dipole–dipole interactions up to ten spins at low temperature.  \n \nII. Entanglement measure a nd reduced density matrix  \n \nThe most natural and widely used quantitative measures of entanglement is entanglement of \nformation  [12] , which is intended to quantify the resources needed to create an  entangled state  of \ntwo spins   \n(mn mn mn F Tr E )ρρ ρ ln ) (=     ,               (1) \nwhere mnρis the reduced density m atrix, which  describes dynam ics of the   -th and   \n-th spins. F or  m -th and   -th spins,  the reduce d density matr ix  m n\nnmnρ  defined by  \n()() )( τρ τρmn mn Tr= ,                                                                                (2) \nwhere    denotes the trace over th e degrees  of freedom  for all spins  excep t the   \n-th and   -th ones,  ()...mnTr m\nn )(~τρmn   is the com plex conjugation of mnρ. The analytic expression \nfor    is given by  FE\n()(x x x x xEF − −−−= 1 log 1 log )(2 2 ) ,       (3) \nwhere  ()2\n211 1 C x −+=   and    is the con curren ce be tween two spins [12]. For \nmaximally entangled states, the concurrence is  C\n1=C   while for separable states  0=C . \nThe concurrence between the two spins    and    is expressed by the formula  m n\n()\n⎭⎬⎫\n⎩⎨⎧− = ∑\n=k\nmn\nkmn mnC λλ4\n12 ,0 max ,                 (4) \nwhere  (){}k\nmn mn λ λ max=   and  ()( )4,3,2,1=kk\nmnλ   are the square roots of the eigenvalues of \nthe product  \n()()y y mn y y mn mnR σστρσστρτ ⊗ ⊗ = )(~)( )( .      (5) \nHere    is th e Pauli m atrix.  ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−=00\nii\nyσ\n         \n \nIII. MQ NMR dynamics and tw o-spin coheren ces at low \ntemperature \nLet us consider an MQ NMR experim ent on a system  of nuclear spins coupled by the \ndipole-dipole interaction (DDI) in a strong external m agnetic field  0Hr\n at low  \ntemperature.  At initia l tim e  0=t   the spin system is assum ed to be in therm al \nequilibrium  with the lattice and the equilib rium  spin de nsity operator  eqρ  has the following form \n()\n([)]H TrH\neqββρ−−=expexp                        (6) \nwhere  H  is the Ham iltonian of  the sy stem , kTh=β   ,  T  is the Zeem an tem perature.   \nThe basic s cheme of MQ NMR experim ents consists of  four distinct periods of tim e: \npreparation, evolution, m ixing and detecti on [10]. There are m any radiofrequency (RF)  \npulse sequences exciting MQ cohe rences dur ing the preparation period. For a dipolar-\ncoupled spin system , the m ultiple p ulse sequen ce with an eight-pu lse cycle is kno wn to  \nbe very efficient [10]. It creates th e double-quan tum effective Ha miltonian \n,2)( (2) −+= H H HMQ                      (7) \nwhere  ±±\n<±∑−=kj jk kj IID H41 )2( ,  ) cos31(2\n32\njk\njkjkrD θγ−=h  is the coupling constant \nbetween spins    and   ,  j kγ  is the gyrom agnetic ratio,    is the distan ce between  \nspins    and  k ,  jkr\njjkθ  is the angle be tween th e internuclear vecto r  jkrr  and t he exter nal \nmagnetic field  0Hr\n  which is directed along the z-axis,   and    are the ra ising and \nlowering operators of spin   .   +\njI−\njI\nj\n      The density m atrix of the spin system ,  )(τρ  , at the end of the preparation period is  \n                      (8)),( )( )( τρττρ+= U Ueq\nwhere  ) exp()(MQHi U τ τ −=  . Then the evolution period without any pulses follows. The  \ntransfer of the inform ation about MQ cohere nces to the observable magnetization occurs \nduring the m ixing period. The res ulting signal  ),(tSτ stored as population infor mation \nreads [10, 14] \n()τ τω kt ik\nkJ e tS∆−∑=),(                       (9) \nwhere  ω∆  is the RF offset , chosen to be larg er than the local dipolar  field frequency,  \ndω[10].  ()τkJ   is the spectral intensities of order  k  [14] \n()[].)()(τρτρτzk\nk Tr J=                         (10) Here  is de termined in the f ollowin g way: f irst we calcu late    \n(  is the z-com ponent of the spin angular m omentum  operator) a nd then the terms \nof()τρzk)( )( )( τττρ+= UI Uz z\nzI\n)(τρzare g rouped accord ing to their MQ order k: ()∑=\nkzk\nz τρτρ )(  [10, 14].  \n         The intensities given by (10) are in tegrated ch aracteris tics describ ing MQ \ncoherences of the order k in a spin system . At the sam e time, the von Neum ann entropy \nand concurrence describe entanglem ent between two selected spins  m  and . It would \nbe m ore adequate to co mpare co rrelations between spins, which have the direct attitude to \nentanglem ent and correlations between the sam e spins whic h lead to  occurren ce of MQ \ncoherences.  One could  expect that the spect ral intensity of the MQ coherence of the \nsecond order given by the reduced density  matrix characteri zes all the possible \ncoherences in which the spins    and  are involved. Therefore, we will use the sam e \nreduced density m atrix n\nm n\nmnρto calculate both  the MQ coherence intensities an d \nconcurr ence between tw o spins in a multi-sp in system . \n         The spectral intensities  of the coherences of two spins m and  n can be \ndeterm ined using the reduced  density m atrix (5) as:  )(τmn\nkJ\n()[ ].)( )(τρτρτzk\nmn mnmn\nk Tr J= ,                        (11) \nwhere ()()τρ τρzk\nmnzk\nmn Tr=)( . Obviously, that the configurat ion of MQ coherences (11) \nconsists of the coherences of th e zeroth and second orders only.  \n \nVI. Two-spin MQ dynami cs and evolution of entanglement \n         W e will study a spin system  which is initially in therm odynam ic equilibrium  state at \nlow tem perature and is described by density matrix (6) . We restrict ourse lves to \nnumerical sim ulations of MQ and entanglem ent dynam ics for one-dim ensional and \ncircu lar (rin g) sp in system s. Ring of si x dipolar-coupled proton spins of a b enzene \nmolecule [15], hydroxyl proton chains in calcium hydroxyapatite    [16] \nand fluorine chains in calcium  fluorapatite    [16] are examples of suitable \nobjects to study MQ dynam ics by NMR techni que. In our num erical sim ulations, the  \ndipolar coupling constant of the near est neighbors is chosen to be   . We 34 5 ) )( ( PO OH Ca\n34 5 ) (POFCa\n1\n1,  1−\n+=s Djjassum e also that the an gles jkθ are the sam e for all pairs of spins and the distances  \nbetween nearest neighbors  are equal. Then the coupling constants of spins    and    \nare  jkr j k\n()\n()( )[3\n/ sin/ sin\n1, NkjN\njjD− +ππ]  for the ring and    for the ch ain, respectively .  3\n1, | |/ kj Djj−+\nThe num erical sim ulations of the MQ and entanglem ent dynam ics are perform ed using \nthe software based on th e MATLAB package, al lowing us to investigate spin system s up \nto ten spins.  Along with evol ution of the MQ coherences, ()τkJ , given by (10), w e \nexam ine the tim e dependence of  the two-spin coherences, ()τmn\nkJ , given by (11), and \nconcurrence,  ()τmnC  given by (4), between the spins    and    when the spin sys tem \nevolves under the Ham iltonianm n\nH. In our calculations, the param eter Hβ  (here  ...  \ndenotes a norm  of the operator H) which determ ines the temperature dependence of \nintensities of MQ coherenc es is taken 10. For protons in the external m agnetic \nfield T, this value corresponds  to the tem perature of 1 mK. Fig. 1 shows tim e \ndependence of the \"integrated\"  and \"dif ferentiated\"   intensitie s of the MQ  \ncoherences, and concurrences  between various spin pairs in the four spin chain. The \ninitial pe riod of evolution is charac terized by the creation of en tanglem ent only between \nthe nearest neighbors in the chain (Fig. 1a). Then, entanglem ent develops between distant \nspins (Fig. 1b and c). The longer is the di stance between spins, the m ore tim e the \nappearan ce of their en tanglem ent takes. One can see from  Fig. 1a that the concurrence \n and the spe ctral intensity  have qualitatively close dynam ics. They reach their \nmaximal and m inimal values at the sam e moments of tim e. Sim ilar behavior was found \nfor all nearest neighbors. This dynam ics consistency with the large stretch can be \nattributed to concurrence and spectral intensity of the next-n ext neighbors, but in any way \nit is im possible to attrib ute to next neighbors. It is wort h noticing that the results of the \nnumerical calculations show no consistency between concu rrence and  integrated sp ectral \nintens ity   (Fig. 1). 50=H\n2JmnJ2\nmnC\n2,1C2,1\n2J\n2J\n         The results of the numerical calculation of the tim e dependence of the concurrences \nand spectral intensities in a circle of six di polar-coupled spins and in  a chain of ten spins \nare presented in Figs 2 and 3, respectively. One can see from  Figs. 2a and 3a that there is consistency  between s pectral inte nsities  of the coherences of two spins   and \nconcurrencies  for the neares t neigh bors while these chara cteristics for rem ote spins \ndo not show any consistency (F igs 2b, 2c an d 3b). The sam e conclusion can be done f or \nany spin pairs. 2,1\n2J\n2,1C\n            Note one interesting featur e in behavior of the c oncurrence between next \nneighbors ( ) in a six sp in circ le: the  entangled quantum  states arise pra ctically at the \ninitial stage  of evolutio n (Fig. 2b) and grow more ra pidly than the intensities  and \n.  For ten s pin chains, entanglem ent between  the first spin and spins lo cated at th e \nmiddle of the chain disappears, in spite of  the dir ect in teraction between these spins.  \nSimilar results have been obtained for the 8- an d 9- spin chains [17]. Surprisingly, that, at \nthe s ame time, the concurren ces  between the ends of the chain,  i.e. between the \nmost rem ote spins, are non-zero (Fig. 3b).  3,1C\n2J\n3,1\n2J\n10,1C\n \nConclusions \nIn orde r to adequate ly compare entan glement between different spins and MQ coherences \nin a sp in system , we have propo sed to u se the d ifferentia ted ch aracteristic of the \ncoherences between cho sen spin  pairs and calculate the d iffere ntiated inte nsities us ing the \nreduced  density m atrix. Our num erical calcu lations show that the consistency betw een \nspin coherence and entanglem ent appears only for the near est neighbors w hile for remote \nspins no consistency w as observed. Since the c hosen spin pair is always involved into \nformation of coherence intensity of the highest  order of MQ coherenc e (all the sp ins of  a \nsystem  change their state sim ultaneously), it is interes ting to analyze con sisten cy between  \nthis coheren ce and the concurren ce. Such c onsistency is observed only f or of the most  \nremote spins in the ring of  six spins f or tim es greater than \n2,15\nD=τ  (six th order  \ncoherence intensity is shown by the bl ue dash-doted line in Fig. 2c). Howe ver, for a in \none-dim ensional linear ten-sp in chain, the consistency of  the highest order of MQ \ncoherence possessing the intensity  with concurrence  is displayed only for their \nfirst ex tremums (Fig. 3 c). Thus, ev en the d ifferentiated  approach f or calculation  of the 6J\n10J10,1CMQ coherence intensity correspond ing to the chosen remote spin  pair is not revealed any \ngeneral consistency between dynamics of the coherence and entanglement. \n \nReferences \n1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information \n(Cambridge University Press, Cambridge, 2000). \n2. G. Benenti, G. Casati, and G. Strini , Principles of Quan tum Computation and \nInformation, Volume I and II, (World Scientific, 2007). \n3. L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80, 517 (2008). \n4. R. Horodecki, P. Horodecki, M. Horodeck i, K.Horodecki, http ://arxiv.org/abs/quant-\nph/0702225v1. \n5. C. H. Bennett and D. P. DiVincenzo, Nature 404, 247--255 (2000). \n6. C. H. Bennett, G. Brassard ,  C. Cr epeau , R. Jozsa , A. Peres , W.K. Wootters, Phys.        \n      Rev. Lett. 70, 1895 (1993).  \n7. C. F. Roos, K. Kim, M. Riebe, R. Blatt, Nature, 443, 316 (2006). \n8. P. Cappellaro, J. Emerson, N. Boulant, C. Ramanathan, S. Lloyd, and D. G. Cory, \nPhys. Rev. Lett., 94, 020502 (2005). \n9. T. Konrad, F. de Melo, M. Tiersch, C.  Kasztelan, A. Aragà £ o, A. Buchleitner, \nNature Physics 4, 99  (2008). \n10. J. Baum, M. Munovitz, A. N. Garroway, A. Pines, J. Chem. Phys. 83, 2015 (1985). \n11. J.Tang and A.Pines, J.Chem.Phys., 73, 2512  ( 1980). \n12. W. K.Wootters, Phys. Rev. Lett  80, 2245 (1998)  \n13. C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters,  Phys. Rev. A \n54, 3824, (1996) \n14. E. B. Fel'dman and I. I. Maximov, J. Magn. Reson. 157, 106 (2002). \n15. J.-S. Lee and A. K. Khitrin, Phys. Rev. A,  70, 022330 (2004)  \n16. G. Cho, J. P. Yesinowski, J. Phys. Chem. 100, 15716 (1996) \n17. G. B. Furman, V. M. Meerovich, and V. L. Sokolovsky, Phys. Rev. A, 78, 042301  \n (2008). \n \nFigures  02468 100.00.30.60.902468 100.00.20.402468 100.00.20.40.60.8\n(c)(b)J2 ,   Jmn\n2 ,  Cmn\nD12τ(a)\n \nFig. 1 (Color online)  Tim e depende nce of the differentiated  (black so lid line ) and \ninteg rated  (red dashed line) intensities of the MQ coherences, and concurrences \n(green dotted line) in a f our-spin chain. (a) m =1, n=2; (b) m =1, n=3, ; (c) \nm=1, n=4, .  mnJ2\n2J\nmnC 203,1\n2×J\n104,1\n2×J02468 100.00.20.402468 100.00.20.402468 100.00.10.20.30.4Jk ,  J2mn , Cmn\nD12τ(c)(b)(a)\n \nFig. 2 (Color online) Tim e depende nce of the differentiated  (black so lid line ) and \ninteg rated  (red dashed line) inte nsities and concurrences (green dotted line) in the \nsix spin circle. (a) m =1, n=2; (b) m =1, n=3, ; (c) m =1, n=4, . Blue \ndash-dotted line shows intensity  of the MQ coherence of the sixth order. mnJ2\n2JmnC\n203,1\n2×J 104,1\n2×J\n6J\n \n \n \n 02468 10-0.20.00.20.40.602468 100.00.10.20.30.402468 100.00.20.40.6Jk ,  J2mn , Cmn\nD12τ\n \nFig. 3 (Color online) Tim e depende nce of the differentiated  (black so lid line ) and \ninteg rated  (red dash line) intensities of th e MQ coherences and concurrences \n(green do tted line ) in th e ten spin  chain. (a) m =1, n=2; (b) m =1, n=3, ; (c) \nm=1, n=10, . Blue dash-dotted line shows intensity  of the MQ \ncoherence of the tenth order. mnJ2\n2J\nmnC 202,1\n2×J\n3 10,1\n2 10×J3\n1010×J\n \n \n \n \n \n "    },    {        "title": "1409.7885v1.Spin_electron_acoustic_waves__The_Landau_damping_and_ion_contribution_in_the_spectrum.pdf",        "content": "arXiv:1409.7885v1  [physics.plasm-ph]  28 Sep 2014Spin-electron acoustic waves: The Landau damping and ion co ntribution in the\nspectrum\nPavel A. Andreev∗\nFaculty of physics, Lomonosov Moscow State University, Mos cow, Russian Federation.\n(Dated: June 18, 2021)\nSeparated spin-up and spin-down quantum kinetics is derive d for more detailed research of the\nspin-electron acoustic waves. Kinetic theory allows to obt ain spectrum of the spin-electron acoustic\nwaves including effects of occupation of quantum states more accurately than quantum hydrody-\nnamics. We apply quantum kinetic to calculate the Landau dam ping of the spin-electron acoustic\nwaves. We have considered contribution of ions dynamics in t he spin-electron acoustic wave spec-\ntrum. We obtain contribution of ions in the Landau damping in temperature regime of classic\nions. Kinetic analysis for ion-acoustic, zero sound, and La ngmuir waves at separated spin-up and\nspin-down electron dynamics is presented as well.\nPACS numbers: 52.30.Ex, 52.35.Dm\nKeywords: quantum plasmas, quantum kinetics, wave dispers ion, Landau damping, spin-electron acoustic\nwave\nI. INTRODUCTION\nRecently developed separate spin evolution quantum\nhydrodynamic (SSE-QHD) model [1], giving separated\ndescription of spin-up and spin-down electrons, allowed\nus to discover new type of longitudinal collective excita-\ntions in degenerate quantum plasmas. This excitation is\ncalled the spin-electron acoustic wave (SEAW). In this\nmodel, spin-up and spin-down electrons are considered\nas two different species. The SEAW exists in magne-\ntised plasmas due to difference of the Fermi momentum\nof spin-up and spin-down electrons at presence of an ex-\nternal magnetic field.\nPropagation of the SEAWs parallel and perpendicular\nto an external magnetic field was considered in Ref. [1].\nFurther research of the SEAWs was performed in Refs.\n[2] and [3]. Dispersion of the SEAWs in different two di-\nmensional structures was studied in Ref. [3]. Plane-like\ntwo dimensional electron gas in a magnetic field perpen-\ndicular to the sample, and conducting nanotubes, having\ncylindrical geomentry, in an external magnetic field par-\nallel to the cylinder axis were considered in Ref. [3].\nMore generalcase of oblique wavepropagationin three\ndimensional structures was considered in Ref. [2]. It was\ndemonstrated that at oblique propagation we have two\nbranches of the SEAWs instead of one existing in limit\ncases of parallel and perpendicular propagation.\nIn paper [1], derivation of the separated spin evolu-\ntion QHD with two different species of spin-up and spin-\ndown electrons was demonstrated on a simple example\nof the single-particle Pauli equation. It rather obvious\nthat a single-particle equation has nothing to do with a\nplasma description. A full derivation should be based\nupon a many-particle theory, as, for instance, the many-\nparticle QHD (MPQHD) developed by Kuz’menkov and\n∗Electronic address: andreevpa@physics.msu.rucoauthors [4], [5], [6], [7], [8], [9]. The final equations,\npresented in Ref. [1], were obtained by the correspond-\ning modification of the MPQHD. Hence they have more\ngeneral form than the result of the separate spin-up and\nspin-down fluidisation of the single-particle Pauli equa-\ntion. Nevertheless application of the single-particle Pauli\nequation was a simple way to demonstrate the correct\nstructure of the SSE-QHD. This is a generalisation of\nfamous fluidisation of the single-particle Pauli equation\nperformed by Takabayasi [10].\nConsequences of separate spin evolution for the Lang-\nmuir [1], [2], [3] and Trivelpiece–Gould [2] waves were\nalso studied in mentioned parers.\nThis paper is dedicated to further analysis of the spin-\nelectron acoustic waves and influence of separate spin\nevolution of electrons on ion acoustic and zeroth sound\nwaves. In this paper we focus our attention on waves\npropagating parallel to the external magnetic field. Here\nwe develop the separate spin evolution quantum kinetics.\nKinetic theory gives a background for more careful anal-\nysis of distribution of electrons over different quantum\nstates and contribution of these effects in the quantum\nplasma properties.\nSince we have an example of derivation of SSE-QHD\nfrom the single particle Pauli equation, we stress atten-\ntion on a many-particle derivation of separate spin evo-\nlution quantum kinetics.\nThe separate spin evolution quantum kinetics allows\nto calculate the Landau damping of the SEAWs. We\napply the separate spin evolution quantum kinetics to\nobtain the Landau damping of the SEAWs and other ex-\ncitations in two different regimes: regime of intermediate\ntemperatures, when electronsaredegenerateandionsare\nclassical,andregimeoflowtemperatureswhenallspecies\nare degenerate.\nSome topics in quantum plasmas were discussed in re-\nviews [11], [12], [13].\nThis paper is organized as follows. In Sec. II we de-\nscribe basic definitions of quantum kinetics and describe2\nquantum mechanic background essential for derivation\nof the quantum kinetics. Sec. III contains closed set of\nseparate spin evolution quantum kinetic equations. In\nSec. IV equilibrium state is described. Linearised kinetic\nequations for small perturbations of the equilibrium are\npresented in Sec. IV as well. In Sec. V a general form\nof dispersion equation for oblique propagating longitudi-\nnal waves is obtained. In Sec. VI we present detailed\nanalysis of spectrum of longitudinal waves propagating\nparallel to the external magnetic field. In Sec. VII a\nbrief summary of obtained results is presented.\nII. METHOD OF DERIVATION OF\nSEPARATED SPIN-UP AND SPIN-DOWN\nQUANTUM KINETICS\nA. Structure of many-particle N-spinor wave\nfunction\nIf we have a single particle with no spin degree of free-\ndom it can be described by the wave function ψ(r,t),\nwhich is a complex function of three space coordinates\nand time. If we have two particles of that kind we need\ntoapplythetwo-particlewavefunction ψ(r1,r2,t), which\nis a complex function in six dimensional configuration\nspace. It is hard to make assumptions for this function\nwhen we consider two interacting particles. However if\ninteraction is weak, or we have two non-interactingparti-\ncles, we can represent a two-particle wave function as the\nproduct of single particle wave functions ψ(r1,r2,t) =\nψ(r1,t)ψ(r2,t), or including antisymmetry of fermion\nwave function ψ(r1,r2,t) =1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingleψa1(r1,t)ψa1(r2,t)\nψa2(r1,t)ψa2(r2,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nNext focus our attention on the spin-1/2 particles, as\nelectrons, which are main subject of this paper. A single\nspin-1/2 particle is descried by the spinor (the first-rank\nspinor) wave function, which is a two component wave\nfunctionψ=/parenleftbigg\nψu\nψd/parenrightbigg\n(see for instance [14] section 56).\nNextsteponthewayofconstructionofthemany-particle\ntheory is obtaining of the wave function for two spin-1/2\nparticles. Thiswavefunctionappearstobeasecondrank\nspinor [14].\nSecond-rank spinor is a four component quantity ψςτ\n(see for instance [14] section 56 after formula 56.13).\nComponents of ψςτare transformed as products ψςψτ\nof components of two first-rank spinors. The 2 ×2 unit\nmatrixˆItogetherwith three sigma(Pauli) matrixesform\na basis in space of the second-rank spinors.\nSummarisingallwrittenabovewecanpresentascheme\nof generalisation:\n\nψ(r,t)⇒ψ(r1,r2,t)\n⇓ ⇓/parenleftbigg\nψu(r,t)\nψd(r,t)/parenrightbigg\n⇒Ψ(r1,r2,t),\n(1)where\nΨ(r1,r2,t) =/parenleftbigg\nψu1(r1,r2,t)ψu2(r1,r2,t)\nψd1(r1,r2,t)ψd2(r1,r2,t),/parenrightbigg\n(2)\nis the second rank spinor having presentation of 2 ×2\nmatrix. The two-particle wave function of spin-1/2 par-\nticles, being a matrix, should wear spinor subindexes,\nhence we write Ψ( r1,r2,t) = Ψs1s2(r1,r2,t).\nSpin and coordinate parts of the many-particle wave\nfunction can be separated in absence of the spin-current\nand spin-orbit interactions\nΨ(R,t) = ΨS(R,t) = Λ(R,t)·χS, (3)\nwhereR={r1,...,ri,...,rN}is the set of coordinates of\nNparticles,S={s1,...,si,...,sN}is the set of spinor\nsubindexes of Nparticles. Full many-particle wave func-\ntion Ψ(R,t) is antisymmetric to permutation of argu-\nments. Hence if orbital part Λ( R,t) is antisymmetric,\nthen the spin part χSis symmetric. In opposite case or-\nbital part Λ( R,t) is symmetric, then the spin part χSis\nantisymmetric.\nB. Many-particle Pauli equation as the starting\npoint of derivation of kinetic equations\nThus we start with the many-particle Pauli equation\nı¯h∂tΨ(R,t) =/parenleftbiggN/summationdisplay\ni=1/parenleftbigg1\n2miˆD2\ni+qiϕext\ni−γiσiBi(ext)/parenrightbigg\n+1\n2N/summationdisplay\ni,j/negationslash=i(qiqjGij−γiγjGαβ\nijσα\niσβ\nj)/parenrightbigg\nΨ(R,t),(4)\nwhere Ψ(R,t) = Ψ S(R,t), we can also rewrite\nterms containing spin operators with detail\ndescription of spinor indexes σiΨ(R,t) =\n(σiΨ(R,t))S= (σiΨ(R,t))s1,...,si,...,sN=\nσsisi′Ψs1,...,si′,...,sN(R,t), and σα\niσβ\njΨ(R,t) =\n(σα\niσβ\njΨ(R,t))S= (σα\niσβ\njΨ(R,t))s1,...,si,...,sj,...,sN=\nσα\nsisi′σβ\nsjsj′Ψs1,...,si′,...,sj′,...,sN(R,t).\nEquation (4) governs evolution of N-spinor wave func-\ntion Ψ(R,t). In equation (4) miis the mass of parti-\ncle with number i, below we consider system of parti-\ncles with equal masses, qiis the charge of particle, γi\nis the gyromagnetic ratio, for electrons it can be writ-\nten asγi≈1.00116µB,µB=qi¯h/(2mic) is the Bohr\nmagneton, the difference of |γe|from the Bohr mag-\nneton includes contribution of the anomalous magnetic\ndipole moment, ϕext\niis the scalar potential of an exter-\nnal electromagnetic field acting on particle with number\ni,Bi(ext)is the external magnetic field, ( ˆDiψ)(R,t) =\n((−ı¯h∇i−qi\ncAi,ext)ψ)(R,t), withAi,extis the vector\npotential of an external electromagnetic field acting on3\nparticle, σiis the Pauli matrixes describing spin of par-\nticles, ¯his the reduced Planck constant, cis the speed of\nlight,Bi(ext)=∇i×Ai(ext),Ei(ext)=−∇iϕext(ri,t)−\n1\nc∂tAext(ri,t),Gpn=1\nrijis the Green function of the\nCoulomb interaction containing module of the interpar-\nticle distance rij=ri−rj, and\nGαβ\nij= 4πδαβδ(rij)+∇α\ni∇β\ni1\nrij(5)\nis the Green function of the spin-spin interaction, δijis\nthe Kroneckers delta.\nLet us present the explicit form of the Pauli matrixes\n/hatwideσx=/parenleftbigg\n0 1\n1 0/parenrightbigg\n,/hatwideσy=/parenleftbigg\n0−ı\nı0/parenrightbigg\n,/hatwideσz=/parenleftbigg\n1 0\n0−1/parenrightbigg\n.\n(6)\nThe commutation relation for spin-1/2 matrixes is\n[/hatwideσα,/hatwideσβ] = 2ıεαβγ/hatwideσγ, (7)\nwithεαβγis the Levi-Civita symbol.\nC. Explicit form of the Pauli equation for two\nspin-1/2 particles\nAs the first simple example let us rewrite the Pauli\nequation (4) for a single particle in more explicit form\nı¯h∂tψ↑=/parenleftbigg(¯h\nı∇−qe\ncA)2\n2m+qeϕ−γeBz/parenrightbigg\nψ↑\n−γe(Bx−ıBy)ψ↓, (8)\nand\nı¯h∂tψ↓=/parenleftbigg(¯h\nı∇−qe\ncA)2\n2m+qeϕ+γeBz/parenrightbigg\nψ↓\n−γe(Bx+ıBy)ψ↑. (9)\nThe single particle wave spinor can be easily presented\nas a linear combination of the spin-up and spin-down\nstates described corresponding unit spinors\nψs(r,t) =ψ↑/parenleftbigg\n1\n0/parenrightbigg\n+ψ↓/parenleftbigg\n0\n1/parenrightbigg\n. (10)\nIt have allowed us to rewrite the Pauli equation in a\nrather more explicit form given be formulae (8) and (9).\nApplying wave functions describing spin-up ψ↑and\nspin-downψ↓states we can write probability density to\nfind the particle in a point rwith spin-up ρ↑=|ψ↑|2\nor spin-down ρ↓=|ψ↓|2. We also see ρ=ρ↑+ρ↓. Di-\nrections up ↑(down↓) corresponds to spins having same\n(opposite) direction as (to) the external magnetic field.\nWhilemagneticmomentshaveoppositetospindirections\nsince we consider negatively charged particles.Let us consider structure of the many-particle wave\nfunction Ψ( R,t) in more details. To our end we will need\ndifferent basis in space of the second-rank spinors:\nτ1=1\n2/parenleftbigg\n/hatwideσ0+/hatwideσz/parenrightbigg\n=/parenleftbigg\n1 0\n0 0/parenrightbigg\n(11)\nτ2=1\n2/parenleftbigg\n/hatwideσx+ı/hatwideσy/parenrightbigg\n=/parenleftbigg\n0 1\n0 0/parenrightbigg\n(12)\nτ3=1\n2/parenleftbigg\n/hatwideσx−ı/hatwideσy/parenrightbigg\n=/parenleftbigg\n0 0\n1 0/parenrightbigg\n, (13)\nand\nτ4=1\n2/parenleftbigg\n/hatwideσ0−/hatwideσz/parenrightbigg\n=/parenleftbigg\n0 0\n0 1/parenrightbigg\n, (14)\nwhere\n/hatwideσ0=ˆI=/parenleftbigg\n1 0\n0 1/parenrightbigg\n(15)\nis the unit second rank spinor.\nItisessentialtorepeatthat thetwo-particlewavefunc-\ntionoftwospin-1/2particlesisa2 ×2matrix(seeformula\n(2)). Consequently we can expand it as a superposition\nofmatrixes/hatwideσ0andσ={/hatwideσx,/hatwideσy,/hatwideσz}orthe set ofmatrixes\n{/hatwideτa}witha= 1, 2, 3, 4. So, the two-particle wave func-\ntion has form of Ψ =/summationtext\naψa/hatwideτa, whereψa=ψa(r1,r2,t)\nare complex functions.\nWave function of two spin-1/2 particles Ψ S(R,t) can\nbe presented via the upper Ψ ↑(R,t), or lower Ψ ↓(R,t)\nline in the 2-rankspinor Ψ S(r1,r2,t) =/parenleftbigg\nΨ↑(r1,r2,t)\nΨ↓(r1,r2,t)/parenrightbigg\n,\nwhere Ψ ↑(r1,r2,t) =/parenleftbigψ1(r1,r2,t)ψ2(r1,r2,t)/parenrightbig\n, and\nΨ↓(r1,r2,t) =/parenleftbigψ3(r1,r2,t)ψ4(r1,r2,t)/parenrightbig\n.\nThe density probability in the six dimensional con-\nfigurational space appears in the traditional form\nρ(r1,r2,t) = Ψ+(R,t)Ψ(R,t). Its explicit form is\nρ(r1,r2,t) =/summationtext\niψi(r1,r2,t). We can separate terms\nin this sum in two groups ρ↑(r1,r2,t) andρ↓(r1,r2,t)\nin the following way. We can introduce a notation\nρ↑(r1,r2,t) = Ψ+\n↑Ψ↑,ρ↓(r1,r2,t) = Ψ+\n↓Ψ↓, which is\nnot a mathematical symbol, but it will be very useful\nto get a compact form of formulae below. Applying\nwave functions describing spin-up Ψ ↑and spin-down Ψ ↓\nstates of two particle systems we can write probability\ndensity to find both particles in point r1andr2with\nspin-upρ↑(r1,r2,t) =|ψ1|2+|ψ2|2or spin-down\nρ↓=|ψ3|2+|ψ4|2. We also see ρ=ρ↑+ρ↓. Di-\nrections up ↑(down↓) corresponds to spins having same\n(opposite) direction as (to) the external magnetic field.\nA compact form of the Pauli equation for two spin-\n1/2 interacting particles can be written easily using the\ngeneral form of the Pauli equation for N particles (4)\nı¯h∂tΨ(R2,t) =/bracketleftbigg2/summationdisplay\ni=1/parenleftbigg1\n2miˆD2\ni+qiϕext\ni−γiσiBi(ext)/parenrightbigg4\n+q1q2G12−γ1γ2Gαβ\n12σα\n1σβ\n2)/bracketrightbigg\nΨ(R,t),(16)\nwhere Ψ(R2,t) = Ψs1,s2(r1,r2,t).\nWe are going to present a more explicit form of equa-\ntion (16). To this end we introduce operator /hatwideΛ as\n/hatwideΛ =ı¯h∂t−2/summationdisplay\ni=1/parenleftbigg1\n2miˆD2\ni+qiϕext\ni/parenrightbigg\n−q1q2G12.(17)Finally we able to present explicit form of the Pauli\nequation for two particles involved in the Coulomb and\nspin-spin interactions, and also interacting with the ex-\nternal electromagnetic field\n/hatwideτ1/braceleftbigg\nΛψ1+γ1[B1zψ1+(B1x−ıB1y)ψ3]+γ1γ2[(Gzx+Gzy+Gzz)ψ1+(Gxx+Gxy+Gxz)ψ3−ı(Gyx+Gyy+Gyz)ψ3]/bracerightbigg\n+/hatwideτ2/braceleftbigg\nΛψ2+γ2[B2zψ2+(B2x−ıB2y)ψ4]+γ1γ2[(Gxz+Gyz+Gzz)ψ2+(Gxx+Gyx+Gzx)ψ4−ı(Gxy+Gyy+Gzy)ψ4]/bracerightbigg\n+/hatwideτ3/braceleftbigg\nΛψ3+γ1[−B1zψ3+(B1x+ıB1y)ψ1]+γ1γ2[−(Gzx+Gzy+Gzz)ψ3+(Gxx+Gxy+Gxz)ψ1+ı(Gyx+Gyy+Gyz)ψ1]/bracerightbigg\n+/hatwideτ4/braceleftbigg\nΛψ4+γ2[−B2zψ4+(B2x+ıB2y)ψ2]+γ1γ2[−(Gxz+Gyz+Gzz)ψ4+(Gxx+Gyx+Gzx)ψ2+ı(Gxy+Gyy+Gzy)ψ2]/bracerightbigg\n= 0.\n(18)\nComparing formula (18) with its analog for a single\nparticle presented by equations (8) and (9) we see that\ntwoparticlesystemisrathermorecomplicate. It isessen-\ntial to mention that two particle Pauli equation reflects\nmany properties of N particle Pauli equation. Hence it\nallows to understand correct structure of quantum kinet-\nics of electrons with separated spin-up and spin down\nevolution.\nD. Basic definitions of quantum kinetics\nMost famous quantum distribution function was sug-\ngested by Wigner [15], however we do not apply it\nhere. We construct our kinetic theory in according with\nthe many-particle quantum hydrodynamic (MPQHD)\nmethod [4], [6], [9]. We start with classic microscopicdis-\ntribution function [16], [17]. We change classic dynamic\nfunctions of position and momentum of particles on the\ncorresponding operators. As the result we find the oper-\nator of many-particle microscopic quantum distribution\nfunction [18], [19]\nˆf=/summationdisplay\niδ(r−/hatwideri)δ(p−/hatwidepi). (19)\nQuantum mechanical averaging of the operator of\nmany-particle distribution function gives us the micro-\nscopic distribution function for system of spinning parti-cles [18], [19]\nfa(r,p,t) =1\n4/integraldisplay/parenleftigg\nΨ+(R,t)/summationdisplay\ni/parenleftbigg\nδ(r−ri)δ(p−/hatwidepi)\n+δ(p−/hatwidepi)δ(r−ri)/parenrightbigg\nΨ(R,t)+h.c./parenrightigg\ndR, (20)\nfor each species of particles a=efor electrons and a=i\nfor ions. In formula (20) we have Ψ+(R,t) = Ψ+\nS(R,t) =\n(ΨS(R,t))∗.\nWe can introduce the distribution function of sub-\nspecies of electrons for spin-up and spin-down electrons:\nfe,s=fe,s(r,p,t) =1\n4/integraldisplay/parenleftigg\nΨ+\ns(R,t)/summationdisplay\ni/parenleftbigg\nδ(r−ri)δ(p−/hatwidepi)\n+δ(p−/hatwidepi)δ(r−ri)/parenrightbigg\nΨs(R,t)+h.c./parenrightigg\ndR,(21)\nfor each subspecies of electrons.\nIn formula (21) we have applied functions Ψ S(R,t),\nwhich are the upper Ψ ↑(R,t), or lower Ψ ↓(R,t) line in\nthe N-rank spinor Ψ S(R,t) =/parenleftbigg\nΨ↑(R,t)\nΨ↓(R,t)/parenrightbigg\n.5\nIII. SET OF KINETIC EQUATIONS\nWe consider quantum plasmas as the set of three\nspecies of particles: spin-up electrons, spin-down elec-\ntrons and ions. Hence we have three kinetic equations\npresented below.\nKinetic equation for spin-up electrons is\n∂tfe↑+v∇rfe↑+qe/parenleftbigg\nE+1\nc[v,B]/parenrightbigg\n∇pfe↑\n+γe∇Bz·∇pfe↑+γe\n2/parenleftbigg\n∇Bx·∇pSe,x\n+∇By·∇pSe,y/parenrightbigg\n=γa\n¯h[Se,xBy−Se,yBx].(22)\nKinetic equation for spin-down electrons has same\nstructure as equation for spin-up electrons, but it has\nsome different coefficients\n∂tfe↓+v∇rfe↓+qe/parenleftbigg\nE+1\nc[v,B]/parenrightbigg\n∇pfe↓\n−γe∇Bz·∇pfe↓+γe\n2/parenleftbigg\n∇Bx·∇pSe,x\n+∇By·∇pSe,y/parenrightbigg\n=−γa\n¯h[Se,xBy−Se,yBx].(23)\nKinetic equation for ions is\n∂tfi+v∇rfi+qi/parenleftbigg\nE+1\nc[v,B]/parenrightbigg\n∇pfi= 0,(24)\nwhere we consider the charge-charge interaction only.\nConsidering the charge-charge and the spin-spin inter-\nactions we apply the self-consistent field approximation\n[13], [20], [17]. The MPQHD equations beyond the self-\nconsistent field approximation can be found in Refs. [5],\n[7], [21].\nKinetic equations for electrons contain spin-\ndistribution functions Se,x(r,p,t) andSe,y(r,p,t).\nThe spin distribution functions for each species ap-\npears as the quantum mechanical average of the corre-\nsponding operator\nˆSα=/summationdisplay\niδ(r−/hatwideri)δ(p−/hatwidepi)/hatwideσα\ni. (25)\nHence explicit form of the spin distribution function is\nSα\na(r,p,t) =1\n4/integraldisplay/parenleftigg\nΨ+\nS(R,t)/summationdisplay\ni/parenleftbigg\nδ(r−ri)δ(p−/hatwidepi)\n+δ(p−/hatwidepi)δ(r−ri)/parenrightbigg\n/hatwideσα\niΨS(R,t)+h.c./parenrightigg\ndR.(26)The spin distribution functions Se,x(r,p,t) and\nSe,y(r,p,t) appear for all electrons inspite the sep-\naration of spin-up and spin-down electrons in the\ndistribution functions.\nDifferentiating explicit forms of SxandSyand apply-\ning the Pauli equation (8) and (9) for the time deriva-\ntives of the N-rank spinor wave function Ψ Swe obtain\nthe following equations for spin-distribution functions of\nelectrons\n∂tSe,x+v∇rSe,x+qe/parenleftbigg\nE+1\nc[v,B]/parenrightbigg\n∇pSe,x\n+γe∇Bx∇p(fe↑+fe↓)−2γe\n¯h/parenleftbigg\nSe,yBz−(fe↑−fe↓)By/parenrightbigg\n= 0,\n(27)\nand\n∂tSy+v∇rSe,y+qe/parenleftbigg\nE+1\nc[v,B]/parenrightbigg\n∇pSe,y\n+γe∇By∇p(fe↑+fe↓)−2γe\n¯h/parenleftbigg\n(fe↑−fe↓)Bx−Se,xBz/parenrightbigg\n= 0.\n(28)\nLet us mention that SxandSydo not wear subindexes\n↑and↓. As we can see from definitions of SxandSy\nthey are related to both projections spin-up Ψ ↑and spin-\ndown Ψ ↓. Operators/hatwideσx\niand/hatwideσy\nimixing upper and lower\ncomponents of N-rank spinor wave function. Whereas\n/hatwideσz\nido not mix them giving Sz(r,p,t) =fe↑(r,p,t)−\nfe↓(r,p,t). Thefulldistributionofallelectrons fe(r,p,t)\nis the sum of distribution functions of spin-up fe↑(r,p,t)\nand spin-down fe↓(r,p,t) electrons fe=fe↑(r,p,t) +\nfe↓(r,p,t).\nElectromagneticfieldsin the QHDequationspresented\nabove obey the Maxwell equations\n∇E= 4πe/parenleftbigg\nni−ne↑−ne↓/parenrightbigg\n, (29)\n∇B= 0, (30)\n∇×E=−1\nc∂tB, (31)\nand\n∇×B=1\nc∂tE+4π∇×Me\n+4π\nc(qeje↑+qeje↓+qiji), (32)\nwhereMe={γe˜Sex,γe˜Sey,γe(ne↑−ne↓)}is the magne-\ntization of electrons in terms of hydrodynamic variables.6\nMaterialfieldsenteringtheMaxwellequationshavethe\nfollowing relations to the distribution functions\nna(r,t) =/integraldisplay\nfa(r,p,t)dp, (33)\nja(r,t) =/integraldisplayp\nmafa(r,p,t)dp, (34)\n˜Sex(r,t) =/integraldisplay\nSex(r,p,t)dp, (35)\nand\n˜Sey(r,t) =/integraldisplay\nSey(r,p,t)dp. (36)\nLet us consider the spin density\n˜Sα(r,t) =/integraldisplay\ndR/summationdisplay\niδ(r−ri)ψ∗(R,t)/hatwideσα\niψ(R,t),(37)\nproportional to the magnetization Ma(r,t), usually\nused in the quantum hydrodynamics [4], [9], and [20]:\nMa(r,t) =γaSa(r,t). Next integrating the spin distri-\nbution function over the momentum we get the spin den-\nsity appearing in the quantum hydrodynamic equations\n[9], [20]\nHere we describe explicit form of hydrodynamic spin\ndensity projections ˜Sα(r,t) for the simple single parti-\ncle case to give a taste of the spin density structure,\nwhich reflects structure of the spin-distribution function.\nHere we need to represent both components of the spinor\nwave function as ψs=aseıφs. These quantities appear\nas follows ˜Sx=ψ∗σxψ=ψ∗\n↓ψ↑+ψ∗\n↑ψ↓= 2a↑a↓cos∆φ,\n˜Sy=ψ∗σyψ=ı(ψ∗\n↓ψ↑−ψ∗\n↑ψ↓) =−2a↑a↓sin∆φ,˜Sz=\nψ∗\n↑ψ↑−ψ∗\n↓ψ↓=a2\n↑−a2\n↓, where ∆φ=φ↑−φ↓.˜Sx,˜Sy, and\n˜Szappear as mixed combinations of ψ↑andψ↓. These\nquantities do not related to different species of electrons\nhaving different spin direction. ˜Sxand˜Sydescribe si-\nmultaneous evolution of both species.\nIV. LINEARISED SET OF SEPARATED\nSPIN-UP AND SPIN-DOWN QUANTUM\nKINETIC EQUATIONS\nOperator[ v,ez]∂pcanberepresentedas1\nm∂ϕ, whereϕ\nis the angle in the cylindrical coordinates in momentum\nspace.\nIn equilibrium the set of kinetic equations (22), (23),\n(24), (27), and (28) has the following form\n∂ϕf0e↑= 0, ∂ϕf0e↓= 0, ∂ϕf0i= 0,(38)\n∂ϕS0e,x=S0e,y, (39)and\n∂ϕS0e,y=−S0e,x. (40)\nWe have included that time and space derivatives of the\ndistribution functions equal to zero. We have also in-\ncluded that electric field in equilibrium equals to zero as\nwell. Equilibrium magnetic field equals to the external\nfield directed parallel to the Z axis Bx=By= 0.\nA. Equilibrium distributions\nWeconsiderdegenerateelectrons. Inabsenceofthe ex-\nternal magnetic field two electrons occupy each quantum\nstate with momentum below the Fermi momentum\nf0e=2\n(2π¯h)3Θ(pFe−p), (41)\nwherepFe= (3π2n0e)1\n3¯h.\nDistribution (41) is a spherically symmetric distribu-\ntion, which does not contain dependence on angle vari-\nablesθ,ϕ.\nIfwehavedegenerateelectronsinanexternalmagnetic\nfieldwhenoccupationofspin-upandspin-downstatesare\ndifferent. Under influence of an external magnetic field\npart of spin-up electrons change direction and transit to\nspin-down states. Thus instead of the Fermi step con-\ntaining fully occupied (two electrons in a state) states,\nwhich is the unmodified Fermi step, we have two dif-\nferent Fermi steps for spin-up and spin-down electrons.\nThe Fermi step for spin-up electrons is shorter than the\nunmodified Fermi step, whereas The Fermi step for spin-\ndown electrons is longer than the unmodified Fermi step.\nEquilibrium distribution function for each subspecies of\nelectrons are\nf0s=1\n(2π¯h)3Θ(pFs−p), (42)\nwherepFs= (6π2n0s)1\n3¯h, ands=↑, or↓.\nDistribution function of all electrons is the sum of f0↑\nandf0↓, hence\nf0e=1\n(2π¯h)3[θ(pF↑−p)+θ(pF↓−p)],(43)\nwhich pass into (41) at B0→0.\nWe consider two limits for ions: classic low tempera-\nture ions and degenerate ions.\nFor classic ions we consider the Maxwell distribution\nfunction for equilibrium distribution\nf0i(p) =n0i\n(√2πmiTi)3exp/parenleftbigg\n−p2\n2miTi/parenrightbigg\n,(44)\nwhereTiis the temperature of classic ions in units of\nenergy, and p=mv.7\npFupFd\npxpz\npy\nFIG. 1: (Color online) The figure shows distribution func-\ntionsnof degenerate spin-up and spin-down electrons being\nin external magnetic field. This distribution function give s\naverage occupation number of quantum states with different\nenergies.\nFor degenerate ions we have\nf0i=2\n(2π¯h)3Θ(pFi−p). (45)\nWe neglect change of ion occupation number for magne-\ntised ions.\nFrom equations (39) and (40) we find general form of\ndependence of equilibrium spin distribution functions on\nmomentum\nS0x=C(p/bardbl,p⊥)cos(ϕ+ϕ0), S0y=C(p/bardbl,p⊥)sin(ϕ+ϕ0).\n(46)\nFurther analysis leads to the following explicit form of\nequilibrium spin distribution functions\nS0x=1\n(2π¯h)3/parenleftbigg\nΘ(pFu−p)+Θ(pFd−p)/parenrightbigg\ncos(ϕ+ϕ0),\n(47)\nS0y=1\n(2π¯h)3/parenleftbigg\nΘ(pFu−p)+Θ(pFd−p)/parenrightbigg\nsin(ϕ+ϕ0).\n(48)Let us mention that corresponding equilibrium hydro-\ndynamic spin densities equal to zero\n/integraldisplay\nS0x(p)dp= 0, (49)\nand\n/integraldisplay\nS0y(p)dp= 0, (50)\nas it should be. These integrals equal to zero due to\nintegration over the angle ϕ.\nB. Linearised kinetic equations\nNow we are ready to present linearised kinetic equa-\ntions.\nLinearised kinetic equation for spin-up electrons reads\nas\n∂tδfe↑+v∂rδfe↑\n+qe/parenleftbigg\nδE+1\nc[v,δB]/parenrightbigg\n∂pf0e↑+qe1\nc[v,B0]∂pδfe↑\n+γe∇δBz·∇pf0e↑+γe\n2/parenleftbigg\n∂αδBx·∂pαS0e,x\n+∇δBy·∇pS0e,y/parenrightbigg\n=γa\n¯h/parenleftbigg\nS0e,xδBy−S0e,yδBx/parenrightbigg\n.(51)\nLinearised kinetic equation for spin-down electrons ap-\npears as\n∂tδfe↓+v∇rδfe↓\n+qe/parenleftbigg\nδE+1\nc[v,δB]/parenrightbigg\n∇pf0e↓+qe1\nc[v,B0]∇pδfe↓\n−γe∇αδBz·∇pαf0e↓+γe\n2/parenleftbigg\n∇δBx·∇pS0e,x\n+∇δBy·∇pS0e,y/parenrightbigg\n=−γa\n¯h/parenleftbigg\nS0e,xδBy−S0e,yδBx/parenrightbigg\n.(52)\nLinearised kinetic equation for ions has the following\nform\n∂tδfi+v∇rδfi\n+qi/parenleftbigg\nδE+1\nc[v,δB]/parenrightbigg\n∇pf0i+qi1\nc[v,B0]∇pδfi= 0.(53)8\nLinearisedkineticequationforx-projectionofthe spin-\ndistribution function of electrons\n∂tδSe,x+v∇rδSe,x\n+qe/parenleftbigg\nδE+1\nc[v,δB]/parenrightbigg\n∇pS0e,x+qe1\nc[v,B0]∇pδSe,x\n+γe∇δBx·∇p(f0e↑+f0e↓)\n=2γe\n¯h/parenleftbigg\nδSe,yB0z+S0e,yδBz−(f0e↑−f0e↓)δBy/parenrightbigg\n,(54)\nand linearised kinetic equation for y-projection of the\nspin-distribution function of electrons\n∂tδSy+v∇rδSe,y\n+qe/parenleftbigg\nδE+1\nc[v,δB]/parenrightbigg\n∇pS0e,y+1\nc[v,B0]∇pδSe,y\n+γe∇δBy·∇p(f0e↑+f0e↓)\n=2γe\n¯h/parenleftbigg\n(f0e↑−f0e↓)δBx−S0e,xδBz−δSe,xB0z/parenrightbigg\n.(55)\nEquations of matter evolution (51)-(55) are coupled\nwith equations of electromagnetic field\n∇δE= 4π/summationdisplay\na=u,d,iqa/integraldisplay\nδfa(r,p,t)dp,(56)\n∇δB= 0, (57)\n∇×δE=−1\nc∂tδB, (58)\nand\n∇×δB=1\nc∂tδE+4π∇×δMe\n+4π\nc/summationdisplay\na=u,d,iqa/integraldisplayp\nmaδfa(r,p,t)dp,(59)\nwhere\nδMx=γe/integraldisplay\nδSe,x(r,p,t)dp, (60)\nδMy=γe/integraldisplay\nδSe,y(r,p,t)dp, (61)\nand\nδMz=γe/integraldisplay\n[δf↑(r,p,t)−δf↓(r,p,t)]dp.(62)\nE kll\nxyz\nEkv\nFIG. 2: (Color online) The figure shows velocity, wave vector ,\nand electric field in oblique propagating longitudinal wave s.\nC. Small amplitude perturbations propagating\nparallel to the external magnetic field\nEquilibriumconditionisdescribedbythenon-zerocon-\ncentrations n0↑,n0↓,n0=n0↑+n0↓,S0x,S0y, and an\nexternal magnetic field Bext=B0ez, butE0= 0. As-\nsuming that perturbations are monochromatic\n\nδfe↑\nδfe↓\nδfi\nδE\nδB\nδSx\nδSy\n=\nFA↑\nFA↓\nFAi\nEA\nBA\nSAx\nSAy\ne−ıωt+ıkr,(63)\nwe get a set of linear algebraic equations relatively to\nFA↑,FA↓,FAi,VA↑,VA↓,EA,BA,SAx, andSAy. Con-\ndition of existence of nonzero solutions for amplitudes of\nperturbations gives us a dispersion equation.\nDifference of spin-up and spin-down concentrations of\nelectrons ∆ n=n0↑−n0↓is caused by external mag-\nnetic field. Since electrons are negative their spins get\npreferable direction opposite to the external magnetic\nfield∆n\nn0= tanh(γeB0\nTe) =−tanh(|γe|B0\nTe). Here we con-\nsider temperature in units of energy, so we do not use the\nBoltzmann constant.\nWe consider plasmas in the uniform constant external\nmagnetic field. We see that in linear approach numbers\nof electrons of each species conserves.\nLinearised set of kinetic equations has rather complex\nform, but we follow results of Refs. [1] and [3], hence\nwe are focused on properties of the longitudinal waves.\nThis assumption makes analysis more simple. We should\nmention that waves in magnetised plasmas are not longi-\ntudinal in most cases. An exeptional regime supporting\npropagation of longitudinal waves is limit of wave prop-\nagation parallel to external magnetic field. Thus this is9\nthe main area of our research. However we obtain an ap-\nproximate dispersion equation for longitudinal waves in\nregime of oblique wave propagation. Longitudinal waves\nrequireE/ba∇dblk. As a consequence we obtain δB= 0.\nV. DISPERSION EQUATION FOR\nLONGITUDINAL WAVES: GENERAL FORM\nGeneral form of dispersion equation for oblique prop-\nagating longitudinal waves in separated spin evolution\nmodel appears as\n1+4π2e2\nk/braceleftigg/summationdisplay\ns=u,d+∞/summationdisplay\nn=−∞p2\nFs\n(2π¯h)3×\n×/integraldisplayπ\n0sinθdθJn/parenleftbigg\nkxvFs\nΩssinθ/parenrightbigg\nkzvFscosθ−ω+nΩs×\n×/bracketleftigg\n2cosαcosθJn/parenleftbiggkxvFs\nΩssinθ/parenrightbigg\n+sinαsinθ/parenleftigg\nJn+1/parenleftbiggkxvFs\nΩssinθ/parenrightbigg\n+Jn−1/parenleftbiggkxvFs\nΩssinθ/parenrightbigg/parenrightigg/bracketrightigg\n++∞/summationdisplay\nn=−∞n0i\n(2πmiTi)3\n21\n2πTi/integraldisplay\ndpJn/parenleftbigg\nkxvTi\nΩisinθ/parenrightbigg\ne−p2\n2miTi\nkzvz−ω+nΩi×\n×/parenleftigg\n2Jn/parenleftbiggkxvTi\nΩisinθ/parenrightbigg\nvzcosα\n+/bracketleftbigg\nJn+1/parenleftbiggkxvTi\nΩisinθ/parenrightbigg\n+Jn−1/parenleftbiggkxvTi\nΩisinθ/parenrightbigg/bracketrightbigg/parenrightigg/bracerightigg\n= 0,\n(64)\nwherevz=vcosθ,v⊥=vsinθ,kz=kcosα,kx=\nksinα, andJn(x) are the Bessel functions.\nDispersionequationforlongitudinalwavespropagating\nparallel to external field in magnetised plasmas rather\nsimpler, in this limit we have k/ba∇dblB0, consequently we\nobtainα= 0,kx= 0,kz=k. This assumption leads to\nmore simplification Jn(0) = 0 ifn/ne}ationslash= 0, andJ0(0) = 1.\nAfter all these simplifications we find\n1+8π2e2\nk2/parenleftigg/summationdisplay\ns=u,dm2\nevFs\n(2π¯h)3/parenleftbigg\n2+ω/integraldisplayπ\n0sinθdθ\nkvFscosθ−ω/parenrightbigg\n+1\n2πTin0i\n(2πmiTi)1\n2/parenleftbigg\n(2πmiTi)1\n2+ω/integraldisplaye−p2z\n2miTi\nkvz−ωdpz/parenrightbigg/parenrightigg\n= 0.\n(65)For the Maxwell distribution in equilibrium state we\nmeet the following integral in the dispersion equation\nZ(α) =1√π/integraldisplay+∞\n−∞exp(−ξ2)\nξ−αdξ\n=1√π/bracketleftbigg\nP/integraldisplay+∞\n−∞exp(−ξ2)\nξ−αdξ/bracketrightbigg\n+ı√πexp(−α2),(66)\nwhereα=ω\nkvTwithvT≡/radicalig\nT\nm, and the symbol Pde-\nnotes the principle part of the integral.\nLet us present assumptions of formula (66). At α≫1\nwe have\nZ(α)≃ −1\nα/parenleftbigg\n1+1\n2α2+3\n4α4+.../parenrightbigg\n+ı√πexp(−α2).(67)\nThis approximate formula will be applied below at de-\nscription of classic ion contribution in spectrum of plas-\nmas.\nVI. SPECTRUM OF LONGITUDINAL WAVES\nPROPAGATING PARALLEL TO EXTERNAL\nFIELD IN MAGNETISED SEPARATED SPIN-UP\nAND SPIN-DOWN QUANTUM PLASMAS\nTaking integrals over angle θandpzin equation (65)\nwe get the following explicit form of dispersion equations\nin regimes of classic and degenerate ions.\nClassic ions\nDegenerate electrons give, in dispersion equation, well-\nknown logarithmic term. Since we have two species of\nelectrons we obtain two logarithmic terms:\n1 =3\n2ω2\nLu\nv2\nFuk2/parenleftbiggω\nkvFulnω+kvFu\nω−kvFu−2/parenrightbigg\n+3\n2ω2\nLd\nv2\nFdk2/parenleftbiggω\nkvFdlnω+kvFd\nω−kvFd−2/parenrightbigg\n+ω2\nLi\nω2/parenleftbigg\n1+3k2v2\nTi\nω2/parenrightbigg\n−/radicalbiggπ\n2ıω2\nLi\nk2v2\nTiω\nkvTiexp/parenleftbigg\n−ω2\n2k2v2\nTi/parenrightbigg\n.(68)\nQuantum degenerate ions\nIfions aredegenerateaswell aselectronswe havethree\nsimilar logarithmic terms\n1 =/summationdisplay\na=u,d,i3\n2ω2\nLa\nv2\nFak2/parenleftbiggω\nkvFalnω+kvFa\nω−kvFa−2/parenrightbigg\n.(69)\nBelow we present approximate formulas we apply to\nsolve dispersion equations analytically.\nAtω≫kvaone finds well-known expansion\nω\nkvalnω+kva\nω−kva= 2/parenleftbigg\n1+1\n3k2v2\na\nω2+1\n5k4v4\na\nω4/parenrightbigg\n.(70)\nAtω≪kvawe obtain another well-known expansion\nω\nkvalnω+kva\nω−kva=−πıω\nkva+2ω2\nk2v2a/parenleftbigg\n1+1\n3ω2\nk2v2a/parenrightbigg\n.(71)10\nA. Langmuir wave\nDispersion equation for high frequency regime at clas-\nsic ions\n1 =ω2\nLd\nω2/parenleftbigg\n1+3\n5k2v2\nFd\nω2/parenrightbigg\n+ω2\nLu\nω2/parenleftbigg\n1+3\n5k2v2\nFu\nω2/parenrightbigg\n+ω2\nLi\nω2/parenleftbigg\n1+3k2v2\nTi\nω2/parenrightbigg\n−/radicalbiggπ\n2ıω2\nLi\nk2v2\nTiω\nkvTiexp/parenleftbigg\n−ω2\n2k2v2\nTi/parenrightbigg\n.(72)\nDispersion equation for high frequency regime at clas-\nsic ions\n1 =ω2\nLu\nω2/parenleftbigg\n1+3\n5k2v2\nFu\nω2/parenrightbigg\n+ω2\nLu\nω2/parenleftbigg\n1+3\n5k2v2\nFu\nω2/parenrightbigg\n+ω2\nLi\nω2/parenleftbigg\n1+3k2v2\nTi\nω2/parenrightbigg\n.(73)\nIn general case the frequency of excitations appears in\ncomplex form ω=ωR+ıωIm.\nSpectrums of the Langmuir waves are\nω2\nR= (ω2\nLd+ω2\nLu+ω2\nLi)\n+3\n5k2/parenleftbiggn0u\nn0ev2\nFu+n0d\nn0ev2\nFd+me\nmiv2\nFi/parenrightbigg\n(74)\nfor degenerate ions, and\nω2\nR= (ω2\nLd+ω2\nLu+ω2\nLi)\n+3\n5k2/parenleftbiggn0u\nn0ev2\nFu+n0d\nn0ev2\nFd+5\n3me\nmiv2\nTi/parenrightbigg\n,(75)\nwith\nωIm=1\n2/radicalbiggπ\n2ıω2\nLi\nk2v2\nTiω2\nLe\nkvTiexp/parenleftbigg\n−ω2\nLe\n2k2v2\nTi/parenrightbigg\n(76)\nfor classic electrons.\nIn formulae (74) and (75) sum of partial Langmuir fre-\nquenciesω2\nLd+ω2\nLugives full Langmuir frequency of elec-\ntronsω2\nLe=ω2\nLd+ω2\nLu, sincen0e=n0u+n0d.\nTheLangmuirwaveindegenerateelectrongasdoesnot\nhave collisionless damping ωIm= 0 if ions are degenerate\nas well. In regime of classic (Maxwellian) ions, there\nis small damping of the Langmuir waves in degenerate\nelectron gas.\nIn Ref. [1] we have obtained ω2= (ω2\nLd+ω2\nLu) +\n1\n3k2(n0u\nn0ev2\nFu+n0d\nn0ev2\nFd). We can see that pressure term\nhas different coefficient. This difference appeared dueto application of the Fermi pressure (for spin-up and\nspin-down separately) as an equation of state PFs=\n1\n5(6π2)2\n3n5\n3s¯h2\nm. The Fermi pressure gives the equation of\nstate for equilibrium, whereas we considered perturba-\ntions of an equilibrium state. To cancel the difference\nbetween hydrodynamic and kinetics of small perturba-\ntions in three dimensional plasmas we can write down\nthe following modified equation of state Pms=1\n5mv2\nFs\nn2\n0sn3\ns\n(see Ref. [13] formula 99), where nsis the full concen-\ntration of particles with sspin projection on z direction,\nandn0sis the equilibrium concentration of particles with\nsspin projection. This formula gives the Fermi pressure\nin equilibrium n=n0, and it gives spectrum coinciding\nwith results of kinetic theory of degenerate electron gas.\nLet us represent the real part of the Langmuir spec-\ntruminapproximate,andmoreexplicitform. Wepresent\nthisspectrumintermsofconventionalvariables ωLe,vFe,\nand ∆n, hence we have\nω2=ω2\nLe+3\n10k2v2\nFe/bracketleftbigg/parenleftbigg\n1−∆n\nn0e/parenrightbigg5\n3\n+/parenleftbigg\n1+∆n\nn0e/parenrightbigg5\n3/bracketrightbigg\n.(77)\nThis dependence on ∆ ncorresponds to results obtained\nin Refs. [5], [21].\nB. Spin-electron acoustic waves\nIn this subsection we present one of main results of\nthis paper. We present the kinetic analysis of the spin-\nelectron acoustic waves. At hydrodynamic description\nwe were able to get spectrum of the SEAWs at all wave\nvectors [1], [2]. Here we can get an analytical solution for\nintermediate frequencies described below (see conditions\n(78)and(86)). Nevertheless,thequantumkineticsallows\nus to study the Landau damping of the SEAWs.\nRegime of high spin polarisation allows to perform an-\nalytic consideration of the spin-electron acoustic wave\nspectrum.\n1. SEAW: Classic ions\nPart of spectrum of spin-electron acoustic wave can be\nderived at the following conditions\nkvTi,kvFu≪ω≪kvFd. (78)\nDispersion equation (68) simplifies at conditions (78).\nIts simple form appears as\n1+3ω2\nLd\nk2v2\nFd/parenleftbigg\n1+π\n2ıω\nkvFd−ω2\nk2v2\nFd/parenrightbigg\n=ω2\nLu\nω2/parenleftbigg\n1+3\n5k2v2\nFu\nω2/parenrightbigg\n+ω2\nLi\nω2/parenleftbigg\n1+3k2v2\nTi\nω2/parenrightbigg11\n−/radicalbiggπ\n2ıω2\nLi\nk2v2\nTiω\nkvTiexp/parenleftbigg\n−ω2\n2k2v2\nTi/parenrightbigg\n.(79)\nIn major order equation (79) gives the following spec-\ntrum\nω2\nR0=/parenleftbigg\nω2\nLu+ω2\nLi/parenrightbigg\n1+3ω2\nLd\nk2v2\nFd. (80)\nIncludingtermsofsecondorderweobtainmoregeneral\ndispersion dependence\nω2\nR=ω2\nLu/parenleftbigg\n1+3\n5k2v2\nFu\nω2\nR0/parenrightbigg\n+ω2\nLi/parenleftbigg\n1+3k2v2\nTi\nω2\nR0/parenrightbigg\n1+3ω2\nLd\nk2v2\nFd−3ω2\nLdω2\nR0\nk4v4\nFd.(81)\nWe also obtain imaginary part of the frequency giving\nLandau damping of the SEAW\nωIm=1\n2ωR3π\n2ω2\nLd\nk2v2\nFdωR0\nkvFd+/radicalbigπ\n2ω2\nLi\nk2v2\nTiωR0\nkvTiexp/parenleftbigg\n−ω2\nR0\n2k2v2\nTi/parenrightbigg\n1+3ω2\nLd\nk2v2\nFd−3ω2\nLdω2\nR0\nk4v4\nFd.\n(82)\nIn long-wavelength regime ωLd≫kvFdformula (80)\nsimplifies to\nω2\nR0=1\n3k2v2\nFd(ω2\nLu+ω2\nLi)\nω2\nLd. (83)\nAt intermediate spin polarisation ω2\nLu≫ω2\nLiwe can\nneglect ion contribution in formula (83) and find\nω2\nR0=1\n3n0u\nn0dk2v2\nFd. (84)\n2. SEAW: Degenerate ions\nDispersion equation for the SEAW has form of\n1+3ω2\nLd\nk2v2\nFd/parenleftbigg\n1+π\n2ıω\nkvFd−ω2\nk2v2\nFd/parenrightbigg\n=ω2\nLu\nω2/parenleftbigg\n1+3\n5k2v2\nFu\nω2/parenrightbigg\n+ω2\nLi\nω2/parenleftbigg\n1+3\n5k2v2\nFi\nω2/parenrightbigg\n.(85)\nEquation (85) arises at conditions\nkvFi,kvFu≪ω≪kvFd. (86)\nEquation (85) gives spectrum of the SEAWs\nω2\nR0=/parenleftbigg\nω2\nLu+ω2\nLi/parenrightbigg\n1+3ω2\nLd\nk2v2\nFd. (87)Landau damping of the SEAW is found to be\nωIm=1\n2ωR3π\n2ω2\nLd\nk2v2\nFdωR0\nkvFd\n1+3ω2\nLd\nk2v2\nFd−3ω2\nLdω2\nR0\nk4v4\nFd.(88)\nAtω2\nLd≫k2v2\nFdwe find simplification of formula (88)\nωIm=π\n4ωR0\nkvFdωR0≪ωR0.\nIn opposite limit ω2\nLd≪k2v2\nFdthe denominator\nin formula (88) equals to 1 and we have ωIm=\n3π\n4ω2\nLd\nk2v2\nFdωR0\nkvFdωR0≪ωR0.\nSo, the Landau damping of the SEAWs always smaller\nthan frequency of the wave. Thus we have found that\nthe SEAW is a weakly damped wave.\nIncluding smaller corrections in equation (85) we can\nfind generalisation of formula (87)\nω2\nR=ω2\nLu/parenleftbigg\n1+3\n5k2v2\nFu\nω2\nR0/parenrightbigg\n+ω2\nLi/parenleftbigg\n1+3\n5k2v2\nFi\nω2\nR0/parenrightbigg\n1+3ω2\nLd\nk2v2\nFd−3ω2\nLdω2\nR0\nk4v4\nFd.(89)\nFormula (80) coincides with the result for Maxwellian\nions (80). Hence its long-wavelength limit ( ωLd≫kvFd)\ncoincides with formula (83).\n3. SEAW: Discussion\nFormulae (80) and (87) can be rewritten in terms of\nωLeandvFe. This representation explicitly shows con-\ntribution of mismatch of the Fermi surfaces of spin-up\nand spin-down electrons.\nω2\nR0=1\n2(1−∆n\nn0e)ω2\nLe+ω2\nLi\n1+3\n2ω2\nLe\nk2v2\nFe(1+∆n\nn0e)1\n3. (90)\nAtωLd≫kvFdandω2\nLu≫ω2\nLiformula (90) simplifies\nand we find\nω2\nR0=1\n3(1−∆n\nn0e)\n(1+∆n\nn0e)1\n3·k2v2\nFe. (91)\nIn this subsection we work under condition vFd≫vFu\n=⇒2>1+∆n\nn0e≫1+∆n\nn0e=⇒∆ncomparable with n0.\nThus we conclude that phase velocity of the SEAW given\nby formula (91) considerablyless than the electron Fermi\nvelocityωR0≈1√\n6/radicalig\n1−∆n\nn0ekvFe≪kvFe.\nC. SEAW: Regime of phase velocity near the\nFermi velocity of spin-up electrons vFu\nLet us consider the limit ω→kvFu, which is the low\nfrequency analog of the zeroth sound. In this case we12\ncan present frequency of oscillations as ω=kvFu+δω.\nDispersion equation arises as\n1 =3\n2ω2\nLu\nk2v2\nFu/bracketleftbiggω|ω≈kvFu\nkvFuln/parenleftbigg2kvFu\nδω/parenrightbigg\n−2/bracketrightbigg\n−3\n2ω2\nLd\nk2v2\nFd/parenleftbigg\n2+n1\n3\n0u\nn1\n3\n0dlnvFd−vFu\nvFd+vFu+πı/parenleftbiggn0u\nn0d/parenrightbigg1\n3/parenrightbigg\n.(92)\nCorresponding dispersion dependence arises as\nω=kvFu/braceleftbigg\n1−2vFd+vFu\nvFd−vFu×\n×exp/bracketleftbigg\n−2−2\n3k2v2\nFu\nω2\nLu−2/parenleftbiggn0d\nn0u/parenrightbigg1\n3/bracketrightbigg/bracerightbigg\n.(93)\nD. Ion-acoustic wave\n1. Classic ions\nIon-acoustic waves exist at the following conditions\nkvTi≪ω≪kvFu,kvFd. (94)\nDispersion equation for ion acoustic waves with classic\nions has the following form\n1+3/parenleftbiggω2\nLd\nk2v2\nFd+ω2\nLu\nk2v2\nFu/parenrightbigg\n+3\n2πıω/parenleftbiggω2\nLd\nk3v3\nFd+ω2\nLu\nk3v3\nFu/parenrightbigg\n=ω2\nLi\nω2−/radicalbiggπ\n2ıω2\nLi\nk2v2\nTiω\nkvTiexp/parenleftbigg\n−ω2\n2k2v2\nTi/parenrightbigg\n.(95)\nEquation (95) gives dispersion dependence of ion-\nacoustic waves\nω2\nR=ω2\nLi\n1+3(ω2\nLd\nk2v2\nFd+ω2\nLu\nk2v2\nFu). (96)\nIfωLs≫kvFswe find long-wavelength limit of disper-\nsion dependence (96)\nω2\nR=1\n3k2v2\nFdv2\nFuω2\nLi\nv2\nFdω2\nLu+v2\nFuω2\nLd. (97)\nImaginary part of frequency of ion-acoustic waves at\nMaxwellian ion appears as\nωIm=−3π\n4ω4\nR\nω2\nLi/parenleftbiggω2\nLd\nk3v3\nFd+ω2\nLu\nk3v3\nFu/parenrightbigg\n−1\n2/radicalbiggπ\n2/parenleftbiggωR\nkvTi/parenrightbigg3\nωRexp/parenleftbigg\n−ω2\nR\n2k2v2\nTi/parenrightbigg\n.(98)2. Degenerate ions\nConditions of existence of the ion-acoustic waves in\nplasmas of degenerate electrons and ions are\nkvFi≪ω≪kvFu,kvFd. (99)\nIn this regime we can obtain the dispersion equation\n1+3/parenleftbiggω2\nLd\nk2v2\nFd+ω2\nLu\nk2v2\nFu/parenrightbigg\n+3\n2πıω/parenleftbiggω2\nLd\nk3v3\nFd+ω2\nLu\nk3v3\nFu/parenrightbigg\n=ω2\nLi\nω2.(100)\nEquation (100) gives the following solution in leading\norder on small parameters\nω2\nR=ω2\nLi\n1+3(ω2\nLd\nk2v2\nFd+ω2\nLu\nk2v2\nFu). (101)\nAtωLs≫kvFs(long-wavelength regime) we find sim-\nplification of solution (101) as follows\nω2\nR=1\n3k2v2\nFdv2\nFuω2\nLi\nv2\nFdω2\nLu+v2\nFuω2\nLd. (102)\nIncluding smaller corrections to the spectrum (101)\nfind decrement of the Landau damping for ion-acoustic\nwaves\nωIm=−3π\n4ω4\nR\nω2\nLi/parenleftbiggω2\nLd\nk3v3\nFd+ω2\nLu\nk3v3\nFu/parenrightbigg\n.(103)\n3. Ion acoustic waves: Discussion\nAtkvFs≫ωLsformula (101) gives well-known limit\nω2\nR=ω2\nLi.\nNext let us consider formula (102), which has been\nobtainedfromformula(101)inregimeoflong-wavelength\nωLs≫kvFs, in more explicit form\nω2\nR=2\n3me\nmik2v2\nFe1\n(1−∆n\nn0e)1\n3+(1+∆n\nn0e)1\n3.(104)\nIf magnetic field is small than ∆ n/n0e≪1. In this\nregime we obtain\nω2\nR=1\n3me\nmik2v2\nFe/parenleftbigg\n1+1\n9∆n2\nn2\n0e/parenrightbigg\n.(105)\nIn the long-wavelength limit ωLs≫kvFsthe decre-\nment of Landau damping for ion acoustic wave (103) can\nbe rewritten as\nωIm=−π\n12me\nmikvFe1\n[(1−∆n\nn0e)1\n3+(1+∆n\nn0e)1\n3]2.(106)\nWe can find simplification of formula (106) for regime\nof small magnetic field\nωIm=−π\n48me\nmikvFe/parenleftbigg\n1+2\n9∆n2\nn2\n0e/parenrightbigg\n.(107)13\nE. Zeroth sound\nThe zeroth sound is a well-known high frequency solu-\ntion of the dispersion equation for degenerate ions. Usu-\nally one obtains it for an equal occupation of spin-up and\nspin-down states. Now we consider the zeroth sound in\nregime of high difference in occupation of spin-up and\nspin-down states by degenerate electrons. In this case\nω∼kvFd≫kvFu≫kvFi.\nThe zero-sound (Ref on Silin) appears at ω→kvFd\nω=kvFd+δω, (108)\nwhereδω≪kvFd\nIn this regime dispersion equation takes the following\nform\n1 =3\n2ω2\nLd\nk2v2\nFd/bracketleftbiggω|ω≈kvFd\nkvFdln/parenleftbigg2kvFd\nδω/parenrightbigg\n−2/bracketrightbigg\n.(109)\nEquation (109) gives the following solution\nω0=kvFd+δω=kvFd/bracketleftbigg\n1+2exp/parenleftbigg\n−2−2\n3k2v2\nFd\nω2\nLd/parenrightbigg/bracketrightbigg\n.\n(110)\nLet us present here the Fermi velocity of spin-down\nelectrons via the conventional Fermi velocity\nvFd=vFe/parenleftbigg\n1+∆n\nn0e/parenrightbigg1\n3\n, (111)\nwithvfe= (3π2n0e)1\n3¯h/m.\nMore explicit form of the zeroth sound spectrum (110)\nappears at substitution (111) in formula (110). Hence we\nhave\nω0=kvFe/parenleftbigg\n1+∆n\nn0e/parenrightbigg1\n3\n×\n×/bracketleftbigg\n1+2exp/parenleftbigg\n−2−4k2v2\nFe\n3ω2\nLd/parenleftbigg\n1+∆n\nn0e/parenrightbigg−1\n3/parenrightbigg/bracketrightbigg\n.(112)\nWe can also consider regime of small difference in oc-\ncupation numbers as well. In this case kvFd∼kvFu, and\nω∼kvFd>kvFu≫kvFi.\nω=kvFd+δω, (113)and\nω−kvFu= ∆+δω, (114)\nwhere ∆ = k(vFd−vFu).\nIn this regime the dispersion appears as follows\nω=kvFd/braceleftigg\n1+2exp/bracketleftbigg\n−2−2\n3k2v2\nFd\nω2\nLd\n−n1\n3\n0u\nn1\n3\n0d/parenleftbigg\n2+lnn1\n3\n0d−n1\n3\n0u\nn1\n3\n0d+n1\n3\n0u/parenrightbigg/bracketrightbigg/bracerightigg\n.(115)\nVII. CONCLUSIONS\nMethod of separate spin evolution quantum kinetics,\nwhich separately describes spin-up and spin-down elec-\ntrons, has been developed. This method has been ap-\nplied to rederivation of spectrum of the Langmuir waves\nand the SEAWs obtained earlier in terms of SSE-QHD.\nRegimeofwavepropagationparallelto the externalmag-\nnetic field has been considered at calculations of spec-\ntrum of magnetised spin-1/2 quantum plasmas. Contri-\nbution of ions dynamics in dispersion of SEAW has been\nconsidered. Calculation of the Landau damping of the\nSEAW has been performed. Influence of separated spin\nevolution on real and imaginary parts of spectrums of\nion-acoustic waves and zeroth sound have been found.\nCalculation of the Landau damping of the SEAWs has\ndemonstratedthattheSEAWsareweaklydampedwaves.\nThus we have shown that hydrodynamic calculations of\nthe real part of spectrum of the SEAWs were reasonable.\nWe have presented fundamental applications of the\nseparates spin evolution quantum kinetics method. Fur-\nthermore, this method, along with the developed ear-\nlier SSE-QHD, creates strong background for research of\nspin-1/2 quantum plasmas. It open possibilities for more\ndetailed analysis of various effects in quantum plasmas\nthen usual spin-1/2 QHD or similar quantum kinetics.\nAcknowledgments\nThe author thanks Professor L. S. Kuz’menkov for\nfruitful discussions.\n[1] P. A. Andreev, arXiv:1405.0719.\n[2] P. A. Andreev, L. S. Kuz’menkov, arXiv:1406.6252.\n[3] P. A. Andreev, L. S. Kuz’menkov, arXiv:1408.3662.\n[4] L. S. Kuz’menkov, S. G. Maksimov, and V. V. Fedoseev,\nTheoretical and Mathematical Physics, 126110 (2001).\n[5] L. S. Kuz’menkov, S. G. Maksimov, and V. V. Fedoseev,Theoretical and Mathematical Physics, 126212 (2001).\n[6] P. A. Andreev and L. S. Kuz’menkov, Russian Phys.\nJour.50, 1251 (2007).\n[7] P. A. Andreev, L. S. Kuz’menkov, Physics of Atomic Nu-\nclei71, N.10, 1724 (2008).\n[8] P. A. Andreev, L. S. Kuzmenkov, M. I. Trukhanova,14\nPhys. Rev. B 84, 245401 (2011).\n[9] P. A. Andreev, L. S. Kuzmenkov, arXiv:1210.1090.\n[10] T. Takabayasi, Prog. Theor. Phys. 14, 283 (1955).\n[11] D. A. Uzdensky and S. Rightley, Reports on Progress in\nPhysics, 77, Issue 3, 036902 (2014).\n[12] P. K. Shukla, B. Eliasson, Rev. Mod. Phys. 83, 885\n(2011).\n[13] P. A. Andreev, L. S. Kuz’menkov, arXiv:1407.7770.\n[14] L. D. Landau, E. M. Lifshitz, Quantum Mechanics: Non-\nRelativistic Theory . Vol. 3 (3rd ed.). Pergamon Press,\n(1977).\n[15] M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner,Physics Reports, 106, 121 (1984).\n[16] S. Weinberg, Gravitation and Cosmology (John Wiley\nand Sons, Inc., New York, 1972).\n[17] Yu. L. Klimontovich, Statistical Physics [in Russian],\nNauka, Moscow (1982); English transl., Harwood, New\nYork (1986).\n[18] P. A. Andreev, arXiv:1404.4899.\n[19] P. A. Andreev, arXiv:1308.3715.\n[20] P. A. Andreev, L. S. Kuz’menkov, Int. J. Mod. Phys. B\n261250186 (2012).\n[21] P. A. Andreev, Annals of Physics, 350, 198 (2014)."    },    {        "title": "1607.02406v1.Competition_between_Bose_Einstein_Condensation_and_spin_dynamics.pdf",        "content": "arXiv:1607.02406v1  [cond-mat.quant-gas]  8 Jul 2016Competition between Bose Einstein Condensation and spin dy namics\nB. Naylor1,2, M. Brewczyk3, M. Gajda4, O. Gorceix1,2, E. Mar´ echal1,2, L. Vernac1,2, B. Laburthe-Tolra1,2\n1 Universit´ e Paris 13, Sorbonne Paris Cit´ e, Laboratoire d e Physique des Lasers,\nF-93430 Villetaneuse, France; 2 CNRS, UMR 7538,\nLPL, F-93430 Villetaneuse, France; 3 Wydzia/suppress l Fizyki,\nUniwersytet w Bia/suppress lymstoku, ul. K. Cio/suppress lkowskiego 1L, 15-2 45 Bia/suppress lystok,\nPoland; 4 Institute of Physics, Polish Academy of Sciences,\nAleja Lotnik´ ow 32/46, 02-0668 Warszawa, Poland\n(Dated: July 11, 2016)\nWe study the impact of spin-exchange collisions on the dynam ics of Bose-Einstein condensation,\nby rapidly cooling a chromium multi-component Bose gas. Des pite relatively strong spin-dependent\ninteractions, the critical temperature for Bose-Einstein condensation is reached before the spin-\ndegrees of freedom fully thermalize. The increase in densit y due to Bose-Einstein condensation\nthen triggers spin dynamics, hampering the formation of con densates in spin excited states. Small\nmetastable spinor condensates are nevertheless produced, and manifest strong spin fluctuations.\nPACS numbers: 03.75.Mn , 05.30.Jp, 67.85.-d, 05.70.Ln\nDilute quantum gases are especially suited for the in-\nvestigationofnon-equilibriumdynamicsinclosedoropen\nquantum systems, for example associated to the physics\nof thermalization [1], prethermalization [2], or localiza-\ntion [3]. In particular, they provide a platform to study\nthe kinetics of Bose-Einstein condensation. Soon af-\nter the first Bose-Einstein condensates (BECs) were ob-\ntained, it was for example possible to investigate how\nthe BEC nucleates [4, 5]. More recently, experiments\nperforming a temperature quench below the superfluid\ntransitioninvestigatedthe dynamicsofspontaneoussym-\nmetry breaking [6] and revealed the production of long-\nlived topological defects [7]. The aim of this work is to\nextend the dynamical studies of Bose-Einstein condensa-\ntion to the case of a multi-component Bose gas, in or-\nder to establish the mechanisms to reach both superfluid\nand magnetic orders. While these orders are intrinsically\nconnected due to Bose stimulation [8, 9] (which contrasts\nwith the case of Fermi fluids [10]), it was predicted that\nstrong spin-dependent interactions induce spin ordering\nat a finite temperature abovethe BEC transition [11].\nWe find that the dynamics of Bose-Einstein conden-\nsation is drastically modified due to spin-changing colli-\nsions arising from relatively strong spin-dependent inter-\nactions. Thermalization of the spin degrees of freedom\nis influenced by the occurrence of BEC, and in turns in-\nfluences which multi-component BECs can be produced.\nOur experiment also demonstrates the difficulty to ther-\nmalize the spin degrees of freedom, which has a strong\nimpact on the spin distribution ofthe BECs, and ontheir\nlifetime. This is of particular relevance for large spin\natoms, and most notably for strongly magnetic atoms\nsuch as Cr [12, 13], Er [14], and Dy [15] for which dipolar\nrelaxation strongly limits the lifetime of multicomponent\ngases [16].\nWe induce fast evaporative cooling of a multi-\ncomponent s=3 chromium thermal cloud by lowering the\ndepth of a spin-insensitive optical dipole trap (ODT)\nFIG. 1: a) Experimental sequence, showing the reduction in\nthe ODT intensity in a duration tS. An absorption image is\ntaken after a time tand Stern-Gerlach separation. b) Sim-\nple cartoon of the evolution of the momentum distributions\n(p(k)) of atoms in the three lowest spin excited states, illus-\ntrating the difficulty of achieving BEC (peak on top of the\nbroad thermal kdistribution) in spin excited states due to\nspin dynamics. Absorption pictures showing: (c) a BEC in\nms=−3 and a thermal gas in other spin states for a gas\ninitially prepared with magnetization M=−2.5±0.25; d) a\nsmall multi-component BEC for M=−2.00±0.25.\n(see Fig 1a). When the gas is only slightly depolarized,\nthe thermal gas of the most populated, lowest energy,\nms=−3 state rapidly saturates (i.e. reaches the maxi-\nmal number of atoms in motional excited states allowed\nby Bose statistics [17]) and a BEC is produced for this\nspin state. Saturation is also reached for the thermal\ngas in the second-to-lowest energy state ms=−2. How-\never, surprisingly, this state fails to condense and the\nBEC remains fully magnetized. In contrast, when the\nexperiment is performed with an initially more depolar-\nized thermal gas, spinor (i.e. multi-component) conden-\nsates are obtained, although they remain very small (see2\nFig. 1) and show strong spin fluctuations. Comparison\nwith numerical simulations based on the classical field\napproximation (CFA) [18] reveals that the difficulty to\nobtain a multi-component BEC is due to spin exchange\ncollisions, which rapidly empty the condensates in spin\nexcited states by populating spin states for which the\nthermal gas is not yet saturated. There is an intrigu-\ning interplay between condensation and spin dynamics,\nas the large increase in density associated to BEC trig-\ngers fast spin dynamics which in turn tends to deplete\nthe BEC in spin excited states. The observed spin fluc-\ntuations in the BEC are ascribed to a combined effect of\nphasefluctuations due to symmetry-breakingat the BEC\ntransition, and spin dynamics.\nTo prepare an incoherent spin mixture of thermal\ngases, we start from a thermal gas of 2 .104 52Cr atoms,\natT= 1.1×Tc= 440±20nK, polarized in the Zeeman\nstatems=−3. We adiabatically reduce the magnetic\nfieldBso that the Zeeman energy is of the same or-\nder as the thermal kinetic energy. Depolarization of the\ncloud is driven by magnetization-changing collisions as-\nsociated to dipole-dipole interactions [19, 20]. We obtain\na gas of longitudinal magnetization M=−2.50±0.25,\nwithM≡/summationtexts\ni=−sini(niis the relative population in\nZeeman state ms=i). We then reduce the trap depth\nby applying an approximately linear ramp to the ODT\nlaser intensity in a time tS. This results in fast forced\nevaporative cooling of all Zeeman states (which we re-\nfer to as ”shock cooling”, see Fig. 1a). We study spin\ndynamics and condensation dynamics by measuring both\nthe spinandmomentum distributionsasafunction ofthe\ntimetafter the beginning of the evaporation ramp. This\nmeasurement is performed by switching off all trapping\nlights, and applying an average magnetic field gradient\nof 3.5 G.cm−1during a 6 ms time of flight to perform a\nStern-Gerlach separation of the free-falling atoms.\nFig.1c shows a typical absorption picture. It reveals\na BEC in ms=−3, and a thermal gas in spin ex-\ncited states. We extract the number of thermal and con-\ndensed atoms of each spin state through bi-modal fits\naccounting for Bose statistics. We plot in Fig.2 the ther-\nmal atom numbers as well as the condensate fractions in\nms= (−3,−2) as a function of time tfor a shock cooling\ntimetS= 500 ms. We found similar results for tS= 250\nms andtS= 1 s.\nThe grayarea in Fig.2 highlights a relatively long cool-\ning time during which the ms=−3 andms=−2 gases\nhold approximately the same number of thermal atoms,\nand there is a BEC in the lowest state ms=−3 but\nnot inms=−2. This phenomenon is surprising, be-\ncause it shows that the ms=−2 component fails to un-\ndergo Bose-Einstein condensation although its thermal\ngas is saturated. Indeed, ms=−2 andms=−3 thermal\natoms have the same measured mechanical temperature\n(withinour5%experimentaluncertainty),experiencethe\nsame trapping potential, and both interact through theFIG. 2: a) Number of thermal atoms in ms=−3 (black\ndiamonds) and ms=−2 (red disks) as a function of time\ntfor a shock cooling time tS= 500 ms. b) Corresponding\ncondensate fractions in ms=−3 (black diamonds) and ms=\n−2(red disks). The shaded region highlights whenboth ms=\n−3 andms=−2 thermal clouds are saturated, but only\nms=−3 atoms condense.\nS= 6 molecular potential with the existing ms=−3\nBEC. Therefore the ms=−2 cloud has the same ther-\nmodynamical properties than the ms=−3 thermal gas,\nand like this latter spin component, should condense for\nfurthercooling[17]. However,BECdoesnotoccurin this\nstate, until t≤700 ms. Only for t≥700ms do we distin-\nguish a very small BEC also in ms=−2. This demon-\nstrates that a BEC in ms=−2 hardly forms, although\nthe thermal gas is saturated and cooling proceeds.\nTo interpret these observations, we stress that\nmagnetization-changing collisions occur on a larger time\nscale than shock cooling dynamics and can be neglected,\ncontrarily to [19]. Here, spin dynamics is almost en-\ntirely controlled by spin exchange interactions at con-\nstant magnetization driven by spin dependent contact\ninteractions [9]. A key point is that for an incoherent\nmixture the spin dynamics rate γk,l\ni,jfor the spin chang-\ning collision ( ms=i,ms=j)→(ms=k,ms=l) is\nset by the density of the cloud through γk,l\ni,j=nσk,l\ni,jv\nwithnthe atomic density, vthe average relative atomic\nvelocity and σk,l\ni,jthe relevant cross section within Born\napproximation [21]. This rate is extremely sensitive to\nthe presence of a BEC (which enhances n). Therefore,3\nthe emergence of a BEC in a spin-excited state should\ntrigger faster spin dynamics. In addition to these dis-\nsipative spin-exchange processes, BEC can also trigger\ncoherent spin oscillations due to forward scattering, with\natypicalrateΓk,l\ni,j=4π¯h\nmn/summationtext\nSaS/angbracketlefti,j|S/angbracketright/angbracketleftS|k,l/angbracketrightwherethe\nsum is on even molecular potentials S, with associated\nscattering length aS. Our interpretation for the absence\nof a BEC in the state ms=−2 is thus that a large BEC\ncannot form in this state because fast spin-exchange pro-\ncesses (−2,−2)→(−1,−3) deplete the BEC as soon as\nit is produced. Thus spin dynamics and condensation\ndynamics are strongly intertwined.\nTo check this interpretation, we have performed nu-\nmerical simulations using the Gross-Pitaevskii (GP)\nequationandthe classicalfield approximationto describe\nthermal states. According to CFA, the GP equation de-\ntermines the evolution of the classical field which is a\ncomplex function carrying the information on both the\ncondensed and thermal atoms [18, 22, 23]. The initial\nclassical field corresponds to 13 .103Cr atoms at the crit-\nical temperature of about 400nK and with the experi-\nmental Zeeman distribution. To describe such a sample\nwefollowtheprescriptiongivenin[23]. Evaporativecool-\ning is mimicked by adding a purely imaginary potential\nto the GP equation at the edge of the numerical grid.\nOur simulations confirm the existence of a saturated\nms=−2gasandtheabsenceofacondensateinthisstate\n(see Fig.3). To evaluate the impact of spin-exchange\nprocesses on the dynamics of condensation, we have re-\nproduced these simulations assuming a4=a6. In this\ncase, the rates associated to spin-exchange processes\n(−2,−2)→(−1,−3),γ−1,−3\n−2,−2and Γ−1,−3\n−2,−2, which respec-\ntively scale as ( a6−a4)2and (a6−a4), both vanish. As\nshown in Fig. 3, a BEC then forms in the spin excited\nstatems=−2. This confirms the crucial role of spin-\ndependent interactions in the dynamics of Bose-Einstein\ncondensation.\nIt is interesting to face our observations with the\naccepted scenario for the thermodynamics of non-\ninteracting multi-component Bose gases at fixed magne-\ntization[24]. Inthis picture, aBECpolarizedin the most\npopulated state forms below a first critical temperature;\nall the other thermal spin states saturate simultaneously\nandcondensebelowasecondcriticaltemperature[24]. In\nour situation, our observations indicate that the external\ndegrees of freedom have reached an equilibrium, at an\neffective temperature which we find identical for all spin\ncomponents. However, although the thermal clouds of\nthe two lowest spin components are saturated, the other\nthermal clouds are not saturated. This is in profound\ncontradiction with the prediction of Bose thermodynam-\nics, and shows that the spin degrees of freedom in our\nexperiment remain out of equilibrium.\nThis lack of thermal equilibrium for the spin degrees\nof freedom results from the fact that spin exchange pro-\ncessesfor the thermal gas are slow in regards of con-FIG. 3: Numerical results. Evolution of the condensate frac -\ntions for different values of a4(blue diamonds: ms=−3; red\ncircles:ms=−2). Filled markers correspond to the experi-\nmental case: a4= 64aBanda6= 102.5aB[16] where aBis\nthe Bohr radius. Empty markers correspond to simulations\nwherea4was set equal to a6to suppress spin dynamics. A\nsignificative BEC fraction in ms=−2 is then obtained.\ndensation dynamics. For example, the rate of the dom-\ninant spin exchange term, averaged over density, γ1=\nn0σ−1,−3\n−2,−2v\n2√\n2, withn0the peak atomic density, is typi-\ncally 3s−1for a thermal gas at TC. A much longer\ntimescale would therefore be necessary in order to reach\nspin equilibrium. This rate is slow compared to typical\nthermalization rates of the mechanical degrees of free-\ndom e.g. γ2=n0σ−2,−2\n−2,−2v\n2√\n2≈40 s−1.γ2>> γ 1in-\nsures that the mechanical degrees of freedom thermal-\nize faster than the spin degree of freedom, and that a\nsmallms=−2 BEC can in principle be formed. How-\never, once the ms=−2 BEC is formed, the rates asso-\nciated to ( −2,−2)→(−3,−1) collisions rise to typically\nγ1,BEC≈15 s−1and Γ−1,−3\n−2,−2≈100 s−1(for 500 atoms in\nthe condensate). Spin exchange collisions then deplete\nthems=−2 BEC as fast as it is created and a multi-\ncomponent BEC cannot be sustained due to the lack of\nsaturation of the ms=−1 thermal gas.\nUnder our experimental conditions, non-saturated\nspin-excited states thus act as a reservoir into which\npopulation may be dumped, thus preventing BEC but\nin the stretched state, the only collisionally stable one.\nThe situation bears some analogies to the condensation\nof magnons [25, 26] and polaritons, where BEC is ob-\ntained in the lowest momentum state by collisions of\nhigher states in the lower polariton branch [27]. Like\nfor polaritons, it is likely that spin-exchange interactions\nare increased by Bose-stimulation due to the pre-existing\nms=−3 condensate.\nToproducemulti-component condensatesduring evap-\noration, we performed a second series of shock cool-\ning experiments, with a lower gas magnetization M=\n−2.00±0.25(wheretheuncertaintyisassociatedtodetec-\ntionnoise). Theinitial ms=−3thermalgasisnowdepo-4\n0.10\n0.08\n0.06\n0.04\n0.02\n0.00\n500 400 300 200 100 0a) c)\nCondensate□fractionm =□-3□□□-2□□□-1□□□0s\nt□□(ms)S\nb)\nFIG. 4: a) Absorption images after shock cooling experiment s\nperformed with tS= 50 ms and an initial magnetization\nM=−2±0.25. A small BEC is present in the three lowest\nspin states i.e. ms= -3,-2, and -1. The different images il-\nlustrate the fluctuations of magnetization of the condensat e\nfraction. b) Numerical results after evaporation, for thre e ini-\ntial relative sets of phases. c) Total condensate fraction o f the\nmulti spin component gas for t=tSms andM=−2±0.25 as\na function of tS. We observe small multi-component conden-\nsates in the three lowest energy states. The solid line guide s\nthe eye.\nlarized using a radio-frequency pulse. After decoherence\nofthespincomponents,thisleadstoathermalincoherent\nmixture with initial fractional population in ms=−3,\n−2,−1 and 0 states approximately (31%,40%,21%,6%)\nwitharelativeuncertaintyof10%. Whenshockcoolingis\nperformed fast compared to γ1,BEC, we observe the pro-\nduction of very small multi-component condensates in all\nthree lowest energy states (as illustrated in Fig 1 and 4).\nOur numerical simulations show that spin-dynamics has\nagain a very profound influence on the dynamics of con-\ndensation. In practice, spin excited states with ms>0\nare not saturated. Therefore, spin dynamics tends to\npopulate these non saturated states and empty the con-\ndensates, which thus remain small and short-lived.\nAn important observation is that the spin distribu-\ntionoftheobtainedmulti-componentBECsshowsstrong\nfluctuations (see Fig.4a) compared to the thermal frac-\ntion. Weinterpretthisfeatureinthefollowingway. Bose-\ncondensation of the different spin-excited states intro-\nduce a spontaneous symmetry breaking as the phase of\neach condensate is chosen randomly. This spontaneous\nsymmetry breaking, already observed in [25], can also\nbe interpreted as the production of fragmented BECs\n[28, 29]. As spin-dynamics is sensitive to the relative\nphase between the condensates in the various spin states\n[9, 30], we propose that the observed spin fluctuations\nresult from a combined effect of spin dynamics and of\nthe spontaneous symmetry breaking.\nWe performed numerical simulations to test this sce-\nnario. As the CFA does not provide a direct way to pro-\nvide symmetry breaking at the BEC transition, we chose\nto apply random relative phases to the wave-functions\ndescribing the thermal atoms in different spin compo-nents before condensation. This provides an empirical\nway to simulate symmetry breaking. We performed a se-\nries of numerical simulations for different sets of relative\nphases between the Zeeman components. We then ob-\ntained small condensates with fluctuating magnetization\n(see Fig.4b). Furthermore, we also observe that spin and\ncondensation dynamics are also significantly modified by\nthe applied random phases. Due to large computational\ntime for each run, a systematic study of BEC magnetiza-\ntion as a function of initial phases has not yet been per-\nformed and remains to be thoroughly investigated. How-\never, while the magnetization fluctuations obtained in\nthe numerical simulations are typically five time smaller\nthan the experimental measurements, these preliminary\nresultsthus supportthescenariothatthe combinedeffect\nof spontaneous symmetry breaking and spin dynamics\nlead to the observed spin fluctuations.\nAs a conclusion, our study reveals a strong interplay\nbetween Bose condensation and spin dynamics, which is\nof particular relevance when spin-dependent and spin-\nindependentinteractionstakeplaceonasimilartimescale\n(in contrast to previous studies with alkali atoms, see\n[31]). This interplay can for example result in a delay in\nobtaining a BEC in spin-excited states or alternatively\nto the production of weak metastable spinor gases which\ndecay due to spin-exchange collisions. Our results also\nshow that the difficulty to fully thermalize the spin de-\ngrees of freedom is a prominent effect to be taken into\naccount for very large spin systems (such as Dy [15] and\nEr[14]), whereallspin states must be saturatedfora sta-\nble multi-component BEC to be produced. Finally, we\npoint out that when a multi component BEC is dynami-\ncally produced, spontaneous symmetry breaking leads to\nindependent phases within the BEC components which\ntriggers spin fluctuation.\nThis work was supported by Conseil R´ egional d’Ile-\nde-France under DIM Nano-K/IFRAF, Minist` ere de\nl’Enseignement Sup´ erieur et de la Recherche within\nCPER Contract, Universit´ e Sorbonne Paris Cit´ e\n(USPC), and by the Indo-French Centre for the Promo-\ntion of Advanced Research-CEFIPRA. 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Fern´ andez-Rossier(1)\n(1)Departamento de F´ ısica Aplicada, Universidad de Alica nte, San Vicente del Raspeig, 03690 Spain\n(2) Institut Neel, CNRS, Grenoble, Avenue des Martyr, Franc e\n(Dated: November 12, 2021)\nThe spin dynamics of a single Mn atom in a laser driven CdTe qua ntum dot is addressed theoreti-\ncally. Recentexperimental results1–3showthat it is possible toinduce Mnspin polarization bymea ns\nof circularly polarized optical pumping. Pumping is made po ssible by the faster Mn spin relaxation\nin the presence of the exciton. Here we discuss different Mn sp in relaxation mechanisms. First,\nMn-phonon coupling, which is enhanced in the presence of the exciton. Second, phonon-induced\nhole spin relaxation combined with carrier-Mn spin flip coup ling and photon emission results in Mn\nspin relaxation. We model the Mn spin dynamics under the influ ence of a pumping laser that injects\nexcitons into the dot, taking into account exciton-Mn excha nge and phonon induced spin relaxation\nof both Mn and holes. Our simulations account for the optical ly induced Mn spin pumping.\nPACS numbers:\nI. INTRODUCTION\nThe tremendous progress in the miniaturization of\nelectronic devices has reached the point that makes it\ncrucial to address the effect of a single dopant in a de-\nvice and motivates the study of a single dopant spin to\nstore digital information4. The manipulation of a single\natom spin in a solid state environment has been demon-\nstrated using several approaches, like scanning tunneling\nmicroscope on magnetic adatoms5,6, or optical probing\nof NV centers in diamond7and single magnetic atoms\nin semiconductor quantum dots, the topic of this paper.\nSingle quantumdots doped with asingleMn atom canbe\nprobed by means of single exciton spectroscopy in pho-\ntoluminescence (PL) experiments. This has been done\nboth in II-VI1–3,8–15and III-V16,17materials. In the\ncase of single Mn doped CdTe dots, information about\nthe quantum spin state of a single Mn atom is extracted\nfrom the single exciton quantum dot photoluminescence\ndue to the one on one relation between photon energy\nand polarization and the electronic spin state of the Mn\natom18–26. This has made it possible to measure the\nspin relaxation time of a single Mn atom in a quantum\ndot under optical excitation, using photon autocorrela-\ntion measurements14, and to realize the optical initial-\nization and readout of the spin of the Mn atom1–3.\nThe observation of Mn spin orientation under quasi-\nresonant optical pumping1–3can be accounted for if the\nMn spin relaxation time is shorter in the presence of a\nquantum dot exciton3,27–29. In that situation, resonant\nexcitation of an optical transition associated to a given\nMn spin projection results in the depletion of the laser\ndriven Mn spin state, via Mn spin relaxation in the pres-\nence of the exciton. Whereas theoretical understanding\nof the exchange couplings between electrons, holes and\nMn spin in quantum dots permits to account for the ob-\nserved PL spectra8,10,11,18, a complete understanding of\nthe spin dynamics under the combined action of laser\npumping, incoherent spin relaxation and coherent spin-\nflips is still missing. In this paper wemakeprogressalongthis direction on two counts. First, we discuss different\nMn spin relaxation mechanism, taking fully into account\nthe interplay between incoherent dynamics due to the\ncoupling to a reservoir and the coherent spin flips asso-\nciated to exciton-Mn exchange in the quantum dot. Our\ncalculations show that the most efficient Mn spin relax-\nation channel, in the presence of the exciton, arises from\na combination of phonon-induced hole spin relaxation,\nwhichturnsthebrightexcitonintoadark, followedbyre-\ncombinationtothegroundenabledbydark-brightmixing\ndue to Mn-carrier spin flip exchange. Thus, we provide a\nquantitative basis to a recently proposed scenario29. Sec-\nond,wemodeltheMnspindynamicswitharateequation\nfor the Mn spin and the Mn plus exciton spin states that\nincludes the spin relaxation rates between the few-body\nstates calculated from microscopic theory.\nOurtheorypermitstomodeltheexperimentalobserva-\ntions and, importantly, it identifies the light-hole heavy-\nhole mixing as a crucial parameter that determines not\nonly the PL lineshape10,11,18but also the amplitude of\nseveral spin-relaxation mechanisms at play in this sys-\ntem.\nThe rest of this manuscript is organized as follows.\nIn section II we review the Hamiltonian for a single\nMn spin interacting with a single exciton in a quantum\ndot. The anisotropic Mn-hole coupling is derived from\na simplified18,30single-particle description of the lowest\nenergy quantum dot hole states which affords analyti-\ncal expressions for the critical parameter in the theory,\nthe light-hole heavy hole (LH-HH) mixing30. The de-\npendence of the properties of the Mn-exciton states on\nthe LH-HH mixing are discussed. In section (III) we dis-\ncuss the Mn spin relaxation due to Mn-phonon coupling,\nboth with and without an exciton in the quantum dot.\nWhereas this mechanism is probably dominant for the\nMn spin relaxation in the optical ground state, it is not\nsufficient to account for the rapid Mn spin relaxation in\nthe presence of the exciton. This leads us to consider\nother spin relaxation mechanisms. In section (IV) we de-\nscribethe spin relaxationofholesdue to theircouplingto2\nacoustic phonons, using a Bir-Pikus Hamiltonian. Using\nthe simplified description of hole states, we obtain ana-\nlytical results for the hole spin lifetime in a non-magnetic\ndot, which are in agreement with previous work using a\nmoresophisticateddescriptionofsinglehole states31. We\nthen compute the lifetime of the exciton-Mn states due\nto hole spin relaxation. In section (VI) we present our\nsimulations of the optical pumping process, using rate\nequations for the exciton-Mn quantum states, including\nthe laser pumping, the spontaneous photon emission, the\nMn and hole spin relaxation due to phonons. Our simu-\nlations account for the optical initialization and readout\nobserved experimentally.\nII. EXCITON- MnHAMILTONIAN\nIn this section we describe a minimal Hamiltonian\nmodel that can accounts for the PL spectra of single Mn\ndoped CdTe quantum dots. For that matter we need to\nconsider both the Mn spin in the unexcited crystal and\ntheMn spininteractingwithaquantumdotexciton. The\npeaks in the PL spectra are associated to the energy dif-\nferences between the states of the dot with and without\nthe exciton.\nA. Mn spin Hamiltonian\nMn is a substitutional impurity in the Cd site of CdTe.\nThus, it hasan oxidationstateof2+, sothat the 5 delec-\ntrons have spin S=5\n2, resulting in a sextuplet33whose\ndegeneracy is lifted by the interplay of spin orbit and the\ncrystal field. In an unstrained CdTe, the crystal field\nhas cubic symmetry which should result in a magnetic\nanisotropy Hamiltonian without quadratic terms. Elec-\ntron paramagnetic resonance (EPR) in CdTe strained\nepilayers34showthat Mn hasan uniaxialtermin the spin\nHamiltonian. In a quantum dot there could be some in-\nplane anisotropy as well, which lead us to consider the\nfollowing Hamiltonian:\nH0=DM2\nz+E(M2\nx−M2\ny)+gµB/vectorB·/vectorM(1)\nwhereMaare theS=5\n2spin operators of the electronic\nspin of the Mn. The eigenstates of this Hamiltonian are\ndenoted by φm\nH0|φm∝angb∇acket∇ight=Em|φm∝angb∇acket∇ight=Em/summationdisplay\nMzφm(Mz)|Mz∝angb∇acket∇ight(2)\nwhere|Mz∝angb∇acket∇ightare the eigenstates of Mz. In this paper\nwe neglect the hyperfine coupling to the I=5\n2nu-\nclear spin, which could affect the decay of the electronic\nspin coherence1. The magnetic anisotropy parameters,\nEandDcan not be inferred from PL experiments,\nwhich are only sensitive to the Mn-exciton coupling.\nEPR experiments34in strained layer could be fit with\nD= 12µeV,E= 0 andg= 2.0. Thus, the ground stateshould have Mz=±1\n2, split from the first excited state\nby 2D. At 4 Kelvin and zero magnetic field, we expect\nall the six spin levels to be almost equally populated. We\nrefer to these six states as the ground state manifold , in\ncontrast to the excited state manifold, which we describe\nwith 24 states corresponding to 4 possible quantum dot\nexciton states and the 6 Mn spin states.\nB. Single particle states of the quantum dot\nWe describe the confined states of the quantum dot\nwith a simple effective mass model. In the case of the\nconductionband electrons, we neglectspin orbit coupling\nandwe onlyconsiderthe lowestenergyorbital, with wave\nfunctionψ0(/vector r), which can accommodate 1 electron with\ntwo spin orientations.\nIn the case of holes, spin orbit coupling lifts the sixfold\ndegeneracy of the top of the valence band into a J=3\n2\nquartetanda J=1\n2doubletwhich,inCdTe,ismorethan\n0.8eV below in energy. Confinement and strain lift the\nfourfold degeneracy of the J=3\n2hole states, giving rise\nto a mostly Jz=±3\n2heavy hole doublet and a almost\nJz=±1\n2light hole doublet. Importantly, it is crucial\nto include LH-HH mixing to describe the experimental\nobservation.\n1. Effect of confinement\nThe top of the valence band states are described in\nthe kp approximation with the so called Kohn-Luttinger\nHamiltonian35,36. For that matter, the crystal Hamilto-\nnian is represented in the basis of the 4 topmost J=3\n2\nvalence states of the Γ point. We label them by J=\n3\n2,Jz. The resulting kp Hamiltonian can be written as\nHholes=HKL\nHKL=/summationdisplay\ni,j=x,y,zVKL\nijJiJjkikj+κµBJzB(3)\nwherekiare the wave vectors, Jiare the spin3\n2matrices,\nandVKL\nijare coefficients given in the appendix A1. The\nlasttermaccountsfortheZeemancouplingtoanexternal\nfield along the growth direction.\nThe kp Hamiltonian for states in the presence of a\nquantum dot confinement potential that breaks transla-\ntional invariance reads:\nHkp=−/planckover2pi12/summationdisplay\ni,j=x,y,zVKL\nijJiJj∂i∂j+κµBJzB+V(/vector r)δjz,j′z\n(4)\nIn general, the numerical solution of equation (4) can\nbe very complicated. Following previous work18,30, we\nmake two approximations that permit to obtain analyt-\nical solutions. First, we model the quantum dot with\na hard-wall box-shape potential. The dimensions of the3\nbox areLx,LyandLz. This permits to write the wave\nfunction as a linear combination of |J=3\n2,Jz∝angb∇acket∇ightstates\nmultiplied by the confined waves\nψ/vector n(/vector r) =/radicalbigg\n8\nVSin/parenleftbiggnxπx\nLx/parenrightbigg\nSin/parenleftbiggnyπy\nLy/parenrightbigg\nSin/parenleftbiggnzπz\nLz/parenrightbigg\n(5)\nOur second approximation is to restrict the basis set to\nthe fundamental mode only, nx=ny=nz= 1. As a\nresult, the quantum dot Hamiltonian reads\nHkp=−/planckover2pi12/summationdisplay\ni,j=x,y,zVKL\nijJiJj∝angb∇acketleft∂i∂j∝angb∇acket∇ight (6)\nwhere\n∝angb∇acketleft∂i∂j∝angb∇acket∇ight=/integraldisplay\nψ1,1,1(/vector r)∂i∂jψ1,1,1(/vector r) =δij/parenleftbigg2π\nLi/parenrightbigg2\n(7)\nThus, within this approximation, the quantum dot hole\nstates are described by a 4*4 Kohn Luttinger Hamilto-\nnian where the terms linear in kivanish and the k2\niterms\nare replaced by/parenleftig\n2π\nLi/parenrightig2\n. The resulting matrix Hconfhas\ntwo decoupled sectors denoted by + ( Jz= +3\n2,Jz=−1\n2)\nand−(Jz=−3\n2,Jz= +1\n2). In the (+3\n2,−1\n2,+1\n2,−3\n2)\nbasis we have:\nHconf=/parenleftbigg\nH+0\n0H−/parenrightbigg\n(8)\nwith\nH+=/parenleftbigg\nP+Q−3b\n2R\nRP−Q+b\n2/parenrightbigg\n(9)\nand\nH−=/parenleftbigg\nP−Q−b\n2R\nRP+Q+3b\n2/parenrightbigg\n(10)\nwhereb≡κνBBandP,QandRare given in the ap-\npendix. In order to find the corresponding energies and\nwavefunctions it is convenient to write these matrices as:\nH±=a±+/vectorh±·/vector σwhere/vector σare the Pauli matrices act-\ning on the pseudospin1\n2space of the + and −spaces,\na±=P∓b/2 and\n/vectorh±=/parenleftbig\nR,0,Q∓b/parenrightbig\n=|/vectorh±|(Sinθ±,0,Cosθ ±) (11)\nWe keep only the two ground states (heavy-hole like),\ndenoted by | ⇑∝angb∇acket∇ightand| ⇓∝angb∇acket∇ight, which are several meV away\nfrom the light-hole like states. The ground state doublet\nfor the quantum dot holes states so obtained, neglecting\nstrain, can be written as\n| ⇑∝angb∇acket∇ight=cosθ+\n2|+3\n2∝angb∇acket∇ight+sinθ+\n2|−1\n2∝angb∇acket∇ight\n| ⇓∝angb∇acket∇ight=cosθ−\n2|+−3\n2∝angb∇acket∇ight+sinθ−\n2|+1\n2∝angb∇acket∇ight(12)\nThus, the LH-HH mixing parameters, θ±depend on the\ndot dimension, Li, on the Kohn Luttinger parameters, γi\nand on the applied magnetic field B.2. Effect of homogeneous strain\nWenowconsidertheeffectofthestrainthatarisesfrom\nthe lattice mismatch between the CdTe quantum dot and\nthe ZnTe substrate on the J=3\n2states of the valence\nband. Ithasasimilareffectthatconfinement,resultingin\na splitting of the J=3\n2manifold and a mixing of the LH\nandHHstates. TheHamiltonianthatdescribesthe effect\nof strain, as described by the strain tensor ǫij, on the top\nof the valence band states in zinc-blende semiconductors\nwas proposed by Bir and Pikus. We can write the Bir\nand Pikus (BP) Hamiltonian as37:\nHBP=/parenleftbigg\na−9b\n4/parenrightbigg\n(exx+eyy+ezz)+\n+3b/summationdisplay\ni=x,y,zJ2\nieii+√\n3d((JxJy+JyJx)exy+c.p.) (13)\nwhere c.p.stands for cyclic permutation, and a=\n−3.4eV,b=−1.2eV,d=−5.4eV for CdTe37.\nFor CdTe quantum dots grown in ZnTe, we mainly\nconsider the effects of strain anisotropy in the growth\nplane13and describe the strain by the average values\nofexyandexx−eyy. In this approximation the BP\nHamiltonian is reduced to a block diagonal matrix in the\n(+3\n2,−1\n2,+1\n2,−3\n2) basis:\nHBP=/parenleftbigg\nHBP+0\n0HBP−/parenrightbigg\n(14)\nwhere\nHBP+=/parenleftbigg\n0ρse−2iϕs\nρse2iϕs∆lh/parenrightbigg\n(15)\nHBP−=/parenleftbigg\n∆lhρse−2iϕs\nρse2iϕs0/parenrightbigg\n(16)\nwhere ∆ lhis the HH-LH splitting, ρsthe strained in-\nduced amplitude of the HH-LH mixing and φsthe angle\nbetweenthestrainedinducedanisotropyaxisinthequan-\ntum dot plane and x (100) axis. Importantly, the effect\nof confinement and the effect of strain have a very simi-\nlar mathematical structure. They both split the LH and\nHH levels and mix them. The main difference lies in the\nmixing term, which is real for the confinement Hamil-\ntonian controlled by the shape of the quantum dot and\ncomplex for the BP Hamiltonian depending on the strain\ndistribution in the quantum dot plane.\n3. Combined effect of confinement and strain\nWefinallyconsiderthe combinedactionofconfinement\nand strain described by Hholes=Hconf+HBP. Summing\nthe Hamiltonians of equations (8) and (14) to obtain two\ndecoupled matrices for the + and −subspaces. They can\nbe written as\nHtot,±=A±+/vectorH±·/vector σ (17)4\nwhereA±=P∓b\n2+∆lh\n2and\n/vectorH±=/parenleftbigg\nR+ρscos(2ϕs),±ρssin(2ϕs),Q∓b−∆lh\n2/parenrightbigg\n(18)\nIt is convenient to express the ground state doublet asso-\nciated to Hholesin terms of the spherical coordinates of\nthe vectors /vectorH±,|/vectorH±|,θ±andφ±:\n| ⇑∝angb∇acket∇ight=Cosθ+\n2|+3\n2∝angb∇acket∇ight−Sinθ+\n2eiφ+|−1\n2∝angb∇acket∇ight\n| ⇓∝angb∇acket∇ight=Cosθ−\n2|−3\n2∝angb∇acket∇ight−Sinθ−\n2eiφ−|+1\n2∝angb∇acket∇ight(19)\nwhere\neiφ±=R+ρse±2iϕs\n|R+ρse±2iϕs|(20)\nExpectedly, this expression is formally very similar to\nthat of equation (12).\nFormally, we express eq. (19) as\n|σh∝angb∇acket∇ight=/summationdisplay\njzCh(jz)|jz∝angb∇acket∇ight (21)\nBoth in equations (12) and (19) the (+3\n2,−1\n2) sector\nis decoupled from the ( −3\n2,+1\n2). Whereas, this is not\ntrue in general, it is sufficient to account for the correct\nsymmetry of a variety of exchange couplings between the\nhole and both the Mn and the electrons.\nC. Effective Mn-carrier exchange Hamiltonian\n1. Hole-Mn Hamiltonian\nWenowconsidertheexchangecouplingofholespin( /vectorJ)\nand the Mn spin ( /vectorM). The leading term in the exchange\ninteraction is the Heisenberg operator33,38:\nVexch=βδ(/vector rh−/vector rM)/vectorJ·/vectorM (22)\nwhereβis the hole-Mn exchange coupling constant. For\nMn in CdTe we have βN0=0.88 eV, where N0is the\nvolume ofthe CdTe unit cell33. The exchangeinteraction\nis taken as short ranged, the Mn atom is located at /vector rMn\nand/vectorJare the spin3\n2angular momentum matrices. We\nrepresentthe operator(22) in the product basis |M∝angb∇acket∇ight×σh.\nThus, the exchange operator in the product basis reads:\n∝angb∇acketleftM|∝angb∇acketleftσh|Vexch|M′∝angb∇acket∇ight|σ′\nh∝angb∇acket∇ight=\nβ|ψ0(/vector rMn)|2/summationdisplay\na∝angb∇acketleftM|Ma|M′∝angb∇acket∇ight|∝angb∇acketleftσh|Ja|σ′\nh∝angb∇acket∇ight(23)\nwhereψ0(/vector r) is the envelope part of the heavy hole wave\nfunction, eq. (5), and jh≡β|ψ0(/vector rMn)|2is the hole-\nMn coupling constant, which depends both on a materialdependent constant βand on a quantum dot dependent\nproperty, the probability of finding the hole at the Mn\nlocation.\nAfter a straightforwardcalculation we obtain the effec-\ntive Mn-hole coupling spin model working in the space\n(M,σh) of dimension 12 as a function of the hole wave\nfunction parameter θ:\nVh−Mn=jhxMxσx+jhyMyσy+jhzMzσz(24)\nwherethejhiaredimensionlesscoefficientsgiven,for B=\n0 andθ=θ+=θ−by:\njhx=jh\n2/parenleftig√\n3Sinθ+1−Cosθ/parenrightig\n(25)\njhy=jh\n2/parenleftig√\n3Sinθ−1+Cosθ/parenrightig\n(26)\njhz=jh\n2(1+2Cosθ) (27)\nNotice that for θ= 0 there is no LH-HH mixing and\nwe havejhx=jhy= 0 andjhz=3\n2jh. In this extreme\ncase the Mn-hole coupling is Ising like and Mzandσzare\nconserved. This limit is a good starting point to model\nhole-Mn coupling in CdTe quantum dots18,39\n2. Electron-Mn Hamiltonian\nIn analogy to the hole-Mn bare coupling, the electron-\nMn coupling reads:\nVe−Mn=αδ(/vector re−/vector rM)/vectorS·/vectorM (28)\nwhere/vectorSis the spin of the electron. Since the spin orbit\ncoupling has a very small effect on the slike conduction\nband, theeffectiveexchangeforthe quantumdotelectron\nand the Mn is also a Heisenberg term given by\nVe−Mn=je/vectorS·/vectorM=je(SxMz+SyMy+SzMz) (29)\nwhereje≡α|ψ0(/vector rMn)|2is the electron-Mn coupling\nconstant, which depends both on a material dependent\nconstantαand on a quantum dot dependent property,\nthe probability of finding the electron at the Mn loca-\ntion. In our model we take the same orbital wave func-\ntion for the confined electron and hole, so that the ratio\nje/jh=α/β≃4 for CdTe. A deviation from this ex-\npected ratio is indeed observed in real CdTe quantum\ndots8.\nD. Exciton-Mn wavefunctions and energy levels\n1. Hamiltonian\nThe effective Hamiltonian for the exciton in a single\nMn doped CdTe quantum dot is the sum of the single\nion magnetic anisotropy Hamiltonian, the Mn-electron5\nand Mn-hole exchange coupling and the electron-hole ex-\nchange coupling\nH=HS+Ve−Mn+Vh−Mn+Ve−h (30)\nwhere\nVe−h=jehSzσh (31)\nis the electron hole exchange coupling, neglecting trans-\nverse components. Electron hole exchange is ferromag-\nnetic (jeh<0) and splits the 4 exciton levels into two\ndoublets, the low energy dark doublet ( ⇑↑,⇓↓), denoted\nbyX=±2 and the high energy bright doublet ( ⇑↓,⇓↑)\n(X=±1).\nSince we consider two electron states ( Sz=↑,↓),\ntwo hole states ( σh=⇑,⇓), and six Mn states Mz=\n±5\n2,±3\n2,±1\n2, the Hilbert space for the Mn-exciton sys-\ntem has dimension 24. Whereas we do obtain the exact\neigenstates of Hamiltonian (30) by numerical diagonal-\nization, it is convenient for the discussion to relate them\nto eigenstate of the Ising, or spin conserving part, of the\nHamiltonian:\nH=HIsing+Hflip (32)\nwhere\nHIsing=DM2\nz+jehSzσh+jeSzMz+jhMzσh(33)\nand\nHflip=E(M2\nx−M2\ny)+je(SxMx+SyMy)+\n+(jhxσxMx+jhyσyMy) (34)\nIf we expand jhxandjhyin the series of LH-HH mixing\nparameterθ, they are the same in the first order of θ.\nFor simplicity, we take\njh⊥≡jhx=jhy=jhθ\n2√\n3(35)\nin the following calculation. In the case of a LH-HH\nmixing induced by the anisotropy of the confinement de-\nscribed by a hard-wall box shape potential, we get from\nthe Kohn-Luttinger Hamiltonian:\nθ=π2√\n3γ2|1\nL2x−1\nL2y|\n/radicalbigg\n3π4γ2\n2/parenleftig\n1\nL2\nx−1\nL2\ny/parenrightig2\n+γ2\n1/parenleftig\n−2\nL2\nz+1\nL2\nx+1\nL2\ny/parenrightig2(36)\nThe eigenstates of HIsingare trivially given by the\nproduct basis\n|P∝angb∇acket∇ight ≡ |Mz∝angb∇acket∇ight|Sz∝angb∇acket∇ight|σh∝angb∇acket∇ight (37)\nwith eigenenergies:\nEP=DM2\nz+jehSzσh+jeSzMz+jhMzσh(38)Mz-2-10123Energy (meV)-2\n-1\n+1\n+2\n5\n2-2 2 2 2 23 1- -1 3 5\nFIG. 1: (Color online) Scheme of the energy levels of the\nquantum dot exciton interacting with 1 Mn when spin-flip\nterms are neglected.\nSince the magnetic anisotropyterm DM2\nzis present both\nin the ground state and exciton state manifolds, it does\nnot affect the PL spectra of the bright excitons. Within\nthis picture, for each of the 6 possible values of Mz, there\nare 4 exciton states. We use a short-hand notation to\nrefertotheIsingstates PX(Mz)whereX=±1,±2labels\nthe spin of the exciton, X=Sz+σz. An energy diagram\nfor the exciton levels, within the Ising approximation, is\nshown in figure (1).\nThe PL spectra of a single Mn doped quantum dot\npredicted by the model of Ising excitons, ie, neglecting\nthe spin flip transitions, features 6 peaks corresponding\nto transitions conserving Mz. For the recombination of\nσ+excitons (Sz=−1\n2,σh=⇑) the high energy peak\ncorresponds to Mz= +5\n2and the low energy peak to\nMz=−5\n2on account of the antiferromagnetic coupling\nbetween the hole and the Mn. In the case of σ−excitons\nthe roles are reversed, but the PL spectrum is identical\nat zero magnetic field.\n2. Wave functions\nWhen spin-flip terms are restored in the Hamiltonian,\nthePstates are no longer eigenstates, but they form a\nvery convenient basis to expand the actual eigenstates of\nH, denoted by |Ψn∝angb∇acket∇ight:\n|Ψn∝angb∇acket∇ight=/summationdisplay\nPΨn(P)|P∝angb∇acket∇ight=/summationdisplay\nX,MzΨn(X,Mz)|X,Mz∝angb∇acket∇ight(39)\nIn most cases, there is a strong overlap between Ψ nand\na single state |P∝angb∇acket∇ight. This is expected for several reasons.\nFirst, the single ion in plane anisotropyis probably much\nsmaller than the uniaxial anisotropy, D≫E. Second,\nthe electron-hole exchange, which is the exchange energy\nin the system, splits the dark and bright levels. Thus,6\n0 0.02 0.04\nJh⊥ (meV)-0.4-0.200.2Energy (meV)\n0 0.02 0.04\nJh⊥ (meV)0.40.60.81 IPZ\n0 0.02 0.04\nJh⊥ (meV)-2-1012Energy (meV)\nFIG. 2: (Color online) Left panel: Evolution of the exciton\nlevels as a function of the LH-HH mixing parameter Jh⊥.\nRight panel: Evolution of the IPZ as a function of the LH-\nHH mixing parameter. The inset presents the evolution of\nthe energy for all the 24 exciton levels.\nboth electron and hole spin flip due to the exchange with\nthe Mn spin is inhibited because they involve coupling\nbetween energy split bright and dark excitons. In addi-\ntion, the electron Mn exchange is smaller than the hole\nMn exchange, whose spin-flip part is proportional to the\nLH-HH mixing and approximately 10 times smaller than\nthe Ising part. In order to quantify the degree of spin\nmixing of an exact exciton state Ψ n, we define the in-\nverse participation ratio:\nIPZn≡/summationdisplay\nP|Ψn(P)|4(40)\nThis quantity gives a measure of the delocalization of\nthe state Ψ non the space of product states of eq. (37).\nIn the absence of mixing of different Pstates, we have\nIPZn= 1. In the case of a state equally delocalized in\nthe 24statesofthe Pspace, wewouldhaveΨ n(P) =1√\n24\nandIPZn=1\n24.\nIn figure (2) we plot the evolution of both the energy\n(left panel) and the IPZ(right panel) as a function of\nJh⊥, the LH-HH mixing parameter, of four states de-\nnoted by their dominant component at Jh⊥= 0. For our\nchoice of exchange constants, two of them |+1,−3\n2∝angb∇acket∇ightand\n|−2,−1\n2∝angb∇acket∇ightare almostdegenerateat Jh⊥= 0, which means\nthat the hole-Mn exchange compensates the dark-bright\nsplitting, and couple these states via a hole-Mn spin flip.\nAs a result, their energy levels split linearly as a function\nofJh⊥and the wave-functions have a large weight on the\ntwo product states for finite Jh⊥. In contrast, the other\ntwo levels shown in figure (2), |+1,−1\n2∝angb∇acket∇ightand|−2,−3\n2∝angb∇acket∇ight\nare not coupled via hole-Mn spin flip. As a result, their\nenergies shift as a function of Jh⊥due to coupling to\nother states, and their IPZ undergoes a minor change,\nreflecting moderate mixing.3. Exchange induced dark-bright mixing\nThe most conspicuous experimentally observable con-\nsequence of the exchange induced mixing, is the transfer\nof optical weight from the bright to the dark exciton,\nwhich results in the observation of more than 6 peaks in\nthe PL. This can be understood as follows. The spin-flip\npart of the hole-Mn interaction couples the bright exci-\nton|+1,Mz∝angb∇acket∇ightto the dark exciton |−2,Mz+1∝angb∇acket∇ight. Thus,\na state with dominantly dark character | −2,Mz+ 1∝angb∇acket∇ight\nand energy given, to first order, by that of the dark ex-\nciton, has a small but finite probability of emitting a\nphoton through its bright component, via a Mn-hole co-\nherent spin-flip. Thus, PL is seen at transition energy\nof the dark exciton. Reversely, nominally bright excitons\nloose optical weight due to their coupling to the dark sec-\ntor. Importantly, the emission of a photon from a dark\nexciton with dominant Mn spin component Mz, entails\ncarrier-Mn spin exchange, so that the ground state has\nMz±1.\nAccording to previous theory work18the rate for the\nemissionofacircularlypolarizedphotonfromthe exciton\nstate Ψ nto the ground state φmreads:\nΓ±\nn,m= Γ0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nMzφm(Mz)Ψ∗\nn(Mz,X=±1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n(41)\nwhere\nΓ0≡3ω3d2\ncv\n4πǫ/planckover2pi1c3(42)\nis the recombination rate of the bare exciton, ωis the\nfrequencyassociatedto the energydifference between the\nexciton state nand the ground state m,cis the speed\nof light,ǫis the dielectric constant of the material, dcv\nis the dipole matrix element. From the experiments, we\ninfer Γ 0=0.5 ns−1\nIn the absence of spin-flip terms, the matrix Γ±\nn,m\nwouldhaveonlynon-zeroelements for n=|X=±1,Mz∝angb∇acket∇ight\nstates connected to m=Mzstates. The presence of\nspin-fliptermsintheHamiltonianenablestherecombina-\ntion from exciton states with dominant dark component.\nIn figure (3) we represent the matrix elements Γ±\nn,m/Γ0\nforjeh=−0.73meV,jh= 0.36meV,je=−0.09meV,\njh⊥= 0.036meV,D= 0.01meVandE= 0meV. It\nis apparent that the recombination rates from the dark\nstates are, at least, 2 times smaller (a and b) than those\nof the bright states. For Γ 0= 0.4 ns−1, the lifetime of\nthe dark excitons (a and b) are in the range of 3 ns.\nThus, this provides a quite efficient Mn spin relaxation\nmechanism, provided that a dark exciton is present in\nthe quantum dot.\nThe recombinationrate matrix, together with the non-\nequilibrium occupation of the exciton states, Pn, deter-\nmines the PL spectrum18for±circular polarization:\nI±\nPL(ω) =/summationdisplay\nn,m,PnΓ±\nn,mδ(/planckover2pi1ω−En−Em) (43)7\nFIG. 3: (Color online) Recombination rates of the excitons\nlevels in a Mn doped quantum dot Γ i/Γ0:a:|+ 2,+1\n2>→\n|+3\n2>,b:| −2,−1\n2>→ |−3\n2>,c:| −1,+3\n2>→ |+3\n2>,\nd:|+ 1,−3\n2>→ | −3\n2>, states from e to n are bright\nexcitons. Other states are mainly dark excitons with a small\nbright component.\nIn a typical PL spectrum3, the dark peaks are, at most, 2\ntimessmallerthanthebrightpeaks. SinceΓ±\nn,misatleast\n2 times smaller for dark states, this implies a larger oc-\ncupation of the dark states. Thus, we can infer a transfer\nfrom the optical ground state to the dark exciton states,\nvia the bright exciton states. This transfer requires an\nincoherent spin flip of either the electron or hole. Below\nwe show that phonon induced hole-spin relaxation pro-\nvides the most efficient channel for this bright to dark\nconversion.\nIII. MN SPIN RELAXATION DUE TO\nSPIN-PHONON COUPLING\nIn this section we discuss the Mn spin relaxation in the\nabsence of excitons. In the absence of carriers and given\nthe fact that Mn-Mn distance is comparable to the dot-\ndot distance (100 nm for a dot and Mn density of about\n1010cm−2), which makes direct super-exchange negligi-\nble, the Mn-phonon coupling should be the dominant,\nalbeit small, Mn spin relaxation mechanism. Transverse\nphonons induce local rotations of the lattice. Since the\ncrystal field, together with spin orbit coupling, deter-\nmines the Mn magnetocrystalline anisotropy, the phonon\ninduced lattice rotation40acts as a stochastic torque on\nthe Mn spin, resulting in spin relaxation.\nThe atomicdisplacement atpoint /vector rin the crystalis ex-\npressed in terms of the phonon operators with wave vec-\ntor/vector q, polarization mode λ=T1,T2,L, frequency ωλ(/vector q)\nand polarization vector /vector eλ(/vector q)37:\n/vector u(/vector r) =/summationdisplay\n/vector q,λUλ(/vector q)/vector eλ(/vector q)/parenleftig\nb†\n/vector q,λ+b−/vector q,λ/parenrightig\nei/vector q·/vector r(44)where\nUλ(/vector q) =/radicaligg\n/planckover2pi1\n2ρωλ(/vector q)V(45)\nandVandρare the volume of the crystal and the mass\ndensity respectively. In a zinc-blende structure there are\ntwo transverse acoustic (TA) phonon branches and one\nlongitudinalacousticbranch(LA).FollowingWoods31we\nhave:\n/vector eTA1=1\nqq⊥/parenleftbig\nqxqz,qyqz,−q2\n⊥/parenrightbig\n(46)\n/vector eTA2=1\nq⊥(qy,−qx,0) (47)\nwhereq≡ |/vector q|andq⊥=/radicalig\nq2x+q2y. These vectors satisfy\n/vector q·/vector eTAi= 0, and/vector eTAi·/vector eTAj=δijThe longitudinal mode\nhas/vector eLA=1\nq/vector q.\nThe lattice rotation vector is given by40\n/vectorδΦ(/vector r) =/vector∇×/vector u(/vector r) (48)\nso that only the transverse modes contribute. Within\nthis picture, the Mn spin-phononcoupling can be written\nas40:\nVM−ph=i/bracketleftig\nH0,/vectorM/bracketrightig\n·/vectorδΦ(/vector rMn) (49)\nWithout loss of generality we can set the Mn position as\nthe origin,/vector rMn= 0. Equation (49) couples the Mn spin\nto a reservoir of phonons whose non-interacting Hamil-\ntonian is\nHph=/summationdisplay\n/vector q,λ/planckover2pi1ωλ(/vector q)b†\n/vector q,λb/vector q,λ (50)\nWithin the standard system plus reservoir master\nequation approach, we have derived the scattering rate\nfrom a state nto a staten′, both eigenstates of the single\nMn Hamiltonian H0, due the emission of a phonon. In\norder to use a general result for that rate (B7), derived\nin the appendix (B), we need to express the spin-phonon\ncoupling (49) using the same notation than in equation\n(B1):\nVn,n′\n/vector q,λ=i2Uλ(/vector q)/vectorfn,n′·(/vector q×/vector eλ(/vector q)) (51)\nwhere\n/vectorfn,n′≡ ∝angb∇acketleftn|/bracketleftig\nH0,/vectorM/bracketrightig\n|n′∝angb∇acket∇ight (52)\nWe compute now the scattering rate due to a single\nphonon emission assuming 3 dimensional phonons de-\nscribed above. The rate reads:\nΓn→n′=|∆|3\n12πρ/planckover2pi14c5(nB(∆)+1)/summationdisplay\nb,b′=x,y,zfb\nn′,n(fb′\nn′,n)∗\n(53)\nwherec= 1.79kms−1is the CdTe speed of sound41,ρ=\n5870kgm−3is the mass density of the CdTe unit cell42\nandnB(∆)≡1\neβ|∆|−1. The|∆|3factor comes from the\ndependence ofthe phonondensityofstatesonthe energy.8\nA. Mn spin relaxation in the optical ground state\nWenowdiscusstherelaxationoftheMnelectronicspin\ndue to spin-phonon coupling without an exciton in the\nquantum dot. According to our experimental results1,3,\nthe Mn spin relaxation time in our samples is at least\n5µs.\nIf we takeE= 0, the transition rate between the ex-\ncited states |φn∝angb∇acket∇ight=|Mz= +5\n2∝angb∇acket∇ightand|φn′∝angb∇acket∇ight=|Mz= +3\n2∝angb∇acket∇ight,\nvia a phonon emission, is given by:\nΓn→n′=640|D|5\n3πρ/planckover2pi14c5(nB(∆)+1) (54)\nThe dependence on D5comes both from the density of\nstates of phonons ρ∝ω3and the square of the Mn\nphononcoupling, which isproportionalto the anisotropy,\nand gives the additional D2factor. Whereas the uniaxial\nanisotropy of Mn in CdTe quantum wells has been de-\ntermined by EPR34, the actual value for Mn in quantum\ndots is not known and can not be measured directly from\nsingle exciton spectroscopy of neutral dots. Therefore, in\nfigure (4) we plot the lifetime for the transition of the Mn\nspin from5\n2to3\n2, due to a phonon emission, as a function\nofD. We takeDin a range around the value reported\nfor CdTe:Mn epilayers, D= 12µeV34. We find that the\nspin lifetime of Mn in the optical ground state can be\nvery large. Even for D= 20µeVthe Mn spin lifetime\nis in the range of 0.1 seconds, well above the lower limit\nfor the Mn spin relaxation reported experimentally1,3.\nWhereas we can not rule out completely that the Mn\nspin lifetimes that long, there are other spin relaxation\nmechanisms that might be more efficient that the Mn-\nphonon coupling considered above, like the coupling of\nthe Mn electronic spin to nuclear spins of Mn and the\nhost atoms1.\nSince part of the ∆5scaling arises from the ω3scaling\nofthe phonondensityofstates, wehaveexploredthe pos-\nsibility that phonons localized in the wetting layer could\nbe more efficient in relaxing the Mn spin. For that mat-\nter we have considered a toy model of two dimensional\nphonons confined in a slab of thickness L= 2nm. The\nresulting Mn spin relaxation rate for those reads:\nΓn→n′=∆2\n16/planckover2pi13c4ρW(nB(∆)+1)/summationdisplay\nb,b′=x,y,zAb,b′fb\nn′,n(fb′\nn′,n)∗\n(55)\nwhere W is the width of the sample and A is a diagonal\nmatrix with Axx= 1,Ayy= 1,Azz= 2.\nIn figure (4) we plot the associated spin lifetime in this\ncase, taking W= 2nmand show how it is at least 100\nshorter than for 3D phonons, but still we would have\nT1≃1 ms forD= 20µeV.\nB. Mn spin relaxation in the presence of an exciton\nHere we discuss how the Mn spin relaxation due to\nMn-phonon coupling is modified when an exciton is in-10 20\nD (µeV)1061081010Tground (ns)3D\n2D\n10 20\nD (µeV)105106107Texcited (ns)3D\n2D\nFIG. 4: (Color online) Left panel: Lifetime of the (+5\n2to\n+3\n2) transition in the optical ground state at zero field as a\nfunction of the magnetic anisotropy energy splitting D. Right\npanel: Lifetime of the same transition in the presence of a +1\nexciton for different values of D. The rates are calculated\nfor a 3 dimensional (3D) and a 2 dimensional (2D) density of\nstates of acoustic phonons.\nteracting with the Mn. The Mn-phonon coupling is still\ngiven by Hamiltonian (49), with H0given by eq. (1). We\nassume that the only effect of the exciton on the Mn is\nto change the energy spectrum and mix the spin wave-\nfunctions, giving rise to larger spin relaxation rates, due\nto the larger exchange-induced energy splittings.\nInthepresenceoftheexciton, the Mn-phononcoupling\nresults in transitions between different exciton-Mn spin\nstates,nandn′. As we did in the case of the Mn without\nexcitons, we need to express the spin-phonon coupling\n(49) using the same notation than in equation (B1).\nFor that matter we define the matrix elements\n/vectorFn,n′≡ ∝angb∇acketleftΨn|/bracketleftig\nH0,/vectorM/bracketrightig\n|Ψ′\nn∝angb∇acket∇ight=\n/summationdisplay\nX,Mz,M′zΨn(X,M)∗Ψn(X,M′)∗/vectorfM,M′(56)\nwhere\n/vectorfM,M′≡ ∝angb∇acketleftM|/bracketleftig\nH0,/vectorM/bracketrightig\n|M′∝angb∇acket∇ight (57)\nMandM′stand for eigenstates of the Mn spin operator\nMz. Thus, in the exciton-Mn spin states basis, the Mn-\nphonon coupling reads:\nVn,n′\n/vector q,λ=i2Uλ(/vector q)/vectorFn,n′(M)·(/vector q×/vector eλ(/vector q)) (58)\nNotice how if we neglect the spin mixing of the exciton\nstates we have /vectorFn,n′=/vectorfM,M′and the only difference in\nthe scattering rates arises from the larger energy split-\ntings in the presence of the exciton.\nUsing the equation (B7) for the phonon induced spin\nrelaxation rate, and in analogy with equation (53) we9\nwrite:\nΓn→n′=|∆|3\n12πρ/planckover2pi14c5(nB(∆)+1)/summationdisplay\nb,b′=x,y,zFb\nn,n′(Fb′\nn,n′)∗\n(59)\nIn figure (4) we see how Mn-phonon spin relaxation is\nmuch faster in the presence of the exciton. Ignoring the\ndifference arising from the spin mixing, we can write the\nratio of the rates as:\nΓn→n′(X)\nΓn→n′(G)=/parenleftbigg∆X\n∆G/parenrightbigg3\n(60)\nThe energy splitting associated to the5\n2to3\n2spin flip\nin the ground state is 4 D. In the presence of the exci-\nton the energy splitting of the same transition would be\n4D+jh−je. If we take D= 12µeV,jh= 360µeV and\nje=−90µeV the ratio yields ≈103. From the experi-\nmental side we know that T1G>5µsand, in the presence\nof the exciton T1≃50ns. Thus, the ratio could be ac-\ncounted for by this mechanism. However, in order to\nhaveT1G= 5µs we would need to assume an unrealisti-\ncally large value for D. Thus, we think that another spin\nrelaxation mechanism must be operative in the system\nwhen the exciton is in the dot which makes it possible\nto control the spin of the Mn in a time scale of 50 ns.\nIn the next sections we discuss the hole spin relaxation\ndue to phonons as the mechanism that, combined with\nMn-carrier exchange, yields a quick Mn spin relaxation\nin the presence of the exciton.\nIV. HOLE SPIN RELAXATION IN NON\nMAGNETIC DOTS\nA. Hole-phonon coupling\nWe now consider the relaxation of the hole spin due\nto hole-phonon coupling. We consider first the case of\nundoped quantum dots. The coupling of the spin of the\nhole to phonons can be understood extending the Bir\nPikus Hamiltonian to the case of inhomogeneous strain\nassociated to lattice vibrations:\nǫij(/vector r)≡1\n2/parenleftbigg∂ui\n∂rj+∂uj\n∂ri/parenrightbigg\n(61)\nIt is convenient to write the strain tensor field as:\nǫij(/vector r) =/summationdisplay\n/vector qei/vector q·/vector rǫij(/vector q) (62)\nso that we write:\nǫij(/vector q) =1\n2/summationdisplay\nλUλ(/vector q)/parenleftig\nb†\n/vector q,λ+b−/vector q,λ/parenrightig/parenleftig\nqjei\nλ(/vector q)+qiej\nλ(/vector q)/parenrightig\n(63)\nWe consider the coupling of the ground state doublet,\nformed by states ⇑and⇓, to the phonon reservoir32. The0 2 4\n∆(meV)00.010.020.030.040.05Rates(ns-1)Lx=5nm Ly=6nm Lz=3nm\nLx=5nm Ly=6.5nm Lz=3nm\n50100Area(nm2)0.00010.0010.010.1Rates(ns-1)\n4 6 8 10\nLx(nm)0.00010.0010.010.1Rates(ns-1)\nTA1\nTA2\nLA\nTOTAL\nFIG. 5: (Color online) Left panel: Hole spin-flip rate as\na function of the dot size and shape. Lyis fixed at 6nm,\nscaning of Lxchange the LH-HH mixing. In the inset the\nratioLy/Lxis fixed at 1.2 so the shape of the dot are fixed,\nonly the size of the dot changes. One can see that the hole\nspin-flip rate is a size sensitive quantum quantity, the rate is a\nsemi-exponential function of the size of the dot. Right pane l:\nHole spin-flip rate as a function of the energy splitting for\ntwo different values of the quantum dot anisotropy, ie LH-HH\nmixing.\neffectivehole-phononHamiltonianisobtainedbyproject-\ning the BP Hamiltonian onto this subspace:\nVh−phonon=/summationdisplay\nij,/vector qσh,σ′\nhIσh,σ′\nh\nij(/vector q)|σh∝angb∇acket∇ight∝angb∇acketleftσ′\nh|ǫij(/vector q) (64)\nHere|σh∝angb∇acket∇ightdenotes the quantum dot state defined in eq.\n(19) and the coupling constant reads\nIσh,σ′\nh\nij(/vector q)≡/summationdisplay\njz,j′zVijC∗\nh(jz)Ch′(j′\nz)∝angb∇acketleftjz|JiJj|j′\nz∝angb∇acket∇ightI/vector q(65)\nwhereI/vector q=/integraltext\n|ψ0(/vector r)|2ei/vector q·/vector rd/vector r. Hamiltonian (64) shows\nhow the absorption or emission of a phonon can induce\na transition between the two quantum dot hole states, ⇑\nand⇓.\nWe now calculate the time scale for the spin relaxation\nof a single hole in a non magnetic dot under the influ-\nence of an applied magnetic field so that the hole ground\nstate doublet is split in energy. In order to compute the\ntransition rate for decay of the hole from the excited to\nthe ground state we use again the general equation (B7).\nFor that matter, we express the hole-spin coupling (64)\nas:\nVh−phonon=/summationdisplay\n/vector q,λσh,σ′\nhVσh,σ′\nh\n/vector q,λ|σh∝angb∇acket∇ight∝angb∇acketleftσ′\nh|/parenleftig\nb†\nλq+bλ,−q/parenrightig\n(66)10\nwhere\nVσh,σ′\nh\n/vector q,λ=i\n2/summationdisplay\ni,jIσh,σ′\nh\nij(/vector q)Uλ(/vector q)/parenleftig\nqiej\nλ(/vector q)+qjei\nλ(/vector q)/parenrightig\n(67)\nB. Calculation of hole spin-flip rates with simple\nmodel\nIn order to illustrate the physics of the phonon-driven\nhole spin relaxation we consider the case of a single hole\nin a non-magnetic dot under the influence of an applied\nmagnetic field. For that matter, we compute the Hamil-\ntonian (67) using the wave functions from the simple\nmodel of confined holes defined in eq. (12). We focus\non the non-diagonal terms in the hole spin index, i.e.,\nthe terms that result in scattering form ⇑to⇓due to\nphonon emission.\nImportantly, the BP Hamiltonian couples hole states\nthat differ in, at most, two units of Jz. Thus, in the\nabsence of LH-HH mixing, the BP Hamiltonian does not\ncouple directly the ⇑and⇓states. Transitions between\n⇑and⇓states, as defined in eq. (19), are only possible,\nthrough one phonon processes, through the ǫyz(JyJz+\nJzJy) andǫzx(JzJx+JxJz) terms in the Hamiltonian.\nAfter a straightforward calculation we obtain:\nV⇑,⇓\n/vector q,λ=i\n2√\n3dSin/parenleftbiggθ1−θ2\n2/parenrightbigg\n(ǫyz(/vector q)−iǫzx(/vector q)) (68)\nTheimportantroleplayedbythe LH−HHmixingθ1,2is\napparent. Using equation (B7) it is quite straightforward\nto compute the rate for the 3 phonon branches. They are\nall proportional to\nΓ0\n⇑→⇓=1\n18πD2\nu′Sin2/parenleftbiggθ1−θ2\n2/parenrightbigg∆3\nρ/planckover2pi14c5(69)\nwith coefficients7\n5, 1 and8\n5for the TA1, TA2 and L\nmodes respectively. Here, Du′stands for the deforma-\ntion potential of Kleiner-Roth43, following reference 44,\nDu′=−3√\n3d\n2.ρfor the mass density of CdTe, cfor its\ntransverse speed of sound, and ∆ for the energy split-\nting between the ⇑and⇓states, which is proportional to\nthe external magnetic field B. In figure (5) we plot the\nrates Γ TA1,ΓTA2,ΓL, as well as their sum as function of\nthe dot size (left panel) and as a function of the energy\nsplitting between the initial and final hole state, ∆ (right\npanel). We see how hole spin relaxation rates can be in\nthe range of Γ ≃1/(40ns).\nThe results of figure (5) suggest that for sufficiently\nhigh ∆, as those provided by the Mn-hole exchange, the\nholespincanrelaxinatimescaleof30 ns. Thesenumbers\nare in the same range than those obtained by Woods et\nal31. As we discuss in the next section, these spin flips,\ntogether with Mn-carrier exchange, can also induce Mn\nFIG. 6: (Color online) Scheme of the Mn spin flip chan-\nnels due to the combined action of hole-phonon coupling and\ncarrier-Mn exchange.\nspin relaxation in a time scale much shorter than the one\ndue to Mn-phonon coupling only.\nImportantly, the rate is finite only if θ1−θ2∝negationslash= 0, which\nis the case in the presence of an applied magnetic field.\nThis indicates that, within the simple model of eq. (12),\nthe non-diagonal terms in the hole-phonon Hamiltonian\n(64) vanishes identically. This is not a general feature\nof (64), but rather an particular property of the simple\nmodel (12). In particular, the non-diagonal term in (12)\nis non-zero at zero field as soon as the (+3\n2,−1\n2) states\nhavealsosomeweightonthe+1\n2components, whichhap-\npens both when a more realistic model for confinement\nis used or when homogeneous strain components eyzand\nezxare included.\nV. SPIN RELAXATION IN MAGNETIC DOTS\nDUE TO HOLE-PHONON COUPLING\nThe results of the previous sections indicate that, be-\ncause of their coupling to phonons, the hole spin life-\ntime in a non-magnetic dot is much shorter than the Mn\nspin lifetime. Here we explore the consequences of this\nphonon-driven hole spin relaxation for the single exciton\nstates in a dot doped with one magnetic atom. The lead-\ning process results in a Mn spin conserving decay from\nthe bright exciton to the dark exciton state, via hole-\nspin flip in a time scale in the 10 ns range. Combined\nwith the optical recombination of the dark state, made\npossible via Mn-hole or Mn-electron spin flip, provide a\npathwayforexciton induced Mn spin relaxationin a time\nscale under 100 ns, as observed experimentally1–3.\nWe also explore the scattering between two bright\nstates enabled by the combination of phonon induced\nhole spin relaxation and Mn-carrier exchange. The life-11\ntimes of these processes is in the range of 103ns and\nhigher, and therefore they are probably not determinant\nfor the optical orientation of the Mn spin in the sub-\nmicrosecond scale.\nA. Exciton-phonon coupling in magnetic dots\nThe Hamiltonian that couples the exciton states Ψ n\nto the phonons is derived by projecting the hole-phonon\ncoupling Hamiltonian (64) onto the exciton states (39) .\nThe result reads:\nVX−phon=/summationdisplay\nn,n′/vector qλ|Ψn∝angb∇acket∇ight∝angb∇acketleftΨn′|Vn,n′\n/vector q,λ/parenleftig\nb†\nλq+bλ,−q/parenrightig\n(70)\nwhere\nVn,n′\n/vector q,λ=/summationdisplay\nMz,σe,σh,σ′\nhVσh,σ′\nh\n/vector q,λΨn(Mz,X)Ψ∗\nn′(Mz,X′) (71)\nwhereX= (σe,σh) andX′= (σe,σ′\nh) (same electron\nspin) and Vσh,σ′\nh\n/vector q,λis given by equation (67).\nB. Qualitative description of the spin relaxation\nprocesses\nIn order to describe qualitatively the variety of dif-\nferent processes accounted for by Hamiltonian (70) it is\nconvenient to consider an initial state ψnas a linear com-\nbination of a dominant component |M∝angb∇acket∇ight| ↓∝angb∇acket∇ighte| ⇑∝angb∇acket∇ighthplus a\nminor contribution of two dark components, which arise\nfrom the coherent exchange of the Mn with either the\nelectron or the hole:\n|ψn∝angb∇acket∇ight=|M∝angb∇acket∇ight| ↓∝angb∇acket∇ighte| ⇑∝angb∇acket∇ighth+\n+ǫe|M−1∝angb∇acket∇ight| ↑∝angb∇acket∇ighte| ⇑∝angb∇acket∇ighth+ǫh|M+1∝angb∇acket∇ight| ↓∝angb∇acket∇ighte| ⇓∝angb∇acket∇ighth(72)\nwhereǫe∝je/jehandǫh∝jh/jehare small dimension-\nless coefficients that can be obtained doing perturbation\ntheory.\nDepending on the elementary process that takes place,\nthere are several possible final states:\n1. Hole spin relaxation. In this case the final state\nwould be dominantly a dark exciton whose wave\nfunction read:\n|ψn′∝angb∇acket∇ight=|M∝angb∇acket∇ight| ↓∝angb∇acket∇ighte| ⇓∝angb∇acket∇ighth+O(ǫ) (73)\nand the scattering rate Γ 0would be proportional\nto|I⇑,⇓|2. This is process II in figure (6).\n2. Hole spin relaxation plus coherent hole-Mn spin\nflip. This is process III in figure (6) This can be re-\nalizedthrough 2 dominant channels. An incoherent\nhole spin flip will couple the dominant component6.0\n5.0\n4.0Lx(nm)\n0.060.030.00\nJh⊥(meV)246101246102246103Time(ns)6.0\n5.0\n4.0Lx(nm)\n0.060.030.00\nJh⊥(meV)103104105106107Time(ns)\nFIG. 7: (Color online) Calculated rates for the transitions\nbetween exciton states in a Mn-doped quantum dot due to\nhole-phonon coupling. Left panel, red line, transition fro m\n| −1,−5\n2>to|+ 2,−5\n2>, Right panel, red line, transition\nfrom| −1,−5\n2>to| −2,−1\n2>. blue line, transition from\n|−1,−5\n2>to|+1,−3\n2>. Lyis fixed at 6nm and scanning\nLxchanges the LH-HH mixing parameter J h⊥.\nof the initial state, |M∝angb∇acket∇ight| ↓∝angb∇acket∇ighte| ⇑∝angb∇acket∇ighthwith a secondary\ncomponent |M∝angb∇acket∇ight| ↓∝angb∇acket∇ighte| ⇓∝angb∇acket∇ighthof the final state\n|ψn′∝angb∇acket∇ight=|M−1∝angb∇acket∇ight| ↓∝angb∇acket∇ighte| ⇑∝angb∇acket∇ighth+\n+ǫh|M∝angb∇acket∇ight| ↓∝angb∇acket∇ighte| ⇓∝angb∇acket∇ighth+O(ǫe) (74)\nIn this case the final state is a bright exciton in the\nsame branch +1 than the initial state but the Mn\ncomponent goes from MtoM−1.\nThe second channel comes from the hole spin flip\nof the minority dark component of the initial state,\nǫh|M+1∝angb∇acket∇ight| ↓∝angb∇acket∇ighte| ⇓∝angb∇acket∇ighthwhich decays into the majority\ncomponent of the final state\n|ψn′∝angb∇acket∇ight=|M+1∝angb∇acket∇ight| ↓∝angb∇acket∇ighte| ⇑∝angb∇acket∇ighth+O(ǫe,h) (75)\nThus, in this second case a hole spin flips due to\nphonons, plus a coherent Mn-electron spin flip con-\nnect theX= +1,Minitial state to the X=\n−1,M+ 1 state. Thus, both the initial and final\nstate in this process are the same than in the first\nchannel, the rates for each would be proportional\ntoǫ2\nhΓ0, but the decay pathways are different, and\ninterferences are expected.\n3. Holespin relaxationplus coherentelectron-Mnspin\nflip. ThisisprocessIinfigure(6)Asintheprevious\ncase, therearetwochannelsforthis type ofprocess.\nIn the first channel, the majority component of the\ninitial state decays into a final state given by:\n|ψn′∝angb∇acket∇ight=|M−1∝angb∇acket∇ight| ↑∝angb∇acket∇ighte| ⇓∝angb∇acket∇ighth+ǫ′\ne|M∝angb∇acket∇ight| ↓∝angb∇acket∇ighte| ⇓∝angb∇acket∇ighth(76)\nThe incoherent hole spin flip connects the initial\nstate (72) to the final state (76) through the mi-\nnority component |M∝angb∇acket∇ight| ↓∝angb∇acket∇ighte| ⇓∝angb∇acket∇ighthof the later.12\n0 200 400 600Time(ns)00.0050.01PL (Arb. Units)Rategeneration=1ns-1\nRategeneration=0.5ns-1\n-2 -1 0 1 2 3\nEnergy (meV)00.511.52PL (Arb. Units)\n          pump from initial state\n   (Mn spin are equally populated)dark period    pump after \n the dark period\nFIG. 8: (Color online) Simulation of PL intensity from state\n(X= +1,Mz=−5\n2) under the influence of a driving laser\npumping the system resonantly from optical ground state\nMz=−5\n2to the excited state ( X=−1,Mz=−5\n2) for\ntwo laser intensities. The inset is the PL spectrum assum-\ning all the states are equally populated. In the calculation ,\nthe quantum dot anisotropy (L x=5nm and L y=6nm) controls\ntheLH-HHmixing. The otherparameters arediscussed in the\ntext.\nThe second channel comes from the hole spin flip\nof the minority dark component of the initial state,\nǫe|M−1∝angb∇acket∇ight| ↑∝angb∇acket∇ighte| ⇑∝angb∇acket∇ighthwhich decays into the majority\ncomponent of the final state\n|ψn′∝angb∇acket∇ight=|M−1∝angb∇acket∇ight| ↑∝angb∇acket∇ighte| ⇓∝angb∇acket∇ighth+O(ǫe,h) (77)\nThus,aholespinflipsduethephonon,plusacoher-\nent Mn-electron spin flip connect the X= +1,M\ninitial state to the X=−1,M−1 state. The scat-\ntering rate of these two process scales as ǫ2\neΓ0\nC. Calculation of the relaxation rates\nIn order to implement equations (70,71) to compute\nscattering rates, we use the single particle basis for the\nholesdonewith equations(12) which leads, atfinite mag-\nneticfieldtothematrixelement(68)thatwouldbeincor-\nporated into equations (71) to compute the rates using\nequation (B7). As discussed above, a zero field model\n(12) yields a zero spin-flip matrix element in equation\n(68). This is a feature of the simple hole model rather\nthan an intrinsic property of the system. Thus, for the\nsake of simplicity, we compute the rates between exci-\nton states by computing the matrix element (68) as if\nthere was a magnetic field that yields the energy split-\nting between the initial and final exciton states equal to\nthe splitting produced by the exchange interaction with\nthe Mn spin.\nIn thecalculationofthe ratesweperformanadditional\napproximation: we only consider spin-flip terms in equa-\ntion (71) and we do exclude spin-conserving terms. The0500 1000\nTime (ns)00.050.10.150.20.25Population\n0500 1000\nTime (ns)00.10.20.30.40.5MzAverage Spin\n2|      >1\n|       >21\n|       >23|      >\n|      >\n|      >23\n25\n25\nFIG. 9: (Color online) Left panel: average magnetization an d\nright panel: occupation of the different spin states under op -\ntical pumping of the state ( X=−1,Mz=−5\n2). Parameters\nare the same as for the calculation presented in figure (8).\nresults for transition rates from the state nwith domi-\nnant (−1,−5\n2) to 3 possible final states with dominant\ncomponents (+2 ,−5\n2), (−2,−1\n2) and (+1,−3\n2) as a func-\ntion of the spin-flip Mn hole exchange Jh⊥, are shown in\nfigure (7). The transition to the (+2 ,−5\n2), which only\ninvolves the irreversible spin flip of the hole via a phonon\nemission is the dominant process and has a lifetime of 30\nns. The transition to the (+1 ,−3\n2) state requires both\nthe hole spin flip and the Mn-hole spin flip and it is 3\norders of magnitude less efficient.\nThus, these calculations indicate that the most likely\nmechanism for Mn spin orientation in the presence of\nan exciton combines a rapid bright-to dark conversion,\nproduced by phonon induced hole spin flip, and a dark to\nground transition, enabled by Mn-carrier spin exchange\nand radiative recombination.\nVI. LASER DRIVEN SPIN DYNAMICS\nA. Summary of scattering mechanisms and master\nequation\nThespindynamicsofasingleMnatominalaserdriven\nquantumdotisdescribedintermsofthe24excitonstates\nΨnand the 6 ground states φm. In the previous sections\nwe have calculated the scattering rates of these states.\nThey can be summarized as follows:\n1. Transitions from the Ψ nto theφm, via photon\nemission (eq. (41)). In the case of bright excitons,\nthis processis the quickest ofall, with a typical life-\ntime of 0.3 ns. In the case of dark excitons the life-\ntime depends on the bright/dark mixing, which is\nboth level and dot dependent. Dark lifetime ranges\nfrom twice the one of bright excitons to 1000 times13\nlarger, ie, between 1 and 300 nanoseconds. In any\nevent, dark recombination involves a Mn spin flip.\n2. Transitions between different φmstates, due to Mn\nspin phonon coupling (eq. (53)). The lifetimes of\nthese transitions are, at least, 1ms (see right panel\nof figure 4).\n3. Transitions between different exciton states Ψ n\nthatflip the spinofthe Mn only, dueto Mn-phonon\ncoupling (eq. (59). The lifetimes of these transi-\ntions are, at least, 0.1ms (see left panel figure 4).\n4. Transitions between exciton states due to hole-\nphonon coupling (eq. (70). The bright to dark\ntransition is the quickest process with a lifetime of\nabout 30ns (see figure 7). Bright to bright tran-\nsitions, combining hole-phonon and Mn-carrier in-\nteractions, have lifetimes in the 10 µsrange.\nIn addition to these dissipative scattering processes,\nwe have to consider driving effect of the laser field, de-\nscribed in the semiclassical approximation. All things\nconsidered, we arrive to a master equation that describes\nthe evolution of the occupations pN, whereN= (n,m)\nincludes states both with and without exciton in the dot.\nThe master equation reads:\ndpN\ndt=/summationdisplay\nN′ΓN′→NpN′−/summationdisplay\nN′ΓN→N′pN(78)\nEq. (78) is a system of 36 coupled differential equations\nthatwesolvebynumericaliteration, startingfromather-\nmal distribution for the initial occupation pN. Since the\ntemperature is larger than the energy splitting in the\nground state, but much smaller than the band gap, at\nt= 0 we have the six ground states with similar occu-\npationPm≃1/6,Pn≃0. As a result, the average\nmagnetization, defined as:\n∝angb∇acketleftMz∝angb∇acket∇ight=/summationdisplay\nmpm∝angb∇acketleftφm|Mz|φm∝angb∇acket∇ight (79)\nis zero, at zero magnetic field, as expected.\nB. Optical Mn spin orientation\nUnder the action of the laser, the exciton states be-\ncome populated and, under the adequate pumping con-\nditions, the average Mn magnetization ∝angb∇acketleftMz∝angb∇acket∇ightacquires a\nnon-zero value. This transfer of angular momentum,\nknown as optical Mn spin orientation has been observed\nexperimentally1and predicted theoretically27. It results\nfrom a decrease of the Mn spin lifetime in the pres-\nence of the exciton in the dot. In that circumstance,\nthe laser transfer population from the Mzstate to the\nX,Mzstate. The enhanced relaxation transfers popula-\ntion fromX,MztoX,M′\nzand the recombination to M′\nz\nstate. Thus, if the laser is resonant with a single Mzto0 200 400\nTime(ns)00.0050.010.015PL (Arb. Units)Rategeneration=1ns-1\nRategeneration=0.5ns-1\nRategeneration=0.2ns-1\n0.5 1\nRategeneration(ns-1)0100200300400500600τpump\nFIG. 10: (Color online) Evolution of Mn spin orientation\nefficiency as a function of the laser power. The pumping is\ndetected in the PL intensity from state ( X= +1,Mz=−5\n2)\nundertheinfluenceofadrivinglaser pumpingthesystemfrom\noptical ground state Mz=−5\n2to the excited state ( X=−1,\nMz=−5\n2). The inset presents the laser power dependence\nofτpump, from where we can see that the efficiency of the\npumping gets higher with the increasing of the laser power.\nQuantity Symbol Value\nHole-Mn exchange jh0.31 meV\nElectron-Mn exchange je-0.09 meV\nElectron-Hole jeh-0.73 meV\nUniaxial Anisotropy D10µeV\nIn plane Anisotropy E 0\nQuantum dot width Ly6nm\nQuantum dot width Lx5nm\nQuantum dot height Lz3nm\nTABLE I: Parameters used in the simulation of the resonant\nPL observed in the time resolved optical pumping experi-\nments.\nX,Mztransition, the Mzstate is depleted, which results\nin a decrease of the PL coming both from the X,Mzand\nthe−X,Mztransitions.\nIn figure (8) we show the result of our simulations for\na dot at thermal equilibrium ( kBT= 4K) at t= 0\nwhich is pumped with a laser pulse resonant with the\nX=−1,Mz=−5\n2transition, which is the high en-\nergy one, since the hole is parallel to Mn spin. The\nlaser pulse has a duration of 300 nanoseconds, so that\nthe spectral broadening is negligible. In the upper panel\nwe plot the PL coming from the counter-polarizedtransi-\ntion,X= +1,Mz=−5\n2, which has lower energy and can\nbe detected without interference with the laser, for two\ndifferent pumping power intensity. It is apparent that af-\nter a rise of the PL in a time scale of 8 ns, corresponding\nthe spin relaxation of the exciton spin from X=−1 to\nX= +1, the PL signal is depleted. The origin of the\ndepletion is seen in figure (9). The occupation of the14\n0.06\n0.03\n0.00Jh⊥(meV)\n5.04.0\nLx(nm)400\n200τpump(ns)0.010\n0.005\n0.000PL(Arb. Units)\n600 400 200 0\nTime(ns) Lx=5nm\n Lx=5.3nm\n Lx=5.6nm\nFIG. 11: (Color online) Evolution of Mn spin orientation effi-\nciency as a function of the valence band mixing controlled by\nthe anisotropy of the confinement potentiel (L y=13nm, vari-\nable L x). The pumping is detected in the PL intensity from\nstate (X= +1,Mz=−5\n2) under the influence of a driving\nlaser pumpingthe system from optical ground state Mz=−5\n2\ntotheexcitedstate ( X=−1,Mz=−5\n2). Theinset shows the\nevolution of τpumpwith L xand J h⊥. The exciton generation\nrate is fixed at 1 ns−1.\nMz=−5\n2spin state in the ground reduced down to zero,\nin benefit of the other Mn spin states.\nAccordingly, theaveragemagnetizationbecomesfinite.\nThus, net angular momentum is transferred from the\nlaser to the Mn spin. The transfer takes place through\nMn spin relaxation enabled in the presence of the exci-\nton. As discussed above, the most efficient mechanism\ncombines hole-spin relaxation due to phonons combined\nwith dark-bright mixing, which involves a Mn spin flip.\nInterestingly, the fact that in the steady state several\nMn spin states are occupied, including the higher energy\nones, is compatible with a picture in which the Mn spin\nis precessing. Thus, a steady supply of spin-polarized\nexcitons in the dot would result in the precession of the\nMn spin, a scenario similar to that of current drive spin-\ntorque oscillators45. Further work necessary to confirm\nthis scenario is outside the scope of this paper.\nThe efficiency of the process increases with the laser\npower, as shown in figure (10). We define the spin ori-\nentation time τpumpas the time at which the PL of the\ncounter polarized transition is half the maximum. We\ncan see that, as expected, τpumpis a decreasing func-\ntion of the laser power. A pumping time τpump≃90ns\nis obtained with a generation rate of about 1ns−1. The\namplitude of the valence band mixing, controlled by the\nanisotropy of the confinement potential or the in-plane\nstrain distribution, is the main quantum dot parame-\nter controlling the efficiency of the optical pumping. As\npresented in figure (11), decreasing the quantum dot\nanisotropy, i.e., decreasing the LH-HH mixing parameterJh⊥, produces a rapid increase of τpump(inset of figure\n(11). This is a direct consequence of the reduction of the\nphonon induced hole spin flip.\nVII. SUMMARY AND CONCLUSIONS\nWehavestudiedthespindynamicsofasingleMnatom\nin a CdTe quantum dot excited by a laser that drives the\ntransition between the 6 optical ground states, associ-\nated to the 2 S+1 states of the Mn spin S=5\n2, and the\n24 single exciton states, corresponding to X=±1,±2\nstates interacting with the Mn spin. The main goal is\nto have a microscopic theory for the Mn spin relaxation\nmechanisms that makes it possible to produce laser in-\nduced Mn spin orientation in a time scale of less than\n100 ns.1–3For that matter, we need to describe how the\nMn and the quantum dot exciton affect each other.\nIn section (II) we describe the different terms in the\nMn spin Hamiltonian, including exchange with the 0-\ndimensional exciton. The symmetry of the exchange in-\nteraction depends on the spin properties of the carriers,\nwhich in the case of holes are strongly affected by the\ninterplay of confinement, strain and spin orbit coupling.\nIn section (II) we also use a model for holes18,23,30in\nquantum dots, which permits to obtain analytical ex-\npressionsfor the wavefunctions ofthe holes, the hole-Mn\nexchange, in terms of the dimensions of the dot and the\nKohh-Luttinger Hamiltonian.\nIn section (III) we study the dissipative dynamics of\nthe Mn spin due to its coupling to phonons, both with\nand without excitons in the dot. The Mn spin-phonon\ncoupling arises from the time dependent stochastic fluc-\ntuations of the crystal field and thereby of the single\nion magnetic anisotropy, induced by the phonon field.\nWhereas the Mn spin relaxation is accelerated by 2\nor 3 orders of magnitude in the presence of the exci-\nton, the efficiency of this mechanism is too low to ac-\ncount for the optical orientation of the Mn spin reported\nexperimentally1–3. The small Mn spin-phonon coupling\ncomes from the small magnetic anisotropy of Mn as a\nsubstituional impurity in CdTe.\nIn section (IV) we describe the interaction between\nthe hole spin and the phonons in non-magnetic dots. Us-\ning the simple analytical model for the holes presented\nin section II we obtain analytical formulas for the hole\nspin relaxation. We find that hole spin lifetime can be\nin the range of 30 ns for a hole spin splitting as large\nas that provided by the hole-Mn coupling. Thus, we ex-\npect that bright excitons will relax into dark excitons\nvia hole-spin relaxation. This provides a microscopic\nmechanism to the scenario for Mn spin relaxation pro-\nposed by Cywinski29: bright excitons relax into dark ex-\ncitons, via carrier spin relaxation, and the joint process\nof Mn-carrier spin exchange couples the dark excitons\nto the bright excitons, resulting in PL from dark states\nwhich implies Mn spin relaxation in a time scale of a few\nnanoseconds. This scenario is confirmed by calculations15\npresented in section V. Finally, in section VI we present\nthe master equation that governs the dynamics of the 30\nstates of the dot, we solve it numerically and we model\nthe optical Mn spin orientation reported experimentally.\nOur main conclusions are:\n•Mn spin-phonon spin relaxation is presumably too\nweak to account for Mn spin dynamics in the pres-\nence of the exciton\n•The Mn spin orientation is possible in a time scale\nof one hundred nanoseconds via a combination of\nphonon-induced hole spin relaxation and the sub-\nsequent recombination of the dark exciton enabled\nby spin-flip exchange of the Mn and the carrier\n•The critical property that governs the hole-Mn ex-\nchange and the hole spin relaxation is the mixing\nbetween light and heavy holes, which depends both\non the shape of the dot and on strain.\n•Our microscopic model permits to account for the\noptically induced Mn spin orientation.\nFuture work should address how the coupling of the\nelectronic Mn spin to the nuclear spin modifies our re-sults. This probably plays a role for the Mn in the dot\nwithout excitons. In addition, future work should study\nthe role played by Mn spin coherence, and the interplay\nbetween optical and spin coherence.\nAcknowledgments\nWe thank F. Delgado, C. Le Gall, R. Kolodka\nand H. Mariette for fruitful discussions. This\nwork has been financially supported by MEC-Spain\n(Grants MAT07-67845, FIS2010-21883-C02-01, and\nCONSOLIDER CSD2007-00010), Generalitat Valen-\nciana (ACOMP/2010/070), Fondation NanoScience\n(RTRA Genoble) and French ANR contract QuAMOS.\nAppendix A: Kohn Luttinger Hamiltonian\nThe Kohn-Luttinger Hamiltonian for the 4 topmost\nvalence bands of a Zinc Blend compound are given by:\nH(kx,ky,kz) =\nP+Q−3\n2κνBB S R 0\nS†P−Q−1\n2κµBB 0 R\nR†0P−Q+1\n2κµBB −S\n0 R†−S†P+Q+3\n2κνBB\n(A1)\nwhere36\nP=/planckover2pi12γ1k2\nz+k2\n⊥\n2m0Q=/planckover2pi12γ1−2k2\nz+k2\n⊥\n2m0(A2)\nS= 2√\n3γ2/planckover2pi12kzk⊥\n2m0(A3)\nand\nR=−√\n3/planckover2pi12\n2m0/parenleftbig\n−γk2\n−+µk2\n+/parenrightbig\n(A4)\nwhereγ1,2,3are dimensionless material dependent pa-\nrameters,γ=1\n2(γ2+γ3),µ=1\n2(γ2−γ3),m0is the free\nelectron mass, k2\n⊥=k2\nx+k2\nyandk±=kx±iky. For the\ndot states the relevant parameters are:\nP=/planckover2pi12\n2m0γ1π2/parenleftbigg1\nL2z+1\nL2x+1\nL2y/parenrightbigg\n(A5)Q=/planckover2pi12γ1\n2m0π2/parenleftbigg−2\nL2z+1\nL2x+1\nL2y/parenrightbigg\n(A6)\nR=−/planckover2pi12π2\n2m0√\n3γ2/parenleftbigg1\nL2x−1\nL2y/parenrightbigg\n(A7)\nAppendix B: General formula for phonon-induced\nspin-flip rate\nIn this appendix we derive a general formula for the\nscattering rate between two electronic state nandn′in-\nduced by a phonon emission. The Hamiltonian of the\nsystem can be split in 3 parts, the electronic states n, the\nphonon states, and their mutual coupling. The phonon\nstates are labelled according to their polarization and\nmomentum, λ,/vector q. We consider the following coupling\nV=/summationdisplay\nm,m′,/vector q,λVm,m′\n/vector q,λ|m∝angb∇acket∇ight∝angb∇acketleftm′|/parenleftig\nb†\nλq+bλ,−q/parenrightig\n(B1)16\nwheremandm′are electronic states. We refer to the\nfree phonon states as the reservoir states. Within theBorn-Markovapproximation,the scatteringratebetween\nstatesnandn′.\nΓn→n′=2π\n/planckover2pi1/summationdisplay\nrPr/summationdisplay\nr′|∝angb∇acketleftnr|V|n′r′∝angb∇acket∇ight|2δ(En−En′+er−er′) (B2)\nwherePris the occupation of the rreservoir state with\nenergyer. This equation can be interpreted as a statis-\ntical average over reservoir initial states rof the Fermi\nGolden rule decay rate of state N,r.\nThe sums over randr′are performed using the fol-\nlowing trick. For a given r, the initial reservoir state, r′\nmust have an additional phonon, since we consider the\nphonon emission case. Thus, we write:\n|r′∝angb∇acket∇ight=1/radicalbignλ′,q′+1b†\nλ′,q′|r∝angb∇acket∇ight (B3)\nso that\n∝angb∇acketleftr|b†\nq,λ+b−q,λ|r′∝angb∇acket∇ight=δ−q,q′δλ,λ′/radicalig\nnr\nλ′,q′+1 (B4)The matrix element\n∝angb∇acketleftnr|V|n′r′∝angb∇acket∇ight=Vn,n′\n/vector q,λ/radicalig\nnr\nλ′,q′+1 (B5)\nWe see how from allthe terms in the sum that defines the\ncoupling, only one survives and fixes the index r′. Thus,\nthe only the sums left are the over the initial reservoir\nstates and the λ,qindex that define the final state. Now\nwe use the definition of the Bose function:\n/summationdisplay\nrPr/parenleftbig\nnr\nλ′,q′+1/parenrightbig\n=nB(ωλ′(q′))+1 (B6)\nand we arrive to the following expression for the rate:\nΓn→n′=2π\n/planckover2pi1/summationdisplay\nλ,q|Vn,n′\n/vector q,λ|2(nB(ωλ(q))+1)δ(En−En′−ωλ(q)) (B7)\nNotice that it is possible to write the rate as a sum\nover different contributions arising from different po-\nlarizations, Γ =/summationtext\nλΓλ. 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We develop a phenomenologi-\ncal description of the coupled dynamics of the superconducting condensate and the spin system,\nand demonstrate that supercurrents produce a reactive spin-orbit torque on the magnetization. By\nperforming a microscopic self-consistent calculation, we show that the spin-orbit torque originates\nfrom a spin-polarization of the Cooper pairs due to current-induced spin-triplet correlations. In-\nterestingly, we \fnd that there exists an intrinsic limitation for the maximum achievable spin-orbit\ntorque, which is determined by the coupling strength between the condensate and the spin system.\nIn proximitized hole-doped semiconductors, the maximum achievable spin-orbit torque \feld is esti-\nmated to be on the order of 0 :16 mT, which is comparable to the critical \feld for current-induced\nmagnetization switching in ferromagnetic semiconductors.\nI. INTRODUCTION\nIn metallic ferromagnets, spin-polarized currents can\ninduce a spin-transfer-torque (STT) on the magneti-\nzation via direct transfer of spin-angular momentum\nfrom the itinerant charge carriers to the local magnetic\nmoments.1This phenomenon has opened the door for\ncurrent-driven manipulation of magnetization in spin-\ntronic devices. However, a limiting factor is the high\ncurrent densities required to switch the magnetization\nand the associated large dissipation and Joule heating.\nA new and alternative current-induced spin-torque\nmechanism has been observed in systems with broken\nspatial inversion symmetry and strong spin-orbit cou-\npling (SOC).2{8Due to the SOC of these systems, an\nelectric current is always accompanied by a net spin-\npolarization of the charge carriers,9{17which via the\nexchange interaction produces a torque on the magne-\ntization. Such relativistic current-induced torques are\ncommonly referred to as spin-orbit torques (SOTs). In\ncontrast to the STTs, the SOTs require neither spin-\npolarizers nor textured ferromagnets to induce magneti-\nzation dynamics. Furthermore, the SOTs show remark-\nably high torque e\u000eciencies,2{4leading to magnetization\nreversal at current densities that are approximately an\norder-of-magnitude smaller than that typically observed\nfor STT-induced switching in metallic systems.\nIn the past few years, there has been a rapidly grow-\ning interest in superconductor-ferromagnet heterostruc-\ntures with strong SOC.18{27These systems are particu-\nlarly intriguing because they are considered to be promis-\ning platforms for realizing topological superconductiv-\nity. This was recently experimentally demonstrated for a\nchain of magnetic atoms placed on a conventional super-\nconductor, where signatures of Majorana fermions at the\nedges of the chain were observed.22Topological super-\nconductivity has also been predicted in two-dimensional\nlattices of ferromagnetically ordered adatoms.23{27\nSo far, there is little knowledge of how supercurrents\nin these systems in\ruence the ordered spins. However,\nkxky\nk + Q/2 \n-k + Q/2 Q/2 \nca\nb\nE( k)\n00\nkxkyFIG. 1: (color online). (a) Ferromagnetically ordered adatom\nspins on a superconductor with strong SOC. (b) Energy dis-\npersion of a system with Rashba SOC and an exchange \feld\nalong they-axis. A typical Fermi surface of this system is\nillustrated in (c). Optimal Cooper pairing occurs for mo-\nmentum states with a \fnite center of mass momentum, i.e.,\nk+Q=2 and\u0000k+Q=2.\nstudies on ferromagnetic Josephson junctions have shown\nthat supercurrents can produce a torque on the ferromag-\nnetic interlayer via the creation of spin-triplet Cooper\npairs.28{34The spin-triplet correlations in the interlayer\nare generated either via magnetic textures, ferromagnet-\nnormal metal-ferromagnet trilayers, or SOC. Supercon-\nductors with broken spatial inversion symmetry and\nstrong SOC will be in a mixed superconducting state\nof singlet and triplet pairings.35The Cooper pairs can\nthus develop a net spin polarization when the time re-\nversal symmetry is broken. Because the superconduc-\ntor/adatom systems have both broken spatial inversionarXiv:1606.08470v1  [cond-mat.mes-hall]  27 Jun 20162\nsymmetry and strong SOC, an interesting question is\nwhether supercurrents in these systems result in current-\ndriven magnetization dynamics.\nIn this work, we consider a lattice of ferromagneti-\ncally ordered spins in contact with a conventional su-\nperconductor with broken spatial inversion symmetry\nand strong SOC (Fig. 1a). We \fnd that supercur-\nrents produce a reactive SOT on the spins and formu-\nlate a phenomenological description of the supercurrent-\ninduced magnetization dynamics. In contrast to nor-\nmal metallic ferromagnets, the back-action of the spin\ndynamics on the superconducting system is important,\nand the resulting equations for the condensate and the\nspin system should be solved simultaneously to provide\na correct description of the dynamics. Furthermore,\nwe study the spin-torque mechanism by using a tight-\nbinding Bogoliubov-de Gennes (BdG) formalism to self-\nconsistently calculate the response of the system to a\nsupercurrent. We show that the SOT originates from\ncurrent-induced spin-triplet correlations, which is deter-\nmined by the orientation of the supercurrent with respect\nto the crystallographic axes. Moreover, we \fnd that there\nexists an intrinsic limitation for the maximum achievable\nSOT and estimate the corresponding e\u000bective SOT \feld\nto be on the order of 0 :16 mT in proximitized hole-doped\nsemiconductors, which is comparable to the critical SOT\n\feld for magnetization reversal in (Ga,Mn)As. We there-\nfore believe that the supercurrent-induced SOTs can lead\nto the development of new e\u000ecient techniques for manip-\nulating magnetization, which minimize the disadvantages\nassociated with dissipation and Joule heating.\nII. PHENOMENOLOGICAL DESCRIPTION\nIn what follows, we develop a phenomenology that cap-\ntures the low-frequency long-wavelength physics of the\nsupercurrent-induced magnetization dynamics. The su-\nperconducting condensate is treated in the framework\nof the Ginzburg-Landau theory, which is valid at length\nscales larger than the superconducting coherence length\n\u00100. We assume a homogeneous ferromagnetic equilibrium\nstate and consider spatial modulations of the ferromag-\nnetic order parameter at length scales much larger than\nthe exchange length lex\u0018p\nJ=K set by the spin sti\u000b-\nnessJand relevant anisotropy constants K. Typically,\nlex\u001810\u0000100 nm and \u00100\u001840\u0000360 nm.1,36The char-\nacteristic frequency !of ferromagnets is on the order\nof!\u00181 GHz,1which is far below the typical energy\ngap of superconductors: \u0016 h! << \u0001\u00180:18\u00001:5 meV.36\nThe magnetization precession will therefore not lead to\nquasi-particle excitations in the superconductor and we\ncan assume that the condensate responds adiabatically\nto the magnetization dynamics.\nWe start by formulating the free energy functional,\nF[m; ;A], of the system:\nF=Z\ndr[Fm(m) +Fme(m; ;A) +Fe( ;A)]:(1)Here, m(r;t) represents the order parameter of the spin\nsystem and is a unit vector parallel to the magneti-\nzation M(r;t) =Msm(r;t), (r;t) is the order pa-\nrameter \feld of the superconductor, and A(r;t) is the\nmagnetic vector potential that yields the magnetic \feld\nB(r;t) =r\u0002A(r;t).FmandFeare the free energy\ndensities of the isolated spin system and superconduct-\ning condensate, respectively,1,36\nFm=X\nijJij\n2@im\u0001@jm+U(m);\nFe=X\nijKij(\u0005i )\u0003(\u0005j ) +\u000bj j2+\f\n2j j4+B2\n8\u0019;\nwhereUdescribes the magnetic anisotropy energy, \u0005=\n\u0000i\u0016hr\u0000(2e=c)Ais the momentum operator of the con-\ndensate, 2eis the charge of the Cooper pairs, and cis the\nspeed of light. JijandKijare second-rank polar tensors,\nwhich are invariant under the symmetry point group of\nthe system.37\nThe termFmedescribes the coupling between the su-\nperconductor and the magnetization. We consider weak\nmodulations of a homogenous ferromagnetic equilibrium\nstate and can thus neglect magnetoelectric coupling ef-\nfects associated with magnetic textures. In this case, Fme\nis governed by the Lifshitz invariant38\nFme=\u0000X\nij\u0014ijmi\u0003j; (2)\nwhere \u0003= \u0003\u0005 + \u0005\u0003 \u0003represents the momentum\ndensity of the superconducting condensate. The tensor\n\u0014ijis linear in the SOC and is an invariant axial ten-\nsor of the point group.37Consequently, the tensor van-\nishes for systems with spatial inversion symmetry. Fme\ncan be derived microscopically by considering an s-wave\nsuperconductor with SOC of the form \u0011so;ij\u001bipj(where\n\u0011so;ij/\u0014ij) and calculate the energy change due to a\nZeeman \feld.38\nSo far, most works have concentrated on the e\u000bects of\nthe Lifshitz invariant (2) in non-centrosymmetric super-\nconductors exposed to an external magnetic \feld. How-\never, two recent studies showed that Fmeleads to persis-\ntent currents in a conventional superconductor with SOC\nwhen magnetic impurities are placed at the surface.39\nFmecouples the momentum of the condensate to the\ndirection of the magnetization and favors a spatial mod-\nulation of in equilibrium. The physical origin of Fme\nis an SOC-induced shift of the Fermi surface, leading to\na \fnite center of mass momentum of the Cooper pairs.\nTo illustrate this phenomenon, consider a system with\nRashba SOC and a Zeeman splitting h0along they-\naxis induced by the magnetization (Fig. 1b): H(k) =\n\u0016h2k2=2m+\u000bR(k\u0002^z)\u0001\u001b+h0\u001by. Here, \u001bis a vector consist-\ning of the Pauli matrices, mis the e\u000bective quasi-particle\nmass, and\u000bRparameterizes the SOC. For this system,\nthe Lifshitz invariant becomes Fme=\u0000\u0014(^z\u0002m)\u0001\u0003. The\nFermi surface of the Hamiltonian is two circles, whose3\ncenters are shifted in opposite directions along the x-axis\n(Fig. 1c). Due to the shift of the Fermi surface, the op-\ntimal Cooper pairing occurs for momentum states with\na \fnite center of mass momentum, i.e., k+Q=2 and\n\u0000k+Q=2. Therefore, the order-parameter \feld gains a\nspatial modulation  \u0018exp(iQ\u0001r) in equilibrium; a state\nthat is referred to as the helical phase.38Phenomenolog-\nically, the helical phase is captured by the Lifshitz in-\nvariantFme\u0018\u0000\u0014(^z\u0002m)\u0001Q, which favors the vector\nQto be perpendicular to the in-plane component of the\nmagnetization.\nIn what follows, we demonstrate that the Lifshitz in-\nvariant also leads to a reciprocal phenomenon of the he-\nlical phase. If  is forced to have a spatial modulation\n\u0018exp(iq\u0001r) such that a supercurrent is induced, then the\ncondensate can via Fmelower its energy by developing a\nnet spin density Sind(and magnetic moment mind) per-\npendicular to the vector q:Fme\u0018\u0000\u0014(^z\u0002mind)\u0001q<0.\nImportantly, we \fnd that the induced spin density Sind\nproduces a novel SOT on the magnetization.\nThe magnetization dynamics is described by the\nLandau-Lifshitz-Gilbert (LLG) equation1\n_m=\u0000\rm\u0002[He\u000b+Hso] +\u000bGm\u0002_m: (3)\nHere, He\u000b=\u0000(1=Ms)\u000eFm=\u000emis the e\u000bective \feld found\nfrom the magnetic free energy functional Fm=R\ndrFm,\n\ris the gyromagnetic ratio, and the term proportional\nto the Gilbert damping parameter \u000bGdetermines the\nmagnetization dissipation. Because of the Lifshitz in-\nvariant (2), the variation of Eq. (1) with respect to the\nmagnetization also yields a reactive SOT-\feld\nHso;i=X\nj\u0014ij\u0003j=Ms; (4)\nwhich is governed by the SOC and the momentum density\nof the superconducting condensate.\nThe magnetization evolves slowly on the characteris-\ntic timescale of the electron dynamics. We can therefore\nassume that the superconducting condensate at time tis\nclose to the equilibrium state with the static magnetiza-\ntionm(t). The equilibrium state, which is determined\nby the Ginzburg-Landau (GL) equations, is obtained by\nvariational minimization of the free energy (1). The vari-\nation with respect to  \u0003yields the equation\nX\nijKij\u0005i\u0005j +\u000b +\fj j2 \u00002\u0014ijmi\u0005j = 0;(5)\nwhereas a variation of Aprovides the equation\njs;i\n2e=X\njKij \u0003\u0005j +Kji \u0005\u0003\nj \u0003\u00002\u0014jimjj j2:(6)\nHere, js= (c=4\u0019) (r\u0002B) is the supercurrent density.\nThe conventional GL equations are obtained for a fully\nisotropic system, in which Kij=K\u000eijand\u0014ij= 0.\nEqs. (3), (5) and (6) give a phenomenological descrip-\ntion of the coupled dynamics of the spin system and thesuperconducting condensate. Via the Lifshitz invariant,\nthe state of the superconducting condensate depends on\nthe direction of the magnetization. The e\u000bects of Fmebe-\ncome crucially important when the length scale 2 \u0019=Q as-\nsociated with the helical wavevector Qis smaller than the\ncharacteristic length scales of the ferromagnetic system.\nIn this case, the condensate is strongly a\u000bected by the\nmagnetization dynamics and its state cannot be consid-\nered as quasi-static for the dynamics. Thus, Eqs. (3), (5)\nand (6) should be solved simultaneously to provide a cor-\nrect description of both the magnetization dynamics and\nthe superconducting condensate. This di\u000bers markedly\nfrom the situation in normal metallic ferromagnets, in\nwhich the back-action of the spin dynamics on the itin-\nerant electron system usually can be disregarded in the\nsolution of the LLG equation.\nFor a two-band model with Rashba SOC, the helical\nwavevector is on the order of Q\u0018\u000eNh 0=\u0016hvFwherevF\nis the Fermi velocity.40Here, the factor \u000eN= (N+\u0000\nN\u0000)=(N+\u0000N\u0000) measures of the di\u000berence between the\ndensity of states N\u0006of the two bands at the Fermi en-\nergy. In the limit \u000bR=vF<< 1, it is determined by\n\u000eN= 2\u000bR=\u0016hvF. To get some insight into the typical\nscale ofQ, let us estimate Qfor the proximitized hole-\ndoped semiconductor system studied in Sec. III. For this\nsystem, we \fnd the helical wavevector Q\u00181:7\u0002107\nm\u00001(material parameters are given in Sec. III A). This\nis about an order of magnitude larger than the wavevec-\ntor observed for the pair potential in a proximitized HgTe\nquantum well system subjected to an in-plane magnetic\n\feld of 1 T.41Thus, it is likely that the spatial modu-\nlation of the order parameter \feld  becomes important\nfor the magnetization dynamics at length scales larger\nthan 0:1\u00001:0\u0016m.\nIII. MICROSCOPIC CALCULATION\nTo gain a better understanding of the underlying phys-\nical mechanisms of the SOT, we will now use the BdG\nformalism to self-consistently calculate the response of\nthe system to a supercurrent.\nA. Model\nWe model the two-dimensional superconductor by the\ntight-binding Hamiltonian\nH=\u0000~tX\nhijicy\nicj\u0000\u0016X\nicy\nici+X\nicy\ni(hi\u0001\u001b)ci+ (7)\niX\nhijicy\ni\u0010\n\u001b\u0001\u0011so\u0001^dij\u0011\ncj+X\ni\u0010\n\u0001icy\ni\"cy\ni#+h:c:\u0011\n:\nHere, cy\ni= (cy\ni\"cy\ni#), wherecy\ni\u001cis a fermionic creation\noperator that creates a particle with spin \u001cat lattice site\ni= (x;y) and the symbol hijiimplies a summation over4\nnearest lattice sites. ~tis the spin-independent hopping\nenergy, hiis the Zeeman splitting induced by the adatom\nspins ( hiandm(i) are collinear), and ^dijis a unit vector\nthat points from site jto site i.\nThe second-rank tensor ( \u0011so)ijparameterizes the SOC.\nWe consider a system described by the C2vpoint group,\nin which the SOC can be decomposed in two terms hav-\ning Rashba and Dresselhaus symmetry, respectively. For\nRashba SOC, the SOC tensor takes the form \u0011so=\n~\u000bRi\u001by, whereas the speci\fc form the Dresselhaus SOC\ndepends on how the coordinate system is \fxed with re-\nspect to the crystallographic axes. If the x-axis is along\none of the two re\rection planes of C2v, then the Dressel-\nhaus SOC tensor is \u0011so= ~\u000bD\u001bx. With respect to this\nreference frame, a rotation of the axes by \u0019=2 degrees\nabout thez-axis leads to a sign change of ~ \u000bD, while a ro-\ntation of\u0019=4 degrees changes the tensor to \u0011so= ~\u000bD\u001bz.\nNote that ( \u0011so)ijand\u0014ijsatisfy the same transforma-\ntion rules and thus have the same tensorial forms, i.e.,\n(\u0011so)ij/\u0014ij.\n\u0001i=Vhci\"ci#idescribes the superconducting s-wave\npairing and is determined by\n\u0001i=\u0000V\n2X\nn\u001c\u001c0(i\u001by)\u001c\u001c0v\u0003\nn\u001c(i)un\u001c0(i) [1\u00002f(\u000fn)]:(8)\nHere,V > 0 is the on-site attractive interaction between\nthe quasi-particles, h:::idenotes the thermal average, f(\u000f)\nis the Fermi-Dirac distribution, and we have inserted\nthe Bogoliubov transformation ci\u001c=P\nn[un\u001c(i)\rn+\nv\u0003\nn\u001c(i)\ry\nn], where\ry\nn(\rn) are the Bogoliubov quasi-\nparticle creation (destruction) operators, which repre-\nsent a complete set of energy eigenstates: H=Eg+P\nn\u000fn\ry\nn\rn.Egis the groundstate energy; the summa-\ntion runs over positive energy eigenstates with an energy\nsmaller than the cut-o\u000b energy \u0016 h!Dset by the Debye\nfrequency!D.\nThe Hamiltonian (7) is transformed to BdG Hamilto-\nnian by using the Bogoliubov transformation36, which is\nthen iteratively solved together with the self-consistency\ncondition (8)42until the Euclidean norm of the pair\npotential (k\u0001k=pP\nij\u0001ij2) reaches a relative error\non the order 10\u00005. In the following, the Hamiltonian\n(7) is scaled with the hopping energy ~tand the chem-\nical potential, the pairing strength, the Rashba (Dres-\nselhaus) SOC, the Zeeman splitting, the thermal energy\nkBT, and the Debye frequency are set to: \u0016=~t=\u00004,\nV=~t= 5, ~\u000bR(D)=~t= 0:5,h0=~t= 0:1,kBT=~t= 0:001,\nand \u0016h!D=~t= 2:0, respectively. In Eq. (7), we use open\nboundary conditions. The hopping and Rashba energies\nin the tight-binding Hamiltonian (7) are related to a cen-\ntral di\u000berence discretization of the corresponding con-\ntinuum model via the relationships ~t= \u0016h2=2ma2and\n~\u000bR(D)=~t=ma\u000bR(D)=\u0016h2, whereais the spacing between\nthe grid points and \u000bR(D)is the SOC parameter in the\ncontinuum model. The parameter values given above\nmodel a lightly hole-doped semiconductor in proximity\nto a conventional s-wave superconductor, in which the\n0.2 \n-0.2 0\n-0.1 0.1 0.3 0.5 Rashba SOC Dresselhaus SOC\n0.2 \n0\n-0.2 \n0.5 \n0.3 \n0.1 \n-0.1 \n××\n× ×××\n× ×a\nb\nc fed\nx\nxx\nxy y\ny yFIG. 2: (color online). (a) The equilibrium state for a sys-\ntem with Rashba SOC and a Zeeman splitting \feld along\nx. The color represents the phase \u001eiof the pair potential\n\u0001i=j\u0001ijexp(i\u001ei), while the black arrows illustrate the lo-\ncal spin density S(i) = (\u0016h=2)hcy\ni\u001bcii. (b) System (a) with\nan enforced superconducting phase di\u000berence of \u0019=2 between\ntwo of the sample edges. (c) Symmetry plot of the induced\nspin density for a Rashba system. The \fgure shows the stereo-\ngraphic projection of the C2vpoint group and the blue arrows\nillustrate the orientation of the induced spin density for a su-\npercurrent along di\u000berent crystallographic directions. (d)-(f)\nShow corresponding plots for the case with Dresselhaus SOC\nand a Zeeman splitting \feld along y. In (a)-(b) and (d)-(e),\nthe size of the system is 31 \u000227 grid points.\ne\u000bective mass is m= 0:6me(meis the electron mass),\nthe SOC is \u000bR(D)= 0:21 eV \u0017A, the Fermi energy is\nEF= 2:47 meV when measured from the bottom of the\nlowest subband, and the Fermi wavelength is \u0015F\u001820\nnm, which is much larger than the discretization con-\nstanta= 3 nm.43\nB. Results and discussion\nFirst, we study the equilibrium spin density S(i) =\n(\u0016h=2)hcy\ni\u001bciiof the superconducting condensate. We\nconsider the two cases with Rashba and Dresselhaus\nSOC separately. Fig. 2a shows the self-consistent solu-\ntion for a Rashba system with an exchange \feld along\nx. The black arrows represent the spin density, while5\nI/Imax\nP/Pmax\n0.20.2\n0.40.6 0.8 1.0 00.40.60.81.0\nFIG. 3: (color online). Current-phase relation and the in-\nduced spin-polarization of the Cooper pairs for the Rashba\nsystem in Figs. 2a,b. The lines represent piecewise polyno-\nmial \fts of the data points.\nthe color illustrates the phase \u001eiof the pair potential\n\u0001i=j\u0001ijexp(i\u001ei). The phase variation perpendicular\nto the exchange \feld (i.e., along y) is a signature of the\nhelical phase. We see that the condensate has a net spin\npolarization anti-parallel to the exchange \feld. This is\nalso the case for a system with Dresselhaus SOC of the\nform \u0011so= ~\u000bD\u001bzand an exchange \feld along y(Fig. 2d).\nNote that in this case, the pair potential has a phase vari-\nation parallel to the exchange \feld, which is in agreement\nwith Eq. (2) when \u0014ij/(\u001bz)ij.\nNext, we investigate the e\u000bects of a supercurrent. A\nsupercurrent is induced along the x-axis by enforcing the\npair potential to have a constant phase in a small region\nclose to each of the two boundaries along x. We set\nthe widths of these two regions to three lattice points.\nThus, the pair potential is solved self-consistently for the\nentire sample except for the two regions at the boundaries\nwhere the phase \u001eiis kept \fxed (however, the magnitude\nj\u0001ijis allowed to optimize itself). These two regions will\ntherefore act as sinks/sources for the supercurrent.\nFig. 2b,e shows the solution for the Rashba and Dres-\nselhaus systems with a phase di\u000berence of \u0019=2 between\nthe two boundaries. In both cases, the spin density is\ntilted away from the equilibrium value. In other words:\nthe supercurrent induces a spin-density Sind. A simi-\nlar inverse spin-galvanic e\u000bect has been theoretically pre-\ndicted for superconductors with Rashba SOC in the ab-\nsence of magnetization.44,45\nSindis solely an e\u000bect of the SOC, and its orientation\nis determined by the direction of the supercurrent rel-\native to the crystallographic axes. Fig. 2c,f shows the\nstereographic projection of the C2vpoint group, and the\nblue arrows illustrate the orientation of Sindfor di\u000berent\ndirections of the supercurrent ( r\u001e >0 along the di\u000ber-\nent directions). Generally, the supercurrent results in a\nspin density Sind;i/\u0014ij\u0003j. Via the exchange coupling,\nSindproduces a torque on the magnetization and is the\n0.1\n-0.10.0\n0.0 0.5 1.0 1.5 2.0\n×10-2 \n16 \n8\n0\n00.080.16812 16 ×10-2 \n00.08 0.16a\nb cFIG. 4: (color online). (a) A microscopic calculation of\nthe anisotropic part Fme(\u0012) =Fs(\u0012)\u0000Fs(0) of the super-\nconductor's free energy Fsfor di\u000berent directions of h=\nh0[cos(\u0012);sin(\u0012);0]. (b) The magnetoelectric anisotropy con-\nstantKme= (jFme(\u0019=2)j+jFme(3\u0019=2)j)=2 for di\u000berent values\nofh0. (c) The average energy gap for di\u000berent values of h0.\nIn all \fgures, the size of the system is 25 \u000223 grid points and\nthe phase di\u000berence between the two boundaries is \u001e= 0:8\u0019.\nThe squares represent the calculated values, while the line in\n(a) is a piecewise polynomial \ft. In (a) the free energy was\ncalculated for h0=~t= 0:1.\nphysical origin of the SOT \feld in Eq. (4): Hso/Sind.\nThe polarization of the condensate originates from\nspin-triplet correlations. Let gT+(i) =h~ci\"~ci\"i(gT\u0000(i) =\nh~ci#~ci#i) denote the amplitude for triplet pair correla-\ntions with spin up (down) along an arbitrary quan-\ntization axis, which is determined by the unitary ro-\ntation operator U\u001c\u001c0. Here, ~ci\u001c=U\u001c\u001c0ci\u001c0are the\nfermionic operators in the rotated frame. The quantity\nP=P\ni[jgT+(i)j2\u0000jgT\u0000(i)j2] represents a measure of the\nspin polarization of the Cooper pairs along the quanti-\nzation axis. In Fig. 3, we consider the Rashba system\nin Fig. 2a-b and plot Pand the supercurrent Ialongx\nas a function of the phase di\u000berence \u001ebetween the left\nand right boundaries. The spin quantization axis is along\ny. It is clear from Fig. 3 that Pis proportional to the\nsupercurrent. We obtain a similar relationship between\nthe current and Pfor the Dresselhaus system in Fig. 2d-\ne when the polarization is measured along x. Thus, we\nconclude that the underlying physical mechanism of the\nSOT \feld (4) is current-induced spin-triplet correlations.\nThe strength of the SOT \feld can be investigated by\nself-consistently calculating the free energy of the con-6\ndensate for di\u000berent directions of h=h0[cos(\u0012);sin(\u0012);0].\nHere,\u0012is the angle with the x-axis, which is parallel to\nthe direction of the supercurrent. The anisotropic part\nof the free energy is then a direct measure of the Lifshitz\ninvariant (2).\nWe consider a system with Rashba SOC. The free en-\nergy of an inhomogeneous superconductor is46\nFs=\u00001\n\fX\nnln\u0014\n2 cosh\u0012\f\u000fn\n2\u0013\u0015\n+1\nVZ\ndrj\u0001(r)j2;(9)\nwhere the sum is over the positive energy eigenstates and\n\f= 1=kBT.\nFig. 4a shows the anisotropic part Fme(\u0012) =Fs(\u0012)\u0000\nFs(0) of the free energy. The angular dependence of Fme\nfollows the functional form Fme\u0018(^z\u0002h)\u0001\u0003, which is\nconsistent with the Lifshitz invariant (2) when a current\nis applied along the x-axis (with extrema at \u0012=\u0019=2 and\n\u0012= 3\u0019=2). The di\u000berent extremum values at Fme(\u0019=2)\nandFme(3\u0019=2) is caused by a change in the momentum\ndensity due to the helical modulation (along the x-axis)\nof the order parameter \feld.\nThe e\u000bect of the Zeeman splitting h0on the SOT is\ntwofold. Firstly, it determines the coupling strength be-\ntween the spin system and the condensate and thus en-\nhances the magnetoelectric coupling Fme. Secondly, it\nsuppresses superconductivity and thus reduces the super-\ncurrent/momentum density. The competition between\nthese two counteracting e\u000bects implies that there ex-\nists an intrinsic limitation for the maximum achievable\nSOT. Fig. 4b shows the magnetoelectric anisotropy con-\nstantKme= (jFme(\u0019=2)j+jFme(3\u0019=2)j)=2. A maximum\nSOT is achieved for h0=~t\u00180:125 withKme\u00180:16~t=\n1:13 meV and corresponds the point where the Zeeman\nsplitting is comparable to the pair potential, i.e., h0\u0018\u0001.\nFor larger values of h0, the suppression of the supercon-\nductivity becomes stronger (Fig. 4c), which leads to a\nlowering of Kme.\nThe e\u000bective SOT \feld induced by the supercurrent is\nHso\u0018Kme=VMs, whereVis the volume of the ferro-\nmagnetic system. Assuming Ms= 70:8 e.m.u. cm\u00003,7\nV= 23\u000225a3, andKme= 1:13 meV, yields an SOT\n\feld on the order of Hso\u00180:16 mT. In the ferromag-\nnetic semiconductor (Ga,Mn)As, current-driven magne-\ntization switching has been observed for e\u000bective SOT\n\felds on the order of 0 :14\u00000:35 mT.2Therefore, it is\nreasonable to believe that the supercurrent-induced SOT\nis strong enough to manipulate the magnetization of the\nferromagnetically ordered spins.\nIV. SUMMARY\nIn summary, we have studied the magnetization dy-\nnamics of a two-dimensional lattice of spins in contactwith a conventional superconductor and have formulated\na phenomenological description of the coupled dynamics\nof the superconducting condensate and the magnetiza-\ntion. Interestingly, we found that supercurrents induce a\nreactive SOT \feld that originates from current-induced\nspin-triplet correlations and whose spatial orientation is\ndetermined by the symmetry of the SOC. Furthermore,\nwe showed that there exists an intrinsic limitation for\nthe maximum achievable SOT, which is determined by\nthe coupling strength between the condensate and the\nspin system. Based on material parameters for a prox-\nimitized hole-doped semiconductor, we estimated the in-\nduced SOT \feld to be on the order of 0 :16 mT.\nAppendix A: Expressions for spin-density, pair\ncorrelations and current density\nThe Hamiltonian (7) can be diagonalized by using the\nBogoliubov transformation36\nci\u001c(r) =X\nn\u0000\nun\u001c(i)\rn+v\u0003\nn\u001c(i)\ry\nn\u0001\n: (A1)\nHere,\ry\nnand\rnare the Bogoliubov quasi-particle cre-\nation and destruction operators, which satisfy fermionic\nanti-commutation relations and represent a complete set\nof energy eigenstates:\nH=Eg+X\nn\u000fn\ry\nn\rn: (A2)\nEgis the ground state energy, and the summation runs\nover positive energy eigenstates with an energy lower\nthan the cut-o\u000b energy set by the Debye frequency. The\nthermal averages of the Bogoliubov quasi-particle ex-\ncitations are given by h\ry\nn\rni=f(\u000fn), wheref(\u000f) =\n1=(exp(\f\u000f) + 1) is the Fermi-Dirac distribution. It is\nalso useful to introduce the distribution function of the\ncorresponding hole states: fh(\u000f) = 1\u0000f(\u000f).\nBy using the Bogoliubov transformation (A1), the spin\ndensity S(i) = (\u0016h=2)hcy\ni\u001bciican be expressed as\nS\u000b(i) =\u0016h\n2X\n\u001c\u001c0(\u001b\u000b)\u001c\u001c0\u001a\u001c\u001c0(i);\n\u001a\u001c\u001c0(i)\u0011X\nn[(u\u0003\nn\u001c(i)un\u001c0(i)\u0000vn\u001c(i)v\u0003\nn\u001c0(i))f(\u000fn)\n+vn\u001c(i)v\u0003\nn\u001c0(i)]:\nThe charge density \u001ai=qhniiat site iis given by the\nthermal average of the number operator ni=cy\nici, where\nqis the charge of the quasi-particles. An expression for\nthe current density js(i) is found from the Heisenberg\nequationdni=dt= (i=\u0016h)[H;n i], which yields7\n(js(i)))k=2q~t\n\u0016hX\nn\u001cIm [u\u0003\nn\u001c(i)Dkun\u001c(i)f(\u000fn) +vn\u001c(i)Dkv\u0003\nn\u001c(i)fh(\u000fn)] +\n2q\n\u0016hX\nn\u001c\u001c0h\nu\u0003\nn\u001c(i)Ak;\u001c\u001c0un\u001c0(i)f(\u000fn) +vn\u001c(i)Ak;\u001c\u001c0v\u0003\nn\u001c0(i)fh(\u000fn)i\n+2q\n\u0016hX\n\u001c\u001c0Imh\n\u0001ii\u001by;\u001c\u001c0hcy\ni\u001ccy\ni\u001c0ii\n:(A3)\nHere,Dkun\u001c(i) = [un\u001c(i+ak)\u0000un\u001c(i\u0000ak)]=2 and\nAk;\u001c\u001c0=\u0010\n\u001b\u0001\u0011so\u0001^di(i\u0000ak)\u0011\n\u001c\u001c0, where akis the lattice\nvector along k2fx;y;zg. Note that the last term van-\nishes when the pair potential satis\fes the self-consistency\ncondition. Otherwise, the term acts as a sink/source.\nEq. 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Elettr., Universit` a ”Federico II”, 80125 Napoli , Italy\n(e)ECE Dept., UMIACS, AppEl Center, University of Maryland, Co llege Park MD 20742, USA\n(Dated: March 1, 2018)\nWe study the stability of magnetization precessions induce d in spin-transfer devices by the injec-\ntion of spin-polarized electric currents. Instability con ditions are derived by introducing a general-\nized, far-from-equilibrium interpretation of spin-waves . It is shown that instabilities are generated\nby distinct groups of magnetostatically coupled spin-wave s. Stability diagrams are constructed as a\nfunction of external magnetic field and injected spin-polar ized current. These diagrams show that\napplying larger fields and currents has a stabilizing effect o n magnetization precessions. Analytical\nresults are compared with numerical simulations of spin-tr ansfer-driven magnetization dynamics.\nPACS numbers: 75.60.Jk, 85.70Kh\nCurrents of spin-polarized electrons can induce large-\namplitude magnetization precessions at microwave fre-\nquencies in small-enough magnetic devices [1, 2]. There\nis mounting experimental evidence that these so-called\nspin-transfer phenomena do occur in nano-pillaror nano-\ncontact devices under current densities of the order of\n106−108A/cm2[3–6]. This discovery has boosted the\nalreadywidespreadinterestinthephysicsoftheinterplay\nbetween magnetism and electron transport, and has trig-\ngered efforts toward the promising development of new\ngenerations of microwave spin-transfer nano-oscillators.\nA spin-transfer device is a non-linear open system,\ndriven far-from-equilibrium by the action of the spin-\npolarized electric current. The excited magnetization\nprecessions represent strong excitations of the magnetic\nmedium, which in principle may giveriseto varioustypes\nof instability and eventually to transitions to chaotic dy-\nnamics. A parallel can be drawn with ferromagnetic-\nresonance Suhl’s instabilities [7], in which certain spin-\nwavescangetcoupledtotheuniformprecessionandstart\nto growto largenon-thermal amplitudes, thus destroying\nthe spatial uniformity of the original state.\nIn this Letter, we demonstrate that spin-wave insta-\nbilities may occur in spin-transfer-driven magnetization\ndynamics as well. However, the system is far from equi-\nlibrium and the classical notion of spin-waves fails. In-\ndeed, it is the large-amplitude magnetization precession\ninduced by spin transfer that plays the role of reference\nstate, and spin-waves only exist in a generalized, non-\nequilibrium sense, as small-amplitude perturbations of\nthat state [8–10]. This scenario emerges with clarity in\nthe time-dependent vector basis in which the reference\nmagnetization precession is stationary. The spin-wave\nequations in this basis are characterized by two features:\n(i) a well-defined dispersion relation ω(q;cosθ0), whose\nnon-equilibrium nature is revealed by its explicit depen-\ndence on the magnetization precession amplitude cos θ0;\nand (ii) the presence of time-periodic coupling terms dueto the magnetostatic fields generated by individual spin-\nwaves. This coupling leads to the appearance of narrow\ninstability tongues around the parametric resonance con-\nditionω(q;cosθ0)∼ω0, whereω0is the magnetization\nprecession angular frequency.\nSpin-wave instabilities occur only for particular com-\nbinations of external magnetic field and injected spin-\npolarized current. In addition, instabilities result in lim-\nitedspatialandtemporaldistortionswhichsomewhatob-\nscure but yet do not completely disrupt the precessional\ncharacterof the originalstate. This robustness of excited\nprecessions with respect to spin-wave instabilities has a\nprecise physical origin. Indeed, the discrete nature of the\nspin-wave spectrum caused by boundary conditions in\nsub-micrometer devices reduces the number of available\nspin-wave modes which can contribute to instabilities.\nOn the other hand, the strength of the magnetostatic ef-\nfects responsible for instabilities is drastically reduced,\ndue to the ultra-thin nature of spin-transfer devices, and\ninstability thresholds are consequently enhanced. Fi-\nnally, spin-transfer-driven precessions are characterized\nby large amplitudes and, as such, are less easily masked\nby the onset of non-uniform modes. In spin-transfer\nnano-oscillators, spin-wave instabilities are expected to\nresult in increased oscillator line-widths, a conclusion\nthat might explain some of the puzzling experimental\nresults obtained in this area [11].\nTo start the technical discussion, consider a ultra-thin\ndisk with negligible crystal anisotropy (e.g., permalloy).\nTypically, this disk will be the so-called free layer of a\nnanopillar spin-transfer device (see inset in Fig. 1). The\ndisk plane is parallel to the ( x,y) plane and is traversed\nby a flow of electrons with spin polarization along the ez\ndirection. The dimensionless equation for the dynamics\nof the normalized magnetization m(r,t) (|m|2= 1) in\nthe disk in the presence of spin transfer is [1, 12]:2\nFIG. 1: (Color online) Stability diagram in (h az,β/α) con-\ntrol plane for aultra-thin permalloy disk. System paramete rs:\nα= 0.02,d= 0.6,R= 23.6,N⊥= 0.02(lengths aremeasured\nin units of the exchange length lEX= 5.72 nm). Magnetiza-\ntion is parallel to spin-polarization in region P; anti-parallel\nto spin-polarization in region A; precessing around the spin-\npolarization axis in regions OandSW. Dashed line is an ex-\nample of line of constant precession amplitude (cos θ0= 0.5)\ncomputed from Eq.(2). Spin-wave instabilities occur in reg ion\nSW. Small framed area is shown in detail in Fig. 2. Inset:\ntypical geometry of a nanopillar spin-transfer device.\n∂m\n∂t−αm×∂m\n∂t= (1)\n−m×/parenleftbig\nhazez+hM+∇2m−βm×ez/parenrightbig\n.\nHere, the external magnetic field h azezand the magneto-\nstatic field hMare measured in units of the spontaneous\nmagnetization M s, time in units of ( γMs)−1(γis the\nabsolute value of the gyromagnetic ratio), and lengths\nin units of the exchange length. The external field is\nperpendicular to the disk plane, while the spin-transfer\ntorque is simply proportional to the sine of the angle be-\ntweenmandez. The parameter βis proportional to the\nspin-polarized current density (see [12] for the detailed\ndefinition), andintypicalsituationsitiscomparablewith\nthe damping constant α.\nWhenever |haz−β/α| ≤Nz−N⊥(NzandN⊥are the\ndisk demagnetizing factors, with Nz+2N⊥= 1), Eq.(1)\nadmits time-harmonic solutions m0(t), corresponding to\nspatially uniform precession of the magnetization around\nthez-axis[13](seeFig. 1). Theprecessionamplitudeand\nangular frequency are respectively equal to:\ncosθ0=haz−β/α\nNz−N⊥, ω0=β\nα. (2)\nTo study the stability of m0(t), consider the perturbed\nmotionm(r,t) =m0(t) +δm(r,t), with|δm(r,t)| ≪1.\nThe correspondingmagnetostaticfield will be: hM(r,t) =\n−Nzm0z−N⊥m0⊥+δhM(r,t), whereδhMrepresents themagnetostatic field generated by δm. Since we are inter-\nested in ultra-thin layers, we shall assume that δmdoes\nnot depend on z:δm(r,t) =δm(x,y,t).\nThe perturbation δmis orthogonal to m0(t) at all\ntimes, since the local magnetization magnitude |m|2= 1\nmust be preserved. Hence, it is natural to represent δm\nin the time-dependent vector basis ( e1(t),e2(t)) defined\nin the plane perpendicular to m0(t), withe2(t) paral-\nlel toez×m0(t) ande1(t) such that ( e1,e2,m0) form\na right-handed orthonormal basis. The perturbation can\nbe written as: δm(r,t) =δm1(r,t)e1(t)+δm2(r,t)e2(t).\nBylinearizingEq.(1)around m0(t)andaveragingthelin-\nearized equation over the layer thickness, one obtains the\nfollowing coupled differential equations in matrix form:\n/parenleftbigg\n1α\n−α1/parenrightbigg∂\n∂t/parenleftbigg\nδm1\nδm2/parenrightbigg\n=/parenleftbigg\n0 1\n−1 0/parenrightbigg/parenleftbigg\n/angb∇acketleftδhM/angb∇acket∇ight1\n/angb∇acketleftδhM/angb∇acket∇ight2/parenrightbigg\n+\n+/parenleftbigg0N⊥+∇2\n⊥\n−N⊥−∇2\n⊥0/parenrightbigg/parenleftbiggδm1\nδm2/parenrightbigg\n, (3)\nwhere∇2\n⊥=∂2/∂x2+∂2/∂y2, while/angb∇acketleft.../angb∇acket∇ightrepresents the\nzaverage over the thickness of the disk, and /angb∇acketleftδhM/angb∇acket∇ight1=\n/angb∇acketleftδhM/angb∇acket∇ight·e1(t),/angb∇acketleftδhM/angb∇acket∇ight2=/angb∇acketleftδhM/angb∇acket∇ight·e2(t).\nTo grasp the physical consequences of Eq.(3), consider\nthe plane-wave perturbation δm(r,t) =a(t)exp(iq·r)\nin an infinite layer ( N⊥= 0). The corresponding magne-\ntostatic field is [14]:\n/angb∇acketleftδhM/angb∇acket∇ight=−sqδmz−(1−sq)δmq;sq=1−exp(−qd)\nqd,\n(4)\nwhereδmz= (δm·ez)ezandδmq= (δm·eq)eq,eqbe-\ning the unit vector in the qdirection. The field −sqδmz\nis generated by the magnetic charges at the layer sur-\nface, whereas the field −(1−sq)δmqis due to volume\ncharges. By taking into account that ∇2\n⊥δm=−q2δm,\nδmz·e1(t) =δm1sin2θ0, andδmz·e2(t) = 0, one\nfinds from Eq.(3) that m0(t) is always stable with re-\nspect to the action of exchange forces and surface mag-\nnetic charges. Only volume charges can make the pre-\ncession unstable. This conclusion follows from the fact\nthat thez-axis, along which the surface-charge magneto-\nstatic field is directed, is a symmetry axis for the prob-\nlem. Surface-charge-driven instabilities may appear in\nnon-uniaxial systems.\nThe two-dimensional and uniaxial character of the\nproblem makes it natural to introduce polar coordinates\n(r,φ) in the disk plane, with the origin at the centre of\nthe disk. The natural boundary condition in polar coor-\ndinates is∂δm/∂r|r=R= 0, where Ris the disk radius.\nThe generic perturbation satisfying this boundary condi-\ntion consists of cylindrical spin-waves of the type:\nδm(r,φ,t) =+∞/summationdisplay\nn=−∞∞/summationdisplay\nk=0ank(t)Jn(qnkr) exp(inφ),(5)3\nwhereJn(z) is then-th order Bessel function. The wave-\nvector amplitude qnkis identified by two subscripts be-\ncause, for each n, it must satisfy the boundary condi-\ntion∂Jn(z)/∂z= 0 forz=qnkR, which has infinite\nsolutionsqn0,qn1,qn2,...of increasing amplitude. The\ncylindrical spin-waves Fnk(r,φ) =Jn(qnkr) exp(inφ) are\na complete orthogonal set of eigenfunctions of the ∇2\n⊥\noperator: ∇2\n⊥Fnk(r,φ) =−q2\nnkFnk(r,φ). The magne-\ntostatic field δhMcan be computed by applying Eq.(4)\nto the plane-wave integral representation: Fnk(r,φ) =\n1/(2πin)/integraltext2π\n0exp(iqnk·r) exp(inψ)dψ, where the polar\nrepresentation of randqnkisr= (r,φ) andqnk=\n(qnk,ψ), respectively. By following these steps, writing\nank(t) asank(t) =cnk,1(t)e1(t)+cnk,2(t)e2(t), and ne-\nglecting small terms proportional to N⊥, Eq.(3) is trans-\nformed into the following system of coupled equations:\ndcnk\ndt=Ankcnk+∞/summationdisplay\np=0∆+\nnk;p\n∆nkRn+2,p(t)cn+2,p(6)\n+∞/summationdisplay\np=0∆−\nnk;p\n∆nkR∗\nn−2,p(t)cn−2,p;cnk≡/parenleftbiggcnk,1\ncnk,2/parenrightbigg\n,\nwhere:\nAnk=1\n1+α2/parenleftbigg1−α\nα1/parenrightbigg/parenleftbigg0 −νnk\nνnk−κnksin2θ00/parenrightbigg\n,\n(7)\nRnk(t) = exp(2 iω0t)1−snk\n4× (8)\n×1\n1+α2/parenleftbigg1−α\nα1/parenrightbigg/parenleftbiggicosθ0−1\n−cos2θ0−icosθ0/parenrightbigg\n,\n∆±\nnk;p=/integraldisplayR\n0rJn(qnkr)Jn(qn±2,pr)dr , (9)\nand ∆ nk=/integraltextR\n0rJ2\nn(qnkr)dr,νnk=q2\nnk+ (1−snk)/2,\nκnk=−1 + 3(1−snk)/2,snkbeing the value of sqin\nEq.(4) forq=qnk.\nThe coupling terms proportional to Rnk(t) in Eq.(6)\nare the consequence of volume-charge magnetostatic ef-\nfects. They are all of the order of (1 −sq). One has that\n(1−sq)≪1 up toq∼1 in ultra-thin layers with d/lessorsimilar1\n(see Eq.(4)). If one neglects these terms altogether, one\nobtains a system of fully decoupled equations for indi-\nvidual cylindrical spin-waves, characterized by the dis-\npersion relation:\nω2(q;cosθ0) =/parenleftbigg\nq2+1−sq\n2/parenrightbigg\n× (10)\n×/parenleftbigg\nq2+sq+1−3sq\n2cos2θ0/parenrightbigg\n,which is obtained from Eq.(7) in the limit α→0. How-\never, the time-periodic coupling terms may give rise to\nparametric instabilities. Interestingly, these instabili-\nties are governed by a small number of dominant terms,\nwhich can be identified by using the asymptotic formula\nJn(z)∼/radicalbig\n2/πzcos(z−nπ/2−π/4) in the equation ex-\npressing boundary conditions. One obtains the estimate\nqnk≃π(2s+ 1)/4R, wheres=|n|+ 2k. When this\napproximate expression is used for qn±2,pin Eq.(9), one\nobtains:\nn≥2 : ∆±\nnk;p≃∆nkδp,k∓1,∆+\nn0;p≃0,\nn=±1 : ∆−\n1k;p= ∆+\n−1,k;p= ∆1kδpk,(11)\nn≤ −2 : ∆±\nnk;p≃∆nkδp,k±1,∆−\nn0;p≃0.\nThese relations have an important physical conse-\nquence, which is best appreciated by rewriting Eq.(5) in\nthe form:δm=/summationtext∞\ns=0δm(s), where:\nδm(s)=/summationdisplay\n|n|+2k=sank(t)Jn(qnkr) exp(inφ).(12)\nUnder the approximation (11), one finds from Eq.(6)\nthatδm(s1)is decoupled from δm(s2)for anys2/negationslash=s1.\nOn the other hand, for each s, the (s+ 1) cylindrical\nwaves (namely, n=s,s−2,...,−s+2,−s) involved in\nδm(s)form a one-dimensional chain, in the sense that\nonly neighboring waves in the above list are coupled.\nTheabsenceofcouplingbetweendistinct chainswouldbe\ncomplete if the approximation qnk≃π(2s+1)/4Rwere\nexact. In that case, all the cylindrical waves in δm(s)\nwould be characterized by exactly the same wave-vector\namplitude.\nInstabilities are governed by the multipliers of the one-\nperiod map [15] associated with the dynamics of δm(s).\nWehaveusedEqs.(6) and(11)tomakeanumericalstudy\nofthese multipliers for different chains, in orderto obtain\nthe instability pattern associatedwith each of them. The\nresults for a permalloy disk with radius R= 135 nm and\nthicknessd= 3.43 nm are shown in Fig. 1. The band ( O\n+SW) in between the PandAregions is where magne-\ntization precessionoccurs. Spin-waveinstabilities appear\nin region SW. Each chain δm(s)provides a distinct in-\nstability channel. In particular, chains s= 1,s= 2,\nands= 3 (s= 0 yields no instability at all) give rise to\nwell-separated instability regions that can be neatly re-\nsolved, as shown in Fig. 2(a). In general, the s-th chain\ngives rise to an instability tongue around the parametric\nresonance condition ω(q;cosθ0)∼ω0, whereω(q;cosθ0)\nis given by Eq.(10) and qis of the order of the wave-\nvector amplitudes involved in the chain (see dashed lines\nin Fig. 2(a)). According to parametric resonance theory,\nresonance occurs for ω=nω0/2,n= 1,2,.... The dom-\ninant, lowest-threshold resonance occurs for n= 1, that4\nFIG. 2: (Color online) (a): Magnification of Fig. 1. Labels\ns= 1,2,3 identify the perturbation chain responsible for the\ncorresponding instability tongue. The pair of dashed lines\naccompanying each of the s= 2 and s= 3 tongues (one\nline only for s= 1) represents the parametric resonance con-\nditionω(qnk;cosθ0) =ω0for the largest and smallest qnk\nin the chain. Horizontal line at β/α= 0.15 is line along\nwhich the computer simulations shown in (b) and (c) were\ncarried out. (b) and (c): Magnitude mavg\n⊥of average in-\nplane magnetization obtained from numerical integration o f\nEq.(1) under decreasing (b) and increasing (c) external mag -\nnetic field. Dashed line represents the prediction of Eq.(2)\nfor sinθ0. Snapshots illustrate magnetization patterns ap-\npearing just after the instability jumps. Vertical dotted l ines\nare guides for the eye to compare thresholds with theoretica l\npredictions obtained from (a).\nis, atω=ω0/2. However, one can see from Eq.(8) that\nthe parametric frequency is 2 ω0rather than ω0, which\nexplains why the resonancecondition is ω∼ω0. This pe-\nculiarity is the consequence of the rotational invariance\nofthe problem, and is expected to disappearin situations\nwith broken rotational symmetry. Interestingly, Fig. 1\nreveals that, for a given precession amplitude cos θ0(see\ndashed line), applying larger fields and currents has a\nstabilizing effect on the precession. Also, larger fields\nstabilize precessions of given frequency ω0=β/α.\nTo test the predictions of the theory, we have car-ried out computer simulations based on the numerical\nintegration of Eq.(1) by the methods discussed in Ref.\n[16]. Simulations were carried out by slowly varying\nthe external magnetic field under constant current. As\nshown in Fig. 2(b), at large fields the magnitude mavg\n⊥\nof the average in-plane magnetization is in full agree-\nment with the prediction of Eq.(2) for spatially uniform\nprecession. Then, under decreasing field mavg\n⊥exhibits\nwell-pronounced jumps, whose positions agree with the\ntheoretical instability thresholds for the s= 2 ands= 3\nchains within ten percent. Beyond these jumps, non-\nuniform modes appear in the dynamics (see Fig. 2(b)),\ncharacterizedbytwo-foldandthree-foldpatternsthatare\nconsistent with the symmetry of the cylindrical waves in-\nvolvedin the s= 2 ands= 3 chains, respectively. Agree-\nment with the theory is also confirmed by the hysteresis\nin the instability thresholds occurring under decreasing\nor increasing external field (Fig. 2(c)).\nThe stability of spin-transfer-driven magnetization\nprecessions has been studied in this Letter under the\nsimplest conditions, namely, uniaxial symmetry and pure\nsinθ0angulardependence ofthespin-torque. Severalfea-\ntures of physical interest have emerged: the role played\nby non-equilibrium spin-waves; the fact that instabilities\nare governed by distinct chains of magnetostatically cou-\npled spin-waves; and the fact that excited precessionsare\nnot completely disrupted but only somewhat obscured\nby spin-wave instabilities. Future work will be devoted\nto extending the present approach to more general, non-\nuniaxial geometries.\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] S. I. Kiselev et al., Nature 425, 380 (2003).\n[4] W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and\nT. J. Silva, Phys. Rev. Lett. 92, 027201 (2004).\n[5] I. N. Krivorotov et al., Phys. Rev. B 76, 024418 (2007).\n[6] C. T. Boone et al., Phys. Rev. Lett. 103, 167601 (2009).\n[7] H. Suhl, J. Phys. Chem. Solids 1, 209 (1957).\n[8] G. Bertotti, I. D. Mayergoyz, and C. Serpico, Phys. Rev.\nLett.87, 217203 (2001).\n[9] A. Kashuba, Phys. Rev. Lett. 96, 047601 (2006).\n[10] D. A. Garanin and H. Kachkachi, Phys. Rev. B 80,\n014420 (2009).\n[11] Q. Mistral, et al., Appl. Phys. Lett. 88, 192507 (2006).\n[12] G. Bertotti et al., Phys. Rev. Lett. 94, 127206 (2005).\n[13] Y. B. Bazaliy, B. A. Jones, and S. C. Zhang, Phys. Rev.\nB69, 094421 (2004).\n[14] G. Bertotti, I. D. Mayergoyz, and C. Serpico, Nonlin-\near Magnetization Dynamics in Nanosystems (Elsevier,\nOxford, 2009), Sect. 8.5.\n[15] L. Perko, Differential Equations and Dynamical Systems\n(Springer, New York, 1996).\n[16] M. d’Aquino, C. Serpico, and G. Miano, J. Comput.\nPhys.209, 730 (2005)."    },    {        "title": "1801.04426v1.Tailoring_multilayer_quantum_wells_for_spin_devices.pdf",        "content": "Pramana –J. Phys. (2018) 123: xxxx\nDOI 12.3456 /s78910-011-012-3c/circlecopyrtIndian Academy of Sciences\nTailoring multilayer quantum wells for spin devices\nS. ULLAH1,*, G. M. GUSEV1, A. K. BAKAROV2and F. G. G. HERNANDEZ1\n1Instituto de F ´ısica, Universidade de S ˜ao Paulo, Caixa Postal 66318, CEP 05315-970 S ˜ao Paulo, SP, Brazil\n2Institute of Semiconductor Physics and Novosibirsk State University, Novosibirsk 630090, Russia\n*Corresponding author. E-mail: saeedullah.phy@gmail.com\nMS received xx December 20xx; revised xx January 20xx; accepted xx January 20xx\nAbstract. The electron spin dynamics in multilayer GaAs /AlGaAs quantum wells, containing high-mobility\ndense two-dimensional electron gases, have been studied using time-resolved Kerr rotation and resonant spin\namplification techniques. The electron spin dynamics was regulated through the wave function engineering and\nquantum confinement in multilayer quantum wells. We observed the spin coherence with a remarkably long de-\nphasing time T∗\n2>13 ns for the structure doped beyond metal-insulator transition. Dyakonov-Perel spin relaxation\nmechanism, as well as the inhomogeneity of electron g-factor, was suggested as the major limiting factors for the\nspin coherence time. In the metallic regime, we found that the electron-electron collisions become dominant over\nmicroscopic scattering on the electron spin relaxation with the Dyakonov-Perel mechanism. Furthermore, the data\nanalysis indicated that in our structure, due to the spin relaxation anisotropy, Dyakonov-Perel spin relaxation mech-\nanism is e fficient for the spins oriented in-plane and suppressed along the quantum well growth direction resulting\nin the enhancement of T∗\n2. Our findings, namely, long-lived spin coherence persisting up to about room tempera-\nture, spin polarization decay time with and without a magnetic field, the spin-orbit field, single electron relaxation\ntime, transport scattering time, and the electron-electron Coulomb scattering time highlight the attractiveness of\nn-doped multilayer systems for spin devices.\nKeywords. Spin coherence, Quantum wells, g-factor, Spin dephasing, Kerr rotation, Spin-orbit field.\nPACS Nos 78.20.Ls; 85.70.Sq; 75.25.-j; 76.60.Es\n1. Introduction\nIn recent years, the spin dynamics in semiconductor\nnanostructures has become the focus of intense research\ndue to the possibility of using the spin degree of free-\ndom in future technology [1]. Among the key require-\nments, for successful implementation of novel spintronic\ndevices, quantum computation, and quantum informa-\ntion processing [2, 3, 4], a suitable system exhibiting\nlow relaxation rate and a long transport length [5, 6, 7]\nis highly desirable. Those applications could benefit\nfrom such systems because they can store and process\nthe information before the decoherence e ffect set in.\nHowever, due to strong coupling to its environment\nin a solid-state system, the spins in low-dimensional\nstructures like quantum wells (QWs) and quantum dots\n(QDs) meet a vital problem of strong dephasing. In\nthis respect, various material structures [8, 9, 10, 11,\n12, 13, 14] have been tried to control this fast decoher-\nence. Among those materials, GaAs-based heterostruc-\ntures have drawn considerable attention because of itsnumerous properties that make it well suited for ap-\nplications in telecommunication, high frequency, and\nhigh-speed electronics [15].\nRecent advances in molecular-beam epitaxy (MBE)\nenable the engineering of new and advanced multilayer\nstructures. By tailoring the sample geometry, thereby\nproducing the environment to confine the carriers wave\nfunction that penetrates into the barriers, one can wit-\nness an internal magnetic field (spin-orbit field). Such\nfield is believed to be the tuning force for the spin ma-\nnipulation [3]. Today, most of the schemes proposed\nfor the generation, manipulation, and detection of spins\nrely on this internal magnetic field [16, 17, 18]. Re-\ncently, the spin-orbit e ffects have been attracted renewed\ninterest due to the emergence of striking phenomenas\nsuch as persistent spin helix (PSH) [19, 20], spin Hall\neffect [23], large spin relaxation anisotropy [24], and\nMajorana fermions [21, 22]. Additionally, such wave\nvector ( k) driven fields induced by bulk inversion asym-\nmetry (Dresselhaus field) [25] or structure inversion asym-\nmetry (Rashba field) [26] can also inherently result in\narXiv:1801.04426v1  [cond-mat.mes-hall]  13 Jan 2018xxxx Page 24 of 30 Pramana –J. Phys. (2018) 123: xxxx\nspin relaxation through DP mechanism [27].\nThe expectations for device applications of spin-\npolarized electrons will become more realistic by un-\nderstanding the microscopic mechanisms responsible\nfor the spin relaxation as well as its manifestation in\ndifferent experimental conditions, for example, applied\nmagnetic field and sample temperature, etc. It is be-\nlieved that such relaxation processes are substantially\nmodified in the two-dimensional systems compared to\nthe bulk [2]. While there is a vast literature on the\nspin relaxation process of electrons in semiconductor\nQWs, there are only a few investigations of carrier spin\nrelaxation in multilayer structures. In present work,\nwe investigate the electron spin dynamics in multilayer\nGaAs /AlGaAs structures. Such structures, in princi-\nple, allow the long-lived spin polarization as well as\nthe manipulation of those spin through the spin-orbit\nfield [5, 6, 7, 24]. Recently, the authors demonstrated\nthat such multilayer QWs could transport coherently\nprecessing electron spins over about half millimeters at\nliquid He temperature [6].\nThe paper is organized as follows. Section 2. presents\nthe material and experimental details. Section 3. is de-\nvoted to experimental results of spin dynamics reported\nin three di fferent samples. Concluding remarks are dis-\ncussed in section 4.\n2. Materials and experiments\nTo explore the spin dynamics, we investigated here three\ndifferent samples (namely A, B, and C) grown by MBE\non a (001)-oriented GaAs substrate. All samples are\nsymmetrically delta-doped beyond the metal-insulator\ntransition (MIT) where the DP spin relaxation has been\nreported to be more e fficient [28, 29, 30]. For all the\nsamples, the density of Si-doping was 2.2 ×1012cm−2\nseparated from the QW by 7 periods of short-period\nAlAs /GaAs superlattices with 4 AlAs and 8 GaAs mono-\nlayers per period. Sample A is a 45-nm-wide GaAs\nQW. Owing to a large well width and high electron den-\nsity the electronic system results in a double quantum\nwell (DQW) configuration by forming a soft barrier in-\nside the well due to the Coulomb repulsion of electrons.\nSample B studied here is a triple quantum well (TQW)\nwith 22-nm-thick central well separated from the side\nTable 1 . Studied structures where nsandµare the total electron\ndensity and mobility in the QW, determined by electrical transport\nmeasurements at low temperature, respectively.\nName Structure QW width (nm) ns(cm−2)µ(cm2/Vs)\nSample A DQW 45 9.2 ×10111.9×106\nSample B TQW 10-22-10 9.0 ×10115.0×105\nSample C TQW 12-26-12 9.6 ×10115.5×105\nFigure 1 . (a) Schematic of time-resolved pump-probe technique.\nThe spin polarization are generated by the circularly polarized pump\nand detected by a time-delayed weak linearly polarized probe pulse.\n(b) Typical Kerr rotation signal as a function of time delay between\npump and probe pulses.\nwells by 2-nm-thick Al0.3Ga0.7Asbarriers. Both side\nwells have an equal width of 10-nm. Sample C is a\nwide TQW having the same structure of sample B with\na barrier thickness of about 1.4 times thinner than that\nof sample B. It contains a 26-nm-thick central well and\ntwo 12-nm-thick lateral wells. For both the TQW sam-\nples, the central well width is kept wider than the lat-\neral wells to be populated because, due to the electron\nrepulsion and confinement, the electron density tends\nto concentrate mostly in the side wells. The estimated\ndensity in the side wells is 35 % larger than that in the\ncentral well. The characteristics of the studied samples\nare summarized in table 1.\nWe employed time-resolved pump-probe Kerr ro-\ntation (KR) [31] and resonant spin amplification (RSA)\n[32] techniques to monitor the spin precession of 2DEGs\nconfined in multilayer structures. A Ti:sapphire laser,\nwith 100 fs pulses and repetition frequency ( frep) of 50\nkHz was used for optical excitation. The light beam\nwas split into the pump and probe beams by a beam\nsplitter. Spin polarization along the structure growth\ndirection were generated by focusing the circularly po-\nlarized pump pulses at nearly normal incidence to ap-Pramana –J. Phys. (2018) 123: xxxx Page 25 of 30 xxxx\nFigure 2 . Spin dynamics in Sample A: (a) DQW band structure and charge density for the two occupied subbands. (b) TRKR traces\nmeasured at T=5 K for di fferent excitation wavelengths. (c) TRKR responses measured at λ=817 nm for di fferent magnetic fields.\nExperimental traces are shown by symbols while the solid lines represent the fitted curves using Eq. 1 (d) ωLand (e) T∗\n2retrieved from the\nfit as a function of Bext. (f) RSA signal measured at ∆t=-0.17 ns. The experimental parameters are given inside corresponding panels.\nproximately 50 µm on the sample surface. The evolu-\ntion of those optically generated spin ensemble can be\nmonitored via rotation of polarization plane of a lin-\nearly polarized probe pulse reflected by the sample. It\nis accomplished with the help of a mechanical delay\nline that varies the optical path length of one beam rel-\native to the other. An external magnetic field Bextis\napplied perpendicular to the structure growth direction\n(V oigt geometry) as shown in Fig. 1(a). The external\nmagnetic field force the spin precesses around it. The\namount of polarization rotation ( ΘK) of the probe beam\nupon the reflection on the sample is a direct measure\nof the amount of spin orientation at that moment. This\nsmall rotation of the direction of linear polarization can\nbe detected using the balanced bridge. A typical os-\ncillatory response of such a TRKR experiment in the\npresence of an applied external magnetic field is shown\nin Fig. 1(b). The frequency of oscillations is a direct\nmeasure of electron g-factor, g=/planckover2pi1ωL/µBBext, while\nthe exponential decay envelope gives a spin dephasing\ntime T∗\n2. The combination of both, spin dephasing and\nspin precession leads to an exponentially decaying co-\nsine function of the Kerr rotation described by:\nΘK=Aexp/parenleftBigg−∆t\nT∗\n2/parenrightBigg\ncos/parenleftBigg|g|µBBext\n/planckover2pi1∆t+ϕ/parenrightBigg\n(1)Where Ais the initial amplitude, µBis the Bohr magne-\nton,Bextis the external magnetic field, /planckover2pi1is the reduced\nPlanck constant,|g|is the electron g-factor,ϕis the ini-\ntial phase, and T∗\n2is the ensemble dephasing time. The\ncosine factor reflects spin precession about the external\nmagnetic field.\n3. Results and Discussion\n3.1Spin dynamics in sample A\nFig. 2(a) depicts the calculated DQW band structure\nand charge density for the two closely spaced popu-\nlated subbands with separation energy ∆12=1.4 meV\nand equal subband density ns[33]. To find the maxi-\nmum Kerr signal with long dephasing time the TRKR\nmeasurement was carried out for di fferent pump-probe\nwavelengths. The time evolution of Kerr signal for the\nDQW as a function of excitation wavelengths is shown\nin Fig. 2(b). For clarity of presentation, the TRKR\ntraces are vertically shifted and normalized to ∆t=0.\nAt higher wavelengths, the decay of spin beat is very\nslow, and the electron spin polarization doesn’t com-\npletely decay during the pulse repetition period ( trep=\n13.2 ns) as a result one can evidence strong negativexxxx Page 26 of 30 Pramana –J. Phys. (2018) 123: xxxx\ndelay oscillation. Because of the maximum signal at λ\n=817 nm, the influence of spin dynamics on the exter-\nnal magnetic field was studied keeping this wavelength\nfor the following discussion. Fig. 2(c) displays the\npump-probe delay scan of KR signal recorded at T=\n5 K for various magnetic fields with pump /probe power\nof 1 mW /300µW. In the presence of an applied mag-\nnetic field, the TRKR signal results in weakly damping\noscillations.\nTo get T∗\n2and electron g-factor, the TRKR traces\nwere fitted (red curves) to Eq: 1. The fitted values of ωL\n(half-filled diamonds) and T∗\n2(open circles) are plotted,\nin Fig 2(d) and (e), as a function of Bext. The precession\nfrequency increases with magnetic field, where, the lin-\near interpolation of the data, shown by solid red line,\nyields|g|=0.452±0.003 which is close to the pub-\nlished value of|g|=0.44 for the bulk GaAs [34] and\nsimilar to the one reported for quasi-two-dimensional\nsystem with two occupied subbands [35]. T∗\n2decreases\nwith growing magnetic field due to the inhomogeneous\nspread of ensemble g-factor [10] and the DP spin re-\nlaxation mechanism [27, 36]. The observed T∗\n2, being\nlimited by variation in the electron g-factor ∆g, follows\n1/B-like behavior. Data analysis allows us to estimate\nthe size of this dispersion in g-factor by fitting the data\nto 1/T∗\n2(B)= ∆gµBB/√\n2/planckover2pi1[10]. Such a fit to the data,\nshown by solid line, yields ∆g=0.002 which is 0.41 %\nof measured g-value.\nThe observation of spin precession at negative ∆t\nindicates that those signal lasts from the previous pump\npulse which overlaps with the signal from the following\npulse and hence complicates the evaluation of T∗\n2. In\nsuch situations, the RSA technique, based on the inter-\nference of spin polarizations generated by subsequent\npulses can be used to retrieve the accurate value of T∗\n2.\nFig. 2(f) shows the KR traces obtained by scanning Bext\nin the range from -100 mT to +100 mT while keeping\n∆tfixed at -0.17 ns. We observed a series of Lorentzian\nresonance peaks with spacing ∆B=h frep/gµB∼12.5\nmT. The line width of those resonance peaks allows to\nevaluate T∗\n2by using Lorentzian model [32]:\nΘK=A//bracketleftBig\n(ωLT∗\n2)2+1/bracketrightBig\n(2)\nwhere T∗\n2=/planckover2pi1/gµBB1/2with half-width B1/2. The fitting\nyields T∗\n2=6.44±0.19 ns which is among the longest\nT∗\n2observed for structures with similar doping levels\n[30, 37]. Based on previous literature [38, 39, 40],\nthe observed RSA signal corresponds to the regime of\nisotropic spin relaxation where all the peaks have the\nsame height, and spin components of carriers oriented\nalong the growth axis and normal to it relax at the same\nrate. In the opposite case, in anisotropic spin relaxation,\nthe spin components of carriers relax at a di fferent rateas a result one can see its influences on the relative am-\nplitudes of RSA peaks.\n3.2Spin dynamics in sample B\nThe calculated band structure and charge density of the\nsymmetric triple quantum well (Sample B) is shown in\nFig. 3(a). The thin barriers separating the wells lead\nto the strong tunneling of electron states into di ffer-\nent wells. As a result, there are three populated sub-\nbands ( i,j=1, 2, 3) with corresponding energy gapes\n∆12=1.0 meV , ∆23=2.4 meV , and ∆13=3.4 meV ,\nobtained from the self-consistent Hartree-Fock calcula-\ntion, which are in complete agreement with periodic-\nity of the magneto-intersubband (MIS) oscillations [41,\n42]. These energy gaps characterize the coupling strength\nbetween the wells. To select the right excitation en-\nergy for this sample, we first measured KR signal vs ∆t\nfor di fferent pump-probe wavelengths [see Fig. 3(b)].\nFrom the experimental traces, it is clearly evident, that\nat lower wavelengths up to 818 nm the signal display-\ning a rapidly damping initial part transforming into a\nslowly decaying oscillatory tail. However, at a higher\nwavelength the signal last longer than trepdemonstrat-\ning that in this structure the signal has maximum inten-\nsity when excitation energy is tuned to a higher wave-\nlength.\nPanel 3(c) shows the TRKR traces measured at λ\n=821 nm for various magnetic field in the range from\n0.4 to 2 T. According to previous literature, in highly-\ndoped QWs the hole contribution to the electron spin\ndynamics can be found as a shift of the center of grav-\nity of the electron spin precession [10, 11, 43]. In our\nstructure, we found such a contribution at Bext=2 T as\nmarked by the arrow in Fig. 3(c). The magnetic field\ndependencies of ωLandT∗\n2are shown in Fig 3(d). The\nobserved linear dependence of ωLon the magnetic field\ngives a g-value of|g|=0.453±0.002. From the Bext\ndependence of T∗\n2, one can witness a strong reduction\ninT∗\n2with growing field, leading to a 1 /B-like depen-\ndence. The observed dependence assume ∆g=0.0005\n(0.10 % of obtained g-value). To see the influence of\nsample temperature on the electron spin dynamics, the\ndelay scan of KR signal was carried out at three di ffer-\nent temperatures. Obviously, the signal lasts longer at\nlow temperature as reflected by strong negatively delay\noscillations. Additionally, the signal is robust against\ntemperature and was traced up to 250 K.\nTo avoid the contribution of variation in the en-\nsemble g-factor to the spin relaxation process, the spin\ndynamics presented in Fig. 3(f) was measured in the\nlimit of lowest possible magnetic fields. For that we\nused the RSA technique by scanning Bextover a range\nof -150 mT to 150 mT while keeping the delay time\nfixed at ∆t=-0.24 ns. Fitting the zero-field RSA max-Pramana –J. Phys. (2018) 123: xxxx Page 27 of 30 xxxx\nFigure 3 . Spin dynamics in Sample B: (a) TQW (sample B) band structure and charge density for the three occupied subbands with subband\nseparation ∆12=1.0 meV and ∆23=2.4 meV and ∆13=3.4 meV . (b) KR signal measured for di fferent excitation wavelengths at T=8 K.\n(c) KR as a function of ∆trecorded for the di fferent magnetic field at λ=821 nm. The red curve on the top of experimental trace (blue) is\na bi-exponential decaying cosine fit to the data. (d) T∗\n2andωLas a function of applied magnetic field. (e) TRKR traces recorded at various\ntemperature in the range up to 250 K. (f) RSA scan measured for λ=821 nm.\nimum, using the Lorentzian model, leads to the out-of-\nplane dephasing time of 13.6 ±2.07 ns. Apart from the\nlong-lived spin coherence, we observed a strong spin\nrelaxation anisotropy between the electron spins ori-\nented in-plane and out-of-plane as apparent from the\nsuppression of zeroth-field peak compared to the side\npeaks. The observed anisotropy has its origin in the\npresence of an internal magnetic field. The magnitude\nand direction of this internal field can be inferred by fit-\nting the data to the model described in Ref. [24, 44].\nSuch a fitting, displayed in a selected range (from -\n50 mT to 50 mT) for clarity, yields the internal field\nmagnitude of B⊥=3 mT. Detailed study of spin relax-\nation anisotropy as a function of experimental param-\neters (namely, sample temperature, pump-probe delay\nand optical power) can be found in Ref. [24].The ob-\nserved long T∗\n2in the out-of-plane direction for both\nsample A and B stipulates that the scattering-induced\nDP mechanism is weak in the studied structures [12].\nHowever, combined with the inhomogeneous spread of\ng-factor it leads to a strong T∗\n2reduction.\n3.3Spin dynamics in sample C\nFinally, we report on the spin dynamics for sample C.\nThe band structure and charge density of this structure,with the three populated subbands, is displayed in Fig.\n4(a). Contrary to the DQW, we noticed that for both the\nTQWs the third subband has the opposite charge distri-\nbution compared to the first and second subbands. The\nthird subband has charge density localized in the cen-\ntral well, while the electrons in the lower subbands are\nmore distributed in the side wells. The TRKR signals\nmeasured by the changing excitation wavelength of the\nlaser pulses over the range from 811 nm to 821 nm,\nwhile keeping the same pump power under an external\nmagnetic field of 1 T, are shown in Fig. 4(b). From\nexperimental curves, it is clearly evident that the decay\nof the Kerr rotation signals is changing with excitation\nwavelengths that is only for lower wavelengths the spin\nprecession lasting up to ∆t=2 ns. To get information\nover spin dephasing time and electron g-factor the pre-\ncessional signal was fitted with mono-exponential de-\ncaying cosine function as shown by red curves plotted\non the top of experimental data. The obtained T∗\n2andg-\nfactor are shown in Fig. 4(c). A clear variation of spin\ndephasing time is observed with the increase of laser\ndetuning having the maximum at 811 nm. Additionally,\nthe electron g-factor shows a variation of 0.028, due to\nthe change in precession frequency as marked by the\ndashed line, in the measured range of wavelengths.xxxx Page 28 of 30 Pramana –J. Phys. (2018) 123: xxxx\nFigure 4 . Spin dynamics in Sample C: (a) Sample C band structure and charge density for the three occupied subbands. (b) TRKR traces\nrecorded at T=10 K for di fferent pump-probe wavelengths (colored) and fits to the data (red). Where the spin beats live longer at lower\nwavelengths (c) The relative T∗\n2andg-factor evaluated from b. (d) Dependence of Kerr signal on external magnetic fields and corresponding\n(e)ωLand (f) T∗\n2.\nTo investigate the dependence of spin dynamics on\nthe applied magnetic field a series of TRKR measure-\nments, for the wavelength with maximum KR signal,\nwere performed at T=10 K. Fig. 4(d) shows TRKR\nscans measured with no magnetic field and in the trans-\nverse magnetic field up to 2 T. In the frame of DP spin\nrelaxation mechanism, the observed signal at Bext=0\ncorresponds to the strong scattering regime [5]. Un-\nlikely, in the weak scattering regime, the electrons spin\nprecess about the spin-orbit field by one or more rev-\nolutions before scattering and hence leading to an os-\ncillatory behavior [45]. From TRKR signals the depen-\ndence of spin dephasing time and Larmor frequency on\nthe applied magnetic field is plotted in Fig. 4(e & f).\nThe linear dependence of Larmor frequency on applied\nmagnetic field yields an e ffective Lande factor of 0.389\n±0.003 which is in agreement with the magnitude of\ng-value reported previously on the same sample [6].\nAdditionally, with growing magnetic field up to 1.5\nT, we observed a monotonous increase in T∗\n2. In such\nsituation, the Bextleads to the cyclotron motion of con-\nduction band electrons which lets the direction of kto\nchange, thereby suppressing the precession around the\nrandom internal magnetic field. As a result, the elec-\ntron spin preserves its initial spin orientation, and thus\ninconsistent with DP mechanism [46], leading to theenhancement of T∗\n2. This is the key di fference com-\npared to sample A and B where cyclotron e ffect is sup-\npressed, and the DP mechanism is more e fficient. The\ndependence of T∗\n2on the applied magnetic field follows\na quadratic dependence given as [46, 37]:\nT∗\n2(B)=T∗\n2(0)//parenleftBig\n1+ω2\ncτ2\np/parenrightBig\n(3)\nHere, T∗\n2(0) is the zero-field spin relaxation time, ωcis\nthe cyclotron frequency, and τpis the single electron\nrelaxation time, which is defined as the inverse sum of\ntransport scattering rate and electron-electron scatter-\ning rate [5]\nτp=/parenleftBig\nτ−1+τ−1\nee/parenrightBig−1(4)\nFrom fit (solid red curve) to the data we retrieved τp\n=0.15 ps which is in agreement with the magnitude\nof quantum lifetime reported by transport for a simi-\nlar TQW sample [41]. The transport scattering time\nwas estimated, using the electron charge eand e ffective\nmass m∗, byτ=µm∗/e=20 ps which is quite di fferent\nfromτp. The observed large di fference was associated\nwith the fact that τincludes only the large-angle scatter-\ning, whileτpis caused by all kind of scattering events.\nThe ratio of measured τandτpdetermine the nature ofPramana –J. Phys. (2018) 123: xxxx Page 29 of 30 xxxx\ndominant scattering mechanism [47]. For GaAs based\nheterostructures, it was assumed that τ/τ p/lessorsimilar10 for\nbackground impurity scattering and τ/τ p/greaterorsimilar10 for the\nremote ionized impurity scattering [47]. The observed\nτ/τ p≈135, indicates that the dominant scattering in\nour structure is caused by remote instead of background\nimpurities. The electron-electron scattering time ( τee)\ncan be approximated by using Eq. 4 which leads to τp≈\nτeedemonstrating the supremacy of electron-electron\nscattering over microscopic scattering mechanisms [48].\n4. Conclusions and outlook\nIn summary, we have studied the electron spin dynam-\nics in high-mobility two-dimensional electron gases us-\ning the pump-probe reflection techniques: time-resolved\nKerr rotation and resonant spin amplification. The DQW\nstructure yields T∗\n2=6.44 ns, while, in the TQW we\nobserved a strikingly long T∗\n2exceeding the laser rep-\netition. In the wide TQW, T∗\n2increases with the ap-\nplied magnetic field but is much smaller than that in\nthe DQW and other TQW. Additionally, we observed\nanisotropic feature due most likely to the presence of an\ninternal magnetic field. The observed long-lived spin\ncoherence persists up to about room temperature, with\nencouraging indication for spin-optoelectronics and par-\nticularly the long spin memories in multilayer GaAs\nQWs. We believe that the determination of all the rele-\nvant time scales will be useful for the future spintronics\ndevices and quantum information processing.\nAcknowledgement\nF.G.G.H. acknowledges financial support from Grant\nNo. 2009 /15007-5, 2013 /03450-7, 2014 /25981-7 and\n2015/16191-5 of the S ˜ao Paulo Research Foundation\n(FAPESP). S.U acknowledges TWAS /CNPq for finan-\ncial support.\nReferences\n[1] S. A. Wolf, Science. 294, 1488 (2001).\n[2] M. I. Dyakonov. Spin Physics in Semiconductors . Springer-\nVerlag, Berlin, 2008.\n[3] D. D. Awschalom, D. Loss, and N. Samarth. Semiconduc-\ntor Spintronics and Quantum Computation . Springer-Verlag,\nBerlin, 2002.\n[4] F. Henneberger and O. Benson, Semiconductor Quantum Bits .\nPan Stanford, Singapore, 2009.\n[5] S. Ullah, G. M. Gusev, A. K. Bakarov, and F. G. G. Hernan-\ndez,J. Appl. Phys. 119, 215701 (2016).\n[6] F. G. G. Hernandez, S. Ullah, G. J. Ferreira, N. M. Kawahala,\nG. M. Gusev, and A. K. Bakarov, Phys. Rev. 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Frandsen,2,3Andrew D.\nChristianson,4Dao-Xin Yao,1Meng Wang,1,∗and Robert J. Birgeneau5,2\n1Center for Neutron Science and Technology, School of Physic s,\nSun Yat-Sen University, Guangzhou, 510275, China\n2Materials Science Division, Lawrence Berkeley National La boratory, Berkeley, California 94720, USA\n3Department of Physics and Astronomy, Brigham Young Univers ity, Provo, Utah 84602, USA\n4Materials Science and Technology Division, Oak Ridge Natio nal Laboratory, Oak Ridge, Tennessee 37831, USA\n5Department of Physics, University of California, Berkeley , California 94720, USA\nWe report transport and inelastic neutron scattering studi es on electronic properties and spin\ndynamics of the quasi-one-dimensional spin chain antiferr omagnet RbFeS 2. An antiferromagnetic\nphase transition at TN≈195 K and dispersive spin waves with a spin gap of 5 meV are obse rved. By\nmodeling the spin excitation spectra using linear spin wave theory, intra and inter-chain exchange\ninteractions are found to be SJ1= 100(5) meV and SJ3= 0.9(3) meV, respectively, together\nwith a small single-ion anisotropy of SDzz= 0.04(1) meV. Comparison with previous results for\nother materials in the same class of Fe3+spin chain systems reveals that although the magnetic\norder sizes show significant variation from 1.8 to 3.0 µBwithin the family of materials, the exchange\ninteractions SJare nevertheless quite similar, analogous to the iron pnict ide superconductors where\nboth localized and delocalized electrons contribute to the spin dynamics.\nPACS numbers:\nI. INTRODUCTION\nThe discovery of iron-based superconductors has at-\ntractedsignificantscientific interests1–3. Mostiron-based\nsuperconductors crystallize in quasi-two-dimensional\n(2D) layered structures consisting of Fe Pn4or FeCh4\n(Pn= pnictogens, Ch= chalcogens) tetrahedra4–8.\nMore recently, the observation of pressure-induced su-\nperconductivity in the quasi-one-dimensional (1D) spin-\nladder compound BaFe 2S3, which also consists of\nFeS4tetrahedra, has drawn attention to the 1D iron-\nchalcogenide materials9,10. Adding to this interest in 1D\nmaterials is a recent investigation of the 1D spin-chain\ncompound TlFeSe 2under pressure, which is metallized\nabove 2 GPa and may support superconductivity above\n30 GPa11. More broadly speaking, iron-based supercon-\nductors and related materials exemplify the close rela-\ntionship between magnetism and superconductivity in\nstructures comprised of 2D layers, 1D ladders, and 1D\nchains. Nevertheless, a complete understanding of the\nimpact of structure on magnetism and superconductiv-\nity remains elusive, motivating continued study in this\nfield.\nInthiscontext, theternarymetalchalcogenides AFeX2\n(A=alkali metal, Tl; X=S,Se) are interesting materials\nbecause they host linear spin chains formed by edge-\nsharing1\n∞[FeX4/2]−tetrahedra along the chain axis, as\nshown in Fig. 1(a)12–14. As previously reported, KFeS 2,\nRbFeS 2, KFeSe 2, and RbFeSe 2crystallize in the mono-\nclinic space group C/2c, while TlFeS 2and TlFeSe 2crys-\ntallize in the monoclinic space group C/2m15–17. CsFeS 2\nis somewhat different, crystallizing in an orthorhombic\nspace group and undergoing a structural transition from\nImmmtoP¯1 upon cooling14,18,19.\nThese spin chain compounds host trivalent iron ionswith a 3 d5electron configuration. The chains of Fe3+\nspins all exhibit antiferromagnetic (AFM) order at low\ntemperature, with the moments aligning perpendicularly\nto the spin chain axis except in the case of KFeS 2and\nRbFeS 2, in which the moments deviate from the spin\nchain axis by relatively small angles (see Table I12). In-\nterestingly, the observed ordered moment sizes are in the\nrange 1.8∼3µB, smaller than the expected size of 5 µB\nforFe3+spins. Furthermore, adirect measurementofthe\nbulk electronic structure by 2 pcore-level hard x-ray pho-\ntoemission spectroscopy and calculation density of states\nin TlFeS 2and TlFeSe 2suggest that the competition be-\ntween the delocalized and localized characters of Fe 3d\nelectrons is important to understand the electronic struc-\nturesofbothcompounds29. Forthesereasons,ithasbeen\nsuggested that the Fe3+3d5electrons exhibit consider-\nabledelocalizationandaspin S= 3/2,eventhoughthese\ncompoundsareshowingsemiconductingbehaviorsatam-\nbient pressure. This delocalization may be related to the\nshort nearest neighbor (NN) intrachain atomic distance\nof Fe-Fe, which approaches the metallic bond distance\nof Fe. This could result in Fe-Fe covalency in addition\nto Fe-S covalency, thus reducing the magnetic moment.\nThe lack of metallic conductance in the AFe X2system is\nlikely related to the 1D character of the structure.\nGiven the short NN Fe-Fe bond, a strong intrachain\nmagnetic exchange interaction should be expected. For\nTlFeS2(space group C/2c), inelastic neutron scattering\n(INS) measurements determined that the NN AFM in-\ntrachain exchange interaction SJis∼65 meV26,27. Ad-\nditionally, the interchain-intrachain exchange ratio is of\norder 10−3, confirming a strong 1D behavior in the mag-\nnetic interactions. However, KFeS 2(space group C/2m)\nis reportedto havea significantlysmallerNN AFM intra-\nchain exchange interaction SJ=25(1) meV20,21,30. The2\nTABLE I: Crystal and magnetic information for the 1D spin cha in compound AFeX2(A= K, Rb, Tl; X= S, Se). EL\ntand\n∆srepresent the zone-boundary energy and the spin gap of the sp in wave along the chain direction.\nCompound Space TNd(˚A) Moment µorder Spin wave Ref.\ngroup (K) (Fe-Fe) orientation ( µB)\nKFeS2C/2c250.0(5) 2.70 13.6◦from chain 2.43(3) EL\nt= 221(4) meV [12,15,16,20,21]\n∆s≈4.5 meV\nRbFeS 2C/2c188(1) 2.71 20◦from chain 1.8(3) [12,22]\nEL\nt≈203meV\n≈195 close to the chain ∆ s≈5 meV [this work]\nKFeSe 2C/2c310(1) 2.81 ⊥chain 3.0(2) - [12]\nRbFeSe 2C/2c249(1) 2.83 ⊥chain 2.66(5) - [12,23,24]\nTlFeS 2C/2m196 2.65 ⊥chain 1.85(6) EL\nt≈260 meV [25–27]\n∆s≈4.3 meV\nTlFeSe 2C/2m290 2.74 ⊥chain 2.1(2) - [13,17,28]\nFIG. 1: Crystal and magnetic structure of RbFeS 2. Rb, Fe,\nand S atoms are drawn as green, blue, and yellow spheres,\nrespectively. The exchange interactions J1andJ3are marked\nwith black and red dashed lines, respectively. The magnetic\nstructure adapts to the published results12, the red arrows\nrepresent the magnetic moment direction.\nresult was deduced only from the low-energy spin excita-\ntions and may not be accurate20. In a later INS experi-\nment with higher incident energies, the spin waves were\nobserved to extend to 221(4) meV at the zone boundary.\nHowever,theNNintrachainexchangeinteractionwasnot\nextracted from the data21. Considering this uncertainty\nin the exchange parameters for KFeS 2(and the complete\nlack of such information for RbFeS 2, which shares the\nC/2mstructure with KFeS 2), it would be valuable toclarify the exchange parameters for the C/2msystems\nand compare them to the representative C/2ccompound\nTlFeS2. Such a comparison would further elucidate the\nrelationship between structure and magnetism in low-\ndimensional iron-based systems.\nIn this paper, we report transport and inelastic neu-\ntron scattering (INS) measurements on RbFeS 2. We find\nan AFM phasetransition takesplace at TN≈195K with\nmagnetic moment oriented close to the chain direction.\nIntheINSspectra, weobservetwobranchesofspinwaves\nalong the HandLdirections and a spin gap located at\nthe Bragg peak positions. The energies of the band-top\nand spin gap are close to those of the isostructural com-\npound KFeS 2, which has a moment size of ∼2.43(3)µB\nfor Fe3+. By modeling these spectra using a linear spin\nwave theory, we find that the spectra can be fully re-\nproduced with an intrachain exchange of SJ1= 100(5)\nmeV, an interchain exchange of SJ3= 0.9(3) meV, and\na small single-ion anisotropy of SDzz= 0.04(1) meV.\nThese results demonstrate that although a wide variety\nof magnetic structures, ordered moment sizes, and elec-\ntron itinerancy are observed among the 1D spin chain\ncompounds, the spin excitations are nevertheless quite\nsimilar acrossthe family of compounds. This suggests an\ninterplaybetween localized and delocalizedmagnetism in\nthe 3dbandsofFe3+reminiscent ofthe 2Diron-basedsu-\nperconductors.\nII. EXPERIMENTAL DETAILS\nSingle crystal samples of RbFeS 2were grown using the\nBridgman method. The sintering procedure is identical\nto that of Rb 0.8Fe1.5S2we grew previously31. The ob-\ntained single crystals are needle-like in shape, consistent\nwith the 1D structure. DC susceptibility and resistivity\ndata were collected on a commercial physical property\nmeasurement system (PPMS, Quantum Design). The\nneedle-like single crystals are difficult to align, we thus3\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s49/s50/s51/s52\n/s50/s56/s48 /s51/s50/s48 /s51/s54/s48 /s52/s48/s48/s48/s49/s48/s50/s48/s40/s98/s41/s84 /s78 /s32 /s49/s57/s53/s32/s75\n/s72/s32 /s61/s32/s49/s48/s48/s48/s48/s32/s79/s101\n/s32/s32/s77/s47/s72 /s32/s40 /s49/s48/s45/s52\n/s32/s101/s109/s117/s32/s79/s101/s45/s49\n/s32/s109/s111/s108/s45/s49\n/s41\n/s84 /s32/s40/s75/s41/s32/s72 /s99/s104/s97/s105/s110\n/s32/s72 /s32 /s99/s104/s97/s105/s110/s40/s97/s41\n/s32/s32/s32/s40/s107 /s32/s99/s109/s41\n/s84 /s32/s40/s75/s41\nFIG. 2: (a) Temperature dependence of the susceptibility of\nRbFeS 2measured under magnetic fields applied parallel and\nperpendicular to the chain direction. (b) Temperature depe n-\ndence of the resistivity measured along the chain direction .\nground 3 g of single crystals into powder for neutron\nscattering experiments. The INS experiment was per-\nformed on the ARCS time-of-flight chopper spectrometer\nat the Spallation Neutron Source, ORNL32. The powder\nsample was sealed in an aluminum can and loaded into\na He top-loading refrigerator. The sample was measured\nwith incident energies of Ei= 30, 50, 150, and 250 meV\nat 4 K with the corresponding energy resolutions ∆ E=\n1.2, 2.2, 6.2, and 11.4 meV, respectively, as determined\nby the full width at half maximum (FWHM) of the\nenergy cuts at E= 0 meV. The data reduction for the\nINS data was performed using the DAVEsoftware33.\nLinear spin wave theory was employed to simulate the\nINS spectra using the SpinWsoftware34.\nIII. RESULTS\nFigure2(a) shows the dc susceptibility measured un-\nder magnetic fields applied parallel ( χ/bardbl) and perpendic-\nular (χ⊥) to the chain direction. Both the χ/bardblandχ⊥exhibit a slop change at TN≈195 K, revealing an AFM\nphase transition. Below TN, theχ⊥is larger than χ/bardbl\nin magnitude, suggesting that the magnetic easy axis is\nclose to the chain direction. The results are in agree-\nment with the previous results that the AFM phase tran-\nsition of RbFeS 2is atTN= 188(1) K and the mag-\nnetic moment arranges in the acplane with an angle\nof 20◦to thecaxis as shown in Fig. 112. The up-\nturns at low temperature reveal the existence of para-\nmagnetic impurities. Above TN, theχ/bardblandχ⊥in-\ncrease linearly with increasing temperature, resembling\nthat of TlFeS 2, TlFeSe 2, and RbFeSe 223,28. The linear\nincreaseof susceptibility with the increasing temperature\nalso appears in some quasi two-dimensional metallic lay-\nered iron pnictides above its AFM transition2, such as in\nBa(Fe1−xCox)2As2(x= 0−0.2)35,36, Ca(Fe 1−xCox)2As2\n(x= 0−0.2)37, andLaFeAsO 1−xFx(x= 0−0.15)38. The\nlinear susceptibility behavior could be attributed to the\nantiferromagneticcorrelationof local moment in a strong\ncoupling description35,39or the antiferromagnetic spin\nfluctuations of itinerant electron40,41. Figure 2(b) shows\nthe temperature dependence of the resistivity down to\n275 K, indicating a semiconducting behavior. Below this\ntemperature, the resistance is out of the limitation of our\ninstrument. The semiconducting behavior was also ob-\nserved in the 1D system such as KFeS 2, TlFeS 2, TlFeSe 2,\nand RbFeSe 223,28,42, where it is ascribed to the fiber-like\nmorphology of the 1D structure which contains defects\nand breaks in the sample.\nIn Figs3(a-d), we present INS spectra obtained from\nthe powder sample at 4 K using incident neutron ener-\ngies ofEi= 30, 50, 150, and 250 meV, respectively. We\nobserve intense excitations at Q= 1.25 andQ= 1.82\n˚A−1, along with a spin gap of ∼5 meV. Additionally,\nexcitations stemming from Q= 3.5˚A−1with much\nweaker intensities appear at Ei= 150 and Ei= 250\nmeV spectra, as shown in Figs. 3(c) and (d). The in-\ntensities of the excitations decrease with increasing Q,\nconsistent with a magnetic origin. The three Qval-\nues from which we observe excitations are correspond to\nthe AFM wave vectors ( H,K,L) = (0,0,1),(1,0,1),and\n(3,0,1), respectively, demonstrating that these are spin\nwave excitations out of the magnetic ground state. Here,\n(H,K,L) are Miller indices for the momentum transfer\nQ= 2π/radicalBig\n(H\nasinβ)2+(K\nb)2+(L\ncsinβ)2−2HLcosβ\nac(sinβ)2, where\nthe lattice parameters are a= 7.162(7),b= 11.566(7),\nandc= 5.453(5)˚A, andβ= 112.75(7)◦obtained from\nrefining the energy cuts of Ei= 30 meV spectrum at\nE= 0 meV at 4 K. The flat excitations below 30 meV\nthat increase in intensity with Qcorrespond to phonon\nfrom the sample and the thin aluminum can.\nTaking a more quantitative view of the spin gap and\nspin wave dispersions, we show constant QandEcuts\nin Fig.4. Panel (a) displays the constant Qcut inte-\ngrated over Q= 1.25±0.2˚A−1withEi= 30 meV. The\nabrupt increase at ∆ E≈5 meV confirms a spin gap of\n∆s≈5 meV. Figures 4(b) and (c) show constant E4\nFIG. 3: INS spectra S(Q,ω) of RbFeS 2at 4 K with Ei= (a) 30, (b) 50, (c) 150, and (d) 250 meV. SpinWsimulated spectra\nS(Q,ω) with the exchange parameters described in the text. The ins trumental resolutions of (e) 1.2, (f) 2.2, (g) 6.2, and (h)\n11.4 meV have been convoluted in the simulation for comparin g to the INS spectra with Ei= 30, 50, 150, and 250 meV,\nrespectively. The color represents intensities.\ncuts with Ei= 50 and 150 meV, respectively. The ex-\ncitation near Q= 1.82˚A−1shows clear dispersion with\nincreasing energy transfer. The excitation correspond-\ning to (H,K,L) = (3,0,1) can also be recognized around\nQ= 3.5˚A−1in Fig.4(c), alsoshowinga continuousevo-\nlution with increasing energy transfer. Constant Qcuts\natQ= 7.4,8.0,8.6 and 9.4 ˚A−1are displayed for the\nenergy range 180 −220 meV ( Ei= 250 meV) in Fig. 4\n(d). The high energyexcitations located around E≈203\nmeV are much higher than the cut-off energy of phonon.\nInstead, these high-energy excitations correspond to the\nband-top of the spin waves at the zone-boundary along\ntheLdirection. The INS spectra are comparable with\nthe spin waves of KFeS 2and TlFeS 2as shown in Table\nI21,26,27.\nHaving measured the spin wave dispersion, we now\nturn to modeling the INS spectra using linear spin wave\ntheory with the following Heisenberg Hamiltonian:\nˆH=/summationdisplay\ni,jJi,jSi·Sj−Dzz/summationdisplay\ni,zS2\ni,z, (1)\nwhereJi,jincludes the NN intrachain exchange inter-\nactionJ1and NN interchain exchange interaction J3as\nmarked in Fig. 1.Dzzis the single ion anisotropy term.\nSince we observethe spin wavesoriginatingonly fromthe\nAFM wave vectors ( H,0,L) in our INS spectra, we only\ntake into accountthe NN exchangeinteractionsalongthe\nHandLdirections. By solving Eq. 1using the linear\nspin wave approximation, the dispersion relations could\nbe written as:E=/radicalBig\nA2\nk−B2\nk,\nAk=S(2J1+2J3+Dzz),\nBk=S(2J1cos(πL)+2J3cos(2πH+πL)),(2)\nFrom Eq. 2, the extremes of the spin waves can be\nextracted. The spin gap ∆ sand the top of the acoustic\nspin wave along the H(EH\nt) andLdirections ( EL\nt) are\nobtained as follows:\n∆s≈S/radicalbig\nDzz(4J1+4J3+Dzz),\nEH\nt≈S/radicalbig\n16J1J3+Dzz(4J1+4J3+Dzz),\nEL\nt≈S(2J1+2J3+Dzz).(3)\nBased on the experimental data, we determine the\nvalues for these extremes as ∆ s≈5,EH\nt≈40,and\nEL\nt≈203 meV. From this, we determine the products\nof the spin Sand the magnetic exchange interactions to\nbeSJ1= 100(5) ,SJ3= 0.9(3), and SDzz= 0.04(1)\nmeV. The errors are estimated by allowing 5% uncer-\ntainty of the experimental determined extremes in Eq.\n3. The small SDzz= 0.04(1) meV of RbFeS 2yields an\nisotropic magnetic behavior, which can be further ver-\nified by the Land´ e gfactor of 2.00064 measured with\nelectron-spin-resonance (ESR). The gfactor closing to\nthe spin only value of g= 2 suggests a small residual\norbital moment30,43.\nTheSpinW software is employed to simulate the\nspherically averaged spin wave spectra based on the5\n/s48 /s53 /s49/s48 /s49/s53 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53\n/s49 /s50 /s51 /s52 /s49/s56/s48 /s49/s57/s48 /s50/s48/s48 /s50/s49/s48 /s50/s50/s48/s32/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s32/s83/s105/s109/s117/s108/s97/s116/s105/s111/s110\n/s40/s100/s41/s40/s99/s41/s40/s98/s41/s69 /s105/s61/s51/s48/s109/s101/s86\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s32\n/s69/s32/s40/s109/s101/s86/s41/s81 /s32/s61/s32/s49/s46/s50/s53 /s48/s46/s48/s50/s32/s197 /s45/s49\n/s32/s83/s112/s105/s110/s32/s103/s97/s112/s40/s97/s41\n/s81 /s32/s40/s197 /s45/s49\n/s41/s32/s81 /s32/s40/s197 /s45/s49\n/s41\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s69 /s105/s61/s53/s48/s109/s101/s86\n/s54/s32/s109/s101/s86/s49/s48/s32/s109/s101/s86/s49/s52/s32/s109/s101/s86/s49/s56/s32/s109/s101/s86\n/s81 /s32/s40/s197 /s45/s49\n/s41\n/s32/s50/s56/s32/s109/s101/s86/s51/s50/s32/s109/s101/s86\n/s50/s52/s32/s109/s101/s86/s69 /s105/s61/s49/s53/s48/s109/s101/s86\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32 /s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115 /s41\n/s50/s48/s32/s109/s101/s86/s69 /s105/s61/s50/s53/s48/s109/s101/s86\n/s32/s32\n/s32/s55/s46/s52\n/s32/s56/s46/s48\n/s32/s56/s46/s54\n/s32/s57/s46/s52/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s69/s32/s40/s109/s101/s86/s41\nFIG. 4: (a) Constant Qcut between 1 .23< Q <1.27˚A−1\nforEi= 30 meV. (b) Constant energy cuts at E= 6, 10,\n14, and 18 meV integrated within E±2 meV for Ei= 50\nmeV. (c) Constant energy cuts at E= 20, 24, 28, and 32\nmeV integrated within E±2 meV with Ei= 150 meV. The\nblack symbols are experimental data while the red solid line s\ndepict simulated results. (d) Constant Qcuts atQ= 7.4, 8,\n8.6, and 9.4 ˚A−1integrated within Q±0.2˚A−1withEi= 250\nmeV.\nabove-determined exchange interactions. The results af-\nterconvolutionwith the experimentalresolutionfunction\nare plotted in Figs. 3(e)-(h). The simulated spectra\nmatch well with the experimental data. The band-top of\nthe spin waves at the zone-boundary along the Ldirec-\ntion is invisible in Fig. 3(h) because the intensities of\nthe spin waves are greatly reduced due to the effect of\nthe magnetic form factor of Fe3+. We also perform iden-\ntical constant energy cuts for the simulated spectra and\nplot them with the experimental data together in Figs.\n4(b) and (c). The energy cuts from the simulated spec-\ntra are in good agreement with the experimental results,\ndemonstrating a high accuracy of the determination of\ntheproducts SJs. Inthehigh Qregion,therearediscrep-\nancies between the simulated and experimental results,\nmore obviousat energytransfer below 30 meV, which are\nattribute to the strong phonon intensities in the experi-\nmental spectra. To visualize the spin waves more clearly,\nwe plot in Fig. 5the dispersions along high-symmetry\ndirections in the [ H,L] Brillouin zone using the obtained\nexchange interactions. The dispersion relations of two\nacousticspin wavebranchesextending to ∼40and∼203\nmeV along the HandLdirections are in good agreement\nwith the experimental observations.\nIV. DISCUSSION\nBy utilizing the cutting edge time-of-flight neutron\nscattering spectrometer, two branches of the acoustic\nspin waves of RbFeS 2have been observed on a pow-\nFIG. 5: Spin waves for single crystals of RbFeS 2simulated\nusingthe SpinWsoftware alongthehigh-symmetrydirections\nin the [H,L] Brillouin zone with the path depicted in the\ninset. An instrumental resolution of 5 meV is convoluted for\nvisualization. The color represents intensities.\nder sample. The spin waves of RbFeS 2and the fitted\nmagnetic exchange interactions resemble the 1D analogs\nKFeS2and TlFeS 221,26. The ordered moment size is re-\nduced on the sample we measured compared with the\nexpectation for the localized Fe3+3delectrons. The re-\nduction may result from the delocalization of the 3 delec-\ntrons of Fe3+and the quantum fluctuations because of\nintrinsic nature of the 1D spin chain27,44.\nOwing to the 1D nature, the single crystals are easily\ndisassembled into thin fibers that result in poor electric\nconductivity28. However, the 2 pcore-level Hard x-ray\nphotoemission spectra of TlFeS 2and TlFeSe 229reveal\nthe delocalization of the Fe3+3delectrons. Based on\nthe reduced magnetic moment of 1.8(3) µBand the sim-\nilarity to TlFeS 2and TlFeSe 2, the delocalization of the\n3delectrons of Fe3+may exist in RbFeS 2as well. In\nthis scenario, the linear magnetic susceptibility of the\nternary metal chalcogenides AFeX2(A=alkali metal, Tl;\nX=S,Se) could be attributed to the antiferromagnetic\nspin correlations of local moments or spin fluctuations\nof itinerant electrons, analogous to the iron-based super-\nconductorswiththeFe2+3delectrons35,39–41. Therobust\nspin excitations in KFeS 2, RbFeS 2, and TlFeS 2in spite\nof the varied magnetic orders and moment sizes suggest\nboth local moments and delocalized electrons contribute\nto spin dynamics. This resembles the spin dynamics of\nthe 2D iron-basedsuperconductors, where localmoments\nand itinerant electrons couple and high energy spin exci-\ntations are robust against electron or hole doping45,46.\nThis inspires more researches on exploring interesting\nphysics, such as insulator-metal transition and supercon-\nductivity on the 1D spin chain system. We note the spin\nwave spectra may deviate from the Heisenberg Hamil-\ntonian due to the existence of the delocalized electrons,\nwhich call for further studies on the spin waves of single\ncrystal samples.6\nV. CONCLUSIONS\nIn summary, we have measured the magnetic trans-\nport and spin waves of the AFM spin chain compound\nRbFeS 2. The susceptibility measurements yield an AFM\nphase transition at TN≈195 K with magnetic moment\noriented close to the chain direction. The reduced mag-\nnetic moment and the similarity of RbFeS 2to TlFeS 2\nand TlFeSe 2, which host both localized and delocalized\ncharacters of electrons indicate the existence of delocal-\nization of the Fe3+3delectrons in RbFeS 2. The spin\nwaves can be successfully modeled using a Heisenberg\nHamiltonian and linear spin wave theory, allowing us to\nextract the single ion anisotropy SDzzand the products\nof the spin and the exchange interactions SJ1,3. The\nsimulated spectra based on the as-determined exchange\nparameters match well with the INS spectra. The vari-\nety of static magnetic orders yet the consistency of the\nspin excitations observed among several related 1D spin\nchain compounds highlight the interplay between local\nmoments and delocalized electrons, resembling the situ-\nation for iron-based superconductors.VI. ACKNOWLEDGEMENTS\nM. W. was supported by the National Natural\nScience Foundation of China (Grant No. 11904414,\n12174454), National Key Research and Development\nProgram of China (No. 2019YFA0705702). D. X. Y. was\nsupported by NKRDPC-2018YFA0306001, NKRDPC-\n2017YFA0206203, NSFC-11974432, GBABRF-\n2019A1515011337, and Leading Talent Program of\nGuangdong Special Projects. Work at University of\nCalifornia, Berkeley and Lawrence Berkeley National\nLaboratory was funded by the U.S. Department of\nEnergy, Office of Science, Office of Basic Energy Sci-\nences, Materials Sciences and Engineering Division\nunder Contract No. DE-AC02-05-CH11231 within the\nQuantum Materials Program (KC2202) and the Office\nof Basic Energy Sciences. The experiment at Oak Ridge\nNational Laboratory’s Spallation Neutron Source was\nsponsored by the Scientific User Facilities Division,\nOffice of Basic Energy Sciences, U.S. Department of\nEnergy.\n∗Electronic address: wangmeng5@mail.sysu.edu.cn\n1Y. Kamihara, T. Watanabe, M. 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Lett. 115,\n117001 (2015)."    },    {        "title": "1801.10148v1.Controlled_spatial_separation_of_spins_and_coherent_dynamics_in_spin_orbit_coupled_nanostructures.pdf",        "content": "Controlled spatial separation of spins and coherent\ndynamics in spin-orbit-coupled nanostructures\nShun-Tsung Lo1y, Chin-Hung Chen1y, Ju-Chun Fan1y, L. W. Smith2z, G. L. Creeth3, Che-Wei\nChang1, M. Pepper,3J. P. Griffiths2, I. Farrer2x, H. E. Beere2, G. A. C. Jones2, D. A. Ritchie2, &\nT.-M. Chen1\u0003\n1Department of Physics, National Cheng Kung University, Tainan 701, Taiwan\n2Cavendish Laboratory, J J Thomson Avenue, Cambridge CB3 0HE, United Kingdom\n3Department of Electronic and Electrical Engineering, University College London, London WC1E\n7JE, United Kingdom\nyThese authors contributed equally to this work.\nzPresent Address: Wisconsin Institute for Quantum Information, University of Wisconsin-Madison,\nMadison, WI 53706, USA\nxPresent Address: Department of Electronic & Electrical Engineering, University of Sheffield,\nMappin Street, Sheffield S1 3JD, United Kingdom\n\u0003To whom correspondence should be addressed; E-mail: tmchen@mail.ncku.edu.tw.\nThe spatial separation of electron spins followed by the control of their individual spin dy-\nnamics has recently emerged as an essential ingredient in many proposals for spin-based\ntechnologies because it would enable both of the two spin species to be simultaneously uti-\nlized, distinct from most of the current spintronic studies and technologies wherein only one\nspin species could be handled at a time. Here we demonstrate that the spatial spin splitting\nof a coherent beam of electrons can be achieved and controlled using the interplay between\nan external magnetic field and Rashba spin-orbit interaction in semiconductor nanostruc-\ntures. The technique of transverse magnetic focusing is used to detect this spin separation.\nMore notably, our ability to engineer the spin-orbit interactions enables us to simultaneously\nmanipulate and probe the coherent spin dynamics of both spin species and hence their cor-\nrelation, which could open a route towards spintronics and spin-based quantum information\nprocessing.\nThe spin-orbit interaction in materials gives rise to a separation of different spin species in\nmomentum space, creating many interesting phenomena such as the spin Hall1–3, the quantum spin\nHall4, 5effects, and the spin-momentum locking6, 7. However, it does not separate the spin up and\nspin down electrons in real space. In other words, even though different spins behave very differ-\n1arXiv:1801.10148v1  [cond-mat.mes-hall]  30 Jan 2018ently they cannot be resolved and tracked in real space, similar to spin-degenerate systems where\nthe spin-orbit interaction is negligible. So far most of the spintronic technologies which require\nspin to be resolved prior to subsequent operations have to rely on the creation of a spin imbalance\nwith, for example, ferromagnets or optical injection. However, these methods are limited in both\nfundamental and practical aspects since only one spin type (i.e., the majority spin) can be utilized.\nFor example, the correlation between different spin types remains experimentally unexplored un-\nless one can resolve and track both spin types simultaneously, for which it is necessary to spatially\nsplit electron spins rather than polarize them. Developing a simple way to spatially separate the\nopposite spin types, then manipulate and track the coherent spin dynamics of both of the two spin\ntypes and, more importantly, their phase correlation is therefore essential and a frontier in current\nresearch.\nThe Stern-Gerlach magnet is well-known for separating spins but is limited to uncharged par-\nticles, and modified proposals for electron spins using inhomogeneous spin-orbit effective fields8–10\nhave yet to be realized. The spin Hall effect geometry1–3can also produce spin separation, where\nthe diffusive electrons that are scattered to opposite edges of a conductor are coupled to spins of\nopposite orientations; however, no control can be exercised in such a random scattering system. A\npromising way to achieve spatial separation of electron spins in a spin-orbit coupled system is to\napply a transverse magnetic field. Spin-up and spin-down electrons have different momenta and\nthus, when moving through a magnetic field, will experience different Lorentz forces and conse-\nquently undergo different cyclotron motions. This concept has been successfully demonstrated,\nusing a hole gas in which the spin-orbit interaction was not tunable11–13, but to manipulate and\nstudy the behaviour of the spatially separated spins remains an outstanding challenge.\nHere we combine this simple concept of spatial spin separation with techniques to coherently\nmanipulate and detect spins, and thereby demonstrate a spatial spin splitting of a coherent electron\nbeam together with full control of the dynamics of these spatially separated spins. The spatial\nseparation, coherent spin dynamics, and phase correlation between the up- and down-spin electrons\ncan all be – electrically and on-chip – controlled and probed. This allows both of two spin types\n(instead of just the majority one as in most previous studies) to be simultaneously probed and\nmanipulated, which promises to advance spintronic technologies that require both spin types to be\noperated together.\n2Results\nSpatial separation of spins Figure 1a captures the operation of our devices. A quantum point\ncontact (QPC) – a one-dimensional constriction created by applying voltages to split gates pat-\nterned on the surface of an InGaAs heterostructure – is used to inject an unpolarized electron beam\ninto a two-dimensional electron gas (2DEG). The 2DEG is formed in the InGaAs quantum well\n(see Methods), wherein the structural inversion asymmetry of the well generates a momentum-\ndependent magnetic field BSO\nRon the spin of every moving electron, the so-called Rashba spin-orbit\ninteraction. This Rashba spin-orbit effective magnetic field BSO\nRlies in the plane of the 2DEG (i.e.,\nthex-yplane in Fig. 1a) and is orientated perpendicular to the electron’s momentum. It lifts the\nspin degeneracy in momentum space and leads to two spin-polarized Fermi circles, parallel and\nantiparallel to BSO\nR(Fig. 1b). Electrons in the parallel and antiparallel spin states (hereafter, we\nrefer to these as the up and down spins, respectively), though moving in the same direction and\nspatially unresolved when injected from a QPC into the 2DEG, have different Fermi wavevectors\nand thus will be deflected along different cyclotron trajectories in the presence of a transverse\nmagnetic field. Spin-selective spatial separation of an electron beam is therefore achieved.\nTo study the spatial separation of the two spin species, another QPC is placed at a distance\nLfrom the QPC emitter to act as a charge collector, forming a geometry (Figs. 1a, 1c, and the\ninset of 1d) known as transverse magnetic focusing11, 12, 14–18. Magnetic focusing occurs when the\nelectrons that leave the QPC emitter are focused into the QPC collector, giving peaks in collector\nvoltage (i.e., focusing peaks) at magnetic fields where an integer multiple of cyclotron diameter is\nequal toL. The two spatially separated spin species travel with different cyclotron radii and thus\nwill require two different magnetic fields\nB\"#=2\u0016hk\"#\neL=2(p2m\u0003EF\u0007m\u0003\u000b=\u0016h)\neL; (1)\nto focus themselves directly into the collector (inset of Fig. 1d), where \u0016his Planck’s constant\ndivided by 2\u0019,eis the elementary charge, m\u0003is the electron effective mass, EFis the Fermi\nenergy,k\"(k#) refers to the Fermi wavevector of spin-up (-down) state, and \u000bparametrizes the\nstrength of Rashba spin-orbit interaction. A spatial splitting of electron spins is therefore visible\nas a peak splitting in the magnetic focusing spectrum, allowing us to easily track and investigate\nthe spatial spin separation.\nFigure 1d shows the magnetic focusing spectrum, with the emitter ( GE) and collector con-\nductance (GC) both set to 100\u0016S (above the quantized plateau at 2e2=h) to allow both spin\n3species to propagate through the one-dimensional channels (Supplementary Notes 1 and 2). For\nB < 0focusing peaks appear periodically at integer multiples of B\u00190:19T, corresponding to\nwhen electrons are focused into the collector. This value is consistent with the cyclotron motion\nB= 2p2m\u0003EF=eL calculated using the two-dimensional electron density. For B > 0electrons\nare directed in the opposite direction, therefore no peaks in collector voltage are observed. The\nsplitting of the focusing peak (hereafter referred to as the focusing peak doublet) is observed on the\nfirst and the third focusing peaks as evidence of spatial spin splitting. The low-field B\"and high-\nfieldB#peak within the doublet corresponds to the spin-up and spin-down electrons, respectively.\nThe Rashba parameter \u000bestimated from the peak splitting using equation (1) is 3:1\u000210\u000011eVm,\nclose to the value estimated from the beating pattern in the Shubnikov-de Haas oscillations (see\nSupplementary Note 1). There is additional structure around the focusing peaks which is likely\ndue to the quantum interference effects19. We note that the focusing peak doublet is not visible\non the second focusing peak. This is consistent with the model20that the electrons are subject to\nspin-flip when they are reflected from the edge of the 2DEG and hence the two spatially separated\nspin branches reunite with each other at the collector (see Supplementary Note 3).\nControl of charge and spin dynamics So far the magnetic focusing spectrum can only show\nthat the electrons leaving from a QPC emitter are spatially spin split, without being able to shed\nany light on the spin dynamics afterwards. An important open question remains on how each\nspin species evolves due to the influence of a rotating BSO\nR(in the reference frame of the spin)\n– which rotates along the cyclotron trajectories as the momentum rotates – in such a spin-orbit\ncoupled 2DEG. For example, it is desirable to understand whether electron spins can maintain\ntheir coherence before reaching the collector, and also whether these spins adiabatically follow\nBSO\nR. To study the binary spin dynamics, we now force the collector to act as a spin analyzer by\nintroducing the lateral spin-orbit interaction21, 22and manipulating the energy and population of the\none-dimensional subbands in the collector. A voltage difference between the two sides of the split\ngate is used to create a lateral inversion asymmetry and consequently a lateral spin-orbit effective\nmagnetic field BSO\nLpointing along the zaxis (Fig. 1a). The electrically tunable BSO\nL+BSO\nRwithin\nthe emitter and collector QPC allows us to respectively prepare and analyze the electron spins\nalong any specific direction in the y-zplane. Additionally, a top gate (gate T in Fig. 1c) covers the\nentire focusing path and is used to vary BSO\nR(and equivalently \u000b) in the 2DEG region.\nThe electron spins transmitted through the QPC emitter stabilize at the state determined by\nBSO\nL+BSO\nRand consequently their orientations are initialized out of the 2DEG plane. In other words,\nthe spin-up (spin-down) electrons are tilted toward negative (positive) z-direction by BSO\nLowing to\nbeing in the one-dimensional BSO\nL+BSO\nRparallel (antiparallel) spin states. After leaving the emitter,\n4the electrons experience only the in-plane BSO\nR(since the focusing transverse magnetic field is\nsmall compared to BSO\nR) and therefore can precess about it as depicted in Fig. 1a if they propagate\ncoherently. We can alter the spin orientation by controlling the spin precession frequency using top\ngate voltage VT, which determines BSO\nR. Here we first demonstrate an electrically-tunable spatial\nspin separation in Fig. 2a, where the evolution of focusing spectrum of the first doublet is measured\nas a function of VTatGE= 160 \u0016S andGC= 100 \u0016S. The two superimposed dashed lines are\nthe calculated B\"andB#focusing fields using the model of spin precession described below. The\nspatial separation between the two spin species, manifested as the peak splitting jB#\u0000B\"j=\n4m\u0003\u000b=\u0016heL, increases with increasing VT.\nWe now move on to study the spin dynamics of the two spin species and the phase correlation\nbetween them. This is achieved by lowering the collector conductance to GC= 20 \u0016S such that\nthe QPC acts as a spin analyzer, as described in Supplementary Note 2. The orientation of incident\nelectron spins is indicated by the magnitude of the collector voltage (i.e., the focusing peak height).\nElectrons can propagate through the collector if their spin is parallel to the polarization direction,\nand cannot pass if their spin is antiparallel. Figure 2b shows that both the B\"andB#focusing\npeaks in collector voltage oscillate with VT. These oscillations are \u0019out-of-phase with each other,\ni.e., each local maximum (minimum) in collector voltage along the B\"focusing peak – which\ncorresponds to the incident spins being parallel (antiparallel) to the polarization direction of the\ncollector – coincides with the local minimum (maximum) along the B#peak. Evidently, both the\nup and down spin coherently precess and maintain their initial \u0019out-of-phase correlation after\nundergoing the action of the rotating BSO\nR.\nWithin the adiabatic approximation in which BSO\nRchanges its direction slowly such that the\nsystem adapts its configuration accordingly, the direction of electron spins with respect to BSO\nR\nremains conserved (i.e., the spinors can be described as a superposition of the adiabatic BSO\nReigen-\nstates with conserved probabilities; see Supplementary Note 4 for more details). Hence, the elec-\ntron spins precess about BSO\nRwith a Larmor frequency of !s\n\"#= 2\u000bjk\"#j=\u0016h. The spin precessional\nangle accumulated by electrons traveling along a semiclassical cyclotron orbit to the collector is\ntherefore given as23\n\u001e(VT) =!s\n\"#t\"#=\u0019m\u0003\u000b(VT)L=\u0016h2; (2)\nwheret\"#is the time interval for the focusing process. This angle is irrespective of spin orientation\nand depends only on the strength of spin-orbit interaction \u000b(VT)for a fixedL, consistent with the\nobservation of antiphase oscillations in the collector voltage for B\"andB#in Fig. 2b. Moreover,\nthe oscillations enable us to calculate the gate-voltage dependent variation of \u000busing equation (2),\n5which is consistent with the value obtained from the splitting of focusing peaks using equation (1).\nLater in this paper we will compare these \u000bvalues derived independently using these two methods.\nSuch a consistency is here evident from the excellent quantitative agreement between the position\nof the focusing peaks measured experimentally and the values calculated using equation (1) in\naccordance with the \u000bvalue derived from the spin precessional motion (dashed lines in Figs. 2a\nand 2b). The fact that the two antiphase oscillations are quantitatively described by considering\nspin precession in the adiabatic limit indicates that both of the spatially separated up and down spin\nadiabatically follow and precess about the rotating BSO\nRas illustrated in Fig. 1a. This also suggests\nthat the phase correlation between the two separate, neighboring spin types can be electrically\ncontrolled via tuning \u000b. It is worth noting that the phase correlation observed in focusing spectra\nis equal to\u0019regardless of the strength of spin-orbit interaction because both spin types are focused\ninto the same collector by different magnetic fields, while in reality opposite spins travel along\ndifferent trajectories and gain different phase shifts determined by equation (2).\nSimilar results are observed in other devices, as shown in Fig. 3 where the data are obtained\nusing device B after illumination (see Methods). Figures 3a and 3b compare the magnitude of\ntheB\"andB#focusing peaks as a function of VT, forGC= 20 and100\u0016S, respectively. For\nGC= 20 \u0016S (Fig. 3a) the collector acts as a spin analyzer. The B\"andB#collector voltages\noscillate with VT, and are\u0019out of phase with each other. In contrast, no oscillations are observed\nwhenGCis raised to 100\u0016S, where the QPC collector acts only as a charge detector (Fig. 3b). The\noscillations also disappear when either the emitter or the collector QPC is biased symmetrically\n(Supplementary Note 5), which is consistent with our spin precession model. When the emitter\nis biased symmetrically (i.e., BSO\nL= 0), the electron spins which are emitted are aligned along\nthe axis of BSO\nRand hence no spin precession shall occur. Also, when BSO\nLis removed from the\ncollector, the spin polarization is analyzed along the stationary BSO\nRspin states, and hence no spin\nprecession can be probed. Note that as with device A, there is a quantitative agreement between the\n\u000b(VT)obtained with equations (1) and (2), which use the peak splitting and oscillatory collector\nvoltage data, respectively.\nOne advantage of device B is that the QPC emitter and collector can be independently con-\ntrolled since they do not share a common middle gate. This enables us to reverse the polarity of\nthe lateral inversion asymmetry of the QPC and hence BSO\nL, simply by reversing the polarity of the\nvoltage difference between the two sides of the split gate. Figure 3c, d presents a comparison of the\nfocusing spectra for \u0001VE= +1:5V and\u00001V . Here the focusing spectra are plotted as a function\nof magnetic field and \u000b(VT)\u0002L, instead ofVT(as in other figures), since when \u0001VEchanges the\ndistanceLbetween the emitter and collector also changes and thus needs to be taken into account\n6(see Supplementary Note 6). A phase inversion in the oscillations for both the B\"andB#focusing\npeaks is apparent as the lateral bias \u0001VEis changed from +1:5V to\u00001V . Such an inversion can be\neasily understood using a schematic in Fig. 3e which illustrates the phase evolution, depicted using\nBloch spheres, of the spin-up (red arrows) and spin-down (blue arrows) electrons travelling along\nthe cyclotron trajectory at positive and negative \u0001VE, respectively. Since the initial phase correla-\ntion between spin-up and spin-down electrons is inverted as the direction of BSO\nLis reversed, the\nobserved phase correlation for the arrivals that undergo the same phase evolution (and equivalently\n\u000bL) must also be inverted.\nFigure 4 summarizes values obtained for the Rashba coefficient \u000b. The values obtained via\nthe focusing peak splitting using equation (1) (open symbols) and via the oscillatory collector volt-\nage using equation (2) (solid symbols) are both shown and are in excellent quantitative agreement\nwith each other. In order to directly compare data before (red symbols) and after (blue symbols)\nillumination, we plot the Rashba coefficient \u000bas a function of carrier density. The value of \u000bfol-\nlows the same trend line both before and after illumination. For comparison, we also plot the val-\nues of\u000b(VT)published in recent work22using a spin field-effect transistor (fabricated on the same\nwafer used here), where \u000bis estimated from spin precession measurements in a steady -instead of\nrotating- BSO\nR. There is excellent quantitative agreement between the values of \u000bobtained from\nthese different devices and methods. The spin focusing technique appears more informative than\nthe conventional SdH beating analysis24which is sometimes difficult to observe (Supplementary\nNote 1), and provides a reliable means for the determination of \u000bvalue in the ballistic transport\nregime.\nDiscussion\nThe ability to manipulate and probe coherent spin dynamics in materials with high spin-orbit in-\nteraction is important for understanding the physics of emerging materials, as well as to having\nimplications for spintronics and (topological) quantum computing25–27. Distinct from most previ-\nous studies22, 28– which rely on the introduction of polarized electrons to break the spin symmetry\nand are limited in that only the majority spin type can be resolved and used – our spin focusing\ntechnique provides a route to probe and manipulate the coherent spin dynamics of both spin species\nand their phase correlation in semiconductor nanostructures, and can be readily extended to ma-\nterials with unusual band structures such as topological insulators6, 7, 29, graphene and its hybrid\nstructures30. A recent study18that used the conventional magnetic focusing technique to probe\nthe properties of graphene is a successful example. From a technological viewpoint, our ability\nto spatially bifurcate the two electron spin types and coherently manipulate them to any specific\n7orientation (through spin precession and the fast manipulation of BSO\nRandBSO\nLusing surface gates)\nmake it possible to prepare two separate, neighbouring spins with an electrically controllable phase\ncorrelation, which has implications for interferometer and quantum logic operations.\nMethods\nDevices. A gated modulation-doped In 0:75Ga0:25As/In 0:75Al0:25As heterostructure is used in this\nwork. The layer sequence is grown by molecular beam epitaxy as follows: 250 nm In 0:75Al0:25As;\n30 nm In 0:75Ga0:25As (quantum well); 60 nm In 0:75Al0:25As (spacer); 15 nm In 0:75Al0:25As (Si-\ndoped); 45 nm In 0:75Al0:25As; and 2 nm In 0:75Ga0:25As (cap). A dielectric layer (27 nm and 40 nm\nfor device A and B, respectively) of SiO 2is deposited on the wafer surface by plasma-enhanced\nchemical vapor deposition. Subsequently, surface gates are defined using electron-beam lithog-\nraphy and thermal evaporation of Ti/Au. There are two device designs, denoted device A and\ndevice B, as shown in Fig. 1c. In device A the lateral biases of the emitter and collector QPCs are\ndefined as \u0001VE=VE\u0000VMand\u0001VC=VC\u0000VM, respectively, whereas in device B \u0001VE=VE1\u0000VE2\nand\u0001VC=VC1\u0000VC2. Note that the emitter is covered by the top gate, such that the Fermi wavevec-\ntor of the focusing electrons that transit from the emitter to the bulk can be reliably controlled with\nthe top gate. The collector is not covered by the top gate so that the spin polarization can be ana-\nlyzed along a fixed axis, independent of the top gate voltage. Data from device A and B are taken\nbefore and after illumination, respectively, which give very different characteristics of the 2DEG.\nMeasurements. Experiments are performed at a base temperature of 25 mK in a dilution refrig-\nerator equipped with a superconducting magnet. The carrier density and mobility of the 2DEG are\nmeasured to be 2:1\u00021011cm\u00002and1:7\u0002105cm2V\u00001s\u00001, respectively, using four-terminal mag-\nnetotransport measurements (Supplementary Note 1). This gives a mean free path of 1:3\u0016m for\nmomentum relaxation. After illumination, they increased to 3:9\u00021011cm\u00002,2:6\u0002105cm2V\u00001s\u00001,\nand2:7\u0016m, respectively. For transverse magnetic focusing experiments, simultaneous lock-\nin measurements of emitter and collector QPC conductances are carried out by supplying two-\nindependent excitation sources of a 77Hz a.c. voltage Vexc= 100 \u0016V to the emitter and a 37Hz\na.c. current Iexc= 1nA to the collector. The magnetic field is applied normal to the 2DEG plane\nto focus electrons into the collector. The focusing signal is measured as a voltage drop developed\nacross the QPC collector in linear response to the 77 Hz a.c. current from the QPC emitter.\nData availability. The data that support the findings of this study are available from the corre-\nsponding author upon reasonable request.\n8References\n1. Hirsch, J. E. Spin Hall Effect. Phys. Rev. Lett. 83, 1834-1837 (1999).\n2. Kato, Y . K., Myers, R. C., Gossard, A. C. & Awschalom, D. D. Observation of the Spin Hall\nEffect in Semiconductors. Science 306, 1910-1913 (2004).\n3. Valenzuela, S. O. & Tinkham, M. Direct electronic measurement of the spin Hall effect. Nature\n442, 176-179 (2006).\n4. Bernevig, B. A. & Zhang, S.-C. Quantum Spin Hall Effect. Phys. Rev. Lett. 96, 106802 (2006).\n5. Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum Spin Hall Effect and Topological\nPhase Transition in HgTe Quantum Wells. Science 314, 1757-1761 (2006).\n6. Hsieh, D. et al. 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Taychatanapat, T., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Electrically tunable trans-\nverse magnetic focusing in graphene. Nature Physics 9, 225-229 (2013).\n19. van Houten, H. et al. Coherent electron focusing with quantum point contacts in a two-\ndimensional electron gas. Phys. Rev. B 39, 8556-8575 (1989).\n20. Usaj, G. & Balseiro, C. A. Transverse electron focusing in systems with spin-orbit coupling.\nPhys. Rev. B 70, 041301 (2004).\n21. Debray, P. et al. All-electric quantum point contact spin-polarizer. Nature Nanotech. 4, 759-\n764 (2009).\n22. Chuang, P. et al. All-electric all-semiconductor spin field-effect transistors. Nature Nanotech.\n10, 35-39 (2015).\n23. Datta, S. & Das, B. Electronic analog of the electro-optic modulator. Appl. Phys. Lett. 56,\n665-667 (1990).\n24. Nitta, J., Akazaki, T., Takayanagi, H. & Enoki, T. Gate Control of Spin-Orbit Interaction in an\nInverted In 0:53Ga0:47As/In 0:52Al0:48As Heterostructure. Phys. Rev. Lett. 78, 1335-1338 (1997).\n25. Dario, B. & Procolo, L. Quantum transport in Rashba spin-orbit materials: a review. Rep.\nProg. Phys. 78, 106001 (2015).\n26. Petersson, K. D. et al. Circuit quantum electrodynamics with a spin qubit. Nature 490, 380-383\n(2012).\n27. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and\ntopological quantum computation. Rev. Mod. Phys. 80, 1083-1159 (2008).\n1028. Choi, W. Y . et al. Electrical detection of coherent spin precession using the ballistic intrinsic\nspin Hall effect. Nature Nanotech. 10, 666-670 (2015).\n29. Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045-\n3067 (2010).\n30. Geim, A. K. & Grigorieva, I. V . Van der Waals heterostructures. Nature 499, 419-425 (2013).\nAcknowledgements\nWe thank S.-C. Ho and C.-T. Liang for discussions. This work was supported by the Ministry of\nScience and Technology (Taiwan), the Headquarters of University Advancement at the National\nCheng Kung University, and the Engineering and Physical Sciences Research Council (UK).\nAuthor Contributions\nS.-T.L., C.-H.C., and J.-C.F. performed the measurements and analysed the data, in which C.-W.C.\nand T.-M.C. participated. L.W.S. and G.L.C. fabricated the devices with contributions from M.P.,\nand T.-M.C; I.F., H.E.B., and D.A.R. provided wafers; J.P.G. and G.A.C.J. performed electron-\nbeam lithography. S.-T.L and T.-M.C. wrote the paper with input from all of the authors; T.-M.C.\ndesigned and coordinated the project.\nAdditional information\nCompeting interests: The authors declare no competing financial interests.\n11Z\nyX\nBSO \nLBSO \nRBSO \nR\nda\ncb\nV E M CT\nVexcIexc\nE1 E2 C1\nC2Tkxky\nSO\nRB|k↑| |k↓|Δk\ndevice A device B\nB↑B↓iiiB>0iB↑iiB↓x\nyFigure 1: Scheme for spatial spin separation and control of spin dynamics. a , Schematic view\nof a spin focusing device. The structural inversion asymmetry gives rise to an in-plane Rashba spin-\norbit field BSO\nRon the spin of every moving electron, illustrated by the inset. We define the spin up,\n\"(spin-down,#), as parallel (antiparallel) to BSO\nR. Spin-up and spin-down electrons have different\nFermi wavevectors and thus will be deflected along different cyclotron trajectories in a transverse\nmagnetic field, resulting in spatial spin separation. Within the QPC constriction, an additional\nlateral spin-orbit field BSO\nLcan be created via laterally biasing the gates to tilt spins toward either\npositive or negative zdirection. The two spatially separated spin species thus precess about BSO\nR\nin the 2DEG region. The spin-orbit fields BSO\nRandBSO\nLare represented by green arrows, while the\nred and blue arrows represent up and down spins, respectively. b, The Fermi surface (red and blue\ncircle of radius k\"andk#for spin up and spin down) is spin-split with a wavevector separation \u0001k\n(=k#\u0000k\") in the presence of Rashba spin-orbit interaction. The arrows are coloured following\nthe same convention as in a.c, SEM images of device A and B, with scale bar of 1 \u0016m. Devices A\nand B are measured before and after illumination, respectively, which gives markedly different\nelectron densities and mobilities (see Methods). Device B contains two pairs of split gates to allow\nindependent control of the QPC emitter (using E1 and E2) and collector (using C1 and C2). d,\nTransverse magnetic focusing spectrum measured from device B. The inset shows representative\ntrajectories for spin-up (red trace) and spin-down (blue trace) electrons at different magnetic fields.\n120.10 0.15 0.20 0.250.00.10.20.3\n  VT (V)\nB (T)    \n0.10 0.15 0.20 0.250.00.10.20.3\n  VT (V) \nB (T)  \n  \n0 6\n    Collector Voltage ( µV)\n0 10\n    Collector Voltage ( µV)\nnπa b B↓B↑ B↓B↑\n(n+2)π\n(n+1)π(n+3)πFigure 2: Magnetic spin focusing spectra. a , Collector voltage as a function of magnetic field B\nand top gate voltage VTfor device A with emitter conductance GE= 160 \u0016S and collector con-\nductanceGC= 100 \u0016S. The lateral bias \u0001VEis fixed at 1:33V (see Methods for the quantification\nof\u0001VE) whereas \u0001VCranges from 2:15V to 2:41V asVTincreases to keep both QPCs at fixed\nconductance values. The solid line illustrates the average Bbetween the spin-up and spin-down\nfocusing peaks (B\"+B#)=2, which can be used to determine the carrier density n2D. The dashed\nlines show the focusing peak positions calculated using the spin precessional motion. b, As in a\nbut withGCreduced to 20\u0016S to turn the collector into a spin analyzer. \u0001VEis fixed at 1:23V\nwhereas \u0001VCranges from 2:02V to 2:30V . The subsequent maxima (minima) of the oscillating\ncollector voltage along the B\"andB#focusing peaks correspond to rotations of the incident spins\nbyn\u0019, wherenis an integer, such that the spin is parallel (antiparallel) to the polarization direction\nof the collector.\n13a\nbc\nenπ (n+1)π (n+2)π\nSO\nRB\nSO\nLBSO\nRB\nΔVE<0zΔVE= 1.5 VΔVE= -1 VB↑ B↑ B↓ B↓\nSO\nLBSO\nRB\nΔVE>0SO\nRB\nz\n R+\n R−\n-0.24 -0.22 -0.20 -0.18 -0.16 -0.143.73.83.94.0\n  \nαL (10-17 eVm2)\nB (T)\n  \n5 12\n   Collector Voltage (µV)\n-0.26 -0.24 -0.22 -0.20 -0.18 -0.163.73.83.9\n  \nαL (10-17 eVm2)\nB (T) \n \n5 15\n   Collector Voltage (µV)dFigure 3: Spin precession in a rotating BSO\nR. a, Collector voltage of the B\"(red) andB#(blue)\nfocusing peaks as a function of VT, withGE= 100 \u0016S andGC= 20 \u0016S. Data for all panels\nin this figure are from device B. The lateral biases of the QPC emitter and collector are set at\n\u0001VE= 0:25V and \u0001VC= 0:5V , respectively. b, As in aexcept with GCincreased to 100\u0016S\nfor comparison. c, Magnetic spin focusing spectrum as a function of \u000bLand magnetic field for\n\u0001VE= 1:5V and \u0001VC= 0:5V .d, As in cbut with \u0001VEchanged to\u00001V to invert the direction\nofBSO\nL. This gives rise to an inverse \u0019out-of-phase oscillation in the B\"andB#focusing peaks\nwith respect to that in panel c. Only the data with the collector voltage above 5 \u0016V are shown\nto highlight the varying focusing peak height. Data for panels canddare obtained in a different\ncooldown to panels aandb. The dashed lines indicate the focusing peak positions calculated using\nthe same method as in Fig. 2. e, A sequence of Bloch spheres illustrate the phase evolution of the\nspin-up (red arrows) and spin-down (blue arrows) electrons moving along the focusing trajectory.\nThe top (bottom) row of spheres represents the phase evolution for \u0001VE>0(\u0001VE<0); the\nvertical (horizontal) axis represents BSO\nR(BSO\nL). Starting with electrons within the QPC emitter,\nthe combination of Rashba and lateral spin-orbit interactions prepares the BSO\nR+BSO\nLparallel and\nantiparallel spin states. After leaving the QPC and entering the 2DEG both spin types experience\nonly the Rashba effective field BSO\nRand precess about it.\n14a bFigure 4: Comparison of the measured Rashba coefficients. The Rashba coefficient \u000bis plotted\nas a function of carrier density n2D. Red and blue data points correspond to data obtained using\nthe magnetic spin focusing technique (illustrated in inset a) before (device A) and after (device B)\nillumination, respectively. Two methods are used to extract \u000b. Open symbols show the values\ngiven by equation (1) in the main article which considers the spatial spin separation of electrons.\nSolid symbols show values obtained using equation (2) which considers the precessional motion of\nthe spin. The dashed line shows a polynomial fit to the data from spin focusing. For comparison,\nthe Rasha coefficient obtained in recent measurements of a spin field effect transistor22(illustrated\nin inset b) are shown by the black solid line.\n15"    },    {        "title": "1701.02006v2.Dynamical_Generation_of_Spin_Current_and_Phase_Slip_in_Exciton_Polariton_Condensates.pdf",        "content": "Dynamical Generation of Spin Current and Phase Slip in\nExciton-Polariton Condensates\nS. D. Guo1and\u0003Bo Xiong1, 2\n1Department of Physics, Nanchang University, 330031 Nanchang, China\n2Skolkovo Institute of Science and Technology,\nNovaya Street 100, Skolkovo 143025, Russian Federation\u0003\n(Dated: September 1, 2021)\nAbstract\nWe show that how to generate propagation of spin degree in spin-symmetric exciton-polariton condensates\nin a semiconductor microcavity. Due to the stimulated spin-dependent scattering between hot excitons and\ncondensates, exciton polaritons form a circular polarized condensate with spontaneous breaking of the spin\nrotation symmetry. The spin antiferromagnetic state is developed evidently from the density and spin flow\npumped by localized laser source. The low energy spin current is identified where the steady state is char-\nacterized by the oscillating spin pattern. Finally, we predict via simulation how to dynamical generation of\nphase slip where ring-shape phase jump shows the behavior of splitting and joining together.\nPACS numbers: 72.25.Rb, 75.30.Ds, 72.70.+m, 71.36.+c\nI. INTRODUCTION.\nRecently, in semiconductor microcavities with\nquantum wells sandwiched between highly reflec-\ntive mirrors, the strong coupling is achieved be-\ntween excitons and photons [1–4]. Such coher-\nentlight-matterparticlescalledexciton-polaritons\nobey the Bose-Einstein statistics and thus con-\ndense at critical temperatures ranging from tens\nKelvin [5–7] till several hundreds Kelvin [8, 9],\nwhich exceeds by many orders of magnitude\nthe Bose-Einstein condensation temperature in\natomic gases. Recently, electrically pumped po-\nlariton laser or condensation was realized based\n\u0003stevenxiongbo@gmail.com; ORCID iD: 0000-0003-2434-\n4898on a microcavity containing multiple quantum\nwells [10, 11]. Considering the high transition\ntemperatures and high tunability from pumping\nsource, semiconductor microcavities are perfectly\nsuited for studies of macroscopically collective\nphenomenon and have initiated the fascinating re-\nsearch on the polariton quantum hydrodynamics.\nThe polaritons have two allowed spin projec-\ntions on the structure growth axis, \u00061, corre-\nsponding to right- and left- circular polarizations\nof photons. In diverse semiconductor materials\nlike GaAs/GaAlAs [12], Si [13], organic single-\ncrystal microcavity SiN x/SiO 2[14] and so on, spin\ninjection and detection has been successfully real-\nized which is one of the key ingredients for func-\ntional spintronics devices. A number of prominent\n1arXiv:1701.02006v2  [cond-mat.mes-hall]  4 Mar 2021spin-related phenomena both in interacting and\nin noninteracting polariton systems have already\nbeen predicted and observed in the microcavities,\nsuch as, spontaneous polarization [15–21], polar-\nization multistability [22–27], optical spin Hall ef-\nfect [28–34] and topological insulator [35–40], spin\nZeeman and Meissner effect [41–43].\nSpin degrees of freedom in two-dimensional\nexciton-polaritons superfluid can drastically\nchange elementary topological vortices referred\nto as half-quantum vortices (HQV) [44–48]\nwhich are characterized by a half-integer value\nof vorticity in contrast to the regular quantum\nvortex [49–56] where the vorticity takes only\ninteger values. Usually HQV carry only one\nhalf-integer topological charge originating both\nfrom the superfluid current proportional to r\u0012,\nand from \u0019spin disclinations superimposed\nas a result of Berry’s phases induced by spin\nrotations [57]. Relevant ideals of half vortices\nhave been discussed in A phase of3He [58–60], in\ntriplet superconductors Sr 2RuO 4[61] and spinor\natomic Bose-Einstein condensates [62–65] with\ntwo different spin components where HQV is just\nresiding in one of components [66–70].\nHowever, precise coherent control of spin po-\nlarization, propagation and topological defects\nin exciton-polariton condensates still remains a\ncore challenge. Here, we address this problem,\nand demonstrate exciton-polariton condensates\nwill not only show spontaneous polarization and\nalso coherent propagation of the pseudospin un-\nder nonlocal spin injection. When taking into ac-\ncount incoherent hot exciton reservoir scatteredinto coherent states, dramatically enhanced spin-\npolarized signal can be observed at the appropri-\nate pumping regime. Moreover, the coherent spin\nantiferromagnetic state can also be identified and\nmanipulated by spin-symmetric pumping source.\nAdditionally, cavity engineering allows us to the\ndynamic generation of phase slip where ring-shape\nphase jump shows the behavior of splitting and\njoining together induced by incoherent reservoir\nas a result of effective gauge field.\nII. PHYSICAL BACKGROUND.\nIn the absence of external magnetic field the\n“spin-up” and “spin-down” states \u001b=\u0006of nonin-\nteracting polaritons, or their linearly polarized su-\nperpositions, are degenerate corresponding to the\nright (\u001b+)and left (\u001b\u0000)circular polarizations of\nexternal photons. The spinor nature of exciton\npolaritons can therefore be manifested since the\nspin are essentially free in semiconductor micro-\ncavities. To illustrate the fully degenerate spinor\nnature, and as a first step, the Zeeman energy\nmust be much smaller than the interaction energy.\nThus we shall consider only the case of zero mag-\nnetic field achieving a good approximation in the\nfollowing. Since the interaction between exciton\npolaritons depends on their total spins (singlet or\ntriplet), their spin states may be changed after the\nscattering. The spin-dependent interactions cause\nthe polariton spin states exchange. Moreover, ad-\nditional mixing may comes from the longitudinal-\ntransverse (LT) splitting of polaritons (referred to\nas the Maialle mechanism) [71] and from struc-\n2tural anisotropies [72].\nThe low energy dynamics is therefore described\nby a pairwise interaction that is spin-rotation in-\nvariant and preserves the spin of the individual\nexciton polaritons. The general form of this inter-\naction is ^V(r1\u0000r2) =\u000e(r1\u0000r2)X2f\nF=0gF\u0001^PF\nwheregF= 4\u0019~2aF=M,Mis the mass of ex-\nciton polaritons, ^PFis the projection operator\nwhich projects the pair 1 and 2 into a total spin\nF state, and aFis the s-wave scattering length\nin the total spin F channel. For exciton po-\nlaritons of f= 1bosons, interaction has form\n^V=g0\u0001^P0+g2\u0001^P2. In terms of nonlinear op-\ntics, the coupling coefficients of polarization inde-\npendentc0and so-called linear-circular dichroism\nc2can be estimated through the matrix elements\nof the polariton-polariton scattering in the singlet\nand triplet configurations.\nIt is convenient to write the Bose condensate\n\ta(r)\u0011<^ a(r)>as\ta(r) =p\nn(r)\u0010a(r), where\nn(r)is the density, and \u0010ais a normalized spinor\n\u0010+\u0001\u0010= 1. It is obvious that all spinors related\nto each other by gauge transformation ei\u0012and\nspin rotationsU(\u000b;\f;\r ) =e\u0000iSx\u000be\u0000iSy\fe\u0000iSz\rare\ndegenerate, where (\u000b;\f;\r )are the Euler angles.\nThe non-equilibrium dynamics of polariton\ncondensates is described by a Gross-Pitaevskii\ntype equation for the coherent polariton field,\nwhich should be coupled to a hot-excitons reser-\nvoir excited by the nonresonant exciting pump.\nThe model is, however, generalized to take into\naccount the polarization degree of freedom of hot\nexciton. In this approach, instead of polarization\nindependent scattering, we must take into accountdichroism scattering between hot exciton and co-\nherent polariton field.\nLet us turn to the pseudospin representation,\nthen the local spin density\u0000 !sat the position r\nand timetis\u0000 !s(r;t) = \ty(r;t)^\u0000 !s\t(r;t), where\n^\u0000 !s= (~=2)^\u0000 !\u001bwith^\u0000 !\u001bbeing the Pauli matrices.\nThe usual definition of the free-particle probabil-\nity current Jn= Reh\n\ty(r;t)^P^I\nm\t(r;t)i\n, where ^I\nis the identity, and probability spin current J\u0000 !s=\nReh\n\ty(r;t)^P\u0000 !s\nm\t(r;t)i\n. In addition, the emer-\ngent magnetic monopoles defined by analogy with\nMaxwell’s equation as r\u0001\u0000 !scan be realized and\ncharacterized by a divergent in-plane pseudospin\npattern, that have been present in magnetically\nfrustrated materials, spin-ice [73–79], magnetic\nnanowires [80] and atomic spinor Bose-Einstein\ncondensates [81, 82]. The dynamics of each spin\nunder the effect of magnetic field is governed by\nthe precession equation @tS=H\u0002S=~. The to-\ntal effective magnetic field Hrepresents the sum\nof the field responsible provided by the spin de-\npendent and independent polariton-polariton in-\nteractions and polariton-hot exciton interaction\n(LT splitting HLTis assumed to be negligible in\nhigh density regime). Very different from those\nisolated or closed system, the dynamic of spin pat-\ntern in such open-dissipative system is crucially\ndetermined by the pump source. We will go into\nfurther details in the following.\nIII. THEORETICAL MODEL.\nIn the following, we study the propagation of\npolarized polariton in the a planar microcavity\n3Figure 1. (Color online) The spontaneously cir-\ncular polarization of spinor condensate non-resonant\npumped by linearly polarized laser. (a) Proposed\nschemetoexperimentallystimulatingspontaneouscir-\ncular polarization by nonpolarized laser beam. (b)\nSpinor is polarized when the laser power is larger than\nfirst threshold value, however, unpolarized after laser\npower is above second threshold value. (c) Density\ndistribution of hot exciton (left picture which has the\nsame profile for both components) and spinor polari-\nton (middle and right pictures for each components)\nin real space. Here, simulations are in the absence\nof disorder for 4 pumping points with a small radius\n1:54\u0016m. The size of profile is 24x24 and the other\nparameters used in the simulations are shown in the\npaper.and generation of spin polarization, spin current\nand the observability of the HQV, in realistic\nstructures. The equation of motion for the spinor\npolariton wave function reads [83–86]\ni~@t \u0006(r) =\u001a\n\u0000~2\n2mr2+i~\n2\u0000\ng2nR\u0006+h2nR\u0007+\f2j \u0006j2+f2j \u0007j2\u0000\rC\u0001\n+Vext(r)\u001b\n \u0006(r)\n+\b\n~\u0000\n\f1j \u0006j2+f1j \u0007j2\u0001\n+VR(r)\t\n \u0006(r); (1)\nwhere \u001brepresents the condensed field, with\n\u001b=\u0006representing the spin state of polaritons\nwith effective mass m.\rCrepresents the coher-ent polariton decay rate. \f1andf1is the spin-\nconserved and spin-exchange polariton-polariton\ninteraction strength, respectively. nR\u001bis the den-\n4sity of the incoherent hot exciton reservoir. And\nhere,VR(r) = ~[g1nR\u0006+h1nR\u0007+ \nP\u0006(r)]rep-\nresents spin-conserved and spin-exchange interac-\ntions with hot exciton reservoir where P\u0006(r)is\nthe spatially dependent pumping rate and g1,h1,\n\n>0are phenomenological coefficients to be de-\ntermined experimentally. Vext(r)represents the\nstatic disorder potential in semiconductor micro-\ncavities, which is typically chosen as the same for\nboth component polaritons. g2nR\u0006andh2nR\u0007are\nrelated with the condensation rate in that growth\nof condensate are stimulated by hot excitons with\nsame spin or cross spin, respectively [87]. \f2and\nf2are the same-spin and cross-spin nonradiative\nloss rates, respectively.\nThe equation 1 of condensate is coupled to a\nrateequationdescribingthetimeevolutionofden-\nsitynR\u001bof incoherent hot exciton as:\n@tnR\u0006=\u0000\u0000nR\u0006\u0000\u0002\ng2j \u0006j2+h2j \u0007j2\u0003\nnR\u0006+P\u0006;\n(2)\nwherethereservoirrelaxationrate \u0000ismuchfaster\nthan that of condensate \u0000\u001d\rCwhere the Gaus-\nsian pump laser P\u0006=Wis assumed nonpolar-\nized (corresponding to linear or horizontal polar-\nization) providing a sufficient large occupation in\nmomentum space of incoherent hot exciton. The\nstimulated emission of the hot exciton reservoir\ninto condensate is taken into account by the term\n[g2j \u0006j2+h2j \u0007j2]nR\u0006. The spatial diffusion rate\nof reservoir density has been neglected. In the\nfollowing, we solve the coupled Eqs. 1 and 2 nu-\nmericallystartingfromasmallrandominitialcon-\ndition. As we can see that, the time evolution of\nthe system has been obtained until a steady state\nFigure 2. (Color online) The spontaneously circu-\nlar polarization of spinor condensate non-resonant\npumped by 6 and 8 linearly polarized laser, respec-\ntively. Top panel: density distribution of hot exci-\nton (left picture which has same profile for both com-\nponents) and spinor polariton (middle and right pic-\nturesforeachcomponents)inrealspacefor6pumping\npoints. Bottom panel: distribution of magnetic polar-\nization along the Z axis for 6 pumping points (left\npicture) and 8 pumping points (right picture, where\ninset shows density distribution of two component po-\nlariton). The size of profile is 24x24 and the other pa-\nrameters used in the simulations are the same as those\nin the Fig. 1.\nis reached independent of the initial noise.\nIV. STEADY STATE.\nA. Spatially homogeneous system\nLet us begin with some analytical considera-\ntion on spinor condensate. In the homogeneous\ncase, i.e., under a spatially homogeneous pump-\ning and in the absence of any external poten-\ntial, Eqs. 1 and 2 admit analytical stationary\nspinor configuration. Below the pumping thresh-\n5old, the condensate remains unpopulated, while\nthe reservoir grows linearly with the pump in-\ntensity asnR\u0006=W=\u0000. At the threshold pump\nintensityWth, the stimulated emission rate ex-\nactly compensates the losses g2nR\u0006+h2nR\u0007=\n\rCand condensate becomes populated dynam-\nically. We notice that threshold pump inten-\nsity becomes Wth= \u0000\rC=(g2+h2). Above the\nthreshold, the reservoir density is homogeneous\nnR\u0006=W=(\u0000 +g2j \u0006j2+h2j \u0007j2), from this, we\nobtain\nZR\u0018\u0000W(g2\u0000h2)\n\u00002+ \u0000 (g2+h2)ncZC;(3)\nhere, we have defined reservoir polarization ZR=\nnR+\u0000nR\u0000, condensate polarization ZC=j +j2\u0000\nj \u0000j2and condensate total density nc=j +j2+\nj \u0000j2. As long as g26=h2, condensate polarization\nis directly proportional to the reservoir polariza-\ntion.\nFrom the Eqs. 1, we find that the condensate\ndensity is\nnc\u0018\u0000\nW\u0000Wth\u0001\n\rC\u00011\n1\u00001\n2\u0010\nW\nWth+\f2+f2\ng2+h2\u0000\n\rC\u0011;(4)\nand condensate polarization satisfy\nMCZC= 0: (5)\nwhere\nMC=\u0012\n4Wg 2h2+ \u00002(\f2\u0000f2)\u0000W\u00002\r2\nC\nWth\u0001Wth\u0013\n:\nExcept very stringent condition MC= 0, other-\nwise, magnetization of condensate is always zero,\ni.e.,ZC= 0seen from the Eq. 5. If assuming\ncross-spin radiative and nonradiative loss rates isnegligible, magnetization condition MC= 0leads\nto following condition for pump laser power\nW=\r2\nC\n\f2(Wth)2=g2\n2\n\f2\u00002; (6)\ntherefore, considering necessary condition W >\nWth, wefindfollowingconditonshouldbesatisfied\nfor spontaneous magnetization of condensate,\ng3\n2\n\f2\rC\u00003>1:\nIf assuming condensate wave function takes\nthe form  \u0006(r) =P k\u0006!\u0006ei(k\u0006\u0001r\u0000!\u0006t)\u0018\n 0\u0006ei(k\u0006\u0001r\u0000!\u0006t), we find spectrum as\n!\u0006=~k2\n\u0006\n2m+~\n\u0006W; (7)\nwhere\n~\n\u0006= \n +(\f1+f1)nc\u0006(\f1\u0000f1)ZC\n2W\n+2 (g1+h1) \u0000 +G\u0001nc\u0006H\u0001ZC\n2 [\u00002+ \u0000 (g2+h2)nc+A];\nhere, wave vector k\u0006and frequency !\u0006re-\nmains so far undetermined, and coefficience\nG=g1g2+g1h2+g2h1+h1h2,H =\n(g1h2+g2h1\u0000g1g2\u0000h1h2). However, from Eq.\n7, we find frequency difference between two com-\nponent is given by\n!+\u0000!\u0000=~\u0000\nk2\n+\u0000k2\n\u0000\u0001\n2m+ \u0001~\n;(8)\nhere,\n\u0001~\n\u0018ZCf(\f1\u0000f1)=W\n\u0000(g1\u0000h1) (g2\u0000h2)=\u0002\n\u00002+ \u0000 (g2+h2)nc+A\u0003\t\n;\nwhereAis high order term of density and polar-\nizationA= (g2+h2)2(n2\nc+Z2\nC)=4which can be\ndominant term for the large density and polariza-\ntion. Interestingly, we can see that energy gap\n6is polarization dependence. In particular, when\n\f1'f1or large enough laser power W, polar-\nization dependence of frequency difference disap-\npears.\nB. Local density and spin approximation\nIn the presence of an inhomogeneous laser\npumpW(r)(or multiple pump Wi(r)), much\nricher phenomena will be represented, such\nas, spin domain formation, emergent magnetic\nmonopole, generation of half vortex and so on.\nUnder inhomogeneous laser pump, we thus look\nfor stationary spinor polariton wave function as\nfollowing form\n\t =0\n@ +\n \u00001\nA=p\n\u001a(r)\u0010(r)e\u0000i(\u001e(r)\u0000!\u0006t);(9)\nwhere\u001a(r)and\u001e(r)are the local density and\nphase of the condensate, and \u0010(r)is spinor func-\ntion. We are going to assume that the local pump\nimposes a boundary condition for the spinor func-\ntion at each pumping spot rp:limr!rp\u0010(r) =\u0015,\nlimr!rpkC(r) = 0, here, we have defined local\ncondensate density wave vector kC(r) =rr\u001e(r).\nIn the following, the dimensionless form of the\nmodelcanbeobtainedbyusingthescalingunitsof\ntime, energy, and length as: T= 1=\rC,E=~\rC,\nL=p\n~=m\rC, respectively.\nInserting Eq. 9 into the Eqs. of motion 1 and\n2, one obtains the following set of conditions forstationary solution:\n!\u0006=\u00001\n2\u0012r2p\u001ap\u001a+r2\u0010\u0006\n\u0010\u0006+ 2rp\u001a\u0001r\u0010\u0006p\u001a\u0010\u0006\u0000k2\nC\u0013\n+1\n\rC\u0000\n\f1j\u0010\u0006j2\u001a+f1j\u0010\u0007j2\u001a+g1nR\u0006+h1nR\u0007\u0001\n+\nW\n\rC;\n(10)\nand\n1\n2\u0000\ng2nR\u0006+h2nR\u0007+\f2j\u0010\u0006j2\u001a+f2j\u0010\u0007j2\u001a\u0000\rC\u0001\n+1\n2r\u0001kC(r) +rp\u001a\u0001kC(r)p\u001a+kC(r)\u0001r\u0010\u0006\n\u0010\u0006= 0;\n(11)\nand\n\u0000nR\u0006+\u0000\ng2j\u0010\u0006(r)j2+h2j\u0010\u0007(r)j2\u0001\n\u001a(r)nR\u0006=W(r):\n(12)\nDifferent from the single component condensate,\nnow in Eq. 10, the quantum pressure terms\nare not only originated from density r2p\u001abut\nalso from the spinor r2\u0010and even spin-density\ncouplingrp\u001a\u0001r\u0010. Moreover, in Eq. 11, be-\nsides the current divergence term, we can see the\nmoretermsappearedwhichisoriginatedfromcou-\npling of superfluid current with density pressure\nrp\u001a\u0001kC(r)or spin pressure kC(r)\u0001r\u0010.\nWe can make local density approximation\n(LDA) and local spin approximation (LSA) if the\nspatialvariationofthelaserpump W(r)issmooth\nenough. In such approximations, the quantum\npressure term in Eq. 10 and 11 can be neglected.\nInterestingly, similartothehomogeneouscase, the\ncondensate density profile and polarization is still\ngiven by the same Eq. 4 and Eq. 5, respectively,\nexcept homogeneous laser pump Wis replaced\nwith local value W(r)in there.\n7Under the Gaussian laser pump profile, we can\nlook for cylindrically symmetric stationary solu-\ntions. The condensate frequency !\u0006is\n!\u0006=~\n\u0006\u0001W\n\rC; (13)\nwhich is determined by the boundary condi-\ntion that the local density wave vector vanishes\nkC(r=rp) = 0at the center of the each pumping\nspot. Here,\n~\n\u0006= \n +(\f1+f1)\u001a\u0006(\f1\u0000f1)\u001aSZ\n2W\n+2 (g1+h1) \u0000 + [G\u0001\u001a\u0006H\u0001\u001aSZ]\n2 [\u00002+ \u0000 (g2+h2)\u001a+B\u0001\u001a2];(14)\nfrom here, we can find frequency difference be-\ntween two component as\n!+\u0000!\u0000=\u0001~\n\u0001W\n\rC; (15)\nhere,\n\u0001~\n = ~\n+\u0000~\n\u0000\n=\u001a(rp)SZ(rp)f(\f1\u0000f1)=W\n\u0000(g1\u0000h1) (g2\u0000h2)=\u0002\n\u00002+ \u0000 (g2+h2)\u001a+A\u001a2\u0003\t\n;\nhere, we have defined condensate polarization\nSZ(rp) =j\u0010+(rp)j2\u0000j\u0010\u0000(rp)j2and coefficient of\ndensity square term B= (g2+h2)2(1 +S2\nZ)=4,\nwhich has maximal value (g2+h2)2=2for the to-\ntal polarization\u00061. Interestingly, we can see that\nenergy gap is polarizationdependence. Inparticu-\nlar, when\f1'f1or large enough laser power, po-\nlarization dependence of frequency difference dis-\nappears.\nLocal density wave vector kC(r)of condensate\nis reaching maximal value with the condensate\ndensity decreased and spin polarized away fromthe pumping center. Polaritons condense at the\nlaser spot position has a large blueshifted energy\ndue to their interactions with uncondensed hot ex-\ncitons, thus within a short time, these interaction\nenergywillleadtothemotionofpolaritoninitially\nlocalized at pumping point. In particular, spon-\ntaneous polarization may happen because polar-\nization may lower the frequency obviously under\nthe laser power is large enough as we can see from\nEq. 14. Therefore, spin domain, spin current and\ntopological defect may be formed under such ap-\npropriate condition.\nIn the following, through extensive numerical\nsimulations of the Eq. 1 coupled to the reser-\nvoir evolution Eq. 2, above analytical results have\nbeen approved, such as, the dynamical formation\nof spin domain, spin current and half vortex for\na wide range of pump parameters obviously avail-\nable within state-of-the-art techniques.\nV. NUMERICAL RESULTS FOR SPONTA-\nNEOUS POLARIZATION.\nEqs. of motion 1 and 2 can be solved numer-\nically with the initial condition nR\u001b(x;y;t )\u00190,\n \u001b(x;y;t )\u00190. The parameters of the pump\nare chosen according to the related experiments\n[32–34, 48] which study the optical spin hall ef-\nfect, tunable spin textures and half solitons. In\nour calculations the following parameters are used\ntypically for state-of-the-art GaAs-based micro-\ncavities: the polariton mass is set to m= 10\u00004\nmewheremeis the free electron mass; the de-\ncay rates are chosen as \rC= 0:152ps\u00001and\u0000 =\n83:0\rC; thus, the scaling units of time, energy and\nlength are 6:58ps,0:1meV, and 1:54\u0016m, respec-\ntively; the interaction strengths are set to ~\f1=\n40\u0016eV\u0016m2,f1=\u00000:1\f1,g1= 2\f1,h1=\u00000:2\f1;\nthe condensation rate are set to ~g2= 0:16meV\n\u0016m2,~h2= 0:016meV\u0016m2, and condensation\nloss rate\u0000~\f2= 0:16meV\u0016m2,~f2= 0:016\nmeV\u0016m2. In our simulation, the dimensionless\nscattering coefficient for each interaction term has\nbeen tuned carefully in order to get the physical\nphenomena we want due to complicated nonlinear\neffects. From an experimental point of view, the\ndimensionless interaction parameters must be ad-\njusted to match pump intensity. The pump inten-\nsity was chosen according to the experimentally\nmeasured blueshift of the polariton condensate,\nand its profile is Gaussian shape as:\nW(r) =w0\n\u0019w2\n1Xn\ni=1e\u0000(x\u0000xi)2\u0000(y\u0000yi)2\nw2\n1;\nhere, for a typical case, w1= 1:0,jxij=jyij= 1:5,\nandw0is tuned accordingly.\nAs expected, Eqs. of motion 1 and 2 tend to\nsettle to a steady state with a spontaneously cir-\ncular polarization under increasing laser power as\nshown in Fig. 1(b). Threshold laser power for\nspontaneously circular polarization is greater than\nthat of starting condensation which can be under-\nstood from our derived Eqs. 5 and 6. The coher-\nent polarized polaritons ballistically fly away from\nthe laser spot due to their interactions converted\ninto kinetic energy of coherent polariton. In par-\nticular, the circular polarization rapidly saturates\nwith increasing the pumping power and may lead\nto an almost full polarization [17–19]. Surpris-ingly, full circular polarization state will change\nimmediately back to the linear polarization with\nfurther increasing the laser power (i.e., the density\nof condensate exceeding a threshold value). Such\nphenomenon can be understood from Eqs. 10 and\n11wherevariousquantumpressuretermswilltake\nimportant roles. Moreover, as shown in the Fig.\n1(c), density profiles of incoherent hot exciton and\npolariton condensate represent linear and circu-\nlarpolarization,respectivelywithinspontaneously\ncircular polarization regime. As we can see that,\nwhile unpolarized hot excitons experience a lim-\nited diffusion, polarized polaritons ballistically fly\naway from the laser spot due to the conversion\nbetween interactions energy and kinetic energy.\nFig. 2 show the density distribution of incoher-\nent hot exciton and coherent polariton condensate\nunder six and eight unpolarized pumping laser\npoints. Interestingly, the neighbouring condensed\npolaritonsarepolarizedwithoppositepolarization\nas can be seen from szdistribution clearly. More-\nover, steady state with magnetic domain wall for-\nmation has been obtained and characterized by\nvanishing total magnetization. Such phenomena\nis fundamentally related with emergent effective\nmagnetic field by the inhomogeneous pump laser\nas we mentioned before. Furthermore, other inter-\nesting magnetic textures may be formed from the\nevolution of Eqs. of motion 1 and 2. In the follow-\ning, we will address the question how to generate\nthe density current, spin current, phase fluctua-\ntion and slip via tuning pumping (or geometrical)\nsource.\n9Figure 3. (Color online) Normalized average density\ncurrentJnxof condensate which is non-resonantly ex-\ncited by 6 points laser with linear polarization. The\ninsets shows the total density profile before and af-\nter shifting position of two middle lasers (see the\nschematic picture) along the x direction, and also\nshowsJnxunder decreasing the pumping power to\n80%. The size of profile is 24x24 and the other param-\neters used in the simulations are the same as those in\nthe Fig. 1.\nVI. DENSITY CURRENT, SPIN CUR-\nRENT, PHASE SLIP.\nPhysically, condensed fluid is a long-range\ncooperative phenomenon characterized by long-\nrange correlation and coherent ordering of the\nmomenta of particle. The various correlation\nfunction may imply net surface currents and or-\nbitalangularmomentumappearinginthissystem.\nTherefore, It is important to study the density\nand spin current, and furthermore, study how to\ngenerate and control them. In the following, we\nwill address these questions by suddenly shifting\npumping laser position by a distance. Interest-ingly, we find that a steady current can be gen-\nerated apparently. In particular, if shifted the\npumping laser is linear or circular polarized, we\nobserve large phase fluctuation where ring-shape\nphase jump shows the behavior of splitting and\njoining together. The above-mentioned behav-\niors may be understood from emergent effective\ngauge field caused by externally pumped incoher-\nent reservoirs.\nA. Density current\nFirst, we numerically simulate time evolution\nof Eqs. of motion 1 and 2 under pumped by six\nlinearly polarized laser and then, suddenly shift-\ning two middle laser’s position. The results are\nshown in the Fig. 3 for the average density cur-\nrent, which are normalized by the total density of\ncondensate. As is shown in the Fig. 3, a large-\namplitude oscillation appears within a short time\nwhen switching on the pumping lasers. With time\nevolution, oscillation decays very quickly and dis-\nappear at 60 unit of time. The appearance of\nsuch oscillation can be understood from the large\noverlap of incoherent hot exciton and coherent po-\nlariton which leads to the large repulsive force in\nthe beginning. Then, with coherent polariton’s\ndiffusion under such repulsive force, condensate\nstay in a steady state with zero averaged current,\nwhich means a balanced configuration in momen-\ntum space of condensate.\nSecond, we want to generate steady current\nwithout decay by breaking above balanced con-\nfiguration. Therefore, we suddenly shift two mid-\n10dle laser’s position at time 1050 (referring to the\nschematic picture in the inset of Fig. 3). Inter-\nestingly, a persistent current with small oscilla-\ntion can be observed clearly and it’s amplitude\nis centred at -0.15. The appearance of such per-\nsistent current can be understood from breaking\nbalanced-momentum configuration due to chang-\ning interaction energy between different part of\ncondensate. Moreover, accompanied fast small-\namplitude oscillation can be understood as sur-\nface oscillation modes which are moved back and\nforth due to confinement by the pumping laser.\nFurthermore, such oscillation can be suppressed\nby lowering the pumping power completely as is\nshownintheinsetofFig. 3, wherepumpingpower\ndrops up to 80 percent of previous case. However,\nwe can not generate persistent spin current by us-\ning above method. Therefore, we will address this\nissue in the following section.\nB. Spin current\nPolariton condensates are excellent candidates\nfor designing novel spin-based devices at room\ntemperature due to their many features, such as\nstrong optical nonlinear response, spin polariza-\ntion properties, and fast spin dynamics. There-\nfore, inthefollowing, wewillshowhowtogenerate\nspin transportation of coherent polariton. In par-\nticular, we observed persistently long-range spin\ntransport without dissipation. We will show our\nresults obtained by numerically simulate time evo-\nlution of Eqs. of motion 1 and 2 in the following.\nFirst, we obtained time evolution of average\nFigure 4. (Color online) Normalized average spin cur-\nrent\nJsx;x\u000b\nof condensate which is non-resonantly ex-\ncited by 6 points laser with linear polarization. The\ninsets shows density profile of the spin current Jsx;x\nat the final stage after shifting position of two middle\nlasers along the x direction, and that for normalized\naveragespincurrentalongtheydirection\nJsx;y\u000b\n. The\nsize of profile is 24x24 and the other parameters used\nin the simulations are the same as those in the Fig. 1.\nspin current\nJsx;x\u000b\nas shown in the Fig. 4, where\npolariton condensate is non-resonantly excited by\n6 pumping lasers with linear polarization. In the\nearly stage, a large-amplitude oscillation appears\nwithin a short time when switching on the pump-\ning lasers. With time evolution, oscillation de-\ncays very quickly and disappear at 60 unit of time.\nAbove phenomena are very similar to those of av-\nerage density current shown in Figures 3. How-\never, it is interesting to point out that such large-\namplitude oscillation has very asymmetric behav-\nior in contrary to symmetric behavior in average\ndensity current. Such asymmetric phenomenon\nmay be understood from the symmetry break-\ning by effective magnetic field stimulated by the\n11Figure 5. (Color online) Total density and each phase\nprofile of condensed polariton excited by 6 points laser\nwith linear polarization. The left column and right\ncolumn correspond to the spatial distributions before\nand after shifting position of two middle lasers along\nthe x direction, respectively. The size of profile is\n24x24 and the other parameters used in the simula-\ntions are the same as those in the Fig. 1.\npumping lasers. Importantly, the remaining ques-\ntionishowtogeneratesteadyspincurrentwithout\ndissipation. Therefore, we try to deal with such\nquestion by manipulating pumping laser.\nInterestingly, persistent spin current is quickly\ndeveloped at time 1050 and it’s amplitude is cen-\ntred at 0.15. Moreover, the fast small-amplitude\noscillation still appear which may be understood\nas stimulating surface oscillation mode by break-\ning symmetry on the spatial distribution of pump-\ning lasers. Next, we compare the spin current\nFigure 6. (Color online) Total density and each phase\nprofile of condensed polariton excited by 6 points laser\nwith circular polarization. The left column and right\ncolumn correspond to the spatial distributions before\nand after shifting position of two middle lasers along\nthe x direction, respectively. The size of profile is\n24x24 and the other parameters used in the simula-\ntions are the same as those in the Fig. 1.\nalong the different directions. Interestingly, aver-\nage spin current\nJsx;y\u000b\nalong the y direction has\ndramatically different behavior as shown in the\ninset of Fig. 4. As we can see that\nJsx;y\u000b\nrep-\nresents very symmetric oscillation centred at zero\nvalue. Suchdifferentbehaviorbetween\nJsx;x\u000b\nand\n\nJsx;y\u000b\nis due to shift lasers’ position along the x\ndirection instead of y direction. Therefore, we can\nconclude that net spin current may be induced by\nbreakingsymmetricdistributionofpumpinglasers\n12along preferred direction. It must be pointed out\nthatlocalspincurrent Sx;xmaybepositiveorneg-\native value as indicated in the insets of Fig. 4.\nSuch nondissipative spin current is induced by the\neffective magnetic field with density- or current-\ndependence function. Importantly, manipulation\nof such effective magnetic field may be utilized to\ngenerate various polarization textures as well as\nspin-polarized vortices. Now, the question is how\nto generate stable topological defects in our stud-\nied system.\nC. Phase Slip\nAs is well known, condensed polariton provides\na very promising platform to generate and control\nspincurrentandvariousspintexturesthroughma-\nnipulating effective gauge fields (like Dresselhaus\nand Rashba fields). In particular, there are many\nkinds of quantum phases in spinor quantum flu-\nids can be accessible experimentally in this plat-\nform. For example, there may generate fascinat-\ning topological defects by manipulating pumping\nlasers [49, 50, 52, 54].\nPhysically, in order to generate topological de-\nfects, large phase fluctuations must be occurred\nby reducing the coherence length and amplitude\nof the order parameter (polariton condensate).\nTherefore, let us first study how to generate large\nphase fluctuations. In order to generate that, we\nsuddenlymovedthepositionofthepumpinglasers\nin the middle site, and then see how the phase\nfluctuations are formed dynamically.\nFigures 5 shows the total density and eachphase profile of condensed polariton before and\nafter moving the lasers in the middle site, where\neach component of condensed polariton is illumi-\nnated with the same laser power. Interestingly,\nwhile condensed polaritons are concentrated on\nthe right part, large phase disturbance has been\ngenerated for each component. In particular, in\nlow density region, there are large phase fluctua-\ntions where ring-shape phase jump shows the be-\nhavior of splitting and joining together. Physi-\ncally, due to energy advantages, topological de-\nfects are initially formed in low-density regions,\nthen, due to the dissipation of energy, these topo-\nlogicaldefectsgraduallymovedtothehigh-density\narea and eventually reached a stable state. There-\nfore, we can expect that it is very promising to\nproduce stable topological defects (such as quan-\ntum vortices) in such system.\nFurthermore, we want to control which com-\nponent of the condensed polaritons will generate\nlarge phase fluctuations. Therefore, each compo-\nnent of condensed polariton is illuminated with\nthe different laser power and then see how the\nphase fluctuations are formed dynamically. Fig-\nures 6 shows the total density and each phase pro-\nfile of condensed polariton before and after mov-\ning the lasers in the middle site. Interestingly, the\nphases of the two components have very different\nshapedistributions. Here, largephasedisturbance\nhas been generated for component one which was\nilluminated with the laser power, however, there is\nnot much change in the phase of the second part.\nMoreover, in component one, large phase fluctua-\ntions are closer to high-density areas where ring-\n13shape phase jump shows the behavior of splitting\nand joining together.\nIt must be admitted that stable topological de-\nfects are not created as they require reconfiguring\na large number of spins and density at a large\nenergy cost. Generally, what kinds of stable topo-\nlogical defects are developed depending on the dy-\nnamics of gauge potential together with vector\nfield, such as Maxwell-Chern-Simons-vector Higgs\nmodel for the the superconductivity of Sr 2RuO 4\n[61]. In our studied non-equilibrium exciton-\npolaritons liquid, spin- and density-dependent ef-\nfective gauge fields play important roles on the\nphase fluctuations and make effective gauge fields\nmore controllable comparing with conventional\nsolid state system and ultracold atoms. Finally,\nwe remark that the physics described in our study\nmay be generally applicable to the recovery of\ncomplex order parameters in other systems. Pho-\ntoinducedphasefluctuationsmaybecrucialtoun-\nderstanding the mechanism of photoinduced su-\nperconductivity in the striped cuprates. These\nphenomena can be conveniently probed by real-\nspace spectroscopy, and phase imaging [46].VII. CONCLUSIONS.\nIn conclusion, we have demonstrated a prac-\ntical way to control spin polarization, gener-\nate density and spin current, and induce large\nphase fluctuations in an exciton-polariton con-\ndensate. For the polariton lifetime, The above-\nmentioned behaviors can be readily excited in\nphotoluminescence experiments and detectable by\nthe time-resolved micro-photoluminescence spec-\ntroscopy [49, 89] or spin noise spectroscopy [90,\n91]. Our results are of particular significance for\ncreating these excitations in experiments and for\nexploring novel phenomena associated with them.\nThis noticeably spin amplification and spin trans-\nport could offer a promising way to optimize spin\nsignals in future devices with using polariton con-\ndensates.\nAcknowledgments We are grateful to N.\nBerloff for discussions. The financial support from\nthe early development program of NanChang Uni-\nversity and Skoltech-MIT Next Generation Pro-\ngram is gratefully acknowledged.\nCompliance with ethical standards\nConflict statement We declare we have no\nconflict of interests.\n[1] C. Weisbuch, M. Nishioka, A. 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Greilich1\n1Experimentelle Physik 2, Technische Universit at Dortmund, 44221 Dortmund, Germany\n2Spin Optics Laboratory, Saint Petersburg State University, 198504 St. Petersburg, Russia\n3P. N. Lebedev Physical Institute, Russian Academy of Science, 119991 Moscow, Russia\n4Resource Center Nanophotonics, Saint Petersburg State University, 199034 St. Petersburg, Russia\n(Dated: June 28, 2022)\nWe discuss the implications of a small indium content (3%) in a GaAs epilayer on the electron-\nand nuclear-spin relaxation due to enhanced quadrupolar e\u000bects induced by the strain. Using the\nweakly perturbative spin-noise spectroscopy, we study the electron-spin relaxation dynamics without\nexplicit excitation. The observed temperature dependence indicates the presence of localized states,\nwhich have an increased interaction with the surrounding nuclear spins. Time-resolved spin-noise\nspectroscopy is then applied to study the relaxation dynamics of the optically pumped nuclear-spin\nsystem. It shows a multi-exponential decay with time components, ranging from several seconds to\nhundreds of seconds. Further, we provide a measurement of the local magnetic \feld acting between\nthe nuclear spins and discover a strong contribution of quadrupole e\u000bects. Finally, we apply the\nnuclear spin di\u000busion model, that allows us to estimate the concentration of the localized carrier\nstates and to determine the nuclear spin di\u000busion constant characteristic for this system.\nI. INTRODUCTION\nSilicon dopants (Si) are known to provide shallow\ndonors in gallium arsenide (GaAs) [1]. Depending on\nthe crystal quality and the concentration of such im-\npurities in n-doped GaAs (GaAs:Si), the electrons can\nshow extremely long times of spin relaxation [2{9] lim-\nited by the Dyakonov-Perel mechanism [10, 11] or spin-\nspin exchange interactions [12], and even experience\nweak localization [13] at high and moderate concentra-\ntions of dopants. At relatively small doping density,\nnD.1015cm\u00003, well below the metal-insulator tran-\nsition (MIT) [14], at which the motion-assisted and spin-\nexchange relaxation mechanisms are suppressed, the hy-\nper\fne interaction of a localized-electron spin with the\nsurrounding spins of lattice nuclei provides the domi-\nnant mechanism of spin relaxation [15]. In addition,\nsince each electron is localized on a shallow impurity,\nthe regime of short correlation time is realized even\nat liquid-helium temperature [16]. On the contrary, in\nsingly-charged quantum dots (QDs), where each elec-\ntron is strongly localized due to the spatial con\fne-\nment and the reduced bandgap of Indium-Gallium Ar-\nsenide (InGaAs), the correlation time is long [17]. In\nthis case, the precession of an electron spin around the\nrandomly oriented \feld of nuclear spin \ructuations re-\nsults in a fast dephasing of the electron spin orienta-\ntion, during a nanosecond time scale, followed by a to-\ntal loss of the spin coherence during a microsecond time\nscale [18, 19].\nResearch on spin systems in the context of quan-\ntum information processing and data storage is guided\nby the need of a technical implementation of a largely\nisolated nuclear-spin system with well controlled nu-\nclear state and reduced \ructuations. Such a con-trol can be achieved when the phase transition to a\nspin-ordered state is realized [20{23]. However, nu-\nclear spin ordering has not yet been observed since\nit requires extremely small nuclear-spin temperatures\n\u00180.1\u0016K [24]. Recent achievements in nuclear-spin\ncooling in GaAs:Si have shown values at least an or-\nder of magnitude larger than the theoretical estima-\ntions demand [25{28]. On the other hand, the nu-\nclear spin-relaxation times observed in InGaAs/GaAs\nQDs were found to be longer than in lattice-matched\nAlGaAs/GaAs QDs due to enhanced quadrupole inter-\nactions, suppressing the nuclear spin \ructuations [29{\n31]. Generally, the nuclear spin system of InGaAs\nis more complex than that of GaAs due to the ad-\nditional indium content. Strong quadrupole e\u000bects\nare observed even in highly annealed QD ensem-\nbles [32, 33] for which the concentration of indium is\nreduced.\nIn this paper, we study both the electron- and nuclear-\nspin dynamics in an epilayer of InGaAs. The dop-\ning by Si provides centers of electron localization sim-\nilar to GaAs:Si. In addition, due to the admixture\nof the indium, the bandgap of epilayer material is\nshifted to the transparency region of the GaAs sub-\nstrate. For studying the electron-nuclear spin dynam-\nics, this allows us to implement the spin-noise spec-\ntroscopy [34] which is an especially powerful method\nfor characterizing spin systems (almost) nonperturba-\ntively [35, 36]. As we will show below, the electron-\nspin dynamics reveals the behavior of well-localized\ncenters, similar to ensembles of singly-charged QDs,\neven at moderate Si doping. At the same time, the\ndynamics of the nuclear spin polarization is similar\nto GaAs:Si but characterized by complex relaxation\nprocesses assisted by the strong quadrupole interac-\ntion.arXiv:2206.13177v1  [cond-mat.mes-hall]  27 Jun 20222\nII. SAMPLE AND SETUP\nWe use ad= 10\u0016m thick InGaAs epilayer with 3%\nof indium content grown by molecular beam epitaxy on\na GaAs substrate. As the comparatively large indium\natoms (atomic number: Z= 49, nuclear spin: I= 9=2)\nreplace smaller gallium atoms ( Z= 31,I= 3=2),\nthe band structure is gently tuned. The sample was\ndoped homogeneously by Si-atoms with a density of\nnD= 3:1\u00021016cm\u00003, measured by a 4-point method\nat temperature T= 77 K in a magnetic \feld B= 0:5 T.\nThis concentration was found to maximize the electron-\nspin relaxation time compared to similar samples with\nsmaller doping concentrations [37].\nAs the carrier concentration plays an important role\nfor further considerations, we performed additional trans-\nport measurements in the Van der Pauw geometry. Si-\nmultaneous measurements of the Hall e\u000bect and resis-\ntivity were performed using two lock-ins with di\u000berent\nfrequencies (73 and 162 Hz respectively) for a transport\ncurrentJ= 1\u0016A, as shown in the insert of Fig. 1(a).\nThe lock-in ampli\fers were checked to avoid interference\ne\u000bects and the transport current was proven to not over-\nheat the system at the lowest temperature. We could\nvary the temperature from T= 2 to 300 K and apply a\nmagnetic \feld up to 8 T. The magnetic \feld perpendic-\nular to the wafer plane was swept from positive to neg-\native values and the longitudinal resistance Rxxand the\nHall resistance Rxydata were symmetrized and antisym-\nmetrized, correspondingly, to compensate for imperfect\ncontact alignment leading to the mutual admixture of\nRxxandRxy.\nThe temperature dependence of the sheet resistivity,\n\u001axx, measured for B= 0 T and the carrier concentration\nextracted from the Hall e\u000bect via Vxy=JB=(qenHalld)\nare shown in Fig. 1(a). As one can see from the \fgure,\nthe resistivity drops and saturates with increasing sam-\nple temperature while the concentration of conducting\nelectrons exhibits a minimum at temperature T'50 K.\nTo further investigate the carrier mobility, we have per-\nformed measurements of the magnetoresistance, MR, and\nthe Hall resistivity. The extracted MR in low magnetic\n\felds was measured at several temperatures, as shown\nin Fig. 1(b). The temperature and magnetic-\feld depen-\ndencies of the resistivity are typical for GaAs-based semi-\nconductors with a doping level above the MIT [38{40],\nin particular, with respect to the low- Tresistivity up-\nturn, the weak localization-caused negative MR at low T\nand the positive MR at higher temperatures. The low-\nTvalue of the resistance ( \u001840 \n) and the Hall slope\n(16 \n/T) imply that only a d= 10\u0016m thick epilayer\ncontributes to the conductivity, at least at low tempera-\ntures. The Hall carrier density saturates at the level of\nn0= 3:9\u00021016cm\u00003, slightly larger than the one mea-\nsured previously for a larger T(3:1\u00021016cm\u00003) [37].\nFigure 2(a) schematically shows the setup for the stud-\nies of Faraday rotation (FR), spin-noise spectroscopy\n(SNS), and time-resolved SNS with optical pumping.\n100 kΩ100 kΩ0.1V 73Hz – xx\n0.1V 162Hz – xyVxy\nVxx\n05010015020025030000.050.1\nT (K)ρ\nxx (Ω·cm)\n33.544.5\nn\nHall (1016 cm−3)\n−1−0.5 00.51−1001020\nB (T)MR (%)(a)\n(b)\nT = 77 K\nT = 20 K\nT = 8 K\nT = 2 KFIG. 1. (a) Temperature dependencies of the low-\feld resis-\ntivity (solid line) and the Hall carrier density (points) with\nthe dashed line as guide to the eye. Inset shows the schematic\nof the transport experiment. (b) Magnetoresistance curves\nmeasured at di\u000berent temperatures.\nThe average electron-spin polarization along the z-\ndirection (optical axis) is detected through the FR mea-\nsured by a linearly polarized continuous wave (CW) laser,\nreferred to as the probe laser. We \fx the probe laser\nwavelength at \u0015pr= 852:63 nm, which is 4.63 nm above\nthe maximum of the photoluminescence signal of the In-\nGaAs sample ( \u0015PL,max (6 K) = 848 nm), and hence, prop-\nagates with reduced absorption through the sample at\nthe local maximum of the Faraday rotation (see below),\nas illustrated in Fig. 2(b). Furthermore, the sample was\npolished from the back side to allow for transmission mea-\nsurements and covered by an antire\rection coating from\nboth sides to reduce the etalon e\u000bects of the laser. Still,\nsome residual e\u000bect was observed in the Faraday rotation\nmeasurements, see the oscillating red line in Fig. 2(b).\nBehind the sample, the induced FR of the probe beam\nis detected using a half-wave plate followed by a Wollas-\nton prism and a balanced photoreceiver. The di\u000berential\nsignal is digitized and fast Fourier transformed by a \feld-\nprogrammable gate array, calculating the power spectral\ndensity (PSD) of the measured signal [41]. The band-\nwidth of the diodes determines the covered spectral range\nand is limited to 100 MHz in our case (Femto HCA-S).\nTo extract the carrier spin-noise from the background\nelectronic and shot noise, the measured power spectrum\nSmeas(f) is subtracted and divided by a reference signal\nSref(f), which is recorded at a di\u000berent external mag-\nnetic \feld ( Bx= 14 mT). The resulting spin noise is\nexpressed in shot-noise units (SNU) as PSD(SNU) =\nSmeas(f)=Sref(f)\u00001.3\nProbe  laser\nλ = 852.63 nm\n84084585085586086587001\nWavelength (nm)PL, FR, AbsorptionProbe laser\nT = 6 K\n(b)(a)\nPump laserB\nxB\nz\nλ/2WP NPBS\nSampleFFT 2\nλ\npu = 785 nmSh\nAbsorption\nFRPL\nFIG. 2. (a) Schematic of the setup used to detect spin noise\nspectra by measuring FR with the probe beam and to polarize\nthe spin system with the pump beam. Sh is the shutter used\nto switch on and o\u000b the pump, and NPBS is a non-polarizing\nbeam splitter, used to combine the linearly polarized probe\nand circularly polarized pump. (b) Normalized photolumi-\nnescence (PL) (blue line), Faraday rotation (red line), and\nabsorption (black line) spectra of the InGaAs epilayer. Ver-\ntical dashed line is the position of the probe laser.\nAn additional circularly polarized CW pump laser at\n\u0015pu= 785 nm is used to create a non-zero electron-spin\npolarization. We apply it in order to measure the lu-\nminescence and conventional Faraday rotation spectra,\nshown in Fig. 2(b) [42]. Due to the high energy excitation\n(above the band gap), the pump-laser beam is absorbed\nby the sample and does not pass to the detection chan-\nnel. This allows us to use a collinear scheme, in which\npump and probe take the same optical path.\nIf not stated di\u000berently, the powers of the laser beams\narePpr= 1 mW for the probe and Ppu= 0:3 mW for the\npump. Both beams are tightly focused onto the sam-\nple surface with spot diameters of Dpr\u001913\u0016m and\nDpu\u001940\u0016m. The pump beam can be blocked by a\nremotely controlled shutter with a switching time resolu-\ntion of\u001810 ms.\nThe InGaAs epilayer is mounted in the center of two\npairs of electromagnetic coils generating corresponding\nmagnetic \felds, BxandBz, see Fig. 2(a). The sample\nis placed into a helium \row cryostat at a temperature of\nT= 6 K.\nIII. ELECTRON-SPIN DYNAMICS\nTo characterize the electron spin relaxation dynam-\nics, we \frst perform FR measurements. To avoid ef-\nfects of nuclear spin polarization, the measurements are\ndone using polarization modulation of the pump light by\nthe electro-optic modulator switched between \u001b+and\u001b\u0000\nlight polarization at frequency fEOM = 200 kHz. The FR\n−400−200020040088.59\nB\nz (mT)θ\nF (mrad)\n−1 −0.5 0 0.5 10246810\nB\nx (mT)θ\nF (mrad)(a)\n(b)Bc = 276 ± 15 mT\nBHWHM     = 0.07 mT(narrow)\nPpu = 0.5 mW\nPpr = 1 mWfEOM = 200 kHz\n10−410−30.010.11100.010.11\nPpu (mW)BHWHM (mT)wide\nnarrow(c)\nτs(narrow) = 610 nsτs(wide) = 120 nsBHWHM  = 0.3 mT(wide)−1−0.500.51B1/2 = 54 µTFIG. 3. (a) Spin polarization recovery in longitudinal mag-\nnetic \feld (points) and its \ft by \u0012F(Bz) =\u00121=[1+(\u001c=\u001cc)(1+\n(Bz=Bc)2)\u00001] (red line). Insert is a close-up for the struc-\nture around zero \feld with a half-width at half-minimum\nB1=2= 54\u0016T. Red line is a Lorentzian \ft. (b) The Hanle de-\npolarization curve (circles) and its \ft by a bi-Lorentzian func-\ntion (red line). Shaded curves show the decomposition with\ntwo characteristic spin relaxation times. (c) Pump power de-\npendence for both extracted components. Gray colored data\nare not following a linear dependence due to saturation e\u000bects\nand are not considered. Black lines are linear \fts.\nof the probe beam is detected by a lock-in technique at\nthe frequency of modulation. Application of the longitu-\ndinalBz\feld recovers the electron spin polarization while\nthe transverse \feld Bxerases it, as shown in Figs. 3(a)\nand 3(b). Note that the widths of the polarization\nrecovery curve (PRC) [Fig. 3(a)] and the Hanle curve\n[Fig. 3(b)] di\u000ber by more than three orders of magni-\ntude. Interestingly, such a behavior is common for semi-\ninsulating samples of GaAs:Si [16]. From the dataset,\nwe extract the half-width at half-minimum of the PRC,\nBc= 276\u000615 mT, corresponding to a correlation time\n\u001cc= 75\u00065 ps. We additionally point out, that the insert\nin Fig. 3(a) demonstrates a narrow dip structure, which\nis a result of competition between nuclear spin cooling\nand nuclear spin warm-up in the oscillating Knight \feld\nof the electrons [16].\nFurthermore, the Hanle curve is best \ftted using two\nLorentzians. Their half-width at half-maximum at cor-\nresponding laser powers is: B(narrow)\nHWHM = 0:07 mT and\nB(wide)\nHWHM = 0:3 mT, see Fig. 3(b). The presence of two4\ncomponents indicates the presence of two sets of elec-\ntrons, contributing to this relaxation with the lifetimes\nofT(narrow)\ns = 286 ns and T(wide)\ns = 67 ns. Here we used:\nTs=~\nge\u0016BBHWHM; (1)\nwithjgej= 0:568 being the electron gfactor (its value\nis taken from the spin noise measurements in magnetic\n\feld at the same sample position, see below), ~is the re-\nduced Planck constant, and \u0016Bthe Bohr magneton. In\ngeneral, the spin lifetime Tscan be de\fned as: 1 =Ts=\n1=\u001cs+G=n 0[36, 43]. Here, Gis the generation rate of car-\nriers that is dependent on the power of the optical excita-\ntion,n0is the carrier concentration, and \u001csthe intrinsic\nlongitudinal spin-relaxation time. Therefore, by extrap-\nolation to zero pump power, it is possible to extract the\n\u001cs. Figure 3(c) demonstrates such an experiment with\n\u001c(narrow)\ns = 610\u000610 ns for the narrow component. The\nbroad component disappears below Ppu= 0:02 mW and\nextrapolates to the value of \u001c(wide)\ns = 120\u000610 ns.\nNext, to avoid unnecessary spin polarization of car-\nriers by pumping, we use SNS. Here, the probe beam\ndetects \ructuations of the electron-spin polarization in\nits ground state without optical pumping of the spin\nsystem [34, 44]. In the usual case, the spontaneously\nappearing spin excitations decay exponentially in time\nand are, therefore, observed as a single Lorentzian peak\nin the spin-noise power-spectrum [45, 46], see Fig. 4(a),\nmeasured at Bx= 3 mT and Ppr= 1 mW. Following\nthe dependence of the peak position ( fL) versus the ex-\nternal transverse magnetic \feld Bx, one can determine\nthe Larmor g-factorge. The inset in Fig. 4(a) shows\nthis dependence, characterized by the Larmor frequency\n!L= 2\u0019fL. The red solid line is a \ft by the equation:\nfL=ge\u0016BBx=h; (2)\nwith the Planck constant h.\nTheB-linear \ft yields jgej= 0:568\u00060:001. The\nelectrongfactor in InGaAs is expected to be slightly\nlower than in bulk GaAs with ge;GaAs\u0019\u00000:44 [36], as\nthe additional indium content contributes with ge;InAs=\n\u000015 [47, 48] (one measures here the absolute value, but\na negative sign is expected). Further investigations of\nthe InGaAs sample indicate a spatial inhomogeneity of\nthe electron gfactor across the sample, varying between\n\u00000:53 and\u00000:6, which can be related to a gradient of\nindium content in the epilayer.\nTo compare the SNS with the preceding Hanle mea-\nsurements, we measure the probe power dependence of\nthe SNS at Bx= 0 mT. Figure 4(b) demonstrates the\nobserved peak centered at zero frequency which is best\n\ftted with two Lorentzian, the sum of which is shown by\nthe red curve. The HWHM of each peak, \u0000 e, is inversely\nproportional to the electron-spin lifetime Tsaccording to:\nTs=1\n2\u0019\u0000e: (3)\n0510152025303500.050.10.15\nFrequency f (MHz)PSD (shot noise units)024681012050100\nBx (mT)fL (MHz)IgeI = 0.568\n(a)\n(b)0510152025303500.020.040.06\nFrequency f (MHz)PSD (shot noise units)Bx = 3 mT\nPpr = 1 mW\n0.1 10.1110\nPpr (mW)Γe (MHz)Ppr = 1 mWBx = 0 mT\nτs(narrow) = 500 nsτs(wide) = 80 nsFIG. 4. (a) Example of a spin-noise spectrum measured\nforBx= 3 mT and Ppr= 1 mW. The power spectral density\n(PSD) is expressed in shot-noise units. Red solid line is a \ft\nby a single Lorentzian function. Inset shows the dependence\nof the spin noise peak on Bx. Red line is a \ft by Eq. (2)\nwithjgej= 0:568\u00060:001. (b) Spin-noise spectra measured\natPpr= 1 mW and Bx= 0 mT. The data are \ftted by a bi-\nLorentzian function, red line. Shaded curves show the two \ft\ncomponents, narrow (blue) and wide (green), correspondingly.\nInset shows the half width at half maximum (\u0000 e) for each\ncomponent versus probe power. Black curves show linear \fts.\nIn the optical SNS, the spin lifetime (or the peak width)\nis also a\u000bected by the non-zero probe power, in a sim-\nilar way as in the Hanle e\u000bect. Experimentally, one\ncan access the intrinsic spin-relaxation time by a power-\ndependent measurement with extrapolation to zero probe\npower, as presented in the inset of the Fig. 4(b). The\nintrinsic widths correspond to \u001c(wide)\ns = 80\u00067 ns and\n\u001c(narrow)\ns = 500\u0006100 ns. These values compare well with\nthe ones measured using the Hanle e\u000bect. The bigger\nerror values are related to the limited sensitivity of the\nspin noise setup at lower probe power. The power range\ncan be potentially extended to much smaller values using\nthe homodyne detection demonstrated in Ref. [49].\nThe electron-spin lifetime of GaAs is well studied for\nvarious doping densities [4, 8, 9]. It depends strongly on\nthe doping concentration and has a maximum right below\nthe Mott-insulator transition (MIT) with \u001cs\u0019800 ns for\nnD= 6:6\u00021015cm\u00003[9]. At doping densities around\nnD= 4:4\u00021016cm\u00003, close to the doping of the studied\nInGaAs epilayer, the relaxation time is found to be one\norder of magnitude shorter. The spin relaxation time\nin InGaAs could be further enhanced by optimizing the5\n0510152025300246810\nTemperature (K)Γ\ne (MHz)05101520253000.10.20.30.4\nTemperature (K)Integral PSD\nBx = 4 mT\nPpr = 1 mW\nFIG. 5. Temperature dependence of the spin-noise peak\nwidth \u0000 emeasured at Bx= 4 mT, using Ppr= 1 mW. Red line\nis a \ft by the Arrhenius equation, allowing us to determine\nthe activation energy Ea= 4:7\u00060:2 meV. Inset demonstrates\nthe spectrally integrated noise power versus temperature.\ndoping concentration.\nIn the absence of externally applied magnetic \felds\n(Bx= 0 mT), the spin-noise signal consists of a peak\ncentered at frequency 0 MHz, which describes the spin\nrelaxation along the z-axis, see Fig. 4(b). This observa-\ntion suggests that the spin noise is produced by electrons,\nwhich are either weakly a\u000bected by the surrounding nu-\nclear spins due to motional narrowing, or the nuclear spin\n\ructuations cannot be considered as frozen and have a\nshort nuclear spin correlation time ( \u001cc) [18, 46, 50, 51].\nSuch a nuclear spin dynamics could be driven by the\nKnight \feld of the electrons, or by the interaction of the\nnuclei quadrupole moments with strain and random elec-\ntric \felds in the structure [52].\nIn the case of a strong coupling to the nuclei or of long\n\u001cc, one would expect to observe an additional peak at\nnon-zero frequency, which would be related to the spin\nprecession in the e\u000bective magnetic \feld produced by the\nrandom \\frozen\" nuclear spin \ructuations with compo-\nnents orthogonal to the z-axis.\nA corresponding observation was reported in Ref. [53],\nwhich discusses spin noise studies of a 10 \u0016m GaAs:Si\nepilayer with a low donor concentration of nD\u00191\u0002\n1014cm\u00003. This represents a situation with a long cor-\nrelation time, leading to a two-peak structure at zero\nexternal magnetic \feld.\nOn the opposite side, at higher donor concentrations\n(nD\u0019(1\u00007)\u00021016cm\u00003), the spin noise is primarily\nproduced by the free electrons \ructuating at the edge of\nthe Fermi sea and no two-peak structure is observed at\nBx= 0 mT, see Ref. [36]. This is additionally supported\nby measurements of the temperature dependence demon-\nstrating a linear increase of the integral spin-noise power\nwith temperature.\nOur observations demonstrate that the spin noise peakmeasured at magnetic \felds above 1 mT can be \ftted well\nwith a single Lorentzian, see Fig. 4(a). This observation\nstill needs further investigations, but will be used here\nto simplify the measurement evaluation. To gain more\ninsight into the degree of localization of the electrons\nwe conduct a temperature-dependent measurement of the\nspin-noise spectra for our sample. Figure 5 demonstrates\nthe variation of the \u0000 eof the peak measured at Bx=\n4 mT. The peak-width is constant up to the temperature\nof\u001810 K and continuously increases above. The inset\nto the \fgure additionally demonstrates that the integral\nnoise power remains about constant up to T= 20 K and\ndrops fast with further increase of the temperature. To\ndescribe it qualitatively, we use the Arrhenius equation:\n\u0000(T) = \u0000g+ \u0000excexp\u0012\n\u0000Ea\nkBT\u0013\n(4)\nwith the activation energy, Ea, the relaxation rate\n2\u0019\u0000g= 15:9\u00060:1\u0016s\u00001of the ground state, and the\nrelaxation rate 2 \u0019\u0000exc= 207\u000613\u0016s\u00001that character-\nizes the strength of carrier-phonon interaction [54, 55].\nThe measurement implies that the spin noise signal is\nproduced by residual electrons that are localized at 6 K,\nas the detected activation energy Ea= 4:7\u00060:2 meV\ncorresponds to a much higher temperature of about\nEa=kB\u001955 K. This activation energy agrees with\ntypical values for GaAs:Si, where electrons are localized\nat donor centers [56].\nWe note here that the localized states, most prob-\nably, originate from pairs of closely situated donors\nscreened by the degenerate electron gas [57]. The\nbound-carrier density is, at least, an order of mag-\nnitude smaller than n0. An estimation yields [57]\nNb= (4=3)\u0019`3n2\n0exp[\u0000(4=3)\u0019`3n0] where`=\n(1=2)[\u0019a3\nB=(3n0)]1=6is the screening length and aB=\n~=p2me\u000bEa= 11 nm is the Bohr radius calculated us-\ning the e\u000bective mass me\u000b= 0:067m0(m0being the\nfree electron mass in vacuum) and the activation energy\nEa= 4:7 meV, which is close to the standard Bohr ra-\ndius for GaAs:Si [58]. This gives Nb'4:2\u00021015cm\u00003\nforn0= 3:9\u00021016cm\u00003, measured at low T.\nIV. NUCLEAR-SPIN DYNAMICS\nDecay of nuclear polarization\nIn the previous section, we have discussed the e\u000bect of\nthe unpolarized nuclear-spin bath on the localized elec-\ntron spins. This can also be seen oppositely, the electron\nLarmor frequency can be used as a sensor for the e\u000bec-\ntive magnetic \felds in the localization area of the elec-\ntron. Here, we use this sensor to study the relaxation\ndynamics of the polarized nuclear spins back to thermal\nequilibrium by the time-resolved version of SNS, as pro-\nposed by Ref. [59]. Again, the electron Larmor frequency\nis tested using SNS with a time-resolution of \u00181 s. This6\nis chosen as a compromise between the required accumu-\nlation time for reliable peak detection and the shortest\ntimescale of the observed nuclear-spin polarization decay.\nWe use the same technique as presented in Ref. [60].\nThe measurement starts with dynamic nuclear-spin po-\nlarization by optically polarized electron spins, produced\nby the circularly polarized pump. Depending on the rel-\native direction of the applied longitudinal magnetic \feld\nand the helicity of the circular polarization of the pump,\none can determine the relative orientation of the nuclear\nspins and the electron spin polarization, or the nuclear\nspin temperature \u0002 N[58]. In our experiments we used\nBz= 10 mT,Bx= 0 mT. After the pumping period,\nthe pump beam was blocked by a shutter, Bzwas set to\nzero, andBx= 4 mT was applied to detect the Larmor\nfrequency with the probe laser. If the switching of the\nmagnetic \feld happens adiabatically, the created nuclear\npolarization follows the direction of the external \feld [61].\nAt the same moment, the detection period was started.\nThe Larmor frequency determined from the peak po-\nsition in the noise spectra is proportional not only to the\ntransverse magnetic \feld, but also to the nuclear-spin po-\nlarizationpNcreated by the optical pumping. It changes\nEq. (2) to:\nfL(t) =ge\u0016B\u0000\nBx+bmaxpN(t)\u0001\n=h: (5)\nThe nuclear-spin polarization induces the Overhauser\n\feldBN(t) =bmaxpN(t) with a maximum value bmax.\nFor InGaAs with 3% of indium we estimate the max-\nimal Overhauser \feld of bmax=P\njIjAj\u001fjnj=ge\u0016B=\n4:125 T, with Ijbeing the nuclear spin of the correspond-\ning isotope, Ajthe hyper\fne constant, \u001fjits abundance,\nandnjthe respective fraction of the nuclei in the mate-\nrial composition [18]. The electron g-factor isjgej= 0:6,\nsee below.\nFigures 6(a) and 6(b) show colormaps with the vari-\nation of the noise spectra versus observation time af-\nter pumping for \fve minutes with a pump power of\nPpu= 0:3 mW. In combination with a right circularly\npolarized pump, the induced Overhauser \feld points ei-\nther in the direction opposite to the magnetic \feld Bx,\nsee Fig. 6(a), or in the same direction, Fig. 6(b). The\npresented spin-noise spectra are taken every second with\na probe power of Ppr= 1 mW. The time dependence\nunveils the Overhauser \feld decay. When the decay is\ncompleted, the Larmor frequency remains constant at\nfL= 33:69 MHz corresponding to the externally applied\n\feldBx= 4 mT [62]. This implies a gfactorge=\u00000:6\nwhich indicates a di\u000berent sample position compared to\nprevious measurements but is well within the range of\ngfactors across the epilayer. Furthermore, the spectra\nat short times demonstrate a broadening of the noise\npeak due to the spatial inhomogeneity of the Overhauser\n\feld distribution. In the discussion below we concentrate\non the case with negative nuclear spin temperature, see\nFig. 6(b).\nThe decay of nuclear polarization is related to the in-\nteraction with the electron-spin system and to the dipole-\n0 100 200 300050100\n00.020.040.060.080.1\nTime (s)Frequency (MHz)\n0 100 200 300050100\nTime (s)Frequency (MHz)\n1 10 100 10000.010.1110100\nTime (s)Frequency (MHz) - 33.69 MHzdata\nτ1 = 7.3 s\nτ2 = 44 s\nτ3 = 210 s\nfit sum\nPSD (SNU)(a)\n(b)\n(c)33.69 MHzΘN > 0\nΘN < 0FIG. 6. Colormaps of time-resolved SNS after optical pump-\ning for \fve minutes. They display the decay of the nuclear\nspin polarization (Overhauser \feld) for two di\u000berent direc-\ntions of the longitudinal \feld Bzwhile pumping, leading ei-\nther to a positive nuclear spin temperature \u0002 Nshown in (a)\nor to a negative \u0002 Nin (b). (c) The decay of the spin-noise\npeak position in (b) can be described by a multi-exponential\nfunction with three characteristic decay times \u001c1;\u001c2, and\u001c3.\nThe frequency 33.69 MHz shown by the white horizontal line,\nmarks the peak position without nuclear polarization.\ndipole interaction between the nuclear spins [58]. Ad-\nditionally, increased quadrupolar e\u000bects in the studied\nsample are expected to in\ruence the relaxation dynam-\nics, see the next chapter [26]. Depending on the domi-\nnating interactions, the decay of the Overhauser \feld can\nbe described in a simpli\fed way by using a sum of expo-\nnential functions with characteristic decay times \u001ciand\nthe corresponding amplitudes ai. In the studied InGaAs\nepilayer, we explicitly detected three di\u000berent relaxation\ntimes, referred to as \u001c1,\u001c2, and\u001c3, leading to:\npN(t) =a1e\u0000t=\u001c1+a2e\u0000t=\u001c2+a3e\u0000t=\u001c3: (6)\nThe need for three decay times is illustrated in Fig. 6(c).7\n0.010.1110100101102\n P\npu (mW)Decay time (s)\n0.010.111000.30.60.9\nP\npu (mW)Probability a\n(a)\n(c)(b)\n(d)τ3\nτ2\nτ1a1~\na2~\na3~\n101102103100101102\n Pumping time  (s)Decay time (s)τ1  τ2  τ3  \n10110210300.30.60.9\n Pumping time  (s)Probability a~ ~a1~\na2~\na3~\nFIG. 7. (a) Overhauser-\feld decay time versus pumping\npower. Varying pump powers between Ppu= 5\u0016W and\nPpu= 5 mW lead to di\u000berent decay time constants of the\nOverhauser \feld. Pumping time is \fxed at 10 min. (b)\nVariation of the decay probability components with pumping\npower. (c) Dependence of the decay times on the pumping\ntime for a \fxed pump power Ppu= 0:3 mW. (d) Variation\nof the decay probabilities with pumping time. Solid lines are\nexponential \fts to the data.\nIn this measurement, the shortest detected time is \u001c1=\n7:3\u00060:1 s, the middle decay time is \u001c2= 44\u00061 s, and\nthe longest one is \u001c3= 210\u00062 s.\nTo get more insight into the relaxation dynamics we\nconducted a series of measurements for varying pump\npower and pump time. For the pump power depen-\ndence,Ppuwas varied from 5 \u0016W to 5 mW. During 10 min\nof pumping with Ppu, the longitudinal magnetic \feld\nBz= 10 mT is applied. The subsequent time-resolved\nspin noise spectroscopy measures the polarization decay\nwith the probe beam applied at the transverse magnetic\n\feldBx= 4 mT. The development of all three decay\ntimes in Fig. 7(a) indicates a rise of times with increasing\npump beam power until it exceeds the power of 0.1 mW.\nAbove this power, the nuclear-spin polarization time sat-\nurates. Below Ppu= 0:1 mW, the decay times decrease\nexponentially as shown by the solid line \fts in Fig. 7(a).\nAbove the critical pump power, between 0.1 mW and\n5 mW, the decay times remain relatively stable with av-\neraged values of\n\u001c1= 7\u00061s; (7a)\n\u001c2= 38\u00066s; (7b)\n\u001c3= 210\u000610s: (7c)\nFigure 7(b) shows the same exponential behavior and\nthe same critical pump power for the decay probabilities.\nThe normalized amplitude ~ ai=ai=(a1+a2+a3) refers to\nthe probability that the i-th of the three decays occurs.\nIn principle, a reduction of the pumping time is ex-\npected to a\u000bect the nuclear-spin system in the same\nway as a decrease of the pump power. To con\frm thisexpectation, di\u000berent pumping times (10 s, 30 s, 1 min,\n5 min, and 10 min) were used for a time-resolved spin-\nnoise experiment with Ppu= 0:3 mW atBx= 4 mT.\nFigures 7(c) and 7(d) demonstrate that the qualitative\ntrends of the decay times \u001ciand the probabilities ~ aire-\nsemble the results of the pump-power dependence. The\npump time-limit required for a saturated nuclear polar-\nization is between 1 min and 5 min. Above this pump\ntime, atPpu= 0:3 mW, the nuclear spin system is sat-\nurated. Together with the pump power limit of 0 :1 mW\nand optical pumping for 10 min, the pump time limit is a\nuseful result for further investigations using time-resolved\nSNS with optical pumping on the saturated nuclear-spin\nsystems.\nWe relate our measurements to similar studies on\nGaAs:Si epilayers with comparable doping, for which\nonly two exponents were observed [60, 63]. More specif-\nically, the authors of Ref. [60] found two decay times\n\u001c1= 30 s and \u001c2= 300 s for the dielectric phase of GaAs\ndoping (nD= 2\u00021015cm\u00003) and only one exponent\n\u001c= 150 s for the metallic phase ( nD= 4\u00021016cm\u00003).\nThe setup was similar to the one used by us, with pump\ntimes of 1 min to 5 min, at Bz= 12 mT, and, most impor-\ntantly,Bx= 4 mT, as the values of the decay times can\nstrongly depend on the chosen Bx[64]. For the dielectric\nphase, the authors argued that the shorter time is as-\nsociated with the electron-assisted spin depolarization of\nnuclei close to the donor centers and the longer time to\nspin di\u000busion governed by dipole-dipole interaction be-\ntween nuclei far away from the donor centers, outside of\nthe Bohr radius.\nApplying this time designation to our data, we can\nsuggest that the \u001c3component belongs to the nuclear\nspin di\u000busion outside the Bohr radius of the electron.\nIt should take a rather long time and requires a rather\nstrong pump power for it to be observed. Further, the\npresented pump-power and pump-time dependence of the\nOverhauser \feld's depolarization support the idea that\nthe shortest time \u001c1should arise from a di\u000berent decay\nmechanism than the longer decay times. Therefore, we\nassign it to electron-assisted depolarization, requiring a\nshorter pump time and weaker pump power for it to be\npresent, as the electron spin has a direct hyper\fne cou-\npling to the nuclear-spin bath. The additional nuclear-\nspin decay time ( \u001c2) in the InGaAs epilayer has a similar\nbehaviour as \u001c3and might originate from depolarization\nthrough quadrupole e\u000bects that are strongly enhanced\ncompared to GaAs due to the indium content. To pro-\nvide an experimental proof for such an enhancement we\ndetermine the local \feld, induced by the \ructuating nu-\nclear spins.\nMeasurement of the local \feld\nOne of the ways to describe the local \feld of interact-\ning nuclear spins is to use the thermodynamics frame-\nwork, particularly the concept of spin temperature and8\n−20−100102030020406080\nB\nx (mT)f\nL (MHz) - g\ne µ\nBB\nx/h100150200250300Time after pump off (s)\nBL = 20 mTBL = 0.2 mT\n(c)ΘNi= -250 µK00.020.040.061/τ\n2 (1/s)\n024681000.0020.0040.006\nB\nx (mT)1/τ\n3(1/s)\nIgeI = 0.54(a)\n(b)\nFIG. 8. (a) and (b) Spin relaxation rates for the second and\nthird time components measured as a function of the exter-\nnal magnetic \feld. Blue dots are data and red lines are \fts\nby Lorentzian functions. (c) Overhauser \feld contribution\nto the frequency of the SN peak position for the adiabatic\ndemagnetisation experiments. Blue dots are the experimen-\ntally determined frequencies after the subtraction of the elec-\ntron Larmor frequency without nuclear contribution. Red\nline is the \ft using the Eq. A12, which gives the values of\nBLand \u0002 Ni. Green dashed line is an example of a \ft with\nBL= 0:2 mT. Top x-scale gives the time relative to the point\nof pump blocking.\nadiabatic demagnetization [61, 65]. In that case, one in-\ntroduces the spin temperature of the nuclear spin sys-\ntem \u0002N. The nuclear spin polarization pNorients itself\nwith external \feld Band can be described by Curie's\nlawpN=\rNBC=\u0002N, with\rNbeing the nuclear gyro-\nmagnetic ratio and Cthe Curie constant. An adiabatic\ndecrease of BfromBitoBfwould conserve the pN, but\nwould reduce the spin temperature by a factor of Bf=Bi.\nThe local \feld sets the limit for that factor at low mag-\nnetic \felds. The dipole-dipole interaction between nu-\nclei determines the local \feld BL, soBL\u00190:2 mT for\nGaAs [66, 67]. For nuclear isotopes having quadrupole\nmoments, the local \feld can increase due to electrical\n\felds or strain in the structure, which can be induced\nby lattice deformations [28, 68]. Once the external \feld\nB\u0019BL, the nuclear polarization is randomized, limit-\ning the adiabatic spin cooling to BL=Bi. However, once\nthe external \feld is increased above BL, the polarization\nrecovers along the applied \feld direction. Therefore, to\ndetermine the BL, it is required to measure the polariza-\ntionpNof the nuclear spin system as a function of anexternal magnetic \feld Bxby slowly ramping it through\nzero. For the nuclear spin polarization being optically\nprepared at a temperature \u0002 NiandBi> BL, we can\ndescribe the pNas [61, 69]:\npN(t) =Bx(t)\n3kB\u0002N(t)~h\rN(I+ 1)i;\n\u0002N(t)\n\u0002Ni=p\nB2x(t) +B2\nLp\nB2\ni+B2\nL;(8)\nwithIbeing the nuclear spin and the angled brackets\nshowing the averaging over the nuclear isotopes.\nFurther, we accommodate the experiments presented\nin Refs. [26, 68]. These references present a thorough\ndescription of a method to determine the BLapplied to\nan-doped GaAs epilayer, supported by the theoretical\nbackground. In our case, the sample was illuminated for\n15 minutes with the pump beam of Ppu= 0:5 mW at\nBz= 10 mT and Bx=\u000030 mT. Then, the pump beam\nwas blocked by a shutter, and the timer of the experi-\nment was started. We waited for about one minute in\ndarkness to allow for fast nuclear depolarization within\nthe Bohr radius of electrons (the time scale of \u001c1, see\nFig. 6(c)) and set the Bzto zero. After that, the SN\nspectra are taken with one-second accumulation while\nstepping the Bxfrom\u000030 to 30 mT with 1 mT steps. To\nextract the evolution of the nuclear polarization pN, we\ndetermined the peak position of the SN spectra at each\nmagnetic \feld and subtracted the electron Larmor fre-\nquency (ge\u0016BBx=h) without nuclear polarization at the\nsame \feld, compare with Eq. 5. The value of the g-factor\njgej= 0:54 was determined independently at the same\nsample position without preceding nuclear polarization.\nThe blue points in Fig. 8(c) represent the extracted data.\nAs discussed above, the nuclear polarization should re-\ncover once the magnetic \feld increases above zero. The\nobserved asymmetry is related to the long accumulation\ntimes so that the spin-lattice relaxation reduces the sig-\nnal, see the top x-scale in Fig. 8(c) for the times after\nthe pump blocking. In Appendix A, we consider the sit-\nuation when remagnetization takes place together with\nspin-lattice relaxation. On the experimental side, it re-\nquires the knowledge of the magnetic \feld dependence\nof the spin relaxation. To establish that, we have done\nmeasurements similar to the one presented in Fig. 6 for\ndi\u000berent \fxed magnetic \felds Bx. Figures 8(a) and 8(b)\nrepresent the extracted values for the rates of the second\nand third \ftting components \u001c2and\u001c3, respectively. A\nLorentzian function gives the best \ft for these rate de-\npendencies on the external magnetic \feld. It allows us to\nobtain the functional dependence of the relaxation rates\nonBx[64].\nFinally, the red line in Fig. 8(c) represents a \ft by\nEq. (A12) with two free \ftting parameters: BLand\n\u0002Ni. Best \ft gives the BL= 20\u00061 mT andj\u0002Nij=\n250\u000610\u0016K. For comparison, we show the curve for\nBL= 0:2 mT by the dashed green line [70]. As one can\nsee, the value of BLis two orders of magnitude larger9\nthan that measured for n-doped GaAs epilayers [26]. In\nRefs. [26, 68], it is also demonstrated that the quadrupole\ne\u000bects are responsible for an increase of BL, leading to\nvalues ofBL\u00192 mT.\nSuch signi\fcant values of BLin our sample indicate\na strong quadrupolar contribution to the local \felds but\nalso raise a concern about the validity of the spin tem-\nperature approach, compare with Ref. [71]. To test it,\nwe have done two additional experiments. At the optical\npumping stage, Bxwas \fxed now at +30 mT in addition\ntoBz= 10 mT. Then, after the pump blocking and wait-\ning for one minute, the \feld is swept to Bx=\u000030 mT at\na rate of (i) 320 mT/s or (ii) 6 mT/s. Once Bx=\u000030 mT\nis reached, the experiments and data processing are done\nsimilarly. This resulted in values of BLand \u0002Nithat are\nidentical within the error bounds to the case without a\nsweep, Fig. 8(c). It indicates that the spin temperature\napproach is still valid in our case.\nSpin di\u000busion model\nTo advance our understanding of the nuclear-spin re-\nlaxation, we additionally analyze the decay of the nu-\nclear polarization using the di\u000busion model, presented in\nRef. [72]. To do that, we calculate the di\u000busion equation:\ndpN(t;r)\ndt=D\u0001pN(t;r)\u0000pN(t;r)\u0010\nT\u00001\n1e(r) +T\u00001\n1;K\u0011\n+\nG(t;r):\n(9)\nHere,pN(t;r) is the nuclear-spin polarization at dis-\ntancerfrom the center measured at time t,T1e(r) =\nT1e(0) exp(4r=aB) is the position-dependent nuclear-spin\nrelaxation time due to interaction with bounded elec-\ntrons,T1;Kis the time characterizing the nuclear-spin\nrelaxation due to interaction with the Fermi-edge elec-\ntrons (the Korringa mechanism [64]), Dis the nuclear-\nspin di\u000busion constant, and G(t;r) is the pumping rate.\nThe nuclear-spin relaxation rate at the donor origin is\nde\fned by\n1\nT1e(0)= \u0000t\n22\u001cc\n1 +!2\u001c2c: (10)\nHere,!=ge\u0016BBx=}is the magnetic \feld given in fre-\nquency units, \u0000 tis the probability of occupation of the\ndonor (for simplicity, we take \u0000 t= 1),\u001ccis the correla-\ntion time measuring the residing time of the electron at\nthe donor interacting with nuclear spins with the magni-\ntude given by:\n\n =Ahf\n2}v0\n\u0019a3\nB(11)\nwhereAhf= 46\u0016eV is the electron-nuclear hyper\fne\nconstant averaged over the atom species in a unit cell,\nv0= (0:283)3nm3is the two-atom unit-cell volume.\n1 10 100 10000.0010.010.1110100\nt − T\npump (s)B\nN (mT)τc = 75 ps, T1,K = 1500 s \nD = 1.6×10−13 [cm2/s]\nnD = Nb = 4.2×1015 [cm−3]\nD = 6.3×10−14 [cm2/s]\nnD = Nb = 4.2×1015 [cm−3]\nD = 1.0×10−13 [cm2/s]\nnD = n0 = 3.9×1016 [cm−3]FIG. 9. Decay dynamics calculated using Eqs. (9) and (13)\n(solid lines) for various nDandD,\u001cc= 75 ps estimated from\nthe polarization recovery curve, and T1;K= 1500 s estimated\nfor the doping density n0. Circles show the experimental data\nretracted from Fig. 6(c).\nEquation (9) has no simple analytical solution, there-\nfore we treat it numerically by substituting pN(t;r) =\n(1=r)PN(t;r) for numerical stability. The pumping rate\nis given by:\nG(t;r) =PeI+ 1\nS+ 11\nT1e(r)[\u0002(t)\u0000\u0002(t\u0000Tpump)];(12)\nwhereS= 1=2 andI= 3=2 are the electron and nu-\nclear spins, Pe=hSzi=Sis the electron-spin polarization\nwhen pumping, and the term in brackets represents the\nswitch-on and switch-o\u000b pumping at the moments t= 0\nandt=Tpump, given by theta-functions. The calcu-\nlation is performed in a region R=n\u00001=3\nD for a given\ndonor concentration with \frst and second-type bound-\nary conditions at r= 0 andr=R, respectively. To\nperform a direct comparison of the calculations with the\nexperiment, the time evolution of the Overhauser \feld is\ncalculated:\nBN(t) =bnZR\n0PN(t;r) exp\u0000\n\u00002r=aB\u0001\nrdr (13)\nwherebnis a scaling factor. The results of the calcula-\ntions for the spin relaxation at times t\u0000Tpump are shown\nin Fig. 9. As one can see from the \fgure, the experi-\nmental data retracted from Fig. 6(c) are reasonably well\nreproduced by the modeling, especially in the limit of\nshort and long times.\nHowever, this becomes only possible if the concentra-\ntion of donors is reduced by an order of magnitude, down\ntonD=Nb= 4:2\u00021015cm\u00003. A relatively small vari-\nation of the di\u000busion constant Dallows one to \ft better\nthe experimental dependence at long times. This pro-\nvides an estimation of DwhennDis known. On the\ncontrary, the calculations done for a nominal concentra-\ntionnD=n0= 3:9\u00021016cm\u00003provide a much faster10\nOverhauser-\feld relaxation dynamics than the one ob-\nserved experimentally (see green curve in Fig. 9). Note\nthat, in principle, the quadrupole interaction could re-\ntard the spin di\u000busion, which would result in a reduced\nDfor \ftting the data. However, we \fnd that a slightly\nhigher value of Dis required to properly \ft the experi-\nmental data at longer times than the one at the initial\nrelaxation stage (see blue lines in Fig. 6). We also \fnd a\nsmall relative sensitivity of the model to a variation of \u001cc\nat small times t. Furthermore, in our modeling, !\u001cc\u001c1\nbecause the external \feld Band the observed BNare\nsmall. In other words, the transition from the short to\nthe long correlation-time regime, where T1eis a\u000bected\nmost by!\u001cc[17], is not achieved.\nV. CONCLUSION\nTo conclude, we have investigated the in\ruence of the\nindium contribution in the GaAs matrix on the donor-\nbounded carrier- and nuclear-spin relaxation dynamics.\nBesides the red shift of the band gap and donor emis-\nsion energies we observe an enhancement of the car-\nrier localization for comparable doping concentrations\nof the GaAs:Si structures. The dynamics of nuclear-\nspin relaxation reveals a complex three-exponential de-\ncay of the Overhauser \feld, which we interpret as re-\nsult of the enhanced quadrupole e\u000bects induced by the\nindium. We provide experimental evidence for this en-\nhancement by measuring the local \feld, which exceeds\nthe quadrupole-free local \feld given by the dipole-dipole\ninteraction by two orders of magnitude. This value of\nthe local \feld puts the studied structure in the range be-\ntween the low-stressed GaAs epilayers with BL\u00192 mT,\nwhere the spin temperature approach is valid, and highly-\nstressed self-assembled InGaAs QD structures with BL=\n300 mT [71], where the spin temperature approach breaks\ndown. The modeling of the nuclear-spin polarization re-\nlaxation by the di\u000busion model suggests that the donor\nconcentration should be reduced by an order of magni-\ntude in order to match the experimental results. The ori-\ngin of this discrepancy is not completely clear, but could\nbe related to donor depletion due to surface charges,\nweak carrier localization, and the need to include the\nquadrupole interaction into the di\u000busion model. Fur-\nther studies including the optimization of the silicon dop-\ning as well as an extension of the di\u000busion model are\nplanned. Additionally, application of external stress to\ncontrol the quadrupole interaction could be an option\nto in\ruence the spin relaxation dynamics and, there-\nfore, provide silicon-doped InGaAs epilayers as an ad-\nvantageous alternative to GaAs:Si for low-temperature\nnuclear-spin ordering and to InGaAs QDs for reduced\nquadrupole e\u000bects.ACKNOWLEDGMENTS\nWe thank D. S. Smirnov and V. V. Belykh for fruit-\nful discussions. The sample was grown by using the\nfacilities of the SPbU Resource Center \\Nanophoton-\nics\". The transport measurements were performed at the\nP. N. Lebedev Physical Institute shared facility center.\nThe optical investigations presented in this work were\ndone at TU Dortmund. The data analysis and represen-\ntation were performed by using the MagicPlot software.\nWe acknowledge the \fnancial support by the Deutsche\nForschungsgemeinschaft in the frame of the International\nCollaborative Research Center TRR 160 (Projects A5\nand A6) and the Russian Foundation for Basic Research\n(Grant No. 19-52-12054 and 19-52-12043). MYP and\nKVK acknowledge Saint Petersburg State University for\nthe research grant 91182694.\nAppendix A: Remagnetization with spin-lattice\nrelaxation\nEquation (8) is usually obtained from thermodynamic\nconsiderations, using entropy balance at equilibrium.\nThat approach does not allow one to take into account re-\nlaxation processes. In order to generalize Eq. (8) for the\ncase of slow (as compared to spin-spin processes) spin-\nlattice relaxation, we re-derive it from the condition of\nenergy balance during a change of the magnetic \feld.\nThe rate of the energy change of the spin system under\nthe action of a changing external magnetic \feld is:\ndE\ndt=@hHi\n@t=\u0000h~Mi@~B\n@t=\u0000Tr(^M2\nB)\fB@B\n@t;(A1)\nwith\f= (kB\u0002N)\u00001,^MBbeing the projection operator\nof the total magnetic moment of the nuclei on the ex-\nternal \feld B. This equality is true, since the external\nmagnetic \feld is the only parameter of the Hamiltonian\nof the spin system that depends on time directly [73]. On\nthe other hand, the energy of the spin system is:\nE=h^Hi=\u0000\fTr(^H2) =\u0000\f(Tr^H2\nZ+ Tr ^H2\nSS)\n=\u0000\fTrM2\nB(B2+B2\nL);(A2)\nwith ^HZ=\u0000MBBbeing the Hamiltonian of Zeeman\ninteraction, ^HSSthe Hamiltonian of spin-spin interac-\ntions, andBL\u0011q\n(TrM2\nB)\u00001Tr(^HSS) is, by de\fnition,\nthe local \feld [65], characterizing the strength of spin-\nspin interactions. Accordingly, the total time derivative\nof the energy, considered as a function of the inverse spin\ntemperature \fand the external magnetic \feld B, is:\ndE\ndt=d\ndt\u0002\n\u0000\f(t)TrM2\nB\u0000\nB2\nL+B2(t)\u0001\u0003\n=\n=\u0000TrM2\nB\u0014\u0000\nB2\nL+B2(t)\u0001d\f(t)\ndt+ 2\f(t)B(t)dB(t)\ndt\u0015\n:\n(A3)11\nSetting Eqs. (A1) and (A3) to be equal, we obtain:\n\u0000\nB2\nL+B2(t)\u0001d\f(t)\ndt+\f(t)B(t)dB(t)\ndt= 0; (A4)\nthat gives a di\u000berential equation for \f:\nd\f(t)\ndt=\u0000\f(t)B(t)\nB2\nL+B2(t)dB(t)\ndt; (A5)\nwhich, in order to account for spin-lattice relaxation,\nshould be complemented by a relaxation term of the form\n\u0000(\f\u0000\fL)=T1, where\fL= (kBT)\u00001is the inverse temper-\nature of the lattice, and the spin-lattice relaxation time\nT1depends, in general, on the magnetic \feld [64]. In\nexperiments with optical cooling of nuclei in weak mag-\nnetic \felds, the nuclear temperatures usually do not ex-\nceed a few millikelvin (otherwise there is no noticeable\nnuclear magnetization), and the lattice temperature is\nseveral kelvins. Therefore, we can set \fLequal to zero,\nand the equation for \ftakes a simple form:\nd\f(t)\ndt=\u0000\f(t)B(t)\nB2\nL+B2(t)dB(t)\ndt\u0000\f(t)\nT1(B); (A6)\npermitting an analytic solution. By dividing both sides\nof Eq. (A6) by \fit is brought to the form:\ndln\f(t)\ndt=\u0000d\ndtlnq\nB2\nL+B2(t)\u00001\nT1(B): (A7)\nIf at the time moment t= 0 the magnetic \feld was equal\ntoBi, and the inverse spin temperature equal to \fi, so-\nlution of Eq. (A7) yields the following time dependence\nof\f:\n\f(t)\n\fi=s\nB2\nL+B2\ni\nB2\nL+B2(t)exp\u0012\n\u0000Zt\n0T\u00001\n1(B(t0)dt0)\u0013\n:\n(A8)\nIn the absence of spin-lattice relaxation, Eq. (A8) gives\nan expression for the inverse spin temperature under an\nadiabatic change in the magnetic \feld, usually obtainedfrom the condition of entropy being constant in the adi-\nabatic process [65]. In view of:\nhMBi= Tr\u0010\n\u001aN^MB\u0011\n\u0019\fBTr(^M2\nB); (A9)\nwith\u001aNbeing the density matrix for an ensemble of spins\nin high-temperature approximation:\n\u001aN=exph\n\u0000\f\u0010\n^HSS+^HZ\u0011i\nTr\u0010\nexph\n\u0000\f\u0010\n^HSS+^HZ\u0011i\u0011\u0019\n\u00191\u0000\f\u0010\n^HSS+^HZ\u0011\n;(A10)\nwe obtain the expression for the magnetization:\nhMB(t)i\nhMB(0)i=B(t)\nBis\nB2\nL+B2\ni\nB2\nL+B2(t)\u0002\n\u0002exp\u0012\n\u0000Zt\n0T\u00001\n1(B(t0)dt0)\u0013\n:(A11)\nIn our experiments, a multi-exponential relaxation is\nobserved, which is presumably related to spin di\u000busion\nin presence of spatially inhomogeneous spin-lattice relax-\nation due to hyper\fne and quadrupole interactions [64].\nThe rigorous way to take these processes into account\nwould be complementing Eq. (A6) with a di\u000busion term\nand considering the spin temperature and T1as functions\nof coordinates. 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Lifshitz, \\Statistical physics,\"\n(Butterworth-Heinemann, 3rd edition, 1980) Chap. 12."    },    {        "title": "1201.3680v1.Spin_selective_Kondo_insulator__Cooperation_of_ferromagnetism_and_Kondo_effect.pdf",        "content": "arXiv:1201.3680v1  [cond-mat.str-el]  18 Jan 2012Spin-selective Kondo insulator: Cooperation of ferromagn etism and Kondo effect\nRobert Peters∗and Norio Kawakami\nDepartment of Physics, Kyoto University, Kyoto 606-8502, J apan\nThomas Pruschke\nDepartment of Physics, University of G¨ ottingen, 37077 G¨ o ttingen, Germany\n(Dated: July 27, 2018)\nWepropose thenotion ofspin-selectiveKondoinsulator, wh ichprovidesafundamentalmechanism\nto describe the ferromagnetic phase of the Kondo lattice mod el with antiferromagnetic coupling.\nThis unveils a remarkable feature of the ferromagnetic meta llic phase: the majority-spin conduction\nelectrons show metallic- while the minority-spin electron s show insulating-behavior. The resulting\nKondogap in the minority spin sector, which is due tothe coop eration of ferromagnetism andpartial\nKondo screening, evidences a dynamically-induced commens urability for a combination of minority-\nspin electrons and parts of localized spins. Furthermore, t his mechanism predicts a nontrivial\nrelation between the macroscopic quantities such as electr on magnetization, spin polarization and\nelectron filling.\nPACS numbers: 71.10.Fd 71.27.+a 71.30.+h 75.20.Hr\nEven 30 years after their discovery heavy-fermion sys-\ntems attract much attention due to their fascinating\nproperties. Apart from being Fermi liquids with ef-\nfective mass thousand times as large as the free elec-\ntron one, they show all kinds of competing or coexisting\nphases, and at the boundaries between these phases one\nfrequently observes quantum phase transitions, accom-\npanied by barely understood non-Fermi liquid behavior\n[1–3]. Heavy fermion compounds usually include lan-\nthanides or actinides with open 4 f- or 5f-shells, which in\nthe simplest theoretical modeling can be viewed as a reg-\nular lattice of local moments coupled to the conduction\nelectrons. This coupling typically leads to twocompeting\nmechanisms: the long-ranged RKKY interaction and the\nlocal Kondo screening. While the RKKY interaction fa-\nvorsa magnetically ordered state, the Kondo screening is\nusuallyconsideredtoformaparamagneticheavy-fermion\nstate. The competition of these two mechanisms can be\neasily understood in terms of the Doniach phase diagram\n[4].\nWhile in most heavy-fermioncompounds the magnetic\norder is antiferromagnetic, there are a certain class of\ncompounds showing ferromagnetic order. For example,\nthe recently discovered YbNi 4P2is a ferromagnetically\norderedheavy-fermioncompound which seems to be very\nclose to a quantum critical point [5]. Taking such fer-\nromagnetic heavy fermion compounds as motivation we\nanalyze in detail the mechanism stabilizing the ferromag-\nnetic state. An interesting question in this context is, if\nand how the Kondo effect accounts for the ferromagnetic\nstate [6–9].\nIn this letter, we propose a spin-selective Kondo in-\nsulator, where the Kondo screening plays an essential\nrole in stabilizing the ferromagneticmetallic state at zero\ntemperature, which elucidates a previously unrecognized\nfeature of the ferromagnetic phase: the majority-spin(minority-spin)conductionelectronsareinametallic(in-\nsulating) state. We claim that this notion is not spe-\ncific to certain choices of system parameters but is fun-\ndamental and ubiquitous for the ferromagnetic phase in\nthe Kondo lattice model. Due to partial Kondo screen-\ning, parts of the local moments are bound to the elec-\ntrons, resulting in a dynamically-induced commensura-\nbility which is essential for producing the gap in the mi-\nnorityspinelectrons. Wefindthatthiscommensurability\ncondition leads to a nontrivial relation between electron\nmagnetization, spin polarization and electron filling.\nThe competition or cooperation between the magnetic\nphase mediated by the RKKY interactionand the Kondo\nscreening can be modeled via a Kondo lattice model with\nantiferromagnetic coupling between the local moments\nand the conduction electrons. The Kondo lattice model\nreads [4, 10, 11],\nH=t/summationdisplay\n<i,j>σc†\niσcjσ+J/summationdisplay\ni/vectorSi/vector si\n/vector si=c†\niσm/vector ρσmσnciσn,\nwherec†\niσcreates an electron on site iwith spin-direction\nσ,/vector ρrepresentsthe vectorofPauli-matrices,and /vectorSirepre-\nsentsthelocalspinswhicharecoupledtotheelectronsvia\nan antiferromagnetic spin-spin interaction with strength\nJ >0.\nTo solve the Kondo lattice model we use the dynam-\nical mean field theory (DMFT) [12–14]. DMFT maps\nthe lattice model onto a quantum impurity model with\na fermionic bath being determined self-consistently. Al-\nthough being an approximation to real systems, DMFT\nhas provided many insights into the physical properties\nand can even captures subtle differences in the lattice\ngeometry. For solving the impurity model, we use the\nnumerical renormalization group (NRG) [15, 16], which2\n-0.5 00.5\nω/W00.511.5ρ(ω)W\n-0.5 00.5\nω/W-0.0500.05\nω/W\nincreasing n↓ increasing n↑n↓n↑\n0.011\n0.028\n0.077\n0.091\n0.1340.054\n0.108\n0.181\n0.197\n0.241n↓n↑\nFigure 1: (Color online) Spin-resolved spectral functions for\nthe ferromagnetic state in the Kondo lattice model J/W=\n0.25. The inset shows a magnification around the Fermi en-\nergy for the spin-down component illustrating the gap in the\nspectral function.\nis able to reliably calculate spectral functions at very low\ntemperatures [17, 18].\nFirst, we briefly summarize the known DMFT results\nfor the Kondo lattice model [10, 11, 19, 20] (A discussion\non the RKKY interaction within DMFT can be found in\n[19].) At half filling there is a pronounced antiferromag-\nnetic N´ eel state for weak coupling, which vanishes with\nincreasingcouplingstrength Jviaacontinuoustransition\nto a paramagnetic insulating state, the Kondo insula-\ntor. Doping slightly away from half filling this transition\nchanges into a transition between an antiferromagnetic\nstate (possibly spin-density-wave) and a paramagnetic\nmetallic state. Especially the paramagnetic state around\nhalf filling is dominated by the Kondo effect, where the\nKondoscreeningoflocalizedspinsresultsin alargeFermi\nsurface accompanied by a narrow band and a gap close\nto the Fermi energy. Away from half filling the effects\nof Kondo screening become less important as there is an\nimbalance between local moments and available conduc-\ntion electrons. Such a tendency might be even stronger\nwhen the system enters a ferromagnetic state realized\nat low fillings, because an additional imbalance between\nspin-up and spin-down electrons arises. Contrary to this\nnaive expectation, however, we demonstrate here that\nthe Kondo screening plays an essential role even in the\nferromagnetic phase. In particular, we reveal that the\ncooperation of ferromagnetism and Kondo screening can\nrealize a novel kind of Kondo insulating state in the fer-\nromagnetic metallic phase.\nFigure1shows the local spin-resolved spectral-\nfunctions calculated in the ferromagnetic phase for a\nBethe lattice with antiferromagnetic Kondo coupling\nJ/W= 0.25 (bandwidth W= 4t). For this coupling\nstrength the ferromagnetic phase extends from a nearly1 2 3 4 56\nW/J0.0010.010.11gap size ∆/Wn↓=0.1\nn↓=0.2\n0.1 0.2n↓0.10.20.3gap size ∆/WJ/W=0.3\nJ/W=0.4\nJ/W=0.5\nFigure 2: (Color online) Gap width ∆ /Win the minority-\nspin spectral function depending on the spin-coupling Jand\nthe occupation nc\n↓. The temperature of the system is T/W=\n3·10−4. The lines in the left panel are fits as ∼exp(−a/J).\nempty system to approximately nc=nc\n↑+nc\n↓= 0.5.\nOne finds a striking difference in the spectral functions,\nwhich has not been recognized previously, for the major-\nity spin ( nc\n↑) and the minority spin ( nc\n↓). While in the\nmajority-spin spectral function a peak at the Fermi en-\nergyω= 0 and a dip for ω >0 can be found, there is\na gap at the Fermi energy in the minority-spin spectral\nfunction. It is important to note that such a gap is not\npresent in the ferromagnetic phase for a Kondo lattice\nmodel with ferromagnetically coupled spins. We propose\nthat this gap in the spectral function is due to a partial\nKondo screening of the localized spins, which results in\nan intriguing state: although the ferromagnetic state is\nmetallic, only the majority-spin electrons contribute to\nthe low-temperature properties, in particular transport.\nTheminorityspins,eventhoughnotcompletelydepleted,\nform an insulator, which we name spin-selective Kondo\ninsulator .\nIncreasing the occupation number, the dip in the\nmajority-spinspectral function movescloser to the Fermi\nenergy and becomes more pronounced. Eventually, the\nferromagnetic state is replaced by a paramagnetic state,\nforwhichthespectralfunctionsforbothspin-components\nsuffer from the typical suppression of the DOS for ω >0\ndue to Kondo screening. Increasing the occupation to-\nwards half filling this dip becomes deeper and finally\nmoves to the Fermi energy, forming the Kondo insula-\ntor.\nClear evidence showing that the above insulating gap\nis indeed caused by the Kondo screening can be found\nin the dependence of the gap width ∆ on the occupation\nand coupling strength, shown in Fig. 2. The left panel\nin Fig.2displays the dependence of the gap width on\nthe coupling strength. For this purpose the minority-\nspin occupancy was kept constant (also resulting in a3\n0 0.2 0.4 0.60.8\nFilling n=n↑+n↓00.51Magnetization\n0.2 0.4 0.60.8\nFilling n=n↑+n↓<m>=<n↑>-<n↓>\n-<Sz>\ncommensurabilityJ/W=0.3 J/W=0.5\nc-electrons\nf-electrons\nsinglet\nnc↓+nf↓= 1\nFigure 3: (Color online) Upper panel: Magnetization and\n“commensurability”, ( nc\n↓+nf\n↓), for two different coupling\nstrengths and different occupation numbers calculated for a\nBethe lattice at T/W= 3·10−4. The electron magnetization\nis shown as /angbracketleftm/angbracketright=nc\n↑−nc\n↓, while the spin expectation value\n/angbracketleftSz/angbracketrightis shown mirrored as −/angbracketleftSz/angbracketright. The commensurability-\ncondition is explained in the text. Lower panel: sketch of th e\nlocal configuration (see text).\nnearly constant majority-spin occupancy). The depen-\ndency on Jperfectly obeys a Kondo temperature-like\nform ∆ ∼exp(−a/J) with a fitting constant a, sug-\ngesting that the Kondo physics is essential for the gap-\nformation. As a function of increasing filling the gap\nwidth decreases monotonically, as shown in the right\npanel of Fig. 2. As soon as the ferromagnetic phase van-\nishes, the gap at the Fermi energy closes, too, and the\nminority and majority spectral functions look similar to\nthe right panel in Fig. 1.\nLet us now elucidate the basic physics behind this fer-\nromagnetic state. In Fig. 3the magnetization of the\nconduction electrons /angbracketleftm/angbracketright=nc\n↑−nc\n↓and the polariza-\ntion of the localized spins −/angbracketleftSz/angbracketrightis shown. Note that\nthe local spin-polarization always has the sign opposite\nto the conduction electron magnetization due to the an-\ntiferromagnetic coupling, thus −/angbracketleftSz/angbracketrighthas the same sign.\nIncreasing the number of conduction electrons, the mag-\nnetization of the electrons first increases due to increas-ing filling, and eventually decreases again due to the\nsuppression of the ferromagnetic state. On the other\nhand, the spins are almost fully polarized for a nearly\nempty lattice, with monotonically decreasing polariza-\ntion for increasing conduction electron number. In the\nspirit of a pseudo-fermion representation, let us assume\nthat the localized spins are actually formed by a local\nhalf-filled and strongly-interacting energy level so that\n/angbracketleftSz/angbracketright= (nf\n↑−nf\n↓)/2 andnf\n↑+nf\n↓= 1 (defining nf\nσas the\nspin-dependent occupation of this level). Remarkably,\nwe find that the following nontrivial commensurability\ncondition holds within the ferromagnetic state:\nnc\n↓+nf\n↓= 1, (1)\nas can be seen in Fig. 3. Note that Eq. ( 1) is equiva-\nlent tonf\n↑=nc\n↓. It should be noticed that this condition\nis nota priori given but is generated dynamically due\nto many-body effects. To clarify the origin of the above\ncommensurability we propose that a partial local Kondo-\nsinglet is formed in which /angbracketleftnc\n↓/angbracketrightmajority- and minority-\nelectrons participate, thus combining all spin-down con-\nduction electrons together with a part of the f-electrons\nand the spin-up conduction electrons to a Kondo spin-\nsinglet. Here, we have assumed that spin-down is the\nminority-spin direction. The remaining majority-spin\nconduction electrons and spin-down f-electrons form a\nferromagnetic state. (see a sketch in the lower panel of\nFig.3). That the number of spin-down electrons in-\ncludingf- and conduction-electrons sums up to unity\ngives a commensurable situation, which results in a gap\nattheFermi energy. Ontheotherhand, forthe majority-\nspins there is not such a commensurabilitycondition, but\nnc\n↑+nf\n↑=nc=nc\n↑+nc\n↓holds. Therefore, this par-\ntial Kondo screening results in an insulating state for\nthe minority-spin, while the majority-spin electrons re-\nmain metallic. For this reason we have called this state\na “spin-selective Kondo insulator”. The commensurabil-\nity condition ( 1) smoothly connects to the Kondo insu-\nlator at half filling, suggesting that this ferromagnetic\nstate should exist up to half filling. However, our results\nclearlyshowthat there is atransition fromthis ferromag-\nnetic phase to a paramagnetic state at electron fillings\nfornc> nferro. This is only possible, if the expectation\nvalue/angbracketleftSz/angbracketrightjumps, leading to a discontinuous phase tran-\nsition at nferro. Our finding of the discontinuous transi-\ntion completely agrees with the recent analytical results\nshowing that non-analytic terms prevent the continuous\ntransition from a ferromagnet to a paramagnet [21].\nA further important consequence deduced directly\nfrom the commensurability condition ( 1) is a nontrivial\nrelation between electron magnetization, spin polariza-\ntion and occupation number:\n2/angbracketleftSz/angbracketright+/angbracketleftm/angbracketright=/angbracketleftnc/angbracketright−1. (2)\nThis formula connects these three quantities which are\notherwise independent from each other. By arranging it-4\nFigure 4: (Color online) Momentum-resolved spectral func-\ntions for the square lattice and J/W= 0.3,n=nc\n↑+nc\n↓=\n0.25,m=nc\n↑−nc\n↓= 0.1. The right side always shows a\nmagnification of the left side around the Fermi energy ω= 0\nrepresented by the green line. From top to bottom the fig-\nures show the majority-spin and the minority-spin spectral\nfunction, respectively.\nFigure 5: (Color online) Momentum-resolved occupation\nnumber n(k) (same parameters as in Fig. 4). Left (right)\npanel shows the majority- (minority-) spin component. The\ndotted line represents the Fermi surface for non-interacti ng\nelectrons (for the majority-spin this lines coincides with the\nshown surface). Note that for improving the contrast, the\noccupation for the minority-spin electrons is displayed in the\nintervaln(k)∈[0,0.3].\nself in this way the system can gain an additional energy\noriginating from the partial Kondo screening. Note that\nit should be possible to verify such a relation experimen-\ntally.\nThe formation of the gap in the spectral function\ndoes not depend on the lattice geometry. For exam-\nple, it can also be found in DMFT calculations for\na two-dimensional square lattice. Figure 4shows the\nmomentum-resolved spectral functions for J/W= 0.3\nandnc\n↑+nc\n↓= 0.25 for a square lattice. The right\npanels are magnifications around the Fermi energy. In\nthe minority-spin spectral function (bottom right) the\ngap can be clearly seen. While the majority-spin elec-\ntrons are renormalized for ω >0 with a finite life-time,\nthe minority-spin electrons are renormalized for ω <0.\nThis behavior is also visualized in Fig. 5, in which themomentum-resolved occupation number is shown. While\nfor the majority-spin electrons the occupation number\ndistribution looks like in the non-interacting case, for the\nminority-spin electrons this function is actually smeared\nout as compared to a reference non-interacting system,\ni.e. we indeed observe a large Fermi volume here.\nUsing DMRG, we have confirmed that the spin-\nselectiveKondoinsulatortogetherwith the commensura-\nbility condition can be found for the ferromagnetic phase\nofthe one-dimensional(1D) Kondolattice model, too. In\nfact, previous calculations for the ferromagnetic ground\nstateobservedamagnetization Stot= 1/2(L−Nc)[22](L\nsystem length, Ncelectron number), which supports the\ncommensurability condition, and two separated bands in\nthe spectral functions [23]. From these precise analyses\nof the 1D model, we conclude that our finding is ubiqui-\ntous for the Kondo lattice model and not an artifact of\nDMFT.\nIn conclusion we have clarified the physics behind the\nferromagnetic metallic phase realized in the Kondo lat-\ntice model. We have demonstrated that the cooperation\nof ferromagnetism and partial Kondo screening results\nin an intriguing phase, here named spin-selective Kondo\ninsulator, where an insulating state is stabilized for the\nminority-spin electrons while the majority-spin electrons\nare still metallic. We believe that the mechanism pro-\nposed here, the dynamically generated commensurabil-\nity, should be generic for the ferromagnetic phase in the\nKondo lattice models. It alternatively provides the non-\ntrivial relation between the electron magnetization, spin\npolarization and occupation number, for which the sys-\ntem can gain a maximum of additional energy. The pro-\nposed relation between the macroscopic quantities might\nbe confirmed in experiments. Good candidates in this\ncontext are ferromagnetic heavy fermion compounds, es-\npecially compounds having a large Kondo temperature.\nFor such compounds a verification of the above stated\nrelation might be possible. Furthermore, spin-resolved\ntransport measurements should show metallic majority-\nspin but insulating minority-spin electrons as well as a\nlarge Fermi surface for the minority-spin component.\nWe acknowledge fruitful discussions with A. Koga.\nRP thanks the Japan Society for the Promotion of\nScience (JSPS) and the Alexander von Humboldt-\nFoundation. TP also gratefully acknowledges support\nby JSPS through the Bridge program. NK is supported\nby KAKENHI (Nos. 21540359, 20102008) and JSPS\nthrough its FIRST Program.\n∗peters@scphys.kyoto-u.ac.jp\n[1]P. Coleman, Handbook of Magnetism and Advanced Mag-\nnetic Materials (John Wiley and Sons, 2007), p. 95.\n[2]P. Coleman and A. Schofield, Nature 443, 226 (2005).\n[3]P. Gegenwart and Q. Si, Nature Physics 4, 186 (2008).5\n[4]S. Doniach, Physica B 91, 231 (1977).\n[5]C. Krellner, S. Lausberg, A. Steppke, M. Brando, L. Pe-\ndrero, H. Pfau, S. Tenc´ e, H. Rosner, F. Steglich, and\nC. Geibel, New Journal of Physics 13, 103014 (2011).\n[6]W. Lee, H. Ku, and R. Shelton, Phys. Rev. B 38, 11562\n(1988).\n[7]N. Perkins, J. Iglesias, M. Nunez-Regueiro, and B. Co-\nqblin, Europhysics Letters 79, 57006 (2007).\n[8]S. Yamamoto and Q. Si, Proc. Natl. Acad. Sci. USA 107,\n15704 (2010).\n[9]G. Li, G. Zhang, and L. Yu, Phys. Rev. B 81, 094420\n(2010).\n[10]C. Lacroix and M. Cyrot, Phys. Rev. B 20, 1969 (1979).\n[11]P. Fazekas and E. Muller-Hartmann, Z. Phys. B: Con-\ndens. Matter 85, 285 (1991).\n[12]W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324\n(1989).\n[13]A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg,\nRev. Mod. Phys. 68, 13 (1996).\n[14]T. Pruschke, M. Jarrell, and J. Freericks, Adv. Phys. 44,187 (1995).\n[15]K. Wilson, Rev. Mod. Phys. 47, 773 (1975).\n[16]R. Bulla, T. Costi, and T. Pruschke, Rev. Mod. Phys.\n80, 395 (2008).\n[17]R. Peters, T. Pruschke, and F. Anders, Phys. Rev. B 74,\n245114 (2006).\n[18]A. Weichselbaum and J. von Delft, Phys. Rev. Lett. 99,\n076402 (2007).\n[19]R. Peters and T. Pruschke, Phys. Rev. B 76, 245101\n(2007).\n[20]J. Otsuki, H. Kusunose, and Y. Kuramoto, J. Phys. Soc.\nJpn.78, 034719 (2009).\n[21]D. V. Efremov, J. J. Betouras, and A. Chubukov,\nPhys. Rev. B 77, 220401(R) (2008).\n[22]H.Tsunetsugu, M. Sigrist, andK.Ueda, Rev.Mod.Phys.\n69, 809 (1997).\n[23]S. Smerat, U. Schollw¨ ock, I. P. McCulloch, and\nH. Schoeller, Phys. Rev. B 79, 235107 (2009)."    },    {        "title": "1305.4843v1.Transport_of_spin_anisotropy_without_spin_currents.pdf",        "content": "arXiv:1305.4843v1  [cond-mat.mes-hall]  21 May 2013Transport of spin-anisotropy without spin currents\nMichael Hell(1,2), Sourin Das(3,1,2), and Maarten R. Wegewijs(1,2,4)\n(1) Peter Gr¨ unberg Institut,\nForschungszentrum J¨ ulich, 52425 J¨ ulich, Germany\n(2) JARA- Fundamentals of Future Information Technology\n(3) Department of Physics and Astrophysics,\nUniversity of Delhi, Delhi 110 007, India\n(4) Institute for Theory of Statistical Physics,\nRWTH Aachen, 52056 Aachen, Germany\n(Dated: September 2, 2018)\nWe revisit the transport of spin-degrees of freedom across a n electrically and thermally biased\ntunnel junction between two ferromagnets with non-colline ar magnetizations. Besides the well-\nknown charge and spin currents we show that a non-zero spin-quadrupole current flows between the\nferromagnets. This tensor-valued current describes the no n-equilibrium transport of spin-anisotropy\nrelating to both local and non-local multi-particle spin co rrelations of the circuit. This quadratic\nspin-anisotropy, quantified in terms of the spin-quadrupol e moment, is fundamentally a two-electron\nquantity. In spin-valves with an embedded quantum dot such c urrents have been shown to result\nin a quadrupole accumulation that affects the measurable qua ntum dot spin and charge dynamics.\nThe spin-valve model studied here allows fundamental quest ions about spin-quadrupole storage\nandtransport to be worked out in detail, while ignoring the detection by a q uantum dot. This\nphysical understanding of this particular device is of impo rtance for more complex devices where\nspin-quadrupole transport can be detected. We demonstrate that, as far as storage and transport\nare concerned, the spin anisotropy is only partly determine d by the spin polarization. In fact, for\na thermally biased spin-valve the charge- and spin-current may vanish, while a pure exchange spin-\nquadrupole current remains, which appears as a fundamental consequence of Pauli’s principle. We\nextend the real-time diagrammatic approach to efficiently ca lculate the average of multi-particle\nspin-observables, in particular the spin-quadrupole curr ent. Although the paper addresses only\nleading order and spin-conserving tunneling we formulate t he technique for arbitrary order in an\narbitrary, spin-dependent tunnel coupling in a way that len ds itself to extension to quantum-dot\nspin-valve structures.\nPACS numbers: 85.75.-d, 73.63.-b, 75.30.Gw\nI. INTRODUCTION\nSpintronics combines the concepts of electronic trans-\nport and spin physics. One of the earliest examples in\nsolid state physics was the tunnel magnetoresistance ef-\nfect, discovered by Julliere in 19751: the charge cur-\nrent through two tunnel-coupled ferromagnets decreases\nwhen their magnetizations are changed from a parallel\nto an antiparallel configuration. The simple explanation\nof Julliere1, based on the spin-dependence of the density\nof states for spin- ↑and↓, has been refined and extended\nby later works. Slonczewski calculated the spin current\nthrough the FM-I-FM junction2, which can be detected\nby a second tunnel junction3. The spin current is respon-\nsible for an exchange coupling between magnetizations of\nthetwoferromagnets2,4). Animportantearlyapplication\nof spin currents is spin injection from ferromagnets into\nnon-magnetic systems, (for a review see5).\nSince then, the frontiers of spintronics have been\npushed more and more towards the nanoscale, in par-\nticular by attaching macroscopic leads to small quantum\ndots. To name only a few interesting effects in which the\ntransport heavily relies on the spin physics, we mention\nthe Kondo effect6,7, Pauli blockade8, and various types\nof spin-blockade effects9. The spintronic features men-tioned for the mesoscopic systems also have a counter-\npart in microscopic quantum dot physics. For instance,\nspin injection into quantum dots and spin currents have\nbeen measured10. Moreover, for non-collinearly magne-\ntized ferromagnets,theabovementionedexchangeeffects\ntranslates into a dipolar exchange field11, which can even\nlift the spin-valve effect.\nBesides these analogies, there are, however, profound\ndifferences when microscopic systems such as quantum\ndots are involved. Due to the spatial confinement of\nelectrons, Coulombelectron-electroninteractionbecomes\nall-important and correlations between electrons play a\nprominent role. Spin correlations are built up due to the\nexchange spin-spin interaction, which results from the\nconcerted action of charging effects and the Pauli princi-\nple. This couples the spin-dipole moments of the individ-\nual electrons to high-spin states ( S/greaterorequalslant1). Such high-spin\nquantum systems have non-trivial higher spin-moments\nbeyondthe averagespin, suchasthe spin-quadrupole mo-\nment(SQM), which is usually the dominant part. In the\nphysical language of atomic and molecular magnetism,\nthe SQM characterizes the quadratic spin anisotropy .\nIt quantifies the preference of pairs of spins that make\nup the large moment S/greaterorequalslant1 to bealignedalong a spe-\ncificaxisirrespective of their orientation along this axis\n(up, down). SQM is also relevant to transport: for ex-2\nample, a spin anisotropy barrier can completely deter-\nmine the signatures of the conductance through molecule\nmagnets12and magnetic adatoms13. However, in these\ndevices the spin anisotropy appears rather as a prop-\nerty “fixed” to the atoms/molecule and not something\nthat could be moved around. This latter idea has been\nintroduced by recent publications14–16, which point out\nthat spin-quadrupole moment, like spin-dipole moment,\ncan beinjectedand accumulated in a high-spin quantum\ndot attached to ferromagnets. Thus, spin anisotropy has\nturned out to be a true transport quantity in some ways\nsimilar to spin-dipole moment. Thus, the transport pic-\nture of spin degrees of freedom needs to be extended be-\nyond that offered by charge and spin currents. This is\nat the heart of this paper, which studies the storageand\ntransport of spin-quadrupole moment in spintronic de-\nvices, merging concepts of spintronics and electron-spin\ncorrelations (for example present in single-moleculemag-\nnets). The aim of this paper is to answer the following\nthree fundamental questions raised by the above cited\nstudies:\ni. How is SQM storedinmacroscopic system, i. e.,\nferromagnets?\nii. How is SQM transported macroscopically between\nsuch reservoirs?\niii. How can one define an SQM current operator and\nwhat is the physical interpretation of its average?\nThe answers are by no means obvious since SQM, un-\nlike charge and spin, is a two-electron quantity. We\ntherefore resort to the simplest possible setting – the\nJulliere model of two tunnel-coupled ferromagnets with-\noutan embedded quantum dot. The idea is to take one\nstep “back”relative to the references14–17and to learn as\nmuch aspossible from this simple spin-valvemodel about\nthe concepts essential to multi-spin transport.\nWe emphasize from the start that we thereby com-\npletely ignore the complications of the measurable effects\nof SQM currents, which seem to occur only when SQM\ncan accumulate in a quantum dot. In the tunnel-junction\nspin-valve the charge current as in14–16does not measure\nthe spin-current, although it displays spin-dependent ef-\nfects. Similarly, this study shows that the charge and\nspin current do not measure the SQM current. Thus,\nour results in no way invalidate results of previous stud-\nies of charge and spin currents; in this simplesetup they\nsimply coexist with the SQM currents. As long as one\nis only interested in the charge current, one can ignore\nSQM currents in this setup. We will therefore not sug-\ngest any concrete “meters” of SQM effects in this paper.\nThese were addressed elsewhere14–16where for instance\nin16the Kondo effect was shown to be sensitive to the\nquadrupolar analogue of the spin-torque.\nStill, the physical insights gained by this study pro-\nvide a sound foundation for the discussion of their coun-\nterparts in more complex, interacting nanoscale devices,\nwhich allow for SQM detection. For this reason, we will\nalso address how SQM transport through the spin-valve\nmay be controlled by various non-linear driving param-eters such as voltage, temperature gradients and mag-\nnetic parameters. Finally, we note that all our results\nare obtained within a modern version of the real-time\ntransport formalism, which we have extended to deal ef-\nficiently with multi-particle spin-degrees of freedom.\nThe paper is structured as follows: in Sec. II we for-\nmulate the spin-valve model and discuss the physical\nsituations it applies to. We define the one- and two-\nparticle densities of states that enter into the results. In\nSec. III we show that simple Stoner ferromagnets pro-\nvide reservoirs of uniaxial spin anisotropy in addition to\nspinpolarization. We introduceaspin-multipole network\npicture extending the idea of a charge and spin trans-\nport network. For multi-electron quantities, such as spin\nanisotropy, this picture is radically different since they\ndescribe local and non-local correlations. In Sec. IV. we\nwill see how this naturally suggests the general defini-\ntion of spin quadrupole current operators. In Sec. V\nthe non-equilibrium averages of these operators are pre-\nsented for our spin-valve model. We discuss the decom-\npositionofthespin-quadrupolecurrentsintoadissipative\npart(spin-quadrupoleinjection/emission)andacoherent\npart (spin-quadrupole torque), similar to the spin dipole\ncurrent. The appendices contain – besides details – a\nsystematic account of some important technical develop-\nments of the real-time transport theory that we employ.\nII. SPIN-VALVE MODEL\nWe start with an overview of the main concepts and\nideas, which are central to our comprehensive analysis,\naimed at answering the three guiding questions posed\nin the introduction. The key to understanding the first\nquestion, i. e., how SQM is stored, is to investigate the\nmicroscopic origin of SQM by considering a system two\ncoupled spin 1/2. This provides a natural link to atomic\nand molecular physics, which will be discussed in IIA.\nNote that we deal here with the spin-quadrupole mo-\nment of a system consisting of electrons and not with the\nelectric nuclear quadrupole moment, which have been\ninvestigated in great detail18.\nWe moreover introduce the Hamiltonian for the spin-\nvalve structure (see Sec. IIB) consisting of two tunnel-\ncoupled ferromagnets, allowing for non-collinear magne-\ntization directions. The ferromagnets are described us-\ning a Stoner model. Importantly, the spin-dependent\none-particle density of states is notsufficient to quantify\nspin-multipole properties of ferromagnets. In IIC, we in-\ntroduce a two-particle density of states (see Eq. (16)),\nwhich is required for the calculation of the average spin-\nquadrupole moment and its current (Secs. IIIA2a\nand V). It can only be calculated if the explicit spin-\ndependence of the dispersion relation is available. For\nall concrete results presented in this paper, we employ\na single, wide, flat band approximation, whose valid-\nity will be discussed in Sect. IID. Throughout we set\n/planckover2pi1=e=c=kB= 1.3\nA. Spin-quadrupole Moment: From Atomic\nPhysics to Spintronics\nTo address the storage of SQM, we consider two elec-\ntrons occupying two different orbitals with the combined\nsystem being in a spin-triplet state. The single-particle\nspin vector operators of these electrons, s1\niands2\ni, add\nup the total spin operator Si=s1\ni+s2\ni(i=x,y,z).\nFrom the operator components of the latter, the spin-\nquadrupole moment tensoroperator Q=/summationtext\nijQijeiej\ncan be constructed,\nQij=1\n2{Si,Sj}−1\n3S2δij, (1)\nwherei,j=x,y,z. In the triplet states |T+∝an}bracketri}ht=| ↑↑∝an}bracketri}ht\nor|T−∝an}bracketri}ht=| ↓↓∝an}bracketri}ht, the average spin dipole moment is non-\nzero:∝an}bracketle{tTm|S|Tm∝an}bracketri}ht=mez, form=±. The average spin-\nquadrupolemomenthasnon-zerocomponentsaswell(see\nApp. A3):\n∝an}bracketle{tT±|Q|T±∝an}bracketri}ht=1\n3ezez−1\n6/summationdisplay\nl/ne}ationslash=zelel.(2)\nSince the largest element of this tensor, given by the\ncomponent ∝an}bracketle{tT± |Qzz|T±∝an}bracketri}ht, is positive, the spins are\nlikely to be alignedwith thez-th axes in state |T±∝an}bracketri}ht\n– irrespective of their orientation . Thus besides spin-\npolarization, spin-quadrupole moment is “ stored” in this\ntwo-electron system. One may object and ask whether\nthe quadrupole moment is not completely determined by\nthe spin-dipole moment since the tensor (2) could be en-\ntirely expressed in terms of ∝an}bracketle{tT±|S|T±∝an}bracketri}ht. However, in a\nquantum system, even without two-particle interactions,\nwe have∝an}bracketle{tSiSj∝an}bracketri}ht ∝ne}ationslash=∝an}bracketle{tSi∝an}bracketri}ht∝an}bracketle{tSj∝an}bracketri}htdue to exchange processes. As\na result a system may be purely “quadrupolarized”, i.e.\n∝an}bracketle{tQ∝an}bracketri}ht ∝ne}ationslash= 0 while ∝an}bracketle{tS∝an}bracketri}ht= 0. An example of this is the triplet\nstate|T0∝an}bracketri}ht=1√\n2(| ↑↓∝an}bracketri}ht+| ↓↑∝an}bracketri}ht) for which the expectation\nvalues of all spin components vanish, ∝an}bracketle{tT0|S|T0∝an}bracketri}ht= 0, but\n∝an}bracketle{tT0|Q|T0∝an}bracketri}ht=−2\n3ezez+1\n3/summationdisplay\nl/ne}ationslash=zelel,(3)\nindicating that this is a “planar” spin state, in contrast\nto the axial spin state (2). In the context of quan-\ntum information, this state is one of the triplet Bell\nstates|Bz∝an}bracketri}ht=|T0∝an}bracketri}ht. The other two Bell states |Bx∝an}bracketri}ht=\n1√\n2(| ↑↑∝an}bracketri}ht−| ↓↓∝an}bracketri}ht ) ,|By∝an}bracketri}ht=1√\n2(| ↑↑∝an}bracketri}ht+| ↓↓∝an}bracketri}ht) further illus-\ntrate that states of zerospin-polarization( ∝an}bracketle{tBk|S|Bk∝an}bracketri}ht= 0\nfor eachk=x,y,z) can be distinguished by their spin\nanisotropy : the latter is quantified by the average of the\nspin-quadrupole tensor (see App. A3), which reads\n∝an}bracketle{tBk|Q|Bk∝an}bracketri}ht=−2\n3ekek+1\n3/summationdisplay\nl/ne}ationslash=kelel.(4)\nSince the largest element of this tensor, ∝an}bracketle{tBk|Qkk|Bk∝an}bracketri}ht,\nis negative in state |Bk∝an}bracketri}ht, the spins lie in the plane per-\npendicular to the k-th axes without any definite orienta-\ntion. Such states appear as eigenstates of bi-axial spinHamiltonians of type H=−DS2\nz+E(S2\nx−S2\ny), which\nare also well-known in molecular magnetism. In general,\nthe average of Qin any triplet superposition state is a\nsymmetric tensor, whose principal values lie in the in-\nterval [−2/2,+1/3]. In fact, a triplet quantum state\nis completely specified by giving the average of boththe\nspin-dipole and the spin-quadrupole moment: formally,\none can show that an arbitrary mixed-state density op-\nerator in the triplet subspace can be decomposed into a\nbases of spin dipole and quadrupole operators14,15. In\nthis sense, the spin-quadrupole moment is thus a degree\nof freedom independent of the spin-dipole moment in any\nsystem of more at least two spins. Quadrupole moments\nare not limited to the spin degree of freedom only. One\nmay define pseudo-spin dipole and -quadrupole operator\nwhenever one deals with a system of at least three levels.\nSuch systems arise, for instance, when combining spin\nand orbital degrees of freedom. Such pseudo-quadrupole\nmoments then express other types of correlations, which\nare inevitably needed to fully characterize the state of\nsuch systems. In this paper we are, however, only con-\ncerned with the spin-quadrupole moment, which is most\nrelevant for spintronics.\nThe above ideas can be extended to one of the ba-\nsic, circuit element of spintronics: a ferromagnetic many-\nelectron system (see Sec. IIIA2a). The average of the\nmacroscopic spin operator Si=/summationtext\nasa\ni, wheresa\niis\nthei-the component of the spin of electron a, quanti-\nfies the magnetization of the ferromagnet. Similar to the\nspin, the macroscopic spin-quadrupole moment can also\nbe decomposed into a sum of microscopic contributions\ncomingfrom electron pairs. By inserting Si=/summationtext\nasa\niinto\nEq. (1), we obtain\nQij=/summationdisplay\na<bqab\nij, (5)\nqab\nij=sa\nisb\nj−2\n3/parenleftbig\nsa·sb/parenrightbig\nδij. (6)\nThe average spin-quadrupole moment thus quantifies\nspincorrelations between all possible electron pairs .\nIt can be shown that Qcaptures the tripletcorrela-\ntions between the spins (see App. A3). Other types\nof spin correlations become important if spin singlet\nstates are additionally considered. This does not only\nconcern spin-singlet correlations, but also correlations\nof Dzyaloshinskii-Moriya type, related to antisymmet-\nric tensors quadratic in the spin, as found in Ref.14.\nFurthermore, observables expressed by higher powers in\nthe spin operators describe spin-multipole correlations of\nhigher rank (e.g., spin octupoles, etc.). Although all of\nthese are of interest, we focus in this paper only on two-\nelectron spin-triplet correlations, which are exclusively\ndetermined by the spin-quadrupole moment, and were\nfound in the simplest possible situation15to be the dom-\ninant spin-multipole moment coupling to the spin-dipole\ndynamics.\nFor a ferromagnet, we will see later that the spin-\ndipolarization induces a spin-quadrupolarization similar4\nto the simple example of two-electrontriplet states |T±∝an}bracketri}ht,\nsee the discussion of Eq. (2). This will become evident\nwhen we identify a classical or directcontribution, which\nis completely determined by the spin polarization. In\naddition, there is a quantum or exchange contribution to\nspin anisotropy, which is independent of spin. The latter\nreveals the two-electron nature of spin-quadrupole mo-\nment and comes as a consequence of the Pauli-principle.\nWe will see that this pure quantum anisotropycan be un-\nderstood as an tensor-valued “Pauli-exlusionhole” in the\ntriplet spin-correlations, accounting for correlations that\nare forbidden by the Pauli principle (see Sec. IIIB4).\nThis in particular makes the spin-quadrupole moment\nan independent degree of freedom that is “stored” in a\nferromagnetin addition to the chargeand the spin-dipole\nmoment. The studies14,15indicate that it is a quantity\nthat must be reckoned with in nanoscale spintronic sys-\ntems with high spin-polarizations.\nWe now turn to the second, central question an-\nnounced in the introduction: howcan one transport spin-\nquadrupolemoment(seeSecs. IVBandV)?Ifwetunnel-\ncoupletwoferromagnetsandapplyafinitevoltagebias,it\niswell-knownthat besidesachargecurrentaspin-current\nwill flow2since electrons carry both charge and spin as\nan intrinsic degree of freedom. But can there also be a\nflowof spin anisotropy? At first sight, one may answer\n“no” because single electrons do not have an intrinsic\nspin-quadrupole moment. However, as an electron spin\ntunnels from one ferromagnet to the other, it retains its\ncorrelations with other electrons. By this, triplet corre-\nlations initially stored locally in one of the ferromagnets\nturn into nonlocal triplet correlations between electrons\nin different ferromagnets. This leads, even by tunneling\nof single electrons, to a nonzero spin-anisotropy current .\nThe aspect of non-locality of SQM is another essential\naspect of its two-particle nature. Even on a macroscopic\nlevel, spin-anisotropytransport can therefore only be un-\nderstood in a network picture accounting for both local\nandnon-local sourcesofSQM.Suchaspin-multipolenet-\nworkpicture–radicallydifferentfromthatforchargeand\nspin – will be developed here. It illustrates that storage\nand transport of SQM cannot be understood indepen-\ndently from each other.\nThesegeneralconsiderationsbringustothethirdmain\nquestion of our paper, namely how to define the SQM\ncurrent operator (see IVB). It can then be identified\nwith the rate of change in SQM stored in these local and\nnon-local sources. To develop a further understanding of\nSQMtransport,weneedtocalculatethe averagecurrents\nfor the simple spin-valve. We will discuss how to decom-\npose the result into various physically meaningful contri-\nbutions. Besides a direct and an exchangepart we find in\nanalogy to the spin-current dissipative and coherent con-\ntributions. The interplay of these contributions causes\nthe SQM current to generate a biaxial spin-anisotropy\nfor non-collinear ferromagnets. This transport of spin-\nanisotropy opens up the interesting possibility to gener-\nate anisotropic magnetic systems starting with isotropicones, in a way similar to creating spin-polarized systems\nby spin-transport. To our knowledge, this has not been\ndiscussed so far, even though the effects of “static” spin\nanisotropy ontransport have been studied extensively in\natomic / molecular magnetism12and spintronics. Based\non this it is expected that spin-quadrupole currents play\na role in many nanoscale spintronics devices with signif-\nicant quantum spin-correlations.\nIn our comprehensive study of the dependence of the\nSQM current on physical parameters we will find a strik-\ning result, highlighting the above mentioned different\nnature of SQM transport as compared to spin trans-\nport. We predict the possibilityofa pure spin-quadrupole\ncurrent, i.e., a quadrupole current not accompanied by\ncharge and spin current. This SQM current is entirely\ndue to quantum exchange processes and is driven by a\ndensity gradient of “Pauli exclusion holes” across the\njunction. A clear notion of the Pauli exclusion holes\nwill be defined in Sec. IIIB4. We find that the spin-\nanisotropy flow direction can be controlled by the direc-\ntion of the thermal bias - a non-trivial result as a deeper\nanalysis of SQM storage will reveal. Transport of spin\ncorrelations is thus possible and controllable without af-\nfecting net spin polarization or charge distribution. This\nremarkable conclusion illustrates most clearly that the\nspin quadrupole moment is really an independent trans-\nport quantity that should be incorporated into spintron-\nics theories. It also indicates possible, promising applica-\ntions: injectionofsuchanSQMcurrentmay,forinstance,\nmodify or even generate spin-anisotropy in an embedded\nsystem without changing its spin-polarization. This may\nperhaps allow for novel ways of performing operation in\nmulti-spin systems.\nB. Spin-valve Hamiltonian\nWe start from a quite general model Hamiltonian\nH=HL\n0+HR\n0+HT (7)\nwith the non-interacting Hamiltonians of subsystem r=\nL,R:\nHr\n0=/summationdisplay\nnkσεr\nnkσc†\nrnkσcrnkσ, (8)\nand the tunneling Hamiltonian with TLR\nσσ′= (TRL\nσ′σ)∗:\nHT=/summationdisplay\nnn′kk′σσ′TLR\nσσ′c†\nLnkσcRnk′σ′+h.c. (9)\nHere the field operators crnkσact on the single particle\nlevelkofbandninsubsystem r=L,Rwithspinσ=↑,↓.\nWeassumethatthesingle-electronspin sforeveryorbital\n(n,k) in the same ferromagnet can be quantized along a\ncommon physical direction ˆJr(with|ˆJr|= 1), i. e.,\n/parenleftBig\n/hatwideJr·s/parenrightBig\n|σ∝an}bracketri}htr=σ|σ∝an}bracketri}htr, (10)5\nFIG. 1: Spin-valve setup: a tunnel junction between (a)\nmacroscopic ferromagnets with Stoner fields Jr=Jr/hatwideJr. (b)\nmesoscopic islands with local magnetic fields Br=Jr/hatwideJr, each\nin equilibrium with energy- and particle- reservoir. A comb i-\nnation of (a) and (b) is another possibility (not shown).\nwith|σ∝an}bracketri}htr=e−iθrˆm·s|σ∝an}bracketri}htezwhere we rotate by the angle\nθrbetween/hatwideJrandezabout the axis perpendicular to\nboth these vectors, defined by ˆm=ez׈Jr/|ez׈Jr|\n. For simplicity, we assume the tunneling amplitudes to\nbe band- (n) and energy-( k) independent; moreover, the\nspin is conserved by the tunneling, [ HT,S] = 0. Never-\ntheless, the tunneling amplitudes in Eq. (9)\nTLR\nσσ′=tL∝an}bracketle{tσ|σ′∝an}bracketri}htR (11)\nare in general spin-dependent because the field oper-\natorscLnkσandcRnkσannihilate electrons with spins\nquantized along non-collinear directions /hatwideJL∦/hatwideJR. The\nspin conservation in the tunneling is reflected by a spin-\nindependence of the “bare” tunneling amplitude t. More\non this can be found in App. E, where we include spin-\nsymmetry breaking tunneling processes in our extension\nof the real-time transport theory.\nWe model the two subsystems as reservoirs, each kept\nin a thermal equilibrium state ρr=e−(Hr\n0−µrNr)/Tr/Zr\nwhereZr= Tre−(Hr\n0−µrNr)/Tris the grand-canonical\npartition function and Nris the particle number opera-\ntor of electrode r. Both electrodes are have fixed electro-\nchemicalpotentials µrandtemperatures Tr, whosegradi-\nents drive the stationary state currents of interest. Notethat even if the tunneling is present, each electrode is\nheld in equilibrium at each instant of time.\nC. Two-Particle Density of States\nIn Sec. III we will calculate the expectation values of\nthe charge and the spin multipoles involving sums over\nthe mode index k. We now indicate which quantities\nparametrize the spin information from the ferromagnetic\nelectrodes. As usual, we take the continuum limit and\nreplace sums over kby a frequency integral. For one-\nparticle quantities such as charge and spin, one can ex-\npresses the results in terms of the spin-dependent one-\nparticle density of states (1DOS):\nνr\nσ(ω) =/summationdisplay\nn,kδ(εr\nnkσ−ω) (12)\n= ¯νr(ω)(1+σnr(ω)), (13)\nwhere ¯νr(ω) is thespin-averaged DOS\n¯νr(ω) =νr\n↑(ω)+νr\n↓(ω)\n2. (14)\nAll the spin-dependence of the 1DOS is contained in the\nspin polarization (of the 1DOS)\nnr(ω) =νr\n↑(ω)−νr\n↓(ω)\nνr\n↑(ω)+νr\n↓(ω). (15)\nImportantly, we will find that the 1DOS (13), although\nformulated for a general one-particle energy spectrum\nεr\nnkσ, is not sufficient to quantify quantum transport of\nspin completely, in particular the spin-spin correlations\ndescribed by the spin-quadrupole moment. We will see\nthat the latter requires an additional, spin-dependent\ntwo-particle exchange DOS (2DOS):\nνr\nσσ′(ω,ω′) =/summationdisplay\nn,kδ(εr\nnkσ−ω)δ(εr\nnkσ′−ω′).(16)\nThe physical meaning of the 2DOS can be understood\nmost easily by considering two identical copiesof the\nsame ferromagnet. The 2DOS is nonzero if there is a\npair of states for an electron with spin σat energyω\nin the first copy and an electron of spin σ′at energy\nω′in the second copy, but within the samek-mode in\nthesamebandn. We emphasize that the latter restric-\ntion requires additional modeling: one cannot make in-\ndependent approximations for the 1DOS and the 2DOS\nsince they are not completely independent of each other:\nfor example, the spin-diagonal components of the 2DOS\nmust satisfy the relation νr\nσσ(ω,ω′) =δ(ω−ω′)νr\nσ(ω).\nYet, the remaining components of the 2DOS, νσ¯σ(ω,ω′),\nwhere ¯σ=−σdenotes the opposite of σ, cannotbe\nconstructed from 1DOS. If the energies of electrons with\nspinσand ¯σare related by a function εr\nnk¯σ=gr\nnσ¯σ(εr\nnkσ)6\n(for example, if the dispersion relation can be solved for\nk), then\nνr\nσσ′(ω,ω′) =νr\nσ(ω)δ(gr\nnσσ′(ω)−ω′) (17)\nwithgr\nnσσ(ω) =ωtrivially. Clearly, more than the 1DOS\nis needed here. As a consequence, in general one has\nto start from the spin dependent dispersion relation and\ncalculate all required components of the 1- and 2DOS\nconsistently. Transport of two-particle transport proper-\nties therefore probes more of the electronic structure of\nthe ferromagnets than the one-particle currents of charge\nand spin.\nD. Stoner Model and Flat Band Approximation\nThe central results of this paper, Eq. (89)-(91), are\nvalid for general case of the above 1-DOS and 2DOS.\nHowever, since we focus on physically understanding\nspin-anisotropy transport, rather than making material-\nspecific predictions, we keep all complications by band\nstructure / dispersion relation features to a minimum.\nStoner model / external magnetic field. As explained\nabove, we must specify the spin-dependence of the dis-\npersion relation for a consistent treatment of the 1DOS\nand 2DOS. We model this by a rigid (i. e., energy-\nindependent) splitting of absolute value Jr\nnbetween the\nspin↑and spin ↓states,\nεr\nnkσ=εr\nnk−σJr\nn/2, (18)\nwhich may be different in each band n. This model can\nbe used to discuss several situations, sketched in Fig. 1\n(a)-(b). In case (a) macroscopicferromagnetsare treated\nwithin the Stoner model. In this case Jr\nncan differ de-\npending on the strength of the electron-electron interac-\ntion in each band. In this case the restriction Jr\nn/greaterorsimilarTr\nmust imposed to avoid the breakdown of ferromagnetism\n(which is not modeled by Eq. (8)). In case (b) we con-\nsider mesoscopic magnetic islands, each in equilibrium\nwith a reservoir. One may now let the Jr\nnmodel an ex-\nternal magnetic field, which may be different locally in\neach electrode, i.e., we identify Jr\nn=Br. The main dif-\nference between (a) and (b) is the relative importance\nof quantum exchange contributions in two-particle spin\nquantities due to the smaller magnetic moment of the\nreservoirs (see below). When considering nanoscopic is-\nlands charging and non-equilibrium effects on the trans-\nport will of course be important, which are neglected\nhere. The main motivation for considering case (b) is\nthat it provides an interesting comparison with results\nfor quantum dot spin-valves where the latter effects are\nfully taken into account14,15. For readability we will dis-\ncuss the results throughout the paper in the language\nof case (a), ferromagnets with Stoner splittings, unless\nexplicitly stated otherwise.\nFlat Band Approximation. We secondly restrict our-\nselves to a single, flat band (sketched in Fig. 2) in eachferromagnet in the limit of large bandwidth 2 Dr= 2D.\nThe latter limit assumes that all other energy scales\n(Tr,Jr,µr) are much smaller than the distance Wof the\nband edge closest to all electro-chemicalpotentials, given\nW:= min\nr,r′,σ,p/parenleftbigg\npD−σJr\n2−µr′/parenrightbigg\n,(19)\nwhich is positive since we assume all µrto lie inside the\nbands. We refer to this in the following shortlyas the flat\nband approximation , keeping in mind that we actually\nrefer to a set of assumptions. For all frequencies ω<W,\nthe spin-dependent DOS is given by\nνr\n↑=νr\n↓=No\n2D, (20)\nwhereNois the total number of orbitals in each subsys-\ntem and\nνr\nσσ′(ω,ω′) =νr\nσ(ω)δ/parenleftBig\nω+σ−σ′\n2J−ω′/parenrightBig\n,(21)\nwhere we used Eq. (18) to rewrite Eq. (17). One may\ncriticize the simplicity of this approximation in that it\ndoes not account for spin-polarization near the Fermi en-\nergy, but only for a Stoner shift, which is only noticeable\nat the band edges. We will see in Sec. VA, however, that\nthis alreadycapturesplenty ofimportantaspects inSQM\ntransport.\nFIG. 2: Spin-dependent density of states of a single flat band\nfor a non-magnetic system (left) and a Stoner ferromagnet\n(right). The Stoner splitting redistributes a fraction of J/4D\nof theNoparticles from spin-down to the spin-up states rel-\native to the non-magnetic case as indicated by the + and −.\nWe omit the electrode and band index for simplicity.\nClearly, the results for the average particle number,\nspin, SQM and their currents have to be independent of\nthe choice of both the coordinate system and the spin-\nquantization axis (spin Hilbert space basis); they may\nonly depend on the physical vectors ˆJrand the scalar\nparameters µr,TrandJr\nnandt(below). A key techni-\ncal result of the paper is that we reformulate real-time\ndiagrammatictransporttheorysuchthecalculationisex-\nplicitly shows this covariance at every stage , which also\nmakes it much more efficient (see App. E).7\nMoreover, the usual modification that implements a\nspin-dependent DOS as νσ= ¯ν(1 +σn) with constant\n¯νandnis only valid as long one deals with single-\nparticleobservables such as the spin (even when account-\ning for many-body effects). For these calculations, all re-\nsults can usually be expressed using the 1DOS. However,\nwhen dealing with two-particle observables relying on the\n2DOS,it is crucialtospecify the dispersionrelationaswe\ndiscussedinSec.IIC.Theabovespin-dependentbut con-\nstantDOSphysicallyarisesfrommixingofdifferenttypes\norbitals in a tight-binding picture, resulting in more than\none band. These additional bands can often be ignored,\nbut this is no longer true for the 2DOS which is sensitive\nto these details. To make this clear, we merely mention\ntwo possible valid alternative models accounting consis-\ntently for a spin-polarization at the Fermi energy: (i) a\nsingle curved band (see Fig. 3 (a)) and (ii) two bands\nwith different bandwidths and a large Stoner splitting\nso thatdifferent bands overlap at the Fermi energy (see\nFig. 3 (b)). Since our single, flat wide band approxima-\ntion is already sufficient to illustrate essential effects of\nspin-quadrupolestorageandtransportwewillnot pursue\nthese band-structure details further here.\nFIG. 3: (a) Spin- and energy-dependent density of states\nfor aone-band Stoner ferromagnet (b) Spin-dependent, con-\nstantdensityofstates for a two-bandStonerferromagnet with\ndifferent bandwidths 2 Dnand Stoner splittings Jnfor band\nn= 1,2.\nIII. SPIN-MULTIPOLE STORAGE\nIn this section, we show that a system of ferromagnets,\neach kept at equilibrium, does not only store charge and\nspin polarization, but also stores spin anisotropy , quan-\ntified by the expectation value of the spin-quadrupole\nmoment operator (1).\nIn Sec. IIIA we will first investigate the simplest case\nof a single ferromagnet at zero temperature. We discuss\nhow the average SQM tensor relates to fluctuations in\na macro-spin picture and relate this to the microscopic\ntriplet spin-spin correlations. We identify an exchangecontribution, which accounts for a “hole” in the quan-\ntum two-particle correlations of the spins due to the\nPauli principle, giving rise to negative or Pauli-forbidden\nanisotropy.\nIn Sec. IIIB we extend these considerations to finite\ntemperatures and multiple electrodes (without coupling\nthem, i.e.,HT= 0), bothofwhichintroducenewaspects.\nThecaseoftwoelectrodesneedstobecarefullyaddressed\nin order to define an SQM current later on: we must un-\nderstand from where andto where SQM flows. It turns\nout that the ferromagnetic electrodes cannotsimply be\nidentified with the sources of SQM and we formalize our\nconsiderationsinaconvenientgeneralspin-multipolenet-\nwork theory in Sec. IIIB2.\nA. Single Electrode at T= 0\nWe first calculate and analyse the average particle\nnumber, spin-dipole moment and spin-quadrupole mo-\nment of an isolated electrode in the simple limit of zero\ntemperature in the approximation of a Stoner-shifted\nflat band (see IID). In this subsection we will omit\nthe electrode index rand band index nand denote by\n∝an}bracketle{t ∝an}bracketri}ht=∝an}bracketle{tψ0| |ψ0∝an}bracketri}httheT= 0 ground state average.\n1. Average Charge and Spin\nWe first review the averagecharge and spin-dipole mo-\nment for later comparison of these one-particle quanti-\nties with the SQM, a two-particle quantity. For zero\ntemperature, all states with energy εkσ/lessorequalslantµbelow the\nelectro-chemical potential µare occupied and all levels\nwithεkσ> µare empty (cf. Fig. 2). Thus, the ground\nstate average of particle number operator\nN=/summationdisplay\nk,σc†\nkσckσ, (22)\ncorresponds to the sum of the green areas below the\nelectro-chemical potential in Fig. 2: with νσ= ¯νwe\nfind\n∝an}bracketle{tN∝an}bracketri}ht=/summationdisplay\nσ¯ν/parenleftBig\nµ+D+σ\n2J/parenrightBig\n(23)\n=No/parenleftBig\n1+µ\nD/parenrightBig\n. (24)\nHereNo= 2D¯νis the number of orbitals in the band-\nwidth 2D. The particle number is independent of the\nStoner splitting Jin this simple approximation.\nThe average of the spin operator,\nS=/summationdisplay\nk,σsσσ′c†\nkσckσ′, (25)\nmeasures the spin- dipolarization of the system, where\n(si)σσ′= (σi)σσ′/2 andσi,i=x,y,zare the Pauli ma-\ntrices. Choosing the coordinate system such that ez=ˆJ,8\nwe obtain for T= 0:∝an}bracketle{tSx∝an}bracketri}ht=∝an}bracketle{tSy∝an}bracketri}ht= 0 and\n∝an}bracketle{tSz∝an}bracketri}ht=1\n2/summationdisplay\nσσ¯ν/bracketleftBig\nµ−/parenleftBig\n−D−σ\n2J/parenrightBig/bracketrightBig\n=1\n2Ns.(26)\nThis equals the difference of the number of spin up and\ndown electrons, i.e., the number of half-filled orbitals\nwith polarized spins,\nNs= ¯νJ=J\n2DNo, (27)\nand corresponds to the difference of the areas under two\nDOS curves below µin Fig. 2.\n2. Average SQM and Spin Anisotropy\nThe average SQM ∝an}bracketle{tQ∝an}bracketri}ht=/summationtext\nij∝an}bracketle{tQij∝an}bracketri}hteiejis a real and\nsymmetric tensor, which can therefore always be diago-\nnalized. With the above choice of the coordinate system\nwithez=ˆJ,∝an}bracketle{tQij∝an}bracketri}htis already diagonal by symmetry with\nrespect torotationsabout ˆJ. The averageofthe non-zero\ntensor operator component\nQzz=2\n3S2\nz−1\n3(S2\nx+S2\ny) (28)\nnow measures the spin anisotropy with respect to the z-\naxis in the ground state: ∝an}bracketle{tQzz∝an}bracketri}ht>0 indicates that the\nspin is aligned (but not necessarily oriented) with the\neasyz-axis, while ∝an}bracketle{tQzz∝an}bracketri}ht<0 indicates an easy-plane con-\nfiguration where the spin preferably lies in the perpen-\ndicularxy-plane. If ∝an}bracketle{tQzz∝an}bracketri}htvanishes, neither alignment\nlongitudinal or transverse to the z-direction is favoured.\nThis is the case, e.g., for a spin-isotropic state for which\n∝an}bracketle{tS2\nx∝an}bracketri}ht=∝an}bracketle{tS2\ny∝an}bracketri}ht=∝an}bracketle{tS2\nz∝an}bracketri}ht; however, it can also be realized\nby states that are anisotropic in xy-plane, for which\n∝an}bracketle{tS2\nx∝an}bracketri}ht ∝ne}ationslash=∝an}bracketle{tS2\ny∝an}bracketri}htwhile∝an}bracketle{tS2\nz∝an}bracketri}ht=1\n2/parenleftbig\n∝an}bracketle{tS2\nx∝an}bracketri}ht+∝an}bracketle{tS2\ny∝an}bracketri}ht/parenrightbig\n. These two\nsituations are thus distinguished by the average of one\nother non-zero SQM tensor components ∝an}bracketle{tQxx∝an}bracketri}htor∝an}bracketle{tQyy∝an}bracketri}ht\n(since/summationtext\ni∝an}bracketle{tQii∝an}bracketri}ht= 0 these are not independent).\nWe now investigate to what extent the average spin\npolarization in a Stoner ferromagnet implies a uniaxial\nanisotropy. Classically, one expects spin polarization to\nalways imply some nonzero spin anisotropy, but the con-\nverse need not be true as our example in Sec. I showed.\nWe now calculate the average SQM in two ways, first\nfocusing an a collective macrospin picture, common in\natomicandmolecularmagnetism, and then disentangling\nit into its microscopic contributions from electron pairs\nrelevant to spintronics.\na. Average Macrospin SQM Thegroundstateofthe\nferromagnet is a maximally polarized pure spin state,\n|ψ0∝an}bracketri}ht=|S,m=S∝an}bracketri}ht(as sketched in Fig. 4).\nThe value of the spin Sis determined from the half-\nfilled orbitals with Nspolarized spins\nS=∝an}bracketle{tSz∝an}bracketri}ht ≈1\n2Ns. (29)\nSince|ψ0∝an}bracketri}htisamaximalspineigenstatetherearenoquan-\ntumfluctuationsinthefirst, longitudinalpartofEq.(28):\nFIG. 4: Schematics of the occupation of the orbitals for an\nelectrode at zero temperature: doubly occupied orbitals fo rm\na zero spin state (Pauli principle) while all spins in the sin gly\noccupied orbitals are parallel, maximizing the total spin ( c. f.\ntext)\n∝an}bracketle{tS2\nz∝an}bracketri}ht=∝an}bracketle{tSz∝an}bracketri}ht2. The second, transverse contribution, how-\never, can be written as S2\nx+S2\ny=S−S+−i[Sx,Sy] =\nS−S++SzusingS±=Sx±iSy. It has a non-vanishing\npart due to the quantum spin commutation relations:\nsinceS+|ψ0∝an}bracketri}ht= 0,∝an}bracketle{tS2\nx+S2\ny∝an}bracketri}ht=∝an}bracketle{tSz∝an}bracketri}ht. TheT= 0 average\nEq. (28) is found to be\n∝an}bracketle{tQzz∝an}bracketri}ht=2\n3∝an}bracketle{tS2\nz∝an}bracketri}ht−1\n3∝an}bracketle{tS2\nx+S2\ny∝an}bracketri}ht (30)\n=2\n3S2−1\n3S, (31)\nThe spin-anisotropy, quantified by the average SQM,\nthus has competing contributions: spin-polarization in-\nduces anisotropy in the z-direction ( ∝S2), but trans-\nverse spin fluctuations tend to suppress it ( ∝S). The\nquantum fluctuations of the spin in the ground state\n“resist” perfect alignment of the spin, despite the max-\nimal spin alignment. In fact, Eq. (31) also holds with\nNs= 1,S= 1/2 in which case the longitudinal term\nis completely cancelled by the transverse fluctuations: a\nspin 1/2 is “so quantum” that it always has zero spin\nanisotropy due to spin fluctuations, in fact, in anystate.\nSince the filled shells do not contribute to the value of S,\nthissuggeststhat ∝an}bracketle{tQzz∝an}bracketri}htatT= 0onlyaccountsfortriplet\ncorrelations between the open shell electrons with paral-\nlel spin. However, a full understanding of the transverse\nfluctuations needs a further refinement of that picture.\nb. Microscopic SQM Storage Above we linked the\nzero-temperature average SQM to the spin anisotropy\nstored in a ferromagnet and related it to its average col-\nlective spin and its transverse quantum fluctuations. We\nwillinvestigatenowhowthesequantumfluctuations tend\nto smear out the spin, reducing the uniaxial anisotropy.\nFor this, we decompose the spin anisotropy into its mi-\ncroscopic contributions from all particles: we start with\nthe longitudinal contribution to Qzzin Eq. (28) and ex-\npress the total spin operator Sz=/summationtext\nasa\nzas the sum of\nthe single- electron spins:\nS2\nz=/summationdisplay\na(sa\nz)2+2/summationdisplay\na<bsa\nzsb\nz (32)\n=/summationdisplay\nkσ1\n4c†\nkσckσ+/summationdisplay\nkk′σσ′σσ′\n4c†\nkσc†\nk′σ′ck′σ′ckσ.(33)9\nThus,S2\nzhas both a one- and a two-electron part.\nWhen averaging, the first term gives/summationtext\nσ1\n4Nσwhere\nNσ= (D+σJ)/2 is the number of orbitals occupied\nwith spinσ. For the two-particle part, we can first treat\nthe electrons as if they were classically distinguishable,\nyielding a contribution whenever the states ( k,σ) and\n(k′,σ′) are occupied. This allows us to factorize the re-\nsulting expression/parenleftbig/summationtext\nσσ\n2Nσ/parenrightbig/parenleftBig/summationtext\nσ′σ′\n2Nσ′/parenrightBig\n=1\n4N2\nsinto\nthe product of averages, i. e., ∝an}bracketle{tSz∝an}bracketri}ht2. We therefore\ncall this a direct (two-particle) contribution . However,\nif (k,σ) = (k′,σ′), we have to be careful: due to Pauli’s\nprinciple, it is forbidden to annihilate electrons in the\nsame state twice, hence we have to excludethis possibil-\nity by a correction term −/summationtext\nσσσ\n4Nσ=−No. We call\nthis the exchange (two-particle) contribution , a denota-\ntion that will become more clear in IIIB3. This yields\naltogether\n∝an}bracketle{tS2\nz∝an}bracketri}ht=1\n4(No+N2\ns−No) =∝an}bracketle{tSz∝an}bracketri}ht2,(34)\nconfirming the result trivially obtained in the macrospin\npicture (since |ψ0∝an}bracketri}htis an eigenstate of Sz). The classical\nintuition is only correct because of a non-trivial cancel-\nlation of a one-particle and “quantum” Pauli-exclusion\nterm on a microscopic level. This importance of this\nsubtlety will become clear later (cf. Sec. IIIB4).\nWe proceed with decomposing the transverse fluctua-\ntions into a one-particle and a two-particle term,\nS2\nx+S2\ny=/summationdisplay\ni=x,y/bracketleftBigg/summationdisplay\na(sa\ni)2+2/summationdisplay\na<bsa\nisb\ni/bracketrightBigg\n(35)\n=/summationdisplay\nkσ1\n2c†\nkσckσ+/summationdisplay\nkk′σσ′1−σσ′\n4c†\nk¯σc†\nk′¯σ′ck′σ′ckσ,(36)\nand averaging over the ground state yields a non-\nvanishing one-particle term1\n2/summationtext\nσNσ, describing trans-\nversesingle-spin fluctuations. For the two-particle part,\nthe direct term vanishes as the individual spins are\nflipped in the modes kandk′so that the ground state\nis not reproduced any more. This agrees with the fact\nthat the averages ∝an}bracketle{tSx∝an}bracketri}ht=∝an}bracketle{tSy∝an}bracketri}ht= 0. However, we must\nagain treat the case k=k′separately: when this mode\nis doubly occupied and we have σ′= ¯σ, the sequence\nof the four field operators together exchanges the spins,\nreproducing the ground state. This gives a correction\n−N↓, which is again due to Pauli’s principle: a con-\nfiguration of two indistinguishable spins and the same\nconfiguration with both spin exchanged cannot be told\napart. This two-electron exchange fluctuations together\nwith the single-spin fluctuations make up for the total\ntransverse fluctuations of the macrospin,\n∝an}bracketle{tS2\nx+S2\ny∝an}bracketri}ht=1\n2No−N↓=∝an}bracketle{tSz∝an}bracketri}ht.(37)\nIf we we next combine the longitudinal and the trans-\nverse term to obtain ∝an}bracketle{tQzz∝an}bracketri}ht, we see that the one-particle\ncontributions drop out:\n∝an}bracketle{tQzz∝an}bracketri}ht=1\n6N2\ns−/parenleftbig1\n6No+1\n3N↓/parenrightbig\n. (38)As the SQM of a spin-1/2 vanishes (cf. the end of Sect.\nIIIA2a), the SQM exclusively measures true two-spin\ncorrelations and does not contain any single-spin infor-\nmation: ThesecondbracketinEq.(38)isapure two-spin\nexchange correction that accounts a kind of “hole” in the\ntriplet correlations. The notion of this “Pauli exclusion\nhole”will be explainedpresciselyinSec. IIIB4. It physi-\ncally arises from exchange contributions bothin∝an}bracketle{tS2\nz∝an}bracketri}htand\n∝an}bracketle{tS2\nx+S2\ny∝an}bracketri}ht29. Eq. (38) can be expressed as\n∝an}bracketle{tQzz∝an}bracketri}ht=1\n3×1\n2Ns(Ns−1). (39)\nThus, in the present case, the SQM counts the number\nof pairs of the parallel spins in different half-filled or-\nbitals. In accordance with the macrospin picture, the\ndoubly occupied orbitals can be simply ignored. How-\never, at finite temperatures, the Fermi edge becomes un-\nsharpandmodesbelowthe theelectro-chemicalpotential\nµalso contribute to SQM. In contrast to the macrospin\npicture, the present microscopic description already in-\ncluded the entire Fermi sea into the description and can\ntherefore be extended to finite temperatures (see Sect.\nIIIB4). For T >0 we will also directly start from Q\nin second-quantized from, which provides a clear way to\ndemonstrate why SQM only senses spin- tripletcorrela-\ntions.\nImportantly, these direct and exchange contribution\nto Eq. (38) scale differently with the number of polarized\nspinsNs=J\n2DNo. For a macroscopic ferromagnet, the\nexchange contribution to the SQM can be be neglected\ndue to the relative unimportance of excluding a single or-\nbital among many. In this case, SQM is entirely induced\nby spin-dipolarization. For Ns→ ∞the SQM per pair of\npolarizedspins hasonlyafinite directcontributionof1 /3\nby Eq. (39), or alternatively, per orbital ( J/2D)2/3. For\nmesoscopic ferromagnetic systems with Ns∼10−100\npolarized spins the exchange corrections start to become\nrelevant, and for magnetic molecular quantum dots in\nmagnetic field Ns∼1−10 and both terms can even be\nof comparable size. In both these cases, the exclusion\nprinciple for a few quantum levels becomes relatively im-\nportant.\nB. Two Electrodes at T >0\nWe now extend the above analysis to two electrodes,\nwhich are, moreover, at finite temperatures TLandTR.\nThis brings in two new aspects. First, in Sec. IIIB1, we\nfind that for finite temperatures that the average SQM\ncannot be expressed anymore in the average spin as for\nT= 0. TheexchangeSQMcontributionisresponsiblefor\nthis difference, quantifying pure quantum contributions\nto the anisotropy as we will see in Sec. IIIB3. This con-\ntribution involvesa two-particleexchangeDOS, which is\nevaluatedanddiscussedinSec.IIIB4. Thisnewquantity\nis used to explain the notion of a “Pauli exclusion hole”\nin the triplet spin correlations, which are encoded in the10\nSQM. This provides the key to understanding how quan-\ntum two-particle exchange processes allow for an SQM\ncurrent in the absence of spin-dipole current, the central\nresult of the paper in Sec. VC.\nThe second new aspect, the subdivision of the sys-\ntem into smaller units, touches upon the seemingly naive\nquestion of how to define an SQM current. Clearly,\nan SQM current cannot quantify the “amount” of spin\nanisotropy that flows througha tunnel barrier as single\ntunneling electrons have zero SQM: this idea only makes\nsense for a one-particle quantity such as charge or spin.\nIn contrast, SQM is a two-particle quantity, i. e., built\nup bypairsof electrons. As the electrons of a pair can\nstay at different sides of the tunnel junction, SQM is not\nonly stored locallyin each ferromagnet, but also non-\nlocallybetween the ferromagnets. The concept ofstorage\nof SQM thus needs to include nonlocal sources ofSQM in\naddition to the local ones discussed so far. In Sec. IIIB2\nwe develop a spin-multipole network theory to aid the\nphysical intuition and which will prove to be very helpful\nfor the discussion of SQM transport later on and which\nhas a wider range of application than the model studied\nin this paper.\n1. Average Charge and Spin\nIn the following we calculate the average charge and\nspin-dipole moment in a more technical way and in some\nmore detail. We illustrate how to rewrite the spin-\ndependent part of expectation values most elegantly in\nterms of expressions independent of the choice of the co-\nordinate system and of the spin quantization axis. This\nservesas agood exampleofthe manipulationswe present\nin App. E where we reformulate the real time diagram-\nmatic transport theory in an explicitly covariant way.\nFirstly, theone-particleoperators(22)forthecharge and\n(25) for the spin (now including the reservoirindex r) are\njointly described by the four-component operator\nRr\nµ=/summationdisplay\nk,σ,σ′(rr\nµ)σσ′c†\nrkσcrkσ′. (40)\nHere (rr\nµ)σσ′=r∝an}bracketle{tσ|rµ|σ′∝an}bracketri}htrdenotes the matrix elements\nofthesingle-particleoperator rµforspinstatesquantized\nalongˆJr. Usingr0= /BDandri=siensures that R0=\nNandRi=Sifori= 1,2,3. We will from hereon\ndistinguish whether the 0-component is included or not\nby using Greek or Latin indices, respectively. Taking the\naverage of Eq. (40) involves\n∝an}bracketle{tc†\nrkσcr′k′σ′∝an}bracketri}ht=fr\n+(εr\nkσ)δrr′δkk′δσσ′(41)\nwith the Fermi function\nfr\n+(ω) =1\ne(ω−µr)/Tr+1. (42)Recasting the sum over all k-modes as an integral over\nall energies by inserting the DOS (see Eq. (13)) yields\n∝an}bracketle{tRr\nµ∝an}bracketri}ht=/summationdisplay\nσ,σ′(rr\nµ)σσ′/integraldisplay\ndωδσσ′νσfr\n+,(43)\nwherewesuppressedthe ω-dependenceforbrevity. Using\nEq. (10), i. e. ( Jr·s)|σ∝an}bracketri}htr=σ|σ∝an}bracketri}htr, we may rewrite\nνr\nσ(ω)δσσ′= ¯νr(ω)r∝an}bracketle{tσ|ˇnr(ω)·ˇr|σ′∝an}bracketri}htr,(44)\nintroducing ˇ r0= /BD/√\n2 andˇr=√\n2sand the\nfour-component vector /hatwidenr=√\n2(1,/hatwideJrnr). The spin-\ndependent part of Eq. (43) can be recast as a trace in\nspin space:\n∝an}bracketle{tRr∝an}bracketri}ht=/integraldisplay\ndω¯νrfr\n+Tr[r(ˇnr·ˇr)].(45)\nThe trace is clearly covariant in the general sense, i.e.,\nform-invariantunderchangesofeitherthecoordinatesys-\ntem and / or quantization axis (it is not related to con-\ncepts from relativity; vectors with four elements are just\nconvenient). We obtain\n∝an}bracketle{tNr∝an}bracketri}ht=/integraldisplay\ndω2¯νr(ω)fr\n+(ω), (46)\n∝an}bracketle{tSr∝an}bracketri}ht=/integraldisplay\ndω2¯νr(ω)sr(ω)/hatwideJr, (47)\nAnalogous to the average occupation number of a single\nlevel at energy ωin (46),fr\n+(ω), we denote\nsr(ω) =fr\n+(ω)1\n2nr(ω) (48)\nin Eq. (47) as the average spin-polarization function\nof electrons at frequency ω, wherenr(ω) is the spin-\npolarization (15). Note that we only needs to use spin\n1/2 operator algebra to calculate the average in Eq. (45)\nin coordinate-free form and the same can be done for all\nthe less transport calculations, see App. E.\n2. Network Picture: Non-Locality\nEqs. (46) and (47) show that each physical electrode\ncorrespondstoasinglesourceofchargeandspin. Wenow\nformalize the concept of particle and spin-dipole storage\nin terms of a network theory , which at first sight may\nseem superfluous. In fact, it will prove to be helpful\nto compare this with the storage and transport of spin-\nquadrupole moment.\nThefollowingconsiderationsareformulatedmorecom-\npactly and hold more generally for a composite system of\nany number of subsystems labeled by an index r. Such\na system may comprise of just two electrodes, each at\nequilibrium, as discussed in this paper (then r=L,R),\nbut it may also include, e.g., strongly interacting quan-\ntum dots out of equilibrium as discussed in14–16and in11\nforthcomingworks. We firstaskhowthetotal chargeand\nspin-dipole moment is distributed over the subsystems.\nThe answer is fairly intuitive for these one-particle quan-\ntities: the total charge (spin) is the sum of the charge\n(spin) stored in each electrode, i. e.,\nRtot\nµ=/summationdisplay\nrRr\nµ. (49)\nWe can simply associate each subsystem shown in Fig. 5\n(a) with a nodeof charge(spin) as depicted in Fig. 5 (b).\nNote that decomposition (49) is even possible if Rtot\nµis\nnotconserved. (We postpone the discussion of the links\nin the network until we defined current operators in Sec\nIV where we complete the network theory.)\nFIG. 5: (a) Physical setup of two ferromagnets and network\npicture for (b) the spin-dipole moment, a one-particle quan -\ntity (like the charge) and for (c) the spin-quadrupole momen t,\na two-particle quantity.\nThis simple correspondence breaks down for SQM.\nWhenweaskhowthis two-particle quantityisdistributed\nover composite system, the answer is radically different.\nWe start from the total SQM of the system, written as\nQtot=Stot⊙Stot(50)\nabbreviating the symmetric, traceless dyadic product of\ntwo vector operators aandbas\n(a⊙b)ij=1\n2(aibj+biaj)−1\n3δija·b(51)\nWe decompose Qtotby inserting Stot=/summationtext\nrSr,\nQtot=/summationdisplay\n/an}bracketle{trr′/an}bracketri}htQrr′, (52)\nwhere∝an}bracketle{trr′∝an}bracketri}htindicates that we sum only over all pairs\nQrr′=Qr′r=grr′Sr⊙Sr′, (53)\nand the factor grr= 1 andgrr′= 2 (r∝ne}ationslash=r′) accounts for\nthe double occurrence of each pair r,r′withr∝ne}ationslash=r′in theexpansion (50). Eq. (53) is symmetric in randr′since\nSr⊙Sr′=Sr′⊙Srand we can write\nQrr′=1\n2grr′(Sr⊙Sr′+Sr′⊙Sr).(54)\nNote that Qrr′is a Hermitian operator and therefore an\nobservablebecause spin operatorsofdifferent subsystems\ncommute: ( Sr⊙Sr′)†=Sr′⊙Sr=Sr⊙Sr′.\nWe now develop a network picture for the SQM by\nassociating to each pairof subsystems ∝an}bracketle{trr′∝an}bracketri}hta single ef-\nfective source or node. For the two-terminal spin-valve\nin Fig. 5(a) that we study, threeSQM-nodes appear in\nthe corresponding network picture of Fig. 5 (c). The\ntotal SQM is stored in two localnodes (QLL,QRR) and\nin onenon-local node (QLR=QRL). The (non)local\nnodes describe spin-triplet correlations between pairs of\nelectrons of the same (different) subsystem(s). This non-\nlocalityofSQMstorageisveryimportantforthe physical\nunderstanding and definition of a SQM current opera-\ntor. It is the injection of SQM currents from these non-\nlocal nodes that drive the measurable local SQM dynam-\nics in embedded quantum dots, as found in Ref.15. For\nthe spin-valve considered here it now becomes clear how\nsingle electron tunneling can transport SQM: first, local\ncorrelations, e. g., in the ∝an}bracketle{tLL∝an}bracketri}ht-node are turned into non-\nlocalcorrelationsin the ∝an}bracketle{tLR∝an}bracketri}ht-node. The transferofSQM\nis then completed by another single-electron tunneling\nevent that re-localizes the pair, but now in the right elec-\ntrode, contributing then to the ∝an}bracketle{tRR∝an}bracketri}ht-node. This picture\nwill be refined once we defined SQM current operators in\nSec. IVB.\n3. Direct and Exchange Contribution to Average SQM\nWe next inquire to which extent the stored SQM is in-\ndependent of the average spin-dipole moment, extending\nthe discussion of Sec. IIIA2b. The average of the SQM\noperatorfor node ∝an}bracketle{trr′∝an}bracketri}htgiven by (53), can be decomposed\nit into a direct and an exchange part using Wick’s the-\norem for the averages of products of field operators (see\nApp. A3 for details):\n∝an}bracketle{tQrr′∝an}bracketri}ht=∝an}bracketle{tQrr′∝an}bracketri}htdir+∝an}bracketle{tQrr′∝an}bracketri}htex.(55)\nDirect SQM . The first possible directcontraction com-\nbines field operators from the same spin operator in\nEq. (30). It can therefore be factorized into the expecta-\ntion values of the spin operators given by Eq. (47):\n∝an}bracketle{tQrr′∝an}bracketri}htdir=/summationdisplay\nkk′σσ′sr\nσσ⊙sr′\nσ′σ′fr\n+(εr\nkσ)fr′\n+(εr′\nk′σ′)(56)\n=∝an}bracketle{tSr∝an}bracketri}ht⊙∝an}bracketle{tSr′∝an}bracketri}ht=qrr′\ndir/hatwideJr⊙/hatwideJr′(57)\nwith\nqrr′\ndir=|∝an}bracketle{tSr∝an}bracketri}ht||∝an}bracketle{tSr′∝an}bracketri}ht|. (58)\nThis direct SQM incorporates the cumulative effect of\nthe energy resolved spin-polarization sr(ω). It quantifies12\nthe uncorrelated contribution of the quantum spins to\nthe spin anisotropy: as intuitively expected, an electrode\nwith a favoured spin direction (polarization) possesses a\nfavoured spin alignment (anisotropy). For a macroscopic\nsystem in equilibrium, the average SQM is dominated by\nthe direct part, which is completely determined by the\naverage spin-dipole moment.\nExchange SQM . For meso- and nanoscopic systems\nthe last statement ceases to be true due to the neglect of\nthe Pauli’s principle in the spin-spin correlations. In the\nsecondexchange contraction field operators of different\nspin operators are contracted, giving a term\n∝an}bracketle{tQrr′∝an}bracketri}htex=δrr′/summationdisplay\nkσσ′sr\nσσ′⊙sr′\nσ′σfr\n+(εr\nkσ)fr\n+(εr\nkσ′),(59)\nwhichaccountsfortrue correlations inthesenseofSpear-\nman’s rank correlation coefficient30. This becomes clear\nwhen rewriting Eq. (59) using Eq. (57):\n∝an}bracketle{tQrr′∝an}bracketri}htex=∝an}bracketle{tSr−∝an}bracketle{tSr∝an}bracketri}ht∝an}bracketri}ht⊙∝an}bracketle{tSr′−∝an}bracketle{tSr′∝an}bracketri}ht∝an}bracketri}ht.(60)\nNote that Eq. (59) involves only onesum overk. Thus,\nthe exchange term indeed scales linearly with the sys-\ntem size in contrast to the direct term (see Eq. (57)) and\ncan be neglected for macroscopic systems (cf. last para-\ngraphin Sect. IIIA2b). Hereit is interestingto consider\nour Hamiltonian as a model for a mesoscopic ferromag-\nnet or a metallic island in a strong external magnetic\nfield, Fig. 1(b). In this case the exchange contribution\nmay even become the dominant part in transport when\nthe spin currentvanishes: then the spin-polarization ∝an}bracketle{tSr∝an}bracketri}ht\nandthereforealso ∝an}bracketle{tQrr′∝an}bracketri}htdirdonotchange, while ∝an}bracketle{tQrr′∝an}bracketri}htex\ndoes. When including a tunnel-coupling between the fer-\nromagnets the transport through the junction correlates\nspins of both systems and non-local exchange SQM cur-\nrentscanindeed arise. For this reason, we keep the ex-\nchange term here and study it in some more detail.\nTensorial structure. Eq. (60) can be expressed as\n∝an}bracketle{tQ∝an}bracketri}htex=−δrr′qrr\nex/hatwideJr⊙/hatwideJr(61)\nwith the positive quantity\nqrr\nex=1\n4/summationdisplay\nk(fr\n+(εr\nk↑)−fr\n+(εr\nk↓))2>0.(62)\n(see App. A2). Clearly, only if εk↑−εk↓≪Tfor allk,\nthe exchange contribution vanishes, i.e., for the Stoner\nmodel ifJ≪T. However, if J <T, each spin-polarized\norbitalkgives a negative correction to the direct spin\nanisotropy. We will refer to this as the Pauli exclusion\nhole, located in orbital kwith a “distribution function”\n(f(εk↑)−f(εk↓))2. We give a microcopic interpetation\nof this below in Sec. IIIB4. A gradient of these Pauli\nexclusion holes across the junction results drives an ex-\nchangeSQM current, which mayeven flow in the absence\nof a spin current, see Sec. VC.\nWe moreover note that the tensor ∝an}bracketle{tQ∝an}bracketri}htexhas the same\nprincipal axes as ∝an}bracketle{tQ∝an}bracketri}htdir(the reason for this is discussedin the following section). Thus, the local SQM ∝an}bracketle{tQrr∝an}bracketri}ht ∝\n/hatwideJr⊙/hatwideJr, has a diagonal representation in any coordinate\nsystem that includes the Stoner field direction, e.g., ez=\n/hatwideJr, with non-zero elements ∝an}bracketle{tQrr\nxx∝an}bracketri}ht=∝an}bracketle{tQrr\nyy∝an}bracketri}ht=−∝an}bracketle{tQrr\nzz∝an}bracketri}ht/2.\nThis shows that the local SQM sources are uniaxially\nanisotropic, and different alignments in the plane per-\npendicular to ˆJrare not preferred. Since the direct con-\ntribution exceeds the exchange contributions, we find as\nexpected ∝an}bracketle{tQrr\nzz∝an}bracketri}ht>0, i.e., an easy-axis anisotropy favoring\nthe collinear orientation of the spins into the z-direction\nover any orientation in the xy-plane,∝an}bracketle{tQrr\nxx∝an}bracketri}ht=∝an}bracketle{tQrr\nyy∝an}bracketri}ht<0.\nThe non-local SQM ∝an}bracketle{tQrr′∝an}bracketri}ht,r∝ne}ationslash=r′, has three\nnon-degenerate principal values: it describes bi-axial\nanisotropy . It has a unique principal axes in which\n∝an}bracketle{tQrr\nzz∝an}bracketri}ht>∝an}bracketle{tQrr\nyy∝an}bracketri}ht>∝an}bracketle{tQrr\nxx∝an}bracketri}ht, i.e., directions perpendicular to\nthe dominant easy axis ( z) are distinguished, see App. B.\n4. Microscopic Picture of SQM Storage\nThe physical meaning of the exchange SQM becomes\ntransparent when revisiting the microscopic picture of\nSQM storage. When calculating the direct SQM by\nEq. (56) one pretends to have to two distinct ferromag-\nnetsrandr′and “counts” triplet correlations by adding\nall cross-correlations between electrons occupying these\ndistinguishable ferromagnets. This procedure gives the\nfull resultforthe non-localSQM(cf. Eq.(59)): for r∝ne}ationslash=r′\n∝an}bracketle{tQrr′∝an}bracketri}ht=∝an}bracketle{tQrr′∝an}bracketri}htdir. (63)\nThis is correct as we we treat the two ferromagnets as\ndistinguishable objects by assumption (the total density\noperator is a direct product).\nThe direct, localSQM (r=r′) also correctly “counts”\nthe local spin anisotropy as long as it concerns correla-\ntions of electrons from different modes k∝ne}ationslash=k′, which are\nalso distinguishable (green lines in Fig. 6). However, this\nprocedure fails for electrons occupying the samemode\nk′=k: a single mode (irrespective of whether being\nsingly or doubly occupied) does not contribute to the to-\ntal SQM (see App. A3). Thus, the local exchange SQM\nhas to cancel the contribution that the direct SQM (56)\nincorrectly ascribes to single modes (indicated by the red\nline in Fig. 6)\nFor establishing an “uncounting” procedure to exclude\nthe single-modeSQM, one may againsimply think oftwo\nidentical, but distinguishable copies of the same mode k\nand calculate the direct SQM generated from all these\nmodes (see Fig. 6). In this picture, exchange SQM rep-\nresents a “spin-anisotropy hole” ascribed to each mode\nand therefore shows a formal analogy to a one-particle\nquantity. This analogy will reemerge when we consider\nthe transport of SQM in Sect. IIIB5. To emphasize this\nmulti-particle exchange aspect, we will refer to this as a\nPauli exclusion hole in the spin triplet correlations.\nAs a consequence, local exchange SQM must have the\nsame tensorial structure as the direct SQM, but with13\nFIG. 6: Microscopic contributions to the local SQM ∝angbracketleftQrr∝angbracketright.\nTwo copies of the same ferromagnet are considered and green\nlines indicate correlations between pairs of distinguisha ble\nelectrons in different orbitals (counted by the direct SQM).\nThe red line indicates the correlations between indistingu ish-\nable electrons in the same orbital that the direct SQM counts\ntoo much : according to Pauli’s principle two electrons can-\nnot form a triplet state in the same orbital. The exchange\nSQM contribution takes care of this and thus represents a\nPauli hole in the correlations, corresponding to the red lin e.\nWhen considering only the 1st copy at finite temperature,\nthe macrospin picture discussed in Sect. IIIA2a is recov-\nered. For finite temperature, the occupation probabilities are\nthermally smeared at the Fermi edge.\nopposite magnitude, which is explicitly conveyed by the\nnegative sign in Eq. (61). Since qrr\ndir>0 (by Eq. (58)),\nit follows also that qrr\nex>0 must hold. This is confirmed\nexplicitly by Eq. (62), which shows that the exchange\nSQM senses the spin alignment, a non-negative quan-\ntity that accumulates when summing over all energies\nork-modes, respectively. This prohibits cancellations of\nsigned contributions as they occur in the spin-dipole mo-\nment. This means that spin-dipole moment may cancel\nwhence SQMs do not. Eq. (62) also shows explicitly that\nexchange corrections become negligible at high tempera-\ntures, i. e., if Tr≫εr\nk↑−εr\nk↓for allk, as expected.\n5. Energy-Resolved Exchange SQM Storage\nSofar, itwashelpfull todiscussthemicroscopicpicture\nof SQM storage in terms of contributions from orbitalsk.\nHowever, to make progress in calculations we replace the\nk-sums by energy integrals. An energy-resolved picture\nof SQM storage will therefore be important for under-\nstanding the key features of SQM transport compared to\ncharge and spin see Sec. VC. For the rest of this chap-\nter, we will only discuss the local exchange SQM, i. e.,\nr′=r, and therefore drop the electrode index for brevity.\nReplacing the sum over kin Eq. (62) by integrals over\nfrequencies ω, ω′and inserting the two-particle density\nof states (16), we can recast the exchange SQM into the\nform of Eq. (59) after carrying out the spin sum (seeApp. A). The SQM exchange magnitude then reads as\nqex=/integraldisplay\ndω¯ν(ω)qex(ω) (64)\nTheaverage exchange spin-quadrupolarization for elec-\ntrons at frequency ω,\nqex(ω) =f+(ω)a(ω), (65)\nwith the spin-anisotropy function\na(ω) =/summationdisplay\nσaσ(ω), (66)\nand\n¯ν(ω)aσ(ω) =/integraldisplay\ndω′f+(ω′)/summationdisplay\nσ′σσ′\n4νσσ′(ω,ω′).(67)\nvalid for general dispersion relations. Note that the in-\ntegrand in (64) is nota positive function, in contrast to\neach term in Eq. (62). For the discussion of the SQM\ncurrents, it will be important to understand the mean-\ning of the function qex(ω): it quantifies the cumulative\nexchange triplet correlation for electrons occupying the\nsame orbital. It is the formal analogue to the average\nspin-polarization function s(ω). To link the above re-\nsult further to the microscopic picture developed in Sec.\nIIIB4 and to simplify the interpretation of the exchange\nSQM current in Sec. VA, we decomposed the spin-\nanisotropy function a(ω) into its spin-dependent contri-\nbutionsaσ(ω): they give the direct single-mode SQM,\nprovided that an electron with spin σis present at fre-\nquencyωin the first copy while summing over the con-\ntributions from the second copy in Fig. 6 (cf. App. A4).\nThis reveals the formal anlogy between a(ω) and average\nspin-polarization function in Eq. (48), given by n(ω)/2.\nThelatterquantifiestheaveragespinatfrequency ω,pro-\nvided we have full occupation is at this frequency. How-\never, in stark contrast to the latter, a(ω) is not solely a\nband structure property as it depends on a Fermi func-\ntion due to its two-particle origin. Note that the ex-\nchange SQM is entirely described by qex(ω) and that the\nspin polarization s(ω) does not enter, in contrast to the\ndirect SQM.These twofunctions haveverydifferent tem-\nperature and energy dependence, again making explicit\nthat the SQM is independent of the spin-polarizationdue\nto the presence of exchange terms.\nThe functions qex(ω) anda(ω) are of key importance\nfor the results of this paper and we will therefore explain\ntheir basic physical meaning using the simple Stoner\nmodelεkσ=εk−σJ/2 and the flat band approxima-\ntion (cf. IID). Then the spin-anisotropy function a(ω)\nhas the spin-resolved contributions\naσ(ω) =1\n4[f+(ω)−f+(ω+σJ)].(68)\nIn Fig. 7 we plot these two contributions and their sum\ntogether with the average spin quadrupolarization qex.14\nFIG. 7: Average local exchange quadrupolarization qex(ω)\n(blue), spin anisotropy a(ω) (red) and its two contributions\na↑(ω)>0 (broken line) and a↓(ω)<0 (broken line) as\nfunction of ( ω−µ)/Jfor two temperatures T/J= 0.1 in\n(a) and T/J= 0.5 in (b). As Tapproaches Jfrom below,\nthe weight of a↑(ω) (a↓(ω)) considerably shrinks (rises) and\nqex(ω) is strongly suppressed. See also Sect. IIIB6.\nWe first discuss the shape of aσ(ω) forσ=↑,↓for\nlow temperature 4 T/J <1 translating the arguments\nof the microscopic picture of Fig. 6 into energy space in\nFig. 8. As mentioned, the function aσ(ω) characterizes\nthe single-orbital SQM for a spin σoccupying a mode at\nenergyωand displays four regimes (in the following ∼\nmeans ‘”up to thermal smearing T”). These are marked\n(a)-(d) in Fig. 7 (a) and correspond to the regimes in\nFig. 8. We discuss them now in detail:\n(a)ω/greaterorsimilarµ+J⇒a↑(ω) =a↓(ω) = 0: there are no\noccupied states at energy ω, so no exchange correction is\nneeded.\n(b)µ/lessorsimilarω/lessorsimilarµ+J⇒a↓(ω)<0,a↑(ω) = 0: if a\nspin-↑is in the first copy, the corresponding mode in\nthe second copy has vanishing probability to be occupied\nwith electrons of anyspin since both εk↑=ω/greaterorsimilarµand\nεk↓=ω+J/greaterorsimilarµ. Thus,similartoregime(a),noexchangecorrection for spin- ↑electrons is needed and we obtain\na↑(ω) = 0. In contrast, if a spin- ↓is in the first copy, the\ncorresponding mode in the second copy is predominantly\noccupied with spin- ↑becauseεk↓=ω/greaterorsimilarµ, butεk↑=\nω−J/lessorsimilarµ. This contributes negatively to direct SQM,\nresulting in a↓(ω)<0.\n(c)µ−J/lessorsimilarω/lessorsimilarµ⇒a↑(ω)>0,a↓(ω) = 0: if in\nthis case a spin- ↑is in the first copy, the corresponding\nmode in the second copy is also mostly occupied with\nspin-↑sinceεk↑=ω/lessorsimilarµ, butεk↓=ω+J/greaterorsimilarµ. This\ngivesapositivecorrectionto the directSQM. In contrast,\na↓(ω) = 0 asεk↓=ωandεk↑=ω−Jrefers to a mostly\ndoublyoccupied mode in the second copy, which has a\nvanishing direct SQM contribution (cf. case (d)).\n(d)ω/lessorsimilarµ−J⇒a↑(ω) =a↓(ω) = 0: the correspond-\ning orbital deep inside the Fermi sea is doubly occupied:\nf(εk↑) =f(εk↓) = 1. By Eq. (62) the direct SQM due\nto both spin- ↑and spin- ↓-electrons cancel each other.\nFIG. 8: Microscopic picture of the spin-quadrupolarizatio n\nfunction aσ(ω),σ=↑,↓characterizing the energyresolved\nspin-anisotropy content of a ferromagnet (cf. Fig. 6), see t ext.\nAltogether, the anisotropy function a(ω) =/summationtext\nσaσ(ω)\nisexactly antisymmetric with respectto theelectrochem-\nical potential (see Fig. 7 and App. A3)\na(µ+ω) =−a(µ−ω). (69)\nAs mentioned at the outset, the average exchange\nquadrupolarization qex(ω) =f+(ω)[a↑(ω) +a↓(ω)] has\nboth positive and negative contributions; however, qex>\n0 as the integrated qex(ω) in Eq. (64) is always positive\nby Eq. (62). At T= 0 only positive correlationsat ω<µ\ncount, and we recover the result (37). For T >0, ther-\nmally excited spins ↓negatively correlate with spins ↑in\nthe same orbital, thus reducing qex(see Fig. 7). Even-\ntually atT≫Jthis cancellation reduces qexexactly to\nzero without ever becoming negative. We now see ex-\nplicitly that the exchange SQM only becomes thermally\nsuppressed for T≫J.\nThe average exchange quadrupolarization makes ex-\nplicit that Pauli-forbidden triplet correlations are stored\nby electrons in an energy window ∼2Jwith opposite\nsigns above and below the Fermi energy. Thus, the in-\ntegrated exchange quadrupolarization is thermally sup-\npressed for T≫Jwhen the occupation probability is\nnearly constant across the energy scale J.15\n6. Parameter Dependence of Average Exchange SQM\nIn the flat band approximation (cf. Sect. IID), the\nintegral (64) can be carried out yielding (see App. A4)\nqex=¯νT\n2/bracketleftbiggJ\n2Tcoth/parenleftbiggJ\n2T/parenrightbigg\n−1/bracketrightbigg\n,(70)\nwhich is positive since xcoth(x)>1, in agreement with\nthe above discussion. In the limit J/T→0,qexvan-\nishes as it should (see above) and in the opposite limit of\nT/J→0,qexscales linearly with Ns, the number of free\nspins in the ferromagnet (cf. Eq. (27)),\nqex|T=0=1\n4¯νJ=1\n4Ns, (71)\nin accordance with the T= 0 result (31)31. The av-\nerage one-particle spin-dipole moment ∝an}bracketle{tSz∝an}bracketri}htT=0=1\n2Ns\nthus basically serves as a reference scale for two-particle\nqex(when multiplied by /planckover2pi1= 1 in our units). The low\nT <Jbehavior is most interesting because in the regime\nJ < Tthe results do not apply to a Stoner ferromag-\nnet, for which the self-consistent magnetization Jwould\nbreakdown (ourmodel Fig. 1(a) assumesfixed J). How-\never, considering our model as a description of meso-\nscopic islands in an external magnetic field of strength\nB=J(Fig. 1 (b), this range may also be relevant. With\nthis in mind we show in Fig. 9 the pronounced temper-\nature dependence of the exchange SQM (normalized to\nthe spin polarization) over the entire range for fixed J.\nThisshouldbecontrastedwiththeaveragespinforwhich\nallT-dependence completely cancels out due to the con-\nstancy of the assumed DOS. As already anticipated in\nSec. IIC, a two-particle quantity, the SQM, probes more\nof the electronic structure of the ferromagnets than the\none-particle charge and spin-dipole moment do.\n0.0 0 .5 1 .0 1 .5 2 .0\nT/J0.10.20.30.40.5qex//angbracketleftSz/angbracketright\nFIG. 9: Magnitude of the local exchange SQM, qex, normal-\nized to∝angbracketleftSz∝angbracketright=Ns/2 as a function of temperature T/Jfor\n¯ν=No/2D,D= 25J,µ= 0.\nIn Fig. 10, we plot the dependence of the exchange\nSQM (70) on the Stoner splitting J, illustrating thatwhile|∝an}bracketle{tS∝an}bracketri}ht|scaleslinearlywith J,∝an}bracketle{tQ∝an}bracketri}htexinitially increases\nquadratically and then approaches a linear asymptote:\nqex=¯νT\n4/braceleftbigg\n(J/T)2/6J/T≪1\n(J/T)−2J/T≫1.(72)\nFIG. 10: Magnitude of the spin ∝angbracketleftSz∝angbracketright ≈Ns/2 =NoJ/2D\n(green)(Eq.(26))andtheexchangeSQM qex(Eq.(70))(blue)\nas a function of Stoner splitting J/Tfor ¯ν=No/(2D),D=\n25T,µ= 0.\nIV. SPIN-MULTIPOLE CURRENT\nOPERATORS\nWe now turn to the third central question posed in the\nintroduction - the proper definition of the SQM current\noperator. In the previous section, we answered the im-\nportant question fromwhere and towhere SQM can be\ntransported in terms of nodesin a spin-multipole net-\nwork, cf. IIIB2. We now investigate the linksbetween\nthe nodes in this network, which correspond to the SQM\ncurrent operators as noted at the end of Sec. IIIB. Their\ndefinition and physical interpretation requires some care\nbecause (i) like the spin, the total SQM is not conserved\ninadevicecomprisingStonerferromagnetsand(ii)unlike\nthe spin, this two-particle quantity does not flow directly\nbetween local nodes, but is buffered in non-local nodes.\nTo tackle point (i), we first revisit the one-particle\ncharge and spin current operator. The spin-dipole cur-\nrent does not have the intuitive definition similar to the\ncharge current (total outgoing current = rate of loss of\ncharge) since the spin is notconserved internally in the\nferromagnets. Starting from continuity equations in in-\ntegral form, the spin currents rather have to be defined\nas the change in spin induced by the tunneling . In close\nanalogy, we deriveSQM current operatorsobeying a con-\ntinuity equation and current conservation laws.\nDue to point (ii), we have also consider SQM current\noperators, accounting for the flow of SQM between local16\nand non-local nodes. These turn out to be of central\nphysical importance and reflect that on a microscopic\nlevel SQM is carried by pairs of correlated spin dipoles.\nTheflowofspinanisotropyinanelectronicsystemisthus\ninherently a two-particle process. We will see that, as a\nresult, the layout of the physical device and the network\nfor SQM transport are different: the 2-terminal spin-\nvalve requires a serial 3-node SQM network. For more\ncomplex devices even the connectivity is different19.\nA. Charge and Spin-Dipole Current\nThe physical quantities of interest are the rates of\nchange in local quantities in the physical subsystems of\nthe circuit due to transport processes. For one-particle\nquantities such as charge and spin the physical subsys-\ntems are in one-to-one correspondence with the nodes of\nthe charge / spin network, cf. Fig. 5. The time deriva-\ntive operator ˙Rr\nµ(t) giving the rate of change in the com-\nbinedcharge-spinone-particleoperator Rr\nµ(cf. Eq.(40)),\nd\ndt∝an}bracketle{tRr\nµ(t)∝an}bracketri}ht=∝an}bracketle{t˙Rr\nµ(t)∝an}bracketri}ht, is given by\n˙Rr\nµ(t) :=i[H,Rr\nµ], (73)\nexploiting the von-Neumann equation ˙ ρ(t) =−i[H,ρ(t)]\nand the cyclic invariance of the trace tr( Rr\nµ˙ρ(t)) =\ntr(i[H,Rr\nµ]ρ(t)). We next decompose the total system\nHamiltonian Hinto the part describing the decoupled\nsubsystems, H0, and the tunneling HT=/summationtext\n/an}bracketle{trs/an}bracketri}htHrs\nTwith\nHrs\nTonly accounting for tunneling processes between a\npair of nodes rands. This yields a continuity equation\nin integral form for operators,\n˙Rr\nµ(t) =˙Rr\nµ(t)|0+/summationdisplay\ns/ne}ationslash=rIrs\nRµ, (74)\nwhich decomposes the total rate of change in the charge\n(spin) operator in node rinto two physically meaningful\ncontributions: The first contribution to Eq. (74) is given\nby\nRr\nµ|0=i[H0,Rr\nµ] (75)\nand accounts for the time-evolution due to internal pro-\ncesses in node r. We will depict this contribution in the\nnetwork picture in Fig. 11 by an external arrowattached\nto noder. The second part Ir\nRµ=/summationtext\ns/ne}ationslash=rIrs\nRµ=i[HT,Rr\nµ]\nquantifies the rate of change induced by tunneling, i. e.,\nthisdefines the current of observable Rr\nµintonoder. In\nthe form of Eq. (74), it has already decomposed into its\nvarious contributions emanating from all other subsys-\ntemss:\nIr\nRµ=/summationdisplay\ns/ne}ationslash=rIrs\nRµ, (76)\nWhenever the operatorIrs\nRµ∝ne}ationslash= 0, we depict this in the\nnetwork picture by an arrow inking the two nodes rands. Note that still the averagecurrent∝an}bracketle{tIrs\nRµ∝an}bracketri}htthat flows be-\ntween the nodes may vanish, e. g., for a some special set\nof parameters. So far, our considerations are quite gen-\neral and also apply to systems including quantum dots.\nFIG. 11: Network picture for the spin with current opera-\ntorsrepresented by links. The network picture for charge is\nsimilar, but without external arrows pointing to the nodes\n(indicated by ˙Sr|0). The latter must be introduced due to\nthe internal violation of spin conservation in each spin-no de\nIn our model (Eqs. (7)-(9)), the particle number Rr\n0=\nNris conserved internally in each electrode individually\nand therefore\nNr|0= [H0,Nr] = 0, (77)\nwhich is the 0-component of Eq. (75) and ˙N=/summationtext\nαIα\nN.\nThese make up for the total change in charge. The spin\nRr\ni=Sr\ni(i∝ne}ationslash= 0), however, is notconserved internally in\nthe ferromagnets:\n˙Sα=˙Sr|0+Iα\nS, (78)\nwhereIα\nSis the operator for spin-current into node α.\nUsing Eq. (8), one finds\n˙Sr|0=/summationtext\nnJr\nnˆJr×Sr\nn∝ne}ationslash= 0 (79)\nforSr\nn=/summationtext\nkσsr\nσσ′c†\nrnkσcrnkσbeingthe contributionfrom\nbandnto the spin of electrode r. This describes a pre-\ncession of Sr\nnabout the Stoner field of electrode r.\nTo obtain an explicit starting point for the real-time\ncalculation of the average charge and spin current (see\nApp. E), we use Eq. (75) and Eq. (9) and recover the\nfamiliar form of the charge and spin current operators,\nIrs\nRµ=/summationdisplay\nnn′kk′σσ′(−irµT)rs\nσσ′c†\nrnkσcsn′k′σ′−h.c.,(80)\nabbreviating the matrix product in spin-space\n(rµT)rs\nσσ′=/summationtext\nτ(rµ)στTrs\nτσ′. The operator (80) de-\nscribes the net current injected from node sintor,\naccounting for tunneling processes from node stor(the\nfirst contribution in Eq. (80)) and the reverse process\n(the second). Only the sum of both terms is a Hermitian\noperator and therefore a physical observable. Since both\nprocesses contribute with an opposite sign to the current\n(80), we obtain the antisymmetry relation Irs\nRµ=−Isr\nRµ.\nThis has an important physical consequence: summing\nup all charge (spin) currents in the system yields the\nzero operator:\n/summationdisplay\nr/summationdisplay\ns/ne}ationslash=rIrs\nRµ= 0. (81)17\nThis charge (spin) current conservation law expresses\nthat charge (spin) is conserved by tunneling , that is\n[HT,Rtot\nµ] = 0. Since the total spin is not conserved\nunder the full time evolution (due to the internal evolu-\ntion˙Stot|0∝ne}ationslash= 0), there is no analogue of Eq. (81) for the\ntotal time derivative ˙Sr. We emphasize furthermore that\nthis conservation law holds on an operator level and not\nonly for expectation values.\nB. Spin-quadrupole Current\nWe now try to proceed analogously for the SQM net-\nworkin Fig.5ofSec.IIIB.Generally,weareinterestedin\nfinding the rate of changein the spin anisotropystored in\nlocal nodes. To this end, we need to consider the change\nin SQM, ˙Qrr′\n, in both the local nodes ( r=r′) and the\nnon-local nodes ( r∝ne}ationslash=r′). Taking the time derivative of\nEq. (54) and using Eq. (74) we obtain:\n˙Qrr′\n=˙Qrr′\n|0+/summationdisplay\n/an}bracketle{tss′/an}bracketri}ht/ne}ationslash=/an}bracketle{trr′/an}bracketri}htIrr′,ss′\nQ,(82)\nwhere∝an}bracketle{tss′∝an}bracketri}htdenotes the sum over pairsof indicesss′(i.e.,\nignoring their order). This is the continuity equation in\nintegral form for the change in SQM in node ∝an}bracketle{trr′∝an}bracketri}ht. The\nfirst term is the change in SQM due to the internal time\nevolution\n˙Qrr′\n|0=grr′\n2/bracketleftBig\n˙Sr|0⊙Sr′+Sr⊙˙Sr′|0+(r↔r′)/bracketrightBig\n,(83)\nwhich involves the non-zero internal time evolution ˙Sr|0\ngiven by Eq. (79). Like the spin, the SQM is thus not\nconserved in any of the nodes in our model. The respon-\nsible Stoner fields also effectively exert a “torque” on the\nSQM, thereby rotating the principal axes of this tensor.\nSimilar to the spin, we depict this in the network picture\n(Fig. 12) by one-sided arrows pointing at this node.\nFIG. 12: Network picture for SQM including the links repre-\nsenting SQM current operators . Similar to spin, SQM is not\nconserved internally in the nodes giving rise to external SQ M\ncurrents.\nThe SQM current Irr′\nQ=/summationtext\n/an}bracketle{tss′/an}bracketri}ht/ne}ationslash=/an}bracketle{trr′/an}bracketri}htIrr′,ss′\nQis given\nby the Hermitian tensor operator\nIrr′\nQ=grr′\n2/bracketleftBig\nIr\nS⊙Sr′+Sr⊙Ir′\nS+(r↔r′)/bracketrightBig\n.(84)This is a central result of the paper. Since the spin and\nthe spin current do in general not commute as operators\non Fock space32, the individual terms in this expression\nare not Hermitian and therefore not observables. The\noperator (84) reflects that in general the average SQM\ncurrentis not simply the product ofspin and spin current\nsince∝an}bracketle{tIr\nSiSr′\nj∝an}bracketri}ht ∝ne}ationslash=∝an}bracketle{tIr\nSi∝an}bracketri}ht∝an}bracketle{tSr′\nj∝an}bracketri}htdue to quantum mechanical\nexchange correlations, interactions, etc. Therefore, SQM\nis not determined by spin-dipole moment: it requires a\nseparate description in spintronics transport theory. For\nthe bilinear tunnel coupling (9), the contribution to the\nnet current from node ∝an}bracketle{tss′∝an}bracketri}htinto in node ∝an}bracketle{trr′∝an}bracketri}htis\nIrr′,ss′\nQ=grr′\n2/bracketleftBig\nIrs\nS⊙Sr′δr′s′+Srδrs⊙Ir′s′\nS/bracketrightBig\n+(r↔s′)]. (85)\nNotably, this SQM current is zero unless one of the in-\ndicess,s′match the indices r,r′. This puts an important\nrestriction on the network connectivity: the local SQM\nnodesare onlylinkedtonon-localnodes, andnottoother\nlocal nodes. Changes of local spin-anisotropy,\n˙Qrr=˙Qrr|0+Irr\nQ, (86)\nwhich are due to ransport thus only occur through\nchanges in non-local spin correlations:\nIrr\nQ=/summationtext\ns/ne}ationslash=r(Irs\nS⊙Sr+Sr⊙Irs\nS),(87)\nwhereIrs\nSis the spin-current operator from node sinto\nr.\nFIG. 13: Illustration of the microscopic picture of SQM tran s-\nport. (a)Considertwoelectronsinelectrode Lthatcontribute\nto the SQM of the local node ∝angbracketleftLL∝angbracketright. (b) By transferring one\nof these electrons to subsystem R, the local spin-spin cor-\nrelation is lost but a new non-local correlations are estab-\nlished, increasing the SQM of node ∝angbracketleftLR∝angbracketright. Thus, emitting\nspin-polarized electrons non-local correlations are set u p in\nthe circuit. (c) When the second electron follows the first,\nlocal SQM correlations are created, but now in subsystem R\n(node∝angbracketleftRR∝angbracketright). Note that this picture is only meant to illus-\ntrate the non-locality aspect of SQM, but incorrectly portrays\nthe spins as distinguishable objects.\nAll the considerations so far in this section were are\ngeneral. For the simple spin-valve we consider n this pa-\nper the general theory above implies that SQM cannot\nbe directly transferred from the local node ∝an}bracketle{tLL∝an}bracketri}htto the\nlocal node ∝an}bracketle{tRR∝an}bracketri}ht; it is rather first “buffered” in the in-\ntermediate, non-local node ∝an}bracketle{tLR∝an}bracketri}ht. This restriction on the\nSQM network connectivity is related to the real-space\npicture of SQM transport sketched in Fig. 13. This\npicture unveils why SQM transport is possible even in\nthe single electron transport limit (leading order in HT),18\nas discussed for spin-values with embedded spin- isotropic\nquantumdots14–16. Inthiscasewehaveontherighthand\nside of Eq. (86) ˙Qrr|0= 0 wherernow labels the quan-\ntum dot embedded in this spin-valve. Then SQM cur-\nrents are responsible for the change in the local QD spin-\nanisotropy of a subsystem r, i. e.˙Qrr=Irr\nQwhereIrr\nQ\nofEq.(87) wasalreadyobtainedin Ref.15foraspin-valve\nwith an embedded quantum dot, however, without using\nthe new network picture. The calculations in15demon-\nstrate that in such more complicated devices the SQM\ncurrentIrr\nQ, averagedover the non-equilibrium state, de-\npends non-trivially on the average accumulated charge\nand spin of the dot. The SQM current thus couples to\nmeasurable charge and spin currents and should in gen-\neral be considered for the description of spin and charge\ndynamics.\nAnalogous to the charge and spin current the SQM\ncurrent (85) is antisymmetric with respect to the node\nindices, i.e., when the two pairs of subsystem-indices rr′\nandss′are interchanged: Irr′,ss′\nQ=−Iss′,rr′\nQ. (Note\nthat the order of indices denoting a pair does not matter,\ni.e.,rr′=r′r). As a consequence, the SQM currents sum\nto the zero tensor operator:\n/summationdisplay\n/an}bracketle{trr′/an}bracketri}htIrr′\nQ=/summationdisplay\n/an}bracketle{trr′/an}bracketri}ht/summationdisplay\n/an}bracketle{tss′/an}bracketri}ht/ne}ationslash=/an}bracketle{trr′/an}bracketri}htIrr′,ss′\nQ= 0.(88)\nSimilar to spin, this SQM current conservation law ex-\npresses the conservation of total circuit SQM (50) in\nthetunneling . This is a direct consequence of the to-\ntal spin-dipole conservation by tunneling:/bracketleftbig\nHT,Qtot/bracketrightbig\n=\nStot⊙[HT,Stot]+[HT,Stot]⊙Stot= 0.\nFinally, we note that the restriction on the network\nconnectivityfoundabovederivesfromtheparticularform\nof our tunneling Hamiltonian (9), which is bilinear in the\nfield operators. The network thus describes the connec-\ntivity on the operator level. The topologyofthis network\nis different when HTis, e.g., an effective exchange cou-\npling that is quartic in the fields. Since such a coupling\nis usually derived from the bilinear tunneling model (9)\nstudied here, we will not dwell on this further.33\nV. AVERAGE SPIN-MULTIPOLE CURRENTS\nIn this section we complete the discussion of the third\nmain question posed in the introduction: we present ex-\nplicit results for the averagespin-quadrupole current cal-\nculated to first order in the tunnel coupling Γ of the two\nferromagnets and compare it to the average charge and\nspincurrent. Thecalculationsofthesearecompactlypre-\nsentedininApp.D,applyingacovariantreformulationof\nthe real-time diagrammatic technique (for a systematic,\nself-containedtechnical presentationseeApp. E). Covari-\nance is used here in the sense that all expressions are\nform-invariant under a change of the spatial coordinate\nsystem and the spin quantization axis. An advantage of\nthis technique is that it can be extended to spin-valves\nwith embedded quantum dots19.In Sec. VA we discuss the results for a general multi-\nband dispersion relation εr\nnkσ, applying the insights ob-\ntained from the spin-multipole network theory developed\nin IIIB2and IVand the microscopicpictureexplained in\nIIIB4. We decompose the SQM current into physically\nmeaningful contributions: direct vs. exchange (Pauli ex-\nclusion hole aspects) and dissipative vs. coherent (quan-\ntum fluctuation aspects). An intimate connection be-\ntween storage (see III) and transport of charge, spin\nand SQM is then established by comparing their energy-\nresolved contributions (see Sec. IIIB5).\nIn Sec. VB we specialize to the flat band approxima-\ntion(cf. IID) toobtaintangibleanalyticalandnumerical\nresults. Even in this simple limit – where the dissipa-\ntive spin-current vanishes due to the energy independent\nDOS – the average SQM current tensor has a non-trivial\nparameter dependence. This concerns both its magni-\ntude (principal values) and its alignment (principal vec-\ntors), which are substantially controllable by magnetic\nand electric parameters for non-collinear Stoner vectors.\nIn Sec. VC we demonstrate that a pureSQM current,\ni. e., not accompanied by a spin current, is in principle\npossible. This spin-anisotropy transfer is entirely carried\nby Pauli exclusion holes, giving rise to a non-vanishing\nexchange SQM current. For a temperature bias between\nferromagnets with collinear Stoner vectors, we even show\nthatapureSQMcurrentsevenpersistsintheabsenceofa\nchargecurrent. Electrodeswithnon-trivialspinstructure\ncan thus “talk” to each other in ways not described by\ncharge and spin currents\nA. Charge, spin and SQM current\nThe average charge, spin and spin-quadrupole current\nassociated with the left electrode read\n∝an}bracketle{tIL\nN∝an}bracketri}ht=IL\nN,0+IL\nN,F/parenleftBig\n/hatwideJL·/hatwideJR/parenrightBig\n, (89)\n∝an}bracketle{tIL\nS∝an}bracketri}ht=EL\nS/hatwideJL+AL\nS/hatwideJR+TL\nS/parenleftBig\n/hatwideJL×/hatwideJR/parenrightBig\n,(90)\n∝an}bracketle{tILL\nQ∝an}bracketri}ht= 2/angbracketleftbig\nSL/angbracketrightbig\n⊙∝an}bracketle{tIL\nS∝an}bracketri}ht−\n/hatwideJL⊙/bracketleftBig\nEL\nex/hatwideJL+AL\nex/hatwideJR+TL\nex/hatwideJL×/hatwideJR/bracketrightBig\n(91)\nThese were calculated in App. D to the first order in the\nenergy-resolved tunneling rate\nΓ(ω) = 2πt2¯νL(ω)¯νR(ω), (92)\nwhere ¯νr(ω) is given by Eq. (14). Here ⊙denotes the\nsymmetric, traceless tensor product (51). The charge\ncurrent coefficients are (we will not write the ωdepen-\ndence unless confusion may arise)\nIN,0=/integraldisplay\ndω2Γ∆, (93)\nIL\nN,F=/integraldisplay\ndω2Γ∆nLnR, (94)19\nwhere the spin-polarization function nr(ω) is given by\nEq. (15). The well-known bias function,\n∆(ω) =fR\n+(ω)−fL\n+(ω), (95)\nis only non-zero in the bias window, µL/greaterorsimilarω/greaterorsimilarµR, up\nto thermal smearing. The occurrence of the factor ∆( ω)\nsignals that a term arises from dissipative processes in\nwhich the energy of initial and final state have to be the\nsame. The spin current coefficients are\nEL\nS=/integraldisplay\ndωΓ∆nL, (96)\nAL\nS=/integraldisplay\ndωΓ∆nR, (97)\nTL\nS=/integraldisplay\ndωΓ/parenleftbigg\nβLnR\n¯νL+βRnL\nνR/parenrightbigg\n,(98)\nthe function βr(ω) incorporates the effect of the spin-\npolarization of the DOS, nr(ω), through the principal\nvalue integral,\nβr(ω) = Re/integraldisplaydω′\nπ¯νr(ω′)nr(ω′)fr\n+(ω′)\nω−ω′+i0,(99)\nintegrating over all virtual state energies ω′. Here and\nbelow such functions, not limited by energy conservation\nastheyinvolvevirtualintermediatestates, appearincon-\ntributions from coherent processes . Finally, the exchange\nSQM emission, absorption and torque coefficients\nEL\nex= 2/integraldisplay\ndωΓ∆aL, (100)\nAL\nex= 2/integraldisplay\ndωΓ∆nR˜aL, (101)\nTL\nex= 2/integraldisplay\ndωΓ(αRfL\n+−βR)aL\n¯νR,(102)\ndepend on the localspin-anisotropyfunction aL, cf. (66),\nand an additional evenspin-anisotropy function,\n˜aL(ω) =/integraldisplay\ndω′fr\n+(ω′)/summationdisplay\nσ,σ′σ′\n4νr\nσσ′(ω,ω′)\n¯νr(ω)(103)\n=/summationdisplay\nσσar\nσ(ω), (104)\nwhereνr\nσσ′(ω,ω′) is the 2DOS (16). Finally, the torque\ncoefficientTL\nQinvolves an additional function similar to\nEq. (99) but without the distribution function fR\n+(ω) un-\nder the principal value integral:\nαR(ω) = Re/integraldisplaydω′\nπnR(ω′)¯νR(ω′)\nω−ω′+i0.(105)\nThe reader should note that the coefficients of Eqs.\n(100)-(102) are defined such that a minus sign appears\nin Eq. (91), which we introduced in agreement with the\nsign convention for exchange SQM in Eq. (61), that is,∝an}bracketle{tQLL∝an}bracketri}ht=−qL\nex/hatwideJL⊙/hatwideJL. Moreover, one obtains the ex-\npressions for ∝an}bracketle{tIRR\nQ∝an}bracketri}htby formally substituting L↔Rin\nEq. (91), and ∝an}bracketle{tILR\nQ∝an}bracketri}ht=−∝an}bracketle{tILL\nQ∝an}bracketri}ht − ∝an}bracketle{tIRR\nQ∝an}bracketri}htfollows from\nthe SQM current conservation law (88) (see also below).\nOne checks that the results are invariant under scalar\ngauge transformations (global energy shifts). Finally, we\nnote that since positive currents are defined as entering\na node, positive (negative) absorption coefficients corre-\nspond to injection (ejection) of particles and vice versa\nfor emission coefficients.\nThe SQM current Eq. (91) is the central result of this\npaper. It depends explicitly (but not exclusively) on the\nspin current (90). Therefore, its physical interpretation\nis aided by first giving a pertinent review of the different\ncontributions to the charge and spin-currents (89) and\n(90), respectively.\n1. Charge Current\nEq. (89) is a well-known and experimentally tested re-\nsult forthe chargecurrent, which accountsforsingle elec-\ntron tunneling processes between the left and right elec-\ntrode. It has only dissipative contributions (i.e., involv-\ning∆(ω)): thefirstpart IN,0in Eq. (89)onlydepends in\nthe average DOS ¯ νrwhereas the second, spin-dependent\ncorrection that depends on the DOS spin-polarizations\nnrthroughIL\nN,Fand on the angle between the Stoner\nvectors through /hatwideJL·/hatwideJR= cosθ. The reduction going\nfromθ= 0→πis the celebrated spin-valve or tunnel\nmagneto resistance (TMR) effect.34\n2. Spin Current\nThe spin current (90) was obtained by Braun et. al\nand we review here its two type of contributions.\na. Dissipative spin emission ∼ˆJLand absorption\n∼ˆJRThe dissipative spin-dipole current (first two\nterms in Eq. (90), containing ∆( ω)) is analogous to the\nparticle current: an electron emitted from the left node\ntransports a spin-dipole moment ˆJL/2 and the electrons\nabsorbed from the right node transports ˆJR/2. The ex-\npression for particle current Eq. (93) simply has to be\nsupplemented by a factor nr(ω)ˆJr/2 to obtain the terms\nfor spin emission (96) and absorption (97).\nb. Coherent spin torque ∼ˆJL׈JRThe coherent\nspin current (last term in Eq. (90)) has no such analogy\nto the particle current and corresponds to a spin-torque.\nThis corresponds to spin flips induced byvirtual fluctua-\ntions between the left and the right electrode restricted\nby energy conservation only in the final state, but not\nin the intermediate state. This is reflected by its depen-\ndenceontheprincipalvalueintegral βr(ω)(cf. discussion\nof Eq. (99)): an electron with spin σoccupying a level at\nenergyωin the left electrode can fluctuate to all empty20\nlevels with energy ω′in the right electrode with an am-\nplitude∝1/(ω−ω′). While in the right electrode, the\nelectron spin ∝ˆJLis not collinear to the Stoner field\n∝ˆJRin the right electrode and precesses about it, ex-\nplaining the factor /hatwideJL×/hatwideJRin the coherent spin current\n(98). Note that a net spin-torque on the magnetization\nof the left electrode only occurs if the 1DOS of bothelec-\ntrodes are spin-polarized.\nFinally, we note that at zero bias, the dissipative spin-\ncurrent vanishes, EL\nS=AL\nS= 0, but the coherent spin-\ncurrent (torque contribution) remains, TL\nS∝ne}ationslash= 0: non-\ncollinear ferromagnets keep interacting by virtual fluc-\ntuations, thereby exerting a torque on each other.\n3. SQM Current\na. SQM current conservation The most immediate\nproperty of the SQM current expression (91) is its formal\nlack of symmetry with respect to interchanging the elec-\ntrodesL↔R. This differs notably from charge and spin\ncurrent, for which the original expression is reproduced\nwith a minus sign when interchanging L↔R. This dis-\ntinction is related to the striking characteristics of the\nSQM network picture compared to the charge and spin\nnetwork(seeSec.VA3): the currentsofthelatter, ∝an}bracketle{tIL\nRµ∝an}bracketri}ht,\ndescribe the net flow into the L-node coming from from\ntheR-node (see Fig. 11). Interchanging LandRyields\nthentheoppositecurrentfrom RtotheL-node,reflecting\nthe current conservation law (81): ∝an}bracketle{tIL\nRµ∝an}bracketri}ht+∝an}bracketle{tIR\nRµ∝an}bracketri}ht= 0. In\ncontrast, ∝an}bracketle{tILL\nQ∝an}bracketri}htdescribes the net flow of SQM from the\nlocal∝an}bracketle{tLL∝an}bracketri}ht-node to the nonlocal ∝an}bracketle{tLR∝an}bracketri}ht-node (see Fig. 12).\nIf we interchange L↔Rin Eq. (91), we obtain the\ncurrent∝an}bracketle{tIRR\nQ∝an}bracketri}htemanating from the ∝an}bracketle{tRR∝an}bracketri}ht-node. Impor-\ntantly,∝an}bracketle{tILL\nQ∝an}bracketri}ht+∝an}bracketle{tIRR\nQ∝an}bracketri}ht=−∝an}bracketle{tILR\nQ∝an}bracketri}ht ∝ne}ationslash= 0 in accordance with\nthe SQM current conservation law (88). This again em-\nphasizes the relevance of the nonlocal node ∝an}bracketle{tLR∝an}bracketri}ht, which\n“buffers” the SQM currents from both local nodes.\nb. Direct and Exchange SQM Current The SQM\ncurrent allows for twophysically meaningful, different\ndecompositions. The first decomposition is given by Eq.\n(91), whichbreaksuptheSQMcurrentintodifferenttwo-\nparticle contributions, the first, directterm Eq. (91) and\nthe second, exchange term. This has no analogue in the\none-particle charge and spin current.\nThe direct current ∝an}bracketle{tILL\nQ∝an}bracketri}htdirquantifies the tunneling-\ninduced change in the direct part of the average SQM\n(57),∝an}bracketle{tQLL∝an}bracketri}htdir:=∝an}bracketle{tSL∝an}bracketri}ht ⊙ ∝an}bracketle{tSL∝an}bracketri}ht, which ignores the Pauli\nexclusion hole (cf. Sec. III). Indeed, using Eq. (87), we\nreproduce the first term in Eq. (91):\n∝an}bracketle{tILL\nQ∝an}bracketri}htdir:=∝an}bracketle{t˙QLL−˙QLL|0∝an}bracketri}htdir (106)\n= 2∝an}bracketle{tSL∝an}bracketri}ht⊙∝an}bracketle{tIL\nS∝an}bracketri}ht. (107)\nSimilar to the average SQM, the direct average SQM\ncurrentis completely determined by a product of av-\nerage spin-dipole properties, here the spin ∝an}bracketle{tSL∝an}bracketri}htand thespin-current ∝an}bracketle{tIL\nS∝an}bracketri}ht, given by Eq. (47) and (90), respec-\ntively. This equation substantiates the classical picture\nof transport SQM or spin-anisotropy sketched in Fig. 13:\nwhen single electrons move, the triplet correlations be-\ntween pairs of electron spins first delocalize and then re-\nlocalize, resulting in a change of the local SQM, the part\ndescribed by ∝an}bracketle{tQLL∝an}bracketri}htdir.\nThe exchange SQM current ∝an}bracketle{tILL\nQ∝an}bracketri}htex, the second term\nin Eq. (91), accounts for the tunnel-induced change\nin∝an}bracketle{tQLL∝an}bracketri}htex, i.e., a negative quantum anisotropy due\nto the Pauli-exclusion holes in the triplet spin correla-\ntions. It cannotbe expressed in the averagespin-current.\nThe above classical picture of SQM transport thus needs\ncorrection: by reducing SQM transport to “spin times\nspin-current” one over-estimates the anisotropy flow, by\ncounting Pauli-forbidden, local triplet correlations(those\ncoming from the same orbital) and accounting for their\ntransformation into non-local correlations when one of\nthe two electrons tunnels out. The SQM exchange cur-\nrent compensates for this: it is an effective back-flow of\nnon-local anisotropy into to the local nodes of the SQM\nnetwork (Fig. 12). We now reach a central conclusion\nof the paper: whenever the average spin-current is made\nto vanish ∝an}bracketle{tIL\nS∝an}bracketri}ht= 0, a non-zero SQM exchange current is\ngenerally present since ∝an}bracketle{tSL⊙IL\nS∝an}bracketri}ht ∝ne}ationslash=∝an}bracketle{tSL∝an}bracketri}ht ⊙ ∝an}bracketle{tIL\nS∝an}bracketri}htdue to\nthe Pauli exclusion holes. In Sec. VC we explicitly verify\nthat the cancellations of the single-particle contributions\nthat cause the spin-current to cancel have no counter-\npart for the two-particle exchange SQM current. This\nindicates the possibility of pure SQM transport, that is,\nwithout spin current.\nThemostprominentdistinctionbetweenthedirectand\nexchange SQM currents is that they differ by a relative\nfactor|∝an}bracketle{tSL∝an}bracketri}ht|. To see this explicitly, we express the SQM\ncurrent (91) as a symmetric, traceless tensor product of\nthe unit vector ˆJLwith a linear combination of ˆJL,ˆJR\nandˆJL׈JRby inserting the explicit spin current (90):\n∝an}bracketle{tILL\nQ∝an}bracketri}ht=/hatwideJL⊙/bracketleftBig\nEL\nQ/hatwideJL+AL\nQ/hatwideJR/bracketrightBig\n+TL\nQ/hatwideJL⊙(/hatwideJL×/hatwideJR), (108)\nEach of the coefficients has a direct and exchange contri-\nbution, respectively:\nEL\nQ= 2EL\nS/parenleftBig\n∝an}bracketle{tSL∝an}bracketri}ht·ˆJL/parenrightBig\n−EL\nex,(109)\nAL\nQ= 2AL\nS/parenleftBig\n∝an}bracketle{tSL∝an}bracketri}ht·ˆJL/parenrightBig\n−AL\nex,(110)\nTL\nQ= 2TL\nS/parenleftBig\n∝an}bracketle{tSL∝an}bracketri}ht·ˆJL/parenrightBig\n−TL\nex.(111)\nSince all coefficients Eq. (96)-(98) and Eq. (100)-(102)\nappearing on the right hand sides are in general of the\nsame order, the ratio of the direct SQM current to\nthe exchange SQM scales linearly with the average spin\n|∝an}bracketle{tSL∝an}bracketri}ht| ∼Ns= (J/D)Noas expected from our analy-\nsis below Eq. (60) in Sec. IIIB2. Consequently, for\namacroscopic ferromagnet, the SQM current is domi-\nnated by its direct part (107) and is thus induced by21\nthe spin-current.Furthermore, since the SQM current ac-\ncounts for the change in the correlations between the\nspin of a transported electron with all otherspins in\nthe system, only the SQM current per electron is ex-\npectedtobeameaningfulquantityinthethermodynamic\nlimit35. As soon as one of the subsystems is of meso- or\nnanoscopic dimensions the relative factor |∝an}bracketle{tSL∝an}bracketri}ht|matters\n(cf. Sec. IIIA2b) and the exchange SQM current should\nbe reckonedwith. For a nanoscopicsystem, the full SQM\ncurrent was already studied in15, while also including the\nrelevantchargingandnon-equilibriumeffects, whichwere\nneglected here. With this in mind, we will in the follow-\ning always first discuss the direct part, dominating the\nSQM current for macroscopic ferromagnets (Fig. 1 (a)),\nandthenseparatelyconsidertheexchangecorrection,rel-\nevant for mesoscopic ferromagnets, (Fig. 1 (b)).\nc. Dissipative and Coherent SQM Current The sec-\nond, alternative decomposition of the SQM current is\nthat into a dissipative and coherent part, the first and\nsecondtermofEq.(108), respectively,similartothespin-\ndipole current. For non-collinear ˆJLandˆJRthese terms\nare linearly independent tensors and their coefficients\nhave very different parameter dependencies. The tenso-\nrial structure of the total SQM current is determined by\ntheir non-trivial interplay. Its discussion requires explicit\nresults and is therefore postponed to Sec. VB3 where we\nuse the flat band approximation.\nThe decomposition of the direct SQM current follows\nby Eq. (107) directly from the decomposition of the spin-\ncurrent (90) into a dissipative emission, dissipative ab-\nsorption and coherent torque part. Since the exchange\nSQMcurrentisacorrectiontothedirectcurrentaccount-\ning for Pauli-forbidden triplet correlations, cf. explana-\ntion below Eq. (107), it must have the same decom-\nposition into emission, absorption and torque part with\ncoefficients given by Eqs. (109)-(111), respectively.\nDissipative SQM emission ∼/hatwideJL⊙/hatwideJLand absorp-\ntion∼/hatwideJL⊙/hatwideJR.The SQM emission can be microscopi-\ncally understood as the delocalization of triplet spin cor-\nrelations from node ∝an}bracketle{tLL∝an}bracketri}htto node∝an}bracketle{tLR∝an}bracketri}ht(see Fig. 5). The\nSQM absorption describes the converse relocalization of\nsuch correlations from node ∝an}bracketle{tLR∝an}bracketri}htto node∝an}bracketle{tLL∝an}bracketri}ht. This is\nreflected by the tensorial structure of these contributions\nto the average SQM currents: they coincide with the av-\nerage SQM stored in the node, from where they are emit-\nted (/hatwideJL⊙/hatwideJL∼ ∝an}bracketle{tQLL∝an}bracketri}ht) or absorbed ( /hatwideJL⊙/hatwideJR∼ ∝an}bracketle{tQLR∝an}bracketri}ht).\nImportantly, there is no SQM absorption in ∝an}bracketle{tILL\nQ∝an}bracketri}htthat\noriginates from the ∝an}bracketle{tRR∝an}bracketri}ht-node (/hatwideJR⊙/hatwideJR∼ ∝an}bracketle{tQRR∝an}bracketri}ht) as\nexpected from the connectivity in the SQM network pic-\nture of Sec. IVB.\nThe relation between the storage and transport is also\nreflected by the integrands of the exchange SQM cur-\nrent in Eqs. (100)-(101): these resemble the average lo-\ncal spin-quadrupolarization qL\nex(ω) =fL\n+(ω)aL(ω) in the\nexpression (64) for ∝an}bracketle{tQLL∝an}bracketri}htex. These can formally be ob-\ntained by a replacement fL\n+(ω)→∆(ω), i.e., similar tothe relation between charge current and the dissipative\nspin current in Sec. VA2. However, the symmetry of\nthe function aL(ω) with respect to ωis is very different\nfrom that of nL(ω) appearing the spin current emission\n(Eq. (96)-(97)). This fact underlies a key result of the\npaper in Sec. VC.\nThe microscopic picture of exchange SQM storage can\nbe extended to capture a precise, physical understanding\nof the exchange SQM currentas follows: as explained in\nSect. IIIB4, the Pauli exclusion holes can be “counted”\nas single-mode cross correlations between two identical\ncopies of the same ferromagnet (cf. Fig. 8). For the\nexchange SQM current, we have to imagine that the\nelectron in the first copy undergoes a tunneling process\n(representing to the spin current in Eq. (87)), and the\nsecond copy is left unchanged (representing the spin in\nEq. (87))36. For the dissipative exchange SQM emission,\nthisbecomesdirectlyclearfromthe expression(100): the\nFermi function in ∆( ω) refers to the electron tunneling\nat energyωin the first copy and the local anisotropy\nfunctionaLcontains the single-mode correlation of that\nelectron with the average(unchanged) second copy. This\nterm therefore describes the flow of Pauli exclusion holes\narising from the spin emission ∼ˆJL. The exchange\ncoefficient (101) corresponding to the spin absorption\n∼ˆJR, the the first factor ∆ nRrelates to the absorbed\nspin from the right electrode and the second factor ˜ aL\nrepresents the correlations of that absorbed spin with\nthe local spins in the left electrode. Notably, the spin-\ndependent contributions aL\nσtoaL=/summationtext\nσaL\nσare added in\n˜aL=/summationtext\nσσaL\nσ>0 such that they always count as posi-\ntive. Here the sign of how to count the Pauli-forbidden\nanisotropy is related to the sign of nR: ifnR>0, mostly\nspin-↑is absorbed and the missing anisotropy generated\nby this is positive, while for nR<0 mostly spin- ↓is\nabsorbed and the missing anisotropy is negative as ex-\nplained in Sect. IIIB4\nCoherent SQM torque ∼/hatwideJL⊙(ˆJL×/hatwideJR) The co-\nherent contribution to the SQM current basically origi-\nnates from the spin torque. This follows by considering\nthedirectcontributionthatderivesfromthespincurrent,\ncf. Eq.(107). Itaccountsforthechangeofthecorrelation\nbetween of the spin of an electron fixed in the left elec-\ntrodewith the spinofanelectronthat virtually fluctuates\ninto the right electrode (spin-flip scattering). Since dur-\ning this fluctuation the latter spin precesses about the\nStoner field and a net conversion of local into non-local\ncorrelations results, i.e., there is an associated SQM cur-\nrent. The exchange SQM torque coefficient TL\nexexcludes\nthe single-mode correlations: in the microscopic picture\nonly the electron in the first copy undergoes a virtual\nfluctuation (indicated by βRandαRin Eq. (102)) , while\nthe second copy is left unchanged (indicated by aLLin\nEq. (102))37. This is the effect of the spin torque on the\nlocal Pauli-exclusion holes.22\nB. Parameter Dependence\nHaving discussed the general structure and physical\nmeaning of the main results (89)-(91), we now simplify\nthem as far as possible by making the flat band approx-\nimation. Although this is a crude approximation, it re-\nveals a general key feature of the exchange SQM, namely\nitssensitivitytothespin alignment , anon-negativequan-\ntity that accumulates when summing over energies / k-\nmodes. ThisprohibitscancellationsintheexchangeSQM\ncurrent as they occur due to signed contributions in\nthe charge and spin-dipole current. (cf. Sec. IIIB4).\nAs noted in Sec. IID the 2DOS appearing in the ex-\nchange expressions requires modelling of the electrodes\nthat goes beyond the 1DOS. However, in the flat-band\napproximation we only need Eq. (A44). We furthermore\napply the bias voltage symmetrically to the ferromag-nets,µL= +V/2 andµR=−V/2 while considering\narbitrary non-collinear Stoner vectors ˆJLandˆJR. We\nassume all further parameters to be symmetric: Jr=J,\nTr=T,Dr=Dandνr\nσ=νσforσ=↑,↓(except for\na temperature gradient discussed in Sec. VC). In this\napproximation the densities of states are fixed by the\nbandwidths, νσ= 1/2D, and the tunneling rate is set by\nthe spin-conserving tunnel amplitude t: Γ = 2π(t/2D)2,\ncf. Eq. (11). Together with the leading order approxi-\nmation in Γ this limits the applicability of the results to\nthe regime Γ ≪T,J,≪W(cf. Eq. (19)). The central\nequations (89)-(91) now simplify to\n∝an}bracketle{tIL\nN∝an}bracketri}ht=IL\nN,0, (112)\n∝an}bracketle{tIL\nS∝an}bracketri}ht=TL\nS/parenleftBig\n/hatwideJL×/hatwideJR/parenrightBig\n, (113)\n∝an}bracketle{tILL\nQ∝an}bracketri}ht= 2/parenleftBig\n∝an}bracketle{tSL∝an}bracketri}ht·/hatwideJL/parenrightBig\n/hatwideJL⊙∝an}bracketle{tIL\nS∝an}bracketri}ht−/hatwideJL⊙/bracketleftBig\nEL\nQ/hatwideJL+TL\nQ/parenleftBig\n/hatwideJL×/hatwideJR/parenrightBig/bracketrightBig\n(114)\n=−EL\nex/hatwideJL⊙/hatwideJL+/parenleftBig\n2/parenleftBig\n∝an}bracketle{tSL∝an}bracketri}ht·/hatwideJL/parenrightBig\nTL\nS−TL\nex/parenrightBig\n/hatwideJL⊙/parenleftBig\n/hatwideJL×/hatwideJR/parenrightBig\n. (115)\nIn this approximation the DOS (13) is not spin-\npolarized in the bias window: nr(ω) = 0 forµL/lessorsimilarω/lessorsimilar\nµR. Thereforethechargecurrent Eq.(112)reducestoits\nnon-magnetic part, i.e., there is no spin-valve effect. By\nthe same token, the dissipative part of the spin current\n(113) vanishesdue to the cancellationofparticleand hole\ncontributions. Thus only the coherent spin torque part\nremains, whose coefficient we now estimate as follows:\ninserting Eq. (99) into Eq. (98), we obtain\nTL\nS= 2t2Re/integraldisplay\ndω/integraldisplay\ndω′/productdisplay\nr(¯νr(ω)nr(ω))\n×fR\n+(ω′)−fL\n+(ω)\nω−ω′+i0. (116)\nFor our DOS approximation, we have\n¯νL(ω)nL(ω)¯νR(ω′)nR(ω′) = sgn( ωω′)/(4D)2if\n|ω| −D/lessorequalslantJ/2 and|ω′| −D/lessorequalslantJ/2 and zero oth-\nerwise. At these energies the Fermi functions are 0 or\n1 and if their difference in Eq. (116) is nonzero, we can\napproximate 1 /|ω−ω′| ≈1/(2D), yielding\nTL\nS≈ −Γ\n4πJ2\nD. (117)\nwhere Γ = 2 π|t|2/(2D)2. The resulting spin current is\nequivalent to a spin-torque exerted by an magnetic field\nBR≈¯νRJ|t|2/D/hatwideJRon the spin on SL(insert Eq. (26)\nfor/vextendsingle/vextendsingleSL/vextendsingle/vextendsingleinto Eq. (113)). Finally, in the SQM current\n(114)-(115) the absorption coefficient AL\nQ– and with\nit, the non-local anisotropy function aLR– drops out inthis approximation because of the vanishing of the spin-\npolarization, nr(ω) = 0 in the bias window.\n1. Dissipative SQM Flow Direction\nWe first discuss the direction of the dissipative spin-\nanisotropy emission, ∝an}bracketle{tILL\nQ∝an}bracketri}ht=−2EL\nQ/hatwideJL⊙/hatwideJL, which en-\ntirely arises from the exchange term in Eq. (114) (the di-\nrect part vanishes in absence of dissipative spin current).\nRemarkably, the emission always results in a lossof lo-\ncal exchange spin-anisotropy ∝an}bracketle{tQLL∝an}bracketri}htex=−qLL\nexˆJL⊙ˆJL\nirrespective of the voltage bias direction, because EL\nQis\nalwaysnegative (unless zero). By interchanging the left\nand right electrode index in all expressions, we observe\nthat the spin-anisotropy of the ∝an}bracketle{tRR∝an}bracketri}ht-node decreases as\nwell. We conclude from the SQM current conservation\nlaw that non-local spin-triplet correlations are built up\nirrespective of the bias direction. This is in accordance\nwith the physical intuition of SQM transport that we\nhave developed using our network picture the tunneling\nof electrons across the junction delocalizes spin-triplet\ncorrelations. However, such a pure delocalization is spe-\ncial to a voltage-biased tunnel junction. When we dis-\ncuss the situation of thermal bias later, we will see that\nan effective transfer of spin-anisotropy between the local\nnodes is still possible.\nThese results can also be clearly understood in terms\nof the microscopic picture of SQM storageintroduced in\nSect. IIIB5. To see this, we first note that the exchange\nSQM emission (100) is obtained from the average (see23\nEqs. (64) -(65)) by replacing the Fermi function fL\n+by\nthe bias function ∆. If µL>µR, electrons leavethe left\nelectrode, which is indicated by ∆( ω)>0 at energies\nµR/lessorsimilarω/lessorsimilarµL. This destroys the positive local exchange\nSQM content at this energy (given by aL(ω)>0, see\nalso Fig. 7). Conversely, for the opposite bias µL<µR,\nwe find ∆(ω)<0 forµL/lessorsimilarω/lessorsimilarµRsince the tunneling\nelectrons enterthe left electrode. Electrons at these fre-\nquencies provide negative exchange SQM (as aL(ω)<0).\nIn both cases this results in a negative change inqL\nex.\n2. Scalar Parameter Dependence\nWe next discuss the dependence of the direct and ex-\nchange SQM current on the scalarparameters J,V=\nµL−µRandT. For the direct SQM current,\n∝an}bracketle{tILL\nQ∝an}bracketri}htdir= 2/parenleftBig\n∝an}bracketle{tSL∝an}bracketri}ht·/hatwideJL/parenrightBig\nTL\nS/hatwideJL⊙/parenleftBig\n/hatwideJL×/hatwideJR/parenrightBig\n,(118)\nthis is simple because TL\nSis nearly independent of Vand\nTandincreasesas J3(useEqs. (117)and(26)), asshown\nin Fig. 14.\n0 1 2 3 4 5 6 7\nJ/t0.00.51.01.52.02.5|ILL\nQ,dir/(No·Γt)|×10−3\nFIG. 14: Dependence of ILL\nQ,dir= 2∝angbracketleftSL∝angbracketright·ˆJLTL\nSon the Stoner\nsplitting J/t, whereNois the fixed number of orbital states\nof the left subsystem and J= 5t,D= 25t,T=t/2.\nThe exchange SQM current, in contrast, shows a\nstronger dependence on the scalar parameters. To see\nthis, we first roughly estimate how its emission and\ntorque coefficients scale with VandJfor low temper-\naturesT/lessorsimilarV,J. ForT= 0 the integrand of (100) is the\nproduct of the anisotropy function aLL(ω), which has a\nsupport of width 2 J, and the bias function ∆( ω) with a\nsupport of width V, and the smaller one of these energy\nscales limits the SQM emission:\nEL\nex≈ −Γ\n2min(|V|,|J|), (119)\nwhere Γ = 2 π|t|2/(2D)2. In contrast, the SQM torque(102) scales in the same way as the spin torque:\nTL\nex≈ −Γ\nπJ2\nD, (120)\nwhere we also set T= 0 and proceeded analogous to the\nestimation of the spin torque (cf. Eq. (117)). The ad-\nditional suppression factor J/Din Eq. (120) relative to\nEq. (119) for V < Jreflects that the SQM torque orig-\ninates from coherent virtual fluctuations to states near\nthe band edges where the spin-polarization is nonzero in\nan energy window proportional to the Stoner splitting J.\nThis gives rise to two regimes, in which the coherent ex-\nchange term (120) is larger (smaller) than the dissipative\nexchangeterm (119) for V≶V∗, whereEL\nex∼TL\nexoccurs\nfor38\nV∗=J2\nπD. (121)\nFIG. 15: Dependence of the magnitude of SQM emission\n|EL\nex|/Γt(blue) and torque |TL\nex|/Γt(green) on the Stoner\nsplitting J/tforT= 0 (dark colors) and T= 0.5t(light col-\nors). The residual parameters are V=t,D= 25t. ForT= 0,\nthe crossover occurs at J∼J∗=√\nπDV≈9t. The initial\nnon-linearity of the SQM emission coefficient for the finite\ntemperature (preceded by a linear regime), and the smaller\nsaturation value compared to the T= 0 case are due to the\nthermal smearing.\na. Stoner-field dependence In Fig. 15 we show a nu-\nmerical calculation of the precise shapes (100) and (102)\nof the exchange emission coefficient EL\nexand the torque\ncoefficientTL\nex, confirming the estimates (119) and (120):\ntheyshowthatthe torqueincreasesquadraticallyandthe\nemission saturates to a constant on the scale of bias V\n(which is smaller than the value predicted by Eq. (119)\ndue to finite temperature).\nb. Bias dependence Fig. 16 shows the same\ncrossover, but now as function of the bias Vfor fixedJ\nandT. The torque is constant and given approximately\nby (120) and the emission saturates at the value set by\n(119) when Vapproaches the scale of J. This voltage24\nFIG. 16: Dependence of the magnitude of SQM emission\n|EL\nex|/Γt(blue) and torque |TL\nex|/Γt(green) on the bias volt-\nageV/tforT= 0 (dark colors) and T= 0.5t(light colors).\nThe residual parameters are J= 5t,D= 25t. For both tem-\nperatures, the torque is constant at |TL\nex| ≈0.15 according to\nestimation (120). The small deviation of this value and the\nsaturation level of |EL\nex|/Γtfor finite temperature compared\ntoT= 0 is due to the thermal smearing. Therefore, the\nrough estimate V∗≈J2/πD≈1/3 for the crossing point at\nEL\nex=TL\nexis exactly fulfilled only for T= 0.\ndependence allows for magnetic and electric control over\nthe orientation of the exchange SQM current tensor, dis-\ncussed in the next Sec. VB3. The saturation at V∼J\nis an interesting, new feature of the dissipative exchange\nSQM current, not present in the charge or spin current.\nIt provides access to the Stoner shiftof the DOS, cf.\nFig. 7, even without spin-polarization of the DOS in the\nbias window. Similar to the spin current a finite coherent\nSQM term remains at zero bias, even though the dissi-\npative SQM current vanishes.\nFIG. 17: Dependence of the magnitude of exchange SQM\nemission |EL\nex|/Γt(blue) and torque |TL\nex|/Γt(green) on tem-\nperature T/tforV=t,J= 5t D= 25t.c. Temperature dependence In Fig. 17 we show the\ntemperature dependence of the exchange SQM emission\nand torque coefficients, keeping VandJfixed. Both co-\nefficients decrease monotonously with temperature, but\nwith verydifferentcharacteristicdependencies on T. The\nreason is that the emission integral (100) incorporates\nFermi functions, which have a much stronger exponential\ndependencewith T−1, whereastorqueintegral(102)com-\nprises the renormalization function βR, which depends\nmuch weaker, namely algebraically on T−1. Moreover,\nFig. 17 shows that exchange SQM emission is strongly\nsuppressed when Tapproaches the voltage V(<J) when\nthe bias function ∆ = fR\n+−fL\n+is largely broadened over\nan energy range of ∼4T∼J(for the parameters of\nFig. 17), so that positive and negative contributions of\nthe spin-anisotropy function aLcancel each other. This\nis similar to the discussion of the exchange SQM storage\nin Sec. IIIB5.\n3. Angle Dependence\nThe exchange SQM current, in contrast to the direct\npart, has a nontrivial dependence on the angle between\nthe two Stoner vectors ˆJLandˆJRdue to the interplay\nof its dissipative and coherent contributions (see Sec.\nVB3). This requires a more extensive analysis since we\nare dealing with a tensor-valued current. There are two\nrelevant questions relating to the orientation of the SQM\ntensors. The first question is whether the orientation of\nthe local SQM ∝an}bracketle{tQLL∝an}bracketri}htis changed by the injected SQM\ncurrent∝an}bracketle{tILL\nQ∝an}bracketri}ht. One can show that if these tensorscom-\nmute, [∝an}bracketle{tILL\nQ∝an}bracketri}ht,∝an}bracketle{tQLL∝an}bracketri}ht] = 0, then the SQM current corre-\nspondsonlytoachangeintheprincipalvaluesofthelocal\nSQMwithoutchanging its principal axes (see App. C).\nNowEq.(91) showsthat fornon-collinear /hatwideJLand/hatwideJRthe\naverage SQM current ∝an}bracketle{tILL\nQ∝an}bracketri}htis a superposition of three\nlinearly independent, symmetric and traceless tensors,\n/hatwideJL⊙/hatwideJL(emission (100)) , /hatwideJL⊙/hatwideJR(absorption (101))\nand/hatwideJL⊙(ˆJL×/hatwideJR) (torque (102)). The latter two ten-\nsors do not commute with ∝an}bracketle{tQLL∝an}bracketri}ht ∝/hatwideJL⊙/hatwideJLfor non-\ncollinear/hatwideJLand/hatwideJRand vanish only for collinear /hatwideJLand\n/hatwideJR. This holds for both the direct and exchange con-\ntributions. Therefore the injected average SQM current\n∝an}bracketle{tILL\nQ∝an}bracketri}htwill tend to change the principal axes of the av-\nerage local SQM ∝an}bracketle{tQLL∝an}bracketri}ht, besides changing its principal\nvalues.\nThe second question is whether the direct and ex-\nchange SQM currents tend to induce the same rotation\nof the principal axis of the SQM, which is equivalent to\nthese tensors commuting, [ ∝an}bracketle{tILL\nQ∝an}bracketri}htdir,∝an}bracketle{tILL\nQ∝an}bracketri}htex] = 0. To\nshow that this is not the case, we now first explicitly\nfind the (different) principal axes and values of ∝an}bracketle{tILL\nQ∝an}bracketri}htdir\nand∝an}bracketle{tILL\nQ∝an}bracketri}htex. This will furthermore allow us to plot and\ndiscuss these average SQM currents in a clear way.25\na. Principal axes and values The following analysis\nholds for any dispersion εr\nnkσ. We first diagonalize the\nfull average SQM current tensor Eq. (91) and then show\nhow the direct and exchange part can be obtained from\nthe result. Since the former is real and symmetric tensor\nit can always be diagonalized:\n∝an}bracketle{tILL\nQ∝an}bracketri}ht=/summationdisplay\nλ=±,0Iλˆvλˆvλ. (122)\nHere the ˆvλdenote the orthonormal principal axes (the\nhat indicating normalization)and the Iλdenoteprincipal\nSQMcurrents ,whichquantifythemagnitudeoftheSQM\ncurrent. (We dropped the superscripts “ LL” onIλand\nˆvλfor brevity). In App. B we show that for non-collinear\nStonervectors(cos( θ) =/hatwideJL·/hatwideJR∝ne}ationslash=±1)theprincipalSQM\ncurrents are\nI0=−1\n3Dθ, (123)\nI±=1\n6Dθ±1\n2Sθ, (124)\nwhich add up to 0 as they should (traceless tensor). The\nunnormalized principal axes read\nv0=/hatwideJL×/parenleftBig\nAL\nQ/hatwideJR+TL\nQ/hatwideJL×/hatwideJR/parenrightBig\n,(125)\nv±= (Dθ±Sθ)/hatwideJL+AL\nQ/hatwideJR+TL\nQ/hatwideJL×/hatwideJR.(126)\nIn Eqs. (123)-(126), we used the abbreviations\nDθ:=/parenleftbig\nEL\nQ+AL\nQcosθ/parenrightbig\n, (127)\nSθ:=/radicalbigg\nD2\nθ+/parenleftBig/parenleftbig\nTL\nQ/parenrightbig2+/parenleftbig\nAL\nQ/parenrightbig2/parenrightBig\nsin2θ.(128)\nThe principal SQM currents obey the inequalities\nI−/lessorequalslantI0/lessorequalslantI+ (129)\nsinceSθ/greaterorequalslantDθ. For collinear Stoner vectors ( θ= 0,π),\ntwo eigenvalues are degenerate,\nI+=/parenleftbigp\n6+1\n2/parenrightbig/vextendsingle/vextendsingleEL\nQ+AL\nQ/vextendsingle/vextendsingle, (130)\nI−=/parenleftbigp\n6−1\n2/parenrightbig/vextendsingle/vextendsingleEL\nQ+AL\nQ/vextendsingle/vextendsingle, (131)\nI0=p\n3/vextendsingle/vextendsingleEL\nQ+AL\nQ/vextendsingle/vextendsingle (132)\nwithp= sgn(D0),ˆv+=ˆJLand any two vectors in the\nplane perpendicular to ˆJLare principal axes of the SQM\ncurrent. We thus see that in general, the principal SQM\ncurrents take three different values, i.e. a bi-axial spin-\nanisotropy is transported whenever the Stoner vectors\nare non-collinear. We emphasize that by itself the av-\nerage spin-current vectorprovides no information about\nthe non-collinearity of the spin-valve. The SQM current\ntensor, incontrast,does: the transportedanisotropyonly\nbecomes uniaxial for collinear Stoner vectors ( θ= 0,π),\nin which case ˆJLis the hard axis39. The possibility of\ninjecting biaxial anisotropy into molecular scale systems\nis of interest since this type of anisotropy it is associated\nwith interesting quantum-spin tunneling effects20.We obtain the diagonal form of the direct and ex-\nchange SQM individually by replacing the coefficients\nEq. (109)-(111) in the above formulas by EL\nQ→EL\ndir=/parenleftBig\nSL·/hatwideJL/parenrightBig\nEL\nSandEL\nQ→ −EL\nex, respectively, etc. Since\nthese two sets of coefficients are in general different func-\ntions of the various parameters, we conclude that the\ndirect and exchange SQM tensors have different princi-\npal axes and do not commute, [ ∝an}bracketle{tILL\nQ∝an}bracketri}htdir,∝an}bracketle{tILL\nQ∝an}bracketri}htex]∝ne}ationslash= 0.\nThey therefore tend to induce the different rotations of\nthe principal axis of the local SQM. As a result the total\nprincipal SQM currents are notthe sum of the direct and\nexchange principal SQM currents.\nb. Flat band approximation So far the considera-\ntionsweregeneral. We nowinvestigatethe magnetictun-\ning of the exchange SQM orientation by the Stoner vec-\ntors. For the flat band approximation, Eq. (112)-(114),\nthe principal SQM currents and the principal axes are\nsymmetric with respect to θ=π/2, i. e.,\nIλ/parenleftbigπ\n2−α/parenrightbig\n=Iλ/parenleftbigπ\n2+α/parenrightbig\n, (133)\nˆvλ/parenleftbigπ\n2−α/parenrightbig\n=ˆvλ/parenleftbigπ\n2+α/parenrightbig\n. (134)\nThis is due to the vanishing SQM absorption coefficient,\nAL\nQ, in this limit. In Fig. 18 we plot both the direct\nand the exchange principal SQM currents as a function\nof the angle θ∈[0,π/2]. We observe that both Idir\n0and\nIex\n0are constant. In both cases, the repulsion of both\nIdir/ex\n±is caused by the respective torque coefficient (see\nEq. (127)), which increases with Jand decreases with D\n(cf. Eq. (117)-(120)).\nTo analyse the angle dependence of the principal axes\nˆvλ, we construct a right handed coordinate system from\nthe non-collinear, non-orthogonal Stoner vectors: let\nez=/hatwideJL,ex=/hatwideJL×/hatwideJR/sinθ,ey=ez×ex. In Fig. 19\nwe show the trajectories of the direct and exchange prin-\ncipal axes as we rotate /hatwideJR=−sinθey+cosθezthrough\nθ∈[0,π/2]. Then both ˆvdir\n0=ˆvex\n0=eyare a fixed direc-\ntions in this coordinate system, i.e., independent of the\nangleθ. The other principal axes ˆvdir/ex\n±lie in thexz-\nplane perpendicular to ˆvdir/ex\n0and are different for the\ndirect and exchange contribution.\nThe direct SQM current only consists of a torque con-\ntribution and therefore its principal axes are indepen-\ndent ofθin the above coordinate system fixed by /hatwideJLand\n/hatwideJR. WithAL\ndir=EL\ndir= 0 andTL\ndir= 2(SL·/hatwideJL)TL\nS,\nEq. (125)-(126) give ˆvdir\n±= (±ez−ex)/√\n2 forθ∝ne}ationslash= 0,π\n(when the direct SQM current is nonzero). In contrast,\nthe exchange SQM current shows a nontrivial competi-\ntion of the torque and emission contributions: in Fig. 19\nwe plot the trajectories of its principal axes in the plane\nperpendicular to ˆvex\n0=eyasθis increased to from 0 to\nπ/2.\nWe emphasize that already in this simple model of\nflat-band ferromagnets the spin-anisotropy flow has non-\ntrivialtensorialstructure: in generalthe principal axesof\nthe exchange SQM tensor are neither collinear to /hatwideJL,/hatwideJR26\nFIG. 18: Angle dependence of the principal SQM currents (a)\nIdir\nλ/(t·Ns) and (b) Iex\nλ/t, respectively (red: I0, green: I−,\nblue:I+). The principal direct SQM currents only depend\non the nonzero spin torque Idir\nλ=λ2|SL||TL\nSsinθ|(λ= 0,±),\nwhence the principal exchange SQM currents depend both\nthe nonzero exchange SQM emission and torque. Parameters:\nJ= 5T,D= 25t,V=T=t, Γ = 2π/2500. Note that the\ninequalities Eq. (129) and/summationtext\nλIdir\nλ=/summationtext\nλIex\nλ= 0 are fulfilled.\nnor collinear to /hatwideJL×/hatwideJR. Only for nearly collinear config-\nurations (θ≈0,π), when the torque contribution is neg-\nligible, do we have such a simple result: /hatwidevex\n+≈ex=/hatwideJL\nand/hatwidev−≈ez=/hatwideJL×/hatwideJR/sin(θ).\nA further striking property of the exchange principal\naxes is that they can not only be tuned magnetically\nby the Stoner vectors but also to a large extent electri-\ncally. As discussed in Sec. VB2 there is a crossover,\nshown in Fig. 16, from a torque-dominated (low bias) to\nan emission-dominated exchange SQM (large bias). As\na result, increasing the voltage results in a change of the\nprincipal axes of the SQM current - an effect that is com-\nparableto that resultingfrom tuning the angle θbetween\nthe magnetizations in Fig. 19.\nThe voltage scale at which the direct and exchange\nFIG. 19: Angle dependence of the exchange SQM tensor\n∝angbracketleftILL\nQ∝angbracketrightex. Each point corresponds an principal vector Iex\n+ˆvex\n+/t\n(blue) and Iex\n−ˆvex\n−/t(green) of ∝angbracketleftILL\nQ∝angbracketrightexfor increasing angle θ\nin steps of π/36 from 0 to π/2. Each pair of vectors Iex\n±ˆvex\n±/t\nfor one angle defines the semi axes of an ellipse, drawn for the\nextremal values θ= 0 and θ=π/2. The axes of this ellipse\nindicate the principal axes of ∝angbracketleftILL\nQ∝angbracketrightexand and the diameter\n2|I±/t|gives the principal SQM currents. The arrows of the\nsemi axes indicate the sign of the principal SQM current: it\nis positive (negative) if the arrow points away from (toward s)\nthe origin. The full tensor can be visualized by including\nthe principal vector Iex\n0ˆvex\n0/t∝ey(pointing into the plane),\ncompleting the ellipse to an ellipsoid. ˆJRis rotated from ˆJL\n(θ= 0) out of the plane when θincreases. The residual pa-\nrameters are D= 25T,J= 5T,V=T=t, Γ = 2π/2500.\nSQM current compete, i.e., |EL\nex| ∼ |TL\ndir|= 2|SL|TL\nS, can\nbe estimated40using|SL| ∼JNo/D∼Ns(cf. Eq. (27))\nto\nV∼(J2/D)Ns. (135)\nSince by Eq. (119) EL\nexhas a voltage dependence only\nforV/lessorsimilarJ, the electric tuneability is feasible only when\nthe crossover scale (135) /lessorsimilarJ: this is the case when the\nnumber of polarized spin Ns/lessorsimilarD/J, i. e. the number\nof orbitals is limited to No/lessorsimilar(D/J)2, which can still\nbe fairly large number. This is a first indication that\nfor mesoscopic ferromagnets this electrical tuneability of\nthe SQM orientation may be possible and may be an\ninteresting topic for future analysis. We emphasize the\ncrudeness of our model here and the importance of inves-\ntigating charging and non-equilibrium effects, see Ref.19.\nC. Transport of spin anisotropy without spin\ncurrent\nFinally, we explore the possibility of a pure spin-\nanisotropy current , i. e. a non-vanishing SQM current in27\nFIG. 20: Voltage dependence of the exchange SQM tensor.\nFor explanation see Fig. 19 where each points corresponds to\na a different voltage V/t, increasing in steps of 1 /10 from 0\nto 1.5. The remaining parameters are θ=π/4,D= 25T,\nJL=JR= 5t,T=t, Γ = 2π/2500.\nthe absence of a spin current, which was anticipated in\nSec. VA3.\n1. Conditions for zero charge and spin current\nWe first discuss the conditions for a vanishing spin cur-\nrent for a general band structure εr\nnkσ. We expect zero\nspin current only for collinear magnetizations. If the\nmagnetizations are non-collinear, the spin current has\nthree non-collinear contributions and demanding that all\nthese vanish requires EL\nS=AL\nS=TL\nS= 0. This might be\npossible, but only for special band structures and pa-\nrameter values ( Tr,µr), but this is beyond the scope\nof this paper. For collinear magnetizations, the spin\ntorque automatically vanishes and the spin current reads\n∝an}bracketle{tIS∝an}bracketri}ht=/integraltext\ndωΓ∆/parenleftBig\nnLˆJL+nRˆJR/parenrightBig\nwith/hatwideJL||/hatwideJR. A generic\nsituation with cancelling spin current is then given for\nantiparallel magnetized ( /hatwideJL=−/hatwideJR) ferromagnets with\nidentical spin-polarization of the 1DOS in the bias win-\ndow, that is, nL(ω) =nR(ω), so that the bracket in the\nabove integrand is zero. This defines a parameter regime\nfor which ∝an}bracketle{tIS∝an}bracketri}ht= 0, as one may still apply any voltage or\ntemperature bias. Again, there might be exotic material\ncombinations, for which the spin current even vanishes\nfor/hatwideJL= +/hatwideJR. For our crude flat band approximation\n(cf. Sect. IID), the dissipative spin current is zero in any\ncase, so that collinearity /hatwideJ=/hatwideJL=±/hatwideJRof the Stoner\nvectors is already sufficient for cancelling spin current.\nWe can even go one step further and envisage a sit-\nuation, for which the charge current vanishes as well:note that it still has a non-magnetic contribution (93)\n∝/integraltextdω∆ (Γ is constant in the bias window and we con-\nsiderTr≪D). For pure voltage bias µL∝ne}ationslash=µR, but\nTL=TR, the bias function ∆( ω) is symmetric and pos-\nitive and we will always have a charge current. However,\nfor a pure temperature bias,TL∝ne}ationslash=TRandµL=µR,\nthe bias function ∆ = fR\n+−fL\n+isantisymmetric and the\ncharge current contributions above and below the com-\nmon electrochemical potential cancel. Since TR> TL\nthe right “hot” electrode has a larger (smaller) occupa-\ntion probability for electrons with energy ω>µ(ω<µ)\nthan the left electrode. Consequently, the particle cur-\nrent flowing from left to right for electrons with energy\nω > µexactly cancels the charge current for electrons\nflowing from the right to the left at energies ω<µ. Inte-\ngrated over all frequencies this gives the zero net charge\ncurrent.\n2. Pure quadrupole current\nStrikingly, in contrast to charge and spin current, the\nSQM current remains non-zero for a pure thermal bias.\nIt comes entirely from the exchange emission part:\n∝an}bracketle{tILL\nQ∝an}bracketri}ht=−2Γ/integraldisplay\ndω∆aL/hatwideJ⊙/hatwideJ∝ne}ationslash= 0,(136)\nwhere Γ can be pulled out of the integral since the DOS\nis constant at energies ωfor whichaL(ω)∝ne}ationslash= 0. Since the\nspin-anisotropy function aL(ω), Eq. (66), is antisymmet-\nricaswellwe integrateanoverallsymmetricfunction and\n∝an}bracketle{tILL\nQ∝an}bracketri}htis non-zero. This is a central result of the paper.\nIn Fig. 21 we plot the total SQM current for a thermal\nbias with collinear Stoner vectors as function of the tem-\nperature difference, as given by Eq. (136). In Fig. 22 we\nplot the dependence on the Stoner field JL.\n−0.6−0.4−0.20.0 0.2 0.4 0.6τR−20246|ILL\nQ/Γt|×10−2\nFIG. 21:/vextendsingle/vextendsingleILL\nQ/vextendsingle/vextendsingle/t(from Eq. (136) with ∝angbracketleftILL\nQ∝angbracketright=−ILL\nQ/hatwideJ⊙/hatwideJas\na function of the temperature bias ratio τR= (TR−TL)/TR\nforTL=t,J= 5t,D= 25tand Γ = 2 π/2500.28\n0 1 2 3 4 5\nJL/t−20246|ILL\nQ/Γt|×10−2\nFIG. 22: Same as Fig. 21, but now showing/vextendsingle/vextendsingleILL\nQ/vextendsingle/vextendsingle/tas func-\ntion of Stoner splitting JL/tfor fixed thermal bias τR=\n(TR−TL)/TR=−0.5,−0.1,0.1,0.5 (from topmost to bot-\ntommost curve). The antisymmetry of the linear result\nEq. (137), ILL\nQ(τR)≈ILL\nQ(−τR), breaks down in the non-\nlinear regime as shown in as Fig. 22 for τR=±0.5.\nThe linear response41in the temperature bias ratio\nτR= (TR−TL)/TR≪1 varied for fixed TLgives for\nthe SQM current magnitude, defined here by ∝an}bracketle{tILL\nQ∝an}bracketri}ht=\n−ILL\nQ/hatwideJ⊙/hatwideJ,\nILL\nQ=Γ\n2(TL−TR)×\nTL\nTR/bracketleftBigg\n1−/parenleftbiggJL/2TL\nsinh(JL/2TL)/parenrightbigg2/bracketrightBigg\n.(137)\nFor fixed, different temperatures, the magnitude of the\nSQM current increases monotonously as a function of\nthe Stoner splitting as shown in Fig. 22. It eventually\nsaturates for JL≈10TLat the asymptotic value of\nILL\nQ≈(Γ/2)τRTL.\nA crude understanding of the above results is the fol-\nlowing. Since the magnitude of local exchange SQM\ndecreases with temperature (Pauli exclusion effects get\nwashed out thermally), cf. Fig. 9 and Eq. (70), the ther-\nmal gradient induces a “gradient in the correlations” re-\nsulting in the SQM flow of Pauli exlusion holes from the\ncolder to the hotter reservoir, roughly speaking.\nWe emphasize, however, that this should not be inter-\npreted as a direct transfer of spin-correlations between\nthe two local SQM nodes since they first have to be con-\nverted into non-local spin-correlations: in the language\nofournetworkpicture, these arefirstbuffered in the non-\nlocal intermediate node. This becomes clearer in view of\nthe SQMconservationlaw(88), whichreadsofourdevice\n(cf. Fig. 12) after averaging ∝an}bracketle{tILR\nQ∝an}bracketri}ht=−∝an}bracketle{tILL\nQ∝an}bracketri}ht −∝an}bracketle{tIRR\nQ∝an}bracketri}ht.\nInterchanging the role of the left and the right electrode\nin Eq. (137), we see that the change in the local spin-\nanisotropy of the ∝an}bracketle{tLL∝an}bracketri}ht- and∝an}bracketle{tRR∝an}bracketri}ht-node have oppositesign. Taking only the O(∆T)- contribution, we may re-\nplaceTLandTR, respectively, by the average tempera-\nture in the second line of Eq. (137): we then find that\nthere is no netcreation of non-local spin-correlations\nonly to first order in in the thermal bias ∆ T, i. e.,\n∝an}bracketle{tILR\nQ∝an}bracketri}ht=O(∆T2).\nA more rigorous explanation of the thermally driven\nSQM current is based on a microscopic point of view (cf.\nSecs. IIIB4 and IIIB5). These considerations may be\nuseful for proposals for more complicated device setups\nthat would allow for the detection a pure SQM current\n(anissuethat isnot coveredhere). The exchangeSQMin\n(136) is quantified by the anisotropy function aL(ω) (cf.\nEq. (66) and Fig. 7), which describes the Pauli exclusion\nhole to which an electron at energy ωcontributes. The\nmicroscopicreasonwhy aL(ω)changessignwasexplained\nin detail in Sec. IIIB5: basically for ω < µ(ω > µ)\na given electron at energy ωmost likely sees a parallel\n(antiparallel) spin at energy ω+J(ω−J). Electrons\nwith opposite energies relative to µthus contribute with\nan opposite sign to the Pauli exclusion hole. Since the\nthermal bias transports electrons above and below the\nFermi edge into opposite directions, the contributions to\nthe local average SQM ∝an}bracketle{tQLL∝an}bracketri}htthusadd up, explaining\nwhy Eq. (136) is finite. Notably, the thermal bias drives\nthis flow of spin correlations between the ferromagnets\nwithout any other one-particle quantity being net trans-\nported. For example, the charge of each electron is in-\ndependent of its energy and therefore the contributions\nabove and below the Fermi energy cancel.\nImportantly, in this case the direction of the spin-\nanisotropy flow can be controlled by the sign of the tem-\nperature gradient: for TL<TR, the SQM current mag-\nnitudeILL\nQis negative, i. e., local planar spin-triplet cor-\nrelations are net delocalized by the tunneling. The left\nlocal node therefore losesPauli exclusion holes. This can\nbe understood following arguments similar to that of the\npurely voltage-biased tunnel junction (see Sect. VB1).\nHowever, in stark contrast to pure voltage bias, the cur-\nrent magnitude ILL\nQbecomes positive if TR<TL. This\nphysically means that net local planar spin-correlations\narecreatedby tunneling. The reasonis that electrons are\ninjectedintotheleftelectrode belowtheFermienergy. As\nthese electron obey Pauli’s principle, they are forcedto\nform new Pauli-exclusion holes. Furthermore, the elec-\ntrons areextracted only abovethe Fermi energyand they\ncarry away positive (axial) spin correlations (leaving a\nnegative contribution behind). For the contrasting situa-\ntionofapurevoltagebias, theenergy-resolvedflowdirec-\ntion has to be opposite: electrons care only net injected\n(extracted) at energies larger (lower) then the Fermi en-\nergy.\nAs a conclusion, the possibility to control the spin-\nanisotropy flow direction by the thermal bias applied to\nthe tunnel junction is a non-trivial result of this paper.\nThis fact and the prediction of a pure spin-quadrupole\nmoment current demonstrate most clearly that triplet-\nspin correlations form an independent degree of freedom29\nwhich is not only stored in a system of ferromagnets, but\ncan also be transported between them.\nVI. SUMMARY AND OUTLOOK\nIn this paper we investigated fundamental questions\nabout the spin anisotropy, as quantified by the spin-\nquadrupole moment (SQM), which arise when it is con-\nsidered as a transport quantity . In the physical language\nof atomic and molecular magnetism, the SQM charac-\nterizes the quadratic spin anisotropy, which is usually\nits dominant part. It quantifies the preference of spins\nto bealignedalong a specific axisirrespective of their\norientation along it (up, down). We addressed three\ncentral questions related to the quantum transport of\nspin-anisotropy: (i) how can SQM be stored in and (ii)\ntransported between two ferromagnets in a spintronic\ncircuit; (iii) how can one define an SQM current ten-\nsor operator , derive SQM continuity equations and SQM\ncurrent conservation laws; how does the non-equilibrium\nsteady-state average of the SQM current relate to the\nspin-current?\nOur work was motivated by studies14–16that indicated\nthat the physical picture of the transport of spin degrees\nof freedom through magnetic nanostructures needs to be\nextended. A refinement of this picture, resulting from\nthis paper, is as follows. Electrons are charged parti-\ncles with an intrinsic spin-dipole moment and vanish-\ning higher spin moments. Therefore, the motion of an\nisolated electron is associated with a charge and spin\ncurrent only. However, in a multi-electron system the\nelectron becomes correlated with other electrons. Mov-\ning this electron therefore implies a change of correla-\ntions. In particular, the transfer of spin-triplet corre-\nlations between different subsystems is quantified by the\nspin-quadrupolecurrent. Thiscomplementstheresultsof\nprior studies14–16, which demonstrated that these tensor-\nvalued currents lead to an accumulation of SQM. The\nlatter couples to the accumulation of spin and charge\nand their measurable currents. In this paper we ignored\nthe complicationsofthisaccumulation, aswellasinterac-\ntion and non-equilibrium effects that appear in nanoscale\nspintronicdevices. Weexclusivelyfocusedonthedescrip-\ntion of transort of SQM between macro- and mesoscopic\ncircuit elements.\nAnswering question (i), we found that macroscopicfer-\nromagnets, the basic elements of spintronic devices, store\na macroscopic SQM, which is generated by their internal\nStoner field. This directspin-anisotropy is of easy-axis\ntype and scales quadratically with the number of half-\nfilled, spin-polarized orbitals Ns. This follows the clas-\nsical intuition that orientation of spins also implies their\nalignment. However, for mesoscopic systems, an addi-\ntionalquantum exchange contribution to the SQM be-\ncomes relevant16, which scales linearly with Ns. It quan-\ntifies the effect of Pauli-exclusion holes that exist in the\ntriplettwo-particle spin correlations, expressing the sim-ple fact that electrons in the same orbital do not form a\ntriplet spin state. This Pauli-forbidden spin-anisotropy\nis of the easy-plane type, countering therefore the direct\neasy-axis anisotropy.\nImportantly, the effect of the Pauli exclusion holes is\ncumulative, i. e., their contributions always add up and\ncannot cancel each other. This is in stark contrast to\nthe average spin-dipole moment, for which contributions\nfrom electrons with opposite spin orientation can cancel\neach other. This shows that the averageSQM is a degree\nof freedom independent of one-particle quantities such as\naverage charge and spin.\nTo answer question (ii) we developed a spin-multipole\ntransport theorywith an associated networkpicture. For\nspin-dipole moment, each ferromagnet is represented by\na node of the network storing spin-dipole moment. How-\never, due to its two-particle nature in electronic systems,\nSQM is also stored as non-local correlations between\nspins from different spin-polarized subsystems. The net-\nworkpictureofSQMthereforeincorporatesalso non-local\nSQM nodes . As a consequence, the SQM networkdiffers\nfrom the physical layout of the system of ferromagnets,\nboth in the number of nodes and in their connectivity.\nFor thetwo-terminal spin-valve that we studied in this\npaper, this network thus consists of threeSQM nodes.\nThis network theory also applies also to spin-valves with\nembedded quantum dots19.\nBased on this microscopic picture, we inferred the\nproper definition of the spin-quadrupole current tensor\noperators , answeringquestion (iii). Byacontinuityequa-\ntion, the SQM currents generate the change of the local\nanisotropy due to quantum transport processes. They\nfurthermore obey a current conservation law expressing\nthe conservation of SQM in the tunneling. For the two-\nterminal spin-valve it reads ILL\nQ+IRR\nQ+ILR\nQ= 0.\nFinally, we found by explicit calculation that the non-\nequilibrium steady-state average of allthese these SQM\ncurrents is non-zero, even for this elementary spintronic\nsetup and analysed these in detail. Similar to the aver-\nage SQM, these average SQM currents have a decompo-\nsition into classical and quantum two-particle contribu-\ntions, similar to the average SQM itself. The direct SQM\ncurrentis implied by a non-zero average spin and spin\ncurrent. It reflects the classical intuition that “orienta-\ntion implies alignment”. In addition to this, we found\nan quantum exchange SQM current , which is profoundly\ndifferent from spin currents.\nIn analogy to the spin, we also distinguished dissipa-\ntive and coherent contributions to the SQM: the spin\nprecession responsible for the spin-torque term in the\nspin-current – lifting the spin out of the plane of the\nStoner vectors – has a counterpart in the SQM current.\nThese spin-torque SQM terms similarly result from co-\nherent fluctuations by virtual tunneling into a ferromag-\nnet (i. e., spin-dependent scattering) which probe the\nspin-dependence of the entire band structure. This effect\nis also responsible for the exchange field21in quantum\ndot spin-valves. The different bias-voltage dependence30\nof the dissipative and coherent terms allows for electric\ncontrol of both the magnitude and the orientation of the\nspin-anisotropy current tensor.\nFurthermore, for non-collinear ferromagnets, the spin-\nanisotropy current was found to be a bi-axialtensor. Its\nthree distinct principal values and axes reflect the low-\nered symmetry of a non-collinear setup, which is not re-\nvealed by the spin current, which is just a vector. We\nshowed that this dependence on the Stoner vectors al-\nlows for substantial magnetic tuning of the SQM cur-\nrent tensor orientation. The possibility of injecting biax-\nial anisotropy into, e.g., molecular magnets is of interest\nsince the intrinsic, spin-orbit generatedanisotropyofthis\ntype is associated with quantum-spin tunneling effects20.\nThe striking central result of this paper, as announced\nby the paper title, is a pure SQM current whenever the\nspin current vanishes by net cancellation of one-particle\ncontributions. This spin-anisotropy flow is driven by a\ngradient of Pauli exclusion holes in the triplet spin-spin\ncorrelations, that is, a true quantum two-particle cur-\nrent. We illustratedthis generalresultforatemperature-\nbiasedjunctionconnectingtwoanti-parallelferromagnets\nwith a flat-band DOS. In this case, a pure SQM current\ngenerates an uniaxial, easy plane anisotropy, i.e., a neg-\native anisotropy that counteract an easy axis anisotropy.\nIt may be of interest to inject such an SQM current into\na single-molecule magnet considered as a memory cell in\norder to temporarily switch off its easy-axis anisotropy\nbarrier in order to put it into “writing” mode. This also\nrelates to the recently studied tunnel-induced renormal-\nizationoftheintrinsicanisotropyofmolecularmagnetsin\ncontact with spin-polarized electrodes16,22,23. One may\neven envisage the utilization of SQM as a resource, as\nan alternative to conventional spintronics, i.e., utilize\nthe storage, transport, manipulation and readout of spin\nanisotropy without transporting or affecting spin polar-\nization. The possibility of pure SQM currents pointed\nout in this paper indicates that this is principle con-\nceivable and warrants further study. Altogether the\nabove indicates that the theory of a generalized “spin-\nmultipoletronics”, is a real possibility, if not a necessity\nwhen spintronics moves to the nanoscale.\nWe acknowledge G. E. W. Bauer, M. B¨ uttiker, D. P.\nDiVincenzo, J. K¨ onig, M. Misiorny, R. Saptsov and J.\nSplettst¨ osser for useful discussions.\nAppendix A: Storage of Spin-quadrupole Moment\nIn this appendix wegivethe calculationofthe local av-\nerage SQM stored in a ferromagnet, cf. Sec. IIIB3. We\npresent three derivations, each of which unveils different\nphysicalandtechnicalaspectsusedinthemainpart. The\nfirst, most straightforward approach is given in Sec. A1.\nIt shows how the Pauli exclusion hole arises in Eq. (A6),\nwhich provides the key to the physical interpretation of\nexchange SQM. Secondly, we give a technically more so-\nphisticatedderivationinSec. A2, whichwillbehelpfultounderstand all steps of our transport calculations. It ex-\npresses the Pauli exclusion hole in a coordinate-free form\nin Eq.(A17). Thirdly, we presentin Sec. A3 aderivation\nthat makes explicit the spin-triplet content of the corre-\nlations by vector coupling of the spins of electron pairs.\nFinally, we discuss the spin-anisotropy function and de-\nrive a closed expression for the exchange SQM in the flat\nband approximation(cf. IID). Throughoutthe appendix\nwe focus on understanding the exchange contributions to\nthe SQM, which we showed in the absence of tunneling\nto appear only locally for ( r=r′), cf. IIIB3. We there-\nfore only consider one electrode rwith one band nand\nsubsequently drop these indices in all expressions below,\ne. g.crnkσ→ckσ, when convenient.\n1. Exchange SQM and the Pauli Principle\nWe furthermore take a coordinate system for which\nez=ˆJand quantize the spin along this vector. The\ncalculation of the average local SQM starts from the op-\nerator Eq. (53) in the main text. We insert the second-\nquantized form (40) of the spin operator and anticom-\nmutec†\nk1σ′\n1twice to the right:\nQ=/summationdisplay\n{kiσ′\niσi}sσ′\n2σ2⊙sσ′\n1σ1c†\nk2σ′\n2ck2σ2c†\nk1σ′\n1ck1σ1(A1)\n=/summationdisplay\n{kiσ′\niσi}sσ′\n2σ2⊙sσ′\n1σ1c†\nk1σ′\n1c†\nk2σ′\n2ck2σ2ck1σ1(A2)\nThis generates a term δk2k1δσ2σ′\n1c†\nk2σ′\n2ck1σ1, which we\nomittedbecauseitvanishesafterperformingthesumover\nthe spin indices by virtue of s⊙s= 0: for all σ′\n2,σ1\n/summationdisplay\nσ2σ′\n1sσ′\n2σ2⊙sσ′\n1σ1δσ2σ′\n1=∝an}bracketle{tσ′\n2|s⊙s|σ1∝an}bracketri}ht= 0.(A3)\nComputingthe averageofEq.(A2) usingWick’s theorem\n∝an}bracketle{tc†\nk′σ′ckσ∝an}bracketri}ht=δkk′δσσ′f+(εkσ) in the standard way with\nf+(ω) denoting the Fermi function,\n∝an}bracketle{tc†\nk1σ′\n1c†\nk2σ′\n2ck2σ2ck1σ1∝an}bracketri}ht (A4)\n=∝an}bracketle{tc†\nk1σ′\n1ck1σ1∝an}bracketri}ht∝an}bracketle{tc†\nk2σ′\n2ck2σ2∝an}bracketri}ht−∝an}bracketle{tc†\nk1σ′\n1ck2σ2∝an}bracketri}ht∝an}bracketle{tc†\nk2σ′\n2ck1σ1∝an}bracketri}ht\nyields a direct and an exchange part:\n∝an}bracketle{tQ∝an}bracketri}ht=/summationdisplay\nk2k1σ2σ1f+(εk2σ2)f+(εk1σ1)×\n(sσ1σ2⊙sσ2σ1−δk2k1sσ2σ2⊙sσ1σ1) (A5)\n=1\n4/summationdisplay\nk2k1(1−δk2k1)\n/summationdisplay\nσ2σ1σ1σ2f+(εk2σ2)f+(εk1σ1)ez⊙ez.(A6)\nHere we used the result\nsσ1σ1⊙sσ2σ2=sσ1σ2⊙sσ2σ1=σ1σ2\n4ez⊙ez.(A7)31\nClearly, sσ1σ1⊙sσ2σ2=σ1σ21\n4ez⊙ez, whereas for\nσ1=−σ2we have sσ1σ2⊙sσ2σ1=1\n2(ex+iσ1ey)⊙\n1\n2(ex−iσ1ey) =1\n4(ex⊙ex+ey⊙ey) =−1\n4ez⊙ez.\nThe last step follows from/summationtext\niei⊙ei= 0 which is just\nthe traceless, symmetric part of the unit tensor by the\ncoordinate-space completeness relation/summationtext\nieiei=I.\nWith the two-particle operator (A2) in the standard\nsecond-quantized form (see also Eq. (A21) below) the\ncontributionstotheaverage ∝an}bracketle{tQ∝an}bracketri}htcanbediscussedasscat-\ntering processes, treating Qas if it were an interaction\n(although it is tensor-valued). In Fig. 23 we represent\nthe contributions to the average SQM (A5) by Feynman\ndiagrams for scattering processes between an initial pair\nof states (1,2) to a final to pair of states (1′,2′).\nFIG. 23: Feynman diagrams for calculating the (a) direct and\n(b) the exchange contribution the the average SQM.\nThe momenta in the final states are the same as for\nthe initial states, ki=ki′, since the SQM operator does\nnot act on the orbital part of the wave function (cf.\nEq. (A22)). Two different scattering processes are per-\nmitted: the first one is a direct scattering, for which the\nelectron in initial state iends up in state i′, restricting\nthe spin indices to σi=σ′\ni(while already ki=ki′).\nThese direct scattering contributions, multiplied with\ntheir tensor-valued amplitudes sσ1σ2⊙sσ2σ1in (A5) add\nup to the direct SQM. This contribution to the spin\nanisotropy is thus generated by two electrons (labelled\nby their states 1 and 2) as if they were distinguishable ,\ni.e., by treating the two-particle scattering classically.\nThe second type of scattering process, in which the\nparticles are exchanged , accounts for the fact that elec-\ntrons are indistinguishable. This is only possible if the\nmomenta are the same,k1=k2, and furthermore, the\nspins are exchanged, σ′\n1=σ2andσ′\n2=σ1. This ex-\nchange contribution to the average SQM entirely cancels\nthe direct contribution for equal k1=k2, correcting for\nthe treatment of electrons as distinguishable particles.\nIn other words, the exchange SQM accounts for Pauli\n“holes” in the triplet spin-spin correlation tensor. The\nPauli principle thus counters the direct classical contri-\nbution to the spin anisotropy.\nIndistinguishability becomes important if we consider\npairs of electrons from the same k-mode. Due to Pauli’s\nprinciple, their wave function must have a symmetric or-\nbital part and an antisymmetric spin part, that is, they\nform a spin singlet with zero SQM, i. e., triplet correla-\ntions are forbidden. This is analogous to the direct andexchange contributions to the average Coulomb interac-\ntion with respect to Slaterdeterminants (e.g., in Hartree-\nFock theory). In that case, nearby electrons with parallel\nspins repel each other due to the exchange potential. We\nmention that asexpected fromthis analogy, thermalfluc-\ntuationssuppresstheeffectofthePauliprincipleonSQM\nas well, c.f. Eq. (A34) below.\n2. Spin-Trace Technique\nWe now reformulate the above calculation in a way\nthat will be used in the transport calculations (cf. Sec.\nE4): the result (A17) then assumes the coordinate-free\nform presented in Sec. IIIB3.\n∝an}bracketle{tQrr∝an}bracketri}htex=/summationdisplay\nkσσ′ττ′(sr\nσσ′⊙sr\nττ′)δστ′δσ′τ\n×fr\n+(εr\nkσ)fr\n−(εr\nkσ′). (A8)\nHere we reintroduced the electrode index. To recast this\nintocovariant expression, we first introduce the two-\nparticle density of states (16),\nνr\nσσ′(ω,ω′) =/summationdisplay\nn,kδ(ω−εr\nkσ)δ(ω′−εr\nkσ′),(A9)\nand rewrite Eq. (A8) in terms of frequency integrals\n∝an}bracketle{tQrr∝an}bracketri}htex=/summationdisplay\nkσσ′ττ′/integraldisplay\ndωdω′/parenleftBig\nsr\nσσ′⊙sr′\nττ′/parenrightBig\nδστ′δσ′τνσσ′(ω,ω′)fr\n+(ω)fr\n−(ω′). (A10)\nThe 2DOS νr\nσσ′(ω) can be expressed as a matrix element\nin spin space by\nνr\nσσ′(ω,ω′)δστ′δσ′τ\n= 2/summationdisplay\nµ1,µ2r∝an}bracketle{tτ′|ˇrµ1|σ∝an}bracketri}htrAr\nµ1µ2(ω,ω′)r∝an}bracketle{tσ′|ˇrµ2|τ∝an}bracketri}htr.(A11)\nHere we used the four component operator ˇ rwith ˇr0=/BD/√\n2 and ˇri=√\n2sifori=x,y,z. The four dimen-\nsional matrix Ar\nµ1µ2incorporates all relevant 2DOS in-\nformation. It decomposes into a scalar, two vectors and\na tensor in coordinate space:\nAr\n00=1\n4/summationdisplay\nσσ′νr\nσσ′(ω,ω′), (A12)\nAr\ni0=1\n4/summationdisplay\nσσ′σνr\nσσ′(ω,ω′)ˆJr\ni, (A13)\nAr\n0j=1\n4/summationdisplay\nσσ′σ′νr\nσσ′(ω,ω′)ˆJr\nj, (A14)\nAr\nij=1\n4/summationdisplay\nσσ′σσ′νr\nσσ′(ω,ω′)ˆJr\niˆJr\nj.(A15)32\nInserting Eq. (A11) into Eq. (A10) and recasting the sum\nover the spin indices as a trace in spin space yields\n∝an}bracketle{tQrr∝an}bracketri}htex=/integraldisplay\ndωdω′fr\n+(ω)fr\n−(ω′)\n2/summationdisplay\nµ1µ2Ar\nµ1µ2(ω,ω′)Tr[ˇrµ1s⊙ˇrµ2s](A16)\n=/integraldisplay\ndωdω′1\n4/summationdisplay\nσσ′σσ′νr\nσσ′(ω,ω′)\n×fr\n+(ω)fr\n−(ω′)/hatwideJr⊙/hatwideJr. (A17)\nThis recovers the results (64)-(67) obtained in the main\ntext from Eq. (A6). The explicit calculation in the last\nstep is now reduced to using spin-1/2 operator algebra\nsisj=1\n4δij\n/BD+1\n2/summationtext\nkiεijkskands⊙s= 0, i.e., without\nusing matrix elements. These steps are analogous to the\nevaluation of the diagrammatic expressions for the SQM\ncurrent in App. D.\n3. SQM and Triplet Spin Correlations\nFinally, we express the SQM tensor operator Qin\nthe second-quantized form. This allows one to perform\nvector-coupling of the pairs of involved spin, thereby\nmaking explicit that only triplet correlations are “mea-\nsured” by ∝an}bracketle{tQ∝an}bracketri}ht, something that did not become clear in\nthe above calculations. This is merely important for the\nphysical understanding, but seems to bring no advantage\nfor calculations. The many-body quadrupole operator is\na sum overquadrupole operatorsof pairs of particles, the\nlatter labeled by a,b= 1,2,3,...\nQ=/summationdisplay\na<bQab(A18)\nwith Cartesian components i,j=x,y,z:\nQab\nij= 2/parenleftBigg\n1\n2(sa\nisb\nj+sa\njsb\ni)−1\n3δij/summationdisplay\nksa\nksb\nk/parenrightBigg\n.(A19)\nHere we inserted the total spin operator S=/summationtext\nasainto\nEq. (1) and used the result Qaa= 0 (‘a ‘single electron\nhas no anisotropy”, cf. Sec. IIA). The SQM from pair\n∝an}bracketle{tab∝an}bracketri}htcan also be expressed by introducing coupling the\ntwo spins to Sab=sa+sb,\nQab\nij=1\n2(Sab\niSab\nj+Sab\njSab\ni)−1\n3δij/parenleftbig\nSab/parenrightbig2,(A20)\nusingsa\nisa\nj=1\n4δij+i1\n2/summationtext\nkεijksa\nk. Note the factor 2 in\nEq. (A19). The general second quantization prescription\nimmediately gives\nQ=/summationdisplay\n{kiσi}1\n2∝an}bracketle{tk′\n2σ′\n2k′\n1σ′\n1|Q12|k2σ2k1σ1∝an}bracketri}ht\nc†\nk′\n1σ′\n1c†\nk′\n2σ′\n2ck2σ2ck1σ1. (A21)We now make explicit use the particular property of the\nmatrix elements of the pair SQM Q12. First, we note\nthatQ12acts only on the spin of the electrons,\n∝an}bracketle{tk′\n2σ′\n2k′\n1σ′\n1|Q12|k2σ2k1σ1∝an}bracketri}ht\n=δk′\n2k2δk′\n1k1∝an}bracketle{tσ′\n2σ′\n1|Q12|σ2σ1∝an}bracketri}ht.(A22)\nIf we inserted this into Eq. (A21) we would recover\nEq. (A2). Instead of this, we now introduce a singlet-\ntriplet basis for each pair of considered spins σ1andσ2\nabove:\n|S∝an}bracketri}ht=1√\n2/summationdisplay\nσσ|σ¯σ∝an}bracketri}ht, (A23)\n|T0∝an}bracketri}ht=1√\n2/summationdisplay\nσ|σ¯σ∝an}bracketri}ht, (A24)\n|Tm∝an}bracketri}ht=|mm∝an}bracketri}ht, m=± (A25)\nwhere ¯σ=−σ. The crucial point is that Q12only has\nmatrix elements in the triplet sector. This follows from\nthe fact that Q12is symmetric under exchange of the\nspins, i.e.,/bracketleftbig\nP,Q12/bracketrightbig\n= 0 wherePis the exchange opera-\ntor. Therefore Q12is block-diagonal with respect to the\neigen spaces of P, which are here the singlet and triplet\nstates satisfying P|S∝an}bracketri}ht=−|S∝an}bracketri}htandP|Tm∝an}bracketri}ht= +|Tm∝an}bracketri}ht.\nThus,∝an}bracketle{tS|Q12|Tm∝an}bracketri}ht=∝an}bracketle{tTm|Q12|S∝an}bracketri}ht= 0. Moreover,\nthe diagonal singlet block is zero, ∝an}bracketle{tS|Q12|S∝an}bracketri}ht= 0 by\nEq. (A20) with a= 1,b= 2 and S12|S∝an}bracketri}ht= 0, completing\nthe proof. As a result\n∝an}bracketle{tσ′\n2σ′\n1|Q12|σ2σ1∝an}bracketri}ht=/summationdisplay\nmm′∝an}bracketle{tσ′\n2σ′\n1|Tm∝an}bracketri}ht∝an}bracketle{tTm|Q12|Tm′∝an}bracketri}ht∝an}bracketle{tTm′|σ2σ1∝an}bracketri}ht.(A26)\nInserting Eq. (A26) into Eq. (A21), we obtain the central\nresult of the appendix,\nQ=/summationdisplay\nmm′1\n2∝an}bracketle{tTm′|Q12|Tm∝an}bracketri}ht/summationdisplay\n{ki}Em′†\nk2k1Em\nk2k1,(A27)\nwith two-particle operators that explicitly generate only\ntriplet pairs :\nEm\nk2k1=/summationdisplay\nσ2σ1∝an}bracketle{tTm|σ2σ1∝an}bracketri}htck2σ2ck1σ1(A28)\n=/braceleftbiggck2mck1mm=±1\n1√\n2/summationtext\nσck2σck1¯σm= 0.(A29)\nConsideredas an interaction, Qthus only scatters triplet\ncorrelated pairs of electrons. Due to the restrictions on\nthe spins in these operators Em\nk2k1, the averages are\n∝an}bracketle{tEm′†\nk2k1Em\nk2k1∝an}bracketri}ht=δmm′×\n/braceleftbiggf(εk2m)f(εk1m)(1−δk2k1), m =±1\n1\n2/summationtext\nσf(εk2σ)f(εk1¯σ)(1−δk2k1), m= 0(A30)\nwith the tensor-valued matrix elements ∝an}bracketle{tT+|Q12|T+∝an}bracketri}ht=\n∝an}bracketle{tT−|Q12|T−∝an}bracketri}ht=−∝an}bracketle{tT0|Q12|T0∝an}bracketri}ht/2 =1\n2ez⊙ezgiven by33\nEqs. (2)-(3) in the main text. These relations follow\nfrom the fact that Q12is traceless in the Hilbert space,/summationtext\nm=0,±1∝an}bracketle{tTm|Q12|Tm∝an}bracketri}ht= 0 and that the m=±states\nhave the identical spin-anisotropy. We recover Eq. (A6):\n∝an}bracketle{tQ∝an}bracketri}ht=1\n4ez⊙ez/summationdisplay\nk2k1(1−δk2k1)\n×/parenleftBigg/summationdisplay\nm=±f+(εk2m)f+(εk1m)\n−/summationdisplay\nσ=±f+(εk2σ)f+(εk1¯σ)/parenrightBigg\n(A31)\nThis derivation, however, shows explicitly that the m=\n±1 terms contribute the same, uniaxial anisotropy ten-\nsor∝an}bracketle{tT± |Q12|T±∝an}bracketri}ht, whereas the m= 0 term con-\ntributes an easy-plane anisotropy tensor ∝an}bracketle{tT0|Q12|T0∝an}bracketri}ht=\n−2∝an}bracketle{tT+|Q12|T+∝an}bracketri}ht. Moreover, the Pauli-exclusion hole\nfactor 1−δk2k1is immediately explicit. Thus, ∝an}bracketle{tQ∝an}bracketri}htcan\nbe calculated by first accounting for triplet correlations\nbetween spins of pairs of electrons in all possible orbitals,\nincluding the same orbital,\n∝an}bracketle{tQ∝an}bracketri}htdir=1\n4/summationdisplay\nk2k1σ2σ1σ2f(εk2σ2)σ1f(εk1σ1)ez⊙ez\n=∝an}bracketle{tS∝an}bracketri}ht⊙∝an}bracketle{tS∝an}bracketri}ht, (A32)\ngiving Eq. (57), and then subsequently cancelling the lat-\nter violation of the Pauli principle by the exchange term,\n∝an}bracketle{tQ∝an}bracketri}htex=−qexez⊙ez (A33)\nwith the positive exchange magnitude\nqex=1\n4/summationdisplay\nk(f+(εk↑)−f+(εk↓))2.(A34)\nWe obtain Eq. (62) from the main text. Finally, we\nshow that the two-particle DOS νσσ′can be decomposed\nexplicitly into tripletDOS components: converting the\nsums in Eq. (A34) to integrals we obtain\nqex=/integraldisplay\ndωdω′f+(ω)f+(ω′)/summationdisplay\nσσ′σσ′νr\nσσ′(ω,ω′).(A35)\nThe only relevant combination of the 2DOS in the above\nexpression can be recast as\n/summationdisplay\nσσ′σσ′νr\nσσ′=νr\nT++νr\nT−−√\n2νr\nT0(A36)\nwith the triplet exchange 2DOS function ( m=±)\nνr\nTm(ω,ω′) :=νr\nmm(ω,ω′), (A37)\nνr\nT0(ω,ω′) :=1√\n2/summationdisplay\nσνr\nσ¯σ(ω,ω′).(A38)\nThis gives a precise decomposition into triplet spin cor-\nrelations that contribute to ∝an}bracketle{tQrr∝an}bracketri}htex.4. Spin-anisotropy Function\nFinally, we further substantiate the physical interpre-\ntation of the anisotropy function, which plays a key role\nin the main text. The basic idea of “quadrupolarization”\nof two triplet-correlated electrons is simply to “count”\nwhether the spins are parallel ( ↑↑or↓↓, counted as +),\nor antiparallel ( ↑↓or↓↑, counted as −). In both cases\ntheirindividual orientations,i.e., their dipolarization ↑or\n↓, is ignored. Eq. (A35) precisely expresses this notion\nfor the exchange SQM. It is instructive to start from the\nk-sum representation (A34) and to write it as\nqex=/summationdisplay\nkσf+(εkσ)akσ. (A39)\nHere,giventhat an electron with spin σoccupies orbital\nk, we “count” by\nakσ=/summationdisplay\nσ′σσ′\n4f(εkσ′) (A40)\nthe average quadrupolarization contribution from elec-\ntrons in that same orbital k: parallel spin σ′=σgives\n+f+(εkσ), antiparallel σ′= ¯σgives−f(εk¯σ).42Convert-\ningthe sum to anintegral, weobtain Eq.(64) ofthe main\ntext:\nqex=/integraldisplay\ndωf+(ω)¯ν(ω)a(ω). (A41)\nThe anisotropy function a(ω) =/summationtext\nσaσ(ω) has two con-\ntributions:\n¯ν(ω)aσ(ω) :=/summationdisplay\nkakσδ(ω−εkσ).(A42)\nThe quantity ¯ ν(ω)aσ(ω) is the exchange quadrupolariza-\ntion of a spin σelectron at energy ω. One should note\nthat the function akσ, defined by Eq. (A40), does not\nonly depend on the energy εkσ, but also on the energy\nεk¯σ. Sinceεk¯σis not necessarily an implicit function of\nεkσfor arbitrary band structures, one can in general re-\nformulate Eq. (A42) only in terms of the 2DOS (16):\n¯ν(ω)aσ(ω) =/integraldisplay\ndω′f+(ω′)/summationdisplay\nσ′σσ′\n4νσσ′(ω,ω′),(A43)\nresulting in Eq. (67) of the main text.\nHowever, for the Stoner model, which we discuss from\nhereon, the simple relation εk¯σ=εkσ−σJ/2 can be\nexploited to express akσas a function of εkσonly. We\ntherefore obtain the simpler result\n¯ν(ω)aσ(ω) =νσ(ω)(f+(ω)−f+(ω+σJ)),(A44)\nwhich only depends on the 1DOS νσ. This unfortu-\nnately hides the underlying two-particle nature of the\nexchange SQM, but aids the interpretation of the total\nspin-anisotropy function: Eq. (A44) shows that the con-\ntributiona↑(ω) from up-spins is positive and comes from34\nthe range of energies µ−J < ω < µ , whereas the con-\ntributiona↓(ω) from down-spins is negative and comes\nfrom energies µ<ω<µ +J(both up to thermal smear-\ning). Adding both contributions yields for the full spin-\nanisotropy function after some manipulations\na(ω) =1\n4[2f+(ω)−f+(ω+J)−f+(ω−J)]\n+1\n4n(ω)[f+(ω+J)−f+(ω−J)].(A45)\nHere it should be noted that n(ω) is not independent of J\nbut is a function of it through Eq. (13), (15) and (18).43\nThe combinations of Fermi-functions are non-zero only\nfor energies |ω−J|/lessorsimilarT.\nIt was pointed out in Sec. VC for a purely thermally\nbiased tunnel junction that the finite SQM current that\nremains whenever the spin current vanishes arises en-\ntirely from the exchange SQM, i.e., from the antisym-\nmetric part of the spin-anisotropy function a(ω) relative\ntoµ. Generally, when assuming a weakly energy depen-\ndent average DOS ¯ ν(ω) in the range |ω−J|/lessorsimilarT, the\nfirst term Eq. (A45) always gives rise to such a term.\nThe second term, in which the spin-polarization n(ω) is\nmultiplied by a symmetric function relative to µ, can\nonly cancel this function if n(ω) is strongly antisymmet-\nric (i.e.,n(ω) =±1 forω≷0, up to thermal smearing).\nIf we further specialize to the approximation of a flat\nband symmetric about µ(cf. Sec. IID), this second term\nis exactly zero because the spin polarization vanishes in\nthe window [ ω−J,ω+J] up to thermal smearing. Only\nthe first line of Eq. (A45) remains and gives a thermally\ninduced pure SQM current as discussed in Sec. VC: sub-\nstitutingx= (ω−µ)/T, Eq. (A41) can be rewritten as\nqex=¯νT\n4/integraldisplay\ndxf(x)[2f(x)−f(x−j)−f(x+j)],(A46)\nwheref(x) = [ex+1]−1andj=J/T. Asa(ω) is nonzero\nin an energy window 2 Jcentered at µ, which is far away\nfrom the band edges, we can replace the 1DOS by its\nconstantvalue ¯ νandextendthelimitsintegrationto ±∞.\nUsing the identity f(x) = 1−f(−x) and the result\n/integraldisplay\ndxf(x)f(−(x−y)) =/braceleftbigg\n1y= 0\nyb(y) else,(A47)\nwhereb(x) = [ex−1]−1is the Bose function, and taking\nthe limity→0 we obtain Eq. (70) of the main text:\nqex=¯ν\n4[−2+jb(j)+(−j)b(−j)] (A48)\n=¯ν\n2/bracketleftbiggj\n2coth/parenleftbiggj\n2/parenrightbigg\n−1/bracketrightbigg\n. (A49)\nAppendix B: Symmetric and Traceless Tensors\nIn this appendix, we collect some relevant results on\nsymmetric, traceless tensors. We first show how a par-\nticular type of such tensors, constructed from two realvectorsaandb,\nA=a⊙b=1\n2(ab+ba)−1\n3(a·b)I,(B1)\ncan be diagonalized using dyadic calculus24, i. e., with-\nout introducing a coordinate system. The tensor Acan\nbe expressed in terms of its principal values λµand nor-\nmalized vectors ˆvµthat define the principal axes as\nA=/summationdisplay\nµ=0,±λµˆvµˆvT\nµ. (B2)\nThe results (123)-(132) of the main text can then be\nobtained from Eq. (115) by setting a=ˆJLandb=\nELˆJL+ALˆJR+TL/parenleftBig\nˆJL׈JR/parenrightBig\n. To diagonalize (B1), we\nhave to distinguish two cases:\nCase (i) a∦b: eigenvalues\nλµ=−1\n3(a·b)+1\n2δµ,±[(a·b)+µab],(B3)\nwhereµ= 0,±and normalized eigenvectors\nˆv0=1\n|a×b|(a×b), (B4)\nˆv±=1\n2ab(ab±(a·b))[ab±ba] (B5)\nwitha=|a|,b=|b|.\nCase (ii) a∝bardblb, i. e.,b=αa: eigenvalues for µ= 0,±\nλµ=/parenleftbig\nδµ,+−1\n3/parenrightbig\nαa2, (B6)\nandˆv+=a/aandˆv0,ˆv−are any two orthonormal\nvectors that span the plane perpendicular to ˆv+.\nTo prove case (i), we first note that one principal axis\nis obviously v0=a×b∝ne}ationslash= 0:\nA·v0=/bracketleftbig1\n2(ab+ba)−1\n3(a·b)I/bracketrightbig\n·(a×b) (B7)\n=−1\n3(a·v)v0=λ0v0. (B8)\nIn order to derive the the remaining principal values, we\nneed to set up the characteristic equation:\n0 = det( A−λI) (B9)\n= det(A)−λspm(A)+λ2tr(A)−λ3,(B10)\nwhere the coefficients are given by the trace, the sum of\nprincipal minors and the determinant of A, respectively:\ntr(A) =/summationdisplay\niAii, (B11)\nspm(A) =1\n2tr/parenleftbig\nA×\n×A/parenrightbig\n, (B12)\ndet(A) =1\n6/parenleftbig\nA×\n×A/parenrightbig\n:A. (B13)\nHere we used the shorthand notations/parenleftbig\nA×\n×A/parenrightbig\ni1i2=εi1j1k1Aj1j2Ak1k2εj2k2i2 and\nB:A=/summationtext\ni,jBijAij. Eqs. (B11)-(B13) are the\n(only) three rotational invariants, i. e. they are invari-\nant under transformations A→R·A·RTwhereR35\nis a rotation matrix: R·RT=RT·R=I. Inserting\nEq. (B1) we obtain\ntr(A) = (a·b), (B14)\nspm(A) =−1\n3(a·b)2−1\n4(a×b)2,(B15)\ndet(A) =2\n27(a·b)3+1\n12(a·b)(a×b)2.(B16)\nInserting these into Eq. (B10), one finds that λ0, given\nby (B3) with µ= 0, is indeed a principal value of A.\nBy polynomial division we obtain a quadratic equation\nfor the remaining principal values λ±, which is solved\nby Eq. (B3) with µ=±. The general solution of\n(A−λ±I)·v±= 0is given by24:\nv±=c·/bracketleftBig\n(A−λ±I)×\n×(A−λ±I)/bracketrightBig\n(B17)\n=−1\n2c·[](a×b) (a×b)\n+/parenleftbig\n2q±(a·b)−q2\n±/parenrightbig\nI−q±(ab+ba)/bracketrightbig\n.(B18)\nwhereq±= (a·b)±|a||b|. Herecis a vector such that\nv±∝ne}ationslash= 0 and either c=aorc=bfulfills this condition,\nyielding the result (B5) after normalization.\nFor case (ii) we have A=a⊙αa=α/parenleftbig\naa−a2I/3/parenrightbig\n.\nSincea×b= 0, the above results cannot be applied.\nThe vector aobviously defines a principal axes since\nA·a= +2\n3αa2a, and for any vector a⊥in the plane\nperpendicular to awe have A·a⊥=−1\n3αa2. This con-\nfirms the principal values (B6), completing the proof.\nAppendix C: Rotation of the SQM by SQM\nCurrents\nIn this appendix we show that a SQM tensor Qis\nrotated if it does not commute with the SQM current\ntensorIQas pointed out in VB3. More formally,\n[Q,IQ]−= 0 implies the both tensors have collinear\nprincipal axes. We outline the analogous situation for\nthe spin vector: it is geometrically clear that the spin\nSdoes not rotate, that is, S=S(t)e3in some time-\nindependent basis if and only if the spin current vector\nIS=˙Sis collinear to Sat all times.\nWe now outline a proof of this statement, which can\nbe extended to the SQM. Assume Sdoes not rotate, then\nS=S(t)e3in sometime independent basis eiand. Then\n˙S=˙S(t)e3=IS. Thus, IS·S=±|S||IS|e3·e3=\n±|S||IS|. Conversely, assume S·IS=±|S||IS|, i. e.\nS=S(t)e3(t) andIS=IS(t)e3(t), but for some time-\ndependent e3(t). One can derive a contradiction from\nthe latter assumption: first, we have ˙S=S(t)e3(t) +\nS(t)˙e3(t) =IS=IS(t)e3(t). Byassumption, the normal-\nization is preserved, that is,d\ndt(e3·e3) = 2e3·˙e3= 0.\nThus, if ˙e3⊥e3, we have ˙S(t) =IS(t) and either\nS(t) = 0 (which implies a trivial spin) or ˙e3= 0, contra-\ndicting our assumption. This completes the proof.\nWe now prove a similar statement for the SQM tensor\nQand its current IQ=˙Q,\nQhas fixed principal axes ⇔[Q,IQ]−= 0.(C1)AssumeQ=/summationtext\niQi(t)eieT\nihasatime-independentbasis,\ni.e., only the principal values Qi(t) change but the prin-\ncipal axes eido not rotate. Thus, ˙Q=/summationtext\ni˙Qi(t)eieT\ni=\nIQ=/summationtext\nijIij(t)eieT\nj, implying Iij(t) =δij˙Qi(t) and\nconsequently [ Q,IQ]−= 0. Now assume the con-\nverse, i. e., IQandQcommute. Since both are real,\nsymmetric tensors, they can be diagonalized and since\nthey commute they have common principal axes: Q=/summationtext\niQi(t)ei(t)eT\ni(t) andIQ=/summationtext\niIi(t)ei(t)eT\ni(t). We\nnextaderiveacontradictionfromtheassumption ˙ei(t)∝ne}ationslash=\n0. We first find ˙Q=/summationtext\ni/bracketleftBig\n˙QieieT\ni+Qi/parenleftbig\n˙eieT\ni+ei˙eT\ni/parenrightbig/bracketrightBig\n=\nIQand note that ˙eieT\niis independent of ekeT\nkfor all\nkbecause ˙ei⊥ek. Thus, we have Ii=˙Qiand either\nQi= 0 for allior˙eieT\ni+ei˙eT\ni= 0, which would implies\n˙ei= 0. Thus, eiis time independent and the proof is\ncomplete.\nAppendix D: Charge, Spin and Spin-quadrupole\nCurrent in O( Γ)\nIn this section, we apply the general diagrammatic\ntechnique developed in App. E accounting only for con-\ntributions up to 1st order in the tunneling rate Γ assum-\ning spin-conserving tunneling. For the charge and spin\ncurrent, the construction as described in E7 gives two\ncontributing diagrams, depicted in Fig. 24 (a).\nFIG. 24: Diagrams representing O(Γ)-contributions to (a)\ncharge and spin current, (b) direct SQM current and (c) ex-\nchange SQM current.\nIn first order, there is only one irreducible contraction\npossible. Furthermore,theh. c. indices ηarefixedbythe\nobservable vertex. However, there are two possibilities\nto chose the charge indices χfor the tunneling double\nvertex. The total sign of the diagram equals χsince we\nhave(i) afactor −1 due to one crossing(ii) a factor ¯ χdue\nto the early and late vertices and (iii) a factor + since\nthere are no intermediate vertices. This results in\n∝an}bracketle{tIL\nRµ∝an}bracketri}ht= 2Im\n/integraldisplay\ndω1dω1′1\ni0−ω1′+ω1/summationdisplay\nρ1ρ1′τ1τ1′\n/summationdisplay\nχχ(FL\nχ)ρ1(ω1)(FR\n¯χ)ρ1′(ω1′)(ˇtτ1ˇtτ2)\ntr(ˇrρ′\n1ˇrτ2ˇrρ1rµˇrτ1)], (D1)36\nwhere (Fi)ρi= ¯νri(ωi)fri\n¯χi(ωi)√\n2[δρi,0+(1−δρi,0)nrˆJr\nρi].\nNote thatrµis associated with the current vertex and\ntherefore has no “check”. For our case, the tunneling\nis spin-independent so that ˇtτ=√\n2tδτ,0. We will use\nfurthermore ˇ r0=r0/√\n2 = /BD/√\n2, ˇri=√\n2ri=√\n2sifor\ni= 1,2,3 and\ntr(ˇrρ1′ˇrρ1ˇrµ) =1√\n2δµ0δρ1ρ1′\n+1√\n2¯δµ,0(δρ10δρ1′µ+δρ1′0δρ1µ+iερ1′ρ1µ).(D2)\nWe obtain the chargecurrent (89) givenin the main text,\n∝an}bracketle{tIL\nN∝an}bracketri}ht= 2Im/bracketleftbigg/integraldisplay\n11′1\nπΓLR\n11′(−∆11′)/parenleftbig\n1+nL\n1·nR\n1′/parenrightbig\n/parenleftbigg\nP1\nω1−ω1′−iπδ(ω1−ω1′)/parenrightbigg/bracketrightbigg\n= 2/integraldisplay\n11′Γ1∆1/parenleftbig\n1+nL\n1·nR\n1/parenrightbig\n, (D3)\nand for the spin current we obtain Eq. (90) of the main\ntext,\n∝an}bracketle{tIL\nSi∝an}bracketri}ht= 2Im/bracketleftbigg/integraldisplay\n11′ΓLR\n11′(−∆11′)\n×/parenleftbigg\nP1\nω1−ω1′−iπδ(ω1−ω1′)/parenrightbigg\n×1\n2/parenleftbig\nnL\n1+nR\n1′+i/parenleftbig\n−nL\n1×nR\n1′/parenrightbig/parenrightbig/bracketrightbigg\n(D4)\n=/integraldisplay\n1Γ1/bracketleftBig\n∆1/parenleftbig\nnL\n1+nR\n1/parenrightbig\n+nL\n1\n¯νR\n1×βR\n1+βL\n1×nR\n1\n¯νL\n1/bracketrightBig\n.(D5)\nHere we introduced the short-hand notations Γ 1= ΓLR\n11,\nnr\n1=nr(ω1) and furthermore\nΓLR\n11′= ΓRL\n11′= 2π|t|2¯νL(ω1)¯νR(ω1′) (D6)\n∆11′=fR\n+(ω1′)−fL\n+(ω1) (D7)\nβr(ε) =−/integraldisplay\ndωPfr\n+(ω)¯νr(ω)nr(ω)\nω−ε(D8)\nHereP/parenleftbig1\nz/parenrightbig\n= Re/parenleftBig\n1\nz+i0/parenrightBig\ndenotes the principal value.The calculation of the SQM current in O(Γ) for spin-\nindependent tunneling proceeds in a similar way. How-\never, due to its two-particle nature, there are two pairs\nof diagrams with different contraction topologies, which\nmake up the direct (Fig. 24 (b)) and the exchange con-\ntribution (Fig. 24 (c)) to the SQM current, respectively.\nEach pair differs with respect to charge indices.\nOnthelevelofdiagrams,oneimmediatelyseesthatthe\nexpressions involving the spin operator and the spin cur-\nrent operator factorize for the direct contribution since\nthe contraction labelled with 2’ in Fig. 24 (b) does not\ncross any other lines. This corresponds to the product of\nthe expectation value of two operators. Therefore, with-\nout only further calculation, we obtain\n∝an}bracketle{tILL\nQ∝an}bracketri}htdir= 2∝an}bracketle{tIL\nS∝an}bracketri}ht⊙∝an}bracketle{tSL∝an}bracketri}ht. (D9)\nwhere∝an}bracketle{tIL\nS∝an}bracketri}htis the previouslycalculated spin current (D5).\nTheevaluationoftheexchangecontributionis morecom-\nplicated because the 2′-contractiondoes crossother lines.\nApplying the diagram rules from E8, we obtain\n∝an}bracketle{tILL\nQ∝an}bracketri}htex= 2·2/summationdisplay\nχ¯χIm/integraldisplay\ndω1dω1′dω2′\n2t2fL\n−(ω2′)fL\nχ(ω1)(FR\n¯χ)ρ1′(ω1′)\ni0+ω1−ω1′\n×AL\nµνtr(ˇrµs⊙ˇrνsˇrρ1′) (D10)\nwhere the 2DOS (16) enters through the matrix AL\nµν\ngiven by (A12)-(A15). The first factor 2 comes from\nHermitian conjugation symmetry, the second factor of\n2 is due to the product rule when applying the deriva-\ntive to the SQM operator, which is quadratic in spin (cf.\nEq. (85)) and the third one in the second line is associ-\nated with the SQM current vertex (see Sec. E8). The\nsign factor is obtained as follows: The number of cross-\nings is even and the intermediate spin vertex has ηe= +,\ngiving no sign, but from the early and late vertex, we ob-\ntain a sign factor ¯ χ. It remains to calculate the spin trace\nby employing the anticommutation relations of spin-1/2\noperatoralgebra and the identity s⊙s= 0 (cf. Eq. (51)):\n∝an}bracketle{tILL\nQ∝an}bracketri}htex= 2/integraldisplay\n1,2′fL\n−(ω2′)ΓLR\n1∆1/bracketleftbigg(AL\n12′)0i\n¯νL\n1nR\n1ei⊙ˆJR+(AL\n12′)ij\n¯νL\n1ei⊙ej/bracketrightbigg\n+2/integraldisplay\n1,1′,2′fL\n−(ω2′)ΓLR\n11′∆11′P1\nω1−ω1′/bracketleftbigg(AL\n12′)ij\n¯νL\n1nR\n1′ei⊙/parenleftBig\nej׈JR/parenrightBig/bracketrightbigg\n(D11)\n=−2/integraldisplay\n1ΓLR\n1∆1/bracketleftBig\n˜aL\n1nR\n1ˆJL⊙ˆJR+aL\n1ˆJL⊙ˆJL/bracketrightBig\n−2/integraldisplay\n1,1′ΓLR\n11′∆11′P1\nω1−ω1′/bracketleftBig\naL\n1nR\n1′ˆJL⊙/parenleftBig\nˆJL׈JR/parenrightBig/bracketrightBig\n.(D12)\nIn the last step, we inserted fL\n−(ω2′) = 1−fL\n+(ω2′) into Eq. (D11) and used that the contribution from 137\nvanishes when summing over all spin indices. Restoring\nall indices explicitly, we obtain Eq. (91), the main result\nof the paper. Moreover, we identified the spin-anisotropy\nfunctions (104) and (66):\naL(ω) =/integraldisplay\ndω′fL\n+(ω′)/summationdisplay\nσσ′σσ′\n4νL\nσσ′(ω,ω′)\n¯νL(ω),(D13)\n˜aL(ω) =/integraldisplay\ndω′fL\n+(ω′)/summationdisplay\nσσ′σ′\n4νL\nσσ′(ω,ω′)\n¯νL(ω).(D14)\nAppendix E: Covariant Real Time Diagrammatics\nIn this appendix, we give a self-contained derivation\nof the technique used for calculating the charge, spin-\ndipole and SQM currents. Using this technique the ac-\ntual calculation becomes very compact and is presented\nSec. VA. The interest in presenting the derivation here\nlies in three factors: (i) The real-time technique is more\ngeneraland can be applied to morecomplex systems con-\ntaining stronglyinteractinglocalizedsystems. Therefore,\nit is not often applied to non-interacting systems since\nother approaches are available in that case. However, for\nthe calculation of multi-particle averages, its practical\nrules of calculation prove to be very convenient. There-\nfore it is of interest to point out how the technique sim-\nplifies when applied to non-interacting problems. (ii) We\nreformulate the real-time technique here such that one\ncan deal more efficiently with any non-trivial spin de-\npendencies. In a forthcoming work we show that this\ngeneralizes to the more complex cases19. (iii) Finally, it\nis also of great help to have these simpler calculations,\nformulated in the same way, for comparison the more\ncomplex ones (for which other approaches do not work\nanymore)19.\nAfter reviewing the compact Liouville space\nnotation25–27in (E1), we indicate how the general\nreal-time approach simplifies and show that the calcula-\ntion of operator expectation values in the long-time limit\nand to any order in the tunnel-coupling Γ = 2 πνLνR|t|2\namounts to evaluation of irreducible diagrams in a\nperturbation expansion (E3). The central technical\nachievement of this paper is a covariant formulation of\nthese diagram rules for the charge, spin-dipole and SQM\ncurrent: the expressions they produce are manifestly\ninvariant under the change of the coordinate system and\nof the spin-quantization axis (cf. Sec. IIIB1 and IIIB).\nThey are thus coordinate-free both in real space and in\nHilbert / Liouville space. This reformulation is crucial\nto keep the calculation of the non-equilibrium steady\nstate average of the spin-quadrupole moment tractable.\nThe required steps are:\n1. Separation of spin and energy dependence in the di-\nagrammatic expressions, recasting the spin part as traces\nover Pauli operators (E4),\n2. Collection of all sign factors (E5),3. Halving the number of diagrams by exploiting their\ncomplex conjugation symmetry (E6),\n4. Identification of observable-specific diagram rules\nfor charge, spin-dipole (E7), and finally\n5. Spin-quadrupole current (E8).\nSteps 1, 4 and 5 are presented for the first time in\nthis paper. We note that the technique formulated is\nmore general than the problem of interest in two ways:\nit applies to (i) arbitrary orders of the tunnel coupling\nand (ii) models with spin-dependent tunnel amplitudes.\n1. Compact Notation in Liouville Space\nThe calculation is formulated entirely in Liouville\nspace, the space of linear operators acting on a Hilbert\nspace of a quantum mechanical system. The linear trans-\nformations of operators, the elements of Liouville space,\nare called superoperators. Although our notation follows\nRef.25and recent developments reported in Ref.27, we\nhave made some further convenient modifications which\nwarrant some discussion.\nSimilar to any usual operator, any superoperator can\nbe expressed in terms of field superoperators , which we\ndefine following27by\nJ1·= (−χ1η1)N/braceleftbigg\ncr1η1n1k1σ1·χ1η1=−\n·cr1η1n1k1σ1χ1η1= +.(E1)\nHere the ·denotes the operator argument of the super-\noperator. Here N·= [N,·] is the particle number super-\noperator. The subscript “1” is an abbreviation for all\nindices\n1 = (χ1,η1,r1,n1,k1,σ1). (E2)\nHereη1is the Hermitian conjugation index, which deter-\nmines whether the field operator is a creation operator\n(cη1=−=c†) or an annihilation operator ( cη1=+=c)\nof an electron in electrode r1in bandn1in modek1\nwith spinσ1. New in this notation is the charge index\nχ, which distinguishes whether physically the total su-\nperparticle number is increased ( χ= +) or decreased\n(χ=−) by the action of J1(noteη1does not have such\nphysical meaning: an annihilation (creation) operator\nacting from the right increases (decreases ) the superpar-\nticle number). We prefer this physically moremeaningful\ncharge index χinstead of the commonly combination of\nthe Keldysh index p=−χη.\nThetimeevolutionofthedensityoperatorinSec.E3is\ngenerated by the Liouvillian superoperator L=L0+LT\ndescribing the internal evolutions, L0·= [H0,·], and the\none due to tunneling, LT= [HT,·]. Using Eq. (E1), we\nobtain\nL0=χ1δη1,−ε1J1J¯1 (E3)\nwithε1=εr1\nn1k1σ1and\nLT=T11′J1J1′, (E4)38\nwhere\nT11′= ¯χ1′δη1¯η1′δχ1¯χ1′δr1Lδr1′R[TLR\nσ1σ1′]η1′(E5)\nHere we extended the use the Hermitian-conjugation in-\ndexηassuperscript to indicate complex conjugation, i.\ne., [TLR\nσ1σ1′]+:=TLR\nσ1σ1′and [TLR\nσ1σ1′]−:=TLR∗\nσ1σ1′. In both\nEqs. (E3) and (E4) we implicitly sum over all indices\ncontained Eq. (E2).\nOur main interest is to deal efficiently with the de-\npendence of expressions on the choice of the spin-\nquantization axis. However, for non-collinear spin-\npolarized systems, there exists no specific choice for\nthe spin-quantization axis that simplifies the calculations\nconsiderably. Thebeststrategyisthereforetocompletely\nremove a reference to the spin quantization axis in all\nexpressions. We start with the tunneling Liouvillian, for\nwhich we allow any type of symmetry breaking tunneling\nprocesses, for example due to a magnetic impurity in the\nbarrier. The most general tunneling amplitudes reads\nTLR\nσσ′=L∝an}bracketle{tσ|ˇt·ˇr|σ′∝an}bracketri}htR, (E6)\nwhereˇr·ˇt=/summationtext3\nµ=0ˇrµˇtµ. Hereˇris a four-component\nvector of operators ˇ r0= /BD/√\n2 and ˇri=√\n2sifor\ni=x,y,zforming an basis for the Liouville space formed\nby operators on the spin-1/2 Hilbert space. The ba-\nsis is orthonormal with respect to the scalar product\n(A,B) = tr(A†B). The tunneling is completely specified\nby a four-component vector ˇt: ifLTis spin-conserving,\nas assumed in the main part of the paper, the spatial\ncomponents of ˇtµ=δµ,0√\n2tmust be zero (and Eq. (E6)\nreduces to Eq. (11) of Sec. II). Spin non-conserving tun-\nneling processes are thus introduced by any further non-\nzero components ˇti=ti/√\n2 fori=x,y,z. Note that\nthe spin-dependence through the bra and ket in Eq. (E6)\nmerely reflects the choice of (different, arbitrary) quanti-\nzation axes for the field operators in the reservoirs con-\nnected by the tunneling: clearly it should cancel out of\nthe final answer. Written in the form (E6), the tunnel-\ning Liouvillian is explicitly covariant: changing either of\nthese quantization axis merely changes the meaning of\nthe dummy indices σ,σ′. There is also no explicit de-\npendence on the coordinate system either, since ˇt·ˇris a\ncoordinate-free expression.\n2. Wick’s theorem for Super Operators\nFor the perturbative calculation of expectation values\nin Sec. E3, we will use Wick’s theorem for the super field\noperators as defined in Eq. (E1), which reads as28:\n∝an}bracketle{tJn...J1∝an}bracketri}ht= tr(Jn...J1ρ0)\n=/summationdisplay\ncontr.(−1)NP/productdisplay\ni>jγij (E7)\nwith the grand canonical distribution\nρ0=/productdisplay\nr1\nZre−(Hr\n0−µNr)/Tr, (E8)the partition function Zr= trr(e−(Hr\n0−µNr)/Tr) and the\ncontraction function\nγ11′= (−χ1′η1′)δ1¯1′|excl.χf1. (E9)\nHeref1=f(¯χ1(εr1\nk1σ1−µr1)/Tr1) with the Fermi func-\ntionf(x) = 1/(ex+ 1). In Eq. (E9), δ1¯1′|excl.χdenotes\na Kronecker symbol for all indices 1 and ¯1′except for\nthe charge indices χ. In agreement with physical intu-\nition, the chargeindex χ1, at the beginning ofthe process\ndetermines the type of distribution function appearing:\nχ1′= +(−)correspondstoaparticle(hole). In Eq. (E7),\nwe sum as usual over all possible pair contractions and\nNPis the signature of the permutation that is needed to\ndisentangle all pairs of contracted superoperators while\nkeeping the order of the contracted operators within a\npair. The easyformofEq.(E7) reliesonthe fact that the\nfield superoperators obey anti-commutation relations:\n[J1,J1′]+= (−χ1η1)δ1¯1′ (E10)\nwithδ1¯1′=δr1r1′δk1k1′δn1n1′δσ1σ1′δχ1χ¯1′δη1η¯1′with the\nconjugate multi-index:\n¯1 = (−χ1,−η1,r1,n1,k1,σ1).(E11)\nThe inclusion of the fermion-parity superoperator\n(−χ1η1)Nin Eq. (E1) is crucial for the validity of Eq.\n(E10) from which Eq. (E7) basically follows, as pointed\nout by Saptsov et. al.27.\n3. Perturbative Calculation of Expectation Values\nThe expectation value of an observable Ais given by\n∝an}bracketle{tA∝an}bracketri}ht(t) = Tr(Aρtot(t)) (E12)\nwhereρtot(t) =e−iL(t−t0)ρ0is the time-dependent den-\nsity operator of the system and L=L0+LTis the Li-\nouvillian. We are interested in the long-time limit of\nEq. (E12), which by virtue of the final value theorem\nA:= lim t→∞tr(Aρ(t)) = (−i)limz→i0z∝an}bracketle{tA∝an}bracketri}ht(z)(E13)\nfollows from the Laplace transform ∝an}bracketle{tA∝an}bracketri}ht(z) =/integraltext∞\nt0dteizt∝an}bracketle{tA∝an}bracketri}ht(t) of∝an}bracketle{tA∝an}bracketri}ht(t) with Im( z)>0. We\nobtain\n∝an}bracketle{tA∝an}bracketri}ht(z) = Tr/parenleftbigg\nAi\nz−L0−LTρ0/parenrightbigg\n.(E14)\nThe trace can be evaluated if we rewrite the resolvent\n(z−L0−LT)−1in terms of a power series in LT, apply\nWick’s theorem, collect diagrams irreducibly contracted\nto theAoperator into a self-energy kernel Σirr\nA(z) and\nresum the series\n∝an}bracketle{tA∝an}bracketri}ht(z) = Σirr\nA(z). (E15)39\nTo compare this simple result to the usual situation con-\nsidered in the real-time approach, we assume for a mo-\nment that the leads are tunnel-coupled to an interacting\nsystemwithafewdegreesoffreedom. Sinceonlythenon-\ninteracting leads can integrated out by applying Wick’s\ntheorem, the objective is to derive an exact effective the-\nory for the reduced density operator of the system ρ=\nTrresρtot. BymerelyreplacingTr( A...)→Trres(...), one\nmay take the same steps as above to express the Laplace\ntransform of the reduced density matrix as\nρ(z) = Tr res/parenleftbiggi\nz−L0−LTρ0/parenrightbigg\n(E16)\n=i\nz−L−Σ(z)ρ(t0) (E17)\nwhere Σ(z) is the (reducible) self-energy. Eq. (E16) is\nused as a starting point for a diagrammatic perturbation\ntheory in the coupling LT. In our case, we have simply\nno “system”, i.e., L= Σ(z) = 0 and ρ(z) =i/zfor\nρ(t0) = Tr resρtot(t0) = 1 when we take the trace over\nleads.\nMost of the steps that follow up can be generalized to\nthe case where there is a non-trivial system coupled to\nthereservoirs19, makingthefollowinganalysisofinterest.\nThe left action of A, considered as a superoperator,\ncan be expressed in terms of field superoperators, i. e.,\nA·=Aa1...ama1′...am′Ja1...JamJa1′...Jam′,(E18)\nwhere we assume rai=Landrai′=R, which can always\nbe achieved by a rearrangement of the field superoper-\nators by virtue of the anticommutation relation (E10).\nUsingLT=T11′J1J1′andL0J1= (L0−x1)J1with\nx1=η1ε1, we can shift all field superoperators to the\nleft. Since ρ0is an eigenstate of the internal Liouvil-\nlian, that is, L0ρ0= 0, we can pull ρ0also to the left,\nsetting allL0to zero in the resolvents and finally apply\nWick’s theorem Eq. (E7). We obtain for the n−th order\ncontribution to the Laplace transform:\nΣirr\nA(z)|n=/summationdisplay\ncontr.,{k}Aa1...am′Tnn′...T11′\n×n/productdisplay\nk=11\nz+Xk(−1)Np/productdisplay\ni<jγij.(E19)\nThe sumincludes allpossiblepaircontractionsandallin-\ndices{k}={a1,....,a m′,n,n′,....,1,1′}. The frequen-\ncies in the propagators read\nXi=/summationdisplay\nj/lessorequalslanti(ηjεj+ηj′εj′). (E20)\nWe represent the expressions contributing to Eq. (E19)\nby diagrams as follows: each tunneling amplitude Tii′\nis associated with a double vertex (two dots) on a line.\nThe line represents the free propagation of the system,\ndirected from right (earlier times) to left (later times), sothatthe orderofthe verticesnaturallycorrespondstothe\norder of the field operators in the expression Eq. (E19).\nTheleft-mostelementisa2m-vertexwith2mdots, which\nrepresents the m-particle observable A. As usual25, con-\ntractions are depicted by lines abovethe propagator line\nconnecting two associateddots iandj. Furthermore, the\nelectrode indices of the dots are fixed: all i’-indices be-\nlong tor=L(filled dots in Fig. 25), whence all i′-indices\nbelong tor=R(empty dots in Fig. 25).\nFIG. 25: Examples for Liouville space diagrams: (a) two con-\ntracted tunneling Liouvillians, (b) Hermitian conjugatio n in-\ndices are expressed by directions of arrows, (c) reading off t he\npropagator for a certain segment (for the segment indicated\nby the vertical dashed line, we obtain X2=ω1−ω2). Charge\nindicesχare denoted by + or −below the propagator line,\nthe factors ηωappearing in Eq. (E20) are denoted are shown\nfor the sake of completeness.\nThe factor Xiof thei-th propagator segment can\nbe readily read off from the diagrams as illustrated in\nFig. 25: firstly, all energies associated with contractions\nthat connect dots that lie on the same side of this seg-\nment do not contribute to Xi(cf. Fig. 25 (c), grey\ncontractions). For “later” dots this is clear as they are\nalways excluded in the sum of definition (E20). For “ear-\nlier” dots the respective energy occurs twice in the sum\nwith opposite sign ηi(enforced to the contraction (E9)).\nThus, only the contraction lines that intersect a verti-\ncal line drawn at the i-th propagator segment contribute\ntoXi(red contractions in Fig. 25 (c)). To determine\nXicompletely, we need to know the h. c. indices ηi,\nwhich we will depict by arrows attached to the contrac-\ntion lines: if a vertex has ηi= +(−), the arrow points\ntowards (away from) this vertex (see Fig. 25 (b)).\nFinally, we indicate the charge index χ=±of ev-\nery dot in the diagram by a sign below the propagator\nline (see Fig. 25 (c)). Note that charge indices χof a\ndouble vertex must be opposite by Eq. (E5). Our dia-\ngrams therefore represent the algebraic expression asso-\nciated with both a fixed contraction structure and fixed\n(η,χ)-indices. The latter is a distinction to the Liouville\nspace diagrams used, e.g, in25: distinct combinations of\n(η,χ) represent distinctdiagrams. Hence, the n-th order\ncontribution ∝an}bracketle{tA∝an}bracketri}htnto the expectation value of Ais repre-\nsented by a set of diagrams covering all allowed combi-\nnations contractions andall combinations for ( η,χ)’s.\nFor the ensuing discussion, we will first ignore any sign\nfactors in ∝an}bracketle{tA∝an}bracketri}htn(z), for example due to the contraction\nfunctions. This discussion and the question which dia-40\ngrams are allowed is most conveniently postponed to the\nvery end of our derivations.\n4. Energy-Spin Separation\nWe now arrive at the crucial part of the derivation\nofcovariant diagram rules: the separation of the spin-\ndependent and energy-dependent parts in Σirr\nA(i0) and\nrecasting the former as a trace in spin space similar to\nSec. A2. For the sake of simplicity, we first replace\nthe observable vertex by a tunneling vertex and discuss\nmodifications afterwards.\nIn Eq. (E19), we have three different spin-dependent\nfactors: (i) the energies εi=εri\nnikiσi, (ii) the contractions\nγij∝δσiσjand (iii) the tunneling amplitudes tσσ′.\n(i)+(ii): Any pair of contracted indices, say iandj\noccur in Eq. (E19) with a sum of the form\n/summationdisplay\nni,kiδσiσjg(εri\nnikiσi), (E21)\nwheregis some function of εri\nnikiσi=εrj\nnjkjσj. We can get\nrid of the spin-dependence of the energies by introducing\nthe spin-dependent DOS, proceeding analogous to the\nderivation of Eq. (45). Eq. (E21) then equals\n/integraldisplay\ndωi¯νri(ωi)g(ωi)[ri∝an}bracketle{tσi|ˇr·ˇnri(ωi)|σ′∝an}bracketri}htri]¯ηi.(E22)\nThe scalar product now involves a new 4-vector with\nˇn(ω) =√\n2/parenleftBig\n1,nr(ω)ˆJr/parenrightBig\n. Furthermore, we artificially\nintroduced the complex conjugation by ¯ ηeven though\nthe matrix elements r∝an}bracketle{tσ|ˇr·ˇnr(ω)|σ′∝an}bracketri}htrare real. This will\nbecome advantageous for later manipulations. We there-\nfore simply replace εibyωiin Eq. (E19) and the sum\nover the index inow abbreviates also an integration over\nωiinstead of a summation over ni,ki.\n(iii): The spin-dependence of the tunneling amplitudes\ncan be rewritten as a matrix element in spin space, too:\n[TLR\nσσ′]η′=/bracketleftbig\nL∝an}bracketle{tσ|ˇt·ˇr|σ′∝an}bracketri}htR/bracketrightbigη′\n.(E23)\nWe next separate the frequency dependent parts (prop-\nagators, Fermi functions, ˇtandˇn-vectors) and the spin-\ndependent matrix elements, which leads to\nA|n=/summationdisplay\n{κ,ρ,ωi}(signs)(prop .)/bracketleftbigg\n...ˇtκi...(ˇFi)ρi\nπ.../bracketrightbigg\n×/summationdisplay\n{σ}/parenleftBig\n...(ˇrκi)ηi′\nσiσi′...(ˇrρi)¯ησk′\nσkσk′.../parenrightBig\n.(E24)\nThe sum over the κi’s andρi’s indicates the scalar prod-\nucts of the respective 4-vectors with the matrix elements\n(ˇrµ)σiσi′of the charge-spin operator ˇ rµ. For simplicity,\nwe do not indicate here the quantization axis for the ba-\nsis states used to express these matrix elements since wewill see below that this choice drops our. We moreover\nintroduced the abbreviation\n(Fi)ρi= ¯νri(ωi)ˇnri\nρi(ωi)fri\n¯χi(ωi).(E25)\nInterestingly, the sum over the spin indices factorizes\ninto sums over the spin indices of vertices that are con-\nnected in the diagrams in a loop(when formally con-\nsidering contraction lines to be connected at each dou-\nble vertex, cf. Fig. 25 (c), which consists of a single\nloop). The contribution of each loop can be evaluated\nindependently as follows: if we start at an arbitrary\ndot and follow the directed contraction line. For ev-\nery loop element we encounter (vertex, contraction), we\nput charge-spin operators in a sequence from the right\nto the left into a trace. For example, starting at the\ndot labelled 1’ in Fig. 25 (c), we obtain the expression\ntr(ˇrκ1′ˇrρ1ˇrκ3′ˇrρ3′ˇrκ4′ˇrρ2ˇrκ2′ˇrρ1′). Thus, Eq. (E24) be-\ncomes\nA|n∼(signs)/integraldisplay\n{ωi}(propagators)\n×/productdisplay\nloopstr/bracketleftbig\n.../parenleftbigˇt·ˇr/parenrightbig\n...(Fi·ˇr).../bracketrightbig\n,(E26)\nwherewemovedthe frequency-dependent4-vectors tand\nFiinto the spin traces and restored the scalar products.\nTherestofthissectionisdedicatedtoprovethissimple\nrule. We therefore start from the sequence of matrix\nelements ˇrσσ′, obtained by writing them down from the\nright to the left in the order in which we encounter down\nwhen following a directed loop. We have to prove that\npairsofthe samespin indicesarenext to eachotherwhen\nwechangeallHermitianconjugationindicesofthematrix\nelements in to in Eq. (E24) to + (except for the left- and\nright-most one), so that the expression can be recast as a\ntrace. We will suppress the µ-indices for convenience and\nwe letσprecdenote a spin index of the preceding element\nwe have already run through and σsuccwill refer to a spin\nindex of the succeeding loop element. We have to insert\nthe following factors:\n(i) For each contraction, put a factor ˇ r¯ηearlyσlateσearlywhere\nηearlyis the h. c. index of the earlier vertex involved.\nWe first note that the h. c. index ηiof the vertex i\nat which we start to follow a contraction line has al-\nwaysηi=−(the arrow points away from the vertex).\nWe have to distinguish two cases: (ia) if we start at the\nearly vertex (e. g. contraction 1′in Fig. 11), we have\nηearly=−and thus ˇ r¯ηearlyσlateσearly= ˇr+\nσsuccσprecwhence\nin case (ib) we start at the late vertex (e. g. con-\ntraction 1 in Fig. 25 (c)), so ηearly= + and thus\nˇr¯ηearlyσlateσearly= ˇr−\nσprecσsucc= ˇr+\nσsuccσprec, using the Her-\nmiticity of the Pauli matrices.\n(ii) For each double vertex, we put a factor ˇ rηearlyσlateσearly.\nAgain, there two cases: (a) if ηearly= +, we arrive at\nthe earlier vertex, so we have ˇrηearlyσlateσearly= ˇr+\nσsuccσprec\n(e. g., later double vertex in Fig. 25 (c)) whence\n(b) ifηearly=−(e. g. earlier vertex in Fig.41\n25 (c)), we arrive at the later vertex, so we have\nˇrηearlyσleftσright= ˇr−\nσprecσsucc= ˇr+\nσsuccσprec. These considera-\ntions prove the simple rule:\n/summationdisplay\n{σi,σi′}ˇr+\nσnσn′...ˇr+\nσ1σ1′ˇr+\nσ1′σn= tr(ˇr...ˇr).(E27)\nHere we used that the matrix elements of adjacent\nPauli operators are taken for spin states with respect\nto thesamequantization axis, i. e., we can com-\nbine/summationtext\nσ1′ˇr+\nσ1σ1′ˇr+\nσ1′σn=r1∝an}bracketle{tσ1|ˆr|σ1′∝an}bracketri}htr′\n1r′\n1∝an}bracketle{tσ1′|ˆr|σn∝an}bracketri}htrn=\nr1∝an}bracketle{tσ1|ˆrˆr|σn∝an}bracketri}htrnand so on.\n5. Sign Factors\nHaving tackled the most tedious part, the spin and\nenergy dependence, it remains to collect all sign factors\nof an expression from its representing diagram, yielding\nsigns = ( −1)#cr./productdisplay\nearly+late\nverticesχe/productdisplay\nintermediate\nverticesηe.(E28)\nHere #cr. is the number of crossing contractions lines\nand “e” always refers to the earlier vertex of each double\nvertex. The meaning of early / intermediate / late ver-\ntices is depicted in Fig. 26 and explained in the proof of\nEq. (E28).\nTo prove Eq. (E28), we first note that there are\nthree origins for signs: (i) an overall permutation fac-\ntor (−1)Npfrom Wick’s theorem (E7), (ii) a factor\n(−χearlyηearly) for every contraction (see Eq. (E9)) and\n(iii) a factor ( −χearly) for every double vertex (see\nEq. (E5)). (i) is readily obtained from the number of\ncrossingcontractionlines in the diagrams, givingthe first\nfactor of Eq. (E28). The signs due to (ii) and (iii) can\nbe determined together: first of all, the minus signs can\nbe omitted since the number of contractions and double\nvertices is always even. For the further procedure, it is\nhelpful to distinguish three types of vertices sketched in\nFig. 26: (a) “late double vertices” where both vertices\nare the later ones in their contractions. Then no signs\ndue to contractions have to be considered and only the\nchargeindex ofthe earliervertexoccurs. We furthermore\nhave (b) “early double vertices” where both vertices are\nthe earlier ones in their contractions. The signs due to\nbothcontractionscancel eachother since chargeand h. c.\nindices ofboth verticesin anydouble vertex areopposite.\nThus, the sign forthis type ofvertexis againgivenby the\nearlier charge index. Finally, there are (c) “intermediate\ndouble vertices” where one of the vertices is earlier and\nthe other one is later in its contraction. The sign factor\nis in this case the h. c. index ηearlyof the earlier vertex\nof the double vertex.\nFIG. 26: Different types of vertices: late double vertex (a),\nearly double vertex (b) and intermediate double vertices (c ),\n(d). The corresponding sign factors are denoted below the\nvertex. Note that χe,ηerefer to the earlier vertex of the\ndouble vertex and notto being earlier in the contractions.\n6. Hermitian Conjugated Partner Diagrams\nSince any observable must have a real expectation\nvalue, the imaginary parts of all diagrams have to cancel\neach other. In fact diagrams come in complex conjugates\npairs, which are obtained from each other by inverting\nthe h. c. indices, or, diagrammatically speaking, by in-\nverting all arrow directions. Therefore, it is sufficient to\ntake only onediagram as a representative of both. We\ntherefore obtain schematically instead of Eq. (E26)\nA|n∼2Re(signs)/integraldisplay\n{ωi}(propagators)\n×/productdisplay\noriented\nloopstr/bracketleftbig\n.../parenleftbigˇt·ˇr/parenrightbig\n...(Fi·ˇr).../bracketrightbig\n.(E29)\nWe will next explicitly show how the complex-\nconjugation symmetry arises in our formulation of real-\ntime diagrammatics. Inverting all h. c. indices, ηi→¯ηi,\ninvolves the following modifications:\n(i) The sign factor is negated. All charge indices may\nbe kept fixed except for those of the current vertex (see\nEq. (E33)). This involves an additional minus sign from\nthe early and late double vertices. The sign due to the\nintermediate vertices does not change since the number\n#IV of this type of vertices is always even. Revert-\ning allηe→¯ηetherefore results in an additional factor\n(−1)#IV= 1 compared to the original sign factor.\n(ii) The orderofthe Pauli operatorsin the spin traceis\ninverted, yielding the complex conjugate of the original\ntrace expression:\ntr(ˇr1...ˇrn)→tr(ˇrn...ˇr1) = tr(ˇr1...ˇrn)∗,(E30)\nwhere we used ˇ r†\n1= ˇr1and (ˇr†\nn...ˇr†\n1) = (ˇr1...ˇrn)†.\n(iii) The propagators are mapped onto their negative\nHermitian conjugate since Xi→ −Xi\n1\ni0+Xi→1\ni0+(−Xi)=−/parenleftbigg1\ni0+Xi/parenrightbigg∗\n.(E31)\nAs the number of propagators is odd, we obtain an ad-\nditional minus sign.\nSince all remaining factors are real (cf. Eq. (E26))\nand have no further η-dependence, we conclude that the42\nmappingηi→¯ηiresults in the positivecomplex conju-\ngate contribution of the initial expression. Adding both\nthese partner diagrams gives two times the real part of\none of the two diagrams, proving Eq. (E29)\n7. Covariant Diagram Rules for Charge and Spin\nCurrent\nWe first describe the modifications of the covariant\ndiagram rules required for the charge and spin cur-\nrent, which can be treated simultaneously as outlined in\nIVA. The expressions for the current operator read (see\nEq. (80)):\nIRr\nµ=/summationdisplay\nnn′kk′σσ′(−irµT)r¯r\nσσ′c†\nrnkσc¯rn′k′σ′−h.c. (E32)\nwith (rT)σσ′=/summationtext\nτrστTr¯r\nτσ′. The structure of the current\nis very similar to the tunneling Hamiltonian with the\nsimple replacement Tr¯r\nσσ′→(−irµT)r¯r\nσσ′. Introducing a\nsuperoperator for the left action IRrµ·=IRrµ·, we may\ntherefore treat the corresponding current vertices similar\nto the tunneling Liouvillian by making the replacement\n¯χ1′/bracketleftbig\nL∝an}bracketle{tσ1|ˇt·ˇr|σ1′∝an}bracketri}htR/bracketrightbigη1′\n→δ¯χ1′η1′,+[(−i)L∝an}bracketle{tσ1|iRrµ|σ1′∝an}bracketri}htR]η1′(E33)withiRrµgiven by Eq. (E36) below. The restriction\nδ¯χ1′η1′,+is due to the fact that all field operators in IRr\nµact from the left whereas the tunneling Liouvillian as\na commutator also possesses a part that acts from the\nright. Consequently, we have account for the following\nmodifications:\n(i) InsertiRrµinstead of/parenleftbigˇt·ˇr/parenrightbig\ninto the spin traces.\n(ii) Setη1′= ¯χ1′= +. The first equality follows from\nthe factorδ¯χ1′η1′,+in Eq. (E33) and the second from the\nfreedom to chose one representative of the complex con-\njugated diagrams, which we fix by an explicit choice for\nη1′. Note that the “missing”factor ¯ χ1′= + in the second\nline of Eq. (E33) does give any further modifications for\nthis choice.\n(iii) Replace Re( ...)→Im(...) in Eq. (E29), which is\ndue to the additional factor ( −i)η1′=+=−iin Eq. (E33).\nWe now present a systematic way to draw all diagrams\nandreadofftherespective n-thordercontributionstothe\nexpectation value IRrµ|n, schematically given by\nIRr\nµ|n∼/summationdisplay\nirr.contr.\n{χ,η}2Im\n\n(signs)/integraldisplay\n{ωi}/parenleftbigg\n...1\ni0+Xi.../parenrightbigg/productdisplay\nloopstr/bracketleftbig\n.../parenleftbigˇt·ˇr/parenrightbig\n...(Fi·ˇr).../bracketrightbig\n\n. (E34)\nThe first sum in Eq. (E34) adds up all possible dia-\ngrams, whereas the residual term corresponds to an in-\ndividual diagram. All allowed diagrams contributing to\nIRrµ|nare obtained by the following drawing instructions\n(an easy example is given in Fig. 25 (c)).\n1.Propagator and vertices. For then-th order\ncontribution, put ndouble vertices on a propagator line.\nThe later (earlier) vertex of each double vertex refers to\nthe left (right) electrode. Mark the latest vertex with\nthe symbol IRrµdesignating this one to be the observable\nvertex. Symbol: a horizontal line represents the prop-\nagator line; double dots, encircled by a line depict the\ndouble vertices; full (empty) dots refer to the left (right)\nelectrode.\n2.Contractions. Construct all possible irreducible\ncontractions. Full (empty) vertices are only permitted to\nbe contracted with full (empty) vertices. Symbol: con-\ntractions depicted full lines above the propagator; attach\nan “i”, denoting the Liouville index of the earlierdot in\neach contraction.\n3.Hermitian Conjugation Indices. Construct allpossibilities for the choice of the h. c. indices, obeying\nthe following rules: (i) the h. c. of one double vertex\nhave to be opposite, (ii) the earlier h. c. index of the\nobservable vertex is +, (iii) the h. c. indices have to\nalternate in each loop. Symbol: Arrow pointing to (away\nfrom) a dot is associated with η= + (η=−).\n4.Charge indices. Construct all possible charge\nindex arrangements, restricted by (i) the charge indices\nof a double vertex have to be opposite, (ii) the charge\nindices of the observable vertex have to be opposite to\nthe h. c. indices. Symbol: +,– below the vertex\nTranslating diagrams into algebraic expressions,\nschematicallyindicated in Eq. (E34), proceedsas follows:\n1.Propagators : Foreachpropagatorsegmentfollow-\ning atunneling vertex write down a factor\n1\ni0+XwithX=/summationdisplay\n(ωearly−ωlate).(E35)\nThe sum in Xinvolves all frequencies associated with\ncontractions that cross over that segment from the left\nto the right (earlier frequency) or from the right to the43\nleft (later frequency), respectively.\n2.Loopwise evaluation of spin traces : Start at\nany point and follow the loop in direction of the arrow.\nInsert from the right to the left into a trace in spin space\nthe following terms when encountering\n- a tunneling vertex: ˇt·ˇr,\n- an observable vertex:\niRrµ=/braceleftbigg\n−/parenleftbigˇt·ˇr/parenrightbig\nrµ, r=R\n+rµ/parenleftbigˇt·ˇr/parenrightbig\n, r=L.(E36)\n- a contraction: Fi·ˇrwhereirefers to the earlier dot,\n(Fi)ρi= ¯νri(ωi)fri\n¯χi(ωi)ˇnriρi(ωi) and\nˇnr\nρ=√\n2/braceleftbigg1ρ= 0\nnrˆJr\nρρ= 1,2,3. (E37)\nRepeat this for all other loops in the diagram.\n3.Signs: Put a factor\n(−1)#cr./producttext\nearly+late\nverticesχe/producttext\nintermediate\nverticesηe,\nwhere #cr. is the number of crossings of contractions,\nthe subscript “ e” refers to the earlier dot of a vertex.\nFor the meaning of late, early and intermediate double\nvertices, see Fig. 26.\n4.Complete expectation value : Multiply all ex-\npressions obtained from steps 1, 2 and 3 and integrate\nover all frequencies {ωi}, whereirefers to the indices\nassociated with the contractions. Take 2Im( ...) of this\nexpression and sum up the contributions of all valid dia-\ngrams.\n8. Covariant Diagram Rules for Spin-quadrupole\nCurrent\nThe calculation of the expectation value of the SQM\ncurrent requires some modifications compared to that of\nthe charge and spin-dipole current. The reason is that\nthe spin-quadrupole current operator for node ∝an}bracketle{trr′∝an}bracketri}ht, i. e.\nIrr′\nQ=i/bracketleftBig\nHT,Qrr′/bracketrightBig\n, is a dyadic of two vector operators:\nIrr′\nQ=1\n2/bracketleftBig\ngrr′ISr⊙Sr′+(r↔r′)+h.c./bracketrightBig\n.(E38)\nWe remind the reader of the factor grr′= 2 ifr∝ne}ationslash=r′\nandgrr= 1. Crudely speaking, the diagrams for the\nspin-quadrupole current have the same structure as for\nthe spin current, but include an additional spin vertex.\nHowever, the spin operator has to be treated differently\nfromthetunnelingorcurrentvertices(E32)asiscontains\nonlyonesum over the k-modes (see Eq. (40)) instead of\ntwo. We will discuss this point later in detail.\nWe will restrict ourselves in the following to spin-\nindependent tunneling. If and only if this is the case,\nwe may replace Ir\nSby 2Ir¯r\nSrwhere\nIr¯r\nSr=/summationdisplay\nk,k′,σ,σ′[−ir∝an}bracketle{tσ|iSr|σ′∝an}bracketri}ht¯r]c†\nrkσc¯rk′σ′(E39)with ¯r=R,Lforr=L,RandiSrdefined in Eq. (E36)\nforµ=x,y,z. Note that Ir¯r\nSronly accounts for the con-\ntribution to the spin current for electrode rdue to tun-\nneling from subsystem ¯ rtor, but not for tunneling from\nrto ¯r. Using Eq. (E38), the expectation value of the\nspin-quadrupole current is then given by\n∝an}bracketle{tIrr′\nQ∝an}bracketri}ht= 2grr′Retr/parenleftBig\nIr¯r\nSr⊙Sr′/parenrightBig\nρ(t)+(r↔r′).(E40)\nInthefollowing,wewilldiscussthemodificationsthatare\nnecessary to calculate Eq. (E40) starting from the spin\ncurrent. First of all, IQrr′contains four field operators\nso that the associated observable vertex is a quadruple\nvertex instead of a double vertex. This is sketched in\nFig. 27: the later two vertices refer to the spin current\noperator (whose dots refer to twoelectrodes) and the\nearlier vertices refer to the spin operator (whose dots\nrefer to only oneelectrode). As all field operators act\nfrom the left, all charge indices are opposite to the h. c.\nindices. The action of the spin operator can be expressed\nby\nSj∼δη1′,+δ−η1′χ1′,+δχ1,¯χ1′δη1,¯η1′δk1k1′(sr\nj)σ1σ1′J1J1′.\n(E41)\nThis fixesηearly= ¯χearly= +.\nFIG. 27: SQM current vertex\nThe technical challenge for implementing the spin-\nquadrupole current vertex is the single k-summation in\nthe spin operator. If we contract the two dots within the\nspin operator this is no problem since the contraction\nalso sets the k-indices to be the same anyway. However,\nif the two dots are contracted with other vertices, their\nk’s are not independent, so we cannot introduce the one-\nparticle DOS for both contractions individually (denoted\nby 1 and 1′in the following). Instead, we obtain an ex-\npression of the form:\n/summationdisplay\nτ1τ1′/summationdisplay\nk1δσ1τ1′δσ1′τ1g(εr′\nn1k1τ1′,εr′\nn1k1τ1)r′∝an}bracketle{tσ1|s|σ1′∝an}bracketri}htr′,\n(E42)\nwheregis some function and the Cartesian index j\nrefers to the spin operator. We may treat this expres-\nsion similar to the calculation of the exchange contri-\nbution to the spin-quadrupole moment in equilibrium\n(see Sec. IIC) by introducing the two-particle density\nof statesνr\nσ1σ1′(ω1,ω1′) (16) and expressing the latter by\nEq. (A11). Then the term in (E42) equals\n/integraldisplay\n{ω1,ω1′}g(ω1,ω1′)2Ar′\nµνr′∝an}bracketle{tσ1|ˇrµsˇrν|σ1′∝an}bracketri}htr,(E43)44\nwhere the 2DOS (16) enters through the matrix Ar\nµν\ngiven by (A12)-(A15). Consequently, the frequency and\nenergy part of the diagrams can be treated without mod-\nification as explained in Sec. E4. Furthermore, the signs\ncan be treated without modifications as well as we can\nsimply omit the factor ¯ χ1′= + in Eq. (E41). In con-\ntrast, the spin-part is altered: we now insert for the spin\nvertex and the two associated contractions the factor√\n2Ar\nµνˇrµsjˇrνinto the spin trace.\nFinally, we mention that the symmetrization in r↔r′\nand multiplying with grr′in Eq. (E40) is equivalent to\nmultiplying withafactorof2andsymmetrizingin r↔r′\nonly forr∝ne}ationslash=r′.\nWe now summarize the modifications of the diagram\nrules for the SQM current compared to the spin current.\nWe first state the drawing rules:\n1.Propagator and vertices: the observable vertex\nconsists of four vertices, the later two depicting ISrand\nthe earlier two depicting Sr′.\n2.Contractions. No changes.\n3.Indices. The h. c. indices of the observable vertexare fixed to (–,+,–,+)and the chargeindices are opposite\nto this.\nThe changes in the rules for translation into algebraic\nexpressions are:\n1.Propagators. No changes.\n2.Evaluation of loops. If the vertices associated\nwithSr′in Eq. (E40) are\n(a) contracted with eachoter, then insert Fr′\n+(ω1′)·ˇrs.\n(b) contracted with other vertices, then insert\n2t2fr1\n¯χ1(ω1)fr1′\n¯χ′\n1(ω1′)Ar′\nµνˇrµsˇrν.\nThis accounts for both the vertices and their contrac-\ntions. Here 1 (1’) refers to the indices of the contractions\nof the later (earlier) vertex.\n3.Signs.For the evaluation of the signs, treat the\nSQM current vertex as two double vertices.\n4.Complete expectation value. Multiply with a\nfactor of 2 and retain the symmetric and traceless part of\nthe tensor-valued result ∼iSr⊗s. Ifr∝ne}ationslash=r′, symmetrize\nthe result in r↔r′.\n1M. Julliere, Phys. Lett. A 54, 225 (1975).\n2J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989).\n3R. M. Potok, J. A. Folk, C. M. Marcus, and V. Umansky,\nPhys. Rev. Lett. 89, 266602 (2002).\n4M. Braun, J. K¨ onig, and J. Martinek, Superlatt. and Mi-\ncrostruct. 37, 333 (2005).\n5G. 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R. Wegewijs, submitted\n(2012).\n17M. Misiorny, I. Weymann, and J. Barna´ s, Phys. Rev. B\n86, 245415 (2012).\n18W. N. Cottingham and D. A. Greenwood, An Introduction\nto Nuclear Physics , vol. 1 (Cambridge University Press,2001).\n19M. Hell and M. R. Wegewijs, in preparation (2013).\n20D. Gatteschi and R. Sessoli, Angew. Chem. Int. Ed. 42,\n268 (2003).\n21J. K¨ onig and J. Martinek, Phys. Rev. Lett. 90, 166602\n(2003).\n22M. Misiorny, I. Weymann, and J. Barna´ s, Phys. Rev. Lett.\n106, 126602 (2011).\n23F. May, M. R. Wegewijs, and W. Hofstetter, Beilstein J.\nNanotechnol. 2, 693 (2011).\n24I. V. Lindell, Methods for electromagnetic field analyi-\nsis, IEEE Series on Electromagnetic Wave Theory (IEEE\nPress, 1992).\n25H. Schoeller, Eur. Phys. Journ. B 168, 179 (2009).\n26M. Leijnse and M. R. Wegewijs, Phys. Rev. B 78, 235424\n(2008).\n27R. B. Saptsov and M. R. Wegewijs, Phys. Rev. B 86,\n235432 (2012).\n28R. B. Saptsov, private notes (2010).\n29The exchange term is only by chance proportional to ∝angbracketleftS2\nx+\nS2\ny∝angbracketrightat zero temperature. At finite temperatures, this does\nnot hold any more, showing that both terms are physically\nquite distinct.\n30Here, one has to treat the spin vector as a stochas-\ntic variable when averaging over the grand-canonical en-\nsemble. Spearman’s rank correlation coefficient CABfor\ntwo random variables A,Bis defined by C(A,B) =\n∝angbracketleft(A−∝angbracketleftA∝angbracketright)(B−∝angbracketleftB∝angbracketright)∝angbracketright. Eq. (60) is a linear combination of\ntheC(Sr\ni,Sr′\nj) for different components of the spin opera-\ntor so that only triplet correlations are extracted. Howeve r,\nif allC(Sr\ni,Sr′\nj) = 0, this implies ∝angbracketleftQrr′∝angbracketrightex= 0\n31For/hatwideJ=ezwe get∝angbracketleftQ∝angbracketrightex=−qexez⊙ez=−qex[2\n3ezez−\n1\n3(exex+eyey)], that is, ∝angbracketleftQzz∝angbracketrightex=−2\n3qex=−2\n3Ns\n4=\n−1\n3∝angbracketleftSz∝angbracketrightT=0, in agreement with (31).\n32/bracketleftbig\nSr\ni,Ir\nSj/bracketrightbig\n∝negationslash= 0 can be explicitly shown by inserting the45\nexpressions for the spin operator (25) and the spin cur-\nrent operator (80), respectively, and applying the anti-\ncommutation relations of the field operators.\n33Although the restrictions on the network connectivity de-\nrives form the bilinearity of the tunneling Hamiltonian (9) ,\nthis does not mean that SQM cannot be exchanged di-\nrectly between local nodes: when calculating the averages,\nincluding coherent processes of higher order in this the bi-\nlinear coupling is indeed found to occur, see19.\n34The current contribution of electrons at energy ωis\nchanged by the factor (1+cos( θ)nL(ω)nR(ω)), i.e., when\nbothspin-polarizations are nonzero. For parallel Stoner\nvectors the fraction of electrons with spin σin electrode\nrwith energy ωis changed from 1/2 to (1+ σnr(ω))/2 as\ncompared to the non-magnetic case. These electrons can\nonly access a fraction (1+ σn¯r(ω))/2 of the states in the\nother electrode ¯ rinstead of 1/2 as in the non-magnetic\ncase. The reason for this is conservation of spin by the\ntunneling (cf. Eq. (11)), so all states with opposite spin in\nthe other electrode are forbidden final states. In total, thi s\nchanges the current by a factor/summationtext\nσ(1+σnr)(1+σn¯r)/2 =\n1 +nLnR>1. In the case of non-collinear Stoner vec-\ntors electrons can access both spin channels of the other\nelectrode due to non-zero spin-overlap factors in Eq. (11),\ngiving an overall additional reduction factor cos( θ).\n35In contrast, charge and spin current are finite in the ther-\nmodynamic limit by themselves and notper electron. This\nhas the same reason as for the averages: the average spin\nper electron , but only the average SQM per electron pair\nhave afinitelimit, cf. discussion of theexchange SQM(59).\n36We also have to account for the situation in which the role\nof the first and the second copy are interchanged; however,\nwhen summing over all contributions this gives the sameresult as in the first case. This yields the additional factor\nof 2 in the SQM current.\n37The analogy between SQM and spin torque becomes ex-\nplicit when rewriting TL\nS=/integraltext\ndωΓ/parenleftBig\nαR\n¯νRfL\n+−βR\n¯νR/parenrightBig\nnLby\ninterchanging ω↔ω′in the first term contributing to\nthe double integral (98). Then TL\nexis obtained from TL\nSby\nreplacing the spin-polarization by the quadrupolarizatio n\nfunction, nL→aL.\n38Eq. (121) holds for V∗< J, i. e.,J < D/π, which is\nvalid for the flat-band approximation (cf. IID). Otherwise\nEq. (119) in already limited by |J|.\n39Ifθ∝negationslash= 0,π, the SQM current could only be uniaxial if\nAL\nQ= 0 and TL\nQ= 0 is fulfilled at the same time (otherwise\ntheIλare clearly angle dependent according to Eqs. (127)\nand (128)). This is in general not expected as the SQM\ntorque senses the band structure over a wide energy range.\n40Assuming Ns/greaterorsimilar10, we neglect the exchange SQM torque\nsinceTL\nS∼TL\nex(cf. Eqs. (117) and (120)).\n41Eq. (137) is obtained by substituting x= (ω−µL)/TL\nin Eq. (136) and expanding the bias function ∆( ω)≈\n−f′(x)xτRwithf(x) = (ex+ 1)−1. This gives ILL\nQ≈\nΓ\n2TLτR/integraltext\ndxf′(x)x/bracketleftbig\n2f(x)−/summationtext\nσf(x+σj)/bracketrightbig\nto which we\napply Eq. (A47) from App. A4.\n42Note that these two electrons are treated here as ifthey\ncould occupy the same k-mode, which is of course forbid-\nden by the Pauli principle. As explained in Sec. A1, this\nonly corrects for the mistake that is made when the direct\nSQM is calculated by treating all electrons as distinguish-\nable objects, that is, when ignoring Pauli principle.\n43In particular, one cannot set n(ω) = 0 for all ωwithout\nsettingJ= 0, since these are equivalent for the Stoner\ndispersion considered."    },    {        "title": "0812.4215v2.Spin_signal_propagation_in_time_dependent_noncollinear_spin_transport.pdf",        "content": "arXiv:0812.4215v2  [cond-mat.mtrl-sci]  6 Jan 2009Spin-signal propagation in time-dependent noncollinear s pin\ntransport\nYao-Hui Zhu,∗Burkard Hillebrands, and Hans Christian Schneider†\nPhysics Department and Research Center OPTIMAS,\nUniversity of Kaiserslautern, 67653 Kaiserslautern, Germ any\nAbstract\nUsing a macroscopic analysis, we demonstrate that time-dep endent noncollinear spin transport\nmay show a wavelike character. This leads to modifications of pure spin-diffusion dynamics and\nallows one to extract a finite spin-signal propagation veloc ity. We numerically study the dynamics\nof a pure spin current pumped into a nonmagnetic layer for pre cession frequencies ranging from\nGHz to THz.\nPACS numbers: 72.25.Ba, 73.40.Jn, 75.47.-m, 85.75.-d\n1I. INTRODUCTION\nTransporting information encoded in electronic spins through layer s of ferromagnetic and\nnormal metals is a central theme of magnetoelectronics.1Structures, in which all spins are\nare essentially collinear, i.e., parallel or antiparallel, have been thoroughly investigated in\nexperimental and theoretical studies. The quasi-static propert ies for the special case of\nstructures with collinear spin and magnetization directions where th e spin-polarized cur-\nrent flows perpendicularly to the plane of the layers,2can be analyzed in terms of a scalar\nspace-dependent spin accumulation for up and down spins.3,4The functionality of collinear\nmagnetoresistive structures can be enhanced by including tunnelin g elements.5,6,7Although\ncollinear spin transport is of importance for certain variants of gian t and tunneling magne-\ntoresistance effects, a non-collinear alignment of spin and magnetization orientations leads\nto additional degrees of freedom for the manipulation of spin angula r momentum and has\nattracted much attention in recent years.1,8For instance, one can exploit the angular de-\npendence of the giant magnetoresistance effect9, or can change the the alignment of spins\nby spin currents, leading to the phenomenon of spin transfer torq ue10,11,12,13and potential\nnovel applications.14. A different method to exploit the freedom of noncollinear spin ori-\nentations in magnetic multilayers is the use of magnetization precess ion in a ferromagnetic\nlayer, which “pumps” a spin currents into an adjacent nonmagnetic metal.15A precessing\nmagnetization, which is necessary for spin pumping, creates the ne ed to deal with a time-\ndependent orientation of the spins in the whole multilayer, so that it b ecomes essential to\nstudy dynamical noncollinear spin transport problems.\nWe are concerned with a theoretical analysis of the propagation of signals encoded in\na spin current, which flows through a multilayer structure with nonc ollinear magnetization\nand spin directions. Most investigations of time-dependent noncollin ear spin transport are\nbased on the Bloch-Torrey diffusion equations for the nonequilibrium magnetization or spin\naccumulation.16Theseequationsessentially describespintransportasadiffusionpr ocessand\ntherefore show the same problem as the spin diffusion equation for c ollinear spins17,18,19,20:\nno finite propagation velocity for a spin signal can be defined becaus e the diffusion equation\nleads to a finite spin current density everywhere as soon as there is a source. Recently, we\nshowed that this difficulty can be resolved for collinear spin transpor t by using a “telegraph”\nequation, which generalizes the diffusion equation, and leads to notic eable differences from\n2the diffusion equation results for frequencies exceeding several 1 00 GHz for metals such as\ncopper.21Importantly, the telegraph equation shows a wave-diffusion duality , which enables\none to define a finite propagation velocity for the spin signal. In this p aper, we use a similar\ntreatment for noncollinear spin transport to show how a finite signa l propagation velocity\narises in this case. We predict that noncollinear spin transport at hig h frequencies shows a\ndynamics that is more complicated than what is expected from an ana lysis using the spin-\ndiffusion equation. We numerically analyze the propagation of a spin cu rrent pumped into\na nonmagnetic metal by a precessing magnetization in an adjacent f erromagnetic layer.\nThis paper is organized as follows. In Sec. II, we present the macro scopic dynamical\nequations governing noncollinear spin transport. In Sec. III, the dynamical equations are\ncombined into a telegraph equation, which is studied analytically to disc uss qualitative\naspects of dynamical noncollinear spin-transport. In Sec. IV, we solve numerically the\ndynamical equations for the spin transport, and the main conclusio ns are summarized in\nSec. V.\nII. DYNAMICAL EQUATIONS\nIn nonmagnetic conductors and some ferromagnetic metals,22the dynamics of conduction\nelectrons under theinfluence ofexternal fields canbedescribed b y ageneralized semiclassical\nBoltzmann equation23,24\ni/planckover2pi1∂ˆρ\n∂t+i\n2/braceleftbigg∂ˆε\n∂/vectork,∂ˆρ\n∂/vector r/bracerightbigg\n−i\n2/braceleftbigg∂ˆε\n∂/vector r,∂ˆρ\n∂/vectork/bracerightbigg\n= [ˆε,ˆρ]+i/planckover2pi1∂ˆρ\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ncol, (1)\nwhich we take as the starting point for our analysis of time-depende nt noncollinear electron-\nspin transport in these systems. In Eq. (1), ˆ ρ(/vector r,/vectork,t) is the single particle density matrix in\nspin space,\nˆρ=\nρ↑↑ρ↑↓\nρ↓↑ρ↓↓\n, (2)\nˆε(/vector r,/vectork,t) is the effective single-particle energy matrix, and {·,·}and [·,·] denote respectively\nthe anticommutator and commutator for matrices in spin space. Fo r completeness, we\nremark that in Eq. (2), the single-particle density matrix\nρss′(/vector r,/vectork,t) =V\n(2π)3/integraldisplay\nd3qei/vector q·/vector r/an}bracketle{tc†\n/vectork−/vector q/2,s′c/vectork+/vector q/2,s/an}bracketri}ht, (3)\n3is defined by a statistical average over creation and annihilation ope ratorsc†andc, with\nnormalizationvolume V. The diagonalmatrix elements ρ↑↑andρ↓↓arethe electron distribu-\ntion functions of the spin-up and spin-down, respectively, wherea s the off-diagonal elements\nρ↑↓=ρ∗\n↓↑represent the spin coherence.25Because the unit matrix ˆIand the Pauli matrices\nˆσx, ˆσy, ˆσzform a basis for 2 ×2 matrices, the spin-density matrix ˆ ρcan be represented by\nˆρ= (1/2)[(ρ↑↑+ρ↓↓)ˆI+/vector u·ˆσ], where/vector u= Tr(ˆσˆρ) = (2Reρ↑↓,−2Imρ↑↓,ρ↑↑−ρ↓↓) is the Bloch\nvector and ˆσthe vector of Pauli matrices.\nBefore proceeding from Eq. (1) for the spin-density matrix to equ ations for macroscopic\nquantities, such as spin currents and spin accumulation, we list a few assumptions made\nabout quantities occurring in Eq. (1). First, we consider only layere d structures whose\nextensions perpendicular to the growth direction ( xaxis) are infinite, and we also assume\nthat the electric fields /vectorE=E/vector x/|/vector x|is oriented along the growth direction x. Second, the\neffect of magnetic fields onthe orbital motionof electrons is neglect ed. These magnetic fields\ninclude the static external magnetic field /vectorBsand the magnetic field generated by induction\ndue to the time-dependent electric field /vectorE(x,t).26We therefore assume that the electric\nfieldE(x,t) =−∂φ(x,t)/∂xcan be derived from a time-dependent electric potential φ(x,t).\nThird, anisotropiceffective mass model for thespin-degenerate c ondutcion electrons is used,\ni.e.,εk=/planckover2pi12k2/(2m∗) =m∗v2/2, where /vectorkand/vector vdenote the the electron wave vector and\nvelocity, respectively. Thus we have to deal with a spin density matr ix ˆρthat depends only\nonxand has cylindrical symmetry around the xaxis inkspace.\nFinally, we make a relaxation-time approximation for the collision term27\n∂ˆρ\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ncol=−ˆρ−/an}bracketle{tˆρ/an}bracketri}hta\nτ−/an}bracketle{tˆρ/an}bracketri}hta−(ˆI/2)Tr/an}bracketle{tˆρ/an}bracketri}hta\nT1, (4)\nwhereτandT1are the momentum and spin relaxation times, respectively. Moreove r,\n/an}bracketle{tˆρ/an}bracketri}hta≡(4π)−1/integraltext\ndΩ/vectorkˆρis the angular average in the momentum space. By using Eq. (4) for\nthe collision term, we have assumed that the longitudinal spin relaxat ion time T1is equal to\nthe transverse one T2. The validity of this approximation is discussed in detail by Ref. 16.\nNote that T1in Eq. (4) is one half of τsfused in Eq. (2) of Ref. 27.\nWith above simplifications, the effective single-particle energy ˆ ε(/vector r,/vectork,t) is simplified to\nˆε(x,|/vector v|,t) =ε0ˆI+ ˆεs, whereε0=/planckover2pi12k2/(2m∗)−eφ(x,t) and ˆεs=−µ·/vectorBs=µBσ·/vectorBs.\nTherefore, Eq. (1) simplifies to\n∂ˆρ\n∂t+vx∂ˆρ\n∂x−eE\nm∗∂ˆρ\n∂vx+1\n2γ(/vector u×/vectorBs)·σ=−ˆρ−/an}bracketle{tˆρ/an}bracketri}hta\nτ−/an}bracketle{tˆρ/an}bracketri}hta−(ˆI/2)Tr/an}bracketle{tˆρ/an}bracketri}hta\nT1,(5)\n4whereγ=gµB//planckover2pi1is the absolute value of the electron ( g≈2) gyromagnetic ratio.\nTo derive macroscopic spin transport equations comparable with th e Bloch-Torrey diffu-\nsion equation, we need to sum over the electron wave vector /vectorkor, equivalently, the velocity\n/vector vin Eq. (5). We first derive an equation for the spin density27,28by multiplying both sides\nof Eq. (5) by ˆσ/V, taking the trace, and summing over /vector v\n∂/vector ns(x,t)\n∂t=−γ/vector ns(x,t)×/vectorBs−/vector ns(x,t)\nT1−∂/vector s(x,t)\n∂x, (6)\nwhere/vector ns(x,t) =V−1/summationtext\n/vector vTr(σˆρ) =V−1/summationtext\n/vector v/vector uand/vector s(x,t) =V−1/summationtext\n/vector vvxTr(σˆρ) =V−1/summationtext\n/vector vvx/vector u\nare the spin density and spin current density, respectively. For th e spin current density,\nwe multiply both sides of Eq. (5) by vxˆσ/V, take the trace, and sum over /vector v. Using the\nexpansion (A2) for the velocity dependence of the spin density mat rix and the procedure in\nAppendix A, we obtain\n/vector s(x,t) =−D∂/vector ns(x,t)\n∂x−µE(x,t)/vector ns(x,t)−τγ/vector s(x,t)×/vectorBs−τ∂/vector s(x,t)\n∂t,(7)\nwhere\nD=v2\nF\n3τ (8)\nis the diffusion constant and µ=eτ/m∗the electron mobility. Note that /vector ns(x,t) and\n/vector s(x,t) defined above are the particle (electron) number densities, which can be converted\nto the charge, spin, and magnetic moment densities by multiplication w ith−e,/planckover2pi1/2, and\n−µB, respectively. The spin density /vector ns(x,t) can also be converted to the chemical potential\ndifference µs(x,t), i.e., the spin accumulation, by the relation /vector ns(x,t) =Nµs(x,t), where\nN= 4πm∗2vF/h3is the density of states at the Fermi level of the electron gas for o ne spin\norientation.29\nEquation (7) resembles the dynamical equation for the spin curren t derived by Qi and\nZhang27using a “mean field” approximation. Our derivation shows that their q uantityv2\nx\nis equal to v2\nF/3. As will be discussed in the next section, this is the wavefront veloc ity for\na spin disturbance, which plays an important role in spin-signal propa gation dynamics21.\nIII. TELEGRAPH EQUATION\nTo see the physical significance of Eqs. (6) and (7) for the time-de pendent noncollinear\nspin transport and compare them with the Bloch-Torrey equation, we combine them by\n5eliminating /vector s(x,t) into a form reminiscent of a telegraph equation21\n∂2/vector ns(x,t)\n∂t2+(1\nτ+1\nT1)∂/vector ns(x,t)\n∂t+/vector ns(x,t)\nτT1+γ/bracketleftbigg\n2∂\n∂t+(1\nτ+1\nT1)/bracketrightbigg\n/vector ns(x,t)×/vectorBs\n+γ2[/vector ns(x,t)×/vectorBs]×/vectorBs\n=c2\ns∂2/vector ns(x,t)\n∂x2+µE(x,t)\nτ∂/vector ns(x,t)\n∂x+µ\nτ∂E(x,t)\n∂x/vector ns(x,t).(9)\nSimilarly, one can also derive a telegraph equation for /vector s(x,t) by eliminating /vector ns(x,t) from\nEqs. (6) and (7). Equation (9) contains a second-order time deriv ative, which is absent in\nthe spin diffusion equation. The second-order time and space deriva tives lead to a wave\ncharacter in addition to its diffusion character, and thus yield a well-d efined propagation\nvelocitycsfor the signal in time-dependent noncollinear spin transport in a simila r way to\nthe collinear case.21\nAssuming the static magnetic field /vectorBsto be oriented along the zaxis and separating the\ncomponents perpendicular (transverse) and parallel (longitudina l) to/vectorBsin Eq. (9), we have\n∂2nx(y)\ns\n∂t2+/parenleftbigg1\nτ+1\nT1/parenrightbigg∂nx(y)\ns\n∂t+nx(y)\ns\nτT1+(−)γBs/bracketleftbigg\n2∂\n∂t+/parenleftbigg1\nτ+1\nT1/parenrightbigg/bracketrightbigg\nny(x)\ns−γ2B2\nsnx(y)\ns\n=c2\ns∂2nx(y)\ns\n∂x2+µE\nτ∂nx(y)\ns\n∂x+µ\nτ∂E\n∂xnx(y)\ns (10)\nand\n∂2nz\ns\n∂t2+/parenleftbigg1\nτ+1\nT1/parenrightbigg∂nz\ns\n∂t+nz\ns\nτT1=c2\ns∂2nz\ns\n∂x2+µE\nτ∂nz\ns\n∂x+µ\nτ∂E\n∂xnz\ns. (11)\nIn the following, only the equation for the transverse component [E q. (10)] will be discussed,\nsince the equation for the longitudinal component is similar to that of the collinear case.21\nFor vanishing electric field, i.e., E= 0, we seek damped and dispersive wave solutions to\nEq. (10) of the form\nnx\ns(x,t) =n0exp[i(kx−ωt)], (12)\nny\ns(x,t) =n0exp[i(kx−ωt+φ)], (13)\nwhereωis the angular frequency and k=kr+ikithe complex wave vector. Substituting\nEqs. (12) and (13) into Eqs. (10), we obtain the dispersion relation\nω2+iω(1/τ+1/T1)−1/(τT1)−c2\nsk2+γ2B2\ns−γBs[2ω+i(1/τ+1/T1)]sinφ= 0,(14)\nwhereφis restricted to φ=±(π/2) + 2nπandnis an integer, because nx\nsandny\nsmust\nsatisfy the system of equations (10) at the same time. According t o Eqs. (12) and (13),\n6φ= +(−)π/2 corresponds to the rotation direction of the transverse compo nent of/vector ns(x,t)\nwithxat timet. For definiteness, we study the case with φ=π/2 in the following.\nSubstituting k=kr+ikiinto Eq. (14) and separating the real and imaginary parts, we have\nk2\nr(i)=1\n2c2s/bracketleftbigg/radicalBig\nb2+ω2\neffα2+(−)b/bracketrightbigg\n, (15)\nwhereωeff=ω−γBsandb=ω2\neff−ξ. Here, the constants α= 1/τ+1/T1andξ= 1/(τT1)\nhave been introduced. The wavelength and damping length can be de fined asλ= 2π/kr\nandld= 1/ki, respectively. The equation of the critical angular frequency ωcrit, above which\nthe wave character is significant, can be derived by setting λ=ld,\nωcrit\neffτ=1\n2/bracketleftBig\nδ(1+η)+/radicalbig\nδ2(1+η)2+4η/bracketrightBig\n≈δ+(δ+1\nδ)η, (16)\nwhereδ=π−1/(4π)≈3.06 andη=τ/T1. Then, we have ωcritτ= 3.06 + 3.4η+τγBs\napproximately.\nIV. DYNAMICS OF PUMPED SPIN CURRENT\nIn this section, we study the evolution of the spin current injected into a nonmagnetic\nlayer by the spin-pumping mechanism.15In a junction composed of a ferromagnetic ( x <0)\nanda nonmagnetic ( x >0)layer, the magnetization precession of theferromagnet aroun dan\nexternal magnetic field /vectorBpumpactsasa“spinpump” which transfersspin angularmomentum\nfrom the ferromagnet to the adjacent nonmagnetic layer. The sp in current density pumped\ninto the nonmagnetic layer is15,29,30\n/vector pump\ns=1\n2πg↑↓\nS/vector m×d/vector m\ndt, (17)\nwhereg↑↓is the spin-mixing conductance and Sthe area of the interface. Here, /vector mis the\nunit vector for the magnetization of the ferromagnet. Note that the pumped spin current\nhas been converted to a particle number current density /vector pump\ns. Since we are interested in the\nspin current pumped into a nonmagnetic layer and not in the dynamics of the ferromagnet,\nwe neglect the back-flow spin current /vectorIback\ns, which flows from the nonmagnetic layer to the\nferromagnet due to the spin accumulation in the nonmagnetic layer.29Although the back-\nflow spin current can limit the achievable spin current into the nonmag netic conductor, we\ndo not approach this limit here. With this simplification, we have /vector pump\ns=/vector s(x= 0,t),\n7where/vector s(x= 0,t) is the spin current density at the left boundary of the nonmagnet ic layer.\nSeparating the components perpendicular and parallel to the magn etic field /vectorBpump, we can\nwrite/vector s(x= 0,t) as\njx\ns(x= 0,t) =g↑↓(4πS)−1ωsin(2θ)cos(ωt) (18)\njy\ns(x= 0,t) =g↑↓(4πS)−1ωsin(2θ)sin(ωt) (19)\njz\ns(x= 0,t) =g↑↓(2πS)−1ωsin2θ, (20)\nwhereωis the angular frequency of both the magnetization precession and the spin current\ndensity/vector s(x= 0,t). Here, ωtis the angle between /vector ⊥\ns(jx\nsandjy\ns) and the x-axis.θis the\nangle between /vector mand/vectorBpump, and meanwhile θis also the angle between /vector s(x= 0,t) and\nxy-plane. The amplitude of /vector ⊥\nsis much larger than jz\ns, sinceθis very small under the usual\nradio-frequency excitation conditions.30Therefore, we will focus on /vector ⊥\nsin the following.\nThe propagation of /vector ⊥\ns(x= 0,t) into the nonmagnetic layer is described by Eqs. (6)\nand (7). In a typical setup for spin pumping, there is no electric or m agnetic field in the\nnonmagnetic layer, i.e., E= 0 and /vectorBs= 0. Now, separating the components perpendicular\nand parallel to the magnetic field /vectorBpump, we can rewrite Eqs. (6) and (7) as\n∂n+\ns\n∂t+∂j+\ns\n∂x=−n+\ns\nT1, (21)\nj+\ns=−D∂n+\ns\n∂x−τ∂j+\ns\n∂t, (22)\nwheren+\ns=nx\ns+iny\nsandj+\ns=jx\ns+ijy\nsare introduced to simplify the notations. The\nequations for the parallel component can be obtained after replac ingn+\nsandj+\nsbynz\nsand\njz\nsin Eqs. (21) and (22), respectively. The method of characteristic s used for the numerical\nsolution to Eqs. (21) and (22) is outlined in Appendix B.\nIn our numerical calculation, Cu and permalloy (Py) are chosen as th e materials for\nthe nonmagnetic and ferromagnetic layers, respectively. The Fer mi velocity of Cu is vF=\n1570nm/ps and thus the wave-front velocity is cs=vF/√\n3 = 906nm/ps. The momentum\nandspinrelaxationtimesare τ= 0.07psand T1= 3.5ps, respectively. Thecriticalfrequency\ncan be estimated to be νcrit=ωcrit/(2π) = 7.11THz from Eq. (16). We study several\npumping frequencies: νa= 1/Ta= 2GHz, νb= 1/Tb= 20GHz, νc= 1/Tc= 200GHz,\nandνd= 1/Td= 8.33THz. For a Py/Cu junction,29g↑↓S−1is on the order of 1015cm−2.\nThe precession cone angle θcan reach 15◦for a sufficiently intense radio-frequency field.30\n8 0 100 200 300 400 500\n 0 200 400 600 800 1000\nx (nm) and jsx (10-5 nm-2 ps-1)Tc=5 ps\nt=5/4 Tc(c) 0 100 200 300 400 500jsy (10-5 nm-2 ps-1) Tb=50 ps\nt=5/4 Tb(b) 0 100 200 300 400 500Ta=500 ps\nt=5/4 Ta(a)\n 0 100 200 300 400 500\nFIG. 1: Snapshots of the spin current density /vector ⊥\nsatt= 5/4Ta, 5/4Tb, and 5/4Tc, for the\nfrequencies, νa,νb, andνc, respectively (see text). /vector ⊥\nsis plotted as vector starting from its x-\ncoordinate.\nTherefore, we choose the amplitude of /vector ⊥\ns, i.e.,g↑↓(4πS)−1ωsin(2θ), to be 5 ×10−3nm−2ps−1\nfor the frequencies mentioned above.\nFigure 1 shows snapshots of the spin current density /vector ⊥\nsatt= (5/4)Ta, (5/4)Tb, and\n(5/4)Tc, for the frequencies, νa,νb, andνc, respectively. According to Eqs. (18), /vector ⊥\ns(x= 0,t)\npoints in the direction of the y-axis att= 5/4Ta(b,c), which can also be seen in Fig. (1).\nFigure 1 (a) shows that /vector ⊥\nspoints along y-axis nearly at all xpoints except that it deviates\nfrom the y-axis slightly at positions far away from x= 0. The results in Fig. 1 (a) are\napproximately consistent with those obtained from the diffusion equ ation in Refs. 29 and\n30, where it is shown that both the spin current and spin accumulatio n point along the same\ndirection at all positions for all frequencies at certain time point t. This agreement means\nthat the diffusion equation provides a good description in the low freq uency range.31The\ndeviation of /vector ⊥\nsfrom the y-axis atx >0 increases with frequency and becomes noticeable\natνb= 1/Tb= 20GHz as shown in Fig. 1 (b). Therefore, the applicability of the diffu sion\nequation is questionable in this frequency region. At even higher fre quency,νc= 1/Tc=\n200 GHz, the deviation becomes significant and the diffusion equation is not applicable.\n9 0 100 200 300 400\n 0  200  400  600  800  1000\nx (nm) and nsx (10-7 nm-3)Tc=5 ps\nt=5/4 Tc(c) 0 100 200 300 400nsy (10-7 nm-3)Tb=50 ps\nt=5/4 Tb(b) 0 100 200 300 400\nTa=500 ps\nt=5/4 Ta(a)\n 0 100 200 300 400\nFIG. 2: Snapshots of the spin density /vector n⊥\nsfor the same parameters as in Fig. 1.\nMoreover, the damping length of /vector ⊥\nsdecreases with frequency due to the ‘skin’ effect.21We\ncan conclude that the spin diffusion equation is applicable only in the low f requency range\nand amounts to an adiabatic approximation: the external perturb ation is assumed to be\nmuch slower than the internal dynamics of the electronic system.\nFigure 2 shows snapshots of the spin density /vector n⊥\nsfor the same parameters as in Fig. 1.\nThe spin density /vector n⊥\nsdeviates from y-axis atx= 0 and is noncollinear with /vector ⊥\nsatx >0 at all\nof the three frequencies. This feature is different from the result of the diffusion equation,\nwhere/vector ⊥\nsand/vector n⊥\nsare collinear.29,30The phase shift and the amplitude of /vector n⊥\nsalso vary with\nfrequency. Moreover, the damping length of /vector n⊥\nsdecreases with frequency again due to the\n‘skin’ effect.\nAccording to Eq. (16), the diffusion character is dominant at the fr equencies considered\nso far, because they are still much smaller than the critical freque ncyνcrit. This conclusion\nis supported by the numerical results presented in Figs. 1 and 2, alt hough Figs. 1 (c) and 2\n(c) have already shown weak wavelike character. The deviation fro m the diffusion equation\ndepends largely on the frequency of the spin signal and momentum r elaxation time, which\nvaries with material, temperature, doping and excitation condition. In the following, we\nshow the numerical results for a frequency νd= 8.33THz, where the wave character is\nsignificant according to Eq. (16).\n10-0.004-0.002 0 0.002 0.004\n 0 50 100 150 200 250 300 350\nx (nm)t=3 Td (c)\n-0.004-0.002 0 0.002 0.004\n 0 50 100 150 200 250 300 350\nx (nm)t=3 Td (c)-0.004-0.002 0 0.002 0.004jsx and jsy (nm-2 ps-1) t=2 Td (b)\n-0.004-0.002 0 0.002 0.004jsx and jsy (nm-2 ps-1) t=2 Td (b)-0.004-0.002 0 0.002 0.004 t=1 Td (a)\n-0.004-0.002 0 0.002 0.004 t=1 Td (a)\n-0.004-0.002 0 0.002 0.004\nFIG. 3: Snapshots of /vector ⊥\nsatt=Td, 2Td, and 3Td, whereTd= 0.12 ps. The solid (dashed) curve is\nforjx\ns(jy\ns).\nFigure 3 shows snapshots of the spin current density /vector ⊥\nsatt= 1Td, 2Td, and 3Td, re-\nspectively. The wave form and wave front are clearly visible in Fig. 3. T he propagation\nvelocity of the spin signal can be estimated by tracking the motion of the wave front. The\nresult is approximately equal to the analytical result cs= 906nm/ps. The phase velocity\ncan also be estimated by measuring the wavelength λand using vp=λ/Td. The result\nis roughly equal to the wave front velocity cs, which also indicates the significance of the\nwave character, albeit on the length scale of the damping length (dy namical spin diffusion\nlength). To demonstrate the wave character more directly, we plo t the results of Fig. 3 (a)\nagain in Fig. 4, where /vector ⊥\nsis shown in a vector plot. Note that νdis beyond the frequency\nrange in which Eq. (17) is valid, because Eq. (17) is only applicable in the adiabatic limit,\nν≪1/τ.15Unfortunately, there is no corresponding theoretical result for the nonadiabatic\nspin-pumping in the literature. However, it is a reasonable guess tha t the pumped spin\ncurrent density in the nonadiabatic regime preserves the basic fea ture of Eq. (17): /vector pump\ns\nrotates with a certain fixed frequency. Therefore, the spin curr ent predicted by our results\nshould be at least qualitatively accurate in this frequency range.\n11-15-10-5 0 5 10 15\n 0  20  40  60  80  100  120jsy (5x10-4 nm-2 ps-1)\nx (nm) and jsx (5x10-4 nm-2 ps-1)\nFIG. 4: Snapshot of /vector ⊥\ns(plotted as vector) at t= 1Td.\nV. SUMMARY\nWe showed that time-dependent noncollinear spin transport exhibit s a wave character for\nmodulation of the spin current on timescales shorter than an invers e critical frequency. A\nfinite propagation velocity for the spin signal can be defined due to t his wave character. The\nspindiffusionequationisrecovered onlyformodulationwithfrequenc ies lessthanthecritical\nfequency, and amounts to an adiabatic approximation of time-depe ndent spin transport.\nAcknowledgments\nWe acknowledge financial support from the state of Rheinland-Pfa lz through the MAT-\nCOR program and a CPU-time grant from the John von Neumann Inst itut for Computing\n(NIC) at the Forschungszentrum J¨ ulich.\nAPPENDIX A: DERIVATION\nEquations (6) and (7) of Ref. 27 are derived using the “mean field” a pproximation\n/summationdisplay\n/vector vv2\nx(∂ˆρ/∂x)≈v2\nx/summationdisplay\n/vector v(∂ˆρ/∂x). (A1)\nHere we show that v2x=c2\nsby evaluating the sums occurring in Eq. (A1). We start with\nthe LHS, which we denote by I1=/summationtext\n/vector vv2\nx(∂ˆρ/∂x). Due to the cylindrical symmetry of the\nsystem around the xaxis in velocity space, ˆ ρcan be expanded in Legendre polynomials of\nu= cosθ, whereθis the angle between /vector vand thexaxis, as\nˆρ=∞/summationdisplay\nn=0ˆρn(v,x)Pn(u). (A2)\n12Transforming the summation into an integral, we have\nI1=2πVm∗3\nh3/integraldisplay1\n−1duu2/integraldisplay∞\n0dvv4∞/summationdisplay\nn=0∂\n∂xˆρn(v,x)Pn(u). (A3)\nUsingu2= [2P2(u)+P0(u)]/3, we write the integral as\nI1=2πVm∗3\nh3/integraldisplay∞\n0dvv4∞/summationdisplay\nn=0∂\n∂xˆρn(v,x)/integraldisplay1\n−1du1\n3[2P2(u)+P0(u)]Pn(u).(A4)\nMaking use of the orthogonality relation of Legendre polynomials, we have\nI1=2πVm∗3\nh3/integraldisplay∞\n0dvv4∂\n∂x/bracketleftbigg4\n15ˆρ2(v,x)+2\n3ˆρ0(v,x)/bracketrightbigg\n. (A5)\nIf the system is weakly anisotropic, we can neglect the second-ord er term ˆρ2(v,x),\nI1≈4πVm∗3\n3h3/integraldisplay∞\n0dvv4∂\n∂xˆρ0(v,x). (A6)\nThis approximation is consistent with Ref. 4, where the second-ord er term of the Legendre\npolynomials is neglected and it is shown that this is valid if/radicalbig\nτ/(2T1)≪1.\nBecause ∂ˆρ0(v,x)/∂xis zero unless vfalls in a small region [ vF−∆v,vF+∆v] around\nthe Fermi velocity vFof a system with a degenerate electron gas, we have approximately\nI1=4πVm∗3\nh3v2\nF\n3/integraldisplayvF+∆v\nvF−∆vdvv2∂\n∂xˆρ0(v,x) =v2\nF\n34πVm∗3\nh3/integraldisplay∞\n0dvv2∂\n∂xˆρ0(v,x).(A7)\nWe now need to evaluate the RHS of Eq. (A1), which we denote by\nI2=v2xVm∗3\nh32π/integraldisplay1\n−1du/integraldisplay∞\n0dvv2∂\n∂xˆρ(/vector v,x) =v2x4πVm∗3\nh3/integraldisplay∞\n0dvv2∂\n∂xˆρ0(v,x).(A8)\nwhere, in the last line, we used that the integral over uprojects the contribution of P0out\nof ˆρ(/vector v,x). Because I1=I2, we conclude that v2\nx=v2\nF/3≡c2\ns.\nAPPENDIX B: NUMERICAL SOLUTION\nThe basics of our numerical method have been outlined in Appendix A4 of Ref. 21.\nFor present calculation, it has to be augmented by a discretized ver sion of the boundary\ncondition on at ferromagnet/nonmagnet interface,\n(∆t/T1+2)n+,l+1\ns,i=−(∆t/T1−2)n+,l\ns,i+1+c−1\ns(∆t/τ−2)j+,l\ns,i+1+c−1\ns(∆t/τ+2)j+,l+1\ns,i\n(B1)\n13where the subscripts iand superscripts lstand for the discrete space-time points, and ∆ tis\nthe numerical time step.\n∗Electronic address: yaohuizhu@gmail.com\n†URL:http://www.physik.uni-kl.de/schneider\n1A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rep. 427, 157 (2006).\n2W. P. Pratt, Jr., S.-F. Lee, J. M. Slaughter, R. Loloee, P. A. S chroeder, and J. Bass, Phys. Rev.\nLett.66, 3060 (1991).\n3P. C. van Son, H. van Kempen, and P. Wyder, Phys. Rev. Lett. 58, 2271 (1987).\n4T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993).\n5G. Schmidt, D. Ferrand, L. W. Molenkamp, A. T. 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Gerrits, M. L. Schneider, and T. J. Silva, J. Appl. Phys. 99, 023901 (2006).\n15"    },    {        "title": "2207.10567v1.Simulating_spin_dynamics_with_quantum_computers.pdf",        "content": "Simulating spin dynamics with quantum computers\nJarrett L. Lancaster<and D. Brysen Allen\nDepartment of Physics, High Point University, One University Parkway, High Point, NC 27268\n(Dated: July 22, 2022)\nIBM quantum computers are used to simulate the dynamics of small systems of interacting quantum spins.\nFor time-independent systems with Nf2spins, we construct circuits which compute the exact time evolu-\ntion at arbitrary times and allow measurement of spin expectation values. Larger systems with spins require\na the Lie-Trotter decomposition of the time evolution operator, and we investigate the case of N= 3explic-\nitly. Basic readout-error mitigation is introduced to improve the quality of results run on these noisy devices.\nThesimulationsprovideaninterestingexperimentalcomponenttothestandardtreatmentofquantumspininan\nundergraduate quantum mechanics course.\nI. INTRODUCTION\nThefieldofquantumcomputationhasexperiencedtremen-\ndous growth in recent years. Originally suggested as the\nonly possible way to simulate quantum mechanical systems\nwith many degrees of freedom,1quantum computing algo-\nrithms have also been shown to play a valuable role in an ef-\nficientfactorizationscheme2whichhasdramaticimplications\nfor modern cryptographic systems.3Though such devices are\nstillfarfromcapableofsolvingtheseclassicproblems,noisy,\nintermediate-scalequantum(NISQ)devices4arewidelyavail-\nable from providers such as Rigetti,5IonQ,6and IBM,7The\nquestion for the foreseeable future is: what kinds of problems\nare these NISQ devices capable of solving?\nWhile many potential uses for NISQ-era hardware have\nbeen proposed,8one under-appreciated use for these devices\nis as a learning tool for quantum mechanics. As it happens,\nquantum computers are ideally suited for probing small spin\nsystems,9such as those that would be explored in an under-\ngraduatecourseonquantummechanics. Recentpapersinthis\njournal have explored some basic problems future quantum\ncomputers can conceivably be used to solve,10employing the\nGrover search algorithm on actual quantum hardware,11us-\ningquantumcomputerstomeasurespincorrelations,9andap-\nplying basic error mitigation to improve results on currently\navailable hardware.12Given the free availability of 1-7 qubit\ndevicesfromIBM,thesimulationofspinsystemsprovidesan\nincredibleopportunitytogivestudentsano-costexperimental\ncomponent which complements the fairly abstract theory that\nis standard treatment in such a course.\nInthispaper,wepresentseveraltypesofspinsimulationsin-\nvolvingthetimeevolutionofarbitrarilypreparedinitialstates.\nThe Qiskit software development kit (SDK)13is used to en-\ncode time evolution in quantum circuits, which are then run\nonthecloud-basedIBMhardware. Timeevolutionofasingle\nspin is simulated by mapping the spin to a qubit and apply-\ning a single unitary gate corresponding to the exact time evo-\nlution operator. Results for two-spin and three-spin systems\nwithnontrivialinteractionsarealsopresented. Theschemefor\nperformingtimeevolutionontwo-spinsystemsisexact,butit\nrequires a more complicated circuit. We are forced to use ap-\nproximate methods when constructing time evolution for sys-\ntems with three or more spins, and it is shown that errors and\nnoiseinherenttocurrently-availabledevicesmakeaccuratere-sults quite difficult to obtain for even systems of three spins.\nThe remainder of this paper is organized as follows. Sec-\ntion II contains a brief review of relevant spin physics and\nthe essential ingredients needed to simulate spin systems on\nIBM’squantumhardware. Theexactdynamicsofsinglespins\nevolving in a magnetic field is explored. Lie-Trotter decom-\nposition, essential for simulating systems with more than two\nspins, is introduced in Sec. III in the simplified context of a\nsingle spin, where the convergence of the method and practi-\ncallimitationsinitsimplementationcanbedemonstratedmost\nclearly. Dynamicsininteractingsystemsoftwoandthreespins\nare shown in Sec. IV, and the Lie-Trotter decomposition is\nshowntobeessentialtoperformingtimeevolutioninthethree-\nspin system. Finally, a brief discussion is included in Sec. V.\nII. SINGLE SPINS\nA single spin-1\n2degree of freedom is a two-level system\nwhosequantumstatecanbeexpressedasalinearcombination\nof basis kets ð ë=\u000bð+ë+\fð*ë, whereð,ëare eigenstates\nof the operator Sz, satisfying\nSzð,ë= ,`\n2ð,ë: (1)\nWefollowstandardnotationforlabelingtheeigenstatesof Sz\n(e.g., Ref. 14), but we caution the reader with experience in\nquantum information theory that these labels refer to eigen-\nstates ofSxin other references.15The complex amplitudes\nareconstrainedbytheunitnormalizationofthequantumstate\nê ð ë=ð\u000bð2+ð\fð2= 1. Astheglobalphaseof ð ëisunob-\nservable, a common parameterization is given by \u000b= cos\u0012\n2,\n\f=ei\u001esin\u0012\n2. To aid in calculations, the basis states can be\nrepresented by ð+ë=.1;0/Tandð*ë=.0;1/Tso that the state\nð ëcan represented by the two-component vector\nð ë=H\ncos\u0012\n2\nei\u001esin\u0012\n2I\n: (2)arXiv:2207.10567v1  [physics.ed-ph]  29 Jun 20222\nBy employing the explicit matrix representations of the spin\noperators,\nSx=`\n20\n0 1\n1 01\n;Sy=`\n20\n0 *i\ni01\n;\nSz=`\n20\n1 0\n0 *11\n; (3)\nonemaycomputetheexpectationvaluesofthespinoperators\nêS\u000bëê ðS\u000bð ëfor the state in Eq. (2),\n\u0010Sx\u0011=`\n2sin\u0012cos\u001e;\n\u0010Sy\u0011=`\n2sin\u0012sin\u001e;\n\u0010Sz\u0011=`\n2cos\u0012: (4)\nThus, the parameters \u0012and\u001ecorrespond to angular coor-\ndinates which map the possible states to points on a unit\nsphere—also known as the Bloch sphere. Time evolution can\nbegeneratedbyinteractionofthespinwithamagneticfield B\nvia the Hamiltonian\nH= *\u0016B\fS; (5)\nwhere the constant \u0016represents the magnetic moment which\nallows the spin to couple to an external field. The time-\ndependent state ð .t/ëis obtained by solving the Schrödinger\nequation,\ni`d\ndtð .t/ë=Hð .t/ë; (6)\ngiven some initial state ð .0/ë. The calculation of ð .t/ëis\ntypically performed by representing Has a2  2matrix and\nextractingseparate,coupled,first-orderequationsforthetime-\ndependentamplitudes \u000b.t/and\f.t/,whichcanbesolvedusing\nstandard methods. With the aim of simulating such dynamics\nonaquantumcomputer,itisconvenienttotakeadifferentap-\nproach. One may write the formal solution as\nð .t/ë= exp\u001c\n*i\n`Ht\u001d\nð .0/ëU.t/ð .0/ë;(7)\nandcalculatingthetime-evolutionoperator U.t/explicitlyasa\n22matrix. Asmatrixexponentiationisnotalwayscoveredin\nintroductory quantum mechanics, let us consider the specific\nexample B=B0xin detail. The time-evolution operator can\nbe written\nU.t/ = exp4i!0t\n2 \u001bx5\n; (8)\nwhere!0\u0016B0. Usingthepropertythat [ \u001bx]2=I,onemay\nexpandtheexponentialasaTaylorseries,groupevenandoddterms separately to obtain,\nU.t/ =I0\n1 *.!0t_2/2\n2@+.!0t_2/4\n4@+51\n+i \u001bx0\n!0t_2 *.!0t_2/3\n3@+.!0t_2/5\n5@+51\n= cos!0t\n2I+isin!0t\n2 \u001bx\n=0cos\u001c\n2isin\u001c\n2\nisin\u001c\n2cos\u001c\n21\n; (9)\nwhere\u001c!0t. Here we have made use of the Taylor expan-\nsions for sin\u001c\n2andcos\u001c\n2to sum the infinite series explicitly.\nFor the specific initial state ð .0/ë=ð+ë, one can employ\nEqs. (2)–(3) to obtain\n\u0010Sx.t/\u0011= 0\n\u0010Sy.t/\u0011=`\n2sin!0t\n\u0010Sz.t/\u0011=`\n2cos!0t: (10)\nEquations(10)arethetypesoftheoreticalpredictionswewish\nto test on a quantum computer. In a quantum computer, the\nfundamental object is a qubit which can exist in a superpo-\nsition of two basis states, typically labeled ð0ëandð1ë. This\nbasisiscommonlyreferredtoasthe“computationalbasis.” A\nquantumspinmayberepresentedbyaqubitwiththemapping\nð+ëð0ë;\nð*ëð1ë; (11)\nUniversal quantum computers execute circuits which consist\nof series of gates that modify the states of the qubits. To ex-\ntract information about the system, some measurement must\nbe performed. Thus, the three basic stages in a quantum cir-\ncuit appropriate for simulating the spin dynamics discussed\naboveare(1)stateinitialization,(2)timeevolutionofthestate,\nand(3)measurementofthespincomponents. IBM’squantum\nhardware automatically initializes each qubit to the state ð0ë.\nAgeneralquantumstateoftheforminEq.(2)canbegenerated\nby applying the following unitary operator,\nU3.\u0012;\u001e;\u0015 /=H\ncos\u0012\n2*ei\u0015sin\u0012\n2\nei\u001esin\u0012\n2ei.\u001e+\u0015/cos\u0012\n2I\n:(12)\nUsers may access quantum hardware through the Python-\nbasedQiskitSDK.13TheQiskittextbook15providesanexcel-\nlent overview to the basics of using Qiskit to construct quan-\ntum circuits, so we only sketch the main steps in translating\nstate preparation and time evolution into a series of quantum\ngates. Thereaderisreferredtootherexcellentresources9,15,16\nfor additional background information. Sample Jupyter note-\nbooks containing programs used in this work are included in\nthe supplemental material.17\nTheU3gateisavailableasathree-parametergateinQiskit,\nu(theta,phi,lambda) . The operator in Eq. (12) is actually3\nmore general than we require, as \u0015can be set to any value.\nHowever, as this operation is available as a built-in quantum\ngate for use on the IBM quantum devices, it provides a con-\nvenient mechanism for generating a qubit in an arbitrary state\nð .0/ë=U3.\u0012;\u001e;0/ð0ë. Time evolution is performed by not-ingthatourexpressionforthetimeevolutionoperator U.t/in\nEq. (8) may also be written in terms of the U3.\u0012;\u001e;\u0015 /gate as\nU=U3\u0018\n\u001c;+\u0019\n2;*\u0019\n2\u0019\n.\n(a)\n|0/angbracketrightU3(τ,π\n2,−π\n2)Ry(−π\n2)\n|0/angbracketrightU3(τ,π\n2,−π\n2)Rz(−π\n2)Ry(−π\n2)\n|0/angbracketrightU3(τ,π\n2,−π\n2)\nc\n(b)\nrepeatntimes\n|0/angbracketrightU3/parenleftBig\nτ\nn√\n2,π,π/parenrightBig\nU3/parenleftBig\nτ\nn√\n2,π\n2,−π\n2/parenrightBig\nRy(−π\n2)\n|0/angbracketrightU3/parenleftBig\nτ\nn√\n2,π,π/parenrightBig\nU3/parenleftBig\nτ\nn√\n2,π\n2,−π\n2/parenrightBig\nRz(−π\n2)Ry(−π\n2)\n|0/angbracketrightU3/parenleftBig\nτ\nn√\n2,π,π/parenrightBig\nU3/parenleftBig\nτ\nn√\n2,π\n2,−π\n2/parenrightBig\nc/bracehtipdownleft /bracehtipupright/bracehtipupleft /bracehtipdownright\n(c)\nN(α,β,γ ) =Rz(π\n2−2γ)• Rz(π\n2)\nRz(−π\n2)•Ry(2α−π\n2)Ry(π\n2−2β)•\n(d)\n|0/angbracketrightU3(π\n6,π\n3,0)\nN(α,β,γ )\n|0/angbracketrightU3(3π\n5,4π\n3)\nc\n(e)\nrepeatntimes\n|0/angbracketrightU3(π\n6,π\n3,0)\nN(α,β,γ )\n|0/angbracketrightU3(3π\n5,2π\n3)\nN(α,β,γ )\n|0/angbracketrightU3(−π\n5,4π\n3)\nc\nFIG.1: Quantumcircuitsusedtoperformvarioustasksdescribedinthemaintext;(a)circuittoperformtimeevolutionandmeasurethreespin\ncomponentsforaspinpreparedinthestate ð+ë. Asqubitsareinitializedinto ð0ëbydefault,noadditionalstatepreparationstageisrequired,and\nthecircuitconsistsonlyoftimeevolutionprovidedby U3\u0018\n\u001c;\u0019\n2;*\u0019\n2\u0019\nfollowedbymeasurement. Forthe x-andy-components,staterotations\narerequiredbeforemeasuringinthecomputationalbasis,asdiscussedinthemaintext;(b)circuittoperformtimeevolutionandmeasurethree\nspin components; (c) circuit for exact exponentiation of exp[i\u000b \u001bxä  \u001bx+i\f \u001byä  \u001by+i\r \u001bzä  \u001bz]; (d) circuit for exact time evolution of the\ntwo-spinstate inEq.(26) withHamiltonianin Eq.(23)for Jx= 0:5J,Jy= *0:45J,Jz= 0:25J; (e)circuitfor approximatetimeevolution of\na three-spin state with Hamiltonian in Eq. (31) for Jx= 0:5J,Jy= *0:45J,Jz= 0:25J.\nWhen using IBM quantum devices, it is only possible to\nmake a direct measurement in the computational basis. That\nis, for a qubit in the state ð ë=\u000bð0ë+\fð1ë, a measurement\nwill only return 0 or 1 corresponding to the states ð0ëorð1ë,\nrespectively. Thisconstraintsuggestsoneisonlyabletomea-\nsureSzdirectly in a spin dynamics simulation. Reference 9\nnotes that one can measure spin along any axis (or equiva-\nlently,thequbitinanybasis)byanappropriaterotationofthe\nsystem before performing the measurement in the computa-\ntional basis. Within Qiskit, one may directly apply rotation\ngates (e.g., Rx.\u0012/for a rotation of angle \u0012about thex-axis)\nto the quantum state. To measure spin about the x-axis, one\nfirstperformsrotationsonthesystemwhichbringthevector x\ninto alignment with the z-axis. Explicitly, z=Ry\u0018\n*\u0019\n2\u0019\nx=\nRy\u0018\n*\u0019\n2\u0019\nRz\u0018\n*\u0019\n2\u0019\ny. Thisrotationprocessisdepictedpicto-\nrially in Fig. 2 for some arbitrary axis, n.\nAs IBM quantum devices are available for use by anyonein the world, jobs are often queued–sometimes for significant\ntimes. With efficiency in mind, it is convenient embed three\nindependent, single-spin circuits into a single three-qubit cir-\ncuittomeasureallthreespincomponentsinasinglejob. The\ncircuitshowninFig.1acreatesthreecopiesofthissinglespin,\nperformsidenticaltimeevolutiononeachspin,andthenmea-\nsures one of the three spin components on each qubit. Using\nQiskit,thecircuitinFig.1acanbeconstructedandsenttoIBM\nquantum devices for execution.\nWhen one executes the quantum circuit in Fig. 1a, the re-\nsult yields a sequence of three digits, each being either 0 or\n1, which is written to the “classical register.” For example,\none might obtain the readout ‘100’, indicating that +`\n2was\nobtainedfor SxandSy,while *`\n2wasobtainedfor Sz.18The\nframework of quantum mechanics does not allow for mean-\ningful predictions about the outcome of a single such exper-\niment. Instead, we are limited in being able to compute sta-\ntisticalquantitiessuchasprobabilitiesandexpectationvalues,4\nz\nyxφθn^z\nyx-φ-θn'^\nFIG. 2: To measure spin about the axis n, one must perform the re-\nquired rotations to bring ninto alignment with the z-axis (the com-\nputationalbasis). Givenpolar-sphericalangles \u0012,\u001ecorrespondingto\nn, one findsn¨z=Ry.*\u0012/Rz.*\u001e/n.\nwhich give us information about the averages of many repeti-\ntionsofanexperimentperformedonidenticallypreparedsys-\ntems. Typically, one executes a circuit for some large number\nof “shots,”Nshots¸1, so that experimental averages can be\nconstructedforcomparisontothetheoreticalpredictions(e.g.,\nEqs.10). InQiskit,thecountsassociatedwithexecutingacir-\ncuitoversomenumberofshotsisreturnedaPythondictionary\nof the form ‘state’ : counts . Here ‘state’ is a string\ncorresponding to a possible three-qubit state (e.g., ‘010’\nð010ë) and countsis the number of shots for which this state\nwas written to the classical register. From this information,\nwe construct spin expectation values as follows. Let nabcbe\nthecountsforstate ðabcë,wherea;b;c Ë^0;1`. TakingêSyë\nas an example, we have eight counts, n000;n001;5;n111. A\nweighted average of this spin component is given by\nêSyë=1\nNshots\u001c\u0000n000+n001+n100+n101\u0001\u0018\n+`\n2\u0019\n+\u0000n010+n011+n110+n111\u0001\u0018\n*`\n2\u0019\u001d\n(13)\nNote that the counts for which the middle qubit is measured\nto beð0ë(corresponding to spin “up”) weight the contribu-\ntion of +`\n2, while those in which the middle qubit is in ð1ë\n(corresponding to spin “down”) weight the contribution of\n*`\n2. Analogous averaging can be performed on the other\nqubitstoobtainexperimentalvaluesfortheexpectedaverages\nêSxë,êSzë. Results from running the circuit in Fig. 1a on\nibmq_casablanca (v 1.2.35, a Falcon r4H processor) with\nNshots= 8192are shown in Fig. 3. Error bars shown are sta-\ntistical estimates given by \u001b_`ù1\n2ù\nNshots.\nThe framework outlined in this section allows one to simu-\nlatethedynamicsofasinglespininanexternalmagneticfield.\nStraightforward generalizations for students could consist of\napplyinganadditional U3gatebeforetimeevolutiontocreate\nanarbitraryinitialstateorapplyingadditionalrotationsbefore\nmeasurement to measure spin about a specified axis. Addi-\ntionally, one could investigate a magnetic field pointing along\nadifferentdirection. Forexample,considerthemagneticfield\nB=B0ù\n2.x+y/. One can repeat the basic steps leading to\n0 /2 3/2 2-0.6-0.4-0.200.20.4FIG. 3: Theory (dashed lines) and actual results (markers with er-\nror bars) for time evolution of spin components with B=B0xand\nð .0/ë=ð+ëusing circuit shown in Fig. 1a on ibmq_casablanca\n(v 1.2.35, a Falcon r4H processor).\nEq. (9) to obtain\nexpL\ni!0t\n2ù\n2. \u001bx+ \u001by/tM\n=U3\u0018\n\u001c;3\u0019\n4;*3\u0019\n4\u0019\n:(14)\nTo simulate the dynamics generated by this new magnetic\nfield,oneonlyneedstoreplace U3\u0018\n\u001c;\u0019\n2;*\u0019\n2\u0019\ninthecircuitin\nFig. 1a byU\u0018\n\u001c;3\u0019\n4;*3\u0019\n4\u0019\n. The experimental and theoretical\nresults for this system are shown in Fig. 4. Theoretical spin\nexpectation values are obtained from direct exponentiation of\nthe left side of Eq. (14) to compute ð .t/ëin Eq. (7). Time-\ndependentexpectationvaluesofthespinoperatorsinEqs.(3)\ngive\n\u0010Sx.t/\u0011= *`\n2ù\n2sin!0t;\n\u0010Sy.t/\u0011=`\n2ù\n2sin!0t;\n\u0010Sz.t/\u0011=`\n2cos!0t: (15)\nNote that despite using a circuit with three qubits, the indi-\nvidualqubitsareuncoupledanddonotinteract. Thecircuitin\nFig.1arepresentsthreecopiesofasinglequbitcircuit,withthe\nindividualqubitsonlyexperiencingdifferentrotationspriorto\nmeasurement so that all three spin expectation values can be\ncomputedwithinasinglejob. Inthenextsections,wedevelop\nthe machinery required to explore time evolution in multispin\nsystems in which the spins interact. This added complexity\nwill necessitate the introduction of multiple-qubit gates. Be-\nfore turning attention to the interaction of multiple spins, we\nfirst introduce a technique for simulating dynamics in more\ncomplex Hamiltonians: the Lie-Trotter decomposition.5\n0 /2 3/2 2-0.6-0.4-0.200.20.4\nFIG. 4: Theory/results for time evolution of spin components us-\ningibm_lagos (v 1.0.06, a Falcon r5.11H processor) with B=\nB0ù\n2.x+y/andð .0/ë=ð+ë.\nIII. LIE-TROTTER DECOMPOSITION\nForarbitrarysystems,itisnotpossibletocomputetheexact\ntimeevolutionoperator U.t/ingeneral. Inthiswork,weonly\nconsidertime-independentinteractions. Whilecertainclasses\nof these models can be solved exactly,19it is generally nec-\nessary to use approximate techniques for interacting systems.\nOne such approximation scheme for computing the time evo-\nlution operator relies on splitting the Hamiltonian into non-\ncommutingpiecesforwhichtimeevolutionofeachsubsystem\ncan be computed exactly. By computing the corresponding\nparts of the time evolution separately and layering the action\nof these operators over a large number of substeps, the ap-\nproximate time evolution will converge to the exact result as\nthe number of substeps approaches infinity. This technique is\nknown as a Lie-Trotter decomposition,20and below we intro-\nduce the method using a particularly simple system.\nConsider a Hamiltonian which is composed of two parts\nwhich do not commute.\nH=A+B; (16)\nwith\u0004A;B\u00050. As a simple example, let us take the single\nspininamagneticfieldwith B=B0ù\n2.x+y/consideredinthe\nprevious section so that\nH=!0ù\n2\u0004 \u001bx+ \u001by\u0005: (17)\nIn this case, A=!0ù\n2 \u001bxandB=!0ù\n2 \u001by, and the commutatorof these pieces is demonstrably nonzero,\n\u0004A;B\u0005=!2\n0\n2\u0004 \u001bx; \u001by\u0005=i!2\n0\n2 \u001bz0:(18)\nThe time-evolution operator for this system is given by\nU.t/ = exp\u0004*iHt_`\u0005= exp\u0004*iAt_`*iBt_`\u0005:(19)\nIn Sec. II, we computed U.t/exactly. In general, such exact\ncomputations are not possible, so here we take an alternative\napproach. Let us define the operators\nUA.t/ = exp\u0004*iAt_`\u0005;UB.t/ = exp\u0004*iBt_`\u0005:(20)\nAsAandBdo not commute, U.t/UA.t/UB.t/. However,\naccording to the Lie-Trotter product formula,20we can say\nexp\u0004i.A+B/t_`\u0005\n= lim\nnØ\u0000exp\u0004*iAt_n`\u0005exp\u0004*iBt_n`\u0005\u0001n;(21)\nor\nU.t/ = lim\nnØ\u0018\nUA\u0018t\nn\u0019\nUB\u0018t\nn\u0019\u0019n\n:(22)\nThatis,ifwesubdivideatimestepinto nsub-steps,theprod-\nuct of exponentials will converge to the true expression for\nU.t/asnØ. The particular example of the magnetic field\nin thexy-plane is instructive, as the exact solution for U.t/\nis available for comparison. For the single-spin Hamiltonian\nin Eq. (17), a circuit based on the Trotter decomposition in\nEq. (22) is shown in Fig. 1b. Results from running this cir-\ncuit on the QASM simulator are shown in Fig. 5a for sev-\neral choices of n. The QASM simulator uses classical com-\nputation to simulate results for a given quantum circuit on a\nnoiselessdevice,clearlydemonstratingtheconvergenceofthe\nTrotter decomposition on “ideal” hardware. We note that in-\ncludingmoreTrotterstepsrequiresmorecircuitgates,leading\nto an increased circuit depth. In actual hardware, larger cir-\ncuit depth is correlated with a larger accumulation of errors.\nWhile more steps lead to better Trotter convergence, adding\nmore steps also increases errors. Results for different choices\nofnobtained from actual quantum hardware ( ibmq_manila ,\nv1.0.10, a Falcon r5.11L processor) are shown in Fig. 5b and\ndepict the initial convergence is eventually washed out by the\naccumulation of errors as nincreases. A choice of ní 10ap-\npears to be the best for balancing errors due to circuit depth\nand errors due to finite Trotter steps in this simple circuit on\nthis particular device.\nOfcourse,thetimeevolutionoperatorcorrespondingtothe Hamiltonian in Eq. (5) with B=B0ù\n2.x+y/has been given6\n023-0.6-0.4-0.200.20.40.6\n023023023023\n023-0.6-0.4-0.200.20.4\n023023023023(b)(a)\nFIG.5: (a)Theory/simulated(QASMsimulator)resultsfortime-dependentspincomponentswith B=B0ù\n2.x+y/andð .0/ë=ð+ëwithvari-\nablenumberofTrottersteps;(b)theory/resultsfor B=B0ù\n2.x+y/andð .0/ë=ð+ëwithvariablenumberofTrotterstepsusing ibmq_manila\nv1.0.10, a Falcon r5.11L processor.\nexactlyinEq.(14). Forevenmodestlymorecomplicatedsys-\ntems, it is generally not possible to obtain an exact form for\nthe time evolution operator, so a similar Trotter decomposi-\ntionwillbeessentialtosimulatingthedynamics. Theaimsof\nthis section have been to illustrate the algorithm in a simple\nsetting while also highlighting the accumulation of errors as-\nsociatedwithincreasingcircuitdepth. Accuratesimulationof\ninterestingsystemsonactualquantumhardwareisachalleng-\ning endeavor as adding Trotter steps aids in the convergence\nof the algorithm while also contributing costly errors to the\ncalculation. Reasonableaccuracyhasbeenobtained21forsys-\ntems ofní 10spins using nomore than n= 5steps for short\ntimes. We note that our computations use exactly nTrotter\nsteps for each time step for conceptual simplicity. In certain\napplications,itmightbedesirabletouseavariablenumberof\nTrotter steps so that circuit depth can be decreased at shorter\ntimes.\nIn the next section, we will explore small systems of inter-\nacting spins and demonstrate how Trotter decomposition may\nbe used to simulate dynamics. As a first step, we will show\nhowthecaseof N= 2spinscanbesimulatedexactly,paving\nthewayforusingtheLie-Trotterdecompositiontoinvestigate\nlarger systems.\nIV. MULTIPLE SPINS\nA.N= 2\nWith the ability to decompose the time evolution operator\napproximately into a product of simpler factors, we now turn\nattention to the problem of multiple spins which interact. Forthesimplestcase, N= 2,afairlygeneralinteractionHamilto-\nnian is given by\nH= *JxSxäSx*JySyäSy*JzSzäSz;(23)\nwhere theJx;y;zare couplings. Semi-classically, Eq. (23)\nis motivated by the classical result that the energy con-\ntained in the electromagnetic field of two magnetic dipoles\n \u00161;2is proportional to * \u00161\f \u00162so that aligned spins are\nenergetically favorable. We will proceed with couplings\nas arbitrarily tunable parameters to explore how to simu-\nlate such a system on a quantum computer. To this end,\nthe basic task is to compute the time evolution operator\nU.t/ = exp\u0004*iHt\u0005. For brevity, we employ units in which\n`1for the remainder of this paper. The circuit shown\nin Fig. 1c depicts a representation of the two-qubit oper-\nation exp[*\u000b \u001bxä  \u001bx*\f \u001byä  \u001by*\r \u001bzä  \u001bz]. The two-\nqubit gates are known as “controlled NOT” (CNOT) gates,\nwhich perform the following operation\nð0ëð1ë;ð1ëð0ë; (24)\non the “target” qubit onlyif the “control” qubit is in the state\nð1ë. In the circuit diagram, the target qubit is denoted by the\n“+”andconnectedtothecontrolqubit. EmployingtheQiskit\nconventionforstatelabels,theactionofthisgateontwoqubit\nstates can be summarized as\nCNOTð00ë=ð00ë;CNOTð01ë=ð11ë;\nCNOTð10ë=ð10ë;CNOTð11ë=ð01ë;(25)\nwith the qubits arranged in the form ðtarget;controlë. Two-\nqubit gates such as CNOT typically lead to errors which are7\n0246810-0.500.5\n0246810-0.500.51,21,21,2\nFIG. 6: Time-dependence of spin expectation values for an interact-\ning, two-spin system using ibmq_bogota to simulate the model in\nEq. (23) with the initial state given by Eq. (26).\nanorderofmagnitudelargerthanthoseassociatedwithsingle-\nqubitgates.21Thusitisdesirabletouseaminimumnumberof\nCNOT gates. The circuit in Fig. 1c actually shows an optimal\nrepresentationofthecorrespondingoperator,usingonlythree\nCNOT gates.21–23Setting\u000b= *Jx_4,\f= *Jy_4and\r=\n*Jz_4, the exact time evolution operator U.t/corresponding\nto Eq. (23) is given by N.\u000b;\f;\r /.\nAsaspecificexample,letustake Jx= 0:5J,Jy= *0:45J,\nJz= 0:25Jfor some arbitrary energy scale Jand simulate\ntime evolution on an actual quantum computer. We take the\nfollowing initial single-qubit states\nð 1ë= cos\u0019\n12ð+ë+ei\u0019\n3sin\u0019\n12ð*ë;\nð 2ë= cos3\u0019\n10ð+ë+ei4\u0019\n3sin3\u0019\n10ð*ë:(26)\nand plot as a function of dimensionless time Jt. The initial\nstatechosenis ð .0/ë=ð 1.0/ëäð 2.0/ë. Thissystemissim-\nulated with the circuit shown in Fig. 1d with results collected\nfrom the device ibmq_bogota v1.4.50, a Falcon r4L proces-\nsor. The theoretical predictions are obtained by calculating\nð .t/ë=e*iHt_`ð .0/ëvia an explicit exponentiation of the\nHamiltonian matrix with the function expm()in MATLAB.24\nAscriptwhichcomputesthetheoreticalpredictionsisincluded\nin the supplemental material.17\nB.N > 2\nIn this last subsection, we sketch how to obtain results for\nlarger systems. For Ng3, it is generally not possible to per-\nformexacttimeevolution. Anexactrepresentationofthetime\nevolution in terms of quantum gates is generally only possi-\nbleformodelswhichcanbediagonalizedanalyticallyandare\n“exactlysolvable.” Thoughsuchmodelsdoexist19,theaimof\nthissectionistoequipthereaderwithtacticsforthemoregen-\neralsituation. Thebasicstrategyformodelswithonlynearest-\nneighbor interactions is to write the Hamiltonian as a sum of\ntwo-spinoperatorsanduseaLie-Trotterdecompositiontorep-\nresent the time-evolution operator approximately as a product\noftwo-spintimeevolutionoperators. ConsideraHamiltonian\nHof the formH=É\ni;jhij; (27)\nwherehijrepresents a direct interaction between qubits iand\nj. MostIBMdevicesallowfortheapplicationofCNOTgates\nbetween small subsets of nearest-neighboring qubits. Apply-\ning CNOT gates to qubits which are not directly connected\nis possible through the application of a series of so-called\nSWAPgates.15Intheinterestofminimizingcircuitdepth,we\nonly consider nearest-neighbor interactions in small systems\nwhich can be mapped to physically-connected qubits. To ap-\nply the Trotter decomposition to the time-evolution operator,\none writes\nU.t/ = expL\n*itÉ\ni;jhijM\nùnÇ\nn=1Ç\ni;je*i.t_n/hij:(28)\nEach factor of e*i.t_n/hijcorresponds to the circuit gate\nN.\u000b;\f;\r /showninFig.1c,whichcontainsthreeCNOTgates.\nOne would like the number of Trotter steps nas large as pos-\nsible for the Trotter expansion to converge. But as the CNOT\ngatesarethemostlargelyresponsibleforerrorsintheresults,it\nisdesirabletouseasfewaspossible. Thecurrentstate-of-the-\nartfortimeevolutionofmany-spinsystemsisthusconstrained\nby some optimal choice of nwhich is maximizes conver-\ngencewhileminimizingerrorsfromCNOTgates. Ithasbeen\nshown21that this optimization results in extremely limited\ntimescales for accurate computation for systems níO.10/.\nWitherrorsaffectingtheresultssosignificantly,onecanle-\ngitimately wonder about the value in today’s NISQ hardware.\nOne approach to squeezing as much power as possible out of\ntoday’s devices is to implement some type of error correc-\ntion scheme tothebareresults.12Aparticularlysimplescheme\ninvolves correcting counts for possible readout errors based\non a short calibration experiment.15Consider a simple circuit\nwhich creates the state ð001ëand then measures the state of\neach qubit. It is likely that most counts will return the correct\nstate, but due to noise and measurement error, there will be\ninstances of other states (e.g., ð000ë,ð011ë, etc.). Convert-\ning the counts into probabilities ( pj=nj_Nshots), we col-\nlect these probabilities as a row vector. Repeating this pro-\ncessforallpossiblethreequbit-statesandcollectingtheserows\nyieldsan 88 calibration matrix C. Thematrixmultiplication\nC.N;0;0;5;0/Tthen gives the actual counts obtained when\nthestateð000ëiscreated. Foracircuitthatsimplycreatesand\nmeasures a given state, it is plausible that a significant source\nof error is the readout process. We can interpret these proba-\nbilities as average probabilities for various readout errors and\napply the calibration to results for anycircuit. If xidealare the\nideal counts that would be obtained in the absence of readout\nerrors and xactualare the actual counts, the relationship is\nxactual=Cxideal: (29)\nRunning a job returns counts contained in xactual, but one is\ninterested in the ideal results, which can be obtained from the\nactual results by inverting the calibration matrix,8\n0 5 10-0.500.5\n0 5 10-0.500.5\n0 5 10-0.500.5\n0 5 10-0.500.5\nFIG. 7: Approximate time evolution of spin expectation values for variable number of Trotter steps using ibm_perth to simulate the model\nin Eq. (31) with initial state given in the main text. Rudimentary measurement error mitigation as described in the main text is applied to the\nresults in the first three panels. Uncorrected results are depicted faintly alongside the main results. The right-most panel .n= 50/is obtained\nfrom the QASM simulator.\nxideal=C*1xactual: (30)\nSome caveats should be mentioned regarding the applica-\ntion of Eq. (30). For a low-noise device, one expects Cto\nbe “close” to the identity matrix in that the diagonal entry is\nthe largest (significantly so) of each row. There is no guar-\nantee that Cis invertible, so one commonly uses a pseudoin-\nverse matrix for such cases. Generally, a strict multiplication\nwillresultinnon-integervaluesforthecorrectedcounts. One\ncouldroundtointegervalues,butthenon-integervaluescause\nno observable problems when used in the computation of ex-\npectation values, especially for a large number of shots. We\nemploythissimplestversionoferrorcorrectioninresultsdis-\ncussed below.\nWe consider three spins, each initialized to the state given\nin Eq. (2) with \u00121=\u0019\n6,\u001e1=\u0019\n3,\u00122=3\u0019\n5,\u001e2=4\u0019\n3,\u00123= *\u0019\n5,\n\u001e3=2\u0019\n3. An “open” chain with interactions between nearest\nneighbors is used with\nH= *Jx\u0004SxäSxäI+IäSxäSx\u0005\n*Jy\u0004SyäSyäI+IäSyäSy\u0005\n*Jz\u0004SzäSzäI+IäSzäSz\u0005(31)Aschematiccircuitforperformingapproximatetimeevolu-\ntionforêSz\nj.t/ëisshowninFig.1e. Theerror-mitigatedresults\nfrom ibm_perth v1.1.14, a Falcon r5.11H processor, are de-\npictedalongsidetheoreticalpredictionsinFig.7for n= 1;2;3\nTrotter steps. The challenge of simulating even a three-spin\nsystem is evident from the observation that n= 2yields no-\ntably better agreement between experiment and theory than\nn= 1, but significant degradation occurs upon increasing the\nnumber of Trotter steps to n= 3. The noiseless QASM simu-\nlatorisunaffectedbyCNOTerrors,asshownbytheexcellent\nagreementbetweensimulationandexperimentwhenthenum-\nber of Trotter steps is increased significantly to n= 50. We\nnotethatthetheoreticalpredictionisobtainedfromexactdiag-\nonalization without Trotter decomposition, so the right-most\npanelinFig.7demonstratesthattheTrotterdecompositionit-\nselfindeedconvergestotheexactresultforasufficientlylarge\nnumber of Trotter steps.\nV. DISCUSSION\nWe have presented schemes for simulating small ( Nf3)\nspin systems on IBM quantum computers and measuring the\ntime-dependent expectation values of various spin compo-\nnents. Systems of size N= 1;2have been treated exactly,\nand the results were subject to modest errors due to the com-\npact circuits used. For larger systems, approximate schemesmust be employed to perform time evolution. A Lie-Trotter\ndecompositionwasusedtoapproximatethetimeevolutionop-\nerator in a three-spin system. Even in a system with as few\nasthreespins,thetradeoffbetweenaddingcircuitelementsto\nyield convergence of the algorithm and reducing circuit ele-\nments to minimize noise becomes apparent.\nThe dynamics of small spin systems are treated frequently\nin introductory quantum mechanics. The use of free, cloud-9\nbasedIBMhardwareprovidesahighlyaccessibleopportunity\nfor students to explore these systems experimentally. In order\nto compare experimental results with theoretical predictions,\nstudents gain valuable experience in extracting experimental\nvaluesofphysicalobservablesbymanipulatingtherawcounts\nwhich are returned from the quantum circuit executions. Stu-\ndents must wrestle with what it means to “measure” a quan-\ntummechanicalobservable,averagingalargenumberofinde-\npendentresultsandcomputingstatisticaluncertainties. Exten-\nsionsoftheproblemspresentedcouldformthebasisofstudent\nprojects in which the basic tools are applied to more complex\nsystems(e.g.,largersystemsusingtheQASMsimulator,time-\ndependent systems, etc.).\nInadditiontoprovidingausefulandaccessibleexperimen-\ntal component to the traditional undergraduate treatment of\nquantum mechanics, these types of simulations give students\nexperience with cutting-edge technology. In this sense, the\nseemingly-abstract content in a quantum mechanics course\nhas direct relevance to a field in industry in which massive\nresources are being committed by IBM and other organiza-\ntions. Excellent resources25are being developed for intro-ducingquantumcomputingtostudentswithoutastrongback-\nground in physics. We hope the present work is a useful\ndemonstration of how quantum hardware can be used to ex-\nplorephysicsthatalsoshowshowsomefamiliaritywithquan-\ntum mechanics can be leveraged to gain expertise using this\nemerging technology.\nAcknowledgments\nTheauthorsacknowledgetheuseofIBMQuantumservices\nfor this work. The views expressed are those of the authors,\nand do not reflect the official policy or position of IBM or the\nIBM Quantum team. Additionally, the authors acknowledge\nthe access to advanced services provided by the IBM Quan-\ntum Researchers Program.26\n<Electronic mail: jlancas2@highpoint.edu\n1R.P.Feynman,“Simulatingphysicswithcomputers,” Int. J. Theor.\nPhys. 21, 467–488 (1982).\n2P.W.Shor,“Polynomial-TimeAlgorithmsforPrimeFactorization\nandDiscreteLogarithmsonaQuantumComputer,” SIAM J. Com-\nput.26(5), 1484–1509 (1997).\n3R. L. Rivest, A. Shamir, and L. Adleman, “A Method for Obtain-\ningDigitalSignaturesandPublic-KeyCryptosystems,” Commun.\nACM 21(2) 120–126 (1978).\n4J. Preskill, “Quantum Computing in the NISQ era and beyond,”\nQuantum 2, 79 (2018).\n5Rigetti: https://www.rigetti.com/\n6IonQ: https://ionq.com/\n7IBM Quantum:\nhttps://quantum-computing.ibm.com/\n8Y. Alexeev, D. Bacon, K. R. Brown, R. Calderbank, L. D. Carr,\nF.T.Chong,B.DeMarco,D.Englund,E.Farhi,B.Fefferman,A.\nV. Gorshkov, A. Houck, J. Kim, S. Kimmel, M. Lange, S. Lloyd,\nM. D. Lukin, D. Maslov, P. Maunz, C. Monroe, J. Preskill, M.\nRoetteler, M. J. Savage, and J. Thompson, “Quantum Computer\nSystems for Scientific Discovery,” PRX Quantum 2(1), 017001\n(2021).\n9J. Brody and G. Guzman, “Calculating spin correlations with a\nquantum computer,” Am. J. Phys. 89(1), 35–40 (2021).\n10D. Candela, “Undergraduate computational physics projects on\nquantum computing,” Am. J. Phys. 83(8), 688–702 (2015).\n11B. Kain, “Seaching a quantum database with Grover’s search al-\ngorithm,” Am. J. Phys. 89(6), 618–626 (2021).\n12S.JohnstunandJ-F.VanHuele,“Understandingandcompensating\nfornoiseonIBMquantumcomputers,” Am. J. Phys. 89(10),935–\n942 (2021).\n13Qiskit: https://qiskit.org\n14D.H.McIntyre,C.A.Manogue,andJ.Tate, Quantum Mechanics ,\n(Pearson, New York, 2012).\n15A. Abbas, S. Andersson, A. Asfaw, A. Corcoles, L. Bello, Y.Ben-Haim, et al.,Learn Quantum Computation Using Qiskit ,\nhttps://community.qiskit.org/textbook, (2020).\n16M. A. Nielsen and I. L. Chuang, Quantum Computation and\nQuantum Information , (Cambridge University Press, Cambridge,\n2010).\n17Supplemental material available at\nhttps://github.com/jllancas/SimulatingSpinsQComp\n18IBMusesaqubitnamingconventionwhichisoppositetothecon-\nvention used for labeling many-particle states in physics. For ex-\nample, in the state ð ë=ð100ëthebottomqubit in the circuit\ndiagram is in the state ð1ërather than the top qubit.\n19A. Cervera-Lierta, “Exact Ising model simulation on a quantum\ncomputer,” Quantum 2, 114 (2018).\n20H.F.Trotter,“Ontheproductofsemi-groupsofoperators,”Proc.\nAm. Math. Soc. 10(4), 545-551 (1959).\n21A. Smith, M. S. Kim, F. Pollmann, and J. Knolle, “Simulating\nquantummany-bodydynamicsonacurrentdigitalquantumcom-\nputer,” npj Quantum Information 5, 106 (2019).\n22F. Verstraete, J. I. Cirac, and J. I. Latorre, “Quantum circuits for\nstrongly correlated quantum systems,” Phys. Rev. A 79, 032316\n(2009).\n23ACNOT(controlled-NOT)gateisatwo-qubitgatewhichapplies\nthe logical NOT operation to a target qubit ifthe control qubit is\nmeasuredtobeinthestate ð1ë,leavingthetargetqubitunchanged\notherwise.Two-qubitgatesareessentialforsimulatingsystemsin\nwhichqubits interact.ThereaderisencouragedtoconsultsRef.15\nfordetailedinformationaboutCNOTgatesandtheiruseinQiskit.\n24MATLAB. version 9.10.0.1669831 (R2021a Update 2). Natick,\nMassachusetts: The MathWorks Inc.; 2021.\n25C. Hughes, J. Isaacson, A. Perry, R. F. Sun, and J. Turner,\nQuantum Computing for the Quantum Curious ,(Springer,Berlin,\n2021).\n26IBM Quantum Researchers Program:\nhttps://quantum-computing.ibm.com/programs/researchers"    },    {        "title": "1112.6404v1.Coherent_spin_dynamics_of_electrons_and_holes_in_semiconductor_quantum_wells_and_quantum_dots_under_periodical_optical_excitation__resonant_spin_amplification_versus_spin_mode_locking.pdf",        "content": "arXiv:1112.6404v1  [cond-mat.mes-hall]  29 Dec 2011Coherent spin dynamics of electrons and holes in semiconduc tor quantum wells and\nquantum dots under periodical optical excitation: resonan t spin amplification versus\nspin mode-locking\nI. A. Yugova1,2, M. M. Glazov3, D. R. Yakovlev1,3, A. A. Sokolova2, and M. Bayer1\n1Experimentelle Physik 2, Technische Universit¨ at Dortmun d, 44221 Dortmund, Germany\n2Physical Faculty of St.Petersburg State University, 19850 4 St. Petersburg, Russia and\n3Ioffe Physical-Technical Institute, Russian Academy of Scie nces, 194021 St. Petersburg, Russia\n(Dated: November 18, 2018, file = rsa-modlock-29dec11.tex, printing time = 6:58)\nThe coherent spin dynamics of resident carriers, electrons and holes, in semiconductor quantum\nstructures is studied by periodical optical excitation usi ng short laser pulses and in an external\nmagnetic field. The generation and dephasing of spin polariz ation in an ensemble of carrier spins,\nfor which the relaxation time of individual spins exceeds th e repetition period of the laser pulses,\nare analyzed theoretically. Spin polarization accumulati on is manifested either as resonant spin\namplification or as mode-locking of carrier spin coherences . It is shown that both regimes have the\nsame origin, while their appearance is determined by the opt ical pump power and the spread of spin\nprecession frequencies in the ensemble.\nPACS numbers: 78.67.-n, 78.47.-p, 71.35.-y\nI. INTRODUCTION\nThe coherent spin dynamics of carriers in semiconduc-\ntor nanostructures attract considerable attention nowa-\ndays due to future quantum information technologies\nbased on spintronics applications [1–3]. With respect to\nfundamental studies this research field delivers exciting\nand unexpected results on the properties of spin systems\nand the possibility to control them by external fields or\nby structural parameters.\nOptical pump-probe techniques for time-resolved mea-\nsuring of Faraday and Kerr rotation are based on ex-\ncitation by trains of laser pulses where the pulse dura-\ntions range from hundreds of femtoseconds to a few pi-\ncoseconds. They have been demonstrated to be among\nthe most reliable tools for investigating coherent spin dy-\nnamics [1, 4–16]. The principle of these magneto-optical\ntechniques is the following: an intense laser pulse of cir-\ncularly polarized light (the pump) is used to orient spins\nand, therefore, to create a macroscopic spin polariza-\ntion [17]. This polarization is probed by the linearly\npolarized probe pulses through rotation of their polariza-\ntion plane after propagation through the spin polarized\nmedium (the Faraday effect) or reflection at this medium\n(the Kerr effect). The probe pulse is time-delayed rela-\ntive to the pump pulse, and by tuning this delay one\ncan measure the spin polarization dynamics. To study\nthe coherent spin dynamics the sample is exposed to an\nexternal magnetic field, typically oriented perpendicu-\nlar to the light wave vector (Voigt geometry), which al-\nlows one to detect the precession of the optically induced\nspin polarization and monitor its decay. Application of\nthese techniques to single spins, which is potentially pos-\nsible [18, 19], is demanding. Studying spin ensembles\nthat contain millions of carrier spins is much more con-\nvenient [14, 15, 20].\nIn pump-probe experiments the spin dynamics evolu-\ntion is typically measured over times shorter than the\nrepetition period of the pump pulses, which is about\n13 ns for commonly used mode-locked Ti:Sapphire lasers\nemitting pulses at a repetition rate of 75 −80 MHz. Ithas been shown experimentally that in bulk semiconduc-\ntors, quantum wells (QWs) and quantum dots (QDs) the\ncarrier spin relaxation time can substantially exceed the\nrepetition period [1, 4]. In this case the spin polarization\ninduced by subsequent pump pulses can accumulate if a\nphase synchronization condition is fulfilled for the pre-\ncessing carrier spins. It results in two effects: resonant\nspin amplification (RSA), observed in bulk and QW spin\nsystems with a relatively small dispersion of precession\nfrequencies, and spin mode-locking (SML) found for an\nensembleofsingly-chargedQDswith alargedispersionof\nLarmorfrequencies(see, e.g., Refs.[20, 21]andreferences\ntherein).\nFor studying the RSA regime experimentally, scanning\nthe magnetic field has been suggestedinstead ofthe com-\nmonly used scan of the pump-probe time delay [5], used\nalso for tracing SML. The probe pulse arrival time in this\ncase is fixed at a small negative delay prior to the pump\npulse. The resulting RSA spectrum is a periodic function\nof magnetic field from which information such as carrier\ngfactor and dephasing time of the spin ensemble can be\nextracted.\nIn this paper we show that the RSA and SML are\ntwo different manifestations of the same phenomenon:\nspin accumulation caused by the periodic excitation with\npump pulse trains. We elaborate the fundamental differ-\nences in conditions for appearance of these two regimes.\nThe most important parameters in this regard are the\npump power and the spin precession frequency spread\ncausing spin dephasing. Differences of the two param-\neters, in turn, lead to different phenomenologies in ex-\nperiment, providing significantly different capabilities for\nanalyzing spin systems quantitatively.\nThe paper is organized as follows. In Section II we re-\ncall the basic concepts and equations for describing spin\ncoherence generation. We discuss the difference between\nthe classical and quantum mechanical approaches to de-\nscribing carrier spin coherence generation for resonant\ntrion excitation. Then we consider generation of long-\nlived spin coherence during the trion lifetime. We de-\nscribe the spin dynamics of charged carriers and trions2\nin magnetic field and discuss the effects ofspin relaxation\nandspinprecessionofthetrionspinonthelong-livedspin\ncoherence of resident carriers. We also consider here the\nlong-lived dynamics after generation and the spin accu-\nmulation caused by the train of pump pulses. Section III\nis devoted to the RSA regime, for which we consider dif-\nferent conditional effects: trion spin relaxation, nuclear\nfield fluctuations, and spin relaxation anisotropy. The\nconditions, which are important for observing RSA, and\nthe characteristics, which one can extract from the anal-\nysis of RSA signals, are collected at the end of Sect. III.\nSectionIV describesthe main featuresofmode-lockingof\nelectron spin coherences. Then in Section V we compare\nthe spin dynamics in the RSA and SML regimes, obtain\nconditions for the SML regime and discuss the transition\nto the RSA regime. In the Conclusions, we give a com-\nparative description of the RSA and SML regimes and\ntheir applicability to investigations of long-lived spin dy-\nnamics in low-dimensional systems.\nII. GENERATION OF SPIN COHERENCE\nIn the following we analyze the long-lived spin coher-\nence of resident carriers (electrons and holes) generated\nby periodic light excitation in semiconductor quantum\nwells and quantum dots. We consider a situation with a\nlow concentration of resident carriers, when the proba-\nbility to have two charge carriers with significantly over-\nlapping wavefunctions is low. In this case, mainly few-\nparticle complexes, excitons (electron-hole pairs) and tri-\nons (three particle complexes) can be optically excited,\nwhile other many-body correlations are negligible. For\nquantum wells this corresponds to typical carrier densi-\nties smaller than 1010cm−2, for which at liquid Helium\ntemperatures carriers are localized on QW width fluctu-\nations with respect to their in-plane motion. Only one\ncarrierper localizedsite is typicalfor such concentrations\nand the distance between the localized carriers exceeds\nthe extensions of neutral and charged exciton wavefunc-\ntions. For quantum dots the low concentration regime\ncorresponds to occupation of a dot with only one resi-\ndent carrier, i.e. to a regime of singly-charged QDs.\nHere we consider the theoretical aspects of the prob-\nlem. We do not discuss the experimental aspects of\nthe observations (measurements) of long-lived spin co-\nherences and features of ellipticity and Faraday rotation\nsignals. Welimitourselvestothedegeneratepump-probe\nregime, when the probe laser has the same photon en-\nergy as the pump one, and to resonant excitation of the\ntrion states. We assume that the pulse duration is signif-\nicantly shorter than all characteristic relaxation times of\nthe considered spin system. Other regimes are studied in\ndetail elsewhere[22–25]. These conditions are typical for\nexperiments with semiconductor nanostructures [1, 4].\nFor low concentrations of resident carriers charged ex-\ncitons (trions) play an important role in the generation\nprocess of carrier spin coherence [9, 15]. A negatively\ncharged exciton (T−trion) is a bound state of two elec-\ntrons and one hole, while a positively charged exciton\n(T+trion) is a bound state of two holes and one elec-tron. The trion ground state at zero magnetic field has a\nsinglet spin configuration, such that the spins of the two\nidentical carriers are aligned opposite to each other and\nthe trion Zeeman splitting is controlled by the gfactor of\nthe unpaired carrier, e.g., the hole in T−. Hereinafter we\nassume that only heavy holes with angular momentum\nprojections ±3/2 onto the growth axis are involved.\nThe theoretical analysis used in this paper can be\nequally applied to structures with resident electrons or\nresident holes. In order to do that we have introduced\nuniversal notations: the resident carrier spin S, the trion\nspinST, the Larmor frequency of the resident carrier\nω=gµBB//planckover2pi1, and the Larmor frequency of the trion\nΩ =gTµBB//planckover2pi1. HereBis the external magnetic field,\nµBis the Bohr magneton, gandgTare thegfactors of\nthe resident carrierand trion, respectively. Similarly uni-\nversal notations are also used in what follows to denote\ncharacteristic time scales.\nInn-type doped structures with resident electrons, S\nis the electron spin, STis a (pseudo) spin of the T−trion\n(ST= +1/2 for +3/2 hole and −1/2 for−3/2 hole),ωis\nthe electron Larmor frequency, and Ω is the T−Larmor\nfrequency determined by the hole gfactor. Correspond-\ningly, in p-type doped structures with resident holes, S\nis the heavy hole pseudospin, STis the spin of the T+\ntrion, whichcorrespondstotheelectronspininthistrion,\nωis the heavy hole Larmor frequency, and Ω is the T+\nLarmor frequency determined by the electron gfactor.\nFor the sake of simplicity we consider in most parts\nof this paper n-type doped structures with resident elec-\ntrons, as there are more experimental data available for\nthese structures. Wherever we analyze p-type doped\nstructures this will be notified. Before we proceed to\nthe analysis of spin precession in magnetic field and spin\ndephasing processes, let us inspect briefly the models of\noptical generation of spin coherence.\nA. Resonant excitation of trion. Classical and\nquantum mechanical approaches to carrier spin\ncoherence generation\nThe important quantity for spin coherence generation\nin the system is the singlet trion resonance that is ex-\ncited optically. The trion generation probability for res-\nonant excitation depends on the light polarization and\nthe spin orientation of the resident carrier. For instance,\ninn-type doped structures a σ+polarized pump gen-\nerates a hole with spin projection +3 /2 onto the light\npropagation axis zand an electron with spin projection\n−1/2. Therefore, trion formation is possible only when\nthe resident electron has spin projection +1 /2. As a re-\nsult, the circularlypolarized pump pulse selects electrons\nwith particularspin orientationfrom the ensemble of res-\nident electrons to form trions. This, in turn, leads to spin\npolarization of the resident electrons.\nTherearetwoapproachesfordescribingthespincoher-\nence generation by circularly polarized light pulses [25].\nThe first one is essentiallyquantummechanical: a singly-\ncharged QD or a QW with a localized resident electron\nis modeled as a two-level system [8, 15, 23, 26]. The\nground state corresponds to the resident electron, while3\nσ+\n|+1/2 > |+3/2 > \n        Ground state \n(resident electron)       Excited state \n(singlet trion T  −)(b) (a)\nσ+\n|+1/2 > |+3/2 > \n        Ground state \n(resident electron)       Excited states \n(singlet trions T  − )τ1\nFigure 1: (a) Scheme of transitions for a strongly localized\nelectron (e.g., in a singly-charged quantum dot). The initi al\nstate for the optical transition is a resident electron and fi nal\nstate is a singlet trion T−. This scheme is consistent with\nthe quantum mechanical approach. (b) Scheme of transitions\nfor the case of weakly localized resident carriers (e.g., in a\nquantum well with a low density electron gas); τ1denotes the\nscattering time between different trion states. This scheme is\nconsistent with the classical approach.\nthe excited state is the singlet trion, see Fig. 1(a).\nThe interaction of the two-level system with the reso-\nnant pump pulse depends on the pulse parameters (po-\nlarization, intensity and pulse duration) and on the level\noccupations. The pump pulse action time, τp, is assumed\ntobetheshortestofalltimescalesintheproblem,namely\nthe trion dephasing and scattering times, the electron\nLarmor precession period, the trion radiative lifetime,\nthe spin dephasing/decoherence times, etc. Under usual\nexperimentalconditionsthe trionlifetime ismuchshorter\nthanthe pump pulse repetition period and, consequently,\ntrion spin polarization is absent shortly before the next\npumppulse, i.e. isnotdetectableatnegativetimedelays.\nIt follows then, that the resident carrier spin pseudovec-\ntorS= (Sx,Sy,Sz) before the pump pulse, Sb, and after\nthepumppulse, Saarerelatedtoeachotherthrough[23]:\nSa\nz=±Q2−1\n4+Q2+1\n2Sb\nz, (1a)\nSa\nx=QcosΦSb\nx±QsinΦSb\ny, (1b)\nSa\ny=QcosΦSb\ny∓QsinΦSb\nx, (1c)\nwhere the signs ±correspond to σ+andσ−polarized\npump pulses in n-type structures and to σ−andσ+\npulses for p-type structures, respectively. This sign def-\ninition is also valid for Eqs. (2), (3) and (4). The pa-\nrameters 0 /lessorequalslantQ/lessorequalslant1 and 0 /lessorequalslantΦ<2πcharacterize the\npump pulse area and the spectral detuning of the pulse\nfrom the trion resonance. The explicit expressions for\nthese quantities are given in Ref. [23]. For the resonant\npump pulse Φ = 0 and Q= cos(Θ /2), where Θ is the\npump pulse area: Θ =/integraltext\n2|∝angbracketleftd∝angbracketrightE(t)|dt//planckover2pi1. Here∝angbracketleftd∝angbracketrightis the\ndipole transition matrix element and E(t) is the smooth\nenvelope of the electric field of the laser pulse. The z\ncomponent of the trion spin pseudovector after, e.g., a\nσ+pump pulse in a n-type system or a σ−pump pulse\nin ap-type system is given by [23]\nST\nz=Sb\nz−Sa\nz=1−Q2\n4/parenleftbig\n2Sb\nz±1/parenrightbig\n.(2)\nSuch an approach has been proven to be appropriate\nfor the description of spin coherence generation in n-type\nsingly-charged QDs [15]. At low pump powers, whereΘ≪1, the additive contribution to the electron spin z\ncomponent equals to\nSa\nz−Sb\nz=∓Θ2\n16∝P, (3)\nwherePis the pump pulse power. One of the main pre-\ndictions of the considered quantum mechanical approach\nis that for high pump powers the electron spin zcompo-\nnent depends periodically on the pump area, i.e., shows\nRabi oscillations inherent to a two-level system, see e.g.\nRefs [14, 15, 27].\nThe experimentally studied situation in n-type QW\nstructures is different [9]. For low pump powers and\nresonant trion excitation the electron spin coherence in-\ncreases linearly with the pump power, see Eq. (3), while\nat high powers the spin zcomponent saturates and Rabi\noscillations are not observed [9]. Clearly, the two-level\nmodel is not sufficient for describing such a behavior.\nThe most probable reason is related to the weaker lo-\ncalization of electrons and trions in quantum wells and,\nhence, to the presence of many trion states. Scattering\nbetween these states becomes possible, as schematically\nillustratedinFig.1(b). Theopticalcoherenceofthetrion\nwith the pump is lost due to this scattering, while spin\ncoherence is preserved. As a result, if the scattering time\nbetween different trion states, τ1, is considerably shorter\nthanτp, the Rabi oscillations at high pump powers van-\nish [28], becausethe populationofthe excited stateofthe\ntwo-level system coupled to the optical pulse is small. At\nthe same time, the spin polarization generated by the\npump pulse can be substantial, because spin does not\nrelax during scattering. With an increase of the pump\npulse power the electron spin saturates at the value\nSz,max=∓N/4, (4)\nwhereNis the total number of resident electrons in the\nsystem. Theamountoftrionsformed forresonantexcita-\ntion of the initially unpolarized electron ensemble cannot\nexceedN/2, since only half of the resident electrons have\nsuitable spin orientation to become excited to trion sin-\nglets. The other N/2 of the resident electrons, which are\nnot captured to trions, have become fully polarized.\nAs will be shown below in Sec. IIIA, the quantum me-\nchanical and classicalapproachesgive the same results at\nlowpumppowers. Subsequently, wewillusethequantum\nmechanical approach because it gives good descriptions\nfor spin coherence generation for QDs in any excitation\npower regime and for QWs in the low power excitation\nregime.\nB. Generation of long-lived spin coherence during\nthe trion life time. Spin dynamics of charged\ncarriers in magnetic field\n1. Spin dynamics of resident carrier and trion\nRight after the excitation pulse the coupled dynam-\nics of resident carrier spin, S, and trion spin, ST=\n(ST\nx,ST\ny,ST\nz), can be described by the following system4\nof equations [9, 15, 23, 26]:\ndST\ndt=µB\n/planckover2pi1[gTB×ST]−ST\nτTs−ST\nτr,(5a)\ndS\ndt=µB\n/planckover2pi1[gB×S]−S\nτs+ST\nzez\nτr.(5b)\nHereezis the unit vector along the zaxis. The mag-\nnetic field Bis assumed to be parallel to the xaxis.τT\ns\nis the trion spin relaxation time, τsis the phenomeno-\nlogical spin relaxation time of the resident carrier [29],\nandτris the trion radiative lifetime. It is worth to men-\ntion that carriers left behind after trion recombination\nare polarized parallel or antiparallel to the zaxis due to\nthe optical selection rules, see the last term ∝ST\nzezin\nEq. (5b).\nFromEqs.(5) the carrierspin projectionontothe mag-\nnetic field, Sx, is conserved. Introducing the trion spin\nlifetime, τT=τT\nsτr/(τT\ns+τr), we arrive at the follow-\ning expression for the transverse carrier spin component\nS+=Sz+iSy[15]:\nS+(t) =S+,0e−iωt−t/τs\n+ST\nz,0/bracketleftBig\n−ξe−iωt−t/τs+e−t/τT(ξcosΩt+χsinΩt)/bracketrightBig\n.\n(6)\nHerethe subscript 0denotesthe spin componentsattime\nt= 0, when the pump pulse is finished, e.g., S+,0=\nSz(0)+iSy(0).\nξ=ξ1+iξ2=iω/γ−1\nγτr[(1−iω/γ)2+(Ω/γ)2],(7)\nχ=χ1+iχ2=Ω/γ\nγτr[(1−iω/γ)2+(Ω/γ)2],(8)\nandγ=τ−1\nT−τ−1\ns>0.\nIn order to have a closed equation system (5), we have\nto relate the carrier and trion spins at t= 0. This can\nbe done through Eqs. (1) and (2). After a single pump\npulse (Sb= 0) one has\nST\nz,0=−Sz,0.\nThe first term in the right hand side of Eq. (6) describes\nthe carrier spin precession. The term proportional to\nST\nz,0e−iωtdescribes the spin polarization of the resident\ncarrier after trion recombination. Below, we consider the\nrelation of these two contributions as a function of spin\nsystem parameters and external conditions.\n2. Effect of trion spin relaxation on spin coherence of\nresident carrier\nIn absence of an external magnetic field the efficiency\nof resident carrier spin coherence generation is solely de-\ntermined by the trion spin relaxation [8, 9, 30]. This\nbecomes clear from Eq. (6), which for B= 0 reduces to\nSz(t) =Sz,0e−t/τs+ST\nz,0ξ/parenleftBig\n−e−t/τs+e−t/τT/parenrightBig\n.(9)It follows from Eq. (7) that ξ=−(τrγ)−1≈ −(1 +\nτr/τT\ns)−1, provided that the carrier spin relaxation time\nexceeds by far both trion recombination time and trion\nspin lifetime. These conditions are readily fulfilled in ex-\nperiment. Hence, the long-lived carrier spin coherence is\ngiven by\nSz(t) = (Sz,0−ST\nz,0ξ)e−t/τs, t≫τT.(10)\nIf spin relaxation in the trion is suppressed, i.e. τT\ns≫τr,\nthenξ→ −1. Therefore, since for a single pump pulse\nST\nz,0=−Sz,0, the contribution of the carrier left behind\nfrom the trion decay compensates exactly the spin polar-\nization of the remaining, non-excited carrier component.\nAs a result, no long-lived spin coherence for resident car-\nriers is generated. In general, when the resident carrier\nhas been polarized before pump pulse arrival, this carrier\npolarization will not be affected by the pump pulse and\nconserved after trion recombination. To conclude, trion\nspin relaxation is required to give rise to a non-zero long-\nlived spin coherence of the resident carriers in absence of\na magnetic field.\n3. Spin precession of resident carrier\nThe carrierspin precessionabout an externalmagnetic\nfield results in an imbalance of resident and returning\nspins. Hence, long-lived spin coherence can be excited\neven in the absence of trion spin relaxation. Provided\nthat the trion spin does not precess [31], Ω = 0, the long-\nlived carrier spin coherence is given by [9]\nSz(t) = sign( Sz,0)|Sz,0−ST\nz,0ξ|e−t/τscos(ωt−ϕ), t≫τT\n(11)\nwhereϕis the initial phase, which can be related to the\nparameter ξ, see Ref. [9] for details. Note, that in Ref. [9]\nthe phase is shifted by π/2 with respect to our definition\nin Eq. (11). The amplitude of the long-lived spin coher-\nenceAsafter a single pump pulse can be recast as\nAs=|Sz,0−ST\nz,0ξ|=|Sz,0(1+ξ)| ≈ |Sz,0||ωτr|/radicalbig\n1+(ωτr)2,\n(12)\nwhere the latter approximate equality is valid for a long\ntrion spin relaxation time fulfilling the relation τT\ns≫τr.\nAccording to Eq. (12) the long-lived spin amplitude first\nincreases with growing magnetic field ∝ωτrand then\nsaturates in strong fields.\nThe general case of arbitrary ωτrandτr/τT\nsis illus-\ntrated in Fig. 2. Panel (a) demonstrates the dependence\nof the long-lived spin coherence amplitude Ason mag-\nnetic field (expressed as ω/γ) for different values of the\nratioτr/τT\ns. Depending on the parameter τr/τT\ns, the\nchange of amplitude Asas function of magnetic field\n(through ω∼B) occurs for different field values since\nγitself is determined by τrandτT\ns.\nThe panels (b) and (c) in Fig. 2 show the carrier spin\ncoherence Sz(t) calculated for fast ( τr/τT\ns= 10) and slow\n(τr/τT\ns= 0.01)spinrelaxationofthetrion. Thesolidand\ndashed lines show Sz(t) in zero and finite magnetic field,\nrespectively. One can see from Fig. 2(b), that at τr/τT\ns=5\n0 1 2 3 40.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 \n(c) (b) \n(a) \nSzSz\n  \nω / γTime ( t/τr ) 0 2 4 6 8-1 01 \n \nAs(240) As(0.24) As(0) ≅\nτr / τT\ns = 0.01 0.5 2\nτr / τT\ns = 10 \n  \n0 2 4 6 8-1 01 \n As(0) As\n0.01 τr / τT\ns = 10 \n  \nFigure 2: (Color online) (a) Dependence of the long-lived sp in\ncoherence amplitude Ason the carrier Larmor precession fre-\nquency for different values of τr/τT\ns. (b),(c) Carrier spin co-\nherenceSz(t) normalized to Sz(0) for two different values of\nτr/τT\ns. The spin dynamics at zero magnetic field are shown\nby the dashed lines. The solid lines show Sz(t) at finite mag-\nnetic field ( ωτr= 2.4). The arrows show the corresponding\namplitudes As(ω/γ) for these conditions.\n10 the amplitude of the long-lived spin coherence ( t≫\nτr) in magnetic field coincides with the one at B= 0.\nIn the graph this corresponds to the coincidence of the\ndashedline (zerofield) with the maximaofthe oscillating\nsolid line (finite field, ω/γ= 0.24). In other words, the\napplication of magnetic field here does not change the\nefficiency of spin coherence generation. This is, however,\nnot the case for the smaller ratio of τr/τT\ns= 0.01 (ω/γ=\n240). As one can see in Fig. 2(c) the dashed line at\nlonger delays has considerably smaller amplitude than\nthe maxima of the solid line, As(0)≪ As(240). This\nmeans that the amplitude of long-lived spin coherence,\nAs, canbe stronglyincreasedbyexternalmagneticfields.\nTo conclude, even in the absence of spin relaxation in the\ntriontheapplicationofanexternalmagneticfieldleadsto\nappearance of long-lived spin polarization of the resident\ncarriers.\n4. Effect of spin precession in trion on spin coherence of\nresident carrier\nSpin precession of the trion, characterized by the fre-\nquencyΩ, alsoprovidesamechanism forgeneratinglong-\nlived carrier spin coherence. Although the in-plane hole\ngfactor in quantum wells and in self-assembled quan-\ntum dots is rather small [32–34], the spin precession of\nthe hole in the T−trion may become important in tilted\nmagnetic fields [35], and in the case of the T+trion ex-\ncited inp-doped structures [36].\nAllowing for Ω ∝negationslash= 0 results in the following expression\nfor the amplitude of the long-lived spin coherence As[c.f.Eqs. (11) and (12)]:\nAs=|Sz,0−ST\nz,0ξ|=|Sz,0(1+ξ)| ≈ |Sz,0|(Ωτr)2\n1+(Ωτr)2,\n(13)\nwhere in the latter equality we assume a trion spin re-\nlaxation time, τT\ns≫τr, and neglect the resident carrier\nspin precession, ω≪Ω. It follows from Eq. (13) that the\nspin precession in the trion acts similar to the trion spin\nrelaxation. Here it does not matter whether the spin of\nthe unpaired carrier in the trion was rotated by the mag-\nnetic field or flipped due to spin relaxation: in both cases\nlong-lived carrier spin polarization arises.\n0.00 0.02 0.04 0.06 ST\nz - S z , S T\nz\nT + ω = 54 ωR , Ω = 243 ωR (b) (a)  \n \n  0.00 0.02 0.04 0.06 ST\nz - S z , S T\nzω = 243 ωR , Ω = 54 ωR T −\nPump - probe delay time, t/TR   \nPump - probe delay time, t/TR\nFigure 3: (Color online) Spin dynamics of resident carriers\nand trions for (a) negatively charged trions T−,n-type and\n(b) positively charged trions T+,p-type. The black curves\nshowthetemporal evolutionof ST\nz−Sz. Thegray (red)curves\ngive only the trion contribution to the signal, ST\nz.\nThe situation becomes richer when spin precession of\nbothresidentcarrierandtrionoccurs. Figure3showsthe\nspin dynamics of T−trion and resident electron [panel\n(a),n-type] and T+trion and resident hole [panel (b),\np-type]. The black curves give the difference ST\nz−Sz\nwhich corresponds to signals commonly measured in ex-\nperiment, see also Eqs. (54) and (57) in Ref.[23]. It is\nclearly seen that at short delay times not exceeding the\ntrionlifetime the ST\nz−Szdynamicsareadditionallymod-\nulated by the trion Larmor frequency. For clarity, the\nspin dynamics of trions, ST\nz, are shown separately by the\ngray (red) lines. They decay relatively fast being limited\nby the trion recombination. The trion radiative lifetimes\nas well as spin relaxation times are taken the same in\nboth panels: TR/τr= 130,TR/τT\ns= 13 and TR/τs= 2.6.\nThe carrier and trion Larmor precession frequencies are\ngiven in the panels. TRis the repetition period of ex-\ncitation pulses and ωR= 2π/TR. For commonly used\nmode-lockedlaserswith arepetition frequencyof75MHz\nTR= 13.3 ns.6\nC. Spin accumulation induced by a train of pump\npulses\nIn experiments on coherent spin dynamics periodic\ntrains of pump pulses are commonly used. When the\nspin relaxation time of the resident carrier is comparable\nor longer than the repetition period of the pump pulses,\ni.e.τs> TR, the steady-statecarrierspin polarizationre-\nsults from the cumulative contribution of multiple pump\npulses. In external magnetic fields applied in the Voigt\ngeometry, the steady-state situation is reached for eachprecessing spin by relatively long trains of pump pulses:\nthe decay of the spin polarization is then balanced by the\npumping. As a result, the carrier spin after each repe-\ntition period, S(TR), given by Eq. (6), should be equal\nto the carrier spin right before the pump pulse arrival,\nwhich we denote by Sb(see Fig. 4). Using the connec-\ntion between the carrier spins before and after the pump\npulse, Eq. (1), and assuming that the pump pulse is res-\nonant with the trion transition, Φ = 0, one immediately\ncomes to the following expression for the carrier spin z\ncomponent before pump pulse arrival:\nSb\nz(ω) =±1\n2K\n1+Qe−2TR/τs−e−TR/τs(1+Q)cos(ωTR)−K, (14)\nwhere the signs ±correspond to different polarizations of optical pumping and differe nt types of resident carriers, cf.\nEqs. (1), and\nK=(1−Q2)e−TR/τs\n2/braceleftBig\n(1+ξ1)[Qe−TR/τs−cos(ωTR)]−ξ2sin(ωTR)/bracerightBig\n.\nEquation (14) shows that the spin zcomponent before\nthe next pump pulse arrival, Sb\nz, is a periodic function\nof magnetic field (see Fig. 5) with maxima of |Sb\nz|at fre-\nquencies ωsatisfyingthephasesynchronizationcondition\n(PSC) [5, 14, 37]:\nω=NωR=2πN\nTR, N= 0,1,2,.... (15)\nHereωR= 2π/TRistherepetitionfrequencyofthepump\npulses. Indeed, as one can see from time-resolved sig-\nnals shown in Fig. 4, if the spin precession period of the\nresident carrier is commensurable with the pump pulse\nrepetition period, then the spin coherence generated by\nthe pump is always in phase with that from the previous\npulse [see signal around zero time delay, Fig. 4(a)], and\ncarrier spin polarization is accumulated. Let this phase,\nφ, be zero. Otherwise, if the spin precession and pump\nrepetition periods are not commensurable, the accumu-\nlation of spin polarization is not efficient, as seen from\nthe comparison of the amplitudes in Figs. 4(a) and 4(b).\nIn general, the electron spin precession has a particu-\nlar phase, see Eq. (11), which we determine here as the\ndifference ( ωTR−2πN), where Nis the largest integer\nsatisfying the condition ( ωTR−2πN)≥0. The phase\ncan be expressed as:\ncos(φ) =−Sb\nz//radicalBig\n(Sbz)2+(Sby)2, (16)\nsin(φ) =Sb\ny//radicalBig\n(Sbz)2+(Sby)2.\nNote, that in Fig. 4 and further on in this paper we\nshow for convenience the inverted signal −Sz(in order to\nhave positive signals for σ+pumping). This sign change\ndoes not affect the obtained results but is more suitable\nfor their graphic presentation. For an ensemble of resi-\ndentcarrierswith differentspinprecessionfrequencies, ω,Eq. (14) should be averaged over their distribution [38],\nsee below Sec. IIID and Eq. (28).\n/s45/s48/s46/s48/s50/s48/s46/s48/s48/s48/s46/s48/s50\n/s32/s83\n/s122/s97\n/s32/s83\n/s122/s98 /s32/s83\n/s122/s98 /s32/s83\n/s122/s97/s84 \n/s82\n/s32/s32\n/s83\n/s122/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s40/s98 /s41/s40/s97/s41\n/s80/s117/s109/s112/s32/s45/s32/s112/s114/s111 /s98/s101/s32/s100/s101/s108/s97/s121 /s44/s32 /s116/s32/s47/s32/s84\n/s82\n/s32/s32\n/s32/s61/s32/s50/s32\n/s82\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s48/s49/s48/s46/s48/s48/s48/s46/s48/s49\n/s32/s32\n/s32/s61/s32/s50/s46/s53/s32\n/s82\n/s32\nFigure 4: (Color online) Dependencies of resident carrier s pin\npolarization Szon pump-probe delay for a carrier spin pre-\ncession frequency which is (a) commensurable with the pump\nrepetition frequency ω= 2ωRand (b) not commensurable\nwith this frequency ω= 2.5ωR. Parameters of calculations\nare:τs= 3TR, Θ = 0 .1π. Thick vertical arrows show the\narrival times of the pump pulses. Phase φof the oscillat-\ning polarization, −Sz, isφ= 0 in panel (a) and φ=πin\npanel (b).\nFigure 5 shows Sb\nzcalculated after Eq. (14) for differ-\nent pump pulse areas Θ in the case of fast trion spin\nrelaxation. As xaxis scale in Fig. 5 we take the ratio\nof spin precession frequency ωandωR, which represents7\n0.00 0.01 0.02 0.03 \n  x4 (a)  τs / T R = 3 \n τs / T R = 0.5 \n  τs / T R = 3 Θ = 0.1 π\n0.0 0.2 0.4 0.6  x10 (b) − Szb (arb. units)  Θ = 0.1 π\n Θ = 0.5 π\n Θ =  π\n  \n-1 0 1-ππ (c) \nω / ωR\n  \n-1 0 10 0\n-ππ  (d) Phase, φ (rad.) \nω / ωR\n  \nFigure 5: (Color online) Dependence of the resident carrier\nspin polarization Sb\nzand its phase on magnetic field expressed\nbyω/ωR=gµBB/(/planckover2pi1ωR). Data are shown for zero time delay\n(right before the pump pulse arrival), calculated for differ ent\nratiosτs/TRat Θ = 0 .1π(a,c) and for different pump pulse\nareas Θ at τs/TR= 3 (b,d).\nthe magnetic field dependence of Sb\nzasω∝B. Inte-\nger numbers on the xaxis correspond to magnetic fields,\nfor which the spin precession frequency satisfies the PSC\nof Eq. (15). At these magnetic fields the amplitude of\nthe resident carrier spin polarization, −Sb\nz, increases res-\nonantly evidencing favorable conditions for spin accumu-\nlation, see Fig. 5(a). It is obvious that the accumulation\nefficiency is controlled by the factor τs/TR, as the accu-\nmulation occurs only when the spin relaxation time of\nthe resident carrier, τs, exceeds considerably the repe-\ntition period of the pump pulses. This is confirmed by\nthe calculations shown in Fig. 5(a). For a fixed value of\nτs/TRan increase of the pump pulse area results in a\nbroadening of the peaks, see Fig. 5(b).\nThe phases of the signals from Figs. 5(a) and 5(b) are\nshowninpanels(c)and(d), respectively. Oneclearlysees\nthat the zeros of the phase correspond to maxima of spin\npolarization, −Sb\nz, and the values φ=±πcorrespond to\nits minima.\nOne should note, that the magnitude of the accumu-\nlated spin polarization, as well as the width of the reso-\nnant peaks in the magnetic field dependence of −Sb\nz, are\ndetermined not only by the pump pulse power and the\ncarrier spin relaxation time, but also by the mechanism\nof long-lived spin coherence generation and the spin de-\nphasing time. We present the analysis of these effects in\nthe following Sections.\nIII. RESONANT SPIN AMPLIFICATION\nWe begin with the classical expression for carrier spin\npolarization under RSA conditions [5, 37]. The underly-\ning assumptions are the following: (i) only carrier spin\npolarization is considered, and (ii) it is supposed thateach pump pulse generates only a zcomponent of spin\npolarization, whose magnitude is S0. All non-additive\neffects of the pump pulse [28] are disregarded. After sin-\ngle pump pulse excitation the carrier spin dynamics are\ndescribed by a decaying cosine function periodic with the\nLarmor precession frequency ωand decay with time τs.\nThe effect of a long train of pump pulses on the carrier\nspin polarization can be calculated as:\nSz(ω,t) =∞/summationdisplay\nk=0S0e−(t+kTR)/τscos[ω(t+kTR)],(17)\nwheretis the pump-probe delay and k= 0,1,2,.... This\nequations can be rewritten [37, 38] as:\nSz(ω,t) =S0\n2e−t/τs×\ne−TR/τscos(ωt)−cos[ω(t+TR)]\ncosh(TR/τs)−cos(ωTR).(18)\nIt follows from Eq. (18) that for sufficiently long decay\ntimesτs/greaterorsimilarTRthe carrier spin has sharp resonances as\nfunction ofmagnetic field. As will be shownby ourcalcu-\nlations, this corresponds to the solid line in Fig. 5(a) and\ngives the RSA signals presented in Fig. 6. The peak posi-\ntionsatzeropump-probedelaycorrespondtospinpreces-\nsionfrequencieswhicharecommensurablewiththepump\nrepetition frequency ωR= 2π/TR. The expression (18)\nnear commensurable frequency ( |ωTR−2πN| ≪1) and\nat a zero time delay can be written as:\nSb\nz∼1\n(ωTR−2πN)2+(TR/τs)2,(19)\nHere we assume that TR/τs≪1. The peak width is\ndetermined by the relaxation time of the electron spin\npolarization. Note, that for the spin ensemble the time\nτsshould be changed to the dephasing time T∗\n2[39]. This\nallows one to measure spin relaxation and spin dephas-\ning times exceeding TR, i.e., for conditions where direct\ndetermination by time-resolved methods becomes inap-\nplicable. The equations (18) and (19) describe a number\nof experiments well, see, e.g., Refs.[5, 37, 40, 41], and fa-\ncilitate evaluation of carrier gfactors and spin dephasing\ntimes [39].\nHowever, one sees that the spin polarization in\nEqs. (18) and (19) increases to infinity if τsbecomes\nlargerand larger. Moreover, such an approachdisregards\ncompletely the spin dynamics of trions and the specifics\nofcarrierspindephasinginexternalmagneticfields. This\ncase requires a special treatment. There are also experi-\nments which reveal a complicated shape of RSA spectra\nor a complete absence of RSA despite of very long spin\nrelaxation times, which cannot be described by this sim-\nple model [36, 42, 43]. The general analysis required for\nsuch cases is presented below.\nA. Fast spin relaxation in trion\nIf the spin relaxation of the unpaired carrier in the\ntrion is fast, τT\ns≪τr, the trion spin dynamics does not8\naffect the spin polarization of the resident carrier, see\nSec. IIB2. In this case the carrier polarization induced\nby the pump pulse is not compensated by the carriersleft\nafter trion recombination, as these carriers are unpolar-\nized. Then ξ= 0 and the parameter Kin Eq. (14) has\nthe simple form [14, 23]\nK=(1−Q2)e−TR/τs\n2/bracketleftBig\nQe−TR/τs−cos(ωTR)/bracketrightBig\n.(20)\nThe detailed analysis of Eqs. (14) and (20) for this case\nis given in Refs. [14, 23]. If, moreover, the pump pulse\nareaΘ is small, so that 1 −Q≪1, Eq. (14) togetherwith\nEq. (20) go over into the classical expression of Eq. (18)\nfor carrier spin polarization under RSA conditions.\nIt follows, that for frequencies near the phase synchro-\nnization condition of Eq. (15), the spin zcomponent of\nthe resident carrier can be recast as [23]:\nSb\nz∼1\n(ωTR−2πN)2+[TR/τs+(1−Q)]2,(21)\nwhere we assume that TR/τs≪1, 1−Q≪1 and|ωTR−\n2πN| ≪1. One sees from Eq. (21) that the RSA peak\nwidth is determined by TR/τsor 1−Q, whichever is\nlarger.\nFigure 6 shows RSA signals calculated for a small\npump power, Θ = 0 .1π, at two different delays. The\nshape of the RSA signal at large negative delay ( t=\n−0.1TR) differs from the one at zero delay due to the\ndifferent phases of the spin precession.\nAn increase of the pump pulse area results in broaden-\ning of the RSA peaks, as was already shown in Fig. 5(b).\nFor increasing pump pulse area the RSA peaks are no\nlongerLorentziansand, therefore, Sb\nzcannotbedescribed\nby Eq. (21). The spin polarization for Θ = πand\nτs/TR= 3 shown in Fig. 5(b) looks similar to the one\nfor Θ = 0 .1πandτs/TR= 0.5 in Fig. 5(a). Hence, under\nstrong excitation the dependence of carrier spin polariza-\ntion on magnetic field becomes cosine-like due to satura-\ntion effects. In this case it is not possible to extract the\ncarrier spin relaxation or the dephasing times from the\nwidth of RSA peaks.\nB. Slow spin relaxation in trion: effect of trion\nspin dynamics\nLet us now turn to the general case, in which the trion\nspin relaxation time can be comparable or even longer\nthan its recombination time. It is instructive to start\nfrom the situation in which τT\ns≫τrand long-lived spin\ncoherenceappearsonlyduetocarrierortrionspinpreces-\nsion about the magnetic field. Clearly, the peaks in the\nSb\nz(ω) dependence aresuppressedfor ωτr,Ωτr≪1due to\ninefficient spin generation, and they increasesignificantly\nwith an increase of magnetic field. This is illustrated\nin Fig. 7, where the calculated RSA signals are shown\nforτT\ns= 30τr. Note, that such unusual RSA spectra\nwith suppression of the peak amplitudes in weak mag-\nnetic fields have been observed experimentally in both\nn-type and p-type QWs [22, 36, 42, 44].0.00 0.01 0.02 − Sz (arb. units)  \n \nω / ωR\n  \n-10 -5 0 5 10 -0.01 0.00 0.01 0.02 \n(b) (a) t = 0  \n t = -0.1 TR\n  \nFigure 6: (Color online) Carrier spin polarization Szas func-\ntion of magnetic field ( ω∝B) at two different pump-probe\ndelays denoted in each panel. t= 0 means that the signal\nis calculated for very small negative delay, just before the\npump pulse arrival. Parameters of calculations: τs= 3TR,\nΘ = 0.1π.\nFigure 7(a) shows the signal calculated in absence of\ntrion spin precession (Ω = 0) shortly before the pump\npulse arrival. The peak amplitude at zero magnetic field\n(ω= 0) is given by the ratio τrandτT\nsand goes to zero\nfor infinite τT\ns. The increase of peak amplitudes with in-\ncreasing magnetic field depends on ξand, therefore, on\nthe ratio ω/γ, similar to the amplitude dependencies in\nFig. 2. The peak shapes at zero delay differ from being\nLorentzian, see for comparison Eq. (21) and Figs. 5(a,b)\nand6(a), becausethespinleftbehindaftertrionrecombi-\nnationchangesthephaseofthecarrierspinprecession[9].\nIt is worth to stress, that we can use the same system\nof equations (5) to describe the spin dynamics in n-type\n(resident electron and T−trion) and p-type (resident\nhole and T+trion) structures. Figure 7(a) illustrates\nthe situation that is typical for n-type QWs [31, 42], in\nwhich trion spin precession is absent. Figure 7(b) shows\nthe RSA signal with a trion spin precession frequency\nΩ = 4ω, which may correspondto the T+trion case in p-\ntype QWs [36, 44]. The analysis shows that small Ω, i.e.\nΩ≤ω, leads to no significant changes of the RSA signal\nshape as compared with one in Fig. 7(b). A fast preces-\nsion of the trion spin results in a faster appearance of\nlong-lived spin coherence with increasing magnetic field,\ncompare Figs. 7(a) and 7(b).\nC. Effect of spin relaxation anisotropy\nTo make our analysis of RSA complete, we briefly dis-\ncuss here another effect, which is relevant for weak mag-\nnetic fields. Itaddressesthe situationin whichthe carrier\nspin relaxation or the dephasing times are anisotropic.\nSpin relaxation anisotropy is an inherent feature of semi-\nconductor quantum wells [45–49]. For simplicity, we con-9\n0.00 0.01 \n  \n(b) (a) \nω / ωR\n  \nΩ = 0 \n-20 -15 -10 -5 0 5 10 15 20 0.00 0.01 Ω = 4 ω \n− Szb (arb. units) \n  \nFigure 7: (Color online) Impact of slow spin relaxation of\nthe unpaired spin in the trion: RSA signals at zero delay\nwithout (Ω = 0) and with (Ω = 4 ω) trion spin precession\n(panels (a) and (b), respectively). Parameters of calculat ions:\nτT\ns= 30τr,τr= 0.01TR,τs= 3TRandωτr= 4.4 atB= 1 T\nand Θ = 0 .1π.\nsider the case, in which the zandyspin components of\nthe resident carriersrelaxat different time constants, τs,z\nandτs,y, respectively. Provided that the long-lived car-\nrier spin coherence is excited by the train of weak pump\npulses, the dependence of the carrier spin zcomponent\non the precession frequency is given by [38]\nSb\nz(ω)∼C(˜ωTR)−e−TR/τs\ncosh(TR/τs)−cos(˜ωTR),(22)\nwhere\n1\nτs=1\n2/parenleftbigg1\nτs,z+1\nτs,y/parenrightbigg\n,˜ω=/radicalBigg\nω2−1\n4/parenleftbigg1\nτs,z−1\nτs,y/parenrightbigg2\n,\n(23)\nand\nC(˜ωTR) = cos(˜ωTR)−1\n2˜ω/parenleftbigg1\nτs,z−1\nτs,y/parenrightbigg\nsin(˜ωTR).\nThe dependence of carrier spin polarization, −Sb\nz,\non magnetic field is shown in Fig. 8 for two cases of\nanisotropic carrier spin relaxation: (a) τs,y= 4τs,zand\n(b)τs,y= 0.25τs,z. The amplitudes of all maxima ex-\ncept the one at zero field are the same, because they are\ndetermined by the effective spin relaxation time, τs, de-\nfined by Eq. (23). The amplitude of the zero-field peak is\ndifferent from the other peaks. If τs,y> τs,z, it is smaller\nas compared with the others. The carrier spin relaxation\nin absence of a magnetic field is governed solely by τs,z\nand is faster than at finite magnetic fields, so that accu-\nmulation of carrier spin polarization is weaker at B= 0.\nIn the opposite case of τs,y< τs,zthe zero-field peak is\nhigher, because the lifetime of the spin zcomponent is\nlonger in absence of magnetic field so that spin accumu-\nlation is more efficient [50].0.00 0.02 0.04 \n  \n(b) (a) \n  \nτs, y  = 4 τs, z \n-3 -2 -1 0 1 2 30.00 0.02 τs, y  = 0.25 τs, z  \n − Szb (arb. units) \nω / ωR\n  \nFigure 8: Effect of an anisotropy of the carrier spin relaxati on\ntimes. Theelectronspin zcomponentrightbefore pumppulse\narrival (t= 0) is calculated as function of magnetic field after\nEq. (22). τs,z= 4TR.\nD. Spin decoherence and dephasing\nThe spin relaxation time of localized carriers can be\nextremely long reaching up to microseconds for electrons\nin QDs, for example [51]. This is related with quenching\nof the orbital motion and the corresponding suppression\nof spin relaxation mechanisms contributed by spin-orbit\ncoupling [52, 53]. The coherence time of an individual\nspin is typically much longer compared with the spin de-\nphasing time of an inhomogeneous spin ensemble. The\ninhomogeneity, which leads to a spread of carrier spin\nprecession frequencies, results in spin dephasing charac-\nterized by the T∗\n2dephasing time. This time measured,\ne.g., from the decay of spin beats in external magnetic\nfield is in the few nanoseconds range for QD ensem-\nbles [14, 15, 54] and in the tens of nanoseconds range\nfor QWs containing diluted carrier gases [9, 22, 30, 40].\nOne ofthe main originsforthe inhomogeneityofa spin\nensemble is related to the gfactor spreadof localized car-\nriers. For electrons the gfactor variation can arise from\nchanges of the effective band gap for different localiza-\ntion sites [14, 55, 56]. For localized holes the variations\nare mainly related to changes in the mixing of heavy-hole\nand light-hole states [57]. The spread of gfactors in a\nspin ensemble, ∆ g, is translated into a spread of spin\nprecession frequencies, ∆ ωg, and, therefore, results in a\nspin dephasing rate [38, 40]\n1\nT∗\n2,∆g∼∆gµBB\n/planckover2pi1≡∆ωg, (24)\nwhich is accelerated with increasing magnetic field.\nAnotheroriginofspindephasingtypicalforelectronsis\nrelatedtorandomnuclearfields inthe quantumdots[58].\nEach localized electron is subject to a hyperfine field of\na particular nuclear spin fluctuation, Bn, and, therefore,\nprecesses about this field at a frequency ωn. These fluc-\ntuations are different for localization sites, causing de-\nphasing of the electron spin ensemble. The dephasing10\nrate can be estimated by the root mean square of the\nelectron spin precession frequency in the field of frozen\nnuclear fluctuations [58]:\n1\nT∗\n2,n∼/radicalbig\n∝angbracketleftω2n∝angbracketright. (25)\nAssuming a normal distribution of BnEq. (25) can be\nrewritten as:\n1\nT∗\n2,n∼gµB∆B\n/planckover2pi1≡∆ωn. (26)\nwhere∆ Bisthe dispersionofthe nuclearspin fluctuation\ndistribution [58].\nEstimates show that T∗\n2,nis on the order of several\nnanoseconds for GaAs quantum dots [58, 59]. Hence,\nin weak magnetic fields (e.g., B/lessorsimilar0.3 T for g= 0.5\nand ∆g= 0.005 [43]) the spin beat decay for resident\nelectrons is determined by the hyperfine interaction, and\nin higher fields the dephasing is caused by the spread of\ngfactors [60].\nIn quantum wells with a diluted electron gas the elec-\ntron localization on well width fluctuations is consider-\nably weaker compared to the QD case. As a result, ∆ gis\nsmaller and the hyperfine interaction is weaker. There-\nfore, the spin dephasing times can reach ∼30÷50 ns\nin weak magnetic fields and at low temperatures [9, 22].\nBelow the effect of a spin precession frequency spread on\nRSA signals is analyzed.\n1. Spread of gfactors\nFor a more realistic approach we need to take into ac-\ncount the precession frequency spread, ∆ ω, in the spin\nensemble. Here for distinctness we consider only the fre-\nquency spread caused by ∆ g(the spread related with\nthe nuclear spin fluctuations is considered below). For\nensemble of carrier spins with a spread of gfactors, ∆ g,\nthe spread of Larmorprecession frequencies, ∆ ωg, is pro-\nportional to the magnetic field:\n∆ωg(B) = ∆gµBB//planckover2pi1 (27)\nTo model the ensemble RSA signal one has to sum the\ncontributionsofthe individual spins [38] overthe gfactor\ndistribution function:\nρ(g) =1√\n2π∆gexp/bracketleftbigg−(g−g0)2\n2(∆g)2/bracketrightbigg\n,(28)\nwhereg0is the average gfactor value in the spin en-\nsemble, resulting in an average Larmor frequency: ω0=\ng0µBB//planckover2pi1.\nRSA spectra calculated by means of Eqs. (14) and (28)\nfor short trion spin relaxation, i.e. dependencies of the\ncarrier spin polarization, −Sz, on magnetic field in terms\nofω0/ωRare shown in Fig. 9 for two negative time de-\nlays. An increase of magnetic field leads to broadening\nof the RSA resonances and decrease of their amplitudes.\nThis reflects the acceleration of the spin dephasing rate\n1/T∗\n2∼B, in accordance with Eq. (24).0.00 0.01 0.02 \n  \n(b) (a) \n  \nt = 0 \nt = -0.1 TR\n-10 -5 0 5 10 -0.01 0.00 0.01  \n − Sz (arb. units) \nω0 / ωR\n  \nFigure 9: (Color online) Dependenciesofcarrier spinpolar iza-\ntionSzon magnetic field at two different pump-probe delays\ntgiven in each panel and short trion spin relaxation time\nτT\ns≪τr. A frequency spread ∆ ωg= 0.02ω0, corresponding\nto 2% dispersion of the carrier gfactor, is assumed in the\ncalculations. The dependence on magnetic field is given by\nω0/ωR=g0µBB/(/planckover2pi1ωR).\nFigures 10(a) and 10(b) show RSA signals for long\ntrion spin relaxation, τT\ns= 30τr, with and without\nspin precession in the trion. An ensemble spread of\n∆ωg= 0.02ω0results in a broadening of the RSA peaks\nand a decrease of their amplitudes with increasing mag-\nnetic field, similar to Fig. 9. This results in the charac-\nteristic bat-like shape of the RSA signal [22, 36, 42, 44]\ncompare with Fig. 7 where the spin dephasing was ab-\nsent, ∆ωg= 0. Accounting for the spread of Ω does not\nchange the signals significantly.\nFigure10(a)correspondstoasituationthatisobtained\nfor resident electrons oriented by excitation of the T−\ntrion inn-type (In,Ga)As/GaAs QWs [22, 42]. In such\nstructures the in-plane hole gfactor is small compared\nwith the electron gfactor, and consequently Ω ≪ω, so\nthat the spin precession ofthe T−trion can be neglected.\nFigure 10(b) corresponds to the long-lived hole spin\norientation for excitation of the T+trion in p-type\nGaAs/(Al,Ga)As QWs [36]. For the T+trion the ratio\nΩ andωis opposite, i.e. Ω ≫ω. In Ref. [36] Ω = 4 .5ω\nand the spin precession in trion affects the RSA signal.\nThe resultsofthe calculationsshownin Figs.10(a)and\n10(b) are in good agreement with available experimental\ndata for quantum well structures [22, 36, 42]. All calcu-\nlations were done for a small pulse area Θ = 0 .1π. The\nanalysis of the case of high pump power, which results\nin saturation effects, shows that an increase of the pump\npower results in an increase of the signal amplitude and\nbroadening of all peaks, similar to the case discussed in\nSec. IIIA, see also Fig. 5. The bat-like shape of the RSA\nsignal envelope is conserved even for Θ = πpump pulses.11\n-0.001 0.000 0.001 0.002 \n(b) (a) \nω0 / ωR− Szb (arb. units)   Ω = 0 \n  \n-20 -15 -10 -5 0 5 10 15 20 -0.002 0.000 0.002 \nΩ = 4 ω0 \n \n  \nFigure 10: (Color online) Effect of slow trion spin relaxatio n.\nRSA signals at zero delay without [(a) Ω = 0] and with\n[(b) Ω = 4 ω0] trion spin precession are shown. The signals\nare calculated assuming a spin precession frequency spread\n∆ω= 0.02ω0of the resident carrier. The parameters in the\ncalculations are: τT\ns= 30τr,τr= 0.01TR,τs= 3TRand\nω0τr= 4.4 atB= 1 T and Θ = 0 .1π.\n2. Nuclear field fluctuations and resonant spin\namplification in weak magnetic fields\nInteraction of the nuclear spins with hole spins is weak\nand in many cases can be neglected. At the same time,\nfor localized electrons the hyperfine interaction with the\nnuclei can considerably contribute to the spin dynam-\nics. Therefore, in this subsection we will focus on n-type\nstructures containing resident electrons.\nIn weak magnetic fields the electron spin dephasing\ntime related to the spread of gfactor values, Eq. (24),\nproportional to 1 /B, becomes very long and nuclear field\nfluctuations play an important role. The hyperfine fields\nacting on the electrons due to these nuclear fluctuations\ncan be as large as Bn∼0.5 mT for GaAs QWs [30] and\nan order of magnitude larger in (In,Ga)As QDs [61].\nForB/greaterorsimilarBnthe only important component of the nu-\nclear field fluctuation is the one parallel to the external\nfieldB. It results in a spread of Larmor precession fre-\nquencies, damping of the spin beats and broadening of\nthe RSA peaks, provided Bn>|∆g/g|B.\nThesituationbecomesdifferentinweakmagneticfields\nB < B n. In this case all components of the nuclear fluc-\ntuation field become important. For illustration we con-\nsider a homogeneous electron spin ensemble (∆ g= 0) in\na magnetic field which is the sum of the external mag-\nnetic field Band the fluctuation field Bn. For simplicity,\nwe consider the regime of fast spin relaxation in the trion\n(τT\ns≪τr). To model the dynamics of the electron spin\nensemble one can assume a normal distribution of Bn:\nρn(Bn) =1\n(√\n2π∆B)3exp/parenleftbigg\n−Bn2\n2(∆B)2/parenrightbigg\n.(29)\nwhere ∆ Bis the isotropic dispersion of the nuclear fluc-\ntuation field distribution (∆ B,x= ∆B,y= ∆B,z). The-6 -4 -2 0 2 4 60.000 0.004 Δω n = 0.5 ωR(c) (b) (a) \n − Szb (arb. units) Δω n = 0.02 ωR\nΔω n = 0.08 ωR\nΔω n = 0.2 ωR\nω0/ωR\n  0.00 0.01 \n  0.00 0.01 \n  \n  \nFigure 11: (Color online) RSA signals at zero pump-probe\ndelay calculated for different spreads of the nuclei fluctuat ion\nfield ∆ B. The frequency spreads (∆ ωn∼∆B) are given in\neach panel. Θ = 0 .1π, ∆g= 0 and τT\ns≪τr.\nspread of the Larmor precession frequencies, ∆ ωn, does\nnot depend on the external magnetic field:\n∆ωn=gµB∆B//planckover2pi1. (30)\nThe average Larmor frequency of the spin ensemble in\nthis case is equal to the spin precession frequency in\nan external magnetic field without nuclear fluctuations:\nω0=gµBB//planckover2pi1.\nFigure 11 shows RSA signals at zero time delay av-\neraged over Bnfor different ∆ Bvalues. One sees that\nindeed, an increase of the frequency spread ∆ ωnleads to\nan increase of the dephasing rate evidenced via broaden-\ning of the RSA peaks. For weak magnetic fields B <∆B\ntheycomponent of the nuclear fluctuation field, Bn,y,\nwhich is perpendicular to Szand to the external field,\ncan additionally destroy the long-lived carrier spin po-\nlarization. This is manifested in an additional broaden-\ning and a decrease of the amplitude of the zeroth RSA\npeak (compared to the ±1 peaks), as is clearly seen in\nFig. 11(a,b). The enhancement of Szin the vicinity of\nzero field for large fluctuations, see Fig. 11(c), is due to\nthe fact that the zcomponent of the spin polarization\ncan not destroy by a parallel component of the nuclear\nfluctuation field Bn,z.\nE. Analysis of RSA signals and evaluation of spin\ndephasing times and gfactors\nTo conclude our analysis of RSA we emphasize that\nin spite of the possibly complex shape of RSA signals,\nespecially in case of a long spin relaxation in the trion,\nthe analysis allowsone to obtain variousparameterswith\nhigh accuracy. This is due to the fact that these param-\neters are responsible for different features in the RSA\nspectrum:12\n•thegfactor of the resident carriers gives the mag-\nnetic field positions of the RSA peaks.\n•Thegfactor spread, ∆ g, determines the amplitude\ndecreaseoftheRSApeakswithincreasingmagnetic\nfield.\n•The spin relaxation/dephasing time τsis related to\nthe RSA peak widths [62].\n•The ratio of spin relaxation time τT\nsand radia-\ntive lifetime τrof the trion determines the possible\nincrease of RSA peak amplitudes with increasing\nmagnetic field. If τris obtained from an indepen-\ndent time-resolved measurement, then τT\nscan be\nextracted from fitting the RSA spectrum.\n•Forlongspinrelaxationin the trion(when theRSA\nsignal has a bat-like shape) the symmetry of the\nRSA peaks at zero pump-probe delay can indicate\nthe fact that the trion gfactor is larger than that\nof the resident carrier ( |gT| ≫ |g|). However, the\nvalue of the trion gfactor should be obtained from\nanother experiment.\n•Finally, the amplitude and the width of the zero-\nfield RSA peak can contain information on the\nanisotropy of the spin relaxation of delocalized car-\nriers and the nuclear effects for localized carriers.\nThe spin dynamics parameters considered above can\nbe extracted only for sufficiently homogeneous ensembles\nand at weak excitation powers (small pump pulse areas),\nwhich is typical for semiconductor QWs.\nIt is worth to mention, that there are other generation\nmechanisms of long-lived spin coherence for nonresonant\noptical excitation [9, 22, 44]. In this case, the RSA signal\ncan change its shape dramatically. However, a detailed\nanalysis allows one to identify the generation and relax-\nation mechanisms of carrier spin polarization and obtain\nthe corresponding quantitative information about relax-\nation processes.\nIV. MODE-LOCKING OF CARRIER SPIN\nCOHERENCES\nNow we turn to strongly inhomogeneous spin systems,\nfor which the spread of the spin precession frequencies is\nso large that\nT∗\n2< TR. (31)\nStill, the spin relaxation time of the resident carrier is\nassumed to exceed by far the repetition period, τs≫TR.\nIn this case the ensemble spin polarization generated by\na pump pulse decays within the time T∗\n2, i.e., disappears\nbefore the next pump pulse arrival. Figure 12 presents\nmodel calculations, which show the dynamics of the car-\nrier spin polarization excited by a train of the pump\npulses. Indeed, the polarization decays quite rapidly af-\nter the pump pulses, but thereafter reemerges at nega-\ntive delays −T∗\n2/lessorsimilart <0. Such a behavior has beenexplained in terms of mode-locking of carrier spin coher-\nencesthataresynchronizedbytheperiodictrainofpump\npulses [14, 20].\n/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s84\n/s82/s32/s83\n/s122 /s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s32\n/s40/s98/s41/s40/s97/s41\n/s80/s117/s109/s112/s32/s45/s32/s112/s114/s111 /s98/s101/s32/s100/s101/s108/s97/s121 /s44/s32 /s116/s32/s47/s32/s84\n/s82\n/s32/s32\n/s32/s61/s32/s49/s48/s32\n/s82\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50\n/s32/s32/s32/s61/s32/s55/s46/s53/s32\n/s82\n/s32/s32\nFigure 12: (Color online) Carrier spin polarization as func -\ntion of pump-probe delay for precession frequencies which ( a)\nsatisfy the PSC of Eq. (15) and (b) do not satisfy it. The fre-\nquency spread is ∆ ω=ωRand Θ = π. Thick vertical arrows\nindicate the arrival times of the pump pulses.\nIf the condition (31) is fulfilled, the pump pulse ex-\ncites a broad distribution of spin precession frequencies,\namong which there are several frequencies satisfying the\nphase synchronization condition of Eq. (15). The carrier\nspins with such precession frequencies are excited much\nmore efficiently, i.e., accumulate more spin polarization\nthan the other ones. As a result, the main contribution\nto the signalis givenbythe commensurablespin beat fre-\nquencies. Inotherwords,thespinssatisfyingthePSCbe-\ncome resonantly amplified, while others are not, and the\nsynchronized spins contribute mostly to the experimen-\ntally measured signal of carrier spin polarization. Such\nbehavior of the spin signals, characteristic for the mode-\nlocking of carrier spin coherences, has been observed in\nn-type singly-charged (In,Ga)As QDs [14, 20, 21].\nThe calculations shown in Fig. 12 are carried out after\nEqs.(14) and (30) assuming, forsimplicity, that the trion\nspin relaxation is fast, τT\ns≪τr, and the spread of the\ncarrier spin precession frequencies ∆ ω=ωRdoes not\ndepend on the magnetic field strength.\nLet us have a closer look on the signals in Fig. 12. It\nis remarkable, that the phase of the spin beats before\nthe next pump pulse arrival is fixed for any magnetic\nfield. The averageprecession frequency of spin ensemble,\nω0, satisfies the PSC in Fig. 12(a) while it does not in\nFig. 12(b). The phase, however, in both cases is exactly\nthe same and it also coincides with the one after the\npump pulse, φ= 0. This is in strong contrast with the\nregime of weak dephasing ( T∗\n2/greaterorsimilarTR), see Fig. 5(c), and\ncan be considered as the principle difference of the SML\nand RSA regimesofcarrierspin accumulation. Note that13\nthe regime of weak dephasing is similar to the dynamics\nof a single spin presented in Figs. 4 and 5.\nIt is worth to mention, that the ratio of the signal am-\nplitudes at negative and positive delays depends strongly\non the generation efficiency and conservation of spin po-\nlarization, i.e., on the pump pulse area, the trion spin\nrelaxation, and the ratio of carrier spin relaxation time\nτstoTR[14, 20].\nV. RSA VERSUS MODE-LOCKING\nIn this Section we discuss how one can distinguish the\nRSA and SML regimes and what parameters are respon-\nsible for separating these regimes. This separation is\nbased on the common basic mechanism of the RSA and\nSML effects, which is the accumulation of carrier spin\npolarization under periodic pump pulse excitation. The\nkey difference between the regimesis the ratio ofthe Lar-\nmor frequency broadening to the repetition frequency of\nthe pump pulses: ∆ ω/ωR. This is schematically illus-\ntrated in Fig. 13(a,b) by the frequency spectrum of the\nspin ensemble in a finite magnetic field. Here few PSC\nmodes satisfying Eq. (15) from ( N−2)ωRto (N+2)ωR\nare indicated in by the dashed vertical lines.\nIn the RSA regime ∆ ω≪ωRand only one PSC mode\n(or even none) can fall into the distribution of Larmor\nfrequencies. When the PSC mode coincides with the dis-\ntribution maximum, as it is shown in Fig. 13(a), one ob-\ntainsa peakin the RSA spectrum. And when the overlap\nbetween the mode and the distribution isabsent the RSA\nspectrum has minimum.\nFor the SML regime involvement of at least two PSC\nmodes is necessary. Therefore, the condition for this\nregime is ∆ ω/greaterorsimilarωR, see Fig. 13(b). The calculations\ngiven in this Section show that in fact the transition to\nthe SML regime happens already for ∆ ω/greaterorsimilar0.5ωR, when\nthe tails of the Larmor frequency distribution overlap\nwith more than one PSC mode.\n/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s40/s78 /s43/s49/s41 /s40/s78 /s45/s49/s41/s48/s32/s61/s32 /s78\n/s82\n/s48/s32/s61/s32 /s78\n/s82\n/s40/s78 /s43/s50/s41/s40/s78 /s45/s50/s41/s78/s32/s32\n/s40/s98/s41/s40/s97/s41\n/s32/s32\n/s50\n/s50\n/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32/s32\n/s32/s32\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s32/s47/s32\n/s82\n/s103 /s32/s47/s32 /s103\n/s48\n/s82/s83/s65/s83/s77/s76/s40/s99/s41/s32\n/s32/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s48/s32/s47/s32\n/s82\n/s32/s32\nFigure13: Larmor frequencydistributionfunction(multip lied\nfor convenience by√\n2π∆ω) of a spin ensemble for RSA (a)\nand SML (b) conditions. (c) Parameter diagram showing\nschematically the regimes where RSA and SML occur, see\ntext for details.\nDeeper insight in the separation between the RSA and\nSML regimes is collected below in Figs. 14, 15, and\n16. Here the carrier spin polarization amplitude, Sb\nz, and\nthe signal phase at zero negative delay are analyzed as\nfunctions of magnetic field, time delay, Larmorfrequencyspread, and pump pulse area. We also consider the effect\nof resident carrier spin relaxation taking it into account\nvia the parameter τs/TR. For most figures a pump pulse\narea Θ = πis chosen as it provides efficient spin accu-\nmulation. Let us go step by step through this data set.\nFirst, for demonstration purposes, we assume again\nthat a spread of the carrier spin precession frequencies is\n∆ω=ωR, and it does not depend on magnetic field. For\nn-type structures this corresponds to the case when the\n∆ωof the resident electrons is dominated by the random\nfields of the nuclear spin fluctuations: ∆ ωn∝Bn,x. For\nB > B nonly the Bn,xcomponent parallel to the external\nmagneticfieldshouldbeconsidered,seeSec.IIID2. Sim-\nilar to the previous Sections, the nuclear spin fluctuation\nis considered to be frozen.\nMagnetic field dependencies of the carrier spin po-\nlarization, −Sb\nz, and the signal phase are shown in\nFigs. 14(a) and 14(b) for different ∆ ωandτs/TR= 300.\nFor a small frequency spread of ∆ ω= 0 and 0 .2ωRthe\npolarization amplitude and phase are periodic functions\nof magnetic field, which is characteristic for the RSA\nregime, for comparison see Figs. 5(b) and 5(d). An in-\ncrease of ∆ ωto 0.5ωRdrastically changes the character\nof these functions: both of them become independent of\nmagnetic field. The spin polarization amplitude has a\nfinite value (in this case it is equal 0.08), while φ= 0.\nThese are characteristics of the SML regime.\nDetails of separating the RSA from the SML regime\nwith increasing frequency spread are presented in\nFig. 14(c). The peak amplitudes of the spin polarization\nat the PSC frequencies are plotted for different pump\npulse areas there. The amplitude initially decreases with\nan increasing spread and approaches a saturation level\nfor larger spreads. Independence of the amplitude on the\nspread is characteristic for the SML regime, therefore,\none can see from the Fig. 14(c) that the regimes cross\nover at ∆ ω∼0.5ωR.\nThe spin polarization amplitude in the SML regime\ndepends critically on the pump pulse area, see also\nFig. 14(d). It is close to zero for Θ <0.3π, but strongly\nincreases for Θ exceeding this value, approaching a max-\nimum at Θ = 2 πfor sufficiently large τs/TR= 3000. The\ndependence of Sb\nzfor a large spread, which corresponds\nto a constant plateau level, can be written as:\nSb\nz=1−Q\n1+Q/bracketleftBigg\n1−/radicalbigg\nM2−1\nL2−1/bracketrightBigg\n, (32)\nwhereM=Qe−TR/τsandL= e−TR/τs(1+Q2)/2. The\ncalculations in Fig. 14(d) show that with increasing elec-\ntron spin relaxation time τsthe maximum signal ampli-\ntude shifts to a pulse areaof 2 π[unlike the dependence of\nspin polarization on pulse area for excitation by a single\npulse, for which Rabi oscillations occur with maximum\nat Θ =π].\nThe fact that the separation between RSA and SML\nis controlled by the ratio ∆ ω/ωRoffers the instructive\nopportunity to realise a changeover between these two\nregimes by tuning the magnetic field. This would be\npossible for the case when the Larmor frequency spread\nis controlled by ∆ g, see Sec. IIID1, because in this case\n∆ωgincreases linearly with B. Results of corresponding14\n/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s32/s61/s32/s48\n/s48/s46/s53\n/s82/s48/s46/s50\n/s82/s32/s83\n/s122 /s98\n/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32\n/s32\n/s32/s32\n/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s80/s101/s97/s107/s32/s97/s109/s112/s108/s105/s116/s117/s100/s101/s80/s101/s97/s107/s32/s97/s109/s112/s108/s105/s116/s117/s100/s101/s40/s97/s41\n/s40/s100/s41/s40/s99/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s115/s112/s114/s101/s97/s100/s44/s32 /s32/s47\n/s82/s115/s32/s47 /s84\n/s82/s32/s61/s32/s51/s48/s48\n/s115/s32/s47 /s84\n/s82/s32/s61/s32/s51/s48/s48\n/s48/s46/s54/s32\n/s48/s46/s51/s32/s32\n/s32/s32/s61/s32/s49/s46/s53/s32\n/s32/s32\n/s48 /s49 /s50/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s80/s117/s109/s112/s32/s112/s117/s108/s115/s101/s32/s97/s114/s101/s97/s44/s32 /s32/s47/s32/s61/s32/s48/s46/s53\n/s82\n/s51/s51/s48/s51/s48/s48/s32\n/s32/s115/s32/s47 /s84\n/s82/s32/s61/s32/s51/s48/s48/s48/s32/s45/s49 /s48 /s49\n/s82/s83/s65/s48\n/s45/s40/s98/s41\n/s32/s61/s32/s48\n/s48/s46/s53\n/s82/s48/s46/s50\n/s82/s80/s104/s97/s115/s101/s44/s32 /s32/s40/s114/s97/s100/s46/s41\n/s32\n/s32/s32/s47\n/s82\n/s32/s32\n/s83/s77/s76\nFigure 14: (Color online) Magnetic field dependencies [in\nterms of ω0(B)/ωR] of (a) the carrier spin polarization ampli-\ntude,−Sb\nz, and (b) the signal phase at zero delay calculated\nfor three different Larmor frequency spreads. Dependencies\nof the spin polarization amplitude for PSC modes, i.e. for\ninteger values of ω0(B)/ωR(c) on the frequency spread for\ndifferent pump pulse areas, at τs/TR= 300; and (d) on the\npump pulse area for various τs/TR, for a precession frequency\nspread ∆ ω= 0.5ωR.\ncalculations for ∆ ωg= 0.1ω0are given in Fig. 15. In\nanalogy with Figs. 14(a) and 14(b), one can identify the\nRSA regime in low magnetic fields ( |ω0/ωR|<3), where\nboth the polarization amplitude and the phase change\nwithB, and the SML regime in larger magnetic fields\n(|ω0/ωR|>5), where these parameters do not vary any-\nmore.\nFigure 13(c) shows the range of parameters in which\nthe different spin accumulation regimes can be obtained.\nThe dashed curve corresponds to the condition ∆ ω=\n0.5ωR, which may serve as approximate boundary be-\ntween the RSA and SML regimes. Indeed, if the gfactor\nspread is small, the spin frequency distribution contains\nonly one phase synchronized mode in a broad range of\nmagnetic fields, the latter are expressed via ω0(B)/ωR.\nIt corresponds to the RSA regime for which the parame-\nter space is placed below the dashed curve in Fig. 13(c).\nOn the contrary, if the g-factor spread is large, several\nphase synchronized modes become involved already at\nweak magnetic fields and, for relatively efficient optical\npumping, SML occurs [the parameter space above the\ndashed curve].\nThetimeevolutionofthespinpolarizationforthemag-/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s32/s83\n/s122 /s98\n/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s52/s51/s50\n/s49/s83/s77/s76 /s83/s77/s76 \n/s40/s98/s41/s40/s97/s41/s82/s83/s65 /s32\n/s32/s32\n/s45/s55 /s45/s54 /s45/s53 /s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55/s48\n/s45/s80/s104/s97/s115/s101/s44/s32 /s32/s40/s114/s97/s100/s46/s41\n/s32/s32/s47\n/s82\n/s32/s32\nFigure 15: (Color online) Magnetic field dependence of\n(a) carrier spin polarization −Sb\nzat zero pump-probe delay\n(shortly before pump pulse arrival), and (b) spin precessio n\nphase of the signal calculated for the same parameters as\nin panel (a). The RSA and SML regimes are shown by ar-\nrows. The labels with numbersare in accordance with Fig. 16.\nτs/TR= 3, Θ = π, ∆ωg= 0.1ω0.\nnetic fields in Fig. 15(a) are given in Fig. 16. Panel (a)\ncorresponds to the RSA regime (weak magnetic fields).\nOne can see that the spin polarization phase and am-\nplitude at small negative delays depends on the relation\nto the PSC. However, in the ML regime, shown in panel\n(b), both values are constant irrespective whether the\nPSC are fulfilled or not.\n/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51\n/s83/s77/s76 /s82/s83/s65 /s32/s32/s32/s32/s49/s32/s32/s32\n/s48/s32/s61/s32/s50/s46/s53/s32\n/s82\n/s32/s32/s32/s32/s50/s32/s32/s32\n/s48/s32/s61/s32/s51/s32\n/s82\n/s32/s32\n/s40/s98/s41/s40/s97/s41\n/s80/s117/s109/s112/s32/s45/s32/s112/s114/s111 /s98/s101/s32/s100/s101/s108/s97/s121 /s44/s32 /s116/s32/s47/s32/s84\n/s82/s32\n/s45/s49/s46/s48 /s45/s48/s46/s56 /s45/s48/s46/s54 /s45/s48/s46/s52 /s45/s48/s46/s50 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s32/s83\n/s122 /s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s32/s32/s32/s51/s32/s32 /s32\n/s48/s32/s61/s32/s53/s32\n/s82\n/s32/s32/s32/s32/s52/s32/s32/s32\n/s48/s32/s61/s32/s53/s46/s53/s32\n/s82\n/s32/s32/s32\nFigure 16: (Color online) Carrier spin polarization as func tion\nof pump-probe delay for different magnetic fields denoted in\nFig. 15(a). Panel (a) corresponds to the RSA regime, and\npanel (b) to the SML regime. τs/TR= 3, Θ = π, ∆ωg=\n0.1ω0.\nFrom the results of Secs. IV and V one can conclude\nabout the two main features of the SML regime. The\nfirst one is a fixed phase of the spin signal at very small\nnegative delays, which is independent of the magnetic\nfield. Thisreflectstheprimaryamplificationofspinswith15\ncommensurable spin beat frequencies in a strongly inho-\nmogeneous ensemble. The second one is a characteristic\nrevivalofthe dephased signalbefore the next pump pulse\narrival shown in Fig. 12.\nIt is also interesting, that contrary to the RSA regime\nin the SML regime the magnetic field dependence of the\nspin polarization at zero negative delay is smooth. The\ndependence is similar to that presented by the dashed\nline in Fig. 11(c). The width of this bell-like curve is\ndetermined by the nuclear field fluctuations and is ap-\nproximately equal to 4∆ B.\nLet us summarize the conditions for the SML regime.\nApart from the obvious condition τs≫TRit requires:\n1. A significant spread of carrier spin precession fre-\nquencies, ∆ ωg>0.5ωR. The spread can be caused\nby the nuclear fluctuation fields or by the spread of\ngfactors.\n2. The frequency spread ∆ ω >0.5ωRleads to a de-\nphasing of the spin signal within the time T∗\n2∼\nTR/π, i.e. faster than the time interval between\nsubsequent pump pulses.\n3. One can see from Figs. 14(c) and 14(d) that the\npump pulse area should be sufficiently large, Θ /greaterorsimilar\nπ/2. Otherwise the frequency spread ∆ ω >0.5ωR\nwould cause only a decay of the spin polarization\nwithout its revival before the next pump pulse ar-\nrival.\nVI. CONCLUSIONS\nTo conclude, we have performed a comprehensive the-\noretical study of carrier spin coherence in spin ensembles\nsubject to periodic optical pumping. The effect of spin\naccumulation has been analysed for singly-charged quan-\ntum dots and quantum wells with a low density carrier\ngas. The accumulation results in two regimes of carrier\nspin coherence: resonance spin amplification and spin\nmode-locking. These regimes, while being different intheir phenomenological appearances and realization con-\nditions, have the same origin and occur for spin ensem-\nbles for which the carrier spin coherence time exceeds by\nfar the pump repetition period. The resonance spin am-\nplification and spin mode-locking are mutually exclusive\nregimes because of the requirement on excitation power\nand precession frequency spread.\nFor the RSA regime sufficiently homogeneous spin en-\nsembles and small excitation powers (small pump pulse\nareas) are required. These conditions are experimentally\nrealized in QW structures with electron or hole resident\ncarriers of low density, i.e. for the regime, where neg-\natively or positively charged trions play an important\nrole. In this case the spin dephasing times for resident\ncarriers can be extracted with high accuracy, even when\nthey exceed the pulse repetition period. The spreads of\ngfactors and nuclear spin fluctuations are less important\nfor the long-lived spin coherence compared to the case of\nstrongly inhomogeneous QD ensembles.\nIncontrasttotheRSAregimetheSMLregimerequires\na strong inhomogeneity of the spin precession frequency\nin the spin ensemble and high excitation powers (pump\nareas close to πand more). By now the SML regime\nhas been observed experimentally and studied in great\ndetail for ensembles of (In,Ga)As/GaAs QDs each singly\ncharged with a resident electron. 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Sorokin, A. A. Toropov, S. V. Ivanov,\nand M. Bayer, Phys. Rev. B 84, 085304 (2011) and Phys.\nRev. B84, 15990(E) (2011).\n[60] If the spreads of gfactors and nuclear spin fluctuation\nfields are described by normal distributions (Gaussians)\nthenthedispersionofelectron-spinprecessionfrequenci es\nin the QD ensemble is [43]: ∆ ω=/radicalbig\n(∆gµBB//planckover2pi1)2+ω2n.This leads to a Gaussian dephasing with a characteristic\ntimeT∗\n2= 1/∆ω.\n[61] M. Yu. Petrov, I. V. Ignatiev, S. V. Poltavtsev, A.\nGreilich, A. Bauschulte, D. R. Yakovlev, and M. Bayer,\nPhys. Rev. B 78, 045315 (2008).\n[62] The zeroth RSA peak [Eq. (19)] has the same form as the\nstandard expression for the Hanle effect [17]: the electron\nspin depolarization in a transversal magnetic field under\ncontinuous wave pumping. The influence of the inhomo-\ngeneous distribution of gfactors on the Hanle effect is\ntypically quite weak and, as a rule [40, 63], the extracted\nspin dephasingtime is controlled by the nuclear spin fluc-\ntuations, i.e., by T∗\n2. As compared with the Hanle effect,\nthe studies of theresonant spinamplification allow one to\ndirectly extract the magnetic field dependence of the spin\ndephasing time, and consequently, evaluate the spread of\ngfactors, ∆ g.\n[63] S. V. Andreev, B. R. Namozov, A. V. Koudinov, Yu. G.\nKusrayev, and J. K. Furdyna, Phys. Rev. B 80, 113301\n(2009)."    },    {        "title": "1301.2513v1.A_new_type_of_nuclear_collective_motion___the_spin_scissors_mode.pdf",        "content": "arXiv:1301.2513v1  [nucl-th]  11 Jan 2013A new type of nuclear collective motion – the spin scissors mo de\nE.B. Balbutsev, I.V. Molodtsova\nJoint Institute for Nuclear Research, 141980 Dubna, Moscow Region,Russia\nP. Schuck\nInstitut de Physique Nucl´ eaire, IN2P3-CNRS,\nUniversit´ e Paris-Sud, F-91406 Orsay C´ edex, France;\nLaboratoire de Physique et Mod´ elisation des Milieux Conde ns´ es, CNRS and Universit´ e Joseph Fourier,\n25 avenue des Martyrs BP166, F-38042 Grenoble C´ edex 9, Fran ce\nThe coupled dynamics of low lying modes and various giant res onances are studied with the help\nof the Wigner Function Moments method on the basis of Time Dep endent Hartree-Fock equations in\nthe harmonic oscillator model including spin-orbit potent ial plus quadrupole-quadrupole and spin-\nspin residual interactions. New low lying spin dependent mo des are analyzed. Special attention is\npaid to the spin scissors mode.\nPACS numbers: 21.10.Hw, 21.60.Ev, 21.60.Jz, 24.30.Cz\nKeywords: spin; collective motion; scissors mode; giant re sonances\nI. INTRODUCTION\nThe idea of the possible existence of the collective\nmotion in deformed nuclei similarto the scissorsmo-\ntion continues to attract the attention of physicists\nwho extend it to various kinds of objects, not nec-\nessary nuclei, (for example, magnetic traps, see the\nreview by Heyde at al [1]) and invent new sorts of\nscissors, for example, the rotational oscillations of\nneutron skin against a proton-neutron core [2].\nThe nuclear scissors mode was predicted [3]–[6] as\na counter-rotationof protonsagainst neutrons in de-\nformed nuclei. However, its collectivity turned out\nto be small. From RPA results which were in quali-\ntative agreement with experiment, it was even ques-\ntioned whether this mode is collective at all [7, 8].\nPurely phenomenological models (such as, e.g., the\ntwo rotorsmodel [9]) and the sum rule approach[10]\ndid not clear up the situation in this respect. Finally\nin a very recent review [1] it is concluded that the\nscissorsmodeis”weaklycollective, butstrongonthe\nsingle-particle scale” and further: ”The weakly col-\nlective scissors mode excitation has become an ideal\ntest of models – especially microscopic models – of\nnuclear vibrations. Most models are usually cali-\nbrated to reproduce properties of strongly collective\nexcitations (e.g. of Jπ= 2+or 3−states, giant reso-\nnances, ...). Weakly-collective phenomena, however,\nforce the models to make genuine predictions and\nthe fact that the transitions in question are strong\non the single-particle scale makes it impossible to\ndismiss failures as a mere detail, especially in the\nlight of the overwhelming experimental evidence for\nthem in many nuclei [11, 12].”\nThe Wigner Function Moments (WFM) or phase\nspace moments method turns out to be very useful\nin this situation. On the one hand it is a purely mi-\ncroscopic method, because it is based on the TimeDependent Hartree-Fock (TDHF) equation. On the\nother hand the method works with average values\n(moments) of operators which have a direct rela-\ntion to the considered phenomenon and, thus, make\na natural bridge with the macroscopic description.\nThis makes it an ideal instrument to describe the\nbasic characteristics (energies and excitation prob-\nabilities) of collective excitations such as, in par-\nticular, the scissors mode. Our investigations have\nshownthatalreadytheminimalsetofcollectivevari-\nables, i.e. phase space moments up to quadratic or-\nder, is sufficient to reproduce the most important\nproperty of the scissors mode: its inevitable coex-\nistence with the IsoVector Giant Quadrupole Reso-\nnance(IVGQR)implyingadeformationoftheFermi\nsurface.\nFurther developments of the Wigner Function\nMoments method, namely, the switch from TDHF\nto Time Dependent Hartree-Fock Bogoliubov (TD-\nHFB) equations, i.e. taking into account pair corre-\nlations,allowedustoimproveconsiderablythequan-\ntitative description of the scissors mode [13, 14]: for\nrare earth nuclei the energies are reproduced with\n∼10% accuracy and B(M1) values were reduced by\nabout a factor of two with respect to their non su-\nperfluid values. However, they remain about two\ntimes too high with respect to experiment. We have\nsuspected, that the reason of this last discrepancy is\nhidden in the spin degrees of freedom, which were so\nfar ignored by the WFM method. One cannot ex-\nclude, that due to spin dependent interactions some\npart of the force of M1 transitions is shifted to the\nenergy region of 5-10 MeV, where a 1+resonance of\nspin nature is observed [7].\nIn a recent paper [15] the WFM method was ap-\nplied for the first time to solve the TDHF equations\nincluding spin dynamics. As a first step, only the\nspin-orbit interaction was included in the consider-2\nation, as the most important one among all possible\nspin dependent interactions because it enters into\nthe mean field. This allows one to understand the\nstructure of necessary modifications of the method\navoiding cumbersome calculations. The most re-\nmarkable result was the discovery of a new type of\nnuclear collective motion: rotational oscillations of\n”spin-up” nucleons with respect of ”spin-down” nu-\ncleons (the spin scissors mode). It turns out that\nthe experimentally observed group of peaks in the\nenergy interval 2-4 MeV corresponds very likely to\ntwo different types of motion: the conventional (or-\nbital) scissors mode and this new kind of mode, i.e.\nthe spin scissors mode.\nThree low lying excitations of a new nature were\nfound: isovector and isoscalar spin scissors and the\nexcitationgeneratedbytherelativemotionoftheor-\nbital angular momentum and the spin of the nucleus\n(theycanchangetheirabsolutevaluesanddirections\nkeeping the total spin unchanged). In the frame of\nthe same approach ten high lying excitations were\nalsoobtained: wellknownisoscalarandisovectorGi-\nant Quadrupole Resonances (GQR), two resonances\nof a new nature describing isoscalar and isovector\nquadrupolevibrationsof”spin-up”nucleonswith re-\nspect of ”spin-down” nucleons, and six resonances\nwhich can be interpreted as spin flip modes of vari-\nous kinds and multipolarity.\nTheobtainedresultsareveryinteresting,however,\ntheyareonlyintermediateinourinvestigationofM1\nmodes. Our finite goal is to get reasonable agree-\nment with experimental data for the conventional\nscissors mode, especially for its B(M1) factors which\nremain about two times too strong. We should keep\nin mind that only the standard spin-orbit potential\nwas taken into account in the paper [15], spin de-\npendent residual interactions being completely ne-\nglected.\nThe aim of this work is to get a qualitative under-\nstandingoftheinfluence ofthespin-spin forceonthe\nnew states analyzed in [15], as, for instance, the spin\nscissors mode. As a matter of fact we will find that\nthe spin-spin interaction does not change the gen-\neral picture of the positions of excitations describedin [15]. It pushes all levels up proportionally to its\nstrength without changing their order. The most in-\nteresting result concerns the B(M1) values of both\nscissors modes – the spin-spin interaction strongly\nredistributes M1 strength in the favour of the spin\nscissorsmode. This isa verypromisingfact, because\nit shows that after taking into account in addition\npairing [16] one may achieve agreement with exper-\niment.\nOne of the main points of the present work will,\nindeed, be that we will be able to giveatentative ex-\nplanationofa recentexperimentalfinding [17] where\nthe B(M1) values in233Th of the two low lying mag-\nnetic states are inverted in strength in favor of the\nlowest, i.e., the spin scissors mode, when cranking\nup the spin-spin interaction. Indeed, the explana-\ntion with respect to a triaxial deformation given in\n[17] yields a stronger B(M1) value for the higher ly-\ning state, contrary to observation, as remarked by\nthe authors themselves.\nThe paper is organized as follows. In Sec. 2 the\nTDHF equations for the 2x2 density matrix are for-\nmulated and their Wigner transform is found. In\nSec. 3 the model Hamiltonian is analyzed and the\nmean field generated by the spin-spin interaction is\nfound. In Sec. 4 the collective variables are defined\nand the respective dynamical equations are derived.\nIn Sec. 5 the results of our calculations of energies,\nB(M1) and B(E2) values are discussed. Lastly, re-\nmarks and the outlook are given in the conclusion\nsection. The mathematical details are concentrated\nin appendices A, B.\nII. WIGNER TRANSFORMATION OF\nTDHF EQUATION WITH SPIN\nThe TDHF equation in operator form reads [16]\ni/planckover2pi1˙ˆρ= [ˆh,ˆρ]. (1)\nLet us consider its matrix form in coordinate space\nkeeping all spin indices:\ni/planckover2pi1<r,s|˙ˆρ|r′′,s′′>=/summationdisplay\ns′/integraldisplay\nd3r′/parenleftBig\n<r,s|ˆh|r′,s′><r′,s′|ˆρ|r′′,s′′>−<r,s|ˆρ|r′,s′><r′,s′|ˆh|r′′,s′′>/parenrightBig\n.(2)\nWe do not specify the isospin indices in order to\nmake the formulae more transparent. They will be\nre-introduced at the end.\nThese equations will be solved by the method of\nphase space (or Wigner function) moments. To thisend we will rewrite the expression (2) with the help\nof the Wigner transformation [16]. To make the for-\nmulae more readable we will not write out the coor-\ndinate dependence ( r,p) of the functions. With the3\nconventional notation\n↑fors=1\n2and↓fors=−1\n2the Wigner transform of (2) can be written as\ni/planckover2pi1˙f↑↑=i/planckover2pi1{h↑↑,f↑↑}+h↑↓f↓↑−f↑↓h↓↑+i/planckover2pi1\n2{h↑↓,f↓↑}−i/planckover2pi1\n2{f↑↓,h↓↑}−/planckover2pi12\n8{{h↑↓,f↓↑}}+/planckover2pi12\n8{{f↑↓,h↓↑}}+...,\ni/planckover2pi1˙f↑↓=f↑↓(h↑↑−h↓↓)+i/planckover2pi1\n2{(h↑↑+h↓↓),f↑↓}−/planckover2pi12\n8{{(h↑↑−h↓↓),f↑↓}}\n−h↑↓(f↑↑−f↓↓)+i/planckover2pi1\n2{h↑↓,(f↑↑+f↓↓)}+/planckover2pi12\n8{{h↑↓,(f↑↑−f↓↓)}}+.... (3)\nwhere the functions h,fare the Wigner transforms of ˆh, ˆρrespectively, {f,g}is the Poisson bracket of the\nfunctions fandgand{{f,g}}is their double Poisson bracket; the dots stand for terms proport ional to\nhigher powers of /planckover2pi1. The remaining two equations are obtained by the obvious change of arrows↑↔↓.\nIt is useful to rewrite the above equations in terms of functions f+=f↑↑+f↓↓,f−=f↑↑−f↓↓. By\nanalogy with isoscalar fn+fpand isovector fn−fpfunctions one can name the functions f+andf−as\nspin-scalar and spin-vector ones, respectively. We have:\ni/planckover2pi1˙f+=i/planckover2pi1\n2{h+,f+}+i/planckover2pi1\n2{h−,f−}+i/planckover2pi1{h↑↓,f↓↑}+i/planckover2pi1{h↓↑,f↑↓}+...,\ni/planckover2pi1˙f−=i/planckover2pi1\n2{h+,f−}+i/planckover2pi1\n2{h−,f+}−2h↓↑f↑↓+2h↑↓f↓↑+/planckover2pi12\n4{{h↓↑,f↑↓}}−/planckover2pi12\n4{{h↑↓,f↓↑}}+...,\ni/planckover2pi1˙f↑↓=−h↑↓f−+h−f↑↓+i/planckover2pi1\n2{h↑↓,f+}+i/planckover2pi1\n2{h+,f↑↓}+/planckover2pi12\n8{{h↑↓,f−}}−/planckover2pi12\n8{{h−,f↑↓}}+...,\ni/planckover2pi1˙f↓↑=h↓↑f−−h−f↓↑+i/planckover2pi1\n2{h↓↑,f+}+i/planckover2pi1\n2{h+,f↓↑}−/planckover2pi12\n8{{h↓↑,f−}}+/planckover2pi12\n8{{h−,f↓↑}}+...,(4)\nwhereh±=h↑↑±h↓↓.\nIII. MODEL HAMILTONIAN\nThe microscopic Hamiltonian of the model, harmonic oscillator with spin- orbit potential plus separable\nquadrupole-quadrupole and spin-spin residual interactions is given by\nH=A/summationdisplay\ni=1/bracketleftbiggˆp2\ni\n2m+1\n2mω2r2\ni−ηˆliˆSi/bracketrightbigg\n+Hqq+Hss (5)\nwith\nHqq=2/summationdisplay\nµ=−2(−1)µ\n\n¯κZ/summationdisplay\niN/summationdisplay\njq2−µ(ri)q2µ(rj)+1\n2κ\nZ/summationdisplay\ni/negationslash=jq2−µ(ri)q2µ(rj)+N/summationdisplay\ni/negationslash=jq2−µ(ri)q2µ(rj)\n\n\n,(6)\nHss=1/summationdisplay\nµ=−1(−1)µ\n\n¯χZ/summationdisplay\niN/summationdisplay\njˆS−µ(i)ˆSµ(j)+1\n2χ\nZ/summationdisplay\ni/negationslash=jˆS−µ(i)ˆSµ(j)+N/summationdisplay\ni/negationslash=jˆS−µ(i)ˆSµ(j)\n\n\nδ(ri−rj),(7)\nwhereNandZare the numbers of neutrons and protons and ˆSµare spin matrices [18]:\nˆS1=−/planckover2pi1√\n2/parenleftbigg0 1\n0 0/parenrightbigg\n,ˆS0=/planckover2pi1\n2/parenleftbigg1 0\n0−1/parenrightbigg\n,ˆS−1=/planckover2pi1√\n2/parenleftbigg0 0\n1 0/parenrightbigg\n. (8)4\nThe quadrupole operator q2µ=/radicalbig\n16π/5r2Y2µ(θ,φ) can be written as the tensor product: q2µ(r) =√\n6{r⊗\nr}2µ,where\n{r⊗r}λµ=/summationdisplay\nσ,νCλµ\n1σ,1νrσrν,\nr−1,r0,r1are cyclic coordinates [18] and Cλµ\n1σ,1νis a Clebsch-Gordan coefficient.\nA. Mean Field\nLet us analyze the mean field generated by this Hamiltonian.\n1. Spin-orbit Potential\nWritten in cyclic coordinates, the spin-orbit part of the Hamiltonian r eads\nˆhls=−η1/summationdisplay\nµ=−1(−)µˆlµˆS−µ=−η/parenleftbiggˆl0/planckover2pi1\n2ˆl−1/planckover2pi1√\n2\n−ˆl1/planckover2pi1√\n2−ˆl0/planckover2pi1\n2/parenrightbigg\n,\nwhere [18]\nˆlµ=−/planckover2pi1√\n2/summationdisplay\nν,αC1µ\n1ν,1αrν∇α (9)\nand\nˆl1=/planckover2pi1(r0∇1−r1∇0) =−1√\n2(ˆlx+iˆly),ˆl0=/planckover2pi1(r−1∇1−r1∇−1) =ˆlz,\nˆl−1=/planckover2pi1(r−1∇0−r0∇−1) =1√\n2(ˆlx−iˆly),\nˆlx=−i/planckover2pi1(y∇z−z∇y),ˆly=−i/planckover2pi1(z∇x−x∇z),ˆlz=−i/planckover2pi1(x∇y−y∇x). (10)\nMatrix elements of ˆhlsin coordinate space can be obviously written as\n<r1,s1|ˆhls|r2,s2>=−/planckover2pi1\n2η(r1){ˆl0(r1)[δs1↑δs2↑−δs1↓δs2↓]\n+√\n2ˆl−1(r1)δs1↑δs2↓−√\n2ˆl1(r1)δs1↓δs2↑}δ(r1−r2). (11)\nThe Wigner transform of (11) reads [15]:\nhs1s2\nls(r,p) =−/planckover2pi1\n2η{l0(r,p)[δs1↑δs2↑−δs1↓δs2↓]+√\n2l−1(r,p)δs1↑δs2↓−√\n2l1(r,p)δs1↓δs2↑},(12)\nwherelµ=−i√\n2/summationtext\nν,αC1µ\n1ν,1αrνpα.\n2. q-q interaction\nThe contribution of Hqqto the mean field potential is easily found by replacing one of the q2µoperators\nby the average value. We have\nVτ\nqq= 6/summationdisplay\nµ(−1)µZτ+\n2−µ{r⊗r}2µ. (13)5\nHere\nZn+\n2µ=κRn+\n2µ+ ¯κRp+\n2µ, Zp+\n2µ=κRp+\n2µ+ ¯κRn+\n2µ, Rτ+\nλµ(t) =/integraldisplay\nd(p,r){r⊗r}λµfτ+(r,p,t) (14)\nwith/integraltext\nd(p,r)≡(2π/planckover2pi1)−3/integraltext\nd3p/integraltext\nd3randτbeing the isospin index.\n3. Spin-spin interaction\nThe analogous expression for Hssis found in the standard way, with the Hartree-Fock contribution g iven\n[16] by:\nΓkk′(t) =/summationdisplay\nll′¯vkl′k′lρll′(t), (15)\nwhere ¯vkl′k′lis the antisymmetrized matrix element of the two body interaction v(1,2). Identifying the\nindicesk,k′,l,l′with the set of coordinates ( r,s,τ), i.e. (position, spin, isospin), one rewrites (15) as\nVHF(r1,s1,τ1;r′\n1,s′\n1,τ′\n1;t) =/integraldisplay\nd3r2/integraldisplay\nd3r′\n2/summationdisplay\ns2,s′\n2/summationdisplay\nτ2,τ′\n2\n<r1,s1,τ1;r2,s2,τ2|ˆv|r′\n1,s′\n1,τ′\n1;r′\n2,s′\n2,τ′\n2>a.s.ρ(r′\n2,s′\n2,τ′\n2;r2,s2,τ2;t).\nLet us consider the neutron-proton part of the spin-spin interac tion. In this case\nˆv=v(ˆr1−ˆr2)1/summationdisplay\nµ=−1(−1)µˆS−µ(1)ˆSµ(2)δτ1pδτ2n,\nwhereˆr1is the position operator: ˆr1|r1>=r1|r1>, <r1|ˆr1|r′\n1>=<r1|r′\n1>r′\n1=δ(r1−r′\n1)r′\n1.\nFor the Hartree term one finds:\n<r1,s1,τ1;r2,s2,τ2|ˆv|r′\n1,s′\n1,τ′\n1;r′\n2,s′\n2,τ′\n2>=δ(r1−r′\n1)δ(r2−r′\n2)v(r′\n1−r′\n2)\n1/summationdisplay\nµ=−1(−1)µ< s1,τ1;s2,τ2|ˆS−µ(1)ˆSµ(2)δτ1pδτ2n|s′\n1,τ′\n1;s′\n2,τ′\n2>,\nVH(r1,s1,τ1;r′\n1,s′\n1,τ′\n1;t) =/integraldisplay\nd3r2/integraldisplay\nd3r′\n2/summationdisplay\ns2,s′\n2/summationdisplay\nτ2,τ′\n2\n<r1,s1,τ1;r2,s2,τ2|ˆv|r′\n1,s′\n1,τ′\n1;r′\n2,s′\n2,τ′\n2> ρ(r′\n2,s′\n2,τ′\n2;r2,s2,τ2;t)\n=δτ1pδτ′\n1p/summationdisplay\ns2,s′\n21/summationdisplay\nµ=−1(−1)µ< s1|ˆS−µ(1)|s′\n1>< s2|ˆSµ(2)|s′\n2>\nδ(r1−r′\n1)/integraldisplay\nd3r2v(r1−r2)ρ(r2,s′\n2,n;r2,s2,n;t).\nThe Fock term reads:\n<r1,s1,τ1;r2,s2,τ2|ˆv|r′\n2,s′\n2,τ′\n2;r′\n1,s′\n1,τ′\n1>=δ(r1−r′\n2)δ(r2−r′\n1)v(r′\n2−r′\n1)\n1/summationdisplay\nµ=−1(−1)µ< s1,τ1;s2,τ2|ˆS−µ(1)ˆSµ(2)δτ1pδτ2n|s′\n2,τ′\n2;s′\n1,τ′\n1>,6\nVF(r1,s1,τ1;r′\n1,s′\n1,τ′\n1;t) =−/integraldisplay\nd3r2/integraldisplay\nd3r′\n2/summationdisplay\ns2,s′\n2/summationdisplay\nτ2,τ′\n2\n<r1,s1,τ1;r2,s2,τ2|ˆv|r′\n2,s′\n2,τ′\n2;r′\n1,s′\n1,τ′\n1> ρ(r′\n2,s′\n2,τ′\n2;r2,s2,τ2;t)\n=−δτ1pδτ′\n1n/summationdisplay\ns2,s′\n21/summationdisplay\nµ=−1(−1)µ< s1|ˆS−µ(1)|s′\n2>< s2|ˆSµ(2)|s′\n1>\nv(r1−r′\n1)ρ(r1,s′\n2,p;r′\n1,s2,n;t).\nTaking into account the relations\n< s|ˆS−1|s′>=/planckover2pi1√\n2δs↓δs′↑, < s |ˆS0|s′>=/planckover2pi1\n2δs,s′(δs↑−δs↓), < s |ˆS1|s′>=−/planckover2pi1√\n2δs↑δs′↓\nandv(r−r′) = ¯χδ(r−r′) one finds for the mean field generated by the proton-neutron pa rt ofHss:\nΓpn(r,s,τ;r′,s′,τ′;t) = ¯χ/planckover2pi12\n4/braceleftBigg\nδτpδτ′p/bracketleftBig\nδs↓δs′↑ρ(r,↓,n;r′,↑,n;t)+δs↑δs′↓ρ(r,↑,n;r′,↓,n;t)/bracketrightBig\n−δτpδτ′n/bracketleftBig\nδs↓δs′↓ρ(r,↑,p;r′,↑,n;t)+δs↑δs′↑ρ(r,↓,p;r′,↓,n;t)/bracketrightBig\n+1\n2δτpδτ′p(δs↑δs′↑−δs↓δs′↓)/bracketleftBig\nρ(r,↑,n;r′,↑,n;t)−ρ(r,↓,n;r′,↓,n;t)/bracketrightBig\n+1\n2δτpδτ′n/bracketleftBig\nδs↑δs′↓ρ(r,↑,p;r′,↓,n;t)+δs↓δs′↑ρ(r,↓,p;r′,↑,n;t)\n−δs↑δs′↑ρ(r,↑,p;r′,↑,n;t)−δs↓δs′↓ρ(r,↓,p;r′,↓,n;t)/bracketrightBig/bracerightBigg\nδ(r−r′)+ ¯χ/planckover2pi12\n4/braceleftBigg\np↔n/bracerightBigg\nδ(r−r′).(16)\nThe expression for the mean field Γ pp(r,s,τ;r′,s′,τ′;t) generated by the proton-proton part of Hsscan be\nobtained from (16) by replacing index nbypand the strength constant ¯ χbyχ. The proton mean field is\ndefined as the sum of these two terms with τ=τ′=p. Its Wigner transform can be written as\nVss′\np(r,t) = 3χ/planckover2pi12\n8/braceleftbig\nδs↓δs′↑n↓↑\np+δs↑δs′↓n↑↓\np−δs↓δs′↓n↑↑\np−δs↑δs′↑n↓↓\np/bracerightbig\n+ ¯χ/planckover2pi12\n8/braceleftbig\n2δs↓δs′↑n↓↑\nn+2δs↑δs′↓n↑↓\nn+(δs↑δs′↑−δs↓δs′↓)(n↑↑\nn−n↓↓\nn)/bracerightbig\n, (17)\nwherenss′\nτ(r,t) =/integraldisplayd3p\n(2π/planckover2pi1)3fss′\nτ(r,p,t). The Wigner transform of the neutron mean field Vss′\nnis obtained\nfrom (17) by the obviouschangeof indices p↔n. The Wigner function fand density matrix ρare connected\nby the relation fss′\nττ′(r,p,t) =/integraldisplay\nd3qe−ipq//planckover2pi1ρ(r1,s,τ;r2,s′,τ′;t), withq=r1−r2andr=1\n2(r1+r2).\nIntegrating this relation over pwithτ′=τone finds:\nnss′\nτ(r,t) =ρ(r,s,τ;r,s′,τ;t).\nBy definition the diagonal elements of the density matrix describe th e proper densities. Therefore nss\nτ(r,t)\nis the density of spin-up nucleons (if s=↑) or spin-down nucleons (if s=↓). Off diagonal in spin elements of\nthe density matrix nss′\nτ(r,t) are spin-flip characteristics and can be called spin-flip densities.\nIV. EQUATIONS OF MOTION\nIntegrating the set of equations (4) over phase space with the we ights\nW={r⊗p}λµ,{r⊗r}λµ,{p⊗p}λµ,and 1 (18)7\none gets dynamic equations for the following collective variables:\nLτς\nλµ(t) =/integraldisplay\nd(p,r){r⊗p}λµfτς(r,p,t), Rτς\nλµ(t) =/integraldisplay\nd(p,r){r⊗r}λµfτς(r,p,t),\nPτς\nλµ(t) =/integraldisplay\nd(p,r){p⊗p}λµfτς(r,p,t), Fτς(t) =/integraldisplay\nd(p,r)fτς(r,p,t), (19)\nwhereς= +,−,↑↓,↓↑.We already called the functions f+=f↑↑+f↓↓andf−=f↑↑−f↓↓spin-scalar and\nspin-vector ones, respectively. It is, therefore, natural to ca ll the corresponding collective variables X+\nλµ(t)\nandX−\nλµ(t) spin-scalar and spin-vector variables. The required expressions forh±,h↑↓andh↓↑are\nh+\nτ=p2\nm+mω2r2+12/summationdisplay\nµ(−1)µZτ+\n2µ(t){r⊗r}2−µ+V+\nτ(r,t),\nh−\nτ=−/planckover2pi1ηl0+V−\nτ(r,t), h↑↓\nτ=−/planckover2pi1√\n2ηl−1+V↑↓\nτ(r,t), h↓↑\nτ=/planckover2pi1√\n2ηl1+V↓↑\nτ(r,t),\nwhere according to (17)\nV+\np(r,t) =−3/planckover2pi12\n8χn+\np(r,t), V−\np(r,t) = 3/planckover2pi12\n8χn−\np(r,t)+/planckover2pi12\n4¯χn−\nn(r,t),\nV↑↓\np(r,t) = 3/planckover2pi12\n8χn↑↓\np(r,t)+/planckover2pi12\n4¯χn↑↓\nn(r,t), V↓↑\np(r,t) = 3/planckover2pi12\n8χn↓↑\np(r,t)+/planckover2pi12\n4¯χn↓↑\nn(r,t) (20)\nand the neutron potentials Vς\nnare obtained by the obvious change of indices p↔n.\nThe integration yields:\n˙L+\nλµ=1\nmP+\nλµ−mω2R+\nλµ+12√\n52/summationdisplay\nj=0/radicalbig\n2j+1/braceleftBig\n11j\n2λ1/bracerightBig\n{Z+\n2⊗R+\nj}λµ\n−i/planckover2pi1η\n2/bracketleftBig\nµL−\nλµ+/radicalbig\n(λ−µ)(λ+µ+1)L↑↓\nλµ+1+/radicalbig\n(λ+µ)(λ−µ+1)L↓↑\nλµ−1/bracketrightBig\n−/integraldisplay\nd3r/bracketleftbigg1\n2n+{r⊗∇}λµV++1\n2n−{r⊗∇}λµV−+n↓↑{r⊗∇}λµV↑↓+n↑↓{r⊗∇}λµV↓↑/bracketrightbigg\n,\n˙L−\nλµ=1\nmP−\nλµ−mω2R−\nλµ+12√\n52/summationdisplay\nj=0/radicalbig\n2j+1/braceleftBig\n11j\n2λ1/bracerightBig\n{Z+\n2⊗R−\nj}λµ−i/planckover2pi1η\n2µL+\nλµ\n−/planckover2pi12\n2ηδλ,1/bracketleftbig\nδµ,−1F↑↓+δµ,1F↓↑/bracketrightbig\n−1\n2/integraldisplay\nd3r/bracketleftbig\nn−{r⊗∇}λµV++n+{r⊗∇}λµV−/bracketrightbig\n−2i\n/planckover2pi1/integraldisplay\nd(p,r){r⊗p}λµ/bracketleftbig\nh↑↓f↓↑−h↓↑f↑↓/bracketrightbig\n,\n˙L↑↓\nλµ+1=1\nmP↑↓\nλµ+1−mω2R↑↓\nλµ+1+12√\n52/summationdisplay\nj=0/radicalbig\n2j+1/braceleftBig\n11j\n2λ1/bracerightBig\n{Z+\n2⊗R↑↓\nj}λµ+1\n−i/planckover2pi1η\n4/radicalbig\n(λ−µ)(λ+µ+1)L+\nλµ+/planckover2pi12\n2ηδλ,1/bracketleftbigg\nδµ,0F−+1√\n2δµ,−1F↑↓/bracketrightbigg\n−1\n2/integraldisplay\nd3r/bracketleftbig\nn↑↓{r⊗∇}λµ+1V++n+{r⊗∇}λµ+1V↑↓/bracketrightbig\n−i\n/planckover2pi1/integraldisplay\nd(p,r){r⊗p}λµ+1/bracketleftbig\nh−f↑↓−h↑↓f−/bracketrightbig\n,\n˙L↓↑\nλµ−1=1\nmP↓↑\nλµ−1−mω2R↓↑\nλµ−1+12√\n52/summationdisplay\nj=0/radicalbig\n2j+1/braceleftBig\n11j\n2λ1/bracerightBig\n{Z+\n2⊗R↓↑\nj}λµ−1\n−i/planckover2pi1η\n4/radicalbig\n(λ+µ)(λ−µ+1)L+\nλµ+/planckover2pi12\n4ηδλ,1/bracketleftBig\nδµ,0F−−√\n2δµ,1F↓↑/bracketrightBig\n−1\n2/integraldisplay\nd3r/bracketleftbig\nn↓↑{r⊗∇}λµ−1V++n+{r⊗∇}λµ−1V↓↑/bracketrightbig\n−i\n/planckover2pi1/integraldisplay\nd(p,r){r⊗p}λµ−1/bracketleftbig\nh↓↑f−−h−f↓↑/bracketrightbig\n,8\n˙F−= 2η/bracketleftBig\nL↓↑\n1−1+L↑↓\n11/bracketrightBig\n,\n˙F↑↓=−η[L−\n1−1−√\n2L↑↓\n10],\n˙F↓↑=−η/bracketleftBig\nL−\n11+√\n2L↓↑\n10/bracketrightBig\n,\n˙R+\nλµ=2\nmL+\nλµ−i/planckover2pi1η\n2/bracketleftBig\nµR−\nλµ+/radicalbig\n(λ−µ)(λ+µ+1)R↑↓\nλµ+1+/radicalbig\n(λ+µ)(λ−µ+1)R↓↑\nλµ−1/bracketrightBig\n,\n˙R−\nλµ=2\nmL−\nλµ−i/planckover2pi1η\n2µR+\nλµ−2i\n/planckover2pi1/integraldisplay\nd(p,r){r⊗r}λµ/bracketleftbig\nh↑↓f↓↑−h↓↑f↑↓/bracketrightbig\n,\n˙R↑↓\nλµ+1=2\nmL↑↓\nλµ+1−i/planckover2pi1η\n4/radicalbig\n(λ−µ)(λ+µ+1)R+\nλµ−i\n/planckover2pi1/integraldisplay\nd(p,r){r⊗r}λµ+1/bracketleftbig\nh−f↑↓−h↑↓f−/bracketrightbig\n,\n˙R↓↑\nλµ−1=2\nmL↓↑\nλµ−1−i/planckover2pi1η\n4/radicalbig\n(λ+µ)(λ−µ+1)R+\nλµ−i\n/planckover2pi1/integraldisplay\nd(p,r){r⊗r}λµ−1/bracketleftbig\nh↓↑f−−h−f↓↑/bracketrightbig\n,\n˙P+\nλµ=−2mω2L+\nλµ+24√\n52/summationdisplay\nj=0/radicalbig\n2j+1/braceleftBig\n11j\n2λ1/bracerightBig\n{Z+\n2⊗L+\nj}λµ\n−i/planckover2pi1η\n2/bracketleftBig\nµP−\nλµ+/radicalbig\n(λ−µ)(λ+µ+1)P↑↓\nλµ+1+/radicalbig\n(λ+µ)(λ−µ+1)P↓↑\nλµ−1/bracketrightBig\n−/integraldisplay\nd3r/bracketleftbig\n{J+⊗∇}λµV++{J−⊗∇}λµV−+2{J↓↑⊗∇}λµV↑↓+2{J↑↓⊗∇}λµV↓↑/bracketrightbig\n,\n˙P−\nλµ=−2mω2L−\nλµ+24√\n52/summationdisplay\nj=0/radicalbig\n2j+1/braceleftBig\n11j\n2λ1/bracerightBig\n{Z+\n2⊗L−\nj}λµ−i/planckover2pi1η\n2µP+\nλµ\n−/integraldisplay\nd3r/bracketleftbig\n{J−⊗∇}λµV++{J+⊗∇}λµV−/bracketrightbig\n−2i\n/planckover2pi1/integraldisplay\nd(p,r){p⊗p}λµ/bracketleftbig\nh↑↓f↓↑−h↓↑f↑↓/bracketrightbig\n,\n˙P↑↓\nλµ+1=−2mω2L↑↓\nλµ+1+24√\n52/summationdisplay\nj=0/radicalbig\n2j+1/braceleftBig\n11j\n2λ1/bracerightBig\n{Z+\n2⊗L↑↓\nj}λµ+1−i/planckover2pi1η\n4/radicalbig\n(λ−µ)(λ+µ+1)P+\nλµ\n−/integraldisplay\nd3r/bracketleftbig\n{J↑↓⊗∇}λµ+1V++{J+⊗∇}λµ+1V↑↓/bracketrightbig\n−i\n/planckover2pi1/integraldisplay\nd(p,r){p⊗p}λµ+1[h−f↑↓−h↑↓f−],\n˙P↓↑\nλµ−1=−2mω2L↓↑\nλµ−1+24√\n52/summationdisplay\nj=0/radicalbig\n2j+1/braceleftBig\n11j\n2λ1/bracerightBig\n{Z+\n2⊗L↓↑\nj}λµ−1−i/planckover2pi1η\n4/radicalbig\n(λ+µ)(λ−µ+1)P+\nλµ\n−/integraldisplay\nd3r/bracketleftbig\n{J↓↑⊗∇}λµ−1V++{J+⊗∇}λµ−1V↓↑/bracketrightbig\n−i\n/planckover2pi1/integraldisplay\nd(p,r){p⊗p}λµ−1/bracketleftbig\nh↓↑f−−h−f↓↑/bracketrightbig\n, (21)\nwhere/braceleftBig\n11j\n2λ1/bracerightBig\nis the Wigner 6 j-symbol and Jς\nν(r,t) =/integraldisplayd3p\n(2π/planckover2pi1)3pνfς(r,p,t) is the current. For the sake\nof simplicity the time dependence of tensors is not written out. It is e asy to see that equations (21)\nare nonlinear due to quadrupole-quadrupole and spin-spin interact ions. We will solve them in the small\namplitude approximation, by linearizing the equations. This procedur e helps also to solve another problem:\nto represent the integral terms in (21) as the linear combination of collective variables (19), that allows to\nclose the whole set of equations (21). The detailed analysis of the int egral terms is given in the appendix A.\nWe are interested in the scissors mode with quantum number Kπ= 1+. Therefore, we only need the part\nof dynamic equations with µ= 1.\nA. Linearized equations ( µ= 1), isovector, isoscalar\nWriting all variables as a sum of their equilibrium value plus a small deviatio n\nRλµ(t) =Req\nλµ+Rλµ(t), Pλµ(t) =Peq\nλµ+Pλµ(t), Lλµ(t) =Leq\nλµ+Lλµ(t)9\nand neglecting quadratic deviations, one obtains the linearized equa tions. Naturally one needs to know the\nequilibrium values of all variables. Evident equilibrium conditions for an a xially symmetric nucleus are:\nR+\n2±1(eq) =R+\n2±2(eq) = 0, R+\n20(eq)/ne}ationslash= 0. (22)\nIt is obvious that all ground state properties of the system of spin up nucleons are identical to the ones of\nthe system of nucleons with spin down. Therefore\nR−\nλµ(eq) =P−\nλµ(eq) =L−\nλµ(eq) = 0. (23)\nWe also will suppose\nL+\nλµ(eq) =L↑↓\nλµ(eq) =L↓↑\nλµ(eq) = 0 and R↑↓\nλµ(eq) =R↓↑\nλµ(eq) = 0. (24)\nLet us recall that all variables and equilibrium quantities R+\nλ0(eq) andZ+\n20(eq) in (21) have isospin indices\nτ=n, p. All the difference between neutron and proton systems is contain ed in the mean field quantities\nZτ+\n20(eq) andVς\nτ, which are different for neutrons and protons (see eq. (14) and ( 20)).\nIt is convenient to rewrite the dynamical equations in terms of isove ctor and isoscalar variables\nRλµ=Rn\nλµ+Rp\nλµ, Pλµ=Pn\nλµ+Pp\nλµ, Lλµ=Ln\nλµ+Lp\nλµ,\n¯Rλµ=Rn\nλµ−Rp\nλµ,¯Pλµ=Pn\nλµ−Pp\nλµ,¯Lλµ=Ln\nλµ−Lp\nλµ.\nIt also is natural to define isovector and isoscalar strength const antsκ1=1\n2(κ−¯κ) andκ0=1\n2(κ+ ¯κ)\nconnected by the relation κ1=ακ0[19]. Then the equations for the neutron and proton systems are\ntransformed into isovector and isoscalar ones. Supposing that all equilibrium characteristics of the proton\nsystem are equal to that of the neutron system one decouples iso vector and isoscalar equations. This\napproximations looks rather crude, nevertheless the possible cor rections to it are very small, being of the\norder (N−Z\nA)2. With the help of the above equilibrium relations one arrives at the follo wing final set of\nequations for the isovector system:\n˙¯L+\n21=1\nm¯P+\n21−/bracketleftBig\nmω2−4√\n3ακ0Req\n00+√\n6(1+α)κ0Req\n20/bracketrightBig\n¯R+\n21−i/planckover2pi1η\n2/bracketleftBig\n¯L−\n21+2¯L↑↓\n22+√\n6¯L↓↑\n20/bracketrightBig\n,\n˙¯L−\n21=1\nm¯P−\n21−/bracketleftBigg\nmω2+√\n6κ0Req\n20−√\n3\n20/planckover2pi12/parenleftBig\nχ−¯χ\n3/parenrightBig/parenleftbiggI1\na2\n0+I1\na2\n1/parenrightbigg/parenleftbigga2\n1\nA2−a2\n0\nA1/parenrightbigg/bracketrightBigg\n¯R−\n21−i/planckover2pi1η\n2¯L+\n21,\n˙¯L↑↓\n22=1\nm¯P↑↓\n22−/bracketleftBigg\nmω2−2√\n6κ0Req\n20−√\n3\n5/planckover2pi12/parenleftBig\nχ−¯χ\n3/parenrightBigI1\nA2/bracketrightBigg\n¯R↑↓\n22−i/planckover2pi1η\n2¯L+\n21,\n˙¯L↓↑\n20=1\nm¯P↓↑\n20−/bracketleftBig\nmω2+2√\n6κ0Req\n20/bracketrightBig\n¯R↓↑\n20+2√\n3κ0Req\n20¯R↓↑\n00−i/planckover2pi1η\n2/radicalbigg\n3\n2¯L+\n21\n+√\n3\n15/planckover2pi12/parenleftBig\nχ−¯χ\n3/parenrightBig\nI1(A1−2A2)¯R↓↑\n20+√\n2(A1+A2)¯R↓↑\n00\nA1A2,\n˙¯L+\n11=−3√\n6(1−α)κ0Req\n20¯R+\n21−i/planckover2pi1η\n2/bracketleftBig\n¯L−\n11+√\n2¯L↓↑\n10/bracketrightBig\n,\n˙¯L−\n11=−/bracketleftBigg\n3√\n6κ0Req\n20−√\n3\n20/planckover2pi12/parenleftBig\nχ−¯χ\n3/parenrightBig/parenleftbiggI1\na2\n0−I1\na2\n1/parenrightbigg/parenleftbigga2\n1\nA2−a2\n0\nA1/parenrightbigg/bracketrightBigg\n¯R−\n21−/planckover2pi1η\n2/bracketleftbig\ni¯L+\n11+/planckover2pi1¯F↓↑/bracketrightbig\n,\n˙¯L↓↑\n10=−/planckover2pi1η\n2√\n2/bracketleftbig\ni¯L+\n11+/planckover2pi1¯F↓↑/bracketrightbig\n,\n˙¯F↓↑=−η/bracketleftBig\n¯L−\n11+√\n2¯L↓↑\n10/bracketrightBig\n,10\n˙¯R+\n21=2\nm¯L+\n21−i/planckover2pi1η\n2/bracketleftBig\n¯R−\n21+2¯R↑↓\n22+√\n6¯R↓↑\n20/bracketrightBig\n,\n˙¯R−\n21=2\nm¯L−\n21−i/planckover2pi1η\n2¯R+\n21,\n˙¯R↑↓\n22=2\nm¯L↑↓\n22−i/planckover2pi1η\n2¯R+\n21,\n˙¯R↓↑\n20=2\nm¯L↓↑\n20−i/planckover2pi1η\n2/radicalbigg\n3\n2¯R+\n21,\n˙¯P+\n21=−2/bracketleftBig\nmω2+√\n6κ0Req\n20/bracketrightBig\n¯L+\n21+6√\n6κ0Req\n20¯L+\n11−i/planckover2pi1η\n2/bracketleftBig\n¯P−\n21+2¯P↑↓\n22+√\n6¯P↓↑\n20/bracketrightBig\n+3√\n3\n4/planckover2pi12χI2\nA1A2/bracketleftbig\n(A1−A2)¯L+\n21+(A1+A2)¯L+\n11/bracketrightbig\n,\n˙¯P−\n21=−2/bracketleftBig\nmω2+√\n6κ0Req\n20/bracketrightBig\n¯L−\n21+6√\n6κ0Req\n20¯L−\n11−i/planckover2pi1η\n2¯P+\n21\n+3√\n3\n4/planckover2pi12χI2\nA1A2/bracketleftbig\n(A1−A2)¯L−\n21+(A1+A2)¯L−\n11/bracketrightbig\n,\n˙¯P↑↓\n22=−/bracketleftBigg\n2mω2−4√\n6κ0Req\n20−3√\n3\n2/planckover2pi12χI2\nA2/bracketrightBigg\n¯L↑↓\n22−i/planckover2pi1η\n2¯P+\n21,\n˙¯P↓↑\n20=−/bracketleftBig\n2mω2+4√\n6κ0Req\n20/bracketrightBig\n¯L↓↑\n20+8√\n3κ0Req\n20¯L↓↑\n00−i/planckover2pi1η\n2/radicalbigg\n3\n2¯P+\n21\n+√\n3\n2/planckover2pi12χI2\nA1A2/bracketleftBig\n(A1−2A2)¯L↓↑\n20+√\n2(A1+A2)¯L↓↑\n00/bracketrightBig\n,\n˙¯L↓↑\n00=1\nm¯P↓↑\n00−mω2¯R↓↑\n00+4√\n3κ0Req\n20¯R↓↑\n20\n+1\n2√\n3/planckover2pi12/bracketleftbigg/parenleftBig\nχ−¯χ\n3/parenrightBig\nI1−9\n4χI2/bracketrightbigg(2A1−A2)¯R↓↑\n00+√\n2(A1+A2)¯R↓↑\n20\nA1A2,\n˙¯R↓↑\n00=2\nm¯L↓↑\n00, (25)\n˙¯P↓↑\n00=−2mω2¯L↓↑\n00+8√\n3κ0Req\n20¯L↓↑\n20+√\n3\n2/planckover2pi12χI2/bracketleftbigg/parenleftbigg2\nA2−1\nA1/parenrightbigg\n¯L↓↑\n00+√\n2/parenleftbigg1\nA2+1\nA1/parenrightbigg\n¯L↓↑\n20/bracketrightbigg\n,\nwhereA1,A2are defined in appendix B, κ0=−m¯ω2/(4Q00) [20] with ¯ ω2=ω2//parenleftbig\n1+2\n3δ/parenrightbig\n,a−1=a1=\nR0/parenleftbigg1−(2/3)δ\n1+(4/3)δ/parenrightbigg1/6\nanda0=R0/parenleftbigg1−(2/3)δ\n1+(4/3)δ/parenrightbigg−1/3\nare semiaxes of ellipsoid by which the shape of nucleus\nis approximated, δ– deformation parameter, R0= 1.2A1/3fm.\nI1=π\n4+∞/integraldisplay\n−∞drr4/parenleftbigg∂n+(r)\n∂r/parenrightbigg2\n, I2=π\n4+∞/integraldisplay\n−∞drr2n+(r)2, n+(r) =n+\np+n+\nn=n0\n1+er−R0\na.\nThe isoscalar set of equations is easily obtained from (25) by taking α= 1 and replacing ¯ χ→ −¯χ.\nV. DISCUSSION AND INTERPRETATION OF THE RESULTS\nTheenergiesandexcitationprobabilitiesobtainedbythesolutionoft heisovector setofequations(25) are\ngiven in Table I. The used spin-spin interaction is repulsive, the values of its strength constants being taken\nfrom the paper [21], where the notation χ=Ks/A,¯χ=qχwas introduced. The results without spin-spin\ninteraction (variant I) are compared with those performed with tw o sets of constants Ks, q(variants II, III).\nThe first set of constants (variant II) was extracted by the aut hors of [21] from Skyrme forces following the\nstandard procedure, the residual interaction being defined in ter ms of second derivatives of the Hamiltonian\ndensityH(ρ) with respect to the one-body densities ρ. Different variants of Skyrme forces produce different\nstrength constants of spin-spin interaction. The most consisten t results are obtained with SG1, SG2 [22]11\nTABLE I: Isovector energies and excitation probabilities o f164Er. Deformation parameter δ= 0.25, spin-orbit\nconstant η= 0.36 MeV. Spin-spin interaction constants are: I – Ks= 0 MeV; II – Ks= 92 MeV, q=−0.8;\nIII –Ks= 200 MeV, q=−0.5. Quantum numbers (including indices ς= +,−,↑↓,↓↑) of variables responsible\nfor the generation of the present level are shown in the first c olumn. For example: (1 ,1)−– spin scissors, (1 ,1)+–\nconventional scissors, etc..\n(λ,µ)ςEiv, MeV B(M1), µ2\nN B(E2), BW\nI II III I II III I II III\n(1,1)−1.61 2.02 2.34 3.54 5.44 7.91 0.12 0.36 0.82\n(1,1)+2.18 2.45 2.76 5.33 4.48 2.98 1.02 1.23 1.26\n(0,0)↓↑12.80 16.81 20.02 0.01 0.01 0.04 0.04 0.13 0.52\n(2,1)−14.50 18.52 21.90 0.01 0.02 0.34 0.03 0.13 4.29\n(2,2)↑↓16.18 20.61 24.56 0.02 0.23 0.03 0.18 3.09 0.44\n(2,0)↓↑16.20 22.65 27.67 0 0.03 0 0 0.39 0.02\n(2,1)+20.59 21.49 22.42 2.78 2.19 1.77 35.45 30.47 27.43\n(1,0)↓↑0.26i 0.26i 0.26i -5.4i -5.4i -5.4i 0i 0i 0i\nand Sk3 [23] forces. We use here the spin-spin constants extract ed from Sk3 force. Another set of constants\n(variant III) was also found by the authors of [21] phenomenologic ally in the calculations with a Woods-\nSaxon potential, when there is not any self-consistency between t he mean field and the residual interaction.\nWe tentatively will use it, because in our case also there is no self-con sistency. The strength of the spin-orbit\ninteraction is taken from [24].\nOne can see from Table I that the spin-spin interaction does not cha nge the qualitative picture of the\npositions of the excitations described in [15]. It pushes all levels up pr oportionally to its strength (20-\n30% in the case II and 40-60% in the case III) without changing their order. The most interesting result\nconcerns the relative B(M1) values of the two low lying scissors mode s, namely the spin scissors (1 ,1)−\nand the conventional (orbital) scissors (1 ,1)+mode. As can be noticed, the spin-spin interaction strongly\nredistributes M1 strength in the favour of the spin scissors mode. We tentatively want to link this fact to\nthe recent experimental finding in isotopes of Th and Pa [17]. The au thors have studied deuteron and3He-\ninduced reactions on232Th and found in the residual nuclei231,232,233Th and232,233Pa ”an unexpectedly\nstrong integrated strength of B(M1) = 11 −15µ2\nNin theEγ= 1.0−3.5 MeV region”. The B(M1)\nforce in most nuclei shows evident splitting into two Lorentzians. ”T ypically, the experimental splitting\nis ∆ωM1∼0.7 MeV, and the ratio of the strengths between the lower and upper resonance components\nisBL/BU∼2”. (Note a misprint in that paper: it is written erroneously B2/B1∼2 whereas it should\nbeB1/B2∼2. To avoid misunderstanding, we write here BLinstead of B1andBUinstead of B2.) The\nauthors have tried to explain the splitting by a γ-deformation. To describe the observed value of ∆ ωM1\nthe deformation γ∼15◦is required, that leads to the ratio BL/BU∼0.7 in an obvious contradiction with\nexperiment. The authors conclude that ”the splitting may be due to other mechanisms”. In this sense,\nwe tentatively may argue as follows. On one side, theory [25] and ex periment [26] give zero value of γ-\ndeformation for233Th. On the other side, it is easy to see that our theory suggests th e required mechanism.\nThe calculations performed for233Th give ∆ ωM1∼0.32 MeV and BL/BU∼1.6 for the first variant of the\nspin-spin interaction and ∆ ωM1∼0.28 MeV and BL/BU∼4.1 for second one in reasonable agreement with\nexperimental values. The inclusion of pair correlations will affect our results, but one may speculate that\nthe agreement between the theory and experiment will be conserv ed at least qualitatively.\nThe energies and excitation probabilities obtained by the solution of t heisoscalar set of equations (25)\nare displayed in the Table II. The general picture of the influence of the spin-spin interaction here is quite\nclose to that observed in the isovector case. The only difference is t he low lying mode marked by (1 ,1)+\nwhich is practically insensitive to the spin-spin interaction. The negligib ly small negative B(M1) value of\nspin scissors appears undoubtedly due to the lack of the self consis tency in our calculations. In ref [17]\nthe assignment of the resonances to be of isovector type is only te ntative based on the assumption that at\nsuch low energies there is no collective mode other than the isovecto r scissors mode. However, from [17]\none cannot exclude that also an isoscalar spin scissors mode is mixed in . From our analysis we see that the\nisoscalar spin scissors where all nucleons with spin up counter-rota te with respect the ones of spin up comes\nmore or less at the same energy as the isovector scissors. So it wou ld be very important for the future to\npin down precisely the quantum numbers of the resonances.12\nTABLE II: The same as in Table I, but for isoscalar excitation s.\n(λ,µ)ςEis, MeV B(M1), µ2\nN B(E2), BW\nI II III I II III I II III\n(1,1)−1.73 2.04 2.40 -0.07 -0.05 0 1.12 0.65 0.39\n(1,1)+0.39 0.37 0.37 0.24 0.24 0.24 117.2 117.9 118.3\n(0,0)↓↑12.83 15.59 18.72 0 0 0 0.66 0.31 0.15\n(2,1)−14.51 17.40 20.65 0 0 0 0.12 0.06 0.03\n(2,2)↑↓16.20 19.43 23.09 0 0 0 0 0.07 0.04\n(2,0)↓↑16.22 20.09 24.80 0 0 0 0.20 0.02 0.01\n(2,1)+10.28 11.92 13.60 0 0 0 66.50 57.78 50.87\n(1,0)↓↑0.20i 0.20i 0.20i -0.1i -0.1i -0.1i 30.0i 29.8i 30.3i\nLet us discuss in more detail the nature of the\npredicted excitations. As one sees, the generaliza-\ntion of the WFM method by including spin dynam-\nics allowed one to reveal a variety of new types of\nnuclear collective motion involving spin degrees of\nfreedom. Two isovector and two isoscalar low lying\neigenfrequencies and five isovector and five isoscalar\nhigh lying eigenfrequencies have been found.\nThreelowlyinglevelscorrespondtothe excitation\nof new types of modes. For example the isovector\nlevel marked by (1 ,1)−describes rotational oscilla-\ntions of nucleons with the spin projection ”up” with\nrespect of nucleons with the spin projection ”down”,\ni.e. one can talk of a nuclear spin scissors mode.\nHaving in mind that this excitation is an isovec-\ntor one, we can see that the resulting motion looks\nrathercomplex–protonspinscissorscounter-rotates\nwith respect to the neutron spin scissors. Thus the\nexperimentallyobservedgroupof1+peaks in the in-\nterval 2-4 MeV, associated usually with the nuclear\nscissors mode, in reality consists of the excitations\nof the ”spin” scissors mode together with the con-\nventional [1] scissors mode (the level (1 ,1)+in our\ncase). The isoscalar level (1 ,1)−describes the real\nspin scissors mode: all spin up nucleons (protons\ntogether with neutrons) oscillate rotationally out of\nphase with all spin down nucleons.\nSuch excitations were, undoubtedly, produced im-\nplicitly by other methods (e.g. RPA [1, 2, 21, 27]),\nbut they never were analysed in such terms. It is\ninteresting to note, for example, that in [2] the scis-\nsors mode was analyzed in so-called spin and or-\nbital components. Roughly speaking there are two\ngroups of states corresponding to these two types of\ncomponents, not completely dissimilar to our find-\ning. Whereas the nature of the orbital, i.e. con-\nventional scissors is quite clear, the authors did not\nanalyze the character of their states which consist\nof the spin component. It can be speculated that\nthose spin components just correspondto the isovec-\ntor spin scissors mode discussed in our work here. It\nwould be interesting to study whether our sugges-tion is correct or not. This could for example be\ndone in analyzing the current patterns.\nOne more new low lying mode (isoscalar, marked\nby (1,1)+) is generated by the relative motion of the\norbital angular momentum and spin of the nucleus\n(theycanchangetheirabsolutevaluesanddirections\nkeeping the total spin unchanged).\nIn order to complete the picture of the low-lying\nstates, it is important to discuss the state which is\nslightly imaginary. Let us first state that the nature\nof this state has nothing to do with neither spin scis-\nsorsnorwith conventionalscissors. It cannamelybe\nseen from the structure of our equations that this\nstate corresponds to a spin flip induced by the spin-\norbit potential. Such a state is of purely quantal\ncharacter and it cannot be hoped that we can accu-\nrately describe it with our WFM approach restrict-\ning the considerationby second ordermoments only.\nFor its correct treatment, we certainly should con-\nsider higher moments like fourth order moments, for\ninstance. The spin-orbit potential is the only term\nin our theory which couples the second order mo-\nments to the fourth order ones. As mentioned, we\ndecoupled the system in neglecting the fourth or-\nder moments. Therefore, it is no surprise that this\nparticular spin flip mode is not well described. Nev-\nertheless, one may try to better understand the ori-\ngin of this mode almost at zero energy. For this,\nwe make the following approximation of our diag-\nonalisation procedure to get the eight eigenvalues\nlisted in Table I. We neglect in (25) all couplings be-\ntween the set of variables X+\nλµ,X−\nλµand the set of\nvariables X↑↓\nλµ,X↓↑\nλµ. To this end in the dynamical\nequations for X+\nλµ,X−\nλµwe omit all terms contain-\ningX↑↓\nλµ,X↓↑\nλµand in the dynamical equations for\nX↑↓\nλµ,X↓↑\nλµwe omit all terms containing X+\nλµ,X−\nλµ.\nIn such a way we get two independent sets of dy-\nnamical equations. The first one (for X+\nλµ,X−\nλµ) was\nalready studied in [15], where we have found that\nsuch approximation gives satisfactory (in compari-\nsonwith theexactsolution)resultsbutmustbe used13\ncautiously because of the problems with the angular\nmomentum conservation. The second set of equa-\ntions (for X↑↓\nλµ,X↓↑\nλµ) splits into three independent\nsubsets. Two of them were already analyzed in [15]\n(it turns out that these subsets can be obtained also\nin the limit η→0, which was studied there), where\nit was shown that the results of approximate calcu-\nlations are very close to that of exact calculations,\ni.e. the coupling between the respective variables\nX↑↓\nλµ,X↓↑\nλµandX+\nλµ,X−\nλµis very weak. The only new\nsubset of equations reads:\n˙¯L↓↑\n10=−/planckover2pi12η\n2√\n2¯F↓↑,\n˙¯F↓↑=−η√\n2¯L↓↑\n10. (26)\nThe solution of these equations is\nE=i/planckover2pi1√\n2η=i0.255 what practically coincides\nwith the number of the full diagonalisation. So the\nnon-zero (purely imaginary) value of this root only\ncomes from the fact that z-component of orbital\nangular momentum is not conserved (only total\nspin J is conserved). However, the violation of the\nconservation of orbital angular momentum is very\nsmall as can be seen from the numbers. In any\ncase, we see that this spin flip state has nothing\nto do with neither the spin scissors nor with the\nconventional scissors.\nTwo high lying excitations of a new nature are\nfound. They aremarkedby(2 ,1)−andfollowingthe\npaper[27]canbecalledspin-vectorgiantquadrupole\nresonances. Theisovectoronecorrespondstothefol-\nlowing quadrupole motion: the proton system oscil-\nlates out of phase with the neutron system, whereas\ninside of each system spin up nucleons oscillate out\nof phase with spin down nucleons. The respective\nisoscalar resonance describes out of phase oscilla-\ntions of all spin up nucleons (protons together with\nneutrons) with respect of all spin down nucleons.\nSix high lying modes can be interpreted as\nspin-flip giant monopole (marked by (0 ,0)↓↑) and\nquadrupole (marked by (2 ,0)↓↑and (2,2)↑↓) reso-\nnances.\nIt is a pertinent place to make following citation\nfrom the review by F. Osterfeld [27]: ”Similar os-\ncillations to those in isospin space are also possible\nin spin space. Nucleons with spin up and spin down\nmay move either in phase (spin-scalar S=0 modes)\nor out of phase (spin-vector S=1 modes). The latter\nclass of states is also referred to as spin excitations\nor spin-flip transitions.” On account of our results\nin this work, the latter statement that all spin ex-\ncitations are of spin-flip nature should be modified.\nWe predict in this paper the existence of spin ex-\ncitations of non spin-flip nature – the isovector and\nisoscalarspin scissorsand the isovectorand isoscalarspin-vector GQR!\nVI. CONCLUDING REMARKS\nThe inclusion of spin-spin interaction does not\nchangequalitativelythe pictureconcerningthespec-\ntrum of the spin modes found in [15]. It pushes all\nlevels up without changing their order. However,\nit strongly redistributes M1 strength between the\nconventional and spin scissors mode in the favour\nof the last one. Our calculations did not fully con-\nfirm the expectations mentioned in the introduction,\nnamelythat essentiallyonlythe lowlyingpart ofthe\nspectrumwillbestronglyinfluencedbythespin-spin\nforce. Nevertheless our results turned out to be very\nuseful, because they demonstrate that the common\neffect of spin-spin interaction and pair correlations\nare able to push a substantial part of the M1 force\nout of the area of the conventional scissors mode\nwhat is required for the reasonable agreement with\nexperimental data.\nIn this respect, we should mention that we did\nnot include pairing in this work. Inclusion of pair-\ning would have complicated the formalism quite a\nbit. It shall be worked out in the future. We here\nwanted to study the features of spin dynamics in a\nmost transparent way staying, however, somewhat\non the qualitative side. That is why we did not try\nto discuss in detail possible relations with experi-\nment or to compare with the results of other the-\nories. Nevertheless we mentioned the quite recent\nexperimental work [17], where for the two low ly-\ning magnetic states a stronger B(M1) transition for\nthe lower state with respect to the higher one was\nfound. A tentative explanation in terms of a slight\ntriaxialdeformation in [17] failed. However, our the-\nory can naturally predict such a scenario with a non\nvanishing spin-spin force. It would indeed be very\nexciting, if the results of [17] had already discovered\nthe isovector spin scissors mode. However, much\ndeeper experimental and theoretical results must be\nobtained before a firm conclusion on this point is\npossible.\nIn the light of the above results, the study of spin\nexcitationswith pairingincluded, will be the natural\ncontinuation of this work. Pairing is important for a\nquantitative description of the conventional scissors\nmode. The same is expected for the novel spin scis-\nsors mode discussed here. The effect of pairing gen-\nerally is to push up the spectrum in energy. There-\nfore, as just mentioned, it can be expected that the\nresults come into better agreement with experiment.14\nAcknowledgments\nThe fruitful discussions with D. Pena Arteaga,\nNguen Van Giai, V. O. Nesterenko, A. I. Vdovin\nand J. N. Wilson are gratefully acknowledged.\nAppendix A:\nAll derivations of this section will be done in the\napproximationof spherical symmetry. The inclusion\nof deformation makes the calculations more cumber-\nsome without changing the final conclusions. Let us\nconsider, as an example, the integral\nIh=/integraldisplay\nd(p,r){r⊗p}λµ[h↑↓f↓↑−h↓↑f↑↓].\nIt can be divided in two parts corresponding to the\ncontributions of spin-orbital and spin-spin poten-\ntials:Ih=Iso+Iss,where\nIso=−/planckover2pi1√\n2η/integraldisplay\nd(p,r){r⊗p}λµ[l−1f↓↑+l1f↑↓],Iss=/integraldisplay\nd(p,r){r⊗p}λµ[V↑↓\nτf↓↑−V↓↑\nτf↑↓],\nVss′\nτbeing defined in (20). It is easy to see that\nthe integral Isogenerate moments of fourth order.\nAccording to the rules of the WFM method [28] this\nintegral is neglected.\nLet us analyze the integral Iss(to be definite, for\nprotons). In this case\nV↑↓\np(r) = 3/planckover2pi12\n8χn↑↓\np(r)+/planckover2pi12\n4¯χn↑↓\nn(r),\nV↓↑\np(r) = 3/planckover2pi12\n8χn↓↑\np(r)+/planckover2pi12\n4¯χn↓↑\nn(r).\nIt is seen that Issis split into four terms of identical\nstructure, so it will be sufficient to analyze in detail\nonly one part. For example\nIss4=/integraldisplay\nd(p,r){r⊗p}λµn↓↑f↑↓=/integraldisplay\nd3r{r⊗J↑↓}λµn↓↑=/summationdisplay\nν,αCλµ\n1ν,1α/integraldisplay\nd3rrνJ↑↓\nαn↓↑, (A1)\nwhereJ↑↓\nα(r,t) =/integraltextd3p\n(2π/planckover2pi1)3pαf↑↓(r,p,t). The variation of this integral reads\nδIss4=/summationdisplay\nν,αCλµ\n1ν,1α/integraldisplay\nd3rrν/bracketleftbig\nn↓↑(eq)δJ↑↓\nα+J↑↓\nα(eq)δn↓↑/bracketrightbig\n. (A2)\nIt is necessary to represent this integral in terms of the collective variables (19). This problem can not\nbe solved exactly, so we will use the approximation suggested in [28] and expand the density and current\nvariations as a series (see appendix B).\nLet us consider the second part of integral (A2). With the help of f ormula (B4) we find\nI2≡/summationdisplay\nν,αCλµ\n1ν,1α/integraldisplay\nd3rrνJ↑↓\nα(eq)δn↓↑\n=−/summationdisplay\nν,αCλµ\n1ν,1α/integraldisplay\nd3rrνJ↑↓\nα(eq)/summationdisplay\nβ(−1)β/braceleftBigg\nN↓↑\nβ,−β(t)n++/summationdisplay\nγ(−1)γN↓↑\nβ,γ(t)1\nr∂n+\n∂rr−βr−γ/bracerightBigg\n.(A3)\nLet us analyze at first the more simple part of this expression:\nI2,1≡ −/summationdisplay\nβ(−1)βN↓↑\nβ,−β(t)/integraldisplay\nd3r/summationdisplay\nν,αCλµ\n1ν,1αrνJ↑↓\nα(eq)n+=−/summationdisplay\nβ(−1)βN↓↑\nβ,−βXλµ. (A4)\nWe are interested in the value of µ= 1, therefore it is necessary to analyze two possibilities: λ= 1 and\nλ= 2.15\nIn the case λ= 1, µ= 1 we have\nX11≡/integraldisplay\nd3rn+/summationdisplay\nν,αC11\n1ν,1αrνJ↑↓\nα(eq) =/integraldisplay\nd3rn+1√\n2/bracketleftBig\nr1J↑↓\n0(eq)−r0J↑↓\n1(eq)/bracketrightBig\n. (A5)\nBy definition\nJss′\nν=/integraldisplayd3p\n(2π/planckover2pi1)3pνfss′(r,p) =−i/planckover2pi1\n2/bracketleftBig\n(∇ν−∇′\nν)ρ(r,s;r′s′)/bracketrightBig\nr′=r\n=i/planckover2pi1\n2/summationdisplay\nkv2\nk[φk(r,s)∇νφ∗\nk(r,s′)−φ∗\nk(r,s′)∇νφk(r,s)], (A6)\nwherek≡n,l,j,m is a set of oscillator quantum numbers, v2\nkare occupation numbers, and\nφnljm(r,s) =Rnl(r)/summationdisplay\nΛ,σCjm\nlΛ,1\n2σYlΛ(θ,φ)χ1\n2σ(s) =Rnlj(r)Cjm\nlm−s,1\n2sYlm−s(θ,φ) (A7)\nare single particle wave functions, χ1\n2σ(s) =δσ,sbeing spin functions. Inserting (A6) into (A5) one finds\nX11=i/planckover2pi1\n21√\n2/summationdisplay\nnljmv2\nnljm/integraldisplay\nd3rn+(r)R2\nnlj(r)Cjm\nlΛ,1\n21\n2Cjm\nlΛ′,1\n2−1\n2[YlΛ(r1∇0−r0∇1)Y∗\nlΛ′\n−Y∗\nlΛ′(r1∇0−r0∇1)YlΛ] (A8)\nwith Λ = m−1\n2and Λ′=m+1\n2. Remembering the definition (10) of the angular momentum ˆl1=\n/planckover2pi1(r0∇1−r1∇0) and using the relation [18] ˆl±1YlΛ=∓1√\n2/radicalbig\n(l∓Λ)(l±Λ+1)YlΛ±1one transforms (A8) into\nX11=−i/planckover2pi1\n21√\n2/summationdisplay\nnljmv2\nnljm/integraldisplay\ndrn+(r)r2R2\nnlj(r)Cjm\nlΛ,1\n21\n2Cjm\nlΛ′,1\n2−1\n22√\n2/radicalbig\n(l−Λ)(l+Λ+1)\n=−i/planckover2pi1/summationdisplay\nnl|l−1\n2|/summationdisplay\nm=1\n2[(l+1\n2)2−m2]\n2l+1/integraldisplay\ndrn+(r)r2/bracketleftBig\nv2\nnll+1\n2mR2\nnll+1\n2(r)−v2\nnl|l−1\n2|mR2\nnl|l−1\n2|(r)/bracketrightBig\n.(A9)\nAs it is seen, the value of this integral is determined by the difference of the wave functions of spin-orbital\npartners ( vR)2\nnll+1\n2m−(vR)2\nnl|l−1\n2|m, which is usually very small, so we will neglect it. The only remarkable\ncontribution can appear in the vicinity of the Fermi surface, where some spin-orbital partners with j=l+1\n2\nandj=|l−1\n2|can be disposed on different sides of the Fermi surface. In reality s uch situation happens very\nfrequently, nevertheless we will not take into account this effect, because the values of the corresponding\nintegrals are considerably smaller than R20(eq), the typical input parameter of our model.\nLet us consider now the integral I2,1(formula (A4)) for the case λ= 2, µ= 1. We have\nX21≡/integraldisplay\nd3rn+/summationdisplay\nν,αC21\n1ν,1αrνJ↑↓\nα(eq) =/integraldisplay\nd3rn+C21\n11,10/bracketleftBig\nr1J↑↓\n0(eq)+r0J↑↓\n1(eq)/bracketrightBig\n. (A10)\nWith the help of formulae (A6) and (A7) one can show by simple algebra ic transformations that\n/integraldisplay\ndΩr1J↑↓\n0(eq) =−/integraldisplay\ndΩr0J↑↓\n1(eq), (A11)\nwhere/integraltext\ndΩ means the integration over angles. As a result X21= 0.16\nLet us consider the second, more complicated, part of integral I2:\nI2,2=−/summationdisplay\nβ,γ(−1)β+γN↓↑\n−β,−γ(t)/summationdisplay\nν,αCλµ\n1ν,1α/integraldisplay\nd3rrνJ↑↓\nα(eq)1\nr∂n+\n∂rrβrγ\n=−/summationdisplay\nβ,γ(−1)β+γN↓↑\n−β,−γ(t)X′\nλµ(β,γ). (A12)\nThe case λ= 1, µ= 1:\nX′\n11(β,γ) =1√\n2/integraldisplay\nd3r1\nr∂n+\n∂r/bracketleftBig\nr1J↑↓\n0(eq)−r0J↑↓\n1(eq)/bracketrightBig\nrβrγ\n=−i/planckover2pi1\n4/summationdisplay\nnljmv2\nnljm/integraldisplay\nd3r1\nr∂n+\n∂rR2\nnlj(r)Cjm\nlΛ,1\n21\n2Cjm\nlΛ′,1\n2−1\n2/radicalbig\n(l−Λ)(l+Λ+1)[ YlΛY∗\nlΛ+Y∗\nlΛ′YlΛ′]rβrγ.(A13)\nThe angular part of this integral is\n/integraldisplay\ndΩ[YlΛY∗\nlΛ+Y∗\nlΛ′YlΛ′]rβrγ=/summationdisplay\nL,MCLM\n1β,1γ/integraldisplay\ndΩ[YlΛY∗\nlΛ+Y∗\nlΛ′YlΛ′]{r⊗r}LM\n=−2√\n3r2C00\n1β,1γ+/radicalbigg\n8π\n15r2/summationdisplay\nMC2M\n1β,1γ/integraldisplay\ndΩ[YlΛY∗\nlΛ+Y∗\nlΛ′YlΛ′]Y2M\n=2\n3r2δγ,−β/braceleftBigg\n1−/radicalbigg\n5\n2Cl0\nl0,20C1β\n1β,20/bracketleftBig\nClΛ\nlΛ,2M+ClΛ′\nlΛ′,2M/bracketrightBig/bracerightBigg\n. (A14)\nTherefore\nX′\n11(β,γ) =−i/planckover2pi1\n6δγ,−β/integraldisplay\ndr∂n+(r)\n∂rr3/summationdisplay\nnljm/braceleftBigg\n1−/radicalbigg\n5\n2C1β\n1β,20Cl0\nl0,20/bracketleftBig\nClΛ\nlΛ,20+ClΛ′\nlΛ′,20/bracketrightBig/bracerightBigg\n×\nv2\nnljmR2\nnlj(r)Cjm\nlΛ,1\n21\n2Cjm\nlΛ′,1\n2−1\n2/radicalbig\n(l−Λ)(l+Λ+1)\n=−i/planckover2pi1\n3δγ,−β/summationdisplay\nnl/braceleftBigg\n1−/radicalbigg\n5\n2C1β\n1β,20Cl0\nl0,20/bracketleftBig\nClΛ\nlΛ,20+ClΛ′\nlΛ′,20/bracketrightBig/bracerightBigg\n×\n|l−1\n2|/summationdisplay\nm=1\n2[(l+1\n2)2−m2]\n2l+1/integraldisplay\ndr∂n+(r)\n∂rr3/bracketleftBig\nv2\nnll+1\n2mR2\nnll+1\n2(r)−v2\nnl|l−1\n2|mR2\nnl|l−1\n2|(r)/bracketrightBig\n.(A15)\nOne seesthat, exactly asin formula(A9), the valueofthis integral is determined bythe differenceofthe wave\nfunctions of spin-orbital partners ( vR)2\nnll+1\n2m−(vR)2\nnl|l−1\n2|mnear the Fermi surface, so it can be omitted\ntogether with X11following the same arguments.\nThe case λ= 2,µ= 1 can be analyzed in full analogy with formulae (A10,A11) that allows u s to take\nX′\n21= 0.\nSo, we have shown that the integral I2can be approximated by zero. Let us consider now the first part of\nthe integral (A2):\nI1=/summationdisplay\nν,αCλµ\n1ν,1α/integraldisplay\nd3rrνn↓↑(eq)δJ↑↓\nα=/summationdisplay\nν,αCλµ\n1ν,1α/integraldisplay\nd3rrνn↓↑(eq)n+(r)/summationdisplay\nγ(−1)γK↑↓\nα,−γ(t)rγ\n=/summationdisplay\nν,αCλµ\n1ν,1α/summationdisplay\nγ(−1)γK↑↓\nα,−γ(t)/integraldisplay\nd3rn↓↑(eq)n+(r)/summationdisplay\nL,MCLM\n1ν,1γ{r⊗r}LM. (A16)17\nThis integral can be estimated in the approximation of constant den sityn+(r) =n0. Then\nI1=n0/summationdisplay\nν,αCλµ\n1ν,1α/summationdisplay\nγ(−1)γK↑↓\nα,−γ(t)/summationdisplay\nL,MCLM\n1ν,1γR↓↑\nLM(eq) = 0. (A17)\nIt is easy to show, that R↓↑\nLM(eq) = 0.Let us consider, for example, the case with L= 2:\nR↓↑\n2M=/integraldisplay\nd(p,r){r⊗r}2Mf↓↑(r,p) =/integraldisplay\nd3r{r⊗r}2Mn↓↑(r) =/radicalbigg\n8π\n15/integraldisplay\nd3rr2Y2Mn↓↑(r).(A18)\nBy definition\nnss′(r) =/integraldisplayd3p\n(2π/planckover2pi1)3fss′(r,p) =/summationdisplay\nkv2\nkφk(r,s)φ∗\nk(r,s′) (A19)\nwithφkdefined in (A7). Therefore\nR↓↑\n2M=/radicalbigg\n8π\n15/integraldisplay\nd3rr2Y2M/summationdisplay\nnljmv2\nnljmR2\nnlj(r)Cjm\nlΛ′,1\n2−1\n2Cjm\nlΛ,1\n21\n2YlΛ′Y∗\nlΛ\n=/radicalbigg\n2\n3/summationdisplay\nnljmv2\nnljm/integraldisplay\ndrr4R2\nnlj(r)Cjm\nlΛ,1\n21\n2Cjm\nlΛ′,1\n2−1\n2Cl0\n20,l0ClΛ\n2M,lΛ′= 0, (A20)\nwhere Λ = m−1\n2and Λ′=m+1\n2. The zero is obtained due to summation over m. Really, the product\nCjm\nlΛ,1\n21\n2Cjm\nlΛ′,1\n2−1\n2=±√\n(l+1\n2)2−m2\n2l+1(forj=l±1\n2) doesnotdependonthesignof m, whereastheClebsh-Gordan\ncoefficient ClΛ\n2M,lΛ′changes its sign together with m.\nSummarizing, we have demonstrated that I1+I2≃0, hence one can neglect the contribution of the\nintegrals Ihin the equations of motion.\n•It is necessary to analyze also the integrals with the weight {p⊗p}λµ:\nI′\nh=/integraldisplay\nd(p,r){p⊗p}λµ/bracketleftbig\nh↑↓f↓↑−h↓↑f↑↓/bracketrightbig\n=I′\nso+I′\nss.\nAgain we neglect the contribution of the spin-orbital part I′\nso, which generates fourth order moments. For\nthe spin-spin contribution, we have\nI′\nss4=/integraldisplay\nd(p,r){p⊗p}λµn↓↑(r,t)f↑↓(r,p,t) =/integraldisplay\nd3rΠ↑↓\nλµ(r,t)n↓↑(r,t), (A21)\nwhere Π↑↓\nλµ(r,t) =/integraltextd3p\n(2π/planckover2pi1)3{p⊗p}λµf↑↓(r,p,t) is the pressure tensor. The variation of this integral reads:\nδI′\nss4=/integraldisplay\nd3r/bracketleftBig\nn↓↑(eq)δΠ↑↓\nλµ(r,t)+Π↑↓\nλµ(eq)δn↓↑(r,t)/bracketrightBig\n. (A22)\nThe pressure tensor variation is defined in appendix B. With formula ( B6) one finds for the first part of\n(A22):\nI′\n1=/integraldisplay\nd3rn↓↑(eq)δΠ↑↓\nλµ(r,t)≃T↑↓\nλµ(t)/integraldisplay\nd3rn↓↑(eq)n+(r)≃T↑↓\nλµ(t)n0/integraldisplay\nd3rn↓↑(eq) = 0.(A23)\nThe last equality follows obviously from the definition of n↓↑(A19).18\nThe second part of (A22) reads:\nI′\n2=/integraldisplay\nd3rΠ↑↓\nλµ(eq)δn↓↑(r,t)\n=−/summationdisplay\nβ(−1)β/integraldisplay\nd3rΠ↑↓\nλµ(eq)/braceleftBigg\nN↓↑\nβ,−β(t)n++/summationdisplay\nγ(−1)γN↓↑\nβ,γ(t)1\nr∂n+\n∂rr−βr−γ/bracerightBigg\n.(A24)\nLet us consider at first the simpler part of this integral\n−/summationdisplay\nβ(−1)βN↓↑\nβ,−β(t)/integraldisplay\nd3rΠ↑↓\nλµ(eq)n+(r). (A25)\nThevalue ofthe lastintegralis determinedbythe angularstructur eofthe function Π↑↓\nλµ(r). We areinterested\ninλ= 2,µ= 1. By definition\nΠ↑↓\n21(r) =/integraldisplayd3p\n(2π/planckover2pi1)3{p⊗p}21f↑↓(r,p) =/summationdisplay\nν,σC21\n1ν,1σ/integraldisplayd3p\n(2π/planckover2pi1)3pνpσf↑↓(r,p)\n= 2C21\n11,10/integraldisplayd3p\n(2π/planckover2pi1)3p1p0f↑↓(r,p) =−/planckover2pi12\n2√\n2[(∇′\n1−∇1)(∇′\n0−∇0)ρ(r′↑,r↓)]r′=r\n=−/planckover2pi12\n2√\n2/summationdisplay\nkv2\nk{[∇1∇0φk(r,↑)]φ∗\nk(r,↓)−[∇1φk(r,↑)][∇0φ∗\nk(r,↓)]\n−[∇0φk(r,↑)][∇1φ∗\nk(r,↓)]+φk(r,↑)[∇1∇0φ∗\nk(r,↓)]} (A26)\nwithφkbeing defined by (A7). Taking into account formulae [18]\n∇±1Ylλ=−/radicalBigg\n(l±Λ+1)(l±Λ+2)\n2(2l+1)(2l+3)l\nrYl+1,Λ±1−/radicalBigg\n(l∓Λ−1)(l∓Λ)\n2(2l−1)(2l+1)l+1\nrYl−1,Λ±1,\n∇0Ylλ=−/radicalBigg\n(l+1)2−Λ2\n(2l+1)(2l+3)l\nrYl+1,Λ+/radicalBigg\nl2−Λ2\n(2l−1)(2l+1)l+1\nrYl−1,Λ\none finds that\n/integraldisplay\nd3rΠ↑↓\nλµ(eq)n+(r) =/planckover2pi12/summationdisplay\nnljmv2\nnljm/integraldisplay\ndrn+(r)R2\nnlj(r)(δj,l+1\n2−δj,l−1\n2)l(l+1)[(l+1\n2)2−m2]\n(2l+3)(2l+1)(2l−1)m= 0 (A27)\ndue to summation over m. The more complicated part of the integral (A24) is calculated in a sim ilar way\nwith the same result, hence I′\n2= 0.\nSo, we have shown that I′\n1+I′\n2≃0, therefore one can neglect by the contribution of integrals I′\nh(together\nwithIh) into equations of motion.\n•And finally, just a few words about the integrals with the weight {r⊗r}λµ:\nI′′\nh=/integraldisplay\nd(p,r){r⊗r}λµ/bracketleftbig\nh↑↓f↓↑−h↓↑f↑↓/bracketrightbig\n=I′′\nso+I′′\nss.\nThe spin-orbital part I′′\nsois neglected and for the spin-spin part we have\nI′′\nss4=/integraldisplay\nd(p,r){r⊗r}λµn↓↑(r,t)f↑↓(r,p,t) =/integraldisplay\nd3r{r⊗r}λµn↓↑(r,t)n↑↓(r,t). (A28)\nThe variation of this integral reads:\nδI′′\nss4=/integraldisplay\nd3r{r⊗r}λµ[n↓↑(eq)δn↑↓(r,t)+n↑↓(eq)δn↓↑(r,t)]. (A29)19\nWith the help of formulae (A19) and (B4) the subsequent analysis be comes quite similar to that of the\nintegral (A16) with the same result, i.e. I′′\nh≃0.\n•The integrals/integraltext\nd(p,r)Wλµ/bracketleftbig\nh−f↓↑−h↓↑f−/bracketrightbig\nand/integraltext\nd(p,r)Wλµ/bracketleftbig\nh−f↑↓−h↑↓f−/bracketrightbig\n, whereWλµis any of\nthe above mentioned weights, can be analyzed in an analogous way wit h the same result.\nAppendix B:\nAccording to the approximation suggested in [28], the variations of d ensity, current, and pressure tensor\nare expanded as the series\nδnς(r,t) =−/summationdisplay\nβ(−1)β∇−β/braceleftBigg\nn+(r)/bracketleftBigg\nNς\nβ(t)+/summationdisplay\nγ(−1)γNς\nβ,γ(t)r−γ\n+/summationdisplay\nλ′,µ′(−1)µ′Nς\nβ,λ′µ′(t){r⊗r}λ′−µ′+...\n\n\n, (B1)\nδJς\nβ(r,t) =n+(r)\nKς\nβ(t)+/summationdisplay\nγ(−1)γKς\nβ,−γ(t)rγ+/summationdisplay\nλ′,µ′(−1)µ′Kς\nβ,λ′−µ′(t){r⊗r}λ′µ′+...\n,(B2)\nδΠς\nλµ(r,t) =n+(r)\nTς\nλµ(t)+/summationdisplay\nγ(−1)γTς\nλµ,−γ(t)rγ+/summationdisplay\nλ′,µ′(−1)µ′Tς\nλµ,λ′−µ′(t){r⊗r}λ′µ′+...\n.(B3)\nPutting these series into the integrals (A2, A22), one discovers imm ediately that all terms containing ex-\npansion coefficients N, K, T with odd numbers of indices disappear due to axial symmetry. Furth ermore,\nwe truncate these series omitting all terms generating higher (tha n second) order moments. So, finally the\nfollowing expressions are used:\nδnς(r,t)≃ −/summationdisplay\nβ(−1)β∇−β/braceleftBigg\nn+(r)/summationdisplay\nγ(−1)γNς\nβ,γ(t)r−γ/bracerightBigg\n=−/summationdisplay\nβ(−1)β/braceleftBigg\nNς\nβ,−β(t)n++/summationdisplay\nγ(−1)γNς\nβ,γ(t)1\nr∂n+\n∂rr−βr−γ/bracerightBigg\n, (B4)\nδJς\nβ(r,t)≃n+(r)/summationdisplay\nγ(−1)γKς\nβ,−γ(t)rγ (B5)\nand\nδΠς\nλµ(r,t)≃n+(r)Tς\nλµ(t). (B6)\nThe coefficients Nς\nβ,γ(t) andKς\nβ,−γ(t) are connected by the linear relations with collective variables Rς\nλµ(t)\nandLς\nλµ(t) respectively.\nRς\nλµ=/integraldisplay\nd3r{r⊗r}λµδnς(r) =2√\n3/bracketleftBig\nA1Cλµ\n1µ,10Nς\nµ,0−A2/parenleftBig\nCλµ\n1µ+1,1−1Nς\nµ+1,−1+Cλµ\n1µ−1,11Nς\nµ−1,1/parenrightBig/bracketrightBig\n,(B7)\nwhere\nA1=√\n2Req\n20−Req\n00=Q00√\n3/parenleftbigg\n1+4\n3δ/parenrightbigg\n,A2=Req\n20/√\n2+Req\n00=−Q00√\n3/parenleftbigg\n1−2\n3δ/parenrightbigg\n, (B8)20\nR20=Q20/√\n6,R00=−Q00/√\n3,Q20=4\n3δQ00,Q00=A < r2>=3\n5AR2\n0.\nNς\n−1,−1=−√\n3Rς\n2−2\n2A2, Nς\n−1,0=√\n6Rς\n2−1\n4A1, Nς\n−1,1=−Rς\n00+Rς\n20/√\n2\n2A2,\nNς\n0,−1=−√\n6Rς\n2−1\n4A2, Nς\n0,0=√\n2Rς\n2,0−Rς\n0,0\n2A1, Nς\n0,1=−√\n6Rς\n21\n4A2,\nNς\n1,−1=Nς\n−1,1, Nς\n1,0=√\n6Rς\n21\n4A1, Nς\n1,1=−√\n3Rς\n22\n2A2. (B9)\nLς\nλ,µ=/integraldisplay\nd3r{r⊗δJς}λµ=1√\n3(−1)λ/bracketleftBig\nA1Cλµ\n1µ,10Kς\nµ,0−A2/parenleftBig\nCλµ\n1µ+1,1−1Kς\nµ+1,−1+Cλµ\n1µ−1,11Kς\nµ−1,1/parenrightBig/bracketrightBig\n.(B10)\nKς\n−1,−1=−√\n3Lς\n2−2\nA2, Kς\n−1,0=√\n3(Lς\n1−1+Lς\n2−1)√\n2A1, Kς\n−1,1=−√\n3Lς\n10+Lς\n20+√\n2Lς\n00√\n2A2,\nKς\n0,−1=√\n3(Lς\n1−1−Lς\n2−1)√\n2A2, Kς\n0,0=√\n2Lς\n2,0−Lς\n0,0\nA1, Kς\n0,1=−√\n3(Lς\n11+Lς\n21)√\n2A2,\nKς\n1,−1=√\n3Lς\n10−Lς\n20−√\n2Lς\n00√\n2A2, Kς\n1,0=√\n3(Lς\n21−Lς\n11)√\n2A1, Kς\n1,1=−√\n3Lς\n22\nA2. (B11)\nThe coefficient Tς\nλµ(t) is connected with Pς\nλµ(t) by the relation Pς\nλµ(t) =ATς\nλµ(t),Abeing the number of\nnucleons.\n[1] K. Heyde, P. von Neuman-Cosel and A. Richter,\nRev. Mod. Phys. 82(2010) 2365.\n[2] D. Pena Arteaga, P. Ring, arXiv:0912.0908v1 [nucl-\nth], 2009.\n[3] R. R. Hilton, A possible vibrational mode in heavy\nnuclei, Int. Conf. on Nuclear Structure (Dubna,\nJune 1976), unpublished.\n[4] R. R. Hilton, Ann. Phys. (N.Y.) 214(1992) 258.\n[5] T. Suzuki, D. J. Rowe, Nucl. Phys. A 289(1977)\n461.\n[6] N. Lo Iudice, F. Palumbo, Phys. Rev. Lett. 41\n(1978) 1532.\n[7] D.Zawischa, J. Phys.G: Nucl. Part. Phys. 24(1998)\n683.\n[8] V. G. Soloviev, A. V. Sushkov, N. Yu. Shirikovaand\nN. Lo Iudice, Nucl. Phys. A 600(1996) 155,\nV. G. Soloviev, A. V. Sushkov, N. Yu. Shirikova,\nPhys. Rev. C 53(1996) 1022.\n[9] N. Lo Iudice, Rivista del Nuovo Cimento 23(Is-\nsue 9) (2000).\n[10] E. Lipparini, S. Stringari, Phys. Rep. 175(1989)\n103.\n[11] U. Kneissl, H. H. Pitz, and A. Zilges, Prog. Part.\nNucl. Phys. 37(1996) 349.\n[12] A. Richter, Prog. Part. Nucl. 34(1995) 261.\n[13] E. B. Balbutsev, L. A. Malov, P. Schuck, M. Urban,\nand X. Vi˜ nas, Phys. At. Nucl. 71(2008) 1012.\n[14] E. B. Balbutsev, L. A. Malov, P. Schuck, and M.\nUrban, Phys. At. Nucl. 72(2009) 1305.\n[15] E. B. Balbutsev, I.V. Molodtsova, P. Schuck, Nucl.Phys. A 872(2011) 42.\n[16] P. Ring and P. Schuck, The Nuclear Many-Body\nProblem (Springer, Berlin, 1980).\n[17] M. Guttormsen, L. A. Bernstein, A. B¨ urger, A.\nG¨ orgen, F. Gunsing, T.W. Hagen, A. C. Larsen,\nT. Renstrøm, S. Siem, M. Wiedeking, J. N. Wilson,\nPhys. Rev. Lett. 109(2012) 162503.\n[18] D. A. Varshalovitch, A. N. Moskalev and V. K.\nKhersonski, Quantum Theory of Angular Momen-\ntum(World Scientific, Singapore, 1988).\n[19] E. B. Balbutsev, P. Schuck, Nucl. Phys. A 720\n(2003) 293;\nE. B. Balbutsev, P. Schuck, Nucl. Phys. A 728\n(2003) 471.\n[20] A. Bohr, B. Mottelson, Nuclear Structure, Vol. 2\n(Benjamin, New York, 1975).\n[21] P. Sarriguren, E. Moya de Guerra, R. Nojarov,\nPhys. Rev. C 54(1996) 690;\nP. Sarriguren, E. Moya de Guerra, R. Nojarov, Z.\nPhys. A 357(1997) 143.\n[22] N. Van Giai, H. Sagawa, Phys. Lett. B 106(1981)\n379.\n[23] M. Beiner, H.Flocard, N. Van Giai, P. Quentin,\nNucl. Phys. A 238(1975) 29.\n[24] V.G.Soloviev, Theory of complex nuclei (Pergamon\nPress, Oxford, 1976).\n[25] Gogny D1S web mass table,\nhttp://www-phynu.cea.fr/HFB-Gogny.htm .\n[26] W. Korten et al, Phys. Lett. B 317(1993) 19.\n[27] F. Osterfeld, Rev. Mod. Phys. 64(1992) 491.21\n[28] E. B. Balbutsev, Sov. J. Part. Nucl. 22(1991) 159."    },    {        "title": "1103.6040v2.Theory_of_spin_noise_in_nanowires.pdf",        "content": "arXiv:1103.6040v2  [cond-mat.mes-hall]  18 Nov 2011Theory of spin noise in nanowires\nM. M. Glazov1and E. Ya. Sherman2,3\n1Ioffe Physical-Technical Institute RAS, 194021 St.-Petersb urg, Russia∗\n2Department of Physical Chemistry, The University of the Bas que Country, 48080 Bilbao, Spain\n3IKERBASQUE Basque Foundation for Science, Bilbao, 48011 Bi zkaia, Spain\nWe develop a theory of spin noise in semiconductor nanowires considered as prospective elements\nfor spintronics. In these structures spin-orbit coupling c an be realized as a random function of\ncoordinate correlated on the spatial scale of the order of 10 nm. By analyzing different regimes of\nelectron transport and spin dynamics, we demonstrate that t he spin relaxation can be very slow\nand the resulting noise power spectrum increases algebraic ally as frequency goes to zero. This\neffect makes spin effects in nanowires best suitable for studi es by rapidly developing spin-noise\nspectroscopy.\nPACS numbers: 72.25.Rb,72.70.+m,78.47.-p,85.35.Be\nNanostructuresarethepromisinghardwareelementsfor\nspintronics [1] – a rapidly developing branch of physics\nand technology aiming at studies and application of spin-\ndependent phenomena in the charge transport and infor-\nmation processing. The quest for the systems with ul-\ntralong spin relaxation times [2] is one of the main chal-\nlenges in this field. Since the dynamical spin fluctuations\n[3] characterized by correlations on the spin relaxation\ntimescale, are seen as a spin noise in the frequency do-\nmain, this search can be done with recently developed\nhighly accurate low-frequency spin noise spectroscopy [4]\naimed at the measurement of intrinsic equilibrium spin\ndynamics. The spin noise spectroscopy allows to study\nthe slow spin dynamics in (110)-grown quantum wells [5]\nand in quantum dots [6]. Theoretical background of this\nmethod is given, e.g., in Refs. [7–9].\nAn interesting class of semiconductor nanostructures\ndemonstrating peculiar and slow spin dynamics are the\nquantum wires [10–12], where e.g. InAs, InSb as well as\nGaAs/AlGaAs systems are the prospective realizations.\nThe effects of spin-orbit (SO) coupling on the transport\nwere clearly demonstrated there [13, 14] and the nanowire\nbased qubits were introduced [15, 16]. A SO coupling in-\nduced effective magnetic field acting on electron spins in\nnanowires is directed parallel or antiparallel to a certain\naxis [17–21] resulting in a giant spin relaxation anisotropy\nsimilar to that expected in some two-dimensional sys-\ntems [22]. Since the SO coupling is a structure- and\nmaterial-dependent property, all sorts of disorder (ran-\ndom doping [23–26], interface fluctuations [27], random\nvariations in the shape, etc.) which cause electron scat-\ntering and nonzero resistivity, can cause local variations\nin the coupling. As a result, in addition to the regular SO\ncoupling, caused by the lack of bulk (Dresselhaus term)\nor structure (Rashba term) inversion symmetry, all low-\ndimensional structures inevitably have the random con-\ntribution in it. The spatial scale of the fluctuations is of\nthe order of 10 nm as determined by the characteristic\ndistances in nanostructure were shown to give rise to a\nnumber of fascinating phenomena [28–30]. However, their\nrole in quantum wires was not studied so far.\nHere we address theoretically the electron spin dynam-\nics in ballistic and diffusive semiconductor nanowires aim-\nFigure 1: Schematic plot of the experimental configuration: a\nquantum wire (dark stripe) is illuminated by a linearly pola r-\nized beam and Kerr rotation angle of its polarization plane θK\nis measured. Polarizations of the beams are marked by double -\nheaded arrows. Dashed arrow corresponds to the polarizatio n\nof the reflected beam in the absence of the Kerr effect.\ning at the study of the spin noise spectrum. Different\nregimes of electron spin relaxation are determined and\nthe crossovers between them are analyzed in detail. In\nparticular, we demonstrate that when the electron mo-\ntion is diffusive and the dominant contribution to the SO\ninteraction is random, the spin relaxation becomes alge-\nbraic rather than exponential and the spin noise power\nspectrum diverges at low frequencies ωas 1/ω1/2, show-\ning colored noise [31–33] well suited for the studies by the\nspin noisespectroscopy. Averyslowspindynamicsresult-\ninginthe low-frequencynoisedivergencemakesnanowires\nan exception among semiconductor systems.\nThe spin noise spectroscopy, reviewed in Ref. [4], is\nbased on the optical monitoring of the spin fluctuations\n[34] in Faraday, Kerr or ellipticity signals measured with\na weak linearly polarized probe beam incident on a single\nwire or a wire array sample, see Fig. 1. It can be shown\nsimilarly to Refs. [4, 9, 35] that for the probe tuned to\nthe fundamental absorption edge, the Kerr rotation angle\nθK∝sz[40], hence its autocorrelation function is directly\nrelated to the spin noise: ∝an}bracketle{tθK(t)θK(t′)∝an}bracketri}ht ∝ ∝an}bracketle{tsz(t)sz(t′)∝an}bracketri}ht,\nwheresz(t) is the density of the z−componentof the total\nelectron spin. As a result, this optical technique measures\nlong-rangecorrelationsofequilibriumspinfluctuationsoc-\ncurring in the illuminated spot.\nWe consider a single channel quantum wire extended2\nalong the x−axis and represent the SO Hamiltonian as:\nHSO=1\n2[α(x)kx+kxα(x)]σλ. (1)\nHerekx=−i∂/∂xis the electron wave vector component\nalong the wire axis, α(x) is the coordinate-dependent SO\ncoupling strength. In Eq. (1) we assumed that the spin\nquantization axis, λ, is fixed, and σλis the component of\nspin operator along this axis. The specific form of the SO\nHamiltonian Eq. (1) implies that the effective field acting\non electron spin points either parallel or antiparallel to\nthe axis λ. This is obvious for a constant α(x) [17–21],\nand holds true provided that the microscopic symmetry\nof the fluctuations forming the SO coupling randomness\nis the same as overall symmetry of the system.\nThe SO coupling is assumed to be the sum of the\ncoordinate-independent contribution, α0, and the Gaus-\nsian random function with zero average, αr(x) such as\nα(x) =α0+αr(x) with the correlation function [29]:\n∝an}bracketle{tαr(x)αr(x′)∝an}bracketri}ht=∝an}bracketle{tα2\nr∝an}bracketri}htFcorr(x−x′), (2)\nwhere∝an}bracketle{tα2\nr∝an}bracketri}htis the mean square of SO coupling fluctuations\nand the range function Fcorr(x−x′). We introduce also\nthe typical correlation length of the SO coupling\nld=/integraldisplay∞\n0Fcorr(x)dx, (3)\ncharacterizing the size of the correlated domain of the\nrandom SO coupling. Details of the models of random\nSO coupling can be found in Ref.[29].\nWe begin with the semiclassical regime, where SO cou-\npling disorder is smooth on the scale of electron wave-\nlength,ld≫λF, where the wavelength of the Fermi level\nelectrons λF= 2π/kF, withkFbeing the Fermi wave vec-\ntor for the degenerate electron gas. The Hamiltonian (1)\nimpliesthat the spin rotationanglearoundthe λ-axisdur-\ning the motion from the point x0tox1is\nθ(x1,x0) =2m\n/planckover2pi12/integraldisplayx1\nx0α(x′)dx′, (4)\nwheremis the electron effective mass. Eq. (4) shows that\nthe angle is solely determined by electron initial and final\npositionsanddoesnotdependonthehistoryofthemotion\nbetween these points. This result, being well established\nfor the systems with regularSO coupling [20, 36–39] holds\nalso for the nanowires with the SO coupling disorder. As\nit follows from Eq. (1) the spin precession rate is pro-\nportional to the electron velocity and given coordinate-\ndependent function. Hence, it does not matter whether\nthe electron starting from the point x0reached the point\nx1ballisticallyor diffusively: all contributionsto spin pre-\ncessionoftheclosedpaths, whereelectronpassesthe same\nconfiguration of α(x) in the opposite directions, cancel\neach other.\nThe temporal evolution of electron spin is directly re-\nlated the electron motion along the wire. We consider\nhere spin projections at given zaxis, perpendicular to the\nspin quantization axis λ. Time dependence of electronspinzcomponent averaged over its random spatial mo-\ntion and over the random precession caused by the field\nα(x) can be most conveniently characterized by the cor-\nrelator∝an}bracketle{tsz(t)sz(0)∝an}bracketri}ht=∝an}bracketle{ts2\nz(0)∝an}bracketri}htCss(t) with the normalized\ncorrelation function:\nCss(t) =/integraldisplay∞\n−∞dx p(x,t)∝an}bracketle{tcos[θ(x,0)]∝an}bracketri}ht,(5)\nwherep(x,t) is the probability that electron travels dis-\ntancexduring the time t. Note, that Css(t) can be un-\nderstood as disorder-averaged electron spin zcomponent\nfound with the initial condition sz(0) = 1. It results from\nthe linearity of the spin dynamics equations: the correla-\ntors∝an}bracketle{tsi(t)sj(0)∝an}bracketri}htsatisfy exactly the same equations as av-\nerage values ∝an}bracketle{tsi(t)∝an}bracketri}ht(i,j=x,y,z). In derivation of Eq. (5)\nwe assumed also that the scattering of electrons, which\ndetermines p(x,t) is not correlated with the random SO\nfieldαr(x), hence, the averaging over the realizations of\nαr(x) denoted by the angular brackets and over the tra-\njectories can be considered independently. This can occur\nin nanowires where random Rashba fields are induced by\ndoping while the momentum scattering is due to the wire\nwidth fluctuations. If the same local disorder determines\nthe electron scattering and random SO fields, in relatively\nclean systems the electron mean free path lexceeds by far\nthe disordercorrelationlength ldin Eq.(3). Hence, spatial\nscales of two random processes: lfor the electron back-\nward scattering in the random potential and ldfor the\nspin precession are strongly different. As a result, on the\nl-scale, the memory of the short-range correlations is lost,\nand Eq. (5) holds. Although Eq. (5) is presented for the\nsmooth SO coupling disorder, where the electron motion\nis semiclassical, ld/λF≫1, a general Green’s function\napproach confirms it for arbitrary ld/λFvalues.\nOur next step is to perform averaging of cos[ θ(x,0)] in\nEq. (5) over the random realizations of the α(x)-field. For\nthis purpose we recast\ncos[θ(x,0)] = Re/braceleftbigg\nexp/parenleftbigg\ni2mα0\n/planckover2pi12x/parenrightbigg\nexp[iϑr(x)]/bracerightbigg\n,(6)\nwhere\nϑr(x) = 2m//planckover2pi12/integraldisplayx\n0αr(x′)dx′(7)\nis the contribution of the random SO coupling into the\nspin rotation angle. We expand last exponent in series\ninϑrassuming the Gaussian SO coupling disorder. In\nthe averaging, odd powers of spin rotation angle vanish,\n∝an}bracketle{tθ2n+1\nr(x)∝an}bracketri}ht= 0, for integer nand even powers can be\nexpressed solely with ∝an}bracketle{tθ2\nr(x)∝an}bracketri}htas\n∝an}bracketle{tθ2n\nr(x)∝an}bracketri}ht=/angbracketleftBigg/bracketleftbigg2m\n/planckover2pi12/integraldisplayx\n0αr(x′)dx′/bracketrightbigg2n/angbracketrightBigg\n= (2n−1)!!∝an}bracketle{tθ2\nr(x)∝an}bracketri}htn.\n(8)\nDirect calculation shows that the mean square ∝an}bracketle{tθ2\nr(x)∝an}bracketri}ht\ncaused by the random SO interaction is given by\n∝an}bracketle{tθ2\nr(x)∝an}bracketri}ht= 2/parenleftbigg2m\n/planckover2pi12/parenrightbigg2\n∝an}bracketle{tα2\nr∝an}bracketri}ht/integraldisplayx\n0dx′/integraldisplayx′\n0dyFcorr(y).(9)3\nFinally, Eq. (5) reduces to\nCss(t) =/integraldisplay∞\n−∞dx p(x,t)cos/parenleftbigg2mα0\n/planckover2pi12x/parenrightbigg\nexp/bracketleftbig\n−∝an}bracketle{tθ2\nr(x)∝an}bracketri}ht/2/bracketrightbig\n.\n(10)\nWhen∝an}bracketle{tθ2\nr(x)∝an}bracketri}htbecomesconsiderablylargerthan one, spins\nare completely dephased. Equation (10) is our central re-\nsult: it relates temporal average spin dynamics with elec-\ntron motion along the wire. Distribution function of elec-\ntron displacements, p(x,t), presented for different regimes\nof electron motion below, enables us to calculate spin evo-\nlutionbyEq.(10). Thespinnoisepowerspectrumisgiven\nby the transform of Css(t) [9]:\n/angbracketleftbig\ns2\nz/angbracketrightbig\nω= 2/integraldisplay∞\n0Css(t)cos(ωt)dt. (11)\nTo get a better insight into the problem, we begin with\nthe key limits [41]. First, for the ballistic electron dynam-\nicsp(x,t) =δ(x−vFt), where vF=/planckover2pi1kF/mis the Fermi\nvelocity. The ballistic motion is realized on the temporal\nscalet≪τ=l/vFwithτbeing the momentum relax-\nation time. We are interested in the spin dynamics on\nthe time scale t≫τd=ld/vF, whereτdis the time dur-\ning which electron passes the correlated interval of the SO\ncoupling fluctuations. Using Eq. (10) we obtain damped\noscillations of the spin z-component:\nCss(t)≈cos(Ω0t)exp(−t/τs,r), (12)\nwith the frequency Ω 0= 2mα0vF//planckover2pi12determined by the\naveraged SO coupling and the decay time caused by the\nSO coupling fluctuations\n1\nτs,r=/parenleftbigg2mvF\n/planckover2pi12/parenrightbigg2\n∝an}bracketle{tα2\nr∝an}bracketri}htτd. (13)\nEquation(13) forthespin relaxationtime τs,ris aresult of\nrandom spin precession [29]. Spin noise power spectrum\ncalculated using Eqs. (11) and (12) reads:\n/angbracketleftbig\ns2\nz/angbracketrightbig\nω= 2τs,rRe1−iωτs,r\nΩ2\n0τ2s,r+(1−iωτs,r)2(14)\nwith the result presented in Fig. 2.\nThis ballistic regime of spin dynamics, however, can be\nrealized only in very clean systems, where Ω 0τ≫1. Oth-\nerwise, electron spin evolution occurs at the time scale,\nwhere electron moves diffusively (Fig.3, upper panel), i.e.\np(x,t) =1\n2√\nπDte−x2/4Dt, (15)\nwhereD=v2\nFτis the diffusion coefficient. In the\nabsence of the SO coupling fluctuations and provided\nthat Ω 0τ≪1 exponential spin relaxation is due to the\nDyakonov-Perel’ mechanism [17, 21] with the relaxation\ntimeτs,DP= 1/(Ω2\n0τ). The spin noise spectrum has a\nLorentzian form ∝an}bracketle{ts2\nz∝an}bracketri}htω= 2τs,DP/(1 +ω2τ2\ns,DP) with the\nwidth determined by the relaxation time.\nNew physical features arise when the SO coupling fluc-\ntuations dominate over the regular contribution. From0123450.00.20.40.60.81.01.2\nΩΤs,r/Lesssz2/GreaΤerΩ/LParen1arb.units/RParen1\nFigure 2: Spin noise power spectrum, /angbracketlefts2\nz/angbracketrightω, for ballistic prop-\nagation, Ω 0τs,r= 2. Due the exponential decay in Eq.(12) it\nis finite at ω= 0 with the width determined by the spin re-\nlaxation time τs,r. The spectrum peaks at the frequency Ω 0\nsince average electron spin rotates in the SO field at the rate\nΩ0and asymptotically decays as ω−2in accordance with the\nfluctuation-dissipation theorem.\nnow on we put α0= 0 and consider the system where SO\ncoupling is purely random and concentrate on the long-\ntime (t≫τd,τ,τs,r) dynamics. At these times, the sys-\ntem in Eq. (10) is characterizedby two length parameters.\nOneparameteristhediffusion length√\nDtinEq.(15), the\nother one\nLs=/integraldisplay∞\n0dxexp/bracketleftbig\n−∝an}bracketle{tθ2\nr(x)∝an}bracketri}ht/2/bracketrightbig\n, (16)\ncharacterizes spin randomization. At sufficiently long\ntimes, when√\nDt≫Ls, one can take p(0,t) instead of\np(x,t) and immediately obtain from Eq. (10) that the re-\nlaxation is algebraic rather than exponential:\nCss(t)≈p(0,t)/integraldisplay∞\n−∞dxexp/bracketleftbig\n−∝an}bracketle{tθ2\nr(x)∝an}bracketri}ht/2/bracketrightbig\n=Ls√\nπDt.(17)\nEquation (17) predicts extremely long spin decoherence\ndescribed by the inverse square root law: ∝an}bracketle{tsz(t)∝an}bracketri}ht ∝1/√\nt.\nThis surprising result has a transparent physical inter-\npretation (see Fig. 3): Indeed, if an electron is displaced\nfrom its initial position by a sufficiently large distance,\nx/greaterorsimilarLs, its spin rotation angle becomes so large, that\nit does not contribute to the total spin polarization ow-\ning to exp/bracketleftbig\n−∝an}bracketle{tθ2\nr(x)∝an}bracketri}ht/2/bracketrightbig\nin Eq. (10). As a result, the spin\npolarization is supported by the electrons located in the\nvicinity of their initial positions, mainly due to the return\nafter multiple scatterings by the random potential. The\nfraction of such electrons, in agreement with the diffu-\nsion distribution, decays as p(0,t)∝1/√\ntresulting in the\nsame behavior in the spin polarization. It is interesting\nto mention that this qualitative argument does not work\nfor the constant SO coupling despite spin of electron is\nrestored upon the return to the origin also here. The rea-\nson is that due the oscillations of the spin on the spatial\nscale of the order of /planckover2pi12/mα0(see Fig. 3, lower panel) in\nEq.(10), the diffusive return of electrons to the origin is\ninsufficient for formation of the algebraic relaxation tail.\nAnother realization of the 1 /√\ntspin decay can be\nachieved for the very strong random SO couping where\nthe spin relaxation occurs within one nanosize domain of\ntheSOcoupling,thatisattheelectrondisplacementmuch4\nFigure 3: Upper panel: Schematic illustration of the dis-\nplacements distribution p(x,t) for two different time moments:\nt0< t1. Lower panels: The quantum wire and spins of diffus-\ning electron for the random and regular SO couplings, respec -\ntively. For the random SO coupling, if the electron is within\ntheLsdistance from its initial point [see Eq. (17)] its spin is\npreserved, when it leaves this interval, the spin dephases.\nless than ld. In this case, spin relaxation rate is due to\nthe Dyakonov-Perel’mechanism and is determined by the\nlocalvalueof α(x)insidethedomain. Spinsofelectronslo-\ncated in the intervals with large α(x) will relax fast, while\nspins of those experiencing weak α(x) will relax slow.\nSlow non-exponential spin relaxation, described by\nEq. (17) manifests itself in the low frequency spin noise\nspectrum. From Eq. (11) it follows:/angbracketleftbig\ns2\nz/angbracketrightbig\nω∝1/√ω, i.e\nthe spin noise diverges at ω→0. Such a non-trivial be-\nhavior is inherent to the quantum wires with random SO\ncoupling, where spin restoresupon return to the origin: in\nmultichannel wires for sufficiently fast interchannel scat-\ntering [42] and in two-dimensionalsystems spin relaxation\nis exponential [29] and/angbracketleftbig\ns2\nz/angbracketrightbig\nω=0is finite.\nToconclude, westudied theoreticallyspinnoisein semi-\nconductor nanowire for different regimes of the electron\npropagation. We demonstrated that if the spin relaxation\nis determined by the randomness in the SO coupling, spin\nrelaxation becomes algebraic being closely related to the\nhigh probability for electron to stay close to its initial po-\nsition as a result of a multiple scatterings in the random\npotential. This behavior can appear in at least two possi-\nble regimes: (i) when the electron motion is diffusive and\n(ii) when the spin relaxation occurs on a small spatial\nscale of the order of 10 nm. In any of these cases, the spin\nnoise power spectrum shows colored 1 /√ωnoise. In ad-\ndition, this observation shows that low-frequency optical\nspin noise spectroscopy is an excellent tool for studying\nspin phenomena in semiconductor nanowires and charac-\nterization of random potential and SO coupling there.\nAcknowledgements MMG is grateful to RFBR and\n“Dynasty” Foundation—ICFPM for financial support.\nThis work of EYS was supported by the University ofBasque Country UPV/EHU grant GIU07/40, MCI of\nSpain grant FIS2009-12773-C02-01, and ”Grupos Consol-\nidados UPV/EHU del Gobierno Vasco” grant IT-472-10.\n∗Electronic address: glazov@coherent.ioffe.ru\n[1] I. Zutic, J. Fabian, and S. DasSarma, Rev. Mod. Phys.\n76, 323 (2004).\n[2] M. Wu, J. Jiang, and M. Weng, Phys. Reports 493, 61\n(2010).\n[3] E.L. Ivchenko, Sov. Phys. Semicond. 7, 998 (1974).\n[4] G. M. M¨ uller, M. Oestreich, M. R¨ omer, and J. H¨ ubner,\nPhysica E 43, 569 (2010).\n[5] G. M. M¨ uller, M. R¨ omer, D. Schuh, W. Wegscheider,\nJ. H¨ ubner, and M. Oestreich, Phys. Rev. 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Solid State 34,\n1815 (1992).\n[41] Thegeneral case istreatedintheSupplementaryMateri al.\n[42] Various regimes of spin dynamics in multichannel\nnanowires are briefly discussed in the Supplementary Ma-\nterial.\nSupplementary Material for “Theory of Spin Noise in Nanowir es”\nSI. SPIN NOISE AT ARBITRARY FREQUENCIES\nHerewedeterminethe spin noisespectrum forarbitrary\nfrequencies ω. Weemploythekineticequationforelectron\ndistributionfunction f(x,vx,t)dependentontheposition,\nvelocityvx, and time. The equation has the form:\n∂f\n∂t+vx∂f\n∂x+f−¯f\nτ= 0, (S1)\nwheref(x,vx,t) satisfies the initial condition f(x,vx,0) =\nδ(x)[δv,vF+δv,−vF]/2 meaning that at t= 0 it is built at\nx= 0 with the equal fractions of electrons with veloc-\nitiesvx=±vF. As a result, the carriers can be sepa-\nrated into the right movers, vx=vF, and left movers,\nvx=−vFwith function ¯f= [f(x,vF,t)−f(x,−vF,t)]/2\nbeing the anisotropic part of the distribution. The dis-\ntribution of electron displacements is given by p(x,t) =\nf(x,vF,t)+f(x,−vF,t). It can be shown that the spatial\nFourier transform and cos( ωt) transform of this distribu-\ntion, ˜p(k,ω) has the form\n˜p(k,ω) = 2Reτ(1−iωτ)\n(kl)2−iωτ(1−iωτ).(S2)\nIn accordance with Eq. (11) in the main text the spin\nnoise spectrum can be presented as\n/angbracketleftbig\ns2\nz/angbracketrightbig\nω=/integraldisplay∞\n−∞dk\n2π˜p(k,ω)T(k), (S3)\nwhere\nT(k) =/integraldisplay∞\n−∞exp/bracketleftbig\nikx−∝an}bracketle{tθ2\nr(x)∝an}bracketri}ht/2/bracketrightbig\ndx.(S4)\nAnalytical result can be obtained in the regime where\nspin rotation angles within each correlated domain of the\nSO coupling are small, that is Ω rτd≡2m/radicalbig\n∝an}bracketle{tα2r∝an}bracketri}htld//planckover2pi1≪1\nwith Ω r≡2/radicalbig\n∝an}bracketle{tα2r∝an}bracketri}htkx//planckover2pi1. Here the spin dynamics occurs\non the spatial scale x≫ld, mean squares of spin rotation\nanglesareproportionaltoelectrondisplacement ∝an}bracketle{tθ2\nr(x)∝an}bracketri}ht ≈\n2(Ωrτd)2|x|/ldbeing valid for x≫ldor att≫τd, and\nfunction T(k) takes the form:\nT(k) =2ld(Ωrτd)2\n(Ωrτd)4+(kld)2. (S5)After lengthy transformations we obtain\n∝an}bracketle{ts2\nz∝an}bracketri}htω= 2Reτd\n(Ωrτd)2/radicalbig\niωτ/(iωτ−1)−iωτd.(S6)\nIt can be seen from Eq. (S6) that at low frequencies, ω≪\nτ−1\nd,τ−1, spin noise spectrum has the form:\n/angbracketleftbig\ns2\nz/angbracketrightbig\nω=√\n2τs,r√ωτ, (S7)\ninagreementwiththeanalysisabove. Forhighfrequencies\nωτ≫1,/angbracketleftbig\ns2\nz/angbracketrightbig\nωis given by 2 /(ω2τs,r) since at τd≪t≪τ\nthe electron motion is ballistic, and spin dephasing is\ncaused by the random fluctuations of the spin-orbit cou-\npling, cf. Eq. (12) of the main text. The entire frequency\ndependence of/angbracketleftbig\ns2\nz/angbracketrightbig\nωis plotted in Fig. 4.\n10/Minus40.0010.010.111010/Minus40.011100104\nFrequency/LParen1ΩΤ/RParen1/Lesssz2/GreaΤerΩ/LParen1arb.units/RParen1\nFigure 4: Spin noise power spectrum for diffusive electron\npropagation, Ω 0≡0, Ωrτd= 0.01,τd/τ= 0.1. Solid line\nshows exact result, calculated according to Eq. (S6). Dotte d\n(with the slope -1/2) and dashed (with the slope -2) lines sho w\nthelow-frequencyandhigh-frequencyasymptotic, respect ively.\nSII. SPIN DYNAMICS AND NOISE IN\nMULTICHANNEL WIRES WITH RANDOM\nSPIN-ORBIT COUPLING\nThe spin evolution in multichannel structures depends\non the additional set of parameters, {τi,j}being the scat-\nteringtimesbetweenthechannels iandj, aswellasonthe6\ndetails of spin dynamics in every channel. For qualitative\nanalysis (a general case requires a separate treatment) we\nconsider a structure with two conducting channels, where\n(i) spin-orbit coupling disorder in different channels is not\ncorrelated and (ii) in each channel Ω[c]\nrτd≪1 (super-\nscripts denote channels), i.e. spin rotation angles in corre-\nlated domains of the spin-orbit coupling are always small.\nHere we can characterize the interchannel scattering by a\nsingle time τcand focus on the most interesting case with\nno regular contribution to the spin-orbit field: α0≡0.\nIn the limit of very rare interchannel scattering events\n(the condition is given below) the channels are indepen-\ndent. Hence, the general results expressed by Eqs. (10),\n(11) of the main text and by Eq. (S6) as well as asymp-\ntoticEqs.(S7)and(17)ofthemaintexthold. Althoughin\nthese equations one has to average over the realizations of\nα[c]\nr(x) in different channels, the low-frequency spin noise\npower spectrum remains 1 /√ω, the same as in a single\nchannel wire.\nNowweturntotheefficientinterchannelscatteringwith\nshortτc. Ifτc≪τdelectron quits given channel faster\nthan it quits the correlated domain. Spin rotations be-\ntweeninterchannelscatteringeventsareuncorrelatedand,\ndue to this randomness, spin dynamics is exponential:\n∝an}bracketle{tsz(t)sz(0)∝an}bracketri}ht ∝exp(−Γct), (S8)\nwhere the relaxation rate Γ cis of the order of/bracketleftBig\nmax/braceleftBig\nΩ[1]\nr,Ω[2]\nr/bracerightBig/bracketrightBig2\nτc.Similar exponential decay of the\nspin correlator remains for τd≪τc≪τ. Here, the mean\nsquare of the spin rotation angle between interchannel\nscatterings can be estimated as ∝an}bracketle{t(δΦ)2∝an}bracketri}ht=∝an}bracketle{tθ2\nr(v[c]τc)∝an}bracketri}ht ∝\nτc, withv[c]being the characteristic velocity in the chan-\nnel. Spin relaxation is governed by the Dyakonov-Perel’-\nlike mechanism, with the rate\nΓ∝ ∝an}bracketle{t(δΦ)2∝an}bracketri}ht/τc∼τ−1\ns,r, (S9)\nwhich isτc-independent for exactlythe same reasonas thespin relaxationrate due to the random spin-orbit [Eq.(13)\nin the main text] does not explicitly depend on the elec-\ntron free path.\nMost interesting physics appears for a very weak in-\nterchannel scattering, τc≫τ,τd. In this case, electron\nmoves diffusively in a given channel before the interchan-\nnel scattering occurs. As we have shown above, the 1 /√\nt\ntailin the spinpolarization(and corresponding1 /√ωspin\nnoise) results from the carriers dwelling around the initial\npoint of their trajectories. Since the tail is formed at long\ntimes√\nDt≫Ls, see Eq. (16) of the main text, it is sup-\nported by electrons which moved many times back and\nforth in the random potential. If the interchannel scat-\ntering is probable, electron may return to the initial point\nviaotherchannels,whereitsspinrotationisnotcorrelated\nwith that in the initial one. Therefore, in general 1 /√\nt\ntail is destroyed and the usual exponential spin relaxation\ntakes place. However, if τcis long enough to assure that\nthe typical electron displacement during the diffusion be-\ntween interchannel scatterings Lc=√Dτc≫Ls, there\nis a time interval L2\ns/D≪t≪τcand the corresponding\nfrequency range, where the spin dynamics and the noise\nare algebraic:\n∝an}bracketle{tsz(t)sz(0)∝an}bracketri}ht ∝1√\nt,∝an}bracketle{ts2\nz∝an}bracketri}htω∝1√ω.(S10)\nIn the regimes of a highly efficient interchannel scattering,\nwith the spin relaxationdescribedbyEqs.(S8)or(S9), the\nprobability of spin components restorationupon return to\nthe initial position is stronglysuppressed, and, as a result,\nspin noise power spectrum at low frequency decreases and\nbecomes finite.\nFor completeness, we mention that if the random spin-\norbit coupling does not depend on the channel, that is\nα[1]\nr(x) =α[2]\nr(x), spinprecessionanglebetweenanypoints\nx1andx2is insensitive to the interchannel scattering, and\nour analysis in the main text holds exactly the same."    },    {        "title": "1303.1250v2.A_Quantum_Approach_of_Meso_Magnet_Dynamics_with_Spin_Transfer_Torque.pdf",        "content": "arXiv:1303.1250v2  [cond-mat.mes-hall]  9 Mar 2013A Quantum Approach of Meso-Magnet Dynamics with Spin Transf er Torque\nYong Wang∗and L. J. Sham†\nCenter for Advanced Nanoscience, Department of Physics,\nUniversity of California, San Diego, La Jolla, California 9 2093-0319, USA\nWe present a theory of magnetization dynamics driven by spin -polarized current in terms of the\nquantum master equation. In the spin coherent state represe ntation, the master equation becomes\na Fokker-Planck equation, which naturally includes the spi n transfer and quantum fluctuation.\nThe current electron scattering state is correlated to the m agnet quantum states, giving rise to\nquantum correction to the electron transport properties in the usual semiclassical theory. In the\nlarge spin limit, the magnetization dynamics is shown to obe y the Hamilton-Jacobi equation or the\nHamiltonian canonical equations.\nPACS numbers: 75.78.-n, 05.10.Gg, 85.75.-d\nI. INTRODUCTION\nThe theory of open quantum systems, which has been\ngreatly developed in the past several decades, plays a\ncritical role in the understanding and control of the dy-\nnamics of quantum systems that are coupled to the sur-\nrounding environments.1–3The basic issues such as dissi-\npation, decoherence, measurement, and noise source, etc.\nof the systems are usually investigated in the open quan-\ntum system framework. This theory has wide applica-\ntions in the fields including quantum optics,1–3ultracold\natoms,4and quantuminformationand computation.5On\nthe other hand, in micromagnetics6,7and spintronics8,9\nthe magnetization dynamics is commonly treated as clas-\nsical even though the control and dissipation parame-\nters are couched as from quantum sources. While these\nmethods have been highly successful in simulating mag-\nnetization reversal and spin-torque driven magnetization\ndynamics,10thereisthequestionwhethertherearequan-\ntum effects not exhibited by these classical treatments,\nwhen the magnet is in the mesoscopic range, of 103−107\nspins, between the molecular magnets and the macro-\nscopic magnets as defined by Ref. 11. Addressing this\nquestion may not only pave the way for the future tech-\nnology developments, but also broaden our vision and\ndeepen our understanding of the emerging mesoscopic\nquantum world between the well established microscopic\nand macroscopic ones. In this paper, we present a the-\noryofasingledomainmesoscopicmagnetasamemberof\nthe family of open quantum systems for its spin-current-\ndriven dynamics.\nWhen the spin-polarized current passes through the\nferromagnetism (FM) layer, the spin angular momentum\nof the current electrons is transferred to the FM layer\nand thus rotate the magnetization12,13. This so-called\n“spin transfer torque (STT)” effect has now become the\nmost important method to control the magnetization dy-\nnamics in the nano-scale structures, where the conven-\ntional Oersted field generated by the electric current is\nless practical8,9. Numerous magnetoelectronics devices\nhave been proposed and fabricated based on the STT-\ndriven magnetization dynamics.8,9In spite of its greatsuccess and importance, the fundamental physics of STT\nin the standard theory is semiclassical. The magneti-\nzation dynamics is described by the classical Landau-\nLifshitz-Gilbert (LLG) equation10, while the STT terms\nin the LLG equation is obtained from the quantum scat-\ntering ofthe spin currentelectrons by the classicalpoten-\ntial of the magnetization12,13. This semiclassical picture\nisexpected to breakdownasthe magnetis further minia-\nturized to the mesoscopic regime. Furthermore, the STT\nhas been used to stimulate and control the spin waves14,\nor magnons. Therefore, a more sophisticated investiga-\ntion of the STT in the full quantum picture is necessary\nin order to adapt the new developments in the field. In\na previous study, we shown that the continuous scatter-\nings between the quantum macrospin state of a magnet\nandspin-polarizedelectronsin a simple model simulation\nnot only induce the STT effect but also generate quan-\ntum fluctuations due to the quantum disentanglement\nprocess.15In this paper, we will show that the quantum\nmacrospinscatteringmodelweexploitedbeforeisexactly\nsolvable,andwillinvestigatethemagnetizationdynamics\nfrom the full quantum picture by applying the standard\ntheoretical techniques for open quantum systems to this\nmodel.\nII. QUANTUM MACROSPIN MODEL\nIn parallel with the original study of STT in the semi-\nclassical picture12,13, we consider the motion of a single-\ndomain magnet driven by the spin-polarized current.\nHowever, the magnet here is not described by the clas-\nsical magnetization vector M, but is represented by the\nquantum operators of the total spin angular momentum\n/hatwideJ. The spin-polarized electrons are injected along the\nx-direction in sequence independent of one another, and\ninteract with the magnet located at x= 0 through the\nexchange interaction. The model Hamiltonian for each\nelectron interacting with the magnet is15\nH=−1\n2∂2\nx+δ(x)/parenleftig\nλ0/hatwideJ0+λ/hatwides·/hatwideJ/parenrightig\n.(1)2\nwhere the first term is the kinetic term of a single current\nelectron, and the second term is the interaction between\nthe electrons and the magnet; /hatwideJ0and/hatwideJare the unit and\ntotal spin operators of the magnet, and /hatwidesis the electron\nspin operator; the parameters λ0andλare the spin-\nindependent and spin-dependent interaction strength re-\nspectively, which are used in the semiclassical model if\nthe operators /hatwideJ0and/hatwideJare replaced by their mean field\napproximation according to the correspondence princi-\nple. Note that the Hamiltonian is that of a free magnet\nwithout the external magnetic field and anisotropic crys-\ntal field. This treatment will simplify the calculations\nbelow without invalidating of the general conclusions.\nThe STT effect originates from the elementary\nentangle-disentangle processes of the scattering states\nbetween the magnet and the spin-polarized electrons.15\nThese scattering states are deduced from the scattering\nmatrixSof the Hamiltonian (1). Unlike the scatter-\ning matrix in the semiclassical picture, which is defined\nonly in the Hilbert space of the electron, the scattering\nmatrixSin this full quantum picture is defined on the\nlarger Hilbert space including both the magnet and the\nelectron (see Appendix A), which gives the STT directly\nand more informations compared with the semiclassical\ncase.\nIII. QUANTUM DYNAMICS EQUATIONS FOR\nMAGNET\nA. Quantum Master Equation\nConsider the scattering of the magnet spins by an in-\njected electron as uncorrelated. The incoming compos-\nite state of the magnet and the current electron is as-\nsumed to be a product of their respective density ma-\ntricesρJ\ninandρe\nin. After scattering, the outgoing states\nof the whole system, ρout=SρJ\nin⊗ρe\ninS†, as a result of\nthe unitary scattering matrix S, has a degree of entan-\nglement. Next, the surrounding environment decoheres\nthe entangled state into a joint probability distribution\nof the possible magnet states and the corresponding elec-\ntron states. Properties of the resultant magnet state or\nthe electron state are characterized by their respective\nreduced density ρJ\noutor electron ρe\noutfrom tracing over\nthe degrees of freedom of the other component in ρout.\nThecorrelated quantum dynamics of the magnet and the\nelectron injected in sequence in the spin-polarized cur-\nrent drives the time evolution of the magnet state ρJand\nthe magnetization-dependent electron transport proper-\nties in the electron density matrix ρe. While the mean\nmagnetization dynamics is within reach of the semiclas-\nsical picture, the magnetization fluctuation is given only\nby the full quantum treatment followed here without ad-\nditional stochastic assumptions.\nForatheoryofdynamicsofthe ferromagnetasanopen\nsystem, we treat the current electrons as the equivalent\nof the environment. The Kraus operator16of the magnetis defined in terms of the scattering matrix Sas the evo-\nlution operatorofeachencounter with acurrent electron,\nKk,s;k′,s′≡ /an}b∇acketle{tk,s|S|k′,s′/an}b∇acket∇i}ht, (2)\nwith a specific basis set {|k,s/an}b∇acket∇i}ht}of an incoming electron\nstate of wave vector kand spin up or down state s=±.\nWe have adopted the simple model (1) for the dynamics\nof the rigid macro-spinstates {|J,m/an}b∇acket∇i}ht}with the total spin\nnumberJand thezcomponent quantum number mand\nleave the effects of spin waves for future study. Then,\nthe current electron kinetic energy is conserved and the\nKraus operators are non-zero only if kandk′are on the\nsame energy shell, given by\nKk,s;±k,s= (ξ±1\n2)/hatwideJ0+sζ/hatwideJz,Kk,−s;±k,s=ζ/hatwideJs,(3)\nwhere the coefficients ξandζare functions of the basic\nparameters λ0,λandk,J. The first operator Kk,s;±k,s\nwiththesamespinindex scomesfromscatteringwithout\nspin transfer, and the second operator Kk,−s;±k,swith a\nchange insrepresents spin transfer. These Kraus oper-\nators are functions of the macro-spin /hatwideJ0and/hatwideJ(see Ap-\npendix B). In the semiclassical picture, these Kraus op-\nerators will reduce to scalars representing effective fields\nfor the magnetization dynamics.\nIn a scattering event, let the initial state of the cur-\nrent electron be given by the density matrix ρe\nin=/summationtext\ns,s′fs,s′(k)|k,s/an}b∇acket∇i}ht/an}b∇acketle{tk,s′|. This simple form may be ex-\ntended to account for a wave vector distribution or quan-\ntum coherence between different wave vectors, but will\nnotbeexploitedheretokeeptheexpositionsimple. With\ntheaboveKrausoperators,thequantummapofthemag-\nnet state from ρJ\nintoρJ\noutin the scattering is\nρJ\nout=/summationdisplay\n±,s,s′,s′′fs,s′(k)K±k,s′′;k,sρJ\nin(K±k,s′′;k,s′)†.(4)\nIf the spin-polarized current is considered as a sequence\nof electrons injected at equal time interval τ(a measure\nof the inverse current), the continuous evolution of the\nmagnetic density matrix driven by Eq. (4), with a coarse\ngraining of a time scale much larger than τ, yields the\nquantumdynamicsofthemagnetgovernedbythe master\nequation,\n∂\n∂tρJ(t) =1\nτ[T0(t)+S(t)·T(t)], (5)\nwhereS= Tr[σρe\nin] is the Bloch vector of the spin-\npolarized current electrons, σbeing the Pauli matrices.\nThe operators T0andTare polynomial functions of /hatwideJ0\nand/hatwideJ(see Appendix C). T0corresponds to the unpolar-\nized part of the current which causes fluctuations of the\nmagnet motion without a net spin transfer effect. On\nthe other hand, Tis induced by the electron spin polar-\nization, giving rise to both STT and the magnetization\nfluctuations.\nNote that the master equation (5) is an exact solu-\ntion from the model (1) for arbitrary J. Thus, Eq. (5)3\ncan be applied to molecular magnets of small J. This is\nin contrast with the simulation of the quantum stochas-\ntic dynamics of the magnet in a previous study15which\nused the same scattering matrix but required the large\nJcondition to keep the approximation of the magnetic\nquantum state as a spin coherent state after scattering.\nB. Fokker-Planck Equation\nTo facilitate computation in the large Jcase and, more\nimportantly, to study the quantum-classical crossover of\nthe magnetization dynamics, we put the master equa-\ntion (5) in the spin coherent state P-representation1,4\nanalogous to the boson case. The chosen basis set is the\novercomplete and non-orthogonal states {|J,Ω/an}b∇acket∇i}ht}, where\n|J,Ω/an}b∇acket∇i}htis the spin coherent state of total spin Jin the\ndirection of Ω = (Θ ,Φ). The density matrix ρJin this\nrepresentation is ρJ(t)≡/integraltext\ndΩPJ(Ω,t)|J,Ω/an}b∇acket∇i}ht/an}b∇acketle{tJ,Ω|, and\nthe spin operators /hatwideJtake the form of the differential\noperators.17Substituting these expressions of ρJand/hatwideJ\ninto Eq. (5) with some algebraic manipulations, we ob-\ntain the Fokker-Planck equation for PJ(see Appendix\nD),\n∂\n∂tPJ(/hatwidem,t) =−∇·(TPJ)+∇2(DPJ),(6)\nwhere the unit vector /hatwidempoints in the direction of the\nmacrospin Ω = (Θ ,Φ), the drift vector T=A(/hatwidem×S)×\n/hatwidem+B/hatwidem×Scontainsthetwowell-knowntermsofSTT,8,9\nthe diffusion coefficient D=A(1−/hatwidem·S)/(2J+1) orig-\ninates from the quantum fluctuation generated by the\nscattering.15Theparameters AandBarefunctionsofthe\nparameters ξandζin the Kraus operators (3), namely,\nA= (2J+ 1)|ζ|2/τ,B= 2ℑ[ξ∗ζ]/τ(ℑdenoting the\nimaginary part of) which can be determined from the\nbasic parameters of the Hamiltonian (1).\nThe quasi-probability distribution function PJin\nEq. (6) is different from the one considered in the classi-\ncal theory. Its value could be negative in some situations\nbecausePJdescribes the quantum state ofthe magnet as\nfaithfully as the density matrix ρJ. As shown in Eq. (6),\nthe STT terms naturally arise from the open quantum\ndynamics of the magnet in the presence of the contin-\nuous scatterings by the spin-polarized electrons. Unlike\nthe semiclassical picture, where the STT terms are indi-\nrectly obtained from the current electron spin polariza-\ntion after potential scattering, in the quantum case, the\nSTTtermsfollowsdirectlyfromthefullquantumscatter-\ning. Furthermore, the full quantum treatment also gives\nthe quantum fluctuations accompanying the spin trans-\nfer processes, which does not exist in the semiclassical\ntreatment.\nThe diffusion coefficient Dexpression shows depen-\ndence on the relative angle between the magnet and the\nelectron spin, with its maximal (minimal) value when /hatwidem\nandSare anti-parallel(parallel). This coincides with our\nprevious simulation15that the quantum magnetizationfluctuation is first enhanced and then suppressed during\nthe STT-driven magnet switching, which is observed in\na recent experiment.18For the special case where the in-\njected electrons are fully unpolarized ( S=0),Dis still\na non-zero constant which again suggests that the unpo-\nlarized current can cause quantum magnetization fluctu-\nations without net spin transfer. The steady solution of\nEq. (6) is a constant, which means a uniform distribution\nfunction in the spin coherent state space and the magne-\ntizationwillvanishonaverage. Bycontrast,thesemiclas-\nsical theory predicts only the zero spin torque but no dif-\nfusion dynamics for the magnet. This STT-induced mag-\nnetization fluctuation becomes dominant over the ther-\nmal magnetization fluctuation at low temperatures. The\ncrossovertemperature is estimated by comparing the dif-\nfusion coefficient Din Eq. (6) with the one for thermal\nmagnetization fluctuation,19,20\nαgγgkBT\n|M|∼A\n2J+1(1−/hatwidem·S), (7)\nwhereMis the magnetic moment of the magnet, αg\nthe Gilbert damping coefficient, γgthe gyromagnetic ra-\ntio,kBthe Boltzmann constant, and Tthe temperature.\nSince|M|=γgJ/planckover2pi1and|ζ|2∼ O(1/J2) forJ≫1, we\nobtainαgκBT∼/planckover2pi1/Jτ, in agreement with the quantum\nnoise estimate.15\nThePJdistribution as the solution of the Fokker-\nPlanck equation (6) gives the exact quantum dynamics\nof the magnet based on the quantum macrospin model\n(1). The expectation values of any observable physics\nqualities, such as the magnetization and its fluctuations,\ncan be calculated from the PJdistribution. An exam-\nple is demonstrated in subsection D. In the semiclassical\npicture, the STT is defined on the mean field level of\nthe magnetization, and quantum correlation between the\nmagnetization states does not exist. The quantum the-\nory includes the quantum correlationand is applicable to\nany possible exotic quantum states of the magnet in the\nmesoscopic or microscopic regime.\nC. WKB Approximation for Large J\nInthelarge Jregime, wedemonstratethatthesolution\nof Eq. (6) leads to classical behavior. The expressions for\nAandBsuggest that T∼ O(1/J) andD ∼ O(1/J2),\nthus the diffusion term is smaller than the drift term in\nEq. (6) by the order of magnitude O(1/J). Then, the\nWKB approximation is applied for large J.21Substitut-\ningPJ(/hatwidem,t) =e−JW(/hatwidem,t)into Eq. (6) and keeping the\nterms to the leading order of 1 /J, leads to the Hamilton-\nJacobi equation for W(/hatwidem,t),\n∂W\n∂t+T·∇W+JD(∇W)2= 0. (8)\nThus, the STT-driven magnet obeys the canonical dy-\nnamics in the classical limit, and the function Wplays4\ntheroleofaction. Fortheconstantspin-polarizedcurrent\ncase, the corresponding Hamiltonian canonical equations\nin the spherical polar coordinates are\ndΘ\ndt=∂H\n∂PΘ,dPΘ\ndt=−∂H\n∂Θ,\ndΦ\ndt=∂H\n∂PΦ,dPΦ\ndt=−∂H\n∂Φ. (9)\nHere, the Hamiltonian function is explicitly written as\nH=TΘPΘ+TΦ\nsinΘPΦ+JDP2\nΘ+JD\nsin2ΘP2\nΦ.(10)\nwiththegeneralizedmomentumcomponents, PΘ=∂ΘW\nandPΦ=∂ΦW, and the STT components TΘandTΦ.\nThe equations for Θ and Φ in (9) show that two more\nterms, which originate from the diffusion term in Eq. (6),\ncontribute to the magnetic dynamics in addition to the\nSTT terms even in the classical limit. Eq. (9) and (10)\ngive a more complete description of the classical magne-\ntization dynamics than the semiclassical STT theory.\nD. Numerical Example\nHere we demonstrate how to apply the approach de-\nveloped above to the STT-driven quantum dynamics of\na magnet. We consider a magnet with J= 104, and the\ninitial distribution function obeys the Boltzmann distri-\nbution, i.e. PJ(/hatwidem,0) =Ce−E/kBT, where the energy of\nthe magnet in a magnetic field BisE=−M·B,Cis\nthe normalization factor, and the magnetic moment is\nM=γg/planckover2pi1J/hatwidem. We set the temperature as T= 1 K, and\nthe magnitude of the magnetic field B= 0.05 T. The di-\nrection of Bis chosen that maximum value of PJlocates\nat the angle Ω 0= (2.8,1.0). The initial distribution is\nshowintheFig.1(a). Thenweapplyaspincurrentpulse,\nwhich includes 1 .5×105electrons. The Bloch vector of\nthe electron spin is S= (0,0,1), and the wavevector is\nk= 13.6 nm−1. To calculate the scattering matrix, the\nparameters λ0andλin Eq. (1) are estimated for a mag-\nnet with effective potentials ∆ += 1.3 V, ∆ −= 0.1 V\nand thickness d= 3 nm.\nIn the simulations, we have used the method of\ncharacteristics22to solve the Hamilton-Jacobi equation\n(8), which gives the time-evolution of W(/hatwidem,t) and then\nthe distribution function PJ(/hatwidem,t) =e−JW(/hatwidem,t). Direct\nsolution of the Fokker-Planck equation (6) or the mas-\nter equation (5) is also practical but not explored here.\nThe time is measured in tN=Nτ. The distribution\nfunctions PJatt= 0.3tN,0.5tN,tNare shown in Fig. 1\n(b)(c)(d) respectively. In order to keep the normaliza-\ntion ofPJ, it is renormalized after every 500 scatterings.\nNote the different scales for Θ in Fig. 1. We found that\nPJis first expanded and then compressed, and its center\nmoves from (2.8,1.0) to (0.05,4.2), which show the effect\nof spin current on the distribution function./s50/s46/s54 /s50/s46/s55 /s50/s46/s56 /s50/s46/s57/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s40/s97/s41\n/s48/s46/s48/s50/s48/s52/s48/s54/s48\n/s49/s46/s50 /s49/s46/s53 /s49/s46/s56 /s50/s46/s49/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s40/s98/s41\n/s48/s46/s48/s50/s48/s52/s48/s54/s48\n/s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53\n/s40/s99/s41\n/s48/s46/s48/s50/s48/s52/s48/s54/s48\n/s48/s46/s48/s48 /s48/s46/s48/s52 /s48/s46/s48/s56 /s48/s46/s49/s50/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48\n/s40/s100/s41\n/s48/s46/s48/s50/s48/s52/s48/s54/s48\nFIG. 1. (color online). Distribution function PJfor the nano-\nmagnet at four different time t. (a)t= 0; (b) t= 0.3tN; (c)\nt= 0.5tN; (d)t=tN. The simulation parameters are set as:\nJ= 104, ∆+= 1.3 V, ∆ −= 0.1 V,d= 3 nm, N= 1.5×105,\nk= 13.6 nm−1,S= (0,0,1). A 200 ×200 lattice in Θ-Φ plane\nis used in the simulations.\n/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s116/s32/s47/s32/s116\n/s78/s40/s97/s41\n/s32/s32/s74 /s32/s47/s32 /s74\n/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53\n/s32/s32/s32 /s74 /s32/s47/s32 /s74\n/s116/s32/s47/s32/s116\n/s78/s40/s98/s41\nFIG. 2. (color online). Time-evolution of Jµ=x,y,zand its\nfluctuation δJµ=x,y,z(x: solid line; y: dash line; z: dotted\nline.) caused by the spin transfer torque obtained from the\nprobability distribution function (thickline) and thequa ntum\ntrajectory method (thin line).\nThe meanvalue ofthe macrospin /hatwideJandits fluctuations\nare calculated from PJas\nJµ=x,y,z(t) =/integraldisplay\ndΩP(Ω,t)/an}b∇acketle{tJ,Ω|/hatwideJµ|J,Ω/an}b∇acket∇i}ht,\nδJ2\nµ=x,y,z(t) =/integraldisplay\ndΩP(Ω,t)/an}b∇acketle{tJ,Ω|δ/hatwideJ2\nµ|J,Ω/an}b∇acket∇i}ht.\nThe results are shown in Fig. 2, where the mean magne-\ntization is switched by the STT, and the magnetization\nfluctuations are first enhanced and finally suppressed.\nComparison with the results obtained from the quantum\ntrajectory method in Ref. 15 gives reasonable agreement.\nThe approachdeveloped here together with the quantum\ntrajectory method15fits in the toolbox for micromagnet-\nics simulations with the added value of accounting for\nrelevant quantum effects.5\nIV. ELECTRON DENSITY MATRIX AFTER\nSCATTERING\nFinally, we briefly discuss the electron states after the\nscattering,whichcontaintheinformationsabouttheelec-\ntric current and current noise, etc. After tracing over\nthe degrees of freedom for the magnet in the total den-\nsity matrix ρout, the reduced density matrix of the elec-\ntron isρe\nout=TrJ[Sρe\nin⊗ρJ\ninS†]. This expression is\na generalization of the electron potential scattering in\nthe semiclassical picture23. Here, the transformation of\nthe electron state is no longer unitary due to the recoil\nof the magnet. For instance, we assume that one elec-\ntron with wavevector k >0 and spin-polarized vector\nS= (0,0,1) is injected, i.e., ρe\nin=|k,+/an}b∇acket∇i}ht/an}b∇acketle{tk,+|, and take\ntheP-representation for the quantum states of the mag-\nnet. With the scattering matrix S, we obtain\nρe\nout=/summationdisplay\nk′,s′;k′′,s′′|k′,s′/an}b∇acket∇i}ht/an}b∇acketle{tk′′,s′′|\n×/integraldisplay\ndΩP(Ω)/an}b∇acketle{tΩ|(Kk′′,s′′;k,+)†Kk′,s′;k,+|Ω/an}b∇acket∇i}ht.(11)\nFor the model (1), the terms in Eq. (11) are non-zero\nonly ifk′=±kandk′′=±k. The electron scattering\nstate is correlatedwith the quantum state of the magnet.\nThe scattering formalism in the semiclassical picture will\nbe reproduced if the Kraus operators are replaced by the\ncorresponding scattering matrix elements there. As the\nmagnets are miniaturized further24and the quantum de-\nscription becomes necessary, the semiclassical scattering\nformalismforthe electrontransportwill breakdown, and\nEq.(11)anditsgeneralizationsshouldbeexploitedasthe\nnew starting point.\nV. CONCLUSION\nIn conclusion, the STT-driven magnetization dynam-\nics has been investigated by treating the magnet as an\nopen quantum system in the exactly solvable quantum\nmacrospin model. A set of dynamical equations is es-\ntablished and the quantum-classical connection is made.\nThe full quantum picture here provides a unified and\ncomplete description of the magnetization dynamics and\nelectron transport, and further explorations of the quan-\ntum physics in spintronics along this line is expected.\nACKNOWLEDGMENTS\nWe acknowledge the support of this work by the U.\nS. Army Research Office under contract number ARO-\nMURI W911NF-08-2-0032 and NSF ECCS-1202583.Appendix A: Scattering Matrix\nFirst, wecalculatethescatteringmatrix Softhemodel\nHamiltonian\nH=−1\n2∂2\nx+δ(x)/parenleftig\nλ0/hatwideJ0+λ/hatwides·/hatwideJ/parenrightig\n.(A1)\nConsidering that one electron in state |ψe\nin/an}b∇acket∇i}htis injected\nalong thex-direction and the initial quantum state of\nthe magnet is |ψJ\nin/an}b∇acket∇i}ht, then the incoming state |Ψin/an}b∇acket∇i}htof the\nwhole system before scattering is the product state of\n|ψe\nin/an}b∇acket∇i}htand|ψJ\nin/an}b∇acket∇i}ht, i.e.,|Ψin/an}b∇acket∇i}ht=|ψe\nin/an}b∇acket∇i}ht ⊗ |ψJ\nin/an}b∇acket∇i}ht. After scat-\ntering, the outgoing state |Ψout/an}b∇acket∇i}htwill be|Ψout/an}b∇acket∇i}ht=S|Ψin/an}b∇acket∇i}ht.\nThe scattering matrix Sare determined by the boundary\nconditions at x= 0,\n(Ψin+Ψout)|x=0−= (Ψin+Ψout)|x=0+,\n/integraldisplay0+\n0−H(Ψin+Ψout)dx=ε/integraldisplay0+\n0−(Ψin+Ψout)dx,(A2)\nwhereεis the total energy of the whole system.\nThe scattering problem (A2) is simplified by utilizing\nthe symmetries of the model Hamiltonian (A1). First,\nthe kinetic energy of the electron is conserved during the\nscattering process. Thus the scattering matrix elements\nofSisnon-zeroonlyiftheabsolutevaluesoftheincoming\nand outgoing wavevectors of the electron are the same.\nSecond,theoperators( /hatwides+/hatwideJ)2and/hatwidesz+/hatwideJzarecommutative\nwith the Hamiltonian (A1), and their eigenstates |J,µ/an}b∇acket∇i}ht\nare given as\n(/hatwides+/hatwideJ)2|J,µ/an}b∇acket∇i}ht=J(J+1)|J,µ/an}b∇acket∇i}ht,\n(/hatwidesz+/hatwideJz)|J,µ/an}b∇acket∇i}ht=µ|J,µ/an}b∇acket∇i}ht,\nwithJ=J±1\n2andµ=−J,...,J. Choosing the basis\nset{|k;J,µ/an}b∇acket∇i}ht}, the scattering problem (A2) reduced to\na set ofδ-potential scattering equations, and gives the\nscattering matrix Sin this representation15. Then after\narepresentationtransformationwith the helpofClebsch-\nGorden coefficients, we obtain the scattering matrix Sin\nthe basis set {|k,s;J,m/an}b∇acket∇i}ht}, which takes a block form\nS=\n...0 0\n0Sk,µ0\n0 0...\n, (A3)\nand the form of each block Sk,µis\nSk,µ=\nt++\nk,µr++\nk,µt+−\nk,µr+−\nk,µ\nr++\nk,µt++\nk,µr+−\nk,µt+−\nk,µ\nt−+\nk,µr−+\nk,µt−−\nk,µr−−\nk,µ\nr−+\nk,µt−+\nk,µr−−\nk,µt−−\nk,µ\n.(A4)\nHere, the element tss′\nk,µ(rss′\nk,µ) is the transmission (re-\nflection) probability amplitude from |k,s′;J,µ−1\n2s′/an}b∇acket∇i}htto\n|k,s;J,µ−1\n2s/an}b∇acket∇i}ht(| −k,s;J,µ−1\n2s/an}b∇acket∇i}ht). The spin transfer6\nis related to those elements with s/ne}ationslash=s′. The explicit\nexpressions for the matrix elements are\nt++\nk,µ= cosηJ,+e−iηJ,+cos2αJ,µ+cosηJ,−e−iηJ,−sin2αJ,µ,\nr++\nk,µ=−i(sinηJ,+e−iηJ,+cos2αJ,µ+sinηJ,−e−iηJ,−sin2αJ,µ),\nt−−\nk,µ= cosηJ,+e−iηJ,+sin2αJ,µ+cosηJ,−e−iηJ,−cos2αJ,µ,\nr−−\nk,µ=−i(sinηJ,+e−iηJ,+sin2αJ,µ+sinηJ,−e−iηJ,−cos2αJ,µ),\nt−+\nk,µ= (cosηJ,+e−iηJ,+−cosηJ,−e−iηJ,−)sinαJ,µcosαJ,µ,\nr−+\nk,µ=−i(sinηJ,+e−iηJ,+−sinηJ,−e−iηJ,−)sinαJ,µcosαJ,µ,\nt+−\nk,µ= (cosηJ,+e−iηJ,+−cosηJ,−e−iηJ,−)sinαJ,µcosαJ,µ,\nr+−\nk,µ=−i(sinηJ,+e−iηJ,+−sinηJ,−e−iηJ,−)sinαJ,µcosαJ,µ.\nHere, the phase shifts are given as ηJ,±= tan−1∆J,±\nk,\nwith the effective potentials ∆ J,+=1\n2(Jλ0+Jλ),∆J,−=\n1\n2[Jλ0−(J+1)λ]. The Clebsch-Gordan coefficients\ncosαJ,µand sinαJ,µare given as cos αJ,µ=/radicalig\nJ+µ+1\n2\n2J+1,\nsinαJ,µ=/radicalig\nJ−µ+1\n2\n2J+1.\nAppendix B: Kraus Operators\nNow we express the Kraus operators Kk,s;k′,s′in the\nbasis set {|J,m/an}b∇acket∇i}ht}based on the scattering matrix Sob-\ntainedabove. Theblockformof Smeansthat Kk,s;k′,s′is\nnon-zero only if kandk′have the same absolute values.\nFor example, we have\nKk,+;k,+\n=/an}b∇acketle{tk,+|S|k,+/an}b∇acket∇i}ht\n=\ntk,J+1\n2···0···0\n......0......\n0 0t++\nk,m+1\n20 0\n......0......\n0···0···tk,−J+1\n2\n,(B1)\nwhich is a (2 J+1)-dimension diagonal matrix. With the\nClebsch-Gordan coefficients, the diagonal elements are\nrewritten as\nt++\nk,m+1\n2= (ξ+1\n2)+ζm,\nwhere\nξ=J+1\n2J+1cosηJ,+e−iηJ,++J\n2J+1cosηJ,−e−iηJ,−−1\n2\n=−i(J+1\n2J+1sinηJ,+e−iηJ,++J\n2J+1sinηJ,−e−iηJ,−)+1\n2,\nζ=1\n2J+1(cosηJ,+e−iηJ,+−cosηJ,−e−iηJ,−)\n=−i1\n2J+1(sinηJ,+e−iηJ,+−sinηJ,−e−iηJ,−).\nConsidering the matrix form of the angular momentum\noperator /hatwideJzin the basis set {|J,m/an}b∇acket∇i}ht}, the matrix (B1)shows that the Kraus operator Kk,+;k,+is just\nKk,+;k,+= (ξ+1\n2)/hatwideJ0+ζ/hatwideJz,\nwhere/hatwideJ0is the unit matrix.\nSimilarly, the other Kraus operators are written in the\ncompact form as\nKk,s;±k,s= (ξ±1\n2)/hatwideJ0+sζ/hatwideJz,Kk,−s;±k,s=ζ/hatwideJs.\nAppendix C: Quantum Master Equation\nThemasterequation(5)inthemaintextisobtainedby\nsubstituting the Kraus operators (3) into Eq. (4) there.\nThe calculations are straightforward, and yield the ex-\nplicit expressions for the operators T0andTas\nT0≡(|ξ|2−1\n4)ρJ+|ζ|2(/hatwideJzρJ/hatwideJz+/hatwideJ+ρJ/hatwideJ−)+h.c.,\nTx≡2ξζ∗ρJ/hatwideJx+|ζ|2(/hatwideJzρJ/hatwideJ+−/hatwideJ+ρJ/hatwideJz)+h.c.,\nTy≡2ξζ∗ρJ/hatwideJy+i|ζ|2(/hatwideJ+ρJ/hatwideJz−/hatwideJzρJ/hatwideJ+)+h.c.,\nTz≡2ξζ∗ρJ/hatwideJz+|ζ|2(/hatwideJ+ρJ/hatwideJ−−/hatwideJ−ρJ/hatwideJ+)+h.c..\nThe terms containing a single angular momentum oper-\nators inTcan be interpreted as a field-like torque, while\nthe terms including two angular momentum operators\nthe Slonczewski-type torque and the quantum fluctua-\ntions. This becomes clearer in the spin coherent state\nrepresentation.\nAppendix D: The Fokker-Planck Equation\nHere we explain the derivation of the Fokker-Planck\nequation (6) in the paper. In the spin coherent state rep-\nresentation {|J,Ω/an}b∇acket∇i}ht}, the density matrix ρJis expressed\nas17\nρJ=/integraldisplay\ndΩPJ(Ω)|J,Ω/an}b∇acket∇i}ht/an}b∇acketle{tJ,Ω|. (D1)\nUsingS= (α,β,γ), and substituting the expression (D1)\ninto the master equation (5) in the paper, and utilizing\nthe differential forms of the operators17/hatwideJi|J,Ω/an}b∇acket∇i}ht/an}b∇acketle{tJ,Ω|/hatwideJj\n(i,j= 0,+,−,z), we derive the differential equation for\nPJ(Ω) as\n∂PJ\n∂t=1\nsinΘ∂(−TΘPJ)\n∂Θ+1\nsinΘ∂(−TΦPJ)\n∂Φ\n+1\nsinΘ∂\n∂Θ[sinΘ∂(DPJ)\n∂Θ]+1\nsin2Θ∂2(DPJ)\n∂Φ2.\n(D2)\nHere,\nTΘ=A(αcosΘcosΦ+ βcosΘsinΦ −γsinΘ)\n+B(αsinΦ−βcosΦ),7\nTΦ=A(−αsinΦ+βcosΦ)\n+B(αcosΘcosΦ+ βcosΘsinΦ −γsinΘ),\nD=A\n2J+1(1−αsinΘcosΦ −βsinΘsinΦ −γcosΘ),\nwith the coefficients A= (2J+1)|ζ|2\nτ,B=2ℑ(ξ∗ζ)\nτ. Fur-\nther analysis shows that TΘandTΦare the components\nof the spin transfer torque T=A(/hatwidem×S)×/hatwidem+B/hatwidem×S\nin the spherical coordinates, where the unit vector /hatwidemdenotes the direction of the macrospin. Replacement of\nthe differential operators in spherical coordinates by the\ndivergence operator ∇and Laplace operator ∇2reduces\nEq. (D2) to the simple form of the Fokker-Planck equa-\ntion (FPE)\n∂\n∂tPJ(/hatwidem,t) =−∇·(TPJ)+∇2(DPJ).(D3)\n∗Present address: Department of Physics, University of\nHong Kong\n†lsham@ucsd.edu\n1H. J. Carmichael, Statistical methods in quantum optics 1:\nMaster equations and Fokker-Planck equations (Springer,\nBerlin, 1999).\n2H. J. Carmichael, Statistical methods in quantum optics 2:\nNonclassical fields (Springer, Berlin, 2008).\n3H.-P. Breuer and F. Petruccione, The Theory of Open\nQuantum Systems (Oxford University Press, 2002).\n4F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas,\nPhys. Rev. A 6, 2211 (1972).\n5C. Weedbrook, S. Pirandola, R. Garca-Patron, N. J. Cerf,\nT. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys.\n84, 621 (2012).\n6A. Aharoni, Introduction to the Theory of Ferromagnetism\n(Clarendon Press, Oxford, 1996).\n7H. Suhl, Relaxation processes in micromagnetics (Oxford\nUniversity Press, 2007).\n8D. Ralph and M. Stiles, Journal of Magnetism and Mag-\nnetic Materials 320, 1190 (2008).\n9A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater 11, 372\n(2012).\n10J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n11W. Wernsdorfer, Advances in Chemical Physics 118, 99\n(2007).\n12J.C.Slonczewski, JournalofMagnetism andMagneticMa-\nterials159, L1 (1996).13L. Berger, Phys. Rev. B 54, 9353 (1996).\n14M. Madami, S. Bonetti, G. Consolo, S. Tacchi, G. Carlotti,\nG. Gubbiotti, F. B. Mancoff, M. A. Yar, and J. ˚Akerman,\nNature Nanotech. 6, 635 (2011).\n15Y. Wang and L. J. Sham, Phys. Rev. B 85, 092403 (2012).\n16K. Krause, States, Effects, And Operations Fundamental\nNotions Of Quantum Theory , edited by A. Bohm, J. Dol-\nlard, and W. Wootters (Springer-Verlag, Berlin, 1983).\n17L. M. Narducci, C. M. Bowden, V. Bluemel, G. P. Gar-\nrazana, and R. A. Tuft, Phys. Rev. A 11, 973 (1975).\n18V. E. Demidov, S. Urazhdin, E. R. J. Edwards, M. D.\nStiles, R. D. McMichael, and S. O. Demokritov, Phys. Rev.\nLett.107, 107204 (2011).\n19W. F. Brown, Phys. Rev. 130, 1677 (1963).\n20Z. Li and S. Zhang, Phys. Rev. B 69, 134416 (2004).\n21H. Risken, The Fokker-Planck Equation (Springer-Verlag,\nBerlin, 1984).\n22R. Courant and D. Hilbert, Methods of Mathematical\nPhysics (Interscience, New York, 1953).\n23J. Foros, A. Brataas, Y. Tserkovnyak, and G. E.W. Bauer,\nPhys. Rev. Lett. 95, 016601 (2005).\n24J. K. W. Yang, Y. Chen, T. Huang, H. Duan, N. Thiya-\ngarajah, H. K. Hui, S. H. Leong, and V. Ng, Nanotechnol-\nogy22, 385301 (2011)."    },    {        "title": "1012.3363v2.Enhanced_Tensor_Force_Contribution_in_Collision_Dynamics.pdf",        "content": "arXiv:1012.3363v2  [nucl-th]  6 Jul 2011Enhanced Tensor-Force Contribution in Collision Dynamics\nYoritaka Iwata1and Joachim A. Maruhn2\n1GSI Helmholtzzentrum f¨ ur Schwerionenforschung, D-64291 Darmstadt, Germany and\n2Institut f¨ ur Theoretische Physik, Universit¨ at Frankfur t, D-60325 Frankfurt, Germany\n(Dated: March 2, 2022)\nThe tensor and spin-orbit forces contribute essentially to the formation of the spin mean field,\nand give rise to the same dynamical effect, namely spin polari zation. In this paper, based on time-\ndependent density functional calculations, we show that th e tensor force, which usually acts like\na small correction to the spin-orbit force, becomes more imp ortant in heavy-ion reactions and the\neffect increases with the mass of the system.\nPACS numbers: 25.70.Jj, 21.60.Jz, 21.30.Fe\nINTRODUCTION\nThe tensor force, which is necessary to explain the\nproperties of the deuteron, attracts special attention re-\ncently, because it has turned out to play an essential\nrole in the existence limit of exotic nuclei, as well as\nthe nuclear shell structure far from the β-stability line\n(for example, see [1–8]). An important feature is that\nthe spatial average of the tensor force is exactly equal to\nzero, so that its effect is spatially localized. On the other\nhand, the spin-orbit force, which is necessary to explain\nthe large spin polarizations of scattered nucleons, plays\na crucial role in the nuclear shell structure. The origin of\nthe tensor force can be found in the one-pion exchange\npotential, and that of the spin-orbit force in the relativis-\ntic aspects of quantum dynamics.\nThus the tensor and spin-orbit forces are quite differ-\nent in their origins, while resulting in the same dynam-\nical effect, namely, spin polarization. Spin polarization,\nwhich arises mostly from the spin-orbit force, sponta-\nneously takes place in the early stage of heavy-ion reac-\ntions, and affects the equilibration process to a large ex-\ntent. As long as the microscopic time-dependent Skyrme\nenergy density functional (Skyrme-EDF) calculationsare\nconcerned, the appearance of spontaneous spin polariza-\ntion even in centralcollisions between β-stable nuclei was\nshown,anditsoriginwasclarifiedtobethetime-oddpart\nof the spin-orbit force [9]. Therefore, the enhancement or\nreduction of spin polarization gives an ideal framework\nto pin down the properties of the tensor force in collision\nsituations.\nIn this paper, the role of the tensor force in heavy-ion\nreactions is investigated based on time-dependent den-\nsity functional calculations with explicitly implemented\ntensor force, where the time-odd part of the spin-orbit\nforce is also fully taken into account. Special attention is\npaid to the effect of the tensor force on time evolution.\nAs a result, some information on the importance of the\ncontribution from the tensor force in heavy-ion reactions\nis presented.THEORETICAL FRAMEWORK\nMean field due to spin-orbit and tensor forces\nThe contribution of the tensor force, whose role was\nunderestimatedandmostlyneglectedforalongtime, was\nsubstantially studied in the context of Skyrme-EDF only\nrecently [5–8]. Here we begin with the functional form\nof the tensor and spin-orbit forces in Skyrme-EDF. Let\nρ,σandJrepresent the number density, spin density,\nand spin-orbit density, respectively. The contribution of\nthe tensor and spin-orbit forces to the energy density\nfunctional has the form\nWq(r)·(−i)(∇×σ) (1)\nwhereq=n,p(nandpstand for neutron and proton,\nrespectively). Wq(r), which is called the form factor of\nthe spin mean field, is decomposed into the contributions\nfrom spin-orbit and tensor forces.\nWq(r) =WLS\nq(r)+WT\nq(r), (2)\nwhereWLS\nq(r) andWT\nq(r) denote the form factors of\nspin-orbit and tensor mean fields, respectively. The con-\ntribution of the spin-orbit force to the functional [10] is\nrepresented by\nWLS\nq(r) =1\n2W0(∇ρ(r)+∇ρq(r))+1\n8(t1−t2)Jq(r).\nNote that the second term on the right-hand side, whose\ncontribution in collision situations was discussed in [11],\nhas never been taken into account for some modern\nSkyrme parameterizations such as SLy4d and SKM*, be-\ncause it makes fitting spin-orbit splittings more difficult.\nAlthough there are several versions of the tensor force,\nwe are concerned with the natural tensor force only. Its\ncontribution to the energy functional is represented by\nWT\nq(r) =αJq(r)+βJq′(r) (3)\nwithq′=n,psatisfying q/negationslash=q′, according to Stancu-\nBrink-Flocard [12], where the unique contribution of the\ntensor force can be found in Jq′(r) toWq(r). The full2\n 0 0 . 0 6 0 . 1 2 0 . 1 8Density [fm   ] -3\n3 .0 x 1 0     s-2 2 4 .5 x 1 0     s-2 2 6 .0 x 1 0     s-2 2 1 .5 x 1 0     s-2 2 7 .5 x 1 0     s-2 2 x\nzSkM* + SV-tl s \nFIG. 1: (color online) Time evolution of40Ca +40Ca at the bombarding energy 130 MeV (c.m.). Snapshots of the d ensity are\nshown in a fixed square (40 ×40 fm2) on the reaction plane, where contour lines are plotted for m ultiples of 0.04 fm−3. The\nforce used is SkM* + SV-tls.\nintroduction of the tensor force requires to refit addi-\ntionally the corresponding central-force parameters. Al-\nthoughthefullintroductionbringsaboutlargelydifferent\nand complicated contributions depending on the choice\nof force parameter sets [13], it has been shown to mostly\nresult in weakeningthe natural contribution [14]. Several\nversions of the tensor force are compared in [14]. Here\nwe restrict discussion to the tensor force as defined by\nEq. (3), because the aim is not a discussion of the exis-\ntence limit of exotic nuclei, but rather the general fea-\ntures of the tensor contribution in reactions. It is read-\nily seen that the effect of the tensor force corresponds\nto a quantitative modification of that due to the spin-\norbit force. Accordingly, the contribution of the tensor\nforce should be discussed in association with the spin-\norbit force.\nTensor-force contribution in collision situations\nA framework for measuring the effects of the tensor\nforce is presented with a focus on collision dynamics.\nConcerning the spin polarization, it is reasonable to be-\ngin with a discussion of spin-orbit coupling. It is defined\nby the scalar triple product\nL·S=−i/planckover2pi1(r×p)·(σ+σ′)\n=−i/planckover2pi1r·(p×(σ+σ′)),(4)\nwhereσandσ′denote the spins of the two nucleons. In\ncollision situations r×pis related to the impact parame-\nter. Comparing Eqs. (1) and (4), Wq(r) in Eq. (1) plays\nthe role of the vector rin Eq. (4), where the momentum\npis replaced approximately by ∇in the Skyrme-EDF.\nIn ordertoevaluatethe contributionofthe tensorforce\nto spontaneous spin polarization, we introduce a proper\ntheoretical setting of heavy-ion collisions. Our starting\npoint is that the tensor and the spin-orbit forces are lo-\ncalized effects, which are not easy to compare in collision\ndynamics, if there is some similarity in their localized\npatterns. Let the reaction plane be ( x,z) with the initial\ncollision direction z, and the direction perpendicular tothe reaction plane be y. For simplicity, the spin direction\nof the initial state is assumed to be parallel to the y-axis.\nIn this setting, because only the z-component of pand\nthey-component of σare non-zero, we have\nL·S=−i/planckover2pi1x/parenleftBig\npy(σ+σ′)z−pz(σ+σ′)y/parenrightBig\n=i/planckover2pi1xpz(σ+σ′)y.(5)\nWe see that only the x-component of the vector r, and\nthus the x-component of Wq(r) play a role. In this set-\nting, the role of the tensor force in the spin polarization\ncan be evaluated by the corresponding x-component of\nWT\nq(r). Accordingly, the tensor and spin-orbit forces\ncan be compared, if there is a certain similarity between\nthex-components of WT\nq(r) andWLS\nq(r) (otherwise at-\ntraction or repulsion happen irregularly from place to\nplace). Note that their similarity, which will be shown to\nbe true, is not trivial. In the following the x-components\nofWT\nq(r) andWLS\nq(r) are simply represented by WT\nq(r)\nandWLS\nq(r), if there is no ambiguity.\nROLE OF THE TENSOR FORCE\nSpontaneous spin polarization\nA systematic three-dimensional time-dependent den-\nsity functional calculation is carried out in a spatial box\n48×48×48 fm3with a spatial grid spacing of 0.8 fm, in\nwhich the Skyrme-force parameter set SV-tls [14] is used\nfor the tensor part, and SkM* and SLy4d [16] for the re-\nmainder including the spin-orbit force: α= 71.102 [MeV\nfm−5] andβ= 35.142 [MeV fm−5]. The parameter set\nSV-tls was lately introduced in the context of the refit-\nted tensor force; it is one of the most reliable parameter\nsets in terms of reproducing the contribution of the form\nfactorWT\nq(r) of the tensor mean field. The relative ve-\nlocity in the collisions is set to 10 % of the speed of light,\nand the initial distance of the colliding nuclei to 20.0 fm;\ntheir initial positions are (0,0,10) and (0,0,-10). In order\nto pay special attention to the mass-dependent general3\n-0.01 0 0.01\n x z-8\n 0\n 8-8\n 0\n 8   [fm  ]-3\nP\nFIG. 2: (color online) Spin distribution (the spin is projec ted\nonto the y-axis) of a compound nucleus. A snapshot of a\ncomposite nucleus, which corresponds to the case at time =\n6.0×10−22s in Fig. 1, is shown on the reaction plane. For\nreference, contours of the density distribution are also sh own\n(contour = 0.01, 0.06, 0.11 and 0.16 fm−3).\nfeatures, we consider central collisions between identical\nN=Znuclei:16O +16O,40Ca +40Ca and56Ni +56Ni.\nThe contributions from JqandJq′in Eq. (3) are not so\ndifferent for collisions between N=Znuclei, therefore the\nparameter dependence mostly arises from the sum of α\nandβ. Some featuresofthe tensorforceactingon N=Z\nbound nuclei were studied in [8].\nFigure 1 shows the time evolution of40Ca +40Ca re-\nsulting in fusion, where the terms associated with the\ntensor force (SV-tls) are explicitly included. The same\ncalculation without the spin-orbit force does not achieve\nfusion. Omitting the tensor force while including the\nspin-orbit force shows no notable difference to the den-\nsityevolutionwithallforcetermsincluded. Thissuggests\nthat large dissipation arises from the spin-orbit force,\nwhile the tensor-force contribution is definitely small.\nThe composite nucleus evolves with a continuing oscil-\nlation; the two nuclei get into contact around time = 4.2\n×10−22s, and the first full-overlap is achieved at 5.6 ×\n10−22s.\nLet us consider the y-projection of spin for each single\nnucleon. The spin distribution of the colliding nuclei is\ncalculated by their superposition:\nP(t,r) =ρ(t,r)↑−ρ(t,r)↓,\nwhereρ(t,r)↑andρ(t,r)↓denote the densities of spin-up\nand spin-down components, respectively. In this defi-\nnition, the density plays the role of weight. The value\nofP(t,x) is positive if the spin-up component is more\nabundant, zero for saturated spins, and negative other-\nwise. As is seen from the presence of ( σ+σ′)yin Eq. (5),\nthe problem of comparing the different role of tensor and\nspin-orbitforcesbecomesmeaninglessifspontaneousspinz\nxProton \nTensor force\nSpin-orbit force\n-A 0 A [MeV fm]Neutron\nA = 2.5A = 2.5\nA = 25A = 25W    pLS\n-8\n0\n80-8\n8W    pT\nW    nT\nW    nLS\nFIG.3: (color online) Snapshotsofthe x-componentof Wq(r)\nprojected on the reaction plane, corresponding to the case\nat time = 6.0 ×10−22s in Fig. 1. The values are plotted\nseparately for thetensor andspin-orbit forces, andfor pro tons\n(q=p) and neutrons ( q=n), respectively.\npolarization is absent. Spin polarization appears for all\nthe reactions and all the force parameter sets used; e.g.,\nin Fig. 2, the presence of spin polarization is shown for\n40Ca +40Ca. As a result, the concept of examining the\nroleofthetensorforceinthepresenceofspinpolarization\nis valid and will be carried out in the following.\nFigure 2 shows that strong spin polarization is located\non the edge of the density distribution. The localized\npattern of the spin structure is complicated, leading to\na complicated localization of attraction and repulsion\ndue to the tensor force. The spin distribution is point-\nsymmetric with respect to the origin, which reflects the\nsymmetry of the central collision. Note that the spatial\naverageofspin polarizationforthe spin-saturatedsystem\nis equal to zero.\nComparison between tensor and spin-orbit forces\nLet us begin with the effect of the tensor force in a\ncompound nucleus formed briefly after the full-overlap\nsituation (time = 6.0 ×10−22s). In case of40Ca +\n40Ca, Fig. 3 compares the x-components of Wq(r) for\nthe tensor and spin-orbit forces. Both distributions are\nantisymmetric with respect to the z-axis, and have sim-\nilar distributions but different signs and amplitudes. It\nis clearly seen that the tensor-force contribution is op-4\nSkM* + SV-tl s \n+ A\n- A0\nA = 30 A = 25A = 2.0 A = 2.5 A = 6.0\nW    pLSW  pT\n  Ni +   Ni56 56A = 25\n  Ca +   Ca40 40  O +   O16 16x\nz[MeV fm]\nFIG. 4: (color online) Snapshots of the x-components of the\nform factors of spin mean field WT\np(r) (upper ones) and\nWLS\np(r) (lower ones) at time = 6.0 ×10−22s are shown in a\nsquare (30 ×20 fm2) on the reaction plane. The maximum\namplitude Aof the function is shown in the lower right-hand\nside of each plot.\nposite to the spin-orbit force contribution, and amounts\nto less than 10 percent of latter. It follows that the to-\ntal contribution from tensor and spin-orbit force is not\nso different from the contribution of the spin-orbit force\nalone. No significant difference is noticed between the\nvalues for protons and neutrons. This is expected for a\ncollision between N=Znuclei.\nThis difference in sign and the smallness of the ten-\nsor force contribution compared to the spin-orbit force\ncontribution is found to hold regardless of the choice of\nforce parameter set and the mass of the colliding nuclei.\nThis difference in sign, however, did not appear for the\nyandz-components. On the other hand, comparing the\nx-components of WT\nq(r) andWLS\nq(r) at time = 6.0 ×\n10−22s for16O +16O,40Ca +40Ca and56Ni +56Ni\nas shown in Fig. 4, there is a highly noticeable increase\nwith mass for the tensor-force contributions, while it is\nonly modest for those of the spin-orbit force.\nLet us move on to the time-dependent features of the\ntensor force. For the reaction shown in Fig 1, the time\nevolution of the ratio between tensor and spin-orbit con-\ntributions\nWT\nq/WLS\nq(t) =maxr(WT\nq(t,r))\nmaxr(WLSq(t,r))(6)\nis shown in Fig. 5. In addition, the corresponding x-\ncomponents of WT\np(r) andWLS\np(r) are also shown at\ntimes 1.5 ×10−22s, 6×10−22s and 15 ×10−22s. The\nisoscalar dipole mode shown in Fig. 5 suggest that the\nfull-overlap is achieved at time = 5.5 ×10−22s, and the\nmaximal elongation of the composite nucleus at time =\n7.25×10−22s. The relaxation of the tensor contribution\nis not stronglycorrelatedwith that of the isoscalardipole\noscillation (density oscillation towardsthe fused system).\nThe contribution of the tensor force is quite small before\nthe contact time (4.2 ×10−22s), increases after the con-\ntact time, achieves local-maximum at times 6.75 ×10−22TABLE I: Enhancement of the tensor-force contribution for\n40Ca +40Ca. Values of Eq. (7) are calculated for different\nforce parameter sets and isospins.\nParameter set Protons ( q=p)Neutrons ( q=n)\nSkM* + SV-tls 5.94 6.23\nSLy4d + SV-tls 6.54 7.11\ns and 9.00 ×10−22s, and relaxes afterwards.\nSeveral points should be remarked here. First, the\ntensor-force contribution is enhanced in collision situa-\ntions, being up to 10 times larger than before the con-\ntact time. Second, the opposite sign and the smallness of\nthe tensor compared to the spin-orbit contributions are\napparent during the heavy-ion collision but not before\ncontact. The opposite sign means that the contribution\nof the tensor force continues to weaken the spin polariza-\ntion during the reaction. Third, the similarity between\nprotons and neutrons is confirmed throughout the reac-\ntion.\nThe tensor-force contribution is compared for different\nforce parameter sets in Table I, where the enhancement\nis calculated by the ratio\nWT\nq/WLS\nq(t= 6.5×10−22s)\nWTq/WLSq(t= 1.5×10−22s). (7)\nwhereWT\nq/WLS\nq(t) is calculated as shown in Eq. (6).\nThis table shows that the enhancement is true indepen-\ndent of the choice of force parameter sets, and no signif-\nicant difference exists between protons and neutrons.\nMass dependence\nAs alreadymentioned, the opposite sign and the small-\nness of the tensor compared to the spin-orbit force con-\ntribution is valid independent of the mass. Note that a\ncalculation using SLy4d + SV-tls showed the same fea-\ntures, hinting that this is probably not strongly force-\ndependent.\nFigure 6 shows the mass dependence of the ratio of\ntensor to spin-orbit contributions for protons and neu-\ntron (Eq. (6)), where the values at time = 6.0 ×10−22s\nare chosen to calculate the ratio. In all cases, this time\ncorresponds to the time briefly after the first full overlap\nand shows a relatively large tensor contribution close to\nthe firstmaximum, sothat it islegitimate tocomparethe\nmagnitude for the three cases. While the values are not\nexactly the same for the two parameter sets, they show\nthe same trend; the tensor force contribution becomes\nlarger for reactions involving a heavier nucleus. For the\nheavier cases, 20 percent contribution from the tensor\nforce compared to the spin-orbit contribution is noticed5\n1 .5 3 4 .5 6 7 .5 9 1 0 .5 1 200 .0 50 .10 .1 5\nTime  [1 0     s]-2 2 \n1 .5 6 1 0.5A = 0 .3 A = 2 .5 A =1 .5\nA = 25 A = 25 A = 25+ A\n- A0\nx\nzq = p\nq = n\nW  pT\nW    pLSis-dipole1 .0\n0 Normarized  is-di pole mode\n[MeV fm]W   /    qTW    qLS\nFIG. 5: (color online) The time evolution of the ratio of cont ributions from tensor force to those of the spin-orbit force is shown\nfor protons and neutrons, respectively (upper panel), wher e the calculated points (at multiples 0.75 ×10−22s) are connected\nby 3rd-order spline functions. This reaction corresponds t o the case shown in Fig 1. For reference, the time evolution of the\nisoscalar dipole (is-dipole) mode is shown by a dotted line ( upper panel). The corresponding reaction-plane snapshots of the\nx-components of WT\nq(r) andWLS\nq(r) are shown in a square (30 ×15 fm2) (lower panel), where the maximum amplitude Aof\nthe function is shown in the lower right hand side of each plot .\nTABLE II: Mass dependence of the growth of spin polariza-\ntion (for an explanation see text). For both parameter sets,\nvalues are normalized by the values obtained for16O +16O.\nParameter set16O +16O40Ca +40Ca56Ni +56Ni\nSkM* + SV-tls 1.000 0.396 0.260\nSLy4d + SV-tls 1.000 0.571 0.085\n(SkM* + SV-tls). This is not a negligible effect consid-\nering the remarkable spin-orbit splitting in the ground\nstates of heavy nuclei. This should have a certain impact\non superheavy synthesis; the tensor force is suggested\nto play a considerable role in whether a heavy compos-\nite nucleus is formed successfully or not. On the other\nhand, the spin polarization becomes smaller for reactions\ninvolving heavier nuclei. The statistical ratio of spin po-\nlarization\nmaxr(ρ↑(t,r)−ρ↓(t,r))/summationtext\nr(ρ↑(t,r)+ρ↓(t,r))\nbetween time = 6.0 ×10−22s and 1.5 ×10−22s, which\ncorresponds to the amplitude of spin polarization due to\nthe collision, is summarized in Table II. Thus the tensor-\nforce contribution tends to survive for the heavier cases,\nwhile the spin-orbit force contribution decreases sharply\nwith mass. Note that there is no serious discrepancy\nbetween neutrons and protons visible in Fig. 6.\nFinally, the validity of the obtained results is also con-\nfirmed by additionally examining an old tensor force pa-\nrameter set proposed by Stancu-Sprung [12, 17] ( α=154.390 [MeV fm−5] andβ= 139.910 [MeV fm−5]). The\nmajor difference is that its amplitude is actually smaller\nthan the spin-orbit force contribution, but reaches as\nmuch as 50% of the spin-orbit contribution in56Ni +\n56Ni. The difference between the two parameters can be\nrelated to the largeness of the αandβvalues proposed\nin the Refs. [12, 17] compared to Ref. [14].\nCONCLUSION\nBased on time-dependent density functional calcula-\ntions with explicitly implemented tensor force, the role\nof the tensor force has been studied in the context of col-\nlision dynamics. It is remarkable that the contribution\nfrom the tensor force is enhanced in collision situations.\nIts contribution is mass-dependent and has considerable\ninfluence on reactions involving a heavier nucleus.\nAs long as heavy-ion reactions between N=Zidenti-\ncal nuclei are concerned, the opposite sign and the small-\nness of the tensor force contribution compared to that\nof the spin-orbit force has been confirmed independent\nof mass. In particular, the opposite sign means that the\nspin polarization, thus the large dissipation due to the\nspin-orbit force, is reduced by the tensor force. We con-\nclude that the tensor-force contribution is rather impor-\ntantinheavy-ionreactionswith respecttothemagnitude\nof dissipation. The results presented in this paper give\na solid starting point for future researches clarifying the\nrole of the tensor force in heavy-ion reactions involving\nexotic nuclei, where the drastically different contribution6\nMass number (A) Mass number (A)20 40 60 80 100 12000.050.10.150.20.25\nS kM * + SV-tls\nS Ly4 d + SV-tls\n20 40 60 80 100 12000.050.10.150.20.25\nS kM * + SV-tls\nS Ly4 d + SV-tlsProton (q = p) Neutron (q = n)W   /    qTW    qLS\nFIG. 6: (color online) The ratios between tensor and spin-\norbit force contributions for protons (left panel; q=p) and\nneutrons (right panel; q=n) as functions of the mass of the\ncomposite nucleus. The values at time = 6.0 ×10−22s are\nchosen to calculate the ratio.\nfromJqandJq′in Eq. (3) might play a significant role.\nThis work was supported by the Helmholtz Alliance\nHA216/EMMIandbytheGermanBMBFundercontract\nNo. 06FY159D. The authors would like to thank Prof.\nP. -G. Reinhard for valuable suggestions, and Prof. N.\nItagaki for fruitful discussion.\n[1] T. Otsuka et. al., Phys. Rev. Lett. 95232502 (2005).[2] T. Otsuka et. al., Phys. Rev. Lett. 97162501 (2006).\n[3] T. Otsuka et. al., Phys. Rev. Lett. 104012501 (2010).\n[4] T. Nakamura et. al., Phys. Rev. Lett. 103262501 (2009).\n[5] B. A. Brown, T. Duguet, T. Otsuka, D. Abe, and T.\nSuzuki, Phys. Rev. C74, 061303(R) (2006).\n[6] T. Lesinski, M. Bender, K. Bennaceur, T. Duguet, and\nJ. Meyer, Phys. Rev. C76014312 (2007).\n[7] G. Colo, H. Sagawa, S. Fracasso, and P. F. Bortignon,\nPhys. Lett. B646(2007) 227.\n[8] M. Bender, K. Bennaceur, T. Duguet, P.-H. Heenen, T.\nLesinski, and J. Meyer, Phys. Rev. C80064302 (2009).\n[9] J. A. Maruhn, P.-G. Reinhard, P. D. Stevenson, and M.\nR. Strayer, Phys. Rev. C74027601 (2006).\n[10] D. Vautherin, and D. M. Brink, Phys. Rev. C53 626\n(1972).\n[11] A.S.Umar, andV.E.Oberacker, Phys.Rev. C73054607\n(1972).\n[12] Fl. Stancu, D. M. Brink, and H. Flocard, Phys. Lett.\nB68108 (1977).\n[13] E. B. Suckling, and P. D. Stevenson, Eur. Phys. Lett. 90\n12001 (2010).\n[14] P. Kl¨ upfel, P.-G. Reinhard, T. J. B¨ urvenich, and J. A.\nMaruhn, Phys. Rev. C79034310 (2009).\n[15] J. Bartel, P. Quentin, M. Brack, C. Guet, and H. B.\nHakansson, Nucl. Phys. A38679 (1982).\n[16] E. Chabanat, P. Bonche, P. Hansel, J. Meyer and R.\nSchaeffer, Nucl. Phys. A635231 (1998); A643441(E)\n(1998).\n[17] D. W. L. Sprung, Nucl. Phys. A18297 (1972)."    },    {        "title": "1411.2779v1.Magnonic_Charge_Pumping_via_Spin_Orbit_Coupling.pdf",        "content": "1 \n Magnonic Charge Pumping via Spin-Orbit Coupling \nAuthors:   Chiara Ciccarelli1†, Kjetil M. D. Hals2,3†, Andrew Irvine1, Vit Novak4\n, Yaroslav \nTserkovnyak5 , Hidekazu Kurebayashi1,6‡, Arne Brataas2* and Andrew Ferguson1. \nAffiliations: \n1Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE \n2 Department of Physics, Norwegian University of Science and Technology, NO-7491, \nTrondheim, Norway  \n3The Niels Bohr International Academy, Niels Bohr Institute, 2100 Copenhagen, Denmark \n4Institute of Physics ASCR, v.v.i., Cukrova rnická 10, 162 53 Praha 6, Czech Republic \n5Department of Physics and Astronomy, University of California, Los Angeles, California \n90095, USA \n6PRESTO, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan \n†These authors contributed equally. \n‡Present address: London Centre for Nanotechnology, University College London, London \nWC1H 0AH, U.K. and Department of Electroni c and Electrical Engineering, University \nCollege London, London WC1E 7JE, U.K. \n*Correspondence to:  arne.brataas@ntnu.no \n \n \n  \n \n  \n \n  \n \n  2 \n The interplay between spin, charge, and or bital degrees of freedom has led to the \ndevelopment of spintronic devices like spin -torque oscillators, spin-logic devices, and \nspin-transfer torque magnetic random-acces s memories. In this development spin \npumping, the process where pure spin-cur rents are generated from magnetisation \nprecession 1, has proved to be a powerful method for probing spin physics and \nmagnetisation dynamics 1, 2, 3, 4, 5, 6, 7. The effect originates from direct conversion of low-\nenergy quantised spin-waves in the magnet, known as magnons, into a flow of spins \nfrom the precessing magnet to adjacent normal metal leads. The spin-pumping \nphenomenon represents a conven ient way to electrically detect magnetisation dynamics \n2, 3, 4, 5, 6, 7, however, precessing magnets have been limited so far to pump pure spin \ncurrents, which require a secondary spin-charge conversion element  such as heavy \nmetals with large spin Hall angle 5, 6, 7 or multi-layer layouts 8 to be detectable. Here, we \nreport the experimental observation of charge pumping in which a precessing ferromagnet pumps a charge current, demons trating direct conversion of magnons into \nhigh-frequency currents via the relativistic sp in-orbit interaction. The generated electric \ncurrent, differently from spin currents ge nerated by spin-pumping, can be directly \ndetected without the need of  any additional spin to charge conversion mechanism\n and \ncontains amplitude and phase information about the relativistic current-driven \nmagnetisation dynamics. The charge-pumping phenomenon is generic and gives a deeper understanding of the recently observed spin-orbit torques, of which it is the \nreciprocal effect and which currently attrac t interest for their potential in manipulating \nmagnetic information. Furt hermore, charge pumping provides a novel link between \nmagnetism and electricity and may find applic ation in sourcing alternating electric \ncurrents. \n \n \nA flow of spin angular momentum without an  accompanying charge current is called a pure \nspin current. A simple way to generate pure spin currents is via spin-pumping \n1. The \nphenomenon originates from direct conversion of low-energy quantised spin-waves in the \nmagnet, known as magnons, into a flow of spins from the precessing magnet to adjacent \nnormal metal leads. The reciprocal effect, in which a spin current is able to excite \nmagnetisation dynamics, is known as the spin -transfer torque. In this case spin-angular \nmomentum is transferred from the carriers to the magnet, applying a torque to the 3 \n magnetisation9. This pair of reciprocal effects underlie s much of the progress in spintroincs to \ndate.  \n \nThe spin-orbit coupling provides an efficient route to electrically generate magnetic torques \nfrom orbital motion, i.e., from an  electric current (Fig. 1a,c) 10, 11, 12, 13, 14, 15, 16, 17. These \nrelativistic spin-orbit torques (SOTs) exist in ferromagnets with broken spatial inversion \nsymmetry. They have been reported in (Ga,Mn)As, a material with a broken bulk inversion symmetry \n18, 19, 20, 21 as well as heterostructures comprising ferromagnetic metals 22, 23, 24, 25, 26, \n27. The SOT has been observed to have both field-like 18, 22 and damping-like 21, 24, 25 \ncontributions. Differently from non-relativistic spin-transfer torques, SOTs do not rely on a \nsecondary element that spin-polarises the currents: rather a spin-polarisation results from the carrier velocity. Despite showing promise for magnetic memory applications, the \nunderstanding of SOTs is, however, still immatu re and a further development of the field \nrequires improved theoretical models and ex perimental techniques to reveal their full \ncomplexity. The Onsager reciprocity relations \n28 imply that, as for spin-pumping/spin transfer \ntorque, there exists a reciprocal phenomenon of the SOT, namely, charge pumping generated from magnetisation precession (Fig. 1b,d) \n14, 29. \n \nThe underlying physics of charge pumping is direct conversion of magnons into charge \ncurrents via the spin-orbit coupling. We will therefore refer to this process as magnonic \ncharge pumping . Any external force that drives magnetisation precession can generate \nmagnonic charge pumping. Examples of potential driving forces are magnetic fields, \nalternating currents, thermal gradients, or ci rcularly polarised light pulses. Magnonic charge \npumping can be a favourable alternative to spin pumping for detection of magnetisation \ndynamics, because the effect does not require an additional conversion mechanism to be \nmeasureable. Moreover, charge pumping contai ns information about the SOTs, and therefore \nopens the door for a novel experimental technique to explore these relativistic torques. Since \nthe coefficients that describe the SOT are related to those that describe charge pumping, via \nthe Onsager relations, it is possible to experime ntally measure the amplitude and symmetry of \nthe spin-orbit torque, in order to determine the expected charge-pumping signal. In our \nexperiment, we do this and compare the result to the experimentally measured charge-\npumping signal.   4 \n A simple explanation of magnonic charge pumping can be found from the Hamiltonian \n \nH = p2/2m + p⋅Λ⋅σ + ∆m⋅σ ,                                                                 (1)                         \n                       \nwhere  σ= (σ1, σ2, σ3) is the carrier’s spin operator represented by the Pauli matrices σi, p is \nthe momentum operator, ∆ is the exchange splitting and m is the unit vector in the direction \nof the magnetisation. The second term in the Hamiltonian (1) represents the spin-orbit \ncoupling, where the matrix Λ parameterises the spin-orbit coupling. The velocity operator \nresulting from the Hamiltonian (1) is \n \nv = ∂H/∂p = p/m + Λ⋅σ .                                                                          (2) \n \nThe last term in Eq. (2) is the anomalous te rm, which mediates a coupling between spin and \nmomentum. In ferromagnets, excitations of magnons result in a net non-equilibrium spin \naccumulation δσ(t) due to the exchange interaction, yielding an average velocity response \nδv(t)=Λ⋅δσ(t) which produces an alternating current density j~Λ⋅δσ(t). Since the \nmagnon frequencies are low compared to the exchange splitting, the spin-density response is \nproportional to the rate of change ∂m/∂t of the magnetisation, i.e., δσ(t)~∂m/∂t. \nConsequently, the induced current density is also proportional to ∂m/∂t, where the coefficient \nof proportionality is directly related to the spin-orbit coupling matrix: j~Λ⋅∂m/∂t.  \n \nWe chose compressively strained (Ga,Mn)As on GaAs as the material in which to \ndemonstrate magnonic charge pumping. (Ga, Mn)As is indeed characterized by crystal \ninversion asymmetry, which together with strain leads to easily identifiable SOTs with both Rashba and Dresselhaus symmetry \n30. Furthermore, the use of (Ga,Mn)As avoids the \ncomplexity associated with a competing torque originating in the spin Hall effect, that is \npresent in the layered metal systems 20, 31. The symmetry of strained (Ga,Mn)As is described \nby the crystallographic point group C 2v, where the two-fold symmetry axis is perpendicular to \nthe epilayer 30. In the frame of reference where x’ is along the crystallographic direction \n[110], z’ is along the two-fold symmetry axis and y’ is perpendicular to x’ and z’, Λ~iσ2 and 5 \n Λ~σ1 parameterise the Rashba and Dresselhaus spin-orbit coupling, respectively, and the \ninduced alternating current density is \n \nj(r) = ΛD(r) σ1⋅∂m||/∂t – iΛR(r) σ2⋅∂m||/∂t.                                                   (3) \n \nHere, m|| = (m x’,my’) denotes the in-plane component of the magnetisation, and the \nparameters ΛR(r) and ΛD(r) characterise the strength of the charge current pumped \nmagnonically via the Rashba and Dresselhaus sp in-orbit coupling, respectively. The current \ndensity in Eq. (3) is reciprocal to the field-like SOT τ~m×hso, where hso is the effective SOT \nfield induced by an applied current density J. The SOT field consists of terms with Rashba \nand Dresselhaus symmetry, i.e., hso = h Dσ1⋅J|| + ih Rσ2⋅J||, where the parameters h R and h D are \nlinked via the reciprocity relations to ΛR(r) and ΛD(r), respectively.  \n \nThe terms in Eq. (3) represent reactive charge-pumping processes because they are even \nunder time reversal. In addition, there are dissipative contributions to the magnonic charge \npumping, which are related via the reciprocity relations to the anti-damping SOT. The in-\nplane component of the dissipative current is  (see Supplementary Information for a detailed \nderivation)   \nj\n(d) = ΛR(d) m|| ∂mz’/∂t + ΛD(d) σ3⋅m|| ∂mz’/∂t ,                                             (4) \n \nwhere the phenomenological parameters ΛR(d) and ΛD(d) characterise dissipative charge \npumping by Rashba and Dresselhaus SOC, respectively. \n When the magnetisation precesses with frequency ω\n0 and amplitude A, there is a reactive \ncontribution to the pumped current oscillating at the same frequency with an amplitude of j ω(r) \n= Aω0ΛR,D(r). The polar plot in Fig. 1d shows the symmetry of the Rashba and Dresselhaus \ncontributions to the pumped current for different directions of the magnetisation. Fig. 1c \nillustrates the symmetry of the reciprocal effect and shows the direction of the reactive \ncomponents of the Rashba and Dresselhaus SOT fields for different directions of the applied \ncurrent. There is also a direct current induced by the magnetisation precession (see Supplementary Information). However, its value is small since it is second order in the 6 \n precession amplitude and proportional to the Gilbert damping constant αG and will not be \ndiscussed further.  \n \nFig. 2a shows a schematic of the measuring apparatus. In order to drive magnetisation \nprecession via the SOT, a microwave current is passed through a micro-bar patterned from an \nepilayer with a nominal 9 % Mn concentrati on. During magnetisation precession, frequency \nmixing between the alternating current and the oscillating magneto-resistance leads to a time-\nindependent voltage, V dc 20. Using V dc, we experimentally determine the components of the \nSOT, introducing a rotated reference frame wher e x is along the bar (current) direction and z \nis perpendicular to the epilayer (Fig. 2). The angle θ refers to the mean position of the \nmagnetisation in the x-y plane and is measured from the x-axis. We focus our experiments on \nthe two bar directions [100] and [010], because the SOT field components h sox and h soy then \noriginate purely from the field-like SOTs which have symmetries that respectively resemble \nthe Dresselhaus and Rashba spin-orbit interactions  (Fig. 1c). Fig. 2b shows the derivative of \nthe rectified voltage (dV dc/dB)B mod for a bar oriented along the [100] direction when an in-\nplane magnetic field B is swept through the ferromagnetic resonance condition. The position of the resonance as a function of the field dire ction follows the modified Kittel's formula for \nan in-plane magnetised material with an additional uniaxial anisotropy \n32. The SOT-field hso \ncan be directly extracted from the angle dependence of the anti-symmetric and symmetric parts of the resonance, and the coefficients are summarised in the following table. The SOT \nfield components h\nsox and h soy correspond to the coefficients h D and h R introduced earlier, \nwhile the angle-dependent h soz terms represent the anti-dampi ng contribution. In accordance \nwith a trend that we previously observed, the presently used material has a weaker SOT than \nin the case of lower Mn concentration 20.  \nTable 1.  The coefficients of SOT measured for samples with current along the [100] and \n[010] directions, normalised to a current density of 106 Acm-2. All values are in ( μT). The \nfirst order (sin θ and cos θ) harmonic components of h soz are extracted from fits to the \nexperimental data. \n \n µ 0hsox  µ 0hsoy  µ 0hsoz    \nsinθ term µ0hsoz  \ncosθ term 7 \n [100] -6.1 -8.7 8.5 -13.6 \n[010] 5.2 -5.5 -5.5 -6.9 \n \nIn addition, the magnonic charge pumping induces an alternating voltage, V ω, across the bar, \nwhich we measure with a field modulation lock-in technique (see Supplementary \nInformation). Fig. 2c shows the derivative of the amplitude of the microwave voltage across \nthe sample, (dV ω/dB)B mod, as the magnetic field is swept along different in-plane directions. \nAt ferromagnetic resonance, a resonance also appears in V ω, which indicates that a \nmicrowave electrical signal is generated within the sample by the precessing magnetisation.  \n \nMagnonic charge pumping is proportional to the rate of change of the magnetisation, hence \nthe induced microwave amplitude should be linearly dependent on the precessional \namplitude. To check this characteristic, we measure the voltage V ω as a function of the \nprecessional amplitude A for a fixed directio n of the magnetic field. The amplitude is \ncontrolled by the value of the applied microwave current. Fig. 2d clearly demonstrates a \nlinear dependence on the amplitude. This excludes the possibility that V ω originates from the \nmixing between the microwave current and the modulated resistance during precession, \nbecause such higher-order terms depend non-linearly on the amplitude (see Supplementary Information).  \n \nNext, we demonstrate that the measured signal is reciprocal to the SOT. To this end, we \nmodel the charge pumping by Eqs. (3)-(4) (see Supplementary Information for further \ndetails). Using the Onsager reciprocity relations, the measured SOT fields h\nsoy and h sox \ndetermine the values of ΛR(r) and ΛD(r), respectively, while the measured h soz component \ndetermines ΛR(d) and ΛD(d). The expression for ∂m/∂t is found from the solution of the LLG \nequation. The resulting voltage signal across the bar is given by the total current pumped \nalong the bar direction multiplied by the resistance. In Fig. 3a-b, we plot the magnitude of the symmetric and anti-symmetric components of the integrated resonances with respect to the \nfield direction. The theoretical curves are represented by the continuous lines and show \nagreement with the experimental data in both symmetry and amplitude. This verifies that the \nmeasured voltage signal satisfies its recipr ocal relationship to the SOT. The different 8 \n symmetries found for the [100] and the [010] bar directions further confirm the crystal, thus \nSOC-related, origin of the effect and exclude the Oersted field and artefacts in the measuring \nset-up as possible origins. Also, a variation of the impedance matching following the ac \nchange in magnetic susceptibility during precession cannot justify the resonance in V ω as in \nthis case the symmetry would be dominated by the symmetry of the anisotropic magneto-\nresistance (refer to the Supplementary Information). The slight discrepancy between the experimental and theoretical curves aris es from higher-order harmonics in the \nphenomenological expansion of the pumped curre nt. Such higher-order features have also \nbeen observed in the SOT \n25. To allow comparison of the magnonic charge pumping between \ndifferent materials, we renormalise the pumped current density by the saturation \nmagnetisation, frequency, and precessional amplitude. For our (Ga,Mn)As samples, we find a magnitude of 600 µA·cm\n-2/T·GHz for the [100] direction and 240 µA·cm-2/T·GHz for the \n[010] direction. In ( 20) the authors reported fluctuations of  30% in the magnitude of the SOT \nfor samples of the same material. Likewise, the magnitude of the charge pumping is expected \nto be sample-dependent, although its symmetry is only determined by the crystalline \norientation of the bar, as also shown in the Supplementary Information.   \n \nIn conclusion, we have demonstrated dire ct conversion of magnons into high-frequency \ncurrents via the spin-orbit coupling. While we chose the ferromagnetic semiconductor (Ga,Mn)As, magnonic charge pumping is also predicted in layered systems like Pt/Co/Al\n2O3 \nand can be quantitatively analysed within the same Onsager framework we provide. In these metallic systems, we expect a large, room-temperature charge pumping effect, the \ninvestigation of which will help distinguish between spin Hall and spin-orbit torques.  \n \nMethods and Materials: Materials : The 18 nm thick (Ga 0.91,Mn 0.09)As epilayer was grown \non a GaAs [001] substrate by molecular beam epitaxy. It was subsequently annealed for 8 \nhours at 200 ºC. It has a Curie temperature of 179 K; a room temperature conductivity of 414 Ω\n-1cm-1, which increases to 544 Ω-1cm-1 at 4 K; and a saturation magnetisation of 70.8 \nemu·cm-3. \nDevices : Two terminal microbars are patterned in different crystal directions by electron \nbeam lithography and have dimensions of 4 µm × 40 µm.  \nExperimental procedure : A 7 GHz microwave signal with a source power of 18 dBm is \ntransmitted to an impedance matching circuit made by a 4-finger interdigitated capacitor and 9 \n a λ/2 micro-strip resonator patterned on a low loss printed circuit board and reaches the \n(Ga,Mn)As bar, which is wire-bonded between the resonator and the ground plane. SOT \nexcites magnetic precession as an external field is swept in the plane of the device. The \nmicrowave voltage generated in the (Ga,Mn)As bar by magnonic charge pumping is \ntransmitted via a directional coupler to an am plifier and mixer, from which we measure the \namplitude of the voltage. 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Kim J, Sinha J, Hayashi M, Yamanouchi M, Fukami S, Suzuki T , et al.  Layer thickness dependence of \nthe current-induced effective field vector in Ta vertical bar CoFeB vertical bar MgO. Nat Mater  2013, \n12(3): 240. \n \n27. Fan X, Wu J, Chen YP, Jerry MJ, Zhang HW, Xi ao JQ. Observation of the nonlocal spin-orbital \neffective field. Nat Commun  2013, 4: 1799. \n \n28. Onsager L. Reciprocal relations in irreversible processes. Phys Rev 1931, 37: 405. \n 29. Tatara G, Nakabayashi N, Lee KJ. Spin motive force induced by Rashba interaction in the strong sd \ncoupling regime Phys Rev B  2013, 87(21): 054403. \n \n30. Jungwirth T, Sinova J, Mase k J, Kucera J, MacDonald AH. Th eory of ferromagnetic (III,Mn)V \nsemiconductors. Rev Mod Phys  2006, 78(3): 809. \n \n31. Liu LQ, Moriyama T, Ralph DC, Buhrman RA. Spin-Torque Ferromagnetic Resonance Induced by the \nSpin Hall Effect. Phys Rev Lett  2011, 106(3): 036601  11 \n  \n32. Liu XY, Furdyna JK. Ferromagnetic resonance in Ga1-xMnxAs dilute magnetic semiconductors. J \nPhys-Condens Mat  2006, 18(13): R245-R279. \n  \n \n \n \n \n \n \nAcknowledgments  \n AJF acknowledges support from a Hitachi research fellowship and CC from a Junior \nresearch fellowship at Gonville and Caius College. VN acknowledges MSMT grant Nr. \nLM2011026. \nAuthor contributions  \nKH and AB developed the theory and suggested the experiment. CC and AJF developed the \nexperimental technique and performed the experimental work. VN grew the materials. ACI \nperformed the nanofabrication. CC, KH, AB & AJF wrote the manuscript. All authors \ndiscussed the results and commented on the paper.  \nCompeting Financial Interests  \nThe authors declare no competing financial interests. \n \n \nFig. 1 . (a) A charge current through (Ga,Mn)As resu lts in a non-equilibrium spin polarisation \nof the carriers, which exchange-couples to the ma gnetisation and exerts a torque. The effect is \ninduced by the spin-orbit coupling, which mediates  the transfer of orbital momentum to spin \nangular momentum. An alternating current generates a time varying torque, which drives \nmagnetic precession resonantly when a magnetic fi eld is applied. (b) The reciprocal effect of \n(a). Magnetisation precession leads to a non-equilibrium spin concentration, which is converted into an alternating charge current by the spin-orbit coupling. (c) Polar plot \nillustrating the direction of the effective magnetic field induced by a charge current along \ndifferent crystal directions. The Rashba (h\nR) and Dresselhaus (h D) spin-orbit coupling \ncontributions are indicated by the red and blue arrows, respectively. (d) Polar plot illustrating \nthe direction of the charge current pumped by magnetisation precession around different 12 \n crystal directions. The Rashba (j R) and Dresselhaus (j D) contributions are indicated by the red \nand blue arrows, respectively. \nFig. 2.  (a) Schematics of the measuring set-up. A 7 GHz microwave signal (red arrow) is \nlaunched towards a (Ga,Mn)As bar via an impedance matching circuit. The microwave current passed through the bar excites magne tisation precession via SOT when an in-plane \nmagnetic field B is swept through the resonance. The orientation of the field is defined with \nrespect to the bar direction, as shown on the Cartesian plot. The microwave voltage generated \nin (Ga,Mn)As by magnonic charge pumping (blue arrow) is transmitted through the same \nimpedance matcher to the microwave circuitry, where the amplitude of the signal is amplified and detected. A low-frequency lock-in fiel d-modulation technique is used: a 3.3 mT \noscillating magnetic field B\nmod is applied at 45° from the bar direction. A directional coupler \nseparates the incoming signal used to excite  magnetic precession (red arrow) from the \noutgoing signal generated both by magnonic charge pumping and the microwave signal \nreflected from the circuit (blue arrow). The im pedance matching circuit also includes a bias \ntee that allows the rectified voltage along the bar to be measured. (b) Derivative of the \nrectified voltage along the a [100] oriented bar, (dV dc/dB)B mod, measured by field modulation \nlock-in technique as the magnetic field is swept along different in-plane directions. (c) \nDerivative of the microwave voltage along a [100] oriented bar, (dV ω/dB)B mod, induced by \nmagnonic charge pumping for the same field directions as in (b). (d) Amplitude of the microwave voltage V\nω as a function of the precessional amplitude A. The value of A in mrad \nis obtained from the amplitude of the rectified voltage |V dc|=|I|ΔRA/2, where I is the \nmicrowave current passed through the bar and ΔR is the anisotropic magneto-resistance \ncoefficient. \nFig. 3.  Symmetric (red circles) and anti-symmetric (black squares) components of the \nintegrated resonances shown in Fig. 2c for diffe rent directions of the external field B. (a) \nshows the results obtained for a bar oriented along the [100] crystal direction, while (b) for a \nbar along the [010] direction. The data is compared to the theoretical curves (symmetric and \nanti-symmetric contributions are equal and are shown by a unique blue line). \n c.FIGURE 1\nb. d.\njR\njD\nj\nhR\nhDa.\nImpedance\nmatch Vωcoupler \n7 GHzVdcFIGURE 2\na.\nc. d.\nFIGURE 3\na. b.[100] [010]b.θ45⁰B\nBmod\nθ\nzxy\nbar40 μm\n60 90 12002040 \n (dV/g90/dB)Bmod (/g80V)\nB (mT) -10\n 0\n 10\n 20\n 30/g84 (deg)\n60 90 12004080\n  (dVdc/dB)Bmod (/g80V)\nB (mT)\n0 90 180 270 360061218\n  \n  theory  anti-sym  symV/g90 (/g80V)\n/g84 (deg)0 90 180 270 3600246\n  \n theory anti-sym symV/g90 (/g80V)\n/g84 (deg)0.09 0.18 0.270.00.91.8\n V/g90 (/g80V)\nA (mrad)"    },    {        "title": "2109.02151v1.Electric_field_effect_on_electron_gas_spins_in_two_dimensional_magnets_with_strong_spin_orbit_coupling.pdf",        "content": "Electric \feld e\u000bect on electron gas spins in two-dimensional\nmagnets with strong spin-orbit coupling\nK.S. Denisov\u0003\nIo\u000be Institute, 194021 St.Petersburg, Russia\n(Dated: September 7, 2021)\nAbstract\nThe recent rise of material platforms combining magnetism and two-dimensionality of mobile\ncarriers reveals a diverse spectrum of spin-orbit phenomena and stimulates its ongoing theoretical\ndiscussions. In this work we use the density matrix approach to provide a uni\fed description\nof subtle microscopic e\u000bects governing the electron gas spin behavior in the clean limit upon\nelectric perturbations in two-dimensional magnets with strong spin-orbit coupling. We discuss\nthat an inhomogeneity of electrostatic potential generally leads to the electron gas spin tilting\nwith the subsequent formation of equilibrium skyrmion-like spin textures and demonstrate that\nseveral microscopic mechanisms of 2DEG spin response are equally important for this e\u000bect. We\nanalyze the dynamics of 2DEG spin upon an oscillating electric \feld with a speci\fc focus on the\nemergent electric dipole spin resonance. We address the resonant enhancement of magneto-optical\nphenomena from the spin precession equation perspective and discuss it in terms of the resonant\nspin generation. We also clarify the connection of both static and dynamic spin phenomena arising\nin response to a scalar perturbation with the electronic band Berry curvature.\n\u0003denisokonstantin@gmail.com\n1arXiv:2109.02151v1  [cond-mat.mes-hall]  5 Sep 2021I. INTRODUCTION\nThe recent advances in the development of spintronics devices extensively use relativistic\nspin-orbit properties of free carriers interacting with magnetic layers. The spin-orbit cou-\npling (SOC) of charge carriers generally opens up the possibility to deal with the magneti-\nzation purely by electrical means; the magnetization orientation can be detected electrically\nby virtue of the anisotropic magnetoresistance e\u000bect [1{3], while electric current-induced\nspin-orbit torque occurs to be a highly e\u000bective tool for switching its direction [4{7]. Non-\nstationary dynamics of carriers in presence of SOC can result in stimulated photon emission,\nas in case of terahertz spintronic light emitter [8{11] and spin Hall nano-oscillators [12, 13].\nApart from kinetic phenomena spin-orbit e\u000bects can modify equilibrium spin con\fgurations\nvia indirect RKKY exchange interaction [14{16] and lead to the formation of magnetic\nskyrmions [17, 18] due to Dzyaloshinskii{Moriya terms [19, 20]. An e\u000ecient charge-to-spin\nconversion wanted for modern spintronics needs is often realized when turning to a two-\ndimensional electron gas, as the reduction of the dimensionality tends to be accompanied\nby the lowering of symmetry and by the subsequent increase in SOC [19, 21]. There are an\nincreasing number of di\u000berent material platforms that allow one to combine systematically\nstronger SOC magnitudes of 2D electrons directly with a magnetic component, the examples\ninclude van der Waals heterostructures [22] either proximitized by magnetic layer [23{27]\nor being intrinsic ferromagnets [28{31], semiconductor nanostructures doped by magnetic\ndopants [32, 33], surface states of magnetic topological insulators [34, 35], or layered mag-\nnetic heterostructures [19, 36]. Moreover, combining magnetism with 2D conductive channels\nadditionally o\u000bers new functionalities, such as spin tunnel \feld-e\u000bect transistors [37], spin\ninversion e\u000bect [24] or novel class of spinterfaces [38].\nIn order to fully bene\ft from two-dimensional magnetic systems it is of key importance to\nhave a comprehensive understanding of how the spin density of electron gas in a 2D channel\nresponds to an applied electric \feld, that is the understanding of free electron gas magne-\ntoelectric properties. However, a complete microscopic treatment of the related phenomena\nappears to be extremely challenging, even despite there is a few theoretical approaches e\u000bec-\ntively dealing with multiband systems (e.g. wave-packet dynamics theory [39{43], diagram-\nmatic and ab-initio calculations [44{47]). The di\u000eculty lies in the fact that in spin-orbital\nsystems multiple microscopic mechanisms of quite a subtle character often contribute on the\n2equal footing, which hinders a simpli\fed consideration. In particular, an exchange inter-\naction induced spin splitting in combination with strong spin-orbit coupling generally lead\nto a geometrical structure of electronic band states featured by nonzero Berry curvature\nin k-space. Treating di\u000berent spin-related phenomena with account for the electronic band\ngeometry remains an ongoing discussion. It covers, for instance, the issues of the Liouville's\ntheorem with account for the Berry phase [48{50], the Hall conductivity modi\fcations in\npresence of real-space magnetic textures [51], or, concerning the anomalous and spin Hall\ne\u000bects, the interplay between Karplus-Luttinger anomalous velocity and disorder-induced\nmechanisms [52{55]; the latters have recently been enriched by the electron scattering on a\npair of impurities [55, 56]. Moreover, when calculating spin-related quantities a speci\fc class\nof coarse graining e\u000bects should be taken into account, as is clearly demonstrated in [40, 42].\nIn this paper we respond to an ever-growing role that two-dimensional magnetic systems\nplays for spintronics and consider in detail a complex pattern of microscopic e\u000bects relevant\nfor the magnetoelectric behavior of 2DEG in the clean limit. Based on the density matrix\napproach we describe the most signi\fcant spin-response mechanisms of two-dimensional spin-\norbital systems within the uni\fed framework, reveal the interconnection between di\u000berent\nmicroscopic e\u000bects and clarify its relation to an electronic band geometry.\nThe theoretical model and the density matrix description are formulated in Sec. II. In\nSec. III we analyze a magnetoelectric e\u000bect in thermal equilibrium, namely we consider the\nformation of equilibrium spin textures and local persistent electric currents arising due to an\ninhomogeneous electrostatic potential. We discuss in detail semiclassical electron dynamics\nwith account for a spin-to-momentum locking and identify microscopic mechanisms respon-\nsible for the magnetoelectric response. Namely, we attribute the generation of an extra-spin\ndensity directed within 2DEG plane both to the non-adiabatic correction to the electron\nspin precession and to the correlated change of charge and spin electron densities, the latter\nscenario is sometimes referred as spin-dipole e\u000bect [40]. We provide a uni\fed treatment of\nthese mechanisms using the density matrix, derive general equations governing the contri-\nbution due to each mechanism independently and reveal the role that the Berry curvature\nplays for the emergent phenomena.\nIn Sec. IV we turn to the dynamical regime and investigate the 2DEG spin dynamics upon\nan oscillating electric \feld. We focus speci\fcally on spin resonance phenomena due to electric\ndipole transitions, also referred as the electric dipole spin resonance (EDSR). We derive the\n3precession equation for 2DEG spin density capturing the spin resonance scenario, and clarify\nthe relation of the band states Berry curvature with the spin response susceptibility. We\nalso discuss the spin resonance in terms of optical conductivity and describe the associated\nmagneto-optical properties of 2DEG. In particular, we describe how the EDSR induced\ngeneration of the in-plane spin density is accompanied by the resonant enhancement of the\nHall conductivity, the latter is responsible for magneto-optical Kerr and Faraday e\u000bects. We\nclassify di\u000berent spin polarizations emerging in the dynamical regime and present analytic\nexpressions for the spin resonance related optical conductivity.\nII. THEORETICAL FRAMEWORK\nA. Model band structure\nWe consider a two-dimensional electron gas with parabolic bands a\u000bected both by the\nRashba e\u000bect and by an exchange interaction with a magnetic host. We assume that the\nmagnetization responsible for the spin splitting is directed along z-axis perpendicular to\nthe electron motion plane. The so-called Rashba ferromagnet model covers all the physics\nrelevant for our consideration and allows one address the related spin phenomena in the\nmost transparent way. The e\u000bective Hamiltonian describing this model is given by\nH=k2\n2m+\nk\u0001^S; (1)\nhere the \frst term describes the parabolic dispersion with an e\u000bective mass m, and \nkis an\ne\u000bectivek-space magnetic \feld acting on the electron spin ^S=^\u001b=2;^\u001bis the vector of Pauli\nmatrices. The \feld \nkleads to a spin splitting of the electronic subbands, in our model \nk\nconsists of two parts\n\nk=\nso(k)\u0000\n0ez; \nso(k) = 2\u0015so[ez\u0002k]; (2)\nwhere \nso(k) describes the spin-orbit Rashba interaction with the coupling constant \u0015so,\nand the second term is due to an exchange interaction with a magnetic background, the\nparameter \n 0describes the corresponding splitting of spin subbands at zero momentum.\nThe eigenstates of Eq. 1 Hamiltonian can be written in the following form\n \u0006\nk=eikrju\u0006\nki;ju+\nki=1p\n20\n@bk\n\u0000iei'ak1\nA;ju\u0000\nki=1p\n20\n@\u0000ie\u0000i'ak\nbk1\nA; (3)\n4where (ak;bk) = (1\u0006\n0=\nk)1=2. We use the notation \u0011= (\u0006) for two electron spin subbands.\nThe states  \u0011\nkare characterized by the electron spin s\u0011\nk=hu\u0011\nkj^Sju\u0011\nkidirected either parallel\nor antiparallel to \nk\ns\u0006\nk=\u00061\n2nk;nk=\nk\n\nk; \nk=q\n\n2\n0+ (2\u0015sok)2; (4)\nwhere the unit vector nkpoints along the direction of \nk.\nThe energy dispersion corresponding to \u0011-subband is \"\u0011\nk=k2=2m+\u0011\nk=2. The presence\nofk-dependent spin splitting leads to the renormalization of e\u000bective masses nearby k\u00190,\nnamelym\u0006=m=(1\u0006\u0018), where the parameter \u0018\u00112m\u00152\nso=\n0. We focus on systems with\nsu\u000eciently strong exchange interaction, when \n 0greatly exceeds the spin-orbital coupling.\nWe thus take the parameter \u0018 <1, at that the e\u000bective mass m\u0000>0 is positive and the\nlower energy branch is a monotonic function of the momentum, see Fig. 4b.\nLet us discuss the role of the spin splitting terms. The presence of the Rashba e\u000bect\ninduced spin-momentum locking directly manifests itself in the velocity operator\n^v=i\n~[H;r] =^k\nm+ 2\u0015so[ez\u0002^S]; (5)\nwhere the second term is sensitive to the instantaneous direction of the electron spin. While\nthe average velocity for the eigen spin states is determined by the unperturbed spin vector s\u0011\nk\nv\u0011\nk\u0011h \u0011\nkj^vj \u0011\nki=k\nm+ 2\u0015so[ez\u0002s\u0011\nk]; (6)\nthe changes in the direction of an electron spin caused by external \felds can directly a\u000bect\nthe average of the velocity operator and, correspondingly, in\ruence the orbital motion.\nThe presence of a magnetic gap due to the magnetization directed perpendicular to 2DEG\nplane leads additionally to the fact that electron band states acquire a geometric structure.\nIndeed, the electron spin direction in k-space forms a hedgehog pattern which underlies the\nappearance of the Berry curvature F\u0011\nk=ihrku\u0011\nkj\u0002jr ku\u0011\nki. For a spin-1 =2 Hamiltonian this\nBerry curvature can be expressed as follows\nF\u0011\nk=\u00111\n4\u0019nk\u0001\u0014@nk\n@kx\u0002@nk\n@ky\u0015\n=\u00112\u00152\nso\n0\n\n3\nk; (7)\nand we keep the notation Fk=jF\u0011\nkjfor its absolute value. The total Berry \rux Q\u0011\nFaccu-\nmulated by electrons from \u0011subband up to the Fermi energy \u0016is given by\nQ\u0011\nF=X\nk<k\u0011\nFF\u0011\nk=\u00111\n4\u0019\u0012\n1\u0000\n0\n\n\u0011\nF\u0013\n; \n\u0011\nF=q\n\n2\n0+ (2\u0015sok\u0011\nF)2; (8)\n5where \n\u0006\nFis the spin splitting energy for \u0011= (\u0006) subbands at the Fermi energy, see Fig. 3b.\nThe strong spin-orbit coupling considered in our work means that we do not account for the\ndisorder-induced smearing of SOC features of electronic bands.\nB. Density matrix approach\nLet us \frstly discuss the structure of the density matrix f0for 2DEG in thermal equilib-\nrium without external perturbations. The general form is f0= (e\f(^H\u0000\u0016)+ 1)\u00001, where ^His\ngiven by Eq. 1, \fis the inverse temperature and \u0016is the Fermi energy. In this work we focus\non zero temparature limit \f!1 . The density matrix ^f0\nkin the momentum representation\nis a 2\u00022 matrix which can be presented as follows (we keep hats for spin indices only)\n^f0\nk=1\n2n0\nk+S0\nk\u0001^\u001b: (9)\nWe note that ^f0\nkis diagonal in the basis of eigen states  \u0006\nk, so we can present it as a sum of\n\u0011= (\u0006) spin subband contributions ^f\u0011\nk\n^f0\nk=^f+\nk+^f\u0000\nk; ^f\u0011\nk=n\u0011\nk\u00121\n2+s\u0011\nk\u0001^\u001b\u0013\n; (10)\nwheren\u0011\nk= (e\f(\"\u0011\nk\u0000\u0016)+ 1)\u00001is the Fermi-Dirac distribution function of electrons in the spin\nsubband with energy \"\u0011\nk. The terms in Eq. 9 are given n0\nk=n+\nk+n\u0000\nk, andS0\nk=n+\nks+\nk+n\u0000\nks\u0000\nk,\nheres\u0011\nkcorresponds to the eigen spin states from Eq. 4. The equilibrium spin density S0is\ndirected perpendicular to the 2DEG plane\nS0=1\n2X\nkSp\u0010\n^f0\nk\u0001^\u001b\u0011\n=X\nk\u0000\nn+\nks+\nk+n\u0000\nks\u0000\nk\u0001\n=ez\n\u0000\nF\u0000\n+\nF\n16\u0019\u00152\nso: (11)\nWe note that when both spin subbands are populated ( \u0016 > \n0=2) the equilibrium spin\ndensity takes value S0=m\n0=4\u0019independent of the Fermi energy, this is speci\fc for\nHamiltonian from Eq. 1.\nThe application of a scalar potential U(r;t) deviates the electron distribution from Eq.10.\nIn this paper we focus on spatially smooth perturbations ( kF\u0001rk\u001c1 and\u0015F\u0001rr\u001c1)\nand study the electron gas response in the classical limit. For this purpose we introduce the\nWigner density matrix ^fk(r;t) in the following form\n^fk(r;t) =1\n2nk(r;t) +Sk(r;t)\u0001^\u001b; (12)\n6wherenk(r;t);Sk(r;t) can be treated as particle and spin distribution functions locally in\nreal space. In particular, the 2DEG spin density perturbation emerging in the real space at\npointrcan be found from\n\u000eS(r;t) =1\n2X\nkSp\u0010\n^fk(r;t)\u0001^\u001b\u0011\n\u0000S0=X\nkSk(r;t)\u0000S0: (13)\nIn the clean limit ^fk(r;t) satis\fes the kinetic equation [57]\n@^fk\n@t+1\n2n\n(^v\u0001rr) ;^fko\n\u0000[\nk\u0002Sk]\u0001^\u001b+ (F\u0001rk)^fk= 0; (14)\nwheref;gstands for the anticommutator, rr;kare the nabla operators, F(r;t) =\u0000rrU(r;t)\ndescribes the dynamical force acting on electrons, and the third term takes into account the\nprecession of the electron spin in the e\u000bective magnetic \feld \nk. Let us draw the atten-\ntion to the anticommutator type of ordering between ^vand ^fkthat appears in the second\nterm. This ordering directly stems from the Wigner transformation procedure [58] and it is\nespecially important to describe accurately the response in the inhomogeneous regime.\nIII. STATIC SPIN TEXTURES\nWe start our analysis by inspecting the redistribution of the 2DEG charge and spin\ndensities nearby smooth electrostatic defects, such as Coulomb centres or gating potential\nperturbations. The geometric character of electronic band states and the associated nonzero\nBerry curvature underline the appearance of chiral spin textures and adjoint persistent\nelectric currents that surround electrostatic potential inhomogeneity, see Fig. 1. In [59]\nwe used the Kubo formalism to address the nonlocal regime of the spin density response\ndue to short-range impurities. In this section, instead, we provide a detailed semiclassical\ndescription of this phenomenon and accompany it by the comprehensive physical analysis.\nA. General mechanisms of the intrinsic spin generation\nLet us qualitatively discuss the e\u000bect of the electron spin non-adiabatic rotation upon the\nprecession in a slowly varying magnetic \feld [60{62]. We start by considering the precession\nequation for an electron spin srotating upon a time-dependent frequency \n(t)\nds\ndt= [\n(t)\u0002s]: (15)\n7δ/vectorS(r)\nU(r)/vectorj(r)FIG. 1. Formation of skyrmion-like spin textures and the distribution of the persistent electric\ncurrents nearby electrostatic defects.\nAssuming the adiabatically slow rotation of \n(t), i.e. that the characteristic time \u001cof its\nvariation satis\fes \n \u001c\u001d1, the zero-order solution of the precession equation simply describes\nthe electron spin s0(t) =\n(t)=2j\n(t)jremaining co-aligned with the instant direction of\n\n(t). However, the adiabatic rotation of s0(t) can be maintained only due to the appearance\nof the non-adiabatic correction \u000es(t) directed perpendicular to the instant vector \n(t).\nNaturally, this correction exists in the \frst order in (\n \u001c)\u00001and it can be found from the\nprecession equation keeping only the leading term due to s0(t) in the time derivative\nds0\ndt= [\n(t)\u0002\u000es(t)]!\u000es(t) =1\n2\n3\u0014\n\n\u0002d\ndt\u0015\n: (16)\nThe appearance of \u000es/(\n\u001c)\u00001s0is a general property of the precession equation. Naturally,\nthis is also valid when a Larmor frequency stems from an e\u000bective magnetic \feld in k-space\ndue to a spin-orbit coupling. In this case, however, the vector \nkthat governs the spin\ndynamics of an electron with momentum kvaries in time only provided that the electron\nmomentum does not remain constant along its trajectory _k6= 0, which is the case if F6= 0.\nThe non-adiabatic spin component acquired by an electron can be estimated from Eq. 16\nby replacing the time derivative by d=dt!_k\u0001rk\n(_k\u0001rk)s0\nk= [\nk\u0002\u000esk]!\u000esk=1\n2\n3\nkh\n\nk\u0002(_k\u0001rk)\nki\n: (17)\nWe conclude that an electron moving along its classical trajectory with \fnite acceleration\nhas its spin always slightly tilted compared to the instantaneous direction of \nk. Moreover,\nin view of the spin-momentum locking such an intrinsically generated extra-spin leads to\nthe change in the electron velocity \u000evk= 2\u0015so(ez\u0002\u000esk).\n8The second spin-related phenomenon being important for the collective response of 2DEG\nconcerns the spin-dipole e\u000bect [40]. This mechanism is relevant when the single electron\ndensityj (r)j2deviates from the homogeneous distribution and acquires some \fnite r-\ndependence nearby an inhomogeneity. Let us consider an electron at the unperturbed plane-\nwave state  \u0006\nkfrom Eq. 3 with the momentum k, its spins\u0006\nkis determined by \nk. The\ncorresponding density j \u0006\nkj2is spatially homogeneous. In fact, the smooth spatial variation\nof the density for such electron is possible only provided that its wave-function gets an\nadmixture of other plane-wave band states  \u0006\nk0with momenta k0slighty di\u000bering from k.\nEssentially, the added states have di\u000berent spin orientation s\u0006\nk06=s\u0006\nk, so the resulting average\nspin density appears to be slightly tilted. In terms of the wave-packet dynamics [40, 63] the\nmixing of spin-orbital states leads to the fact that the charge and spin centers of the electron\nwave-packet do not coincide, which creates an additional spin polarization. This scenario\nis speci\fcally important for localized electron states [64, 65]. We emphasize that the spin-\ndipole e\u000bect is essentially connected with the spatial variation of the electron density. In\nparticular, if a given external \feld keeps an electron gas in the homogeneous state, the spin-\ndipole contribution will be absent. The appearance of the non-adiabatic correction from\nEq. 17, on the contrary, is not connected with the change of an electron density, it simply\ntracks the exact electron spin dynamics along quasiclassical trajectories.\nB. Density matrix in a static inhomogeneous setting\nWe proceed with giving a rigorous description of the outlined phenomena based on the\nkinetic equation for the density matrix. Let us consider an electron gas subjected to an\nelectrostatic potential U(r) smoothly varying in space. Since the unperturbed density matrix\n^f0\nk=^f+\nk+^f\u0000\nkgiven by Eq. 10 has two parts corresponding to \u0011= (\u0006) subband states, the\nlinear response correction \u000e^fk(r) = ^fk(r)\u0000^f0\nkwill be determined independently by two\nsubband terms \u000e^fk(r) =\u000e^f+\nk(r) +\u000e^f\u0000\nk(r). We present the corresponding correction \u000e^f\u0011\nkas\nfollows\n\u000e^f\u0011\nk(r) =1\n2\u000en\u0011\nk(r) +\u000eS\u0011\nk(r)\u0001^\u001b; (18)\nwhere\u000en\u0011\nk(r);\u000eS\u0011\nk(r) are the perturbations of the electron density and spin distribution\nfunctions, respectively.\n9The key suggestion implemented in this paper is to use the following ansats for the linear\nresponse spin density\n\u000eS\u0011\nk(r) =\u000en\u0011\nk(r)s\u0011\nk+n\u0011\nk\u000es\u0011\nk(r) +\u000eS\u0011\nk(r); (19)\nwhere we took into account all possible types of \u000eS\u0011\nk(r) variation. Indeed, the \frst term\ndescribes the change of the electron spin distribution due to the change in the density \u000en\u0011\nk.\nThe second term corresponds to the change of the spin vector \u000es\u0011\nkfor each individual electron\nindependently of the electron number distribution. The third term is the remaining linear-\norder variation, which is essentially neither due to \u000en\u0011\nk(r) or\u000es\u0011\nk(r) separately; thus \u000eS\u0011\nk\ndescribes the correlated change of both the electron spin and charge densities. Naturally,\nthe second and the third terms in this expansion turn out to describe the non-adiabatic spin\ntilting and the spin-dipole e\u000bects, respectively.\nWe proceed with calculating \u000eS\u0011\nk(r) from the kinetic equation 14. In what follows we\nkeep in Eq. 14 only the terms linear in UandF=\u0000rrU. In this limit the change of the\nelectron density \u000en\u0011\nkcan be determined independently from the scalar part of Eq. 14. Taking\nthe trace over Eq. 14 we get\n(v\u0011\nk\u0001rr+F(r)\u0001rk)n\u0011\nk(r) = 0: (20)\nHerev\u0011\nkis the electron group velocity given by Eq. 6. In the linear response regime the\ncorrection\u000en\u0011\nkis given by: \u000en\u0011\nk(r) =U(r)(@n\u0011\nk=@\"), where\"is the electron energy. The\nchange in the overall 2DEG density is \u000en(r) =\u000en+(r) +\u000en\u0000(r), where\u000en\u0011(r) =\u0000\u0017\u0011\nFU(r)\nand\u0017\u0011\nFis the density of states in \u0011subbands taken at the Fermi energy. Correspondingly,\nthe perturbation of the spin density Eq. 13 due to the \frst term in Eq. 19 is given by\n\u000eS(1)(r) =X\nk;\u0011s\u0011\nk\u0001\u000en\u0011\nk(r) =ez\n0\u0012\u0017+\nF\n\n+\nF\u0000\u0017\u0000\nF\n\n\u0000\nF\u0013\nU(r): (21)\nThe term\u000eS(1)(r) is responsible for the change in the out-of-plane spin density component\nand it appears even if there is no spin-orbit interaction. A complex spin-orbital electron\ndynamics is responsible for an extra spin response described by \u000es\u0011\nkand\u000eS\u0011\nk. We notice\nthat\u000es\u0011\nk;\u000eS\u0011\nkare absent in a homogeneous setting, thus the expansion of \u000es\u0011\nk;\u000eS\u0011\nkstarts\nwith the linear term rrU. Taking the trace over Eq. 14 multiplied by ^\u001band keeping only\n10the terms linear in rrgradient we get\n[\nk\u0002\u000es\u0011\nk(r)]\u0000(F(r)\u0001rk)s\u0011\nk= 0; (22)\n[\nk\u0002\u000eS\u0011\nk(r)] + [s\u0011\nk\u0002(s\u0011\nk\u0002\nso(rrn\u0011\nk))] = 0; (23)\nwhere \nso(rrn\u0011\nk) is obtained from Eq. 2 by replacing k!r rn\u0011\nk(r).\nLet us comment on the relation between \u000es\u0011\nk;\u000eS\u0011\nkand the previously described kinematic\ne\u000bects. The \frst equation Eq. 22 can be satis\fed by changing the electron spin vector \u000es\u0011\nkin-\ndependently of a particular density distribution n\u0011\nk, it thus indeed describes the spin rotation\nof individual electrons due to the precession in the e\u000bective magnetic \feld \nk. Naturally, the\nnonzero term \u000es\u0011\nkis exactly the non-adiabatic correction to the instant spin vector s\u0011\nkwhich\nfollows adiabatically the local direction of \nk. The solution of the equation 22 replicates\nthe result from Eq.17\n\u000es\u0011\nk(r) =\u00111\n2\n3\nkh\n\nk\u0002(F(r)\u0001rk)\nki\n: (24)\nIt is worth noting that \u000es\u0011\nkis nonlinear with respect to \nk. The second equation Eq. 23\ndescribes the appearance of \u000eS\u0011\nk, the general form of the solution is given by\n\u000eS\u0011\nk(r) =\u00001\n4\n2\nk[\nk\u0002\nso(rrn\u0011\nk)]: (25)\nImportantly, the additional spin density \u000eS\u0011\nkresponds directly to the spatial gradient of\nthe electron density rrn\u0011\nk(r) entering in \nso. In fact, this allows us to refer \u000eS\u0011\nkas the\ncorrelational term: it is neither due to the independent change in the number of electrons or\ndue to the individual electron spin rotation. Instead, \u000eS\u0011\nkdescribes the simultaneous change\nin the electron spin due to the variation in its spatial density, it is indeed relevant to the\nspin-dipole e\u000bect.\nC. Interplay between microscopic mechanisms and the role of Berry curvature\nThe explicit evaluation of extra-spin density terms from Eq. 24,25 for the Rashba ferro-\nmagnet model gives the following expressions\n\u000es\u0011\nk=\u0011eFk\n2\u0015so\u0001E(r)\u0000\u00112e\u00152\nso\n\n3\nk[k\u0002E(r)]; (26)\n\u000eS\u0011\nk=\u0000Fk\u0001\nk\n4\u0015sorrn\u0011\nk(r) +\u0011\u0015so\n2\n2\nkez(\nk\u0001rr)n\u0011\nk(r); (27)\n11whereFkis the magnitude of the Berry curvature from Eq. 7, and the density gradient\nrrn\u0011\nk(r) =\u0000eE(r)(@n\u0011\nk=@\") is due to the redistribution of electrons in the vicinity of an\nelectrostatic potential inhomogeneity.\nWe note that various terms from Eqs. 26, 27 give rise to quite di\u000berent spin phenomena.\nFor instance, the second terms in \u000es\u0011\nk;\u000eS\u0011\nkdepend on the electron momentum direction and\nthey are particularly important for the generation of spin currents in nonmagnetic systems\n(they survive at \n 0!0); the second term in \u000es\u0011\nkis responsible for the universal spin Hall\nconductivity [62]. Alternatively, it keeps signi\fcance for spin dynamics, see the details in\nSec. IV. Below we focus on the local magnetoelectric e\u000bect, that is the appearance of an\nequilibrium spin density in response to the local electric \feld. This phenomenon stems from\nthe \frst terms in \u000es\u0011\nk;\u000eS\u0011\nk; they can directly generate an additional spin density at a given\npoint in a space as they survive averaging over the electron momentum direction. Moreover,\nthese terms can be explicitly expressed in terms of the Berry curvature, thus they are speci\fc\nfor topological systems.\nThe equilibrium spin density perturbations coupled with the Berry curvature of elec-\ntronic states have only in-plane components; substituting Eqs. 26, 27 to the spin density\nperturbation from Eq. 13 we get\n\u000eSk(r) =X\nk;\u0011n\u0011\nk\u000es\u0011\nk(r) +\u000eS\u0011\nk(r)\u0011(\u001ft+\u001fd)\u0001E(r); (28)\nwhere the magnetoelectric susceptibilities \u001ft;dcorrespond to the non-adiabatic spin tilting\nand spin-dipole e\u000bects, respectively. The evaluated expressions for \u001ft;\u001fdare given by\n\u001ft=e\n2\u0015so\u0000\nQ+\nF+Q\u0000\nF\u0001\n; \u001fd=\u0000e\u0015so\n0\n2\u0012\u0017+\nF\n\n2\nF++\u0017\u0000\nF\n\n2\nF\u0000\u0013\n; (29)\nwhereQ\u0006\nFis the total Berry \rux from Eq. 8. It is important to emphasize that both the non-\nadiabatic spin tilting and the spin-dipole e\u000bects are equally important to describe correctly\nthe emergent spin patterns in 2DEG. In Fig. 2 we plot the dependence of the overall spin-\nresponse coe\u000ecient \u001f\u0011\u001ft+\u001fd(solid lines) along with the partial contributions from \u001ft\nand\u001fd(dotted lines) on the electron gas Fermi energy \u0016. We note that the terms \u001ftand\n\u001fdare generally of the same order of magnitude. Moreover, in case when the electron gas\npopulates both spin subbands \u0016>\n0=2 the overall response entirely disappears \u001ft+\u001fd= 0\n(this feature was previously noted by [59, 66]). In the opposite case when electrons \fll only\nthe lowest spin-subband \u0016<\n0=2 the terms \u001ft;\u001fdhave opposite signs, which results in the\n12𝜉= 0.6(a)\n𝜒 (a.u.) 𝜒𝜒𝑑\n𝜒𝑡\n−1−0.5 0 0.5 1 1.5 2 2.5−1−0.500.511.5\n2𝜇/Ω0\n𝜉= 0.2(b)\n𝜒 (a.u.)\n𝜒𝜒𝑑\n𝜒𝑡\n−1−0.5 0 0.5 1 1.5 2 2.5−0.200.20.4\n2𝜇/Ω0FIG. 2. The dependence of the susceptibility \u001f=\u001ft+\u001fdon the Fermi energy for two values of\n\u0018parameter: (a) \u0018= 0:6 and (b)\u0018= 0:2.\nsign-altering dependence of \u001fon the Fermi energy. We \fnally note that when either the\nspin-orbit coupling or the exchange interaction is absent, the coe\u000ecients \u001ft=\u001fd= 0 turn\nto zero and the corresponding equilibrium spin patterns disappear.\nD. Discussion\nLet us discuss the physical signi\fcance of the described phenomena. We \frstly comment\non the role that intrinsic mechanisms described by Eqs. 26, 27 play for the charge and\nspin transport on distances that greatly exceed the mean free path. The non-adiabatic\nspin precession lies in the basis of the Karplus-Luttinger mechanism of the anomalous Hall\ne\u000bect (AHE) [67{69], of the so-called intrinsic mechanisms of the spin Hall (SHE) [62] and\nspin-galvanic e\u000bects [70]. However, in order to estimate correctly the overall electron gas\nresponse one has to additionally examine the disorder e\u000bects. In particular, the intrinsic\ncontribution to AHE, which is due to the anomalous velocity term \u000ev\u0011\nk/eF\u0011\nk\u0001[ez\u0002E], is\ngenerally cancelled out by the contributions due to side-jump scattering processes [53, 56, 57].\nAlternatively, considering the generation of spin currents upon the applied homogeneous\nelectric \feld one has to carefully account for the emergent nonequilibrium phenomena [70{\n73]; e.g. the spin Hall current due to the intrinsic mechanism is often compensated by the\nnonequilibrium spin current arising nearby the sample boundaries [58, 74, 75].\nHowever, the contributions \u000es\u0011\nk;\u000eS\u0011\nkpreserve the importance in the nondissipative regime,\nwhen the underlying electrostatic perturbation varies at the distances much smaller than the\nmean free path. In particular, this matters for 2DEG charge and spin distribution around an\n13ionized impurity, at that the typical spatial scale under consideration is the Thomas-Fermi\nscreening length. The distribution of an excessive 2DEG spin density emerging around an\naxially symmetric perturbation forms a skyrmion-like vortex pattern which is schematically\nshown in Fig. 1. One concludes that a smooth electrostatic potential disorder in topological\nspin polarized 2DEG inevitably generates chiral spin textures, which can be particularly\nimportant for the transport properties of the corresponding system; the formation of non-\ncollinear spin order generally leads to the topological Hall e\u000bect [76{78]. Moreover, in view of\nthe spin-velocity coupling the formation of a mesoscopic in-plane spin density is accompanied\nby the generation of the persistent electrical current density j(r) =e2\u0015so[ez\u0002\u000eS(r)]. In\nthis regard an axially symmetric perturbation from Fig. 1 is additionally featured by radially\npropagating electric currents. The presence of local equilibrium currents also maintains the\norbital magnetization, this e\u000bect has been considered in [66].\nIt is worth mentioning that the considered magnetoelectric susceptibility of free electrons\ngenerally opens up a possibility to directly a\u000bect the host magnetization by a mesoscopic\nelectric perturbation. The electric \feld-induced 2DEG spin density lies in 2D channel plane\nand it is perpendicular to the orientation of host magnetization, thus it is able to pro-\nduce torque-like e\u000bects. However, these issues remain poorly investigated, even despite its\nimportance for the magnetization control at nanoscales.\nThe microscopic mechanisms under consideration are general for multiband systems. In\nthe appendix A we present the connection of our method with the wave-packet quasiclassical\ntechnique used in [39{41]. In the appendix B we relate \u000es\u0011\nk;\u000eS\u0011\nkto the Kubo formula\nmethod for the charge-spin correlation functions used in [59]. In particular, we show that\nthe non-adiabatic spin precession is described by the interband correlation functions, while\nthe spin-dipole e\u000bect stems from the intraband ones.\nIV. SPIN DYNAMICS AND MAGNETO-OPTICAL EFFECTS\nA. Electric dipole spin resonance\nIn this section we focus on the electron gas spin dynamics in presence of an oscillating\nelectric \feld and describe the corresponding optical properties of a magnetic two-dimensional\nsystem. The optical response of a 2D conductive channel is generally encoded in the optical\n14FIG. 3. The electric dipole spin resonance scheme and the appearance of MOKE due to the\nresonant Hall current generation j!/[ez\u0002\u000eS!] .\nconductivity \u001b(!). In particular, the absorption coe\u000ecient \u000b(!) = (4\u0019=c)Re[\u001bxx(!)] is\nconnected with the longitudinal part of conductivity \u001bxx. Also, since the time-reversal\nsymmetry is broken in presence of magnetism, di\u000berent magneto-optical e\u000bects are possible,\ne.g. the magneto-optical Kerr e\u000bect (MOKE), that is the rotation of the re\rected light\npolarization by the complex Kerr angle \u001eK. MOKE generally appears in a conductive media\ndue to nonzero optical Hall conductivity \u001bH(!), for a 2D layer and normal incidence [79]\none can expess \u001eK=\u001bH=\u001bxxp\n1 + (4\u0019i=! )\u001bxx. Importantly, the considered geometry opens\nup the possibility to realize the resonant enhancement of the Hall conductivity and, thus, of\nthe related magneto-optical e\u000bects.\nCommonly, MOKE is seen to acquire a resonance structure due to interband transitions\na\u000bected by the combined e\u000bect of the spin-orbit coupling and the electron spin polarization;\nthe corresponding intrinsic contributions to the Hall conductivity at \fnite frequencies have\nbeen investigated in a number of papers [44, 80{82]. The general idea that we are going to\nexplore in this paper and which stands in the basis for the enhancement of magneto-optical\nphenomena is that the optical properties of magnetic 2D systems can be understood in\nterms of the electric dipole spin resonance (EDSR). Correspondingly, the part of the optical\nconductivity responsible for the resonant features can be directly related to the resonantly\ngenerated spin density of 2DEG.\nLet us illustrate this process in more detail, see Fig. 3. The exchange interaction \feld gives\nrise to a momentum-independent Zeeman splitting of the electron spin subbands, for the\nconsidered geometry it is directed perpendicular to the 2DEG plane. In fact, the spin-orbit\ninteraction can be viewed as k-dependent e\u000bective magnetic \feld \nso(k) acting on electron\n15spins. The applied in-plane ac-electric \feld E!e\u0000i!tcauses the electron's momentum oscil-\nlations\u000ek/E!e\u0000i!t, so the associated spin-orbital \feld also oscillates with frequency !.\nWe note that \nso(k) is perpendicular to the out-of-plane exchange interaction component\n\n0. Naturally, this makes it possible to induce spin transitions when the electric \feld fre-\nquency coincides with the magnitude of the Zeeman spin splitting ~!= \n 0, which is exactly\nthe EDSR scheme [83]. This spin resonance causes the equilibrium electron spin density\nS0kezfrom Eq. 11 to rotate onto 2DEG plane, thus resonantly generating an excessive\nin-plane spin density \u000eS!. In view of the spin-orbit coupling Eq. 5 between the velocity\nand spin operators, the accumulation of \u000eS!immediately leads to a resonant enhancement\nof the associated electric current density \u000ej!= 2e \u0015so[ez\u0002\u000eS!] and of the corresponding\ncontribution to the optical conductivity. Importantly, the in-plane spin density appears in\ntilted polarization with respect to the vector of the electric \feld, see Fig 3. In particu-\nlar, the manifestation of the nonzero Berry curvature lies in the fact, that there exists the\n\"perpendicular\" polarization of the spin density, which gives rise to the anomalous velocity\n\u000ev\u0011\nk/eF\u0011\nk\u0001[ez\u0002E] directed perpendicular to E!and responsible for the the magneto-\noptical e\u000bects. The resonant generation of the spin density in this polarization leads to the\nenhancement of \u001bH(!).\nB. Density matrix in the dynamical regime\nLet us consider an oscillating electric \feld E!e\u0000i!tapplied in plane of the electron gas.\nWe assume that the system remains homogeneous and present ^fkin the following form\n^fk(t) =1\n2nk(t) +Sk(t)\u0001^\u001b: (30)\nWe keep to the high-frequency regime when !greatly exceeds the typical inverse relaxation\ntime\u001c\u00001\nscdue to the scattering processes. The distribution function nk(t) =nk+\u000enk(!)e\u0000i!t\nsatis\fes the scalar part of the kinetic equation Eq. 14\n@nk(t)\n@t\u0000e(E(t)\u0001rk)nk(t) = 0; (31)\nSince the equilibrium part contains terms from both spin subbands nk=n+\nk+n\u0000\nk, the linear\nresponse perturbation \u000enk(!) =\u000en+\nk(!) +\u000en\u0000\nk(!) generally contains two contributions\n\u000en\u0011\nk(!) =\u0000eE\u0001v\u0011\nk\ni!\u0012\n\u0000@n\u0011\nk\n@\"\u0013\n: (32)\n16The equation governing 2DEG spin dynamics is obtained similarly to Eq. 14 and reads as\n@Sk(t)\n@t\u0000[\nk\u0002Sk(t)] +e(E(t)\u0001rk)Sk(t) = 0: (33)\nAt zero electric \feld this equation describes the electron spin precession around \nk. The\nstatic regime solution in this case corresponds to the equilibrium spin distribution S\u0006\nkk\nk\ndirected parallel or antiparallel to the spin splitting \feld, while the non-stationary solution\ndescribes the electron spin precession around \nkwith an eigenfrequency \n k. The nonzero\nE, in its turn, drives the spin dynamics due to the spin transfer in the momentum space.\nNaturally, when the frequency of an external \feld !coincides with the precession frequency\nof thek-electrons, the EDSR conditions are ful\flled leading to the resonant rotation. This\nrotation occurs with the Rabi frequency !R/\u0015soE, which goes to zero at small electric\n\felds. Naturally, in case of vanishing !Rwe can consider the linear response regime with\nSk(t) =S0\nk+\u000eSk(!)e\u0000i!tdi\u000bering from the equilibrium value S0\nk=n+\nks+\nk+n\u0000\nks\u0000\nkby the\nlinear-order correction \u000eSk(!). This is justi\fed when the ongoing evolution of Sk(t) due to\nthe Rabi oscillations is interrupted by the spin relaxation processes. We thus introduce the\nphenomenological spin relaxation rate \u0000 and assume !R\u001c\u0000\u001c\n0.\nIn the linear response regime we can consider the spin response \u000eSk(!) =\u000eS+\nk(!) +\n\u000eS\u0000\nk(!) independently for each spin subband (recall that S0\nk=n+\nks+\nk+n\u0000\nks\u0000\nk). It is con-\nvenient to present the linearized part in the following way \u000eS\u0011\nk(!) =\u000en\u0011\nk(!)s\u0011\nk+n\u0011\nk\u000es\u0011\nk(!),\nwhere\u000en\u0011\nk(!) is determined by Eq. 32 and the equation for \u000es\u0011\nk(!) is given by\n(\u0000i!+ \u0000)\u000es\u0011\nk(!)\u0000[\nk\u0002\u000es\u0011\nk(!)] +e(E!\u0001rk)s\u0011\nk= 0: (34)\nLet us introduce the notation \u000es\u0011\nk0\u0011\u000es\u0011\nk(!!0) for the additional electron spin density\nfrom Eq. 24 emerging in the static limit, we note that ( \u000es\u0011\nk0\u0001\nk) = 0. The third term\nin this equation can be presented as follows e(E!\u0001rk)s\u0011\nk= [\nk\u0002\u000es\u0011\nk0]. The spin density\nperturbation \u000es\u0011\nk(!) lies in the plane perpendicular to \nk, the two independent polarizations\nfor\u000es\u0011\nk(!) are given by \u000es\u0011\nk0and [nk\u0002\u000es\u0011\nk0], wherenk=\nk=\nkfrom Eq. 4. The solution\nof the precession equation can be written in terms of these two vectors as follows\n\u000es\u0011\nk(!) =\u0000\n2\nk\n(!\u0000\nk+i\u0000) (!+ \nk+i\u0000)\u0012\n\u000es\u0011\nk0+\u0000i!+ \u0000\n\nk[nk\u0002\u000es\u0011\nk0]\u0013\n: (35)\nThe \frst term is directly due to the \fnite-frequency evolution of the non-adiabatic spin\ntilt mechanism. The second term exists only at \fnite frequencies and it arises from the\n17electron spin retardation in the momentum space. The denominator has a pole structure\nwhich re\rects the EDSR with the multiple resonances determined by != \nk.\nThe resulting correction to the density matrix can be presented as a sum of two terms\n\u000e^fk=e\u0000i!t(\u000e^fden\nk+\u000e^fspin\nk), where\u000e^fden;spin\nk take the following form\n\u000e^fden\nk=1\n2\u0000\n\u000en+\nk(!) +\u000en\u0000\nk(!)\u0001\n+\u0000\n\u000en+\nk(!)s+\nk+\u000en\u0000\nk(!)s\u0000\nk\u0001\n\u0001^\u001b; (36)\n\u000e^fspin\nk=\u0000\nn+\nk\u000es+\nk(!) +n\u0000\nk\u000es\u0000\nk(!)\u0001\n\u0001^\u001b: (37)\nC. Resonant spin response and optical conductivity\nWe start the discussion of the optical conductivity. The contribution \u000e^fden\nkis related\nspeci\fcally to the perturbation of the electron density and it gives rise to the dominant part\nof the longitudinal conductivity\nj!=eX\nk;\u0011\u000en\u0011\nk(!)\u0001v\u0011\nk=\u001b0\nxx(!)E!; \u001b0\nxx(!) =ie2\n!v2\nF+\u0017+\nF+v2\nF\u0000\u0017\u0000\nF\n2: (38)\nThis is simply the Drude conductivity at \fnite frequency and it describes nondissipative\nretardation of the 2DEG density in ac-electric \feld. On the contrary, the term \u000e^fspin\nkis due\nto the spin rotation only. This contribution is responsible for the spin resonance related\nphenomena and below we consider its role in more detail.\nThe density of an electric current \u000ej!emerging due to the spin part of the density matrix\n\u000e^fspin\nkis coupled with an induced in-plane spin density \u000eS!of 2DEG\n\u000ej!= 2e\u0001\u0015so[ez\u0002\u000eS!]; (39)\n\u000eS!=1\n2X\nkSp\u0010\n\u000e^fspin\nk\u0001^\u001b\u0011\n=X\nk;\u0011n\u0011\nk\u000es\u0011\nk(!): (40)\nSince\u000es\u0011\nk(!) generally has two polarizations, see Eq. 35, the overall spin \u000eS!and corre-\nspondingly the associated current \u000ej!are also featured by two independent polarizations\n\u000eS!=\u001fl(!) [ez\u0002E!] +\u001fH(!)E!; (41)\n\u000ej!=\u001bl(!)E!+\u001bH(!) [ez\u0002E!]; (42)\nwhere\u001bl;H(!) = 2e\u0015so\u001fl;H(!). By this we identi\fed the contributions to the optical con-\nductivity related to the magnetoelectric spin susceptibility.\n18𝜉= 0.5\n𝜇= 1.3Ω 0(a)\n𝜎𝑙,𝐻 [𝑒2/𝜋/planckover2pi1]\nRe[𝜎𝑙(𝜔)]|𝜎𝐻(𝜔)|←Ω−\n𝐹←Ω+\n𝐹\n0.5 1 1.5 2 2.5 300.511.5\n/planckover2pi1𝜔/Ω0\n(b)ε(k)\nk(−)(+)\nµ\nΩ0Ω−\nF\nΩ+\nFFIG. 4. (a) The dependence of optical conductivities \u001bl;Hon frequency exhibits a resonant struc-\nture due to EDSR. (b) Electron band structure and the transition energies \n\u0006\nFat the Fermi level.\nThe correction to the longitudinal conductivity \u001bl(!) is related to the retardation term\n[nk\u0002\u000es\u0011\nk0] in Eq. 35. Using the formula Eq. 26 for \u000es\u0011\nk0and averaging over momentum\ndirections we get (below we restore the Planck constant ~)\n\u001bl(!) =\u0000ie2\u0001X\nk\u0000\nn\u0000\nk\u0000n+\nk\u0001\n~!\n(~!\u0000\nk+i\u0000) (~!+ \nk+i\u0000)\u00152\nso\n\nk\u0012\n1 +\n2\n0\n\n2\nk\u0013\n: (43)\nThe straightforward calculation of this integral gives\n\u001bl(!) =\u0000ie2\n16\u0019~\u00142\n2\n0\n~!\u00121\n\nmin\u00001\n\n\u0000\nF\u0013\n+\u0012\n1 +\n2\n0\n(~!)2\u0013\nln\u0012~!+ \n\u0000\nF\n~!\u0000\n\u0000\nF\u0001~!\u0000\nmin\n~!+ \n min\u0013\u0015\n;(44)\nwhere \n min= \n 0for\u0016 < \n0=2 and \n min= \n+\nFfor\u0016 > \n0=2. The expression from above\nremains well-de\fned at \u0000 !0. In fact, the poles ~!= \nkin the denominator of \u000es\u0011\nk(!) lie\nin the continuum spectrum, so the overall response of closely lying resonances merges onto\nthe!-regular curve featured by the Van Hove singularities at the edges of the spin splittins\n~!= (\n 0;\n\u0006\nF).\nThe real part of the longitudinal conductivity describes the energy dissipation. The\npresence of the resonant poles in Eq. 35 re\rects the appearance of a \fnite absorption.\nIndeed, the absorption coe\u000ecient is nonzero in the frequency range \n min<~! < \n\u0000\nF(see\nFig. 4b) corresponding to EDSR, the expression is given by\n\u000b(!) =4\u0019\ncRe [\u001bl(!)] =\u0019e2\n4~c\"\n1 +\u0012\n0\n~!\u00132#\n; \nmin<~!<\n\u0000\nF: (45)\nThe Hall conductivity \u001bH(!) stems from the Berry curvature related term in \u000es\u0011\nk0. Taking\n19into account Eqs. 26, 35 and averaging over the momentum direction we express \u001bH(!)\n\u001bH(!) =\u0000e2\n~X\nk(n\u0000\nk\u0000n+\nk) \n2\nk\n(~!\u0000\nk+i\u0000) (~!+ \nk+i\u0000)\u0001Fk: (46)\nThe evaluation of this expression gives the following result\n\u001bH(!) =\u0000e2\n4\u0019~\n0\n~!ln\u0012~!+ \n\u0000\nF\n~!\u0000\n\u0000\nF\u0001~!\u0000\nmin\n~!+ \n min\u0013\n: (47)\nImportantly, the Hall conductivity has the same resonance-aware logarithmic term as \u001bl(!).\nFig. 4 demonstrates the resonant enhancement of the Hall conductivity in the EDSR ab-\nsorption frequency range. Namely, we plot the dependence of Re[ \u001bl(!)] and the absolute\nvaluej\u001bH(!)jon the electric \feld frequency. It is clearly seen from Fig. 4 that the increase\ninj\u001bH(!)jmagnitude occurs exactly in the same frequency range where Re[ \u001bl(!)]6= 0 is\nnonzero. In Fig. 5 we plot the dependences of real and imaginary parts of the spin-resonance\nrelated optical conductivities \u001bl;H(!) on frequency. The parameters are the same as in Fig. 4.\nThe Van Hove singularities give rise to the pronounced peaks in j\u001bl;H(!)jat the boundary\nof the absorption band ~!= \n+\nF;\n\u0000\nF. For the parameters taken in this plot ( \u0016= 1:3\n0) the\nlower boundary is determined by \n+\nF, see Fig. 4, as the electrons populate both spin sub-\nbands. We also note that the behavior of \u001bl;H(!) when approaching the static limit !!0 is\ndi\u000berent, see Fig. 5. While the longitudinal part goes to zero \u001bl!0, the Hall conductivity\nhas a \fnite nonzero limit \u001bH!(e2=~)(Q+\nF+Q\u0000\nF) determined by the total Berry \rux Q\u0006\nF\nfrom Eq. 8 and re\recting the appearance of persistent electric currents associated with the\nmagnetoelectric susceptibility. In the static limit, however, the accurate calculation of \u001bH\nfor a macroscopic sample requires one to take into account the disorder e\u000bect [84].\nD. Discussion\nThe calculations of the optical conductivity of multiband systems is typically performed\nusing the Kubo formula [44, 80{82]. In the Appendix C we relate the spin polarization and\nthe density contributions from the density matrix approach with di\u000berent terms from the\nKubo formalism. In Table I we summarize the correspondence between these approaches;\nnaturally the spin resonance related terms are connected with the interband contributions\n\u001binterto the conductivity.\nLet us comment on the role of spin relaxation and electron scattering. The multiple-peak\nstructure of \u001bl;H(!) visible in Fig. 4 can be well resolved only provided that the spin-orbit\n20(a)\n𝜎𝑙[𝑒2/𝜋/planckover2pi1]\nRe[𝜎𝑙(𝜔)]\nIm[𝜎𝑙(𝜔)]←Ω−\n𝐹\n←Ω+\n𝐹\n0.5 1 1.5 2 2.5 3−0.100.10.2\n/planckover2pi1𝜔/Ω0\n(b)\n𝜎𝐻[𝑒2/𝜋/planckover2pi1]\nRe[𝜎𝐻(𝜔)]Im[𝜎𝐻(𝜔)]←Ω−\n𝐹\n←Ω+\n𝐹\n0.5 1 1.5 2 2.5 3−1−0.500.5\n/planckover2pi1𝜔/Ω0FIG. 5. The dependence of the optical conductivities \u001bl(!) (frame a) and \u001bH(!) (frame b) on the\nfrequency!, the parameters \u0018= 0:5;\u0016= 1:3 \n0.\nKubo formula \u001bintra\nxx \u001binter\nxx \u001binter\nH\nDensity matrix \u000e^fden\nkn\u0011\nk\u0002\nnp\u0002\u000es\u0011\nk0\u0003\n\u0001^\u001bn\u0011\nk\u000es\u0011\nk0\u0001^\u001b\nTABLE I. Density matrix and Kubo formula correspondence\ninteraction splitting ( j\n\u0006\nF\u0000\n0j\u001d\u001c\u00001\nsc) exceeds the energy broadening due to scattering pro-\ncesses. This requires rather strong spin-orbit coupling. In the opposite case, the resonance\npro\fle of\u001bl;H(!) will merge onto the single resonant-peak structure centered at \n 0with the\nline-shape sensitive to particular scattering and spin relaxation processes, in analogy with\nEDSR due to an electron gas in nonmagnetic semiconductors [85]. Interestingly, the Hall\nconductivity can possess an additional information on spin relaxation times.\nWe note that the \fnite absorption due to the electric dipole spin resonance in 2DEG\nis not strictly limited to the case when the Zeeman \feld has an out-of-plane component.\nIn fact, most of the EDSR experiments with 2DEG in nonmagnetic semiconductors [86{\n88] were carried out for the in-plane magnetic \feld geometry. This is particularly useful\nwhen one aims to suppress the orbital quantization e\u000bects and to focus on the spin-related\nresponse only. On the contrary, combining spin-orbital electronic channels with magnetism\nallows one to orient the Zeeman \feld perpendicular to the 2DEG plane without breaking the\nspectrum onto Landau levels. Moreover, in this setting the electron band states are featured\nby the appearance of a topological structure. Studying experimentally the electronic spin\nresonance phenomena in these systems seems of high interest as EDSR has an extra degree\nof freedom that is the strong enhancement of the adjoint magneto-optical e\u000bects.\n21Finally, the presented interpretation of the magneto-optical e\u000bects enhancement in terms\nof spin resonance is equally relevant for other two-dimensional models beyond Rashba fer-\nromagnets. For instance, e.g. massive Dirac metals [89], honeycomb lattices [90] or Haldane\nmodel [91] demonstrate similar resonant features of the Hall conductivity.\nSUMMARY\nIn summary, we have considered various spin-orbital phenomena leading to a nontriv-\nial behavior of an electron gas spin density upon application of the electric \feld in two-\ndimensional magnets. Based on the density matrix formalism we identi\fed di\u000berent micro-\nscopic mechanisms responsible for the 2DEG spin tilting in presence of an inhomogeneous\nelectrostatic potential, and described microscopic features of spin resonance upon oscillating\nelectric \feld with speci\fc focus on optical conductivity and magneto-optical phenomena.\nWe traced the connection of the considered spin phenomena with the Berry curvature of\nelectronic band states thereby specifying the role of electrons band topology. The presented\nanalysis clari\fes the basics of the electron gas magnetoelectric response in two-dimensional\nmagnets and contributes to the ongoing discussion of its spintronics applications.\nACKNOLEDGMENTS\nThe Author thanks I.V. Rozhansky, M.M. Glazov, P.S. Alekseev and N.S. Averkiev from\nthe Io\u000be Institute for a very fruitful discussion of the results and for giving useful advices.\nThe work has been carried out with the \fnancial support of the Russian Science Foun-\ndation (project 18-72-10111). K.S.D. also thanks the Foundation for the Advancement of\nTheoretical Physics and Mathematics \\BASIS\".\n22Appendix A: Wave-packet dynamics semiclassical approach\nThe semiclassical theory of band electrons moving in a spatially varying adiabatic per-\nturbationU(r) can be built by considering the wave-packet dynamics [42]. Let us introduce\nthe wave packetjWn\nkiconsisting of the n-th band Bloch states jun\nki, its centre of mass co-\nordinates in real and momentum spaces are located at ( rc;k). The average of the physical\nquantityQdescribed by the operator ^Qcan be expressed in the following way [40]\nQ=X\nk;nfn(k;r)\u0001hWn\nkj^QjWn\nkijr=rc\u0000r r\u0001X\nk;nfn(k;r)\u0001hWn\nkj^Q\u0001(^r\u0000r)jWn\nkijr=rc;(A1)\nwhere the \frst term treats the wave packet as a point particle with the distribution function\nfn(k;r), and the second term is the \frst-order correction due to the wave-packet \fnite\nsize e\u000bects. The great advantage of this consideration is that it allows one to describe the\nelectron dynamics in terms of semiclassical equations. For instance, in the nondissipative\nregimefn(k;r) satis\fes the Liouville's equation\ndfn\ndt=@fn\n@t+ffn;Hg= 0; (A2)\nwhereH=\"n\nk+U(r) is the classical Hamiltonian function in n-th electron band with\nenergy\"n\nk. The Poisson bracket fA;BgforA;B physical quantities depending on ( r;k)\ntakes into account the kinematic Berry phase [43, 49, 50]\nfA;Bg=!\u000b\f\u0001(@\u000bA)(@\fB); !\u000b\f=0\n@\"\u000b\f\r\nn\n\r\u000e\u000b\f\n\u0000\u000e\u000b\f 01\nA; (\u000b;\f) = (r;k); (A3)\nwhere!\u000b\fis the antisymmetric Poisson matrix, \"\u000b\f\ris the Levi-Civita tensor, and \nnis the\nBerry curvature in n-th Bloch band de\fned as follows \nn=rk\u0002An\nk=ihrkun\nkj\u0002jr kun\nki;\nwhereAn\nkis the Berry connection. The expression for the Liouville's equation with account\nfor the explicit form of !\u000b\fis given by:\n@fn\n@t+\u0012@\"n\nk\n@k+h\n_k\u0002\nni\u0013\n\u0001@fn\n@r+_k\u0001@fn\n@k= 0 (A4)\nwhere _k=\u0000rrH=\u0000rrU(r). The second term in brackets describes a full electron\nvelocityv=fH;rg=vn\nk\u0000[rrU;\nn], herevn\nk=rk\"n\nk.\nLet us apply this technique to calculate the emerging spin density nearby the electrostatic\ninhomogeneity. We focus on the linear response regime. Following Eq. A1 we present the\n23spin densityS(r) as follows\nS(r) =X\nk;nfn(k;r)\u0001hWn\nkj^SjWn\nki\u0000r r\u0001X\nk;nfn(k;r)hun\nkj^S(irk\u0000An\nk)jun\nki: (A5)\nIn the second term we took into account that the wave packet jWn\nkiis strongly localized\nnearbykin the momentum space and we can approximate it as follows jWn\nki\u0019eikrjun\nki,\nwhich leads us directly to the expression in Eq. A5. The unperturbed spin density S0\ncorresponds to U(r) = 0, at that fn(k;r) =f0\nn(k) andS0is given by\nS0=X\nk;nf0\nn(k)hun\nkj^Sjun\nki: (A6)\nThe linear order deviations from S0arise from three di\u000berent origins. Firstly, the distribution\nfunctionfn(k;r) =f0\nn(k) +\u000efn(k;r) in presence of Uis modi\fed according to Eq. A4\n(vn\nk\u0001rr)\u000efn(k;r) +F(r)\u0001@f0\nn\n@k= 0; \u000efn(k;r) =\u0000U(r)\u0012\n\u0000@f0\nn\n@\"\u0013\n: (A7)\nTaking into account the redistribution of the electron density in the \frst term in Eq. A5 and\napproximatinghWn\nkj^SjWn\nki\u0019hun\nkj^Sjun\nkiwe obtain the contribution identical with Eq. 21 in\nthe density matrix approach\n\u000eS(1)(r) =X\nk;n\u000efn(k;r)hun\nkj^Sjun\nki: (A8)\nAlso, the inhomogeneous structure of fngives rise to the spin-dipole contribution, that is\nthe second term in Eq. A5\n\u000eS(r) =\u0000F(r)\u0001X\nk;n\u0012\n\u0000@f0\nn\n@\"\u0013\nhun\nkj^S(irk\u0000An\nk)jun\nki: (A9)\nThe straightforward evaluation of this expression for the Rashba ferromagnet model leads\nto the susceptibility \u001fdgiven by Eq. 29. Finally, there is also the linear order perturbation\nwhich is not associated with the change in the electron distribution. In fact, the \frst term\nin Eq. A5 is determined by the average spin of an electron wave packet sn\nk(t) =hWn\nkj^SjWn\nki,\nwhich satis\fes the precession equation\ndsn\nk\ndt= [\nk\u0002sn\nk]: (A10)\nAccording to our discussion from III A, the wave-packet spin acquires a non-adiabatic cor-\nrection\u000esn\nklinear inFand given by Eq. 17. This term gives rise to the spin perturbation\n\u000eS=P\n(k;n)f0\nn\u000esn\nkidentical to \u001ftcontribution to the spin susceptibility from Eq. 29.\n24Appendix B: Kubo formula in the static limit\nIn this appendix we relate the semiclassical description of magnetoelectric susceptibility in\nterms of the density matrix with the Kubo formula for the charge-spin correlation functions,\nconsidered in detail in [59]. The spin density induced in 2DEG by the change in the potential\nenergyU(r) is given in linear response by\n\u000eS(r) =Zdq\n(2\u0019)2eiqrQ(q)U(q); (B1)\nwhereU(q) is the Fourier component of U(r) and the static charge-spin correlation function\nQ(q) can be computed from the Kubo formula\nQ(q) =X\nm;nQmn(q); (B2)\nQmn(q) =X\nkfm\nkhum\nkj^Sjun\nk+qihun\nk+qjum\nki\n\"m\nk\u0000\"n\nk+q+i0\u0000fm\nk+qhun\nkj^Sjum\nk+qihum\nk+qjun\nki\n\"n\nk\u0000\"m\nk+q+i0:\nThe terms Qnnwithm=ndescribe the intraband contributions, while Qmnwithm6=n\ncorrespond to the interband ones.\nThe Kubo formula B2 has been explicitly evaluated for an arbitrary wavevector qin [59]\nfor Rashba ferromagnet and Dirac models. Here we focus on the semiclassical regime when\nthe potential Uchanges smoothly on the Fermi wavelength \u0015Fscale, so the following relation\nis ful\flled\u0015F\u0001rrU\u001cU. In this case the spin response becomes local and the correlation\nfunction for the Rashba ferromagnet model takes the following form Q=iq\u0001\u001f, where\u001fis\ntheq-independent coe\u000ecient describing the susceptibility \u000eS(r) =\u001f\u0001E(r).\nWe now proceed with considering the role of intra- and interband terms. In the intraband\ncontribution Qnnwe replace ( fn\nk\u0000fn\nk+q)=(\"n\nk\u0000\"n\nk+q+i0)\u0019@fn\nk=@\"and keep only the q-linear\nterms in the matrix elements. At that the expression takes the following form\nQnn=\u0000iq\u0001X\nk\u0012\n\u0000@f0\nn\n@\"\u0013\nhun\nkj^S(irk\u0000An\nk)jun\nki; (B3)\nwhereAn\nk=ihun\nkjrkun\nkiis the Berry connection. When taking the Fourier transform Eq. B1\nQnngives exactly the spin perturbation \u000eSin form of Eq. A9 corresponding to the spin-\ndipole term within the semiclassical wave-packet approach. We thus conclude that the\nspin-dipole e\u000bect from Eq. 29 is related to the intraband terms in the Kubo formula.\n25In the interband contributions Qmnwe also keep only the linear terms with respect to q,\nwhich brings us to the following expression\nQmn(q) =iq\u0001X\nkfm\nkRe\u0012hun\nkj^\u001bjum\nki\u0001Amn\nk\n\"m\nk\u0000\"n\nk\u0013\n; (B4)\nwhereAmn\nk=ihum\nkjrkun\nki. The straightforward calculations for the Rashba ferromagnet\nmodel gives\nQmn(q) =iqX\nkfm\nk\u0001Fmn\nk\n2\u0015so; (B5)\nwhereFmn\nk=rk\u0002Amn\nkis the Berry curvature. The interband terms are related exactly\nto the non-adiabatic spin tilt e\u000bect described by \u000es\u0011\nkin the density matrix formalism and\ngiven by\u001ftsusceptibility from Eq. 29.\nAppendix C: Kubo formula in the dynamical regime\nIn this appendix we relate the Kubo formula calculations of the optical conductivity with\nthe spin resonance related terms emerging in the density matrix approach. Kubo formula\nfor the conductivity is given by\n\u001b\u000b\f(!) =ie2\nSX\nk;m;nfm\nk\u0000fn\nk+q\n\"m\nk\u0000\"n\nk+q\u0001v\u000b\n(k;m);(k+q;n)v\f\n(k+q;n);(k;m)\n\"m\nk\u0000\"n\nk+q+~!+i0; (C1)\nwhereq!0 andvijis the proper matrix element of the velocity operator between i;j\nstates. We consider \frstly the longitudinal conductivity \u001bxx(!). The contribution to \u001bxx(!)\ndue to intraband terms has the form\n\u001bintra\nxx(!) =ie2\n~!X\nmZ\nd\" \u0017m(\")\u0012\n\u0000@fm\nk\n@\"\u0013\nhjvx\nk;mj2i; (C2)\nwhere\u0017mis the density of states in the corresponding band mandhjvx\nk;mj2iis the angular\naveraged square of the matrix element modulus. This part describes the Drude conductivity\nat!\u001csc\u001d1 due to the perturbation of the electron density and it corresponds to Eq. 38\nfrom the main text. For the Rashba ferromagnet model the evaluation of the integral gives\n\u001bintra\nxx(!) =ie2\n!\u0001v2\nF+\u0017+\nF+v2\nF\u0000\u0017\u0000\nF\n2: (C3)\nThe contribution to \u001bxx(!) due to interband terms in case of the Rashba ferromagnet\nmodel has the following form\n\u001binter\nxx(!) =ie2\nSX\nkf\u0000\nk\u0000f+\nk\n\u0000\nk\u0001hjvx\n(k;\u0000);(k;+)j2i\n~!\u0000\nk+i0+f+\nk\u0000f\u0000\nk\n\nk\u0001hjvx\n(k;\u0000);(k;+)j2i\n~!+ \nk+i0: (C4)\n26The angular averaged term is hjvx\n(k;\u0000);(k;+)j2i= (\u00152\nso=2)(1 + \n2\n0=\n2\nk). Using this formula and\ncombining the denominators in \u001binter\nxxwe get the following expression\n\u001binter\nxx(!) =\u0000ie2\u0001X\nk\u0000\nf\u0000\nk\u0000f+\nk\u0001\n~!\n(~!\u0000\nk+i\u0000) (~!+ \nk+i\u0000)\u00152\nso\n\nk\u0012\n1 +\n2\n0\n\n2\nk\u0013\n; (C5)\nwhich repeats Eq. 43 for \u001bl(!). We thus conclude that \u001binter\nxx(!) is related to [ nk\u0002\u000es\u0011\nk0]\npolarization in terms of the in-plane spin density (see Eqs. 35, 26 from the main text). It is\ninstructive to analyze the energy absorption due to the spin resonance. For this purpose we\nwrite down explicitly the expression for the real part of the longitudinal conductivity due\nto the interband terms\nRe[\u001binter\nxx(!)] =\u0019e2\n~!SX\nk\u0000\nf\u0000\nk\u0000f+\nk\u0001\f\fvx\n(k;\u0000);(k;+)\f\f2\u0001\u000e\u0000\n\"\u0000\nk\u0000\"+\nk+~!\u0001\n: (C6)\nThe expression has the form of the Fermi golden rule, its straigthforward calculation leads\nto the Eq. 45.\nWe now turn to the transversal component of the conductivity. 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B 101, 045426 (2020).\n34"    },    {        "title": "0804.1514v1.Time_dependent_current_density_functional_theory_of_spin_charge_separation_and_spin_drag_in_one_dimensional_ultracold_Fermi_gases.pdf",        "content": "Time-dependent current-density-functional theory of spin-charge separation and spin\ndrag in one-dimensional ultracold Fermi gases\nGao Xianlong,1Marco Polini,2,\u0003Diego Rainis,2M.P. Tosi,2and G. Vignale3\n1Department of Physics, Zhejiang Normal University, Jinhua, Zhejiang Province, 321004, China\n2NEST-CNR-INFM and Scuola Normale Superiore, I-56126 Pisa, Italy\n3Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA\n(Dated: September 4, 2021)\nMotivated by the large interest in the non-equilibrium dynamics of low-dimensional quantum\nmany-body systems, we present a fully-microscopic theoretical and numerical study of the \\charge\"\nand \\spin\" dynamics in a one-dimensional ultracold Fermi gas following a quench. Our approach,\nwhich is based on time-dependent current-density-functional theory, is applicable well beyond the\nlinear-response regime and produces both spin-charge separation and spin-drag-induced broadening\nof the spin packets.\nPACS numbers: 71.15.Mb, 03.75.Ss, 71.10.Pm\nIntroduction |The dynamics of low-dimensional quan-\ntum many-body systems is a topic of great theoretical\nand experimental interest. With the recently acquired\ncapability to store and control ultracold atomic gases [1],\nit has become possible to study in the laboratory some\nideal systems which were once accessible only to theo-\nretical investigation. In particular, it is now possible to\ncon\fne atoms along a waveguide, thus realizing for the\n\frst time the elusive one-dimensional Fermi and Bose\ngases [2].\nOne-dimensional (1D) Fermi systems are theoretically\ninteresting because they present us with a breakdown of\nthe Landau Fermi liquid paradigm [3]. The low-energy\nexcitations of such systems are collective, while those of a\nconventional Fermi liquid are single-particle like; further-\nmore there is complete separation between charge and\nspin excitations, which in an ordinary Fermi liquid are\ntied together in the \\quasiparticle\".\nFully-microscopic calculations of collective spin and\ncharge dynamics in 1D Fermi systems have recently\nbeen performed by Kollath et al. [4] by means of the\ntime-dependent density-matrix renormalization-group\nmethod. This powerful numerical method seems however\nto be limited to inhomogeneous lattice systems at zero\ntemperature. In Ref. 5 we have pointed out a novel aspect\nof the 1D spin-charge separation phenomenon, which ap-\npears only at \fnite temperature. Namely, we have shown\nthat the propagation of a spin-density packet becomes\nintrinsically di\u000busive due to the spin drag e\u000bect [6, 7, 8]\n{ the transfer of momentum between fermions of oppo-\nsite spin orientations. We have developed a macroscopic\napproach, formulated entirely in terms of collective vari-\nables, for calculating the propagation of spin-density or\njust density packets, and we believe that the di\u000berence\nbetween the two cases can be experimentally detected.\nHowever, some di\u000eculties persist, which limit our abil-\nity to calculate theoretically the dynamics of spin and\ndensity packets: (i) single-particle thermal excitations,\nwhich may a\u000bect the shape of a spin-density packet inways that are di\u000ecult to distinguish from the e\u000bect of\nthe spin drag, are beyond the reach of the macroscopic\ntreatment of Ref. 5; (ii) the macroscopic theory, be-\ning restricted to long wavelengths and low frequencies,\nmisses the interaction of the wavepacket with microscopic\nFriedel-like oscillations, which are induced, for example,\nby the sharp ends of the waveguide; and, \fnally, (iii) the\nmacroscopic theory is restricted to the linear-response\nregime (LRR), whereas any experiment that is likely to\nbe performed with ultracold atomic gases would start\nwith a strong local disturbance [4].\nIt might appear that these di\u000eculties are formidable\nenough to prevent any further progress, but it is not so.\nIt turns out that a theoretical tool already exists, which\ncan take care of all three problems in a relatively simple\nmanner. It is called \\time-dependent spin-current den-\nsity functional theory\" (TD-SCDFT) [9, 10, 11] and it is\nbased on the idea of mapping the time-dependent many-\nbody problem into an e\u000bective single-particle problem,\nwith an e\u000bective potential that is designed to produce\nthe correct evolution of the spin density and the spin-\ncurrent density.\nSingle-particle aspects are naturally included in TD-\nSCDFT because the spin-resolved densities are repre-\nsented as a sum of contributions from one-particle wave-\nfunctions, which are fully microscopic objects that are\npopulated according to a Fermi-Dirac distribution func-\ntion (so thermal e\u000bects and Friedel oscillations are auto-\nmatically taken into account). Collective e\u000bects are in-\ncluded through an exchange-correlation (xc) \feld, which\nis expressed in terms of collective variables { the spin\ndensity and the spin-current density { and is derived\nnon-empirically from a suitable homogeneous reference\nsystem. Finally, the theory is not restricted to the LRR.\nIn this paper we present the \frst application of TD-\nSCDFT to the study of the collective dynamics of\nwavepackets in 1D Fermi gases, and we compare the\nresults with those of Ref. 5. We show that the sim-\nplest xc potential, constructed from the adiabatic local-arXiv:0804.1514v1  [cond-mat.str-el]  9 Apr 20082\nspin-density approximation (ALSDA), is already su\u000e-\ncient to produce spin-charge separation. But we also ob-\nserve that the density packet gets progressively \\contam-\ninated\" with microscopic Friedel-like oscillations coming\nfrom the ends of the waveguide, which were absent in\nthe macroscopic calculation. Going beyond the ALSDA,\nwe show that the spin-drag e\u000bect is nicely captured and\ncontinues to be the dominant cause of di\u000busion. In short,\nwe demonstrate the feasibility of a novel and fully micro-\nscopic computational approach, which leads to detailed\npredictions to be compared with future experiments.\nThe model |We consider a two-component Fermi gas\n(e.g.a mixture of two hyper\fne states of6Li [12]) with N\natoms con\fned inside a tight waveguide along the ^ xdirec-\ntion. The two species of atoms are assumed to have the\nsame mass mand di\u000berent \\spin\" (hyper\fne state label)\n\u001b=\"or#. The fermions interact viaa zero-range s-wave\nrepulsive potential v(x) =g1D\u000e(x) [13], whose strength\ncan be tuned e.g.by using a magnetic \feld-induced Fes-\nhbach resonance between the two di\u000berent spin states [2].\nThe Fermi gas is con\fned in the waveguide by a static\ntrapV(x).\nSimilarly to what done in Refs. 4 and 14, we now\nimagine that a local external potential, which couples\nonly to atoms of one spin species (hereby referred to\nas \\the accumulator \"), acts on the Fermi gas for times\nt\u00140 creating an accumulation of, say, up-spin atoms\nclose to the trap center. The state of the system for\ntimest\u00140 is denoted by j\t0i, which is the ground\nstate of the Hamiltonian ^H0=^HY+^V+^Vacc. Here\n^HY=P\ni^p2\ni=(2m) +g1DP\ni<j\u000e(^xi\u0000^xj) is the Yang\nHamiltonian [15], ^V=P\niV(^xi) describes the trap-\nping potential, and, \fnally, ^Vacc=P\ni;\u001bW\u001b(^xi) with\nW\u001b(x) =A\u001bexp [\u0000x2=(2w2)], is the potential that de-\nscribes the accumulator [16] ( A\"<0 andA#= 0). We\ndo not make any assumption on the strength A\"of the\naccumulator ( i.e.we do not assume to be in the LRR).\nAt timet= 0+the accumulator is suddenly turned o\u000b\nand the \\charge\", n(x;t) =n\"(x;t)+n#(x;t), and \\spin\",\ns(x;t) =n\"(x;t)\u0000n#(x;t), densities are allowed to prop-\nagate along the waveguide in the presence of V(x). Here\nn\u001b(x;t) =h\t(t)j^ y\n\u001b(x)^ \u001b(x)j\t(t)iare the spin-resolved\ndensities, withj\t(t)ithe state of the system at time t[17]\nand ^ y\n\u001b(x) a \feld-operator that creates a fermion with\nspin\u001bat position x.\nDynamics from TD-SCDFT |According to TD-\nSCDFT [9, 10, 11], the spin-resolved densities n\u001b(x;t)\nand the associated current densities j\u001b(x;t) at times\nt > 0 can be found by solving the time-dependent\nKohn-Sham (KS) equations\ni~@t \u000b;\u001b(x;t) =\u0014\n\u0000~2\n2m@2\nx+V(\u001b)\nKS[n\u001b;j\u001b](x;t)\u0015\n \u000b;\u001b(x;t);\n(1)\nwhereV(\u001b)\nKS[n\u001b;j\u001b](x;t) =V(x) +V(\u001b)\nH[n\u001b](x;t) +V(\u001b)\nxc[n\u001b;j\u001b](x;t) is the KS potential, which includes the\ntrapping potential V, the Hartree mean-\feld potential\n[V(\u001b)\nH[n\u001b](x;t) =g1Dn\u0016\u001b(x;t), where \u0016\u001b=\u0000\u001b], and the xc\npotential { the latter a functional of the spin and cur-\nrent densities. Although, in general, the xc e\u000bects in\nTD-SCDFT are represented by a vector potential [10],\nit turns out that in 1D a vector potential can be trans-\nformed into a scalar potential by an appropriate gauge\ntransformation: here we have already taken advantage of\nthis possibility. The densities are self-consistently deter-\nmined via the usual relation\nn\u001b(x;t) =X\n\u000bj \u000b;\u001b(x;t)j2\nexp [(\"\u000b;\u001b\u0000\u0016)=(kBT)] + 1;(2)\nwhere\"\u000b;\u001bare the static KS energies of the initial state\nwith chemical potential \u0016. These energies are found by\nsolving a static KS self-consistent problem corresponding\nto^H0. The current densities are related to the densi-\nties by the continuity equations @xj\u001b(x;t) =\u0000@tn\u001b(x;t).\nNote that due to the time-dependence of n\u001bandj\u001b, the\nKS Hamiltonian is time- dependent .\nIn order to proceed we need a sensible approximation\nfor the xc potential. In order to grasp the physically rel-\nevant facts that occur after the quench described above,\nwe make use of the following approximate expression:\nV(\u001b)\nxc[n\u001b;j\u001b](x;t)'V(\u001b)\nALSDA [n\u001b](x;t) +V(\u001b)\ndyn[n\u001b;j\u001b](x;t).\nIn this equation V(\u001b)\nALSDA [n\u001b](x;t) represents the ALSDA\ncontribution, which depends only on the densities [18]:\nV(\u001b)\nALSDA [n\u001b](x;t) =@[n\"hom\nxc(n\";n#)]\n@n\u001b\f\f\f\f\nn\u001b!n\u001b(x;t);(3)\nwhere\"hom\nxcis the xc energy (per particle) correspond-\ning to the Yang model ^HY, which can be easily found\nfrom the Bethe- Ansatz equations in the thermodynamic\nlimit [15, 19].\nThe dynamical (or non-adiabatic) contribution to the\nxc potential, V(\u001b)\ndyn[n\u001b;j\u001b](x;t), is given by\nV(\u001b)\ndyn(x;t) =\u0000Zx\n\u00001dx0F(\u001b)\nsd(x0;t); (4)\nwhereF(\u001b)\nsdis the spin-drag-related force [6] exerted by\nthe atoms with spin \u0016 \u001bon the atoms with spin \u001b,\nF(\u001b)\nsd(x;t) =\u0000mn\u0016\u001b\nn\u001csd(v\u001b\u0000v\u0016\u001b)\f\f\f\f\nn\u001b!n\u001b(x;t):(5)\nDue to Galileian invariance this force depends on the rel-\native velocity between the two atom species, v\u001b\u0000v\u0016\u001b=\nj\u001b=n\u001b\u0000j\u0016\u001b=n\u0016\u001b. In Eq. (5) \u001csdis the spin-drag relaxation\ntime { the inverse of the rate of momentum transfer be-\ntween atoms of opposite spin orientation { which has re-\ncently been calculated in 1D [20]. We will use the results3\nof Ref. 20 as input for our numerical calculations. Using\nEq. (5) and the continuity equation in Eq. (4) we \fnd\nV(\u001b)\ndyn(x;t) =\u0000Zx\n\u00001dx0m\u001c\u00001\nsd\f\f\nn\u001b!n\u001b(x0;t)\nn(x0;t)\n\u0002X\n\u001b0\u001b\u001b0n\"(x0;t)n#(x0;t)\nn\u001b(x0;t)n\u001b0(x0;t)F\u001b0(x0;t);(6)\nwhereF\u001b0(x0;t) =Rx0\n\u00001dx00@tn\u001b0(x00;t).\nQualitative analysis of spin-charge separation | Before\nproceeding with the numerical analysis we want to clar-\nify the mechanism by which the KS equations (1) pro-\nduce independent evolutions of the charge and spin den-\nsity. To see this, it is not even necessary to go beyond\nthe ALSDA. The essential point is that the KS equa-\ntion guarantees not only the continuity equation but also\nthe continuity equation for the momentum density, which\nreads@tj\u001b(x;t) =\u0000m\u00001@xP\u001b(x;t), where the quantum\npressureP\u001b(x;t) can be expressed in terms of KS or-\nbitals and is therefore an implicit functional of the den-\nsities. Combining the two conservation laws we arrive\nat@2\ntn\u001b(x;t) =m\u00001@2\nxP\u001b(x;t), which looks almost like\na classical wave equation. Indeed a classical wave equa-\ntion is immediately obtained in the LRR (small deviation\nfrom homogeneous, unpolarized state) since the quantum\npressure can then be approximated, in ALSDA, as a lin-\near functional of the densities: P\u001b=P\n\u001b0f\u001b\u001b0\u000en\u001b0where\n\u000en\u001b0are deviations from equilibrium. Then, after simple\nalgebraic transformation we arrive at two independent\nwave equations for n(x;t) ands(x;t) with two di\u000berent\nvelocities,vn(s)=p\n(f\"\"\u0006f\"#)=m, respectively. Admit-\ntedly this analysis pertains to the LLR. Spin and charge\nare not expected to be truly independent in the nonlin-\near regime. But since the correct linear-response limit\nis built into the KS equation we expect that a strong\nsignature of spin-charge separation will be seen also in\nthe nonlinear regime. Our numerical calculations con-\n\frm this expectation.\nNumerical results and discussion | We have solved\nEqs. (1)-(2) with a two-step predictor-corrector Crank-\nNicholson scheme. While the scheme outlined above is\ncompletely general, for simplicity, in the numerical cal-\nculations we have taken a simple box-shaped trapping po-\ntential,V(x) = 0 for\u0000L=2<x<L= 2 andV(x) = +1\nelsewhere. We use L?=L=N as unit of length and\nE?=~2=(mL?2) as unit of energy. For de\fniteness, we\n\fxN\"=N#= 40 (N= 80 fermions in total), A\"=\u0000E?,\nw=L?=p\n2,g1D= 2:0E?L?, andT= 0:5E?=kB. In\nFig. 1 we compare the evolution calculated in the ALSDA\n(which does not include spin drag) with the evolution\nobtained from the macroscopic theory [5]. Both theo-\nries predict a splitting of the initial peak into two peaks\nthat propagate with di\u000berent velocities (spin being slower\nthan charge) in agreement with the general picture of\nFIG. 1: (Color online) Top panel: charge density n(x; t) (in\nunits of 1 =L?) as a function of distance x(in units of L?), after\nthe subtraction of the background constant nb=N=L. The\ninitial density pro\fle is the wavepacket centered at x= 0\nand labeled by \\ t= 0\". The (red) solid line labeled by\n\\MACRO\" represents the corresponding n(x; t) at a later time\nt= 5~=E?, calculated according to the macroscopic theory\nof Ref. 5. The (blue) solid line labeled by \\MICRO\" rep-\nresents n(x; t) at the same time but calculated according to\nTD-SCDFT [Eqs. (1)-(2)] without spin-drag contribution [ i.e.\nwith V(\u001b)\ndyn(x; t) = 0]. Bottom panel: spin density s(x; t) (in\nunits of 1 =L?) as a function of x=L?. Color coding and label-\ning are the same as in the top panel.\nspin-charge separation. However, in the microscopic cal-\nculation the magnitude of the peaks decreases far more\nrapidly with time. It must be appreciated that this hap-\npens in spite of the fact that the total spin and parti-\ncle number are conserved quantities in both approaches.\nNote also that the width of the microscopic results in\nFig. 1 is much larger than that of the macroscopic re-\nsults: the reason is that the microscopic theory includes\ndi\u000busive-like mechanisms related to single-particle ther-4\nFIG. 2: (Color online) Same as in the bottom panel of Fig. 1\nbutwith spin-drag contribution. In the inset we show a 3D\nplot of the TD-SCDFT result for the spin density s(x; t) (in\nunits of 1 =L?) as a function of x=L?andtE?=~.\nFIG. 3: (Color online) A 3D plot of the TD-SCDFT result\nfor the charge density n(x; t) (in units of 1 =L?) as a function\nofx=L?andtE?=~.\nmal excitations [see Eq. (2)], which operate both in the\ncharge and spin channels and are completely missed by\nthe macroscopic theory. Finally, notice the asymmet-\nric forward-leaning shape of the density pulse calculated\nfrom the microscopic theory. This is a nonlinear e\u000bect,\nlikely due to the fact that the local velocity, proportional\nto the density, is higher at the center of the pulse than\nat its edges [21].Fig. 2 shows the e\u000bect of spin drag, which enters the\nmacroscopic calculation as a dissipative term in the wave\nequation for s(x;t) [see Eq. (10) in Ref. 5], the micro-\nscopic one through the current-dependent part of the xc\npotential [Eq. (6)]. It is evident that the e\u000bect of the spin\ndrag is much more pronounced in the microscopic cal-\nculation, where the double-peak structure is completely\nlost. This happens in spite of the fact that the spin-drag\ncoe\u000ecient\u001csdis generally smaller in the microscopic cal-\nculation, due to the e\u000bect of the spin polarization [20].\nClearly, the inclusion of microscopic excitations weakens\nthe collective behavior of the spin density. The large\ndi\u000berence between the macroscopic and microscopic re-\nsults demonstrates the importance of relying on the latter\nwhen performing quantitative calculations.\nIn Fig. 3 we show a 3D plot of the time evolution of\nthe density packet. What is notable here is that already\nat short times some density waves appear, coming from\nthe sharp edges of the waveguide: at larger time they\nmix with the original packet. These Friedel-like oscilla-\ntions are completely beyond the power of the macroscopic\napproach. In conclusion, the above calculations amply\ndemonstrate the versatility of the TD-SCDFT method in\nproducing detailed results which can be compared with\nfuture experiments on the propagation of charge and spin\npulses in ultracold Fermi gases.\nAcknowledgments | G.X. was supported by NSF of\nChina under Grant No. 10704066 and by DOE Grant\nNo. DE-FG02-05ER46203. G.V. was supported by NSF\nGrant No. DMR-0705460.\n\u0003Electronic address: m.polini@sns.it\n[1] For a recent review see e.g.I. Bloch, J. Dalibard, and W.\nZwerger, arXiv:0704.3011v2.\n[2] See for example B. Paredes et al. , Nature 429, 277\n(2004); T. Kinoshita, T. Wenger, and D.S. Weiss, Sci-\nence 305, 1125 (2004); H. Moritz et al. , Phys. Rev. Lett.\n94, 210401 (2005); A.H. van Amerongen et al. ,ibid. 100,\n090402 (2008).\n[3] See e.g.T. Giamarchi, Quantum Physics in One Dimen-\nsion (Clarendon Press, Oxford, 2004).\n[4] C. Kollath, U. Schollw ock, and W. Zwerger, Phys. Rev.\nLett. 95, 176401 (2005); C. Kollath, J. Phys. B 39, S65\n(2006).\n[5] M. Polini and G. Vignale, Phys. Rev. Lett. 98, 266403\n(2007).\n[6] I. D'Amico and G. Vignale, Phys. Rev. B 62, 4853 (2000).\n[7] K. Flensberg, T.S. Jensen, and N.A. Mortensen, Phys.\nRev. B 64, 245308 (2001).\n[8] C.P. Weber et al. , Nature 437, 1330 (2005).\n[9] Z. Qian, A. Costantinescu, and G. Vignale, Phys. Rev.\nLett. 90, 066402 (2003); Z. Qian and G. Vignale, Phys.\nRev. B 68, 195113 (2003).\n[10]Time-dependent density functional theory , ed. by M.A.L.\nMarques et al. (Springer-Verlag, Berlin, 2006), Ch. 5.\n[11] G.F. Giuliani and G. Vignale, Quantum Theory of5\nthe Electron Liquid (Cambridge University Press, Cam-\nbridge, 2005).\n[12] See e.g.M.W. Zwierlein et al. , Science 311, 492 (2006).\n[13] M. Olshanii, Phys. Rev. Lett. 81, 938 (1998).\n[14] A. Recati et al. , Phys. Rev. Lett. 90, 020401 (2003); L.\nKecke, H. Grabert, and W. H ausler, ibid. 94, 176802\n(2005).\n[15] C.N. Yang, Phys. Rev. Lett. 19, 1312 (1967).\n[16] We have assumed a simple Gaussian form for W\u001b(x) only\nfor de\fniteness: the TD-SCDFT scheme outlined below\nis completely general.\n[17] Note that the state j\t(t)iafter the sudden quench\nof the accumulator is formally given by j\t(t)i=exp [\u0000i(^HY+^V)t]j\t0i: the time evolution is due to\nthe non-equilibrium initial condition. The propagation in\ntime is in fact controlled by a time- independent Hamilto-\nnian, ^HY+^V6=^H0.\n[18] In writing Eq. (3) we are neglecting the temperature de-\npendence of the ALSDA xc potential, which we expect to\nbe unimportant for the physics we are describing here.\n[19] S.H. Abedinpour et al. , Phys. Rev. A 75, 015602 (2007).\n[20] D. Rainis et al. , Phys. Rev. B 77, 035113 (2008).\n[21] This e\u000bect is absent in the spin channel, con\frming that\nspin-charge separation holds with good accuracy in spite\nof nonlinearity."    },    {        "title": "1008.2191v2.Semiconductor_Spin_Noise_Spectroscopy__Fundamentals__Accomplishments__and_Challenges.pdf",        "content": "Semiconductor Spin Noise Spectroscopy:\nFundamentals, Accomplishments, and Challenges\nGeorg M. M ¨uller, Michael Oestreich, Michael R ¨omer, Jens H ¨ubner\u0003\nInstitut f¨ ur Festk¨ orperphysik, Leibniz Universit¨ at Hannover, Appelstraße 2, D-30167 Hannover, Germany\nAbstract\nSemiconductor spin noise spectroscopy (SNS) has emerged as a unique experimental tool that utilizes spin fluctuations to pro-\nvide profound insight into undisturbed spin dynamics in doped semiconductors and semiconductor nanostructures. The technique\nmaps ever present stochastic spin polarization of free and localized carriers at thermal equilibrium via the Faraday e \u000bect onto the\nlight polarization of an o \u000b-resonant probe laser and was transferred from atom optics to semiconductor physics in 2005. The inim-\nitable advantage of spin noise spectroscopy to all other probes of semiconductor spin dynamics lies in the fact that in principle no\nenergy has to be dissipated in the sample, i.e., SNS exclusively yields the intrinsic, undisturbed spin dynamics and promises optical\nnon-demolition spin measurements for prospective solid state based optical spin quantum information devices. SNS is especially\nsuitable for small electron ensembles as the relative noise increases with decreasing number of electrons. In this review, we first\nintroduce the basic principles of SNS and the di \u000berence in spin noise of donor bound and of delocalized conduction band electrons.\nWe continue the introduction by discussing the spectral shape of spin noise and prospects of spin noise as a quantum interface\nbetween light and matter. In the main part, we give a short overview about spin relaxation in semiconductors and summarize\ncorresponding experiments employing SNS. Finally, we give in-depth insight into the experimental aspects and discuss possible\napplications of SNS.\nContents\n1 Introduction 1\n2 Fundamentals of Spin Noise Spectroscopy 2\n2.1 Measurement Principle and Experimental Real-\nization . . . . . . . . . . . . . . . . . . . . . . 2\n2.2 Spin Noise of Donor Bound Electrons . . . . . 3\n2.3 Spin Noise of Delocalized Conduction Band Elec-\ntrons . . . . . . . . . . . . . . . . . . . . . . . 4\n2.4 Spectral Shape of Spin Noise . . . . . . . . . . 5\n2.5 Spurious Noise Contributions . . . . . . . . . . 6\n2.6 Spin Noise in Atomic Gases: A Quantum Inter-\nface between Light and Matter . . . . . . . . . 6\n3 Spin Dynamics in Semiconductor Structures 7\n3.1 Spin Dephasing in Semiconductors . . . . . . . 7\n3.2 Conventional Experimental Probes . . . . . . . 10\n3.3 Investigations by SNS . . . . . . . . . . . . . . 12\n4 Experimental Aspects of SNS 16\n4.1 Shot Noise Subtraction . . . . . . . . . . . . . 16\n4.2 Data Acquisition and Spectrum Analysis . . . . 17\n4.3 GHz Spin Noise Spectroscopy . . . . . . . . . 17\n\u0003Corresponding author\nEmail address: jhuebner@nano.uni-hannover.de (Jens H ¨ubner)5 Applications 18\n5.1 Quantum Random Number Generator . . . . . 18\n5.2 Spatially Resolved Measurements . . . . . . . 18\n6 Outlook 19\n1. Introduction\nThe physical properties of the electron spin in semiconduc-\ntors have been an active field of research since the pioneering\npublications of Lampel [1] and Parsons [2] in the end of the\n1960s. The theoretical proposal of the spin field e \u000bect transis-\ntor by Datta and Das in 1990 [3] has boosted the number of\nresearch papers in this field, giving rise to the notion of semi-\nconductor spintronics [4, 5, 6, 7]. Some years later, localized\nelectronic spins in semiconductors were suggested as possible\nqubits for quantum computation [8, 9, 10].\nAs a matter of fact, investigations of the electron spin de-\nphasing or relaxation times have been an integral part of semi-\nconductor spintronics since the very beginning [1, 2, 11]. How-\never, most experimental probes of semiconductor spin dynam-\nics rely on generation of a non-equilibrium spin polarization\n[1, 2, 12, 13, 14, 15]. This creation of a spin polarized elec-\ntron ensemble away from thermal equilibrium is necessarily\naccompanied by energy transfer to the system which modifies\nthe e\u000bectiveness of the di \u000berent mechanisms of spin dephasing\nand can—in the worst case—totally obstruct the measurement.\nFortunately, this fundamental problem can be circumvented ac-\ncording to the fluctuation-dissipation theorem [16, 17] which\nPreprint submitted to Physica E October 22, 2018arXiv:1008.2191v2  [cond-mat.mes-hall]  25 Mar 2011states that the response of a system to a perturbation is directly\nlinked to its fluctuations at thermal equilibrium. In other words,\nthe spin fluctuations of an electron ensemble deliver informa-\ntion about the dynamics of the spin system under an infinitesi-\nmal external perturbation which is not realizable in an experi-\nment [18]. Such spin noise, i.e., a time-varying stochastic spin\npolarization, was first predicted by Bloch [19] for a nuclear spin\nsystem and first measured by Sleator et al. [20]. Later, other\ngroups experimentally verified the fluctuation-dissipation theo-\nrem via magnetometric [21, 22, 23] and electrical [24] measure-\nments of the magnetic noise of spin glasses. More recently, Ru-\ngar and co-workers extended the use of spin fluctuations to very\nsmall nuclear and electronic spin ensembles by means of mag-\nnetic force microscopy [25, 26, 27, 28, 29]. In 2006, M ¨uller and\nJerschow utilized nuclear spin noise for magnetic resonance\nimaging [30].\nMeasurement of spin noise by optical means, i.e, spin noise\nspectroscopy (SNS), was first carried out in atom optics by\nAleksandrov and Zapasskii [31] in 1981. Spin-orbit coupling\ngives rise to an interaction between the electron spin and the\nphoton helicity via the dipole selection rules and thereby maps\nthe spin noise onto the light polarization of an o \u000b-resonant probe\nlaser via the Faraday e \u000bect [32]. Application of o \u000b-resonant\nprobe light is an established concept in atom optics to avoid\noptical pumping and to study optically thick samples [33] and\nseveral experiments in alkali metal vapors (see, e.g., Refs. [34,\n35]) clearly prove that for su \u000ecient detuning from the reso-\nnance SNS can be viewed as a quantum non-demolition mea-\nsurement [36, 37, 38] of the atomic spin.\nIn 2005, Oestreich and co-workers were the first to demon-\nstrate SNS in a semiconductor system, introducing this sensitive\nand nearly perturbation-free technique to study semiconductor\nspin physics [39]. SNS has been transferred to semiconduc-\ntor physics rather late since the shorter spin relaxation times in\nsemiconductors compared to atoms require much more sophis-\nticated experimental means [40]. Since the pioneering work\non semiconductor SNS, a considerable number of theoretical\n[41, 42, 43] as well as experimental [40, 44, 45, 46, 47, 48, 49]\npapers have been published demonstrating SNS in three-, two-,\nand zero-dimensional semiconductor systems.\n2. Fundamentals of Spin Noise Spectroscopy\n2.1. Measurement Principle and Experimental Realization\nThe basic idea of SNS is to map the stochastic spin fluc-\ntuations in the sample onto the polarization of the laser light.\nIts experimental realization is straightforward and depicted in\nFig. 1. Linearly polarized, below band gap laser light is trans-\nmitted through the investigated sample, which is mounted in a\ncryostat. The carriers in the sample are at thermal equilibrium\nand, accordingly, the expectation value of the spin polarization\nmz=(N\"\u0000N#)=(N\"+N#) vanishes.1Here, N\"andN#are\n1In the following, the axis of quantization for the electron spin is the z-\naxis, i.e., the direction of light propagation (see Fig. 1). For the sake of brevity\nand relevance for the most of the reviewed experiments, s=1=2 electrons are\nconsidered. However, SNS is also applicable to hole spins.\nB\nFFT spectrum analysislinear polarized\nlaser source\nθF(t)\nbalanced receiver--polarizing\nbeam splittersample in optional trans-\nverse magnetic field\nzx\nyFigure 1: Basic experimental setup for semiconductor SNS. Linearly polar-\nized laser light is transmitted through the sample. The Faraday rotation of the\nprobe light induced by a stochastic spin polarization is measured via an optical\npolarization bridge. The signal is analyzed in the frequency domain via FFT\nspectrum analysis.\nthe number of electrons with spin-up and spin-down, respec-\ntively, and N=N\"+N#denotes the total number of probed\nspins. As a finite electron ensemble is considered, the stan-\ndard deviation \u001bmzis non-zero. Hence, the electron ensemble\nshows a stochastic spin polarization at a given time. This re-\nsults in a spin dependent bleaching of the probed optical tran-\nsition and the spin imbalance becomes manifest in a di \u000berence\nin absorption \u000b\u0006of right (\u001b\u0000) and left (\u001b+) circularly polarized\nlight which in turn translates via the Kramers-Kronig relations\n[50, 51] to a di \u000berence of the dispersive part of the refractive\nindices for the two circular light components. Due to this cir-\ncular birefringence, the linearly polarized probe light, which\nis composed out of \u001b+and\u001b\u0000light, acquires a rotation of its\nlinear polarization direction, which is known as the Faraday ef-\nfect [32]. Thereby, the spin noise in the sample is projected\nonto the direction of the linear light polarization. These fluctu-\nations of the Faraday rotation of the probe light are measured in\nSNS via an optical polarization bridge consisting of a polarizing\nbeam splitter and balanced photodiodes as depicted in Fig. 1.\nThe time-continuous electrical signal from the balanced pho-\ntoreceiver is pre-amplified and—to avoid undersampling (see\nSec. 4.2)—sent through a frequency bandpass filter before the\nanalysis in the frequency domain is performed. This spectral\nanalysis reveals the correlations of the underlying spin dynam-\nics. Figure 2 shows a typical power spectrum of the measured\ntime signal with other noise contributions already subtracted\n(see Sec. 2.5). As expounded in detail in Sec. 2.4, this spin\nnoise spectrum contains essential information about the spin\ndynamics: The peak position yields the Larmor frequency !L,\ni.e., the precessional frequency of the electronic spins in a trans-\nverse magnetic field that is often—but not necessarily—applied\nin SNS to modulate the spin noise and to shift it from zero\nfrequency. The width of the curve wFWHM =(\u0019T2)\u00001scales\nwith the spin dephasing rate T\u00001\n2, and the area under the curve\ngives the spin noise power Pwhich is determined by the number\nof probed electrons and their degree of localization. Note that\npure homogeneous, i.e., exponential, spin dephasing results in\na Lorentzian line shape in the spin noise spectrum whereas in-\n2Figure 2: Typical spin noise spectrum with the shot noise background already\nsubtracted. The peak position gives the Larmor frequency and, hence, the e \u000bec-\ntive electron g-factor. The curve width scales inverse with the spin dephasing\ntime and the spin noise power, i.e, the area under the curve, yields valuable\ninformation about the underlying electron statistics.\nhomogeneous mechanisms yield a Gaussian broadening of the\nspin noise curve. To sum up, contrary to conventional exper-\nimental probes in which a depolarization or a decay of an ar-\ntificial spin orientation is measured (see Sec. 3.2), in SNS, no\nenergy has to be deposited in the sample and unperturbed spin\ndynamics can be experimentally accessed very close at ther-\nmal equilibrium. This measurement principle can be quite uni-\nversally realized for all kinds of doped semiconductor systems.\nThe probed optical transition is required to be polarizable which\nmeans that absorption of circularly polarized light would cre-\nate a spin orientation. The ratio of the degrees of optical spin\norientation and circular light polarization \u0018of the investigated\ntransition is determined by the dipole selection rules. In the\nabsence of spin-orbit coupling, every allowed transition would\nshow vanishing \u0018and the resulting spin noise power would be\nzero. Figure 3 schematically depicts the dipole selection rules\nfor the valence to conduction band transition in bulk GaAs. The\ndi\u000berent optical transition probabilities of the heavy hole and\nthe light hole to the conduction band with a ratio of 3 : 1 yield\n\u0018=\u00000:5 since heavy and light hole bands are degenerate at\nthe\u0000-point and the split-o \u000bband is far away in energy and can\ntherefore be neglected.\nIn the following two sections, two complementary systems—\na low doped semiconductor with localized, non-interacting elec-\ntron spins and a highly doped system with delocalized conduc-\ntion band electrons—are considered. We calculate the mean\ndeviation of mzand the corresponding spin noise power P. We\nassume that the equilibrium spin polarization due to magnetic\nfields is negligible, i.e., kBT\u001dg\u0003\u0016BBwhich holds so far for all\npublished semiconductor SNS measurements.\nCB\ns=1/2\nVB\nj=3/23131+1/2 -1/2\nLH +1/2 -1/2HH +3/2 -3/2Figure 3: Dipole selection rules for the valence band (VB) to conduction band\n(CB) transition in bulk GaAs. Left (right) circularly polarized light, shown\nas light (dark) arrows, creates a spin polarization in the conduction band of\nmz=\u00000:5 (mz= +0:5). The same selection rules also hold for the emission\nwhen spin polarized carriers recombine.\n2.2. Spin Noise of Donor Bound Electrons\nIn a low doped semiconductor system, where the donor elec-\ntrons are localized at the impurity atoms and non-interacting\nwith each other, the donor bound exciton transition ( ~!D0X=\nEG\u0000Eb\nD0X) is probed. Here, EGis the fundamental absorp-\ntion edge and Eb\nD0Xis the binding energy of the exciton-neutral\ndonor complex D0X [52]. Since the donor electrons do not in-\nteract with each other, the number of spin-up(down) electrons\nN\"(#)follows a binomial distribution with mean hN\"(#)i=0:5N\nand standard deviation \u001bN\"(#)=0:5p\nN. Here, N=N\"+N#=\nnDVis the total number of donor electrons within the probe vol-\nume Vwith nDbeing the impurity density. Thus, the standard\ndeviation of the stochastic spin polarization mzof the donor-\nbound electrons is given by\n\u001bmz=p\nN=N: (1)\nIn the absence of spin polarization, the Drude-Lorentz oscillator\nmodel describes this optical transition as a sum of Nharmonic\noscillators with oscillator strength f. Within this model (see,\ne.g., Ref. [53]), the absorption constant \u000band the dissipative\npart of the refractive index \u0014can be written as\n\u000b=\u00142!\nc0\n=e2f nD\n8m\u000f0!D0Xp\u000fB\u0000\n(!\u0000!D0X)2+ \u00002=42!\nc0; (2)\nwhere dispersive and dissipative contributions from far-detuned\ntransitions are subsumed by the background dielectric constant\n\u000fB, and \u0000is the width of the optical transition. The correspond-\ning dispersive part of refractive index is given by\nn=nB+˜n\n=p\u000fB\u0000e2f nD\n4m\u000f0!D0Xp\u000fB!\u0000!D0X\n(!\u0000!D0X)2+ \u00002=4;(3)\nThus, the deviation of the real part of the refractive index from\nnB=p\u000fBdecreases linearly with inverse detuning and the ab-\nsorbed energy decreases with the inverse detuning squared. In\nother words, the change of the refractive index is finite even at\nnegligible absorption.\n3Figure 4: Dissipative and dispersive part of the refractive index of an optical\ntransition according to the Drude-Lorentz oscillator model.\nThe stochastic spin imbalance mzyields a circular dichro-\nism due to the optical selection rules, i.e., di \u000berent constants of\nabsorption\u000b+and\u000b\u0000for\u001b+and\u001b\u0000light,\n\u0001\u000b=\u000b+\u0000\u000b\u0000\n=\u0018mz\u000b; (4)\nas well as a circular birefringence, i.e., di \u000berent indices of re-\nfraction n+andn\u0000for\u001b+and\u001b\u0000light,\n\u0001n=n+\u0000n\u0000\n=\u0018mz˜n: (5)\nIf the probe beam waist w0within the sample can be viewed\nas constant on the length scale of the sample thickness l, the\ncircular birefringence amounts to a Faraday rotation angle of\n\u0012F=\u0019\u0001n l=\u00150: (6)\nAccording to Eq. (1), the observed integrated spin noise power\nreads:\nP=\u001b2\n\u0012F=\u00192\u00182˜n2l\n\u00152\n0nDA; (7)\nwhere A=\u0019w2\n0is the laser spot area and \u0012Fis assumed to be\nsmall. Of course, Eq. (7) is only valid if the probe volume Vis\nlarge compared to the inverse of the doping concentration nD,\nwhich is nearly always the case. Note that Faraday rotation is\nusually independent of the laser spot area A, but in the case of\nSNS the stochastic spin polarization and thereby the Faraday\nrotation angle becomes larger for smaller probe volumes, re-\nsulting in the 1 =Arelation in the above equation. This behavior\nof the observed integrated spin noise power was experimentally\ndemonstrated for atomic vapors as well as for semiconductor\nsystems by Crooker et al. [54, 45] as it is depicted in Fig. 5.\n2.3. Spin Noise of Delocalized Conduction Band Electrons\nNext, we consider a semiconductor system with a doping\nlevel well above the metal-to-insulator transition [55, 56, 57].\nFigure 5: Double logarithmic plot of the integrated spin noise power Pas a\nfunction of the probe laser spot size Afor an n-type bulk GaAs sample ( nD=\n1:4\u00021016cm\u00003) at 10 K. The data is taken from Ref. [45] and shows the 1 =A\ndependence as expected from Eq. (7).\nIn such a sample system, the Fermi energy lies within the con-\nduction band, a behavior that is reported to occur for n1=3\nDaB>\n0:43, where aBis the e \u000bective Bohr radius of the donor electron-\nimpurity system, while the critical density of the metal-to-insulator\ntransition is given by n1=3\nDaB\u00190:25:::0:33 [56]. The two main\ndi\u000berences to very low doped samples are that (i)electrons are\ndegenerate and Fermi-Dirac statistics have to be applied and\nthat(ii)the interband transition is described by a sum of Drude-\nLorentz oscillators with di \u000berent resonant energies. This semi-\nconductor system is defined by its density of states D(E) and\nits doping density nD. The Fermi level EFfor electrons in the\nconduction band is calculated via the integral equation\nnD=Z1\nEGf(E)D(E)dE; (8)\nwhere f(E) is the Fermi-Dirac distribution function\nf(E)=1=\u0010\ne[E\u0000EF]=kBT+1\u0011\n: (9)\nThe stochastic spin polarization in this systems varies with the\nenergy position in the conduction band. According to the Pauli\nprinciple, the variance of the spin imbalance is determined by\nthe number of occupied and unoccupied electronic states:\n\u001b2\nmz(E)/f(E)\u00021\u0000f(E)\u0003: (10)\nFor a more rigorous calculation, the given absorption spectrum\n\u000b(E) has to be modeled by a sum of Drude-Lorentz oscillators\nwith di \u000berent resonance energies and an energy dependent spin\npolarization or even by a fully microscopic model. From a more\npractical viewpoint—due to the term f(E)\u00021\u0000f(E)\u0003—a spin\nnoise contribution is at low temperatures only expected from\nelectrons within a width of kBTaround the Fermi energy EF.\nAlso, the optical absorption sets in at energies around EF+EG\n(see Fig. 6), again with a width of kBT. Hence, the signal\n4CB\nVB+ 1/2 - 1/2Figure 6: Schematic of the fluctuating occupation numbers for delocalized spin-\nup and spin-down electronic states in bulk GaAs.\nstrength can be approximated at low temperatures by calculat-\ning the optical transitions by a single Drude-Lorentz oscilla-\ntor centered around EFand assuming noise contributions from\nelectrons with a carrier density reduced by a factor F[40, 45]:\nF\u0002nD=Z1\nEGf(E)\u00021\u0000f(E)\u0003D(E)dE: (11)\nThe temperature and probe light wavelength dependence are\nwell described for two- (see Fig. 12) [44] as well as for three-\ndimensional systems [40] by using this reduced carrier density\nin Eqs. (3) and (7) and modeling the optical transition as de-\nscribed above. Besides using the two-dimensional density of\nstates in Eqs. (8) and (11), also \u0018has to be adapted in order\nto describe a two-dimensional system, in which the optical se-\nlection rules for the three-dimensional case are modified since\nthe degeneracy of heavy and light hole is lifted due to the quan-\ntum confinement. Reference [58] gives values for \u0018for di \u000berent\nGaAs /AlGaAs based quantum well systems. Note that through-\nout this section the stochastic spin imbalance is assumed to be\nsmall so that the position of the absorption edge is independent\nof the spin polarization—unlike to magneto-optical measure-\nments in very high magnetic fields.\nAt this stage, it is important to note that the amount of ob-\nserved spin noise power gives information about the underly-\ning electron statistics, i.e., the integrated spin noise power of\nlocalized non-interacting electrons is temperature independent\nwhile the spin noise power of fully delocalized electrons van-\nishes at zero temperature due to Pauli spin blockade [48]. In\nother words, the amount of spin noise power extrapolated to\nzero temperature is a measure of the degree of electron localiza-\ntion which is especially interesting for comparative SNS studies\nin the vicinity of the metal-to-insulator transition (see Sec. 3.3\nand Refs. [45] and [48]).\n2.4. Spectral Shape of Spin Noise\nFigure 2 shows a typical spin noise spectrum, i.e., the fre-\nquency power spectrum of the spin fluctuations recorded in the\ntime domain. The Wiener-Chintchin theorem [59, 60] states\nthat this power spectrum corresponds to the Fourier transform\n0\nsignal in time domain signal in frequency domainauto-correlation function\n|FFT|²    convolution    \nFFTFigure 7: Visualization of the Wiener-Chintchin theorem for totally uncorre-\nlated noise, like laser shot noise, (left panels) and spin noise (right panels).\nof the auto-correlation function of the time signal. Sophisti-\ncated calculations of the spin noise spectrum can be found in\nRefs. [41] and [43]. Braun and K ¨onig give a fully quantum me-\nchanical density matrix formulation of spin noise and also con-\nsider the special case of an oscillating external magnetic field\n[41]; Kos et al. [43] explicitly take the orbital motion of the\nelectrons into account in their work and also consider the case\nof non-negligible transverse spin polarization due to high mag-\nnetic fields [43].2In this paragraph we pursue a more classical\napproach and consider a single spin precessing in a transverse\nmagnetic field; its auto-correlation function is given by [41, 43]\nhsz(0)sz(t)i/cos!Lte\u0000t=T2fort>0; (12)\nwhere!LandT2are the Larmor frequency and the spin de-\nphasing time, respectively. According to the Wiener-Chintchin\ntheorem, which is visualized in Fig. 7 for the case of completely\nuncorrelated, i.e., white noise (left panels) and spin noise (right\npanels), the Fourier transform of this expression directly yields\nthe spin noise spectrum. The Fourier transform of such an ex-\nponentially damped spin oscillation yields a Lorentzian shaped\nspectral spin noise power density\nS(f)=2P\n\u0019wFWHM\n4\u0010\nf\u0000!L\n2\u0019\u00112+w2\nFWHM: (13)\nHere, wFWHM =(\u0019T2)\u00001gives the full width at half maximum\n(FWHM) and Pis the integrated spin noise power as calcu-\nlated in Eq. (7). The experimental data in Fig. 2 fits well to\na Lorentzian and, hence, the Larmor frequency as well as the\nspin dephasing time can be extracted from the experimental\nspin noise spectrum.\nNevertheless, the single spin correlation function in Eq. (12)\nis only valid for localized spins. Since the probe volume is\nalways finite, also spatial correlations have to be considered.\nThe fact that time of flight broadening modifies the observed\nspin lifetime in SNS is a known fact from atom optics [61].\nM¨uller et al. were the first to experimentally demonstrate that\nthese transit e \u000bects also play an important role in semiconduc-\ntor SNS [44]. Time of flight broadening is included in the spin\n2It directly follows from the spin commutation relations and the Heisen-\nberg uncertainty principle that a transverse spin polarization increases the un-\ncertainty of the investigated z-component (see Sec. 2.6). On the other hand, a\nlongitudinal spin polarization reduces the noise power.\n5noise spectra by incorporating the spatial degrees of freedom\nin the spin correlation function hsz(t0;r0)sz(t;r)i. This prob-\nlem was tackled first in a straightforward approach in Ref. [44]\nand recently in a more rigorous treatment by Kos et al. [43].\nHowever, extracting the spin lifetime and transit time simulta-\nneously from the experimental spin noise spectra is still a very\ntedious task and more theoretical work is clearly needed here.\nThe essential point is that in many cases experimental access to\nintrinsic spin lifetimes is only granted by enlarging the probe\nlaser spot [44, 45, 48]. In other experimental techniques where\noriented spins are continuously optically injected, these e \u000bects\ngenerally play a less important role [45] (see Sec. 3.2). Never-\ntheless, these transit e \u000bects give the unique possibility to study\nspatial electron or spin dynamics at thermal equilibrium in the\nabsence of any gradients in electron or spin density [44]. It\nshould be noted that in principle also pure spatial spin dynam-\nics without accompanying charge dynamics can contribute to\nthis time of flight broadening. Such dynamics were studied by\nmeans of spin density gratings, however, necessarily in the pres-\nence of spin density gradients [62, 63].\n2.5. Spurious Noise Contributions\nThe balanced detection scheme in Fig. 1 is employed to ef-\nficiently reject classical noise of the laser system due to inten-\nsity fluctuations. In addition to this suppressible classical noise,\nquantum mechanical shot noise, which is a consequence of the\nphoton nature of light, contributes to the measured polarization\nnoise. In contrast to spin noise, photon shot noise is uncorre-\nlated and therefore results in white noise. Figure 7 illustrates\nhow shot noise becomes manifest in theory as a constant o \u000b-\nset in the noise spectrum. In practice, the white shot noise is\ndistorted due to the frequency-dependent sensitivity of the de-\ntection system (see Sec. 4.1 for resulting experimental impli-\ncations). The optical shot noise level is calculated by Poisson\nstatistics, i.e., the photon flux fluctuates with a standard devi-\nation ofpPlaser=~!laserresulting in a shot noise power density\nat the detector of 2 \u00172~!laserPlaserwhere\u0017[V=W] is the conver-\nsion gain of the detector and Plaserand!laserare the probe laser\npower and energy, respectively. Checking this linear relation-\nship between the background noise level and the laser power\nis a quick test that the experimental accuracy is shot noise lim-\nited, i.e, at the standard quantum limit. Furthermore, the elec-\ntronic components, like detector and the amplifier, introduce\nadditional electrical noise. However, in experiments at moder-\nately high laser powers, this noise contribution is usually sig-\nnificantly smaller than the optical shot noise level.\nThe peak spin noise power density at the detector is accord-\ning to Eq. (13) given by S(!L=2\u0019)\u0002(\u0017Plaser)2=2PT2\u0002(\u0017Plaser)2\nwhere the Faraday rotation is converted into units of the detec-\ntor output by multiplication with the detector gain \u0017and the\nprobe laser power Plaser. Note that the spin noise power is con-\ntrary to the shot noise power quadratic in laser power and a\nhigher laser power accordingly increases the signal strength \u0011\nwhich is quantified by the ratio between peak spin noise power\ndensity and shot noise power level:\n\u0011=PT2Plaser=~!laser: (14)Table 1 gives a survey on recent semiconductor SNS experi-\nments regarding the orders of magnitude of the integrated spin\nnoise power at the detector P, the spin dephasing time T2, the\npeak spin noise power density S(!L=2\u0019), the probe laser power\nPlaser, and the signal strength \u0011and also shows estimated val-\nues for prospective SNS measurements in a single quantum dot\nand in bulk GaAs at room temperature. The last column of\nTab. 1 reveals that e \u000ecient data averaging is crucial for SNS in\nsemiconductors, especially for systems with high spin dephas-\ning rates. Therefore, the data acquisition and the subsequent\nspectral analysis are further discussed in Sec. 4.2.\n2.6. Spin Noise in Atomic Gases: A Quantum Interface be-\ntween Light and Matter\nThis section gives a brief and not comprehensive survey on\nthe progress of SNS in atomic vapors of alkali metals that has\nbeen achieved since the seminal work of Aleksandrov and Za-\npasskii [31]. This discussion mainly aims at the aspect of the\nquantum non-demolition nature of this experimental method, in\nwhich sample excitations are clearly reduced by probing with\no\u000b-resonant light [33]. While Aleksandrov and Zapasskii ob-\nserved Raman like scattering events in their experiment [31, 66]\nindicating light-induced spin flips, several more recent experi-\nments [34, 67, 35] showed that energy absorption is negligible\nat su\u000ecient detuning so that these experiments can be viewed\nas quantum non-demolition measurements [36, 37, 38] of the z-\ncomponent of the atomic ensemble spin jzwith the z-axis being\nthe axis of light propagation. Accordingly, the Hamiltonian of\nthe interaction between light and matter can be written as\nH/jz\u0001\u001bz; (15)\nwhere\u001bzis the component of the quantum Stokes operator de-\nscribing the circular light polarization (see, e.g., Ref. [34, 68,\n69]). The important feature is that jzdoes commute with the\ninteraction Hamiltonian. For the actual measurement, a meter\nvariable is needed that does not commute with H. Obviously,\nin SNS the meter variable is the linear light polarization \u001by.\nSolving the Heisenberg equation of motion for the given inter-\naction by taking advantage of the commutation relation for the\ncomponents of jand\u001bdelivers (see, e.g, Ref. [35])\n\u001bout\nz=\u001bin\nz; (16)\n\u001bout\ny=\u001bin\ny+\f\u001bin\nxjin\nz; (17)\njout\nz=jin\nz; (18)\njout\ny=jin\ny+\f0jin\nx\u001bin\nz: (19)\nHere, Eq. (17) describes the Faraday rotation of the linearly\npolarized probe light, the quantum non-demolition nature mani-\nfests in (16) and (18), and the measurement induced back-action\non the transverse spin component is given by Eq. (19). This\nback-action on the y-component of spin system becomes max-\nimal if the x-component of the spin system is fully polarized,\ni.e., if, according to the Heisenberg uncertainty principle, the\nincommensurability of jyandjzbecomes maximal. Even in the\nabsence of spin fluctuations due to the very long spin lifetimes\n6Table 1: (a) Magnitude of spin noise power P, spin dephasing time T2, peak spin noise power density S(!L=2\u0019), probe laser power Plaser, and ratio of S(!L=2\u0019)to\nthe background shot noise level, i.e., the signal strength \u0011, for di \u000berent semiconductor systems at cryogenic temperatures. All quantities are detector independent\nand mostly apply to the weakly perturbing detection regime, i.e., strong detuning and large laser spots (contrary, e.g., to the spectrum in Fig. 2). Electrical noise\nis neglected for calculating \u0011. (b) Estimated values for prospective measurements. In the case of the single spin, additional electrical noise equivalent to 0.04 mW\nprobe laser power is assumed.\nInvestigated system Ref. Ph\n\u0016rad2i\nT2[ns]S(!L=2\u0019)hnrad2\nHzi\nPlaser[mW]\u0011\n(a)\nn-doped Bulk GaAs (1 :8\u00021016cm\u00003,l\u0019340\u0016m) [48] 10 100 1 1 10\u00002\nn-doped Bulk GaAs (2 :7\u00021015cm\u00003,l\u0019500\u0016m) [48] 100 10 1 1 10\u00002\nInGaAs quantum dots [47] 1000 1 1 1 10\u00002\nn-doped Bulk GaAs (8 :8\u00021016cm\u00003,l\u0019370\u0016m) [48] 10 10 0.1 1 10\u00003\nModulation n-doped multiple quantum well [44] 10 10 0.1 1 10\u00003\nn-doped Bulk GaAs (1014cm\u00003,l\u00192\u0016m) [48] 10 1 0 :01 1 10\u00004\n(b)\nRoom temperature SNS in bulk GaAs [64] 10 0.1 0 :001 1 10\u00005\nSingle electron spin in optical cavity [65] 100 10 1 0.01 10\u00005\ncompared to the measurement time, the measured Faraday ro-\ntation is still subject to noise which results from the projective\nmeasurement and is known as projection noise [70]. A continu-\nous measurement of jzresults in an increase of the uncertainty\nofjy:\u0010\n\u000ejout\ny\u00112=\u0010\n\u000ejin\ny\u00112+\u0010\u0010\n\u000e\u001bin\nz\u00112and, subsequently, due to\nthe Heisenberg principle, in a decrease of the uncertainty of\njz. Thus, SNS allows measurement of a spin component with\naccuracy beyond the standard quantum limit as shown theoreti-\ncally and experimentally by Kuzmich and co-workers [67, 71].\nIn other words, by undergoing a SNS measurement, the co-\nherent spin state evolves into a squeezed spin state [72, 73]\nin analogy to the notion of squeezed light [74]. This concept\nof spin squeezing was also applied to the atom clock levels of\na mesoscopic ensemble of cold caesium atoms resulting in a\nmetrologically relevant noise reduction of 3.4 dB beyond the\nstandard quantum limit [75]. Julsgaard et al. demonstrated\nthat this quantum non-demolition measurement, consecutively\ncarried out on two di \u000berent alkali vapor samples, yields entan-\nglement of these macroscopic objects [35]. Comprehensively,\nthe interaction Hamiltonian in Eq. (15) represents a so-called\nquantum interface between matter and light [76, 77] which, of\ncourse can transfer information in both ways. Via this inter-\nface, non-classical light states can be used for spin squeezing\n[78, 79, 80], the atomic spin ensemble can also be utilized as a\nquantum memory of light [81], and quantum teleportation be-\ntween matter and light becomes feasible [82].\nA transfer of these exciting experiments to semiconductor\nspin physics is clearly desirable. Besides possessing very sharp\nresonances, easy tunable optical density, and a longer spin life-\ntime, metal atoms in a vapor gas cell also carry the advantage of\nan easy optical access from all three spatial directions. Anyhow,\nin consideration of the experimental results that are reviewed in\nthis section, it is surprising that in theoretical proposals of light\nand semiconductor spin entanglement [83, 84, 85, 86, 87, 88],\nmostly single spins and not ensemble spins are considered. Ad-\nditionally, isotropic exchange interaction in an electronic spin\nensemble in a semiconductor [89] and the resulting correlated\nspin relaxation may allow for measurement precision beyond\nthe standard quantum limit for timescales longer than the spinrelaxation time [90]. Application of squeezed light for SNS in\nsemiconductors is discussed in Ref. [91].\n3. Spin Dynamics in Semiconductor Structures\nThe main purpose of this section is to highlight the insights\nSNS has already given to the understanding of electron spin dy-\nnamics in semiconductors as well as to discuss the potential of\nSNS as an ultrasensitive probe. To this end, we first discuss the\nmost important mechanisms contributing to the spin decay in\nsemiconductors in Sec. 3.1; this discussion illustrates that spin\norientation as well as electron heating due to above bandgap\nlight absorption can significantly alter the observed spin dy-\nnamics. In Sec. 3.2, a survey of the conventional experimen-\ntal probes for semiconductor spin dynamics follows and reveals\nin which cases only SNS grants experimental access to intrin-\nsic spin dynamics. A detailed overview of SNS experiments in\nsemiconductors closes this section.\n3.1. Spin Dephasing in Semiconductors\nThere are several publications surveying the physical mech-\nanisms of spin decay in semiconductors in great detail [5, 11,\n89, 92, 93]. On that account, we only name the key facts of\nspin dephasing in semiconductors which play a role for the un-\nderstanding of the SNS experiments.\nThe Heisenberg picture and the spin commutation relationh\nsx;syi\n=iszdirectly disclose the precessional motion of the\nspin observable sin a magnetic field Bwhere the Hamilton op-\nerator is given by H/B\u0001s. Extending this treatment to the po-\nlarization of an ensemble of spins m, the equations of motion—\nknown as Bloch equations [19]—become for B=B\u0001ex\n@mx\n@t=\u0016Bg\u0003\n~(m\u0002B)x\u0000mx\u0000m0\nx\nT1;\n@my\n@t=\u0016Bg\u0003\n~(m\u0002B)y\u0000my\nT2;\n@mz\n@t=\u0016Bg\u0003\n~(m\u0002B)z\u0000mz\nT2; (20)\n7where g\u0003is the e \u000bective electron Land ´eg-factor which can in\nconsequence of spin-orbit coupling significantly vary from the\nfree electron g-factor. The quantity m0\nxdescribes the equilib-\nrium spin polarization along the external field. Relaxation to\nequilibrium m0\nxis accompanied by energy dissipation while the\nspin polarization transverse to the magnetic field usually de-\ncays with the energy of the spin system being conserved. The\nspin dephasing time T2as well as the spin relaxation time T1\nare introduced phenomenologically in Eq. (20).3SNS is sen-\nsitive to T2if no magnetic field is applied along the direction\nof light propagation. This section mainly focuses on the dis-\ncussion of spin dephasing as there has not been an investigation\nof the longitudinal spin relaxation time via semiconductor SNS\nyet. However, the physical di \u000berence between spin dephasing\nand relaxation blurs at the relatively low magnetic fields used\nin most SNS experiments, resulting in T1'T2. Spin dephas-\ning can either be of homogeneous or of inhomogeneous nature.\nInhomogeneous spin dephasing is for instance observed when\nan electronic ensemble is probed in which all electrons expe-\nrience di \u000berent magnetic fields or have di \u000berent e \u000bective g-\nfactors. Inhomogeneous—contrary to homogeneous—spin de-\nphasing is reversible which means that it could be eliminated in\nspin echo experiments; the corresponding inhomogeneous spin\ndephasing time is denoted by T\u0003\n2.\nIn order to discuss the di \u000berent mechanisms of spin dephas-\ning, we adopt the random walk formalism of Pines and Slichter\n[94], which lucidly displays the main features of the particu-\nlar mechanisms. Pines and Slichter consider a spin in interac-\ntion with its environment. This interaction results in an average\nchange of the spin direction by an angle of \u000e\u001ein the time span\n\u001ccwhere\u001ccis the correlation time of the given interaction. The\nchange of the angle varies its sign with this time constant due\nto scattering events. The mean square of the rotational phase\nchange of the spin after time tis given by\nD\n\u0001\u001e2E\n\u0018(\u000e\u001e)2t=\u001cc: (21)\nPines and Slichter define T2to be the time after whichD\n\u0001\u001e2E\nreaches unity, a definition that is closely related to the one in\nEq. (20). In the following, three cases have to be considered:\nIn the first case (i), the change of the rotational frequency oc-\ncurs during the scattering event itself. The spin dephasing time\nbecomes [95]\n1=T2\u0018(\u000e\u001e)2=\u001cc: (22)\nIn the second case (ii), the interaction occurs during the whole\ntime span of \u001ccresulting in a change of the precession frequency\n!by\u000e!\u0018\u000e\u001e=\u001c c. The consequent spin dephasing time reads\n1=T2\u0018(\u000e!)2\u001cc: (23)\nThereby, the spin dephasing becomes less e \u000ecient at shorter\ncorrelation times \u001cc. This concept of motional narrowing was\nfirst put forward by Bloembergen et al. to account for the nar-\nrow linewidths found in the nuclear magnetic resonance spec-\ntra of liquids [96]. In the third case (iii), the correlation time is\n3In this review, the definitions of T1,T2, and T\u0003\n2according to Ref. [5] are\nused.large compared to 1 =\u000e!, i.e., the spin polarization has decayed\nbefore the first scattering event occurs and \u001cchas to be replaced\nbyT2in Eq. (23), resulting in\n1=T2\u0018\u000e!: (24)\nIn general, all mechanisms of homogeneous spin dephasing can\nbe assigned to one of the three above cases. In the following,\nwe discuss the most relevant processes.\nElliott-Yafet mechanism. The Elliott-Yafet (EY) mechanism [97,\n98] is based on the fact that electronic Bloch states are because\nof spin-orbit coupling not pure spin-up or spin-down states but\nsuperpositions of both, e.g., \tkn\"=[akn(r)j\"i+bkn(r)j#i]eik\u0001r.\nThis admixture of the other spin species is small ( jbj\u001c1). Nev-\nertheless, scattering into another k-state comes along with a fi-\nnite possibility of a spin-flip. Correspondingly, the EY mecha-\nnism is of the form given by Eq. (22) [5]:\n1=TEY\n2\u0018hb2i=\u001cp; (25)\nwhere\u001cpis the momentum scattering time. Qualitatively, it\ndoes not matter which process gives the main channel for mo-\nmentum relaxtion. Either scattering due to impurity atoms [97],\nphonons [98], or electron-electron interaction [99] lead to spin\nrelaxtion via the EY mechanism. Obviously, all of these un-\nderlying scattering mechanisms obey a strong energy or tem-\nperature dependence and, consequently, optical excitation will\nalter the e \u000eciency of spin dephasing. Nevertheless, not only\nthe correlation time is energy dependent, but also the size of the\nspin-down admixture bvaries with the electronic energy. For\nIII-V semiconductors, Eq. (25) becomes [100, 95]\n1=TEY\n2(Ek)/E2\nk=\u001cp(Ek): (26)\nRecently, Jiang and Wu theoretically studied the relative strength\nof the EY mechanism to other mechanisms concluding that the\nEY mechanism is unimportant in most III-V semiconductors at\nzero magnetic field [101].\nDyakonov-Perel mechanism. In non-centrosymmetric semicon-\nductor structures, spin-orbit coupling becomes also manifest in\nspin-split energy bands, i.e., Ek\"=E\u0000k#,Ek#. The lack of\ninversion symmetry can either result from bulk inversion asym-\nmetry as in III-V semiconductors (Dresselhaus spin-splitting)\n[102, 103], from structure inversion asymmetry as in asymmet-\nrically doped quantum wells (Rashba spin-splitting) [104, 105],\nor from interface inversion asymmetry (see, e.g., Ref. [106]).\nThis spin splitting is described by an e \u000bective, wave vector\ndependent magnetic field \n(k)=g\u0016B(H=~s\u0001\n(k)). Hence,\nspins of electrons in di \u000berent k-states precess around di \u000berent\ne\u000bective magnetic field vectors and, subsequently, a spin po-\nlarization dephases due to the so-called Dyakonov-Perel (DP)\nmechanism [107]. The correlation time of this interaction is\nagain given by the momentum scattering rate including scatter-\ning due to impurities, phonons, and electron-electron interac-\ntion. The relevance of electron-electron scattering to the DP\nmechanism was pointed out by Wu and Ning [108, 109] as well\nas by Glazov and Ivchenko [110, 111].\n8For\u001cp\u001c1=!, which is usually the case, the DP mechanism\nis of the type given by Eq. (23):\n1=TDP I\n2(Ek)\u0018h\n2i\u001cp(Ek): (27)\nThe Dresselhaus spin-splitting for bulk semiconductors is cubic\nink[102]. Thus, as for the EY mechanism, not only the mo-\nmentum relaxation time but also the strength of the spin-orbit\ncoupling becomes energy dependent:\n1=TDP I\n2(Ek)/E3\nk\u001cp(Ek): (28)\nThe Dresselhaus spin splitting is modified in quantum wells\nwhere kin the growth direction is given by momentum quan-\ntization [112]. A special case results for (110) grown quantum\nwells where the Dresselhaus field has no in-plane component\nand, hence, spins aligned along the growth direction do not de-\nphase due to bulk inversion asymmetry [112, 113, 114]. In sys-\ntems with very low momentum scattering rates as in high mo-\nbility quantum wells at ultralow temperatures [115, 116], the\nDP process is described by Eq. (24):\n1=TDP II\n2\u0018h\ni: (29)\nBir-Aronov-Pikus process. The interaction between electrons\nand holes leads to electron spin dephasing via momentum scat-\ntering and the resulting EY mechanism. However, Bir et al.\nshowed that in presence of holes electron spin dephasing due to\nexchange interaction between electron and holes is much more\ne\u000ecient [117]. The strength of this Bir-Aronov-Pikus (BAP)\nmechanism depends on the hole density, the electron-hole over-\nlap, and the fact whether holes are bound or delocalized. The\nBAP process shows distinct regimes with di \u000berent dependen-\ncies on the hole density [11, Ch. 3]. Qualitatively, in all of\nthese regimes, the e \u000eciency of electron spin dephasing is in-\ncreasing with hole density. Also, the temperature dependence\ndoes not follow a simple expression and varies for the di \u000ber-\nent regimes: Nevertheless, for fixed hole density, the electron-\nhole overlap increases with decreasing temperatures and, ac-\ncordingly, the BAP mechanism gets more e \u000ecient. While it\nis clear experimental evidence that the BAP mechanism signif-\nicantly contributes to spin dephasing in the absence of other\nspin dephasing processes [114], its relative strength compared\nto other mechanisms has become subject of scientific discus-\nsion (see Refs. [118] and [119]). The Pauli blockade strongly\nsuppresses the BAP spin flip mechanism at low temperatures\naccording to Zhou and Wu [119].\nSpin dephasing by hyperfine coupling. While in natural sili-\ncon only roughly 5% of the silicon nuclei carry a spin angular\nmomentum, in GaAs all lattice nuclei have a finite spin. In\nany case, the electronic spin interacts with the spins of the lat-\ntice nuclei due to the Fermi contact interaction. This hyperfine\ncoupling represents an interface between electronic and nuclear\nspins, as proposed by Overhauser in 1953 [120] for the case\nof metals. In a semiconductor, an electronic spin polarization\ncreates a nuclear spin polarization on the laboratory timescale\nwhich in turn strongly influences the electronic spin dynam-\nics [1, 121, 122]. Spin dephasing due to hyperfine interac-\ntion in semiconductors was first theoretically investigated byDyakonov and Perel [121] and later extensively discussed by\nMerkulov et al. [123]. Depending on the number of magnetic\nlattice nuclei and the extension of the donor wavefunction, a\nlocalized electronic spin interacts with a certain number of nu-\nclear spins NL. Recalling the prediction of nuclear spin noise\nby Bloch [19], an average stochastic polarization ofpNLnu-\nclear spins is present at thermal equilibrium. This hyperfine\ninteraction leads to an electronic spin precession with an aver-\nage frequencyh\nHFiin the nuclear magnetic field, the so called\nOverhauser field. An expression to calculate this field is given\nin Ref. [123] (see also Refs. [48, 124]). The nuclear spins\nthemselves precess in the magnetic field of the electron, the so-\ncalled Knight field, which is a factor ofpNLsmaller than the\nOverhauser field. Hence, in the first step, the nuclear spin polar-\nization can be viewed as frozen. The correlation time \u001ccof the\nhyperfine interaction is determined by the strength of electronic\nlocalization, i.e, by the time an electronic spin resides at a cer-\ntain donor site. Spin di \u000busion via exchange interaction occurs\norders of magnitude faster than electronic hopping in the low\ndoping regime [89] so that \u001cc\u0019~=J[125, 89] is given by the\nexchange integral between remote donor states J. In the inter-\nmediate doping regime below the metal-to-insulator transition,\nwhereh\nHFi\u001cc\u001c1, a spin polarization dephases according to\nEq. (23) with a rate of [89]\n1=THF I\n2\u0018D\n\n2\nHFE\n\u001cc: (30)\nTherefore, spin dephasing based upon hyperfine interaction be-\ncomes less e \u000ecient with increasing doping concentrations and\nis completely negligible in the metallic state. In the regime of\nvery low doping and low temperatures, where electrons are con-\nsidered as non-interacting and strongly localized, no motional\nnarrowing occurs, i.e., h\nHFi\u001cc\u001d1 and, due to the stochastic\nnuclear spin polarization, a spin ensemble is subject to inhomo-\ngeneous spin dephasing according to Eq. (24):\n1=THF II\n2\u0003\u0018h\nHFi: (31)\nHowever, Eqs. (30) and (31) only describe the decay of the\nspin components perpendicular to the Overhauser field at the\nparticular donor sites. In the absence of an external magnetic\nfield, the angle between electronic and nuclear magnetic field is\nconserved during the electronic spin precession period. Hence,\none third of the spin polarization of a spin ensemble does not\ndephase on the timescale of the electronic, but of the nuclear\nprecession period as theoretically proposed by Merkulov et al.\n[123] and experimentally demonstrated by Braun et al. [124].\nDue to the spatial variation of the electronic wavefunction, the\nKnight field is spatially inhomogeneous and di \u000berent nuclei\nat a given donor site have di \u000berent precessional frequencies.\nSubsequently, the angle between electronic and nuclear spin is\nnot conserved on the timescale of the nuclear spin precession.\nThus, the spin component randomly aligned with the nuclear\nfield undergoes spin dephasing with a roughly estimated rate of\n1=THF III\n2\u0018h\nHFi=p\nNL: (32)\nSpin dephasing by anisotropic exchange interaction. The ex-\nchange interaction is mentioned as an origin of motional nar-\n9rowing of the hyperfine induced spin dephasing in the last para-\ngraph. However, in semiconductors without spatial inversion\nsymmetry, the exchange interaction itself is in connection with\nspin-orbit coupling a source of spin dephasing for localized\nelectronic spins. Due to spin-orbit coupling and a crystalline\nstructure lacking spatial inversion, the exchange interaction be-\ntween two spins is not described by a Hamiltonian of the form\ns1\u0001s2, but by means of a second rank tensor. The antisym-\nmetric part of this tensor gives rise to an anisotropic exchange\ninteraction or the so called Dzyaloshinskii-Moriya (DM) inter-\naction [126, 127]. Kavokin was the first to suggest in 2001 that\nspin tunneling from one donor site to another will in average\nencounter a finite rotation of \rdue to this anisotropic exchange\ninteraction [128]. Hence, the DM interaction gives rise to spin\ndephasing of the type of Eq. (22):\n1=TDM\n2\u0018\r2=\u001cc; (33)\nwhere the time between two spin di \u000busion events \u001cc\u0019~=J\n[125, 89] is, as in the previous paragraph, given by the isotropic\npart of the exchange interaction J. Electron hopping contributes\nto spin dephasing analogously [89, 129].\n3.2. Conventional Experimental Probes\nSNS was for the first time applied to investigate spin dy-\nnamics in semiconductors in the year 2005 [39]. Decades be-\nfore, quite a consistent picture on spin dynamics in semiconduc-\ntors already existed. Besides on exhaustive theoretical work,\nthis picture had been mainly based on optical experiments of the\nsteady state depolarization carried out in the 1960s and 1970s\n(see Ref. [11]) long before time-resolved measurement tech-\nniques relying on (sub) ps laser light pulses were introduced.\nInvestigation of semiconductor spin dynamics in the time do-\nmain became feasible with the increasing usage of these new\nlaser light sources in the early 1990s (see Ref. [130]). Never-\ntheless, up to the year 2005, all optical techniques for investigat-\ning spin dynamics in semiconductors were based on optical ori-\nentation of the electron spins and, hence, move the sample sys-\ntem away from thermal equilibrium. Besides these optical tech-\nniques, also, electron spin resonance has evolved into a valu-\nable tool to study electron spin dynamics in semiconductors.\nThe first semiconductor system to be studied was n-type silicon\nin the 1950s [131, 132, 133]. Later, electron spin resonance\nwas transferred to other semiconductos like InSb [134, 135],\nGaAs [136, 137], and InAs [138]. However, due to dynamic\nnuclear e \u000bects and low signal strength extracting spin relax-\nation times from spin resonance measurements is often a very\ndi\u000ecult task [135, 137]. Thus, resonance is often detected by\nmeasuring the degree of depolarization via photoluminescence\n[139], electrical transport [140], or below band gap Kerr ro-\ntation [141]. Especially, electrically detected spin resonance\nis highly sensitive and promises even quantum non-demolition\nmeasurement of a single spin [142, 143]. In general, however,\nelectron spin resonance is a depolarization measurement and,\naccordingly, the sample is not at thermal equilibrium during the\nexperiment [133]. Other experimental probes for spindynam-\nics in semiconductors are also based on transport and are leftout in this section as they—contrary to the optical techniques—\nrequire device fabrication. A survey of these electrical tech-\nniques can be found in Ref. [5]. In the following, we discuss the\noptical techniques in view of the di \u000berent spin dephasing mech-\nanisms (Sec. 3.1) and show that SNS is the experimental probe\nof choice for certain sample systems, like, e.g., bulk semicon-\nductors with a doping density below the metal-to-insulator tran-\nsition.\nOptical Measurements of the steady state depolarization. As\ndiscussed in Sec. 2.1, irradiation of an intrinsic semiconductor\nwith circularly polarized above band gap light leads to a spin\npolarization in the conduction band along the z-axis, i.e, the\ndirection of light propagation. The maximum degree of polar-\nization mmax\nz\u0011\u0018is determined by the dipole selection rules\n(see Fig. 3). A closer look at the corresponding rate equations\nreveals that the actual degree of spin polarization in undoped\nsemiconductors reads (see Refs. [11, Ch. 2] and [5])\nmz=\u0018\n1+\u001c=T2; (34)\nwhere\u001cis the electron-hole recombination time. Hence, the\nsteady state electron spin polarization is an indirect measure\nof the free electron spin dephasing time [144, 145]. Since the\ndipole selection rules are not only relevant for light absorption\nbut also for light emission, the steady state electron spin po-\nlarization is experimentally accessed by the degree of circu-\nlar light polarization of the photoluminescence under the as-\nsumption that the hole spin is unpolarized due to very e \u000bective\nhole spin dephasing [146]. The precessional motion of an elec-\ntron spin polarization in an external transverse magnetic field\n[see Eq. (20)] yields further depolarization of the optically in-\njected spins along the z-axis and the value of spin polarization\nin Eq. (34) becomes [11, Ch. 2]\nmz(B)=mz(0)\n1+\u0010\u0016Bg\u0003\n~BT\u00112; (35)\nwhere the measured spin lifetime Tis composed of the actual\nspin dephasing time and the electron recombination time:\n1\nT=1\nT2+1\n\u001c: (36)\nThis impact of a magnetic field on the polarization state of lu-\nminescent light in mercury vapor was discovered by Wood and\nEllett [147] and explained by Hanle in 1924 [148]. In the year\n1969, Parsons was the first to measure the quantity Tfor free\nelectrons in GaSb by means of this Hanle e \u000bect [2].\nHanle type measurements can also be carried out to mea-\nsure the spin dynamics of donor electrons in weakly n-doped\nsemiconductors. Here, the recombination of bound electrons is\nusually more e \u000ecient than free electron recombination and the\nspin lifetime is usually longer than the carrier lifetime so that\nthe spin polarization of the optically generated free electrons\nyields a spin polarization of the donor electrons [149, 150]. In\n10this case, the electron recombination rate 1 =\u001cin Eq. (36) is sub-\nstituted by the rate of replacement of donor electrons by opti-\ncally generated spin polarized electrons:\n1\nT=1\nT2+G\nnD; (37)\nwhere Gis the excitation rate of free carriers (see Refs. [151],\n[125] and [11, Ch. 2]). Correspondingly, the measured quantity\nTbecomes strongly dependent on the power of the light excita-\ntion. According to the above reasoning, 1 =T2can be extracted\nfrom the linear extrapolation of the power dependence of 1 =T\nto vanishing excitation. Nevertheless, this evaluation method\nrequires that T2is independent of the excitation power—an as-\nsumption that is not generally valid since carrier injection alters\nthe spin dynamics. Especially, at low temperatures, where polar\noptical phonons cannot be activated for carrier momentum re-\nlaxation (see, e.g., Refs. [152, 153]), and at low doping concen-\ntrations in the non-degenerate regime, in which—according to\nequipartition theorem—the electronic energy scales linear with\ntemperature, the change of the electronic temperature by op-\ntical excitation may have a drastic influence on the observed\nspin lifetime. Additionally, due to the continuous electron-hole\npair generation, the e \u000eciency of the BAP process, the mech-\nanisms based on the DM interaction as well as the hyperfine\nspin dephasing in the motional narrowing regime is altered be-\ncause of the presence of free electrons and holes. While the\nBAP spin dephasing is enhanced because of the increased hole\ndensity, the e \u000eciency of the hyperfine and the DM mechanism\nis reduced due to averaging of the Overhauser field and the\nanisotropic exchange interaction, respectively. This averaging\nover several donor atoms occurs via exchange interaction medi-\nated spin di \u000busion. Exploiting this e \u000bect by flooding the semi-\nconductor with free electrons, Dzhioev et al. demonstrated that\nthe hyperfine interaction induced spin dephasing can basically\nbe switched o \u000b[151, 154]. Additionally, Paget showed in 1981\nthat the presence of optically created electrons also significantly\nalters the observed spin dynamics via exchange averaging be-\ntween the localized and free electronic states [155]. Again, the\noverall influence of free electrons is largest for low-doped sam-\nples at low temperatures. In this carrier regime, short spin de-\nphasing times due to hyperfine interaction may also require ex-\ncitation densities that are comparable to the equilibrium carrier\ndensity to achieve a su \u000eciently high spin polarization. At these\nintensities, also spin di \u000busion may modify the depolarization\ncurves and yield a Hanle width that is—contrary to Eq. (37)—\nquadratic in G[156]. To sum up, because of the temperature\nand free carrier density dependence of the various spin dephas-\ning mechanisms and the non-equilibrium spin polarization, the\nlinear extrapolation of Eq. (37) to zero excitation density may\nin many cases not be justified and the equilibrium spin lifetime\nis not accessible in Hanle-type experiments.\nAdditionally, Hanle-type measurements in contrast to SNS\ndeliver no independent information about the e \u000bective g-factor\nand the time constant T[see Eq. (35)], i.e, one of these two\nquantities has to be known to determine the other. In III-V\nbulk semiconductors, the e \u000bective electron Land ´e factor g\u0003is\nknown to exhibit an energy dependence (see Ref. [157] andreferences therein) and, hence, a doping level dependence as\nwell as a temperature dependence (undoped GaAs: Refs. [158,\n159, 160, 161, 162], n-type GaAs: Ref. [49]). Therefore, pre-\ncision measurements of the spin lifetime via the Hanle e \u000bect\nrequire knowledge about the g-factor which has to be gath-\nered by another experiment. However, if the recombination\nand spin dephasing rates are known or negligible, Hanle mea-\nsurements allow to determine the e \u000bective electron Land ´eg-\nfactor as demonstrated by Snelling et al. with the first study\nofg\u0003in GaAs /AlxGa1\u0000xAs quantum wells in dependence of\nthe quantum well thickness [163] (see Ref. [164] for a more\nrecent study). Furthermore, while Hanle type measurements\nin dependence on a longitudinal magnetic field (see Refs. [11,\nCh. 3] and [125]) yield important insight regarding the corre-\nlation time \u001cc(see Sec. 3.1) of the underlying spin dephasing\nmechanism, these depolarization experiments do not allow to\nstudy the influence of a transverse magnetic field on the spin\ndynamics.\nThe degree of polarization of the luminescent light has not\nnecessarily to be examined in Hanle-type measurements. For\ninstance, the depolarization of the optically created spin ori-\nentation can also be measured by means of below band gap\nFaraday rotation of an additional probe beam as carried out by\nCrooker et al. to directly compare spin dephasing times mea-\nsured by SNS and Hanle measurements [45]. However, the dif-\nferent influence of spin and electron di \u000busion in both experi-\nments (see Sec. 2.4) makes the experimental data hard to com-\npare.\nTime-resolved optical measurements. The spin dephasing time\ncan be accessed more directly in a time and polarization-re-\nsolved measurement of photoluminescence by means of a pulsed\ncircularly polarized pumping laser and a streak camera system\nas first carried out in 1994 [12]. This technique has several ad-\nvantages over continuous-wave Hanle type measurements since\nthe e\u000bective electron g-factor, the recombination rate, and the\nspin dephasing rate are independently of each other extracted\nfrom the experiment. Furthermore, measurements in zero mag-\nnetic field as well as studies of the transverse magnetic field\ndependence are feasible. Time-resolved Kerr [165, 14] and\nFaraday rotation [13, 166] techniques, both also introduced in\n1994, sample the birefringence, which results from the initial\nspin orientation by the pump pulse, via the polarization rota-\ntion of a time-delayed, transmitted (Faraday) or reflected (Kerr)\nprobe pulse [167]. While photoluminescence directly reveals\nthe energy position of the carriers whose spin dynamics are\nprobed, time-resolved Kerr and Faraday rotation data usually\nneeds more interpretation [92, Chap. 2]. In general, electron re-\nlaxation dynamics are accessible in these experiments by means\nof the transient change of the reflectivity and the transmission,\nrespectively. It was shown in several publications that interpre-\ntation of the time-resolved Kerr rotation data is in many cases\nnot possible without also studying the dynamics of electron re-\nlaxation [168, 169].\nAgain, as carrier relaxation proceeds usually faster than re-\ncombination, these three methods work for doped as well as for\nundoped semiconductor systems. The initial optical spin orien-\n11tation comes along with all the disadvantages that are discussed\nin the previous paragraph for Hanle-type measurements. In\ntime-resolved photoluminescence experiments, free holes and\nelectrons are—like in Hanle measurements—present in the sam-\nple during the whole data acquisition time resulting in spin de-\nphasing due to the BAP process [114]. Recently, Krauß et al.\ndemonstrated the strong influence of various electronic scat-\ntering and screening mechanisms that determine the observed\nspin dynamics in time-resolved experiments at elevated exci-\ntation conditions [170]. In doped samples with spin lifetimes\nmuch longer than the recombination time, time-resolved Kerr\nand Faraday rotation techniques could in principle allow mea-\nsurements of the spin dynamics after the electronic system has\nequilibrated, as it is realized, e.g., in Refs. [15, 171, 172]. In\nthese experiments, however, the repetition rate of the laser sys-\ntem significantly exceeds the spin dephasing rate so that spin\ndephasing times are extracted via resonant spin amplification,\nan extension of the time-resolved measurement principle intro-\nduced by Kikkawa and Awschalom in 1998 [15], in which a\ntransverse magnetic field is swept while the time delay between\npump and probe pulse is kept constant. During some fraction\nof the data acquisition time, free carriers are still present due to\noptical pumping and modify the observed dynamics such that\nthe optical excitation has to be included for explaining the ex-\nperimental outcome [173]. Also, the rapid optical excitation\nleads to carrier heating, as in the case of Hanle measurements.\nEspecially, a high fluence of the pump pulse can lead to gener-\nation of a large number of non-equilibrium longitudinal optical\nphonons which further hinders carrier cooling [174]. Therefore,\ndue to ine \u000bective cooling of the electron system by phonons\nat low temperatures [152, 153], ultralow electron temperatures\nare generally not accessible in time-resolved measurements (see\nRef. [162] where the temperature of the measured e \u000bective g-\nfactor levels o \u000bat low temperatures indicating insu \u000ecient cool-\ning power of the electron system). Possible pitfalls of these\nexperiments are further listed in Ref. [89]. The initial optical\norientation can also modify the observed spin dynamics due to\nenthralling e \u000bects resulting from the Hartree-Fock contribution\nto the electron-electron interaction [116].\n3.3. Investigations by SNS\nSince 2005 SNS has been used to study spin dynamics in\nbulk semiconductors [39, 40, 45, 48, 49] as well as in two [44]\nand zero dimensional [47] semiconductor systems. SNS does\nnot rely on artificial spin orientation, which can—as discussed\nin previous section—conceal the equilibrium spin dynamics. At\npresent, all publications on SNS in semiconductors are focused\non III-V-based samples, which can be viewed as quintessential\nsystems for spintronic research. SNS should, however, be appli-\ncable to a large group of di \u000berent semiconductor systems, direct\nas well as indirect semiconductors, provided that the probed\ntransition obeys appropriate selection rules (see Sec. 2.1). In\nthe following paragraphs, we give a survey on the existing in-\nvestigations via SNS on n-type bulk GaAs, modulation-doped\n(110) GaAs /AlGaAs quantum wells and unintentionally p-doped\nself assembled (In,Ga)As /GaAs quantum dots.\nFigure 8: Low temperature spin dephasing times as a function of doping den-\nsity. Measurements by SNS are indicated by asterisks. All other data is acquired\nby means of resonant spin amplification and time-resolved Faraday rotation\n[175], Hanle measurements [125, 176], optically detected electron spin reso-\nnance [177], and time-resolved photoluminescence [178, 179]; the data point\nat 1014cm\u00003of Ref. [125] is measured in a 0 :1\u0016m thick bu \u000ber layer of a\nGaAs /AlGaAs stack.\nBulk GaAs. SNS was utilized to study the temperature [40, 45,\n48] and transverse magnetic field [49] dependence of electron\nspin dephasing in n-type bulk GaAs in di \u000berent doping regimes.\nAlso, spatially-resolved measurements of spin dynamics that\nare feasible via SNS for all three spatial dimensions (see Sec.\n5.2) delivered a better understanding of inhomogeneous spin\ndephasing mechanisms for doping regimes close to the metal-\nto-insulator transition [49].\nFigure 8 summarizes measured low temperature spin de-\nphasing times as a function of the dopant density, following\nFig. 3 of Ref. [125]. The plotted data is acquired via SNS and\nvarious other experimental techniques—like Hanle type mea-\nsurements, time resolved experiments, resonant spin amplifica-\ntion, and optically detected electron spin resonance (see Sec. 3.2).\nThe spin dephasing in bulk GaAs can be divided into three\nregimes: (i)At very low densities of nD.1014cm\u00003, all donor\nelectrons can be viewed as non-interacting and the ensemble\nspin dephasing time is determined by the inhomogeneous dis-\n12tribution of nuclear fields [see Eq. (31)]. SNS has delivered\nan inhomogeneous spin lifetime of T\u0003\n2=2:8(7) ns [48] which\nis very close to the theoretical value of THF II\n2\u0003=3:6 ns [123].\nThe presence of free carriers in other experimental techniques\nleads to averaging of the nuclear fields and, hence, conventional\nmeasurements can only deliver an upper bound of the spin de-\nphasing time. (ii)With augmenting doping densities, the ex-\nchange interaction yields enhanced motional narrowing of the\nhyperfine interaction induced spin dephasing [see Eq. (30)] and\nthe spin dephasing times increase. Again, the spin dephasing\ntime at nD=2:7\u00021015cm\u00003[48] acquired via SNS is signif-\nicantly smaller than the values that are measured via the vari-\nous other techniques in this doping regime, ranging from 40 to\n600 ns. It is quite probable that this significant deviation does\nnot result from sample specifics like the exact doping regime\nor the degree of compensation but again from avoiding exci-\ntation of free electrons which mitigate spin dephasing by nu-\nclear fields via exchange averaging. The corresponding corre-\nlation times that can be acquired in Hanle-like measurements\nin a longitudinal field [125] are several orders of magnitude\nsmaller than theoretically expected and rather correspond to\nvalues expected in the case of interaction with free electrons\n[89] which explains the more e \u000ecient motional narrowing [see\nEq. (30)]. Further, the strong temperature dependence reported\nin Refs. [178] and [179] is not found via SNS [48] and may\nalso be an indication of insu \u000ecient cooling of the carrier sys-\ntem. With increasing doping density, spin dephasing originat-\ning from hyperfine interaction becomes more and more inef-\nficient while the processes based on anisotropic exchange in-\nteraction (see Sec. 3.1) and possibly other processes become\nmore e \u000bective. At the metal-to-insulator transition ( nMIT\nD=\n1:::2\u00021016cm\u00003) low temperature spin dephasing times at-\ntain their maximum. The e \u000eciency of the various mechanisms\nof spin dephasing at the metal-to-insulator transition has been\ndebated recently [89, 129, 180, 181] since the spin dephas-\ning times observed in conventional experiments seem to be too\nlow to be explained by the known mechanisms. Nonetheless,\nthe value acquired by means of SNS of T2=267 ns at nD=\n1:6\u00021016cm\u00003[48] fits well to the theoretical values around\n300 ns that were put forward by Gorkov and Krotkov [180]. In\nthis doping regime, the energy deposition in the sample that\nnecessarily accompanies conventional experimental probes ob-\nviously reduces the measured spin dephasing times. This asser-\ntion is backed up by Fig. 9 which displays the spin dephasing\ntime measured by SNS in two n-type bulk GaAs samples above\nthe metal-to-insulator transition as a function of the laser power\nthat is deposited in the sample. The sample with a dopant con-\ncentration of nD=3:7\u00021016cm\u00003, only slightly above the\nmetal-to-insulator transition, shows a very drastic dependence\non the amount of absorbed laser power that does not converge\neven for the lowest tested values of deposited probe laser power.\nAccordingly, long spin lifetimes in bulk GaAs at the metal-to-\ninsulator transition as given in Ref. [48] can only be acquired\nby means of SNS with strong detuning from the resonance as\nwell as increased probe laser spot size. (iii) In the metallic\nregime, at doping densities above the metal-to-insulator tran-\nFigure 9: Spin dephasing time measured by spin noise spectroscopy in two n-\ntype bulk GaAs samples above the metal-to-insulator transition as a function of\nthe absorbed probe laser power. The amount of absorbed probe laser power is\nvaried either by tuning the laser power or the laser wavelength from 827 nm to\n850 nm. Data is taken from Ref. [45].\nFigure 10: Temperature dependence of the spin dephasing rates in n-type bulk\nGaAs for di \u000berent doping densities. All data is acquired via SNS and taken\nfrom Ref. [48].\nsition, spin dephasing is predominantly determined by the DP\nprocess. The electronic energy and, hence, the spin splitting rel-\nevant for the e \u000eciency of the DP process increases with higher\ndoping densities [see Eq. (28)]. Sample excitations due to op-\ntical orientation play a less important role in this degenerate\ndoping regime (see Sec. 3.2). Therefore all experimental tech-\nniques deliver similar results in this regime. The data for the\nsample with nD=7:1\u00021016cm\u00003in Fig. 9 proves this reason-\ning since the spin dephasing time is signifcantly less dependent\non the amount of absorbed laser power than at lower doping\nintensities.\nThe equilibrium sample temperature is a well defined ex-\nperimental parameter in SNS since carrier heating is avoided.\n13Figure 11: Spin quality factor Q=g\u0003\u0016BBT\u0003\n2=has a function of the transverse\nmagnetic field in n-type bulk GaAs for di \u000berent doping densities measured via\nSNS. The lines are guides to the eye. The data is reproduced from Ref. [49].\nThus, SNS is an ideal tool to study the temperature dependence\nof the spin dephasing rates in the di \u000berent doping regimes. Spin\ndephasing times in n-type GaAs ranging from liquid helium\ntemperatures up to 80 K are given in Ref. [48] and are repro-\nduced in Fig. 10. The spin dephasing at low temperatures is\nfound to depend only weakly on temperature or—in the case of\nthe investigated sample with the lowest doping concentration—\nis even independent of temperature ( nD=2:7\u00021015cm\u00003).\nIn the sample at the metal-to-insulator transition ( nD=1:8\u0002\n1016cm\u00003), the electrical conductivity shows a very similar tem-\nperature behavior as the spin dephasing rate, indicating a quite\ndirect relation between the spatial electron dynamics and spin\ndephasing [48]. A substantial amount of the donor atoms in\nlow doped samples is ionized at elevated temperatures and spin\ndephasing is governed by the DP process for all doping con-\ncentrations. Here, conventional experimental methods are ex-\npected to be equally sensitive as SNS. In any case, the data in\nFig. 10 reveals that at elevated temperatures the spin lifetimes\nare no more the longest at the metal-to-insulator transition, but\nat higher doping concentrations where motional narrowing via\nmomentum scattering at impurity atoms is more e \u000ecient [see\nEq. (28)]. Scattering at ionized impurities further becomes\nmanifest in the T3=2temperature dependence of the dephas-\ning rates [5] which is observed in Fig. 10. The doping and\ntemperature dependence of n-type bulk GaAs is comprehen-\nsively studied in Ref. [101] by means of fully microscopic cal-\nculations which further include electron-electron and electron-\nphonon scattering. These theoretical studies give a more de-\ntailed temperature dependence as indicated by the fits in Fig. 10.\nA spread of the e \u000bective g-factor in the sample may yield an\ninhomogeneous broadening of the spin dephasing in high trans-\nverse magnetic fields [15, 108, 173, 182, 183]. Such e \u000bects can\nbe investigated via SNS because of the recently achieved ad-\nvancement of SNS to GHz frequencies (see. Sec. 4.3). M ¨uller\net al. examined the spin dynamics in high transverse mag-netic fields in two n-type bulk GaAs samples: one very close\nat the metal-to-insulator transition ( nD=1:8\u00021016cm\u00003), the\nother well above ( nD=8:8\u00021016cm\u00003) [49]. Spin dephasing\nin high magnetic fields is quantitatively well characterized by\nmeans of the spin quality factor Q=g\u0003\u0016BBT\u0003\n2=h[15] which is\nplotted as a function of the applied magnetic field in Fig. 11:\nIn the metallic doping regime, the Q-factor increases with the\napplied field and, hence, no inhomogeneous broadening is ob-\nserved in accord with existing studies [15] and the theoretical\nexpectation that delocalized electronic states average out all in-\nhomogeneities [183]. The inhomogeneous broadening close to\nthe metal-to-insulator transition, which becomes manifest in a\ntransition from a Lorentzian to a Gaussian line shape in the\nspin noise spectra (see Fig. 16) as well as in the formation of\naQ-factor plateau, is around a factor of three less pronounced\nthan in a similar sample examined by resonant spin amplifi-\ncation [15]. The higher temperature used in the SNS experi-\nments of 25 K compared to the resonant spin amplification ex-\nperiment at liquid helium temperatures directly disproves the\nassertion that the inhomogeneous broadening in these cases re-\nsults from a spread of the e \u000bective electronic g-factor due to\na thermal spread of the electronic energy. Instead, a spatial g-\nfactor variation is found in the investigated sample [49] mea-\nsured by SNS with spatial depth resolution (see Sec. 5.2): The\nabsolute value of the g-factor is increased at the sample sur-\nfaces which may result from surface depletion. Delocalized\nelectrons would average over such spatial inhomogeneities and\nno increase of the spin dephasing rate would be observed, but\nSNS directly reveals that the electrons in the investigated sam-\nple ( nD=1:6\u00021016cm\u00003) are to some extent localized [49].\nThis can be deduced from the temperature dependence of the\nobserved spin noise power as already mentioned in Sec. 2.3.\nIn general, the temperature dependence of the spin noise\npower can be divided in three distinct regimes, of which all are\nfound in the experiment [48]. An extrapolation to zero tem-\nperature should deliver vanishing spin noise power in the case\nof a degenerate electron gas in which spin flips are suppressed\nat 0 K due to the Pauli principle. For localized electrons, the\nspin noise power is independent of the temperature and in the\nintermediate doping regime close to the metal-to-insulator tran-\nsition, where electron transport proceeds via hopping, a mixed\nbehavior with residual spin noise power at zero temperature is\nfound, proving partial localization of electrons.\nGaAs /AlGaAs quantum wells. Semiconductor quantum wells\nattract a lot of attention in the context of spintronics since they\nallow to tailor spin orbit fields (see Sec. 3.1). GaAs based quan-\ntum wells with an (110) growth axis are especially interesting\nsince the Dresselhaus field points along the growth axis for\nallk-states such that electronic spins aligned with the growth\naxis do not dephase according to the DP mechanism [112].\nThe longer spin dephasing times in (110) grown quantum wells\ncompared to equivalent (001) structures were experimentally\nshown by Ohno et al. in 1999 [113]. Later, D ¨ohrmann and\nco-workers demonstrated that spins in the quantum well plane\nstill undergo spin dephasing via the DP process and that, sub-\nsequently, spin dephasing is anisotropic in (110) grown struc-\n14Figure 12: Spin noise power Pand e \u000bective spin dephasing rate 1 =Tk\n2measured\nin an (110) grown GaAs /AlGaAs multiple quantum well structure (each quan-\ntum well: nD=1:8\u00021011cm\u00002, thickness 16.8 nm) as a function of the probe\nlaser energy at T=20 K. The solid line is a calculation of the spin noise power\naccording to the modeling described in Sec. 2.3. As expected from Eq. (7), the\ndata resembles nicely the square of the real part of a refractive index within the\nLorentz oscillator model (see Fig. 4). The spin dephasing rate increases signifi-\ncantly by tuning to the resonance due to optical creation of holes. Time of flight\ne\u000bects also contribute to the measured spin dephasing rate. The dashed curve\nis a guide to the eye. All data is taken from Ref. [44].\ntures [114]. However, in this investigation via time and polar-\nization resolved photoluminescence, the anisotropy of the spin\ndephasing is diminished at low temperatures due to the BAP\nmechanism. Like in all experimental probes that rely on optical\nspin orientation (see Sec. 3.2), the presence of optically created\nholes yields additional spin dephasing which becomes domi-\nnant because of the enhanced exchange interaction at low tem-\nperatures and the absence of other e \u000ecient mechanisms of spin\ndephasing. In 2007, Couto et al. spatially separated [184] opti-\ncally created holes from electrons by means of surface acoustic\nwaves. Nevertheless, the influence of these acoustic waves to\nthe spin dephasing had not been established yet and, hence, the\ndominant process of spin dephasing and the corresponding spin\nlifetimes in (110) GaAs quantum wells at low temperatures re-\nmained unknown.\nIn 2009, application of SNS to a modulation doped (110)\ngrown GaAs /AlGaAs multiple quantum well structure (each\nquantum well: nD=1:8\u00021011cm\u00002, thickness 16.8 nm) [44]\nenabled the investigation of electron spin dephasing at low tem-\nperatures in the absence of optically generated electron-hole\npairs. The potentially strong influence of the BAP process be-\ncomes manifest in a drastic increase of the spin dephasing rates\nby tuning the probe laser close to the optical transition (see\nFig. 12) while the measured spin noise power resembles the\nsquare of the real part of a refractive index within the Lorentz\noscillator model (see Fig. 4) as described in Sec. 2.3. In this\nSNS experiment, also the finite transit times of the probed elec-\ntrons through the laser spot play an important role. However,\nwith an enlarged laser spot, to avoid these time of flight e \u000bects,\nand with the laser detuned from the resonance, to avoid exci-\nFigure 13: Spin dephasing time T2measured in an (110) grown GaAs /AlGaAs\nmultiple quantum well structure (each quantum well: nD=1:8\u00021011cm\u00002,\nthickness 16.8 nm) as a function of an applied in-plane magnetic field at T=\n20 K. The magnetic field rotates the spins aligned along the growth axis into the\nquantum well plane and, hence, they are subject of spin dephasing according\nto the DP mechanism. The line represents the spin dephasing time calculated\nby means of kinetic spin Bloch equations where random spin orbit fields are\nadditionally taken into account. The experimental data is taken from Ref. [44]\nand the theory curve from Ref. [185].\ntation of holes, SNS delivers the intrinsic spin dephasing times\nof the investigated sample structure. A spin dephasing time for\nspins aligned along growth axis of Tk\n2=24(2) ns is measured\nat 20 K. The anisotropic spin dephasing that is concealed in\ntime and polarization-resolved photoluminescence experiments\nat these temperatures [114] is also recovered via SNS, where a\nratio of Tk\n2=T?\n2=7:4(1:0) between the dephasing times of spins\naligned along and perpendicular to the growth direction is mea-\nsured by application of an in-plane magnetic field (see Fig. 13).\nThe observed lifetimes Tk\n2cannot be limited by one of the well\nstudied spin dephasing mechanisms (Sec. 3.1). Also, the re-\ncently discovered intersubband spin relaxation [114, 186] can-\nnot completely account for the experimental findings according\nto the microscopic calculations by Zhou and Wu [187]. M ¨uller\net al. [44] suggested that the observed lifetimes Tk\n2are lim-\nited by a mechanism that was initially put forward by Sher-\nman in 2003 [188] which results from random spin-orbit fields\narising from electrical fields due to inevitable spatial fluctua-\ntions of the impurity atoms in the \u000e-doping sheets (see also\nRef. [189]). Recently, several theoretical investigations on spin\ndephasing due to random spin-orbit fields as well as on spin\ndynamics in (110) grown GaAs quantum wells were published\n[187, 185, 190, 191, 192, 193]. While the microscopic calcula-\ntion from Ref. [185] agrees well with the experimental findings,\nespecially with the measured magnetic field dependence (see\nFig. 13), the work by Glazov et al. , where also spin-flip colli-\nsions of electrons from di \u000berent quantum wells of the multiple\nquantum well structure are explicitly considered [192], implies\nthat still additional, even unknown processes may contribute to\nthe observed spin dephasing. The aforementioned transit time\ne\u000bects obviously pose a challenge for acquiring the intrinsic\n15Figure 14: Spin noise spectra of holes confined in self-assembled\n(In,Ga)As /GaAs quantum dots. A strong inhomogeneous broadening in a trans-\nverse magnetic field is observed. The spectra are shifted for clarity. Data is\ntaken from Ref. [47].\nspin lifetimes. However, this time of flight broadening also im-\nplicates a great potential since it uniquely allows to study spatial\nelectron dynamics at thermal equilibrium [44].\n(In,Ga)As /GaAs quantum dots. The recent work by Crooker\net al. [47] represents a compelling proof-of-principle experi-\nment revealing that SNS is by far sensitive enough to detect\nspin dynamics of electrons and holes e \u000bectively confined to\nzero dimensions. Besides the first SNS experiment on self-\nassembled quantum dots, Ref. [47] also contains the first spin\nnoise measurements of hole spins and by above bandgap il-\nlumination intentionally optically created electrons, of which\nboth are neatly identified by the specifics of the e \u000bective g-\nfactor. The investigated sample structure consists of 20 lay-\ners of (In,Ga)As /GaAs grown by molecular beam epitaxy on a\n(100) GaAs substrate where each layer has a quantum dot den-\nsity of around 1010cm\u00002. The relatively large number of probed\nquantum dots and the large inhomogeneous spread of the con-\nfinement energy around 0 :2 eV, which also results in the strong\nbroadening of the spin noise curves in high transverse magnetic\nfields (see Fig. 14), preclude demolition free application of SNS\nby detuning from the probed resonance. Still, this experiment\nmay pave the way for application of SNS on single quantum\ndots. The detection of a single electron spin by below band gap\nFaraday [194] and Kerr [65, 195] rotation has already been es-\ntablished which shows that SNS of a single electron spin should\nbe feasible.\n4. Experimental Aspects of SNS\nOver the last years, semiconductor SNS has developed into\na very sensitive tool to study spin dynamics in semiconductors.\nThe sensitivity has significantly increased and reached a level\nthat allows to apply SNS to quantum wells [44], quantum dot\narrays [47], and even only a few microns thick epilayers of bulksemiconductor material [48]. Recently, the technical limitation\nof SNS to frequencies within the bandwidth of the balanced\nphotoreceiver has been overcome and SNS was demonstrated\nat frequencies of several GHz [49]. This section is devoted\nto rather technical aspects that are only parenthetically men-\ntioned in the corresponding research papers, but are crucial for\nthe achieved advancements. In Sec. 4.1, several possibilities\nto separate the actual spin noise from other noise contributions\nare discussed. E \u000ecient data averaging, which is of great im-\nportance to semiconductor SNS, is discussed in Sec. 4.2. In\nSec. 4.3, the rather new advancement of SNS to GHz frequen-\ncies is presented.\n4.1. Shot Noise Subtraction\nSpin noise is not the only noise contribution that is de-\ntected in SNS. While classical noise is eliminated by stable\nlaser sources and balanced detection, optical shot noise is al-\nways present and exceeds the amount of spin noise by several\norders of magnitude as discussed in Sec. 2.5. Additionally,\ncommercial detectors with the necessary bandwidth for semi-\nconductor SNS of 100 MHz to 1 GHz exhibit electrical noise\nthat is not negligible at low probe powers. Laser shot noise is\nwhite noise and usually adds as a constant noise floor to the\nspin noise. However, the frequency response of the detector\nand an optional pre-amplifier can generally not be viewed as\nconstant within the frequency intervals given by the spin noise\nwidth. Subtraction of the background noise floor is therefore\nnecessary to avoid distortion of the spin noise spectra. To this\nend, a reference noise curve that does not contain spin noise has\nto be acquired. This can be achieved by shifting the spin noise\npeak in frequency by variation of the applied magnetic field as\ndemonstrated in Fig. 2 and in Refs. [39, 40, 45, 47, 49].\nAlternatively, a reference noise spectrum can be acquired\nby switching the optical bridge setup from detection of circu-\nlar birefringence, i.e., Faraday rotation, to linear birefringence\nand thereby suppressing the spin noise signal contained in the\nprobe light, while keeping the photon shot noise background.\nThe suppression of spin noise can be achieved by two distinct\nschemata: (i)The polarization state of the probe laser light can\nbe changed from linearly polarized light to circularly polarized\nlight before it is transmitted through the sample [44]. Here, the\ncircularly polarized light does not acquire a Faraday rotation\nand is split in equal parts into the two orthogonal linear polar-\nization states via the polarizing beam splitter cube in front of\nthe detector (see Fig. 1). This scheme, however, has some dis-\nadvantages, e.g., if the sample exhibits linear dichroism due to\nstrain or magnetic e \u000bects. (ii)The second scheme eliminates\nthe acquired Faraday rotation behind the sample [46, 48]. To\nthis end, the fast axis of a variable retarder behind the sample\nis aligned along the linear light polarization and the retardation\nis switched from \u0015=2 (no change) to \u0015=4 (suppression of spin\nnoise). The variable retardation can be either implemented by\na motorized Soleil-Babinet compensator [44] or a liquid crys-\ntal retarder [46, 48]. The usage of the latter is convenient be-\ncause of the higher switching speed between the two polariza-\ntion states. Switching the liquid crystal retarder, however, also\nintroduces a slight change of the light transmission due to a\n16spin noise power densityprobe wavelength  λ = 826 nm\nprobe wavelength  λ = 850 nm\n3 hours 3 minutesFigure 15: Spin noise spectra ( n-type GaAs, 10 K) acquired by di \u000berent spec-\ntrum analyzers. (a)-(d) Comparison of commercial analyzers employing a ref-\nerence oscillator (probe wavelength \u0015=826 nm, averaging time 4 minutes):\n(a) Hewlett & Packard 4395a, (b) Hewlett & Packard PSA, (c) Tektronix RSA\n3408a, (d) Rhode & Schwarz FSU 8. (e),(f) Comparison of a sweeping spec-\ntrum analyzer (e, Rhode & Schwarz FSU 8, averaging time 3 hours) with a\nreal-time FFT analyzer (averaging time 3 minutes) at a probe wavelength of\n\u0015=850 nm.\nchange in the absolute refractive index of the waveplate that\nhas to be accounted for in the experiment.\nIn some cases, best results are achieved if a double di \u000ber-\nence scheme is utilized by changing both the light polarization\nand the magnetic field [44]. Additionally, the frequency re-\nsponse of the detection has to be taken into account for reliable\nmeasurements of the correct value for the spin noise power.\n4.2. Data Acquisition and Spectrum Analysis\nIn the first paper on semiconductor SNS [39], a sweeping\nspectrum analyzer was utilized for transforming the acquired\ntime signal into the frequency domain. E \u000ecient data averag-\ning is of great importance for flattening the shot noise back-\nground due to the low ratio of peak spin noise power to back-\nground noise density \u0011(see Tab. 1). However, a spectrum ana-\nlyzer with a sweeping local oscillator measures the noise only\nat the reference frequency and thereby disregards the majority\nof the available data stream. Sweeping over 1 GHz bandwidth\nwith a resolution of 1 MHz simply means that around 99 :9%\nof the acquired signal remain unused at a time. Still, di \u000berent\ncommercial spectrum analyzers show a significant di \u000berence in\nsensitivity as depicted in Figs. 15 (a)-(d). Fundamentally, this\nproblem is circumvented by digitizing the data stream and sub-\nsequent realtime spectrum analysis via fast Fourier transforma-tion (FFT). The FFT algorithm allows simultaneous detection\nof spin noise at all frequencies within the detection bandwidth\nand, hence, with no dead time as long as all digitized data can\nbe further processed, i.e., 100% of the signal acquired in the\ntime domain enter into the data processing and averaging. In\norder to comprehend the compelling increase of detection sen-\nsitivity, Figs. 15 (e) and (f) show two SNS spectra acquired by\nmeans of a commercial sweeping spectrum analyzer as well as a\nFFT spectrum analyzer. This advance of the SNS setup was first\nrealized by R ¨omer et al. [40] and employed in all subsequent\npublications on semiconductor SNS [44, 45, 46, 48, 47, 49].\nThe actual realtime FFT analysis is perfectly suitable for paral-\nlel computing and, therefore, scalable to high throughput. As\nthe computer’s PCI Express bus allows data transmission with\nrates of up to 16 GByte =s and multicore CPUs become more\nand more e \u000ecient, software based realtime FFT on the CPU\nyields an extremely high data transmission; currently, our group\nroutinely processes the noise signal with a sampling rate of\nfS=1 GSamples /s. In a similar approach, Crooker et al. imple-\nmented the FFT routine by means of a digitizer incorporating\nfield programmable gate array processors ( fS=2 GSamples /s)\n[47]. According to the Nyquist-Shannon theorem [196, 197],\nthe SNS setup in Fig. 1 can only detect spin noise at frequen-\ncies smaller than the detection bandwidth which is given by\nhalf of the sampling rate: B=fS=2. It is important to cut\no\u000ball shot noise at frequencies larger than Bby means of low\npass frequency filters. Otherwise, undersampling of these fre-\nquency components would result in an increased background\nnoise level within the detection bandwidth.\nThe bit depth Ris another figure of merit for an analog-to-\ndigital converter and specifies together with the sampling rate\nthe data transmission rate of the digitizer I=fS\u0002R(see, e.g.,\nRef. [198]). The bit depth determines the quantization error of\na digitized signal, i.e., the di \u000berence between analog input and\ndigital output. In the case of uniform quantization and avoid-\nance of overload of the digitizer, the variance of the quantiza-\ntion error reads \u0001\u00002=12 according to Bennett’s famous approx-\nimation [199].4Here, \u0001/2\u0000Rgives the size of the least sig-\nnificant bit. Thus, the variance of the quantization error scales\nexponentially with the utilized number of bits per sample. Inter-\nestingly, the signal-to-noise ratio in SNS is not limited by this\nquantity: The ever present shot noise floor (see Sec. 4.1) rep-\nresents an additive dither (see, e.g., Refs. [201, 202, 203, 204,\n205]) to the spin noise signal which facilitates quite e \u000ecient av-\neraging of the quantization error. A detailed understanding of\nthe interplay of averaging and quantization errors is necessary\nto achieve the maximal sensitivity for SNS. An in-depth inves-\ntigation on the sensitivity of SNS due to quantization errors will\nbe published elsewhere [206].\n4.3. GHz Spin Noise Spectroscopy\nSNS utilizing continuous-wave lasers as in Fig. 1 can only\nmeasure spin noise at frequencies below the detector bandwidth\nand has so far been only been demonstrated at frequencies smaller\n4For a discussion of the validity of this approximation, see, e.g., Ref. [200].\n17Figure 16: Spin noise spectra acquired by GHz SNS [49]. The repetition rate\nof the probe laser is set to frep=160 MHz. Spin dynamics at frequencies\nsignificantly higher than the detector bandwidth are measured without any loss\nof sensitivity. The investigated system is n-type bulk GaAs at the metal-to-\ninsulator transition ( nD=1:8\u00021016cm\u00003). A crossover from homogeneous\nto inhomogeneous spin dephasing, i.e., from a Lorentzian to a Gaussian line\nshape, occurs at high magnetic fields (see Sec. 3.3). Spectra are shifted for\nclarity. The negative spin noise peak results from background noise subtraction.\nthan 1 GHz. Recently, this limitation has been overcome by re-\nplacing the continuous-wave laser in Fig. 1 with an ultrafast\npulsed laser light source [49]. Thereby, the spin-spin correla-\ntion in Eq. (12) is only probed when an ultrashort laser pulse tra-\nverses the sample and the relevant correlator additionally con-\ntains the probing pulse train:\nhsz(0)sz(t)i!h sz(0)sz(t)i\u0002X\nn\u000e\u0010\nt\u0000n=frep\u0011\n; (38)\nwhere frepis the repetition rate of the laser source. Thus, the\nspin noise spectrum, which is given by a peak S(f) around the\nLarmor frequency !L=2\u0019in conventional SNS, evolves into a\nsum of peaks all shifted by the repetition rate of the laser:\nS(f)!X\n\u0006mS\u0010\nf\u0000m frep\u0011\n: (39)\nAccordingly, spin noise at frequencies much higher than the\nbandwidth of the detector appears to slow down due to this stro-\nboscopic sampling and can still be detected. This new experi-\nmental technique of GHz SNS is applied in Ref. [49] to detect\nspin noise at Larmor frequencies up to 16 GHz (see Fig. 11).\nGHz SNS is limited to dynamics on timescales that are long\nwith respect to the pulse length. Thus, sub ps pulses allow to\naccess the THz regime. It is important to note, that this ultrafast\nsampling does not per se introduce any further noise and, cor-\nrespondingly, does not show a reduced sensitivity compared to\nconventional SNS. From a technical point of view, pulsed laser\nlight sources generally tend to a higher degree of instability than\ncontinuous-wave lasers; nevertheless, for the here discussed ex-\nperiment, the resulting classical noise occurs on the frequency\nscale well below 1 Hz and is, hence, irrelevant to the experi-\nmental sensitivity. The maximal spin dephasing rates that canbe resolved by this technique are limited by half of the laser\nrepetition rate as well as the bandwidth of the detector.\nStarosielec and H ¨agele suggested ultrafast SNS, also em-\nploying pulsed laser light [42]. In their proposal, the spin-spin\ncorrelation function is not investigated by means of frequency\nanalysis, but in a more direct fashion by varying the time delay\nbetween two subsequent probe pulses . Experimental realiza-\ntion of this proposal would allow to detect spin dynamics with\nprecessional frequencies and dephasing rates both only limited\nby the inverse pulse length.\n5. Applications\nThe original motivation to transfer SNS to semiconductors\nwas to implement a perturbation-free experimental probe to gain\na better understanding of semiconductor spin dynamics that may\nhelp to realize spintronic devices. Besides from that, a new ex-\nperimental method often carries some potential in itself to find\nits way from the laboratory towards applications. The technique\nof nuclear magnetic resonance is of course a great example for\nsuch a transfer and shows that it is in any case worthwhile to\nthink about the potential of SNS. In this section, two potential\napplications of semiconductor SNS are reviewed. In the first\napplication, SNS is employed as a quantum random number\ngenerator (Sec. 5.1). Secondly, the spatial resolution of SNS\ncan be utilized for sample characterization by acquiring three-\ndimensional images of the doping concentration (Sec. 5.2).\n5.1. Quantum Random Number Generator\nPseudorandom numbers that are generated in deterministic\ncomputer algorithms may lead to erroneous results in numer-\nical simulations [207]. This problem can be circumvented by\napplication of physical random number generators. Of course,\nactual randomness can only be achieved if the number genera-\ntor relies on a truly unpredictable physical process. Quantum\nmeasurements are known to be inherently unpredictable and,\nhence, produce real random numbers. Katsoprinakis et al. im-\nplemented a quantum random number generator based on spin\nnoise measurements of Rubidium vapor where the bit rate of\ngenerated random numbers is given by the spin dephasing rate\n[208]. Hence, they argue that a quantum random number gener-\nator based on semiconductor SNS may produce relatively high\nbit rates on the order of 10 Mbit /s.\n5.2. Spatially Resolved Measurements\nThe spatial distribution of impurity atoms crucially deter-\nmines the functional capability of semiconductor devices. With\ndecreasing device size, even the stochastic dopant fluctuations\ncan become relevant. However, the most often used method\nto determine dopant concentrations are Hall measurements that\nhave almost no spatial resolution. Secondary ion mass spec-\ntroscopy allows to map the impurity distribution, but is destruc-\ntive. Scanning tunneling microscopy facilitates non-destructive\ninvestigations of the impurity distribution with atomic resolu-\ntion, though, it is limited to the sample surface. Now, SNS\n18laser\nsample(a) (b)Figure 17: (a) Proof-of-principle experiment by R ¨omer et al. [46]: The thick-\nnesses of the two di \u000berent wafers A and B are measured by depth-resolved\nSNS. (b) SNS allows to produce three dimensional images of the doping con-\ncentration in a given semiconductor sample.\npromises to close the gap between those methods that lack three-\ndimensional resolution and those that are destructive.\nSNS is not only sensitive to the spin dynamics at the sample\nsurface as other optical techniques since SNS employs below\nband gap light. Furthermore, most of the spin noise signal is\nacquired within the Rayleigh range of the focused probe laser\nlight. These two facts allow to spatially resolve semiconductor\nspin dynamics in all three dimensions of space via SNS with\nstrongly focused probe light. In GaAs, the e \u000bective g-factor\n(see Ref. [157] and references therein), the spin dephasing time\n(see Sec. 3.3), as well as the spin noise power (see Secs. 2.2 and\n2.3) depend on the local doping concentration. These quantities\nspecify the detected spin noise spectra and, therefore, spatially\nresolved SNS should allow to produce three dimensional im-\nages of the impurity concentration in a semiconductor sample\n(see Fig. 17 (b)). In 2009, this feature was demonstrated in a\nproof-of-principle experiment [46]: R ¨omer and co-workers ac-\nquired a series of spin noise spectra in a sample stack consisting\nof two di \u000berent commercial n-doped GaAs wafers. The probe\nlight was focused by high aperture optics and the sample was\naxially scanned by varying the focus position (see Fig. 17 (a)).\nThe contributions of the two individual samples can be recov-\nered from the spin noise spectra by a fitting routine. That way,\nthe thickness of the two wafers can be correctly reproduced,\ni.e., a spatial doping profile is reconstructed. Even a three-\ndimensional mapping of the doping concentration can, in prin-\nciple, be achieved by simultaneous laterally and depth-resolved\nmeasurements. The spatial resolution may be extended well be-\nyond the limits of the laser focus and the Rayleigh range if the\nlaser spot scans the sample by small steps in conjunction with\nsophisticated data processing.\n6. Outlook\nSNS allows in principle perturbation-free investigation of\nspin dynamics in semiconductors and is in this regard a unique\nexperimental tool. Application of SNS is primarily useful for\nsample systems in which excitations strongly change the inves-\ntigated dynamics, as in low doped semiconductors at low tem-peratures and in systems where all well-known spin dephasing\nprocesses are known to be ine \u000ecient. So far SNS has been ap-\nplied to n-type bulk GaAs [40, 46, 45, 48, 49], GaAs /AlGaAs\nbased quantum wells [44], and ensembles of (In,Ga)As /GaAs\nquantum dots [47]. Nevertheless, SNS can be universally uti-\nlized in other semiconductor materials—with direct as well as\nwith indirect optical transitions—and should also work in other\nsolid state material classes, e.g, in materials with magnetic or-\nder where collective magnetic modes are thermally excited.\nSNS probes the spin fluctuations in the investigated sample\nsystem, i.e., the spin-spin correlation function. Spin correla-\ntions of higher order, which are also contained in the acquired\ntime signal, can reveal further information about the underlying\nspin dynamics. For example, third-order correlations may allow\nfor separation of homogeneous spin dephasing from inhomoge-\nneous processes in prospective SNS experiments on very small\nelectron ensembles [214].\nThe experimental sensitivity of semiconductor SNS is in\nthe case of large electron ensembles, where moderately high\nprobe laser powers can be utilized, mostly limited by optical\nshot noise. Here, application of squeezed light as probe light\ncan further increase the signal-to-noise ratio [79]. In the case of\nsmall electron ensembles, significant noise contributions from\nthe detector cannot be avoided and further enhancement of the\nexperimental sensitivity may be achieved by a Mach-Zehnder\ninterferometer like setup [75, 209, 210, 211] or cavity enhance-\nment of the Faraday rotation [212, 213].\nThe role of residual sample excitations, e.g., considering\noptical excitation of deep centers [49] or spin flip light scat-\ntering [66, 194], clearly needs further attention. This question\nis of particular interest with respect to a possible transfer of\nthe quantum non-demolition experiments on atomic gases (see\nSec. 2.6) to semiconductor physics as well as for the detec-\ntion of a single electronic spin confined in a quantum dot. Only\nif spin flip scattering of the probe light occurs on timescales\nlonger than the spin lifetime, SNS can fulfill a meaningful mea-\nsurement of a single electronic spin. The realization of such a\nquantum non-demolition measurement in a semiconductor has\nrecently gained a lot of research interest (see, e.g., Refs. [143,\n194, 215, 216, 217]). Especially, an implementation based on\noptical detection via Faraday rotation—as in SNS—is desirable\nsince such a measurement is employed as building block in sev-\neral schemes for photon-spin and spin-spin entanglement (see,\ne.g., Refs. [83, 84, 85, 86, 87, 88]).\nAcknowledgements\nThis work was supported by the German Science Foun-\ndation (DFG priority program 1285 ‘Semiconductor Spintron-\nics’), the Federal Ministry for Education and Research (BMBF\nNanoQUIT), and Centre for Quantum Engineering and Space-\nTime Research in Hannover (QUEST). G.M.M. acknowledges\nsupport from the Evangelisches Studienwerk.\n19References\nReferences\n[1] G. Lampel, Phys. Rev. Lett. 20 (1968) 491.\n[2] R. R. Parsons, Phys. Rev. Lett. 23 (1969) 1152.\n[3] S. Datta, B. Das, Appl. Phys. Lett. 56 (1990) 665.\n[4] S. A. Wolf, D. 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It can be e xplained with the complicated  spin configurations and \ndynamics during the switching process. The de tail spin configurations and dynamics are \ndetermined by spin wave excitation with the finite wave vector, wh ich is related with \nthe exchange coupling energy and junction shape.\n \nPACS: 75.76.+j, 72.25.-b, 85.75.Dd, 75.78.Cd \n \n    \n                                            \na)Author to whom correspondence should be addressed. E-mail : mhjung@sogang.ac.kr   2 \n I. INTRODUCTION \nThe switching current density is one of the important quantities in the development of \nthe spin transfer torque magnetic random  access memory (STT-MRAM) for the next \ngeneration non-volatile memory applicatio ns. The switching current density is \ndetermined by the instability conditions of the spin wave, which is excited by the STT. In addition, it is widely accepted that the swit ching current density is determined by the \nvarious physical parameters such as spin  polarization, saturati on magnetization, and \nshape of magnetic tunneling junction (MTJ),  based on the macro sp in model [1,2,3]. In \nthe macro spin model, the contribution of th e exchange stiffness is usually ignored, \nbecause the contribution of exchange energy is zero. However, we have recently found \nthat the switching current dens ity is a sensitive function of the exchange stiffness \nconstant by using the micromagnetic simula tions [4]. We have also found that the \ndetailed spin configuration and dynamics are important in the realistic switching \nprocesses. The detailed spin configur ation and dynamics are determined by the \nexchange stiffness constant and the shapes of  the MTJs, which play an important role in \nthe determination of switching current density.  \nIn this study, we investigate the depe ndence of switching current density on the \njunction sizes for various exchange stiffn ess constants. By using public domain \nmicromagnetic simulator, Object-Oriente d MicroMagnetic Framework (OOMMF) [5] \nwith the public STT extension module [6,7], we calculate the switching current density of typical MTJ structure with various lateral junction sizes.  We find that the dependence \nof switching current density on the junction sizes is not a simple monotonic function, \nbut it shows complicated behaviors. The results  can be explained with the detailed spin \nconfigurations and weakly quantized spin wave with finite wave vector.  3 \n II. SIMULATIONS \nWe consider the typical STT-MRAM structure with an insulating barrier between \nexchange-biased synthetic ferrimagnet layer of F 3/NM/F 2 and free ferromagnetic layer \nF1 as shown in Fig. 1 [8,9]. The saturation magnetization and the ferromagenet \nthicknesses of F 1, 2, 3 layers are 1.3 106 A/m and 2 nm, respectiv ely. The thicknesses of \nnormal metal (NM) and insulator (I) layers are both 1 nm. The cross section of the \nnanopillar is an ellipse of ab nm2, where the long axis length is a and the short axis \nlength is b. In this simulation, we vary a from 30 to 120 nm with b = 20, 30, and 40 nm, \nand the cell size is 1 11 nm3. The exchange stiffness constants are considered to be  \nAex = 1.0, 2.0 and 3.0 10-11 J/m, because the reported experimental data of most \ninteresting materials such as CoFeB are in  these ranges, depending on the composition \nand fabrication details [10,11,12]. For simplic ity, no crystalline anisotropy energy is \nconsidered and the Gilbert damp ing constant is fixed to be  = 0.02. The exchange \nbias field of 4 105 A/m is assigned to the l ong axis of the ellipse (+ x-direction) for the \nF3 layer. We consider only the in-plane STT contribution and i gnore the out-of-plane \nSTT contribution. We apply positive current , where electrons flow from free to \nreference layer, which prefers antiparallel st ate. The pulse duration time is 10 ns, and we \ncheck the switching status af ter 2 ns. By repeating the pr ocedure, we determine the \nswitching current density for the parallel to antiparallel states. All micromagnetic \nsimulations are done at zero temperature, and thus the thermal exc ited contributions are \nignored in this study. More details of micromagnetic simulations can be found \nelsewhere [6].  \n \nIII. RESULTS AND DISCUSSIONS 4 \n The simulation results of the switching current densities Jc for various junction sizes \nare depicted in Fig. 2 (a)~(c) and Fig. 3 (a)~(c), as  a function of the long axis length of \nthe ellipse ( a = 30 ~ 120 nm) for different short axis length of the ellipse ( b = 20, 30, \nand 40 nm) with different exchange stiffness constant ( Aex = 1.0, 2.0, and 3.0 10-11 J/m). \nFigure 2 shows the Jc variations for different b values with a fixed Aex value at each \npanel, while Figure 3 shows the Jc variations for different Aex values with a fixed b \nvalue at each panel as the same  data set. In Fig. 2 (a) with Aex = 1.010-11 J/m, Jc \noscillates strongly for b = 30 nm, oscillates weakly for b = 20 nm, and increases \nmonotonically for b = 40 nm. These patterns are changed for Aex = 2.0 10-11 J/m in \nFig. 2 (b), and they are slowly increased for Aex = 3.0 10-11 J/m in Fig. 2 (c). Such \ntendencies are not easy to be explained with simple macro spin model. In order to get better insight, we replot the same data in Figs. 3 (a)~(c) for fixed b values. The J\nc \nvariations are quite different for the short axis length of b, in spite of the small change \nof b. The variation is most serious for b = 30 nm seen in Fig. 3 (b), and it is somewhat \nmonotonic for b = 40 nm seen in Fig. 3 (c).  Such strong dependence of Jc on the \njunction size and exchange stiffness is our ma in finding in this study. It should be noted \nthat the switching cu rrent density without considerati on of junction size and exchange \nstiffness, which is calculated from macro spin model, is Jc0 = 1.841011 A/m2. This \nvalue is much smaller than all of our micromagnetic simulation results. \nLet us explain the physical re asons of such variation of Jc based on the macro spin \nmodel including the contribution of exchange s tiffness term with finite spin wave vector \nk [13,14,15,16,17,18]. The switching current density Jc is given by \n2\n102 1~22ex\nce f f y z x s\nsAJHN N N M kaM\n  .    (1) 5 \n Here, ,,xyzN  denote the demagnetization factors of the free layer (zy xNN N  for \nthin ellipse), and effHis the effective field including external field, stray field, Oersted \nfield, and perpendicular STT field-like term.  is the Gilbert damping constant and \n1\n02p\nssaeM d. Here, p, ds , Ms, 0, and  are the spin polarization of the \npolarizer layer, thickness of the free layer, saturation magne tization, permeability of the \nvacuum, and reduced Plank’s constant, respectively. According to Eq. (1), the contribution of exchange stiffness with spin wave vector k is clear for thin films. The \neasiest excitation occurs with k = 0, which is a uniform mode. However, we calculate \nwith the finite size nanopill ar structures, and thus the k value is limited by the junction \ndimension. Therefore, the k value is weakly quantized to minimize the exchange and \ndemagnetization energies. Such weak quantiz ation can roughly expl ain the oscillatory \nbehaviors of J\nc. \nNow, we discuss the details of the spin c onfigurations during th e switching process in \norder to support above explanations. Let us focus a 60 b nm2 ellipse with Aex = 1.010-\n11 J/m, which indicates the red dashed circ le in Fig. 2 (a). The switching current \ndensities Jc vary from 2.68 to 4.75 1011 A/m2, when the short axis lengths b are \ndifferent. The 60 40 nm2 case shows the lowest Jc value, and the 60 30 nm2 case \nshows the highest Jc value. To reveal the reason, we apply J = 3.01011 A/m2, smaller \nthan Jc for b = 20 and 30 nm but larger than Jc for b = 40 nm, which po ints out a blue \nhorizontal arrow in Fig. 2 (a). We de pict the time dependence of normalized x-\ncomponent of magnetization Mx at positions A, B, C, and D (see Fig. 1 for the definition \nof each position), and total Mx in Fig. 4 (a)~(c). For b = 20 and 30 nm, the \nmagnetization at positions A and B shows large oscillations for whol e time. For example, 6 \n the amplitude of oscillation at position A for b = 20 nm in Fig. 4 (a) changes between -\n1.0 to 0.8, indicating that the magnetization of position A is almost switched. Then, the \namplitude of oscillation decreases at positions B and C. Finally, at position D which is the center of the ellipse, no oscillation is obser ved in spite of the strong oscillation of \nthe off-center positions. As a result, the total oscillation is finite and the switching is not \noccurred. This situation is similar to the b = 30 nm case in Fig. 4 (b). Even though the \napplied current density J is smaller than J\nc for b = 20 and 30 nm, because J is larger \nthan Jc0 (= 1.181011 A/m2), the observed large oscillati ons are not surprising. On the \nother hand, for the b = 40 nm case in Fig. 4 (c) the magnetization is switched around 9 \nns.  \nFor more details, we perform the fast Fourier transform (FFT) of time dependent \nmagnetization as shown in Figs. 5 (a)~(c).  The corresponding frequency dependencies \nare shown in Figs. 4 (a)~(c). There are several points we shoul d address about the \nFourier analysis. First, it is clearl y shown that the ellipse with larger b shows more \ncomplicated modes, implying that the oscill ation is not coherent. This result is \nreasonable when we consider the junction size. It  is more manifested in Fig. 6, that will \nbe discussed later. Second, the main peak frequencies decrease from 22, 20, to 17 GHz \nfor b = 20, 30, and 40 nm, respectively. Since all physical parameters are same except \nthe short axis length b, we conjecture that the effective field of each junction is changed \nwith b, leading to the change of peak freque ncy [19]. Since the larger junction can \nreduce demagnetization energy more easily by fo rming complicated spin configurations, \nthe effective field of larger junction becomes smaller. This is consistent with the main peak frequencies. Finally, let us pay our attention to the sp ectra at the center position D. \nAccording to Fig. 4 (c), th e switching occurs abruptly, and the corresponding Fourier 7 \n transform is broad spectra as shown in Fig. 5 (c). The Fourier spectra of position D \nshows no noticeable peak except broad stru cture in low frequency region (< 5 GHz), \nwhich is found in all position spectra. This is somewhat surprising results. Since the \nmain peaks around 20 GHz correspond to the spin wave excitation, which is predicted by the macro spin model, it must play an important role in the switching processes. \nHowever, the switching process at the cen ter position D has no specific frequency \ndependence. Therefore, we can conclude that  the macro spin model is too simple to \nunderstand the real magnetiz ation reversal process. \nFigs. 6 (a)~(f) show the snapshots of specific times seen in Figs. 4 for 60 b nm\n2. \nFigs. 6 (a) and (b) are the snapshots of 60 20 nm2 at t = 9.18 and 9.24 ns. As already \ndiscussed, the magnetization at positions A and B oscillates strongly, and at C it \noscillates weakly. Furthermore, there is no oscillation at the cen ter position D. Such \nbehavior starts before 2 ns, and keeps stea dy oscillation till turn the current off. The \nmotions of left and right sides are very asym metry, so that the motion of center part is \nsuppressed. Figs. 6 (c) and (d) of 60 30 nm2 also show similar asymmetry along the \nlong axis. However, there is asymmetry breakin g along the short axis as shown in Fig. 6 \n(d). The asymmetry breaking is more significant for 60 40 nm2 in Figs. 6 (e) and (f). At \nt = 8.20 ns, the asymmetry is already broken and the center magnetization starts to move, \nand at t = 9.24 ns, the center magnetization shows finite rotation. One possible reason of \nsuch asymmetry breaking is easy formation of complicated spin conf iguration due to the \nlarger junction size.  \nHowever, the dependence of switching curr ent density on the junction sizes is not \nsimple as already shown in Fig. 2. The switching process require s asymmetry breaking \nat the center position, and it is strongly coupl ed with the detail sp in configuration and 8 \n dynamics which depends on the exchange  stiffness and the junction size.  \nWe investigate more details of the spin configurations during the switching process, \nbut it is difficult to find simple relations hip between the switching current and junction \nsize. Based on our observations, however , we can make some conjectures. The \nmagnetization dynamics along the long axis are asymmetric so that the STT effect on \nthe center part is suppressed. Therefore, the Jc dependence on the junction sizes is \nrelated with the formation of complicated sp in configuration, which is determined by \nthe exchange length, 2\n0 ~ ~ 2 ~ 10 nmex ex u ex s exlA K A f M A f  , where f is \nthe correction factor of sh ape anisotropy (< 1) and Aex is in unit of 10-11 J/m. Therefore, \nthe exchange length varies 10~20 nm, which depends on the exchange stiffness and the \njunction size in this study, and it is compar able to the junction dimension. This implies \nthat the length scale of a few 10 nm can lead to noticeable variation of the shape anisotropy energy and it causes the limitation of spin wave vector k. This limitation is \nwhat we call weak quantization of spin wave vector k. Therefore, irregular variation of \nswitching current for the exchange stiffne ss and the junction size is understandable. \nIt must be addressed that the asymmetry br eaking can be more easily achieved in real \nexperiments due to the non-uniform current density, which is not implemented in our \nsimulations. Typical MTJ device has an or der of ~100% tunneling magnetoresistance \n(TMR), it leads to the rapid variation of TMR from place to place. Since the electrodes \nare metallic, the potential across the insulating layer is equal, so that the local current \ndensity will vary from place to place. It  depends on the relative orientation of \nmagnetization between free and reference laye rs. Furthermore, imperfect junction shape \nintroduced by the lithography processes also  leads to the asymmetry breaking more 9 \n easily. Therefore, the junction size depende nce may be weaker than that expected. \nWe also investigate other cases in Figs. 2, however, the detail spin dynamics is too \ncomplicate to be explained with simple model. What we can claim is that the Jc \nvariation is much stronger than that es timated by macro spin model, and it requires \nmore careful analysis. \n \nIV. CONCLUSIONS \nWe investigate the effect of junction size on the switching current density by \nemploying micromagnetic simulations with STT.  It is found that the dependence of the \njunction size is much stronger than that esti mated by macro spin model. The variation of \nswitching current dens ities can be explained by the form ation of asymmetry breaking of \nspin configurations, which is determined by the exchange stiffness and shape anisotropy \nenergies. Based on our micromagnetic simula tions, we can conclude that the main \nreason of the large variation of the switching current densities is that the junction dimension is comparable to the exchange lengt h of the system. Therefore, there is more \nchance to reduce the switching current density by optimization of the exchange stiffness \nand junction size. \n \nAcknowledgements  \nThis work was supported by the NRF funds (Grant Nos.  2010-0023798 and 2010-10 \n 0022040, Nuclear R&D Program) of the MEST of Korea, and by the IT R&D program \nof MKE/KEIT (10043398).  11 \n References \n                                            \n[1] J. Z. Sun, Phys. Rev. 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Matsukura, and H. Ohno, IEEE \nMagn. Lett. 2, 3000304 (2011). \n[13] W. H. Rippard, M. R. Pufall, S. Kaka, S.  E. Russek, and T. J. Silva, Phys. Rev. Lett. \n92, 027201 (2004). \n[14] J. C. Slonczewski: J. Magn. Magn. Mater. 195, L261 (1999). \n[15] Y . Zhou, J. Åkerman, and J. Z. Sun: Appl. Phys. Lett. 98, 102501 (2011). 12 \n                                                                                                                                 \n[16] H. Sato, M. Yamanouchi, K. Miura, S. Ikeda, H. D. Gan, K. Mizunuma, R. \nKoizumi, F. Matsukura, and H. Ohno: Appl. Phys. Lett. 99, 042501 (2011). \n[17] J. Z. Sun, P. L. Trouill oud, M. J. Gajek, J. Nowak, R.  P. Robertazzi, G. Hu, D. W. \nAbraham, M. C. Gaidis, S. L. Brown, E. J. O’Sullivan, W. J. Gallagher, and D. C. \nWorledge: J. Appl. Phys. 111, 07C711 (2012). \n[18] D. V . Berkov and J. Miltat: J. Magn. Magn. Mater. 320, 1238 (2008). \n[19] J. Yoon, C.-Y . You, Y . Jo, S.-Y . Park, and M.–H. Jung, J. Kor. Phys. Soc. 57, 1594 \n(2010). 13 \n Figure Captions \n \nFig. 1 Typical MTJ structures with syntheti c antiferromagnetic fixed layer. The length \nof the long (short) axis of the ellipse is denoted by “ a” (“b”). The labels “A”~“D” \nindicate the specific position of  the junction along the long axis. “A” is the left end and \n“D” is the center of the ellipse.  Fig. 2 Switching current densities as a function of long axis length  a for various short \naxis length b (= 20, 30, and 40 nm)  with different exchange stiffness constant A\nex of (a) \n1.0, (b) 2.0, and (c) 3.0 10-11 J/m. \n Fig. 3 Switching current densities as  a function of long axis length a for various \nexchange stiffness constant A\nex ( = 1.0, 2.0, and 3.0 10-11 J/m) with the length of short \naxes b of (a) 20, (b) 30, and (c) 40 nm. \n Fig. 4 Time dependent normalized M\nx for each position A, B, C and D, and total Mx. The \nexchange stiffness constant is Aex = 1.010-11 J/m and the junction dimensions are (a) \n6020, (b) 6030, and (c) 60 40 nm2 with the current density of 3.0 1011A/m2. \n \nFig. 5 Fourier transform of the normalized Mx of Fig. 4 for each position A, B, C, and D \nindicated in Fig. 1. The exchange stiffness constant is Aex =1.010-11 J/m and the \njunction dimensions are (a) 60 20, (b) 60 30, and (c) 60 40 nm2 with the current \ndensity of 3.0 1011A/m2. \n 14 \n Fig. 6 Snapshots of magnetizati on configurations at specific times of Fig. 4. (a, b) are \nfor 6020, (c, d) are for 60 30, and (e, f) are for 60 40 nm2.  a b \nF1 \nI \nF2 \nNM \nF3 \nAFM  \ncurrent  \n J >0 \nx z \nA B C D  \nFig. 1  20 40 60 80 100 1202.02.53.03.54.04.55.05.5\n20 40 60 80 100 120 20 40 60 80 100 120(c) Aex = 3.0x10-11 J/m (b) Aex = 2.0x10-11 J/m  Jc (x1011 A/m2)\na (nm)b = \n 20 nm \n 30 nm\n 40 nm(a) Aex = 1.0x10-11 J/m\n  \na (nm)\n  \na (nm)Fig. 2  Fig. 3  \n20 40 60 80 100 1202.02.53.03.54.04.55.05.5\n40 60 80 100 120 60 80 100 120(c) b = 40 nm (b) b = 30 nm  Jc (x1011 A/m2)\na (nm)Aex =   1.0x10-11 J/m,   \n      =   2.0x10-11 J/m,  \n      =   3.0x10-11 J/m \n(a) b = 20 nm\n  \na (nm)\n  \na (nm)0.0 2.0 4.0 6.0 8.0 10.0 12.0-2.00.02.04.06.08.010.012.0\n(c) Aex=1.0x1011 J/m, 60x40 nm2\n  Mx (arb. unit)\nt (ns)Fig. 4  \n0.0 2.0 4.0 6.0 8.0 10.0 12.0-1.00.01.02.03.04.05.06.0\nDTotal\nC\nB(a) Aex=1.0x1011 J/m, 60x20 nm2\n  Mx (arb. unit)\nt (ns) A      B      C     D     Total\nA\n0.0 2.0 4.0 6.0 8.0 10.0 12.0-1.00.01.02.03.04.05.06.0 (b) Aex=1.0x1011 J/m, 60x30 nm2\n  Mx (arb. unit)\nt (ns)10 20 30 40050010001500(a) Aex=1.0x1011 J/m, 60x20 nm2\n  FFT of Mx (arb. unit)\nf (GHz) A      B      C     D \n10 20 30 400500(c) Aex=1.0x1011 J/m, 60x40 nm2\n  FFT of Mx (arb. unit)\nf (GHz)Fig. 5  \n10 20 30 40050010001500\n(b) Aex=1.0x1011 J/m, 60x30 nm2\n  FFT of Mx (arb. unit)\nf (GHz)(a) t = 9.18 ns (b) t = 9.24 ns \n(c) t = 9.13 ns \n(d) t = 9.18 ns 60 x 20 nm2 \n60 x 30 nm2 \n(e) t = 8.20 ns (f) t = 9.24 ns 60 x 40 nm2 \nFig. 6  "    },    {        "title": "1706.02879v2.Spin_wave_analysis_for_Kagome_triangular_spin_system_and_coupled_spin_tubes__low_energy_excitation_for_the_cuboc_order.pdf",        "content": "arXiv:1706.02879v2  [cond-mat.str-el]  4 Sep 2017Journal of the Physical Society of Japan DRAFT\nSpin-Wave Analysis for Kagome-Triangular Spin System and C oupled Spin Tubes:\nLow-Energy Excitation for the Cuboc Order\nMasahiro Ochiai, Kouichi Seki, and Kouichi Okunishi1\nGraduate School of Science and Technology, Niigata Univers ity, Niigata 950-2181, Japan\n1Department of Physics, Niigata University, Niigata 950-21 81, Japan\nThe coupled spin tube system, which is equivalent to a stacke d Kagome-triangular spin system, exhibits the cuboc\norder – a non-coplanar spin order with a twelve-sublattice s tructure accompanying spontaneous breaking of the transla -\ntional symmetry – in the Kagome-triangular plane. On the bas is of the spin-wave theory, we analyze spin-wave excita-\ntions of the planar Kagome-triangular spin system, where th e geometric phase characteristic to the cuboc spin structur e\nemerges. We further investigate spin-wave excitations and dynamical spin structure factors for the coupled spin tubes ,\nassuming the staggered cuboc order.\nKEYWORDS: coupled spin tubes, Kagome-triangular lattice, Cuboc order, spin wave\n1. Introduction\nFrustration effects in spin systems often induce nontriv-\nial spin orders, such as a 120◦structure with spin chi-\nrality or a non-coplanar spin order, accompanying sponta-\nneous breaking of the translation and spin rotational sym-\nmetries. Among various interesting spin orders, the cuboc\norder formed in a quasi-two-dimensional (quasi-2D) plane\nwith frustrating interactions has recently attracted much inter-\nest.1–3)The cuboc order is defined as a non-coplanar spin or-\nder with a twelve-sublattice structure, which can be specifi ed\nby a triple-wavevector structure in the momentum space. Re-\ncent experiments on Kagome-lattice-based spin systems suc h\nas NaBa 2Mn 3F114)and Cu 3Zn(OH) 6Cl25)actually suggest the\npossibility of the cuboc order. Moreover, the phase transit ion\nassociated with the cuboc order provides fascinating physi cs\nfrom the theoretical viewpoint.6)\nAnother interesting compound for the cuboc order is the\ncoupled spin tube system CsCrF 4.7, 8)AC-susceptibility and\nneutron diffraction experiments9, 10)suggest signatures of\nnontrivial spin order below T<4 K. Nevertheless, the order\nrealized in CsCrF 4has not been experimentally specified yet.\nThen, an essential point for CsCrF 4is that a small but non-\nnegligible inter-tube coupling forms the Kagome-triangul ar\nlattice11)in a cutting plane of the coupled spin tubes (see Fig.\n1). Since the exchange coupling in the tube-leg direction is\nnot frustrating, the Kagome-triangular lattice structure plays\na significant role in forming the cuboc order. Monte Carlo\nsimulations for the classical Heisenberg model of spin tube s\ncoupled with the weak ferromagnetic inter-tube coupling ha ve\ndemonstrated that the cuboc order actually emerges at a fi-\nnite temperature.6)However, the physical properties (involv-\ning quantum effect) of the cuboc order for the coupled spin\ntubes have not been quantitatively investigated yet.\nIn this paper, using the spin-wave theory, we study low-\nenergy excitations of the coupled spin tubes, or, equivalen tly,\nof a stacked Kagome-triangular system. We first analyze dis-\npersion relations of the spin-wave Hamiltonian for the 2D\nKagome-triangular plane. Then, a key point is that the spin\nquantization axes are nontrivially tilted in the cuboc spin con-\nfiguration, for which the hopping matrix elements may ac-\nquire geometric phases. We next investigate spin-wave exci -bc\na(a)(b)\na\nb\nFig. 1. (Color online) (a) Lattice structure of a spin tube. Jdenotes the\nexchange coupling in the unit triangle and Jcis the coupling in the tube-leg\ndirection along the c-axis. (b) Kagome-triangular lattice structure of coupled\nspin tubes in the ab-plane. Spins on triangles of the tubes are coupled with\nthe inter-tube coupling J′(broken lines). The inter-tube coupling of J′has a\nKagome lattice structure, whereas the intra-tube coupling s ofJ(triangles of\nsolid lines) correspond to next-nearest couplings of the Ka gome lattice. The\nlattice translation vectors in the ab-plane are defined as aandb, whiler1,r2,\nandr3denote the vectors indicating the nearest-neighbor sites.\ntations of the coupled spin tubes with 3D couplings, assumin g\nthe staggered cuboc order in the c-axis (tube-leg) direction. In\nparticular, we calculate the dynamical spin structure fact ors\nfor the coupled spin tubes in addition to the spin-wave dis-\npersion relations. On the basis of these spin-wave results, we\nclarify features of the spin-wave excitations for the coupl ed\nspin tubes with the cuboc order and discuss their relevance t o\nCsCrF 4.\nThis paper is organized as follows. In the next section, we\nsummarize basic properties of the coupled spin tubes and the\ncuboc order. In§3, we analyze the spin-wave Hamiltonian for\nthe 2D Kagome-triangular plane, which is represented as a\n24×24 matrix reflecting the twelve sublattice structure of\nthe cuboc order. In §4, we perform the spin-wave analysis of\nthe coupled spin tubes with the full 3D interactions, assum-\ning the staggered cuboc order in the tube-leg direction. In §5,\nwe summarize our results and then discuss their relevance to\nrecent experiments.\n2. Coupled Spin Tubes and Cuboc Order\nLet us introduce the coupled spin tube model. As shown\nin Fig. 1(a), we assume that the unit triangle of a spin tube\n1J. Phys. Soc. Jpn. DRAFT\n120°Structure\n120°Structureq=0\n3 ×3Ferro\nFig. 2. (Color online) Ground-state phase diagram of the classical Heisen-\nberg model on the coupled tube lattice. The ground state basi cally depends\nonly on the couplings JandJ′in the ab-plane since Jccauses no frustration\neffect.\nis located in the ab-plane and the tube-leg direction is along\nthec-axis. Then, the Hamiltonian of the coupled spin tubes is\nwritten as\nH=J/summationdisplay\n/an}bracketle{ti,j/an}bracketri}htintraSi·Sj+J′/summationdisplay\n/an}bracketle{ti,j/an}bracketri}htinterSi·Sj+Jc/summationdisplay\n/an}bracketle{ti,j/an}bracketri}htlegSi·Sj,(1)\nwhereSdenotes a vector spin for the classical case or a spin\nmatrix with magnitude Sfor the quantum spin. JandJcre-\nspectively represent the exchange couplings in the unit tri an-\ngle and in the tube-leg direction. Note that for CsCrF 4the\nintra-tube and tube-leg couplings are antiferromagnetic a nd\nJc∼2J≫|J′|is expected.12)Although interesting ground-\nstate properties of the single S=1/2 quantum spin tube\nhave been clarified so far,13–17)here, we emphasize the im-\nportance of the small but finite inter-tube coupling J′in the\ncontext of the cuboc order. As shown in Fig. 1(b), the cou-\npled spin tubes basically have a triangular lattice structu re in\ntheab-plane. However, an interesting aspect is that the lattice\nstructure of the inter-tube coupling is topologically equi va-\nlent to the Kagome lattice, and the intra-tube coupling corr e-\nsponds to a next-nearest-neighbor interaction of the Kagom e\nlattice. We therefore call the lattice structure of the coup led\nspin tubes in the ab-plane the Kagome-triangular lattice.11)\nThis Kagome-triangular lattice structure is essential for the\ncuboc order, while the tube-leg coupling Jcbasically causes\nno frustration.\nThe ground-state phase diagram of the coupled spin tubes\nbased on the classical Heisenberg model is shown in Fig. 2.\nNote that CsCrF 4is located around J′∼0 with the antiferro-\nmagnetic intra-tube coupling J>0, but the determination of\nJ′is an experimentally subtle problem. In this paper, we ba-\nsically assume the ferromagnetic inter-tube coupling, J′<0,\nand particularly focus on the weak inter-tube coupling regi on,\n−1<J′/J<0, where the cuboc order appears. For the classi-\ncal coupled spin tubes, the cuboc order has actually been con -\nfirmed to be realized at a finite temperature by Monte Carlo\nsimulations.6)As|J′/J|increases, the incommensurate order\nemerges in−2<J′/J<−1, and finally the ferromagnetic\norder becomes stable for J′/J<−2.\nThe cuboc order is defined as a non-coplanar spin order\nwith a twelve-sublattice structure, as in Fig. 3. The strong an-\ntiferromagnetic intra-tube coupling basically imposes a r igid\nplanar 120◦structure on the unit triangles of the tubes. How-12 34\n5\n67\n89\n1011\n12\n(a) (b)A\nB\nDC123\n456\n789\n101112a\nb\nxyz\nFig. 3. (Color online) Spin structure of the cuboc order. (a) Twelve -\nsublattice structure of the cuboc order, where the translat ional symmetry is\nbroken by 2×2 in the unit of the tube triangle. Then, the 120◦structures\ndefined for the triangles labeled by A, B, C, and D form a tetrah edron in the\nspin space, as shown in (b). The numbers assigned for the 12 su blattices in\n(a) correspond to those of the spins in (b). x,y, and zassigned to the edges\nof the box frame in (b) denote the directions of the unit vecto rs in the spin\nspace.\never, the ferromagnetic inter-tube coupling also causes fr us-\ntration. In order to reduce the energy due to the inter-tube\ncoupling, the rigid 120◦planes tilt with each other, accom-\npanying the translational-symmetry breaking of 2 ×2 in the\ntriangle unit. Then, the four tilting 120◦planes labeled by A,\nB, C, and D form a tetrahedron in the spin space, as depicted\nin Fig. 3(b). Here, the number indices for the 12 spins in Fig.\n3(a) respectively correspond to those for spins on the tetra he-\ndron in Fig. 3(b). Another important feature of the cuboc or-\nder is that spin chiralities can be defined for the tetrahedro n;\nthe spins on each surface triangle of the tetrahedron have a\nvector spin chirality pointing to the center or the outer sid es\nof the tetrahedron. For example, the vector spin chirality i n\nthe case of Fig. 3(b) has “ +”. Meanwhile, the scalar spin chi-\nrality can also be defined for the three spins at each vertex of\nthe tetrahedron.\nThe spin configuration of the cuboc order can be explicitly\nrepresented as\nSµ(r)=cos/parenleftigg1\n2qb·r/parenrightiggex√\n2−cos/parenleftigg1\n2qa·r/parenrightiggey√\n2,\nSν(r)=cos/parenleftigg1\n2(qa−qb)·r/parenrightiggez√\n2−cos/parenleftigg1\n2qb·r/parenrightiggex√\n2,\nSη(r)=cos/parenleftigg1\n2qa·r/parenrightiggey√\n2−cos/parenleftigg1\n2(qa−qb)·r/parenrightiggez√\n2,(2)\nwhereqaandqbare the reciprocal vectors in the ab-plane. The\nvectorsex,ey, andezrepresent the unit vectors in the spin\nspace along the edges of the box frame in Fig. 3(b) and the\nvector spin is normalized as |S|=1. In addition, the su ffixes\nof the spins in Eq. (2) take µ={1,4,7,10},ν={2,5,8,11},\nandη={3,6,9,12}, depending on which sublattice point the\nposition vector rindicates in Fig. 3(a). An important feature\nof the cuboc order is that it has the triple- qstructure, and then\nthe Fourier transformation of the classical Hamiltonian (1 )\nhas the energy minimum of the ground state at the M point\nin the momentum space. This triple- qstructure plays an es-\nsential role in analyzing soft modes of spin-wave excitatio ns.\n2J. Phys. Soc. Jpn. DRAFT\n3. Spin-Wave Analysis for the Kagome-Triangular Plane\nIn general, 3D couplings are necessary for realizing a finite -\ntemperature order with spontaneous breaking of the spin ro-\ntational symmetry. For the coupled spin tubes, however, the\ncompetition between JandJ′on the Kagome-triangular plane\nplays a central role in the formation of the cuboc order.\nThus, we first consider the Hamiltonian for the 2D Kagome-\ntriangular plane,\nH2D=J/summationdisplay\n/an}bracketle{ti,j/an}bracketri}htintraSi·Sj+J′/summationdisplay\n/an}bracketle{ti,j/an}bracketri}htinterSi·Sj, (3)\nfor which we basically discuss ground-state properties. He re,\nSin Eq. (3) is a quantum spin with the magnitude S.\nFor the construction of the spin-wave Hamiltonian, we\nshould carefully take account of the twelve-sublattice str uc-\nture; the spin quantization axis for each spin embedded in th e\ntwelve-sublattice structure should be locally defined alon g the\ndirection of the classical spin arrow in Fig. 3(b). Then, we\nperform the Holstein-Primako fftransformation for the spin of\neach quantization axis, and rewrite it as the unified spin coo r-\ndinate defined by ex,ey, andez, corresponding to the edges\nof the box frame in Fig. 3(b). The resulting spin-wave Hamil-\ntonian is written as\nH2D=H0+HS+O(S1/2), (4)\nwhereH0≡−6(J−2J′)S2Nis the classical energy and HS\ndenotes the linear-spin-wave Hamiltonian. Here, Ndenotes\nthe total number of magnetic unit cells in the system. The\nlinear spin-wave term can be represented as a standard matri x\nform,\nHS=S\n8/summationdisplay\nqX†\nqHqXq, (5)\nwhere X†\nq(Xq) is a vector array containing 24 boson operators\nand Hqis a 24×24 matrix representing the hopping ampli-\ntudes of the bosons.\nThe array of boson operators is explicitly defined as\nX†\nq≡(A†\nq,B†\nq,C†\nq,D†\nq,At\n−q,Bt\n−q,Ct\n−q,Dt\n−q),(6)\nwhere A†\nq,B†\nq,C†\nq, and D†\nqrespectively represent a set of bo-\nson creation operators carrying momentum qfor the unit tri-\nangle spins labeled by A, B, C, and D in Fig. 3. For instance,\nthe operators A†\nq(At\n−q) contain three boson operators,\nA†\nq=(a†\n1,q,a†\n2,q,a†\n3,q),At\n−q=(a1,−q,a2,−q,a3,−q),(7)\nwhere a†\nν,q(aν,q) is the boson creation (annihilation) operator\nin the momentum space with the sublattice index ν={1,2,3}\nfor the unit triangle labeled by A.18)Explicitly, we have a†\nν,q=\n1\nN/summationtext\nra†\nν(r)e−iq·r, whereris a lattice translation vector in the\nunit of the magnetic unit cell with the 12 sublattices, and th e\nmomentum qruns in the magnetic Brillouin zone.\nFor the above array of boson operators, we can write the\ntransition amplitude matrix as\nHq≡/parenleftiggPqQq\nQ†\nq¯Pq/parenrightigg\n, (8)\nwhere PqandQqdenote 12×12 matrices. Straightforwardcalculations lead to\nPq=K+αq−3γ1,q−3γ2,q−3γ3,q\n−3γ1,qK+αq−3γ3,q−3γ2,q\n−3γ2,q−3γ3,qK+αq−3γ1,q\n−3γ3,q−3γ2,q−3γ1,qK+αq, (9)\n¯Pq=K+αq−3¯γ1,q−3¯γ2,q−3¯γ3,q\n−3¯γ1,qK+αq−3¯γ3,q−3¯γ2,q\n−3¯γ2,q−3¯γ3,qK+αq−3¯γ1,q\n−3¯γ3,q−3¯γ2,q−3¯γ1,qK+αq, (10)\nQq=−3 ¯αqη1,qη2,qη3,q\nη1,q−3 ¯αqη3,qη2,q\nη2,qη3,q−3 ¯αqη1,q\nη3,qη2,qη1,q−3 ¯αq, (11)\nwhere K,αq( ¯αq),γi,q(¯γi,q), andηi,qare 3×3 matrices of the\nspin-wave hoppings in the unit of triangles. Noticing the ex -\nplicit forms of these matrices in Appendix A, we briefly de-\nscribe their meanings. First, Kcontains the diagonal coupling\nindependent of q, which corresponds to a chemical potential\nfor the spin-wave bosons. Secondly, αqrepresents the hop-\nping of the bosons within the unit triangle having the 120◦\nstructure on the tetrahedron in Fig. 3(b). Thirdly, the γand\n¯γmatrices represent the hopping of the bosons between two\nadjacent triangles among A, B, C, and D in the tetrahedron.\nFinally, ¯αqandηdescribe the matrix elements for a†\nν,qa†\nµ,q,\na†\nµ,qb†\nν,q···, oraµ,qaν,q,···, and so forth, with µ/nequalν.\nFor the non-coplanar spin order, an interesting point is tha t\nthe spin-wave hopping terms often acquire a geometric phase\nattributed to rotations of quantization axes. For the cuboc or-\nder, the geometric phase is actually included as e±iφin ¯αq,γν,q\nandην,q. Explicitly, we have\nφ=arctan(2√\n2), (12)\nwhich corresponds to the relative angle between two surface\ntriangles on the tetrahedron in Fig. 3(b). This geometric ph ase\nin the spin-wave Hamiltonian induces a nontrivial quantum\neffect on spin-wave dispersion relations.\n3.1 Spin-wave dispersion relation\nWe diagonalize the spin-wave Hamiltonian (5) with the Bo-\ngoliubov transformation,\nXq=TqX′\nq, (13)\nwhere T qis a 24×24 transformation matrix and X′\nqrepre-\nsents a set of new boson operators diagonalizing the spin-\nwave Hamiltonian. Following a general formulation in Ref.\n19, we can construct the Bogoliubov transformation matrix\nTqthrough the eigenvalue equation\ngHqTq=TqgΛq, (14)\nwhereΛqis the eigenvalue matrix whose entry is λν,qand\ng≡/parenleftiggI 0\n0−I/parenrightigg\n(15)\nspecifies the commutators of bosons. Here, Iis the 12×12\nidentity matrix. Taking account of the contribution of ±qin\nEq. (5), we obtain the spin-wave spectrum as ων,q≡2λν,q.\nWe numerically diagonalize the asymmetric matrix gHqwith\nZGEEV of LAPACK to obtain ων,qfor Eq. (5). Also, the trans-\n3J. Phys. Soc. Jpn. DRAFT\nFig. 4. (Color online) Schematic diagram of the Brillouin zone. The inner\nhexagon indicates the magnetic Brillouin zone for the twelv e-sublattice struc-\nture of the cuboc order, while the outer hexagon represents t he Brillouin zone\nfor the Kagome-triangular plane defined by Fig. 1(b). The arr ows define the\nmomentum path for spin-wave dispersion relations.\nformation matrix T qcan be basically obtained as the eigen-\nvectors of gHq. Nevertheless, the eigenvectors computed un-\nder the normalization convention of ZGEEV should be renor-\nmalized, so as to satisfy TqgT†\nq=g.\nBefore presenting numerical results, we comment on the\nBrillouin zone for the cuboc order in the Kagome-triangular\nplane. In Fig. 4, the inner hexagon represents the reduced\nBrillouin zone for the cuboc order with the broken transla-\ntion symmetry, whereas the outer hexagon represents the Bri l-\nlouin zone for the Kagome-triangular lattice. In the follow ing,\nwe basically adopt the outer Brillouin zone of the original\nKagome-triangular lattice defined by Fig. 1(a). In particul ar,\nwe use the pathΓ→K→M→Γ→K′→M′→Γin Fig.\n4 to show spin-wave dispersion relations, for which the Γ, M\nand M′points are equivalent. Here, it should be noted that the\nK′point is not equivalent to the K point, reflecting the three\nfold axis symmetry of the Kagome-triangular plane.\nIn Fig. 5, we plot the spin-wave dispersion relations for\nJ′/J=−0.1,−1.0, and−1.25. In Fig. 6, we also show\nthe lowest-energy excitation in the 2D momentum space for\nJ=−J′=1.0. In these figures, we can see that the spin-wave\nexcitations have soft modes at the Γ, M, and M′points, re-\nflecting the triple- qstructure of Eq. (2) for the cuboc order.\nAs|J′|increases, the asymmetry between the K and K′points\nbecomes significant. A higher-energy mode at the K′point\ngradually decreases, while the modes at the K points main-\ntainω∼0.5 in Figs. 5(b) and (c). At J′/J=−1.25, we find\nthat the lowest-energy branch at the K′point touchesω=0,\nwhere its low-energy behavior turns out to be ωq∼(q−qK′)2.\nThis behavior suggests that the cuboc order is destabilized at\nJ′/J=−1.25 within the linear spin-wave theory. We have\nactually checked that for J′/J<−1.25, the spectrumων,q\nformally becomes complex around the K′point.\nAs in Fig. 2, the transition line between the cuboc and in-\ncommensurate orders for the classical case is given by J=\n−J′. However, we observe no singular behavior at the clas-\nsical transition point of J′/J=−1.0, and the cuboc phase\nboundary obtained by the spin-wave theory is extended up\ntoJ′/J=−1.25. This is an interesting feature of the non-\ncoplanar spin order, where the geometric phase factor e±iφ\nmay induce a nontrivial quantum e ffect even at the linear-spin-\nwave level.20)\nHere, we would like to comment on a connection to\nDomenge’s spin-wave result in Ref. 1, where the same cuboc\nspin configuration for a similar Kagome spin system with the0.00 \u000020\u000140.8\n(a)\n(b)\n0.02.04.06.0\n(c)\n0.02.04.06.0\nFig. 5. (Color online) Spin-wave dispersion relations for the plan ar\nKagome-triangular system: (a) J=1.0 and J′=−0.1, (b) J=1.0 and\nJ′=−1.0, (c) J=1.0 and J′=−1.25. The horizontal axis is along the mo-\nmentum path defined in Fig. 4. Note that the scale of the vertic al axis in (a)\nis different from those in (b) and (c).\nnearest-neighbor couplings was investigated. A di fference be-\ntween the Kagome-triangular lattice and Domenge’s model\nin Ref. 1 is in the connectivity of the next-nearest-neighbo r\ncoupling (in the sense of the Kagome lattice); in the present\nKagome-triangular lattice, half of the nearest neighbor si tes\nare connected with the Jcoupling. Then, the hopping paths\namong sublattices A, B, C, and D in Fig. 3 have one-to-\none correspondence to the triangles of the tetrahedron in th e\nspin configuration space. Meanwhile, for Domenge’s model\nin Ref. 1, all of the nearest-neighbor sites are connected wi th\neach other. Then, we can define the 120◦structure with the op-\nposite vector spin chirality for the surface triangle of the tetra-\nhedron corresponding to the other half of the nearest-neigh bor\nbonds [e.g.,{2,4,12}spins in Fig. 3(b)]. The cuboc orders for\nthese two models may be adiabatically connected with each\nother. However, the Kagome-triangular lattice may contain\nthe minimal couplings to realize the cuboc order. We also not e\nthat in Ref. [1], the rotating frame based on the triple- qstruc-\nture is used to reduce the matrix dimension of the spin-wave\nHamiltonian, where the role of the geometric phase is not vis -\nible in the Hamiltonian level.\n4J. Phys. Soc. Jpn. DRAFT\nFig. 6. (Color online) The lowest-energy branch of spin-wave dispe rsion\nrelations for J=1.0 and J′=−1.0. The outer(green) hexagon represents the\nBrillouin zone of the planar Kagome-triangular lattice.\n4. Spin-Wave Analysis for the Coupled Spin Tubes\nWe investigate the coupled spin tube system (1) with full\n3D couplings, which can also be regarded as the stacked\nKagome-triangular system. Since there is no frustration al ong\nthec-axis direction, the staggered pattern of the cuboc or-\nder can be assumed. Then, the magnetic unit cell contains\n24 spins, implying that we deal with 24 kinds of Holstein-\nPrimakoffbosons. We write a vector array of the bosons for\nthe staggered cuboc order as\nX†\nq=(A†\nq,B†\nq,C†\nq,D†\nq,¯A†\nq,¯B†\nq,¯C†\nq,¯D†\nq,\nAt\n−q,Bt\n−q,Ct\n−q,Dt\n−q,¯At\n−q,¯Bt\n−q,¯Ct\n−q,¯Dt\n−q), (16)\nwhere ¯Aq,¯Bq,···denote sets of bosons corresponding to the\nstaggered spins of Aq,Bq,···, respectively. Then, the spin-\nwave Hamiltonian can be written as\nH3D\nS=S\n8/summationdisplay\nqX†\nqH3D\nqXq, (17)\nwhere H3D\nqis a 48×48 matrix. For the coupled spin tubes,\ntedious but straightforward calculations yield\nH3D\nq=Pq 0 QqQc\nq\n0 ¯PqQc\nqQ†\nq\nQ†\nqQc\nq¯Pq 0\nQc\nqQq 0 Pq, (18)\nwhereqis the momentum in the 3D reciprocal lattice vec-\ntor space. In this equation, Pq,¯Pq, and Qqare respectively\nthe same as Eqs. (9), (10), and (11) with Kreplaced by K3D.\nIn addition, Qcrepresents inter-layer hoppings of the bosons\nin the c-axis direction, which are given by the real diagonal\nmatrix\nQc\nq=βq 0 0 0\n0βq 0 0\n0 0βq 0\n0 0 0βq. (19)\nHere,βqis a 3×3 diagonal matrix, which is also given in\nAppendix A.\nThe cuboc spin configurations in the adjacent Kagome-\ntriangular layers have the staggered structure. This sugge sts0.01.02.03.0\n(a)\n0.02.04.06.0(b)\n(c)\n0.02.04.06.0\nFig. 7. (Color online) Spin-wave dispersion relations of the coupl ed spin\ntubes with J=1.0 and Jc=1.0 in the qc=0 sector: (a) J′=−0.1, (b)\nJ′=−1.0, and (c) J′=−1.25. The horizontal axis is along the momentum\npath defined in Fig. 4. Note that the scale of the vertical axis in (a) is different\nfrom those in (b) and (c).\nthat the matrix elements in Eq. (18) basically also have the\nsame structure within each layer. However, the relative ang le\nwhen rotating the quantization axis in the cuboc configurati on\nacquires the opposite sign due to the spin inversion, implyi ng\nthat the signs of the geometric phase factors for the two ad-\njacent layers are also alternating. Here, it should be remar ked\nthat the vector chirality on the unit triangle is invariant u nder\nthe spin inversion, which suggests that the sign of the geo-\nmetric phases/scalar spin chirality rather than the vector spin\nchirality is intrinsic to the non-coplanar spin structure o f the\ncuboc order.\n4.1 Spin-wave dispersion relation\nLet us discuss the spin-wave dispersion for the coupled spin\ntubes. The numerical Bogoliubov transformation for Eq. (17 )\nis straightforwardly constructed as in the 2D case. In the fo l-\nlowing, the exchange coupling in the c-axis direction is fixed\natJc=1.0. In Fig. 7, we show spin-wave dispersion relations\nwith qc=0 fixed, where the momentum path is the same as\nthat in Fig. 4. This is because the dispersion relations in th e\nc-direction are basically described by the usual curves for t he\nstaggered spin configurations.\nIn Fig. 7, the spin-wave dispersion relations have soft\n5J. Phys. Soc. Jpn. DRAFT\nmodes at theΓ, M, and M′points in consistent with the 2D\ncase. An important di fference from the 2D result is that the K\nand K′points turn out be equivalent for the 3D case. As was\nmentioned above, the staggered configuration of the cuboc or -\nder has the opposite sign to the geometric phase, for which th e\nroles of the K and K′points are alternated. Then, the qc=0\nmode for the staggered cuboc order is described by the super-\nposition of the spin waves attributed to the staggered config -\nurations, implying that the K and K′points in Fig. 7 become\nequivalent at the dispersion level.\nAnother important feature is that in Figs. 7(b) and (c), the\nhigher energy modes at the K and K′points are lowered, as in\nthe 2D result. In particular, we find that these modes touche\nthe zero energy at J′/J=−1.25 in Fig. 7(c), which is the same\nphase boundary as in the 2D case. However, the low-energy\nbehavior in the vicinity of the K or K′point is modified to\nων,q∼ |q−qK/K′|. In addition, the formal solution of ων,q\nforJ′<−1.25 abruptly becomes complex around the K /K′\npoint, where the staggered cuboc order is unstable. Thus, it\ncan be concluded that the transition to the incommensurate\nphase for the 3D case is modified to the first order within the\nlinear spin-wave level.\n4.2 Dynamical spin structure factor\nWe discuss the dynamical spin structure factors for the cou-\npled spin tubes, which are relevant to neutron scattering ex -\nperiments. In this paper, we consider the dynamical spin str uc-\nture factor for the net spins in the magnetic unit cell, which is\ndefined as\n˜Sxx(q,ω)≡/integraldisplay∞\n−∞/an}bracketle{t˜Sx\nq(0)˜Sx\n−q(t)/an}bracketri}hte−iωtdt, (20)\nwhere ˜Sx\nq≡/summationtext24\nν=1Sx\nν,qand/an}bracketle{t···/an}bracketri}ht indicates the canonical en-\nsemble average. The xdirection of the spin space is defined\nin Fig. 3(b). We also calculate ˜Syy(q,ω) and ˜Szz(q,ω). Tech-\nnical details are briefly summarized in Appendix B. Below,\nwe present results for a finite temperature, T/J=0.1. How-\never, note that the results for T/J=0.1 are qualitatively the\nsame as those for the ground state.\nFigure 8 shows dynamical spin structure factors in the\nqc=0 sector for J=1.0,J′=−1.0 and Jc=1.0. In Fig.\n8(a), we can see that ˜Sxx(q,ω) captures partial branches of\nthe dispersion curves in Fig. 7(b), although the Γ, M, and M′\npoints are equivalent at the dispersion level. Also, ˜Syy(q,ω) in\nFig. 8(b) exhibits strong intensity at branches di fferent from\n˜Sxx(q,ω). A similar situation can be seen for ˜Szz(q,ω) in Fig.\n8(c), where modes shifted from those in ˜Sxx/yy(q,ω) are ob-\nserved. This is partly because the dispersion curves with th e\ntripleqcontain the four modes folded by the translational\nsymmetry breaking in the ab-plane. In addition, for example,\nthe cuboc spin configurations have anisotropic behaviors in\nthexandydirections, where the π/2-rotation with respect to\nthezaxis is not compatible with the cuboc order in Fig. 3(b).\nWe think that the symmetries in the real space and the spin\nspace may cause such complex selection rules in the dynami-\ncal spin structure factors.\nIn experimental situations, the directions of the spin con-\nfiguration space are di fficult to control. In particular, CsCrF 4\nis synthesized as powder samples. Thus, the angle-averaged\ndynamical structure factors may be relevant for actual expe r-\niments. We also comment that the strong intensity observed\n0.02.04.00.02.04.00.02.04.0\n01020304050\nFig. 8. (Color online) Dynamical spin structure factors for the cou pled spin\ntubes of J=1.0,J′=−1.0, and Jc=1.0 atT/J=0.1: (a) Sxx(q,ω), (b)\nSyy(q,ω), and (c) Szz(q,ω). The relative value of the color-map intensity is\nmeaningful.\nfor higher-energy branches may be a characteristic feature for\ndetecting the cuboc order in experiments.\nAs was seen for the dispersion relations in Fig. 7, the soft-\nening at the K or K′points is characteristic to the transition\nto the incommensurate order. In Fig. 8, however, ˜Sxx(q,ω)\nhas no intensity for the lowest-energy branch at the K and K′\npoints, implying that ˜Sxx(q,ω) is not appropriate for detecting\nthe transition point. Meanwhile, ˜Syy(q,ω) or ˜Szz(q,ω) have\nstrong intensity at the midpoint of the Γand K (K′) points [i.e.\nK (K′) point of the inner hexagon in Fig.4]. Thus, we should\nsee˜Syy/zz(q,ω) rather than ˜Sxx(q,ω) to capture the signature\nof the transition to the incommensurate phase.\n5. Summary and Discussion\nWe have investigated low-energy excitations for coupled\nspin tubes with the cuboc order. First, we have pointed out\nthe importance of the Kagome-triangular lattice structure in\ntheab-plane. We then performed the spin-wave analysis for\nthe 2D Kagome triangular spin system to extract spin-wave\nexcitations, where the soft modes appear at the Γ, M, and M′\npoints, reflecting the triple- qstructure of the cuboc order. We\nhave also found that the classical phase boundary between th e\ncuboc and incommensurate ordered phases is shifted up to\nJ′/J=−1.25, where the geometric phase characteristic to the\n6J. Phys. Soc. Jpn. DRAFT\ncuboc spin structure is essential. It is an interesting rema ining\nproblem to clarify how the spin-wave interaction of higher- S\nterms affects the present linear-spin-wave results.\nWe have further examined the spin-wave excitations for the\ncoupled spin tubes or equivalently for the stacked Kagome-\ntriangular system, assuming the staggered cuboc order in th e\nc-axis direction. As shown in Fig. 7, the spin-wave excita-\ntions in the qc=0 sector also have soft modes at the Γ, M and\nM′points. In contrast to the 2D case, the spin-wave disper-\nsion relations show symmetric behavior between the K and\nK′points because of the contributions from the adjacent stag-\ngered layers where the sign of the geometric phase factors\nis also alternating. Finally, we have calculated the dynami cal\nspin structure factors ˜Sαα(q,ω) withα∈{x,y,z}for the\ncoupled spin tubes. We have then found that ˜Sαα(q,ω) have\nnontrivial x-,y-, and z-dependence, reflecting the anisotropy\nof the cuboc order in the spin configuration space. In exper-\nimental situations, thus, we should carefully take account of\nthe direction dependences of ˜Sαα(q,ω).\nFinally, we would like to comment on the relevance to ex-\nperiments associated with the cuboc order. As was mentioned ,\na neutron scattering experiment on CsCrF 4suggests the exis-\ntence of a nontrivial order below 4 K, but its details have not\nbeen identified yet.10)We believe that the present spin-wave\nresults for the cuboc order provide an important reference\nto identify the order of CsCrF 4. In addition, we should note\nthat CsCrF 4may contain the Dzyaloshinskii-Moriya interac-\ntion.7, 21)Thus, how the Dzyaloshinskii-Moriya interaction af-\nfects the present spin-wave results is an important problem .\nWe also think that our spin-wave results could be helpful for\nanalyzing interesting low-energy behaviors observed for p la-\nnar Kagome-based spin systems such as NaBa 2Mn 3F114)and\nCu3Zn(OH) 6Cl2.5)\nWe would like to thank Y . Akagi and H. Katsura for valu-\nable discussions. This work was supported by Grants-in-Aid\n26400387, 16J02724, and 17H02931 from the Ministry of Ed-\nucation, Culture, Sports, Science and Technology of Japan.\n1) J.-C. Domenge, P. Sindzingre, C. Lhuillier, and L. Pierre , Phys. Rev. B\n72, 024433 (2005).\n2) J.-C. Domenge, C. Lhuillier, L. Messio, L. Pierre, and P. V iot, Phys. Rev.\nB77, 172413 (2008).\n3) L. Messio, C. Lhuillier, and G. Misguich, Phys. Rev. B 83, 184401\n(2011). The cuboc order dealt with in this paper corresponds to\n“cuboc2” in this reference.\n4) H. Ishikawa, T. Okubo, Y . Okamoto, and Z. Hiroi, J. Phys. So c. Jpn. 83,\n043703 (2014).\n5) B. Fåk, E. Kermarrec, L. Messio, B. Bernu, C. Lhuillier, F. Bert, P.\nMendels, B. Koteswararao, F. Bouquet, J. Ollivier, A. D. Hil lier, A. Am-\nato, R. H. Colman, and A. S. Wills, Phys. Rev. Lett. 109, 037208 (2012).\n6) K. Seki and K. Okunishi, Phys. Rev. B 91, 224403 (2015)\n7) H. Manaka, Y . Hirai, Y . Hachigo, M. Mitsunaga, M. Ito, and N . Terada,\nJ. Phys. Soc. Jpn. 78, 093701 (2009).\n8) H. Manaka, T. Etoh, Y . Honda, N. Iwashita, K. Ogata, N. Tera da, T.\nHisamatsu, M. Ito, Y . Narumi, A. Kondo, K. Kindo, and Y . Miura , J.\nPhys. Soc. Jpn. 80, 084714 (2011).\n9) H. Manaka and Y . Miura, J. Korean Phys. Soc. 62, 2032, (2013).\n10) M. Hagihala, T. Masuda, and H. Manaka, private communica tion.\n11) The Kagome-triangular lattice was introduced in Ref. 4.\n12) H.-J. Koo, J. Magn. Magn. Mater. 324, 2806 (2012).\n13) K. Kawano and M. Takahashi, J. Phys. Soc. Jpn. 66, 4001 (1997).\n14) D. C. Cabra, A. Honecker, and P. Pujol, Phys. Rev. B 58, 6241 (1998).\n15) S. Nishimoto and M. Arikawa, Phys. Rev. B 78, 054421 (2008).16) T. Sakai, M. Sato, K. Okunishi, Y . Otsuka, K. Okamoto, and C. Itoi,\nPhys. Rev. B 78, 184415 (2008).\n17) T. Sakai, M. Sato, K. Okamoto, K. Okunishi, and C. Itoi, J. Phys.: Con-\ndens. Matter 22, 403201 (2010).\n18) The sublattice indices for triangles B, C, and D respecti vely correspond\ntoν={4,5,6},{7,8,9}and{10,11,12}. In addition, superscript tindi-\ncates the transpose of vectors and matrices.\n19) R. M. White, M. Sparks, and I. Ortenburger, Phys. Rev. 139, A450\n(1965).\n20) If we omit the phase factor e±iφin Eq. (5), the phase boundary corre-\nsponds to the classical one, J′/J=−1.0.\n21) I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958); T. Moriya, Phys.\nRev. 120, 91 (1960).\nAppendix A: Matrix Elements\nWe explicitly write down matrices for the spin-wave\nHamiltonians in§3 and§4. The matrices correspond to three\nspins in the triangle unit. Below, r1,r2, andr3represent the\nvectors indicating the nearest-neighboring sites in Fig. 1 (b),\nrcis the unit vector of the lattice translation in the c-axis\n(tube-leg) direction for the 3D case, and qdenotes the mo-\nmentum. The phase factor e±iφoriginates from the geometric\nphase defined by the angle between two triangle planes of the\ntetrahedron.\nK=4J−2J′0 0\n0 J−2J′0\n0 0 J−2J′\nK3D=4J−2J′+Jc 0 0\n0 J−2J′+Jc 0\n0 0 J−2J′+Jc\nαq=0 Je−iq·r3Je−iq·r2\nJeiq·r3 0 Je−iq·r1\nJeiq·r2 Jiq·r1 0\n¯αq=0 Je−iq·r3eiφJe−iq·r2eiφ\nJeiq·r3eiφ0 Je−iq·r1eiφ\nJeiq·r2eiφJiq·r1eiφ0\nβq=4Jccos(q·rc) 0 0\n0 Jccos(q·rc) 0\n0 0 Jccos(q·rc)\nγ1q=0 0 J′eiq·r1eiφ\n0 0 J′eiq·r2eiφ\nJ′e−iq·r1e−iφJ′e−iq·r2e−iφ0\nγ2q=0 J′e−iq·r1eiφ0\nJ′eiq·r1e−iφ0 J′e−iq·r3e−iφ\n0 J′eiq·r3eiφ0\nγ3q=0 J′eiq·r2e−iφJ′eiq·r3e−iφ\nJ′e−iq·r2eiφ0 0\nJ′e−iq·r3eiφ0 0\n¯γ1q=0 0 J′eiq·r1e−iφ\n0 0 J′eiq·r2e−iφ\nJ′e−iq·r1eiφJ′e−iq·r2eiφ0\n¯γ2q=0 J′e−iq·r1e−iφ0\nJ′eiq·r1eiφ0 J′e−iq·r3eiφ\n0 J′eiq·r3e−iφ0\n7J. Phys. Soc. Jpn. DRAFT\n¯γ3q=0 J′eiq·r2eiφJ′eiq·r3eiφ\nJ′e−iq·r2e−iφ0 0\nJ′e−iq·r3e−iφ0 0\nη1q=0 0 J′eiq·r1\n0 0 J′eiq·r2\nJ′e−iq·r1J′e−iq·r2 0\nη2q=0 J′e−iq·r1 0\nJ′eiq·r1 0 J′e−iq·r3\n0 J′eiq·r3 0\nη3q=0 J′eiq·r2J′eiq·r3\nJ′e−iq·r2 0 0\nJ′e−iq·r3 0 0\nNote that ¯αqisnotHermitian.\nAppendix B: Calculation of S(q,ω)\nWe briefly summarize computational details of the dynam-\nical structure factor for the net spins in the magnetic unit c ell,\nwhich is defined by Eq. (20). Using the Holstein-Primako ff\nbosons, we have\n˜Sx\nq=/summationdisplay\nνξνaν,q+ξ∗\nνa†\nν,−q+ζν(S−a†\nν,qaν,q) (B·1)\nwithinO(S0). Here,ξcontains coefficients ofO(S1/2) due to\nthe Holstein-Primako fftransformation as well as geometric\ncoefficients attributed to the orientation of spins in the mag-\nnetic unit cell. Meanwhile, ζis determined only by the orien-\ntation angle of spins. In the following, moreover, we assume\nthat the average spin of the classical order in the magnetic u nit\ncell is zero, implying\n/summationdisplay\nνζν=0, (B·2)\nwhich is actually the case for the cuboc order. Then, it is suf -\nficient to consider\n˜Sx\nq≃/summationdisplay\nνξνaν,q+ξ∗\nνa†\nν,−q (B·3)for the calculation of S(q,ω) up toO(S). Introducing a vector\narray of the coefficients,\nvt=(ξ1,···,ξ∗\n1,···), (B·4)\nwe may write\n˜Sx\nq(0)˜Sx\n−q(t)=/parenleftig\nXt\nq(0)v/parenrightig/parenleftig\nvtX−q(t)/parenrightig\n. (B·5)\nUsing the relation\nXt\nq(0)v=X†\n−q(0)v∗, (B·6)\nwe can write Eq. (B ·5) as the standard form,\n˜Sx\nq(0)˜Sx\n−q(t)=X†\n−q(0)LxxX−q(t), (B·7)\nwhere Lxx≡v∗vt. Using the Bogoliubov transformation Xq=\nTqX′\nq, we then have\n/an}bracketle{t˜Sx\nq(0)˜Sx\n−q(t)/an}bracketri}ht=/an}bracketle{tX′†\n−q(0)Mxx\n−qX′\n−q(t)/an}bracketri}ht, (B·8)\nwhere Mxx\n−q≡T†\n−qLxxT−qis Hermitian. Here, X′\n−qdefines\nBogoliubov particles diagonalizing the Hamiltonian such a s\nαν,−qandα†\nν,q. In Eq. (B·8), the diagonal elements of Mxx\n−q\ncontribute to the average. The Heisenberg equation of motio n\nyieldsαν,q(t)=αν,qe−iων,qt, etc. Then, we can perform the t\nintegral in Eq. (20) and arrive at\nSxx(q,ωq)=π/summationdisplay\nν[Mxx\n−q]ν,νfB(βων,−q)δ(ω+ων,−q)\n+π/summationdisplay\nν[Mxx\n−q]N+ν,N+ν(fB(βων,−q)+1)δ(ω−ων,−q)),(B·9)\nwhere fBis the Bose distribution function. This expression\nis useful for numerical computations. Note that in Fig. 8, we\napproximate theδ-function in Eq. (B ·9) with the Lorentzian\nof the broadening factor ǫ=0.02.\n8"    },    {        "title": "0806.3805v1.Spin_dynamics_in_point_contacts_to_single_ferromagnetic_films.pdf",        "content": "arXiv:0806.3805v1  [cond-mat.mtrl-sci]  24 Jun 2008\n/CB/D4/CX/D2 /CS/DD/D2/CP/D1/CX\r/D7 /CX/D2 /D4 /D3/CX/D2 /D8 \r/D3/D2 /D8/CP\r/D8/D7 /D8/D3 /D7/CX/D2/CV/D0/CT /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /AS/D0/D1/D7/C7/BA /C8 /BA /BU/CP/D0/CZ /CP/D7/CW/CX/D21/B8 /CE/BA /CE/BA /BY/CX/D7/D9/D21,2/B8 /C1/BA /C3/BA /CH /CP/D2/D7/D3/D21/B8 /C4/BA/CH /D9/BA /CC /D6/CX/D4/D9/D8/CT/D21/B8 /BT/BA /C3 /D3/D2/D3 /DA /CP/D0/CT/D2/CZ /D32/B8 /CP/D2/CS /CE/BA /C3 /D3/D6/CT/D2/CX/DA/D7/CZ/CX2\n1/BU/BA /CE /CT/D6/CZ/CX/D2 /C1/D2/D7/D8/CX/D8/D9/D8/CT /CU/D3/D6 /C4 /D3/DB /CC /CT/D1/D4 /CT/D6 /CP/D8/D9/D6 /CT /C8/CW/DD/D7/CX\r/D7 /CP/D2/CS /BX/D2/CV/CX/D2/CT /CT/D6/CX/D2/CV/B8/C6/CP/D8/CX/D3/D2/CP/D0 /BT \r /CP/CS/CT/D1/DD /D3/CU /CB\r/CX/CT/D2\r /CT/D7 /D3/CU /CD/CZ/D6 /CP/CX/D2/CT/B8 /BI/BD/BD/BC/BF 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/C2/BA /BU/CP/D7/D7/B8 /CF/BA /BV/BA /BV/CW/CX/CP/D2/CV/B8 /CE/BA /CC /D7/D3/CX/B8 /CP/D2/CS /C8 /BA /CF /DD/CS/CT/D6/B8 /C6/CP/D8/D9/D6/CT/B8 /BG/BC/BI/B8 /BG/BI/B4/BE/BC/BC/BC/B5/BA/BD/BG"    },    {        "title": "1607.01640v1.Quantum_dynamics_of_nuclear_spins_and_spin_relaxation_in_organic_semiconductors.pdf",        "content": "arXiv:1607.01640v1  [cond-mat.mes-hall]  6 Jul 2016Quantum dynamics of nuclear spins and spin relaxation in org anic semiconductors\nV. V. Mkhitaryan and V. V. Dobrovitski\nAmes Laboratory, Iowa State University, Ames, Iowa 50011, U SA\nWe investigate the role of the nuclear spin quantumdynamics in hyperfine-inducedspin relaxation\nof hopping carriers in organic semiconductors. The fast hop ping regime with a small carrier spin\nprecession during a waiting time between hops is typical for organic semiconductors possessing long\nspin coherence times. We consider this regime and focus on a c arrier random walk diffusion in one\ndimension, where the effect of the nuclear spin dynamics is ex pected to be the strongest. Exact\nnumerical simulations of spin systems with up to 25 nuclear s pins are performed using the Suzuki-\nTrotter decomposition of evolution operator. Larger nucle ar spin systems are modeled utilizing\nthe spin-coherent state P-representation approach developed earlier. We find that th e nuclear\nspin dynamics strongly influences the carrier spin relaxati on at long times. If the random walk is\nrestricted to a small area, it leads to the quenching of carri er spin polarization at a non-zero value\nat long times. If the random walk is unrestricted, the carrie r spin polarization acquires a long-time\ntail, decaying as 1 /√\nt. Based on the numerical results, we devise a simple formula d escribing the\neffect quantitatively.\nPACS numbers: 72.25.Dc, 75.76.+j, 85.75.-d\nI. INTRODUCTION\nOne of the most remarkable characteristics of or-\nganic semiconductors is long spin coherence times, which\nmakes these materials suitable for various applications\nin spintronics1–3and magnetotransport devices.4–12The\nreason behind this valuable property is that the light\nelements as hydrogen and carbon, from which organic\nsemiconductors are composed, have very weak spin-orbit\ninteraction. For the same reason, the hyperfine cou-\npling between the charge carrier and hydrogen nuclear\nspins is often the dominant source of the carrier spin\nscattering,7–9,13–15and the resulting spin relaxation.\nDue to the inherently present disorder, charge trans-\nport in organic semiconductors occurs via incoherent dif-\nfusive random walk overthe chargecarryingmolecules or\nπ-conjugated segments of polymers. During the waiting\ntime between two consecutive hops the carrier spin and\nthe local hydrogen nuclear spins couple by a hyperfine\ninteraction. Existing theories of hyperfine-induced spin\nrelaxation and more general spin-dependent phenomena\nin organic semiconductors approximate the hyperfine in-\nteraction by locally random static magnetic fields, acting\non the carrier spin.16–21This provides a semiclassical ap-\nproximation to the quantum spin dynamics. The distri-\nbution ofrandommagneticfields istakentobe Gaussian.\nThe hyperfine interaction strength is then controlled by\nthe mean square deviation of the Gaussian distribution,\nbhf. Experimentally established values of bhfare of order\nof few mT.\nThe semiclassical approximation neglects the action of\nthe carrier spin on the local nuclear spin environment.\nThus it is good for relatively slow nuclear spin dynamics\nand fast carrier hopping. However, the semiclassical ap-\nproximation may be inadequate if the carrierspends long\ntimes at the molecular sites, leading to sizeable changes\nin the local spin environments. This can happen, e.g.,\nif the carrier moves in effectively lower dimensions, sothat its random walk trajectory undergoes multiple self-\nintersections. In the case of one-dimensional diffusive\nrandomwalk ofa total duration t, the averagetime spent\nby a carrier on a site on its trajectory is ∼√tτ0, and in\nd= 2 dimensions this time is ∼τ0ln(t/τ0), whereτ0is\nthe average waiting time, i.e., the average time between\nconsecutive hops.22In both these cases the total time\nspent by the carrier at a site can become sufficiently long\nfor the quantum dynamics of local spin environment to\nbe important.\nIn this paper we investigate the effect of nuclear spin\nquantum dynamics on the spin relaxation in d= 1. We\nutilize numerical simulations based on a Monte Carlo\nsampling of random walk trajectories. To simulate the\nspin dynamics of small systems with up to 25 total\nnumber of nuclear spins we employ the Suzuki-Trotter\ndecomposition of evolution operator.23,24For systems\nwith larger number of nuclear spins we make use of the\ncoherent-state P-representation approach for quantum\ncentral spin dynamics.25,26\nWestudythetime decayofaveragespinpolarizationof\nacarrierspin, P(t), initiallyinjected intoanorganiclayer\nin the spin-up state [ P(0) = 1]. The spin polarization\nevolves as the carrier walks randomly over a linear chain\nofLmolecularsites, andits spininteractswith Nnuclear\nspins at each site. The decay of P(t) is controlled by\nthe dimensionless combination, η= (bhfτ0),17,20,21which\nis the average precession angle of carrier spin between\ntwo consecutive hops (we take /planckover2pi1= 1 and the electron\ngyromagnetic ratio, γe= 1, making the magnetic fields\nand the Larmor frequencies equivalent). We focus on\nsmallη≪1, characteristic for organic semiconductors\nfeaturing long spin coherence time.\nOur calculations prove that the initial, dominant de-\ncay of spin polarization, down to about P(t) = 0.05, is\ninsensitive to the dynamics of nuclear spins and can be\nobtained very accurately from the semiclassical approx-\nimation. Therefore the results of previous studies based2\non the semiclassical approach20,21remain valid for the\ninitial decay of P(t).\nThe effect of quantum dynamics of nuclear spins de-\nvelops at longer times. For relatively small LandN, we\nobserve a plateau-like long-time behavior of P(t), where\nits value is about ( LN)−1, independently from η. This\nis in sharp contrast to the semiclassical behavior. For\nfiniteNandL→ ∞we find yet another unique long-\ntime dependence, namely P(t)≈α/parenleftbig\nN/radicalbig\nt/τ0/parenrightbig−1, where\nα≈0.43. On the other hand, for finite Land increasing\nN, the long-timepolarizationdisappearsand P(t)simply\nregains the semiclassical form.\nThe paper is organized as follows. In the next Section\nwe discuss the hyperfine interaction between the carrier\nand hydrogennuclear spins, and its semiclassicaldescrip-\ntion in terms of random static magnetic fields. Our re-\nsults on the spin relaxation by a quantum nuclear spin\nbath and their comparisonwith those of the semiclassical\napproximation are presented in Sect. III. We discuss our\nresultsand provideexplanationsforthe commonfeatures\nand differences of the quantum and semiclassical results\nin Sec. IV.\nII. THE HYPERFINE INTERACTION AND ITS\nSEMICLASSICAL DESCRIPTION\nWe consider a diffusive random walk of a carrier over\na linear chain of Lmolecular sites. Accordingly, the sites\nare enumerated by the scalar coordinate, r= 1,..,L. At\neach molecular site r, the carrier spin S= 1/2 couples\ntok= 1,2,..,Nnuclear spins Ir,k= 1/2 by a hyper-\nfine interaction. We will consider an isotropic hyperfine\ncoupling, governed by the Hamiltonian\nHr=B0Sz+SN/summationdisplay\nk=1akIrk, (1)\nwhereB0is the external magnetic field along the z-axis,\nand{ak}N\nk=1are the hyperfine coupling constants. This\ndescription implies that the carrier-host molecular sties\nare identical, so that akdo not depend on r.\nQuite generally, theories of spin related phenomena in\norganic semiconductors approximate the quantum Over-\nhauser field, given by the sum in Eq. (1), by a constant\nclassical vector,\nbr=N/summationdisplay\nk=1akIrk, (2)\nwhich is not affected by the interaction.16The approx-\nimation Eq. (2) can be justified as follows. With the\ninteraction Eq. (1), the nuclear spin precession period\nscales as√\nN.27Therefore, for large None can neglect\nthe slow dynamics and consider the nuclear spins being\nstatic. For large Nit is also reasonable to approximate\nthe distribution of brby the Gaussian with the standard\ndeviation,bhf=1\n2/radicalbig/summationtext\nka2\nk.While one expects that the approximation Eq. (2) is\ngood for increasingly large N, its accuracy for finite N\nis questionable. In organic materials N≈10 is the ex-\npected number of hydrogen nuclei coupled to a carrier\nspin at a molecular site.9,17With this large N, the relax-\nation of a diffusing carrier spin can still be sensitive to\nthe quantum dynamics of nuclear spins.\nIII. CALCULATION OF SPIN RELAXATION\nCAUSED BY A QUANTUM NUCLEAR SPIN\nBATH\nIn this Section we calculate the spin relaxation of a\ncarrier performing a diffusive random walk over a linear\nchain ofLsites. At each site the carrier spin couples to\nNnuclear spins according to the Hamiltonian (1). We\nassume that the random hops occur between the near-\nest neighbor sites with equal probability. Accordingly,\nthe random waiting times have the average, τ0, uniform\nfor all sites and the waiting time distribution at each\nsite is Poissonian, N(τ) =τ−1\n0exp(−τ/τ0). For finite\nL, boundary conditions for the random walk should also\nbe specified. However, simulations with periodic or re-\nflective boundaries yield very close results, so that we\npresent the ones with periodic boundary conditions.\nOur calculation of the spin relaxation is based on the\nfollowing simple consideration. A carrier, initially in-\njected atr=r0in the spin-up state | ↑∝an}bracketri}ht, diffusively\nmoves over the available molecular sites and suffers spin\nflips. The system thus begins its evolution from the ini-\ntial spin state, |ψ(0)∝an}bracketri}ht=| ↑∝an}bracketri}ht⊗|χ∝an}bracketri}ht, where|χ∝an}bracketri}htis the initial\nwavefunction of all ( L·N) nuclear spins. In the course\nof carrier random walk with the trajectory r(t), the spin\nstate of the system evolves according to the Schr¨ odinger\nequation,i∂t|ψ(t)∝an}bracketri}ht=Hr(t)|ψ(t)∝an}bracketri}ht. Its solution can be for-\nmally written in terms of the time-ordered exponent,\n|ψ(t,[r(t)])∝an}bracketri}ht=Texp/parenleftbig\n−i/integraldisplayt\n0dt′Hr(t′)/parenrightbig\n| ↑∝an}bracketri}ht⊗|χ∝an}bracketri}ht.(3)\nBy writing [ r(t)] in the argument we emphasize that\n|ψ(t,[r(t)])∝an}bracketri}htis the spin wavevector after time t, provided\nthat the carrier underwent a random walk with the tra-\njectoryr(t). The carrier spin polarization can be defined\nasP(t) = 2Sz(t) with the bar standing for a triple av-\nerage, where the first is the quantum mechanical aver-\nage, the second is the average over the possible quantum\nstatesofthenuclearspinbath, andthethirdextendsover\ndifferent realizations of random walk trajectories. Thus\nP(t) = 2/angbracketleftBig/angbracketleftBig\n∝an}bracketle{tψ(t,[r(t)])|Sz|ψ(t,[r(t)])∝an}bracketri}ht/angbracketrightBig\n{χ}/angbracketrightBig\nrw.(4)\nThe calculation of P(t) from Eq. (4) can be car-\nried out numerically by utilizing the Suzuki-Trotter\ndecomposition23–25of the time-ordered exponent in\nEq. (3), combined with a Monte Carlo sampling of ran-\ndom walk trajectories. However, as is well known, mem-3\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s40/s97/s41/s76/s61/s53/s44/s32/s78/s61/s52\n/s32/s32\n/s80\n/s116/s47\n/s48/s32/s66\n/s48/s61/s32/s48\n/s32/s66\n/s48/s61/s32/s48/s46/s56/s98\n/s104/s102\n/s32/s66\n/s48/s61/s32/s51/s98\n/s104/s102\n/s48 /s56/s48 /s49/s54/s48 /s50/s52/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s40/s98/s41/s76/s61/s55/s44/s32/s78/s61/s51\n/s32/s32\n/s80\n/s116/s47\n/s48/s32/s66\n/s48/s61/s32/s48\n/s32/s66\n/s48/s61/s32/s48/s46/s57/s98\n/s104/s102\n/s32/s66\n/s48/s61/s32/s51/s46/s54/s98\n/s104/s102\nFIG. 1: (Color online) Spin relaxation in various applied ma g-\nnetic fields, obtained by the numerically exact simulation o f\nquantum spin dynamics. P(t) is plotted with open symbols.\nBlack lines are the results from the semiclassical approxim a-\ntion. (a): System of L= 5 sites, N= 4 nuclear spins per\nsite, with η≡(bhfτ0) = 0.1 and the hyperfine couplings,\na1= 0.83bhf,a2= 0.9bhf,a3= 1.05bhf,a4= 1.18bhf. Plotted\nare the results for relaxation in the applied fields, B0= 0\n(red),B0= 0.8bhf(green), and B0= 3bhf(blue). (b): Sys-\ntem ofL= 7 sites and N= 3 nuclear spins per site, with\nη= 0.167 and a1= 1.3bhf,a2= 1.15bhf,a3= 0.99bhf. Red,\ndark green, and violet symbols represent the applied fields,\nB0= 0,B0= 0.9bhf, andB0= 3.6bhf, respectively.\nory requirements impose severe limitations on the num-\nber of spins that can be modeled.25This is because the\nrequired memory grows exponentially with the number\nof spins. In our case this means that the total number of\nnuclear spins, ( L·N), is restricted to only few tens.\nBased on the above scheme, we have simulated the\nquantum spin dynamics of various systems with up to 25\ntotal number of nuclear spins, uniformly distributed at\nL= 3–25 molecular sites. Representative results of such\nsimulations, for a system of L= 5 sites and N= 4 spins\nper site with η≡(bhfτ0) = 0.1, and ofL= 7 sites and\nN= 3 spins per site with η= 0.167, are demonstrated in\nFigs. 1 (a) and (b), respectively. The plots also contain\nnumerical results for the same systems, found from the\nsemiclassical approximation Eq. (2) [for the details of\nsemiclassical calculation of spin relaxation see Ref. 21].\nAtsmallt, thequantumandsemiclassicalresultsinFig.1\ncoincide. This indicates that the initial decay of P(t) is\nnearly insensitive to the quantum dynamics of nuclear\nspins, so that it can be well described semiclassically. On\nthe other hand, there is a strongdiscrepancy between the\nquantum and semiclassical results at longer times. Thus\nthe true long-time behavior of P(t) is not captured by\nthe semiclassical approximation; while the latter predicts\na vanishing spin polarization, the decay of P(t) showsa long-time plateau behavior, quenching at a small but\nfinite value.\nDefinitely, the quenching of P(t) at long times, seen\nin Fig. 1, is a consequence of the nuclear spin dynamics.\nHowever, further analysis of the effect of nuclear spin dy-\nnamicsrequiresamodelingoflargerspinsystems. Tothis\nendweemploythe P-representationmethod forquantum\nspinsystemsimulation, proposedinRef. 25(see alsoRef.\n26). The method is based on a Monte Carlo sampling of\nthe density matrix in the spin coherent-sate basis, allow-\ning an efficient modeling of large quantum systems with\nthousands of bath spins. It also allows to assess the spin\ndynamics at considerably longer times.\nTo check the accuracy of the P-representation method\nfor the problem at hand, we compare its outcome for\nsystems of up to 25 nuclear spins with the results of ex-\nact simulation. The comparison clearly verifies the ef-\nficiency of P-representation approach to our problem.\nWe demonstrate typical examples of such comparison\nin Fig. 2, by plotting the quantum simulation results of\nFig. 1 together with the P-representation curves for the\nsame systems. The P-representation curves in Fig. 2 vir-\ntually coincide with the exact simulation results both in\nshort- and long-time domains, and only a minor devia-\ntion can be observed at the intermediate times.\nWith theP-representationmethod at hand we areable\nto investigate both larger spin systems and longer time\ndynamics. In particular, we examine the long-time be-\nhaviorofthe spinrelaxationin theregimeofsmall η≪1,\nwhere the relaxation is slow. Our P-representation sim-\nulations reinforce the previous conclusion that at long\ntimes the decay of P(t) is very slow, so that the spin po-\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s40/s97/s41/s76/s61/s53/s44/s32/s78/s61/s52\n/s32/s32\n/s80\n/s116/s47\n/s48/s32/s66\n/s48/s61/s32/s48\n/s32/s66\n/s48/s61/s32/s48/s46/s56/s98\n/s104/s102\n/s32/s66\n/s48/s61/s32/s51/s98\n/s104/s102\n/s48 /s56/s48 /s49/s54/s48 /s50/s52/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s40/s98/s41/s76/s61/s55/s44/s32/s78/s61/s51\n/s32/s32\n/s80\n/s116/s47\n/s48/s32/s66\n/s48/s61/s32/s48\n/s32/s66\n/s48/s61/s32/s48/s46/s57/s98\n/s104/s102\n/s32/s66\n/s48/s61/s32/s51/s46/s54/s98\n/s104/s102\nFIG. 2: (Color online) Numerically exact calculation resul ts\nfrom Fig. 1 (open symbols) are plotted together with the\nresults obtained from the P-representation method (black\nlines). The comparison confirms the high accuracy of the\nP-representation approach to this problem.4\n/s48 /s49/s48/s48/s48 /s50/s48/s48/s48 /s51/s48/s48/s48/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48\n/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48\n/s76/s61/s53/s44/s32/s78/s61/s52\n/s32/s32\n/s80\n/s116/s47/s32 /s61/s48/s46/s50/s53\n/s32 /s61/s48/s46/s49/s54/s55\n/s32 /s61/s48/s46/s49\n/s32 /s61/s48/s46/s48/s55\n/s32 /s61/s48/s46/s48/s53\nFIG. 3: (Color online) Long-time behavior of P(t) forL= 5,\nN= 4, and η= 0.25 (orange), η= 0.167 (green), η= 0.1\n(red),η= 0.07 (blue), and η= 0.05 (magenta).\nlarizationnearlyquenches toa few percentsofits original\nvalue. We also find that the value of P(t) at the long-\ntime plateau is almost insensitive to η. This is shown in\nFig. 3, where P(t) is plotted for various values of η, for\nthe system with L= 5,N= 4. Moreover, as seen in\nFigs. 1 and 2, the plateau value of P(t) is almost inde-\npendent on the external magnetic fields of the order of\nbhf. Therefore the plateau value is universal in the sense\nthat it is determined by LandN, and does not depend\nonηandB0.\nFurther numerical analysis of the long-time dynam-\nics shows that the amplitude of the long-time plateau\nofP(t) decreases both with LandN. Specifically, for\nnot too large LandNthe amplitude of the long-time\nplateau scales as ( LN)−1(excludingL= 1). Forη= 0.1,\nwe demonstrate this dependence in Fig. 4. For fixed L\nand increasing N,P(t) eventually saturates at the curve\nobtained from the semiclassical description Eq. (2) [see\nFig. 4(a)]. This saturation means a total elimination of\nthe effect of nuclear spin dynamics for large N. The de-\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53\n/s40/s97/s41/s105/s110/s99/s114/s101/s97/s115/s105/s110/s103/s32 /s78/s61/s48/s46/s49/s44/s32/s76/s61/s53\n/s32/s32\n/s80\n/s116/s47\n/s48/s32/s78/s61/s51\n/s32/s78/s61/s52\n/s32/s78/s61/s53\n/s32/s78/s61/s49/s48\n/s32/s78/s61/s50/s48\n/s32/s108/s97/s114/s103/s101/s45 /s78\n/s40/s98/s41/s105/s110/s99/s114/s101/s97/s115/s105/s110/s103/s32 /s76/s61/s48/s46/s49/s44/s32/s78/s61/s52\n/s32/s32\n/s80\n/s116/s47\n/s48/s32/s76/s61/s52\n/s32/s76/s61/s53\n/s32/s76/s61/s56\n/s32/s76/s61/s49/s48\n/s32/s76/s61/s50/s48\n/s32/s108/s97/s114/s103/s101/s45 /s76\nFIG. 4: (Color online) Dependence of the long-time plateau\nvalue ofP(t)on the numberof molecular sites, L, and number\nof spins per site, N, forη= 0.1. (a)P(t) is plotted for L= 5\nand five different Nranging from 3 to 20. For N→ ∞, the\ncurves saturate at the dashed line, which is the semiclassic al\napproximation result for the same L. (b)P(t) is plotted for\nfive different L-values ranging from 4 to 20, and N= 4 fixed.\nThe dashed line is the saturation curve, N= 4,L→ ∞./s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52\n/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52\n/s61/s48/s46/s49 /s44/s32/s32/s108/s97/s114/s103/s101/s45 /s76\n/s32/s32\n/s80\n/s116/s47/s32/s78/s61/s49\n/s32/s78/s61/s50\n/s32/s78/s61/s51\n/s32/s78/s61/s52\nFIG. 5: (Color online) Large-L saturation curves for η= 0.1\nandN= 1 (magenta), N= 2 (blue), N= 3 (green), N= 4\n(red). Black lines are plotted from Eq. (6). The statistical\nerror bars are of the order of the symbols size.\ncay regime of P(t) with a fixed small Nand increasingly\nlargeLis less obvious. The saturation curve of P(t)\nin Fig. 4(b) retains the long-time tail and thus differs\nfrom the result of semiclassical approximation qualita-\ntively. This is an indication that the long-time behavior\nofP(t) remains sensitive to the quantum dynamics of\nnuclear spins for L→ ∞.\nTo understand this effect, we investigate the large- L\nsaturation curves for different N. We infer that in this\nregime the long-time tail of P(t) slowly decays with time\nas 1/√\nt, and scales with Nas 1/N. Figure 5 illustrates\nthis dependence for η= 0.1. We further checked that\nthis result is almost insensitive to η, at least for η= 0.2\n– 0.01. However, we note that the overall amplitude of\nthe effect is small; for N= 2, for example, the 1 /√\nt\ndecay sets up at P(t)≈0.012 (see Fig. 5).\nIV. DISCUSSION\nAs we have shown above, the quantum dynamics of\nnuclear spins does not influence the initial relaxation of\nP(t) but shows up at long times, as a plateau for finite\nL·N, or as a slow decay ∼1/√\ntforL→ ∞.\nThe fact that the initial decay of P(t) is not affected\nby the nuclearspin dynamics can be understood from the\nfollowing reasoning. Defining the cumulant expansion,\nP(t) = exp/bracketleftbig/summationtext∞\nn=1Kn(t)/bracketrightbig\n, whereKn∝bn\nhf, from Eq. (4)\none can find the first non-vanishing cumulant, K2, to be\nK2(t) =−2b2\nhft/integraldisplay\n0dt1t1/integraldisplay\n0dt2GL(0,t1−t2),(5)\nwhereGL(r,t) is the Green function of the random walk\noverLsites. Equation (5) exactly coincides with the\nsecond cumulant function which can be found from the\nsemiclassicalapproximation.21This explains why the ini-\ntial relaxation is insensitive to the quantum dynamics of\nnuclear spin, even in the ultra-quantum case N= 1.5\nSimilarly to the semiclassical result,21the approxima-\ntionP(t)≈exp/bracketleftbig\nK2(t)/bracketrightbig\ncorrectly describes the dominant\nspin relaxation, so that the spin relaxation time, τS, is\nset byK2(t) and is insensitive to the nuclear spin dy-\nnamics. For large number of sites ( L≫η−2/3) the form\nP(t)≈exp/bracketleftbig\n−(t/τS)3/2/bracketrightbig\nwithτS≃τ0/η4/3follows from\nthe results of previous semiclassical treatments.20,21\nOur findings for the long-time behavior of P(t) can be\nsummarized in the formula,\nP(t)≈1\nN/parenleftBigg\n1\nL+α/radicalbig\nt/τ0/parenrightBigg\n, t≫τS,(6)\nwhereα≈0.43 is a constant. The combination in the\nbrackets of Eq. (6) is approximately the inverse number\nof sites visited by the carrier. Indeed, for small Land\nt/τ0≫L2, all theLsites are visited equally many times\nand the inverse number of visited sites is 1 /L. ForL→\n∞, on the other hand, the averagenumber ofsites visited\nby the carrier is ∼/radicalbig\nt/τ0, and thus its inverse, α//radicalbig\nt/τ0.\nThen Eq. (6) is the inverse number of nuclear spins that\ncouple to the carrier spin during a long-term diffusion.\nTo further elucidate this behavior we note that the\nHamiltonian (1) conserves the z-component of the to-\ntal spin. Therefore the initial carrier spin polarization\ndoes not average to zero but gets redistributed between\nthe carrier and nuclear spins. It is reasonable to expect\nthat for systems with smaller number of nuclear spins\nP(t)canbeessentiallynon-zeroforarbitrarilylongtimes.\nHowever, the fact that this redistribution leads to a non-\noscillatory, fixed or slowly changing P(t) at long times\nis highly nontrivial. Moreover, the peculiar dependenceEq. (6) means that the polarization becomes evenly dis-\ntributed between the carrierspin and nuclearspins which\ncouple to the carrier spin in the course of diffusion.\nIt is also remarkable that the amplitude of the tail is\nalmost independent of η, but is determined by Land\nN. This independence resembles the central spin prob-\nlem (L= 1, largeN), where the central spin polarization\nevolves into 1 /3 of its initial value regardless of the in-\nteraction strength,16,27and relaxes from this value very\nslowly.25,26,28\nIn conclusion, we have demonstrated that the quan-\ntum dynamics of nuclear spins leads to the long-time\nsteady or slowly decreasing carrier spin polarization; a\nfeature which is not captured by the semiclassical ap-\nproximation. The long-time behavior extends up to the\ntimes when other relaxation mechanisms become impor-\ntant (e.g., carrier spin-lattice relaxation times, or hydro-\ngen nuclear spin dephasing times). The effect can be\nstrong for carriers diffusing over fewer molecular sites,\ne.g., in the situation realizing in tunnel magnetoresis-\ntance experiments in organic spin valves, at the onset of\nmultiple-step tunneling.7\nAcknowledgments\nWe thank J. Shinar and M. E. Raikh for helpful dis-\ncussions. Also we are grateful to H. Terletska for reading\nthe manuscript. Work at the Ames Laboratory was sup-\nported by the US Department of Energy, Office of Sci-\nence, Basic Energy Sciences, Division of Materials Sci-\nences and Engineering. The Ames Laboratory is oper-\nated for the US Department of Energy by Iowa State\nUniversity under Contract No. DE-AC02-07CH11358.\n1W. J. M. Naber, S. Faez, and W. G. van der Wiel, J. Phys.\nD40, R205 (2007).\n2S. Sanvito, Nature Mater. 6, 803 (2007).\n3Organic Spintronics , edited by Z. V. Vardeny (CRC Press,\nHeidelberg, 2010).\n4V. Dediu, M. Murgia, F. C. Matacotta, C. Taliani, and S.\nBarbanera, Solid State Commun. 122, 181 (2002).\n5Z. H. Xiong, D. Wu, Z. V. Vardeny, and J. Shi, Nature\n(London) 427, 821 (2004).\n6Y. Sheng, D. T. Nguyen, G. Veeraraghavan, O. Mermer,\nM. Wohlgenannt, S. Qiu, and U. Scherf, Phys. Rev. B 74,\n045213 (2006).\n7J. J. H. M. Schoonus, P. G. E. Lumens, W. Wagemans,\nJ. T. Kohlhepp, P. A. Bobbert, H. J. M. Swagten, and B.\nKoopmans, Phys. Rev. Lett. 103, 146601 (2009).\n8V. A. Dediu, L. E. Hueso, I. Bergenti, and C. Taliani, Nat.\nMater.8, 850 (2009).\n9T. Nguyen, G. Hukic-Markosian, F. Wang, L. Wojcik, X.\nLi, E. Ehrenfreund, Z. Vardeny, Nat. Mater. 9, 345 (2010).\n10M. Gr¨ unewald, M. Wahler, F. Schumann, M. Michelfeit,\nC. Gould, R. Schmidt, F. W¨ urthner, G. Schmidt, and L.\nMolenkamp, Phys. Rev. B 84, 125208 (2011).\n11M. Gr¨ unewald, R. G¨ ockeritz, N. Homonnay, F. W¨ urthner,\nL. W. Molenkamp, and G. Schmidt, Phys. Rev. B 88,085319 (2013).\n12A. Riminucci, M. Prezioso, C. Pernechele, P. Graziosi, I.\nBergenti, R. Cecchini, M. Calbucci, M. Solzi, and V. A.\nDediu, Appl. Phys. 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Theor.\nPhys.58, 1377 (1977).\n24P. deVries andH.De Raedt, Phys.Rev.B 47, 7929 (1993).\n25W. Zhang, N. Konstantinidis, K. A. Al-Hassanieh, and\nV. V. Dobrovitski, J. Phys.: Condens. Matter 19, 083202\n(2007).26K. A.Al-Hassanieh, V. V. Dobrovitski, E. Dagotto, and B.\nN. Harmon, Phys. Rev. Lett. 97, 037204 (2006).\n27I. A. Merkulov, Al. L. Efros, and M. Rosen, Phys. Rev. B\n65, 205309 (2002).\n28S. I. Erlingsson and Yu. V. Nazarov, Phys. Rev. B 70,\n205327 (2004)."    },    {        "title": "1802.06911v1.Strong_enhancement_of_the_spin_Hall_effect_by_spin_fluctuations_near_the_Curie_point_of_FexPt1_x_alloys.pdf",        "content": " 1 Strong enhancement of the spin Hall effect  by spin fluctuations  near the Curie point of \nFexPt1-x alloys  \nYongxi Ou1*, D.C. Ralph1,2 and R.A. Buhrman1* \n1Cornell Universi ty, Ithaca, New York 14853, USA , \n2Kavli Institute at Cornell, Ithaca, New York 14853, USA   \n*email: yo84@cornell.edu ; rab8@cornell.edu . \nAbstract  \nRobust  spin Hall effect s (SHE ) have recently  been observed in non-magnetic heavy metal \nsystems  with strong spin -orbit interactions . These SHE are either attributed to an intrinsic band -\nstructure effect  or to extrinsic  spin-dependent  scattering  from impurities , namely side -jump or \nskew scattering. Here we report on an extraordinarily strong spin Hal l effect , attributable to spin \nfluctuations, in ferromagnetic Fe xPt1-x alloys near their Curie point, tunable with x. This results in  \na damping -like spin -orbit torque being exerted on an adjacent ferromagnetic layer that is strongly \ntemperature dependent in this transition region, with a peak value that indicates a lower bound \n0.34 ± 0.02 for the peak spin Hall ratio within the FePt. We also observe a pronounced peak in \nthe effective spin -mixing conductance of the FM/FePt interface, and determine  the spin diffusion \nlength in these FexPt1-x alloys . These results establish new opportunities for fundamental studies \nof spin dynamics and tran sport in ferromagnetic systems with strong spin fluctuations, and a new \npathway for efficiently generating strong spin currents for applications.  \n  2 The m anipulation of the magnetization in ferromagnet ic (FM) nanostructure s with pure \nspin current  densities Js has become a primary tool  in spintronics since the  demonstrations of \nnanomagnet  switching  driven by  large spin orbit torques (SOT) originating from the spin Hall \neffect  (SHE ) [1–4] in an adjacent heavy metal carrying a longitudinal electrical current density Je.  \nMost SOT efforts so far  have focused  on the utilization of large  intrinsic  spin Hall ratio s, \n, for certain  heavy metal s that are  compatible with  the requirements of  a \nsuccessful spintronics technology  [5–9]. An alternative approach to enhance  \nSH  is to introduce \ndopants into a metallic system where by strong spin -orbit interaction s can strenghten  the spin \nHall effect , either by enhanced extrinsic  spin-dependent skew  or side -jump  scattering or by the \nintrinsic  effect through a beneficial modification of the electronic band structure at the Fermi \nlevel  [10]. Typically the dopant has been a heavy element without a strong magnetic moment, e.g. \nIr, Bi, Au  [11–14], and the  result ing enhancement, typically attribu ted to  skew -scattering, has \nbeen measured by the inverse spin Hall effect or by a non -local spin accumulation technique. \nRece ntly Wei et al.  [15] have reported a moderate , but notable,  temperature -dependent \nenhance ment of the  inverse spin Hall effect  in dilute NiPd alloys , attributed to spin-fluctuation -\nenhanced skew scattering by the Ni ions in the vicinity of the ferromagnetic transition. We note \nthat NiPd is one of a class of ferromagnetic alloys that have long been know to exh ibit giant \nmagnetic moments per ferromagnetic solute, particularly in the dilute limit  [16]. Reference  [15] \nprovides strong motivation for examining the  direct  SHE  in other ferromagnetic alloys in which \nthere can be a stronger spin -orbit interaction  between the conduction electrons  and the \nferromagnetic component. Here we demonstrate  that for Fe-doped Pt alloys, FexPt1-x, the \neffective spin Hall angle as measured  directly  from the damping -like torque exerted on an \nadjacent ferromagnetic layer is incr eased  by more than a factor of 3 in the vicinity of  its Curie  3 temperature Tc in comparison to the already -substantial value it has well above Tc, and at its peak \nhas a value at least comparable to that of beta -W [5], and with a much lower electrical resistivity .  \n FexPt1-x alloys are well known for their unusual ly robust  magnetic anisotropy properties  \narising from the strong  conduction electron  spin-orbit interaction  with the Fe orbital \nmoment  [17,18] , and  also for the dependence of the magnetic state on the chemical order. For \nexample, well -ordered Fe0.25Pt0.75 exhibits antiferromagnetism , while chemical ly disorde red \nFe0.25Pt0.75 is ferromagnetic  [19,20] . Ferromagnetic Fe xPt1-x films also exhibit  quite large \nanomalous Hall effects  (AHE)  [21–23], which suggests that FexPt1-x can be a promising material \nfor the generation of spin currents by the extrinsic SHE .  \n To investigate this possibility we  prepared  multilayers containing  two different  sets of  \nFexPt1-x thin films  made  by co -deposition  at room temperature  via dc magnetron sputtering ; in \none case the nominal composition was Fe 0.15Pt0.85 and in the other Fe 0.25Pt0.75. Multilayer  stacks \nconsisting of  substrate/Ta/IrMn/Fe xPt1-x/MgO/Ta were used for thin film characterization and \nsubstrate/Ta/IrMn/Fe xPt1-x/Hf/FeCoB/Hf/MgO/Ta  stacks were used  for the SOT measurements . \nThese samples  were prepared by direct current ( dc) sputtering (wit h rf magnetron sputtering for \nthe MgO layer) in a deposition chamber with a base pressure . The dc sputtering \ncondition was 2mTorr Ar pressure. The Fe xPt1-x alloy was grown by co -sputtering from two pure \nsputtering targets (i.e. Fe and Pt targets). All sa mples in this work had a Ta(1nm) seed layer to \nprovide a smooth base layer and a Ta(1nm) top layer to provide an oxidized protection layer for \nthe stack. All samples were annealed in an in -plane magnetic field (2000 Oe) in a vacuum \nfurnace at 300C for 1 h to enhance the PMA. For measurements of the AHE and SOT, Hall bar  \ndevices with lateral dimensions of  were patterned  via photolithography and ion \nmilling (see the sample schematic in Fig. 1d) .   \n85 10 Torr\n25 60μm 4 We perform ed x-ray diffraction (XRD) measurements on two multilayer samples  with the \nlayer structures : IrMn 3(10)/Fe 0.15Pt0.85(10)/MgO and IrMn 3(10)/Fe 0.25Pt0.75(10)/MgO  (the number \nin parentheses is  the thickness in nanometers) . The IrMn 3 layer was included to provide \nantiferromagnetic pinning of the FexPt1-x layers when cooled to well below their Curie points for \nresearch that will be presented elsewhere ; the Fe xPt1-x layers are thick enough that  in the \nexperiments to be considered below the IrMn 3 does not contribute any significant SOT on the \nfree magnetic l ayer. We show  in Fig . 1a the (111) XRD peaks for the Fe 0.15Pt0.85 and the  Fe0.25 \nPt0.75 samples , and also for  separate  10 nm Pt , Fe0.50Pt0.50 and IrMn f ilms for comparison. The \n(111) peak is reasonably narrow, shift ing to higher\n2  angle  as the Fe content increases,  \nindicating a decrease in the unit cell  size with increased Fe content . As expected from the use of \nroom temperature deposition the re wa s no evidence of a (110) peak in the XRD of  these samples  \nthat would indicate significant chemical order  [24] (The small peak at  \n2 41o  in Fig.1a is due \nto the IrMn 3 base layer ). Finally  as expected for a disorde red metal  the resistivity of the films \nwas only weakly temperature dependent, decreasing by less than 10% from room temperature to \n160 K , indicating the dominance of impurity scattering. The  resistivity of the films did vary  with \nFe content, from \nPt(10) 15 cm\n  to \n0.15 0.85Fe Pt (10) 55 cm \n  to \n0.25 0.75Fe Pt (10) 75 cm \n , \nindicating an increased electron scattering rate with increased Fe content.   \n To further confirm the chemical disorder  and the ferromagnetic character of these alloys \nwe made  temperature -dependent vibrating sample magnetometry (VSM)  measurements of the \nsamples  (Fig. 1b). Both  Fe0.15Pt0.85 and Fe 0.25Pt0.75 were  found to be ferromagnetic at sufficiently \nlow temperature, with fits of the spontaneous  magnetization \n  Ms(T) to the empirical function  5 \n( ) (0) (1 ( / ) )s s cM T M T T   [25] yielding  a Curie temperature  of \n174cTK for the Fe0.15Pt0.85 \nsample and \n288cTK  for the Fe0.25Pt0.75 sample.   \n In Fig. 1 c we show the temperature dependence of the “anomalous Hall angle”   \n/xy xx\n of the Fe0.15Pt0.85(10) and Fe 0.25Pt0.75(10) sample s as measured in  a magnetic  field Hz \n= 2 kOe  applied perpendicular to the plane of the  film.  (Here  \nxx  is the resistivity in the \ndirection of current flow and \nrxy  is the transverse Hall resistivity). As can be seen in the \nFe0.15Pt0.85(10)  sample, t here is a significant AHE at high temperature that increases gradually as  \nthe temperature is decr eased toward  \n 200K  and then increases more rapidly as the magnetization \nin the film develops  as T decreases below  Tc, qualitatively as would be expected for the case of \nstrong skew scattering from the Fe ions.  In the inset of Fig. 1c, there is a similar temper ature \ndependence trend for the Fe 0.25Pt0.75(10) sample below its Curie temperature.  In the AHE what is \ndetected is the charge flow in the direction perpendicular to the plane defined by the bias current \ndirection y and the direction of the internal magneti c field z. This transverse charge flow is \naccompanied by  a diffusive spin current arising from  the spin-dependent scattering. The resulting \nVAHE, or equivalently \nrxy , scales, for the extrinsic case, with the rate of skew scattering , but  it \nalso depends on the strength of the internal magnetization of the material and its spin dependent \ncharge transport properties . This makes it challenging to quantify the underlying spin flow based \nonly on AHE  measurements.  \n To achieve better quantitativ e measurements of  the spin currents produced by an \nelectrical current in the FePt alloys we employed the harmonic response SOT technique  [26,27]  \nwhereby we measured the magnetic deflection of an ad jacent, perpendicularly magnetized \nferromagnetic thin film that occurs as the result of the  spin torque arising  from the  absor ption of   6 the transverse polarized component of the spin current emanating from the spin source, the FePt \nalloys in this case.  Such measurement s of SOT effective fields  usually only set a lower bound on \nqSH\n due to the expected  less than perfect spin transparency of the interface between the spin \nsource and spin sink  [28,29] . \n For the harmonic response SOT measurements we fabricate d two sets of FePt -based \nmultilayer samples IrMn 3(10)/Fe 0.15Pt0.85(10)/Hf(1)/FeCoB(1)/Hf(0 .35)/MgO (A), and \nIrMn 3(10)/Fe 0.25Pt0.75(10)/Hf(0.8)/FeCoB(1)/Hf(0.35)/MgO (B), where FeCoB represents \nFe60Co20B20. The amorphous Hf insertion layer (1nm and 0.8 nm for (A) and (B) respectively) \nbetween the FePt and the FeCoB  was employed to counter the de trimental effect of the fcc \ncrystal structure of the FePt on obtaining  perpendicular magnetic anisotropy (PMA) in the thin \nFeCoB layer , while the very thin (0.35 nm) Hf insertion layer between the FeCoB and the MgO \nenhanced the interfacial magnetic anisotro py energy density, strengthening the PMA  [30].  \n In Fig. 2a, we show the response of the anomalous Hall resistance of one of the \nFe0.15Pt0.85 heterostructure Hall bars (sample A) to an applied out -of-plane field Hz at different \ntemperature s between 300 K and 140 K.  The sharp field -induced switching events , with an \nincreasing coercivity  upon decreasing temperature, are from the  PMA  FeCoB layer .  When the \ntemperature is lower than the Curie temperature of Fe 0.15Pt0.85 there is also a  quasi -linear \nbackground  evident for Hz greater than the coercive fields  that is much larger than can be \nexpect ed from the ordinary Hall effect , and instead is due to  the AHE of  the in -plane magnetized \nFePt layer . The AHE resistance for sample (B) is similar  [31].  \n We determined t he damping -like (DL) and field -like (FL) effective fields  (\nDLH  and \nFLH\n respectively ) arising from the SOT  by measuring the first and second harmonic transverse  7 Hall signals \nV  and \n2V  [27], from which we obtain \n  DHL(T)=-2(¶V2w/¶HL(T))/(¶2Vw/¶HL(T)2)\n and hence \n  DHDL=(DHL+2d×DHT)/(1-4d2) , and  \n  DHFL=(DHT+2d×DHL)/(1-4d2)\n. Here \n()LTH  is the external bias field applied parallel to \n(transverse to) the current direction and \n is the ratio of the planar Hall resistance to the \nanomalous  Hall resistance  [26,27] .   \nIn Fig. 2b we show the temperature dependence s of the DL effective fiel ds for both \nsample (A)  and sample (B), plotted as a function of T/T c, with Tc determined from the fit s to the \nmagnetization of the samples (Fig. 1b)  (See  [31] for discussion of the FL SOT behavior of these \nsamples) . For sample (A) (Fe 0.15Pt0.85) the measurement is from room temperature 293 K to T = \n160 K , and for sample (B) (Fe 0.25Pt0.75) from 33 0 to 275 K. For sample (A) for which we have \nmeasurements starting around 100 degrees above Tc, we see that the  DL effective field  per \ncurrent density   is more or less constant until  T/T c ≈ 1.44 (250 K), below which it \nincreases, at first gradually , then very rapidly  reaching a peak near Tc (172K) more than 3 times \nits 293 K value . This  behavior  is dramatically different from that of the DL torque found  with \nconventional heavy metal systems  [32,33] . Below this peak  then drops off even \nmore quickly until below  T/T c = 0.92 (160 K ) the observable development of spin torque from  \nthe PMA FeCoB on the emerging strong  ferromagnetism of the in-plane polarized Fe0.15Pt0.85 \nmakes further  quantitative  harm onic response measurements untenable (see  [31] for more \ninformation).  \n As also shown in Fig. 2b the behavior of sample (B) over the same scaled temperature \nrange above and below Tc, is quite similar , with the peak value of being less that 20% \n  DHDL/DJe\n  DHDL/DJe\n  DHDL/DJe 8 different than that of sample (A) , and even less if we take into account the spin attenuati on effect \nof the different Hf spacer thickness (1 nm for sample A and 0.8 nm for sample B) where Hf has a \nspin diffusion length of approximately 1 nm  [34]. This close similarity in the values of the peak \nanti-damping spin torque is observed  despite th e 35% difference in resistivity , and 67% \ndifference in Fe concentrat ion. This is consistent with skew scattering being the dominant spin \nHall effect  in these materials, at least in the ferromagnetic transition region , but more study will \nbe needed to confirm that attribution . \n Some years ago Kondo  [35] developed a theory for the scattering of conduction electrons \nby localized orbital moments to explain an anomaly in the magnetoresistance and AHE of \nferromagnetic Ni and Fe near  their Curie point s [36], with Kondo attributing  the anomaly to \nincreasingly strong er spin fluctuations as \nT®Tc  from below . Recently Gu et al. [37] extended \nthis theory to explain results by Wei et al. [15] on inverse spin Hall effect ( ISHE ) measurements \nof NiPd alloys near  their Tc, including the effect of correlations between neighboring localized \nmomen ts. We surmise that spin fluctuations are also the origin of the strong peak in the SOT \ntorque (spin current) that we observe with the FePt alloys, although our results , in addition to \nbeing a direct measure of the Js generated by the spin Hall effect , differ from the earlier work  by \nthe strength of the effect , which we  attribute to the exceptionally strong spin -orbit in teraction in \nthe Fe -Pt system. Our results are also distinctive in that the peak in the SOT effective field \n(emitted spin current) is followed by a sharp decline  with decreasing temperature  that we \nattribute  to the effect of the internal exchange field  in the FePt  that develops as T is lowered \nbelow Tc, which , once well established , acts to quickly dephase a spin current that is polarize d in \na direction not collinear with the internal magnetization  [23].   9  If spin fluctuations are indeed the origin of the enhanced SHE in the FePt alloys near Tc, \nthen it is predicted  [38] that there  should also be an  enhance ment of  the effective interfacial spin -\nmixing conductance\nffeg  between the FeCoB and FePt alloy in the vicinity of the  latter’s  Curie \npoint, as has been recently observed with antiferromagnetic  spin sinks near their N eel point by \ninverse spin Hall measurements  [39] and spin pumping  [40]. In the latter case the interfacial \nenhancement of damping \nDaºa(tFM)-a0 , wher e \na0  is the Gilbert damping parameter for the \nbulk FM material, can be related to \nffeg  by  [41]. We do  indeed \nobserve a  pronounced peak  in magnetic dampin g of the FeCoB layer  in our sample s as they are \ncooled through the Tc of the FePt . In Fig. 3a w e show the effective spin mixing conductance,  \nffeg\n as determined by resonant line width measurements made by flip-chip field -modulated FMR , \non a  Fe0.25Pt0.75(10)/ Hf(0.25)/ FeCoB(7.3)/MgO sample  (See  [31] for details of measurements) . \nAs can be seen \nffeg  increases rapidly as T moves below Tc, and then drops abruptly by more than \na factor of three to a value  (\n-230nm ) much closer to that expected for a typical FM/Pt \ninterface  [29]. The temperature -dependent behavior observed here is distinctly different from the \ntemperature -insensitive \nffeg  in Pt/ferromagnet bilayer systems  [42]. Note that the peak in \nffeg\ndoes not occur simultaneously with the peak in which indicates that the latter’s peak \nis not just due to an enhanced  \nffeg . \n To better  quantify  the peak strength of SHE  in the FePt  alloys  and to account for the spin \ncurrent attenuation in the Hf spa cer layer , we prepared another Fe 0.25Pt0.75 heterostructure, \nIrMn 3(10)/Fe 0.25Pt0.75(10)/Hf(0.5)/FeCoB( 0.9)/Hf(0.35)/MgO,  i.e. with a thinner (0.5 nm) Hf \nspacer layer. In Table I we compare the peak \n  DHDL/DJe  value that we obtained with a Hall bar \n  DHDL/DJe 10 measurement of this sample to that previously measured  [33] for a Pt(4)/Hf(0.5)/FeCoB(1)/MgO \nsample at 293 K  (room temp erature) . Assuming the same spin current attenuation in both cases \nfrom the 0.5 nm Hf/FeCoB interface this indicates that the peak spin Hall effect in the Fe 0.25Pt0.75 \nis approximately 5.5x larger than in Pt(4). More quantitatively, with the torque -field relation \n2DL\nDL s FeCoB\neH eMtJ\n [29], we can calculate the DL spin torque efficiency  =\n0.34 0.02DL   \nfor the sample  Fe0.25Pt0.75(10)/Hf(0.5)/FeCoB( 1). Considering the attenuation of the spin current \nas it passes through the 0.5 nm of  Hf, and the likely less than ideal spin transparency of the \nHf/FeCoB interface, this only sets the lower bound  for the peak spin Hall ratio of the Fe 0.25Pt0.75 \nmaterial as \n0.34 .  \n   Another key parameter for understanding and optimizing the effectiveness of SOTs \narising from the SHE is the spin diffusion length  \ns within the material. We  obtained a measure \nof \ns  by producing a  series of PM A samples  without the IrMn layer \nFe0.25Pt0.75(tFePt)/Hf(0.8)/FeCoB( 1)/Hf(0.35)/MgO/Ta(1) where the thickness tFePt of the FePt \nalloy was varied from 2 to 10 nm.  The measured damping -like effective fields for these samples \nare plotted in Fig. 3b as a function of tFePt for two different temperatures 293  and 330 K, i.e. in \nthe near vicinity of Tc and somewhat above it. The solid lines are a fit of the function \nDHDL(tFePt)/DJe=(DHDL(¥)/DJ)(1-sech(tFePt/ls)\n  [43] to the data.  The results  at the two \ntemperatures  are quite similar,  with \ns 1.5 nm .  \n We further confirm the strength of the SHE in the FePt alloys with current -induced \nswitching measurements. In Fig. 4 we show  the switchi ng behavior of sample (B) as measured at \n293K\n, in close vicinity to  Tc. The direction of current -induced switching is reversed upon  11 changing the sign of a small in -plane applied magnetic field, in a way that is characteristic of \nantidamping torque SHE switching  [44]. The switching current density for this case was \n626 10 A/cm\n. \n In summary, we have studied the spin -orbit torques resulting from the SHE in chemically \ndisordered FePt alloys, above and through their ferromagnetic transition points Tc. The SOTs \nexerted by these  materials on an adjacent FeCoB thin film exhibit a striking temperature \ndependence in which the DL SOT displays a strong maximum in the vicinity of  Tc. We attribute \nthis pronounced SOT behavior to spin fluctuation enhancement of the spin Hall effect  arising \nfrom the strong spin -orbit interaction between the conduction electrons  and the localized Fe \nmoments. There is also a strong T-dependent enhancement of the effe ctive spin -mixing \nconductance of the FeCoB/FePt interface that we similarly attribute to spin  fluctuations in the \nFePt ferromagnetic transition region . The peak strength of the DL SOT indicates an \nexceptionally large  spin Hall angle , > 0.34 near the Curie point of the FePt  alloys . We also \nrealized current -induced magnetization switching by the DL SOT in close vicinity to  the Curie \npoint of these ferromagnetic FePt alloys and measured the spin diffusion length to be quite \nsimilar, ≈ 1. 5 nm, both above and in close vicinity to  Tc. This fluctuation enhanced spin Hall \neffect, which is tunable through the composition of the FePt alloy, provides new opportunities \nfor the study of spin -dependent scattering and transport in systems with very strong spin -orbit \ninteractions, and for applicat ions where a very strong spin current from a relatively low \nresistivity material can be particularly beneficial.  \n \n \n  12 Acknowledgements  \nY.O. thanks Shengjie Shi for the assistance in low temperature flip -chip FMR measurements. \nThis research was supported by ONR (N000014 -15-1-2449) and by NSF/MRSEC (DMR -\n1120296) through the Cornell Center for Materials Research (CCMR), and  by NSF through use \nof the Cornell NanoScale Facility, an NNCI member (ECCS -1542081).  \nCompeting Financial Interests  \nThe authors  hold patents and have patent applications filed on their behalf regarding some \naspects of the spin -orbit torque research  discussed  in this report . \n   13 REFERENCES  \n[1] I. M. Miron, G. 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Lett. 107, 046601 (2011).  \n[43] P. M. Haney, H. -W. Lee, K. -J. Lee, A. Manchon , and M. D. Stiles, Phys. Rev. B 87, \n174411 (2013).  \n[44] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett.  17 109, 096602 (2012).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  18 Figure 1   \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nd \n38 39 40 41 420.00.51.0\n  Normalized count\n2 Pt\n Fe0.15Pt0.85\n Fe0.25Pt0.75\n Fe0.50Pt0.50\n IrMna\n150 200 250 3000.000.050.100.150.20\n250 275 30002Fe0.25Pt0.75AH anlge (x10-3)\nT (K)\n  anomalous Hall angle (x10-3)\nT (K)Fe0.15Pt0.85c\n0 100 200 300 4000100200300400b\nMs(T)=Ms(0)[1-(T/Tc)]\n  \n Fe0.25Pt0.75\n Fe0.15Pt0.85Ms  (emu/cm3)\nT (K) 19 Figure 1 . (a) XRD measurements on the samples: IrMn 3(10)/F 0.50Pt0.50(10)/MgO, \nIrMn 3(10)/Fe 0.25Pt0.75(10)/MgO and IrMn 3(10)/Fe 0.15Pt0.85(10)/MgO, and two control samples: \nIrMn 3(10)/Pt(1)/MgO and  IrMn 3(10)/Pt(8)/MgO. ( b) Temperature dependent VSM \nmeasurements on the samples IrMn 3(10)/Fe 0.25Pt0.75(10)/MgO and \nIrMn 3(10)/Fe 0.15Pt0.85(10)/MgO. The dashed lines are fits to the empirical equation\n( ) (0) (1 ( / ) )s s cM T M T T  \n. (c) Temperature dependence of t he anomalous Hall angle of the \nsample s IrMn 3(10)/Fe 0.15Pt0.85(10)/MgO  (main) and IrMn 3(10)/Fe 0.25Pt0.75(10)/MgO  (inset).  (d) \nSchematic of the Hall bar device.  \n \n \n \n \n \n \n \n \n \n \n  20 Figure  2 \n \n \n \n \n \n \n \n \n \nFigure 2 . (a) Temperature -dependent AHE resistance of sample (A) Fe 0.15Pt0.85. (b) Dampinglike \neffective fields of sample (A) Fe 0.15Pt0.85 and (B) Fe 0.25Pt0.75 as a function of normalized \ntemperature . Thei r temperature dependent magnetizations are also plotted here for \ncomparison.  \n \n \n \n \n/cTT\n-100 0 100-0.06-0.030.000.030.06 Fe0.15Pt0.85 293K\n 250K\n 210K\n 183K\n 140K\n  Rxy ()\nHz (Oe)a\n0.0 0.5 1.0 1.5 2.00200400600\nHDLJe   x10-6 Oe/(A/cm2) Ms (emu/cm3)\nT/TCb\n0246\n Fe0.25Pt0.75\n Fe0.15Pt0.85\n  21 Figure 3  \n \n \n \n \n \n \n \n \nFigure 3 . (a) The effective spin mixing conductance of an in -plane magnetized \nFe0.25Pt0.75(10)/Hf(0.25)/FeCoB(7.3) sample  as determined by  a flip-chip FMR measurement of \nthe damping parameter for the FeCoB r esonance. The temperature dependence of the DL \neffective field of sample (B) is also plotted here for comparison. ( b) Spin diffusion length \nmeasurement of the samples Fe 0.25Pt0.75(t)/Hf(0.8)/FeCoB( 1). \n \n \n \n \n0 3 6 9 120246810\n330K=1.4 nm293K=1.5 nm\n  \n 293 K\n 330 KHDL/Je x10-6 Oe/(A/cm2)\nFePt3 thickness (nm)b\nFe0.25Pt0.75\n0.8 0.9 1.0 1.1 1.2060120180\nHDLJe   x10-6 Oe/(A/cm2) g\neff (nm-2)\nT/Tca\n02468\n Fe0.25Pt0.75 22 Figure 4 \n \n \n \n \n \n \n \n \nFigure 4 . Current -induced magnetization switching of sample (B) IrMn 3(10)/Fe 0.25Pt0.75 \n(10)/Hf(0.8)/FeCoB(1 )/Hf(0.35)/MgO at room temperature under an external magnetic field \nalong the current direction.  \n \n \n \n \n \n \n-8 -4 0 4 8-0.08-0.040.000.040.08\n  Rxy ()\nIdc (mA)Hext= +100 Oe T= 293Ka\n-8 -4 0 4 8-0.08-0.040.000.040.08b\nT= 293K Hext= -100 Oe\n  Rxy ()\nIdc (mA) 23 Table I  \nSample  Fe0.25Pt0.75(10)/Hf(0.8)/FeCoB(1)  Fe0.25Pt0.75(10)/Hf(0.5)/FeCoB( 1) Pt(4)/Hf(0.5)/FeCoB(1)  \nDL effective field  \n6210 Oe/(A/cm )\n 5.6 12.2 2.3 \nReference  This work  This work  Ou et al.  [33] \n \nTable I: Comparison of the dampinglike effective field s as measured at 293 K  for two \nFe0.25Pt0.75/Hf/FeCoB samples and a Pt/Hf/FeCoB sample .  \n \n \n \n \n \n \n \n \n \n \n  24 Strong enhancement of the spin Hall effect by spin fluctuations near the Curie point of \nFexPt1-x alloys -Supplementary Materials  \nYongxi Ou1*, D.C. Ralph1,2 and R.A. Buhrman1* \n1Cornell Universi ty, Ithaca, New York 14853, USA , \n2Kavli Institute at Cornell, Ithaca, New York 14853, USA   \n*email: yo84@cornell.edu ; rab8 @cornell.edu . \n \nTable of contents:  \nS1. Temperature -dependent anomalous Hall measurements of the PMA sample \nFe0.25Pt0.75(10)/Hf(0.8)/FeCoB(1)Hf(0.35)/MgO  \nS2. Spin-orbit -torque effective fields arising from the FePt alloys as determined by the harmonic \nresponse technique  \nS3. Second harmonic signals at low temperatures  \nS4. FMR measurement of FeCoB magnetic damping for in–plane magnetized Fe Pt/Hf/FeCoB \nmultilayers , and  determination of  the effective spin mixing conductance  \n \n \n \n \n \n \n \n \n  25  \nS1. Temperature -dependent anomalous Hall me asurements of  the PMA sample Fe0.25Pt0.75 \n(10)/Hf(0.8)/FeCoB(1)/ Hf(0.35)/ MgO  \n \n \n \n \n \n \n \n \n \n \n \nS1. Anomalous Hall resistance of IrMn 3(10)/  Fe0.25Pt0.75 (10)/Hf(0.8)/FeCoB(1)/ Hf(0.35)/ MgO  at \nthree different temperatures near and below the Curie temperature Tc = 288 K of the Fe 0.25Pt0.75. \n  \nFigure S1 shows anomalous Hall  (AH)  resistance measurements of a Hall bar of the  \nsample (B)  heterostructure  IrMn 3(10)/Fe 0.25Pt0.75(10)/Hf(0.8)/FeCoB(1)/ Hf(0.35)/ MgO  as a \nfunction of an external magnetic field applied perpendicular to the plane of the sample, as \nmeasured from room temperature (293K) to 260K. Just as for the data from sample (A) discussed \nin the main text, these anomal ous Hall signals have contributions from both the perpendicularly -\nmagnetized FeCoB(1nm) layer and the in -plane magnetized Fe 0.25Pt0.75(10nm) layer. For the \nPMA FeCoB layer, the coercive field increases as the temperature T is reduced but there is only a \nminimal increase in its contribution to the AH resistance. The Fe 0.25Pt0.75 layer contributes a \nbackground that is linear in applied field, with a slope that increases in magnitude as T is \ndecreased below Tc due to the increasing magnetization of the Fe 0.25Pt0.75 layer.  \n-100 -50 0 50 100-0.10-0.050.000.050.10\n  Rxy ()\nHz (Oe) 293K\n 275K\n 260K 26 S2. Spin -orbit -torque effective fields  arising from the FePt alloys  as determined by the \nharmonic response  technique  \n \n \n \n \n \n \n \n \n \nS2. Measured values of the damping -like and field -like effective fields as a function of \ntemperature for samples (a) Fe 0.15Pt0.85(10)/Hf(1)/FeCoB(1) and (b) Fe 0.25Pt0.75 \n(10)/Hf(0.8)/FeCoB(1). Also indicated are the corresponding values of magnetization as a \nfunction of temperature for the FePt layers.  \n  \nIn addition to the damping -like effectiv e fields discussed in the main text, we also \ndetermined field -like effective fields using the harmonic response technique, for both samples (A) \nand (B), as shown in Fig. S2a and Fig. S2b. To illustrate the corresponding Curie temperatures, \nwe also plot in both panels the temperature dependence of the magnetization of the appropriate \nFePt alloy, determined by VSM. In Fig. S2a, for the Fe 0.15Pt0.85 sample, the field -like effective \nfield decreases as temperature is reduced from 293 K, consistent with the tempe rature \ndependence of the field -like term observed previously in pure Pt  [1], except that there is a small \npeak at approximately 170K, coincident with the large  peak in the damping -like field. We \nattribute both peaks to a maximum in the spin current density produced by the FePt alloy. For \nsample (B) the field -like terms is much stronger overall than that for sample (A), but it also has a \n0 100 200 3000100200300400\n|H/J| x10-6 Oe/(A/cm2) Ms (emu/cm3)\nT (K)Fe0.15Pt0.85(10)/Hf(1)/FeCoB(1)\n0.02.55.0\n Damping-like\n Field-like\n a\n0 100 200 300 4000200400600b Fe0.25Pt0.75(10)/Hf(0.8)/FeCoB(1) Ms (emu/cm3)\nT (K)0246 Damping-like\n Field-like\n|H/J| x10-6 Oe/(A/cm2)  27 local maximum at 293K, co incident with that of the damping -like field. We attribute the more \ncomplicated behavior for the field -like effective field evident in Fig. S2b to temperature -\ndependent changes in the strength of perpendicular anisotropy for the FeCoB layer in sample (B) \n-- the anisotropy field decreases from 1693 Oe to 909 Oe as T increases from 293K to 330K.  As \nhas been previously reported  [1] there is in general an apparent inverse correlation between the \nstrength of the field -like term for perpendicularly magnetized HM/FM samples as measured by \nthe harmonic response technique and the FM anisotropy field  . \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  28 S3. Second harmonic signals at low temperatures  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nS3. (a) and (b) The second harmonic signals of the samples IrMn(10)/Fe 15Pt85(10)/MgO and \nIrMn(10)/Fe 15Pt85(10)/Hf(1)/FeCoB(1)/MgO in the vicinity of the Curie temperature. The a b \nc \n-2000 -1000 0 1000 2000-300-1500150300\n  V2\nxy (nV)\nHL (Oe) 172 K\n 150 K\n 140 K\n-1600 -800 0 800-40-2002040\n  V2\nxy (nV)\nHL (Oe) Ta(1)/IrMn(10)/Fe15Pt85(10)/MgO\n Ta(1)IrMn(10)/Fe15Pt85(10)/Hf(1)/FeCoB(1)/MgO\nT=210 K\n-1600 -800 0 800-40-2002040\nT=180 K\n  V2\nxy (nV)\nHL (Oe) Ta(1)/IrMn(10)/Fe15Pt85(10)/MgO\n Ta(1)IrMn(10)/Fe15Pt85(10)/Hf(1)/FeCoB(1)/MgO 29 straight lines are linear fit to the data. (c) Second harmonic voltage for sample (A) Fe 0.15Pt0.85 \n(10)/Hf(1)/FeCoB(1) below its Curie temperature from 172K to 140K.  \n  \nIn order to confirm that the harmonic measurements in our samples indeed measure the \nspin torque exerted on the perpendicularly magnetized FeCoB layer due to the spin cur rent from \nthe FePt, instead of a SOT on the FePt exerted by the IrMn layer, we measured the second \nharmonic signal of a sample without the FeCoB layer, IrMn(10)/Fe 15Pt85(10)/MgO, in the \nvicinity of the Curie temperature and compared it with the signal of t he sample with the FeCoB \nlayer as in the main text. The results under the longitudinal field sweep are shown in Fig. S3 (a) \nand (b). It can be clearly seen that in the sample without the FeCoB layer, there is negligible \nsecond harmonic signal, while the sa mple with FeCoB shows an obvious field -dependent signal. \nThis is also true in the transverse field sweep (not shown here). These results demonstrate that \nthe harmonic signals measured in Fig. 2b in the main text are indeed from the perpendicularly \nmagnetiz ed FeCoB layer.   \nAs mentioned in the main text, when the samples are cooled to well below the Curie \ntemperature of the FePt alloys, the emergence of strong ferromagnetism in the FePt layer makes \nit difficult to measure the spin -orbit effective fields usin g the harmonic response technique. In \nFig. S3 (c) we show as an example the second harmonic voltage for sample (A) measured from \n172K down to 140K. For the higher temperature (172K) measurement the data have a simple \nlinear dependence on applied in -plane m agnetic field, as expected from the harmonic response \ntechnique for a PMA (FeCoB) system in response to spin -orbit torque  [2]. The two parallel \nbranches correspond to the up and down magnetization states of the FeCoB layer. This indicates \nthat at 172K, the dominant signal in the second harmonic response is the usual Hall signal from \nthe FeCoB layer. When the heterostructure is cooled to a lower tem perature (150K, then to 140 \nK), there is an additional signal near zero magnetic field that increases in magnitude as \ntemperature is reduced. This extra second harmonic signal, which adds to the FeCoB signal, \narises from the now -ferromagnetic Fe 0.15Pt0.85 layer and is generated on account of the planar \nHall effect in the Fe 0.15Pt0.85 layer as a response to the torques from the other layers. The \nsignature of this effect is a diverging \n1/LH  field-dependence  [2] for the magnitude of the Hall  30 signal near HL = 0, due to deflections of the Fe 0.15Pt0.85 magnetization within the sample plane, \nthat is distinctly different from the linea r field dependence of signals due to the PMA FeCoB \nlayer. One can still clearly see the switching of the FeCoB in the signal at around 700 Oe for all \ntemperatures in Fig. S3 (c), but the strongly nonmonotonic field dependence from the Fe 0.15Pt0.85  \nlayer o bscures the linear signal from the FeCoB layer. As a result, the more -complicated second \nharmonic signal can no longer be used to make quantitative measurements of spin -orbit torques \nacting on the PMA FeCoB layer. For the 150K and 140K data at fields above  1000 Oe, the \nsecond harmonic signals also begin to deviate from the small angle approximation  [3], which is \nbeyond the scope of the work discussed here.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  31 S4. FMR measurement of FeCoB magnetic damping for in–plane magnetized \nFePt/Hf/FeCoB multilayers , and  determination of the effective spin mixing conductanc e. \n \n \n \n \n \n       \n \n \n \n \nS4. (a) Resonance linewidth for sample Fe 0.25Pt0.75 (10)/Hf(0.25)/FeCoB(7.3) as a function of \nmicrowave frequency at 293K. (b) Magnetic damping as a function of the reciprocal of the \nFeCoB thickness for a series of Fe 0.25Pt0.75(10)/Hf(0. 25)/FeCoB( tFeCoB ) samples.  \n  \nWe use d a flip -chip technique to measure the ferromagnetic resonance signals for a  series \nof Fe0.25Pt0.75(10)/Hf(0.25)/FeCoB( tFeCoB )/MgO chips, where tFeCoB   ranged from 2.5nm to 7.3nm.  \nIn this technique  a microwave waveguide optimized for transmission in the 1 -20 GHz range \ncarries a 1 0 dBm rf power generated by a signal generator (Agilent E8257). The sample is placed \non top of this waveguide such that the magnetic layers face the waveguide. A dc magnetic  field \nis scanned using an external electromagnet to detect the resonance condition. A small ac field \ngenerated by Helmholtz coils, which provides an ac signal for lock -in detection , is added to the \ndc bias field . When the resonance condition is satisfied,  microwave power is absorbed into the \nuniform precession mode. The changes in the absorbed power (d P/dH) are detected using a \nrectifying diode, at the ac field modulation frequency.  \n0 3 6 9 12 15 18050100150\n  Linewidth (Oe)\nFrequency (GHz)a\n293 K\n0.0 0.1 0.2 0.3 0.40.000.010.020.030.040.05\n293 K\n  \n1/tFeCoB (nm-1)b 32 Figure S 4a shows the resonance linewidth as measured at 293 K as a functio n of the \nmicrowave frequency  for the Fe0.25Pt0.75(10)/Hf(0.25)/ FeCoB(7.3)/MgO sample, as obtained by \nfitting the field -swept signal to the derivative of a  Lorentzian . The slope of the  linear fit to these \npoints gives us the magnetic damping of the free la yer. Repeating this measurement for samples \nwith different tFeCoB  (Fig. S4b) provides the data needed to obtain \na0  and \nffeg  via the spin \npumping theory prediction, \n  [4]. The linear fit in Figure S4b gives \n00.004 0.001\n and \n2\neff65 2nm g  . The temperature dependence of \nffeg  was then \ndetermined by measuring the enhanced damping via the resonance linewidth of the \nFe0.25Pt0.75(10)/Hf(0.25)/ FeCoB(7.3)/MgO sample as a function of T. Linewidth measurements \nof a FeCoB layer without an adjacent FePt film showed no significant difference in linewidth \nover the same temperature range.  \n \n \n \n \n \nReference:  \n[1]  Y. Ou, C. -F. Pai, S. Shi, D. C. Ralph, and R. A. Buhrman, Phys. Rev. B 94, 140414 \n(2016).  \n[2]  M. Hayashi, J. Kim, M. Yamanouchi, and H. Ohno, Phys. Rev. B 89, 144425 (2014).  \n[3]  K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y. Mokrousov, S. Blügel, S. Auffret, O. \nBoulle, G. Gaudin, and P. Gambardella, Nat. Nanotechnol. 8, 587 (2013).  \n[4]  Y. Tserkovnyak, A. Brataas, and G. Bauer, Phys. Rev. B 66, 224403 (2002).  \n "    },    {        "title": "1311.1979v2.Ultra_long_spin_decoherence_times_in_graphene_quantum_dots_with_a_small_number_of_nuclear_spins.pdf",        "content": "arXiv:1311.1979v2  [cond-mat.mes-hall]  8 Jan 2014Ultra long spin decoherence times in graphene quantum dots w ith a small number of\nnuclear spins\nMoritz Fuchs,1John Schliemann,2and Bj¨ orn Trauzettel1\n1Institut f¨ ur Theoretische Physik und Astrophysik,\nUniversit¨ at W¨ urzburg, D-97074 W¨ urzburg, Germany\n2Institut f¨ ur Theoretische Physik, Universit¨ at Regensbu rg, D-93053 Regensburg, Germany\n(Dated: August 14, 2018)\nWe study the dynamics of an electron spin in a graphene quantu m dot, which is interacting with\na bath of less than ten nuclear spins via the anisotropic hype rfine interaction. Due to substantial\nprogress in the fabrication of graphene quantum dots, the co nsideration of such a small number\nof nuclear spins is experimentally relevant. This choice al lows us to use exact diagonalization to\ncalculate the long-time average of the electron spin as well as its decoherence time. We investigate\nthe dependence of spin observables on the initial states of n uclear spins and on the position of\nnuclear spins in the quantum dot. Moreover, we analyze the eff ects of the anisotropy of the hyperfine\ninteraction for different orientations of the spin quantiza tion axis with respect to the graphene plane.\nInterestingly, we then predict remarkable long decoherenc e times of more than 10ms in the limit of\nfew nuclear spins.\nPACS numbers: 76.20.+q, 76.60.Es, 85.35.Be, 03.65.Yz, 81. 05.ue\nI. INTRODUCTION\nIn recent years, spin qubits hosted in solid state\nnanostructures have been under extensive research due\nto their possible applications in quantum information\nprocessing and computation. Among the host mate-\nrials of spin qubits, quite different approaches can be\nfound, forinstance, III-V-semiconductorandcarbonnan-\notube quantum dots1(QD) as well as nitrogen vacan-\ncies in diamond2. These host materials show promis-\ning prospects but, unfortunately, also come with certain\ndrawbacks.\nA precise control of the qubit state is the major advan-\ntageofIII-V-semiconductorQDs basedonAl(Ga)As het-\nerostructures. Preparation and readout of the qubit with\nhigh fidelity via electrostatical gates has been demon-\nstrated in many ground-breaking experiments3–13. How-\never, the disadvantageofthis materialsystem is the pres-\nenceofmanynuclearspinsinherenttotheatomsofgroup\nIII and group V elements of the periodic table. These\nnuclear spins give rise to a fast decoherence of the elec-\ntron spin14,15. Elaborate design of experiments includ-\ning pulse sequences and methods to polarize the nuclear\nspins16–21such as dynamic nuclear polarization may help\nto compensate the effect of the spin bath. Nevertheless,\nit seems desirable to reduce the number of nuclear spins\nwhich can be achieved on the basis of other host materi-\nals.\nObvious candidates are carbon and silicon, since their\nspin carrying isotopes have only very low natural abun-\ndances of about 1% and 5%, respectively. In silicon, the\nqubits can be fabricated22,23either using donor impuri-\nties or by confining a single electron via electrostatical\ngates. However, a controlled localization of the donor\nimpurities is still a challenging task and electrostati-\ncally confined Si QDs often involve nanostructures with\nothermaterialslikeGe, which potentiallyintroduceaddi-tionalnuclearspins23. CarbonbasedQDscanbe realized\nby confining an electron spin in carbon nanotubes24–31\n(CNT) via electrical gates allowing for a good control\nin the few electron regime. However, the curvature of\nthe CNTs gives rise to a sizable spin-orbit coupling, yet\nanother intrinsic source of decoherence to the electron\nspin. A different approach to a carbon based QD is the\nuse of nitrogen vacancies in diamonds2,32–35, which show\ntremendously long coherence times. Unfortunately, con-\ntrol and readout of the qubit have to be done optically,\nwhich is disadvantageous for the realization of future on-\nchip electric circuits.\nThese examples illustrate a more general issue of de-\nsigning qubits, where an easy (electric) control and scal-\nability seem to compete with noiseless environmentsand,\nhence, longdecoherence times. A system potentially pro-\nviding the best of both worlds is a graphene QD36,37,\nwhich offers very interesting electronic properties38and\na small spin-orbit coupling39–42, as well as the possibil-\nity to control the number of nuclear spins by isotopic\npurification25,43,44. Moreover, the hyperfine interaction\nbetweentheremainingnuclearspinsandtheelectronspin\nis much smaller than in GaAs orSi. Additionally, the hy-\nperfine interaction in grapheneis anisotropicwhich could\nprovide interesting applications as we discuss at the end\nof this article.\nExperimentally, graphene QDs are, for instance, re-\nalized by confining electrons with gates in bilayer\ngraphene45,46and graphene nanoribbons47,48, respec-\ntively, or by etching the QD structure out of graphene\nflakes49–59. Typical diameters are of the order of tens to\nhundredsof nanometersresultingin K= 15toK= 1500\nnuclear spins assuming a natural abundance of spin car-\nrying13C of 1%. Thus reducing the abundance of13C\nby only two orders of magnitude leads to very small spin\nbaths even in the case of rather large QDs. Recently,\nultra small graphene QDs with diameters in the 1nm2\nrange were made by electroburning60. Altogether, these\nconsiderations show that the study of few nuclear spin\nmodels with K <10 as considered in this work is highly\nrelevant for future research in the field.\nIn this paper, we aim to set the basis for forthcoming\ninvestigations of the spin dynamics in graphene nanos-\ntructures. Besides quantum information theory, espe-\ncially ongoing research on magnetism on edges61–66and\nvacancies67,68in graphene can benefit from a detailed\nknowledge of the properties of the anisotropic hyperfine\ninteraction (AHI). Moreover, we intend to complement\nour previous analytic study69of the electron spin dy-\nnamics. Considering a large nuclear spin bath, we in-\nvestigated the coherence of the electron spin in a non-\nMarkovianapproachusingageneralizedmasterequation.\nIn this work, however, we were limited to large external\nmagnetic fields in order to justify the perturbative treat-\nment of the hyperfine interaction.\nSincewerestrictourselvestolessthantennuclearspins\nin the present work, we can apply exact diagonalization\nto the hyperfine Hamiltonian, which offers a powerful\ntool to investigate the dynamics of the electron spin for a\nwide parameter regime14,70–77. In particular, we analyze\nthe role of the number of nuclear spins K, their position\nwithintheQD,aswellastheirinitialspinstate. Thereby,\nwe use the long-time average /an}bracketle{tSz/an}bracketri}htTof the longitudinal\nelectron spin and its decoherencetime TDto quantify the\ninfluence of these different aspects. Moreover, we investi-\ngate the dependence of /an}bracketle{tSz/an}bracketri}htTandTDon the orientation\nofthe spin quantizationaxiswith respecttothe graphene\nplane. For the long-time average, we find a continuous\ncrossoverfrom ainitial state dominated regimefor K <5\nto a regime more affected by the configuration of the nu-\nclear spins for K >6 where the relative positions of the\nnuclear spins with respect to each other matter. As we\nwill show below, this behavior can be understood by an\nanalysis of the Hilbert space dimensions as well as of the\nspatial distribution of the nuclear spins in the QD. Be-\nsidesthisregimechange,agrowingnuclearspinbathsup-\npresses fluctuations around the long-time average more\nand more effectively, while the average itself is almost\nconstant for all K <9 with/an}bracketle{tSz/an}bracketri}htT≈/planckover2pi1/4 in the out-of-\nplane orientation and /an}bracketle{tSz/an}bracketri}htT≈0 in the in-plane case. By\nresolving the orientation dependence in more detail, we\nfind good agreement with /an}bracketle{tSz/an}bracketri}htT(β) =/an}bracketle{tSz/an}bracketri}htT(0)·cos(β)2,\nwhereβ= 0 andβ=π/2 correspond to the out-of-plane\nand in-plane orientation, respectively, cf.Fig. 1 (a).\nEvidently, the decoherence time TDstrongly depends\non the number of nuclear spins K. We observe that the\nconfigurationofthe nuclear spins is decisive even for very\nsmall numbers of nuclear spins, whereas the initial states\nplay only a minor role. Depending on the relative po-\nsitions of the nuclei, the decoherence times may deviate\nover several orders of magnitude. This behavior can be\ntraced back to changes of the spectrum of eigenvalues of\nthe full Hamiltonian. Moreover, the decoherence times\nsignificantly differ between out-of-plane and in-plane ori-\nentation. For K= 3 andβ= 0, the majority of inves-tigated configurations show very long decoherence times\nabove 10ms where, in many cases, even no decoherence\natall wasfound. For β=π/2, in contrast, wealwaysfind\ndecoherence, which predominantly occurs within 500 µs.\nWithincreasingbathsize,thedecoherencetimesdecrease\nfor both orientations of the quantization axis. Then, de-\ncoherence times below 500 µs are most common.\nThe article is organized as follows. In Sec. II, we ex-\nplain our model of the QD and discuss all relevant in-\nteractions of the spins with each other. Subsequently, in\nSec. III, we present the method used to obtain both the\nlong-timeaverageoftheelectronspinanditsdecoherence\ntimes. All results are shown and analyzed in Sec. IV.\nBased on a summary in Sec. V, we give an outlook on\npossible applications of few nuclear spin graphene QDs\nand on interesting future projects in this field.\nII. MODEL\nWe study the spin dynamics in a graphene quantum\ndot, where one electron spin is in contact with a bath of\nnuclear spins hosted by the13C atoms. Due to the con-\nfinement, the electron can occupy a discrete spectrum of\nbound states, with an energy splitting between different\nstates36,78–82. If the temperature is small compared to\nthe level spacing ∆ Eof these bound-state energies, the\nelectron resides in the ground state, which we describe\nby an envelope function φ(/vector r). Hence, the probability to\nfind the electron in a certain region of the QD can be\ndescribed by its absolute square |φ(/vector r)|2. In this paper\nwe define the “center“ of the dot as the region around\n/vector rmax, where the envelope function is maximal |φ(/vector rmax)|2.\nFar away from this center, the envelope function has to\nvanish:\n|φ(/vector rmax+∆/vector r)|2→0 for|∆/vector r|→∞.(1)\nIn this work, we model a graphene QD by the set of\natomic sites{/vector rk}Nsites\ni=1obeying\n|φ(/vector ri)|2\n|φ(/vector rmax)|2>C, (2)\nwhereC= 10−6is a constant cut-off. A plot of a QD\nrealized in this way is shown in Fig. 1. The choice of this\nfinite system of discrete sites /vector riimposes a normalization\ncondition\nNsites/summationdisplay\ni=1|φ(/vector ri)|2= 1, (3)\nsince we want to find the electron with probability 1\nsomewhere within the dot. Effectively, we ignore every-\nthing outside the barrier defined by the cut-off, which is\njustified by the vanishing probability to find the electron3\n-30-20-100102030\n-30 -20 -10 0 10 20 30-30-20-100102030\n-30 -20 -10 0 10 20 302 nm2 nm\n-30-20-10010203010-810-610-410-2\nFIG. 1: (Color online) (a): The graphene (( x1,x2,x3)) and quantization axis (( x,y,z )) reference frames. (b): A graphene QD\n(red sites) for a Gaussian envelope function with K= 10 uniformly random distributed13C atoms (blue squares) carrying a\nnuclear spin 1 /2. The extent of the dot over the graphene lattice is defined vi a the electron envelope function, where all sites\nwithin the dot obey the cut-off relation defined in Eq. (2). (c): The envelope function for fixed x1= 0 and x1= 22aNN,\nrespectively. The dashed line indicates the cut-off defined i n Eq. (2).\nthere. This choice will become more clear when we dis-\ncuss the hyperfine interaction between the electron and\na single nuclear spin below.\nA possible choice for the envelope function is a Gaus-\nsian\nφ(/vector r) =φ0exp/bracketleftbigg\n−1\n2/parenleftigr\nR/parenrightig2/bracketrightbigg\n=φ(r),(4)\nwherer=|/vector r|is the absolute value of the electron po-\nsition and the norm φ0is chosen to satisfy the normal-\nization condition in Eq. (3). This assumption is also\nin agreement with a recent experiment investigating the\nwave function of a graphene QD with soft confinement83.\nNote that the envelope function is not the exact electron\nwave function, but should give a good approximation to\nthe precise solution. This can be seen, for instance, in\ngraphene QDs based on semi-conducting armchair nano-\nribbons36. The most important aspects, which are cap-\ntured by this specific choice, are the absence of nodes\nin the ground state, a peak of the wave function in the\ncenter as well as a strong decay at the edges of the dot,\nwhich is illustrated by Fig. 1 (c).\nHavingdefinedthe shapeofourdot, wearenowableto\nintroduce the nuclear spins present in the system. Since\nwe do not have further knowledge about the distribution\nof13C within the dot, we randomly place the nuclear\nspins on the sites defined by Eq. (2), where each site is\nchosen with equal probability. An example of a configu-\nration of ten nuclear spins is shown in Fig. 1.\nWe now proceed to the interactions between the nu-\nclearspins and the electron spin and between themselves.\nThe most relevant spin-spin interaction in our system isthe hyperfine interaction between the electron spin /vectorSand\nKnuclear spins /vectorIklocated at sites /vector rk,\nˆHHI=AHIK/summationdisplay\nk=1/summationdisplay\nµ,ν← →Aµν|φ(/vector rk)|2ˆSµˆIk,ν,(5)\nwhere the indices µandνrun over spatial coordinates\nx,y,z. The energy scale of this interaction is given by\nAHI= 0.6µeVand← →Aµνisasphericaltensor84ofrank2,\nwhich takes into account the anisotropy of the hyperfine\ninteraction in graphene85. Remarkably, this interaction\nis strongly modulated by the envelope function, a fact\nwhich arises from the on-site nature of the hyperfine in-\nteraction. Thus, making the boundary of the dot smooth\nby taking a very small cutoff C→0 in Eq. (2), would\nonly add vanishingly small contributions to the interac-\ntion in Eq. (5). Therefore we choose a small, but finite\ncutoff for simplicity.\nBesides the anisotropic hyperfine interaction (AHI),\nthere is also a dipole-dipole interaction between pairs\nof nuclear spins. In the parameter regime considered\nin this work, this interaction, however, is about five or-\nders of magnitude smaller than the AHI and, thus, ne-\nglected. We also proved its irrelevance by a numerical\nstudy, which we will not present here.\nSince external magnetic fields allow to manipulate\nspins experimentally, we include a Zeeman-Hamiltonian\nto account for this:\nˆHZE=/planckover2pi1γSBzˆSz+/planckover2pi1γ13CBzK/summationdisplay\nk=1ˆIk,z≈AZEˆSz,(6)\nwhere we used the fact that the electron gyromagnetic4\nratioγS= 1.76×1011s−1T−1is much larger than the\ngyromagnetic ratio of the nuclear spins γ13C= 6.73×\n107s−1T−1to justify the right-hand side of Eq. (6).\nIn the presence of an external magnetic field, an in-\nterplay of the spin orbit coupling with acoustic phonons\ncanleadtospinrelaxationtimes T1rangingfrommillisec-\nonds to seconds86–89for small external magnetic fields.\nThe exact value of T1significantly varies with the spec-\ntrum of the phonons which depends on the details of the\ndot nanostructure. Providing that the graphene flake is\nflat throughout the spatial extent of the QD, however,\nthe spin orbit interaction should be small. Thus, this\nassumption justifies to neglect the influence of the spin\norbit interaction on our problem.\nIn the following, we aim to simulate a model experi-\nment consisting of a preparation of the spins and the ac-\ntual measurement of the spin dynamics. For the prepa-\nration, one can think of two different scenarios. First,\nthe states of both the electron spin and the nuclear\nspins can be prepared in the presence of a strong ex-\nternal magnetic field B0≫K·AHI|φ(rmax)|2//planckover2pi1γS=\nK·|φ(rmax)|2·5.7mT, which imprints a well defined\nquantization axis. At the beginning of the actual mea-\nsurement, this field is turned off or reduced to a finite\nvalue and the time evolution of the quantity of interest\nis recorded. However, since the tuning of magnetic fields\nis typically slow, this preparation scheme might not be\nadequate for a real experiment and we have to look for\nother solutions. In this case, we can think of injecting\na spin polarized electron via a spin dependent tunnel-\ning process from a normal lead or via a spin conserving\ntunneling process from a spin polarized lead.\nAnyhow, in both considered scenarios our system fea-\ntures two natural reference frames, the first defined by\nthe graphene plane and the second one by the quanti-\nzation axis of the electron as depicted in Fig. 1 (a).\nIn order to clarify the notation, the graphene coordinate\nsystem(GCS) iswritten as /vector˜v= (˜vx1,˜vx2,˜vx3) with allob-\njects being marked with a tilde, whereas a vector in the\nquantization axis coordinate system (QCS) is labeled by\n/vector v= (vx,vy,vz)cf.Fig. 1. If one chooses, without loss of\ngenerality, the x2- and they-axis to coincide, both coor-\ndinate systems are connected via a rotation ˆD(β) by an\nangleβaround this common axis.\nDue to the symmetries of the carbon orbitals85, the\nspherical tensor← →Aµνof the AHI in Eq. (5) takes its\nsimplest form in the GCS, namely\n← →˜A=\n−1\n20 0\n0−1\n20\n0 0 1\n, (7)\nwhile the spin operators, ˆSµandˆIk,ν, and the spin states\nare most conveniently defined in the QCS.\nIn the main part of this work, we are interested in the\ntime-dependent expectation value of an arbitrary opera-\ntorˆO\n/an}bracketle{tO/an}bracketri}ht(t) =/an}bracketle{tψ0|ˆU†(t)ˆOˆU(t)|ψ0/an}bracketri}ht, (8)where|ψ0/an}bracketri}htdescribes the initial state of the total spin\nsystem and ˆU(t) = exp[−i/planckover2pi1−1tˆH] is the time evolu-\ntion operator determined by the total Hamiltonian ˆH=\nˆHHI+ˆHZE. Note, thatthisisthetotalHamiltonianwith\nrespect to the QCS, where the Zeeman Hamiltonian is al-\nwaysdiagonaland the AHI Hamiltonian is obtained from\nits simple GCS formˆ˜HHIby\nˆHHI=ˆD(β)ˆ˜HHIˆD†(β). (9)\nLikewise, we could keep the Hamiltonian fixed in its GCS\nform and instead transform the operators ˆOandˆHZEas\nwell as the initial state |ψ0/an}bracketri}htfrom the QCS to the GCS.\nFor technical reasons, however, we choose to transform\nthe Hamiltonian, while keeping the operator and the ini-\ntial state fixed for arbitrary β.\nIII. METHOD\nIn order to numerically compute the time evolution in\nEq. (8), we need a basis to represent the state of our\nsystem and the operators acting on it. A natural choice\nforN= 1+Kspins is given by the tensor product states\nof the electron spin and the nuclear spin eigenstates\n|n/an}bracketri}ht=|mn\nS/an}bracketri}ht⊗K/circlemultiplydisplay\nk=1|mn\nk/an}bracketri}ht=|⇓↑↓↓↑.../an}bracketri}ht,(10)\nwhere the electron spin is represented by |mn\nS/an}bracketri}ht,mn\nS=⇓\n,⇑and the nuclear spin states by |mn\nk/an}bracketri}ht,mn\nk=↓,↑.\nFor convenience, we have ordered the nuclear spins/vextendsingle/vextendsinglemn\nSmn\nKmn\nK−1.../angbracketrightbig\naccording to the value of the envelope\nfunction at the corresponding site:\n|φ(rK)|2≥|φ(rK−1)|2≥... (11)\nWithin this basis, an arbitrary state is given by a linear\nsuperposition of these 2Nstates\n|ψ/an}bracketri}ht=2N−1/summationdisplay\nn=0αn|n/an}bracketri}ht,2N−1/summationdisplay\nn=0|αn|2= 1 (12)\nwith complex coefficients αn, while all operators are rep-\nresented by 2N×2Nmatrices. By diagonalizing the total\nHamiltonian ˆMˆHˆM†= diag(λ0,λ1,...,λ 2N−1) we are\nable to re-express the time evolution operator\nˆV(t) =ˆMˆU(t)ˆM†= diag(exp[−i/planckover2pi1−1λ0t],...),(13)\nwhereˆMis an unitary operator formed by the eigenvec-\ntors ofˆHand theλnare the corresponding eigenvalues.\nFinally, we re-write Eq. (8) and find\n/an}bracketle{tO/an}bracketri}ht(t) =/an}bracketle{tψ0|ˆM†ˆV†(t)ˆMˆOˆM†ˆV(t)ˆM|ψ0/an}bracketri}ht,(14)\nwhich we will evaluate for different parameter regimes in\nthe following section. The numerical diagonalization is\nperformed using the EIGEN90package for C++.5\nFIG. 2: (Color online) Exemplary time evolution of the longi -\ntudinal electron spin component ∝angbracketleftSz∝angbracketright(t) as a function of time\nfor out-of-plane orientation β= 0, cf. Fig. 1 (a). For a\ncertain range of time [ Tmin,Tmax] using a resolution ∆ T, we\ncalculate the long-time average ∝angbracketleftSz∝angbracketrightTand the standard de-\nviationσSzand find the maximal deviation ∆ Szaround this\nvalue. These quantities as well as the details of the oscilla -\ntions including the beating structure depend on the choice o f\nthe parameters. The decoherence time TDis determined by\na constant threshold CS.\nAs onecannoticefrom the explanationsabove, wedeal\nwith a quite big parameterspacein which we can analyze\nthe outcome of Eq. (14). First, we control the shape of\nthe dot by means of the envelope function |φ(r)|2, sec-\nondly the number of nuclear spins Kis variable and fi-\nnally these spins can have different positions or configu-\nrationsCwithin the dot. All of these parameters change\nthe AHI Hamiltonian in Eq. (5). Moreover, we will in-\nvestigate different initial states |ψ0/an}bracketri}htof the electron and\nthe nuclear spins affecting Eq. (14). Additionally, the\neigenvector matrix ˆM, appearing in this equation is a\nfunction of the twisting angle βbetween the GCS and\nthe QCS. Note that the spectrum of eigenvalues λnis\nunaffected by a change of β. Finally, we can also modify\nthe absolute value of the external magnetic field, which\nwe will parametrize by the resulting Zeeman energy of\nthe electron AZE.\nIV. RESULTS\nIn this section, we present our findings for the model\nsystem defined above. All calculations were carried out\nusing an envelope function of the Gaussian type in Eq.\n(4) withR= 7aNNand a cut-off C= 10−6. This cor-\nresponds to a dot with diameter D≈7.2nm containing\nNsites≈103carbonatoms, suchthat K= 9atomscorre-\nspond to the natural abundance nI= 0.01 of13C. In or-\nder to investigatethe impact ofdifferent initial states, we\nchoose random complex (RC) initial states14,70. These\nstates were created by drawing complex coefficients αnfrom Re[αn],Im[αn]∈[−1,1] with equal probability and\nnormalizing them according to Eq. (12). Moreover, we\nchoose the electron spin always to point down resulting\nin initial states consisting of |⇓.../an}bracketri}htstates only, which\nmeans that αn= 0 forn≥2K.\nIn order to determine qualitatively and quantitatively\nthe impacts of the parameters, we investigate the time\ndependent expectation value /an}bracketle{tSz/an}bracketri}ht(t) of the longitudinal\nelectron spin component, which is calculated using Eq.\n(14). A typical time evolution of /an}bracketle{tSz/an}bracketri}ht(t) is plotted in Fig.\n2. Within the decoherencetime TD, the initial amplitude\nof the electron spin of /planckover2pi1/2 decays to its long-time average\nvalue, where still finite oscillations and beatings occur.\nThis can be traced back to the finite size of the spin bath\nconsidered here.\nIts long-time average value is calculated by\n/an}bracketle{tSz/an}bracketri}htT=1\nNTNT/summationdisplay\ns=0/an}bracketle{tSz/an}bracketri}ht(Tmin+s·∆T),(15)\nwhere we average over NT= (Tmax−Tmin)/∆Ttime\nsteps of width ∆ T. In order to investigate the oscilla-\ntions of/an}bracketle{tSz/an}bracketri}ht(t) quantitatively, we consider the standard\ndeviation\nσSz=/radicaltp/radicalvertex/radicalvertex/radicalbt1\nNTNT/summationdisplay\ns=0/parenleftig\n/an}bracketle{tSz/an}bracketri}ht(Tmin+s·∆T)−/an}bracketle{tSz/an}bracketri}htT/parenrightig2\n(16)\nas well as the sample range\n∆Sz= max\nt∈[Tmin,Tmax][/an}bracketle{tSz/an}bracketri}ht(t)]−min\nt∈[Tmin,Tmax][/an}bracketle{tSz/an}bracketri}ht(t)].\n(17)\nThe latter quantity is a measure for the occurrence of\noscillations with a big amplitude which originate from ei-\nther recurrences of the signal, beatings or an entire lack\nof decoherence. While for beatings one expects rather\nsmall sample ranges ∆ Sz< Sz(0), the former two cases\nshould give values on the order of the initial amplitude,\n∆Sz∼O(Sz(0)) in the out-of-plane case β= 0 and\n∆Sz∼2·O(Sz(0)) in the in-plane case β=π/2.\nBesides these quantities characterizing the long time\naverage of the electron spin, we are also interested in\nthe amount of time it takes to decohere the system. In\nordertobeindependentfromspecificmodelsofthedecay,\nsuch as exponential or power-law decoherence, and to\naccount for the characteristics of the numerics, we find\nthis decoherencetime TDbythe first minimum exceeding\na certain threshold CS. For clarity, CSis also illustrated\nin Fig. 2. This approachis similarto the one used in Ref.\n77 to find the decoherence times. Of course, the choice of\nthis constant CSchanges the value of TD. However, its\norder of magnitude and its dependence on the different\nparametersisratherindependentfromaspecificchoiceas\nlong asCSis not too close to /an}bracketle{tSz/an}bracketri}htT, which we confirmed\nfor different values of CS.\nIn the following, we analyze both the decoherence time\nand the long-time average of the longitudinal electron6\n01020304050RC initial states\n0 10 20 30 40 50\nConfigurations\n0 10 20 30 40 50\nConfigurations\n-0.3-0.25-0.2-0.15\nFIG. 3: (Color online) Plot of the long-time average ∝angbracketleftSz∝angbracketrightT\nfor out-of-plane orientation β= 0 and K= 3 and K= 6 nu-\nclear spins without an external magnetic field. We considere d\n51 different RC initial states and 51 random configurations.\nFor both numbers of nuclear spins, the electron spin looses\nroughly one half of its amplitude to ∝angbracketleftSz∝angbracketrightT≈ −0.22/planckover2pi1. The\nhorizontal stripes for K= 3 indicate the importance of the ini-\ntial states for few nuclear spins, while configurations seem less\nrelevant signaled by weaker vertical structures. This chan ges\nforK= 6 nuclear spins, where the configurations dominate\nover initial states indicated by the vertical structures in the\nright plot.\nspin component for different parameter sets. For each\nnumberKof nuclear spins many initial states and con-\nfigurations are created and labeled by numbers 0 ,1,2...\nfor later comparison of the results. Note, that for differ-\nentnuclearspinnumbers Ktheselabelsdescribedifferent\ninitial states and configurations. Moreover, we concen-\ntrateontwoorientationsofthequantizationaxis, namely\nout-of-plane orientation for β= 0 and in-plane orienta-\ntion withβ=π\n2.\nWe investigated the effect of finite magnetic fields for\nexemplary initial states, configurations and K= 2,4,6\nnuclear spins, where we varied the resulting Zeeman con-\nstant from AZE/AHI≪1 toAZE/AHI≫1. For in-\ncreasingAZE, we find a continuous crossoverto a perfect\nalignment of the electron spin in the case of a very strong\nmagnetic field. In the following, weput AZE= 0 because\nwewouldliketobetter understandthe low-magneticfield\nbehaviorofthe spin dynamics in the presenceofthe AHI.\nA. Dependence of the long-time average on\ndifferent initial states, configurations, and the\nnumber of nuclear spins\nFirst, we investigate the consequences of both differ-\nent RC initial states and different configurations of the\nnuclei within the dot. We calculated /an}bracketle{tSz/an}bracketri}htT,σSzand,\n∆Szfor different parameter sets and found stable results\nforTmin= 0.5×109τHI,Tmax= 1.5×109τHI, and01020304050RC initial states\n0 10 20 30 40 50\nConfigurations\n0 10 20 30 40 50\nConfigurations\n-0.002-0.00100.0010.002\nFIG. 4: (Color online) Same plot as in Fig. 3 but for in-\nplane orientation β=π/2. For all parameters the long-time\naverage of the longitudinal electron spin component sharpl y\nsaturates at ∝angbracketleftSz∝angbracketrightT≈0. In contrast to the out-of-plane case,\nthe results seem independent of the initial state even in the\nfew spin regime. For certain configurations, however, we find\na non negligible dependence on the initial state.\n-0.5-0.2500.250.50.751\n2 3 4 5 6 7 8 9[ ]\nK<Sz>T\nσSz\n∆Sz\nFIG. 5: (Color online) Plot of the long-time average ∝angbracketleftSz∝angbracketrightT,\nthe standard deviation σSzand, the sample range ∆ Szas a\nfunction of the number of nuclear spins Kfor in-plane ( β=\nπ/2, red) and out-of-plane orientation ( β= 0, black). The\nvalues are obtained by averaging over 51 RC initial states\nand 51 different configurations, see Figs. 3 and 4. Error bars\nare given by the standard deviation with respect to averagin g\nover all 51 ×51 results. While the long-time average is almost\nconstant, the averaged standard deviation σSzas well as the\naveraged sample range ∆ Szstrongly decrease for larger K\nindicating the reduction of fluctuations and of the occurren ce\nof beating or recurrence events.\n∆T= 104τHIwithτHI=/planckover2pi1/AHI≈1ns. In Fig. 3, we\nplotthe long-timeaverage /an}bracketle{tSz/an}bracketri}htTasafunction ofdifferent\nRC states and configurations for K= 3 andK= 6, re-\nspectively, in out-of-plane orientation. The color map in\nFig. 4 was created for the same parameters with in-plane\norientation.\nFor a small number of nuclear spins K= 3 and\nβ= 0, we observe strong fluctuations for both different\nRCstates and different configurations around an aver-\nage value of/an}bracketle{tSz/an}bracketri}htT≈−0.22/planckover2pi1as depicted in the color\nmap of Fig. 3. The horizontal stripes dominate over the\nvertical structures indicating, that the choice of the RC\ninitial states has a greater influence on the results than7\nthe spatial configuration of the nuclear spins within the\ndot.\nMoreover, we find large oscillations around this long-\ntime average value for many configurations and initial\nstates. This results in both sizable sample ranges ∆ Sz\nand standard deviations σSz. By averaging over all 51 ×\n51 results, we find /an}bracketle{t/an}bracketle{tSz/an}bracketri}htT/an}bracketri}ht= (−0.22±0.06)/planckover2pi1,/an}bracketle{tσSz/an}bracketri}ht=\n(0.13±0.04)/planckover2pi1, and/an}bracketle{t∆Sz/an}bracketri}ht= (0.52±0.11)/planckover2pi1, which is also\nshown in Fig. 5. The large average value of the sample\nrange/an}bracketle{t∆Sz/an}bracketri}htshows that for most cases analyzed, there\nwas at least one big change in amplitude. However, no\ntotal spin flip to + /planckover2pi1/2 is achieved. The occurrence of\nsizable standard deviations indicates that there are on\naverage many of these events. Thus in the few nuclear\nspin regime, coherent oscillations of the electron spin are\nthe dominant dynamics, where recurrences of the initial\nvalue take place with a period TP=/planckover2pi1/maxi(|λi|)∼\n/planckover2pi1/(AHI·|φ(rK)|2)≥100ns.\nIf we consider a larger environment of nuclear spins\nas presented in Fig. 3 with K= 6, the behavior of the\nlong-time averagechanges. First ofall, the result is much\nmore uniform with respect to both the RC initial states\nand the configurations. In addition, the remaining differ-\nences in/an}bracketle{tSz/an}bracketri}htTdepend on the configurations rather than\non the initial states, which is obvious from the vertical\nlines present in this colormap. Averagingoverall 51 ×51\nresults gives/an}bracketle{t/an}bracketle{tSz/an}bracketri}htT/an}bracketri}ht= (−0.22±0.02)/planckover2pi1, which is essen-\ntially the same as for K= 3. However, the standard\ndeviation/an}bracketle{tσSz/an}bracketri}ht= (0.06±0.03)/planckover2pi1and the sample range\n∆Sz= (0.37±0.06)/planckover2pi1clearlydecrease. Weconfirmedthis\ntrend of decreasing fluctuations by repeating the above\naveraging procedure for other numbers of nuclear spins.\nThese results are presented as a function of Kin Fig. 5.\nWhile the long-time average value is constant, both the\nstandarddeviationandthesamplerangebecomesmaller.\nEspecially, the pronounced decay of the sample range\nclearly indicates that recurrences occur much less and,\nhence, that the corresponding recurrence times are in-\ncreasing with more nuclear spins.\nThus, the major effect of an increased number of nu-\nclearspins is to suppress the oscillationsaround the long-\ntime average and changing the system from an initial\nstate dominated regime to a regime where the configu-\nration of the nuclear spins is important. This behavior\ncanbe understood byanalyzingthe impact ofthe nuclear\nspin number on the dimension of the Hilbert space and\non the strength of the hyperfine interaction.\nFor a small number of nuclear spins, the dimension\nof the corresponding Hilbert space D= 2K+1is small\nand, hence, we draw our RC initial states from a rather\nlimited set, where individual single product states |n/an}bracketri}ht\nlead to very different dynamics of the electron spin. Due\nto the combination of only 2Kstates|n/an}bracketri}htto a RC initial\nstate, it is not unlikely that one of these states dominates\nover the rest leading to rather diverse results.\nBy increasing K and, thus, the Hilbert space dimen-\nsion this situation is changed. Since the individual state\n|⇓...↑↓/an}bracketri}htof nuclear spins at the border of the dot is al-most irrelevant due to a small |φ(/vector rk)|2, groups of effec-\ntively equivalent states are superposed. Thus, a more ef-\nfective averaging is achieved suppressing the dependence\non a specific initial state. As a consequence, it is very\nunlikely for a single state to dominate over the rest.\nThe coupling strength of nuclei is the key in un-\nderstanding the dependence of the results on the con-\nfiguration. Its energy scale is given by the product\nAHI|φ(rK)|2of the hyperfine coupling constant and the\nmaximal value of the envelope function at the sites of\nthe nuclear spins. For a small number K, the prob-\nability to find two or more nuclear spins, which cou-\nple almost equally with the electron spin, is low due to\nthe large gradient of the envelope function. Hence, ef-\nfectively only one nuclear spin strongly interacts with\nthe electron leading to simple oscillations. This behav-\nior can be also easily derived by diagonalizing the re-\nsulting, effective 4 ×4 matrix of the AHI Hamiltonian\ngiven in Eq. (5). In doing so, one finds a discrete spec-\ntrum of frequencies given by the degenerate eigenvalues\n{λi}={−1/2,0,1/4,1/4}·AHI|φ(rK)|2. This fact is\nresponsible for the rather uniform dynamics with respect\nto different configurations in a small Kregime.\nThis situation can of course also occur for larger nu-\nclearspinenvironments,asshowninFig. 6( a)forK= 6.\nIt is, however, rather the exception from the more prob-\nable case of several nuclei coupling comparably to the\nelectron, where a almost continuous spectrum is found\nas depicted in Fig. 6 ( b). If we characterize these spectra\nquantitatively by counting the number of distinct eigen-\nvalues, i.e., eigenvalues which differ significantly, we can\nmap the configuration of the nuclei to the spectra as de-\npicted in Fig. 6 ( c).\nFor the in-plane case, our findings are quite different\nfromthe formerones. The electronspin saturatesaround\n/an}bracketle{tSz/an}bracketri}htT= 0 for both K= 3 andK= 6 as shown in Fig.\n4. Interestingly, we find already for K= 3, that this\naverage is reached very precisely with smaller fluctua-\ntions than in the out-of-plane case. This fact becomes\nalso clear from averaging the longitudinal electron spin\n/an}bracketle{t/an}bracketle{tSz/an}bracketri}htT/an}bracketri}ht= (0.000±0.004)/planckover2pi1over all results. Moreover,\nthe results are independent from the choice of the RC\ninitial state. Some single configurations, however, give\nrise to deviations from this, where also a dependence\non the initial state is restored. It seems, that this is\nthe case, where several nuclear spins couple comparably\nto the electron spin explaining the sensitivity on initial\nstates. The size of the fluctuations is on averagegiven by\n/an}bracketle{tσSz/an}bracketri}ht= (0.15±0.02)/planckover2pi1. The mean value of the sample\nrange of/an}bracketle{t∆Sz/an}bracketri}ht= (0.92±0.07)/planckover2pi1close to 1 indicates, that\nin most cases, the electron spin is at least once almost\ncompletely flipped to +1\n2/planckover2pi1in contrastto the out-of-plane\norientation. The K= 6 study shows qualitatively the\nsame result with /an}bracketle{t/an}bracketle{tSz/an}bracketri}htT/an}bracketri}ht= (0.000±0.001)/planckover2pi1, where the\nfluctuations/an}bracketle{tσSz/an}bracketri}ht= (0.057±0.004)/planckover2pi1are further sup-\npressed. Moreover, the appearance of recurrences and8|φ(rK-2)|2/ |φ(rK)|2\n|φ(rK-1)|2/ |φ(rK)|210-410-21\n10-410-213579111315\n# of distinct λiλi/ (AHI· |φ(rK)|2)\n-0.500.5\nFIG. 6: (Color online) ( a): Eigenvalues λiof the AHI Hamil-\ntonian for for K= 6 nuclear spins and configuration C= 10\nin normalized units of AHI|φ(rK)|2. (b): Eigenvalues λifor\nK= 6 and C= 3. (c): Number of distinct eigenvalues λi\nas a function of the relative probabilities |φ(rK−1)|2/|φ(rK)|2\nand|φ(rK−2)|2/|φ(rK)|2forK= 6 nuclear spins. If both\n|φ(rK−1)|2/φ(rK)|2≈1 and|φ(rK−2)|2/φ(rK)|2≈1 at least\nthree nuclear spins are strongly interacting with the elect ron\nspin causing a spectrum with many different eigenvalues as\ndepicted in ( b). If only the most central nuclear spin couples\nstrongly to the electron spin (lower left part), the spectru m\nis highly degenerate showing only three different eigenvalu es\nas shown in ( a). The upper limit for the number of 15 has no\ndeeper meaning besides distinguishing both types of spectr a.\ntotal spin flips is also strongly decreased for K= 6 as\nis clear from the sample range /an}bracketle{t∆Sz/an}bracketri}ht= (0.51±0.09)/planckover2pi1.\nAnalyzing this observable as a function of the number of\nnuclear spins, we observe again a prominent suppression\nof the fluctuations for growing Kas is apparent in Fig.\n5.\nIn order to understand the differences between the in-\nplane and out-of-plane dynamics of the electron spin in\nmore detail, an analytic analysis of the dynamics in the\ncase of only one nuclear spin is very useful. Calculating\nthe long-time average analytically for K= 1 yields\n/an}bracketle{tSz/an}bracketri}htT(β) = lim\n∆T→∞1\n2∆TT−∆T/integraldisplay\nT+∆T/an}bracketle{tSz/an}bracketri}ht(t,β)\n=−/planckover2pi1\n4cos(β)/bracketleftig\n2ρ↓↓cos(β)+(ρ↑↓+ρ↓↑)sin(β)/bracketrightig\n,(18)-0.3-0.2-0.100.1\n0 0.2 0.4 0.6 0.8 1<Sz>T[ ]\nβ/ π\nFIG. 7: (Color online) Dependence of the long-time average\n∝angbracketleftSz∝angbracketrightT(β) on the orientation of the quantization axis with re-\nspect to the graphene plane for K= 6 nuclear spins. The ex-\nample shown here was calculated for the configuration C= 3,\nwhose continuous spectrum is presented in Fig. 6 (b). The\nonly parameter used to fit the numerical values to the ana-\nlytic curve ∝angbracketleftSz∝angbracketrightT(β) =∝angbracketleftSz∝angbracketrightT(0)·cos2(β) is the out-of-plane\nvalue∝angbracketleftSz∝angbracketrightT(0).\nwhere the initial density matrix\nρ0=|ψ0/an}bracketri}ht/an}bracketle{tψ0|=\nρ↓↓ρ↓↑0 0\nρ↑↓ρ↑↑0 0\n0 0 0 0\n0 0 0 0\n (19)\nis only non-zero for the electron spin pointing down as\nforthe RC initial states. Formorenuclearspins involved,\nthe resulting equations become much more complicated.\nHowever, for the special case of only one strongly cou-\npling nuclear spin, the structure of the AHI Hamiltonian\nremains the same and Eq. (18) still holds.\nWe investigated the βdependence of the long-time\naverage numerically for some configurations and initial\nstates and K= 2,4,6 and 9 nuclear spins, where we\nfind good agreement of our results with /an}bracketle{tSz/an}bracketri}htT(β) =\n/an}bracketle{tSz/an}bracketri}htT(0)cos2(β) with increasing K. Particularly, we ob-\nserved this behavior also for configurations with several\nnuclearspinscouplingalmostequallytotheelectronspin.\nAs an example, we plot in Fig. 7 the βdependence of\n/an}bracketle{tSz/an}bracketri}htTforK= 6 and configuration C= 3. Its spectrum\nis shown in Fig. 6 (b). This fact is also supported by\nour results presented in Figs. 3 and 4, where we find on\naverage/an}bracketle{tSz/an}bracketri}htT≈−0.22/planckover2pi1forβ= 0 and/an}bracketle{tSz/an}bracketri}htT≈0 for\nβ=π/2. The deviation of /an}bracketle{tSz/an}bracketri}htT(β= 0) from−/planckover2pi1/4 orig-\ninates from the finite time window [ Tmin,Tmax] used in\nthe numerical calculations, which misses recurrences of\nthe full initial value of /an}bracketle{tSz/an}bracketri}ht(t= 0) =−/planckover2pi1/2.\nFrom this numerical findings and Eqs. (18) and (19),\nwe supposethat contributionsfrom the offdiagonalparts\ncancel each other almost completely and that the ele-\nments of the diagonal parts of the density matrix ρ↓↓,ρ↑↑\nhave approximately equal weight of 1 /2K, which seems\nreasonable for random complex initial states.9\nB. Decoherence times\nIn this section, we want to investigate the decoherence\ntimes of the longitudinal electron spin Szfor different\ninitial states and different configurations. We chose the\nthreshold to be always about 0 .1/planckover2pi1below the obtained\nlong-time average, which gives CS=−0.325/planckover2pi1for the\nout-of-plane case β= 0 andCS=−0.1/planckover2pi1for the in-\nplane caseβ=π/2. Moreover, we used exactly the same\ninitial states and configurationsforall Kasfor the calcu-\nlation of the long-time average. The decoherence times\nwere estimated for times up to 107τHI≈10ms with\na time resolution ∆ T= 102τHI, which yields at least\nP= 2π/(∆T·λmax)≈20pointsperperiodofthe highest\nabsolute frequency max i(|λi|). ForK= 6 we extended\nthe investigated time regime to 108τHI≈100ms using\nthe same time resolution ∆ T.\nAs it turns out, the decoherencetimes obtained by this\nmethod are rather independent from the initial states.\nSeveral factors are important for this fact. First of all,\nforlargernumbersofnucleiofcoursethe samearguments\nconcerning the Hilbert space dimensions as for the long-\ntimeaveragehold. However,wealsofindforsmall Konly\nlittle dependence on the initial states. One reason for\nthis is probably, that our method is robust against small\nchanges of the longitudinal electron spin caused by dif-\nferent initial states, since we measure when the signal is\nabove a certain threshold, but not how much. Finally, as\nweshowbelow, the decoherenceseemsstronglyrelatedto\nthepresenceofmanyincommensuratefrequencies. These\nfrequencies are proportional to the eigenvalues of the hy-\nperfine Hamiltonian and, hence, independent from the\ninitial state.\nTherefore, we focus in the following on the conse-\nquences of different configurations on the decoherence\ntimes for different numbers of nuclear spins. In principle,\nthere are two relevant aspects concerning the positions\nof the nuclei, the absolute value of the envelope func-\ntion|φ(/vector rK)|2at the site of the strongest coupling nuclear\nspin and the relative position of the nuclei with respect\nto each other. The importance of the former is obvious,\nsince the envelope function sets the maximal energyscale\nofthe AHI in Eq. (9) to AHI·|φ(rK)|2and, consequently,\nrescales all times by a factor |φ(rK)|−2. Therefore, if we\nwant to analyze the influence of the relative positions,\nwe have normalize the decoherence times according to\nTD→TD·|φ(rK)|2.\nWe begin our discussion with investigating these nor-\nmalized decoherence times for K= 6 nuclei in more de-\ntail and then turn to absolute decoherence times as a\nfunction of Kafterwards.\nA color map of the normalized decoherence times for\n51 initial states and 51 configurations is shown in Fig. 8.\nFor the out-of-plane case, we find that the decoherence\ntimesarealmostindependentoftheinitialstate, butvary01020304050Random initial states\n0 10 20 30 40 50\nConfigurations0 10 20 30 40 50\nConfigurations11.522.533.544.55\nlog10(TD· |φ(rK)|2)\nFIG. 8: (Color online) Normalized decoherence time TD·\n|φ(rK)|2as a function of 51 RC initial states and 51 ran-\ndom configurations for K= 6 nuclear spins in in-plane and\nout-of-plane orientation. While the decoherence time is al -\nmost the same for different initial states, it strongly depen ds\non the configurations showing deviations over several order s\nof magnitude. White spaces indicate the total lack of de-\ncoherence up to absolute times of 0 .1 s given a threshold of\nCS=−0.325/planckover2pi1. For special configurations, C= 10,32 and\n35, and β= 0 there is no decoherence at all, but coherent\noscillations of the electron spin.\nover several orders of magnitude for different configura-\ntions. If we plot the normalized times as a function of\nthe number of distinct eigenvalues, cf. Fig. 6, we find\na direct connection between these times and the config-\nuration of the nuclei in the dot. As is clear from Fig.\n9, long decoherence times can be only found for the dis-\ncrete spectra, which are realized if only one nuclear spin\nstrongly interacts with the electron. The configurations\nwithout any decoherence, which are indicated by white\nspaces in Fig. 8, exhibit discrete spectra with the mini-\nmal number of distinct eigenvalues of 3. An example of\nsuch a spectrum is shown in Fig. 6 (a). In these cases the\ndynamics of the longitudinal electron spin are coherent\noscillations, where recurrences appear with a period of\nTP=|φ(rK)|−2·τHI≥100ns. In contrast to this, short\nnormalized decoherence times are a consequence of con-\ntinuous spectra as presented in Fig. 6 (b). Thus, by the\nconfigurations studied, we can proof a direct relation be-\ntween the relative positions of the nuclear spins and their\nrelative coupling strengths, respectively, and the order of\nmagnitude of the decoherence times.\nFor the in-plane case, the qualitative picture is similar,\nhowever, with shorter normalized decoherence times over\nall, such that we find decoherence within the investigated\ntimes for all configurations. In contrast to the out-of-\nplane case, also discrete spectra can show rather short\ndecoherencetimesforspecificconfigurations. Altogether,\nthis demonstrates a much faster decoherence due to the\nbroken symmetry in the in-plane orientation.\nTurning from normalized times to absolute decoher-\nence times, the value of the envelope function |φ(rK)|210\nTD > 0.1s\n 1 2 3 4 5\n 3 5 7 9 11 13 15log10(TD ´ |φ(rK)|2)\n# of distinct λi 0 0.02 0.04 0.06 0.08 0.1\nrelative frequency\nFIG. 9: (Color online) Normalized decoherence time TD·\n|φ(rK)|2as a function of the number of distinct eigenvalues of\nthe AHI Hamiltonian for K= 6 nuclear spins in out-of-plane\norientation. For this plot all out-of-plane results presen ted in\nFig. 8 are considered. Obviously, long decoherence times oc -\ncur only for a small number of distinct eigenvalues. Togethe r\nwith the results of Fig. 6 the importance of the relative cou-\npling strengths ∝ |φ(rK−1)|2/|(φ(rK))|2, ... and, hence, of\nthe relative position of different nuclei becomes evident.\n00.20.40.60.81n(TD∈[ Tmin, Tmax[)\n2 3 4 5 6 7 8 9\nK2 3 4 5 6 7 8 9\nK[0,5 µs[\n[5µs,500 µs[\n[500 µs,10 ms[\n> 10 ms\nFIG. 10: (Color online) Relative number of absolute decoher -\nence times TDfalling in a certain time interval [ Tmin,Tmax[\nfor different numbers Kof nuclei in in-plane and out-of-plane\norientation. For each K51 RC initial states and 51 configu-\nrations were considered leading to Ncalc= 2601 calculations\nin total. ( a): In the out-of-plane case, long decoherence times\nare clearly dominating for few nuclear spins. Increasing K\nleads to a quick decay of the decoherence times such that\nshort times TDare common. Very short decoherence times\nstart to become relevant for K > 6. (b): In the in-plane case,\neven for few nuclear spins short relaxation times are the rul e.\nFor increasing K, the percentage of short decoherence times\nis growing further.\nat the site of the strongest coupling nuclear spins addi-\ntionally becomes relevant, since it sets the order of mag-\nnitude of all times. Putting a larger and larger number\nof nuclear spins on a QD of constant area increases the\naverage value of|φ(rK)|2, since it is more likely to find a\nspin very close to the center. Moreover, as we discussed\nabove, an increased Kmakes it much more probable to\nhave several nuclear spins coupling almost equally to the\nelectron spin. Altogether, this lets us expect a prominentdecay of long decoherence times as a function of growing\nK, which is confirmed by Fig. 10. For β= 0 and very\nfew nuclear spins K= 3, we find that the majority of de-\ncoherence times is longer than 10ms, whereas very short\nTDare almost completely irrelevant. For K= 8 the per-\ncentages of short and long times are inverse. Now, only\nless than 7% of the decoherence times are longer than\n10ms, while most of the decay of the electron spins takes\nplace within 500 µs. However, for K= 6, surprisingly,\nstill about one-fifth of the cases shows ultra long deco-\nherence times.\nIn the in-plane orientation, long decoherence times\nmake up only a small fraction even for few nuclear spins.\nShort decoherence times in the range of 5 µs to 500µs\nare significantly increasing for more spins. Notably, ul-\ntra short times below 5 µs do not become much more\nimportant.\nInsummary,typicaldecoherencetimesareontheorder\nof ms under ideal conditions of small nuclear spin num-\nbers and out-of-plane orientation. In the case of such\nlong decoherence times, of course, other effects like spin\norbit coupling could become relevant. In the presence of\nacoustic phonons and small external magnetic fields, this\nspin orbit coupling86,89can lead to spin relaxation times\nofT1∼1ms below the decoherence times found here.\nFor larger numbers of nuclear spins and, generally,\nfor in-plane orientation, decoherence times are smaller,\nbut still above 5 µs. Typical decoherence times of GaAs\nQDs under spin echo4lie in theT2,echo∼1µs regime,\nwhereas the current record of T2,CPMG≈200µs was\nmeasured using the Carr-Purcell-Meiboom-Gill (CPMG)\npulse sequence10. Pure dephasing times T∗\n2are below\n50ns forGaAs. Although all our estimates for the de-\ncoherence times are done for a model without any effort\nto improve the coherence of the electron spin like pulse\nsequences or strong magnetic fields, in almost all consid-\nered cases, we areabovethe GaAsspin echo time T2,echo.\nFor smaller nuclear spin numbers, graphene even outper-\nforms the CPMG time, which lets us expect very long\ndecoherence times in graphene QDs when using pulse se-\nquences.\nV. SUMMARY AND CONCLUSION\nStarting from a generic model of a graphene QD, we\nstudied the dynamics of the electron spin caused by the\nhyperfine interaction with the nuclear spins present in\nthe dot. The number of nuclei was varied from K= 2 to\nK= 9, where the upper limit corresponds to the natural\nabundance of spin carrying13C for the dot size consid-\nered in this article. Besides the role of the number of\nnuclei, we also investigated the influence of the initial\nconditions as well as the impact of different configura-\ntions of the nuclei in the dot. Moreover, we explored the\nconsequences of the orientation of the spin quantization\naxiswith respect to the grapheneplane. In orderto char-\nacterize and quantify these effects, we analyzed both the11\nlong-time average /an}bracketle{tSz/an}bracketri}htTof the longitudinal electron spin\ncomponent and its decoherence time TD.\nSince nuclear spins are usually very hard to control in\nthe envisioned experiments, we chose the initial states to\nbe random complex (RC) superpositions of single prod-\nuct states. For this class of initial states, we found an ap-\npreciable effect on the long-time average only in the case\nof very few nuclear spins with K <5. Upon increasing\nthe number of nuclear spins the effects of quantum par-\nallelism and amplitude averaging14,70reduce the differ-\nencesbetween individualRCinitial statesmoreandmore\neffectively. Inthis parameterregime, theresultsaredom-\ninated by the configuration of the nuclear spins within\nthe dot, i.e., by their relative positions with respect to\neach other. For different configurations, the spectrum\nof eigenvalues of the hyperfine interaction varies from a\nhighly discrete one with many degenerate eigenvalues to\na continuous spectrum with many incommensurate fre-\nquencies.\nFor allK, a pronounced dependence of the long-time\naverageontheorientationangle βbetweenthespinquan-\ntization axisand the normal vectorof the graphene plane\nwas found. It saturates at approximately one-half of its\ninitial value of/an}bracketle{tSz/an}bracketri}htT≈−/planckover2pi1/4 forβ= 0 and at/an}bracketle{tSz/an}bracketri}htT≈0\nin the in-plane case with β=π/2. While the long-time\naverage of the electron spin is surprisingly almost con-\nstant with respect to K, we observed a strong reduction\nof fluctuations around it for larger nuclear spin baths.\nIn contrast to the long-time average, the decoherence\ntimesTDnever showed a recognizable dependence on the\ninitial states. Instead, the decoherence times depended\ndecisively on the configuration of the nuclear spins in\nthe dot. Long decoherence times were observed for only\none nuclear spin strongly interacting with the electron\nspin, while severalalmostequally couplednuclei leadto a\nvery fast decoherence. Moreover, the decoherence times\nshowed a strong dependence on the number of nuclear\nspins as well as on the orientation of the quantization\naxis. In the out-of-plane case, about 75% of our results\nexperienced decoherence times longer than Tmax= 10ms\nforK= 3. ForK= 8, instead, less than 10% showed no\ndecoherence within this time frame while, in most cases,\ntheelectronspindecayedin lessthan500 µs. Considering\nthe in-plane orientation, already for K= 3 the majority\nof investigated initial state / configuration sets decohere\nwithin 500µs.\nAlthough ourresultswereobtainedfora specificmodel\nof the graphene quantum dot using a Gaussian enve-\nlope function, they could be generalized quite naturally.\nIn our model, the QD was comparably small with a\nsharp boundary. This choice resulted in a steep enve-\nlope function. Physically, this situation corresponds ap-\nproximately to an etched QD. Thinking of larger QDs\nwith smoother boundaries, we expect a flatter envelope\nfunction which gives rise to more nuclear spins interact-\ning comparably with the electron spin. Consequently,\nit becomes more likely to end up with rather low fluc-\ntuations around the long-time average and to find quiteshort decoherence times. In contrast, the realization of\neven smaller dots60with diameters of about 1nm causes\na very steep envelope function. This case should result\nin, at most, one nuclearspin interactingwith the electron\nspin.\nBoth scenarios seem experimentally interesting in or-\nder to engineer QDs for different applications. A13C\nenriched QD could potentially be used to prepare the\nelectron spin very precisely in a certain superposition of\nspin up and down for subsequent experiments. A very\nsmall QD, in contrast, could serve as a storage for the\nelectron spin where very long decoherence times are to\nbe expected.\nBesides these technical points, graphene QDs could\nalso serve as a rich playground to test fundamental as-\npects of quantum mechanics and quantum information\ntheory in an interesting system-bath setup. If we con-\nsider the hosted electron spin as the system, we are able\nto control both its spatial size and its state electrostat-\nically. Moreover, the electron spin can be straightfor-\nwardly addressed via external magnetic fields or in an\noptical way. In contrast, a direct preparation of the state\nof the nuclear spin bath seems challenging. However,\nthe size of the nuclear spin bath can be modified sys-\ntematically by isotopic purification of either12C or13C.\nFinally, as we argued above, the design of the QD en-\nables the experimentalist to manipulate the strength as\nwell as the nature of the system-bath interaction. Thus,\nthese considerationsrendergrapheneQDsto be aflexible\nsystem-bath realization offering a controllable, fermionic\nbath of spin 1 /2 nuclei.\nGiven these opportunities, our setup not only seems\nvery promising for studying a quantum to classical\ncrossover as a function of the bath size in a fermionic en-\nvironment, but also for more advanced concepts of quan-\ntum information theory such as quantum Darwinism91.\nWith the notion of “quantum Darwinism,” W. H. Zurek\nsummarizes his ideas of emergent classicality in a pure\nquantum universe, where the formation of classical and\nquantum system bath correlations, as well as the ac-\ncompanied information exchange matter. Since measure-\nmentsaremostoftenindirect, relyingontheenvironment\nas a mediator, it seems to be evident that the environ-\nment plays the crucial role in the creation of objective\nproperties. However, as far as we know, this theory is\nup to now tested only in few experiments92,93in a rather\nindirectway. Inouropinion, takingadvantageofthecon-\ntrollable spin bath in graphene QDs, this system offers a\nunique opportunity to reveal new insights in the role of\nthe environment in the classical world we experience.\nVI. ACKNOWLEDGEMENTS\nWe would like to thank Manuel Schmidt for valuable\ndiscussions. M.F. and B.T. acknowledge support from\nthe Priority Program 1459 “Graphene” of the DFG and\nfrom the Euro-GRAPHENE Program of the ESF. 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Speci\fc examples of the latter include ferro-\nmagnets, collinear and noncollinear antiferromagnets, general ferrimagnets, and spin glasses. We\nstudy the limit of the exchange-dominated interactions, so that the full system is isotropic in spin\nspace, apart from a possible symmetry-breaking order. A general such interface yields three coe\u000e-\ncients (corresponding to three independent generators of rotations) generalizing the well-established\nnotion of the spin-mixing conductance, which pertains to the collinear case. We develop a nonequi-\nlibrium thermodynamic description of the emerging interfacial spin transfer and its e\u000bect on the\ncollective spin dynamics, while circumventing the usual discussion of spin currents and net spin dy-\nnamics. Instead, our focus is on the dissipation and work e\u000bectuated by the interface. Microscopic\nscattering-matrix based expressions are derived for the generalized spin-transfer coe\u000ecients.\nIntroduction. |The problem of interfacial spin trans-\nfer, along with the associated spin torque [1] and spin\npumping [2, 3], has been central to the \feld of metal-\nbased spintronics for over twenty years [4, 5]. For much\nof its history, the focus has been on the dynamics of\ncollinear ferromagnets. In this case, the spin-mixing con-\nductance has become the key quantity for describing both\nthe spin torque [6] and the spin pumping [2], which have\nsubsequently being recognized as Onsager-reciprocal pro-\ncesses [7, 8]. Recently, a straightforward generalization to\nthe dynamics of collinear antiferromagnets has been put\nforward [9, 10]. In particular, it has been argued [11] that\nat frequencies much smaller than the exchange energy,\nthe interfacial spin transfer is dominated by the rigid\nN\u0013 eel-order dynamics. As such, it can be parametrized\nby an antiferromagnetic spin-mixing conductance [9], in\nclose analogy to the ferromagnetic case, yielding only\nsmall corrections due to the internal canting dynamics.\nIn this Letter, we generalize the description of the low-\nfrequency torque and pumping to noncollinear magnetic\ncon\fgurations. The main underlying assumption is that\nthe interactions near the interface are dominated by the\nspin-isotropic exchange coupling (of arbitrary form, al-\nlowing, in particular, for frustration). At low frequen-\ncies, the associated spin dynamics near the interface can\nbe captured in terms of rigid SU(2) rotations, with the\nspin-mixing conductance generalized to a 3 \u00023 positive-\nde\fnite matrix. (See Fig. 1 for a schematic.) The latter,\nwhen diagonalized along certain principal axes locked to\nthe magnet's spin rotations, can be parametrized by three\nindependent coe\u000ecients. The theory naturally lends it-\nself to noncollinear antiferro- and ferrimagnets as well as\nspin glasses [12{14].\nWe argue that the most streamlined description of spin\ntransfer in this generalized setting is accomplished by\ndeparting from the usual analysis of the interfacial spin\ncurrents and instead focusing on energy. Namely, the\ncentral object of the theory is the Rayleigh dissipation\nfunction for the magnetic heat pumping into the normal\nmetal, o\u000bset by the appropriate work on the collective\nµspin-split normal metal\nrigid spin rotations!interfaceˆG\nwork\ndissipationFIG. 1. A schematic of the magnetic system (right) in contact\nwith a normal metal (left). The nonequilibrium spin state of\nthe metal is parametrized by the (vectorial) spin accumula-\ntion\u0016. The magnet, whose spin arrangement is determined\nby some isotropic exchange Hamiltonian, is described, near\nthe interface ( x= 0), by uniform (rigid) rotations of all spins.\nIts instantaneous nonequilibrium state is thus characterized\nby the (vectorial) frequency of SO(3) rotation !. The 3\u00023\nmatrix ^G, which is governed by the electron re\rection am-\nplitudes at the interface, generalizes the concept of the spin-\nmixing conductance pertinent to the collinear case. The cen-\ntral object of the theory is the modi\fed Rayleigh dissipation\nfunction (2), expressed in terms of ^G,!, and\u0016.\nmagnetic dynamics (either ordered or disordered) by the\nspin-transfer torque. Our perspective is thus based on\nenergetics rather than spin conservation (albeit the lat-\nter is recovered in the appropriate cases). Following a\ngeneral construction, we will check the new methodol-\nogy against the known spin-torque/pumping results for\nthe collinear (anti)ferromagnets, and then apply it to the\ncase of spin glasses.\nPhenomenology. |The collective magnetic dynamics\nnear the interface are parametrized as a rigid rotation\nof spins. This corresponds to the low-frequency limit,\nwhen all the relevant energy scales in the magnet (asso-\nciated with anisotropies, Dzyaloshinsky-Moriya interac-arXiv:1703.04020v2  [cond-mat.mes-hall]  11 Aug 20172\ntions, magnetic \feld, as well as the driving frequency) are\nmuch lower than the microscopic exchange interaction.\nIn this limit, the largest-amplitude dynamics correspond\nto the spin rotations as a whole, along with smooth spa-\ntial textures thereof [11]. The latter are inconsequential\nto our interfacial analysis. For simplicity, we start by\nassuming the magnet is insulating.\nAt low frequencies, the instantaneous dissipation rate\nassociated with the magnetic dynamics depicted in Fig. 1\ncan generally be written as [15, 16]\nP=!T^G!=!iGij!j; (1)\nsumming over the repeated indices. ^Gis a positive-\nde\fnite (symmetric) 3 \u00023 matrix parametrizing heat \row\ninto the normal metal, which (microscopically) depends\non the strength of the exchange coupling across the in-\nterface. The (spin) frame of reference can be rotated to\ndiagonalize ^G! fG1;G2;G3g, whereGi\u00150 are the\n(generally) anisotropic damping parameters for rotations\nabout the corresponding (principal) axes.\nThe usual Rayleigh dissipation function would be given\nby half of the dissipation power (1) [15]. In the presence\nof a spin accumulation \u0016in the metal, however, the in-\nterfacial energy \row gets modi\fed, due to the work done\nby\u0016on the magnetic system [17]. In the special case of\n\u0016=~!, in particular, we see, from the corotating frame\nof reference, that the combined system is in the state of a\nmutual equilibrium [18]. Indeed, the spin accumulation\nis cancelled by the \fctitious \feld ~!due to the rotation,\nwhile the spins in the magnet are static. In this case,\nthe electrons of the metal should not exert any torque\non the magnetic dynamics. The correspondingly modi-\n\fed Rayleigh dissipation function, which accounts for the\nwork done by the spin-accumulation induced torque, is\nthus deduced to be\nR=1\n2~!T^G~!; (2)\nwhere ~!\u0011!\u0000\u0016=~vanishes in the aforementioned state\nof the mutual (dynamic) equilibrium [19]. We will now\ndevelop a microscopic, scattering-matrix based theory for\ncalculating ^G, before applying Eq. (2) to some concrete\nexamples of the (Lagrangian) magnetic dynamics.\nScattering-matrix formalism. |To introduce the rele-\nvant microscopic concepts in the simplest setting, we\nstart with the case of a single quantum transport chan-\nnel in the normal metal. The re\rection matrix thus has\ndimensions 2\u00022:\n^r\u0011fr\u001b\u001b0g; (3)\nwithr\u001b\u001b0standing for the interfacial electron scattering\ncoe\u000ecients for spin \u001b0into\u001b. Having obtained the re-\n\rection matrix in a certain (spin) frame of reference at\ntimet= 0, it would become\n^r(t) =^U(t)^r^Uy(t); (4)at a later time t[denoting ^r(0) by ^r]. The time-\ndependent SU(2) transformation ^U(t) describes the in-\nstantaneous state of the magnet, corresponding to a\nthree-dimensional rotation of the ( t= 0) reference state.\nThe equation of motion for the rotation matrix is\ni~d\ndt^U(t) =!(t)\u0001^ s^U(t); (5)\nwith the initial condition of ^U(0) = 1. ^ sis the electron\nspin operator (i.e., ~=2 times a vector of the Pauli matri-\nces^\u001b) and!is the (vectorial) angular velocity.\nThe energy dissipation rate, for a given instantaneous\nfrequency of rotation !, is given by [20, 21]\nP=~\n4\u0019Trh\n_^r_^ryi\n; (6)\nwhere _^r\u0011d^r=dt is the rate of change of the re\rection\nmatrix. Substituting Eq. (4) into (6), we \fnd\nP=~\n8\u0019Tr\u0002\n!2\u0000^r(!\u0001^\u001b)^ry(!\u0001^\u001b)\u0003\n; (7)\nwhere!\u0011^R\u00001!,^Rbeing the SO(3) rotation matrix\ncorresponding to the SU(2) spin rotation ^U, at timet.\nWe thus conclude, according to the de\fnition (1), that\nGij=~\n4\u0019\u0012\n\u000eij\u00001\n2Trh\n^r^Rii0^\u001bi0^ry^Rjj0^\u001bj0i\u0013\n;(8)\nor in matrix form,\n^G=~\n4\u0019^R^g^R\u00001; (9)\nwhere\ngij\u0011\u000eij\u00001\n2Tr\u0002^r^\u001bi^ry^\u001bj\u0003\n: (10)\nNote that in order to retain only the relevant symmet-\nric part of ^G, the matrix ^gentering Eq. (9) needs to be\nsymmetrized [i.e., ^g!(^g+^gT)=2], which should be un-\nderstood as implicit in the above de\fnition [22].\nIn the simplest case of a collinear (ferro-, antiferro-, or\nferri-)magnet with the magnetic order oriented along the\nzaxis, the matrix (10) simpli\fes tremendously to\n^g!gmixf1;1;0g(collinear order) ; (11)\nwhere g mix\u00111\u0000Rer\"\"r##\u0003is the (real part of the)\nspin-mixing conductance for a single quantum channel\n[2]. The gzzmatrix element is zero as rotations around\nthezaxis commute with the collinear order.\nMultichannel leads. |It is straightforward to generalize\nour treatment to an arbitrary number Nof transverse\nquantum channels in the normal-metal lead. In this case,\nthe rotation matrix ^Uintroduced in Eq. (4) should be\nthought of as 2 N\u00022Nblock-diagonal with the usual3\nSU(2) rotations along the diagonal. Repeating our steps,\nwe reproduce Eq. (9) for the 3 \u00023 dissipative tensor ^G,\nbut with the 3\u00023 matrix ^gnow given by\ngij=N\u000eij\u00001\n2X\nmnTr\u0002^rmn^\u001bi^ry\nmn^\u001bj\u0003\n: (12)\nHere, ^rmnis the 2\u00022 re\rection matrix for electrons scat-\ntering from channel ninto channel m, which run from\n1:::N . As before, a symmetrization with respect to the\ni;jindices is implicit on the right-hand side of Eq. (12).\nThis equation, along with Eqs. (1), (2), and (9), forms a\ncentral result of the present work.\nFor the special case of a collinear order, this again gives\nEq. (11), with the familiar expression for the spin-mixing\nconductance [2, 21]:\ngmix=N\u0000ReX\nmnr\"\"\nmnr##\u0003\nmn (collinear order) :(13)\nIn the ferro- or ferrimagnetic cases, this spin-mixing con-\nductance is generically nonzero, so long as electrons expe-\nrience some exchange upon re\rection, which would make\nr\"\"\nmn6=r##\nmn. In the antiferromagnetic case, the spin-\nmixing conductance is also generally \fnite, but is domi-\nnated by the umklapp scattering channel, in the simplest\ncase of an ideal compensated interface with a transla-\ntional antiferromagnetic sublattice symmetry [9].\nFor a general noncollinear and multichannel case, ^g\ncan be diagonalized to yield three non-negative eigenval-\nues. The corresponding principal axes de\fne a natural\nmagnet-\fxed frame of reference for the analysis of the\ninterfacial spin torque and pumping. We can suppose\nthat our laboratory coordinate system is chosen to di-\nagonalize ^g(corresponding to the magnetic orientation\natt= 0), with subsequent dynamics yielding a rotated\ndamping tensor (9).\nCollinear order. |Equipped with the (torque-\nmodi\fed) Rayleigh dissipation function (2), we can\nreadily construct the boundary conditions for the\nappropriate magnetic dynamics. To that end, we\nneed to start with the bulk Lagrangian of the mag-\nnet. For a collinearly-ordered material, the general\n(low-temperature) Lagrangian density is given by [24]\nL=\u0000sa(n)\u0001@tn+\u001f\n2(@tn\u0000\rn\u0002B)2\u0000A\n2(@in)2\u0000E(n);\n(14)\nwhere nis the directional order parameter (s.t., jnj\u00111),\nslongitudinal (along n) spin density, \rgyromagnetic ra-\ntio,Bmagnetic \feld, Aorder-parameter sti\u000bness, index i\nruns over spatial (Cartesian) coordinates, \u001fis related to\nthe transverse (to n) spin susceptibility, and E(n) is the\nlocal energy density, including anisotropies and Zeeman\ncoupling\u0000\rsn\u0001Bto the longitudinal magnetic moment.\na(n) is a vector potential produced on a unit sphere by a\nmagnetic monopole of unit charge. Antiferromagnets cor-\nrespond to s= 0, while low-frequency dynamics in ferro-\nand ferrimagnets can be obtained by setting \u001f!0.The Euler-Lagrange equation of motion is then given\nby\n@\u0017@L\n@(@\u0017n)\u0000@L\n@n+@R\n@(@tn)= 0; (15)\nwhere\u0017runs over all space-time coordinates. R \u0011\nR\u000e(x) should be understood as the spatial density of the\nRayleigh dissipation function (2), with Rhere de\fned\nper unit area of the interface placed at x= 0 (with the\nmagnet corresponding to x>0; see Fig. 1). For the case\nof a collinear order,\nR=~gmix\n8\u0019(@tn\u0000\u0016\u0002n=~)2(collinear order) ;(16)\nwheregmixis the interfacial spin-mixing conductance per\nunit area. Using Lagrangian (14), we \fnd for the equa-\ntion of motion (taking care to respect the constraint\njnj\u00111):\n@t(sn+m)\u0000\rm\u0002B\u0000n\u0002\u0000\nA@2\nin\u0000@nE\u0001\n=\u001c\u000e(x);(17)\nwhere m\u0011\u001fn\u0002(@tn\u0000\rn\u0002B) is an auxiliary variable\ncorresponding physically to the transverse spin density\n(obtained from @BL=\r, which corresponds also to the\ngenerators of rotations dictated by the Lagrangian (14)\n[13]). The net spin density is thus given by sn+m. The\nright-hand side,\n\u001c\u0011~gmix\n4\u0019n\u0002(\u0016\u0002n=~\u0000@tn); (18)\nis understood as the dissipative torque (spin-current den-\nsity) produced by the electrons scattering o\u000b the inter-\nface. Equations (17) and (18) reproduce and connect\nthe standard ferromagnetic [4] and antiferromagnetic [11]\nlimits (corresponding respectively to setting m!0 and\ns!0). Integrating the equation of motion (17) near the\ninterface, we \fnally get\n\u0000An\u0002@xn=\u001c; (19)\nre\recting the spin continuity at the interface [25]. The\nwork done by the torque (18), per unit area and time, is\n@tw\u0011\u001c\u0001n\u0002@tn=~gmix\n4\u0019(\u0016\u0002n=~\u0000@tn)\u0001@tn:(20)\nThe second term, /\u0000(@tn)2, here, is just the ordinary\nGilbert damping endowed by the metallic reservoir [2].\nThe \frst term re\rects the antidamping nature of the\nspin-transfer torque, for the appropriate orientation of\nthe spin accumulation.\nSpin glass. |We consider now the opposite extreme of\na disordered magnet, in which the orientation of the in-\ndividual spins are randomly distributed due to frustrated\nexchange interactions. The full SO(3) group of spin ro-\ntations is broken in the ground state, characterized by a\nmatrix or Edwards-Anderson-like order parameter [26].4\nSlow (in a hydrodynamic sense) deviations from equi-\nlibrium are represented by a vector \u0012= (\u0012x;\u0012y;\u0012z) of\nrotation angles along the principal axes of ^Gde\fning the\nlaboratory frame. The linearized dynamics is captured\nby the Lagrangian density [12, 13, 22]\nL=\u001f\n2(@t\u0012+\rB)2+\u001f\n2@t\u0012\u0001(\rB\u0002\u0012)\u0000A\n2(@i\u0012)2\u0000E(\u0012):\n(21)\nIn the absence of anisotropies and net magnetization at\nequilibrium ( B= 0), Eq. (21) predicts 3 independent\npolarizations of spin waves with a linear dispersion [23].\nFor a macroscopically isotropic spin con\fguration, we\nexpect ^G/^1 in the presence of an exchange-dominated\ncoupling with the normal-metal electrons. The linearized\nRayleigh function (per unit area of the interface) then\nreads (at\u0012!0)\nR=~g\n8\u0019\u0010\n@t\u0012\u0000\u0016\n~\u00112\n(spin glass) ; (22)\nwhereg\u0011g1=g2=g3are the eigenvalues of ^gin\nEq. (12), normalized by the area. The equation of motion\n(for a static B) reduces to\n@tm\u0000\rm\u0002B\u0000A@2\ni\u0012+@\u0012E=~g\n4\u0019\u0010\u0016\n~\u0000@t\u0012\u0011\n\u000e(x);(23)\nwhere m\u0019\u001f(@t\u0012+\rB) is the spin density ( \u0011@BL=\r).\nAs before, this may be interpreted as a continuity equa-\ntion for spin \row, subject to local precession and in-\nterfacial spin transfer. Notice that the pairs ( \u0012\u000b;m\u000b)\nare canonically conjugate, a consequence of the fact that\nthe spin-density components de\fne generators of the in-\n\fnitesimal rotations in the magnetic system. Integrating\nEq. (23) near the interface leads to the spin-\rux continu-\nity at the interface:\n\u0000A@x\u0012=~g\n4\u0019\u0010\u0016\n~\u0000@t\u0012\u0011\n: (24)\nThis generalized phenomenology enables the study of\nspin signals transmitted through disordered magnets,\nwhich can be probed in a set-up like the one shown in\nFig. 2. The spin accumulation \u0016induced by the spin Hall\ne\u000bect in one of the metals triggers the coherent precession\nof randomly oriented spins in the glass phase, while the\nsignal is collected in a second terminal by means of the\nreciprocal pumping e\u000bect. The steady-state precession\nfrequency \n=@t\u0012is proportional to the nonequilibrium\nspin density, \u001f\n, induced in the system. In the geom-\netry of Fig. 2(a), the frequency is easily obtained [28]\nby balancing the boundary conditions (24) with the bulk\nGilbert damping: ~\n=\u0016=(2 + 4\u0019\u000bsL= ~g). In the ab-\nsence of anisotropies, the signal decays only algebraically\nwith the distance between the terminals L, due to the\nbulk damping \u000b, in contrast to the (thermal) spin waves\nin a collinear magnet [27]. Spin glasses provide a (po-\ntentially) more versatile platform for long-ranged signal\n(a)\n(b)FIG. 2. Schemes for lateral (a) and vertical (b) spin injec-\ntion/detection in electrically insulating spin glasses. The sig-\nnal is sustained by the coherent precession of randomly ori-\nented spins, triggered by the spin accumulation \u0016in the left\nterminal and collected in the right terminal by the reciprocal\npumping e\u000bect. The steady-state solution for the precession\nangle about\u0016reads\u0012(t;x) = \nt+\u0012(x), where\u0000A@x\u0012corre-\nsponds to the spin-current density in the bulk of the magnet.\nThe (minus) divergence, A@2\nx\u0012, of the spin current in the bulk\nof the magnet balances the spin damping rate, \u000bs\n,\u000bbeing\nthe Gilbert-damping constant and sthe high-\feld saturated\nspin density. The precession frequency \n is proportional to\nthe nonequilibrium spin density along \u0016and is determined by\nthe boundary condition in Eq. (24). The measured electrical\ndrag signal is negative in (a) and positive in (b), and would\nfollow the numerical estimates of Ref. [28].\ntransmission, in comparison to a spin-super\ruid state in\neasy-plane magnets [28]. In particular, they o\u000ber \rexibil-\nity regarding the spin injection and detection geometries,\nas illustrated in Fig. 2.\nDiscussion. |The key element of the theory is the\nmodi\fed Rayleigh dissipation function (2), which cap-\ntures the e\u000bects of both the spin pumping into the metal\nreservoir and spin torque by its spin accumulation. The\nformer is directly linked to the dissipation of energy into\nthe normal lead, while the latter to the work on the mag-\nnetic dynamics by the spin-polarized electrons. When\nthe spin accumulation \u0016exceeds the natural precession\nfrequency ~!, this work can e\u000bectively reverse the damp-\ning, potentially leading to magnetic instabilities and self-\noscillations [1, 5]. (Additional bulk damping of the ma-\nterial would raise the threshold for such instabilities.) In\ngeneral, the spin accumulation \u0016needs to be calculated\nself-consistently with the spin-current density js=\u0000\u001c\n\rowing into the normal metal.\nWhile our focus has been on electrically insulating\nmagnets, a generalization to conducting magnets is pos-\nsible by considering transmission as well as re\rection of5\nelectrons [2]. For the case of su\u000eciently thick magnets,\nhowever, the transmission can generally be expected to\nlead to a full dephasing of spin transport [4], bringing us\nback to Eq. (12), which is governed by the re\rection co-\ne\u000ecients only. Finally, we remark that through Eqs. (1)\nand (2) we invoked only the dissipative coupling between\nthe magnet and the normal-metal reservoir. Such dissi-\npative spin transfer is known to be the most prominent\ninterfacial process for collinear ferromagnets [4, 21] and\nantiferromagnets [9], which is responsible for dynamic\ninstabilities [5], thermal-magnon and super\ruid spin in-\njection [9, 28], as well as the spin Seebeck physics trig-\ngered by heat biases [29]. We expect this to naturally\nextend to the noncollinear case. The nondissipative cou-\npling, which is quanti\fed through the imaginary part of\nthe spin-mixing conductance in the collinear case, can be\nformally captured by rede\fning the e\u000bective Lagrangian\n(or, equivalently, Hamiltonian or free energy) of the cou-\npled system and renormalizing the reactive coupling coef-\n\fcients [4]. While it is in principle possible to account for\nthis both phenomenologically and microscopically in the\nscattering-matrix formalism [30], it is beyond our imme-\ndiate interests. Future works should also address gener-\nalizations of our theory to nonrigid exchange dynamics in\nsoft magnets, which may also pump spin and contribute\nto dissipation, and the role of strong spin-orbit interac-\ntions at the interface.\nWe are grateful to Se Kwon Kim and Pramey Upad-\nhyaya for insightful discussions. This work was supported\nby the U.S. Department of Energy, O\u000ece of Basic Energy\nSciences under Award No. DE-SC0012190.\n[1] J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989); J.\nMagn. Magn. Mater. 159, L1 (1996); L. Berger, Phys.\nRev. B 54, 9353 (1996).\n[2] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[3] S. Mizukami, Y. Ando, and T. Miyazaki, Jpn. J. Appl.\nPhys. 40, 580 (2001); R. Urban, G. Woltersdorf, and\nB. Heinrich, Phys. Rev. Lett. 87, 217204 (2001); B. Hein-\nrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Ur-\nban, and G. E. W. Bauer, Phys. Rev. Lett. 90, 187601\n(2003).\n[4] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005), and refer-\nences therein.\n[5] D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater.\n320, 1190 (2008), and references therein.\n[6] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys.\nRev. Lett. 84, 2481 (2000).\n[7] Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77,\n134407 (2008).\n[8] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and P. J.\nKelly, in Spin Currents , edited by S. Maekawa, S. O.\nValenzuela, E. Saitoh, and T. Kimura (Oxford Univer-\nsity Press, Oxford, 2012) pp. 87{135.[9] S. Takei, B. I. Halperin, A. Yacoby, and Y. Tserkovnyak,\nPhys. Rev. B 90, 094408 (2014).\n[10] R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Phys. Rev.\nLett. 113, 057601 (2014).\n[11] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama,\nT. Ono, and Y. Tserkovnyak, \\Antiferromagnetism:\nThe next \ragship magnetic order for spintronics?\"\narXiv:1606.04284.\n[12] B. I. Halperin and W. M. Saslow, Phys. Rev. B 16, 2154\n(1977).\n[13] A. F. Andreev and V. I. Marchenko, Sov. Phys. Uspekhi\n23, 21 (1980).\n[14] H. V. Gomonay, R. V. Kunitsyn, and V. M. Loktev,\nPhys. Rev. B 85, 134446 (2012).\n[15] L. D. Landau and E. M. Lifshitz, Statistical Physics, Part\n1, 3rd ed., Course of Theoretical Physics, Vol. 5 (Perga-\nmon, Oxford, 1980).\n[16] Notice that the Rayleigh function is a quadratic form of\nthe angular velocity of the order parameter. The work\ncarried out by the dissipative force \u001c=\u0000@R=@!equals\n\u001c\u0001!=\u00002R.\n[17] S. K. Kim, S. Takei, and Y. Tserkovnyak, Phys. Rev. B\n92, 220409 (2015).\n[18] Y. Tserkovnyak and A. Brataas, Phys. Rev. B 71, 052406\n(2005).\n[19] When Ris di\u000berentiated with respect to magnetic dy-\nnamics, we will obtain a term /!\u0000\u0016=~in the resultant\nEuler-Lagrange equation of motion. This establishes an a\nposteriori proof of our modi\fed dissipation function (2).\n[20] M. Moskalets and M. B uttiker, Phys. Rev. B 66, 035306\n(2002).\n[21] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008).\n[22] See the Supplementary Material for a detailed\nparametrization of the single-channel scattering problem\nand a derivation of the spin-glass Lagrangian (21).\n[23] In the presence of a \feld saturated spin density s=\u001f\rB,\nthe linear spin-wave mode polarized along ssurvives,\n!k=p\nA=\u001fjkj, whereas the two remaining modes are\nhybridized, leading to a gapped, !+\n?\u0018 jsj=\u001f, and a\nquadratically dispersing, !\u0000\n?\u0018Ajkj2=jsj, branches, sim-\nilarly to a ferrimagnet. The long-ranged spin transmission\nproposed in Fig. 2 remains when \u0016is aligned with s(sus-\ntaining also a possible collinear anisotropy along the same\ndirection).\n[24] A. Auerbach, Interacting Electrons and Quantum Mag-\nnetism (Springer-Verlag, New York, 1994).\n[25] Note that even though our theory is constructed based\non energetics, we are in the end able to deduce the as-\nsociated spin currents from the equation of motion, once\nthe spin density is identi\fed according to the Lagrangian\ndescription. As illustrated by the above examples, the\nappropriate spin density can be deduced even if it is not\nexplicitly included among the original Lagrangian vari-\nables [13].\n[26] S. F. Edwards and P. W. Anderson, J. Phys. F: Metal\nPhys. 5, 965 (1975).\n[27] L. S. Cornelissen, J. Liu, R. A. Duine, J. Ben Youssefm,\nand B. J. van Wees, Nature Phys. 11, 1022 (2015).\n[28] S. Takei and Y. Tserkovnyak, Phys. Rev. Lett. 112,\n227201 (2014).\n[29] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Na-\nture Mater. 11, 391 (2012); S. Ho\u000bman, K. Sato, and6\nY. Tserkovnyak, Phys. Rev. B 88, 064408 (2013). [30] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. B 84, 054416 (2011)."    },    {        "title": "1607.03263v1.Thermal_spin_dynamics_of_yttrium_iron_garnet.pdf",        "content": "Thermal spin dynamics of yttrium iron garnet\nJoseph Barker1and Gerrit E.W. Bauer1, 2, 3\n1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n2WPI-AIMR, Tohoku University, Sendai 980-8577, Japan\n3Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\nYttrium Iron Garnet is the prototypical material used to study pure spin currents. It is a complex material with\n20 magnetic atoms in the unit cell. Almost all theories and experimental analysis approximates this complicated\nmaterial to a simple ferromagnet with a single spin wave mode. We use the method of atomistic spin dynamics\nto study the temperature evolution of the full 20 mode exchange spin wave spectrum. Our results show a strong\nfrequency dependence of the modes in quantitative agreement with neutron scattering experiments. We find this\ncauses in a reduction in the net spin pumping due to the thermal occupation of optical modes with the opposite\nchirality to the FMR mode.\nIntroduction – Spin transport in magnetic insulators has at-\ntracted much interest since the experimental demonstration\nof the Spin Seebeck effect (SSE) [1]. The field of study is\nbroadly termed spin caloritronics and encompasses the cou-\npling between spin, charge and heat currents [2]. Typical ex-\nperimental setups consist of bilayers made of a ferrimagnetic\ninsulator coated with a thin metallic film possessing a large\nspin Hall angle. Of special interest is the ferrimagnetic insu-\nlator Yttrium Iron Garnet (YIG) due to its very low damping,\n\u000b\u001910\u00005and therefore long-lived spin waves [3, 4]. The\nmagnetism is carried by localised Fe moments in 8 tetrahe-\ndral (minority) and 12 octahedral (majority) oxygen cages per\nunit cell, with an anti-parallel ferrimagnetic state between the\ntwo coordinations. However, most recent theories and exper-\niments treat YIG as a ferro -magnet with a single, parabolic\nspin wave mode [5, 6], simply because the influence of YIG’s\ncomplex electronic and magnetic structure on spin transport\nis not known. Ohnuma et al. [7] introduced a simple two\nmode model to describe the basic aspects of ferrimagnetic dy-\nnamics, but its spectrum bares little resemblance to that of\nYIG [3]. In this Letter we show that the frequencies and line\nwidths of spin waves in YIG are strongly temperature depen-\ndent. We find that at room temperature higher frequency spin\nwave modes are significant occupied. Optical modes with op-\nposite chirality to the acoustic mode turn out to have a dis-\nproportionate effect on spin transport and must be taken into\naccount when modelling or interpreting, for example, the spin\nSeebeck effect.\nExperimentally, techniques such as Brillouin light scatter-\ning give access to the long wavelength, GHz frequency, dipole\nspin waves [8]. Studying the THz frequency ‘exchange’ spin\nwave modes requires expensive inelastic neutron scattering\nexperiments [9]. The role of the high frequency ‘thermal’\nmagnons remains poorly understood, despite their importance\nin interpreting recent experiments [10–14]. Since spin wave\nspectra are material specific, improving the understanding of\ngeneral characteristics could aid in the selection of materi-\nals to progress towards applications such as sensing and heat\nscavenging [15, 16].\nAtomistic model – YIG is an insulator with a large elec-\ntronic band gap. The Fe3+ion d-shells are half-filled with\nspin value of S= 5=2and can be modeled with the Heisen-berg Hamiltonian\nH=\u00001\n2X\nijJijSi\u0001Sj\u0000X\ni\u0016s;iB\u0001Si: (1)\nHereJijis the isotropic exchange energy between (normal-\nized) spins withjSij= 1, where the indices i;jenumerate\nsites on the YIG crystal lattice. The Fe3+magnetic moment is\n\u0016s=gp\nS(S+ 1)\u0016B= 5:96\u0016B(\u0016Bis the Bohr magneton).\nAn external field Bz= 0:01T defines the spin quantization\n(z) axis. The crystal magnetic anisotropy and dipolar interac-\ntion of YIG are negligibly small compared to THz frequencies\nof thermal spin waves and disregarded here. Neutron scat-\ntering [9] indicates that nearest neighbor exchange dominates\nand we adopt Jad=\u00009:60\u000210\u000021J,Jdd=\u00003:24\u000210\u000021J\nandJaa=\u00000:92\u000210\u000021J [3], where the subscripts refer\nto the majority ( d) and minority ( a) spins. All couplings are\nantiferromagnetic, which in principle could produce a frus-\ntrated state, but the dominating Jadis so large that a per-\nfectly anti-collinear ground state is stable. By Metropolis\nMonte-Carlo calculations [17] we compute the magnetization\nm=1\nNaP\niSa;i+1\nNdP\niSd;ias a function of temperature\nand find a Curie temperature of 520K close to the experimen-\ntal value of 559K [18] and in agreement with other calcula-\ntions [19]. Throughout this work we address a bulk system by\nperiodic boundary conditions for a supercell of repeated unit\ncells.\nThe spin wave spectrum can be obtained by diagonalizing\nthe Hamiltonian (1) [3] but non-linear effects such magnon-\nmagnon interactions, thermal noise and damping require a\ndifferent approach. At finite temperatures, the spin moments\nfluctuate around the local equilibrium state. The exchange\ncoupling correlates the motion of all moments, giving rise to\ncollective spin waves. The dynamical structure factor (or spin\nwave power spectrum) as measured by inelastic neutron scat-\ntering is the Fourier transform of spatio-temporal spin-spin\ncorrelation functions. Here we compute the local spin dy-\nnamics, and from them the structure factor, thereby avoiding\na magnon ansatz and including magnon-magnon interactions\nto all orders.\nThe spin dynamics is described by the atomistic Landau-arXiv:1607.03263v1  [cond-mat.mtrl-sci]  12 Jul 20162\n0510152025\nNΓHf(THz)\nNΓH\nT=1 0 0K\nNΓH\nT=2 0 0K\nNΓH\nT=3 0 0KT=1 K\nkBT\nFIG. 1. (Color online) YIG spin wave spectrum as calculated for different temperatures. The dashed lines mark 2\u0019~f=kBT. The red/blue\ncoloring denotes the +/- mode chirality relative to the magnetization direction.\nLifshitz-Gilbert equation (LLG) based on the Hamiltonian (1)\n@Si\n@t=\u0000j\rj\n(1 +\u00152)\u0016s;i(Si\u0002Hi+\u0015Si\u0002Si\u0002Hi):(2)\nThe local effective field is Hi=\u0018i\u0000@H=@Siwhere\u0018iis a\nstochastic term with\nh\u0018i(t)i= 0;h\u0018i(t)\u0018j(t0)i=\u000eij\u000e(t\u0000t0)2\u0015\u0016skBT=\r: (3)\nwhereh\u0001\u0001\u0001i denotes the statistical time average. The gyro-\nmagnetic ratio is \r= 1:76\u00021011rad s\u00001T\u00001. The mea-\nsured Gilbert damping \u000bof ferrimagnets is a combination\nof the damping parameters of the sublattices. According to\nWangsness’ formula \u000b= (\u0015aMa+\u0015dMd)=(Md\u0000Ma)[20].\nHere we use the damping parameter \u0015=\u0015a=\u0015d= 2\u000210\u00005\ngiving\u000b= 10\u00004, which is a typical (although not record)\nvalue. We solve the stochastic Langevin equation (2) with\nthe Heun method, using a time step of \u0001t= 0:1fs. The\nlow damping requires careful equilibration. We first use a\nMetropolis Monte-Carlo algorithm to converge to the equilib-\nrium magnetization. We then integrate the LLG dynamically\nfor1ns, which is sufficient to achieve a steady state regime in\nthe presence of noise. Finally we collect data for 0:1ns which\nis Fourier transformed in space and time to distill the spectral\ninformation.\nIn our classical formalism the thermal noise is white, and\nthrough equipartition the system obeys Rayleigh-Jeans rather\nthan Planck or Bose-Einstein statistics for magnons. Our re-\nsults at low temperatures and high energies do not capture\npossible quantum effects. However, at elevated temperatures\nquantum effects are suppressed by magnon-magnon interac-\ntions and classical spin models can be expected to give a good\nagreement with experiments.\nThe spin wave spectrum is revealed in terms of structures in\nthe space-time Fourier transform of the spin fluctuations. TheFourier representation of the spin dynamics reads\nSk(q;!) =1p\n2\u00191\nNcNcX\nn=1X\nreiq\u0001(r\u0000pn)Z+1\n\u00001ei!tSk;n(r;t)dt\n(4)\nwhere pnis the position of the n-th spin (of a total of Nc=\n20)in the unit cell and k=x;y;z . Theq-space resolution\nis determined by the system size of 64\u000264\u000264unit cells\n(5,242,880 spins).\nThermal spin motive force is proportional to the trans-\nverse dynamical susceptibility or equal-time spin correla-\ntion function [5] which is related to the correlation functionD\n_Sy;i(0)Sx;i(0)\u0000_Sx;i(0)Sy;i(0)E\nthat is equivalent to the\nwave vector and frequency integral of the power spectrum\nh!Sx(q;!)S\u0003\ny(q;!)i\u0000h!Sy(q;!)S\u0003\nx(q;!)i:This correla-\ntion function is a Stoke’s parameter V=\u00002Im(SxS\u0003\ny)and\nthe sign identifies the chirality or polarization of the eigenvec-\ntors [21]. The label ‘+’ chirality implies counter-clockwise\nrotation i.e. the precession direction of a magnetic moment in\nan applied field e.g. under FMR.\nSpin wave spectrum – In Fig. 1 we display the calculated\nspin wave spectra for different temperatures. The coloring in-\ndicates the chirality of the modes. Red modes have the ‘+’\nchirality, while the blue modes precess in the opposite direc-\ntion. The latter (optical) modes are energetically costly due\nto the strong exchange field between the two sublattices, so\nemerge only at frequencies above the exchange splitting.\nAt the lowest temperature considered (1 K) the ampli-\ntude of the excitations (or number of magnons) is small and\nmagnon interactions are very weak. The calculated spec-\ntrum therefore agrees well with the linearized spin wave the-\nory [3]. The following discussion is focused on the two\nnearly rigidly shifted parabolic modes with opposite chirality.\nThe lowest frequency mode is the ferromagnetic-like acous-\ntic mode. The second mode is blue shifted by a spin wave\ngap caused by the exchange field between the two sublattices\n\u0001 = 3JadhSz;ai\u00002JadhSz;di\u001910THz and is the optical,3\n0246810\n0 100 200 300 400 500\n∆ω(THz)\nT (K)ASDPlant 1977\nFIG. 2. (color online). Temperature dependence of the spin wave\ngap. Blue circles are the calculations in this work, red points are the\nneutron scattering data of Plant [9]. The shaded area marks ~! <\nkBT. The dashed line is the reduction in exchange field due to spin\nfluctuation.\nantiferromagnetic-like mode between the two sublattices. We\nobserve 5 additional flat modes in the 5 and 10 THz range that\nare thermally excited at room temperature. Since their mass is\nvery high, they are expected to only weakly contribute to spin\ntransport than the highly dispersive ones.\nThermal fluctuations reduce the magnetic order hSz;ai,\nhSz;diand thereby the exchange field \u0001, as observed in Fig. 1.\nOur calculations agree very well with neutron scattering ex-\nperiments [9] (Fig. 2). We emphasize that our exchange con-\nstantsJijare not temperature dependent - the effect is caused\nby statistical mechanics alone. Below ~!=kBT(dashed\nline) spin waves modes are thermally occupied. The occupa-\ntion above this line is small and technically governed by quan-\ntum statistics, but unimportant for (near to) equilibrium prop-\nerties. Far below room temperature only the ferromagnetic-\nlike acoustic mode is significantly occupied and the use of\na single parabolic spin wave model is justified. However, at\nroom temperature and above, this approach breaks down and\nthe effects reported here must be taken into account in order\nto understand the properties of YIG.\nOur method allows a comprehensive treatment of the non-\nlinear thermodynamics from low temperatures up to the mag-\nnetic phase transition. The magnon ansatz often used to de-\nscribe spin dynamics is based on the Holstein-Primakoff trans-\nformation expanded to low order in the number of magnons.\nA larger number of thermally excited magnons initially can\nbe captured by magnon-magnon interactions. These reduce\nthe magnon lifetime as reflected by an increased broaden-\ning of the spectral function. When the broadening becomes\nlarger than the spin wave energy splittings, the magnon con-\ncept breaks down completely. This picture is illustrated by\nFig. 1 in which we observe that with increasing temperature\nthe flat modes completely melt into an incoherent background\nthat can no longer be interpreted in terms of spin waves. How-\never, the coherence of the fundamental acoustic and optical\na)b)\n0510152025\nNΓHf(THz)\nNΓH\nFIG. 3. (color online). Examples of the on-site correlations from\ndifferent points in the unit cell a) FeA at (0;1=2;0)and b) FeA at\n(1=2;0;0)in fractional coordinates.\nmodes turns out to be remarkably robust against thermal fluc-\ntuations: the line widths as well as the parabolic curvature,\na metric of spin wave stiffness, hardly change with tempera-\nture. Our formalism can provide information about individual\nlocal moment fluctuations irrespective of the coherence of the\nspin wave excitations, which can be used to shed light into this\nremarkable behavior. In Fig. 3 we plot the site-resolved con-\ntributions to the power spectrum. We clearly see that the fun-\ndamental acoustic and optic modes are homogeneously spread\nover the unit cell, while the other modes are strongly localized\non one of the local moments that are consequently coupled by\nthe smallJaaandJddexchange constants. The spin waves\nwith a flat dispersion are slack and susceptible to thermal agi-\ntation.\nThe resilience of both fundamental modes agrees with ob-\nservations [22] but appears to contradict common wisdom\nthat the spin waves stiffness decreases linearly with tem-\nperature [23]. The reason for the anomalous behaviour of\nYIG have not been well understood, especially in the higher\ntemperature regime where the optical magnons become in-\nvolved [3]. The present results indicate that the decoupling\nof the fundamental extended modes from the floppy localized\nmodes, as discussed above, protects the spin wave stiffness as\nwell as the lifetime.\nSpin pumping – is the emission of spin currents from a mag-\nnetic material into metal contacts by the magnetization dy-\nnamics. The latter can be driven for example by microwaves\nand ferromagnet resonance or by thermal excitations. In the\npresence of a temperature difference, the imbalance between\nthermal spin pumping and spin transfer torque from the metal\ncauses net spin currents that can be converted into voltages by\nthe inverse spin Hall effect, i.e. the spin Seebeck effect. Spin\npumping and SSE are interface effects that are proportional\nto the spin-mixing interface conductance and equal time and\nspace spin correlation functions of the magnet at the interface\n[5]. When the interface correlations are identical to that in4\nthe bulk, i.e. are not perturbed by the presence of the inter-\nfaces (Golden Rule or tunneling approximation), our simula-\ntions provide direct access to the spin Seebeck effect except\nfor a constant that contains the spin mixing conductance, spin\nHall angle and temperature gradient. Our approach does not\naddress surface spin wave states or spin-pumping induced en-\nhanced damping. For samples much thicker than the magnon\nrelaxation lengths thermal gradients induce magnon spin and\nheat transport in the bulk of the ferromagnet, which have been\nheld responsible for e.g. YIG layer thickness dependence of\nthe SSE [6, 10, 12]. These effects are beyond the scope of the\npresent study, however.\nThe spin mixing conductance of an interface between a\nmetal and a magnetic insulator with local moments is gov-\nerned by the exchange integrals between the local moments\nand the conduction electrons [24]. The DC pumped spin cur-\nrent then reads\nhIiA=~\n4\u0019Reg\"#\nNANAX\nihSi\u0002_Sii (5)\nwhereNAare the number of moments at the interface A. Av-\neraging over the (weak) dependence of the mixing conduc-\ntance on the specific interface [24] and to leading order in the\ntransverse dynamics hIi=~Reg\"#S=(4\u0019);where the equal\ntime and space correlation function\nS=1\nNcNcX\nih_Sy;iSx;i\u0000_Sx;iSy;ii (6)\ncan be obtained either directly from the simulations or by in-\ntegrating the spectral power functions discussed above over\nfrequency and momenta. At equilibrium this current is ex-\nactly compensated by the thermal spin transfer torque-induced\ncurrents from the metal. The observed spin Seebeck voltage\nthen reads, to leading order, \u0001VSSE\u0018S@T; where@Tis\nthe temperature gradient over the sample. We do not address\nthe proportionality constant here. The correlation functions\nthat govern electric spin injection, chemical potential-driven\ntransport [13], which might be relevant for the spin Seebeck\neffect [25], as well as bulk magnonic transport that dominates\nthe signal for thick ferromagnets, are subject of ongoing study.\nIn Fig. 4 we plotSas a function of temperature for YIG as\nwell as for a hypothetical ferromagnet with parallel moments\non a simple BCC lattice (FM) that is tuned to the same satura-\ntion magnetization and Curie temperature as YIG.\nBoth the ferrimagnet YIG and the FM show similar fea-\ntures. An increasing temperature initially increases spin\npumping by enhanced thermal agitation. Close to the criti-\ncal temperature spin pumping collapses to zero at TCtogether\nwith the net magnetization. The YIG spin pumping is max-\nimized close to 300 K, i.e. far from the critical region of\nthe phase transition. This is caused by the increasing ther-\nmal occupation of the high frequency optical mode with op-\nposite chirality, which plays a significant role above 300 K\n(see Fig. 2). A similar effect causes the low temperature sign\n020406080100120140\n0 100 200 300 400 500 600\nS(s≠1)T( K )FM\nYIGFIG. 4. (color online). Spin pumping (Eq. 6) of YIG and a hypothet-\nical ferromagnet into a metal contact as a function of temperature.\nchange of the spin Seebeck signal in GdIG [14]. On the other\nhand, the thermal occupation of the optical modes actually en-\nhances the net magnetization rather than decreasing it [3].\nUchida et al. [26] observed a power law \u0001VSSE\u0018(TC\u0000\nT)3close to the Curie temperature TC;while the magnetiza-\ntion scales\u0018(TC\u0000T)1\n2as expected from mean-field theory.\nHere we find the same critical exponent1\n2for both the mag-\nnetization and SSE effect, which can be rationalized in terms\nof the spin wave gap that is closed in proportion with the ex-\nchange field. The anomalous scaling found experimentally\nmust be attributed to physical effects not related to the dy-\nnamic susceptibility. A strong suppression of spin transport in\nthe ferrimagnet by large thermal fluctuations is a possible ex-\nplanation. On the other hand, the present spin Seebeck theory\nis based on the Landau-Lifshitz-Gilbert equation which might\nnot hold close to the phase transition. More study is needed to\nunderstand the spin Seebeck effect close to the critical regime.\nConclusion – We present atomistic simulations of the spin\ndynamics of the electrically insulating ferrimagnet yttrium\niron garnet with application to the spin Seebeck effect. The\ncalculations transcend previous theories by taking fully into\naccount (i) the complicated crystal and ferrimagnetic struc-\nture and (ii) the non-linearities caused by magnon-magnon in-\nteractions at elevated temperatures. We observe a remarkable\nresilience of the fundamental acoustic and optical modes with\nrespect to thermal agitation, which is explained by their large\ndispersion and spatial isolation from numerous floppy modes\nwith large heat capacity. At room temperature and above, the\nferrimagnetic optical mode is significantly occupied. Its neg-\native chirality leads to a suppression of thermal spin pumping\nand spin Seebeck effect. The critical exponent observed for\nthe spin Seebeck effect [26] remains as yet unexplained.\nThis work was supported by JSPS KAKENHI Grant Nos.\n25247056, 25220910, 26103006 and the Tohoku University\nGraduate Program in Spintronics. We thank Jiang Xiao and\nHiroto Adachi for useful discussions.5\n[1] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda,\nT. Ota, Y . Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer,\nS. Maekawa, and E. Saitoh, Nature Mater. 9, 894 (2010).\n[2] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nature Mater.\n11, 391 (2012).\n[3] V . Cherepanov, I. Kolokolov, and V . L’vov, Phys. Rep. 229, 81\n(1993).\n[4] Y . Sun, Y .-Y . Song, H. Chang, M. Kabatek, M. Jantz,\nW. Schneider, M. Wu, H. Schultheiss, and A. Hoffmann, Appl.\nPhys. Lett. 101, 152405 (2012).\n[5] J. Xiao, G. E. W. Bauer, K.-c. Uchida, E. Saitoh, and\nS. Maekawa, Phys. Rev. B 81, 214418 (2010).\n[6] U. Ritzmann, D. Hinzke, and U. Nowak, Phys. Rev. B 89,\n024409 (2014).\n[7] Y . Ohnuma, H. Adachi, E. Saitoh, and S. Maekawa, Phys. Rev.\nB87, 014423 (2013).\n[8] S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Phys. Rep.\n348, 441 (2001).\n[9] J. S. Plant, J. Phys. C: Solid State Phys. 10, 4805 (1977).\n[10] T. Kikkawa, K.-i. Uchida, S. Daimon, Z. Qiu, Y . Shiomi, and\nE. Saitoh, Phys. Rev. B 92, 064413 (2015).\n[11] H. Jin, S. R. Boona, Z. Yang, R. C. Myers, and J. P. Heremans,\nPhys. Rev. B 92, 054436 (2015).\n[12] E.-J. Guo, A. Kehlberger, J. Cramer, G. Jakob, and M. Kläui,\narXiv (2015), 1506.06037v1.\n[13] L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J.van Wees, Nat Phys 11, 1022 (2015).\n[14] S. Gepraegs, A. Kehlberger, F. Della Coletta, Z. Qiu, E.-J.\nGuo, T. Schulz, C. Mix, S. Meyer, A. Kamra, M. Altham-\nmer, H. Huebl, G. Jakob, Y . Ohnuma, H. Adachi, J. Barker,\nS. Maekawa, G. E. W. Bauer, E. Saitoh, R. Gross, S. T. B. Goen-\nnenwein, and M. Klaeui, Nat. Commun. 7, 10452 (2016).\n[15] K. Uchida, M. Ishida, T. Kikkawa, A. Kirihara, T. Murakami,\nand E. Saitoh, J. Phys.: Condens. Matter 26, 389601 (2014).\n[16] K. Uchida, H. Adachi, T. Kikkawa, A. Kirihara, M. Ishida,\nS. Yorozu, S. Maekawa, and E. Saitoh, Proc. IEEE PP, 1\n(2016).\n[17] D. Hinzke and U. Nowak, Comput. Phys. Commun. 121, 334\n(1999).\n[18] E. E. Anderson, Phys. Rev. 134, A1581 (1964).\n[19] J. Oitmaa and T. Falk, J. Phys.: Condens. Matter 21, 124212\n(2009).\n[20] R. K. Wangsness, Phys. Rev. 91, 1085 (1953).\n[21] A. B. Harris, Phys. Rev. 132, 2398 (1963).\n[22] R. C. LeCraw and L. R. Walker, J. Appl. Phys. 32, S167 (1961).\n[23] R. Bastardis, U. Atxitia, O. Chubykalo-Fesenko, and\nH. Kachkachi, Phys. Rev. B 86, 094415 (2012).\n[24] X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, Europhys. Lett. 96,\n17005 (2011).\n[25] L. J. Cornelissen, K. J. H. Peters, R. A. Duine, G. E. W. Bauer,\nand B. J. van Wees, arXiv (2016), 1604.03706.\n[26] K.-i. Uchida, T. Kikkawa, A. Miura, J. Shiomi, and E. Saitoh,\nPhys. Rev. X 4, 041023 (2014)."    },    {        "title": "1101.3888v1.Many_body_singlets_by_dynamic_spin_polarization.pdf",        "content": "arXiv:1101.3888v1  [quant-ph]  20 Jan 2011Many-body singletsby dynamicspinpolarization\nWang Yao\nDepartment of Physics and Center of Theoretical and Computa tional Physics, The University of Hong Kong, Hong Kong, Chin a\n(Dated: June 21, 2018)\nWeshowthatdynamicspinpolarizationbycollectiveraisin gandloweringoperatorscandriveaspinensemble\nfrom arbitrary initial state to many-body singlets, the zer o-collective-spin states with large scale entanglement.\nFor an ensemble of Narbitrary spins, both the variance of the collective spin an d the number of unentangled\nspins can be reduced to O(1)(versus the typical value of O(N)), and many-body singlets can be occupied\nwith a population of ∼20%independent of the ensemble size. We implement this approac h in a mesoscopic\nensemble of nuclear spins through dynamic nuclear spin pola rization by an electron. The result is of two-fold\nsignificanceforspinquantum technology: (1)aresourceofe ntanglement fornuclearspinbasedquantum infor-\nmation processing; (2) a cleaner surrounding and less quant um noise for the electron spinas the environmental\nspinmoments areeffectively annihilated.\nPACS numbers: 76.70.Fz,42.50.Dv,03.67.Bg,71.70.Jp\nMany-body singlets (MBS) are the zero-collective-spin\nstates of a spin ensemble with large scale quantum entangle-\nment and zero spin uncertainties. They appear in a variety of\ncontextsinquantumphysicsandincondensedmatterphysics ,\ne.g. as horizon states of the quantum black hole [1], and as\ngroundstatesofquantumantiferromagneticmodels[2]. The ir\nspecialcharacteristicsplacethematthecenterofattenti onfor\nquantumapplications. First,MBSareinvariantunderasimul-\ntaneous unitary rotation on all spins. This makes MBS suit-\nable for spanning a decoherence-freesubspace [3], for quan -\ntum communications without a shared reference frame [4],\nandformetrologyofthespatialgradientorfluctuationsofe x-\nternal fields [5]. Second, MBS is an extreme example for the\nsqueeze of spin uncertainties [6–9]. The collective spin ha s\nzero variance in all directions and thus a source of quantum\nnoise is removed, e.g. in the context of a quantum object af-\nfectedbyaspinbath. Third,MBScontainlargescalequantum\nentanglement: everyspinisentangledwiththerestpartoft he\nensemble. An example of a pure MBS is the product of two-\nqubitsinglets(Bell pairs). Inthemaximallymixedstateof all\nMBS, the distillable bipartite entanglement is logarithmi c in\ntheensemblesize[1].\nDespite the successful generation of photonic analog of 4-\nqubitsingletsbyparametricdownconversion[3, 10],reali za-\ntionofMBSinageneralspinensembleisanoutstandinggoal\nawaiting technically feasible approaches. Theoretical st udy\nshows that spin squeezing based on quantum non-demolition\nmeasurement can reduce the total collective spin variance o f\nan atomic ensemble by a factor of 5 in the lossless case [5].\nHowever, in such squeezed state the weighting of MBS is\nsmall andvanishesinlarge Nlimit.\nHereweintroduceaconceptuallynewapproachforsqueez-\ning of collective spin uncertainties and generation of larg e\nscaleentanglement. Theapproachusescollectivespinrais ing\nand lowering operations only, and is applicable to an ensem-\nble ofNarbitrary spins initially on arbitrary state. The state\nafter squeezing is significant in figures of merit: in the low\nlosslimit, MBS are occupiedwith an N-independentpopula-\ntion of∼20%, and both the variance of the total collectivespin and the number of spins unentangled with the rest are\nO(1)(versusthetypicalvaluesof O(N)). We implementthis\napproach in a mesoscopic ensemble of nuclear spins, a spin\nsystem of extensive interests either as a noise source or as a\nsuperiorinformationstoragein quantumtechnology. The im -\nplementation uses only generic features of dynamic nuclear\nspin polarization processes by an electron, and is applicab le\nto variouselectron-nuclearspin systems. Distillation of MBS\ncan be realized by post-selection based on measurement of\nthe electron spin. MBS can be a valuable resource of quan-\ntum entanglement for nuclear spin quantum information pro-\ncessing[11–13]. Inelectronspinbasedquantumcomputatio n\nschemes, preparingthe peripheralnuclearspinsinto MBS re -\nsults in a cleaner surrounding and hence improved quantum\ncoherenceoftheelectronspin.\nWe refer to the definition of spin squeezing in the general-\nized sense [5, 9], where the degree of squeezing is quantified\nby/an}b∇acketle{tˆJ2/an}b∇acket∇i}ht, withˆJ≡/summationtextN\nn=1ˆInbeing the total collective spin\nfor an ensemble of Nparticles with equal or different spins.\n/an}b∇acketle{tˆJ2/an}b∇acket∇i}ht= 0indicates perfect squeezing where the Nspins are\nin the MBS./an}b∇acketle{tˆJ2/an}b∇acket∇i}ht(¯s)−1gives an upper boundon the number\nof spins unentangled with others where ¯sis the average spin\nperparticle[5,9]. Statesofthespinensemblecanbegroupe d\ninto multiplets, i.e. irreducible invariant subspaces of t he to-\ntal spin. A multiplet {/vextendsingle/vextendsingleJ,M,αk\nJ/angbracketrightbig\n,M=−J,...,J}will be\ndenoted in short as {J,αk\nJ}, whereαk\nJis a general index for\ndistinguishingthesetoforthogonal(2 J+1)-dimensionalmul-\ntiplets. Theaimistotransferpopulationfromallmultiple tsto\nthosesingletswith J= 0.\nKey to our squeezingapproach is to apply raising operator\nof the form ˆj+\nA−ˆj+\nBon the spin coherent states |J,−J,αJ/an}b∇acket∇i}ht.\nHeretheensembleispartitionedarbitrarilyintotwosubse tsA\nandBwithcollectivespin ˆjAandˆjBrespectively(Fig.1(a)).\nWe findthekeyidentity\n/an}b∇acketle{tJ+1,−J+1,αJ+1|(ˆj+\nA−ˆj+\nB)|J,−J,αJ/an}b∇acket∇i}ht\n/an}b∇acketle{tJ,−J,αJ|(ˆj+\nA−ˆj+\nB)|J+1,−J−1,αJ+1/an}b∇acket∇i}ht∗\n=−[(J+1)(2J+1)]−1\n2. (1)\nMoreover,/an}b∇acketle{tJ′,M′,αJ′|ˆj+\nA−ˆj+\nB|J,−J,αJ/an}b∇acket∇i}ht= 0for|J′−2\n5 3 ,2 2 A B j j   = =     M = 4\n3\n2\n1\n0\n-1\n-2\n-3\n-4J = 4 J = 3 J = 2 J = 1(b) (a) (c) A   B \nC\nDC\nD\nA   B J+1,1\n1q\nJα+\n+J+1,1q\nJα+J+1,1\n1q\nJα−\n+\nJ,1k\nJα+J,k\nJα\nFIG. 1: (a) The red dashed line partitions the spin ensemble i nto\ntwo subsets AandBwith collective spin ˆjAandˆjBrespectively.\nThegreendashedlinegivesadifferentpartitionoftheense mble into\nsubsetCandD. (b) An example of how the operator ˆj+\nA−ˆj+\nBcou-\nples various basis states |J,M,j A,jB/an}bracketri}ht(solid arrows). The absolute\nvalue squared of the transition matrix elements are indicat ed with\nthe thickness of the arrows. The transitions related to the s pin co-\nherent states |J,−J/an}bracketri}htare highlighted. Hollow vertical arrows show\nthe coupling by ˆJ−. (c) Schematics of the population transfer rates\nbetween multiplets under the condition that each multiplet is initial-\nized on the spin coherent state |J,−J/an}bracketri}htwhen the ˆj+\nA−ˆj+\nBoperator\nis applied. The transfer rate of {J,αk\nJ} → {J,αk′\nJ}is identical to\nthe rate of the backward transfer {J,αk′\nJ} → {J,αk\nJ}. The rate of\n{J+1,αq\nJ+1} → {J,αk\nJ}is bya factor of (J+1)(2J+1)larger\nthanthat of the backward transfer {J,αk\nJ} → {J+1,αq\nJ+1}.\nJ|>1orM′/ne}ationslash=−J+1. Thus,underthe conditionthateach\nmultiplet is initialized on the spin coherent state, applic ation\noftheˆj+\nA−ˆj+\nBoperatortendstotransferpopulationsfrommul-\ntiplets of largerdimensionto multipletsof smaller dimens ion\n(Fig. 1(c)). The transfer rate of {J+1,αq\nJ+1}→{J,αk\nJ}is\nbyafactorof (J+1)(2J+1)largerthanthatofthebackward\ntransfer{J,αk\nJ}→{J+1,αq\nJ+1}.\nSqueezing of collective spin uncertainties can therefore\nbe realized by dynamic spin polarization with the lower-\ning operator ˆJ−and raising operators of the form ˆj+\nA−\nˆj+\nB. Consider the use of two such operators ˆj+\nA−ˆj+\nB\nandˆj+\nC−ˆj+\nDwhereCandDconstitute a different bi-\npartition of the ensemble (Fig. 1(a)). The Hilbert space\ncan be divided into independent subspaces according to\nthequantumnumbers {jA∩D,jB∩D,jB∩C,jA∩C}conserved\nby the raising/lowering operations, and their values deter -\nmine the number of (2 J+1)-dimensional multiplets n(J).\nIfˆJ−is applied more frequently such that the system\nis in spin coherent states every time ˆj+\nA−ˆj+\nBorˆj+\nC−(c) 0τ2τ(a) \n3τ t 4τ(b) \nˆˆ\nA B j j + + −\nˆˆ\nC D j j + + −ˆJ−\n5τM resolution \n0.04 \n0.00 J=2\nJ=1 0.04 \n0.00 \n0.04 \n0.00 J=0p(J, j A, j B)\n0\n707jAjB\n0.04 \n0.00 J=3\n0.2 \n0.0 0.6 \n0.4 P(J)\n0\n2\nJ4\n6\n8\n10 \n12 \n14 5τ4τ3τ2ττ0\nt\nFIG. 2: (a) An example of the squeezing control by dynamic spi n\npolarization with the operators ˆJ−,ˆj+\nA−ˆj+\nBandˆj+\nC−ˆj+\nD. (b)\nand (c) show simulation results of the control in (a). The sys -\ntem is initially in the completely mixed state in the subspac e with\n{jA∩D= 7/2,jB∩D= 7/2,jB∩C= 7/2,jA∩C= 7/2}. The\npolarization rates: 10−3Λh= Λo, and the delay time τ= 2/Λo.\n(b)p(J,jA,jB)gives the population of the multiplet {J,jA,jB}in\nthe state at t= 4τ. (c)P(J)≡/summationtext\njA,jBp(J,jA,jB)gives the in-\ntegratedprobabilityoffindingthesystemwithtotalcollec tivespinJ\nat various time. At t= 5τ, MBS are occupied with the population\nP(J= 0) = 0 .21, and/an}bracketle{tˆJ2/an}bracketri}ht= 2.42.\nˆj+\nDis applied, we find the steady-state in each subspace:\nρ=/summationtext\nJf(J)/summationtextn(J)\nk=1|J,−J,αk\nJ/an}b∇acket∇i}ht/an}b∇acketle{tJ,−J,αk\nJ|wheref(J) =\n(J+ 1)(2J+ 1)f(J+ 1). Most subspaces contain at\nleast one MBS [14], and n(J)≤n(0)(2J+1). Thus,\nin the steady state, MBS are occupied with a population\nn(0)f(0)≥[/summationtext\nJg(J)]−1= 0.20, and the variance/an}b∇acketle{tˆJ2/an}b∇acket∇i}ht≤\n[/summationtext\nJg(J)]−1/summationtext\nJJ(J+1)g(J) = 2.44, whereg(J)≡\n(2J+1)/bracketleftBig/producttextJ−1\ni=0(i+1)(2i+1)/bracketrightBig−1\n.\nDynamic spin polarization by the collective raising and\nlowering operators can be described by Lindblad terms in\nthe master equation ˙ρ=−1\n2/summationtext3\nm=1(ˆL†\nmˆLmρ+ρˆL†\nmˆLm−\n2ˆLmρˆL†\nm)whereˆL1≡√ΛhˆJ−,ˆL2≡√Λo(ˆj+\nA−ˆj+\nB),\nandˆL3≡√Λo(ˆj+\nC−ˆj+\nD). TheˆJ−andˆj+\nA−ˆj+\nBoperators\nareappliedwiththerates Λh|/an}b∇acketle{tψf|ˆJ−|ψi/an}b∇acket∇i}ht|2andΛo|/an}b∇acketle{tψf|ˆj+\nA−\nˆj+\nB|ψi/an}b∇acket∇i}ht|2respectively, and the squeezing scheme requires\nthat the former rate shall always be larger. We note that\n/an}b∇acketle{tJ,M,j A,jB|(ˆj−\nA−ˆj−\nB)(ˆj+\nA−ˆj+\nB)|J,M,j A,jB/an}b∇acket∇i}htincreases\nwith the decrease of Jand reaches the maximal value of\n∼(jA+jB)2for smallJ, while/an}b∇acketle{tJ,M|ˆJ+ˆJ−|J,M/an}b∇acket∇i}ht∼J2.\nThus we find the requirement Λh/Λo>(jA∩D+jB∩D+\njB∩C+jA∩C)2, the latter quantity ∼4Ns2in an ensemble\nofNspinsparticles. Spindecoherencecausesthepopulation\ndecay of MBS with a rate ∼Nγnwithγnbeing the single\nspindecoherencerate. Thelowlossconditionistherefored e-\nfined as1\n4Ns2Λh>Λo≫γn, where spin decoherence has\nnegligibleeffectonthesqueezingefficiency[15].3\nWenumericallydemonstrateasqueezecontrolwhere ˆj+\nA−\nˆj+\nBandˆj+\nC−ˆj+\nDare applied in alternating fashion (see\nFig.2(a)). Withspindecoherenceneglectedunderthelowlo ss\ncondition, calculation can be significantly simplified for t his\nchoice of control and a moderately large spin system can be\nsimulated. Intheintervalwhen ˆJ−andˆj+\nA−ˆj+\nB(orˆj+\nC−ˆj+\nD)\nare applied, the relevant Hilbert space can be further divid ed\ninto independent subspaces according to the quantum num-\nbers{jA,jB}(or{jC,jD}). Iftheduration τofeachinterval\nissufficientlylargetoensurethereachofsteadystate,wes im-\nply need to solve for steady state in each small subspace and\nkeep track of the basis transform between {|J,M,j A,jB/an}b∇acket∇i}ht}\nand{|J,M,j C,jD/an}b∇acket∇i}ht}upon the switch of raising operators.\nOff-diagonal coherence is found to be negligible which can\nfurthersimplify the calculation. An exampleof thesimulat ed\nsqueezing dynamics is given in Fig. 2(b-c). The initial den-\nsity matrix is the completely mixed one in the subspace with\n{jA∩D= 7/2,jB∩D= 7/2,jB∩C= 7/2,jA∩C= 7/2}.\nThe polarization rates are 10−3Λh= Λo, andτ= 2/Λo.\nFig. 2(b) shows that the population distribution p(J,jA,jB)\namongthemultipletsindeedapproachesthesteadystateval ue\nafter a time of 4τ. Att= 5τ, MBS are occupiedwith a pop-\nulationof 0.21and/an}b∇acketle{tˆJ2/an}b∇acket∇i}ht= 2.42.\nInanensembleofnuclearspins,theapplicationsofthetwo\ntypes of operators ˆJ−andˆj+\nA−ˆj+\nBare realized in the pro-\ncess of dynamic nuclear spin polarization (DNSP), a major\ntool for manipulation of nuclear spins [16–29]. We consider\nthehyperfineinteraction ˆH0=/summationtext\nn|ψ(rn)|2ˆIn·← →A·ˆScou-\npling the electron spin ˆSto peripheral lattice nuclear spins\nˆIn.← →Ais the hyperfinecouplingconstantin tensor form,and\nthe position dependenceof coupling enters throughthe enve -\nlope function ψ(r)of the electron only. ˆH0describes gen-\nerally the hyperfine interaction of electron or hole system i n\nquantum dots or shallow donors formed in group IV or III-\nV materials [30, 31]. In most DNSP schemes, ˆH0induces\nthe electron-nuclear flip-flop in passing electron spin pola r-\nization to the nuclei and the energy cost is compensated by\nemission/absorption of phonons or photons [19–23]. These\nDNSP schemes are termed as the dctype hereafter. Alterna-\ntively,DNSPcanalsoutilizethe accorrectiontothehyperfine\ncoupling: ˆHac=/summationtext\nn/parenleftBig\ndω·∇|ψ(rn)|2/parenrightBig\ncos(ωt)ˆIn·← →A·ˆS\nwhen an acelectric field induces an electron displacement\ndωcos(ωt), withenergycostforelectron-nuclearflip-flopdi-\nrectly supplied by acfield [27–29]. Such DNSP process is\ntermedhereafterasthe actype.\nFor nuclear spins on the periphery of an electron, MBS\ncan be realized by combining dcandacDNSP processes\nwhich polarize nuclear spins in opposite directions with th e\noperators/summationtext\nn|ψ(rn)|2ˆI−\nnand/summationtext\nn∂\n∂µ|ψ(rn)|2ˆI+\nnrespec-\ntively. Here µis the direction of acelectric field. The lat-\ntice sites with equal electron density |ψ(r)|2are grouped\ninto coordinationshells. On each shell,/summationtext\nn|ψ(rn)|2ˆI−\nnand/summationtext\nn∂\n∂µ|ψ(rn)|2ˆI+\nnare of the character of ˆJ−andˆj+\nA−ˆj+\nB\nrespectively. Under the influence of incoherent electron sp in\ndynamicsin theDNSP process,the largeshell-to-shelldiff er-\n(b) (a) (c)\n1\n2H0\nHac Hac \nHac x\ny\ndc DNP \nac DNP ( x)\nac DNP ( y)… …… …… … … …(d) \n0 τ 2τ 3τ t\nFIG.3: (a)Schematicsofanelectroninanuclearspinbath. T hefirst\ncoordination shell has 4 nuclear spins (green color) and the second\nshell has 8 nuclear spins (blue color). (b) Upper part: dchyperfine\ncoupling coefficients at the various lattice sites. Lower pa rt:achy-\nperfinecouplingcoefficientswith acelectricfieldin xdirection. The\nheights of the bar give the magnitude. (c) The achyperfine coupling\ndecomposed into two terms ˆHac=ˆH1\nac+ˆH2\nac.ˆH1\nac= ˜a1(ˆj+\nA−\nˆj+\nB)ˆS−eiωtEx+c.c., andˆH2\nac= ˜a2(ˆj+\nC−ˆj+\nD)ˆS−eiωtEx+c.c..\nThe 4sites withpositive coupling coefficients intheupper p art form\nsubsetAandthe4siteswithpositivecouplinginthelowerpartform\nsubsetC, andB(D) is the complement of A(C). (d) Schematic\nof the DNSP control where the switching between dcandacDNSP\nis concatenated with the switching of the acelectric field between x\nandydirections.\nenceinthe dchyperfinecouplingstrengthcauseslossofinter-\nshell coherence in a timescale much faster than the squeez-\ning. Thus different coordination shells can be independent ly\nsqueezedtowardsMBS.\nFig. 3(a) shows the schematic of an electron with a 2D\nGaussianenvelopefunction. The12latticenuclearspinsfo rm\ntwo coordination shells according to the dchyperfine cou-\npling strength (Fig. 3(b)). For the green shell with 4 lattic e\nnuclear spins, ˆHac= ˜aˆS−(ˆj+\nA−ˆj+\nB)eiωtEx+ ˜aˆS−(ˆj+\nC−\nˆj+\nD)eiωtEy+c.c., whereEx(y)is theacelectric field in the\nx(y) direction. Fig. 3(d) shows the schematic of the DNSP\ncontrolwherethe switchingbetween dcandacDNSP is con-\ncatenatedwiththeswitchingofthe acelectricfieldbetween x\nandydirections. Numerical simulation of such a control has\nbeen given in Fig. 2. The blue shell represents the more gen-\neral case where nuclear spins are polarized in the acDNSP\nprocess by ˜a1(ˆj+\nA−ˆj+\nB) + ˜a2(ˆj+\nC−ˆj+\nD), a linear superposi-\ntion of raising operators of the desired form (Fig. 3(c)). Th e\nidentity in Eq. (1) obviously holds if ˆj+\nA−ˆj+\nBis replaced\nbyˆj+\nC−ˆj+\nD, thus we have this same identity for their linear\nsuperpositionaswell. Numericalsimulationsconfirmthato p-\neratorsof this newformare equallyefficient in the squeezin g\neffect[15].\nInteraction between neighboring nuclear spins causes spin\ndiffusionandspindephasingwhichcanresultinlossofMBS.\nThedipolarinteractionbetweenneighboringlattice sites is of\nthe strength∼10Hz. Nuclear spin diffusion by the ˆI+\nnˆI−\nm\ncoupling terms is efficiently suppressed when the shell-to-4\nshellinhomogeneityinthehyperfinecouplingislarge. Byth e\nˆIz\nnˆIz\nmterm,thenuclearspinsaresubjecttoadipolarmagnetic\nfielddependentontheconfigurationoftheirneighbors,whic h\nleads to dephasing with a rate γn∼10−100Hz. To realize\nefficientsqueezing,fast DNSP mechanismsaredesired.\nFor optically controllable electron spin, e.g. in quantum\ndotorimpurityinIII-Vsemiconductors,fast dcDNSP canbe\nrealized by the hyperfine-mediatedoptical excitation of sp in-\nforbidden excitonic transitions [21, 32]. Assuming the ele c-\ntronZeemansplitting ωe∼0.2GHz,theintrinsicbroadening\nof charged exciton γt∼0.2GHz, and an optical Rabi fre-\nquencyΩ∼3GHz for the excitonic transition, we estimate\nthe DNSP rate: Λh=a2Ω2\nω2eγt∼10MHz on a coordination\nshellwithhyperfinecoupling a= 3MHz. Forotherelectron-\nnuclear spin system, fast dcDNSP may be realized through\nthe bath-assisted electron-nuclear flip-flop in the presenc e of\nanefficientenergydissipationchannel,e.g. anelectronFe rmi\nsea innearbyleads[20].\nacDNSP is of the rate Λo=˜a2\nγswhereγsis the broad-\nening of the electron spin resonance [28]. The magnitude of\ntheachyperfine interaction, ˜a, dependson the strength of ac\nelectric field and the inhomogeneity of the electron envelop\nfunction. Givingthe phosphorusdonorin silicon asan exam-\nple, we have the first several shells: (A,6.0,6),(B,4.5,12),\n(C,3.3,4),(D,2.2,12)and(F,1.7,12)where the first letter\nis the label of the shell by convention, the second number is\nthe hyperfine coupling strength in unit of MHz, and the third\nis the number of equivalent sites on the shell [33]. The dis-\ntancebetweenneighboringshellsis in the orderof ∼0.1nm.\nThus, we estimate ˜a∼MHz by a moderate displacement of\nthe electron dω∼0.1nm. Assuming γs∼0.1GHz,Λocan\nbe of∼10kHz on these shells. Thus, we conclude that the\nlowlossconditioncanindeedbesatisfiedfornuclearspinso n\ntheperipheryofa stronglyconfinedelectron.\nThe work was supported by the Research Grant Council\nof Hong Kong under Grant No. HKU 706309P. The author\nacknowledges helpful discussions with L. J. Sham, H. Y. Yu\nandX.D. Xu.\n[1] E.R.Livine andD. R.Terno,Phys. Rev. A 72, 022307 (2005).\n[2] G. Misguich and C. 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Neumann, N. Mizuochi, F. Rempp, P. Hemmer, H. Watan-\nabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, and\nJ. Wrachtrup, Science 320, 1326 (2008).\n[14] The states unconnected with MBS are of ∼1%of the entire\nHilbert space if two raising operators of the form ˆj+\nA−ˆj+\nBare\nused, and can be reduced to ∼0.01%if three such operators\nare used inthe dynamic spinpolarization.\n[15] H. Y.Yuand W.Yao,unpublished.\n[16] A. Greilich, A. Shabaev, D. R. Yakovlev, A. L. Efros, I. A . Yu-\ngova,D.Reuter,A.D.Wieck,andM.Bayer,Science 317,1896\n(2007).\n[17] D. J.Reilly,J.M. Taylor,J.R.Petta,C.M.Marcus, M.P. Han-\nson, andA.C. Gossard, Science 321, 817 (2008).\n[18] X.Xu,W.Yao,B.Sun,D.G.Steel,A.S.Bracker,D.Gammon ,\nand L.J.Sham, Nature 459, 1105 (2009).\n[19] A. Tartakovskii, T. Wright, A. Russell, V. Falko, A. Van kov,\nJ. Skiba-Szymanska, I. Drouzas, R. Kolodka, M. Skolnick,\nP.Fry,et al.,Phys.Rev. Lett. 98, 026806 (2007).\n[20] J. Danon andY. Nazarov, Phys. Rev. Lett. 100, 056603 (2008).\n[21] V. Korenev, Phys. Rev. Lett. 99, 256405 (2007).\n[22] A. S. Bracker, E. A. Stinaff, D. Gammon, M. E. Ware, J. G.\nTischler, A. Shabaev, A. L. Efros, D. Park, D. Gershoni, V. L.\nKorenev, etal.,Phys. Rev. Lett. 94, 047402 (2005).\n[23] B. Urbaszek, P. Braun, T. Amand, O. Krebs, T. Belhadj,\nA.Lemaitre,P.Voisin,andX.Marie,Phys.Rev.B 76,201301R\n(2007).\n[24] K. Onoand S.Tarucha, Phys.Rev. Lett. 92, 256803 (2004).\n[25] C. Latta, A. Hogele, Y. Zhao, A. N. Vamivakas, P. Maletin sky,\nM. Kroner, J. Dreiser, I. Carusotto, A. Badolato, D. Schuh,\net al.,Nat.Phys. 5,758 (2009).\n[26] I. T. Vink, K. C. Nowack, F. H. L. Koppens, J. Danon, Y. V.\nNazarov, and L.M.K. Vandersypen, Nat.Phys. 5, 764 (2009).\n[27] E. Laird, C. Barthel, E. Rashba, C. Marcus, M. Hanson, an d\nA. Gossard, Phys.Rev. Lett. 99, 246601 (2007).\n[28] M. Rudner andL.Levitov, Phys. Rev. Lett. 99, 246602 (2007).\n[29] E.Rashba, Phys.Rev. B 78, 195302 (2008).\n[30] B. Eble, C. Testelin, P. Desfonds, F. Bernardot, A. Balo cchi,\nT.Amand,A.Miard,A.Lemaitre,X.Marie,andM.Chamarro,\nPhys. Rev. Lett. 102, 146601 (2009).\n[31] J. Fischer, W. Coish, D. Bulaev, and D. Loss, Phys. Rev. B 78,\n155329 (2008).\n[32] E. A. Chekhovich, M. N. Makhonin, K. V. Kavokin, A. B.\nKrysa, M. S. Skolnick, and A. I. Tartakovskii, Phys. Rev. Let t.\n104, 066804 (2010).\n[33] E.B. Haleand R.L.Mieher, Phys. Rev. 184, 739 (1969)."    },    {        "title": "1608.02230v1.Spin_orbit_coupling_in_Fe_based_superconductors.pdf",        "content": "Noname manuscript No.\n(will be inserted by the editor)\nSpin-orbit coupling in Fe-based superconductors\nM.M. Korshunov \u0001Yu.N. Togushova \u0001I. Eremin \u0001P.J. Hirschfeld\nReceived: date / Accepted: date\nAbstract We study the spin resonance peak in re-\ncently discovered iron-based superconductors. The res-\nonance peak observed in inelastic neutron scattering\nexperiments agrees well with predicted results for the\nextendeds-wave (s\u0006) gap symmetry. Recent neutron\nscattering measurements show that there is a dispar-\nity between longitudinal and transverse components of\nthe dynamical spin susceptibility. Such breaking of the\nspin-rotational invariance in the spin-liquid phase can\noccur due to spin-orbit coupling. We study the role of\nthe spin-orbit interaction in the multiorbital model for\nFe-pnictides and show how it a\u000bects the spin resonance\nfeature.\nKeywords Fe-based superconductors \u0001Spin-resonance\npeak\u0001Spin-orbit coupling\nThe nature of the superconductivity and gap sym-\nmetry and structure in the recently discovered Fe-based\nsuperconductors (FeBS) are the most debated topics\nin condensed matter community [1]. These quasi two-\ndimensional systems shows a maximal Tcof 55 K plac-\ning them right after high- Tccuprates. Fe d-orbitals form\nM.M. Korshunov\nE-mail: korshunov@phys.u\r.edu\nL.V. Kirensky Institute of Physics, Krasnoyarsk 660036, Rus-\nsia\nM.M. Korshunov and Yu.N. Togushova\nSiberian Federal University, Svobodny Prospect 79, Krasno-\nyarsk 660041, Russia\nP.J. Hirschfeld\nDepartment of Physics, University of Florida, Gainesville,\nFlorida 32611, USA\nI. Eremin\nInstitut f ur Theoretische Physik III, Ruhr-Universitat\nBochum, D-44801 Bochum, Germany\nKazan Federal University, Kazan 420008, Russiathe Fermi surface (FS) which in the undoped systems\nconsists of two hole and two electron sheets. Nesting\nbetween these two groups of sheets is the driving force\nfor the spin-density wave (SDW) long-range magnetism\nin the undoped FeBS and the scattering with the wave\nvector Qconnecting hole and electron pockets is the\nmost probable candidate for superconducting pairing in\nthe doped systems. In the spin-\ructuation studies [2,3,\n4], the leading instability is the extended s-wave gap\nwhich changes sign between hole and electron sheets\n(s\u0006state) [5].\nNeutron scattering is a powerful tool to measure\ndynamical spin susceptibility \u001f(q;!). It carries infor-\nmation about the order parameter symmetry and gap\nstructure. For the local interactions (Hubbard and Hund's\nexchange),\u001fcan be obtained in the RPA from the bare\nelectron-hole bubble \u001f0(q;!) by summing up a series of\nladder diagrams to give \u001f(q;!) = [I\u0000Us\u001f0(q;!)]\u00001\u001f0(q;!),\nwhereUsandIare interaction and unit matrices in or-\nbital space, and all other quantities are matrices as well.\nScattering between nearly nested hole and electron\nFermi surfaces in FeBS produce a peak in the normal\nstate magnetic susceptibility at or near q=Q= (\u0019;0).\nFor the uniform s-wave gap, sign \u0001k= sign\u0001k+Qand\nthere is no resonance peak. For the s\u0006order parameter\nas well as for an extended non-uniform s-wave symme-\ntry,Qconnects Fermi sheets with the di\u000berent signs\nof gaps. This ful\flls the resonance condition for the in-\nterband susceptibility, and the spin resonance peak is\nformed at a frequency below \nc= min (j\u0001kj+j\u0001k+qj)\n(compare normal and s\u0006superconductor's response in\nFig. 1) [6,7,8]. The existence of the spin resonance in\nFeBS was predicted theoretically [6,7] and subsequently\ndiscovered experimentally with many reports of well-\nde\fned spin resonances in 1111, 122, and 11 systems\n[9,10,11].arXiv:1608.02230v1  [cond-mat.supr-con]  7 Aug 20162 M.M. Korshunov et al.\n 0 1 2 3 4 5 6 7 8 9\n 0  1  2  3  4  5Im χ(q=[π,0],ω)\nω/∆0non-SC, χ+-\nnon-SC, 2 ×χzz\ns++, χ+-\ns++, 2×χzz\ns±, χ+-\ns±, 2×χzz\nFig. 1 Fig. 1. Calculated Im \u001f(Q;!) in the normal state, and\nfor thes++ands\u0006pairing symmetries. In the latter case, the\nresonance is clearly seen below != 2\u00010. Spin-orbit coupling\nconstant\u0015= 100 meV, intraorbital Hubbard U= 0:9 eV,\nHund'sJ= 0:1 eV, interorbital U0=U\u00002J, and pair-hopping\ntermJ0=J.\nOne of the recent puzzles in FeBS is the discov-\nered anisotropy of the spin resonance peak in Ni-doped\nBa-122 [12]. It was found that \u001f+\u0000and 2\u001fzzare dif-\nferent. This contradicts the spin-rotational invariance\n(SRI)hS+S\u0000i= 2hSzSziwhich have to be obeyed in\nthe disordered system. One of the solution to the puzzle\nis the spin-orbit (SO) interaction which can break the\nSRI like it does in Sr 2RuO 4[13]. Here we incorporate\nthe e\u000bect of the SO coupling in the susceptibility cal-\nculation for FeBS to shed light on the spin resonance\nanisotropy.\nThe simplest model for pnictides in the 1-Fe per\nunit cell Brillouin zone comes from the three t2gd-\norbitals. The xzandyzcomponents are hybridized and\nform two electron-like FS pockets around ( \u0019;0) and\n(0;\u0019) points, and one hole-like pocket around \u0000= (0;0)\npoint. Thexyorbital is considered to be decoupled from\nthem and form an outer hole pocket around \u0000point.\nThe one-electron part of the Hamiltonian is given by\nH0=P\nk;\u001b;l;m\"lm\nkcy\nkl\u001bckm\u001b, wherelandmare orbital\nindices,ckm\u001bis the annihilation operator of a particle\nwith momentum kand spin\u001b. This model for pnic-\ntides is similar to the one for Sr 2RuO 4and, in particu-\nlar, thexyband does not hybridize with the xzandyz\nbands. Keeping in mind the similarity to the Sr 2RuO 4\ncase, for simplicity we consider only the Lz-component\nof the SO interaction [13]. Due to the structure of the\nLz-component, the interaction a\u000bects xzandyzbands\nonly.\nFollowing Ref. [14], we write the SO coupling term,\nHSO=\u0015P\nfLf\u0001Sf, in the second-quantized form asHSO= i\u0015\n2P\nl;m;n\u000flmnP\nk;\u001b;\u001b0cy\nkl\u001bckm\u001b0^\u001bn\n\u001b\u001b0, where\u000flmnis\nthe completely antisymmetric tensor, indices fl;m;ng\ntake valuesfx;y;zg$fdyz;dzx;dxyg$f 2;3;1g, and\n^\u001bn\n\u001b\u001b0are the Pauli spin matrices.\nThe matrix of the Hamiltonian H=H0+HSOis\nthen\n^\"k\u001b=0\n@\"1k 0 0\n0\"2k\"4k+ i\u0015\n2sign\u001b\n0\"4k\u0000i\u0015\n2sign\u001b \" 3k1\nA (1)\nAs for Sr 2RuO 4, eigenvalues of ^ \"k\u001bdo not depend on\nspin\u001b, therefore, spin-up and spin-down states are still\ndegenerate in spite of the SO interaction.\nWe calculated both + \u0000(longitudinal) and zz(trans-\nverse) components of the spin susceptibility and found\nthat in the normal state \u001f+\u0000>2\u001fzzat small frequen-\ncies, see Fig. 1. As expected, for the s++supercon-\nductor (conventional isotropic s-wave) there is no res-\nonance peak and the disparity between \u001f+\u0000and 2\u001fzz\nis very small. For the s\u0006superconductor, however, the\nsituation is opposite { we observe a well de\fned spin\nresonance and \u001f+\u0000is larger than 2 \u001fzzby about 15%\nnear the peak position (Fig. 1).\nIn summary , we have shown that the spin resonance\npeak in FeBS gains anisotropy in the spin space due\nto the spin-orbit coupling. This result is in qualitative\nagreement with experimental \fndings. We do not ob-\nserve changes in the peak position but this may be due\nto the simple model that we studied.\nAcknowledgements Partial support was provided by DOE\nDE-FG02-05ER46236 (P.J.H. and M.M.K.) and NSF-DMR-\n1005625 (P.J.H.). M.M.K. acknowledge support from RFBR\n(grants 09-02-00127, 12-02-31534 and 13-02-01395), Presid-\nium of RAS program \\Quantum mesoscopical and disordered\nstructures\" N20.7, FCP Scienti\fc and Research-and-Educational\nPersonnel of Innovative Russia for 2009-2013 (GK 16.740.12.0731\nand GK P891), and President of Russia (grant MK-1683.2010.2),\nSiberian Federal University (Theme N F-11), Program of SB\nRAS #44, and The Dynasty Foundation and ICFPM. I.E. ac-\nknowledges support of the SFB Transregio 12, Merkur Foun-\ndation, and German Academic Exchange Service (DAAD PPP\nUSA No. 50750339).\nReferences\n1. P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Rep.\nProg. Phys. 74, 124508 (2011).\n2. S. Graser et al. , New. J. Phys. 11, 025016 (2009).\n3. K. Kuroki et al. , Phys. Rev. Lett. 101, 087004 (2008).\n4. S. Maiti et al. , Phys. Rev. Lett. 107, 147002 (2011).\n5. I. I. Mazin et al. , Phys. Rev. Lett. 101, 057003 (2008).\n6. M. M. Korshunov and I. Eremin, Phys. Rev. B 78,\n140509(R) (2008).\n7. T. A. Maier and D. J. Scalapino, Phys. Rev. B 78,\n020514(R) (2008).\n8. T. A. Maier et al. , Phys. Rev. B 79, 134520 (200).Spin-orbit coupling in Fe-based superconductors 3\n9. A. D. Christianson et al. , Nature 456, 930 (2008).\n10. D. S. Inosov et al. , Nature Physics 6, 178 (2010).\n11. D. N. Argyriou et al., Phys. Rev. B 81, 220503(R) (2010).\n12. O. J. Lipscombe et al. , Phys. Rev. B 82, 064515 (2010).\n13. I. Eremin, D. Manske, and K. H. Bennemann, Phys. Rev.\nB65, 220502(R) (2002).\n14. K. K. Ng and M. Sigrist, Europhys. Lett. 49, 473 (2000)."    },    {        "title": "0903.4772v2.Comment_on_recent_papers_regarding_next_to_leading_order_spin_spin_effects_in_gravitational_interaction.pdf",        "content": "arXiv:0903.4772v2  [gr-qc]  31 Oct 2009Comment on recent papers regarding next-to-leading order\nspin-spin effects in gravitational interaction\nJan Steinhoff and Gerhard Sch¨ afer\nTheoretisch-Physikalisches Institut, Friedrich-Schill er-Universit¨ at,\nMax-Wien-Platz 1, 07743 Jena, Germany\n(Dated: August 9, 2021)\nAbstract\nIt is argued that the tetrad in a recent paper by Porto and Roth stein on gravitational spin-spin\ncoupling should not have the given form. The fixation of that t etrad was suggested by Steinhoff,\nHergt, and Sch¨ afer as a possible source for the disagreemen t found in the spin-squared dynamics.\nHowever, this inconsistency will only show up in the next-to -leading order spin-orbit dynamics and\nnot in the spin-squared dynamics. Instead, the disagreemen t found at the next-to-leading order\nspin-squared level is due to a sign typo in the spin-squared p aper by Porto and Rothstein.\nPACS numbers: 04.25.-g, 04.25.Nx, 04.30.Db\n1In recent papers, Steinhoff, Hergt, and Sch¨ afer derived the ne xt-to-leading order (NLO)\nspin-squared dynamics for binary black holes [1, 2]. The result led to a spin-precession\nequation which disagreed with an earlier result by Porto and Rothste in (PR) [3] based on\nthe formalism of [4]. The suggestion was given way in [1] that a different choice of the tetrad\neµ\namay cure the disagreement. It is well known that a tetrad is only fixe d by the metric up\nto a local Lorentz transformation, which made it a plausible source f or the disagreement.\nBoth the correct fixation of the tetrad and the disagreement in th e spin-squared dynamics\nwill be clarified here.\nBy evaluating, say for particle with index 1, uµ\n1uν\n1gµνin both the local and coordinate\nframes, one gets the consistency condition (metric signature -2 a s in the papers by PR)\n(˜va=0\n1)2−˜va=i\n1˜va=i\n1=g00(x1)+2g0i(x1)vi\n1+gij(x1)vi\n1vj\n1, (1)\nwhere ˜va\n1=ea\nµ(x1)uµ\n1relates to the local frame and vi\n1=ui\n1,u0\n1= 1 to the coordinate frame.\nThis condition is not fulfilled for Eqs. (22) and (23) in [4], using leading or der terms of the\nregularized metric for spinning binary black holes in harmonic coordina tes on the right-hand\nside of this condition. This inconsistency will show up at the NLO spin-o rbit dynamics via\nthe spin supplementary condition S0i\n1˜va=0\n1+Sij\n1˜va=j\n1= 0. In passing we note that a choice\nconsistent with (1) and sufficient for the NLO is given by\n˜va=0\n1= 1−GNm2\nr, (2)\n˜va=i\n1=vi\n1+GNm2\nr(vi\n1−2vi\n2)+GN\nr2Sij\n2nj. (3)\nThevi\n2term in (3) arises from the boosted Schwarzschild metric in harmonic coordinates.\nHere the tetrad eµ\nawas obtained from Eq. (34) in [5] (notice eI\nµ=ea\nµΛI\na), which is the fixation\nof the tetrad that enters the derivation of the Feynman rules.\nThe disagreement at the spin-squared level is indeed not due to a fu rther modification\n(via a local Lorentz transformation) of the tetrad. Instead, by comparing Eq. (73) in [4]\nwith Eq. (62) in [3], a simple calculation reveals that the signs of the last terms are not\nthe same. By redoing the corresponding calculations one can check that Eq. (73) in [4] is\ncorrect, i.e., there is a sign typo in (62) of [3]. The spin-precession eq uations of [3] and [1]\nnow coincide after the spin transformation\nSNW\n1=SSHS\n1−1\n2m3\n1[(P1×S1)×˙P1]×S1, (4)\n2has been performed, where the index “SHS” refers to the spin exp ression in [1] and “NW” to\nthe corresponding one in [3]. P1differs from p1in [1] only by higher order terms, cf., Eq. (5)\nin [1]. However in this multisheeted domain, it was quite a cumbersome ta sk to check the\ncorrectness of the other terms (taking for granted the correc tness of the Feynman-diagram\nexpressions) or to deduce the reason for the disagreement by co mparing with our result.\nIt should be noted that Newton-Wigner (NW) variables are originally o nly defined in flat\nspacetime, andthatageneralizationtocurvedspacetime isnotuniq ue. Inourunderstanding\nNW variables should have a standard canonical meaning [7], i.e.,\n{zi\na,Paj}=δij, (5)\n{Sa(i),Sa(j)}=ǫijkSa(k), (6)\nzero otherwise , which is true in our papers. In the papers by PR the NW variables are\nconstructed such that, besides agreement with the usual NW var iables in flat spacetime,\nthe spin has constant length, i.e., the spin equation of motion manifes tly describes a spin\nprecession. While at the spin-orbit and spin(1)-spin(2) level this imp lies that the spin is\nalso standard canonical, this is not true at the spin(1)-spin(1) leve l; see Eq. (13) in [1].\nIndeed, the spin-squared term in Eq. (4) is not related to a canonic al transformation and\nshould in our understanding, where NW stands for “standard cano nical”, be included into\nthe definition of the NW variables. However, the spin equations of mo tion in [3] and [1] are\nphysically equivalent, so the discrepancy in the understanding of NW variables is a matter\nof taste only.\nA comparison of the center-of-mass motion is still missing. This is nec essary because all\nS2\n1terms in the potential do not contribute to the spin equation of mot ion and are therefore\nnot verified yet. Here possible higher order corrections, analogou s to our Eq. (4), to Eqs.\n(39) and (59) in [4] may be needed to arrive at standard canonical v ariables for position and\nlinear momentum.\n3Acknowledgments\nThis work is supported by the Deutsche Forschungsgemeinschaft (DFG) through\nSFB/TR7 “Gravitational Wave Astronomy”.\n[1] J. Steinhoff, S. Hergt, and G. Sch¨ afer, Phys. Rev. D 78, 101503(R) (2008).\n[2] S. Hergt and G. Sch¨ afer, Phys. Rev. D 78, 124004 (2008).\n[3] R. A. Porto and I. Z. Rothstein, Phys. Rev. D 78, 044013 (2008).\n[4] R. A. Porto and I. Z. Rothstein, Phys. Rev. D 78, 044012 (2008).\n[5] R. A. Porto, Phys. Rev. D 73, 104031 (2006).\n[6] T. D. Newton and E. P. Wigner, Rev. Mod. Phys. 21, 400 (1949).\n[7] Our understanding of NW variables emphasizes a slightly different aspect from their original\nsetting [6], where locality of the position variables, not c anonicity of the independent variables,\nis the defining property. However, in flat spacetime the canon icity of the variables follows from\nthe former and we think this is the important property that sh ould be promoted to curved\nspacetime.\n4"    },    {        "title": "2112.15301v2.Spin_injection_generated_shock_waves_and_solitons_in_a_ferromagnetic_thin_film__the_spin_piston_problem.pdf",        "content": "Spin-injection-generated shock waves and solitons in a ferromagnetic thin film: the\nspin piston problem\nMingyu Hu,1,\u0003Ezio Iacocca,2and Mark Hoefer1\n1Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA\n2Department of Physics & Energy Science, University of Colorado, Colorado Springs, CO 80918, USA\n(Dated: January 4, 2022)\nThe unsteady, nonlinear magnetization dynamics induced by spin injection in an easy-plane ferro-\nmagnetic channel subject to an external magnetic field are studied analytically. Leveraging a disper-\nsive hydrodynamic description, the Landau-Lifshitz equation is recast in terms of hydrodynamic-like\nvariables for the magnetization’s perpendicular component (spin density) and azimuthal phase gra-\ndient (fluid velocity). Spin injection acts as a moving piston that generates nonlinear, dynamical\nspin textures in the ferromagnetic channel with downstream quiescent spin density set by the ex-\nternal field. In contrast to the classical problem of a piston accelerating a compressible gas, here,\nvariable spin injection and field lead to a rich variety of nonlinear wave phenomena from oscillatory\nspin shocks to solitons and rarefaction waves. A full classification of solutions is provided using\nnonlinear wave modulation theory by identifying two key aspects of the fluid-like dynamics: sub-\nsonic/supersonic conditions and convex/nonconvex hydrodynamic flux. Familiar waveforms from\nthe classical piston problem such as rarefaction (expansion) waves and shocks manifest in their\nspin-based counterparts as smooth and highly oscillatory transitions, respectively, between two dis-\ntinct magnetic states. The spin shock is an example of a dispersive shock wave, which arises in\nmany physical systems. New features without a gas dynamics counterpart include composite wave\ncomplexes with \"contact” spin shocks and rarefactions. Magnetic supersonic conditions lead to two\npronounced piston edge behaviors including a stationary soliton and an oscillatory wavetrain. These\ncoherent wave structures have physical implications for the generation of high frequency spin waves\nfrom pulsed injection and persistent, stable stationary and/or propagating solitons in the presence\nof magnetic damping. The analytical results are favorably compared with numerical simulations.\nI. Introduction\nSpin transport in magnetic materials has been in-\ntensely studied, due, in part, to its potential spin-\ntronic applications in information technology. A promis-\ning means for long-distance transport of angular mo-\nmentum is by way of large-amplitude, fluid-like exci-\ntations [1, 2]. A useful approach to study these non-\nlinear spin dynamics is the hydrodynamic framework.\nFirst proposed by Halperin and Hohenberg [3] to de-\nscribe spin waves in anisotropic ferro- and antiferromag-\nnets under long-wavelength assumptions, the hydrody-\nnamic perspective—essentially a transformation of the\nLandau-Lifshitz equation to a set of fluid-like variables—\nhas since been utilized by a number of researchers to in-\nvestigate a variety of novel spin textures and dynamics,\nsometimes referred to as superfluid spin transport [4–13].\nActually, magnetic damping implies energy dissipation,\nwhich must be compensated if sustained superfluid-like\nspinstatesaredesired. Anadequatecompensationmech-\nanism is the injection of spin into material boundaries via\nthe spin-Hall effect, spin-transfer torque [6–8, 10, 14, 15],\nor the quantum spin-Hall effect [16]. Recent experimen-\ntal observations of superfluid-like spin transport indicate\nthat such dynamics are possible [14, 16].\nThe analytical study of fluid-like spin transport in fer-\nromagnetic materials can be conveniently formulated in\n\u0003mingyu.hu@colorado.eduterms of dispersive hydrodynamics (DH) in which large\nscale, nonlinear wave motion in a dispersive medium is\ndescribed by conservation laws subject to dispersive cor-\nrections [17]. Just such a DH formulation of magneti-\nzation dynamics was proposed in [8] as an exact trans-\nformation of the Landau-Lifshitz (LL), a standard con-\ntinuum, micromagnetic model of ferromagnetic materi-\nals. It recasts the three-component magnetization vec-\ntorm= (mx;my;mz), constrained to normalized unit\nlength, in terms of two dependent variables: the lon-\ngitudinal spin density n=mzand the fluid velocity\nu=\u0000rarctan(my=mx). The latter is proportional to\nthe longitudinal component of the spin current. Since\nmy;mx!0whenmz!\u0006 1,uis undefined when the\nmagnetization is saturated in the perpendicular direc-\ntion so this is referred to as the vacuum state. When\nthe local fluid speed exceeds the critical value ucr= p\n(1\u0000n2)=(1 + 3n2)(different from the local speed of\nsound due to broken Galilean invariance), the flow can\nbe understood as supersonic [8], resulting, for example,\nin the generation of magnetic vortices and antivortices\nwith vacuum states at their core [18, 19]. Since the\ntransformation is exact, the DH formulation captures\nall of the essential physics that are involved: exchange,\nanisotropy, and damping, manifesting as dispersion, non-\nlinearity, and viscous effects, respectively. Under condi-\ntions in which anisotropy and exchange dominate, such\nas when the spin density exhibits a large gradient, the\nspin system can develop rapidly oscillatory structures in-\ncluding solitons and spin shocks, also known as disper-arXiv:2112.15301v2  [cond-mat.mes-hall]  3 Jan 20222\nFIG. 1. Temporal development of a spin shock (DSW) solution to the piston problem in a 1D easy-plane ferromagnetic channel.\n(a) Initial state prior to spin piston acceleration with constant spin density n=h0= 0:8and fluid velocity u= 0. (b) The\npiston velocity accelerates to 99:9%ofu0= 0:2leading to a compressive wave. (c) A spin shock is under development. (d) A\nspin shock is fully developed.\nsive shock waves (DSWs), observed in the envelope of\nmagnetostatic spin waves in Yttrium Iron Garnet films\n[20]. Such DSWs are known to occur in a variety of other\nDH media including Bose-Einstein condensates (BECs)\n[21], nonlinear spatial [22] and fiber [23] optics, and fluid\ndynamics [24, 25]. Highly oscillatory, unsteady DSWs\ncontrast sharply with nonlinear, dissipation-dominated,\nnon-oscillatory, steady shock waves in other systems such\nas a compressible gas [26].\nWith the DH interpretation of spin dynamics in mind,\nwe will focus on the analytical study of the canonical\nproblem of spin injection into an easy-plane ferromagnet\nas a feasible mechanism to generate large-amplitude, un-\nsteady spin textures with fluid-like features. This prob-\nlem was recently considered by us in [12] by way of nu-\nmerical simulations of the LL equation. We showed that\nthe presence of a perpendicular, uniform, external mag-\nnetic field and the rapid onset of spin injection resulted\nin three evolutionary stages: 1) injection rise leading to\nthe generation of fluid-like expansion and/or compres-\nsion waves, 2) pre-relaxation in which the dynamics are\ndominated by exchange and anisotropy resulting in rar-\nefaction and shock waves, and 3) relaxation to steady\nstate where damping, exchange and anisotropy result in\nsteady, precessional dissipative exchange flows [10], also\nknown as spin superfluids [5, 6].\nIn the present work, we focus on the pre-relaxation\nstage where magnetic damping is negligible relative to\nexchange and anisotropy. We interpret spin injection at\none material boundary as a “spin piston” whose resul-\ntant spin current is analogous to the piston velocity. The\nrapid onset of spin injection causes the acceleration of\nthe spin piston, which in turn leads to the development\nof a large spin density gradient in the sample. Such con-\nditions result in dispersive hydrodynamics. The piston\ndrives fluid-like excitations into an otherwise static mag-\nnetic configuration whose spin density is determined by\na perpendicular external magnetic field. Different spininjection and field strengths lead to a variety of spin rar-\nefaction waves, spin shocks, and solitons.\nThe development of a spin shock generated by the spin\npiston is shown in Fig. 1. As demonstrated in [12], by\nconsidering the problem on short enough time scales,\nwe can neglect magnetic damping. Therefore, in this\nwork, we use nonlinear wave/Whitham modulation the-\nory [17, 27, 28] to analytically classify the dynamic spin\ntextures generated by the spin piston with fixed velocity\nu0and fieldh0with negligible damping. Our primary re-\nsult is the phase diagram depicting the various solution\ntypes in the injection-field ( u0-h0) plane of Fig. 2. This\ndiagram demonstrates the rich selection of fluid-like spin\ntextures that can be generated in this system. Moreover,\nthese dispersive hydrodynamic waves have physical im-\nplications for the generation of spin waves from pulsed\ninjection and stable solitons coincident with dissipative\nexchange flows.\nIn addition to the piston problem, another canonical\nhydrodynamic problem is the space-time evolution of an\ninitial, sharp gradient, known as the Riemann problem\n[29]. We highlight the work of [30] in which the Riemann\nproblem for polarization waves in a two-component BEC\nis classified. As it turns out, the governing equations\nstudied there are equivalent to the LL equation in one\nspatial dimension that we study here in dispersive hy-\ndrodynamic form absent magnetic damping. As such, we\nrely heavily upon the analysis carried out in [30]. Never-\ntheless, the piston problem studied here introduces new\nboundary effects that do not occur in Riemann problems\nsuch as supersonic flow conditions that generate a soliton\nattached to the piston or a partial DSW that emanates\nfrom it. Moreover, the spin piston problem is a physi-\ncally plausible setting to generate spin shocks and other\ndispersive hydrodynamic spin textures in magnetic ma-\nterials.\nRelated superfluid and superfluid-like piston problems\nhave been studied theoretically [31, 32] and experimen-3\nFIG. 2. Classification of spin piston dynamics in terms of the piston velocity u0(spin injection) and background spin density h0\n(external magnetic field). The acronyms used in the Figure and throughout the text are defined in Table I. Dotted black curve:\ndivide between convex (left) and nonconvex (right) regimes. Solid black lines: rL\n+=rR\n+(see main text). The pink-shaded region\nimplies the existence of the vacuum state jnj= 1within the oscillatory solution. The subsonic regime is identified in white.\nSector I: RW; Sector II: DSW+; Sector III: DSW+CDSW+; Boundary III =IV between sector III and IV: CDSW; Sector IV:\nRW CDSW+. The supersonic regime is in the gray, shaded region. Sector V: S+jRW, supersonic condition v\u0000< v +<0; Sector\nVI:S\u0000jRW CDSW+, supersonic condition v\u0000< v +<0; Sector VII: PDSW+jDSW+, supersonic condition 0< v\u0000< v +.\nTABLE I. List of acronyms and symbols.\nRW rarefaction wave\nDSW dispersive shock wave\nCDSW contact dispersive shock wave\nPDSW partial dispersive shock wave\nS soliton\n| constant plateau separating waves\n\u0006superscripts +: elevation soliton, \u0000: depression soliton\ntally [33, 34] in BECs and optics. While they reveal\nintriguing dispersive hydrodynamic features such as the\ngeneration of an oscillatory wake at the piston accompa-\nnied by vacuum points [31, 34], there are a number of\nnew effects predicted by the spin piston problem stud-\nied here. This is because the hydrodynamic flux of the\nspin system is nonconvex whereas the flux in BEC and\noptics is convex. Nonconvexity manifests in the spin ana-\nlogues of conservation of mass and momentum with non-\nmonotonic hydrodynamic fluxes are in the spin density\nand fluid velocity. Mathematically, the long-wavelength\nhydrodynamic system loses strict hyperbolicity and/or\ngenuine nonlinearity. This leads to new types of disper-\nsive hydrodynamics [35]. In addition to expanding rar-\nefactionwaves(RWs)andcompressiveDSWsinFigures1\nand 2, nonconvexity results in hybrid spin textures com-\nposed of a RW and a special kind of contact spin shock\nor contact DSW (CDSW)—the dispersive hydrodynamic\nanalogue of a contact discontinuity in gas dynamics—\nwhosevelocitycoincideswithalongwavelengthmagneticTABLE II. Physical and mathematical properties of the solu-\ntion sectors in the phase diagram of Fig. 2.\nI: RW subsonic, expansive, convex\nII: DSW+subsonic, compressive, convex\nIII: DSW+CDSW+subsonic, compressive, nonconvex\nIV: RW CDSW+ subsonic, expansive/compressive,\nnonconvex\nV: S+|RW supersonic, expansive, convex\nVI: S\u0000|RW CDSW+supersonic, expansive/compressive,\nnonconvex\nVII: PDSW+|DSW+supersonic, compressive,\nconvex/nonconvex\nspeed of sound. Table I lists the acronyms and symbols\nused throughout the main text and in Fig. 2. We identify\nthe supersonic transition at the piston as coincident with\neither the rapid generation of a stationary soliton or a\npartial DSW. Finally, sufficiently large external field and\npositive piston velocity result in the generation of vac-\nuum points within the oscillatory solution. Although we\nfocus on the early, dissipationless spin dynamics, each of\nthese distinct dispersive hydrodynamic excitations have\nimplications for the long-time, steady-state evolution of\nthe spin system subject to magnetic damping [12]. These\nimplications are discussed in our concluding remarks Sec-\ntion VIII.\nThe rest of the paper is organized as follows. Section\nII describes the spin piston problem setup. Section III\nprovides a summary of the results of Whitham modula-4\ntiontheoryfrom[30]sothattheanalysisisself-contained.\nSomeadditionalanalyticaldetailsareprovidedintheAp-\npendix. InSectionV,solutionswithzeroappliedfieldare\npresented and analyzed in both the subsonic and super-\nsonic regimes. In Sections VI and VII, solutions arising\nin the presence of a uniform perpendicular applied field\nare analyzed in the subsonic and supersonic regimes, re-\nspectively. Finally, we present the conclusion in Section\nVIII.\nII. Model\nConsider a one-dimensional, easy-plane ferromagnetic\nchannel oriented in the ^xdirection with length L. Spin\ninjection is applied to the left edge where x= 0. The\nright edge at x=Lcorresponds to a free spin boundary.\nThe governing equation is the non-dimensional, dissipa-\ntionless LL equation, given by\n@tm=\u0000m\u0002he\u000b; x2(0;L);t> 0;(1)\nwhere\nhe\u000b=@xxm\u0000mz^z+h0^z: (2)\nHere, m=M=Ms= (mx;my;mz)is the normalized\nmagnetization vector, and Msis the saturation magneti-\nzation. The effective field (2) is also normalized by Ms\nand consists of exchange, easy-plane anisotropy, and a\nuniform externally applied magnetic field with constant\nmagnitude h0along the perpendicular-to-plane ( ^z) di-\nrection. The non-dimensionalization leading to Eq. (1) is\nachieved by scaling time by j\rj\u00160Msand space by \u0015\u00001\nex,\nwhere\ris the gyromagnetic ratio, \u00160is the vacuum per-\nmeability, and \u0015exis the exchange length. The dissipa-\ntionless LL serves as a valid model here considering the\ntimescale within which damping is not a key factor in\nthe development of the dynamical structures [12]. We\nwill discuss the role of damping on longer time scales in\nthe conclusion.\nThe following analysis is based on the DH formulation\nof Eq. (1) in terms of the hydrodynamic variables\nspin density: n=mz;\nfluid velocity: u=\u0000@x\b =\u0000@xarctan(my=mx);\nwhere \bis the azimuthal phase angle. The DH formula-\ntion is given by [18]\n@tn=@x\u0002\n(1\u0000n2)u\u0003\n; (3a)\n@tu=@x\u0002\n(1\u0000u2)n\u0003\n\u0000@x\u0012@xxn\n1\u0000n2\u0000n(@xn)2\n(1\u0000n2)2\u0013\n;\n(3b)\n@t\b =h0\u0000(1\u0000u2)n+1p\n1\u0000n2@x\u0012@xnp\n1\u0000n2\u0013\n;(3c)\nwhere (3b) follows from the negative gradient of (3c).\nThese equations result from an exact transformation ofthe LL equation (1). Equation (3) is analogous to the\nmass, momentum, and Bernoulli equations for an invis-\ncid, irrotational, compressible fluid. Owing to a phase\nsingularity, the vacuum state occurs when jnj= 1. Equa-\ntion (3) is invariant to the reflection transformation\nh0!\u0000h0; n!\u0000n;\b!\u0000\b; u!\u0000u:(4)\nThe boundary conditions (BCs) for Eq. (3) are\n@xn(0;t) = 0; @xn(L;t) = 0; (5a)\nu(0;t) =ub(t); u(L;t) = 0; (5b)\nwhereub(t)models the time dependence of a perfect\nspin injection source that increases from 0 to the max-\nimum intensityju0jmonotonically and smoothly with\nthe rise time t0. We adopt a hyperbolic tangent profile\nto model the injection rise: u(0;t) = (u0=2)ftanh[(t\u0000\nt0=2)=(t0=10)] + 1g, wheret0= 80is the time that the\ninjection magnitude reaches 99:99%ofju0j. For a typi-\ncal Permalloy, this hyperbolic tangent profile produces a\nrelatively sharp change in the hydrodynamic variables—\nabout 2ns—when compared to the typical precessional\nperiodofspin-injectedDEFs, ontheorderof10–20ns[9].\nThe modulationally stable region, consisting of velocities\nuin the interval [\u00001;1], corresponds to stable fluid-like\nconfigurations, so we restrict ju0j<1[8]. The initial\ncondition (IC) is given by\nn(x;t= 0) =h0; (6a)\nu(x;t= 0) = 0; (6b)\nwithjh0j<1. Thus, the spin injection problem can be\nreduced to a piston problem: a piston at x= 0, initially\nwith velocity u= 0, is accelerated to u=u0, generat-\ning a flow to the right ( u0>0) or left (u0<0) into\nthe quiescent fluid with density n=h0. In the rest of\nthis work, we will refer to this piston analogy for our\ninterpretation of the spin dynamics that result from the\ninitial-boundary value problem (3)–(6). We focus on the\nclassification of solutions when they are fully developed\nsuch as in Fig. 1(d).\nIII. Nonlinear Wave Dynamics and Whitham\nModulation Theory\nIn this section, we provide some necessary background,\nprimarily following [30], on Whitham modulation theory,\na powerful tool for studying multiscale nonlinear wave\ndynamics [17, 27, 28]. Modulation theory results in equa-\ntions that describe the slow variation of nonlinear, peri-\nodic traveling wave solutions.\nA. Traveling Wave Solutions\nConsider the traveling wave solutions of Eq. (3) in the\nformn(x;t) =n(\u0018)andu(x;t) =u(\u0018)with the moving5\ncoordinate \u0018=x\u0000Vt, where\nV2=1\n20\n@1 +4X\ni<jninj+4Y\nini+vuut4Y\ni(1\u0000n2\ni)1\nA(7)\nis the square of the wave’s phase speed and ni,i=\n1;2;3;4are wave parameters. The positive/negative\nsquare root corresponds to right/left-going wave solu-\ntions, respectively. By insertion of n(\u0018),u(\u0018)into (3)\nand direct integration, the traveling wave satisfies the\nordinary differential equation (ODE)\n\u0012dn\nd\u0018\u00132\n=\u0000R(n); (8)\nwhereR(n) = (n\u0000n1)(n\u0000n2)(n\u0000n3)(n\u0000n4)is the\npotential function, a quartic polynomial with zeros at\nni,i= 1;2;3;4. The velocity field u(\u0018)can be obtained\nin terms of n(\u0018)and the roots ni[30] so we focus on\nthe modulation analysis with n. For real, ordered n1\u0014\nn2\u0014n3\u0014n4in the interval [\u00001;1], the traveling wave\nsolution for neither oscillates within [n1;n2]or[n3;n4]\nwith wavelength given by\nL=4K(m)p\n(n3\u0000n1)(n4\u0000n2); (9)\nwhereK(m)is the complete elliptic integral of the first\nkind withmgiven by\nm=(n4\u0000n3)(n2\u0000n1)\n(n4\u0000n2)(n3\u0000n1): (10)\nWhennoscillates within [n1;n2], the traveling wave\nsolution is\nn(\u0018) =n2\u0000(n2\u0000n1)cn2(W;m )\n1 +n2\u0000n1\nn4\u0000n2sn2(W;m );(11)\nwhere\nW=p\n(n3\u0000n1)(n4\u0000n2)\u0018=2 (12)\nandcn,snare Jacobi elliptic functions [36]. The velocity\nis given in terms of nby\nu=\u0000A1+Vn\n1\u0000n2; (13)\nwhere\nA2\n1=1\n20\n@1 +4X\ni<jninj+4Y\nini\u00074Y\niq\n1\u0000n2\ni1\nA;\nV=vuuut1\n20\n@1 +4X\ni<jninj+4Y\nini\u00064Y\niq\n1\u0000n2\ni1\nA:\n(14)The positive square root of A1is taken here. The upper\n(lower) sign in (14) gives the fast (slow) branch of wave.\nVin Eq. (7) is the fast branch. We can also describe the\nsolution in terms of an alternative set of physical wave\nparameters (\u0016n;\u0016u;a;k ), equivalent to ni, corresponding to\nthe mean spin density \u0016n, mean velocity \u0016u, the amplitude\na=n2\u0000n1, and the wavenumber k= 2\u0019=L.\nWhenn3!n2andm!1, the solution limits to a\ndepression soliton\nn=n2\u0000n2\u0000n1\ncosh2W+n2\u0000n1\nn4\u0000n2sinh2W;(15)\nwithbackgroundmeandensity \u0016n=n2. Thefluidvelocity\nin the soliton limit is\nu=\u0000B+csn\n1\u0000n2; (16)\nwhere\nB= \u0016u(\u0016n2\u0000a\u0016n+ 1) + \u0016n\u0016;\ncs= \u0016u(2\u0016n\u0000a) +\u0016;\n\u0016=\u0006p\n(1\u0000(\u0016n\u0000a)2)(1\u0000\u0016u2);(17)\nwherecsis the soliton speed and \u0016uis the background\nmean velocity. The sign +(\u0000) gives the fast (slow) soli-\nton. Intermsoftheroots fnig4\ni=1,csand\u0016uforthesoliton\ncan be obtained by taking the limit n3!n2in (13) and\n(14).\nWhenn4!n3,m!0, there are two possible limiting\nsolutions. If n2\u0000n1\u001cn3\u0000n1, then the solution limits\nto a small-amplitude harmonic wave. If n2!n3=n4,\nthe solution limits to a depression algebraic soliton\nn(x;t) =n2\u0000n2\u0000n1\n1 +1\n4(n2\u0000n1)2\u00182:(18)\nThe background mean density for the algebraic soliton\nis\u0016n=n2=n3=n4. The algebraic soliton amplitude\nisa=n2\u0000n1. The background mean velocity \u0016uand\nalgebraic soliton speed are obtained by setting n2=n3=\nn4in (13) and (14).\nWhennoscillates within [n3;n4], the traveling wave\nsolution is\nn(\u0018) =n3+(n4\u0000n3)cn2(W;m )\n1 +n4\u0000n3\nn3\u0000n1sn2(W;m );(19)\nwhereWis given in (12). The velocity is (13) with A1\ntaking the negative square root and the same Vin (14).\nWhenn3!n2andm!1, the solution limits to an\nelevation soliton\nn=n3+n4\u0000n3\ncosh2W+n4\u0000n3\nn3\u0000n1sinh2W:(20)\nThe background mean density for the soliton is \u0016n=n2=\nn3. The soliton amplitude is a=n4\u0000n3. The back-\nground mean velocity \u0016uis obtained by taking the limit6\nn3!n2in(13)and(14). Alternatively, thefluidvelocity\nin the soliton limit is (16) with\nB= \u0016u(\u0016n2+a\u0016n+ 1) + \u0016n\u0016;\ncs= \u0016u(2\u0016n+a) +\u0016;\n\u0016=\u0006p\n(1\u0000(\u0016n+a)2)(1\u0000\u0016u2);(21)\nwhere +(\u0000) gives the fast (slow) soliton.\nWhenn2!n1,m!0, there are two possible limiting\nsolutions. If n4\u0000n3\u001cn4\u0000n1, then the solution limits\nto a small-amplitude harmonic wave. If n3!n2=n1,\nthe solution limits to an elevation algebraic soliton\nn(x;t) =n3+n4\u0000n3\n1 +1\n4(n4\u0000n3)2\u00182:(22)\nThe background mean density for the algebraic soliton\nis\u0016n=n3=n2=n1and the soliton amplitude is a=\nn4\u0000n3. The background mean velocity \u0016uand algebraic\nsoliton speed are obtained by setting n1=n2=n3in\n(13) and (14).\nB. Whitham Modulation Equations\nThemodulationequationscanbeexpressedindiagonal\nform by introducing new modulation variables \u0015iknown\nas Riemann invariants\n@\u0015i\n@t+vi@\u0015i\n@x= 0; i= 1;2;3;4; (23)\nwhere the Riemann invariants are ordered as \u00151\u0014\u00152\u0014\n\u00153\u0014\u00154, and theviare the Whitham velocities\nvi=1\n24X\ni=1\u0015i\u0000L\n2@L=@\u0015i; i21;2;3;4;(24)\nwhereL=4K(m)p\n(\u00153\u0000\u00151)(\u00154\u0000\u00152)andm=(\u00154\u0000\u00153)(\u00152\u0000\u00151)\n(\u00154\u0000\u00152)(\u00153\u0000\u00151).\nThe transformation between the niand\u0015iis provided\nin the Appendix.\nThe LL-Whitham modulation equations (23) are non-\nconvex, namely they can lose strict hyperbolicity (two\nWhitham velocities coalesce) and/or they can lose gen-\nuine nonlinearity where@vi\n@\u0015i= 0in certain parameter\nregimes, resulting in a nonmonotonic dependence of the\nWhitham velocity on the Riemann invariant.\nC. Piston Sonic and Convexity Conditions\nThe modulation equations (23) exhibit two important\nlimiting simplifications. By comparing (9) and (10), we\nfind that the soliton limit (L!1andm!1) occurs\nwhen\u00152!\u00153and the spin wave harmonic limit (a!0\nandm!0) occurs when either \u00151!\u00152or\u00153!\u00154.\nIn these limits, two of the modulation equations coincidewith the long-wave, dispersionless limit of Eqs. (3a) (3b)\n@t\u0016n=@x\u0002\n(1\u0000\u0016n2)\u0016u\u0003\n; (25a)\n@t\u0016u=@x\u0002\n(1\u0000\u0016u2)\u0016n\u0003\n: (25b)\nThe remaining two modulation equations merge and cor-\nrespond to modulations of either the soliton amplitude or\nthe spin wave wavenumber. The limiting velocities deter-\nmine the motion of DSW edges. In the soliton limit, we\nhave\ns\u0000\u0011lim\n\u00152!\u00153v2= lim\n\u00152!\u00153v3=1\n2(\u00151+ 2\u00153+\u00154):(26)\nIn one of the harmonic limits, we have\ns+\u0011lim\n\u00153!\u00154v3= lim\n\u00153!\u00154v4= 2\u00154+(\u00152\u0000\u00151)2\n2(\u00151+\u00152\u00002\u00154):\n(27)\nThe velocities s\u0000<s+are the trailing and leading edges\nof the DSW.\nThe dispersionless equations (25) describe the evolu-\ntion of the mean density \u0016nand mean velocity \u0016u. These\nequations can be expressed in diagonal form\n@r\u0006\n@t+v\u0006@r\u0006\n@x= 0; r\u0006= \u0016u\u0016n\u0006p\n(1\u0000\u0016u2)(1\u0000\u0016n2);\n(28)\nwhere the dispersionless Whitham velocities v+=\n1\n2(3r++r\u0000) = 2\u0016u\u0016n+p\n(1\u0000\u0016u2)(1\u0000\u0016n2),v\u0000=1\n2(r++\n3r\u0000) = 2\u0016u\u0016n\u0000p\n(1\u0000\u0016u2)(1\u0000\u0016n2)are also the long-\nwavelengthspinwavevelocities. Thesevelocitiesareused\nto identify the magnetic sonic condition [8]. The piston\nis subsonic if v\u0000<0<v+, when\nj\u0016uj<ucr(\u0016n) =r\n1\u0000\u0016n2\n1 + 3\u0016n2; (29)\nand supersonic if v\u0000< v +<0(\u0016u <\u0000ucr(\u0016n)) or\n0< v\u0000< v +(\u0016u > u cr(\u0016n)). Consequently, two differ-\nent boundary behaviors will arise.\nThe dispersionless system (25) has simple wave solu-\ntions where only one of the Riemann invariants changes:\n(+)-waves when r\u0000is constant and (\u0000)-waves when r+\nis constant. These solutions require the hyperbolic sys-\ntem of equations (25) to remain genuinely nonlinear [37],\nwhich holds so long as\n\u0016u6=\u0006\u0016n;j\u0016uj6= 1;j\u0016nj6= 1: (30)\nThese are the convexity conditions .\nIV. Phase Diagram of Figure 2\nIn this section, we provide a qualitative description of\nthe solution types depicted in Fig. 2 as well as a quan-\ntitative description of the boundaries between the differ-\nent sectors. Each distinct solution type originates from7\nthe prevailing physical and mathematical properties of\nthe hydrodynamic equations (3) at the piston boundary:\nsubsonic/supersonic flow, compression/expansion waves,\nand convexity. These properties determine the various\ncurves partitioning the phase diagram in Fig. 2. The so-\nlution type acronyms and symbols are defined in Tab. I.\nNote that the reflection symmetry (4) implies that the\nphase diagram can be reflected in u0andh0to obtain\nthe classification for h0<0. A more detailed, quantita-\ntive description of each solution type is developed in the\nnext three sections.\nThe Whitham modulation equations (23) are a set of\nhyperbolic equations that we will solve in order to deter-\nminethestructureofsolutionsinthephasediagram. The\noscillatory solutions we obtain here exhibit the follow-\ning fundamental feature: they terminate when either the\nwave amplitude goes to zero (the harmonic limit) or the\nwavelength goes to infinity (the soliton limit). In both\ncases, the dispersionless equations (25) govern the mean\ndensity and velocity. A general property of hyperbolic\nequations such as (25) is that any dynamic front adjacent\nto a constant region is a simple wave [38]. Therefore, we\ncan determine a relationship between the constant states\nto the left and right of the RW, DSW, CDSW, etc., by\nholding one dispersionless Riemann invariant constant.\nFor the spin piston located at the left boundary, we will\nexcitethefastestwave,a (+)-wave,inwhichtheRiemann\ninvariantr\u0000in Eq. (28) is constant across the wave\n(+)-wave:nLuL\u0000q\n(1\u0000(uL)2)(1\u0000(nL)2)\n=nRuR\u0000q\n(1\u0000(uR)2)(1\u0000(nR)2):(31)\nThe superscripts LandRdenote the constant (mean)\nstates to the left and right of the wave, respectively. In\norder for a (+)-wave to solve the spin piston problem, we\nalso require the RW or DSW to propagate to the right of\nthe boundary. Namely, we require the leftmost edge of\nthe wave to have positive velocity\nadmissibility: 0<(\nv+(rL\n\u0000;rL\n+); RW;\ns\u0000(rL\n\u0000;\u00152=\u00153;rL\n+);DSW:\n(32)\nIt turns out that all the solutions depicted in Fig. 2 are\nadmissible except in the supersonic sector VII.\nThe right state is constant, determined by the exter-\nnal magnetic field and free spin boundary condition (6a),\n(6b)\nnR=h0; uR= 0: (33)\nThe constant left state is achieved after the piston veloc-\nity has saturated at t\u0019t0(ub(t)!u0), provided the\nadmissibility condition (32) is satisfied. When the left\nstate is subsonic, we use (31), (33), and (5b) to obtain\nthe spin density on the left\nsubsonic: nL=h0q\n1\u0000u2\n0\u0000u0q\n1\u0000h2\n0; uL=u0:\n(34)The flow is subsonic so long as (29) with \u0016u!uLand\n\u0016n!nLis satisfied. The transition from subsonic to\nsupersonic in the phase diagram Fig. 2 occurs when\njuLj=ucr(nL): (35)\nUsing (34), there are multiple solutions of eq. (35). The\nregion of parameters corresponding to subsonic condi-\ntions at the x= 0boundary is the interior of the follow-\ning four curves\nu0=\u0006s\n2 +h2\n0\u0006h0p\nh2\n0+ 8\n6;\nu0=s\n\u00002 + 3h2\n0\u0006h0p\n9h2\n0\u00008\n2;(36)\nIn other words, (36) are the sonic curves. The subsonic\nregion is reflected in Fig. 2 by the unshaded and pink-\nshaded regions containing sectors I–IV.\nConsequently, sectors V–VII are supersonic and we\nneedanalternativewaytodetermine nLbecausenL6=u0\nat the piston boundary. Sectors V and VI are associated\nwith the supersonic condition v\u0000<v+<0and the way\nto resolve this was first identified in [10] where a sta-\ntionary spin soliton was introduced with its extremum\nin density and velocity centered at the piston boundary.\nThus, only half the soliton is within the domain and it\nwasreferredtoasa contact soliton . Thesolitonsolutions,\ngiven by the fast branch of (15) and (20), provide for a\nrapid transition from supersonic conditions at the piston\nto subsonic conditions in the soliton far-field (n;u). In\norder to uniquely determine the soliton, we invoke three\nassumptions. First, we identify the soliton far-field with\n(n;u) = (nL;uL). Then, for a ( +)-wave, we can use\nEqs. (31) and (33) to determine\nnL=\u0000uLq\n1\u0000h2\n0+h0q\n1\u0000(uL)2:(37)\nSecond, byequatingtheextremesolitonvelocity u(\u0018= 0)\nin (16) to the piston velocity u0, we have\nu0=\u0000B\n1\u0000(nL\u0006a)2; (38)\nwhereBis given in Eqs. (17), (21), a >0is the soliton\namplitude. Finally, the soliton is stationary so that cs=\n0in (17) or (21), giving\nuL(2nL\u0006a) +q\n(1\u0000(nL\u0006a)2)(1\u0000(uL)2) = 0:(39)\nThe+(\u0000)in (38) and (39) correspond to a bright (dark)\nsoliton. For example, in the supersonic sector V, the\nsoliton is of elevation type so (21) applies and the +sign\nis taken in (38) and (39). The three conditions (37), (38),\nand (39) uniquely determine the soliton amplitude aand\nits far-field (nL;uL).8\nIn [12], it was shown that this problem gives rise to\ncompression or expansion waves emanating from the pis-\nton depending upon the input parameters (u0;h0). This\nis determined by whether or not the ( +)-wave speed v+\nis increasing or decreasing from left to right during the\npiston acceleration period.\ncompression : v+(nL(t);ub(t))>v+(h0;0)(40)\nimplies compression waves and expansion waves other-\nwise. The pure compression region is reflected in Fig. 2\nby the solid black lines u0= 0andu0=h0. When\n0< u0< h0, the subsonic solutions involve only DSWs.\nWhen 0< h 0< u 0, the subsonic solutions involve both\nRWs and DSWs. When u0<0orh0= 0, the subsonic\nsolutions are RWs.\nWhenu0>0, there is another effect at play: loss of\nconvexity (30) when uL=jnLj. For the subsonic regime,\nuL=u0and (34) implies convexity is lost when\nloss of convexity: h0= 2u0q\n1\u0000u2\n0:(41)\nThis is the dotted curve in Fig. 2. To the right of\nthis curve, the solutions exhibit hybrid waves involving\nCDSWs, and either DSWs (when 0< u 0< h 0) or RWs\n(whenu0>h0).\nOne more feature of the solutions is depicted in Fig. 2:\nvacuum points. When jnj= 1, the velocity uis unde-\nfined and corresponds to the absence of fluid or vacuum.\nWe find that only oscillatory solutions such as DSWs and\nCDSWs, i.e., u0>0, can result in the generation of iso-\nlated points at which jnj= 1. The threshold for this\nbehavior is determined by equating the extrema of the\noscillation density (11) or (19) with n=\u00061, namely\nnj= (\u00001)jfor some root nj,j2f1;2;3;4g. A quanti-\ntative determination of this threshold requires the solu-\ntion of the Whitham modulation equations (23), which\nwe undertake in the next several sections. The vacuum\nthresholdisdepictedinthephasediagram2byasolidred\ncurve, above which the solutions exhibit vacuum points.\nIn the following sections, we solve the Whitham mod-\nulation equations to obtain the detailed structure of the\nshock, rarefaction, and soliton solutions.\nV. Zero Applied Field\nWhenh0= 0, all of the dynamics are governed by\nthe dispersionless limit (3) with additional treatment if\nthe solution is supersonic. This case corresponds to the\nhorizontal axis in the phase diagram 2. The (+)-wave for\nr+=r+(\u0018)satisfiesv+(rR\n\u0000;r+) =\u0018=x=(t\u0000\u0016t), where \u0016t\nis a constant time shift, rR\n\u0000=\u00001, andrL\n+<r+(\u0018)<rR\n+.\nA. Subsonic Regime: RW\nThe subsonic solution when h0= 0is a RW. The (+)-\nwave assumption leads to the background spin density onthe left given by (34)\nnL=\u0000u0: (42)\nThe system is always convex because ju0j<1in\n(41). The admissibility condition (32) is satisfied until\nv+(rL\n\u0000;rL\n+) = 0, which is also the sonic condition (35)\nleading to u0=\u0006ucr=\u00061p\n3. Thus, for a RW solu-\ntion to be admissible, the piston velocity u0is required\nto be subsonic with u2\n0<1\n3. We have additionally con-\nfirmed that there are no admissible (\u0000)-wave solutions\nwithrL\n+=rR\n+. The Riemann invariant configuration and\nan example solution is shown in Fig. 3(a). In the the-\noretical solution, a time delay \u0016t= 30(recall,t0= 80)\nis introduced to account for the piston acceleration time.\nThis choice of time delay is consistently applied to all the\ntheoretical solutions in the following sections. Across the\nsubsonic domain, our theoretical predictions on nLagree\nexcellently with simulation results, shown in Fig. 4(a).\nFIG. 3. Riemann invariant configurations in the upper panels\ncorresponding to theoretical (dotted) and numerical (solid)\nsolutions. The vertical dashed lines correspond to x\u0006=\ns\u0006(t\u0000\u0016t), where \u0016t= 30is the time delay introduced to ac-\ncount for the piston acceleration time. (a) RW solution with\nu0=\u00000:3, satisfying the subsonic condition ju0j<1p\n3; (b)\nsoliton|RW solution with u0=\u00000:7, satisfying the supersonic\ncondition1p\n3<ju0j<1\nB. Supersonic Regime: S+|RW\nWhen the piston velocity is supersonic with u2\n0>1\n3, a\ncontactsolitondevelopsatthepistonboundarysmoothly\nconnected to a RW via an intermediate constant state.\nThe Riemann invariant configuration of the solution is\nshown in the top panel of Fig. 3(b). The soliton is rep-\nresented by the Riemann invariants \u00152=\u00153.\nThis soliton is theoretically determined by (37)-(39)\nfor a bright soliton. It is verified that when the piston9\nFIG. 4. Theory and simulation results of the left constant\nstate density nLfor subsonicju0j<1p\n3(a) and supersonic\n1p\n3<ju0j<1(b). The soliton amplitude ais also shown in\n(b).\nis moving at the sonic speed u0=ucr=\u00061p\n3, the soli-\nton does not exist, i.e. a= 0. Therefore, the soliton\nat the piston boundary only emerges in the supersonic\nregime. A representative supersonic solution is shown\nin the bottom panel of Fig. 3(b), exhibiting good agree-\nment with the numerical simulation. Across the super-\nsonic domain, theoretical predictions of nLandaof the\nsoliton demonstrate excellent agreement with simulation\nresults, shown in Fig. 4(b). Herein we have confirmed\nour assumptions proposed in Sec. IV on the characteri-\nzation of the solitonic supersonic solutions. In addition,\nour analysis found that no vacuum state, where jnj= 1\noccurs. Thelargestmagnitudeof nisreachedatthepeak\n(crest) of the elevation (depression) soliton at the piston\nboundary and this magnitude is always less than 1 for\n1\n3<u2\n0<1.\nFIG.5. Riemanninvariantconfigurationsandthecorrespond-\ning theoretical (dotted) and numerical (solid) solutions. The\nvertical dashed lines correspond to x\u0006=s\u0006(t\u0000\u0016t), where\n\u0016t= 30is the time delay introduced to account for the pis-\nton acceleration time. (a) Sector I: RWwith u0=\u00000:1and\nh0= 0:8. (b) Sector II: DSW+with u0= 0:2andh0= 0:8,\nthe dotted curve is the theoretical DSW envelope.VI. Nonzero Applied Field, Subsonic Regime\nIn this section, we present subsonic solutions with\nnonzero applied field. The solution map is the white re-\ngion of the phase diagram Fig. 2, including sectors I-IV.\nWe consider each sector in turn.\nA. Sector I: RW\nIn sector I, the system satisfies the convexity condition\n(30) and yields simple wave solutions. Again, the solu-\ntion is an expansive RW satisfying r\u0000=rR\n\u0000,r+=r+(\u0018),\nandv+(r\u0000;r+) =\u0018=x=(t\u0000\u0016t). An example Riemann\ninvariant configuration and solution in sector I is shown\nin Fig. 5(a). Good agreement between theory and simu-\nlation is demonstrated.\nThe admissibility condition (32) has been verified\nacross sector I. The sonic condition is determined by\nv+= 0, yielding the boundary between sector I and V.\nAgain, the sonic condition coincides with the admissibil-\nity threshold, indicating that only subsonic solutions are\nadmissible in sector I. On the boundary between sector\nI and II which is the h0-axis, there are no induced dy-\nnamics because u0= 0. Furthermore, no vacuum state\nis present in sector I since the largest magnitude of n\nis at the piston boundary where the piston velocity is\nrestricted toju0j<1.\nB. Sector II: DSW+\nSector II is to the left of the convexity curve (dotted\nblack curve) in the phase diagram Fig. 2, so the system\nis convex, yielding simple wave solutions. Furthermore,\nv+(rR\n\u0000;rL\n+)> v+(rR\n\u0000;rR\n+)leads to compressive DSW so-\nlutions that satisfy \u00153=\u00153(\u0018),v3(rR\n\u0000;rR\n+;\u00153;rL\n+) =\u0018=\nx=(t\u0000\u0016t). The DSW solutions satisfy the admissibility\ncondition (32). An example Riemann invariant config-\nuration and solution in sector II is shown in Fig. 5(b).\nNeartheDSW’sharmonicedge, thenumericalsimulation\nand the predicted envelope amplitude deviate somewhat.\nThis is a common feature of the asymptotic (large t) be-\nhaviorofDSWs[17]. Fig.6showsthat, despitethepiston\naccelerationperiod, thetheoreticallypredictedtrajectory\nof the DSW’s soliton edge differs from the simulation re-\nsult by a constant time shift.\nThe DSW solution exhibits vacuum in the pink-shaded\nregion in the phase diagram Fig. 2. The vacuum state\nis first reached when the maximum of the trailing edge\nsoliton density n=n4reaches 1. We evaluate n4in\nthe soliton limit, which is a function of the Riemann in-\nvariants, to determine this threshold (see Appendix). As\ntime progresses, the vacuum point will move inside the\noscillatory structure [17]. Example DSWs with vacuum\nwill be shown in Fig. 8. We point out that the vacuum\nthreshold determination is the same across all subsonic10\nsectors whose solution contains a DSW structure, despite\nthe convexity of the system.\nFIG. 6. Space-time contour plot of a DSW+solution in sector\nII with u0= 0:2,h0= 0:8. The black solid line is the pre-\ndicted trailing edge soliton location for an ideal piston with\ninstantaneous acceleration.\nC. Sector III: DSW+CDSW+\nSector III is to the right of the convexity curve (dot-\nted black curve) in the phase diagram Fig. 2, so the so-\nlution breaks the convexity condition (30), manifested\nas the coalescence of two Riemann invariants \u00153=\u00154\nand Whitham velocities v3=v4. The Riemann invari-\nant configuration and an example solution are shown\nin Fig. 7(a), satisfying r\u0000=rR\n\u0000,\u00153=\u00153(\u0018), andv3(rR\n\u0000;rR\n+;\u00153;rL\n+) =\u0018=x=(t\u0000\u0016t). The spin injection u0\nsatisfies (40), leading to a compressive DSW+CDSW+\ncomposite wave where rL\n+> rR\n+gives the DSW portion\nand the coalescence of Riemann invariants \u00153=\u00154gives\nthe CDSW portion.\nA CDSW is a degenerate DSW solution whose soli-\nton limit is an algebraically decaying soliton where three\nRiemann invariants, \u00152,\u00153, and\u00154, coincide. The al-\ngebraic soliton travels at the speed of a dispersionless\n(long-wave) characteristic velocity, mimicking a contact\ndiscontinuity in viscous hydrodynamics. It is observed\nnumerically that CDSWs generally require a longer time\nthan DSWs to develop. Therefore, a larger discrepancy\nbetween the simulated CDSW portion and the analytical\nwaveenvelopeisobservedcomparedtotheDSWportion.\nThe admissibility of the composite wave solution in sec-\ntor III, 0< s(1)\n\u0000< s(2)\n\u0000< s+, has been confirmed. The\nregion where a vacuum state is present in the solution is\nshaded in pink in Fig. 2 and a typical solution is shown\nin Fig. 8(c).\nD. Sector IV: RWCDSW+\nBefore moving on to sector IV, we discuss the solution\non the boundary between sector III and IV, where rL\n+=\nrR\n+as shown in the Riemann invariant configuration in\nFig. 7(b). The system is nonconvex and the solution is a\nsingle CDSW+because\u00153=\u00154across the shock.\nFIG. 7. Riemann invariant configurations and example solutions when h06= 0for (a) Sector III: DSW+CDSW+withu0= 0:55\nandh0= 0:7; (b) Boundary of sectors III and IV: CDSW+with u0= 0:6andh0= 0:6. (c) Sector IV: RW CDSW+with\nu0= 0:73andh0= 0:6; The vertical dashed lines separate different components of the composite wave solutions based on\npredicted edge velocities. The dotted curves are the predicted envelopes of the DSW structure in the solution. In (b), the\ndotted curve also predicts the dispersionless RW portion of the solution. All modulation solutions include the time delay \u0016t= 30\nto account for the piston acceleration time.11\nSector IV is to the right of the convexity threshold\n(dotted black curve, Eq. (41)) in Fig. 2, so the system is\nnonconvex. During piston acceleration, compressive dy-\nnamics are induced, then followed by expansive dynam-\nics. Thus, the solution is a RW CDSW+composite wave\nthat satisfies r\u0000=rR\n\u0000,v3(rR\n\u0000;rR\n+;\u00153;\u00153) =\u0018=x=(t\u0000\u0016t),\nand\u00154=\u00153. The Riemann invariant configuration and\nan example solution are shown in Fig. 7(c). The admissi-\nbility (32) of the solutions have been verified in the sector\nwith the threshold s(1)\n\u0000= 0coinciding with the sonic con-\nditionv+= 0and Eq. (35). The vacuum region, shaded\nin pink in Fig. 2, is determined by evaluating the wave\nenvelopen4in the algebraic soliton limit. The onset of\nvacuum is found to be independent of u0in this case and\nhappens at h0= 1=p\n2. Representative solutions con-\ntaining a vacuum point are shown in Fig. 8(c), (d).\nIntheexamplesolutionsshowninFig.7(b), (c), weob-\nserve that there is a smooth tail at the algebraic soliton\nlimit of the CDSW when it connects to the dispersion-\nless portion of the solution. This phenomenon is most\nevidently shown in Fig. 7(b) with a single CDSW. This\nbehavior does not occur in DSWs where the exponential\nsolitonedgeterminatesdirectlyatthedispersionlessedge\nstate (see the bottom panels of Figs. 5(b) and 7(a)). This\nphenomenon serves as a distinguishing feature to identify\nthe soliton edge of a CDSW.\nVII. Nonzero Applied Field, Supersonic Regime\nSectors V and VI satisfy the supersonic condition (35)\nin whichv\u0000<v+<0and a contact soliton is developed\nat the piston boundary. Sector VII also satisfies the su-\npersonic condition where 0< v\u0000< v +and a PDSW\nemanates from the piston boundary.\nA. Sector V: S+jRW\nThe contact soliton is uniquely determined by (37)–\n(39). With the determined soliton far-field (nL;uL) =\n(\u0016n;\u0016u), the modulation solution is an expansive RW, sat-\nisfying (31) and r+=r+(\u0018)wherev+(r\u0000;r+) =\u0018=\nx=(t\u0000\u0016t). The admissibility condition (32) is satisfied.\nSimilar to the zero field supersonic solution, no vacuum\npoint is attained. A representative solution is shown in\nFig. 9(a) with quantitative agreement between the theo-\nretical prediction and the numerical simulation.\nB. Sector VI: S\u0000jRWCDSW+\nThe depression contact soliton is uniquely determined\nby (37)–(39) with far-field (nL;uL) = (\u0016n;\u0016u)breaking the\nconvexity condition (30), leading to a RW CDSW+com-\nposite wave satisfying r\u0000=rR\n\u0000,v3= (rR\n\u0000;rR\n+;\u00153;\u00153) =\nFIG. 8. Example solutions with vacuum states when h06= 0.\n(a) Sector II: DSW+with u0= 0:45andh0= 0:92; (b) Bor-\nder of sectors II and III: CDSW+withu0= 0:8andh0= 0:8;\n(c) Sector III: DSW+CDSW+with u0= 0:6andh0= 0:85;\n(d) Sector IV: RW CDSW+with u0= 0:8andh0= 0:75.\nThe dotted curves are the predicted envelopes of the DSW\nstructure. In (d), the dotted curve includes the predicted dis-\npersionless RW portion in the solution. The vertical dashed\nlines separate different components in composite modulation\nsolutions. The modulation solutions includes the time delay\n\u0016t= 30to account for the piston acceleration time.\nFIG. 9. Supersonic solutions when h06= 0. (a) Sector\nV:S+jRWwith u0=\u00000:6andh0= 0:8; (b) Sector VI:\nS\u0000jRW CDSW+with u0= 0:92andh0= 0:6; (c) Sector\nVI: S\u0000jRW CDSW+with a vacuum point, u0= 0:96and\nh0= 0:75. The dotted curves trace the predicted piston edge\nsoliton, the dispersionless portion of the solution, and the en-\nvelope of the DSW-type portion of the solution. The vertical\ndashed lines divide the different components in a composite\nwave based on the predicted edge velocities. The time delay\n\u0016t= 30is used in theoretical plotting to account for the piston\nacceleration time.\n\u0018=x=(t\u0000\u0016t), and\u00154=\u00153. A representative solution12\nis shown in Fig. 9(b). Same as in sector IV, the on-\nset of vacuum, when the algebraic soliton of the CDSW\nportion reaches 1, is independent of u0and happens at\nh0= 1=p\n2. This thresh The pink-shaded region in Fig. 2\nindicates a vacuum state is present in the solution. A\nvacuum solution from this sector is shown in Fig. 9(c).\nC. Sector VII: PDSW+|DSW+\nThe supersonic condition ju0j>ucr(nL)in sector VII\nis0< v\u0000< v +. This positive velocity configuration\nis different from all other supersonic sectors with nega-\ntive dispersionless velocities. It gives rise to a PDSW\n[39] at the piston edge. For this sector, we focus on\nthe qualitative identification of the solution features with\nthe support of simulations. As we observed numerically\n(see Fig. 10(a) for example), the PDSW is led by a soli-\nton at its right edge and terminates on the left at the\npiston boundary without reaching the small amplitude\nlimit. The intermediate state connecting the PDSW and\na DSW-type wave demonstrates slow oscillations that\npossibly is not a constant plateau and requires additional\nanalysis. WithoutthePDSWfar-fielddetermined,weare\nunable to determine the modulation solution. Note that\na vacuum point is present inside the solution, consistent\nwith our prediction in Fig. 2.\nWe have numerically confirmed that along the sonic\ncurve bounding the subsonic sector II, where the system\nremains convex, there is no PDSW emerging from the\npiston boundary. However, within the nonconvex sub-\nsonic sector III when near the sonic curve at the sector\nVII boundary, we numerically observed that a PDSW\ndevelops at the piston boundary as shown in Fig. 10(b).\nConsequently, the predicted sonic boundary between sec-\ntors III and III does not precisely explain this phase\nchange. We have not been able to quantitatively iden-\ntify the threshold for the occurrence of this phase tran-\nsition using modulation theory. However, all simulations\nthat we have performed in sector VII exhibit this PDSW\nstructure.\nFIG. 10. (a) PDSWjDSW+solution in sector VII with u0=\n0:45andh0= 0:98. (b) Supercritical solution in sector III\nwith u0= 0:8andh0= 0:95.VIII. Conclusion\nUsing the dispersive hydrodynamic framework, we\nhave analytically classified the piston-like dynamics of a\ndissipationless easy-plane ferromagnetic channel subject\nto spin injection at one channel boundary. This frame-\nwork enables the analytical description of noncollinear\nmagnetic textures beyond the small-amplitude, weakly\nnonlinear regime.\nTwo properties of the system is fundamental to our\nanalysis. First, the piston analogy naturally leads to the\ninvestigation of magnetic sub- to supersonic conditions,\ncorresponding to distinct piston boundary behavior: ei-\nther a constant hydrodynamic flow in the subsonic case,\nor a soliton or a non-stationary partial DSW (PDSW)\nboth in the supersonic case. We provided quantitative\ncharacterization of the solutions using the modulation\ntheory and qualitative identification of the PDSW solu-\ntions, where a sharp threshold for this behavior is yet to\nbe determined.\nSecond, the modulation equations exhibit nonconvex-\nity where the modulation velocities coalesce. Adopting\nthe method developed in [30], a non-classical dispersive\nshock wave solution, a contact DSW (CDSW), is pre-\ndicted when the system exhibits nonconvexity as a single\nwave or one component of a composite wave. A distin-\nguishing feature is a short, smooth ramp at the algebraic\nsoliton edge of a CDSW where the soliton connects to a\ndispersionless (non-oscillatory) portion of the solution.\nWhile our analysis was developed for conservative spin\ndynamics applicable over short enough time scales, it has\nintriguing implications for longer times wherein magnetic\ndamping leads to relaxation of the dynamics to a steady\nconfiguration. First, rarefaction waves expand in time\nwith negligible oscillations. This implies that such a so-\nlution minimizes the excitation of spin waves in the sys-\ntem. On the contrary, spin shocks exhibit pronounced\noscillations that can reflect many times in the channel\nbefore being quenched by magnetic damping. While this\ncan be seen as a disadvantage, it is also important to note\nthat the spin waves excited by a spin shock are launched\nwithin a specific spectral band that is determined by the\ntransition between the left and right states [40], opening\nopportunities for controllable transport of angular mo-\nmentum by means of pulsed injection. Second, we find\nthat a stationary soliton established in the conservative\nregimecanremainafterstabilizationviadamping, result-\ning in the contact soliton-dissipative exchange flow [10].\nThird, numerical simulations in [12] show that it is also\npossible to excite propagating soliton trains that persist,\noscillatingbackandforthinthechannel, eveninthepres-\nence of damping. In additional simulataions, we observe\nhere that such solitons are excited precisely when the\noriginating spin shock contains a CDSW. These are ex-\namplesofsituationswherethetransientdynamicsimpact\nthe transport characteristics of the dissipative exchange\nflow in equilibrium.\nThe dispersive hydrodynamic interpretation of ferro-13\nmagnetic dynamics allows one to adopt a large pool of\nanalytical tools that are traditionally used for classical\nfluids, which provides new perspectives on the study and\nunderstanding of spin dynamics. The dynamical problem\nstudied here has a problem setup that is designed to be\nexperimentally accessible and we expect our methodol-\nogy to aid the experimental realization of superfluid-like\nspin transport in the form of nonuniform magnetic tex-\ntures.\nIX. Acknowledgment\nM. Hu thanks T. Congy for the helpful discussions. All\nauthors acknowledge support from the U.S. Department\nof Energy, Office of Science, grant DE-SC0018237. M.\nHoefer acknowledges partial support from National Sci-\nence Foundation DMS-1816934 and M. Hu acknowledges\nsupport from the National Institute of Standards and\nTechnology Professional Research Experience Program.\n. Appendix: Determination of the Physical Wave\nPattern Given Riemann Invariants\nIn this appendix, we present additional information on\nthe characterization of periodic traveling wave solutions\nto the LL equation (1). The LL-Whitham equations in\nterms of the Riemann invariants \u0015= (\u00151;\u00152;\u00153;\u00154)have\nbeen given in (23). The family of traveling waves dy-\nnamics satisfy Eq. (8). The quartic polynomial R(n)can\nbe written in terms of four roots fnig4\ni=1. It can also\nbe expressed in terms of the Riemann invariants \u0015[30]\nwhere\nR(n) =n4+s1+s3\nf1n3+s2n2+ (f1s1\u0000s1+s3\nf1)n\n+1\n4(s2\n1\u00004\u00004s2+ 4f2\n1);\n(A.1)\ns1=4X\ni\u0015i; s2=4X\ni<j\u0015i\u0015j; s3=4X\ni<j<k\u0015i\u0015j\u0015k;\ns4=\u00151\u00152\u00153\u00154;(A.2)\n\u00150\ni=q\n1\u0000\u00152\ni; s0\n4= \u00054\ni\u00150\ni: (A.3)\nFor a given set of Riemann invariants \u0015, there are four\npossible physical wave patterns corresponding to four\npossible choices of f1:\nf1=\u0006q\n(1 +s2+s4+s0\n4)=2; (A.4a)\norf1=\u0006sgn(s1+s3)q\n(1 +s2+s4\u0000s0\n4)=2;(A.4b)This4-valuedmappingofRiemanninvariantstotraveling\nwave profiles implies that the LL-Whitham modulation\nsystem is nonconvex. Later, we denote f1aasf1taking\nthe positive expression in (A.4a) and f1basf1taking the\npositive expression in (A.4b). The fluid velocity can be\ncomputed from the density as\nu(\u0018) =\u0000f1+s1\n2n\n1\u0000n2: (A.5)\nThe multi-valued mapping from the Riemann invari-\nants to the roots of the potential function are [30]:\nn1=\u00001\n2f1(\u00153\u0000\u00152)~s1+ (\u00153\u0000\u00151)~s2\u0000(\u00152\u0000\u00151)~s3\n(\u00153\u0000\u00152)\u00150\n1+ (\u00153\u0000\u00151)\u00150\n2\u0000(\u00152\u0000\u00151)\u00150\n3;\nn2=\u00001\n2f1(\u00153\u0000\u00152)~s1+ (\u00153\u0000\u00151)~s2+ (\u00152\u0000\u00151)~s3\n(\u00153\u0000\u00152)\u00150\n1+ (\u00153\u0000\u00151)\u00150\n2+ (\u00152\u0000\u00151)\u00150\n3;\nn3=\u00001\n2f1(\u00153\u0000\u00152)~s1\u0000(\u00153\u0000\u00151)~s2\u0000(\u00152\u0000\u00151)~s3\n(\u00153\u0000\u00152)\u00150\n1\u0000(\u00153\u0000\u00151)\u00150\n2\u0000(\u00152\u0000\u00151)\u00150\n3;\nn4=\u00001\n2f1(\u00153\u0000\u00152)~s1\u0000(\u00153\u0000\u00151)~s2+ (\u00152\u0000\u00151)~s3\n(\u00153\u0000\u00152)\u00150\n1\u0000(\u00153\u0000\u00151)\u00150\n2+ (\u00152\u0000\u00151)\u00150\n3;\n(A.6)\nwhere ~si= (s1\u0000\u0015i)\u00150\ni+s4\u00150\ni\n\u0015i\u0007s0\n4\u0015i\n\u00150\ni. 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Thesis (2015)."    },    {        "title": "1902.00055v3.Chaos_preserving_reduction_of_the_spin_flip_model_for_VCSELs__failure_of_the_adiabatic_elimination_of_the_spin_population_difference.pdf",        "content": "1\nChaos-preserving reduction of the spin-flip model\nfor VCSELs: failure of the adiabatic elimination of\nthe spin-population difference\nMartin Virte, Member, IEEE, Francesco Ferranti, Senior Member, IEEE\nAbstract —When studying the dynamics of Vertical-Cavity\nSurface-Emitting Lasers, and their polarization properties, the\nspin-flip model appears to be the simplest model qualitatively\nreproducing all dynamical features that have been observed\nexperimentally. Nonetheless, because of the fast time-scale of the\nspin-relaxation processes, the specific role and the importance\nof the spin-population difference - which is one of the specific\nfeature of the spin-flip model - has been continuously questioned.\nIn fact, the debate regarding the possible adiabatic elimination\nof the spin-population remains fully open.\nIn this paper, our goal is to bring new light into this issue by\ndemonstrating that this variable is essential to preserve the most\ncomplex dynamical features predicted by the spin-flip model, such\nas polarization chaos, and, therefore, needs to be conserved. To do\nso, we first perform a detailed analysis, focusing on the chaotic\ndynamics, to determine the minimal embedding dimension for\nthe spin-flip model. As the latter confirms that a reduction of\nthe model could be envisaged, we then consider the adiabatic\nelimination of the spin-population difference to highlight and\nexplain its failure to reproduce essential dynamical features\nobtained in the original model.\nIndex Terms —Semiconductor Laser, VCSELs, chaos, optical\nchaos, polarization dynamics.\nI. I NTRODUCTION\nDue to their intrinsic advantages over standard edge-\nemitting semiconductor lasers, Vertical-Cavity Surface-\nEmitting Lasers (VCSELs) are surely making their way to-\nwards a widespread use not only in telecom but also for\nillumination - as in the iPhone X - or gesture recognition\nand other smart sensing applications. Their main drawback\nhas been identified early on as polarization instabilities [1]:\nthe most striking example being polarization switching (PS)\nevents where the linearly polarized laser light suddenly rotates\nby90\u000e[2]–[6]. But a wide range of behaviour have been\nreported including elliptical polarization [7], noise-induced\nhopping [8], [9] and even chaotic dynamics with the so-called\npolarization chaos [10], [11]. Although these instabilities have\nbeen first studied to be avoided [12], [13], it was also remarked\nManuscript received XXX\nThis work has been supported by the Research Foundation - Flanders\n(FWO) through project REFLEX (krediet aan Navorsers, 1530318N) and the\npost-doctoral fellowship of MV , and by the METHUSALEM program of the\nFlemish Governments.\nMartin Virte is with the Brussels Photonics (B-PHOT), Dept. of Applied\nPhysics and Photonics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel,\nBelgium. He is a post-doctoral Fellow from the Research Foundation -\nFlanders (FWO). Email: mvirte@b-phot.org\nF. Ferranti is with the Microwave Department, Institut Mines-T ´el´ecom\n(IMT) Atlantique, CNRS UMR 6285 Lab-STICC, 29238 Brest CEDEX 3,\nFrance. Email: francesco.ferranti@imt-atlantique.frthat polarization dynamics could be embraced instead: thus\nleading to the tuning of polarization oscillations for high-speed\nspin-VCSELs [14], [15], polarization switching as optical\nmemory mechanism [16]–[18] or random bit generation using\npolarization chaos [19].\nThe polarization features of VCSELs are accurately described\nusing the so-called spin-flip model (SFM) [20], [21]. The main\nfeatures of the SFM are that it takes into account: 1) the\nphase and amplitude anisotropies of the gain medium; 2) two\ndistinct pools of carrier depending on their spin, each of them\nconnected to circularly polarized light of a certain handedness.\nThe spin relaxation processes connecting the two carrier pools\nare then modeled by a simple coefficient commonly called\nthe spin-flip rate. The SFM is, so far, the simplest model\nqualitatively predicting all the dynamical polarization features\nexperimentally observed in VCSELs. Indeed, other models\nhave been proposed for a given behaviour [8], [22], but\nnone - apart from the SFM - reproduced the wide variety\nof features observed experimentally. On the other hand, the\nnonlinearities of the SFM hinder the physical interpretation.\nTo get a better understanding and gain further physical insight,\ndifferent model reductions - based on various approximations\n- have been proposed [23]–[25]. Experimentally, a rather large\nvalue for the spin-flip rate has typically been reported [26]–\n[28]: the spin relaxation rate typically seems to be much\nfaster than the other time-scales of the system, in particular\nthe carrier and field decay rates. As a result, adiabatically\nremoving the spin-population difference, i.e. only considering\none carrier pool, was often considered to be a reasonable\nand realistic approximation, especially when focusing on the\nsystem steady-states [23]–[25]. On the contrary, a rather low\nvalue of the spin-flip rate seems to be a crucial ingredient to\nobtain polarization chaos in solitary VCSELs [29]. Obviously,\nthis observation suggest that, at least for these peculiar exper-\nimental cases [10], [11], the spin-population difference could\nactually play a significant role in the laser dynamics. Hence,\nwe can naturally wonder where is the limit of validity for the\napproximation allowing the adiabatic elimination of the spin-\npopulation difference.\nIn this work, we first aim at confirming that the SFM dimen-\nsion, i.e. the number of independent variables, could indeed\nbe reduced below the current state of the art. We determine\nthe minimal embedding dimension of the system, i.e. the\nminimal number of variables required to accurately represent\nits behaviour, using numerical processing of simulated data\nbased on Principal Component Analysis (PCA) [30] alongarXiv:1902.00055v3  [physics.optics]  28 May 20192\nwith the ”False Neighbors” technique [31], [32]. On top of\nthis important insight, which indicates that a reduction could\nindeed be considered, we also identify from our analysis data\nthe spin-population difference as a potential candidate for such\nreduction. Yet, we then show that an adiabatic elimination\nof the corresponding variable performs, in fact, very poorly\nat preserving system behavior such as polarization chaos,\nbut also the stability of steady-state solutions. We therefore\nhighlight that the spin-flip model could indeed be further\nreduced without qualitative loss, but the model-order reduc-\ntion approach exploiting the adibatic elimination of the spin-\npopulation difference is not a suitable solution to accomplish\nthis goal.\nII. S PIN-FLIP MODEL AND PARAMETERS\nIn its original version, the SFM was a 6-dimension model\nwhich has first been introduced in [20]. When investigating\nits dynamical properties [21], it was quickly observed, how-\never, that only the phase difference between the two linear\npolarization modes played a significant dynamical role. Using\nphase-amplitude decomposition, it was then proposed to use\nthis phase-difference to get a 5-dimensional model [25]. This\nversion of the SFM is the one we exploit here and reads as\nfollows:\ndR+\ndt=\u0014(N+n\u00001)R+\u0000(\racos(\b) +\rpsin(\b))R\u0000(1)\ndR\u0000\ndt=\u0014(N\u0000n\u00001)R\u0000\u0000(\racos(\b)\u0000\rpsin(\b))R+(2)\nd\b\ndt=2\u0014\u000bn\u0000\u0012R\u0000\nR+\u0000R+\nR\u0000\u0013\n\rpcos(\b)\n+\u0012R+\nR\u0000+R+\nR\u0000\u0013\n\rasin(\b) (3)\ndN\ndt=\u0000\r\u0000\n\u0000\u0016+ (N+n)R2\n++ (N\u0000n)R2\n\u0000\u0001\n(4)\ndn\ndt=\u0000\rsn\u0000\r\u0000\n(N+n)R2\n+\u0000(N\u0000n)R2\n\u0000\u0001\n(5)\nwithR\u0006the amplitude of the right and left circular\npolarizations, \bthe phase difference between them, N\nthe total carrier population and nthe carrier population\ndifference between the two carrier reservoirs for each circular\npolarization [20]. Because the carriers in these two reservoirs\nhave different spins, the carrier population difference is often\nreferred to as the spin-population difference. The parameters\nare as follows: \u0014and\rare the field and carrier decay\nrates respectively, \rsis the spin flip relaxation rate, \u000bis\nthe linewidth enhancement factor, \rpand\raare the phase\nand amplitude anisotropies respectively. For simplicity, no\nmisalignment between amplitude and phase anisotropies is\nconsidered [3], [33], [34].\nUnless stated otherwise, we use the same parameter values as\nin previous work [10], [35]: \u0014= 600ns\u00001,\rs= 100ns\u00001,\n\ra=\u00000:7ns\u00001,\r= 1ns\u00001and\u000b= 3. We also consider\ntwo distinct cases of chaotic dynamics as discussed in\n[35] which are obtained for \rp= 4ns\u00001(case 1) and\n\rp= 25ns\u00001(case 2). The injection current is normalized\nby the current at threshold, i.e. \u0016= 1 corresponds to the\nlaser threshold when no anisotropies are considered [21].The normalized injection current \u0016is varied from 1 to 10;\nthe higher limit corresponds, in this theoretical framework,\nto 10 times the laser threshold without anisotropies which is\nalready largely above the current range commonly considered\n[11]. Although more complex dependencies appear to be\nneeded for an accurate estimation of the laser threshold, see\ne.g. eq. 11 in [27], we believe that this higher limit provides\na reasonable first order approximation for comparison with\nexperimental observations. Finally, the linearly polarized\n(LP) steady-state stable at threshold for the parameters\nconsidered here will be identified as X-LP (which is also\nthe low-frequency eigenmode) while the orthogonal LP\nsteady-state will be identified as Y-LP (the high-frequency\neigenmode). This notation is identical to the one used in\nprevious work [10], [35].\nIII. O PTIMAL EMBEDDING OF POLARIZATION CHAOS\nThe first part of this work is dedicated to the determination\nof the minimal number of variables required to accurately\ndescribe the polarization chaos dynamics. As such, this will\nalso tell us whether our model could potentially be further\nreduced and simplified without any loss of generality. A\ncommon approach to do so is to use the so-called Principal\nComponent Analysis (PCA) technique [30]. This technique\nperforms a linear mapping of the data to a lower-dimensional\nspace in such a way that the variance of the data in\nthe low-dimensional representation is maximized. It is a\npopular technique for dimensionality reduction. The original\nvariables are transformed into new variables, called principal\ncomponents (PCs), by an orthonormal linear transformation\nand are ordered by decreasing variance. Using the 5-dimension\nsimulated data generated by the SFM model, we generate\na set of 5 linearly independent PCs describing the system\ntrajectory in the variable space. Here, it is however important\nto note that different pre-processing steps might applied to\nthe data before applying PCA. As any pre-processing step\non the data can influence the outcome of the PCA, the user\nneeds to carefully choose them. In this paper, we will use the\ncommon centering pre-processing step, i.e. the removal of\nthe mean value of the different variables, and will compare\nresults of the PCA with and without this centering.\nThe PCA technique provides a decomposition of the original\nSFM data matrix Xof size #time samples x#states (i.e.\n#time samples x#variables ) as\nP=XW (6)\nwhere the matrix Whas dimension #states x#states .\nThe symbol #is equivalent to ”number of”. The number of\ncolumns of PandWcan be truncated and a reconstructed\nXreccan be computed as\nXrec=P(:;1 :r)W(:;1 :r)T=PtruncWT\ntrunc (7)\nwhereP(:;1 :r);W(:;1 :r)denote the first rcolumns of the\ncorresponding matrices. It is important to highlight that each3\n-150-100-500\nPC var. (dB)\n1.4 1.45 1.5 1.55\nNormalized Injection Current-150-100-500\n3 5 7 9a.1\na.2b.1\nb.2\nFig. 1. Variance of the 5 PCs of the PCA decomposition for case 1 (a) and\n2 (b) for PCA with (a.1, b.1) and without centering (a.2, b.2). In order of\ndecreasing variances, and thus from top to bottom, the PCs are represented\nin blue, orange, yellow, purple and green. The evolution of the 5th PC is\nemphasized by a thick line.\nprincipal component P(:;k);k= 1;:::;5can be expressed as\na linear combination as\nP(:;k) =XW(:;k) (8)\nwhereP(:;k);W(:;k)denote the k\u0000thcolumn of the\ncorresponding matrices. Here, we obtain a set of five linear\ncombinations of variables that are linearly independent of each\nother in order of decreasing variance. Considering both case\n1 (\rp= 4ns\u00001) and case 2 ( \rp= 25ns\u00001) for increasing\ninjection currents, we perform PCA independently for each\ncurrent step. We thus obtain Fig. 1 that shows the variance\nassociated with each PC after a PCA decomposition with and\nwithout centering of the initial data. The regions for which\npolarization chaos is observed are highlighted by the light gray\nbackground. The results are very consistent between case 1\nand case 2, but we also see an influence of the centering.\nIn all panels, we clearly see that at least one PC exhibits\na variance significantly smaller than the others, more than\n50 dB weaker than the variance of the dominant PCs. This\ntherefore suggests that there would indeed be some potential\nto reduce the number of variables of the model without loss\nof generality. When the centering pre-processing is used, two\nPCs - instead of only one - show smaller variance than the\ndominant PCs. Interpreting this result as a possible reduction\nof two orders, i.e. possible reduction to a 3D model, might\nbe an overstatement, but these results strongly suggest that at\nleast one variable could be removed.\nWhen dealing with chaotic dynamics, it is obvious that\nany small changes of the system should be considered with\nthe highest level of caution as these could have a dramatic\nimpact on the system behaviour. To verify the suitability of\nan embedding for a chaotic systems, a common test is the\nso-called ”False Neighbors” test [31], [32]. If the embedding\ndimension is too small, the chaotic attractor will necessarily be\nfolded, i.e. some parts of the chaotic trajectory will artificially\nbecome close and thus create the so-called ”False Neighbors”.\nTo detect such folding, we use the following approach. After\napplying PCA, we reconstruct the 5-D states trajectories using\n-505\nMDG (log10)\n1.4 1.45 1.5\nNormalized Injection Current020406080\nFN (%)\n4 6 8 10a.1\na.2b.1\nb.2Fig. 2. Mean Distance Growth and False Neighbor test results with reconstruc-\ntion based on PCA without centering for case 1 (a) and case 2 (b). For each\npanel, we show the embedding of 2 (blue, top curve), 3 (red, middle curve)\nand 4 (yellow, bottom curve) PCs for an increasing injection current. (a.1, b.1)\nMean Distance Growth minus unity shown in logarithmic scale. 0corresponds\nto a doubling of the nearest neighbors distance while \u00001 correspond to no\ndistance increase. The dotted horizontal lines indicate the threshold value used\nto detect false neighbors. (a.2, b.2) Ratio of False Neighbors detected over\nthe whole time-series considered.\nonly a subset of the calculated PCs using (7). We then compare\nthe distance between the data point and its closest neighbor\nin the initial and reconstructed 5-dimension state trajectories\nspace. From these data, we derive two figures of merit:\n\u000fthe Mean Distance Growth (MDG): the average growth\nfor all selected closest neighbors for all data points\nconsidered. Obviously, a large MDG >>1indicates a\nprobable folding while a MDG very close to 1 will tend\nto confirm that the embedding dimension is large enough\nto contain the chaotic trajectory.\n\u000fthe amount of “False Neighbors” (FN): the number of\nclosest neighbors - for all data points - for which the\ndistance growth exceed a given threshold Rtol. The\namount of FN of course depends on the threshold value,\nbut, as discussed in [31], the approach is quite robust\nagainst the choice of threshold. Here, we show results\nforRtol= 20 , but verified that the same conclusions\nhold for threshold values from 2 to 30.\nHere, we consider 10000 data point per time-series with a\ntime-step of 25 ps. We take the single closest neighbor for\neach data point [31] and impose a minimal time separation\nof 1000 ps between them to avoid correlation in time. The\nevolution of these two figures of merit for increasing currents\nand for different embedding dimensions are shown in Figs. 2-3\nwithout and with centering, respectively. Considering both data\nsets, it is clear that considering only the first 2 PCs is largely\ninsufficient as the amount of false neighbors is quite large:\nfor both cases, inside the chaotic region, the closest neighbor\nafter embedding is most of the time a ”false” neighbor. Adding\nthe third PC induces a significant improvements, and only\na few false neighbors are detected - between 1 to 4 %\nwithout centering (Fig. 2) and less than 0.5 % with centering\n(Fig. 3) - and only inside the chaotic region. But it is only\nwhen considering the first 4 PCs that the amount of detected\nFalse Neighbors effectively reach and remain at 0 for all\nconditions. From a system analysis viewpoint, the imperfect4\n-505\nMDG (log10)\n1.4 1.45 1.5\nNormalized Injection Current020406080\nFN (%)\n4 6 8 10a.1\na.2b.1\nb.2\nFig. 3. Mean Distance Growth and False Neighbor test results with recon-\nstruction based on PCA with centering. The rest of the caption is identical to\nthe one of Fig. 2\nembedding using only 3 PCs could be considered for a first\napproximation even though it would miss some details of\nthe dynamics. On the other hand, from a nonlinear dynamics\nviewpoint, a 4 dimension embedding using the first 4 PCs\nseems to be sufficient to contain the whole chaotic attractor\nwithout inducing any detectable folding. Indeed, in this case\nabsolutely no false neighbors is detected and low value of the\nmean distance growth is systematically observed. In essence,\nthese results confirm that the 5th PC, i.e. the PC with the\nsmallest variance, does not provide essential information over\nthe system behaviour or chaotic features. Therefore, we can\nconclude that these results confirm that the spin-flip model\ncould be reduced from 5 to 4 variables while preserving all\ncomplex features including chaos.\nConsidering this perspective, we can further analyze the\ndifferent outputs of the PCA to identify the variables that could\neffectively be dismissed. Such information can be provided\nby the Wmatrix which describes the linear combination\nof the original variables leading to the PC description. To\nput it differently, the Wmatrix shows the contribution of\neach variable to the different PCs. Most variables are actually\ncontributing to different PCs in a rather complex way. In Fig.\n4, we show the contribution of the spin-population difference\nnto each of the PCs. Thus, it is rather striking that the spin-\npopulation difference is, in fact, almost only contributing to\nthe 5th PC, with the exception of Fig. 4(b.1) that we discuss\nbelow. It is however important to understand that this does\nnot mean that the 5th PC is almost perfectly equal to the\nnvariable, since the other SFM variables can also have an\nimpact on the 5th PC in the linear combination formula (8).\nThe result shown in Fig. 4 suggests that the spin-population\ndifferencencould be a good candidate to be considered\nfor the model reduction as we will do in the next section\nusing an adiabatic elimination. The failure of this adiabatic\nelimination approach, which we will discuss in what follows,\nshould however not lead to the conclusion that the SFM cannot\nbe reduced to 4 variables despite the outcome of the PCA and\n”False Neighbors” analyses, but only to the fact that another\napproach needs to be considered to obtain a suitable model-\norder reduction. Finally, in Fig. 4(b.1), i.e. for PCA with\ncentering of case 2: in this case, the spin-population difference\n-0.500.51\n1.4 1.45 1.5 1.55\nNormalized Injection Current-0.500.51\n5th row of W\n3 5 7 9a.1 b.1\nb.2 a.2Fig. 4. Evolution of the contribution of the spin-population difference nto\neach of the 5 PCs - i.e. the fifth row of the Wmatrix - for case 1 (a) and 2\n(b). In order of decreasing variances, the PCs are represented in blue, orange,\nyellow, purple and green. The 5th PC is emphasized by a thick line. The top\npanels (a.1) and (b.1) show the evolution for the PCA with centering, while\n(a.2) and (b.2) give their counterpart obtained without centering.\nalso appears to contribute to the 4th PC shown in purple. Yet,\nthis result needs to be put in perspective with the variance of\nthis 4th PC as shown in Fig. 1(b.1): there, we see that the\n4th and 5th PC both have very low variance, more than 50\ndB below the three other PCs. Hence it does not invalidate\nthe proposed interpretation as the spin-population difference\nnnever contributes to one of the dominant PC. Nevertheless,\nthis might eventually be seen as a hint of the failure of the\nadiabatic elimination reduction, but this interpretation would\nbe far too speculative at this stage.\nIV. A DIABATIC ELIMINATION OF THE CARRIER\nPOPULATION DIFFERENCE\nAs discussed above, the dynamical analysis that we\nperformed suggests that the spin-population difference n\ncould be a good candidate for the reduction of the spin-flip\nmodel. Since a similar conclusion can be reached from\nphysical considerations [23], [24], this approach obviously\nneed to be further investigated.\nBased on the rate equations describing the spin-flip model,\nand similarly to what has been done in previous works\n[23], [24], we can eliminate the carrier population difference\nadiabatically. Starting with (5) and considering that the spin-\nflip rate\rsis large enough for nto reach a steady-state much\nfaster than the other variables, we directly obtain that:\nn=N(R2\n\u0000\u0000R2\n+)\n\rs=\r+R2\n++R2\n\u0000(9)\nThe differences with the expression found in [24] are only\ndue to the particular approximation made therein and that\nwe do not apply here. We can then easily confirm that\nthis expression of the carrier population difference is an\naccurate approximation by comparing it to the simulated\nvalues. By doing so, we obtain that the error is typically\ntwo-orders of magnitude smaller than the value taken by the\ncarrier population difference variable. As such, although such\nsmall error will have an obvious impact on the simulated5\nFig. 5. Comparison of bifurcation diagrams between the full SFM (blue)\nand the reduced model (orange) after adiabatic elimination of the carrier\npopulation difference for case 1 (a) and 2 (b), respectively. The diagram\nshows the maxima of the intensity of X-LP, i.e. the linear polarization stable\nat threshold.\nchaotic time-series, we would expect at this point a negligible\nqualitative impact on the laser behaviour. On the contrary,\nwhen computing bifurcation diagrams - as shown in Fig. 5 -\nusing this reduced model, we can only observe rather dramatic\nqualitative changes. In particular, in case 2, the initially wide\nchaotic region is shifted and shrunk into a narrow range\nof injection current. Thus, for currents above 2.7, the laser\nonly exhibits a stationary behaviour corresponding to Y-LP,\ni.e. the linearly polarized steady-state orthogonal to the\nsteady-state stable at threshold X-LP. This is particularly\npuzzling considering that, in case 2, the Y-LP steady-state\nis supposed to be unstable up to extremely high values of\ncurrent (\u0016>50) [35].\nTo clarify this issue, we have re-investigated the stability\nof both the X-LP and Y-LP steady-state for the full SFM and\nfor the reduced model. To compute it, we linearize the rate\nequations around the steady-states and numerically extract the\neigenvalues of the linear system. By adiabatically eliminating\nthe carrier population difference, we obviously remove one\nequation. But using the expression of ngiven in (9) also\ncreates several additional dependencies that are nonexistent in\nthe full model. While no impact of the adiabatic elimination\nis observed on the stability of the X-LP state (not shown),\nwe observe that the stability of the Y-LP state is not well-\npreserved. As displayed in Fig. 6, the lower boundary of the\nstability region with respect to the injection current appears\nto be independent of the birefringence for the reduced model\nwhile a clear dependence can be observed in the full-model.\nAs could be expected, this discrepancy is reduced as the value\nof the spin-flip rate is increased, but this agreement is still\nimperfect and restricted to low values of the birefringence \rp.\nWe now investigate analytically the stability of the Y-LP\nsteady-state. As described in the appendix, the linearized\nsystem has two pairs of eigenvalues \u00151and\u00152of the form\nA\u0006p\nB. Because\u00152always corresponds to a pair of complex\nconjugated eigenvalues with a negative real part for the typical\nrange of parameter values that we consider here, we will focus\non\u00151which effectively limits the stability of Y-LP. The latter\ncan be expressed as follows:\n\u00151= (\u00002\ra\u0000\u001f)\u0006q\n\u001f2\u00004\u000b\rp\u001f\u00004\r2p (10)\nFig. 6. Stability map of the Y-LP steady-state as a function of the birefrin-\ngence and the injection current for the full SFM (1, left) and the reduced model\n(2, right) for three distinct values of spin-flip rate: 100ns\u00001(a),500ns\u00001\n(b) and 1000 ns\u00001(c). The white (grey) regions indicate where the Y-LP\nsteady state is stable (unstable).\nwith the variable \u001fbeing defined as:\n\u001f=\r\u0014\n\rs\u0010\n\u0016\u00001 +\ra\n\u0014\u0011\n(11)\nIn practice,\u001fcorresponds to the injection current, normalized\nby the Y-LP steady-state threshold \u0016Y\u0000LP;th = 1\u0000\ra=\u0014-\nmeaning that \u001fis necessarily positive - and with a peculiar\nscaling. Inside the square root term in (10), we have a convex\n2ndorder polynomial equation in \u001f. The roots of the equation\nare2\rp(\u000b\u0000p\n\u000b2+ 1) and2\rp(\u000b+p\n\u000b2+ 1) , which are\nrespectively negative and positive for \rp>0. In this case,\nthe second term represents a crucial threshold with respect to\n\u001f: above it, we have real eigenvalues, while, below it, we have\na pair of complex conjugated eigenvalues. Thus, it is important\nto remark that, with the set of parameters considered in this\nwork, this threshold is increasing along with \rpand is already\nabove\u0016= 5for\rp= 2ns\u00001. As a result, for most cases, the\nlatter will apply, and the stability is then defined by the first\nterm of ( 10), i.e. the real part of the eigenvalue pair. From its\nexpression, we immediately get that the real part is negative\nif:\n\u0016>1\u0000\ra\n\u0014\u00002\ra\rs\n\u0014\r(12)\nThis equation is almost the same as eq. 55 of [25], except\nfor the 2nd term \ra=\u0014which is however expected to have\na negligible impact as we typically have \ra<< \u0014 . This is\nclearly the stability limit for the Y-LP state visible in Fig. 6\nwhich correspond to a well-known Hopf bifurcation [21], [25].\nWhen we have two real eigenvalues, since the square root term\nis necessarily positive, the largest eigenvalue will necessarily\nbe the one for which the two terms are added. In this case, it\ncan only be negative if the following condition is fulfilled:\nq\n\u001f2\u00004\u000b\rp\u001f\u00004\r2p<2\ra+\u001f (13)6\nTo have a real eigenvalue, the left hand side must be positive.\nThis implies that 2\ra+\u001f>0- which corresponds to (12) - is\na necessary condition for the previous inequality to be verified.\nIf both sides are positive, the previous equation is equivalent\nto:\n0<(\r2\na+\r2\np) + (\ra+\u000b\rp)\u001f (14)\nIf(\ra+\u000b\rp)>0, this inequality is obviously always verified.\nFor\ra<0and\rp>0, this comes down to a threshold on\nthe birefringence: \u000b\rp>j\raj, which, with our typical set of\nparameters, corresponds to values of the birefringence above\napproximately 0:23. On the other hand, when \ra+\u000b\rp<0\nthe Y-LP steady-state appears to be only stable when:\n\u001f<\u0000\r2\na+\r2\np\n\u000b\rp+\ra(15)\ni.e.\u0016<1\u0000\ra\n\u0014\u0000\rs(\r2\na+\r2\np)\n\r\u0014(\u000b\rp+\ra)(16)\nApart from the extra \ra=\u0014term which is expected to be\nnegligible, this expression is similar to eq. 53 found in [25]\nbut for the X-LP steady state and with a change of sign. This\ndifference is easily explained by the fact that X-LP and Y-LP\nare interchangeable by simultaneously changing the sign of \ra\nand\rp. This leads to the same change of sign, as shown in\nfig. 1 (a) and (b) of [25], where the same curve is obtained for\n\u0016xsand\u0016ys. Using the same approach as for Fig. 6, we could\nalso confirm that an excellent agreement is obtained between\nthe reduced and complete model for \rp<\ra=\u000b.\nTo conclude, for the case of \ra<0and\rp>0, the stability\nof the Y-LP steady-state is largely impacted by the adiabatic\nelimination of the spin-population inversion. While for suf-\nficiently small values of the birefringence \rp, an acceptable\nagreement between the reduced and complete model can be\nobtained, significant discrepancies arise when \rpis increased.\nIn the reduced model as soon as \rpis larger than \ra=\u000b,\nthe Y-LP state is always stable when the condition of eq.\n12 is met. This means that, intrinsically, the stability of Y-\nLP becomes independent of \rp. Although this reduction could\nbe a sufficient but limited approximation for very large spin-\nflip rates\rs>1000ns\u00001and small birefringence values\n\rp\u00195ns\u00001, we showed in our previous work [29] that these\nare, by far, conditions not suitable to generate polarization\nchaos dynamics.\nV. D ISCUSSION\nLooking at all in-depth investigations of VCSELs dynamics,\nit is undeniable that the spin-flip rate is an essential parameter\ncontrolling the dynamical behaviour of VCSELs. Yet, the spin-\nflip rate only plays a role of the laser dynamics through\nits influence on the spin population difference. Often, it is\nassumed - also considering the typically large values of the\nspin-flip rate reported experimentally - that the adiabatic elim-\nination of the spin-population difference could be a suitable\napproximation and would retain all essential non-linearities\nin the SFM model. Yet, in this work, we highlight that the\nimpact of this adiabatic elimination on the polarization chaos\nand stability of the system steady-states is far too large tobe neglected, even for large values of the spin-flip rate and,\nespecially, for birefringence values in the order of tens of ns\u00001\nand above [14], [15]. However, our analysis using the PCA and\n”False Neighbors” techniques confirm that there is potential\nfor model order reduction, but it needs to be performed\nusing methods that, unlike the adiabatic elimination of the\nspin-population, would preserve all key dynamical features of\nthe model. However, is is an aspect that we will leave for\nfuture work. Model order reduction is a vast field [36]–[39].\nDepending on the complexity of the system representation\n(e.g., linear time-invariant, time-variant, non linear, etc.) to be\nreduced, the model order reduction approach has an increasing\ncomplexity especially if system properties need to be preserved\nin the reduced model. Model order reduction schemes very\noften use specific mathematical transformation to transform\nthe original state vector space into a reduced space, while\nkeeping accuracy and properties with respect to the original\nmodel. The property-preserving model order reduction is much\nmore complex than just an accuracy-preserving model order\nreduction. Already in the case of linear time-invariant systems,\nmodel order reduction techniques can provide accurate reduced\norder models of the transfer function behavior in a frequency\nbandwidth of interest, but they can fail in preserving prop-\nerties such as stability and passivity. For nonlinear systems,\nit is obviously even more complex. Therefore, a dedicated\ninvestigation of such techniques that can perform model order\nreduction of the original SFM equations are beyond the scope\nof this contribution and deserves a dedicated effort in a\nseparated work.\nFinally, we would like to discuss why the work presented here\nleads to conclusions that significantly differ from previous\nreports, in particular [23], [24]. We believe that this can\nbe mostly attributed to the different focus of our analysis:\nwhile the authors of [23], [24] were working to get a better\nphysical understanding of the stability boundaries of the\nlinearly polarized steady-states, we look here at the modelling\naccuracy of the dynamical features. Similarly, while strong\napproximations are taken in [23], [24], we tried to keep a\nvery general viewpoint, thus did not use similar approximation.\nIn the end, we are convinced that these two aspects are of\ncourse complementary. However, as highlighted in this work,\nwhen considering dynamical behaviours, the spin-population\ndifference appears as an essential piece of the puzzle that must\nbe taken into account to its full extent to ensure qualitative\nrelevance of the modelling.\nAPPENDIX A\nSTABILITY OF THE Y-LP STEADY -STATE IN THE\nADIABATICALLY REDUCED SPIN -FLIP MODEL\nThe Y-LP steady-state, is defined as follows:\nN= 1\u0000\ra\n\u0014;\u001e=\u0019 ;R =r\n\u0016\u0000N\n2N(17)\nFor simplicity, we will approximate the expression of the\nadiabatically eliminated population difference by considering\nthat the denominator can be directly expressed as \rs=\r. Since\nthis term is typically of the order of 100 to 1000 while the\namplitude terms are expected to be of the order of 1, the7\napproximation is small for the practical cases considered here.\nAfter linearization of the equations, we obtain the followingsystem whose eigenvalues will determine the stability of the\nsteady-state.\n2\n664_\u000eR+\n_\u000eR\u0000\n_\u000e\u001e\n_\u000eN3\n775=2\n664\u0014(N\u00001)\u00002\u0014\rNR2=\rs\ra+ 2\u0014\rNR2=\rs\rpR \u0014\rR\n\ra+ 2\u0014\rNR2=\rs\u0014(N\u00001)\u00002\u0014\rNR2=\rs\u0000\rpR \u0014\rR\n\u00004\u0014\r\u000bNR=\r s\u00002\rp=R 4\u0014\r\u000bNR=\r s+ 2\rp=R\u00002\ra 0\n\u00002\rNR \u00002\rNR 0\u0000\r(1 + 2R2)3\n7752\n664\u000eR+\n\u000eR\u0000\n\u000e\u001e\n\u000eN3\n775(18)\nUsing a symbolic computation software, we obtain two pairs\nof eigenvalues of the form \u00151=A1\u0006p\nB1and\u00152=A2\u0006p\nB2. Using the following change of variable\n\u001f=\r\u0014\n\rs\u0010\n\u0016\u00001 +\ra\n\u0014\u0011\n(19)\nthe first eigenvalue pair can be efficiently expressed as follows:\n\u00151= (\u00002\ra\u0000\u001f)\u0006q\n\u001f2\u00004\u000b\rp\u001f\u00004\r2p (20)\nThe second pair of eigenvalue can be expressed as:\n\u00152=\r\u0014\u0016\n2(\ra\u0000\u0014)\u0006s\n\r2\u00142\u00162\n4(\ra\u0000\u0014)2\u00002\r\u0014\u0016+ 2\r\u0014(\ra\n\u0014\u00001)\n(21)\nAs mentioned in the core of the article, \u00151is the interesting\npair of eigenvalue and is therefore further described therein.\nThe impact of \u00152is far more limited as will be described here.\nSince we typically have j\raj<< \u0014 with\u0014being positive by\ndesign, the first term will always be negative for the considered\nparameter range. 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Ferranti, “Chaos in Solitary VCSELs: Exploring\nthe Parameter Space with Advanced Sampling,” Journal of Lightwave\nTechnology , vol. 36, no. 9, pp. 1601–1607, 2018. [Online]. Available:\nhttp://ieeexplore.ieee.org/document/8226754/\n[30] I. Jolliffe, Principal Component Analysis (Springer Series in Statistics) .\nSpringer, 2002.\n[31] M. B. Kennel, R. Brown, and H. D. I. Abarbanel,\n“Determining embedding dimension for phase-space reconstruction\nusing a geometrical construction,” Physical Review A , vol. 45,\nno. 6, pp. 3403–3411, mar 1992. [Online]. Available:\nhttps://link.aps.org/doi/10.1103/PhysRevA.45.3403\n[32] H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring,\n“The analysis of observed chaotic data in physical systems,” Reviews\nof Modern Physics , vol. 65, no. 4, pp. 1331–1392, oct 1993. [Online].\nAvailable: https://link.aps.org/doi/10.1103/RevModPhys.65.1331\n[33] M. Virte, M. Sciamanna, E. Mercier, and K. Panajotov, “Bistability\nof time-periodic polarization dynamics in a free-running VCSEL,”\nOpt. Express , vol. 22, no. 6, p. 6772, mar 2014. [Online]. Available:\nhttp://www.opticsinfobase.org/abstract.cfm?URI=oe-22-6-6772\n[34] M. Virte, E. Mirisola, M. Sciamanna, and K. Panajotov, “Asymmetric\ndwell-time statistics of polarization chaos from free-running VCSEL,”\nOpt. Lett. , vol. 40, no. 8, pp. 1865–1868, 2015.\n[35] M. Virte, K. Panajotov, and M. Sciamanna, “Bifurcation to nonlinear\npolarization dynamics and chaos in vertical-cavity surface-emitting\nlasers,” Phys. Rev. A , vol. 87, no. 1, p. 013834, jan 2013.\n[36] A. Antoulas, Approximation of Large-Scale Dynamical Systems . Society\nfor Industrial and Applied Mathematics, 2005.\n[37] W. H. A. Schilders, H. A. van der V orst, and J. Rommes, Model order\nreduction : theory, research aspects and applications . Berlin: Springer,\n2008.\n[38] A. Quarteroni and G. Rozza, Reduced order methods for modeling and\ncomputational reduction . Cham: Springer, 2014.\n[39] P. Benner, M. Ohlberger, A. Cohen, and K. Willcox, Model Reduction\nand Approximation . Philadelphia, PA: Society for Industrial and\nApplied Mathematics, 2017.\nMartin Virte received the master in engineering with a major in Photonics\nfrom the French Grande Ecole Sup ´elec (now CentraleSup ´elec, Universit ´e de\nParis-Saclay, France) and the M.Sc. in Physics from Sup ´elec and the Univer-\nsit´e de Lorraine both in 2011. He simultaneously obtained the PhD degree\nin Photonics from Sup ´elec and in engineering from the Vrije Universiteit\nBrussels, Brussels, Belgium in 2014 in the frame of a joint PhD. In 2015, he\nreceived a three-year post-doctoral fellowship from the Research Foundation\nFlanders (FWO) to pursue his research at the Vrije Universiteit Brussel,\nBelgium. Since 2018 he is research professor with the Brussels Photonics\nTeam (B-PHOT) of the Vrije Universiteit Brussel.\nMartin authored and co-authored more than 16 papers in international refereed\njournals and several contributions in national and international conferences.\nMartin received the Graduate Student Fellowship Award from the IEEE\nPhotonics Society in 2014 in recognition of his PhD work. His researchis currently focusing on laser dynamics, more competition, optical chaos\nand their applications in sensing and communication. In addition, he acts\nas a reviewer for several international journals including journal of lightwave\ntechnology, chaos, applied physics letters and others. He is a member of SPIE,\nIEEE (IEEE Photonics Society) and OSA.\nFrancesco Ferranti (M’10-SM’17) received the Ph.D. degree in electrical\nengineering from Ghent University, Ghent, Belgium, in 2011. He is currently\nan Associate Professor at the Microwave Department at Institut Mines-\nT´el´ecom (IMT), Brest, France. He has been awarded the Anile-ECMI Prize for\nMathematics in Industry 2012 and the Electromagnetic Compatibility Society\nPresident’s Memorial Award 2012.\nHe has authored and co-authored 56 papers in international peer-reviewed\njournals, 52 papers in international peer-reviewed conferences and 2 books\nchapters. He has given invited lectures and chaired several sessions at inter-\nnational conferences. He serves as a regular reviewer for several international\njournals. He is a Senior IEEE member.\nHis research interests include parametrized modeling and model order\nreduction, dynamical systems, machine learning, sampling techniques, design\nspace exploration, uncertainty quantification, optimization, and behavioral\nmodeling."    },    {        "title": "2306.07556v1.Angle_dependence_of____15__N_nuclear_spin_dynamics_in_diamond_NV_centers.pdf",        "content": "Angle dependence of15N nuclear spin dynamics in diamond NV centers\nYusuke Azuma∗and Shintaro Nomura\nDivision of Physics, Univ. of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8571, Japan\nHideyuki Watanabe\nNational Institute of Advanced Industrial Science and Technology\n(AIST) Central2, Umezono, Tsukuba, Ibaraki, 305-8568, Japan\nSatoshi Kashiwaya\nDepartment of Applied Physics, Nagoya Univ. Chikusa-Ku, Nagoya, Aichi, 464-8571, Japan\nWe report on the dynamics of the Rabi oscillation and the Larmor precession of15Nnuclear\nspin using nonselective short microwave pulses for initialization of15Nnuclear spins. We observe\nthe Larmor precession of15Nnuclear spin depending on the angle between the applied magnetic\nfield and the axis of the nitrogen vacancy center. We propose to utilize the change of the Larmor\nfrequency of the nuclear spins to detect static magnetic fields at high sensitivity. Our results may\ncontribute to enhancing the sensitivity of dc magnetic fields and devising novel protocols using15N\nnuclear spin in nitrogen vacancy centers in diamonds.\nI. INTRODUCTION\nHigh-sensitive measurements of magnetic fields are de-\nsired for applications in various fields, such as materials\nscience and biomedicine. Quantum sensing is a method\nof measuring physical quantities using a quantum sys-\ntem and is attracting much attention for its potential to\nachieve higher sensitivity than conventional methods.[1]\nIn particular, nitrogen-vacancy (NV) centers in diamonds\n[2, 3] have attracted considerable attention as an out-\nstanding system for quantum information processing at\nroom temperature and atmospheric pressure. The NV\ncenter in diamonds consists of two adjacent carbon atoms\nreplaced by nitrogen and a vacancy. Both electron spin\nand nuclear spins in diamond can be used for quantum\ninformation processing, and are controlled by microwave\n(MW) or radio-frequency waves (RF) pulses. A nuclear\nspin is highly isolated from the environment, and can be\nused to store information as a quantum memory.[4] An\nelectron spins are used in sensing because they interact\nmore strongly with the surrounding environment.\nIt has been demonstrated that a hybrid system of an\nelectron and nuclear spins in diamonds enhances the sen-\nsitivity of magnetic field. [5–8] The electron spin accu-\nmulates the phase from the magnetic field by the cou-\npling to the external magnetic field, and the accumulated\nphase is transferred and stored in the nuclear spin. This\nmethod effectively increases the sensitivity of the sensor\nby exploiting long coherence time of the nuclear spin. An\nunderstanding of the hyperfine interaction in the electron\nstate of the NV center [9] is essential to utilize the nuclear\nspins to enhance the sensitivity.\nA drawback of utilizing nuclear spins to enhance the\nsensitivity is that the speed of nuclear spin state control\nis often slow, and initialization takes time. One approach\n∗azuma.yusuke.sj@alumni.tsukuba.ac.jpis to use the interlevel anticrossing of the ground state or\nexcited state of the NV center electron spin as a method\nto initialize the N nuclear spin of the diamond NV center.\nThis method has drawbacks in that the bias magnetic\nfield have to be fixed at ≈51 or≈102 mT and that a small\nchange in the bias magnetic field may affect the N nuclear\nspin polarization. [10–12] The other approach is to use a\nmicrowave pulse with a narrow linewidth to selectively\nexcite hyperfine-split levels.[7] To this end, the duration\nof the microwave pulse τMWhas to be sufficiently long,\ntypically, longer than τMW≥1µs. The N nuclear spin\nstate is read out either optically [4–12] or electrically [13,\n14] through the NV electron spin.\nWe have recently proposed a method for initialization\nof15N nuclear spins using nonselective microwave pulses\nto significantly reduce the time to control the nuclear spin\nstate. [15] This method enables us fast quantum control\nof nuclear spins. By using this method, we report on the\ndynamics of the Rabi oscillation and the Larmor preces-\nsion of15N nuclear spin by varying the angle between\nthe applied static magnetic field and the axis of the NV\ncenter. We measure the Larmor frequency of the nuclear\nspin depending on the external transverse magnetic field,\nand compare with a model calculation that takes into\naccount the hyperfine interactions. Finally, we propse a\nhigh-sensitive static magnetic field sensing utilizing nu-\nclear spins.\nII. MODEL CALCULATION OF EFFECTIVE\nMAGNETIC ROTATION RATIO OF THE\nNUCLEAR SPIN\nWe study ensemble of NV centers formed in diamonds\nby implantation of15N ions. [16, 17] The implanted15N\nis distinguished from naturally abundant14N with an\nisotopic abundance ratio of 99.6%, originally contained\nin the crystal as an impurity.15N has spin I= 1/2 andarXiv:2306.07556v1  [quant-ph]  13 Jun 20232\nFIG. 1. NV center in diamond. (a) Crystal structure of the\nNV center. (b) Energy levels in the ground state of the NV\ncenter. In a zero magnetic field, the energy at |mS=±1⟩is\nhigher than at |0⟩. In addition, an external magnetic field\ninduces Zeeman splitting. The hyperfine interactions cause\nfurther splitting.\ntakes on a simpler spin structure than14N with I= 1.\nUnder an external magnetic field B= (Bx,By,Bz),\nthe ground state Hamiltonian ˆHof the15NV centers can\nbe written as [2, 18]\nˆH/ℏ=DˆS2\nz+γeB·ˆS−γnB·ˆI+ˆS·A·ˆI.(1)\nwhere ˆS=\u0010\nˆSx,ˆSy,ˆSz\u0011\nandˆI=\u0010\nˆIx,ˆIy,ˆIz\u0011\nare the\nelectron spin and nuclear spin operators, respectively,\nwith electron and nuclear gyromagnetic ratios γe=\n2π×28.0 MHz /mT and γn= 2π×(−4.32) kHz /mT. The\ndirection of the NV axis is set to the zaxis. The zero field\nsplitting along the NV axis is D= 2π×2.87 GHz. The\nhyperfine interaction is described by the diagonal tensor\nA=\nA⊥0 0\n0A⊥0\n0 0 A∥\n, (2)\nwith transverse and longitudinal components A⊥= 2π×3.65 MHz and A∥= 2π×3.03 MHz, respectively [Fig.1].\nThe angle between the zaxis and the direction of the\nexternal magnetic field Bis defined as θ. Expanding the\nHamiltonian in Eq. (1), assuming that the magnetic field\nangle θlies in the x-zplane, we obtain\nˆH/ℏ=DˆS2\nz+γe\u0010\nBxˆSx+BzˆSz\u0011\n−γn\u0010\nBxˆIx+BzˆIz\u0011\n+A∥ˆSzˆIz+A⊥\u0010\nˆSxˆIx+ˆSyˆIy\u0011\n.(3)\nFollowing the descriptions given in Refs. [19] and [20], we\ntransform the Hamiltonian using a perturbation theory.\nAfter the unitary transformations to the double-rotating\nelectron and the rotating spin frames, the Hamiltonian\nˆ˜Hin the rotational coordinate frame becomes\nˆ˜H=−γn(βind+β(mS))·ˆI, (4)\nwhere βindis the magnetic field, independent of the elec-\ntron spin, along the z′-axis as redefined for the rotating\nsystem,\nβind=βindˆz′=s\nB2z+\u0012\n1 + 2γeA⊥\nγnD\u00132\nB2xˆz′,(5)\nandβ(mS) is the magnetic field, which depends on the\nspin state of the electron,\nβ(mS) =1\nβind\nBxn\nmS\u0010\n1 + 2γeA⊥\nγnD\u0011A∥\nγn−3m2\nSγeA⊥\nγnDBzo\n0\n−mSA∥\nγnBz−3m2\nSγeA⊥\nγnD\u0010\n1 + 2γeA⊥\nγnD\u0011\nB2\nx\n.\nThe effective nuclear Larmor frequency is then given\nby,\nωmS=|γn(βind+β(mS))|=−γns\nB2x\u0012\n1 + 2γeA⊥\nγnD−3m2\nSγeA⊥\nγnD\u00132\n+\u0012\nmSA∥\nγn−Bz\u00132\n. (6)\nIII. EXPERIMENTAL\nWe used a (100)- oriented ultra-pure diamond chip (El-\nement Six Ltd., electronic grade) with a size of 2 .0×2.0×\n0.5 mm3[16, 17, 21]. After ion implantation of15N+\n2, the\ndiamond chip was annealed at 800◦C. NV centers were\ncreated about 10 nm below the surface of the diamond\nchip. Inhomogeneous dephasing time T∗\n2of the diamond\nchip was estimated to be 0.8 µs.\nFigure 2(b) shows the schematics of the experimen-\ntal setup. Magnetic field Bwas applied by a pair\nof Nd 2Fe14B permanent magnets. The external mag-netic field was 4 .0 mT. The photoluminescence from NV\ncenters was imaged at room temperature with a wide-\nfield microscope equipped with a cooled scientific CMOS\ncamera (Zyla5.5, Andor) and a 100 ×objective with\nan NA of 0.73 and a working distance of 4 .7 mm af-\nter passing through a long-wavelength optical pass filter\nwith a cutoff wavelength of 650 nm. A double-balanced\nmixer (IQ-1545, Marki) upconverts the baseband I and\nQ pulses from an arbitrary waveform generator (33622A,\nKeysight) by mixing with a microwave from a local os-\ncillator (SMC100A, Rhodes-Schwarz). The upconverted\nsignals are amplified with an amplifier (ZHL-16W-43+,3\nFIG. 2. Experimental setup. (a) Relationship between the NV center axis and the magnetic field angle θ. The direction of\nthe NV axis is set to be parallel to the z-axis, The angle θis the angle between the z-axis and the static magnetic field B.\n(b) Schematics of the measurement system. Laser pulses are incident to the (001) surface of the diamond and the emitted\nphotoluminescence is collected by an objective lens and read by a scientific CMOS camera. Permanent magnets are mounted\non a stage that can be rotated in two directions to change the direction of the external magnetic field.\nMini-Circuits) and fed to a microwave planar ring an-\ntenna [22] placed above a diamond chip. The antenna\napplys a spatially uniform microwave field in the field of\nview of the microscope image. The microwave π/2 pulse\nlength was 10 ns. A pulse sequencer (Pulse Blaster ESR\nPro, Spincore) drove the pulsed laser diode, the arbi-\ntrary wave generator (33622A, Keysight), the microwave\nswitch, and the scientific CMOS camera. RF from an\nRF generator (33120A, Keysight) was applied to a 10 µm\nwide Au/Cr wire on a Si chip [17, 23].\nIV. RESULTS\nThe application of the magnetic field leads to a split-\nting of the degenerate energy levels into the states\n|mS, mI⟩, where mS(= 0 ,±1) and mI(=↑,↓) are elec-\ntron spin and15N nuclear spin magnetic quantum num-\nbers, respectively. Two resonant frequencies f↑for\n|mS= 0, mI=↑⟩ ↔ |− 1,↑⟩andf↓for|0,↓⟩ ↔ |− 1,↓⟩are\ndetermined by pulsed optically detected magnetic reso-\nnance (ODMR).\nWe measure Rabi oscillations of15N nuclear spins by\nvarying the angle θbetween the magnetic field and the\naxis of the NV. Figure 3 shows a schematics of a pulse\nsequence for observation of15N nuclear spin Rabi oscilla-\ntion and Larmor precession. We use non-selective pulses\nfor the separation of nuclear spins by state [15].\nWe consider a rotating coordinate system that rotates\nat a frequency fM= (f↑+f↓)/2. First, the electron\nspins are initialized to the state mS= 0 by a green laser\npulse. Then, the electron spins are rotated by π/2 around\nthex-axis by irradiating MW at the frequency fM. The\napplication of off-resonance pulses induces precession of\nthe electron spins around the z-axis. Depending on the\ndirection of the electron spin, the electrons in the states|↑⟩and|↓⟩precess around the z-axis at f↑−fMorf↓−\nfM. Note here that the directions of the rotations are\nopposite because the sign of the frequencies f↑−fMand\nf↓−fMare different. Then, a MW pulse at frequency\nfMis applied to rotate the electron spins by π/2 around\nthey-axis. The above procedure of MW pulse irradiation\nand precession of the electron spins is called a Ramsey\nprocess. Next, Rabi oscillations of15N nuclear spins are\ninduced by an RF pulse at the resonant frequency fR\nof|−1,↑⟩ ↔ |− 1,↓⟩. Finally, the spin state of the15N\nnuclear spins is read after initializing the electron spin\nby a green laser pulse. The Ramsey process is applied\nagain, which transfers the15N spin state to the electron\nspin state. Then the electron spin state is read out by the\nPL by a green laser pulse excitation. The PL intensity\nreflects the15N spin state projected to the z-axis. In our\nmeasurement, the number of readouts Nwas set to 4.\nAt a magnetic field angle θ= 0◦, the rotation axis\nof the nuclear spins is parallel to the z-axis, and hence\na Larmor precession of the15N nuclear spins is not ob-\nserved. In the case of θ̸= 0◦, the oscillation of the15N\nspin state projected to the z-axis is observed because a\nLarmor precession of the15N nuclear spins occurs around\nthe static magnetic field axis away from the z-axis.\nThe observed dynamics of15N nuclear spins are shown\nin Fig. 4. The frequency fMwas adjusted at each angle\nθby observing pulsed-ODMR spectra. The frequency\nof the RF pulses were determined from ODMR spectra\nas a function of RF frequency to be fR= 3.012 MHz.\nThe Larmor frequency for the magnetic quantum number\nmS=−1 is estimated from Eq. (6) to be 3 .012 MHz with\nsmall variations between θ= 0◦and 10◦, in agreement\nwith the experimentally obtained value. The data points\nin Fig. 4 were measured at the RF pulse duration integer\nmultiple of a period of 1 /3.012µsin order to eliminate the\neffect of the rapid Larmor oscillation at mS=−1.4\nFIG. 3. Schematics of a pulse sequence for15N nuclear spin Rabi oscillation and Larmor precession. The Bloch sphere on the\ntop of the figure is represented in the rotating coordinate system at fM= (f↑+f↓)/2. The NV center is initialized to |0⟩\nby a green laser pulse. The electron spin is rotated by π/2 around the x-axis by irradiating with a microwave pulse at fM.\nThe electron spins precess around the z-axis by π/2 at f↑−fMfor|↑⟩andf↓−fMfor|↓⟩. The second microwave pulse\nrotates the electron spin by π/2 around the y-axis. Then Rabi oscillation of15N nuclear spins is induced by RF at a resonance\nfrequency fR. Finally, the nuclear spin state is transferred to the electron spins without destroying the15N nuclear spin state.\nThe electron spin state is measured by excitation by a green laser pulse that follows.\nFIG. 4. (a)-(f) Measured results by varying the angle between the axis of the NV center and the external static magnetic field\n(θ) between 0-10◦, showing15N nuclear spin Rabi oscillation and Larmor precession. The solid red curves are experimentally\nobtained results. The blue dashed curves are the best fitted curves to Eq. (7). Only Rabi oscillations of the15N nuclear spins\nare observed at θ= 0◦, while both the Rabi oscillation and the Larmor precession are observed at θ̸= 0◦. (g) The Larmor\nprecession frequency of the15N nuclear spin as a function of the angle θ. The red points are experimentally obtained values\nand the blue curve is the theoretical value calculated from Eq. (6) without any fitting parameters.\nThe best fitted curves to\nf(t) =Ae−Bt(1−Ccos(ωRt)−Dcos(ωLt)) +E.(7)\nare also shown in Figs. 4(a)- 4(f). The Rabi frequency\nωRand the Larmor frequency ωLare obtained from the\ncurve fittings.The Rabi frequency ωRobtained from the fittings is\n2π×4.30 kHz for all the traces at 0◦≤θ≤10◦in Figs.\n4(a)-4(f). This is reasonable because the incident RF\namplitude was kept constant. On the other hand, a small\nchange in θis found to change the Larmor frequencies ωL\nsignificantly as shown in Fig. 4(g). The experimentally5\nobtained Larmor frequencies and a theoretical curve (Eq.\n(6)) agree well without any fitted parameters.\nV. DISCUSSION\nWe have shown that the Larmor precession of the nu-\nclear spins changes depending on the electron spin state\ndue to the hyperfine interaction. A small change in the\nangle ( δθ= 10◦) between the axis of the NV center and\nthe external static magnetic field leads to a significant\nchange in the Larmor frequency of the nuclear spins from\n18 to 55 kHz due to the anisotropy in the hyperfine in-\nteraction. The change in the Larmor frequency of the\nnuclear spin may be used to measure the magnitude of\nthe lateral static magnetic field at high sensitivity.\nThe electron-spin-nuclear-spin hybrid system has been\nexperimentally demonstrated to allow highly sensitive ac\nmagnetic field sensing [7, 8]. The coherence time of the\nnuclear spins is typically 103times longer than that of\nthe electron spins. High sensitivity is achieved by trans-\nferring the electronic spin state to the nuclear spin state\nand holding it for a long time limited by T1,n≈52 ms [7].\nWe propose to utilize the change of the Larmor fre-\nquency of the nuclear spins due to the lateral magnetic\nfield to detect static magnetic fields at high sensitivity.\nFrom Eq. (6), the Larmor frequency of15N nuclear spins\natmS= 0 is given by\nωmS=0=−γnBzs\n1 + tan2θ\u0010\n1 + 2γeA⊥\nγnD\u0011\n.(8)\nThe accumulated phase at an optimum precession pe-\nriodT∗\n2,n/2 is given by\nωmS=0T∗\n2,n/2≈ −γnBzp\n1 + 52 .4tan2θT∗\n2,n/2,(9)\nwhere the coherence time of15N (T∗\n2,n) was estimated\nto be 9 ms. [24] This surpasses the accumulated phase\nof the NV electron spin by a Ramsey measurement as\ngiven by ωmeT∗\n2/2 = γeBzT∗\n2/2 where T∗\n2is typically\n1µs. The superiority of the method to utilize15N\nnuclear spin rises sharply as θincreases because of thelarge coefficient 52.4 in Eq. (9), which leads to further\nlowering of the minimum detectable magnetic fields.\nMoreover, this method has the advantage of being able\nto read out the accumulated phase of the nuclear spins\nrepeatedly by transferring it to the electron spins. The\ncoherence of15N nuclear spin is disturbed by the spin\nflip of the NV electron spin, and hence, is limited by the\npopulation decay time T1of the NV electron spin. T∗\n2,n\ncan be lengthened by decreasing the lattice temperature\nor by optical pumping into the ms= 0 electron spin\nstate by 594 nm laser illumination. [25] Whereas the\ninterrogation time and the contrast of read-out have\nto be taken into consideration practically, the above\ncomparison indicates that the nuclear spin-based mea-\nsurement method is promising.\nVI. CONCLUDING REMARKS\nPolarization and initialization of15N nuclear spins in a\nshort time have been demonstrated by utilizing a method\nby separating the nitrogen nuclear spins using nonselec-\ntive microwave pulses. The Larmor frequency of the15N\nnuclear spins has been measured by changing the angle\nθbetween the axis of the NV center and the external\nmagnetic field. The Larmor frequency changes signifi-\ncantly with an increase in the angle θdue to the hyper-\nfine interaction in accordance with a model calculation\nwithout any fitting parameters. We propose a method to\nlower the minimum detectable static magnetic fields by\nthe Larmor precession of the15N nuclear spins. Our re-\nsults may contribute to a wide range of applications, such\nas magnetic field sensing and quantum information pro-\ncessing using nuclear spin. An accurate understanding\nof the behavior of the nuclear spin in the presence of the\nhyperfine interactions contributes to enhancing the sen-\nsitivity of quantum sensing and devising novel protocols\nusing NV centers in diamonds.\nACKNOWLEDGMENTS\nThis work was partly supported by a Grant-in-Aid for\nScientific Research (Nos. 21H01009 and 22K18710) from\nJapan Society for the Promotion of Science.\n[1] C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum\nsensing, Rev. Mod. Phys. 89, 035002 (2017).\n[2] M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko,\nJ. Wrachtrup, and L. C. Hollenberg, The nitrogen-\nvacancy colour centre in diamond, Phys. Rep. 528, 1\n(2013).\n[3] J. F. Barry, J. M. Schloss, E. Bauch, M. J. Turner, C. A.\nHart, L. M. Pham, and R. L. Walsworth, Sensitivity\noptimization for nv-diamond magnetometry, Rev. Mod.Phys. 92, 015004 (2020).\n[4] L. Jiang, J. S. Hodges, J. R. Maze, P. Maurer, J. M.\nTaylor, D. G. Cory, P. R. Hemmer, R. L. Walsworth,\nA. Yacoby, A. S. 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Mizuochi, Room tem-\nperature electrically detected nuclear spin coherence of\nnv centres in diamond, Sci. Rep. 10, 792 (2020).\n[14] M. Gulka, D. Wirtitsch, V. Iv´ ady, J. Vodnik, J. Hruby,\nG. Magchiels, E. Bourgeois, A. Gali, M. Trupke, and\nM. Nesladek, Room-temperature control and electrical\nreadout of individual nitrogen-vacancy nuclear spins,\nNat. Commun. 12, 4421 (2021).\n[15] Y. Azuma, H. Watanabe, S. Kashiwaya, and S. Nomura,\nRapid control of15N nuclear spin within diamond nv cen-\nters, Extended Abstracts of the 2022 International Con-ference on Solid State Devices and Materials , 195 (2022).\n[16] B. Ofori-Okai, S. Pezzagna, K. Chang, M. Loretz,\nR. Schirhagl, Y. Tao, B. Moores, K. Groot-Berning,\nJ. Meijer, and C. Degen, Spin properties of very shal-\nlow nitrogen vacancy defects in diamond, Phys. Rev. B\n86, 081406 (2012).\n[17] G. Mariani, S. Nomoto, S. Kashiwaya, and S. Nomura,\nSystem for the remote control and imaging of mw fields\nfor spin manipulation in nv centers in diamond, Sci. Rep.\n10, 1 (2020).\n[18] J. R. Maze, J. M. Taylor, and M. D. Lukin, Electron\nspin decoherence of single nitrogen-vacancy defects in di-\namond, Phys. Rev. B 78, 094303 (2008).\n[19] L. Childress, M. Gurudev Dutt, J. Taylor, A. Zibrov,\nF. Jelezko, J. Wrachtrup, P. Hemmer, and M. Lukin,\nCoherent dynamics of coupled electron and nuclear spin\nqubits in diamond, Science 314, 281 (2006).\n[20] J. T. Oon, J. Tang, C. Hart, K. Olsson, M. Turner,\nJ. Schloss, and R. Walsworth, Ramsey envelope mod-\nulation in NV diamond magnetometry, Bulletin of the\nAmerican Physical Society (2022).\n[21] S. Nomura, K. Kaida, H. Watanabe, and S. Kashiwaya,\nNear-field radio-frequency imaging by spin-locking with a\nnitrogen-vacancy spin sensor, J. Appl. Phys. 130, 024503\n(2021).\n[22] K. Sasaki, Y. Monnai, S. Saijo, R. Fujita, H. Watanabe,\nJ. Ishi-Hayase, K. M. Itoh, and E. 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You3\n1The Key Laboratory for Advanced Materials and Devices,\nDepartment of Optical Science and Engineering, Fudan Unive rsity, Shanghai 200433, China\n2Department of Physics, Auburn University, Auburn, Alabama 36849, USA\n3School of Physics, Georgia Institute of Technology, Atlant a, Georgia 30332-0430, USA\n(Dated: November 15, 2018)\nWe propose to localize spin mixing dynamics in a spin-1 Bose- Einstein condensate by a tem-\nporal modulation of spin exchange interaction, which is tun able with optical Feshbach resonance.\nAdopting techniques from coherent control, we demonstrate the localization/freezing of spin mixing\ndynamics, and the suppression of the intrinsic dynamic inst ability and spontaneous spin domain for-\nmation inaferromagnetically interactingcondensate of87Rbatoms. This work points toapromising\nscheme for investigating the weak magnetic spin dipole inte raction, which is usually masked by the\nmore dominant spin exchange interaction.\nPACS numbers: 03.75.Mn, 03.75.Kk, 05.45.Gg, 42.65.-k\nDynamic localization is ubiquitous in nonlinear sys-\ntems, both for classical dynamics as in an inverted pen-\ndulum with a rapidly modulating pivot [1] or an ion in\na Paul trap [2], and for quantum dynamics like a one-\nor two-dimensional soliton in a Bose-Einstein condensate\n(BEC) whenthe attractiveinteractionstrength israpidly\nmodulated [3–6]. It is often used to stabilize a dynami-\ncally unstable system.\nSpin mixing dynamics of a spin-1 atomic condensate\nare dynamically unstable [7] when the spin exchange in-\nteraction is ferromagnetic, i.e., favoring a ground state\nwith all atomic spins aligned. When confined spatially,\nthe unstable dynamics is known to cause formation of\nspin domain structures [8, 9]. For many applications of\nspinor condensates, from quantum simulation to preci-\nsion measurement [10], it is desirable that spin domain\nformation is suppressed. In addition, atomic spin dipo-\nlarinteractions, althoughweakwhen comparedtotypical\nspin exchange interactions, induce intricate spin textures\nthat are difficult to probe when masked by spin domain\nstructures. Thus the suppressing/freezing of the unde-\nsirable dynamics from spin exchange interaction is also\nimportant for investigating the effect of dipolar interac-\ntion [11, 12].\nCompared to conventional magnets in solid states, a\nspin-1 BEC has one unsurpassed advantage: its spin ex-\nchange interaction between individual atoms can be pre-\ncisely tuned through optical (as well as magnetic) Fes-\nhbach resonances [13–19]. By adjusting the two s-wave\nscattering lengths a0anda2of two colliding spin-1 atoms\nviaopticalmeans,thespinexchangeinteractionstrength,\ncharacterizedby c2= 4π/planckover2pi12(a2−a0)/3MwithMthemass\nof the atom, is tunable. Analogous to an inverted rigid\npendulum with a rapidly oscillating pivot, a fast tem-\nporal modulation of the spin exchange interaction can\nlocalize the spin mixing dynamics, equivalent to a sup-\npressing/nulling of the spin exchange interaction.\nThis study is devoted to a theoretical investigation ofspin dynamics in a spin-1 BEC under the temporal mod-\nulation of the spin exchange interaction. As an applica-\ntion, we illustrate the suppression of the dynamic insta-\nbility and the resulting prevention of spin domain forma-\ntion in a condensate with ferromagnetic interaction. The\nproposed scheme to controlthe spin exchangeinteraction\nwill potentially provide a substantial improvement to the\naccuracy of several envisaged magnetometer setups and\nto enable cleaner detections of dipolar effects.\nFor both spin-1 atoms87Rb and23Na, popular ex-\nperimental choices, their spin-independent interaction\nstrength, characterized by c0= 4π/planckover2pi12(2a2+a0)/3M, is\ntwo to three orders of magnitude larger than |c2|[20–\n22]. This ensures the validity of single spatial mode ap-\nproximation (SMA) [23–27] when the number of atoms\nis small and the magnetic field is low. The spin degrees\nof freedom and the spatial degrees of freedom become\nseparated within the SMA. This allows one focus on the\nmost interesting spin dynamics free from density depen-\ndent interactions.\nWithin the meanfield framework,the spin dynamicsof\na spin-1 condensate under the SMA is described by [28]\n˙ρ0=2c\n/planckover2pi1ρ0/radicalbig\n(1−ρ0)2−m2sinθ, (1)\n˙θ=2c\n/planckover2pi1/bracketleftBigg\n(1−2ρ0)+(1−ρ0)(1−2ρ0)−m2\n/radicalbig\n(1−ρ0)2−m2cosθ/bracketrightBigg\n,\nwhereρi(i= +,0,−)isthe fractionalpopulationofcom-\nponent|i/angbracketright, (/summationtext\niρi= 1),m=ρ+−ρ−is the magnetiza-\ntion in a spin-1 Bose condensate, a conserved quantity.\nθis the relative phase [28]. φ(/vector r) is a unit normalized\nspatial mode function under the SMA determined from a\nscalar Gross-Pitaevskii equation with an s-wave scatter-\ning length of a2. As before, the effective spin exchange\ninteraction is given by c(t) =c2(t)N/integraltextd/vector r|φ(/vector r)|4, albeit\nthetimedependence, with Nthetotalnumberoftrapped\natoms. Although the system dynamics (1) does not con-2\nserve the total spin energy\nE(t) =c(t)ρ0/bracketleftBig\n(1−ρ0)+/radicalbig\n(1−ρ0)2−m2cosθ/bracketrightBig\n,(2)\ndue to the temporal modulation, the transversal spin\nsquared f2\n⊥=f2\nx+f2\ny= 2E(t)/c(t) remains conserved\nand isdetermined solelyby the initial condition. Because\nE(t) andc(t) are modulated exactly in the same manner,\nreplacing θwithf2\n⊥, equation (1) is further simplified to\n(˙ρ0)2=4c2\n/planckover2pi12f2(ρu−ρ0)(ρ0−ρd), (3)\nwhereρu,d=f2\n⊥/parenleftBig\n1±/radicalbig\n1−f2/parenrightBig\n/2f2withf2=f2\n⊥+m2.\nρu(d)takes the +( −) sign, denoting the largest (smallest)\nvalue ofρ0along the orbit and satisfies ˙ ρ0|ρu,d= 0. It is\nstraightforward to find the solution\nρ0(t) =ρu+ρd\n2+ρu−ρd\n2sin[γ+/integraldisplayt\n0β(s)ds],(4)\nwhereβ(t) =±2c(t)f//planckover2pi1is the frequency of the periodic\nspin evolution and the ±sign denotes the forward and\nbackward evolutions, respectively, and\nγ= atan/parenleftBigg\nρ0(0)−[(ρu+ρd)/2]/radicalbig\n[ρu−ρ0(0)][ρ0(0)−ρd]/parenrightBigg\nis given by the initial values of ρ0andθ. The above\nsolution (4) is valid for an arbitrary temporal modulation\nfunction c(t).\nTo control the spin dynamics, we consider several sim-\nple but practical modulations in the following. Based\non these examples, we demonstrate that spin dynamics\nwith a modulated cis very different from the free dy-\nnamics and understand how a temporal modulated c(t)\naffects the spin dynamics.\nFirst we consider a sinusoidal modulation with c(t) =\ndcos(Ωt).dand Ω are respectively the modulation am-\nplitudeandfrequency. ThesolutionEq.(4)thenbecomes\nρ0(t) =ρu+ρd\n2+ρu−ρd\n2sin[γ+ηsin(Ωt)],(5)\nwithη=±2df//planckover2pi1Ω. The corresponding results are il-\nlustrated in Fig. 1 for Ω = 0, 1 /2, 1, and 2. The case\nof Ω = 0 is simply the free evolution without modula-\ntion with a period T0= 2π/βdetermined by the initial\ncondition [29]. For other cases, irrespective of the values\nfor Ω, the frequency of oscillation is always Ω and the\ncorresponding period is 2 π/Ω. As shown in Fig. 1, the\noscillation amplitudes show two distinctive regions: one\nfor Ω≤Ωc≡2df/π/planckover2pi1(i.e.η≥π) where the amplitude is\nthe same with/without modulation; another for Ω >Ωc\n(i.e.η < π) where the amplitude decreases with mod-\nulation frequency Ω. In this latter region, it is easy to\ncheck that A= (ρu−ρd)(1−cosη)/2 for the case shown\nin Fig. 1.0 2 40.30.50.7\nTime( π )ρ0\n0 1 200.10.20.30.4\nΩA↑\nΩC\nFIG. 1: (Color online) Left panel: Time dependent fractiona l\npopulation ρ0(t). The black dotted line refers to the free\nevolution without modulation, while the blue dash-dotted,\nred dashed, and blue solid lines are for Ω = 1 /2, 1, and 2,\nrespectively. The parameters and initial conditions are /planckover2pi1= 1,\nd=−1,m= 0,ρ0(0) = 0.7,θ(0) = 0, and Ω c≈0.58. Right\npanel: The dependence of oscillation amplitude Aofρ0on\nthe modulation frequency Ω.\n0.30.40.50.60.7ρ0(a) (b)\n−0.2 00.20.30.40.50.60.7\nθ( π )ρ0(c)\n−0.2 00.2\nθ( π )(d)\nFIG. 2: (a) Orbits without modulation; Orbits with modu-\nlation for (b) Ω = 1 /2, (c) Ω = 1, and (d) Ω = 2. Parameters\nare the same as in Fig. 1.\nFigure 2 illustrates the orbits for the corresponding\nspin dynamics. A full orbit is occupied if Ω <Ωcbut\nonly partial orbits are occupied when Ω >Ωc. The occu-\npied portion deceases when Ω increases. Spin dynamics\nfor the first half period is reversed during the second half\nperiod evolution, irrespective of the values for Ω /negationslash= 0.\nThis reversal is responsible for a more robust modulated\ndynamicsagainstvariousnoisesasnoisesarenotreversed\nand their effect can be averaged zero according to coher-\nent control theory [30].\nNext we consider a periodic square function modula-\ntion with\nc(t) =/braceleftbigg\nd, n(w+τ)≤t < n(w+τ)+w,\n0, n(w+τ)+w≤t <(n+1)(w+τ),(6)\nforn= 0,1,2,···. The spin dynamics is halted com-\npletely if n(w+τ) +w≤t <(n+ 1)(w+τ) and is\nunmodulated if n(w+τ)≤t < n(w+τ)+w. The corre-\nsponding plots for ρ0andθdisplay interesting step like3\n0123−0.200.2\nTime (π)θ (π)\n(b)\n01230.30.40.50.60.7\nTime (π)ρ0\n(a)\nFIG. 3: (Color online) (a) Time-dependent fractional pop-\nulationρ0and (b) time-dependent relative phase θfor the\nperiodic square modulation Eq. (6) with τ=w= 1/4 and\nd=−1.\nfeatures as shown in Figs. 3(a) and 3(b).\nThe modulation dynamics considered above offers\nmany interesting possibilities. A direct application is to\nremove a dynamical instability observed in a ferromag-\nnetically interacting spin-1condensate [7–9, 31, 32]. This\ninstability is removedwheneverthe imaginarypart ofthe\neigenfrequency for the corresponding Bogoliubov excita-\ntion becomes zero in a modulation cycle. We show this\ninstability is indeed suppressed in the following for the\ncosine modulation c2(t) =dcos(Ωt). This suppression\ninhibits the spontaneous formation of spin domains.\nWe now consider our system of a homogeneous87Rb\nspin-1 condensate starting from an off-equilibrium initial\nstate [7]. The averaged spin /vectorf=/angbracketleftFx/angbracketrightˆx+/angbracketleftFy/angbracketrightˆy+/angbracketleftFz/angbracketrightˆz\nis conserved where Fx,y,zare spin-1 matrices. Starting\nfrom any stationary point, the evolution of the collective\nexcitations takes a compact form\ni/planckover2pi1∂/vector x\n∂t=M(t)·/vector x, (7)\nwhere/vector x= (δΨ+,δΨ0,δΨ−,δΨ∗\n+,δΨ∗\n0,δΨ∗\n−)Twith devi-\nationsδΨjandδΨ∗\njfrom the stationary solution [7].\nThe general solution to Eq. (7) is /vector x(t) =U(t,0)/vector x(0)\nwhereU(t2,t1) =Texp[−(i//planckover2pi1)/integraltextt2\nt1dsM(s)] withTthe\ntime ordering operator. For periodic modulation, the so-\nlution during t∈[pT,(p+ 1)T] is further simplified to\n/vector x(t) =U(t,pT)(UT)p/vector x(0), where p= 0,1,2,···indexes\nthe number of periods. T= 2π/Ω is the period of modu-\nlation, and UT=U(T,0) =Texp[−(i//planckover2pi1)/integraltextT\n0dsM(s)] is\nthe evolution operator for a complete period. |/vector x(t)|will\ngrow (decay) exponentially with pif the modulus of UT\nare larger (smaller) than unity. Diagonalizing UTand\nrewriting it as UT=V†exp(−iTF)V, where the Fluo-\nquet operator Fis a 6-by-6 diagonal matrix, the crite-\nria for stable dynamics reduces to a vanishing imaginary\npart ofFq(q= 1,2,···,6). Unstable dynamics arises\nif the imaginary part is not zero, while stable dynamics\nemerges if the imaginary part of all Fqis exactly zero.\nThe inset of Fig. 4 shows the imaginary part of a typ-\nical spectrum for the system under a cosine modulation\nofc2. We focus on the most unstable mode which in\nprinciple dominates the unstable dynamics. Compared10−110010110200.10.20.30.4\nΩ / Ω0k− / k0\n0 0.05 0.1−202x 10−3\nk / k0Im(F) / F0\n↓k−\nFIG. 4: (Color online) The dependence of the wave vector\nk−for the most unstable mode on the modulation frequency\nΩ. The inset illustrates the imaginary part of a typical Bo-\ngoliubov spectrum under modulation (blue double contours) .\nFor comparison, black dashed lines denote the results witho ut\nmodulation.\nto the modulation free results (in dashed lines), we find\nthe most unstable mode (in double contours) is not only\nsuppressed in amplitude but also shifted to larger wave\nvector. The dependence of k−on the modulation fre-\nquency Ω is illustrated in Fig. 4. The almost indepen-\ndence of k−on Ω at small values contrasts with a strong\nmonotonic increase at large values of Ω.\nTheemergenceofspindomainsisprohibitedduetothe\nmodulation. On one hand, the suppressionof Fimplies a\nsmaller effective spin exchange interaction thus a longer\neffectivespinhealinglength ξ; Ontheotherhand, theup-\nshift ofk−means shorter wave length λ= 2π/k−. If the\nunstable mode wave length (potentially domain width)\nis smaller than the spin healing length, the condensate is\nable to heal by itself. The domain structure would never\nappear if ξexceedsλ.\nBecause of the modulation, however, the resulting dy-\nnamics becomes completely different from the case of\nc2= 0 or no spin dynamics at all. The modulation does\nnot stop spin mixing dynamics, i.e., as we continue to\nobservespin waveswhich is nominally disguisedin exper-\niments by the spontaneously formed spin domains [8, 9].\nFurthermore, we expect the modulated spin dynamics\nto be robust against various experimental noises because\nthe periodic modulation effectively cancels uncorrelated\nnoises from alternating modulation periods.\nFinally we confirm the above conclusions for a trapped\nspin-1 condensate with full numerical simulations. We\nadopt experimental parameters as in Ref. [7]: The initial\nconditions are as in the experiment [8], with87Rb con-\ndensates [ ρ0(0) = 0.744,θ(0) = 0, N= 2.0×105, and\nm= 0], in a trap Vext(/vector r) = (M/2)(ω2\nxx2+ω2\nyy2+ω2\nzz2)4\nFIG. 5: (Color online) (a) Localization of the spin dynam-\nics for a trapped spin-1 condensate in 3 cases: (i) mod-\nulation starts at t= 0 (red dashed line); (ii) no modu-\nlation (black dash-dotted line); and (iii) modulation star ts\natt= 6 (blue solid line). Spatial distribution m(z) =/integraltext\ndr2πr(|Φ+(r,z)|2− |Φ−(r,z)|2) at different times for the\nabove three cases: (b) — case (i); (c) — case (iii); (d) —\ncase (ii). az=/radicalbig\n/planckover2pi1/Mωz. Trivial and flat m(z) at early times\n(t <250) has been omitted. Dynamical instability induced\nspontaneous domain formation is prohibited by the modu-\nlated spin exchange interaction.\nwithωx=ωy= (2π)240 Hz and ωz= (2π)24 Hz. The\nmodulation function is c2(t) =dcos(Ωt) withd=c2,\nand Ω = ωzwhich is about 3 times larger than the free\nspin evolution frequency 2 π/T0. Two cases will be con-\nsidered: (1) The modulation is applied immediately ( ρ0\noscillates around ρu); (2) The modulation is turned on\natt= 6 (1/ωz) (ρ0oscillates around ρd).\nThe results from numerical simulations are shown in\nFig. 5. With modulation, the spin dynamics is clearly\nlocalized as ρ+(same for ρ−) oscillates with a smaller\namplitude aroundits initial value, incontrasttothe large\namplitude unmodulated result [panel (a)]. In addition,\nthe unmodulated dynamics shows domain structures af-\ntert≈300, due to the intrinsic dynamical instability.\nWhile for both modulated cases, no spin domains are\nobserved. Thus temporal modulation of spin exchange\ninteraction does suppress the intrinsic dynamical insta-\nbilityandprohibitspontaneousdomainformation. Inour\nextensive numerical simulations, we find that when addi-\ntional white noises are added, spin domains are found to\narise quicker for the unmodulated case; yet for the two\ncases with modulations, almost the same behaviors are\nobserved as if no noise were added.\nAlthough the life time of the condensate at opti-\ncal Feshbach resonance is reduced dramatically in87Rb\ngases [15], we notice that there exists at least one magic\nwindow of relative frequency, e.g., between resonance βandγin Fig. 7 of Ref. [15], where the condensate life\ntime lasts several hundred milliseconds and c2changes\nsign. On the other hand, the spin domain emerges in a\ntime scale typically shorter than 100 ms [33]. Therefore\nthe suppression of the domain formation in87Rb conden-\nsate is experimentally feasible.\nIn summary, we propose to localize the spin mixing\ndynamics in a spin-1 condensate by temporally modulat-\ning the spin exchange interaction. For condensed atoms,\nthe modulation can be facilitated with the technique of\noptical Feshbach resonance [13, 15]. We demonstrate the\nsuppression of the intrinsic instability thus the inhibition\nof spontaneous spin domain formation in a ferromagnet-\nically interacting spin-1 Bose condensate, such as87Rb\ncondensate in the F= 1 manifold. In addition, the ef-\nfective freezing of spin mixing dynamics due to spin ex-\nchange interaction provides a cleaner approach to inves-\ntigate magnetic spin dipolar interaction effect in a87Rb\nBose condensate [11, 12].\nW.Z.acknowledgessupportbythe973ProgramGrant\nNo. 2009CB929300, the National Natural Science Foun-\ndation of China Grant No. 10904017, and the Program\nfor New Century Excellent Talents in University.\n[1] L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon,\nOxford, 1960).\n[2] G. Horvath, R.Thompson, andP.Knight, Contemporary\nPhysics38, 25 (1997).\n[3] H. Saito and M. Ueda, Phys. Rev. Lett. 90, 040403\n(2003).\n[4] F. 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Lett. 99, 070403 (2007).[26] Y. Liu, S. Jung, S. E. Maxwell, L. D. Turner, E. Tiesinga,\nand P. D. Lett, Phys. Rev. Lett. 102, 125301 (2009).\n[27] Y. Liu, E. Gomez, S. E. Maxwell, L. D. Turner,\nE. Tiesinga, andP. D. Lett, Phys. Rev.Lett. 102, 225301\n(2009).\n[28] W. Zhang, D. L. Zhou, M.-S. Chang, M. S. Chapman,\nand L. You, Phys. Rev. A 72, 013602 (2005).\n[29] H. Pu, C. K. Law, S. Raghavan, J. H. Eberly, and N. P.\nBigelow, Phys. Rev. A 60, 1463 (1999).\n[30] C. P. Slichter, Principles of Magnetic Resonance\n(Springer-Verlag, New York, 1992).\n[31] J. Mur-Petit, M. Guilleumas, A. Polls, A. Sanpera,\nM. Lewenstein, K. Bongs, and K. Sengstock, Phys. Rev.\nA73, 013629 (2006).\n[32] Q. Gu and H. Qiu, Phys. Rev. Lett. 98, 200401 (2007).\n[33] S. R. Leslie, J. Guzman, M. Vengalattore, J. D. Sau,\nM. L. Cohen, and D. M. Stamper-Kurn, Phys. Rev. A\n79, 043631 (2009)."    },    {        "title": "1602.05253v2.Fractional_Spin_Fluctuation_as_a_Precursor_of_Quantum_Spin_Liquids__Majorana_Dynamical_Mean_Field_Study_for_the_Kitaev_Model.pdf",        "content": "arXiv:1602.05253v2  [cond-mat.str-el]  31 Aug 2016APS/123-QED\nFractional Spin Fluctuation as a Precursor of Quantum Spin L iquids:\nMajorana Dynamical Mean-Field Study for the Kitaev Model\nJunki Yoshitake1, Joji Nasu2, and Yukitoshi Motome1\n1Department of Applied Physics, University of Tokyo, Bunkyo , Tokyo 113-8656, Japan\n2Department of Physics, Tokyo Institute of Technology, Megu ro, Tokyo 152-8551, Japan\n(Dated: May 23, 2018)\nExperimental identification of quantum spin liquids remain s a challenge, as the pristine nature\nis to be seen in asymptotically low temperatures. We here the oretically show that the precursor of\nquantumspinliquidsappearsinthespindynamicsinthepara magnetic stateoverawidetemperature\nrange. Using the cluster dynamical mean-field theory and the continuous-time quantum Monte\nCarlo method, which are newly developed in the Majorana ferm ion representation, we calculate\nthe dynamical spin structure factor, relaxation rate in nuc lear magnetic resonance, and magnetic\nsusceptibility for the honeycomb Kitaev model whose ground state is a canonical example of the\nquantum spin liquid. We find that dynamical spin correlation s show peculiar temperature and\nfrequency dependence even below the temperature where stat ic correlations saturate. The results\nprovide the experimentally-accessible symptoms of the fluc tuating fractionalized spins evincing the\nquantum spin liquids.\nPACS numbers: 71.10.Fd, 71.27.+a, 75.10.-b\nThe quantum spin liquid (QSL) has attracted much\nattention for decades, as a new state of matter in in-\nsulating magnets stabilized by quantum fluctuations [1].\nAlthough several candidate materials have been studied,\nexperimental identification of QSLs still remains a chal-\nlenge in modern condensed matter physics [2, 3]. This\nis mainly because of the absence of conventional order\nparameters: it is hard to prove the lack of any symmetry\nbreaking down to the lowest temperature ( T). Many at-\ntempts were made also on the low- Tbehavior of thermo-\ndynamic quantities, for instance, the specific heat [4–6],\nwhich reflect the low-energy excitations specific to QSLs.\nAll these efforts are nonetheless extremely difficult, as\nthe asymptotically low- Tphysics might be sensitively af-\nfected by extrinsic factors, such as impurities and subor-\ndinate interactions.\nOntheotherhand, QSLsareestablishedinseveralthe-\noreticalmodels. Amongthem, theKitaevmodelprovides\na canonical example of exact QSLs with fractional exci-\ntations in the ground state [7]. The model is believed to\ndescribe the anisotropic exchange interactions realized in\ninsulating magnets with strong spin-orbit coupling, such\nasIr oxides[8]. This has stimulated anew trend ofexplo-\nration of QSLs in real materials [9–12]. Recently, several\nexperimental effortshavebeen made on the identification\nof the fractional excitations in the paramagnetic state\nabove the N´ eel temperature as a precursor to QSLs [13–\n16]. Indeed, such a signature in a wide Trange was the-\noretically predicted for thermodynamic quantities [17].\nHowever,thesignatureoffractionalizationismostclearly\nvisibleinthedynamics, forwhichtheoreticalstudieswere\nlimited to the ground state [18, 19]. Thus, the ‘missing\nlink’betweentheoryandexperimentexistsinthedynam-\nical properties in the experimentally-accessible Trange.\nThis is, however, a theoretical challenge as it requires tohandle both quantum and thermal fluctuations simulta-\nneously.\nIn this Letter, we present numerical results on the dy-\nnamical properties of the Kitaev model at finite T. To\ntake into account quantum and thermal fluctuations on\nan equalfooting, wedevelop the cluster dynamical mean-\nfield theory(CDMFT) and the continuous-timequantum\nMonteCarlomethod (CTQMC) inthe Majoranafermion\nrepresentation of this quantum spin model. We calculate\nthe experimentallymeasurablequantities: the dynamical\nspin structure factor, S(q,ω),which is measured in the\nneutron scattering experiment, the relaxation rate in nu-\nclearmagneticresonance(NMR), 1 /T1, andthemagnetic\nsusceptibility χ, for both ferromagnetic (FM) and anti-\nferromagnetic(AFM) cases. Weshowthatthedynamical\nspin fluctuations in the paramagnetic state are strongly\ninfluenced by the thermal fractionalization of quantum\nspins.S(q,ω) exhibits the growth of inelastic and quasi-\nelastic responses at very different Tscales. Also, 1 /T1\nbeginstoincreasebelowthetemperaturewherethestatic\nspin correlations saturate, and shows a peak at very low\nT, despite the suppression of χfrom the Curie-Weiss\nbehavior. These unconventional features will provide a\nsmoking gun for fractionalized spins in the Kitaev-type\nQSLs.\nWe consider the Kitaev model on a honeycomb lattice,\nwhose Hamiltonian is given by [7]\nH=−Jx/summationdisplay\n/an}bracketle{tj,k/an}bracketri}htxSx\njSx\nk−Jy/summationdisplay\n/an}bracketle{tj,k/an}bracketri}htySy\njSy\nk−Jz/summationdisplay\n/an}bracketle{tj,k/an}bracketri}htzSz\njSz\nk,\n(1)\nwhereSp\njis thep(=x,y,z) component of the S= 1/2\nspin at site j. The sum of ∝an}bracketle{tj,k∝an}bracketri}htpis taken for the nearest-\nneighbor (NN) sites on three inequivalent bonds of the\nhoneycomb lattice, as indicated in Fig. 1(a).2\nThe exact solution for the ground state of the model\n(1) is obtained by introducing Majorana fermions [7]. A\nformulation, which is suitable for the following numeri-\ncal calculations at finite T, is obtained by applying the\nJordan-Wigner transformation to the one-dimensional\nchains composed of the JxandJybonds and introducing\ntwo types of Majorana fermions cjand ¯cj[20–22]. Then,\nthe Hamiltonian in Eq. (1) is rewritten as\nH=iJx\n4/summationdisplay\n(j,k)xcjck−iJy\n4/summationdisplay\n(j,k)ycjck−iJz\n4/summationdisplay\n(j,k)zηrcjck,\n(2)\nwhere (j,k)pis the NN pair satisfying j < kon thep\nbond. Here, ηr=i¯cj¯ckis defined on each zbond (ris the\nbond index); ηris aZ2variable taking ±1, asη2\nr= 1 and\nit commutes with the Hamiltonian as well as all the other\nηr′. The ground state is given by all ηr= 1, dictating a\nQSL with gapless or gapful excitations depending on the\nanisotropy in the coupling constants [7].\nAt finite T, however, the configuration of {η}is\ndisturbed by thermal fluctuations. Hence, the model\nin Eq. (2) describes itinerant Majorana fermions cou-\npled to thermally-fluctuatingclassicalvariables ηr, which\ncan be regarded as a variant of the double-exchange\nmodel. This allows one to utilize theoretical tools\ndeveloped for fermion systems, such as the quantum\nMonte Carlo method [23]. In this study, we construct\nthe CDMFT [24] for this Majorana fermion problem.\nBy following the formulation for the double-exchange\nmodel [25] and using the path-integral representation for\nMajorana fermions [26], the effective action for a cluster\nembedded in a bath [see Fig. 1(a)] is given by\nS{η}\neff=−T/summationdisplay\nj,k,n≥0χj,−ωn(G0(iωn))−1\nj,kχk,ωn\n+iJz\n2T/summationdisplay\n/an}bracketle{tj,k/an}bracketri}htz,nηrχj,−ωnχk,ωn, (3)\nwhereχj,ωnis the Grassmann number correspond-\ning to the Majorana operator cj, andG0represents\nthe Weiss function including the effect of bath. For\na given configuration of {η}, the impurity problem\nis exactly solvable and Green’s function is obtained\nas (G{η}(iωn))−1= (G0(iωn))−1−h{η}, where h{η}\nis the matrix representation of the second term in\nEq. (3). Local Green’s function is also exactly cal-\nculated through G(iωn) = P( {η})G{η}(iωn), where\nP({η}) =Z{η}//summationtext\n{η}Z{η}withZ{η}=e−S{η}\neff=/producttext\nn≥0det[−G{η}(iωn)], as we can compute G{η}(iωn)\nand P({η}) for all2Nc/2configurationsof {η}in aNc-site\ncluster [27]. The self-consistent equations in CDMFT\nare given as Σ( iωn) = (G0(iωn))−1−(G(iωn))−1and\n(G0(iωn))−1=/parenleftbig1\nN/summationtext\nk[iωn−2H0(k)−Σ(iωn)]−1/parenrightbig−1+\nΣ(iωn), where Σ is the self-energy, Nis the number of(a)\n(b)\nCDMFTQMCTH TL\n0.00.20.40.60.81.0\n10-210-1100\nT\nFIG. 1: (a) Schematic picture of the Kitaev model on the\nhoneycomb lattice [Eq. (1)] and the mapping to a 26-sites\ncluster used in the Majorana CDMFT. (b) The specific heat\nCvand equal-time spin correlations for NN sites, /angbracketleftSz\njSz\nk/angbracketright, ob-\ntained by the Majorana CDMFT for the isotropic FM case.\nQMC data in Ref. [17] are plotted by gray symbols for com-\nparison.\nclusters in the whole lattice, and H0(k) is the Fourier\ntransform of the first and second terms in Eq. (2).\nThus, the Majorana CDMFT provides the exactre-\nsults for thermodynamic quantities of the quantum spin\nmodel (1), except for the cluster approximation. This\nis a distinct advantage of the Majorana representation:\nthe original quantum spin representation does not ad-\nmit the exact enumeration. In addition, the cluster ap-\nproximation works quite well in the current system with\nextremely short-range spin correlations [28], as demon-\nstrated below. In the following calculations, we take the\n26-sites cluster shown in Fig. 1(a), and consider 60 ×80\narray of the unit cell [29]. Typically, the CDMFT loop is\nrepeated for ten times until convergence.\nA benchmark of the Majorana CDMFT is shown in\nFig. 1(b). We compare the specific heat and the equal-\ntime NN spin correlations ∝an}bracketle{tSp\njSp\nk∝an}bracketri}htobtained by CDMFT\nwith those by QMC in Ref. [17]. The data are calculated\nfor the isotropic FM case, Jx=Jy=Jz= 1 (the sign of\n∝an}bracketle{tSp\njSp\nk∝an}bracketri}htis reversed for AFM). As indicated by two broad\npeaks in the specific heat in the QMC results, the system\nexhibits two crossovers at TH∼0.375 and TL∼0.012.\nThe spin correlations grow down to T∼TH, while they\nsaturate below THand do not show significant changesat\nTL[17]. These behaviors are excellently reproduced by\nCDMFT, except for the low- Tpeak in the specific heat.\nThe sharp anomaly in the CDMFT result at T≃0.014is3\ndue to a phase transition by ordering of ∝an}bracketle{tη∝an}bracketri}ht, which is an\nartifact of the mean-field nature of CDMFT. The com-\nparison, however, shows that the CDMFT gives qualita-\ntively correct results in a wide Trange above the low- T\ncrossover, i.e., T/greaterorsimilar0.015. We note that the quantum\nspin liquid state at sufficiently low T, where all ηr= 1,\nis also reproducible [29]. Thus, the present CDMFT en-\nables to calculate the physical properties with sufficient\nprecision in the wide Trange except for the vicinity of\nTL. In the following, we apply the CDMFT in this qual-\nifiedTrange above TLto the study of spin dynamics,\nwhich one cannot compute by QMC.\nIn the calculations of dynamical quantities, we need\nan additional effort beyond the exact enumeration in the\nMajoranaCDMFT. This isbecausethe calculationofdy-\nnamical spin correlations ∝an}bracketle{tSp\nj(τ)Sp\nk∝an}bracketri}htrequires the imag-\ninary time evolution of the ¯ cvariables that compose\nthe conserved quantities η. To compute ∝an}bracketle{tSp\nj(τ)Sp\nk∝an}bracketri}ht, we\nadopt the CTQMC based on the strong coupling ex-\npansion [30]. ∝an}bracketle{tSz\nj(τ)Sz\nk∝an}bracketri}hton anr0bond is calculated\nas∝an}bracketle{tSz\nj(τ)Sz\nk∝an}bracketri}ht=/summationtext\n{η}′,ηr0=±1P({η}′,ηr0)∝an}bracketle{tSz\nj(τ)Sz\nk∝an}bracketri}ht{η}′\nby using the CDMFT solutions; here, {η}′represents\nthe configurations of ηrexcept for the r0bond, and\n∝an}bracketle{tSz\nj(τ)Sz\nk∝an}bracketri}ht{η}′is calculated for a given {η}′by CTQMC.\nAs the interaction between Majorana fermions lies only\non ther0bond, it is sufficient to solve the two-site impu-\nrity problem in CTQMC. In the CTQMC calculations,\nwe typically run 107steps with measurement at every\n20 steps, after 105steps of initial relaxation, for each\n∝an}bracketle{tSz\nj(τ)Sz\nk∝an}bracketri}ht{η}′.∝an}bracketle{tSp\nj(τ)Sp\nk∝an}bracketri}htforp=x,yare obtained by\ntaking the lattice coordinate so that p=z. In the fol-\nlowing, we present the results for the isotropic coupling,\nJx=Jy=Jz=J, where the ground state is a gapless\nquantum spin liquid [7]. We compute both FM and AFM\ncases [32] with setting |J|= 1 as the energy unit. The\nsystematic study of the anisotropiccases will be reported\nelsewhere.\nUsing the Majorana CDMFT+CTQMC, we calcu-\nlate the dynamical spin structure factor, NMR re-\nlaxation rate, and magnetic susceptibility. The dy-\nnamical spin structure factor is defined as S(q,ω) =\n1/(3NNc)/summationtext\np/summationtext\nj,keiq·(rj−rk)Sp\nj,k(ω), where Sp\nj,k(ω) is\nobtained by solving ∝an}bracketle{tSp\nj(τ)Sp\nk∝an}bracketri}ht=/integraltext\ndωSp\nj,k(ω)e−ωτby\nthe maximum entropy method [31]. We confirmed the\nvalidity of the procedures by the fact that the low- Tre-\nsult with ∝an}bracketle{tη∝an}bracketri}ht ≃1 (beyond the quantified Trange) re-\nproduces the T= 0 solution [18]. The NMR relaxation\nrate is obtained by using the relation, 1 /T1∝Sx\nj,k(ω=\n0) +Sy\nj,k(ω= 0); we compute the contributions from\nonsite and NN-site correlations separately, as the hy-\nperfine coupling is unknown. The magnetic susceptibil-\nity is calculated as χp= 1/(NNc)/summationtext\nj,k/integraltext\ndτ∝an}bracketle{tSp\nj(τ)Sp\nk∝an}bracketri}ht;\nχx=χy=χz=χfor the isotropic coupling.\nFigure 2 shows the Tdependences of the dynamical\nspin structure factor S(q,ω) for both FM and AFM\n(a) (b)\n(c) (d)\n(e) (f)\n(g) (h)\nM \u0001K K M \u0000K KFM AFM\nFIG.2: Thedynamicalspinstructurefactor S(q,ω)obtained\nby the Majorana CDMFT+CTQMC for the (a)(c)(e)(g) FM\nand (b)(d)(f)(h) AFM cases at (a)(b) T≃0.018, (c)(d) T≃\n0.094, (e)(f) T≃0.24, and (g)(h) T≃2.4.\ncases. At high T > T H,S(q,ω) shows only a diffusive\nresponse at ω∼0 with less qdependence in both FM\nand AFM cases, as shown in Figs. 2(g) and 2(h). The q-\nωdependence begins to develop while lowering Tbelow\nTH∼0.375; the diffusive weight shifts to a positive ωre-\ngion ranging to ω∼JbelowTH[Figs. 2(e) and 2(f)], and\nat the same time, the quasi-elastic component at ω∼0\ngrows gradually [Figs. 2(c) and 2(d)]. The quasi-elastic\nresponse is large around the Γ point in the FM case,\nwhereas it is distributed on the Brillouin zone bound-\nary (along the K-M line) in the AFM case. With fur-\nther decreasing T, the intensity of the quasi-elastic peak\ncontinues to increase while approaching TL∼0.012, as\nshown in Figs. 2(a) and 2(b). The low- Tbehavior con-\nverges on the ground state solution, which has a sharp\npeak atω∼0.12Jtogether with an incoherent weight at\nω∼J[18].\nTo see these behaviors more clearly, we show the data\nat the Γ and K points, denoted as S(Γ,ω) andS(K,ω),\nrespectively, in Fig. 3. Note that S(K,ω) is the same\nfor the FM and AFM cases by symmetry. In the FM\ncase,S(Γ,ω) andS(K,ω) show qualitatively similar T-ω4\n0 \u0002 \u0003\n\u0004 \u0005 \u0006\n\u0007 \b \t\n\n \u000b \f\n\r \u000e \u000f\n-1.5-1.0-0.50.00.51.01.52.0\nω0.00.20.40.60.8\n-1.5-1.0-0.50.00.51.01.52.0\nω0.00.20.40.60.8\n-1.5-1.0-0.50.00.51.01.52.0\nω0.00.10.20.30.4\n-1.5-1.0-0.50.00.51.01.52.0\nω0.00.10.20.30.4\n-1.5-1.0-0.50.00.51.01.52.0\nω0.00.10.20.30.4\n-1.5-1.0-0.50.00.51.01.52.0\nω0.00.20.40.60.81.01.2\n-1.5-1.0-0.50.00.51.01.52.0\nω0.00.20.40.60.81.01.2\n-1.5-1.0-0.50.00.51.01.52.0\nω0.00.20.40.60.81.01.2\n-1.5-1.0-0.50.00.51.01.52.0\nω(a)\n(c)\n(e)(d)(b)\n(f)TL TH\nFIG. 3: S(Γ,ω) for the (a) FM and (c) AFM cases, and (e)\nS(K,ω) at several T. (e) is common to the FM and AFM\ncases. The corresponding contour plots in the T-ωplane are\nshown in (b)(d)(f). The arrows indicate the temperatures\nused for the data in (a)(c)(e). The dashed curves represent\nthe average frequency of S(q,ω) (see the text for details). In\n(a)(c)(e), the errorbars are shown for every ten data along t he\nωaxis.\ndependence, as shown in Figs. 3(a)(b) and 3(e)(f); the\ninelastic response at ω∼Jappears below TH, and the\nquasi-elastic one at ω∼0 rapidly grows as approaching\nTL. On the other hand, in the AFM case, the strong\nquasi-elastic intensity at low Tis absent, while the in-\nelastic response at ω∼Jarises below TH, as in the FM\ncase, as shown in Figs. 3(c)(d).\nDespite the different qdependence reflecting the sign\nofJ,S(q,ω) exhibits common characteristic ω-Tdepen-\ndence: the emergence of the inelastic response at ω∼J\nforT/lessorsimilarTH, and the rise of the quasi-elastic response\nasT→TL. These peculiar behaviors are regarded as\nthe signatures of the fractionalization of quantum spins\ninto two types of Majorana fermions, itinerant “matter\nfermions” and localized “fluxes”, which correspond to c\nandηin Eq.(2), respectively. The previousQMCstudies\nrevealed that the matter fermions and fluxes affect the\nthermodynamics at very different Tscales [17, 23]; the\nkinetic energy of matter fermions, which is equivalent to\nthe equal-time spin correlations ∝an}bracketle{tSp\njSp\nk∝an}bracketri}ht[see Eqs. (1) and\n(2)], is gained at T∼TH, whereas the fluxes shows a\ncondensation at T∼TLto the flux-free state with all\nηr= 1. Our results of S(q,ω) obtained above indicateonsite\nNN-siteTH TL TH TL\n(a) (b)\nFMAFM\n0246810\n10-210-11000.00.10.20.30.4\nχ\nT0.00.20.40.60.81.0\n10-210-11001 /T1\nT\nFIG. 4: Tdependences of (a) the NMR relaxation rate 1 /T1\nand (b) the magnetic susceptibility χ. In (b), the dashed\ncurves represent the Curie-Weiss behavior χCW.\nthat the former is closely related to the evolution of the\ninelastic response at ω∼JbelowTH, while the latter to\nthe rise of the quasi-elastic response as approaching TL.\nThe relation between the inelastic response and the\nmatter fermions is grasped through two sum rules for\nthe dynamical spin structure factor. For instance, at the\nΓ point, the sum rules read/integraltext\nSp(Γ,ω)dω=∝an}bracketle{tSp\njSp\nk∝an}bracketri}ht+1/4\nand/integraltext\nωSz(Γ,ω)dω= (Jx∝an}bracketle{tSx\njSx\nk∝an}bracketri}ht+Jy∝an}bracketle{tSy\njSy\nk∝an}bracketri}ht)/2 (simi-\nlarly for p=x,y), where ∝an}bracketle{tSp\njSp\nk∝an}bracketri}htdenotes the equal-time\nNN spin correlation [33]. As mentioned above, in the Ki-\ntaev model, ∝an}bracketle{tSp\njSp\nk∝an}bracketri}htcorresponds to the kinetic energy of\nthe matter fermions. Hence, when we compute the av-\nerage frequency of Sp(Γ,ω) by the ratio of the two sum\nrules, ¯ω≡/integraltext\nωSz(Γ,ω)dω//integraltext\nSz(Γ,ω)dω, the kinetic en-\nergy gain of the matter fermions at T∼THresults in the\nshift of ¯ωfrom almost zero to a nonzero positive value.\nThis is indeed seen in Figs. 3(b) and 3(d), where ¯ ωis\nplotted by the dashed curves. The difference of the val-\nues of low- T¯ωbetween the FM and AFM cases is also\naccountedforbytheoppositesignoftheNNcontribution\nin the first sum rule.\nOn the other hand, below T∼TL, the quasi-elastic\nresponse converges on the sharp peak at ω∼0.12Jwith\na flux gap ∆ ≃0.065J[7] in the T= 0 solution [18]. As\nthe fluxes are proliferated above T∼TL[17], the decay\nof the quasi-elastic response for T/greaterorsimilarTLis considered as\na consequence of the thermally excited fluxes. Indeed,\nthe flux gap is smeared out in our results above T∼TL.\nWe find that the influence of excited fluxes is more\nclearly visible in the NMR relaxation rate 1 /T1. Fig-\nure4(a)showstheCDMFT+CTQMCresultsof1 /T1: we\nplot the value of Sx\nj,k(ω= 0)+Sy\nj,k(ω= 0) in the FM case\n(theNNcomponentchangesitssignintheAFMcase). In\nthe high- Tregion above TH, as expected for the conven-\ntional paramagnets [34], the onsite component is nearly\nTindependent, while the NN-site one increases gradu-\nally with decreasing T, reflecting the growth of equal-\ntime spin correlations ∝an}bracketle{tSp\njSp\nk∝an}bracketri}htin Fig. 1(b). Below TH,\nhowever, both components increase and show a peak at\nslightly above TL, despite the saturation of equal-time\ncorrelations [35]. The pronounced peak is regarded as\nthe consequence of thermally excited fluxes above TL, as5\nthe suppression of 1 /T1forT/lessorsimilarTLis due to the for-\nmation of the flux gap in the low- Tlimit [18]. The un-\nexpected behavior below THis also seen in comparison\nwith the magnetic susceptibility χin Fig. 4(b); despite\nthe enhancement of1 /T1,χis suppressedfromthe Curie-\nWeiss behavior, χCW= 1/(4T−J), which is obtained\nby the standard mean-field approximation in the origi-\nnal spin representation. These Tdependences of 1 /T1,\nχ, and∝an}bracketle{tSp\njSp\nk∝an}bracketri}htbelowTHare highly unusual; in conven-\ntional quantum magnets, the dynamical spin correlations\ngrow with the static ones. The dichotomy between the\nstatic and dynamical correlations is a clear signature of\nfractionalization of quantum spins [29].\nIn summary, we have presented a comprehensive set\nof theoretical results for dynamical and static spin cor-\nrelations, which evince fluctuating fractionalized spins in\nthe Kitaev QSLs. The results are unveiled by using the\nMajorana CDMFT+CTQMC method developed in the\ncurrent study. Experimentally, an unusual inelastic re-\nsponse, similar to our results of S(q,ω), was observed\nin the recent neutron scattering experiment for a Ki-\ntaev candidate, α-RuCl 3[16]. Also, a similar peak in\n1/T1to our results was observed for another candidate,\nLi2RhO3[36]. The deviation of χfromχCWwas already\nreported in many materials [9–11]. Obviously, further\nsystematic studies for the Kitaev candidate materials are\nhighly desired to test our predictions. Our results will\nstimulate the rapidly-evolving “pincer attack” by theory\nand experiment for the long standing issue—the identifi-\ncation of fractionalized spins in QSLs.\nThe authors thank M. Imada, Y. Kato, M. Udagawa,\nand Y. Yamaji for fruitful discussions. This research was\nsupported by KAKENHI (No. 24340076 and 15K13533),\ntheStrategicProgramsforInnovativeResearch(SPIRE),\nMEXT, and the Computational Materials Science Initia-\ntive (CMSI), Japan.\n—Supplemental Material—\nCluster size dependence\nThe CDMFT is an approximation which replaces the\ninfinite-size system by a finite-size cluster embedded in a\nbath [see Fig. 1(a) in the main text]. It becomes exact\nwhen the cluster size is increased to infinity. Thus, it is\ncrucial how the results converge to the thermodynamic\nlimit as a function of the cluster size.\nFigure S1 shows the cluster size dependence\nof the magnetic susceptibility obtained by the\nCDMFT+CTQMC method. The data are plotted\nas functions of the cluster width in the xydirection, not\nthe total number of lattice sites included in the cluster.\nThis is because the width in the xydirection is rather\nrelevant compared to that in the zdirection in thepresent CDMFT, presumably due to the Majorana rep-\nresentation based on the Jordan-Wigner transformation\nalong the xychain. Other physical quantities calculated\nin the main text behave in a similar manner.\nAs shown in Fig. S1, for both FM and AFM cases, the\nresults show good convergence when increasing the clus-\nter size. In fact, the convergence is very quick, except for\nthe low-Tregion in the vicinity of the artificial transition\ntemperature Tc≃0.014, which is close to TL≃0.012\n[see Fig. 1(b) in the main text]. For instance, the data\natT≃0.038, which is sufficiently high compared to Tc,\nare almost unchanged while increasing the cluster width\nlarger than ∼4 in all the different series of the clusters.\nOn the other hand, while lowering temperature and ap-\nproaching Tc, the cluster size dependence becomes sub-\nstantial, as shown in the data at T≃0.017 in the figure.\nNevertheless, the data for the 26-site cluster, which is\nused in the main text [the width is ∼4.3 in the series of\nFig. S2(a)], give sufficiently converged results: the rem-\nnant relative errors are /lessorsimilar3% for both FM and AFM\ncases. Note that the remnant errors become discernible\nonly in the very vicinity of Tc. From these observations,\nwe confirmed that our data in the wide range of Tabove\nTc≃0.014 are quantitatively correct and well reproduce\nthe behaviors expected in the thermodynamic limit.\n(a) (b)\n4.55.05.56.06.5\n7 \u0010 \u00112 3 4 5 6 \u0012 8 9 1 \u0013 \u0014 \u0015 \u0016 \u0017χFM\ncluster width0.270.280.290.300.310.32\n23456789101112χAFM\ncluster width\nFIG. S1: Cluster size dependence of the magnetic suscep-\ntibility for (a) the FM and (b) AFM cases. The different\nsymbols represent the different series of the clusters: tria n-\ngles, circles, and squares correspond to Fig. S2(a), S2(b), and\nS2(c), respectively. The data in the dashed circles with ar-\nrows indicate the results for the 26-site cluster used in the\ncalculations in the main text. The definition of the cluster\nwidth is described in the caption of Fig. S2.\nCDMFT+CTQMC results for T < T L\nAs mentioned in the main text, the CDMFT calcula-\ntion exhibits a phase transition by ordering of ∝an}bracketle{tηr∝an}bracketri}htat\nT≃0.014 because of the mean-field nature of CDMFT.\nBelow the critical temperature, ∝an}bracketle{tηr∝an}bracketri}htbecomes almost 1,\nnamely, the state is almostsimilar to the flux-free ground\nstate. In the ground state, S(q,ω) shows a small gap\n≃0.065Jdue to the gapped flux excitation [18]. In\nFig. S3, we show the CDMFT+CTQMC results for the\ndynamical spin structure factor at T= 0.00825, which is6\n(a)\nwidth\n( \u0018 \u0019\nwidth(c)\nwidth\nFIG. S2: Schematic picture of three series of the clusters us ed\nin Fig. S1. In each series of clusters, the cluster size is var ied\nby the cluster width in the xydirection, while keeping the\nwidth in the zdirection. The cluster width in the xydirection\nis given by the average number of the z-bonds (indicated by\nthe red lines in the figures), which is used in the plots in\nFig. S1. Intheseexamples, thecluster widthis (a)13 /3≃4.3,\n(b) 4, and (c) 5.\nwell below the critical temperature as well as TL. The\nresult exhibits the flux gap, consistent with the previous\nresults at T= 0. This further supports the validity of\nour CDMFT+CTQMC calculations. In Figs. 2 and 3 in\nthe main text, the flux gapis smearedout and not clearly\nvisible as the fluxes are excited by thermal fluctuations\naboveTL.\n0.00.51.01.52.02.5\n-1.5-1.0-0.50.00.51.01.52.0\nω\n(a) (b)\nM\u001a K \u001b\nFIG. S3: (a) S(Γ,ω) and (b) S(q,ω) atT= 0.00825 for the\nFM case. In (a), the errorbars are shown for every ten data\nalong the ωaxis.\nPlots of Fig. 4 in the T-linear scale\nInFig.4inthemaintext, weshowtheNMRrelaxation\nrate 1/T1and the magnetic susceptibility χas functions\nof lnT. For reference, we present them in the T-linear\nscale in Fig. S4.\n[1] P. W. Anderson, Resonating valence bonds: a new kind\nof insulator , Mater. Res. Bull. 8, 153 (1973).\n[2] L. Balents, Spin liquids in frustrated magnets , Nature\n464, 199 (2010).\n[3] C. Lacroix, P. Mendels, and F. Mila, Introduction to\nFrustrated Magnetism , Springer Series in Solid-State Sci-\nences (Springer, Heidelberg, 2011).\n[4] J. S. Helton, K. Matan, M. P. Shores, E. A. Nytko, B.\nM. Bartlett, Y. Yoshida, Y. Takano, A. Suslov, Y. Qiu,onsite\nNN-siteTH TH\n(a) (b)\nFMAFM\n0.00.20.40.60.81.0\n0.0 0.6 1.2 1.8 2.41 /T1\nT0246810\n0.0 0.6 1.2 1.8 2.40.00.10.20.30.4\nχ\nT\nFIG. S4: The same plots as Fig. 4 in the main text in the\nT-linear scale.\nJ.-H. Chung, D. G. Nocera, and Y. S. Lee, Spin Dy-\nnamics of the Spin-1/2 Kagome Lattice Antiferromagnet\nZnCu3(OH)6Cl2, Phys. Rev. Lett. 98, 107204 (2007).\n[5] Y. Okamoto, M. Nohara, H. Aruga-Katori, and H. Tak-\nagi,Spin-Liquid State in the S= 1/2Hyperkagome An-\ntiferromagnet Na 4Ir3O8, Phys. Rev. Lett. 99, 137207\n(2007).\n[6] S. Yamashita, Y. Nakazawa, M. Oguni, Y. Oshima, H.\nNojiri, Y. Shimizu, K. Miyagawa, and K. Kanoda, Ther-\nmodynamic properties of a spin-1/2 spin-liquid state in a\nκ-type organic salt , Nature Phys. 4, 459 (2008).\n[7] A. Kitaev, Anyons in an exactly solved model and beyond ,\nAnn. Phys. 321, 2 (2006).\n[8] G. Jackeli and G. Khaliullin, Mott Insulators in the\nStrong Spin-Orbit Coupling Limit: From Heisenberg to a\nQuantum Compass and Kitaev Models , Phys. Rev. Lett.\n102, 017205 (2009).\n[9] Y. Singh and P. Gegenwart, Antiferromagnetic Mott in-\nsulating state in single crystals of the honeycomb lattice\nmaterial Na 2IrO3, Phys. Rev. B 82, 064412 (2010).\n[10] Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale,\nW. Ku, S. Trebst, and P. Gegenwart, Relevance of the\nHeisenberg-Kitaev Model for the Honeycomb Lattice Iri-\ndatesA2IrO3, Phys. Rev. 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Plumb, Y.-J. Kim, and\nK. S. Burch, Scattering Continuum and Possible Frac-\ntionalized Excitations in α-RuCl 3, Phys. Rev. Lett. 114,\n147201 (2015).\n[16] A. Banerjee, C. A. Bridges, J-Q. Yan, A. A. Aczel, L. Li,\nM. B. Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu,\nJ. Knolle, D. L. Kovrizhin, S. Bhattacharjee, R. Moess-7\nner, D. A. Tennant, D. G. Mandrus, S. E. Nagler, Prox-\nimate Kitaev quantum spin liquid behaviour in a honey-\ncomb magnet , Nature Materials, nmat4604 (2016).\n[17] J. Nasu, M. Udagawa, and Y. Motome, Thermal\nfractionalization of quantum spins in a Kitaev model:\nTemperature-linear specific heat and coherent transport\nof Majorana fermions , Phys. Rev. B 92, 115122 (2015).\n[18] J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moess-\nner,Dynamics of a Two-Dimensional Quantum Spin Liq-\nuid: Signatures of Emergent Majorana Fermions and\nFluxes, Phys. Rev. Lett. 112, 207203 (2014).\n[19] J. Knolle, G.-W. Chern, D. L. Kovrizhin, R. Moessner,\nand N. B. 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Rep. 269, 133 (1996).\n[32] We note that S(q,ω)AFM=−S(q,ω)FM+2S(q= K,ω),\nwhereS(q= K,ω) =Sp\nj,j(ω) is common to the FM and\nAFM cases.\n[33] P. C. Hohenberg and W. F. Brinkman, Sum rules for the\nfrequency spectrum of linear magnetic chains , Phys. Rev.\nB10, 128 (1973).\n[34] T. Moriya, Nuclear Magnetic Relaxation in Antiferro-\nmagnetics, II , Prog. Theor. Phys., 16, 641 (1956).\n[35] The difference between the two components corresponds\ntoS(Γ,ω= 0)intheAFMcase [see Fig. 3(d)].Itbecomes\nsmall after the growth of spin correlations below TH, as\nit is proportional to the probability of aligning two spins\non azbond parallel to the zdirection without an energy\ncost.\n[36] P. Khuntia, S. Manni, F. R. Foronda, T. Lancaster, S.\nJ. Blundell, P. Gegenwart, and M. Baenitz, Local Mag-\nnetism and Spin Dynamics of the Frustrated Honeycomb\nRhodate Li 2RhO3, preprint (arXiv:1512.04904)."    },    {        "title": "0806.0019v1.Time_resolved_Dynamics_of_the_Spin_Hall_Effect.pdf",        "content": "Time-resolved Dynamics of the Spin Hall E\u000bect\nN. P. Stern, D. W. Steuerman, S. Mack, A. C. Gossard, and D. D. Awschalom\u0003\nCenter for Spintronics and Quantum Computation,\nUniversity of California, Santa Barbara, CA 93106 USA\n(Dated: November 11, 2021)\n1arXiv:0806.0019v1  [cond-mat.mes-hall]  30 May 2008The generation and manipulation of carrier spin polarization in semiconduc-\ntors solely by electric \felds has garnered signi\fcant attention as both an in-\nteresting manifestation of spin-orbit physics as well as a valuable capability for\npotential spintronics devices1,2,3,4. One realization of these spin-orbit phenom-\nena, the spin Hall e\u000bect (SHE)5,6, has been studied as a means of all-electrical\nspin current generation and spin separation in both semiconductor and metallic\nsystems. Previous measurements of the spin Hall e\u000bect7,8,9,10,11have focused\non steady-state generation and time-averaged detection, without directly ad-\ndressing the accumulation dynamics on the timescale of the spin coherence\ntime. Here, we demonstrate time-resolved measurement of the dynamics of\nspin accumulation generated by the extrinsic spin Hall e\u000bect in a doped GaAs\nsemiconductor channel. Using electrically-pumped time-resolved Kerr rotation,\nwe image the accumulation, precession, and decay dynamics near the channel\nboundary with spatial and temporal resolution and identify multiple evolution\ntime constants. We model these processes using time-dependent di\u000busion anal-\nysis utilizing both exact and numerical solution techniques and \fnd that the\nunderlying physical spin coherence time di\u000bers from the dynamical rates of spin\naccumulation and decay observed near the sample edges.\nTheories have predicted5,6,12,13, and experiments con\frmed7,9, that an electric current in\na crystal with spin-orbit coupling gives rise to a transverse spin current via the spin Hall\ne\u000bect. Spin-dependent scattering of carriers by charged impurities (the extrinsic SHE)5,6,14\nor the direct e\u000bect of spin-orbit coupling on the band structure (intrinsic SHE)12,13causes\nspin-dependent splitting in momentum space and a resulting pure spin current. Although\nnot directly observable, the presence of this spin current can be inferred from the existence\nof non-equilibrium spin accumulation near sample boundaries. While extrinsic spin Hall\ncurrents generated by impurity scattering evolve on momentum scattering timescales ( <\n1 ps), spin Hall accumulation is expected to develop on the much slower spin coherence\ntimescale\u001c(\u00181 ns). Since this timescale is of the same order as that desired for fast\nelectrical manipulation of spin polarization in spintronics devices, understanding dynamics\non this timescale is critical for both physical and practical insights to the extrinsic SHE\nprocesses.\nSteady-state observations of electrically-generated spin accumulation7,15,16are e\u000bective\n2for inferring \u001c, but they cannot directly access the dynamical processes on the nanosec-\nond timescale. In contrast, time-resolved spin dynamics with picosecond resolution are rou-\ntinely measured using ultra-fast optical pump-probe techniques17,18. Time-resolution of bulk\ncurrent-induced spin polarization (CISP) was achieved using a photoconductive switch15,\nbut only precessional dynamics were observed due to short duration of the ultra-fast current\npulse. Further, in contrast to SHE, CISP is a bulk phenomenon and consequently neither\nthe steady-state4,15nor the time-resolved15measurements investigated spatial dynamics near\nthe sample edge. In the present work, we combine the spatial resolution a\u000borded by scanning\nKerr microscopy19with an optical probe pulse delayed relative to the electrical pump pulse\nto achieve both temporal and spatial resolution of spin polarization generated electrically\nby the SHE in an n-doped GaAs channel. Details of the experimental technique are shown\nin Figure 1 and are discussed in the Methods section below.\nFigure 2b-d shows the Kerr rotation \u0012K(t) as a function of delay time tin a \fxed magnetic\n\feldBapplied along the y-axis. The spin polarization is generated by the SHE from a\nvoltage pulse V(t) with length tp= 6 ns and amplitude V0= 2 V (Fig. 2a). The laser is\npositioned at y= 126\u0016m so as to be close to the boundary spin generation with minimal\nclipping of the spot by the edge. During the pulse (0 < t < tp), spin polarization builds\nup due to the SHE. After the pulse has passed ( t>tp), the accumulated spin polarization\nundergoes decay and spin precession (Fig. 2b-d). We \ft \u0012K(t) fort>tpto an exponentially\ndecaying cosine to extract an inhomogeneous depolarization time \u001c\u0003and Larmor precession\nfrequency\u0017L=g\u0016BB=h wheregis the Lande g-factor, \u0016Bthe Bohr magneton, and his\nPlanck's constant. Fits to \u0017L(B) givejgj= 0:346\u00060:002 (Fig. 2e), which is consistent with\ngexpected for this doping level20. The depolarization time \u001c\u0003= 2:8\u00060:1 ns measured at a\nspeci\fc spatial location should di\u000ber from the physical spin decoherence time \u001c. Because spin\npolarization can di\u000buse due to accumulation gradients, there is a second pathway for spin\ndepolarization beyond decoherence which depends strongly on the measurement location.\nWe reconcile the dynamically measured \u001c\u0003with the intrinsic decoherence time \u001clater in our\ndiscussion.\nTypical optical studies of electrically generated spin accumulation measure the time-\naveraged projection of spin precession as a function of applied transverse magnetic \feld B.\nIn analogy with the Hanle e\u000bect of depolarization of luminescence, szshould depolarize for\nincreasingBwhen 2\u0019\u0017L\u00181=\u001c.21Therefore, coherence times in steady-state experiments are\n3typically extracted from the linewidths of sz(B). Near the sample edge, sz(B) is a Lorentzian\nlineshape analogous to the Hanle e\u000bect7, whereas it becomes more complicated away from\nthe edges due to the interplay of spin precession and di\u000busion11,21. In the current experiment,\nmeasurement of \u0012K(B) does not represent a time-averaged steady-state accumulation, but\nrather a snapshot at a \fxed time tof the dynamic behavior of an electrically-generated spin\nensemble in a magnetic \feld.\nRepresentative scans of \u0012K(B) aty= 126\u0016m are shown for t= 1, 5, and 9 ns in Fig.\n2f-h with the magnetic \feld applied along the yaxis. For small t,\u0012K(B) grows in a broad\npeak that narrows as tincreases (Fig. 2f). Only for t\u0018tp> \u001cdoes\u0012K(B) approach the\nLorentzian lineshape expected from a conventional Hanle analysis (Fig. 2g). For t > tp,\n\u0012K(B) is primarily governed by spin precession, displaying characteristic periodic lobes of\ndecreasing amplitude away from B= 0 (Fig. 2h).\nIn Fig. 3a, we use a longer pulse tp= 15 ns to investigate accumulation dynamics with\nthe current \rowing for various V0. We \ft\u0012K(B= 0) to an exponential saturation with a\ntime constant \u001cacc. For each V0,\u001caccis around 40% of the \u001c\u0003measured from decay of the\nspin polarization (Fig. 3b). Both \u001c\u0003and\u001caccdecrease weakly with V0, which is expected\ndue to electron heating11,22.\nWe characterize the magnetic \feld lineshapes by their inverse half-width B\u00001\n1=2, which\nincreases with tbefore quickly saturating (Fig. 3a). We can understand this evolution of\nB1=2in a simple physical picture21. Soon after the pulse turns on at t= 0, spins are all\nrecently generated at the sample edge and have had little time for spin precession about\nB. For later times, spins have a larger spread in generation times (up to tp) and have\ncorrespondingly more time for precession; hence, there is more depolarization for a given B\nand the Hanle curve narrows (Fig. 2g). For t\u001d\u001c, the average precession time is governed\nby\u001crather than tand the Hanle width becomes constant as in a steady-state measurement.\nThe coherence times \u001c1=2calculated from the saturation of B1=2neart\u0018tp, agree with\ndecay times \u001c\u0003and are consequently also longer than the accumulation times \u001cacc(Fig. 3b).\nDi\u000busion analysis of the SHE accumulation is necessary to reconcile the observed di\u000berences\nbetween the timescales \u001cacc,\u001c\u0003, and\u001c1=2.\nElectrically-generated spin accumulation in GaAs can be modeled using drift-di\u000busion\nequations21,23. We treat the channel as in\fnitely long since its length lis much larger than\nthe spin di\u000busion length Ls= 3:9\u0016m found from steady-state measurements at B= 0.\n4This assumption reduces the problem to only one spatial dimension and precludes the need\nfor general two-dimensional modeling including spin drift11. The SHE generates a spin\ncurrent transverse to the in-plane electric \feld E=E^xand proportional to the spin Hall\nconductivity \u001bSH,ji\nj=\u001bSH\u000fijkEk. The total current of the ispin component along yfrom\nboth di\u000busion and SHE is ji\ny=\u0000D@ysi\u0000\u001bSHE\u000eiz, whereD=L2\ns=\u001cis the spin di\u000busion\nconstant. Spin conservation at sample edges y=\u0006w=2 is enforced with hard-wall boundary\nconditions normal to the edge ( ji\ny=\u0000D@ysi\u0000\u001bSHE\u000eiz= 0). Di\u000busive boundary conditions\naccounting for spin-orbit e\u000bects at the sample edge (such as in the intrinsic SHE) would\nrequire modi\fcation for e\u000bects on the scale of the mean free path21. Assuming slow spin\ndecoherence (relative to momentum scattering) at rate 1 =\u001c, the spin polarization will obey\na continuity equation for s(y;t) including decay and precession terms:\n@si\n@t(y;t) =\u0000@ji\ny\n@y(y;t)\u0000si(y;t)\n\u001c+ (g\u0016B=\u0016h)B\u0002s(y;t) (1)\nWe \frst develop an intuitive picture of the time-dependent spin Hall processes using the\nexact solution to Eq. 1 under the simplest conditions. For B= 0, the components of s\nare uncoupled and Eq. 1 can be solved using a Green's function to obtain an in\fnite series\nsolution for sz(y;t). The di\u000busion equation for szcan be written as:\n@sz\n@t(y;t)\u0000D@2sz\n@y2(y;t) +sz\n\u001c(y;t) =F(y;t) (2)\nF(y;t)\u0011r\u0001 (\u001bSHE) =\u001bSH@E\n@y\nF(y;t) is a source function which contains all SHE terms. The homogeneous F= 0 Green's\nfunction for Eq. 2 is:\nG(y;y0;t;t0) =X\nmsin[kmy] sin[kmy0]e\u0000\u0015m(t\u0000t0)\u0002(t\u0000t0) (3)\nwheremis an integer, \u0002( t) is the Heaviside step function, km= (2m+ 1)\u0019=w and\u0015m=\n1=\u001c+k2\nmL2\ns=\u001c. For time-independent E, integrating Eq. 3 yields a series representation\nof the one-dimensional steady-state solution to the spin Hall di\u000busion equation.7To obtain\na time-dependent solution, we assume an ideal square electric \feld pulse of width tpand\namplitudeE0,E=E0[\u0002(y+w=2)\u0000\u0002(y\u0000w=2)][\u0002(t)\u0000\u0002(t\u0000tp)]. The corresponding source\nfunction is F(y;t) =\u001bSHE[\u000e(y+w=2)\u0000\u000e(y\u0000w=2)][(\u0002(t)\u0000\u0002(t\u0000tp)] and the solution is\n5found by integrating:\nsz(y;t) = 2=wZ1\n\u00001dy0Z1\n\u00001dt0G(y;y0;t;t0)\n=4\u001bSHE0\nwX\nm(\u00001)m\n\u0015msin[kmy]\u0002Tm(t)\nTm(t)\u00118\n>>>><\n>>>>:0; t< 0\n(1\u0000e\u0000\u0015mt); 0<t<tp\ne\u0000\u0015m(t\u0000tp)(1\u0000e\u0000\u0015mtp); t>tp(4)\nThe three regimes of the term-by-term time-dependence function Tm(t) in Eq. 4 can each\nbe observed in Fig. 2b. The lines in Fig. 3a, represent \fts of \u0012K(t) to Eq. 4 keeping the \frst\n200 terms in Eq. 4 and convoluting the solution with the Gaussian pro\fle of the laser spot.\nSinceLs= 3:9\u00060:2\u0016m was found independently from steady-state spatial measurements,\nthe only \ft parameters are \u001cand an overall amplitude scaling. The best \ft values for the\nparameter \u001care plotted in Fig. 3b and are signi\fcantly longer than the experimentally\nmeasured timescales \u001cacc,\u001c\u0003, and\u001c1=2.\nIn Fig. 3c, we plot \u0012K(t) and calculations from Eq. 4 convoluted with the laser pro\fle for\ny= 126, 124, 122, and 120 \u0016m forV0= 2 V using \u001c= 4:2 ns obtained from the earlier \fts.\nWe \fx the amplitude of the calculation from a \ft to y= 126\u0016m, and the remaining curves\nhave no free parameters. For yaway from the edge, \u0012K(t) does not grow exponentially in\ntand we cannot de\fne the time constant \u001caccas in Fig. 3a. Comparison of calculations\nfrom Eq. 4 with and without the spot size averaging reveals that the apparent asymmetry\nbetween the growth and decay times \u001caccand\u001c\u0003near the edge is primarily due to spatial\naveraging of these di\u000busion pro\fles over the Gaussian laser spot. The di\u000berence between \u001c\u0003\nand\u001cis real, however; dynamically measured spin polarization near the sample edge evolves\nwith a faster time constant than the underlying spin coherence time.\nWe can understand the fast evolution of spin polarization from the interplay of di\u000busion\nand spin decoherence. Since polarization gradients cause spins to di\u000buse away from the\nsample boundary, spin depolarization must occur faster than decoherence of the electrically-\ngenerated spins. These dynamics are captured in the di\u000busion analysis of Eq. 4 by the fast\ndecay rate \u0015mof terms with large min Eq. 4. Higher mterms are primarily responsible\nfor the discrepancy between the best \ft value for \u001cand the faster timescales \u001caccand\u001c\u0003\nobserved for spin accumulation and decay in Fig. 3b, but they only contribute signi\fcantly\n6toszneary=w=2 where all terms are in phase. In this boundary region, timescales should\ndi\u000ber most from the coherence time \u001c.\nWe numerically calculate @sz=@yaty= 126\u0016m from spatial scans in Fig. 3d. The\nspatial derivative of sz(y) is proportional to the di\u000busive spin current and is non-zero at the\nsample edge in the presence of a compensating spin Hall current for 0 < t < tp. After the\nspin Hall current disappears at t=tp,@sz=@yrelaxes with time constant \u001cj= 1\u00060:1 ns to\nsatisfy the di\u000busive jz\ny= 0 boundary condition (Fig. 3d, inset). Since V(t) evolves faster\nthan\u001c\u00001, we expect the di\u000busive spin current at the sample edge to relax faster as well.\nIn\fnitesimally close to the boundary, @sz=@yshould respond to changes in V(t) as fast as\nthe momentum scattering time. Taking into account the \fnite laser spot size, we calculate\n\u001cj= 1:15 ns from our model, in agreement with the measured value.\nIntroducing the magnetic \feld B=B^ycouples the spin components sxandsz. For\nthis regime, we perform numerical solutions to the system of coupled time-dependent linear\ndi\u000berential equations represented by Eq. 1 using the best \ft values for Lsand\u001cobtained\nfrom the earlier \feld independent analysis. For the numerical solutions, we use the exact\npulse pro\fle E(t) =V(t)=lmeasured from the oscilloscope (Fig. 2a) as a source. The curves\nin Fig. 2b-d and Fig. 2f-h are numerical calculations of \u0012K(t) and\u0012K(B). There are no free\nparameters except an overall scaling to match the amplitude of \u0012K. The full experimental\ndata set\u0012(t;B) aty= 126\u0016m fortp= 6 ns is shown in Fig. 4a. Figure 4b shows the full\ncalculation of spin accumulation from the numerical solution to Eq. 1 for y= 126\u0016m.\nThe agreement between the experiments and calculations demonstrates that a single ho-\nmogeneous decoherence time \u001ccaptures the various timescales observed in time-resolved\nmeasurements of the accumulation, decay, and di\u000busive dynamics of boundary spin polar-\nization due to the extrinsic spin Hall e\u000bect. While di\u000busive timescales are set by the spin\ncoherence time, evolution near sample boundaries can be limited by the faster response of\nthe spin current. This spatial dependence of timescales could prove helpful for utilizing\nelectrically-generated spin polarization in high-frequency semiconductor devices.\nMethods\nChannels of width wand length lare processed from a 2- \u0016m thick silicon-doped GaAs\nepilayer on 200 nm of undoped Al 0:4Ga 0:6As grown on a semi-insulating (001) GaAs substrate\n7by molecular beam epitaxy (Fig. 1b). The n-GaAs has doping density n= 1\u00021017cm\u00003\nand mobility \u0016= 3800 cm2=Vs atT= 30 K. The sample is mounted in a helium \row\ncryostat so that the channel ( x-direction) is perpendicular to the externally applied in-plane\nmagnetic \feld B(y-direction). A voltage V(t) applied across annealed Ni/Ge/Au/Ni/Au\nOhmic contacts creates an in-plane electric \feld E(t) =V(t)=lalongx. We desire an\nimpedance of\u001850 \n in order to deliver the maximum broadband electrical power to the\ndevice; choosing w= 256\u0016m andl= 130\u0016m yields a device with dc resistance R= 48 \natT= 30 K.\nTime resolution of SHE accumulation is achieved by electrically-pumped Kerr rotation\nmicroscopy using a mode-locked Ti:sapphire laser tuned to 1.51 eV that emits a 76-MHz\ntrain of\u0018150-fs pulses. The pulse repetition rate is reduced to 38 MHz by pulse picking\nwith an electro-optic modulator. Each laser pulse is divided into a trigger and a linearly\npolarized probe pulse. Spin polarization is generated at the sample edges by the SHE due\nto the current from a square electrical pulse of width tp, amplitude V0, and 0.8 ns rise\ntime applied to the sample from a pulse pattern generator triggered by the optical pump\npulse. The linearly polarized probe beam is focused through a microscope objective to a 1\n\u0016m spot on the surface of the sample which can be scanned with submicron resolution. A\nbalanced photodiode bridge measures the Kerr rotation of the linear polarization axis \u0012Kof\nthe re\rected beam which is proportional to the spin polarization along the z-axissz. The\nleading edge of the electric pulse pro\fle V(t) arrives at an electronically programmable delay\ntimetbefore the arrival of the optical pulse. All reported measurements are taken at the\ncenter of the length of the channel ( x= 0).\nAn absorptive RF switch alternates the center conductor of the coaxial cable between\nthe two complementary outputs of the pulse generator at frequency fV= 1:337 kHz. The\nRF switch only passes ac components of V(t), so the 0 V baseline is restored by adding\na square wave at frequency fVback onto the switched pulse train (Fig. 1c) resulting in a\nmodulation of the pulse amplitude + V0and\u0000V0at frequency fV.\u0012Kis then measured with\na lock-in ampli\fer analogous to the ac detection used in Refs. 7,10,11 but with a de\fnite\nphase relationship between electrical and optical pulses.\nThe electrical pulse induces a time-dependent re\rectivity modulation \u0001 R=R of the optical\nbeam during the pulse duration due to electron heating24,25that tracks the pro\fle V(t)\nmeasured by an oscilloscope (Fig. 2a). We use this e\u000bect to calibrate t= 0 and con\frm that\n8the device acts as a proper 50 \n termination due to the minimal temporal pulse distortion\nat the sample.\nAcknowledgments.\nWe thank NSF and ONR for \fnancial support. N.P.S. acknowledges the support of the\nFannie and John Hertz Foundation and S.M. acknowledges support through the NDSEG\nFellowship Program.\nThe authors declare no competing \fnancial interests.\n\u0003Correspondence and requests for materials should be addressed to D.D.A. (e-mail:\nawsch@physics.ucsb.edu).\n1Datta, S. and Das, B. Electronic analog of the electro-optic modulator. Appl. Phys. Lett. 56,\n665-667 (1990).\n2Wolf, S. A. et al. Spintronics: a spin-based electronics vision for the future. Science 294, 1488-\n1495 (2001).\n3\u0014Zuti\u0013 c, I., Fabian, J., and Das Sarma, S. Spintronics: Fundamentals and applications. Rev. Mod.\nPhys. 76, 323-410 (2004).\n4Kato, Y. K., Myers, R. C., Gossard, A. C. and Awschalom, D. D. Electrical initialization and\nmanipulation of electron spins in an L-shaped strained n-InGaAs channel. Appl. Phys. Lett. 87,\n022503 (2005).\n5D'yakonov, M. I. and Perel, V. I. Current-induced spin orientation of electrons in semiconduc-\ntors. Phys. Lett. 35A, 459-460 (1971).\n6Hirsch, J. E. Spin Hall e\u000bect. Phys. Rev. Lett. 83, 1834-1837 (1999).\n7Kato, Y. K., Myers, R. C., Gossard, A. C. and Awschalom, D. D. Observation of the spin Hall\ne\u000bect in semiconductors. Science 306, 1910-1913 (2004).\n98Wunderlich, J., Kaestner, B., Sinova, J. and Jungwirth, T. Experimental observation of the\nspin-Hall e\u000bect in a two-dimensional spin-orbit coupled semiconductor system. Phys. Rev. Lett.\n94, 047204 (2005).\n9Valenzuela, S. O. and Tinkham, M. Direct electronic measurement of the spin Hall e\u000bect. Nature\n442, 176-179 (2006).\n10Sih, V., Lau, W. H., Myers, R. C., Horowitz, V. R., Gossard, A. C., and Awschalom, D. D.\nGenerating spin currents in semiconductors with the spin Hall e\u000bect. Phys. Rev. Lett. 97, 096605\n(2005).\n11Stern, N. P., Steuerman, D. W., Mack, S., Gossard, A. C., and Awschalom, D. D. Drift and\ndi\u000busion of spins generated by the spin Hall e\u000bect. Appl. Phys. Lett. 91, 062109 (2007).\n12Murakami, S., Nagaosa, N. and Zhang, S. C. Dissipationless quantum spin current at room\ntemperature. Science 301, 1348-1351 (2003).\n13Sinova, J. et al. Universal intrinsic spin Hall e\u000bect. Phys. Rev. Lett. 92, 126603 (2004).\n14Engel, H.-A., Halperin, B. I., and Rashba, E. I. Theory of spin Hall conductivity in n-doped\nGaAs. Phys. Rev. Lett. 95, 166605 (2005).\n15Kato, Y. K., Myers, R. C., Gossard, A. C. and Awschalom, D. D. Current-induced spin polar-\nization in strained semiconductors. Phys. Rev. Lett. 93, 176601 (2004).\n16Crooker, S. A. et al. Imaging spin transport in lateral ferromagnet/semiconductor structures.\nScience 301, 2191-2195 (2005).\n17Awschalom, D. D., Halbout, J.-M., von Molnar, S., Siegrist, T., and Holtzberg, F. Dynamic\nspin organization in dilute magnetic systems. Phys. Rev. Lett. 55, 1128-1131 (1985);\n18Baumberg, J. J. et al. Ultrafast Faraday spectroscopy in magnetic semiconductor quantum\nstructures. Phys. Rev. B 50, 7689-7699 (1994).\n19Stephens, et al. Spatial imaging of magnetically patterned nuclear spins in GaAs. Phys. Rev. B\n68, 041307(R) (2003).\n20Yang, M. J., Wagner, R. J., Shanabrook, B. V., Waterman, J. R., and Moore, W. J. Spin-\nresolved cyclotron resonance in InAs quantum wells: a study of the energy-dependent gfactor.\nPhys. Rev. B 47, 6807-6810R (1993).\n21Engel, H.-A. Hanle e\u000bet near boundaries: Di\u000busion-induced lineshape inhomogeneity. Phys.\nRev. B 77, 125302 (2008).\n22Beck, M., Metzner, C., Malzer, S., and D ohler, G. H. Spin lifetimes and strain-controlled spin\n10precession of drifting electrons in GaAs. Eurphys. Lett. 75, 597-603 (2006).\n23Tse, W., Fabian, J., \u0014Zuti\u0013 c, I., and Das Sarma, S. Spin accumulation in the extrinsic spin Hall\ne\u000bect. Phys. Rev. B 72, 241303(R) (2005).\n24Batz, B. Re\rectance modulation at a Germanium surface. Solid State Commun. 4, 241-234\n(1966).\n25Berglund, C. N. Temperature-modulated optical absorption in semiconductors. J. Appl. Phys.\n37, 3019-3023 (1966).\n11Figures\n12a\nTi:sap p\nPhoto\nBrid\nBb\nB\nl\n100 μm\nPulse Gen e~150 fs\nBS\nEOM\n phireTrigger\nPbPD\nProbeSample RF \nswitc hPolarizing BS\nBSPD\nPD diode \ndge\nz \nc Si t h O t\nz\nxy V = 0c Switch Outp\n0ts)\nx\n(0 0) (0 126)FG Output\n0b. unit\nw(0, 0)E(0 ,126)\n \nV(t)0\n0V (arb\nw\nV(t)\n0 10V\n t (ms)0 1erator\nh\nFG\ntfV\nputFIG. 1: Experimental design for time-resolved measurement of electrically-generated\nspin polarization. a, Schematic diagram of time-resolved measurement of SHE (EOM, electro-\noptic modulator, BS, Beam splitter, PD, photodiode, FG, Function Generator). b,Optical micro-\nscope image of the sample with coordinate system de\fned (units in microns) showing the origin\n(black circle) and the location for most of the measurements (red circle). The bright yellow regions\nare the gold contacts and the grey region is the GaAs channel. cIllustration of the ac pulse scheme\nfor lock-in detection. The pure ac components of the switch output (red) is added to a square wave\natfVto create a triggered pulse train with amplitude modulation at fV\u00191 kHz.\n132t) (V) V(\n0\n6d)0\n6K(μrad\n0θK\n3ad)\n0K(μraθ\n1rad) 0θK(μr\n0θ\n0 a\nHz)0.5e\n∆R/Rscope trace\nL(GH \n   νL\n|g|= 0.346 ±-0.5\n \nB0Tb\nad)3 |g|\nf t = B= 0 mT\nK(μra\n  θK\n0\nB=2 0m Tc\nad)6 t = g B= 20 mT\nθK(μra\n  θ0\nB= 60 mTd  rad)3t = h \n θK(μr\n0\n10 20\nθ\n-200 0\nt(ns)            10            20 B(mT)200          0     0.002\n= 2 ns\n= 5 ns\n= 9 ns\n200\n)      200FIG. 2: Time-resolved measurement of the spin Hall accumulation. a, Voltage pulse\npro\fleV(t) measured by an oscilloscope (red line) and by re\rectivity modulation \u0001 R=R (black\ndiamonds). The arbitrary units of \u0001 R=R are scaled to match the scope voltage. The vertical\ndotted lines mark the time region 0 <t<tp.b - d, Representative scans of Kerr rotation \u0012(t) for\ntp= 6 ns and y= 126\u0016m. The blue lines are calculations from Eq. 1 with \u001c=4.2 ns,Ls= 3:9\n\u0016m, andy= 126\u0016m. (e)\u0017Lextracted from cosinusoidal \fts to \u0012K(t).f - h, Kerr rotation \u0012K(B)\nas a function of magnetic \feld at three representative times. The blue lines are calculations from\nEq. 1.\n14arad)8aθK(μr T-1)θ\n0  0.11(mT\n0B1/2-1\n0\nc\n6cμrad)6  θK(μ\n3\n0\n00b2 V\n1.5 Vb)6τ 1 V\n05V\ne (ns)6\n0.5 V\nTime\n4τ1/2τ* t=1 5n sτaccτ1/2\n0 10 20tp= 15 ns\n1 02acc\nt(ns) V0(V)\nd\n) \nt6\n)6d\n(a.u.)1 tp= 6 ns\nV0 =2  V126 μm\n(μrad6\n jz\ny\n0\n6 12124 μm\n θK(\n3t(ns)6 12\n7n s 122 μm\n7 ns\n120μm\ny(μm)1200\n t()0 6 12120 μm\ny(μm) t(ns)2\n5 ns\n9 ns\n11 ns\n12511 nsFIG. 3: Dynamics of spin Hall accumulation. a, Kerr rotation \u0012K(t) (upper panel), and\ninverse \feld width B\u00001\n1=2(lower panel) with tp= 15 ns at y= 126\u0016m forV0= 2:0 V (black),\n1.5 V (red), 1.0 V (blue), and 0.5 V (green). The solid lines are \fts from our model. b,The\naccumulation time \u001cacc(green triangles), Hanle time \u001c1=2(blue circles), decay time \u001c\u0003(black\nsquares), and coherence time \u001c(red diamonds) as function of voltage. c,\u0012K(t) fory= 126\u0016m\n(black), 124 \u0016m (red), 122 \u0016m (blue), and 120 \u0016m (green). The solid lines are calculations with\namplitude \fxed by a \ft at y= 126\u0016m.d,Spatially-resolved Kerr rotation \u0012K(y) near the edge\nfort= 5 ns (black), 7 ns (red), 9 ns (blue), and 11 ns (green). The inset shows the decay of\njz\ny/@sz(y)=@yextracted at y= 126\u0016m after the voltage pulse is complete ( tp= 6 ns).\n150.2\n (T)0\n B(0\n-0.2\nData a\n M b\n \n \nθK\n0             10           20\n 0             10      \n-4\nt(ns) t(ns)\nModel\nK(μrad)\n     20\n8FIG. 4: Modeling boundary accumulation from the spin Hall e\u000bect. a, Kerr rotation\n\u0012K(B;t) aty= 126\u0016m fortp= 6 ns. (b) Calculations of sz(B;t) from Eq. 1 with \u001c= 4:2 and\nLs= 3:9\u0016m aty= 126\u0016m.\n16"    },    {        "title": "1702.05330v2.Arbitrary_Nuclear_Spin_Gates_in_Diamond_Mediated_by_a_NV_center_Electron_Spin.pdf",        "content": "Arbitrary Nuclear Spin Gates in Diamond Mediated by a NV-center Electron Spin\nJ. Casanova, Z.-Y . Wang, and M. B. Plenio\nInstitute of Theoretical Physics and IQST, Albert-Einstein-Allee 11, Universit¨ at Ulm, D-89069 Ulm, Germany\nWe show that arbitrary N-qubit interactions among nuclear spins can be achieved e \u000eciently in solid state\nquantum platforms, such as nitrogen vacancy centers in diamond, by exerting control only on the electron\nspin coupled to the nuclei. This allows to exploit nuclear spins as robust quantum registers and the direct\nmeasurement of nuclear many-body correlators. The method takes advantage of recently introduced dynamical\ndecoupling techniques and avoids the necessity of external, slow, control on the nuclei. Our protocol is general,\nbeing applicable to other nuclear spin based platforms with electronic spin defects acting as mediators as silicon\ncarbide.\nI. INTRODUCTION\nNuclear spins in solid-state platforms such as diamond or\nsilicon carbide are natural and reliable quantum registers with\nexceptional coherence times [1]. In order to realise the po-\ntential of this resource for quantum computing [2], quantum\nsimulations [3], and quantum sensing [4, 5], it is necessary\nto achieve both individual nuclear spin rotations and arbitrary\ncoherent coupling between nuclear spins. In close proxim-\nity, nuclear spins exhibit a natural coupling because of their\nintrinsic nuclear dipolar interactions which may be exploited\nfor quantum simulations [6]. However, this natural coupling\nis generally small as a consequence of the weak nuclear mag-\nnetic moments [3], while the available closest internuclear dis-\ntances are always lower bounded by the lattice constants of the\nhost material. Furthermore, the spin active nuclear isotopes\nappear in the platforms of interest with a low natural abun-\ndance, e.g. 1 :1% and 4:7% for the cases of13C and29Si nuclei\nwhich are the relevant species for diamond [7, 8] and silicon\ncarbide [9, 10]. Furthermore, the positions of the nuclei are\nfixed which make the modulation of the internuclear coupling\nchallenging.\nThe tuning of the direct coupling of nuclear spins is ex-\ncessively demanding, although it can be suppressed (up to a\ncertain degree) by the application of suitably tuned radio fre-\nquency (rf) fields [6, 11, 12]. However, nuclear spins can be\ncoupled to external magnetic fields through Zeeman interac-\ntions [13], and more importantly, they can couple to nearby\nelectron spins. In this respect, the electron spin of a NV center\nis a promising nano-scale device to detect and control nuclear\nspins using the electron-nuclear hyperfine interaction [7, 8].\nThe large magnetic moment of electron spins allows a sin-\ngle NV center to couple strongly to many nuclear spins. In\naddition, with dynamical control achieved due to the applica-\ntion on the NV center of decoupling sequences such as Carr-\nPurcell-Meiboom-Gill (CPMG) [14, 15], pulse arrangements\nof the XY kind [16, 17], and adaptive XY sequences [18–20],\none can entangle the NV electron to individual nuclear spins\n(including the14N nucleus inherent to the NV center) for the\nsake of electron-nuclear two-qubit quantum gates and quan-\ntum algorithms [21–32].\nParticularly, the set of techniques developed in [18–20] al-\nlows for highly selective and robust electron-nuclear quan-\ntum gates evolving according to the Hamiltonians \u001bzIxor\u001bzIy\nby using sequences of non-equidistant microwave decouplingpulses. Note that \u001bz=jms=\u00061ihms\u00061j\u0000jms=0ihms=0j\ncorresponds to an e \u000bective electronic spin-1\n2operator [7, 8],\nwhile Ix,Iyare nuclear spin operators with I=1\n2. In addition,\nin [33] it is shown how the judicious application of a delay\nwindow achieves an interaction of the kind \u001bzIz. Further-\nmore, these techniques can incorporate a decoupling rf field\n[19, 20, 33] to combine electron-nuclear entangling gate gen-\neration with a suppression of the internuclear decoupling.\nIn this work we show that arbitrary single, and N-body nu-\nclear spin interactions can be realised e \u000eciently in the frame\nof electron spin defects with nearby nuclear spins through\na specific combination of selective electron-nuclear gates.\nThe latter can be achieved when the natural hyperfine cou-\nplings between the electronic- and nuclear-spins are appropri-\nately modulated with dynamical decoupling techniques. Our\nmethod combines the two key advantages of electron and nu-\nclei qubits, namely fast electronic control and the long nuclear\nspin coherence times. In addition, we demonstrate how the\nsame techniques allow to measure directly nuclear many-body\ncorrelators. To exemplify the protocol we use NV centers in\ndiamond as the model system, but our method is general and\ncan be used in other solid-state quantum platforms such as\nsilicon carbide.\nII. ELEMENTARY ENTANGLING GATES\nWith the dynamical decoupling techniques in [18–20, 33],\none can achieve highly selective entangling quantum gates of\nthe form\nQ\u000b\nj(')=exp\u0010\ni'\u001bzI\u000b\nj\u0011\n; (1)\nbetween the NV electron spin (with Pauli operator \u001bz) and\nthej-th nuclear spin (with the spin operator I\u000b\njand\u000b=x;y;z)\nwhile prolonging the electron spin coherence. Another impor-\ntant feature is that, as opposed to standard dynamical decou-\npling methods [14–17], the phase 'is fully tunable. We will\nsee later that this is a crucial requirement for designing our\nquantum algorithm. In addition, with microwave control, sin-\ngle qubit gates can be applied to the NV electron spin. These\nare, for instance, X\u001e=ei\u001e\u001bx\n2i.e. a rotation of an arbitrary\nand controllable phase \u001earound the xaxis, although any other\ndirection is available.arXiv:1702.05330v2  [quant-ph]  12 Sep 20172\nIII. PROTOCOL FOR N-BODY NUCLEAR\nINTERACTIONS\nThe elementary gates presented before permit the imple-\nmentation of N-body interactions (denoted in the following\nbyU\u001e) between the nuclear spins that can be individually ad-\ndressed by the NV center. In the last section we will demon-\nstrate numerically that the individual nuclear addressing is\npossible even when a certain target spin is surrounded by other\nnuclei interacting with the NV center. The latter is of great\nbenefit when dealing with dense samples because they con-\ntain a potentially large number of available nuclear qubits.\nFor example, by having the entangling gates Q\u000b1\nj1('1)Q\u000b2\nj2('2)\nat hand, one can demonstrate the following equality (up to an\nirrelevant global phase):\nU\u001e=Q\u000b1\nj1('1)Q\u000b2\nj2('2)X2\u001e+\u0019Q\u000b1\nj1('1)Q\u000b2\nj2('2)X\u0019\n=ei\u001e\u001bx(cos'1\u00002isin'1\u001bzI\u000b1\nj1)(cos'2\u00002isin'2\u001bzI\u000b2\nj2); (2)\nwhich contains two-qubit, and many-body (in this specific\ncase three-body) interactions involving the electron and the\nnuclear spins, see Appendix for a detailed derivation of\nEq. (2).\nWe want to note that an especially interesting situation is\nrealised when '1='2=\u0019\n2, i.e. when we are making interact\nwith the same phase the electron spin with di \u000berent nuclei. In\nthis case we have U\u001e=exph\n\u0000i4\u001e\u001bxI\u000b1\nj1I\u000b2\nj2i\nnamely a three-\nbody time evolution operator where the phase \u001ecorresponds\nto the one of the central gate, X2\u001e+\u0019, in the first line of Eq. (2).\nNow it is possible to generalise the results in Eq. (2) and\ndemonstrate that with the following sequence of gates\nU\u001e=QNX2\u001e+\u0019QNX\u0019; (3)\nwhere QN=QN\nn=1Q\u000bn\njn('n),Nlabels the total number of\naddressable nuclear spins, \u000bn=x;y;z, and by noting that\n[Q\u000bn\njn('n);Q\u000bm\njm('m)]=0 for n,m, i.e. the di \u000berent entan-\ngling gates commute, one can find that, for 'n=\u0019\n28n, and in\nthe case N=2M(i.e. we are subsequently addressing an even\nnumber of nuclear spins) the time evolution operator is\nUe\n\u001e=exp26666664(\u00001)Mi22M\u001e\u001b x2MY\nn=1I\u000bn\njn37777775: (4)\nIn the same manner, for N=2M+1 (an odd number of nuclear\nspins are addressed) we have\nUo\n\u001e=exp26666664(\u00001)(M+1)i22M+1\u001e\u001b y2M+1Y\nn=1I\u000bn\njn37777775: (5)\nNote that in both cases we have a ( N+1)-body evolution that\ninvolves the electron and nuclear spins.\nNow, if the Ue\n\u001eorUo\n\u001eoperators act on an initial state such\nthat, for the even case, we have \u001ae\nt=0=jx\u0006ihx\u0006j\n\u001aNwhere\n\u001bxjx\u0006i=\u0006jx\u0006i, while the initial state for the odd case is \u001ao\nt=0=jy\u0006ihy\u0006j\n\u001aNwith\u001byjy\u0006i=\u0006jy\u0006iand\u001aNrepresents an initial\nnuclear state, we have the following two possibilities\nUe\n\u001e\u001ae\nt=0(Ue\n\u001e)y=jx\u0006ihx\u0006j\n˜Ue\n\u0006\u001e\u001aN(˜Ue\n\u0006\u001e)y(6)\nand\nUo\n\u001e\u001ao\nt=0(Uo\n\u001e)y=jy\u0006ihy\u0006j\n˜Uo\n\u0006\u001e\u001aN(˜Uo\n\u0006\u001e)y; (7)\nwhere the N-body nuclear operators ˜Ue\n\u0006\u001e,˜Uo\n\u0006\u001eread\n˜Ue\n\u0006\u001e=exp26666664\u0000(1)Mi22M(\u0006\u001e)2MY\nn=1I\u000bn\njn37777775; (8)\nand\n˜Uo\n\u0006\u001e=exp26666664\u0000(1)(M+1)i22M+1(\u0006\u001e)2M+1Y\nn=1I\u000bn\njn37777775: (9)\nIt is also important to note that the gates X2\u001e+\u0019andX\u0019in\nEq. (3) can be replaced by Y2\u001e+\u0019andY\u0019where Y\u001e=ei\u001e\u001by\n2,\nor for any other gate that implies a rotation on the XY plane.\nThe latter would give rise to a set of results similar to the ones\nin Eqs. (4), (5), (8), and (9). Finally, note that, during gate\nperformance, the electron spin gets selectively coupled with\ndi\u000berent target nuclei of the sample but will also be a \u000bected\nby di \u000berent noise sources (see later for a description of the\ntypical error sources in the case of NV centers in diamond).\nTherefore, since the electron spin is the mediator of nuclear\ninteractions, its quantum state has to be protected against er-\nrors during the protocol which we achieve by means of dy-\nnamical decoupling techniques.\nIn this manner we have realised an e \u000bective N-body inter-\naction acting on a set of nuclei, while the electron spin gets un-\ncoupled after the process. We would like to note that Eqs. (8)\nand (9) allows one to couple distant nuclear spins and, remark-\nably, the achieved phase \u001ecan be easily extended without af-\nfecting significantly to the total time of the protocol. In this\nrespect note that \u001eis introduced through a single-qubit gate\non the electron spin that can be implemented in a time on the\norder of nanoseconds. Furthermore, an electron spin rotation\nto changejx\u0006i!j y\u0006iallows us to subsequently combine the\nfinal results in Eqs. (8), (9) in order to concatenate a set of\nN-qubit quantum gates upon di \u000berent nuclei.\nIn the same manner, and using again the NV center spin\nas the interaction mediator, a single entangling gate Q\u000bn\njn('n)\ncan be used to individually rotate any addressable spin by an\narbitrary phase. This is achieved by simply noting that\nQ\u000bn\njn('n)jz\u0006ihz\u0006j\n\u001aN(Q\u000bn\njn)y=jz\u0006ihz\u0006j\ne\u0006\u0010\ni'nI\u000bn\njn\u0011\n\u001aNe\u0007\u0010\ni'nI\u000bn\njn\u0011\n;\n(10)\nwhere\u001bzj\u0006zi=\u0006j\u0006 zi. Hence, single nuclear spin rotations\ncan be applied without having to introduce weak control rf\nfields that would require further calibration of the system.\nIn the end of the computing process a selective SWAP\ngate between the electron and a target nuclear spin allows\nto transfer the nuclear spin quantum state to the NV cen-\nter and reconstruct the nuclear spin state, or measure a3\nnuclear spin operator, by optical readout applied on the\nNV center which, at low temperatures, can achieve fideli-\nties exceeding 95% [32]. In this respect, we want to\nnote that a SWAP gate can be performed, up to a global\nphase, as SWAP e;jn=exp [ i\u0019\n2(\u001bzIz\njn+\u001bxIx\njn+\u001byIy\njn)]=\nexp [ i\u0019\n2\u001bzIz\njn] exp [ i\u0019\n2\u001bxIx\njn] exp [ i\u0019\n2\u001byIy\njn], where each of the\nprevious two-qubit gates can be realised with [18–20, 33]\nplus additional single-qubit gates on the electron spin.\nNote also that a selective iSWAP gate (with iSWAP e;jn=\nexp [ i\u0019\n2(\u001bxIx\njn+\u001byIy\njn)]=exp [ i\u0019\n2\u001bxIx\njn] exp [ i\u0019\n2\u001byIy\njn]) is also\nvalid to retrieve the nuclear state information.\nIV . MEASURING NUCLEAR MANY-BODY\nCORRELATIONS\nThe same techniques can be applied to directly measure\nthe expectation value of delocalised N-body nuclear opera-\ntors. This can be done through the next equality (for the sake\nof simplicity we develop here the case for Ue\n\u001ewhile the odd\ncase, i.e. the case for Uo\n\u001e, is similar). After a set of N-body\noperations we have that the final state is \u001a=\u001ae\n\u001an(t), with\n\u001ae=j\u000b\u0006ih\u000b\u0006jand\u000b=x;yorzdepending on the sequence\nof gates we performed. Now we can take, with an electron\nspin flip, the state \u001aeto an eigenstate of \u001bzi.e.\u001ae!j1ih1j\nwith\u001bzj1i=j1i, apply an additional gate Ue\n\u001eand measure, for\nexample, the \u001byelectronic operator. This process leads to\nh\u001byi=Tr\u0014\n\u001ae\n\u001an(t)(Ue\n\u001e)y\u001byUe\n\u001e\u0015\n=Tr\u0014\n\u001ae\n\u001an(t)e\u0002\n(\u00001)M(\u0000i) 22M(2\u001e)\u001bxQ2M\nn=1I\u000bn\njn\u0003\n\u001by\u0015\n:(11)\nNow, if we select the phase \u001esuch that (\u00001)M2\u001e=(\u0019\n2+m2\u0019)\nwith m2Z, we find thath\u001byi=Tr\u0014\n\u001ae\n\u001an(t)\u001bzQ2M\nn=1\u001b\u000bn\njn\u0015\nand one can write\nh\u001byi=Trn\u0014\n\u001an(t)2MY\nn=1\u001b\u000bn\njn\u0015\n=\u001c2MY\nn=1\u001b\u000bn\njn\u001d\n; (12)\nwhere Tr n\u0002\u0001\u0003denotes the trace over the nuclear degrees of\nfreedom. In this manner a highly delocalised nuclear operator\ngets encoded into an easy to measure electronic expectation\nvalue.\nV . IMPLEMENTATION WITH NV CENTERS\nIn order to demonstrate the working principle of our proto-\ncol we will consider diamond technologies, i.e. a NV center\nin the presence of a nuclear spin bath, as the target system to\nnumerically study our method. When a strong magnetic field\nBzis aligned with the NV axis, the ˆ zdirection, the Hamilto-\nnian of the coupled system in the rotating frame of the free\nenergy electronic spin Hamiltonian H0=DS2\nz\u0000\reBzSzreads\n(~=1)\nH=AkSz\u0000X\nj\rjBzIz\nj+SzX\nj~Aj\u0001~Ij+Hc: (13)Here, the first term AkSzwith Ak\u0019 \u0000 (2\u0019)\u00022:162 MHz,\nsee [34], and the spin-1 operator Sz=j1ih1j\u0000j\u0000 1ih\u00001j,\ncorresponds to the longitudinal component of the coupling\nwith the14N nucleus adjacent to the vacancy. In our simu-\nlations, instead of using the14N nucleus as a resource qubit,\nwe take it as the origin of a large detuning error with magni-\ntudejAkj[35] for demonstrating the robustness of our protocol.\nHowever if the14N is polarised at the beginning of the opera-\ntional process that detuning error is negligible. The constants\n\re=\u0000(2\u0019)\u00022:8024MHz\nGand\rj\u0011\r13C=(2\u0019)\u00021:0705kHz\nG\n8jrepresent the electronic and nuclear (in this case for the13C\nnucleus) gyromagnetic ratios.\nThe interaction between the NV center and the nuclear\nspins is mediated by the hyperfine vector that we will consider\nas dipolar like in the simulations, i.e. ~Aj=\u00160\re\rj\n4\u0019j~rjj3[ˆz\u00003(ˆz\u0001~rj)~rj\nj~rjj2],\nwhere~rjis the vector that connects the NV center and each\nenvironmental nuclei. Note also that, because of recently de-\nveloped positioning methods [19] we will take ~Ajas known\nquantities. The large zero field splitting D=(2\u0019)\u00022:87 GHz\nhad allowed us to neglect non-secular components in Eq. (13).\nFurthermore, an external microwave control can be introduced\nin Eq. (13) through the term Hc= \n(j1ih0jei#+j0ih1je\u0000i#) with\n\nbeing the Rabi frequency of the microwave field. In this\nmanner, we are selecting the electronic spin subspace j0i,j1i\nto define our qubit. In our numerical simulations we will addi-\ntionally introduce an error in \nfor considering realistic exper-\nimental conditions. More specifically, if the required time for\na\u0019-flip rotation of the electron spin is t\u0019=1\n4\n, we will e \u000bec-\ntively introduce a Rabi frequency \n(1+\u000f) with\u000fthe relative\nerror that we will set as 1% [36]. As we will see in the Ap-\npendix, we consider this error as constant because a realistic\nestimation for the correlation time of amplitude fluctuations\nfor microwave fields is \u00191ms [36], which is a large quan-\ntity when compared with the time to execute each individual\nunit of our dynamical decoupling sequence. For more details\nsee Appendix B (numerical simulations). Finally #is a phase\nthat, for the sake of robustness, remains constant during each\nmicrowave pulse but changes between pulses [18, 38, 39], see\nmore details in the Appendix B.\nHence, under the action of the microwave control pulses the\nfinal Hamiltonian is\nH=\u0000X\nj!jˆ!j\u0001~Ij+F(t)\u001bzX\nj~Aj\u0001~Ij; (14)\nwhere\u001bz=j1ih1j\u0000j0ih0j,F(t)=\u00061 is the modulation\nfunction appearing because of the action of \u0019-pulses upon\nthe electron spin, and ~!j=\r13CBzˆz\u00001\n2~Ajwith!j=j~!jj\nand ˆ!j=~!j=j~!jj. In this ideal description the detuning term\nAkSzhas been eliminated because of the external microwave\ndriving, however our simulations will incorporate this detun-\ning term since they are performed assuming the Schr ¨odinger\nequation associated to the Hamiltonian in Eq. (13). In addi-\ntion, we will not consider electron relaxation processes be-\ncause, at low temperatures around 4 K, measurements of the\nrelaxation time ( T1) on the order of seconds have already been\nreported [31, 40] which largely exceeds the time for perform-\ning entangling nuclear gates. It is noteworthy that, in the case4\nh\u0000z1⌦\u0000z2⌦\u0000z3i\u0000h\u0000z1i\u000011\u000002⇡4⇡b)a)\nFIG. 1: Expectation value of a) the \u001bz\n1operator. Solid line, ideal re-\nsult according to the propagator in the second line of Eq. (15) while\nthe squares correspond to the result when the sequence of gates in\nthe first line of Eq. (15) is applied. b) Expectation value of the de-\nlocalised operator \u001bz\n1\n\u001bz\n2\n\u001bz\n3. The solid line represents the ideal\nevolution while diamonds are the result of applying our method. All\nthe points in the plot (squares and diamonds) correspond to the ap-\nplication of 3202 imperfect microwave pulses with a \u0019pulse time of\n12:5 ns, and a total evolution time of \u00190:7 ms to generate the propa-\ngator in Eq. (15). Note that this value is independent of the achieved\nphase\u001e.\nof [31] NV coherence times larger than 25 ms are achieved\nin high purity IIa-type diamond sample at 4.2 K. Other ex-\nperiments, as the one in [37], report NV T2times of\u00190:6\nseconds at 77 K and, again, in samples with low nitrogen con-\ncentration. In this manner, and according to the previously\ncommented experimental results, we will not consider the NV-\nelectron coupling because of P1centers in the diamond lattice.\nVI. NUMERICAL RESULTS\nWe have numerically simulated the three-body nuclear time\nevolution operator exp ( i23\u001eI1\nxI2\nxI3\nx), that gives rise to maxi-\nmally entangled GHZ-like states [41] when the nuclear reg-\nister is initialized into the state j ### i . The latter can\nbe prepared by polarisation transfer from the electron spin,\nsee for example [29–31], or by dynamical nuclear polarisa-\ntion (DNP) [42–44]. The three-body nuclear propagator can\nbe achieved by applying the sequence in Eq. (3) with QN=Q3\nn=1Q\u000bn\njn('n) where\u000b1=\u000b2=\u000b3=xand'1='2='3=\u0019\n2.\nMore specifically, this is\nU\u001e=Qx\n1\u0012\u0019\n2\u0013\nQx\n2\u0012\u0019\n2\u0013\nQx\n3\u0012\u0019\n2\u0013\nX2\u001e+\u0019Qx\n1\u0012\u0019\n2\u0013\nQx\n2\u0012\u0019\n2\u0013\nQx\n3\u0012\u0019\n2\u0013\nX\u0019\n=\u0000exp\u0012\ni23\u001e\u001b yIx\n1Ix\n2Ix\n3\u0013\n=\u0000exp\u0012\ni\u001e\u001b y\u001bx\n1\u001bx\n2\u001bx\n3\u0013\n:(15)\nWe selected a three qubit nuclear register such that ~A1=\n(2\u0019)\u0002(\u000056;\u000032;\u000045) kHz,~A2=(2\u0019)\u0002(\u00007:6;39;52) kHz,\n~A3=(2\u0019)\u0002(\u000022;13;96) kHz, all of them corresponding to\nnuclei located in available positions of the diamond lattice,\nand the static magnetic field is Bz=0:65 T, for more de-\ntails see Appendix. Furthermore, and although not included\nin our theoretical description in Eqs. (13) and (14), we have\nalso taken into account in the simulations the e \u000bect of inter-\nnuclear interactions. These produce a coupling between the\ni-th and j-th nuclei of the form gi;j=~\u00160\r2\n13C\n2d3\ni;j[1\u00003(nz\ni;j)2] withdi;jthe relative distance between nuclei and nz\ni;jthe amplitude\nof the projection in ˆ zon their relative positioning vectors. In\nour case we have g1;2\u0019\u0000(2\u0019)\u000220 Hz, g1;3\u0019\u0000(2\u0019)\u000210\nHz, and g2;3\u0019(2\u0019)\u00027:5 Hz. Figure 1 shows the evolution of\nthe expectation value of a single nucleus \u001bz\n1and of the delo-\ncalised operator \u001bz\n1\n\u001bz\n2\n\u001bz\n3. The solid line corresponds to\nthe ideal behavior while squares and diamonds represent the\nresult when our method is applied. Furthermore, we computed\nthat the fidelity for the creation of a three-qubit GHZ-like nu-\nclear state of the form j\ti=exp ( i\u0019\n4\u001by\u001bx\n1\u001bx\n2\u001bx\n3)jy+ij###i =\njy+i1p\n2(j###i +ij\"\"\"i ) is 98:8%. This state has been prepared\nemploying the same number of imperfect pulses, 3202, than\nthe one used for Fig 1. In addition, in Appendix B (Numerical\nsimulations) we have included a plot that presents the impact\nof pulse-phase inaccuracy in our method.\nVII. FURTHER APPLICATIONS\nThe generation of these kind of gates, single- and N-qubit,\nallows to deal with problems involving fermionic interactions.\nIt is known that any creation or annihilation fermionic opera-\ntor admits a form in terms of tensorial products of Pauli ma-\ntrices when the Jordan-Wigner transformation is applied [45].\nHence, an appropriate application of our techniques would\nbe of benefit to implement dynamics that include interacting\nfermions, e.g. quantum chemistry problems, in a solid-state\nquantum platform. Furthermore, the access to arbitrary multi-\nqubit spin propagators is of interest for simulating spin models\nwith topological order [46], as well as to generate dynamics\nand to perform measurements in di \u000berent models of quantum\ncomputing as the case of deterministic quantum computation\nwith one quantum bit, the DQC1 protocol, that do not require\nto initially polarise the nuclear register [47, 48].\nVIII. CONCLUSIONS\nWe presented a protocol that allows the generation of single\nandN-qubit quantum gates between nuclear spins in a solid\nstate register such as diamond, as well as to measure highly\ndelocalised nuclear spin correlators. These gates are medi-\nated by an e \u000bective electron spin, for example a NV center,\nexternally controlled with microwave radiation in a dynam-\nical decoupling scheme to assure selective entangling gates\nand electron spin state protection. The method is general and,\ntherefore, applicable to other lattice defects as silicon carbide\nor germanium vacancy centers.\nIX. ACKNOWLEDGEMENTS\nThe authors acknowledge J. F. Haase, T. Theurer, and R.\nPuebla for their useful comments on the manuscript. This\nwork was supported by the Alexander von Humboldt Foun-\ndation, the ERC Synergy grant BioQ, the EU projects DI-\nADEMS, EQUAM and HYPERDIAMOND as well as the5\nYX#x1#x2#x3#x4#x5#y5#y4#y3#y2#y1t1t2t1t2t1t1t2t2\nFIG. 2: Internal structure of each X or Y composite pulses in terms\nof the 5 elementary \u0019-pulses. These \u0019-pulses are arranged symmetri-\ncally, see the tunable time distances t1andt2that distribute the pulses,\nwith respect to the central \u0019-pulse.\nDFG via the SFB TRR /21 and the SPP 1601. J. C. acknowl-\nedges Universit ¨at Ulm for a Forschungsbonus.\nX. APPENDIX\nA. Electron-nuclei many body gate\nHere we show how to derive Eq. (2).\nU\u001e=Q\u000b1\nj1('1)Q\u000b2\nj2('2)X2\u001e+\u0019Q\u000b1\nj1('1)Q\u000b2\nj2('2)X\u0019\n=\u0014\nexp ( i'1\u001bzI\u000b1\nj1) exp ( i'2\u001bzI\u000b2\nj2)\u0015\nexp ( i\u001e\u001bx)\ni\u001bx\u0014\nexp ( i'1\u001bzI\u000b1\nj1) exp ( i'1\u001bzI\u000b2\nj2)\u0015\ni\u001bx\n=ei\u0019exp\u0002i\u001ee(i'1\u001bzI\u000b1\nj1)e(i'2\u001bzI\u000b2\nj2)\u001bxe(\u0000i'1\u001bzI\u000b1\nj1)e(\u0000i'2\u001bzI\u000b2\nj2)\u0003\n=ei\u0019exp\u0002i\u001e\u001b xe(\u0000i2'1\u001bzI\u000b1\nj1)e(\u0000i2'2\u001bzI\u000b2\nj2)\u0003\n=ei\u0019exp\u001a\ni\u001e\u001b x[cos ('1)\u00002isin ('1)\u001bzI\u000b1\nj1]\n\u0002[cos ('2)\u00002isin ('2)\u001bzI\u000b2\nj2]\u001b\n: (16)\nNow, if the global phase factor ei\u0019is neglected, we get the\nresult at Eq. (2). Note that we have used Q\u000b\nj(')=exp\u0010\ni'\u001bzI\u000b\nj\u0011\nandX\u001e=ei\u001e\u001bx\n2in concordance with the definitions in the main\ntext.\nB. Numerical simulations\nTo obtain Fig. [1], we chose a electron nuclear configura-\ntion such that the hyperfine vectors for the three13C nucleiare\n~A1=(2\u0019)\u0002(\u000056;\u000032;\u000045) kHz;\n~A2=(2\u0019)\u0002(\u00007:6;39;52) kHz;\n~A3=(2\u0019)\u0002(\u000022;13;96) kHz: (17)\nThe static Bzfield is aligned with the NV axis (the zaxis)\nand has a value of 0 :65 T. We drive the electron spin with mi-\ncrowave pulses in the form of top-hat functions with a \u0019-pulse\ntime of 12:5 ns. The microwave sequence is made of three\ndi\u000berent steps, one for each of the operations Qx\n1(\u0019\n2),Qx\n2(\u0019\n2)\nandQx\n3(\u0019\n2), driven by an appropriate dynamical decoupling\nsequence. Note that each of these steps is repeated twice be-\ncause the gates Qx\n1(\u0019\n2),Qx\n2(\u0019\n2) and Qx\n3(\u0019\n2) appear in front and\nbehind the central gate X2\u001e+\u0019in the first line of Eq. (15) in the\nmain text.\nTo implement each step, we use repetitively several AXY-\n8 blocks [18] where each block has the following struc-\nture XYXYYXYX with X (or Y) being a composite pulse\ncontaining 5 \u0019-pulses, see Fig. 2. In addition one should\nnote that each \u0019-pulse is applied along an axis in the x-y\nplane determined by the phase #x;y\nj. This can be seen not-\ning that each \u0019-pulse is generated through the Hamiltonian\nHc= \n(j1ih0jei#+j0ih1je\u0000i#). To assure robustness, see [18],\nwe set these phases as #x\n1=\u0019=6,#x\n2=0,#x\n3=\u0019=2,#x\n4=0,\nand#x\n5=\u0019=6, while the #y\njare shifted by an amount \u0019=2 with\nrespect to#x\nj. That is#y\nj=#x\nj+\u0019=2 The gate Qx\n1(\u0019\n2) required\n\u001969\u0016sto be displayed (we have used the 11-th harmonic of\nthe decoupling sequence and 440 microwave pulses, i.e. 88\nrobust composite pulses. The other gates Qx\n2(\u0019\n2) and Qx\n3(\u0019\n2) are\nimplemented in\u0019107\u0016s(440 microwave pulses, i.e. 88 ro-\nbust composite pulses) and \u0019177\u0016s(720 microwave pulses,\ni.e. 144 robust composite pulses) respectively by making use\nof the 17-th harmonic in both cases. Each block has a distinct\ninterpulse spacing to assure that the final achieved phase for\neach of the the Qx\njgates is\u0019=2. In addition, one can calculate\nthat the largest time to execute an AXY-8 block is \u00199:8\u0016s,\nwhich corresponds to the case of the Qx\n3(\u0019\n2) gate. One can\n00.20.40.60.811.20.930.940.950.960.970.980.99\n✓00.20.40.60.81.01.20.930.950.970.99F\nFIG. 3: Fidelity of state preparation, F, under the presence of a\nrandom error\u0006\u0012(in degrees) in the pulse-phases #x;y\njsee Fig. 2. We\nobserve a fidelity decrease for larger values of \u0012. Each point in the\nplot has been taken by averaging 100 runs of the scheme in Eq. 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Hammel1;\u0003\nMay 26, 2022\nSpintransportelectronics–spintronics–focuses\non utilizing electron spin as a state variable for\nquantum and classical information processing and\nstorage [1]. Some insulating materials, such as di-\namond, offer defect centers whose associated spins\nare well-isolated from their environment giving\nthem long coherence times [2–4]; however, spin\ninteractions are important for transport [5], en-\ntanglement [6], and read-out [7]. Here, we report\ndirect measurement of pure spin transport – free\nof any charge motion – within a nanoscale quasi\n1D ‘spin wire’, and find a spin diffusion length\n\u0018700nm. We exploit the statistical fluctuations\nof a small number of spins [8] (p\nN < 100net spins)\nwhich are in thermal equilibrium and have no im-\nposed polarization gradient. The spin transport\nproceeds by means of magnetic dipole interactions\nthat induce flip-flop transitions [9], a mechanism\nthat can enable highly efficient, even reversible\n[10], pure spin currents. To further study the dy-\nnamics within the spin wire, we implement a mag-\nnetic resonance protocol that improves spatial res-\nolution and provides nanoscale spectroscopic in-\nformation which confirms the observed spin trans-\nport. This spectroscopic tool opens a potential\nroute for spatially encoding spin information in\nlong-lived nuclear spin states. Our measurements\nprobe intrinsic spin dynamics at the nanometre\nscale, providing detailed insight needed for prac-\ntical devices which seek to control spin.\nyThese authors contributed equally to this work\n1Department of Physics, Ohio State University, Columbus,\nOhio 43210, USA.\n\u0003Corresponding Author: hammel@physics.osu.eduUnderstanding and controlling spin transport is a cen-\ntralchallengeinspintronicsandhasbeenextensivelystud-\nied in systems where the spin density is large and treated\nas a continuum variable [11]. However, an understand-\ning at the few spin level is needed for miniaturization and\nquantum applications. Appropriate and well-controlled\ncoupling between spins can allow for efficient spin trans-\nport, while preserving long spin lifetimes. For example,\ntwo-spin flip-flop transitions – the simultaneous flipping\nof a pair of anti-aligned neighbouring spins due to their\ndipolar interaction – conserve magnetization, and succes-\nsive flip-flops can result in a pure, diffusive spin transport\n[9,12], as reported here. In contrast to non-equilibrium\nexperiments [13,14] which measure thermal polarization\nrecovery, we monitor the spin noise [8,15], which arises\nfrom statistical fluctuations. This allows us to observe\nthe intrinsic thermal equilibrium spin dynamics of the en-\nsemble [16].\nIn macroscopic systems containing a large number of\nparticles, Fick’s law of diffusion describes the time evolu-\ntion of particles moving from regions of high to low con-\ncentration, resultinginasmoothevolutiontowardequilib-\nrium. By contrast, in few-spin ensembles fluctuations can\nleadtotransientpolarizationsmuchlargerthantheBoltz-\nmann (thermal) polarization, and even flow “against” the\npolarization gradient [17]. Our measurements focus not\non the polarization itself, but on the time auto-correlation\nof the polarization in a nanoscale spin detection volume.\nIn our experiments, these correlations are lost as a con-\nsequence of spin transport out of our measurement vol-\nume. We model this process using numerical simulations\nthat incorporate this inherent randomness of few-spin,\nstatistically-polarized ensembles, and our model corrob-\norates our measurements.\nIn order to study flip-flop-dominated spin dynamics, we\n1arXiv:1309.3199v3  [cond-mat.mes-hall]  8 Jan 2014Figure 1: MRFM spin wire setup, schematic of spin dynamics in wire, and spin noise measurements .a,\nMRFM setup for measuring spin noise in the 200 nm \u0002250 nm\u00024\u0016m spin wire of P1 centres in diamond. The coil\nexcites resonance along a contour of the magnetic particle’s field (plus an external field), called the ‘resonant slice’.\nAs the cantilever oscillates, the field due to the magnetic particle varies and the slice sweeps out the spin detection\nvolume. Spins toward the centre of the detection volume contribute to the spin signal more strongly (solid red) than\nthose near the edge. b-e, Schematic showing the detection volume as it is walked into the spin wire, the rectangular\nstripe region. The thick double arrows represent flip-flop-mediated spin diffusion in and out of the detection volume.\nf, The spin signal increases as a function of walk-in distance until the volume is fully inside the spin wire as in ( e).\ng, The correlation time of the measured spins. At first (as in b) spins are able to quickly diffuse out of the volume,\nbut as the measurement volume is walked in further (as in c), spins must diffuse a longer distance to exit the volume,\ncausing the correlation time to increase. Once the measurement volume is fully inside the stripe ( dande), spins can\ndiffuse out on both sides, decreasing the correlation time to roughly half the peak value.\n2fabricated a nanoscale channel of implanted nitrogen (P1)\ncentres in diamond at a density of 6 ppm. At this den-\nsity the strength of the dipolar coupling between spins is\nstrong, leading to a flip-flop time, T\u000b\u001810\u00004s, several\norders of magnitude shorter than T1\u00181s. The density\noutside the spin wire ( 0:3ppm) is sufficiently small such\nthat flip-flop transitions out of the wire are very rare, thus\nconfining the spin transport to the wire.\nWe use magnetic resonance force microscopy (MRFM)\nto probe the spins within the wire (Fig. 1a). The force\ndetector is an IBM-style ultra-soft cantilever [18] with a\nSmCo 5magnetic particle glued onto the tip. To reso-\nnantly detect spins, we implement the iOSCAR timing\nprotocol [8,19]. A superconducting niobium coil is used\nto drive the spin resonance at a frequency of !rf= 2:18\nGHz. This frequency defines a ‘resonant slice’ where the\ntotal magnetic field (tip field plus external field) experi-\nenced by sample spins is equal to!rf\n\r= 778G. By driving\nthe cantilever at its self-determined resonant frequency to\nan amplitude of 150nm, the resonant slice sweeps out a\nvolume of this width, hereafter termed the detection vol-\nume.\nWe scan the detection volume into the spin wire (Fig.\n1b-e) and measure the force exerted on the cantilever by\nthe selected spins. From this measurement we can ex-\ntract two quantities: spin signal (Fig. 1f, top), and the\nforce correlation time \u001cm(Fig. 1f, bottom). The spin sig-\nnal is obtained from the variance in the time record of\nthe force signal and is directly related to the number of\nmeasured spins [19]. \u001cmdescribes the characteristic time\nfor the net moment of the detected spins to decorrelate.\nThis can be viewed as the time needed to transport spin\nout of the detection volume via a random-walk diffusion\nprocesses, with each step taking an average time T\u000b. Once\nthedetectionvolumeentersthespinwire, thesignalgrows\nwith the number of detected spins, eventually reaching a\nplateau when completely within the wire. The correlation\ntime shows a complex behaviour as the detection volume\nmoves into the wire. These results can be understood\nwithin the framework of flip-flop-mediated spin transport:\nensemble correlation is lost as spins diffuse into or out of\nthe detection volume. Just inside a displacement of 0 nm\n(Fig. 1b), the spins inside the volume easily and rapidly\ninteract with nearby outside neighbours, resulting in a\nrelatively short correlation time. With increasing over-\nlap of the detection volume and the spin wire, spins mustdiffuse further to exit the detection volume and change\nthe overall magnetization of the ensemble, thus increasing\n\u001cm. The correlation time peaks when approximately 68%\nof the detection volume has entered the spin wire (Fig.\n1c). This is attributed to the force sensitivity profile of\nthe detection volume: spins in the middle of the volume\nare most sensitively detected (solid red indicates highest\nsensitivity in Fig. 1), while the edges are least sensitive\n(see SI). As the detection volume approaches 100%over-\nlap (Fig. 1d), spins are able to diffuse out either side of\nthe most sensitive region, reducing the correlation time\nby roughly a factor of 2. Scanning deeper into the spin\nwire results in no further change, since the measurement\ngeometry becomes translationally invariant (Fig. 1e).\nA global measurement of the spin polarization, en-\ncompassing the entire spin wire, would not uncover this\nmagnetization-preserving spin transport process. In the\nopposite limit of measuring a single spin, T1-type spin re-\nlaxation becomes indistinguishable from flip-flop induced\npolarization changes. By systematically varying the size\nof the overlap of the detection volume with the spin wire,\nand hence the number of detected spins, our measure-\nment of\u001cmreveals the observed signature of spin trans-\nport. This is similar to measurements of spin lifetime by\nelectrically- or optically-detected Hanle signals [11,20,21],\nwhere diffusion of spins away from the detection volume\n(electricalcontactoropticalprobespot)caninfluencespin\ndephasing.\nSince the average polarization gradient vanishes at\nthermal equilibrium, the conventional model of diffusion\ndrivenbypolarizationgradientpredictsthat \u001cmwillbein-\ndependent of detection volume position within our ther-\nmal equilibrium wire. Such a model neglects the domi-\nnant role spin fluctuations play in nanoscale ensembles.\nIn order to fit the measured data in Fig. 1f, we use a\nMonte Carlo simulation which models the flip-flops be-\ntween individual spins as a Markov process (see SI). From\nthis fit we can extract the flip-flop time, T\u000b= 0:21ms,\nand the corresponding diffusion constant of D=a2\nTff=\n4:6\u000210\u00009cm2\ns, wherea=n\u00001=3= 9:82nm is the aver-\nage nearest-neighbour separation of spins with a density\nn= 1:06\u00021018cm\u00003(6 ppm). We find good agree-\nment comparing this to the theoretical flip-flop time [12]\nT\u000b;th= 30T2= 0:36ms, where we have used T 2= 11:9\n\u0016s, which arises from dipolar interactions (at a density\nof 6 ppm) and has been experimentally verified [22]. We\n3Figure 2: Hyperfine spectrum measurement for spin wire .a, Energy level diagrams for three hyperfine-split\nspin populations, which are simultaneously on resonance, yet spatially separated by the gradient. b, Schematic of the\nthree volumes which are walked into the spin wire. c, The spin signal increases in a step-like fashion as each volume\nenters the wire. d, A nanovolume EPR spectrum can be obtained by deconvolving the spin signal with the known\nforce sensitivity (see SI) and the three hyperfine peaks are resolved.\n4furthermore find a diffusion length L=pDT1= 720nm,\nwhich is significantly larger than the lateral dimensions\nof the wire (depth 250nm, width 200nm, and length 4\n\u0016m). This diffusion length is competitive with metallic\nspin transport devices [23].\nThe above expression for T\u000b;thassumes zero magnetic\nfield gradient for the system, but in our experiment we\nhave a strong gradient ( \u00181:3G/nm). In the presence\nof a field gradient, neighbouring spins experience a field\ndifference \u0001B, and thus different Zeeman energy split-\ntings. This can suppress flip-flops because they no longer\nconserve energy [24–26]. However, inhomogeneous line\nbroadening, if on the order of \u0001B, can make up this en-\nergy difference and allow flip-flops to proceed. Since the\nmeasured flip-flop time T\u000bagrees closely to the expected\nzero-gradient value T\u000b;th, we expect a spectral linewidth\n\u0001B&6:5G.\nTo measure the EPR spectrum of the spin wire, we im-\nplemented a modified iOSCAR protocol which improves\nspatial resolution and therefore provides the capability to\nresolve spectral features (since spatial and spectral resolu-\ntion are directly related in magnetic resonance imaging).\nA conventional iOSCAR measurement utilizes the entire\ndetection volume swept out by the cantilever oscillation,\nas this couples to the largest number of spins and cre-\nates the largest force signal. By truncating the detection\nvolume to the most-sensitive portion, we couple only to\nspins in this smaller region while maintaining high force\nsensitivity. Utilizing this technique (further details in SI),\nwhich we call partially-interrupted OSCAR (piOSCAR),\nwe again scan the detection volume into the spin wire.\nThe improved resolution enables us to resolve a staircase\nstructure with three steps (Fig. 2c), corresponding to the\nthree peaks of the P1 center’s hyperfine spectrum as dis-\ncussed below.\nThe P1 centre is a substitutional nitrogen defect with\nan unpaired spin-1\n2electron hyperfine coupled to the spin-\n1nitrogen nucleus. The hyperfine interaction results in a\ntriplet of peaks in the EPR spectrum (Fig. 2a) corre-\nsponding to the three nuclear spin states, mI=\u00001;0;1.\nThe hyperfine splitting for our diamond crystal orienta-\ntion relative to the external field is 33 G (see SI and\nref. [27]). In the magnetic field gradient of our probe mag-\nnet,thissplittingresultsinthreespatiallydistinctspinde-\ntection volumes, with each volume defined by the contour\nof magnetic field which satisfies the resonance conditionfor a particular hyperfine transition (Fig. 2b). We ob-\nserve a step-like increase in the spin signal as each volume\nenters the wire and the cantilever becomes coupled to an\nadditional hyperfine transition (Fig. 2c). The spacing be-\ntween steps (s = 25 nm) provides an accurate means of\nmeasuring the probe field gradient, because the spacing is\nset by the ratio of the known hyperfine splitting ( 33G)\nto the gradient, which we find to be 1:3Gnm. This is con-\nsistent with our estimations of gradient obtained using a\nstandard technique [28]. The staircase structure in figure\n2c is a convolution of the implanted spin density (approx-\nimately a step function), our force sensitivity profile, and\nthe EPR spectrum of spins in the sample; deconvolution\nrevealstheEPRspectrumshowninFig. 2d(detailsinSI).\nFrom the spectrum we find an average linewidth of 7:6G,\nwhich agrees with the expected linewidth from above, and\ncorroborates the transport we measure in the spin wire.\nIn conclusion, we have measured flip-flop-mediated spin\ntransport in an insulating diamond spin wire in the com-\nplete absence of charge transport. This transport arises\nfrom the intrinsic spin dynamics of dipole-coupled P1 cen-\ntres at thermal equilibrium, which we observe by measur-\ning spin noise without imposing a polarization gradient.\nBy spatially resolving the three electron spin populations\n(corresponding to the three spin states of the hyperfine-\ncoupled nitrogen nucleus) we obtained the EPR spectrum\nof less than 100net spins in the spin wire. The resulting\nlinewidth confirms that flip-flop mediated spin diffusion\nis responsible for the observed spin transport. The spec-\ntrum also enables accurate measurement of the applied\nmagnetic field gradient. These measurements provide in-\nsight into the mechanisms of spin transport relevant for\nthe development of nanoscale spin device elements.\nMethods The ultra-soft silicon cantilever has a spring\nconstant of 100\u0016N/m, resonance frequency of about 3\nkHz, and approximate dimensions of 90microns in length,\n1micron width, and 100nm thickness. Displacement of\nthe cantilever is measured through a 1550nm laser inter-\nferometer. The MRFM measurements were taken at tem-\nperatures of 4:2K. The measured moment of the SmCo 5\nmagnetic particle was 3:6\u000210\u000012J/T. The niobium res-\nonator coil had about 2:5turns and a 300micron diam-\neter, and the resonance bowl had a diameter of about\n3microns. The diamond sample studied in this experi-\nment had a background nitrogen concentration of 0:3ppm\n(5:27\u00021016cm\u00003). To create the high spin-density wire,\n5the sample was first prepared using electron beam pat-\nterning and then exposed to nitrogen ion implantation at\na variety of energies to create a uniform spin density. Fi-\nnally the sample was annealed to yield an approximately\nuniform, channel ( 6ppm or 1:06\u00021018cm\u00003) with width,\ndepth, and length of 200nm,250nm, and 4microns, re-\nspectively. The sample was purchased commercially from\nElement Six, grown via chemical vapor deposition, and\nthe nitrogen ion implantation was performed by Leonard\nKroko Inc.\nAcknowledgements The research presented in this\nmanuscript was financially supported by the Army Re-\nsearch Office (grant number W911NF-09-1-0147), the Na-\ntional Science Foundation (grant number DMR-0807093),\nand the Center for Emergent Materials (CEM), an NSF\nfunded MRSEC (grant number DMR-0820414). Techni-\ncal support was provided by the NanoSystems Laboratory\nat the Ohio State University. The authors would like to\nthank R. J. Furnshahl for valuable discussions related to\nthe simulations.\nReferences\n[1] Wolf, S. et al.Spintronics: A spin-based electronics\nvision for the future. Science294, 1488–1495 (2001).\n[2] Hanson, R., Dobrovitski, V. V., Feiguin, A. E.,\nGywat, O. & Awschalom, D. D. Coherent dy-\nnamics of a single spin interacting with an ad-\njustable spin bath. 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Physical review letters 87,\n277602 (2001).\n7"    },    {        "title": "1204.0619v1.Spin_Turbulence_in_a_Trapped_Spin_1_Spinor_Bose__Einstein_Condensate.pdf",        "content": "arXiv:1204.0619v1  [cond-mat.quant-gas]  3 Apr 2012APS/123-QED\nSpin Turbulence in a Trapped Spin-1 Spinor Bose–Einstein Co ndensate\nKazuya Fujimoto1and Makoto Tsubota1,2\n1Department of Physics, Osaka City University, 3-3-138 Sugi moto, Sumiyoshi-ku, Osaka 558-8585, Japan\n2The OCU Advanced Research Institute for Natural Science and Technology (OCARINA),\nOsaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Os aka 558-8585, Japan\n(Dated: June 14, 2018)\nWe numerically study spin turbulence in a two-dimensional t rapped spin-1 spinor Bose–Einstein\ncondensate, focusingontheenergyspectrum. Thespinturbu lenceinthetrappedsystemisgenerated\nby instability of the helical structure of the spin density v ector in the initial state. Our numerical\ncalculation finds that in the trapped system the spectrum of t he spin-dependent interaction energy\nin the ferromagnetic case exhibits a −7/3 power law, which was confirmed in a uniform system by\nour previous study. The relation between the −7/3 power law and the motion of the spin density\nvector is discussed by investigating the orbits of dynamica l variables in the spin space.\nPACS numbers: 03.75.Mn, 03.75.Kk\nI. INTRODUCTION\nTurbulence in classical fluids has been studied for a\nlong time. It is an important theme in modern physics,\nbut is not yet understood sufficiently [1]. Great progress\nin the study of turbulence was made by Kolmogorov [2],\nwho suggested that the spectrum of the kinetic energy\nexhibits a −5/3 power law in fully developed isotropic\nhomogeneous turbulence. This is called the Kolmogorov\n−5/3 power law.\nRecently, quantum turbulence (QT) in one-component\natomic Bose–Einstein condensates (BECs) has been in-\nvestigated. Some theoretical and numerical studies show\nthat the Kolmogorov −5/3 power law appears in QT [3].\nQT in an atomic BEC has been experimentally realized\nby Henn etal.[4], but the Kolmogorov −5/3 power law\nhas not yet been observed.\nMulti-component atomic BECs can be realized [5],\ndescribed by more than two macroscopic wave func-\ntions, which means that the degrees of freedom in\nmulti-component BECs is larger than in one-component\nBECs. Thus, various behaviors that do not appear in\none-component BEC are expected to occur in this sys-\ntem. The hydrodynamical instability in two-component\nBECs, such as Rayleigh–Taylor, Kelvin–Helmholtz and\nRichtmyer–Meshkov instability, has been studied using\nthe Gross–Pitaevskii (GP) equation [6–8]. The turbu-\nlence in a two-component BEC has been numerically and\ntheoretically investigated, and a tangle of two kinds of\nquantizedvorticeswasfoundtoforminthesystem[9,10].\nAnother multi-component BEC system is the spinor\nBEC, comprised of condensates corresponding to differ-\nent hyperfine states of an atom. We investigated the\nturbulence in a spinor BEC, namely spin turbulence, in\nour previous work [11]. In the prior research, we gen-\nerated the spin turbulence by a counterflow instability,\nfinding that, in the ferromagnetic case, the spectrum of\nthe spin-dependent interaction energy exhibits a −7/3\npower law. This power law can be understood by scaling\nanalysis with some assumptions. One assumption is that\nthe total density of the condensate is uniform. However,in the actual experiments, the condensates are trapped\nby a harmonic potential that is not uniform. Therefore,\nwhether the −7/3 power law appears in the trapped sys-\ntem is an important issue.\nIn this paper, we report the numerical observation of\nthe−7/3power law in a trapped spin-1 spinor BEC. Our\nstudy isbasedon anumericalcalculationofthe GPequa-\ntion. The spin turbulence in the trapped system arises\nfrom the instability of the helical structure of the spin\ndensity vector in the initial state. This helical structure\nhas been experimentally realized by Vengalattore etal.\n[12]. We find that, in the ferromagnetic case, the −7/3\npower law also appears in the trapped system if the sys-\ntem size is much larger than the spin coherence length.\nThis paper is organized as follows. Section II describes\nthe formulation. The results of the numerical simulation\nof the GP equation are described in Sec. III. Here we\nshow how the helical structure of the spin density vector\nbecomes unstable to the development of spin turbulence\nand the −7/3 power law appears in the spin-dependent\ninteraction energy. In Sec. IV, in order to understand\nthe relation between the energy spectra and the motion\nof the spin density vector, we discuss the characteristic\ntime of the spin, the orbits in the spin space and the size\nof the BEC cloud for sustaining the power law. Section\nV is devoted to conclusions.\nII. FORMULATION\nA. Gross–Pitaevskii equation\nWe consider a trapped spin-1 spinor BEC at zero tem-\nperature, which is well described by the macroscopic\nwave functions ψm(m= 1, 0, −1). Then, the wave\nfunctionsψmobey the GP equation [13, 14]:\ni/planckover2pi1∂\n∂tψm= (−/planckover2pi12\n2M∇2+V)ψm+c0nψm+c11/summationdisplay\nn=−1s·Smnψn,\n(1)2\nwhereMandVare the mass of a particle and the trap-\nping potential. In our study, the potential is given by\nV=Mω2(x2+y2)/2 +Mω2\nzz2/2. The total density n\nand the spin density vector si(i=x,y,z) are given\nbyn=/summationtext1\nm=−1|ψm|2andsi=/summationtext1\nm,n=−1ψ∗\nm(Si)mnψn,\nwhere (Si)mnare the spin-1 matrices. The parameters\nc0andc1are the coefficients of the spin-independent and\ndependentinteractions,whichareexpressedby4 π/planckover2pi12(a0+\n2a2)/3Mand 4π/planckover2pi12(a2−a0)/3Mwithswave scattering\nlengthsa0anda2.\nThe total energy Eis given by\nE=/integraldisplay1/summationdisplay\nm=−1[ψ∗\nm(−/planckover2pi12\n2M∇2+V)ψm]dr\n+c0\n2/integraldisplay\nn2dr+c1\n2/integraldisplay\ns2dr.(2)\nThe spin-dependent interaction energy is the last term\nwith the coefficient c1of the right hand side of Eq. (2),\nwhich is characteristic of the spinor BEC. The ground\nstate in a uniform system without a magnetic field is\nferromagnetic for c1<0 and polar for c1>0. This\ninteraction exchanges the particles between the different\ncondensates, so the particle number in each component\nis not conserved. Thus, the instability and turbulence in\nspinor BECs are expected to have properties that one-\nand two-component BECs do not.\nWe consider that the condensate is strongly confined\nin thezdirection(ω≪ωz). In this case, we can approx-\nimately separate the degrees of freedom of the macro-\nscopic wavefunctions as ψm(x,y,z,t) =¯ψm(x,y,t)f(z).\nWe assume f(z) = (1/2πa2\nhz)1/4e−z2/4a2\nhzwithahz=\n(/planckover2pi1/2Mωz)1/2. Then, the GP equation (1) is reduced to\ni/planckover2pi1∂\n∂t¯ψm= (−/planckover2pi12\n2M¯∇2+¯V)¯ψm+¯c0¯n¯ψm+¯c11/summationdisplay\nn=−1¯s·Smn¯ψn\n(3)\nwith¯∇2=∂2/∂x2+∂2/∂y2,¯V=Mω2(x2+y2)/2, ¯c0=\nc0/2√πahz, ¯c1=c1/2√πahz, ¯n=/summationtext1\nm=−1|¯ψm|2and\n¯si=/summationtext1\nm,n=−1¯ψ∗\nm(Si)mn¯ψn. We consider experiments\nwith87Rb, which exhibits a ferromagnetic interaction.\nThus, we use M= 1.42×10−25kg,a0= 5.39×10−9m,\na2= 5.31×10−9m,N= 3×105,ω= 2π×20 /s and\nωz= 2π×600 /s.\nB. Numerical method\nWe use the Crank–Nicholson method to numerically\ncalculate the GP equation (3). The coordinate is nor-\nmalized by the length ah= (/planckover2pi1/2Mω)1/2∼1.72µm and\nthe box size is 80 ah×80ah. Space in the xandydirec-\ntions is discretized into 1024 ×1024 bins. The time is\nnormalized by the frequency ωof the trapping potential\nin thexandydirections.\nFIG. 1: (Color online) Profiles of each component of the spin\ndensity vector in the initial state. The box size is 40 ah×40ah,\nwhereahis (/planckover2pi1/2Mω)1/2∼1.72µm.\nC. Initial state\nWe use an initial state with a helical structure of the\nspin density vector to obtain the spin turbulence in the\ntrapped spin-1 spinor BEC. In the following, we show the\nmathematical expression of this state. The macroscopic\nwave functions ¯ψ= (¯φ,0,0) has spin density vector ¯s=\n|¯φ|2ˆez, where ˆej(j=x,y,z) is the unit vector in the\njdirection. Here, ¯φcan be obtained by the imaginary\ntime step of Eq. (3). By multiplying this wave function\nby the rotation matrix ˆU=e−iαˆSze−iβˆSye−iγˆSzin the\nspin space, we obtain\n\n¯ψ1¯ψ0¯ψ−1\n=¯φe−iγ\ne−iαcos2β\n21√\n2sinβ\neiαsin2β\n2\n, (4)\nwhereα,β,γare the Euler angles. Then, the spin\ndensity vector is expressed by ¯s=|¯φ|2(sinβcosαˆex+\nsinβsinαˆey+ cosβˆez). Therefore, using α=π/2,\nβ=khxandγ= 0, the spin density vector has a he-\nlical structure with wave number kh. In this paper, we\nconsider the case of kh∼1.22 /µm. Thus, Eq. (4) be-\ncomes\n\n¯ψ1¯ψ0¯ψ−1\n=¯φ\n−icos2(khx/2)\n1√\n2sin(khx)\nisin2(khx/2)\n. (5)\nFigure 1 shows the profile of each component of the spin\ndensity vector in the initial state of Eq.(5). The heli-\ncal structure has been experimentally realized by Ven-\ngalattore etal.[12], where they prepared the structure by\nmeansofamagneticfield, investigatinghowthestructure\nbecomes unstable to changes into some disordered state\nthrough observing the spin density vector. In our calcu-\nlations, we add some small white noise to the initial state\nof Eq. (5). The noise causes the particle number of each\ncomponent to fluctuate by 0 .1–0.3%. This is consistent\nwith experimental results [15].3\nx yωt= 0(a)\n(c)(b)\n(d)ωt= 5\nωt= 60 ωt= 180\nFIG. 2: (Color online) Profiles of the spin density vector at\nωt= 0, 5, 60, 180. The shading of the arrows denotes the\namplitude of the spin density vector. The box size is 40 ah×\n40ah.\nD. Spectrum of spin-dependent interaction energy\nWe derive an expression for the spectrum of the spin-\ndependent interaction energy. The spin-dependent inter-\naction energy Esis given by\nEs=c1\n2/integraldisplay\n¯s(¯r)2d¯r (6)\nwith¯r= (x,y). We expand the spin density vector ¯s(¯r)\nwith plane waves: ¯s(¯r) =/summationtext\n¯k˜s(¯k)ei¯k·¯rwith¯k= (kx,ky).\nThen, the spin-dependent interaction energy Esis repre-\nsented by ˜s(¯k) as\nEs=c1A\n2/summationdisplay\n¯k|˜s(¯k)|2, (7)\nwhereAis the area of the system. Therefore, the en-\nergy spectrum of the spin-dependent interaction energy\nis given by\nEs(k) =c1A\n2∆k/summationdisplay\nk<|¯k1|<k+∆k|˜s(¯k1)|2,(8)\nwhere ∆kis 2π/Lfor a system size L.\nIII. NUMERICAL RESULTS\nWe show our numerical results with the GP equation\n(3) starting from the initial state of Eq. (5) [16]. Ourcalculation finds that the helical structure in the trapped\nsystem leads to spin turbulence and the spectrum of\nthe spin-dependent interaction energy exhibits the −7/3\npower law.\nThe helical structure of the spin density vector in the\ntrapped system is unstable, generating the spin turbu-\nlence. Figure 2 (a) shows the profile of the spin density\nvector in the initial state. This helical structure immedi-\nately collapses as shown in Fig. 2 (b). As time goes by,\nthe spin density vector is significantly disturbed, point-\ning in various directions, as shown in Figs. 2 (c) and (d).\nThus, spin turbulence in the trapped system is realized.\nThespectrum ofthespin-dependent interactionenergy\nexhibits the −7/3power law in the spin turbulence in the\ntrapped system. Figures 3 show the time development\nof the spectrum, which is numerically calculated by Eq.\n(8). In the initial state, the spectrum in Fig. 3 (a) has a\nsharppeak at kah∼2 correspondingto the wavenumber\nkhof the helical structure. As the instability develops,\nmodulation of the spin density vector with various wave\nnumbers is excited. Thus, as shown in Fig. 3 (b), the\nspectrum loses the sharp peak, letting the energy flow\ninto both the low and high wave number regions. As\ntime passes, the spin density vector is significantly dis-\nturbed and the system exhibits spin turbulence. Then,\nthe spectrum develops to exhibit the −7/3 power law in\nthe wave number region kTF<k<k s, as shown in Fig.\n3 (c). Here kTFandksare the wave numbers corre-\nsponding to the Thomas–Fermi radius RTFand the spin\ncoherence length ξs=/planckover2pi1//radicalbig\n2M|c1|n0. This−7/3 power\nlaw in the spectrum continues at least until ωt= 180.\nFigure 3 (d) shows the spectrum at ωt= 180.\nIV. DISCUSSION\nA. Motion of the spin density vector and the −7/3\npower law\nThe appearance of the −7/3 power law is accompanied\nby characteristic behavior of the spin density vector. We\nshould note the time scale of the motion as well as the\nrandomness of the spin density vector. After the ini-\ntial helical structure becomes unstable, the spin vectors\nexhibit relatively large-scale, slow motion. The helical\nstructure then disappears and the vectors become disor-\ndered. As the −7/3power law appears, the vectors begin\nto perform rapid, fine oscillations around random direc-\ntions; the motion of each spin vectorappears to be frozen\nto a random direction.\nThis can be understood qualitatively by the follow-\ning discussion. In our previous study [11], we performed\nKolmogorov-typescaling analysisfor the equation of mo-\ntion of the spin density vector and thus obtained the\n−7/3 power law. In the wave number region sustaining\nthe−7/3 power law, the characteristic time ts(k) of the\nmode of wave number kis expressed by\nts(k)∼ǫ−1/3k−3/4, (9)4\nωt= 0\n10-1110-1010-910-810-710-610-510-410-3\n0.010.1110100\nkahk−7/3:\nkTF ks\n10-1110-1010-910-810-710-610-510-410-3\n0.010.1110100\nkahk−7/3:\nkTF ks(a)\n(c)(b)\n(d)|Es(k)|/¯hωah |Es(k)|/¯hωahωt= 5\nωt= 60 ωt= 180\nFIG. 3: (Color online) Spectrum of the spin-dependent inter action energy at ωt= 0, 5, 60, 180. ksandkTFare the wave\nnumbers corresponding to the spin coherence length ξs=/planckover2pi1//radicalbig\n2M|c1|n0and the Thomas–Fermi radius RTF, wheren0is the\ntotal density at the center in the initial state. Their expre ssions are ks= 2π/ξsandkTF= 2π/RTF.\nwhereǫis the energy flux flowing from low to high wave\nnumbers. Equation (9) is obtained from ǫ∼k−4t−3\ns,\nwhich means that the energy flux is independent of the\nwave number [11]. Equation (9) shows that the charac-\nteristic time tsshortens when the wave number is high.\nAs the spin density vectors become disordered and the\nhighkmodes (small scale) are excited, the characteristic\ntime scale becomes short, leading to the rapid, fine oscil-\nlations. On the other hand, the motion with low kmodes\n(large scale) has a long characteristic time ts. Thus, the\nroughdirectionofthe spin densityvectorateachposition\nis largely fixed and changes only slowly.\nThus the motion of the spin density vector is closely\nrelated to the advent of the −7/3 power law.\nB. Orbits in spin space\nIn order to reveal these properties, it is useful to\nconsider the orbits of the dynamical variables in phase\nspace, which is an approach often used in the re-\nsearch field of nonlinear dynamics. Figure 4 shows\nthe orbits of the normalized spin density vectors in\nthe (¯sxa2\nh/N,¯sya2\nh/N,¯sza2\nh/N) space for different time\nstages. Thetwoorbitscorrespondtothespindensityvec-\ntor at the positions (0 ,0) and (0,RTF/2) inside the BEC\ncloud. The amplitude of the spin density vector depends\non the position inside the BEC cloud and is maximumat the center; such a dependence of the amplitude is re-\nflected in the orbits of the spin at different positions. We\ndiscuss the time development of the orbits of Fig. 4 with\nreference to Fig. 2 and Fig. 3. The periodic orbits in the\nearly stage (Fig. 4 (a)) indicate some coherent motion of\nthe initial helical structure. Some of the orbits become\nunstable (Fig. 4 (b)) after the helical structure become\nunstable. Then, the spin density vectors start to become\ndisordered(Fig. 2 (b)) and the energyspectrum becomes\nbroad but does not yet exhibit the −7/3 power law (Fig.\n3(b)). After a time, the two orbits become separated, as\nshown in Fig. 4(c), and independently localized in some\nplaces in the spin space, generating rapid, fine oscilla-\ntion in those areas. During this process the spin density\nvectors become disordered and frozen (Fig. 2(c)) and\nthe energy spectrum exhibits the −7/3 power law (Fig.\n3(c)). This behavior continues until ωt= 180, while the\norbits appear to shrink from Fig. 4 (b) through (c) to\n(d). After the power law is established, the orbits move\nvery slowly in the spin space, as shown from Fig. 4 (c)\nto (d).\nIn order to measure the shrinkage quantitatively, we\nintroduce the size lof the orbit in the spin space, where\nthe sizelat timeτ=ωtis defined by the maximum\ndistance between two points in an orbit during τtoτ+2.\nFigure 5 shows the sizes l(τ) of orbits at three different\npositions (0 ,0), (0,RTF/2) and (0,3RTF/4) inside the\nBEC cloud. The sizes of the orbits are found to fluctuate5\n2\n0\n×10-3-2\n12-1-2×10-3×10-3\n101\n-2-1012::\n¯sxa2\nh/N ¯sya2\nh/N¯sza2h/N\n2\n0\n×10-3-2\n12-1-2×10-3×10-3\n101\n-2-1012::\n¯sxa2\nh/N ¯sya2\nh/N¯sza2h/N\n2\n0\n×10-3-2\n12-1-2×10-3×10-3\n101\n-2-1012::\n¯sxa2\nh/N ¯sya2\nh/N¯sza2h/N\n2\n0\n×10-3-2\n12-1-2×10-3×10-3\n101\n-2-1012::\n¯sxa2\nh/N ¯sya2\nh/N¯sza2h/N(a)ωt= 0∼2 ωt= 5∼7\nωt= 60∼62 ωt= 180∼182(b)\n(c) (d)dk (148)\n(149)\nm,nmn (150)\n(sincos sinsin cos (151)\n(sin( cos( (152)\n¯s(0,0) (153)\n(0 4) (154)\n(0 2) (155)\n(0 4) (156)dk (148)\n(149)\nm,nmn (150)\n(sincos sinsin cos (151)\n(sin( cos( (152)\n(00) (153)\n(0 4) (154)\n¯s(0,RTF/2) (155)\n(0 4) (156)\ndk (148)\n(149)\nm,nmn (150)\n(sincos sinsin cos (151)\n(sin( cos( (152)\n¯s(0,0) (153)\n(0 4) (154)\n(0 2) (155)\n(0 4) (156)dk (148)\n(149)\nm,nmn (150)\n(sincos sinsin cos (151)\n(sin( cos( (152)\n¯s(0,0) (153)\n(0 4) (154)\n(0 2) (155)\n(0 4) (156)dk (148)\n(149)\nm,nmn (150)\n(sincos sinsin cos (151)\n(sin( cos( (152)\n¯s(0,0) (153)\n(0 4) (154)\n(0 2) (155)\n(0 4) (156)dk (148)\n(149)\nm,nmn (150)\n(sincos sinsin cos (151)\n(sin( cos( (152)\n(00) (153)\n(0 4) (154)\n¯s(0,RTF/2) (155)\n(0 4) (156)dk (148)\n(149)\nm,nmn (150)\n(sincos sinsin cos (151)\n(sin( cos( (152)\n(00) (153)\n(0 4) (154)\n¯s(0,RTF/2) (155)\n(0 4) (156)dk (148)\n(149)\nm,nmn (150)\n(sincos sinsin cos (151)\n(sin( cos( (152)\n(00) (153)\n(0 4) (154)\n¯s(0,RTF/2) (155)\n(0 4) (156)\nFIG. 4: (Color online) Orbits of the spin density vector at tw o positions (0 ,0) (red filled squares) and (0 ,RTF/2) (blue empty\nsquares) inside the BEC cloud for the periods (a) ωt= 0–2, (b) 5–7, (c) 60–62 and (d) 180–182. The spin density vec tor is\nnormalized by N/a2\nh.\nand shrink as time passes; the size is generally smaller at\nouter positions in the cloud, presumably because of the\nsmaller amplitude of the spin density vector.\nIt is interesting that the freezing of the separated spin\norbits can determine the power law of the energy spec-\ntrum.\nC. Size of condensate and the −7/3power law\nWe discuss the relation between the size of the con-\ndensate and the −7/3 power law. We confirm that if\nthe Thomas–Fermi radius RTFis not sufficiently larger\nthan the spin coherence length ξs, the spectrum of the\nspin-dependent interaction energy does not exhibit the\n−7/3 power law. This is confirmed by a numerical cal-\nculation with RTF/ξs∼5. For the calculations in this\npaper, the condition RTF/ξs∼10 is satisfied. Therefore,\nin the experiments, the condition RTF/ξs≥10 could be\nnecessary for observing the −7/3 power law.\nV. CONCLUSIONS\nWe study spin turbulence in a trapped spin-1 spinor\nBEC by numerical calculation of the GP equation (3).The spin turbulence in the trapped system is generated\nby instability of the helical structure of the spin density\nvector. Our numerical calculation finds that the spec-\ntrum of the spin-dependent interaction energy exhibits\nthe−7/3 power law, which is consistent with our pre-\nvious result for a uniform system [11]. We discuss the\nrelation between the −7/3 power law and the motion of\nthe spin density vector.\nWe believe that the −7/3 power law can be experi-\nmentally observed because of the following reasons. As\ndiscussed in Sec. IV C, the spin turbulence in a large\ntrapped system ( RTF/ξs∼10) sustains this power law\nfor a long time. All the components of the spin density\nvectorhavebeenexperimentallyobservedbyaphasecon-\ntrast imaging method [12, 15]. The expressions for the\nspectrum of the spin-dependent interaction energy show\nthat it is possible to obtain a spectrum if the spin density\nvector is observed everywhere. This is because Eq. (8)\ncontains only the Fourier component of the vector ˜s(k).\nTherefore, it is possible to observe this power law.\nACKNOWLEDGMENT\nM. T. acknowledges the support of a Grant-in-Aid for\nScience Research from JSPS (Grant No. 21340104).6\n:\n150200 0501005\n0123×10-3\n4\nωtl:\n:dk (148)\n(149)\nm,nmn (150)\n(sincos sinsin cos (151)\n(sin( cos( (152)\n¯s(0,0) (153)\n(0 4) (154)\n(0 2) (155)\n(0 4) (156)dk (148)\n(149)\nm,nmn (150)\n(sincos sinsin cos (151)\n(sin( cos( (152)\n(00) (153)\n(0 4) (154)\n¯s(0,RTF/2) (155)\n(0 4) (156)dk (148)\n(149)\nm,nmn (150)\n(sincos sinsin cos (151)\n(sin( cos( (152)\n(00) (153)\n(0 4) (154)\n(0 2) (155)\n¯s(0,3RTF/4) (156)\n 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045\n 0  50  100  150  200\"spin_phase_space_1size.txt\"\n\"spin_phase_space_2size.txt\"\n\"spin_phase_space_3size.txt\"\n 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045\n 0  50  100  150  200\"spin_phase_space_1size.txt\"\n\"spin_phase_space_2size.txt\"\n\"spin_phase_space_3size.txt\"\n 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045\n 0  50  100  150  200\"spin_phase_space_1size.txt\"\n\"spin_phase_space_2size.txt\"\n\"spin_phase_space_3size.txt\"\nFIG. 5: (Color online) Time development of the size of the\norbits of the spin density vector in the spin space of Fig.\n4. The three curves correspond to the three positions (0 ,0),\n(0,RTF/2) and (0 ,3RTF/4) inside the BEC cloud. The size l\nat timeτ=ωtrefers to the maximum distance between two\npoints in an orbit during τtoτ+2.\n[1] U. Frisch, Turbulence (Cambridge University Press,\n1995).\n[2] A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 9 (1941);\n32, 19 (1941); Proc. R. Soc. Ser. A 434, 9 (1991); 434,\n15 (1991).\n[3] M. Kobayashi and M. Tsubota, Phys. Rev. A 76, 045603\n(2007).\n[4] E. A. L. Henn, J. A. Seman, G. Roati, K. M. F. Maga-\nlhaes, and V. S. Bagnato, Phys. Rev. Lett. 103, 045301\n(2009).\n[5] K. Kasamatsu, M. Tsubota, and M. Ueda, Int. J. Mod.\nPhys. B 19, 1835 (2005).\n[6] K. Sasaki, N. Suzuki, D. Akamatsu, and H. Saito, Phys.\nRev. A80, 063611 (2009).\n[7] H. Takeuchi, N. Suzuki, K. Kasamatsu, H. Saito, and M.\nTsubota, Phys. Rev. B 81, 094517 (2010).\n[8] A. Bezett, V. Bychkov, E. Lundh, D. Kobyakov, and M.Marklund, Phys. Rev. A 82, 043608 (2010).\n[9] H. Takeuchi, S. Ishino, and M. Tsubota, Phys. Rev. Lett.\n105, 205301 (2010).\n[10] S. Ishino, M. Tsubota, and H. Takeuchi, Phys. Rev. A\n83, 063602 (2011).\n[11] K. Fujimoto and M. Tsubota, Phys. Rev. A 85, 033642\n(2012).\n[12] M. Vengalattore, S. R. Leslie, J. Guzman, and D. M.\nStamper-Kurn, Phys. Rev. Lett. 100, 170403 (2008).\n[13] T. Ohmi, and K. Machida, J. Phys. Soc. Jpn. 67, 1822\n(1998).\n[14] T. -L. Ho, Phys. Rev. Lett. 81, 742 (1998).\n[15] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore ,\nand D. M. Stamper-Kurn, Nature 443, 312 (2006).\n[16] You can enjoy the movie in http://www.sci.osaka-\ncu.ac.jp/phys/eep/top-e.html."    },    {        "title": "2109.12578v1.Monte_Carlo_simulation_of_ultrafast_nonequilibrium_spin_and_charge_transport_in_iron.pdf",        "content": "arXiv:2109.12578v1  [cond-mat.mtrl-sci]  26 Sep 2021Monte Carlo simulation of ultrafast nonequilibrium\nspin and charge transport in iron\nJ. Briones, H. C. Schneider and B. Rethfeld\nDepartment of Physics and OPTIMAS Research Center, TU Kaise rslautern,\nErwin Schroedinger Str. 46, 67663 Kaiserslautern, Germany\nE-mail:briones@physik.uni-kl.de\nAbstract. Spin transport and spin dynamics after femtosecond laser pu lse\nirradiation of iron (Fe) are studied using a kinetic Monte Ca rlo model. This\nmodel simulates spin dependent dynamics by taking into acco unt two interaction\nprocesses during nonequilibrium: Elastic electron - latti ce scattering, where only\nthe direction of the excited electrons changes neglecting t he energy loss, and\ninelastic electron - electron interaction, where secondar y electrons are generated.\nAn analysis of the particle kinetics inside the material sho ws that a smaller elastic\nscattering time affects the spin dynamics by leading to a larg er spatial spread of\nelectrons in the material, whereas generation of secondary electrons affects the\nspin transport with a larger time of propagation ofhomogene ous spinpolarization.2\n1. Introduction\nFollowing the discovery of ultrafast demagnetization\nin metallic ferromagnets and its connection to hot\nelectron spin transport, the interplay of optical\nexcitation, magnetization dynamics and transport is\nstill under active investigation. For the optical\nexcitation of ferromagnets the electronic system is\nbrought out of equilibrium, which will be later\nthermalized to the surrounding environment with\nmany possible scattering mechanisms [1, 2, 3, 4, 5].\nThe dynamics of spin transport were first studied\nby Battiato, et al. [6] identifying an intermediate\nregime of the spin transport, labeled as superdiffusive\ntransport. The contribution to demagnetization\ndynamics was later supported by experiments [7, 8].\nAnotherrecentstudyofthiseffectintroducedaparticle\nin cell simulation [9] to solve the Boltzmann equation\nfor spin dependent hot-electron transport.\nIn order to understand the influence of the\ndifferent scattering interactions in spin transport we\nwill analyze the spatio-temporal dynamics by tracing\nspin and charge in dependence of depth by considering\nfree electronstatesforenergiesabovethe Fermienergy.\nFor this reason we propose for this type of system an\napplication of the kinetics Monte Carlo technique.\nMonte Carlo simulations have been widely used\nin studies of hot carriers dynamics, such as radiation\nbiology [10, 11, 12], nuclear physics [13] and particle\ntransport [14] among many others. In this paper we\npresent a Monte Carlo approach and its capabilities\nin analyzing the influence of secondary electrons\ngeneration in spin dynamics and spin transport.\nThe outline of the paper is as follows: we will first\npresent a model of excitation process, thereby briefly\nintroduce how the kinetic Monte Carlo technique\nworks. We explain how an electron is treated during\nlaser excitation and discuss the role of the spin\ndependent density of states. The next section will\nfocus on the possible scattering processes that are\ntaken into account in our simulation. Finally we\npresent the results in depth-dependence for different\nscattering times in ferromagnetic iron after ultrafast\nlaser excitation with 6 eV photon energy, as well\nas some results showing the influence of secondary\nelectrons generation in the particle kinetics.\n2. Excitation Process\nIn this section we discuss the algorithm used for the\nsimulations presented in this paper, as well as an\nanalysisofthe energydensity of excited electronswhen\nonly photoexcitation is considered.2.1. Monte Carlo algorithm\nThe asymptotic Monte Carlo trajectory method [15]\nis a statistical technique that models binary collision\ninteractions by random sampling a very large number\nof trajectories until a result converges. The algorithm\nused for random sampling of a variable xis done using\nprobabilitytheory. Inprobabilitytheoryoneintegrates\nthe probabilities of all possible events p(x), where\nxmin≤x≤xmax, into a variable called cumulative\ndistribution function (CDF) F(x) [16]. One can map\nthe CDF onto the range of random variables R, where\nR∈[0,1] and R is distributed uniformly:\nR =F(x)−F(xmin)\nF(xmax)−F(xmin)=/integraltextx\nxminp(x)dx/integraltextxmax\nxminp(x)dx, (1)\nThe variable xis then uniquely determined in\ndependence on R. Now we use this general idea to\nrandom-samplinganyrequiredvariableforthedifferent\ntypes of interactions.\nRandom sampling is also used to decide whether\ncertain interaction or excitation processes happen. In\nthese cases we make use of the discrete form of eq. (1).\nExamples of random sampling are shown in Figs. 1\nand 4 for the excitation of electrons. The treatment of\ninteractions in this simulation is discussed in section 3\nof this work.\n2.2. Laser and material parameters\nWe simulate the excitation of a ferromagneticiron (Fe)\nlayer with a 25 nm depth with a temporally Gaussian\nlaser pulse with full width half maximum (FWHM) of\n25 fs and 6 eV photon energy. The simulation works\nfor any spin dependent density of states. Here, we take\nthe data for iron from Ref. [17].\nThe excitation of an electron from an occupied\nband is modeled by choosing for eq. (1) a probability\nof excitation according to the material’s density of\nstates below EF. Fig. 1 shows the calculated energy\ndependent cumulative distribution function (solid line)\nfor the density of states for spin up electrons (dash-\ndotted line). States above EF, will be considered as\nfree electron states with a constant effective mass.\nAs a first test we study only photoexcitation,\nwithout any scattering processes for the excited\nelectrons. Fig. 2 shows the color-coded electron\ndensity of excited spin up (or majority) and spin\ndown (or minority) electrons after photoexcitation in\ndependence on time and kinetic energy above EF. The\nlaser pulse with 25 fs duration is centered around\nt= 0. The results show that a larger density of\nelectrons is excited from 3d↑states in comparison to\nthe 3d↓, i.e., predominantly majority electrons are\nexcited. The spin-dependent density of states, which\nis sketched in the background of Fig. 2, shows that3\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8\n-6 -4 -2  0  2  4  6 0 0.2 0.4 0.6 0.8 1DOS [arb. units]  \nCumulative Distribution Function [unitless] \nE - Ef [eV]CDF\nFe\nFigure 1. Energy dependent cumulative distribution function\n(solid line - right axis) behavior when it is weighted accord ing\nto the spin up density of states (doted line - left axis) of\nferromagnetic .\nthe excitedenergy-resolvedelectrondensityreflectsthe\nspin dependent density of states of the occupied bands\nin ferromagnetic iron.\nFigure 2. Density of excited electrons vs. time for\nphotoexcitation with a 25 fs Gaussian laser pulse, but no\nscattering. Left: Spin up excited electrons. Right: Spin do wn\nexcited electrons. Middle panel: Sketch of the occupied par t\nthe spin-dependent density of states shifted up by the excit ation\nphoton energy.\n3. Different scattering processes\nWetracethedynamicsofparticlesafterlaserexcitation\nby pure jump processes [18, 19, 12]. In our approach,\nwe consider free electron states above EFas essentially\nfree and focus on the influence of high energy electrons\nin spin transport. Fig. 3 shows a schematic\nrepresentation of how we treat the electron kinetics\nafter ultrafast pulse excitation in a ferromagnet here.\nDuring the simulation any electron interaction process\nis treated by random sampling. For the excitationprocess already discussed above, the initial direction of\nthe excited electronswill be takenas randomdirection.\nWe next consider the probability that an electron\nhas not suffered any collision between ˜t= 0 and tas\np(t) = exp/bracketleftBig\n−/integraldisplayt\n0ν/parenleftbig\nE(˜t)/parenrightbig\nd˜t/bracketrightBig\n. (2)\nHere,νis the total scattering rate defined as the sum\nof all possible scattering transitions. It is dependent\non the energy Eof the considered electrons, which\nmay vary in time. In fact, this equation should be\nsolved for each scattering event. However, we simplify\nthe calculation as proposed in Ref. [20] by assuming a\nconstant scattering rate ν0. Then, the time of free\nflightτcan be sampled with the random variable\nR∈[0,1] as\nτ=−ν−1\n0log(R). (3)\nHowever, assuming a constant total scattering rate,\nindependent of energy, is a statistical overestimation.\nTo compensate, we introduce a further “collision“\nwithanenergy-dependentprobability,whichallowsthe\nparticle to continue its trajectory unperturbed.\nWe randomly sample which collision takes place\nby using the individual collision frequencies and form\nthe cumulative distribution function in order to solve\neq. (1). In the following subsections we analyze the\ninteractions that will be considered and the equations\nor parameters used.\nFigure 3. Schematic figure of hot electron dynamics optically\nexcited. Electrons undergo different interactions in time s uch\nas electron-nucleus interaction, where only the direction of the\nparticle is changed (elastic scattering), and electron-el ectron\nimpact ionization, where secondary electrons are generate d\n(impact ionization).\n3.1. Elastic scattering: Electron - nucleus interaction\nHigh-energy electrons are scattered nearly elastically\nby nuclei, i.e., they change their direction of motion.\nHere, weparametrizethis interactionbytwoimportant\nquantities: The angle of deflection ( θ) and the elastic\nscattering rate ( τ−1\nel). We will follow a procedure4\nwhich has been applied to different materials, see Refs.\n[21, 22, 23]. In order to obtain the angle of deflection\nθfor solving eq. (1) we choose as probability function\nthe differential cross section (dσ\ndΩ), described here by\nthe Mott cross section [24]:\ndσ\ndΩ=|f(θ)|2+|g(θ)|2, (4)\nwhereθis the scattering angle, f(θ) andg(θ) are the\nscattering amplitudes which can be obtained from the\nfollowing expressions:\nf(θ) =1\n2iK∞/summationdisplay\nl=0/braceleftbigg\n(l+1)/bracketleftBig\ne(2iδl)−1/bracketrightBig\n+\nl/bracketleftBig\ne(2iδ−l−1)−1/bracketrightBig/bracerightbigg\nPl(cosθ),(5)\ng(θ) =1\n2iK∞/summationdisplay\nl=1/braceleftbigg\n(l+1)/bracketleftbigg\n−e(2iδl)+\ne(2iδ−l−1)/bracketrightbigg/bracerightbigg\nP1\nl(cosθ).(6)\nHere,K2=W2−1, where Wis the total energy of the\nincident electron in atomic units, δlis the phase shift\nforthel-thpartialwaveand PlandP1\nlaretheordinary\nand associated Legendre Polynomials, respectively.\nWe have now a set of equations that will allow\nus to obtain values of the angle of deflection by using\neq. (1). The next step is to obtain an expressionfor the\nelastic scattering rate. In principle, it can be deduced\nfrom the total scattering cross section, accessible by\nthe integration of eq. (4), together with the density of\ncollision partners. It will depend on energy, material\nand temperature [25, 26, 27, 28]. Since we trace non-\ninteractingelectronsinourapproach,wehavenoaccess\nto integrated quantities like temperature. Therefore,\nwe focus hereon the net effect ofthe scatteringtime τel\nbetween electrons and nuclei by assuming a constant\nvalue. We will analyze the dynamics for two values:\nτel= 25 fs from Ref. [29] and τel= 12 fs from Ref. [30].\nWe will not take into account the loss of energy due to\nrecoil for this simulation.\n3.2. Inelastic scattering: Impact Ionization\nWhen a high-energy electron interacts with electrons\nbelow and close to EFin the occupied band and\ngeneratessecondaryelectrons(SE),wewilllabelthisas\nan inelastic collision. A high-energy primary electron\nwith energy Ecan lose an amount of energy ∆ Eto a\nsecond electron, thereby ionizing the latter. The SE\nproduced in this process can ionize further secondary\nelectrons in a cascade process. The generation of SE\nby cascade process is believed to have an important\ninfluence on ultrafast spin transport [31]. The process\nof secondary electrons generation has been studied in\ndifferent materials, going back to Refs. [32, 33].Our numerical treatment of the SE generation\nprocess is as follows. In order to select the\nnewly excited electron we follow the same procedure\nexplained in subsection 2.2, where the probability of\nexcitation depends on the material’s density of states,\nwith the constraint that two particles cannot be in the\nsame state and thus we avoid selecting two electrons\nfromthesameoccupiedstatetwice. We distinguishthe\nspin of the newly excited electrons by selecting them\nfrom the spin-resolved density of states and weight the\nprobabilities accordingly.\nThe energy lost by the primary electron ∆ Eis\nused to ionize a secondary electron from the occupied\nband. Here we define a binding energy I, which is the\namount of energy necessary to reach Fermi level from\nan occupied state. Taking this into account, the final\nenergy of the ionized electron Esabove Fermi level, is\nthen ∆E=Es+I. The amount of transferred energy\n∆Eis assumed to be half of the energy of the incident\nelectron ( E/2) as it was done in Ref. [34].\nFor the inelastic scattering rate ( τ−1\nee) we will use\nthe energy-dependent collision rate [35] of an excited\nelectron at temperature Te,\n1\nτee(E,Te)=π2√\n3\n128ωp\nE2\nf(πkBTe)2+(E−Ef)2\ne/parenleftBig\n−E−Ef\nkBTe/parenrightBig\n+1,(7)\nwhereEFis the Fermi energy and ωpis the plasma\nfrequency. For the derivation of eq. (7) it is\nassumed that the band electrons have a Fermi-Dirac\ndistribution.\nFor the angle of deflection after an inelastic\nscattering, we use the classical binary collision model\nwhich can be derived from momentum and energy\nconservation. In collisions between two identical\nparticles with identical masses, the angles of deflection\nfor the primary ( θ) and secondary electron ( α) are\ngiven by\nθ= arcsin/parenleftBigg/radicalbigg\n∆E\nEs+∆E/parenrightBigg\n, (8)\nα= arcsin/parenleftBigg/radicalbigg\nEs\n∆E−Isinθ/parenrightBigg\n. (9)\nSince electrons are not distinguishable, we also\ninclude a probability that an electron effectively flips\nits spin. This is an exchange scattering process,\ndue to the Coulomb interaction with electrons in the\noccupied part of the spin-split band structure, which\nhas been analyzed in terms of the spin-flip self energy\nby Hong and Mills [36]. While magnons and Stoner\nexcitations can contribute, this scattering processes\nis called Stoner excitation in Ref. [37]. We follow\nthat paper and employ the energy-dependent spin-flip\nprobability in our simulation, see Fig. 2 in Ref. [37].5\n4. Space dependence\nThe absorption profile of the optical excitation\nimprints a spatial dependence on the initial electron\ndistribution. We take this into account by an\nexcitation probability derived from the Beer-Lambert\nlaw. We select the initial position of the excited\nelectrons in the material by randomly sampling the\nposition of excitation from the profile shown in Fig. 4.\nThe initial direction of movement is sampled randomly\nas well. The kinetic energies are between Fermi\nenergyEFandEF+6 eV, according to the excitation\nof each individual electron. Then, each electron is\ntraced individually in time and we can monitor its\ndisplacement throughout the material.\nsampling probability\n024681012141618\n0 5 10 15 20 250.20.40.60.81Probability [%] \nCumulative Distribution Function [unitless]\nDepth [nm]Probability\nCDF\nFigure 4. Probability density and cumulative distribution\nfunction for electron excitation from penetration depth of the\nlaser pulse according to the Beer-Lambert equation. Red lin es\nrepresent how the sampling of the initial position of the par ticles\nis taken from the values of the Cumulative distribution.\nThe motion of electrons can be characterized by\nthemeansquaredisplacement(MSD) [38]. TheMSDis\ndefined as the spatial spread of the distribution, which\noccurs in the transport direction z. As the electrons\nmove, the MSD of their distribution becomes time\ndependent/angbracketleftbig\n(∆z)2/angbracketrightbig\n∝tα, where αis known as the\ngeneralized diffusion exponent.\nThediffusionexponentcharacterizestheelectronic\nmotion under the influence of the excitation and\nscattering processes. In the ballistic regime, essentially\nno collisions occur, leading to α= 2. Superdiffusive\nbehavior occurs in the intermediate regime with 2 >\nα >1. On longer timescales, the motion of particles\nis randomized by multiple scattering processes, leading\nto a diffusive behavior described by α= 1.\n5. Results\nThe different scattering mechanisms that may occur\nafter fs-laser excitation in ferromagnets affect thenonequilibrium dynamics. We will focus first on\nthe influence of different scattering rates for elastic\nscatterings and then discuss the effects of secondary\nelectrons generation. We will analyze the following\nphysical quantities: Displacement of the particles,\nparticle velocities, diffusion and spin current.\n5.1. Influence of different elastic scattering times\nFigure 5 shows the time evolution of the particle\ndensity in iron with an open boundary at 25nm for\na simulation where all the scattering processes are\nincluded for two different elastic scattering times. The\nspatio-temporal dynamics of the density of excited\nelectrons is shown in colour code. The surface is at\ndepth zero and the laser is centered at time zero. The\nupper subplots are for an elastic scattering time of\nτel= 12 fs, the lower ones for τel= 25 fs. One\nfirst observes that for the smaller elastic scattering\ntime (upper subplots) the particles remain close to\nthe surface longer. For the smaller elastic scattering\ntime the excited particle density is observable inside\nthe material for larger times. This is because elastic\nscattering processes occur more often and change the\ndirection of the particles, contributing to the spreading\nin the material. The red areas in the spin-up channel\n0510152025Spin upDepth [nm]Spin down\nτel = 12 fs\n0510152025\n050100 150Depth [nm]\nTime [fs]\n 0  0.2  0.4  0.6  0.8  1  1.2\nElectron density [1027 m-3]050100 150\nτel = 25 fs\nTime [fs]\nFigure 5. Evolution of particle density through the material\nfor a simulation including all scatterings processes with d ifferent\nscattering times of elastic scatterings: τel= 12 fs (top figure)\nandτel= 25 fs (lower figure) and different spins: Spin up (lhs)\nand Spin down (rhs).\nindicate a higher density of excited spin-up electrons,\nwhich isdue tothe band structurefeaturesdiscussedin\nthe previous section 2 on the photoexcitation process.\nAfter about 75 fs the signature of the spatial laser\npenetration profile has been washed out by scattering\nprocesses and transport. The transport characteristics\nof the dynamics shown here will be analyzed in more6\ndetail using the mean square displacement (MSD) in\nthe following subsection.\nFigure 6 shows the mean velocity in zdirection\nat a depth of 12nm. Both kinds of spin show only\nstatistical differences, here we show the results for spin\nup electrons. The mean velocity is calculated as the\naverage velocity in + zdirection of all excited particles\nat the given depth. Apart from the scattering times\ndiscussed in subsection 3.1 we also show results for a\nthird, shorter one ( τel= 2 fs) as a way to observe\nhow the system is influenced by lower values of τel.\nDuring the first femtoseconds one can observe higher\naverage velocities with direction into the depth of the\nmaterial (velocityup\nz>0), but decreasing in magnitude\nfor lowerelastic scattering times. In all three cases, the\ndifferent scattering times keep the velocity of particles\non average pointing into the material. After 60 fs\nthe average velocity for all three cases approaches the\nsame value and continues decreasing at the same rate\nbut without reaching zero, this means that a large\nnumber of particles travel in + zdirection. During\nthe first femtoseconds a smaller magnitude of average\nvelocities towards the depth can be observed for lower\nelastic scattering time. This is due to a larger number\nof scatterings that occur influencing the dynamics of\nelectrons traveling through the solid.\n0.00.51.0\n-20 0 20 40 60 80 100depth = 12 nmvelocityzup [nm/fs]\nTime [fs]τel = 25 fs\nτel = 12 fs\nτel = 2 fs\nFigure 6. Mean velocity of particles with spin up in z-direction\nat 12 nm for different elastic scattering times ( τel). The graph\nfor spin down electrons was omitted since it showed a similar\ntendency.\n5.2. Influence of secondary-electron generation\nFor the analysis of the influence of secondary electrons\n(SE) we compare calculations with and without SE. In\nFig. 7 the displacement of particle density in the ma-\nterial for spin up (left hand side) and spin down (right\nhand side) is presented. The lower subplots shows\ncalculations without including secondary electrons, la-\nbeled ”elastic scattering”, whereas the upper subplotsshows simulations including secondary electrons gen-\neration, labeled ”inelastic scattering” (it is the same\nas the upper panel of Fig. 5, repeated here for conve-\nnience). During the first femtoseconds, whether sec-\nondary electrons are generated or not, one observes a\nlargerconcentrationofparticlesnearthesurface. Later\nin time particles spread fast from the surface into the\nmaterial because more scatterings take place. This in-\ndicates that the generation of SEs increases the spread\nof the particles into the material. The increase in dis-\nplacement throughout the material can be examined\nbetter with the analysis of the motion regimes using\nthe mean square displacement (MSD).\n0510152025Spin upDepth [nm]Spin down\nIncluding secondary \n electron generation\n0510152025\n050100150200Depth [nm]\nTime [fs]\n 0  0.2  0.4  0.6  0.8  1  1.2\nElectron density [x1027 m-3]050100150200\nOnly elastic \n Scattering\nTime [fs]\nFigure 7. Evolution of particle density of spin up (lhs)\nand spin down (rhs) particles in the material. Lower figure:\nParticles travel through the material and they change their\ndirection of flight only (Elastic scattering). Top figure: Pa rticles\ntravel experiencing two scatterings, elastic scattering a nd impact\nionization which generates secondary electrons. Simulati on for\nτel= 12 fs.\nFigure 8 shows a comparison in the evolution\nof the transport exponent αfor two different elastic\nscattering times τelwith and without the inclusion of\nsecondary electron generation. Only the analysis for\nspinupelectronsisshownsincethespindownelectrons\npresent onaveragea similarbehaviorwith only slightly\ndifferent magnitudes. The data from Figs. 5 and 7\nare analyzed now as described in section 4 using the\nMSD with the transport exponent α. One can observe\nforτel= 12 fs and τel= 25 fs distinctively all three\nmotion regimes. Starting from ballistic, going through\nsuperdiffusive and finally becoming diffusive. Since the\nparticles can in principle be initially excited with an\narbitrary initial direction pointing into the material,\nin Fig. 8 during the first femtoseconds the motion is\nnot entirely ballistic.\nThe approach of the diffusive regime occurs at\ndifferent times, it is faster for smaller scattering\ntime. When the secondary electrons come into7\nplay, transition from superdiffusive into diffusive\nregime is delayed. The generation of secondary\nelectrons effectively increases the duration of electron\nexcitation, influencing the system and keeping it in\nthe superdiffusive regime ( α >1) for a larger time in\ncomparison with the other calculations. These results\nare in agreement with those in Refs. [9, 39].\n1.01.21.51.82.0\n1 10 100τel = 25 fs\nτel = 12 fsTransport exponent α\nTime [fs]Only elastic scattering\nIncluding secondary electron generation\nFigure 8. Analysis of the transport exponent αwhen using\ndifferent elastic scattering times and the influence of secon dary\nelectrons for spin up electrons. When the system has a lower\nelastic scattering time τelit relaxes faster into the diffusive\nregime whereas secondary electrons make this transition lo nger.\nInthestudyofspintransport, spincurrentdensity\nis one of the main features to be analyzed. The spin\ncurrent density jsis defined as\njs(z,t)∝q[∝angbracketleftη↑v↑∝angbracketright−∝angbracketleftη↓v↓∝angbracketright], (10)\nwhereqis the charge of the electron, η↑(η↓) andv↑\n(v↓)aretheparticledensityandthevelocityforspinup\n(spin down), respectively. With this definition the spin\ncurrent is positive if effectively more spin up electrons\nmoveintopositive zdirection. Thespincurrentdensity\nat a fixed depth of 12 nm with (solid line) and without\n(dashed line) secondary electron generation is shown\nin Fig. 9. One can observe that the spin current\ndensity changes quantitatively due to the continuous\ngeneration of secondary electrons, which feeds excited\nelectrons into the dynamics. Fig. 9 showsalso achange\nin the time of maximum intensity in the spin current\ndensity when secondary electrons are generated. As\na result, the propagation time is extended. In the\nbulk of the ferromagnet, the spin current does not\nchange sign during the whole simultation. We note\nthat this is different from spin-polarized transport in\nnormalmetalswheretheexcitationconditionstogether\nwith the transport characteristics can lead to a bipolar\nspin-current signal [9].-5051015202530\n-25 0 25 50 75 100τel = 25 fs\nτel = 12 fs\ndepth = 12 nmSpin current density \n [arb. units]\nTime [fs]Only elastic scattering\nIncluding secondary electron generation\nFigure 9. Spin current density at a depth of 12 nm for\ndifferent τelfor a simulation where it is compared two modeling\nassumptions: Including generation of secondary electrons (solid\nline) and not including them (dotted line).\n6. Summary\nIn conclusion, we developed a kinetic Monte Carlo\nmethod to study the influence of different electron-\nnucleus collision rates and generation of secondary\nelectrons in the ultrafast nonequilibrium spin and\ncharge transport in Iron. This method simulates\nkinetics of individual particles based on random\nsampling, making it a powerful tool for tracing\nelectrons throughout the material. In this simulation\nwe used the probability of excitation according to\nthe material’s density of state to excite and electron\nfrom an occupied band. The displacement of the\nparticles in the material were plotted, along with the\nmean velocity (velocityup\nz) to analyze the dynamics of\nexcited electrons for different elastic scattering times\nat certain depth. We found that lower scattering\ntimes increase the average velocities pointing into\nthe depth. Regarding the influence of the secondary\nelectron generation, we have focused on their impact\nin different regimes of motion and spin current density.\nGeneration of secondary electrons effectively delay the\nexcitation of free electrons. Therefore, they influence\nthe system by delaying the transition from one motion\nregime to another. 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B 86(2) 024404 URL\nhttps://link.aps.org/doi/10.1103/PhysRevB.86.024404"    },    {        "title": "1106.0355v1.Kinetics_of_Spin_Relaxation_in_Wires_and_Channels__Boundary_Spin_Echo_and_Tachyons.pdf",        "content": "Kinetics of Spin Relaxation in Wires and Channels: Boundary Spin Echo and\nTachyons\nValeriy A. Slipko1, 2and Yuriy V. Pershin1,\u0003\n1Department of Physics and Astronomy and USC Nanocenter,\nUniversity of South Carolina, Columbia, SC 29208, USA\n2Department of Physics and Technology, V. N. Karazin Kharkov National University, Kharkov 61077, Ukraine\nIn this paper we use a spin kinetic equation to study spin polarization dynamics in 1D wires\nand 2D channels. This approach is valid in both di\u000busive and ballistic spin transport regimes and,\ntherefore, more general than the usual spin drift-di\u000busion equations. In particular, we demonstrate\nthat in in\fnite 1D wires with Rashba spin-orbit interaction the exponential spin relaxation decay can\nbe modulated by an oscillating function. In the case of spin relaxation in \fnite length 1D wires, it\nis shown that an initially homogeneous spin polarization spontaneously transforms into a persistent\nspin helix. An interesting sound waves echo-like behavior of initially localized spin polarization\npacket is found in \fnite length wires. We show that a propagating spin polarization pro\fle re\rects\nfrom a system boundary and returns back to its initial position similarly to the re\rectance of sound\nwaves from an obstacle. Green's function of spin kinetic equation is found for both \fnite and\nin\fnite 1D systems. Moreover, we demonstrate explicitly that the spin relaxation in 2D channels\nwith Rashba and Dresselhaus spin-orbit interactions of equal strength occurs similarly to that in 1D\nwires of \fnite length. Finally, a simple transformation mapping 1D spin kinetic equation into the\nKlein-Gordon equation with an imaginary mass is found thus establishing an interesting connection\nbetween semiconductor spintronics and relativistic quantum mechanics.\nPACS numbers: 72.15.Lh, 72.25.Dc, 85.75.2d, 14.80.-j\nI. INTRODUCTION\nDynamics of electron spin polarization in semiconduc-\ntor structures1has attracted a lot of attention recently\nin the context of spintronics2. Typically, the most im-\nportant mechanism of spin relaxation in semiconductors\nlacking inversion symmetry is the D'yakonov-Perel' spin\nrelaxation mechanism3,4, intimately related to the spin\nsplitting of the electronic states. Numerous studies of\nD'yakonov-Perel' spin relaxation in the past have dom-\ninantly concentrated on spin relaxation in in\fnite two-\ndimensional (2D) systems3{16, while the electron spin\nrelaxation in con\fned geometries is not yet well under-\nstood. This problem, however, is of crucial importance\nsince actual spin-based electronic devices17are normally\nof a \fnite size.\nThere are only several examples in the literature where\nthe e\u000bects of boundary conditions on D'yakonov-Perel'\nspin relaxation have been explored theoretically and/or\nexperimentally. In particular, investigations of spin re-\nlaxation in systems with boundaries include spin re-\nlaxation in 2D channels18{23, 2D half-space24, 2D sys-\ntems with antidots25, large quantum dots26,27, and one-\ndimensional (1D) \fnite-length wires28and rings29. The\ncommon conclusion of these studies is that in the di\u000busive\nregime of spin transport the introduction of boundary\nconditions results in an increased spin lifetime. In par-\nticular, in Ref. 28 the present authors studied spin relax-\nation in \fnite length quantum wires using drift-di\u000busion\nequations approach. It was demonstrated that a persis-\ntent spin helix30,31spontaneously emerges in course of\nrelaxation of homogeneous spin polarization.\nIn this paper, we extend the results of our previousstudy28applying spin kinetic equation approach to the\nproblem of spin relaxation in 1D wires and 2D channels\nwith spin-orbit interaction. The formalism of spin kinetic\nequation describes both di\u000busive and ballistic regimes\nof spin transport at any value of spin rotation angle\nper mean free path. Therefore, the spin kinetic equa-\ntion is more general than traditional spin drift-di\u000busion\nequations6{8,32,33and can be strictly justi\fed by using\nquasiclassical Green's functions19,34. Basically, the spin\nkinetic equation describes the spin polarization dynam-\nics on shorter time and space scales than those of spin\ndrift-di\u000busion equations. Consequently, the application\nof spin kinetic equation to the problem of spin relaxation\nprovides new insights into the spin polarization dynam-\nics. In particular, in this paper we report the bound-\nary spin echo e\u000bect in which a localized spin polariza-\ntion pro\fle re\rects from a sample boundary and returns\nwith a decreased amplitude to its initial position simi-\nlarly to the usual spin echo of sounds waves. This type\nof spin echo is essentially di\u000berent from the spin echo\nin nuclear magnetic resonance. The later is related to\nnuclear spins dephasing in an applied magnetic \feld35.\nMoreover, we \fnd a simple transformation that maps\nthe spin kinetic equation into the Klein-Gordon equation\nwith an imaginary mass, which is a relativistic analog of\nthe Schr odinger's equation. It is worth mentioning that\nthe Klein-Gordon equation with an imaginary mass de-\nscribes tachyons36{39- hypothesized particles that travel\nfaster than light. Consequently, we suggest that semi-\nconductor spintronics structures have a potential to be\nused as a laboratory test bed for relativistic quantum\nmechanics.\nOur paper is organized as follows. In Sec. II we intro-arXiv:1106.0355v1  [cond-mat.mes-hall]  2 Jun 20112\nduce a spin kinetic equation and derive a set of equations\nfor spin polarization components and boundary condi-\ntions. We also demonstrate that the spin kinetic equa-\ntion can be transformed into the Klein-Gordon equa-\ntion. Next, in Sec. III, we employ the spin kinetic\nequation to study spin relaxation in in\fnite and \fnite\nlength wires with Rashba spin-orbit interaction. In par-\nticular, in Sec. III D, we present the boundary spin echo\ne\u000bect. The spin relaxation in two-dimensional channels\nwith Rashba and Dresselhaus40spin-orbit interactions of\nequal strength is investigated in Sec. IV. We demonstrate\nthat such a problem, for channels in a speci\fc direction,\ncan be mapped into the problem of spin relaxation in\n\fnite length wires with Rashba interaction only. The\nresults of our investigations are summarized in Sec. V.\nII. THEORETICAL FRAMEWORK\nA. Spin kinetic equation\nThe main goal of this paper is to study the kinetics\nof spin relaxation in wires made of 2D quantum well\nor heterostructure with Rashba spin-orbit interaction41.\nTherefore, we \frst introduce the Hamiltonian for an elec-\ntron in 2D space in the presence of spin-orbit interac-\ntion and all important parameters that will be used later\nin the spin kinetic equation. The standard Hamiltonian\nwith the Rashba41term is given by\nH=H0+HR=p2\n2m+\u000b(\u001b\u0002p)\u0001z; (1)\nwhere p= (px;py) is the 2D electron momentum oper-\nator,mis the e\u000bective electron's mass, \u001bis the Pauli-\nmatrix vector, \u000bis the spin-orbit coupling constant and\nzis a unit vector perpendicular to the con\fnement plane.\nIt is not di\u000ecult to demonstrate33that in the case of\nHamiltonian (1) the quantum mechanical evolution of a\nspin of an electron with a momentum pcan be reduced\nto a spin rotation with the angular velocity \n = 2 \u000bp=~\nabout the axis determined by the unit vector n=p\u0002z=p.\nIn this way, the spin-orbit coupling constant \u000benters into\nequations through the parameter \u0011= 2\u000bm~\u00001, which\ngives the spin precession angle per unit length.\nBesides this evolution, 2D electrons experience dif-\nferent bulk scattering events such as, for example, due\nto phonons or impurities. These scatterings random-\nize the electron trajectories. Correspondingly, the di-\nrection of spin rotation becomes \ructuating what causes\naverage spin relaxation (dephasing). This is the famous\nD'yakonov-Perel' spin relaxation mechanism.3,4The time\nscale of the bulk scattering events can then be character-\nized by a single rate parameter, the momentum relax-\nation time \u001c. It is connected to the mean free path by\n`=v\u001c, wherev=p=m is the mean electron velocity.\nTo take into account these scatterings we use a kinetic\nmodel of spin transport presented below. When a char-\nacteristic system size L\u001d`and the time scale of interestis much longer than \u001c, the spin kinetic model yields the\nspin drift-di\u000busion equations (such as, e.g., reported in\nRef. 33).\nIn the semi-classic approximation the kinetic equation\nfor electron spin polarization can be written as (see, e.g.,\nRefs. 26, 10, 19, 34)\n\u0012@\n@t+p\nm\u0001r\u0013\nSp=\np\u0002Sp+StfSpg; (2)\nwhere Sp(r;t) is the vector of spin polarization of elec-\ntrons moving with momentum p, andStfSpgis the colli-\nsion integral describing electron scattering processes. In\nthe\u001c-approximation the collision integral is given by\nStfSpg=\u00001\n\u001c(Sp\u0000hSpi); (3)\nwhere the angle brackets denote averaging over direction\nof electron momentum. The conditions of applicability\nof Eq. (2) can be found in Ref. 19. The collision integral\n(3) corresponds to the elastic scattering of electrons by\nstrong scatterers with a characteristic time \u001cbetween the\ncollisions. Note that the collision integral (3) conserves\nthe total spin polarization redistributing spin polariza-\ntion between electrons moving in di\u000berent directions. For\n1D case the average spin polarization simpli\fes to the fol-\nlowing expression hSpi= (S++S\u0000)=2, where S+andS\u0000\nare the spin polarizations of electrons moving along the\nwire in the positive (with momentum p=mvex), and\nnegative ( p=\u0000mvex) directions with the average veloc-\nityv. Thus, the kinetic equation (2) for 1D wire takes\nthe form of the system of two vector equations\n\u0012@\n@t+v@\n@x\u0013\nS+=\u0000\ney\u0002S+\u00001\n2\u001c(S+\u0000S\u0000);(4)\n\u0012@\n@t\u0000v@\n@x\u0013\nS\u0000= \ney\u0002S\u0000\u00001\n2\u001c(S\u0000\u0000S+):(5)\nThis system of equations should be complimented by ini-\ntial conditions for spin densities S+andS\u0000\nS+(x;t= 0) = S+\n0(x); (6)\nS\u0000(x;t= 0) = S\u0000\n0(x); (7)\nand by boundary conditions for \fnite length wires. The\nboundary conditions conserving the spin polarization in\nelastic scatterings at the boundary \u0000 = [ x= 0;x=L]\nhave the form\n(S+\u0000S\u0000)j\u0000= 0: (8)\nTaking the sum and di\u000berence of Eqs. (4, 5) we easily\nobtain\n@S\n@t+v@\u0001\n@x+ \ney\u0002\u0001= 0; (9)\n@\u0001\n@t+v@S\n@x+ \ney\u0002S+\u0001\n\u001c= 0; (10)3\nwhere the following notations are used: S=S++S\u0000and\n\u0001=S+\u0000S\u0000. As we are mainly interested in \fnding\nthe total spin polarization S,\u0001can be eliminated from\nEqs. (9), (10) via the following transformation. First of\nall, we multiply Eq. (10) by et=\u001cand rewrite it as\n@\u0000\net=\u001c\u0001\u0001\n@t+vet=\u001c@S\n@x+ \net=\u001cey\u0002S= 0: (11)\nThen, Eq. (9) is multiplied by et=\u001cand is di\u000berentiated\nwith respect to time\n@\n@t\u0012\net=\u001c@S\n@t\u0013\n+v@\n@x@\u0000\net=\u001c\u0001\u0001\n@t+ \ney\u0002@\u0000\net=\u001c\u0001\u0001\n@t= 0:\n(12)\nFinally, we substitute @\u0000\net=\u001c\u0001\u0001\n=@tfrom Eq. (11) into\nEq. (12). The resulting equations for spin polarization\ncan be presented as\n@2S\n@t2+1\n\u001c@S\n@t\u0000v2@2S\n@x2\u00002\nvey\u0002@S\n@x\n+ \n2(S\u0000Syey) = 0;(13)\nwhereSyis they-component of S.\nNext, we would like to reformulate the boundary con-\ndition given by Eq. (8) in terms of the function Sonly.\nIt is easy to notice that Eq. (8) corresponds to \u0001j\u0000= 0.\nSubstituting this boundary value of \u0001into Eq. (10) we\nobtain\n\u0012\nv@S\n@x+ \ney\u0002S\u0013\f\f\f\f\n\u0000= 0: (14)\nPreviously, the same form of boundary condition was de-\nrived using the Green's function method42.\nMoreover, since Eq. (13) is the second order di\u000beren-\ntial equation with respect to time, one must specify both\nthe spin polarization and its time derivative at the initial\nmoment of time t= 0\nS(x;t= 0) = S0(x);\u0012@S\n@t\u0013\nt=0=_S0(x): (15)\nNote that if we know \u0001at the initial moment of time\nt= 0 then we can \fnd the \frst time derivative of Sat\nt= 0 using Eq. (9). In particular, if \u0001( x;t= 0) = 0\nthen it follows from Eq. (9) that _S0(x) = 0.\nConsidering y-components of Eqs. (13)-(14) we \fnd\nthatSyis not coupled to any other component of spin\npolarization. Speci\fcally, Eqs. (13)-(14) for Sycan be\nrewritten as\n@2Sy\n@t2+1\n\u001c@Sy\n@t\u0000v2@2Sy\n@x2= 0; (16)\n@Sy\n@x\f\f\f\f\n\u0000= 0: (17)\nConsequently, selecting Sy(x;t= 0) = 0 we can safely\ntake outSyfrom our consideration.Let us introduce a complex polarization\nS=Sx+iSz: (18)\nIt is straightforward to show that Eq. (13), and boundary\nconditions (14) can be rewritten in a more compact form\nusingS:\n@2S\n@t2+1\n\u001c@S\n@t\u0000\u0012\nv@\n@x\u0000i\n\u00132\nS= 0; (19)\n\u0012\nv@S\n@x\u0000i\nS\u0013\f\f\f\f\n\u0000= 0: (20)\nMoreover, it follows from Eqs. (19) and (20) that they\nhave only one stationary solution\nS=S0ei\u0011x; (21)\nwhereS0is an arbitrary complex constant. This solu-\ntion is so-called spin helix9,31, which is persistent in 1D\nRashba wires despite electron collisions. Taking the real\nand imaginary parts of Eq. (21) with S0=Aei\u000e, we \fnd\nthe usual representation of spin helix\nSx=Acos(\u0011x+\u000e); Sz=Asin(\u0011x+\u000e); (22)\nwhereAis the spin helix amplitude, and \u000eis its phase.\nSince the solution (22) of Eqs. (19) and (20) is the only\nstationary one, any initial spin polarization distribution\neventually transforms into the spin helix28, in some cases,\nhowever, having zero amplitude A= 0.\nB. Klein-Gordon equation\nWe can exclude rotations of spin polarization vector\nthat are still present in Eq. (19) and Eq. (20) by intro-\nducing a complex \feld uvia\nu(x;t) =e\u0000i\u0011xet\n2\u001cS(x;t): (23)\nIt can be shown that Eq. (19) and Eq. (20) transform\ninto\n@2u\n@t2\u0000v2@2u\n@x2\u00001\n4\u001c2u= 0; (24)\n@u\n@x\f\f\f\f\n\u0000= 0: (25)\nThe transformation given by Eq. (23) is useful, in partic-\nular, since it allows us writing down the Green's function\nSG(x;t) of Eq. (19) through the known Green's function\nof Eq. (24) (we will take advantage of this fact below).\nWe note that Eq. (24) coincides with the well-known\nKlein-Gordon equation (see, for example, Ref. 43), but\nwith an imaginary \"mass\" (see the sign of the last term).\nIn relativistic quantum mechanics such an equation de-\nscribes hypothetical particles called tachyons36{39. Be-\ncause of the transformation (23), the tachyon physics4\nis somewhat 'hidden' in the dynamics of spin polariza-\ntion and can be recovered by an inverse transforma-\ntion. Analogies between certain exotic particles and con-\ndensed matter systems have attracted a strong attention.\nFor example, electrons in graphene44and graphite45ex-\nhibit properties of Dirac fermions, signatures of mag-\nnetic monopoles were spotted in spin ice materials46,\nand some exotic quasi-particle excitations in a variety\nof condensed-matter systems show a similarity with Ma-\njorana fermions47. Therefore, we believe that our ob-\nservation that is exciting in itself will stimulate further\ntheoretical and experimental work in this area.\nIII. SPIN RELAXATION IN WIRES\nIn this section we present our studies of kinetics of\nspin relaxation in 1D in\fnite and \fnite length wires with\nRashba spin-orbit coupling using the spin kinetic equa-\ntion approach. Our analysis is essentially based on Eq.\n(19) supplemented by initial and, where appropriate, by\nboundary conditions (20). It is assumed that a thin\nnarrow wire is made of a semiconductor heterostructure\nwith spin-orbit coupling and the conduction electrons in\nthe wire occupy the lowest size-quantization level cor-\nresponding to transverse con\fnement. In addition, the\nlength of \fnite length wires is considered to be much\nlonger than the phase coherence length. Therefore, the\nelectron transport in the direction along the wire is de-\nscribed in terms of the classical transport regime.\nA. Relaxation of homogeneous polarization in\nin\fnite wires\nLet us start the analysis of solutions of Eq. (19) with\nthe most simple case, namely, with a problem of relax-\nation of homogeneous initial spin polarization in an in-\n\fnite wire,\u00001< x <1. In this case, assuming that\nsolutions of Eq. (19) do not depend on x, we can rewrite\nthis equation and initial conditions as\nd2S\ndt2+1\n\u001cdS\ndt+ \n2S= 0; (26)\nS(t= 0)\u0011S(0) =Sx(0) +iSz(0);dS\ndt\f\f\f\f\nt=0= 0:(27)\nThe solution of Eq. (26) with initial conditions (27) can\nbe represented in the form\nS(t) =S(0)e\u0000t\n2\u001c\u0012\ncosh\u0014t+sinh\u0014t\n2\u001c\u0014\u0013\n; (28)\nwhere\u0014=p\n1=(4\u001c2)\u0000\n2. It should be emphasized that\nthe parameter \u0014can take both real and imaginary values\ndepending on system parameters. Moreover, in addition\nto describing spin polarization decay in in\fnite wires, Eq.\n(28) also describes spin relaxation at long distances from\nboundaries of \fnite length (and semi-in\fnite) wires.Generally, we can distinguish two spin relaxation\nregimes. It follows from Eq. (28) that when 2 \u001c\n<1\n(Im(\u0014) = 0), the spin relaxation is described by two ex-\nponents with relaxation rates 1 =(2\u001c)\u0006\u0014. In the limit of a\nsmall spin precession angle per mean free path, \u001c\n\u001c1,\nwe \fnd\nS(t) =S(0)e\u0000\u001c\n2t: (29)\nThe above expression coincides with the spin relaxation\ntime predicted by D'yakonov-Perel' theory for 1D relax-\nation. It is clearly seen that in this situation the relax-\nation of spin polarization is characterized by the time\nconstant (\u001c\n2)\u00001, which is much longer than \u001c.\nThe second regime of spin relaxation is realized when\n2\u001c\n>1. In this case \u0014is purely imaginary meaning that\nthe spin relaxation decay described by Eq. (28) consists\nof an exponential decay with a rate of 1 =(2\u001c) modulated\nby oscillating functions. In the limiting case \u001c\n\u001d1 we\nobtain\nS(t) =S(0)e\u0000t=(2\u001c)cos \nt: (30)\nBoth regimes of spin relaxation are presented in Fig. 1.\nB. Relaxation of inhomogeneous polarization in\nin\fnite wires\nAs it is mentioned below Eq. (24), the Green's func-\ntion of Eq. (19), SG(x;t), can be obtained performing a\nback transformation (from utoS) of the known48Green's\nfunction of Eq. (24) (the forward transformation is given\nby Eq. (23)). Following this procedure we \fnd\nSG(x;t) =1\n2v\b(vt\u0000jxj)ei\u0011x\u0000t\n2\u001cI0 p\nv2t2\u0000x2\n2v\u001c!\n;(31)\nwhere \b(t) is the Heaviside step function, and I0(t) is\nthe modi\fed Bessel function of zero order. We note that\nthe Green's function (31) of Eq. (19) describes the evo-\nlution of a point excitation of spin polarization with the\nfollowing initial conditions\nSG(x;t) = 0 fort\u00140;@SG\n@t\f\f\f\f\nt=0=\u000e(x): (32)\nThe Green's function (31) can be employed to determine\nthe spin polarization at any moment of time for any given\ninitial conditions using the relation\nS(x;t) =\u0014@\n@t+1\n\u001c\u00151Z\n\u00001d\u0018SG(x\u0000\u0018;t)S(\u0018;0)\n+1Z\n\u00001d\u0018SG(x\u0000\u0018;t)_S(\u0018;0):(33)5\n0 40 80 120 160 2000.00.20.40.60.81.0\n  Sz/Sz(0)\nt/τ  τκ = 0.45\n  τκ = 0.48\n  τκ = 0.49(a)\n0369 1 2-0.50.00.51.0\n  Sz/Sz(0)\nt/τ  τκ = 3 i\n  τκ = 5 i(b)\nFIG. 1: (Color online) Dynamics of spin relaxation of ho-\nmogeneous spin polarization in in\fnite wires. (a) and (b)\ncorrespond to two di\u000berent regimes of spin relaxation of Eq.\n(28) as discussed in the text. The values of \u0014are indicated\non the plots.\nIn order to better understand the meaning of Eq. (31),\nlet us consider the spin dynamics of the following initial\nexcitation of spin polarization in zdirection:\nS(x;t= 0) = 0;@S(x;t)\n@t\f\f\f\f\nt=0=i\u000e(x): (34)\nAccordingly to Eq. (33), at any t\u00150 the spin polariza-\ntion in the system is given by\nSx(x;t) =\u0000\b(vt\u0000jxj)\n2vsin\u0011xe\u0000t\n2\u001cI0 p\nv2t2\u0000x2\n2v\u001c!\n;(35)\nSz(x;t) =\b(vt\u0000jxj)\n2vcos\u0011xe\u0000t\n2\u001cI0 p\nv2t2\u0000x2\n2v\u001c!\n:(36)\nThis solution describes a propagation of the initial ex-\ncitation of spin polarization in both directions from the\nexcitation point with well de\fned fronts. Such a propaga-\ntion also involves the spin precession and spin relaxation.\n-3 -2 -1 0 1 2 30.00.20.40.60.81.01.2 t = 0.001τ\n t = 1τ\n t = 2τ \n  Sz\nx/l(a)\n-3 -2 -1 0 1 2 3-0.04-0.020.000.020.04\n t = 0.001τ\n t = 1τ\n t = 2τ \n  Sx\nx/l(b)FIG. 2: (Color online) Dynamics of Sz(a) andSx(b) com-\nponents of spin polarization of an initial Gaussian spin polar-\nization pro\fle with spin polarization pointing in zdirection\n(S(x;t= 0) =iexp(\u0000100x2=l2)). The spin polarization com-\nponents are calculated using Eq. (33). This plot was obtained\nusing the parameter value \u0011l= 0:1:\nFig. 2 shows the dynamics of spin polarization in an in-\n\fnite wire found using Eq. (33) with di\u000berent initial con-\nditions. Speci\fcally, we assume a Gaussian spin polariza-\ntion pro\fle and zero time derivative of spin polarization\natt= 0. This plot shows that the initial spin polarization\npro\fle splits into left- and right-moving packets whose\namplitude in zdirection decreases in time. At the same\ntime, the amplitude of Sxin packets increases because of\nthe spin precession. It should be emphasized that such\nmoving packets of spin polarization are not captured by\nthe drift-di\u000busion schemes of spin transport6{8,32,33. In\naddition, we note that an area with a \rat Szestablishes\nbetween the moving packets. This region of \rat distri-\nbution ofSzcan be considered as a precursor of di\u000busive\ndynamics.\nMoreover, the expression for the Green's function (31)6\ntakes a simpler form at some speci\fc points. For example,\nif we consider the position of the moving front, then it is\neasy to obtain\nSG(x=vt\u00000;t) =1\n2vei\u0011xe\u0000t\n2\u001c: (37)\nThis expression basically means that the amplitude of\nthe moving front exponentially decreases with the time\nconstant 2\u001c. Another interesting point is x= 0. We \fnd\nthat atx= 0 the Green's function (31) is a real function\nof time:\nSG(0;t) =1\n2ve\u0000t\n2\u001cI0\u0012t\n2\u001c\u0013\n: (38)\nIts asymptotic values at short and long times can be cal-\nculated. In particular, at short times t\u001c\u001c\nSG(0;t) =1\n2v\u0012\n1\u00001\n2t\n\u001c+3\n16t2\n\u001c2+O\u0012t3\n\u001c3\u0013\u0013\n;(39)\nand at long times t\u001d\u001c\nSG(0;t) =1\n2vr\u001c\n\u0019t\u0010\n1 +O\u0010\u001c\nt\u0011\u0011\n; (40)\nwhich corresponds to the well-known di\u000busive behavior.\nC. Relaxation of homogeneous polarization in\n\fnite length wires\nLet us consider the problem of spin relaxation in wires\nof \fnite length L, 0<x<L , when at the initial moment\nof time the electron spins are homogeneously polarized\nalongz-axis,\nS(x;0) =iS0;@S\n@t\f\f\f\f\nt=0= 0: (41)\nWe \fnd the solution of Eq. (19) with the boundary con-\nditions (20) and initial conditions (41) using the standard\nmethod of separation of variables. A straightforward ap-\nplication of this method leads to the following expression\nfor the complex spin polarization\nS(x;t)\nS0=isin(\u0011L=2)\n\u0011L=2ei\u0011(x\u0000L=2)+ 2\u0011Lei\u0011x\u0000t=(2\u001c)\n\u0002+1X\nn=11\u0000(\u00001)ne\u0000i\u0011L\n(\u0011L)2\u0000(\u0019n)2\u0012\ncosh\u0014nt+sinh\u0014nt\n2\u001c\u0014n\u0013\ncos\u0019nx\nL;(42)\nwhere\u0014n=p\n1=(4\u001c2)\u0000\u00192n2v2=L2.\nPreviously, we investigated the spin relaxation in \f-\nnite length wires in the di\u000busive regime and found that\nan initially homogeneous spin polarization in zdirection\ntransforms into a persistent spin helix at long times28.\nThe same qualitative result follows from Eq. (42): note\nthat only the \frst term in the right-hand side of Eq.\n0.0 0.5 1.0 1.5 2.0-0.50.00.51.0\n t = 0τ       t = 0.1τ\n t = 0.3τ    t = 1τ\n t = 5τ       t = 50τ\n  Sz/S0\nx/l(a)\n0.0 0.5 1.0 1.5 2.0-1.0-0.50.00.51.0\n t = 0τ       t = 0.1τ\n t = 0.3τ    t = 1τ\n t = 5τ       t = 50τ\n  Sx/S0\nx/l(b)FIG. 3: (Color online) Dynamics of formation of persistent\nspin helix in the ballistic regime of spin transport. This plot\nwas obtained using the parameter values \u0011l= 2 andL= 2l:\n(42) survives at long times. However, Eq. (42) by itself\nis more complex compared to those reported in Ref. 28\nfor the case of di\u000busive spin transport because it incor-\nporates both di\u000busive and ballistic spin transport. It can\nbe shown that Eq. (42) gives the same result28in the\ndi\u000busive limit. Indeed, when `=v\u001c\u001cLandt\u001d\u001cit is\neasy to demonstrate that\nlim\n\u001c!0;`!0e\u0000t=(2\u001c)\u0012\ncosh\u0014nt+sinh\u0014nt\n2\u001c\u0014n\u0013\n=e\u0000\u00192n2Dt\nL2;(43)\nwhereD=`2=\u001c. Substituting Eq. (43) into Eq. (42)\nwe obtain the same expression for S(x;t) as in Ref. 28.\nThe dynamics of formation of persistent spin helix in\nthe ballistic regime of spin transport is depicted in Fig.\n3. This \fgure is obtained plotting Eq. (42) at di\u000berent\nmoments of time. One notices that the persistent spin\nhelix is almost entirely formed within a short \u00185\u001ctime\ninterval.\nLet us consider the ballistic limit in more details. In\nthis case assuming `\u001dL, 2\u0014n\u001c=i\u00192\u0019n`=L\u001d1 the\nsecond term in the big round brackets in Eq. (42) can be7\nomitted (since it involves a small multiplier 1 =(2\u0014n\u001c)). Then, we obtain\nS(x;t)\nS0=isin(\u0011L=2)\n\u0011L=2ei\u0011(x\u0000L=2)+\u0011Lei\u0011x\u0000t=(2\u001c)+1X\nn=11\u0000(\u00001)ne\u0000i\u0011L\n(\u0011L)2\u0000(\u0019n)2\u0012\ncos\u0019n(x+vt)\nL+ cos\u0019n(x\u0000vt)\nL\u0013\n: (44)\nWe can sum the series in Eq. (44) taking into account the\nfact that at the initial moment of time, t= 0, the right-\nhand side of Eq. (44) is equal to i. Moreover, since these\nFourier series are even and 2 L-periodic, we can \fnally\nobtain a very simple expression for the spin polarization\nin the ballistic limit:\nS(x;t)\nS0=isin(\u0011L=2)\n\u0011L=2ei\u0011(x\u0000L=2)\u0010\n1\u0000e\u0000t=(2\u001c)\u0011\n+i\n2e\u0000t=(2\u001c)\u0010\nei\u0011(x\u0000j]x\u0000vtj)+ei\u0011(x\u0000j]x+vtj)\u0011\n;(45)\nwhereez=z+ 2Ln, andnis an integer number selected\nin such a way that for any real z,\u0000L<ez\u0014L.\nD. Relaxation of inhomogeneous polarization in\n\fnite length wires\nNext, we investigate the dynamics of relaxation of an\narbitrary spin polarization pro\fle in \fnite length wires.\nFor this purpose, we obtain an expression for the Green's\nfunctionG(x;t;\u0018) that allows \fnding the spin polariza-\ntion pro\fle at any moment of time for given initial con-\nditions. The Green's function G(x;t;\u0018) is calculated by\nmapping Eq. (23) into Eq. (24) taking into account the\nboundary conditions (25) by an even mapping of the ini-\ntial condition on an in\fnite wire. It is straightforward to\nshow thatG(x;t;\u0018) can be written as\nG(x;t;\u0018) =ei\u0011(x\u0000\u0018)\u0000t=(2\u001c)\n2v+1X\nn=\u00001[\b(vt\u0000jx\u0000\u0018\u00002Lnj)\n\u0002I0 p\nv2t2\u0000(x\u0000\u0018\u00002Ln)2\n2v\u001c!\n+\n\b(vt\u0000jx+\u0018\u00002Lnj)I0 p\nv2t2\u0000(x+\u0018\u00002Ln)2\n2v\u001c!#\n:(46)\nThe spin polarization distribution along the wire at any\nmoment of time can be calculated using the following\nequation:\nS(x;t) =\u0014@\n@t+1\n\u001c\u0015ZL\n0d\u0018G(x;t;\u0018)S(\u0018;0)\n+ZL\n0d\u0018G(x;t;\u0018)_S(\u0018;0): (47)As an example of spin dynamics calculation in \fnite\nlength wires, let us consider the evolution of a Gaussian\ninitial spin polarization pro\fle. The time derivative of\nspin polarization at t= 0 is set equal to zero (this is the\nsecond initial condition). Fig. 4 shows results of our cal-\nculations based on Eqs. (46) and (47). Speci\fcally, it is\nclearly seen that the initial spin polarization pro\fle splits\ninto two packets moving in opposite direction similarly\nto the result shown in Fig. 2 for the case of in\fnite wire.\nThe amplitude of Szin these packets decreases in time.\nAtt= 2\u001cthe left-moving packet reaches the wire bound-\nary and is re\rected back changing its direction of motion.\nWe call such a re\rected packet of spin polarization as the\nboundary spin echo to distinguish it from the spin echo\nin nuclear magnetic resonance (NMR)35. The boundary\nspin echo is indeed much closer reassembles the echo of\nsound waves than those in NMR. We also note that at\nlong times any initial spin polarization pro\fle (including\none shown in Fig. 4) shapes into the persistent spin helix\ncharacterized by a certain amplitude and phase.\nIV. SPIN RELAXATION IN 2D CHANNELS\nLet us consider a two-dimensional electron gas in the\npresence of linear in the wave vector Rashba and Dres-\nselhaus spin-orbit couplings. The electron Hamiltonian\nis written as\nH=p2\n2m+\u000b(py\u001bx\u0000px\u001by) +\f(px\u001bx\u0000py\u001by);(48)\nwhere\u000band\fare strengths of Rashba and Dresselhauss\nspin-orbit couplings correspondingly. In the speci\fc case\nthat we consider below, namely when \u000b=\f, the Hamil-\ntonian (48) takes the form\nH=p2\n2m+\u000b(\u001bx\u0000\u001by)(px+py): (49)\nNext, consider a rotation about zaxis by an angle '=\n\u0019=4. Then,xis transformed into x0= (x+y)=p\n2 and\ny0= (y\u0000x)=p\n2. The transformation of the spin part\nof the Hamiltonian (49) is performed using the standard\nunitary transformation\nU=\u0012\nei'=20\n0e\u0000i'=2\u0013\n: (50)8\n012345601234\nt =3.5τt =3τt =2.5τt =2τt =1.5τt =τt =0.5τ\n  Sz\nx/l(a)\nt =0\n01234560.00.10.20.30.4\nt =3.5τt =3τt =2.5τt =2τt =1.5τt =τt =0.5τ\n  Sx\nx/l(b) t =0\nFIG. 4: (Color online) The boundary spin echo in a \fnite\nlength wire. The evolution of an initially Gaussian spin\npolarization pro\fle pointing in zdirection (S(x;t= 0) =\niexp(\u0000100x2=l2)) was found using Eqs. (46) and (47). It is\nclearly seen that the spin polarization packet initially mov-\ning in \u0000zdirection moves back after the re\rection from the\nboundary. This plot was obtained using the parameter values\n\u0011l= 0:1,L= 6l. The curves are displaced for clarity.\nIn the new basis the Hamiltonian (49) takes the form\nH0=UHUy=(p)2\n2m\u00002\u000bpx0\u001b0\ny0; (51)\nwhere\u001b0\ny0=\u001bycorresponds to the projection of spin\noperator in the rotated basis on y0. Under the action\nof the Hamiltonian H0, the spin of an electron precesses\nwith an angular velocity \n0\np=\u0000\n0ey0cos\u0012, where\u0012is\nthe angle between the direction of electron momentum\nandx0axis and \n0= 4\u000bp=~.\nNow, let us consider electron spin dynamics in a chan-\nnel of width Linx0direction and of in\fnite length in\ny0direction. Our goal is to show that the evolution of\nspin polarization (homogeneous in y0direction) in such achannel is similar to the evolution of spin polarization in\n\fnite length wires considered in Sec. III. Assuming that\nthe spin polarization in the channel depends only on x0\nandSy0= 0, we can combine Eqs. (2) and (3) as\n@S(\u0012)\n@t+vcos\u0012@S(\u0012)\n@x0=\u0000\n0cos\u0012ey0\u0002S(\u0012)\n\u00001\n\u001c\u0014\nS(\u0012)\u0000Z2\u0019\n0d\u0012\n2\u0019S(\u0012)\u0015\n;(52)\nwhere the following shorthand notation is used: S(\u0012) =\nS(\u0012;x0;t). Introducing a complex spin polarization\nS(\u0012) =Sx0(\u0012)+iSz(\u0012), we arrive to the following integral\nequation\n@S(\u0012)\n@t+vcos\u0012@S(\u0012)\n@x=i\n cos\u0012S(\u0012)\n\u00001\n\u001c\u0014\nS(\u0012)\u0000Z2\u0019\n0d\u0012\n2\u0019S(\u0012)\u0015\n(53)\ncomplimented by the boundary condition\n(S(\u0012)\u0000S(\u0019\u0000\u0012))j\u0000= 0: (54)\nWe seek the solution of Eq. (53) in the form of Fourier\nseries\nS(\u0012) =\u001b0++1X\nn=1(\u001bncosn\u0012+ \u0016\u001bnsinn\u0012); (55)\nwhere\u001bnand \u0016\u001bnare functions of x0andt. In particular,\nit is clear that \u001b0=R2\u0019\n0d\u0012S(\u0012)=(2\u0019) is proportional to\nthe spin polarization.\nSubstituting Eq. (55) into Eq. (53) and using the\ncompleteness of the set of function f1;cosn\u0012;sinn\u0012gwe\n\fnd in\fnite series of interconnected equations. The two\n\frst equations in the series are:\n@\u001b0\n@t+1\n2\u0012\nv@\n@x0\u0000i\n0\u0013\n\u001b1= 0; (56)\n@\u001b1\n@t+\u0012\nv@\n@x0\u0000i\n0\u0013\u0010\n\u001b0+\u001b2\n2\u0011\n+\u001b1\n\u001c= 0:(57)\nFor a weakly anisotropic electron momentum distribu-\ntionj\u001b2j\u001cj\u001b0j, therefore, we can truncate the series\nby setting\u001b2= 0 in Eq. (57). Thus, Eq. (57) can be\nrewritten as\n@\u001b1\n@t+\u0012\nv@\n@x0\u0000i\n0\u0013\n\u001b0+\u001b1\n\u001c= 0 (58)\nExcluding\u001b1from Eqs. (56), (58), we obtain the follow-\ning equation for the total spin polarization \u001b0:\n@2\u001b0\n@t2+1\n\u001c@\u001b0\n@t\u0000\u0012vp\n2@\n@x0\u0000i\n0\np\n2\u00132\n\u001b0= 0 (59)\nwhich is of the similar form with Eq.(19). This is the\nmain result of Sec. IV proving similarity of spin relax-\nation in \fnite length wires and 2D channels.9\nThe boundary condition for spin dynamics in the chan-\nnel can be obtained in the following way. Substituting the\napproximate solution of Eq. (53),\nS(\u0012) =\u001b0+\u001b1cos\u0012+ \u0016\u001b1sin\u0012; (60)\ninto the boundary condition Eq. (54), we immediately\nnotice that the function \u001b1on the channel boundary must\nturn to zero, namely\n\u001b1j\u0000= 0: (61)\nCombining Eq. (61) with (58), we \fnd the corresponding\nboundary condition for the complex spin polarization \u001b0,\n\u0012@\u001b0\n@x0\u0000i\n0\nv\u001b0\u0013\f\f\f\f\n\u0000= 0; (62)\nwhich is similar to the boundary condition in \fnite length\nwires given by Eq. (20).\nV. CONCLUSIONS\nIn this paper, we studied spin relaxation in 1D and 2D\nsystems with spin-orbit interaction. For this purpose, we\nused the spin kinetic equation that takes into account\nboth ballistic and di\u000busive spin transport regimes. This\nformalism was applied to several interesting problems of\nspin relaxation that were solved analytically. In partic-\nular, we analyzed dynamics of homogeneous and inho-\nmogeneous spin polarizations in in\fnite and \fnite length1D wires. Moreover, it was explicitly demonstrated that\nthe problem of spin relaxation in appropriately oriented\n2D channels with Rashba and Dresselhaus interactions\nof equal strength can be reduced to the problem of spin\nrelaxation in 1D wires. This result establishes a solid\nfoundation for a recently suggested method of creation\nof persistent spin helical con\fgurations in semiconduc-\ntors28.\nOne of the main results of the paper is a prediction of\nthe spin echo in the ballistic regime of spin transport. We\nfound that in \fnite length wires an initially localized spin\npolarization pro\fle re\rects from a sample boundary re-\nturning to its initial position. 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Shabunin, Problems\nin Partial Di\u000berential Equations (FIZMATLIT, 2003)."    },    {        "title": "2211.15091v1.Superfluid_like_spin_transport_in_the_dynamic_states_of_easy_axis_magnets.pdf",        "content": "Super\ruid-like spin transport in the dynamic states of easy-axis magnets\nJongpil Yun and Se Kwon Kim\nDepartment of Physics, Korea Advaced Institute of Science and Technology, Daejeon 34141, Republic of Korea\n(Dated: November 29, 2022)\nThe existing proposals for super\ruid-like spin transport have been based on easy-plane magnets\nwhere the U(1) spin-rotational symmetry is spontaneously broken in equilibrium, and this has been\nlimiting material choices for realizing super\ruid-like spin transport to restricted class of magnets. In\nthis work, we lift this limitation by showing that super\ruid-like spin transport can also be realized\nbased on easy-axis magnets, where the U(1) spin-rotational symmetry is intact in equilibrium but\ncan be broken in non-equilibrium. Speci\fcally, we \fnd the condition to engender a non-equilibrium\neasy-cone state by applying a spin torque to easy-axis magnets, which dynamically induces the\nspontaneous breaking of the U(1) spin-rotational symmetry and thereby can support super\ruid-like\nspin transport. By exploiting this dynamic easy-cone state, we show theoretically that super\ruid-\nlike spin transport can be achieved in easy-axis magnets under suitable conditions and con\frmed the\nprediction by micromagnetic simulations. We envision that our work broadens material library for\nrealizing super\ruid-like spin transport, showing the potential utility of dynamic states of magnets\nas venue to look for spin-transport phenomena that do not occur in static magnetic backgrounds.\nI. INTRODUCTION\nSpintronics is a \feld that harnesses spin degree of free-\ndom of electrons to store, transport, and process infor-\nmation in order to go beyond the conventional electronics\nwhere only charge degree of freedom has been used [1{3].\nSince information is encoded in the form of spin, it is im-\nportant to achieve e\u000ecient spin transport in spintronics,\nwhich demands us to identify and employ low-dissipation\nmagnetic materials and spin transport therein. In this\nregard, magnetic insulators have emerged as energy-wise\ne\u000ecient material platforms for spintronics since they have\nno Joule heating associated with an electric current and\ntherefore can enable low-power spin-based information\ntransport and processing. In a magnetic insulator, spin is\ntransported by its collective excitations, i.e., spin waves,\nwhose quanta are called magnons [4{7]. Since magnon-\nbased information transport and processing can be real-\nized without the Joule heating in principle, generating\nand controlling magnons have been actively investigated\nto realize magnonic devices [8{10], which includes the ex-\nperimental demonstration of long-distance spin transport\nin certain magnets [11{13]. Most of the previously stud-\nied magnonic spin transport have been based on di\u000busion\nof magnons, which has a critical problem in that the spin\ncurrent exponentially decays away from the spin-current\nsource [11]. To circumvent this problem of rapidly decay-\ning spin current of di\u000busive magnons, a novel type of spin\ntransport referred to as super\ruid-like spin transport has\nemerged as an alternative for e\u000ecient spin transport [14].\nSuper\ruid-like spin transport is a spin-analogue of\nmass super\ruidity. The mass super\ruidity can occur\nwhen the wave function de\fned by  =pnei\u0012, where\nnis a particle density and \u0012is an arbitrary phase, breaks\nthe U(1) phase symmetry spontaneously [15]. Like-\nwise, super\ruid-like spin transport can occur in mag-\nnetically ordered systems when the magnetic order pa-\nrameter breaks the U(1) spin-rotational symmetry spon-\ntaneously [14, 16]. In contrast to the exponential de-\n<latexit sha1_base64=\"HA1GJXdAB5Hew9Hgsu3zSpKlO1E=\">AAACDHicbVDLSgNBEJyNrxhfUY9eBoPgxbArQT1GvHiMYKKQBOmddHTI7Owy0ysJSz7Ai7/ixYMiXv0Ab/6NszEHXwUDRVV1D11hoqQl3//wCjOzc/MLxcXS0vLK6lp5faNl49QIbIpYxeYyBItKamySJIWXiUGIQoUX4eAk9y9u0VgZ63MaJdiN4FrLvhRATroqVzqEQ8oQ7GgPhtLy3EeSgkttUwUUm7FL+VV/Av6XBFNSYVM0rsrvnV4s0gg1CQXWtgM/oW4Gxu1VOC51UosJiAFcY9tRDRHabjY5Zsx3nNLj/di4p4lP1O8TGUTWjqLQJSOgG/vby8X/vHZK/aNuJnWSEmrx9VE/VZxinjfDe9KgIDVyBISReQfiBgwIcv2VXAnB75P/ktZ+NTio1s5qlfrxtI4i22LbbJcF7JDV2SlrsCYT7I49sCf27N17j96L9/oVLXjTmU32A97bJ+CvnCc=</latexit>easy-axis magnetic insulator\n<latexit 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Schematics of the experimental setup for realizing\nsuper\ruid-like spin transport in a magnetic insulator having\nthezaxis as an easy axis subjected to a spin torque, denoted\nby a spin-torque oscillator. In the magnetic insulator, the\nblue arrows depict the spatially varying magnetization and\nthe dashed cones represent the precession trajectories (form-\ning cones with tilting angle \u0012from the easy axis) of the local\nspins driven by the spin torque. The black arrow of the left\nmetal represents the charge current, which injects the spin\ncurrent (JL) to the left end of the ferromagnet via the spin\nHall e\u000bect. The red arrows of the left and the right metals are\nthe direction of spin accumulations, at the interfaces between\nthe metals and the ferromagnet. The spin accumulation \u0016R\nat the right metal generated by the nonlocal spin transport\nfrom the left metal can be detected via the inverse spin Hall\ne\u000bect.\ncaying of di\u000busive spin transport, super\ruid-like spin\ntransport has the characteristic of algebraically decay-\ning spin current, which enables long-distance spin trans-\nport in certain magnets [17{26]. The existing research\nof super\ruid-like spin transport has been focused only\non easy-plane magnets in which the system breaks the\nU(1) spin-rotational symmetry spontaneously in equilib-\nrium [27{30].\nIn this work, for a potential material platform for\nsuper\ruid-like spin transport, we consider easy-axis mag-\nnets, where the magnetization aligns with easy-axis and\nthus does not break U(1) spin-rotational symmetry spon-\ntaneously in equilibrium. Instead of using the equilib-\nrium U(1) symmetry breaking as done for previous pro-arXiv:2211.15091v1  [cond-mat.mes-hall]  28 Nov 20222\nposals based on easy-plane magnets, we turn to the dy-\nnamic breaking of the U(1) spin-rotational symmetry.\nMore speci\fcally, to break U(1) spin-rotational symme-\ntry spontaneously in easy-axis magnets, we apply a spin\ntorque and engender a dynamic easy-cone state to sup-\nport super\ruid-like spin transport [31{37]. The system is\nschematically illustrated in Fig. 1. The easy-axis magnet\nsubjected to a suitable spin torque forms a spin-torque\noscillator, in which the local magnetization (shown as\nthe blue arrows) precesses within cones (depicted by the\ndashed black lines with the cone angle \u0012) by breaking the\nU(1) spin-rotational symmetry dynamically. Applying a\ncharge current in the left metal injects a spin current JL\nfrom the left metal to the left end of the magnet via the\nspin Hall e\u000bect [34, 38{40]. The injected spin current is\ntransported through the magnet by super\ruid-like spin\ntransport in the form of the spatially varying order pa-\nrameter. The spin transport generates a spin accumula-\ntion\u0016Rat the interface between the ferromagnet and the\nright metal, which can be probed either by spin pump-\ning [41, 42] or by the inverse spin Hall e\u000bect. In our\nsystem, non-local spin transport refers to the generation\nof the spin accumulation \u0016Rat the right end of the fer-\nromagnet by the spin-current injection from the left end.\nBy the theoretical analysis based on the Landau-Lifshitz-\nGilbert (LLG) equation [43, 44] and the micromagnetic\nsimulations, we show that the spin accumulation \u0016Rde-\ncays algebraically as the ferromagnet length increases,\nexhibiting super\ruid-like spin transport.\nThe paper is organized as follows. In Sec. II, we de-\nscribe the model system, namely the easy-axis ferromag-\nnet subjected to a spin torque, and identify the condi-\ntion with which super\ruid-like spin transport can be re-\nalized by theoretical analysis and micromagnetic simula-\ntions. In Sec. III, we theoretically and numerically show\nthat our system can indeed support algebraically decay-\ning spin current, i.e., super\ruid-like spin transport. In\nSec. IV, we summarize our results.\nII. MODEL\nOur model system which is illustrated in Fig. 1 is a\nquasi-one-dimensional ferromagnetic wire whose energy\nis given by\nU=Z\ndV\u0014Am02\u0000Ke\u000bm2\nz+K2m4\nz\n2\u0000H\u0001m\u0015\n;(1)\nwhere mis the three-dimensional unit vector in the di-\nrection of the magnetization, the prime (0) is the gradient\nwith respect to the z-coordinate along the wire, Ais the\nexchange coe\u000ecient, Ke\u000b>0 is the \frst-order e\u000bective\nanisotropy which combines the shape anisotropy and the\n\frst-order easy-axis crystalline anisotropy, K2>0 is the\nsecond-order anisotropy [45{50], and His the external\nmagnetic \feld. We assume that the system is quasi-one-\ndimensional so that the magnetization varies only along\nthezdirection: m(z;t). The external magnetic \feld isapplied along the easy-axis direction: H=H^z. We con-\nsider the cases where the ground state is given by the\nuniform magnetization in the zdirection m(z;t)\u0011^z,\nwhich is satis\fed when K2<(Ke\u000b+H)=2. Note that\nthe energy Uis invariant under uniform rotations of the\nmagnetization about the z-axis, i.e., m7!Rzmwith an\narbitrary rotation matrix Rzabout thezaxis, indicating\nthat the system possess the U(1) spin-rotational symme-\ntry about the zaxis. Since the ground state m(z;t)\u0011^z\nis invariant under the spin rotations, it does not break\nthe U(1) spin-rotational symmetry.\nThe equation of motion for the dynamics of the magne-\ntization msubjected to a spin torque is given by the LLG\nequation [43, 44] augmented by the spin-torque term :\ns_m\u0000\u000bsm\u0002_m=\u0000m\u0002he\u000b+\u001cST; (2)\nwheresis the saturated (scalar) spin density, the dot\n( _ ) denotes di\u000berentiation with respect to time, \u000b > 0\nis the dimensionless Gilbert damping parameter, he\u000b=\n\u0000\u000eU=\u000emis the e\u000bective \feld, and \u001cST=\u001cSTm\u0002(m\u0002^z)\nis a externally-applied spin torque polarized along the\nzdirection, which can be realized either by the spin-\ntransfer torque [51] or by spin-orbit torque [52]. We as-\nsume that this spin torque \u001cSTis exerted uniformly on\nthe ferromagnet.\nTo endow the ferromagnet with the capability to sup-\nport super\ruid-like spin transport, it is necessary to in-\nduce the ferromagnet to break the U(1) spin-rotational\nsymmetry dynamically, which can be done by driving it\ninto a self-oscillatory mode with the su\u000eciently strong\nspin torque. The detailed condition for this oscillating\nphase can be obtained as follows. The LLG equation in\nterms of the polar angle ( \u0012) and the azimutal angle ( \u001e)\nwithm= sin\u0012cos\u001e^x+ sin\u0012sin\u001e^y+ cos\u0012^zis given by\ns\u0010\n_\u0012sin\u0012+\u000b_\u001esin2\u0012\u0011\n=A\u0000\n\u001e0sin2\u0012\u00010+\u001cSTsin2\u0012;(3)\ns\u0010\n_\u001esin\u0012\u0000\u000b_\u0012\u0011\n=A\u0000\n\u001e02sin\u0012cos\u0012\u0000\u001200\u0001\n+Hsin\u0012+Ke\u000bsin\u0012cos\u0012\n\u00002K2sin\u0012cos3\u0012:(4)\nEquation (3) has a clear physical meaning: It is the\nspin continuity equation: The \frst term and the sec-\nond term on the left-hand side are the time evolution\nof thezcomponent of spin density and the damping\nterm, respectively. The \frst term on the right-hand\nside of Eq. (3) is the divergence of spin current density:\njs=\u0000Asin2\u0012@x\u001eand the second term is the spin cur-\nrent coming from the bulk spin torque.\nNow, we look for a condition under which the spin\ntorque induces a dynamic easy-cone state and thus the\nspontaneous breaking of the U(1) spin-rotational sym-\nmetry. A steady-state solution of Eqs. (3) and (4) with\nconstant polar angle with _\u0012= 0 satis\fes\n\u001cST=\u000b\u0000\nKe\u000bcos\u0012+H\u00002K2cos3\u0012\u0001\n: (5)3\nThen, the condition that the system is in a dynamic easy-\ncone state with 0 <\u0012<\u0019 is given by\n\u000b(Ke\u000b+H\u00002K2)<\u001cST<\u000b \n2\n3r\nKe\u000b\n6K2Ke\u000b+H!\n;\n(6)\nKe\u000b\n6<K2<Ke\u000b+H\n2; (7)\nwhere\u001cc;1=\u000b(Ke\u000b+H\u00002K2) is the lower critical torque\nthat is given by the value of the right-hand side of Eq. (5)\nat\u0012= 0 and\u001cc;2=\u000b\u0010\n2\n3q\nKeff\n6K2Ke\u000b+H\u0011\nis the upper\ncritical torque. The lower critical torque \u001cc;1is the mini-\nmum torque that is required to drive the ferromagnet into\nthe dynamic easy-cone state, i..e, self-oscillatory state.\nWhen we apply the torque lower than the lower critical\ntorque, the magnetization is kept along the zaxis with-\nout breaking the U(1) spin-rotational symmetry. The\nphysical meaning of the upper critical \feld is as follows.\nWhen we apply the spin torque larger than the upper\ncritical torque \u001cc;2, the magnetization reverses into the\nnegativezdirection completely, i.e, m=\u0000^z, which does\nnot show the dynamic oscillation of the magnetization.\nEquation (7) states the condition for the system param-\neters,Ke\u000b;K2, andH. WhenK2is smaller than Ke\u000b=6,\nthe system does not possess a dynamic easy-cone state,\nbut is saturated along either the positive zdirection (for\nweak spin torques) or the negative zdirection (for strong\nspin torques). The condition K2<(Ke\u000b+H)=2 is to\nensure that the unperturbed ground state of the ferro-\nmagnet is given by the zdirection, as discussed above.\nWhen the spin torque and the system parameters sat-\nisfy Eqs. (6) and (7), the system is driven into a dynamic\neasy-cone state, where the magnetization precesses about\nthezaxis with arbitrary initial values for the azimuthal\nangle\u001eand thereby spontaneously break the U(1) spin-\nrotational symmetry.\nTo con\frm the spontaneous U(1) spin-rotatinal sym-\nmetry breaking under conditions [Eqs. (6) and (7)], we\nrun the micromagnetic simulation using MuMax3[53]\nwith the following material parameters of NdCo 5:Ke\u000b=\n2:4\u0002106J/m3,H= 1T; Ms= 1:1\u0002106A/m; A =\n1:05\u000210\u000011J/m and\u000b= 0:1 [54{57]. The demagne-\ntization e\u000bects are captured by the \frst-order e\u000bective\nanisotropyKe\u000bthrough the shape anisotropy in our sim-\nulations, as done analytically in Refs. [46, 49, 50, 58],\nwhich is expected to be a good approximation when\nthe system is quasi-one-dimensional cylindrical wire [59{\n61]. We run the simulation with three di\u000berent val-\nues of second-order anisotropy: K2= 0:4\u0002106J/m3,\n1\u0002106J/m3and 1:6\u0002106J/m3. For the polar angle in\nthe presence of a spin torque, the analytical result (which\ncan be obtained by solving Eq. (5) for \u0012) and the simu-\nlation results are shown in Fig. 2(a) as the lines and the\nsymbols, respectively, which agree with each other. In\nFig. 2(a), the blue and the red symbols correspond to the\ncases withK2= 1:6\u0002106J/m3andK2= 1\u0002106J/m3,\n<latexit sha1_base64=\"6o24vuTY54UwtoL+zl+HXursiqM=\">AAAB6nicbVBNS8NAEJ34WetX1aOXxSLUS0mkqMeCF48V7Qe0oWy2m3bpZhN2J0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekEhh0HW/nbX1jc2t7cJOcXdv/+CwdHTcMnGqGW+yWMa6E1DDpVC8iQIl7ySa0yiQvB2Mb2d++4lrI2L1iJOE+xEdKhEKRtFKD5Xgol8qu1V3DrJKvJyUIUejX/rqDWKWRlwhk9SYrucm6GdUo2CST4u91PCEsjEd8q6likbc+Nn81Ck5t8qAhLG2pZDM1d8TGY2MmUSB7YwojsyyNxP/87ophjd+JlSSIldssShMJcGYzP4mA6E5QzmxhDIt7K2EjaimDG06RRuCt/zyKmldVr2rau2+Vq7X8zgKcApnUAEPrqEOd9CAJjAYwjO8wpsjnRfn3flYtK45+cwJ/IHz+QOMhI1S</latexit>(b)\n<latexit 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(a) The polar angle ( \u0012) of the magnetization as a func-\ntion of the bulk spin torque ( \u001cST). The lines show the theoret-\nical results [Eq. (5)] and the symbols represent the simulation\nresults. The circles, the triangles, and the squares correspond\ntoK2= 2Ke\u000b=3,K2= 5Ke\u000b=12, andK2=Ke\u000b=6, respec-\ntively. (b) The spin accumulation ( j\u0001\u0016Rj) at the right bound-\nary of the ferromagnet induced by a spin-current injection JL\nfrom the left boundary as a function of the ferromagnet length\n(L). The lines show the theoretical result [Eq. (13)] and the\nsymbols correspond to the simulations results. The circles,\nthe triangles, and the squares correspond to input spin cur-\nrentJL= 2\u000210\u00006J/m2, 6\u000210\u00006J/m2and 10\u000210\u00006J/m2,\nrespectively.\nrespectively. For these cases, the ferromagnet is in a\nsteady state with nontrivial polar angle \u00126= 0;\u0019, i.e.,\nin a dynamic easy-cone state under suitable spin-torque\nvalues. The black symbols correspond to the cases with\nK2=Ke\u000b=6, where the polar angle is either 0 or \u0019re-\ngardless of the spin-torque values as discussed above and\nthe dynamic easy-cone state is not available.\nThe obtained dynamic easy-cone state can be inter-\npreted in the framework of the Gross-Pitaevskii equa-\ntion [62, 63] as follows. The LLG equation (2) for\nthe magnetization mcan be recast into the equation\nfor the complex order parameter de\fned by  (x;t) =\nmx(x;t)\u0000imy(x;t) =p\u001aei\u001e, where\u001a= sin\u0012and\u001eare\nanalogous to the density and the phase of the conden-\nsate [23, 64]. The LLG equation in terms of the complex\norder parameter  is given by\nis@ \n@t=\u0012\n(Ke\u000b\u00002K2+H) +\u0012\n\u00001\n2Ke\u000b+ 3K2\u0013\nj j2\u0013\n \n+s\u000b\u0012\n1\u00001\n2j j2\u0013@ \n@t+i\u001cST\u0012\n1\u00001\n2j j2\u0013\n ;\n(8)4\nup to the third order in  . The \frst term on the right-\nhand side of the equation originates from the potential\nenergy of our system [Eq. (1)], in which ( Ke\u000b\u00002K2+\nH) can be interpreted as the single particle potential\nand (\u0000Ke\u000b=2 + 3K2) can be regarded as the interaction\nstrength. The condition to possess a vacuum ground\nstate and the condition to have a stable condensate under\npumping (i.e., the repulsive interaction) are respectively\ngiven byK2<(Ke\u000b+H)=2 andKe\u000b=6<K 2, which are\nidentical to the conditions in Eq. (7) that were obtained\ndirectly from the LLG equation.\nThere are two additional terms on the right-hand side\nin the equation that break the time-reversal symmetry:\nThe second term is the damping term and the third term\nrepresents the particle pumping by the spin torque. To\nhave a \fnite condensate  6= 0 in a steady state, the\npumping/\u001cSTshould be su\u000eciently strong to dominate\nthe damping term. This condition is given by \u001cST>\n\u000b(Ke\u000b+H\u00002K2), which is identical to the lower critical\ntorque that we obtained above. The upper critical torque\nis not available in Eq. (8), since it is truncated to the third\norder in the order parameter and thus cannot capture the\ndynamics of the dense condensate.\nIII. SUPERFLUID-LIKE SPIN TRANSPORT\nNow, let us investigate the nonlocal spin transport be-\nhavior of an obtained dynamic easy-cone state of the\neasy-axis ferromagnet by assuming that our system sat-\nis\fes Eqs. (6) and (7) so that it breaks the U(1) spin-\nrotational symmetry dynamically. The situation that we\nconsider is depicted in Fig. 1. To inject a spin current JL\nto the ferromagnet through the left end, one heavy metal\nwith a \fnite charge current is attached to the left end of\nthe ferromagnet. To detect a spin accumulation \u0016Rat\nthe right end of the ferromagnet, the other heavy metal\nwith no external current is attached to the right end of\nthe ferromagnet. When we attach the metals to the left\nand right boundaries of the ferromagnet, there arises two\ne\u000bects: the spin-current injection from the metal with a\n\fnite current and spin pumping from the ferromagnet to\nthe metals [24], which determines the boundary condi-\ntions for the spin current at the left end x= 0 and the\nright endx=L:\njs(0) =JLsin2\u0012\u0000\rsin2\u0012_\u001e(0); (9)\njs(L) =\rsin2\u0012_\u001e(L); (10)\nwherejs=\u0000Asin2\u0012@x\u001eis the spin current of the ferro-\nmagnet,Lis the length of the ferromagnet, \u0012is the polar\nangle of the magnetization, \r=~g\"#=4\u0019, andg\"#is the\ne\u000bective interfacial spin-mixing conductance between the\nferromagnet and the normal metal [39, 65]. The \frst term\n/JLon the right-hand side of Eq. (9) is the spin current\ninjected from the left metal to the ferromagnet by the\nspin Hall e\u000bect, where JLis proportional to the product\nof the charge current \rowing in the left metal and thee\u000bective spin Hall angle of the interface between the fer-\nromagnet and the left metal. The second term /\ron\nthe right-hand side is the spin current ejected from the\nferromagnet to the left metal by the spin pumping. The\nright-hand side of Eq. (10) is the spin current ejected from\nthe ferromagnet to the right metal by the spin pumping.\nBy solving the bulk LLG Eq. (3) with the boundary\nconditions [Eqs. (9) and (10)] for a steady state, we ob-\ntain the following spin current density and the preces-\nsional velocity of the azimuthal angle:\njs(x;t) = (JL\u0000(\r+\u000bsx)!+\u001cSTx) sin2\u0012; (11)\n_\u001e(x;t)\u0011!=JL+\u001cSTL\n2\r+\u000bsL; (12)\nwhere the value of the polar angle \u0012changes from the\nvalue obtained from Eq. (5) due to the additional input\nspin current from the left boundary [66]. The precession\nof the magnetization induces a \fnite spin accumulation\ngiven by\u0016R=\u0000~^z\u0001m\u0002_m=\u0000~sin2\u0012_\u001ewith ~the\nreduced Planck constant, which can be measured exper-\nimentally [20, 41, 42].\nTo investigate the non-local spin transport from the\nleft endx= 0 to the right end x=Lthrough the dy-\nnamic ferromagnet, we employ the spin accumulation \u0016R\nat the interface between the ferromagnet and the right\nheavy metal and extract the component that is induced\nby the spin-current injection JLfrom the left metal. In\nother words, we use the di\u000berence of the spin accumu-\nlation\u0016Rbetween the two cases: with spin-current in-\njection from the left end ( JL6= 0) and without the\nspin-current injection ( JL= 0). Using _\u001eof Eq. (12),\n\u0001\u0016R=\u0016R(JL6= 0)\u0000\u0016R(JL= 0) is given by\n\u0001\u0016R=\u0000\u0012JL\n2\r+\u000bsLsin2\u0012\u0013\n~\n\u0000\u001cSTL\n2\r+\u000bsL\u0000\nsin2\u0012\u0000sin2\u00120\u0001\n~;(13)\nwhere\u00120is the polar angle obtained from Eq. (5) in the\nabsence of an input spin current JL= 0. The \frst term\non the right-hand side is the spin accumulation at the\nright end induced by injecting a spin current JLat the\nleft end. It decays algebraically \u00181=Lfor su\u000eciently\nlong samples as a function of the ferromagnet length\nL, which is the characteristic super\ruid-like spin trans-\nport. The second term can be interpreted as the e\u000bect\nof the polar-angle change (from \u00120to\u0012) induced by the\nspin-current injection JLfrom the left end. The alge-\nbraic decaying behavior of the second term is not evident\nfrom the analytical expression above, but we can show\nthat, by linearizing Eq. (4) with respect to the injected\nspin current JL, the second term is approximately given\nby [2 ~scos\u00120=(6K2cos2\u00120\u0000Ke\u000b)]JL\u001cSTL=(2\r+\u000bsL)2,\nwhich decays as 1 =Lfor su\u000eciently long samples. There-\nfore, the spin accumulation \u0001 \u0016Rat the right end induced\nby the spin-current injection from the left end decays as\n1=Las the ferromagnet length Lincreases, exhibiting\nsuper\ruid-like spin transport.5\nTo con\frm our theoretical prediction of super\ruid-\nlike spin transport in a dynamic cone state of ferro-\nmagnets, we perform the micromagnetic simulations and\ncompare the simulation results against the theoretical re-\nsults [Eq. (13)]. In simulations, we use the same mate-\nrial parameters that were mentioned above and the \fxed\nsecond-order anisotropy K2= 1:6\u0002106J/m3. For sim-\nplicity, we assumed that the spin pumping e\u000bect is neg-\nligible by setting \r= 0J/m2. Figure 2(b) plots the spin\naccumulation di\u000berence \u0001 \u0016RbetweenJL6= 0 andJL= 0\nas a function of the ferromagnet length Lfor several dif-\nferent values of the input spin current JL. The non-local\nspin transport \u0001 \u0016Rdecays algebraically, not exponen-\ntially, as the ferromagnet length Lincreases. Our ana-\nlytical [Eq. (13)] and simulation results [Fig. 2(b)] show\nthat we can realize super\ruid-like spin transport using\ndynamic states of easy-axis ferromagnets. These are our\nmain results.\nIV. SUMMARY\nTo go beyond the previous works on super\ruid-like spin\ntransport that have been restricted to easy-plane mag-\nnets, we have investigated the possibility of super\ruid-\nlike spin transport in an easy-axis ferromagnet driven to\na spin-torque oscillating regime. We have identi\fed the\ncondition for the spin torque with which the system can\nbe stabilized to a dynamic easy-cone state that breaks the\nU(1) spin-rotational symmetry spontaneously. By com-bining the theoretical analysis and the micromagnetic\nsimulations, we have shown that the spin current injected\nfrom one end of the ferromagnet decays algebraically,\nrather than exponentially, as the system length increases,\nwhereby demonstrating that super\ruid-like spin trans-\nport can be achieved in an easy-axis ferromagnet under\nsuitable dynamic biases. We hope that our work stimu-\nlates further investigations of super\ruid-like spin trans-\nport and other unconventional spin transport in various\ntypes of magnets, departing from simple easy-plane or\neasy-axis magnets.\nACKNOWLEDGMENTS\nWe acknowledge the discussion with Yaroslav\nTserkovnyak. 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Spin-orbit\ninteraction and elastic scattering result in anisotropic relaxation of electron spin polarization. The\noverall spin dynamics is described by a superposition of spin modes in the system. The relaxation\ntime of the most long-living mode depends quasi-periodically on the inverse electric \feld. The spin\nmodes can be conveniently revealed by means of spin noise spectroscopy. It is demonstrated that\nthe spectrum of spin \ructuations consists of peaks with the low-frequency peak much narrower than\nsatellite ones, and the widths of the peaks are determined by the decay times of the modes.\nPACS numbers: 72.25.Hg, 72.25.Rb, 72.70.+m, 73.63.Hs, 78.47.db\nI. INTRODUCTION\nA moderately strong electric \feld applied to a semicon-\nductor system with low carrier concentration can provide\nthe streaming regime of electron transport [1]. In this\nregime, each free charge carrier accelerates quasiballisti-\ncally in the passive region of the momentum space, where\nits energy is smaller than the optical-phonon energy ~!0.\nAs soon as the carrier energy amounts to ~!0, it emits\nan optical phonon and scatters to a state with a small\nenergy. This is the end of the period, after which the\nnext cycle of acceleration starts. The electron momen-\ntum changes periodically from zero to a value p0corre-\nsponding to the electron energy equal to ~!0, Fig. 1. The\nperiod of oscillations is the electron travelling time from\np= 0 top=p0:\nttr=p0=jeEj; (1)\nwheree <0 is the electron charge and Eis the electric\n\feld strength. The electron distribution in the momen-\ntum space is strongly anisotropic, i.e. its length in the\n\feld direction, p0, is much larger than its width. The\ncorresponding region of the momentum space is called a\nneedle and is shown by the red solid line in Fig. 1.\nThis regime of carrier transport has been studied in\ndetail in three-dimensional systems. A semiconductor\nlaser in p-Ge has been realized based on the streaming\ne\u000bect [2]. Many interesting features can be revealed by\nstudying the electric current \ructuations in the stream-\ning regime [3, 4]. Recent theoretical investigations of\nstreaming in two-dimensional systems demonstrate col-\nlective wave-like excitations of the electrons with multi-\nbranch spectra and considerable spatial dispersion [5].\nBallistic transport with dominant optical phonon scat-\ntering has been realized in graphene [6], and a number\nof interesting theoretical proposals have been made for\ngraphene [7{9].\nThe inclusion of electron spin degree of freedom into\nthe two-dimensional streaming-regime kinetics gives rise\nto rich spin-related phenomena. Due to electron drift in\nthe electric \feld and linear in momentum Rashba and\nFIG. 1: Illustration of the streaming regime in the momentum\nspace. During a ballistic motion inside the needle (red solid\nline), an electron spin rotates with the frequency \n(p) (blue\narrows). The electron can be elastically scattered by an im-\npurity (star and dashed arrow), then it reaches the active area\n(red arrow), instantaneously emits an optical phonon (wavy\narrow) and returns to p= 0 (dotted arrow).\nDresselhaus spin-orbit interactions, electron spin precess\nwith an average frequency \n dr. Electrically induced spin\nbeats and long spin relaxation times as well as high de-\ngree of the current induced spin polarization have been\npredicted for such a system [10, 11]. However the compre-\nhensive study of spin dynamics in the regime of substan-\ntial spin rotations in each acceleration period (\n drttr\u00181)\nhas not been made yet. As we show, the spin dynamics is\nnot reduced to a simple exponential relaxation, and the\nelectron spin polarization at resonant conditions persists\ndespite of multiple elastic scatterings.arXiv:1505.04745v1  [cond-mat.mes-hall]  18 May 20152\nEstimations show that the streaming regime in GaAs\nbased heterostructures can be realized in reasonable \felds\nE\u00181 kV/cm, where the parameter \n drttris of the order\nof unity. This shows a possibility of experimental inves-\ntigations of spin-dependent phenomena in the streaming\nregime.\nThe spin-dependent streaming kinetics has a few tem-\nporal ranges: ttr, \n\u00001\ndr, the elastic scattering time \u001cp, and\nthe spin relaxation time \u001cs. Therefore it is natural to\nstudy spin dynamics in the frequency domain. This can\nbe done by means of spin noise spectroscopy being a\nmodern and very e\u000ecient tool for investigation of spin\nproperties in various systems [12{14]. This method is\nbased on the measurement of \ructuating spin signals in\nthe ensemble of unpolarized carriers and allows simul-\ntaneous resolving di\u000berent time ranges [15]. The spin\nnoise of free two-dimensional carriers in electric \felds has\nbeen considered in nearly equilibrium conditions where\nthe electric \feld is weak [16, 17]. Here we investigate spin\n\ructuations in heterostructures in the streaming regime\nwhich represents the opposite situation of strongly non-\nequilibrium electron gas. We demonstrate that the spec-\ntrum of spin \ructuations in this case consists of a series\nof peaks with di\u000berent widths.\nThe paper is organized as follows. In Sec. II, the gen-\neral theory of spin dynamics is developed for the two-\ndimensional streaming regime with account for spin-orbit\ninteraction and elastic scattering. In Section III, we con-\nsider the spin dynamics in the presence of either longi-\ntudinal or transverse e\u000bective \feld. In Sec. IV, the spin\nnoise in the streaming regime is investigated. Concluding\nremarks are given in Sec. V.\nII. SPIN DYNAMICS\nWe address the streaming regime accounting for the\nspin-orbit interaction and elastic scattering by impuri-\nties. We consider a semiconductor A 3B5heterostructure\ngrown along the axis zk[001] and choose the in-plane\naxes asxk[1\u001610] andyk[110]. In this coordinate frame,\nthe linear in momentum Hamiltonian of spin-orbit inter-\naction takes the form [18]\nHSO=\fyx\u001bypx+\fxy\u001bxpy=~\n2\u001b\u0001\n(p); (2)\nwherepis the two-dimensional electron momentum, \u001bi\n(i=x;y) are the Pauli matrices, \fxy;\fyxare the spin\nsplitting constants, and \n(p) is the e\u000bective frequency\nof spin precession caused by Rashba and Dresselhaus ef-\nfects.\nElectron spin dynamics is described by a kinetic theory.\nThe kinetic equation for the spin distribution functionSp(t) in the passive region reads [10, 11]:\n\u0012@\n@t+1\n\u001cp+eE@\n@px\u0013\nSp(t)\n=Z\nd2p0Wpp0Sp0(t) +\n(p)\u0002Sp(t):(3)\nHereWpp0=\u000e(p2\u0000p02)=(\u0019\u001cp) is the probability of elastic\nscattering from p0topwith\u001cpbeing the elastic scatter-\ning time, and we assume the electric \feld to be applied\nopposite to the xk[1\u001610] axis. After each scattering, an\nelectron accelerates in the x > 0 direction until reach-\ning the border of the passive region, and then returns to\np= 0. Such a trajectory is shown by arrows in Fig. 1.\nSpin relaxation in the streaming regime can be caused\nby both electron penetration into the active region\n(p>p 0) and by elastic scattering. The latter mechanism\nis shown to be somewhat more e\u000ecient [11], therefore,\nfor the sake of simplicity, we will neglect the \frst mech-\nanism in this study. To that end we assume that the\noptical phonon emission time is in\fnitely short, so the\nspin distribution is nonzero only in the passive region\n(p < p 0). Accordingly, electrons have zero energy and\nzero momentum immediately after optical phonon emis-\nsion. Hence we can separate the two contributions to the\nspin distribution function\nSp(t) =\u000e(py)\u0012(px)Sn\npx(t) +Sout(p;t): (4)\nThe \frst contribution describes the spin density in the\nneedle (py= 0), while the second one stands for the spin\ndistribution in all the passive region out of the needle.\nThe streaming regime can be realized only if the elastic\nscattering is weak ( \u001cp\u001dttr), therefore we will consider\nspin dynamics up to the \frst order in the small parameter\nttr=\u001cp\u001c1. Moreover, we are aimed to solve the problem\nat the timescale\u0018\u001cpor longer, therefore we assume that\nthe majority of the carriers are in the needle at the time\nt= 0. Anyway this situation always establishes during\nthe time\u00182ttrafter the electric \feld is switched on.\nHence the kinetic equations for the two components of\nthe spin distribution read\n\u0012@\n@t+1\n\u001cp+eE@\n@px\u0013\nSn\npx(t) =\n(p)\u0002Sn\npx(t);(5a)\n\u0012@\n@t+eE@\n@px\u0013\nSout(p;t) =\n(p)\u0002Sout(p;t) +Sn\np(t)\n2\u0019p\u001cp:\n(5b)\nSince the electron trajectories are closed in the p-space,\nthe general solution of Eq. (5a) can be presented as a\nsuperposition of discrete spin modes:\nSn\npx(t) =1X\nn=\u000011X\nl=\u00001g(l)\nn\np0^R(px)e(l)\nn\n\u0002exph\n\u0000i!(l)\nn(t\u0000pxttr=p0)\u0000pxttr=(p0\u001cp)i\n:(6)3\nHerenenumerates the modes, l=\u00001;0;1 distinguishes\ndi\u000berent orientations of the normalized eigenvectors e(l)\nn,\n!nare complex eigenfrequencies of the system, g(l)\nnare\nthe coe\u000ecients, and ^R(px) is the operator of rotation\naround the yaxis by the angle\n\b(p2\nx) =pxZ\n0dpx\neE\ny(px) =\fyxttrp2\nx=(~p0): (7)\nThe appearance of the operator ^R(px) is caused by\nthe fact that inside the needle the precession frequency\n\ny(px) is nonzero, and the electron spins are rotated in\nthe (zx) plane. Once the spin distribution in the needle\nis known at t= 0, the coe\u000ecients g(l)\nncan be calculated\nas\ng(l)\nn=p0Z\n0dpxe\u0000i!(l)\nnttrpx=p0e(l)\nn\u0003^R\u00001(px)Sn\npx(0);(8)\nwhere we have omitted the terms proportional to the \frst\nand higher powers of ttr=\u001cp. The coe\u000ecients g(l)\nndepend\non excitation conditions: At resonant spin excitation in\nthe vicinity of p= 0,g(l)\nnare of the same order for all n,\nwhile at non-resonant excitation the spin distribution at\nt= 0 is a smooth function of p, andg(l)\nndrop withnas\n/1=n2.\nThe spin distribution outside the needle, Sout(p;t),\ncan be readily found by integration of Eq. (5b):\nSout(p;t) =p0xZ\n\u0000p0xdp0\nx^Gpp0Sn\np0(t\u0000ttr(px\u0000p0\nx)=p0)\n2\u0019\u001cpq\np02x+p2y:(9)\nHere\np0x=q\np2\n0\u0000p2y;\nand the tensor operator ^Gpp0denotes Green function of\nthe ordinary di\u000berential equation\neE@\n@pxG\u000b\f\npp0\u0000\u000f\u000b\r\u000e\n\r(p)G\u000e\f\npp0=\u000e(px\u0000p0\nx)\u000e\u000b\f;(10)\nwhere the Greek subscripts and superscripts denote the\nCartesian components, and \u000f\u000b\f\ris the Levi-Civita sym-\nbol. The electrons are immediately taken o\u000b the left\nsemicircle in Fig. 1 by the electric \feld, thus\nSout(\u0000p0x;py;t) = 0:\nIn order to satisfy this boundary condition we take the\nGreen function, ^Gpp0, to be zero at px<p0\nx.\nIn order to \fnd the eigenfrequencies and spin modes\nin the streaming regime, one has to substitute general\nEqs. (6) and (9) into the boundary condition\nSn\n0(t)\u0000Sn\np0(t) =p0Z\n\u0000p0dpySout(p0x;py;t):(11)This condition re\rects the fact of immediate optical\nphonon emission at reaching the border of the active re-\ngion (the right semicircle in Fig. 1). It is violated if the\nelectron-phonon scattering is spin-dependent, but this ef-\nfect is estimated to be negligible in the streaming regime\nfor typical structure parameters [10]. From the coupled\nset of Eqs. (9) and (11) we obtain:\nSn\n0(t)\u0000Sn\np0(t) =1\n2\u0019\u001cpZd2p\np^Gp0pSn\np(t\u0000ttr(p0x\u0000px)=p0);\n(12)\nwherep0= (p0x;py).\nWe solve this equation using the perturbation theory\nin the small parameter ttr=\u001cp. In the absence of elas-\ntic scattering ( ttr=\u001cp= 0), it follows from Eq. (6) that\nthe eigenfrequencies ~ !(l)\nnand eigenvectors ~e(l)\nnsatisfy the\nequation\n^R(p0)~e(l)\nn= e\u0000i~!(l)\nnttr~e(l)\nn: (13)\nHere ^R(p0) is an operator of the electron spin rotation\nafter the travel through all the needle. The corresponding\nrotation angle, see Eq. (7), is \b( p2\n0) = \n drttrwith\n\ndr=\fyxp0=~ (14)\nbeing the average precession frequency in the needle,\nhereafter we assume that \fyx\u00150. As a result, we \fnd\nthat the eigenfrequencies are combinations of multiples\nof the travel and drift frequencies:\n~!(l)\nn=l(2\u0019n=t tr\u0000\ndr);\nand ~e(0)\nn=^y,~e(\u00061)\nn = (^z\u0006i^x)=p\n2 with ^x;^y;^zbeing\nunit vectors along the Cartesian axes. Physically these\nfrequencies re\rect the fact that Sn\ny(px) does not precess,\nbut oscillates in time with the period ttr. By contrast, the\nspin polarization in ( zx) plane precess with the frequency\n\ndrin addition to the periodic oscillations.\nIn the \frst order of the perturbation theory the eigen-\nfrequencies become complex and can be presented as\n!(l)\nn= ~!(l)\nn+\u000e(l)\nn[19]. Provided\n\ndr\u001cp\u001d1 (15)\nto account for the elastic scattering, we substitute the\neigenvectors ~e(l)\nninto Eq. (6) \fnd Sn\npx, and then Eq. (12)\nyields the corrections \u000e(l)\nn. In the opposite case \n dr\u001cp.1,\nthe perturbation theory for degenerate levels should be\nused.\nThe spin dynamics can be probed by Faraday, Kerr\nor ellipticity signals which are determined by the total\nelectron spin, S(t). In the streaming regime most of the\nparticles are localized inside the needle, therefore we have\nS(t) =p0Z\n0dpxSn\npx(t) =1X\nn=\u000011X\nl=\u00001g(l)\nns(l)\nne\u0000i!(l)\nnt:(16)4\nHere we have introduced the average spin polarization in\nthenth mode\ns(l)\nn=p0Z\n0dpx\np0^R(px)~e(l)\nnei~!(l)\nnttrpx=p0: (17)\nThe results of this Section describe spin dynamics at\narbitrary strong and anisotropic spin-orbit splitting.\nIII. RESULTS AND DISCUSSION\nIn general case the Green function ^Gpp0can not be\nfound analytically, therefore in the next subsections we\nconsider separately two limits: (i) the e\u000bective \feld \n(p)\nis oriented along x-axis (\fyx= 0;\fxy6= 0) and (ii) \n(p)k\ny(\fxy= 0;\fyx6= 0). In the end of this Section, we\nbrie\ry analyze spin dynamics in the presence of both\ncomponents in \n(p).\nA. Longitudinal e\u000bective \feld \n (p)kx\nFirst we consider the limit of \fyx= 0, when the ef-\nfective \feld \n(p) is parallel to the electric \feld. Since\n\nx/py, the spin precession in the needle is absent, and\nEq. (13) is simpli\fed to exp( \u0000i~!(l)\nnttr) = 1, which yields\n~!(l)\nn= 2\u0019n=t tr: (18)\nClearly in this limit the condition Eq. (15) fails, and be-\nlow we apply the perturbation theory for degenerate lev-\nels.\nOut of the needle, electrons move ballistically con-\nserving the pymomentum component. Therefore, the\nelectron spin rotates in the ( yz) plane with the con-\nstant frequency 2 \fxypy=~. Hence the Green function of\nEq. (10), ^Gpp0, is the identity operator for Sx, and it\nmultipliesSy(p;t)\u0006iSz(p;t) in Eq. (12) by the factors\nexp [\u0006i\nx(px\u0000p0\nx)=(eE)]. On average, scattered elec-\ntrons have zero momentum py, therefore the spin polar-\nization does not precess even with account for scattering\no\u000b the needle. Hence we can choose the basis vectors as\ne(l)\nn=^y;^x;^zforl=\u00001;0;1, respectively. Substituting\nthesee(l)\nnand ~!(l)\nnfrom Eq. (18) into Eqs. (6) and (12),\nwe obtain the corrections to the eigenfrequencies in the\nform\n\u000e(l)\nn=\u0000i\n\u001cp+i\n2\u0019\u001cpZd2p\npp0exp [2\u0019in(p0x\u0000px+p)=p0]\n\u0002cos [2l\fxypy(p0x\u0000px)ttr=(~p0)]:(19)\nHereafter we assume that all integrations are performed\nover the passive region p<p 0only.\nDespite the developed formalism is quite cumbersome,\nits interpretation is straightforward. Let us consider the\ndynamics of xspin component. It corresponds to l= 0,\nFIG. 2: (a) The spin distribution in the logarithmic scale\nfor\u001cp= 3ttr,\u001cz\ns= 10\u001cpat the times t= 0 (black solid\ncurve), 1:8\u001cp(red dashed curve), 10 \u001cp(blue dotted curve)\nand 20\u001cp(green dash-dotted curve). (b) The decay rates of\nthe \frst three z-spin modes for n= 0 (black solid curve),\nn= 1 (red dashed curve), and n= 2 (blue dotted curve) as\nfunctions of \u0001 \u001e, Eq. (21). The green dash-dotted line denote\nthe spin splitting corresponding to the panel (a). The inset\nshows the decay rates of the x-modes (dots) and the analytical\napproximation Eq. (20) (solid line).\nand for a homogeneous distribution ( n= 0) one \fnds\n\u000e(0)\n0= 0, i.e. the eigenfrequency is zero. This re\rects\nthe fact that the total spin polarization along the xaxis\nis conserved, because it is not a\u000bected by the spin-orbit\ninteraction. Therefore its dynamics is the same as for the\nparticle distribution function. The decay rates of the ex-\ncitedx-modes [Eq. (19) for l= 0 andn6= 0] are nonzero.\nThey are presented in the inset to Fig. 2(b) which shows\nthat all of them are of the same order ( \u00181=\u001cp). This\nmeans that the spin distribution relaxes during the time\n\u001cpto a constant (zero mode). In the limit of jnj\u001d1 one\ncan \fnd from Eq. (19):\n\u0000Im\u000e(0)\nn\u00191\n\u001cp \n1\u00001p\n8jnj!\n: (20)5\nThis analytical expression describes the decay of all\nmodes except for n= 0 with accuracy of 4 %, see the\ninset to Fig. 2(b).\nThe spin components Sout\np;zandSout\np;yprecess, which re-\nsults in spin relaxation. First we consider the limit where\ncharacteristic spin rotation angles out of the needle\n\u0001\u001e=\fxyp0ttr=~ (21)\nare small: \u0001 \u001e\u001c1. In this case all the excited modes\nrelax with approximately the same rates as the x-modes,\ni.e.\u000e(\u00061)\nn\u0019\u000e(0)\nn\u00181=\u001cpforn6= 0. The decay rates of the\nhomogeneous spin distribution 1 =\u001cz;y\ns=\u0000Im\u000e(\u00061)\n0 are\ngiven by [20]\n1\n\u001czs=1\n\u001cy\ns=7\n30(\u0001\u001e)2\n\u001cp: (22)\nOne can see that the spin relaxation time in this limit\nis much longer than \u001cp. This means that the spin distri-\nbution relaxes in two stages. First, after a time \u0018\u001cpall\nthe spin modes except for n= 0 decay, so the spin dis-\ntribution in the needle becomes nearly uniform. In the\nsecond stage, this uniform spin distribution decays with\nthe rate 1=\u001cz\ns. This two-stage relaxation is illustrated in\nFig. 2(a). Interestingly, since the decay rates of all the\nexcited modes are of the same order, the spin distribution\nalmost conserves its shape during the relaxation, so that\nafter the time\u00182\u001cpthe initial shape of the distribution\nis still visible.\nThe advantage of the presented theory is that it can\ndescribe spin dynamics and relaxation for arbitrary char-\nacteristic rotation angle \u0001 \u001e. If this parameter is of the\norder of unity, then all the spin modes decay with the\nrate\u00181=\u001cp. The decay rates of the \frst three modes\ncalculated by Eq. (19) are presented in Fig. 2(b). In the\nlimit of \u0001\u001e!1 ,\u000en=\u0000i=\u001cpfor anyn, because a single\nelastic scattering is su\u000ecient for complete spin dephas-\ning.\nOne can see that for the simple eigenfrequencies\nEq. (18), the average spin polarization is nonzero only\nin the zeroth mode, since only the term with n= 0\ncontributes to Sz(t), see Eq. (17). Therefore the total\nspin decays monoexponentially with the rate \u0000Im\u000e(1)\n0\nfor arbitrary values of \fxydespite in this case all the spin\nmodes decay at di\u000berent timescales. This spin relaxation\nrate is shown by the black solid curve in Fig. 2(b) as a\nfunction of the spin-orbit coupling strength. Note that\nthe decay rates of all the modes are always smaller than\n\u001c\u00001\npbecause at least one scattering is needed to random-\nize spin orientation. The series of local maxima in the\nspin relaxation rate presented in Fig. 2(b) roughly cor-\nresponds to the condition \u0001 \u001e=n\u0019. Qualitatively, these\nmaxima and the oscillations in the decay rates of the\nother modes, are related to the destructive interference\nof the spin rotation angles for scattered electrons.B. Transverse e\u000bective \feld \n (p)ky\nIf the e\u000bective \feld is perpendicular to the electric \feld,\nthe dynamics of yspin component is the same as for the\nspin-independent distribution function. By contrast, the\nspin components lying in the ( zx) plane feel the e\u000bective\n\feld \ny/px, so the characteristic precession angles are\nnot small even inside the needle where px\u0014p0. Therefore\nwe concentrate on these components.\nAt\fxy= 0, one can use the unperturbed basis vectors\nand eigenfrequencies ~e(l)\nnand ~!(l)\nn. We \fnd the Green\nfunction of Eq. (10) in the form ^Gpp0=^R(px)^R\u00001(p0\nx),\nand from Eq. (12) we obtain\n\u000e(l)\nn=\u0000i\n\u001cp+ ieil\ndrttr\n2\u0019\u001cp(23)\n\u0002Zd2p\np0pexp [il(2\u0019n\u0000\ndrttr) (p0x\u0000px+p)=p0]:\nOne can see that real parts of \u000e(l)\nnhave opposite signs\nforl=\u00061, while their imaginary parts coincide.\nImportantly, it follows from Eq. (23) that for all l\n\u000e(l)\nn(\ndr) =\u000e(l)\n0(\ndr\u00002\u0019n=t tr) and, in particular,\n\u000e(l)\nn(2\u0019n=t tr) = 0:\nThis means that the nth spin mode does not decay at\n\ndr= 2\u0019n=t tr. On the other hand, it follows from Eq. (6)\nthat, at \n dr6= 0, the average spin polarization is nonzero\nin any mode. Therefore the decay time for the spin lying\nin the (zx) plane can be in\fnitely long.\nThe fact that the spin relaxation time is in\fnite for\n\ndrttr= 2\u0019ncan be explained by considering an elec-\ntron's motion from pxtop0\nxin the ballistic area. Its spin\nis rotated by the angle \b( p0\nx2)\u0000\b(p2\nx) around the y-axis,\nwhere \b\u0018p2\nxis given by Eq. (7). Since pymomentum\ncomponent is constant at this motion, the rotation an-\ngle can be rewritten as \b( p02)\u0000\b(p2). Moreover, during\nspin-independent elastic scattering, p2is conserved, and\nthe spin direction is not changed. Therefore, if sis a spin\nof a given electron, then the quantity\nI= (sz+ isx)e\u0000i\ndrttrp2=p2\n0 (24)\nis conserved during an electron's motion in the passive\nregion, and this is correct in all orders in ttr=\u001cp. Here we\nassume, as before, that the optical phonon emission time\nis in\fnitely short. Equation (24) demonstrates that when\nan electron reaches the active region, its spin is rotated\naround the yaxis by the angle \n drttrirrespectively to a\nnumber of elastic scatterings. Therefore at \n drttr= 2\u0019n\nthe electron spin always returns to its initial direction\nafter an optical phonon emission, i.e. spin relaxation is\nabsent. This situation is similar to the persistent spin\nhelix [21], but here it is realized in energy domain. The\nanalogy comes from the \fxed relation between the elec-\ntron displacement along the \feld direction, \u0001 x, and the6\nFIG. 3: (a) The decay rates of the \frst four spin modes,\n\u0000Im(\u000e(1)\nn\u001cp) forn= 0 (red solid curve), n= 1 (blue dashed\ncurve),n= 2 (green dotted curve) and n= 3 (magenta dash-\ndotted curve). (b) Spin relaxation rate of the most long-living\nmode as a function of electric \feld (black solid curve) and its\napproximation Eq. (27) (red dashed curve). The spin relax-\nation rate in the presence of both components of the e\u000bective\n\feld is shown by the blue dotted curve at \fxy=\fyx=3.\ngain of its energy \u0001 E=jeEj\u0001xindependent of the elec-\ntron trajectory.\nIn general case the spin relaxation time, \u001czx\ns, corre-\nsponds to the smallest decay rate of all the modes:\n1=\u001czx\ns= min(\u0000Im\u000e(1)\nn)\u0011\u0000Im\u000e(1)\nn\u0003; (25)\nwhere\nn\u0003= [\n drttr=(2\u0019) + 1=2] (26)\nis the number of the most long living mode with [ x] being\nthe integer part of x. The decay rates of the spin modes\ncalculated after Eq. (23) are presented in Fig. 3(a). From\nthis \fgure one can see that the spin relaxation time de-\n\fned by Eq. (25) is a periodic function of \n drttr. Despite\n\ndris the structure parameter, the travel time ttrcan be\nFIG. 4: Average spin sn\u0003in the most long-living mode as\na function of \n drttr(black solid curve) and its asymptotic,\nEq. (28) (blue dotted curve). Inset: Average spin in the nth\nmode,sn, forn= 0 (red solid curve), 1 (blue dashed curve),\n2 (green dotted curve) and 3 (magenta dash-dotted curve).\nchanged by the electric \feld. Figure 3(b) illustrates the\noscillations of the spin relaxation rate as a function of\nthe applied electric \feld in the streaming regime. In the\nlimit \n drttr\u001c1 we have from Eq. (23):\n\u000e(\u00061)\n0\u0019\u0006\ndrttr\n2\u0019\u001cp(2 + 4G\u0000\u0019)\u0000i(\ndrttr)2\n3\u0019\u001cp(2\u0019+ 1\u00006G);\n(27)\nwhereG\u00190:9159:::is the Catalan constant. This\nasymptotic for the spin relaxation time is plotted in\nFig. 3(b) by a red dashed curve. Comparison with the\nnumerical calculation shows that this expression is cor-\nrect up to \n drttr\u00181.\nAfter decay of all the modes except for the n\u0003th one,\nthe total spin polarization is de\fned by two vector coef-\n\fcients,s(\u00061)\nn\u0003, see Eq. (16), and it lies in the ( zx) plane.\nTheir absolute values coincide and decrease as a function\nofn\u0003. The inset to Fig. 4 shows the quantity\n\f\f\fs(1)\nn\f\f\f=r\f\f\fs(1)\nn;z\f\f\f2\n+\f\f\fs(1)\nn;x\f\f\f2\nas a function of the parameter \n drttrfor the \frst four\nmodes. For \n drttr= 0, average spin is present only in\nthe zeroth mode. With increase of \n drttr, the non-zero\nspin polarization arises in all the other modes due to a\ndependence of the spin precession frequency on px. Fig-\nure 4 showsjs(1)\nn\u0003jas a function of \n drttr. One can see\nthat the average spin polarization in the most long living\nmode exhibits damped oscillations. The jumps in the de-\npendencejs(1)\nn\u0003jare related to the switching between the\nmodesnandn+ 1 at the points \n drttr= 2\u0019(n+ 1=2),\nsee Eq. (26). For large \n drttrthe conserved spin decays\nas\njs(1)\nn\u0003j\u00191\n2r\u0019\n\ndrttr: (28)7\nThis dependence is plotted in Fig. 4 by a dotted line.\nIf both components, \n xand \nyare nonzero, a numeri-\ncal solution of Eq. (10) is needed in order to describe the\nspin dynamics. This situation is qualitatively similar to\nthe case of electric \feld orientation at arbitrary angle to\nthe main axes. The results of calculations for \fxy6= 0\nare shown in Fig. 3(b) by a blue dotted line. This de-\npendence demonstrates that the nth mode decays even at\n\ndrttr= 2\u0019nif \nxis nonzero. Spin relaxation is switched\non due to an additional phase /\fxyacquired by electrons\nwhich violates invariance of I, Eq. (24). However, pro-\nnounced drops in the decay rate at resonant conditions\n\ndrttr= 2\u0019nare still present.\nIV. SPIN NOISE\nThe analysis of the spin dynamics performed in the\nprevious Sections reveals multiple timescales in the\nstreaming regime:\nttr\u001c\u001cp<\u001cs:\nTherefore studies in the frequency domain are useful, and\nthe spin noise spectroscopy serves as an excellent tool to\nthat end. In the steady state the majority of the electrons\nare in the needle, so the spin \ructuations are character-\nized by the correlation functions\nD\n\u000eSn\npx;\u000b(t)\u000eSn\np0x;\f(t+\u001c)E\n; (29)\nwhere angular brackets denote averaging over the time t\nfor the given delay \u001c. In the steady state, the distribu-\ntion function in the needle is uniform, so the one-time\ncorrelator is given by [22]\nD\n\u000eSn\npx;\u000b(t)\u000eSn\np0x;\f(t)E\n=N\n4p0\u000e(px\u0000p0\nx)\u000e\u000b\f: (30)\nHereNis the electron two-dimensional density, and we\nignore an electric \feld induced spin polarization because\nits value does not exceed a few per cents in the streaming\nregime [10, 11].\nUltimately we are interested in the total spin corre-\nlation function,hSz(t)Sz(t+\u001c)i, where we assume the\nprobe beam to propagate along zdirection. Since a cor-\nrelator of any physical quantity obeys the same linear\nkinetic equation as the quantity itself [3, 4, 22], the spin\ncorrelation function for \u001c > 0 can be presented in the\nform\nh\u000eSz(t)\u000eSz(t+\u001c)i\n=Z\ndpxZ\ndp0\nxZ\ndp00\nxD\n\u000eSn\npx;z(t)Tz\u000b\np0xp00x(\u001c)\u000eSn\np00x;\u000b(t)E\n;\nwhereT\u000b\f\npp0(\u001c) (\u000b;\f=x;y;z ) is the Green function of the\nkinetic equation (3). Due to linearity of ^Tpxp0xand usingEq. (30), this expression can be recast as\nh\u000eSz(t)\u000eSz(t+\u001c)i=X\npxp0xTzz\np0xpx(\u001c)N\n4p0\u00111\nNS0Sz(\u001c):\n(31)\nHereS0=N=2, andSz(\u001c) is given by Eq. (16), where\nthe spin distribution Sn\npx;z(\u001c) is found using the initial\ncondition\nSn\npx;z(0) =S0=p0: (32)\nAccordingly the coe\u000ecients g(l)\nncan be calculated after\nEq. (8):g(l)\nn=S0s(l)\u0003\nn;z, wheres(l)\nnis given by Eq. (17).\nThe spin noise spectrum is de\fned by\n(\u000eS2\nz)!=1Z\n\u00001hSz(t)Sz(t+\u001c)iei!\u001cd\u001c: (33)\nSince the correlator is an even function of \u001c[3, 4, 22], we\nobtain from Eqs. (31) and (33):\n(\u000eS2\nz)!=N\n41X\nn=\u000011X\nl=\u00001\f\f\fs(l)\nn;z\f\f\f2\nIm\u00121\n!\u0000!(l)\nn+1\n!+!(l)\nn\u0013\n:\n(34)\nThis equation demonstrates that the spectrum consists\nof the series of Lorentzian peaks centered at the eigenfre-\nquencies. The areas of the peaks are proportional to the\nsquared total spin in the corresponding mode, and the\nwidths are determined by the decay rates of the modes.\nAt\n(p)kx, the average spin polarization along z-\naxis is nonzero only in the mode characterized by n= 0\nandl= 1. As it follows from Eq. (19), the correspond-\ning eigenfrequency !(1)\n0is imaginary, and s(1)\n0;z= 1, see\nEq. (17). Hence the spin noise spectrum simply reads\n(\u000eS2\nz)!=N\n2\u001cz\ns\n1 + (\u001czs!)2; (35)\nwhere 1=\u001cz\ns=\u0000Im!(1)\n0is shown by black solid line in\nFig. 2(b). At \u0001 \u001e\u001c1,\u001cz\nsis given by Eq. (22).\nIn the opposite case of transverse e\u000bective \feld\n\n(p)ky, all the spin modes with l=\u00061 contribute to\nthe spin noise spectrum. Figure 5 demonstrates the spin\nnoise spectra for various values of \n drttr. One can see\nthat the spectrum consists of a series of peaks of di\u000ber-\nent widths.\nIf \n drttr\u001c2\u0019, the dominant contribution to the spin\nnoise spectrum is given by the terms with n= 0 and\nl=\u00061. Accordingly, at ! > 0 the spin noise spectrum\nreads\n(\u000eS2\nz)!\u0019N\n4\u001cz\ns\n1 + [(!\u0000\ndr)\u001czs]2; (36)\ni.e. it has a Lorentzian shape centered at the frequency\n\ndr. The width is 1 =\u001cz\ns=\u0000Im\u000e(1)\n0, where\u000e(1)\n0is given8\n02Π4Π\n0Π2Π1\n10-2\n10-4\nFIG. 5: Spin noise spectra in the streaming regime calculated\nafter Eq. (34) for \u001cp= 3ttrand\fxy= 0.\nby Eq. (27). The other peaks are centered at the fre-\nquencies 2\u0019n=t tr\u0006\ndrand have the widths of the order\nof 1=\u001cp; their amplitudes decrease as 1 =n4. The shift\nof the main peak is caused by the nonzero e\u000bective pre-\ncession frequency inside the needle acting as a constant\nmagnetic \feld. According to the \ructuation-dissipation\ntheorem, this shift is equivalent to the electric-current\ninduced shift of the electron spin resonance spectra [23].\nFor \n drttr=2\u0019\u00181, the widths of all the peaks have an\norder of 1=\u001cp, but the amplitudes of the \frst few peaks\nare comparable to each other, see Fig. 5. However, when\n\ndrttrapproaches 2 \u0019n, the spin relaxation time tends to\nin\fnity, see Sec. III B. As a result, the peak centered at\n!=j2\u0019n=t tr\u0000\ndrjbecomes very high and narrow.\nLet us analyze the situation when the spin-orbit inter-\naction is absent. In this limit the total spin is conserved,\nand there are no spin \ructuations. Nevertheless, the spin\nnoise spectroscopy has an access to spin dynamics even\nin this case. Indeed, in transmission or re\rection experi-\nments, the measured spin Faraday or Kerr signals, \u0002( t),\nare related to the spin distribution function via\n\u0002(t) =Z\nd2pK(p)Sp;z(t): (37)\nHereK(p)/(p2+a2)\u00001, whereais determined by the\ndetuning between the probe beam frequency and the en-\nergy gap [11]. The resonant dependence K(p) re\rects\nthe fact that the electrons with larger energy give smaller\ncontribution to the spin signals, and it gives rise to \ruc-\ntuations of \u0002( t). Analysis shows that the spectrum \u00022\n!de\fned in analogy with Eq. (33) has the form:\n(\u000e\u00022)!=N\n41X\nn=\u00001j\u0002nj2Im\u00121\n!\u0000!(0)\nn+1\n!+!(0)\nn\u0013\n;\n(38)\nwhere\n\u0002n=p0Z\n0dp\np0K(p)e2\u0019inp=p 0: (39)\nThe eigenfrequencies !(0)\nn= 2\u0019n=t tr+\u000e(0)\nn, where\u000e(0)\nn\nshould be calculated after Eq. (19). One can see that the\nnoise spectrum \u00022\n!has a structure of Lorentzian peaks\ncentered at multiples of the travel frequency and having\nthe widths of the order of 1 =\u001cp(except for n= 0). This\nspectrum is di\u000berent from the electric current \ructuation\nspectrum [3, 4, 19] by the zero frequency peak: since the\nspin relaxation time is in\fnite in this limit, the width of\nthe peak is zero. The above analysis demonstrates that\nthe proposed method extends the spin noise spectroscopy\ntechnique to the case when the total spin is conserved and\ndoes not \ructuate. The same approach allows measur-\ning, e.g. energy relaxation rate of free electrons if spin\nrelaxation is slow enough.\nV. CONCLUSIONS\nWe have developed a kinetic theory of spin dynamics\nin the streaming regime with account for elastic scat-\ntering and spin-orbit interaction. The spin eigenmodes\nare identi\fed and their decay rates are calculated. We\nhave shown that electron spin dynamics is strongly dif-\nferent in the limits of small and large spin rotation angle\nduring the time ttr. If it is small, then the spin distribu-\ntion becomes uniform inside the needle on the timescale\nof one elastic scattering event. Afterwards, the average\nspin polarization monoexponentially decreases with the\ndecay time \u001cs\u001d\u001cp. In the opposite limit of large rota-\ntion angles, the spin relaxation time has an order of \u001cp.\nHowever, the spin relaxation time oscillates as a function\nof the electric \feld and in\fnitely increases when \n drttr\napproaches a multiple of 2 \u0019. We have demonstrated that\nthis e\u000bect is robust against elastic scattering, and, in fact,\nit is the energy space analogue of the persistent spin helix.\nThe pronounced oscillations exist even in the presence of\nthe transverse component of the e\u000bective magnetic \feld.\nThe spin noise spectrum in the streaming regime is cal-\nculated. The spectrum consists of a series of the peaks\ncorresponding to the di\u000berent spin modes, and the widths\nof the peaks correspond to the lifetimes of the modes.\nThe present study demonstrates that the spin noise spec-\ntroscopy applied to nonequilibrium electron systems re-\nveals the parameters of spin dynamics and, in particular,\nthe spin-orbit splittings. In the range of low frequencies\ninherent to the traditional spin noise spectroscopy, evo-\nlution of the spectrum re\rects the strong oscillations of9\nthe spin relaxation rate. The advantage of the ultrafast\nspin noise spectroscopy [24] paves the way for observation\nof peaks in the spin \ructuation spectrum in the stream-\ning regime. Moreover, if the spin-orbit coupling is small,\nthe resonant measurement of the Faraday or Kerr angle\n\ructuations allow investigating the particle distribution\nfunction dynamics.\nAs an outlook we note, that spin dynamics in the\nstreaming regime is extremely interesting to investigate\nin topological insulators. One of the reasons is a high\nvalue of the current-induced spin polarization. It is estab-\nlished that the spin polarization in topological insulators\nis proportional to a ratio of the drift and the Fermi mo-\nmenta which is small in weak \felds [25{28]. By contrast,in the streaming the needle-like electron distribution re-\nsults in a large drift momentum, and the current-induced\nspin polarization is 100 %.\nAcknowledgments\nWe thank M. M. Glazov for fruitful discussions. Partial\nsupport from RFBR and RFBR-DFG ICRC TRR160,\nDynasty Foundation, RF President Grant No. SP-\n643.2015.5 and Programmes of RAS is gratefully ac-\nknowledged.\n[1] A. A. Andronov, in Spectroscopy of nonequilibrium elec-\ntrons and phonons , edited by C. V. Shank and B. P.\nZakharchenya (Elsevier Science Publishers B.V., 1992),\np. 169.\n[2] E. Gornik and A. A. Andronov, Special Issue on Far-\nInfrared Semiconductor Lasers , Opt. Quant. Electron.\n23(1991).\n[3] V. Bareikis, R. Katilius, J. Pozhela, S. V. Gantsevich,\nand V. L. Gurevich, in Spectroscopy of nonequilibrium\nelectrons and phonons , edited by C. V. Shank and B. P.\nZakharchenya (Elsevier Science Publishers B.V., 1992),\np. 327.\n[4] Sh. Kogan, Electronic noise and \ructuations in solids\n(Cambridge University Press, 2008).\n[5] V. V. Korotyeyev, V. A. Kochelap, and L. Varani, Wave\nexcitations of drifting two-dimensional electron gas under\nstrong inelastic scattering, J. Appl. Phys. 112, 083721\n(2012).\n[6] A. Barreiro, M. Lazzeri, J. Moser, F. Mauri, and A.\nBachtold, Transport properties of graphene in the high-\ncurrent limit, Phys. Rev. Lett. 103, 076601 (2009).\n[7] T. Fang, A. Konar, H. Xing, and D. Jena, High-\feld\ntransport in two-dimensional graphene, Phys. Rev. B 84,\n125450 (2011).\n[8] S. Sekwao and J.-P. Leburton, Hot-electron transient\nand terahertz oscillations in graphene, Phys. Rev. B 83,\n075418 (2011).\n[9] S. Sekwao and J.-P. Leburton, Soft parametric resonance\nfor hot carriers in graphene, Phys. Rev. B 87, 155424\n(2013).\n[10] L. E. Golub and E. L. Ivchenko, Spin-dependent phe-\nnomena in semiconductors in strong electric \felds, New\nJ. Phys. 15, 125003 (2013).\n[11] L. E. Golub and E. L. Ivchenko, in Advances in Semi-\nconductor Research: Physics of Nanosystems, Spintron-\nics and Technological Applications , edited by D. Per-\nsano Adorno and S. Pokutnyi, (Nova Science Publishers,\n2014), p. 93.\n[12] V. S. Zapasskii, Spin-noise spectroscopy: from proof\nof principle to applications, Adv. Opt. Photon. 5, 131\n(2013).\n[13] J. H ubner, F. Berski, R. Dahbashi, and M. Oestreich,\nThe rise of spin noise spectroscopy in semiconductors:\nFrom accoustic to GHz frequencies, Phys. Status SolidiB251, 1824 (2014).\n[14] V. S. Zapasskii, A. Greilich, S. A. Crooker, Yan Li, G. G.\nKozlov, D. R. Yakovlev, D. Reuter, A. D. Wieck, and\nM. Bayer, Optical spectroscopy of spin noise, Phys. Rev.\nLett. 110, 176601 (2013).\n[15] D. S. Smirnov and M. M. Glazov, Exciton spin noise in\nquantum wells, Phys. Rev. B 90, 085303 (2014).\n[16] F. Li, Y. V. Pershin, V. A. Slipko, and N. A. Sinitsyn,\nNonequilibrium spin noise spectroscopy, Phys. Rev. Lett.\n111, 067201 (2013).\n[17] V. A. Slipko, N. A. Sinitsyn, and Y. V. Pershin, Hybrid\nspin noise spectroscopy and the spin hall e\u000bect, Phys.\nRev. B 88, 201102 (2013).\n[18] N. S. Averkiev and L. E. Golub, Spin relaxation\nanisotropy: microscopic mechanisms for 2D systems,\nSemicond. Sci. Tech. 23, 114002 (2008).\n[19] I. B. Levinson and A. Yu. Matulis, Current \ructuations\nin strong electric \felds, Sov. Phys. JETP 27, 786 (1968).\n[20] Equation (22) corrects a misprint in a factor of two made\nin Ref. [11].\n[21] B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Exact\nSU(2) symmetry and persistent spin helix in a spin-orbit\ncoupled system, Phys. Rev. Lett. 97, 236601 (2006).\n[22] E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics\n(Butterworth-Heinemann, 2012).\n[23] Z. Wilamowski, H. Malissa, F. Sch a\u000fer, and W. Jantsch,\ng-factor tuning and manipulation of spins by an electric\ncurrent, Phys. Rev. Lett. 98, 187203 (2007).\n[24] F. Berski, H. Kuhn, J. G. Lonnemann, J. H ubner,\nand M. Oestreich, Ultrahigh bandwidth spin noise spec-\ntroscopy: Detection of large g-factor \ructuations in\nhighly-n-doped GaAs, Phys. Rev. Lett. 111, 186602\n(2013).\n[25] L. E. Golub and E. L. Ivchenko, Spin orientation by elec-\ntric current in (110) quantum wells, Phys. Rev. B 84,\n115303 (2011).\n[26] T. Misawa, T. Yokoyama, and S. Murakami, Electromag-\nnetic spin polarization on the surface of topological insu-\nlator, Phys. Rev. B 84, 165407 (2011).\n[27] D. Pesin and A. H. MacDonald, Spintronics and pseu-\ndospintronics in graphene and topological insulators,\nNat. Mater. 11, 409 (2012).\n[28] C. H. Li, O. M. J. van't Erve, J. T. Robinson, Y. Liu,\nL. Li, and B. T. Jonker, Electrical detection of charge-10\ncurrent-induced spin polarization due to spin-momentum\nlocking in Bi 2Se3, Nat. Nanotech. 9, 218 (2014)."    },    {        "title": "1602.04611v1.Effects_of_Dephasing_on_Spin_Lifetime_in_Ballistic_Spin_Orbit_Materials.pdf",        "content": "E\u000bects of Dephasing on Spin Lifetime in Ballistic Spin-Orbit Materials\nAron W. Cummings1and Stephan Roche1;2\n1Catalan Institute of Nanoscience and Nanotechnology (ICN2),\nCSIC and The Barcelona Institute of Science and Technology,\nCampus UAB, Bellaterra, 08193 Barcelona, Spain\n2ICREA, Instituci\u0013 o Catalana de Recerca i Estudis Avan\u0018 cats, 08070 Barcelona, Spain\n(Dated: June 23, 2021)\nWe theoretically investigate spin dynamics in spin-orbit-coupled materials. In the ballistic limit,\nthe spin lifetime is dictated by dephasing that arises from energy broadening plus a non-uniform\nspin precession. For the case of clean graphene, we \fnd a strong anisotropy with spin lifetimes that\ncan be short even for modest energy scales, on the order of a few ns. These results o\u000ber deeper\ninsight into the nature of spin dynamics in graphene, and are also applicable to the investigation of\nother systems where spin-orbit coupling plays an important role.\nPACS numbers: 72.80.Vp, 72.25.-b, 71.70.Ej, 72.25.Rb\nIntroduction . Following the description of Rashba spin-\norbit coupling (SOC) in two-dimensional electron gases\n(2DEGs) [1], understanding the spin dynamics in these\nsystems has been essential for proposing spintronic de-\nvices [2] and predicting fundamental physical phenomena\n[3{5]. Rashba SOC allows for the electrostatic manipu-\nlation of spin states, paving the way towards non-charge-\nbased computing and information processing [6]. Beyond\ntraditional semiconductor quantum wells, 2D materials\nincluding graphene and MoS 2monolayers have gener-\nated signi\fcant interest. In addition to their predicted\nlong spin lifetimes [7{11], the possibility to harness prox-\nimity e\u000bects or to couple the spin and valley degrees of\nfreedom makes these materials interesting both funda-\nmentally and technologically [12{15].\nFrom a practical perspective, understanding spin life-\ntimes in clean materials is a prerequisite to realizing spin-\ntronic devices, since they determine the upper time and\nlength scales of operation. In Rashba SOC materials, the\nspin lifetime is normally dictated by the Dyakonov-Perel\n(DP) mechanism [16], where SOC induces spin preces-\nsion of charge carriers. After many scattering events the\nrandomization of precession leads to dephasing and a loss\nof the spin signal, such that the spin lifetime \u001csscales in-\nversely with the momentum scattering time \u001cp. This con-\ntrasts with the Elliot-Yafet (EY) mechanism [17, 18], for\nwhich charge carriers can \rip their spin upon scattering,\ngiving\u001cs/\u001cp. The EY mechanism usually dominates\nin disordered metals, but its contribution has been also\ndiscussed for graphene [19].\nThe SOC in graphene is predicted to be small, on the\norder of\u0016eV [20{24], leading to estimates of \u001csin the\nmicro- to millisecond range [7{9]. In contrast, experi-\nmental spin lifetimes range from hundreds of ps to a few\nns for non-local Hanle measurements [25{32]. Various\nextrinsic mechanisms have been proposed to explain this\ndiscrepancy, including lattice deformations [33], metal-\nlic adsorbates [34, 35], or magnetic resonances [36, 37].\nIn experiments and theories that assume the DP or EYmechanism, the loss of spin polarization is controlled by\nmomentum scattering and is applicable when \u001cs\u001d\u001cp.\nHowever, impurity scattering might cease to dominate\nthe spin relaxation in high-mobility materials. To date,\nthere is a lack of theoretical description of spin decoher-\nence in this regime, where charges can propagate ballis-\ntically over long distances.\nThis Letter presents a study of spin dynamics in\nRashba SOC materials in the absence of momentum scat-\ntering. In this regime, the spin lifetime is limited by\ndephasing arising from a combination of energy broaden-\ning and nonuniform spin precession. Using graphene as\nan example, we show that its particular band structure\ncan yield short spin lifetimes, even for modest values of\nbroadening and SOC. The spin dephasing is also shown\nto be strongly anisotropic, and the spin lifetime exhibits\na characteristic dependence on the charge density, medi-\nated by the spin-split band structure. Taken together,\nthese features o\u000ber insight into the fundamental nature\nof spin dynamics in ballistic graphene, and suggest ap-\nproaches to control spin lifetimes by material and device\ndesign. Beyond graphene, we brie\ry consider the spin\ndynamics of the surface state of a 3D topological insula-\ntor, and derive a temperature-dependent spin lifetime in\nthe ballistic limit.\nSpin lifetime in clean systems . When momentum scat-\ntering is negligible, a \fnite spin lifetime can arise from\ndephasing, where an oscillating signal loses strength by\nmixing with other signals of di\u000berent phase or frequency.\nIn the presence of SOC, a charge carrier's spin will pre-\ncess around an e\u000bective magnetic \feld ~Beff. If~Beffis\nenergy- or momentum-dependent, and if the charge car-\nriers are distributed in energy or momentum, the total\nspin signal will undergo dephasing and will decay. As a\nsimple example, consider a system whose precession fre-\nquency varies linearly with energy, !(E) =!0+\u000bE, and\nwhose carriers occupy a Lorentzian energy distribution,\nL(E) =\u0011=[\u0019\u0001(E2+\u00112)], where\u0011is the half-width atarXiv:1602.04611v1  [cond-mat.mes-hall]  15 Feb 20162\nhalf-maximum. The total spin signal is\ns(t) =L(E)\u000ecos(!(E)t) =e\u0000\u000b\u0011t\u0001cos(!0t);(1)\nwhere\u000erepresents the convolution integral [38]. In gen-\neral, Eq. (1) shows that energy broadening plus nonuni-\nform spin precession leads to a decay of the spin due to\ndephasing, with a decay rate \u001c\u00001\ns=\u000b\u0011. For a continuous\nenergy distribution the decay is irreversible; the magni-\ntude of the signal will never recover to its original value.\nIn reality, a \fnite number of charge carriers will occupy\na discrete set of energies, but this also yields irreversible\ndecay if the carriers are randomly distributed in energy.\nThe decay is not necessarily exponential, but depends on\nthe broadening and the variation of the precession. For\nexample, replacing the Lorentzian with a Fermi distribu-\ntion gives a decay of \u0018t=sinh(\u0018t), where\u0018=\u0019\u000bkT and\nkTis the thermal energy [39], while a \fnite bias window\nyields a decay of sinc( \u000bVSDt), whereVSDis the source-\ndrain bias.\nBand structure of graphene with SOC . As discussed\nabove, a nonuniform precession can lead to spin decay in\na clean system. With that in mind, we examine the band\nstructure of graphene in the presence of SOC. Consider-\ning a single \u0019-orbital per carbon atom, the tight-binding\nHamiltonian is\n^H=\u0000tX\nhijicy\nicj+iVRX\nhijicy\ni~ z\u0001(~ s\u0002~dij)cj\n+i2p\n3VIX\nhhijiicy\ni~ s\u0001(~dkj\u0002~dik)cj;(2)\nwhere~ sare the spin Pauli matrices, tis the nearest-\nneighbor hopping, VIis the intrinsic SOC, and VRis the\nRashba SOC, induced by a transverse electric \feld or\nsubstrate [40]. Putting Eq. (2) into the spin+pseudospin\nbasis and taking the Fourier transform yields\n^H=2\n664\f \u0014 0i\r+\n\u0014\u0003\u0000\f\u0000i\r\u0003\n\u00000\n0i\r\u0000\u0000\f \u0014\n\u0000i\r\u0003\n+0\u0014\u0003\f3\n775; (3)\nwhere\u0014=\u0000teiky\u00012b=3\u0001[1 + 2e\u0000ikybcos(kxa)],\r\u0006=\nVR\u0001eiky\u00012b=3\u0001[1 + 2e\u0000ikybcos(kxa\u00062\u0019=3)],\f=\u0000VI\u0001\n[2 sin(2kxa)\u00004 cos(kyb) sin(kxa)],kxandkyare the mo-\nmenta along the x- andy-axes,a=p\n3=2\u0001acc,b=\n3=2\u0001acc, andaccis the carbon-carbon distance [41, 42].\nIn Fig. 1 we plot the band structure, assuming t= 2.7\neV,VI= 2.31\u0016eV, andVR= 25\u0016eV [20{24]. Figure 1(a)\nshows the conduction band over the full Brillouin zone,\nwith Dirac cones at the corners and trigonal warping at\nhigher energies. In Fig. 1(b) we plot the spin splitting\nof the conduction band, which is zero at the \u0000 and M\npoints and more complex near the K and K' points. A\nzoom of this is shown in Fig. 1(c) around the K point,\nindicating a nonuniform and anisotropic splitting. This\nFIG. 1. (color online) Band structure of graphene with SOC.\n(a) The conduction band and (b) spin splitting of the con-\nduction band over the entire Brillouin zone. (c) Splitting of\nthe conduction band near the K point. (d) Splitting of the\nconduction band near the K and K' points for the armchair\nand zig-zag directions. Inset: slice of the band structure near\nthe K point (bands are labeled 1 to 4).\nis shown in more detail in Fig. 1(d), where the splitting\nis plotted along the zig-zag and armchair directions for\nthe K and K' valleys. Along the armchair direction, the\nsplitting increases rapidly away from the Dirac point and\nsaturates at a constant value. Along the zig-zag direction\nthe splitting does not saturate, but instead varies slowly\nafter the initial rise.\nSpin dynamics of graphene with SOC . To understand\nthe connection between the band structure and spin de-\nphasing, we \frst consider the spin dynamics in a clean\nsystem. Starting with ^Hj\u001eii=\u000fij\u001eii, where\u000fiand\nj\u001eiiare the eigenvalues and eigenvectors of ^H, the time-\ndependent spin polarization of an initial state j 0iis\n~ p(t) =X\ni~Aii+X\ni>j[~Aijcos(!ijt) +~Bijsin(!ijt)];(4)\nwhere~Aij(~Bij) is the real (imaginary) part of\nh 0j\u001eiih\u001eij~ sj\u001ejih\u001ejj 0iand the sums run over all eigen-\nstates at a given momentum k. The spin polarization\nconsists of a constant term that depends on the polar-\nization of each band, h\u001eij~ sj\u001eii, and an oscillating term\nwhose frequencies are determined by the band splitting,\n!ij= (\u000fi\u0000\u000fj)=\u0016h. The weights of the oscillating terms are\ndetermined by the spin-mediated band overlap, h\u001eij~ sj\u001eji.\nAs illustrated in Fig. 1, ^Hdepends strongly on the mo-\nmentumk, and therefore so will the precession. This is\nshown in Fig. 2, where we plot the weights, frequencies,3\nFIG. 2. (color online) Spin dynamics in graphene with SOC.\n(a) The cosine weights, (b) the sine weights, and (c) the pre-\ncession frequency vs. momentum k, starting from the K val-\nley and moving along the zig-zag direction. (d)-(f) Time-\ndependent spin polarization for selected values of k. The spin\nis projected along the z-axis.\nand spin dynamics for kalong the zig-zag direction near\nthe K point. Since each eigenstate is polarized in the xy-\nplane, we consider polarization along the z-axis to study\nthe precession. In Fig. 2, there are two clear regimes of\nbehavior. At large k, the dynamics are dominated by os-\ncillations between the two valence bands (bands 1 and 2,\nsee the inset of Fig. 1(d)) and the two conduction bands\n(bands 3 and 4). In this regime, !21and!43are nearly\nidentical, leading to the regular oscillation in Fig. 2(f).\nAt smaller k,!21and!43diverge due to the electron-\nhole asymmetry induced by the intrinsic SOC, giving the\nbeating pattern in Fig. 2(e). As k!0, the dominant\nfrequencies switch from !21and!43to!32and!41. Near\nthe transition, the dynamics are governed by a combina-\ntion of all frequencies, giving the complex precession in\nFig. 2(d).\nThe transition between the low- and high- kregimes\ncan be understood from the eigenstates of ^H. To il-\nlustrate we consider a continuum model, ^H= \u0016hvF~ \u001b\u0001\n~k+\u0015R(~ \u001b\u0002~ s), wherevFis the Fermi velocity, ~ \u001bare\nthe pseudospin Pauli matrices, and \u0015Ris the Rashba\nstrength. Assuming kalong the zig-zag (+ x) direction,\nthe eigenstates at large karej\u001eji\u0019[1\u0017ji\u0010ji\u0010j\u0017j]T,\nwhere\u0017j=\u00001(+1) for bands 1 and 2 (3 and 4), and\n\u0010j=\u00001(+1) for bands 2 and 3 (1 and 4). The spin\npolarization of each eigenstate is (0 ;\u0010j;0) and the pseu-\ndospin polarization is ( \u0017j;0;0). From Eq. (4), the\nweights of the oscillating terms are then proportional to\nh\u001eijszj\u001eji= (1 +\u0017i\u0017j)(1\u0000\u0010i\u0010j). Thus, in the high- k\nregime precession only occurs between eigenstates with\nthe same pseudospin and opposite spin, i.e., only between\nthe two conduction ( !43) or valence ( !21) bands. At\nsmallkthe Rashba term dominates and the eigenstates\nbecomej\u001e1;4i\u0019[0\u00071i0]Tandj\u001e2;3i\u0019[1 0 0\u0006i]T.Here spin-pseudospin coupling is strong, as the spin-up\nand spin-down components of each eigenstate are located\non opposite sublattices [35]. The conduction-valence\nbands no longer overlap, while the overlap between bands\n1 and 4 (2 and 3) dominate the spin dynamics. This in-\nterband coupling, in conjunction with the broadening, is\nwhat can yield fast spin dephasing near the Dirac point.\nSpin lifetime in graphene with SOC . We can now make\nsome predictions about dephasing-induced spin lifetimes\nin clean graphene. Based on the spin dynamics, dephas-\ning should be fast near the Dirac point and slower at\nhigher energies. We can also predict a strong anisotropy\nin the spin lifetime. Along the zig-zag direction, the pre-\ncession away from the Dirac point varies continuously\nwith energy, and \u001csshould be constant. Along the arm-\nchair direction the precession frequency is constant, such\nthat\u001csshould diverge at high energies. For transport in\nall directions simultaneously, the anisotropy should pro-\nduce increased dephasing due to the mixing in k, thus\nreducing\u001cs.\nTo test these predictions, we turn to numerical calcu-\nlations of spin dynamics in clean graphene. We compute\nthe time- and energy-dependent spin polarization of an\ninitial statej 0ias [35, 43]\n~ p(E;t) =P\nk[h (t)j~ s\u000e(E\u0000^H)j (t)i+ h.c.]\n2\u0001P\nkh (t)j\u000e(E\u0000^H)j (t)i;(5)\nwherej (t)i=U(t)j 0i,U(t) =P\njj\u001ejih\u001ejje\u0000i\u000fjt=\u0016h,\n\u000e(E\u0000^H) =P\njj\u001ejih\u001ejjg(E\u0000\u000fj), andgis a broadening\nfunction that can be Lorentzian, a Fermi distribution,\netc. The sum represents a sample over k-space, along a\nsingle direction or over the entire Brillouin zone, and in-\ncludes both valleys. To extract \u001cs, we examine the time\ndependence of ~ p(E;t) at each energy. In general, the\ncomplex spin dynamics near the Dirac point preclude a\n\ft to a simple decaying cosine; instead, we de\fne \u001csas\nthe time when the envelope of ~ p(E;t) falls below e\u00001.\nThe numerical results are shown in Fig. 3(a), where\nwe plot\u001csas a function of energy, assuming Lorentzian\nbroadening with \u0011= 13:5 meV and polarization along the\nz-axis. Along the armchair direction, \u001csdiverges with\nincreasing energy, reaching 4 \u0016s at 300 meV. However,\nnear the Dirac point the dephasing limits \u001csto 14 ns.\nAlong the zig-zag direction, \u001cssaturates to 5-6 ns with\na slightly lower value of 4 ns at the Dirac point. For\ntransport in all directions, dephasing is much stronger\ndue to the anisotropic spin dynamics, giving \u001csbetween\n380 ps and 1.2 ns. A characteristic M-shape is observed,\nwith the high-energy downturn of \u001cs(E) resulting from\nthe increased anisotropy of the spin splitting, as pictured\nin Fig. 1.\nThe Lorentzian broadening in Fig. 3(a) highlights the\nmain features of the dephasing, and the magnitude coin-\ncides with energy scales and defect densities that are com-\nmon in experiments. However, this broadening is usually4\nFIG. 3. (color online) Spin lifetime in graphene with SOC.\n(a) The energy-dependent spin lifetime is strongly anisotropic.\n(b) The spin lifetime can also be limited by thermal broaden-\ning or a \fnite source-drain bias (zig-zag direction).\nenergy-dependent, and \u0011= 13:5 meV corresponds to an\ninelastic scattering time of 50 fs, much shorter than the\nspin precession time. Therefore, we also consider two\nother sources of broadening that may occur in the ballis-\ntic limit. Local and nonlocal measurements have demon-\nstrated that the electronic temperature Telcan be much\nlarger than the lattice temperature, such that thermal\nbroadening could dominate the spin dynamics without\nelectron-phonon scattering [44, 45]. In two-terminal mea-\nsurements a source-drain bias also serves as a source of\nbroadening, with transport occurring over a \fnite energy\nwindow. In Fig. 3(b) we show the impact of these types\nof broadening for transport along the zig-zag direction.\nForTel= 160 K (kT= 13:5 meV) or a source-drain bias\nof 100 mV, \u001cs\u00194 ns, similar to the Lorentzian broad-\nening. The spin dynamics are quite complex near the\nDirac point, such that the energy dependence depends\non the type of broadening and the speci\fc de\fnition of\n\u001cs. However, in general the quantitative features of Fig.\n3(a) are independent of the type of broadening.\nDiscussion and conclusions . To summarize, we have\nshown that the combination of energy broadening and\nnonuniform spin precession leads to dephasing that dic-\ntates spin lifetimes in the ballistic regime. It is important\nto note that without precession there will be no dephas-\ning. As shown in Eq. (4), the spin signal consists of static\nand oscillating components, and only the oscillating com-\nponents decay in time. Experimentally, spin is usually\ninjected in the plane and perpendicular to the transport\ndirection, resulting in an in\fnite lifetime in the ballistic\nlimit. Thus, for dephasing to occur, an extrinsic e\u000bect\nis needed to rotate the spin polarization and allow for\nprecession. This suggests that non-local Hanle measure-\nments, which employ a perpendicular magnetic \feld [25],\nbring a source of dephasing not present in two-terminal\nexperiments [46]. This di\u000berence may not matter for fast\nmomentum scattering, but could become important in\nvery clean samples. Fig. 3(a) showed that the spin life-\ntime is highly anisotropic, indicating that graphene spin-\ntronic devices could be optimized with proper lattice ori-entation, or by collimating the injected current [47]. The\nanisotropy is likely washed out in most experiments, with\n\u001cp\u001910 fs much smaller than the typical spin precession\ntime (\u001850 ps in this work). The results of Fig. 3(b) il-\nlustrate the impact of hot carriers and \fnite bias on spin\ndephasing, and suggest that these e\u000bects can impose fun-\ndamental limits on \u001cs. For a source-drain bias of 100 mV\nwe \fnd\u001cs\u00194 ns along the zig-zag direction. From Eq.\n(1),\u001csscales inversely with the Rashba strength and the\nbias, so reducing VRto 10\u0016eV andVSDto 10 mV would\nyield a lifetime of 100 ns. Note that due to the ballistic\ntransport, the spin relaxation length would be very long.\nThe generality of spin dephasing can also be appreci-\nated by studying the spin dynamics of the surface state\nof a 3D topological insulator. In the simplest approxi-\nmation, this state is characterized by a single Dirac cone\nwith the Hamiltonian ^H= \u0016hvF(^z\u0002~ \u001b)\u0001~k. Consider-\ning thermal broadening and applying Eqs. (1) and (4),\nthe out-of-plane spin dynamics are pz(t) =\u0018t=sinh(\u0018t)\u0001\ncos(!0t), with\u0018= 2\u0019kT= \u0016hand!0= 2EF=\u0016h. This yields\na lifetime of \u001cs=\u001d=T, with\u001d= 3:3 ps-K, giving \u001cs= 11\nfs at room temperature. Recent theoretical work on topo-\nlogical surface states found \u001cs=\u001ctr, where\u001ctris the\ncharge transport time, suggesting that the charge trans-\nport properties can be read directly from the spin dy-\nnamics [52]. However, this may only be true in the low\ntemperature limit, while thermal broadening may domi-\nnate the spin lifetime at higher temperatures.\nTo conclude, we have shown that even for reasonable\nvalues of broadening (meV) and Rashba SOC ( \u0016eV), spin\nlifetimes in clean graphene can still be very short. This\nsuggests that spin lifetimes in Rashba SOC materials, in\nthe absence of extrinsic e\u000bects, may have a fundamen-\ntal limit related to the intrinsic bandstructure. While\nthese results were for the limiting case of ballistic trans-\nport, they can o\u000ber insight into the nature of dephas-\ning and spin lifetimes in disordered systems. They could\nalso impact the observability of phenomena such as the\nanomalous Hall e\u000bect [48]. Beyond graphene, the ap-\nproach and methodology are applicable to other SOC\nmaterials, including 2DEGs [49{51], 2D transition metal\ndichalcogenides [10{12], or topological insulators [52, 53].\nThe authors thank the referees for insightful com-\nments, and are indebted to D. Soriano, J.E. Barrios-\nVargas, S. Valenzuela, J. Fabian, and D. Van Tuan for\nuseful discussions. This work is supported by the Euro-\npean Union Seventh Framework Programme under grant\nagreement 604391 Graphene Flagship, the Severo Ochoa\nProgram (MINECO, Grant SEV-2013-0295), the Spanish\nMinistry of Economy and Competitiveness (MAT2012-\n33911), and Secretar\u0013 \u0010a de Universidades e Investigaci\u0013 on\ndel Departamento de Econom\u0013 \u0010a y Conocimiento de la\nGeneralidad de Catalu~ na.5\n[1] Y. A. Bychkov and E. I. Rasbha, Pis'ma Zh. Eksp. Teor.\nFiz.39, 66 (1984) [JETP Lett. 39, 78 (1984)].\n[2] S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).\n[3] J. Sinova, D. Culcer, Q. Niu, N.A. 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Institut f ur Theoretische Physik, Universit at Hamburg, Jungiusstra\u0019e 9, 20355 Hamburg, Germany\nPACS 75.20.Hr { Local moment in compounds and alloys; Kondo e\u000bect, valence \ructuations,\nheavy fermions\nPACS 75.78.Jp { Ultrafast magnetization dynamics and switching\nPACS 03.65.Sq { Semiclassical theories and applications\nPACS 45.40.Cc { Rigid body and gyroscope motion\nAbstract { Spin dynamics in the Kondo impurity model, initiated by suddenly switching the\ndirection of a local magnetic \feld, is studied by means of the time-dependent density-matrix\nrenormalization group. Quantum e\u000bects are identi\fed by systematic computations for di\u000berent\nspin quantum numbers Sand by comparing with tight-binding spin-dynamics theory for the\nclassical-spin Kondo model. We demonstrate that, besides the conventional precessional motion\nand relaxation, the quantum-spin dynamics shows nutation, similar to a spinning top. Opposed to\nsemiclassical theory, however, the nutation is e\u000eciently damped on an extremely short time scale.\nThe e\u000bect is explained in the large- Slimit as quantum dephasing of the eigenmodes in an emergent\ntwo-spin model that is weakly entangled with the bulk of the system. We argue that, apart from\nthe Kondo e\u000bect, the damping of nutational motion is essentially the only characteristics of the\nquantum nature of the spin. Qualitative agreement between quantum and semiclassical spin\ndynamics is found down to S= 1=2.\nIntroduction. { The paradigmatic system to study the\nreal-time dynamics of a spin-1 =2 coupled to a Fermi sea is\nthe Kondo model [1]. It is mainly considered as a generic\nmodel for the famous Kondo e\u000bect [2], namely screening\nof the impurity spin by a mesoscopically large number of\nelectrons in a thermal state with temperature below the\nKondo temperature TK\u0018exp(\u00001=J\u001a), whereJis the\nstrength of the exchange coupling and \u001ais the density of\nstates. The Kondo e\u000bect is a true quantum e\u000bect which\noriginates from the two-fold spin degeneracy and is pro-\ntected by time-reversal symmetry. Longitudinal spin dy-\nnamics, such as the time-dependent Kondo screening, has\nbeen studied recently [3,4] by starting from an initial state\nwith a fully polarized spin, which can be prepared with the\nhelp of local magnetic \feld. The longitudinal dynamics is\ninitiated by suddenly switching o\u000b the \feld.\nTransversal spin dynamics, on the other hand, appears\nas a more classical phenomenon: It can be induced, for\nexample, by suddenly tilting a strong \feld B\u001dTKfrom,\nsay, ^xto ^zdirection. In \frst place this induces a precession\nof the spin around the new \feld direction with Larmor fre-\nquency!L/B. ForJ= 0, the equation of motion for theexpectation value of the spin, ( d=dt)hSit=hSit\u0002Bwith\nB=B^zhas the same form as the Landau-Lifschitz equa-\ntion for a classical spin [5]. When coupling the spin to the\nFermi sea with a \fnite J, energy can be transferred to the\nelectronic system and dissipated into the bulk. Hence, the\nspin must relax and align to the new \feld direction as is\nnicely seen in numerical studies of the Kondo model out of\nequilibrium [6]. For B\u001dTKthe spin precession and relax-\nation is qualitatively well described by semiclassical tight-\nbinding spin dynamics (TB-SD) (cf. e.g. Ref. [7]) where\nthe spin is assumed to be a classical dynamical observable.\nIn many cases, even the simple Landau-Lifschitz-Gilbert\n(LLG) equation [8,9] including a non-conserving damping\nterm, proportional to the \frst time derivative of the spin,\nseems to capture the essential (classical) physics.\nA major purpose of the present study is to check if there\nare quantum e\u000bects which are overlooked by the semiclas-\nsical approach to transversal spin dynamics (i.e., apart\nfrom the Kondo e\u000bect). To this end we compare numerical\nresults from exact quantum-classical hybrid theory [10,11],\ni.e., the TB-SD [7,12], with those of exact quantum theory,\ncomputed with time-dependent density-matrix renormal-\np-1arXiv:1609.05526v1  [cond-mat.str-el]  18 Sep 2016Mohammad Sayad Roman Rausch Michael Pottho\u000b\nization group (t-DMRG) [13, 14], for di\u000berent spin quan-\ntum numbers S. It turns out that even for S= 1=2 there\nis a surprisingly good qualitative agreement of quantum\nwith semiclassical dynamics. However, we also identify a\nphysical phenomenon, namely nutational motion, where\nremarkable di\u000berences are found:\nClassical and quantum nutation. { Besides precession\nand damping, inertia e\u000bects are well known in classical\nspin dynamics [15, 16] and can be described by an addi-\ntional term to the LLG equation with second-order time\nderivative of the spin. The resulting nutation of the spin\nmotion has been introduced and studied phenomenologi-\ncally [17,18] or with realistic parameters taken from \frst-\nprinciples calculations [19] but can also be derived on a\nmicroscopic level [20{22] within the general framework of\nsemiclassical spin dynamics [23{25].\nIn case of a quantum spin, inertia e\u000bects have not yet\nbeen studied. As compared to spin precession and damp-\ning, nutation is a higher-order e\u000bect [21], so that it is\nnot a priori clear whether or not spin nutation is sup-\npressed by quantum \ructuations. Here, by applying the\nt-DMRG to the spin- SKondo impurity model in a mag-\nnetic \feld, we are able to show for the \frst time that\nnutation also shows up in the full quantum spin dynam-\nics. Remarkably, however, quantum nutation turns out to\nbe strongly damped and shows up on a much shorter time\nscale as compared to the relaxation time. On a fundamen-\ntal level, this pinpoints an unconventional new quantum\ne\u000bect in transversal spin dynamics but is also relevant for\nexperimental studies suggesting, e.g., inertia-driven spin\nswitching [26,27] opposed to standard precessional switch-\ning [28,29].\nModel. { Using standard notations, the Hamiltonian of\nthe Kondo impurity model reads:\nH=\u0000Tn:n:X\ni<jX\n\u001b=\";#(cy\ni\u001bcj\u001b+ H.c.) +Jsi0S\u0000BS:(1)\nHere,ci\u001bis the annihilator of an electron with spin projec-\ntion\u001b=\";#at sitei= 1;:::;L of an open one-dimensional\nchain of length L. The hopping T= 1 between nearest-\nneighboring (n.n.) sites de\fnes the energy and the time\nscale (\u0016h\u00111). We assume a half-\flled band with N=L\nconduction electrons. The impurity spin Sis coupled an-\ntiferromagnetically with exchange coupling constant Jto\nthe local spin si0of the itinerant conduction-electron sys-\ntem at the \frst site of the chain, i0= 1. With the vector\nof Pauli matrices \u001c, we havesi=P\n\u001b\u001b0cy\ni\u001b\u001c\u001b\u001b0ci\u001b0=2.\nSis a quantum spin characterized by quantum num-\nberS=1\n2;1;3\n2;:::, and forS > 1=2, Eq. (1) is the un-\nderscreened Kondo model. Alternatively, Sis considered\nas a classical spin with \fxed length jSj=Scl:where\nScl:=p\nS(S+ 1) for a meaningful comparison with re-\nsults for a quantum spin.\nReal-time dynamics. { To initiate spin dynamics we con-\nsider a local magnetic \feld Bwhich, at time t= 0, is\nsuddenly switched from B=Bini^x, forcing the spin topoint in ^xdirection, to B=B\fn^z. This addresses, e.g.,\nspin-resolved scanning-tunneling microscope experiments\n[30{34]. We choose Bini=1to initially fully polar-\nize the impurity spin. Note that the conduction-electron\nspinsi0in the initial state is also polarized, but typi-\ncally much weaker, depending on the internal Weiss \feld\nBe\u000b\u0011JSproduced by the exchange interaction and the\nimpurity spin. The dynamics is (predominantly) transver-\nsal ifB\fn\u001dTKwhich ensures that the Kondo singlet\nremains broken and that there are no (signi\fcant) longi-\ntudinal spin \ructuations.\nFort! 1 we expect complete relaxation. This is\nachieved if the classical spin S(t) or, in the quantum case,\nS(t)\u0011hSit=h\t(t)jSj\t(t)ifully aligns with the ^ zaxis.\nLikewise the expectation value si0(t)\u0011hsi0itof the local\nconduction-electron spin at i0is expected to orient itself\nantiparallel to S(t) fort!1 .\nTime-dependent DMRG. { To study the (quantum)\ntime-evolution of S(t) andsi0(t) after the sudden switch\nof the \feld, we employ the time-dependent density-matrix\nrenormalization-group technique (t-DMRG) in the frame-\nwork of matrix-product states and operators [13]. The\nimplementation of a quantum spin with arbitrary Sis\nstraightforward. For an impurity model with the spin at-\ntached to the \frst site of the chain, the numerical e\u000bort is\nessentially independent of Sas only the dimension of the\nlocal Hilbert space at i0scales with 2 S+ 1. Due to the\nglobalU(1)\u0002U(1) symmetry of H, the total particle num-\nber and the zcomponent of the total spin are conserved.\nFor a sudden \feld switch from ^ xto ^zdirection, how-\never, only particle-number conservation can be exploited\nin the t-DMRG calculation. As compared to a purely lon-\ngitudinal dynamics, this implies an increased computa-\ntional e\u000bort. The time evolution of matrix-product states\nis computed using the two-site version of the algorithm\nas suggested in Ref. [14, 35] which is based on the time-\ndependent variational principle. The maximum bond di-\nmension reached during the propagation is about 2000.\nQuantum-spin dynamics. { We start the discussion with\nthe t-DMRG results, see the red lines in Fig. 1. The cal-\nculations have been performed for a chain with L= 80\nsites. For a quantum spin S= 1=2 (Fig. 1, top panel),\nand forJ= 1 andB\fn= 2, the dynamics is su\u000eciently\nfast, i.e., the main physical e\u000bects take place on a time\nscale shorter than the time where \fnite-size artifacts show\nup. In the bulk of the non-interacting conduction-electron\nsystem, wave packets typically propagate with group ve-\nlocityvF=d\"(k)=dk=\u00062Tat the Fermi wave vectors\nk=kF=\u0006\u0019=2 for half \flling. This roughly deter-\nmines the maximum speed of the excitations and de\fnes\na \\light cone\" [36, 37]. Hence, a local perturbation at\ni0= 1 starts to show arti\fcial interference with its re\rec-\ntion from the opposite boundary at i=Lafter a time of\nabouttinter= 2L=vg=L=T, i.e., after about 80 inverse\nhoppings { which is well beyond the time scale covered by\nFig. 1.\nThe most obvious e\u000bect in the time dependence of S(t)\np-2Inertia e\u000bects in the real-time dynamics of a quantum spin coupled to a Fermi sea\n\u00001.0\u00000.50.00.51.0S/Smax-0.10.00.1si0\u00001.0\u00000.50.00.51.0S/Smax-0.4-0.20.00.20.4si0\n10\u00001100101timet-0.050.000.050.10S=1/2\nS=5\nS=50t-DMRGTB-SD\n|S|/Smax\n|S|/Smaxt-DMRGTB-SD\nSzsi0,zTB-SDt-DMRGxxzzzxxz\nFig. 1: Top panel, upper part: Dynamics of S(t)=Smaxfor the\nKondo impurity model, Eq. (1), for J= 1 and B=B\fn^zwith\nB\fn= 2. Only xandzcomponents are shown. At t= 0,\nthe system is prepared with S(0)=jS(0)j= ^x. Time units are\n\fxed by the inverse hopping 1 =T\u00111. Red lines: t-DMRG\ncalculations for a quantum spin, S(t)\u0011 h\t(t)jSj\t(t)i, and\nS= 1=2 (Smax=S). Blue lines: semiclassical dynamics (TB-\nSD) with a classical spin S(t) of length Scl:=p\nS(S+ 1) =p\n3=2 (Smax=Scl:).Top panel, lower part: Local conduction-\nelectron moment si0(t)\u0011hsi0it.Middle : The same for S= 5.\nBottom :zcomponents of S(t) andsi0(t) forS= 50.\n(see upper part of the top panel) is the precessional motion\naround the ^ zaxis:Sx(t) (and likewise of Sy(t) which is not\nshown in the \fgure) oscillate with Larmor frequency !L\u0019\nB\fn. Note thatjS(t)j=jh\t(t)jSj\t(t)ijis nearly constant,\ni.e., there are no substantial longitudinal \ructuations or\nKondo screening.\nIn addition to the spin precession, there is damping:\nThe spin relaxes to its new equilibrium direction /^zon\nthe relaxation time scale \u001crel\u001950. Despite the fact that\nthe total energy and the zcomponent of the total spin are\nconserved (as is also checked numerically), this is the ex-\npected result: At t= 0 the system is locally in an excited\nstate; for large t, spin relaxation is achieved by dissipation\nof energy into the bulk of the chain. The dynamics does\nnot stop until the excitation energy \u0018SB\fnis fully dissi-\npated into the bulk, and the system is { locally, close to\ni0{ in its ground state.\nConduction-electron dynamics. { In the ground state of\nthe system at time t= 0, the local conduction-electronspin ati0is partially polarized in \u0000^xdirection, i.e., an-\ntiparallel toS(t= 0) due to the internal magnetic \feld\nJS(0) (see top panel of Fig. 1, lower part). For t>0 we\n\fnd thatsi0(t) follows the dynamics of the impurity spin\nS(t)almost adiabatically, i.e., at a given instant of time tit\nis slightly behind the (instantaneous) ground-state expec-\ntation valuehsi0ig:s:\"#S(t) for the conduction-electron\nsystem with a \\given\" Weiss \feld JS(t). This slight re-\ntardation e\u000bect is clearly visible in Fig. 1 (compare the\nlocation of the \frst minimum of Sx(t) with the \frst maxi-\nmum ofsi0x(t), for instance). In the semiclassical picture\nretardation has been identi\fed to drive the relaxation of\nS(t) [7].\nQuantum nutation. { In addition to the expected pre-\ncessional motion and relaxation of si0(t), there is a weak\nadditional superimposed oscillation visible in si0z(t). For\nS= 1=2 the frequency is close to the precession frequency.\nHowever, the results for higher spin quantum numbers (see\nlower part of the middle panel, S= 5) show that these os-\ncillations have a characteristic frequency !Nand hence a\nphysical cause which may require but is independent of\nthe precessional motion.\nThezcomponent of the impurity spin actually shows\noscillations with the same frequency and almost the same\namplitude (which can hardly be seen in the \frst two panels\nof Fig. 1 due to the rescaling of S(t) bySmax) but becomes\nobvious in the bottom panel (no rescaling, S= 50). By\ncomparing with the semiclassical spin dynamics, we will\nargue that this is in fact nutation of the quantum spin.\nTight-binding spin dynamics. { Most (but not all) fea-\ntures of the transversal quantum dynamics are qualita-\ntively captured by the numerically much cheaper \\tight-\nbinding spin dynamics\" (TB-SD) [7, 12], i.e., quantum-\nclassical hybrid or Ehrenfest dynamics. TB-SD originates\nfrom the Hamiltonian Eq. (1) by treating the impurity\nspinS(t) as a classical dynamical observable which cou-\nples to the (quantum) system of conduction electrons. Its\nequation of motion is derived from the canonical equation\n_S=fS;hHitg(see Refs. [7,10] for the Poisson bracket of\nspin systems), which has the form of a Landau-Lifschitz\nequation,\n_S(t) =S(t)\u0002B\u0000JS(t)\u0002si0(t): (2)\nTo also getsi0(t) =1\n2tr2\u00022\u001ai0i0(t)\u001b, it must be comple-\nmented, however, by a von Neumann equation, id\ndt\u001a(t) =\n[T(t);\u001a(t)], for the reduced one-particle density matrix\n\u001a(t) of the electron system whose elements are de\fned\nas\u001aii0;\u001b\u001b0(t)\u0011hcy\ni0\u001b0ci\u001bit. Here, the elements of the ef-\nfective hopping matrix are Tii0;\u001b\u001b0(t) =\u0000T\u000ehii0i\u000e\u001b\u001b0+\n\u000eii0\u000ei0i0J\n2(S(t)\u001b)\u001b\u001b0. The numerical solution using a high-\norder Runge-Kutta method is straightforward [38].\nResults of the semiclassical approach. { TB-SD results\nare shown by light blue lines in Fig. 1. To make contact\nwith the t-DMRG data, we again consider L= 80 sites al-\nthough much larger systems could be treated numerically\n(see for instance Ref. [7]). Overall, the semiclassical theory\np-3Mohammad Sayad Roman Rausch Michael Pottho\u000b\n10\u0000210\u00001100101102timet0.98⇡0.99⇡⇡angle\u0000(t)S(t)Bsi0(t)\u0000(t)S=1/2S=1S=2S=5S=20S=50\nJ=1\nFig. 2: Angle \r(t) between S(t) and si0(t) in the spin dy-\nnamics after the sudden switch of the \feld from ^ xto ^zdi-\nrection. TB-SD results for J= 1,B\fn= 0:1 and di\u000berent\nScl:=p\nS(S+ 1) as indicated. Inset: schematic illustration\nof the nutational motion, see text.\nproduces qualitatively very similar results as compared to\nthe quantum dynamics. This concerns the precessional\nmotion, the relaxation time scale and also the occurrence\nof nutation and the nutation frequency and amplitude.\nHowever, we can identify basically three quantum e\u000bects\nwhich are di\u000berent or even absent in the TB-SD:\n(i) Initially the local conduction-electron spin at i0is\nless polarized in the quantum case, and this has some\nquantitative consequences for the subsequent spin dynam-\nics. The reason is that with Scl:=p\nS(S+ 1) the classical\nWeiss \feld is stronger: JScl=Jp\n3=2>J= 2 =JS.\n(ii) Opposed to the classical-spin case, which exclusively\ncomprises transversal dynamics, we \fnd jS(t)j6= const in\nthe quantum case, i.e., there are residual longitudinal \ruc-\ntuations (see top panel, upper part). Due to the suppres-\nsion of the Kondo e\u000bect by the magnetic \feld, these are\nmoderate, such that the deviations from the TD-SD are\nsmall. One should note, however, that nevertheless (weak)\nlongitudinal \ructuations are essential for true quantum\nspin dynamics: Assuming the complete absence of longitu-\ndinal \ructuations, we would have hSit=S^n(t) with some\nunit vector ^ n(t). Aligning the momentary quantization\naxis to ^n(t), the quantum state at time tis a product state\nwith zero impurity-bath entanglement. For the impurity-\nspin equation of motion, dhSit=dt=hSit\u0002B\u0000JhS\u0002si0it,\nthis implies the factorization hS\u0002si0it=S(t)\u0002si0(t), re-\nsulting in Eq. (2). With the analogous factorization in the\nequations of motion for the conduction-electron degrees of\nfreedom, this implies classical spin behavior. Hence, lon-\ngitudinal \ructuations produce entanglement and quantum\ne\u000bects.\n(iii) The nutational motion is strongly damped in the\nquantum-spin case. Oscillations of Sz(t) and ofsi0z(t)\nwith frequency !Ndecay on a \fnite time scale \u001cNwhile\nthere is no visible damping of the nutation for a classical\nspin on the scale displayed in Fig. 1. This is most obvious\nforS= 50 (bottom panel), but also for S= 5 (middle\npanel, lower part).Sdependence. { For large spin quantum numbers, one\nexpects that the quantum-spin dynamics becomes equiv-\nalent with that of a classical spin of length Scl:=p\nS(S+ 1) [39{43]. Indeed, the agreement constantly im-\nproves with increasing S, see Fig. 1. The common trends\nfound with increasing Sare the following:\n(i) There is a stronger and stronger initial polarization\nof the local conduction-electron spin at i0due to the in-\ncreasing magnitude of the Weiss \feld Be\u000b\u0011JScoupling\ntosi0. ForS= 5 it is more than 80% polarized.\n(ii) The relaxation time \u001crelincreases with increasing S.\nForS= 5 (see Fig. 1, middle panel) Sz(t) has reached only\n50% of its \fnal saturation value, and for S= 50 (bottom\npanel) there is hardly any damping visible on the time\nscale accessible to the t-DMRG computations. Within\nweak-Jperturbation theory and assuming that the spin\ndynamics is slow as compared to the electronic time scales,\nwe expect\u001crel/Sin the large- Slimit, as is detailed in\nthe Supplemental Material [44]. However, for both the\nsemiclassical and the quantum theory, we \fnd \u001crel/S2\nfrom the data. This is at variance with LLG theory and\ncan be traced back to the breakdown of the Markov ap-\nproximation (see [44]).\n(iii) For the nutation frequency we \fnd !N/Sin the\nlarge-Slimit (see also the discussion below). The am-\nplitude of the nutation vanishes for S! 1 in both,\nthe quantum- and the classical-spin case. In this way\nquantum- and classical-spin dynamics become equivalent\nin the large- Slimit despite the absence of damping of the\nnutational motion in the classical case.\n(iv) We \fnally note that jS(t)j=Smaxbecomes constant\nin the quantum case as S!1 .\nMicroscopic cause of nutation. { The nutational motion\ncan be understood easily within the semiclassical approach\n(except for damping): Recall that the impurity spin pre-\ncession with frequency !L\u0019B\fnis mainly caused by the\ntorque due to the magnetic \feld and note that the second\nterm on the right-hand side of Eq. (2) is small if si0(t)\nandS(t) are nearly collinear. In fact, in the instanta-\nneous ground state at time t, the conduction-electron lo-\ncal momentsi0(t) would be perfectly aligned antiparallel\ntoS(t) due to the antiferromagnetic exchange coupling J\nsuch thatsi0(t) exhibits a precessional motion with the\nsame frequency !L\u0019B\fn. Fig. 2 demonstrates that the\nstronger the e\u000bective \feld JS, the smaller is the devia-\ntion of the angle \r(t) betweenS(t) andsi0(t) from\r=\u0019.\nGenerally, however, \r(t)< \u0019 (for allt) since, due to the\ndamping, it takes a \fnite time for si0(t) to react to the\nnew position of S(t) (see the inset of Fig. 2). Note that for\nvery largeSonly the time average \r(t) is smaller than \u0019\n(for instance, see S\u001520 in Fig. 2). This retardation e\u000bect\nresults in a \fnite (average) torque JS(t)\u0002si0(t) acting on\nsi0(t), as can be seen from its equation of motion:\nd\ndtsi0(t) =JS(t)\u0002si0(t) +TImX\n\u001b\u001b0hcy\ni0\u001b\u001c\u001b\u001b0ci0+1\u001b0it:\n(3)\np-4Inertia e\u000bects in the real-time dynamics of a quantum spin coupled to a Fermi sea\n05101520S05101520nutation frequency!N\n0.00.20.40.60.81.0JJ=1.0S=20t-DMRGTB-SDtwo-spin model\nFig. 3: Nutation frequency !Nas a function of SforJ= 1\n(left) and as a function of JforS= 20 (right). Dynamics\ninitiated by a switch of the \feld from ^ xto ^zdirection with\nB\fn= 0:1. Results for di\u000berent Scl:orS, respectively, as ob-\ntained by TB-SD (crosses) and t-DMRG (circles) in comparison\nwith the classical two-spin model (\flled dots).\nThe second term on the right-hand side is important for\nenergy and spin dissipation into the bulk of the system\nand causes the usual damping of the precession of si0(t)\n(and ofS(t)) aroundB. The \frst term, however, leads to\nnutational motion.\nThis is most easily understood if there is a separation\nof time scales, i.e., if the nutation frequency !Nis large\ncompared to the Larmor frequency !L\u0019B\fn. In this\nlimit, Eq. (3) implies that si0(t) precesses with frequency\n!N\u0019JScl:approximately around the momentary direc-\ntion ofS(t) (which itself slowly precesses around the \feld\ndirection). Actually, however, due to the retardation, si0\nprecesses around an axis which is slightly tilted as com-\npared to the momentary direction of S(t). This is nicely\ndemonstrated by the oscillations of \r(t) with time-average\n\r(t)< \u0019 as displayed in Fig. 2. Furthermore, the equa-\ntions of motion, Eq. (2) and Eq. (3), with the second term\ndisregarded, imply that Sz(t) +si0z(t) = const and, there-\nfore, the impurity spin shows the same nutational motion,\nbut with opposite amplitude.\nIn the middle panel of Fig. 1 we in fact observe a fast\noscillation ofsi0(t) with a frequency almost perfectly given\nbyJScl:(withJ= 1 andS= 5). Note that the nutation of\nS(t) is hardly visible due to the rescaling with Smax:. The\nthird panel for S= 50 nicely demonstrates the nutational\nmotion of both, si0(t) andS(t), with opposite amplitudes\nand common frequency !N\u001d!L.\nFig. 3 displays the results of systematic TB-SD calcu-\nlations which demonstrate the linear dependence of !N\nonJandSfor largeJS. These calculations have been\nperformed for a much weaker \feld B\fn= 0:1 resulting\nin a much slower precession of S(t) aroundB. Note the\nnearly perfect agreement between classical- and quantum-\nspin calculations also for smaller JSwhere there is a sig-\nni\fcant deviation from a linear behavior.\nThe mechanism described above also explains that the\namplitudes of the nutational oscillations vanish in the limit\n100101102spin quantum numberS0.00.20.40.60.81.01.21.4entanglement entropySi0\nMany-Body Systems  and Quantum-Statistical Methodsquantum fluctuations\nps(s+ 1)\nL= 80\ni0=1\nJ=1\nBini=0.5ex)Bfinal=2.0ez\nS=0.5\nS=1.0\nps(s+ 1)\nL= 80\ni0=1\nJ=1\nBini=0.5ex)Bfinal=2.0ez\nS=0.5\nS=1.0quantum spin S •weak longitudinal fluctuations Kondo effect suppressed •S=1/2: reasonable agreement with semiclassical theory •improves for higher S •higher S: weaker dampinglonger reversal timequantum spinclassical spin\nTS(t)B\nJ\nt=0t=20Fig. 4: Entanglement entropy of the two-spin subsystem (im-\npurity spin and site i0= 1, see dashed ellipse inset) in the\nenvironment ( i= 2;:::;L ) as a function of SforJ= 1 and at\ndi\u000berent times t= 0 andt= 20. t-DMRG results for B\fn= 2\nandL= 50.\nS!1 : An increasing internal Weiss \feld JSmore and\nmore alignssi0(t) toS(t), i.e.,\r(t)!\u0019. Consequently,\ntorqueJS(t)\u0002si0(t) acting onsi0(t) vanishes in the large-\nSlimit.\nTwo-spin model. { Fig. 3 additionally presents the re-\nsults for!Nas obtained by a semiclassical two-spin model:\nH2\u0000spin=JsS\u0000BS: (4)\nThis model disregards the coupling of the site i0to the\nbulk of the conduction-electron system and thus cannot\ndescribe the damping of the precessional motion. Due to\nthe absence of damping, the time-averaged angle is \r(t) =\n\u0019.\nFrom the numerical solution of Eq. (4) we also learn that\nit does not predict any damping of the nutational motion.\nThe nutational oscillations themselves, however, are qual-\nitatively captured by H2\u0000spinand, in fact, the whole line\nof reasoning explaining the inertia e\u000bect also applies to\nthis model. The nutation frequencies as computed from\nH2\u0000spin\ft the TB-SD and t-DMRG results rather well\nfor strong e\u000bective \felds Be\u000b\u0011JS\u001dT= 1; stronger\ndeviations are found for JS!2 (see Fig. 3). For JS < 2,\nthere are clear nutational oscillations in the spin dynamics\nof the full model (1), as is seen in the top panel of Fig. 1,\nbut!Ncannot be de\fned accurately.\nBound states. {Be\u000b;cr= 2 is actually the critical value\nof the local e\u000bective \feld Be\u000b\u0011JSwhich couples to the\nlocal conduction-electron spin at i0. ForBe\u000b> B e\u000b;cr\nthere are two one-particle eigenenergies of the Hamiltonian\n(1) corresponding to bound states which symmetrically\nsplit o\u000b the continuum at the lower and at the upper band\nedge, respectively. Note that Be\u000b;crvanishes for a site i0\nin the bulk of an in\fnite chain as is well known for one-\ndimensional systems. Contrary, at the edge ( i0= 1) there\nis a \fnite critical \feld, as is reminiscent of the physics in\nhigher dimensions.\nThe sudden switch of the \feld excites the system lo-\ncally ati0. Consequently, if JS >B e\u000b;cr, the subsequent\np-5Mohammad Sayad and Roman Rausch and Michael Pottho\u000b\ndynamics is predominantly local since the excitation is\nmainly carried by a state whose amplitude is exponen-\ntially suppressed with increasing distance from i0. The\ndynamics should be understood in this case as a weak\nperturbation of the dynamics of the two-spin model Eq.\n(4).\nThat this also applies to the quantum-spin case is\ndemonstrated with Fig. 4 which shows the entanglement\nentropySi0of the subsystem consisting of the quantum\nimpurity spin and the conduction-electron site i0. In the\nground state at t= 0, the entropy decreases with increas-\ning e\u000bective \feld JS. ForJS= 50 it nearly vanishes\nwhich implies that ground-state expectation values of local\nobservables at i0are almost perfectly described with the\n(quantum version of the) two-spin model Eq. (4). With\nincreasing time t, the entropy generally increases, while\nfor strong e\u000bective \felds JSis stays close to zero, i.e., the\ntwo-spin model also well captures the dynamics of local\nobservables in this case.\nDamping of quantum nutation. { To explain the e\u000e-\ncient damping of the nutational motion on a very short\ntime scale\u001cNin the quantum-spin case, we \frst consider\nthe quantum variant of the two-spin model Eq. (4), i.e.,\nboth,Sands, are considered as quantum spins with spin\nquantum numbers Sand 1=2, respectively. The time-\ndependent expectation value Sz(t) after the sudden switch\nof the \feld is readily computed and shows oscillations with\nfrequency!N. Already in the two-spin model those are\ndamped on a time scale \u001cNwhich agrees with that seen in\nthe results of the full model in Fig. 1 for S\u00155. Writing\nSz(t) =hSzit=P\nm;ncm;nexp(i(Em\u0000En))twith energy\neigenstates mandnofH2\u0000spinand coe\u000ecients cm;nde-\npending on the preparation of the initial state, it becomes\nobvious that this damping results from the dephasing of\noscillations with the excitation energies Em\u0000Enof the\nsystem.\nDue to the small Hilbert-space dimension of the two-\nspin model, however, there are strong revivals of the oscil-\nlations occurring at \fnite revival times. In fact, for S= 5,\nthe \frst revival of nutational oscillations of si0z(t) can be\nseen in the t-DMRG result around t= 20 (Fig. 1, middle\npanel, lower part). With increasing Sand thus with in-\ncreasing Hilbert space, however, the revival times quickly\nexceed the time scale accessible to t-DMRG in the full\nmodel. Furthermore, as the example for S= 5 in Fig. 1\nshows, the revivals themselves are strongly damped in the\nfull theory, opposed to the nearly perfect revivals in the\ntwo-spin-model dynamics. As this (secondary) damping\nof nutation is caused by the residual e\u000bective coupling of\nthe two-spin model to the bulk of the system, it becomes\nless and less e\u000ecient with increasing S, while at the same\ntime the revival time strongly increases and the amplitude\nof the oscillations decreases.\nConclusions. { Inertia e\u000bects in spin dynamics have been\ndiscussed intensively in the recent years, mainly in the\ncontext of applications for magnetic devices [15{27]. The\nmost fundamental system which covers the essentials ofspin dynamics, however, namely a single spin coupled to a\nFermi sea has not yet been addressed in this respect. Ap-\nplying exact quantum and semiclassical numerical tech-\nniques to the Kondo impurity model, we could demon-\nstrate that the real-time dynamics, initiated by switching\nthe direction of a magnetic \feld coupled to the spin, not\nonly exhibits spin precession and spin relaxation but also\nnutational motion known from a gyroscope. The e\u000bect\nnot only shows up in the impurity-spin dynamics but also\nin the dynamics of the conduction-electron local magnetic\nmoments. It is very robust and found in a large regime\nof coupling constants using tight-binding spin dynamics\nand treating the spin as a classical observable. We \fnd\nthat nutation amplitudes are small as compared to am-\nplitudes in precessional motion. The frequency is, in the\nstrong-coupling limit, linear in JandScl:.\nOur study has demonstrated that nutational motion\nis not restricted to classical-spin systems but is robust\nagainst quantum \ructuations. Despite the fundamental\ndi\u000berences between semiclassical and quantum dynamics,\nquantum-spin nutation is found to be very similar to the\nclassical-spin case in many respects. There is a qualitative,\nand with increasing spin-quantum numbers also quanti-\ntative agreement between quantum and semiclassical dy-\nnamics. Kondo screening of the impurity spin represents\nan important exception which, however, in the present\nstudy plays a minor role only as Kondo-singlet formation\nis inhibited by the external \feld.\nThe main e\u000bect of the quantum nature of the spin is\na very e\u000ecient damping of the nutational motion on a\nvery short (femtosecond) time scale which is basically in-\ndependent of the relaxation time scale for the precessional\nmotion. In the strong-coupling ( JS!1 ) limit, the spin\ndynamics is essentially local and captured by an emergent\ntwo-spin model which has served to understand the physi-\ncal origin of the damping of quantum nutation, namely de-\nphasing of local spin excitations with revivals suppressed\nby the coupling to the bulk of the system.\nAn important implication of our study is that direct ob-\nservation of nutational motion, e.g., of magnetic nanopar-\nticles with a (quantum) macrospin Scoupled to the\nconduction-electron band of a nonmagnetic metallic sur-\nface, requires a sub-picosecond time resolution. On the\nother hand, inertia-driven spin switching in antiferromag-\nnets [26,27] has already been demonstrated successfully.\n\u0003\u0003\u0003\nWe would like to thank Christopher Stahl for instruc-\ntive discussions. 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P. andPetruccione F. ,The Theory of Open\nQuantum Systems (Oxford University Press, New York)\n2002.\n[46] L. D. Landau and E. M. Lifshitz, Physik. Zeits. Sowje-\ntunion 8,153 (1935); T. Gilbert, Phys. Rev. 100, 1243\n(1955); T. Gilbert, Magnetics, IEEE Transactions on 40,\n3443 (2004).\n[47]Kikuchi R. ,J. Appl. Phys. ,27(1956) 1352.\np-7Mohammad Sayad Roman Rausch Michael Pottho\u000b\nInertia e\u000bects in quantum and classical\ndynamics of a spin coupled to a Fermi sea\n| Supplemental material |\nMohammad Sayad, Roman Rausch and Michael Pottho\u000b\nI. Institut f ur Theoretische Physik, Universit at Hamburg,\nJungiusstra\u0019e 9, 20355 Hamburg, Germany\nClassical spin-only theory. { It is instructive to discuss\ntheScl:dependence of the spin damping and nutation in\nan e\u000bective classical spin-only theory, i.e., after integrating\nout the electron degrees of freedom. Following Ref. [21],\nan e\u000bective equation of motion for S(t) is obtained for the\nclassical-spin case by (i) lowest-order perturbation theory\ninJand (ii) assuming that the spin dynamics is slow:\n(i) In the weak- Jregime, we can use the Kubo formula\nto \fnd the linear response si0(t) =JRt\n0dt0\u001floc(t\u0000t0)S(t0)\nof the local conduction-electron magnetic moment at site\ni0and timetcaused by the time-dependent e\u000bective \feld\nBe\u000b(t0)\u0011JS(t0) at timet0. Here, the time-homogeneous\nresponse function \u001floc(t\u0000t0) is the retarded local spin\nsusceptibility of the electron system at i0. This is a\nrank-two tensor which, for the present case, is diago-\nnal and constant: \u001floc(t) =\u0000i\u0002(t)h0j[si0z(t);si0z(0)]j0i\nwherej0iis the initial ground state at time t=\n0, where \u0002 is the Heaviside step function and where\nsi0z(t) = exp(iHet)si0z(0) exp(\u0000iHet) with the tight-\nbinding Hamiltonian He[\frst term in Eq. (1)]. Inserting\ninto Eq. (2), we \fnd\n_S(t) =S(t)\u0002B\u0000J2S(t)\u0002Zt\n0dt0\u001floc(t\u0000t0)S(t0):(5)\n(ii) Assuming that the classical spin is slow on the\nmemory time scale set by \u001floc(t) and expanding S(t0) =\nS(t)+_S(t)(t0\u0000t)+S(t0)(t\u0000t0)2=2+\u0001\u0001\u0001under the integral,\none \fnds [21] the LLG equation with an additional inertia\nterm:\n_S=S\u0002B\u0000\u000bS\u0002_S+IS\u0002S; (6)\nwhere, after sending the upper integral limit to in\fnity as\nusual [7,21,45],\n\u000b=\u0000J2Z1\n0d\u001c\u001c\u001f loc(\u001c) (7)\nand\nI=\u0000J2\n2Z1\n0d\u001c\u001c2\u001floc(\u001c) (8)\nare the Gilbert damping constant and the moment of in-\nertia, respectively ( \u000b;I > 0). Eq. (6) constitutes a purely\nclassical spin-only theory which, after some extensions,\ncan serve as a starting point for microscopic spin-dynamics\ncalculations [46]. The inertia term is known to give rise to\nnutation (see, e.g., [21,22] for a detailed discussion).\nDependence on Scl:.{ The equation of motion (6) has\na simple scaling property: One can easily verify that ifS(t) solves the equation for parameters B;\u000bandI, then\nS0(t)\u0011\u0015S(t) solves the same equation with rescaled pa-\nrameters\u000b=\u0015 andI=\u0015. We conclude that the damping\nparameter\u000band the inertia constant Ihave a stronger\ne\u000bect on the dynamics of an elongated spin ( Scl:>1).\nNamely, with smaller e\u000bective parameters\n\u000b0=\u000b=S cl:; I0=I=Scl:; (9)\none obtains the same dynamics as for a spin of unit length.\nAs is obvious from the de\fning equations (7) and (8),\nthe parameters \u000b;Ido not depend on Scl:but are proper-\nties of the conduction-electron system only. The equation\nof motion for a given system is therefore independent of\nScl:.\nFor \fxed\u000b, the relaxation time can be calculated [47]\nand is given by \u001crel/(1 +\u000b2S2\ncl:)=(\u000bScl:B). In the large-\nScl:limit, we thus have\n\u001crel/Scl:: (10)\nIdentifying Scl:with the modulus of the angular mo-\nmentumLof a fast-spinning gyroscope, elementary theory\n(see, e.g., Ref. [15]) tells us L=I!N, and hence\n!N/Scl:: (11)\nThis recovers the numerical result found in the Scl:!1\nlimit and corroborates the interpretation given in the main\ntext. It also appears more general and does not depend\non the special form of the underlying equations of motion\nas long as the motion of a magnetic moment is concerned\n[21,22].\nDiscussion. { In fact, Eq. (11), is almost perfectly veri-\n\fed within the framework of the full semiclassical TB-SD,\nsee Fig. 3 for large Scl:. For the dependence of the re-\nlaxation time \u001crelonScl:, however, we \fnd a quadratic\nrelation in the full TB-SD, \u001crel/S2\ncl:, rather than the lin-\near trend predicted by Eq. (10). This indicates that the\ne\u000bective theory is of limited use for reproducing the exact\nresults of TB-SD on the semiclassical level for large Scl:\nand is furthermore inconsistent with the quantum dynam-\nics as well.\nBoth assumptions (i) and (ii) are in fact questionable\nfor the parameter regime studied here: One may reject\nthe Markov-type approximation (ii) and describe the spin\ndynamics with the linear-response theory and Eq. (5). The\nsame scaling argument as above again tells us that an\nelongated ( \u0015 > 1) spinS0(t)\u0011\u0015S(t) solves the same\nintegro-di\u000berential equation,\nS0(t;B;J0) =S(t;B;J); (12)\nbut with rescaled exchange coupling\nJ0=J=p\n\u0015: (13)\nThis means that a weaker interaction J0< J (for\u0015 >1)\nleads to the same dynamics.\np-8Inertia e\u000bects in the real-time dynamics of a quantum spin coupled to a Fermi sea\nNow, from the numerical evaluation of Eq. (5) it is well\nknown [7, 12] that, for \fxed Scl:, the damping becomes\nstronger and the relaxation time shorter with increasing J.\nHence, for \fxed J, with the argument leading to Eq. (13),\nthe dynamics of an elongated classical spin must therefore\nshow a stronger damping, i.e., a shorter relaxation time.\nAs this con\ricts with our observations here (see Fig.\n1), we must conclude that approximation (i), i.e., lowest-\norder perturbation theory in Jor linear-response theory,\nis no longer valid for the parameter regime studied here.\nThis furthermore implies that, besides damping, also the\nnutational motion of a spin exchange coupled to an unpo-\nlarized Fermi sea cannot be captured by the perturbative\napproach (despite the fact that the linear trend Eq. (11)\nis reproduced). This is not too surprising in view of the\nexplanation for the inertia e\u000bect given in the main text,\nnamely the formation of a bound state of the impurity\nspin with the exchange-coupled conduction-electron spin\nand the weak interaction of this bound state with the bulk\nof the system. Those details of the electronic structure are\nobviously not accounted for in a simple e\u000bective spin-only\ntheory, such as Eq. (5), where the electron dynamics only\nenters via the J= 0 spin susceptibility.\np-9"    },    {        "title": "0705.0277v3.Charge_current_driven_by_spin_dynamics_in_disordered_Rashba_spin_orbit_system.pdf",        "content": "arXiv:0705.0277v3  [cond-mat.mes-hall]  28 Jan 2008Charge current driven by spin dynamics in disordered Rashba spin-orbit system\nJun-ichiro Ohe∗\nI. Institut f¨ ur Theoretische Physik, Universtit¨ at Hambu rg, Jungiusstrasse 9, 20355 Hamburg, Germany\nAkihito Takeuchi\nDepartment of Physics, Tokyo Metropolitan University, Hac hioji, Tokyo 192-0397, Japan\nGen Tatara\nDepartment of Physics, Tokyo Metropolitan University, Hac hioji, Tokyo 192-0397, Japan\nPRESTO, JST, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Jap an\n(Dated: August 23, 2021)\nPumping of charge current by spin dynamics in the presence of the Rashba spin-orbit interaction\nis theoretically studied. Considering disordered electro n, the exchange coupling and spin-orbit\ninteractions are treated perturbatively. It is found that d ominant current induced by the spin\ndynamics is interpreted as a consequence of the conversion f rom spin current via the inverse spin\nHalleffect. Wealso foundthatthecurrenthasanadditionalc omponentfrom afictitiousconservative\nfield. Results are applied to the case of moving domain wall.\nRecent spintronics studies aim at manipulation both of charge and sp in degrees of freedom [1]. Central roles\nare played by the spin-orbit interaction and the exchange interact ion between the conduction electrons and local\nspins [2, 3, 4, 5, 6]. It has been shown that the exchange coupling is u seful for electrical control of magnetization\ndynamics via spin transfer torque [2, 3]. It can also be used to pump s pin current from the precession of the\nmagnetization [7, 8, 9]. The spin-orbit interaction has been recently found to induce a magnetism by the application\nof electric voltage (the spin Hall effect) [10, 11, 12].\nBy combining the exchange and the spin-orbit interactions, various phenomena are expected, and the subject\ndiscussed in this paper is one of them; pumping of charge current by dynamical magnetization. The idea is to convert\nthe pumped spin current into a charge current by using of the spin- orbit interaction as proposed by Saitoh et al. [13].\nThis currentdue tothe inversespin Hall effect wasindeed observed [13, 14, 15] in metallicsystems wherethe spin-orbit\ninteraction is induced by Pt atom.\nTheoretically, generation of the electric field or voltage due to the d ynamical spin structure was discussed by\nStern [16]. The field is not a real electric field, but an effective one due to a spin Berry phase acting only on charge\ndegrees of freedom with spin (like the electron). The mechanism is sim ilar to the Faraday’s law, but a magnetic flux\nis replaced by a fictitious field from the spin Berry phase. The induced field is described by ∇×Eeff=−˙b, wherebis\nfield of spin Berry phase, and current is divergenceless; ∇·j= 0. The theory was recently applied to a domain wall\nby Barnes and Maekawa [17], and the effect of the spin relaxation was studied by Duine [18], where the relaxation\nwas introduced by a phenomenological term ( β-term). All these studies have been done in the adiabatic limit, where\nthe exchange coupling between the local spin and the conduction ele ctrons is strong.\nChargecurrent as a result of inversespin Hall effect was theoretic allystudied by Zhang and Niu [19] and Hankiewicz\net al. [20] (the effect was called reciprocal spin Hall effect). The calc ulation was done as a response to applied spin-\ndependent chemical potential, which would not be easy to control e xperimentally. In contrast, our study tries to\nderive direct relation between physically accessible quantities, curr ent and magnetization (local spin).\nAnother type of a voltage generated at an interface of a ferroma gnet-nonmagnet contact was predicted by Wang et\nal. [21] in order to explain the experimental observation [22]. They ha ve pointed out that a net charging is due to a\nback flow of spin current at the interface, and the spin accumulatio n at the interface is essential.\nIn this letter, we theoretically predict another mechanism of a spin- induced charge battery realized in disordered\nconductors. We consider the perturbative regime of the exchang e coupling, that is in the opposite limit of adiabatic\ncases [16, 17, 18]. Proposed mechanism does not rely on the interfa ce, and the pumped current arises simply when\nthe exchange interaction and the spin-orbit interaction exist simult aneously.\nWeconsiderthetwo-dimensionalelectrongassystem(2DEGs)with theRashbaspin-orbitinteractionthatoriginates\nfrom the lack of the inversion symmetry [5, 6, 23]. The conduction ele ctrons couples to the local magnetic moment via\ntheexchangecoupling. Suchasystemcanbeachievedbythe2DEGs attachedtotheferromagneticcontact[24, 25, 26],\nor the 2DEGs in magnetic semiconductors (e.g., CdTe/CdMnTe) [27, 2 8].\nLet us consider the current representation by the simple argumen t. The current jµ, proportional to the average\ntr∝angb∇acketleftkµ∝angb∇acket∇ightof the electron wave vector with respect to electron states (tr is trace over spin indices), vanishes if the\nsystem is spatially symmetric. The Rashba interaction, proportiona l to (k׈σ)z, breaks the spatial symmetry, but2\njS\ns-o\nFIG. 1: Diagrammatic representation of currents at the first order in Jexand the lowest order in Ω, which turns out to vanish.\nDotted lines and wavy lines denote local spins Sand the Rashba interaction, respectively, and thick line re presents the diffusion\nladder,Dq.\ncharge current does not arise since tr ∝angb∇acketleftkµ(k׈σ)z∝angb∇acket∇ight= 0 (only spin current arises [7]). Charge current appears when\nwe introduce the exchange coupling, proportional to S·ˆσwhereSis a local spin. We would have the current\njµ∝tr∝angb∇acketleftkµ(k׈σ)zS·ˆσ∝angb∇acket∇ight ∝ǫµνzSνto the first order exchange interaction, and jµ∝ǫµνz(S×S′)νat the second oder\ninteraction with different spins, SandS′.\nIn order to obtain the precise expression of the current, we perf orm analytical calculations by using a diagrammatic\ntechnique. Although the argument in the previous paragraph gives the qualitative idea of pumping charge current,\nit turns out that the linear term in Svanishes identically in the Rashba case, and the second order term n eeds a\ndynamical part as S×˙S. We will demonstrate that the pumped current has two component s. One describes the\ninverse spin Hall effect [13], and the other is a conservative curren t which written as a divergence of a scalar potential.\nWe will show that the current due to the inverse spin Hall effect is dom inant in various cases, and the conservative\ncurrent is relatively small. However, the conservative one is also impo rtant from the view of the fundamental physics,\nbecause it provides the fictitious scalar potential which acts only on the particle having both charge and spin.\nWe consider a disordered 2DEGs with the Rashba spin-orbit interact ion. The 2DEGs also interact with the local\nspin,Sx(t), via the exchange coupling. The local spin is treated as classical, an d slowly varying in space and time.\nThe system is represented by a Hamiltonian H=H0+Hex(t) +Hso+Himp, whereH0≡/summationtext\nkεkc†\nkckdescribes free\nelectrons with εk=/planckover2pi12k2/2m(mbeing the effective mass). The exchange and the Rashba spin-orbit interactions are\ngiven by\nHex(t) =−Jex/summationdisplay\nk,q,ΩSq(Ω)eiΩtc†\nk+qˆσck,\nHso=−α/summationdisplay\nkǫµνzkµ(c†\nkˆσνck), (1)\nwhereJexis the exchange coupling constant, Sq(Ω) denotes the Fourier transform of the local spin structure\nandαis the strength of the Rashba spin-orbit interaction. Spin-indepen dent disorder is represented by Himp=/summationtextni\ni=1/summationtext\nk,k′u\nVei(k−k′)·ric†\nk′ck, which gives rise an elastic electron lifetime τ= (2πρniu2/V)−1, whereρis the density\nof states, niis the number of the impurities, uis the strength of the impurity scattering and Vis the volume of the\nsystem.\nThe charge current density of this system is given by\njµ(x,t) =ie\nV/summationdisplay\nk,k′ei(k−k′)·xtr/bracketleftbigg/parenleftbigg(k+k′)µ\n2m−αǫµνzˆσν/parenrightbigg\nG<\nk,k′(t,t)/bracketrightbigg\n. (2)\nG<\nk,k′(t,t′) is a lesser Green function which is a 2 ×2 matrix in spin space with components G<\nkσ,k′σ′(t,t′) =\ni∝angb∇acketleftc†\nk′σ′(t′)ckσ(t)∝angb∇acket∇ight(σ,σ′=±), where ∝angb∇acketleft···∝angb∇acket∇ightis the expectation value estimated by the total Hamiltonian H.\nWecalculatethecurrentbytreatingbothRashba(tothefirstord er)andexchangeinteractionsperturbatively,which\nis valid if Jexτ≪1 andαkFτ≪1[29]. Successiveimpurity scatteringsare denoted by ladderappro ximation, resulting\ninadiffusionpropagatoratsmallmomentumtransfer( q),Dq≡(Dq2τ)−1, whereD≡k2\nFτ/2m2isadiffusionconstant.\nDominant contributions are from diagrams that include a maximal num ber of diffusion propagators. Contributions\nfrom the first order in Jexare shown in Fig. 1, that turn out to vanish identically. The leading con tribution coming\nfrom the second order in Jexare shown in Fig. 2.3\n(a) (b)\nFIG. 2: Leading contribution from the second order in Jex. Contributions from (a) and (b) are denoted by jφ\nµandjISH\nµin\nEq. (4), respectively. Contributions from other diagrams c ancel out or are smaller by O(1/εFτ).\nAfter straightforward calculations, the current in the slowly vary ing limit (Ω τ≪1) is obtained as\njµ(x,t) =4eαJex2\niπmVǫνηz(niu2\nV)/summationdisplay\nq,Qe−iQ·xDq(SQ−q×˙Sq)η\n×/bracketleftBig\n(niu2\nV)DqDQAµ\nQBν\nqCqQ+DqEµ\nqQBν\nq−DQAµ\nQFν\nqQ/bracketrightBig\n, (3)\nwhereAµ\nQ≡/summationtext\nkkµgr\nk−Q\n2ga\nk+Q\n2,Bν\nq≡/summationtext\nkkν(gr\nk)2ga\nk+q,CqQ≡Re/summationtext\nkgr\nkga\nk+qga\nk+Q,Eµ\nqQ≡\niIm/summationtext\nk/parenleftBig\nk+Q\n2/parenrightBig\nµgr\nkga\nk+qga\nk+QandFν\nqQ≡Re/summationtext\nk(k+q)νgr\nk(ga\nk+q)2ga\nk+Q(gr\nk= (ga\nk)∗= (εF−εk+i\n2τ)−1(εFbeing\nthe Fermi energy)). The second and third terms in Eq. (3) (propo rtionalto Eµ\nqQandFν\nqQ, respectively) are the second\nleading term with a long range limit ( q,Q→0). It is, however, physically the most essential term as we see belo w.\nFor a spatially smooth structure of spins, i.e., q,Q≪ℓ−1(ℓis the electron mean free path), we can approximate\nAµ\nQ∼2πiρτmDQ µ,Bν\nq∼4πρτ3εFqν,CqQ∼ −ρ/2ε2\nF,Eµ\nqQ∼ −πiρτ2qµandFν\nqQ∼πρτ3Qν. Then, the current is\nobtained as [30] jµ(x,t) =jISH\nµ(x,t)+jφ\nµ(x,t), where\njISH\nµ(x,t) =−3emαJex2τ2\nπǫµηz/integraldisplayd2x1\na2Dx−x1(Sx×˙Sx1)η,\njφ\nµ(x,t) =2eαJex2τ3\nπ2ǫνηz∂\n∂xµ/integraldisplayd2x1\na2/integraldisplayd2x2\na2Dx−x1∂D(2)\nx1−x2\n∂x1ν(Sx1×˙Sx2)η. (4)\nHere,Dx≡a2\nV/summationtext\nqe−iq·xDq,D(2)\nx≡a2\nV/summationtext\nqe−iq·x(Dq)2andais the lattice constant. (Note that singular behavior at\nq→0 is cut off at q∼L−1, whereLis a system size.)\nThe first term in Eq. (4), jISH, describes a current whose direction is correlated with the magnet ization direction,\nperpendicular to S×˙S, and represents inverse spin Hall effect [13]. In order to make clear the physical meaning of\nthis current, we compare with the spin currents pumped by magnet ization. In the absence of spin-orbit interaction,\nwe obtain the pumped spin current at the lowest order in Jexas\njsµ(x,t) =JexεFτ2\n2π∂\n∂xµ/integraldisplayd2x1\na2Dx−x1/bracketleftbigg\n˙Sx1−2Jexτ/integraldisplayd2x2\na2Dx1−x2(Sx1×˙Sx2)/bracketrightbigg\n. (5)\nThe polarizationofthe spin currentisin both directions, ∝ ∝angb∇acketleft˙S∝angb∇acket∇ightand∝angb∇acketleftS×˙S∝angb∇acket∇ight, where∝angb∇acketleft···∝angb∇acket∇ightdenotes averageoverdiffusive\nelectron motion. This spin current is a gradient of a certain spin pote ntial,jsµ=−∇µφs. The result of Eq. (5) is\nconsistent with the observation by Tserkovnyak et al. [8], where t he spin current appears at the interface between\nferromagnetandnormalmetalsassociatedwiththephenomenolog icalparameterofspin-mixingconductance. (Wenote\nthat there is also an equilibrium component of spin current [31]. This co mponent, js(eq)\nµ(x) =Jex2\n24π2ε2\nFτ(∇µSx)×Sx, is\nfree from diffusion poles. Hence, it is local and therefore small comp ared with dynamical contributions.) The meaning\nof pumped spin current is understood by taking a divergence:\n∇·js(x,t) =−mJex\n2π/bracketleftbigg\n˙Sx−2Jexτ/integraldisplayd2x1\na2Dx−x1(Sx×˙Sx1)/bracketrightbigg\n. (6)4\nFIG. 3: (Color online) Two typical geometries with ferromag nets attached to two-dimensional electron system. Left: Da tta-\nDas geometry [24] and Right: perpendicular geometry like in Ref. [13]. In both cases, inverse spin Hall current is given b y\njISH\nµ∝ǫµνz(S×˙S)ν(µ=x,y).\nComparing the second term ( ∇ ·js(2)ν) tojISH, we see that jISH\nµ=γISHǫµνz(∇ ·js(2)ν), where γISH=−3eατ. This\nexpression is the Rashba-version of inverse spin Hall effect, j∝js׈σ, proposed in Ref. [13]. (Note that the spin\ncurrent considered in Ref. [13] is the one flowing through the interf ace that enables the spin current to enter without\ndivergence.) Eq. (6) representsa conservationlawofspin, and co rrectlydescribesa fact that the Gilbert-type damping\n(∝angb∇acketleftS×˙S∝angb∇acket∇ight) results in a flow of spin current or ˙S.\nIn contrast, the second term in Eq. (4), jφ, is a gradient of a scalar quantity. It can be interpreted as a curre nt\narisingfrom apotential or aconservedforce. The fictitious electr ic field, defined by Eind=jφ/σ0, whereσ0=e2nτ/m\nis Boltzmann conductivity, is written as Eind=−∇φind. The scalar potential is obtained as\nφind(x,t) =−2αJex2τ2\nπeεFǫνηz/integraldisplayd2x1\na2/integraldisplayd2x2\na2Dx−x1∂D(2)\nx1−x2\n∂x1ν(Sx1×˙Sx2)η. (7)\n(Note that these scalar potential and field are fictitious ones, act ing only on charge having spin degrees of freedom.)\nThe current jφis in the direction where magnetization changes. It contributes to t he perpendicular current in the\nDatta-Das spin transistor geometry [24]. However, it is not in-plane current in the layer geometry [13] as shown in\nFig. 3. It does not contribute to the case of moving domain walls as we will show below.\nLet us apply our results to a case of a moving domain wall (as would be r ealized by using magnetic semiconduc-\ntors [27, 28]). We define polar angles ( θ,φ) with respect to the easy and hard axis of spin as cos θ=Seasy/S,tanφ=\nShard/Smid, whereSeasy,ShardandSmiddenote spin components in easy-, hard- and medium-anisotropy dir ections.\nRigid and one-dimensional (in the x-direction) domain wall solution is represented by two dynamical var iables,X(t)\nandφ(t) [32], as cos θ(x,t) = tanhx−X(t)\nλand sinθ(x,t) = [coshx−X(t)\nλ]−1, whereλis wall thickness. By assuming\na dirty case λ≫ℓ(ℓbeing the mean free path), and noting that Sx×˙Sx′vanishes if |x−x′| ≫λ, we can approx-\nimateSx×˙Sx′by local value as Sx×˙Sx∼sinθ(x,t)(˙φeθ−˙X\nλeφ). Here, eθ= (cosθcosφ,cosθsinφ,−sinθ) and\neφ= (−sinφ,cosφ,0), indicating that the damping ( S×˙S) on the translational motion ( ˙X) is in the φ-direction,\nwhile it is within the wall plane when φvaries. The pumped inverse spin Hall current, jISH\nµ∼ −j0ǫµηz∝angb∇acketleftS×˙S∝angb∇acket∇ightη, where\nj0≡3\nπemαJex2τ2D0(D0≡L\na2/integraltextλ\nℓdxDx) depends much on the wall geometry. We consider three types of t he domain\nwall as shown in Fig. 4, a Neel wall (N) and two Bloch walls ((B-i) and (B- ii)). By estimating ∝angb∇acketleftS×˙S∝angb∇acket∇ightinside the wall\n(by using ∝angb∇acketleftcosθ∝angb∇acket∇ight=∝angb∇acketlefttanhx\nλ∝angb∇acket∇ight= 0 etc.), we obtain\njISH\nx(N)∼ −j0˙X\nLsinφ, jISH\nx(B−i)∼j0˙X\nLcosφ, (8)\nwhich are driven by translational motion, and\njISH\nx(B−ii)∼j0λ\nL˙φ, (9)\nwhich is driven by tilt of the wall. In the case of the Neel and the out-o f-plane Bloch wall, the current is a constant\n(if wall velocity is constant) at small speed and shows an oscillation (s inφor cosφ) when the domain wall is above\nWalker’s break down. In contrast, no current is induced for the in- plane Bloch wall at small velocity and finite but\nsteady current arises above breakdown (as long as ˙φis more or less constant). The gradient part of the current, jφ,\non the other hand, is averaged out for any type of the domain wall.\nLet us briefly see the magnitude of current (Eqs. (8)(9)). We use εF= 10meV, Jex= 10meV, a= 10nm and\nα= 0.3×10−11eVm [6]. Using D0∼(L3/a2ℓ2λ)ln(L/λ) (our diffusive result depends much on sample size L), the\ncurrent is estimated as j∼4×10−14[C/m]×L2λ2\na4lnL\nλ×Ω[Hz]. If we choose Ω = 100MHz, L= 1µm andλ∼10a, we\nobtainj∼10[A/m], i.e., current is I≡jL∼10µA, which would be detectable experimentally.5\nFIG. 4: (Color online) Three domain walls in the xy-plane, Neel, Bloch-i and Bloch-ii.\nWe have considered a case of an uniform Rashba interaction. If we a llowαto be position dependent, α(x), we have\nother contributions. For instance, charge current linear in Sarises proportional to ∝angb∇acketleft(∇α)˙S∝angb∇acket∇ight. Thus, various currents\nare expected in the case of finite-size Rashba system (finite size is a lways the case in experiments).\nInconclusion, wehavetheoreticallyshownthatchargecurrentisp umpedbymagnetizationdynamicsinthepresence\nof the Rashba spin-orbit interaction. The dominant part was found to be due to the inverse spin Hall effect, i.e.,\nconversion of spin current into charge current by spin-orbit inter action. In addition to the inverse spin Hall current,\nwe found a conservative current flowing basically along the gradient of the magnetization damping. This current\nis rotation free, and should be distinguished from the inverse spin Ha ll current and from the divergenceless current\npredicted by Stern and others [16, 17, 18]. It would be extremely int eresting if one could experimentally determine\nthe type of the pumped current.\nThe authors are grateful to E. Saitoh, B. Kramer, R. Raimondi, M. Yamamoto, S. Kettemann, J. Shibata, H.\nKohno, S. Murakami, and T. Ohtsuki for valuable discussions. This w ork has been supported by the Deutsche\nForschungsgemeinschaft via SFBs 508 and 668 of the Universit¨ at Hamburg.\nNoted added in proof. —After finishing the manuscript, we found optically induced inverse s pin Hall effect was\nobserved in GaAs[33].\n∗Present address: Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan; Electronic address:\njohe@imr.tohoku.ac.jp\n[1] S.A. Wolf et al., Science 294, 1488 (2001).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n[4] H. Ohno, Science 281, 951 (1998).\n[5] E.I. Rashba, Sov. Phys. Solid State 2, 1109 (1960).\n[6] J. Nitta et al., Phys. Rev. Lett. 78, 1335 (1997).\n[7] A. Brataas, Y.V. Nazarov, and G.E.W. Bauer, Phys. Rev. Le tt.84, 2481 (2000).\n[8] Y. Tserkovnyak, A. Brataas, and G.E.W. Bauer, Phys. Rev. Lett.88, 117601 (2002).\n[9] Y. Tserkovnyak et al., Rev. Mod. Phys. 77, 1375 (2005).\n[10] S. Murakami, N. Nagaosa, and S.-C. 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Phys. Lett. 62, 3010 (1993).\n[28] F. Takano et al., Physica B 298, 407 (2001).\n[29] J. I. Inoue, G.E.W. Bauer, and L.W. Molenkamp, Phys. Rev . B70, 041303(R) (2004).\n[30] Correctly speaking, even without spin-orbit interact ion, small charge current arises at the second order of diffus ion pole.\nThis current, proportional to ∝angbracketleftSx·˙Sx′∝angbracketright, is of non-magnetic origin, i.e., due to time-dependent pot ential, and is of minor6\nimportance, besides being very small when Sis slowly varying in space, S·˙S∼0.\n[31] G. Tatara and H. Kohno, Phys. Rev. B 67, 113316 (2003).\n[32] J.C. Slonczewski, Int. J. Magn. 2, 85 (1972).\n[33] H. Zhao et al., Phys. Rev. Lett. 96, 246601 (2006)."    },    {        "title": "2012.03873v3.Continuous_dynamical_decoupling_of_spin_chains__Inducing_two_qubit_interactions_to_generate_perfect_entanglement.pdf",        "content": "arXiv:2012.03873v3  [quant-ph]  11 Dec 2023Continuous dynamical decoupling of spin chains: Inducing t wo-qubit interactions to\ngenerate perfect entanglement\nAbdullah Irfan,1Syed Furqan Abbas Hashmi,1Syeda Neha Zaidi,1\nMuhammad Usman Baig,1Wahaj Ayub,1and Adam Zaman Chaudhry1,∗\n1School of Science and Engineering, Lahore University of Man agement Sciences (LUMS),\nOpposite Sector U, D.H.A, Lahore 54792, Pakistan\nEfficient control over entanglement in spin chains is useful f or quantum information processing\napplications. In this paper, we propose the use of a combinat ion of two different configurations of\nstrong static and oscillating fields to control and generate near-perfect entanglement between any\ntwo spins in a spin chain, even in the presence of noise. This i s made possible by the fact that\nour control fields not only decouple the spin chain from its en vironment but also selectively modify\nthe spin-spin interactions. By suitably tuning these spin- spin interactions via the control fields, we\nshow that the quantum state of any two spins in the spin chain c an be made to be a Bell state. We\nillustrate our results for various spin chains, such as the X Y model, the XYZ model, and the Ising\nspin chain.\nI. INTRODUCTION\nQuantum spin chains have been the subject of exten-\nsive theoretical and experimental studies [ 1]. They are\nparadigmaticsystems providingconvenientand tractable\nmodels that yield insights into a range of physical phe-\nnomena [ 2–4]. In particular, spin chains have attracted\nconsiderable attention due to their potential use in quan-\ntum information applications such as quantum com-\nputation and short-distance quantum communication\n[1]. Achieving entanglement between spins is impera-\ntive when it comes to quantum information applications,\nand has been a long-standing goal and a focus of many\nstudies [5–9]. For instance, long-range Ising-type inter-\nactions [10], local effective magnetic fields [ 11], staggered\nmagnetic fields [ 12], and dynamical decoupling [ 13] have\nbeen made to realize this goal. Driven spin chains as\nhigh-quality quantum routers, which generate highly en-\ntangled states over arbitrary distances in spin chains by\napplying external fields, have also been proposed [ 14].\nSpin chains may also act as quantum mediators in or-\nder to achieve perfect long-range entanglement between\nremote spins in bulk and on the surfaces of magnetic\nnanostructures [ 15].\nLike any physical system, spin chains interact with\ntheir environment, which eventually leads to the loss\nof the quantum character of the system, a phenomenon\nknown as decoherence [ 16]. Moreover, entangled states\nare very fragile when exposed to the environment [ 17].\nThis poses a serious problem when it comes to using\nthe quantum state of the spin chain for quantum infor-\nmation processing tasks [ 18–21]. One reliable method\nof suppressing the system-environment interaction of an\nopen quantum system is dynamical decoupling [ 22–32].\nDynamical decoupling works by averaging out unwanted\neffects of the environment interaction through the ap-\nplication of control fields which effectively modulate the\n∗adam.zaman@lums.edu.pksystem-environment interaction [ 13]. In the context of\nspin chains, Ref. [ 13] has shown that the application\nof strong static and oscillating fields not only protects\nthe spin chain from decoherence, but also modulates the\nspin-chain interaction such that the effective Hamilto-\nnian of the spin chain includes interaction terms that are\nnot present in the original spin chain Hamiltonian. This\nshows that in addition to protecting the spin chain from\nthe environment, control fields can potentially be used\nto achieve perfect quantum state transfer and improved\ntwo-spin entanglement generation in the spin chain.\nOur aim in this paper is to look for a configuration\nof control fields that selectively suppresses the spin-spin\ninteractions in addition to decoupling the chain from the\nenvironment. If such control fields exist, can we devise a\nscheme that allows us to perfectly entangle spins in the\nchain while protecting it from decoherence? We answer\nthis question by showing that we can apply different con-\ntrol fields to the even and odd indexed sites in a chain\nin order to decouple the spins from the environment and\nremove spin-spin interactions in the chain, effectively ob-\ntaining a chain of isolated spins or qubits. We refer to\nthis configuration of fields as a staggered configuration.\nThis then opens up the possibility of inducing two-qubit\ninteractions in the spin chain by applying the same con-\ntrol fields to two spins and staggered fields to the rest\nof the spin chain. We propose a scheme that allows us\nto use such two-qubit interactions to entangle any two\nspins in the spin chain. We note that simply decoupling\nthe spin chain from the environment using control fields\nmay also result in a modulated spin chain Hamiltonian\nthat generates entanglement in the chain [ 13]. Such a\nmodulated Hamiltonian may be considered to be a con-\nsequence of decoupling the spin chain from the environ-\nment, and therefore entangles different spins to varying\ndegrees depending on the their positions and on the orig-\ninal spin chain Hamiltonian. Generally speaking, the en-\ntanglement generation is far from ideal. The technique\nwe discuss in this paper generates entanglement by first\nsuppressing spin-spin interactions in the chain, and then2\nusingasequenceoftwo-qubitinteractionstoentangleany\ntwo spins of our choice. As a result, we have greater con-\ntrol overthe entanglement process, allowingus to predict\nthe duration of each interaction, and the state of the spin\nchain when it is perfectly entangled.\nWe begin by considering a general one-dimensional\nspin chain that interacts with its environment [ 33]. We\nshow that a staggered configuration of control fields can\nnotonlyprotectthespinchainfromtheenvironment,but\nit can also suppress spin-spin interactions in the chain.\nThis is achieved by proving that the time-averaged ef-\nfective Hamiltonian is zero when staggered control fields\nare applied. Applying the same control fields to spins is\nreferred to as a constant configuration. By applying a\nconstant configuration of control fields to two neighbor-\ning spins, and a staggered configuration of control fields\nto the rest of the spin chain, we can induce a two-qubit\ninteraction that can perfectly entangle the two neigh-\nboring spins. We then propose a scheme that uses a\nsequence of two-qubit interactions between neighboring\nspins to perfectly entangle any two spins in the chain.\nAs an example, we solve this scheme for the XY chain,\nshowing analytically that any two spins in the chain can\nbe perfectly entangled by choosing appropriate interac-\ntion durations. Since the interaction is only between two\nspins at any point in time, we essentially solve a system\nof two spins evolving under the action of the effective\nHamiltonian. We find a way to remarkably simplify the\nquantum state of the spin chain at the end of each two-\nqubit interaction- the generationofentanglement is then\nevident. Thereafter, we apply our scheme to other spin\nchains such as the XYZ model and present numerical re-\nsults conclusively demonstrating the effectiveness of our\nscheme.\nThis paper is organized as follows. In Sec. IIwe show\nhow a staggered configuration of static and oscillating\ncontrol fields can be used to decouple a spin chain from\nthe environment and also suppress spin-spin interactions\nin the chain. Sec. IIIconsiders the XY model and de-\nscribeshowacombinationofconstantandstaggeredcon-\ntrol fields can be used to induce two-qubit interactions,\nwhich can be used to generate perfect entanglement in\nthe spin chain. Sec. IVprovides results from numerical\nsimulations that corroborate our presented scheme, and\nshows how our proposed method can be used for other\nspin chains such as the Ising chain and the XYZ model.\nWe conclude the paper in Sec. V.\nII. FORMALISM\nWe begin by showing how strong static and oscillating\ncontrol fields can be used to decouple a spin chain from\nits environment. Instead of using the same control fields\nfor every site in the spin chain, we propose a configura-\ntion of controlfields such that there is one type of control\nfield for spins at odd numbered sites, and another type\nof control field for spins at even numbered sites. We callthis a staggered configuration. This achieves a two-fold\ntask. First, it decouples the spin chain from its environ-\nment to lowest order. Second, it suppresses the spin-spin\ninteractions in the chain, effectively producing a chain\nof non-interacting qubits. Note that if the same field is\napplied to all the spins, the spin-spin interactions are not\nremoved [ 13].\nConsider then a spin chain with nearest neighbor in-\nteractions. We write the Hamiltonian as (we take /planckover2pi1= 1\nthroughout)\nH0=N−1/summationdisplay\nj=13/summationdisplay\nk=1ζjkσ(j)\nkσ(j+1)\nk, (1)\nwhereζjkare the coupling strengths between the spins,\njlabels the sites, and k= 1,2,3 denotesx,y, andz,\nrespectively. The Pauli spin operators follow the usual\ncommutation relations [ σ(p)\nj,σ(m)\nk] = 2iδpmǫjklσ(p)\nl. Note\nthat we are not using cyclic boundary conditions. We\nassume the spin chain interaction with its environment\nto be given by\nHSB=N−1/summationdisplay\nj=13/summationdisplay\nk=1B(j)\nkσ(j)\nk, (2)\nwhereB(j)\nkare arbitrary environment operators; they\ncan also represent randomly fluctuating noise terms for\na classical bath. Our first task is to find periodic con-\ntrol fields that decouple the spin chain from its environ-\nment, at least to lowest order. Corresponding to the\ncontrol fields is a unitary operator Uc(t) that satisfies\ni∂Uc(t)\n∂t=Hc(t)Uc(t), whereHc(t) is the Hamiltonian\nthat describes the action of the control fields on the sys-\ntem. Moreover, the fields we consider for this task are\nperiodic; the unitary operatorsatisfies Uc(t+tc) =Uc(t),\nwheretcis the time period. In order for the control fields\nto decouple the spin chain from the environment to low-\nest order, we must have that [ 33–36]\n/integraldisplaytc\n0dtU†\nc(t)HSBUc(t) = 0 (3)\nKeeping in mind the form of the interaction between the\nspinchainandtheenvironment,weguessthatafieldcon-\nfiguration represented by the following unitary operator\nmay decouple the spin chain from its environment:\nUc(t) =/parenleftBigg\n/producttextN\ni=1,3,5,...eiωnxσ(i)\nxteiωnyσ(i)\nyt/parenrightBigg\n×/parenleftBigg\n/producttextN−1\ni=2,4,6,...eiωmxσ(i)\nxteiωmyσ(i)\nyt/parenrightBigg\n.(4)\nFor concreteness, here we have considered Nto be odd;\nthe case ofeven Nis dealt with in a similar fashion. Here\nω= 2π/tc, and the integers nx,ny(for spins at odd in-\ndexed sites) and mx,my(for spins at even indexed sites)3\ndifferentiate between the two types of control fields ap-\nplied to alternate spins. In other words, Uc(t) represents\na staggered field configuration in which different con-\ntrol fields are applied to spins located at odd and even-\nnumbered sites in the chain. Now, it is obvious that our\nunitary operator Uc(t) satisfiesUc(t+tc) =Uc(t). Our\ntask then is to show that this configuration of control\nfields satisfies the decoupling condition given in Eq. ( 3).\nWe expect this to be the case - loosely put, eiωnxσ(i)\nxtav-\nerages out the contribution of the noise due to σ(i)\nyand\nσ(i)\nz, whileeiωnyσ(i)\nyttakescareofthe remainingnoise. To\nverify that this is indeed the case, it is useful to define\nhj,k,p(t) =U†\nc(t)σ(j)\nkUc(t),\nwherek= 1,2,3,σ(j)\n1=σ(j)\nx,σ(j)\n2=σ(j)\ny,σ(j)\n3=σ(j)\nz,\nandp=jmod2. The index ptells us if the constants\nused in the control field at site jarenx,nyormx,my;\np= 0 means that the field represented by nx,nyis being\napplied, and p= 1 means that the field represented by\nmx,myis being applied. Using the commutation rela-\ntions for the Pauli spin operators, it is straightforward to\nshow that\nhj,1,0(t) =cos(2ωnyt)σ(j)\nx−sin(2ωnyt)σ(j)\nz,\nhj,2,0(t) =sin(2ωnxt)sin(2ωnyt)σ(j)\nx+cos(2ωnxt)σ(j)\ny\n+sin(2ωnxt)cos(2ωnyt)σ(j)\nz,\nhj,3,0(t) =cos(2ωnxt)sin(2ωnyt)σ(j)\nx−sin(2ωnxt)σ(j)\ny\n+cos(2ωnxt)cos(2ωnyt)σ(j)\nz,\nhj,1,1(t),hj,2,1(t), andhj,3,1(t) are the same as hj,1,0(t),\nhj,2,0(t), andhj,3,0(t) respectively, except that nxis re-\nplaced bymxandnybymy. With these relations, it\nis easy to show that Eq. ( 3) is satisfied if nx/ne}ationslash=nyand\nmx/ne}ationslash=my, meaning that Uc(t) effectively decouples the\nspinchainfromits environment(atleasttolowestorder).\nThe control field Hamiltonian corresponding to Uc(t) is\nfound from the Schrodinger equation to be\nHc(t) =N/summationdisplay\ni=1,3,5,...{ωny/bracketleftbig\nsin(2ωnxt)σ(i)\nz\n−cos(2ωnxt)σ(i)\ny/bracketrightbig\n−ωnxσ(i)\nx}\n+N/summationdisplay\ni=2,4,8,...{ωmy/bracketleftbig\nsin(2ωmxt)σ(i)\nz\n−cos(2ωmxt)σ(i)\ny/bracketrightbig\n−ωmxσ(i)\nx},(5)\nwithnx/ne}ationslash=nyandmx/ne}ationslash=my. We must point out that the\nspin chain is decoupled from the environment only if the\nfields oscillate fast enough. More precisely, decoupling\noccurs iftc≪τcwhereτcis the environment correlation\ntime [37].\nIn addition to decoupling the spin chain from its en-\nvironment, our staggered control field configuration canalso suppress the spin-spin interactions in the chain. To\nshow this, we must find the effective Hamiltonian of the\nspin chain when control fields are applied. It is known\nthat if the control fields are strong enough and oscillate\nfast enough (that is, faster than the typical timescale of\nthe evolution due to the spin chain Hamiltonian itself),\nthe effective Hamiltonian can be written as [ 33,35]\n¯H=1\ntc/integraldisplaytc\n0dtU†\nc(t)H0Uc(t), (6)\nwhereH0is the original spin chain Hamiltonian given in\nEq. (1). Forour case, the effective Hamiltonian simplifies\nto\n¯H=1\ntcN−1/summationdisplay\nj=1/integraldisplaytc\n0dt3/summationdisplay\nk=1ζjkhj,k,p(t)hj+1,k,p′(t).(7)\nNote thatp′= (j+1)mod2. Because one of the jand\nj+1 sites is odd while the other is even, one of pandp′\nmust be equal to one while the other must be zero. To\nsimplify Eq. ( 7) further, we define the operators\nI(j)\nk=1\ntc/integraldisplaytc\n0hj,k,p(t)hj+1,k,p′(t)dt. (8)\nThese allow us to write the effective Hamiltonian suc-\ncinctly as\n¯H=N−1/summationdisplay\nj=13/summationdisplay\nk=1ζjkI(j)\nk. (9)\nTo explicitly obtain an expression for ¯H, we must eval-\nuate the integrals I(j)\nk. To remove the spin-spin inter-\nactions in the spin chain, at least to lowest order, one\npossible choice, which we stick to in this paper, is to\nchooseny= 2nx,my= 2mx, andnx/ne}ationslash=mx. The inte-\ngrals inI(j)\nkthen evaluate to zero for all jandk, leading\nto¯H= 0. Notice that since nx/ne}ationslash=mx, the suppression of\nthe spin-spin interactions depends crucially on the fact\nthat we apply different control fields to odd and even-\nnumbered sites. On the other hand, let us examine what\nhappens if we apply the same fields to two neighboring\nspins in the chain. In other words, we have a ‘constant’\nconfiguration of the control fields for the two spins. In\nthis case, we find that\nI(j)\n1=1\n2/parenleftbig\nσ(j)\nxσ(j+1)\nx+σ(j)\nzσ(j+1)\nz/parenrightbig\n, (10)\nI(j)\n2=1\n4/parenleftbig\nσ(j)\nxσ(j+1)\nx+2σ(j)\nyσ(j+1)\ny+σ(j)\nxσ(j+1)\ny\n+σ(j)\nyσ(j+1)\nx+σ(j)\nzσ(j+1)\nz/parenrightbig\n, (11)\nI(j)\n3=1\n4/parenleftbig\nσ(j)\nxσ(j+1)\nx+2σ(j)\nyσ(j+1)\ny−σ(j)\nxσ(j+1)\ny\n−σ(j)\nyσ(j+1)\nx+σ(j)\nzσ(j+1)\nz/parenrightbig\n. (12)\nThe effective Hamiltonian is now not zero. Moreover,\nsinceny/ne}ationslash=nx, the two spins are still decoupled from4\ntheir environment. To sum up then, both the constant\nand staggered configurations of control fields can decou-\nple the spin chain form its environment. While the stag-\ngered configuration suppresses spin-spin interactions, the\nconstant configuration modulates the spin-spin interac-\ntions, giving us an effective Hamiltonian that describes\nthe interaction of the spins. We now show how we can\nuse both both staggeredand constantfield configurations\nin conjunction to generate entanglement between specific\nspins in the chain. Note that the effect of the environ-\nment is also suppressed.\nIII. SCHEME FOR ENTANGLEMENT\nGENERATION\nFor concreteness, we focus on generating perfect en-\ntanglement between the first spin and an arbitrary spin\nin the spin chain. Our considerations can be generalized\nin a very straightforward manner to any two spins in the\nchain. We start by noting that the expression for I(j)\nk\ncontains terms from two neighboring spins, namely hj,k,p\nandhj+1,k,p′wherejis the site of the spin. Whether or\nnotI(j)\nkand subsequently the effective Hamiltonian for\nthese two neighboring spins is zero depends on whether\nthe integers defining the control field at both sites are\nequal or not. Keeping this fact in mind, we propose a\nfield configuration such that for the first two spins of the\nchain, the integers defining the control field are equal,\nand for the rest of the chain, no two neighboring spins\nhaveacontrolfield defined by the sameintegers. In other\nwords, we have a constant configuration for the first two\nspins and a staggered configuration for the rest of the\nchain. This allows the first two spins to interact while\nthe rest of the spin chain is effectively dormant and non-\ninteracting. The effective Hamiltonian can be written as\n¯H=/parenleftBigg3/summationdisplay\nk=1ζ1kI(1)\nk/parenrightBigg\n⊗1(3)⊗1(4)⊗...⊗1(N),(13)\nwhere 1denotes the identity operator. Depending on the\ninteraction, the effective Hamiltonian ¯Hmay entangle\nthe first two spins to some degree. For example, one can\nprove that if we consider the XY chain with constant\ncoupling strengths ( ζj1=ζj2= 1 for convenience, ζj3=\n0), the first two spins become perfectly entangled if the\nspin is left to evolve under ¯Hfor a certain duration. We\nlabel this duration τ1. We then make a change. We\nnow set the integers for the second and third spins in\nthe chain equal while keeping the integers of neighboring\nsites different for all other neighboring spins in the chain.\nFollowing our previous reasoning, it is clear that now the\nsecondandthirdspinsinteract, whiletheremainingspins\nare non-interacting. The effective Hamiltonian is now\n¯H=1(1)⊗/parenleftBigg3/summationdisplay\nk=1ζ2kI(2)\nk/parenrightBigg\n⊗1(4)⊗...⊗1(N).(14)The idea is to now allow the spin chain to evolve under\nthis effective Hamiltonian until the first and the third\nspins are perfectly entangled. The time required for this\ninteraction is labeled τ2. This process can be continued\nuntil the first spin is perfectly entangled with any spin\nthat we wish to entangle it with.\nLet us now demonstrate this scheme of generating en-\ntanglement in detail for the XY spin chain with Nsites.\nWe setζj1=ζj2= 1 andζj3= 0 in Eq. ( 1). We begin\nby setting the initial state of the system as\n|ψ0(0)/an}bracketri}ht=|0/an}bracketri}ht1⊗|0/an}bracketri}ht2⊗...⊗|0/an}bracketri}htN.\nWe define the states |0/an}bracketri}htand|1/an}bracketri}htas the eigenstates of σz\nwith eigenvalues +1 and −1 respectively. The effective\nHamiltonian that describes the interaction between two\nneighboringspins iandi+1 under the action of the same\ncontrolfields appliedtothe twospinsis[seeEqs.( 10)and\n(11)]\n¯Hi=1\n4/bracketleftbigg\n3σ(i)\nxσ(i+1)\nx+2σ(i)\nyσ(i+1)\ny+3σ(i)\nzσ(i+1)\nz+\nσ(i)\nxσ(i+1)\ny+σ(i)\nyσ(i+1)\nx/bracketrightbigg\n. (15)\nTo find the evolution generated by this two-qubit in-\nteraction, we find that the eigenstates of this effective\nHamiltonian,writtenintermsofthebasisstates |00/an}bracketri}hti,i+1,\n|01/an}bracketri}hti,i+1,|10/an}bracketri}hti,i+1, and|11/an}bracketri}hti,i+1are\n|e1/an}bracketri}ht=\n0\n−β\nβ\n0\n,|e2/an}bracketri}ht=\nα∗\n0\n0\n−β\n,|e3/an}bracketri}ht=\n0\nβ\nβ\n0\n,|e4/an}bracketri}ht=\nα∗\n0\n0\nβ\n,\nwhereα= (1 + 2i)/√\n10 andβ= 1/√\n2. The corre-\nsponding eigenvalues are λ1=−2,λ2= (3−√\n5)/4,λ3=\n1/2 andλ4= (3+√\n5)/4. Consequently,\n|00/an}bracketri}hti,i+1=α|e2/an}bracketri}ht+α|e4/an}bracketri}ht\n|01/an}bracketri}hti,i+1=−β|e1/an}bracketri}ht+β|e3/an}bracketri}ht\n|10/an}bracketri}hti,i+1=β|e1/an}bracketri}ht+β|e3/an}bracketri}ht\n|11/an}bracketri}hti,i+1=−β|e2/an}bracketri}ht+β|e4/an}bracketri}ht.\nWe now return to our original problem. With the first\ntwo spins interacting, the state at time tis\n|ψ(t)/an}bracketri}ht=/parenleftBig\nγ1(t)|00/an}bracketri}ht1,2+γ2(t)|11/an}bracketri}ht1,2/parenrightBig\n|0/an}bracketri}ht,(16)\nwhere\nγ1(t) =|α|2/parenleftBig\ne−iλ4t+e−iλ2t/parenrightBig\n, (17)\nγ2(t) =αβ/parenleftBig\ne−iλ4t−e−iλ2t/parenrightBig\n, (18)\nand|0/an}bracketri}htdenotes that the other spins are all in the state\n|0/an}bracketri}ht. This evolution is consistent with the fact that5\n|00/angbracketright\nγ2(τ1)|110/angbracketright\nγ2(τ1)η2(τ2)|1100/angbracketright\nγ2(τ1)η2(τ2)γ2(τ3)|11110/angbracketrightγ2(τ1)η2(τ2)γ1(τ3)|11000/angbracketrightγ2(τ1)η1(τ2)|1010/angbracketright\nγ2(τ1)η1(τ2)η2(τ3)|10100/angbracketrightγ2(τ1)η1(τ2)η1(τ3)|10010/angbracketrightγ1(τ1)|000/angbracketright\nγ1(τ1)γ2(τ2)|0110/angbracketright\nγ1(τ1)γ2(τ2)η2(τ3)|01100/angbracketrightγ1(τ1)γ2(τ2)η1(τ3)|01010/angbracketrightγ1(τ1)γ1(τ2)|0000/angbracketright\nγ1(τ1)γ1(τ2)γ2(τ3)|00110/angbracketrightγ1(τ1)γ1(τ2)γ1(τ3)|00000/angbracketright\nFIG. 1. A tree diagram is a useful way to keep track of the evolu tion of the spin chain. Each state vector in this diagram\nshould be appended with |0/angbracketright, meaning that the other spins are in the state |0/angbracketright. As we continue evolving the spin chain using\nthe two-qubit Hamiltonian, the state of the chain becomes a l inear combination of a larger number of different state vecto rs.\nThe complete state just after each interaction is given by th e sum of the states in each column. For example, the sum of the\nfour terms in the third column represents the state of the sys tem when spins 1 and 3 have interacted.\nσ(i)\nzσ(i+1)\nzcommutes with the effective Hamiltonian. The\nidea nowis to continue this evolutionuntil time τ1, which\nis the time required for spins 1 and 2 to become per-\nfectly entangled. That this is indeed possible can easily\nbe shown by calculating the concurrence [ 38]; this can\nalso be seen in the results that we present in the next\nsection. The next step is to make spins 2 and 3 interact.\nTo find the subsequent evolution, let us first note that\nwe can write\n|ψ(τ1)/an}bracketri}ht\n=/bracketleftBig\n|0/an}bracketri}ht1⊗/parenleftBig\nγ1(τ1)|00/an}bracketri}ht2,3/parenrightBig\n+|1/an}bracketri}ht1⊗/parenleftBig\nγ2(τ1)|10/an}bracketri}ht2,3/parenrightBig/bracketrightBig\n|0/an}bracketri}ht.\nNow,\ne−i¯Hit|00/an}bracketri}hti,i+1=γ1(t)|00/an}bracketri}hti,i+1+γ2(t)|11/an}bracketri}hti,i+1,\ne−i¯Hit|10/an}bracketri}hti,i+1=η1(t)|01/an}bracketri}hti,i+1+η2(t)|10/an}bracketri}hti,i+1,whereγ1(t) andγ2(t) are given by Eqs. ( 17) and (18),\nandη1(t) andη2(t) are given by\nη1(t) =−β2/parenleftBig\ne−iλ1t−e−iλ3t/parenrightBig\n, (19)\nη2(t) =β2/parenleftBig\ne−iλ1t+e−iλ3t/parenrightBig\n. (20)\nThese results can then be used to work out the evolu-\ntion of the spin chain due to the interaction between the\nsecond and third spins. Thereafter, the third spin and\nthe fourth spin interact, and so on. The state of the spin\nchain becomes increasingly complicated. Noticing that\ne−i¯Hitacts on both |00/an}bracketri}hti,i+1and|10/an}bracketri}hti,i+1to produce a\nlinear combination of |00/an}bracketri}hti,i+1and|10/an}bracketri}hti,i+1, we can rep-\nresent the evolution of the spin chain by using the tree\ndiagram shown in Fig. 1.\nWe now use the worked out evolution of the spin chain\nto find suitable interaction durations τ1,τ2,...,τisuch\nthat spins1and i+1areperfectly entangled. We propose\na question that could help us find out how this can be\ndone: can the interaction durations be chosen such that\nthe two-qubit state of the spins 1 and i+1 ends up in a\nBell state with the rest of the spins in the |0/an}bracketri}htstate? For\nthis to be possible, we can see in our tree diagram that\nonly two terms in the complete spin chain state must be\nnon-zero. These are the terms given by\n\ni/productdisplay\nj=1γ1(τj)\n|0/an}bracketri}ht1⊗i/productdisplay\nj=2|0/an}bracketri}htj⊗|0/an}bracketri}hti+1,and\nγ2(τ1)\ni/productdisplay\nj=2η1(τj)\n|1/an}bracketri}ht1⊗i/productdisplay\nj=2|0/an}bracketri}htj⊗|1/an}bracketri}hti+1.\nIf the coefficients of all the other terms in the state are\nzero, we are simply left with the state\n|ψf/an}bracketri}ht=/bracketleftBigg\ni/productdisplay\nj=1γ1(τj)\n|00/an}bracketri}ht1,i+1+\nγ2(τ1)\ni/productdisplay\nj=2η1(τj)\n|11/an}bracketri}ht1,i+1/bracketrightBigg\n⊗|0/an}bracketri}ht.(21)\nThe two-qubit state of the first spin and the i+1 spin is6\nthen obvious. This state is fully entangled if\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleγ1(τ1)γ2(τ1)/parenleftBigi/productdisplay\nj=2γ1(τj)η1(τj)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.5.(22)\nNotice that γ1(τ1)γ2(τ1) comes from the first two-qubit\ninteraction. Each additional two-qubit interaction is re-\nsponsible for a γ1(τj)η1(τj) term. Fig. 2shows that τ1\ncan be chosen such that |γ1(τ1)γ2(τ1)|= 0.5. Moreover,\nFig.3shows that the interaction duration can always be\nchosen such that |γ1(τj)η1(τj)|= 1 withj≥2. As a\nresult, we conclude that Eq. ( 22) can always be satis-\nfied by choosing suitable interaction durations. We have\nalso checked that if we choose such interaction durations,\nthe coefficients of all other terms in the complete state\nof the spin chain are zero, justifying writing the state of\nthe spin chain in the form given by Eq. ( 21). Since the\nfirst interaction duration is obtained by finding a maxi-\nmum of|γ1(τ1)γ2(τ1)|and all other interaction durations\nare obtained by finding a maximum of |γ1(τj)η1(τj)|, we\nexpect the first interaction to require a duration differ-\nent from the following interactions, all of which may be\nchosen to be the same. Also, we see from our graphs\nthat|γ1(τ1)γ2(τ1)|and|γ1(τj)η1(τj)|are periodic. For\nthe numerical simulations that follow, we will take the\ndurations that give the first maximum.\n051015200.00.10.20.30.40.5\ntD(t)\nFIG. 2. A plot of D(t) against time, where D(t) =\n|γ1(t)γ2(t)|.0102030405060700.00.20.40.60.81.0\ntD(t)\nFIG. 3. A plot of D(t) against time, where now D(t) =\n|γ1(t)η1(t)|.\nIV. RESULTS\nTo test our predictions of perfect entanglement for the\nXY spin chain, we now perform numerical simulations.\nTo measure the degree of entanglement, we calculate the\nconcurrence between the two spins [ 38]. We calculate the\npartial trace of the density matrix of the system over all\nspins other than the two spins whose concurrence we aim\nto find and refer to it as ρ2. We then calculate\nR=/radicalBig√ρ2/tildewideρ2√ρ2, (23)\nwhere/tildewideρ2= (σy⊗σy)ρ∗\n2(σy⊗σy). The concurrence is\ngiven by\nC(ρ) = max(0,λ1−λ2−λ3−λ4),(24)\nwhereλ1,λ2,λ3, andλ4are the eigenvalues of the Rin\ndescending order. We evolve the spin chain in two differ-\nent ways. First, we apply control fields to a spin chain\nthat interacts with the environment and plot the concur-\nrencebetweentwospinsasafunction oftime. Wereferto\nthis as the complete Hamiltonian picture, with the com-\nplete Hamiltonian given by H0+HSB+Hc(t). Second,\nwe show that the dynamics can be reproduced using the\ntime-averaged effective Hamiltonian ¯H. We refer to this\nas the effective Hamiltonian picture. In both approaches,\nweusetheschemediscussedpreviously. Thatis, wemake\nspins 1 and 2 interact until they are entangled and then\nturn off the interaction by switching the control fields.\nThen we make spins 2 and 3 interact until spins 1 and 3\nare entangled and then turn off the interaction. We con-\ntinue this process until we reach the spin that we wish to\nentangle with the first spin. Fig. 4illustrates the scheme\nfor a chain with five spins.\nWe now present our numerical results illustrating the\nentanglementattheendofeachstep. Recallthatthespin\nchainis initialized in the state/producttextN\ni=1|0/an}bracketri}hti. The noiseterms\nB(j)\nkin the expression for HSB[see Eq. ( 2)] are taken to\nbethesameforeachsite forsimplicity. Thesenoiseterms7\nStep 1\nStep 2\nStep 3\nStep 4Effective Hamiltonian Picture Complete Hamiltonian Picture\nFIG. 4. The diagram shows the general scheme we use to\nentangle the ends of a spin chain with N= 5. In the com-\nplete Hamiltonian picture, a white circle at site irepresents\na control field Hamiltonian at site idescribed by the con-\nstantsnx,nyand a black circle at site irepresents a control\nfield Hamiltonian at site idescribed by the constants mx,my.\nNotice that at each step, the two neighboring spins that are\ninteracting have the same control field Hamiltonian at their\nsites, while the control field Hamiltonian for the rest of the\nchain is staggered. In the effective Hamiltonian picture, th e\ntwo gray circles in each step represent the two-qubit intera c-\ntion¯Hi[see Eq. ( 15)] that is caused by the same control field\nHamiltonians at those sites. The white circles represent no\ninteraction, which is caused by staggered fields at those sit es.\ntCt\nFIG.5. Step1. Spins1and2inanXYchain( N= 5,ζj,k= 1)\ninteract for τ1(≈1.4) to become perfectly entangled, after\nwhich the interaction is effectively turned off. The concur-\nrence between spins 1 and 2 is given by C(t). The solid blue\ncurve is the concurrence using the effective Hamiltonian, th e\ndotted-dashed gray curve is the concurrence using the com-\nplete Hamiltonian, and the dashed black curve is the concur-\nrence without the control fields. We use dimensionless units\nthroughout with /planckover2pi1= 1.\nare generated via independent Ornstein-Uhlenbeck pro-\ncesses with zero mean, correlation time τ= 0.5, and\nstandard deviation σ= 2.0. Results for the concur-\nrence are presented in Figs. 5,6,7, and8. Each plot\nshows the concurrence C(t) between the first spin and\nthe most recent spin to have experienced an interaction\nwith its neighboring spin. The solid, blue curve shows\nthe concurrence using the effective Hamiltonian, the dot-\ndashed gray curve shows the concurrence using the com-\nplete Hamiltonian, and the black, dashed curve showstCt\nFIG. 6. Step 2. Spins 2 and 3 interact for τ2(≈11.3), after\nwhich spins 1 and 3 are perfectly entangled, as shown by\ntheir concurrence C(t). Notice that C(t) = 0 for the duration\nτ1since that is the duration for which spins 1 and 2 were\ninteracting while all other spins were decoupled from each\nother.\ntCt\nFIG. 7. Step 3. Spins 3 and 4 interact for τ3(≈11.3), after\nwhich spins 1 and 4 are perfectly entangled, as shown by their\nconcurrence C(t).C(t) begins to increase after ( τ1+τ2) since\nthat is the duration during which the first two steps of the\nscheme take place.\nthe concurrence using only H0+HSB(that is, no control\nfields are applied). Due to the noise, it is not surprising\nthat the black, dashed curve largely overlaps with the\nhorizontal axis. The overlap of the solid blue and dot-\ndashed grey curves shows that the effective Hamiltonian\napproach captures the exact dynamics very well. No-\ntice that by the end of step 4, the concurrence between\nthe first and last spins is practically one. We have also\nchecked that the state of spins 1 and 5 at the end of step\n4 is very close to1√\n2(|00/an}bracketri}ht+i|11/an}bracketri}ht). ForN >5, simulat-\ning the chain via the complete Hamiltonian requires long\ndurations. Since we have already shown the equivalence\nof the complete and effective Hamiltonian pictures, we\nsimply use the effective Hamiltonian to simulate longer\nspin chains. Our scheme works, as expected, for a longer\nspin chain as well [see Fig. 9].8\ntCt\nFIG. 8. Step 4. Spins 4 and 5 interact for τ4(≈11.3), after\nwhich spins 1 and 5 are perfectly entangled, as shown by\ntheir concurrence C(t). We have checked that when C(t) is\napproximately one, the state of spins 1 and 5 is very close to\n1√\n2(|00/angbracketright+i|11/angbracketright), which is a perfectly entangled state.\n0204060800.00.20.40.60.81.0\ntC(t)\nFIG. 9. C(t) is the concurrence between spins 1 and 10 in\nan XY chain ( N= 10,ζjk= 1). From t= 0 tot≈80,\nconsecutive pairs of neighboring spins interact. Thereaft er,\nspins 9 and 10 are made to effectively interact until spins 1\nand 10 are perfectly entangled.\nTheentanglementschemewehavepresentedisnotjust\nrestrictedtotheXYmodel. Anecessaryconditionforour\nscheme to work is that the effective Hamiltonian must be\nabletogenerateperfectentanglementbetweentwoneigh-\nboring spins, given a suitable initial state. Let us then\nexamine the isotropic XYZ model (or the XXX model)\nwithζj1=ζj2=ζj3= 1 in Eq. ( 1). We now choose the\ninitial state of the spin chain to be\n|ψ(0)/an}bracketri}ht=|10/an}bracketri}ht1,2⊗|0/an}bracketri}ht.\nThe effective Hamiltonian describing the spin-spin inter-\nactions between two spins when the same control fields\nare applied to the two spins is now\n¯Hi=σ(i)\nxσ(i+1)\nx+σ(i)\nyσ(i+1)\ny+σ(i)\nzσ(i+1)\nz.(25)\nAs in Sec. III, we find the evolution due to this effec-\ntive interaction by finding the eigenstates of the effec-tive Hamiltonian in the {|00/an}bracketri}hti,i+1,|01/an}bracketri}hti,i+1,|10/an}bracketri}hti,i+1,\n|11/an}bracketri}hti,i+1}basis. These very familiar eigenstates are\n|e1/an}bracketri}ht=1√\n2\n0\n1\n−1\n0\n,|e2/an}bracketri}ht=\n1\n0\n0\n0\n,\n|e3/an}bracketri}ht=1√\n2\n0\n1\n1\n0\n,|e4/an}bracketri}ht=\n0\n0\n0\n1\n,\nwith eigenvalues λ1=−3,λ2= 0,λ3= 1 andλ4= 1\nrespectively. We can then also write\n|00/an}bracketri}hti,i+1=|e2/an}bracketri}ht,\n|01/an}bracketri}hti,i+1=−1√\n2|e1/an}bracketri}ht+1√\n2|e3/an}bracketri}ht,\n|10/an}bracketri}hti,i+1=1√\n2|e1/an}bracketri}ht+1√\n2|e3/an}bracketri}ht,\n|11/an}bracketri}hti,i+1=|e4/an}bracketri}ht.\nWe now let the first two spins interact. After time t,\n|ψ(t)/an}bracketri}ht=/parenleftBig\nχ1(t)|10/an}bracketri}ht1,2+χ2(t)|01/an}bracketri}ht1,2/parenrightBig\n|0/an}bracketri}ht,(26)\nwhere\nχ1(t) =1\n2/parenleftBig\ne−it+ei3t/parenrightBig\n, (27)\nχ2(t) =1\n2/parenleftBig\ne−it−ei3t/parenrightBig\n. (28)\nAs before, we select a time τ1such that spins 1 and 2 are\nperfectly entangled. Then, we let the next spins interact\npairwise for time τj. We arrive at a similar condition to\nEq. (22) for spins 1 and i+1 to be fully entangled:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleχ1(τ1)χ2(τ1)i/productdisplay\nj=2χ2(τj)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.5. (29)\nNote that this is a somewhat simpler condition as com-\npared to Eq. ( 22) since we do not have to consider the\nevolution of |00/an}bracketri}htj,j+1since it is an eigenstate of the effec-\ntive Hamiltonian. To fulfill this condition, we can choose\n|χ1(τ1)χ2(τ1)|= 0.5 and|χ2(τj)|= 1 with all τjequal\ntoτ2. We can work out the required values of τ1and\nτ2by first finding that |χ1(τ1)χ2(τ1)|=1\n2sin(4τ1) and\n|χ2(τ2)|= sin(2τ2). It then follows that we can choose\nτ1=π\n8andτ2=π\n4. With these times, spins 1 and i+1\nare in a Bell state, while all the other spins are in the\nstate|0/an}bracketri}ht. Fig.10is an illustration of the first and last\nspins of a XX spin chain being entangled via our scheme.\nLet us now consider the quantum Ising spin chain with\nζj1= 1 andζj2=ζj3= 0. The effective Hamiltonian in\nthis case is\n¯Hi=1\n2/bracketleftBig\nσ(i)\nxσ(i+1)\nx+σ(i)\nzσ(i+1)\nz/bracketrightBig\n.(30)9\n0 2 4 6\nt0.00.20.40.60.81.0C (t)\nFIG. 10. C(t) is the concurrence between spins 1 and 10\nfor the XXX model with N= 10. From t= 0 tot≈5.9,\nconsecutive pairs of neighboring spins interact. From t≈5.9\nonwards, spins 9 and 10 are made to interact until spins 1 and\n10 are perfectly entangled.\nThis is really the usual XX interaction since, by perform-\ning a rotation of the coordinate axes, we can write the\neffective Hamiltonian in terms of σyinstead. That is,\n¯H′\ni=1\n2/parenleftbig\nσi\nxσi+1\nx+σi\nyσi+1\ny/parenrightbig\n. (31)\nIn this rotated basis, we choose the initial state to be\n|10/an}bracketri}ht1,2⊗ |0/an}bracketri}ht. The same eigenbasis |ei/an}bracketri}htis the same as\nthat for the XXX model with eigenvalues λ1=−1,λ2=\n0,λ3= 1,λ4= 0. Letting the first two spins interact we\nhave\n|ψ(t)/an}bracketri}ht=/parenleftBig\nα1(t)|10/an}bracketri}ht1,2+α2(t)|01/an}bracketri}ht1,2/parenrightBig\n|0/an}bracketri}ht,(32)\nwhere\nα1(t) =1\n2/parenleftBig\ne−it+eit/parenrightBig\n= cos(t), (33)\nα2(t) =1\n2/parenleftBig\ne−it−eit/parenrightBig\n=−isin(t).(34)\nTo get spins 1 and i+1 in a maximally entangled state,\nwe now have the condition\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleα1(τ1)α2(τ1)i/productdisplay\nj=2α2(τj)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.5, (35)\nTo fulfill this, we can choose τ1such that\n|cos(τ1)sin(τ1)|= 0.5, and setting all the τjto be\nequal toτ2such that |sin(τ2)|= 1. With these times,\nspins 1 and i+ 1 are in a Bell state and all the oth-\ners spins are in the state |0/an}bracketri}ht. The generation of the\nentanglement is illustrated in Fig. 11.\nFinally, let us consider an anisotropic XYZ model. We\ntakeζj1= 1.1,ζj2= 1,ζj3= 1. By now, it should be0.0 2.5 5.0 7.5 10.0 12.5\nt0.00.20.40.60.81.0C (t)\nFIG. 11. C(t) is the concurrence between spins 1 and 10 for\nthe quantum Ising model with N= 10 following our scheme.\nobvious how to proceed with our scheme. The effective\nHamiltonian is\n¯Hi= 1.05[σ(i)\nxσ(i+1)\nx+σ(i)\nzσ(i+1)\nz]+σ(i)\nyσ(i+1)\ny.(36)\nWe choose the initial state to be |0/an}bracketri}ht. Spins 1 and i+ 1\nare now maximally entangled if\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleµ1(τ1)µ2(τ1)/parenleftBigi/productdisplay\nj=2µ1(τj)ν1(τj)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.5,(37)\nwhere\nµ1=1\n2/parenleftBig\ne−iλ4t+e−iλ2t/parenrightBig\n, (38)\nµ2=1\n2/parenleftBig\ne−iλ4t−e−iλ2t/parenrightBig\n, (39)\nν1=1\n2/parenleftBig\ne−iλ1t−e−iλ3t/parenrightBig\n, (40)\nandλ1=−3.1, λ2= 1, λ3= 1 andλ4= 1.1. To ful-\nfill this condition, we can set |µ1(τ1)µ2(τ1)|= 0.5, and\n|µ1(τ2)ν1(τ2)|= 1, where all the τjhave been set equal\ntoτ2. Figs.12and13show that these two conditions\ncan indeed be satisfied. Proceeding further, Figs. 14,\n15,16, and17show how entanglement is generated be-\ntween the first spin and the rest of the spins one by one.\nAs before, the dot-dashed gray curve shows the concur-\nrence obtained by using the complete Hamiltonian, the\nsolid, blue curve are the results with the effective Hamil-\ntonian approach, and the black, dashed curve is the con-\ncurrence in the absence of any control fields. These plots\nnot only illustrate the validity of the effective Hamilto-\nnian approach with highly efficient removal of the effect\nof the environment, but also the fact that maximal bi-\npartite entanglement can be generatedbetween twospins\nin the spin chain.10\n0 50 100 150\nt0.00.10.20.30.40.5D (t)\nFIG. 12. A plot of D(t) against time, where D(t) =\n|µ1(t)µ2(t)|.\n0 20 40 60\nt0.00.20.40.60.81.0D (t)\nFIG. 13. A plot of D(t) against time, where D(t) =\n|µ1(t)ν1(t)|.\ntCt\nFIG. 14. Step 1. Spins 1 and 2 in an anisotropic XYZ chain\n(N= 5,ζj1= 1.1,ζj2= 1,ζj3= 1) interact for τ1(≈15.7).tCt\nFIG. 15. Step 2. Spins 2 and 3 interact for τ2(≈0.77).C(t)\nis the concurrence between spins 1 and 3.\ntCt\nFIG. 16. Step 3. Spins 3 and 4 interact for τ3(≈0.77).C(t)\nis the concurrence between spins 1 and 4.\ntCt\nFIG. 17. Step 4. Spins 4 and 5 interact for τ4(≈0.77).\nC(t) is the concurrence between spins 1 and 5. At the point\nwhen the interaction stops, C(t) = 0.997 rounded off to three\ndecimal places.11\nV. CONCLUSION\nIn conclusion, we have shown that by applying stag-\ngeredfieldstoaspinchain,notonlycanwelargelydecou-\nple the spin chain from the environment, but we can also\nsuppress the spin-spin interactions, effectively obtaining\na chain of non-interacting spins. 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Lett. 80, 2245 (1998) ."    },    {        "title": "2011.03246v2.Acoustic_spin_Hall_effect_in_strong_spin_orbit_metals.pdf",        "content": "arXiv:2011.03246v2  [cond-mat.mes-hall]  9 Nov 2020Acoustic spin Hall effect in strong spin-orbit metals\nTakuya Kawada,1Masashi Kawaguchi,1,∗Takumi\nFunato,2Hiroshi Kohno,2and Masamitsu Hayashi1,†\n1Department of Physics, The University of Tokyo, Tokyo 113-0 033, Japan\n2Department of Physics, Nagoya University, Nagoya 464-8602 , Japan\n(Dated: November 10, 2020)\nWe report on the observation of the acoustic spin Hall effect t hat facilitates\nlattice motion induced spin current via spin orbit interact ion (SOI). Under ex-\ncitation of surface acoustic wave (SAW), we find a spin curren t flows orthogonal\nto the propagation direction of a surface acoustic wave (SAW ) in non-magnetic\nmetals. The acoustic spin Hall effect manifests itself in a fie ld-dependent acous-\ntic voltage in non-magnetic metal (NM)/ferromagnetic meta l (FM) bilayers.\nThe acoustic voltage takes a maximum when the NM layer thickn ess is close to\nits spin diffusion length, vanishes for NM layers with weak SO I and increases\nlinearly with the SAW frequency. To account for these result s, we find the\nspin current must scale with the SOI and the time derivative o f the lattice\ndisplacement. Such form of spin current can be derived from a Berry electric\nfield associated with time varying Berry curvature and/or an unconventional\nspin-lattice interaction mediated by SOI. These results, w hich imply the strong\ncoupling of electron spins with rotating lattices via the SO I, show the potential\nof lattice dynamics to supply spin current in strong spin orb it metals.\n∗masashi.kawaguchi@phys.s.u-tokyo.ac.jp\n†hayashi@phys.s.u-tokyo.ac.jp\n1Spin current represents a flow of spin angular momentum carried by electrons. The\nspin Hall effect[1] allows electrical generation of spin current in mate rials with strong spin\norbit interaction (SOI)[2]. The spin Hall angle, a material parameter that characterizes\ncharge to spin conversion efficiency, scales with the longitudinal res istivity and the spin\nHall conductivity[3]. For the intrinsic spin Hall effect, the spin Hall con ductivity is de-\ntermined by the electron band structure[4, 5] ( i.e., the Berry curvature of the bands near\nthe Fermi level) and the SOI of the host material. As spin current ca n be used to control\nthe direction of magnetization of a ferromagnetic layer placed adja cent to the spin source,\ndeveloping materials and means to create it with high efficiency are the forefront of modern\nSpintronics[6–8].\nRecent studies have shown that not only electrons but other degr ees of freedom can\ngenerate spin current. Precessing magnetization pumps out spin c urrent from magnetic\nmaterials, a mechanism known as spin pumping[9–11]. In the spin Seebe ck effect[12, 13], a\ntemperature gradient applied to a magnetic material induces a magn on population gradient\nand the associated diffusion spin current. Spin current can also be p roduced from exchange\nof angular momentum between a rotating body and electrons, an eff ect referred to as spin-\nrotation coupling[14]. The effect has been observed in liquid metals[15] and non-magnetic\nlight metals ( e.g., Cu)[16]. Generation of spin current via spin pumping, spin Seebeck e ffect\nand spin-rotation coupling do not require large SOI of the host mate rial.\nHere we show a profoundly different approach to generate spin cur rent. We find a spin\ncurrent directly emerges from the dynamics of lattice via SOI. Similar to the spin Hall\neffect where a spin current flows transverse to electrical curren t, a spin current develops\northogonal to the propagation direction of a surface acoustic wa ve (SAW) in non-magnetic\nmetals. Theefficiency togeneratespincurrent isproportionaltot hespinHallangleandmay\nbeinfluenced byafactorthatdepends onthefilmstructure. To ac count fortheexperimental\nresults, we findthespin current must scalewiththeSOIandthetime derivative ofthelattice\ndisplacement.\nThin film heterostructures are grown on piezoelectric LiNbO 3substrates using radio fre-\nquency(rf)magnetronsputtering. Thefilmstructureissub./X( d)/CoFeB(1)/MgO(2)/Ta(1)\nwith X=W, Pt, Ta and Cu (thickness in unit of nanometers). The hete rostructures are re-\nferred to as X/CoFeB bilayers hereafter. Standard optical lithog raphy is used to pattern\nHall bars from the film and electrodes/interdigital transducers (I DTs)[17] from conducting\n2FIG. 1. Experimental setup to probe the acoustic spin Hall effect. (a) Schematic illus- \ntration of the experimental setup including the substrate, the fil m, the IDTs and the VNA. The \nbottom image illustrates lattice motion induced spin current, i.e. , the acoustic spin Hall effect. \nSpin current flows orthogonal to the SAW propagation. (b) Representative optical microscopy \nimage of the device. The bright regions are the electrodes and the gray s quare at the center is the \nl bar made of the film. (c) SAW transmission amplitude from IDT1 to IDT2 (IDT2 to IDT1) \nis plotted as a function of frequency ( ) by the blue (red) line. A Hall bar made of W(2.4)/CoFeB \nbilayer is placed between the IDTs. \nmetals (see Methods for the details of sample preparation).\nThe experimental setup and the coordinate system are schema tically illustrated in \nFig. 1(a). The Hall bar is placed between the two IDTs. Figure 1(b ) shows a representative \noptical microscope image of the device. A vector network ana lyzer (VNA) is connected \nto the IDTs to excite a Rayleigh-type SAW from one end and to de tect its transmission \nat the other end. Figure 1(c) shows typical transmission spec tra with a W/CoFeB bilayer \nplaced between the IDTs. The transmission amplitude takes a maximum at 194 MHz, \nwhich corresponds to the fundamental excitation frequency of the SAW ( SAW ) defined by \nthe geometry of the IDTs and the sound velocity of the substra te. \nThe acoustoelectric properties of the films are studied as a f unction of magnetic field. A FIG. 1. Experimental setup to probe the acoustic spin Hall effect. (a) Schematic illus-\ntration of the experimental setup including the substrate, the film, the IDTs and the VNA. The\nbottom image illustrates lattice motion induced spin curre nt,i.e., the acoustic spin Hall effect.\nSpin current flows orthogonal to the SAW propagation. (b) Rep resentative optical microscopy\nimage of the device. The bright regions are the electrodes an d the gray square at the center is the\nHall bar made of the film. (c) SAW transmission amplitude from IDT1 to IDT2 (IDT2 to IDT1)\nis plotted as a function of frequency ( f) by the blue (red) line. A Hall bar made of W(2.4)/CoFeB\nbilayer is placed between the IDTs.\nmetals (see Methods for the details of sample preparation).\nThe experimental setup and the coordinate system are schematic ally illustrated in\nFig. 1(a). The Hall bar is placed between the two IDTs. Figure 1(b) s hows a representative\noptical microscope image of the device. A vector network analyzer (VNA) is connected\nto the IDTs to excite a Rayleigh-type SAW from one end and to detec t its transmission\nat the other end. Figure 1(c) shows typical transmission spectra with a W/CoFeB bilayer\nplaced between the IDTs. The transmission amplitude takes a maximu m at∼194 MHz,\nwhich corresponds to the fundamental excitation frequency of t he SAW ( fSAW) defined by\nthe geometry of the IDTs and the sound velocity of the substrate .\nThe acoustoelectric properties of the films are studied as a functio n of magnetic field. A\n3continuous rf signal with fixed frequency fand power Pis fed from one of the VNA ports\nto the corresponding IDT, which launches a SAW along xthat propagates to the film and\ninduces lattice motion. The longitudinal (along x) and transverse (along y) voltages of the\nHall bar, defined as VxxandVyx, respectively, are measured during the SAW excitation.\nSinceVxxandVyxcontain similar information, here we focus on the results of Vxx; see\nsupplementary material section I for the characteristics of Vyx. In order to extract the\nvoltage originating from the SAW, we subtract the average voltage measured under off-\nresonance conditions ( f/negationslash=fSAW) and obtain the acoustic voltage ∆ Vxx≡Vxx− /angbracketleftVoff\nxx/angbracketright.\n/angbracketleftVoff\nxx/angbracketrightis the average value of Vxxwhenfis set far from fSAW(see Methods for the details).\nWe apply an in-plane magnetic field of magnitude Hduring the voltage measurements. The\nangle between the field and the x-axis is defined as ϕH. As the magnetic easy axis of the\nCoFeB layer points along the film plane and the in-plane magnetic anisot ropy is weak, we\nassume the orientation of the magnetization follows that of the mag netic field.continuous rf signal with fixed frequency and power is fed from one of the VNA ports \nto the corresponding IDT, which launches a SAW along that propagates to the film and \ninduces lattice motion. The longitudinal (along ) and transverse (along ) voltages of the \nHall bar, defined as xx and yx , respectively, are measured during the SAW excitation. \nSince xx and yx contain similar information, here we focus on the results of xx ; see \nsupplementary material section I for the characteristics o f yx . In order to extract the \nvoltage originating from the SAW, we subtract the average vol tage measured under off- \nresonance conditions ( SAW ) and obtain the acoustic voltage ∆ xx xx −/angbracketleftoff \nxx \noff \nxx is the average value of xx when is set far from SAW (see Methods for the details). \nWe apply an in-plane magnetic field of magnitude during the voltage measurements. The \nangle between the field and the -axis is defined as . As the magnetic easy axis of the \nCoFeB layer points along the film plane and the in-plane magne tic anisotropy is weak, we \nassume the orientation of the magnetization follows that of the magnetic field. \nFIG. 2. Field angle dependence of the acoustic voltage. (a-d) Magnetic field angle ( \ndependence of ∆ xx when a rf signal of SAW and 10 dBm is applied to IDT2 (a,c,d) \nand IDT1 (b). Films placed between the IDTs are W(1.8)/CoFeB (a,b), Pt (2.0)/CoFeB (c), \nand Cu(1.8)/CoFeB (d) bilayers. The error bars represent standard dev iation of the repeated \nmeasurements. The black lines show fit to the data with Eq. (1). FIG. 2. Field angle dependence of the acoustic voltage. (a-d) Magnetic field angle ( ϕH)\ndependence of ∆ Vxxwhen a rf signal of f∼fSAWandP∼10 dBm is applied to IDT2 (a,c,d)\nand IDT1 (b). Films placed between the IDTs are W(1.8)/CoFeB (a,b), Pt(2.0)/CoFeB (c),\nand Cu(1.8)/CoFeB (d) bilayers. The error bars represent st andard deviation of the repeated\nmeasurements. The black lines show fit to the data with Eq. (1) .\n4Figures 2(a,c,d) show the field angle ( ϕH) dependence of ∆ Vxxfor W/CoFeB, Pt/CoFeB\nand Cu/CoFeB bilayers when a rf signal of f∼fSAWandP∼10 dBm is applied to IDT2.\nFor W/CoFeB and Pt/CoFeB bilayers, ∆ Vxxshows a sinusoidal variation with a period of\n90◦. Note that the sign ( i.e., the phase) of the sinusoidal variation is the same for the\ntwo bilayers although the sign of the spin Hall angle is opposite betwee n Pt and W[3]. In\ncontrast, no such variation is found for the Cu/CoFeB bilayer. Figu re 2(b) shows ∆ Vxxvs.\nϕHof the W/CoFeB bilayer when the rf signal is applied to IDT1. Clearly, t he mean offset\nvoltage and the sinusoidal variation change their signs as the SAW pr opagation direction is\nreversed. Similar features are observed for the Pt/CoFeB bilayer s.\nWe fit the ϕHdependence of ∆ Vxxwith the following function:\n∆Vxx= ∆V0\nxx+∆V2ϕ\nxxcos2ϕH+∆V4ϕ\nxxsin22ϕH, (1)\nwhere ∆Vnϕ\nxx(n=2,4) represents the coefficient of the sinusoidal function with a pe riod of\n360◦/nand ∆V0\nxxis theϕH-independent component. ∆ V0\nxxis proportional to what is known\nas the acoustic current, which originates from rectification of the localized electric field and\ncharge density[18].\nThefdependence of ∆ V0\nxxis plotted in Fig. 3(a). ∆ V0\nxxtakes a peak at f∼194 MHz,\nwhich corresponds to fSAW(see Fig. 1(c)), and changes its sign as the SAW propagation\ndirection is reversed[19]. The fdependence of ∆ V2ϕ\nxxand ∆V4ϕ\nxxare shown in Figs. 3(b)\nand 3(c), respectively. ∆ V4ϕ\nxxis significantly larger than ∆ V2ϕ\nxxand shows a clear peak at\nf∼fSAW, suggesting that its appearance is associated with the excitation o f SAW. The rf\npower (P) dependence of ∆ V4ϕ\nxxis shown in Fig. 3(d). ∆ V4ϕ\nxxincreases linearly with P.\nTo identify the origin of ∆ V4ϕ\nxx, we have studied its dependence on the X layer thickness\n(d). Hereafter, we use ∆ V0\nxxand∆V4ϕ\nxxto represent the corresponding value at f∼fSAW. As\nthe transmittance of the SAW slightly varies from device to device du e to subtle differences\nintheIDTs, wenormalize∆ V4ϕ\nxxwith∆V0\nxxanddefine v4ϕ\nxx≡∆V4ϕ\nxx/∆V0\nxx. Figure4(a)shows\nthed-dependence of v4ϕ\nxxfor W/CoFeB bilayers. We find v4ϕ\nxxtakes a maximum at d∼2\nnm. Interestingly, such d-dependence of v4ϕ\nxxresembles that of the spin Hall magnetoresis-\ntance (SMR)[20, 21]. The d-dependence of the SMR ratio, r2ϕ\nxx≡ |∆R2ϕ\nxx/R0\nxx|is plotted in\nFig. 4(b). ∆ R2ϕ\nxxrepresents the resistance change when the magnetization ofthe CoFeB layer\nis rotated in the xyplane[22] and R0\nxxis the base resistance that does not vary with ϕH.\nClearly, the d-dependence of v4ϕ\nxxandr2ϕ\nxxare similar. According to the drift-diffusion model\n5FIG. 3. Resonant excitation of the acoustic voltage. (a-c) RF frequency ( ) dependence of \nxx (a), ∆ xx (b) and ∆ xx (c). The blue (red) triangles represent results when the rf sign al is \napplied to IDT1 (IDT2). The rf power ( ) is fixed to 10 dBm. (d) dependence of ∆ xx when \nis varied. The solid lines show fit to the data with a linear function . Upper and lower panels \nshow results when a rf signal is applied to IDT1 and IDT2, respectiv ely. (a-d) The error bars show \nfitting errors of ∆ xx with Eq. (1). Data presented are obtained using W(2.4)/CoFeB bilayer.\nof spin transport in non-magnetic metal (NM)/ferromagnetic metal (FM) bilayers[21, 22], \nthe maximum of xx is proportional to the square of the NM layer spin Hall angle ( SH ), \nand the NM layer thickness at the maximum is close to its spin di ffusion length ( ). Using \nthe model (see Methods), we obtain SH 23 and 73 nm from the -dependence \nof xx for W, which are in good agreement with previous studies[22].\nThesimilarityinthe -dependenceof xx and xx suggeststhataspincurrentisgenerated \nin the X layer. The fact that xx is almost absent for Cu/CoFeB bilayers (see Fig. 2(d)) \nfurther supports this notion: the spin Hall angle of Cu is sign ificantly smaller than that of Pt \nand W. Note, however, that there are a few differences between th e acoustic voltage and the \nSMR.First, thefield-angledependenceofthetwoisdifferent. Typicallytheresistancedueto FIG. 3.Resonant excitation of the acoustic voltage. (a-c) RF frequency ( f) dependence of\n∆V0\nxx(a), ∆V2ϕ\nxx(b) and ∆ V4ϕ\nxx(c). The blue (red) triangles represent results when the rf s ignal is\napplied to IDT1 (IDT2). The rf power ( P) is fixed to ∼10 dBm. (d) Pdependence of ∆ V4ϕ\nxxwhen\nfis varied. The solid lines show fit to the data with a linear fun ction. Upper and lower panels\nshow results when a rf signal is applied to IDT1 and IDT2, resp ectively. (a-d) The error bars show\nfitting errors of ∆ Vxxwith Eq. (1). Data presented are obtained using W(2.4)/CoFe B bilayer.\nof spin transport in non-magnetic metal (NM)/ferromagnetic met al (FM) bilayers[21, 22],\nthe maximum of r2ϕ\nxxis proportional to the square of the NM layer spin Hall angle ( θSH),\nand the NM layer thickness at the maximum is close to its spin diffusion len gth (λN). Using\nthe model (see Methods), we obtain θSH∼0.23 andλN∼0.73 nm from the d-dependence\nofr2ϕ\nxxfor W, which are in good agreement with previous studies[22].\nThesimilarity inthe d-dependence of v4ϕ\nxxandr2ϕ\nxxsuggests thataspincurrent isgenerated\nin the X layer. The fact that v4ϕ\nxxis almost absent for Cu/CoFeB bilayers (see Fig. 2(d))\nfurther supports this notion: the spin Hall angle of Cu is significantly smaller thanthat of Pt\nand W. Note, however, that there are a few differences between t he acoustic voltage and the\nSMR. First, thefield-angledependence ofthetwo isdifferent. Typic ally theresistance dueto\n6FIG. 4. X layer thickness, magnetic field and resonance frequency depende nce of the \nacoustic voltage. (a) Normalized acoustic voltage xx = ∆ xx xx plotted against W layer \nthickness ( ) for W/CoFeB bilayers. The rf frequency ( ) and power ( ) are set to SAW and \n10 dBm, respectively. (b) -dependence of xx of the same system shown in (a). The black line \nis a fit to the data with Eq. (4). (c) The field angle ( ) dependence of the acoustic voltage ∆ xx \nobtained using various field magnitude ( ). Purple, green and orange lines are for 8.0 mT, \n14 mT, and 55 mT, respectively. (d) ∆ xx plotted as a function of . (c,d) Data are obtained \nusing rf signal of SAW and 10 dBm applied to IDT1. (e) dependence of xx . (f) \nThe SAW resonance frequency ( SAW ) dependence of ∆ xx . The rf power ( ) is fixed to 10 dBm. \nThe solid lines show linear fits passing through the origin. (c-f) Data presented are obtained using \nW(2.4)/CoFeB bilayer. The blue (red) triangles in (a,f) represent r esults when the rf signal is \napplied to IDT1 (IDT2). The error bars in (a,d,f) show fitting errors of ∆ xx with Eq. (1). FIG. 4.X layer thickness, magnetic field and resonance frequency dependence of the\nacoustic voltage. (a) Normalized acoustic voltage v4ϕ\nxx= ∆V4ϕ\nxx/∆V0\nxxplotted against W layer\nthickness ( d) for W/CoFeB bilayers. The rf frequency ( f) and power ( P) are set to ∼fSAWand\n∼10 dBm, respectively. (b) d-dependence of r2ϕ\nxxof the same system shown in (a). The black line\nis a fit to the data with Eq. (4). (c) The field angle ( ϕH) dependence of the acoustic voltage ∆ Vxx\nobtained using various field magnitude ( H). Purple, green and orange lines are for H∼8.0 mT,\n14 mT, and 55 mT, respectively. (d) ∆ V4ϕ\nxxplotted as a function of H. (c,d) Data are obtained\nusing rf signal of f∼fSAWandP∼10 dBm applied to IDT1. (e) Hdependence of |∆R2ϕ\nxx|. (f)\nThe SAW resonance frequency ( fSAW) dependence of ∆ V4ϕ\nxx. The rf power ( P) is fixed to 10 dBm.\nThe solid lines show linear fits passing through the origin. ( c-f) Data presented are obtained using\nW(2.4)/CoFeB bilayer. The blue (red) triangles in (a,f) rep resent results when the rf signal is\napplied to IDT1 (IDT2). The error bars in (a,d,f) show fitting errors of ∆ Vxxwith Eq. (1).\n7SMR variesascos2 ϕH(see forexample, Ref.[20]), whereas thedominant contribution to the\nacoustic voltage ∆ Vxxvaries as sin22ϕH. Second, v4ϕ\nxxis more than one order of magnitude\nlarger than r2ϕ\nxx. Third, we find a striking difference in the magnetic field magnitude ( H)\ndependence between the two. In Fig. 4(c), we show the Hdependence of ∆ Vxxvs.ϕHfor\nW/CoFeB bilayer. As evident, the offset voltage (∆ V0\nxx) hardlychanges with H. In contrast,\nthe magnitude of ∆ V4ϕ\nxxincreases with decreasing H. TheHdependence of ∆ V4ϕ\nxx, plotted\nin Figs. 4(d), shows that ∆ V4ϕ\nxxscales with 1 /H. As a reference, we show in Fig. 4(e) the H\ndependence of |∆R2ϕ\nxx|. Contrary to ∆ V4ϕ\nxx,|∆R2ϕ\nxx|is nearly constant against H.\nTo account for these results, we modify the drift-diffusion model o f spin transport that\nis used to describe SMR[21]. First, we include SAW-induced straining of the FM layer\nand magnetoelastic coupling[23, 24], which cause changes in the magn etization direction\nwith respect to the magnetic field[25, 26]. Consequently, ∆ Vxxacquires an extra factor of\n1\nHsin2ϕHcompared to the resistance change that originates from SMR. (Se e supplementary\nmaterial section I where we show that ∆ V4ϕ\nxxis absent for W/NiFe bilayer due to the small\nmagnetoelastic coupling of NiFe.) Next, to generate a (rectified) dc current, the spin current\nmust vary in time and space such that it couples to the motion of magn etic moments driven\nby the SAW-induced strain. We find the following form of spin current jy\ns,z(electron spin\norientation along yand flow along z) produces a rectified dc current and accounts for the\nexperimental results:\njy\ns,z=A∂ux\n∂t, (2)\nwhereuxis the lattice displacement along the wave propagation direction ( x).Ais a\nprefactor that determines the spin current generation efficiency and is proportional to λso,\nthe SOI.\nThe spin current jy\ns,zgenerated in the NM layer drifts to the NM/FM interface and\ncauses spin accumulation. The accumulated spin at the interface ca uses a back flow of spin\ncurrent within the NM layer, which is converted to electrical curren t via the inverse spin\nHall effect[11]. The amount of spin accumulation at the interface dep ends on the direction\nof the FM layer magnetization due to the action of spin transfer tor que[20, 21], thus causing\ntheϕHdependent acoustic voltage. The resulting acoustic voltage reads (see supplementary\nmaterial section II)\n∆Vxx≈cλ2\nsoK(d)fSAWPsgn(k)b\nHMSsin22ϕH, (3)\n8wherecis a constant that depends on the material and the geometry of th e device, K(d)\ncharacterizes the d-dependence similar to that of the SMR (see Eq. (4)), kis the wave vector\nof the Rayleigh-type SAW (sgn( x) takes the sign of x), andbandMSare, respectively, the\nmagnetoelastic coupling constant and the saturation magnetizatio n of the FM layer.\nEquation (3) captures many features of the acoustic voltage fou nd in the experiments.\nAs evident, ∆ Vxxvaries as sin22ϕH. The coefficient of sin22ϕHin Eq. (3), equivalent to\n∆V4ϕ\nxx, changes its sign upon reversal of the wave propagation direction (defined by the sign\nofk), scales with1\nHandP, and is proportional to the square of the spin orbit coupling of the\nNM layer, and thus independent of the sign of the NM layer spin Hall an gle. The thickness\ndependence of ∆ V4ϕ\nxx, coded in K(d), is in relatively good agreement with the experimental\nresults. We have also studied the fSAWdependence of ∆ V4ϕ\nxxfor W/CoFeB bilayer; the\nresults are plotted in Fig. 4(f). As evident, ∆ V4ϕ\nxxscales with fSAW. We emphasize that\nEq. (2) is the only form of spin current that can account for these results. Note that the\nlinear dependence of ∆ V4ϕ\nxxwithfSAWexcludes contributions from spin-dependent inertial\nforce[27] and related effects in the presence of SOI[28], which are p roportional to higher\norder offSAW.\nTheseresultsthereforedemonstratethatthelatticemotionindu cesaspincurrent. Recent\nstudies have shown that spin-rotation coupling[14, 15] can induce s pin accumulation in the\nNM layer, which results in generation of spin current if the NM layer th ickness is largerwhere is a constant that depends on the material and the geometry of the device, \ncharacterizes the -dependence similar to that of the SMR (see Eq. (4)), is the wave vector \nof the Rayleigh-type SAW (sgn( ) takes the sign of ), and and are, respectively, the \nmagnetoelastic coupling constant and the saturation magne tization of the FM layer. \nEquation (3) captures many features of the acoustic voltage found in the experiments. \nAs evident, ∆ xx varies as sin . The coefficient of sin in Eq. (3), equivalent to \nxx , changes its sign upon reversal of the wave propagation dire ction (defined by the sign \nof ), scales with and , and is proportional to the square of the spin orbit coupling of the \nNM layer, and thus independent of the sign of the NM layer spin Hal l angle. The thickness \ndependence of ∆ xx , coded in ), is in relatively good agreement with the experimental \nresults. We have also studied the SAW dependence of ∆ xx for W/CoFeB bilayer; the \nresults are plotted in Fig. 4(f). As evident, ∆ xx scales with SAW . We emphasize that \nEq. (2) is the only form of spin current that can account for th ese results. Note that the \nlinear dependence of ∆ xx with SAW excludes contributions from spin-dependent inertial \nforce[27] and related effects in the presence of SOI[28], whi ch are proportional to higher \norder of SAW \nTheseresultsthereforedemonstratethatthelatticemotio ninducesaspincurrent. Recent \nstudies have shown that spin-rotation coupling[14, 15] can induce spin accumulation in the \nNM layer, which results in generation of spin current if the NM l ayer thickness is larger \nFIG. 5. Efficiency to generate lattice motion induced spin current. (a,b) Maximum \nvalues of the normalized acoustic voltage xx, max xx xx max (red bars) and the maximum \nSMR ratio xx, max xx /R xx max (green bars) (a) and their ratio xx, max \nxx, max (b) obtained for \noFeB (X=Ta, W, and Pt) bilayers and CoFeB/W bilayers. W( ) and W( ) represent \nW/CoFeB and CoFeB/W bilayers, respectively. FIG. 5. Efficiency to generate lattice motion induced spin current. (a,b) Maximum\nvalues of the normalized acoustic voltage v4ϕ\nxx,max≡/vextendsingle/vextendsingle∆V4ϕ\nxx/∆V0\nxx/vextendsingle/vextendsingle\nmax(red bars) and the maximum\nSMR ratio r2ϕ\nxx,max≡/vextendsingle/vextendsingle∆R2ϕ\nxx/R0\nxx/vextendsingle/vextendsingle\nmax(green bars) (a) and their ratio γ≡v4ϕ\nxx,max\nr2ϕ\nxx,max(b) obtained for\nX/CoFeB (X=Ta, W, and Pt) bilayers and CoFeB/W bilayers. W( β) and W( β+α) represent\nW/CoFeB and CoFeB/W bilayers, respectively.\n9than the SAW decay length (typically, of the order the SAW waveleng th, which is a few µm\nhere)[16]. To clarify the role of spin-rotation coupling, we have stud ied ∆V4ϕ\nxxof inverted\nstructures, CoFeB/W bilayers. In both W/CoFeB and CoFeB/W bilay ers, spin-rotation\ncoupling induces spin density in the W layer, which can cause a flow of sp in current toward\nthe CoFeB layer as the latter can act as a spin sink. If such spin curr ent were to flow,\nthe flow direction will be opposite for the normal (W/CoFeB) and inve rted (CoFeB/W)\nstructures and consequently results in ∆ V4ϕ\nxxwith opposite sign. We find that the signs of\n∆V4ϕ\nxxfor W/CoFeB and CoFeB/W bilayers are the same, demonstrating th at spin-rotation\ncoupling is not the source of spin current (see supplementary mate rial sections I and III).\nFor the same reason, we can rule out SAW-induced spin pumping[25, 2 9] from the CoFeB\nlayer and the inverse spin Hall effect of the W layer. This is also suppor ted by the fact that\nthe signs of ∆ V4ϕ\nxxfor W/CoFeB and Pt/CoFeB bilayers are the same (see Fig. 2) albeit t he\ndifference in the sign of θSHfor W and Pt.\nInFig.5(a), wesummarize themaximum valueof v4ϕ\nxxandr2ϕ\nxxwhendisvaried, denotedas\nv4ϕ\nxx,maxandr2ϕ\nxx,max, respectively, foreachbilayer (X=Ta, W,Pt). ResultsfromtheCo FeB/W\nbilayers are included. Note that the structure of W depends on the growth condition: from\nthe film resistivity[30, 31], we consider W forms a highly-resistive β-phase in W/CoFeB\nbilayer whereas it is a mixture of the β-phase and the low-resistivity crystalline α-phase in\nCoFeB/W bilayer. Consequently, the SMR ratio ( r2ϕ\nxx,max) is smaller for the latter due to\nthe smaller θSH[31–33]. Interestingly, we find that v4ϕ\nxx,maxtakes nearly the same value for\nthe two bilayers, indicating that there are factors other than θSHthat sets the magnitude of\nv4ϕ\nxx,max. In Fig. 5(b), we plot the ratio γ≡v4ϕ\nxx,max\nr2ϕ\nxx,maxto characterize such contribution. We find\nγis significantly larger for bilayers with Pt and ( β+α)-W (CoFeB/W) than that with β-W\n(W/CoFeB) and Ta. Since the former two layers are textured wher eas the latter two are\nhighly disordered ( i.e., amorphous-like), we consider the texture of the films may influenc e\nγ. Little correlation is found between γand the bulk modulus of the X layer.\nFinally, we discuss the source of spin current that scales with the tim e derivative of lattice\ndisplacement (Eq. (2)). First, a conventional mechanism would be t o consider internal\nelectric field associated with the SAW and the resulting spin Hall effect of the NM layer.\nThere are two major sources of internal electric field. One is the pie zoelectric field ( Ep)\nlocalized at the film/substrate interface. Spin current generated fromEpcan only reach the\nNM/FM interface when the film thickness is smaller than λN. The thickness dependence\n10ofv4ϕ\nxx(Fig. 4(a)) rules out such contribution. The other source is the tim e varying electric\nfield (Eb) caused by the motion of ions[34–36]. Ebis uniform along the film normal as long\nas the film thickness is significantly smaller than the SAW decay length. In general, Ebis\nscreened by the conduction electrons in metallic films: we infer it gene rates negligible spin\ncurrent. With the current understanding, we consider it is difficult t o quantitatively account\nfor the experimental results with the combination of the SAW induce d electric field and the\nspin Hall effect. Second, Eq. (2) can be derived assuming the followin g interaction[37, 38]:\nHint=su·(p×σ), wheresis a constant, uis the lattice displacement vector, and pandσ\nare electron momentum and spin orientation, respectively. This inte raction derives from the\nSOI[37, 38] and the coefficient sis proportional to λso, similar to the relation between θSH\nandλso.Hintresembles the Rashba Hamiltonian[39] but can exist here since the inv ersion\nsymmetry is broken by the dynamical lattice displacement u. Further studies are required,\nhowever, to justify the presence of such Hamiltonian. Third, the t ime derivative of the\nlattice displacement can cause changes in the Berry curvature of e lectron wave function.\nIndeed, theoretical studies have identified the right hand side of E q. (2) as the Berry electric\nfield[40, 41]. It remains to be seen whether spin current emerges fr om the Berry electric\nfield under strong SOI. Finally, the phonon angular momentum[42–44 ] may contribute to\nthe generation of spin current. Similar to the spin Seebeck effect[12 ], where the spin angular\nmomentum of magnons are transferred to electrons, the angular momentum of phonons (i.e.\nsound waves) can be transferred to the electrons and induce spin current. The efficiency of\nsuch process must be addressed to assess its contribution.\nIn summary, we have shown that spin current is directly created fr om lattice motion\nassociated with surface acoustic wave (SAW). Such acoustic spin H all effect is observed in\nnon-magnetic metal (NM)/ferromagnetic metal (FM) bilayers thr ough a field-dependent dc\nacousticvoltage. Theacousticvoltageroughlyscaleswiththesqua reofthespinHallangleof\ntheNMlayer andisproportionaltotheSAWfrequency. The NMlayer thickness dependence\nof the acoustic voltage is similar to that of the spin Hall magnetoresis tance. Using a diffusive\nspin transport model, we show that such characteristics of the ac oustic voltage can be\naccounted for when a spin current that scales with thetime derivat ive oflattice displacement\nis generated in the NM layer. Possible sources of such spin current in clude a Berry electric\nfield associated with time varying Berry curvature and/or an uncon ventional SOI-mediated\nspin-lattice interaction that resembles the form of Rashba Hamilton ian. The efficiency to\n11generate spin current, represented by the maximum acoustic volt age, also seems to depend\non a factor related to the film texture; the efficiency is nearly the sa me for amorphous-like\nβ-W and textured Pt despite the difference in their spin Hall angle. The finding of the\nacoustic spin Hall effect thus implies a mechanism that facilitates an SO I mediated coupling\nofelectronspins andarotatinglattice. Further studies arerequir ed tounveil themicroscopic\nmechanism to describe such coupling.\nI. MATERIALS AND METHODS\nA. Sample preparation\nRadio frequency magnetron sputtering is used to deposit the films o n piezoelectric\nY+128◦-cut LiNbO 3substrates. The film structure is sub./X( d)/CoFeB(1)/MgO(2)/Ta(1)\nwith X=W, Pt, Ta and Cu (thickness in unit of nanometers). The inver ted structure is\nsub./MgO(2)/CoFeB(1)/X( d)/MgO(2)/Ta(1) with X=W. The MgO(2)/Ta(1) layers serve\nas a capping layer to prevent oxidation of the films. For bilayers with X =Pt and Cu, a 0.5\nnm thick Ta layer is inserted before deposition of X to promote their s mooth growth. Hall\nbars are formed from the films using optical lithography and Ar ion et ching. Subsequently,\nwe use optical lithography and a liftoff process to form interdigital t ransducers (IDTs) and\nelectrodes made of Ta(5)/Cu(100)/Pt(5).\nSchematic illustration of the SAW device and definition of the coordina te system are\nshown in Fig. 6. The distance of the two IDTs is ∼600µm and each IDT has 20 pairs of\nsingle-type fingers. The width and gap of the fingers are set to a: the corresponding SAW\nwavelength is ∼4a. The finger overlap, i.e., the SAW aperture ( La), is fixed to ∼450µm.\nA Hall bar made of the film is placed at the center of the two IDTs. The length and width\nof the Hall bar are set to ∼450µm and∼400µm, respectively.\nWe vary ato change the SAW resonance frequency ( fSAW).ais fixed to ∼5µm for\nmost of the results shown, which gives fSAW∼194 MHz. In Fig. 4(f), we vary ato change\nfSAW:ais set to ∼5,∼4,∼3,∼2µm to obtain fSAWof∼194,∼242,∼321,∼479 MHz,\nrespectively.\n12FIG. 6. Schematic illustration of the SAW device. The orange structur e represent the IDTs and \nthe dark gray area show the film. \nB. Voltage measurements \nThe longitudinal (along ) and transverse (along ) voltages, defined as xx and yx \nrespectively, are measured during the SAW excitation. To ex tract the voltage originating \nfrom the SAW, we subtract the average voltage measured under o ff-resonance conditions, \ndefined as off \nxx yx off \nxx yx is obtained as follows. Under a fixed magnetic field and rf \npower, we study the frequency ( ) dependence of xx yx xx yx takes a peak when \nSAW . We choose frequencies ( off ) that are outside the peak structure of xx yx , typically a \nfewtensofMHzawayfrom SAW (seeFig.1(c)foratypicaltransmissionspectra). off \nxx yx is \ntheaveragevalueof xx yx measuredatseveral off off \nxx yx issubtractedfromthemeasured \nvoltage xx yx at frequency to obtain the acoustic voltage ∆ xx yx xx yx −/angbracketleftoff \nxx yx \noff \nxx yx is always measured prior to the measurement of xx yx at frequency . Voltage \nmeasurements at each condition are repeated 5-100 times to i mprove the signal to noise \nratio. \nC. Spin Hall magnetoresistance \nIn the main text, we have used ∆ xx , the resistance change when the magnetization of \nthe CoFeB layer is rotated in the xy plane, to estimate SMR. ∆ xx is equal to the sum \n13 FIG. 6. Schematic illustration of the SAW device. The orange structure represent the IDTs and\nthe dark gray area show the film.\nB. Voltage measurements\nThe longitudinal (along x) and transverse (along y) voltages, defined as VxxandVyx,\nrespectively, are measured during the SAW excitation. To extract the voltage originating\nfrom the SAW, we subtract the average voltage measured under o ff-resonance conditions,\ndefined as /angbracketleftVoff\nxx(yx)/angbracketright./angbracketleftVoff\nxx(yx)/angbracketrightis obtained as follows. Under a fixed magnetic field and rf\npower, we study the frequency ( f) dependence of Vxx(yx).Vxx(yx)takes a peak when f∼\nfSAW. We choose frequencies ( foff) that are outside the peak structure of Vxx(yx), typically a\nfewtensofMHzawayfrom fSAW(seeFig.1(c)foratypical transmissionspectra). /angbracketleftVoff\nxx(yx)/angbracketrightis\ntheaveragevalueof Vxx(yx)measuredatseveral foff./angbracketleftVoff\nxx(yx)/angbracketrightissubtractedfromthemeasured\nvoltageVxx(yx)at frequency fto obtain the acoustic voltage ∆ Vxx(yx)≡Vxx(yx)−/angbracketleftVoff\nxx(yx)/angbracketright.\n/angbracketleftVoff\nxx(yx)/angbracketrightis always measured prior to the measurement of Vxx(yx)at frequency f. Voltage\nmeasurements at each condition are repeated 5-100 times to impro ve the signal to noise\nratio.\nC. Spin Hall magnetoresistance\nIn the main text, we have used ∆ R2ϕ\nxx, the resistance change when the magnetization of\nthe CoFeB layer is rotated in the xyplane, to estimate SMR. ∆ R2ϕ\nxxis equal to the sum\n13of the SMR and the anisotropic magnetoresistance (AMR). Since th e latter is significantly\nsmaller than the former for the system under study[22], we assume ∆R2ϕ\nxxrepresents the\nSMR. To obtain the SMR more accurately, it is customary to measure the resistance change\nwhen the magnetization of the CoFeB layer is rotated in the yzplane[20], defined as ∆ Rsmr\nxx.\nWe have verified that ∆ R2ϕ\nxxand ∆Rsmr\nxxtake similar value, justifying the assumption that\n∆R2ϕ\nxx/R0\nxxrepresents the SMR.\nThe X layer thickness dependence of the spin Hall magnetoresistan ce is fitted using the\nfollowing equation[20, 21]:\n∆R2ϕ\nxx\nR0\nxx=θ2\nSH\n1+ζK(d),\nK(d)≡λN\ndtanh2d\n2λNtanhd\nλN,(4)\nwhereζ≡ρNtF\nρFd,ρFandtFare the resistivity and thickness of the FM (=CoFeB) layer,\nrespectively and ρNis the resistivity of the X layer. Here we have assumed a transparen t\nX/FMinterface forspin transmission andneglected the effect of lon gitudinal spin absorption\nof the FM layer[22]. 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Chudnovsky, “Angular momentum in spin-phonon processes,” Phys.\nRev. B92, 024421 (2015).\n[44] J. Holanda, D. S. Maior, A. Azevedo, and S. M. Rezende, “D etecting the phonon spin in\nmagnon-phonon conversion experiments,” Nat. Phys. 14, 500 (2018).\n17"    },    {        "title": "1108.5239v1.Direct_observation_of_correlation_time_of_dynamic_nuclear_polarization_in_single_quantum_dots.pdf",        "content": "arXiv:1108.5239v1  [cond-mat.mtrl-sci]  26 Aug 2011Direct observationofcorrelationtimeofdynamicnuclear p olarizationinsinglequantum dots\nR. Kaji,∗S. Adachi, and S. Muto\nDepartment of Applied Physics, Hokkaido University, N13 W8 , Kitaku, Sapporo 060-8628, Japan\nH. Sasakura\nResearch Institute for Electronic Science, Hokkaido Unive rsity, N21 W10, Kitaku, Sapporo 001-0021, Japan\n(Dated: June 24, 2018)\nThe spin interaction between an electron and nuclei was inve stigated optically in a single self-assembled\nInAlAs quantum dot (QD). In spin dynamics, the correlation t ime of the coupled electron-nuclear spin system\nand the electron spin relaxation time play a crucial role. We examined on a positively charged exciton in a QD\nto evaluate these key time constants directly via the tempor al evolution measurements of the Overhauser shift\nandthedegree ofcircular polarization. Inaddition, theva lidityof our usedspindynamics model was discussed\ninthe context of the experimentally obtained keyparameter s.\nPACS numbers: 73.21.La, 78.67.Hc,71.35.Pq, 71.70.Jp\nI. INTRODUCTION\nThe hyperfine interaction in semiconductor quantum dots\n(QDs)isenhancedowingtothestrong3Dconfinementofthe\nelectron wave function; consequently, this has attracted c on-\nsiderable attention from the fundamentaland practical poi nts\nofview. Thesophisticatedcontrolofnuclearspinpolariza tion\n(NSP) is required for fascinating applications such as a lon g-\nlived memory at an atomic level1and qubit conversion be-\ntweenanelectronspinandaphoton2. InsemiconductorQDs,\nthe enhancedhyperfineinteractionprovidesthe possibilit y of\npolarizingnuclearspins(n-spins)inonedirectionwithth eop-\nticallyselectiveexcitationoftheelectronspin(e-spin) . Infact,\na large NSP of up to 30 −60% was observedrecently in inter-\nface GaAs QDs3, self-assembled InAlAs QDs4–6, In(Ga)As\nQDs7–9, and InP QDs10. In these QDs, the confined electron\nis subjectto a largenuclearfield (Overhauserfield: BN) up to\nseveralTesla. Inaddition,afundamentalinterestisthekn owl-\nedge of the decay of the e-spin polarization induced by the\nrandom fluctuation of the Overhauser field. This fluctuation\ninduces an additional e-spin precession around the e ffective\nmagnetic field, and it imposes an inevitable contribution to\nthe e-spin relaxation and decoherence through the transver se\nand longitudinal components of BNfluctuation, as predicted\nina previouswork11.\nFrom these points of view, it is necessary to examine the\nspin dynamics of a coupled electron-nuclei (e-n) system tha t\niswellisolatedinaQD.Intheframeworkofthecurrentsemi-\nclassicaldynamicsmodelofNSPformation6,12,thefollowing\ntwo physicalquantitiesplay a significantrole: the correla tion\ntime of the coupled e-n spin system and the e-spin relaxation\ntime.\nThecorrelationtime τcindicatesthecharacteristictimedur-\ning which the randomly fluctuating e ffective field is consid-\nered to be constant in magnitude and direction according to\nthe traditionalspin relaxationtheory13,14, andit inducesa ho-\nmogeneousbroadening (2/planckover2pi1/τc)of the target e-n levels. Since\nthe NSP formation rate is very sensitive to the degree of en-\nergymismatchinthee-nspinflip-flopprocess,whichisdeter -\nmined by the splitting and the broadeningof the correspond-ing e-n energy levels, τchas a crucial influence on the NSP\ndynamics. Despite its importance, this key quantity has bee n\ngenerallyusedasafittingparameterinthemodelcalculatio ns\nto reproduce the observed results, and the direct experimen -\ntal estimation ofτchas not been reported thus far in any QD\nmaterials. Meanwhile, the e-spin relaxation time τsalso af-\nfectsNSP formationbychangingthe longitudinalcomponent\nofthee-spinpolarization. Asmentionedinalatersection, the\ne-spin relaxation rate (1 /τs) can generally be expressed as a\nLorentzian function of the e ffective magnetic field, and it is\ninfluencedbytheaforementionedcorrelationtimethroughi ts\nwidth of 2/planckover2pi1/τc. Accordingly, we measure the e-spin relax-\nation time at a zero e ffective fieldτs0, which gives the maxi-\nmumamplitudeof1 /τs;thisamplitudecanbeevaluatedinde-\npendentlyofτc.\nA secondary interest is the possibility of the measurement\nof NSP in a QD structure through the degree of circular po-\nlarization(DCP)ofpositivelychargedexcitonemissions. The\nDCP ofthetime-integratedphotoluminescence(PL)hasbeen\nused as a powerfultool to detect NSP or the Overhausershift\n(OHS), which is the energy shift in the electronic level in-\nduced by BN, in bulk and quantum well structures for a long\ntime. However, the method to probe NSP in single QDs has\nbeen limited only to the change in the energy splitting of the\nPL lines; this is the simplest way to evaluated OHS, but its\naccuracyhasbeenlimitedbythespectralresolutionofthee x-\nperimentalsetup. Inthecouplede-nsystem,byusingtheDCP\nofpositivelychargedexcitonemission,whichdirectlyrefl ects\nthe e-spin polarization, it may be possible to follow not onl y\nthe e-spin but n-spin dynamics, and the study of the coupled\ne-n system may act as a tool for sensitive measurements of\nQD-NSP.\nIn this study, we investigatedthe e-n spin dynamicsin QD\nstructures by using the DCP of the positively charged exci-\nton (X+). The DCP of X+PL, which is mainly determined\nby the e-spin polarization, changed in synchronization wit h\nthe OHS or the energy splitting of the e-spin levels, and this\nphenomenonprovides the possibility of the sensitive probi ng\nof the QD-NSP. By taking advantage of this feature, the key\nquantities (τcandτs0) were evaluated directly from the ex-\nperimentaldata. In addition,we extendedthedynamicmodel2\nof NSP by including the dynamics of the X+state, and we\nconfirmed the validity of our model by comparing the time-\nresolvedOHS andDCP measurementswiththecalculatedre-\nsults.\nII. SAMPLEAND SETUP\nSelf-assembled In 0.75Al0.25As QDs, which were embedded\nwith an Al 0.3Ga0.7As layer grownon an undoped(100)GaAs\nsubstrate by molecular beam epitaxy, were used in the exper-\niments. Assuming a lens-shaped QD with a typical diameter\nof∼20 nm and a height of ∼4 nm derived from atomic force\nmicroscopy (AFM) measurements, the number of nuclei in\na single QD was estimated roughly to be ∼3×104. Micro-\nPL measurements were performed at 6 K under longitudinal\nmagnetic fields ( Bz) of up to 5 T. A cw-Ti:sapphire laser of\n∼728 nm, which provides the transition energy at the wet-\ntinglayerofInAlAsQDs,wasemployedtoilluminatetheQD\nsample. TheQDemissionspectraweredetectedusingatriple\ngratingspectrometerandaliquidN 2-cooledSi-CCDdetector.\nThoughtheenergyresolutionofoursetupwas ∼12µeV,itcan\nbe improvedto 5µeV by spectral fitting. The polarization of\ntheexcitationlightwascontrolledusinganelectro-optic mod-\nulator (EOM), and the Zeeman splitting energy and the DCP\nofa targetsingleQD spectrawere evaluated.\nIII. RESULTSAND DISCUSSION\nA. Electron andnuclearspinpolarization inacoupledsyste m\nFirst, we investigate the availability of X+DCP as a pow-\nerful measure of the electron and nuclear spin polarization in\naQD.Fig. 1(a)showsthePLspectraobtainedfromthetarget\nsingle QD under a zero magnetic field. Regardlessof the un-\ndopedsampleweused,thePLspectrafromthevariouscharge\nstatesthatoriginatedfromthesamesingleQDwereobserved .\nThechargestatesofthreepeakswithhighintensitieswerea s-\nsigned to a neutral biexciton (XX0), a neutral exciton (X0),\nand a positively charged exciton (X+) from the lower energy\nside, respectively. Since X+had the strongest PL intensity in\nourQDsampleandbecausetherewere nodarkexcitonstates\npresent, it is expected to be the dominant contributor in NSP\nformation.\nFig.1(b) depicts the X+PL spectra at Bz=5 T for the\nlinearly (denoted by grey squares) and the circularly ( σ+\nandσ−: denoted by open and solid circles) polarized exci-\ntations. In the X+states composed of the spin-paired two\nholes and an electron, the exchange interactions between th e\nelectron and hole spins play no role, and the energy split-\nting of the PL lines ( ∆EZ) is determined solely by the Zee-\nman interaction of the spins with the (e ffective) magnetic\nfields. Underthiscondition, ∆EZcanbeexpressedasfollows:\n∆EZ=gh\nzµBBz+ge\nzµB(Bz±BN),wherege(h)\nzdenotestheelec-\ntron (hole) g-factor in the growth direction, µBdenotes the\nBohr magneton, and BNdenotes the Overhauser field. Since\nthe holespinhasa low probabilityofexistenceat the nucleu s/X45/X78/X74/X65/X72/X6E/X61/X6C/X20/X6D/X61/X67/X6E/X65/X74/X69/X63/X20/X66/X69/X65/X6C/X64/X20/X28/X54/X29 /X58/X2B/X20/X44/X43/X50 \n/X58/X30/X20/X44/X43/X50 ΔEZ/X20/X28µ/X65/X56/X29/X36/X34/X30 \n/X36/X30/X30 \n/X35/X36/X30 \n/X35/X32/X30 \n/X30\n/X2D/X32/X30 \n/X2D/X34/X30 \n/X2D/X36/X30 \n/X2D/X38/X30 ΔEe/X20/X28µ/X65/X56/X29 /X44/X43/X50 /X31/X2E/X30 \n/X30/X2E/X38 \n/X30/X2E/X36 \n/X30/X2E/X34 \n/X30/X2E/X32 \n/X30/X2E/X30 \n/X34/X2E/X30 /X34/X2E/X32 /X34/X2E/X34 /X34/X2E/X36 /X34/X2E/X38 /X35/X2E/X30 /X42/X7A/X2D/X42/X4E/X7E/X30 /X42/X7A/X2D/X42/X4E/X3E/X30 ( gz +gz ) µBBze h/X50/X4C/X20/X49/X6E/X74/X65/X6E/X73/X69/X74/X79/X20/X28/X61/X2E/X20/X75/X2E/X29 \n/X50/X4C/X20/X45/X6E/X65/X72/X67/X79/X20/X28/X65/X56/X29/X31/X2E/X36/X34/X30 /X31/X2E/X36/X34/X32 /X31/X2E/X36/X34/X34 /X31/X2E/X36/X34/X36 X+\nX0XX 0/X36/X20/X4B/X2C/X20/X30/X20/X54 /X28/X61/X29 /X28/X63/X29 \n/X28/X62/X29 /X50/X4C/X20/X49/X6E/X74/X65/X6E/X73/X69/X74/X79/X20/X28/X61/X2E/X20/X75/X2E/X29 /X42/X7A/X3D/X35/X20/X54 \n/X2D/X34/X30/X30 /X2D/X32/X30/X30 /X32/X30/X30 /X34/X30/X30 /X30\n/X5A/X65/X65/X6D/X61/X6E/X20/X73/X68/X69/X66/X74/X20/X28 µ/X65/X56/X29σ/X2B/X20/X50/X4C σ/X2D/X20/X50/X4C \nFIG. 1: (a) PL spectra from the target single QD at zero magnet ic\nfield. (b) PL spectra of X+state atBz=5 T with linearly (gray\nsquares) and circularly ( σ+/σ−: open/solid circles) polarized excita-\ntion. Theσ−(+)PLcomponent is positioned atthe higher (lower) en-\nergy side. The difference between the Zeeman splitting ∆EZfor the\ncircularly and linearly polarized excitations is defined as the Over-\nhauser shift (∆EOHS), and it is evaluated as ∆EOHS=98µeV (−21\nµeV) forσ−(σ+) excitation. (c) Bzdependences of∆EZof the X+\nPLline(upperpanel),theenergysplittingoftheelectroni clevel∆Ee\n(middle panel), and the DCPs of X+and X0(denoted by the black\nand gray symbols in lower panel). The solid line in the upper p anel\nisthe calculated∆EZatBN=0.\nsite, the effect ofBNon the hole spin could be neglected ex-\ncept for the special case15. Under a large Bzof a few Tesla,\nBNmanifests itself as an OHS defined as ∆EOHS=ge\nzµBBN.\nSinceBNis essentially zero for the linearly polarized excita-\ntion, OHS is deduced from the di fference between∆EZfor\nthecircularlyandlinearlypolarizedexcitations,anditi seval-\nuated in Fig. 1(b) as∆EOHS=98µeV (−21µeV) withσ−\n(σ+) excitation. As per our definition, the σ−(σ+) excitation\ngenerates BNin the opposite(same) directionto Bz, andit in-\nduces an apparent increase (decrease) in ∆EZbecause of the\nrelationge\nz·gh\nz<0. Hereafter,we focuson the σ−case where\nthe compensationof BzviaBNis achieved; consequently,the\nbistabilities of NSP have been observed for external param-\neters such as the excitation power8, excitation polarization7,\nandexternalmagneticfield9.\nFigure1(c) summarizes the e ffects of NSP on the X+PL\nobserved in the Bzdependence measurement. In the experi-\nment, the excitation polarization was fixed at σ−, and the ex-\nternal field was swept from 4.0 T to 5.0 T with a sweeping\nrate of 0.11 T/min. The symbols and the solid line in the up-\nper panel indicate the observed ∆EZand the calculated ∆EZ\nundertheconditionwhen BN=0. Thedifferencefromthezero\nBNlineistheOHSat each Bz. Ascanbeclearlyobserved,an\nabrupt decrease in ∆EZwas observed at Bz=4.31 T owing\nto the bistable nature of NSP. In order to measure the degree\nofBzcompensation via BN, we introduce the e ffective mag-\nneticfieldasexperiencedbye-spin; Beff(=Bz−BN). Byusing\nthe previouslyobtainedvaluesof gh\nz=+2.54andge\nz=−0.375,3\nwededucedtheelectronicsplittingenergy ∆Ee=ge\nzµBBeff,as\nshown in the middle panel of Fig. 1(c). In the region where\nBz<4.31T, the absolute valueof ∆Eenearlyreducesto zero,\nand the Overhauser field fully compensates for the external\nfield. With increasing Bz, the magnitude of BNshows a clear\nreductionand|∆Ee|increasesabruptly.\nHere, we focus on the DCP of the X+PL (the lower\npanel of Fig. 1(c)). In this work, the DCP is defined as\n(I−−I+)/(I−+I+)(I+(−)denotes the integrated PL intensity\nof theσ+(−)component). It is noteworthy that the DCP of\nX+PL shows a clear jump from ∼0.6 to∼0.9; this transition\nsynchronizeswith the decrease in ∆Ee. As mentionedabove,\nthe DCP of X+is essentially determined solely by the e-spin\npolarization/angbracketleftSz/angbracketright,anditcanbeexpressedasDCP =2/angbracketleftSz/angbracketright. Ac-\ncordingly,a high(low)valueofDCP indicatesasmall (large )\nreductionine-spinpolarization(i.e.,e-spinrelaxation ). Since\nthe e-spin relaxation rate depends on |∆Ee|(see Eq.1in the\nnextsection),thechangeintheX+DCPpresentsthepossibil-\nity of a direct measurements of the electron and nuclear spin\npolarizationsina couplede-nsystem.\nIt is noteworthy that the DCP observed in the other charge\nstates show different behaviors. The OHS observed in the\nX0and XX0PLs show changes similar to that observed in\nX+; this is one of the evidences that these PL lines originate\nfrom the same QD16. In contrast, the tendencies of DCP are\nquitedifferentforthese otherexcitoncomplexpeaks. TheX0\nDCP stays constant ( ∼0.7), thereby signifying independence\nfrom∆Ee, as shown in the lower panel of Fig. 1(c) (denoted\nby the gray symbols). This can be attributed to the contri-\nbution of the unpolarized X0supplied from the XX0. XX0\ndecays to X0by emittingσ+andσ−photons with identical\nprobabilities, and therefore, XX0DCP is basically zero (not\nshown here). The DCP of X0is approximately calculated as\n[(n+n/4)−n/4]/[(n+n/4)+n/4]∼0.67, if QDs are ex-\ncited under the power at which nelectron-holepairs (X0) are\ngenerated in each QD (in this case, XX0/X0=1/2 according\nto Poisson statics). Furtherstudies in this directionrequ iresa\nclose examination of the fine structures of the exciton level s\nbecauseofthe electron-holeexchangeinteraction.\nB. Experimental estimation ofcorrelation timeandelectro n\nspinrelaxation time\nWe estimated the keyquantities( τcandτs0) in the e-n spin\ndynamics from the experimental data. The e-spin relaxation\nrate underthe effectivemagnetic field in frequencyunit Ωe(=\n∆Ee//planckover2pi1)cangenerallyexpressedas13,14\n1\nτs=1\nτs0·1\n1+(Ωeτc)2. (1)\nThis equationrepresentsa Lorentzianshape with a full widt h\nat thehalfmaximum(FWHM)of2 /planckover2pi1/τcin energyandanam-\nplitude of 1/τs0. Here,τcandτs0correspond to the e-n cor-\nrelation time and the e-spin relaxationtime at Ωe=0, respec-\ntively. Asmentionedabove,thee-spinrelaxationrateappe ars\ndirectly in the reduction in X+DCP, and these key quantities\ncanbeobtainedfromthefittingwith theinverseofEq. 1.\n/X44/X43/X50 \n/X45/X6C/X65/X63/X74/X72/X6F/X6E/X20/X65/X6E/X65/X72/X67/X79/X20/X73/X70/X6C/X69/X74/X74/X69/X6E/X67/X20/X28 µ/X65/X56/X29/X30/X2E/X39 \n/X30/X2E/X38 \n/X30/X2E/X37 \n/X30/X2E/X36 \n/X30/X2E/X35 \n/X30/X2E/X34 \n/X2D/X36/X30 /X2D/X34/X30 /X2D/X32/X30 /X30 /X32/X30 /X20/X32/X20/X54\n/X20/X32/X2E/X35/X20/X54\n/X20/X33/X20/X54\n/X20/X33/X2E/X35/X20/X54\n/X20/X34/X20/X54\n/X20/X35/X20/X54/X20/X34/X2E/X35/X20/X54/X42/X7A\nFIG.2: X+DCPsasafunctionoftheelectronicenergysplittingatthe\ndifferentexternalfield. Symbolsandcolorsindicatetheexperi mental\ndata and corresponding Bz. Each of the data values was obtained\nfromthetime-resolvedmeasurementsofOHSandDCP.Theabse nce\nof the data points around ∆Ee∼−15µeV is attributed to the abrupt\nchanges inOHSandDCP.The solidcurve represents theLorent zian\nfittingwithawidthof ∼15µeV and anoffset of∼0.8.\nFigure2shows the DCPs as a function of ∆Eeat differ-\nent values of external field strength (2 T ≤Bz≤5 T). The\ndata were obtained from the time-resolved measurements, as\nshown in a later section of the paper (Fig. 3(b)). As clearly\nshown, a definite dip is observedat ∆Ee≃0. The Lorentzian\nfitting (FWHM∼15µeV) with an offset (∼0.8) depicted by a\nsolid curve in the figure was able to reproduce the entire ex-\nperimental data. It should be noted that the X+DCPs at the\ndifferentBzvaluesdepictauniquecurve,andthisfactjustifies\ntheassumptionthat τcandτs0areindependentof Bz.\nFirst, we estimate the e-n correlationtime τc. By using the\nrelationof FWHM =2/planckover2pi1/τc, we evaluatedthe correlationtime\nto beτc∼80 ps. The obtained τccoincides with the values\nthat Maletinsky et al.9(35 ps) and Braun et al.7(50 ps) used\nin the calculations to reproduce their observations in sing le\nIn(Ga)AsQDs. Tothebestofourknowledge,anexperimental\nestimation of this correlation time has not been reported th us\nfaralthoughτcofseveraltensofpicosecondshasbeenusedin\nthedatafitting. Inaddition,thevalueisinthesamemagnitu de\nas the X+decoherence time ( ∼43 ps) that was measured by\nFourierspectroscopyofthesame InAlAsQD17.\nSecondly, we focus on the e-spin relaxation time under a\nzero effective magnetic field. This characteristic time can be\nexpressed asτs0=τR//bracketleftBig\nSop/Sz(0)−1/bracketrightBig\n13, whereSopdenotes\nthe initial e-spin polarization injected into the QD ground\nstate, and Sz(0)is one when Beff=0. Here,τRdenotes the\nrecombination time, and it is found to be ∼0.75 ns by other\nindependent time-resolved measurements. From the DCP at\n∆Ee=0,Sz(0)is evaluated to be ∼0.6. In contrast, precise\nevaluation of Sopis difficult because the e-spin polarization\ncreated with theσ−excitation is lost partially during the en-4\nergy relaxation process from the wetting layer to the QD\nground state. For simplicity, Sopis replaced by the o ffset\nvalue of the DCP curve ( ∼0.8) because the e-spin relaxation\nprocess is strongly suppressed in the region of large Beffac-\ncording to Eq. 1. By using these values, we evaluated the\ne-spinrelaxationtimeunder ∆Ee=0tobeτs0∼3τR. Thisvalue\nisingoodagreementwiththeexcitonspinrelaxationtimeob -\ntained in other measurements for resonant and nonresonant\nexcitations18–20.\nThus far, we were able to estimate τcandτs0directly from\nthe experimental data. Next, it is required to determine the\nfactors influencing these characteristic times. Here, we co n-\nsider the effects of the randomly fluctuating Overhauser field\ninduced by the n-spin ensemble21. The random fluctuation\nofBN(denotedby∆BN)inducesadditionale-spinprecession,\nandthememoryofthee-spinpolarizationislostfromtheini -\ntialvalue. Accordingtothetraditionalspinprecessionmo del,\nthe longitudinal component of ∆BN(∆BN,/bardbl) induces a loss in\nthetransversecomponentofe-spinpolarization( i.e.,decoher-\nence). On the other hand, the transverse component of ∆BN\n(∆BN,⊥) gives rise to a loss in the longitudinal e-spin polar-\nization(i.e., relaxation). Assuminga randomvariablein NSP\nwith a Gaussian distribution of a width√\nN(N: the number\nof the nuclei in a QD), the fluctuationof BNcan be estimated\nas∆BN/simequalA/√\nNgeµB(ge: an isotropic electron g-factor, A:\nthe hyperfinecoupling constant). By using the typical value s\nfor a InAlAs QD ( ge,A,N)=(−0.37, 50µeV, 3×104), we\ncanroughlyestimatethefluctuationof BNtobe∆BN≈15mT.\nThecorrespondingdecoherencetime( T∗\n2,n)andtherelaxation\ntimeτen0aregivenasthe functionsof ∆BN.\nTo begin with, we compare T∗\n2,nand the observed correla-\ntiontimeτc22. Thee-spindecoherencetimeinducedby ∆BN,/bardbl\ncan be expressed as T∗\n2,n=/planckover2pi1//bracketleftbigg\nge\nzµB/radicalBig\n2∆B2\nN/3/bracketrightbigg\n,23and it is\nestimated to have an approximate value of ∼3 ns. The fact\nthat the estimated T∗\n2,nis fairly longer than the observed τc\nmayindicatethepresenceofanotherscatteringprocesswhi ch\nis responsiblefor the shorter decoherencetime T∗\n2. In our as-\nsumption,1/τcisgivenasalinearcouplingof1 /T∗\n2,nand1/T∗\n2\nand is dominantly determined by 1 /T∗\n2. One of the plausible\ncauses for 1/T∗\n2is the charge fluctuation in the QD region.\nLaiet al. have mentioned the e ffect of the e-spin tunneling\nbetweentheQDandthen-dopedlayerintheircharge-tunable\nQD structure24. AlthoughourQD sample hasno diodestruc-\ntureandthereisnointeractionwiththeelectrode,unliket heir\ncharge-tunable QD sample, similar phenomena such as car-\nriertunnelingbetweenQD anditssurroundingsoccurevenin\nour sample; consequently various types of the charge states\nappearduringthe exposuretime of the CCD detector( ∼0.1-1\ns),asshowninFig. 1(a). Thechangesofthechargestatemay\ninterrupt the e-spin precession, and it may a ffect the correla-\ntion time of an e-n spin system. Further0researchis require d\nregardingtheoriginof1 /T∗\n2.\nNext, we compare τen0and the observed e-spin relaxation\ntime under Beff=0. Assuming an uniform electron wave\nfunction in the QD region, the e-spin relaxation time in-\nduced by∆BN,⊥underBeff=0, which is given as τen0=/bracketleftBig\nNτc(A/N/planckover2pi1)2/bracketrightBig−1, to be∼50 ns25. Since the estimated τen0h(  ) \nX+(  ) \nh(  ) σ− σ+ σ− X+(  ) 1/ τs\n1/ τh1/ τR1/ τR\n/X45/X78/X70/X2E/X20/X44/X61/X74/X61/X20\n/X43/X61/X6C/X63/X2E /X2B/X20/X70/X75/X6D/X70 Bz=3.5  T\n/X30/X2E/X38 \n/X30/X2E/X34 \n/X30/X2E/X30 \n/X38/X30 \n/X36/X30 \n/X34/X30 \n/X32/X30 \n/X30/X44/X43/X50 /X4F/X48/X53/X20/X28 µ/X65/X56/X29\n/X54/X69/X6D/X65/X20/X28/X73/X65/X63/X2E/X29 /X30 /X32 /X34 /X36Bz~BN\nIzd        /dt /X28/X61/X29 \n/X28/X62/X29 \nFIG. 3: (a) Current dynamics model of e-spin system includin g X+\nand the single hole states, and the corresponding PL polariz ations.\n(b) Transient evolutions of DCP and OHS at Bz=3.5 T. The exci-\ntation polarizations are depicted in the upper side of the pa nel. The\nsolidcurves are the calculated resultsinthe coupled e-nsy stem, and\nthey were able to reproduce all the behaviors of the experime ntal\nresults. Right inset indicates a schematic of the n-spin pol arization\n(black solid curve) and depolarization (red dashed line) te rms for an\nexplanation of the transient OHS.\nis one order of magnitude longer than the observed τs0, we\nintroduce the e-spin flip term (1 /τe0) separately from the e-\nn flip-flop term (1/τen0), and the total e-spin relaxation rate\nobtained from the measurements is assumed to be 1 /τs0=\n1/τen0+1/τe0. In our assumption, these two spin relaxation\nprocesseshavethesame Ωedependencyasthatrepresentedby\na commonLorentzian function. Note that the large reduction\nin theDCP shownin Fig. 2cannotbe reproducedwithoutthe\nintroductionof1/τe0.\nInthissection,thee-ncorrelationtimeandthee-spinrela x-\nationtimeat Beff=0,twokeyquantitiesine-nspindynamics,\nwere evaluated from the experimental data. Since the esti-\nmated spin decoherence and relaxation time induced by the\nrandomfluctuationof BNareoffairlylongerdurationthanthe\nvaluesobtainedfrom the measurements,other scattering pr o-\ncesses are required to explain these durations. Although th e\nsource that decides τcandτs0has not yet been identified at\nthepresentstage,adirectevaluationprovidesthevaluabl ein-\nformationforthemodelingofe-nspindynamics,asdiscusse d\ninthenextsection.5\nC. Dynamics model of coupledelectron-nuclear spinsystem\nFinally, we test the validity of the dynamics model of the\ncouplede-n spin system. Thetemporalevolutionof the mean\nNSP/angbracketleftIz/angbracketrightisdescribedbythefollowingrateequation26:\nd/angbracketleftIz/angbracketright\ndt=1\nTNF[Q(/angbracketleftSz/angbracketright−S0)−/angbracketleftIz/angbracketright]−1\nTND/angbracketleftIz/angbracketright(2)\nwhereS0denotes the thermal e-spin polarization, 1 /TNF\nand 1/TNDdenote the n-spin polarization and depolarization\nrates27, respectively, and Q=I(I+1)/[S(S+1)] denotes\nthe momentum conversion coe fficient from the e-spin to n-\nspin system. It is noteworthythat the n-spin polarization r ate\nis also responsible for the e-spin relaxation, and it appear s in\nthe e-spin dynamicsvia 1 /τen0=N/TNF0(1/TNF0: the n-spin\npolarization rate at Beff=0). Although Eq. 2, with constant\nvaluesofthe averagede-spinpolarization /angbracketleftSz/angbracketright, hasexplained\nthe observed OHS qualitatively in previous studies, the ac-\ntual/angbracketleftSz/angbracketrightinthedynamicsisexpectedtochangealongwiththe\nevolutionof/angbracketleftIz/angbracketright. In ourmodelcalculation, /angbracketleftSz/angbracketright, which drags\nthe randomly-oriented n-spin ensemble to the highly polar-\nizedstate,isdeterminedbythedynamicsinthefollowingfo ur\nstates, as shown in Fig. 3(a): X+with the spin-up/downelec-\ntron (n↑andn↓: the populationsof the correspondingstates),\nandthe spin-up/downsingle holestates (similarlydenotedby\nn⇑andn⇓),and/angbracketleftSz/angbracketrightisgivenas/parenleftbign↑−n↓/parenrightbig//bracketleftbig2/parenleftbign↑+n↓/parenrightbig/bracketrightbig. These\nfourstatesareconnectedwiththeratesoftheopticalpumpi ng\nwithσ−light,theradiativerecombination(1 /τR),andthespin\nflip ofelectronandhole(1 /τsand1/τh).\nThe direct observation of the temporal evolutions of OHS\nand DCP can provide a better understanding of the e-n spin\ndynamics. Typical transients obtained from the target X+PL\natBz=3.5 T are shown in Fig. 3(b). In orderto set the initial\nNSPtozero,theexcitationpolarizationbeforethetimeofo ri-\ngin was modulated between σ+andσ−, with a frequency of\n10 Hz. The temporalevolutionof OHS ( =A/angbracketleftIz/angbracketright) is explained\nschematically by the di fference between the n-spin polariza-\ntion(ablacksolidcurve)andthedepolarization(areddash edline) rates, as shown in the right inset of Fig. 3(b). After\nswitchingtoσ−excitation,OHSincreasesgraduallyinthere-\ngion of small difference and increases explosivelyaroundthe\npeak of the n-spin polarization rate. Under this experimen-\ntal condition, the OHS jumps clearly to the saturated value\nwithin3s,and Bzcompensationvia BNisachievedwithinthe\nhomogeneous broadening of the n-spin polarization rate. At\ntheexactmomentoftheabruptincreaseintheOHS,theDCP\noftheX+PLdropssuddenlyfrom0.8to0.7. Thesolidcurves\nare the calculated results obtained from the abovementione d\ndynamics model. In the calculation, the key parameters ( τc\nandτs0) were in the same order as the experimentally evalu-\nated values reported in a previous section of the paper. The\nfactthatthecalculationcouldreproducetheobservedDCP a s\nwell asOHS showsthe validityofourdynamicsmodel.\nIV. CONCLUSION\nIn conclusion, we investigated the spin dynamics of the\ncoupled electron-nuclearspin system in a single InAlAs QD.\nThe DCP of X+PL, which is basically determined by e-\nspinpolarization,showedsynchronizedchangeswiththeel ec-\ntronic energy splitting, and this fact o ffers the possibility of\nNSPprobingviaX+DCPinaQDstructure. Bytakingadvan-\ntageofthisfeature,thecorrelationtimeofthee-nspinsys tem\nand the e-spin relaxation time, which play a crucial role in\nspin dynamics, were evaluated as τc∼80 ps andτs0∼3τR,\nrespectively. Theexperimentallyobtained τcagreeswellwith\ntheresultsobtainedbycalculationsorthedecoherencetim eof\nX+PL;further,τs0agreeswellwiththeexcitonspinrelaxation\ntime obtained from other experiments. Although the definite\nsourceofthesekeyquantitieswerenotidentifiedatthissta ge,\na direct estimation from measurements is very important in\nthe characterization of the e-n spin dynamics. The spin dy-\nnamics model used in this study successfully reproduces the\nobservationsof DCP as well as the OHS, and we believe that\nthemodelcansignificantlycontributetothe understanding of\ne-nspindynamics.\n∗Electronic address: r-kaji@eng.hokudai.ac.jp\n1J.M.Taylor,C.M.Marcus, andM.D.Lukin,Phys.Rev.Lett. 90,\n206803 (2003).\n2S.Muto,S.Adachi,T.Yokoi,H.Sasakura,andI.Suemune,App l.\nPhys.Lett. 87, 112506 (2005).\n3D.Gammon,Al.L.Efros,T.A.Kennedy,M.Rosen,D.S.Katzer,\nD. Park, S. W. Brown, V. L. Korenev, and I. A. Merkulov, Phys.\nRev. Lett. 86, 5176 (2001).\n4T. Yokoi, S. Adachi, H. Sasakura, S. Muto, H. Z. Song, T. Usuki ,\nS.Hirose, Phys.Rev. B 71, R041307 (2005).\n5R. Kaji, S. Adachi, H. Sasakura, and S. Muto, Appl. Phys. Lett .\n91, 261904 (2007).\n6R. Kaji, S. Adachi, H. Sasakura, and S. Muto, Phys. Rev. B 77,\n115345 (2008).\n7P.-F.Braun,B.Urbaszek, T.Amand,X.Marie,O.Krebs,B.Ebl e,\nA.Lemaitre, andP.Voisin, Phys.Rev. B 74, 245306 (2006).\n8A. I. Tartakovskii, T. Wright, A. Russell, V. I. Fal’ko, A. B.Van’kov, J. Skiba-Szymanska, I. Drouzas, R. S. Kolodka, M. S .\nSkolnick, P. W. Fry, A. Tahraoui, H.-Y. Liu, and M. Hopkinson ,\nPhys.Rev. Lett. 98, 026806 (2007).\n9P. Maletinsky, A. Badolato, and A. Imamoglu, Phys. Rev. B 75,\n035409 (2007).\n10E. A. Chekhovich, M. N. Makhonin, J. Skiba-Szymanska, A. B.\nKrysa, V. D. Kulakovskii, M. S. Skolnick, and A. I. Tartakovs kii,\nPhys.Rev. B 81, 245308 (2010).\n11I. A. Merkulov, Al. L. Efros, and M. Rosen, Phys. Rev. B, 65,\n205309 (2002).\n12R.Kaji,Doctoral Thesis (Hokkaido University. 2011)\n13Optical Orientation , Modern Problems inCondensed Matter Sci-\nences Vol. 8, Chaps. 2 and 5, edited by F. Meier and B. Za-\nkharchenya (North-Holland, NewYork, 1984).\n14Spin Physics in Semiconductors , Springer Series in Solid-State\nSciences Vol. 157, Chaps. 1 and 11, edited by M. I. Dyakonov\n(Springer, VerlagBerline Heidelberg, 2008).6\n15B. Eble, C. Testelin, P. Desfonds, F. Bernardot, A. Balocchi , T.\nAmand, A. Miard, A. Lemaˆ ıtre, X. Marie, and M. Chamarrom,\nPhys.Rev. Lett. 102, 146601 (2009).\n16H. Sasakura, R. Kaji, S. Adachi, and S. Muto, Appl. Phys. Lett .\n92, 041915 (2008).\n17S. Adachi, N. Yatsu, R. Kaji, and S. Muto, Appl. Phys. Lett. 91,\n161910 (2007).\n18T. Watanuki, S. Adachi, H. Sasakura, and S. Muto, Appl. Phys.\nLett.86, 063114 (2005).\n19H. Kumano, S. Kimura, M. Endo, H. Sasakura, S. Adachi, S.\nMuto, and I. Suemune, J. Nanoelectron. Optoelectron. 1, 39\n(2006).\n20R. Kaji, S. Adachi, T. Shindo, and S. Muto, Phys. Rev. B 80,\n235334 (2009).\n21P. Maletinsky, Optical Orientation of Nuclear Spins in an In-\ndividual Quantum Dot , Chap. 3 (S¨ udwestdeutcher Verlag f¨ ur\nHochschulschriften, 2008).\n22Althoughτcis limited by the shortest decoherence time of thecoupled e-n spin system, the decoherence time of the decoupl ed\ne-spinsystemisthought todetermine τcmainly.Thedecoherence\ntime of n-spin ensemble was estimated as the order of hundred s\nof milliseconds by the n-spindi ffusion constant measurements.\n23I.Merkulov, Physics Uspekhi 45, 1293 (2002)\n24C. W. Lai, P. Maletinsky, A. Badolato, and A. Imamoglu, Phys.\nRev. Lett. 96, 167403 (2006).\n25Strictly speaking, τen0also includes the factor of finite electron\noccupancy in QD ( fe≤1), andfeτen0is used in the calculation.\nThus,the estimated τen0of∼50ns maygive the lower limitof the\ne-spinrelaxation timedue tothe hyperfine interaction.\n26A. Abragam, The Principles of Nuclear Magnetism (Clarendon,\nOxford, 1961).\n27Forsimplicity,themagnetic fielddependence of1 /TNDisignored\nhere and it is not necessary to introduce this dependence at t he\npresent stage."    },    {        "title": "0806.1220v1.Quantum_Fluctuation_Driven_Coherent_Spin_Dynamics_in_Small_Condensates.pdf",        "content": "arXiv:0806.1220v1  [cond-mat.other]  6 Jun 2008Quantum-Fluctuation-Driven Coherent Spin Dynamics in Sma ll Condensates\nXiaoling Cui1,2, Yupeng Wang1and Fei Zhou2\n1Beijing National Laboratory for Condensed Matter Physics a nd Institute of Physics,\nChinese Academy of Sciences, P. O. Box 603, Beijing 100190, C hina\n2Department of Physics and Astronomy, The University of Brit ish Columbia, Vancouver, B. C., Canada V6T1Z1\n(Dated: November 19, 2018)\nWe have studied quantum spin dynamics of small condensates o f cold sodium atoms. For a\ncondensate initially prepared in a mean field ground state, w e show that coherent spin dynamics are\npurelydriven by quantum fluctuations of collective spin coordinat es and can be tuned by quadratic\nZeeman coupling and magnetization. These dynamics in small condensates can be probed in a high-\nfinesse optical cavity where temporal behaviors of excitati on spectra of a coupled condensate-photon\nsystem reveal the time evolution of populations of atoms at d ifferent hyperfine spin states.\nRecently, single-atom detection in optical cavities has\nbeen realized in experiments by having atoms and cav-\nity photons in a strongly coupling regime[1, 2]. This re-\nmarkableachievementhasbeenappliedtostudyoptically\ntransported atoms in cavities[3]; furthermore the cou-\npling between a small Bose-Einstein condensate (BEC)\nand cavity photons and resultant collective excitations\nhave also been successfully investigated[4]. The sensitiv-\nity that a cavity-based atom detector has, together with\na translating optical lattice which can effectively trans-\nport ultra cold atoms from a magnetic-optical trap to\na cavity make it possible to study the physics of small\nBECs. Especially, this potentially opens the door to\nexplore coherent dynamics of ultra-cold atoms in rela-\ntively small condensates. The physics of BECs of small\nnumbers of atoms can be qualitatively different from\nthe physics of big condensates and represents a new do-\nmain of cold-atom research. In small condensates, vari-\nous intrinsic beyond-mean-fielddynamics can be relevant\nwithin an experimentally accessible time scale. These\nnew physical phenomena however have been quite diffi-\ncult to study using the standard absorption-imaging ap-\nproach to cold atoms because of relatively fewer atoms\nare involved in small condensates. Cavity electrodynam-\nics in a strong coupling regime and high sensitivities to\nintra-cavity atoms on the other hand are ideal for in-\nvestigating small condensates where the beyond-mean-\nfield dynamics are mostly visible. In this letter, we fo-\ncus on the basic concepts of beyond-mean-field coherent\nspin dynamics in BECs with typically a few tens to a\nfew hundreds of atoms and detailed analysis of detect-\ning these fascinating properties of small condensates in\noptical cavities with high-finesse. Research on this sub-\nject could substantiallyadvance ourunderstandingofthe\nnature of quantum-fluctuation dynamics[5], in this par-\nticular case, dynamics purely driven by fluctuations with\nwavelengths of the size of condensates. Secondly, results\nobtained can help to better recognize limitations of pre-\ncise measurements of various interaction constants based\non mean-field coherent dynamics[6]. Thirdly, our results\nshould shed some light on the feasibility of investigating\nfluctuation dynamics of small condensates using opticalcavities and also pave the way for future studies of dy-\nnamics of coupled small condensates.\nTounderstandspindynamicsofasmallcondensate, we\nfirst study the evolution of a condensate of Nhyperfine\nspin-one sodium atoms which is initially prepared in a\nmean field ground state,\n|n/angbracketright=(n·ψ†)N\n√\nN!|0/angbracketright. (1)\nHerenis a unit director and three components of ψ†,\nψ†\nα,α=x,y,zare creation operators for three spin-one\nstates,|x/angbracketright= (|1/angbracketright −| −1/angbracketright)/√\n2,|y/angbracketright= (|1/angbracketright+| −1/angbracketright)/i√\n2\nand|z/angbracketright=|0/angbracketrightrespectively. And in this representation,\nSα=−iǫαβγψ†\nβψγis the total spin operator. States in\nEq.1 with n=ezminimize the interaction energy of the\nfollowing Hamiltonian for spin-one atoms in the presence\nof a quadratic Zeeman coupling along the z-direction,\nH=c2\nNS2+q(ψ†\nxψx+ψ†\nyψy). (2)\nHerec2is a spin interaction constant and qis the\nquadratic Zeeman coupling[7, 8, 9, 10]. Mean field\nground states are stationary solutions to the multi-\ncomponentGross-Pitaevskiiequationsforspin-oneatoms\nand dynamics of these initial states demonstrated below\nare therefore a beyond-mean field phenomenon. When\nderiving Eq.2 for a trapped condensate, we assume\nthat spin dynamics are described by a single mode, i.e.\nψα(r,t) =/radicalbig\nρ(r)ψα(t); for a small condensate of less\nthan one thousand weakly interacting atoms, this ap-\nproximation is always valid. c2is typically a few nano\nkelvin for sodium atoms; q= (µBB)2/(4∆hf) and the\nhyperfine splitting is ∆ hf= (2π)1.77GHz(µBis the\nBohr magneton and ¯ his set to be unity).\nTo illustrate the nature of non-mean-field dynamics\nand crucial role played by quantum fluctuations, we ex-\npand the full Hamiltonian in Eq.2 around a mean field\nground state. In the lowest order expansion, we approx-\nimateψ†≈√\nNez+ψ†\nxex+ψ†\nyey, andψ†\nx,yare much2\nless than√\nN; the Hamiltonian then can be expressed in\nterms of the bilinear terms\nHB=/summationdisplay\nα=x,yq+4c2\n2NP2\nα+qN\n2θ2\nα+... (3)\nwhere forα=x,y,θα=1√\n2N(ψ†\nα+ψα) andPα=\ni/radicalBig\nN\n2(ψ†\nα−ψα) are pairs of conjugate operators which\nsatisfy the usual commutation relations [ θα,Pβ] =iδα,β.\nSemiclassically, collective coordinates θα,α=x,yrep-\nresent projections of ψ†or order parameter nin thexy\nplane, and Px(y)∼Sy(x)is the spin projection along the\ny(x)-direction. The bilinear Hamiltonian is equivalent to\na harmonic oscillator moving along the direction of θx,y\nwith a mass meff=N\nq+4c2, a harmonic oscillator fre-\nquencyω=/radicalbig\nq(q+4c2) and effective spring constant\nqN; the mass at q= 0 is induced by scattering between\natoms. The excitation spectrum is En= (n+ 1/2)ω,\nn= 0,1,2.... Whenq= 0, the Hamiltonian describes a\nparticle moving in a free space.\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48 /s49/s48 /s50/s48 /s51/s48/s48/s49/s50\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s51/s50 /s54/s52 /s49/s50/s56/s50/s52/s56\n/s40/s97/s41/s78\n/s48/s47/s78\n/s99\n/s50/s116/s78\n/s48/s47/s78\n/s32/s113/s61/s48\n/s32/s113/s61/s99\n/s50/s47/s52/s48\n/s32/s113/s61/s99\n/s50/s47/s50/s48\n/s32/s113/s61/s99\n/s50/s47/s49/s48\n/s32/s32\n/s32/s32/s40/s49/s48/s45/s52\n/s41\n/s32/s32\n/s32/s109/s61/s48/s46/s57/s56/s40/s98/s41\n/s32/s32\n/s32/s109/s61/s48/s46/s49\n/s32/s109/s61/s48/s46/s50/s32\n/s32/s78/s99\n/s50/s116\n/s99\n/s32/s32\nFIG. 1: a) (color online) Time evolution of N0(t), the atom\npopulation at |1,0/angbracketrightstate for different quadratic Zeeman cou-\nplingq. Initially, all N= 200 atoms occupy |1,0/angbracketrightstate which\ncorresponds to a mean field ground state. The inset is tc, the\ntime for the first dip in the q= 0 data, as a function of the\nnumber of atoms N. b) Time evolution of N0(t) for different\nmagnetization m(hereq= 0). All initial states are again\nmean-field ground states for given m. Inset is for m= 0.98.\nIn this and Fig.2,3, c2= (2π)50Hz.\nIn the ground state of the bilinear Hamiltonian of\nEq.3,/angbracketleftθα/angbracketright=/angbracketleftPα/angbracketright= 0 and nand/angbracketleftS/angbracketrighthave no projec-\ntions in the xyplane. However, quantum fluctuations ofθx,y-coordinates in the ground state can be estimated as\n/angbracketleftθαθα/angbracketright=1\n2N/radicalBig\nq+4c2\nq. This is a measure of how strongly\nnfluctuatesinthe xy-plane. Asexpected, thesequantum\nfluctuations are substantial only when qis small and are\nsuppressed by a quadratic Zeeman field which effectively\npins the order parameter along the z-direction. A direct\ncalculation also shows that the amplitude of quantum\nfluctuations /angbracketleftθ2\nα/angbracketrightMFin themean field ground state de-\nfined in Eq.1 is 1 /2N. This indicates that the mean field\ngroundstateis agoodapproximationonlywhen q≫4c2.\nOn the otherhand, as qdecreasesand the effective spring\nconstant gets smaller, the deviation becomes more and\nmore severe. When qapproaches zero, quantum fluctua-\ntionsθαin the harmonic oscillator ground state become\ndivergent implying that the mean field ground state is no\nlonger a good approximation.\nIndeed, the the energy of mean field ground state is\nEMF=q\n2+c2which is much higher than1\n2ωwhenq≪\nc2; such a state corresponds to a highly excited wave\npacket, because of a relatively narrow spread along θα-\ndirections and consequently an enormous kinetic energy\nassociated with momenta Pα. We therefore expect that\ndynamics in this limit could dramatically differ from a\nstationary solution. Since the total number of atoms N\nis equal to/summationtext\nαψ†\nαψα, the population of atoms at |z/angbracketright(or\n|1,0/angbracketright) stateN0=/angbracketleftψ†\nzψz/angbracketrightis directly related to quantum\nfluctuations of θαandPα,\nN0=N+1−/summationdisplay\nα/parenleftbiggN\n2/angbracketleftθ2\nα/angbracketright+1\n2N/angbracketleftP2\nα/angbracketright/parenrightbigg\n.(4)\nEq.4 shows that the time evolution of N0(t) is effectively\ndriven by quantum fluctuations in θαandPα; a study of\nN0(t) probes underlying quantum- fluctuation dynamics.\nFor an initial state prepared in a mean field ground\nstate with n=ezwhere all atoms condense in |1,0/angbracketright\nstate, one finds that /angbracketleftθ2\nα/angbracketright=1\n2Nand/angbracketleftP2\nα/angbracketright=N\n2. The evo-\nlution of such a symmetric Gaussian wave packet subject\nto the bilinear Hamiltonian can be solved exactly using\nthe standard theory for harmonic oscillators. The wave\npacket will remain to be a Gaussian one with the width\noscillating as a function of time. Qualitatively, because\nof the symmetry, only harmonic states with even-parity\nare involved in dynamics and therefore the oscillation\nfrequency is 2 ω. Furthermore during oscillations, the ki-\nnetic energy stored in initial wave packets is converted\ninto the potential one and vice versa . Especially when\nq≪c2, oscillations are driven by the enormous initial\nkinetic energy associated with Pα; the oscillation ampli-\ntude can be estimated by equaling the total energy EMF\nto the potential energy which leads to /angbracketleftθ2\nα/angbracketright ∼c2/(Nq).\nA straightforwardcalculation yields the time dependence\nof/angbracketleftθ2\nα/angbracketrightand/angbracketleftP2\nα/angbracketrightthat leads to3\nN0\nN= 1−8c2\n2\nq(q+4c2)Nsin2wt. (5)\nThe oscillating term in Eq.5 shows the deviation from\nthe stationary behavior due to quantum fluctuations in\nθα-coordinates. The deviation is insignificant when qis\nnot too small; howeverwhen qis of the order of c2/N, we\nexpect that the non-mean field dynamics becomes very\nvisible. Note that the approach outlined here neglects\nall higher order anharmonic interactions and therefore is\nonly valid when the relative amplitude of fluctuations is\nsmall; that is when q≫c2/N.\nWhenqapproaches zero, the short time dynamics fol-\nlowingthe bilinearHamiltonianisequivalenttoaparticle\nof a massmeff=N/4c2that is initially localized within\na spread /angbracketleftθ2\nα/angbracketright= 1/2Nhaving a ballistic expansion with\na typical velocity give as /angbracketleftv2\nα/angbracketright= 8c2\n2/N. The time de-\npendence of spread /angbracketleftθ2\nα/angbracketrighttherefore is 1 /2N+(8c2\n2/N)t2.\nSo att∼√\nN/c2, the number of atoms not occupy-\ning the initially prepared |1,0/angbracketrightstate becomes of order\nofN. This limit was first addressed by Law et al.in the\ncontext of four-wave-mixing theory[9], and also in early\nworks[11, 12]; to describe the physics after this charac-\nteristic time scale requires analysis of full quantum dy-\nnamics. This time scale however becomes quite long for\na few million atoms which makes it difficult to observe\nquantum dynamics in large condensates.\nIn the following, we are going to present our numeri-\ncal results on dynamics and focus on its dependence on\nquadratic Zeeman coupling qand magnetization m. For\na condensateof N= 200atoms, we numerically integrate\nthe time-dependent N-body Schrodinger equation of the\nquantum Hamiltonian in Eq.2. The time evolution of N0\ndriven by quantum fluctuations is shown in Fig.1a). As q\nincreases far beyond 0 .2c2,N0oscillates as a function of\ntime with frequency 2 ωand the amplitude of oscillations\ndecreases; the damping is not visible over tens of oscilla-\ntions. When qis below 0.2c2, anharmonic effects become\nsubstantial and oscillations are no longer perfect; when\nq=c2/40, oscillations are strongly damped after a few\ncycles and revived afterwards. For q= 0,N0drops to a\nminimum of about 0 .38Nwhent=tc= 0.53√\nN/c2and\nremains to be a constant before reviving to be 0 .8Nat\nabout 10tc. For sodium atoms with a typically density\n2×1014cm−3,c2= (2π)50Hz;tc= 23.8msforN= 200\nand increases to a few seconds when Nreaches 2 ×106.\nWe have also studied the quantum dynamics of a mean\nfield condensate with a finite magnetization along the z-\ndirection, m=mez. States which minimize the mean\nfield energy of the Hamiltonian in Eq.2 with q= 0 are\n|m/angbracketright=[(cosηex+isinηey)·ψ†]N\n√\nN!|0/angbracketright (6)where sin2η=m,m(∈[−1,1]) is the normalized magne-\ntization. By expanding the Hamiltonian around these\nmean field states, one obtains a harmonic oscillator\nHamiltonian defined in terms of conjugate operators,\nθz=1√\n2N(ψ†\nz+ψz) andPz=i/radicalBig\nN\n2(ψ†\nz−ψz). The effec-\ntive mass is meff=N\n2c2(1+√\n1−m2)and the harmonic os-\ncillator frequency ω= 2|m|c2. States shown in Eq.6 have\na narrow width along the direction of θz,/angbracketleftθ2\nz/angbracketright= 1/2N\nand thereforecarrylargeconjugate momenta Pz; the cor-\nrespondinglargekinetic energydrivesauniquenon-mean\nfield quantum spin dynamics. The harmonic expansion\nagain is only valid when mis large and fluctuations are\nweak. Simulations of the full Hamiltonian have been car-\nried out in this case; in Fig.1b), we show the time depen-\ndence ofN0(t) for different magnetization. Only when m\nisclosetounity, weaklydampedoscillationsareobserved.\n/s48\n/s53/s48\n/s49/s48/s48\n/s49/s53/s48\n/s50/s48/s48/s45/s56/s48\n/s45/s52/s48\n/s48\n/s52/s48\n/s56/s48/s45/s49/s52/s52/s45/s49/s51/s54/s45/s49/s50/s56/s45/s49/s50/s48/s48\n/s53/s48\n/s49/s48/s48\n/s49/s53/s48\n/s50/s48/s48/s45/s56/s48\n/s45/s52/s48\n/s48\n/s52/s48\n/s56/s48/s45/s52/s53/s45/s52/s48/s45/s51/s53/s45/s51/s48/s45/s50/s53\n/s49/s54/s50/s49/s54/s52/s49/s54/s54\n/s45/s52/s50/s45/s51/s54/s45/s51/s48\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s45/s49/s51/s50/s45/s49/s50/s56/s45/s49/s50/s52/s48\n/s53/s48\n/s49/s48/s48\n/s49/s53/s48\n/s50/s48/s48/s45/s56/s48\n/s45/s52/s48\n/s48\n/s52/s48\n/s56/s48/s49/s54/s48/s49/s54/s53/s49/s55/s48/s49/s55/s53\n/s99\n/s50/s116\n/s67/s47/s50\n/s40/s100/s41/s80/s51/s47/s50\n/s40/s99/s41/s99\n/s50/s116\n/s67/s47/s50/s80/s50/s47/s50\n/s32/s112/s49\n/s32\n/s32/s32\n/s112/s50\n/s32\n/s32/s112/s51/s32\n/s32\n/s99\n/s50/s116/s80/s47/s50 /s32/s40/s77/s72/s122/s41\n/s32\n/s32/s40/s98/s41/s32\n/s32/s32\n/s67/s47/s50\n/s40/s77/s72/s122/s41/s80/s49/s47/s50\n/s40/s77/s72/s122/s41\n/s40/s97/s41/s99\n/s50/s116\nFIG. 2: (a,b,c) (color online) Eigenfrequencies ∆ pas a func-\ntionoftfordifferentdetuning∆ cwhentherelativepopulation\nat state|1,0/angbracketright,ρ0(t) evolves. At t= 0, all atoms occupy |1,0/angbracketright\nstate and N= 200. d) ∆ pas a function of time tfor ∆c= 0.\nWe propose a method to probe quantum spin dynam-\nics of a small condensate of spin-one sodium atoms using\ncavity quantum electrodynamics. For a Bose-Einstein\ncondensate with N atoms coupled to a quantized field\nof a cavity, a single cavity photon can coherently inter-\nact with atoms which leads to a collective coupling of\ng√\nN[13]. In experiments[3, 4], atoms are transported\ninto a cavity via a moving optical lattice; excitations are\nmeasured by individual recordings of cavity transmission\nwhen frequencies of an external probe light are scanned.\nHere we consider a multi-component BEC coupled to a\nsingle cavity mode; the eigenfrequencies of the coupled\nsystem uniquely depend on populations at three hyper-\nfine states. By measuring the energy spectrum of this\ncoupled system, one obtains temporal behaviors of atom4\npopulations at different states.\nWe restrictourselvesto excitationswhich involveasin-\ngle cavity photon interacting with atoms in a BEC. We\nstudy atomic transitions from 3 S1\n2→3P1\n2in sodium\natoms. The Hamiltonian consists the following terms,\nHcavity=/summationdisplay\ni¯hwgiˆg†\niˆgi+/summationdisplay\nj¯hwejˆe†\njˆej+/summationdisplay\np¯hwcˆc†\npˆcp\n−i¯h/summationdisplay\np/summationdisplay\ni,jgp\nijˆe†\njˆcpˆgi+h.c., (7)\nwhereilabels three states |F= 1,mF= 0,±1/angbracketrightin 3S1\n2\norbital and jeight states |F′= 1,mF′/angbracketright,|F′= 2,mF′/angbracketrightin\n3P1\n2orbital. ˆg†\niand ˆe†\njcreate atoms in one of 3 S1\n2and\n3P1\n2states respectively with corresponding frequencies\nwgi,wej. ˆc†\npcreates a photon with frequency wcand po-\nlarizationpin the cavity mode. gp\nij(=Dp\nij/radicalbig\n¯hwc/2ǫ0V)\nis the coupling strength for a transition i→jdriven by a\ncavity photon with polarization p, which depends on the\ndipole matrix element Dp\nij, the effective mode volume V.\n/s49/s54/s50/s49/s54/s51\n/s45/s51/s50/s45/s51/s49/s45/s51/s48\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s45/s49/s51/s50/s45/s49/s51/s49/s45/s49/s51/s48/s112/s51/s47/s50\n/s112/s50/s47/s50\n/s112/s49/s47/s50 /s40/s77/s72/s122/s41\n/s32\n/s99\n/s50/s116\n/s32/s32\n/s32/s32\n/s32/s32\nFIG. 3: Eigenfrequencies ∆ pas a function of time tdriven\nby dynamics of population ρ0(t) in the presence of quadratic\nZeeman coupling q= 0.05c2(orB= 95mG). Again at t= 0,\nall atoms occupy |1,0/angbracketrightstate and N= 200, ∆ c= 0.\nFor simplicity, we set the energy of 3 S1\n2states to\nbe zero, i.e. wgi= 0; the energy of excited states is\nw1,2\ne=wa±∆ with 2∆ being the hyperfine splitting\nbetweenF′= 2 andF′= 1 states. For atomic tran-\nsitions induced by left-circularly( σ+) polarized photons,\nthe selection rule is ∆ F= 0,±1,∆mF= 1. In a cavity,\na state with a cavity photon (labeled as 1c), NmFatoms\nat 3S1\n2|1,mF/angbracketrightstates and no atoms at excited states (la-\nbeled as0 j) is expressedas |1c;N1,N0,N−1;0j/angbracketright; it iscou-\npled to the following states with one of atoms excited to\n3P1\n2states (labeled 1 F′) and no cavity photons (as 0 c),\n|0c;N1−1,N0,N−1;1F′=2/angbracketright,|0c;N1,N0−1,N−1;1F′=1/angbracketright,\n|0c;N1,N0−1,N−1;1F′=2/angbracketright,|0c;N1,N0,N−1−1;1F′=1/angbracketright,\n|0c;N1,N0,N−1−1;1F′=2/angbracketright. We diagonalize the Hamil-\ntonian matrix and obtain six eigenfrequencies ωpfor this\ncoupled system. Three are 3 P1\n2orbitals without mix-\ning with 3S1\n2states,wp=wa±∆; the other three de-\npend on relativepopulations of atoms at each spin state,ρmF=NmF/N,mF= 0,±1. The latter three eigen\nfrequencies are determined by the eigen value equation,\n(∆p−∆c)(∆2\np−∆2)−∆pNg2\n1F1−∆Ng2\n1F2= 0.(8)\nHere ∆ p=wp−wa,m=ρ+1−ρ−1is the normal-\nized magnetization, and g1is the coupling between 3 S1\n2\n|F= 1,mF= 1/angbracketrightand 3P1\n2|F′= 2,mF′= 2/angbracketrightbyσ+\nlight;F1= (2+m)/3 andF2= (1+m)/2−ρ0/6. Ap-\nparently eigenfrequencies ∆ p1,p2,p3are a function of ρmF\nand therefore can be used to probe the variation in ρ0,±1\ndue to coherent spin dynamics. ∆ p1,p2,p3depend on a\ndimensionless parameter r=√\n6\n3√\nNg1\n∆. When detuning\n∆c= 0 and asr→ ∞, ∆p1,p2,p3arearound 0 ,±√\n6\n3√\nNg1\nrespectively. Eigenfrequency ∆ p2varies from −3∆/4 to\n−∆/2 whenρ0increases from 0 to 1; the variation am-\nplitudeδ= ∆p(ρ0= 1)−∆p(ρ0= 0) reaches a satu-\nrated value ∆ /4. For sodium atoms, ∆ = (2 π)94.4MHz;\ncavity parameters are chosen according to Ref.[4] and\ng1= (2π)10MHz. In Fig. (2), we show the evolution\nof ∆pin time for different detuning ∆ cwhen atoms are\ninitially prepared at state |1,0/angbracketrightof 3S1/2. The evolution\nof ∆p(t) which can be probed by a σ+beam maps out\npopulation N0(t) driven by underlying quantum fluctu-\nations. In Fig. (3), we further show the time depen-\ndence of ∆ pdue to oscillatory quantum spin dynamics\nforq= 0.05c2(B≈95mG),N= 200 and ∆ c= 0.\nIn conclusion, we have illustrated the nature of co-\nherent spin dynamics driven by quantum-fluctuations in\nsmall condensates. The time evolution of population of\natoms at different hyperfine spin states is shown to re-\nveal intrinsic dynamics of quantum fluctuations of or-\nder parameters and spin projections. These dynamics\ncan be probed by studying eigenfrequencies of a coupled\ncondensate-photon system in a high-finesse optical cav-\nity available in laboratories. We thank Gerard Milburn,\nJunliang Song for stimulating discussions. This work is\nin part supported by NSFC, 973-Project (China), and\nNSERC (Canada), CIFAR and A. P. Sloan foundation.\n[1] H. Mabuchi, Q. A. Turchette, M. S. Chapman and H. J.\nKimble, Optics Letters 21, 1393 (1996).\n[2] C. J. Hood et al., Phys. Rev. Lett. 80, 4157 (1998).\n[3] J. A. Sauer et al., Phys. Rev. A 69, 051804(2004).\n[4] F. Brennecke et al., Nature 450, 268 (2007).\n[5] Other fluctuation-driven spin dynamics were studied in\nJ. L. Song, F. Zhou, Phys. Rev. A 77, 033628 (2008).\n[6] M. S. Chang et al., Nature Physics 1, 111 (2005).\n[7] T. L. Ho, Phys. Rev. Lett. 81, 742 (1998).\n[8] T. Ohmi, K. Machinda, J.Phys.Soc.Jpn. 67, 1822 (1998).\n[9] C. K. Law et al., Phys. Rev. Lett. 81, 5257 (1998).\n[10] J. Stenger et al., Nature 396, 345(1998)\n[11] F. Zhou, Phys. Rev. Lett. 87, 080401 (2001).\n[12] R. B. Diener and J. L. Ho, cond-mat/0608732(2006).\n[13] M. Tavis, F. W. Cummings, Phys. Rev. 170, 379 (1968)."    },    {        "title": "1312.5529v1.Dynamics_of_a_localized_spin_excitation_close_to_the_spin_helix_regime.pdf",        "content": "Dynamics of a localized spin excitation close to the spin-helix regime\nG. Salis,1,\u0003M. P. Walser,1P. Altmann,1C. Reichl,2and W. Wegscheider2\n1IBM Research{Zurich, S aumerstrasse 4, 8803 R uschlikon, Switzerland\n2Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland\nThe time evolution of a local spin excitation in a (001)-con\fned two-dimensional electron gas\nsubjected to Rashba and Dresselhaus spin-orbit interactions of similar strength is investigated the-\noretically and compared with experimental data. Speci\fcally, the consequences of the \fnite spatial\nextension of the initial spin polarization is studied for non-balanced Rashba and Dresselhaus terms\nand for \fnite cubic Dresselhaus spin-orbit interaction. We show that the initial out-of-plane spin\npolarization evolves into a helical spin pattern with a wave number that gradually approaches the\nvalueq0of the persistent spin helix mode. In addition to an exponential decay of the spin po-\nlarization that is proportional to both the spin-orbit imbalance and the cubic Dresselhaus term,\nthe \fnite width wof the spin excitation reduces the spin polarization by a factor that approaches\nexp(\u0000q2\n0w2=2) at longer times.\nI. INTRODUCTION\nThe spin-orbit interaction (SOI) in atoms is a relativis-\ntic correction of the orbital energy levels. It can be under-\nstood as the interaction of the electron spin with a mag-\nnetic \feld that has its origin in the Lorentz-transformed\nelectrostatic \feld of the atomic core. In a crystalline\nsolid, this interaction in\ruences the band energies; specif-\nically, it leads to a spin splitting of electronic states with\n\fnite momentum in crystals with an inversion asymme-\ntry. A crystallographic inversion asymmetry as it exists,\ne.g., in bulk zincblende semiconductors leads to the Dres-\nselhaus SOI1for conduction-band electrons. In addition,\na Rashba SOI can be induced2,3in a layered structure\nby applying an electric \feld perpendicular to the layers.\nFor electrons con\fned in a quantum well, the Dresselhaus\nspin splitting depends on the quantum con\fnement4. For\na typical material like a GaAs-based quantum well, it\nis on the order of 100 \u0016eV5at the Fermi energy, which\ntranslates into a large e\u000bective magnetic \feld of 5-10 T.\nThis makes SOI an interesting tool for coherent manip-\nulation of electronic spin states in solids. On the other\nhand, the large \feld is a signi\fcant source for the decay\nof the spin polarization by the so-called Dyakonov-Perel\nmechanism6. It has been shown theoretically7,8that by\nbalancing the Dresselhaus and the Rashba contribution\nto SOI, the interaction attains a special symmetry with\nrespect to the size and direction of the electron momen-\ntum and leads to the preservation of a helical spin mode.\nFor (001)-oriented GaAs-based quantum wells, the spin\npolarization of this helical mode rotates about an in-\nplane axis when the position is varied along the perpen-\ndicular in-plane direction. The measured decay rate of\nimprinted spin gratings of variable wave number9follows\nthe theoretical prediction with a minimum decay rate at\nthe wave number q0of a persistent spin helix8.\nIn a typical experimental con\fguration, spin polariza-\ntion is injected locally into the non-magnetic semicon-\nductor by, e.g., spin injection contacts10or optical ori-\nentation11. It is interesting to consider the evolution of\nsuch spin polarization into a spin helix pattern. In gen-eral, the initial spin polarization can be described by a\nsuperposition of helical spin modes with di\u000berent wave\nvectors q= (qx;qy) and decay rates \u0000( qx;qy)12. Mea-\nsurements show that for a spatially con\fned spin excita-\ntion, this superposition evolves into a helical spin pattern\nthat di\u000busively expands with time and whose wave num-\nber evolves towards q013. For a balanced SOI, where the\nRashba SOI coe\u000ecient \u000bis equal to \f=\f1\u0000\f3(with\n\f1and\f3the linear and cubic Dresselhaus coe\u000ecients as\nde\fned below), the evolution of a spatially delta-shaped\nspin excitation can be analytically described by the prod-\nuct of a Gaussian function and a helical mode with wave\nnumberq012. In real situations, the spin excitation has\na \fnite extension and SOI may not be balanced. To un-\nderstand the experimentally observed spin dynamics, it is\nimportant to analyze to what extend such non-idealities\nalter this description.\nHere, we theoretically study the e\u000bects that occur\nunder realistic experimental conditions where (i) the\nRashba and Dresselhaus SOI are not balanced and cubic\nDresselhaus terms are present, (ii) the initial out-of-plane\nspin polarization has a \fnite spatial extension along the\ndirectionyof the helical spin precession but is constant\nalong the perpendicular direction x, and (iii) spin po-\nlarization is localized in both in-plane directions. The\nmodel derived bridges the gap between a delta-shaped\nand a spatially broad excitation, i.e., between the forma-\ntion of a long-lived helical spin mode and a spatially ho-\nmogeneous spin decay described by the Dyakonov-Perel\nmechanism. We \fnd an exponential decay with a rate\nproportional to Ds((\u000b\u0000\f)2+ 3\f2\n3), whereDsis the spin\ndi\u000busion constant. An initial spin polarization Sz(y) with\n\fnite extension along ywill be further reduced because\nof the transition of the relevant wave numbers from zero\ntoq0. This latter e\u000bect is also responsible for a grad-\nual decrease of the period of the evolving helical spin\npolarization. Analytical results are derived on the as-\nsumption of an initial spin polarization Sz(y) that is in-\ndependent of x. If the initial spin polarization is also\ncon\fned along x, the spin eigenmodes and their disper-\nsion exhibit anticrossings for \u000b6=\f. We show that the\nlocalization along x, however, does not appreciably alterarXiv:1312.5529v1  [cond-mat.mes-hall]  19 Dec 20132\nthe behavior of the spin evolution apart from a trivial\ndi\u000busive expansion along that direction. Finally, we ver-\nify the predicted transients of the polarization amplitude\nand the mode wave number in an experiment in which\nspin polarization in a GaAs quantum well is initialized\nin con\fned areas of di\u000berent extensions.\nII. MODEL\nA. Spin-Di\u000busion equation\nWe \frst de\fne the coordinate system and the spin-\norbit coe\u000ecients. We consider a two-dimensional elec-\ntron gas con\fned along the [001] crystalline direction z\nof a zincblende crystal, such as GaAs. We de\fne the two\nin-plane directions xandyalong [1 10] and [110], respec-\ntively. The Hamiltonian for an electron with in-plane\nwave vector k= (kx;ky) and e\u000bective mass m\u0003is\nH=~2k2\n2m\u0003+~\n2\nSO\u0001\u001b; (1)\nwith the SOI de\fned by\n\nSO=2\n~0\nBB@\u0010\n\u000b+\f1+ 2\f3k2\nx\u0000k2\ny\nk2\u0011\nky\u0010\n\u0000\u000b+\f1\u00002\f3k2\nx\u0000k2\ny\nk2\u0011\nkx\n01\nCCA: (2)\nThe spin is represented by the three Pauli matrices\n\u001b= (\u001b1;\u001b2;\u001b3). The Dresselhaus SOI coe\u000ecients \f1\nand\f3are related to the bulk Dresselhaus coe\u000ecient \r\nby\f1=\u0000\rhk2\nziand\f3=\u0000\rk2=4. The expectation\nvalue ofk2\nzwith respect to the QW ground-state enve-\nlope wave-function is denoted by hk2\nzi. Whereas \u000band\n\f1give rise to SOI that is linear in k=jkj, the cubic co-\ne\u000ecient\f3itself depends quadratically on kand in total\naccounts for SOI that is cubic in k. We consider a degen-\nerate electron gas with kBT\u001c~2k2\nF=(2m\u0003), wherekFis\nthe Fermi wave number and kBthe Boltzmann constant.\nAlso, we assume that the SOI is small compared with\nthe Fermi energy and that the initial spin polarization\ndensity is small compared with the electron sheet den-\nsity. The relevant electronic states are then centered at\nthe Fermi energy, and kcan be replaced by kFin Eq. (2)\nand in the de\fnition of \f3.\nWe investigate a situation close to the balanced SOI\nof a perfect spin helix and with the same signs for \u000band\n\f. We characterize this condition by r2\u001c1, introducing\nthe parameters r1= (\u000b\u0000\f)=(\u000b+\f),r2=\f3=(\u000b+\f), and\nr2=r2\n1+r2\n2. To describe the evolution of a general spin\npolarization\u001a(x;y) in direct space, it is favorable to con-\nsider its Fourier components b\u001a(qx;qy) that harmonically\noscillate in space and time:\n\u001a(x;y;t ) =ab\u001a(qx;qy)eiqxx+iqyy\u0000i!t: (3)For the Fourier-space spin polarization b\u001a, it is possi-\nble to derive a spin-di\u000busion equation in the presence of\nSOI. Using a density-matrix response function with stan-\ndard perturbation theory14{16, or starting from a semi-\nclassical spin kinetic equation17, the following equation\nis obtained:\n\u0010\n\u0000i!+Ds(q2\nx+q2\ny+q2\n0~D)\u0011\nb\u001a= 0: (4)\nWe de\fned the spin helix wave number q0=2m\u0003\n~2(\u000b+\f)\nand the spin di\u000busion constant Ds=~2k2\nF\u001c=(2m\u00032) that\ndepends on the e\u000bective electron momentum scattering\ntime\u001c. Note that \u001ccontains also contributions from\nelectron-electron scattering18, which is not the case for\ncharge di\u000busion. The term q2\n0Ds~Daccounts for the spin\ndynamics related to SOI. The diagonal elements of this\nmatrix yield the Dyakonov-Perel dephasing rates. The\nnon-diagonal terms arise from correlations between the\nmomentum and the spin-orbit \feld and drive the heli-\ncal spin modes. Equation (4) is valid in the weak spin-\norbit regime where the scattering time is small com-\npared to the spin precession period, \n SO\u001c\u001c1. It does\nnot account for additional spin scattering mechanisms as\nElliott-Yafet16or Bir-Aronov-Pikus19. The matrix ~Dis\ngiven by\n~D=0\nB@r2\n1+r2\n2 0\u0000ir12qx\nq0\n0 1 +r2\n2\u0000i2qy\nq0\nir12qx\nq0i2qy\nq01 +r2\n1+ 2r2\n21\nCA: (5)\nBy determining the eigenvectors b\u001an(qx;qy) and eigen-\nvalues\u0015nof~D, one obtains for each pair ( qx;qy) three\neigenmodes n= 1;2;3 that solve Eq. (4) and that decay\nexponentially with a rate\ni!n=Ds(q2\nx+q2\ny+q2\n0\u0015n): (6)\nThe time evolution of an arbitrary spin polarization\n\u001a(x;y;t ) in direct space can then be expressed in terms\nof these eigenmodes by using the Fourier integral\n\u001a(x;y;t ) =Z1\n\u00001Z1\n\u000013X\nn=1anb\u001ane\u0000i!nt+iqxx+iqyydqxdqy:\n(7)\nHere,an(qx;qy) are the amplitudes of the excited\neigenmodes.\nB. Evolution of a spin excitation with \fnite\nextension along the helix direction\nWe \frst discuss the situation where only eigenmodes\nwithqx= 0 are excited, i.e. where the initial spin polar-\nization does not vary as a function of x. Settingqx= 0\nin Eq. (5), the eigenvalues of ~Dare163\n\u00151=r2\n1+r2\n2=r2(8)\nand\n\u00152;3= 1 +1\n2r2\n1+3\n2r2\n2\u00061\n2s\n16q2y\nq2\n0+r4: (9)\n\u00151corresponds to a mode with a unidirectional spin\npolarization along the x-direction. \u00152and\u00153are the\neigenvalues of two helical modes with opposite helicity.\nIn the special situation of a perfect spin helix ( r1= 0,\nr2= 0), one obtains12\u00152;3= 1\u00062qy=q0, and thusi!2;3=\nDs(qy\u0006q0)2. The spin decay of mode 3 is completely\nsuppressed for qy=q0. Note that the same is true for\nmode 2 at qy=\u0000q0. As we will see below, also the\neigenvectors of modes 2 and 3 interchange when the sign\nofqyis inverted.\nIn the general case where r16= 0 orr26= 0, the square\nroot term in Eq. (9) can be approximated by 2 qy=q0as\nlong asqy\u001dr2q0=4. This yields\n\u00152;3= 1 +1\n2r2\n1+3\n2r2\n2\u00062qy=q0: (10)\nIn the other limit where qy= 0,\u00152and\u00153are split by\nr2, which is however much smaller than \u00152\u0019\u00153\u00191. As\na consequence, Eq. (10) is a good approximation for all\nqy. The dispersion of modes 2 and 3 can thus be written\nas\ni!2;3=Ds(qy\u0006q0)2+ \u0000s; (11)\nwith\n\u0000s=1\n2Dsq2\n0\u0000\nr2\n1+ 3r2\n2\u0001\n: (12)\nForr16= 0 orr26= 0, in addition to the exponential\ndecay rate Ds(qy\u0006q0)2, a decay with rate \u0000 soccurs.\nNote that in \u0000 s, the cubic Dresselhaus term r2\n2is weighted\nmore (by a factor of three) than the imbalance term r2\n1/\n(\u000b\u0000\f)2. The proportionality of \u0000 sto 3\f2\n3for\u000b=\fwas\nderived in Ref. 12.\nFor illustration purpose, we will assume the follow-\ning parameters for the SOI: \u000b= 1:7\u000110\u000013eVm,\f1=\n3:4\u000110\u000013eVm,\f3= 0:7\u000110\u000013eVm,\u001c= 0:5 ps and\nan electron sheet density of ns= 5\u00011015m\u00002. Fig-\nure 1 compares the mode dispersion that follows from\nEqs. (6) and (9) with the approximations Eqs. (11) and(12). Even for the relatively large deviation from bal-\nanced SOI ( r2= 0:077), the splitting of modes 2 and 3\natqy= 0 is small (inset of Fig. 1) and the dispersion\nis well approximated by the parabolic functions given in\nEq. (11).\n-2 -1 0 1 2\nqy / q 0iωn (10 10  s -1 )\n012345\n-0.1 0 0.1 1.4 1.8 1\n2\n3mode n \nFIG. 1: Dispersion of the eigenmodes of the spin di\u000busion\nequation for qx= 0 with parameters as given in the main\ntext. The quantity i!nindicates the spin decay rate of mode\nnand is shown as a function of the wave number qy. At\nqy=q0(qy=\u0000q0), the decay rate of mode 3 (2) is at its\nminimum, corresponding to the persistent spin helix. Symbols\ncorrespond to the exact values calculated from Eqs. (6) and\n(9), whereas the solid lines are the approximations based on\nEqs. (11) and (12). For r2\u001c1, the two solutions only deviate\naroundqy= 0 where a small splitting of modes 2 and 3 is\nneglected in the approximation (see inset).\nWe consider a spin polarization that at time t= 0\nis oriented along the z-direction. The spatial distri-\nbution is assumed to be uniform along xand Gaus-\nsian along ywith a width of wy, and an amplitude A:\n\u001a(x;y;0) = (0;0;Aexp(\u0000y2=2w2\ny)). This corresponds to\na spin polarization in Fourier space at t= 0 of\nab\u001a(qy) =0\nB@0\n0\nAwyp\n2\u0019exp\u0010\n\u0000q2\nyw2\ny\n2\u00111\nCA: (13)\nBecause of the uniform distribution along x, we can\nomit the Fourier transformation along qxin Eq. (7). For\neachqy,ab\u001ais decomposed into three eigenvectors b\u001anwith\namplitudes an. With the initial spin polarization along z,\nonly modes 2 and 3 are excited; the polarization of mode\n1 is uniformly pointing along x. Calculating the eigen-\nvectors of ~Dand using the approximation of Eq. (11),\nthe two modes are given by4\na2;3b\u001a2;3=Awy\n2p\n2\u0019exp \n\u0000q2\nyw2\ny\n2!\u0012\n1\u0006r2q0\n4qy\u00130\n@0\n\u0006i(1\u0007r2q0\n4qy)\n11\nA: (14)\nRe (a nρnz).q0\n \nqy / q 0Im (a nρny).q0\n 1\n2\n3mode n ^ ^00.4 0.8 \n-2 -1 0 1 2-0.4 -0.2 00.2 0.4 \nFIG. 2: Spin polarization of the three eigenmodes in q-space\ndirectly after excitation by a spin polarization oriented along\nzand with a Gaussian pro\fle along the y-direction. The\ninitial width of the pro\fle is set to wy= 2=q0. Only the real\npart of the z-components (a) and the imaginary part of the\ny-components (b) of modes 2 and 3 have a \fnite size. Mode\n1 is not excited.\nFigure 2 displays the real and imaginary parts of anb\u001anversusqy. Thez-component of mode 3 has a positive\nreal value and the y-component a positive imaginary one.\nIn direct space, this corresponds to the z-component\nbeing proportional to cos qyy, and they-component to\n\u0000sinqyy, see Eq. (3). This constitutes a helical spin\nmode where the spin polarization rotates counterclock-\nwise in the y\u0000zplane when moving towards the positive\nyaxis (for a positive qyand observed from the positive x-\naxis). In contrast, the negative imaginary value of a2b\u001ay\n2\nleads to a clockwise rotating helical spin mode. Note\nthat if the sign of qyis reversed, also the helicity of the\nrespective mode is \ripped.\nEach modeb\u001an(qx;qy) decays exponentially with a de-\ncay ratei!n(qx;qy). In Fig. 3 we plot the time evolution\nof mode 3 exemplarily. As mode 3 has a minimum decay\nrate atqy=q0, the original excitation centered at qy= 0\nwill shift with time towards qy=q0. In the same way,\nthe weight of mode 2 will displace towards \u0000q0.\nThe spin dynamics in direct space is found from\nEq. (7), where we consider only the physically meaning-\nful real part of \u001a. Making use of the symmetry of the\nmode dispersion with respect to qy= 0, we obtain\n\u001a=Awyp\n2\u0019Z1\n\u00001e\u0000q2\nyw2\ny=2\u0000\u0000st\u0000Ds(q0\u0000qy)2t0\nBB@0\nsinqyy\u0010\n1\u0000r4q2\n0\n16q2y\u0011\ncosqyy\u0010\n1\u0000r2q0\n4qy\u00111\nCCAdqy: (15)\nThe terms of the integrand that contain rare only rele-\nvant around qy= 0 and can be neglected after integrationbecause they are odd in qy. This results in the following\nexpression for the spin polarization in direct space:\n\u001a(x;y;t ) =Awy\nw0yexp \n\u0000y2\n2w0y2\u0000Dsq2\n0w2\ny\nw02yt\u0000\u0000st!0\n@0\nsinq0\n0y\ncosq0\n0y1\nA: (16)\nThe spin polarization establishes a helical oscillation\nwith wave number q0\n0=\u0010\n1\u0000w2\ny\nw02y\u0011\nq0. The envelope\nof this oscillation is given by a Gaussian distributionof widthw0\ny=q\nw2y+ 2Dst. Equation (16) therefore\ndescribes the following modi\fcations compared with a\ndelta-shaped excitation: (1) The amplitude of the heli-5\n-1.5 -1 -0.5 0 0.5 1 1.5 200.04 0.08 0.12 \nqy / q 0Re (a 3ρ3z e -iωt).q0^0 ns \n0.2 ns \n0.4 ns \n2 ns \nFIG. 3: Time evolution of mode 3 in q-space after the same\nexcitation as shown in Fig. 2. With time, the excitation cen-\ntered around qy= 0 shifts its weight towards qy=q0, where\nthe decay rate is smallest, transforming the initial spin polar-\nization along zinto a helical spin mode.\ncal state decays not only with rate \u0000 s, but also with an\nadditional time-varying rate Dsq2\n0w2\ny=w02\ny. (2) The spa-\ntial oscillation period of the spin polarization decreases\nwith time and approaches 2 \u0019=q0only asymptotically. (3)\nThe di\u000busive expansion reduces the signal proportionaltowy=w0\ny= 1=q\n1 + 2Dst=w2y, instead of just propor-\ntional to 1=p\nt.\nFort\u001dw2\ny=2Ds,q0\n0\u0019q0and the time-varying de-\ncay rate suppresses the spin polarization by a constant\nfactor of exp(\u0000q2\n0w2\ny=2). This means that if wy\u001c1=q0,\nthen the evolution of \u001ais equivalent to that of a delta-\nshaped excitation, at least for times t >(Dsq2\n0)\u00001. For\nlarger widths or shorter times, the modi\fcations dis-\ncussed above need to be considered to interpret exper-\nimental data.\nC. Spin excitation with \fnite extension along two\ndirections\nIf the initial spin polarization is localized along both\nthex- and they-direction, qxcannot be set to zero in\n~D, and the eigenvalues di\u000ber from Eq. (9). However, the\nmatrix elements that contain qxare much smaller than\nthose withqybecauser1\u001c1. It is therefore tempting to\nuse the same mode spectrum as for qx= 0, but to include\nthe nonzero qxin Eq. (6). For a Gaussian excitation of\nwidthwx=wy=w, this yields the following result\n\u001a(x;y;t ) =Aw2\nw02exp\u0012\n\u0000x2+y2\n2w02\u0000Dsq2\n0w2\nw02t\u0000\u0000st\u00130\n@0\nsinq0\n0y\ncosq0\n0y1\nA; (17)\nwith\nw02=w2+ 2Dst (18)\nand\nq0\n0=\u0012\n1\u0000w2\nw02\u0013\nq0: (19)\nIn the exact solution that takes nonzero qxin~Dinto\naccount, anticrossings of the eigenvalues occur close to\nqy=q0=2. Figure 4 shows numerically calculated (ex-\nact) values of the mode dispersion !nforqx=q0=2.\nIndependent of qy, di\u000busion along the xdirection in-\ncreases the decay rate by Dsq2\nx, weakening the relevance\nof such avoided crossings for the overall spin dynamics.\nIn Fig. 5, the calculated nonzero components of b\u001anare\nshown for the speci\fc case of qx=q0=2. All three eigen-\nmodes are excited with \fnite components along all three\ndirections. For qy>0, the dispersion of mode 2 ex-\nhibits no anticrossing, and therefore the mode spectrum\nis excited similarly as for qx= 0, see Fig. 2. However,modes 1 and 3 anticross around qy=q0=2, and therefore\nthose modes share the weight of the initial excitation.\nForqy<0, mode 3 stays una\u000bected by the anticrossing,\nwhereas modes 2 and 3 share the weight. Interestingly,\nalso thex-components of all modes are excited. As re-\nquired, the x-component of the sum of all modes is zero\nat timet= 0, whereas a \fnite spin polarization along\nxemerges at \fnite times because of the di\u000berent decay\ntimes of the three modes.\nFigure 6 compares the di\u000berence in the evolution\nof spin polarization in direct space for the two cases,\nnamely, whether the approximation qx= 0 is used in\nthe di\u000busion matrix ~Dor not. In the former case, \u001azis\ncalculated using Eq. (17), in the latter case, \u001azis numer-\nically obtained by a Fourier transformation of the eigen-\nmodes according to Eq. (7). In Fig. 6(a), the evolution of\nthe local excitation into the helical spin pattern is visible\nin the color-scale representation of the spin polarization\n\u001az(x= 0;y;t). Cross sections of \u001az(y) at di\u000berent times\ntare superposed as data points (full calculation) and as\nsolid lines (approximation of qx= 0). No signi\fcant dif-\nference is seen. In Fig. 6(b), the amplitude of \u001az(0;0;t)\nis shown versus ton a logarithmic plot for three di\u000berent6\niωn (10 10  s -1 )qx  = q 0 / 2 \n1\n23\nqy / q 0-2 -1 0 1 2012345\nmode n \nFIG. 4: Dispersion of the eigenmodes at qx= 0:5q0. Inde-\npendent of qy, the spin decay rate is increased by Dsq2\nx. In\naddition, the \fnite qxcomponent leads to an anticrossing of\nthe modes around qy=\u00060:5q0.\nwidths of the initial spin polarization. Supposing that\nan equal number of spin-polarized electrons are excited\nin the three cases, the amplitude Ais scaled with 1 =w2\nin the plot. The spin polarization is slightly larger for\nthe full calculation, but the temporal behavior is very\nsimilar to the approximation. The relative di\u000berence be-\ntween the two calculations increases for wider spin ini-\ntialization. It completely disappears in the case r1= 0\n(\u000b=\f), even for r2>0, which is consistent with the\ndisappearance of the o\u000b-diagonal matrix elements of ~D\nthat contain qx.\nIII. COMPARISON WITH EXPERIMENT\nTo verify the two predicted consequences the \fnite\nspatial extension of the initial excitation has, namely,\n(1) the additional decay rate Dsq2\n0w2=w02and (2) the\ntime dependence of q0\n0[Eq. (19)], we compare the model\nwith time- and spatially resolved measurements of the\nspin dynamics in a (001)-grown GaAs/AlGaAs quantum\nwell. We use the experimental technique described in\nRef. 13. In brief, spin polarization is created in the con-\nduction band by a circularly polarized pump laser pulse\nvia the optical orientation e\u000bect. A second laser pulse\n(probe) maps the evolving spin polarization Sz(x;y;t )\nusing the magneto-optical Kerr e\u000bect. The pump and\nprobe beams are focused onto the sample surface, where\nthe intensity pro\fle of the probe beam reaches a width\nof 1\u0016m (Gaussian sigma). The pump beam spot is set to\ntwo di\u000berent sizes: One comparable to the probe beam,\nand the other twice as large. Spatial maps are recorded\nby scanning the pump beam over the sample surface.\nThe time evolution is monitored by varying the time\ndelay between pump and probe. The quantum well in-\nvestigated is characterized by the following parameters:\nqx = q 0 / 2 Re (a nρnz).q02\n Im (a nρny).q02\n 1\n2\n3mode n ^\n^\n00.1 0.2 0.3 0.4 \n-0.1 00.1 \nqy / q 0Im (a nρnx).q02^\n-2 -1 0 1 2-0.1 00.1 FIG. 5: Spin polarization of the three eigenmodes excited by\na spin polarization along zwith a Gaussian pro\fle of width\nw= 2=q0along both the x- andy-directions. Modes are\nshown as a function of qy=q0atqx=q0=2. In contrast to\nthe case where the spin polarization is independent of the\npositionx, all modes have \fnite spin components along all\nthree directions, and all modes are excited simultaneously.\nThe real parts of the xandycomponents, as well as the\nimaginary part of the zcomponent are not excited.\nsheet density ns= 4\u00011015m\u00002,\f1= 4:1\u000110\u000013eVm,\n\f3= 0:7\u000110\u000013eVm and\u000b= 2:8\u000110\u000013eVm.\nThe color-scale plot in Fig. 7(a) shows a map of\n\u001az(0;y;t) recorded by scanning the position of the in-\ncident pump beam along the y-direction at various times\ntafter excitation (for the small pump spot). The con-\ntour lines mark the positions of \u001az(0;y;t)\u00190 of a model\nbased on Eq. (17) that is globally \ftted to the full data set\n\u001az(0;y;t). The parameters w0andq0\n0have been modeled\naccording to Eqs. (18) and (19), respectively. In the case\nof a small (large) excitation spot the following \ft param-\neters were obtained: w= 1:00 (2.02)\u0016m, \u0000\u00001\ns= 1:12 ns\n(1.01),q0= 1:08\u0016m\u00001(1.05), and Ds= 267 cm2/s (262).\nNote that the values obtained for ware determined by\nthe dynamics of \u001az(0;y;t) after excitation, speci\fcally by\nq0\n0(t) andA0(t), and were not \fxed to a prede\fned value.\nNext we examine the measured dynamics of the wave7\n-10 010 \nt (ns) y ( µm) ρz (x=y=0) (a) \n(b) \n0 0.5 1 1.5 10 -3 10 -2 10 -1 10 0\n0.25 \n1\n4w2.q02 =full calculation \napproximation q x = 0 \nFIG. 6: Evolution of the spin polarization \u001azin direct space,\nafter inital excitation of a Gaussian shape along xandy.\nIn (a), cross sections of \u001az(x= 0;y) are shown at di\u000berent\ntimestafter excitation, with normalized amplitudes. Symbols\ncorrespond to the approximation where qxis set to zero in ~D,\nwhereas for the solid lines the full matrix was considered. In\n(b), the amplitude of \u001azatx=y= 0 is shown versus time\nfor the two cases and for di\u000berent inital widths w.\nvectorq0\n0and the spin polarization amplitude \u001az(0;0;t)\nin Figs. 7(a) and (b). The symbols show q0\n0andA0ob-\ntained from individual \fts of A0exp(\u0000y2=2w02)\u0002cos(q0\n0y)\nto experimental data at various t. The e\u000bect of the probe\nbeam size is included in the \ft by a convolution of the \ft\nfunction with a Gaussian of sigma width 1 \u0016m. The solid\nand dashed lines represent the modeled time dependen-\ncies. It can be seen from Fig. 7(a) that q0\n0approaches q0\nsigni\fcantly faster in the case of a small spot (dots) than\nin the case of a large spot (squares), thereby con\frming\nstatement (2). Also the measured amplitudes A0(t) are in\nexcellent agreement with the model [Fig. 7(b)]. For the\nlarger excitation spot, the initial signal decay is stronger\nthan for the smaller spot. This supports the existence\nof the additional decay as expected from statement (1)\nabove.\nIV. CONCLUSION\nA model has been developed for the spatial and tempo-\nral evolution of the spin polarization \u001a(x;y;t ) in a (001)-\noriented quantum well in a zincblende semiconductor. A\nsituation in which Rashba and Dresselhaus SOI are of\nsimilar size has been considered. Speci\fcally, in this sys-\ntem we investigated the spin dynamics after an initial\nlocal spin excitation with polarization along zand of lat-\n0 1 2 3\nt (ns) 0 1 2 3\nt (ns) q0’ ( µm-1 )\nA’/(Aw 2)  (arb. units) \nw’ 2 ( µm2)\nt (ns) (a) (b) \n00.2 0.4 0.6 0.8 1\n10 -4 10 -3 10 -2 10 -1 10 0\n0 0.5 040 \n0 1 2 0\n-10 +10 +20 -20 \ny ( µm) \nw= \n2µm\n 1µmw= 1 µm\n2µm\n w= 1 µmFIG. 7: Dynamics of a local spin excitation of size w= 1\u0016m\n(dots) and 2 \u0016m (squares). The experimentally observed\ndynamics (symbols) is compared with a theoretical model\n(curves). In (a) the time dependence of the wave number\nq0\n0(t) and in (b) the spin polarization amplitude A0=(Aw2) is\nshown. The color-scale (contour) plot in the inset of (a) shows\nthe measured (modeled) map of \u001az(0;y;t) for a small pump\nspot size (w= 1\u0016m). The inset in (b) contains the exper-\nimentally obtained w02(t) (dots) and the \ft from the global\nmodel (dashed line). The solid line shows a linear \ft to w02(t)\nrestricted to t<1 ns, yielding a slightly larger di\u000busion con-\nstant ofDs= 334 cm2/s, possibly caused by a decrease of ns\nwith time.\neral spatial extension w. For a spatially delta-shaped ex-\ncitation (w!0), the persistent spin helix mode at wave\nnumberq0is fully excited, and \u001azis given by a helical\nspin mode with constant wave number q0and wrapped\ninto a Gaussian envelope that expands di\u000busively. The\nspin polarization decays exponentially with a decay rate\ngiven by \u0000 s= 2Dsm\u00032=~4\u0010\n(\u000b\u0000\f)2+ 3\f2\n3\u0011\n. For \fnite\nw, there are two important modi\fcations: First, there is\nan additional decay mechanism that can be described by\na time-dependent decay rate Dsq2\n0w2=w02. This rate sup-\npresses the spin polarization at times t\u001dw2=2Dsby a\nfactor exp(\u0000q2\n0w2=2), which corresponds to the weight of\nthe initial excitation at q0in Fourier space. Second, the\nwave number q0\n0of the helical spin polarization becomes\ntime dependent and reaches q0only asymptotically for\nt\u001dw2=2Ds. Both modi\fcations are observed in ex-\nperimental data obtained by time-resolved Kerr rotation\nmeasurements on GaAs/AlGaAs quantum-well samples.\nThese results are also relevant in the case of electrical\nspin injection from, e.g., ferromagnetic contacts.\nAn important consequence of our \fnding is that the de-\ntermination of \u0000 sfrom experimental data requires some\ncare because of the signal suppression and the slower for-\nmation of the helical spin mode for \fnite w. As an ex-\nample, the time scale where q0\n0reaches 0:9q0is 9w2=2Ds,\nwhich amounts to 670 ps for w= 2\u0016m in our experiment.\nThis e\u000bect is even more pronounced for smaller Ds, for\nexample in samples with smaller electron sheet densities.\nAlso the prefactor w2=w02in Eq. (17) in\ruences the spin\ntransient signi\fcantly.\nIn the limit of w! 1 , Eq. (17) predicts an\nexponential spin decay rate of the out-of-plane spin8\npolarization of Dsq2\n0+ \u0000s. This is in agreement\nwithi!(qx=qy= 0) in Eq. (11) and equiva-\nlent to1\n2(\u001c\u00001\nz+\u001c\u00001\ny) with the usual Dyakonov-Perel\nrates20\u001c\u00001\nz= 8Dsm\u00032=~4\u0000\n\u000b2+\f2+\f2\n3\u0001\nand\u001c\u00001\ny=\n4Dsm\u00032=~4\u0000\n(\u000b+\f)2+\f2\n3\u0001\n. Note that in this limit, the\ndecay rate of a spin polarization along zis not given by\n\u001c\u00001\nzalone, but by the average of \u001c\u00001\nzand\u001c\u00001\ny. This is\na consequence of the correlation of position and spin ori-\nentation of each individual electron in the balanced SOI\nsituation with r\u001c1. Even though the spin ensemble\ndecays rapidly because of the spatially broad excitation,individual electron spins form a helix and thereby rotate\nin they-zplane.\nAcknowledgements\nFinancial support from NCCR Nano and NCCR QSIT\nis acknowledged. We thank R. Allenspach, Y. S. Chen,\nA. Fuhrer, D. Loss and R. Warburton for fruitful discus-\nsions.\n\u0003Electronic address: gsa@zurich.ibm.com\n1G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n2F. T. Vasko, Sov. Phys. - JETP Lett. 30, 541 (1979).\n3Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039\n(1984).\n4M. I. Dyakonov and V. Y. Kachorovskii, Sov. Phys. Semi-\ncond. 20, 110 (1986).\n5M. P. Walser, U. Siegenthaler, V. Lechner, D. Schuh, S. D.\nGanichev, W. Wegscheider, and G. Salis, Phys. Rev. B 86,\n195309 (2012).\n6M. I. Dyakonov and V. I. Perel, Sov. Phys. 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Rev. B 86, 174301 (2012).\n17M. C. L u\u000be, J. Kailasvuori, and T. S. Nunner, Phys. Rev.\nB84, 075326 (2011).\n18C. P. Weber, N. Gedik, J. E. Moore, J. Orenstein, J.\nStephens, and D. D. Awschalom, Nature 437, 1330 (2005).\n19R. V olkl, M. Griesbeck, S. A. Tarasenko, D. Schuh, W.\nWegscheider, C. Sch uller, and T. Korn, Phys. Rev. B 83,\n241306 (2011).\n20J. Kainz, U. R ossler, and R. Winkler, Phys. Rev. B 68,\n075322 (2003)."    },    {        "title": "0812.4750v1.Jordan_Wigner_Fermionization_and_the_Theory_of_Low_Dimensional_Quantum_Spin_Models__Dynamic_Properties.pdf",        "content": "arXiv:0812.4750v1  [cond-mat.str-el]  27 Dec 2008December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nCHAPTER 1\nJORDAN-WIGNER FERMIONIZATION AND THE THEORY\nOF LOW-DIMENSIONAL QUANTUM SPIN MODELS.\nDYNAMIC PROPERTIES\nOleg Derzhko\nInstitute for Condensed Matter Physics\nof the National Academy of Sciences of Ukraine\n1 Svientsitskii Street, L’viv-11, 79011, Ukraine\nE-mail: derzhko@icmp.lviv.ua\nThe Jordan-Wigner transformation is known as a powerful too l in con-\ndensed matter theory, especially in the theory of low-dimen sional quan-\ntum spin systems. The aim of this chapter is to review the appl ication\nof the Jordan-Wigner fermionization technique for calcula ting dynamic\nquantities of low-dimensional quantum spin models. After a brief intro-\nduction of the Jordan-Wigner transformation for one-dimen sional spin\none-half systems and some of its extensions for higher dimen sions and\nhigher spin values we focus on the dynamic properties of seve ral low-\ndimensional quantum spin models. We start from the famous s= 1/2\nXXchain. As a first step we recall well-known results for dynami cs\nof the z-spin-component fluctuation operator and then turn to the dy -\nnamics of the dimer and trimer fluctuation operators. The dyn amics\nof the trimer fluctuations involves both the two-fermion (on e particle\nand one hole) and the four-fermion (two particles and two hol es) excita-\ntions. We discuss some properties of the two-fermion and fou r-fermion\nexcitation continua. The four-fermion dynamic quantities are of inter-\nmediate complexity between simple two-fermion (like the zzdynamic\nstructure factor) and enormously complex multi-fermion (l ike the xxor\nxydynamic structure factors) dynamic quantities. Further we discuss\nthe effects of dimerization, anisotropy of XYinteraction, and additional\nDzyaloshinskii-Moriya interaction on various dynamic qua ntities. Finally\nwe consider the dynamic transverse spin structure factor Szz(k, ω) for\nthes= 1/2XXmodel on a spatially anisotropic square lattice which\nallows one to trace a one-to-two-dimensional crossover in d ynamic quan-\ntities.\n1December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n2 O. Derzhko\n1. Introduction (Spin Models, Dynamic Probes etc.)\nThe subject of quantum magnetism dates back to 1920s. E. Isin g1sug-\ngested a simplest model of a magnet as a collection of Nspins which\nmay acquire two values σ=±1 and interact with nearest neighbors on\na lattice as/summationtextJσiσjand with an external magnetic field as −h/summationtextσi. To\nexplain the properties of the model we have to calculate the p artition func-\ntionZ= Tr exp( −βH) which yields the Helmholtz free energy per site\nf= lim N→∞(−TlnZ/N) (in what follows we set kB= 1 to simplify the\nnotations). In one dimension the problem was solved by E. Isi ng. Later\nL. Onsager solved the square-lattice Ising model2and we know the solu-\ntion in two dimensions3. There is no solution of the Ising model in three\ndimensions until now.\nAnother version of interspin interaction was suggested by P . A. M. Dirac\nand W. Heisenberg. The Heisenberg exchange interaction rea ds/summationtextJ/vector σi·/vector σj=/summationtextJ/parenleftbig\nσx\niσx\nj+σy\niσy\nj+σz\niσz\nj/parenrightbig\nwhere the Pauli matrices /vector σ= (σx,σy,σz) are\ndefined as\nσx=/parenleftbigg0 1\n1 0/parenrightbigg\n, σy=/parenleftbigg0−i\ni 0/parenrightbigg\n, σz=/parenleftbigg1 0\n0−1/parenrightbigg\n. (1)\nDenoting the halves of the Pauli matrices as sα=σα/2 (in what follows we\nset/planckover2pi1= 1 to simplify the notations) we consider the following Hami ltonian\nH=/summationdisplay\n/angbracketlefti,j/angbracketright/parenleftbig\nJxsx\nisx\nj+Jysy\nisy\nj+Jysz\nisz\nj/parenrightbig\n−h/summationdisplay\nisz\ni. (2)\nWe note that the Hamiltonian of the anisotropic XYZ Heisenberg model\n(2) covers in the limiting cases some specific models like the Ising model\n(Jx=Jy= 0), the isotropic XY(orXXorXX0) model (Jx=Jy,Jz=\n0), the anisotropic XYmodel (Jx∝ne}ationslash=Jy,Jz= 0), the isotropic ( XXX )\nHeisenberg model ( Jx=Jy=Jz), and the Heisenberg-Ising ( XXZ ) model\n(Jx=Jy=J,Jz= ∆J).\nAgain we would like to calculate the partition function Zof the spin-\n1/2 model (2). Unfortunately, this task is very complicated even in one\ndimension. Due to H. Bethe we know how to find the eigenstates o f the\nspin-1/2 linear chain Heisenberg model4. The famous Bethe ansatz for theDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 3\nwave function has the form\n|ψ∝an}b∇acket∇i}ht=/summationdisplay\n1≤n1<...<n r≤Na(n1,...,n r)s−\nn1...s−\nnr| ↑↑...↑∝an}b∇acket∇i}ht,\na(n1,...,n r) =/summationdisplay\nP∈Srexp\nir/summationdisplay\nj=1kPjnj+i\n2/summationdisplay\ni<jθPiPj\n,\n2 cotθij\n2= cotki\n2−cotkj\n2, Nk i= 2πλi+/summationdisplay\nj/negationslash=iθij, (3)\nwhere the sum in the definition of coefficients a(n1,...,n r),P ∈Sr, runs\nover allr! permutations of the labels {1,2,...,r },Pjis the image of junder\nthe permutation P. For further details see, e.g., Ref.5.\nLet us briefly recall the quantities of interest in the statis tical mechanical\nstudies of the spin models. As we have mentioned already the t hermody-\nnamic quantities like the entropy, the specific heat, the mag netization etc.\nfollow from the partition function Z=/summationtext\nλexp(−Eλ/T) = Trexp ( −βH),\nthe sum runs over all states λof the system with energy Eλ. Usually\nwe are also interested in the equal-time spin correlation fu nctions, e.g.\n∝an}b∇acketle{t/vector si·/vector sj∝an}b∇acket∇i}ht,∝an}b∇acketle{t(...)∝an}b∇acket∇i}ht= Tr(exp( −βH)(...))/Z; their nonzero limiting values,\n(e.g., lim |i−j|→∞∝an}b∇acketle{t/vector si·/vector sj∝an}b∇acket∇i}ht) may indicate the existence of long-range order in\nthe system.\nWithin a linear response regime we add to the Hamiltonian H0a small\nperturbation H0→H0−b(t)B, where the external field b(t) couples to\nthe dynamical variable B, and observe a response of a dynamical vari-\nableA,∝an}b∇acketle{tA(t)∝an}b∇acket∇i}ht − ∝an}b∇acketle{tA∝an}b∇acket∇i}ht0=/integraltext∞\n−∞dt′χAB(t−t′)b(t′) withχAB(t−t′) = iθ(t−\nt′)∝an}b∇acketle{t[A(t),B(t′)]∝an}b∇acket∇i}ht0(hereθ(x) is the Heaviside step function). The Fourier-\ntransform of the dynamic susceptibility χAB(t−t′),ℜχAB(ω)+iℑχAB(ω),\nis the quantity which can be measured experimentally. We not e that the\nreal and imaginary parts of the dynamic susceptibility are c onnected via\nthe dispersion (or Kramers-Kronig) relation. On the other h and, the imag-\ninary part of the dynamic susceptibility can be expressed wi th the help\nof the fluctuation-dissipation theorem through another dyn amic quantity,\nthe dynamic structure factor. Thus, SAA(ω) =/integraltext∞\n−∞dtexp (iωt)∝an}b∇acketle{tA(t)A∝an}b∇acket∇i}ht=\n2ℑχAA(ω)/(1−exp(−βω)).\nUsually, the operator Ais constructed from the local operator of the\nconsidered system Anas follows:Ak= (1/√\nN)/summationtextN\nn=1exp(ikn)An. We canDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n4 O. Derzhko\nalso rewrite the dynamic structure factor in the following f orms\nSAA(k,ω) =N/summationdisplay\nl=1exp(−ikl)/integraldisplay∞\n−∞dtexp(iωt)∝an}b∇acketle{tAn(t)An+l(0)∝an}b∇acket∇i}ht\n= 2π/summationdisplay\nλ,λ′exp (−βEλ′)\nZ|∝an}b∇acketle{tλ′|Ak|λ∝an}b∇acket∇i}ht|2δ(ω−Eλ+Eλ′). (4)\nSometimes it is convenient to make the following change in th e first line\nin Eq. (4): An(t)→An(t)− ∝an}b∇acketle{tA∝an}b∇acket∇i}ht,An+l(0)→An+l(0)− ∝an}b∇acketle{tA∝an}b∇acket∇i}ht. We also note\nthat in the zero-temperature limit T= 0 (orβ→ ∞ ) the second line\nin Eq. (4) becomes simpler, SAA(k,ω) = 2π/summationtext\nλ|∝an}b∇acketle{tGS|Ak|λ∝an}b∇acket∇i}ht|2δ(ω−ωλ),\nωλ=Eλ−EGS.\nIn what follows we discuss mainly the dynamic properties of s pin-1/2\nXYchains; just for this class of spin models application of the Jordan-\nWigner fermionization approach is most fruitful. We notice here that re-\ncently it has been found that Cs 2CoCl 4is a good realization of the spin-1/2\nXXchain6and calculations of the dynamic quantities for the correspo nd-\ning spin models might be important for the interpretation of the data from\ndynamic experiments7. As an example of earlier studies we may mention\ndynamic experiments on the spin-1/2 XXchain compound PrCl 38.\nThe rest of this chapter is organized as follows. At first we br iefly in-\ntroduce the Jordan-Wigner transformation (Sec. 2) and conc isely discuss\nsome of its generalizations (Sec. 3). Then we consider in det ail the dynamic\nstructure factors for the spin-1/2 isotropic XYchain in a transverse field\ndistinguishing the quantities which probe two-fermion, fo ur-fermion and\nmany-fermion excitations (Sec. 4). Next we examine the dyna mics for two\nslightly more complicated chains: the dimerized isotropic XYchain (Sec. 5)\nand theXYchains with the Dzyaloshinskii-Moriya interaction (Sec. 6 ).\nThe results obtained for one-dimensional XYspin models do not involve\nany approximation. This is not true in the two-dimensional c ase for which\nthe Jordan-Wigner approach provides only approximate expr essions for dy-\nnamic quantities. We illustrate the Jordan-Wigner fermion ization approach\nin two dimensions examining some dynamic quantities for the square-lattice\nspin-1/2 isotropic XYmodel (Sec. 7). We end up with a brief summary\n(Sec. 8).December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 5\n2. The Jordan-Wigner Transformation\nTo be specific, we consider the one-dimensional spin s= 1/2XXZ Heisen-\nberg chain with the Hamiltonian\nH=N/summationdisplay\nn=1J/parenleftbig\nsx\nnsx\nn+1+sy\nnsy\nn+1+ ∆sz\nnsz\nn+1/parenrightbig\n−hN/summationdisplay\nn=1sz\nn; (5)\nwe imply either periodic or open boundary conditions in Eq. ( 5). Here the\nspin operators sα\nisatisfy the commutation relations/bracketleftig\nsα\ni,sβ\nj/bracketrightig\n= iδijǫαβγsγ\ni,\nǫαβγis the totally antisymmetric Levi-Civita tensor with ǫxyz= 1. In par-\nticular,/bracketleftbig\nsx\ni,sy\nj/bracketrightbig\n= iδijsz\nietc. Obviously, sαcan be viewed as the halves\nof the Pauli matrices (1). After introducing the spin raisin g and lowering\noperators (or the ladder operators) s±\nn=sx\nn±isy\nn(sx\nn= (s+\nn+s−\nn)/2,\nsy\nn= (s+\nn−s−\nn)/2i) the Hamiltonian (5) becomes\nH=/summationdisplay\nnJ/parenleftbigg1\n2/parenleftbig\ns+\nns−\nn+1+s−\nns+\nn+1/parenrightbig\n+∆/parenleftbigg\ns+\nns−\nn−1\n2/parenrightbigg/parenleftbigg\ns+\nn+1s−\nn+1−1\n2/parenrightbigg/parenrightbigg\n−h/summationdisplay\nn/parenleftbigg\ns+\nns−\nn−1\n2/parenrightbigg\n. (6)\nWe note that the spin raising and lowering operators satisfy commutation\nrelations of Fermi type at the same site, i.e.\n/braceleftbig\ns−\nn,s+\nn/bracerightbig\n= 1,/braceleftbig\ns−\nn,s−\nn/bracerightbig\n=/braceleftbig\ns+\nn,s+\nn/bracerightbig\n= 0 (7)\nand of Bose type at different sites\n/bracketleftbig\ns−\nn,s+\nm/bracketrightbig\n=/bracketleftbig\ns−\nn,s−\nm/bracketrightbig\n=/bracketleftbig\ns+\nn,s+\nm/bracketrightbig\n= 0, n∝ne}ationslash=m. (8)\nWe may use the Jordan-Wigner transformation9to introduce Fermi\noperators according to the following formulas\nc1=s−\n1, cn= (−2sz\n1)(−2sz\n2).../parenleftbig\n−2sz\nn−1/parenrightbig\ns−\nn, n= 2,...,N, (9)\nc†\n1=s+\n1, c†\nn= (−2sz\n1)(−2sz\n2).../parenleftbig\n−2sz\nn−1/parenrightbig\ns+\nn, n= 2,...,N. (10)\n(Sometimes one can find in Eqs. (9), (10) instead of −2szthe identical\nexpressions 1 −2s+s−= exp ( ±iπs+s−).) Really, the operators introduced\nalways satisfy the Fermi commutation relations\n/braceleftbig\ncn,c†\nm/bracerightbig\n=δnm,{cn,cm}=/braceleftbig\nc†\nn,c†\nm/bracerightbig\n= 0. (11)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n6 O. Derzhko\n(To check this one has to note that ( −2sz)2= 1 and that szs±=−s±sz.)\nThe inverse transformation to the one given by Eqs. (9), (10) reads\ns−\n1=c1, s−\nn= exp\n±iπn−1/summationdisplay\nj=1c†\njcj\ncn, n= 2,...,N, (12)\ns+\n1=c†\n1, s+\nn= exp\n±iπn−1/summationdisplay\nj=1c†\njcj\nc†\nn, n= 2,...,N. (13)\nMoreover, the Hamiltonian (6) in terms of the Fermi operator s (9), (10)\nhas the following form\nH=/summationdisplay\nnJ/parenleftbigg1\n2/parenleftig\nc†\nncn+1−cnc†\nn+1/parenrightig\n+∆/parenleftbigg\nc†\nncn−1\n2/parenrightbigg/parenleftbigg\nc†\nn+1cn+1−1\n2/parenrightbigg/parenrightbigg\n−h/summationdisplay\nn/parenleftbigg\nc†\nncn−1\n2/parenrightbigg\n(14)\n(we usec†\njc†\nj+1=s+\nj/parenleftbig\n−2sz\nj/parenrightbig\ns+\nj+1=s+\njs+\nj+1etc.). In the case of periodic\nboundary conditions implied for the spin Hamiltonian (6) th e transformed\nHamiltonian (14) obeys either periodic or antiperiodic bou ndary conditions\ndepending on the parity of the number of fermions. However, i n what fol-\nlows the calculated quantities in the thermodynamic limit N→ ∞ will\nbe insensitive to the boundary conditions implied (for furt her details see\nRef.10).\nFrom Eq. (14) it becomes clear that the spin-1/2 isotropic XYchain in\na transverse ( z) magnetic field with the Hamiltonian\nH=/summationdisplay\nnJ/parenleftbig\nsx\nnsx\nn+1+sy\nnsy\nn+1/parenrightbig\n+ Ω/summationdisplay\nnsz\nn (15)\nin the Jordan-Wigner picture is represented by the Hamilton ian\nH=/summationdisplay\nnJ\n2/parenleftig\nc†\nncn+1−cnc†\nn+1/parenrightig\n+ Ω/summationdisplay\nn/parenleftbigg\nc†\nncn−1\n2/parenrightbigg\n(16)\nand therefore is an exactly solvable model11,12. Moreover, the XYex-\nchange interaction may be anisotropic; then the intersite i nteraction has\nthe form\nJxsx\nnsx\nn+1+Jysy\nnsy\nn+1\n→J\n2/parenleftig\nc†\nncn+1−cnc†\nn+1/parenrightig\n+γ\n2/parenleftig\nc†\nnc†\nn+1−cncn+1/parenrightig\n(17)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 7\nwithJ= (Jx+Jy)/2,γ= (Jx−Jy)/2. We can also consider an additional\nintersite interaction, the so-called Dzyaloshinskii-Mor iya interaction, which\ndoes not spoil a simple fermionic Hamiltonian13\nD/parenleftbig\nsx\nnsy\nn+1−sy\nnsx\nn+1/parenrightbig\n→iD\n2/parenleftig\nc†\nncn+1+cnc†\nn+1/parenrightig\n. (18)\nMoreover, within the Jordan-Wigner fermionization approa ch we can ex-\namine rigorously some types of multi-spin interactions14, for example,\nsx\nnsz\nn+1sx\nn+2+sy\nnsz\nn+1sy\nn+2→ −1\n4/parenleftig\nc†\nncn+2−cnc†\nn+2/parenrightig\n;\nsx\nnsz\nn+1sy\nn+2−sy\nnsz\nn+1sx\nn+2→ −i\n4/parenleftig\nc†\nncn+2+cnc†\nn+2/parenrightig\n. (19)\nWithin the frames of the Jordan-Wigner approach we can also g eneral-\nize simple spin-1/2 XYchains assuming regularly alternating Hamiltonian\nparameters15or some types of random Hamiltonian parameters16and still\nface exactly solvable models.\nOn the other hand, as can be easily seen from Eq. (14) the Ising interac-\ntion between zspin components leads to interacting spinless fermions and as\na result the advantages of fermionization are less evident. (Obviously, we can\nsplit the interaction term in the spirit of the Hartree-Fock approximation17,\nhowever, the resulting theory will be only an approximate on e. On the other\nhand, in the low-energy limit we can bosonize the fermionic H amiltonian\nobtaining an exact low-energy effective theory18,19,20.) We cannot exam-\nine rigorously within the Jordan-Wigner fermionization ap proach the case\nof the next-nearest-neighbor interaction since\nsx\nnsx\nn+2+sy\nnsy\nn+2→c†\nn/parenleftig\n1−2c†\nn+1cn+1/parenrightig\ncn+2−cn/parenleftig\n1−2c†\nn+1cn+1/parenrightig\nc†\nn+2.(20)\nIt is worthwhile to note here that recently the Jordan-Wigne r fermioniza-\ntion approach has been applied to the spin-1/2 isotropic XYmodel on a\ndiamond chain21, however, the authors of that paper apparently missed\nsome interaction terms in the fermionic Hamiltonian and the ir statement\nabout rigorous results for such a model is wrong. Finally we n ote that an\nexternal magnetic field directed along xoryaxes has an enormously com-\nplicated form in the Jordan-Wigner picture.\n3. Generalization of the Jordan-Wigner Transformation\nThe Jordan-Wigner fermionization is a powerful tool for the study of quan-\ntum spin chains. Since the late 1980s there were several atte mpts to extend\nthis approach to two (and three) dimensions22,23,24,25as well as to spinDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n8 O. Derzhko\nvaluess>1/226,27,28. For a review on the two-dimensional Jordan-Wigner\nfermionization approach see also Ref.29.\nBearing in mind the Jordan-Wigner transformation in one dim ension as\na guideline we consider in the two-dimensional case the foll owing relation\nbetween spin s= 1/2 and Fermi operators\nd/vectori= exp/parenleftbig\n−iα/vectori/parenrightbig\ns−\n/vectori, d†\n/vectori= exp/parenleftbig\niα/vectori/parenrightbig\ns+\n/vectori,\ns−\n/vectori= exp/parenleftbig\niα/vectori/parenrightbig\nd/vectori, s+\n/vectori= exp/parenleftbig\n−iα/vectori/parenrightbig\nd†\n/vectori,\nα/vectori=/summationdisplay\n/vectorj(/negationslash=/vectori)B/vectori/vectorjd†\n/vectorjd/vectorj. (21)\nHered,d†are the Fermi operators, the operators s±defined according to\n(21) commute at different sites if the c-number matrix B/vectori/vectorjsatisfies the\nrelation\nexp/parenleftig\niB/vectori/vectorj/parenrightig\n=−exp/parenleftig\niB/vectorj/vectori/parenrightig\n. (22)\nThere are many choices of the matrix B/vectori/vectorjwhich realize the two-\ndimensional Jordan-Wigner transformation. Following Y. R . Wang23we\nuse the Cartesian coordinates /vectori= (ix,iy) to construct a complex number\nτ/vectori=ix+ iiy=|τ/vectori|exp/parenleftbig\ni arg(τ/vectori)/parenrightbig\nand then choose\nB/vectori/vectorj= arg/parenleftig\nτ/vectorj−τ/vectori/parenrightig\n=ℑln/parenleftig\nτ/vectorj−τ/vectori/parenrightig\n=ℑln (jx−ix+ i(jy−iy)).(23)\nIndeed, for such a choice Eq. (22) is satisfied, exp/parenleftig\niB/vectorj/vectori/parenrightig\n=\nexp/parenleftig\ni arg(τ/vectori−τ/vectorj)/parenrightig\n= exp/parenleftig\ni/parenleftig\narg(τ/vectorj−τ/vectori)±π/parenrightig/parenrightig\n=−exp/parenleftig\niB/vectori/vectorj/parenrightig\n. Another\nchoice of the matrix B/vectori/vectorjhas the following form24\nB/vectori/vectorj=π/parenleftbig\nθ(ix−jx)(1−δix,jx) +δix,jxθ(iy−jy)/parenleftbig\n1−δiy,jy/parenrightbig/parenrightbig\n; (24)\nhereθ(x) is the Heaviside step function (see also Ref.29).\nAfter performing the Jordan-Wigner transformation (21) fo r the two-\ndimensional spin-1/2 XXZ Heisenberg Hamiltonian one gets\nH=/summationdisplay\n/angbracketleft/vectori,/vectorj/angbracketright/parenleftbiggJ/vectori/vectorj\n2/parenleftig\nd†\n/vectoriexp/parenleftig\ni/parenleftig\nα/vectorj−α/vectori/parenrightig/parenrightig\nd/vectorj+d/vectoriexp/parenleftig\ni/parenleftig\nα/vectori−α/vectorj/parenrightig/parenrightig\nd†\n/vectorj/parenrightig\n+J/vectori/vectorj∆/parenleftbigg\nd†\n/vectorid/vectori−1\n2/parenrightbigg/parenleftbigg\nd†\n/vectorjd/vectorj−1\n2/parenrightbigg/parenrightbigg\n(25)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 9\nwith\nα/vectorj−α/vectori=/integraldisplay/vectorj\n/vectorid/vector r·/vectorA(/vector r),\n/vectorA(/vector r) =/vector∇α/vector r=−/summationdisplay\n/vector r′(/negationslash=/vector r)/vector nz×(/vector r′−/vector r)\n(/vector r′−/vector r)2d†\n/vector r′d/vector r′ (26)\n(we have used Eq. (23) for α/vector r(21)). We need further approximations to\nproceed with statistical mechanics calculations for the Ha miltonian (25).\nWithin the mean-field description one assumes d†\n/vector rd/vector r→ ∝an}b∇acketle{td†\n/vector rd/vector r∝an}b∇acket∇i}ht=∝an}b∇acketle{tsz\n/vector r∝an}b∇acket∇i}ht+\n1/2→1/2. We expect such an approximation to be valid in the case of\nzero magnetic field. For the mean-field description in the cas e of nonzero\nmagnetic field and an analysis of the magnetization processe s in the spin\nsystem see Ref.30. We also note that a more sophisticated (self-consistent\nsite-dependent) mean-field treatment has been suggested as well25.\nAfter adopting the mean-field approach we face the problem of particles\nin a magnetic field with the flux per elementary plaquette Φ 0=π. We may\nchange the gauge preserving the flux per elementary plaquett e to make the\nHamiltonian more convenient for further calculations. For example, for a\nsquare lattice we have\nH=/summationdisplay\n/angbracketleft/vectori,/vectorj/angbracketright/parenleftbiggJ/vectori/vectorj\n2/parenleftig\nd†\n/vectorid/vectorj−d/vectorid†\n/vectorj/parenrightig\n+J/vectori/vectorj∆/parenleftbigg\nd†\n/vectorid/vectori−1\n2/parenrightbigg/parenleftbigg\nd†\n/vectorjd/vectorj−1\n2/parenrightbigg/parenrightbigg\nJix,iy;ix+1,iy=−J,\nJix,iy;ix,iy+1=Jix+1,iy;ix+2,iy=Jix+1,iy;ix+1,iy+1=J.(27)\nIn the one-dimensional case when either vertical or horizon tal bonds vanish\nthe Hamiltonian (27) transforms into Eq. (14) (with h= 0).\nRecently A. Kitaev has suggested a new exactly solvable two-\ndimensional quantum spin model31. This is a spin-1/2 model on a honey-\ncomb lattice with interactions between different component s of neighboring\nspins along differently directed bonds. An alternative repr esentation of the\nhoneycomb lattice is a brick-wall lattice (see Fig. 1). The H amiltonian of\nthe model reads\nH=/summationdisplay\nj+l=even/parenleftig\nJ1sx\nj,lsx\nj+1,l+J2sy\nj−1,lsy\nj,l+J3sz\nj,lsz\nj,l+1/parenrightig\n; (28)\njandldenote the column and row indices of the lattice. We discuss i n what\nfollows a fermionic representation for the Kitaev model32. Let us performDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n10 O. Derzhko\nFig. 1. A honeycomb lattice (up) with its equivalent brick-w all lattice (down). The\nbonds J1run from south-west to north-east, the bonds J2run from south-east to north-\nwest, the bonds J3run from south to north.\nthe Jordan-Wigner transformation\ns+\nj,l=a†\nj,lexp\niπ\n/summationdisplay\ni/summationdisplay\nk<la†\nikaik+/summationdisplay\ni<ja†\nilail\n\n (29)\n(compare with Eqs. (21), (24)). As a result we find that\nJ1sx\nj,lsx\nj+1,l→J1\n4/parenleftig\na†\nj,la†\nj+1,l+a†\nj,laj+1,l−aj,la†\nj+1,l−aj,laj+1,l/parenrightig\n,\nJ2sy\nj−1,lsy\nj,l→J2\n4/parenleftig\n−a†\nj−1,la†\nj,l+a†\nj−1,laj,l−aj−1,la†\nj,l+aj−1,laj,l/parenrightig\n,\nJ3sz\nj,lsz\nj,l+1→J3/parenleftbigg\na†\nj,laj,l−1\n2/parenrightbigg/parenleftbigg\na†\nj,l+1aj,l+1−1\n2/parenrightbigg\n.(30)\nNext we introduce the following operators\ncj,l=a†\nj,l+aj,l, d j,l= i/parenleftig\na†\nj,l−aj,l/parenrightig\n, j+l= odd;\ncj,l= i/parenleftig\na†\nj,l−aj,l/parenrightig\n, d j,l=a†\nj,l+aj,l, j+l= even. (31)\nIn terms of these operators the Hamiltonian reads as follows\nH=/summationdisplay\nj+l=even/parenleftbigg\n−iJ1\n4cj,lcj+1,l+ iJ2\n4cj−1,lcj,l+ iJ3\n4Dj,lcj,lcj,l+1/parenrightbigg\n.(32)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 11\nSinceDj,l= idj,ldj,l+1are good quantum numbers the Hamiltonian (32)\ncorresponds to a model of spinless fermions with local stati cZ2gauge fields.\nThus, Eq. (32) explains a hidden simple structure of the spin model (28).\nThe generalizations of the Jordan-Wigner transformation f or arbitrary\nspin values were discussed by several authors26,27,28, however, these\nmappings have not yet provided a substantial break-through for difficult\nstrongly correlated problems.\n4. Spin-1/2 Isotropic XYChain in a Transverse Field:\nDynamic Quantities\nWe start with the simplest spin-1/2 XY model, the transverse\nXXchain, with the Hamiltonian (15). After performing the Jord an-\nWigner transformation we arrive at a tight-binding model fo r spinless\nfermions (16) and after performing the Fourier transformat ion,ck=/parenleftig\n1/√\nN/parenrightig/summationtextN\nn=1exp(ikn)cn(k= 2πn/N if the number of fermions is odd\nork= 2π(n+1/2)/Nif the number of fermions is even, n=−N/2,−N/2+\n1,...,N/ 2−1 ifNis even orn=−(N−1)/2,−(N−1)/2+1,...,(N−1)/2\nifNis odd), the Hamiltonian (16) becomes diagonal\nH=/summationdisplay\nkΛk/parenleftbigg\nc†\nkck−1\n2/parenrightbigg\n,Λk= Ω +Jcosk. (33)\nAs it has been mentioned above, for the analytical calculati ons discussed\nbelow we may consider only periodic boundary conditions for the fermionic\nHamiltonian (i.e. k= 2πn/N in Eq. (33)).\n4.1.Two-fermion excitations\nWe begin with the transverse dynamic structure factor Szz(k,ω)\n(4)33,34,35. The calculation of the zztime-dependent spin correlation\nfunction is straightforward. After exploiting the Jordan- Wigner trans-\nformation we have ∝an}b∇acketle{tsz\nn(t)sz\nn+l∝an}b∇acket∇i}ht − ∝an}b∇acketle{tsz\nn∝an}b∇acket∇i}ht∝an}b∇acketle{tsz\nn+l∝an}b∇acket∇i}ht=∝an}b∇acketle{tc†\nn(t)cn(t)c†\nn+lcn+l∝an}b∇acket∇i}ht −\n∝an}b∇acketle{tc†\nncn∝an}b∇acket∇i}ht∝an}b∇acketle{tc†\nn+lcn+l∝an}b∇acket∇i}ht. Herec†\nn(t) =/parenleftig\n1/√\nN/parenrightig/summationtext\nkexp(ikn)c†\nk(t) andc†\nk(t) =\nc†\nkexp(iΛ kt). Next we have to use the Wick-Bloch-de Dominicis theorem,\n∝an}b∇acketle{tc†\nk1ck2c†\nk3ck4∝an}b∇acket∇i}ht=∝an}b∇acketle{tc†\nk1ck2∝an}b∇acket∇i}ht∝an}b∇acketle{tc†\nk3ck4∝an}b∇acket∇i}ht − ∝an}b∇acketle{tc†\nk1c†\nk3∝an}b∇acket∇i}ht∝an}b∇acketle{tck2ck4∝an}b∇acket∇i}ht+∝an}b∇acketle{tc†\nk1ck4∝an}b∇acket∇i}ht∝an}b∇acketle{tck2c†\nk3∝an}b∇acket∇i}ht, and\nto calculate the elementary contractions introducing the F ermi function\nnk= 1/(1 + exp (βΛk)),∝an}b∇acketle{tc†\nk1ck2∝an}b∇acket∇i}ht=δk1k2nk1,∝an}b∇acketle{tc†\nk1c†\nk2∝an}b∇acket∇i}ht= 0. As a result, theDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n12 O. Derzhko\nfinal expression for the zztime-dependent spin correlation function reads\n∝an}b∇acketle{tsz\nn(t)sz\nn+l∝an}b∇acket∇i}ht − ∝an}b∇acketle{tsz∝an}b∇acket∇i}ht2=1\nN2/summationdisplay\nk1,k2exp (−i (k1−k2)l)\n·exp (i (Λ k1−Λk2)t)nk1(1−nk2),\n∝an}b∇acketle{tsz∝an}b∇acket∇i}ht=1\nNN/summationdisplay\nn=1∝an}b∇acketle{tsz\nn∝an}b∇acket∇i}ht=−1\n2N/summationdisplay\nktanhβΛk\n2. (34)\nPlugging Eq. (34) into Eq. (4) we get the desired transverse d ynamic struc-\nture factor\nSzz(k,ω) =N/summationdisplay\nl=1exp(−ikl)/integraldisplay∞\n−∞dtexp(iωt)∝an}b∇acketle{t(sz\nn(t)− ∝an}b∇acketle{tsz∝an}b∇acket∇i}ht)/parenleftbig\nsz\nn+l− ∝an}b∇acketle{tsz∝an}b∇acket∇i}ht/parenrightbig\n∝an}b∇acket∇i}ht\n=/integraldisplayπ\n−πdk1nk1(1−nk1+k)δ(ω+ Λk1−Λk1+k)\n=/summationdisplay\nk⋆nk⋆(1−nk+k⋆)\n2/vextendsingle/vextendsingleJsink\n2cos/parenleftbigk\n2+k⋆/parenrightbig/vextendsingle/vextendsingle(35)\nwhere −π≤k⋆< π are the solutions of the equation ω=\n−2Jsin(k/2)sin(k/2 +k⋆).\nThezzdynamic structure factor (35) is governed exclusively by a t wo-\nfermion (one particle and one hole) excitation continuum. T he properties of\nthe two-fermion excitation continuum were discussed by G. M ¨ uller et al34;\nwe present these results briefly below. The boundaries of the two-fermion\ncontinuum in the plane wave-vector k– frequency ω(we assume ω≥0,\n−π≤k<π) are determined by the equations\nω=−Λk1+ Λk2, k=−k1+k2( mod (2π)),Λk= Ω +Jcosk,(36)\nwhere −π≤k1< π. Moreover, in the ground state we have to require in\nadditionnk1>0 and 1 −nk2>0, i.e. Λ k1≤0 and Λ k2≥0.\nWe start with the zero-temperature case. In this case the two -fermion\nexcitation continuum exists as long as |Ω|<|J|. Let us introduce the\nparameterα= arccos( |Ω|/|J|) and the following characteristic lines in theDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 13\nk–ωplane\nω1(k)\n|J|= 2/vextendsingle/vextendsingle/vextendsingle/vextendsinglesink\n2sin/parenleftbigg|k|\n2−α/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (37)\nω2(k)\n|J|= 2/vextendsingle/vextendsingle/vextendsingle/vextendsinglesink\n2sin/parenleftbigg|k|\n2+α/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (38)\nω3(k)\n|J|= 2/vextendsingle/vextendsingle/vextendsingle/vextendsinglesink\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (39)\nThe two-fermion dynamic quantities in the ground state may h ave non-zero\nvalues only within a restricted region of the k–ωplane with the lower bound-\naryωl(k) =ω1(k) and the upper boundary ωu(k) =ω2(k) if|k| ≤π−2αor\nωu(k) =ω3(k) ifπ−2α≤ |k|. Obviously, the two-fermion dynamic quan-\ntities may have only three soft modes k0={0,±2α}. Moreover, there is a\nmiddle boundary of the two-fermion excitation continuum ωm(k) =ω2(k)\nifπ−2α≤ |k|along which the two-fermion dynamic quantities exhibit a\njump increasing their values by 2. Finally, the two-fermion dynamic quan-\ntities show one-dimensional square-root van Hove divergen cies along the\ncurveωs(k) =ω3(k). In Fig. 2 we display the characteristic lines (37), (38),\nFig. 2. The two-fermion excitation continuum which governs the ground-state two-\nfermion dynamic quantities. |J|= 1,|Ω|= 0.1 (a), |Ω|= 0.9 (b). We show the lower\nboundaries (bold lines), the middle boundaries (dashed lin es), the upper boundaries (thin\nlines) and the lines of potential singularities (dotted lin es).\n(39) which give the boundaries of the two-fermion continuum and potential\nsoft modes and singularities.\nAs temperature increases the lower boundary is smeared-out and finally\ndisappears, the upper boundary becomes ω3(k) along which van Hove sin-\ngularities occur. In the high-temperature limit the two-fe rmion dynamic\nstructure factor becomes Ω-independent.December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n14 O. Derzhko\nIn Fig. 3 we display the transverse dynamic structure factor Szz(k,ω)\nFig. 3. Szz(k, ω) (gray-scale plots) for the chain (15) with J=−1, Ω = 0 (a), Ω = 0 .3\n(b), Ω = 0 .6 (c) at T= 0 and at T→ ∞ (d).\n(35) at zero temperature (panels a, b, c) and in the high-temp erature limit\n(panel d).\nThere are other dynamic quantities which probe the two-ferm ion exci-\ntation continuum. Let us consider the dimer operator\nDn=sx\nnsx\nn+1+sy\nnsy\nn+1→1\n2/parenleftig\nc†\nncn+1−cnc†\nn+1/parenrightig\n. (40)\nThat operator is related to a perturbation to the Hamiltonia n (15) which\nmimics dimerization, ǫ/summationtext\nncos(πn)Dn. The dynamics of fluctuations of the\ndimer operator can be measured experimentally: the corresp onding dimer\ndynamic structure factor is relevant to phonon-assisted op tical absorption\nprocesses in magnetic-chain compounds36.\nThe calculation of the time-dependent dimer-dimer correla tion function\nrepeats all steps discussed above while deriving (34) and en ds up with\n∝an}b∇acketle{tDn(t)Dn+l∝an}b∇acket∇i}ht − ∝an}b∇acketle{tD∝an}b∇acket∇i}ht2=1\nN2/summationdisplay\nk1,k2cos2k1+k2\n2exp (−i (k1−k2)l)\n·exp (i (Λ k1−Λk2)t)nk1(1−nk2),\n∝an}b∇acketle{tD∝an}b∇acket∇i}ht=1\nNN/summationdisplay\nn=1∝an}b∇acketle{tDn∝an}b∇acket∇i}ht=−1\n2N/summationdisplay\nkcosktanhβΛk\n2.(41)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 15\nInserting Eq. (41) into Eq. (4) we get the dimer dynamic struc ture factor\nSDD(k,ω) =N/summationdisplay\nl=1exp(−ikl)/integraldisplay∞\n−∞dtexp(iωt)∝an}b∇acketle{t(Dn(t)− ∝an}b∇acketle{tD∝an}b∇acket∇i}ht) (Dn+l− ∝an}b∇acketle{tD∝an}b∇acket∇i}ht)∝an}b∇acket∇i}ht\n=/integraldisplayπ\n−πdk1cos2/parenleftbigg\nk1+k\n2/parenrightbigg\nnk1(1−nk1+k)δ(ω+ Λk1−Λk1+k)\n=/summationdisplay\nk⋆cos2/parenleftbigk\n2+k⋆/parenrightbig\nnk⋆(1−nk+k⋆)\n2/vextendsingle/vextendsingleJsink\n2cos/parenleftbigk\n2+k⋆/parenrightbig/vextendsingle/vextendsingle.(42)\nWe can also calculate the zDandDzdynamic structure factors\nSzD(k,ω) = exp/parenleftbigg\nik\n2/parenrightbigg/integraldisplayπ\n−πdk1cos/parenleftbigg\nk1+k\n2/parenrightbigg\n·nk1(1−nk1+k)δ(ω+ Λk1−Λk1+k),\nSDz(k,ω) = (SzD(k,ω))∗. (43)\nIn Fig. 4 we display the dynamic structure factors SDD(k,ω) (42) and\nFig. 4. SDD(k, ω) (a, b) and SzD(k, ω) (multiplied by exp ( −ik/2)) (c, d) (gray-scale\nplots) for the chain (15) with J=−1.SDD(k, ω): Ω = 0 .3,T= 0 (a), T→ ∞ (b).\nSzD(k, ω): Ω = 0 .3,T= 0 (c), Ω = 0 .3,T= 0.1 (d).\nSzD(k,ω) (43) at zero temperature (panels a, c), at low temperature ( panel\nd) and in the high-temperature limit (panel b).December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n16 O. Derzhko\nComparing Eqs. (35), (42), (43) we immediately recognize th at all these\ndynamic quantities are governed exclusively by the two-fer mion excitation\ncontinuum (for other two-fermion dynamic quantities see be low and also\nRefs.37,7) and therefore all of them exhibit generic properties inher ent\nin the two-fermion dynamic quantities. However, they also e xhibit some\nspecific properties originating from additional factors in the integrands in\nEqs. (35), (42), (43) (e.g. singularities may be suppressed etc., compare\nFig. 3b and Fig. 4a). In general, the dynamic structure facto r governed by\nthe two-fermion excitation continuum can be written in the f ollowing form\nSAB(k,ω) =/integraldisplayπ\n−πdk1dk2CAB(k1,k2)\n·nk1(1−nk2)δ(ω+ Λk1−Λk2)δk+k1−k2,0,\nCzz(k1,k2) = 1,\nCDD(k1,k2) = cos2k1+k2\n2,\nCzD(k1,k2) =1\n2(exp(−ik1) + exp (ik2)),\nCDz(k1,k2) = (CzD(k1,k2))∗. (44)\nAll these quantities show generic properties (spectral bou ndaries, soft\nmodes, singularity structure) and specific properties cont rolled by\nCAB(k1,k2).\n4.2.Four-fermion excitations\nWe proceed by considering more complicate dynamic quantiti es. Namely,\nconsider a trimer operator38\nTn=sx\nnsx\nn+2+sy\nnsy\nn+2\n→1\n2/parenleftig\nc†\nncn+2−cnc†\nn+2−2c†\nnc†\nn+1cn+1cn+2+ 2cnc†\nn+1cn+1c†\nn+2/parenrightig\n.(45)\nThat operator enters as a perturbation to the Hamiltonian (1 5) which mim-\nics trimerization, ǫ/summationtext\nncos(2πn/3)Tn. The dynamics of fluctuations of the\ntrimer operator, although it can be analyzed rigorously, is less evident from\nthe experimental point of view. Its importance, however, is justified as a\nquantity of intermediate complexity between the zzand thexxandxy\ndynamic quantities.\nThe calculation of the time-dependent trimer-trimer corre lation function\ncontains no new conceptual ideas but is somewhat tedious. Th e final resultDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 17\nfor the time-dependent trimer correlation function reads\n∝an}b∇acketle{tTn(t)Tn+l∝an}b∇acket∇i}ht − ∝an}b∇acketle{tT∝an}b∇acket∇i}ht2=1\nN2/summationdisplay\nk1,k2C(2)\nTT(k1,k2)exp ( −i (k1−k2)l)\n·exp(i (Λ k1−Λk2)t)nk1(1−nk2)\n+1\nN4/summationdisplay\nk1,k2,k3,k4C(4)\nTT(k1,k2,k3,k4)exp ( −i (k1+k2−k3−k4)l)\n·exp (i (Λ k1+ Λk2−Λk3−Λk4)t)nk1nk2(1−nk3)(1−nk4),\n∝an}b∇acketle{tT∝an}b∇acket∇i}ht=1\nNN/summationdisplay\nn=1∝an}b∇acketle{tTn∝an}b∇acket∇i}ht=c2+ 2c2\n1−2c0c2(46)\nwith\nC(2)\nTT(k1,k2) = (1 −2c0)2cos2(k1+k2)\n+4c1(1−2c0)/parenleftbigg\ncos2/parenleftbigg\nk1+k2\n2/parenrightbigg\n+ cos2/parenleftbiggk1\n2+k2/parenrightbigg/parenrightbigg\n+4c2\n1/parenleftbig\ncos2k1+ cos2k2/parenrightbig\n+8/parenleftbig\n−c2+c2\n1+ 2c0c2/parenrightbig\ncos2k1+k2\n2+ 8c2\n1cos2k1−k2\n2\n+4c1(1−2c0−4c2)/parenleftbigg\ncos2k1\n2+ cos2k2\n2/parenrightbigg\n+4c2−8c1−8c2\n1+ 4c2\n2+ 16c0c1−8c0c2+ 16c1c2, (47)\nC(4)\nTT(k1,k2,k3,k4)\n= 16 sin2k1−k2\n2sin2k3−k4\n2cos2k1+k2+k3+k4\n2≥0 (48)\nandcp= (1/N)/summationtext\nkcos(pk)nk. Obviously, the calculation of such an aver-\nage as ∝an}b∇acketle{tc†\nk1c†\nk2ck3ck4c†\nk5c†\nk6ck7ck8∝an}b∇acket∇i}htaccording to the Wick-Bloch-de Dominicis\ntheorem is rather complicated. Substituting (46) into Eq. ( 4) we obtain the\nfollowing result for the trimer dynamic structure factor\nSTT(k,ω) =N/summationdisplay\nl=1exp(−ikl)/integraldisplay∞\n−∞dtexp(iωt)∝an}b∇acketle{t(Tn(t)− ∝an}b∇acketle{tT∝an}b∇acket∇i}ht)(Tn+l− ∝an}b∇acketle{tT∝an}b∇acket∇i}ht)∝an}b∇acket∇i}ht\n=S(2)\nTT(k,ω) +S(4)\nTT(k,ω) (49)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n18 O. Derzhko\nwith\nS(2)\nTT(k,ω) =/integraldisplayπ\n−πdk1C(2)\nTT(k1,k1+k)nk1(1−nk1+k)δ(ω+ Λk1−Λk1+k),(50)\nS(4)\nTT(k,ω) =1\n4π2/integraldisplayπ\n−πdk1dk2dk3C(4)\nTT(k1,k2,k3,k1+k2−k3+k)\n·nk1nk2(1−nk3)(1−nk1+k2−k3+k)δ(ω+ Λk1+ Λk2−Λk3−Λk1+k2−k3+k).(51)\nFor further details see Ref.38.\nAs can be seen from Eqs. (49), (50), (51) the trimer dynamic st ruc-\nture factor is governed both by the two-fermion (one particl e and one\nhole) excitation continuum discussed above, the term S(2)\nTT(k,ω), and by\nthe four-fermion (two particles and two holes) excitation c ontinuum, the\ntermS(4)\nTT(k,ω). The four-fermion excitation continuum is determined by\nthe conditions\nω=−Λk1−Λk2+ Λk3+ Λk4, k=−k1−k2+k3+k4( mod (2π)),(52)\nwhere −π≤k1,k2,k3<πand Λ k= Ω +Jcosk. Moreover, in the ground\nstate we have to require in addition nk1>0,nk2>0, 1−nk3>0, 1−nk4>\n0, i.e. Λ k1≤0, Λ k2≤0, Λ k3≥0, Λ k4≥0.\nWe start with the zero-temperature case. The lower boundary is given\nby one of the following curves\nω(1)\nl(k)\n|J|= 2 sin|k|\n2sin/parenleftbigg\nα−|k|\n2/parenrightbigg\n,\nω(2)\nl(k)\n|J|= 4 cosk\n4cos/parenleftbigg\nα+|k|\n4/parenrightbigg\n,\nω(3)\nl(k)\n|J|=−2 sin/parenleftbigg\nα+|k|\n2/parenrightbigg\nsin/parenleftbigg\n2α+|k|\n2/parenrightbigg\n,\nω(4)\nl(k)\n|J|=−2 sin/parenleftbigg\nα−|k|\n2/parenrightbigg\nsin/parenleftbigg\n2α−|k|\n2/parenrightbigg\n,\nω(5)\nl(k)\n|J|=−4 sin|k|\n4sin/parenleftbigg\nα−|k|\n4/parenrightbigg\n(53)\ndepending on the value of Ω, |Ω| ≤ |J|and the value of k,π≤k < π as\nis shown in the left panel in Fig. 5. The boundary between the r egioni\n(whereω(i)\nl(k) is the lower boundary) and the region j(whereω(j)\nl(k) is\nthe lower boundary) (see the left panel in Fig. 5) is given by t he formula\n|k|=lij(α) wherel12(α) = 4 arctan/parenleftig/parenleftig\ntanα−√\ntan2α−3/parenrightig\n/3/parenrightig\n,|k| ≤\n2π/3;l13(α) =π−α,π/2≤ |k| ≤2π/3;l14(α) = 2α;l23(α) = 2π−December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 19\nFig. 5. The lower boundary ωl(k) (left panel) and the upper boundary ωu(k) (right\npanel) of the four-fermion excitation continuum in the plan e wave-vector k– transverse\nfield Ω, |J|= 1.\n4α;l34(α) =|k|+ cosα−1/2, 2π/3≤ |k| ≤π;l45(α) = 4α(for further\ndetails see Ref.38). The four-fermion contribution to dynamic quantities\nmay exhibit soft modes |k0|={0,2π−4α,2α,4α}. Next we pass to the\nupper boundary of the four-fermion excitation continuum wh ich is given by\none of the following curves\nω(1)\nu(k)\n|J|= 4 cosk\n4, (54)\nω(2)\nu(k)\n|J|= 4 cosk\n4cos/parenleftbigg\nα−|k|\n4/parenrightbigg\n(55)\ndepending on the value of Ω, |Ω| ≤ |J|and the value of k,π≤k < π as\nis shown in the right panel in Fig. 5. The boundary between the regions\n1 and 2 is given by the curve |k|= 4α. In Fig. 6 we compare the ground-\nstate two-fermion and four-fermion excitation continua fo r two values of the\ntransverse field Ω. The four-fermion excitation continuum a lways contains\nthe two-fermion excitation continuum. The lower boundarie s may coincide\n(e.g. in the zero-field case the lower boundary is |J|sin|k|for both continua,\npanel a in Fig. 6a) whereas the upper boundaries are different .\nNext we turn to the van Hove singularities inherent in the fou r-fermion\ndynamic quantities. Evidently, the quantity\nS(k,ω) =/integraldisplayπ\n−πdk1/integraldisplayπ\n−πdk2/integraldisplayπ\n−πdk3S(k1,k2,k3,k)\n·δ(ω− |J|cosk1− |J|cosk2+|J|cosk3+|J|cos(k+k1+k2−k3))(56)\nmay exhibit van Hove singularities characteristic to the th ree-dimensionalDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n20 O. Derzhko\nFig. 6. Lower boundaries and upper boundaries of the two-fer mion and four-fermion\nexcitation continua for |J|= 1 and Ω = 0 (a) and Ω = 0 .3 (b) at T= 0. The two-fermion\ncontinuum is shown shaded.\ndensity of states. The lines of potential singularities are as follows\nω(1)\ns(k)\n|J|= 2 sin|k|\n2,\nω(2)\ns(k)\n|J|= 4 sin|k|\n4,\nω(3)\ns(k)\n|J|= 4 cosk\n4. (57)\nThe four-fermion dynamic quantities may exhibit cusp singu larities (akin to\ndensity-of-states effects in thee dimensions) along these c urves. We illustrate\npotential singularities in the frequency profiles of S(k,ω) (56) at different\nkin Fig. 7.\nFor nonzero temperatures the lower boundary is smeared out a nd fi-\nnally disappears. The upper boundary is given by Eq. (54). In the high-\ntemperature limit the properties of the four-fermion excit ation continuum\nbecome Ω-independent.\nAfter discussing some generic properties of the four-fermi on dynamic\nquantities (inherent in any four-fermion dynamic quantity ) we illustrate\nsome specific properties conditioned by the function C(4)\nTT(k1,k2,k3,k4) (48).\nIn Fig. 8 we display the trimer dynamic structure factor (49) . The con-\ntributions of the two-fermion excitation continuum and the four-fermion\nexcitation continuum to this quantity can be easily disting uished.\nWe may formally introduce the polymer operator\nP(l)\nn=sx\nnsx\nn+l+sy\nnsy\nn+l(58)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 21\nFig. 7. S(k, ω) (56) vs ωatk= 2π/3 (a) and k=π(b) for S(k1, k2, k3, k) = 1 (bold\ncurves) and S(k1, k2, k3, k) =nk1nk2`\n1−nk3´ `\n1−nk1+k2−k3+k´\n,T= 0 for Ω = 0\n(solid curves), Ω = 0 .3 (long-dashed curves), Ω = 0 .6 (short-dashed curves), Ω = 0 .9\n(dotted curves). Vertical lines denote the values of ω(j)\ns(k),j= 1,2,3 (57).\nFig. 8. STT(k, ω) (49) (gray-scale plots) of the chain (15) with J=−1, Ω = 0 (a),\nΩ = 0 .3 (b), Ω = 0 .6 (c) at T= 0 and at T→ ∞ (d).\n(evidently P(1)\nn=DnandP(2)\nn=Tn). Now the dynamic polymer structure\nfactorSPP(k,ω) will involve 2 m-fermion excitations with m= 1,2,...,l .\nThese quantities are of moderate complexity in comparison w ithSxx(k,ω)\nandSxy(k,ω) which are enormously complex (see below). We also note that\nin the limit l→ ∞\n∝an}b∇acketle{tP(l)\nn(t)P(l)\nn+m∝an}b∇acket∇i}htl→∞−→2∝an}b∇acketle{tsx\nn(t)sx\nn+m∝an}b∇acket∇i}ht2+ 2∝an}b∇acketle{tsx\nn(t)sy\nn+m∝an}b∇acket∇i}ht2. (59)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n22 O. Derzhko\nThe last term in Eq. (59) is nonzero only if Ω ∝ne}ationslash= 0.\nIn passing, we note that the multimagnon continua of quantum spin\nchains have been discussed recently on general ground by T. B arnes39. Ob-\nviously, there is an essential difference in comparison with our case, since\nthe Jordan-Wigner fermions obey the Fermi statistics and th is point has im-\nportant consequences for the four-fermion excitation cont inuum considered\nin some detail above.\n4.3.Many-fermion excitations\nWe pass to dynamic structure factors which are governed by ma ny-fermion\nexcitations. Let us recall that according to the Jordan-Wig ner transforma-\ntion we have\nsx\nn=1\n2/parenleftig\n1−2c†\n1c1/parenrightig\n.../parenleftig\n1−2c†\nn−1cn−1/parenrightig/parenleftbig\nc†\nn+cn/parenrightbig\n=1\n2ϕ+\n1ϕ−\n1...ϕ+\nn−1ϕ−\nn−1ϕ+\nn,\nsy\nn=1\n2iϕ+\n1ϕ−\n1...ϕ+\nn−1ϕ−\nn−1ϕ−\nn,\nsz\nn=−1\n2ϕ+\nnϕ−\nn (60)\nwhere we have introduced the operators ϕ±\nm=c†\nm±cm. Obviously the\noperatorsϕ±\nmare linear combinations of the operators ck,c†\nkin terms of\nwhich the Hamiltonian is diagonal (see Eq. (33)).\nConsider now the xxtime-dependent spin correlation function\n4∝an}b∇acketle{tsx\nj(t)sx\nj+n∝an}b∇acket∇i}ht=∝an}b∇acketle{tϕ+\n1(t)ϕ−\n1(t)...ϕ+\nj−1(t)ϕ−\nj−1(t)ϕ+\nj(t)\n·ϕ+\n1ϕ−\n1...ϕ+\nj−1ϕ−\nj−1ϕ+\njϕ−\njϕ+\nj+1ϕ−\nj+1...ϕ+\nj+n−1ϕ−\nj+n−1ϕ+\nj+n∝an}b∇acket∇i}ht.(61)\nIt contains a product of 2(2 j+n−1)ϕ±operators (in contrast to\n4∝an}b∇acketle{tsz\nj(t)sz\nj+n∝an}b∇acket∇i}ht=∝an}b∇acketle{tϕ+\nj(t)ϕ−\nj(t)ϕ+\nj+nϕ−\nj+n∝an}b∇acket∇i}htwhich contains the product of only\nfourϕ±operators). Therefore the calculation of xxandxydynamic quan-\ntities (which are governed by many-fermion excitations) is essentially more\ncomplicated.\nExact analytical results for xxandxydynamic quantities are rather\nscarce. At the high-temperature limit T→ ∞ we know40,41that\n4∝an}b∇acketle{tsx\nj(t)sx\nj+n∝an}b∇acket∇i}ht=δn,0cos(Ωt)exp/parenleftbigg\n−1\n4J2t2/parenrightbigg\n,\n4∝an}b∇acketle{tsx\nj(t)sy\nj+n∝an}b∇acket∇i}ht=−δn,0sin(Ωt)exp/parenleftbigg\n−1\n4J2t2/parenrightbigg\n. (62)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 23\nAt zero temperature the xxandxytime-dependent correlation functions\nare extremely simple only when |Ω|>|J|42. Consider for example the case\nΩ>|J|when the ground state is completely polarized |GSs∝an}b∇acket∇i}ht=/producttextN\nn=1| ↓n∝an}b∇acket∇i}ht\n(in spin language) or completely empty ck|GSc∝an}b∇acket∇i}ht= 0 (in fermionic lan-\nguage). Owing to the simplicity of the ground state s+\nm|GSs∝an}b∇acket∇i}ht=c†\nm|GSc∝an}b∇acket∇i}ht=/parenleftig\n1/√\nN/parenrightig/summationtext\nkexp (ikm)c†\nk|GSc∝an}b∇acket∇i}ht,s−\nm|GSs∝an}b∇acket∇i}ht= 0. Therefore\n4∝an}b∇acketle{tsx\nj(t)sx\nj+n∝an}b∇acket∇i}ht=∝an}b∇acketle{tGSs|s−\nj(t)s+\nj+n|GSs∝an}b∇acket∇i}ht\n=1\nN/summationdisplay\nkexp (ikn−i (Ω +Jcosk)t),\n4∝an}b∇acketle{tsx\nj(t)sy\nj+n∝an}b∇acket∇i}ht=−i∝an}b∇acketle{tGSs|s−\nj(t)s+\nj+n|GSs∝an}b∇acket∇i}ht=−4i∝an}b∇acketle{tsx\nj(t)sx\nj+n∝an}b∇acket∇i}ht (63)\nand as a result\nSxx(k,ω) = iSxy(k,ω) =π\n2δ(ω−Ω−Jcosk). (64)\nMany results at T= 0 refer to the asymptotic behavior of the xxorxy\ntime-dependent spin correlation functions43,44. From the paper by A. R. Its\net al we know the long-time asymptotic behavior at nonzero te mperatures\n∝an}b∇acketle{ts+\nj(t)s−\nj+n∝an}b∇acket∇i}ht ∼/braceleftigg\nexp (f(n,0)),n\nJt>1,\nt2(ν2\n−+ν2\n+)exp (f(n,t)),n\nJt<1,\nf(n,t) =1\n2π/integraldisplayπ\n−πdp|n+Jtsinp|ln/vextendsingle/vextendsingle/vextendsingle/vextendsingletanhβ(Ω−Jcosp)\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nν±=1\n2πln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingletanhβ/parenleftbigg\nΩ∓J/radicalig\n1−/parenleftbign\nJt/parenrightbig2/parenrightbigg\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.(65)\nOn the other hand, we can obtain the xxandxydynamic quantities\nnumerically45,46,47,48,49,50,51. Consider the slightly more complicated in-\nhomogeneous spin-1/2 anisotropic XYchain in a transverse field with the\nHamiltonian\nH=N/summationdisplay\nj=1Ωjsz\nj+N−1/summationdisplay\nj=1/parenleftbig\nJxx\njsx\njsx\nj+1+Jxy\njsx\njsy\nj+1+Jyx\njsy\njsx\nj+1+Jyy\njsy\njsy\nj+1/parenrightbig\n→ −1\n2N/summationdisplay\nj=1Ωj+N/summationdisplay\ni,j=1/parenleftbigg\nc†\niAijcj+1\n2/parenleftig\nc†\niBijc†\nj−ciB∗\nijcj/parenrightig/parenrightbigg\n(66)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n24 O. Derzhko\nwhere\nAij= Ω iδij+J+−\niδj,i+1+J−+\ni−1δj,i−1=A∗\nji,\nBij=J++\niδj,i+1−J++\ni−1δj,i−1=−Bji,\nJ+−\nj=1\n4/parenleftbig\nJxx\nj+Jyy\nj+ i/parenleftbig\nJxy\nj−Jyx\nj/parenrightbig/parenrightbig\n=/parenleftbig\nJ−+\nj/parenrightbig∗,\nJ++\nj=1\n4/parenleftbig\nJxx\nj−Jyy\nj−i/parenleftbig\nJxy\nj+Jyx\nj/parenrightbig/parenrightbig\n=/parenleftbig\nJ−−\nj/parenrightbig∗. (67)\nTo diagonalize a form bilinear in Fermi operators like (66) w e perform the\nlinear canonical transformation\nηk=N/summationdisplay\nn=1/parenleftbig\ngkncn+hknc†\nn/parenrightbig\n, η†\nk=N/summationdisplay\nn=1/parenleftbig\ng∗\nknc†\nn+h∗\nkncn/parenrightbig\n. (68)\nThe resulting Hamiltonian reads as follows\nH=N/summationdisplay\nk=1Λk/parenleftbigg\nη†\nkηk−1\n2/parenrightbigg\n,\n/braceleftig\nηk′,η†\nk′′/bracerightig\n=δk′k′′,{ηk′,ηk′′}=/braceleftig\nη†\nk′,η†\nk′′/bracerightig\n= 0 (69)\nif the coefficients gkn,hknsatisfy the set of equations\n/parenleftbig\ngkhk/parenrightbig\nM= Λ k/parenleftbig\ngkhk/parenrightbig\n,\ngk=/parenleftbig\ngk1...g kN/parenrightbig\n,hk=/parenleftbig\nhk1...h kN/parenrightbig\n,M=/parenleftbiggA B\n−B∗−A∗/parenrightbigg\n.(70)\nFurther it may be convenient to introduce the linear combina tions Φ kn=\ngkn+hknand Ψ kn=gkn−hknwhich enter the relations\nϕ+\nj=c†\nj+cj=N/summationdisplay\np=1/parenleftbig\nΦpjη†\np+ Φ∗\npjηp/parenrightbig\n,\nϕ−\nj=c†\nj−cj=N/summationdisplay\np=1/parenleftbig\nΨpjη†\np−Ψ∗\npjηp/parenrightbig\n. (71)\nWe calculate the time-dependent spin correlation function s using the Wick-\nBloch-de Dominicis theorem. For example,\n4∝an}b∇acketle{tsz\nn(t)sz\nn+m∝an}b∇acket∇i}ht=∝an}b∇acketle{tϕ+\nn(t)ϕ−\nn(t)ϕ+\nn+mϕ−\nn+m∝an}b∇acket∇i}ht\n=∝an}b∇acketle{tϕ+\nnϕ−\nn∝an}b∇acket∇i}ht∝an}b∇acketle{tϕ+\nn+mϕ−\nn+m∝an}b∇acket∇i}ht\n−∝an}b∇acketle{tϕ+\nn(t)ϕ+\nn+m∝an}b∇acket∇i}ht∝an}b∇acketle{tϕ−\nn(t)ϕ−\nn+m∝an}b∇acket∇i}ht+∝an}b∇acketle{tϕ+\nn(t)ϕ−\nn+m∝an}b∇acket∇i}ht∝an}b∇acketle{tϕ−\nn(t)ϕ+\nn+m∝an}b∇acket∇i}ht.(72)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 25\nThe r.h.s. of Eq. (72) may be compactly written as the Pfaffian o f the 4 ×4\nantisymmetric matrix\n4∝an}b∇acketle{tsz\nn(t)sz\nn+m∝an}b∇acket∇i}ht\n= Pf\n0 ∝an}b∇acketle{tϕ+\nnϕ−\nn∝an}b∇acket∇i}ht ∝an}b∇acketle{tϕ+\nn(t)ϕ+\nn+m∝an}b∇acket∇i}ht ∝an}b∇acketle{tϕ+\nn(t)ϕ−\nn+m∝an}b∇acket∇i}ht\n−∝an}b∇acketle{tϕ+\nnϕ−\nn∝an}b∇acket∇i}ht 0 ∝an}b∇acketle{tϕ−\nn(t)ϕ+\nn+m∝an}b∇acket∇i}ht ∝an}b∇acketle{tϕ−\nn(t)ϕ−\nn+m∝an}b∇acket∇i}ht\n−∝an}b∇acketle{tϕ+\nn(t)ϕ+\nn+m∝an}b∇acket∇i}ht −∝an}b∇acketle{tϕ−\nn(t)ϕ+\nn+m∝an}b∇acket∇i}ht 0 ∝an}b∇acketle{tϕ+\nn+mϕ−\nn+m∝an}b∇acket∇i}ht\n−∝an}b∇acketle{tϕ+\nn(t)ϕ−\nn+m∝an}b∇acket∇i}ht −∝an}b∇acketle{tϕ−\nn(t)ϕ−\nn+m∝an}b∇acket∇i}ht −∝an}b∇acketle{tϕ+\nn+mϕ−\nn+m∝an}b∇acket∇i}ht 0\n.(73)\nSimilarly (see Eq. (61)), for the more complicated xxtime-dependent spin\ncorrelation function we have\n4∝an}b∇acketle{tsx\nn(t)sx\nn+m∝an}b∇acket∇i}ht\n= Pf\n0 ∝an}b∇acketle{tϕ+\n1ϕ−\n1∝an}b∇acket∇i}ht ∝an}b∇acketle{tϕ+\n1ϕ+\n2∝an}b∇acket∇i}ht...∝an}b∇acketle{tϕ+\n1(t)ϕ+\nn+m∝an}b∇acket∇i}ht\n−∝an}b∇acketle{tϕ+\n1ϕ−\n1∝an}b∇acket∇i}ht 0 ∝an}b∇acketle{tϕ−\n1ϕ+\n2∝an}b∇acket∇i}ht...∝an}b∇acketle{tϕ−\n1(t)ϕ+\nn+m∝an}b∇acket∇i}ht\n......... · · ·...\n−∝an}b∇acketle{tϕ+\n1(t)ϕ+\nn+m∝an}b∇acket∇i}ht −∝an}b∇acketle{tϕ−\n1(t)ϕ+\nn+m∝an}b∇acket∇i}ht −∝an}b∇acketle{tϕ+\n2(t)ϕ+\nn+m∝an}b∇acket∇i}ht... 0\n,(74)\ni.e.∝an}b∇acketle{tsx\nj(t)sx\nj+n∝an}b∇acket∇i}htcan be written as a Pfaffian of a 2(2 j+n−1)×2(2j+n−1)\nantisymmetric matrix. The elementary contractions involv ed in (73), (74)\nread\n∝an}b∇acketle{tϕ+\nj(t)ϕ+\nm∝an}b∇acket∇i}ht=N/summationdisplay\np=1/parenleftbig\nΦpjΦ∗\npmF(Λp) + Φ∗\npjΦpmF(−Λp)/parenrightbig\n,\n∝an}b∇acketle{tϕ+\nj(t)ϕ−\nm∝an}b∇acket∇i}ht=N/summationdisplay\np=1/parenleftbig\n−ΦpjΨ∗\npmF(Λp) + Φ∗\npjΨpmF(−Λp)/parenrightbig\n,\n∝an}b∇acketle{tϕ−\nj(t)ϕ+\nm∝an}b∇acket∇i}ht=N/summationdisplay\np=1/parenleftbig\nΨpjΦ∗\npmF(Λp)−Ψ∗\npjΦpmF(−Λp)/parenrightbig\n,\n∝an}b∇acketle{tϕ−\nj(t)ϕ−\nm∝an}b∇acket∇i}ht=−N/summationdisplay\np=1/parenleftbig\nΨpjΨ∗\npmF(Λp) + Ψ∗\npjΨpmF(−Λp)/parenrightbig\n,\nF(Λp) =exp(iΛ pt)\n1 + exp (βΛp). (75)\nIt is worthwhile to recall some properties of the Pfaffians whi ch are\nused for calculating them. In the first numerical studies the authors used\nthe relation\n(PfA)2= detA (76)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n26 O. Derzhko\nand computed numerically the determinants which gave the va lues of Pfaf-\nfians according to (76). On the other hand, the Pfaffian may be co mputed\ndirectly46noting that\nPf/parenleftbig\nUTAU/parenrightbig\n= detUPfA (77)\nand that\nPf\n0R120 0...0\n−R120 0 0 ...0\n0 0 0 R34...0\n0 0 −R340...0\n..................\n0 0 0 0 ...0\n=R12R34.... (78)\nWe use the approach described to calculate the xxandxydynamic\nstructure factors for the spin-1/2 transverse XXchain numerically52. To\nestimate the quality of the numerical procedure we compare o ur numer-\nical findings with exact analytical results in the high-temp erature limit\nand with exact asymptotics for finite temperatures in Fig. 9. Knowing the\nFig. 9. Panel a: Time-dependence of the autocorrelation fun ction /angbracketleftsx\nj(t)sx\nj/angbracketright,j= 51\nat infinite temperature obtained numerically (symbols) and analytically (see Eq. (62))\n(solid curves). Ω = 2 ,1 (downward and upward triangles), Ω = 0 .5 (open circles), Ω =\n0.1 (squares), Ω = 0 (filled circles). Panel b: Time-dependence of the real part of the\nautocorrelation function /angbracketleftsx\nj(t)sx\nj/angbracketright,j= 51 at Ω = 0 for various temperatures obtained\nnumerically (symbols) in comparison with asymptotics (65) .β= 5,1,0.1,0.00001 (from\ntop to bottom). The exact analytical result for β= 0 is also shown (the lowest curve).\nEvidently only the slopes of the asymptotics should be compa red with the numerical\nresults.\ntime-dependent correlation functions we obtain the corres ponding dynamicDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 27\nstructure factors according to\nSxx(k,ω) =/summationdisplay\nn=0,±1,...exp(−ikn)2ℜ/parenleftbigg/integraldisplay∞\n0dtexp(i (ω+ iǫ)t)∝an}b∇acketle{tsx\nj(t)sx\nj+n∝an}b∇acket∇i}ht/parenrightbigg\n,\nSxy(k,ω) =/summationdisplay\nn=0,±1,...exp (−ikn)2iℑ/parenleftbigg/integraldisplay∞\n0dtexp (i (ω+ iǫ)t)∝an}b∇acketle{tsx\nj(t)sy\nj+n∝an}b∇acket∇i}ht/parenrightbigg\n(79)\nwithǫ→+0. In practice we consider chains of N= 400 sites, take j=\n41,51,61,nup to 50 or up to 100, and set ǫ= 0...0.001...0.1 (see\nRef.52). The results of our calculations for Sxx(k,ω) andSxy(k,ω) are\nillustrated in Figs. 10, 11.\nFig. 10. Sxx(k, ω) (gray-scale plots) for the chain (15) with J=−1, for Ω = 0 .0001\n(a), Ω = 0 .3 (b), Ω = 0 .6 (c) at low temperature β= 20 and for Ω = 0 .6 in the\nhigh-temperature limit β= 0 (d).\nLet us recall, the transverse dynamic structure factor Szz(k,ω) probes\ntwo-particle excitations, i.e. it is governed by the excita tions which are\ncomposed of two Jordan-Wigner spinless fermions. The two-f ermion exci-\ntation continuum has a sharp upper frequency cutoff at which Szz(k,ω)\nmay diverge. At T= 0 it has also a sharp lower frequency cutoff which\ntouchesω= 0 atk0(soft modes). Szz(k,ω) is almost structureless (apart\nfrom upper boundary singularities) and exists only for |Ω|<|J|. In the\nhigh-temperature limit T→ ∞Szz(k,ω) becomes Ω-independent. All these\nfeatures are nicely seen in Fig. 3.December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n28 O. Derzhko\nFig. 11. i Sxy(k, ω) (gray-scale plots) for the chain (15) with J=−1 for Ω = 0 .1 (a, b),\nΩ = 0 .3 (c, d) at low temperature β= 20. Positive parts are shown in panels a and c,\nnegative parts are shown in panels b and d.\nIn contrast, the dynamic structure factors Sxx(k,ω) andSxy(k,ω) are\nmany-particle quantities in terms of the Jordan-Wigner spi nless fermions.\nThe frequency range of these quantities is not a priori restricted, however,\nin the low-temperature limit T→0Sxx(k,ω) andSxy(k,ω) are rather small\n(but nonzero) outside the two-fermion excitation continuu m. These quan-\ntities show washed-out excitation branches roughly follow ing the boundary\nof the two-fermion excitation continuum. [Although the res ults presented\nin Figs. 10, 11 refer to the case J <0 (the ferromagnetic sign of the XXex-\nchange interaction) the results for J >0 (the antiferromagnetic sign of the\nXXexchange interaction) follow by symmetry. In fact, while ch anging the\nsign ofXXexchange interaction, + J→ −J, we getSxx(k,ω),Sxy(k,ω)\ngiven by Eq. (79) in which the wave-vector is changed k→k∓π.] From the\nexact calculation in the strong-field zero-temperature lim it (64) we know\nthatSxx(k,ω),Sxy(k,ω) are proportional to δ(ω−Λk), Λk= Ω +Jcosk.\nIn the high-temperature limit T→ ∞Sxx(k,ω) andSxy(k,ω) becomek-\nindependent, but depend on Ω. All the features described can be seen in\nFigs. 10, 11.\nIt is instructive to compare our precise numerical findings i n the low-\ntemperature limit with the results for the ground-state dyn amic structure\nfactors obtained by bosonization18,19,20. In the case Ω = 0 within theDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 29\nframework of the bosonization approach we have\nSαα(k,ω)∼θ(ω− |vk|)\n/parenleftig\nω2−(vk)2/parenrightig1−ηα\n2, α=x,z. (80)\nHerev=Jis the velocity and ηx= 1/2,ηz= 2 are the exponents which\ndescribe correctly the singularity at the lower continuum b oundaryωl(k) =\n|Jsink| → |Jk|ask→0 ork→ ±π. In the case of nonzero transverse fields\nΩ∝ne}ationslash= 0 the values of the Fermi momentum and Fermi velocity are cha nged.\nIn Fig. 12 we compare the predictions of the bosonization app roach (80)\nFig. 12. Sxx(k, ω) for the chain (15) with J=−1; frequency profiles at k=\n0,0.1,0.2,0.3 (from left to right) at Ω = 0 (panel a) and Ω = 0 .3 (panel b). Bosoniza-\ntion results which follow from Eq. (80) are shown by thin line s (v= 1 for Ω = 0 and\nv= 0.9539. . .for Ω = 0 .3); numerical results at low temperature β= 100 are shown by\nsolid lines.\nwith the numerical results at low temperatures.\nTo summarize, in this section we have discussed dynamic prop erties of\nthe spin-1/2 transverse XXchain within the Jordan-Wigner fermioniza-\ntion approach. Within this scheme the spin Hamiltonian corr esponds to\nthe Hamiltonian of noninteracting spinless fermions. The t ransverse dy-\nnamic structure factor corresponds to the fermionic densit y dynamic struc-\nture factor and probes the two-fermion excitation continuu m. There are\nmore dynamic structure factors which probe the two-fermion excitation\ncontinuum, e.g., the dimer dynamic structure factor. All tw o-fermion dy-\nnamic quantities have common features (spectral boundarie s, potential soft\nmodes and singularities) and specific features. There are al so dynamic quan-\ntities which probe the four-fermion excitation continuum; as an example of\nsuch a quantity we have discussed the trimer dynamic structu re factor.December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n30 O. Derzhko\nRemarkably, the dynamic structure factors which are associ ated with the\ndynamics of fluctuations of the xoryspin components (in contrast to\nthe transverse dynamic structure factor which is associate d with dynamics\nof fluctuations of the zspin component) are enormously complex within\nthe Jordan-Wigner description since they probe many-fermi on excitations.\nNevertheless the two-fermion excitation continuum is stil l important for\nthese dynamic quantities at low temperatures. As we have obs erved in our\nnumerics, most of the spectral weight is concentrated along the boundaries\nof the two-fermion excitation continuum (it was also noted e arlier for the\nXXZ Heisenberg chain53). This is not the case in the high-temperature\nlimit when these dynamic structure factors show Gaussian ri dges. In the\nnext two sections we shall follow to what extent our observat ions survive\nfor more complicated spin-1/2 XYchains.\n5. Dimerized Spin-1/2 Isotropic XYChain in a Transverse\nField\nNow we pass to the dimerized spin-1/2 XXchain in a transverse field. The\nHamiltonian of the model reads\nH=/summationdisplay\nnJ(1−(−1)nδ)/parenleftbig\nsx\nnsx\nn+1+sy\nnsy\nn+1/parenrightbig\n+ Ω/summationdisplay\nnsz\nn\n→/summationdisplay\nnJ\n2(1−(−1)nδ)/parenleftig\nc†\nncn+1−cnc†\nn+1/parenrightig\n+ Ω/summationdisplay\nn/parenleftbigg\nc†\nncn−1\n2/parenrightbigg\n,(81)\nwhereδis the dimerization parameter (0 < δ < 1). After performing\nconsequently the Fourier transformation, c†\nn=/parenleftig\n1/√\nN/parenrightig/summationtext\nkexp (ikn)c†\nk,\nk= 2πp/N,p=−N/2,...,N/ 2−1 (Nis even), and the Bogolyubov\ntransformation, ck=uk+πηk+ ivkηk+π,uk=/parenleftbig\n1/√\n2/parenrightbig/radicalbig\n1 +|cosk|/ǫk,\nvk= sgn(sin(2 k))/parenleftbig\n1/√\n2/parenrightbig/radicalbig\n1− |cosk|/ǫk,ǫk=/radicalbig\ncos2k+δ2sin2k, the\nHamiltonian becomes diagonal, H=/summationtext\nkΛk/parenleftig\nη†\nkηk−1/2/parenrightig\nwith the ele-\nmentary excitation energy Λ k= Ω +λk,λk= sgn(cosk)Jǫk(for further\ndetails see Refs.35,54).\nThe calculation of the transverse dynamic structure factor follows the\nscheme described in Sec. 4 and ends up with the following resu lt\nSzz(k,ω)\n=/integraldisplayπ\n−πdk1/parenleftig\n(uk1uk1+k+vk1vk1+k)2nk1(1−nk1+k)δ(ω+λk1−λk1+k)\n+ (uk1vk1+k−vk1uk1+k)2nk1(1−nk1+k+π)δ(ω+λk1−λk1+k+π)/parenrightig\n(82)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 31\n(see also Refs.35,55,56,54).\nAgain for the xxandxydynamic structure factors exact analytical\nresults are rather scarce. In the high-temperature limit on ly the autocorre-\nlation functions survive41\n∝an}b∇acketle{tsx\nj(t)sx\nj∝an}b∇acket∇i}ht=1\n4ℜZj(t),∝an}b∇acketle{tsx\nj(t)sy\nj∝an}b∇acket∇i}ht=1\n4ℑZj(t),\nZj(t) =Θ1/parenleftig\nJ+t,J−\nJ+/parenrightig\nΘ1/parenleftig\n0,J−\nJ+/parenrightigH1/parenleftig\nJ+t,J−\nJ+/parenrightig\nH1/parenleftig\n0,J−\nJ+/parenrightig\n·exp\n−iΩt−\n1−E/parenleftig\nJ−\nJ+/parenrightig\nK/parenleftig\nJ−\nJ+/parenrightig\nJ2\n+t2\n. (83)\nHere the Jacobian theta and eta functions are defined as follo ws\nΘ1(u,k) =∞/summationdisplay\nn=−∞qn2exp (2niz),\nH1(u,k) =∞/summationdisplay\nn=−∞q(n+1\n2)2\nexp ((2n+ 1)iz),\nq= exp/parenleftigg\n−πK/parenleftbig√\n1−k2/parenrightbig\nK(k)/parenrightigg\n, z=πu\n2K(k)(84)\nand the complete elliptic integrals of the 1st and the 2nd kin ds are given\nby\nK(k) =/integraldisplayπ\n2\n0dθ/radicalbig\n1−k2sin2θ,E(k) =/integraldisplayπ\n2\n0dθ/radicalbig\n1−k2sin2θ. (85)\nMoreover,J2\n±=J2(1±δ)2/4.\nIn the strong-field limit |Ω|>|J|atT= 0 we can repeat the calculation\nof the previous section to find, for example, for the xxtime-dependent\ncorrelation function and the corresponding dynamic struct ure factor the\nfollowing result\n4∝an}b∇acketle{tsx\nj(t)sx\nj+n∝an}b∇acket∇i}ht=1\nN/summationdisplay\nkexp (ikn)/parenleftbig\nu2\nkexp (−iΛkt) +v2\nkexp (−iΛk+πt)\n−i (−1)j+nukvk(exp (−iΛkt)−exp (−iΛk+πt))/parenrightig\n,(86)\nSxx(k,ω) =π\n2/parenleftbig\nu2\nkδ(ω−Λk) +v2\nkδ(ω−Λk+π)/parenrightbig\n.(87)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n32 O. Derzhko\nFor arbitrary values of temperature and transverse field the xxandxy\ndynamic structure can be obtained numerically54.\nLet us discuss the dynamic quantities of the dimerized trans verseXX\nchain. We begin with the transverse dynamic structure facto r which can be\nwritten as (compare with Eq. (44))\nSzz(k,ω) =/integraldisplayπ\n−πdk1dk2C(1)(k1,k2)\n·nk1(1−nk2)δ(ω+λk1−λk2)δk+k1−k2,0\n+/integraldisplayπ\n−πdk1dk2C(2)(k1,k2)\n·nk1(1−nk2)δ(ω+λk1−λk2)δk+k1−k2+π,0,\nC(1)(k1,k2) = (uk1uk2+vk1vk2)2\nC(2)(k1,k2) = (uk1vk2−vk1uk2)2. (88)\nAs can be seen from Eq. (88) the transverse dynamic structure probes two-\nfermion excitations. Szz(k,ω) may have nonzero value within a restricted\nregion of the k–ωplane when there is such a wave-vector k1,−π≤k1<π\nthatω=−λk1+λk1+korω=−λk1+λk1+k+π. Moreover, at zero tem-\nperature there are additional restrictions arising from th e Fermi functions.\nThe lines of potential singularities follow from the analys is of the equa-\ntions dλk1/dk1−dλk1+k/dk1= 0 and dλk1/dk1−dλk1+k+π/dk1= 0. The\ncharacteristic lines in the k–ωplane which determine the behavior of the\ntransverse dynamic structure factor were reported for the fi rst time by\nJ. H. Taylor and G. M¨ uller35.\nIn Fig. 13 we show the region of the k–ωplane in which the two-\nFig. 13. Location of the roots of equations ω=−λk1+λk1+kandω=−λk1+λk1+k+π\n(−π≤k1< π) in the k–ωplane for δ= 0 (panel a) and δ= 0.1 (panel b): light region:\nno roots, light gray region: two roots, dark gray region: fou r roots.December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 33\nfermion dynamic quantity may have nonzero values. In Fig. 14 we show the\nFig. 14. Szz(k, ω) (gray-scale plots) for the chain (81) with J=−1,δ= 0.1 at different\ntemperatures β=∞(a, b), β= 20 (c, d), β= 1 (e, f) for Ω = 0 (left panels a, c, e)\nand Ω = 0 .3 (right panels b, d, f). Note that the results at β= 1 for Ω = 0 and Ω = 0 .3\n(panels e and f) are practically indistinguishable.\ntransverse dynamic structure factor at different temperatu res. Comparing\nFig. 13b and Fig. 14 one can see the van Hove singularities and the effects\nof the Fermi functions and the C(1)- andC(2)-functions. In Figs. 15, 16 we\nshowSzz(k,ω) at various values of the transverse field for two temperatur es\nβ=∞andβ= 20.\nNext we pass to the xxdynamic structure factor obtained numerically.\nTypically we consider chains of N= 400 sites assume in (4) n= 41,lup\nto 50, consider tup totc= 200 and take ǫ= 0.00154. In Fig. 17 we show\nSxx(k,ω) of the dimerized transverse XXchain at low temperatures for\ndifferent values of the transverse field.December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n34 O. Derzhko\nFig. 15. Szz(k, ω) (gray-scale plots) for the chain (81) with J=−1,δ= 0.1 at zero\ntemperature β=∞and different values of the transverse field Ω = 0 .1 (a), Ω = 0 .11\n(b), Ω = 0 .3 (c) and Ω = 0 .9 (d).\nFig. 16. The same as in Fig. 15 for β= 20.\nIn contrast to the zzdynamic structure factor which is a two-fermion\ndynamic quantity, the xxdynamic structure factor is a many-particle dy-December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 35\nFig. 17. Sxx(k, ω) (gray-scale plots) for the chain (81) with J=−1,δ= 0.1 at low\ntemperature β= 20 and for Ω = 0 .1 (a), Ω = 0 .3 (b), Ω = 0 .6 (c) and in the high-\ntemperature limit β= 0 for Ω = 0 .6 (d).\nnamic quantity within the Jordan-Wigner fermionization ap proach. There-\nfore, nonzero values of Sxx(k,ω) far above the two-fermion excitation con-\ntinua may be expected. However, as can be seen in Fig. 17 the op posite is\ntrue: At low-temperatures Sxx(k,ω) shows several well-defined excitation\nbranches which follow roughly the boundaries of the two-fer mion excita-\ntion continua. Although we can describe the low-energy phys ics also using\nthe bosonization treatment, high-frequency features cann ot be reproduced\nwithin such an approach.\nFinally we note that the dimerized XXchain does not show bound-\nstate branches; within the fermionization approach this ma y be related to\nthe absence of interactions between fermions. In contrast, a particle-hole\nbound state can be observed in the dimerized Heisenberg chai n57.\n6. Spin-1/2 XYChains with the Dzyaloshinskii-Moriya\nInteraction\nIn this section we examine the effect of the Dzyaloshinskii-M oriya interac-\ntion (actually, the zcomponent of the vector of the Dzyaloshinskii-Moriya\ninteraction, see Eq. (18)). The Hamiltonian of the transver seXXchainDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n36 O. Derzhko\nwith the Dzyaloshinskii-Moriya interaction reads\nH=/summationdisplay\nn/parenleftbig\nJ/parenleftbig\nsx\nnsx\nn+1+sy\nnsy\nn+1/parenrightbig\n+D/parenleftbig\nsx\nnsy\nn+1−sy\nnsx\nn+1/parenrightbig/parenrightbig\n−h/summationdisplay\nnsz\nn.(89)\nInterestingly, the Dzyaloshinskii-Moriya interaction ca n be eliminated\nfrom the Hamiltonian (89) resulting in renormalization of t he isotropic XY\nexchange interaction58. To see this, consider the following spin axes rotation\nsx\nn→˜sx\nn=sx\nncosφn+sy\nnsinφn,\nsy\nn→˜sy\nn=−sx\nnsinφn+sy\nncosφn,\nsz\nn→˜sz\nn=sz\nn,\nφn= (n−1)ϕ,tanϕ=D\nJ. (90)\nAfter such a unitary transformation the Hamiltonian (89) be comes\nH=/summationdisplay\nn˜J/parenleftbig\n˜sx\nn˜sx\nn+1+ ˜sy\nn˜sy\nn+1/parenrightbig\n−h/summationdisplay\nn˜sz\nn,\n˜J= sgn(J)/radicalbig\nJ2+D2. (91)\nUsing the unitary transformation (90) we can examine the effe ct of\nthe Dzyaloshinskii-Moriya interaction using the dynamic q uantities of the\ntransverseXXchain without the Dzyaloshinskii-Moriya interaction dis-\ncussed already in Sec. 4. First of all we note that the zzdynamic structure\nfactor is given by Eq. (35), however, with Λ k=−h+˜Jcosk. The for-\nmulas determining the two-fermion excitation continua bou ndaries are still\ngiven by Eqs. (37), (38), (39) but with ˜Jinstead ofJon the l.h.s. of these\nequations and in the definition of the parameter α.\nExploiting (90) we find the following relations between the xxandxy\ndynamic structure factors of the model (89) (l.h.s. of Eq. (9 2)) and the\ndynamic structure factors of the model (91) (r.h.s. of Eq. (9 2))59\nSxx(k,ω) =1\n2(Sxx(k−ϕ,ω)|˜J+Sxx(k+ϕ,ω)|˜J\n+i/parenleftbig\nSxy(k−ϕ,ω)|˜J−Sxy(k+ϕ,ω)|˜J/parenrightbig/parenrightbig\n,\nSxy(k,ω) =1\n2/parenleftbig\nSxy(k−ϕ,ω)|˜J+Sxy(k+ϕ,ω)|˜J\n−i (Sxx(k−ϕ,ω)|˜J−Sxx(k+ϕ,ω)|˜J)). (92)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 37\nTherefore, using Eq. (62) we obtain for the model (89)\nSxx(k,ω) =√π\n4˜J/parenleftigg\nexp/parenleftigg\n−(ω+h)2\n˜J2/parenrightigg\n+ exp/parenleftigg\n−(ω−h)2\n˜J2/parenrightigg/parenrightigg\n,\niSxy(k,ω) =√π\n4˜J/parenleftigg\nexp/parenleftigg\n−(ω+h)2\n˜J2/parenrightigg\n−exp/parenleftigg\n−(ω−h)2\n˜J2/parenrightigg/parenrightigg\n,(93)\ni.e. in the high-temperature limit Sxx(k,ω) andSxy(k,ω) arek-independent\nand display a single Gaussian ridge at ω=|h|.\nIn the zero-temperature and strong-field limit ( T= 0,|h|>√\nJ2+D2)\naccording to (92) and (64) we find\nSxx(k,ω) =−sgn(h)iSxy(k,ω)\n=π\n2δ/parenleftig\nω− |h| −˜Jcos(k+ sgn(h)ϕ)/parenrightig\n. (94)\nFor arbitrary values of temperature and transverse field we u se Eq.\n(92) and numerical results for the xxandxydynamic structure fac-\ntors of the transverse XXchain (91) (see Sec. 4) to reveal the effect of\nthe Dzyaloshinskii-Moriya interaction. Some of our finding s are plotted in\nFig. 18 where we show Sxx(k,ω) forD= 0 (left panels) and D∝ne}ationslash= 0 (right\nFig. 18. Sxx(k, ω) (gray-scale plots) for the model (89) with J=−1,D= 0 (left panels\na, c) and D= 1 (right panels b, d) for h= 0 (upper panels a, b) and h=−0.6 (lower\npanels c, d) at low temperature β= 20.December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n38 O. Derzhko\npanels) at different values of the transverse field h.\nWe recall that in the low-temperature limit when J <0 andD= 0\nSxx(k,ω) andSxy(k,ω) are concentrated in the k–ωplane along the curves\n(37), (38), (39) which determine the boundaries of the two-f ermion excita-\ntion continuum ωl(k),ωm(k) andωu(k) (see Sec. 4, Figs. 10, 11). [For the\nantiferromagnetic sign of XXexchange interaction J >0 these dynamic\nquantities are concentrated along the curves ωl(k±π),ωm(k±π) and\nωu(k±π) as it follows from simple symmetry arguments.] In the case w hen\nthe Dzyaloshinskii-Moriya interaction is present, D∝ne}ationslash= 0, the two-fermion\nexcitation continuum splits into two continua (see Fig. 18) , the ‘left’ one\nwith the boundaries ωl(k−ϕ),ωm(k−ϕ) andωu(k−ϕ) and the ‘right’\none with the boundaries ωl(k+ϕ),ωm(k+ϕ) andωu(k+ϕ). (The ‘left’\nand the ‘right’ continua are connected by symmetry operatio n.) The larger\nDis the larger is the splitting controlled by ϕ= arctan(D/J). At fixed\nD∝ne}ationslash= 0 andh= 0 the spectral weight is equally distributed between the\nleft and the right continua (panel b in Fig. 18). While |h|increases from\n0 to√\nJ2+D2the spectral weight moves from one continuum to another\ncontinuum (panel d in Fig. 18).\nWe note in passing that the discussed peculiarities of the xxdynamic\nstructure factor may be used for an unambiguous determinati on of the\nDzyaloshinskii-Moriya interaction in chain compounds for example, in res-\nonance experiments60,50,59,61.\nIn the case of the anisotropic XYchain the Dzyaloshinskii-Moriya in-\nteraction cannot be eliminated by the transformation (90). Now we face the\nHamiltonian\nH=/summationdisplay\nn/parenleftbig\nJxsx\nnsx\nn+1+Jysy\nnsy\nn+1+D/parenleftbig\nsx\nnsy\nn+1−sy\nnsx\nn+1/parenrightbig/parenrightbig\n+ Ω/summationdisplay\nnsz\nn\n→/summationdisplay\nn/parenleftbiggJ+ iD\n2c†\nncn+1−J−iD\n2cnc†\nn+1+γ\n2/parenleftig\nc†\nnc†\nn+1−cncn+1/parenrightig\n+Ω/parenleftbigg\nc†\nncn−1\n2/parenrightbigg/parenrightbigg\n(95)\nwithJ= (Jx+Jy)/2 andγ= (Jx−Jy)/2. This Hamiltonian can\nbe put into a diagonal form by performing the Fourier transfo rma-\ntion,ck=/parenleftig\n1/√\nN/parenrightig/summationtext\nnexp (ikn)cn,cn=/parenleftig\n1/√\nN/parenrightig/summationtext\nkexp(−ikn)ck,\nand the Bogolyubov transformation, ck=−iukβk+vkβ†\n−k,βk=\niukck+vkc†\n−k,uk= sgn (γsink)/parenleftbig\n1/√\n2/parenrightbig/radicalbig\n1 + (Ω +Jcosk)/λk,vk=\n/parenleftbig\n1/√\n2/parenrightbig/radicalbig\n1−(Ω +Jcosk)/λk,λk=/radicalig\n(Ω +Jcosk)2+γ2sin2k. The fi-December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 39\nnal result reads H=/summationtext\nkΛk/parenleftig\nβ†\nkβk−1/2/parenrightig\n, Λk=Dsink+λk. We no-\ntice that the elementary excitation energy spectrum is gapl ess when Ω2≤\nJ2+D2−γ2andγ2≤D2or when Ω2=J2andγ2>D2.\nThe calculation of the transverse dynamic structure factor follows the\nlines explained in some detail in Sec. 4 and ends up with62\nSzz(k,ω) =3/summationdisplay\nj=1S(j)\nzz(k,ω),\nS(j)\nzz(k,ω) =/integraldisplayπ\n−πdk1B(j)(k1,k)C(j)(k1,k)δ/parenleftig\nω−E(j)(k1,k)/parenrightig\n,\nB(1)(k1,k) =B(3)(k1,k) =1−f(k1,k)\n4, B(2)(k1,k) =1 +f(k1,k)\n2,\nf(k1,k) =/parenleftbig\nΩ +Jcos/parenleftbig\nk1−k\n2/parenrightbig/parenrightbig/parenleftbig\nΩ +Jcos/parenleftbig\nk1+k\n2/parenrightbig/parenrightbig\n−γ2sin/parenleftbig\nk1−k\n2/parenrightbig\nsin/parenleftbig\nk1+k\n2/parenrightbig\nλk1−k\n2λk1+k\n2,\nC(1)(k1,k) =/parenleftig\n1−nk1+k\n2/parenrightig/parenleftig\n1−n−k1+k\n2/parenrightig\n,\nC(2)(k1,k) =/parenleftig\n1−nk1+k\n2/parenrightig\nnk1−k\n2,\nC(3)(k1,k) =nk1−k\n2n−k1−k\n2,\nE(1)(k1,k) = Λk1+k\n2+ Λ−k1+k\n2,\nE(2)(k1,k) = Λk1+k\n2−Λk1−k\n2,\nE(3)(k1,k) =−Λk1−k\n2−Λ−k1−k\n2.(96)\nThe transverse dynamic factor, as it follows from Eq. (96), i s shown in panel\nc in Fig. 19 for a typical set of parameters.\nFrom Eq. (96) we see that the transverse dynamic structure fa ctor is\ngoverned by three two-fermion excitation continua. Let us d iscuss some\nproperties of these continua (see Fig. 20 where we show two-f ermion exci-\ntation continua for a specific set of parameters J=−1,γ= 0.5,D= 1,\nΩ = 0.5). We begin with the high-temperature limit when the Fermi f ac-\ntors are not essential (left panels in Fig. 20). The two-ferm ion dynamic\nstructure factor may have nonzero values in the k–ωplane if the equation\nω−E(j)(k1,k) = 0 has at least one solution k⋆\n1,−π≤k⋆\n1< π. Next, theDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n40 O. Derzhko\nFig. 19. Sxx(k, ω) (a), Syy(k, ω) (b), Szz(k, ω) (c) for the spin chain (95) with J=−1,\nγ= 0.5,D= 1, Ω = 0 .5 at low temperature β= 50. Note that these quantities are\nshown for kthat varies from −πto 3π.\nlower and the upper boundaries are given by\nω(j)\nl(k) = min\n−π≤k1<π/braceleftig\n0,E(j)(k1,k)/bracerightig\n,\nω(j)\nu(k) = max\n−π≤k1<π/braceleftig\nE(j)(k1,k)/bracerightig\n. (97)\nThe two-fermion dynamic quantities may exhibit van Hove sin gularities\nalong the line ω(j)\ns(k) =E(j)(k1,k) wherek1satisfies the equation\n∂\n∂k1E(j)(k1,k) = 0. (98)\nIf for the solution of Eq. (98) we also have ∂2E(j)(k1,k)/∂k2\n1∝ne}ationslash= 0 the\ntwo-fermion dynamic structure factor shows the well-known square-rootDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 41\nFig. 20. Two-fermion excitation continua ( j= 1 (a, b), j= 2 (c, d), j= 3 (e, f)) for\nJ=−1,γ= 0.5,D= 1, Ω = 0 .5. Left panels: T→ ∞; right panels: T= 0.\nsingularity. However, it may happen that ∂2E(j)(k1,k)/∂k2\n1= 0 but\n∂3E(j)(k1,k)/∂k3\n1∝ne}ationslash= 0. Then a van Hove singularity is characterized by\nthe exponent 2/3. That is really the case, for example, for J= 1,γ= 0.5,\nD= 1, Ω = 0 .5 forj= 2 atk= 1.0784.... We have∂E(2)(k1,k)/∂k1=\n∂2E(2)(k1,k)/∂k2\n1= 0,∂3E(2)(k1,k)/∂k3\n1∝ne}ationslash= 0 atk1=k⋆\n1= 2.1648....\nTherefore in the ǫ-vicinity of ω= 0.7859...the two-fermion dynamic struc-\nture factor should be proportional to |ǫ|−2/3. We mention that the singu-\nlarity with this exponent is also present for D= 062.\nIn the zero-temperature case the effect of the Fermi function s involved\nin theC(j)-functions (see Eq. (96)) becomes important (right panels i nDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n42 O. Derzhko\nFig. 20). As a result the region of possible values of k1is contracted. Further\ndetails can be found in Ref.62.\nFinally, we mention the role of the B(j)-functions (see Eq. (96)) for\nthe two-fermion dynamic structure factors which are respon sible for some\nspecific features of the transverse dynamic structure facto r (compare panels\nb, d, f in Fig. 20 and panel c in Fig. 19).\nWe pass to the xxandyydynamic structure factors (see panels a and\nb in Fig. 19). These dynamic structure factors are many-ferm ion dynamic\nquantities and although they are not restricted to some regi on in thek–ω\nplane, they are rather small outside the two-fermion excita tion continua\n(compare panels a and b with panel c in Fig. 19). In the low-tem perature\nlimit thexxandyydynamic structure factors show several washed-out exci-\ntation branches which are in correspondence with character istic lines of the\ntwo-fermion excitation continua. In the high-temperature limitSxx(k,ω)\nandSyy(k,ω) becomek-independent.\nIt should be stressed that the constant frequency or constan t wave-\nvector scans of the dynamic structure factors clearly manif est the presence\nof the Dzyaloshinskii-Moriya interaction and some easily r ecognized fea-\ntures of these quantities can be used for determining the Dzy aloshinskii-\nMoriya interaction.\n7. Square-Lattice Spin-1/2 Isotropic XYModel\nLet us discuss what kind of results for spin models can be obta ined in two\ndimensions after applying the Jordan-Wigner transformati on (Sec. 3). We\nconsider the spin-1/2 isotropic XYmodel on a spatially anisotropic square\nlattice with the Hamiltonian\nH=∞/summationdisplay\ni=0∞/summationdisplay\nj=0/parenleftbiggJ\n2/parenleftbig\ns+\ni,js−\ni+1,j+s−\ni,js+\ni+1,j/parenrightbig\n+J⊥\n2/parenleftbig\ns+\ni,js−\ni,j+1+s−\ni,js+\ni,j+1/parenrightbig/parenrightbigg\n, (99)\nwhereJandJ⊥are theXXexchange interactions in the horizontal and\nvertical directions. Our aim is to calculate the transverse dynamic structure\nfactor\nSzz(k,ω) =∞/summationdisplay\np=0∞/summationdisplay\nq=0exp(i (kxp+kyq))/integraldisplay∞\n−∞dtexp(iωt)\n·/parenleftbig\n∝an}b∇acketle{tsz\nn,m(t)sz\nn+p,m+q∝an}b∇acket∇i}ht − ∝an}b∇acketle{tsz\nn,m∝an}b∇acket∇i}ht∝an}b∇acketle{tsz\nn+p,m+q∝an}b∇acket∇i}ht/parenrightbig\n. (100)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 43\nWe notice that the transverse dynamic structure factor Szz(k,ω) (100) for\nthe spin model (99) is related to the density-density dynami c structure\nfactor of hard-core bosons on a square-lattice63.\nWe apply the two-dimensional Jordan-Wigner transformatio n (21), (23)\nto the spin Hamiltonian (99). Moreover, we adopt the mean-fie ld approach\nfor the phase factors and change the gauge leaving the flux per elementary\nplaquette Φ 0to be equal to π. As a result we arrive at the Hamiltonian like\n(27), i.e.\nH=∞/summationdisplay\ni=0∞/summationdisplay\nj=0/parenleftbiggJ\n2(−1)i+j/parenleftig\nd†\ni,jdi+1,j−di,jd†\ni+1,j/parenrightig\n+J⊥\n2/parenleftig\nd†\ni,jdi,j+1−di,jd†\ni,j+1/parenrightig/parenrightbigg\n. (101)\nThe Hamiltonian (101) contains the correct results in the on e-dimensional\nlimit when either J⊥= 0 orJ= 0 (in the former case to recover the one-\ndimensional Hamiltonian (16) (with Ω = 0) one has to perform i n addition\na gauge transformation d†\ni,j= exp (iπψi)f†\ni,j,ψ0= 0,ψi+1=ψi+i).\nThe Hamiltonian (101) can be diagonalized by performing 1) t he Fourier\ntransformation, di,j=/parenleftbig\n1//radicalbigNxNy/parenrightbig/summationtext\nkx,kyexp (i (kxi+kyj))dkx,ky,kα=\n2πnα/Nα,nα=−Nα/2,−Nα/2 + 1,...,N α/2−1,α=x,y,Nx=Ny=√\nN→ ∞ is even, which yields\nH=1\n2/summationdisplay\nk|Ek|/parenleftig\ncosγk/parenleftig\nb†\nkbk−a†\nkak/parenrightig\n+ i sinγk/parenleftig\nb†\nkak−a†\nkbk/parenrightig/parenrightig\n,\n|Ek|=/radicalig\nJ2\n⊥cos2ky+J2sin2kx,\ncosγk=J⊥cosky\n|Ek|,sinγk=Jsinkx\n|Ek|(102)\nwithbk=dkx,kyandak=dkx±π,ky±πand 2) the Bogolyubov transforma-\ntion,αk= cos(γk/2)bk+i sin(γk/2)ak,βk= sin (γk/2)bk−i cos(γk/2)ak,\nwhich yields\nH=/summationdisplay\nk′Λk/parenleftig\nα†\nkαk−β†\nkβk/parenrightig\n,\nΛk=|Ek|=/radicalig\nJ2\n⊥cos2ky+J2sin2kx≥0 (103)\n(the prime near the sum in Eq. (103) means that kvaries in the thermo-\ndynamic limit in the region −π≤ky≤π,−π+|ky| ≤kx≤π− |ky|).\nThe calculation of the transverse dynamic structure factor repeats the\nsteps elaborated in some detail for the one-dimensional cas e. First, we useDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n44 O. Derzhko\nthe Wick-Bloch-de Dominicis theorem to obtain the zztime-dependent spin\ncorrelation function\n∝an}b∇acketle{tsz\nn,m(t)sz\nn+p,m+q∝an}b∇acket∇i}ht − ∝an}b∇acketle{tsz\nn,m∝an}b∇acket∇i}ht∝an}b∇acketle{tsz\nn+p,m+q∝an}b∇acket∇i}ht\n=∝an}b∇acketle{td†\nn,m(t)dn+p,m+q∝an}b∇acket∇i}ht∝an}b∇acketle{tdn,m(t)d†\nn+p,m+q∝an}b∇acket∇i}ht,\n∝an}b∇acketle{td†\nn,m(t)dn+p,m+q∝an}b∇acket∇i}ht=1\n2N/summationdisplay\nkexp(i (kxp+kyq))\n·/parenleftig/parenleftig\ncos2γk\n2−i (−1)n+m/parenleftbig\n(−1)p+q−1/parenrightbig\ncosγk\n2sinγk\n2+ (−1)p+qsin2γk\n2/parenrightig\n·nkexp (iΛ kt)\n+/parenleftig\nsin2γk\n2+ i (−1)n+m/parenleftbig\n(−1)p+q−1/parenrightbig\ncosγk\n2sinγk\n2+ (−1)p+qcos2γk\n2/parenrightig\n·(1−nk)exp ( −iΛkt)),\n∝an}b∇acketle{tdn,m(t)d†\nn+p,m+q∝an}b∇acket∇i}ht=1\n2N/summationdisplay\nkexp (−i (kxp+kyq))\n·/parenleftig/parenleftig\ncos2γk\n2+ i (−1)n+m/parenleftbig\n(−1)p+q−1/parenrightbig\ncosγk\n2sinγk\n2+ (−1)p+qsin2γk\n2/parenrightig\n·(1−nk) exp( −iΛkt)\n+/parenleftig\nsin2γk\n2−i (−1)n+m/parenleftbig\n(−1)p+q−1/parenrightbig\ncosγk\n2sinγk\n2+ (−1)p+qcos2γk\n2/parenrightig\n·nkexp(iΛ kt)).(104)\nThen we plug Eq. (104) into Eq. (100) to obtain the following e xpression\nfor thezzdynamic structure factor\nSzz(k,ω) =π/integraldisplayπ\n−πdk1y\n2π/integraldisplayπ\n−πdk1x\n2π\n·/parenleftbigg\ncos2γk1+k−γk1\n2(nk1(1−nk1+k)δ(ω+ Λk1−Λk1+k)\n+ (1−nk1)nk1+kδ(ω−Λk1+ Λk1+k))\n+ sin2γk1+k−γk1\n2(nk1nk1+kδ(ω+ Λk1+ Λk1+k)\n+ (1−nk1) (1−nk1+k)δ(ω−Λk1−Λk1+k))).(105)\nOne can easily convince oneself that Eq. (105) contains the c orrect result in\nthe one-dimensional limit (35) (with Ω = 0) when either J⊥= 0 orJ= 0.\nIn the two-dimensional case Eq. (105) is an approximate form ula for the\ntransverse dynamic structure factor of the spin-1/2 isotro picXYmodel on\na spatially anisotropic square lattice.\nLet us discuss the dynamic quantity obtained in some detail64. In\nFig. 21 we show gray-scale plots for Szz(k,ω) and in Fig. 22 we show fre-December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 45\nFig. 21. The zzdynamic structure factor Szz(kx,0, ω) (gray-scale plots) for the square-\nlattice s= 1/2XXmodel (99) as it follows from Eq. (105) at T= 0 (left column) and\natT= 10 (right column). J=−1,J⊥=−0.1 (a, b), J⊥=−0.5 (c, d).\nFig. 22. Frequency dependence of the zzdynamic structure factor (105) for momentum\ntransfer along the chain kx=π/2 as the interchain interaction changes ( J=−1,J⊥=\n−0.1 (solid curves), J=−1,J⊥=−0.5 (dashed curves), J=−1,J⊥=−0.9 (dotted\ncurves)) at zero temperature T= 0 (a) and high temperature T= 10 (b).\nquency profiles of Szz(k,ω) for a representative set of parameters. Formula\n(105) implies the interpretation of Szz(k,ω) as a two-fermion excitation\nquantity. As can be seen from Figs. 21, 22 Szz(k,ω) exhibits several washed-\nout excitation branches which can be generated by two spinle ss fermions\nin accordance with (105). We begin with the low-temperature limit when\nonly the fourth term in Eq. (105) (which contains (1 −nk1)(1−nk1+k))December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n46 O. Derzhko\nsurvives. Consider, for example, two fermions with k1= (0,π/2)−k/2 and\nk1+k= (0,π/2) +k/2 with the energy of the pair\nωk= 2/radicalbigg\nJ2sin2kx\n2+J2\n⊥sin2ky\n2(106)\nor two fermions with k1= (π/2,0)−k/2 andk1+k= (π/2,0)+k/2 with\nthe energy of the pair\nωk= 2/radicalbigg\nJ2cos2kx\n2+J2\n⊥cos2ky\n2. (107)\nThese modes are the well-known spin waves65clearly present at low tem-\nperatures (panels a and c in Fig. 21). Further, one can recogn ize the high\nfrequency modes [ k1=−k/2,k1= (π/2,π/2)−k/2] with the dispersion\nrelations\nωk= 2/radicalbigg\nJ2sin2kx\n2+J2\n⊥cos2ky\n2, (108)\nωk= 2/radicalbigg\nJ2cos2kx\n2+J2\n⊥sin2ky\n2. (109)\nAnother set of high-frequency modes [ k1= 0,k1= (π/2,π/2)] have the\ndispersion relations\nωk=J⊥+/radicalig\nJ2sin2kx+J2\n⊥cos2ky, (110)\nωk=J+/radicalig\nJ2cos2kx+J2\n⊥sin2ky. (111)\nThe low-frequency mode [ k1= (0,π/2)] with the dispersion relation\nωk=/radicalig\nJ2sin2kx+J2\n⊥sin2ky (112)\n(it is composed of two fermions, the energy of one of which equ als zero)\nforms the low-frequency cutoff at zero temperature. Compari ng left and\nright panels in Figs. 21 and 22 one can also see the modes which become\nvisible only as temperature increases (at zero temperature they are for-\nbidden because of the Fermi factors in Eq. (105)). Putting k1x=−kx,\nk1y=π/2−kxforky= 0 andk1x=ky,k1y=π/2−kyforkx= 0 we get\nωkx,0=/parenleftbigg/radicalig\nJ2+J2\n⊥−J⊥/parenrightbigg\n|sinkx|,\nω0,ky=/parenleftbigg/radicalig\nJ2+J2\n⊥−J/parenrightbigg\n|sinky|. (113)December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 47\nThis excitation branch contains most of the spectral weight at high tem-\nperatures (see panel d in Fig. 21 and panel b in Fig. 22).\nThe established modes (106) – (113) manifest themselves as p eaks, cusps\nor cutoffs in the frequency or wave-vector profiles of Szz(k,ω). The fre-\nquency profiles depicted in Fig. 22 may be almost symmetric or asymmet-\nric, they may resemble δ-peaks or result from two coalesced peaks, they\nmay gradually disappear or may be abruptly cut off.\nIt is worthwhile to mention here some experimental studies o n dynamic\nproperties of two-dimensional quantum spin models, in part icular, the neu-\ntron scattering experiments on Cs 2CuCl 466(for a theory of dynamic cor-\nrelations in the spin-liquid phase in Cs 2CuCl 4see Ref.67). Cs 2CuCl 4is\na two-dimensional low-exchange quantum magnet. It has a lay ered crys-\ntal structure; in each layer the exchange paths form a triang le lattice with\nnonequivalent interactions along chains J= 0.374(5) meV and along zig-zag\nbondsJ′= 0.34(3)J. The interlayer coupling is small J′′= 0.045(5)Jand it\nstabilizes the long-range magnetic order below TN= 0.62(1) K. The neutron\nscattering measurements in the spin-liquid phase (i.e. abo veTNbut below\nJ,J′when the two-dimensional magnetic layers are decoupled) cl early in-\ndicate that the dynamic correlations are dominated by highl y dispersive\nexcitation continua which is a characteristic signature of fractionalization\nof spin-1 spin waves into pairs of deconfined spin-1/2 spinon s. Linear spin-\nwave theory including one- and two-magnon processes cannot describe the\ncontinuum scattering. The proposed theories67are based either on a quasi-\none-dimensional approach (that immediately introduces sp inon language)\nor on the explicitly two-dimensional resonating-valence- bond picture.\nAs a final remark we recall that Eq. (105) contains the exact re sult (35)\nin the one-dimensional limit. On the other hand, Eq. (105) gi ves an ap-\nproximate result in the two-dimensional case because of the mean-field de-\nscription of the phase factors which arise after fermioniza tion. The adopted\nmean-field treatment neglects a complicated interaction be tween spinless\nfermions. In the case of the XXZ Heisenberg model the interaction be-\ntween fermions is present even within the adopted mean-field procedure\ndue to the interaction between zspin components. The quartic terms in\nthe fermionic Hamiltonian may be treated after making furth er approxima-\ntion (see references cited in Ref.29and also Ref.68).December 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n48 O. Derzhko\n8. Conclusions\nThe Jordan-Wigner transformation which realizes a spin-to -fermion map-\nping was suggested as a rigorous framework for the descripti on of spin-1/2\nXYchains in the early 1960s. In general, the Jordan-Wigner fer mioniza-\ntion permits to map a system of interacting spins s= 1/2 onto a system\nof spinless fermions. It may happen that the spinless fermio ns are nonin-\nteracting. In this case this approach reveals an exactly sol vable spin model.\nHowever, even for exactly solvable spin models not all ‘simp le’ quantities of\ninterest in spin language remain simple in fermionic langua ge. For example,\nthezspin component attached to the site j,sz\nj, becomes the product of\ntwo Fermi operators attached to this site, c†\njcj−1/2. In contrast, the local\nspin operators sx\nj,sy\nj,s±\njbecome nonlocal objects in fermionic description\ninvolving a string of sites 1 ,2,...,j (see Eqs. (12), (13)). This leads to\nsome complications in studying the dynamics of fluctuations of these op-\nerators: the dynamics of fluctuations of operators which see m to be rather\nsimple in spin language may be governed by many-particle cor relations in\nfermionic language. As we have discussed in sections 4, 5, 6, the Jordan-\nWigner fermionization approach permits to establish a numb er of rigorous\nresults for the dynamics of spin-1/2 XYchains. Especially easy are the cases\nof two- and four-fermion dynamic quantities which are amena ble mostly to\nanalytical calculations. The case of many-fermion dynamic quantities is\nmore complicated, however, these quantities can be examine d numerically\nat very high precision.\nFor more realistic spin-1/2 XXZ Heisenberg chains the Jordan-Wigner\nfermionization approach leads to a system of interacting sp inless fermions.\nThe simplest way to proceed in this case is to apply Hartree-F ock-like\napproximations17. If we are interested in low-energy physics only it might\nbe helpful to apply the bosonization approach18,19,20.\nThe results for one-dimensional quantum spin systems obtai ned using\nthe Jordan-Wigner fermionization can be compared with the o utcomes of al-\nternative approaches: field-theoretic bosonization techn iques18,19,20valid\nin the low-energy limit (see Fig. 12), Bethe ansatz calculat ions (for calcula-\ntion of dynamic structure factors of spin-1/2 XXZ chains see Refs.69,70)\nor exact diagonalization computations which, however, are restricted to\nsmall finite systems.\nFor two-dimensional quantum spin models achievements are r ather mod-\nest. In this case the Jordan-Wigner fermionization approac h can provide\nan approximate theory; the simplest one treats in the mean-fi eld spirit theDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\nJordan-Wigner fermionization 49\nphase factors which arise after fermionization.\nWe end up with a brief comment about the experimental relevan ce\nof some of the dynamic quantities calculated. They may be use d for in-\nterpretation of the energy absorption in electron spin reso nance (ESR)\nexperiments60. Consider an ESR experiment in which the static magnetic\nfield is directed along the zaxis and the electromagnetic wave with the\npolarization in α⊥zdirection (say α=x) are applied to a magnetic sys-\ntem which is described as a spin-1/2 XYchain (ESR experiment in the\nstandard Faraday configuration). In such an ESR experiment o ne measures\nthe intensity of the radiation absorption I(ω) as a function of frequency\nω >0 of the electromagnetic wave. Within the linear response th eory the\nabsorption intensity is written as\nI(ω)∝ωℑχαα(0,ω), (114)\nwhere ℑχαα(0,ω) is the imaginary part of the ααcomponent of the dynamic\nsusceptibility χαα(k,ω) at zero wave-vector k= 0. We notice that\nℑχαα(0,ω) =1−exp (−βω)\n2Sαα(0,ω), (115)\nwhere the dynamic structure factor is defined by Eq. (4). Thus , the pe-\nculiarities of the dynamic structure factor Sαα(0,ω) caused, e.g., by the\nXYexchange interaction anisotropy, Dzyaloshinskii-Moriya interaction or\ndimerization should manifest themselves in ESR experiment s. The time-\ndependent spin correlation functions taken at the same site or at the neigh-\nboring sites manifest themselves in the spin-lattice relax ation rate 1 /T1\nmeasured by nuclear magnetic resonance (NMR)61.\nThe activity in the field of the Jordan-Wigner fermionizatio n approach\nhas expanded much over the last few decades. Despite some lim itations,\nthe Jordan-Wigner fermionization approach has a wide range of applica-\nbility. Particularly attractive is that it allows one to han dle complicated\nproblems of low-dimensional quantum spin systems armed wit h relatively\nsimple tools. It thus seems quite likely that it will continu e to be used\nsuccessfully in the coming years.\nAcknowledgments\nThe author would like to thank T. Krokhmalskii, T. Verkholya k, J. Stolze,\nG. M¨ uller and H. B¨ uttner in collaboration with whom the stu dy of the\ndynamics was performed. He is grateful to T. Krokhmalskii fo r preparingDecember 27, 2008 16:39 WSPC/Trim Size: 9in x 6in for Review V olume oderzh˙070612\n50 O. Derzhko\nall figures for the paper, many interesting conversations an d helpful com-\nments and suggestions. He thanks J. Stolze and T. Verkholyak for a criti-\ncal reading of the manuscript. 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Exp. , P12013 (2006)."    },    {        "title": "0711.1048v2.Hamiltonian_of_two_spinning_compact_bodies_with_next_to_leading_order_gravitational_spin_orbit_coupling.pdf",        "content": "arXiv:0711.1048v2  [gr-qc]  16 Apr 2008Hamiltonian of two spinning compact bodies with next-to-le ading order gravitational\nspin-orbit coupling\nThibault Damour∗\nInstitut des Hautes ´Etudes Scientifiques, 91440 Bures-sur-Yvette, France\nPiotr Jaranowski†\nFaculty of Physics, University of Bia/suppress lystok, Lipowa 41, 15 –424 Bia/suppress lystok, Poland\nGerhard Sch¨ afer‡\nTheoretisch-Physikalisches Institut, Friedrich-Schill er-Universit¨ at, Max-Wien-Pl. 1, 07743 Jena, Germany\n(Dated: October 24, 2018)\nA Hamiltonian formulation is given for the gravitational dy namics of two spinningcompact bodies\nto next-to-leading order ( G/c4andG2/c4) in the spin-orbit interaction. We use a novel approach\n(valid to linear order in the spins), which starts from the se cond-post-Newtonian metric (in ADM\ncoordinates) generated by two spinless bodies, and compute s the next-to-leading order precession, in\nthis metric, of suitably redefined “constant-magnitude” 3- dimensional spin vectors S1,S2. We prove\nthe Poincar´ e invariance of our Hamiltonian by explicitly c onstructing ten phase-space generators\nrealizing the Poincar´ e algebra. A remarkable feature of ou r approach is that it allows one to derive\ntheorbitalequations of motion of spinning binaries to next-to-leadin g order in spin-orbit coupling\nwithout having to solve Einstein’s field equations with a spi n-dependent stress tensor. We show\nthat our Hamiltonian (orbital and spin) dynamics is equival ent to the dynamics recently obtained\nby Faye, Blanchet, and Buonanno, by solving Einstein’s equa tions in harmonic coordinates.\nPACS numbers: 04.25.-g, 04.25.Nx\nI. INTRODUCTION\nIn view of the needs of upcoming gravitational-wave observations, it is crucial to be able to describe in detail the\ndynamics of spinning compact binaries. We think that this aim will be fullfilled by combining the k nowledge acquired\nby analytical techniques with that obtained by numerical ones. The present paper is devoted to a new, Hamiltonian\nanalytical treatment of the general relativistic dynamics of spinnin g binaries.\nThe dynamics of spinning bodies in general relativity is a rather complic ated problem which has been the subject of\nmany works over many years (starting from the pioneering contrib utions of Mathisson [1], Papapetrou [2], Pirani [3],\nTulczyjew [4], and others). This paper focusses on (gravitational) spin-orbit effects, i.e. dynamical effects which are\nlinearin the spins of a binary system. The spin-orbit interaction can be ana lytically obtained as a post-Newtonian\n(PN) expansion. The leading-order contribution of this expansion is proportional to G/c2, while the next-to-leading\norder one contains two sorts of terms: G/c4andG2/c4(hereGdenotes Newton’s gravitational constant and cthe\nspeed of light). The first complete derivation of leading-order (LO) spin-orbit effects in comparable-mass binary\nsystems is due to Barker and O’Connell [5, 6]. These authors derive d the spin-orbit interaction by considering the\nquantum scattering amplitude of two spin-1\n2particles. This curious fact prompted several authors to give pur ely\nclassical derivations of LO spin-orbit effects (see, e.g., Refs. [7, 8, 9]). For a discussion of LO spin-orbit effects in\ncoalescing binary systems see Refs. [10, 11].\nThe next-to-leading order(NLO) spin-orbit interaction was analyt ically tackled only overthe last few years. After a\nfirst incomplete attack due to Tagoshi, Ohashi, and Owen [12], comp lete results were obtained very recently by Faye,\nBlanchet, and Buonanno [13] and Blanchet, Buonanno, and Faye [14 ]. Reference [13] calculated the translational\nequations of motion, as well as the rotational equations of motion f or compact spinning binaries to NLO (as here,\nonly terms linear in spin were considered). For their derivation, Blanc het et al., working in harmonic coordinates,\nintroduced the pole-dipole energy-momentum tensor due to Tulczy jew [4] in the Einstein field equations. They also\nused the general-relativistic-covariant spin supplementary condition (SSC) of Tulczyjew [4] or, equivalently in the\n∗Electronic address: damour@ihes.fr\n†Electronic address: pio@alpha.uwb.edu.pl\n‡Electronic address: gos@tpi.uni-jena.de2\nlinear-in-spin approximation, of Pirani [3].\nThe new derivation of NLO spin-orbit interactions in the present pap er is based on a novel approach, and is totally\nindependent from the results of Refs. [13, 14]. At the end, we shall be able to connect our results to those of [13, 14],\nthereby giving us confidence in the correctness of both investigat ions. We do not use Tulczyjew’s pole-dipole energy-\nmomentum tensor. We do not either make use ofthe Papapetrou(o r, more completely, Mathisson-Papapetrou-Pirani)\ntranslational equations of motion. Our starting point consists of t he second post-Newtonian (2PN) metric generated\nby spinless point masses in ADM coordinates, say g(2PN)o. The crux of our approach then consists in noting that (to\nlinear order in the spins) it is enough to compute the NLO spin precess ion equations in g(2PN)oto derive the spin-\norbit NLO contribution in the Hamiltonian, say HNLO\nso(x1,x2,p1,p2,S1,S2). Then, from HNLO\nso(x1,x2,p1,p2,S1,S2)\nwe can derive the NLO spin-dependent terms in the translational eq uations of motion (simply by using Hamilton’s\ncanonical evolution equations). Technically, we shall derive the spin precession equations by starting from the 4-\ndimensional parallel transport equation for the spin 4-vector (wit h covariant spin supplementary condition), and then\nby rewriting them in terms of a suitably defined 3-dimensional spin vec tor, having a constant Euclidean magnitude.\n(This method is essentially that used in Ref. [7] at the LO.) We shall th en check the Poincar´ e invariance of our\nHamiltonian by explicitly constructing ten phase-space generators realizing the Poincar´ e algebra (similarly to the\nproof of the Poincar´ e invariance of the 3PN orbital Hamiltonian give n in [15]). After our construction, we shall give\nthe relation with the results obtained in Refs. [13, 14] in the form of e xplicit transformation formulae.\nWe leave to a sequent paper a discussion of the physical consequen ces of our Hamiltonian formulation, and notably\nits use for improving the description of spin effects within the effective one-body approach [16].\nII. 3-DIMENSIONAL EUCLIDEAN SPIN VECTOR IN CURVED SPACETIM E, AND ITS ANGULAR\nVELOCITY\nWhen working to linear order in the spin, the translational and rotat ional equations of motion of a spinning particle\nin curved space [1, 2, 3, 4] (see also [17] and [13]) read1\nmDuµ\ndτ=1\n2ǫαβλρ\n√−g˜SαuβuνRν\nµλρ, (2.1)\nD˜Sµ\ndτ= 0, (2.2)\nwhereuµis the normalized 4-velocity of the spinning particle, uµuµ=−1,mits conserved mass, and ˜Sµits 4-\ndimensional spin vector; in addition, τdenotes the proper time parameter, d xµ/dτ=cuµ, D the 4-dimensional\ncovariant derivative, Rµ\nνλρthe Riemann curvature tensor, and gthe determinant of the 4-dimensional metric gµν.\nAn important feature of our approach is that we shall not need to c onsider the translational equations of motion\n(2.1). It will be enough to consider the rotational ones (2.2). One immediate consequence of (2.2) is that the\n4-dimensional length of ˜Sµis preserved along the world line\ngµν˜Sµ˜Sν=s2, s2= const, (2.3)\nwheregµνgνλ=δµ\nλ. The constant scalar smeasures the proper magnitude of the spin. The Eqs. (2.1) and (2.2 ), to\nlinear order in spin, are compatible with the covariant SSC\n˜Sµuµ= 0. (2.4)\nAt the same approximation, this (Pirani [3]) SSC is equivalent to the T ulczyjew [4] one Sµνpkin\nν= 0, where pkin\nµ=\nmcuµ+O(s2) is thekinematical momentum (which differs from the canonical momentum we shall use b elow), and\nwhereSµνis the antisymmetric spin tensor (see, e.g., [13]).\nMore explicitly, Eq. (2.2) reads, when expressed in terms of the coo rdinate time t≡x0/c,\nd˜Sµ\ndt=cΓν\nµρ˜Sνvρ, (2.5)\n1In this paper, Greek indices run over the numbers 0 ,1,2,3, Latin indices over 1 ,2,3,ǫλραβis the completely antisymmetric (flat\nspacetime) Levi-Civita symbol with ǫ0123= 1. Note that Ref. [17] uses an opposite sign convention for ǫλραβ, which leads to an\nopposite sign on the right-hand-side of (2.1).3\nwhere Γνµρare the Christoffel symbols and vµ≡c−1dxµ/dt=uµ/u0= (1,vi). Note that, in this paper, we normalize\nthe “velocity” vi≡c−1dxi/dtso that it is dimensionless.\nIn addition, we can use Eq. (2.4) to compute the covariant time comp onent of the spin vector in terms of its\n(covariant) spatial components:\n˜S0=−˜Sivi. (2.6)\nSubstituting this result into Eq. (2.3) one finds that the constancy of the 4-dimensional spin magnitude takes the\n3-dimensional form\nGij˜Si˜Sj=s2, (2.7)\nwhereGijis the symmetric matrix:\nGij≡gij−g0ivj−g0jvi+g00vivj. (2.8)\nNow a technically very useful fact is that a positive-definite symmetric matrix such as the one just defined, Gij, admits\naunique positive-definite symmetric square root , sayHij=Hji, such that\nGij=HikHkj. (2.9)\nThis uniqueness result (in some given coordinate system) then natu rally leads us to defininga constant-in-magnitude\n3-dimensional Euclidean spin vector Si≡Sias2\nSi≡Hij˜Sj, SiSi=s2. (2.10)\nUpon further use of the spin supplementary condition (2.6), the sp atial covariant component of the rotational\nequation of motion (2.5) yields\nd˜Si\ndt=˜Vij˜Sj, (2.11)\nwhere\n˜Vij≡c/parenleftBig\nΓji0+Γjikvk−Γ0i0vj−Γ0ikvjvk/parenrightBig\n. (2.12)\nMaking use of Eqs. (2.10) and (2.11) one can now easily derive an evolu tion equation for the constant-magnitude\n3-dimensional spin vector Si(dot means differentiation with respect to the coordinate time t):\n˙Si=VijSj, Vij≡˙Hik(H−1)kj+Hik˜Vkl(H−1)lj. (2.13)\nThe constancy of the Euclidean magnitude of Siimplies that the matrix Vijdetermining the “rotational velocity” of\nSi=Siisantisymmetric :Vij=−Vji(a result which is easily checked to hold for the explicit expression of Vijgiven\nabove). It is then convenient to “dualize” Vijand to replace it by the 3-dimensional Euclidean (pseudo-) vector\nΩi≡ −1\n2εijkVjk. (2.14)\nWith this notation the rotational equation of motion (2.13) reads\n˙Si= +εijkΩjSk. (2.15)\nIn other words, we get a Newtonian looking spin precession equation ˙S=Ω×S.\nIn summary, the angularvelocityofrotation Ωofthe constant-magnitudespin 3-vector(2.10) isdirectlycomput able\nfrom the spacetime metric (and its Christoffel symbols) by using the explicit formulas (2.12), (2.13), and (2.14). (For\nthe self-gravitating spinning particles we are considering, one will ne ed, as usual, to regularize the self-interaction\nterms hidden in the formal results written above. See below.) Note t hatΩdepends, in general, both on the positions\nand the velocities of all the particles in the system. Indeed, from the explicit formulas above, one sees that Ωdepends\non the velocity of the considered spinning particle. Moreover, the m etric and Christoffel symbols at the location of\nsome particle will depend on the positions and velocities of the otherparticles.\n2A slightly more geometrical way of phrasing this definition w ould consist in saying that, starting from a given coordinat e system, we\nare constructing a well-defined orthonormal “rep` ere mobil e” (or “vierbein”) along the worldline of a spinning particl e, with respect to\nwhich the covariant spin 4-vector has components (0 ,Si). By definition, the spatial components of the metric in this local orthonormal\nframe take the standard Euclidean values δij, so that we can trivially raise or lower indices on our spin 3- vector.4\nIII. DERIVING THE SPIN-ORBIT INTERACTION HAMILTONIAN FROM THE ANGULAR\nVELOCITY OF THE EUCLIDEAN SPIN 3-VECTOR\nLet us now show how the knowledge of the just discussed spin angula r velocity vector Ωallows one to derive the\nspin-orbit interaction Hamiltonian Hso, i.e. the part of the Hamiltonian which is linear in the spin variables.\nLet us first recall that a basic result in Hamiltonian dynamics is Darbou x’s theorem which says that any (non\nsingular) symplectic form ωon an even-dimensionalmanifold can alwaysbe (locally) rewritten (af ter a suitable change\nof phase-space coordinates) in the canonical form ω=/summationtext\nAdqA∧dpA. When considering N(interacting) spinning\nparticles, the dimension of phase space is N(3 + 3 + 2) = 8 N, because the description of each particle requires: 3\nspatial coordinates, 3 momenta and 2 spin degrees of freedom , such as two angles θ,φneeded to parametrize the\ndirection of the (constant-magnitude) spin 3-vector Si. Darboux’s theorem then means, in this case, that it is always\npossible to redefine phase-space coordinates such that the symp lectic form takes the form\nω=/summationdisplay\na/parenleftBig/summationdisplay\nidqi\na∧dpai+sad(−cosθa)∧dφa/parenrightBig\n.\nHerea= 1,...,Nlabels the various particles (with N= 2 in our case), while i= 1,2,3 labels the spatial dimensions.\nWe have written ωin the form it is known to take in special relativity [18, 19]. In the latte r case (and, say for\nsimplicity, in the case of free particles), the spin-dependent term in ωwas shown to take (globally) the form indicated,\nwithsadenoting the magnitude of the conserved spin of the ath particle, in the sense of (2.3), and with θaandφa\ndenoting the polar angles of the flat-space limit of the above-introd uced constant-magnitude Euclidean spin vector\nSi\na, (2.10). When considering the interacting case (i.e. turning on a non-zero value of G/c2), and when keeping, for\nsimplicity, only the terms linear in spin (so that one can expand the dyn amics in powers of both Gandsa), it is\neasily checked (by a perturbation analysis3) that it is alwayspossible to construct Darboux-type canonical co ordinates\nwhere the spin degrees of freedom are simply the polar angles (in a loc al orthonormal frame) of the above-introduced\nconstant-magnitude Euclidean spin vector Si\na.4\nFinally, we can transcribe this result in the language of Poisson brack ets (instead of that of a symplectic form), by\nstating that there exist phase-space variables x= (xi\na),p= (pa\ni), andS= (Sa\ni) (witha= 1,...,N, andi= 1,2,3),\nwhereSa\niare, say, the constant-magnitude vectors (2.10) such that the usual (Newtonian-like) Poisson brackets\n{xi\na,pb\nj}=δb\naδi\nj,{Sa\ni,Sb\nj}=δabεijkSa\nk,zero otherwise , (3.1)\napply to the case of a general-relativistically interacting sytem of Nspinning particles.\nNote, however, that this result is essentially kinematical , and has nearly no dynamical content. To describe the\ndynamicsofinteractingspinningparticles, weneed toknowtheexpr essionoftheHamiltonian intermsofthecanonical\nvariables: H=H(xa,pa,Sa). As we work linearly in the spins, we look for an Hamiltonian of the gene ral form:\nH(xa,pa,Sa) =Ho(xa,pa)+Hso(xa,pa,Sa). (3.2)\nHere,Hodenotes the orbital part of H, whileHsocontains all the linear-in-spin terms, and can be called the “spin-\norbit part”. The orbital Hamiltonian Hois explicitly known up to the 3PN order [15, 20]. Our aim here is to comput e\nthe spin-orbit Hamiltonian Hsoto NLO. Because Hsois, by definition, linear in the spins we can always write it in\nthe general form\nHso(xa,pa,Sa) =/summationdisplay\naΩa(xb,pb)·Sa, (3.3)\nwhereΩa= (Ωi\na) depends on (all) the orbital degrees of freedom ( xb,pb), but does not depend on the spins Sb. The\nscalar product in the Eq. (3.3) is the usual Euclidean one.\nIn Eq. (3.3) Ωais a priori just a notation for the coefficient of SainHso. But let us now show that it is equal to\nthe quantity computed in the previous section, i.e. the angular veloc ity with which the ath spin vector Saprecesses.\nIndeed, the general principles of Hamiltonian dynamics, together w ith the canonical Poisson brackets (3.1) and the\nform (3.3), yield\n˙Sa={Sa,Hso(xb,pb,Sb)}=Ωa(xb,pb)×Sa. (3.4)\n3I.e. by considering general coordinate changes of the form q′=q+O(s),p′=p+O(s) and working linearly in the spins s.\n4As we shall discuss below, we can still modify Si\naby a rather general local rotation, but the important point i s that our definition of\nSi\na, (2.10), is a smooth deformation of the correct flat-spaceti me limit.5\nThe only difference between (3.4) and the previous result (2.15) is th at, in (3.4), Ωais expressed in terms of canonical\npositionsandmomenta,whilein(2.15) Ωitwascomputedintermsof(sayADM)coordinatesandcoordinatev elocities.\nBecause we are working only to linear order in the spin, and because ( as was explained above) the canonical phase-\nspace coordinates appearing in (3.3) and (3.4) differ from the usual ADM-type coordinates used to express the metric\n[and thereby to compute the angular velocity Ωa(xADM\nb,vADM\nb) by means of (2.12), (2.13), and (2.15)] only by terms\nproportionalto the spins, it sufficesto use the known[15, 20] spin lesslink between ADM momentaand ADM velocities\nto compute Ωa(xb,pb) fromΩa(xADM\nb,vADM\nb).\nIn the previous section we introduced a specific, well-defined “cons erved” spin 3-vector Sito parametrize the two\ndegrees of freedom of a spinning particle. Our choice had the nice fe atures of being universally associated to the\nchoice of a coordinate system, and of reducing to the choice made in the flat spacetime limit [19]. However, it was by\nno means physically unique.\nLet us now show that the freedom in the choice of conserved spin ve ctor is simply a “gauge freedom” (local rotation\ngroup) which does not change the physical results one can deduce from the Hamiltonian. Indeed, the condition\nSiSi=s2leaves as ambiguity in the definition of the conserved spin variable Sia local 3-dimensional Euclidean\nrotation Si→S′\ni, with\nS′\ni=RijSj, (3.5)\nwhereRis an arbitrary rotation matrix. It is sufficient to consider the case o f an infinitesimal rotation, say\nRij=δij−θij, (3.6)\nwhereθijis a small antisymmetric matrix. This leads to an infinitesimal change\nδS=θ×S, (3.7)\nwhere we introduced the dual vector θsuch that θij=εijkθk.\nLet us show that such a change can be considered as being induced b y an infinitesimal canonical transformation g\nin the full phase space ( x,p,S). (Canonical transformations are symmetries of Hamiltonian dyna mics. In particular\nthey preserve the basic Poisson brackets written above.) We reca ll that such a canonical transformation acts on any\nphase-space function faccording to\nδf={f,g}. (3.8)\nIt is then easily checked that a transformation of the form\ng(x,p,S) =θ(x,p)·S (3.9)\ntransforms the spin vector according to\nδS={S,g}=θ×S, (3.10)\nwhich exactly reproduces the effect of an infinitesimal local rotatio n written above. However, we have learned that\nsuch a local rotation must be accompanied by a corresponding tran sformation of the orbital degrees of freedom ( x,p)\nof the form: δx={x,g},δp={p,g}. Then, under the simultaneous changes of x,p,Sinduced by the canonical\ntransformation g(and the corresponding change of the spin angular velocity Ω′≃Ω+ dθ/dt) one finds that the\nnumerical value (evaluated at corresponding phase-space points ) of the Hamiltonian is invariant.\nWe havethereforeshownthat the arbitrarinessin the “rotationa lstate”ofthe conservedspin issimply (as expected)\na “gauge symmetry” (under a local SO(3) group).\nIV. DERIVATION OF THE SPIN-ORBIT HAMILTONIAN IN ADM COORDIN ATES\nLet us now sketch the computation of the NLO angular velocity Ω iin ADM coordinates [which will then give us\nthe NLO spin-orbit Hamiltonian according to Eq. (3.3)].\nAs usual we split the four-dimensional metric gµνinto three-dimensional objects ( α,βi,γij), where\nα≡(−g00)−1/2, βi≡g0i, γij≡gij. (4.1)\nOne can show, using the definitions βi=γijβj,γijγjk=δik, that the following exact formulas hold:\nΓ00i=1\nα/parenleftBig\nα,i+Kijβj/parenrightBig\n, (4.2a)6\nΓ0\nij=1\nαKij, (4.2b)\nΓij0=1\n2γikγkj,0−1\nαβiα,j+1\n2γik(βk,j−βj,k)−1\nαβiβkKkj, (4.2c)\nΓi\njk=3Γi\njk−1\nαβiKjk, (4.2d)\nwhereKijis the extrinsic curvature of the constant time slice. Note that, fo r convenience, we use the Kij∼+ ˙γij\nsign convention (instead of the −˙γijconvention used e.g. in Ref. [17]). In terms of the field momenta πijit reads,\nKij=16πG\nc31√γ/parenleftBig\nγikγjl−1\n2γijγkl/parenrightBig\nπkl, (4.3)\nwhereγ= det(γij). The Christoffel symbols related with the 3-metric γijare denoted by3Γi\njk. Let us also note\nthat the dimensionless coordinate velocity vican be expressed in terms of the bare kinematical linear momenta\npbare\ni=mcui, in full generality, as follows\nvi=αγijpbare\nj\n(m2c2+γklpbare\nkpbare\nl)1/2−γijβj. (4.4)\nNote, however, that the latter result applies to the canonical momentum only modulo corrections proportional to the\nspin.\nWe employ the ADMTT (ADM transverse-traceless) coordinate con ditions [21]\nγij=/parenleftbigg\n1+1\n8φ/parenrightbigg4\nδij+hTT\nij, πii= 0, (4.5)\nand recall that\nπij= ˜πij+πij\nTT (4.6)\nwithπij\nTTbeing of the order 1 /c5[22].\nLet us now expand all quantities in a post-Newtonian (PN) expansion . Here and below the subscript ( n) indicates\nthe part of a quantity which is of the nth post-Newtonian order, i.e. which is proportional to (1 /c2)n. For instance\nwe decompose:\nΩi= Ω(2)i+Ω(4)i+O(c−6). (4.7)\nHere Ω (2)i∝G/c2is the well-known LO contribution [5, 7, 8, 9], while Ω (4)i∝G/c4+G2/c4is the NLO contribution\nthat we wish to compute. These contributions are more explicitly give n in terms of the “precession velocity” ˜Vijof\nthe “coordinate spin vector” ˜Si, which entered Eq. (2.11). Inserting in Eq. (2.12) the Christoffel s ymbols (4.2), and\nthen inserting the result in Eq. (2.13) [where Hijis computed from Eqs. (2.8), (2.9), (4.1), (4.3), (4.5), and (4.6)], w e\nobtain the following more explicit formulas for the 3-vectors Ω (2)iand Ω (4)ifrom Eq. (4.7):\nΩ(2)i/c=1\n2εijk/parenleftbigg\nβ(3)j,k+/parenleftBig\nα(2),j−1\n2φ(2),j/parenrightBig\nvk/parenrightbigg\n, (4.8a)\nΩ(4)i/c=1\n2εijk/parenleftbigg\nβ(5)j,k+β(3)kα(2),j−1\n2φ(2)β(3)j,k+1\n16φ(2)φ(2),jvk−1\n2φ(4),jvk−hTT\n(4)kl,jvl\n+(α(4),j−α(2)α(2),j)vk+ ˜πjl\n(3)vkvl−1\n2α(2),kvjvlvl+1\n4˙vj\ncvkvlvl/parenrightbigg\n. (4.8b)\nAt this point, it only remains to implement three technical steps: (i) t o insert the explicit form of the 2PN-accurate\nmetric describing two spin-less particles in ADMTT coordinates (from [23] and [22]), (ii) to replace the velocities viby\ntheir 1PN-accurate expression in terms of the canonical momenta pi, and, finally, (iii) to regularize the self-interaction\nterms that arise when evaluating Eqs. (4.8).\nThe explicit expressions for the metric functions entering Eqs. (4.8 ) can be found e.g. in Appendix A of Ref. [22]\n(where the functions φ(2),φ(4), ˜πij\n(3), andhTT\n(4)ijcan be found) and in Ref. [23] (where the functions α(2),α(4)and\nβ(3)i,β(5)iare given).7\nAs for reexpressing the velocities in terms of momenta, it yields a fur ther PN-expansion of the form vi\na=vi\na(1)+\nvi\na(3)+O(1/c5), where5vi\na(1)is the coordinate velocity of the ath particle expressed in terms of the canonical variables\nxaandpaat the Newtonian accuracy, i.e., vi\na(1)=pai/(mac), andvi\na(3)is the 1PN correction to pai/(mac). The\nlatter 1PN correction explicitly reads\nvi\n1(3)=G(n12·p2)\n2c3r12ni\n12+/parenleftBig\n−p2\n1\n2m3\n1c3−3Gm2\nm1c3r12/parenrightBig\np1i+7G\n2c3r12p2i, (4.9)\nthe expression for vi\n2(3)can be obtained from the above by exchanging the particles’ labels.\nThe final step then consists in evaluating (note that the meaning of Ωi\na(2)and Ωi\na(4)is now slightly different because\nof the re-expansion of velocities in a PN expansion)\nΩi\na(2)/c≡1\n2εijkRega/parenleftbigg\nβ(3)j,k+/parenleftBig\nα(2),j−1\n2φ(2),j/parenrightBig\nvk\na(1)/parenrightbigg\n, (4.10a)\nΩi\na(4)/c≡1\n2εijkRega/parenleftbigg\nβ(5)j,k+β(3)kα(2),j−1\n2φ(2)β(3)j,k+1\n16φ(2)φ(2),jvk\na(1)−1\n2φ(4),jvk\na(1)−hTT\n(4)kl,jvl\na(1)\n+(α(4),j−α(2)α(2),j)vk\na(1)+ ˜πjl\n(3)vk\na(1)vl\na(1)−1\n2α(2),kvj\na(1)vl\na(1)vl\na(1)+1\n4˙vj\na(1)\ncvk\na(1)vl\na(1)vl\na(1)\n+/parenleftBig\nα(2),j−1\n2φ(2),j/parenrightBig\nvk\na(3)/parenrightbigg\n, (4.10b)\nwhere Rega/parenleftbig\nf(x)/parenrightbig\nindicates that one must regularize the limit x→xa. At the level at which we are working,\nthis regularization is not ambiguous and can, for instance, be simply p erformed by using Hadamard’s “partie finie”\nregularization (as explained, e.g., in Appendix B of Ref. [22]). The final results we got read\nΩ1(2)=G\nc2r2\n12/parenleftbigg3m2\n2m1n12×p1−2n12×p2/parenrightbigg\n, (4.11a)\nΩ1(4)=G2\nc4r3\n12/parenleftBigg/parenleftbigg\n−11\n2m2−5m2\n2\nm1/parenrightbigg\nn12×p1+/parenleftbigg\n6m1+15\n2m2/parenrightbigg\nn12×p2/parenrightBigg\n+G\nc4r2\n12/parenleftBigg/parenleftbigg\n−5m2p2\n1\n8m3\n1−3(p1·p2)\n4m2\n1+3p2\n2\n4m1m2−3(n12·p1)(n12·p2)\n4m2\n1−3(n12·p2)2\n2m1m2/parenrightbigg\nn12×p1\n+/parenleftbigg(p1·p2)\nm1m2+3(n12·p1)(n12·p2)\nm1m2/parenrightbigg\nn12×p2+/parenleftbigg3(n12·p1)\n4m2\n1−2(n12·p2)\nm1m2/parenrightbigg\np1×p2/parenrightBigg\n.(4.11b)\nThe expressions for Ω2(2)andΩ2(4)can be obtained from the above formulas by exchanging the particle s’ labels.\nFrom these results we can then explicitly write the spin-orbit Hamilton ian to leading and next-to-leadingPN orders.\nIndeed,\nHso(xa,pa,Sa) =/summationdisplay\naΩa(xb,pb)·Sa=/summationdisplay\na/parenleftbig\nΩa(2)(xb,pb)+Ωa(4)(xb,pb)/parenrightbig\n·Sa. (4.12)\nMore explicitly, the separate LO and NLO contributions in the PN expa nsion of the spin-orbit interaction term,\nHso(xa,pa,Sa) =1\nc2HLO\nso(xa,pa,Sa)+1\nc4HNLO\nso(xa,pa,Sa)+O/parenleftbigg1\nc6/parenrightbigg\n, (4.13)\n5Here and below a,b= 1,2 are the particles’ labels, so ma,xa= (xi\na), andpa= (pai) denote, respectively, the mass parameter, the\nposition vector, and the linear momentum vector of the ath body; for a/ne}ationslash=bwe also define rab≡xa−xb,rab≡ |rab|,nab≡rab/rab;\n|·|stands here for the Euclidean length of a 3-vector.8\nread,\nHLO\nso(xa,pa,Sa) =c2/summationdisplay\naΩa(2)(xb,pb)·Sa, (4.14a)\nHNLO\nso(xa,pa,Sa) =c4/summationdisplay\naΩa(4)(xb,pb)·Sa. (4.14b)\nFinally, note a remarkable feature of our Hamiltonian approach to sp in-orbit effects: the sole computation of the\nrotational velocity of the (“conserved”) spin vector (given by parallel trans port in the 2PN-accurate metric of N\nspinlessbodies) determines the NLO spin-dependent terms in the translational equations of motion of Nspinning\nparticles. Indeed, the sole knowledge of Ωa(xb,pb) yields that of the total spin-dependent Hamiltonian (3.2) with\n(3.3), so that the general principles of Hamiltonian dynamics (with ca nonical Poisson brackets) yield\n˙xa= +∂H(xb,pb,Sb)\n∂pa,˙pa=−∂H(xb,pb,Sb)\n∂xa. (4.15)\nIn view of the availability of algebraic manipulation programmes, there is no need to write down explicitly the trans-\nlational equations of motion (4.15), with NLO accuracy in spin-orbit t erms (and 3PN accuracy in spin-independent\nterms [15, 20]). We shall verify below that the Hamiltonian, ADM-coor dinate translational equations of motion (4.15)\nare equivalent to the harmonic-coordinate ones recently derived in [13, 14] by a more complex calculation which\ninvolved the explicit consideration of spin-dependent contributions in the metric.\nV. POINCAR ´E INVARIANCE\nThe general relativistic dynamics of an isolatedN-body system should admit the full Poincar´ e group as a global\nsymmetry (because it is a symmetry which preserves asymptotic fla tness). On the other hand, this symmetry is\nnot manifest in the Hamiltonian ADM approach to the N-body dynamics because it splits space and time, and uses\nnon-Lorentz covariant coordinate conditions. In a previous pape r [15], treating non-spinning particles, the authors\nshowed how to bypass this technical mismatch: the basic idea is that , in the Hamiltonian formalism, the global\nPoincar´ e symmetry is realized in phase-space in a non-linear manner . However, one can efficiently detect the presence\nof this symmetry by proving the existence of ten phase-space gen eratorsH(xa,pa,Sa),Pi(xa,pa,Sa),Ji(xa,pa,Sa),\nGi(xa,pa,Sa) (depending on all phase-space variables) whose Poisson bracket s reproduce the standard Poincar´ e\nalgebra. In the case of non-spinning particles, Ref. [15] construc ted the ten generators of the Poincar´ e group at the\n3PN level of approximation. We shall show here how to extend this co nstruction to the more involved case of a system\nof spinning particles.\nLet us first recall the explicit Poisson-bracket form of the Poincar ´ e algebra that should be realized:\n{Pi,Pj}= 0,{Pi,H}= 0,{Ji,H}= 0, (5.1a)\n{Ji,Pj}=εijkPk,{Ji,Jj}=εijkJk, (5.1b)\n{Ji,Gj}=εijkGk, (5.1c)\n{Gi,H}=Pi, (5.1d)\n{Gi,Pj}=1\nc2H δij, (5.1e)\n{Gi,Gj}=−1\nc2εijkJk. (5.1f)\nThe translation, Pi, and rotation, Ji, generators are simply realized as\nPi(xa,pa,Sa) =/summationdisplay\napai, (5.2a)\nJi(xa,pa,Sa) =/summationdisplay\na/parenleftbig\nεikℓxk\napaℓ+Sai/parenrightbig\n. (5.2b)9\nNote the very simple, additive, form of these generators, and, in p articular, how our Hamiltonian “conserved spin”\nvariables appear as Newtonian-like (but relativistically correct) con tributions.\nAs for the Hamiltonian H, we already know that (in our linear-in-spin approximation), it is a sum of an orbital\npart,Ho, and of the above-determined spin-orbit part, Hso, Eqs. (4.12), (4.13), and (4.14):\nH(xa,pa,Sa) =Ho(xa,pa)+Hso(xa,pa,Sa). (5.3)\nThe orbital Hamiltonian Ho(including the rest-mass contribution) is explicitly known up to the 3P N order [15, 20]:\nHo(xa,pa) =/summationdisplay\namac2+HoN(xa,pa)+1\nc2Ho1PN(xa,pa)+1\nc4Ho2PN(xa,pa)+1\nc6Ho3PN(xa,pa)+O/parenleftbigg1\nc8/parenrightbigg\n.(5.4)\nThe most delicate generator to consider is the boost (or center-o f-mass) vector G. It can be represented as a sum\nof “orbital” and “spin-orbit” parts\nG(xa,pa,Sa) =Go(xa,pa)+Gso(xa,pa,Sa), (5.5)\nwhere, as everywhere in this paper, we call “spin-orbit” the part w hich is linear in the spin variables. The orbital\npart,Go, was explicitly determined up to the the 3PN order in Ref. [15]:\nGo(xa,pa) =/summationdisplay\namaxa+1\nc2Go1PN(xa,pa)+1\nc4Go2PN(xa,pa)+1\nc6Go3PN(xa,pa)+O/parenleftbigg1\nc8/parenrightbigg\n.(5.6)\nThe spin-orbit partcan be decomposedin leading-order(LO), next -to-leading-order(NLO), and further contributions:\nGso(xa,pa,Sa) =1\nc2GLO\nso(xa,pa,Sa)+1\nc4GNLO\nso(xa,pa,Sa)+O/parenleftbigg1\nc6/parenrightbigg\n. (5.7)\nThe leading-order term in (5.7) is known from the special-relativistic lim it (by replacing the special-relativistic energy\nin the results of, e.g., Refs. [19, 25], by the rest-mass contribution )\nGLO\nso(xa,pa,Sa) =−S1×p1\n2m1+(1↔2), (5.8)\nwhere the operation “+(1 ↔2)” denotes the addition to each displayed term of another one obt ained by exchanging\nthe particles’ labels.\nThe real difficulty lies in constructing the NLO contribution to the boo st generator (and in proving that it satisfies\nthe correct Poincar´ e algebra displayed above). We solved this pro blem by using (as in our previous work [15])\nthe method of undetermined coefficients. The most general form o fGNLO\nsocan a priori depend on eight unknown\ndimensionless numerical coefficients g1,...,g 8:\nGNLO\nso=p2\n1\n8m3\n1S1×p1+Gm2\nr12/parenleftBigg\ng1S1×p1\nm1+g2S1×p2\nm2+/parenleftbigg\ng3(n12·p1)\nm1+g4(n12·p2)\nm2/parenrightbigg\nn12×S1\n+/parenleftBig\ng5/parenleftbig\nS1,n12,p1/parenrightbig\nm1+g6/parenleftbig\nS1,n12,p2/parenrightbig\nm2/parenrightBig\nn12/parenrightBigg\n+Gm2\nr2\n12/parenleftBigg\ng7/parenleftbig\nS1,n12,p1/parenrightbig\nm1+g8/parenleftbig\nS1,n12,p2/parenrightbig\nm2/parenrightBigg\nx1+(1↔2),(5.9)\nwherewehaveintroducedthefollowingnotationfortheEuclideanmix edproductof3-vectors: ( V1,V2,V3)≡V1·(V2×\nV3) =εijkVi\n1Vj\n2Vk\n3. Note that the coefficient of the first term on the right-hand-side is determined by considering\nthe special-relativistic limit [18, 19, 25]. We have also used some struc tural information coming from a conceivable\nfield-theory computation of G(say as the space integral of the 0 icomponent of some effective stress-energy tensor).\nIndeed, such a computation could be thought of in terms of some Fe ynman-like diagrams, where the interaction terms\n(i.e. those containing a power of G) would all be proportional to some basic “source” term involving eith erS1and\nm2(connected by a propagator, and possibly some power of the veloc ities,v1∼p1/m1orv2∼p2/m2), or similar\nterms involving S2andm1. The main point being that pure “self-interaction” terms (say prop ortional to S1andm1)\ncannot appear.\nLet us now consider the explicit Poincar´ e algebra requirements of E qs. (5.1a)–(5.1f). It is easily verified that the\ngenerators Pi,Ji,H, andGi, in the forms given above, exactly satisfy the relations (5.1a), (5.1 b), and (5.1c). We now\nconsider whether the center-of-mass vector Gwith the 2PN spin-orbit part given by Eq. (5.9) can satisfy the three10\nrelations (5.1d)–(5.1f). This requirement yields many equations tha t have to be satisfied by the unknown coefficients\ng1,...,g 8. We have first found that there exist uniquevalues of the coefficients g1,...,g 8ensuring the fulfillment of\nthe sole relation (5.1d). These values are\ng1=5\n4, g2=−3\n2, g3= 0, g4=−1\n2, g5=−1\n4, g6= 1, g7=3\n2, g8=−2. (5.10)\nThen we have checked that the solution (5.10) also guarantees the fulfillment of the remaining relations (5.1e) and\n(5.1f).\nIn summary, we succeeded in proving the Poincar´ e invariance of th e above-defined NLO spin-orbit interaction\n[determined by Eqs. (4.12), (4.13), and (4.14)] by explicitly constru cting ten phase-space generators satisfying the\nPoincar´ e algebra brackets of Eqs. (5.1a)–(5.1f).\nVI. COMPARISON WITH HARMONIC-COORDINATE-BASED RESULTS\nReferences [13, 14] recently computed, by means of two separat e calculations and in harmonic coordinates , both\nthe NLO spin-dependent contributions in the translational equatio ns of motion of two spinning particles, and the\ncorresponding NLO terms in the spin precessional equations of mot ion. In the present Section we shall prove that\nour results are physically equivalent to the results of Refs. [13, 14] by finding the explicit form of the transformation\nthat match the ADM variables used by us with the harmonic variables u sed in Refs. [13, 14]. Let us start by warning\nthe reader that in the whole paper [13] and in most of the paper [14] Blanchet et al. chose to express their results\nin terms of some “non-conserved” spin variables SBBF\na, i.e. variables whose Euclidean magnitudes are not conserved\nin time. It is only in Sec. VII of [14] that redefined spin variables with co nserved Euclidean lengths, say ScBBF\na, are\nintroduced and used.\nOur task here will be to exhibit the explicit transformation between t he “ADM variables” ( xa,pa,Sa) used in our\nwork, and the “harmonic variables” ( ya,va≡˙ya,ScBBF\na) used in [13, 14], and to prove that this transformation maps\nthe two sets of results into each other. (It is more convenient for us to exhibit the link with the “conserved” version\nof the harmonic spin variable used by Blanchet et al. The relation betw een their two spin variables, SBBF\naandScBBF\na,\nis given in Eq. (7.4) of [14].)\nWe write the transformation of variables in the general form6\nya(t) =Ya(xb(t),pb(t),Sb(t)), (6.1a)\nScBBF\na(t) =Σa(xb(t),pb(t),Sb(t)). (6.1b)\nLet us first find the transformation Σabetween spin variables. Section VII of Ref. [14] gives (see Eq. (7.6 ) there)\nthe explicit result for the angular velocity vector ΩBBF\naof their conserved harmonic spin variable, yielding a spin\nprecessional equation of motion of the form\ndScBBF\na\ndt=ΩBBF\na×ScBBF\na, a= 1,2. (6.2)\nThey give the NLO expression of ΩBBF\nain terms of the harmonic orbital coordinates ( ya,va). We have re-expressed\nΩBBF\na(yb,vb) in terms of ADM coordinates and momenta, to 1PN accuracy (using the well-known link between the\ntwo sets of variables7). We then compared the result with our results (4.11). We have fou nd\nΩBBF\na(2)(yb,vb) =Ωa(2)(xb,pb), (6.3a)\nΩBBF\na(4)(yb,vb) =Ωa(4)(xb,pb)+dθa\ndt, (6.3b)\nwhere\nθ1=G\nc4r12/parenleftbigg\n−(n12·p2)\n4m1n12×p1+(n12·p2)\nm2n12×p2−9\n4m1p1×p2/parenrightbigg\n. (6.4)\n6Here, both sides refer to the same numerical value of their re spective coordinate times.\n7We recall that harmonic and ADM coordinates coincide at 1PN, but that one must transform velocities into momenta by means of the\n1PN transformation Eq. (4.9). See [24] for the 3PN-accurate version of this transformation.11\nFrom the results (6.3)–(6.4) it is easy to deduce that the two sets o f spin precession equations of motion are physically\nequivalent8, and that the two sets of spin variables are related as in the genera l transformation links written above\nwith a spin transformation Σaof the explicit form:\nΣa(xb,pb,Sb) =Sa+θa(xb,pb)×Sa. (6.5)\nIn other words, our conserved spin variable differs from the conse rved spin variable defined in Eq. (7.4) of [14] by a\nsmall (time-dependent) rotation of angle θa(xb,pb). Such a difference was a priori to be expected because constant -\nmagnitude spin vectors are not uniquely defined. We have shown abo ve that to each choice of coordinate system is\ncanonically associated a particular choice of local orthonormal fra me (along the worldline of a spinning particle), and\nthereby a particular choice of “conserved” spin 3-vector. We hav e investigated whether the conserved spin 3-vector\ndefined by Blanchet et al. does correspond to applying our general definition to the case of harmonic coordinates. The\nanswer is “no”. We found that if Blanchet et al. had used our genera l definition (2.10) in their harmonic coordinate\nsystem, the angular velocity Ωathat they would have obtained would differ from our ADM spin vector b y a rotation\nvectorθadiffering from the result above by having the factor 9 replaced by 1 in the last term of Eq. (6.4). There\nis nothing surprising in such a difference as the spin re-definition used by Blanchet et al. was somewhat arbitrary.\nAnyway, as already mentioned above physical results will not depen d on such “gauge choices.”\nLet us now turn to the determination of the transformation Yabetween ADM and harmonic orbitaldegrees of\nfreedom. As usual we can decompose Yainto spin-independent, Yo\na, and spin-dependent (and linear-in-spin), Yso\na,\nterms:\nYa(xb,pb,Sb) =xa+Yo\na(xb,pb)+Yso\na(xb,pb,Sb), (6.6)\nwhere the spin-dependent term is of the form\nYso\na(xb,pb,Sb) =Yso\na(2)(xb,pb,Sb)+Yso\na(4)(xb,pb,Sb)+O(c−6). (6.7)\nThe spin-independent part of the transformationwas explicitly give n, up to the 3PN order, in Ref. [24]. The leading\norder spin-dependent part has been known for many years (see, e.g., Ref. [26]), and equals\nYso\na(2)(xb,pb,Sb) =Sa×pa\n2m2ac2. (6.8)\nWe havedetermined the next-to-leadingorderspin-dependent pa rt,Yso\na(4), by using againthe method of undetermined\ncoefficients. We haveconsidered the most generaltemplate for Yso\na(4)which depends (after using the special relativistic\nlimit to determine the 1 /c4term which remains in the G→0 limit, and structural information of the same type as\nthat explained above in the case of Gso)9on 12 unknown coefficients. It reads\nYso\n1(4)(xa,pa,Sa) =−p2\n1\n8c4m4\n1S1×p1+Gm2\nc4r121\nm1/parenleftBigg\na1S1×p1\nm1+a2S1×p2\nm2\n+/parenleftbigg\na3(n12·p1)\nm1+a4(n12·p2)\nm2/parenrightbigg\nn12×S1+/parenleftbigg\na5/parenleftbig\nS1,n12,p1/parenrightbig\nm1+a6/parenleftbig\nS1,n12,p2/parenrightbig\nm2/parenrightbigg\nn12/parenrightBigg\n+G\nc4r12/parenleftBigg\nb1S2×p1\nm1+b2S2×p2\nm2+/parenleftbigg\nb3(n12·p1)\nm1+b4(n12·p2)\nm2/parenrightbigg\nn12×S2\n+/parenleftbigg\nb5/parenleftbig\nS2,n12,p1/parenrightbig\nm1+b6/parenleftbig\nS2,n12,p2/parenrightbig\nm2/parenrightbigg\nn12/parenrightBigg\n. (6.9)\nOne can now think of two different ways of determining whether ther e exists a set of coefficients a1,...,a 6;b1,...,b 6\nsuch that our translational Hamiltonian equations of motion (with NL O spin-dependent terms), Eq. (4.15), are\n8As a further check, we have also explicitly verified that the H amiltonian time derivative (computed with our dynamics, na mely\n{SBBF\na,H}) of the originally defined (non-conserved) spin vector SBBF\naof [13] coincides with the NLO spin precession law given\nby Eqs. (6.1)–(6.3) there. To do this calculation we defined t he phase-space quantity SBBF\na(xb,pb,Sb) by inserting Eqs. (6.3), (6.4) into\nEq. (7.6) of [14].\n9More specifically we required that, say, m1Yso\n1be proportional (modulo some velocity-dependent factors i nvolving va∼pa/ma) either\ntom1S2or tom2S1.12\nphysically equivalent to the corresponding translational harmonic e quations of motion derived in [13]. (1) A first way\nwould consist of inserting the putative general transformation Ya(a1,...,a 6;b1,...,b 6) directly into the translational\nequations of motion derived in [13] (using the fact that we have alrea dy determined how their spin variables are linked\nto ours), and to compare the result to the explicit form of our tran slational Hamiltonian equations of motion, Eq.\n(4.15). This approach is, however, computationally heavy. (2) The refore, we have instead used a simpler approach\nconsisting in comparing the ten conserved quantities derived in harm onic coordinates in Ref. [13], namely the energy\nE(ya,va,SBBF\na), the total linear momentum P(ya,va,SBBF\na), the total angular momentum J(ya,va,SBBF\na), and\nthe center-of-mass vector G(ya,va,SBBF\na), with the ten phase-space Poincar´ e generators constructed above within\nour Hamiltonian formalism. To do this comparison explicitly, we first nee d to perform two replacements: (i) to\nreplace the non-conserved spin variable SBBF\naused in [13] in terms of the conserved one ScBBF\naintroduced in [14],\ntherebyobtainingnewexpressions E(ya,va,ScBBF\na),P(ya,va,ScBBF\na),J(ya,va,ScBBF\na),G(ya,va,ScBBF\na)fortheten\nconserved quantities, and then (ii) to replace the harmonic-coord inate velocities va= dya/dtin terms of Hamiltonian\ntime-derivatives, namely Va={Ya,H}. Finally, the values of the coefficients a1,...,a 6andb1,...,b 6must fulfill\nthe equations\nE/parenleftbig\nYa(xb,pb,Sb),Va(xb,pb,Sb),Σa(xb,pb,Sb)/parenrightbig\n=H(xa,pa,Sa), (6.10a)\nP/parenleftbig\nYa(xb,pb,Sb),Va(xb,pb,Sb),Σa(xb,pb,Sb)/parenrightbig\n=/summationdisplay\napa, (6.10b)\nJ/parenleftbig\nYa(xb,pb,Sb),Va(xb,pb,Sb),Σa(xb,pb,Sb)/parenrightbig\n=/summationdisplay\na/parenleftbig\nxa×pa+Sa/parenrightbig\n, (6.10c)\nG/parenleftbig\nYa(xb,pb,Sb),Va(xb,pb,Sb),Σa(xb,pb,Sb)/parenrightbig\n=G(xa,pa,Sa). (6.10d)\nBy considering the first three of these equations (i.e. by comparing the two expressions for the energy, the total linear\nmomentum, and the total angular momentum), we obtained a unique set of values for all the unknown coefficients\na1,...,a 6;b1,...,b 6. We then verified that these values satisfy also the fourth of Eqs. (6.10) (thereby giving us\nconfidence in the correctness of our Hamiltonian, and providing man y non-trivial checks of the previous results\n[13, 14]).\nOur unique solution for the spin-dependent transformation of orb ital coordinates Yso\na(2)+Yso\na(4)reads:\nYso\n1(2)(xa,pa,Sa)+Yso\n1(4)(xa,pa,Sa) =S1×p1\n2c2m2\n1−S1×p1\nc4m2\n1/parenleftBigg\np2\n1\n8m2\n1+Gm2\nr12/parenrightBigg\n+G\n2c4m2r12/parenleftBig\n3S2×p2+2(n12·p2)n12×S2+/parenleftbig\nS2,n12,p2/parenrightbig\nn12/parenrightBig\n.(6.11)\nNote that the first three terms on the right side of Eq. (6.11) (i.e. t he terms proportional to S1) have the same\nstructure as the exact special relativistic value [18, 19, 25] for th e shiftYso\n1between the canonical10orbital coordinate\nx1and the usual Lorentz-covariant (harmonic) orbital coordinate y1, namely\ny1=x1+S1×p1\nm1(m1c2+E1), (6.12)\nwhereE1=/radicalbig\n(m1c2)2+(p1c)2is the relativistic energy (including the rest-mass contribution). Th ep1-dependent\nterms in the first three terms of Eq. (6.11) correspondto the NLO expansion of the special-relativistic result, while the\nadditional G-dependent contribution can be roughly understood as a gravitat ional addition to the special-relativistic\nenergyE1(though it does not have the correct coefficient to be really interpr eted so simply).\nBy contrast, the terms proportional to S2in Eq. (6.11) do not have correspondants in the special-relativistic ( i.e.\nG→0) limit. As was to be expected they vanish in the limit where the second body (of mass m2) is heavy and\nfixed (p2/m2→0). (Indeed, if we consider a non-spinning test particle, m1,S1= 0, moving in the background of\na fixed, heavy spinning mass, m2,S2, the harmonic-coordinate geodesic action of m1will already yield a canonical\nHamiltonian action.) We leave to future work a direct derivation of the se terms from the perturbative construction\nof canonical coordinates (of the type q=y+O(s),p=pbare+O(s)) alluded to above.\n10The classical canonical variables, here denoted xa,pa,Sa, correspond, at the quantum level, to the so-called Pryce-N ewton-Wigner\nvariables.13\nFinally, as a further check on the algebra, we have also used the “dir ect” method (1) mentioned above (the first\nmethod we couldhaveused todetermine the values ofthe coefficient sa1,...,a 6;b1,...,b 6). Moreexplicitly, we started\nfrom the harmonic-coordinatetranslational equations of motion w ith NLO spin-orbit effects given in Eqs. (5.3) of Ref.\n[13]. We then replaced in these equations the non-conserved spin ve ctorSBBF\naby its expression (as given in Eq. (7.4)\nof [14]) in terms of their conserved spin vector ScBBF\na. This yields 2PN-accurate translational equations of motion of\nthe form\ndva\ndt=AN\noa(yb,vb)+1\nc2/parenleftBig\nA1PN\noa(yb,vb)+ALO\nsoa(yb,vb,ScBBF\nb)/parenrightBig\n+1\nc4/parenleftBig\nA2PN\noa(yb,vb)+ANLO\nsoa(yb,vb,ScBBF\nb)/parenrightBig\n+O(c−6). (6.13)\nWe then compared the right-hand-side of Eq. (6.13), let us denote it byAa=Aa(yb,vb,ScBBF\nb), to its direct\nHamiltonian recomputation by means of our Hamiltonian flow, i.e.\nAa={Va,H}=/braceleftbig\n{Ya,H},H/bracerightbig\n, (6.14)\ntogether with the needed transformations (6.1) (determined abo ve) between harmonic and canonical variables. Again,\nthis verificationworkedperfectly and (together with the similar dire ct verificationofthe NLO spin precessionequation\nmentioned above) gives us confidence that bothsets of results (harmonic and Hamiltonian) are correct.\nAcknowledgments\nThis work is supported in part by the KBN Grant no 1 P03B 029 27 (to P .J.) and by the Deutsche Forschungsge-\nmeinschaft (DFG) through SFB/TR7 “Gravitational Wave Astrono my.” T.D. thanks ICRANet (Pescara, Italy) for\npartial support.\n[1] M. Mathisson, Acta Phys. Pol. 6, 163 (1937).\n[2] A. Papapetrou, Proc. R. Soc. A 209, 248 (1951).\n[3] F. A. E. Pirani, Acta Phys. Pol. 15, 389 (1956).\n[4] W. Tulczyjew, Acta Phys. Pol. 18, 37 (1959); 18, 393 (1959).\n[5] B. M. Barker and R. F. O’Connell, Phys. Rev. D 12, 329 (1975).\n[6] B. M. Barker and R. F. O’Connell, Gen. Relativ. Gravit. 11, 149 (1979).\n[7] T. Damour, “Note on the spin precession effect in a relativ istic binary system,” in Physics and Astrophysics of Neutron\nStars and Black Holes , Enrico Fermi Course Vol. LXV, edited by R. Giacconi and R. Ru ffini (North-Holland, Amsterdam,\n1978) pp. 547–549.\n[8] G. B¨ orner, J. Ehlers, and E. Rudolph, Astron. Astrophys .44, 417 (1975).\n[9] P. D. D’Eath, Phys. Rev. D 12, 2183 (1975).\n[10] L. E. Kidder, C. M. Will, and A. G. Wiseman, Phys. Rev. D 47, R4183 (1993).\n[11] L. E. Kidder, Phys. Rev. D 52, 821 (1995).\n[12] H. Tagoshi, A. Ohashi, and B. Owen, Phys. Rev. D 63, 044006 (2001) [arXiv:gr-qc/0010014].\n[13] G. Faye, L. Blanchet, and A. Buonanno, Phys. Rev. D 74, 104033 (2006) [arXiv:gr-qc/0605139].\n[14] L. Blanchet, A. Buonanno, and G. Faye, Phys. Rev. D 74, 104034 (2006) [arXiv:gr-qc/0605140]; Erratum: ibid.75,\n049903(E) (2007).\n[15] T. Damour, P. Jaranowski, and G. Sch¨ afer, Phys. Rev. D 62, 021501(R) (2000) [arXiv:gr-q/0003051]; Erratum: ibid.63,\n029903(E) (2000).\n[16] A. Buonanno and T. Damour, Phys. Rev. D 59, 084006 (1999) [arXiv:gr-qc/9811091]; T. Damour, P. Jaran owski,\nand G. Sch¨ afer, Phys. Rev. D 62, 084011 (2000) [arXiv:gr-qc/0005034]; T. Damour, Phys. Re v. D64, 124013 (2001)\n[arXiv:gr-qc/0103018].\n[17] C. M. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973).\n[18] J. M. Souriau, Structure des Syst` emes Dynamiques (Dunod Universit´ e, Paris, 1970).\n[19] L. Bel and J. Martin, Ann. Inst. Henri Poincar´ e 33, 409 (1980).\n[20] T. Damour, P. Jaranowski, and G. Sch¨ afer, Phys. Lett. B 513, 147 (2001) [arXiv:gr-qc/0105038].\n[21] R. Arnowitt, S. Deser, and C. M. Misner, in Gravitation: An Introduction to Current Research , edited by L. Witten (John\nWiley, New York, 1962), p. 227.\n[22] P. Jaranowski and G. Sch¨ afer, Phys. Rev. D 57, 7274 (1998) [arXiv:gr-qc/9712075]; Erratum: ibid.63, 029902(E) (2000).\n[23] T. Ohta, H. Okamura, T. Kimura, and K. Hiida, Prog. Theor . Phys.51, 1598 (1974).14\n[24] T. Damour, P. Jaranowski, and G. Sch¨ afer, Phys. Rev. D 63, 044021 (2001) [arXiv:gr-qc/0010040]; Erratum: ibid.66,\n029901(E) (2002).\n[25] A. J. Hanson and T. Regge, Ann. Phys. (N.Y.) 87, 498 (1974).\n[26] T. Damour and G. Sch¨ afer, Nuovo Cimento B 101, 127 (1988)."    },    {        "title": "2302.14206v1.Magnetic_field_control_of_light_induced_spin_accumulation_in_monolayer_MoSe__2_.pdf",        "content": "arXiv:2302.14206v1  [cond-mat.mes-hall]  28 Feb 2023Magnetic field control of light-induced spin accumulation i n monolayer MoSe 2\nRafael R. Rojas-Lopez,1,2,∗Freddie Hendriks,2Caspar H. van der\nWal,2Paulo S. S. Guimar˜ aes,1and Marcos H. D. Guimar˜ aes2,†\n1Departamento de F´ ısica, Universidade Federal de Minas Ger ais, 31270-901, Belo Horizonte, Brazil.\n2Zernike Institute for Advanced Materials, University of Gro ningen, 9747 AG Groningen, The Netherlands.\nSemiconductor transition metal dichalcogenides (TMDs) ha ve equivalent dynamics for their two\nspin/valley species. This arises from their energy-degene rated spin states, connected via time-\nreversalsymmetry. Whenanout-of-planemagneticfieldis ap plied, time-reversalsymmetryisbroken\nand the energies of the spin-polarized bands shift, resulti ng in different bandgaps and dynamics in\nthe K +and K −valleys. Here, we use time-resolved Kerr rotation to study t he magnetic field\ndependence of the spin dynamics in monolayer MoSe 2. We show that the magnetic field can control\nthe light-induced spin accumulation of the two valley state s, with a small effect on the recombination\nlifetimes. We unveil that the magnetic field-dependent spin accumulation is in agreement with hole\nspindynamics at thelonger timescales, indicatingthatthe electron spins havefaster relaxation rates.\nWe propose a rate equation model that suggests that lifting t he energy-degeneracy of the valleys\ninduces an ultrafast spin-flip toward the stabilization of t he valley with the higher valence band\nenergy. Our results provide an experimental insight into th e ultrafast charge and spin dynamics\nin TMDs and a way to control it, which will be useful for the dev elopment of new spintronic and\nvalleytronic applications.\nAtomically-thin transition metal dichalcogenides\n(TMDs) offer the possibility to optically address the\nspin and valley degrees-of-freedom of charge carriers.\nThrough circularly polarized light, σ+orσ−, one can\nexcite electron-hole pairs at opposite points at the edges\nof the Brillouin zone, known as K+andK−valleys\n[1–3]. The high spin-orbit coupling in these materials\nfurther causes a spin splitting of the levels, connect-\ning the spin and valley degrees-of-freedom and their\ndynamics. For this reason, TMDs are very attractive\nfor (opto)valleytronic and (opto)spintronic applications\n[4–8]. One of the main bottlenecks for such applications\nis the control of these degrees-of-freedom. Magnetic\nfields have been shown to strongly affect the valley\npolarization. This has been demonstrated by photo-\nluminescence (PL) measurements of TMD monolayers\nunder high out-of-plane magnetic fields, showing a linear\nshift of the emission peaks reflecting the reduction of\nthe band gap for one valley and the increase of the other\n[9–13]. This effect became known as the valley-Zeeman\neffect. These experiments extract an exciton and trion\ng-factor of ∼ −4, which is in agreement with theoretical\nexpectations [14–16]. Different works have reported the\ndynamics of electrons, excitons, and trions by perform-\ning time-resolved photoluminescence (TRPL) [17–19],\ntime-resolved differential reflectivity (TRDR) [20–22],\nand time-resolved Kerr rotation (TRKR) [23–28], with\nmost works focusing on either zero or in-plane external\nmagnetic fields. More recently, Zhang et al. [29] showed\nthe effect of out-of-plane magnetic fields on polarized\nTRPL, but without any clear difference between the\nresults obtained at different circularly polarized excita-\ntion and high magnetic fields. Nonetheless, PL is solely\n∗rrlopez@fisica.ufmg.br\n†m.h.guimaraes@rug.nlsensitive to the valley polarization and radiative decay\nof excitons and trions. Therefore, PL is insensitive\nto light-induced spin polarization of resident carriers\nwhich can sustain the spin information over much longer\ntimes. The understanding of the decay processes and\npossibilities of control of resident carriers are of huge\nimportance for the engineering of a new generation\nof opto-spintronic devices [8, 30–32]. However, the\nuse of an out-of-plane magnetic field to control the\nlong-lived spin dynamics in monolayer TMDs – beyond\nthe radiative recombination times – remains largely\nunexplored.\nHere we show that the spin information of long-lived\ncarriers in monolayer MoSe 2can be effectively con-\ntrolled via the valley-Zeeman effect. Using time-resolved\nmagneto-optic Kerr effect, we show that magnetic fields\ncan induce an ultrafast spin-valley scattering and en-\nhancement of a light-induced spin accumulation with a\npreference for the valley with higher valence band en-\nergy. While we find that the spin scattering rates do\nnot show a clear trend for the external magnetic field\nstrength, the magneto-optical signal strength shows a\nclear linear behavior. Particularly, we observe a nearly\nfield-independent long spin lifetime of ∼2 ns. The mag-\nnetic field dependence of the light-induced spin accumu-\nlation we measure agrees with hole spin dynamics and\nresident carrier recombinationat longer timescales, while\nwe argue that electron spins may have faster relaxation\nrates. Our results are well described by a simple rate-\nequation model, which takes into consideration a field-\ndependent carrier recombination and intervalley scatter-\ning. Our model indicates that the magnetic field has a\nstronger effect on the hole transfer between valleys than\nin electrons, resulting in a spin imbalance with a larger\npopulation of carriers from the valley with the smaller\nbandgap. In this way, we demonstrate here that an ap-\nplied magnetic field can be used to effectively control the2\nSpin-flip \nof holesK- K+B > 0B = 0\nFIG. 1. (a) Schematics of the TRKR measurements with an exter nal magnetic field. (b) Valley-Zeeman effect on the K +and\nK−spin states of monolayer MoSe 2. Dashed lines indicates the zero magnetic field spin states w hile solid lines represent the\nmagnetic field shifted states. The Valley-Zeeman effect resu lts in larger energy shifting in the valence band ∆ Evthan in the\nconduction band ∆ Ec. (c) TRKR signal for σ+(red) and σ−(blue) polarized pump, at different magnetic fields: B = -5 T, 0\nT (d), +5 T (e), for λ= 755 nm and T = 6 K.\noptically-generated spin accumulation in TMDs.\nOur samples were prepared by mechanical exfoliation\nof a bulk MoSe 2crystal (HQ Graphene) onto a poly-\ndimethylsiloxane (PDMS - Gel Pack) and then trans-\nferred to a 285nm SiO 2/Si substrate. The spin dynamics\nin oursamples was measuredby TRKR, which presents a\nhigher sensitivity to spin-related phenomena, compared\nwith TRPL or TRDR. We perform a single-color (degen-\nerated) pump-probe technique in a dual-frequency mod-\nulation (see supplementary information) using a similar\nexperimental setup as described elsewhere [25]. When\na high intensity circularly polarized ( σ±) laser pulse\n(pump) excites the monolayer, electrons of a single val-\nley (K±) are excited to the conduction band and can\ngenerate a spin imbalance. By shining a linearly polar-\nized (probe) pulse at a certain time-delay after the pump\npulse, the spin/valley imbalance is measured by a change\non the polarization axis of the reflected beam, which is\nrotated by an angle θK(Fig. 1a). Finally, the time evo-\nlution of the spin imbalance in the sample is measured\nby changing the delay time (d t) between the pump and\nprobe pulses. All measurements were performed at low-\ntemperature T = 6 K.\nFigure 1d shows the TRKR at zero magnetic field and\nhowthe signoftheKerrrotationdependsonwhichvalley\nwe are exciting: a positive (negative) Kerr rotation is\nobserved upon shining σ+(σ−) polarized pump pulses.\nThis behavior is a fingerprint of the symmetry and spinselectivity of the two valleys of the TMD, when excited\nwith circularly polarized light [23–25, 28, 33]. Two decay\ntime constants are visible in our measurements, a faster\n(τ1∼1.5 ps) and a longer one ( τ2∼2 ns) that goes\nbeyond the measurement range.\nWe observe a drastic change on the TRKR dynamics\nupon an applied magnetic field. Figure 1c and e show the\nTRKR signal of our sample for an applied magnetic field\nof -5 T and +5 T, respectively. Within the first 3 ps after\nexcitation we observe a reversal of the TRKR signal for\nonepumppolarization,whileastabilizationfortheother.\nThis can be explained by an out-of-plane magnetic field\nlifting the valley degeneracy causing an opposite shift on\nthe TMD bands of opposite valleys [9–12] (Fig 1b). Due\nto the different net angular momenta of the valence and\nconduction bands, the energy shifts for the conduction\n(∆Ec) and valence bands (∆ Ev) are also different. The\nresulting effect is that the lowest energy state for holes\nlies within one valley, while for electrons lies in the other.\nThese observations reveal that our TRKR signal is\ndominated by hole relaxation. Upon populating the K−\nvalley through a σ−-polarized pump, we observe a re-\nversal of the signal from negative to positive for positive\nmagnetic fields. For this case, the electron ground state\nlies in the conduction band of the K−valley, while the\nground state of holes is located at the K+valley. The\nfact that we observe a positive TRKR signal after ∼3\nps, indicates a higher spin population of the K+valley,3\nB = -5T B = +5T(b) (a)\nFIG. 2. (a) TRKR signal for σ+(red) and σ−(blue) polarized\npump, at B = -5 T (a) and +5 T (b) at long time scales.\nThe orange dashed lines show the fit that extracts the slow\nrelaxation times. (c) Extracted slow decay times ( τ2) and\n(d) amplitudes (A 2) at different magnetic fields for the two\nexcitation polarizations.\nimplying a fast intervalley scattering and that the spin\naccumulation measured in our experiments arises from\nthe hole spin. When the direction of the magnetic field\nis reversed, similar behavior is observed but with an op-\nposite TRKR signal.\nThe long decay times we observedo not showany clear\ndependence with the magnetic field. The two decay time\nconstants are obtained through a biexponential decay of\nthe form θK=A1e−t/τ1+A2e−t/τ2, whereAnare the\nKerr rotation angles at a delay time d t= 0 and τnthe\ndecay times at the fast and slow decay processes n=\n1,2, respectively. This is done for measurements using\nboth directions of pump polarization and various values\nofappliedmagneticfields, fromB=-5Tto+5T.Figures\n2a and 2b show the TRKR signals at B = -5 T to +5 T\noveralongd trange,upto1.55ns. TheTRKRsignalsare\nstill clearly measurable even at these long-time delays.\nThe magnetic field dependence for τ2is shown in Figure\n2c (see the supplementary material for τ1). For zero, and\nalsohigh magnetic fields, we find non-zeropump-induced\nTRKR signals. However, for 0 <|B|<3 T, the signal for\none of the pump polarizations is within our experimental\nnoise, and therefore only measurements for one pump\npolarizationwereusedwithinthisrange. Nonetheless, we\ndo not observe any striking features on the TRKR decay\ntimes for the whole range of magnetic fields studied, even\nthough additional spin scattering channels are predicted\ntoarisefromthebreakingoftime-reversalsymmetry[34].\nThis indicates that, while the intervalley scattering rate\ncould be modified by the magnetic field, the main source\nof spin scattering remains unaffected for our samples.\nThe amplitude of our TRKR signals shows a clear lin-\near behavior with the magnetic field, Fig. 2d. Strikingly,\nthe slopes for the two pump polarizations are slightly\ndifferent, resulting in a non-symmetric response with the\nFIG. 3. (a) TRKR Polarization at different wavelengths and\nzero external magnetic fields. Measurements were shifted in\nthe vertical axis for clarity. (b) Polarized photoluminesc ence\nspectra of our sample at B = 0 T. Inset: Intensity profile\nof the TRKR Polarization at different delay times: 2, 4, 6,\n10, 100, and 200 ps. (c) TRKR for σ+(red) and σ−(blue)\npolarized pump at λ= 765 nm for B = -5 T (c) and +5 T\n(d) at T = 6 K.\nmagnetic field direction at long time delays. Previous\nmeasurement runs of the same sample also presented a\nlinear trend with the magnetic field, with different slopes\nforeachexcitationpolarization, see supplementarymate-\nrialfordetails. Althoughsmallexperimentalartifactsare\nnot discarded, a phenomenological origin of these results\nwould be inconsistent with a simple description involv-\ning solely a Zeeman energy shift of the energy states. We\ncurrently do not have a clear understanding of the origin\nof such an effect, which should be explored in more detail\nin later studies.\nBystudyingthedependenceoftheTRKRsignalonthe\nexcitation wavelength, we can probe the contribution of\nexcitons and trions to the spin signal. To reduce spurious\neffects which are independent of the valley/spin polariza-\ntion, here we use the polarization of the Kerr rotation,\nnamely the difference of the TRKR measurements for\nthe two polarizations of excitation ( θK(σ+)−θK(σ−))/2\n[25]. As expected, our signal is strongly modulated upon\na change of the wavelength when going over the exciton\nand trion resonances (Fig. 3a). We observe that when\nexciting with λ/lessorequalslant745 nm, KR polarization decays in\nthe first couple of picoseconds. On the other hand, when\nexciting at λ/greaterorequalslant750 nm, the spin imbalance remains fi-\nnite, within the time range of our measurements. The\nlargest signal is seen at λ= 755 nm, see inset of Figure\n3b. A comparison to photoluminescence measurements\n(Fig. 3b) reveals that our findings are consistent with\nshort-lived signals coming from excitations at and below\nthe wavelength of the exciton resonance (X), while the\nlong-lived signals come from energies close to the peak\nof trion emission (T). Our observations are in agreement4\n(c)(a)\nB > 0B = 0\nFIG. 4. (a) Experimental (left) and theoretical (right) res ults for B = 0 T with τ+=τ−= 20 ps, τc+−=τc−+= 1.8 ps and\nτv+−=τv−+= 5 ns. (b) Experimental (left) and theoretical (right) resu lts for B = +5 T with an asymmetric hole scattering\ntimeτv+−= 5 ps, τv−+= 5 ns. (c) Representation of the dynamics of the spin populat ion under positive magnetic fields:\nthe ground state of the system with an initial hole populatio n is photo-excited ( σ−) at the valley K−. Later, electrons are\nscattered between valleys in the conduction band followed b y the scattering of holes from K−toK+. Subsequently, excited\nstates recombine radiatively. Finally, resident holes in K+reach thermal equilibrium with K−. The represented transfer times\nindicate the dominant process at each panel.\nwithpreviousreportsinliterature[17,18,24,35]showing\nthat trions dominate the spin signal in TMDs monolay-\ners.\nWe observe an interesting effect arising from the mag-\nnetic field-induced reduction of the bandgap for one val-\nley with respect to the other. For high magnetic fields (B\n=±5 T) and with the lowest excitation energy resulting\non a measurable signal ( λ= 765 nm), we only observe a\nTRKR signal above our noise level for one pump polar-\nization, Figure 3c and d. For the positive magnetic field,\nthe bandgap of the K +valley is reduced and produces\na spin accumulation when excited with σ+. Neverthe-\nless, as the bandgap at K −is increased, the energy of\nthe pumping photons is not enough to excite the elec-\ntrons and generate any spin imbalance, resulting in no\nTRKR signal for σ−excitation. A similar behavior is\nobserved for negative magnetic fields, but with opposite\nspin accumulation in the K −valley and no signal for σ+\nexcitation. Additional measurements of wavelength de-\npendence with the magnetic field can be found in the\nsupplementary material.\nTo quantitatively describe the spin dynamics and to\nelucidate the relaxation processes behind our measure-\nments, we use a rate-equation model similar to what has\nbeen proposed before [24], but including the effect of an\nexternal magnetic field. We consider three main carrier\nscattering processes: direct radiative recombination atthe same valley, the spin-valley flip of electrons in the\nconduction band, and the spin-valley flip of holes in the\nvalence band. The breaking of the energy degeneracy of\ntheK+andK−valleys is represented by different inter-\nvalley scattering rates. Therefore, considering the more\ngeneralcase, where conductionand valence band transfer\nrates, and also radiative recombination of both valleys,\nare different, we get to the set of equations\ndn+\ndt=−n+p+\nτ+−n+\nτc+−+n−\nτc−+(1)\ndn−\ndt=−n−p−\nτ−−n−\nτc−++n+\nτc+−(2)\ndp+\ndt=−n+p+\nτ+−p+\nτv+−+p−\nτv−+(3)\ndp−\ndt=−n−p−\nτ−−p−\nτv−++p+\nτv+−(4)\nwhereniandpiare the photo-excited populations at\ntheKivalley for electrons and holes, respectively. The\nfirst term at the right side of each equation represents\nthe radiative recombination at a specific valley with re-\ncombination time τ±. The second term represents the\npopulation of carriers that is transferred to the opposite\nvalley in the conduction or valence bands with an in-\ntervalley scattering time τc,v±∓. Finally, the last term5\nrelates to the carrier transfer coming from the oppo-\nsite valley (source term). As Kerr rotation measures the\nspin/valley imbalance of the system, we model our mea-\nsurements as the difference between the valley popula-\ntionsθK= (n++p+)−(n−+p−). The initial conditions\nallow us to consider the two types of excitation, for in-\nstance,n+(0) =N/2,p+(0) =N/2,n−(0) = 0, and\np−(0) = 0 for σ+excitation, where N/2 is the number of\nexcited carriers.\nThe results obtained by solving our rate-equation sys-\ntem are in good agreement with the experimental mea-\nsurements with and without magnetic fields. In Figure\n4a we compare our experimental and theoretical results\nat zero magnetic field, where valley degeneracy is still\npresent. We noticed that to properly reproduce our mea-\nsurements, the radiative recombination rate could not be\neither the shorter nor the longer decay process, and in-\nstead, good results were achieved by using decay times\n(τ) of tens of picoseconds. The latter agrees with previ-\nousreportsofthe trionlifetime forMoSe 2[17, 18, 26, 29].\nTherefore, as our results point to a spin accumulation of\nholes at the long time range ( τv= 5ns), the fast decay\n(τc= 1.8ps) is assigned to electron transfer at the con-\nduction band. This is consistent with previous reports\npointing out to the fast depolarization of electrons in the\nconduction band when compared to the holes, and even\nradiative recombination of trions [24, 36].\nTo simulate our results at high magnetic fields (Figure\n4b) we introduce an asymmetry in the scattering times\nof the different considered processes (see supplementary\ninformation for details). Our model indicates that the\nmain process responsible for the fast switch of spin po-\nlarization is a strong reduction of the hole inter-valley\nscattering through the valley with higher valence band\nenergy. In the right panel of Figure 4b we show the\nresult of the TRKR calculation just with the effect of\nan asymmetric scattering process at the valence band,\nby considering τv−+= 5 ps, three orders of magnitude\nsmaller than τv+−. Meanwhile, the introduction of dif-\nferent scatteringtimes for the electronsin the conduction\nband and different radiative recombination times at thethe two valleys can help to refine the result at the short\nlifetimes but they have no major impact on the total de-\ncay profile. We point out that in both calculations in\nFigures 4a and 4b, we did not obtain a long-lived decay\nprocess, suchasthe onesobservedin the experimentalre-\nsults, evenconsideringlargerscatteringlifetimesforholes\nτv. Therefore, we associate this long-lived component to\nresident charge carriers in agreement with previous re-\nports [24, 26, 27, 35]. In Figure 4c, we illustrate the\nobtained spin dynamics for positive magnetic field and\nan optical excitation of the K−valley.\nOur results show that optically-generated spins in\nTMDs can be very effectively controlled by magnetic\nfields, which is explained by a strong field-induced asym-\nmetry on the intervalley scattering rates. We envision\nthat the external magnetic field used in our experiments\ncould be replaced by magnetic proximity to van der\nWaals magnets. These heterostructures have already\nshown to result on a significant control over valley polar-\nization in photoluminescence measurements [30], and our\nwork implies that a significant polarization of long-lived\nresident carriers can potentially be efficiently obtained\nand controlledvia proximityeffects, with lifetimes ofsev-\neral ns opposed to a few ps for excitons/trions. We be-\nlievethatthiswillopennewdirectionsforopto-spintronic\ndevices based on TMDs which do not depend on the sta-\nbilization of bound electron-hole pairs, but make use of\nthe full potential of the long spin lifetimes offered by res-\nident carriers in these materials.\nSee Supplementary Information for additional details\nregarding the experimental setup, photoluminescence\nmeasurements with magnetic field, TRKR data at differ-\nent wavelengths with an external magnetic field, TRDR,\nand further discussion of the rate-equation model.\nWe thank J. G. Holstein, H. de Vries, F. van der\nVelde, H. Adema, and A. Joshua for their technical\nsupport. This work was supported by the Dutch Re-\nsearch Council (NWO—STU.019.014), the Zernike In-\nstitute for Advanced Materials, and the Brazilian fund-\ning agencies CNPq, FAPEMIG and the Coordena¸ c˜ ao de\nAperfei¸ coamento de Pessoal de N´ ıvel Superior - Brasil\n(CAPES) - Project code 88887.476316/2020-00.\n[1] K. F. Mak, K. He, J. 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Experimental setup used for our time-resolved me asurements.\nFormeasuringtime-resolveddataweusedasingle-color(degenera ted)pump-probetechniqueinadoublemodulation\nconfiguration. Figure S5 pictures our experimental setup with the key elements represented on it. For excitation, we\nused a tunable Ti:sapphire laser (Mai Tai - Spectra Physics) with a re petition rate of 80 MHz and a pulse width <\n300 fs. With a beam-splitter and a set of neutral density filters (no t shown in Figure S5), we set the pump and probe\nbeams to a 4:1 fluence ratio with a ∼6µJ/cm2probe fluence. The pump is set to circularly polarized light ( σ±)\nwith a linear polarizer (P1) and a quarter wave plate (QWP) after pas sing through a chopper with a frequency of\n70 Hz. For blocking the pump beam after the reflection on the sample we use the set of lenses L1, L3, and L4 that\nhelp focus the beam in a different place between L3 and L4, on its way b ack from the cryostat. The lens L1 also\nensures a larger spot size for the pump beam at the sample, due to a different focal point. The probe beam is linearly\npolarized (P2) and its polarization is modulated by a photoelastic modu lator (PEM) at a frequency Ω = 50 kHz.\nThe beam is detected on the way back from the sample by an amplified p hotodetector (PD) after passing through a\nlinear polarizer (A), set cross-polarized with the probe for detect ing the Kerr rotation signal.\nThe presence of the chopper in the pump path and the PEM in the pro be path has the purpose of improving the\nsignal-to-noise ratio, also known as the double modulation technique . The PEM is set with its fast axis at 45◦to\nthe axis of polarization of the incident beam that makes the polarizat ion of the probe oscillate between circularly\nand linearly polarized with a frequency 2Ω. The difference in the modula tion frequencies allows us to separate the\nsignal coming from each beam. With this configuration, the reflecte d signal is sent to one lock-in amplifier with the\nreference to the PEM oscillation frequency, which helps clean the sig nal coming from the probe beam. The output of\nthe filtered signal is sent to a second lock-in amplifier that has its ref erence on the chopper frequency and helps to\nimprove the signal from the probe which is affected by the pump. Usin g the Jones matrix formalism, it is possible to\ndetermine that the intensity of the signal at the second harmonic o f the PEM oscillation frequency is related to the\nKerr rotation as I2Ω=θKRJ2(ρ)cos(2Ωt) [25, 37], where θKthe Kerr rotation, Ris the average reflected intensity,\nJ2the spherical Bessel function of the first kind and ρ= 3.041 is the retardation given by the PEM to optimize the\nsignal. Therefore, the first lock-in amplifier is set to detect and amp lify the second harmonic which is proportional to\nthe Kerr rotation signal.8\nII. PHOTOLUMINESCENCE AT HIGH MAGNETIC FIELDS\nFigure S6. Photoluminescence spectra of 1L MoSe 2at different magnetic fields for the two polarization excitat ions. (b) Valley-\nZeeman splitting for the trion peak, calculated as the differ ence of the energies Eσ+−Eσ−. (c) Degree of circular polarization\n(Pc) of excitons and trions at different magnetic fields.\nPhotoluminescence measurements were performed with λ= 690 nm laser diode with a power of 100 µW at a\ntemperature of 6 K. For controlling the polarization, we set the fas t axis of a quarter wave plate (QWP) aligned\nto the polarization of the excitation light. When the emitted light from the sample, E+ˆσ++E−ˆσ−, with ˆσ+the\nJones matrix associated to the circularly polarized light, passes aga in through the waveplate, the fast axis is no longer\naligned with the polarization of the light. The latter will decompose mos t of the signal of the circularly polarized\nstates at ±π/4 with respect to the axis of the QWP. Therefore, by placing an analy zer (polarizer) at those angles, we\ncan independently measure the intensity |E+|2and|E−|2.\nFigure S6a shows the photoluminescence spectra of our sample at d ifferent magnetic fields B = 0, ±5 T. Each\nspectrum was measured at the two polarizations, σ+(red) and σ−(blue). The two characteristic peaks of MoSe 2are\nobserved: one coming from exciton recombination (1.664 eV) and th e other from the emission of trions (1.630 eV).\nFor analyzing the valley-Zeeman shifting we fitted the trion peak for both polarizations and plotted the difference\nof the peak centers Eσ+-Eσ−as a function of the magnetic field (Figure S6b). We obtained a linear b ehavior with\na slope of ( −0.26±0.01) meV/T, indicating a g-factor of -4.5, which is in good agreement with previous reports in\nthe literature [9, 13, 16]. Figure S6c shows the degree of circular po larization of the exciton and trion peaks as a\nfunction of the magnetic field calculated as the difference of the inte grated intensity at the two different polarizations\nPc= (Iσ+−Iσ−)/(Iσ++Iσ−).9\nIII. FAST DECAY PROCESS IN TRKR VS. MAGNETIC FIELD\nFigure S7 shows the fast decay times τ1and corresponding amplitudes A1obtained from the biexponential decay\nfits of the Kerr rotation angles measured at different magnetic field s.\nFigure S7. Extracted fast decay times τ1and (b) amplitudes A 1from the fits of TRKR presented in the main manuscript.\nIV. SPIN DYNAMICS: WAVELENGTH DEPENDENCE\nFor keeping track of the wavelength dependence of the Kerr rota tion with the magnetic field, we also measured\nthe TRKR at the two polarizations ( σ±) for different wavelengths at B = 0 T, B = ±5 T. To reduce spurious effects\nand simplify the analysis, we calculate the Kerr rotation polarization d efined as ( θK(σ+)−θK(σ−))/2. The results\nare shown in Figure S8. In all the cases, we observe that the stron ger and long-lived signals are associated with the\nwavelengths between λ= 755 nm and 760 nm. Therefore, the magnetic field does not induce c lear changes in the KR\nsignal close to exciton resonances nor change (significantly) the r esonance close to the trion energy peak.\nB = 5 T B = 0 T\n0.2 ps\n0.5 ps\n1.4 ps\n1.7 ps740 nm\n745 nm\n750 nm\n755 nm\n760 nm\n765 nm0.3 ps\n0.9 ps\n1.3 ps\n1.9 psB = -5 T\n0.4 ps\n1.1 ps\n1.7 ps\nFigure S8. Kerr rotation polarization of our sample at differ ent wavelengths studied with an out-of-plane magnetic field B=±5\nT and at B = 0 T. Extracted lifetimes are shown at each plot.10\nV. RATE EQUATION MODEL -DIFFERENT PARAMETERS\nFigure S9a shows all the different decay paths considered in our mod el. When a magnetic field is applied, valley\ndegeneracy breaking can affect the recombination and scattering times of excited spins at each valley. To start\nexploring the effect of such asymmetry, we first pursued the repr oduction of our results at zero magnetic field, Figure\nS9b. In Figure S9(c-h) we present the TRKR calculated for differen t sets of parameters. As for zero magnetic\nfield valley degeneracy is still present, equal scattering and recom bination rates are considered for both valleys:\nτ+=τ−=τr,τv+−=τv−+=τv, andτc+−=τc−+=τc.\n05 \u0000 \u0001 1000 150010.500.51Ka.u.\nDelay time (ps)0 500 1000 150010.500.51K- K+\n0 500 1000 150010.500.51\n0 500 1000 150010.500.51\n0 500 1000 150010.500.51\n0 500 1000 150010.500.51(c)\nKa.u.\nDelay time (ps)Delay time (ps) Delay time (ps)\nDelay time (ps) Delay time (ps)(d) (e)\n(f) (g) (h)B > 0B = 0\nFigure S9. (a) Representation of the valley-Zemman shiftin g and carrier transfer process of the bands at the vallleys K±under\na positive magnetic field. (b) Experimental result of TRKR in MoSe2at zero magnetic field for the two pump polarizations\nσ±at T = 6 K. (c-h) Simulation of the TRKR using the model present ed in the main text. The parameters for the calculation\nare presented in each figure. When two sets of parameters are p resented on the same figure, it indicates that the same result\nis obtained for both cases.\nWith the obtained parameters at zero magnetic field (Figure S9h), w e proceed to study the effect of the magnetic\nfield introducing the assymetry in the decay times. Figure S10 show t he results of TRKR calculted with different\nparameters for reproducing our experimental results at B = ±5 T.11\n0\u0002 \u0003 \u0004 1000 150010.500.51\n0 500 1000 150010.500.51\n0 500 1000 150010.500.51\n0 500 1000 150010.500.51Delay time (ps)\nDelay time (ps) Delay time (ps)Delay time (ps)Exp.\nExp.B = -5 TB = +5 T(a) (b) (c)\n(d) (e) (f)\nFigure S10. (a) Experimental TRKR of MoSe 2at B = 5 T. (b-c) Calculated TRKR for modeling (a). (d) Experim ental result\nat B = -5 T. (e-f) Simulated TRKR for modeling the results show n in (d)\nVI. TRKR VS. B - DIFFERENT SET OF MEASUREMENTS\nFigure S11. (a-b) TRKR decay times and (c-d) amplitudes extr acted at a different point of the sample, in a different cool dow n\nof the same sample, with a different alignment with λ= 755 nm, Fpump= 100µJ/cm2,Fprobe= 10µJ/cm2, T = 6 K.12\nVII. FURTHER DISCUSSION ON FITTING THE DECAY PROCESS\nFor fitting the spin dynamics in the main document we considered a simp le approach of two exponential decay, as\nit is within the scope of our paper. For the fits, we use the data meas ured in two ranges with different acquisition\nparameters. One for short-time scale analysis up to 20 ps, acquire d at 0.1 ps/s, and a long one for the slow decay\nprocess up to 1550 ps, with a velocity of acquisition of 3 ps/s. For ea ch data set is possible to accurately fit either\nthe fast or the slow decay process. Nevertheless, fits for interm ediate decay times (tens of ps) can fail for some curves\n(see Figure S12a). Here, we present an alternative three-decay process fit ( A1e−t/τ1+A2e−t/τ2+A3e−t/τ3) over the\nlong-range data set that helps to elucidate the dynamics of the inte rmediate process and its effect on the results of\nthe two-exponential decay.\nB = 5 T\nB = -5 TB = 0 T\n(a) (b)\nB = 5 T\nB = -5 TB = 0 T\nFigure S12. (a) TRKR at different magnetic fields for a σ+excitation fitted with a bi-exponential decay and (b) a three -\nexponential decay (black dashed line).\nFigure S13. Fitting parameters extracted from a three expon ential decay fit.\nFigure S12b presents the fits of the TRKR when exciting with polariza tionσ+at three different magnetic fields. In\nFigure S13 we present the extracted lifetimes and amplitudes of eac h decay process. We observe a consistent result for\nthe fast and slow decay processes when compared with the data pr esented with the two-exponential decay case. For\nτ2we observed lifetimes of tens to a hundred picoseconds, which is con sistent with the trion recombination lifetime13\nreported in the literature [17, 18, 24, 35]. Also, for the amplitude A2, we observe a linear behavior with the magnetic\nfield as observed for A3.\nVIII. TIME-RESOLVED DIFFERENTIAL REFLECTIVITY AT B = 0 T\nFigure S14. (a) Time-resolved differential reflectivity stu died at different wavelengths and their fits (red dashed lines ). Inset:\nextracted amplitudes Aiand; (b) extracted lifetimes τiof the bi-exponential fit."    },    {        "title": "1303.4478v3.Reading_charge_transport_from_spin_dynamics_on_the_surface_of_topological_insulator.pdf",        "content": "arXiv:1303.4478v3  [cond-mat.mes-hall]  30 Sep 2013Reading charge transport from spin dynamics on the surface o f topological insulator\nXin Liu1,2and Jairo Sinova1,3\n1Department of Physics, Texas A&M University, College Stati on, TX 77843-4242, USA\n2Department of Physics, The Pennsylvania State University, University Park, PA 16802-6300, USA\n3Institute of Physics ASCR, Cukrovarnick´ a 10, 162 53 Praha 6 , Czech Republic\n(Dated: November 5, 2018)\nResolving the conductance of the topological surface state s (TSSs) from the bulk contribution has\nbeen a great challenge for studying the transport property o f topological insulators. By developing a\nnon-purturbativediffusion equation that describes fully t he spin-charge dynamics in the strong spin-\norbit coupling regime, we present a proposal to read the char ge transport information of TSSs from\nits spin dynamicswhich can be isolated from thebulkcontrib ution bytime-resolved second harmonic\ngeneration pump-probe measurement. We demonstrate the qua litatively different Dyaknov-Perel\nspin relaxation behavior between the TSSs and the two-dimen sional spin-orbit coupling electron gas.\nThe decay time of both in-plane and out-of-plane spin polari zation is naturally proved tobe identical\nto the charge transport time. The out-of-plane spin dynamic s is shown to be in the experimentally\nreachable regime of the femtosecond pump probe spectroscop y and thereby we suggest experiments\nto detect the charge transport property of the TSSs from thei r unique spin dynamics.\nPACS numbers: 73.25.+i, 72.25.Rb, 72.10.-d\nTopologicalinsulators(TIs) are a class of time-reversal\ninvariant matter [1, 2]. Recently, both theory [3–5] and\nexperiments [6–12] focus on the transportproperty ofthe\nTIs which have spin-momentum locking surface states.\nThe helical topological surface states (TSSs) break the\nFermi doubling and have very strong spin-orbit coupling\n(SOC) which gives their unique charge and spin trans-\nport properties [13, 14]. However, in most TIs the bulk\nstates also contribute the conductance, which makes it a\ngreat challenge to extract the transport properties of the\nTSSs, such asmobility, from the experimental data [6, 7].\nOn the other hand, the spin polarization of the TSSs has\nbeen detected by the time-resolved second harmonic gen-\nerationpump-probemeasurementwhichcanseparatethe\nsurface response from the bulk [8, 9]. However the com-\nplete spin dynamical theory of TSS is still absent due to\nthe strong SOC of TSSs which can not be treated pertur-\nbatively as it is usually done in the traditional diffusive\nequation. Here the very strong SOC means Ω soτp≫1\nwhere Ω sois the spin precession frequency due to the\nSOC andτpis the momentum scattering time.\nIn this work, we developed a non-perturbative spin dy-\nnamic theory to fully capture the spin dynamics of the\nTSSs which reveal the qualitative difference to that in\nother SOC 2DEGs such as GaAs quantum well [15–18].\nThe spin polarization along x,y,z direction are decoupled\nto each other even though the TSSs have strong SOC.\nThis is much different tothe spin dynamicsin the 2DEGs\nwherethe different spin componentsarecoupled forfinite\nspin wave length in both weak and strong SOC regime\nand the coupling strength is proportional to the SOC\n[15–18]. The decay time of both in-plane and out-of-\nplane spin polarization is predicted to be exactly equal\ntothe chargetransporttime, τtr, insteadofbeing propor-\ntional to the inverse of Ω2\nsoτp, which is commonly the la-belofDyakonov-Perel(DP) mechanism. Weestimatethe\ntime evolution of the out-of-plane spin dynamics which\nwe show to be in the experimentally reachable regime\nof the femtosecond pump probe spectroscopy [9]. Be-\ncause the surface spin dynamics can be isolated from the\nbulkcontributionbyrecentsecondharmonicpump probe\nmeasurement [8, 9], we propose to read charge transport\nproperty of TSSs from the spin by the pump probe mea-\nsurement.\nWe assume that the low energy TSS has the Dirac-like\nHamiltonian [1, 2]\nˆH=v(z×σ)·k+H′(r), (1)\nwherevis the constant velocity, zis the unit vector per-\npendiculartothesurfaceofTI, kisthewavevectorofthe\nelectron,σis pauli matrix and H′(r) is the perturbative\nHamiltonian which is dependent on the spacial coordi-\nnater. Here we assume /planckover2pi1= 1 and consider only spin\nindependent scattering. The velocity operator has the\nform\nˆV=vz×σ, (2)\nand is proportional to the in-plane spin polarization per-\npendicular to the velocity direction. This special velocity\noperator makes the charge transport of the TSSs qual-\nitatively different to the charge transport in the SOC\n2DEG where the velocity operator is dominated by the\nmomentum. As we know that the conductivity can be\ncalculated from the current-current correlation function\n[19] and the spin dynamics can be obtained by eval-\nuate the spin-spin correlation function[15] even in the\npresence of electron-electron interaction[20]. In the tra-\nditional 2DEG system, the velocity operator is propor-\ntional to the momentum and thereby the conductivity is\nactually calculated from the momentum-momentum cor-2\nrelation function[19]. However for the TSSs, the current-\ncurrent correlation function is proportional to spin-spin\ninstead of momentum-momentum correlation function\nwhich make it possible to get the charge transport infor-\nmation from the spin dynamics. This statement should\nnot be dependent on the special form of the perturba-\ntion Hamiltonian such asthe disorderpotential, electron-\nphonon and electron-electron interaction because the ve-\nlocity operator is defined as ˆV=−i[r,H] butrcom-\nmutes with H′(r). In current work, to simplify our dis-\ncussion, we only focus on the non-magnetic short range\nimpurity in the zero temperature.\nThedynamicsofthespin-chargepolarization,asanon-\nequilibrium process, can be described by quantum Boltz-\nmann equation [21, 22]\n∂tˆg+∇R·{1\n2ˆV,ˆg}+i[ˆH0,ˆg]+ˆg\nτp=ˆρ\nτp,(3)\nwhereˆH0(k) =v(z×σ)·k, ˆg(θ,R,t) is the angular dis-\ntribution function of the 2 ×2 matrix of spin-1\n2Keldysh\nGreen’s function [21–23], θis the angle between the mo-\nmentum and the xaxis, ˆρ(R,t) =/integraltext\nˆgdθ/2πis the densitymatrix and τpis the momentum scattering time at Fermi\nsurface. The difficulty of obtaining the spin dynamics\nequation in the strong SOC regime lies on the fact that\nthespinprecessionangle2 vkfτp, istoolargetobe treated\nperturbatively. To circumvent this problem, we multiply\nσjwherej= 0,x,y,zon both sides of Eq. 3 and cal-\nculate the trace. Using the fact that Tr( σjσk)/2 =δjk,\nthe quantum Boltzmann equation can be written in the\nclassical spin-charge 4D space after integrating out Eas\n[22, 24]\nˆKjkgk=iρj, (4)\nwheregj(ρj) = Tr[ˆ g(ˆρ)σj/2]. To simplify our discussion,\nwe assume the uniform spin polarization along ydirec-\ntion, say∂yˆρ= 0. In this case, by multiplying the inverse\nof the matrix ˆKon both sides of Eq. 4 and integrating\nout the variable θ, we obtain the non-purturbative spin-\ncharge dynamic equation\nρj=ˆDjkρk, (5)\nwhereˆD=/integraltextdθ\n2πˆK−1andˆK−1has a complicated but\nanalytically and numerically tractable form\n\n˜Ω/parenleftig\n˜Ω2+˜Ω2\nso/parenrightig\nicos(θ)sin(θ)˜∆x˜Ω2\nso−i˜∆x/parenleftig\n˜Ω2+cos2(θ)˜Ω2\nso/parenrightig\n−i˜Ωsin(θ)˜∆x˜Ωso\nicos(θ)sin(θ)˜∆x˜Ω2\nso˜Ω/parenleftig\n˜Ω2+˜∆2\nx+sin2(θ)˜Ω2\nso/parenrightig\n−˜Ωcos(θ)sin(θ)˜Ω2\nsocos(θ)/parenleftig\n˜Ω2+˜∆2\nx/parenrightig\n˜Ωso\n−i˜∆x/parenleftig\n˜Ω2+cos2(θ)˜Ω2\nso/parenrightig\n−˜Ωcos(θ)sin(θ)˜Ω2\nso˜Ω3+cos2(θ)˜Ω2\nso˜Ω ˜Ω2sin(θ)˜Ωso\ni˜Ωsin(θ)˜∆x˜Ωso −cos(θ)/parenleftig\n˜Ω2+˜∆2\nx/parenrightig\n˜Ωso −˜Ω2sin(θ)˜Ωso˜Ω/parenleftig\n˜Ω2+˜∆2\nx/parenrightig\n\n/parenleftig\n˜Ω2+˜Ω2so/parenrightig\n˜Ω2+˜∆2x/parenleftig\n˜Ω2+cos2(θ)˜Ω2so/parenrightig (6)\nEq. (5) is a generalized spin diffusion equation and\nvalid from the weak to the strong SOC regime. Because\nof the helical Hamiltonian, the denominator of the ma-\ntrixˆK−1is the function of cos2θwhose periodicity is π.\nBy using the property of trigonometric functions with-\nout calculating the integral of θ, we can easily prove\nthat the spin polarizations along x,yandzdirection are\ncompletely decoupled even though the TSSs have very\nstrong SOC. This is very different to the spin wave of\n2DEG where in the case of finite spin wave vector along\nxdirection, the SOC makes the spin polarization being\nrotated inx−zplane shown in Fig 1a which is called\nspin helix(SH)[16]. On the TI surface, taking the spin\npolarization along zdirection for example, it is decou-\npled to any other spin components and therefore the\nSH is absent Fig 1b. This qualitative difference is be-\ncause 2DEG respect the Fermi doubling but TSSs does\nnot. For the spin polarization of the 2DEG shown in\nFig. 1(a), there are two spin sub-bands on the Fermi\nsurface which allow two spin helical modes(SHMs) withtwo different relaxation times[16, 25, 26]. In Fig. 1(b),\nbecause the non-equilibrium state is around the Fermi\nsurface and Ef≫kT, the spin sub-band below the\nDirac point is still completely filled. Accordingly, at the\nky= 0 channel the terms such as/summationtext\nk>0C†\nk,↓Ck+q,↑and/summationtext\nk<0C†\nk−q,↑Ck,↓, which add electrons to the spin sub-\nband below the Dirac point, will not contribute to spin\npolarization [24]. Here ↑(↓) indicates spin along +( −) y\ndirection. Therefore, the non-equilibrium spin polariza-\ntion of TSSs around Fermi surface is constructed by the\nsum of the term/summationtext\nk>0C†\nk+q,↑Ck,↓, which gives the clock-\nwise SHM, and the term/summationtext\nk<0C†\nk,↓Ck−q,↑which gives\ncounter-clockwise SHM. Therefore the total spin polar-\nization is along zdirection shown in Fig. 1b.\nNow, let us focus on the spin polarization along xdi-\nrection which is completely decoupled from other spin\ncomponents. Its non-purturbative dynamic equation de-\ntermining its complex frequency takes the form\n/parenleftig\n1−ˆDxx/parenrightig\nρx= 0, (7)3\nEf\nx (b)E\nkk-q+CkC k+q+CkC\n+ (a)\n(2)(3)\n(2) (3)k-q+CkC\nk+q+CkCk+q+CkC\nk-q+CkC(1)+(2) (3)+(4)\nkE\nFIG. 1. (a) Spin polarization configure at ky= 0 channel in\n2DEG. The solid(hollow) circle means creating(annihilati ng)\nan electron into the system which is corresponding to C†(C)\noperator. The clockwise SHM (the red arrows) in x−zplane\nis constructed by/summationtext\nkc†\nk+q,↑ck,↓+ h.c.(the red creation and\nannihilation operators) and the counter-clockwise SHM (th e\nblue arrows) is constructed by/summationtext\nkc†\nk−q,↑ck,↓+h.c.(the blue\ncreation and annihilation operators). Here ↑(↓) indicates\nspin along +( −) y direction. (b) Spin polarization configure\natky= 0channel. The collective spin polarization arises from\nthe summation of the half clockwise SHM and half counter-\nclockwise.\nwhere\nˆDxx=/radicalig\n˜Ω2+˜∆2x/radicalig\n˜Ω2+˜∆2x+˜Ω2so\n˜∆2x/radicalig\n˜Ω2+˜Ω2so−˜Ω\n˜∆2x,(8)\nlis the mean free path, ˜Ω = 1−iωτp,˜Ωso= 2vkfτp\nand˜∆x(y)=lqx(y). Here we have Fourier transformed ∂t\nand∂x(y)to−iωandiqx(y)which are the frequency and\nwavelength of the spin polarized wave. The in-plane spin\npolarization of the TSSs has a very unique property: it is\nproportional to the charge current perpendicular to it on\nthe TI surface. Therefore the charge current relaxation\ntime must be equal to the in-plane spin relaxation time.\nTo keep this argument in mind, we study the spin polar-\nizationalong x-direction. Firstweassumethat |˜Ω| ≪˜Ωso\nand˜∆x≪˜Ωso. The eigen-frequency of the spin dynam-\nics takes the form\niωτp=1+˜∆2\nx\n2, (9)\nwhich can be Fourier transform to the real space as\n∂tρx=−Ds∂2\nxρx−ρx\n2τp, (10)\nwhereDs=v2\nfτp/2. The prior theory of spin dynam-\nics of the TSSs (Ref.3) is based on the Kubo formulaand needs to expand the spin-charge kinetic equation in\nterms ofωτpand˜∆xto their leading order. Making the\nsame approximation, our dynamic equation of the spin\npolarization along xdirection becomes [24]\niωτp= 1+˜∆2\nx\n4= 0, (11)\nwhich is equivalent to\n∂tρx=−1\n2Ds∂2\nxρx−ρx\nτp. (12)\nEq. (12) is consistent with the spin diffusion equation in\nRef. 3. However we have found that when ω= 0, Eq.\n(12) gives ˜∆x=q2l2=−4 which is of the same order\nbut four times larger than the exact solution ˜∆2\nx=−1.\nBoth˜∆x=−4 or˜∆x=−1 are inconsistent with the\nassumption that |˜∆x| ≪1. On the other hand, the\nnon-purtabative result of Eq. (10) shows that the spin\nrelaxation time is τs= 2τp, instead of τs=τpindi-\ncated by Eq. (12) based on the purtabative calculation\nwhereτs= 1/|Im(ω)|is the spin life time. To better\nunderstand this result, let us compare it with the tra-\nditional DP mechanism where spin life time is inverse\nproportional to τp. We assume an initial electron state\nhas momentum state ψi,p= exp(ikfx) and spin state\nψi,s= (1,i)T/√\n2 on the Fermi surface. The disorder\npotential scatters the momentum which can be consid-\nered as rotating the effective magnetic field due to SOC\nwith speed on the order of 1 /τp. For the traditional DP\nmechanism, Ω soτp≪1 indicates the spin precesses too\nslow to follow the rotation of the SOC field. In the limit\nofτp= 0 the spin have no time to precess and the spin\nlife time is approaching infinite. This is called motional\nnarrowing and is the unique property of the DP in the\nweak SOC regime. However in the strong SOC regime,\nΩsoτp≫1 the rotation can be considered as an adiabatic\nprocess so that after disorder scattering, the momentum\nof the final state is ψf,p= exp(ikfxcosθ+ikfysinθ)\nand spin will be adiabatically rotated to the eigenstate\nψf,s= (1,exp(iθ+π/2))T/√\n2 whereθ/ne}ationslash= 0. The initial\nand final momentum states are orthogonal to each other\nindicates that the momentum memory is completely lost\nafter the disorder scattering. However, the initial and\nfinal spin states has finite overlap, |/an}bracketle{tψf,s|ψi,s/an}bracketri}ht|2= cos2θ,\nwhose average value is 1 /2 and consistent to our conclu-\nsion that the spin life time is two times longer than the\nmomentum life time. On the other hand, it is well known\nthat charge transport time τtrfor steady current in the\nsystem with Dirac-like Hamiltonian is double of the mo-\nmentum scattering time τpdue to the spin-momentum\nlocking [4]. Therefore, our non-perturbative theory sys-\ntemically provesthat the spin relaxationtime and charge\ntransport time τtrare equal for the TSSs, τs=τtrwhich\nisqualitativelydifferent tothe traditionalDP mechanism\nand consistent with one of the unique properties of the4\nDirac-like Hamiltonian, that the spin operator is propor-\ntional to the velocity operator.\nIn the case of uniform in-plane spin polarization, with-\nout assuming |˜Ω| ≪˜Ωso, there is another solution of\nEq. 7\niωτp=3\n4±i˜Ωso, (13)\nwhich gives a damped oscillatory spin dynamic mode.\nDue to the strong SOC, this spin oscillation is accompa-\nnied by an AC current with the same frequency. This AC\ncurrent may provide another way to detect the transport\nproperty of TSSs.\nNow, we focus on the spin polarization perpendicular\nto the surface. Based on Eq. (5,6), the dynamic equation\nof the spin polarization along z direction takes the form\n1−ˆDzz= 0, (14)\nwhere\nˆDzz=/radicalig\n˜Ω2+˜∆2x/radicalig\n˜Ω2+˜Ω2so/radicalig\n˜Ω2+˜∆2x+˜Ω2so.(15)\nWhenωz= 0 for the steady spin polarization, we have\n˜∆x=i/radicalig\n2+˜Ω2so≈2ikfl, (16)\nwhich means the diffusive length of out of plane spin\nis of the same order as the Fermi wave length. This\nis in contrast to the diffusive length in other materials\nsuch as the GaAs/AlGaAs where it is of the same order\nof (or much smaller) then the mean free path lin the\nstrong (weak) SOC regime [15, 17, 18, 27]. When the\nspin is uniformly polarized perpendicular to the surface,\nthe eigenfrequency has the form\niωτp=1\n2(1−/radicalig\n1−4˜Ω2so)≈1\n2±i˜Ωso,(17)\nwhere the approximationis valid when ˜Ωso≫1. Eq. (17)\nindicates that the spin along the z-direction will oscillate\nin a damped form with the frequency equal to ˜Ωso=\n2vkfτpand the relaxation time given as τs=τtr= 2τp,\nwhich is the same to the decay rate of the in-plane spin\npolarization. We notice that although the nonuniform\nspin dynamics on the TI surface are very different to\nthat in 2DEG with strong SOC, the uniform out-of-\nplane spin dynamics follow the same dynamic equation.\nExperiments[28] have observed τs= 2τpin the 2DEG\nwith strong SOC which is independent on the Fermi en-\nergy, the strength of SOC and temperature. Therefore,\nwe expect the similarresults forthe spin dynamicson the\nTI surface and therefore provide a reliable way to mea-\nsure the charge transport time τtrof TSSs. Because the\nspin along z-direction is not coupled to any other spin\nand charge density, this oscillation will not induce cur-\nrent oscillation or charge density oscillation. The experi-\nments have been able to freely tune the Fermi surface of\n(a)\n(b)0\n0\nFIG. 2. (a) The second harmonic pumping probe measure-\nment. The circularized pump light with the frequency ω0is\nfollowed by the linear polarized probe light with the same\nfrequency with the time delay t. The reflected light with 2 ω0\nfrequency has a θrotated angle of the linear polarization due\nto the Kerr effect. (b) The evolution of the uniform out of\nplane spin polarization. According to our theory, we take th e\nexperimental parameters from the sample Q2 in table 1 of the\nRef.[6] as kf= 0.032˚A−1,Ef= 84meV and kfl= 69.\nTI close to or away from the Dirac point in a wide range\n[29]. A circular polarized pump beam can generate spin\nnon-equilibriumpolarizationofTSSswhichhasbeen pre-\ndicted theoretically [30] and verified experimentally [31].\nThe spin dynamics of the TSSs can be detected by the\ntime-resolved second harmonic optical pump-probe mea-\nsurements. AsonlyTSSs contributethesecondharmonic\ngeneration[8, 9], this type of measurement can isolate the\nsurface response from the bulk. If the Fermi surface is\nturned to 84 meV [6] above the Dirac point, the exper-\niment should observe about 24 THz oscillation through\nthe Kerr rotation angle. The femtosecond pump probe\nspectroscopy can be used to measure this oscillation. At\nthe same time, the decay rate of this spin oscillation is\nτtrwhich gives us the charge transport time of the TSSs\n(Fig.2).\nIn conclusion, we develop a non-purtabative method\nto study the generalized spin dynamic of the TSSs which\nis shown to be qualitatively different to the traditional\nDP mechanism. We found the in-plane spin relaxation\ntime of the pure exponential decay mode is exactly equal\nto the charge transport time. We also predict two fast\noscillatory modes of spin polarization perpendicular and\nparallel to the surface of a topological insulator, which\ncannot be obtained from the prior spin diffusion equa-\ntions, based on perturbative approaches. At last, we\nshown how to read the charge transport properties of\nTSSs from the out of plane spin dynamics.\nWe acknowledge Chia-Ren Hu, Artem Abanov, Cenke5\nXu, Jainendra Jain and Chao-Xing Liu for very help-\nful discussion and support from DMR-1105512, NHARP,\nand ONR-000141110780. X.L. acknowledges partial sup-\nport by the DOE under Grant No. de-sc0005042.\n[1] M. Z. Hasan and C. L. Kane,\nRev. Mod. Phys. 82, 3045 (2010).\n[2] X.-L. Qi and S.-C. Zhang,\nRev. Mod. Phys. 83, 1057 (2011).\n[3] A. A. Burkovand D. G. Hawthorn, Phys. Rev. Lett. 105,\n066802 (2010).\n[4] D. Culcer, E. H. Hwang, T. D. Stanescu, and\nS. Das Sarma, Phys. Rev. B 82, 155457 (2010).\n[5] J.-H. Gao, J. Yuan, W.-Q. Chen, Y. Zhou, and F.-C.\nZhang, Phys. Rev. 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Gan,\nJOURNAL OF APPLIED PHYSICS 110(2011), 10.1063/1.3633228."    },    {        "title": "1504.03792v3.Acoustic_study_for_dynamical_molecular_spin_state_without_undergoing_magnetic_phase_transition_in_spin_frustrated_ZnFe__2_O__4_.pdf",        "content": "arXiv:1504.03792v3  [cond-mat.str-el]  16 Oct 2015Acoustic study of dynamical molecular-spin state without u ndergoing magnetic phase\ntransition in spin-frustrated ZnFe 2O4\nTadataka Watanabe1,∗Shota Takita1, Keisuke Tomiyasu2, and Kazuya Kamazawa3\n1Department of Physics, College of Science and Technology (C ST),\nNihon University, Chiyoda, Tokyo 101-8308, Japan\n2Department of Physics, Tohoku University, Sendai, Miyagi 9 80-8577, Japan and\n3Comprehensive Research Organization for Science and Socie ty (CROSS), Tokai, Ibaraki 319-1106, Japan\n(Dated: January 23, 2018)\nUltrasound velocity measurements were performed on a singl e crystal of spin-frustrated ferrite\nspinel ZnFe 2O4from 300 K down to 2 K. In this cubic crystal, all the symmetric ally-independent\nelastic moduli exhibit softening with a characteristic min imum with decreasing temperature be-\nlow∼100 K. This elastic anomaly suggests a coupling between dyna mical lattice deformations and\nmolecular-spin excitations. In contrast, the elastic anom alies, normally driven by the magnetostruc-\ntural phase transition and its precursor, are absent in ZnFe 2O4, suggesting that the spin-lattice\ncoupling cannot play a role in relieving frustration within this compound. The present study infers\nthat, for ZnFe 2O4, the dynamical molecular-spin state evolves at low tempera tures without un-\ndergoing precursor spin-lattice fluctuations and spin-lat tice ordering. It is expected that ZnFe 2O4\nprovides the unique dynamical spin-lattice liquid-like sy stem, where not only the spin molecules but\nalso the cubic lattice fluctuate spatially and temporally.\nPACS numbers: 72.55.+s, 75.20.-g, 75.40.Gb, 75.50.Xx\nI. INTRODUCTION\nCubic spinels AB2O4with magnetic Bions have at-\ntracted considerable interest in light of the geometri-\ncal frustration which is inherent in the B-site sublat-\ntice of corner-sharing tetrahedra (pyrochlore lattice).1\nOne of the most extensively studied spinel systems is\nchromite spinels ACr2O4withA= Mg and Zn, for\nwhich the magnetic properties are fully dominated by\nthe Jahn-Teller (JT)-inactive Cr3+with spin S= 3/2\n(Fig. 1(a)) residing on the pyrochlore network.2ACr2O4\nwith Weiss temperature Θ W≃ −390 K undergoes an\nantiferromagnetic (AF) ordering at TN≃13 K along\nwith a cubic-to-tetragonal structural distortion.3–5Fer-\nrite spinels AFe2O4withA= Zn and Cd are another\nJT-inactive spinel system with Fe3+showing a high spin\nofS= 5/2 (Fig. 1(b)).6ForAFe2O4with Θ W≃120 K\n(A= Zn) and ≃ −50 K (A= Cd), neutron scattering\nexperiments in the high-purity single crystals observed\nneither long-range magnetic ordering nor a structural\ntransition down to low temperature (1.5 K) although an\nAF-transition-like anomaly occurs in the magnetic sus-\nceptibility at T∗≃13 K (Fig. 1(c)).7,8Additionally, it\nis noted that the magnetic susceptibility of ZnFe 2O4ex-\nhibits a deviation from the Curie-Weiss law below ∼100\nK (Fig. 1(d)),7which implies the enhancement of the AF\ninteractions at low temperatures.9Thus, the frustrated\nmagnetism of AFe2O4should be different in nature from\nthat ofACr2O4.\nForACr2O4, the phase transition to spin-lattice or-\ndering is explained by the spin-JT mechanism via spin-\nlattice coupling, where local distortions of the tetra-\nhedra release the frustration in the nearest-neighbor\nAF interactions.10–12In the frustrated paramagnetic\n(PM) phase of ACr2O4, inelastic neutron scatter-ing (INS) experiments provided evidence of quasielas-\ntic magnetic scattering, indicating the presence of\nstrongspinfluctuationsbecauseofspinfrustration.3,13–15\nThis quasielastic mode involved the fluctuations of AF\nhexagonal spin molecules (AF hexamers) in the py-\nrochlore lattice (Fig. 1(e)).13,15Further, ultrasound ve-\nlocity measurements of ACr2O4suggested the coexis-\ntence/crossover of the precursor spin-lattice fluctuations\ntowards a phase transition (spin-JT fluctuations) and\nthe gapped molecular-spin excitations also coupled with\nthe lattice,16,17which is compatible with the recent ob-\nservation of finite-energy molecular-spin excitations in\ntime-of-flight INS experiments in the PM phase of this\ncompound.18\nForAFe2O4, whereas spin-lattice ordering is absent\ndown to low temperature, the INS experiments observed\nmagnetic diffuse scattering and its very soft dispersion\nrelation in the energy range below ∼2 meV, arising pos-\nsibly from the dynamical molecular-spin state.7,8Thus,\nin the absence of the spin-JT effect, the frustrated mag-\nnetism in AFe2O4is expected to be mainly governed by\nthe dynamical molecular-spin state. For ZnFe 2O4, the\nobserved diffuse scattering was attributed to the fluctua-\ntions of AF twelve-membered spin molecules (AF dode-\ncamers illustrated in Fig. 1(f)).19The formation of the\ndifferenttypesofspinmoleculesinbetweenZnFe 2O4(the\nAF dodecamers) and ACr2O4(the AF hexamers) is con-\nsidered to arise from the difference in the dominant ex-\nchange paths, specifically, the third-neighbor AF interac-\ntionsJ3with additional nearest-neighbor ferromagnetic\n(FM)J1forZnFe 2O4,7,19butAFJ1forACr2O4,13,15, re-\nspectively. For CdFe 2O4, the INS experiments produced\nscattering patterns, which resembles that of ACr2O4, in-\ndicative of the dominant AF J1.8\nInterestingly,thediffuse-neutron-scatteringpatternsof2\nH = 1000 Oe T* 0.5\n0.4\n0.3\n0.2M/H  (emu/mol Oe)\n T (K)50403020100(c) \nZnFe 2O4\nH || [001] \n(e) Cr 3+ AF hexamer Cr 3+\n(f) Fe 3+ AF dodecamer Fe 3+(a) Cr 3+ (3d 3 )eg\nt2g \n(b) Fe 3+ (3d 5 )eg\nt2g \n3002001000 50 150 250 \n T (K)40\n30\n20\n10\n0H/M  (mol Oe/emu)\nH = 1000 Oe ZnFe 2O4\nH || [001] (d) \nFIG. 1: (Color online) (a) and (b) show the spin states of\nCr3+(3d3) and high-spin Fe3+(3d5) in the octahedral crystal\nfield, respectively. (c) and (d) depict respectively the mag -\nnetic susceptibility and the inversed magnetic susceptibi lity\nof single-crystalline ZnFe 2O4as functions of temperature. (e)\nand (f) illustrate respectively a Cr3+AF hexamer and a Fe3+\nAF dodecamer in the pyrochlore lattice. The dotted cube in\n(e) and (f) depicts a unit cell of the cubic spinel structure.\nZnFe2O4vary with temperature,7while those of ACr2O4\nand CdFe 2O4are independent of temperature.8,13,15\nThis temperature dependence in ZnFe 2O4was explained\nby the competition between the third-neighbor AF J3\nand the temperature-dependent nearest-neighbor FM\nJ1, where the nearest-neighbor FM J1are weakened\nwith decreasing temperature due to the temperature-\ndependent bond angle of the nearest-neighbor Fe3+-O-\nFe3+.7,9The neutron scattering experiments in ZnFe 2O4\nsuggested that the deviation from the Curie-Weiss law\nbelow∼100 K in the magnetic susceptibility of this\ncompound (Fig. 1(d)) arises from the AF component,\nwhereas the FM component leads to the Curie-Weiss be-\nhavior with Θ W≃120 K at high temperatures.7And\nit is considered that, for ZnFe 2O4, the formation of the\nspin molecules (the AF dodecamers) is realized at low\ntemperatures, where the AF component generates the\nfrustration.19\nNotably, molecular-spin excitations were observed in\ntheINSexperimentsinthefrustratedmagnetsofnotonly\nACr2O4andAFe2O4but also HgCr 2O4,20GeCo2O4,21\nLiV2O4,22and Tb 2Ti2O7,23where the number of mag-\nnetic ions, shape, and symmetry of spin molecules are\nconsideredto varyfrom compound to compound depend-\ning on the dominant exchange path. These observationsimply that the dynamical molecular-spin state can uni-\nversally emerge in the frustrated magnets. Thus a com-\nparative study among the spinel magnets using a variety\nof experimental probes must provide a root for under-\nstanding the nature of dynamical molecular-spin state.\nIn this paper, we present an analysis of ultra-\nsound velocity measurements for the zinc ferrite spinel\nZnFe2O4. The sound velocity or the elastic modulus\nis a useful probe enabling symmetry-resolved thermo-\ndynamic information to be extracted from frustrated\nmagnets.16,17,24–32As mentioned earlier, the observed\nelasticanomaliesin the chromitespinels ACr2O4inferred\nthe coexistence of spin-JT fluctuations and molecular-\nspin excitations in the PM phase.16,17The present study\nfinds elastic anomalies in ZnFe 2O4that suggest the\nevolution of a dynamical molecular-spin state at low\ntemperatures without undergoing precursor spin-lattice\nfluctuations and spin-lattice ordering, which is a be-\nhavior uniquely different from another spin-frustrated\nmolecular-spinsystem ACr2O4. Moreover,ourstudyalso\nsuggests that, for ZnFe 2O4, the molecular-spin excita-\ntions consist of multiple gapped modes and sensitively\ncouple to the trigonal lattice deformations, which is a\nbehavior similar to ACr2O4although the details are dif-\nferent in between ZnFe 2O4andACr2O4.\nII. EXPERIMENTAL\nThe ultrasoundvelocitymeasurementswereperformed\non a high-purity single crystal of ZnFe 2O4grown by the\nflux method.7Figure 1(c) and (d) plots respectively the\ntemperature dependence of the magnetic susceptibility\nand the inversed magnetic susceptibility of the ZnFe 2O4\nsingle crystal used in the present study, where the AF-\ntransition-like anomaly at T∗∼13 K and the deviation\nfrom the Curie-Weiss law below ∼100 K occur.7The\nultrasound velocities were measured using the phase-\ncomparison technique with longitudinal and transverse\nsound waves at a frequency of 30 MHz. The ultrasonic\nwaves were generated and detected by LiNbO 3transduc-\ners glued to the parallel mirror surfaces of the crystal.\nThe measurements were performed at temperatures from\n300 K to 2 K for all the symmetrically-independent elas-\ntic moduli in the cubic crystal, specifically, compression\nmodulus C11, tetragonal shear modulusC11−C12\n2≡Ct,\nand trigonal shear modulus C44. The respective mea-\nsurements of C11,Ct, andC44were performed using lon-\ngitudinal sound waves with propagation k/bardbl[001] and po-\nlarization u/bardbl[001], transverse sound waves with k/bardbl[110]\nandu/bardbl[1¯10], and transverse sound waves with k/bardbl[110]\nandu/bardbl[001]. The sound velocities of ZnFe 2O4measured\nat room temperature (300 K) are 6480 m/s for C11, 2930\nm/s forCt, and 3740 m/s for C44.3\nIII. RESULTS AND DISCUSSION\nFigure 2(a)-(c) presents the temperature dependence\nof the elastic moduli, C11(T),Ct(T), andC44(T) in\nZnFe2O4. On cooling from room temperature (300 K)\nto∼100 K, all the elastic moduli exhibit ordinary hard-\nening consistent with the background C0\nΓ(T) taken from\nan empirical evaluation of the experimental CΓ(T) in 100\nK< T <300 K (dotted curves in Fig. 2(a)-(c))33. Here,\nthe background values at T= 0 K,C0\n11(0)≃233.8 GPa,\nC0\nt(0)≃46.7 GPa, and C0\n44(0)≃75.9 GPa give the re-\nspective sound velocities of v0\n11≃6620 m/s, v0\nt≃2960\nm/s, and v0\n44≃3770m/s, which are ∼2%,∼1%, and ∼\n0.8% largerthan the measuredsoundvelocitiesat300K,\nrespectively. The values of v0\n11,v0\nt, andv0\n44give the aver-\naged sound velocity ¯ v∼3700 m/s,34which is compatible\nwith the Debye temperature Θ D∼250 K for ZnGa 2O4,\na nonmagnetic reference compound for ZnFe 2O4, giving\nthe averaged sound velocity ¯ v∼3500 m/s.35\nIn Fig. 2(a)-(c), below ∼100 K, all the elastic mod-\nuli of ZnFe 2O4exhibit an anomalous temperature vari-\nation, specifically, the deviation from the ordinal hard-\nening indicated as the dotted curves in Fig. 2(a)-(c).\nC11(T) exhibits softening with decreasing temperature\nbelow∼50 K but turns to hardening below ∼6 K. Sim-\nilarly,Ct(T) andC44(T) exhibit softening with decreas-\ning temperature below ∼80 K, which turn to harden-\ning below ∼6 K and ∼4 K, respectively. Taking into\naccount the absence of orbital degeneracy in the B-site\nhigh-spin Fe3+in ZnFe 2O4(Fig. 1(b)), the anomaly in\nCΓ(T) should have a magnetic origin where the spin de-\ngrees of freedom play a significant role.6Note that the\nelasticanomaliesemergeat temperatureswherethe mag-\nnetic susceptibility exhibits the deviationfrom the Curie-\nWeiss law (Fig. 1(d)). This correspondence implies that\nthe elastic anomalies are driven by the generation of\nfrustration below ∼100 K, which is compatible with the\ntemperature-dependent J1/J3suggested from the neu-\ntron scattering experiments.7\nThe elastic anomalies in the JT-inactive magnets like\nZnFe2O4are attributed to magnetoelastic coupling act-\ning on the exchange interactions. In this mechanism, the\nexchange striction arises from a modulation of the ex-\nchange interactions by ultrasonic waves.34Both the lon-\ngitudinal and transverse sound waves couple to the spin\nsystem via the exchange striction mechanism, which de-\npends on the directions of both polarization uand prop-\nagationkof sound waves relative to the exchange path.\nSimilar to ZnFe 2O4, the softening-with-minimum elas-\ntic anomaly in CΓ(T) is also observed in other frustrated\nmagnets of ACr2O4,16,17SrCu2(BO3)2,25,26GeCo2O4,29\nand MgV 2O4,32the origin of which is considered to be\nthe coupling of the lattice to the gapped magnetic ex-\ncitations via the exchange striction mechanism. Recall-\ning that molecular-spin excitations, i.e., the excitations\nof AF dodecamers, were observed in the INS experi-\nments for ZnFe 2O4,7,19the softening-with-minimum ex-\nhibited in CΓ(T) for ZnFe 2O4arises from the coupling(e) Ct\n40 %\nMgCr2O4ZnFe 2O4\nTN\nΔCt / Ct\n(f) C44 \n4 %MgCr2O4\nTNZnFe 2O4\nΔC44 / C44 (d) C11 \nZnFe 2O4\nMgCr2O410 %TNΔC11 / C11 T*\n3002001000\nT (K)ZnFe 2O4(a) C11 \n232\n228\n224~1.8 %C11  (GPa)\n76 \n75 \n74 \nT (K)C44  (GPa)(c) C44 ZnFe 2O4 ~1.3 %\n30025020015010050046.6\n46.2\n45.8Ct (GPa)ZnFe 2O4(b) Ct\n~0.9 %k||u||[001] \nAF Dodecamer \nk||[110] u||[110] -\nk||[110] u||[001] T*\nT*\nFIG. 2: (Color online) (a)-(c) Elastic moduli of ZnFe 2O4as\nfunctions of temperature. (a) C11(T), (b)Ct(T), and (c)\nC44(T). The dotted curves in (a)-(c) indicate the background\nC0\nΓ(T) in each modulus taken from an empirical evaluation\nof the experimental CΓ(T) in 100 K < T < 300 K.33The\ninsets to (a)-(c) illustrate single AF dodecamers in the Fe3+\npyrochlore lattice with the propagation kand polarization u\nof sound waves in the respective elastic modes. (d)-(f) The\ndependence on temperature of the elastic moduli of ZnFe 2O4\n[from Fig. 2(a)-(c)] and MgCr 2O4.17(d)C11(T), (e)Ct(T),\nand (f)C44(T). The curves are vertically shifted for clarity.\nT∗andTNin (d)-(f) indicate the temperature at which the\nAF-transition-like anomaly occurs in the magnetic suscept i-\nbility for ZnFe 2O4seen in Fig. 1(c), and the AF ordering\ntemperature for MgCr 2O4, respectively.\nof the lattice to the molecular-spin excitations, which\nis similar to the softening-with-minimum observed in\nCΓ(T) ofACr2O4.16,17The insets to Fig. 2(a)-(c) il-\nlustrate single AF dodecamers in the Fe3+pyrochlore\nlattice with the propagation kand polarization uof\nsound waves in the respective elastic modes. From the\nsymmetry point of view, the AF-dodecamer excitations\nshould couple more sensitively to the trigonal lattice de-\nformations generated by sound waves with k/bardbl[110] and\nu/bardbl[001] (the inset to Fig. 2(c)). For ZnFe 2O4, as shown\nin Fig. 2(a)-(c), the magnitude of the softening in CΓ(T)4\nis indeed largest in the trigonal shear modulus C44(T),\nspecifically, ∆ C11/C11∼0.4 %, ∆Ct/Ct∼1.8 %, and\n∆C44/C44∼2.8%, whichiscompatiblewiththe trigonal\nsymmetry of the AF dodecamer.\nFigure 2(d)-(f) compares the relative shifts of C11(T),\nCt(T), andC44(T), respectively, in between ZnFe 2O4\n[from Fig. 2(a)-(c)] and MgCr 2O4.17Note here that\nCΓ(T) for MgCr 2O4in the PM phase ( T > T N) ex-\nhibits not only softening-with-minimum in C44(T) from\nmolecular-spin excitations but also a huge Curie-type\n−1/Tsoftening in C11(T) andCt(T), being a precursor\nto the spin-JT transition.17This coexistence of twotypes\nof elastic anomalies in MgCr 2O4infers the coexistence of\nmolecular-spin excitations and spin-JT fluctuations. For\nZnFe2O4, in contrast, CΓ(T) exhibits solely softening-\nwith-minimum, as is clearly seen in the expanded view\nofCΓ(T) (Fig. 3(a)-(c) [open circles, from Fig. 2(a)-(c)]),\nwhich infers the presence of molecular-spin excitations\nbut the absence of spin-JT fluctuations.\nWe also note here that, whereas CΓ(T) for MgCr 2O4\nexhibits a discontinuity at TN(Fig. 2(d)-(f)), CΓ(T) for\nZnFe2O4exhibits no discontinuity at T∗(Figs. 2(d)-\n(f) and 3(a)-(c)). Thus, the experimental CΓ(T) for\nZnFe2O4indicates that the AF-transition-like anomaly\natT∗in the magnetic susceptibility seen in Fig. 1(c)\nis not a phase transition. This inference is compati-\nble with the absence of long-range magnetic ordering at\nleast down to 1.5 K as revealed by the neutron scattering\nexperiments.7Regarding CΓ(T) for ZnFe 2O4, the conti-\nnuity in elasticity at T∗(the absence of a phase transi-\ntion) is compatiblewith the absenceofCurie-typesoften-\ning (the absence of a precursorfor the magnetostructural\ntransition), indicating that the spin-lattice coupling can-\nnot produce the spin-JT transition because the strength\nof the exchange interactions is not large enough to over-\ncome the cost in elastic energy involved in the static\nlong-range lattice deformation. As a result, it is consid-\nered that, for ZnFe 2O4, solely the dynamical molecular-\nspin state emerges without undergoing spin-lattice fluc-\ntuations and spin-lattice ordering, which is different from\nthecoexistenceofthedynamicalmolecular-spinstateand\nthe spin JT effect in ACr2O4. However, the precise na-\nture of the magnetic state of ZnFe 2O4belowT∗remains\ntobediscovered. Freezingofthespinmoleculesisapossi-\nbility that might occur at T∗in ZnFe 2O4. Furthermore,\nalthough the spin-lattice coupling in ZnFe 2O4is much\nweaker than that in ACr2O4as indicated in Fig. 2(d)-\n(f), there remains a possibility of spin-JT transition at\nlowT <1.5 K for ZnFe 2O4.\nAs is clear from a comparison between CΓ(T) for\nZnFe2O4(Fig. 2(a)-(c)) and C44(T) for MgCr 2O4\n(Fig. 2(f)), the softening in CΓ(T) begins to occur be-\nlow∼50 K or ∼80 K in ZnFe 2O4but above 300 K in\nMgCr2O4. The softening occurring at lower tempera-\ntures for ZnFe 2O4indicates the evolution of the dynam-\nical molecular-spin state at lower temperatures, which is\ndriven by the generation of frustration below ∼100 K be-\ncauseofthetemperature-dependent J1/J3. Furthermore,the magnitude of the softening in CΓ(T) for ZnFe 2O4is\nsmaller than that in C44(T) for MgCr 2O4. As discussed\nlater in conjunction with Eq. (1) and Table I, the reason\nfor the smaller magnitude of the softening for ZnFe 2O4\nrelativetoMgCr 2O4isbecausethe couplingisweakerbe-\ntween the dynamical lattice deformations and molecular-\nspin excitations. Additionally, as is also clear from a\ncomparison between CΓ(T) of ZnFe 2O4(Fig. 3(a)-(c))\nandC44(T) of MgCr 2O4(Fig. 2(f)), the former exhibits\nits minimum at ∼5 K, which is lower than the minimum\npoint of ∼50 K in the latter. As also discussed later in\nconjunction with Eq. (1) and Table I, the lower temper-\nature at which the minimum point occurs for ZnFe 2O4\nrelative to MgCr 2O4is due to the smaller gap associated\nwith its molecular-spin excitations.\nSoftening-with-minimum in CΓ(T) driven by the\nmolecular-spin excitations is generally explained as the\npresence of a finite gap for the excitations, which is sen-\nsitive to strain.17In the mean-field approximation, the\nelastic modulus CΓ(T) of the molecular-spin system is\nwritten as17:\nCΓ(T) =C0\nΓ(T)−G2\n1,ΓNχΓ(T)\n{1−KΓχΓ(T)},(1)\nwhereC0\nΓ(T) is the background elastic constant, Nthe\ndensity of spin molecules, G1,Γ=|∂∆1/∂ǫΓ|the coupling\nconstant for a single spin molecule measuring the strain\n(ǫΓ) dependence of the excitation gap ∆ 1,KΓthe inter-\nspin-molecule interaction, and χΓ(T) the strain suscepti-\nbility of a single spin molecule. From Eq. (1), the mini-\nmum in CΓ(T) appears when this elastic mode strongly\ncouples to the excited state at ∆ 1; on cooling, CΓ(T) ex-\nhibits softening roughly down to T∼∆1, but recovery\nof the elasticity (hardening) roughly below T∼∆1.\nAs explained above, the softening-with-minimum\nanomaly in CΓ(T) for ZnFe 2O4should arise from a gap\nin the molecular-spin excitations that is sensitive to the\nstrain. This interpretation helps to understand the INS\nresults for ZnFe 2O4.7The broad magnetic scattering\nspectrum should observe gapped molecular-spin excita-\ntions that are considerably smeared. The smeared INS\nspectra at 1.5 K, only one-third of the minimum po-\nsition of ∼5 K inCΓ(T), are probably due to strong\nspin frustration as well as some kind of quantum ef-\nfect in the molecular-spin system. Recall that, for\nother frustrated magnets of ACr2O4and SrCu 2(BO3)2,\nobservations have been reported of the softening-with-\nminimum in CΓ(T) and the T-dependent observation\nof the gapped magnetic excitations in the INS spectra.\nForACr2O4, whereas CΓ(T) exhibited the minimum at\n∼50 K [Fig. 2(f)],17the INS experiments observed broad\nquasielastic magnetic scattering spectrum at tempera-\ntures down to TN≃13 K but observed distinct ∼4-meV\ngapped excitations below TN.3,13,15For the dimer-spin\nsystem SrCu 2(BO3)2, softening-with-minimum in CΓ(T)\nwas observed,25,26whereas the INS experiments demon-\nstrated∼3-meVgapped excitations at temperatures only\nbelow∼10 K.36For ZnFe 2O4, it is expected that the5\nZnFe 2O4(b) Ct\nT*46.6\n46.4\n46.2\n46.046.8 Ct (GPa) AF Dodecamer \nN = 1.73x10 27  m -3 \nEq. (1) ( Δ1 and Δ2)Eq. (1) ( Δ1)AF Dodecamer \nN = 1.73x10 27  m -3 \nEq. (1) ( Δ1 and Δ2)Eq. (1) ( Δ1)ZnFe 2O4(a) C11 \nT*233.6\n233.4\n233.2\n233.0233.8C11  (GPa)\nZnFe 2O4(c) C44 \nT*\n50403020100\n T (K)75.5\n75.0\n74.5\n74.076.0 C44  (GPa)AF Dodecamer \nN = 1.73x10 27  m -3 \nEq. (1) ( Δ1 and Δ2)Eq. (1) ( Δ1)\nFIG. 3: (Color online) Expanded view of CΓ(T) below 50 K\nin ZnFe 2O4[open circles, from Fig. 2(a)-(c)]. (a) C11(T), (b)\nCt(T), and (c) C44(T).T∗in (a)-(c) indicates the temper-\nature at which the AF-transition-like anomaly occurs in the\nmagnetic susceptibility for ZnFe 2O4seen in Fig. 1(c). The\nsolid curves (dotted curves) in (a)-(c) are fits of the exper-\nimental CΓ(T) to Eq. (1) with two singlet-triplet gaps ∆ 1\nand ∆ 2(single singlet-triplet gap ∆ 1) for the AF-dodecamer\nexcitations.19\nINS experiments at low T <1.5 K show clear gapped\nmolecular-spin excitations.\nWe now give a quantitative analysis of the experi-\nmentalCΓ(T) in ZnFe 2O4using Eq. (1) assuming ex-\ncitations of the AF dodecamers in the Fe3+pyrochlore\nlattice [Fig. 1(f)].19Here, the value of the density of\nAF dodecamers, Nin Eq. (1), is assumed to be N=\n1.73×1027m−3, which is one-twelfth of the density of\nFe3+ions in ZnFe 2O4.19We fit Eq. (1) to the experi-\nmentaldata CΓ(T)below50K.Althoughthebackground\nC0\nΓ(T) in Eq. (1) generally exhibits hardening with de-\ncreasing temperature,33we here assume that C0\nΓ(T) is\nconstant because, for ZnFe 2O4, the hardening of the\nbackground below ∼50 K is negligibly small compared\nwith the softening-with-minimum in CΓ(T), as indicated\nin Fig. 2(a)-(c).\nFor ZnFe 2O4, taking into account the vanishing total\nspinStot= 0inthegroundstateoftheAFdodecamers,19\n∆1in Eq. (1) is assumed to be singlet-multiplet excita-TABLE I: Values of the fitting parameters in Eq. (1) with two\nsinglet-triplet gaps ∆ 1and ∆ 2for the experimental CΓ(T) of\nZnFe2O4[from Fig. 3(a)-(c)] (upper column) and ACr2O4(A\n= Mg and Zn) (lower column).17For ZnFe 2O4(upper col-\numn), values of the fitting parameters in Eq. (1) with singlet -\ntriplet ∆ 1and singlet-nonet ∆ 2are also shown in parentheses.\n∆1(K)G1,Γ(K) ∆ 2(K)G2,Γ(K)KΓ(K)\nZnFe2O4C11 890 2630 -6\n[Dodec.] (1190) (3180) (-18)\nCt5 980 15 2910 -10\n(5) (1230) (15) (3470) (-23)\nC44 1350 3570 -6\n(2038) (4370) (-27)\nMgCr 2O4C4439 3600 136 10200 -19\nZnCr2O4C4434 3160 111 9290 -19\n[Hex.]\ntions of the single AF dodecamers. We note that, to\nreproduce the softening in the experimental CΓ(T) for\nZnFe2O4using Eq. (1), we must assume the coupling\nof the lattice to not only the lowest excitations ∆ 1but\nalso the higher excitations ∆ i(i= 2, 3, 4, ...). Dot-\nted curves in Fig. 3(a)-(c) are examples of the fits us-\ning Eq. (1) with the single singlet-triplet gap ∆ 1= 5\nK, where we assume the inner-AF-dodecamer excitations\nfrom the ground state with Stot= 0 to the excited state\nwithStot= 1; if we include only ∆ 1in Eq. (1), the soft-\nening in the experimental CΓ(T) for ZnFe 2O4cannot be\nreproduced. The gradient of the softening in CΓ(T) pro-\nduced by Eq. (1) becomes steeper at low temperatures\nand more gentle at high temperatures than the experi-\nmentaldata. Hence the experimental CΓ(T) forZnFe 2O4\nsuggests that the molecular-spin excitations consist of\nmultiple gapped modes.\nThe level scheme of the AF-dodecamer excitations are\nnotclarifiedsofar,whichshouldincludeinner-andouter-\nAF-dodecamer excitations. However, by assuming the\ncoupling of the lattice to not only the lowest excita-\ntions ∆ 1but also the higher excitations ∆ i(i= 2, 3,\n4,...), Eq. (1) reproduces well the experimental CΓ(T)\nfor ZnFe 2O4. Solid curves in Fig. 3(a)-(c) are examples\nof the fits using Eq. (1) with the lowest singlet-triplet\nexcitations ∆ 1, and the next higher singlet-triplet exci-\ntations ∆ 2. This assumption is similar to that applied to\nthe AF-hexamer excitations in ACr2O4.17Given the val-\nuesofthefitting parameterslisted inthe uppercolumnof\nTable I, the fits to Eq. (1) arein excellent agreementwith\nthe experimental data of ZnFe 2O4(Fig. 3(a)-(c)), repro-\nducing the softening-with-minimum in CΓ(T). We note\nhere that the fitted curves obtained by assuming ∆ 2to\nbe the singlet-quintet, -septet, and -nonet gaps, respec-\ntively, also give excellent agreement with the experimen-\ntal data of ZnFe 2O4(the fitted curves are not shown in\nFig. 3(a)-(c)). Thus, although the present study reveals\nthe coupling of the lattice to the multiple AF-dodecamer\nexcitations for ZnFe 2O4, the nature of the excitations ∆ 2\ncannot be identified so far.6\nIn the fitting of Eq. (1) to the experimental data of\nZnFe2O4, we find that the magnitudes of G1,Γ,G2,Γ, and\n|KΓ|increase with increasing the degree of ∆ 2degener-\nacy. This is exemplified by the values of the fitting pa-\nrameters with the singlet-nonet ∆ 2shown in parentheses\nin the upper column of Table I. On the other hand, we\nalso find three qualitative features of G1,Γ,G2,Γ, andKΓ,\nwhich are common regardless of the degree of ∆ 2degen-\neracy and thus should be intrinsic features for ZnFe 2O4.\nFirst, the KΓvalues are negative for all the elastic\nmodes, indicating that the inter-AF-dodecamer interac-\ntion is antiferrodistortive. Second, the coupling constant\nG2,Γ=|∂∆2/∂ǫΓ|is larger than G1,Γ=|∂∆1/∂ǫΓ|, indi-\ncating that the higher excitations ∆ 2couple to the dy-\nnamical lattice deformations more strongly than the low-\nest excitations ∆ 1. The larger value of G2,ΓthanG1,Γ\nmight suggest the coupling of the lattice to the higher\nmultiple excitations ∆ i(i= 2, 3, 4, ...). Third, both\nthe coupling constants G1,ΓandG2,Γexhibit the largest\nvalues for the trigonal shear modulus C44(T) of the three\nelastic moduli, and hence are compatible with the trigo-\nnal symmetry of the AF dodecamer.\nIn the lower column of Table I, the values of the\nfit parameters in Eq. (1) for the experimental C44(T)\nofACr2O4(A= Mg and Zn) are also listed for\ncomparison.17ForACr2O4, the value of Nin Eq. (1)\nis assumed to be N= 3.45×1027m−3(one-sixth of the\ndensityofCr3+ionsinACr2O4), wherethespin molecule\nis assumedto be the Cr3+AF hexamer(Fig. 1(e)).13,15,17\nAs described before in conjunction with Fig. 2(a)-(c) and\nFig. 2(f), the magnitude of the softening in CΓ(T) for\nZnFe2O4is smaller than that in C44(T) for MgCr 2O4. In\naccordance with Eq. (1), this difference in the magnitude\nbetween ZnFe 2O4andACr2O4arises from the difference\nin the coupling strength between the dynamical lattice\ndeformations and molecular-spin excitations. From Ta-\nble I, the coupling constants G1,ΓandG2,ΓforC44(T) of\nZnFe2O4aresmallerthanthoseof ACr2O4. Additionally,\nalong with Figs. 3(a)-(c) and 2(f), CΓ(T) for ZnFe 2O4\nexhibits a minimum at ∼5 K, which is lower than the\nminimum point of ∼50 K inC44(T) for MgCr 2O4. From\nEq. (1), this difference in the minimum point between\nZnFe2O4andACr2O4arises from the difference in the\nmagnitudes of the gaps ∆ 1and ∆ 2in the molecular-spin\nexcitations.\nFor ZnFe 2O4, the magnitudes of the inter-spin-\nmolecule interactions |KΓ|listed in Table I are com-\nparable to those of ∆ 1and ∆ 2, which is in contrast\nto the weaker magnitudes of |KΓ|than ∆ 1and ∆ 2for\nACr2O4. The comparable magnitudes of |KΓ|, ∆1, and\n∆2for ZnFe 2O4imply that the inter-spin-molecule in-teractions are not completely negligible, which is com-\npatible with the presence of a very weak dispersive fea-\nture of the molecular-spin excitations in the INS spectra\nof ZnFe 2O4, suggesting the presence of the inter-spin-\nmolecule correlations.7Taking into account the smaller\nvalues of ∆ 1and ∆ 2for ZnFe 2O4than for ACr2O4,\nthe very weak dispersion observed in the INS spectra of\nZnFe2O4might be a result of frustration occurring in the\neffectively weaker exchange interactions.\nWe finally note that the recent time-of-flight INS ex-\nperiments in the PM phase of MgCr 2O4revealed the\npresence of multiple modes associated with finite-energy\nmolecular-spin excitations, where it is suggested that not\nonly the ground state but also the excited states are\nhighly frustrated in this compound.18Hence, although\nthe assumption oftwo gaps∆ 1and ∆ 2givesfrom Eq. (1)\na successful agreement with experimental data for CΓ(T)\nof both ZnFe 2O4andACr2O4, it is expected that the\nlevel schemes of the spin molecules in these compounds\nconsist of not only the excited levels of ∆ 1and ∆ 2but\nalsohigherexcitedlevels. SimilartoMgCr 2O4,thefuture\ntime-of-flight INS experiments in ZnFe 2O4are expected\nto reveal complex and exotic molecular-spin excitations.\nIV. SUMMARY\nIn summary, ultrasound velocity measurements of\nZnFe2O4revealed the elastic anomalies, which strongly\nsuggest the evolution of the molecular-spin excitations at\nlow temperatures. Additionally, the present study also\nrevealed that the elastic anomalies driven by the magne-\ntostructural phase transition and its precursorare absent\nin ZnFe 2O4, suggesting that the spin-JT mechanism can-\nnot play a role in releasing frustration within this com-\npound. The present study infers that, for ZnFe 2O4, the\ndynamical molecular-spin state evolves at low temper-\natures without undergoing precursor spin-lattice fluctu-\nations and spin-lattice ordering. Further experimental\nand theoretical studies are indispensable if the dynami-\ncal molecular-spin state in ZnFe 2O4, which is expected\nto govern the magnetic properties, is to be understood.\nV. ACKNOWLEDGMENTS\nThis work was partly supported by a Grant-in-Aid\nfor Scientific Research (C) (Grant No. 25400348) from\nMEXT of Japan, and by Nihon University College of Sci-\nence and TechnologyGrants-in-Aidfor ProjectResearch.\n∗tadataka@phys.cst.nihon-u.ac.jp\n1S. H. Lee, H. Takagi, D. Louca, M. Matsuda, S. Ji, H.\nUeda, Y. Ueda, T. Katsufuji, J. H. Chung, S. Park, S. W.\nCheong, and C. Broholm, J. Phys. Soc. Jpn. 79, 011004(2010).\n2H. Ueda, H. Mitamura, T. Goto, and Y. Ueda, Prog.\nTheor. Phys. 159, 256 (2005).\n3S. H. Lee, C. Broholm, T. H. Kim, W. Ratcliff II, and S.7\nW. Cheong, Phys. 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Lett. 84, 5876 (2000)."    },    {        "title": "0707.2600v2.Subnanosecond_switching_of_local_spin_exchange_coupled_to_ferromagnets.pdf",        "content": "Subnanosecond switching of local spin-exchange coupled to ferromagnets\nJonas Fransson\u0003\nTheoretical Division and Center for Nonlinear Studies,\nLos Alamos National Laboratory, Los Alamos, New Mexico 87545 and\nDepartment of Physics and Materials Science, Uppsala University, SE-751 21 Uppsala, Sweden\n(Dated: September 6, 2021)\nThe dynamics of a single spin embedded in a the tunnel junction between ferromagnetic contacts\nis strongly a\u000bected by the exchange coupling to the tunneling electrons. Using time-dependent\nequation of motion for the spin under in\ruence of the spin-polarized tunneling current, it is shown\nthat the magnetic \feld induced by bias voltage pulses allows for sub-nanosecond switching of the\nlocal spin and the possibility of spin reversal is illustrated. Furthermore, it is shown that the time-\nevolution of the Larmor frequency sharply peaks around the spin-\rip event, and it is argued that\nthis feature can be used as an indicator for the spin-\rip.\nPACS numbers: 73.40.Gk, 73.43.Fj, 03.65.Yz, 67.57.Lm\nDetection and manipulation of single spins is an im-\nportant \feld of science since it pushes the limits of quan-\ntum measurements. Single spins would be an object for\nqubits and would, thus, be crucial for quantum informa-\ntion technology. Potentially, spintronics will replace con-\nventional electronics devices with spin analogues where\nmanipulation, control, and read-out of spins enable func-\ntionality with no or little charge dynamics.1Of major im-\nportance is to understand how spins can be manipulated\nthrough electric \felds, and questions of the time-scales\ninvolved.\nCurrent-induced magnetic switching have been ad-\ndressed for planar2,3,4,5,6and magnetomechanical7sys-\ntems, as well as questions concerning spin-transfer\ntorque8,9,10,11and decoherence.12Recently, the dynam-\nics of a local spin coupled to superconducting leads was\nconsidered and found non-trivial in the sense that a \f-\nnite charge current introduces a nutation of the spin\nmotion.15,16Analogous nutations were found for a local\nspin embedded in the tunnel junction between ferromag-\nnetic leads biased with a harmonic voltage.17Concern-\ning single spin manipulations, such kind of nutation pro-\nvides signi\fcant implications for electrically controlling\nthe dynamics of local spins. Spin-\rip transitions of tun-\nneling electrons generated by spin-orbit coupling is also\na source for direct manipulations of the local spin.18Spin\nnutations and spin echo were observed under in\ruence of\nexternal microwave magnetic \feld using force detection\ntechniques for studies of electron-spin resonance.19\nThe purpose of this paper is to consider some implica-\ntions of the nutations of the local spin that are generated\nby a time-dependent bias voltage across the junction.17\nIt is argued that that the nutations of the local spin can\nbe electrically controlled and its amplitude can be made\nsu\u000eciently large for the spin to undergo spin-\rip transi-\ntions, by application of short bias voltage pulses across\nthe junction, e.g. pulses of time span less than 1 ns.\nIn an arrangement of ferromagnetic leads with unequal\nmagnetization and/or in non-collinear alignment, the lo-\ncal spin tends to line up along the magnetic moment of\nthe source lead, and by reversing the bias across the junc-tion stimulates reversal of the local spin. In this respect,\nthe spin-polarized current can be considered to be gen-\nerating an anisotropy \feld that stabilize the orientation\nof the local spin, as long as the current \rows across the\njunction. It is further argued that the time-evolution\nof the spin Larmor frequency sharply peaks around the\nevent of the spin-\rip transition. This feature can thereby\nbe used as a signal for detection of the spin-\rip event.\nFollowing the setup given in Ref. 17, a local spin S\nembedded in the tunnel junction between two ferromag-\nnetic leads, see Fig. 1, is considered. The leads are di-\nrectly coupled through tunneling and indirectly through\nexchange interaction with the local spin. The Hamilto-\nnian for the system can then be written as\nH=HL+HR+HS+HT: (1)\nThe \frst two terms HL(R)=P\nk\u001b2L(R)\"k\u001bcy\nk\u001bck\u001bde-\nscribe the electrons in the leads, where an electron is cre-\nated (destroyed) in the left ( L) or right (R) lead at the\nenergy\"k\u001band spin\u001b=\";#, bycy\nk\u001b(ck\u001b). The Hamilto-\nnian for a free spin Sin the presence of a magnetic \feld\nB0is given by\nHS=\u0000g\u0016BB0\u0001S; (2)\nwheregand\u0016Bare the gyromagnetic ratio and Bohr\nmagneton, respectively. The two leads are weakly cou-\npled via the tunneling Hamiltonian\nHT=X\npq\u000b\f\u0000\ncy\np\u000b[T0\u000e\u000b\f+T1S\u0001\u001b\u000b\f]cq\f+H:c:\u0001\n:(3)\nFM L FM R\nVy z\nxB0\nFIG. 1: (Color online) Local spin embedded in the tunnel\njunction between two ferromagnetic leads.arXiv:0707.2600v2  [cond-mat.mes-hall]  16 May 20082\nHere and henceforth, let electrons in the left (right) lead\ncarry the subscript p(q). Here also, \u001b\u000b\fis the vec-\ntor of Pauli spin matrices, with spin indices \u000b;\f. The\nrate of the direct tunneling between the leads is denoted\nbyT0whereas the tunneling with rate T1is in\ruenced\nby the presence local spin, both of which are assumed\nto be spin-independent. Spin-\rip transitions in the di-\nrect tunneling between the leads are neglected in this\nmodel, and it is furthermore implied that the level of\nthe localized electron is far below the Fermi surface of\nthe conduction electrons in both electrodes. Therefore,\nthe voltage bias should not exceed this energy level dif-\nference. For convenience the respective amplitudes are\ntaken to be momentum independent (although it is not\nrequired). Typically, from the expansion of the work\nfunction for tunneling, T1=T0\u0018J=U, whereJandUis\nthe spin-spin exchange interaction parameter and spin-\nindependent tunneling barrier, respectively13,14Further,\nan external magnetic \feld B0(t) may applied to the sys-\ntem, see Fig. 1.\nIt should be noticed that the model should in prin-\nciple also contain exchange interaction between the lo-\ncal spin and the electrons in the leads, e.g. terms like\ncy\nk\u000bS\u0001\u001b\u000b\fck\f, wherek2LorR. Such terms will, how-\never, not contribute to the current driven spin dynamics\nconsidered here, and are therefore omitted.\nEstimates of the signal-to-noise ratio has been pre-\nsented by Balatsky et al.13and Nussinov et al.20for\nthe present model. It was then found that the spectral\npower densityhI2!i\u0018I2\n0(T1=T0)2\u001f(!), where\u001f(!) is a\nLorentzian associated with power spectrum of the local\nspin. The power spectrum of the shot noise is approxi-\nmatelyhI2\nshot(!)i\u0018I0. UsingI0\u00181=\u001ce, where 1=\u001ceis\nthe spin-independent scattering rate. Thus, the signal-to-\nnoise ratio becomes hI2!i=hI2\nshot(!)i\u0018(T1=T0)2\u001f(!)=\u001ce,\nwhich in the present system is bounded both from above\nand below by a number of order unity.20Hence, the signal\nmay be quite sizable, which enables the further discus-\nsions in this paper.\nWhen a time-dependent voltage bias is applied across\nthe tunneling barrier, such that V(t) =Vdc+Vac(t),\nwhereVdcandVacare the dc and ac components, a\ndipole forms around the barrier region through the ac-\ncumulation or depletion of electron charge. This pro-\ncess results in the time dependence of single-particle en-\nergies asEp(q)=\u000fp(q)+WL(R)(t), with the constraint\nWL(t)\u0000WR(t) =eVac(t). However, the occupation of\neach state in the respective contact remains unchanged\nand is determined by the distribution established before\nthe time dependence is turned on. Therefore, the chem-\nical potentials on the left \u0016Land on the right lead \u0016R\ndi\u000bers by the dc component of the applied voltage bias,\n\u0016L\u0000\u0016R=eVdc. The tunnel junction is then character-\nized by two time scales, the Larmor precession frequency\nof the spin!L=g\u0016BjBjand the characteristic time scale\nof the ac \feld.\nThe dynamics of the local spin was derived in Ref.\n17 and was found to satisfy the Landau-Lifshitz-Gilbertequation22,23\ndn\ndt=\u000b(t)dn\ndt\u0002n+g\u0016Bn\u0002B(t); (4)\nunder the assumption that the dynamics of the local spin\nis much slower than the electronic processes. This as-\nsumption is motivated by the fact that the energy as-\nsociated with the spin dynamics, ~!L\u00181\u0016eV, is much\nsmaller that the typical electronics energy on the order of\n1 meV.24In Eq. (4) the spin S(t) =Sn(t), whereS=jSj,\nwhereas n=S=S. The damping factor \u000b(t)/ST2\n1=D,21\nat zero temperature, where 2 Dis the band widths of the\nleads,17and it is reasonable to believe this is the case\nalso for \fnite temperatures. Thus, by assuming large D,\nthe damping \u000bbecomes negligible.\nThe e\u000bective magnetic \feld B(t) =B0(t) +B(1)\nind(t) +\nB(2)\nind(t), where the induced magnetic \felds can be ex-\npressed as17\ng\u0016BB(1)\nind(t) =\u00002T0T1X\npq\u001b\u001bz\n\u001b\u001b[f(\u0018p\u001b)\u0000f(\u0018q\u001b)]\n\u0002ImZt\n\u00001ei[(\u0018p\u001b\u0000\u0018q\u001b)(t\u0000t0)+'(t)\u0000'(t0)]dt0^z (5)\ng\u0016BB(2)\nind(t) =SZt\n\u00001n\nK(2)\nxy(t;t0)(ny^x\u0000nx^y)\n+[K(2)\nxx(t;t0)\u0000K(2)\nzz(t;t0)]nz^zo\ndt0(6)\nwhere\nK(2)\nxy(t;t0) = 2\u0019T2\n1ReX\npq\u001b[f(\u0018p\u001b)\u0000f(\u0018q\u0016\u001b)]\n\u0002ei[(\u0018p\u001b\u0000\u0018q\u0016\u001b)(t\u0000t0)+'(t)\u0000'(t0)](7)\nand\nK(2)\nxx(t;t0)\u0000K(2)\nzz(t;t0) =\n\u00002\u0019T2\n1ImX\npq\u001bn\n[f(\u0018p\u001b)\u0000f(\u0018q\u0016\u001b)]ei(\u0018p\u001b\u0000\u0018q\u0016\u001b)(t\u0000t0)\n\u0000[f(\u0018p\u0016\u001b)\u0000f(\u0018q\u001b)]ei(\u0018p\u0016\u001b\u0000\u0018q\u001b)(t\u0000t0)o\nei['(t)\u0000'(t0)](8)\nHere'(t) =Rt\nt0V(t0)dt0, for some initial time t0, such\nthat'(t)\u0000'(t0) =Rt\nt0V(t00)dt00, whereas\u0018p(q)\u001b=\"p(q)\u001b+\n\u0016L(R).\nThese induced \felds are direct responses to the bi-\nasing of the spin-polarized leads. The spin-imbalances\n(NL(R)\"\u0000NL(R)#) in the leads generate a \feld along the\nz-direction which modi\fes the Larmor frequency of the\nlocal spin. More important for the switching, however,\nis whether or not the magnetic moments of the leads\nare equal and/or parallel. Unequal and/or non-parallel\nmagnetic moments of the leads generate a magnetic \feld\nwhich tends to force the local spin into the magnetic di-\nrection of the source lead, as is shown below.3\n10 20 30 40 \n0.5 1 1.5 \ntime (ns) b) Kxy (2) dt /old_integral\n-∞ t0.2 0.5 0.8 θ(t)/ πa) \nc) ωL (GHz)ind -0.03 00.03 \nFIG. 2: (Color online) Time-dependent variation of a) \u0012(t),\nb)Rt\n\u00001K(2)\nxydt0, and c) the induced Larmor frequency !ind\nL(t),\nfor a square bias voltage pulse, c.f. dotted line in a). Here,\nVac= 1 mV,T1N0= 0:1,pL=\u0000pR= 1=2.\nIn order to better understand the motion of the lo-\ncal spin, consider the corresponding classical equation\nof motion, which is consistent with the parametriza-\ntion of the spin on the unit sphere, i.e. Sn=\nS(cos\u001esin\u0012;sin\u001esin\u0012;cos\u0012). With B0=B0^ztime-\nindependent, the equations can be written\nd\u001e\ndt=\u0000g\u0016B[B0+B(1)\nind(t)]\n+SZt\n\u00001[K(2)\nxx(t;t0)\u0000K(2)\nzz(t;t0)]dt0cos\u0012;\nd\u0012\ndt=SZt\n\u00001K(2)\nxy(t;t0)dt0sin\u0012:(9)\nThe equation for the polar angle \u0012can be analytically\nsolved and in the stationary regime it gives\n\u0012(t) = 2 arctan\u0012\ntan\u00120\n2e\u00002\u0019eVdcT2\n1P\n\u001b\u001bz\n\u001b\u001bNL\u001bNR\u0016\u001b(t\u0000t0)\u0013\n;\n(10)\nwhere\u00120=\u0012(t0). This expression clearly shows the pos-\nsibility to switch the local spin by means of a stationary\nbias voltage. For example, con\fguring the leads such\nthatNL\"NR#\u0000NL#NR\">0,25and biasing the system\nbyVdc>0, gives an exponential decay of the argument\nwhich leads to that the polar angle \u0012(t)!0 (local spin\nbecomes\"), ast!1 , for any given initial polar angle\n\u00120. By the same argument \u0012(t)!\u0019(local spin becomes\n#) if the bias Vdc<0 is applied.\nUsing the result for the polar angle within the sta-\ntionary regime, the characteristic time-scale ~=\u001cc'\n2\u0019eVT2\n1P\n\u001b\u001bz\n\u001b\u001bNL\u001bNR\u0016\u001bfor the polar angle motion of\nthe spin is obtained. Parametrizing the spin-polarizedDOSNL(R)\u001b=N0(1+\u001bpL(R))=2,26where\u00001\u0014pL(R)\u0014\n1, assuming T1N0\u00180:1,pL=\u0000pR=\u00001=2, andV\u00181\nmV gives the characteristic time scale \u001cc\u00195 ps, which\nis su\u000eciently short to switch the spin from being #to\"\nwithin 1 ns, see Fig. 2 a).\nIn a realistic set-up of the system it would be desirable\nto obtain the current induced switching by application of\nbias pulses of some time span \u001cs. A sudden onset of a bias\npulse leads to transient e\u000bects in the induced magnetic\n\felds B(n)\nind(t); n= 1;2, which transfer into the motion\nof the spin. Therefore, consider application of the bias\nVdc+Vac[\u0002(t\u0000\u001c0)\u0000\u0002(t\u0000\u001c1)], where\u001c1=\u001c0+\u001cs. We also\nassume that B0= 0 since we are interested in the local\nspin dynamics generated by the spin-polarized current.\nAs shown above, a stationary bias will eventually line up\nthe local spin in the magnetic direction of the source lead\nand since the main interest lies in a switching obtained\nby a pulsed bias, henceforth Vdc= 0. Then, the equation\nof motion for \u0012(t) can be written\nd\u0012\ndt=\u00002\u0019ST2\n1X\n\u001b\u001bz\n\u001b\u001bNL\u001bNR\u0016\u001beVac\u001a\u0014\n1\u0000e\u0000(t\u0000\u001c0)=\u001c\u0015\n\u0002[\u0002(t\u0000\u001c0)\u0000\u0002(t\u0000\u001c1)] +\u0014\ne\u0000(t\u0000\u001c1)=\u001c(11)\n\u0000e\u0000(t\u0000\u001c0)=\u001c\u0015\ncos [eVac(t\u0000\u001c1)]\u0002(t\u0000\u001c1)\u001b\nsin\u0012\nwhere the time-scale \u001crelates to the electronic tunneling\nprocesses. Such processes are of the order of 1 fs, which is\nmuch faster than the characteristic time-scale \u001ccand the\nLarmor frequency ~!Lfor biases between 1 - 100 mV.24\nFor e.g. a 1 ns bias pulse of 1 mV, the critical time scale\nis\u0018100 fs, which may be achieved within present experi-\nmental state-of-the-art-technology for nanoscale systems.\nPhysically the time-scale \u001cimplies that the induced mag-\nnetic \feld is a retarded response to the bias across the\njunction.\nThe calculated time-dependence of the polar angle is\nplotted in Fig. 2 a) for a square bias voltage pulse, and\nit is readily seen that the angle goes from \u0019to 0 within\nthe time span of the pulse, hence, the pulse is su\u000eciently\nlong to \rip the spin from #to\". The plot in Fig. 2\nb) shows the induced magnetic \feld which is acting to\nalign the local spin along the magnetic direction of the\nsource lead. At the onset (termination) of the bias pulse,\nthe amplitude of the induced magnetic \feld grows (de-\ncays) exponentially, as expected from Eq. (11). At the\npulse termination, however, there are additional oscilla-\ntions in the induced \feld, as a reaction to the removed\nbias. These oscillations, are not visible in the motion of\nthe polar angle, since they are exponentially suppressed,\nc.f. Eq. (10).\nThe Larmor frequency !Lof the precession of the lo-\ncal spin is a\u000bected by the time-dependent variation of \u0012\nthat gives rise to a momentary change of its value. By4\n0.20.50.8θ(t)/π\n0.5 110203040ωLind (GHz)a)\nb)θ0=5π/6\n3π/4\n2π/3\nπ/2\nπ/3\nπ/4\nπ/6\nt-t  (ns)0\nFIG. 3: (Color online) Time-dependent variation of a) \u0012(t)\nand b)!ind\nL(t), for di\u000berent initial polar angles \u00120, under the\nbiasV(t) =Vdc+Vacf[\u0002(t\u0000t0)\u0000\u0002(t\u0000t1)]\u0000[\u0002(t\u0000t1)\u0000\n\u0002(t\u0000t2)]g, witht1\u0000t0=t2\u0000t1= 1 ns. For clarity, the plots\nin panel b) have been shifted 5 GHz upwards for increasing\n\u00120. Other parameters are as in Fig. 2.\nde\fnition\n!2\nL(t)\u0011(g\u0016B)2jB(t)j2=\u0012\nB0+B(1)\nind(t)\n+SZt\n\u00001[K(2)\nxx(t;t0)\u0000K(2)\nzz(t;t0)]dt0cos\u0012(t)\u00132\n:\n+\u0012\nSZt\n\u00001K(2)\nxy(t;t0)dt0\u00132\nsin2\u0012(t) (12)\nThe time-dependent parts of the induced \felds in the\n\frst term are equal and can be written as ( Vdc= 0)\neVac\u0014\ne\u0000(t\u0000\u001c0)=\u001c\u0000e\u0000(t\u0000\u001c1)=\u001c\u0015\nsin [eVac(t\u0000\u001c1)]\u0002(t\u0000\u001c1):\n(13)\nHence, this \feld does not a\u000bect the Larmor frequency\nuntil after the bias pulse has terminated, and then it gen-\nerates an oscillatory variation of !Lwith an exponential\ndecay, see Fig. 2 c). These oscillations are su\u000eciently\nlarge to enable read-out of the time instant when the\npulse is terminated.\nThe last term in Eq. (12), containing the \feld that\ngenerates the spin-\rip, vanishes as long as \u0012(t) = 0;\u0006\u0019,\nthat is, when the spin is either \"or#in the global spin\nquantization axis. This term gives a contribution, how-\never, during the short time interval when the spin \rips,\nsince then 0 <j\u0012(t)j< \u0019. Hence, one would expect a\nsharp peak in the time-evolution of !L(t), which indeed\nis seen in Fig. 2 c). This peak in the time-evolution of the\nLarmor frequency is a \fngerprint of an actual occurrenceof a spin-\rip and would be measurable. Using the values\ncomparable to those in Fig. 2, results in an amplitude of\nthe peak of the order of 50-500 GHz in case of vanishing\nexternal magnetic \feld, for biases 1 - 10 mV.\nIn the discussion we have neglected in\ruences from e.g.\nanisotropy \felds. The absence of such or similar \felds\nthat would stabilize the spin direction, cause the spin\norientation to drift randomly in equilibrium. It has been\nshown that the spin-polarized current between two ferro-\nmagnetic leads has such e\u000bect on the local spin, i.e. the\ne\u000bect of the spin-polarized current on the local spin is to\nstabilize its orientation. The plots in Fig. 3 a) show \u0012(t)\nfor several di\u000berent initial polar angles \u00120, given the bias\nV(t) =Vdc+Vacf[\u0002(t\u0000t0)\u0000\u0002(t\u0000t1)]\u0000[\u0002(t\u0000t1)\u0000\n\u0002(t\u0000t2)]g, witht1\u0000t0=t2\u0000t1= 1 ns. The plots\nillustrate how the spin-polarized current forces the local\nspin to be aligned with the majority spin of the source\nlead,\u0012(t)!0 fort0\u0014t\u0014t1. Furthermore, by reversing\nthe bias voltage, the spin orientation can be reversed at\nwill, which is illustrated by the plots in Fig. 3 a) where\n\u0012(t)!\u0019, in the interval t1\u0014t\u0014t2. Panel b) in Fig.\n3 further illustrates how the Larmor frequency peaks at\nthe instant of the spin-\rip, both when its polar angle ap-\nproaches zero and \u0019. The increasing amplitude of the\npeak with increasing di\u000berence between the \fnal and ini-\ntial polar angle is expected, because of the increasing\nreorientation path the spin has to traverse.\nThe result displayed in Eq. (11) shows that it would\nbe possible to \rip the spin by having only one of the leads\nferromagnetic and the other non-magnetic. Thus, a set-\nup of an scanning tunneling microscope (STM) with a\nspin-polarized STM tip and non-magnetic substrate sur-\nface would correspond to having, say, pL6= 0 andpR= 0.\nHence, letting pL= 1=2 and using the same parameters\nas in Fig. 2, one \fnds a time scale for the spin \rip of\nroughly\u001cc'10 ps, that is, doubled the time scale com-\npared to the case discussed above. The spin \rip e\u000bect\nshould nonetheless be likewise observable as in the case of\ntwo ferromagnetic leads in anti-parallel alignment, how-\never, on a larger time scale.\nThe result in Eq. (11) also shows that if both leads\nare being ferromagnetic and in parallel alignment, the\nresulting spin \rip e\u000bect on the local spin decreases since\nthe induced \felds from the left and right leads tend to\ncancel each other. The polar angle becomes a constant of\nmotion whenever the two leads are equally strong ferro-\nmagnets in parallel con\fguration, since then pL=pR=p\nwhich yieldsP\n\u001b\u001bz\n\u001b\u001bNL\u001bNR\u0016\u001b= (N0=2)2[(1 +p)(1\u0000p)\u0000\n(1\u0000p)(1 +p)] = 0.\nAnisotropy \feld acting on the local spin has been omit-\nted in the present paper for simplicity. In a realistic sys-\ntem, however, the local spin is likely subject to some\ntype of anisotropy \feld which stabilize de\fnite spin di-\nrections. Such \felds may be small enough to not substan-\ntially increase the time scale of the spin-\rip processes in\nthe present situation. Without any type of anisotropic\n\feld acting on the local spin, its orientation will in equi-\nlibrium, i.e. V(t) = 0, most likely describe a random drift5\ndue to exchange interaction with equilibrium \ructuations\nin the current between the leads. Such a current will lack\nany de\fnite spin-polarization that the non-equilibrium\nsituation can present.\nIn summary, it is demonstrated that the dynamics of a\nlocal spin embedded in the tunnel junction between fer-\nromagnetic leads can be manipulated at will by means\nof a time-dependent bias voltage. Especially, spin-\rip\ntransitions can be stimulated at a sub-nanosecond time-\nscale by bias voltage pulses, even for a moderate spin-polarization of the leads. The time-evolution of the Lar-\nmor frequency of the local spin sharply peaks at the spin-\n\rip event, something that determines actual occurrence\nof a spin-\rip. These \fndings would be useful in the ad-\nvent spintronics and quantum information technology.\nThe author thanks A. V. Balatsky and J. -X. Zhu for\nuseful discussions. This work has been supported by US\nDOE, LDRD and BES, and was carried out under the\nauspices of the NNSA of the US DOE at LANL under\nContract No. DE-AC52-06NA25396.\n\u0003Electronic address: Jonas.Fransson@fysik.uu.se\n1D. D. Awschalom, M. E. Flatte, and N. Samarth, Spin-\ntronics (Scienti\fc American, New York, 2002), pp 67-73.\n2J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996);\nJ. Magn. Magn. Mater. 195, L261 (1999).\n3L. Berger, Phys. Rev. B 54, 9353 (1996).\n4I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev,\nD. C. Ralph, and R. A. Burman, Science 307, 228 (2005).\n5D. M. Edwards, F. Federici, J. Mathon, and A. Umerski,\nPhys. Rev. B 71, 054407 (2005).\n6T. Nozaki, N. Tezuka, and K. Inomata, Phys. Rev. Lett.\n96, 027208 (2006).\n7A. A. Kovalev, G. E. W. Bauer, and A. Brataas, Phys.\nRev. Lett. 94, 167201 (2005).\n8Y. Ji, C. L. Chien, and M. D. Stiles, Phys. Rev. Lett. 90,\n106601 (2003).\n9Y. Liu, Z. Zhang, P. P. Freitas, and J. L. Martins, Appl.\nPhys. Lett. 82, 2871 (2003).\n10G. E. W. Bauer, A. Braatas, Y. Tserkovnyak, and B. J.\nvan Wees, Appl. Phys. Lett. 823928 (2003).\n11B.Ozyilmaz, A. D. Kent, D. Monsma, J. Z. Sun, M. J.\nRooks, and R. H. Koch, Phys. Rev. Lett. 91067203 (2003).\n12H. Katsura, A. V. Balatsky, Z. Nussinov, and N. Nagaosa,\nPhys. Rev. B 73, 212501 (2006).\n13A. V. Balatsky, Y. Manassen, and R. Salem, Phys. Rev. B\n66, 195416 (2002).\n14J. -X. Zhu and A. V. Balatsky, Phys. Rev. B, 67174505\n(2003).15J.-X. Zhu, Z. Nussinov, A. Shnirman, and A. V. Balatsky,\nPhys. Rev. Lett. 92, 107001 (2004).\n16Z. Nussinov, A. Shnirman, D. P. Arovas, A. V. Balatsky,\nand J. -X. Zhu, Phys. Rev. B 71, 214520 (2005).\n17J. Fransson and J. -X. Zhu, New J. Phys. 10, 013017\n(2008).\n18J. -X. Zhu and J. Franson, J. Phys.: Condens. Matter 18,\n9929 (2006).\n19K. Wago, D. Botkin, C. S. Yannoni, and D. Rigur, Phys.\nRev. B 57, 1108 (1998).\n20Z. Nussinov, M. F. Crommie, and A. V. Balatsky, Phys.\nRev. B 68, 085402 (2003).\n21For a stationary bias voltage V(t) =Vdcand non-magnetic\nleads, the damping factor can be written, using the results\nin17,\u000b(t) = 8\u00192N2\n0T2\n1eVdc=D, whereN0is the density of\nelectron states in the leads.\n22L. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8, 153,\n(1935).\n23T. L. Gilbert, Phys. Rev. 100, 1243 (1955); IEEE Trans.\nMagn. 40, 3443 (2004).\n24J.-X. Zhu and A. V. Balatsky, Phys. Rev. Lett. 89, 286802\n(2002).\n25P\n\u001b\u001bz\n\u001b\u001bNL\u001bNR\u0016\u001b>0 either means i) both leads being fer-\nromagnetic in a non-parallel con\fguration , or ii) only left\n(right) lead is ferromagnetic. In both cases there is a spin\n\"(#) majority in the left (right) lead.\n26J. Fransson, Europhys. Lett. 70, 796 (2005)."    },    {        "title": "1401.7572v1.Initial_correlation_in_a_system_of_a_spin_coupled_to_a_spin_bath_through_an_intermediate_spin.pdf",        "content": "arXiv:1401.7572v1  [quant-ph]  29 Jan 2014Initial correlation in a system of a spin coupled to a spin bat h\nthrough an intermediate spin\nV. Semin∗\nQuantum Research Group, School of Chemistry and Physics,\nUniversity of KwaZulu-Natal, Durban, 4001, South Africa\nI. Sinayskiy†and F. Petruccione‡\nQuantum Research Group, School of Chemistry and Physics and National Institute for Theoretical Physics,\nUniversity of KwaZulu-Natal, Durban, 4001, South Africa\n(Dated: October 8, 2018)\nThe strong system-bath correlation is a typical initial con dition in many condensed matter and\nsome quantum optical systems. Here, the dynamics of a spin in teracting with a spin bath through\nan intermediate spin are studied. Initial correlations bet ween the spin and the intermediate spin are\ntaken into account. The exact analytical expression for the evolution operator of the spin is found.\nFurthermore, correlated projection superoperator techni ques are applied to the model and a time-\nconvolutionless master equation to second order is derived . It is shown that the time-convolutionless\nmaster equation to second order reproduces the exact dynami cs for time-scales of the order 1 /γ,\nwhereγis the coupling of the central spin to the intermediate spin. It is found that there is a strong\ndependence on the initial system-bath correlations in the d ynamics of the reduced system, which\ncannot be neglected.\nPACS numbers: 03.65.-w, 42.50.Ar, 03.65.Yz, 75.10.Jm\nI. INTRODUCTION\nThe open quantum dynamics [1] of spin systems finds\nwide application in various fields of physics, e.g., quan-\ntum theory of magnetism [2], quantum information pro-\ncessing [3], quantum biology [4] and quantum dots [5–\n7]. Several models were proposed to study decoherence\nof single and multi-spin systems interacting with a sur-\nrounding environment [8]. The model considered here is\nquitetypicalforvarioussituations,e.g., coupledquantum\ndots with one of them strongly interacting with an exter-\nnal spin environment or a system of two spin-spin inter-\nactingelectronswithoneofthem stronglycoupledtosur-\nrounding nuclear spins. Very often, the derivation of the\nreduced dynamics involves complications and difficulties\nthat can be overcomein many cases by the application of\napproximation techniques. In particular, the Markovian\napproximation together with the quantum master equa-\ntion approach turns out to be very useful [9, 10]. How-\never, any approximation method is inevitably based on\nsome assumptionswhich do not necessarilyreflect the ac-\ntual propertiesofthe composite system. Moreover,many\nrealistic spin systems exhibit non-Markovian behaviour\nfor which the standard derivation of the quantum mas-\nter equation ceases to be applicable. The non-Markovian\ndynamics of a spin-system coupled to a spin environment\nhas been extensively investigated [11–16].\nThere are only few solvable models of open quantum\n∗Electronic address: semin@ukzn.ac.za\n†Electronic address: sinayskiy@ukzn.ac.za\n‡Electronic address: petruccione@ukzn.ac.zasystem dynamics, which allow for the exact analytical\ndynamics of the reduced system. Some examples are the\ndamped harmonic oscillator [17], the spontaneous emis-\nsionofatwolevelatomintoazerotemperaturebath[18],\nthe pure decoherence of a two-level system [19, 20] and\nspin star models [16]. From this point of view, develop-\ning new exactly solvable non-perturbative models plays a\ncrucial role for the deeper understanding of the realistic\nsystems dynamics. Furthermore, exactly solvable models\nmake it possible to test and develop new approximation\ntechniques. A typical spin environment is characterized\nby non-Gaussian fluctuations and strong memory effects\nfor a wide range of parameters [21]. Even for the sim-\nplest case of a two-level system interacting with a bath\nof spins in a spin star configuration the Markov limit for\nthe quantum master equation does not exists [16, 22].\nRecently, a correlated projection operator approach was\ndeveloped [23–25]. This approximation technique was\nshown to be an efficient tool in the description of various\nphysical systems [25–27].\nTypically, in the derivation of the quantum master\nequation it is assumed that initially the system and the\nbath are uncorrelated. However, this is not the case in\nmany condensed matter and quantum optical systems.\nTaking into account correlations between system and\nbath gives raise to inhomogeneous terms in the quan-\ntum master equation [1]. Recently, initial correlations in\ntheJaynes-Cummingsmodelwerestudied usingthetrace\ndistance [28]. It was found that initial correlation plays\nan important role in the reduced dynamics of the two-\nlevel system. The influence of initial correlation on the\ndissipation of a qubit interacting with a bosonic reservoir\nat zero temperature was studied in Ref. [29]. The role of\ninitial correlations in the decoherence process of a qubit2\ninteractingwith abosonicbathfordifferent bathspectral\ndensities was investigated in [30, 31], and it was shown\nthatthe evolutionofthediagonalelementsofthe reduced\nsystem exhibits a strong dependence on initial correla-\ntions. The exact non-Markovian dynamics of open quan-\ntum systems in the presence of initial system-reservoir\ncorrelations for a photonic cavity system coupled to a\ngeneral non-Markovian reservoir [32] shows that the ini-\ntial two-photon correlation between the cavity and the\nreservoircan induce nontrivialsqueezingdynamics to the\ncavity field. However, in all present research the role of\ninitial system bath correlation was investigated for pre-\nviously known exactly solvable models [1].\nIn this paper, we study the reduced dynamics of a spin\ncoupledtoaspinbaththroughanintermediatespin. The\nmain goals of this work are the following: Firstly, to use\nthe exact solution of the model derived here to show the\nimportance of initial system-bath correlations. Secondly,\nto demonstrate that in the case where initial system-\nbath correlations are present, a rigorously derived mas-\nter equation will contain an inhomogeneous term which\nsubstantially affects the dynamics of the reduced system.\nThirdly, by comparing the exact and approximate solu-\ntions of the master equation, to understand the limita-\ntions of the approximate correlated projector method in\nthe presence of an inhomogeneous term in the quantum\nmaster equation.\nThis paper is organized as follows. In Sec. II we de-\nscribe the model of a spin coupled to a spin bath through\nan intermediate spin. In Sec. III we present the ana-\nlytical solution for the evolution operator and build the\nreduced density matrix for the spin. In Sec IV we ap-\nply the correlation projection operator method to obtain\nthe quantum master equation in the time-convolutionless\nform (TCL2) and compare approximate and exact solu-\ntion of the studied model. Finally, in Sec. V we discuss\nthe results and conclude.\nII. MODEL\nWe consider the model of a spin coupled to a spin bath\nthrough an intermediate spin (see Fig. 1). The interac-\ntion Hamiltonian is given by\nH=HSI+HIB, (1)\nwhere\nHSI=γ\n2(σ+τ−+σ−τ+)\nis the Hamiltonian describing spin-spin interactions and\nσ±,τ±are the creation (annihilation) operators for the\nspin and the intermediate one, respectively. The param-\neterγdenotes the strength of the spin-spin interaction.\nThesecondtermintheinteractionHamiltoniandescribes\nthe interactions between the intermediate spin and the\nspins in the bath\nHIB=α\n2√\nN(τ+J−+τ−J+).Intermediate spinSpin\nBath\nFIG. 1: (Color online) The model of a spin coupled to a spin\nbath through an intermediate spin.\nIn the above equation, J±=N/summationtext\ni=1σi\n±, andσi\n±are creation\n(annihilation) operators of i-th spin of the bath, αis the\nstrength of the interaction and Ndenotes the numbers of\nbath spins. The factor 1 /√\nNis introduced in the above\nHamiltonian as usual [33] to obtain the correctbehaviour\nin the thermodynamic limit ( N→ ∞). The uniform cou-\npling of the Hamiltonian HIBis a simplification allowing\nto build an analytical solution for the model. In this\npaper units are chosen such that kB=/planckover2pi1= 1.\nIII. EXACT SOLUTION\nTo describe the exact dynamics of the total system we\nneed to specify an initial state of the total system which\nis given by the density operator ρtot(0) and to find an\nevolution operator of the total system U(t) in an explicit\nform, as\nU(t) = exp[−iHt]. (2)\nWith the knowledge of the evolution operator and the\ninitial state of the total system the reduced dynamics of\nthe spin can be found as\nρS(t) = trIB{U(t)ρtot(0)U†(t)}. (3)\nHerewewillconsideraninitiallycorrelatedstatebetween\nthe spin and the intermediate spin, while the rest of the\nbath is assumed to be unpolarized.\nThe initial state of the total system reads,\nρtot(0) =ρSI(0)⊗ρB(0), (4)\nwhereρSI(0) is the density matrix of the spin and the\nintermediate one is given by a generic X-like initial two-\nqubit density matrix, as\nρSI(0) =\nρ0\n110 0ρ0\n14\n0ρ0\n22ρ0\n230\n0ρ0\n32ρ0\n330\nρ0\n410 0ρ0\n44\n, (5)3\nwhileρB(0) is the density matrix describing the unpolar-\nized state of Nparticle with spin 1/2, as\nρB(0) =IB\n2N. (6)\nThe density matrix (5) contains any two spin states of\nthe form\n|Φ±\nθ,β/an}bracketri}ht= sin(θ)|+/an}bracketri}htS|−/an}bracketri}htI±exp(iβ)cos(θ)|−/an}bracketri}htS|+/an}bracketri}htI,(7)\n|Ψ±\nθ,β/an}bracketri}ht= sin(θ)|+/an}bracketri}htS|+/an}bracketri}htI±exp(iβ)cos(θ)|−/an}bracketri}htS|−/an}bracketri}htI,(8)\nwhere the |±/an}bracketri}htSand|±/an}bracketri}htIare the eigenvectors of σzand\nτz, respectively. We notice that |Φ±\nπ\n4,0/an}bracketri}htand|Ψ±\nπ\n4,0/an}bracketri}htare\nthe usual Bell states.\nA. Analytical expression for the evolution operator\nLetUijdenote the components of the evolution op-\neratorUin the basis {|− −/an}bracketri}ht,|−+/an}bracketri}ht,|+−/an}bracketri}ht,|+ +/an}bracketri}ht}of\neigenvectors of the operator σz⊗τz,whereσzis a diag-\nonal operator of the spin, and τzis a diagonal operator\nof the intermediate spin. We can write\nU|++/an}bracketri}ht=U11|++/an}bracketri}ht+U21|+−/an}bracketri}ht+U31|−+/an}bracketri}ht+U41|−−/an}bracketri}ht,(9)\nU|+−/an}bracketri}ht=U12|++/an}bracketri}ht+U22|+−/an}bracketri}ht+U32|−+/an}bracketri}ht+U42|−−/an}bracketri}ht,(10)\nU|−+/an}bracketri}ht=U13|++/an}bracketri}ht+U23|+−/an}bracketri}ht+U33|−+/an}bracketri}ht+U43|−−/an}bracketri}ht,(11)\nU|−−/an}bracketri}ht=U14|++/an}bracketri}ht+U24|+−/an}bracketri}ht+U34|−+/an}bracketri}ht+U44|−−/an}bracketri}ht.(12)\nOn the other hand, the operator Usatisfies the\nSchr¨ odinger equation\nid\ndtU|i,j/an}bracketri}ht=HU|i,j/an}bracketri}ht, (13)wherei,j= + or−.\nSubstituting Eqs. (9)-(12) into Eq. (13) yields the\nfollowing system of coupled differential equations\n\n\ni˙U1j=α\n2√\nNJ−U2j,\ni˙U2j=α\n2√\nNJ+U1j+γ\n2U3j,\ni˙U3j=α\n2√\nNJ−U4j+γ\n2U2j,\ni˙U4j=α\n2√\nNJ+U3j.(14)\nHere,j= 1,2,3,4 is the number of the column of the\nevolution operator Uin the chosen basis.\nDifferentiating the second and third equations in (14)\nand combining with the first and fourth equations in (14)\nwe obtain\n\n\ni˙U1j=α\n2√\nNJ−U2j,\ni¨U2j=−iα2\n4NJ+J−U2j+γ\n2˙U3j,\ni¨U3j=−iα2\n4NJ−J+U3j+γ\n2˙U2j,\ni˙U4j=α\n2√\nNJ+U3j.(15)\nAll terms in this system of the differential equations are\ndiagonalin the common eigenbasisofthe J2andJzoper-\nators of the bath. Hence, the standard method of solving\nsystems of differential equations can be applied.\nInitialconditionsfortheevolutionoperatorfollowfrom\nits definition (2). Clearly, we have\n∂n\n∂tnU(t)|t→0= (−iH)n. (16)\nUsing the previous relation we can determine that the\nsystem (15) admits the following solution\nU11= 1−α2\n4NJ−/braceleftbig\n(−1+cosh( A−))C+F−1G−2\n−−(−1+cosh( A+))C−F−1G−2\n+/bracerightbig\nJ+,\nU12=iα\n2√\n2NJ−F−1/braceleftbig\nsinh(A+)C−G−1\n+−sinh(A−)C+G−1\n−/bracerightbig\n,\nU13=αγ\n4√\nNJ−F−1{−cosh(A+)+cosh( A−)},\nU14=i√\n2α2γ\n8NJ−F−1/braceleftbig\nsinh(A+)G−1\n+−sinh(A−)G−1\n−/bracerightbig\nJ−,\nU21=iα\n2√\n2NF−1/braceleftbig\nsinh(A+)C−G−1\n+−sinh(A−)C+G−1\n−/bracerightbig\nJ+,\nU22=1\n2F−1{−cosh(A+)C−+cosh(A−)C+},\nU23=iγ\n2√\n2F−1{−sinh(A+)G++sinh(A−)G−},\nU24=αγ\n4√\nNF−1{−cosh(A+)+cosh( A−)}J−,\nU31=αγ\n4√\nNF−1{−cosh(A+)+cosh( A−)}J+,\nU32=iγ\n2√\n2F−1{−sinh(A+)G++sinh(A−)G−},\nU33=1\n2F−1{−cosh(A+)C−+cosh(A−)C+},\nU34=iα\n2√\n2NF−1/braceleftbig\nsinh(A+)C−G−1\n+−sinh(A−)C+G−1\n−/bracerightbig\nJ−,\nU41=i√\n2α2γ\n8NJ+F−1/braceleftbig\nsinh(A+)G−1\n+−sinh(A−)G−1\n−/bracerightbig\nJ+,\nU42=αγ\n4√\nNJ+F−1{−cosh(A+)+cosh( A−)},\nU43=iα\n2√\n2NJ+F−1/braceleftbig\nsinh(A+)C−G−1\n+−sinh(A−)C+G−1\n−/bracerightbig\n,\nU44= 1−α2\n4NJ+/braceleftbig\n(−1+cosh( A−))C+F−1G−2\n−−(−1+cosh( A+))C−F−1G−2\n+/bracerightbig\nJ−.(17)4\nIn the above system of equations we have used the nota-\ntion\nG±= 1/2/radicalig\n−α2\nN(J+J−+J−J+)−γ2±4F,\nF=1\n4/radicalig\n4α4\nN2J2z+2α2γ2\nN(J+J−+J−J+)+γ4,\nC±= (2α2Jz±4FN+γ2N)/(4N),\nA±=G±t/√\n2.(18)\nB. Reduced density matrix\nUsing the exact analytical expression for the evolution\noperator (17) after the trace over the intermediate spin\nvariables we find explicitly the reduced dynamics of the\nspin (see Eq. (3)), as\nρS\n11(t) =Tr B/bracketleftig\nρ0\n11U11U†\n11+(ρ0\n22U12+ρ0\n32U13)U†\n12\n+ρ0\n44U14U†\n14+(ρ0\n23U12+ρ0\n33U13)U†\n13\n+ρ0\n11U21U†\n21+(ρ0\n22U22+ρ0\n32U23)U†\n22(19)\n+ρ0\n44U24U†\n24+(ρ0\n23U22+ρ0\n33U23)U†\n23/bracketrightig\n/2N,\nρS\n22(t) = 1−ρS\n11(t),\nρS\n12(t) =ρS\n21(t) = 0. (20)\nThe relation (17), (19), (20) are the main result of this\narticle.\nTo perform the trace over the collective bath variables\none needs to take into account the degeneracy of states\nof the bath as\ntrBρ=N/2/summationdisplay\nj=0,1\n2j/summationdisplay\nm=−jν(N,j)/an}bracketle{tj,m|ρ|j,m/an}bracketri}ht,(21)where the degeneracy is given by\nν(N,j) =/parenleftbiggN\nN\n2+j/parenrightbigg\n−/parenleftbiggN\nN\n2+j+1/parenrightbigg\n.(22)\nThe vectors |j,m/an}bracketri}htare eigenvectors of the bath operators\nJ2andJz, namely\nJz|j,m/an}bracketri}ht=m|j,m/an}bracketri}ht,(23)\nJ2|j,m/an}bracketri}ht=j(j+1)|j,m/an}bracketri}ht,(24)\nJ±|j,m/an}bracketri}ht=/radicalbig\nj(j+1)−m(m±1)|j,m±1/an}bracketri}ht,(25)\nwhere the eigenvalues vary from j= 1/2 toN/2 for odd\nN,or fromj= 0 toN/2 for even N, andm=−j,...,j.\nUsing the exact expression for the reduced density ma-\ntrix, we can analyse the dynamics of the spin. The dy-\nnamics of the probability to find the spin in the state |+/an}bracketri}ht\nis shown in Figs. 2 – 5.\nC. The limit of large number of bath spins\n(N→ ∞)\nThis subsection is devoted to the case of an infinite\nnumber of spins in the bath, i.e., the case N→ ∞. The\nlimit of large number of spins can be obtained using the\nfollowing formula [33]\nlim\nN→∞2−NtrB/braceleftig\nf/parenleftig\nJ±J∓\nN,Jz√\nN/parenrightig/bracerightig\n(26)\n=/parenleftbig2\nπ/parenrightbig3/2∞/integraltext\n−∞dm/integraltext\nCdzdz∗f(|z|2,m)e−2(m2+|z|2).\nAn exact calculation shows that\nρS\n11(t)N→∞= 1/2/parenleftbigg\n1+(ρ0\n11+ρ0\n22−ρ0\n33−ρ0\n44)cos/parenleftbiggγt\n2/parenrightbigg\n+i(ρ0\n23−ρ0\n32)sin/parenleftbiggγt\n2/parenrightbigg/parenrightbigg\n−/parenleftbigg\n(ρ0\n11+ρ0\n22−ρ0\n33−ρ0\n44)cos/parenleftbiggγt\n2/parenrightbigg\n+i(ρ0\n23−ρ0\n32)sin/parenleftbiggγt\n2/parenrightbigg/parenrightbigg\nf(t) (27)\n+/parenleftbigg\n(ρ0\n11−ρ0\n22+ρ0\n33−ρ0\n44)sin/parenleftbiggγt\n2/parenrightbigg\n+i(ρ0\n23−ρ0\n32)cos/parenleftbiggγt\n2/parenrightbigg/parenrightbigg\ng(t),\nρS\n22(t)N→∞= 1−ρS\n11(t)N→∞, (28)\nwhere the functions f(t) andg(t) are given by\nf(t) =γ2\nα2eγ2/2α2∞/summationdisplay\nn=1(γt)2n\n2n+2(2n)!∂n−1\n∂An−1e−A/2\nA|A→γ2/α2,and\ng(t) =iγ\n4αeγ2/2α2−α2t2/8/radicalbigπ\n2\n×/parenleftig\nerf/parenleftig\n2γ−iα2t\n2√\n2α/parenrightig\n−erf/parenleftig\n2γ+iα2t\n2√\n2α/parenrightig/parenrightig\n.\nUsing the exact expression for the reduced density ma-5\n0 5 10 15 200.00.20.40.60.81.0\n2ΓtΡ11S\nFIG. 2: Dynamics of the population of the upper state ρS\n11\nbased on the exact solution of the Schr¨ odinger equation (19 )\nfor differentcouplingstrengths between intermediate spin and\nenvironmental spins. Solid, dashed, dot-dashed and dotted\nlines correspond to α= 0,α=γ/4,α=γandα= 100γ, re-\nspectively. The initial state is the Bell state |Φ+\nπ/2,0/angbracketright; number\nof spins in the bath N= 50.\n0 5 10 15 200.30.40.50.60.7\n2ΓtΡ11S\nFIG. 3: Dynamics of the population of the upper state ρS\n11\nbased on the exact solution of the Schr¨ odinger equation (19 )\nfor differentcouplingstrengths between intermediate spin and\nenvironmental spins. Solid, dashed, dot-dashed and dotted\nlines correspond to α= 2γ,α=γ,α=γ/2 andα=γ/4, re-\nspectively. The initial state is the Bell state |Φ+\nπ/3,0/angbracketright; number\nof spins in the bath N= 50.\ntrix we can analyse the dynamics of the spin in the ther-\nmodynamic limit, Eq. (27). The dynamics of the proba-\nbility to find the spin in the state |+/an}bracketri}htis shown in Fig. 4\nand Fig. 6.\nIV. APPROXIMATION TECHNIQUES\nA. Projection operator techniques\nProjection operator techniques are a powerful tool of\nstatistical physics [35–39]. The application of projection\noperator techniques in the theory of open quantum sys-\ntems is basedonconsideringthe operationoftracingover\nthe environment as a formal projection ρ/ma√sto→ Pρin the0 5 10 15 200.30.40.50.60.7\n2ΓtΡ11S\nFIG. 4: Dynamics of the population of the upper state ρS\n11\nbased on the exact solution of the Schr¨ odinger equation (19 )\nfor different number of spins in the environment. Dotted,\ndashed and solid lines correspond to 10, 50 and ∞number of\nthe environmental spins, respectively. The initial state i s the\nBell state |Φ+\nπ/3,0/angbracketright; coupling strength α=γ/2.\n0 5 10 15 200.20.40.60.81.0\n2ΓtΡ11S\nFIG. 5: Dynamics of the population of the upper state ρS\n11\nbased on the exact solution of the Schr¨ odinger equation (19 )\nfor different initial system-environment correlations. So lid,\ndashed, dotted and dot-dashed lines correspond to initial\nstates|Φ+\nπ/3,0/angbracketright,|Φ+\nπ/3,π/2/angbracketright,|Φ+\nπ/3,π/4/angbracketrightand|Φ+\nπ/3,π/6/angbracketright, respec-\ntively. Number of spins in the bath N= 50; coupling strength\nα=γ.\nstate space of the total system [1, 34]. The superoper-\natorPhas the property of a projection operator, that\nisP2=P,and the density matrix Pρis said to be the\nrelevant part of the density ρof the total system. Cor-\nrespondingly, a projector I −Porρ/ma√sto→ Qρis defined as\na projection onto the irrelevant part of the total density\nmatrix.\nB. Time-convolutionless master equation\nOne of all possible ways of deriving an exact master\nequation for the relevant part of ρis to remove the de-\npendence of the system’s dynamics on the full history\nof the system and to formulate a time-local equation of6\n0 5 10 15 200.20.40.60.81.0\n2ΓtΡ11S\nFIG. 6: Dynamics of the population of the upper state ρS\n11\nbased on the exact solution of the Schr¨ odinger equation Eq.\n(27) in the thermodynamic limit for different initial system -\nenvironment correlations. Solid, dashed and dotted lines c or-\nresponds to initial states |Φ−\nπ/3,0/angbracketright,|Φ−\nπ/3,π/4/angbracketrightand|Φ−\nπ/3,π/2/angbracketright,\nrespectively. The coupling strength is α=γ/4.\nmotion, which is given by\n∂\n∂tPρ(t) =K(t)Pρ(t)+I(t)Qρ(0).(29)\nThis equation is called the time-convolutionless (TCL)\nmaster equation, and K(t) is a time-dependent superop-\nerator, which is referred to as the TCL generator. As for\nthe Nakajima-Zwanzig equation [35, 36], in general there\nis also an inhomogeneous term proportional to Qρ(0) on\nthe right-hand side of Eq. (29).\nThe expansion to second order in Hof the TCL gen-\nerator and the inhomogeneity are given by [1]\nK(t) =PL(t)P+t/integraldisplay\n0dsPL(t)L(s)P,(30)\nI(t) =PL(t)Q+t/integraldisplay\n0dsPL(t)L(s)Q. (31)\nC. Correlated projection superoperators\nThe starting point of the projection operator tech-\nniques is the introduction of a superoperator Pwhich\nacts on the total system’s density matrix, and which is\nusually defined by\nPρ= (trEρ)⊗ρ0, (32)\nwhereρ0is some fixed state of the environment. Obvi-\nously, the map Psatisfies the condition of a projector,\nnamelyP2=P.\nThe complementary map is defined via Q=I− P,\nwhereIdenotes the identity. Note that Pρcontains all\ninformation about the open system in the sense that forthe expectation value of any observable OSof the open\nsystem the relation tr {OSρ}= tr{OSPρ}holds.\nThe projector (32) is not the only possible choice [23].\nIn fact, a general class of projection superoperators can\nbe represented as follows,\nPρ=/summationdisplay\nitrE{Aiρ}⊗Bi, (33)\nwhere{Ai}and{Bi}are two sets of linear independent\nHermitian operators on HEsatisfying the relations\ntrE{BiAj}=δij, (34)/summationdisplay\ni(trEBi)Ai=IE, (35)\n/summationdisplay\niAT\ni⊗Bi≥0. (36)\nOncePis chosen, the dynamics of the open system is\nuniquely determined by the dynamical variables\nρi(t) = trE{Aiρ(t)}. (37)\nThe connection to the reduced density matrix is simply\ngiven by\nρS(t) =/summationdisplay\niρi(t), (38)\nand the normalization condition reads\ntrSρS(t) =/summationdisplay\nitrSρi(t) = 1. (39)\nThe TCL equation (29) together with the projection\noperator (33) leads to a coupled system of time-local dif-\nferential equations,\nd\ndtρi(t) =/summationdisplay\njKij(t)ρj(t)+/summationdisplay\njIij(t)ρj(0),(40)\nwith superoperators defined by\nKij(t)OS≡trE{AiK(t)(OS⊗Bj)},(41)\nIij(t)OS≡trE{AiI(t)(OS⊗Bj)}.(42)\nIn the subsequent discussion we assume that KandIare\ndefined through Eqs. (30) and (31)\nD. Application to the model\nAsit wasindicatedbyFisherandBreuer[23] thecorre-\nlated projection approach is most efficient if projections\non subspaces corresponding to some conserving quantity\nare considered. For the spin-star models the most ap-\npropriate conserving quantity is the z-projection of the\ntotal angular momentum of the total system. Explain-\ning the symmetries of the model under consideration we\ngeneralise the correlated projection operator as follows7\nPρ=/summationdisplay\njj/summationdisplay\nm=−jtrIB/parenleftbig\nΠ+\njmρ/parenrightbig\n⊗Π+\njm\nNj(43)\n+/summationdisplay\njj/summationdisplay\nm=−jtrIB/parenleftbig\nΠ−\njmρ/parenrightbig\n⊗Π−\njm\nNj,\nwhere Π±\njm=|±/an}bracketri}ht/an}bracketle{t±|⊗| j,m/an}bracketri}ht/an}bracketle{tj,m|.The|±/an}bracketri}htare eigenvec-\ntorsτzand|j,m/an}bracketri}htare eigenvectors of the bath operators\nJzandJ2. The corresponding eigenvalues are mand\nj(j+1). We introduce the notation\nNj= trΠ±\njm=/parenleftbiggN\nN\n2+j/parenrightbigg\n−/parenleftbiggN\nN\n2+j+1/parenrightbigg\n.(44)\nNaturally, the projection on the irrelevant part is defined\nas\nQρ=ρ−Pρ. (45)\nCombining Eqs. (43),(45) and (40) with the Hamilto-\nnian (1) and for the initial state (4), using (30) and (31),\nwegetasystemofdifferentialequationsforthequantities\nRjm= trIB/parenleftbig\nΠ+\njmρ/parenrightbig\n,andrjm= trIB/parenleftbig\nΠ−\njmρ/parenrightbig\n,namely\n˙rjm=−α2\n2Nb(j,−m)(rjm−Rjm−1)t (46)\n−γ2\n4(rjmσ+σ−+σ+σ−rjm−2σ+Rjmσ−)t+Λ1,\n˙Rjm=−α2\n2Nb(j,m)(Rjm−rjm+1)t (47)\n−γ2\n4(Rjmσ−σ++σ−σ+Rjm−2σ−rjmσ+)t+Λ2,\nwhereb(j,m) = (j−m)(j+m+1).The inhomogeneities\nin the above equation are defined as\nΛ1=/parenleftbigg\niγ\n2N+1(ρ0\n23−ρ0\n32) 0\n0 0/parenrightbigg\n, (48)\nΛ2=/parenleftbigg\n0 0\n0−iγ\n2N+1(ρ0\n23−ρ0\n32)/parenrightbigg\n. (49)\nThe initial conditions for Eqs. (46) and (47) are given\nby\nRjm(0) =Nj/2N/parenleftbigg\nρ0\n110\n0ρ0\n33/parenrightbigg\n, (50)\nrjm(0) =Nj/2N/parenleftbigg\nρ0\n220\n0ρ0\n44/parenrightbigg\n. (51)/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus\n0.0 0.2 0.4 0.6 0.8 1.00.40.50.60.70.80.91.0\n2ΓtΡ11S\nFIG. 7: Comparison of the dynamics of the population of\nthe upper state ρS\n11occurring in the TCL2 master equation\n(52)-(55) and the exact solution of the Schr¨ odinger equati on\n(19). For the initial states |Φ+\nπ/3,π/2/angbracketrightthe TCL2 solution is\nrepresented by the cross symbol and the exact solution by\nthe solid curve. For the initial state |Φ+\nπ/3,π/3/angbracketrightthe TCL2\nsolution is represented by the square symbol and the exact\nsolution by the dashed curve. For the initial state |Φ+\nπ/3,π/angbracketright\nthe TCL2 solution is represented by the triangle symbol and\nthe exact solution by the dotted curve. The coupling strengt h\nisα=γ/2; the number of spins in the bath is N= 20.\nFrom Eqs. (46) and (47) we find a closed system of\nequations for the matrix elements\n˙R11\njm−1=α2\n2Nb(j,−m)(r11\njm−R11\njm−1)t, (52)\n˙r11\njm=−α2\n2Nb(j,−m)(r11\njm−R11\njm−1)t (53)\n+γ2\n2(R22\njm−r11\njm)t+iγ\n2N+1(ρ0\n23−ρ0\n32),\n˙R22\njm= 2α2\nNb(j,m)(r22\njm+1−R22\njm)t (54)\n−γ2\n2(R22\njm−r11\njm)t−iγ\n2N+1(ρ0\n23−ρ0\n32),\n˙r22\njm+1=−2α2\nNb(j,m)(r22\njm+1−R22\njm)t. (55)\nWe notice that ˙R11\njm−1+ ˙r11\njm+˙R22\njm+ ˙r22\njm+1= 0.This\nis a consequence of the choice of the projection superop-\nerator (Eq. (43)) as a projection on a conserved quantity\nof the total system, i.e., the total angular momentum.\nThe populations can be expressed through the projec-\ntionsrandR,as\nρS\n11=/summationdisplay\njj/summationdisplay\nm=−j(R11\njm+r11\njm), (56)\nρS\n22=/summationdisplay\njj/summationdisplay\nm=−j(R22\njm+r22\njm). (57)\nThe system (52) can be easily solved numerically. We\nuse the convention rj|j+1|=Rj|j+1|= 0.The TCL2 dy-\nnamics of the probability to find the spin in the excited\nstate following from Eqs. (52)-(55) is shown in Fig. 7.8\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus\n0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.50.6\n2ΓtΡ11S\nFIG. 8: Comparison of the dynamics of the population of the\nupper state ρS\n11occurring in the TCL2 master equation (52)-\n(55) and the exact solution of the Schr¨ odinger equation (19 ).\nForα=γ/2 the TCL2 solution is represented by the cross\nsymbol and the exact solution by the solid curve. For α= 2γ\nthe TCL2 solution is represented by the square symbol and\nthe exact solution by the dashed curve. For α= 10γthe\nTCL2 solution is represented by the triangle symbol and the\nexact solution by the dotted curve. The initial state is the\nBell state |Φ+\nπ,0/angbracketright; a number of spins in the bath N= 20.\n/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus/Plus\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square\n0 2 4 6 8 100.00.20.40.60.8\n2ΓtΡ11S\nFIG. 9: Comparison of the long-time dynamics of the popu-\nlation of the upper state ρS\n11occurring in the TCL2 master\nequation (52)-(55) and the exact solution of the Schr¨ oding er\nequation (19). For α=γ/2 the TCL2 solution is repre-\nsented by the cross symbol and the exact solution by the\nsolid curve. For α= 2γthe TCL2 solution is represented\nby the square symbol and the exact solution by the dashed\ncurve. For α= 10γthe TCL2 solution is represented by the\ntriangle symbol and the exact solution by the dotted curve.\nThe initial state is the Bell state |Φ+\nπ,0/angbracketright; a number of spins in\nthe bath N= 20.\nV. RESULTS AND DISCUSSION\nThe exact dynamics of the spin is analysed in Figures\n2 to 6. Figs. 2 and 3 show the population of the upper\nstate of the spin in dependence of the strength of the\ninteraction between the bath spins and the intermediate\nspin. Figure 2 shows the dynamics for limiting cases.\nWhen the strength of the interaction is small, α≪γ,\nthe influence of the bath on the spin dynamics is veryweak and the spin evolution is very similar to the case\nα= 0.For the case α≫γthe population oscillates\nnear the initial state with small amplitude. The most\ninteresting case is α∼γ: for this domain we observe a\nstrong dependence on the value of α(see Figs. 2 and 3).\nThe dynamics of the population of the upper state for\ndifferent numbers of bath spins is shown in Figure 4.\nWe can see that the dynamics of the spin depends very\nweakly on the number of spins in the bath. Already for\n50 spins a minimal difference from the thermodynamic\nlimit can be observed.\nFig. 5 shows the strong dependence on the initial cor-\nrelation between the reduced spin and the intermediate\nspin. The initial state of the total system is given by Eq.\n(4) with ρSI(0) taken as correlated Bell-like pure states\n|Φ+\nπ/3,β/an}bracketri}ht(Eq. (7)). The different curves in Fig. 4 corre-\nspond to different values ofthe initial phase parameter β.\nFollowing the explicit expressions for the reduced density\nmatrix in the thermodynamics limit, Eq. (27), it is clear\nthat for the model considered here initial system-bath\ncorrelations affect the dynamics of the reduced system\nonly ifℑρ23(0)/ne}ationslash= 0. For the initial state analysed in Fig.\n5 this corresponds to the case β/ne}ationslash=πn(n∈Z).\nFrom Fig. 5 one can clearly see that the initial correla-\ntions give a non-negligible contribution to the dynamics\nof the reduced system for all times. The dependence of\nthe dynamics of the reduced system on the initial system\nbath correlations in the thermodynamic limit is analysed\nin Fig. 6. The initial state of the total system is given by\nEq. (4) with ρSI(0) chosen to be correlatedBell-likepure\nstate|Φ−\nπ/3,β/an}bracketri}ht(Eq. (7)). Similar to the case presented in\nFig. 5, one can see that initial system-bath correlations\nplay an important role in the dynamics of the reduced\nsystem.\nUsing the explicit formula for the density matrix for\nthe reduced system in the thermodynamic limit, Eq.\n(27), one can explicitly analyse limiting cases for the ra-\ntio of coupling strengths α/γ. From Eq. (27) one can\nsee that an influence of the bath on the dynamics of the\nspin is described by the functions f(t) andg(t). In the\ncaseα≪γthe expansion of the functions f(t) andg(t)\nreads,\nf(t)≈sin2γt\n4+α2\nγ2/parenleftbigg\n−2sin2γt\n4+γt\n4sinγt\n2/parenrightbigg\n+O/parenleftbiggα4\nγ4/parenrightbigg\nand\ng(t)≈1\n2exp/parenleftig\n−α2t2\n8/parenrightig/parenleftbig\nsinγt\n2+\nα2\n4γ2/parenleftig\nγtcosγt\n2+/parenleftig\nγ2t2\n4−2/parenrightig\nsinγt\n2/parenrightig\n+O/parenleftig\nα4\nγ4/parenrightig/parenrightig\n.\nFrom the expansion for the functions f(t) andg(t) one\ncan see that in the case α≪γthe dynamics is domi-\nnated by the unitary oscillations with the frequency γ/2\nwhich corresponds to unitary evolution of the spin and\nintermediate spin.\nIn the other limiting case, α≫γ, one obtains that\nf(t) =g(t) = 0, which explicitly shows the limitation of9\nthe equation for the thermodynamic limit given by the\nEq. (26). As one can see from Fig. 2 in the case α≫γ\n(Fig. 2 dotted curve) the system is very weakly coupled\nto an environment and in the limit α/γ→ ∞remains in\nits initial state.\nThe limitation of Eq. (26) follows from the asymp-\ntotic character of this equation. This equation gives the\ncorrect thermodynamic limit only if the constant of the\nsystem bath interaction is small with respect to all the\nother characteristic parameters of the total system (like\nin the case α≪γ).\nFigures 7-9 show the dynamics of the population of\nthe upper state. In order to benchmark the approxima-\ntion technique the dynamics of this observable is derived\nfrom the solution of the TCL2 master equation with cor-\nrelated projection operator (Eq. (46) and Eq. (47)) and\nthe exact solution of the Schr¨ odinger equation (19). The\ncomparison is performed for different initial conditions,\nparameters α, γand different time frames. From Fig.\n7 one can see that the TCL2 approximation technique\ngives good results for sufficiently large time. A deviation\nfrom the exact solution is not exceeded by 5% for time\nscale of 1 /γ. In Fig. 8 the dependence on the coupling\nstrength γ/αis analysed. One can see that the correla-\ntion projection operator technique gives good agreement\nforγ/α/greaterorsimilar1. The last inequality can be understood if\ntake into account that the equations are perturbative\nboth in alpha and in gamma. The relevant order of the\nexpansion is the maximal value of two constants αorγ.\nHowever, the evolution of the relevant system defines γ.\nFor this reason, the most accurate approximation of the\ndynamics is obtained for α/lessorsimilarγ.\nFig 9. addresses the long time behaviour of the re-\nduced system dynamics. It is clear that for all ranges\nof the parameters γ/αconsidered here the approxima-\ntion technique shows convergence to a false equilibrium\nvalue. However, this behaviour has no correlation with\nthe dynamics described by the exact solution. From Fig.\n9 one can also see that increasing the ratio γ/α, the dis-\ncrepancybetween approximationtechnique and exact so-\nlution is growing. A more adequate description of the\nlong time dynamics and parameter ratios γ/α >1 might\nrequire higher orders in the TCL expansion.The fact that the TCL2 master equation with corre-\nlatedprojectionoperatorgivesthe satisfactoryresultsfor\ntime-scales of the order 1 /γindicates that this approxi-\nmationtechniquefitsthedescriptionofspin-bathsystems\nmuch better than the traditional form of the projection\noperatorin the form(32). Asit wasindicated inRef. [16]\na TCL2 master equation for spin systems gives adequate\nresults only for time scales of the order of 0 .05/γ.\nForthesituationconsideredinthispaperifonechooses\na projection operator in the form Pρ= (tr IBρ)⊗\nIN+1/2N+1, whereNis number of bath spins. The cor-\nresponding master equation reads\n˙ρ=−γ2t/4/parenleftbig\nσ−σ+ρ+ρσ−σ+−2σ−ρσ++h.c./parenrightbig\n.\nIt is clear that the above equation does not contain any\ninformation about system-bath correlation and cannot\ngive an adequate description of the reduced system dy-\nnamics.\nIn conclusion, we have found an exact solution for a\nsimple spin system coupled to a spin bath through an\nintermediate spin. We have studied the dynamics of the\nsystem and have shown that the initial correlations be-\ntween the spin and the intermediate spin have a strong\ninfluenceonthedynamicsofthespin. Ontheotherhand,\nthe dynamics of the spin are weakly dependent on the\nnumber of bath spins. In addition to the exact solution,\nan approximate TCL2 master equation was derived with\nthe help of the correlation projection operator technique.\nThe derived equation explicitly takes into account initial\ncorrelations between the spin and the intermediate spin.\nThe solution of the approximate master equation was\ncompared with the exact solution. It was shown that the\napproximate technique gives good results for short time\ndynamics.\nAcknowledgments\nThis work is based upon research supported by the\nSouth African Research Chair Initiative of the Depart-\nment of Science and Technology and National Research\nFoundation.\n[1] H.-P. Breuer and F. Petruccione, The Theory of Open\nQuantum Systems (Oxford University Press, 2002).\n[2] J.B. Parkinson, D.J.J. Farnell, An Introduction to Quan-\ntum Spin Systems, Lect. Notes Phys. 816 (Springer,\nBerlin Heidelberg, 2010).\n[3] A. Nielsen and I.L. Chuang, Quantum Computation\nand Quantum Information (Cambridge University Press,\nCambridge, 2000).\n[4] I. Sinayskiy, A. Marais, F. Petruccione and A. Ekert,\nPhys. Rev. Lett 108, 020602 (2012).\n[5] R. Hanson et al, Rev. Mod. Phys., 79, 1217 (2007).\n[6] M. D. Petrovi´ c and N. Vukmirovi´ c, Phys. Rev. B 85,195311 (2012).\n[7] M. Borhani and X. Hu, Phys. Rev. B 85, 125132 (2012)\n[8] W. Zhang, N. Konstantinidis, K. Al-Hassanieh, and V.\nV. Dobrovitski, J. Phys.: Cond. Mat. 19083202 (2007).\n[9] C. W.Gardiner, QuantumNoise(Springer, Berlin, 1991).\n[10] D. F. Walls and G. J. Milburn, Quantum Optics\n(Springer-Verlag, Berlin, 2008, 2nd ed.).\n[11] Y. Hamdouni, M. Fannes, andF.Petruccione, Phys. Rev.\nB73, 245323 (2006).\n[12] Z. Huang, G. Sadiek, and S. Kais, J. Chem. Phys. 124,\n144513 (2006).\n[13] D. D. Bhaktavatsala Rao, V. Ravishankar, and V. Sub-10\nrahmanyam, Phys. Rev. A 74, 022301 (2006).\n[14] X. Z. Yuan, H. S. Goan, and K. D. Zhu, Phys. Rev. B\n75, 045331 (2007).\n[15] E. Ferraro, H.P. Breuer, A. Napoli, M. A. Jivulescu and\nA. Messina, Phys. Rev. B 78, 064309 (2008)\n[16] H.-P. Breuer, D. Burgarth, F. Petruccione, Phys. Rev. B\n70, 045323 (2004).\n[17] B.L. Hu, J.P. Paz and Y. Zhang, Phys. Rev. D 45, 2843\n(1992).\n[18] B.M. Garraway, Phys. Rev. A 55, 2290 (1997).\n[19] W.G. Unruh, Phys. Rev. A 51, 992 (1995).\n[20] G.M. Palma, K.A. Suominen and A.K. Ekert Proc. R.\nSoc. Lond., A452, 567 (1996).\n[21] N.V. Prokofiev and P.C.E. Stamp, Rep. Prog. Phys. 63,\n669 (2000).\n[22] N. Arshed, A.H. Toor, D.A. Lidar, Phys. Rev. A. 81,\n062353 (2010)\n[23] J. Fischer, H.-P. Breuer, Phys. Rev. A 76, 052119 (2007).\n[24] H.-P. Breuer, Phys. Rev. A 75, 022103 (2007).\n[25] H.-P. Breuer, J. Gemmer, M. Michel, Phys. Rev. E 73,\n016139 (2006).\n[26] E. Barnes, L. Cywinski, S. DasSarma Phys. Rev. B 84,\n155315 (2011).\n[27] M. Michel, R. Steinigeweg, H. Weimer, Eur. Phys. J.Special Topics 151, 13 (2007).\n[28] A. Smirne, H.-P. Breuer, J. Piilo, B. Vacchini, Phys. Re v.\nA82, 062114 (2010).\n[29] M. Bana, S. Kitajima, F. Shibataa, Physics Letters A\n375, 24, 2283 (2011).\n[30] C. Uchiyama, Phys. Rev. A 85, 052104 (2012).\n[31] V. G. Morozov, S. Mathey and G.R¨ opke, Phys. Rev. A\n85, 022101 (2012).\n[32] H.-T. Tan and W.-M. Zhang, Phys. Rev. A 83, 032102\n(2011).\n[33] Y. Hamdouni, F. Petruccione, Phys. Rev. B 76, 174306\n(2007).\n[34] M. Richter, A. Knorr, Ann. Phys. 325, 711 (2010).\n[35] S. Nakajim, Prog. Theor. Phys. 20, 948 (1958).\n[36] R. Zwanzig, J. Chem. Phys. 33, 1338 (1960).\n[37] H. Grabert, Projection Operator Techniques in Nonequi-\nlibrium Statistical Mechanis , (Vol. 95 of Springer Tracts\nin Mod. Phys., Springer-Verlag, Berlin, 1982)\n[38] R.Kubo, M. TodaandN.Hashitsume, Statistical Physics\nII. Non-equlibrium Statistical Mechanics , (Springer-\nVerlag, Berlin, 1985)\n[39] I. Prigogine, Non-equilibrium Statistical Mechanics (In-\nterscience Publishers, New York, 1962)"    },    {        "title": "1604.05810v2.Second_post_Newtonian_Lagrangian_dynamics_of_spinning_compact_binaries.pdf",        "content": "arXiv:1604.05810v2  [gr-qc]  22 Apr 2016Second post-Newtonian Lagrangian dynamics of spinning com pact binaries\nLi Huang and Xin Wu∗\nDepartment of Physics &Institute of Astronomy, Nanchang University, Nanchang 330 031, China\nThe leading-order spin-orbit coupling is included in a post -Newtonian Lagrangian formulation of\nspinning compact binaries, which consists of the Newtonian term, first post-Newtonian (1PN) and\n2PN non-spin terms and 2PN spin-spin coupling. This makes a 3 PN spin-spin coupling occur in the\nderived Hamiltonian. The spin-spin couplings are mainly re sponsible for chaos in the Hamiltonians.\nHowever, the 3PN spin-spin Hamiltonian is small and has diffe rent signs, compared with the 2PN\nspin-spin Hamiltonian equivalent to the 2PN spin-spin Lagr angian. As a result, the probability of\nthe occurrence of chaos in the Lagrangian formulation witho ut the spin-orbit coupling is larger than\nthat in the Lagrangian formulation with the spin-orbit coup ling. Numerical evidences support the\nclaim.\nPACS numbers: 04.25.Nx, 04.25.-g, 05.45.-a, 45.20.Jj\nI. INTRODUCTION\nOn February 11, 2016, the LIGO Scientific Collabora-\ntion and Virgo Collaboration announced the detection of\ngravitational wave signals (GW150914), sent out from\nthe inspiral and merger of a pair of black holes with\nmasses 36 M⊙and 29M⊙[1]. The gravitational wave dis-\ncovery directly confirmed a major prediction of Albert\nEinstein’s 1915 general theory of relativity. The LIGO\nprojectwasoriginallyproposedinthe1980sanditsinitial\nfunding was approved in 1992. Since then, the dynamics\nofspinningcompactbinarieshasreceivedmoreattention.\nA precise theoretical waveform template is necessary to\nmatchwithgravitationalwavedata. Asakindofdescrip-\ntionofthewaveformsanddynamicalevolutionequations,\nthe post-Newtonian (PN) approximation to general rel-\nativity was used. Up to now, the PN expansion of the\nrelativistic spinning two-body problem has provided the\ndynamicalnon-spinevolutionequationsandthe spinevo-\nlution equations to fourth post-Newtonian (4PN) order\n[2-7].\nA key feature of the gravitational waveforms from a\nchaotic system is the extreme sensitivity to initial condi-\ntions [8-10], and therefore the chaos is a possible obstacle\nto the method of matched filtering. For this reason, sev-\neral authors were interested in the presence or absence\nof chaotic behavior in the orbital dynamics of spinning\nblack hole pairs [11-17]. There were three debates about\nthis topic.\nSixteen years ago fractal methods were used to show\nthat the 2PN harmonic-coordinates Lagrangian formula-\ntion of two spinning black holes admits chaotic behavior\n[11]. Here the Lagrangian includes contributions from\nthe Newtonian, 1PN and 2PN non-spin terms and the ef-\nfects of spin-orbit and spin-spin couplings.[41] However,\nthe authors of [12] made an opposite claim on ruling out\nchaos in compact binary systems by finding no positive\nLyapunov exponents along the fractal of [11]. The au-\n∗Electronic address: xwu@ncu.edu.cnthors of [18, 19] refuted this claim by finding positive\nLyapunovexponents, and pointed out that the reasonfor\nthe false Lyapunov exponents obtained in Ref. [12] lies\nin using the Cartesian distance between two nearby tra-\njectories by continually rescaling the shadow trajectory.\nThis is a debate with respect to Lyapunov exponents re-\nsulting in two different claims on the chaotic behavior\nof comparable mass spinning binaries. In fact, the true\nreason for the discrepancy was found in Ref. [20] and\nshould be that different treatments of the Lyapunov ex-\nponents were given in the three papers [12, 18, 19]. The\nauthors of [12] used the stabilizing limit values as the\nvalues of Lyapunov exponents, whereas those of [18, 19]\nused the slopes of the fit lines. It is clear that obtaining\nthe limit values requires more CPU times than obtaining\nthe slopes.\nSecond debate is different descriptions of chaotic re-\ngions and parameter spaces. It was reported in Ref.\n[21] that increasing the spin magnitudes and misalign-\nments leads to the transition to chaos, and the strength\nof chaos is the largest for the spins perpendicular to the\norbital angular momentum. However, an entirely differ-\nent description from Ref. [22] is that chaosoccursmainly\nwhen the initial spins are nearly antialigned with the or-\nbital angular momentum for the binary configuration of\nmasses 10 M⊙and 10M⊙. These descriptions seem to\nbe apparently conflicting, but they can all be correct,\nas mentioned in Ref. [23]. This is because a compli-\ncatedcombinationofallparametersandinitialconditions\nrather than single physical parameter or initial condition\nis responsible for yielding chaos. No universal rule can\nbe given to dependence of chaos on each parameter or\ninitial condition.\nThird debate relates to different dynamical behav-\niors of PN Lagrangian and Hamiltonian conservative\nsystems at the same order. In other words, the de-\nbate corresponds to a question whether the two formu-\nlations are equivalent. The equivalence of Arnowitt-\nDeser-Misner (ADM) Hamiltonian and harmonic coor-\ndinate Lagrangian approaches at 3PN order were proved\nby two groups [24, 25]. This result is still correct for ap-\nproximation accuracy to next-to-next-to-leading (4PN)2\norder spin1-spin2 couplings [4]. Recently, the authors\nof [26] resurveyed this question. In their opinion, this\nequivalence means that all the known results of the\nADM Hamiltonian approach can be transferred to the\nharmonic-coordinates Lagrangian one, or those of the\nharmonic-coordinatesLagrangianapproachcanbe trans-\nferred to the ADM Hamiltonian one. In fact, the two\napproaches are not exactly equal but are approximately\nrelated. In this case, dynamical differences between the\ntwoapproachesarenotavoided. Foratwo-blackholesys-\ntem with twobodies spinning, the PN Lagrangianformu-\nlation of the Newtonian term and the 1.5PN spin-orbit\ncoupling is non-integrable and possibly chaotic, whereas\nthe PN Hamiltonian formulation of the Newtonian term\nand the 1.5PN spin-orbit coupling is integrable and non-\nchaotic [26]. This is due to the difference between the\ntwo formulations, 3PN spin-spin coupling. That is, the\nLagrangian is exactly equivalent to the Hamiltonian plus\nthe 3PN spin-spin term. As a result, the 3PN spin-spin\neffect leads to the non-integrability of the Hamiltonian\nequivalent to the Lagrangian. Ten years ago, similar\nclaims were given to the two PN formulations of the\ntwo-black hole system with one bodies spinning [21, 27-\n29]. An acute debate has arisen as to whether the com-\npact binaries of one bodies spinning exhibit chaotic be-\nhaviour. A key to this question in Ref. [30] is as follows.\nThe construction of canonical, conjugate spin variables\nin Ref. [31] showed directly that four integrals of the\ntotal energy and total angular momentum in an eight-\ndimensional phase space of a conservative Hamiltonian\nequivalent to the Lagrangian determine the integrability\nof the Lagrangian. Precisely speaking, no chaos occurs\nin the PN conservative Lagrangian and Hamiltonian for-\nmulations of comparable mass compact binaries with one\nbody spinning.\nOne of the main results of [26, 32] is that a PN La-\ngrangian approach at a certain order always exists a for-\nmal equivalent PN Hamiltonian. This is helpful for us\nto study the Lagrangian dynamics using the Hamilto-\nnian dynamics. Following this direction, we shall re-\nvisit the 2PN ADM Lagrangian dynamics of two spin-\nning black holes, in which the Newtonian, 1PN and 2PN\nnon-spin terms and the 1.5PN spin-orbit and 2PN spin-\nspin contributions are included. A comparison between\nthe Lagrangian and related Hamiltonian dynamics will\nbe made, and a question of how the orbit-spin coupling\nexerts an influence on chaos resulted from the spin-spin\ncoupling in the Lagrangianwill be particularlydiscussed.\nThe present investigation is unlike the work [11], where\nthe onset of chaos in the 2PN Lagrangian formulation\nwas mainly shown. It is also unlike the paper [33], where\nthe effect of the orbit-spin coupling on the strength of\nchaos caused by the spin-spin coupling was considered\nbut the 1PN and 2PN non-spin terms were not included\nin the Lagrangian.\nIn our numerical computations, the velocity of light\ncand the constant of gravitation Gare taken as ge-\nometrized units, c=G= 1.II. POST-NEWTONIAN APPROACHES\nSuppose that compact binaries have masses M1and\nM2, and then their total mass is M=M1+M2. Other\nparameters are mass ratio β=M1/M2, reduced mass\nµ=M1M2/Mand mass parameter η=µ/M=β/(1+\nβ)2. In ADM center-of-mass coordinate system, ris a\nrelative position of body 1 to body 2, and vis a veloc-\nity of body 1 relative to the center of mass. Let unit\nradial vector be n=r/rwith radius r=|r|. The evolu-\ntion of spinless compact binaries can be described by the\nfollowing PN Lagrangian formulation\nL0=LN+1\nc2L1PN+1\nc4L2PN. (1)\nThe above three parts are the Newtonian term LN, first\norder PN contribution L1PNand second order PN con-\ntribution L2PN. They are expressed in Ref. [34] as\nLN=1\nr+v2\n2, (2)\nL1PN=1\n8(1−3η)v4+1\n2[(3+η)v2+η(n·v)2]1\nr\n−1\n2r2, (3)\nL2PN=1\n16(1−7η+13η2)v6+1\n8[(7−12η−9η2)v4\n+(4−10η)η(n·v)2v2+3η2(n·v)4]1\nr\n+1\n2[(4−2η+η2)v2+3η(1+η)(n·v)2]1\nr2\n+1\n4(1+3η)1\nr3. (4)\nIn fact, these dimensionless equations deal with the use\nof scale transformation: r→GMr,t→GMtandL0→\nµL0. In terms of Legendre transformation H0=p·v−\nL0with momenta p=∂L/∂v, we have the following\nHamiltonian\nH0=HN+1\nc2H1PN+1\nc4H2PN, (5)\nwhere these sub-Hamiltonians are\nHN=p2\n2−1\nr, (6)\nH1PN=1\n8(3η−1)p4−1\n2[(3+η)p2+η(n·p)2]1\nr\n+1\n2r2, (7)\nH2PN=1\n16(1−5η+5η2)p6+1\n8[(5−20η−3η2)p4\n−2η2(n·p)2p2−3η2(n·p)4]1\nr+1\n2[(5+8η)\np2+3η(n·p)2]1\nr2−1\n4(1+3η)1\nr3.(8)3\nThe two Hamiltonians H1PNandH2PNare the results\nof [34]. Besides them, other higher-order PN terms can\nbe derived from the Lagrangian L0. For example, a third\norder PN sub-Hamiltonian was given in [26] by\nH3PN=3\n16(−η+7η2−12η3)p8+1\n8r[(2−7η+3η2\n+30η3)p6+(4η−11η2+36η3)(n·p)2p4\n+6(η2−3η3)(n·p)4p2]+1\n4r2[(5+18η\n+21η2−9η3)p4+4(5η2+2η3)(n·p)4\n+(14η−7η2−27η3)(n·p)2p2]\n+1\n2r3[(−3−31η−7η2+η3)p2\n+(−η−η2+7η3)(n·p)2]. (9)\nIt is obtained from coupling of the 1PN term L1PNand\nthe 2PN term L2PN. Since the difference between the\n2PN Lagrangian L0and the 2PN Hamiltonian H0is at\nleast 3PN order, the two PN approaches are not exactly\nequivalent. Clearly, a Hamiltonian that is equivalent to\nthe 2PN Lagrangian[42] cannot be at second order but\nshould be at a higher enough order or an infinite order.\nThis is one of the main results of [26].\nWhen the two bodies spin, some spin effects should be\nconsidered. Now, the leading-orderspin-spincouplingin-\nteraction L2ss[35], as one kind of spin effect, is included\nin the Lagrangian L0. In this sense, the obtained La-\ngrangian becomes\nL1=L0+L2ss, (10)\nwhere\nL2ss=−1\n2r3[3\nr2(S0·r)2−S02],(11)\nS= (2+3\n2β)S1+(2+3β\n2)S2,\nS0= (1+1\nβ)S1+(1+β)S2.\nNote that each spin variable is dimensionless, namely,\nSi=SiˆSiwith spin magnitude Si=χiMi2/M2(0≤\nχi≤1)and3-dimensionalunitvector ˆSi. Because L2ssis\nindependent of velocity, it does not couple the 1PN term\nL1PNor the 2PN term L2PNvia the Legendre trans-\nformation. It is only converted to a spin-spin coupling\nHamiltonian\nH2ss=−L2ss. (12)\nAdding this term to the Hamiltonian H0, we have the\nfollowing Hamiltonian\nH1=H0+H2ss. (13)\nWhen another kind of spin effect, the leading-order\nspin-orbit coupling L1.5so[35], is further included in theabove-mentionedLagrangian L1, we obtain a Lagrangian\nformulation as follows:\nL2=L1+L1.5so, (14)\nL1.5so=−1\nr3S·(r×v). (15)\nUnlikeL2ss,L1.5sodepends on the velocity. Therefore,\nthe Legendre transformation results in the appearance of\nthe leading-orderspin-orbit Hamiltonian H1.5soobtained\nfrom the coupling of the Newtonian term LNand the\nspin-orbit term L1.5so, the next-order spin-orbit Hamil-\ntonianH2.5soobtainedfromthecouplingofthe1PNterm\nL1PNand the spin-orbit term L1.5soand the next-order\nspin-spinHamiltonian H3ssobtainedfromthecouplingof\nthe leading-order spin-orbit term L1.5soand itself. These\nHamiltonians are written in [26] as\nH1.5so=−L1.5so|v→p, (16)\nH2.5so=1\nr3(3η−1\n2p2−3+η\nr)S·(r×p),(17)\nH3ss=1\n2r6[r2S2−(S×r)2]. (18)\nWe take three Hamiltonians:\nH2=H1+H1.5so, (19)\nH3=H2+H2.5so+H3PN, (20)\nH4=H3+H3ss. (21)\nForthethreeHamiltonians, H4isthe bestapproximation\nto the Lagrangian L2although the former is not exactly\nequivalent to the latter. As an important point to note,\nL2exhibits chaos when the two objects spin and the spin\neffects are L1.5sobut are not L2ss. This is because many\nhigher-order spin-spin couplings (such as H3ss), included\nin a higher enough order Hamiltonian equivalent to L2\n(with its equations of motion to another higher enough\norder), make this equivalent Hamiltonian non-integrable\n[26]. However,anyPNconservativeLagrangianapproach\nis always integrable and non-chaotic when only one body\nspins and the spin effects are not restricted to the spin-\norbit couplings. This is due to integrability of the equiv-\nalent Hamiltonian [30, 31].\nClearly, the above-mentioned same PN order La-\ngrangian and Hamiltonian approaches (e.g. L2andH2)\naretypically nonequivalent, and thereforethey both have\ndifferent dynamics to a large extent. Now, we are mainly\ninterested in knowing how the spin-orbit coupling L1.5so\naffects the chaotic behavior yielded by the spin-spin cou-\nplingL2ssin the above Lagrangians. In other words,\nwe should compare differences between the L1andL2\ndynamics. Considering that the 2PN spin-spin coupling\nH2ssequivalent to L2ssand the 3PN spin-spin coupling\nH3ssassociated to L1.5soplay an important role in the\nonset of chaos in the Hamiltonians, we should also focus\non differences between the H3andH4dynamics. These\ndiscussions await the following numerical simulations.4\nIII. NUMERICAL COMPARISONS\nLet us take initial conditions r(0) = (17 .04,0,0),\nv(0) = (0,0.094,0), dynamical parameters χ1=χ2= 1,\nβ= 0.79, and initial unit spin vectors\nˆS1= (0.1,0.3,0.8)//radicalbig\n0.12+0.32+0.82,\nˆS2= (0.7,0.3,0.1)//radicalbig\n0.72+0.32+0.12.\nAn eighth- and ninth-order Runge-Kutta-Fehlberg algo-\nrithm of variable step sizes [RKF8(9)] is used to solve the\nrelated Lagrangian and Hamiltonian systems. This algo-\nrithm is highly precise. In fact, it gives an order of 10−11\nto the accuracy of the Lagrangian L2and an order of\n10−12to the accuracy of the Hamiltonian H2, as shown\nin Fig. 1. Here, the errors of L2andH2were estimated\naccording to two different integration paths. The error\nofL2, i.e., the error of H2, was obtained by applying the\nRKF8(9) to solve the 2PN Lagrangian equations of L2,\nwhile the errorof H2was given by applying the RKF8(9)\nto solvethe 2PNHamiltonian equations of H2. Although\ntheerrorshavesecularchanges,theyaresosmallthatthe\nobtained numerical results should be reliable.\nThe evolution of the orbit in Fig. 2 demonstrates that\nthesameorderLagrangianandHamiltonianformulations\nL2andH2diverge quickly from the same starting point.\nThis supports again the general result of [26] on the non-\nequivalence of the PN Lagrangian and Hamiltonian ap-\nproaches at the same order.\nFor the given orbit, we investigate dynamical differ-\nences among some Lagrangian and Hamiltonian formu-\nlations using several methods to find chaos.\nA. Chaos indicators\nA power spectral analysis method is based on Fourier\ntransformation and gives the distribution of frequencies\nto a certain time series. It can roughly detect chaos from\norder. Discrete frequencies are usually regardedas power\nspectra of regular orbits, whereas continuous frequencies\nare generally referred to as power spectra of chaotic or-\nbits. In light of this criterion, distributions of continuous\nfrequency spectra in Figs. 3(a) and (d)-(f) seem to show\nthe chaoticity of the Lagrangian L1and the Hamiltoni-\nansH2,H3andH4. On the other hand, distributions of\ndiscrete frequency spectra in Figs. 3(b) and (c) describe\ntheregularityoftheHamiltonians H1andtheLagrangian\nL2. It is sufficiently proved that the same PN order La-\ngrangian and Hamiltonian approaches ( L1andH1,L2\nandH2)havedifferentdynamics. Itisworthemphasizing\nthat the method of power spectra is ambiguous to differ-\nentiate among complicated periodic orbits, quasiperiodic\norbitsandweaklychaoticorbits. Therefore,morereliable\nqualitative methods are necessarily used.\nAs a common method to distinguish chaos from or-\nder, a Lyapunov exponent characterizes the average ex-\nponential deviation of two nearby orbits. The varia-tional method and the two-particle one are two algo-\nrithms for calculating the Lyapunov exponent [36, 37].\nAlthough the latter method is less rigorous than the for-\nmer method, its application to a complicated system is\nmore convenient. For the use of the two-particle method,\nthe principal Lyapunov exponent is defined as\nλ= lim\nt→∞1\ntlnd(t)\nd(0), (22)\nwhered(0) andd(t) are separations between two neigh-\nboring orbits at times 0 and t, respectively. The best\nchoice of the initial separation d(0) is an order of 10−8\nunder the circumstance of double precision [36]. In addi-\ntion, avoiding saturation of orbits needs renormalization.\nA global stable system[43] is chaotic if λreaches a stabi-\nlizing positive value, whereas it is ordered when λtends\nto zero. In terms of this, it can be seen clearly from Fig.\n4 that the four approaches L1,H2,H3andH4with pos-\nitive Lyapunov exponents are chaotic, and the two for-\nmulations H1andL2with zero Lyapunov exponents are\nregular. NotethateachofthesevaluesofLyapunovexpo-\nnents is given after integration time 2 ×105. Obtaining\na reliable stabilizing limit value of Lyapunov exponent\nusually costs an extremely expensive computation [20].\nA quicker method to find chaos is a fast Lyapunov\nindicator (FLI) [38, 39]. It was originally calculated us-\ning the length of a tangential vector and renormalization\nis unnecessary. Similar to the Lyapunov exponent, this\nindicator can be further developed with the separation\nbetween two nearby trajectories [40]. The modified indi-\ncator is of the form\nFLI = log10d(t)\nd(0). (23)\nAn appropriate choice for renormalization within a rea-\nsonable amount of time span is important. See Ref. [40]\nformoredetailsofthisindicator. TheFLIincreasesexpo-\nnentially with time for a chaotic orbit, but algebraically\nfor a regular orbit. The completely different time rates\nare used to distinguish between the two cases. Based on\nthis point, the dynamical behaviors of the six approaches\nL1,L2,H1,H2,H3andH4can be described by the FLIs\nin Fig. 5. These results are the same as those given by\nthe Lyapunov exponents in Fig. 4. Here each FLI was\nobtainedafter t= 5×104. In this sense, the FLI isindeed\na faster method to identify chaos than the Lyapunov ex-\nponent. Because of this advantage, the method of FLIs\nis widely used to sketch out the global structure of phase\nspaceorto providesomeinsight into dependence ofchaos\non single physical parameter or initial condition [23].\nB. Effects of varying the mass ratio on chaos\nFixing the above initial conditions, initial spin config-\nurations and spin parameters ( χ1andχ2), we let the\nmass ratio βrun from 0 to 1 in increments of 0.01. For5\nTABLE I: Chaotic parameter spaces of each approach\nApproach Mass Ratio β\nL1 0.05, 0.06, 0.08, [0.11,0.16], [0.18,0.83],\n0.86, 0.86, [0.89,0.92], 0.94\nH1 [0.12,0.19], [0.23,0.27], 0.30,\n[0.33,0.34], [0.40,0.42], [0.62,0.64]\nL2 0.03, 0.05, [0.09,0.41], [0.43,0.45], 0.47, 0.54\nH2 0.09, [0.17,0.22], [0.24,0.99]\nH3 [0.09,0.22], [0.24,0.65], [0.67,0.97]\nH4 [0.09,0.10], [0.15,0.17], 0.19, 0.48, 0.62, 0.65,\n0.68, 0.75, 0.79, 0.82, [0.84,0.91], 0.93\neach value of β, FLI is obtained after integration time\nt= 5×104. In this way, we plot Fig. 6 in which de-\npendence of FLI for each PN approach on βis described.\nIt is found through a number of numerical tests that 7.5\nis a threshold of FLI to distinguish between regular and\nchaoticorbits. A globalstable orbit is chaoticif its FLI is\nlargerthanthe threshold, but orderedifits FLI issmaller\nthanthethreshold. Inthissense, this figureshowsclearly\nthe correspondence between the mass ratio and the or-\nbital dynamics.\nIt is shown sufficiently that the PN Lagrangian and\nHamiltonian approximations at the same order, ( L1,H1)\nand (L2,H2), have different dynamical behaviors in sig-\nnificant measure. More details on the related differ-\nences are listed in Table 1. It can also be seen that\nthe chaotic parameter space of L1is larger than that\nofL2. That means that the spin-orbit coupling L1.5so\nmakes many chaotic orbits in the system L1evolve into\nordered orbits.[44] In other words, the probability of the\noccurrence of chaos in the system L1without L1.5sois\nlarge, but that in the system L2withL1.5sois small.\nThus,L1.5soplays an important role in weakening or\nsuppressing the chaotic behaviors yielded by L2ss. Here,\nthe chaotic behaviors weakened (or suppressed) do not\nmeanthatanindividualorbitmustbecomefromstrongly\nchaotic to weakly chaotic (or from chaotic to nonchaotic)\nwhenL1.5sois included in L1. This orbit may become\nmore strongly chaotic, or may vary from order to chaos.\nFor example, β= 0.03 in Table 1 corresponds to the\nregularity of L1but the chaoticity of L2. In fact, the\nchaotic behaviors weakened or suppressed mean decreas-\ning the chaotic parameter space. The result obtained in\nthe generalcasewith the PNterms L1PNandL2PNis an\nextension to the special case without the PN terms L1PN\nandL2PNin Ref. [33]. As the authors of [33] claimed,\nthis result is due to different signs of H2ss(equivalent to\nL2ss) andH3ss(associated to L1.5so). The two spin-spin\nterms are responsible for causing chaos in the Hamilto-\nnianH4, andH2sshas a more primary contribution tochaos. It is further shown in Fig. 6 and Table 1 that H4\nwith the inclusion of H3sshas weak chaos and a small\nchaotic parameter space, compared with H3with the ab-\nsence ofH3ss. This is helpful for us to explain why L1.5so\ncan somewhat weaken or suppress the chaoticity caused\nbyL2ssin the PN Lagrangian system L2.\nIV. CONCLUSIONS\nWhen a Lagrangian function at a certain PN order\nis transformed into a Hamiltonian function, many addi-\ntionalhigher-orderPNtermsusuallyoccur. Inthis sense,\nthe PN Lagrangian and Hamiltonian approaches at the\nsame order are generally nonequivalent. The equivalence\nbetween the Lagrangian formulation and a Hamiltonian\nsystem often requires that the Euler-Lagrangian equa-\ntions and the Hamiltonian should be up to higher enough\norders or an infinite order.\nFor the Lagrangian formulation of spinning compact\nbinaries, which includes the Newtonian term, 1PN and\n2PNnon-spintermsand2PNspin-spincoupling,theLeg-\nendre transformation gives not only the same order PN\nHamiltonian but also many additional higher-order PN\nterms, such as the 3PN non-spin term. Therefore, the\nsameorderLagrangianandHamiltonianapproacheshave\nsome different dynamics. This result is confirmed by nu-\nmerical simulations. This Lagrangian is non-integrable\nand can be chaotic under an appropriate circumstance\ndue to the absence of a fifth integral of motion in the\nequivalent Hamiltonian. When the 1.5PNspin-orbit cou-\npling is added to the Lagrangian, the 3PN spin-spin cou-\npling appear in the derived Hamiltonian. 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In fact, the binaries move in a bounded\nregion.\n[44] The chaoticity of the system L1is due to the spin-spin\ncoupling L2ss.L2minusL1isL1.5so.7\n/s48/s46/s48 /s49/s46/s48 /s50/s46/s48 /s51/s46/s48 /s52/s46/s48 /s53/s46/s48/s48/s46/s48/s49/s46/s48/s50/s46/s48/s51/s46/s48/s52/s46/s48/s53/s46/s48/s54/s46/s48\n/s40/s97/s41/s32/s32/s32/s32/s32/s76/s50\n/s32/s32/s72/s32/s40/s49/s48/s45/s49/s49\n/s41\n/s116/s32/s40/s49/s48/s52\n/s77/s41/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s40/s98/s41/s32/s32/s32/s32/s32/s32/s72/s50\n/s32/s32/s72/s32/s40/s49/s48/s45/s49/s50\n/s41\n/s116/s32/s40/s49/s48/s52\n/s77/s41\nFIG. 1: (color online) Hamiltonian errors of the 2PN Lagrang ian and Hamiltonian approaches, L2andH2. (a) The RKF8(9)\nis used to solve the Euler-Lagrangian equations of L2so as to obtain the difference ∆ Hbetween the Hamiltonian H2(t) at time\ntand the initial Hamiltonian H2(0). (b) The integrator directly solves the Hamiltonian H2and obtain the difference ∆ H.\n/s45/s50/s48 /s45/s49/s53 /s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48 /s49/s53 /s50/s48/s45/s50/s48/s45/s49/s53/s45/s49/s48/s45/s53/s48/s53/s49/s48/s49/s53/s50/s48\n/s32/s76/s50\n/s32/s72/s50\n/s32/s32/s89\n/s88/s40/s97/s41\n−10010 −10010−505 \nXY\n ZL2\nH2(b)\nFIG. 2: (color online) Comparison of orbits in the 2PN Lagran gian and Hamiltonian approaches, L2andH2. (a) The orbits\nprojected onto the X-Yplane, and (b) the orbits in the three-dimensional Euclidea n space.8\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s40/s97/s41/s32/s32/s32/s32/s32/s76/s49\n/s32/s32/s80/s111/s119/s101/s114/s32/s115/s112/s101/s99/s116/s114/s117/s109\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s49/s48/s45/s50\n/s41/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56/s48/s50/s52/s54/s56\n/s40/s98/s41/s32/s32/s32/s32/s72/s49\n/s32/s32/s80/s111/s119/s101/s114/s32/s115/s112/s101/s99/s116/s114/s117/s109\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s49/s48/s45/s50\n/s41/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s40/s99/s41/s32/s32/s32/s32/s76/s50\n/s32/s32/s80/s111/s119/s101/s114/s32/s115/s112/s101/s99/s116/s114/s117/s109\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s49/s48/s45/s50\n/s41\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s49/s50/s51/s52/s53\n/s40/s100/s41/s32/s32/s32/s32/s32/s32/s32/s72/s50\n/s32/s32/s80/s111/s119/s101/s114/s32/s115/s112/s101/s99/s116/s114/s117/s109\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s49/s48/s45/s50\n/s41/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56/s48/s49/s50/s51/s52/s53\n/s72/s51/s40/s101/s41\n/s32/s32/s80/s111/s119/s101/s114/s32/s115/s112/s101/s99/s116/s114/s117/s109\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s49/s48/s45/s50\n/s41/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54/s48/s49/s50/s51/s52/s53\n/s40/s102/s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s52\n/s32/s32/s80/s111/s119/s101/s114/s32/s115/s112/s101/s99/s116/s114/s117/s109\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s49/s48/s45/s50\n/s41\nFIG. 3: Power spectra of six PN approaches.\n/s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48/s45/s52/s46/s53/s45/s52/s46/s48/s45/s51/s46/s53/s45/s51/s46/s48/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53\n/s40/s97/s41\n/s72/s49/s72/s51/s76/s49\n/s32/s32/s108/s111/s103\n/s49/s48/s32/s124 /s124\n/s108/s111/s103\n/s49/s48/s32/s116/s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48/s45/s54/s45/s53/s45/s52/s45/s51/s45/s50/s45/s49\n/s40/s98/s41\n/s72/s50\n/s72/s52\n/s76/s50\n/s32/s32/s108/s111/s103\n/s49/s48/s32/s124 /s124\n/s108/s111/s103\n/s49/s48/s32/s116\nFIG. 4: (color online) Lyapunov exponents λof six PN approaches.9\n/s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s40/s97/s41\n/s72/s51\n/s72/s49/s76/s49\n/s32/s32/s70/s76/s73\n/s108/s111/s103\n/s49/s48/s32/s116/s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48\n/s40/s98/s41\n/s76/s50/s72/s52/s72/s50\n/s32/s32/s70/s76/s73\n/s108/s111/s103\n/s49/s48/s32/s116\nFIG. 5: (color online) Fast Lyapunov indicators (FLIs) of si x PN approaches.\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53\n/s76/s49/s40/s97/s41\n/s32/s32/s70/s76/s73\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s51/s54/s57/s49/s50/s49/s53/s49/s56\n/s72/s49/s40/s98/s41\n/s32/s32/s70/s76/s73\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48/s52/s53\n/s40/s99/s41/s32/s32/s32/s32/s32/s32/s76/s50\n/s32/s32/s70/s76/s73\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48\n/s40/s100/s41\n/s72/s50\n/s32/s32/s70/s76/s73\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s40/s101/s41/s32/s32/s32/s32/s72/s51\n/s32/s32/s70/s76/s73\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s40/s102/s41/s32/s32/s32/s32/s32/s72/s52\n/s32/s32/s70/s76/s73\nFIG. 6: (color online) Dependence of FLI on the mass ratio βfor each approach."    },    {        "title": "2301.02707v1.Manipulation_of_nonequilibrium_spin_dynamics_of_an_ultracold_gas_in_a_moving_optical_lattice.pdf",        "content": "Manipulation of nonequilibrium spin dynamics of an ultracold gas in a moving optical\nlattice\nZ. N. Hardesty-Shaw,1Q. Guan,2, 3, 4J. O. Austin,1D. Blume,2, 3R. J. Lewis-Swan,2, 3,∗and Y. Liu1,†\n1Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA\n2Homer L. Dodge Department of Physics and Astronomy,\nThe University of Oklahoma, Norman, Oklahoma 73019, USA\n3Center for Quantum Research and Technology, The University of Oklahoma, Norman, Oklahoma 73019, USA\n4Department of Physics and Astronomy, Washington State University, Pullman, WA 99164, USA\n(Dated: January 10, 2023)\nThe isolation and control of disparate degrees of freedom underpin quantum simulators. We\nadvance the programmability of cold atom quantum simulators with a first realization of the dynamic\ninterplay of spatial and spin degrees of freedom. We experimentally demonstrate that violent spatial\nevolutions tune long-lived coherent spin dynamics and develop a model of quantum spin-mixing\nincorporating the spatial evolution via time-dependent spin-spin interactions. Our results open new\npaths towards the simulation of quantum spin models with tunable interactions via tailored spatial\ndynamics.\nIntroduction – Ultracold quantum gases that feature spa-\ntial and spin degrees of freedom offer a powerful plat-\nform for simulating quantum magnetism in controlled,\nisolated settings [1–5]. When combined with optical lat-\ntices, these simulation capabilities are exemplified by ex-\nperimental studies featuring tunable dimensionality and\nfilling factors [6–9]. Possessing long coherence times,\nthese systems also provide an ideal platform for study-\ning out-of-equilibrium phenomena such as spin-mixing\n[10–13], transport [14, 15], dynamical phases of matter\n[16], and critical dynamics across quantum phase tran-\nsitions [7, 8, 17]. Simultaneously, advances in spin- and\nspatially-resolved probes [7, 9, 18, 19] and the control of\ntime- and spin-dependent lattice potentials are opening\nup new opportunities, including the study of multi-state\ntunneling physics [20] and driven-dissipative phases [21],\nin the presence of the spin degree of freedom.\nTypically, the energy scales of the spin and spatial de-\ngrees of freedom are disparate. This has been exploited\nto obtain a reduced description of the spin dynamics that\ndepends only on a spatial profile that remains frozen due\nto, e.g., a strong confining potential [11, 16, 22–28]. In\nthe context of spinor Bose-Einstein condensates (BECs),\nthis decoupled regime has received significant attention\n[29–31] and, amongst other applications, has been uti-\nlized to generate entangled states in the highly control-\nlable spin degree of freedom [32–37], which can also be\nmapped to the motional degrees of freedom [38–40].\nIn contrast, the interplay of spatial and spin degrees of\nfreedom remains largely unexplored, and, although typi-\ncally weak, can provide a powerful avenue for controlling\nthe spin dynamics through tailored dynamical manipula-\ntion of the spatial properties of the gas [41–45]. We inves-\ntigate this interplay, providing a first example of how spa-\ntial degrees of freedom can be utilized to manipulate the\nspin dynamics. We experimentally observe that a one-\ndimensional (1D) moving lattice, combined with a skew\noptical dipole trap (ODT), induces violent transient spa-tial motion which is nevertheless accompanied by long-\nlived spin-mixing dynamics. We develop a theoretical un-\nderstanding of these observations based on a dynamical\nsingle spatial-mode approximation (dSMA), which leads\nto an effective spin model with a time-dependent spin-\nspin interaction coefficient that depends on the tempo-\nral evolution of the BEC density profile. Experimental\nobservations — including a robust critical regime featur-\ning divergent timescales for the spin dynamics, which is\ntuned by the applied moving optical lattice and associ-\nated spatial motion — are qualitatively described by our\nmodel.\nOur results open the way for the exploitation of classi-\ncal spatial dynamics for simulating many-body quantum\nspin dynamics with highly tunable, time-dependent in-\nteractions [46, 47], thereby enhancing the class of quan-\ntum spin models accessible in spinor BECs. In addition,\nour findings imply spatial dynamics can provide new con-\ntrol knobs for the nonequilibrium generation of entangled\nspin states for, e.g., quantum-enhanced sensing [38].\nExperimental setup – Each experimental cycle begins\nwith a sodium spin-1 BEC at quadratic Zeeman energy\nqin an ODT (see Supplemental Materials [48]). A key\nfeature of spinor BECs is their spin degree of freedom\ncharacterized by the spin-dependent interaction coeffi-\ncientc2. Spin-mixing and other nonequilibrium phenom-\nena driven by a static c2have been studied in various\ncontexts [7, 8, 11, 23, 24, 28, 31]. Here, in contrast, we\ndemonstrate that c2can be tuned dynamically by uti-\nlizing a moving lattice to change the BEC’s spatial den-\nsity profile. We construct a 1D moving lattice with two\nnearly orthogonal optical beams whose frequency differ-\nence ∆fdetermines the moving lattice speed [48]. Our\ninitial BEC has a fractional population ρ0= 0.5 of atoms\nin the|S= 1,m= 0/angbracketrightstate and zero magnetization (equal\npopulations in the |S= 1,m=±1/angbracketrightstates). The BEC\nis then adiabatically loaded into the lattice, which is sta-\ntionary at time t= 0 and quenched to the desired speedarXiv:2301.02707v1  [cond-mat.quant-gas]  6 Jan 20232\n0.6\n0.4\n0.2ρ0\n(a) (b) (c)40 \n35 \n30 \n25 \n20 \n15 \n10 \n5Spin oscillation period T (ms) \nq/h (Hz)Δf = 4.6 ER/h\n Markers: Experiment \n Line: Theory \n70 60 50 40 30 20 10 (g)\nc2(t)  / c 2(0) \n(h)\n0\nBEC tFt1 0 Time(i)\ntF (ms) tF (ms) tF (ms)tF (ms)30 20 10 Δf = 0 \nΔf = 4.6 ER/h1.2 \n0.8 \n0.4 \n0.0 \nΔf\n0uLρ0\n30 20 10 30 20 10 30 20 10 0.6\n0.4\n0.2\n(d) (e) (f)q < q* q ≈ q* q > q* \nFIG. 1. (a)-(f) Exemplary time traces of ρ0for spinor BECs. Panels (a)-(c) show experimental results for ρ0(red markers)\natq/h= 15 Hz (a), 25 Hz (b), and 65 Hz (c) as well as sSMA predictions (dotted lines) with c2,fit/h= 23.7(1)Hz extracted\nfrom Fig. 1(g). Panel (a) compares static (black) and moving (red) lattice results. Solid lines are sinusoidal fits to guide the\neye. Panels (d)-(f) compare predicted ρ0forq/h= 15 Hz (d), 22 Hz (e), and 65 Hz (f) from 1D GP simulations (solid lines) to\nsSMA (dotted lines) with c2obtained through fits to the GP data [48], and dSMA (dashed lines) with c2(t) obtained from GP\ndata. The chosen qvalues exemplify the interaction dominated ( q<q∗) and Zeeman ( q>q∗) regimes separated by the critical\nregionq≈q∗. (g) Observed T(markers) versus qfit by analytical sSMA expressions (solid line) with the fitting parameter\nc2,fit/h= 23.7(1)Hz [48]. (h) Red (black) lines are evolution of c2(t)/c2(0) forq/h= 15 Hz obtained from 1D GP simulations\nof the moving (static) lattice. All moving lattice data use a lattice depth uL= 2.3ER,t1= 1.43ms, and fixed ∆ f= 4.6ER/h.\n(i) Timeline of the lattice depth (lower panel) and moving lattice speed (upper panel).\nfort>0 (see Fig. 1(i)). We study the ensuing non-trivial\nspin (Fig. 1) and spatial (Fig. 2) dynamics of the atoms\nby holding them in the moving lattice for a time tFbefore\nreleasing them for ballistic expansion and imaging [9, 48].\nSpin dynamics – We first study the nonequilibrium spin\ndynamics generated by experimental sequences (Fig. 1(i)\n∆f= 4.6ER/h) which near-resonantly couple the initial\nstationary BEC with the p= 2/planckover2pi1kLmomentum state.\nHere,ERis the recoil energy, h(/planckover2pi1) is the (reduced)\nPlanck constant, and kLis the lattice vector [6, 48].\nSpin-mixing oscillations, arising from coherent intercon-\nversions among two m= 0 atoms and a pair of atoms\nin them=±1 Zeeman states [3], constitute a useful\ntool in understanding the spin dynamics. The periods\nTof these oscillations are determined by the competi-\ntion between c2andq, illustrated by typical examples of\nthe interaction dominated region (Fig. 1(a)) and Zeeman\ndominated region (Fig. 1(c)). We also see convincing\nexperimental signatures of a critical separatrix regime\nnearq=q∗whereTdiverges (see Fig. 1(b)). These\nobservations are qualitatively consistent with expecta-\ntions based on established theory formulations referred\nto as a static single spatial-mode approximation (sSMA)\nin this paper, which assumes c2is time-independent\n[3, 7, 8, 11, 23, 24, 28, 31]. Fig. 1(g) shows the ob-\nservedTcan be used to estimate the effective static spin-\nspin interaction c2,fit/h= 23.7(1) Hz as sSMA predicts\nq∗≈c2,fitfor our initial state [6, 11]. However, direct\ncomparisons to the sSMA predicted time traces (dotted\nlines in Figs. 1(a)-(c)) demonstrate that the model fails\nto capture experimentally observed features such as thedamping of the oscillation amplitude and the drift of the\noscillations. Another notable observation that cannot be\nexplained by sSMA is the shift of the separatrix location\ninduced by the moving lattice, as shown by a compari-\nson between static (∆ f= 0) and moving lattice results\nin Fig. 1(a).1These experimental observations suggest\nsSMA provides an incomplete description of our system.\nTheoretical model – To explain the sSMA’s shortcomings,\nwe develop the following dSMA model which assumes c2\nvaries with time and describes our system with the spin\nHamiltonian [6, 22, 39, 48],\nˆHeff(t) =c2(t)\n2NˆS·ˆS+q(ˆn1+ ˆn−1). (1)\nHere, ˆS=/summationtextN\ni=1ˆsiwhere ˆsidenotes the spin-1 operator\nfor theith of the total N atoms and ˆ nmis the number\noperator for the Zeeman state m. The time-dependent\nc2(t) arises from the temporal evolution of the BEC’s\nspatial density profile and, in turn, modulation of the\neffective interaction strength of the spin model, driven\nby the moving lattice. Formally, c2(t) emerges from the\ntime-dependence of the Gross-Pitaevskii (GP) orbitals\nψm(r,t) that describe the spatial dynamics of the mth\nZeeman component. By assuming the spatial density\n1Fig. 1(a) indicates that for static lattices at an identical qin the\ninteraction dominated regime Tis smaller and thus, using the\nsame sSMA interpretation, would lie on a curve shifted to higher\nq(indicating a larger characteristic c2) relative to the moving\nlattice data shown in Fig. 1(g).3\n-4 -2 02\n-4 -2 0 2\n-4 -2 02\n-4 -2 0 2tF = 1.22 ms\ntF = 3.72 msL R1 R2 \npx/ħkLpz/ħkLpz/ħkL(a)\n-4 -2 02\n-4 -2 0 2tF = 2.22 msLR1 pz/ħkL\n(b)\n(c)Experimental Theoretical\n-4 -2 02\n-4 -2 0 2tF = 1.22 ms\n-4 -2 02\n-4 -2 0 2tF = 2.22 msLR1 \n-4 -2 02\n-4 -2 0 2\npx/ħkL\ntF = 3.72 msLR1 R2 (d)\n(e)\n(f)\n0.6\n0.4\n0.2\n0.0\n0.6\n0.4\n0.2\n0.0\n0.8\n0.4\n0.0Optical density Optical density Optical density \nFIG. 2. Time-of-flight snapshots of 2D integrated momentum\ndistribution with ∆ f= 4.6ER/h,uL= 1.2ER,t1= 0.72 ms,\nandtF= 1.22 ms (a), 2 .22 ms (b), and 3 .72 ms (c) in a shal-\nlow ODT with LODT,z= 20µm [48] atq/h= 42Hz. (d)-(f)\nAnalogous theoretical results of in-situ momentum distribu-\ntions based on 2D GP simulations [48]. The colorbar scale\nindicates the optical density of the images for each row.\nprofiles|ψm(r,t)|2for the different mstates are the same\nbut time-dependent (as we find in theory calculations dis-\ncussed in more detail later) it is implicit that while the\nspatial degree of freedom may contribute to the spin dy-\nnamics the converse is not true, i.e., the spin degree of\nfreedom does not feed back onto the evolution of the spa-\ntial profile. Setting |ψm(r,t)|2∝|φ(r,t)|2and integrat-\ning out the spatial degrees of freedom leads to Eq. (1)\nwithc2(t)∝(N−1)/integraltext\nd3r|φ(r,t)|4[48]. A key result\nof the presented experiment-theory work is that dSMA\nenables a transparent understanding of the non-trivial\nspin dynamics triggered by violent spatial evolution of\nthe BEC that occurs on faster characteristic time scales\nthan the spin dynamics.\nInterplay of spin and spatial dynamics – To illustrate\nthe typical spatial dynamics driving the spin-mixing ob-\nserved in Fig. 1, we show experimental BEC momentum\ndistributions in Figs. 2(a)-(c), which capture the emer-\ngence of violent spatial motion due to the interplay of\nmomentum kicks generated by the moving lattice and\nthe shallow ODT harmonic confinement on a timescale\nsignificantly shorter than the observed spin dynamics.\nThe rapid appearance of many of discrete momentum\npeaks and associated spatial dynamics shown in Fig. 2simultaneously suggests that the deviations from sSMA\npredictions in Figs. 1(a)-(c) are to be expected but also\nentices us to reconcile elements of the good qualitative\nagreement between the experimental data and sSMA cal-\nculations in Fig. 1(g). We note that the creation of the\ndiscrete momentum peaks is a coherent process and does\nnot conflict with the assumption of the single spatial-\nmode approximation. In fact, the Supplemental Materi-\nals [48] show that Figs. 1(a)-(c) are replicated if different\nmomentum components are used to construct ρ0, thereby\nproviding experimental support for dSMA.\nTo gain further insight, we use numerical GP calcu-\nlations [48], which provides a mean-field description of\nthe full spinor BEC dynamics including both spatial and\nspin degrees of freedom. The complexity of the experi-\nmental system, in particular the disparate timescales of\nspin and spatial dynamics, precludes a full quantitative\n3D GP treatment. Instead, we use a reduced dimension-\nality 1D spinor GP calculation with parameters tuned to\ncapture essential aspects of the experimental 3D system.\nThis simplified treatment enables us to develop a qual-\nitative understanding of the experimental results [48].\nThe GP simulations (solid lines in Figs. 1(d)-(f)) quali-\ntatively replicate the coherent spin dynamics, including\na diverging oscillation period for q≈q∗(Fig. 1(e)) and\nrobust harmonic oscillations for q < q∗(Fig. 1(d)) and\nq > q∗(Fig. 1(f)) with damped amplitude and drift-\ning mean value, respectively. We note that the up-\nward drifting mean value in the experimental data for\nq > q∗(Fig. 1(c)), notably not captured by the 1D GP\ntheory, may be induced by a subtle resonance mecha-\nnism between the spin and spatial dynamics [31] that\ndepends sensitively on the dimensionality of the system.\nHigher dimensionality calculations will be presented else-\nwhere [49].\nWe use GP calculations to make a more fine-grained\ntheoretical investigation of the relationship between the\nspatial and spin dynamics and, in particular, certify that\nwhile the BEC undergoes violent motion on fast time-\nscales: i) all Zeeman components are described by a com-\nmon spatial density profile |φ(r,t)|2, and ii) the sophis-\nticated interplay of the moving lattice and ODT drive\ncomplex dynamics of c2(t). These observations lead us\nto self-consistently compare the 1D GP results to dSMA\npredictions, i.e., mean-field dynamics based on Eq. (1)\nwithc2(t) computed via the GP density |φ(r,t)|2[48].\nThe GP and dSMA time traces in Figs. 1(d)-(f) show\nexcellent agreement with each other. This implicitly\ndemonstrates the spin dynamics do not feed back into the\nspatial evolution, in agreement with expectations based\non the disparity of energy scales. The dSMA results also\nshow significantly improved qualitative agreements with\nexperimental data than sSMA results, providing further\nsupport for our dSMA model.\nGP calculations of the spatial dynamics in the pres-\nence of a moving lattice lead to appreciable variation of4\nc2(t) (red line in Fig. 1(h)) at t/lessorsimilar3 ms. Over longer\ntimescalesc2(t) features an overall decrease, which we un-\nderstand as being driven by the relaxation of the spatial\ndensity profile as the BEC fractures into many momen-\ntum components. This behavior is in stark contrast with\npredictions for a static lattice (black line in Fig. 1(h))\nthat indicate c2(t) instead fluctuates around a well de-\nfined time-averaged value with small oscillations due to\nexcitations created during the loading phase. After the\ninitial transient behavior in the moving lattice, our re-\nsults (Fig. 1(h)) indicate that the decay of c2(t) is slow\nrelative to the characteristic time of the spin dynamics,\nand therefore observables such as the spin oscillation pe-\nriod are captured by sSMA [48]. Our calculations show\nthat the precise details of the spin dynamics (e.g., quali-\ntative features including damping of the spin oscillations\nin Fig. 1(a)) can depend greatly on the temporal vari-\nation ofc2(t) and hence a more rigorous description is\nprovided by the GP and dSMA models.\nSingle-particle resonances – The precise evolution of\nthe spatial density profile, seen in Figs. 2(a)-(c) as a\nmenagerie of seemingly irregularly distributed wavepack-\nets in momentum space, can be understood with the aid\nof GP simulations (Figs. 2(d)-(f)). To more precisely\ncapture the impact of gravity and the finite trap depth,\nthe simulations in Fig. 2 are performed using an axially\nsymmetric 2D setup. This numerical treatment is feasi-\nble due to the relatively short time scales over which the\nspatial dynamics are studied in detail. Using 2D simula-\ntions also enables us to capture key details of the momen-\ntum kicks that 1D simulations, such as those employed\nin Fig. 1, miss. At short times, the lattice kicks atoms\nfrom the initial BEC with momentum p= 0 to the near-\nresonant state with momentum p= 2/planckover2pi1kL(Figs. 2(a)\nand 2(d)). Subsequently, an additional momentum com-\nponent, referred to as the lead peak (L-peak), splits from\nthep= 2/planckover2pi1kLpeak and decelerates as it travels away\nfrom the minima of the relatively shallow ODT poten-\ntial. As the L-peak slows, its momentum evolves until\nit sweeps through the approximate resonance region cen-\ntered on the line pz≈0.86px+ 0.20/planckover2pi1kL(see the gray\nshaded region in Fig. 3(b) and Supplemental Materi-\nals [48]), where the lattice couples two nearly resonant\nmomentum states, corresponding to the L-peak and a\nnew peak labelled R1, that are separated by another\n2/planckover2pi1kLmomentum kick (Figs. 2(b) and 2(e)). Similarly,\nthe R1-peak also decelerates until it crosses the resonance\nregion and the lattice generates a new peak labelled R2\n(Figs. 2(c) and 2(f)). This pattern continues and the\nBEC fractures into a multitude of momentum states.\nFigure 3(a) confirms our prior analysis, which suggests\na dependence on both the confining ODT potential and\nmoving lattice, by tracking the position of the L- and\nR1-peaks in time and momentum space in a compressed\nODT. The position of the L-peak initially follows a tra-\njectory consistent with a Lissajous curve derived from a\n-1 01\n2 1 0 -1 pz/ħkL\npx/ħkL(b)Red: R1- Peak Blue: L- Peak \n-1 01pz/ħkLpx/ħkL\n-1 01\n1 2 3 4\ntF (ms)5\n(a)FIG. 3. (a) Time evolution trajectories of the mean position\nof the L-peak (circles) and R1-peak (squares) in the pz-px\nplane taken in a compressed ODT with LODT,z= 33µm [48],\nextracted from experimental images similar to those shown in\nFigs. 2(a)-(c) at q/h= 42Hz. With increasing time, markers\nappear larger due to movement in the pxaxis. Blue (red)\nsolid lines are the L-peak (R1-peak) trajectories correspond-\ning to Lissajous curves (see text) [48]. (b) Trajectories of\npanel (a) projected into the pz-pxplane. The gray shaded\nregion encompasses an approximate resonance region where\ndecelerated atoms are kicked by the lattice (to create, e.g.,\nthe R1-peak) [48].\nsimplified classical treatment of a single particle initially\nmoving with momentum 2 /planckover2pi1kLin the ODT (see Supple-\nmental Materials [48]). A sudden momentum kick im-\nparted by the lattice couples the L- and R1-peaks as the\nformer crosses the approximate resonance region, shown\nby the gray region in Fig. 3(b) [48]. As time increases\n(i.e., deeper into the trajectory of each peak), the agree-\nment between the experimentally observed and the the-\noretically predicted trajectories for the L- and R1-peaks\ndeteriorates, as shown near the end of the time axis in\nFig. 3(a). The deterioration is most clearly seen when\ncomparing an ideal (effectively LODT,i=∞) harmonic\ntrap with our typical shallow ODT depth that features a\ncomparatively small curvature (see Supplemental Mate-\nrials [48]). In Fig. 3, we use compressed ODTs instead of\nshallow ODTs, trading reduced visibility and condensate5\nfraction, to better illustrate the bending of the trajectory\nin thepx−pzplane [48].\nConclusion – Our results open a new direction for the\nexploitation of spatial dynamics as a control knob for\nquantum simulations of many-body quantum spin models\nwith tunable, time-dependent interactions [46, 47]. Tai-\nlored modulation of the spatial profile could be used to\ncontrol the precise time dependence of the spin-spin inter-\nactions and realize Floquet-driven spin dynamics [59, 60].\nThis can have immediate applications for the dynam-\nical generation of entangled spin states for quantum-\nenhanced sensing [36, 39, 61–63]. The observed short\ntime dynamics also raise intriguing questions about equi-\nlibration of spinor BECs. For example, future studies\nmight utilize the time dependence of c2(t) to force these\nsystems along different equilibration trajectories.\nAcknowledgements – D. B. acknowledges support by the\nNational Science Foundation (NSF) through grant No.\nPHY-2110158. R. J. L-S. acknowledges support by NSF\nthrough Grant No. PHY-2110052 and the Dodge Family\nCollege of Arts and Sciences at the University of Okla-\nhoma (OU). Z. N. H-S., J. O. A., and Y. L. acknowledge\nsupport by the Noble Foundation and the NSF through\nGrant Nos. PHY-1912575 and PHY-2207777. This work\nused the OU Supercomputing Center for Education and\nResearch (OSCER).\n∗lewisswan@ou.edu\n†yingmei.liu@okstate.edu\n[1] Polkovnikov, A., Sengupta, K., Silva, A. & Vengalattore,\nM. Colloquium: Nonequilibrium dynamics of closed in-\nteracting quantum systems. Rev. Mod. 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Su-\npersensitive quantum sensor based on criticality in an\nantiferromagnetic spinor condensate. Phys. Rev. A\n101, 043609 (2020). URL https://link .aps.org/doi/\n10.1103/PhysRevA .101.043609 .\n[63] Sundar, B. et al. Bosonic pair production and squeez-\ning for optical phase measurements in long-lived dipoles\ncoupled to a cavity (2022). arXiv:2204.13090.Supplemental Material: Manipulation of nonequilibrium spin dynamics of an ultracold\ngas in a moving optical lattice\nZ. N. Hardesty-Shaw,1Q. Guan,2, 3, 4J. O. Austin,1D. Blume,2, 3R. J. Lewis-Swan,2, 3and Y. Liu1\n1Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA\n2Homer L. Dodge Department of Physics and Astronomy,\nThe University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma 73019, USA\n3Center for Quantum Research and Technology, The University of Oklahoma,\n440 W. Brooks Street, Norman, Oklahoma 73019, USA\n4Department of Physics and Astronomy, Washington State University, Pullman, WA 99164, USA\n(Dated: January 10, 2023)arXiv:2301.02707v1  [cond-mat.quant-gas]  6 Jan 2023S2\nI. EXPERIMENTAL DETAILS\nA. Experimental sequence\nOur experimental sequences start by creating a S= 1 spinor BEC of up to 105sodium (23Na) atoms in a crossed,\nanisotropic harmonic optical dipole trap (ODT) at a particular quadratic Zeeman shift qtuned by external magnetic\nfields, similar to our previous work [1–3]. We apply a resonant radio-frequency (RF) pulse to prepare an initial\nstate with fractional population ρ0=/angbracketleftˆn0/angbracketright/N= 0.5 in the|S= 1,m= 0/angbracketrightstate and zero magnetization, M=\n/angbracketleftˆn1−ˆn−1/angbracketright/N= 0. We then adiabatically load the initial state into a one-dimensional moving optical lattice. The\nlattice is constructed from two nearly orthogonal lattice beams originating from a single-mode laser with wavelength\n1064 nm and characterized by the potential Vlat(r,t) =uLcos2[kL·r−2π∆f(t)t/4], with lattice vector kLoriented at\napproximately 40◦from thez-axis defined by gravity. The resulting standing wave potential has a lattice spacing of\nλL/2 = 0.81µm. The time-dependent frequency difference ∆ f(t) =|fH−fV|, wherefHandfVare the corresponding\nlattice beam frequencies, determines the velocity vof the moving lattice, v=λL(fH−fV). The velocity vis\nmanipulated via a linear ramping rate α=h(∆f(t2)−∆f(t1))\nt2−t1such that when v <0 (v >0) the atoms move in\nthep= 2/planckover2pi1kL(p=−2/planckover2pi1kL) direction. In the main text, the data in Fig. 1 were taken with positive velocities, while\nthe data in Figs. 2 and 3 were taken with negative velocities. The value of ∆ fis initially set to zero and, after an\nadiabatic ramp of the lattice depth uLto its final value at t=t1, is quenched to its final value. The total time the\natoms spend in the lattice is denoted by tF(for details see Fig. 1(i) of the main text). At the conclusion of each\nsequence, the trapping potentials are turned off so that the atoms can ballistically expand and be captured using a\ntwo-step microwave imaging after a given time of flight (TOF) [1]. Each data point in this paper is an average of at\nleast 8 repeated measurements and all error bars reported are estimated one standard deviations.\nB. Optical dipole trap\nAn essential element of our experimental setup is a harmonic confinement potential skew to the moving lattice\npotential. The interplay of these two potentials triggers the nontrivial spatial dynamics that are key to our findings.\nThe harmonic confinement is provided by a crossed optical dipole trap (ODT) constructed by two orthogonal beams\nwith wavelength λ= 1064 nm. One ODT beam (ODT1) is orthogonal to gravity while the other (ODT2) is at a\n76 degree angle relative to gravity (see Fig. S1). ODT1 is along the xODT axis, while the projection of ODT2 into\nthe plane normal to gravity falls along the yODT axis (see Fig. S1). The moving lattice lies 72 degrees horizontally\nfrom ODT1 and is tilted at a 40 degree angle relative to gravity. Due to experimental considerations, our theoretical\ncalculations therefore occur in three distinct coordinate systems that share a common zaxis defined by gravity: the\ncoordinate systems defined by the ODT potential, the moving lattice, and the imaging plane, as illustrated in Fig. S1.\nxODT \npx\nyODT py\nkL,x \nkL,y 72 °30 °\npz  & kL,z  zODT \nxODT yODT pz  & kL,z  zODT \nkL,x kL,y px py14 °\n76 °\nga) b)\nFIG. S1. a) The three coordinate systems used in our theoretical calculations projected into the plane spanned by pxandpy.\nBlue (red) [black] axes refer to the ODT (moving lattice, with associated lattice vector kL) [imaging plane, with associated\nmomentum vector p] coordinate system. b) Similar to a) but projected into the plane spanned by yODT andzODT. The green\nvector labeled by gindicates the direction of gravity. The projections in this figure are to scale.S3\nThe potential generated by our crossed ODT can be parameterized to a good approximation by\nV3D(xODT,yODT,z) =−V0\n\n1\n1 +x2\nODT\nz2\n0exp\n−2(y2\nODT+z2)\nw2\n0/parenleftBig\n1 +x2\nODT\nz2\n0/parenrightBig\n+1\n1 +y2\nODT\nz2\n0exp\n−2(x2\nODT+z2)\nw2\n0/parenleftBig\n1 +y2\nODT\nz2\n0/parenrightBig\n\n\n,(S1)\nwhereV0=P0hαfsλ2λNa2\n2π3mec2w2\n0(λ2−λNa2),w0= 33µm is the ODT beam waist, z0=πw2\n0\nλis the associated Rayleigh length,\nP0is the ODT power, λNais the D2 line of sodium atoms, λis the wavelength of the ODT beam, meis the mass\nof an electron, h(/planckover2pi1) is the (reduced) Planck constant, αfs≈1\n137is the fine structure constant, gis the gravitational\nacceleration, and cis the speed of light. The ODT power, P0, can be varied to change the effective trap depth and\nsize.\nShallow ODTCompressed ODT \nLODT,z LODT,z 40 \n30 \n20 \n10 \n0\n-10\n-60 -40 -20 0 20 ODT Potential (kHz)\nzODT ( μm) \nFIG. S2. Cross-sectional cut of the ODT potential after accounting for gravity in the zODT-direction at a local minimum in\nthexODT andyODT directions. The effective length of the ODT along the zODT-direction is LODT,z= 20µm for the shallow\nODT (red solid line) and LODT,z= 33µm for the compressed ODT (blue dashed line). Similar plots can be made for the ODT\npotential in the yODT-direction, giving an effective length of LODT,y= 31µm (LODT,y= 40µm) for the shallow (compressed)\nODT.\nIn Fig. S2 we utilize Eq. (S1) and take into account the effects of gravity to generate cross-sectional cuts for the\ncompressed ODT trap ( P0≈35 mW), which was utilized for the experiments discussed in Fig. 3 of the main text\nand Fig. S3 of the Supplementary Information, and the shallow ODT trap ( P0≈17 mW), which was utilized for all\nother experimental figures and discussions. The effective trap length LODT,iis defined as the difference between the\nvalues for which V3Dtakes on a local maximum and local minimum (see Fig. S2) in the i-coordinate axis. Thus, the\nshallow trap (red solid line) is characterized by a smaller effective trap length LODT,ithan the compressed trap (blue\ndashed line), i.e., atoms with high momentum exit the shallow trap more easily than the compressed trap. Since the\ntrapping extends over a larger spatial region for the larger LODT,i, the compressed trap extends the time scales over\nwhich the complex spatial dynamics can be observed. However, the extended trapping times come at the cost of an\nincreased average atom temperature, which in turn tends to reduce the coherence and condensate fractions. This is\ndemonstrated in Fig. S3, which exhibits less coherent peaks than those shown in the analogous time-of-flight (TOF)\nimage in Fig. 2(c) of the main text.\nC. Spin-mixing oscillations\nAll spin oscillation data presented in the main text are extracted from considering just the zero momentum, p= 0,\npeak. For completeness, Fig. S4 presents spin oscillation data derived from the same data sets as Figs. 1(a)-(c) of the\nmain text but obtained by counting all trapped atoms. Using this data, we obtain similar results to Figs. 1(a)-(c);\nspecifically, the respective fitted periods agree within the margins of the errors. The agreement between the distinct\ncounting methods provides further support for the assumption of a dynamical single spatial-mode approximation\n(dSMA), as it indicates that coherence is retained between different momentum components as they evolve under the\ndynamics driven by the lattice potential.\nIn Fig. 1(g) of the main text we see good agreement between the experimental data and the sSMA predictions in\nthe Zeeman-dominated region. This is because we expect the dynamics of c2(t) to be less important in this region.\nTo support this, Fig. S5 shows experimental data for time traces of ρ0, the fractional population of the |S= 1,m= 0/angbracketright\nstate, obtained for a large range of ∆ fatq/h= 42 Hz, in the Zeeman-dominated region. We observe no significantS4\n-4 -2 024\n-4 -2 0 2L\nR1 tF = 3.72 ms\npx/ħkLpz/ħkL\nOptical Density \n0.0 0.1 0.2 0.3 \nR2 \nFIG. S3. Typical experimental TOF images obtained with the compressed ODT ( LODT,z= 33µm, LODT,y= 40µm) displayed\nin Fig. S2. Data are taken following the same experimental sequence as in Fig. 2(c) of the main text, which instead used the\nshallow ODT ( LODT,z= 20µm, LODT,y= 31µm). The colorbar scale on the right indicates the optical density of the image.\nρ0\ntF (ms) tF (ms)q < q* q ≈ q* q > q* \ntF (ms)0.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0.035 30 25 20 15 10 50.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0.035 30 25 20 15 10 50.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0.035 30 25 20 15 10 5\n(a) (b) (c)\nFIG. S4. Example time traces of ρ0, the fractional population of the |S= 1,m= 0/angbracketrightstate, obtained by counting all trapped\natoms (see text) for: (a) q/h= 15 Hz, (b) q/h= 25 Hz, and (c) q/h= 65 Hz. Here, qis the quadratic Zeeman energy. All\ndata is obtained from an experimental sequence illustrated in Fig. 1(i) of the main text with final lattice depth uL= 2.3ER\nand fixed ∆ f= 4.6ER/h. Sinusoidal fits (lines), which are used to extract the spin oscillation period T, are shown to guide\nthe eye.\ndependence of the period or amplitude of the oscillations on the actual value of ∆ f(i.e., the speed of the moving\nlattice), and thus c2(t) associated with the resulting spatial dynamics.\nII. DETAILS ON THEORETICAL TREATMENT OF SPIN- 1GASES\nA. Gross-Pitaevskii treatment of spin- 1condensates\nWe consider a spin-1 condensate of Nsodium atoms of mass MNaunder external confinement in the presence of a\nmoving lattice. The two-body interactions are characterized by the spin-independent and spin-dependent interaction\ncoefficients g0andg2,\ng0=4π/planckover2pi12(aS=0+ 2aS=2)\n3MNa(S2)\nand\ng2=4π/planckover2pi12(aS=2−aS=0)\n3MNa, (S3)\nwhereaS=0= 48.9a0andaS=2= 54.5a0are, respectively, the s-wave scattering lengths for the S= 0 andS= 2\nstates with a0the Bohr radius [1, 4]. Our treatment includes the quadratic Zeeman shift term (proportional to q) but\nnot the linear Zeeman shift, which does not play a role for the dynamics since it is conserved.\nAt the mean-field level, the coupled dynamics of the spin and spatial degrees of freedom is described by the time-\ndependent spinor Gross-Pitaevskii (GP) equation,S5\n0.8\n0.6\n0.4\n0.2\n25 20 15 10 5\ntF (ms)ρ0\n Δf = 4.6 ER/h Δf = 7 ER/h Δf = 9.3 ER/hq/h  = 42 Hz \nFIG. S5. Time traces of ρ0at fixedq/h= 42Hz and a range of lattice speeds, set by ∆ f= 4.6ER/h(blue), 7ER/h(green),\nand 9.3ER/h(black). All data is for a final lattice depth uL= 2.3ER. Sinusoidal fits (lines), which are used to extract the\nspin oscillation period T, are shown to guide the eye. The ∆ f= 7ER/h(∆f= 9.3ER/h) curves are offset by 0 .2 (0.4) in the\nydirection for visual clarity.\ni/planckover2pi1∂\n∂t\nψ−1\nψ0\nψ1\n=/bracketleftbigg\n−/planckover2pi12∇2\n2MNa+V(r,t) +g0(N−1)/parenleftbig\n|ψ−1|2+|ψ0|2+|ψ1|2/parenrightbig/bracketrightbigg\nψ−1\nψ0\nψ1\n+\nq0 0\n0 0 0\n0 0q\n\nψ−1\nψ0\nψ1\n (S4)\n+g2(N−1)\n|ψ−1|2+|ψ0|2−|ψ1|2ψ∗\n1ψ0 0\nψ1ψ∗\n0|ψ−|2+|ψ−1|2ψ−1ψ∗\n0\n0 ψ∗\n−1ψ0|ψ1|2+|ψ0|2−|ψ−1|2\n\nψ−1\nψ0\nψ1\n,\nwhereV(r,t) includes contributions from the moving optical lattice Vlat(r,t), the confining potential (see Sec. I A),\nand gravity. Within the GP formalism, each Zeeman component of the BEC is described by a mean-field wavefunc-\ntionψm(r,t), such that both spin and spatial degrees of freedom, and their interplay, are simultaneously captured.\nThe model ignores quantum fluctuations, which are expected to contribute minimally for the regimes in which the\nexperiment operates.\nFor the scenarios reported in the main text, solution of the GP equation for the full 3D system by direct numerical\nintegration is not feasible. This is due to both the disparate timescales for the spin and spatial degrees of freedom\nas well as the large spatial region occupied by the BEC when it is kicked by the lattice. Specifically, the repeated\nmomentum kicks imparted by the moving optical lattice on the fractured BEC lead to intricate spatial structure\nas well as a rapidly expanding cloud that must be tracked over comparatively long time scales (36 ms in Fig. 1 of\nthe main text), requiring a large simulation box with good spatial resolution. Thus, the GP result presented in the\nmain text are from numerical simulations of reduced dimensionality 1D (Fig. 1) or 2D (Fig. 2) models, wherein the\ninteraction coefficients g0andg2, mean spatial density of the BEC and trapping potential V(r,t) are adjusted to\ncapture the features of the experimental setup. The GP results presented are generated by numerically integrating\nthe coupled GP equations using the XMDS2 software package [5].\nThe dynamics shown in Figs. 2-3 of the main text are restricted to a few ms, i.e., much shorter times than those on\nwhich the spin oscillation dynamics occurs (see Fig. 1 of the main text). These dynamics out to 3 ms are amenable\nto 2D GP equations. To this end, we define a “2D simulation plane” by the direction of the px−pzimaging plane.\nThe reduced dimensionality trap V2Dis obtained by projecting the 3D trap V3Donto the 2D simulation plane. Our\nsimulations capture the role of the ODT trap (including, e.g., non-harmonic corrections to the confining potential)\nand its interplay with gravity. Approximating the azimuthal and polar angles introduced in Sec. I B by 30◦and 76◦,\nrespectively, the 2D trap is given by\nV2D(y,z) =V3D[cos(76◦) cos(30◦)y,−sin(30◦)y,sin(76◦)z]. (S5)\nThe projection of the moving lattice in the px−pyplane is described by kL,x(see Fig. S1). Neglecting the 12◦angle\nbetween the kL,xaxis and the negative pxaxis, the moving lattice is in the px−pzplane. Given the fact that theS6\nlattice has a 40◦angle relative to the pzaxis (see the discussion above Fig. S1), the lattice vector is described by\nkL=kL[−sin(40◦),0,cos(40◦)]. The reduction from 3D to 2D does change the mean density and, correspondingly,\nthe chemical potential. This is accounted for by introducing the effective 2D interaction strength geff\n2D; similar to the\n1D case,g0(N−1) andg2(N−1) are replaced by geff\n2Dandgeff\n2D/28.06, respectively. The coupling constant geff\n2Dis set\nby enforcing that the chemical potential of the 2D system, obtained by solving the scalar GP equation, is equal to\nthat of the 3D system at t= 0.\nThe procedure for mapping the full 3D system to a reduced dimensionality model is not unique. Figure S6(c)\nillustrates this exemplary for the 1D case. Since the chemical potential µis the characteristic energy scale of the\ntime-independent scalar GP equation and c2the characteristic energy scale of the effective spin Hamiltonian, it is\nnatural to demand that the reduced dimensionality scalar GP model reproduces these two energy scales. The solid\nand dashed horizontal lines in Fig. S6(c) show the chemical potential µand the spin-spin interaction strength c2of\nthe 3D system in the presence of a static optical lattice with uL= 2.3ER(note that the initial state preparation\nwithin the full 3D framework is significantly simpler than tracking the time evolution); this is the final lattice depth\nused in the experiment. Fixing the parameters of the scalar 1D GP equation requires setting the values of /planckover2pi1ωz/ER\nandgeff\n1D/(ERk−1\nL). The 1D scalar GP equation is given by Eq. (S7) with the r-vector replaced by zandg0(N−1)\nreplaced by geff\n1D, wheregeff\n1Dhas units of “energy times length”. The 1D spinor GP equation is obtained analogously,\ni.e.,g0(N−1) andg2(N−1) are replaced by geff\n1Dandgeff\n1D/28.06, respectively. The black and red dashed lines in\nFig. S6(c) show c2andµas a function of geff\n1D/(ERk−1\nL) for fixed /planckover2pi1ωz/ER(plugging in uL= 2.3ER, this corresponds\ntoωz= 197 Hz). It can be seen that the dashed lines cross the solid lines at geff\n1D/(ERk−1\nL)≈7 and 12.5, respectively,\ni.e., for the /planckover2pi1ωz/ERvalue chosen there exists no unique geff\n1D/(ERk−1\nL) at which the values of c2andµcalculated within\nthe 1D framework agree with the respective values calculated within the 3D framework. Since we do not find a unique\ngeff\n1D/(ERk−1\nL) for other values of /planckover2pi1ωz/EReither, there exists—unless additional conditions are added—an arbitrariness\nin the chosen 1D simulation parameters. With this in mind, we select parameters that place the divergence of Tat\nabout the same qvalue as observed experimentally (see Fig. S6(a)). Specifically, the simulations shown in Figs. 1(d)-\n1(f) of the main text (and in Fig. S6(a)) use the same final lattice depth as the experiment (namely, uL= 2.3ER),\nωz= 197 Hz, and geff\n1D/(ERk−1\nL) = 12.0.\n(c)\nFIG. S6. (a) Spin oscillation period Tas a function of Zeeman shift q. The black markers show Textracted from 1D spinor\nGP calculations that account for the spin-independent and spin-dependent interactions. The spin oscillation period diverges\naroundq∗/h= 22 Hz. Emulating the analysis of the experimental data shown in Fig. 1(g) of the main text, the solid line is\nobtained by fitting the 1D GP data using the analytical sSMA expressions and treating the spin-spin interaction as a fitting\nparameter (namely, c2,fit). (b) Time evolution of the overlap of the Zeeman states (see Eq. (S10)) as a function of the Zeeman\nshiftq. An overlapFof one indicates that all Zeeman states share a common spatial density profile, consistent with the\ndSMA. The only significant deviation from unity is observed for relatively long times ( t/greaterorsimilar20 ms) when q≈q∗. (c) Parameter\ncalibration of reduced dimensionality 1D GP equation. The black and red solid horizontal lines show the interaction energy\nc2(left axis) and chemical potential µ(right axis), respectively, for a 3D scalar BEC in the presence of a static lattice with\nuL= 2.3ERforN= 80,000 and angular trapping frequencies ωx= 125 Hz,ωy= 125 Hz, and ωz= 155 Hz. The red and\nblack dashed lines show c2andµ, obtained by solving the 1D scalar GP equation as a function of the 1D coupling constant\ngeff\n1D. The 1D calculations use ωz= 197 Hz and uL= 2.3ER.\nB. Single spatial-mode approximation and effective spin model\nWe now motivate and introduce an approximate treatment that decouples the spin and spatial degrees of freedom.\nTypically, the energy scales associated with the spin-independent terms of the Hamiltonian (i.e., the energy scales\nof the external harmonic and lattice confinement, the interactions that are proportional to g0, and the chemical\npotentialµof the system) are of the order of kilohertz and much larger than those of the spin-dependent terms ofS7\nthe Hamiltonian (i.e., the value of the Zeeman shift qand the interactions that are proportional to g2), which are of\nthe order of hertz. This scale separation motivates an approximate treatment wherein the spin and spatial degrees of\nfreedom are treated independently [6], with the spatial degree of freedom controlled solely by the spin-independent\nterms of the Hamiltonian and the spin dynamics governed by the spin-dependent terms of the Hamiltonian.\nFollowing the literature [7], we make a single spatial-mode approximation (SMA) wherein the bosonic field operators\nare decomposed as\nˆψm(r) = ˆamφ(r,t), (S6)\nwhere ˆam(ˆa†\nm) is a bosonic operator that destroys (creates) a particle in Zeeman state min a spatial mode defined\nby the spatial mean-field wavefunction φ(r,t), which is normalized to one, i.e.,/integraltext\nd3r|φ(r,t)|2= 1; in Eq. (S6), m\nlabels the Zeeman states ( m= 0 and±1). The key assumption of the SMA is that φ(r,t) is identical for all three\nZeeman states. The mean-field wave function φ(r,t) is the solution to the time-dependent scalar Gross-Pitaevskii\n(GP) equation\ni/planckover2pi1∂φ(r,t)\n∂t≈/bracketleftbigg\n−/planckover2pi12∇2\n2MNa+V(r,t) +g0(N−1)|φ(r,t)|2/bracketrightbigg\nφ(r,t), (S7)\nwhereV(r,t) is the same as Eq. (S4). The spin dynamics, in turn, is governed by the effective Hamiltonian ˆHeff(t),\nˆHeff(t) =c2(t)\n2NˆS·ˆS+q(ˆa†\n1ˆa1+ ˆa†\n−1ˆa−1), (S8)\nwhere the quantity ˆSdenotes a collective spin operator. The time-dependent interaction strength c2(t),\nc2(t) = (N−1)g2/integraldisplay\nd3r|φ(r,t)|4, (S9)\nis driven by the time dependence of the spatial mean-field wave function φ(r,t), i.e., the spin dynamics is governed—\nthrough the coefficient c2(t)—by the spatial dynamics. In our experiment, the spatial dynamics is, to a large degree,\ninduced by the moving optical lattice potential. The time dependence of the interaction coefficient c2(t) is distinct\nfrom prior works (e.g., Refs. [2, 8, 9]), which assumed that the condensate is prepared in the ground-state of a static\nconfining potential such that subsequent spatial motion is minimal and to a good approximation |φ(r,t)|2=|φ(r,0)|2.\nFor this reason, we use the distinguishing nomenclature of dynamical SMA (dSMA; time-dependent c2) and static\nSMA (sSMA; time-independent c2) for our and prior works, respectively. The latter is recovered from Eq. (S8) by\nassumingc2(t) =c2. To zeroth-order, the dynamics of the moving lattice experiments during the first few milli-\nseconds is dominated by spatial dynamics. At later times, however, the spin degrees of freedom become increasingly\nimportant as evidenced by the observation of spin oscillations (see Fig. 1 of the main text).\nThe dSMA is supported by our 1D spinor GP calculations. First, Fig. 1 of the main text shows good agreement\nbetween dSMA and GP predictions for ρ0(t). Second, we can explicitly validate the assumption that each Zeeman\nstate occupies a single common spatial mode by computing the overlap,\nF=|/integraltext\ndxψ∗\n0(x)ψ1(x)|/radicalBig/integraltext\ndx|ψ0(x)|2/radicalBig/integraltext\ndx|ψ1(x)|2. (S10)\nThe time evolution of this quantity over a range of Zeeman shifts is plotted in Fig. S6(b). We observe that Fremains\nnear unity across the interaction dominated regime ( q<q∗) throughout the timescales we investigate (up to 40 ms in\nthe GP calculations, which is longer than the 30 ms covered by the experiment). In the Zeeman regime, the overlap\nremains close to unity for t/lessorsimilar20 ms before minor deviations appear. As might be expected naively, a substantial\nbreakdown of the single spatial-mode approximation occurs in a narrow region around the critical regime, q≈q∗.\nThe experimental spin oscillation data in Fig. 1 of the main text are analyzed using the sSMA, i.e., the mean-\nfield equations associated with ˆHeff(t) for a time-independent spin-spin interaction coefficient. Specifically, the spin\noscillation period Tis extracted by fitting the experimentally measured fractional population ρ0(t) for various q\nwith sinusoidal functions, which provide good approximations to the spin oscillation dynamics away from the critical\nregime where the period diverges. All fits include at least one full period of oscillation. To determine the spin-spin\ninteraction coefficient from the extracted periods, we perform a nonlinear least squares fit of the T-versus-qdata using\nthe analytical solutions to the mean-field equations associated with ˆHeff[10]. The fit uses ρ0(0) = 0.5 andθ(0) = 0\nand treats c2as a free parameter (we denote the fit result by c2,fit). Whileρ0(0) is measured experimentally, θ(0)\nis not. However, based on the experimental sequence used to prepare the initial Zeeman populations we expect that\nθ(0) is equal to 0. As reported in the main text, our fitting procedure yields c2,fit= 23.7(1) Hz.S8\nTo gain additional insights, we perform an analogous analysis for the spin oscillation data obtained by solving the\n1D spinor GP equations. Specifically, we obtain ρ0(t) and the associated period Tby solving the spinor GP equations\n(see markers in Fig. S6(a)), and then extract c2,fitfrom theT-versus-qdata using the sSMA (solid line in Fig. S6(a)).\nFor the 1D spinor GP simulations shown in Fig. S6(a), the spin oscillation period diverges at q∗/h≈22 Hz, i.e.,\nat roughly the same value of the Zeeman energy as in the experiment presented in the main text. The qualitative\nagreement between the experimental and theoretical analysis, particularly the consistency with the sSMA results, is\nencouraging and suggests that the 1D spinor GP simulations provide a qualitatively correct description of the moving\nlattice experiments.\nThe GP simulations also enable us to better understand why the spin oscillation period extracted from the exper-\nimental data is consistent with the predictions of sSMA, even though the time traces show significant discrepancies.\nIn the regime q≤q∗, the sSMA theory predicts that the spin oscillation period should be strongly correlated with\nthe spin-spin interaction strength [3, 10, 11], which the GP results (see, e.g., Fig. 1(h) of the main text) predict to\ndecay relatively slowly, compared to the typical timescales of the spin degree of freedom, apart from initial transient\ndynamics for t/lessorsimilar3 ms. This suggests that the spin oscillation period obtained from the experimental data for q≤q∗\nshould be interpreted as being reflective of the characteristic scale of c2(t) over the experimental sequence. On the\nother hand, in the regime q/greatermuchq∗the oscillation period is expected to be dominated by the quadratic Zeeman shift.\nThus, the experimentally observed spin oscillation periods are fit well by the sSMA predictions as the time dependence\nofc2(t) is less relevant for larger q.\nIII. CLASSICAL INTERPRETATION OF SPATIAL DYNAMICS: LISSAJOUS CURVES\nA. Model\nFigure 3 of the main text presents an analysis of the experimentally observed spatial dynamics of the condensate.\nSpecifically, the plot tracks the positions of the L- and R1-peaks in momentum space. These peaks are extracted\nby fitting each momentum peak in a TOF image with a 2D Gaussian and extracting the fitted position. Positions\nare then converted to quasimomentum. The main text compared the experimental results to the predicted Lissajous\ncurves. This section introduces the classical trajectory model that generates these Lissajous curves, using a single\nconfined particle and initial conditions that are determined self-consistently from the properties of the experimental\noptical lattice and confining potential.\n-0.8 -0.6 -0.4 -0.20.0-0.20.00.20.40.60.8\npcom,x/ℏkLpcom,y/ℏk\nFIG. S7. Example trajectory of the center-of-mass momentum pcomextracted from a 2D GPE simulation via pcom=/integraltext\nd2p p/summationtext\nm|ψm(p,t)|2, for a lattice depth of uL= 1.2ERand ∆f= 4.6ER/h.\nOur model is underpinned by the insight that the spatial motion of the condensate atoms is predominantly controlled\nby the approximately harmonic trap and the main effect of the optical lattice is to stroboscopically impart sudden\n(instantaneous) kicks to the atoms when a resonance condition is satisfied dynamically. This is well supported by\nan inspection of the time evolution of the center-of-mass (COM) momentum of the entire atomic cloud according\nto a 2D GP calculation, as shown in Fig. S7. The phase-space trajectory of the COM momentum is dominated by\nsmooth evolution – as different momentum components move freely within the confining potential V2D(x,z) – with\nthe exception of a sequence of regular abrupt reversals, indicating the sudden imparting of 2 /planckover2pi1kLto a subset of atoms\nby the moving optical lattice. Our classical model constructs analogous Lissajous curves for an ensemble of individual\n(non-interacting) momentum components – defined with respect to their initial position and velocity within the trapS9\n– that are treated as classical particles traversing the 3D confining potential V3D(r). Note that our classical trajectory\nsimulations employ the full 3D potential since these calculations are not numerically challenging. The initial position\nand momentum for each particle are determined by solving for the dynamically appearing resonances at which the\nmoving lattice couples different momentum components.\nFor concreteness, we introduce the specifics of our model by using it to describe the trajectory of the L-peak, which\nis assumed to be instantaneously populated at t=t1(i.e., when the lattice detuning is suddenly quenched) due to\na near-resonant coupling mediated by the moving lattice between the initial BEC at p= 0 and a second non-zero\nmomentum state p= 2/planckover2pi1kLwhose kinetic energy differs by 4 ER/h≈∆f. Already, we note that the assumption\nthat the L-peak is “born” instantaneously at t=t1is an oversimplification, but we argue that for lattice depths uL\non the order of ERthe coupling between momentum states generated by the moving lattice is much faster than any\nsubsequent spatial motion. Based on this description, the L-peak is initially located at the central minimum of the\ntrap, which herein we define as the origin of our co-ordinate system, rL(t=t1) = (0,0,0), as it is directly outcoupled\nfrom the stationary BEC, and has initial momentum pL(t=t1) = 2 /planckover2pi1kL.\nWe expect the L-peak to quickly separate from the stationary BEC and then to begin climbing the walls of the\nconfining potential and decelerate. To describe this evolution, we assume that the mean position and momentum of\nthe L-peak follow a trajectory defined by an equivalent classical particle subject to the confining potential V3D. Such\na trajectory is described by solving the equations,\nMNad2r(t)\ndt2=−∇V3D[cos(30◦)x−sin(30◦)y,cos(76◦) sin(30◦)x+ cos(76◦) cos(30◦)y+ sin(76◦)z,sin(76◦)z−cos(76◦)y]\n(S11)\nand\np(t) =MNadr(t)\ndt, (S12)\nfor the mean position and momentum of the corresponding classical particle, respectively. At this point, we highlight\nthat we are assuming that the moving lattice is effectively invisible for the L-peak’s motion; this is justified since\nthe L-peak’s motion rapidly changes its “status” from being on resonance with the p= 0 BEC at t=t1to being\noff-resonant with the p= 0 BEC for tjust a bit larger than t1.\nThe L-peak moves freely within the confining potential until it dynamically satisfies a resonance condition where\nthe optical lattice instantaneously couples the L-peak with momentum pL(t) to a new state – the R1-peak – with\nmomentum pR1(t) =pL(t) + 2/planckover2pi1kL. The resonance condition is defined by equating the difference between the kinetic\nenergies of these states (it is assumed that the potential energy is unchanged as the new momentum component is\ncreated at the same location within the trap) with the energy imparted by the moving lattice, i.e., 4 .6ER,\n|pL(t) + 2 /planckover2pi1kL|2\n2MNa−|pL(t)|2\n2MNa= 4.6ER. (S13)\nFor the lattice vector kLused in our moving-lattice experiment, this resonance condition can be written as a relation\nbetween different momentum components of the L-peak,\npL(t)·kL= (0.6/4)/planckover2pi1k2\nL. (S14)\nIn the experiment, we approximately have kL=kL[−sin(40◦),0,cos(40◦)], which leads to the resonance condition\nofpL,z= 0.86pL,x+ 0.20/planckover2pi1kL. The critical time when this resonance is fulfilled is estimated to be tcr= 34.6/planckover2pi1/ER=\n1.67 ms. The time tcralso defines the time at which we assume the R1-peak is “born”. The initial condition for the R1-\npeak is then straightforward to compute as rR1(tcr) =rL(tcr) = (−65.0,0,55.1)k−1\nLandpR1(tcr) =pL(tcr) + 2 /planckover2pi1kL=\n(−0.42,0,−0.35)/planckover2pi1kL+ 2/planckover2pi1kL. The computation of the Lissajous curve for the R1-peak, and subsequent creation of,\ne.g., the R2-peak, follows the same scheme as discussed above.\nWe reiterate that these theoretical calculations should only be relied upon as a qualitative guide to help understand\nthe mechanism of trap-induced resonances that lead to the experimentally observed dynamical fracturing of the BEC\nin momentum space. The Lissajous curves can only provide an approximate guide to the observed trajectories of\nthe fractured components in momentum space for a number of reasons. Primarily, these are that the phase-space\ntrajectories generated by Eqs. (S11) and (S12) ignore the role of: i) interactions and ii) time-dependent corrugation\nof the confining potential due to the moving lattice.\nLastly, we comment on the width of the approximate resonance. As previously discussed, the optical lattice couples\nmomentum states separated by 2 /planckover2pi1kLover a finite energy window, i.e., states that do not exactly fulfill the resonance\ncriterion Eq. (S14). While quantifying the width of this energy window, or resonance region, is not straightforward,\nand further complicated by the role of interactions and motion of the BEC components in the ODT, we can provide\na rough estimate.S10\nWe use that the coupling of the initial p= 0 BEC to the p= 2/planckover2pi1kLmomentum state is off-resonance (the\nenergy difference between these states is 4 ERrather than 4 .6ER) to determine an estimate of a minimum width of\nthe resonance region. Specifically, the fact that we see atoms transfer from the zero momentum state to the 2 /planckover2pi1kL\nmomentum state at t≈t1for a detuning of 4 .6ER, even though the true resonance for these momentum states occurs\nfor a detuning of 4 ER, indicates that the lattice-induced coupling of momentum states is near resonance for a window\nof at least 4 .6ER±0.6ER. Correspondingly, the upper and lower boundaries of the resonance width can be derived\nby replacing the right hand side of Eq. (S13) with 5 .2ERand 4ER, respectively. This width is what is plotted in\nFig. 3(b) of the main text and Fig. S8 as gray shaded regions.\nB. Experimental comparison with Lissajous curves\npz/ħkL\npx/ħkLRed (Blue) : R1-Peak (L-Peak) \nSolid lines: theor y Markers: experime nt -2 -1 01\n-2 -1 0 1\nFIG. S8. Trajectories of the mean position of L-peak (blue triangles) and R1-peak (red squares), extracted similarly to Fig. 3\nof the main text. Markers represent data taken in a typical ODT ( LODT,z= 20µm). Blue (red) solid lines are L-peak (R1-peak)\npositions according to Lissajous curves based on a calculation of a classical point particle in an ideal (effectively, LODT,i=∞)\nharmonic trap. Preceding L-peak formation, atoms occupy the p= 0 state at pz=px= 0 (prepared BEC) and the p= 2/planckover2pi1kL\nstate. The gray shaded region marks the approximate resonance region where decelerated atoms are kicked by the lattice (to\ncreate, e.g., the R1-peak).\nAs illustrated in Fig. 3 of the main text, theoretical calculations of the Lissajous trajectories qualitatively reproduce\nthe experimental trajectories that utilize the compressed ODT. However, as discussed in the main text, ODT com-\npression is accompanied by lower condensate fractions and reduced visibility (see also Fig. S3). In Fig. S8 we compare\nthe Lissajous curves with the ODT typically employed in our experiment (uncompressed trap with LODT,z= 20µm).\nWe find stark differences between experiment and theory curves for these parameters, including an early exiting of\natoms out of the ODT trap due to gravity. The disagreements found in the compressed trap of Fig. 3 and Fig. S8\nillustrate that fully capturing all the aspects of the experiment is necessarily beyond the simplified classical model,\nwhich is designed to minimally capture the dynamical appearance of resonances between different momentum states.\nA closer quantitative comparison would be provided by, e.g., a full 3D GP simulation and analysis of momentum\nspace dynamics analogous to what is carried out in Figs. 3 and S8 for the experimental data.\n[1] Chen, Z. et al. Quantum quench and nonequilibrium dynamics in lattice-confined spinor condensates. Phys. Rev. Lett.\n123, 113002 (2019). And references therein.\n[2] Zhao, L., Jiang, J., Tang, T., Webb, M. & Liu, Y. Dynamics in spinor condensates tuned by a microwave dressing field.\nPhys. Rev. A 89, 023608 (2014).\n[3] Zhao, L., Jiang, J., Tang, T., Webb, M. & Liu, Y. Antiferromagnetic spinor condensates in a two-dimensional optical\nlattice. Phys. Rev. Lett. 114, 225302 (2015).S11\n[4] Knoop, S. et al. Feshbach spectroscopy and analysis of the interaction potentials of ultracold sodium. Phys. Rev. A 83,\n042704 (2011). URL https://link.aps.org/doi/10.1103/PhysRevA.83.042704 .\n[5] Dennis, G. R., Hope, J. J. & Johnsson, M. T. XMDS2: Fast, scalable simulation of coupled stochastic partial differential\nequations. Computer Physics Communications 184, 201–208 (2013). URL https://www.sciencedirect.com/science/\narticle/pii/S0010465512002822 .\n[6] Law, C. K., Pu, H. & Bigelow, N. P. Quantum spins mixing in spinor Bose-Einstein condensates. Phys. Rev. Lett. 81,\n5257–5261 (1998). URL https://link.aps.org/doi/10.1103/PhysRevLett.81.5257 .\n[7] Stamper-Kurn, D. M. & Ueda, M. Spinor Bose gases: Symmetries, magnetism, and quantum dynamics. Rev. Mod. Phys.\n85, 1191 (2013).\n[8] Yi, S., M¨ ustecaplıo˘ glu, O. E., Sun, C. P. & You, L. Single-mode approximation in a spinor-1 atomic condensate. Phys.\nRev. A 66, 011601 (2002). URL https://link.aps.org/doi/10.1103/PhysRevA.66.011601 .\n[9] Liu, Y. et al. Quantum phase transitions and continuous observation of spinor dynamics in an antiferromagnetic condensate.\nPhys. Rev. Lett. 102, 125301 (2009).\n[10] Zhang, W., Zhou, D. L., Chang, M.-S., Chapman, M. S. & You, L. Coherent spin mixing dynamics in a spin-1 atomic\ncondensate. Phys. Rev. A 72, 013602 (2005). URL https://link.aps.org/doi/10.1103/PhysRevA.72.013602 .\n[11] Kronj¨ ager, J., Becker, C., Navez, P., Bongs, K. & Sengstock, K. Magnetically tuned spin dynamics resonance. Phys. Rev.\nLett.97, 110404 (2006). URL https://link.aps.org/doi/10.1103/PhysRevLett.97.110404 ."    },    {        "title": "2004.08435v1.Spectroscopic_signatures_of_next_nearest_neighbor_hopping_in_the_charge_and_spin_dynamics_of_doped_one_dimensional_antiferromagnets.pdf",        "content": "Spectroscopic signatures of next-nearest-neighbor hopping in the charge and spin\ndynamics of doped one-dimensional antiferromagnets\nUmesh Kumar,1, 2, 3Gregory Price,4, 5Kenneth Stiwinter,5\nAlberto Nocera,6, 7Steven Johnston,1, 2,\u0003and Trinanjan Datta5,y\n1Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37966, USA\n2Joint Institute of Advanced Materials at The University of Tennessee, Knoxville, Tennessee 37996, USA\n3Theoretical Division, T-4, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA\n4Department of Mathematics, Augusta University, 1120 15thStreet, Augusta, Georgia 30912, USA\n5Department of Chemistry and Physics, Augusta University, 1120 15thStreet, Augusta, Georgia 30912, USA\n6Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4\n7Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1\n(Dated: April 21, 2020)\nWe study the impact of next-nearest-neighbor (nnn) hopping on the low-energy collective excita-\ntions of strongly correlated doped antiferromagnetic cuprate spin chains. Speci\fcally, we use exact\ndiagonalization and the density matrix renormalization group method to study the single-particle\nspectral function, the dynamical spin and charge structure factors, and the Cu L-edge resonant\ninelastic x-ray scattering (RIXS) intensity of the doped t-t0-Jmodel for a set of t0values. We \fnd\nevidence for the breakdown of spin-charge separation as jt0jincreases and identify its \fngerprints\nin the dynamical response functions. The inclusion of nnn hopping couples the spinon and holon\nexcitations, resulting in the formation of a spin-polaron, where a ferromagnetic spin polarization\ncloud dresses the doped carrier. The spin-polaron manifests itself as additional spectral weight in\nthe dynamical correlation functions, which appear simultaneously in the spin- and charge-sensitive\nchannels. We also demonstrate that RIXS can provide a unique view of the spin-polaron, due to its\nsensitivity to both the spin and charge degrees of freedom.\nI. INTRODUCTION\nA central problem in condensed matter physics is to\nunderstand how charge and spin carriers couple to collec-\ntive excitations in strongly correlated materials. For ex-\nample, the behavior of a small number of holes introduced\ninto an antiferromagnetic background of spins lays at the\nheart of unconventional superconductivity in the high-\ntemperature (high-T c) superconducting cuprates [1{4].\nBut it is still an open question as to how superconduc-\ntivity emerges from the complex interplay of the spin,\ncharge, and lattice excitations [1, 2, 5{11].\nTo address this problem, the community has developed\npowerful numerical approaches for simulating single- and\nmulti-band Hubbard models and several techniques are\nnow available for computing their ground and excited-\nstate properties [5, 8{10, 12{21]. It is now possible\nto make detailed predictions of dynamical correlation\nfunctions in many cases, which can be compared di-\nrectly with spectroscopies like angle-resolved photoemis-\nsion spectroscopy (ARPES) [22, 23], inelastic neutron\nscattering (INS) [24], and resonant inelastic x-ray scat-\ntering (RIXS) [25, 26].\nAlgorithmic advances have produced signi\fcant new\ninsights into the physics of the Hubbard model itself,\nand a conceptual picture of competition has come into\nfocus in recent years [27, 28]. Here, the strong electronic\n\u0003Corresponding author:sjohn145@utk.edu\nyCorresponding author:tdatta@augusta.educorrelations produce multiple nearly-degenerate states,\nwhere subtle perturbations can stabilize one state over\nanother. For example, state-of-the-art numerical simula-\ntions of the singleband Hubbard model have found that\nits ground state lies very close in energy to various charge-\nand spin-orders (i.e. stripes) that compete with super-\nconductivity [8{10, 20, 29]. As a result, the pure Hubbard\nmodel with only nearest-neighbor (nn) hopping tdoes not\nappear to have a superconducting ground state for inter-\naction strengths that are physically relevant for the high-\nTccuprates [9]. This conclusion is extremely sensitive to\nperturbing factors [9, 10, 20, 29], however. For example,\nthe inclusion of next-nearest-neighbor (nnn) hopping t0\ncan frustrate the stripes and stabilize d-wave supercon-\nductivity [9, 10]. It is, therefore, important to study the\ne\u000bects of nnn hopping and other realistic factors like dis-\norder, lattice interactions, and additional orbitals on the\nproperties of correlated electron models. It is also neces-\nsary to elucidate their e\u000bects on the dynamical properties\nof the model to be able to identify their relevance in real\nmaterials with spectroscopies. A non-zero t0, for instance,\nhas a measurable in\ruence on the spin response of the 2D\nHubbard model [18], which can be studied using INS or\nRIXS.\nQuasi-one-dimensional (1D) systems have attracted\nconsiderable attention { both from a theoretical [3, 9, 30{\n38] and experimental [39{47] perspective { as these sys-\ntems have traditionally been more amenable to theoreti-\ncal modeling and analysis [31]. Our current understand-\ning of the correlated 1D spin chains described by the t-J\nor Hubbard model with nn hopping is built on the idea\nof an exotic quantum liquid that can support spinlessarXiv:2004.08435v1  [cond-mat.str-el]  17 Apr 20202\ncharge (holons) and chargeless spin (spinons) excitations.\nBosonization calculations supplemented with renormal-\nization group analysis suggest that the universal \fxed\npoint for the fermionic 1D t-Jchain is the Luttinger liq-\nuid [31], which can support spin-charge separated (frac-\ntionalized) holon and spinon modes. But the introduc-\ntion of the nnn hopping t0can spoil this clean separation\nof the spin and charge degrees of freedom such that it\nis no longer possible to write down a wave function fac-\ntorized into its constituent charge and spin excitation\ncomponents.\nThe nature of the quantum state in the 1D t-Jchain\nwith further neighbor hopping and interactions has been\nstudied via exact diagonalization [32, 33, 35] and per-\nturbative [48] methods. The ground state of a 1D\nHubbard chain with nnn hoppings was also studied\nwith the density matrix renormalization group method\n(DMRG) [49]. These studies indicate the presence of a\nferromagnetic spin-polaron state in a suitable parame-\nter regime [32, 33, 35, 48]. The perturbative approach\npoints to the existence of either a ferromagnetic or an-\ntiferromagnetic state depending on the relative sign of t\nandt0[35]. Treating the hopping of the hole to third-\norder in perturbation theory also suggests an e\u000bective\nexchange constant that scales as Je\u000b\u0018t(t0)2\n\u000f2. (Here,tis\nthe nn hopping, t0is the nnn hopping, and \u000fis the on-site\nenergy, di\u000berent from Hubbard U). The 1Dt-t0-Jmodel\ncan be mapped onto a zig-zag chain. Hence, the bound\nstate of the triplet state (two up spins) with a single hole\ncan be visualized to live on the triangular plaquette [33].\nThe physics of spin-polarons has also recently received\nrenewed interest in the context of the 2D cuprates [50].\nEarly calculations of the dynamical properties of the\nt-t0-Jmodel and spin-polaron were carried with ED for\nshort chains [33]. Here, we study the problem more\nbroadly with improved momentum resolution and includ-\ning the RIXS response. Our goal is to determine how nnn\nhopping alters the physics of a strongly-correlated anti-\nferromagnetic spin chains and how this might be detected\nspectroscopically. To this end, we use density matrix\nrenormalization group (DMRG) to compute the single-\nparticle spectral function A(k;!), which can be mea-\nsured using ARPES, and the dynamical spin- and charge-\nstructure factors S(q;!) andN(q;!), respectively, which\ncan be measured using e.g. INS or EELS (electron\nenergy-loss spectroscopy). RIXS also encodes informa-\ntion about spin- and charge- excitations, but making di-\nrect links between the RIXS intensity and S(q;!) and\nN(q;!) is not straightforward [18, 51]. We, therefore,\nalso explicitly compute the Cu L-edge RIXS response of\nthe model using ED and the Kramers-Heisenberg formal-\nism. In doing so, we provide predictions for the dynami-\ncal properties of the t-t0-Jmodel for a range of t0=tvalues.\nWe \fnd evidence for the breakdown of spin-charge sepa-\nration and the formation of a spin-polaron with increas-\ningjt0j, but which is sensitive to the sign and magnitude\nof the nnn hopping. Furthermore, t0also has an (some-\ntimes drastic) impact on the collective excitations. Wealso show how the spin-polaron's presence can be identi-\n\fed through the appearance of additional spectral weight\nthat forms simultaneously in the spin- and charge chan-\nnels of the system's dynamical response functions.\nThis paper is organized as follows: Sec. II presents the\nmodel and methods used to study the dynamical corre-\nlation functions and RIXS cross-section. Next, Sec. III\npresents our results, beginning with a review of the non-\ninteracting limit in Sec. III A. Sec. III B focuses on\nthe single-particle spectral function, Sec. III C focuses\non the dynamical spin- and charge-structure factors, and\nSec. III D focuses on the Cu L-edge spectra. Finally, Sec.\nIV provides some additional discussion and presents our\nconclusions.\nII. MODEL AND METHODS\nOur goal is to understand the momentum-resolved low-\nenergy (\u00141 eV) collective excitations of 1D AFM cuprate\nspin chains. These materials have been studied success-\nfully in the past using the t-JHamiltonian, where high-\nenergy charge and orbital excitations are integrated out\nof a complete multi-orbital model. For example, the t-\nJmodel reproduces the low-energy RIXS [40, 41] and\nINS [52] spectra reported for the undoped corner-shared\ncuprate Sr 2CuO 3, as well as the INS data reported for\nthe zig-zag system SrCuO 2[53]. The use of an e\u000bective\nt-Jmodel to describe the spin-chain cuprates is also sup-\nported by a recent DMRG study that explicitly compared\nthe RIXS spectra obtained with this model to the spectra\ncomputed from a four-orbital pd-model for Sr 2CuO 3[54].\nThis study found that the two models agree at low-energy\n(\u00141 eV), apart from a scaling factor in their intensity\nthat was attributed to covalency e\u000bects [54].\nThe results mentioned above indicate that the t-J\nmodel and its extensions can provide reliable predic-\ntions of the low-energy properties of strongly correlated\ncuprate spin-chains. As outlined in the introduction,\nhowever, it is also important to understand how longer-\nrange hopping in\ruences the results obtained from the\nmodel. To address this issue, we adopted a computa-\ntional framework similar to the one used in Refs. 36 and\n40, but extended to include nnn hopping. The Hamilto-\nnian is given by\nH=X\ni;j;\u001btij~cy\ni;\u001b~cj;\u001b+JX\ni\u0012\nSi\u0001Si+1\u00001\n4nini+1\u0013\n:(1)\nHere,tij=tandt0are the nn ( j=i\u00061) and nnn\n(j=i\u00062) hopping integrals, respectively, and tij= 0\notherwise; ~cy\ni;\u001b(~ci;\u001b) is the creation (annihilation) oper-\nator for a spin- \u001b(=\";#) hole at site iunder the con-\nstraint of no double occupancy; Jis the exchange cou-\npling;ni=P\n\u001b~cy\ni;\u001b~cj;\u001bis the number operator; and Siis\nthe spin operator at site i. To facilitate comparisons with\nprevious work, we adopt model parameters t= 0:4 eV,3\nJ= 0:25 eV, which are typical for corner-shared cuprate\nmaterials [36], and vary t0between [\u0000t\n2;t\n2].\nWe access the model's collective excitations by com-\nputing several dynamical correlation functions. The \frst\nis the single-particle spectral function\nA(k;!) =X\nf;\u001b\f\f\fhfj~cy\nk;\u001bjgi\f\f\f2\n\u000e(Ef\u0000Eg\u0000!);(2)\nwhere ~cy\nk;\u001b=1p\nNP\nie\u0000ikRi~cy\ni;\u001bis the Fourier transform\nof the creation operator, and Riis the lattice vector for\nthe magnetic atom in unit cell i, which is associated with\nCu in our case. We also considered the two-particle dy-\nnamical spin and charge structure factors, which are de-\n\fned as\nS(q;!) =X\nf\f\f\fhfj^S\u000b\nqjgi\f\f\f2\n\u000e(Ef\u0000Eg\u0000!) (3)\nand\nN(q;!) =X\nf\f\f\fhfj^Nqjgi\f\f\f2\n\u000e(Ef\u0000Eg\u0000!); (4)\nrespectively. Here, qand!are the 1D momentum and\nenergy transfer to the chains, respectively, and ^S\u000b\nq=\n1p\nNPN\ni=1e\u0000iqRiS\u000b\niand ^Nq=1p\nNPN\ni=1e\u0000iqRiniare the\nFourier transforms of the S\u000b\nilocal spin ( \u000b=z;\u0006) and\nni=P\n\u001b~cy\ni;\u001b~ci;\u001bis the number operator, respectively.\nThe RIXS intensity I(q;!) is computed using the\nKramers-Heisenberg formalism, and is given by\nI(q;!)/X\nfjMfgj2\u000e(Ef\u0000Eg\u0000!); (5)\nwhere\nMfg=X\nnhfjDy\nkoutjnihnjDkinjii\nEg+!in\u0000En+i\u0000n:\nIn the above expression, the incoming (outgoing) pho-\ntons have energy !in(!out) and momentum kin(kout);\n!=!in\u0000!outandq=kin\u0000koutare the energy and\nmomentum transferred along the chain direction, respec-\ntively;jgi,jni, andjfiare the initial, intermediate, and\n\fnal states of the RIXS process with energies Eg,En,\nandEf, respectively; Dkis the dipole operator describ-\ning the 2p!3datomic transition; and \u0000 nis related to\nthe inverse core-hole lifetime. For our numerical calcula-\ntions we used \u0000 n= 0:3 eV, independent of the value of\nn, as is appropriate for the Cu L-edge [18].\nAt the CuL-edge, the dipole operator takes the form\nDk=X\ni\u001beik\u0001Rih\n~di;\u001bpy\ni;\u001b+ h:c:i\n; (6)\nwherepy\ni;\u001b(pi;\u001b) creates (annihilates) a spin \u001bhole in a\nCu 2porbital located at site i. Note that we have ne-\nglected an orbital-dependent prefactor in Eq. (6) thatdepends on the photon polarization and scattering ge-\nometry. In what follows, we will consider the spin-\nconserving and non-spin-conserving channels individu-\nally. We, therefore, also take into account the e\u000bect of\nthe spin-orbit coupling in the description of the 2 pcore\nstates.\nFor all of the RIXS spectra shown in this work, we set\nthe incident photon !into coincide with the resonance\n(maximum intensity) of the XAS obtained for the same\nmodel. Here, IXAS(!) is computed using Fermi's golden\nrule and is given by\nIXAS(!)/X\nnjhnjDk=0jgij2\u000e(En\u0000Eg\u0000!):(7)\nThe single particle spectral function and dynamical\nspin and charge structure functions are computed using\nthe DMRG correction-vector method [55] and the Krylov\ndecomposition [56], as implemented in the DMRG++\ncode [57]. This approach requires real-space represen-\ntations of Eqs. (2) { (4), which can be found in Ref.\n58. Our DMRG calculations were carried out on N= 80\nsite chains with open boundary conditions with a \fxed\nnumber of holes ( Nh= 76) such thathni=Nh\nN= 0:95.\nWe also kept up to m= 1000 DMRG states to maintain\na truncation error below 10\u00007and introduced a spec-\ntral broadening in the correction-vector approach \fxed at\n\u0011= 0:08t. To compute the XAS and RIXS intensities, we\ndiagonalized Eq. (1) exactly on N= 20 site chains with\nperiodic boundary conditions and used the eigenstates to\nevaluate Eqs. (5) and (7). When numerically evaluating\nEq. (5), we approximated the energy-conserving delta\nfunction with a Gaussian function \u000e(!)\u00191\n\rp\n2\u0019e\u0000!2\n2\r2,\nwith\r=t\n10.\nIII. RESULTS AND DISCUSSION\nA. Results in the non-interacting limit\nBefore proceeding to our main results, it is useful to\nexamine the single- and two-particle responses for the\nnon-interacting model. Our aim here is to remind the\nreader of how t06= 0 alters the bare band structure and\nthe topology of the Fermi surface [59].\nFigure 1 shows results for the single-particle disper-\nsion and dynamical charge structure factor N(q;!). The\nexcitation spectrum in this case is determined by the dis-\npersion\u000f(k) = 2tcos (ka) + 2t0cos (2ka), which is plotted\nas thin black lines in Figs. 1(a)-(c) for t0=t= 0,\u00001\n2,\nand1\n2, respectively. When t0= 0, the band structure\nhas a simple cosine shape, with a local minimum lo-\ncated at the zone boundary, consistent with the use of\nhole language in Eq. (1). The band structure is modi-\n\fed fort06= 0 (throughout, we assume that jt0j\u0014jt=2j).\nFor example, the band maxima shift from zone center\ntokmin=\u00061\nacos\u00001(t\n4t0) whent0<\u0000t\n4[Fig. 1(b)],\nwhile the local minima shifts from the zone boundaries4\nFIG. 1. Results for the noninteracting 1D t-t0model, cal-\nculated on an N= 400 site chain. Panels (a) - (c) show\nthe band dispersion of the non-interacting model \u000f(k) =\n2tcos(ka) + 2t0cos(2ka) fort0= 0,\u0000t\n2, andt\n2, respectively.\nThe thick red overlays indicate the location of the \flled states\nin the dilute limit and the dashed line indicates the resulting\nFermi energy. Panels (d) - (f) show the corresponding dynam-\nical charge structure factors N(q;!) calculated on an N= 400\nsite cluster and a total \flling of hni= 0:05. The overlays in\npanels (d) - (f) are given by !(q) =\u000f(k)\u0000\u000f(kmin) and plotted\nas a function of q=k\u0000kmin.\ntokmin=\u00061\nacos\u00001(\u0000t\n4t0) whent0>t\n4[Fig. 1(c)]. These\nchanges in the locations of the band extrema alter the\ntopology of the Fermi surface when the band is nearly\nempty or nearly \flled. This fact is illustrated by the thick\nred lines in Fig. 1, which indicate the occupied states in\nthe case of a nearly empty band. One can see that the\nFermi surface transitions from a single pocket centered\nat the zone boundary for t0\u0014t\n4to two Fermi surfaces\ncentered at kmin6=\u0006\u0019\nawhent0>t\n4.\nThe topological change of the Fermi surface can have\na profound e\u000bect on the excitation spectrum of the sys-\ntem. To illustrate this, Figs. 1(d)-(f) plots N(q;!) ob-\ntained whenhni= 0:05 spinless fermions/unit cell oc-\ncupy the bands shown in Figs. 1(a)-(c), respectively.\n(We consider spinless fermions to make a connection to\nholon excitations in the doped interacting chains, see\nbelow and also App. B of Ref. [36].) The overlays in\nFigs. 1(d)-(f) are guides to the eye; they are given by\n!(k\u0000kmin) =\u000f(k)\u0000\u000f(kmin), wherekminis the momentum\nof the lowest \flled state in each case. In other words, the\noverlay represents scattering from the band minimum toanotherkvalue. In this case, N(q;!) measures intraband\nscattering and provides indirect information on the un-\nderlying band structure. The charge response in panels\n(d) and (e) consist of a single dispersing narrow contin-\nuum. In contrast, panel (f) has two distinct branches,\nwhich occurs because scattering to the left and the right\nfrom each of the Fermi surfaces is no longer equivalent\nwhent0=t\n2and the band minima shift away from the\nhigh-symmetry points. Many of these features persist\nin the interacting system (see below), but the electronic\ninteractions alter the underlying band dispersions.\nB. The single-particle spectral function for the\ninteracting case\nWe now turn our attention to the electronic proper-\nties and spectral functions of the interacting chains. The\nphysical behavior of the 1D t-Jmodel without next-\nnearest neighbor hopping is well established [48, 60{62].\nIts elementary excitations are collective spin and charge\ndensity excitations that can be mapped onto fraction-\nalized quasiparticle excitations. The spin density exci-\ntations map to chargeless spin-1\n2quasiparticles known\nas spinons with a dispersion relation given by !s(k) =\n\u0019J\n2sin(ka). Similarly, the charge density excitations map\nto spinless charge- equasiparticles known as holons with\na dispersion !h(k) = 2t[1\u0000cos(ka)].\nFigure 2 shows our DMRG results for the single-\nparticle spectral function A(k;!) forhni= 0:95 and val-\nues oft0, as indicated. When t0= 0,A(k;!) has two\ndistinct dispersing features, which correspond to the ex-\npected spinon and holon excitations. The dispersions\nof these quasiparticles are highlighted using the solid\ngreen and dashed red lines, which trace the expected\npath. These spectral features are a \fngerprint of spin-\ncharge separation and have been directly observed in\nSrCuO 2and other 1D spin chains using ARPES [23, 63{\n66]. Moreover, the spectrum resembles the main features\nof the spectral function of the 1D Hubbard model at\nU=t\u001d1 [67, 68]. In particular, we can attribute the spec-\ntral weight above the Fermi level to the spinon-antiholon\ncontinuum of empty states due to the anomalous spectral\nweight transfer upon hole-doping (electron doping in this\nwork since we are assuming hole language).\nThe spectral function for t0=\u0000t\n2is qualitatively sim-\nilar in that two distinct sets of excitations are observed\nthat resemble the spinon and holon excitations in the\nt0= 0 case. Their dispersions, however, are modi\fed\nfrom the pure cosine and sine forms, and a slight kink-\nlike feature appears in the dressed dispersion relationship\nwhere the spinon and holon excitations overlap. We at-\ntribute these changes to a coupling between the spin and\ncharge degrees of freedom introduced by the non-zero t0\n[see Fig. 1(b)]. We will show later that the changes in\nthe quasiparticle dispersions can also be observed in the\ntwo-particle correlation functions.\nThe situation is much di\u000berent when t0=t\n2, as shown5\nFIG. 2. DMRG results for the single particle spectral function\nA(k;!) of thet-t0-Jmodel. The spectra were calculated on an\nN= 80 site chain with t= 0:4 eV,J= 0:25 eV,hni= 0:95,\nand values of t0as indicated in each panel. The solid green\nand dashed red lines in panel (a) are guides for the eye for\nthe spinon and holon excitations, respectively.\nin Fig. 2(c). In this case, the spectral weight is com-\npletely reorganized, and one can no longer make out\nspectral features for the spinons and holons. Instead,\nwe \fnd two new distinct sets of features. The \frst is\na sharp peak located near EFand centered at the zone\nboundary. This excitation is quasi-particle-like and has\na very narrow bandwidth, whose spectral weight is sup-\npressed ask!0. The second feature is a more incoher-\nent but still quasi-particle-like excitation located between\n!\nt2[\u00004;0]. This excitation has a cosine-like dispersion\nwith a period of\u0019\na. These changes are a clear indicator\nthat the spin-charge separation picture has broken down\nfor this value of t0. In the next section, we will show that\nthe dynamical spin and charge structure factors support\nthis conclusion. Our results agree with several previousstudies [32, 33, 35] that found evidence for binding of the\nspinons and holons using similar models and parameters.\nThe introduction of t0can be viewed as converting\nthe lattice geometry from a chain to a triangular lad-\nder, thus creating an intermediate structure between\n1D and 2D. In such a system, the physical properties\nare known to deviate signi\fcantly from the spin-charge\nseparation scenario, which is strictly valid in 1D only\n(t0= 0). From this point of view, it is interesting to\ncompare our results with those obtained in 2D. For ex-\nample, our results in Fig. 2c resemble the main features\nof theU=t\u001d1 spectral function of the 2D Hubbard\nmodel [14, 69{71], where a prominent quasiparticle band\nappears due to the anomalous spectral weight transfer\nfrom the upper Hubbard band to the lower Hubbard band\nupon hole doping [72]. A similar feature was also iden-\nti\fed as a \rat band in early studies in the 2D t-Jmodel\n(witht0= 0) upon hole doping [73, 74]. In our case,\nwe attribute the narrow band around the Fermi level\n(!= 0) with quasiparticle excitations with weakly ferro-\nmagnetic character (as shown below in our ground state\nanalysis for ferromagnetic-polaron correlation functions\nin Fig.3b). Fluctuations arising from the Nagaoka fer-\nromagnetism [75] could dominate the antiferromagnetic\n\ructuations near the Mott transition upon doping, in the\nextremely large U=t!1 limit of the Hubbard model.\nAs we will see below when discussing the spin excitation\nspectrum, this feature appears as a faint excitation below\nthe two-spinon continuum for small momentum transfer\nq'0. In the charge channel, it appears as a sharp\nlow energy mode with spectral weight concentrated at\nzone center, with a width of the order of the exchange\ninteraction J. In summary, our results for t0=t=2 devi-\nate substantially from strictly spin-charge separated 1D\ncharacter, indicating that the doped charge is no longer\nfractionalizing, or that t06= 0 induces an e\u000bect similar to\na dimensional crossover towards 2D.\nAs previously mentioned, the introduction of a non-\nzerot0is expected to produce a spin-polaron [32, 33, 35].\nHere, the doped electron is dressed by a spin polarization\ncloud whose spatial extent depends on the strength of the\nexchange coupling. Speci\fcally, the size of the ferromag-\nnetic polarization cloud around a single hole grows as J=t\ndecreases, eventually extending across the entire system\nwhenJ=t!0 [32]. To con\frm the presence of the spin-\npolaron in our case, and to determine its spatial extent,\nwe computed the spin-polaron correlation function\nC(i) =\u001a\nhSc\u00001\u0001nh\ncSc+1i=hnh\nciifi= 0\nhnh\ncSi\u0001Si+1i=hnh\nci ifi>0(8)\nusing DMRG. Here, nh\nc= 1\u0000ncmeasures the electron\noccupation at the reference site index c(taken to be the\ncenter site of the chain, unless otherwise stated), and i\nis a site index measured relative to c. Notice that by\nde\fnition,C(i= 0) indicates the spin correlation across\nthe doped electron in the chain[76, 77]. Fig. 3(a) shows\nresults for a single doped electron added to an N= 41\nsite chain. When t0<t\n2,C(i) is negative at all distances,6\nFIG. 3. DMRG results for the spatial dependence of the spin-\npolaron correlation function for t= 0:4,J= 0:25, and as a\nfunction of t0. Results are shown for (a) Nh= 40 on aN= 41\nsite chain and (b) Nh= 76 on aN= 80 site chain.\nindicating that the spins form an antiferromagnetic back-\nground that is largely independent of the hole's position.\nHowever,C(i) increases slightly within two unit cells of\nthe doped electron and when t0=t\n2,C(i) even changes\nsign at these distances. This behavior re\rects the for-\nmation of a small ferromagnetic polarization cloud sur-\nrounding the charge. It also con\frms that the spinons\nand holons become weakly coupled for t0<t\n2but more\nstrongly coupled when t0=t\n2, leading to the formation\nof a spin-polaron. These di\u000berences also explain why we\nsee a more drastic reorganization of the spectral weight\nin Fig. 2 when t0=t\n2. Fig. 3(b) shows analogous results\nfor anN= 80 site chain at 5% doping. The similar-\nity between the two panels shows that the spin-polaron\npicture persists at this doping level.\nC. The dynamical spin and charge structure factors\nWe now consider the e\u000bects of t0on the dynamical spin\nS(q;!) and charge N(q;!) structure factors. These two-\nparticle correlation functions can provide crucial infor-\nmation about the nature of the excitations in the model.\nFIG. 4. DMRG results for the dynamical spin structure factor\nS(q;!) for thet-t0-Jmodel. The spectra were calculated on\nanN= 80 site chain with t= 0:4 eV,J= 0:25 eV,hni= 0:95,\nand values of t0as indicated in each panel.\nWe can also compare and contrast their behavior with a\nmore complicated RIXS response.\nFigure 4 shows DMRG results for S(q;!) athni= 0:95\nwith the same parameters used to produce Fig. 2. Panel\n(a) shows results for t0= 0, which exhibits the usual\ntwo-spinon continuum [41, 52, 61, 78]. We \fnd that the\ngapless excitation near q= 0 broaden somewhat rela-\ntive to the undoped case. At the same time, the gap-\nless excitations near q=\u0006\u0019\nain the undoped case shift\nto\u0006(\u0019\na\u00002kF), wherefkF=\u0019\n2a(1\u0000hni) is the Fermi\nmomentum measured relative to the band minimum, as\nexpected for a Luttinger liquid. These results are fully\nconsistent with prior results for the doped t-Jand Hub-\nbard models [30, 79].\nFigures 4(b) and 4(c) show results for the same system\nbut now with t0=\u0000t\n2andt\n2, respectively. In both cases,\nthe spin excitations manifest as a continuum, but appear7\nFIG. 5. DMRG results for the dynamical charge structure\nfactorN(q;!) for thet-t0-Jmodel. The spectra were calcu-\nlated on an N= 80 site chain with t= 0:4 eV,J= 0:25 eV,\nhni= 0:95, and values of t0as indicated in each panel.\nto harden (soften) for t0=\u0000t\n2(t\n2) relative to the t0= 0\ncase, also in agreement with prior studies [30, 33, 80].\nHowever, we also \fnd that the gapless excitations lo-\ncated at\u0006(\u0019\na\u00002kF) fort0= 0 shift back to \u0006\u0019\nawhen\nt0=t\n2, indicating that the system is deviating from the\nexpectations for a Luttinger liquid. We also observe ad-\nditional dispersing features with weak intensity outside of\nthe boundaries of the two-spinon continuum when t06= 0.\nSpeci\fcally, for t0=\u00001\n2(1\n2), there is a faint excitation\nlocated just above (below) the spinon continuum. Both\nsets of excitations disperse toward zero energy as q!0\nand have corresponding features in the dynamical charge\nstructure factor. We will return to these features in a\nmoment.\nFigure 5 shows the dynamical charge structure factor\nN(q;!), again forhni= 0:95 andt0as indicated in each\npanel. Fort0= 0 [Fig. 5(a)], the charge excitations agreewell with the N(q;!) obtained from the spinless model\nwithhni\u00190:05 [Fig. 1(d)]. This result is to be expected\nif the charge quantum number of the small number of\ndoped carriers are carried by spinless holons.\nThe situation appears to be qualitatively similar when\nt0=\u0000t\n2[Fig. 5(b)] in that N(q;!) has a sharp dis-\npersing feature that loosely resembles the spectrum in\n1(e). A closer inspection, however, reveals two key dif-\nferences: First, the precise dispersion relation in Fig. 5(b)\ndi\u000bers from the non-interacting case [1(e)]. Speci\fcally,\nthe local maximum remains at the zone boundaries and\nthe overall dispersion in N(q;!) has a slight kink-like\nbend near ( q;!) =\u0000\u0019\n2a;3t\u0001\nin the interacting case. In-\nterestingly, these features can also be made out in the\nrenormalized holon dispersion shown in Fig. 2(b), indi-\ncating that the sharp mode in N(q;!) can be viewed as\na renormalized holon excitation. The second signi\fcant\ndi\u000berence is an additional weak continuum of spectral\nweight located below !\u00142tand near the zone bound-\naries, which overlaps with the continuum of spin excita-\ntions seen in S(q;!). The phase space overlap of these\nfeatures in the two correlation functions re\rects the fact\nthat the spin and charge degrees of freedom are coupled\nto some extent, consistent with the formation of a weakly\ndressed spin-polaron.\nThe charge excitations are more signi\fcantly reorga-\nnized when t0=t\n2, as shown in Fig. 5(c). In this case,\nwe observe two distinct sets of excitations, as well as\na large amount of incoherent spectral weight at higher\nenergy. Interestingly, the dispersions of two sharper fea-\ntures inN(q;!) can be inferred from the features noted in\nthe corresponding spectral function shown in Fig. 2(c).\nFor example, the sharp low-energy mode located near\nthe zone center can be linked to particle-hole scatter-\ning within the \rat features in the spectral function near\nEF. These correspond to the quasi-particle-like excita-\ntions with a weakly ferromagnetic character identi\fed in\nthe spectral function analysis of the previous section.\nTheir bandwidth is of the same order of magnitude of\nthe exchange coupling J= 0:25 eV, which is somehow\nis reminiscent of the quasiparticle band found in 2D t-\nJand Hubbard models upon hole doping [69, 73, 74].\nSimilarly, the higher energy feature in N(q;!) can be as-\nsociated with particle-hole excitations from the branch\nof the spectral function at high binding energies to the\nportion near the Fermi level. These results indicate that\nthe substantial reorganization of the spectral function,\nin this case, can also be found in the dynamical charge\nstructure factor. Later, we will show that they also ap-\npear in the spin-conserving channel of Cu L-edge RIXS\nas well.\nIt is important to emphasize that several excitations\nobserved in Fig. 5(b)-(c) overlap with the ones observed\nin Figs. 4(b)-(c), indicating that these excitations have\nboth spin and charge character. The fact that the same\nexcited state appears in both the spin and charge re-\nsponse re\rects the coupling between the fractionalized\nspinon and holon modes and the resulting spin-polaron.8\nFIG. 6. Exact diagonalization results for the Cu L-edge RIXS spectra of the the t-t0-Jmodel in the non-spin-conserving\n(\u0001S= 1), calculated on an N= 20 site chain with t= 0:4 eV andJ= 0:25 eV. Panel (a) shows results for half-\flling\nhni= 1 witht0= 0, whereas panels (b)-(h) show results for hni= 0:95, andt0as indicated. The solid black lines in each panel\nindicate the boundaries of two-spinon continuum expected at half-\flling when t0= 0. Fort0=t > 0 (<0), the continuum of\nspin excitations appear to soften (harden) with increasing jt0j. At the same time, new weak excitations appear outside of the\nspinon continuum when jt0jbecomes large. All panels in this \fgure have been plotted with the same color scale.\nThis interpretation also helps explain why the reorgani-\nzation ofS(q;!) andN(q;!) is stronger when t0=t\n2, as\nthis case has a more well de\fned spin-polaron. These re-\nsults con\frm the predictions made in Refs. 32 and 33 but\nnow on much larger chains with a signi\fcantly improved\nmomentum resolution.\nD. CuL-edge RIXS Spectra\nRIXS is capable of measuring collective charge and spin\nexcitations in a single experiment [25]. This technique\ncan provide a unique view of fractionalized [36, 40, 41] ex-\ncitations in quasi-1D systems. We, therefore, computed\nthe CuL-edge RIXS response for the t-t0-Jmodel for a\nset oft0=f0;t\n10;t\n4;t\n2gto provide a guide for future ex-\nperiments seeking to study the phenomena discussed in\nthe previous sections.\nThe RIXS response at Cu L-edge can be decom-\nposed into spin-conserving (SC) (\u0001 S= 0) and non-spin-\nconserving (NSC) (\u0001 S= 1) channels [81{85], and each\nof these can be resolved by exploiting the polarization of\nthe incoming and outgoing photons [86]. We evaluated\nboth in what follows since each gives distinct information\nabout the system's excitations.1. CuL-edge RIXS in the non-spin-conserving channel\nWe \frst consider the NSC channel. Physically, this\nchannel is most relevant when there is a strong spin-orbit\ncoupling in the core 2 p-orbital of the Cu site [87], and it\nusually dominates the Cu L-edge RIXS spectra. The\nRIXS spectra recorded in this channel also compare well\nwithS(q;!) at the Cu L-edge [51].\nFig. 6 shows our ED results for the Cu L-edge RIXS\nspectra in the NSC channel for the t-t0-Jmodel, which\nwere obtained using ED on an N= 20 site chain. Panel\n(a) shows the spectra for half-\flling ( hni= 1) to ap-\nprise our readers that the spectra closely resemble S(q;!)\nfor the undoped chain. Here, the thin black lines indi-\ncate the boundaries of the two-spinon continuum, and\nour spectrum is con\fned within these boundaries. Panel\n(b) shows the RIXS spectra for hni= 0:95 andt0= 0,\nwhich compares well S(q;!) shown in Fig. 4(a). Specif-\nically, the sharpest feature in the spinon continuum ap-\npears to have hardened and a gapless excitation appears\natk=\u0006(\u0019\na\u00002kF), where 2kFis the Fermi momentum\nas de\fned in Sec. III C.\nFigs. 6(c)-(e) show results hni= 0:95 andt0=t> 0, as\nindicated in each panel. Here, the weight of the spinon\nexcitations near the upper boundary of the spinon contin-\nuum appears to soften for increasing values of t0. At the\nsame time, new spectral weight begins to appear outside9\nFIG. 7. Exact diagonalization results for the Cu L-edge RIXS spectra of the t-t0-Jmodel in the spin-conserving (\u0001 S= 0)\nchannel, calculated on an N= 20 site chain with t= 0:4 eV andJ= 0:25 eV. As with Fig. 7, panel (a) shows the spectra for\nhalf-\flling, whereas (b)-(h) show the spectra for hni= 0:95, and values of t0as indicated. The solid black lines indicate the\nboundaries of two-spinon excitations. For t0=t> 0, the dispersion of the holon excitation appears to soften slightly until t0=t\n2\nwhere the spectral weight completely reorganizes. For t0=t < 0, the holon exictation remains well de\fned and harden as the\nmagnitudejt0jincreases. All panels in this \fgure have been plotted with the same color scale.\nof the two-spinon continuum. This additional weight is\nparticularly pronounced at low-energies. It also becomes\nmore prominent for t0\u0015t\n4, where underlying bare Fermi\nsurface changes from a pocket at k=\u0006\u0019\nato two pock-\nets centered at kmin[see Fig. 1(c)]. We also note that\nthe new low-energy feature near q= 0 coincides with the\nlow-energy charge excitation observed in Figs. 4(c) and\n5(c). The fact that this feature can be resolved both in\nthe NSC channel of RIXS and in N(q;!) supports the\ninterpretation that it has both spin and charge compo-\nnents.\nFigs. 6(f)-(h) show spectra for t0=t< 0, where the spec-\ntra appear to harden slightly on increasing jt0=tj. (The\nRIXS spectra for jt0j=t\n4shown in panels (d) and (g)\ncompare well with the S(q;!) results reported in Fig. 3\nof Ref. [30] forjt0=tj= 0:3.) Interestingly, for intermedi-\nate values of t0, the spectra largely resemble the t0= 0\ncase. It is not until t0=\u0000t\n2that a new excitation appears\nabove the spinon continuum. These results rea\u000erm that\nthe formation of the spin-polaron is sensitive to the sign\noft0, and the picture emerging from Fig. 6 is analogous\nto the one obtained by examining S(q;!) andN(q;!) in\nthe previous sections.2. CuL-edge RIXS in the spin-conserving channel\nIn the case of AFM cuprates, the SC channel (\u0001 S= 0)\nis dominated by charge [51] and double spin-\rip excita-\ntions [88]. Contrasting this channel with the NSC chan-\nnel can, therefore, provide another avenue to probe spin\nand charge excitations in these systems selectively. Fig. 7\nshows our results for the Cu L-edge RIXS spectra in this\nchannel. As with the previous section, the thin black lines\nindicate the boundaries of the two-spinon continuum.\nFig. 7(a) shows results obtained at half-\flling to re-\nmind the reader of what occurs in the undoped case.\nHere, the spectrum is dominated by double spin-\rip ex-\ncitations, where the spectral weight is mostly con\fned to\nthe two-spinon continuum but with a node in the inten-\nsity near the zone boundaries q=\u0006\u0019\na[37, 85, 88, 89].\nOur results are in agreement with earlier studies [36, 90];\nhowever, we also observe a feature with a very weak\nintensity centered near ( q;!) = (0;t), which resembles\nfour-spinon excitations uncovered previously at the oxy-\ngenK-edge [40] but with diminished intensity. Fig. 7(a)\nthus con\frms that these excitations can also be resolved\nat the CuL-edge, albeit with an overall weaker intensity\ndue to the shorter lifetime of the Cu 2 pcore-hole [40].\nFig. 7(b) shows the SC RIXS response for the doped\ncasehni= 0:95 witht0= 0. Compared to the undoped\ncase, one sees an additional dispersing feature with a10\nbandwidth of 4 t. A similar excitation was predicted at\nthe oxygen K-edge [36] and was attributed to a holon\nexcitation. This interpretation is supported by the fact\nthat the dispersion of this excitation agrees well with\nthat expected for a holon !(q) = 2t[1\u0000cos(qa)], and\nwithN(q;!) computed for a dilute spinless chain [see\nFig. 1(d)].\nFigs. 7(c)-(e) show spectra for hni= 0:95 andt0=t> 0\nas indicated, while panels (f)-(h) show the cases with\nt0=t < 0. Forjt0j<t0\n4, the spectra look similar to the\nt0= 0 case, but with a slight softening (hardening) of\nthe holon excitation for t0>0 (t0<0). However, for\nt0=t\n2[panel (e)], the charge excitations break up into an\ntwo distinct sets of excitations with additional incoherent\nweight at high energy, similar to what was observed for\nthe dynamical charge structure factor shown in Fig. 5(c).\nWe, therefore, associate them with the same particle-hole\nscattering processes within the spectral function shown\nin Fig. 2(c).\nIV. CONCLUSIONS\nWe have studied the e\u000bects of nnn hopping on the\nevolution of the low-energy charge and spin dynamics\nof quasi-1D AFM cuprate spin chains within the doped\nt-t0-Jmodel. Speci\fcally, we presented DMRG results\nfor the single-particle spectral function and the dynami-\ncal spin and charge structure factors. We found evidence\nthatt0couples the fractionalized spinon and holon exci-\ntations. This coupling can lead to a spin-polaron state,\nwhere a ferromagnetic spin polarization cloud dress the\ndoped electron. In this respect, our results are consistent\nwith an earlier ED study that was carried on N= 16\nsite chains [33] but with higher momentum resolution\nand covering a wider parameter regime. As such, we can\nobtain a complete picture of the breakdown of fraction-\nalization with the inclusion of nnn hopping. To the best\nof our knowledge, the charge dynamics of the doped 1D\nt-t0-Jmodel has not been widely explored in the existing\nliterature.\nOur results provide details predictions for the dynami-\ncal response functions of doped cuprate spin chains withnnn hopping, which can be probed by spectroscopies such\nas INS or ARPES. We also provided predictions for the\nCuL-edge RIXS spectra. RIXS has emerged as a novel\nspectroscopic method for probing both charge and spin\nexcitations in quantum magnets within a single experi-\nment. It is, therefore, an ideal probe for exploring frac-\ntionalization in 1D [36, 40, 41]. We evaluated both the\nNSC and SC channels at this edge and identi\fed exci-\ntations related to fractionalized spinons and holons, as\nwell as the spin-polaron as a function of t0. Based on\nthis, we propose that Cu L-edge RIXS measurements\ncan be used to identify the presence of spin-polarons in\nvarious spin-chain systems with appropriate values of t0.\n1D chain cuprates compound such as SrCuO 2[91] and\nSr2CuO 3[92] can be doped; however, the magnitude of t0\nin the corner-shared system Sr 2CuO 3should be small and\nso the zig-zag SrCuO 2may be a better candidate. In this\ncontext, the edge-shared cuprates may be another prefer-\nable system as the Cu-O-O-Cu hybridization pathways\nshould result in larger e\u000bective t0than the corner-shared\ncuprates, which tend to be dominated by nn Cu-O-Cu\nhopping processes. Another possibility may be to engi-\nneer 1D iridates with long-range hopping, which has re-\ncently been shown to be possible in arti\fcial heterostruc-\ntures [93].\nACKNOWLEDGMENTS\nU. K. acknowledges partial support from US DOE\nNNSA under contract No. 89233218CNA000001 through\nthe LDRD Program. G. P. acknowledges funding support\nfrom 2018 Augusta University CURS summer scholars\nprogram. A. N. acknowledges support from the Canada\nFirst Research Excellence Fund. S. J. acknowledges sup-\nport from the National Science Foundation under Grant\nNo. DMR-1842056. T. 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Seo, Advanced\nMaterials 29, 1603798."    },    {        "title": "1609.05725v1.Characterization_of_spin_orbit_fields_in_InGaAs_quantum_wells.pdf",        "content": "Characterization of spin-orbit \felds in InGaAs quantum wells\nT. Henn, L. Czornomaz, and G. Salis\u0003\nIBM Research - Z urich, S aumerstrasse 4, 8803 R uschlikon, Switzerland\n(Dated: March 25, 2022)\nAbstract\nCoherent electron spin dynamics in 10-nm-wide InGaAs/InAlAs quantum wells is studied from\n10 K to room temperature using time-resolved Kerr rotation. The spin lifetime exceeds 1 ns at\n10 K and decreases with temperature. By varying the spatial overlap between pump and probe\npulses, a di\u000busive velocity is imprinted on the measured electron spins and a spin precession in\nthe spin-orbit \feld is measured. A Rashba symmetry of the SOI is determined. By comparing\nthe spatial precession frequency gradient with the spin decay rate, an upper limit for the Rashba\ncoe\u000ecients \u000bof 2\u000210\u000012eVm is estimated.\n\u0003gsa@zurich.ibm.com\n1arXiv:1609.05725v1  [cond-mat.mes-hall]  19 Sep 2016Spin-orbit interaction (SOI) couples the electron's spin to its motion and is of great sig-\nni\fcance for spin manipulation by electric \felds and currents, as proposed and implemented\nin spintronics devices [1, 2] and spin-based quantum bits [3]. Much interest has recently\nbeen shown in using materials with large SOI to implement helical modes that are spin-\npolarized and counterpropagating, and that are expected to be protected against elastic\nback-scattering [4]. Such systems are also important for Majorana zero modes [5, 6] and\ngenerally require semiconductor materials with small band gaps, such as InAs or InSb. The\nRashba-type SOI [7], especially, increases dramatically when decreasing the band-gap of the\nsemiconductor.\nSOI in two-dimensional electron gases con\fned in semiconductor quantum-well (QW)\nstructures can be measured using transport or optical techniques. Time-resolved optical\npump-probe methods [8{10] have proven especially useful to directly access the SOI and\nhave thereby provided rich information about the size and symmetry of SOI [11, 12]. Up to\nnow, such optical investigations were limited to the wavelength range below 1000 nm, based\non the convenience of available lasers and detectors. To investigate materials with larger\nSOI, an extension of such techniques to larger wavelengths is desired.\nIn Ref. 10, an all-optical technique based on imprinting a di\u000busive spin motion was\ndeveloped to measure SOI in GaAs quantum wells. Here, we adapt this technique to the\n1500 nm telecom wavelength region. Investigating the coherent electron spin dynamics in\nIn0:53Ga0:47As QWs placed in the intrinsic region of a n-i-p structure, we \fnd spin decay\ntimes of 1.2 ns at 10 K and 120 ps at room temperature. SOI is found to be dominated by the\nRashba type. By comparing the variation of spin precession frequency versus pump-probe\noverlap position with the spin decay rate, we estimate an upper limit for the Rashba SOI\ncoe\u000ecient\u000bof 2\u000210\u000012eVm. In terms of energy splitting, this corresponds to 400 \u0016eV for\na typical Fermi wavenumber of 1 \u0002108m\u00001.\nThe sample studied contains three 10-nm-wide In 0:53Ga0:47As QWs, each embedded in\n20-nm-wide In 0:52Al0:48As barriers. The QWs are placed in between a 200-nm-thick n-\ndoped region on the surface side and an equally thick p-doped region on the substrate side.\nThe n- and p-doped regions were each divided into 100 nm of lower doping concentration\n(2\u00021017cm\u00003) on the QW side, and higher doping concentration (1 \u00021019cm\u00003) on the outer\nside. The sample was grown lattice-matched by metal-organic chemical vapor deposition on\nan InP substrate. The opposite doping on the two sides generates an electric \feld across\n20 200 40002468\n0.00 T0.38 T0.72 T1.00 T\nDelay (ps) Kerr rotation (normalized)data\nfit\n50100150t (ps)\n0 0.5 102040\nMagnetic field (T)n (GHz)|g| =  2.6(a)\n(c)(b)FIG. 1: (Color online). (a) Time-resolved Kerr rotation measurements (symbols) taken at room\ntemperature track the dynamics of the optically excited electron spins at four di\u000berent magnetic\n\feldsB. Red lines are \fts to an exponentially decaying cosine oscillation. (b) The \ftted decay times\n\u001cdo not depend signi\fcantly on the applied magnetic \feld. (c) The \ftted precession frequency \u0017\nincreases linearly with B, with the slope indicating an electron g-factor of 2.6.\nthe three QWs of about 8 V/ \u0016m which is expected to establish a large Rashba SOI for the\nQW electrons.\nThe electron spin dynamics was measured using time-resolved Kerr rotation (TRKR) [13].\nPulses of tunable wavelength \u0015were generated by an optical parametric oscillator (APE\nBerlin) pumped by a mode-locked Ti:sapphire laser at a repetition rate of 80 MHz. After\npassing through a single-mode \fber, the pulse length was on the order of a few picoseconds.\nPump pulses were circularly polarized and focused onto the sample surface (sigma-width of\nspot size 10 - 12 \u0016m, time-averaged power 2 mW). Linearly polarized and time-delayed probe\npulses (power 200 \u0016W, time delay t) were focused to a similar size. By choosing \u0015close to\nthe absorption edge of the QW, the circularly polarized pump pulses excite spin-polarized\nelectrons into the conduction band of the QW layer, while a maximum Kerr rotation of the\nlinearly polarized probe pulses is achieved. The Kerr rotation is measured using a balanced\nInGaAs photodiode bridge and is proportional to the transient out-of-plane component of\nthe electron spin polarization at time t. Using a motorized mirror, the relative spatial\ndisplacement between pump and probe spots was controlled along both dimensions of the\nsample surface. The sample was placed in a cryostat at temperatures between 10 and 296 K.\nA magnetic \feld Bof up to 1 T was applied in the plane of the sample.\nData of TRKR at room temperature is shown in Fig. 1(a). At B= 1 T, the optically\n3excited spin polarization precesses with a frequency \u0017close to 40 GHz and decays within\na time\u001cof a little more than 100 ps. At B= 0 T, no spin precession is observed and the\nsignal decays within a similar time constant. The data is \ft (red curves) by an exponentially\ndecaying cosine function, aexp(\u0000t=\u001c) cos(2\u0019\u0017t), to obtain the decay time \u001c, the precession\nfrequency\u0017, and the signal amplitude a. Time-resolved di\u000berential re\rectivity measure-\nments (not shown) indicate a charge recombination time much longer than 1 ns in the whole\ntemperature range, such that \u001crepresents the lifetime of the electron spin. For magnetic\n\feldsBbetween 0 and 1 T, we observe a \u001cof around 120 ps, approximately independent of\nB[Fig. 1(b)]. This indicates that \u001cis not limited by inhomogeneities of spins probed within\nthe extension of the laser spots like, e.g. a distribution of gfactors. Also, the high quality\nof the single-frequency \ft indicates that the g-factors of the electrons measured in the three\nQWs are not signi\fcantly di\u000berent. As expected, \u0017increases linearly with B[Fig. 1(c)].\nFrom the slope (red curve) we determine an electron g-factor of -2.6. The negative sign is\ninferred from the theoretical model in which band-gap dependent corrections of the g-factor\nare subtracted from a positive value of 2 [14].\nFigure 2 summarizes the dependence of the spin dynamics on temperature T. As the band\ngap increases with decreasing T, we need to lower the laser wavelength \u0015as we go to smaller\nT. Figure 2(a) shows the \u0015used for TRKR measurements (symbols). These match the\ntemperature dependence of a Varshni model with parameters obtained for lattice-matched\nInGaAs on InP [15], see line in Fig. 2(a). For each T, we measure the electron g-factor\n[Fig. 2(b)]: we observe a small increase of the negative g-factor towards zero, which is in\nagreement with observations in GaAs that were explained by a temperature dependence of\nthe dipole matrix elements between the conduction and valence bands [16]. The spin decay\ntime\u001cmonotonically increases if Tis lowered, reaching more than 1 ns at 10 K. Figure 2(c)\nshows the corresponding decay rate 1 =\u001c. For the sample studied, the spin lifetime is expected\nto be limited by the Dyakonov Perel mechanism [17], whose temperature dependence [18]\nis given by an increase of the spin di\u000busion constant Dsproportional to kBTfor a non-\ndegenerate electron gas. In addition, Dsvaries linearly with the electron scattering time\nrelevant for spin di\u000busion.\nWe use the technique \frst presented in Ref. 10 to investigate the SOI in the QWs.\nElectrons with a di\u000busive velocity v=Dsx=\u001b2are selected by displacing the probe pulse by\na distancexwith respect to the position of the pump pulse. This velocity induces an e\u000bective\n4−2.64−2.62−2.6−2.58g−factor\n05101/t (GHz)\nTemperature (K)0 50 100 150 200 250 300(a)\n(b)\n(c)14001450150015501600 l (nm)Varshni model\nTRKR measurementsFIG. 2: (Color online). (a) The QW absorption edge depends on temperature; the wavelength\n\u0015of the laser used for TRKR (symbols) is adapted accordingly. Solid line shows the wavelength\npredicted by a Varshni model taking parameters from InGaAs grown lattice-matched on InP. (b)\nThe g-factor slightly increases towards positive values as Tis increased, similar to the dependence\nfound in GaAs. (c) The decay rate 1 =\u001cmonotonically increases with temperature. At 10 K, the\nspin lifetime exceeds 1 ns.\nmagnetic \feld through the SOI, which modi\fes the electron spin precession frequency. The\nvector components of the e\u000bective spin-orbit magnetic \feld are determined by applying an\nexternal magnetic \feld B. Thereby, only components along the external magnetic \feld\nchange\u0017linearly, whereas perpendicular components lead to quadratic modi\fcations that\ncan be neglected if the e\u000bective spin-orbit \feld is small compared to B. Figure 3(a) shows a\nseries of TRKR traces for di\u000berent pump-probe overlap positions xbetween -30 and 28 \u0016m,\ntaken at room temperature and with B= 1 T applied perpendicular to the overlap scan\ndirection. As expected, the signal amplitude peaks at maximum overlap, as seen in Fig. 3(b).\nThe Kerr amplitude ais well \ftted by a Gaussian pro\fle of sigma-width \u001b=14\u0016m. This\nwidth corresponds to the convoluted spot sizes of pump and probe beams. In Fig. 3(c), we\n50 200 400\nDelay (ps)Kerr rotation (arb. units)\ndata fit00.10.20.30.40.5Kerr amplitude\n−20 0 20−0.500.5\nPosition (mm)Dn (GHz)along B \nperp. to B (a) (b)\n(c)FIG. 3: (Color online). (a) TRKR traces taken at room temperature and B= 1:0 T, with the\nlaser probe spot displaced perpendicularly to Bfrom -30\u0016m (lowest curve) to 28 \u0016m (uppermost\ncurve). (b) The \ftted amplitude of the Kerr signal represents the convoluted pump and probe\nspot pro\fles and yields a sigma-width of \u001b=14\u0016m. (c) The \ftted precession frequency depends\non the overlap position. Shown is \u0001 \u0017, de\fned as half of the di\u000berence between frequencies taken\natB=\u00061 T. If the probe spot is displaced perpendicularly to the \feld, a linear change of \u0001 \u0017is\nobserved, whereas \u0001 \u0017remains approximately constant if the probe beam is scanned along the \feld\ndirection. The change in \u0001 \u0017represents the addition of the e\u000bective spin-orbit magnetic \feld along\nthe external \feld direction.\npresent \u0001\u0017= (\u0017(B)\u0000\u0017(\u0000B))=2, the di\u000berence between measurements taken with opposite\ndirections of B. For overlap scans perpendicular to B(black dots), \u0001 \u0017varies linearly with\nthe overlap position. For overlap scans along B, \u0001\u0017is \rat, indicating that { as expected {\nthe dominant spin-orbit \feld points perpendicularly to the di\u000busive velocity.\nIn Fig. 4(a), the slopes d\u0001\u0017=dx are summarized for di\u000berent T. We present the data\nmultiplied by \u001b2. A monotonic increase with Tis observed. The squares and diamonds cor-\nrespond to the external magnetic \feld oriented along the sample's [110] and [1 10] directions.\nThe similar values and the same sign for the two directions indicates that the Rashba SOI\nis much stronger than the Dresselhaus part. The Dresselhaus SOI alone would reverse the\nsign for the two directions [10], and a superposition of a signi\fcant Dresselhaus contribution\nto the Rashba one would change the value of the frequency slope d\u0001\u0017=dx for the two direc-\ntions. The model presented in Ref. 10 connects the slopes with the SOI. In the Rashba-only\ncase, we obtain d\u0001\u0017=dx\u0001\u001b2= 2Dsm\u000b=(\u0019~2). Here,mis the e\u000bective electron mass and ~\n6050010001500dn/dx◊s2  (Hz◊m)\n \n vertical\nhorizontal\n0 50 100 150 200 250 3000123\nTemperature (K) a (10-12 eVm)(a)\n(b)FIG. 4: (Color online). Normalized slopes of \u0001 \u0017versus temperature. This quantity is proportional\nto the relevant spin-orbit coe\u000ecient and to the spin di\u000busion constant Ds. Measurements for the\ntwo sample orientations (squares and diamonds) show similar values with the same sign, indicating\na predominant Rashba contribution to SOI. (b) By comparing the slope of \u0001 \u0017with the spin decay\nrate 1=\u001c, values for the Rashba coe\u000ecient \u000bare obtained.\nthe reduced Planck constant. In order to get quantitative values for the Rashba coe\u000ecient\n\u000b,Dsneeds to be known. This can be obtained from the temporal expansion of the spin\npro\fle measured [19{21]. In our experiment, this expansion cannot be resolved because it\nis too small compared with the laser spot sizes. Nevertheless, we can estimate values for\n\u000bby taking into account the information contained in the measured values of \u001c. Assum-\ning a Dyakonov Perel spin relaxation mechanism [22, 23], 1 =\u001cis given by 6 Dsm2=~4\u0002\u000b2.\nThe equation holds for both the degenerate and non-degenerate regime. The dependence\non temperature enters through Dsvia the electron distribution and the dominant electron\nscattering mechanism. Since 1 =\u001calso depends linearly on Ds, we can remove Dsby dividing\n1=\u001cbyd\u0001\u0017=dx\u0002\u001b2. This ratio is equal to 3 \u0019m=~2\u0002\u000b. Figure 4(b) summarizes the values\nfor\u000bobtained in this way (for the \u0001 \u0017slopes we take the average between the two crystal di-\nrections). They represent an upper limit for \u000bif one considers that additional contributions\nmay a\u000bect the electron spin decay. We obtain values for \u000bthat start around 2 \u000210\u000012eVm\nat low temperatures and decrease to around 1 \u000210\u000012eVm at room temperature. Such a\nlarge value is especially promising regarding the measured spin ensemble coherence times\nabove 1 ns. Considering an electric \feld Eacross the QWs of about 8 V/ \u0016m, the ratio \u000b=E\n7amounts to 25 e \u0017A2. This compares well to calculated values [24] of r6c6c\n41of 5.2 e \u0017A2for GaAs\nand 117 e \u0017A2for InAs.\nIn conclusion, we have demonstrated pump-probe based all-optical measurements of SOI\nin low-band-gap InGaAs QWs. The SOI is found to be of Rashba type with an upper limit\nfor the coe\u000ecient \u000bof 2\u000210\u000012eVm. These experiments can be extended to materials such\nas InAs, GaSb and InSb with smaller band gaps and expectedly much stronger SOI, since\na spatial resolution of the spin dynamics on a scale of the spin-orbit length (expected to be\nmuch smaller than 1 \u0016m in these materials) is not required. These experiments will allow\nus to better understand and characterize the SOI in low-band-gap semiconductor quantum\nstructures used for creating topological insulators, helical modes and Majorana fermions.\nThe work is \fnancially supported by the Swiss National Science Foundation within the\nNCCR QSIT, and by the EU project SiSpin. We acknowledge P. Altmann and D. Widmer\nfor experimental assistance, as well as M. Tschudy, R. Grundbacher and U. Drechsler for\nhelp with sample fabrication.\n[1] I. \u0014Zuti\u0013 c, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004).\n[2] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Rev. Mod. Phys.\n87, 1213 (2015).\n[3] K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, and L. M. K. Vandersypen, Science 318,\n1430 (2007).\n[4] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).\n[5] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Phys. Rev. Lett. 104, 040502 (2010).\n[6] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010).\n[7] Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984).\n[8] L. Meier, G. Salis, I. Shorubalko, E. Gini, S. Sch on, and K. Ensslin, Nat Phys 3, 650 (2007).\n[9] J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig, S.-C. Zhang, S. Mack, and D. D.\nAwschalom, Nature 458, 610 (2009).\n[10] M. Kohda, P. Altmann, D. Schuh, S. D. Ganichev, W. Wegscheider, and G. Salis, Applied\nPhysics Letters 107, 172402 (2015).\n[11] P. Altmann, M. Kohda, C. Reichl, W. Wegscheider, and G. Salis, Phys. Rev. B 92, 235304\n8(2015).\n[12] P. Altmann, F. G. G. Hernandez, G. J. Ferreira, M. Kohda, C. Reichl, W. Wegscheider, and\nG. Salis, Phys. Rev. Lett. 116, 196802 (2016).\n[13] W. H. Lau, V. Sih, N. P. Stern, R. C. Myers, D. A. Buell, A. C. Gossard, and D. D. Awschalom,\nApplied Physics Letters 89, 142104 (2006).\n[14] L. M. Roth, B. Lax, and S. Zwerdling, Phys. Rev. 114, 90 (1959).\n[15] D. K. Gaskill, N. Bottka, L. Aina, and M. Mattingly, Applied Physics Letters 56, 1269 (1990).\n[16] J. H ubner, S. D ohrmann, D. H agele, and M. Oestreich, Phys. Rev. B 79, 193307 (2009).\n[17] M. I. Dyakonov and V. Y. Kachorovskii, Sov. Phys. Semicond. 20, 110 (1986).\n[18] J. Kainz, U. R ossler, and R. Winkler, Phys. Rev. B 70, 195322 (2004).\n[19] C. P. Weber, N. Gedik, J. E. Moore, J. Orenstein, J. Stephens, and D. D. Awschalom, Nature\n437, 1330 (2005).\n[20] H. Zhao, M. Mower, and G. Vignale, Phys. Rev. B 79, 115321 (2009).\n[21] M. P. Walser, C. Reichl, W. Wegscheider, and G. Salis, Nature Phys. 8, 757 (2012).\n[22] N. S. Averkiev and L. E. Golub, Phys. Rev. B 60, 15582 (1999).\n[23] G. Salis, M. P. Walser, P. Altmann, C. Reichl, and W. Wegscheider, Phys. Rev. B 89, 045304\n(2014).\n[24] R. Winkler, Spin-Orbit Coupling E\u000bects in Two-Dimensional Electron and Hole Systems\n(Springer, Berlin, 2003).\n9"    },    {        "title": "1109.0481v2.Spin_resolved_bunching_and_noise_characteristics_in_double_quantum_dots_coupled_to_ferromagnetic_electrodes.pdf",        "content": "arXiv:1109.0481v2  [cond-mat.mes-hall]  16 Mar 2013Spin-resolved bunching and noise characteristics in doubl e quantum dots coupled to\nferromagnetic electrodes\nJunYan Luo,1,∗HuJun Jiao,2BiTao Xiong,1Xiao-Ling He,1and Changrong Wang1\n1School of Science, Zhejiang University of Science and Techno logy, Hangzhou 310023, China\n2Department of Physics, Shanxi University, Taiyuan, Shanxi 030006, China\n(Dated: November 13, 2018)\nWe study spin-resolved noise in Coulomb blockaded double qu antum dots coupled to ferromagnetic\nelectrodes. The modulation of the interdot coupling and spi n polarization in the electrodes gives\nrise to an intriguing dynamical spin ↑-↑(↓-↓) blockade mechanism: Bunching of up (down) spins\ndue to dynamical blockade of an up (down) spin. In contrast to the conventional dynamical spin\n↑-↓bunching (bunching of up spins entailed by dynamical blocka de of a down spin), this new\nbunching behavior is found to be intimately associated with the spin mutual-correlation, i.e., the\nnoise fluctuation between opposite spin currents. We furthe r demonstrate that the dynamical spin\n↑-↑and↑-↓bunching of tunneling events may be coexistent in the regime of weak interdot coupling\nand low spin polarization.\nPACS numbers: 72.25.-b, 72.70.+m, 73.63.Kv, 73.23.Hk\nI. INTRODUCTION\nThemeasurementofsignal-to-noiseratioinmesoscopic\ntransport devices is of vital importance as it enables\naccess to intriguing information about the statistics of\nquasiparticles and various intrinsic dynamics that are\nnot available from conventional current measurements\nalone [1, 2]. For transport through a localized state,\nthe nonequilibrium noise is generally suppressed due to\nthe Pauli exclusion principle, leading thus to a sub-\nPoissonian statistics [3–5]. However, in systems of multi-\nple nonlocal states, such as double quantum dot devices,\nthe intrinsic quantum coherence and many-particle in-\nteractions give rise to different sources of correlations [6].\nElectrontransportcanexhibit a unique dynamicalblock-\nade mechanism, leading thus to a super-Poisson fluctua-\ntion [7–12].\nIn spintronic structures, transport is governednot only\nby the charge flow, but more importantly, by the spin\ntransfer [13–20]. The involving spin degrees of freedom\nintroduce additional correlated mechanisms. Study of\nspin current fluctuations is crucial for possible applica-\ntions in control and manipulation of individual spins.\nA number of investigations have been devoted to the\nnoisecharacteristicsofspin-dependent transportthrough\nvarious nanostructures, such as quantum dots [19–22],\nmolecules [23–25], and nanotubes [26–29]. Different tun-\nneling processes like sequential tunneling, cotunneling\n[30, 31], etc. were revealed to have vital roles to play\nin spin transport. In order to distinguish various spin\ndynamics it is instructive to unravel the charge noise\ninto spin-resolved components. Consider a general meso-\nscopic device of a quantum dot (QD) system connected\nto terminals α,α′.... The chargecurrent through the ter-\n∗Electronic address: jyluo@zust.edu.cnQD2 \nQD1  FM  FM  VL VR \nFIG. 1: Schematic setup for spin-dependent transport\nthrough double quantum dots, in which only QD1 is tunnel-\ncoupled to ferromagnetic electrodes.\nminal “α” is/an}bracketle{tIα/an}bracketri}ht=/an}bracketle{tIσ\nα/an}bracketri}ht+/an}bracketle{tI−σ\nα/an}bracketri}ht, where/an}bracketle{tIσ\nα/an}bracketri}htis the spin- σ\ncomponent of the current. The temporal correlation of\nspin transport is characterized by the spin-resolved cor-\nrelation function Cσσ′\nαα′(t−t′) =1\n2/an}bracketle{t{∆Iσ\nα(t),∆Iσ′\nα′(t′)}/an}bracketri}ht,\nwith ∆Iσ\nα(t) =Iσ\nα(t)−/an}bracketle{tIσ\nα(t)/an}bracketri}ht. Straightforwardly, the in-\ndividual spin-resolved noise power is given by Sσσ′\nαα′(ω) =/integraltext∞\n−∞dteiωtCσσ′\nαα′(t). By choosing an arbitrary spin axis\nz, the total charge current noise can be readily unrav-\neled into Sch\nαα′=S↑↑\nαα′+S↓↓\nαα′+S↑↓\nαα′+S↓↑\nαα′. Here,\nthe spin self-correlation S↑↑\nαα′(S↓↓\nαα′) represents fluctua-\ntion between the same spin currents, which was recently\nshown to be capable of serving as a sensitive tool to iden-\ntify the dynamical spin ↑-↓(↓-↑) bunching, i.e., bunching\nof up (down) spins due to dynamical blockade of a down\n(up) spin [32–34]. It then naturally comes to a ques-\ntion: What can we infer from the spin mutual-correlation\nnoise (fluctuation between opposite spin currents) S↑↓\nαα′\norS↓↑\nαα′?\nIn the context of spin-dependent transport through a\nsystem of multiple nonlocal states, we investigate in this\nworkthese spin-resolvednoisecomponents and their con-\nnections to spin-resolvedbunching behavior. Specifically,\nwe consider a double quantum dot, where only one of the\ndots is tunnel-coupled to the ferromagnetic (FM) elec-2\n/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s32/s32/s100 /s78/s76\n/s32/s32/s100 /s78/s76\n/s84/s105/s109/s101/s32/s91 /s32/s116/s32/s47/s32/s49/s32\n/s93 /s84/s105/s109/s101/s32/s91 /s32/s116/s32/s47/s32/s49/s32\n/s93/s40/s102/s41/s40/s101/s41/s40/s100/s41/s40/s99/s41/s40/s98/s41/s99 /s44/s49\n/s99 /s44/s32/s50/s99 /s44/s49/s99 /s44 /s32/s50\n/s32/s32/s32\n/s84/s105/s109/s101/s32/s91 /s32/s116/s32/s47/s32/s49/s32\n/s93/s40/s97/s41\n/s32 /s32\n/s40 /s32/s108 /s32/s41/s40/s107/s41/s40/s106/s41/s40/s105/s41/s40/s104/s41\n/s32/s40/s103/s41\n/s32 /s32/s32\n/s32 /s32 /s32 /s32 /s32 /s32/s32/s32/s32\n/s32\n/s32/s32\n/s32/s32\n/s32/s32\n/s32/s32/s32\n/s32\n/s32/s32\n/s32/s32\n/s32/s32/s32\n/s32\n/s32/s32/s32\n/s32\n/s32/s32/s32\n/s32/s32\n/s32/s32\n/s32\n/s32/s32/s32\n/s32/s40/s109/s41\n/s32/s32\n/s32/s32/s40/s110/s41\n/s32/s32\n/s32/s32/s40/s48/s41\n/s32/s32\n/s32/s32/s40/s112/s41\n/s32/s32\n/s32/s32/s40/s113/s41\n/s32/s32\n/s32/s32/s40/s114/s41\nFIG. 2: Sets of typical detection record (up and down spin tun neling events) and corresponding real-time quantum states under\nparallel magnetic alignment. (a)-(f) Ω /Γ = 1.0 andp= 0.8, (g)-(l) Ω /Γ = 0.2 andp= 0, and (m)-(r) Ω /Γ = 0.1 andp= 0.25.\nEach quantum dot has only one level (assumed to be in resonanc e, i.e.,E1=E2) within the bias window V=VL−VR. In the\nCoulomb blockade regime and for temperature kBT≪V, the Fermi functions can be approximated by either one or zer o, so\nthe temperature is not involved here.\ntrodes (see Fig.1). The system, which can be mapped\nto the one investigated recently in experiments [35–38],\nis of particular interest, as it is arranged in such a con-\nfiguration that can maximize locality versus nonlocality\ncontrast [39–42]. In the Coulomb blockade regime, we\nobserve a unique dynamical spin ↑-↑(↓-↓) blockade phe-\nnomenon, namely, bunching of up (down) spins due to\ndynamical blockade of an up (down) spin. Different from\nthe conventional spin ↑-↓bunching, it is revealed that\nthis new mechanism is intimately associated with pos-\nitive spin mutual-correlation. We further demonstrate\nthat the spin ↑-↑and↑-↓bunching of tunneling events\nmay be coexistent in the regime of low spin polarization\nand weak interdot tunnel-coupling.\nThe paper is organized as follows. In Sec.II, we de-\nscribe the double quantum dot system tunnel-coupled\nto FM electrodes. In Sec.III, a Monte Carlo approach\nis introduced to simulate real-time single spin tunnel-\ning events. The spin-resolved noises, together with spin-\nresolved bunching of tunneling events will be discussed\nin detail. It is then followed by the conclude in Sec.IV.\nII. MODEL DESCRIPTION\nThe system under study is sketched in Fig.1, in which\nonly QD1 is connected to the FM electrodes, whereas\nQD2 is side-connected to the QD1. The Hamiltonian\nof the entire system is ˆH=ˆHB+ˆHS+ˆH′, where\nˆHB=/summationtext\nα=L,R/summationtext\nk,σεαkσc†\nαkσcαkσdenotes the left and\nright FM electrodes; cαkσ(c†\nαkσ) is the electron annihila-\ntion (creation) operatorofthe electrode α= L or R, withspinσ=↑or↓. The ferromagnetism of the electrode α\nis accounted for by the spin-dependent density of states\ngασ(ω). Throughout all of our calculations presented\nhere, we approximate the density of states to be energy\nindependent, gασ(ω) =gασ. (Real ferromagnets have a\nstructured density of states, which only modifies details\nof our results but not the general physical picture.) The\nasymmetryin the densityofstatesischaracterizedbythe\ndegree of spin polarization pα= (gα↑−gα↓)/(gα↑+gα↓)\nwith−1≤pα≤1, in which pα= 0 denotes a non-\nmagnetic electrode and pα=±1 corresponds to a half-\nmetallic electrode.\nThe Hamiltonian for the coupled dots reads\nˆHS=/summationdisplay\nℓ=1,2\n/summationdisplay\nσ=↑,↓Eℓˆnℓσ+U0ˆnℓ↑ˆnℓ↓\n+U′ˆn1ˆn2\n+Ω/summationdisplay\nσ(d†\n1σd2σ+d†\n2σd1σ), (1)\nwheredℓσ(d†\nℓσ) is the creation (annihilation) operator\nof an electron with spin σ=↑or↓in QD1 ( ℓ=1) or\nQD2 (ℓ=2); ˆnℓσ=d†\nℓσdℓσand ˆnℓ=/summationtext\nσˆnℓσare the\ncorresponding particle number operators. Each QD has\na spin-degenerate level within the bias window V=\nVL−VR.U0andU′are respectively the intradot and\ninterdot Coulomb repulsions; Ω denotes the strength of\ninterdot tunnel-coupling. Hereafter, we consider double-\ndot Coulomb blockade regime [43, 44], i.e., U0andU′are\nlarge enough such that states with two or more electrons\nin the double dots are not allowed. The involved states\nare both dots empty |0/an}bracketri}ht, one electron with spin σin the3\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s49/s50/s51\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s49/s50/s51/s83 /s32/s47/s50/s101/s83 /s32/s47/s50/s101\n/s32/s32\n/s80/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s32/s32 /s112/s32 /s61/s48/s46/s49\n/s32 /s61/s48/s46/s50\n/s32 /s61/s49/s46/s48\n/s32 /s61/s53/s46/s48/s40/s97/s41\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s49/s50/s51\n/s32/s32\n/s80/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s32/s32 /s112/s40/s98/s41\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s83/s99/s104\n/s32/s47/s50/s101\n/s80/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s32/s32 /s112/s83 /s32/s32/s40/s83 /s41 /s32/s47/s50/s101\n/s32/s32\n/s80/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s32/s32 /s112/s40/s99/s41\n/s32/s32\n/s32/s32/s40/s100/s41\nFIG. 3: Spin self-correlations S↑↑\nααandS↓↓\nαα, spin mutual-correlation S↑↓\nαα(S↓↑\nαα), as well as the total charge current noise Sch\nαα\nvs polarization pfor various interdot couplings Ω /Γ = 0.1, 0.2, 1.0, 5.0. The left and right electrodes are of P configuration.\nThe other parameters are the same as those in Fig. 2.\nQD1|1σ/an}bracketri}htorQD2|2σ/an}bracketri}ht. Experimentally, it canberealized\nby properly tuning the gate and bias voltages [6, 45–47].\nTunneling between QD1 and electrodes is described\nbyˆH′=/summationtext\nαkσ/parenleftbig\ntαkσc†\nαkσd1σ+ H.c./parenrightbig\n. The FM elec-\ntrodes result in spin-dependent tunnel couplings Γ ασ=\n2π/summationtext\nk|tαkσ|2δ(εαkσ−ω). In what follows, we consider\ntwo magnetic configurations. (i) The parallel (P) align-\nment, when the majority of electrons in both electrodes\npoint in the same direction. (ii) The antiparallel (AP)\ncase, in which the magnetization of the two electrodes\nare opposite to each other. Thus for the P alignment we\nhave\nΓL↑/L↓=1\n2(1±pL)ΓLand Γ R↑/R↓=1\n2(1±pR)ΓR,(2a)\nwhile for the AP configuration we set\nΓL↑/L↓=1\n2(1±pL)ΓLand Γ R↑/R↓=1\n2(1∓pR)ΓR.(2b)\nHere Γ α= (Γα↑+ Γα↓) is the total coupling strength\nregardless the spin orientation.\nIII. RESULTS AND DISCUSSIONS\nIn order to get a deep understanding of the spin dy-\nnamics and fluctuations in transport, a Monte Carlo\nmethod is employed to simulate the real-time single spin\ntunneling events in the quantum-jump regime. We intro-\nduce two stochastic point variables d NLσ(t) and dNRσ(t)\n(with values either 0 or 1) to stand for, respectively,\nthe numbers of spin- σ(σ=↑,↓) electron tunneled to the\ndouble dots from left electrode and that from the dou-\nble dots to the right electrode, during the time interval\ndt. The conditional quantum master equation for the\nreduced density matrix reads [48]\ndρc=−iLρc(t)dt−/summationtext\nσ=↑,↓{ΓLσA[d†\n1σ]+ΓRσA[d1σ]\n−PLσ(t)−PRσ(t)}ρc(t)dt\n+/summationtext\nσ=↑,↓dNLσ/bracketleftBig\nΓLσJ[d†\n1σ]\nPLσ(t)−1/bracketrightBig\nρc(t)\n+/summationtext\nσ=↑,↓dNRσ/bracketleftBig\nΓRσJ[d1σ]\nPRσ(t)−1/bracketrightBig\nρc(t),(3)where the attached subscript “c” is to indicate that the\nquantum state is conditioned on previous measurement\nresults. Here the superoperators are defined as L(···)≡\n[ˆHS,···],J[X]ρc≡XρcX†andA[X]ρc≡1\n2(X†Xρc+\nρcX†X). The involving stochastic point variables satisfy\nE[dNLσ(t)] =PLσ(t)dt= Tr{J[√ΓLσd†\n1σ]ρc}dt,(4a)\nE[dNRσ(t)] =PRσ(t)dt= Tr{J[√ΓRσd1σ]ρc}dt,(4b)\nwhereE[(···)] denotes an ensemble average of a large\nnumber of quantum trajectories. In this quantum trajec-\ntory approach, spin tunneling events condition the fu-\nture evolution of the system state [see Eq.(3)], while\nthe instantaneous quantum state conditions the mea-\nsured spin tunneling events through the double dots [see\nEq.(4a) and Eq.(4b)] in a self-consistent manner. One\nthus is propagatingin parallelthe informationofthe con-\nditioned (stochastic) state evolution ( ρc) and detection\nrecord (d Nασ) in a single realization of the readout mea-\nsurement experiment.\nThe spin tunneling events (d Nασ) leads straight-\nforwardly to the spin- σdependent current Iσ\nα(t) =\nedNασ(t)/dt, and consequently the total charge current\nIch\nα=I↑\nα+I↓\nα. Hereafter, we will use ¯I≡E[Ich\nα(t)]t→∞\nto represent the ensemble stationary current. The spin-\nresolved time correlation function [ Cσσ′\nαα′(t)] and its cor-\nresponding noise spectrum [ Sσσ′\nαα′≡Sσσ′\nαα′(ω= 0)] can be\nevaluated following [49], or by using alternatively a spin-\nresolved quantum master equation approach [50, 51].\nIn what follows, noise between different electrodes are\nnot considered as it simply satisfies Sσσ′\nLR=−Sσσ′\nααfor\nthe present two-terminal device (Note such a relation\ngenerally does not hold for a multi-terminal structure\n[32, 33, 52]). For simplicity, we assume symmetric tun-\nnel couplings (Γ L= ΓR= Γ/2) and equal magnitude of\nspin polarization in the two electrodes ( pL=pR=p).\nFirstletusfocusonthePalignment. Fig.2(a)-(f)show\nthe set of measured spin tunneling events (d NL↑/L↓) and\nthe corresponding real-time quantum state ( ρc) for po-\nlarization p= 0.8 and interdot coupling Ω /Γ = 1.0. We\nobserve unambiguously the bunching of up-spin tunnel-\ning events. When a down spin is injected into the double4\ndots, it will stay there and experiences some oscillations\nbetween the two dots, until it finally escapes to the right\nelectrode [see Fig.2 (e) and (f) the instantaneous quan-\ntum states of a down spin in QD1 and QD2]. The up\nspins can flow only in short time windows where the cur-\nrent is not blockaded by a down spin [see Fig.2(a)-(c)],\nleading thus to the conventional dynamical spin block-\nade, as discussed in Refs. [32–34]. For clarity, we refer to\nthis mechanism asdynamical spin ↑-↓blockade to specify\nthe bunching of up spins due to dynamical blockade of a\ndown spin. The dynamical spin ↑-↓blockade gives rise to\na prominent super-Poisson spin self-correlation S↑↑\nαα, as\nshown in Fig.3(a). Note here we only need to consider\nS↑↑\nααdue to the fact that S↑↑\nααandS↓↓\nααaresymmetric with\nrespect to the spin polarization p, i.e., Fig.3(a) vs (b).\nThe spin self-correlation S↑↑\nααincreases monotonically\nwith the polarization as displayed in Fig.3(a). Only\nfor sufficient spin polarization that the “ ↑-↓” competi-\ntion mechanism takes place, which leads eventually to\nthe super-Poisson spin self-correlation S↑↑\nαα. Yet, it is\nalso instructive to investigate the noise at low polariza-\ntion, for instance p=0. The total charge current noise\nSch\nααas shown in Fig.3(d) exhibits unambiguously super-\nPoisson statistics for Ω /Γ = 0.2 (dashed line). It im-\nplies bunching of charge tunneling events regardless of\nthe spin orientations. However,neither ofits components\n(S↑↑\nααorS↓↓\nαα) exceedsthe Poissonvalue [ S↑↑\nαα/(2e¯I)|p=0=\nS↓↓\nαα/(2e¯I)|p=0= 0.46]. It means that the super-Poisson\nchargenoisedoesnotstemfromthedynamicalspin ↑-↓or\n↓-↑bunching. In other words, the spin self-correlations\nS↑↑\nααandS↓↓\nααdo not capture the whole picture of spin\nbunching.\nWe ascribe the occurrence of the super-Poisson charge\nnoise to a unique dynamical spin ↑-↑or↓-↓bunching,\nwhich is confirmed by the numerical simulation of real-\ntime spin tunneling as shown in Fig.2(g)-(j) for p=0 and\nΩ/Γ = 0.2. When a down spin is injected into QD1,\nit cycles to the QD2, where it is localized due to weak\ninterdot tunnel-coupling. The current is thus blockaded\nuntil the down spin finally tunnels to the right electrode,\nwhich is then followed by a bunch of down spins flowing\nthrough the system, i.e., bunching of downspins due to\ndynamical blockade of a downspin [see Fig.2(k)-(l)]. We\ncall this new mechanism the dynamical spin ↓-↓blockade\nto distinguish it from the ↑-↓one. Similarly, dynami-\ncal spin ↑-↑bunching is observed as shown in Fig.2(g)-\n(i). However, the dynamical spin ↑-↑or↓-↓blockade\ndoes not necessarily give rise to super-Poissonian spin\nself-correlations. We will reveal this new spin bunching\nmechanismisintimatelyassociatedwith thespin mutual-\ncorrelation S↑↓\nαα, which thus can be utilized as an addi-\ntional diagnostic tool to the dynamics and fluctuations\nin spin transport.\nFor a quantitative analysis, we first evaluate some fun-\ndamentaltimescalesinvolvedintransport(forΩ /Γ = 0.2\nandp=0). By using 2000 independent trajectories analo-\ngoustothe onesshownin Fig.2(g)-(l), wegetthe average\ndelay between the occupancy of the dots by two consecu-/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s45/s49/s46/s50/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s32/s32/s67/s116/s67\n/s84/s105/s109/s101 /s32/s32/s32/s116 /s32/s47/s32/s49/s32 /s44/s32 /s112 /s61/s48\n/s32 /s44/s32 /s112 /s61/s48/s46/s56\nFIG. 4: Time dependence of correlation function C↑↓\nαα(t) for\nΩ/Γ = 0.2,p= 0 (solid line) and Ω /Γ = 1.0,p= 0.8 (dotted\nline).\ntive up spins τ0= 1.01Γ−1and the average dwell time of\nup spins on the double dots τ↑= 4.00Γ−1. The average\nduration of the “bunch” of up spins is then obtained as\nτb= 6.01Γ−1. (An alternative approach to obtain these\nquantities can be found in Ref. [32].) The above time\nscales are able to reveal the intrinsic dynamics in spin\ntransport. Consider, for instance, the spin-resolved time\ncorrelation function C↑↓\nαα(t), as shown by the solid line in\nFig.4. It is negative for times shorter than τ0. It then\nrises, becomes positive and reaches a maximum at a time\ncomparable to τ↑. Finally, it decreases on a time scale of\nτ↑+τb. For the time scales of tunneling of down spins,\nanalogous analysis can be applied. The above charac-\nteristic times in the correlation function thus allow us to\nattribute the positive S↑↓\nααto the dynamical spin ↑-↑or↓-\n↓blockade mechanism. In comparison, these unique time\nfeatures can not be identified in the case of large interdot\ncoupling (Ω /Γ = 1.0) and strong magnetic polarization\n(p=0.8), as shown by the dotted line in Fig.4.\nThe requirements to observe the dynamical spin ↑-↑\nor↓-↓bunching of tunneling events thus can be inferred\nfrom the spin mutual-correlation. For the P alignment,\nthe analytic expression is give by\nS↑↓\nαα= 2e¯I(1−p2)Γ2−16Ω2\n200Ω2. (5)\nIt might be either positive or negative, depending on\nthe degree of spin polarization and the strength of inter-\ndot tunnel-coupling. For Ω <1\n4Γ, positive spin mutual-\ncorrelationisobservedprovidedtheelectrodesareweakly\npolarized,implyingthustheoccurrenceofdynamicalspin\n↑-↑or↓-↓bunching [cf. Fig.2(g)-(l)]. It is worth noting\nthat the presented spin transport behavior persists over\na wide range of polarization as long as the interdot cou-\npling is weak enough. It thus offers an opportunity to\nobserve the coexistence of spin ↑-↑and↑-↓bunching of\ntunneling events, as displayed in Fig.2(m)-(r) for spin\npolarization p=0.25 and interdot coupling Ω /Γ = 0.1. In\nthe opposite regime of Ω >1\n4Γ, the interdot coupling is5\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s83/s99/s104\n/s32/s47 /s50/s101\n/s32\n/s83 /s32/s32/s40/s83 /s41 /s32/s47/s50/s101/s83 /s32/s47/s50/s101/s83 /s32/s47/s50/s101\n/s32/s32\n/s80/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s32/s32 /s112/s32\n/s32/s40/s97/s41\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52\n/s32/s32\n/s80/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s32/s32 /s112/s40/s98/s41\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s80/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s32/s32 /s112/s40/s100/s41/s32/s32\n/s80/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s32/s32 /s112/s40/s99/s41\n/s32/s32/s32\n/s32\nFIG. 5: Spin self-correlations S↑↑\nααandS↓↓\nαα, spin mutual-correlation S↑↓\nαα(S↓↑\nαα), as well as the total charge current noise Sch\nαα\nvs polarization pfor various interdot couplings Ω /Γ =0.1, 0.2, 1.0, and 5.0. The left and right electrodes are of AP alignment.\nThe other parameters are the same as those in Fig. 2.\nlarge enough to overcome electron localization in QD2,\nleading thus to the disappearance of the dynamical spin\n↑-↑or↓-↓bunching, as confirmed by our numerical real-\ntime simulation (not shown explicitly). The resultant\nspin mutual-correlation is negative at arbitrary strength\nof spin polarization [see the dotted (Ω /Γ = 1) and dash-\ndotted (Ω /Γ = 5) lines in Fig.3(c)].\nLet us now turn to the situation where the electrodes\nare of AP alignment. The self-correlations S↑↑\nααandS↓↓\nαα\nvs spin polarization pare plotted in Fig.5(a) and (b),\nrespectively. Again, we take S↑↑\nααfor illustration. If the\nleft electrodeisfully spin-downpolarized, transportofup\nspins are completely suppressed, which leads to a vanish-\ningS↑↑\nααasp→ −1. In the opposite limit of p→1, only\nup spinsareallowedtotunnelintothe coupleddots; how-\never, under the AP alignment the rate of tunneling out\nto the right electrode is strongly inhibited. In this case,\ntunneling of up spins is in close analogy to electron tun-\nneling through an extremely asymmetric double barrier\nstructure [3]. The up spin tunneling events are thus un-\ncorrelated, and the resultant noise correlation turns out\nto be Poisson ( S↑↑\nαα→1), independent of interdot cou-\npling strength Ω. In a wide range in between, the noise is\nvery sensitive to the interdot coupling strength. Particu-\nlarly, we observe a super-Poissonian spin self-correlation\nS↑↑\nαα, as shown by the solid line in Fig.5(a). The occur-\nrence of dynamical spin ↑-↓bunching relies on two condi-\ntions: (i) Appropriate spin polarization in the electrodes,\nand (ii) strong localization of a down spin in QD2, which\nis fulfilled at weak interdot coupling Ω (Ω <1\n9Γ accord-\ning to our calculation). A rising interdot coupling leads\nto delocalization of the down spin. The dynamical spin\n↑-↓blockade is lifted, which results eventually in a sub-\nPoissonself-correlationforarbitrarypolarization[see, for\ninstance, the dashed line in Fig.5(a) for Ω /Γ = 0.2].\nAlthough neither of the two spin self-correlations ( S↑↑\nαα\nandS↓↓\nαα) exhibits super-Poisson statistics for Ω /Γ = 0.2,\nthe total charge current noise displays unambiguously\nsuper-Poisson characteristics [dashed line in Fig.5(d)].\nAnalogous to situation of P configuration, the super-\nPoisson noise here arises from the dynamical spin ↑-↑or↓-↓bunching, as one can infer from the positive spin\nmutual-correlation [dashed line in Fig.5(c)]. A decrease\nin the strength of interdot tunnel-coupling leads to en-\nhancementofthedynamicalspin ↑-↑or↓-↓bunching,and\nfinally increases the spin mutual-correlation [cf. Fig.5].\nThus, if the interdot tunnel-coupling is low enough one\nmay observe the coexistence of dynamical spin ↑-↓(↓-↑)\nand↑-↑(↓-↓) bunching of tunneling events, as indicated\nby the super-Poisson spin self-correlation and positive\nmutual-correlation Fig.5 (a) and (b), respectively. Yet,\ndifferent from the P alignment, the dynamical spin ↑-↑\nblockade survives even for a strong interdot coupling Ω,\nas long as the electrodes are not weakly polarized. In\nthis case, an up spin injected into the double dots may\nexperience some oscillations between the two dots before\nit can tunnel out to the right electrode owing to a small\nΓR↑. Its dwell on the double dots serves an “effective”\ndynamical spin ↑-↑blockade mechanism, yielding thus a\npositive spin mutual-correlation. On the other hand, the\nsmall Γ R↑also inhibits tunneling of up spins, which ex-\nplains the suppressed spin mutual-correlation observed\nin Fig.5(c).\nWe note that current transport through similar struc-\nture has also been investigated recently in [31, 53] and\nsuper-Poissonian noise was reported. There, the exis-\ntence of a very lower tunneling rate between QD2 and\nthe electrodes than that between QD1 and the elec-\ntrodes leads to finite occupation of the QD2, which gives\nrise eventually to the super-Poissonian fluctuations. Al-\nthough the mechanisms are different, the finally results\nhappen to be consistent qualitatively. That is, the total\ncharge current noise is larger in the P configuration than\nin the AP one for a large polarization, see Fig.3(d) and\nFig.5(d).\nIV. CONCLUSION\nIn the context of spin-dependent transport though a\nCoulomb blockaded double quantum dot system, we re-\nvealed unambiguously unique dynamical spin ↑-↑and↓-↓6\nbunching of tunneling events, as confirmed by the real-\ntime Monte Carlo simulation of spin tunneling. Different\nfrom the conventional dynamical spin ↑-↓bunching, this\nnew mechanism is found to be intimately associated with\nthe spin mutual-correlation for both (parallel and an-\ntiparallel) magnetic configurations. Under conditions of\nlow spin polarization and weak interdot tunnel-coupling,\nwe demonstrated the coexistence of the dynamical spin\n↑-↓and↑-↑bunching events. 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Brandes,\nLANL e-print arXiv:1209.1226 (2012)."    },    {        "title": "0807.0379v2.Dynamics_of_nonequilibrium_thermal_entanglement.pdf",        "content": "arXiv:0807.0379v2  [quant-ph]  5 May 2009Dynamics of non-equilibrium thermal entanglement\nIlya Sinaysky∗\nQuantum Research Group, School of Physics, University of Kw aZulu-Natal, Durban, 4001, South Africa\nFrancesco Petruccione†\nQuantum Research Group, School of Physics and National Inst itute for Theoretical Physics,\nUniversity of KwaZulu-Natal, Durban, 4001, South Africa\nDaniel Burgarth‡\nMathematical Institute, University of Oxford, 24-29 St Gil es’ Oxford OX1 3LB, UK\n(Dated: November 3, 2018)\nThe dynamics of a simple spin chain (2 spins) coupled to boson ic baths at different temperatures\nis studied. The analytical solution for the reduced density matrix of the system is found. The\ndynamics and temperature dependence of spin-spin entangle ment is analyzed. It is shown that the\nsystem converges toasteady-state. Iftheenergy levels oft he twospins are different, thesteady-state\nconcurrence assumes its maximum at unequal bath temperatur es. It is found that a difference in\nlocal energy levels can make the steady-state entanglement more stable against high temperatures.\nPACS numbers: 03.67.Mn, 03.65.Yz, 65.40.G-\nI. INTRODUCTION\nIn describing real physical systems one should always\ntake into account the influence of the surroundings. The\nstudy of open systems is particularly important for un-\nderstanding processes in quantum physics [1]. Whereas\nin most cases the interaction with an environment de-\nstroys quantum correlations within the system, it is well\nknown that in some situations it can also build up en-\ntanglement [2] and in principle even prepare complex\nentangled states [3]. The dynamics of entanglement in\nopen systems provides many interesting insights into re-\nlaxation and transport situations, in particularif the sys-\ntem dynamics involves many-body interactions (such as\nspin chains, see [4] for a review). In order to understand\nthe role of the various parameters that compete in this\nsetup, it is useful to find exactly solvable models. Here\nwe study the dynamics of a model that was recently in-\ntroduced by L. Quiroga [5]. It consists of a simple spin\nchain in contact with two reservoirs at different temper-\natures. In such a non-equilibrium case most studies are\nrestricted to the steady-state to which the system con-\nvergesinthe limit oflongtimes[5, 6, 7, 8]. The dynamics\nfor the model in the zero-temperature limit was studied\nin [9]. In the following, we study the dynamics of this\nmodel for generic temperatures.\nThis paper is organized as follows. In Sec. II we de-\nscribe the model of a spin chain coupled to bosonic baths\nat different temperatures as introduced in Ref. [5]. For\ncompleteness we follow [5] in deriving the master equa-\ntion for the reduced density matrix in the Born-Markov\n∗Electronic address: ilsinay@gmail.com\n†Electronic address: petruccione@ukzn.ac.za\n‡Electronic address: daniel.burgarth@maths.ox.ac.ukapproximation. In Sec. III we present the analytical so-\nlution for the system dynamics and showthe convergence\nof the obtained solution to the density matrix of the non-\nequilibrium steady state solution. One should note that\nin [5] this steady-state was found only in the case when\nthe energy levels of the spins are equal. Finally, in Sec.\nIV we discuss the results and conclude.\nII. MODEL\nWe consider the simplest spin chain consisting of two\nspins, with each spin coupled to a separate bosonic bath.\nIn the derivation of the master equation we follow the\nformalism suggested in Ref. [5]. The total Hamiltonian\nis given by\nˆH=ˆHS+ˆHB1+ˆHB2+ˆHSB1+ˆHSB2,\nwhere\nˆHS=ǫ1\n2ˆσz\n1+ǫ2\n2ˆσz\n2+K(ˆσ+\n1ˆσ−\n2+ ˆσ−\n1ˆσ+\n2)\nis the Hamiltonian describing spin-to-spin interactions\nand ˆσz\ni,ˆσ±\niare the Pauli matrices. In this paperunits are\nchosen such that kB=/planckover2pi1= 1. The constants ǫ1andǫ2\ndenote the energy of spins 1 and 2, respectively, whereas\nKdenotes the strength of spin-spin interaction. We will\nseelaterthat the energydifference ∆ ǫ=ǫ1−ǫ2hasacru-\ncial role in determining the entanglement of the thermal\nstate. We refer to the case ∆ ǫ= 0 studied in [5] as the\nsymmetric case. Ourstudyfocusesonthenon-symmetric\ncase ∆ǫ/ne}ationslash= 0. The Hamiltonians of the reservoirs for each\nspinj= 1,2 are given by\nˆHBj=/summationdisplay\nnωn,jˆb+\nn,jˆbn,j.2\nThe interaction between the spin subsystem and the\nbosonic baths is described by\nˆHSBj= ˆσ+\nj/summationdisplay\nng(j)\nnˆbn,j+ ˆσ−\nj/summationdisplay\nng(j)∗\nnˆb†\nn,j≡/summationdisplay\nµˆVj,µˆfj,µ.\nThe operators ˆVj,µare chosen to satisfy [ ˆHS,ˆVj,µ] =\nωj,µˆVj,µ,and the ˆfj,µact on bath degrees of freedom\n(this is always possible; their explicit form will be given\nlater on). Physically, the index µcorresponds to transi-\ntionsbetween eigenstates of the system induced by the\nbath. The whole system (spin chain with reservoirs) is\ndescribed by the Liouville equation\nd\ndtˆα=−i[ˆH,ˆα].\nWe assume that the evolution of the dynamical subsys-\ntem (coupled spins) does not influence the state of the\nenvironment (bosonic reservoirs) so that the density op-\nerator of the whole system ˆ α(t) can be written as:\nˆα(t) = ˆρ(t)ˆB1(0)ˆB2(0)\n(irreversibility hypothesis), where each bosonic bath\nis described by a canonical density matrix ˆBj=\ne−βjˆHBj/tr[e−βjˆHBj] and ˆρ(t) denotes the reduced den-\nsity matrix of the spin chain.\nIn Born-Markov approximation the equation for the\nevolution of the reduced density matrix [10] is:\ndˆρ\ndt=−i[ˆHS,ˆρ]+L1(ˆρ)+L2(ˆρ)\nwith dissipators\nLj(ˆρ)≡/summationdisplay\nµ,νJ(j)\nµ,ν(ωj,ν){[ˆVj,µ,[ˆV†\nj,ν,ˆρ]]−\n−(1−eβjωj,ν)[ˆVj,µ,ˆV†\nj,νˆρ]}\nand where the spectral density is given by\nJ(j)\nµ,ν(ωj,ν) =/integraldisplay∞\n0dseiωj,νs/an}bracketle{te−isˆBjˆf†\nj,νeisˆBjˆfj,µ/an}bracketri}htj.\nTo find a solution we go to the basis of the eigenvectors\n|λi/an}bracketri}htwith eigenvalues λiof the Hamiltonian ˆHS,\n|λ1/an}bracketri}ht=|0,0/an}bracketri}ht, λ1=−ǫ1+ǫ2\n2,\n|λ2/an}bracketri}ht=|1,1/an}bracketri}ht, λ2=ǫ1+ǫ2\n2,\n|λ3/an}bracketri}ht= cos(θ/2)|1,0/an}bracketri}ht+sin(θ/2)|0,1/an}bracketri}ht, λ3=κ,\n|λ4/an}bracketri}ht=−sin(θ/2)|1,0/an}bracketri}ht+cos(θ/2)|0,1/an}bracketri}ht, λ4=−κ,whereκ≡/radicalig\nK2+(∆ǫ)2\n4and tanθ≡2K/(∆ǫ). In this\nrepresentation the dissipative operator Li(ˆρ) becomes\nLj(ˆρ) =2/summationdisplay\nµ=1J(j)(−ωµ)(2ˆVj,µˆρˆV†\nj,µ−{ˆρ,ˆV†\nj,µˆVj,µ}+)+\nJ(j)(ωµ)(2ˆV†\nj,µˆρˆVj,µ−{ˆρ,ˆVj,µˆV†\nj,µ}+),\nwith transition frequencies\nω1=λ2−λ3,\nω2=λ2+λ3\nand transition operators\nˆV1,1= cos(θ/2)(|λ1/an}bracketri}ht/an}bracketle{tλ3|+|λ4/an}bracketri}ht/an}bracketle{tλ2|),\nˆV1,2= sin(θ/2)(|λ3/an}bracketri}ht/an}bracketle{tλ2|−|λ1/an}bracketri}ht/an}bracketle{tλ4|),\nˆV2,1= sin(θ/2)(|λ1/an}bracketri}ht/an}bracketle{tλ3|−|λ4/an}bracketri}ht/an}bracketle{tλ2|),\nˆV2,2= cos(θ/2)(|λ3/an}bracketri}ht/an}bracketle{tλ2|+|λ1/an}bracketri}ht/an}bracketle{tλ4|).\nIn this paper we consider the bosonic bath as an infinite\nsetofharmonicoscillators,sothespectraldensityhasthe\nformJ(j)(ωµ) =γj(ωµ)nj(ωµ), where nj(ωµ) = (eβjωµ−\n1)−1andJ(j)(−ωµ) =eβjωµJ(j)(ωµ). For simplicity we\nchoosethecouplingconstanttobefrequencyindependent\nγ1(ω) =γ1andγ2(ω) =γ2.In the basis |λi/an}bracketri}htthe equation\nfor the diagonal elements of the reduced density matrix\nis given by\nd\ndt\nρ11(t)\nρ22(t)\nρ33(t)\nρ44(t)\n=B\nρ11(t)\nρ22(t)\nρ33(t)\nρ44(t)\n,\nwhereBis a 4×4 matrix with constant coefficients. The\ntime-dependence for the non-diagonal elements has the\nfollowing form\nρi,j(t) =etsi,jρi,j(0),\nwheresi,jis a complex number. For the initial state of\nsystem in the computational basis {|00/an}bracketri}ht,|01/an}bracketri}ht,|10/an}bracketri}ht,|11/an}bracketri}ht}\nwe choose\nˆρ(0) =p0|00/an}bracketri}ht/an}bracketle{t00|+p1|01/an}bracketri}ht/an}bracketle{t01|+p2|10/an}bracketri}ht/an}bracketle{t10|\n+(1−p0−p1−p2)|11/an}bracketri}ht/an}bracketle{t11|+c12|01/an}bracketri}ht/an}bracketle{t10|+c∗\n12|10/an}bracketri}ht/an}bracketle{t01|.3\nIII. EXACT SOLUTION\nThe analytical solution in the basis of eigenvectors |λi/an}bracketri}ht\nis given by:\nρii(t) =1\nX1Y2\na11a12a13a14\na21a22a23a24\na31a32a33a34\na41a42a43a44\n\nρ11(0)\nρ22(0)\nρ33(0)\nρ44(0)\n,\nwhere the coefficients aijare given by:\na11= (X+\n1+X−\n1e−tX1)(Y+\n2+Y−\n2e−tY2),\na12= (1−e−tX1)(1−e−tY2)X+\n1Y+\n2,\na13= (1−e−tX1)X+\n1(Y+\n2+Y−\n2e−tY2),\na14= (X+\n1+X−\n1e−tX1)(1−e−tY2)Y+\n2,\na21= (1−e−tX1)(1−e−tY2)X−\n1Y−\n2,\na22= (X−\n1+X+\n1e−tX1)(Y−\n2+Y+\n2e−tY2),\na23= (X−\n1+X+\n1e−tX1)(1−e−tY2)Y−\n2,\na24= (1−e−tX1)X−\n1(Y−\n2+Y+\n2e−tY2),\na31= (1−e−tX1)X−\n1(Y+\n2+Y−\n2e−tY2),\na32= (X−\n1+X+\n1e−tX1)(1−e−tY2)Y+\n2,\na33= (X−\n1+X+\n1e−tX1)(Y+\n2+Y−\n2e−tY2),\na34= (1−e−tX1)(1−e−tY2)X−\n1Y+\n2,\na41= (X+\n1+X−\n1e−tX1)(1−e−tY2)Y−\n2,\na42= (1−e−tX1)X+\n1(Y−\n2+Y+\n2e−tY2),\na43= (1−e−tX1)(1−e−tY2)X+\n1Y−\n2,\na44= (X+\n1+X−\n1e−tX1)(Y−\n2+Y+\n2e−tY2).\nTaking into account the initial conditions, the non-\nvanishing non-diagonal elements are:\nρ34(t) =e−i2tλ3−t(X1+Y2)\n2ρ34(0),ρ43(t) = ¯ρ34=ei2tλ3−t(X1+Y2)\n2ρ43(0).\nIn the present solution we have introduced some con-\nstants:\nXi=X+\ni+X−\ni,\nYi=Y+\ni+Y−\ni,\nX∓\ni= 2cos2(θ/2)J(1)(±ωi)+2sin2(θ/2)J(2)(±ωi)\nY∓\ni= 2sin2(θ/2)J(1)(±ωi)+2cos2(θ/2)J(2)(±ωi)\nor\nX∓\ni= (J(1)(±ωi)+J(2)(±ωi))\n+∆ǫ/radicalbig\n4K2+(∆ǫ)2(J(1)(±ωi)−J(2)(±ωi))\nY∓\ni= (J(1)(±ωi)+J(2)(±ωi))\n−∆ǫ/radicalbig\n4K2+(∆ǫ)2(J(1)(±ωi)−J(2)(±ωi)).\nOne can easily see that this solution converges with in-\ncreasing time to a diagonal density matrix which does\nnot depend on the initial conditions:\nlim\nt→∞ρii(t) =1\nX1Y2\nX+\n1Y+\n2\nX−\n1Y−\n2\nX−\n1Y+\n2\nX+\n1Y−\n2\n,\nlim\nt→∞ρ34(t) = 0.\nIn the symmetric case (∆ ǫ= 0) the above limit in repro-\nduces the result obtained by Quiroga in [5]. In order to\nquantify the entanglement between the spins we consider\nthe concurrence [11]. In the steady-state ( t→ ∞) it is\ngiven by\nC∞=2\nX1Y2Max(0,sinθ\n2|X+\n1Y−\n2−X−\n1Y+\n2|\n−/radicalig\nX−\n1X+\n1Y−\n2Y+\n2).\nIV. RESULTS AND DISCUSSION\nThe dynamics of entanglement is analyzed in Figures\n1 and 2. In Figure 1 the dynamics of the concurrence\nbetween the two qubits is shown. For the model con-\nsidered here the spin chain Hamiltonian ˆHScan entan-\ngle the qubits for specific times, which gives rise to the\noscillations of concurrence one observes for short times4\nFIG. 1: Dynamics of the concurrence C(t) for the initial re-\nduced density matrix ˆ ρ0=|1,0/angbracketright/angbracketleft1,0|. The parameters of the\nmodel chosen to be γ1=γ2= 0.02,ǫ1= 2,ǫ2= 1,K= 1\nfor different temperatures of baths: curve (1) corresponds t o\nT1= 0,5;T2= 0,2; curve (2) T1= 1;T2= 0,5; curve (3)\nT1= 1,5;T2= 1.\nFIG. 2: Dynamics of the concurrence C(t) for different ini-\ntial states of the reduced density matrix of qubits; T1= 1,\nT2= 0.5,γ= 0.02,ǫ1= 2,ǫ2= 1,K= 1. The curve (1)\ncorresponds to ˆ ρ0= (|1,0/angbracketright − |0,1/angbracketright)(/angbracketleft1,0| − /angbracketleft0,1|)/2; curve\n(2) corresponds to ˆ ρ0=|1,0/angbracketright/angbracketleft1,0|; curve (3) corresponds to\nˆρ0=|1,1/angbracketright/angbracketleft1,1|.\nFIG. 3: Steady-state concurrence C∞(TM,∆T) as a func-\ntion of the mean bath temperature TM= (T1+T2)/2 and\ntemperature difference ∆ T=T1−T2in the symmetric case\nǫ1=ǫ2= 3 with K= 1.\nFIG. 4: Steady-state concurrence C∞(TM,∆T) as a function\nof the mean bath temperature TM= (T1+T2)/2 and the\ntemperature difference ∆ T=T1−T2in the case ǫ1= 3,\nǫ2= 1,K= 1.\nFIG. 5: Maximally possible steady-state concurrence\nCMax\n∞(ǫ1,ǫ2) in the strong coupling case ǫ1,ǫ2< Kwith\nK= 1.\n(note that the initial state is chosen to be separable\nˆρ0=|1,0/an}bracketri}ht/an}bracketle{t1,0|). For large times, the system converges\nto its steady-state. One can see the disappearance of en-\ntanglement with increasing temperatures of the bosonic\nbaths which was shown for the steady-state in [5]. In\nFigure 2 the dynamics of the concurrence for different\ninitial states of the qubits in shown. For all cases the\nsystem converges to one and the same value of entan-\nglement. The plots in Figures 1 and 2 show clearly the\ncompetition between unitary and dissipative dynamics.\nIf the qubits starts from a “symmetric state”, i.e. |1,1/an}bracketri}ht,\nno oscillations in the concurrence dynamics are observed\nand if the qubits start from a “non-symmetric state”, i.e.\n|1,0/an}bracketri}ht/an}bracketle{t1,0|or(|1,0/an}bracketri}ht−|0,1/an}bracketri}ht)/√\n2, onecanseeoscillationsof\ntheconcurrencewhichcorrespondtotheenergyexchange\nbetween the qubits in the unitary evolution. Both figures\nreveal that after time of order t∼2/γthe concurrence5\nFIG. 6: Maximally possible steady-state concurrence\nCMax\n∞(ǫ1,ǫ2) in the weak coupling case ǫ1,ǫ2> KforK= 1;\nIn the corner: profile of the 3D surface at the line ǫ1+ǫ2= 4.\n“forgets” about initial conditions and converges to the\nsame value, given by C∞from the end of the Sec. III.\nThe steady-state concurrence C∞(ǫ1,ǫ2,K,T1,T2) is an-\nalyzed in Figures 3, 4, 5 and 6. In Figures 3 and 4 we\nplot the steady-state concurrence for the symmetric and\nnon-symmetric cases as a function of the mean tempera-\nture (TM= (T1+T2)/2) and the temperature difference\n(∆T=T1−T2) of the baths. In the symmetric case one\ncaneasilyseethatthemaximalvalueoftheentanglement\nis reached for equal bath temperatures (∆ T= 0)\nCeq\nsym=sinh(1/T)−1\n2cosh(ω1/2T)cosh(ω2/2T).\nThe critical temperature in units of Kabove which\nthe steady-state becomes separable is given by TC=\narcsinh(1)−1(TC≈1.136) [5]. It is interesting to note\nthat in the non-symmetric case (Figure 4) the maxi-\nmal entanglement is reached in the non-equilibrium case\n(∆T/ne}ationslash= 0). In particular, the maximal entanglement is\nlargerthanthecorrespondingnon-symmetricequilibrium\nconcurrence\nCeq\nnon−sym=sinθsinh((ω2−ω1)/2T)−1\n2cosh(ω1/2T)cosh(ω2/2T).\nThe temperature at which entanglement disappears is a\nfunction of the energy difference ∆ ǫbetween qubits:\nTC=/radicalbig\n∆ǫ2/4+1\narcsinh/radicalbig\n∆ǫ2/4+1.It is easy to see that this function reaches its minimum\nvalue in the symmetric case (∆ ǫ= 0). In Figures 5 and 6\nwe show the maximally reachable value of entanglement\nas a function of qubits energies in the strong and weak\ncoupling case. For every pair of energies ( ǫ1,ǫ2) we max-\nimize the value of the concurrence for the different tem-\nperatures of the baths ( T1,T2). One can see that in the\nstrong coupling case ( ǫ1,ǫ2< K; Figure 5) the maximal\nvalue of the entanglement corresponds to the symmetric\ncase. In Figure 6 one can see that in the weak coupling\ncase (ǫ1,ǫ2> K) the maximal value of the entanglement\nis reached in the non-symmetric case.\nIn conclusion, we have found an analytical solution for\na simple spin system coupled to bosonic baths at differ-\nenttemperatures. Westudiedthedynamicsofthesystem\nand showed that on the long term the system converges\nto the steady-state solution. Resolving the entanglement\ndynamicsallowedustodistinguishbetweenentanglement\ncreated by the system and by the bath. For the symmet-\nric case ( ǫ1=ǫ2) we reproduced the steady-state found\nby [5]. We focused on the non-symmetric case ( ǫ1/ne}ationslash=ǫ2)\nwhere we found that the steady-state concurrence as-\nsumes its maximal value for unequal bath temperatures.\nThis is corresponds to a dynamical equilibrium, where\nthe spin chain transfers heat between the baths. We also\nfound that a difference in local energy levels can make\nthe steady-state entanglement more stable against high\ntemperatures. These analytical results motivate further\nnumerical studies on longer spin chains.\nAcknowledgments\nThis work is based upon research supported by the\nSouth African Research Chair Initiative of the Depart-\nment of Science and Technology and National Research\nFoundation.\nDBwouldliketothankFPforthehospitalityatUKZN\nand acknowledges support by the QIP-IRC and Wolfson\nCollege, Oxford.\n[1] H.-P.Breuer and F.Petruccione, The Theory of Open\nQuantum Systems (Oxford University Press, 2002).\n[2] D. Braun, Phys. Rev. Lett. 89, 277901 (2002).[3] S. Diehl et al., arXiv:0803.1482v1 [quant-ph].\n[4] L. Amico, R. Fazio, A. Osterloh and Vlatko Vedral, Rev.\nMod. Phys. 80, 517 (2008).6\n[5] L. Quiroga, F.J. Rodriguez, M.E. Ramirez, R. Paris,\nPhys. Rev. A75, 032308 (2007).\n[6] T. Prosen, New J. Phys. 10, 043026 (2008).\n[7] C. Mejia-Monasterio and H. Wichterich, Eur. Phys. J.\nSpec. Top. 151, 113 (2007).\n[8] D. Burgarth and V. Giovannetti, Phys. Rev. A 76,062307 (2007).\n[9] M. Scala, R. Migliore, A. Messina, arXiv:0806.4852v1\n[quant-ph].\n[10] M. Goldman, J.Magn.Reson. 149, 160 (2001).\n[11] W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)"    },    {        "title": "0709.2054v1.Spin_polarization_in_biased_Rashba_Dresselhaus_two_dimensional_electron_systems.pdf",        "content": "arXiv:0709.2054v1  [cond-mat.other]  13 Sep 2007Spin polarization in biased Rashba-Dresselhaus two-dimen sional\nelectron systems\nP. Kleinert\nPaul-Drude-Intitut f¨ ur Festk¨ orperelektronik,\nHausvogteiplatz 5-7, 10117 Berlin, Germany\nV.V. Bryksin\nA.F. Ioffe Physical Technical Institute,\nPolitekhnicheskaya 26, 194021 St. Petersburg, Russia\n(Dated: October 27, 2018)\nAbstract\nBased on spin-charge coupled drift-diffusion equations, whi ch are derived from kinetic equations\nfor the spin-density matrix in a rigorous manner, the electr ic-field-induced nonequilibrium spin\npolarization is treated for a two-dimensional electron gas with both Rashba and Dresselhaus spin-\norbit coupling. Most emphasis is put on the consideration of the field-mediated spin dynamics for\na model with equal Rashba and Dresselhaus coupling constant s, in which the spin relaxation is\nstrongly suppressed. Weakly damped electric-field-induce d spin excitations are identified, which\nremind of space-charge waves in crystals.\nPACS numbers: 72.25.Dc, 72.25.Rb, 72.10.-d\n1I. INTRODUCTION\nSpin-dependent transport phenomena are of great interest to b oth basic research and\ndevice applications. Especially, semiconductor-based spin electron ics has been the subject\nof numerous investigations. In this field, spin rather than charge is exploited for signal\nprocessing. The spin-orbit interaction (SOI) opens the possibility o f manipulating the spin\nof carriers using purely electrical means. Unfortunately, the ver y same SOI has the unde-\nsired effect of causing spin relaxation due to precession in a wave-ve ctor dependent effective\nmagnetic field, which is traced back to the SOI. For a quantum well gr own on a [001]\nsubstrate, the Dyakonov-Perel spin relaxation1is the most dominant effect. In general,\nhowever, spin relaxation depends on the details of the band struct ure and the relevant scat-\ntering mechanisms (see Ref. [2] and references therein). For an a symmetric quantum well,\nbulk-inversion asymmetry and structure-inversion asymmetry giv e rise to Dresselhaus and\nRashba spin-orbit contributions to the Hamiltonian, respectively. T he interplay between\nthe linear Rashba and Dresselhaus terms causes a spin relaxation an isotropy that has been\ntreated in a number of theoretical works.3–7For a quantum well grown along the [001] di-\nrection, the main axes of the spin-relaxation-time tensor are given by [110] and [1 10]. Most\ninteresting is the observation that under idealized conditions and wh en the linear Rashba\nand Dresselhaus terms have equal strength, the relaxation of sp in oriented along the [110]\naxis is totally suppressed.3In this particular case, a conserved quantity exists, which hinders\nspin randomization.5,6This remarkable behavior of the spin-relaxation time lead to the pro-\nposal of a nonballistic spin-FET.5Other studies of the combined Rashba-Dresselhaus model\nreferred to the spin- and charge-Hall effect.8–13\nRecently, the field received a fresh impetus by the identification of a n exact SU(2) sym-\nmetry of the model, when reaching the condition of equal Rashba an d Dresselhaus coupling\nstrength.14,15From a theoretical point of view, an equivalent system is the Dresse lhaus [110]\nmodel. The revealed symmetry gives rise to a massless modewith infinit e lifetime at nonzero\nwave vector. Qualitative features of the associated spin pattern have been experimentally\nconfirmed16by optical techniques that probe spin relaxation rates. Furtherm ore, it has been\npredicted17,18that coherent spatial oscillations of the spin polarization develop in s uch a\nsystem under appropriate injection conditions.\nIn this paper, we extend these interesting studies by treating spin effects under the\n2influence of an applied in-plane electric field. For the combined Rashba -Dresselhaus model\nwith a SOI that is linear in k, spin-charge coupled drift-diffusion equations are derived in a\nsystematic manner. Special results are presented and discussed for the special model with\nequal Rashba and Dresselhaus coupling strengths.\nII. SPIN-CHARGE COUPLED DRIFT-DIFFUSION EQUATIONS\nThe effective Hamiltonian of our approach\nH0=/planckover2pi12k2\n2m+α(kyσx−kxσy)+β(kxσx−kyσy) (1)\nincludestheRashbaspin-orbitterm, whichisduetotheinversionasy mmetryoftheconfining\npotentialofthequantumwell. Inaddition,thereistheDresselhaus coupling, whichispresent\nin semiconductors lacking bulk inversion symmetry. The model Hamilto nian refers to a two-\ndimensional semiconductor nanostructure grown along the [001] d irection. In Eq. (1), k,\nm, andσi(i=x,y,z) denote the in-plane wave vector, the effective electron mass, an d the\nusual Pauli matrices, respectively. αandβare the strengths of the Rashba and Dresselhaus\nspin-orbit couplings. Our total Hamiltonian encompasses also contr ibutions stemming from\nthe short-range spin-independent elastic scattering on impurities and the in-plane electric\nfieldE= (Ex,Ey,0). Its explicit form together with related kinetic equations for the spin-\ndensity matrix has been published recently.19It should be noted that for the combined\nRashba-Dresselhaus model, the field-induced spin polarization depe nds on the orientation\nof the in-plane electric field.12,13\nThe main quantity for the theoretical analysis of spin-related phen omena is the spin-\ndensity matrix fλ\nλ′(k,k′,t), which is calculated from quantum-kinetic equations.20From this\nset of equations, spin-charge coupled drift-diffusion equations ar e derived for the physical\ncomponents f= Tr/hatwidefandf= Trσ/hatwidef, which are integrated over the polar angle of k(denoted\nby the bar). In the case of weak SOI ( αkFτ//planckover2pi1,βkFτ//planckover2pi1≪1, withkFandτbeing the Fermi\nwave vector and elastic scattering time, respectively), the followin g ansatz is justified21\nf(k,q,t) =−F(q,t)dn(εk)/dεk\ndn/dε F,f(k,q,t) =−F(q,t)dn(εk)/dεk\ndn/dε F, (2)\nwheren(εk) denotes the Fermi function and εk=/planckover2pi12k2/(2m). Furthermore, we introduced\nthe electron density n=/integraltext\ndερ(ε)n(ε), withρ(ε) being the density of states of the two-\ndimensional electron gas. Applying this straightforward calculation al scheme21, spin-charge\n3coupled drift-diffusion equations are obtained, which read in spatial coordinates rj\n∂Fi\n∂t+∂Jij\n∂rj+MijFj=2mτ\n/planckover2pi13τsLiF, (3)\nwith the expression for the spin flux\nJij=/parenleftbigg\nµEj−D∂\n∂rj/parenrightbigg\nFi+4Dm\n/planckover2pi12εiklQkjFl−4mDτ\n/planckover2pi13(α2−β2)/bracketleftbigg2m\n/planckover2pi12Qij−εijkµEk\n2D/bracketrightbigg\nF.(4)\nThe diffusion coefficient Dand the mobility µsatisfy the Einstein relation µ=\n(eD/n)dn/dε F. Other quantities, which enter Eqs. (3) and (4), are defined by\nL=2m\n/planckover2pi12τs(−(αµEy+βµEx),(αµEx+βµEy),0),1\nτs= 4Dm2\n/planckover2pi14(α2+β2),(5)\nMij=Aij\nτs+1\nτsεikjLk+2αβ\n(α2+β2)τsSij, (6)\n← →A=\n1 0 0\n0 1 0\n0 0 2\n,← →Q=\nβ α0\n−α−β0\n0 0 0\n,← →S=\n0 1 0\n1 0 0\n0 0 0\n. (7)\nεijkis the totally antisymmetric tensor in three dimensions. The spin flux Jij, which gives\nthei’th component of the particle flow with spin polarization along the axis j(i,j=x,y,z),\nhas the general form that is in accordance with symmetry requirem ents.22Neglecting the\nspin-charge coupling and the influence of the electric field, we obtain a result\n∂Fi\n∂t=−m\n/planckover2pi12QjlεjikJkl, Jkl=4Dm\n/planckover2pi12εkmnQmlFn, (8)\nin which the spin flux Jklappears as a source of nonequilibrium spin polarization.22,23The\nset of basic equations (3) and (4), which restore published results14in the case of vanishing\nelectricfield, aresolvedanddiscussedintheFourierspacewithresp ecttospatialcoordinates.\nFor the sake of a better readability of the paper, these equations are summarized in the\nAppendix.\nIII. RESULTS AND DISCUSSION\nA number of quite interesting electric-field effects on the spin polariz ation occurring\nin systems with Rashba and Dresselhaus SOI can be studied on the ba sis of the coupled\nspin-charge drift-diffusion Eqs. (3) and (4) [or (A1) and (A2)]. To k eep the presentation\ntransparent, we restrict ourselves to spatially infinite systems by omitting any boundary\neffects.\n4A. Charge current\nThe current density of charge carriers is defined by the time deriva tive of the dipole\nmoment, which can be expressed by\nj(t) =−ie∇κ∂\n∂tF(κ,t)|κ=0. (9)\nAs we are mainly interested in the charge transport along the [110] a nd [110] directions, the\nspatial coordinates are rotated by introducing new wave vectors κ±= (κx±κy)/√\n2. In\nthese coordinates, the charge current along the κ+direction is given by\nj+(t) =−ie∂2\n∂t∂κ+F(κ+,κ−= 0,t)|κ+=0. (10)\nTaking into account Eq. (A1), we immediately obtain for the Laplace t ransformed current\ndensity\nj+(s) =eµE+F\ns+e\n/planckover2pi1(α+β)F−(s)−2meτ\n/planckover2pi13(α2−β2)µE−Fz(s), (11)\nwhereE±= (Ex+Ey)/√\n2 andF±= (Fx±Fy)/√\n2. The components of the density matrix,\nwhich enter this equation, refer to a spatially homogeneous system . The related ( κ=0)\ndrift-diffusion Eqs. (A1) and (A2) are easily solved. Inserting the s olution into Eq. (11) and\ntaking into account only contributions linear to the electric field, we g et\nj+(ω) =eµE+F/braceleftbigg\n1+α2τ\n/planckover2pi12D1−γ2\niωτs−(1−γ)2/(1+γ2)/bracerightbigg\n, (12)\nwithγ=β/α. As the current is aligned along the electric field direction, there is no Hall\ncurrent, the appearance of which would require the treatment of the nonlinear field term in\nEq. (11), which is of higher order in the spin-orbit coupling constant . In Eq. (12), the pure\ncharge current eµE+Fis complemented by a spin contribution, which exhibits a Drude-like\nfrequency dependence governed by the spin-relaxation time τsthat may be much larger than\nthe elastic scattering time τ. Consequently, the frequency dispersion of the spin-mediated\ncurrent contribution can set in at lower frequencies as in the ordina ry Drude formula. The\nspin-induced current contribution disappears, when the Rashba a nd Dresselhaus coupling\nconstants are equal ( α=β). For the Rashba model ( β= 0), Eq. (12) resembles the results\nderived previously for small polarons.24\n5B. Out of plane spin polarization for α=β\nAs another application of the spin-charge drift-diffusion Eqs. (A1) and (A2), we treat the\nevolution of an initial spin lattice produced at t= 0 along the κ+direction\nfz0(κ+) =1\n2fz0δ(κ+−κ0)+1\n2f∗\nz0δ(κ++κ0), fz0=|fz0|eiϕ, (13)\nwithκ0being a given wave vector. Restricting ourselves to the special cas eα=β, we obtain\nthe exact solution\nFz(κ+,s) =fz0(κ+)σ\nσ2+2Ω2κ+, (14)\nwith the effective Laplace variable\nσ=s−iµE+κ++Dκ2\n++2/τs, (15)\nand the frequency\nΩκ+= 2√\n2mα\n/planckover2pi12(µE++2iDκ+). (16)\nThe inverse Laplace and Fourier transformations of Eq. (14) are e asily calculated and we\nobtain for the asymptotic dynamics at large times\nFz(r+,t) =|fz0|\n2exp/bracketleftbig\n−D(κ0−2K)2t/bracketrightbig\ncos[κ0r++ϕ+µE+(κ0−2K)t].(17)\nForκ0= 2K, theoriginalstaticspin latticesurvives andretainsitsshapeinthes teady state.\nUnder this excitation condition, a long lived spin pattern is expected t o occur. If κ0slightly\ndeviates from 2 K, the electric field drives a propagating spin wave, the amplitude of wh ich\ndiminishes with time exponentially. The frequency of this wave is given b yµE+(κ0−2K).\nC. Field-induced spin accumulation and Hanle effect\nIn this subsection, the electric field-induced out-of-plane spin acc umulation is studied for\na different initial spin preparation. First, we focus on the contribut ion, which appears in\na homogeneous electron gas ( κ=0). From Eqs. (A7) to (A10) given in the Appendix, we\nobtain the steady-state solution\nFz=−4αβmτ\nD/planckover2pi13µ2(E2\nx−E2\ny)\n2/τs+(µE)2/DF, (18)\n6which appliesforasemiconductor withdifferent Rashba andDresselh aus SOIconstants( α/negationslash=\nβ). This field-mediated spin accumulation depends on the orientation o f the electric field\nwithin the plane and has the character of a second-order field effec t. A spin accumulation,\nwhichisproportionaltotheelectricfield, occursonlyintheplane.12Thequadraticfieldeffect\nin Eq. (18) disappears for the pure Rashba ( β= 0) and Dresselhaus ( α= 0) system as well\nas under the condition |Ex|=|Ey|. For sufficiently high electric fields [2 /τs≪(µE)2/D],\nthe out of plane spin polarization approaches the constant field-ind ependent value Fz=\n−4αβmτcos(2ϕ)F//planckover2pi13, withϕbeing the polar angle of the electric field.\nIn addition to this field contribution of a homogeneous system, we st udy the response to\na permanent harmonic spin generation of the form\nG(r+,t) =Geiωt+iκ+r+. (19)\nDisregarding the spin-charge coupling, the drift-diffusion Eqs. (A8 ) to (A10) are analytically\nsolved by\nFz(κ+,ω) =G(iΩ+2/τs+)(iΩ+2/τs−)\n(iΩ+2\nτs+)(iΩ+2\nτs−)(iΩ+2\nτs)+T2\n+(iΩ+2\nτs−)+T2\n−(iΩ+2\nτs+),(20)\nwhere the short-hand notations\niΩ =iω−iµκ+E++Dκ2\n+, T+= 2K+(µE++2iDκ+), T−= 2K−µE−, K±= (α±β)m\n/planckover2pi12(21)\nwere used. This solution becomes more transparent for the specia l caseα=β, where we\nobtain\nFz(κ+,ω) =Giω−iµκ+E++Dκ2\n++2/τs\nN+N−, N±=iω−iµE+(κ+±2K)+D(κ+±2K)2.(22)\nThe dispersion relation of eigenmodes ω=µE+(κ+−2K) is derived from the denominator\nN−. This mode has the character of free carrier oscillations complemen ted by a spin part\nthat gives rise to a soft mode at κ+= 2K.\nFurther information about the solution in Eq. (20) is obtained for α≈βand Ω≪1/τs.\nTwo limits can be distinguished in this case. Under the condition Ω ≪1/τs−, which applies\nto the steady state, Eq. (20) simplifies to\nFz=G\n2/τs+(µE)2/D, (23)\n7which can be interpreted as the electric-field analogy of the Hanle eff ect.21,25Another result\nis obtained in the opposite case 1 /τs−≪Ω, when the out-of-plane spin polarization depends\non the orientation of the electric field\nFz=G\n2/τs+[µ(Ex+Ey)]2/(2D). (24)\nThis solution dictates the behavior of the spin polarization in the limit α→β. Both results\nagree for the special field configuration Ex=Ey=E/√\n2.\nD. Spin remagnetization waves for α=β\nFinally, we treat the relaxation of an initial homogeneous spin moment F0. In this case,\nwe useκ=0in our basic equations so that spin and charge degrees of freedom d ecouple\nfrom each other. Most interesting results are expected, when th e Rashba and Dresselhaus\nSOI couplings coincide ( α=β). The set of Eqs. (A8) to (A10) is easily solved for the\nLaplace-transformed functions. The result\nFx(t) =Fx0−Fy0\n2+e−2t/τs/braceleftbiggFx0+Fy0\n2cos(ΩEt)−Fz0√\n2sin(ΩEt)/bracerightbigg\n, (25)\nFy(t) =−Fx0−Fy0\n2+e−2t/τs/braceleftbiggFx0+Fy0\n2cos(ΩEt)−Fz0√\n2sin(ΩEt)/bracerightbigg\n,(26)\nFz(t) =e−2t/τs/braceleftbiggFx0+Fy0√\n2sin(ΩEt)+Fz0cos(ΩEt)/bracerightbigg\n, (27)\ndescribes spin rotations with the frequency Ω E=√\n2Kµ(Ex+Ey). Such rotations of the\nmagnetic moment were studied previously both for hopping of small p olarons24,26and for\nextended states in a two-dimensional electron gas.27,28In analogy to space-charge waves,\nthese eigenmodes are called spin-remagnetization waves. Typically, the amplitude of these\nexcitations exponentially decrease with increasing time. It is a peculia rity of the special\nRashba-Dresselhaus model that the in-plane magnetic moment is co nserved: Fx(t)−Fy(t) =\nFx0−Fy0. This observation provides a further example for the occurrence of undamped spin\nexcitations, whenever the coupling constants αandβare equal.\nIV. SUMMARY\nBothfortheRashba-Dresselhausmodelwithequalcouplingconst antsandfortheDressel-\nhaus [110] model it has been recently realized that the relaxation of spin oriented in the [110]\n8axis is totally suppressed. As the spin lifetime becomes arbitrarily long within these models,\nthe existence of a persistent spin grating has been predicted. Rec ent experiments16,29,30in-\ndeed revealed evidence supporting this interesting peculiarity of slo w spin relaxation rates.\nThe extreme suppression of spin relaxation in the special theoretic al models for semicon-\nductor nanostructures with SOI is due to a spin symmetry that give s rise to a soft mode at\nκ= 2K. This weakly damped eigenmode leads also to interesting phenomena in a biased\nspin-orbit coupled system that was studied in this paper. Based on r igorous spin-charge\ncoupled drift-diffusion equations for the components of the spin-d ensity matrix, a number\nof field-mediated spin effects were considered:\n(i) A Drude-like spin-induced contribution was identified in the longitud inal charge cur-\nrent that disappears in the special Rashba-Dresselhaus model wit h equal coupling strengths.\nThe frequency positionand thewidth of theresonance aredeterm ined by the spin-relaxation\ntime and the coupling constants αandβ.\n(ii) A persistent spin pattern created at κ= 2Kis not destroyed by the electric field.\nHowever, under the condition of slight-off resonance, the field for ces the pattern to move.\n(iii) In a homogeneous electron gas with Rashba and Dresselhaus SOI , an out-of-plane\nspin polarization develops, which has a nonlinear field character. In a ddition, long-lived\nremagnetization waves can be excited, whose frequency is given by ω=µE+(κ+−2K).\nThese eigenmodes can be studied in a similar manner as space-charge waves in crystals. In\naddition, the electric-field analogy of the Hanle effect changes its ch aracter at α≈β.\n(iv) The in-plane spin polarizationof the special Rashba-Dresselhau s model with α=βis\nconserved and the spins rotate due to the electric field with the fre quency Ω E=√\n2Kµ(Ex+\nEy).\nThe experimental confirmation of the field-mediated spin effects pr edicted here would\nstimulate further progress both in basic research and technologic al innovations.\nAcknowledgments\nPartial financial support by the Deutsche Forschungsgemeinsch aft and the Russian Foun-\ndation of Basic Research is gratefully acknowledged.\n9APPENDIX A: FOURIER TRANSFORMED DRIFT-DIFFUSION EQUATIONS\nThe Fourier transformed version of our basic spin-charge coupled drift-diffusion equations\n(3) and (4) has the form\n/bracketleftbigg∂\n∂t−iµEκ+Dκ2/bracketrightbigg\nF+iωκ·F−2imτ\n/planckover2pi13(α2−β2)([κ×µE]·F) = 0,(A1)\n/bracketleftigg\n∂\n∂t−iµEκ+Dκ2+← →T\nτs/bracketrightigg\nF−[H×F]−iχ\n/planckover2pi1[κ×µE]F+χHF= 0,(A2)\nwith the abbreviations\nHx=2m\n/planckover2pi12[α(µEy+2iDκy)+β(µEx+2iDκx)], (A3)\nHy=−2m\n/planckover2pi12[α(µEx+2iDκx)+β(µEy+2iDκy)], Hz= 0, (A4)\n← →T=\n12αβ\nα2+β20\n2αβ\nα2+β21 0\n0 0 2\n, χ=2mτ\n/planckover2pi13(α2−β2). (A5)\nFor the Rashba model ( β= 0), these equations agree with results derived previously.21\nLet us treat these equations in another representation charact erized by the components\nκ±=κx±κy√\n2, F±=Fx±Fy√\n2. (A6)\nFor excitations of the spin polarization that propagate exclusively a long the κ+direction\n(κ−= 0), Eqs. (A1) and (A2) are written in the form\n/bracketleftbigg∂\n∂t−iµE+κ++Dκ2\n+/bracketrightbigg\nF−i\n/planckover2pi1(α+β)κ+F−+2imτ\n/planckover2pi13(α2−β2)µE−κ+Fz= 0,(A7)\n/bracketleftbigg∂\n∂t−iµE+κ++Dκ2\n++2\nτs+/bracketrightbigg\nF++2m\n/planckover2pi12(α+β)(µE++2iDκ+)Fz\n−4m2τ\n/planckover2pi15(α−β)2(α+β)µE−F= 0, (A8)\n/bracketleftbigg∂\n∂t−iµE+κ++Dκ2\n++2\nτs−/bracketrightbigg\nF−+2m\n/planckover2pi12(α−β)µE−Fz\n+4m2τ\n/planckover2pi15(α+β)2(α−β)(µE++2iDκ+)F= 0, (A9)\n10/bracketleftbigg∂\n∂t−iµE+κ++Dκ2\n++2\nτs/bracketrightbigg\nFz+2imτ\n/planckover2pi13(α2−β2)µE−κ+F\n−2m\n/planckover2pi12(α+β)(µE++2iDκ+)F+−2m\n/planckover2pi12(α−β)µE−F−= 0, (A10)\nwithE±= (Ex±Ey)/√\n2 and\n2\nτs+=(α+β)2\nτs(α2+β2),2\nτs−=(α−β)2\nτs(α2+β2). (A11)\nFor the particular case α=β, the coupling between spin and charge degrees of freedom\ndisappears and the secular equation for the spin components gives the following dispersion\nrelations for eigenmodes of the biased spin system\nω1=−µE+κ+−iDκ2\n+, ω2,3=−(κ+±2K)[µE++iD(κ+±2K)], (A12)\nwithK= 2mα//planckover2pi12. This result implies that there appears a field-induced undamped sof t\nmode at κ+= 2K, which reflects the presence of the spin rotation symmetry.14\n1M. I. Dyakonov and V. I. Perel, JETP Lett. 13, 467, (1971), [Sov. Phys. JETP 33, 1053 (1971)].\n2J. L. Cheng and M. W. Wu, J. Appl. Phys. 101, 073702 (2007).\n3N. S. Averkiev and L. E. Golub, Phys. Rev. B 60, 15582 (1999).\n4N. S. Averkiev, L. E. Golub, and M. Willander, J. Phys.: Conde ns. Matter 14, R271 (2002).\n5J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801 (2003).\n6J. Schliemann and D. Loss, Phys. Rev. B 68, 165311 (2003).\n7Y. V. Pershin, Physica E 23, 226 (2004).\n8N. A. Sinitsyn, E. M. 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Lett. 98, 036603\n(2007).\n12"    },    {        "title": "2104.08341v1.PT_symmetry_enabled_spin_circular_photogalvanic_effect_in_antiferromagnetic_insulators.pdf",        "content": "1 \n PT-symmetry enabled spin circular photogalvanic effect in \nantiferromagnetic insulators  \nRuixiang Fei,1,† Wenshen  Song ,1 Lauren Pusey -Nazzaro,1 Li Yang 1, 2, ‡ \n1 Department of Physics, Washington University in St Louis, St Louis, Missouri 63130, United \nStates  \n2 Institute of Materials Science and Engineering, Washington University in St. Louis, St. Louis, \nMissouri 63130, United States   \n \nAbstract  \nThe short timescale spin dynamics in antiferromagnets  is an attractive feature from the standpoint \nof ultrafast spintronics . Yet generating highly polarized spin current s at room temperature remains \na fundamental challenge  for antiferromagnets . We propose a spin circular photogalvanic effect  \n(spin -CPGE) , in which  circularly polarized light  can produce  a spin current without net charge  \ncurrent a t room temperature , through an “injection -current -like” mechanism in parity -time(PT)-\nsymmetric antiferromagnetic (AFM) insulators . We demonstrate this effect  by first-principles \nsimulations of bilayer CrI 3 and room -temperature AFM  hematite.  Our calculations show that the \nspin-CPGE is significant , and the  magnitude of spin photo -current is comparable with  the widely \nobserved charge photocurrent in ferroelectric materials. Interesti ngly, this spin photocurrent is not \nsensitive to spin -orbit interactions, which were regarded as fundamental mechanism s for \ngenerating spin current . Given the fast response of light -matter interactions , large energy scale , \nand insensitivity to spin-orbit interactions , our work gives hope to realizing  a fast -dynamic and \ntemperature -robust pure spin current in a wide range of PT-symmetric AFM materials, including \nweak -relativistic magnetic  insulators  and topological axion insulators . \n  2 \n Introductio n:  Spintronics  based on antiferromagnetic (AFM) materials has great potential in \nreducing device scale and power consumption  [1–3]. In contrast to ferromagnets, antiferromagnets \nare robust against perturbed magnet ic fields, produce no stray fields, and are capable of generating \nlarge magneto -transport effects  [1], making them promising for next -generation spintronics. \nParticularly , antiferromagnets exhibit unique advantages  in generating spin  current , which is the \nbasis for spintronic devices . Compared with gigahertz microwave pulses used in ferromagnets, \nterahe rtz pulse s can pu mp spin current by exciting the left- and right -hand mode  of magnon s in \nantiferromagnets  [4–6] due to the strong exchange interaction , giving rise to  ultrafast spin \ndynamics  [7].  Besides , spin caloritronics based on magnon s in AFM materials , such as the spin \nSeebeck and spin Nernst effect s [8–11], was proposed to generate pure spin current  in the same \ndevice . However , because of the int rinsically small energy scales, magnon -based spin current s \ndecay  rapidly with increasing temperature   [5,12] ,  making it difficult  for de vice applications  at \nroom temperature .   \nOptical pump -induced spin dynamics  may be a promising approach to surmounting such \ndifficulties  because of its intrinsically larger energy scale  [13,14]  and observed strong light -matter \ninteractions in magnetic materials  [15–17]. Notably , second -order light-matter interactions, such \nas the circular photogalvanic effect  (CPGE ) or bulk photovoltaic effect  (BPVE) , are known to  \ncreate  charge current in polar materials at room temperature without bias  [18–23]. Recently,  This \nidea has been expanded to two-dimensional (2D) party -time (PT) symmetric AFM insulators  [24] \nand topological axion insulator s [25,26] , in which  a sizeable DC charge current was predicted to \nbe generated by linearly polarized light  [24,25,27] .  \nBeyond charge current, light can also drive  spin current . For example,  spin photo current  \nwas predicted in  AFM hematite  [28] and bilayer CrCl 3 [29], via the shift-current mechanism  under 3 \n linearly polarized  light, denoted as the spin -BPVE  [28]. Unfortunately , linearly -polarized light  \nsimultaneously excit es significant  charge current  because of the broken SU( 2) symmetry due to \nspin-orbit coupling (SOC)  [24,25] , substantially diluting spin polarization  of the overall  current . It \nremains a fundamental challenge to generate a highly polarized  spin current in AFM  materials  at \nroom temperature .   \nIn this work, we predict a spin circular photogalvanic effect  (spin -CPGE), in which  \ncircularly polarized light  can generate pure spin current in PT-symmetric antiferromagnets. This \neffect arises from a second -order light -matter interaction , i.e., an inject -current -like mechanism . \nThe net charge current is zero, being a  PT-even contribution and having a trivial  non-abelian Berry \nconnection , while the spin current is nonzero since it is a  PT-odd contribution . We further \ndemonstrate this effect  by first -principles simulations  of two typical PT-symmetric \nantiferromagnets : bilayer CrI 3 and three -dimensional (3D)  hematite. Our result confirms enhanced \npure spin current and unexpectedly show s that SOC is inessential for generating spin current in \nPT-symmetric antiferromagnets . Our work  suggest s that collinear PT-symmetric AFM insulators, \neven weak -relativistic AFM insulators , can serve as effective  and temperature -robust  spin \ngenerators for both spin orientations via ultrafast light -matter interaction s. \nPT-symmetry induced chiral spin photocurrent  \nLet us consider an insulating AFM crystal that breaks both time -reversal symmetry ( T) and \ninversion symmetry ( P) but respects their combination PT. In such a case, the Hamiltonian satisf ies \n𝐻(𝑘,↑)=𝐻(𝑘,↓)  for any 𝑘 wavevector of reciprocal space, resulting in exactly double spin-\ndegenerate band s, as shown in Figure 1(a). Because of the se degenerate band s and zero  Abelian  \nBerry curvature, very subtle  longitudinal  and transverse  spin current can be o bserved  in transport  \nmeasurement s. Nevertheless, after involving the second -order light-electron interaction for the 4 \n non-biased case, we will show that these degenerate spins exhibit  different behaviors, enabl ing a \ndesired spin-polarized current.  \nFor a coherent light illumination, the general form of a second -order nonlinear DC \nphotocurrent is 𝐽𝑐=𝜒𝑎𝑏𝑐(0;𝜔,−𝜔)𝐸𝑎(𝜔)𝐸𝑏(−𝜔), where  a and b the polarization directions of \nlight, c is the outgoing current direction , and  𝜒𝑎𝑏𝑐(0;𝜔,−𝜔) is the DC photoconductivity  in the \nc-direction . For circularly polarized light, the  inject -current -like photoconductivity 𝜂𝑎𝑏𝑐𝛼≡\n𝜒𝑎𝑏𝑐(0;𝜔,−𝜔) under  the relaxation time approximation is  [19,30]  \n𝜂𝑎𝑏𝑐𝛼=−𝜋𝑒3\nℏ2𝜔2∑ ∫𝑑3𝑘𝐼𝑚(𝑣𝑚𝑛,𝛼𝑎(𝑘)𝑣𝑛𝑚,𝛼𝑏(𝑘))𝑓𝑚𝑛(⟨𝑚|{𝜎𝛼,𝑣𝑐}|𝑚⟩𝜏𝑚− 𝑚𝑛\n⟨𝑛|{𝜎𝛼,𝑣𝑐}|𝑛⟩𝜏𝑛)𝛿(𝜔−𝜔𝑚𝑛),     (1) \nwhere 𝐼𝑚(𝑣𝑚𝑛,𝛼𝑎(𝑘)𝑣𝑛𝑚,𝛼𝑏(𝑘))=1\n2(𝑣𝑚𝑛,𝛼𝑎𝑣𝑛𝑚,𝛼𝑏(𝑘)−𝑣𝑚𝑛,𝛼𝑏(𝑘)𝑣𝑛𝑚,𝛼𝑎(𝑘)) is the imaginary part of \noptical oscillator strength. 𝑣𝑚𝑛,𝛼𝑎(𝑘) and 𝑣𝑛𝑚,𝛼𝑏(𝑘) are the a-direction and b-direction interband \nvelocity  matrix element s between the 𝑚-th and 𝑛-th bands  with 𝛼 spin at the same wave vector k, \nrespectively.  𝑓𝑚𝑛=𝑓(𝜖𝑚𝑘)−𝑓(𝜖𝑛𝑘) is the occupation number difference between the 𝑚-th and \n𝑛-th bands.  𝜏𝑚 and 𝜏𝑛 are the minimum value of spin -relaxation time and free -carrier relaxation \ntime of the 𝑚-th and 𝑛-th bands , respectively . We define  {𝜎𝛼,𝑣𝑐} =1\n2(𝑣𝑐𝜎𝛼+𝜎𝛼𝑣𝑐), whence \n⟨𝑚|{𝜎𝛼,𝑣𝑐}|𝑚⟩ is the 𝑐-direction velocity matrix of the 𝑚-th band electron with the 𝛼-direction \nspin.  \nFrom PT-symmetry, it follows that the electronic bands are spin degenerate , and the group \nvelocity matrix of two spins should be in the same direction , namely , the spin velocity matrix \nsatisfy  {𝜎↑,𝑣𝑐}={𝜎↓,𝑣𝑐}. However, the PT-symmetry enforces  a constraint  on the imaginary part \nof the optical oscillator strength : 𝐼𝑚(𝑣𝑚𝑛,↑𝑎(𝑘)𝑣𝑛𝑚,↑𝑏(𝑘))=−𝐼𝑚(𝑣𝑚𝑛,↓𝑎(𝑘)𝑣𝑛𝑚,↓𝑏(𝑘)). Therefore , \nthe spin-“up” photo conductivity  of Eq. 1  is opposite to the spin-“down” photoconductivity  at the 5 \n same k point . Since  the overall spin photoconductivity is defined as 𝜂𝑠=𝜂(↑)−𝜂(↓), we expect \na net spin contribution  without  charge c ontribution  at each k point . Finally, t he overall spin-current \nis obtained by  an integral over the whole reciprocal space.  As shown in Figure 1(b) , if the energy \ncontour of light-pumped  free carrier s is no t symmetric , which is the typical case in non-\ncentrosymmetric  materials, the integral is nonzero, as is the overall spin current . Intuitively , the \nlight-driven electron transport ations  are helical channel s where the electron spin projection is \nconnected with its transp ort direction, i. e., electrons characterized by spin “up” is traveling in one \ndirection, while electrons characterized by spin “down” is moving in the opposite direction , \ngenerating a pure spin current  shown in Figure 1 ( c).   \nIt is worth noting that ligh t will create electrons and holes simultaneously, and total spin \ncurrent is composited by the hole and electron quasiparticle contributions in Eq. 1 since the \nsummation includes both conduction and valance bands. We plot Figure 1( d) to further illustrate \nthe roles of electrons and holes and how to measure the spin current. Under circularly polarized \nlight, the excited holes and electrons with the same spin direction should travel in the opposite \ndirection because of their opposite transport direction. As sh own in Figure 1 ( d), the “spin-up” \nelectrons and “spin-down” holes are accumulated at the left boundary of the AFM insulator. If \nattached to a ferromagnetic  material, in which the majority is assumed to be spin “up”, the spin \n“down” hole s will annihilate with the minority spin “down” electron s in the ferromag netic material \nwhile the spin -up electrons remain . This ultimately results in a measured spin -polarize d current.  \nWe must  also remark  that there are  mainly  two typical second -order light -matter \ninteractions : the inject -current mechanism and the shift -current mechanism. In addition to the \ninject ion current discussed above, shift current can be excited by the  circularly polarized light \nsimultaneously. However, because the magnitude of shift current is usually two orders smaller 6 \n than that of inject ion current  [24,25,31] , the overall observed photocurrent is dominated by \ninject ion current in this work.  \nLarge  photo -driven spin current in  two-dimensional  materials . \nIn the following, we employ  first-principles simulation s (see supplementary information \nfor simulation details  [32]) [33–35] to quantitatively calculate the spin photocurrent of PT-\nsymmetric AFM insulators based on the aforementioned inject ion-current mechanism. We first \nchoose bilayer AFM CrI 3 as the prototy pe 2D material. The top view of bilayer CrI 3 is illustrated \nby Figure 2(a) . To address the essential hexagonal structurem, we only plot the Cr atoms. The blue \nand red atoms r epresent the spin-up-layer and spin-down -layer Cr atoms, respectively. Our first -\nprinciples calculations indicate that  the ground state of bilayer CrI 3 has an interlayer AFM order , \nin agreement with previous theoretical and experimental results  [17,36,37] . Such a g round -state \nstructure preserves PT-symmetry , so that bilayer CrI 3 has doubl y degenerate bands at any k -point \nin momentum space , as confirmed by  our first-principles density functional theory (DFT) result  in \nFigure 2(b).  \nFigure 2(c) shows the calculated spin photoconductivity according to Eq. 1 . We set the \npolarization of the circularly polarized light to be in the xy-plane and measure the Sz component  of \nspin current . The grey solid line and dashed line show the spin current along both x-direction ( 𝜂𝑥𝑆𝑧) \nand y -direction ( 𝜂𝑦𝑆𝑧), respectively . It is worth mentioning that the quantitative photoconductivity \nneeds an estimated carrier /spin lifetime 𝜏 (Eq. 1). There is no widely  accepted  value of the \ncarrier/spin lifetime of CrI 3. Previous works  [24] adopted a value  of  ~0.4 ps , which is smaller than \nthose of typical transition metal dichalcogenides (TMDs), such as MoS 2 (~ 1ps)  [38]. In this work, \nwe choose a more c onservative value of 0.1 ps.  7 \n This spin  photoconductivity is sig nificant and expected to be dete cted by expe riments in \nnear future. As shown b y the dashed line spectrum in Figure 2(c), the magnitude of spin \nphotoconductivity  (𝜂𝑦𝑆𝑧) can reach up to 40 μA/V2. This is roughly one order larger than the widely \nobserved charge photoconductivit y of BaTi O3 [39] and 2D GeS and its analogs  [40] due to BPVE \nand is the same order as that of mag netic injection charge -current by linearly polarized light in \nbilayer CrI 3 [24]. Moreover, the magnitude of this  intrinsic spin photocurrent is comparable with \nthose of obse rved photocurrents in CdSe and GaAs quantum well structures  [41–43].  \nFinally , we find that the direction of spin photo current can be switched by the Neel vector , \nas shown in Figure 2( d). Although both configurations in Figure 2( d) are AFM, their spin currents \nare opposite to each other  due to opposite Neel vectors . This is be cause the magnetic atomic \nstructure indicated in the blue dashed square  (denoted as the “up” Neel vector ) can be tra nslated \nto that in the red dashed square  by the time -reversal symmetry operator. Accordingly, the  direction \nof the current should be switched by time -reversal symmetry . This  correlation between the Neel \nvector direction  and spin photocurrent might be us eful for detecting the Neel vector of AFM \nmaterials, which has been regarded as a challenge for years  [44].   \nSizeable photo -driven spin current without SOC.  \nSOC has been viewed as the fundamental mechanism for the  spin-current because it can \ngenerate spin -polarized currents from charge currents, char acterized as the spin Hall effect  [45–\n47]. Besides, the SOC lies at the heart of the photo -driven charge current in PT-symmetric AFM  \nmaterials , e.g. bilayer CrI 3 [24] and even -septuple layer MnBi 2Te4 [25], to break the  SU(2) spin-\nrotation symm etry. Therefore, it is necessary to reveal the origin of such an enhanced spin \nphotocurrent in PT-symmetric AFM insulators  and the role of SOC . 8 \n We have calculated the spin photoconductivity 𝜂 of bilayer AFM CrI 3 by artificially  \nreducing the spin -orbital interaction strength ( λso). We take the y-direction spin photo current as \nan example  (see x-direction spin photocurrent  in supplementary information  [32]), and the results \nare presented in Figure 3(a): the solid l ine is the spin photoconductivity with  the intrinsic SOC \nstrength  of CrI 3, and the dashed line is that with the negligible SOC ( λ=0.001λ so). The \ncorresponding DFT -calculated band structures are provided in the supplementary information. \nDespite the differences from the changes of band structures due to SOC, the magnitud es of these \ntwo spectra are similar; the spin photoconductivity of the dashed line can reach 45 μA/V2, which \nis slightly larger than that of spin  photoconductivity with full SOC. This surprisingly indicate s that \nSOC is not necessar ily responsible for  the enhanced spin photo current.  \nTo further understand this result, w e show the spin -texture of the valence band  𝐸=\n−0.17𝑒𝑉 (with zero energy set to be the top of valence bands) with full SOC for  bilayer CrI 3 in \nFigure 3(b). Importantly, t he bands  are not symmetrical in reciprocal space, e.g. the oval-shaped \nenergy contour  of the inner band , a consequence of  broken SU(2) spin -rotation symmetry by the \nSOC. Therefore, the spin photoconductivity at the k point should not cancel with that at th e -k \npoint. We illustrate this by plotting the  distribution of spin photoconductivity in reciprocal space, \nshown in Figure 3(c).  There is a non-odd-parity symmetry of this distribution, which results in a \nsizable non -zero spin current  after integration .  \nOn the other hand, after reducing the SOC ( λ=0.001λ so), we observe a different picture: \nthe band  structure with negligible  SOC is symmetrical in momentum space  because of the \npreserved SU(2) symm etry, as shown in Figure 3(d) . However, the spin  texture  does not exhibit \nany symmetry in reciprocal space . Therefore,  the spin current contributed by the k and -k point in \nthe momentum space still cannot cancel each other , although the bands are symmetric , resulting 9 \n in a non -zero spin current.  Figure 3( e) confirms  this non-odd-parity symmetry of the spin \nphotoconductivity distribution  for photon energy at 1.2 eV .  \n \nLarge photo -driven spin current in 3D materials . \nWe take bulk 𝛼-Fe2O3, i.e. hematite , as a 3D example to demonstrate the predicted spin \nphoto current , in which long -distance spin transport has been reported  [48]. Bulk 𝛼-Fe2O3 is the \nmost stable form of iron oxide s, exhibiting an AFM  order with the space group 𝑅3̅𝑐. The magnetic \nmoments of adjacent Fe3+ layers couple in an antiparallel fashion , forming  an overall AFM \nordering  [49]. Below the Morin te mperature ( 𝑇𝑀≈263 K),  the direction of the magnetic moments \nis parallel to the z-axis [50] as shown in Figure 4(a) . Hematite  below 𝑇𝑀 preserve the PT symmetry , \nwhere the symmetry center is labeled as a dashed circle in Figure 4(a). Therefore, it is an excellent \ncandidate to study  the photo -driven spin current  in weak -relativistic system s at room temperature . \nAlthough  spin photocurrent under linearly polarized light was predicted  in hematite  by considering \nthe shift -current mechanism  [28], a simultaneously excited  and strong  charge current substantially \nreduces the spin polarization. In the following, we will show that circularly polarized light can \nexcite enhanced spin current without charge motion via the inject -current -like mechanism  \npredicted in this work .    \nWe have calculat ed the photoconductivity 𝜂 of the spin current according to Eq. 1 by using \na conservative relaxation time  of τ=0.1 ps. Figure 4 (b) shows the Sz-component spin current \nwhen  the polarization of the circularly polarized light is set in the yz-plane , and Figure 4(c) is that \ngenerated by the  xy-plane -polarized circularly polarized light . Interestingly, t he magnitude of spin \nphotoconductivity  𝜂 is in the same order or even higher than that of the aforemention ed large SOC \nmaterial, e.g. bilayer CrI 3. For example, as  shown by the dot-dashed line spectrum in Figure 4(b), 10 \n the magnitude of photoconductivity  (𝜂𝑦𝑆𝑧) can reach up to 200 μA/V2 by using  τ=0.1 ps in Eq. \n1. Such  sizeable spin photoconductivity is about one order larger than that of the charge \nphoto conductivity  of BaTiO 3 due to BPVE  [39], and the same order as that of charge  \nphoto conductivity  because  of CPGE in polar insulators , e.g. GeSe  [31] and CdSe  [43]. Figure 4(d) \nshows the spin photo conductivity of y-direction current under a xy-plane circularly polarized light  \nthat is distributed in  𝛤𝑇𝑈  plane of the reciprocal space   (see the plane in supplementary \ninformation  [32] ). Similar to Figure 3( e) for AFM CrI 3, we do not observe any symmetry  of the \nspin-conductivity  in the momentum space, leading to such a non-zero spin photo current .  \nSummary and outlook.  \nIn this work, we have demonstrated a spin-CPGE effect for generating enhanced  spin \nphoto current in PT-symmetric AFM insulators. We demonstrate that s uch spin current is protected \nby PT symmetry and is not sensitive to SOC  because of the non-odd parity of spin current in the \nmomentum space . We quantitatively show the generation of nonlinear DC spin current under \ncircularly polarized light using quantum perturbation theory and first -principles calculation s. \nUsing prototypical 2D AFM insulator bilayer , CrI 3, and 3D  weak -relativistic  AFM material  𝛼-\nFe2O3, we predict that circularly polarized light can excite pure spin current with the same order \nof magnitude as the widely observed charge photocurrent in nonmagnetic  materials. Such  \npredicted spin -CPGE is accessible in many traditional and emerging AFM insulators. Among them  \nare NiO [51] and Cr 2O3 [52], which have been used to generate magnon spin current  [5], emerging \n2D AFM insulators, such as the MnPS 3 family materials  [8,16] , and topological axion insulators, \nsuch as the even -layer Mn 2Bi2Te4 family materials  [53,54] . Our prediction will not only be helpful \nto und erstand recent important measurements  of vertical -direction photocurrent  or spin photo -11 \n current  in CrI 3 junction device s [55], but also build a  general framework to search for efficient \nspin-pumping via light -matter interactions  at room temperature . \nACKNOWLEDGMENT  \nR.F. thank Zhiqiang Bao for  fruitful discussions.  R. F.   and  L.Y.  are supported by the Air Force \nOffice of Scientific Research (AFOSR) grant No. FA9550 -20-1-0255 and the National Science \nFoundation (NSF) CAREER grant No.  DMR -1455346.  The computational resources a re provided \nby the Stampede of Teragrid at t he Texas Advanced Computing Center (TACC) through XSEDE.  \n \nEmail:  †ruixiangfei@gmail.com  \n‡lyang@physics.wustl.edu  \n \n \n \n \n \n \n  12 \n Figures  \n \nFigure 1 . Circularly polarized light -driven spin current in an AFM insulator. (a) Doubly \ndegenerate bands are due to PT-symmetry. (b) The transport direction of two opposite directions \nof spins after pumping. The blue and red ovals represent the contour plot of tw o spin components \nof conduction bands, and the black arrows show the overall spin current direction. (c) The \nschematic of circularly polarized light -induced spin current under circularly polarized light. (d) \nThe proposed experimental setup for measuring both electron and hole spin current s. Solid and \nopen circles represent electrons and holes, respectively.  \n \n \n13 \n  \nFigure 2 . Spin CPGE in bilayer  CrI 3. The simplified atomic structure (a) and DFT band structure \n(b) of bilayer AFM  CrI 3. (c) The photo -driven spin current conductivity along the in -plane x and \ny-direction for the spin component of  Sz. (d) The spin current direction (black arrows) for  AFM \nbilayer CrI 3 (side view of Cr atoms) with opposite Neel vector directions.  \n \n \n \n14 \n  \n \nFigure 3 . Spin photocurrent of bilayer CrI 3 without SOC . (a)The y-direction photo -driven spin \ncurrent of S z component with full SOC (grey solid line) and extremely weak SOC (dashed line),  \ni.e. the SOC strength 𝜆=0.001 𝜆𝑠𝑜. The spin texture of valence bands at 𝐸=−0.17𝑒𝑉 in Figure \n2(b) with full SOC (b) and extremely weak SOC (d). The spin photo -conductivity distribution in \nthe momentum space with full SOC (c)  and extremely weak SOC (e) . \n \n \n15 \n     \nFigure 4 . Spin CPGE in  a 3D material. (a) The atomic structure of bulk 𝛼-Fe2O3, the dashed circle \nbeing  the PT-symmetry center. 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Nath b \na Department of Physics, University of Kalyani, Kalya ni 741235, Nadia, W.B., India \nb School of Physics, Indian Institute of Science Educ ation and Research, Thiruvananthapuram 695016, \nKerala, India \nc Department of Physics, National Dong Hwa University , Da Hsueh Road, Shoufeng, Hualien 97401, \nTaiwan  \n \n \nABSTRACT \nWe report the low-temperature magnetothermal proper ties and spin dynamics of a mixed pyrochlore, \nDy 2GaSbO 7 through the coordination of measurements of dc mag netization, ac susceptibility, and heat \ncapacity, and CF computation. In Dy 2GaSbO 7, the spins freeze at temperature Tice  ≈ 3.1 K, corresponding \nto dynamical spin-freezing into a disordered state in compliance with the spin-ice rules, but with \nenhanced zero-point entropy, unlike cannonical dipo lar spin-ice materials, Dy 2Ti 2O7 and Ho 2Ti 2O7, due to \nrandom disorder and varied chemical pressure of B-s ite (GaSb tetrahedra) ions.  \n \n \n \n \n \n \n \n \n \n \n \n 2 \n I.  INTRODUCTION \nGeometrically frustrated magnetic systems, in which  the topology of the spin lattice leads to a frustr ation \nof the spin-spin interactions, have drawn continuin g research interest because of the realization of a  wide \nvariety of novel low-temperature quantum states [1, 2], e.g., spin-ice [3,4], spin-liquid-like, and spi n-glass \nstates, unconventional long-range-ordered and exoti c low-energy excitations [5]. Pyrochlore oxides, \nhaving general formula unit (f.u.) A 2B2O7, where A is the trivalent rare-earth (R 3+ ) ion and B tetravalent \ntransition metal ion, are the robust examples of fr ustrated system. These materials contain an infinit e net-\nwork of corner-sharing tetrahedra where the magneti c R3+  ions (spins) reside at the corners of tetrahedra \nand hence cannot satisfy pair-wise AF interactions simultaneously. The exotic magnetic ground state is , \ntherefore, achieved in pyrochlores by three competi ng interactions: nearest-neighbor (n.n.) AF exchang e \ninteractions, FM long-range dipolar interactions an d directional anisotropies of crystal-field (CF) \ninteraction, along with several other weak perturba tions, e.g., further n.n. exchange interactions, th ermal \nand/or quantum fluctuations (‘order-by-disorder’ me chanism) [6], external pressure [7], random disorde r \nin the form of off-stoichiometry or intersite mixin g [8], which have dramatic effects on the low-\ntemperature properties of  these systems.     \nOf particular interest, spin ices, which are direct ly mapped onto common ‘water ice’, have emerged as \nexemplary of highly frustrated systems in three dim ensions [3,9,10]. In these systems, the spins are \narranged in Pauling’s “two-spins in, two-spins out”  configurations onto the tetrahedron by the effecti ve \nferromagnetic interactions in the presence of stron g CF induced Ising-type anisotropy. As a result, th e \nmagnetic systems are highly degenerate in the groun d state in macroscopic scale and enter into a froze n \ndisordered with a finite residual (‘zero-point’) en tropy S0 = 1/2 R ln (3/2) = 1.68 J K −1mol −1 below spin-\nfreezing temperature. Consequently, the entropy rec overed at high temperature should fall short of tha t \nexpected for a two-level pseudo-spin-1/2 system [ S = kB ln(2 N) = R ln 2], resulting in a total entropy SP = \n0.71 Rln 2 = 4.1 J K −1mol −1 [1,10,11]. Dy 2Ti 2O7 [11], Dy 2Sn 2O7 [1,12] and also Ho 2Ti 2O7 [13,14] are \ncategorized as ‘dipolar spin-ices’ (DSI), in which FM n.n. dipolar interactions dominate over the weak  AF \nexchange interactions [10,15] among the magnetic mo ments, oriented along the local <111> Ising \ndirections of tetrahedral units by the trigonal CF.  However, recent heat capacity measurement on \nDy 2Ti 2O7  [16] have revealed the amount of residual entropy is significantly reduced from the Pauling \nentropy in Dy 2Ti 2O7 over much longer time scale, signaling disappearan ce of the spin-ice state and a first-\norder transition to the long-range ordered phase ou t of the spin-ice manifold, caused by the perturbat ions, \nviz., long-range nature of dipolar interactions bey ond nearest neighbors and spin-phonon coupling due to \nlattice distortions [17]  etc. Dy 2Ge 2O7 [18-20]  is arguably another DSI which lies more close to AF -DSI 3 \n phase boundary [20]. On the other side, Dy 2Zr 2O7 crystallizes into disordered fluorite structure an d \nexhibits spin-liquid-like characteristics with comp lete disappearance of zero-point entropy [21]. \nThe A 2B2O7 pyrochlore family can be extended to include the mi xed pyrochlores, A 2B′3+ B′′5+ O7, in \nwhich B′and B′′ ions are distributed randomly at six-coordinated B  sites [22], provided the ionic radius \nratio ( rA/rB) [23] lies within the range of 1.46 −1.8 for the structurally ordered pyrochlores [22]. A \nrenewed interest in the mixed pyrochlores emerged f rom the report of spin-ice-like state observed in \nDy 2ScNbO 7 [12] with reduced zero-point entropy value than th e canonical DSI, Dy 2Ti 2O7. The cation \ndisorder arising due to the mixed occupancy at B-si te in Dy 2NbScO 7 could break the strict ‘spin-ice’ for \nthe arrangements of the Dy 3+  spins leading to a partial release of the zero-poi nt entropy [12]. On the other \nhand, in Dy 2FeSbO 7, spin-ice like configurations of Dy 3+  spins has been observed, excluding the effect of \nFe 3+  moments, which order ferromagnetically, on the nei ghboring Dy 4 tetrahedra [24]. Similarly in \nanother B-site diluted compound, Dy 2Ti 2−yFe yO7, the spin dynamics are suppressed due to the Dy-Fe  \ninteractions and altered CF interactions [25]. The underlying principle in the backdrop of spin-ice ph ysics \nis that the variations of lattice constant, ( rA/rB) ratio, n.n. spin-spin (A-A) distance, etc. impose d by the \nchemical substitution of the B-site ions over any p yrochlore spin-ice series induce change in chemical  \npressure on the rare-earth network [20], which in e ffect leads to the variation of the relative streng th and \nmagnitude of the competing interactions over the sp in-ice series. Another intriguing observation in th e \ncontext of spin-ice dynamics is the emergence of ma gnetic (anti-) monopole-like excitations in, e.g., \nDy 2Ti 2O7 [2, 26-30], and Dy 2Ge 2O7 [19], generated by the flipping of one spin out of  “two-in/two-out” \nspin-ice configurations.  \nGiven this background, the search for new spin-ices  is, therefore, an active field of condensed-matter \nphysics research and has drawn the attention of the  broader science community. In this work, we have \ninvestigated the low-temperature properties and spi n dynamics of a mixed pyrochlore, Dy 2GaSbO 7 (DGS) \nthrough the coordination of measurements of dc magn etization, ac susceptibility, and heat capacity, an d \nCF computation, and found that DGS is a new spin-ic e compound, but with increased zero-point entropy, \nunlike pure DSIs. \n \nII. EXPERIMENTAL \n   The polycrystalline Dy 2GaSbO 7 (DGS) sample were prepared by conventional solid-s tate reaction \nmethod using high-purity powders of Dy 2O3, Ga 2O3 and Sb 2O5. All starting materials were preheated to \n200°C for 6 hours to remove any moisture. The stoic hiometric amounts of all precursor powders were \nthen ground thoroughly, pelletized, and subsequentl y calcined for a total ∼44 hours in a high-temperature \nmuffle furnace in presence of air at 1320°C, with i ntermittent grinding and pelletizing. To identify t he 4 \n crystalline phase of the as-prepared powder samples , the X-ray powder diffraction (XRD) pattern was \nrecorded (Fig. 1) at room temperature by scanning w ith a speed of 3° /min. over a range of 2 θ = 10° − 90° \nat 0.02° step-intervals using a Rigaku Miniflex 600  bench-top powder diffractometer which operates at 40 \nkV and 15 mA. The positions of the first peak (111)  and the most intense peak (222) at (15.15 °, 30.42 °), \nrespectively, are consistent with the pyrochlore st ructure and no impurity peaks were found. The struc ture \nanalysis has been done by the Rietveld refinement m ethod using Fullprof program on the XRD data. The \nrefinement results indicated a good agreement betwe en the observed and calculated Bragg's intensities of \nDGS (Fig. 1) for the pyrochlore-type structure (spa ce group Fd 3m). The occupancy ratio of Dy, Ga, Sb \nand O atoms are changed very nominally from their i deal values and found to be 0.99:0.48:0.52:1 in \nrefinement. The lattice parameter a0 and 48 f oxygen positional x-parameter were found to be 10.207(1) Å \nand 0.3275(2), respectively, in agreement with the literature [31]. The detailed structure analysis an d \ncharacterization of DGS sample by optical spectrosc opy were presented in our previous paper, Ref. [32] . \n \n \nFIG. 1. XRD profiles of Dy 2GaSbO 7: observed (red points), calculated (solid lines), and the difference (green lines). \nVertical marks (in blue) correspond to the allowed Bragg reflections for cubic space group Oh7 − Fd 3m. Most intense \nand significant ( hkl ) reflections are also denoted. \n \nThe magnetization M(T,H) of the sample in the shape of a pellet (mass ~ 28 .8 mg) was measured \nusing a vibrating sample magnetometer attachment to  physical property measurement system (PPMS: \nQuantum Design). Zero-field-cooled (ZFC) warming an d field-cooled (FC) warming modes of M(T) \nmeasurements were carried out between T = 2–300 K with applied magnetic field of µ0H = 0.1 T and 1 T. \nIsothermal magnetization M(H) data at different temperatures, T = 2, 5, 10, 20, 50 and 100 K,  were also \n5 \n recorded with varied fields up to µ0H = ±9 T. The AC magnetic susceptibility χac (f,T ) was measured as a \nfunction of temperature within the range of 2 K − 40 K in the excitation frequency range f = 110 Hz − 10 \nkHz using the ACMS option of the PPMS at ac driving  field,  Hac  = 8 Oe. \nHeat capacity CP of the pellet-shaped sample (mass ~ 8.2 mg) at cons tant pressure p ≈ 1.2 mPa was \nmeasured at different fields, H = 0-6T down to 0 .39 K, employing a standard thermal-relaxation \ntechnique using the heat capacity attachment to the  PPMS above 2 K and a 3He option for measurements \nbelow 2 K. The accuracies of heat capacities are wi thin ± 2% over the range from 10 K to 270 K, ± 3% \nover the temperature range, T = 2 −10 K, and ± 2% below 2 K. \n \nIII. RESULTS AND ANALYSIS   \nDC magnetization :  \nThe temperature-dependent dc magnetic susceptibilit y, χ(T), of polycrystalline DGS for applied \nmagnetic field of µ0H = 0.1 T and 1 T are plotted in Fig. 2 after demagne tization correction to the applied \nfield using the expression, µ0Heff  = H − 4πNdM. The demagnetization factor, Nd, is taken to be 1/3 \nassuming spherical shape of the pellet sample [33].  The χ(T) data increase smoothly and do not show any \ndistinct anomaly down to T = 2 K, which is evidence of paramagnetism of Dy 3+  magnetic ions in DGS. \nThe values of χ(T) for H = 0.1 T almost coincide with the data for the fiel d of 1 T in the temperature \nrange of 300 −20 K, implying that the temperature-dependent magne tization M(T) increases linearly by ten \ntimes as the external field increases from 0.1 T to  1 T.  Below T = 20(±4) K, χ|1T  becomes lower in \nmagnitude than χ|0.1T (e.g., χ|0.1T /χ|1T  ~ 4.7 at T = 2 K), implying the occurrence of a spin-spin corr elation \nin the Dy tetrahedral network below  T § 20 K at high field in competition with Zeeman inte raction. Such \nmagnetic feature was also observed in Dy 2Ge 2O7 [18], and Dy 2Zr 2O7 [34]. Thermal hysteresis is not \nobserved in the ZFC and FC modes of susceptibilitie s in the range T = 2–300 K, which rules out the \nformation of any magnetic cluster or spin-glass-lik e freezing in DGS.  \nThe χ(T) data were fitted to a Curie-Weiss (CW) law, χ(T) = C/( T−θCW ) + χ0, where C = Beff kN3 /2µ  \nis the Curie constant, and χ0 (= χV +χdia ) measures the contributions from temperature-indep endent Van \nVleck paramagnetic ( χV) and inner-shell diamagnetic susceptibility ( χdia ) to the measured dc \nsusceptibility. The value of χdia  of DGS is estimated to be −0.71 ×10 −4 emu/mol-Dy by adding the \ncontributions from Dy 3+ , Ga 3+ , Sb 5+  and O 2− ions [35]. A slight downward curvature in the χ−1(T) data is \nobserved around 50 K, and therefore, shown in inset  to Fig.2 , the CW fit was performed at two differe nt \ntemperature regimes [36]: (i) the fit of χ−1(T) data for H = 0.1 T within the high-temperature  range of 6 \n 300 −100 K yields an effective magnetic moment µeff  = 10.52(2) µB/Dy and θCW  = − 4.93(1) K. The \nnegative θCW  is indicative of net AF interactions among Dy spin s at higher temperatures. (ii) on the other \nhand, the χ−1(T) data when fitted over 2 K § T § 50 K yields µeff  = 10.0(1) µB/Dy and θCW  = +1.62(1) K. \nThe positive value of θCW  is a ‘robust measure’ [9] of effective FM interact ions among the Dy 3+ spins \ndeveloped at low- T(K). The value of µeff /Dy 3+  ion extracted from the high- T(K) fit is close to the free-ion \nvalue of Dy \nfi µ= 10.646 µB/Dy expected for the 6H15/2  ground multiplet of Dy 3+ -ion, and the low \ntemperature value is equal to the value of µ = 10.0 µB/Dy for a ground state with Jz = ±15/2 double. Thus \nthe ground doublet state plays a major role in tuni ng the magnetic properties at low temperature. The \nvariation of µeff  with temperature is attributed to the CF mixing up  of the J = 15/2 ground states with \ncontributions from higher states within the J manifold by intermediate-coupling mechanism. The v alue of \nχ0 ≈ χV was found to be 0.002(1) emu/mol-Dy from the CW fi t and matches with the value obtained for \nDy 2Ti 2O7 [33]. \n \nFIG.2. DC magnetic susceptibility, χ(T), of Dy 2GaSbO 7 measured in the magnetic field of H = 0.1 T and 1 T with \nZFC and FC mode. Straight lines are fit to χ(T) using CF theory (blue) and within the frame work of mean-field \napproximation (CF+MF) (black).  Inset: CW fit to th e inverse susceptibility. \nTo make a closer inspection on the variation of sus ceptibility data with temperature, we have \nexamined the experimental χT of DGS against 1/ T. The χT values for both fields coincide at high \ntemperatures and exhibit a trough at 1/ T ≈ 0.02 K −1 or T = 50 K, above which the CF effect dominates and \nIsing approximation loses its validity thereafter [ 37]. The χT data show a distinctly linear behavior for \nµ0H = 0.1 T and 1 T within lower temperature region of 20 −2 K; for the lower field, χT data increase \n7 \n linearly above  1/ T ≈ 0.05 K −1, while for the higher field, χT decreases smoothly. Similar signatures are \nalso obtained for Ho 2Ti 2O7 and Yb 2Ti 2O7, and Siddharthan et al  [37] attributed such behavior to the \ncompeting effect and the relative strength of the F M long-ranged dipolar interactions and AF n.n. \nsuperexchange interactions among the magnetic Ising  spins within tetrahedral network. \nThe isothermal magnetization M(H) of Dy 2GaSbO 7 as a function of H at some selected temperatures \nare drawn in Fig. 3. At T =2 K, the M(H) isotherm exhibits a steep increase below 1 T [31]  and tends to \nattain thereafter a saturation value of Msat  = 5.06 µB/Dy above H ≈ 8 T. This value is nearly 48% of the \ntheoretical µeff  = 10.646 µB/Dy value or half of the Brillouin value ( gJ Jz = 10 µB/Dy) of free Dy 3+  ion and \nis comparable to the Msat  value obtained for other Dy-based pyrochlores Dy 2Ti 2O7 [33], Dy 2−xCa xTi 2O7 \n[38], Dy 2Zr 2O7 [21] and Dy 2Ge 2O7 [18]. The magnetization isotherms can be explained  with powder \naveraging the polycrystalline data by considering t he effective Ising-type spin-1/2 ground state with \ntransverse and longitudinal g-factors (g ⊥ = 0 and g|| = gz), where Z-axis is along local <111> axes of the \nelementary tetrahedron. The magnetic moment per ion  for this model at any temperature T is given by \n[33], \nM(H) =   ∫+ ⋅\n\n\nx\nV\nBB\nzHdx xxHTk\ng02\n)tanh( 2χµ           (1) \nwhere,  x = TkHg BBz2 /µ . Eq.(1) describes the measured data very well (sol id curves in Fig.3) with gz \n= 19.3 ±0.2, which is close to the gz = 2 gJJz = 20, as expected for the pure Kramers doublet Jz = ±15/2 \nstates of Dy 3+ . \n \nFIG. 3. Field dependencies of isothermal magnetizat ion of Dy 2GaSbO 7 at several temperatures. Staright lines are \nfitting within Ising spin ½ approximation. \n8 \n  \n   The M2 vs  Heff  /M isotherms, namely Arrott plot, for DGS are analyzed  using Arrott equation of state, \nMHeff /0µ = a + bM 2 ,  where a and b are temperature-dependent coefficients. The value of b which \nmeasures the slope of the curves is found to be pos itive (b = +0.004 T(ion/ µB)3 at 2 K increasing to +0.06 \nT(ion/ µB)3 at 100 K), in the overall magnetic measurements, i mplying that a second-order transition \noccurs at CW temperature, T ∼ O(θCW ), in Dy 2GaSbO 7 in accordance with the Banerjee criterion [39].  \n \nCrystal-field computation \nThe dc magnetic susceptibility was calculated using  a Van Vleck equation within the framework of \nsingle-ion crystal-field (CF) theory in combination  with the molecular field (MF) [24,40]. In this mod el, \nthe CF Hamiltonian ( HCF ) is expressed by six CF interaction parameters (CF Ps) in trigonal symmetry and \nthe Zeeman interaction Hamiltonian contains spin-sp in exchange tensors, e.g., λ|| and ( λ⊥,x and λ⊥,y), \nwhich are longitudinal and transverse to the R-R bo nd direction. Calculated susceptibility with CF+MF \nmodel was plotted in Fig.2. Detailed description of  the computation was given in Appendix A. We found \nthat the ground CF state, | Ψg〉, is a well-isolated by an energy separation, ΔCF  ≈ 192 K, from the first \nexcited doublet and has major contribution from the  JZ = ±15/2 state with g|| = 19.1(1) and g⊥ = 0. The \nlow-temperature magnetic properties of the material  can, therefore, be modeled by an effective pseudo-\nspin S~= 1/2 system [1,3,10], oriented along local <111> a xes. \nThe interaction Hamiltonian between the n.n. Ising- spins (σi = ±1 at site i) can, therefore, be expressed \nby the following form, Heff  = Jeff (1) σ1σ2 [9,10], where Jeff (1)  = Jnn  + Dnn  is defined as the effective energy \nscale of n.n. interactions [9,10]; Jnn  = 3 / ) 2 / )( 2 (2||,|| Bxgµ λλ⊥−  [40] and Dnn  = 32|| / ) 2 / )( 3 / 5 ( nn Br gµ  \nare, respectively, the effective n.n. exchange and dipolar coupling [10, 40]. Using the values of the \nparameters derived above, we obtained Jnn  = −0.752(1) K and Dnn  = +2.01 K at low temperatures, and \nhence Jeff (1)  = +1.26(1) K. Thus though the n.n. exchange intera ctions between Dy 3+-spins is AF, due to \ndominance of the FM dipolar interactions in the n.n . scale, Jeff (1)  >0 signifying effective FM interactions \nfor n.n. spin-ice criterion [9,10] and giving rise to a positive θCW  in DGS. Typically the frustrated \npyrochlores in which n.n. dipolar interactions are the leading energy interactions (due to large µeff ) with \nn.n. FM coupling Jeff (1)  and a critical range of the ratio Jnn /Dnn  > −0.91 and Jeff (1) /Dnn  > 0.095 [9, 15] are \nreferred to as ‘standard dipolar spin-ice’ (s-DSI) [10,15,41]. In the present case, Jnn /Dnn  = −0.374 in \ncomparison to the values, −0.46, −0.5, −0.56, respectively, for Dy 2Sn 2O7, Dy 2Ti 2O7 [20] and Dy 2Pt 2O7, \n[42] and −0.27, −0.35 for Ho 2Ti 2O7 and Ho 2Ge 2O7 [20], which reveals that DGS is more inside the DS I 9 \n region of the magnetic phase boundary than its Dy c ounterparts [20]. Table I enlists some selected \nstructural and magnetic parameters of these DSIs. T he variation in the relative strength of magnetic \ninteractions through any set of pyrochlore DSI seri es is attributed to ‘chemical pressure effect’ due to B-\nion substitution [20]. Using the above values of Jnn  and Dnn , one could obtain a rough estimate of the \nsecond and third n.n. interactions, in terms of Jeff (2)  = )315 /( nn nn J D , and Jeff (3)  = )40 /( nn nn JD\n [17] to \nconsider the long-range nature of exchange interact ions (g-DSI model) [41], taking account of the effe ct \nof distortions in the lattice sites which causes th e local changes in magnetic correlations and are re ferred \nto as d-DSI model [17]. The values are Jeff (2)  = −0.103 K and Jeff (3)  = −0.067 K, which are anti-\nferromagnetic and much weaker than Jeff (1) . An estimate of the dipolar and exchange contribut ions to the \nCW temperature can also be obtained from the relati ons θdip  = 16 π3\n02|| 9 / ) 7607 . 3 ( ) ( aN g d B − µ  and θex  \n= 6 / ) )( 2 (2||,|| Bxgµ λλ⊥−  [40] and found to be, respectively, +3.29 K and −1.51 K. Hence in the low-\ntemperature zone (where θCF  ≈ 0 [24]), the value of θCW  = θdip  + θex  = +1.78 K which agrees with the \nvalue ( θCW  ≈ +1.62 K) obtained from the CW analysis of the low- temperature susceptibility.  \n \nTABLE I. Lattice parameters and selected magnetic p arameters for Dy-based pyrochlore spin-ices. \n \n a0 (Å) rA/rB x θCW (K)  Jnn  (K) Dnn  (K) Jeff (1)  (K)  Jnn /Dnn  Jeff (2)  (K) Jeff (3) (K) Tsi  (K) c \nDy 2Sn 2O7 \n[20]   10.4 1.488 0.337 +1.7 −1.0 2.15 +1.16 −0.46   5.2 [12] \nDy 2Pt 2O7 \n[43]  10.19 1.643  +0.77 −1.28 2.29 1.01 −0.56    \nDy 2Ti 2O7 \n[20]  10.124 1.697 0.323 1.1 −1.15 2.35 +1.20 −0.49 −0.14 a 0.025 a 4.5 [61] \nDy 2Ge 2O7 \n[20] 9.93 1.938  0 −1.83 2.5 +0.67 −0.73    \nDy 2FeSbO 7 \n[24]  10.261 1.65 0.33 +2.42 −0.313  1.93 +1.62 −0.16 −0.237 −0.154 12.1 \nDy 2GaSbO 7 10.207 1.684 0.327 +1.62 −0.752  +2.01 +1.26 −0.374 \n−0.383 b −0.103 −0.067 9.57 \na Taken from [41]. \nbThe value of Jnn /Dnn , obtained from the variation of  Tpeak , at which heat capacity peak appears, against  \nJnn /Dnn   [15] for DSI model, is found −0.383 at Tpeak / Dnn  ≈ 0.676 for DGS and hence  Jnn  = −0.770 K. \ncTsi  is the temperature at which integrated magnetic en tropy reaches to spin-ice value of 4.1 J K −1(mol-\nDy) −1 when the field of H = 1T is applied.  Tsi  ≈ 7.5 K for Dy 2NbScO 7 [12]. \n 10 \n     \nFIG.4. Real χ′ (a) and imaginary χ″ (b) parts of the ac susceptibility χac (T) vs temperature for Dy 2GaSbO 7 sample \nspanning the frequency range f = 111−10000 Hz. Inset of (b) shows spin-freezing a nomaly at Tice  ∼ 3.13 K at f = 111 \nHz. \nAC magnetic susceptibility : \n   To investigate the spin-ice dynamics, the temperatu re-dependent ac susceptibility χac (T) data of DGS \nwere plotted in Fig. 4 at different frequencies bet ween 111 Hz – 10 kHz in the temperature range 2 K t o \n40 K. The real part ( χ′) of χac (T) increases as temperature decreases and does not s how any anomaly \nwithin our measurement frequency window (Fig. 4a) d own to 2 K, similar to χdc. The χ′ is independent of \nfrequency down to T ∼ 8 K, below which frequency dependence is clearly d eveloped for higher \nfrequencies and χ′(T) gradually reduces. Unlike paramagnetic-like behav ior of χ′(T), frequency \ndependency becomes more pronounced in the imaginary  part χ″(T) of χac (T) which increases differently \nwith lowering temperature as frequency increases. A  very small spin-freezing anomaly develops at lower  \ntemperature, (e.g. Tice  ∼ 3.13 K at f = 111 Hz), shown in inset to Fig.4(b), which shift s towards higher \ntemperatures (e.g., at 3.32 K for 500 Hz, at 3.43 K  for 1 kHz) with increasing frequency and may \ndisappear at higher frequency, f ≥ 2.5 kHz. In contrast to DGS, it may be recalled th at in canonical Dy-\nbased DSIs, e.g., Dy 2Ti 2O7 (DT) [43-45]  and Dy 2Sn 2O7 (DS) [12, 46], two dynamic spin freezings are \nobserved in χac (T) curves at high temperature (at ∼16 K for DT, ∼25 K for DS), followed by a lower- T \npeak (at ∼2-4K for DT and DS). The high- T peak corresponds to occurrence of spin-spin correl ations in \nshort-range scale [45] or single spin freezing [47] , while low- T freezing is associated with the formation \nof spin-ice state [48,49]. The frequency dependence  of the spin-freezing temperature, Tice , of DGS is \nroughly fitted to an Arrhenius formula, f = f0 exp( −Ea/Tice ), where f0 is characteristic frequency, and Ea is \nactivation energy for spin relaxation [12,43]. The value of Ea is found to be 126±4 K which is lower than \n11 \n the obtained values for Dy 2Ge 2O7 (162 K) [18], Dy 2FeSbO 7 (∼165 K) [24], Dy 2Ti 2O7 (∼200 K) [43], \nDy 2Sn 2O7 (≈ 220 K) [12], and Dy 2Pt 2O7 (∼267 K) [42]. In contrast, much lower value of Ea was reported \nfor a mixed pyrochlore compound, e.g., 42 K for Dy 2Sn 1.75 Sb 0.25 O7.125  [12]. The spin relaxation \nmechanism and the energy barrier Ea are associated with the crystal-field perturbation s [43], and \ntherefore, cation disorder from the random occupanc y at B site elements may cause the lower value of Ea \nin the mixed pyrochlores [12]. \n \n \nFIG.5. Heat capacity Cp versus T(K) for Dy 2GaSbO 7 measured at H = 0 T (symbol). Solid lines represents the \ncalculated values considering the contributions fro m Debye, Einstein and CF (Schottky) heat capacities  (see \nAppendix B). Inset shows the lattice heat capacity using Debye T3 formula (line) below 20K. \nHeat capacity : \nThe total heat capacity Cp(T) of DGS   has been displayed in Fig. 5 for H = 0 T between the \ntemperature range of 0.35 K and 270 K. The field-de pendence of Cp(T) are also plotted in Fig.6 between \n0.35 K and 30 K for the external magnetic field of H = 0, 1, 2, 3, 6 T. The zero-field heat capacity sh ows \na rounded peak at Tpeak ≈ 1.36(2) K, which is of the order of Jeff (1) , as were reported for other Dy-based \ndipolar spin-ice pyrochlores, e.g., at 1.2 K for Dy 2Sn 2O7 [20], at 1.13 K for Dy 2Pt 2O7 [42], at 1.25 K for \nDy 2Ti 2O7 [11,20], at ∼1 K for Dy 2Sn 2−xSb xO7+x/2 [12]. Thus one may notice that the peak position Tpeak  \nshifts to higher temperature as rA/rB ratio increases for the Dy series (Table I), thoug h the dependence is \nnot linear. The peak amplitude, Cpeak  ≈ 2.1 J K −1(mol-Dy) −1, is considerably lower than the mean-field \nprediction of 12.5 J K −1(mol-Dy) −1 expected for a long-range second-order magnetic tr ansition for the \neffective spin S~=1/2 system [50].  The broad maximum and the absence of any sharp feat ure in Cp(T) is a \n12 \n strong indication that the system does not develop any long-range magnetic order via a thermodynamic \nphase transition [9]. The Tpeak shifts to the higher temperatures and broadens as applied magnetic field \nincreases, and finally smears out at higher field. In addition, shown in inset to Fig.5, a kink at T ∼3.25(2) \nK is also observed which can be identified with the  spin-freezing Tice  observed in the ac susceptibility \nχ″(T) and vanishes when field is applied.  \nWe extract the magnetic specific heat, Cmag , by subtracting the ( Clat  + CSch ) from the measured Cp(T) \ndata, and Cmag /T vs  T of DGS are drawn in inset to inset (a) to Fig.6. T he estimation of lattice ( Clat ) and \nSchottky ( CSch ) components of heat capacity is illustrated in App endix B. In addition to two anomalies at \n1.32(2) K and 3.28(3) K, the Cmag /T at zero-field exhibits an upturn at 0.95(1)K, foll owed by a round-\nshaped very low peak at 0.69(1) K, which disappears  at H =1 T. Interestingly, similar upturn was also \nreported for Dy 2Ti 2O7 at T ≤ 0.6 K [16,17], though the peak height is much smal ler for DGS. This peak is \nnot associated with nuclear hyperfine heat capacity , which peaks at 20 mK and extends up to T ∼0.4 K, as \nfound for Dy 2Ti 2O7 [51,52]. The random disorder, either in the form o f low level of stuffing of rare-earth \nions (A site) onto the B-sites [53-55], or oxygen v acancies [56], disorder in the B site ions in mixed  \npyrochlores [12], may cause the small distortions i n the lattice connectivity between neighbors [17]. Such \nrandom disorder may, by lowering the local symmetry  of the crystal-field, destroy the local Ising natur e of \nthe moments and induce quantum fluctuations within t he Dy-tetrahedra, and hence is the most likely \nsource of the upturn [52].  \n \nFIG.6. Heat capacity Cp against T(K) for Dy 2GaSbO 7 at different magnetic fields (symbol). Inset: (a) the Cmag /T at H \n= 0,1 T; (b) integrated values of magnetic entropy versus temperature in  H = 0, 1, 2 T. \n \n13 \n The Cmag /T curve shifts to the higher temperature as field in creases, indicating that the area under the \ncurve (i.e., entropy) increases with the field. The  magnetic entropy change DSmag  for H = 0, 1, and 2 T are \ncalculated (inset b to Fig.6) by integrating Cmag /T within the temperature range, T ≤20 K, ΔSmag  =\n0( ) /T\nmag \nTC dT T∫, by crudely extrapolating Cmag /T to its zero value. We found that the integrated en tropy \nincreases with increasing temperature up to 3.25 K (∼ Tice ) and finally saturates at T ≥ 10 K to the value of \nDSsat  ≈ 2.82(1) J K −1(mol-Dy) −1. The difference between the ΔSsat  and the expected value of S = R ln2 = \n5.76 J K −1 mol −1 corresponding to the full 2 N states available to N Ising S~= ½ spins system in the high-\ntemperature regime ( T>> θCW ) [9] is the zero-point entropy  (S0), and this difference constitutes the \nprimary thermodynamic evidence for the existence of  a spin-ice state. Hence the calculated zero-point \nentropy of DGS for H = 0 T is 2.94 J K −1(mol-Dy) −1, which is much larger than the Pauling spin-ice va lue \nof S0 = 1.68 J K −1 mol −1, indicates additional degeneracy in DGS, which may  be induced by disorder, \ncompared to the macroscopic degeneracy ( Ω0 = 2 N (6/16) N /2 ) in s-DSI system. We hypothesize that out of \npossible 24 spin configurations on a tetrahedron, if four conf igurations remains thermally inactive or \nenergetically constrained due to disorder, and six obeys the conventional ‘two-in/two-out’ spin-ice ru le \nper tetrahedron, then following Pauling-like argume nt [9,47], we could obtain S0 = kBln[2 N (6/12) N /2] ≈ \n2.88 J K −1 mol −1, which agrees with the estimated zero-point entrop y for DGS. Application of a magnetic \nfield increases the Zeeman energy of the spins, ther eby effectively lowering the energy barriers for sp in \norientation among the different manifolds of the gr ound state spin configuration and lifts the degener acy \nof the ground state [11], which restores long-range  order to gain some entropy [53]. We note that the spin-\nice entropy ( ∼4.1 J K −1mol-Dy −1) is recovered at the temperature Tsi  = 9.57(1) K on applying a field of 1T, \nwith partial release ( ∼ 41%) of residual entropy, and attains the saturati on value of 4.50(1) J K −1(mol-\nDy) −1 and 5.0 J K −1(mol-Dy) −1 at 20 K for H = 1T and 2T, respectively (Fig.7). That is, the sa turation \nentropy is 22% and 13% smaller than the value of Rln2 at H = 1 T and 2T, respectively, in contrary to the \npure pyrochlore DSIs, e.g., Dy 2Ti 2O7, Dy 2Sn 2O7 and Ho 2Ti 2O7, for which the saturation entropy at H = 1T \nis 15% less than Rln2 [11,12,14,57]. In other words, high magnetic fi eld is required for DGS to make it \nentropic equivalent to the pure DSIs. Further, in T able I, we note that the values of Tsi  for the mixed \npyrochlores, Dy 2NbScO 7, Dy 2FeSbO 7, and Dy 2GaSbO 7, are greater than for, e.g., Dy 2Ti 2O7 and \nDy 2Sn 2O7, which reveals that the thermal barrier or constra int for spin reorientation is stronger in mixed \npyrochlores than pure DSIs. \n \nIV.   DISCUSSION AND CONCLUSION 14 \n The dc and ac magnetic susceptibility, and heat cap acity with and without magnetic field have been \ninvestigated on a B-site mixed pyrochlore Dy 2GaSbO 7 to explore its spin dynamics. The salient \nobservations may be summarized below: \n(i) Dy magnetic moments in DGS behave as an ensembl e of Ising S~= ½ spins in the ground state. The \nspins are oriented along the local <111> axes of ea ch elementary tetrahedron due to strong single-ion \ncrystal-field anisotropy in coordination with ferro magnetic dipolar and anti-ferromagnetic exchange \ninteractions. The dipolar interactions are strong e nough to counterbalance an AF exchange for DGS. \n(ii) The spins freeze at temperature Tice  ≈ 3.1 K corresponding to dynamical spin-freezing int o a \ndisordered state in compliance with the spin-ice ru les, like cannonical dipolar spin-ices. However unl ike \nDSIs, no single-spin freezing is observed for DGS. The energy barrier, Ea ≈ 126 K, (which determines the \ntunneling barrier of the spin states through quantu m and/or thermal processes [45,47]), associated wit h the \nspin relaxation mechanism, is much less than the en ergy separation ( ΔCF  ≈192 K) of the first excited CF \nlevel from the ground state doublet. That is, the s pin freezing transition in DGS is mainly controlled  by \nthe quantum mechanical tunneling [58] constrained b y the inherent cation disorder at B sites. \n(iii) The spin ensemble has a finite zero-point ent ropy amounting to 2.94 J K −1(mol-Dy) −1, which is much \nlarger than the Pauling residual entropy for common  ‘water ice’ or cannonical DSI pyrochlores. It is \nrelevant to mention that the zero-point entropy dep ends nonmonotonically on dilution of Dy-spins at A-\nsite by nonmagnetic ions and approaches to the valu e of Rln2 for free spins in the limit of high dilution \n[45,47], as well as on stuffing of A-site ions onto  the B-sites. Further, the thermal constraint for s pin \nreorientation is much stronger for the mixed pyroch lores than the pure pyrochlore DSIs. Since the B ′ and \nB′′ ions have different ionic radius in mixed pyrochlo res, due to random occupancy of B-site ions, the \nequality of n.n. distance, A-A = B-B, is lost in th e mixed pyrochlores (similarly for the stuffed DSI \nmaterials [17]) which induces the local changes in the magnetic interactions beyond nearest-neighbors as \nwell as causes variation of chemical pressure on sp in system.  We argue that these effects have signif icant \nimpact on the low-temperature magnetic and heat cap acity properties of the present system. \n(iv) The lattice constant, magnetic moment, and Jeff , which define the monopole contact distance and \nelementary monopole charge [19], of DGS are nearly identical for Dy 2Ti 2O7, and Ho 2Ti 2O7. Therefore, \nDGS is proposed as another ideal system to study mo nopole-antimonopole-pair excitations through the \nstudy of broadband spectroscopic measurements of di electric function and field-dependent thermal \nconductivity. In this context, it would be very int eresting to elucidate how these excitations are aff ected \nby the large thermal constraint and disorder in DGS . 15 \n To conclude, Dy 2GaSbO 7 is a spin-ice compound with complex dipolar and ex change interactions and \nhaving enhanced zero-point entropy, in variance wit h pure DSIs, due to random disorder and chemical \npressure of B-site ions.  \n \nACKNOWLEDGEMENTS \nThis work was supported by SERB funded project (fil e no. CRG/2018/000171) and partially by UGC-\nDAE-CSR (Indore) project (No. CSR-IC-249/2017-18/13 30). The authors also acknowledge the facilities \navailable through DST-PURSE & DST-FIST program of D ept. of Physics, University of Kalyani. \nAuthors are also thankful to UGC-DAE-CSR, Kolkata c enter, for magnetic measurements. \n \nAppendix A \nCombined crystal-field calculation: \nSingle-ion crystal-field (CF) theory in coordinatio n to the molecular field (MF) [40] has proved to be  \nan efficient treatment to describe the temperature dependency of magnetization data and the nature of \nground state of rare-earth ions depending on its lo cal symmetry in the crystal. The CF at the R3+ -ion sites \nin the pyrochlores has trigonal point symmetry m3 (D3d ), with the local 3 axes parallel to the <111>-\ntype directions of the crystal lattice. The effecti ve single-ion Hamiltonian has the form [40]: \nint ( ) ( ) ( ) ext \nt FI CF Z Z H i H i H i H H = + + + .                                                                 (A1) \nHFI  is free-ion (FI) Hamiltonian that operates in the 520 |SLJM J〉 intermediate-coupled states of the 4 f 9 \nelectronic configuration of Dy 3+ -ions [24].   \nThe CF Hamiltonian ( HCF ) in the local system of coordinates with the Z-axis // [111] trigonal \nsymmetry axis at the Dy 3+ site is expressed as,  \n  HCF  = B20 U20  + B40 U40  + B60 U60  + B43 (U43  − U4 −3) + B63 (U63  −U6−3) + B66 (U66  + U6− 6),                 (A2 )                           \nwhere U’s are the one-electron intra-configuration unit te nsor operators and Bkq ’s are even parity CF \ninteraction parameters (CFPs) [59,60]. The last two  terms in Eq. (A1) represent the perturbation energ y \ndue to the Zeeman coupling, e.g., (i) between the R -moment  and the effective magnetic field eff Hr\n, ext \nZH \n= eff Hµ− ⋅ − ⋅ − ⋅ − ⋅ rr, and (ii) between the R-ion  and the local interna l magnetic field at the R-site appearing due to \nlong-range magnetic dipole-dipole and anisotropic e xchange interactions,  int \nZH = int ˆH⋅µr = Hdip  + Hexch , 16 \n described in Ref. [40]. The dipolar terms at R-site s j are given by Hdip, j = ∑\n′′′\njjj jQ ) () , (µ, where the \ncomponents of the tensor Q are the corresponding dipole lattice sums which hav e been computed in the \ncrystallographic system of coordinates in Ref. [40] . The spin-spin exchange tensor has three principal  \nindependent components which couple the n.n. spins along ( λ||) and normal ( λ⊥,x and λ⊥,y) to the R-R bond \ndirection. The values of the exchange coupling cons tants, λ||, λ⊥,x and λ⊥,y, and six CFPs, were obtained \nfrom the fit to the dc susceptibility using a Van V leck equation, \n(1) 2 (0) 2\n, , (2) \n,\n,( ) \n{ 2 }exp( ) n m n m s a B \nj n m \nn m B B E E NE\nZ k T k T µχ= − − ∑ .                                                  (A3)  \nIn order to fit the susceptibility data, the starti ng CFPs, e.g., B20  = −544.4, B40  = 2438.5, B60  = − 687.2, \nB43  = 1392.2, B63  = −795.4, B66  = −1360.0 (in cm −1), in Eq.(A2) were taken from the inelastic neutron  \nscattering results of Dy 2Ti 2O7 [61] and varied these initial parameters first ind ividually and then \ncollectively in a self-consistent manner. The CFPs,  B20  = −395.0, B40  = 1840.0, B60  = − 565.0, B43  = 650.0, \nB63  = 670.0, B66  = −1190.0 (in cm −1) are finally obtained from the best-fit to the sus ceptibility. The ±1% \nvariation in CFPs varies the calculated susceptibil ity by only ±0.03% and the energy levels by ±0.6%.  \nThe calculated energies of the eight doublet sublev els arising from the splitting of 6H15/2  ground multiplet \nof Dy 3+ -ion are 0, 191.6, 307.0, 323.8, 400.0, 649.0, 744. 5, 751.8 (all in K).  \n \nFIG. A1. The contributions of CF, dipolar and n.n. exchange interactions to the longitudinal ( χ||) and transverse ( χ⊥) \ncomponents of dc magnetic susceptibility. \n17 \n  \nThe wave function of the ground CF doublet is appro ximated by | Ψg〉 ≈ ∓ 0.94|±15/2 〉 + 0.2 |±9/2 〉 ± \n0.1|±3/2 〉, with 〈Ψg| JZ |Ψg〉 = 7.16(1). Values of the Landé g-factors of | Ψg〉 are highly anisotropic, g|| = \n19.1(1) and g⊥ = 0, which match with the values obtained in M(H) calculation. The calculated magnetic \nmoment of Dy 3+  ion in the ground state is µg = J Z g J = 9.55(1) µB/Dy which gives the ground state \ncontribution to the effective magnetic moment deriv ed from the CW fit of the susceptibility. The low-\ntemperature magnetic properties of the DGS can, the refore, be approximated by an effective pseudo-spin  \nS~= 1/2 system with a large magnetic moment. \nCalculated single-ion CF susceptibility, χCF (T), (and also 1/ χCF ) shows excellent agreement with the \nexperiment (within 1.5% at 60 K), in the temperatur e range 60 K < T < 300 K (Fig. 2), but decreases \nremarkably below 40 K (by 6% and 36% at 40K and 5 K , respectively). The dipolar and AF exchange \ninteraction with coupling constants, λ|| =  − 0.039(1) T/ µB, λ⊥,x = − 0.042(1) T/ µB, and λ⊥,y = 0, were, \ntherefore, introduced to describe the experimental χ(T); the calculated values (CF+MF) match very well \n(within 2-4% between 40 −5K) with the experimental data down to 5 ±1 K. The components of the \neffective site susceptibility tensors along and nor mal to <111> axes,  χ|| and χ⊥, and the contributions of \nCF, dipolar and n.n. exchange interactions to χ’s are displayed in Fig. A1. The values of χ|| are found \nlarger than χ⊥, which imply that the magnetic anisotropy at Dy si te is Ising-like along the local <111> \naxes of the tetrahedron due to the strong  CF axial  anisotropy combined with spin-spin exchange and \ndipolar interactions among Dy-spins.  \n \nAppendix B \nCalculation for lattice and Schottky heat capacity:  \nThe Cp(T) of Dy 2GaSbO 7 can be expressed as  Cp(T) = Clat  + CSch  + Cmag  within the temperature range \nstudied here, where the components are, respectivel y, lattice specific heat ( Clat ), Schottky anomalies ( CSch ) \nand magnetic specific heat ( Cmag ). In the low- T(K) zone, the lattice contribution is composed of t he lattice \nvibrations (phonons) at constant volume ( Cv ≈ Cp) and electronic part, and is usually expressed by the \nDebye model [62] as, Clat (T) = γ T + βT3, where, β and γ  are coefficients to lattice and the electronic \ncontributions to the heat capacity, respectively. T he value of β is related to the Debye temperature, θD \n=\u0004\u0005\u0006\u0007\b\t\n \n\u000b\f\r\u000e\n\u000f, where n =11 is the number of atoms per f.u. The Cp data of DGS in T < 20 K are fit to \nDebye model, displayed in inset to Fig.5, and found  γ = 0 implying insulating character of DGS and β = \n5 × 10 \u0014\u0015 JK −4/mol-Dy. Using the value of β,  the value of θD of DGS was estimated to be 350(±2) K, 18 \n which may be compared with the reported value for D y 2Ti 2O7 (θD = 283 K below 30 K) [63] and \nDy 1.8 Ca 0.2 Ti 2O7 (θD = 295 K below 19 K) [38]. Since the above θD value could not fit the heat capacity in \nhigh-T regime due to presence of optic modes of lattice v ibrations, Cp(T) data were analyzed using a \ncombination of Debye model (Eq.B1) and Einstein mod el (Eq.B2) of heat capacity.  \nDebye \nph C = ∫−T\nxx\nDD\ndx \nee xTnR /\n0243\n) 1()/ (9θ\nθ                                                          (B1) \nand  \nEinstein \nph C= 22\n] 1) / [exp( ) / exp( ) /(3\n−TTTnR \nEEE\nθθθ                                                   ( B2)                       \n \nwhere θE is Einstein temperature. 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Ther modyn. 120, 174 (2018). \n "    },    {        "title": "2401.00259v1.Ultrafast_X_ray_Diffraction_Probe_of_Coherent_Spin_state_Dynamics_in_Molecules.pdf",        "content": "arXiv:2401.00259v1  [physics.optics]  30 Dec 2023Ultrafast X-ray Diffraction Probe of Coherent\nSpin-state Dynamics in Molecules\nXiaoyu Mi,†Ming Zhang,†and Zheng Li∗,†,‡,¶\n†State Key Laboratory for Mesoscopic Physics and Collaborat ive Innovation Center of Quantum\nMatter, School of Physics, Peking University, Beijing 1008 71, China\n‡Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006,\nChina\n¶Peking University Yangtze Delta Institute of Optoelectron ics, Nantong, Jiangsu 226000, China\nE-mail: zheng.li@pku.edu.cn\n1Abstract\nWe propose an approach to probe coherent spin-state dynamic s of molecules using cir-\ncularly polarized hard x-ray pulses. For the dynamically al igned nitric oxide molecules in a\ncoherent superposition spin-orbit coupled electronic sta te that can be prepared through stim-\nulated Raman scattering, we demonstrate the capability of u ltrafast x-ray diffraction to not\nonly reveal the quantum beating of the coherent spin-state w ave packet, but also image the\nspatial spin density of the molecule. With circularly polar ized ultrafast x-ray diffraction signal,\nwe show that the electronic density matrix can be retrieved. The spatio-temporal resolving\npower of ultrafast x-ray diffraction paves the way for track ing transient spatial wave function\nin molecular dynamics involving spin degree of freedom.\nTOC Graphic\n2Molecular reactions are driven by the motion and reorganiza tion of valence electrons in an ultra-\nshort time scale.1–4Understanding coherent valence electron dynamics involvi ng spin degrees of\nfreedom in ultrashort time scale is of fundamental importan ce to the study of spintronics5and spin\ndynamics in molecules such as spin crossover and spin transf er.6,7In the last decades, methods\nsuch as attosecond streaking,8–10transient absorption,11,12strong field ionization13,14and high-\nharmonic generation (HHG)15have been introduced to probe the valence electron wave pack ets,\nvalence hole migration, and quantum beating between low-ly ing electronic states in coherent su-\nperposition.16It is especially intriguing to equip the probe of ultrafast e lectron dynamics with spin\nresolution.\nX-ray scattering is a powerful tool to achieve spin resoluti on. As an indispensable technique\nin imaging the magnetic structure in condensed matter physi cs,17resonant inelastic x-ray scatter-\ning (RIXS) can be combined with phase retrieval (PR) algorit hms18–20to attain the structure of\nthe magnetic domain.21,22In the RIXS theory for solids, a solid sample is regarded as mu ltiple\nions,23–25and the PR algorithm of the scattering pattern will provide v alues for each ions, which\nrepresent the spin magnetic moments at different locations . In the hard x-ray range, non-resonant\nx-ray scattering is also exploited in different types of mat erials.26,27This technique is often com-\nbined with magnetic circular dichroism (MCD) and magnetic l inear dichroism (MLD) to study the\nspin and magnetic properties of materials.28–30\nIn this Letter, we propose that x-ray scattering technique h as the capability of spatio-temporally\nresolving the spin dynamics of valence electrons in the gas p hase molecules. With aligned gas\nphase neutral nitric oxide molecules in a coherent wave pack et consisting of spin-orbit coupled\ndoublet states, we demonstrate that the circular dichroism (CD) of ultrafast non-resonant hard\nmagnetic x-ray scattering (MXS) can be used to reconstruct t he electronic density matrix and to\nretrieve the transient shape of valence electron orbitals o ccupied by the unpaired electron. The ap-\nproach proposed in this work could become an important appli cation of x-ray free electron lasers\n(XFEL) in probing spin-resolved valence electron dynamics of excited state molecules with ultra-\nfast x-ray diffraction, since XFEL can deliver ultrashort c ircularly polarized x-ray pulses.31The\n3ultrafast MXS method is versatile and can be sensitive to the spin dynamics in magnetic as well as\nnon-magnetic molecules.\nAs shown in Figure 1a, a spin-state wave packet of nitric oxide (NO) molecule wit h an unpaired\nelectron, which is adiabatically aligned with infrared las er and consists of low-lying electronic\nstates/vextendsingle/vextendsingle/vextendsingle2Π1/2/angbracketrightBig\nand/vextendsingle/vextendsingle/vextendsingle2Π3/2/angbracketrightBig\n, can be prepared through impulsive stimulated Raman scatte ring. The\nstimulated Raman scattering not only electronically excit es the molecules, but also dynamically\naligns them as a result of the selection rule. The consequent quantum beating proceeds with a\nperiod of T=278 fs that corresponds to energy spacing ∆E=120 cm−1of the two states. In\nNO, the transfer from/vextendsingle/vextendsingle/vextendsingle2Π1/2/angbracketrightBig\nto/vextendsingle/vextendsingle/vextendsingle2Π3/2/angbracketrightBig\nis only weakly allowed, such that 0 .1%–0.2% of the\nmolecules can be excited.16,32The quantum beating is a spin dynamic process since the/vextendsingle/vextendsingle/vextendsingle2Π3/2/angbracketrightBig\nhas the same spatial electron density as/vextendsingle/vextendsingle/vextendsingle2Π1/2/angbracketrightBig\n. The difference between the two states lies in the\nspin parts of unpaired electrons, thus it demands an ultrafa st spin-resolved probe. We demonstrate a\nmethod to reconstruct the electronic density matrix from th e CD signal of MXS, and the capability\nof non-resonant MXS in resolving the electron dynamics on bo th spatial and spin dimensions\nsimultaneously by x-ray scattering patterns of the NO molec ule in coherent spin-state.\nThe/vextendsingle/vextendsingle/vextendsingle2Π1/2/angbracketrightBig\nand/vextendsingle/vextendsingle/vextendsingle2Π3/2/angbracketrightBig\nstates are the products of spatial and spin parts, depending on the\nprojection quantum numbers of the electron orbital angular momentum ( Λ) and spin angular mo-\nmentum ( Σ) along the molecular axis:/vextendsingle/vextendsingle/vextendsingle2Π1/2,+/angbracketrightBig\n= (|π+β/an}bracketri}ht+|π−α/an}bracketri}ht)/√\n2 and/vextendsingle/vextendsingle/vextendsingle2Π3/2,+/angbracketrightBig\n=\n(|π+α/an}bracketri}ht+|π−β/an}bracketri}ht)/√\n2. Here the notation +in/vextendsingle/vextendsingle/vextendsingle2Π1/2,+/angbracketrightBig\nand/vextendsingle/vextendsingle/vextendsingle2Π3/2,+/angbracketrightBig\nrefers to the parity of\nthe electronic levels. The complex-valued spatial states/vextendsingle/vextendsingleπ±/angbracketrightbig\nrepresent |Λ=±1/an}bracketri}htand/vextendsingle/vextendsingleα(β)/angbracketrightbig\nrepresent/vextendsingle/vextendsingleΣ=±1/2/angbracketrightbig\n. They can be further expanded with real-valued spatial orbi tals|πx/an}bracketri}ht=\n(|π+/an}bracketri}ht+|π−/an}bracketri}ht)/√\n2 and/vextendsingle/vextendsingleπy/angbracketrightbig\n= (|π+/an}bracketri}ht−|π−/an}bracketri}ht)/(√\n2i). The unpaired electron density of |πx/an}bracketri}ht,|π±/an}bracketri}ht\nand/vextendsingle/vextendsingleπy/angbracketrightbig\nare shown in Figure 1b for t=0(T),t=T/4(3T/4), and t=T/2 of the quantum\nbeating, respectively. The stimulated Raman scattering ca n prepare a coherent superposition state\n/vextendsingle/vextendsingleΨ+(t)/angbracketrightbig\n=N(/vextendsingle/vextendsingle/vextendsingle2Π1/2,+/angbracketrightBig\n+c1/vextendsingle/vextendsingle/vextendsingle2Π3/2,+/angbracketrightBig\ne−i∆Et/¯h), (1)\n4Figure 1: (a) The scheme of reconstructing density matrix in the ultrafast molecular dynamics of\nNO from ultrafast non-resonant magnetic x-ray scattering ( MXS). First, aligned NO molecules in\nthe ground state/vextendsingle/vextendsingle/vextendsingle2Π1/2,+/angbracketrightBig\nare partially excited to/vextendsingle/vextendsingle/vextendsingle2Π3/2,+/angbracketrightBig\nby stimulated Raman scattering.\nThen the coherent state is probed by two incident x-ray pulse s with right circular ( σ+) and left\ncircular ( σ−) polarization to obtain the circular dichroic signal, whic h cancels the scattering pattern\ncontributed by electric interaction. By comparing the expe rimental result with ab initio calculation,\nthe off-diagonal elements of the density matrix is finally re constructed. (b) The time-dependent\nelectron density for the unpaired electron of eq 1forc1=1, which is taken for the purpose of\nvisualization. Red and blue spheres represent the N and O ato ms respectively. The spatial orbital\nof the unpaired electron goes through |πx/an}bracketri}ht,|πx/an}bracketri}ht±/vextendsingle/vextendsingleπy/angbracketrightbig\n,/vextendsingle/vextendsingleπy/angbracketrightbig\n,|πx/an}bracketri}ht∓/vextendsingle/vextendsingleπy/angbracketrightbig\nand|πx/an}bracketri}htin a complete\nperiod T=278 fs of the quantum beating process. The theoretical deriv ation for the quantum\nbeating process is in Supplementary Material.\n5where N=1/(/radicalbig\n1+|c1|2), and the quantum beating can persist for picoseconds.16In non-resonant\nMXS, the total differential scattering cross section (DSCS ) of an initial electronic state |i/an}bracketri}htis com-\nposed of elastic and inelastic parts (see Sec. I of Supplemen tary Material, SM)33\n(dσ\ndΩ)tot=(e2\nmec2)2/an}bracketle{ti|∑j j′eiq·rj,j′|i/an}bracketri}ht|e1·e∗\n2|2+(e2\nmec2)22¯hω\nmec2\n[ℜ/an}bracketle{ti|∑j j′eiq·rj,j′(c\nωpj′׈K1−ˆK2\n¯h)·P∗\n1(e1·e∗\n2)|i/an}bracketri}ht\n+ℜ/an}bracketle{ti|∑j j′eiq·rj,j′iσj′\n2·P∗\n2(e1·e∗\n2)|i/an}bracketri}ht], (2)\nwhere ωis the frequency of incident x-ray, q=K1−K2is the momentum transfer of incident\nx-ray photon, e1ande2are the polarization vectors of the incident and outgoing x- ray photon,\nrj,j′=rj−rj′is the coordinate difference between the jth and j′th electrons, pj′andσj′are the\nmomentum and the spin operators of the j′th electron. The two polarization factors P1=e1×e∗\n2\nandP2=e∗\n2×e1−(ˆK2×e∗\n2)×(ˆK1×e1)−(e∗\n2·ˆK1)(ˆK1×e1)+(e1·ˆK2)(ˆK2×e∗\n2)only rely\non the x-ray parameters.\nThe DSCS in eq 2consists of the contribution from Thomson charge scatterin g, where ∑j j′eiq·rj,j′\nindicates the two-electron correlation density function.34The second and the third terms repre-\nsent the orbital and spin-dependent contributions.35For hard x-ray of 0 .2 Å wavelength, the spin-\ndependent cross section is¯hω\nNemec2≈9×10−3smaller than the charge scattering, where Neis the\nnumber of electrons in one molecule and ¯hωis the energy of incident x-ray photon. The charge\nscattering contribution in DSCS can be eliminated by probin g the circular dichroism signal.36\nThe ultrafast non-resonant MXS of spin-state quantum beati ng is investigated using wave func-\ntion from the ab initio calculation at restricted active spa ce self-consistent field (RASSCF) level.37\nThe electronic wave function Ψ(S,MS)\nkis represented in the basis of configuration state functions\n(CSFs)/vextendsingle/vextendsingle/vextendsingleΦ(S,MS)\nk/angbracketrightBig\n:\n/vextendsingle/vextendsingleΨ(t)/angbracketrightbig\n=∑\nkak(t)/vextendsingle/vextendsingle/vextendsingleΦ(S,MS)\nk/angbracketrightBig\n, (3)\n6with the total spin Sand its projection MSon the molecular axis. The corresponding magnetic\nx-ray scattering patterns from the aligned NO molecules in t he coherent superposition of spin-\nstates are simulated using eq 2and wave function in the CSF basis (see Sec. II of SM for the ab\ninitio calculation of MXS from the coherent spin-state wave packet). In Figure 2a, the dichroic\nsignals are shown for the MXS from the NO molecules in the spin -state/vextendsingle/vextendsingleΨ+(t)/angbracketrightbig\n=(/vextendsingle/vextendsingle/vextendsingle2Π1/2,+/angbracketrightBig\n+\n/vextendsingle/vextendsingle/vextendsingle2Π3/2,+/angbracketrightBig\ne−i∆Et/¯h)/√\n2 within a period of quantum beating. In this case, the orbita l contribution\nof the magnetic scattering is proved to be zero with ab initio calculation. Thus the dichroic patterns\nonly depend on the spin-dependent contribution of the magne tic scattering. The CD signal is\ndominated by the transitions between CSFs/vextendsingle/vextendsingle/vextendsingleΦ(S,MS)\nk/angbracketrightBig\nof same spin projection MSin the x-ray\nscattering. The transitions between CSFs of different spin projections through σ±don’t contribute\nto the circular dichroism because the molecule has cylindri cal symmetry around the NO axis (see\nSec. IV of SM for details of vanishing MXS signal contributed from σ±). In real space, the\nalignment of NO molecular axis with an impulsive laser is non -perfect. The alignment degree can\nreach/an}bracketle{tcos2η/an}bracketri}ht ∼0.4–0.616for the angle ηbetween NO axis and the Zaxis of lab frame, which\nis perpendicular to the laser linear polarization directio n. And the initial excitation acts selectively\non the electron in the πxorbital, which aligns parallel with the laser polarization .\nIn Figure 2a, the petaloid-shaped patterns at time t=T/4 and t=3T/4 possess nodes along\ntheqxandqyaxes, it reflects spatial symmetry of the transition matrix e lement/an}bracketle{tπx|SR/vextendsingle/vextendsingleπy/angbracketrightbig\n, where\nSRis the real part of the scattering operator S=∑j j′eiq·rj,j′iσj′\n2·∆[P∗\n2(e1·e∗\n2)], and ∆[P∗\n2(e1·e∗\n2)]\nis the polarization factor in the circular dichroic signal, as shown in Sec. I and III of SM. In eq 4\nand Figure 2a, the selective πxorbital excitation for the dynamically aligned molecule av oids the\naveraging over πxandπyorbitals, otherwise MXS CD signal could vanish. In this case , the operator\nSin eq 2has non-zero amplitude only between CSFs with the same πxorπyorbital occupation. In\nfact, it can be proved that the time-dependent DSCS between t he two circularly polarized x-ray of\n7eq1is (see Sec. V in SM)\n∆dσ\ndΩ(q,t)=\n\n0 t<0\n2|c1|/angbracketleftbig\nπyα/vextendsingle/vextendsingleSR(q)|πxα/an}bracketri}ht\n1+|c1|2sin(ω0t−ϕ)t≥0, (4)\nwhere c1=|c1|eiϕis the complex amplitude in eq 1andω0=2π/Tis the frequency of quantum\nbeating. eq 4implies the oscillatory cross section presented in Figure 2b. From the practical\nperspective, we also estimated photon number counts of MXS c ircular dichroism. Typically 0 .1%–\n0.2% of the NO molecules are electronically excited in the stim ulated Raman scattering.16Thus\nwe assume |c1|=√\n0.002, and the maximum scattering photon number per second in F igure 2b\nis estimated to be Nphoton,0.2%=1.8×103s−1, as shown in Sec. VI of SM. For XFEL in the\nself-amplified spontaneous emission (SASE)38or seeded39regime, the light intensity fluctuation\ncan introduce error between the left and right circularly po larized x-ray pulses. To balance the\nlight intensity for different pulses, we suggest repeated e xperiment as a necessity for MXS circular\ndichroism. For an experiment with unknown c1in NO superposition state, the ab initio calculated\n/angbracketleftbig\nπyα/vextendsingle/vextendsingleSR(q)|πxα/an}bracketri}htcan be used to retrieve |c1|andϕfrom the observed time-dependent MXS CD\nsignal. During long time evolution, quantum decoherence le ads to a decreasing |ρ12|, where|1/an}bracketri}ht\nand|2/an}bracketri}htrepresent/vextendsingle/vextendsingle/vextendsingleΠ1/2,+/angbracketrightBig\nand/vextendsingle/vextendsingle/vextendsingleΠ3/2,+/angbracketrightBig\n. In principle, one can reconstruct the off-diagonal term\nin the density matrix of/vextendsingle/vextendsingle/vextendsingleΠ1/2(3/2),+/angbracketrightBig\nwith MXS in the case with quantum decoherence.\nNon-perfect alignment of the NO rotational wave packet is a n on-negligible effect, and the\nsimulated MXS CD signal involving non-perfect alignment as well as the quantum decoherence is\nshown in Figure 3. By solving time-dependent Schrödinger equation, the degr ee of the molecular\nalignment /an}bracketle{tcos2η/an}bracketri}ht(t)in Figure 3a is calculated for an 800 nm alignment pulse of 60 fs FWHM\nand the peak intensity is 6 ×1013W/cm2. The time interval between revivals of coupled rotational\nand electronic wave packet is about 5 ps. Figure 3b shows the MXS CD signal at time t=T/4,\nand for other time t, the circular dichroic signal exhibits similar oscillator y pattern but decaying\nintensity. Non-perfect alignment of molecules leads to dif fusion of the rotationally averaged spin\n8Figure 2: (a) Simulation of ultrafast MXS circular dichrois m pattern at time t=T/4,t=0 &t=\nT/2 and t=3T/4 for perfectly aligned NO molecules. The superposition sta te is chosen as/vextendsingle/vextendsingleΨ+(t)/angbracketrightbig\n=1√\n2(/vextendsingle/vextendsingle/vextendsingle2Π1/2,+/angbracketrightBig\n+/vextendsingle/vextendsingle/vextendsingle2Π3/2,+/angbracketrightBig\ne−i∆Et/¯h). The incident X-ray is along the NO axis and\nhas a wavelength of 0 .2 Å. (b) The time-dependent differential scattering cross s ection (DSCS)\n∆dσ\ndΩfor the perfectly aligned molecules and the corresponding s ignal strength in photon number\n(Nphoton ) per unit solid angle at position where DSCS reaches its maxi mum. Nphoton is estimated\nwith XFEL parameters in Supplementary Material.\n9Figure 3: (a) The degree of the molecular alignment by solvin g time-dependent Schrödinger equa-\ntion. The NO molecules are at the temperature 20 K. The 800 nm a lignment pulse has a full width\nhalf maximum (FWHM) of 60 fs and the peak intensity is 6 ×1013W/cm2. (b) Simulated ultrafast\nMXS circular dichroism pattern at time t=T/4 for the model with non-perfect alignment and\ndecoherence. The incident x-ray is along the NO axis and has a wavelength of 0 .2 Å. (c) The\ntime-dependent DSCS ∆dσ\ndΩand the corresponding signal strength in photon number ( Nphoton ) per\nunit solid angle at position where DSCS reaches its maximum. The damped amplitude of the os-\ncillatory MXS CD signal exhibits the influence of quantum dec oherence.\n10density in the laboratory frame. Thus the MXS CD pattern exhi bits larger petaloid area than that\nin Figure 2a when perfect alignment is considered. Compared with the ca se of perfect alignment,\nFigure 3b has lower intensity since the averaged expectation of the s cattering operator Sis affected\nby cos ηand quantum decoherence.\nIn Figure 3c, we present the MXS CD signal at position where DSCS reaches its maximum in\nthe reciprocal space. The MXS CD signal has an approximate fo rm of a decaying sine function\nfor the time period between the rotational revivals. Accord ing to Sec. V in SM, the decaying sine\nfunction of ∆dσ\ndΩis proportional to off-diagonal density matrix elements. B y comparing the value\nof this decaying sine function with the simulation under the same degree of alignment, it is possible\nto estimate the off-diagonal density matrix elements in the presence of non-perfect alignment.\nApart from reconstructing the evolving density matrix in ul trafast molecular spin dynamics,\nMXS also shows the capability in resolving spatial distribu tion of spin density. In the quantum\nbeating process of Figure 1b, the spin density is defined as the difference of the densiti es of elec-\ntrons with αandβspin\nρs(r)=ρα(r)−ρβ(r)\n=/angbracketleftbig\nΨ+(t)/vextendsingle/vextendsingle∑\njδ(r−rj)σj/2/vextendsingle/vextendsingleΨ+(t)/angbracketrightbig\n=/angbracketleftbig\nπy/vextendsingle/vextendsingle∑\njδ(r−rj)/2|πx/an}bracketri}htsin(ω0t), (5)\nwhere jis the index of electrons. According to eq 5, the spin density only changes its amplitude\nin the quantum beating process, but stays the same in shape. I n this case, only the z-component\nof the spin density is non-zero, thus we will denote ρs(r)as its z-component without ambiguity\nin the following statements. As discussed in the Sec. II of SM , the MXS dichroic signal can be\napproximately viewed as the product of electron density of t he inner shell electrons and the total\nspin density in the momentum space. We retrieve the 2-dimens ional spin density ˜ρs(qx,qy)in the\nreciprocal space, as shown in Figure 4a. With Fourier transformation of qxandqy, the normalized\n2-dimensional spin density |ρs(x,y)|/max{|ρs|}in the real space is calculated in Figure 4b. In\n11Figure 4b, the 2-dimensional spin density is consistent with eq 5, which shows the spatial overlap\nbetween the unpaired electrons of |πx/an}bracketri}htand/vextendsingle/vextendsingleπy/angbracketrightbig\n. By comparing the reconstructed spin density with\nthe reference one, it shows the capability of MXS dichroism i n probing the spin density with high\naccuracy.\nFigure 4: Spin density of NO molecule retrieved from MXS dich roic signal in Figure 2a att=T/4.\n(a) 2-dimensional spin density ˜ρs(qx,qy)in the momentum space. (b) Normalized 2-dimensional\nspin density |ρs(x,y)|/max{|ρs|}in the real space. The spin density is distributed in the rang e of\n[−2.0,2.0]au in both xandydirection. The left one of |ρs(x,y)|/max{|ρs|}is reconstructed from\nMXS dichroic signal, the right one is directly calculated fo r reference.\nTo conclude, we propose an approach to probe the ultrafast sp in dynamics in molecules and ap-\nply it to the aligned NO molecules in the simulation. The circ ular dichroism of MXS can eliminate\nthe charge scattering signal between the incoming x-ray and molecules, and probes electrons via\nthe magnetic interaction. We have proposed that by fitting th e experimental MXS result with ab\ninitio calculation, MXS circular dichroism can reconstruc t the electronic density matrix on ultra-\nfast time scale. We have also shown that MXS is capable of prob ing the spin density by resolving\nthe spatial distribution. The technique is promising in pro bing molecular dynamics since it pro-\nvides information about spin-polarized electronic with hi gh temporal and spatial resolution. Given\nthe recent progress of XFEL approaching the attosecond regi me38,40and x-ray pulses with circu-\nlar polarization.41The experimental implementation of the ultrafast MXS probe for electron spin\ndynamics is within reach.\n12SUPPORTING INFORMATION\ntheoretical derivation of differential cross section, cal culation of NO molecular dynamics and mag-\nnetic x-ray scattering signal, photon number rate estimati on\nACKNOWLEDGMENTS\nWe gratefully acknowledge Haitan Xu, R. J. 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Photon. 2016 ,10, 468–472.\n17arXiv:2401.00259v1  [physics.optics]  30 Dec 2023Supplementary Material for\nUltrafast X-ray Probe of Coherent Spin-state\nDynamics in Molecules\nXiaoyu Mi,†Ming Zhang,†and Zheng Li∗,†,‡,¶\n†State Key Laboratory for Mesoscopic Physics and Collaborat ive Innovation Center of Quantum\nMatter, School of Physics, Peking University, Beijing 1008 71, China\n‡Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006,\nChina\n¶Peking University Yangtze Delta Institute of Optoelectron ics, Nantong, Jiangsu 226000, China\nE-mail: zheng.li@pku.edu.cn\nS1I. Analytical Derivation of Differential Scattering Cross Sec-\ntion of Nonresonant Magnetic X-ray Scattering (MXS)\nIn nonresonant MXS, the double differential scattering cro ss sectiond2σ\ndΩ2d/planckover2pi1ω2from initial state |i/an}bracketri}ht\nto final state |f/an}bracketri}htis1,2\n(d2σ\ndΩ2d/planckover2pi1ω2)|i/angbracketright→|f/angbracketright=(e2\nmec2)2(ω2\nω1)|/an}bracketle{tf|/summationdisplay\njei(K1−K2)·rj|i/an}bracketri}ht(e1·e∗\n2)−i/planckover2pi1\nmec2/an}bracketle{tf|/summationdisplay\njei(K1−K2)·rj\n[ic(pj׈K1−ˆK2\n/planckover2pi1)·P1+D·σj\n2]|i/an}bracketri}ht|2δ(Ei−Ef+/planckover2pi1ω1−/planckover2pi1ω2), (S1)\nwhereω1(2),K1(2)ande1(2)are frequencies, wave vectors and polarization vectors of i ncident and\noutgoing x-ray. And rj,pjandσjare position, momentum and spin operators of the jth electron.\nThe vector functions P1andDare expressed as\nP1=e1×e∗\n2 (S2)\nD=ω1+ω2\n2[e∗\n2×e1−(ˆK2×e∗\n2)×(ˆK1×e1)]−[ω1(e∗\n2·ˆK1)(ˆK1×e1)−\nω2(e1·ˆK2)(ˆK2×e∗\n2)]. (S3)\nIn nonresonant MXS, the photon energy of hard x-ray is much hi gher than the energy difference\nbetween molecular electronic states. Thus it is reasonable to assume /planckover2pi1ω=/planckover2pi1ω1≈/planckover2pi1ω2≫ |Ei−\nEf|. The integral of the δ-function on the RHS of eq S1resultsω2/ω1= 1, which also affects the\nform ofD. Denoteq=K1−K2, one obtains\n(dσ\ndΩ2)|i/angbracketright→|f/angbracketright=(e2\nmec2)2|/an}bracketle{tf|/summationdisplay\njeiq·rj|i/an}bracketri}ht|2|e1·e∗\n2|2+(e2\nmec2)22/planckover2pi1ω\nmec2ℜ{/an}bracketle{ti|/summationdisplay\nj′e−iq·rj′[c\nω(pj′×\nˆK1−ˆK2\n/planckover2pi1)·P∗\n1+iP∗\n2·σj′\n2]|f/an}bracketri}ht/an}bracketle{tf|/summationdisplay\njeiq·rj|i/an}bracketri}ht(e1·e∗\n2)}, (S4)\nS2where the square of the magnetic part is neglected since it is by(/planckover2pi1ω\nNemec2)2≈8×10−5smaller than\nthe electric part (the first term in eq S4) and/planckover2pi1ω\nNemec2≈9×10−3smaller than the interference part\n(the second term in eq S4), whereNeis the number of electrons in a molecule, and\nP2=e∗\n2×e1−(ˆK2×e∗\n2)×(ˆK1×e1)−(e∗\n2·ˆK1)(ˆK1×e1)+(e1·ˆK2)(ˆK2×e∗\n2).(S5)\nSumming over all final states |f/an}bracketri}ht, we obtain the expression for the differential scattering c ross\nsection (DSCS) of non-resonant MXS as\n(dσ\ndΩ)tot=(e2\nmec2)2/an}bracketle{ti|/summationdisplay\njj′eiq·rj,j′|i/an}bracketri}ht|e1·e∗\n2|2+(e2\nmec2)22/planckover2pi1ω\nmec2ℜ/an}bracketle{ti|/summationdisplay\njj′eiq·rj,j′\n[(c\nωpj′׈K1−ˆK2\n/planckover2pi1)·P∗\n1+iσj′\n2·P∗\n2](e1·e∗\n2)|i/an}bracketri}ht. (S6)\nII. MXS Calculation with ab initio wave functions\nIn this section we present procedure to calculation the spin -dependent contribution to dichroic\nsignal of MXS, using electronic wave functions of the molecu le represented in the basis of config-\nuration state functions (CSF), which are obtained from the r estricted active space self-consistent\nfield (RASSCF) method. The CSF |Φ(S,MS)\nk/an}bracketri}htcan be written as a linear combination of different\nSlater determinants |ψm/an}bracketri}htcomposed of spin orbitals |φp/an}bracketri}htof molecules. Thus the expectation of\nspin scattering operator S=/summationtext\njj′eiq·rj,j′iσj′\n2·∆[P∗\n2(e1·e∗\n2)]of state|Ψ/an}bracketri}htcan be calculated as\n/an}bracketle{tΨ|S(q)|Ψ/an}bracketri}ht=i\n2∆[P∗\n2(e1·e∗\n2)]·/summationdisplay\nk,l/summationdisplay\nm,na∗\nkalb∗\nk,mbl,n{/an}bracketle{tψm|/summationdisplay\njσj|ψn/an}bracketri}ht\n+1\n2/an}bracketle{tψm|/summationdisplay\nj/negationslash=j′[eiq·rj,j′σj′+eiq·rj′,jσj]|ψn/an}bracketri}ht}\n=i\n2∆[P∗\n2(e1·e∗\n2)]·[/integraldisplay\ndr1/summationdisplay\np,qσ1,pqγpq(r1)\n+/integraldisplay\ndr1dr2/summationdisplay\np,q,r,s/parenleftbig\neiq·r12σ2,qs+eiq·r21σ1,pr/parenrightbig\nΓpqrs(r1,r2)] (S7)\nS3whereaiandbi,jare linear combination coefficients respectively from |Φ(S,MS)\ni/an}bracketri}htto|Ψ/an}bracketri}htand from\n|ψj/an}bracketri}htto|Φ(S,MS)\ni/an}bracketri}ht,σi,pqis the spin part of matrix elements /an}bracketle{tφp(ri)|σ|φq(ri)/an}bracketri}ht.γpq(r1)andΓpqrs(r1,r2)\nare one- and two-electron reduced density matrix elements, and\n/summationdisplay\npqγpq(r1) =ρ(r1), (S8)\n/summationdisplay\npqrsΓpqrs(r1,r2) =ρ(2)(r1,r2). (S9)\nIn eq ( S7), onlyz-component of σi,pqcontributes to the final result because of the rotational\naveraging of molecular xandyspin axis. By denoting ρ(r,α(β))as spin-up(down) electron\ndensities and ρs(r) =ρ(r,α)−ρ(r,β)as spin density, the integral in eq ( S7) can be further\nexpressed as\n/integraldisplay\ndr1/summationdisplay\np,qσ1,z,pqγpq(r1) =/integraldisplay\ndr1[ρ(r1,α)−ρ(r1,β)] =/integraldisplay\ndrρs(r), (S10)\n/integraldisplay\ndr1dr2/summationdisplay\np,q,r,seiq·r12σ2,z,qsΓpqrs(r1,r2)\n=/integraldisplay\ndr1dr2eiq·r12[ρ(2)(r1,α;r2,β)+ρ(2)(r1,β;r2,α)−ρ(2)(r1,α;r2,β)−ρ(2)(r1,β;r2,β)],\n=Ne−1\n2Ne/integraldisplay\ndr1dr2eiq·r12[ρ(r1,α)+ρ(r1,β)][ρ(r2,α)−ρ(r2,β)]\n=Ne−1\n2Ne/integraldisplay\ndr1dr2eiq·r12ρ(r1)ρs(r2), (S11)\nwhereρ(2)(r1;r2) =Ne−1\n2Neρ(r1)ρ(r2)by neglecting the electronic correlation effects, and Neis the\nnumber of electrons. This approximation is valid when elast ic scattering plays a dominant role in\nthe MXS signal.\nIn the quantum beating of NO, eq ( S10) makes no contribution since the population of spin-\nup and spin-down electrons are equal. Thus, the spin density can be retrieved from eq ( S11) by\ndividing the MXS dichroic signal by electron density of the i nner shell electrons.\nIn the ab initio calculation, ρs(r)andρ(2)(r1,α(β);r2,α(β))are expanded with Gaussian type\nS4orbitals (GTO). The integrals in eq ( S10) and ( S11) can be calculated by\n/integraldisplay\ndx(x−x1)L1(x−x2)L2e−γ1(x−x1)2e−γ2(x−x2)2=C/integraldisplay\ndx(x−x1)L1(x−x2)L2e−γ(x−p)2\n/integraldisplay\ndx(x−x1)L1(x−x2)L2e−γ1(x−x1)2e−γ2(x−x2)2eiqxx\n=eiqxp−q2x\n4γC/integraldisplay\ndx(x−x1)L1(x−x2)L2e−γ(x−iqx\n2γ−p)2, (S12)\nwhereC=eγ1γ2\nγ1+γ2(x1−x2)2,γ=γ1+γ2andp=γ1x1+γ2x2\nγ1+γ2.\nIII. Calculation of ∆[P∗\n1(2)(e1·e∗\n2)]\nIn this section we present the explicit expression of ∆[P∗\n1(2)(e1·e∗\n2)]. We first express e1(2),R(L)and\nK1(2)in lab frame coordinates. Assuming that the incident x-ray w ave vector is along the Zaxis\nof the lab frame, the unit vector of incident wave vector K1and left (right) circular polarization\ne1,L(R)are\nˆK1=/parenleftbigg\n0 0 1/parenrightbiggT\nˆe1,L=√\n2\n2/parenleftbigg\n1i0/parenrightbiggT\nˆe1,R=√\n2\n2/parenleftbigg\n1−i0/parenrightbiggT\n. (S13)\nThe wave vector of outgoing x-ray has a deviation from the inc ident one and can be expressed with\npolar angleθand azimuth angle φas\nK2=/parenleftbigg\nK2,xK2,yK2,z/parenrightbiggT\n−→ˆK2=/parenleftbigg\nsinθcosφsinθsinφcosθ/parenrightbiggT\ncosθ=K2,z/radicalig\nK2\n2,x+K2\n2,y+K2\n2,zsinθ=/radicalig\nK2\n2,x+K2\n2,y/radicalig\nK2\n2,x+K2\n2,y+K2\n2,z\ncosφ=K2,x/radicalig\nK2\n2,x+K2\n2,ysinφ=K2,y/radicalig\nK2\n2,x+K2\n2,y. (S14)\nS5Because the circular polarization of the x-ray remains unch anged in nonresonant MXS process,\nthe polarization vector of outgoing x-ray e2,L(R)is\ne2,L(R)=Rz(φ)Ry(θ)e1,L(R)\n=\ncosφ−sinφ0\nsinφcosφ0\n0 0 1\n\ncosθ0 sinθ\n0 1 0\n−sinθ0 cosθ\n\n√\n2\n2\n±i√\n2\n2\n0\n\n=√\n2\n2/parenleftbigg\ncosφcosθ∓isinφsinφcosθ±icosφ−sinθ/parenrightbiggT\n. (S15)\nThe corresponding ( e1·e∗\n2),P∗\n1andP∗\n2are\n(e1·e∗\n2)L(R)=1\n2(1+cosθ) (S16)\nP∗\n1,L(R)=1\n2/parenleftbigg\n±isinθsinθ(±icosφ+sinφ)(cosθ+1)/parenrightbiggT\n(S17)\nP∗\n2,L(R)=1\n2sinθ\n±i\n1\n0\n+1\n2sinθ\n−sinφ∓icosθcosφ\ncosφ∓icosθsinφ\n±isinθ\n. (S18)\nFinally, we obtain the important functions for the MXS calcu lation∆[P∗\n2(e1·e∗\n2)]and∆[P∗\n2(e1·\ne∗\n2)]as\n∆[P∗\n1(e1·e∗\n2)] =i\n2\nsinθ(cosθ+1)cosφ\nsinθ(cosθ+1)sinφ\n(1+cosθ)2\n, (S19)\n∆[P∗\n2(e1·e∗\n2)] =i\n2\nsin3θcosφ\nsin3θsinφ\nsin2θ(1+cosθ)\n, (S20)\nS6whereω≈ω1≈ω2is the frequency of the hard x-ray field. From eq S20, it is obvious that\n∆[P∗\n2(e1·e∗\n2)]is purely imaginary, this fact will be used in the next sectio nIV ..\nIV . Analysis for the Vanishing Part of Interference Term in\nMXS of Nitric Oxide Molecule\nIn the electron dynamics of NO, the electronic states are represented by spin orbitals |α(β)/an}bracketri}htand\nreal-valued spatial orbitals |πx(y)/an}bracketri}ht. The spin part of differential scattering cross section (DS CS) of\nMXS circular dichroism is proportional to\nℜ/an}bracketle{ti|/summationdisplay\njj′eiq·rj,j′iσj′\n2·∆[P∗\n2(e1·e∗\n2)]|i/an}bracketri}ht. (S21)\nSince|i/an}bracketri}htrepresents the electronic state, i.e., the superposition o f|Π1/2/an}bracketri}htand|Π3/2/an}bracketri}ht, which is probed\nthe MXS and conforms to Hund case (a), the total spin of electr ons is quantized along the aligned\nmolecular axis, jandj′are electron indices. This can be also understood by the magn etic field\nproduced by orbital current of |π±/an}bracketri}ht, which is a ring current around the molecular axis. The mag-\nnetic field leads to an energy splitting for different electr on spin along the N-O axis, determining\nthezaxis for the aligned spin, which coincides with the molecula r axis. The stimulated Raman\nprocess determines the xandyaxes of spin in the molecular coordinate system, which is the same\nfor different stimulated molecules without rotational ave rage.3However, the spin of NOhas no\npreference for any direction perpendicular to the aligned m olecular axis. Thus the spin operators\nσxandσyin eq S21should be angularly averaged. Suppose the relative angle be tweenX(Y)of\nthe lab frame for ∆[P∗\n2(e1·e∗\n2)]andx(y)axis forσx(y)isθ′, then eq S21should be multiplied by\ncosθ′. It leads to a zero-valued spin part after averaging since/integraltext2π\n0cosθ′dθ′= 0. The zero-valued\nexpectation of the σxandσycomponents in eq S21is important since they correspond to scatter-\ning between different spin orbitals. In other words, only sc attering between the same spin orbital\ncontributes to the final circular dichroism signal.\nS7When the incident x-ray is parallel to the molecular axis tha t is aligned in the lab frame, the z\naxis is the same as Zaxis. Since ∆[P∗\n2(e1·e∗\n2)]is purely imaginary, the product between iσzand\n∆[P∗\n2(e1·e∗\n2)]Zis a real-valued function. eq S21can be written as\nℜ/an}bracketle{ti|/summationdisplay\nj<j′(eiq·rj,j′iσz,j′\n2+e−iq·rj,j′iσz,j\n2)∆[P∗\n2(e1·e∗\n2)]z|i/an}bracketri}ht. (S22)\nand\nℑ/an}bracketle{ti′|/summationdisplay\nj<j′eiq·rj,j′iσj′\n2·∆[P∗\n2(e1·e∗\n2)]|i/an}bracketri}ht= 0. (S23)\nwhere|i/an}bracketri}htand|i′/an}bracketri}htare the electronic states.\nV . Electron Dynamics and the Expression of DSCS in Terms of\nDensity Matrix\nWith|πx(y)/an}bracketri}htand|α(β)/an}bracketri}ht, the electronic states |Π1/2(3/2),+/an}bracketri}htare\n|2Π1\n2,+/an}bracketri}ht=1√\n2(|π+β/an}bracketri}ht+|π−α/an}bracketri}ht) =1\n2[(|πx/an}bracketri}ht+i|πy/an}bracketri}ht)|β/an}bracketri}ht+(|πx/an}bracketri}ht−i|πy/an}bracketri}ht)|α/an}bracketri}ht] (S24)\n|2Π3\n2,+/an}bracketri}ht=1√\n2(|π+α/an}bracketri}ht+|π−β/an}bracketri}ht) =1\n2[(|πx/an}bracketri}ht+i|πy/an}bracketri}ht)|α/an}bracketri}ht+(|πx/an}bracketri}ht−i|πy/an}bracketri}ht)|β/an}bracketri}ht]. (S25)\nThe quantum beating dynamics for |Ψ+(t)/an}bracketri}ht=1√\n2(|2Π1\n2,+/an}bracketri}ht+|2Π3\n2,+/an}bracketri}hte−i∆Et//planckover2pi1)is represented by\n|Ψ+(t)/an}bracketri}ht=1\n2√\n2|α/an}bracketri}ht[(1+e−i2πt/T)|πx/an}bracketri}ht−i(1−e−i2πt/T)|πy/an}bracketri}ht]\n+1\n2√\n2|β/an}bracketri}ht[(1+e−i2πt/T)|πx/an}bracketri}ht+i(1−e−i2πt/T)|πy/an}bracketri}ht]. (S26)\nS8Thus the wave functions of the superposition state at t= 0,T\n4,T\n2,3T\n4,Tare\n|Ψ+(0)/an}bracketri}ht=1√\n2(|α/an}bracketri}ht+|β/an}bracketri}ht)|πx/an}bracketri}ht\n|Ψ+(T/4)/an}bracketri}ht=1−i\n2√\n2[|α/an}bracketri}ht(|πx/an}bracketri}ht+|πy/an}bracketri}ht)+|β/an}bracketri}ht(|πx/an}bracketri}ht−|πy/an}bracketri}ht)]\n|Ψ+(T/2)/an}bracketri}ht=−i√\n2(|α/an}bracketri}ht+|β/an}bracketri}ht)|πx/an}bracketri}ht (S27)\n|Ψ+(3T/4)/an}bracketri}ht=1+i\n2√\n2[|α/an}bracketri}ht(|πx/an}bracketri}ht−|πy/an}bracketri}ht)+|β/an}bracketri}ht(|πx/an}bracketri}ht+|πy/an}bracketri}ht)]\n|Ψ+(T)/an}bracketri}ht=1√\n2(|α/an}bracketri}ht+|β/an}bracketri}ht)|πx/an}bracketri}ht.\nBy defining SR(I)=ℜ(ℑ){/summationtext\njj′eiq·rj,j′iσj′\n2·∆[P∗\n2(e1·e∗\n2)]}, the four integrals corresponding to\nthe elements in density matrix of two electronic states |2Π1\n2,+/an}bracketri}htand|2Π3\n2,+/an}bracketri}htare\nℜ[/an}bracketle{t2Π1\n2,+|S|2Π1\n2,+/an}bracketri}ht] =1\n4\n/an}bracketle{tαπx|\n/an}bracketle{tβπx|\n/an}bracketle{tαπy|\n/an}bracketle{tβπy|\nT\n(SR\n1 1 0 0\n1 1 0 0\n0 0 1 −1\n0 0−1 1\n+SI\n0 0 1 −1\n0 0 1 −1\n−1−1 0 0\n1 1 0 0\n)\n|απx/an}bracketri}ht\n|βπx/an}bracketri}ht\n|απy/an}bracketri}ht\n|βπy/an}bracketri}ht\n\n(S28)\nℜ[/an}bracketle{t2Π1\n2,+|S|2Π3\n2,+/an}bracketri}ht] =1\n4\n/an}bracketle{tαπx|\n/an}bracketle{tβπx|\n/an}bracketle{tαπy|\n/an}bracketle{tβπy|\nT\n(SR\n1 1 0 0\n1 1 0 0\n0 0−1 1\n0 0 1 −1\n+SI\n0 0 −1 1\n0 0 −1 1\n−1−1 0 0\n1 1 0 0\n)\n|απx/an}bracketri}ht\n|βπx/an}bracketri}ht\n|απy/an}bracketri}ht\n|βπy/an}bracketri}ht\n\n(S29)\nℜ[/an}bracketle{t2Π3\n2,+|S|2Π1\n2,+/an}bracketri}ht] =1\n4\n/an}bracketle{tαπx|\n/an}bracketle{tβπx|\n/an}bracketle{tαπy|\n/an}bracketle{tβπy|\nT\n(SR\n1 1 0 0\n1 1 0 0\n0 0−1 1\n0 0 1 −1\n+SI\n0 0 1 −1\n0 0 1 −1\n1 1 0 0\n−1−1 0 0\n)\n|απx/an}bracketri}ht\n|βπx/an}bracketri}ht\n|απy/an}bracketri}ht\n|βπy/an}bracketri}ht\n\n(S30)\nS9ℜ[/an}bracketle{t2Π3\n2,+|S|2Π3\n2,+/an}bracketri}ht] =1\n4\n/an}bracketle{tαπx|\n/an}bracketle{tβπx|\n/an}bracketle{tαπy|\n/an}bracketle{tβπy|\nT\n(SR\n1 1 0 0\n1 1 0 0\n0 0 1 −1\n0 0−1 1\n+SI\n0 0 −1 1\n0 0 −1 1\n1 1 0 0\n−1−1 0 0\n)\n|απx/an}bracketri}ht\n|βπx/an}bracketri}ht\n|απy/an}bracketri}ht\n|βπy/an}bracketri}ht\n,\n(S31)\nwhere the expectation of SIis0in light of the analysis in eq S23. For the expectation of SR,\nthe sum of diagonal elements in the 4×4matrix is always 0since the result of spin-up orbital\ncancels out with the result of spin-down orbital. Further, t he4×4matrix can be divided as four\n2×2matrices. Each off-diagonal element in these 2×2matrices is also 0since it corresponds to\nthe term with σxandσy. eqS28and eq S31are diagonal terms in density matrix of |Π1/2(3/2),+/an}bracketri}ht,\nwhich have no contribution to the circular dichroism. For th e off-diagonal terms eq S29and eq S30\nof the density matrix, the complex coefficient c12=a12+ib12in front of the /an}bracketle{t2Π3\n2,+|S|2Π1\n2,+/an}bracketri}ht\ngives an effective contribution to the DSCS, since\nℜ[c12/an}bracketle{t2Π3\n2,+|S|2Π1\n2,+/an}bracketri}ht] =1\n4\n/an}bracketle{tαπx|\n/an}bracketle{tβπx|\n/an}bracketle{tαπy|\n/an}bracketle{tβπy|\nT\nSR\na12a12b12−b12\na12a12b12−b12\nb12b12−a12a12\n−b12−b12a12−a12\n\n|απx/an}bracketri}ht\n|βπx/an}bracketri}ht\n|απy/an}bracketri}ht\n|βπy/an}bracketri}ht\n\n+1\n4\n/an}bracketle{tαπx|\n/an}bracketle{tβπx|\n/an}bracketle{tαπy|\n/an}bracketle{tβπy|\nT\nSI\n−b12−b12a12−a12\n−b12−b12a12−a12\na12a12b12−b12\n−a12−a12−b12b12\n\n|απx/an}bracketri}ht\n|βπx/an}bracketri}ht\n|απy/an}bracketri}ht\n|βπy/an}bracketri}ht\n.\n(S32)\nAnd in eq S32,/an}bracketle{tαπy|b12SR|απx/an}bracketri}ht−/an}bracketle{tβπy|b12SR|βπx/an}bracketri}ht+/an}bracketle{tαπx|b12SR|απy/an}bracketri}ht−/an}bracketle{tβπx|b12SR|βπy/an}bracketri}ht=\n4/an}bracketle{tαπy|b12SR|απx/an}bracketri}ht. Forc21=c∗\n12in front of the term /an}bracketle{t2Π3\n2,+|S|2Π1\n2,+/an}bracketri}ht, the result is the same.\nFinally, the result of circular dichroism is 2×1\n4×4/an}bracketle{tαπy|b12SR|απx/an}bracketri}ht= 2/an}bracketle{tαπy|b12SR|απx/an}bracketri}ht.\nPrevious discussion indicates a zero-valued signal in the c ircular dichroism before the stimu-\nS10lated Raman scattering excites NOto the electronic state |2Π3\n2,+/an}bracketri}ht. AfterNOmolecules are excited\nto the superposition state, b12(t)can be retrieved from time-dependent circular dichroism pa ttern.\nFor the superposition state\n|Ψ+(t)/an}bracketri}ht=1/radicalbig\n1+|c1|2(|2Π1\n2,+/an}bracketri}ht+|c1|eiϕ|2Π3\n2,+/an}bracketri}hte−i∆Et//planckover2pi1), (S33)\nandb12=|c1|sin(ω0t−ϕ)\n1+|c1|2, whereω0=∆E\n/planckover2pi1. The density matrix of the electronic states can be\nreconstructed from the time-dependent MXS since both ϕand|c1|can be retrieved with b12(t). For\nlong time evolution, quantum decoherence leads to a decreas ing|ρ12|(|1/an}bracketri}htand|2/an}bracketri}htare respectively\n|Π1/2,+/an}bracketri}htand|Π3/2,+/an}bracketri}ht), which results in the damped intensity of the magnetic x-ra y scattering\ncircular dichroism. Following decoherence, the measured t ime-dependent b12(t)is a decaying\nfunction. One can still extract the phase of the decaying fun ction to retrieve c12fromb12, and the\noff-diagonal density matrix elements can be reconstructed .\nVI. Estimate of the Photon Number with XFEL\nThe ultrafast nonresonant MXS can be implemented by x-ray fr ee electron laser (XFEL). In this\nsection, we demonstrate the feasibility to use MXS circular dichroism as a probe of coherent spin\ndynamics, we calculate the rate of scattered photon number w ith XFEL parameters. Within unit\nsolid angle, the total rate of scattered photon number Nphoton is\nNphoton=dσ\ndΩνpulseNpulseNmol, (S34)\nwhereνpulse is the beam rate, Npulse is the incident photon number for a single pulse and Nmolis\nthe number of irradiated molecules per unit area at this mome nt. Further, Nmolcan be expressed\nasnmolL. Herenmolis molecular density and Lis the diameter of the molecular beam. Using\nS11practical experimental parameters of XFEL,4,5the photon number rate is estimated to be\nNphoton=∆(dσ\ndΩ)νpulseNpulsenmolL\n=2×10−5barn×106Hz×1012×1016cm−3×0.1 cm = 2 ×104s−1. (S35)\nConsidering only 0.2%of the NO molecules are typically electronically excited af ter stimulated\nRaman scattering, the final result of photon number rate is\nNphoton,0.2%=√\n0.002\n1+0.0021+1√\n1Nphoton≈1.8×103s−1(S36)\nReferences\n(1) Blume, M. Magnetic Scattering of X-rays. J. Appl. Phys. 1985 ,57, 3615–3618.\n(2) Sch¨ ulke, W. Electron Dynamics by Inelastic X-Ray Scattering ; Oxford University Press: New\nYork, 2007.\n(3) Kraus, P. M.; Zhang, S. B.; Gijsbertsen, A.; Lucchese, R. R.; Rohringer, N.; W¨ orner, H. J.\nHigh-Harmonic Probing of Electronic Coherence in Dynamica lly Aligned Molecules. Phys.\nRev. Lett. 2013 ,111, 243005.\n(4) Emma, P.; Akre, R.; Arthur, J.; Bionta, R.; Bostedt, C.; B ozek, J.; Brachmann, A.; Bucks-\nbaum, P.; Coffee, R.; Decker, F. J. et al. First Lasing and Ope ration of an Angstrom-wavelength\nFree-electron Laser. Nat. Photon. 2010 ,4, 641–647.\n(5) Barty, A.; K¨ upper, J.; Chapman, H. N. Molecular Imaging Using X-ray Free-electron Lasers.\nAnnu. Rev. Phys. Chem. 2013 ,64, 415–435.\nS12"    },    {        "title": "1512.02557v1.Spin_Directed_Momentum_Transfers_in_SIDIS_Baryon_Production.pdf",        "content": "  \n \nSpin-Directed Momentum Transfers in \nSIDIS Baryon Production \n \nDennis Sivers \nPortland Physics Institute \nUniversity of Michigan \n \nAbstract \n \nThe measurement of transverse single-spin asymmetries for baryon production in the \ntarget fragmentation region of semi-inclusive deep-inelastic scattering (SIDIS), can produce important insight into those nonperturbative aspects of QCD directly associated with confinement and with the dynamical breaking of chiral symmetry.  We discuss here, in terms of spin-directed momentum transfers, the powerful quantum field- theoretical constraints on the spin-orbit dynamics underlying these transverse spin observables.  The \nτΑ-odd spin-directed momentum shifts, originating either in the target nucleon ()  \nor in the QCD jets (TNkδ\n)TNpδ produced in the deep inelastic scattering process, represent \nsignificant quantum entanglement effects connecting information from current fragmentation with observables in target fragmentation. \n  \n \n \n \n Instrumentation of the “target fragmentation” region at an electron-ion collider (EIC) can provide a broad range of new tools for the study of nonperturbative dynamics in QCD. [1,2]  The specific observables disc ussed here are the transverse single-spin \nasymmetry measurements  \n  (1.1) ( ' ) , , , , ...\n( ' ) , , ...NAd e p eB X B pn\nPd ep e B X Bσ\nσ↑⇒ = Λ Σ Δ\n⇒↑ ↑ = Λ ↑ Σ ↑ Ξ ↑\n \nThe systematic, organized program of such measurements should also be done at the Jefferson Lab CEBAF 12 GeV machine [3] due to the importance of single-spin observables in understanding spin-orbit dynamics in a full range of kinematics.  As described below, these baryon asymmetry measurements both supplement and enhance the understanding of measurements of target spin asymmetries for meson production \n \n 00( ' ) , , , , ...NAd e p eM X M K K0σ ππ η±±↑⇒ =  (1.2) \n \nfound in the “current fragmentation” region.  It is convenient to parameterize the baryon \nobservables of (1.1) in terms of the transverse-momentum dependent (TMD) “fractured functions” introduced in references [1] and [2].  For the target-spin asymmetries we can define the Fractured Orbital Functions (FOF) 2\n/( , ):(, ;, ; )Nq B\nTN T T Bq qpM xk z p k Q↑Δ⋅GG, and the \nFractured Collins, Heppelman, Ladinsky Functions (FCHLF) \n 2\n/{ , } :(, ;, ; )Nq B B\nTT T N Bq q pM xk p z p Q↑\n↑↑Δ⋅GG. Correspondingly, for the final-state polarization \nobservables we can define the Polarizing Fractured Functions (PFF) \n2\n/( , ):(, ;, ; )Nq B B\nTT T N Bq q pM xk p z p Q↑Δ⋅GG and the Fractured Boer Mulders Functions (FBMF) \n2\n/{ , } :(, ;, ; )Nq\nTN T T Bq qpM xk zk p Q↑\n↑↑ΔGG⋅ .  The content of the expressions for these functions is \ndense and the labeling complicated so the names are attached to connect these transverse \nspin-odd fractured functions involving quantum diquark structures to the classification of \ntransverse-spin odd distributions and frag mentation functions for quarks created by \nMulders and Tangerman. [4]  The four fractured functions presented here describe the spin-dependent differences of the conjoint probabilities for detecting a quark jet with \nkinematics defined by \n2/2 .bj xxQ p== q and transverse momentum TkG\nand a final-state \nbaryon with kinematics defined by the Feynman variable /BB zz p pp q==⋅ ⋅  and \ntransverse momentum TpG in the same deep-inelastic scattering event.  A more complete \ndiscussion of these fractured functions can be found in references [1,2].  For FOF and \nFCHLF the symbol indicates a target transverse spin asymmetry, NΔ\n \n /(,): /(,): /(,):.Nq q q\nBp Bp BpMM M↑ ↑Δ= −↓ (1.3) \nFor the fractured functions PFF and FBMF the symbol NΔdefines a final-state \npolarization asymmetry from an unpolarized target, \n \n  (1.4) /(,): /(,): /(,):.Nq q q\nBp Bp BpMM M↑↑↓Δ= −\nThe set of conjoint probability densities  [/(,): /(,):,qq\nBp B pMM↑↓ ↑↓ ,…] can capture the full \nrange of quantum information contained in the measurements as constrained in the Bell’s \ninequalities [5,6] and used in the study of quantum entanglement.  This connection is important when viewed in the context of the powerful quantum-field-theoretical superselection principles applicable in the measurement of single-spin asymmetries involving the hard scattering of light quarks. [7,8].  For valence SIDIS these arguments also introduce quantum entanglement connecting the target fragmentation region to information in current fragmentation. \n All single-spin asymmetries are odd under an operator, Ο, such that \n 1{; } {; }ij i jkkσ σ−ΟΟ = −G G G G, (1.5) \nwhere  are momentum 3-vectors and ikG\njσGare spin axial vectors.  This operator can be \nseen to be the 3-D Hodge dual of the parity operator, P, \n 1{; } { ; }ij ij Pk P kσ σ−=−G G G G. (1.6) \nThe operator product  then has the action PτΑ=Ο\n 1{; } { ; } .ij i j kkττσ σ−ΑΑ = − −G G G G (1.7) The operator τΑnow designated “naïve time reversal” [9,10] is odd for transverse, parity \nconserving single-spin asymmetries.  In this note we will denote the τΑ-odd spin-\ndirected momentum, \n ˆˆ(TN Tkk Qσ ) =⋅×G\n, (1.8) \nas the observable that defines a single-spin observable in SIDIS and other hard-scattering \nprocesses. \n The explicit connection of the spin-directed  momentum shift that can be produced \nby one unit of = and the asymmetries defined in Eq. (1.1) is indicated in the sketches of \nFig. 1  \n \nFig. 1  Spin-directed momentum transfers provide an alternate, rigorous, definition of single-spin asymmetries. \n \nIn the discussion below we will designate a spin-directed momentum shift generated \nwithin the target nucleon by 2(, )TNkxδμ  and shift generated within a QCD jet \nfragmentation by 2(, )TNpxδμ  and treat them as directly observable asymmetries. \nThe observation that transverse single-spin asymmetries are prohibited as  in \nhard-scattering processes of perturbative QCD [11] can be designated Kane-Pumplin-\nRepko [12] factorization.  For processes su ch as SIDIS this allows the use of field-\ntheoretical idempotent projection operators, 0qm→\n 1\n2τ ±\nΑ±Α⎛Π=⎜⎝⎠⎞\n⎟. (1.9) \nThese operators create superselection rules that isolate the spin-directed momentum shift.  \nIn processes involving the hard scattering of massless quarks τΑ-odd dynamical effects \ncan only be produced in the wave function of the target nucleon or in the color re-\narrangement that incurs in the fragmentation dyna mics leading to final-state hadrons.      \nThe superselection rules allow for the calculation of these shifts within the nucleon by lattice gauge simulations [13].  The application of these operators to a valence-quark SIDIS process is indicated in Fig. 2,  \n \n \nFig. 2  The hard scattering in valence SIDIS separates nonperturbative jet structure from \nnonperturbative dynamics in the target nucleon. \n \nThe hard scattering process in the central part of this diagram that is calculable \nin QCD perturbation theory for a valence u or d quark is necessarily eq eqX⇒\nτΑ-even so that the \nτΑ-odd dynamics leading to the spin-directed momentum shift must appear in the wave \nfunction of the target nucleon or in the color-rearrangement of the fragmentation processes leading to final-state hadrons.  The phenomenological study of spin-directed \nmomentum transfers in the current fragmentation region of this diagram [14,15] has already proved to be very instructive in isolating interesting dynamics. The diagram in Fig. 2 does not explicitly show the final-state interaction between the struck quark and \nthe diquark structure that creates the process dependence of \nτΑ-odd distribution \nfunctions.  The quantum entanglement produced by these final-state interactions can be \nisolated by the projection operator −\nΑΠof (1.9)  The fact that the SIDIS process \ndemonstrates TMD factorization [16] combin ed with the fact that SIDIS kinematics \nallows for a distinction between τΑ-odd dynamics occurring in the target nucleon from \nτΑ-odd effects in fragmentation. [17]   The same kinematic restrictions occur for the \nproduction of baryons in the target fragmentation region of SIDIS as can be demonstrated \nby the three intersecting planes shown in Fig. 3 \n \n \n  \n \nFig. 3  The three independent planes in SIDIS can be used to differentiate between τΑ-\nodd dynamics in the nucleon and in jet fragmentation. \n \nA significant feature of spin-observables  involving baryons is that polarization \nasymmetries and target spin asymmetries can be studied in tandem.  For example the \ncomparison of the FOF  with the PFF  in the same range of \nkinematics allows for the calibration of the /[ , ]:Nu\nud pMΛΔ↑ /[ , ]:Nu\nud pMΛ↑Δ\n2(, )TNkxδμ  associated with the Fractured \nOrbital Function with the 2(, )TNpzδμ  generated in the Polarizing Fractured Function. \nSince these momentum shifts do not evolve under TMD evolution [11,18], the data \ncollected from spin asymmetries can be used to study the evolution of spin-averaged TMD’s. \n Most of the familiar examples of quantum entanglement involve spin observables measured in widely separated kinematical regions. [5,19]  The hard scattering of a valence quark in a polarized proton as pictured in Fig. 2  can be seen to resolve the structure of a complicated internal quantum state.  The Fractured Orbital Function, FOF, resolves this state in terms of an orbiting configuration involving quark-diquark and virtual meson degrees of freedom confined w ithin the polarized proton.  The final-state \ninteractions of the struck quark that lead to a spin-directed momentum transfer involve the attractive chromo-electric force between the struck quark and the diquark spectator.  This force leads to a fraction of the transverse momentum shift of the quark to be transmitted to the diquark,  \n  () () ()22\n{,} [,]\n() ()22\n{,}( ,) ( ,) 2 ( ,)\n(, ) (, ) .OO\nTN TN u TNud ud u\nOO\nTN d TNuu dkx kx kx\nkx kxδμ δμ η δμ\nδμ η δμ+= −\n=−2O\n (1.10) \n \nIn these equations the parameters 0 , 1udηη≤≤ provide a measure of the interaction of \nthe quark-diquark system with the remaining internal constituents.  These equations allow \nthe prediction of   and /[ , ]: /[ , ]:,Nu Nu\npu dp nu dpMM↑↑ΔΔ/[ , ]:Nu\nud pMΛ ↑Δ  from existing data on pion \nspin asymmetries.  This type of long-range correlation tests the underlying understanding \nof orbital distributions in the same manner that the Collins conjugation prediction [20] relating the asymmetry in SIDIS and the Drell-Yan process, \n () ()2(, ) (, ) ,O\nTN TNSIDIS DYkx kx δμ δμ =−2O (1.11) \nwhen expressed in terms of spin-directed momentum transfers.  \n The spin state of a diquark by itself represents a quantum-entangled system and \nthe rank dependence of the fragmentation process in the FCHLF is closely \nrelated to that of the Collins Function, /{ , '} :Nq\nBq q pM↑\n↑↑Δ\n/N\nMqD↑Δ .  That is to say, the τΑ-odd flux rupture \nleading to {, ' }qq B q↑⇒ + ↑  necessarily involves \n 2(, ) (, ) .TN TNqpz pzδμ δμ\n↑=−2\nB (1.12) \nThis means that there will be spin-asymmetrie s associated with the production of mesons \nwithin the target fragmentation region as well as baryons.  We will not discuss here the \nformulation of dihadron fractured functions that can be used to study spin-oriented correlations between hadrons in a full range of kinematics.  At this conference Harut Avakian presented Jefferson Lab data on spin-dependent correlations between baryons and mesons. [21]  I believe these data support the basic structure of the formalism presented here. \nThe author is grateful for informative talks and advice from Gary Goldstein and Simonetta Liutti. \n \nREFERENCES \n \n1. D. Sivers, Phys. Rev. D79, 085008 (2009). \n2. D. Sivers, Phys. Rev. D80, 034029 (2010). \n3. See, for example, M Battaglieri et al. Acta Phys. Polon. B46, 2, 257 (2015). \n4. R.D. Tangerman and P.J. Mulders, Phys. Rev. D51, 3357 (1995); Nucl. Phys. \nB461 , 197 (1996). \n5. J.S. Bell, Physics 1, 195 (1964); Rev. Mod. Phys. 38, 447 (1966). \n6. P.H. Eberhard, Nuovo Cim. B38 75 (1977). \n7. D. Sivers, Phys. Rev. D74, 094008 (2006). \n8. D. Sivers, arXiV:0704.1791 (hep-ph) unpublished \n9. R. Jaffe, in  Proceedings on the Workshop on Deep Inelastic Scattering off \nPolarized Targets, DEZY-Zeuthen, Germany 1997, edited by J.Blumlein et al., \n167. 10. D. Sivers, in Proceedings of the Workshop on Deep Inelastic Scattering off \nPolarized Targets, DEZY-Zeuthen, Germany 1997,  edited by J. Blumlein et al., \n383. \n11. D. Sivers, Int. J. Mod. Phys. Conf. Ser. 25, 1460002 (2014). \n12. G. Kane, J. Pumplin and W. Repko, Phys. Rev. Lett. 41, 1689 (1978). \n13.  B. U. Musch, et al., Phys. Rev. D85, 094510 (2012). \n14.  M. Anselmino et al. , arXiV:1107.4446 (2011). \n15.  Z.-B. Kang, et al., arXiV:1505.05589 (2015). \n16.   J. Collins, arXiV:1307.2920 (2013). \n17.  A. Bacchetta et al., Phys. Rev. D70, 117504 (2011). \n18.  D. Sivers, in EBJ Web. Conf. 85, 2007 (2015). \n19.  A. Einstein, B Podolsky and N. Rosen, Phys. Rev. D47, 777 (1935).  \n20.   J. Collins, Phys. Lett. B536 , 43 (2002). \n21.  H. Avakian, this conference. \n \n \n \n "    },    {        "title": "0911.2452v2.Theory_of_Electron_Spin_Relaxation_in_n_Doped_Quantum_Wells.pdf",        "content": "arXiv:0911.2452v2  [cond-mat.mes-hall]  27 Jan 2010Theory ofElectron Spin Relaxationinn-Doped Quantum Wells\nN.J. Harmon,1W.O. Putikka,1and R. Joynt2\n1Department of Physics, Ohio State University, 191 W. Woodru ff Ave., Columbus, OH 43210, USA\n2Department of Physics, University of Wisconsin-Madison, 1 150 University Ave. Madison, WI 53705, USA\n(Dated: July4, 2018)\nRecentexperimentshavedemonstratedlongspinlifetimesi nuniformlyn-dopedquantumwells. Thespindy-\nnamics of exciton, localized, and conduction spins are impo rtant for understanding these systems. We explain\nexperimental behavior by invoking spin exchange between al l spin species. By doing so we explain quanti-\ntatively and qualitatively the striking and unusual temper ature dependence in (110)-GaAs quantum wells. We\ndiscuss possible future experiments to resolve the pertine nt localized spin relaxation mechanisms. In addition,\nour analysis allows us to propose possible experimental sce narios that will optimize spin relaxation times in\nGaAsand CdTequantum wells.\nI. INTRODUCTION\nIn recent years, uniformly doped quantum wells (QWs)\nhave generated increasing interest due to the long relaxati on\ntimes measured therein.1–3The long relaxation times are due\nto spins localized on donor centers. While similar relaxati on\ntimeshavebeenmeasuredinmodulationdopedsystems,their\nduration has not been as reliable due to the weaker binding\nenergy of localized states and potential fluctuations from r e-\nmote impurities.4–6Localization is either not seen at all4or\nlocalization centers thermally ionize rapidly with increa sing\ntemperaturedueto a smallbindingenergy.5,6QWsuniformly\ndoped within the well have the advantage of being character-\nized by well defined impurity centers with a larger binding\nenergy. Theexperimentalcontrolintheamountofdopingand\nwellsizemakedopedQWsparticularlyappealingtothestudy\nofquasi-two-dimensionalspindynamics.\nMuchofthetheoreticalstudyofspinrelaxationinsemicon-\nductingsystems(QWsin particular)haseitherfocusedsole ly\non itinerant electrons7–9or solely on localized electrons10,11\nwithout regard for either the presence of the other state or\nthe interaction between the two states. Recently the exis-\ntence and interaction between itinerant and localized stat es\nhas been dealt with in bulk systems by Putikka and Joynt12\nand Harmon et al.13. The results of these calculations are in\nvery good quantitative and qualitative agreement with expe r-\nimental observations14,15in bulk n-GaAs and n-ZnO. In this\npaper,thetheoryoftwointeractingspinsubsystemsisappl ied\ntoQWs.\nThe paper is structuredas follows: Section II describesthe\noptical generation of spin polarization in QWs; Section III\nintroduces a set of modified Bloch equations to model spin\ndynamics; Section IV calculates the equilibrium populatio ns\noflocalizedandconductionsstates; SectionV determinest he\nrelaxationratesforallpertinentmechanismsforlocalize dand\nconduction electrons; Sections VI and VII compare our re-\nsults to two GaAs QWs (uniformly doped and undoped) and\none uniformly doped CdTe QW; Section VIII discusses our\nfindings, suggests future work, and proposes QWs for spin\nliftimeoptimization;weconcludeinSectionIX.II. SPINPOLARIZATIONIN QUANTUMWELLS\nIn QWs at low temperatures the creation of non-zero spin\npolarization, in the conduction band and donor states, pro-\nceedsfromtheformationoftrions(chargedexcitons, X±)and\nexciton-bound-donorcomplexes( D0X)respectively,fromthe\nabsorptionofcircularlypolarizedlight.\nPolarization via the trion avenue is most relevant for mod-\nulation doped QWs where donor centers in the well are\nsparse.3,4Due to the modulation doping outside the well, the\nnumber of conduction electrons in the well may be plenti-\nful. In such cases, assuming incident σ+pump pulse, a +3\n2\nhole and −1\n2electron are created. These bind with a resi-\ndent electron fromthe electron gasin the QW to form a trion\n(X−\n3/2). The‘stolen’electronwillbe +1\n2toformasingletstate\nwith the exciton’s electron. Hence, the electron gas will be\nleft negatively polarized since the excitons are preferent ially\nformed with spin up resident electrons. If the hole spin re-\nlaxesfaster than the trion decaysthe electrongas will rema in\npolarized.4Selectionrulesdictate +3\n2(−3\n2)holeswillrecom-\nbineonlywith −1\n2(+1\n2)electrons. Thereforeiftheholespins\nrelaxrapidly,thereleasedelectronswill havenonet polar iza-\ntionandthepolarizedelectrongaswillremainpredominant ly\nnegativelyoriented.\nAverysimilarpictureisgivenforthepolarizationofdonor\nbound electrons in uniformly doped QWs where the donor\nbound electrons play the role of the resident electrons.1,2At\nlow temperatures the donors are nearly all occupied and the\ndensity of the electron gas will be negligible. When excita-\ntionsaretunedatthe exciton-bound-donorresonance,inst ead\nof photo-excitonsbindingwith the resident electrongas, t hey\nbindwith neutral donorsto formthe complexes D0X3/2. This\nnotation implies that a +3\n2hole -−1\n2electron exciton bound\ntoa+1\n2donorboundelectron. Onceagainforveryshorthole\nrelaxationtimes,thedonorboundelectronscanbespinpola r-\nized.\nThe measured long spin relaxation times in uniformly\ndoped QWs imply that spin polarization remains after short\ntime processes such as XandD0Xrecombination have com-\npleted. In other words, the translational degrees of freedo m\nthermalize much more quickly than the spin degrees of free-\ndom. The occupational statistics of itinerant and localize d2\nelectrons are important and can be determined from equilib-\nriumthermodynamics.\nAs the temperature is increased, the electrons bound to\ndonorsthermallyionizeandbecomeitinerant. Inanalogywi th\nthetrioncase,iftheexcitationenergyismaintainedatthe D0X\nfrequency,theinitialpolarizationshoulddecreaseasthe reare\nfewerD0Xcomplexes allowed.6However as the number of\nelectrons in the conduction states increases, the spin that ex-\nists on the donors will equilibrate by cross relaxing to con-\nduction states by the isotropic exchange interaction. If cr oss\nrelaxation is rapid enough, the total spin, which is conserv ed\nbyexchange,willnowexistinthedonorandconductionstate s\nweightedbytheirrespectiveequilibriumdensities.12,13,16The\npolarizedelectronmomentswillthenproceedtorelaxviadi f-\nferent processes for the localized and itinerant states. Si nce\ntrion binding energies ( ∼2 meV)5,6are smaller than donor-\nexciton binding energies ( ∼4.5 meV),17polarization of itin-\nerantelectronsviatrionformationshouldbe negligibleas the\ntemperatureisincreased.\nThe above description is complicated when the photoexci-\ntation energy is at the exciton resonance and not the exciton -\nbound-donor resonance. In such a case, the excitons may\nrecombine or the electron-in-exciton spin may relax before\nbinding to a donor so one expects the low temperature spin\nrelaxationto reflect also the excitonspin dynamicsinstead of\nthedonorelectronspindynamicsalone.6Inessence,theelec-\ntrons in an exciton represent a third spin environment with a\ncharacteristic spin relaxationtime scale differentfrom t hat of\nthe localized donor and itinerant electrons. Because of the\nelectron’sproximitytoahole,relaxationmayresultfroms pin\nexchangeorrecombination.\nTherefore to understand the spin dynamics in QWs, it is\nimperativeto examinethe relaxationprocesses that affect the\npolarizedspinmomentsofthevariousspinsystems.\nIII. MODIFIEDBLOCHEQUATIONS\nAfter rapid exciton-donor-bound complex formation, re-\ncombination,andholerelaxation,wemodelthezerofieldspi n\ndynamicsofthesystemintermsofmodifiedBlochequations:\ndmc\ndt=−/parenleftBig1\nτc+nl\nγcr\nc,l/parenrightBig\nmc+nc\nγcr\nc,lml\ndml\ndt=nl\nγcr\nc,lmc−/parenleftBig1\nτl+nc\nγcr\nc,l/parenrightBig\nml (1)\nwheremc(ml) are the conduction(localized) magnetizations,\nnc(nl) are the conduction (localized) equilibrium occupation\ndensities, τc(τl) are the conduction (localized) spin relax-\nation times, and γcr\nc,lis a parameter describing the cross re-\nlaxation time between the two spin subsystems. Mahan and\nWoodworth16have shown the cross relaxation time between\nimpurity and conduction electron spins to be much shorter\nthan any of the other spin relaxation times relevant here. We\nshall assume below that the same is true for the cross relax-\nation between electrons bound in an exciton and conduction\nor impurity electron spins. The motivation of these modifiedBloch equations is set forth in Refs. (12) and (13). Eqs. (1)\nisvalidforphotoexcitationenergiesthatdonotcausefree ex-\nciton formation (only two relevant spin systems). It is im-\nportant to note that Eqs. (1) hold only for intermediate time\nscales. Thesescalesarelongcomparedwithlaserpulsetime s,\nenergyrelaxationtimesthatdeterminesubsystempopulati ons\nand donor-boundexciton formation times. Fortunately, the se\nintermediate time scales are the ones probed in the experi-\nments.\nStandard methods can be used to solve these differential\nequations with initial conditions mc(0)andml(0). We as-\nsume that the initial spin polarization is perpendicular to the\nQW’sgrowthplaneandthattheexcitationdensity, Nx,issmall\nenough such that the resultant spin relaxation time, τs, will\nnotdependstronglyon Nx.2Thesolutionsyieldatimedepen-\ndence of the total magnetization m(t)=mc(t)+ml(t)to be a\nsumoftwoexponentials-oneofwhichisexp (−t/τs)andthe\nother of which has a time constant proportional to the cross\nrelaxation time. In the case of rapid cross relaxation (fast er\nthan all spin relaxation mechanisms), only one exponential\nsurvivesandweexpressthetotalrelaxationrateas\n1\nτs=nl\nnimp1\nτl+nc\nnimp1\nτc(2)\nwherenimp=nl+ncis the total impurityconcentration. This\nmodel, or variations of it, has been successfully applied to\nbulkn-GaAsandbulkn-ZnO.12,13\nIfthephotoexcitationenergyisset neartheexcitonenergy ,\ntheBlochequationsmustbemodifiedtotakeintoaccountex-\nciton spin relaxation and multiple cross relaxations: γi,jfor\ni,j∈c,l,xfor conduction, localized, and excition spins re-\nspectively. We model exciton spin relaxation as electron-i n-\nexcitonspinrelaxation18andassumethatholespinrelaxation\nisveryrapid. Eq. (1)generalizesto\ndmc\ndt=−/parenleftBig1\nτc+nl\nγcr\nc,l+nx\nγcrc,x/parenrightBig\nmc+nc+Nx−nx\nγcr\nc,lml+nc+Nx−nx\nγcrc,xmx\ndml\ndt=nl\nγcr\nc,lmc−/parenleftBig1\nτl+nc+Nx−nx\nγcr\nc,l+nx\nγcr\nl,x/parenrightBig\nml+nl\nγcr\nl,xmx\ndmx\ndt=nx\nγcrc,xmc+nx\nγcr\nl,xml−/parenleftBig1\nτx+nc+Nx−nx\nγcrc,x+nl\nγcr\nl,x/parenrightBig\nmx,\n(3)\nwhereτxrepresents spin lifetime of an electron bound to a\nhole.nx(mx) is the number (magnetization) of electrons\nbound in an exciton. Nxis the initial density of photoexcited\nelectrons and the quantity Nx−nxis the number of photoex-\ncited electrons that do not participate in an exciton. We as-\nsume quasi-equilibriumsuch that nxis determinedfrom ther-\nmodynamics(see SectionIV). It shouldbe statedthat Eq. (3)\nis valid only for times shorter than the recombination time;\nin other words, on a time scale where Nxcan be assumed\nto not change significantly. Recombination times have been\nmeasured19in similar systems as to those studied here to be\nlongerthantheobservedspinrelaxationtimessothisappro x-\nimationseemsjustified. InSectionVI,wefindthattheeffect s\nof recombination of free carriers can be added to 1 /τcto ob-\ntainexcellentagreementwiththeexperimentaldata.3\nIfwesolvethesystemofequationsinEq. (3)aswedidfor\nEq. (1),we obtaintherelaxationrate\n1\nτs=nl\nnimp+Nx1\nτl+nc+Nx−nx\nnimp+Nx1\nτc+nx\nnimp+Nx1\nτx.(4)\nForbothEqs. (2)and(4),weallow τl,τc, andτxtobephe-\nnomenologicalparametersoftheform τ−1\ni=∑j1/τjwherej\nrefers to a type of spin relaxation mechanism. From the ex-\nperimental constraints and results, we can determine which\nrelaxationmechanismsareimportant.\nIV. OCCUPATION CONCENTRATIONS\nAs shown above, the relative occupations of localized and\nitinerant states play an important role in our theory. Fortu -\nnately,in two dimensionalsystems, the occupationprobabi li-\ntiesofthetwostates( nl/nimpandnc/nimp)canbedetermined\nexactly. The densities we are interested in are dilute enoug h\nsuch that the non-degeneratelimit (Boltzmann statistics) can\nbeutilized.\nFIG.1: Occupationprobabilitiesoflocalized(solidline) andconduc-\ntion (dash-dotted line) states with impurity density nimp=4×1010\ncm−2determined from Eqs. (7, 8). Other parameters for GaAs are\na∗\nB=10.4 nm and m∗=0.067m.\nThe probabilityfora donorto be singly occupied(onlythe\ngroundstate needstobeconsidered20) is21\nnl\nnimp=1\n1\n2e(EB−µ)/kBT+1. (5)\nThedensityofitinerantstatesisgivenby\nnc=Nceµ/kBT(6)\nwhereNc=m∗kBT//planckover2pi12πand the conduction band edge is\ntakentobezeroenergy. Thechemicalpotential µcanbefound\nusingtheconstraint\nnl\nnimp+nc\nnimp=1. (7)Usingtheresultfor µ,oneobtains\nnl\nnimp=/radicalbig\n1+Q(T,nimp)−1/radicalbig\n1+Q(T,nimp)+1, (8)\nwhere\nQ(T,nimp)=8nimp\nNce−Eb/kBT. (9)\nAn example of the temperature dependence of these oc-\ncupation probabilities is shown for a GaAs QW in Figure 1\nwherenimp=4×1010cm−2. At the lowest temperatures,the\ndonors are fully occupied. As the temperature increases, nl\ndecreases and ncincreases to where at around 50 K, the two\noccupationprobabilitiesare equal. From Eqs. (2, 4), it is e v-\nident that these occupational statistics have ramification s in\nthe measured spin relaxation times. The results here are als o\nappliedtothe excitonsinquasi-equilibrium.\nV. SPINRELAXATION\nWe now discuss the relevant spin relaxation mechanisms\nfor both localized and conductionelectrons. The electron- in-\nexciton spin relaxation, τx, is a combination of electron-hole\nrecombinationand electron-holeexchangerelaxation. Due to\nits complicatednaturewe defer the calculationof τxto future\nwork. Herewe treatit asaphenomenologicalparameter.\nA. Localized SpinRelaxation\nFirst wediscussspinrelaxationviatheanisotropicspinex -\nchange for donor bound electrons. This has been treated ex-\ntensively elsewhere.11,22–24Most recently it has been exam-\ninedbyKavokininRef. (10). Itishistreatmentthatwedetai l\nbelowforsemiconductingQWs.\nKavokin argues10that some portionof localized relaxation\nresultsfromspindiffusionduetotheexchangeinteraction be-\ntween donors. Anisotropic corrections to the isotropic ex-\nchange Hamiltonian cause a spin to rotate through an an-\ngleγi,jwhen it is transferred between two donor centers lo-\ncated at positions riandrj. The angle-averaged rotation an-\ngle is/angbracketleftγ2\ni,j/angbracketright1/2=/angbracketleftr2\ni,j/angbracketright1/2/Ls.o.whereLs.o.is the spin orbit\nlength.10The spin is relaxed when the accumulated rotation\nangleΓbecomesontheorderofunitysuchthat Γ2=∑/angbracketleftγ2\ni,j/angbracketright=\n∑/angbracketleftr2\ni,j/angbracketright/L2\ns.o.=2Dexτex/L2\ns.o.=1whereDexisthediffusionco-\nefficientandtherelaxationtimeis\nτex=L2\nso\n2Dex. (10)\nIn quasi-2D (100) QWs where Dresselhaus bulk inversion\nasymmetry(BIA)termsdominate,11\nLs.o.=/parenleftBig2α/planckover2pi1/radicalbig2m∗Eg/angbracketleftk2\nz/angbracketright/parenrightBig−1\n, (11)4\nwhereαis a dimensionlessmeasureof the spin orbitstrength\nand/angbracketleftk2\nz/angbracketrightisduetothequasi-2Dconfinementandisoftheform\nβ2/L2. For infinite well confinement β=π. The diffusion\ncoefficientisapproximately10\nDex=1\n2/angbracketleftr2\ni,j/angbracketright/angbracketleftJ/angbracketright//planckover2pi1. (12)\nwithexchangeconstant25in 2D\nJ2D=15.21Eb/parenleftBigri,j\naB/parenrightBig7/4\ne−4ri,j/aB (13)\nwhereEbis the binding energy: Eb=/planckover2pi12/(2m∗a2\nB). Howri,j\nistobe determinedwill bediscussedinSectionVI.\nThese resultscan be combinedto obtain therelaxationrate\nintermsofa dimensionlessimpurityseparationscale, x:\n1\nτex=15.21α2/planckover2pi13/angbracketleftk2\nz/angbracketright2\nEgm∗2/angbracketleftx2/angbracketright/angbracketleftx7/4e−4x/angbracketright(14)\nwherex=ri,j/aB.\nLocalized electron spins may also relax due to nuclear\nfields. A localized electron is coupled to many nuclear spins\nby the hyperfine interaction. To the electrons, these nuclea r\nspins appear as a randomly fluctuating field but these nu-\nclear fields can be assumed quasi-stationarysince the nucle ar\nevolution time is much longer than electron evolution time\ndue to the contrast in magnetic moments.10What governs\nthe electron spin evolution is the electron correlation tim e,\nτcorr. Ifτcorrislongsuchthat µBg∗//planckover2pi1/angbracketleftB2\nN/angbracketright1/2τcorr>1(where\n/angbracketleftB2\nN/angbracketright1/2=Bmax\nN/√NLis the root-mean-square field, Bmax\nNis\nthe maximum nuclear field, and NLis the number of nuclei\nintheelectron’slocalizationvolume),thentheelectronp olar-\nization decays due to ensemble dephasing; there will be ran-\ndom electron precession frequencies due to a random distri-\nbution of frozennuclear fields.26If the mechanism contribut-\ning to the electron correlation time is exchange induced spi n\ndiffusion, τcorris estimated to be (n1/2\nimpDex)−1in quasi-two\ndimensions.10\nMerkulovet al.26finda dephasingrateforquantumdotsto\nbe\n1\nτnuc=/radicalBigg\n16∑jIj(Ij+1)A2\nj\n3/planckover2pi12NL(15)\nwhere the sum over jis a sum overall nuclei in the unit cell,\nIjis the nuclear spin, Ajis the hyperfine constant, and NLis\nthe number of nuclei in the electron’s localized volume. It\nis important to state that this spin dephasing does not decay\nexponentiallybutdecreasesto 10%oftheoriginalspin pola r-\nization in τnucand then increases to 33% of the original spin\npolarizationin2 τnucwhereitwillthendecayatamuchslower\nrate.26,27\nIfτcorris short such that µBg∗//planckover2pi1/angbracketleftB2\nN/angbracketright1/2τcorr≪1, then the\nrelaxationwill beofthemotionalnarrowingtype.10B. ConductionSpinRelaxation\nConduction band states undergo ordinary impurity and\nphonon scattering. Each scattering event gives a change in\nthe wave vector k, which in turn changes the effective mag-\nnetic field on the spin that comes from spin-orbit coupling.\nThis fluctuating field relaxes the spin. This is known as the\nD’yakonov-Perel’ (DP) spin relaxation mechanism.28,29The\neffectivefield strength is proportionalto the conductionb and\nsplitting. In this article, we are interested in conduction spin\nrelaxation in (001) and (110) oriented QWs. For (001) QWs\nthe spin relaxation rate results from a spin-orbit term in th e\nHamiltonian, Hs.o=/planckover2pi1\n2Ω(k||)·σwhere9\nΩ(k||)=2γ\n/planckover2pi1\nkx(k2\ny−/angbracketleftk2\nz/angbracketright)\nky(/angbracketleftk2\nz/angbracketright−k2\nx)\n0\n.\nThe angularbracketsdenotespatial averagingacrossthe we ll\nwidth.γisabandparameterthatgovernsthemagnitudeofthe\nspin-orbitsplitting. ForGaAs, γ∼17meVnm3.30Weassume\ntheQWshavebeengrownsymmetricallyandthereforeignore\nanyRashbacontribution.31\nThe resulting spin relaxationhas been worked out in detail\nby Kainz et al. in Ref. (9). For the experiment32we compare\nto,wefindthenon-degeneratelimittobeapplicableandhenc e\nusetherelaxationrateforspinorientedinthez-direction ,\n1\nτz=4\n/planckover2pi12τp(T)/bracketleftBigg\nγ2/angbracketleftk2\nz/angbracketright22m∗kBT\n/planckover2pi12kBT−γ2/angbracketleftk2\nz/angbracketright\n2/parenleftBig2m∗kBT\n/planckover2pi12/parenrightBig2\nj2+\nγ21+τ3/τ1\n16/parenleftBig2m∗kBT\n/planckover2pi12/parenrightBig3\nj3/bracketrightBigg\n(16)\nwherej2≈2 andj3≈6 depend on the type of scattering\nmechanism. We assume Type I scattering as defined in Ref.\n(9). The ratio τ3/τ1is unity for Type I scattering. τp(T)is\nthe momentum relaxation time which can be extracted from\nmobilitymeasurements.\nA more interesting case is that of (110) QWs where the\nspin-orbitHamiltonianis33\nHs.o.=−γσzkx/parenleftbig1\n2/angbracketleftk2\nz/angbracketright−1\n2(k2\nx−2k2\ny)/parenrightbig\n(17)\nwhichisobtainedfromthe(001)Hamiltonianbytransformin g\nthecoordinatesystemsuchthat x||[110],y||[001],andz||[110].\nAs can be seen from the form of this Hamiltonian, the ef-\nfective magnetic field is in the direction of the growth plane .\nHence,spinsorientedalongtheeffectivefieldwillexperie nce\nnospinrelaxation.\nConduction spins also relax due to the Elliott-Yafet (EY)\nmechanism34,35which arises from spin mixing in the wave-\nfunctions. Due to spin-orbit interaction, when a conductio n\nelectron is scattered by a spin-independent potential from\nstatektok′, the initial and final states are not eigenstates\nof the spin projection operator Szso the process relaxes the\nspin. In bulk, the relaxation rate is known to be of the form5\n1/τEY=αEYT2/τp(T)whereαEYisamaterial-dependentpa-\nrameterand τpisthemomentumrelaxationtime.36\nHowever the EY mechansim in quasi-two dimensions will\nnottakethesameformsince kwillbequantizedinonedirec-\ntion (the direction of confinement). The treatment in bulk37\nhasbeenextendedtoQWs toobtain38\n1\nτEY≈/parenleftBigΔs.o.\nΔs.o.+Eg/parenrightBig2/parenleftBig\n1−m∗\nm/parenrightBig2EckBT\nE2g1\nτp(T),(18)\nwhereΔs.o.isthespin-orbitsplittingenergyand EcistheQW\nconfinementenergy.\nSpins may also relax due to the Bir-Aronov-Pikus (BAP)\nmechanism39whicharisesfromthescatteringofelectronsand\nholes. This relaxation mechansim is commonly considered\nefficient only in p-type materials when the number of holes\nis large.40We fit the experimental data in Section VI without\nconsiderationofthismechanism.\nWewillnowexaminehowtheserelaxationmechanismsare\nmanifestintwodifferentQWs.\nVI. RESULTSFORGaAs/AlGaAs QUANTUMWELL\nWe apply our method to measured spin relaxation times of\ntwo GaAs/AlGaAs QWs by Ohno et al.:32,41(100) n-doped\nQW with doping nimp=4×1010cm−2, well width L=7.5\nnm;anda(110)undopedQWwithwellwidth L=7.5nm. In\nboth (pump-probe)experiments,the pump or photoexcitatio n\nenergywastunedtotheheavyholeexcitonresonanceandnor-\nmallyincidentonthesample. AsmentionedinSectionII,the\nexciton spin becomes important at low temperaturesfor such\nexcitation energies. The experimental spin relaxation tim es\nas a function of temperature are displayed (solid circles) i n\nFigures2and3.42\nFortheundoped(110)QW, Eq. (4)ismodifiedtobecome\n1\nτs=Nx−nx\nNx1\nτc+nx\nNx1\nτx. (19)\nFor this sample, at low temperatures, nx=Nxso theτs=\nτx≈0.15 ns. At higher temperatures,recombination(in time\nτr) and EY act to relax conduction spins since DP relax-\nation is significantly reduced for the (110) QW orientation.\nTo account for the quasi-two dimensional nature of the QW,\nwe use an intermediate value (between 2D and 3D values)\nfor the exciton’s binding energy.43Eq. (19) (solid line) fits\nthe data (points) with excellent agreement in Figure 2 when\nNx=1.5×1010cm−2andτr=2 ns which are near the ex-\nperimentallyreportedvalues19(Nx≈1010cm−2andτr≈1.6\nns). Thecontributionsfromtheexcitonsandconductionele c-\ntrons are also shown (dashed and dash-dotted lines respec-\ntively). The trend in the data is well described by our theory\n- at low temperatures excitons predominate and the spin re-\nlaxationtimeis τx. When thetemperatureincreases,the exci-\ntonsthermallyionizeleadingtonetmomentintheconductio n\nband. Sincetheconductionbandspinrelaxationtimeislong er\nthantheexcitonspinrelaxationtime,themeasuredrelaxat ion\ntime increaseswith temperatureas described in Eq. (19). We\nFIG. 2: Spin relaxation versus temperature in undoped (110) GaAs\nQW. Points are experiment of Ref. (41). Dash-dotted line: Us -\ning only conduction portion of Eq. (19) and 1 /τc=1/τEY+1/τr.\nDashed line: using only excitonic portion of Eq. (19). Solid line:\nEq. (19). Spin relaxation rate of excitons decreases with te mpera-\nture increase due to thermal ionization. Conduction spin re laxation\nis longer in (110) QW than in other oriented QWs due to vanishi ng\nDPmechanism.\nexpecttherelaxationtimesto eventuallyleveloutastheex ci-\ntons disappear. Eventually, the relaxation time will decre ase\nasthetemperaturedependenceofEYtakeseffect.\nForthedoped(100)QW,Eq. (4)shouldbeusedtodescribe\nthe temperature dependence of the relaxation rate. Using th e\nvaluesfromaboveand nimp=4×1010cm−2,τs=0.35ns,we\ncanextracttheapproximatevalueof τl. Indoingsoweobtain\nτl≈0.5 ns. We stress that this value has considerable uncer-\ntainty due to the uncertainty in the parameters (namely Nx)\nthat determine τl. The presence of impurities has lengthened\nthe observed low temperature spin relaxation time by more\nthana factorof two. The relaxationtime in thedopedsample\ncan be further increased by reducing the excitation density .\nAs the temperature is increased, donors become unoccupied\nand conduction electrons will play a larger role in relaxati on\nas expressed in Eq. (4). We can determine the main conduc-\ntion spin relaxation mechanism by investigating its temper a-\nturedependence.\nWearenowleftwiththetaskofdeterminingwhatthelocal-\nizedandconductionspin relaxationmechanismsare. We plot\nthe relaxationrate forthe n-dopedGaAs QWas a functionof\ntemperaturein Figure 3. The dashed, dotted, and dash-dotte d\nlinesrefertothe threetermsofEq. (4)- the densityweighte d\naverageoftherespectiverelaxationrates. Thesolidlinei sthe\nsumofall threeterms.\nWe begin by calculating spin relaxation due to spin ex-\nchangediffusioninEq. (14). Thisisdifficultduetotheexpo -\nnentialdependenceon ri,j. ForGaAs, α=0.06,Eg=1.52eV,\nm∗=0.067m, andaB=10.4 nm. To calculate /angbracketleftk2\nz/angbracketright=β2/L2,\nwe need to know the band offsets and assume a finite square\nwell. ThepotentialdepthforaAlGaAsQWisabout V0=0.23\neV. This comes fromΔEc\nΔEg=0.62 andΔEg=0.37 eV in\nGaAs.44From this we can determine βwhich will also de-6\nFIG. 3: Spin relaxation versus temperature in n-doped (100) GaAs\nQW. Points are from Ref. (32). Dashed line: excitonic contri bution\nin Eq. (4). Dotted line: localized contribution in Eq. (4). D ash-\ndotted line: conduction contributioninEq. (4). Solidblac kline: Eq.\n(4). Both exciton and localized spin relaxation contribute to the ob-\nserved low temperature spin relaxation. Conduction spin re laxation\nis the most strong contributor to the observed relaxation at higher\ntemperatures.\npend on the well width L. ForL=7.5 nm,β=2.19. Of\ncourse in the limit of V0→∞,β→π. What remains to be\ndeterminedis ri,jwhichisproportionaltotheinter-donorsep-\narationri,j=γn−1/2\nimp. For average inter-donor spacing in two\ndimensions, γav=0.564. Whenweallow γtobefittingparam-\neter,weobtain ri,j=19.5nmwhichcorrespondsto γ=0.4.\nWe now determine the relaxation rate due to the hyperfine\ninteraction. Since µBg∗//planckover2pi1/angbracketleftB2\nN/angbracketright1/2τcorr≫1 whennimp=4×\n1010cm−2, the hyperfinerelaxation is described by Eq. (15).\nSincenearlyallnucleihavethesamespin45(I=3/2),wecan\nexpressEq. (15)as\n1\nτnuc=2/radicalBigg\n5∑jA2\nj\n/planckover2pi12NL, (20)\nwith∑jA2\nj=1.2×10−3meV2andNL∼2.1×105.26This\nyieldsτnuc=3.9 ns. Due to the donor’s confinement in the\nQW, its wavefunctionmay shrink thereby reducing the local-\nizationvolumeandthereforealsoreducing NLandτnuc.43\nIn Figure 3, we find find excellent agreement with experi-\nment over a large temperature range when τp(T)in Eq. (16)\nis madea factorof threesmaller thanwhat is reportedin Ref.\n(9). We attain approximately the same quantitative accurac y\nas in Ref. (9) but since we also take into account the local-\nized spins, we find excellent qualitative agreementas well. It\nshould be emphasized that the quadratic and cubic terms of\nEq. (16) are important in the high temperature regime. The\nEY rate is qualitatively and quantitatively different from the\ndata. For instance, 1 /τEY≈0.1 ns−1at 300 K so we rule it\nout of contention. We also now ignore recombinationof car-\nrierssinceanappreciableamountofequilibriumcarrierse xist\n(n-dopedsystem) leading to recombinationof primarily non -\npolarizedspins.Onewouldnotexpecttheseresultstoagreewithspinrelax-\nation measurements in modulation doped QWs. In modula-\ntiondopedsystems, the occupationdensities nlandnccannot\nbecalculatedaswehavedonehere. Insuch systemsdifferent\nspinrelaxationdependenciesareseen.19,46\nVII. RESULTSFOR CdTe/CdMgTeQUANTUMWELL\nThe experiment by Tribollet et. al. on a n-CdTe QW of-\nfers an instructive complement to the previous experiments\non GaAs. In their experiment, Tribollet et al. measure spin\nrelaxation times τs≈20 ns for CdTe/CdMgTe QWs with\nnimp=1×1011cm−2. Importantly, they excited with laser\nenergies at the donor bound exciton frequency instead of the\nheavyholeexcitonfrequency.\nForCdTe, Eg=1.61eV,m∗=0.11m,andaB=5.3nm. The\nspin-orbitparameter, αisnotknownbutweapproximateitby\nnoting that the spin-orbit splitting energy in CdTe is Δs.o=\n0.927 eV whereas in GaAs, it is Δs.o=0.34 eV. Since αis\napproximately proportional to Δs.o, we obtain α=0.164 for\nCdTe.\nTo obtain potential well depth for CdTe QW, use\nEg(xMg) =1.61+1.76xMgwherexMggives fraction of Mg\nin Cd1−xMgxTe.47If we use xMg=0.1, we get V0=0.12eV\nwhichleadsto β=2.18.\nWe now determine the relaxation rate due to the hyperfine\ninteraction. Since all nuclei with non-zero spin will have t he\nsamespin45(I=1/2),wecanexpressEq. (15)as\n1\nτnuc=2/radicalBigg\n∑jA2\njPj\n/planckover2pi12NL, (21)\nwherePjhas been addended to account for isotopic\nabundances.1The natural abundancies of spin-1/2 Cd and Te\nnuclei dictate that PCd=0.25 andPTe=0.08. The remain-\ning isotopes are spin-0. NL=1.8×104,ACd=31µeV, and\nATe=45µeVwhichyields τnuc=4.4ns.1Theconfineddonor\nwavefunction in CdTe should shrink less than in GaAs since\ntheeffectiveBohrradiusishalfaslarge.\nWe see that this value is within an order of magnitude of\nwhat we have calculated for relaxation due to the hyperfine\ninteraction. We can also compare the experimental time to\nwhat we obtain for spin exchange diffusion. When we allow\nγto be a fitting parameter, we obtain ri,j=19.3 nm which\ncorrespondsto γ=0.61. Thisisinreasonableagreementwith\nγav.\nUnfortunately no relaxation measurements have been per-\nformedat highertemperaturesin n-dopedCdTe QWs that we\nareawareof. Wearealsonotawareofmobilitymeasurements\ninn-dopedCdTeQWs. Theprevalentmechansim(DPorEY)\nwilldependonthemobilitysoweforgodeterminingthemore\nefficient rate. However, in analogy to bulk systems, we ex-\npect the CdTe QW mobilities to be less than the GaAs QW\nmobilities.12,48InthenextsectionweanalyzeCdTe’sspinre-\nlaxationratefor(110)growncrystalso DP canbeignored.7\nVIII. COMPARISONOF GaAs ANDCdTe QUANTUM\nWELLS\nFirst we discuss the low temperature spin relaxation. In-\nterestingly, the localized relaxation time in CdTe is about 20\ntimes longerthanin GaAs. Thiscan be explainedby the spin\nexchangerelaxation despite the largerspin orbit paramete rin\nthe CdTe. This is more than offset by the smaller effective\nBohr radius in CdTe (5 .3 nm vs. 10 .4 nm) and the exponen-\ntial behavior of the anistropic exchange relaxation. Howev er\ndue to the exponential factor, any discrepancy between the\ntwo QWs can be explained by adjusting their respective γs\nappropriately, though the fitted γs do fall near γavg. The dis-\ncrepancyintimesisdifficulttoexplainbythehyperfineinte r-\naction since the two calculated relaxation times are very ne ar\neach other. Additionally, no plateau effect is seen that is i n-\ndicativeofhyperfinedephasing.1,27Anotherpossibilityisthat\none QW is governed by relaxation from spin exchange and\nthe other from hyperfine interactions. Without experimenta l\ndata,answeringthese questionsisdifficult. It isourhopet hat\nfurther experiments will be done to sort out these questions .\nHowever,wecanproposewaysinwhichtheseanswerscanbe\ndiscovered.\nRelaxationbyanisotropicspinexchangeisstronglydepen-\ndent on the impuritydensity. By alteringthe impuritydopin g\nwithin the well, one should see large changes in the spin re-\nlaxation time if this mechanism is dominant. From Eq. (14)\nwe see that this mechansim will also depend on the confine-\nment energy. Hence this mechanism should also be affected\nby changing the well width. The hyperfine dephasing mech-\nanismshouldbe largelyunaffectedbyimpurityconcentrati on\ndifferencesaslongastheyarenotsoextensiveastocauseth e\ncorrelation time to become very short and enter a motional\nnarrowing regime. Varying the well width will have an ef-\nfect on the donor wavefunctions, but as long as they are not\nsqueezedtoothintheeffectshouldnotbedramatic.\nFIG. 4: Spin relaxation in GaAs (100) QWs with different well\nwidths (all other parameters, including τlandτx, do not change).\nPoints are from Ref. (32) where L0=7.5 nm. Dotted: 2 L0; dash-\ndotted: 3L0/2; solid:L0;dashed: L0/2.\nFor spin relaxation at higher temperatures, DP prevails in(100)GaAs QWs as mentionedearlier. Whether DP or EY is\nmore efficient in CdTe depends on the momentum relaxation\ntime. By changes in momentum relaxation times (by chang-\ning well width or impurity concentration), we predict the th e\npossibility to induce a clear ‘dip’ in the temperature depen -\ndencewhichweseeinFigure4. Thissamenon-monotonicity\nhasbeenobservedbulkGaAsandZnO.12–15\nUsing our results we propose that n-doped (110) QWs\nshould optimize spin lifetimes (when excited at exciton-\nbound-donor frequency) since DP is suppressed. Figure 5\ndisplays our results for GaAs and CdTe (110) QWs as impu-\nritydensities nimp=4×1010cm−2andnimp=1×1011cm−2\nrespectively. The decrease seen in GaAs is now due to de-\npopulation of donor states instead of exciton thermalizati on.\nThe depopulationis muchslower in CdTe since the dopingis\nhigher. Theup-turnin the CdTecurveasroomtemperatureis\nreached is due to EY which is too weak to be seen in GaAs.\nWeplotthedatapointsfromtheundoped(110)GaAsQWfor\ncomparison. By avoiding the creation of excitons and their\nshortlifetimes,longspinrelaxationtimescanbeachieved .\nFIG. 5: Spin relaxation in (110) GaAs ( nimp=4×1010cm−2):\ndashed-dotted line. Spin relaxation in (110) CdTe ( nimp=1×\n1011cm−2): solid line. Points from undoped (110) GaAs QW\nexperiment41are included for comparison. For both systems, τp(T)\nfrom Ref. (9) were used. EY istoo weakover the temperature ra nge\ndepictedtobe seenintheGaAs system. However EYisthe cause of\nthe increase inspinrelaxation ratefor the CdTe system.\nIX. CONCLUSIONS\nWe find that the spin relaxation times in n-dopedQWs can\nbewelldescribedbyatheoryinvokingspinexchangebetween\nspin species. Inundoped(110)QWs, whereDP is absent,we\nfind that exciton spin relaxation is important and leads to th e\nobservedsurprisingtemperaturedependence. We predict th at\na similar temperature dependence (though with longer relax -\nation times) should be observedin n-doped(110)QWs when\nexcitedat theexciton-bound-donorfrequency.\nWe have suggested future experimental work to resolve\nwhat mechanisms relax spin localized on donors in n-doped8\nGaAs and CdTe QWs. 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Rev. 129, 2471\n(1963)."    },    {        "title": "0811.0716v1.Multiple_Quantum_NMR_Dynamics_in_Pseudopure_States.pdf",        "content": "arXiv:0811.0716v1  [cond-mat.other]  5 Nov 2008Multiple Quantum NMR Dynamics in Pseudopure States\nG. B. Furman\nDepartment of Physics, Ben Gurion University, Beer Sheva 84 105, Israel and\nOhalo College, Qazrin, 12900, Israel\nAbstract\nWe investigate numerically the Multiple Quantum (MQ) NMR dy namics in systems of nuclear\nspins 1/2 coupled by the dipole-dipole interactions in the c ase of the pseudopure initial state.\nSimulations of the MQ NMR with the real molecular structures such as six dipolar-coupled proton\nspinsof a benzene, hydroxyl proton chains in calcium hydrox yapatite and fluorinechains in calcium\nfluorapatite open the way to experimental NMR testing of the o btained results. It was found that\nmultiple-spin correlations are created faster in such expe riments than in the usual MQ NMR\nexperiments and can be used for the investigation of many-sp in dynamics of nuclear spins in solids\n1I. INTRODUCTION\nMethod of multipe-quantum nuclear magnetic resonance (MQ NMR) in solids [1] is a\npowerful tool for the investigations of structure and spin dynam ics in solids [2]. Significant\nprogress has been made in the simplification of ordinary NMR spectra [3]. Recently, higher-\norder MQ NMR experiments have been utilized to address one of the f undamental issues in\nquantum information processing, the preparation of the pseudop ure state [4]. A quantum\ncomputation includes as an initial step preparation of the input stat e [5]. Conveniently, the\nquantum algorithms start with a pure ground state where populatio ns of all states except\nthe ground state are equal to zero. The realization of a pure stat e in a real quantum system,\nsuch as a spin system, requires extremely low temperatures and ve ry high magnetic fields.\nTo overcome this problem, a so-called “pseudopure” state was intr oduced [6, 7]. The density\nmatrix ofthe spin system inthis state canbe partitionedinto two par ts. The first part of the\nmatrix is a scaled unit matrix, and the second part corresponds to a pure state. The scaled\nunit matrix does not contribute to observables, and it is not change d by unitary evolution\ntransformations. Therefore, the behavior of a system in the pse udopure state is exactly the\nsame as it would be in the pure state.\nRecently, it has been suggested [8, 9] to consider the dipolar order ed state as the initial\nstate for such experiments. As a result of using the dipolar ordere d initial condition, many-\nspin clusters andcorrelationsappear faster [8, 9] thanintheordin aryMQ NMR experiments\n[1] in solids and some peculiarities of multiple-quantum (MQ) dynamics ca n be investigated\nwith these experiments.\nIn the present article we consider MQ NMR dynamics when the initial co ndition is de-\ntermined by the pseudopure state. Our motivation of performanc e of this work is defined\nfirst of all by that the many-spin correlations are created faster in such experiments then in\nstandard MQ NMR methods and can be used for the investigation of m any-spin dynamics\nof nuclear spins in solids. We consider dynamics of MQ NMR in systems pr epared in the\npseudopure state and calculate the resulting signal in MQ experimen ts. Note, that it is not\nnecessary to make any changes in the scheme of the standard exp eriment in order to obtain\nnon-zero signals of MQ coherences in the pseudopure state. Comp uter simulations of such\nexperiments for six spin ring and eight spin linear chain are presented .\n2II. MQ NMR WITH THE INITIAL PSEUDOPURE STATE\nWe consider a system of nuclear spins ( s= 1/2) coupled by the dipole-dipole interaction\n(DDI) in a strong external magnetic field− →H0which is directed along the axis z. The secular\npart of the DDI Hamiltonian [10] has the following form\nHdz=/summationdisplay\nj<kDjk/bracketleftbigg\nIjzIkz−1\n4/parenleftbig\nI+\njI−\nk+I−\njI+\nk/parenrightbig/bracketrightbigg\n, (1)\nwhereDjk=γ2ℏ\nr3\njk(1−3cos2θjk) is the coupling constant between spins jandk,γis the\ngyromagnetic ratio, rjkis the distance between spins jandk,θjkis the angle between the\ninternuclear vector− →rjkand the external magnetic field. Ijαis the projection of the angular\nspin momentum operator on the axis α(α=x,y,z);I+\njandI−\njare the raising and lowering\noperators of spin j.\nThere are many pulse sequences exciting MQ coherences during the preparation period.\nFor a dipolar-coupled spin system, the multipulse sequence with an eig ht-pulse cycle [1]\nis known to be very efficient. The basic scheme of the standard MQ NM R experiments\nconsists of four distinct periods of time: preparation, evolution, m ixing and detection [1].\nIn the rotating reference frame [10] the average Hamiltonian desc ribing the MQ dynamics\nat the preparation period can be presented as [1]\nHMQ=H(2)+H(−2), (2)\nwhereH(±2)=−1\n4/summationtext\nj<kDjkI±\njI±\nk. Then the evolution period without any pulses follows.\nThe density matrix of the spin system, ρ(τ), at the end of the preparation period is\nρ(τ) =U(τ)ρ(0)U+(τ), (3)\nwhereU(τ) = exp(−iτ(H(2)+H(−2))) is the evolution operator on the preparation period\nandρ(0) is the initial density matrix of the system. The transfer of the in formation about\nMQ coherences to the observable magnetization occurs during the mixing period. Usually\nthe thermodynamical equilibrium density matrix is used as the initial on e for MQ NMR\nexperiments. In the high temperature approximation the equilibrium density matrix takes\na form:ρ(0) =Iz. Here we consider MQ NMR dynamics with the initial pseudopure state\nwhen the density matrix can be described as:\nρ(0) =|1/an}bracketri}ht1⊗|1/an}bracketri}ht2⊗....⊗|1/an}bracketri}htN, (4)\n3where|1/an}bracketri}htkrepresents a k−th spin that is up and Nis the number of spins in the system.\nThe method of creating the highly polarized spin states (4) in cluster s of coupled spins\nwas described previously [4, 11]. It is based on filtering multiple-quant um coherence of the\nhighest order, followed by a time-reversal period and partial satu ration. It is convenient\nto expand the density matrix, ρ(τ), at the end of the preparation period of the MQ NMR\nexperiment as [12]\nρ(τ) =/summationdisplay\nnρn(τ), (5)\nwhere the term ρn(τ) is responsible for the MQ coherence of the n-th order. One can find\ne−iδIztρn(τ)eiδIzt=e−inδtρn(τ). (6)\nOn the mixing period the spin system is irradiated with sequence of the pulses shifted on\naπ/2−phase regarding the pulses of the preparation period [1]. As a resu lt the average\nHamiltonian describing evolution of the spin system on the mixing period changes a sign on\nopposite to the sign of the Hamiltonian (2), and the evolution operat or,U(τ),is replaced by\nthe operator U+(τ).Starting with the initial conditions (4), the density matrix, ρ(t), after\nthe three periods of the standard MQ NMR experiment can be writte n as\nρ(t) =U+(τ)e−iδt1Izρ(τ)eiδt1IzU(τ), (7)\nwhereρ(τ) is the density matrix at the end of the preparation period accordin g to Eq.\n(3) andt= 2τ+t1,δis the frequency offset on the evolution period of the duration t1\nwhich is a result of applying the time proportional phase incrementat ion (TPPI) method\n[1]. The transfer of the information about MQ coherences to the ob servable magnetization\noccurs during the mixing period. The last unitary transformation in ( 7) with operator\nU(τ) describes this period. The resulting signal after the mixing period, the longitudinal\nmagnetization, Mz(t) , is\nMz(t) =Tr{ρ(t)Iz} (8)\nUsing Eqs. (3) and (7) and the initial condition (4) it is convenient to p resent the formula\n(8) for the longitudinal magnetization, Mz(t),as follows\nMz(t) =Tr/braceleftbig\ne−iδIztρ(τ)eiδIztρMQ(τ)/bracerightbig\n(9)\nwhere\nρMQ(τ) =U(τ)IzU+(τ), (10)\n4coincides with the density matrix at the end of the preparation perio d of the standard MQ\nNMR experiment with the thermodynamical equilibrium density matrix a s the initial condi-\ntion at high temperature approximation [1]. The density matrix ρMQ(τ) can be represented\nin the following form [12]\nρMQ(τ) =/summationdisplay\nnρMQ\nn(τ) (11)\nwhere the term ρMQ\nn(τ) is responsible for the MQ coherence of the n-th order and satisfies\nto the relationship of Eq. (6). By using Eqs. (6), (7) and (9) one ca n rewrite the expression\nfor the observable signal in terms of the intensities of MQ coherenc es\nMz(t) =/summationdisplay\nne−inδtJn(τ). (12)\nwhereJn(τ) are the normalized intensities of MQ coherences are\nJn(τ) = 2Tr/braceleftBig\nρn(τ)ρMQ\n−n(τ)/bracerightBig\n/N. (13)\nOne can find from Eq. (13) that\nJn(τ) =J−n(τ). (14)\nIt is well known that in the usual MQ NMR experiments, the sum of the intensities of all\nMQ coherences does not depend on time/summationtext\nnJn(τ) = 1. This is also right for the MQ NMR\nin the pseudopure state.\nIII. THE NUMERICAL ANALYSIS OF THE TIME EVOLUTION OF MQ CO-\nHERENCES IN THE PSEUDOPURE STATE\nWe restrict ourselves to numerical simulations of MQ NMR dynamics of one-dimensional\nsystems. For example, quasi-one-dimensional six dipolar-coupled p roton spins of a benzene\nmolecule [4] and hydroxyl proton chains in calcium hydroxyapatite Ca5(OH)(PO4)3[13]\nand fluorine chains in calcium fluorapatite Ca5F(PO4)3[13] are suitable objects to study\nMQ dynamics in pseudopure states. The numerical calculations were performed for MQ\nNMR dynamics of linear chains and rings of 6 and 8 spins. The DDI couplin g constant of\nthe nearest neighbors is chosen to be Dj,j+1=D= 1s−1. Then the coupling constants of\nspinsjandkareD/bracketleftBig\nsin(π/N)\nsin(π(j−k)/N)/bracketrightBig3\nfor ring and D/|j−k|3for chain, respectively. In order to\ncompare the results of the numerical simulations with the analogous ones for the ordinary\n5MQ NMR dynamics we introduce the normalized intensities of MQ cohere nces for the n\norder,Jord\nn(τ) [1]\nJord\nn(τ) =Tr/braceleftBig\nρMQ\nn(τ)ρMQ\n−n(τ)/bracerightBig\n/Tr/braceleftbig\nI2\nz/bracerightbig\n. (15)\nThe dependences of the intensities of MQ coherences on the dimens ionless time, t=Dτ,\nof the preparation period in spin ring containing six spins are present ed in Fig. 1 for the\nboth initial states, equilibrium (a) and pseudopure (b). One can com pare the intensities of\nMQ coherences in the suggested experiment with the standard one s. It is evident that the\nsuggested method can be considered as a useful addition to the st andard MQ NMR method.\nFig. 1 demonstrates that the suggested method yields the intensit ies of MQ coherences of\nthe fourth and six orders which several times higher than the analo gous coherence in the\nstandard MQ NMR experiment. It is clear from the insets of Fig. 1 tha t MQ coherence\nof the fourth and sixth orders, obtained with the initial condition (4 ), appear little earlier\nthan in the usual MQ NMR in a ring of six spins. For example, or the pseu dopure state at\ntimeDτ= 2 the MQ intensity of the fourth order coherence is 0 .05 (red dash line) while\nfor the usually used initial condition up to Dτ= 3 the intensity J4is closed to zero (black\nsolid line). A tendency of the faster growth of MQ coherences of hig h orders takes place also\nfor the linear chain containing eight spins (Fig. 2). This peculiarity is co nnected with the\ninitial pseudopure state (4). As a result, many-spin clusters and c orrelations connected with\nMQ coherences appear faster than in the standard MQ NMR with the initial condition,\nρ(0) =Iz. The numerical calculations confirm also the results obtained in the p revious\nsection. In particular, the growth of MQ coherences occurs in acc ordance with Eq. (14).\nOne can see from Fig. 2 that the observable intensities of the fourt h and sixth orders\nin the linear chain containing eight spins can be negative in contrast to the ordinary MQ\nNMR experiments at high temperatures [1, 12, 14]. At the same time, it was shown [15]\nthat the intensities of MQ coherences can be negative in the standa rd NMR experiments\nat low temperatures and in the dipolar ordered state at high temper atures. In fact, in MQ\nNMR experiments the observable quantity is the longitudinal magnet ization modulated by\nrf pulses. The distinct frequency components of the magnetizatio n can have an arbitrary\nsign.\n6IV. CONCLUSIONS\nThe MQ NMR method for the detection of MQ coherences starting fr om the pseudopure\nstate is proposed. Investigations of MQ NMR dynamics in the pseudo pure states can be\nconsidered as a supplementary method which complements the usua l NMR in order to study\nstructures and dynamical processes in solids. Many-spin clusters and many-spin correlations\narecreatedfasterinsuchexperimentsthanintheusualMQNMRwit htheinitialequilibrium\ncondition without any correlation between the spins. In this paper w e focused on simple\none-dimensional examples but the physical picture obtained here is not limited to perform\nsimulations and experiments in two-dimensional and three dimensiona l systems and open\nnew ways for the study of many-spin systems.\nAcknowledgements\nAuthor would like to thank Ben Gurion University of the Negev for sup porting this work.\n[1] J. Baum, M. Munovitz, A. N. Garroway, A. Pines, J. Chem. Ph ys.83, 2015 (1985).\n[2] J. Baum, K. K. Gleason, A. Pines, J. Chem. Phys., 56, 1377 (1986).\n[3] W. S. Warren, D. P. Weitekamp, A. Pines, J. Chem. Phys., 73, 2084 (1980).\n[4] J.S. Lee and A.K. Khitrin. Phys. Rev. A 70, 022330 (2004).\n[5] G. Benenti, G. Casati, and G. Strini, Principles of Quant um Computation and Information,\nVol. I and II (World Scientific, 2007).\n[6] D. G. Cory, A. F. Fahmy, and T. F. Havel, Proc. Natl. Acad. S ci. U.S.A. 94, 1634 1997 .\n[7] N. Gershenfeld and I. L. Chuang, Science 275, 350 1997 .\n[8] G. B. Furman, S. D. Goren, J. Phys.: Condens. Matter, 17, 4501 (2005).\n[9] S. I. Doronin, E. B. Fel’dman, E. I. Kuznetsova, G. B. Furm an, S. D. Goren, Phys. Rev. B,\n76, 144405 (2007).\n[10] M.Goldman. Spin Temperature and Nuclear Magnetic Resonance in Solids , Oxford.Clarendon\nPress. 1970.\n[11] G. B. Furman, J. Phys. A 39, 15197 (2006) .\n[12] E. B. Fel’dman, S. Lacelle, J. Chem. Phys., 107, 7067 (1997).\n7[13] G. Cho, J. P. Yesinowski, J. Phys. Chem., 100, 15716 (1996).\n[14] S. I. Doronin, I. I. Maximov, E. B. Fel’dman, Zh. Eksp. Te or. Fiz., 118, 687 (2000), JETP,\n91, 597 (2000).\n[15] E. B. Fel’dman, I. I. Maximov, J. Magn. Reson., 157, 106 (2002).\nCaption figures\nFig. 1 (Color online) The time dependence of the intensities of the MQ c oherences in\na ring of six spins coupled by the DDI: (a) termal equilibrium initial stat e and (b) the\npseudopure initial state (4). Black solid line: intensities of the zerot h order coherence J0.\nRed dashline: intensities of thesecond order coherence J2. Green dot line: intensities of the\nfourth order coherence J4. Blue dash-dot line: intensities of the sixth order coherence J6.\nThe insets show that the MQ coherences of fourth (a) and sixth (b ) orders in the pseudopure\nstate (red dash line) appear little earlier than in the usual MQ NMR (bla ck solid line).\nFig. 2 (Color online) The time dependence of the intensities of the MQ c oherences in a\nlinear chain of eight spins coupled by the DDI: (a) termal equilibrium init ial state and (b)\nthe pseudopure initial state (4). Black solid line : intensities of the ze roth order coherence J0\n. Red dash line: intensities of the second order coherence J2. Green dot line: intensities of\nthe fourth order coherence J4. Blue dash-dot line: intensities of the sixth order coherence\nJ6. The insets show that the MQ coherences of fourth (a) and sixth ( b) orders in the\npseudopure state (red dash line) appear little earlier than in the usu al MQ NMR (black\nsolid line).\n8/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53/s48/s46/s51/s48/s48/s46/s51/s53\n/s68/s74\n/s32/s54\n/s40/s98/s41/s74\n/s110\n/s68/s40/s97/s41\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53\n/s74\n/s32/s52\n/s68\n/s68/s74\n/s110/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s45/s48/s46/s48/s48/s50/s48/s46/s48/s48/s48/s48/s46/s48/s48/s50/s48/s46/s48/s48/s52/s48/s46/s48/s48/s54\n/s68/s74\n/s54/s74\n/s110\n/s68/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52\n/s68/s74\n/s52/s74\n/s110"    },    {        "title": "1408.0930v2.Dynamics_of_a_Mn_spin_coupled_to_a_single_hole_confined_in_a_quantum_dot.pdf",        "content": "Dynamics of a Mn spin coupled to a single hole con\fned in a quantum dot\nB. Varghese,1, 2H. Boukari,1, 2and L. Besombes1, 2,\u0003\n1CNRS, Institut N\u0013 eel, F-38042 Grenoble, France.\n2Universit\u0013 e Grenoble Alpes, Institut N\u0013 eel, F-38042 Grenoble, France\n(Dated: May 24, 2022)\nUsing the emission of the positively charged exciton as a probe, we analyze the dynamics of the\noptical pumping and the dynamics of the relaxation of a Mn spin exchange-coupled with a con\fned\nhole spin in a II-VI semiconductor quantum dot. The hole-Mn spin can be e\u000eciently initialized in a\nfew tens of nsunder optical injection of spin polarized carriers. We show that this optical pumping\nprocess and its dynamics are controlled by electron-Mn \rip-\rops within the positively charged\nexciton-Mn complex. The pumping mechanism and its magnetic \feld dependence are theoretically\ndescribed by a model including the dynamics of the electron-Mn complex in the excited state and\nthe dynamics of the hole-Mn complex in the ground state of the positively charged quantum dot. We\nmeasure at zero magnetic \feld a spin relaxation time of the hole-Mn spin in the \u0016srange or shorter.\nThis hole-Mn spin relaxation is induced by the presence of valence band mixing in self-assembled\nquantum dots.\nI. INTRODUCTION\nIn ferromagnets currently used in data storage devices,\nthe magnetic anisotropy determines the stability of the\norientation of the magnetization at a given temperature\nand eventually in the presence of an external applied\nmagnetic \feld. This magnetic anisotropy is an intrin-\nsic property of the magnetic material which depends on\nthe local environment of the magnetic atoms. In the\nlast ten years, experimental breakthroughs have laid the\nfoundations for atomic scale data storage, showing the\ncapability to read and manipulate the spin of a single\nmagnetic atom. In particular, it has been shown that the\nisotropic magnetic moment of a free atom can develop a\nmagnetic anisotropy energy when it interacts with the\nordered surface of a metal1,2. An individual magnetic\natom can also present a large magnetic anisotropy when\ninteracting with ligands in a molecular magnet3. In semi-\nconductors, it has been shown that the optical properties\nof a quantum dot (QD) can be used to control the spin\nstate of individual4{6or pairs7,8of magnetic atoms. In-\ncluding a magnetic atom in a QD o\u000bers additional degree\nof freedom. One can, for example, tune the environment\nof the localized spin by controlling the charge state of the\nQD. This electrical control can in\ruence the magnetic\nanisotropy of the atom making these nano-sized systems\nattractive for basic investigations as well as for miniatur-\nized data-storage applications.\nIn II-VI semiconductors, a Mn atom incorporated in\na neutral self-assembled QD presents a small magnetic\nanisotropy of a few tens of \u0016eV resulting from a strained\ninduced modi\fcation of the local crystal \feld9,10. This\nmagnetic anisotropy is responsible for the Mn spin mem-\nory of a few \u0016s observed at zero magnetic \feld11,12. It\nhas been shown recently that the \fne structure splitting\nof the spin of a Mn can be modi\fed optically14,15or de-\ncreased signi\fcantly in strain-free QDs16. It is also ex-\npected that a Mn atom could develop a large anisotropy\nenergy in the meV range when it is exchange-coupled\nwith a single con\fned heavy-hole spin13. The two lowenergy hole-Mn states, with total spin \u0005 z=\u00061, should\nbehave like an Ising spin system forming an atomic fer-\nromagnet similar to what was observed for instance for\nsingle Co atoms deposited on a Cu surface2. In addition,\nin a semiconductor QD, the exchange induced Mn spin\nmagnetic anisotropy could be electrically controlled by\nchanging the Mn / heavy-hole overlap with an applied\ngate voltage.\nIn this paper, we analyze the dynamics of an individual\nMn spin exchange coupled with a con\fned hole spin in\nCdTe/ZnTe self-assembled QDs. We \frst show that the\ncomplex formed by the Mn spin and the resident hole\nspin can be initialized within a few tens of nanoseconds\nunder the optical injection of spin polarized carriers. We\ndemonstrate that the spin pumping of the hole-Mn com-\nplex is controlled by the electron-Mn \rip-\rops induced by\ntheir exchange coupling within the positively charged ex-\nciton. This initialization process is modelled considering\nthe exchange induced coherent dynamics of the electron-\nMn complex in the excited state of the charged QD (pos-\nitively charged exciton) and the dynamics of the hole-Mn\ncomplex in the ground state of the QD.\nThe transverse magnetic \feld dependence of the opti-\ncal pumping signal reveals a magnetic anisotropy of the\npositively charged exciton-Mn complex. We show that\nthis anisotropy is controlled by the strain induced \fne\nstructure of the Mn atom and a residual exchange cou-\npling of the two holes with the Mn spin.\nUsing this optical pumping technique, we measure, at\nzero magnetic \feld and at T=7K, a spin relaxation time\nof the hole-Mn complex of about 1 \u0016s. This is shorter\nthan for a Mn spin in a neutral QD. We show that the\nsource of this shortening of the spin life-time is the va-\nlence band mixing always present in strain induced II-VI\nself-assembled QDs.\nThe rest of this paper is organized as follows. In Sec.\nII we show how to extract all the parameters controlling\nthe spin structure of positively charged Mn-doped QDs.\nIn Sec. III we demonstrate that the optical injection of\nspin polarized carriers can be used to initialize the spin ofarXiv:1408.0930v2  [cond-mat.mes-hall]  29 Aug 20142\nthe resident hole-Mn complex. In Sec. IV we describe the\nmechanism which is responsible for this e\u000ecient optical\npumping. In Sec. V, we analyze the dynamics of the\noptical pumping and its excitation power and transverse\nmagnetic \feld dependence. In Sec. VI, we present a\nmodel for the spin dynamics of positively charged QDs\nunder optical injection of spin polarized carriers. Finally\nin Sec. VII, we present spin relaxation measurements\nand discuss the stability of the hole-Mn spin memory.\nII. POSITIVELY CHARGED MN-DOPED\nQUANTUM DOTS.\nSingly Mn-doped Cd(Mn)Te/ZnTe QDs containing a\nresident hole coming from a background p-type doping of\nthe ZnTe barriers and ZnTe surface states can be isolated\nunder quasi-resonant optical excitation below the ZnTe\nbarrier energy. As presented in Fig. 1(a), the probability\nto \fnd an individual con\fned holes in these QDs can be\nincreased by applying a positive voltage between a back\ncontact on the p-ZnTe substrate and a 6 nm thick semi-\ntransparent gold Schottky gate deposited on the surface\nof the sample.\nPositively charged magnetic QDs can be identi\fed by\nanalysing the structure of their emission spectra. The\nemitting state in the X+-Mn transition has two valence\nband holes and one electron coupled to the Mn. In a \frst\napproximation the e\u000bect of the two spin-paired holes on\nthe Mn is zero. Thereby, the spin structure of the X+-Mn\nstate is governed by the ferromagnetic coupling between\nthe electron and the Mn. The spin Hamiltonian of this\nsystem reads:\nHeMn=IeMn~S\u0001~ \u001b (1)\nwithIeMn the exchange integral between the Mn and\nthe electron with respective spins\u0000 !Sand\u0000 !\u001b. The twelve\neigenstates of the electron-Mn complex are split into a\nground state septuplet (total spin M= 3) and a \fve-\nfold degenerate manifold with M= 2 (see the inset of\nFig. 1(b)). We label them all as jM;Mzi.\nThe recombination of one of the hole with the elec-\ntron of the X+-Mn leaves a \fnal state with a single hole\ncoupled to the Mn. The hole-Mn exchange interaction is\ndescribed by the spin Hamiltonian:\nHhMn =IhMn~S\u0001~J (2)\nwhere IhMn is the exchange integral between the hole and\nthe Mn atom. The hole spin operators, ~J, represented in\nthe basis of the two low energy heavy-hole states, are re-\nlated to the Pauli matrices \u001cbyJz=3\n2\u001czandJ\u0006=\u0018\u001c\u0006\nwith\u0018=\u00002p\n3e\u00002i\u0012\u001as=\u0001lh.\u001asis the coupling energy\nbetween heavy-holes and light-holes split by the energy\n\u0001lhand\u0012the angle relative to the [110] axis of the prin-\ncipal axis of the anisotropy responsible for the valence\nband mixing18,19.\n PL. Intensity\n2020 2015 2010 2005 2000 1995\nEnergy (meV)\n300\n200\n100\n0Polarization angle (°)\n2004.0 2003.0 2002.0 2001.0\nEnergy (meV)200420022000\nEnergy (meV) πx\n πye Mn \nh MnM=2\nM=3\n \n+5/2-5/2\nJz=+3/2 Jz=-3/2\n-1.0 0.0 1.0\nEnergy (meV)-2-1012\nEnergy (meV)  πx\n  πy-4-202Bias voltage (V)\n2014 2012 2010 2008 2006 2004 2002 2000 1998\nEnergy (meV)\nExp.\nTh.(a)\n(b)\n(c)\nExp. Th.QD1σ+ σ−X-Mn\nX+-MnX--Mn\nSz\n...Sz\n-5/2+5/2\n...FIG. 1: (a) Color-scale plot of the PL intensity of a Mn-doped\nQD (QD1) in a Schottky structure as a function of emission\nenergy and bias voltage for a CW excitation at \u0015exc= 594nm.\nThe series of emission lines can be assigned to the recombi-\nnation of the neutral exciton (X-Mn), positively charged ex-\nciton (X+-Mn) and negatively charged exciton (X\u0000-Mn). (b)\nPL under co-circularly polarized excitation and detection of a\npositively charged Mn-doped QD. The left inset shows the ex-\nperimental and calculated linearly polarized spectra along two\northogonal directions. The right inset is a simpli\fed scheme\nof the energy levels of the ground (h-Mn) and excited states\n(e-Mn) of a positively charged Mn-doped QD. For h-Mn, the\ndotted arrows show the states coupled two by two by the va-\nlence band mixing. The states j\u00001=2;\"eiandj+ 1=2;#ei\n(green arrow) are mixed and split. (c) Experimental (left)\nand calculated (right) color-scale plot of the dependence of\nthe PL of X+-Mn on the direction of a linear analyzer. Four\nlines in the center of the structure are linearly polarized. The\nparameters used in the model are listed in table I.\nIf we neglect the valence band mixing ( \u001as=\u0001lh=0), the\ntwelve eigenstates of HhMn are organized as six doublets\nwith well de\fned SzandJz(Mn and hole spin projection\nalong the z axis). We label these states as jSz;Jzi. For\neach level of X+-Mn with either M= 2 orM= 3,\nthere are six possible \fnal states after annihilation of3\nan electron-hole pair. From this consideration alone, we\nwould expect twelve spectrally resolved lines. As the Mn\nspin is not a\u000bected by the optical transition, the weight\nof each PL line is given by optical and spin conservation\nrules.\nWe consider, for instance, \u001b+ recombination where the\nj#e;*hie-h pair is annihilated. After the e-h annihila-\ntion, the resulting state is jSz;+hiwhich is an eigenstate\nofHh\u0000Mn. The intensity of the optical transition to a\ngiven \fnal state jSz;+hiis proportional to the overlap\nhM;MzjSz;#ei, which is nothing but a Clebsh-Gordan\ncoe\u000ecient which appears in the composition of a spin 1/2\nwith a spin 5/2. The lowest energy transition, with \u001b+\npolarization, would correspond to the high energy \fnal\nstatej\u00005=2;+hiand the initial state j*h;+hi\u0002j\u0000 5=2;#ei\nwhich is identical to j3;\u00003iand thereby gives the high-\nest optical weight. In contrast, transition to that \fnal\nstate fromj2;Mziis forbidden. The other \fve doublets\nhave optical weights lying between 1/6 and 5/6 with both\nj2;Mziandj3;Mziinitial states. The number of spec-\ntrally resolved lines in this model is 11 and the PL of X+-\nMn can then be seen as a superposition of two substruc-\ntures: six lines with intensities decreasing with increasing\ntheir energy position (transitions from M=3 states) and\n\fve lines with intensities increasing with increasing their\nenergy position (transitions from M=2 states).\nAs presented in Fig. 1(b) and 1(c), the central lines\nofX+-Mn are linearly polarized. This results from spin-\n\rip interaction between the Mn and the hole induced by\nthe presence of valence band mixing17. Provided that\n\u001as=\u0001lh<<1, the e\u000bect of this interaction is small both\non the wave function and on the degeneracy of all the\nhole-Mn doublets except the third (green arrow in the\ninset of Fig. 1(b)) which is split. The split states are the\nbonding and antibonding combinations of j\u00001=2;*hi\nandj+ 1=2;+hi. These states are coupled, via linearly\npolarized photons, to the j2;0iandj3;0ielectron-Mn\nstates and four linearly polarized lines are observed on\nthe emission spectra as shown in Fig. 1(c). Polarization\ndirections are controlled by the QD anisotropy (strain\nand shape) responsible for the valence band mixing19.\nTo describe in detail the emission spectra, we have also\nto take into account the perturbation of the wave function\nof the charged exciton in the initial state and of the hole\nin the \fnal state by the hole-Mn exchange interaction.\nThis perturbation depends on the value of the exchange\nenergy between the Mn spin S zand the hole spin J zand\nit can be represented, using second order perturbation\ntheory, by an e\u000bective spin Hamiltonian16,20,21\nHscat=\u0000\u0011S2\nz (3)\nwith\u0011 > 0. The perturbation of the wave function is\ntwice for the positively charged exciton where two holes\ninteracts with the Mn. This perturbation is responsible\nfor the irregular energy spacing of the X-Mn and X 2-Mn\nlines reported in previous work on self-assembled20,21and\nstrain-free Mn-doped QDs16.\nWe can obtain numerical values of IhMn,IeMn,\u001as=\u0001lhTABLE I: Values of the parameters used for the modelling of\nthe \fve QDs discussed in this paper. I eMn, IhMn and\u0011are\nexpressed in \u0016eV. For all the QDs, we use \u0012=0. The width\nat half maximum of the depolarization curves obtained in\nthe time resolved optical pumping measurements for X+-Mn\n(BX+\n1=2) and X-Mn ( BX\n1=2) is expressed in Tesla. The relaxation\ntime of the hole-Mn spin, \u001cr, is in\u0016s.\nQD1 QD2 QD3 QD4 QD5\nIeMn\u000085\u0000100\u0000140\u000085\u0000150\nIhMn 220 230 340 195 285\n\u0011 23 17 25 25 23\n\u001as=\u0001lh0:06 0:15 0:07 0:1 0:17\nBX+\n1=20:11 0:07 0:10 0:13 0:12\nBX\n1=20:07\u0000 0:06\u0000 \u0000\n\u001cr\u0000 1\u0000 0:9 0:7\nand\u0011comparing the transition probabilities calculated\nwith the model to the experimental data (Fig. 1(c)). Val-\nues obtained for the \fve QDs discussed in this paper are\nlisted in table I. A valence band mixing is observed in all\nthe QDs but the value of the coe\u000ecient \u001as=\u0001lhshows\nthat the hole-Mn exchange interaction remains highly\nanisotropic. These values will be used in the model of\nthe spin dynamics in the positively charged Mn-doped\nQDs.\nIII. EVIDENCE OF THE HOLE-MN OPTICAL\nPUMPING.\nSimilarly to what has been proposed22and recently\ndemonstrated for an individual magnetic atom in neutral\nQDs11,12,23,24, we show here that the optical injection of\nspin polarized carriers can be used to initialize the spin\nstate of the hole-Mn complex.\nThe emission spectra of a Mn-doped QD under circu-\nlarly polarized continuous wave (CW) excitation below\nthe ZnTe barrier25is presented in Fig. 2(a) for three dif-\nferent charge states of the dot ( i.e.di\u000berent gate volt-\nages). For a positively charged or a neutral QD the emis-\nsion is strongly co-circularly polarized with the excita-\ntion. In contrast, the emission of the negatively charged\nQD is almost unpolarized.\nThe polarization of the negatively charged QD is con-\ntrolled by the spin of the injected hole. The weak circular\npolarization of X\u0000-Mn suggests that the hole spin is not\nconserved during the energy relaxation of the optically\ninjected e-h pair or completely lost during the lifetime\nof the charged exciton. The strong co-circular polariza-\ntion of X-Mn and X+-Mn shows, on the other hand, that\nspin polarized electrons are e\u000eciently injected in the QD\nand conserved during the lifetime of the exciton or the\npositively charged exciton.\nAs shown on the top inset of Fig. 2(a), the intensity\ndistribution on the di\u000berent lines on the emission of X+-4\n2014 2012 2010 2008 2006 2004 2002 2000 1998\nEnergy (meV)QD1\n σco\n σcross\n \nV=-4.25VPL Intensity (arb units)QD1\n σco\n σcross\n \nV=-2.25VQD1\n σco\n σcross\n \nV=0V\n20042002 2000\nEnergy (meV)0.8\n0.6\n0.4\n0.2(ILE-IHE)/(ILE+IHE)\n0.001246\n0.01246\n0.1246\n1\nPump Power (arb. unit)200420022000\nEnergy (meV)2004 2003 2002 2001 2000\nEnergy (meV) σext./ σdet.\n  πext./ σdet.\n LE\nHE\n0.250.200.150.100.050.00\nBx (T) X+Mn (V=0V)\n X Mn (V=-2.25V)2004 2002 2000\nEnergy (meV) Bx=0T\n Bx=0.23T(a)\n(b) (c)X-MnX+-Mn\nX--Mn\nFIG. 2: (a) Co and cross-circularly polarized PL spectra of\na Mn-doped QD (QD1) inserted in a Schottky structure for\nthree di\u000berent gate voltages. The top inset shows the cir-\ncularly polarized PL spectra obtained under circular (black)\nand linear (blue) excitation. (b) Excitation power depen-\ndence of the ratio of intensity of the high and low energy\nlines (IHE\u0000ILE)=(IHE+ILE) of X+-Mn under co-circularly\npolarized CW excitation and detection. (c) Transverse mag-\nnetic \feld dependence of the ratio of intensity of the high and\nlow energy lines for X-Mn and X+-Mn. The inset shows the\nco-circularly polarized spectra of X+-Mn with and without\ntransverse magnetic \feld.\nMn strongly depends on the polarization of the excita-\ntion. For a linearly polarized excitation and a circular\ndetection, the PL intensity re\rects the oscillator strength\nof the transitions. For a circularly polarized excitation,\nmost of the PL intensity is co-polarized and emitted on\nthe low energy line of the X+-Mn spectra. This highly\nout of equilibrium distribution of the PL intensity is a\nsignature of the optical pumping of the hole-Mn com-\nplex.The presence of optical pumping of the hole-Mn spin\nis con\frmed by the excitation power dependence and the\ntransverse magnetic \feld dependence of the intensity dis-\ntribution in the positively charged exciton emission spec-\ntra. The normalized intensity ratio of the high energy\n(HE) and low energy (LE) lines of X+-Mn under circu-\nlarly polarized excitation (I LE-IHE)/(ILE+IHE) is pre-\nsented in Fig. 2(b) as a function of the excitation power\nand in Fig. 2(c) as a function of a transverse magnetic\n\feld. The increase of the intensity ratio with the excita-\ntion power con\frms the presence of a cumulative pump-\ning process which is improved by the reduction of the\ndelay between the successive injection of spin polarized\ncarriers. On the other hand, the signi\fcant reduction of\nthis intensity ratio in a weak transverse magnetic \feld is\na signature of the precession of the spins involved in an\noptical pumping process.\nWith these optical pumping conditions, most of the\nlight is emitted from the low energy line of X+-Mn. The\nlow energy PL line of X+-Mn corresponds to a transition\nfrom the electron-Mn state j3; +3i=j+ 5=2;\"eito the\nhole-Mn statej+5=2;*hiin\u001b\u0000polarization and from the\nelectron-Mn state j3;\u00003i=j\u00005=2;#eito the hole-Mn\nstatej\u00005=2;+hiin\u001b+ polarization. Consequently, the\nPL intensity distribution observed experimentally shows\nthat just after the optical recombination, the hole-Mn\ncomplex is prepared in the spin state j+ 5=2;*hiunder\n\u001b\u0000excitation and in the state j\u00005=2;+hiunder\u001b+\nexcitation.\nAs illustrated in Fig. 1(b), the states j+ 5=2;*hiand\nj\u00005=2;+hiare the high energy states of the hole-Mn\ncomplex: an evolution of the system towards the ground\nstates,j\u00005=2;*hiandj+ 5=2;+hi, will take place. A\nhole spin relaxation time of about 4 nswas observed in\nCdTe/ZnTe QDs at zero magnetic \feld using as a probe\nthe time evolution of the polarization rate of the neg-\natively charged exciton26. The hole spin relaxation can\neven be faster in the presence of an external applied mag-\nnetic \feld: the interaction with acoustic-phonons is en-\nhanced when the spin states of the hole are split by a few\nhundreds\u0016eV27{30. Similarly, the hole spin split by the\nexchange interaction with the Mn is expected to relax\ntowards the low energy states in a nstime scale. On the\nother hand, a Mn spin relaxation of a few \u0016sis observed\nfor a Mn spin at zero magnetic \feld11and this relaxation\ntime is even longer under magnetic \feld24. We can then\nexpect that the Mn spin is conserved during the hole\nspin relaxation. After this relaxation, the hole-Mn spin\nis prepared in one of the two low energy states, either\nj+ 5=2;+hifor a\u001b\u0000excitation orj\u00005=2;*hifor a\u001b+\nexcitation.\nIV. MECHANISM OF THE HOLE-MN\nOPTICAL PUMPING.\nA scheme of the optical pumping process that we pro-\npose for the initialization of the hole-Mn system is pre-5\nsented in Fig. 3 for a \u001b+ excitation. We consider \frst\nthat the hole-Mn complex is initially in one of the two\nground states, either j+5=2;+hiorj\u00005=2;*hi, after ther-\nmalisation in the dark. This hypothesis is made for clar-\nity of the demonstration: starting from any other hole-\nMn spin state would lead to the same conclusion. It is as-\nsumed that, after the injection of excitons by the optical\nexcitation, the spin of the hole can \rip but the spin of the\nelectron is conserved. This is consistent with the polari-\nsation rates observed in Fig. 2(a). Consequently, for a \u001b+\noptical excitation, either bright excitons j+1i=j*h;#ei\nor dark onesj\u00002i=j+h;#eican be captured by the\nQD.\nLet's consider \frst the situation where the hole-Mn\ncomplex is initially in the state j+ 5=2;+hi. In this situ-\nation, only a bright exciton can be captured by the dot to\nform the positively charged exciton j+5=2;+hi\u0002j*h;#ei.\nThe isotropic coupling between the electron and the Mn\ninduces electron-Mn spin \rip-\rops. After one \rip-\rop,\nthe spin up electron can recombine with the resident spin\ndown hole emitting a \u001b\u0000photon. The hole-Mn complex\nis left in the state j+ 3=2;*hi. This high energy hole-Mn\nstate can relax within a few hundreds psto the lower\nenergy statej+ 3=2;+hithough a hole spin-\rip induced\nby acoustic phonons. Then, a second bright exciton can\nbe injected in the dot containing a spin down hole. An\nelectron-Mn \rip-\rop followed by the recombination of\na\u001b\u0000exciton leave the hole-Mn complex in the state\nj+1=2;*hi. Alternatively, a dark exiton can be captured\nbefore the hole spin relaxation. An electron-Mn \rip-\rop\nand an exciton recombination in \u001b\u0000polarization will also\nleave the hole-Mn spin in the state j+ 1=2;*hi. Repeat-\ning \fve times this process of injection of spin polarized\nelectron (eitherj+ 1iorj\u00002iexciton) with electron-Mn\n\rip-\rops will \fnally drive the hole-Mn system from the\ninitial statej+ 5=2;+hito the statej\u00005=2;*hi.\nWhen the hole-Mn complex is initially in the state\nj\u00005=2;*hi, the QD can only capture a dark exciton. The\npositively charged exciton is in the state j\u00005=2;*hi\u0002j+h\n;#ei. The electron-Mn \rip-\rops are blocked because\nboth the electron and Mn spin zcomponents are min-\nimum. The injected spin down electron can recombine\nwith the resident spin up hole emitting a co-circularly\npolarized\u001b+ photon on the low energy line of X+-Mn.\nAfter recombination, the hole-Mn system is in the state\nj\u00005=2;+hi. After a fast hole spin relaxation, the hole-\nMn complex returns to the state j\u00005=2;*hi. Under CW\n\u001b+ excitation, this process of capture of a dark exciton\nfollowed by an optical recombination and a spin \rip of\nthe hole is repeated, and the hole-Mn complex remains\nblocked in the state j\u00005=2;*hi.\nThe sequence presented here for a \u001b+ excitation leads\nto the preparation of the hole-Mn complex in the state\nj\u00005=2;*hi. Therefore, as observed in the experiment,\nunder\u001b+ excitation the positively charged QD emits pre-\ndominantly co-circularly polarized \u001b+ photons on the low\nenergy line of X+-Mn.\nA similar scheme can be obtained under \u001b\u0000excitation.\nFIG. 3: Scheme of the optical pumping mechanism of the spin\nof the hole-Mn complex under \u001b+ excitation. The electron-\nMn \rip-\rops controlled by He\u0000Mnand the electron-Mn coher-\nence timeTeMn\n2 prepare the Mn spin in the state Sz=\u00005=2.\nSee the text for a detailed description of the pumping process.\nIn this con\fguration, after the pumping cycle induced by\nthe injection of bright and dark excitons with j\"eielec-\ntrons, the hole-Mn spin is blocked in the state j+5=2;+hi\nand the QD emits co-circularly polarized \u001b\u0000photons on\nthe low energy line of X+-Mn.\nThe proposed optical pumping scheme is compatible\nwith all the experimental observations under CW circu-\nlarly polarized excitation. It suggests that the optical\npumping of the hole-Mn spin is induced by the exchange\ninteraction of the Mn spin with the injected spin po-\nlarized electrons. Its dynamics should be controlled by\nHeMn and the generation rate of excitons.\nV. DYNAMICS OF THE HOLE-MN OPTICAL\nPUMPING.\nTo extract the dynamics of this optical pumping pro-\ncess, we modulate the circular polarisation of the CW ex-\ncitation with and electro-optic modulator combined with\na quarter-wave plate (circular polarization switching time\nof about 10 ns) and analyze the time evolution of the in-\ntensity distribution on the X+-Mn spectra.6\nThe time evolution of the circularly polarized PL of\neach lines of a X+-Mn that can be spectrally resolved\nare presented in Fig. 4(b). In the co-circular polarization\ncon\fguration for the excitation and the detection we ob-\nserve two PL intensity transients: \frst an abrupt one\nlimited by the 10 ns time resolution of the polarization\nswitching. It re\rects the population change of the spin-\npolarized exciton. Then, a slower transient is observed\nwith opposite signs on the two extreme PL lines (lines\n1 and 8 which correspond to \fnal states with parallel or\nanti-parallel hole and Mn spins). The characteristic time\nof this exponetial transient in the tens of nsrange is, at\nlow excitation power, inversely proportional to the pump\nintensity and then saturates (Fig. 4(c)). This is a sig-\nnature of the optical pumping process which realizes an\norientation of the hole-Mn spin.\nAs presented in Fig. 4(d), an in-plane magnetic \feld,\nBx, of a few tens of milli-Tesla induces a decrease of\nthe e\u000eciency of the optical pumping as measured on the\namplitude of the pumping PL transient of the high en-\nergy line (see inset of Fig. 4(a)). This results is at \frst\nsight surprising as one could expect a large magnetic\nanisotropy for the hole-Mn system controlled by the large\nanisotropy of the hole gfactor.\nUnder CW excitation, the QD is occupied a signi\f-\ncant part of the time by a positively charged exciton.\nWithin the X+-Mn complex, the coupling of the Mn spin\nwith the carriers is dominated by the exchange interac-\ntion with the electron which results in an isotropic hybrid\nspin system. In the absence of magnetic anisotropy of the\nMn spin, this hybrid spin could freely precess in a trans-\nverse weak magnetic \feld suppressing any possibility of\nspin transfer from the injected spin polarized electron to\nthe hole-Mn system. However, it is known from previous\nstudies of spin dynamics in Mn-doped neutral QDs that\nthe ground state of the Mn presents a \fne structure. In\nepitaxial structures, this \fne structure is dominated by\na magnetic anisotropy with an easy axis along the QD\naxis due to built-in strains in the QD plane. The \fne\nstructure splitting of the Mn is described by the spin\nHamiltonian:\nHMn=D0S2\nz+E[S2\nx\u0000S2\ny] +1\n6a[S4\nx+S4\ny+S4\nz] (4)\nwhereD0is proportional to the biaxial strain, Ede-\nscribes an anisotropy of the strain in the QD plane and\nais the intrinsic splitting induced by the tetrahedral en-\nvironment of the Mn in the crystal ( a= 0:32\u0016eV for a\nMn in CdTe). The magnetic anisotropy term D0is re-\nsponsible for the transverse magnetic \feld dependence\nof the optical pumping of the Mn observed in a neutral\nQD11.D0can vary from 0 \u0016eV for a strain free QD16to\nabout 12\u0016eV for a fully strained CdTe layer matched on\na ZnTe substrate11. This magnetic anisotropy term in\ru-\nences the electron-Mn spin dynamics in a weak transverse\nmagnetic \feld.\nWithin the positively charged exciton, the weak in-\nteraction of the Mn with the two holes, described by\nPL Intensity\n1986 1984 1982 1980 1978 1976 1974 1972\nEnergy (meV) (1)\n (2)\n (3) (4)\n (5) (6) (8)\n (7)QD2\n σco\n σcross2000\n1000\n0PL Intensity (Cts)\n4003002001000\nTime delay (ns)σCoσCross\n∆Ι\nΙ\n200150100500\nTime delay (ns)Bx=0T\nBx=0.3T\nBx=0.045T\nBx=0.07T\nBx=0.1T\nBx=0.12T\nBx=0.15T\nBx=0.18T\nBx=0.23T2.5\n2.0\n1.5\n1.0\n0.5\n0.0∆Ι/Ι\n0.200.100.00\nBx (T)PL Int. (arb. unit)\n4003002001000\nTime delay (ns) (1)\n (2)\n (3)\n (4)\n (5)\n (6)\n (7)\n (8)\n4003002001000\nTime delay (ns)P=0.15mWP=0.25mWP=0.5mWP=1mWP=1.5mWP=2mWP=2.5mW60\n40\n20\n0τpump(ns)\n2.01.00.0\nPump Power (mW)(a)\n(b) (c) (d)FIG. 4: (a) Co and cross-circularly polarized PL of X+-Mn\nin QD2. The inset shows the time evolution of the PL of the\nhight energy line (8) under polarization switching of the ex-\ncitation. (b) Time evolution of the circularly polarized PL of\nthe di\u000berent lines of X+-Mn (labelled from (1) to (8)) under\npolarization switching of the excitation. (c) Excitation power\ndependence of the PL transient measured on the high energy\nline (8). The inset shows the power dependence of the pump-\ning time. (d) Transverse magnetic \feld dependence of the PL\ntransient measured on the high energy line of X+-Mn. The\ninset shows the magnetic \feld dependence of the amplitude\nof the optical pumping transient \u0001 I=I.\nHscat, also introduces a magnetic anisotropy. The in-\n\ruence of this parameter is revealed by the comparison\nof the transverse magnetic \feld dependence of the opti-\ncal pumping e\u000eciency in neutral and positively charged\nQDs. The transverse magnetic \feld dependence of the\npumping transient detected on the high energy line of X-\nMn and on the high energy line of X+-Mn for the same\nQD are presented in Fig. 5(b). The half width at half\nmaximum of the depolarization curve for X+-Mn (BX+\n1=2)\nis signi\fcantly larger than for X-Mn ( BX\n1=2). The values\nofBX+\n1=2andBX\n1=2measured for the di\u000berent QDs studied\nin thispaper are listed in Table I. Whereas the depolariza-\ntion curve of X-Mn is only controlled by D011, the total\nanisotropy of the electron-Mn system induced by both7\nPL Intensity\n1980 1975 1970 1965 1960 1955\nEnergy (meV)X-MnQD3\n σco\n σcross\n \nV=-2.5VPL IntensityX+-Mn\nQD3\n σco\n σcross\n \nV=+5V\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0Norm ∆Ι/Ι\n0.250.200.150.100.050.00\nBx (T)QD3\n X -Mn\n X+-Mn\nB½XB½X+1.0\n0.8\n0.6\n0.4\n0.2\n0.0Norm ∆Ι/Ι\n0.250.200.150.100.050.00\nBx (T) QD1\n QD2\n QD3\n QD4\n QD5(a)\n(b) (c)\nFIG. 5: (a) Co and cross-circularly polarized PL of X+-Mn\nand X-Mn in a Mn-doped QD (QD3). (b) Transverse mag-\nnetic \feld dependence of the amplitude of the optical pumping\ntransients measured on the high energy line of X-Mn and on\nthe high energy line of X+-Mn in QD3. (c) Transverse mag-\nnetic \feld dependence of the amplitude of the optical pump-\ning transients measured on the high energy line of X+-Mn\ncompared for \fve di\u000berent QDs from QD1 to QD5. The cor-\nresponding half width at half maximum of the depolarization\ncurves are listed in table I.\nD0and\u0011is responsible for the transverse magnetic \feld\ndependence of the optical pumping signal of X+-Mn.\nVI. MODEL OF THE HOLE-MN OPTICAL\nPUMPING.\nWe propose a model to better understand this optical\npumping mechanism and its transverse magnetic \feld de-\npendence. It is based on the calculation of the time evolu-\ntion of the populations of the twelve X+-Mn states in the\nexcited state of the QD and the twelve hole-Mn states in\nthe ground state. In this model, we neglect the hyper\fne\ncoupling between the electronic and nuclear spin (I=5/2)\nof the Mn atom which is, in the investigated QDs, weaker\nthan the electron-Mn and hole-Mn exchange interactions.\nWe use the density matrix formalism (24 x 24 density\nmatrix\u001a) to calculate the population in this multi-level\nsystem. The master equation which governs the evolu-\ntion of\u001acan be written in the Lindblad form as:@\u001a\n@t=\u0000i=\u0016h[H;\u001a] +L\u001a (5)\nwhereHis the Hamiltonian of the complete system\n(X+-Mn and hole-Mn) and L\u001adescribes the coupling\nor decay channels resulting from an interaction with the\nenvironment31,32. The coherent evolution of this multi-\nlevel system is controlled by the Hamiltonian HMn+\nHeMn+ 2HscatforX+\u0000MnandHMn+HhMn+Hscat\nfor the hole-Mn complex. For the magnetic \feld depen-\ndence, Zeeman terms for the hole, the electron and the\nMn are also included.\nIn agreement with the polarized PL measurements pre-\nsented in Fig.2(a), we consider that we generate spin po-\nlarized electrons and random holes ( i.e.a\u001b+ excitation\ngeneratesj+ 1iexcitons orj\u00002iexcitons with the same\nprobability). The only recombination channel for the op-\ntically created X+\u0000Mn is a radiative recombination\nwith a characteristic lifetime \u001cr= 0:3nswhich conserves\nthe Mn spin.\nWe take into account a spin relaxation time of the Mn\nin the exchange \feld of the hole, \u001cMn, describing relax-\nation channels with a change of the Mn spin by one unit.\nWe choose \u001cMn= 5\u0016swhich is the typical value of re-\nlaxation time for a Mn spin in an empty QD at zero\nmagnetic \feld and we neglect here the possible decrease\nof this relaxation time under the injection of free carrier\nin the vicinity of the QD33. We include a relaxation of\nthe hole spin coupled with the Mn, \u001ch, in thensrange.\nThe transition rates \u0000 \r!\r0between the di\u000berent states of\nthe hole-Mn complex depend on their energy separation\nE\r\r0=E\r0\u0000E\r. Here we use \u0000 \r!\r0=1/\u001ciifE\r\r0<0\nand \u0000\r!\r0=1/\u001cie\u0000E\r\r0=kBTifE\r\r0>0 (\u001cicorresponding\neither to\u001cMnor\u001ch)22. This describes a thermalization\namong the twelve hole-Mn levels. A pure dephasing time\nfor the hole-Mn system ( ThMn\n2) and for the electron-Mn\nsystem (TeMn\n2) are also included in the model. The inco-\nherent optical excitation, the optical recombination, the\nrelaxation and pure dephasing terms are inserted in L\u001a\nfollowing the method presented in reference15.\nWith this level scheme, we can calculate the popula-\ntion of each X+-Mn state ( i.eelectron-Mn states jSz;\u001bzi)\nunder CW or alternate circularly polarized excitation.\nThese populations are proportional to the intensity of\nthe X+-Mn PL lines. For instance, in \u001b+ polarization,\nthe low energy line corresponds to the transition from the\nlow energy electron-Mn state j3;\u00003i=1p\n6[p\n6j\u00005=2;#ei]\nto the high energy hole-Mn state j\u00005=2;+hiand the high\nenergy line corresponds to the transition from the high\nenergy levelj2; +2i=1p\n6[p\n1j+ 3=2;\"ei\u0000p\n5j+ 5=2;#ei]\nto the low energy level j+ 5=2;+hi. The normalized in-\ntensity ratio of the low and high energy lines, which mea-\nsures the di\u000berence of population of the Mn spin states\nSz=\u00005=2 andSz= +5=2, is then given by:\nILE\u0000IHE\nILE+IHE=\u001aj\u00005=2;#ei\u00005=6\u001aj+5=2;#ei\n\u001aj\u00005=2;#ei+ 5=6\u001aj+5=2;#ei(6)8\nρii\n6040200\nTime delay (ns)Bx=0T\nBx=0.05T\nBx=0.1T\nBx=0.15T\nBx=0.2T\nBx=0.25T\nBx=0.3T|+5/2;-1/2>ρii\n150100500\nTime delay (ns)|-5/2;-1/2>\n|-3/2;-1/2>\n|-1/2;-1/2>\n|+1/2;-1/2>\n|+3/2;-1/2>\n|+5/2;-1/2>\nσ+ σ− σ−015\n10\n5\n0∆Ι/Ι\n0.250.200.150.100.050.00\nBx(T)(c)\n(d)1.0\n0.8\n0.6\n0.4\n0.2\n0.0(ILE-IHE)/(ILE+IHE)\n0.001246\n0.01246\n0.1246\n1\ng (ns-1)1.0\n0.8\n0.6\n0.4\n0.2\n0.0ρii|-5/2>\n|+5/2>|+1/2>\n|-1/2>|+3/2>\n|-3/2>(a)\n(b)\nFIG. 6: Optical pumping signal calculated with the parame-\nters of QD2 (see table I). (a) Calculated intensity ratio of the\nhigh and low energy lines of X+-Mn and population of the Mn\nspin states presented as a function of the generation rate of\nexcitons under \u001b+ excitation ( g=gj+1i=gj\u00002i= 1=\u001cg). (b)\nTime evolution of the population of the di\u000berent electron-Mn\nstates under excitation with alternate circular polarization\n(\u001cg= 4ns). The curves are shifted for clarity. (c) Trans-\nverse magnetic \feld dependence of the pumping transient ob-\nserved on the high energy line ( \u001cg= 4ns). For (a), (b) and\n(c) the parameters used in the calculation are \u001cMn= 5\u0016s,\n\u001ch= 1ns,\u001cr= 0:3ns,ThMn\n2 = 0:5ns,TeMn\n2 = 0:1ns, T=10K,\na= 0:32\u0016eV, D 0=6\u0016eV, E=0.5\u0016eV,gMn= 2,gh= 0:6\nandge=\u00000:4. (d) Amplitude of the optical pumping tran-\nsient as a function of transverse magnetic \feld calculated with\nthe previous parameters (black), with D 0=0\u0016eV, E=0\u0016eV,\n\u0011= 0\u0016eV (red), with TeMn\n2 =1(i.e. lifetime limited co-\nherence of e-Mn) (blue) and with TeMn\n2 =1,ThMn\n2 = 1ns\n(dotted blue).\nThe calculated excitation power dependence of the in-\ntensity ratio of the high and low energy lines is presented\nin Fig. 6(a) for the exchange parameters of QD2 (see ta-\nble I) and a CW \u001b+ excitation. As observed in the exper-\niment (Fig. 2(b)), a variation of three orders of magnitude\nof the exciton generation rate is necessary to probe all the\nrange of variation of the PL intensity ratio. At high exci-tation intensity, the population of the di\u000berent Mn spin\nstatesjSzideduced from the model con\frms that the Mn\nspin is highly polarized with, for a \u001b+ excitation, most of\nthe population in the spin state Sz=\u00005=2 (Fig. 6(a)).\nFig. 6(b) presents the calculated time evolution of\nthe population of the electron-Mn states giving rise to\na\u001b+ emission (spin states jSz;#ei) under CW excita-\ntion with alternate circular polarization. The intensity\nof the low and high energy lines of X+-Mn are given by\n\u001aj\u00005=2;\"eiand 5=6\u001aj+5=2;#eirespectively. The main fea-\nture of the time resolved optical pumping experiments\n(see Fig. 4(b)) are well reproduced by the model. For\ninstance, under \u001b+ excitation and \u001b+ detection, a large\nincrease of the intensity of the low energy line is obtained\ntogether with a decrease of the intensity of the high en-\nergy line. This corresponds to the progressive transfer\nof population from Sz= +5=2 toSz=\u00005=2 by the\npumping process induced by the optical injection of j#ei\nelectrons and electron-Mn \rip-\rops controlled by HeMn\nandTeMn\n2.\nThe in\ruence of a transverse magnetic \feld, Bx, on\nthe optical pumping transient can also be qualitatively\ndescribed by this model. A transverse magnetic \feld de-\npendence of the optical pumping transient calculated for\nthe high energy line of X+-Mn is presented in Fig. 6(c).\nAs in the experiment, a signi\fcant decrease of the ampli-\ntude of the pumping transient is observed for a transverse\n\feld of a few tens of mT.\nCalculated depolarization curves ( i.etransverse mag-\nnetic \feld dependence of the parameter \u0001 I=Ide\fned in\nFig. 4(a)) are presented in Fig. 6(d) for di\u000berent values\nof the parameters D0,\u0011,ThMn\n2 andTeMn\n2. In the ab-\nsence of magnetic anisotropy ( D0=0 and\u0011=0, red line\nin Fig. 6(d)), the transverse magnetic \feld dependence is\ndetermined by the precession of the electron and Mn spin\ninterrupted by a dephasing process (controlled by TeMn\n2):\na transverse \feld of 20 mTcompletely destroys the op-\ntical pumping. In the presence of D0and\u0011(black line\nin Fig. 6(d)) the precession is blocked in a weak trans-\nverse magnetic \feld and the optical pumping can take\nplace. A transverse \feld of 200 mTis now required to\ncompletely suppress the optical pumping. D0and\u0011, that\nboth introduce an anisotropy in the electron-Mn system,\nappear as the main parameters that control the depo-\nlarization curve of X+-Mn. The in\ruence of these two\nindependent parameters that changes from dot to dot is\nresponsible for the dispersion of BX+\n1=2measured in di\u000ber-\nent dots (Fig. 5(c)). The in\ruence of \u0011on the magnetic\nanisotropy of X+\u0000Mn also explains the di\u000berence of\nmagnetic anisotropy measured for X+\u0000MnandX\u0000Mn,\nthe latter being only in\ruenced by D0(Fig. 5(b)).\nThis modelling also shows that the amplitude of the\npumping transient and its transverse magnetic \feld de-\npendence are in\ruenced by the value of the coherence\ntime of the electron-Mn and hole-Mn systems (Fig. 6(d)).\nThese dephasing times are not directly accessible from\nour measurements. Despite the determination of \u0011from\na modelling of the PL of X+\u0000Mn, we cannot inde-9\npendently determine D0from the depolarization curves.\nNevertheless, the shape of the depolarization curve of\nQD2 (Black curve in Fig. 4(d)) can be well reproduced\nby the model with reasonable order of magnitudes for\nall these parameters: \u0011= 17\u0016eV (obtained from a mod-\nelling of the PL of X+\u0000Mnin QD2),ThMn\n2 = 0:5ns,\nTeMn\n2 = 0:1nsandD0= 6\u0016eV (partial relaxation of the\nstrain at the Mn location).\nVII. HOLE-MN SPIN RELAXATION.\nHaving established a method to initialize the spin of\nthe hole-Mn complex and identi\fed the mechanism re-\nsponsible for the optical pumping, we performed pump-\nprobe experiments to observe how the hole-Mn spin is\nconserved in the absence of optical excitation. In this\nexperiment (Fig. 7(a)), the QD is excited with trains of\n300 ns circularly polarized pulses separated by a variable\ndark time ( \u001cdark). At the end of the excitation pulse, an\nout of equilibrium population for the hole-Mn system is\nprepared by optical pumping.\nWe measure after the time \u001cdark the amplitude of the\noptical pumping transient observed at the beginning of\nthe excitation pulse. The presence of this pumping tran-\nsient re\rects a partial relaxation of the hole-Mn polar-\nization during the dark time. The amplitude of the tran-\nsient saturates as \u001cdarkincreases, demonstrating the full\nrelaxation of the hole-Mn spin. An exponential \ft of the\nvariation of this amplitude with \u001cdarkis used to estimate\nthe relaxation time of the hole-Mn spin at zero magnetic\n\feld and T=7K. A relaxation time of about 0.7 \u0016sis\nobtained for the example presented in Fig. 7(a) (QD5).\nWe always found a relaxation time in the 1 \u0016srange or\nshorter for all the studied positively charged QDs (see\ntable I). This relaxation time for the Mn coupled with a\ncon\fned hole spin is shorter than the typical relaxation\ntime observed at zero magnetic \feld for a Mn atom in a\nneutral QD (about 5 \u0016s)11.\nThe exchange interaction of the Mn with a heavy-hole\nspin creates a large magnetic anisotropy for the Mn and\none could expect to increase the Mn spin memory at zero\nmagnetic \feld. However, in the presence of valence band\nmixing,Hh\u0000Mncouples two by two the di\u000berent hole-Mn\nlevels (inset of Fig. 1(b)). This coupling induces a trans-\nfer of population between the di\u000berent hole-Mn levels.\nThe transfer of population becomes irreversible in the\npresence of dephasing in the hole-Mn system and could\nsigni\fcantly contribute to the hole-Mn spin relaxation.\nTo analyze more in detail this possible spin relaxation\nprocess, we calculate the time evolution in the dark of\nthe population of the spin states of the Mn initially pre-\npared by optical pumping under CW circularly polarized\nexcitation. The time evolution of the population of the\nstate Sz=-5/2 prepared by a \u001b+ excitation is presented in\nFig. 7(b). Without valence band mixing, \u001aj\u00005=2idecays\nwith the Mn spin relaxation time \u001cMn= 5\u0016s. A decay\nof\u001aj\u00005=2iin a few hundreds of nsis obtained with a\n3.0\n2.5\n2.0\n1.5\n1.0\n0.5\n0.0Norm. PL Int.\n-4000 -3000 -2000 -1000 0\nTime delay (ns)τdark=8µsτdark\nσco\n∆I\nQD5I1.2\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0∆I/I\n8000 6000 4000 2000 0\nτdark(ns)τrelax=0.7µs\nT=7K\nB=0T\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0ρ|−5/2>\n400 300 200 100 0\nTime delay (ns)σ+\nρs/∆lh=0.0; T2hMn=1ns\nρs/∆lh=0.05; T2hMn=1ns\nρs/∆lh=0.05; T2hMn=0.5nsρs/∆lh=0.1; T2hMn=1ns(a)\n(b)FIG. 7: (a) Optical pumping transient obtained on the high\nenergy line of X+-Mn in QD5 under excitation with circularly\npolarized light trains of 300 ns separated by a dark time of\n\u001cdark=8\u0016s (the excitation sequence is displayed on the top).\nThe inset shows the evolution of the amplitude of the opti-\ncal pumping transient with \u001cdarkfor B = 0T and T=7K. (b)\nCalculated time evolution in the dark of the population of the\nMn spin state Sz=\u00005=2 initially pumped by a \u001b+ CW exci-\ntation (excitation sequence displayed at the top). The calcu-\nlation are performed with the exchange couplings parameters\nof QD5 (see table I), \u001cMn= 5\u0016s,\u001ch= 1ns,\u001cr= 0:3ns,\nTeMn\n2 = 0:1ns, T=10K,a= 0:32\u0016eV, D0=6\u0016eV, E=0.5\u0016eV\nand di\u000berent amplitude of valence band mixing \u001as=\u0001lhand\nhole-Mn coherence time ThMn\n2.\nsmall valence band mixing term \u001as=\u0001lh= 0:05. As illus-\ntrated in Fig. 7(b), this population decay is also slightly\nin\ruenced by the coherence time of the hole-Mn system\nThMn\n2.\nUsing in this model the valence band mixing parame-\nter deduced from the PL spectra, the hole-Mn spin relax-\nation rate seems to be overestimated. However, a quan-\ntitative description is di\u000ecult with this model because\nof the simpli\fed description of the coherent dynamics of\nthe hole-Mn system and the uncertainty in the parame-\nterThMn\n2. In addition, we have to notice that even if we\nchoose a gate voltage and excitation conditions for which\nwe only observe the positively charged exciton in the PL10\nspectra, we cannot exclude to have \ructuation of the hole\nin the dark. An empty dot during a fraction of the dark\ntime would arti\fcially increase the observed relaxation\ntime. Further investigations are required to understand\nthe details of this dynamics.\nVIII. CONCLUSION\nTo conclude, we have shown that we could store a sin-\ngle hole in a singly Mn-doped II-VI semiconductor QD.\nThe spin of the hole-Mn system can be e\u000eciently ini-\ntialized by optical pumping under quasi-resonant excita-\ntion with circularly polarized light. We demonstrated\nthat the optical pumping of the hole-Mn system and\nits transverse magnetic \feld dependence are controlled\nby the electron-Mn exchange coupling within the posi-\ntively charged exciton-Mn complex. Despite the mag-\nnetic anisotropy of the hole-Mn system, the relaxation\ntime of the hole-Mn spin is shorter than a micro-second.This fast spin relaxation is a consequence of the valence\nband mixing always present in II-VI self-assembled QDs.\nIn di\u000berent QD systems with a larger hole con\fnement\nand weaker valence band mixing one could expect a spin\nmemory controlled by the electrically tunable magnetic\nanisotropy of the hole-Mn complex. Nevertheless, a re-\nlaxation time in the micro-second range is su\u000ecient to ex-\nploit these positively charged Mn-doped QDs for an opti-\ncal ultrafast coherent control of the Mn spin34,35. Pulsed\nresonant excitation could be used to create and annihi-\nlate the positively charged exciton and deterministically\ncontrol the electron-Mn \rip-\rops to manipulate the Mn\nspin.\nAcknowledgments\nThis work was realized in the framework of the CEA\n(INAC) / CNRS (Institut N\u0013 eel) joint research team\n\"NanoPhysique et SemiConducteurs\".\n\u0003Electronic address: lucien.besombes@grenoble.cnrs.fr\n1P. Gambardella, S. Rusponi, M. Veronese, S.S. Dhesi, C.\nGrazioli, A. Dallmeyer, I. Cabria, R. Zeller, P.H. Dederich,\nK. Kern, C. Carbone, H. Brune, Science 300, 1130 (2003).\n2J. C. Oberg, M. Reyes Calvo, F. Delgado, M. Moro-\nLagares, D. Serrate, D. Jacob, J. Fernandez-Rossier, C.\nF. Hirjibehedin, Nature Nano. 9, 64 (2014) and references\ntherein.\n3L. Bogani, W. Wernsdorfer, Nat. Mater. 7, 179 (2008) and\nreferences therein.\n4L. Besombes, Y. Leger, L. Maingault, D. Ferrand, H. Ma-\nriette and J. Cibert, Phys. Rev. Lett. 93, 207403 (2004).\n5A. Kudelski, A. Lemaitre, A. 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Aguado,\nPhys. Rev. B 78125324 (2008).\n34D.E. Reiter, T. Kuhn, V.M. Axt, Phys. Rev. B 85, 045308\n(2012).35D.E. Reiter, V.M. Axt, T. Kuhn, Phys. Rev. B 87, 115430\n(2013)."    },    {        "title": "1704.04303v2.Spin_pumping_into_superconductors__A_new_probe_of_spin_dynamics_in_a_superconducting_thin_film.pdf",        "content": "arXiv:1704.04303v2  [cond-mat.mtrl-sci]  14 Jul 2017Spin pumping into superconductors:\nA new probe of spin dynamics in a superconducting thin film\nMasashi Inoue,1Masanori Ichioka,2and Hiroto Adachi2\n1Department of Physics, Okayama University, Okayama 700-85 30, Japan\n2Research Institute for Interdisciplinary Science, Okayam a University, Okayama 700-8530, Japan\n(Dated: March 5, 2022)\nSpinpumpingrefers tothemicrowave-drivenspincurrentin jection from aferromagnet intothead-\njacent target material. We theoretically investigate the s pin pumping into superconductors by fully\ntaking account of impurity spin-orbit scattering that is in dispensable to describe diffusive spin trans-\nport with finite spin diffusion length. We calculate temperat ure dependence of the spin pumping\nsignal and show that a pronounced coherence peak appears imm ediately below the superconducting\ntransition temperature Tc, which survives even in the presence of the spin-orbit scatt ering. The\nphenomenon provides us with a new way of studying the dynamic spin susceptibility in a supercon-\nducting thin film. This is contrasted with the nuclear magnet ic resonance technique used to study\na bulk superconductor.\nI. INTRODUCTION\nInvestigation of the interplay of superconductivity and\nmagnetism has a long history. The seminal theoretical\nwork by Abrikosov and Gor’kov [1] describing the effects\nofmagnetic impurities on superconductivity initiated the\nresearch field of gapless superconductivity [2]. Besides,\nthe theoretical prediction on the appearance of nonuni-\nform superconductivity by paramagnetic depairing ef-\nfects [3, 4] has still had a major impact on the current\nresearch of superconductor/ferromagnetjunctions [5]. In\nthe contextofspintronics, the superconducting tunneling\nexperiment has been applied to the measurement of the\nspin polarization in a number of materials [6, 7], and an\nelectricalspininjectionintoasuperconductoriscurrently\nunder active investigation [8–11]. Furthermore, the in-\nterplay of superconductivity and magnetism has been a\nsubject of intense debate [12].\nIn those investigations mentioned above, the magnetic\ndegrees of freedom are assumed to be in thermal equi-\nlibrium, and the nonequilibrium dynamics of the magne-\ntization does not play any active role. In recent years,\nhowever, this nonequilibrium magnetization dynamics in\nmagnetic heterostructures has drawn great attention as\na new means for the spin current generation, which is\nnow known as spin pumping [13]. In this method, the\nnonequilibrium magnetization dynamics in a ferromag-\nnet is driven by ferromagnetic resonance (FMR), and it\ngives rise to the spin injection into the adjacent target\nmaterialbytransferringspinangularmomentumthrough\nthes-dexchange interaction at the interface. Because\nthe spin pumping relies only on the spin dynamics and\nthus enables a charge-free spin injection [14], by now it\nis widely used as a versatile spin injection method. In-\ndeed, the target materials range from normal metals [15–\n22], semiconductors [14, 23–25], magnetic metals [26–\n28] and insulators [29], to more exotic systems such as\ngraphene [30, 31], organic materials [32, 33], a topologi-\ncal insulator [34], and a Rashba system [35]. Given this\nbackground, it is worth investigating and clarifying thenature of spin pumping into superconductors.\nExperimentally, the spin pumping into a supercon-\nducting material was studied almost a decade ago in a\nNi80Fe20/Nb bilayer system [36], and a decrease in the\nspin pumping signal was observed below the supercon-\nducting transition temperature Tc. On the theoretical\nside, oneoftherecentprogressesisthe theoreticalfinding\nthat the spin pumping signal is intimately related to the\ndynamicspinsusceptibilityoftargetmaterials[37],which\nwas derived in a close analogy to the linear-response ap-\nproachto the spin Seebeck effect [38, 39]. Indeed, a num-\nberofexperimentssuggestingthecorrelationbetweenthe\nspin susceptibility and the spin pumping have been accu-\nmulating [40–42]. Therefore, from the current theoretical\npoint of view, the problem of calculating spin pumping\ninto superconductors is reduced to the evaluation of the\ndynamic spin susceptibility in the superconducting state.\nThe dynamic spin susceptibility in superconductors\nhas long been studied using nuclear magnetic resonance\n(NMR) [43], especially by focusing on the behavior of\nnuclear-spin relaxation rate [44, 45]. Then, a naive ex-\npectation would be that in the literature one could find a\ndetailed calculation of the dynamic spin susceptibility in\na superconducting state. In actual fact, however, exist-\ning theories discussing the NMR nuclear-spin relaxation\nrate focus only on a system without impurity spin-orbit\nscattering where the spin diffusion length diverges [46],\nwhereas we do not encounter such a situation even in a\nmaterial with relatively long spin diffusion length, such\nas Al [47]. Therefore, in order to obtain a physically rea-\nsonable result with nondiverging spin diffusion length, it\nis of crucial importance to deal with the impurity spin-\norbit scattering in an adequate manner. As far as we\nknow, however, no such theoretical calculation has been\nreported in the literature.\nIn this paper we present a theory of spin pumping into\nsuperconducting materials, by fully taking account of the\nimpurity spin-orbit scattering. Making use of the previ-\nous theoretical finding [37] that the strength of the spin\npumping is proportional to the imaginary part of the dy-2\nSISS\nFIG. 1. (Color online) Schematic view of the system con-\nsidered in this paper, where a spin sink (SS) is placed on top\nof a spin injector (SI). The SS is an s-wave superconductor\nin the present paper, whereas it was assumed to be a weak\nitinerant ferromagnet in Ref. [37]. In both cases, the SI is a n\ninsulating magnet.\nnamic spin susceptibility oftargetmaterials, we calculate\ntemperature dependence of the spin pumping signal in a\nbilayer composed of an insulating ferromagnet and an s-\nwave superconductor. By evaluating the dynamic spin\nsusceptibility of the superconducting target material, we\nshow that a pronounced coherence peak appears in the\nsignal immediately below the superconducting transition\ntemperature Tc, which survives even in the presence of\nthe impurity spin-orbit scattering. Since existing NMR\ntechnique is not suitable for thin film samples, we fur-\nther argue that the phenomenon under discussion can be\nused as a new method for detecting the spin dynamics in\na superconducting thin film.\nThe outline of the paper is as follows. In Sec. II, our\nmodel is given. In Sec. III, the formulation to derive\nthe dynamic spin susceptibility of a superconductor is\npresented, by fully taking account of the vertex correc-\ntions due to impurity spin-orbit scattering. In Sec. IV,\ntemperature dependence of the spin pumping into an s-\nwave superconductor is numerically evaluated for various\nstrengths of the impurity spin-orbit scattering. Finally,\nin Sec. V we discuss and summarize our results. We use\nunits/planckover2pi1=kB= 1 throughout this paper.\nII. MODEL\nAsinRef.[37], weconsiderabilayercomposedofaspin\ninjector (SI) and a spin sink (SS), as shown in Fig. 1.\nIn Ref. [37], the SS was assumed to be a weak itiner-\nant ferromagnet such as NiPd. In the present paper, by\ncontrast, we consider a situation where the SS is an s-\nwave superconductor such as Nb. In both cases, the SI is\na magnetic insulator, most typically yttrium iron garnet\n(YIG). We assumethata staticmagneticfield H0=H0ˆz\nis applied to the bilayer in the lateral direction, and that\nthe anisotropy field is much weaker than H0such that it\ncan be disregarded.\nWe focus on the situation wherean externalmicrowavewith the angularfrequency ωacis applied to the SI/SS bi-\nlayer to drive the ferromagnetic resonance (FMR) of the\nSI. In the magnon language the external microwave reso-\nnantly excites uniform magnon mode inside the SI, with\nthe angular frequency ωac. Without the attachment of\nthe SS, the magnon has an intrinsic damping rate α0ωac,\nwhereα0isthe (dimensionless)intrinsicGilbert damping\nconstant. In the presence of the SS, because there arises\nan additional spin dissipation channel, the magnon ac-\nquires an additional Gilbert damping so that the total\nGilbert damping constant αis given by\nα=α0+δα, (1)\nwhereδαis the additional Gilbert damping constant. In\nFMR experiments, the additional Gilbert damping con-\nstant is obtained from the FMR linewidth ∆ Hthrough\nthe relation\nγ∆H=2√\n3(α0+δα)ωac, (2)\nwhereγis the gyromagnetic ratio. The appearance of\nthe additional spin dissipation channel means that spins\nare pumped into the SS, and thus this phenomenon is\ntermedspinpumping. Becausethespinspumpedintothe\nSS diffuse in the form of a spin current, this phenomenon\nhasdrawn much attention as anew meansof spin current\ngeneration.\nNow we briefly review the linear-response approach to\nthe spin pumping that has been developed in Refs. [37,\n48]. As already mentioned, the additional Gilbert damp-\ning is regarded in the magnon language as the additional\ndamping rate of the collective mode, i.e., the magnon.\nAccording to the framework of many-body theory [49],\nthe damping of a collective excitation can be calculated\nfrom the corresponding selfenergy ΣR(ω). In the present\ncase, the selfenergy is diagrammatically expressed in\nFig. 2, from which we see that it is related to the dy-\nnamic spin susceptibility χR\nq(ω) of the SS as\nΣR(ω) =−/angbracketleft/angbracketleftJ2\nsd/angbracketright/angbracketright\n/planckover2pi12/summationdisplay\nqχR\nq(ω), (3)\nwhereJsdis thes-dinteraction at the SS/SI interface,\n/angbracketleft/angbracketleftJ2\nsd/angbracketright/angbracketright= 2J2\nsdS0Nint/(NSINSS) withNint,NSI, andNSS\nbeing the number of localized spins at the interface, and\nthe number of lattice sites in the SI and SS, respectively.\nUsing the relation δαωac=−ImΣR(ωac), we arrive at\nthe final result:\nδα=/angbracketleft/angbracketleftJ2\nsd/angbracketright/angbracketright\n/planckover2pi12/summationdisplay\nq1\nωacImχR\nq(ωac). (4)\nTherefore, the remaining task is to calculate the dynamic\nspin susceptibility in the superconducting state.\nAs mentioned in Sec. I, when calculating the dynamic\nspin susceptibility of superconductors, it is quite impor-\ntant to take account of the spin-orbit scattering by im-\npurities in a proper way, because it is indispensable to3\nJsd Jsd/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1χSS\nSI\nFIG. 2. (Color online) Diagrammatic representation of the\nmagnon selfenergy leading to the additional Gilbert dampin g.\nA solid line is the electron Green’s function, a wavy line the\nmagnon propagator, χthe spin susceptibility, and a black dot\nthes-dinteraction at the SS/SI interface.\ndescribe the spin dissipation that produces a finite spin\ndiffusion length. As for the staticspin susceptibility in\nsuperconductors such a calculation has been known [50],\nwhich reveals that the spin-orbit scattering by impurities\ncompletely modifies the behavior of the static suscepti-\nbility, and the resultant temperature dependence of the\nsusceptibility deviates substantially from that of a pure\nsuperconductor represented by the Yoshida function [51].\nRegarding the dynamic spin susceptibility in the super-\nconducting state, on the other hand, there has been\nno such calculation reported so far except for Ref. [52],\nwhere the analysis is limited only to a narrow gapless\nregion in the immediate vicinity of the superconducting\ntransition temperature Tcand in the presence of pair-\nbreaking perturbation. Therefore, we adopt such a route\nthat the method of Ref. [50] for the staticspin suscepti-\nbility is extended to the dynamic spin susceptibility, by\nemploying the procedure of analytic continuation from\nthe Matsubara susceptibility to the retarded susceptibil-\nity.\nWe begin with the following Hamiltonian for the SS:\nH=HBCS+Himp, (5)\nwhere the first term,\nHBCS=/summationdisplay\npc†\npξpcp−|g|\nV/summationdisplay\np,p′,kc†\np+k↑c†\n−p↓c−p′+k↓cp′↑,\n(6)\nis the BCS Hamiltonian with attractive interaction |g|.\nHere,c†\np= (c†\np,↑,c†\np,↓)istheelectroncreationoperatorfor\nspin projection ↑and↓,ξpis the kinetic energymeasured\nfrom the chemical potential, and Vis the system volume.\nThe second term,\nHimp=/summationdisplay\np,p′c†\npˆUp,p′cp′, (7)\ndescribes the scattering by impurities, where\nˆUp,p′=u0(p−p′)+iuso(p−p′)ˆσ·(p×p′) (8)\nis the impurity potential with ˆσbeing the Pauli matri-\nces, and u0(p−p′) anduso(p−p′) are the amplitude ofthe momentum scattering and the spin-orbit scattering,\nrespectively. Note that in the present paper, we use a\nnotation to represent an operator in the spin space by\nusing “hat” such as ˆO.\nFollowing Ref. [53], the Gor’kov equation for the\npresent system is given by\n/bracketleftBig\nˇLp+ˇVp,p′/bracketrightBig\nˇGp,p′(iεn) = (2π)3δ(p−p′)ˇ1,(9)\nwhere\nˇLp=/parenleftbigg\niεn−ξp,iˆσy∆\niˆσy∆,−iεn−ξp/parenrightbigg\n(10)\ndefines the Green’s function in the pure system, εn=\n2πT(n+1/2) is a fermionic Matsubara frequency with n\nbeing an integer, ∆ is the superconducting gap, and we\ndenote a matrix in the particle-hole (Nambu) space by\n“check” accent. The effect of impurities is described by\nˇVp,p′=/parenleftbiggˆUp,p′,0\n0,ˆUt\np′,p/parenrightbigg\n, (11)\nwhereˆUtmeans the transpose of a matrix ˆUin the spin\nspace. AsinRef.[53],theimpurity-averagedGreen’func-\ntion,\nˇGp(iεn) =/parenleftbiggGp(iεn),Fp(iεn)iˆσy\nF†\np(iεn)iˆσy,G†\np(iεn)/parenrightbigg\n,(12)\nplays the role of the zeroth-order Green’s function in the\npresent approach. Using the selfconsistent Born approx-\nimation for the impurity potential, we obtain\nˇGp(iεn) =−1\n/tildewideε2n+/tildewide∆2+ξ2p/parenleftbigg\n(i/tildewideεn+ξp),/tildewide∆iˆσy\n/tildewide∆iˆσy,(−i/tildewideεn+ξp)/parenrightbigg\n,(13)\nwheretheMatsubarafrequency εnandthesuperconduct-\ning gap ∆ have the common selfenergy corrections in the\npresent situation [50]:\n/tildewideεn=εnη, (14)\n/tildewide∆ = ∆η, (15)\nη= 1+Γ(+)/radicalbig\nε2n+∆2. (16)\nIn the above equation, the scattering rate Γ (+)has two\ncontributions as\nΓ(+)=1\n2/parenleftbigg1\nτ0+1\nτso/parenrightbigg\n, (17)\nwhereτ0is the momentum relaxation time and τso\nis the spin-orbit relaxation time, which are respec-\ntively given by 1 /τ0= 2πN(0)nimp|u0(0)|2and 1/τso=\n2πN(0)nimp|uso(0)|2/angbracketleft(p×p′)2/angbracketrightFS. Here,N(0) is the den-\nsity of states of electrons at the Fermi level, nimpthe\nnumber density of impurities, and /angbracketleft···/angbracketrightFSmeans the av-\nerage over the Fermi surface.4\n/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1\n/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1\nFIG. 3. Diagrammatic representation of the vertex correc-\ntion forδˇG. A dashed line with a cross means the scattering\nby impurities.\nFinally, the superconducting gap is determined self-\nconsistently by the gap equation,\n∆ =|g|\nV/summationdisplay\nεn/summationdisplay\npFp(iεn). (18)\nBecauseofthe common formofthe selfenergycorrections\ngiven by Eqs. (14)-(16), the superconducting transition\ntemperature Tcas well as the gap equation remains the\nsame as in the pure case, i.e.,\nln/parenleftbiggT\nTc/parenrightbigg\n∆ = 2πT/summationdisplay\nεn>0/parenleftBigg\n∆/radicalbig\nε2n+∆2−∆\nεn/parenrightBigg\n,(19)\nwhere we used the relation 1 /|g|= ln(T/Tc) +\n2πT/summationtext\nεn>0(1/εn). This is a consequence of Anderson’s\ntheorem [54] because, unlike the case of magnetic impu-\nrity scattering, the momentum scattering as well as the\nimpurity spin-orbit scattering preservesthe time-reversal\nsymmetry of the electron system.\nIn Appendix A, we briefly review how the present for-\nmalism is applied to the calculation of the static suscep-\ntibility of a superconductor with sizable impurity spin-\norbit scattering [50]:\nχ0=N(0)/parenleftBigg\n1−/summationdisplay\nεnπT∆2\n(ε2n+∆2)1/radicalbig\nε2n+∆2+2\n3τso/parenrightBigg\n.\n(20)\nThe above result is an extension of the Yoshida func-\ntion [51] to the case with impurity spin-orbit scattering,\nwhich leads both to a nondiverging spin diffusion length\nand to a finite susceptibility even at zero temperature.\nIII. DYNAMIC SPIN SUSCEPTIBILITY IN\nSUPERCONDUCTING STATES\nAs explained in the previous section, temperature de-\npendence of the additional Gilbert damping constant δα\nis obtained by calculating the dynamic spin susceptibil-\nityχR\nq(ωac) [see Eq. (4)]. Then, our strategy is to extend\nthe method in Ref. [50] for the calculation of the static\nsusceptibility χ0into that of the dynamic susceptibility\nχR\nq(ωac). The procedure consists of two steps. First, we\ngeneralize the uniform andstaticexternal magnetic field\nassumed in Ref. [50] into a space-time dependent exter-\nnal magnetic field, and consider the following externalHamiltonian\nδH(τ) =−/integraldisplay\nd3rσ(r,τ)·h(r,τ),(21)\nwhereh(r,τ) =hac/hatwidezexp(−iΩντ+ iq·r). Here, the\nBohr magneton µBis absorbed into the definition of\nhac=µBHac, Ων= 2πTνis a bosonic Matsubara fre-\nquency with a positive integer ν >0, and the time evo-\nlution is defined along the imaginary time τasc(r,τ) =\neτHc(r,τ)e−τH. Next, we calculate the resultant dy-\nnamic Matsubara susceptibility χq(iΩν), and perform an\nanalytic continuation from the Matsubara susceptibility\ninto the retarded susceptibility using the relation\nχR\nq(ωac) =χq(iΩν→ωac+i0+),(22)\nwhich is detailed below.\nIn the presence of the space-time dependent external\nmagnetic field, the matrix Green’s function is, up to the\nlinear order in hac, written as\nˇGp,q(iεn,iΩν) =ˇGp(iεn)+δˇGp,q(iεn,iΩν),(23)\nwhereεnandpare internal frequency and momentum\nwhereas Ω νandqare the external frequency and mo-\nmentum. From the linear-response contribution to the\nGreen’s function δˇGp,q(iεn,iΩν), the Matsubara suscep-\ntibility is calculated to be\nχq(iΩν) =−∂\n∂hacT\n2/summationdisplay\nεn/integraldisplay\npTr/bracketleftBig\nˆσzδˆGp,q(iεn,iΩν)/bracketrightBig\n,(24)\nwhereδˆGp,q(iεn,iΩν) is the (1 ,1) component of the ma-\ntrix Green’s function δˇGp,q(iεn,iΩν), and we have intro-\nduced the shorthand notation/integraltext\np=/integraltextd3p\n(2π)3. In the above\nequation, the factor 1 /2 arises because of the relation\n2χ=χzzin the paramagnetic phase, where χzzis the\nlongitudinal susceptibility. The linear-response contribu-\ntion to the Green’s function can be expressed as\nδˇGp,q(iεn,iΩν) =ˇGp−q(iεn−iΩν)ˇΛq(iεn,iΩν)ˇGp(iεn).\n(25)\nHere, the vertex function ˇΛp(iεn) representing the effects\nof impurity ladder shown in Fig. 3 satisfies the following\nequation:\nˇΛq(iεn,iΩν) =hacˆσz+nimp/integraldisplay\np′ˇVp,p′ˇGp′−q(iεn−iΩν)\nסΛq(iεn,iΩν)ˇGp′(iεn)ˇVp′,p. (26)\nNow we perform the analytic continuation from the\nMatsubara susceptibility into the retarded susceptibility\n[Eq. (22)], by using the formula\nT/summationdisplay\nεng(iεn) =/integraldisplay\nCdz\n4πitanh/parenleftBigz\n2T/parenrightBig\ng(z),(27)\nwhere the contour is shown in Fig. 4. Following the stan-\ndard procedure [49], we find that the contour C1andC35\nFIG. 4. The contour to transform the Matsubara sum into\nan integral over ǫ= Re(z).\nresult in the nondissipative part Re χR\nq(ωac) which is not\nof interest to us because of Eq. (4). By contrast, the\ncontourC2gives us the dissipative part:\nImχR\nq(ωac) =∂\n∂hac1\n2/integraldisplay∞\n−∞dǫ\n4π/integraldisplay\np/bracketleftBig\ntanh/parenleftBigǫ\n2T/parenrightBig\n−tanh/parenleftbiggǫ−ωac\n2T/parenrightbigg/bracketrightbigg\nTr/bracketleftBig\nˆσzδˆGRA\np,q(ǫ,ωac)/bracketrightBig\n,(28)\nwhereδˆGRAis the (1,1) component of\nδˇGRA\np,q(ǫ,ωac) =ˇGR\np(ǫ)ˇΛRA\nq(ǫ,ωac)ˇGA\np−q(ǫ−ωac).(29)\nIn the above equation, ˇGR\np(ǫ) =ˇGp(iεn→ǫ+ i0+),\nˇGA\np(ǫ) =ˇGp(iεn→ǫ−i0+), and the vertex function\nˇΛRAdefined on the real-frequency axis satisfies\nˇΛRA\nq(ǫ,ωac) =hacˆσz+nimp/integraldisplay\np′ˇVp,p′ˇGR\np′(ǫ)\nסΛRA\nq(ǫ,ωac)ˇGA\np′−q(ǫ−ωac)ˇVp′,p.(30)\nThe representation used in Ref. [53],\nˇΛRA\nq(ǫ,ωac) =/parenleftBigg\nΛ(1)RA\nq(ǫ,ωac)ˆσz,Λ(2)RA\nq(ǫ,ωac)ˆσx\n−Λ(2)RA\nq(ǫ,ωac)ˆσx,Λ(1)RA\nq(ǫ,ωac)ˆσz/parenrightBigg\n,\n(31)\nwhich is validated to apply even in the present case,\ntransforms Eq. (30) into a set of linear equations for\nΛ(1)RAand Λ(2)RA:\n/parenleftbigg/tildewideΛ(1)RA\n/tildewideΛ(2)RA/parenrightbigg\n=/parenleftbigg\n1\n0/parenrightbigg\n+/parenleftbigg\nA,−B\nC,D/parenrightbigg/parenleftbigg/tildewideΛ(1)RA\n/tildewideΛ(2)RA/parenrightbigg\n,(32)\nwhere we have introduced the normalization Λ(i)RA=\nhac/tildewideΛ(i)RA(i= 1,2) for the later convenience. From\nEq. (30), the coefficients from AtoDin the above equa-tion are calculated to be\nA=Γ(−)\nπN(0)/integraldisplay\np/braceleftBig\nGR\np(ǫ)GA\np−(ǫ−)+FR\np(ǫ)F†A\np−(ǫ−)/bracerightBig\n,(33)\nB=Γ(−)\nπN(0)/integraldisplay\np/braceleftBig\nGR\np(ǫ)F†A\np−(ǫ−)+FR\np(ǫ)GA\np−(ǫ−)/bracerightBig\n,(34)\nC=Γ(−)\nπN(0)/integraldisplay\np/braceleftBig\nGR\np(ǫ)FA\np−(ǫ−)−FR\np(ǫ)G†A\np−(ǫ−)/bracerightBig\n,(35)\nD=Γ(−)\nπN(0)/integraldisplay\np/braceleftBig\nGR\np(ǫ)G†A\np−(ǫ−)−FR\np(ǫ)FA\np−(ǫ−)/bracerightBig\n,(36)\nwherep−=p−qandǫ−=ǫ−ωac. Here, the scattering\nrate Γ (−)arising from the vertex corrections is given by\nΓ(−)=1\n2/parenleftbigg1\nτ0−1\n3τso/parenrightbigg\n, (37)\nwhere the factor 3 in front of τsoshould not be forgotten.\nAfter integrating over the momentum, we obtain\nA= Γ(−)|/tildewiderWǫ|2+|/tildewideǫ|2+|/tildewide∆ǫ|2\n|/tildewiderWǫ|2(2Im/tildewiderWǫ)/parenleftBigg\n1−v2\nFq2/3\n(2Im/tildewiderWǫ)2/parenrightBigg\n,(38)\nB=C= Γ(−)/tildewideǫ/tildewide∆∗\nǫ+/tildewideǫ∗/tildewide∆ǫ\n|/tildewiderWǫ|2(2Im/tildewiderWǫ)/parenleftBigg\n1−v2\nFq2/3\n(2Im/tildewiderWǫ)2/parenrightBigg\n,(39)\nD= Γ(−)|/tildewiderWǫ|2−|/tildewideǫ|2−|/tildewide∆ǫ|2\n|/tildewiderWǫ|2(2Im/tildewiderWǫ)/parenleftBigg\n1−v2\nFq2/3\n(2Im/tildewiderWǫ)2/parenrightBigg\n,(40)\nwherevFis the Fermi velocity, /tildewideǫ= i/tildewideεn|iεn→ǫ+i0+,/tildewide∆ǫ=\n/tildewide∆|iεn→ǫ+i0+,/tildewiderWǫ=/radicalBig\n/tildewideǫ2−/tildewide∆2ǫ. Substituting these ex-\npressions into Eq. (28) and comparing the expression of\nδˇGR[Eq. (29)] to the second term on the right hand side\nof Eq. (30), the dynamic spin susceptibility is calculated\nto be\n1\nωacImχR\nq(ωac) =N(0)\n4/integraldisplay∞\n−∞dǫ\n2T1\ncosh2/parenleftbigǫ\n2T/parenrightbig\n×/parenleftBigg/tildewideΛ(1)RA\nq(ǫ,0)−1\nΓ(−)/parenrightBigg\n,(41)\nwhere/tildewideΛ(1)RAin the integrand is obtained from Eq. (32),\nand we have used the small ωaclimit,ωac≪min(T,∆),\nwhich is satisfied above 50mK for a 10GHz resonance fre-\nquency. Combining the above expression for Im χR\nq(ωac)\nwith Eq. (4), we can calculate the additional Gilbert\ndamping constant:\nδα=/angbracketleft/angbracketleftJ2\nsd/angbracketright/angbracketright\n/planckover2pi12N(0)\n4Γ(−)/summationdisplay\nq/integraldisplay∞\n−∞dǫ\n2T1\ncosh2/parenleftbigǫ\n2T/parenrightbig\n×A(1−D)−BC\n(1−A)(1−D)+BC, (42)\nwhich is a manifestation of the spin pumping into super-\nconducting materials.6\n~λsd = 0.2\nλsd~ = 0.5\nλsd=~ε/∆0NS(ε) / Ν(0)\nT/TcT/Tc=0.95T/Tc=0= 0.1l~\ntrχ0χ0(T)(Tc)/\n0 1 20246810\n 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6 1\n 0.6\n 0 0.4 0.8\n 0.2\nFIG. 5. (Color online) The uniform susceptibility χ0\n[Eq. (20)] as a function of temperature, for a fixed value of\n/tildewideltr= 0.1 and for /tildewideλsd=∞(solid line), 0 .5 (dashed line),\nand 0.2 (dash-dotted line), where /tildewideltrand/tildewideλsdare defined by\nEq. (43). Inset: Density of states [Eq. (46)] for T/Tc= 0 and\nT/Tc= 0.95.\nIV. RESULTS FOR SPIN PUMPING INTO\nSUPERCONDUCTORS\nIn this section, we calculate temperature dependence\nof spin pumping into superconductors using the formal-\nism developed in the previous section, and show that a\npronounced coherence peak appears in the signal imme-\ndiately below Tceven in the presence of the impurity\nspin-orbit scattering.\nFirst of all, let us briefly comment on the choice of\nparameters that characterizes the system under discus-\nsion. The present formalism contains two parameters\nrepresenting the strength of scattering amplitudes: the\nmomentum relaxation time τ0and the spin-orbit relax-\nation time τso. Because it is almost impossible in ex-\nperiments to tune τ0andτsoindependently, it is more\nconvenient to use parameters having a direct connec-\ntion with physical observables. Thus, we chose to use\nthe following two lengths ltrandλsdto characterize the\nsample inhomogeneity. The first one, ltr=vFτtr, is the\nmean free path which enters into the charge conductivity\nσ=e2N(0)vFltr/3, where eis the electron charge and\nτtr= 1/(τ−1\n0+τ−1\nso) is the transport lifetime. The second\none,λsd=√Dτsf, is the spin diffusion length which can\nbe measured experimentally for example by the nonlo-\ncal spin valve [55], where τsf= (3/4)τsois the spin-flip\nrelaxation time [46, 56] and D=v2\nFτtr/3 is the diffu-\nsion coefficient. In our numerical calculation, we normal-\nize the above two lengths by the BCS coherence length\nξBCS=vF/π∆0, where ∆ 0= 1.76Tcis the superconduct-\ning gap at T= 0. Thus, to characterize the system, we\nuse the followingdimensionless parametersin our numer-\nical calculation:\n/tildewideltr≡ltr\nξBCS,and/tildewideλsd≡λsd\nξBCS,(43)\nwhere these two quantities are expressed by Γ (+)andλsd~T/Tc= 0.1l~\ntr = 0.1l~\ntr(Tc)δα/δα(T)λsd = 0.2~\nλ = 0.5sd~\nλsd = 2.0~Peak value\nPeak value(a) (b)\n 0 1 2 3\n 0  0.4  0.8  1.2 0 1 2 3\n 0  1  2\nFIG. 6. (Color online) (a) The additional Gilbert damping\nconstant δα[Eq. (42)] is shown as a function of temperature\nfor a fixed value of /tildewideltr= 0.1, by varying /tildewideλsd= 2.0 (solid line),\n0.5 (dashed line), and 0 .2 (dash-dotted line). Here /tildewideltrand\n/tildewideλsdare defined by Eq. (43). (b) Peak value versus /tildewideλsdfor\n/tildewideltr= 0.1.\nΓ(−)as\n/tildewideltr=π\n2∆0\nΓ(+), (44)\nand\n/tildewideλsd=π√\n12∆0/radicalbig\nΓ(+)(Γ(+)−Γ(−)). (45)\nIn the case of Nb, the BCS coherence length is estimated\nto be of the order of ξBCS≈200 nm by using [57] Tc≈\n9 K and vF≈1.4×106m/s. Note that in the dirty limit\n(ltr≪ξBCS)whichisusuallythecaseinthin-filmsystems\nunder discussion, the coherencelength ξthat is extracted\nfrom the upper critical field becomes comparable to the\nmean free path ( ξ∼ltr), and hence ξdiffers considerably\nfrom the BCS coherence length ξBCS, i.e.,ξ≪ξBCS[58].\nBefore presenting results for the spin pumping, it is\ninstructive to examine the behavior of the uniform spin\nsusceptibilityin orderto seethe roleofthe impurityspin-\norbit scattering. Considering the fact that a thin-film\nsample usually has a rather short mean free path ltrin\ncomparison to the BCS coherence length ξBCS, it is real-\nistic to work in a relatively dirty limit /tildewideltr<1.0 [10]. The\nmain panel of Fig. 5 shows temperature dependence of\nthe uniform spin susceptibility χ0, calculated for a fixed\nvalue of/tildewideltr= 0.1 and for several choices of /tildewideλsd. In the\nabsence of impurity spin-orbit scattering, the curve coin-\ncides with the Yoshida function [51] which is suppressed\nexponentially at low temperatures as χ0∼exp(−∆0/T).\nUpon introducing the impurity spin-orbit scattering and\nreducing the spin diffusion length λsd, there appears a fi-\nnite susceptibility χ0even at the zero temperature limit.\nIn the inset of Fig. 5, the density of states\nNS(ǫ) =N(0)Re/bracketleftbigg/tildewideǫ\n/tildewiderWǫ/bracketrightbigg\n(ǫ >0),(46)7\nltr  ~ = 0.01\nltr  ~ = 0.05\nltr  ~ = 0.9~λ  = 1.0sd~λ  = 1.0sd(Tc)δα/δα(T)\ntrl~T/TcPeak value\nPeak value(a) (b)\n 0 1 2 3\n 0  0.4  0.8 0 1 2 3\n 0  0.4  0.8  1.2\nFIG. 7. (Color online) (a) The additional Gilbert damping\nconstant δα[Eq. (42)] is shown as a function of temperature\nfor a fixed value of /tildewideλsd= 1.0, by varying /tildewideltr= 0.9 (solid line),\n0.5 (dashed line), and 0 .2 (dash-dotted line). (b) Peak value\nversus/tildewideltrfor/tildewideλsd= 1.0.\nis plotted as a function of energy ǫforT/Tc= 0 and\nT/Tc= 0.95. As was already mentioned near the end of\nSec. II, the momentum scattering as well as the impurity\nspin-orbit scattering preserve the time-reversal symme-\ntry of the electron system. Therefore, unlike the case\nwith magnetic impurity scattering, the present system\nsatisfies Anderson’s theorem [54] and thus a clear super-\nconducting gap appears in the density of states irrespec-\ntive of the strength of the impurity scattering. This is in\nstark contrast to gapless superconductors [2], where the\ntime-reversal-symmetry-breaking perturbation destroys\nthe energy gap even when the system is in a supercon-\nducting state.\nLet us now discuss the additional Gilbert damping\ncaused by the spin pumping. Again, we work in a rel-\natively dirty case /tildewideltr<1 and examine the effect of impu-\nrityspin-orbitscattering. Figure6(a)showstemperature\ndependence of the additional Gilbert damping constant,\ncalculated from Eq. (42) for a fixed value of /tildewideltr= 0.1\nand for several values of /tildewideλsd. First we see that, unlike\nthe ballistic limit calculation ( /tildewideltr=∞,/tildewideλsd=∞) [44],\nthere remains a finite signal even in the zero temperature\nlimit. This behavior is consistent with that of the uni-\nform spin susceptibility (Fig. 5), where a nonzero value\nsurvivesin the T→0 limit. Second, we find that, even in\nthe presence of the spin-orbit impurity scattering, a clear\ncoherencepeak appearsimmediately belowthe supercon-\nducting transition temperature Tc. With increasing the\nimpurity spin-orbit scattering and reducing the spin dif-\nfusion length λsd, the height of the coherence peak is\ngradually enhanced as is seen in the comparison of “peak\nvalue” there. This tendency is best visible in Fig. 6 (b),\nwhere peak value is plotted as a function of normalized\nspin diffusion length /tildewideλsd. Thus, the impurity spin-orbit\nscatteringenhancesthe heightofthe coherencepeak. We\nnote again that such effects on the coherence peak due\nto the impurity vertex corrections have not been investi-ltr  ~ = 90\n~λ sd =100~λ sd =100(Tc)δα/δα(T)\nT/Tc tr l~ltr  ~ltr  ~ = 5\n = 25Peak value\nPeak value(a) (b)\n 0 1 2 3\n 0.4  0  0.8  1.2 0 1 2 3\n 0  40  80\nFIG. 8. (Color online) (a) The additional Gilbert damping\nconstant δα[Eq. (42)] is shown as a function of temperature\nfor a fixed value of /tildewideλsd= 100, by varying /tildewideltr= 90 (solid line),\n25 (dashed line), and 5 (dash-dotted line). (b) Peak value\nversus/tildewideltrfor/tildewideλsd= 100.\ngated in the literature.\nA similar conclusion can be derived for the depen-\ndence of the coherence peak on the mean free path ltr.\nIn Fig. 7 (a), temperature dependence of the additional\nGilbert damping constant is plotted by varying /tildewideltrfor a\nfixed value of /tildewideλsd= 1.0. As one can see in Fig. 7 (b),\nagain the coherence peak is enhanced by increasing the\nstrength of scattering events, which causes a decrease of\nthe mean free path ltr. Therefore, the results displayed\nin Figs. 6 and 7 show that the coherence peak in the\nadditional Gilbert damping constant is enhanced by the\nmomentumscatteringaswellasthespin-orbitscattering.\nNext, we investigate the case of a weak spin-orbit scat-\ntering and hence a very long spin diffusion length, which\nmay be valid when the SS is made of Al [47]. Figure\n8 (a) shows temperature dependence of the additional\nGilbert damping constant, calculated for a fixed value of\n/tildewideλsd= 100. As in the previous case, we find that there\nappears a coherence peak below Tc, and the height of the\ncoherence peak is an increasing function of the strength\nof the impurity scattering. However, a crucial difference\nfrom the previous results with sizable spin-orbit scatter-\ning (Figs. 6 and 7) is that the additional Gilbert damping\nis largely suppressed at low temperatures. This result is\nin line with the suppression of the uniform spin suscepti-\nbilityχ0at low temperatures (solid curve in Fig. 5), and\nalso consistent with the ballistic limit calculation [44].\nFinally, it is worth mentioning that once we recall a\nsimilarity of the quantity under discussion to the spin-\nlattice relaxation rate in NMR, the above conclusion on\nthe impurity dependence of the coherence peak is consis-\ntent with an experimental result of Ref. [59], where the\ncoherence peak measured by the NMR spin-lattice relax-\nation rate in Cu/Nb multilayers was found to increase\nwith the reduction of the mean free path.8\nV. DISCUSSION AND CONCLUSION\nThe main message of the present paper is that the spin\npumping intosuperconductorscan be usedasanew tech-\nnique for probing the spin dynamics in a superconduct-\ningthin film. This is contrasted with the NMR technique\nused to study a bulksuperconductor. When applied to\nthin film superconductors, the spin pumping has the fol-\nlowing advantages over NMR. First, while NMR suffers\nfrom a lowering of signal-to-noise ratio in thin film, the\nspin pumping does not because it is based on a thin-film\ntechnologyfrom the beginning. Second, NMR sometimes\nrequires an isotope substitution to secure probe nuclei,\nbut there is no such issue in the spin pumping as the\nsystem is a ferromagnet/target bilayer such that the fer-\nromagnetic resonance condition is ensured by the choice\nof ferromagnet. Thus, we hope that the present method\nis applied to a wide range of superconducting materials.\nIt would be informative to discuss the difference be-\ntween the present work and the previous theoretical\nstudy[60]. Inshort, the maindifferenceliesin the model-\ning of the ferromagnet/superconductor interface. In the\npresent work, because an insulating ferromagnet is used\nas the SI, it is assumed that the exchange interaction\nJsdat the interface is weak enough, such that it can be\ndealt with by a perturbative approach (see the last para-\ngraph of Sec. II in Ref. [48]). In this case the resultant\nsuperconducting gap ∆ as well as the anomalous corre-\nlationF(ǫ)F†(ǫ) survives at the interface, which gives\nrise to the coherence peak in the spin pumping signal as\nwe have seen in Figs. 6-8. In Ref. [60], by contrast, be-\ncause ametallic ferromagnet is used as the SI (see also\nRef. [36]), it is assumed that the exchange interaction at\nthe interface is so strong that the superconducting gap\nis completely suppressed there [61]. The latter condi-\ntion would result in the vanishing of the coherence peak.\nNote that one data set of Ref. [60] shows a nonmono-\ntonic behavior of the additional Gilbert damping below\nTc, but such a behavior is visible only in a system con-\ntaining pair-breakingmagnetic impurities not considered\nin the present work. Under a condition assumed in the\npresent work, i.e., without magnetic impurities, the addi-\ntional Gilbert damping calculated in Ref. [60] only shows\na monotonic decrease below Tc. This suggests that the\norigin of the nonmonotonic behavior found in Ref. [60]\nis different from that of the coherence peak found in the\npresent work.\nBefore conclusion, we add a few remarks on the key\npoints in performing experiments. First, unlike the\nprevious experiment [36] where a metallic ferromagnet\nNi80Fe20was used as the spin injector, we consider in the\npresent paper an insulating magnet as the spin injector\nto simplify the physics involved. Second, it is assumed in\nour analysis that any spin backflow from the supercon-\nducting spin sink is negligibly small. This means that,\nfor a YIG/Nb system, the Nb thickness should be larger\nthan the Nb spin diffusion length ( λsd∼50-100 nm for\nNb [47]), while the YIG film should be as thin as possiblein order to enhance the spin pumping signal.\nTo conclude, we have theoretically studied the spin\npumping into superconductors, and predicted that its\ntemperature dependence exhibits a pronounced coher-\nence peak just below Tceven in the presence of the im-\npurity spin-orbit scattering. Besides, we have revealed\nthat the height of the coherence peak increases upon the\nincrease of the momentum scattering rate as well as the\nspin-orbit scattering rate. We propose that the present\nphenomenon can be used as a new probe for the spin\ndynamics in a superconducting thin film. Because the\npresent theory fully takes account of the vertex correc-\ntions by impurity spin-orbit scattering, it offers a proper\ndescription of the diffusive spin dynamics in s-wave su-\nperconductors. Moreover, since we can draw parallel be-\ntween the spin pumping signal and the NMR spin-lattice\nrelaxation rate, the present result can also be applied to\nan analysis of the NMR data when we discuss the effects\nof the impurity spin-orbit scattering.\nACKNOWLEDGMENTS\nWearegratefultoS. Onari,K.Tanaka, andK.Matano\nfor valuable discussions, and W. Belzig for useful com-\nments. This work was financially supported by JSPS\nKAKENHI Grant No. 15K05151.\nAppendix A: Review of the derivation of static\nsusceptibility\nIn this Appendix, we briefly review the procedure [50,\n53] to calculate the staticspin susceptibility that fully\ntakes account of the impurity spin-orbit scattering. We\nconsider the uniform and static limit of Eq. (21) for the\nexternal Hamiltonian:\nδH=−/parenleftBig/summationdisplay\npc†\npˆσzcp/parenrightBig\n·h0, (A1)\nwhereh0=h0/hatwidezis an uniform and static external mag-\nnetic field with the Bohr magneton µBbeing absorbed\ninto the definition of h0=µBH0. The uniform spin sus-\nceptibility is calculated from the q→0, iΩν→0 limit\nof Eq. (24):\nχ0=−∂\n∂h0T\n2/summationdisplay\nεn/integraldisplay\npTr/bracketleftBig\nˆσzδˆGp(iεn)/bracketrightBig\n.(A2)\nHere,δˆGp(iεn) is the (1 ,1) component of the matrix\nGreen’s function\nδˇGp(iεn) =ˇGp(iεn)ˇΛ(iεn)ˇGp(iεn),(A3)\nwhich is proportional to h0. In the above equation, the\nvertexfunction ˇΛ(iεn)duetotheimpurityladdersatisfies9\nthe following equation,\nˇΛ(iεn) =h0ˆσz+nimp/integraldisplay\np′ˇVp,p′\nסGp′(iεn)ˇΛ(iεn)ˇGp′(iεn)ˇVp′,p.(A4)\nThe representation similar to Eq. (31) transforms\nEq. (A4) into a set of linear equations for Λ(1)and Λ(2):\nΛ(1)(iεn) =h0+AΛ(1)(iεn)−BΛ(2)(iεn),\nΛ(2)(iεn) =CΛ(1)(iεn)+DΛ(2)(iεn),(A5)\nwhere the coeffients A,B,C, andDare expressed as\nA=Γ(−)\nπN(0)/integraldisplay\np/braceleftbig\nGp(iεn)Gp(iεn)+Fp(iεn)F†\np(iεn)/bracerightbig\n,(A6)\nB=Γ(−)\nπN(0)/integraldisplay\np/braceleftbig\nGp(iεn)F†\np(iεn)+Fp(iεn)Gp(iεn)/bracerightbig\n,(A7)\nC=Γ(−)\nπN(0)/integraldisplay\np/braceleftbig\nGp(iεn)Fp(iεn)−Fp(iεn)G†\np(iεn)/bracerightbig\n,(A8)\nD=Γ(−)\nπN(0)/integraldisplay\np/braceleftbig\nGp(iεn)G†\np(iεn)−Fp(iεn)Fp(iεn)/bracerightbig\n,(A9)and the scattering rate Γ (−)is given in Eq. (37). Af-\nter integrating over the momentum p, we obtain A=\nΓ(−)/tildewide∆2/(/tildewideε2\nn+/tildewide∆2)3/2,B=C= iΓ(−)/tildewide∆/tildewideεn/(/tildewideε2\nn+/tildewide∆2)3/2,\nD= Γ(−)/tildewideε2\nn/(/tildewideε2\nn+/tildewide∆2)3/2.\nIn order to calculate the susceptibility, it is convenient\nto use the relation between Eq. (A3) and the last term\nof Eq. 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Belzig, private communication."    },    {        "title": "1401.4554v1.Position_and_Spin_Control_by_Dynamical_Ultrastrong_Spin_Orbit_Coupling.pdf",        "content": "arXiv:1401.4554v1  [cond-mat.mes-hall]  18 Jan 2014Position and Spin Control by Dynamical Ultrastrong Spin-Or bit Coupling\nC. Echeverr´ ıa-Arrondo1and E. Ya. Sherman1,2\n1Department of Physical Chemistry, Universidad del Pa´ ıs Va sco UPV/EHU, 48080 Bilbao, Spain\n2IKERBASQUE, Basque Foundation for Science, Bilbao, Spain\n(Dated: May 24, 2021)\nFocusing on the efficient probe and manipulation of single-pa rticle spin states, we investigate\nthe coupled spin and orbital dynamics of a spin 1/2 particle i n a harmonic potential subject to\nultrastrong spin-orbit interaction and external magnetic field. The advantage of these systems is\nthe clear visualization of the strong spin-orbit coupling i n the orbital dynamics. We also investigate\nthe effect of a time-dependent coupling: Its nonadiabatic ch ange causes an interesting interplay of\nspin and orbital motion which is related to the direction and magnitude of the applied magnetic\nfield. This result suggests that orbital state manipulation can be realized through ultrastrong\nspin-orbit interactions, becoming a useful tool for handli ng entangled spin and orbital degrees of\nfreedom to produce, for example, spin desirable polarizati ons in time interesting for spintronics\nimplementations.\nPACS numbers: 72.25.Rb,73.63.Kv,71.70.Ej\nSpin-orbitinteractionshavebeenprovenveryusefulfor\nrealization of spintronics [1] with electrons in nanosys-\ntems. On one hand, it has been demonstrated in the-\nory and experiment that spins can be tuned by various\nelectric means.[2–7] On the other hand, since spin-orbit\ncoupling entangles spin and orbital motion, spin read-\nout is reachable by electric means.[8] Such combination\npoints to the possibility of probing and manipulating\nspins hosted by semiconductor quantum dots [9, 10] us-\ning only electric fields.[5] The spin-orbit control of qubits\nis a promising tool that suggests to investigate the ul-\ntrastrong spin-orbit coupling regime to see all features of\nthis technique. Extreme spin-orbit interactions can be\nachieved at the surfaces of semiconductors coated with\nheavy metals (see, e.g. [11, 12]), which allow for spin ma-\nnipulation by electric fields.[13, 14] Rashba performed a\ndetailed analysis of two-dimensional quantum dots with\nthe ultrastrong spin-orbit coupling [15]. Very recently,\nit was recognized that fully controllable strong interac-\ntions, greatly beyond the range reachable in semicon-\nductors, can be produced in ultracold atomic Bose and\nFermi gases by optical means.[16, 17] Similar to electrons\nin quantum dots, these systems are located in harmonic\ntraps and can demonstrate changeable in time spin-orbit\ncoupling, opening new venues for studies of related dy-\nnamics.\nThe spin dynamics in these systems can be studied\ntheoreticallybyanalyzingtheinteractionsbetweenahar-\nmonic oscillatorand a two-levelspin, makingit similar to\nthe Jaynes-Cummings model in quantum optics, as sug-\ngested by Debald and Emary.[18] This is a wide-purpose\nmodel (see e.g. [19, 21, 22]) applied in different fields\nof condensed matter and quantum optics such as cavity\nquantum electrodynamics, trapped ions, and supercon-\nducting qubits (see [23, 24] for recent results). In ad-\ndition, carbon nanotubes holding electron spins deeply\ncoupled to the vibrational modes [25] can be describedFIG. 1: (Color online) (a) Setup scheme for external gen-\neration of spin-orbit coupling through a bias voltage. Bold\nsolid curves indicate the confining parabolic potential. (b )\nA non-adiabatic time-dependent field can cause electron dis -\nplacement with spin rotation.\nwith the Jaynes-Cummings model.\nThe systems with spin-orbit couplings have several\nadvantages not applicable elsewhere. We mention just\ntwo. First, the coordinate dependence of the spin density\nmakes it possible to visualize the effects of strong cou-\npling in terms of particle position and measurable spin\ndensities. Second, spin-orbit interaction and the Zeeman\nfield can be made time-dependent [26–28] makingthe rel-\nevant dynamicsboth in spin and coordinatespacesacces-\nsible. Theseeffects providestronglynontrivialextensions\nof the conventional Jaynes-Cummings model.\nIn a quantum dot, a spin-orbit strength α=ξeEz,\nwhereξis the material- and structure-dependent con-\nstant, can be induced by applying an external electric\nfieldEz, [29]; aschemefor the experimentalsetup is given\nin Fig. 1. In cold atomic gases this time-dependent mod-2\nification can be reached by changing amplitudes and ge-\nometries of the corresponding laser fields.\nIn this Letter, we present a description of the spin dy-\nnamics of an electron in a semiconductor quantum dot or\nconfined coldatoms, both systemssubject to strongspin-\nobit interactions. We focus on a one-dimensional har-\nmonic oscillator with spin degree of freedom, capturing\nmain physics of the systems of interest. This oscillator is\nunderanappliedmagneticfieldwhichonecanrotatewith\nrespect to the coordinate axes. We consider two types of\nspin-orbit couplings, constant and time-dependent; the\nresults given below thereby acquiring wider application.\nThe eigenstates can be obtained from the following\nHamiltonian:\nˆH(t) =/planckover2pi12k2/2m+mω2x2/2+α(t)σxk+∆\n2σ∆,(1)\nwheremis the particle effective mass, ∆ and θare the\nZeeman splitting and tilt angle of the magnetic field as\napplied inxzplane, respectively, σ∆=σxsinθ+σzcosθ\nis the corresponding spin projection, and σx,σzare the\nPauli matrices. In the second quantization this Hamilto-\nnian reads\nˆH=/planckover2pi1ω(ˆa†ˆa+1/2)+i/planckover2pi1ωg(t)(ˆa†−ˆa)σx+∆\n2σ∆,(2)\nwhere ˆa†and ˆaare the creation and annihilation opera-\ntors, respectively, and g=α/radicalbig\nm/2/planckover2pi13ωis a dimensionless\ncoupling constant, which can be understood as the ratio\nof the characteristic anomalous spin-dependent velocity\nα//planckover2pi1to the characteristic quantum velocity spread in the\nground state of the harmonic oscillator/radicalbig\n/planckover2pi1ω/m, or as\nthe ratio of the quantum oscillator length l0=/radicalbig\n/planckover2pi1/mω\nto the spin precession length /planckover2pi12/mα.We use the basis of\nspin orbitals |n/angbracketright|σ/angbracketright, composed of the eigenstates of ˆ a†ˆa,\n|n/angbracketright, and those of σz,|σ/angbracketright ≡ |↑/angbracketrightzand|↓/angbracketrightzwith respect\nto thez-axis. Numerical values of gstrongly vary from\nsystem to system. For InSb-based quantum dots, where\nαcan reach 10−5meVcm,mis of the order of 0.02 of\nthe free electron mass, and ω∼1012s−1, one can expect\ng≈1. For cold fermions such as40K, whereα//planckover2pi1can be\nof the order of 10 cm/s and ω∼103s−1,gcan be of the\norder of 10.\nTo make connection to previous works on the ultra-\nstrong regime (see, e.g. [19]), first we investigate the\neffect of a constant coupling. The eigenstates of the full\nHamiltonian |φi/angbracketrighthave the form/summationtext\nn|n/angbracketright(cu\nn|↑/angbracketrightz+cd\nn|↓/angbracketrightz)\nwith the expansion coefficients cu\nnandcd\nn; the normalized\norbitals are expressed in the x-representation as\n/angbracketleftx|n/angbracketright=4/radicalBigg\n1\nπl2\n022n(n!)2exp/bracketleftbigg\n−x2\n2l2\n0/bracketrightbigg\nHn/bracketleftbiggx\nl0/bracketrightbigg\n,(3)\nwhereHnis then-th order Hermite polynomial. In the\nlimit of a very weak coupling, g≪1, in an arbitrar-\nily directed magnetic field, |φi/angbracketrightcontains five main con-\ntributions. The main one is the direct product of |n/angbracketrightFIG. 2: (Color online) Nonzerospin densities ρx,zfor different\ngvalues when the electron is at the ground state. Note that\nρxandρzmerge for smaller gcoupling. Here and below we\nuse a truncated Hilbert space of 64 states, sufficient to study\nultrastrong spin-orbit coupling. The Zeeman splitting in a ll\ncalculations is taken as ∆ = 0 .5/planckover2pi1ω.\nand eigenstate of σ∆. The other four are perturbative\nterms of direct products of |n±1/angbracketrightand eigenstates of\nσ∆. Atθ= 0, corresponding to the conventional Jaynes-\nCummings model, spin selection rules exclude two of\nthese four states. Moreover, at θ= 0, in the eigenstates\nthe contributions with different spatial parities have op-\nposite spins. The expectation value of the velocity in the\neigenstates is zero, /angbracketleftv/angbracketright=d/angbracketleftx/angbracketright/dt=0; however, the mean\nmomentum is finite:\n/angbracketleftv/angbracketright=i\n/planckover2pi1/angbracketleft[H,x]/angbracketright=/planckover2pi1\nm/angbracketleftk/angbracketright+α\n/planckover2pi1/angbracketleftσx/angbracketright= 0,\n/angbracketleftk/angbracketright=−√\n2g\nl0/angbracketleftσx/angbracketright. (4)\nCorrespondingly, for the coordinate /angbracketleftφi|x|φi/angbracketright= 0. Equa-\ntion (4) corresponds to zero mechanical momentum /planckover2pi1k−\nAfor the gauge [20] A=−mασx//planckover2pi1. The total spectrum\nresults from the magnetic-field mixing of two parabolic\nbranches, that for /angbracketleftk/angbracketright=−√\n2g/l0when the spin state is\n|↑/angbracketrightx, and that for /angbracketleftk/angbracketright=√\n2g/l0when it is |↓/angbracketrightx.\nTo show the advantages of systems with spin-orbit\ncoupling, we calculate the spatially resolved spin den-\nsitiesρj(x),j=x,y,z, providing valuable info about\nthe system.[33] For an arbitrary state |ψ/angbracketright, presented in\nthe form/summationtext\nn|n/angbracketright(au\nn|↑/angbracketrightz+ad\nn|↓/angbracketrightz), these functions are\ndefined as\nρj(x) =/summationdisplay\nn,m/angbracketleftn|x/angbracketright/angbracketleftx|m/angbracketright(au∗\nn,ad∗\nn)σj/parenleftbiggau\nm\nad\nm/parenrightbigg\n.(5)\nWe focus on a particle in the ground state and take\nθ=π/4 as an example. The densities ρj(x) are presented\nin Fig. 2. The integrals of ρj(x) over thex-coordinate\nare the spin expectation values.3\nFIG. 3: (Color online) Spin densities (a) ρxand (b) ρyas a\nfunction oftime for an electron spin antiparallel totheapp lied\nmagnetic field with θ=π/4; here we take the marginal case\ng=1.\nNext, weanalyzethe coupleddynamicsofasystemput\naway from an eigenstate. For this purpose, we choose as\na typical example an eigenstate of σ∆antiparallel to the\nmagnetic field:\n|ψ(0)/angbracketright=|0/angbracketright(−sin(θ/2)|↑/angbracketrightz+cos(θ/2)|↓/angbracketrightz),(6)\nThe time dependence is obtained from |ψ(t)/angbracketright=/summationtext\niζi|φi/angbracketrighte−iEit//planckover2pi1,whereζiare the corresponding expan-\nsion coefficients. We study the dynamics of spin densi-\nties forθ=π/4 andg= 1. The particle oscillations are\nshown in Fig. 3(a). The Gaussian-like shape of ρxis ro-\nbustagainsttime; however,thoseof ρyandρz(notshown\nin the Figure) are fully changed. As a consequence, we\nsee a strong correlation between spin state and position\nof the particle even in the dynamical regime.\nOne of the main advantages of quantum dots and cold\natoms is the ability to manipulate the strength of spin-\norbit coupling and, thus, to cause dynamics in the or-\nbital and spin channels. The time-dependent spin-orbit\ncoupling can be used for generation of spin currents in\ntwo-dimensional electron gas [30] and spin separation\nin two-electron quantum dots similar to that predicted\nin Ref.[31]. Here we take a single-period perturbationg(t) =g0sin(Ωt) at frequency Ω, fast enough to yield an\nappreciable nonadiabatic behavior, but being sufficiently\nslow to allow using the available electrical and optical\nmeans to generate the spin-orbit interaction. To clearly\nsee the effects of strong spin-orbit coupling, we compare\nbelowthe obtained numericallyexactresult with the per-\nturbation theory.\nTo use the perturbation theory we take the basis of\nfirst four eigenstates\nψ1=|0/angbracketright| ↓/angbracketrightz, ψ2=|0/angbracketright| ↑/angbracketrightz, ψ3=|1/angbracketright| ↓/angbracketrightz, ψ4=|1/angbracketright| ↑/angbracketrightz\n(7)\nwhere time-dependent wavefunction becomes\nψ(t) =a1(t)ψ1+a3(t)ψ3e−iωt+a4(t)ψ4e−i(ω+∆)t.(8)\nThe assumed time-dependence yields the expansion co-\nefficients\na3(t) =−g0sinθ/integraldisplayt\n0sin(Ωτeiωτdτ (9)\na4(t) =g0cosθ/integraldisplayt\n0sin(Ωτ)ei(ω+∆)τdτ.(10)\nThe expectation values of coordinate and spin projec-\ntion onto the magnetic are expressed as:\n/angbracketleftx(t)/angbracketright ≡ /angbracketleftψ(t)|/hatwidex|ψ(t)/angbracketright=1√\n2/parenleftbig\na3(t)e−iωt+c.c./parenrightbig\n(11)\n/angbracketleftσ∆(t)/angbracketright ≡ /angbracketleftψ(t)|σ∆|ψ(t)/angbracketright=−1+2/vextendsingle/vextendsinglea2\n4(t)/vextendsingle/vextendsingle.(12)\nThese perturbation theory formulas show the role of the\ndirectionofmagneticfieldonthespinandspatialdynam-\nics, not present in the conventional Jaynes-Cummings\nmodel and allowing to extend the abilities for coordinate\nand spin manipulation.\nWe treat the problem numerically for a single period\nT= 2π/Ω [32], beginning with θ= 0, where Zeeman and\nspin-orbit fields are orthogonal, similar to the Jaynes-\nCummings model. Figure 4 demonstrates time depen-\ndenceof/angbracketleftσ∆(t)/angbracketrightfordifferentcouplings g0andcomparison\nwith the perturbation result for g0= 0.5 in the inset. As\nit can be seen in Fig. 4, spin projection at the magnetic\nfield changes strongly with time, corresponding to the\nspin rotation due to the spin-orbit coupling, and remains\nconstant, as expected, after the change stops. The value\nof this projection after the end of the perturbation cor-\nresponds to the degree of nonadiabaticity of this process.\nIt is interesting to mention the appearance of plateaus at\nstrong spin-orbit coupling regime. These plateaus show\nthat even a low-frequency dynamics is strongly nonadi-\nabatic. The reason is the following. In the presence of\nspin-orbit coupling and magnetic field with θ= 0, the\nsplitting of the ground-state doublet, ∆exp( −2g2)≪∆,\nis small due to weakly overlapping eigenstates in the mo-\nmentum space, as discussed after Eq.(4). As a result,\neven very slow changes in the system parameters cannot4\nFIG. 4: (Color online) Dynamics of spins under a time-\ndependent g(t)=g0sin(Ωt). Lines are marked by correspond-\ningg0values. Inset shows comparison of the exact (dashed\nline) and perturbation theory (solid line) results for g0= 0.5.\nbe treated adiabatically. Here spatial motion does not\noccur (/angbracketleftx(t)/angbracketright= 0), in agreement with Eq.(11).\nNext, we take the state of Eq.(6), with θ=π/4 as the\ninitial oneto demonstratethe qualitativeroleofthe mag-\nnetic field direction. First qualitativeeffect is nonzeroos-\ncillations of the coordinate, nonmonochromatic at t<T,\nasdepicted in Fig.5. The oscillationsin /angbracketleftx(t)/angbracketrightarecaused\nmainly by transitions between ground and first excited\neigenstates of the model Hamiltonian; these states get\nmixed by the strong coupling. The amplitude of oscilla-\ntionsincreaseswith g0, asexpected, but alsowith Ω. The\nbehavior strongly depends on the frequency and ampli-\ntude of spin-orbit coupling, as can be seen by comparison\nofadiabaticandnonadiabaticplots. After the endofper-\nturbation, the coordinate oscillates at the frequency ω.\nNext qualitative difference is in the behavior of /angbracketleftσ∆(t)/angbracketright.\nHere, in the case of a strong static spin-orbit coupling,\nthe splitting of the lowest doublet is ∆cos θ. We see the\neffects of nonadiabatic perturbation and strong coupling\nin Fig. 6. Dependent on g0and Ω, different regimes in\nthe dynamics of /angbracketleftσ∆(t)/angbracketrightare possible: the system either\nreturns to the initial state after the end of perturbation\nor switches to other states. The switching effect depends\non the frequency of the field. At a small frequency, the\ndynamics is more adiabatic and perturbative, while for\nthe larger frequency the switching can occur. The in-\nset shows that the perturbation theory works at short\ntime, while at longer times more states participate, and\nthe results become different. From this plot we can for-\nmulate the condition of strong spin-orbit coupling for a\ngiven system as a transition from peak-like (“weak” or\n“moderate” coupling) to step-like (“strong”) evolution of\n/angbracketleftσ∆(t)/angbracketright. It is interesting to mention that the transition\nand, therefore, the definition of the coupling strength inFIG. 5: (Color online) Dynamics of particle mean coordi-\nnate under time-dependent strength g(t)=g0sin(Ωt). Lines\nare marked by corresponding g0values. Inset shows compari-\nson of the exact (dashed line) and perturbation theory (soli d\nline) results for g0= 0.5.\na dynamical system depends on the frequency of the ap-\nplied external field.\nIn summary, we have investigated how strong dynam-\nical spin-orbit couplings can be applied to probe and\nmanipulate spins of electrons in semiconductor quantum\ndotsandcoldatomsinparabolicconfinementthroughthe\ncorrelated spin and orbital motion. We reveal the impor-\ntanceofthetiltangleoftheappliedmagneticfield, theef-\nfect strongly beyond the conventional Jaynes-Cummings\nmodel. The obtained dynamics shows that, under a\nstrong constant coupling, a particle oscillates in correla-\ntion with its spin orientation. The motion of the particle\ncan be influenced by time-dependent coupling with the\nresult strongly dependent on all parameters. We observe\na transition from periodic to step-like behavior of spin\ncomponent parallel to the magnetic field with increas-\ning the coupling strength. This fact clarifies the way\nto define qualitative effect if the ultrastrong spin-orbit\ncoupling. The present work widens the applicability of\nthe spin-orbit control, as it covers different strengths for\nthe induced interaction and tilt angles for the applied\nmagnetic field. It also emphasizes the usability of elec-\ntric and optical fields for spin probe and manipulation,\nwhich are crucial for spintronics. These results may also\nbe of interest for quantum optics and quantum informa-\ntion realizations.\nWe gratefully acknowledge fruitful discussions with E.\nIl’ichev, G. Romero, E. Solano, and, especially, with J.\nC. Retamal. We acknowledge support of the MINECO\nof Spain (grant FIS 2009-12773-C02-01), the Govern-\nment of the Basque Country (grant ”Grupos Consolida-\ndos UPV/EHU del Gobierno Vasco” IT-472-10), and the\nUPV/EHU (program UFI 11/55).5\nFIG. 6: (Color online) Dynamics of /angbracketleftσ∆/angbracketrightunder time-\ndependent strength g(t)=g0sin(Ωt). 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Sherman, EPL 9027010\n(2010)."    },    {        "title": "1309.2365v1.Spin_rectification_induced_by_dynamical_Hanle_effect.pdf",        "content": "arXiv:1309.2365v1  [cond-mat.mes-hall]  10 Sep 2013Spin rectification induced by dynamical Hanle effect\nHiroto Sakimura, Takahiko Matsumoto, and Kazuya Ando∗\nDepartment of Applied Physics and Physico-Informatics,\nKeio University, Yokohama 223-8522, Japan\nAbstract\nDynamic responseof spinaccumulation to a time-dependent m agnetic field has been investigated\nin a ferromagnetic/nonmagnetic bilayer under ferromagnet ic resonance. In this system, magneti-\nzation precession driven by a microwave generates direct-c urrent (dc) and alternate-current (ac)\nspin accumulation in the nonmagnetic layer by the spin pumpi ng. The ac spin accumulation is\ncoupled with the microwave magnetic field through a dynamica l Hanle spin precession, giving rise\nto rectified spin accumulation comparable with the dc spin ac cumulation directly generated by the\nspin pumping.\n∗ando@appi.keio.ac.jp\n1M(t)\nzyxh(t)\nmy(t)δmzδHanle \nFIG. 1: A schematic illustration of the spin pumping and dyna mical Hanle effect. M(t) andh(t)\nare the magnetization and microwave magnetic field, respect ively. The spin pumping creates ac\nspin accumulation δmy(t), which is rectified through the spin precession induced by h(t), giving\nrise to dc spin accumulation δmHanle\nz.\nRectification effects are fundamental in electrical, optical, and mag netic systems. An\nelectrical rectifier is used to convert an alternating current to a d irect current. The\nrectifier essentially strips the high-frequency or alternating part from the incoming cur-\nrent and delivers a low-frequency current, which is based on the tr igonometric relation:\ncosω1tcosω2t= (1/2)[cos(ω1−ω2)t+cos(ω1+ω2)t]; when two waves of frequencies ω1and\nω2are combined, the difference-frequency and sum-frequency ter ms appear. If the frequen-\ncies satisfy ω1=ω2, thetermcos( ω1−ω2)tgives risetoadirect-current (dc) signal. Asimilar\ndynamic response to the product of alternate-current (ac) spin accumulation and a time-\ndependent magnetic field is the origin of a spin rectification effect indu ced by a dynamical\nHanle effect presented in this paper.\nTheHanleeffect refersto thevariationofspin accumulation δminresponse toa magnetic\nfieldHapplied perpendicular to the spin-polarization direction1–3; when a nonmagnet is\nexposed to a magnetic field, spins in the nonmagnet start to preces s around Has∂δm/∂t=\n−γδm×H.\nIn this letter, we show rectification of ac spin accumulation through spin precession in-\nduced by a time-dependent magnetic field: a dynamical Hanle effect. The dynamical Hanle\neffect is discussed in a ferromagnetic/nonmagnetic ( F/N) junction under ferromagnetic res-\nonance, where uniform magnetization precession is driven by a micro wave magnetic field.\nIn this system, the magnetization precession generates dc and ac spin accumulation, or spin\ncurrents, in the Nlayer by the spin pumping.4The dc-component of the spin pumping has\nbeen intensely studied inrecent years.5–11Incontrast, the ac-component ofthe spinpumping\n20200 400 600 800 1000 mzdc \nδ myac \nδ /\n10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 \nτ(s) 10 -3 10 -2 10 -1 0246810 mz    (10 3)dc \nδ myac \nδ /\nατ=10 -13 s\nFIG. 2: The ratio |δmy(t)|/δmdc\nzof the magnitude of the ac spin accumulation |δmy(t)|to that\nof the dc spin accumulation δmdc\nzdirectly induced by the spin pumping as a function of the spin\nrelaxation time τin the nonmagnetic layer for 4 πMs= 0.745 T,h= 0.01 mT,α= 0.01, and\nf= 9.4 GHz. The inset shows Gilbert damping constant αdependence of |δmy(t)|/δmdc\nz.\nhas been directly observed only recently.12Here, we show that the ac-component of the spin\npumping not only generates an ac electric voltage through the inver se spin Hall effect but\nalso generates, through the dynamical Hanle spin precession, larg e dc spin accumulation\ncomparable with the dc spin accumulation directly generated by the d c-component of the\nspin pumping. This finding offers a way to generate a large dc electric v oltage from the\nac-component of the spin pumping using the inverse spin Hall effect.\nThe spin pumping in a F/Njunction generates a spin current js(t) and spin accumulation\nδm(t) in the Nlayer.4–11,13–18The spin current density created by the spin pumping is\nexpressed as13,14\njs(t) =/planckover2pi1\n4πg↑↓\neff1\nM2s/parenleftbigg\nM(t)×dM(t)\ndt/parenrightbigg\n, (1)\nwhereg↑↓\neffis the spin pumping conductance and Msis the saturation value of magnetization\nM. Equation (1) shows that the spin pumping generates two types of spin currents; a dc\nspin current with the spin polarization direction along the zaxis, i.e. the magnetization\nprecession axis, and an ac spin current whose spin polarization direc tion oscillates in the\nx-yplane under the ferromagnetic resonance (FMR) condition [see Fig. 1]. In aF/Nthin\nfilm, where the magnetocrystalline anisotropy can be neglected and a microwave magnetic\nfieldh(t) = (hcosωt,0,0) is applied along the xaxis, the dc and ac spin currents with the\nspin polarization direction along zandyaxis at the resonance condition are obtained from\n3Eq. (1) with the Landau-Lifshitz-Gilbert equation by ignoring the se cond-order contribution\nof the precession amplitude as19\njz\ns=g↑↓\neffh2γ2/planckover2pi1/parenleftBig\n2πMsγ+/radicalbig\n4π2Ms2γ2+ω2/parenrightBig\n16πα2/parenleftbig\n4π2Ms2γ2+ω2/parenrightbig , (2)\njy\ns(t) =jy\ns1cosωt−jy\ns2sinωt, (3)\nwhere\njy\ns1=g↑↓\neffhγ/planckover2pi1/parenleftBig\n2πMsγ+/radicalbig\n4π2Ms2γ2+ω2/parenrightBig\n8πα/radicalbig\n4π2Ms2γ2+ω2, (4)\njy\ns2=g↑↓\neffhγ/planckover2pi1ω\n8π/radicalbig\n4π2Ms2γ2+ω2. (5)\nHere,γandαare the gyromagnetic ratio and the Gilbert damping constant, resp ectively. In\ntheNlayer, the dynamics of spin accumulation δm(x,t) = (δmx(t),δmy(t),δmz(t)) induced\nby the spin pumping is obtained from20\n∂δm(x,t)\n∂t=−γδm(x,t)×Heff(t)−δm(x,t)\nτ+D∇2δm(x,t)+js(t)δ(x),(6)\nwhereHeffis the effective magnetic field, τis the spin relaxation time, and Dis the diffusion\ncoefficient in the Nlayer. For simplicity, in the following discussions, we neglect the exter nal\nmagnetic field and consider the system where the Nlayer is thin enough so that the spin\ndiffusion term can be neglected.\nThe magnitude of the ac and dc spin accumulation directly generated by the spin pump-\ning, i.e., thespinaccumulationintheabsenceofthespinprecession: δm×Heff=0, depends\ncritically on the spin relaxation time τin theNlayer. In the absence of the spin precession,\nthe ac spin accumulation δmy(t) induced by the spin pumping is obtained from Eqs. (3) and\n(6) as\nδmy(t) =τ\n1+(ωτ)2[(jy\ns1+ωτjy\ns2)cosωt+(ωτjy\ns1−jy\ns2)sinωt]. (7)\nUsing Eq. (7), we plot the ratio of the ac to dc spin accumulation, |δmy(t)|/δmdc\nz, in Fig. 2,\nwhere the dc spin accumulation δmdc\nzdirectly generated by the spin pumping is obtained\nfrom Eq. (6): dδmzdc/dt= 0 =−δmzdc/τ+jz\ns. Here, 4 πMs= 0.745 T,h= 0.01 mT,\nγ= 1.86×1011T−1s−1,α= 0.01, andf= 9.4 GHz were used for the calculation, where\nω= 2πfandfis the microwave frequency. Figure 2 shows that |δmy(t)|/δmdc\nzchanges\ncritically around τ≃1/ω= 1.7×10−11s. In a material with τ≫1/ω, where the time scale\n410 -13 10 -12 10 -11 10 -10 10 -9 10 -8 \nτ(s) mzdc \nδ mzHanle \nδ (%) /α\n0.001 \n  0.005 0.01 0.02 0.04 0.06 0.08 = 0.1 \n04\n268\nFIG. 3: The ratio δmHanle\nz/δmdc\nzof the dc spin accumulation induced by the Hanle effect δmHanle\nz\nto that induced by the spin pumping δmdc\nzcalculated for different Gilbert damping constants α\nwith 4πMs= 0.745 T,h= 0.01 mT, and f= 9.4 GHz.\nof the oscillation of the ac spin accumulation 1 /ωis much faster than the spin relaxation\ntimeτ, the ac spin accumulation induced by the spin pumping is almost averag ed out,\nwhereas the dc spin accumulation can be accumulated in the time scale ofτ. In contrast,\nin a material with τ≪1/ω, the spins injected into the Nlayer relaxes before cancelling\nout the ac spin accumulation. Here note that the ac spin current jy\nsis much larger than\nthe dc spin current jz\ns, since the cone angle of the magnetization precession is typically\nless than 1◦, resulting the large ac spin accumulation |δmy(t)|compared with the dc spin\naccumulation δmdc\nzin this system. In fact, as shown in the inset to Fig. 2, |δmy(t)|increases\nwith the Gilbert damping constant α, showing that the ac spin accumulation is significant\nat small amplitudes of magnetization precession, since the precess ion amplitude decreases\nwith increasing α.\nThe acspin accumulation δmy(t) induced by the spin pumping creates rectified spin accu-\nmulation δmHanle\nzwith thespin polarizationdirection along the zaxis throughthe dynamical\nHanleeffect; δmy(t)isrectifiedbytheappliedmicrowave magneticfield h(t) = (hcosωt,0,0)\nbecause of the spin-precession term in Eq. (6): −γδm(x,t)×h(t). By neglecting a contri-\nbution from δmzonδmy, thezcomponent of the spin accumulation due to the precession\nofδmy(t) induced by h(t) is obtained from dδmz(t)/dt=γδmy(t)hcosωt−δmz(t)/τ. The\nspin precession term γδmy(t)hcosωtwith the term δmy(t)∝cosωtgives rise to rectified dc\n5010 20 30 40 50 60 70 mzdc \nδ mzHanle \nδ (%) /\n6 4 21.5 1 0.8 0.6 f = 0.45 GHz \n8 10 \n10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 \nτ(s) 2 4 6 8 10 030 60 \nf (GHz) mzdc \nδ mzHanle \nδ (%) /\nFIG. 4: δmHanle\nz/δmdc\nzcalculated for different microwave frequencies fas a function of the spin\nrelaxation time τ. 4πMs= 0.745 T,h= 0.01 mT, and α= 0.05 were used for the calculation. The\ninset shows fdependence of the maximum value of δmHanle\nz/δmdc\nz, i.e.,δmHanle\nz/δmdc\nzatτ= 1/ω.\nspin accumulation δmHanle\nz= (ω/2π)/integraltext2π/ω\n0δmz(t)dt:\nδmHanle\nz=hγτ2(jy\ns1+ωτjy\ns2)\n2/bracketleftbig\n1+(ωτ)2/bracketrightbig. (8)\nIn Fig. 3, we show the ratio of the dc spin accumulation δmHanle\nzinduced by the dynamical\nHanleeffecttothedcspinaccumulation δmdc\nz=τjz\nsdirectlyinducedbythedcspinpumping:\nδmHanle\nz\nδmdc\nz=ατ/parenleftbig\n4π2Ms2γ2+ω2/parenrightbig/bracketleftBig\n2πMsγ+ατω2+/radicalbig\n4π2Ms2γ2+ω2/bracketrightBig\n/bracketleftbig\n1+(ωτ)2/bracketrightbig/bracketleftBig\nω2+2πMsγ/parenleftBig\n2πMsγ+/radicalbig\n4π2Ms2γ2+ω2/parenrightBig/bracketrightBig (9)\nfor 4πMs= 0.745 T and f= 9.4 GHz. Figure 3 shows that δmHanle\nz/δmdc\nzis maximized\naroundτ≃1/ω= 1.7×10−11s. Here note that τ≃1/ωis the spin relaxation time where\nthe ratio of the ac to dc spin accumulation, |δmy(t)|/δmdc\nz, changes drastically [see Fig. 2],\nsuggesting that it is important to consider separately the two situa tions:τ≪1/ωand\nτ≫1/ω. In a material with τ≪1/ω, spins injected into the Nlayer relax before the\nprecession of δmy(t) due toh(t), resulting the suppression of δmHanle\nzas Fig. 3. In contrast,\nin a material with τ≫1/ω,δmHanle\nzis suppressed by the suppression of |δmy(t)|/δmdc\nz\n[see Fig. 2], resulting the peak structure of δmHanle\nz/δmdc\nzwith respect to τ. Therefore,\nthe dynamical Hanle effect is maximized in a material with the spin relaxa tion time of\nτ≃1/ω, showing that the rectification can be controlled by tuning the micro wave frequency\nf=ω/(2π).\n6The rectification effect is also sensitive to the Gilbert damping consta ntαof theF\nlayer as shown in Fig. 3; the dynamical Hanle spin precession is efficient in a system with\nlargeα. This is due to the fact that the ac spin accumulation δmy(t) induced by the spin\npumping becomes significant compared with the dc spin accumulation δmdc\nzwith increasing\nαas shown in the inset to Fig. 2. At first sight, this result seems to imply that the spin\nrectification depends also on the amplitude of the microwave magnet ic fieldh, since the\ncone angle of the magnetization precession decreases with decrea singh, or|δmy(t)|/δmdc\nz\nincreases with decreasing h. However, this compensates the decrease of the spin-precessio n\nterm; by decreasing h, the spin-precession term δm×Heffalso decreases, resulting that the\nrectification of the spin accumulation is independent of has described in Eq. (9).\nThe magnitude of the rectified spin accumulation δmHanle\nzcan be comparable with that\nof the dc spin accumulation δmdc\nzdirectly induced by the dc spin pumping at low microwave\nfrequencies. Figure 4 shows δmHanle\nz/δmdc\nzat different microwave frequencies fcalculated\nusing Eq. (9) with 4 πMs= 0.745 T and α= 0.05. Figure 4 shows that the spin relaxation\ntimeτmaxat which δmHanle\nzis maximized increases with decreasing fbecause of τmax≃1/ω.\nNotably, the maximum value of δmHanle\nz/δmdc\nzincreases with decreasing f[see also the inset\nto Fig. 4], and δmHanle\nzexceeds 50%of δmdc\nzatf= 0.45 GHz. Since the rectification depends\non the magnetic damping αas shown in Fig. 3, δmHanle\nzcan be further enhanced by selecting\na spin injector ferromagnet with a large magnetic damping α, providing a route for realizing\nδmHanle\nz/δmdc\nz>100%.\nIn summary, we have shown that ac spin accumulation generated by the spin pumping\nis rectified through spin precession induced by a microwave magnetic field: the dynamical\nHanle effect. The magnitude of the rectified spin accumulation can be comparable with\nthat of the dc spin accumulation directly induced by the spin pumping. The rectification is\nsensitive to the Gilbert damping constant and microwave frequency , providing a route for\ngenerating giant dc spin accumulation through the dynamical Hanle e ffect.\nThisworkwassupportedbytheCabinetOffice, Government ofJapa nthroughitsFunding\nProgram for Next Generation World-Leading Researchers, the As ahi Glass Foundation, the\nNoguchi Institute, the Murata Science Foundation, and the Mitsu bishi Foundation.\n1F. J. Jedema, A. T. Filip, and B. J. van Wees, Nature 410, 345 (2001).\n72S. P. Dash, S. Sharma, R. S. Patel, M. P. de Jong, and R. Jansen, Nature462, 491 (2009).\n3Y. Fukuma, L. Wang, H. Idzuchi, S. Takahashi, S. Maekawa, and Y. Otani, Nature Mater. 10,\n527 (2011).\n4H. Jiao and G. E. W. Bauer, Phys. Rev. Lett. 110, 217602 (2013).\n5M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der Wal, and B . J. van Wees, Phys. Rev.\nLett.97, 216603 (2006).\n6K. Ando, T. Yoshino, and E. Saitoh, Appl. Phys. Lett. 94, 152509 (2009).\n7K. Ando, J. Ieda, K. 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Saitoh, Nature Mater. 10, 655 (2011).\n8"    },    {        "title": "1712.03489v2.Effective_gauge_field_theory_of_spintronics.pdf",        "content": "arXiv:1712.03489v2  [cond-mat.mes-hall]  21 Feb 2018Effective gauge field theory of spintronics\nGen Tatara\nRIKEN Center for Emergent Matter Science (CEMS), 2-1 Hirosa wa, Wako, Saitama 351-0198, Japan\nAbstract\nThe aim of this paper is to present a comprehensive theory of spintr onics phenomena based on the concept\nof effective gauge field, the spin gauge field. An effective gauge field g enerally arises when we change a basis\nto describe system and describes low energy properties of the sys tem. In the case of ferromagnetic metals we\nconsider, it arisesfrom structures of localized spin (magnetization ) and couples to spin current of conduction\nelectron. The first half of the paper is devoted to quantum mechan ical arguments and phenomenology. We\nshow that the spin gauge field has adiabatic and nonadiabatic (off-dia gonal) components, consisting an\nSU(2) gauge field. The adiabatic component gives rise to spin Berry’s phase, topological Hall effect and spin\nmotive force, while nonadiabatic components are essential for spin -transfer torque and spin pumping effects\nby inducing nonequilibrium spin accumulation. In the latter part of the paper, field theoretic approaches\nare described. Dynamics of localized spins in the presence of applied s pin-polarized current is studied in\na microscopic viewpoint, and current-driven domain wall motion is disc ussed. Recent developments on\ninterface spin-orbit interaction are also mentioned.\nKeywords: Spintronics, Gauge field, Berry’s phase,\nContents\n1 Introduction 2\n1.1 Symmetry and conservation law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n2 Unitary transformation and gauge field 4\n2.1 Berry’s phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n3 Localized spin 6\n3.1 Spin dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n3.2 Spin relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n3.3 Domain wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n3.4 Domain wall dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n4 Adiabatic spin gauge field in ferromagnetic metal 12\n4.1 Phase from spin texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13\n4.2 Spin electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14\n4.3 Topological monopole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15\n4.4 Detection of spin electromagnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n5 Minimal coupling of spin gauge field 17\n5.1 Perturbative picture of spin gauge field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20\n5.2 Momentum space monopole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22\n6 Spin-transfer effect : Phenomenology 22\nPreprint submitted to Physica E: Low-dimensional Systems a nd Nanostructures February 22, 20187 Spin pumping effect 24\n7.1 Generated spin current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27\n7.2 Adiabatic or nonadiabatic? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28\n7.3 Spin accumulation in ferromagnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29\n7.4 Historical background and adiabatic pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . 30\n8 Brief remark on thermal transport 31\n9 Field theoretical approach 32\n9.1 Field operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33\n9.2 Field Lagrangian for sdmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34\n10 Effective Lagrangian for localized spin 35\n11 Landau-Lifshitz-Gilbert (LLG) equation with electron e ffects 37\n11.1 Induced electron spin density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39\n12 Current-driven domain wall motion 44\n12.1 Threshold current for domain wall motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45\n13 Magnon gauge field 46\n14 Interface Rashba spin-orbit effects 47\n14.1 Effective magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48\n14.2 Rashba-induced spin gauge field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49\n15 Anomalous optical properties of Rashba conductor 50\n15.1 Electromagnetic metamaterial property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51\n15.2 Directional dichroism of magnetic Rashba conductor . . . . . . . . . . . . . . . . . . . . . . . 51\n16 Summary 52\nAppendix A Summary of path-ordered Green’s function 53\nAppendix A.1 Obseervable and path-ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . 53\nAppendix A.2 Interaction representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54\nAppendix A.3 Free electron Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 55\nAppendix A.4 Calculation of lesser component . . . . . . . . . . . . . . . . . . . . . . . . . . . 55\n1. Introduction\nElectromagnetism is absolutely essential for the present technolo gies. Electromagnetism is described by\nthe two field, electric field, E, and magnetic field, B. They satisfy four equations called the Maxwell’s\nequations,\n∇·B= 0\n∇×E=−∂B\n∂t, (1)\nand\n∇·E=ρ\nǫ0\n∇×B=µ0j+ǫ0µ0∂E\n∂t, (2)\n2where where ρandjare density of charge and current, respectively and ǫ0andµ0are dielectric constant\nand magnetic permeability of vacuum, respectively. The first two eq uations (1) allows us to write the two\nfields by a scalar and vector potential, φandA, respectively as\nB=∇×A\nE=−∇Φ−˙A. (3)\nThe six components of vectors EandBare therefore described by the four components of Φ and A. The\nequations for EandBare similar, but not completely symmetric, because they represent different features\nofAand Φ. The fields Φ (scalar potential) and A(vector potential) are called (electromagnetic) gauge\nfield. In terms of the gauge field, the four equations reduces to ev en simpler two equations if we introduce\na relativistic notation (see textbooks such as Ref. (Ryder, 1996) ).\nElectromagnetic effects on charged particles are represented co nveniently in terms of the gauge field.\nThe electric force and the Lorentz force acting on free electrons with charge eand massmis represented by\nthe electron Hamiltonian\nH=1\n2m(p−eA)2+eΦ, (4)\nwherepis momentum. The coupling obtained by replacing pin the kinetic energy by p−eAis called the\nminimal coupling.\n1.1. Symmetry and conservation law\nGauge fields arise from symmetries. The symmetry for the electrom agnetism is the invariance under\nlocal phase transformations, called U(1) symmetry, and it ensure s the conservation of electric charge. A\ngauge field couples to a current that corresponds to the conserv ation law. In the case of electromagnetic\nfield, it is charge current.\nWe demonstratethis fact using field representationfor clearness . Let us denote the field and its conjugate\nbyψandψ†, and denote the Lagrangian density by L(ψ†,ψ). The Lagrangian density contains field\nderivatives only to the linear order with respect to each field ψandψ†. The equation the field satisfies is\ngiven by the condition of least action (the time-integral of the Lagr angian),S≡/integraltext\nd4xL(ψ†(r,t),ψ(r,t)),\nwhere/integraltext\nd4x≡/integraltext\ndt/integraltext\nd3r. Namely, for any small variation δψ†(r,t) andδψ(r,t) of the fields, the action\nremains the same (stationary condition) (δL\nδψ†is a functional derivative);\nδS=/integraldisplay\nd4x/bracketleftbigg\nδψ†δL\nδψ†+δL\nδψδψ+δ∂µψ†δL\nδ∂µψ†+δL\nδ∂µψδ∂µψ/bracketrightbigg\n=/integraldisplay\nd4x/bracketleftbigg\nδψ†/bracketleftbiggδL\nδψ†−∂\n∂xµδL\nδ∂µψ†/bracketrightbigg\n+/bracketleftbiggδL\nδψ−∂\n∂xµδL\nδ∂µψ/bracketrightbigg\nδψ+∂\n∂xµ/parenleftbigg\nδψ†δL\nδ∂µψ†+δL\nδ∂µψδψ/parenrightbigg/bracketrightbigg\n,(5)\nwhere we used δ∂µψ=∂µδψandxµ=r,t,∂µ≡∂\n∂xµ. The last total derivative term of the right-hand side\nvanishes, and we obtain field equation of motions,\nδL\nδψ†−∂\n∂xµδL\nδ∂µψ†= 0,δL\nδψ−∂\n∂xµδL\nδ∂µψ= 0. (6)\nEquation (5) is used to find a conservation law, as known as the Noet her’s theorem (Noether, 1918).\nSuppose that the Lagrangian density is invariant under a certain tr ansformation and that the variation δψ\nandδψ†are those for the invariant transformation. As a result of equatio n of motion, Eq. (5) (without\nintegrals) then indicates that\n∂\n∂xµJµ= 0, (7)\n3where\nJµ≡/parenleftbigg\nδψ†δL\nδ∂µψ†+δL\nδ∂µψδψ/parenrightbigg\n. (8)\nNamely, there is a conserved current Jµassociated with the symmetric transformation δψandδψ†. We note\nthat the expression (8) is a conserved current for internal degr ees of freedom. Original Noether’s current is\ngeneral one including for example the one for translation in time and s pace.\nAn example of the conserved current is the electric charge and cur rent. Physical quantities of electron\nlike density are invariant by phase transformation,\nψ(r,t)→e−iǫψ(r,t), ψ†(r,t)→eiǫψ†(r,t), (9)\nwhereǫis a real constant independent of position and time. For small ǫ, we haveδψ=iǫψandδψ†=−iǫψ†.\nFor the case of free electron, the conserved current, Eq. (8), for the phase transformation is (multiplying by\ne/ǫ)\nJt=eψ†ψ, J i=−i/planckover2pi1e\n2mψ†↔\n∇iψ, (10)\nwhich are electric charge and current. Therefore conservation o f electric charge is a result of invariance\nunder a global (i.e., independent of position and time) phase transfo rmation.\nA gauge field arises if we impose a stronger requirement that the sys tem if invariant under local trans-\nformation. In the case of phase transformation, it is to require th e invariance under\nψ(r,t)→e−iǫ(r,t)ψ(r,t), ψ†(r,t)→eiǫ(r,t)ψ†(r,t), (11)\nfor phase factor depending on the space time. Derivative of field th en becomes ∂µψ→eiǫ(r,t)[∂µ−i(∂µǫ)]ψ,\nand the Lagrangian as it is is modified. The Lagrangian is kept invariant , if we introduce another field\nAcoupled to the field derivative as ∂µψ→Dµψ, whereDµ≡∇µ+iAµis a covariant derivative. The\nLagragian is invariant if we define the field Aµto be transformed as Aµ→Aµ+∂µǫ. This field Aµis a\ngauge field, which defines relative relations among local coordinates (how to define the origin of the phase\nin the case of phase transformation).\nThe nature appears to possess symmetries under local transfor mations, gauge symmetries. In condensed\nmatter, various symmetries other than U(1) symmetry of charge exist approximately for low energy phe-\nnomena. Such gauge fields are called the effective gauge fields. The c oncept of gauge field is highly useful\nfor describing low energy transport effects in condensed matter.\nThe objective of this paper is to demonstrate that spintronics effe cts in ferromagnetic metals are beau-\ntifully described in the framework of electromagnetism by introducin g an effective spin gauge field that\ncouples to electron spin current (Fig. 1). The effective gauge field h as three components corresponding\nto three components of spin, forming an SU(2) gauge field. Its adia batic component is a U(1) gauge field\nhaving the same mathematical structure as charge electromagne tism. Spin Berry’s phase, spin motive force,\nspin transfer effect and spin pumping effects are discussed in detail, and roles of the adiabatic and nonadi-\nabatic components in inducing these effects are clarified. Various sp in-charge conversion effects arise when\nspin-orbit interaction, approximated as another gauge field, is intr oduced. Coupling of the spin gauge field\nto electromagnetism results in anomalous optical properties.\n2. Unitary transformation and gauge field\nAn effective gauge field is a general concept that arises naturally wh en we diagonalize quantum systems.\nLet us consider an electron with mass mdescribed by a Hamiltonian H, given by a sum of free electron\nkinetic energyand apotential VasH=−/planckover2pi12∇2\n2m+V. An eigenstate|ψ(t)∝an}b∇acket∇i}htsatisfiesthe Schr¨ odinger’sequation\ni/planckover2pi1∂\n∂t|ψ∝an}b∇acket∇i}ht=H|ψ∝an}b∇acket∇i}ht. (12)\n4Figure 1: Scope of the paper.\nUsually the state |ψ∝an}b∇acket∇i}htcannot be fully solved, and so we may like to define the state using a un itary transfor-\nmationUfrom a state we are familiar with (like the states for non-interacting or uniform potential cases),\n|φ∝an}b∇acket∇i}ht, as\n|ψ∝an}b∇acket∇i}ht≡U|φ∝an}b∇acket∇i}ht. (13)\nA typical example is the case of slowly changing potential as function of time or space. One should then\nsolve for uniform potential to obtain |φ∝an}b∇acket∇i}htand include the temporal and/or spatial dependence in terms of\nthe matrix U. The Sch¨ odinger’s equation for state |φ∝an}b∇acket∇i}htreads\ni/planckover2pi1/parenleftbigg∂\n∂t+U−1∂\n∂tU/parenrightbigg\n|φ∝an}b∇acket∇i}ht=/parenleftbigg\n−/planckover2pi12\n2m/parenleftbig\n∇+U−1∇U/parenrightbig2+˜V/parenrightbigg\n|φ∝an}b∇acket∇i}ht, (14)\nwhere˜V≡U−1VU. We see that now derivatives are replaced by a ’covariant’ one,\nDµ≡∂µ±i\n/planckover2pi1Aµ, (15)\nwhereµ=t,x,y,zand sign + and−correspondsto µ=tandµ=x,y,z, respectively. (The sign convention\nis chosen in accordance with Lorentz invariance.) In this paper, gre ek letters suffix denotes space and time\nand roman letters are used for spatial suffix. The quantity\nAµ≡∓i/planckover2pi1U−1∂µU, (16)\ndescribes a modification of derivative is an effective gauge field. It co uples to the matter in the same manner\nas the electromagnetic gauge field (Eq. (4)). The spatial compone nt may be called an effective vector\npotential, as it modifies the kinetic energy, and time-component is an effective scalar potential. Effective\ngauge field couples to a current corresponding to the unitary tran sformation Uvia the minimal coupling.\nExplicit example for the spin case is described in Sec. 4.1.\n2.1. Berry’s phase\nIn the presence of slowly changing potential, time-development of q uantum system is represented by\nsolely by a phase factor called the Berry’s phase (Berry, 1987). Le t us briefly mention the effect. The\ncondition assumed is so called the adiabatic condition,\nˆH(t)|ψ∝an}b∇acket∇i}ht=E(t)|ψ∝an}b∇acket∇i}ht, (17)\nwhereˆH(t) is the Hamiltonian operator with slowly varying potential and E(t) is a real number. The\nadiabatic condition means that the state |ψ∝an}b∇acket∇i}ht, defined with respect to eigenstates of H(t), remains to be the\n5eigenstate of the Hamiltonian at each instance. This condition puts a constraint which allows only a phase\nas dynamic variable.\nThe phase arises because of a change of the frame (basis to descr ibe quantum states) as a result of a\ntime-development of the Hamiltonian. What is essential for the phas e appearance is that we do not notice a\nslow change of the frame, and observe the system based on the init ial frame. A change of frame is described\nby a unitary transformation. We denote the state |ψ∝an}b∇acket∇i}htas the one in the ’correct’ basis defined with respect to\nthe Hamiltonian H(t) at each time t, and the state|φ∝an}b∇acket∇i}htas the state of the observer, i.e., the one represented\nusing the basis defined at t= 0. They are related by a unitary matrix Uas in Eq.(13). For the observer’s\nstate|φ∝an}b∇acket∇i}ht, the Schr¨ odinger equation (14) reads\ni(/planckover2pi1∂t+iAt)|φ∝an}b∇acket∇i}ht=/tildewideH|φ∝an}b∇acket∇i}ht, (18)\nwhereAtis a gauge field defined in Eq. (16) and /tildewideH≡U−1HU. Because of the adiabatic condition (17), we\nhave/tildewideH|φ∝an}b∇acket∇i}ht=E(t)|φ∝an}b∇acket∇i}ht, and thus the equation reduces to\n∂t|φ∝an}b∇acket∇i}ht=−i\n/planckover2pi1(E(t)+At)|φ∝an}b∇acket∇i}ht. (19)\nThe termE(t) on the right-hand side describes the standard time-development with energy E(t), while the\ntermAtdescribes the effects of variation of the Hamiltonian. In general Atis a matrix including off-diagonal\ncomponents that causes transition to different states. In the ad iabatic limit, the off-diagonal components\nare neglected, because they give rise to rapidly oscillating term like e−it∆//planckover2pi1at long time ( t), where ∆ is\nexcitation energy for transitions. (In the case of spin, ∆ = Mis energy of spin splitting.) Thus Atcan be\ntreated as a constant in the adiabatic limit, and Eq. (19) is integrate d to obtain\n|φ(t)∝an}b∇acket∇i}ht=eiγ(t)e−i\n/planckover2pi1/integraltextt\n0dt′E(t′)|φ(0)∝an}b∇acket∇i}ht, (20)\nwhere\nγ(t)≡1\n/planckover2pi1/integraldisplayt\n0dt′At(t′), (21)\nis the Berry’s phase arising from the change of the frame, while the s econd phase factor of Eq. (20) is the\nordinary dynamic phase. The gauge field describing the Berry’s phas e is written explicitly as\nAt(t) =−i/planckover2pi1/angbracketleftbig\nφ(0)|U−1(t)∂tU(t)|φ(0)/angbracketrightbig\n. (22)\nIt is written using a derivative of the state as (neglecting higher ord ers of time derivative)\nAt(t) =−i/planckover2pi1∝an}b∇acketle{tψ(t)|∂t|ψ(t)∝an}b∇acket∇i}ht. (23)\nThe Berry’s phase is therefore a result of an effective gauge field Atarising from a unitary transformation\n(16).\n3. Localized spin\nIn this section, theoretical treatments of ferromagnetism are b riefly summarized.\n3.1. Spin dynamics\nMagnetism is collective property arising from an ensemble of many loca lized spins. Each spin, ˆS=\n(ˆSx,ˆSy,ˆSz)1, is a quantum object governed by commutation relation\n[ˆSi,ˆSj] =ˆSi,ˆSj−ˆSj,ˆSi=i/planckover2pi1ǫijkˆSk, (24)\n1In this section, quantum operators are denoted by ˆ.\n6Figure 2: Magnetization Mprecesses anticlockwise around\nB, while S, pointing opposite to M, precesses clockwise\naround direction −B.Figure 3: LLG equation describes the damping of spin, which\ntends to point the spin along −Bdirection.\nwherei,j,k=x,y,z,/planckover2pi1is Planck constant divided by 2 π2andǫijkis a totally antisymmetric tensor that\nsatisfiesǫijk=ǫjki=ǫkij,ǫjik=−ǫijkandǫxyz= 1. Summation over repeated index is assumed (Einstein’s\nconvention but for spatial index, x,y,z.). Spin is an angular momentum and thus create magnetic moment\nˆm=e/planckover2pi1\nmˆS, (25)\nwhich couples to an magnetic field by the interaction\nˆHS=−B·ˆm=−/planckover2pi1γB·ˆS, (26)\nwhereγ≡e\nm(<0) is gyromagnetic ratio. Because the electron charge eis negative, magnetization and spin\npoints opposite, and spin tends to point antiparallel to the magnetic field (Fig. 2). The dynamics of spin is\ndescribed by the Heisenberg equation\n∂\n∂tˆSi=i\n/planckover2pi1[ˆHS,ˆSi], (27)\nwhich reads\n∂ˆS\n∂t=−γB׈S. (28)\nThis is a quantum mechanical equation, but interestingly, this form e quivalent to the one describing torque\non classical objects, applies to macroscopic magnetization.\nThe equation of motion of spin (28) is derived from a Lagrangian\nLS=/planckover2pi1S˙φ(cosθ−1)−HS, (29)\nwhereθ,φare polar angles of S, namely S≡Sn,\nn≡(sinθcosφ,sinθsinφ,cosθ), (30)\nand˙O=∂\n∂tOdenotes time derivative of field O. In the Lagrangian representation, variables θ,φandn\nare classical variables and quantum nature is embedded in the first t ime-derivative term of Eq. (29). The\nequation of motion derived from the Lagrangian is\nd\ndtδL\nδ˙θ−δL\nδθ= 0,d\ndtδL\nδ˙φ−δL\nδφ= 0, (31)\nnamely,\n/planckover2pi1Ssinθ˙θ=δHS\nδφ\n−/planckover2pi1Ssinθ˙φ=δHS\nδθ. (32)\n2In most part of this paper, /planckover2pi1is set to unity.\n7Using\nδHS\nδθ= cosθ/parenleftbigg\ncosφδHS\nδnx+sinφδHS\nδny/parenrightbigg\n−sinθδHS\nδnz=eθ·δHS\nδn=−/planckover2pi1γSeθ·B\nδHS\nδφ= sinθ/parenleftbigg\n−sinφδHS\nδnx+cosφδHS\nδny/parenrightbigg\n= sinθeφ·δHS\nδn=−/planckover2pi1γSsinθeφ·B, (33)\nwe see that the equations (32) leads to\n˙n=˙θeθ+sinθ˙φeφ=γ[eφ(eθ·B)−eθ(eφ·B)] =γn×B, (34)\nwhich is Eq. (28).\n3.2. Spin relaxation\nIn ferromagnets, large number of localized spins moves coherently , and this case is described by replacing\nquantum variable ˆSby a classicalvector Swhose magnitude is proportionalto the number of coherentspins.\nThe magnetization M, commonly used to describe macroscopic magnetism, is related to it a s\nM=µ0/planckover2pi1γ\na3S, (35)\nassuming that each localized spin contribute independently to the ma gnetization, where ais the lattic\nconstant. In this paper, we discuss in terms of the localized spin. Th e fundamental equation of motion for\nclassical spin Sis the same as the quantum one, Eq. (28), with Bthe total magnetic field. In solids, there\nare various microscopic sources for B, such as conduction electron in metals and lattice vibration (phonon s).\nThe total magnetic field acting on each localized spin is therefore not simply written as B=µ0H+M, the\nsum of external magnetic field Hand macroscopic magnetization. Instead, magnetization is taken a ccount\nof by considering microscopic exchange interaction (and dipole inter actions). In fact, the Hamiltonian with\nan exchange interaction J0and an external magnetic field H,\nH=−J0/summationdisplay\nijSi·Sj−/planckover2pi1γµ0/summationdisplay\niH·Si, (36)\nleads, under a mean field approximation, to\nHmf=−/planckover2pi1γ/summationdisplay\niSi·B, (37)\nwith the total magnetic field of B≡µ0H+3J0\n/planckover2pi1γ∝an}b∇acketle{tS∝an}b∇acket∇i}ht, where∝an}b∇acketle{tS∝an}b∇acket∇i}htis average of localized spin. Therefore\nmagnetization in this case is M=3J0\n/planckover2pi1γ∝an}b∇acketle{tS∝an}b∇acket∇i}ht. Effects from uncontrollable magnetic interaction such as spin\nflip scattering by magnetic impurities or phonons lead to a relaxation ( damping) of localized spins. Damping\nis essential in magnetism, as we are familiar with magnetic moment point ing along an applied magnetic field\n(Fig. 3), which does not occur in the absence of damping.\nThere is a long history how to incorporate damping in equation of motio n (28). One way proposed by\nGilbert is to modify it to be\n∂S\n∂t=−γB×S−α\nS/parenleftbigg\nS×∂S\n∂t/parenrightbigg\n, (38)\nwhere the coefficient αis dimensionless, positive and is called the Gilbert damping constant. Th e equation\nis called the Landau-Lifshitz-Gilbert (LLG) equation. When −Bis alongzaxis, small amplitude oscillation\nofnaroundzaxis obtained from Eq. (38) is\nφ=γ|B|t\nθ=θ0eαγ|B|t, (39)\n8indicating that the equation describes a relaxation process to the e quilibrium direction with the period of\nprecession2π\n|γB|and the decay time of1\nα|γB|.\nThere are other ways to introduce damping, like the one called the La ndau-Lifshitz equation,\n∂S\n∂t=−γB×S−αγ\nS[S×(S×B)]. (40)\nThose equations including damping implicitly assume weak damping, and E qs. (40) and (38) are equivalent\nif quantities of the order of α2are neglected. From microscopic viewpoint, LLG equation treating d amping\nby introducing time-derivative is natural, as such a damping term is de rived systematically by a gradient\nexpansion (Kohno et al., 2006), as we shall mention later (Sec. 11) .\nTo treat spin damping in Lagrangian formulation, we use the Rayleigh’s method, and introduce a dissi-\npation function,\nWS≡/integraldisplayd3r\na3/planckover2pi1α\n2S˙S2. (41)\nThe modified equation of motion,\nd\ndtδLS\nδ˙q−δLS\nδq=−δWS\nδ˙q, (42)\nwhereq=θ,φ, turns out to be the LLG equation.\nDampingleadstoanenergydissipationasconfirmedbycalculatingthe time-derivativeoftheHamiltonian\nusing the LLG equation (38);\ndHB\ndt=−αS/parenleftbiggdn\ndt/parenrightbigg2\n+O(α2). (43)\n3.3. Domain wall\nDomain wall is a spatial structure between magnetic domains having d ifferent localized spin directions.\nIn the wall, localized spins rotates as function of spatial coordinate (Fig. 4) . The thickness of the wall,\nλ, is determined by the competition between the exchange energy an d magnetic anisotropy energy, and is\ntypically 10-100 nm. We consider an one-dimensional and rigid wall, neg lecting deformation. We consider\nfirst a system with only an easy axis magnetic anisotropy energy, wh ich is necessary for creation of a wall.\nFor discussing dynamics, we shall later include a hard-axis anisotrop y energy. Choosing the easy axis along\nthezdirection, the anisotropy energy is represented by the Hamiltonian\nHK≡−KS2\n2/integraldisplayd3r\na3cos2θ. (44)\nwhereKis the easy-axis anisotropy energy ( K >0). Including the ferromagnetic exchange coupling, the\nHamiltonian in the continuum expression reads\nH=/integraldisplayd3r\na3/bracketleftbiggJS2\n2[(∇θ)2+sin2θ(∇φ)2]+KS2\n2sin2θ/bracketrightbigg\n. (45)\nThe total energy is minimized by the conditions\nλ2∇2θ−sinθcosθ(1+λ2(∇φ)2) = 0\n∇(sin2θ∇φ) = 0, (46)\nwhere\nλ=/radicalbigg\nJ\nK, (47)\n9turns out to be the thickness of the wall. A static domain wall solution is obtained as\ncosθ=±tanhx−X\nλ\nsinθ=1\ncoshx−X\nλ, (48)\nandφisanyconstant. Wechosethewalldirectionalongthe xaxis,butthechoiceismathematicallyarbitrary\nas far as the spin space and coordinate space are decoupled, i.e., if s pin-orbit interaction is neglected. Value\nofφis also arbitrary in the present system without hard-axis anisotrop y. Historically, a wall with φ= 0\nin Eq. (48), where the localized spins in the wall has a component perp endicular to the wall plane (the\nyz-plane), is called the N´ eel wall, while the case of φ=π\n2with localized spins rotating in the wall plane is\ncalled the Bloch wall. In wires, anisotropy axis tends to be along the wir e direction (here x) to reduce the\nmagnetostatic energy, and another type of N´ eel wall is realized.\nFigure 4: Domain wall structures. Choosing the wall directi on as the xaxis, left figure corresponds to a N´ eel wall with Eq.\n(48) and φ= 0, the middle figure is a Bloch wall with φ=π\n2. Right figure is another type of N’eel wall realized in wires.\n3.4. Domain wall dynamics\nWe describe here the wall dynamics when a magnetic field is applied along the easy axis. It may appear\nthat dynamics wall is described simply by replacing the wall center coo rdinateXin Eq. (48) by a time-\ndependent variable, X(t). This is not, however, sufficient, and we need to introduce φas another dynamical\nvariable,φ(t) (Slonczewski, 1972; Tatara et al., 2008). A hard-axis anisotrop y energy, therefore, plays an\nessential role in the wall dynamics. We introduce it choosing the hard axis as theyaxis. Anisotropy energies\nwe consider are thus\nHK≡KS2\n2/integraldisplayd3r\na3sin2θ(1+κsin2φ), (49)\nwhereκ≡K⊥\nKwithK⊥being the hard-axis anisotropy energy. The external magnetic fie ldHalongzaxis\nis represented by the Hamiltonian\nHH=/integraldisplayd3r\na3µ0/planckover2pi1γH(Sz(X(t))−Sz(X= 0)), (50)\nwhere we subtracted an irrelevant constant. We consider a rigid wa ll, namely, the wall structure does not\nchange when dynamic, which requires that K≫K⊥. The low energy dynamics of the wall is thus described\nby the wall profile of\nnz(x,t) = tanhx−X(t)\nλ, n±(x,t)≡nx±iny=e±iφ(t)\ncoshx−X(t)\nλ, (51)\nwhere two dynamics variables, X(t) andφ(t) are called the collective coordinates. Rewriting the spin\nLagrangian using Eq. (51), we obtain\nL=/planckover2pi1NwS\nλ/bracketleftbigg\n−˙φX−K⊥λS\n2/planckover2pi1sin2φ+µ0γHX/bracketrightbigg\n, (52)\n10where we used/integraltext\ndx1\ncosh2(x/λ)= 2λandNw≡2Aλ\na3is the number of localized spins in the wall, with Abeing\nthe cross sectional area of the system. The dissipation function is written using collective coordinates as\nWS=α/planckover2pi1NwS\n2/bracketleftigg˙X2\nλ2+˙φ2/bracketrightigg\n. (53)\nThe equations of motion obtained from Eqs. (52) (53) read\n˙X−αλ˙φ=vcsin2φ\n˙φ+α˙X\nλ=µ0γH, (54)\nwhere\nvc≡K⊥λS\n2/planckover2pi1. (55)\nThese equations (neglecting dissipation) are Hamilton’s equation for position and canonical momentum\n(P),\n˙X=δH\nδP, ˙P=−δH\nδX. (56)\nThis means that the canonical momentum of domain wall is φand not simply proportional to ˙Xlike an\nparticle. This fact is obviously seen in the Lagrangian (52), where th e first term describes the canonical\nrelation between variables. A domain wall therefore has an intriguing property that angle φneeds to be\nfinite to have a translational motion. This feature can be understo od based on the spin dynamics (Fig. 5).\nFigure 5: Mechanism of domain wall motion when an external ma gnetic field His applied along the easy axis (top figure).\nThe magnetic field induces a precession of localized spins an d out-of-plane component (middle figure). This results in an\nout-of-plane magnetic field Bs, which induces a precession within the plane, which is equiv alent to a translational motion of\nthe wall (bottom figure).\nAs spin dynamics is always a precession around a magnetic field, localize d spins in the wall tends to be\nout-of the wall plane when magnetic field is applied along the easy axis, namely,φnecessarily develops. The\nout-of the plane component then induces another precession with in the wall plane, and this is equivalent to\na translational motion of the whole wall. This complex behavior is expre ssed theoretically by a single term,\n˙φX, in the Lagrangian.\nThe solution for Eq. (54) shows different behavior for two regimes o f magnetic field, H < H wand\nH≥Hw, where (γ <0)\nHw≡−µ0αvc\nλγ. (57)\n11In the weak field regime, wall has a constant speed\n˙X=µ0λγH\nα, (58)\nand the angle φis determined by the speed as ˙X=vcsin2φ. This means that the torque necessary for wall\nmotion is supplied by tilting the wall plane by the finite angle φ. WhenB >B w, the static tilt of the wall\ncannot support the wall motion, resulting in an oscillating motion of Xandφ(Walker’s breakdown). The\nsolution in this case is\nsin2φ=H\nHw+1−/parenleftig\nH\nHw/parenrightig2\nH\nHw+sin2ωt(59)\nω=vc\nλα\n1+α2/radicaligg/parenleftbiggH\nHw/parenrightbigg2\n−1. (60)\nThe wall speed is\n˙X=vc\nH\nHw+1\n1+α21−/parenleftig\nH\nHw/parenrightig2\nH\nHw+sin2ωt\n, (61)\nand its time-average is\n/angbracketleftig\n˙X/angbracketrightig\n=vc\nH\nHw−1\n1+α2/radicaligg/parenleftbiggH\nHw/parenrightbigg2\n−1\n. (62)\nAverage speed is plotted in Fig. 6. In the limit of high field, H≫Hw,/angbracketleftig\n˙X/angbracketrightig\n→−µ0λγHα\n1+α2.\n 0 0.5 1\n 0  0.02  0.04\nHz / H0α=0.01\nα=0.02\nα=0.04v/vc\nFigure 6: Left: Domain wall averaged speed for α= 0.01,0.02,0.04 under an easy axis external magnetic field H. Speed\nand magnetic field are normalized by vcandH0≡vc\nµ0γλ, respectively. Right: High-field regime for α= 0.1, where linear\ndependence on the field is seen.\n4. Adiabatic spin gauge field in ferromagnetic metal\nIn this section, we include conduction electron to describe a ferrom agnetic metal and study transport\nproperties. The ferromagnetic metal is modeled by a simple Hamiltonia n of a free electron with an sd\nexchange interaction with localized spin, S(r,t) =Sn(r,t);\nH=p2\n2m−Mn·σ, (63)\n12wherenis a unit vector along SandM≡JsdSis the spin energy splitting ( Jsdis thesdexchange\nconstant) and σ≡(σx,σy,σz) is the vector of Pauli matrix representing the electron spin opera tor. In most\n3dmetallic ferromagnets, sdexchange interaction is strongest energy scale for spin dynamics. (Compared\nto Fermi energy ǫF, the ratio is M/ǫF/greaterorsimilar0.1 in most cases (Kittel, 1996).) The expectation value of electron\nspin,∝an}b∇acketle{tσ∝an}b∇acket∇i}ht, is therefore locked to the direction nalmost perfectly. This limit is called the adiabatic limit.\n4.1. Phase from spin texture\nFigure 7: In the presence of localized spin structure, condu c-\ntion electron spin feels a position-dependent effective mag -\nnetic field.\nFigure 8: Unitary transformation Uis defined to connect up\nspin state|↑∝angbracketrightand a state pointing n.\nTransport of conduction electrons in the adiabatic limit is theoretica lly studied by calculating the quan-\ntum mechanical phase attached to the wave function of electron s pin. Let us consider a conduction electron\nhopping from a site rto a different site at r′(Fig. 7). The localized spin direction at those sites are\nn(r)≡nandn(r′)≡n′, respectively, and the electron spin’s wave function at the two site s are\n|n∝an}b∇acket∇i}ht= cosθ\n2|↑∝an}b∇acket∇i}ht+sinθ\n2eiφ|↓∝an}b∇acket∇i}ht\n|n′∝an}b∇acket∇i}ht= cosθ′\n2|↑∝an}b∇acket∇i}ht+sinθ′\n2eiφ′|↓∝an}b∇acket∇i}ht, (64)\nwhere|↑∝an}b∇acket∇i}htand|↓∝an}b∇acket∇i}htare spin up and downs states, respectively and θ,φandθ′,φ′are the polar angle of n(r)\nandn(r′), respectively (Fig. 8). The wave functions are concisely written b y use of matrices, U(r), which\nrotates the spin up state |↑∝an}b∇acket∇i}htto|n∝an}b∇acket∇i}ht, as|n∝an}b∇acket∇i}ht=U(r)|↑∝an}b∇acket∇i}ht(Fig. 8). The rotation is a combination of rotation of\nangleθaround +yaxis followed by a rotation of φaround +zaxis, represented by a matrix e−i\n2φσze−i\n2θσy.\nWe add irrelevant phase factors and define the rotation matrix as\nU≡eiπ\n2e−i\n2φσze−i\n2θσye−i\n2(π−φ)σz=/parenleftbiggcosθ\n2e−iφsinθ\n2\neiφsinθ\n2−cosθ\n2/parenrightbigg\n=m·σ, (65)\nwhere\nm≡/parenleftbigg\nsinθ\n2cosφ,sinθ\n2sinφ,cosθ\n2/parenrightbigg\n. (66)\nThe overlap of the electron wave functions at the two sites is thus ∝an}b∇acketle{tn′|n∝an}b∇acket∇i}ht=∝an}b∇acketle{t↑|U(r′)−1U(r)|↑∝an}b∇acket∇i}ht. When\nlocalized spin texture is slowly varying, we can expand the matrix prod uct with respect to a≡r′−ras\nU(r′)−1U(r) = 1−U(r)−1(a·∇)U(r)+O(a2) to obtain\n∝an}b∇acketle{tn′|n∝an}b∇acket∇i}ht≃1−∝an}b∇acketle{t↑|U(r)−1(a·∇)U(r)|↑∝an}b∇acket∇i}ht=eiϕ+O(a2), (67)\nwhere\nϕ≡ia·∝an}b∇acketle{t↑|U(r)−1∇U(r)|↑∝an}b∇acket∇i}ht≡1\n/planckover2pi1a·As. (68)\n13Since (U−1∇U)†=−U−1∇U,ϕis real. A vector Ashere plays a role of a gauge field, similarly to that of\nthe electromagnetism, and it is called (adiabatic) spin gauge field. By u se of Eq. (65), this gauge field reads\n(including the factor of1\n2representing the magnitude of electron spin)\nAs≡i/planckover2pi1∝an}b∇acketle{t↑|U(r)−1∇U(r)|↑∝an}b∇acket∇i}ht=−/planckover2pi1\n2(1−cosθ)∇φ. (69)\nFor a transport between general points connected by a path C, the phase is written as an integral along C\nas\nϕ=1\n/planckover2pi1/integraldisplay\nCdr·As. (70)\n4.2. Spin electromagnetic field\nExistence of path-dependent phase means that there is an effect ive magnetic field, Bs, defined when the\ncontourCis a closed path. In fact, the contour integral is written by use of t he Stokes theorem as a surface\nintegral as\nϕ=e\n/planckover2pi1/integraldisplay\nSdS·Bs, (71)\nwhere\nBs≡∇×As, (72)\nrepresents a curvature or an effective magnetic field. This phase ϕ, arising from strong sdinteraction,\ncouples to electron spin, and is called the spin Berry’s phase.\nTime-derivative of phase is equivalent to a voltage, and thus we have an effective electric field, too.\nApplying the argument of Eq.(67) to the case where spin direction is c hanging with time, the phase factor\nattached during t= 0 tot=ton the electron wave function is eiϕ(t)with\nϕ(t) =1\n/planckover2pi1/integraldisplayt\n0dtAs,t, (73)\nwhere\nAs,t≡/planckover2pi1\n2(1−cosθ)∂tφ, (74)\nis a scalar potential arising from spin dynamics. The sum of the two co ntributions to the phase, Eqs.\n(70)(73) leads to the time-derivative of the total phase as\n˙ϕ=−1\n/planckover2pi1/integraldisplay\nCdr·Es, (75)\nwhere\nEs≡−˙As−∇As,t, (76)\nis the effective electric field. The definitions of the fields (72)(76) ar e the same as the electromagnetic field\nof Eq. (3). The two fields EsandBscoupling to the electron spin are called spin electromagnetic fields ( As\nis spin gauge field).\nIn terms of vector nthe effective fields read\nEs,i=/planckover2pi1\n2n·(˙n×∇in)\nBs,i=−/planckover2pi1\n4/summationdisplay\njkǫijkn·(∇jn×∇kn). (77)\n14Figure 9: Magnetization structures, n(r), of a hedgehog monopole having a monopole charge of Qm= 1 and the one with\nQm= 2 . At the center, n(r) has a singularity and this gives rise to a finite monopole cha rge.\nIn terms of polar coordinates, the magnetic component reads\nBs,i=−/planckover2pi1\n2/summationdisplay\njkǫijksinθ(∇jθ)(∇kφ), (78)\nindicating that it has a geometrical meaning of the area defined by th e magnetization structure (sin θδθδφ\nis the area element for small angle variation δθandδφ). The classical effect of the spin electromagnetic field\nfor electron is given by the same as the conventional electromagne tism (’charge1\n2is included in definition\nofEsandBs);\nm¨r=±Es±˙r×Bs, (79)\nwhere the sing±denotes spin direction. This is obvious from the minimal coupling form, which we shall\nargue later in Eqs. (94) (164).\n4.3. Topological monopole\nAs a trivial consequence of the definition, they appear to satisfy t he Faraday’s law and condition of no\nmonopole,\n∇×Es+˙Bs= 0, ∇·Bs= 0, (80)\nbecause spin vector with fixed length has only two independent varia bles, and therefore/summationtext\nijkǫijk(∇in)·\n(∇jn×∇kn) = 0. They are correct as a local equation. The nature, however, sometimes exhibit surprising\npossibilities that we may not imagine straightforwardly. In the prese nt case, those equations may be broken\nglobally due to a topological reason. Let us define a spin magnetic cha rge (monopole charge) as\n∇·Bs≡−ρm, (81)\nwhich appears to vanish locally, and show that its volume integral, Qm≡/integraltext\nd3rρm, can be finite. In fact,\nusing the Gauss’s law we can write\nQm=/integraldisplay\nr=∞dS·Bs=h\n4π/integraldisplay\ndΩ, (82)\nwhere/integraltext\nr=∞dSdenotes integral over surface at spatial infinity and the last integ ral,/integraltext\ndΩ≡/integraltext\nsinθdθdφ, is\nover the spin direction at the infinity. It thus follows that\nQm=h×integer (83)\n15since1\n4π/integraltext\ndΩ is a winding number, an integer, of a mapping from a sphere in the coo rdinate space to a\nspherein spinspace. Ifthemapping istopologicallynon-trivial, the mo nopolechargebecomesfinite. Typical\nnontrivial structures of nare shown in Fig. 9. The singular structure with a single monopole char geis called\nthe hedgehog monopole from its shape. In a local picture, such top ological monopole arises because the spin\nconfigurationshaving monopole alwayscontain at least one singularp oint where the derivative ∂µndiverges.\nIn the case of as symmetric hedgehog monopole, singularity is at the center of monopole. Such singularities\ncannot be removed by continuous deformation of spin configuratio n, and is therefore topologically stable in\na continuum. One should notice, however, that the topological sta bility is not exact in solids; since localized\nspins are on a discrete lattice, singularities can be annihilated or crea ted with a finite excitation energy.\nThis fact reduces mathematical beautifulness, but is essential fo r applications, since ’topological’ objects\nlike domain wall or vortex can be created externally.\nSimilarly, the Faraday’s law reads\n(∇×Es)i+˙Bsi=−/planckover2pi1\n4/summationdisplay\nijkǫijk˙n·(∇jn×∇kn)≡−jm, (84)\nwhich vanishes locally but is finite when integrated, allowing a topologica l monopole current jmto be finite.\nThe gauge field has a constraint arising from the requirement that a gauge field covering the whole space\nwithout singularity be constructed by patching together locally-de fined gauge fields. In fact, a definition\n(for spatial component)\nAN\ns,i=−g\n4π(1−cosθ)∇iφ, (85)\nwhere superscript N denotes north and gis the monopole charge, is not well-defined at θ=π(south pole).\nA gauge field that is regular at the south pole is defined as\nAS\ns,i=g\n4π(1+cosθ)∇iφ. (86)\nThis field has a singularity at the north pole ( θ= 0), but represents the same effective magnetic field\n(∇×AN\ns=∇×AS\nsexcept at poles). To cover the whole space by either of the gauge fi elds, we have\nnecessarily a singularity. The singularity is the Dirac string. Instead of playing with singular gauge field,\nwe can cover the whole space by patching two gauge fields, AS\ns,iandAN\ns,i. They are related by a gauge\ntransformation\nAN\ns,i=AS\ns,i+i/planckover2pi1Θ−1∇iΘ, (87)\nwhere\nΘ≡e−ig\nhφ(88)\nis a gauge transform function. This function must be single-valued, i.e., be invariant under φ→φ+ 2π.\nThus, a condition\ng= 2π/planckover2pi1n, (89)\nwherenis an integer is imposed (Dirac’s quantization condition). This condition imposing the magnitude\nof spin to ben\n2is quantization of spin.\nThe other two Maxwell’s equations describing ∇·Esand∇×Bsare derived by evaluating the induced\nspin density and spin current based on linear response theory (Tak euchi and Tatara, 2012; Tatara et al.,\n2012). It may appear surprising that the complete Maxwell’s equatio ns for electromagnetism is derived by\ndiscussionofspin-polarizedelectron. Itis, however,justnatura l,becauseelectromagnetismarisesfromacon-\nservation law of charge, which corresponds to spin angular moment um in our adiabatic context. Therefore,\nelectromagnetism is automatically derived if we carry out a correct c alculation that keeps the conservation\nlaw.\n16Es Bs\nFigure 10: Spin electric field Esand spin magnetic field Bsact oppositely for electrons with opposite spin, generatin g spin\ncurrent.\nThe electromagnetism of spin gauge field was discussed by G. Volovik ( Volovik, 1987), and SU(2) hedge-\nhog monopole was argued. The mechanism for emergence of spin gau ge field and monopole is identical to\nthe one pointed out by G. t’Hooft and A. M. Polyakov (Hooft, 1974; Polyakov, 1974) for the case of larger\nnon-Abelian group, introduced for explaining the emergence of elec tromagnetism as a result of a symmetry\nbreaking of a grand unified theory (GUT).\nThe idea of spin gauge field (spin vector potential) was introduced to describe Heisenberg models by P.\nChandra et al. (Chandra et al., 1990). SU(2) gauge description of spin and charge transport was discussed\nin the Boltzmann equation approach in Ref. (Raimondi et al., 2012). E ffective gauge field (artificial gauge\nfield) plays important roles also in cold atom systems (Phuc et al., 2015 ; Aidelsburger et al., 2017).\n4.4. Detection of spin electromagnetic fields\nAlthough spin magnetic field is often called a ’fictitious’ magnetic field, s pin magnetic fields are real fields\ndetectable in transport measurements. They couple to the spin po larization of the electrons according to\nEq. (79) (Fig. 10), and so they are measurable by spin current mea surements. Fortunately, in ferromagnetic\nconductors, conventional electric measurements are sufficient f or detection, because spin current jsis always\naccompanied with electric current jasjs=Pj, wherePis the spin polarization. The electric component\nEsis therefore directly observable as a voltage generation from magn etization dynamics. In experiments,\nvoltage signals of µV order have been observed for the motion of domain walls and vortic es (Yang et al.,\n2009; Tanabe et al., 2012). The spin magnetic field causes an anoma lous Hall effect of spin, i.e., the spin\nHall effect or the topological Hall effect. The spin electric field arises if magnetization structure carrying\nspin magnetic field becomes dynamical due to the Lorentz force fro mBsaccording to Es=v×Bs, wherev\ndenotes the electron spin’s velocity. The topological Hall effect due to skyrmion lattice turned out to induce\nHall resistivity of 4nΩcm (Neubauer et al., 2009; Schulz et al., 2012) . Although those signals are not large,\nexistence of spin electromagnetic fields is thus confirmed experimen tally. It was recently shown theoretically\nthat spin magnetic field couples to helicity of circularly polarized light (t opological inverse Faraday effect)\n(Taguchi et al., 2012), and an optical detection is thus possible.\n5. Minimal coupling of spin gauge field\nSo far we have considered adiabatic component of effective gauge fi eld, starting from a phase factor\nattached to electron spin. As we see from its construction, the ga uge field has originally three components\ncorresponding to spin space, and thus is an SU(2) gauge field. It re duced to an effective U(1) gauge field\nwhen an expectation value was taken in Eq. (68). Here we discuss th e effect of the effective gauge field\ntaking account of its SU(2) nature.\nThe effective gauge field arising from a unitary transformation U, defined in Eq. (65), is (negative and\npositive signs correspond to µ=tandµ=x,y,z, respectively)\nAs,µ=∓i/planckover2pi1U−1∂µU, (90)\nand is expressed by use of Pauli matrices as\nAs,µ=Aα\ns,µσα≡As,µ·σ. (91)\n17The three spin components of vector As,µare\nAs,µ=±/planckover2pi1\n2\n−∂µθsinφ−sinθcosφ∂µφ\n∂µθcosφ−sinθsinφ∂µφ\n(1−cosθ)∂µφ\n. (92)\nIt can be represented as\nAs,µ=±/planckover2pi1\n2n×∂µn−As,µn, (93)\nwhereAs,µ≡Az\ns,µis the adiabatic spin gauge field we discussed in Sec. 4.\nBeing a gauge field for electron spin, the spin gauge field couples to th e spin current via the minimal\ncoupling. To the first order, the coupling reads (see Sec. 9.2 for de rivation)\nHA=−/integraldisplay\nd3r/bracketleftbig\njα\ns,iAα\ns,i−sαAα\ns,t/bracketrightbig\n. (94)\nHerejα\ns,iandsαdenote spin current and spin density in the frame after unitary tra nsformation, i.e., in the\nrotated frame, respectively. (The effective gauge field is a quantit y defined in the rotated frame.) Let us\nwrite the rotated frame spin current and density as\njα\ns,i=js,iˆzα+(j⊥\ns,i)α, sα=sˆzα+(s⊥)α, (95)\nwhere (j⊥\ns,i)α≡jα\ns,i−ˆzαjs,iand (s⊥)α≡sα−ˆzαsrepresent nonadiabatic components, with js,i≡jz\ns,iand\ns≡sz. The gauge coupling then reads\nHA=−/integraldisplay\nd3r/bracketleftbig\njs,iAs,i−sAs,t+(j⊥\ns,i)α(A⊥\ns,i)α−(s⊥)α(A⊥\ns,t)α/bracketrightbig\n, (96)\nwhere (A⊥\ns,µ)α≡Aα\ns,µ−ˆzαAs,µis the nonadiabatic gauge field.\nWe now argue that the gauge coupling directly indicates the following s everal important effects.\n1. Effects on electron transport\n(a) Adiabatic spin electromagnetic field\n(b) Spin current generation\n2. Effects on magnetism\n(a) Spin-transfer effect\n(b) Antisymmetric exchange (DM) interaction\nThe adiabatic spin gauge field was already explained in Sec. 4. Let us br iefly explain other effects.\nSpin current generation. Application of spin gauge field, Aα\ns,µ, leads to generation of spin current and\ndensity. Of particular interest is the first term of Eq. (93). Its sp atial component induces equilibrium\nspin current proportional to n×∇in(Fig. 11). This spin current represents a torque acting between n on\ncollinear localized spins (Tatara et al., 2008). If in a junction of two f erromagnets with localized spin, Si\n(i= 1,2), the current reduces to a discrete form of jα\ns,i∝(S1×S2)α(e12)iproportional to vector chirality\n(e12represents the vector connecting site 1 to site 2). The time-comp onentAs,tcouples to spin density, and\nformsa spin accumulation; It is an effective chemicalpotential for e lectron spin. In a junction of ferromagnet\nand normal metal, the accumulation at the interface leads to a spin c urrent generationinto the normal metal\nproportional to n×˙n(Fig. 11), i.e., spin pumping effect occurs (Silsbee et al., 1979; Tserk ovnyak et al.,\n2002; Tatara and Mizukami, 2017). This effect is explained in detail in S ec. 7.\n18njs=n×∇n\nF F F Nn(t)js∝n×˙n\nFigure 11: Spin current ( js) generation from magnetization. Left figure represetns an e quilibrium spin current due to magne-\ntization structure, while right figure describes the spin pu mping effect in a FN junction. In spin pumping, spin accumulat ion\nis generated by effective chemical potential n×˙nat the interface, resulting in a spin current proportional t on×˙nin N. The\neffects are described by the minimal coupling between the spi n current and spin gauge field arising from the magnetization\ntexture or dynamics.\nSpin-transfer torque. The opposite effects of the interaction Eq. (96) is the electrons’ e ffects on magnetiza-\ntion. In the adiabatic limit, Eq. (96) reduces to\nHz\nA=−/integraldisplay\nd3r[As·js−sAs,t]. (97)\nNoting that the expression for As,tcoincides with that of spin Berry’s phase term of the spin Lagrangian ,\nLB, the spin Lagrangian taking account of Eq. (97) reads\nLB−Hz\nA=/integraldisplay\nd3r/bracketleftbigg\n−2\na3As,0S−sAs,t+As·js/bracketrightbigg\n=/integraldisplayd3r\na3/bracketleftbigg\n/planckover2pi1¯S(cosθ−1)/parenleftbigg∂\n∂t+vs·∇/parenrightbigg\nφ/bracketrightbigg\n, (98)\nwhere¯S≡S+sa3\n2is the magnitude of the total spin and\nvs≡a3\n2¯Sjs. (99)\nThis Galilean invariant form indicates that any spin structure under s pin current flows along the current\nwith velocity vsin the adiabatic limit. This is the spin-transfer effect, which can be app lied to drive\nmagnetization structure by injecting electric current. (As the to tal spin measured in experiments always\ncontains the adiabatic component of electron spin, the localized spin magnitude S(like the on in Sec. 3)\nshould be regarded as ¯S, although we use notation Sfor the total spin for simplicity.)\nDzyaloshinskii-Moriya (DM) interaction. In contrast, nonadiabatic spin current, ( j⊥\ns,i)α, induces antisym-\nmetric exchange interaction, the Dzyaloshinskii-Moriya (DM) intera ction. This is seen from an identity\nRαβ(A⊥\ns,i)β=−/planckover2pi1\n2(n×∇in), (100)\nwhereRαβ≡2mαmβ−δαβis a rotation matrix, mbeing defined in Eq. (66). In fact, using this identity,\nthe spatial nonadiabatic terms of Eq. (96) turns out to be the DM in teraction,\nH⊥\nA≡−/integraldisplay\nd3r(j⊥\ns,i)α(A⊥\ns,i)α=/integraldisplayd3r\na3Dα\ni(n×∇in)α, (101)\nwhere\nDα\ni=−/planckover2pi1a3\n2Rαβ(j⊥\ns,i)β. (102)\n19WehavethereforeaninterestingidentitythatDM coefficientisthe m agnitudeofnonadiabaticspincurrentin\nthe laboratory frame, ( j⊥,(L)\ns,i)β≡Rαβ(j⊥\ns,i)β(Kikuchi et al., 2016). (Here the spin current density is defined\nwithout electric charge eand spin magnitude1\n2). This simple formula tells us a microscopic mechanism for\nemergenceofDMinteraction,namely, inversionsymmetrybreaking givesrisetoafiniteintrinsicspincurrent,\nand DM interaction arises as a result of ’Doppler shift’ (Kikuchi et al., 2016). The form (102) is unique in\nthe sense that the DM coefficient is not described by a correlation fu nction like most physical parameters\nlike exchange interaction. The formula thus enables us numerical ev aluation with less computing time than\nprevious formula. For strong spin-orbit interaction, deviation fro m Eq. (102) is theoretically predicted\n(Freimuth et al., 2017).\nThe Doppler shift picture becomes clear if we regard the DM interact ion as a modification of ferromag-\nnetic exchange interaction due to spin current. In fact, in a moving frame, a spatial derivative is replaced\nby a covariant form (Kim et al., 2013; Kikuchi et al., 2016)\nDinα=∇inα+ηǫαβγ(j(L)\ns,i)βnγ, (103)\nwhereηis a coefficient. Similar Doppler shift for a vector in a moving medium has b een known in the case\nof the velocity vector of sound wave (Landau and Lifshitz, 1987). The magnetic exchange energy induced\nby electron, proportional to ( ∇n)2in the rest frame, is then modified to be ( Din)2= (∇n)2+2η/summationtext\nij(L)\ns,i·\n(n×∇in)+O(η2), resulting in a DM interaction\nIt has been known that in the presence of DM interaction spin waves around uniform ferromagnetic\nstate show nonreciprocal propagation as a result of Doppler shift (Kataoka, 1987), as confirmed in recent\nexperiments (Iguchi et al., 2015; Seki et al., 2016). This effect is na tural from our theory, because DM\ninteraction itself is a sign of internal flow of spin.\nThe spin current that determines the DM interactionby Eq. (102) c an be the equilibrium one orthe non-\nequilibriumonesuchastheoneinjectedexternally. Ourformula(102 )thusindicatesaninterestingpossibility\nto modulate DM interaction by injecting spin current. For a spin curr ent density of ejs= 1012A/m2, the\nmodulation is estimated to be δD=/planckover2pi1a3\n2js= 2.6×10−33Jm= 0.16 meV˚A fora= 2˚A. This value is an\norder of magnitude smaller than the one in natural strongly chiral m aterials such as MnFeGe. In this sense,\nintrinsic spin current induced by atomic spin-orbit interaction is large r than what we can do. Nevertheless,\nexternal controlof DM interaction by current application would be useful for weakly chiral magnets. Voltage\ncontrol of DM interaction has been experimentally demonstrated r ecently (Nawaoka et al., 2015).\n5.1. Perturbative picture of spin gauge field\nWe discussed emergence of spin gauge field so far in the adiabatic limit. The concept of spin gauge\nfield exists also in the weak sdexchange interaction regime. In fact, adiabatic condition justifies gradient\nexpansions and adiabatic limit can be realized even in the weak sdcoupling case if the gradient is small\nenough. (See Eq. (113).) In this subsection, we present a pertur bative picture of spin gauge field in the\nweaksdlimit.\nThesdexchange interaction with a localized spin SisHsd=−JsdS·σ. Let us consider interaction with\ntwo localized spins S1andS2. As a second order contribution, the electron spin wave function a cquires a\nphase proportional to\nV2≡(Jsd)2(S1·σ)(S2·σ) = (Jsd)2[(S1·S2)+i(S1×S2)·σ]. (104)\nThe first term on the right-hand side describes the amplitude of cha rge part, in other words, magnetore-\nsistance effect. The second term containing Pauli matrix indicates t hat spin current (and/or density) are\ninduced as a result of non collinear localized spin as (Fig. 12)\njα\ns,i∝(Jsd)2(S1×S2)α(e12)i, (105)\nwheree12is a unit vector representing relative spatial position of S1andS2. (Here spin current is in the\nlaboratory frame, as rotating frame description is not valid in the pe rturbative regime.) Let us consider a\n20n1 n2jsn2\nn3 n1\nj\nFigure 12: Equilibrium currents generated by localized spi n structures, represented by ni. Non collinear spin structure (vector\nspin chirality) induces a spin current with polarization pr oportional to the vector chirality, js∝n1×n2, while non coplanar\nspin structure (scalar spin chirality) leads to a charge cur rentj∝n1·(n2×n3) as a result of breaking of time-reversal\nsymmetry.\njunction of two ferromagnets with localized spins S1andS2(Fig. 11). The spin current (105) in this case\nflows between the two ferromagnetic layers. It is an equilibrium curr ent that arises even in the static spin\nconfiguration, and is a kind of persistent or super current if spin re laxation effect is neglected. Spin current\nindicates that dynamics is induced as a result of angular momentum ch ange. The equations of motion for\nthe two localized spins read (neglecting external magnetic field)\n˙S1=−α\nS(S1×˙S1)+c(S1×S2)\n˙S2=−α\nS(S2×˙S2)−c(S1×S2), (106)\nwherecis a constant. Equation (106) indicates that the two ferromagnet s tends to align S1andS2parallel\nor anti parallel. This is natural because localized spin S2acts as an effective magnetic field for localized spin\nS1and vice versa. The equilibrium spin current of Eq. (105) therefore represents the torque mediated by\nthe conduction electron between the two localized spins. For smoot h localized spins, the expression reduces\nto a continuum form of\njα\ns,i∝(Jsd)2(S×∇iS)α. (107)\nIf we regard the two localized spins as the one at different time, Eq. ( 105) is a dynamic spin current,\nnamely, we have a spin pumping effect. In the slow change of localized s pins, the current is proportional to\n(Jsd)2(S×˙S), (108)\nwhich is a perturbative picture of spin pumping effect (Tatara and Miz ukami, 2017).\nWe sawthat the secondordercontribution ofthe sdexchangeinteraction is governedby a vectorchirality\nof spins, ( S1×S2) for two spins. This quantity is the non-adiabatic component of the spin gauge field in\nthe adiabatic limit, as seen in Eq. (93). The spin gauge field, therefor e, arises from the non-commutative\nalgebra of spins.\nWe can extend the discussion to the third order. The charge part o f the third order amplitude is\ntr[V3] = (Jsd)3tr[(S1·σ)(S2·σ)(S3·σ)] = 2i(Jsd)3S1·(S2×S3), (109)\nnamely, proportionaltothe scalarchirality S1·(S2×S3) ofthe threespins. This indicatesthataspontaneous\nchargecurrentisinducedbythescalarchiralityoflocalizedspinsasa resultofbrokentime-reversalsymmetry\n(Fig. 12) (Loss and Goldbart, 1992; Tatara and Kohno, 2003). Th is effect is in fact the spin Berry’s phase\neffect as seen by noticing that the continuum limit of the scalar chiralit y isS·(∂iS×∂jS) (iandjare\ndirection of relative spin positions), which agrees with the spin magne tic field, Eq. (77). The persistent\ncurrent represented by Eq. (109) is described as the Amp’ere’s law ,∇×Bs=j(Takeuchi and Tatara,\n2012).\n21Spin chirality persistent charge current gives rise to an anomalous H all effect (Tatara and Kawamura,\n2002; Tatara and Kohno, 2003). It was predicted that spin chiralit y alsoaffects optical response to circularly\npolarized light (topological inverse Faraday effect) (Taguchi et al., 2012). Direct observation of persistent\ncurrent was carried out recently for the case of neutron (Tatar skiy et al., 2016).\nThe spin current (107) is an equilibrium one and cannot be ’converted ’ into a charge current by use of\nthe inverse spin Hall effect, as was mentioned based on a microscopic analysis (Takeuchi et al., 2010); As\nfor magnetically induced spin current, the inverse spin Hall effect ac ts only for non equilibrium one, where\ndynamics is involved. In the case of junction of two ferromagnets ( Fig.11), inverse spin Hall signal shall\narise when the magnetizations start to precess following Eq. (106) . The excess magnetic energy the initial\nstate had is dissipated as joule heat associated with the charge cur rent.\nEquilibriumspincurrenthasbeenpointed outtoinduceelectricpolariz ationin insulators(Katsura et al.,\n2005). Thismagnetoelectriceffectduetomagneticinhomogeneity( spinvectorchirality)waspredictedearlier\nin Ref. (Bar’yakhtar et al., 1983).\n5.2. Momentum space monopole\nIn Sec. 4 and in the previous subsection, we discussed spin gauge fie ld in the real space picture. On the\nother hand, it has been known that the Berry’s curvature in the mo mentum space plays essential roles in\ntransport phenomena such as anomalous Hall effect (Thouless et a l., 1982; Nagaosa et al., 2010). In clean\nfrustrated magnets anomalous Hall conductivity has been shown t o arise from monopoles in the momentum\nspaceasa resultofa non-coplanarspin structure. In this moment um picture, roleofrealspacespin magnetic\nfieldBsis not clear. In contrast, chirality-induced anomalous Hall conduct ivity in disordered metals was\nshown to be governed by real space chirality (Tatara and Kawamur a, 2002; Nakazawa and Kohno, 2014).\nThese features are understood as follows (Onoda et al., 2004). In the clean limit, electrons form bands\ndefined including effects of sdexchange interaction, Jsd. The effect of localized spin structures such as\nchirality are contained in each bands as monopole density. In the diso rdered limit, Jsdτe//planckover2pi1≪1 (τeis elastic\nlifetime of electron), in contrast, bands smeared by energy scale o f/planckover2pi1/τeno longer keep the information of\nspin structure; Instead, the real space spin structure affects the electron hopping amplitude and transport.\n6. Spin-transfer effect : Phenomenology\nAs we have seen in Sec. 5, spin transfer-effect is a direct conseque nce of minimal coupling between\nspin current and adiabatic spin gauge field representing magnetizat ion structure. Here let us present a\nphenomenological theory for the effect for the case of transmiss ion through a domain wall based on quantum\nmechanics. The issue here is how the angular momentum is transfere d between conduction electron and\nlocalized spin via the sdexchange interaction. The thickness of the wall, λ, in typical ferromagnets is\nλ= 10−100nm, and is much larger than the typical length scale of electron, the Fermi wavelength, 1 /kF,\nwhich is atomic scale in metals. The wall is therefore a slowly varying spin structure for conduction electron.\nWe choose the zaxis along the direction of localized spins’ change, and magnetic easy axis for localized\nspins is chosen as along zaxis.3Atz=∞the localized spin is Sz=S, and isSz=−Satz=−∞.\nWe describe electron states with spin along + zand−zdirection by→and←, respectively. Because of the\ndomain wall, the electron is in a potential barrier,\nV→(z) =−JsdSz(z),V←(z) =JsdSz(z). (110)\nNamely, for←electron, the potential in the left regime is low, while that in the right r egion is high (dotted\nlines in Fig. 13).4Considering the domain wall centered at x= 0 having profile of\n3The mutual direction between the localized spin and directi on of spin change is irrelevant in the case without spin-orbi t\ninteraction.\n4We choose the sign of sdexchange interaction as positive, but the sign does not matt er for the spin-transfer effect.\n22Figure 13: Potential energy V(z) arising from sdexchange interaction for conduction electron with spin →and←. The energy\ngaps is 2 M= 2SJsd. Dotted lines are the cases neglecting spin flip inside the wa ll, while solid lines are with spin flip.\nFigure 14: Conduction electron incident on a domain wall may go through the wall or get reflected. The former process\noccurs for a thick wall (adiabatic limit) and electron spin i s rotated, resulting in a spin transfer effect. The latter pro cess is an\nnonadiabatic effect, and leads to a force on a domain wall and e lectric resistance.\nSz(z) =Stanhz\nλ, Sx(z) =S\ncoshz\nλ, Sy= 0, (111)\nconduction electron’s Schr¨ odinger equation with energy Ereads\n/bracketleftbigg\n−/planckover2pi12\n2md2\ndz2−JsdS/parenleftbigg\nσztanhz\nλ+σx1\ncoshz\nλ/parenrightbigg/bracketrightbigg\nΨ =EΨ, (112)\nΨ(z) = (Ψ→(z),Ψ←(z)) begin the two-component wave function. If the spin direction of the conduction\nelectron is fixed along the zaxis, the potential barrier represented by the term proportiona l toσzleads\nto reflection of electron, but in reality, the electron spin can rotat e inside the wall as a result of the term\nproportional to σxin Eq. (112). The mixing of ←and→electron leads to the smooth potential barrier\nplotted as solid lines in Fig. 13.\nLet us consider an incident ←electron from the left. If the electron is slow, the electron spin can keep\nthe lowest energy state by gradually rotating its direction inside the wall. This is the adiabatic limit. As\nthere is no potential barrier for the electron in this limit, no reflectio n arises from the domain wall, resulting\nin a vanishing resistance (Fig. 14(a)) In contrast, if the electron is fast, the electron spin cannot follow the\nrotation of the localized spin, resulting in a reflection and finite resist ance (Fig. 14(b)). The condition for\nslowand fast is determined by the relationbetween the time for the e lectron to pass the wall and the time for\nelectron spin rotation. The former is λ/vFfor electron with Fermi velocity vF(=/planckover2pi1kF/m) (spin-dependence\nof the Fermi wave vector is neglected and mis the electron mass). The latter time is /planckover2pi1/JsdS, as the electron\nspin is rotated by the sdexchange interaction in the wall. Therefore, if\nλ\nvF≫/planckover2pi1\nJsdS, (113)\nis satisfied, the electron is in the adiabatic limit (Waintal and Viret, 200 4). The condition of adiabatic\nlimit here is the case of clean metal (long mean free path); In dirty me tals, it is modified (Stern, 1992;\nTatara et al., 2008).\n23∆X\n∆XSa∆XSa−\nFigure 15: The shift of the domain wall by a distance ∆ Xresults in a change of the spin of the localized spins∆X\naS−/parenleftBig\n−∆X\naS/parenrightBig\n= 2S∆X\na. The angular momentum change is therefore /planckover2pi1if ∆X=a\n2S.\nThe transmission of electron through a domain wall was calculated by G. G. Cabrera and L. M. Falicov\n(Cabrera and Falicov, 1974), and its physical aspects were discus sed by L. Berger (Berger, 1978, 1986).\nLinear response formulation and scattering approach were prese nted in Refs. (Tatara and Fukuyama, 1997;\nTatara, 2000, 2001).\nAs we have seen above, in the adiabatic limit, the electron spin gets ro tated after passing through the\nwall (Fig. 14(a)). The change of spin angular momentum, 2 ×/planckover2pi1\n2=/planckover2pi1, must be absorbed by the localized\nspins. (Angular momentum dissipation as a result of spin relaxation is s low compared to the exchange of\nthe angular momentum via the sdexchange interaction.) To absorb the spin change of /planckover2pi1, the domain wall\nmust shift to the right, resulting in an increase of the spins ←. We consider for simplicity the case of cubic\nlattice with lattice constant a. The distance of the wall shift ∆ Xnecessary to absorb the electron’s spin\nangular momentum of /planckover2pi1is then [/planckover2pi1/(2/planckover2pi1S)]a(Fig. 15). When we apply a spin-polarized current through\nthe wall with the spin current density js(spin current density is defined to have the unit of 1/(m2s) and\nwithout spin magnetitude of1\n2), the rate of the change of spin angular momentum of conduction e lectron\nper unit time and unit area is /planckover2pi1js. As the number of the localized spins in the unit area is 1 /a2, the wall\nmust keep moving a distance of js(a3/2S) per unit time. Namely, when a spin current density is applied,\nthe wall moves with the speed\nvs≡a3\n2Sjs\nwhich agrees with the speed we obtained in Eq. (99). It should be not ed that a simple Lagrangianargument\nof Eq. (98), even without physical argument, is sufficient to draw t he conclusion.\nThe effect was pointed out by L. Berger (Berger, 1986) in 1986, an d is now called the spin-transfer effect\nafter the papers by J. Slonczewski (Slonczewski, 1996).\nFrom the above considerations in the adiabatic limit, we found that a d omain wall is driven by spin-\npolarized current, while the electrons do not get reflected and no r esistance arises from the wall. These two\nfacts naively seem inconsistent, but are direct consequence of th e fact that a domain wall is a composite\nstructure having both linear momentum and angular momentum. The adiabatic limit is the limit where\nangular momentum is transfered between the electron and the wall, while no linear momentum is transfered.\n7. Spin pumping effect\nSpin pumping effect is a method to generate spin current in a junction of a ferromagnet (F) and a normal\nmetal (N) (Fig. 11) by exciting magnetization precession by applying an oscillating magnetic field. The\n24generated spin current density has two independent components , proportional to ˙nandn×˙n, wherenis\na unit vector describing the direction of localized spin, and thus is rep resented phenomenologically as\njs=1\n4π(Arn×˙n+Ai˙n), (114)\nwhereArandAiare phenomenological constants having unit of 1 /m2. (Spin current here is obviously\nin the laboratory frame, as it is the one in the normal metal.) Spin pump ing effect was theoretically\nformulated by Tserkovnyak et al. (Tserkovnyak et al., 2002) by us e of scattering matrix approach. This\napproach, widely applied in mesoscopic physics, describes transpor t phenomena in terms of transmission\nand reflection amplitudes (scattering matrix), and provides quant um mechanical pictures of the phenomena\nwithout calculating explicitly the amplitudes (Moskalets, 2012). Tser kovnyak et al. applied the scattering\nmatrix formulation of general adiabatic pumping (Bttiker et al., 199 4; Brouwer, 1998) to the spin-polarized\ncase. The spin pumping effect was described in Ref. (Tserkovnyak e t al., 2002) in terms of spin-dependent\ntransmissionandreflectioncoefficientsatthe FNinterface,anditw asdemonstratedthatthetwoparameters,\nArandAi, are the real and the imaginary part of a complex parameter called t he spin mixing conductance.\nThe spin mixing conductance, which is represented by transmission a nd reflection coefficients, turned out to\nbe a convenient parameter for discussing spin current generation and other effects like the inverse spin-Hall\neffect. At the same time, scattering approach hides microscopic ph ysical pictures of what is going on, as the\nscattering coefficients are not fundamental material parameter s but are composite quantities of Fermi wave\nvector, electron effective mass and the interface properties. Fo rmulation of spin pumping effect based on the\nGreen’sfunction method were presentedin Refs. (Chen et al., 2009 ; Mahfouzi et al., 2012; Chen and Zhang,\n2015; Tatara, 2016; Tatara and Mizukami, 2017). In this section, we describe the effect from a standard\nmicroscopic view point, following the approach of Ref. (Tatara and M izukami, 2017).\nSpin pumping effect is experimentally observed for both metallic and ins ulating ferromagnets. From\nphysical viewpoints, these two cases appear very different. In th e metallic case, conduction electron in\nthe ferromagnet is excited by spin gauge field arising from spin dynam ics, leading to a spin accumulation\nat FN interface and spin current generation in the normal metal. In contrast, in the case of insulator\nferromagnet, the coupling between the magnetization and the con duction electron in normal metal occurs\ndue to a magnetic proximity effect at the interface (Kang et al., 2017 ) and the pumping effect is a locally-\ninduced perturbative effect. In this paper, we consider the metallic case. The insulator case is discussed in\nRef. (Tatara and Mizukami, 2017).\nThe model we consider is a junction of metallic ferromagnet (F) and a normal metal (N). The mag-\nnetization (or localized spins) in the ferromagnet is treated as spat ially uniform but changing with time.\nThe frequency of magnetization precession is of the order of 10GH z, and is far low frequency compared\nto conduction electron spin’s frequency determined by the sdexchange interaction; For Jsd= 0.1ǫF, the\nfrequency is Jsd//planckover2pi1∼2×104GHz ifǫF= 1 eV. As a result, the conduction electron’s spin follows instan-\ntaneous directions of localized spins, i.e., the system is in the adiabatic limit. Adiabatic limit is described\nstraightforwardly by introducing a unitary transformation that r epresents the time-dependence. For the\nferromagnet, we consider a simple quantum mechanical Hamiltonian,\nHF=−/planckover2pi12∇2\n2m−ǫF−Mn(t)·σ, (115)\nwheremis the electron’s mass, σis a vector of Pauli matrices, Mrepresents the energy splitting due to the\nsdexchange interaction and n(t) is a time-dependent unit vector denoting the localized spin direction . The\nenergy is measured from the Fermi energy ǫF. For simplicity, we consider the case 0 <M <ǫF.\nAs a result of the sdexchange interaction, the electron’s spin wave function is given by ( Sakurai, 1994)\n|n∝an}b∇acket∇i}ht≡cosθ\n2|↑∝an}b∇acket∇i}ht+sinθ\n2eiφ|↓∝an}b∇acket∇i}ht (116)\nwhere|↑∝an}b∇acket∇i}htand|↓∝an}b∇acket∇i}htrepresent the spin up and down states, respectively, and ( θ,φ) are polar coordinates for\nn. To treat slowly varying localized spin, we switch to a rotating frame w here the spin direction is defined\n25with respect to instantaneous direction n. This corresponds to diagonalizing the Hamiltonian at each time\nby introducing a unitary matrix U(t) as\n|n(t)∝an}b∇acket∇i}ht≡U(t)|↑∝an}b∇acket∇i}ht, (117)\nwhereU(t) is a unitary matrix determined by polar angles θandφas in Eq. (65), but angles are time-\ndependent. The Hamiltonian in the rotated frame is diagonalized as (in the momentum representation)\n/tildewideHF≡U−1HFU=ǫk−Mσz, (118)\nwhereǫk≡/planckover2pi12k2\n2m−ǫFis the kinetic energy. As a result of unitary transformation, there arises in a rotated\nHF/tildewidestHFU\nA±s,t\nFigure 16: Unitary transformation Ufor conduction electron\nin ferromagnet converts the original Hamiltonian HFinto a di-\nagonalized uniformly spin-polarized Hamiltonian /tildewideHFand an in-\nteraction with spin gauge field, As,t·σ.\nFigure 17: For uniform magnetization, the non-adiabatic co m-\nponents of the gauge field, A±\ns,t, induce a spin flip conserving\nthe momentum. Excitation thus has an energy of M.\nframe a time-component of a gauge field with three spin components ,As,t≡−iU−1∂\n∂tU(Fig. 16). Including\nthe gauge field in the Hamiltonian, the effective Hamiltonian in the rotat ed frame reads\n/tildewideHeff\nF≡/tildewideHF+As,t·σ=/parenleftbiggǫk−M−Az\ns,tA−\ns,t\nA+\ns,tǫk+M+Az\ns,t/parenrightbigg\n(119)\nwhereA±\ns,t≡Ax\ns,t±iAy\ns,t. We see that the adiabatic ( z) component of the gauge field, Az\ns,t, acts as a\nspin-dependent chemical potential (spin chemical potential) gene rated by dynamic magnetization, while\nnon-adiabatic ( xandy) components causes spin mixing.\nTheHamiltonianEq. (119)isdiagonalizedtoobtainenergyeigenvalues of˜ǫkσ=ǫk−σ/radicalig\n(M+Az\ns,t)2+|A⊥\ns,t|2,\nwhere|A⊥\ns,t|2≡A+\ns,tA−\ns,tandσ=±represents spin (↑and↓correspond to + and −, respectively). We are\ninterested in the adiabatic limit, and so the contribution lowest-orde r, namely, the first order, in the per-\npendicular component, A⊥\ns,t, is sufficient. In the present rotating-frame approach, the gaug e field is treated\nas a static potential, since it already include time-derivative to the line ar order (Eq. (92)). Moreover,\nthe adiabatic component of the gauge field, Az\ns,t, is neglected, as it modifies the spin pumping only at the\nsecond-order of time-derivative. The energy eigenvalues, ǫkσ≃ǫk−σM, are thus unaffected by the gauge\nfield.\nIn the case of uniform magnetization we consider, the mixing due to t he gauge field is between the\nelectrons with different spin ↑and↓but having the same wave vector k, because the gauge field A±\ns,tcarries\nno momentum. This leads to a mixing of states having an excitation ene rgy ofMas shown in Fig. 17. In\nlow energy transport effects, what concern are the electrons at the Fermi energy; The wave vector kshould\nbe chosen as kF+andkF−, the Fermi wave vectors for ↑and↓electrons, respectively. The eigenstates we\nconsider therefore read\n|kF↑↑∝an}b∇acket∇i}htF=|kF↑↑∝an}b∇acket∇i}ht−A+\ns,t\nM|kF↑↓∝an}b∇acket∇i}ht\n|kF↓↓∝an}b∇acket∇i}htF=|kF↓↓∝an}b∇acket∇i}ht+A−\ns,t\nM|kF↓↑∝an}b∇acket∇i}ht. (120)\n267.1. Generated spin current\nSpin pumping effect is now studied by taking account of the interface hopping effects on states in Eq.\n(120). The interface hopping amplitude of electron in F to N with spin σis denoted by ˜tσand the amplitude\nfrom N to F is ˜t∗\nσ. We assume that the spin-dependence of electron state in F is gove rned by the relative\nangle to the magnetization vector, and hence the spin σis the one in the rotated frame. Assuming moreover\nthat there is no spin flip scattering at the interface, the amplitude ˜tσis diagonal in spin. Taking account of\nspin-dependent interface hopping, the non-adiabatic spin density (in the rotated frame) generated in the N\nregion at the interface was calculated field-theoretically by Tatara and Mizukami (2017). The result is\n/tildewides(N)= (πνN)2χF/parenleftbig\nRe[T↑↓]A⊥\ns,t+Im[T↑↓](ˆz×A⊥\ns,t)/parenrightbig\n(121)\nwhereA⊥\ns,t= (Ax\ns,t,Ay\ns,t,0) =As,t−ˆzAz\ns,tis the transverse (non-adiabatic) components of spin gauge field,\nTσσ′≡˜t∗\nσ˜tσ′, (122)\nνNis electron density of states in N and χF≡n↑−n↓\n2Mis susceptibility ( nσis spin-resolved electron density\nin F).\nThe spin polarization in the laboratory frame is obtained by a rotation matrixRij, defined by\nU−1σiU≡Rijσj, (123)\nas\ns(N)\ni=Rij/tildewides(N)\nj. (124)\nExplicitly,\nRij= 2mimj−δij, (125)\nwheremis in Eq. (66). Using identities\nRiz=ni\nRij(A⊥\ns,t)j=/planckover2pi1\n2(n×˙n)i\nRij(ˆz×A⊥\ns,t)j=/planckover2pi1\n2˙ni, (126)\nthe induced interface spin density is finally obtained as\ns(N)= Re[ζs](n×˙n)+Im[ζs]˙n (127)\nwhere\nζs≡/planckover2pi1(πνN)2n↑−n↓\n2MT↑↓. (128)\nSince the N electrons contributing to induced spin density is those at the Fermi energy, the spin current\nis simply proportional to the induced spin density as jsN=/planckover2pi1kF\nms(N), resulting in\nj(N)\ns=/planckover2pi1kF\nm[Re[ζs](n×˙n)+Im[ζs]˙n]. (129)\nThis is the result of spin current at the interface. The pumping efficie ncy is determined by the product\nof hopping amplitudes t↑andt∗\n↓. The spin mixing conductance defined in Ref. (Tserkovnyak et al., 2 002)\ncorrespondsto T↑↓. In the scatteringapproach(Tserkovnyak et al., 2002) basedon adiabaticpumping theory\n(Bttiker et al., 1994; Brouwer, 1998; Moskalets, 2012), the exp ression for the spin mixing conductance in\n27terms of scattering matrix element is exact as for the adiabatic con tribution. Our result (129), in contrast,\nis a perturbative one valid to the second order in the hopping amplitud e. To take full account of the hopping\nin the self energy is possible numerically in a field-theoretical approac h.\nIn bulk systems without spin-orbit interaction and magnetic field, th e hopping amplitudes tσare chosen\nas real, while at interfaces, this is not the case because inversion sy mmetry is broken. Nevertheless, in\nmetallic junctions such as Cu/Co, Cr/Fe and Au/Fe, first principles c alculations indicate that imaginary\npart of spin mixing conductance (our ζs) is smaller than the real part by 1-2 orders of magnitude (Xia et al.,\n2002; Zwierzycki et al., 2005). Large spin current proportional to˙nwould therefore suggest existence of\nstrong interface spin-orbit interaction, which gives rise to the imag inary part of ζs.\n7.2. Adiabatic or nonadiabatic?\ntn(t)ǫ(a)2M\ntn(t)ǫ(b)\nFigure 18: Schematic figures of electron energy ǫunder precessing localized spin, n(t), in the adiabatic limit (a) and with\nnonadiabaticity (b). Top figures represent energy levels wi th separation of 2 Min the rotated frame. In the perfectly adiabatic\ncase (a), the electron state keep the minimum energy state as n(t) changes. Spin pumping does not occur in this limit. Case (b)\nis with nonadiabaticity taken into account. A perpendicula r spin polarization along n×˙nis induced by a temporal change of\nlocalized spin ˙n, resulting in a high energy state (shown in red). This nonadi abatic effect is essential for spin current generation.\nInourapproach,spinpumpingeffectatthelinearorderintime-deriv ativeismappedtoastaticproblemof\nspin polarization formed by a static spin-mixing potential in the rotat ed frame. The rotated frame approach\nemployed here provides clear physical picture, as it grasps the low e nergy dynamics in a mathematically\nproper manner. In this approach, it is clearly seen that pumping of s pin current arises as a result of\noff-diagonal components of the spin gauge field that causes electr on spin flip (Fig. 18).\nIf so, is spin pumping an adiabatic effect or nonadiabatic one? Conven tional adiabatic processes are\nthose where the system under time-dependent external field rem ains to be the lowest energy state at each\ntime (Fig. 18(a)). In the spintronics context, electron passing th rough a thick domain wall seems to be in\nthe adiabatic limit in this sense; The electron spin keeps the lowest ene rgy state by rotating it according to\nthe magnetization profile at each spatial point as was argued in Sec. 6. In contrast, as is seen from the above\nanalysis, spin pumping effect does not arise in the same adiabatic limit; I t is induced by the nonadiabatic\n(off-diagonal) spin gauge field, A±\ns,t, which changes electron spin state in the local rotated frame with a\ncost ofsdexchange energy (Fig. 18(b)). For spin pumping effect, therefor e, nonadiabaticity is essential, as\nindicated also in a recent full counting statistics analysis (Hashimoto et al., 2017).\nA careful microscopic description indicates that a nonadiabaticity is essential even in spin-transfer effect.\nIn fact, electron spin injected into a domain wall along xdirection is polarized along n×∇xnas a result\nof nonadiabatic gauge field (Tatara et al., 2007, 2008), as shown in E q. (213). This non equilibrium spin\npolarization is perpendicular to the wall plane, and thus induces tran slational motion of the wall. This\nis the physical mechanism of spin-transfer effect. At the same time , spin-transfer effect can be discussed\nphenomenologically using conservation law of angular momentum. One should not forget, however, that\nnonadiabaticity is implicitly assumed because spin rotation is caused on ly by a perpendicular component.\nPhysically, the spin pumping effect is essentially the same as electron t ransmission through domain wall\nif we replace a spatial coordinate xand the time, as summarized in Fig. 19. In the case of domain wall,\n28including the nonadiabatic gauge field to the next order leads to cons ideration of domain wall resistance\nand nonadiabatic βtorque (Tatara, 2000, 2001; Tatara and Kohno, 2004).\nSpin-transfer effect for a domain wall Spin pumping due to magnetiza tion precession\nδs∝n×∇xn∝RA±\ns,x δs∝n×˙n∝RA±\ns,t\nFigure 19: Comparison of electron transmission through a do main wall and spin pumping effect. Large arrows represent the\nlocalized spins, n, as function of position x(left figure) or time t(right figure), and electron spin is denoted by a small arrow\nwith a circle. A nonadiabatic spin polarizations δsinduced by the nonadiabatic gauge field A±\ns,µare represented by yellow\narrows. For the domain wall it is always perpendicular to the wall plane.\n7.3. Spin accumulation in ferromagnet\nThe spin current pumping is equivalent to the increase of spin damping due to magnetization precession,\nas was discussed in Refs. (Berger, 1996; Tserkovnyak et al., 200 2). The damping effect is discussed by\ncalculating the torque by evaluating the spin polarization of the cond uction electron spin in F region. To\ndo this, a field theoretic method is convenient, as it enables a direct e stimate of position-dependent spin\ndensity. Details are shown in Ref. (Tatara and Mizukami, 2017), an d we here present only the result. The\ninduced spin density in the ferromagnet is obtained as\ns(F)(r,t) =m2νNa2\n2kF+kF−/summationdisplay\nσ/bracketleftbigg\n(n×˙n)Tσ,−σe−iσ(kF+−kF−)x+˙n(−iσ)Tσ,−σe−iσ(kF+−kF−)x/bracketrightbigg\n,(130)\nwhereνNis electron density of states in the normal metal and kFσ(σ=±) denotes the Fermi wave vector\nfor spin±in the ferromagnet. The induced spin accumulation density in the who le ferromagnet is\ns(F)≡1\nd/integraldisplay0\n−ddxs(F)(x)\n=m2νNa2\nkF+kF−(kF+−kF−)d/bracketleftig\n(n×˙n)/parenleftig\n−Im[T↑↓](1−cos˜d)+Re[T↑↓]sin˜d/parenrightig\n+˙n/parenleftig\nRe[T↑↓](1−cos˜d)+Im[T↑↓]sin˜d/parenrightig/bracketrightig\n,\n(131)\nwhere˜d≡(kF+−kF−)d, anddis the thickness of ferromagnet. As a result of this induced electro n spin\ndensity,s(F), the equation of motion for the averaged magnetization is modified t o be (Berger, 1996)\n˙n=−αn×˙n−γB×n−Mn×s(F), (132)\nwhereBis the external magnetic field.\nLet us first discuss thick ferromagnet case, d≫|kF+−kF−|−1, where oscillating part with respect to ˜d\nis neglected in Eq. (131). The equation of motion then reads\n(1+δ)˙n=−(α+δα)n×˙n−γB×n, (133)\nwhere\nδα=m2νNa2M\nkF+kF−(kF+−kF−)dRe[T↑↓], (134)\n29is the Gilbert damping enhancement by the effect of normal metal an d\nδ=m2νNa2M\nkF+kF−(kF+−kF−)dIm[T↑↓], (135)\nrepresents the shift of the precession angular frequency ωBas\nωB=γB\n1+δ. (136)\nThis is equivalent to the modification of the gyromagnetic ratio, γ, or theg-factor.\nFor most 3d ferromagnets, we may approximatem2νNaMǫF2\n2kF+kF−(kF+−kF−)≃O(1) (askF+−kF−∝M),\nresulting in δα∝a\ndRe[T↑↓]. When interface spin-orbit interaction is taken into account, we ha veT↑,↓=\n˜t0\n↑˜t0\n↓+i/tildewideγxz(˜t0\n↑+˜t0\n↓) +O((/tildewideγ)2), where ˜t0\nσand/tildewideγxzhave usually small imaginary part compared to the real\npart (Xia et al., 2002; Zwierzycki et al., 2005). Moreover, Re[ ˜t0\nσ] can be chosen as positive in most cases\nand thusT↑,↓>0. Equations (134) and (136) indicate that the strength of the ho pping amplitude ˜t0\nσand\ninterface spin-orbit interaction /tildewideγxzare experimentally accessible by measuring Gilbert damping and shift o f\nresonance frequency as has been known (Tserkovnyak et al., 200 2). A significant consequence of Eq. (134)\nis that the enhancement of the Gilbert damping,\nδα∼a\nd1\nǫF2˜t0\n↑˜t0\n↓, (137)\ncan exceed in thin ferromagnetsthe intrinsic damping parameter α, as the two contributions are governedby\ndifferent material parameters. In contrast to the positive enhan cement of damping, the shift of the resonant\nfrequency or g-factor can be positive or negative, as it is linear in the interface spin -orbit parameter /tildewideγxz.\nExperimentally, enhancement of the Gilbert damping and frequency shift has been observed in many\nsystems (Mizukami et al., 2001). In the case of Py/Pt junction, en hancement of damping is observed to be\nproportional to 1 /din the range of 2nm <d<10nm, and the enhancement was large, δα/α≃4 atd= 2 nm\n(Mizukami et al., 2001). These results appear to be consistent wit h our analysis. Same 1 /ddependence was\nobserved in the shift of g-factor. The shift was positive and magnitude was about 2% for Py/ Pt and Py/Pd\nwithd= 2nm, while it was negative for Py/Ta (Mizukami et al., 2001). The ex istence of both signs suggests\nthat the shift is due to the linear effect of spin-orbit interaction, an d the interface spin-orbit interaction we\ndiscuss is one of possible mechanisms.\nFor thin ferromagnet, ˜d(= (kF+−kF−)d)/lessorsimilar1, the spin accumulation of Eq. (131) leads to\nδα=m2νNa2M\n2kF+kF−Im[T↑↓]\nδ=−m2νNa2M\n2kF+kF−Re[T↑↓]. (138)\nThus, for weak interface spin-orbit interaction, positive shift of r esonancefrequency is expected (if Re[ T↑↓]>\n0). Significant feature is that the damping can be reduced or even b e negative if strong interface spin-orbit\ninteraction exists with negative Im[ T↑↓]. Our result indicates that ’spin mixing conductance’ description of\nRef. (Tserkovnyak et al., 2002) breaks down in thin metallic ferroma gnet.\n7.4. Historical background and adiabatic pumping\nSpin current generation due to magnetization precession was point ed out before Tserkovnyak theory by\nR. H. Silsbee et al.(Silsbee et al., 1979), where the effect of interface spin accumulatio n on the electron spin\nresonance in FN junction was focused on. Enhancement of Gilbert d amping constant in FN junction was\ntheoretically studied by L. Berger (Berger, 1996), and developed by other authors ( ˇSim´ anek and Heinrich,\n2003;ˇSim´ anek,2003). Experimentalstudieswerealsocarriedoutandr esultswereinagreementwith thoeries\n(Mizukami et al., 2001; Urban et al., 2001).\n30In 2002, Tserkovnyak et al.presented a novel interpretation to those effects in terms of spin current\ngeneration, which they called the spin pumping effect (Tserkovnyak et al., 2002). Based on the adiabatic\npumping theory, they showed that the spin current generated ar e determined by so-called the spin mixing\nconductance, which is written by use of scattering amplitudes.\nAdiabatic pumping theory started by the seminal paper by Thouless , where he discussed that a current\nis induced in quantum system by applying a periodic modulation of a pote ntial (Thouless, 1983). The study\nwas described by use of scattering theory. Current dynamically ge nerated in electron system is generally\nwritten as (Moskalets, 2012)\nI=e\nh/integraldisplay\ndE(fout(E)−f(E)), (139)\nwheref(E) is the equilibrium distribution with energy Eandfout(E) is a nonequilibrium distribution\nfunction for the outgoing electron in the presence of external pe rturbation. Functions fout(E) andf(E) are\nrelated by scattering matrix element, Sαβ, that also connects the outgoing and incoming electron operator\nasaout,α=Sαβain,β, whereα,βare indeces of leads. Sαβis therefore reflection or transmission amplitudes\nbetween leads αandβ. When the perturbation is periodic with period T, the current in the slow variation\n(adiabatic) limit is given by\nI=ie\n2π/integraldisplay\ndE/parenleftbigg\n−∂f\n∂E/parenrightbigg/integraldisplayT\n0dt\nTtr/bracketleftbigg\nS†(E,t)∂\n∂tS(E,t)/bracketrightbigg\n=ie\n2π/integraldisplayT\n0dt\nTtr/bracketleftbigg\nS†(ǫF,t)∂\n∂tS(ǫF,t)/bracketrightbigg\n. (140)\nThe expression is written using the integral over the scattering ma trix as\nI=ie\n2π/contintegraldisplay\ntr/bracketleftbig\nS†(ǫF,t)dS(ǫF,t)/bracketrightbig\n. (141)\nInthecaseofasingletime-dependent parameter,theintegralist rivialandvanishes, whilefortwoparameters\np(t) = (p1(t),p2(t)), it reads using Stokes theorem\nI=ie\n2π/contintegraldisplay\ntr[∇p×v(p)], (142)\nwhere∇p≡d\ndpis a derivative in the parameter space. The pumped current is thus d etermined by the flux\n∇p×v(p) in the parameter space (Brouwer, 1998; Moskalets, 2012).\n8. Brief remark on thermal transport\nLet us briefly mention transport driven by temperature gradient. In metals, a temperature gradient gives\nrise to a force on electrons in the same manner as external electric field, and thus thermal transport effects\nappear to be discussed in parallel to the electrically-induced effects phenomenologically speaking. Strictly\nspeaking, however, there is no rigorous formalism to incorporate t emperature gradient in quantum systems,\nas the system is non-equilibrium and also because temperature is a co ncept defined in a macroscopic scale.\nNevertheless, as thermal transport effects are important for a pplications like Peltier effect, some theoretical\napproaches were proposed in the 1960’s.\nOf those, Luttinger’s method (Luttinger, 1964) is commonly used n owadays. He introduced a scalar\npotential to describe temperature gradient. The potential coup les to the energy density of the system and\nwas called the gravitational potential, perhaps because gravitatio nal field couples to the energy density of\nthe system in theory of general relativity. Emergence of such a po tential may be understood as follows.\nA quantum system with Hamiltonian Hat equilibrium is described by the partition function, tr[ e−βH],\nwhereβ= 1/(kBT) is the inverse temperature. When temperature is inhomogeneous ,T(r) =T0+δT(r),\n31we may expand (without justification) the partition function to the lowest order of δTto obtain (His the\nHamiltonian density)(Matsumoto et al., 2014)\ntr/bracketleftbigg\nexp/parenleftbigg\n−1\nkB/integraldisplay\nd3rH\nT/parenrightbigg/bracketrightbigg\n≃tr/bracketleftbigg\nexp/parenleftbigg\n−1\nkB/integraldisplay\nd3rH\nT0/parenleftbigg\n1−δT\nT0/parenrightbigg/parenrightbigg/bracketrightbigg\n. (143)\nWe see that temperature inhomogeneity looks like a scalar potential\nψT≡−δT\nT0, (144)\nwhich couples to the Hamiltonian density. Based on this ’gravitational’ potential, Luttinger gave a prescrip-\ntion to calculate thermal transport coefficients in the framework o f linear response theory.\nFor such treatment, temperature needs to be defined locally. In o ther words, the system needs to be in\nlocal equilibrium, satisfying energy conservation law of\n˙E+∇·jE= 0, (145)\nwhereEandjEare energy density and energy current density, respectively. Th is indicates that there is\napproximately a U(1) gauge invariance for energy, similarly to that f or charge. (Correctly speaking, the\nenergy conservation arises from translational invariance in time, a nd the corresponding symmetry is not\nthe U(1) symmetry. For small variation, however, it is approximate d as U(1) gauge invariance.) Then the\ntemperature gradient can be expressed in therms of an effective v ector potential (thermal vector potential)\n(Moreno and Coleman, 1996; Shitade, 2014; Tatara, 2015b,a), wh ich satisfies\n˙AT=∇T\nT0. (146)\nBased on the above approaches, thermal transport can be desc ribed in parallel to the case of electric\ncases by formally replacing the electric charge by energy density. T his feature, however, requires careful\ncalculationofphysicalquantitiesbecauseofenhancementathigh e nergy,asnoticedin somecases(Qin et al.,\n2011; Kohno et al., 2016).\nLuttinger’sapproachhasbeenemployedtostudythermally-induce delectrontransports(Smrcka and Streda,\n1977;Oji and Streda,1985;Qin et al.,2011;Eich et al.,2014),ma gnontransport(Matsumoto and Murakami,\n2011) and thermally-induced torque (Kohno, 2014). Vector pote ntial form was applied in Refs. (Shitade,\n2014; Tatara, 2015a). Thermal transport is particularly importa nt for magnon spintronics in insulator ferro-\nmagnets (magnonics) (Murakami and Okamoto, 2017), as temper ature gradient is most convenient driving\nfield for magnons with no electric charge.\n9. Field theoretical approach\nSo far we discussed in a quantum mechanical picture. Such descript ion may, however, lack transparency\nbecause we always have to think in terms of wave functions. In cont rast, in field theoretic formalisms,\nphysical observables are represented by fields, i.e., operators de fined at each point in space and time, which\nhave their own dynamics. For instance, the Berry’s phase is repres ented in quantum mechanics as an\namplitude of a state change (Eq. (23)), while it has a clear physical m eaning of an effective gauge field in\nfield theory. Most importantly, field theory enables us to evaluate d irectly physical observables and provides\nclear theoretical scenario. In this section, we introduce field theo retical description and discuss spintronics\neffects in the following sections. It turns out that physics becomes clear and consistent in the field-theoretic\nformulation.\n329.1. Field operators\nQuantum particlescanbe createdand annihilated byquantum fluctu ation and accordinglytheir numbers\nfluctuate. This fluctuation is neglected in quantum mechanics where a condition that the particle number\nin the whole space is always unity is imposed. This constraint is removed by introducing creation and\nannihilation operators for the particle, which we denote here by ˆ a†and ˆa, respectively5. Creation and\nannihilation may occur at any position and any time, and so the operat ors are functions of space and time\ncoordinates, i.e., fields. The field operators acts on states which sp ecifies how many particles exist at each\nspace time point. Any states are therefore constructed by apply ing necessary particle creation operators\non a vacuum state |0∝an}b∇acket∇i}ht. The creation and annihilation operators are, by definition, not com mutative with\nparticle number operator, ˆ n, because particle numbers before and after creation have a differ ence of 1. To\nput in equation, we need impose\nˆnˆa†−ˆa†ˆn= ˆa†(147)\nand\nˆnˆa−ˆaˆn=−ˆa (148)\nThese conditions are satisfied if we choose ˆ nas\nˆn= ˆa†ˆa (149)\nand impose either\n[ˆa,ˆa†] = 1,[ˆa,ˆa] = [ˆa†,ˆa†] = 0, (150)\nor\n{ˆa†,ˆa}= 1,{ˆa,ˆa}={ˆa†,ˆa†}= 0, (151)\nfor the operators. Here [ A,B]≡AB−BAis a commutator and {A,B}≡AB+BAis an anti commutator.\nWe have therefore either bosons described by Eq. (150) or fermio ns satisfying Eq. (151). For spintronics,\nthe field of most interest is fermionic conduction electron, which we d enote byc†\nσ(r,t) andcσ(r,t), where\nσ=±denotes spin degrees of freedom. For field operators, the commu tator and anticommutator become\nδ-function in space and time as\n[ˆa(r,t),ˆa†(r′,t′)] =δ(r−r′)δ(t−t′) or{ˆa†(r,t),ˆa(r′,t′)}=δ(r−r′)δ(t−t′),(152)\nbecause operators at different space time coordinates simply comm ute or anticommute.\nIn the absence of correlation effects between fields, the total ma ny particle Hamiltonian is simply a\nsingle-body Hamiltonian of quantum mechanics multiplied by particle num ber density, ˆ n= ˆc†ˆcfor the case\nof electron. We use a vector representation for two spin compone nts of electron operator, i.e., ˆ c= (ˆc+,ˆc−).\nFor the 1-particle quantum mechanical Hamiltonian with mass mand potential V,Hqm=−/planckover2pi12∇2\n2m+V(r),\nthe field version is\nH=/integraldisplay\nd3rˆc†(r,t)/parenleftbigg\n−/planckover2pi12∇2\n2m+V(r)/parenrightbigg\nˆc(r,t). (153)\nHereelectrondensity ˆ c†ˆcissplittomaketheHamiltonianhermitian. Correlationeffectsarestra ightforwardly\nincluded by replacing particle density by ˆ c†ˆc.\n5In this section, field operators are denoted with ˆ, although it shall be suppressed in the later sections.\n33The time-dependence of field operators are governed by the Hamilt onian by the Heisenberg equation,\n∂tˆc=i\n/planckover2pi1[H,ˆc], ∂ tˆc†=i\n/planckover2pi1[H,ˆc†]. (154)\nThe equation motions are derived from a field Lagrangian\nL=/integraldisplay\nd3ri/planckover2pi1ˆc†∂tˆc−H. (155)\nThe first time-derivative term represents the canonical relation b etween creation and annihilation operators.\nIn fact, Eq. (155) indicates that the canonical variable for ˆ cis/planckover2pi1ˆc†, and canonical commutation relation of\n{ˆc,ˆc†}= 1 is derived.\n9.2. Field Lagrangian for sdmodel\nThe field representation of the Lagrangian for conduction electro n interacting with localized spin is\nˆL=/integraldisplay\nd3r/bracketleftbigg\ni/planckover2pi1ˆc†˙ˆc−ˆc†(r,t)/parenleftbigg\n−/planckover2pi12∇2\n2m−M(n·σ)/parenrightbigg\nˆc(r,t)/bracketrightbigg\n. (156)\nConsidering general case of inhomogeneous localized spin structur e, we carry out a unitary transformation\nto choose electron spin’s quantization axis along z-axis. This assumes that the sdexchange coupling is\nstrong and certain adiabatic condition is satisfied. For the spatial v ariation, the condition turns out to be\nEq. (113) in the clean case, while for time-dependent localized spin wit h angular frequency of ω, it would\nbeωτ≪1, whereτis the electron elastic lifetime. In the field representation, the unita ry transformation\ncorresponds to define a new electron operators, aanda†as\nˆa(r,t) =U(r,t)ˆc(r,t). (157)\nA 2×2 matrixU(r,t) is chosen to satisfy\nU−1(n·σ)U=σz, (158)\nat each point, and it is thus as given in Eq. (65). Now the sdinteraction is diagonalizedfor the new electron,\naanda†, as\nˆHsd=−M/integraldisplay\nd3rˆa†σzˆa, (159)\nand thus this electronin the rotatedframe is a goodvariable fordes cribing low energybehavior. The unitary\ntransformation affects, however, the kinetic term is modified as\n∂µˆc=U/parenleftbigg\n∂µ±i\n/planckover2pi1As,µ/parenrightbigg\nˆa, (160)\nwhereAs,µis defined in Eq. (92) and positive and negative signs correspond to µ=tandµ=x,y,z,\nrespectively. The Lagrangian for a-electron is therefore the one with minimal coupling to the gauge field\n(using integral parts,/integraltext\nd3rˆc†(∇2ˆc) =−/integraltext\n(∇ˆc†)(∇ˆc))\nˆL=/integraldisplay\nd3r/bracketleftbigg\ni/planckover2pi1ˆa†/parenleftbigg\n∂t+i\n/planckover2pi1As,t/parenrightbigg\nˆa−/planckover2pi12\n2m/bracketleftbigg\nˆa†/parenleftbigg\n←\n∇+i\n/planckover2pi1As/parenrightbigg/bracketrightbigg/bracketleftbigg/parenleftbigg\n→\n∇−i\n/planckover2pi1As/parenrightbigg\nˆa/bracketrightbigg\n+ǫFˆa†ˆa+Mˆa†σzˆa/bracketrightbigg\n,(161)\nwhere←\n∇and→\n∇(=∇) act on the field on the left and right side, respectively. Defining spin density and\nspin current density operators in the rotated frame (without spin magnitude of1\n2) as\nˆsα≡ˆa†σαˆa\nˆjα\ns,i≡−i/planckover2pi1\n2mˆa†↔\n∇iσαˆa, (162)\n34it reads\nˆL=/integraldisplay\nd3r/bracketleftbigg\ni/planckover2pi1ˆa†˙ˆa−/planckover2pi12\n2m|∇ˆa|2+ǫFˆa†ˆa+Mˆa†σzˆa+ˆjα\ns,iAα\ns,i−ˆn\n2mA2\ns−ˆsαAα\ns,t/bracketrightbigg\n. (163)\nHere it is clear that the spatial and time components of the gauge fie ld,As,i(i=x,y,z) andAs,t, couples\nto spin current density and spin density, respectively.\nThe gauge fieldAα\ns,µis a SU(2) gauge field that have three spin components ( α=x,y,z) and four space-\ntime components ( µ=x,y,z,t). In the adiabatic limit, it reduces to a single spin component Az\ns,µ≡As,µ\n, i.e., to a U(1) gauge field we discussed in Sec. 4. In fact, in this limit, th e minority spin electron can be\nneglected due to a large electron spin polarization energy M. Thus electron field reduces to a→/parenleftbigg\na↑\n0/parenrightbigg\nand we end up with the Lagrangian equivalent to the one with electrom agnetic gauge field,\nL=/integraldisplay\nd3r/bracketleftbigg\ni/planckover2pi1ˆa†\n↑/parenleftbigg\n∂t+i\n/planckover2pi1As,t/parenrightbigg\nˆa↑−/planckover2pi12\n2m/bracketleftbigg/parenleftbigg\n∇+i\n/planckover2pi1As/parenrightbigg\nˆa†\n↑/bracketrightbigg/bracketleftbigg/parenleftbigg\n∇−i\n/planckover2pi1As/parenrightbigg\nˆa↑/bracketrightbigg\n+Mˆa†\n↑ˆa↑/bracketrightbigg\n.(164)\n10. Effective Lagrangian for localized spin\nOnce we know the field Lagrangian, field theory provides in principle an y information on the system we\nwant. We discuss in term of Lagrangian, as Lagrangian contains info rmation about canonical relations in\nthe time-derivative term as we saw in Sec. 9.1, while in the Hamiltonian ap proach, canonical relations need\nto be imposed, although both approaches lead to the same result if c alculated correctly.\nWe first show how effective Lagrangianfor localized spin is derived fro m thesdmodel within equilibrium\nquantum statistical physics. Effective Lagrangian is the one obtain ed by integrating out, in other words,\nevaluating quantum trace of, other quantum degrees of freedom (Sakita, 1985), which is conduction electron\nin our case. All the effects of electrons are formally contained in the effective Lagrangian. In Sec. 5,\nwe discussed that spin current induces Dzyaloshinskii-Moriya intera ction (Eq. (102)). In the effective\nLagrangian study, this fact is discussed systematically on the equa l footing as ferromagnetic exchange\ninteraction induced by conduction electron.\nThe effectiveLagrangianforlocalizedspinis obtainedbyevaluatingth e expectationvalueforthe electron\ntreating the spin gauge field perturbatively. The spatial componen t of the spin gauge field is taken account\nto the second order, while only the first order of temporal compon ent needs to be retained. By use of\npath-integral method, the effective action, time-integral of effe ctive Lagrangian, is obtained as\n∆S=/integraldisplay\ndτ/integraldisplay\nd3r/bracketleftbigg\n−sαAα\ns,0+jα\ns,iAα\ns,i−/planckover2pi12\n2mA2\nsn/bracketrightbigg\n−1\n2/integraldisplay\ndτ/integraldisplay\ndτ′/integraldisplay\nd3r/integraldisplay\nd3r′Aα\ns,i(τ,r)Aβ\ns,j(τ′,r′)χαβ\nij(τ,r,τ′,r′),\n(165)\nwheresα≡∝an}b∇acketle{tˆs∝an}b∇acket∇i}htandjα\ns,i≡/angbracketleftig\nˆjα\ns,i/angbracketrightig\nare expectation values of spin density and spin current, respectiv ely (∝an}b∇acketle{t∝an}b∇acket∇i}ht\nis quantum statistical average). The second term on the right han d side contains spin current correlation\nfunctions\nχαβ\nij(τ,r,τ′,r′)≡/angbracketleftig\nˆjα\ns,i(τ,r)ˆjβ\ns,j(τ′,r′)/angbracketrightig\n=1\nβV/summationdisplay\nqωℓe−iωℓ(τ−τ′)eiq·(r−r′)χαβ\nij(iωℓ,q), (166)\nwhere\nχαβ\nij(iωℓ,q) =/planckover2pi12\nβV/summationdisplay\nkωnkikj\nm2tr[σαGk−q\n2,ωnσβGk+q\n2,ωn+ωℓ], (167)\n35As\nsAs\njs jsAsAs\nn\nFigure 20: Feynman diagram representaion of contributions to the effective Lagrangian to the second order in spatial der ivative\nand linear order in time derivative. The spin gauge field Asrepresented by wavy lines is linear in the derivative of loca lized\nspin. Solid lines represent electron Green’s functions.\nin terms of the imaginary time free Green’s function ( ωn≡(2n−1)π/βis fermionic thermal frequency, with\nnan integer),\nGk,ωn≡1\niωn−ǫk+Mσz. (168)\nThe effective action is diagrammatically represented in Fig. 20.\nThe summation over thermal frequency in Eq. (167) is evaluated us ing contour integration as\n1\nβ/summationdisplay\nωnGk,ωn,σσβGk′,ωn+ωℓ,σ′=−/integraldisplay\nCdz\n2πif(z)1\nz−ǫkσ1\nz+iωℓ−ǫk′σ′\n=f(ǫkσ)−f(ǫk′σ′)\nǫkσ−ǫk′σ′+iωℓ, (169)\nwherez≡iωnis a complex thermal frequency and Cis a contour surrounding the imaginary axis. As\nthe electrons connects localized spins at different position and time, the contribution of the effective action\ncontaining correlation functions are nonlocal in general. For our pu rpose of looking into the second-order\nderivatives, however, it is sufficient to consider the local componen ts, because the spin gauge field contains\na first-order derivative. The correlation functions are therefor e approximated as\nχαβ\nij(τ,r,τ′,r′) =δ(r−r′)δ(τ−τ′)χαβ\nij(iωℓ= 0,q→0), (170)\nwhereχαβ\nij(iωℓ= 0,q→0) denotes that the limit of q→0 is taken after setting ωℓ= 0. The effective action\nis then represented by a local effective Lagrangian as ∆ S=/integraltext\ndτLeff(τ), where\nLeff=/integraldisplay\nd3r/bracketleftbigg\n−sαAα\ns,t+jα\ns,iAα\ns,i−/planckover2pi12\n2mA2\nsn−1\n2χαβ\nij(0,0)Aα\ns,iAβ\ns,j/bracketrightbigg\n. (171)\nWe here see that spin accumulation of electron contributes to an ad ditional spin Berry’s phase term ( Aα\ns,0),\nand linear term in spatial component Aα\ns,iarises if expectation value of spin current is finite. The terms\nsecond order in the gauge field describes exchange interaction aris ing from electron conduction. They are\nin general anisotropic in space and spin, such as, Jαβ\nij∇inα∇jnβ(Jαβ\nijis a coefficient), if symmetry of the\nsystem is broken.\nBelow, we look into the exchange interaction focusing on the isotrop ic electron dispersion and neglecting\nspin-orbit interaction. Spin current vanishes in the absence of ext ernal current, and the spin density is\ndiagonal,sα≡sδαz, where (sis defined without the factor of1\n2of spin)\ns≡1\nV/summationdisplay\nk(fk+−fk−), (172)\nis the electron spin polarization density. The correlation function is w ritten as\nχαβ\nij(0,0) =δij[(δαβ−δαzδβz)χxx(0,0)+δαzδβzχzz(0,0)+ǫαβzχxy(0,0)], (173)\n36and each component is evaluated as\nχxx\nij(ωℓ= 0,q→0) =−/planckover2pi12(k5\nF+−k5\nF−)\n30π2m2Mδij\nχxy\nij(ωℓ= 0,q→0) = 0\nχzz\nij(0,0) =−n\nmδij, (174)\nwherekF±≡/radicalbig\n2m(ǫF±M)//planckover2pi1is the Fermi wave vector of spin ±electron. The effective Lagrangian is\ntherefore obtained as\nLeff=/integraldisplay\nd3r/bracketleftbigg\n−sAz\ns,0−n\n2m/parenleftbigg\n1−/planckover2pi12(k5\nF+−k5\nF−)\n30π2mnM/parenrightbigg/bracketleftbig\n(Ax\ns,i)2+(Ay\ns,i)2/bracketrightbig/bracketrightbigg\n=/integraldisplay\nd3r/bracketleftbiggs/planckover2pi1\n2˙φ(cosθ−1)−Je\n2(∇n)2/bracketrightbigg\n, (175)\nwhere\nJe=n/planckover2pi12\n4m/bracketleftbigg\n1−/planckover2pi12(k5\nF+−k5\nF−)\n30π2mnM/bracketrightbigg\n, (176)\nis the exchange interaction induced by electron, which is positive in th e present model.\nIn the presence of applied current, spin current along localized spin (zdirection in the rotated frame) is\nfinite. In this case we retain the spin current term to obtain\nLeff=/integraldisplay\nd3r/bracketleftbiggs/planckover2pi1\n2[(∂t+vs·∇)φ](cosθ−1)−Je\n2(∇n)2/bracketrightbigg\n, (177)\nwherevsis given by Eq. (99) with spin magnitude Sreplaced by s/2.\nIn the presence of spin-orbit interaction in systems with broken inv ersion symmetry, perpendicular spin\ncurrent,j⊥\ns,arisesingeneral. ThenwehaveaLagrangianwithDzyaloshinskii-Mor iyainteractioninteraction,\nLeff=/integraldisplay\nd3r/bracketleftbiggs/planckover2pi1\n2(∂t+vs·∇)φ(cosθ−1)−Je\n2(∇n)2−Dα\ni(n×∇in)/bracketrightbigg\n, (178)\nwith DM constant given by Eq. (102).\nHere we presented a simple case of single band electron. Contributio ns from many bands need to be\nincluded for quantitative estimates for real materials. Evaluation o f strengths of exchange interaction and\nDM interactionis importantin studies ofmagneticstructures (Kats nelson et al., 2010; Freimuth et al., 2014;\nMikhaylovskiy et al., 2015; Belabbes et al., 2016). Effective Hamiltonia n approach gives a straightforward\nmethodtoevaluateinteractionstrengthswhencombinedwith thefi rstprinciplescalculations(Kikuchi et al.,\n2016).\n11. Landau-Lifshitz-Gilbert (LLG) equation with electron effects\nInthissectionweconsidereffectsofconductionelectrononlocalize dspindynamicsbydirectlycalculating\nthe electron spin polarization and torque. The Hamiltonian we conside r is\nH=HS−M/integraldisplay\nd3rn·ˆs+He, (179)\nwhereˆs≡c†σcis the electron spin field operator and Herepresents the conduction electron Hamiltonian.\nEffectsotherthanelectronsareincluded asaneffective magneticfi eldinHS. Todescribethe effect ofapplied\n37AsAsA\nAAs\nA\nFigure 21: Diagrammatic representation of interaction ver teces including spin gauge field and electromagnetic gauge fi eld.\nWavy, dotted and solid lines represent the spin gauge field As, electromagnetic gauge field, A, and electron, respectively.\nelectric current, we include an electromagnetic gauge field (vector potential), A. The electron Hamiltonian\nis (see Eq. (4))\nH0\ne≡/integraldisplay\nd3r/planckover2pi12\n2m/bracketleftig\nˆc†/parenleftig←\n∇+ie\n/planckover2pi1A/parenrightig/bracketrightig/bracketleftig/parenleftig\n∇−ie\n/planckover2pi1A/parenrightig\nˆc/bracketrightig\n=/integraldisplay\nd3r/bracketleftbigg/planckover2pi12\n2m|∇ˆc|2+ie/planckover2pi1\n2mA·ˆc†↔\n∇iˆc+e2A2\n2mˆn/bracketrightbigg\n.(180)\nThe total current density operator including the ’diamagnetic part ’ due to the electromagnetic vector po-\ntential is\nˆj≡−δHe\nδA=−ie/planckover2pi1\n2mˆc†↔\n∇ˆc−e2\nmˆnA. (181)\nWe do not consider electron spin relaxations like due to spin-orbit inte raction. Relaxation effects are only\nbriefly mentioned later (See. Ref. (Kohno et al., 2006)).\nIncluding the effect of the electron, the equation of motion for loca lized spin (LLG equation) reads\n˙n=−γBS×n+Ma3\n/planckover2pi1Sn×s, (182)\nwhereγBS≡−1\n/planckover2pi1SδHS\nδn,ais the lattice constant, and\ns≡∝an}b∇acketle{tˆs∝an}b∇acket∇i}ht, (183)\nis the conduction electron spin polarization density, which contains a ll the electron effects such as spin-\ntransfer torque.\nCalculation is carried out in the rotating frame that diagonalizes the sdexchange interaction, where low\nenergy behavior of electron spin is correctly described. The electr omagnetic interaction in Eq. (180) with\nspatial derivative is modified by the transformation, resulting in\nˆHe≡H0\ne+Hsd (184)\n=/integraldisplay\nd3r/bracketleftbigg/planckover2pi12\n2m|∇ˆa|2−ǫFˆa†ˆa−Mˆa†σzˆa−ˆjα\ns,iAα\ns,i+1\n2mA2\nsˆn−ˆj·A+e2\n2mA2ˆn+e\nmˆa†AiAs,iˆa/bracketrightbigg\n,(185)\nwhereM≡JsdS, and\nˆj0\ni≡−ie/planckover2pi1\n2mˆa†↔\n∇iˆa, (186)\nis the bare (paramagnetic) current density of the rotated electr on (superscript0is for bare part). The\ninteraction verteces in Eq. (185) are shown diagrammatically in Fig. 2 1.\nThe electron spin density in the rotated frame, ˜s≡/angbracketleftbig\nˆa†σˆa/angbracketrightbig\n, is related to the laboratory frame one by\nrotation matrixRas\nsi=Rij˜sj. (187)\n38It is represented by lesser Green function, defined in the rotated frame by\nG<\nσσ′(r,t,r′,t′)≡i\n/planckover2pi1/angbracketleftig\nˆa†\nσ′(r′,t′)ˆaσ(r,t)/angbracketrightig\n, (188)\nas\n˜s(r,t)≡−i/planckover2pi1tr[σG<(r,t,r,t)]. (189)\nOur objective is to evaluate the Green’s function including the effect of localized spin structure, which\nis expressed by the spin gauge field, As,, and electromagnetic gauge field, A, represented in Fig. 21. The\napplied electric field is written as E=−˙A. We consider spatially uniform EandA. We consider DC case\n(static electric field) by treating the angular frequency Ω of Aas finite during the calculation and taking\nthe limit of Ω→0 at the end (Rammer and Smith, 1986). (The electric field is express ed also by use of a\nscalar potential Φ as E=−∇Φ, but the calculation is easier if we use vector potential.)\nThe lesser Green’s function is calculated by solving for the path-ord ered Green’s function defined by\nGσ,σ′(r,t,r′,t′)≡−i\n/planckover2pi1/angbracketleftbig\nTCaσ(r,t)a†σ′(r′,t′)/angbracketrightbig\n, (190)\nwheret,t′are defined on a contour Cwhich goes from−∞to∞on the upper plane and comes back from\n∞to−∞on the lower plane in a complex time plane. This Green’s function satisfie s\ni/planckover2pi1∂tGkk′(t,t′) =δ(t−t′)/angbracketleftbig\n{ak(t),a†k′(t′)}/angbracketrightbig\n+i\n/planckover2pi1/angbracketleftbig\nTC[H,ak(t)]a†k′(t′)/angbracketrightbig\n, (191)\nwhereGkk′is the Fourier transform of G(r,r′) andHis the total Hamiltonian. We are interested in the\nadiabatic limit, and treat the spin gauge field perturbatively to the line ar order. The effect of the applied\ncurrent is also discussed to the linear order in A. Choosing the initial and final wave vectors as k+q\n2and\nk−q\n2, respectively, the Dyson equation on time contour Creads\nGk−q\n2,k+q\n2(t,t′) =gk(t−t′)δq,0+/planckover2pi1/summationdisplay\nα/integraldisplay\nCdt1/bracketleftbigg/planckover2pi1\nmkiAα\ns,i(q,t1)+Aα\ns,t(q,t1)/bracketrightbigg\ngk−q\n2(t−t1)σαgk+q\n2(t1−t′)\n−e/planckover2pi1\nmki/summationdisplay\nαi/integraldisplay\nCdt1Ai(t1)gk(t−t1)gk(t1−t′)+e\nm/summationdisplay\nαi/integraldisplay\nCdt1Ai(t1)Aα\ns,i(q,t1)gk−q\n2(t−t1)σαgk+q\n2(t1−t′)\n+e/planckover2pi12\nm2/summationdisplay\nαikj/integraldisplay\nCdt1/integraldisplay\nCdt2/bracketleftig/parenleftig\nk+q\n2/parenrightig\niAi(t2)Aα\ns,j(q,t1)gk−q\n2(t−t1)σαgk+q\n2(t1−t2)gk+q\n2(t2−t′)\n+/parenleftig\nk−q\n2/parenrightig\niAi(t1)Aα\nj(q,t2)gk−q\n2(t−t1)gk−q\n2(t1−t2)σαgk+q\n2(t2−t′)/bracketrightig\n+O((As)2,A2), (192)\nwheregk(t) is the free path-ordered Green’s function.\n11.1. Induced electron spin density\nElectron spin density, written in terms of the lesser Green’s functio n at equal time and position,\nG<(r,t,r,t) =/summationtext\nkk′ei(k−k′)·rG<\nkk′(t,t), is represented by bubble diagrams starting from and ending at\nr,tas\n˜s(r,t) =r,t +A+As+As\nA+ +As\nA+···.(193)\nThe first and second diagrams on the right-hand side are trivial one without effects of spin structure,\nand are neglected. The third term containing As(represented by wavy line) only represents the equilibrium\n39contribution in the presence of spin structure. It was already disc ussed in Sec. 10 in the effective Lagrangian\napproach (the term of sin Eq. (175)), but we here argue in a different approach. The fourt h, fifth and sixth\ncontributions including both Asand electromagnetic gauge field A(represented by dotted line) represent\nthe current-induced contribution, which is of our most interest.\nEvaluation of lesser Green’s function from the path-ordered one is carried out using projection formula\n(called the Langreth formula),\n/bracketleftbigg/integraldisplay\nCdt1A(t,t1)B(t1,t′)/bracketrightbigg<\n=/integraldisplay∞\n−∞dt1(Ar(t,t1)B<(t1,t′)+A<(t,t1)Ba(t1,t′))\n/bracketleftbigg/integraldisplay\nCdt1A(t,t1)B(t1,t′)/bracketrightbiggr\n=/integraldisplay∞\n−∞dt1Ar(t,t1)Br(t1,t′), (194)\nwhererandadenote the retarded and advanced components, respectively.\nThe perpendicular components of the spin density is defined as ˜ s±≡1\n2(˜sx±i˜sy). Its equilibrium\ncontribution (denoted by ˜ s(0)) as function of wave vector qis given as ( σ±≡1\n2(σx±iσy) andA±\ns≡\n1\n2(Ax\ns±iAy\ns))\n˜s±,(0)(q) =A±s,tσ∓k−q\n2\nk+q\n2σ±\n=−i/planckover2pi1\nV/integraldisplaydω\n2π/summationdisplay\nkA±\ns,t(q)/bracketleftig\ngk−q\n2,∓,ωgk+q\n2,±,ω/bracketrightig<\n. (195)\nAs the spin gauge field is static, all the electron Green’s functions ca rries the same angular frequency, ω,\nwhile the wave vector is shifted by that for the gauge field, q. Using Eq. (194) and\ng<\nkσ(ω) =fkσδ(/planckover2pi1ω−ǫkσ), (196)\nwherefkσ≡1\neβǫkσ+1is the Fermi distribution function, it reads\n˜s±,(0)(q) =−i/planckover2pi1\nV/integraldisplaydω\n2π/summationdisplay\nk/summationdisplay\nµA±\ns,t(q)/bracketleftig\ngr\nk−q\n2,∓,ωg<\nk+q\n2,±,ω+g<\nk−q\n2,∓,ωga\nk+q\n2,±,ω/bracketrightig\n=2\nVA±\ns,t(q)/summationdisplay\nkfk+q\n2,±−fk−q\n2,∓\nǫk+q\n2,±−ǫk−q\n2,∓. (197)\nThe wave vector qfor the localized spin structure is neglected in the adiabatic limit for ev aluating the\nelectron energy, resulting in\n˜s±,(0)(q)≃−s\n2MA±\ns,t(q), (198)\nwheresis the equilibrium conduction electron spin density defined in Eq. (172) .\n40Current-induced contributions are calculated as\n˜s±,(1a)(q) =A±s\nAσ∓ σ±q\nΩ+\n=−ie/planckover2pi12\nm2Vlim\nΩ→0/integraldisplaydω\n2π/summationdisplay\nk/summationdisplay\nijAi(Ω)A±\ns,j(q)kj/bracketleftbigg/parenleftig\nk+q\n2/parenrightig\ni/bracketleftig\ngk−q\n2,∓,ω−Ω\n2gk+q\n2,±,ω−Ω\n2gk+q\n2,±,ω+Ω\n2/bracketrightig<\n+/parenleftig\nk−q\n2/parenrightig\ni/bracketleftig\ngk−q\n2,∓,ω−Ω\n2gk−q\n2,∓,ω+Ω\n2gk+q\n2,±,ω+Ω\n2/bracketrightig</bracketrightbigg\n˜s±,(1b)(q)≡As\nA\n=−ie\nmVlim\nΩ→0/integraldisplaydω\n2π/summationdisplay\nkAi(Ω)A±\ns,i(q)/bracketleftig\ngk−q\n2,∓,ω−Ω\n2gk+q\n2,±,ω+Ω\n2/bracketrightig<\n. (199)\nEvaluating the lesser component, the first contribution reads\n˜s±,(1a)(q) =−ie/planckover2pi12\nm2Vlim\nΩ→0/integraldisplaydω\n2π/summationdisplay\nk/summationdisplay\nijAi(Ω)A±\ns,j(q)kj\n×/bracketleftbigg\n−f/parenleftbigg\nω+Ω\n2/parenrightbigg/bracketleftig/parenleftig\nk+q\n2/parenrightig\nigr\nk−q\n2,∓,ω−Ω\n2gr\nk+q\n2,±,ω−Ω\n2gr\nk+q\n2,±,ω+Ω\n2+/parenleftig\nk−q\n2/parenrightig\nigr\nk−q\n2,∓,ω−Ω\n2gr\nk−q\n2,∓,ω+Ω\n2gr\nk+q\n2,±,ω+Ω\n2/bracketrightig\n+f/parenleftbigg\nω−Ω\n2/parenrightbigg/bracketleftig/parenleftig\nk+q\n2/parenrightig\niga\nk−q\n2,∓,ω−Ω\n2ga\nk+q\n2,±,ω−Ω\n2ga\nk+q\n2,±,ω+Ω\n2+/parenleftig\nk−q\n2/parenrightig\niga\nk−q\n2,∓,ω−Ω\n2ga\nk−q\n2,∓,ω+Ω\n2ga\nk+q\n2,±,ω+Ω\n2/bracketrightig\n+/bracketleftbigg\nf/parenleftbigg\nω+Ω\n2/parenrightbigg\n−f/parenleftbigg\nω−Ω\n2/parenrightbigg/bracketrightbigg\n×/bracketleftig/parenleftig\nk+q\n2/parenrightig\nigr\nk−q\n2,∓,ω−Ω\n2gr\nk+q\n2,±,ω−Ω\n2ga\nk+q\n2,±,ω+Ω\n2+/parenleftig\nk−q\n2/parenrightig\nigr\nk−q\n2,∓,ω−Ω\n2ga\nk−q\n2,∓,ω+Ω\n2ga\nk+q\n2,±,ω+Ω\n2/bracketrightig/bracketrightig\n.\n(200)\nWe expand with respect to Ω and take the limit of Ω →0 using identities\ngr\nk+q\n2,±,ω−Ω\n2gr\nk+q\n2,±,ω+Ω\n2= (gr\nk+q\n2,±,ω+Ω\n2)2−(gr\nk+q\n2,±,ω+Ω\n2−gr\nk+q\n2,±,ω−Ω\n2)gr\nk+q\n2,±,ω+Ω\n2\n= (gr\nk+q\n2,±,ω+Ω\n2)2+Ω(gr\nk+q\n2,±,ω+Ω\n2)3+O(Ω2) (201)\nand\n/planckover2pi12/parenleftbig\nk+q\n2/parenrightbig\ni\nm(gr\nk+q\n2,σ,ω)2=∂\n∂kigr\nk+q\n2,σ,ω. (202)\n41After integral by parts with respect to k, we obtain\n˜s±,(1a)(q) =ie\nmVlim\nΩ→0/integraldisplaydω\n2π/summationdisplay\nk/summationdisplay\niAi(Ω)\n×/bracketleftbigg\nAs,is±(q)/braceleftbigg\n−f/parenleftbigg\nω+Ω\n2/parenrightbigg\ngr\nk−q\n2,∓,ω−Ω\n2gr\nk+q\n2,±,ω+Ω\n2+f/parenleftbigg\nω−Ω\n2/parenrightbigg\nga\nk−q\n2,∓,ω−Ω\n2ga\nk+q\n2,±,ω+Ω\n2/bracerightbigg\n+Ω\n2A±\ns,i(q)f(ω)/bracketleftig\ngr\nk−q\n2,∓,ω(gr\nk+q\n2,±,ω)2−(gr\nk−q\n2,∓,ω)2gr\nk+q\n2,±,ω−(ga\nk−q\n2,∓,ω(ga\nk+q\n2,±,ω)2−(ga\nk−q\n2,∓,ω)2ga\nk+q\n2,±,ω)/bracketrightig\n+/summationdisplay\nj/planckover2pi12Ω\n2mA±\ns,j(q)f(ω)kjqi[(gr\nk−q\n2,∓,ω)2(gr\nk+q\n2,±,ω)2−(ga\nk−q\n2,∓,ω)2(ga\nk+q\n2,±,ω)2]\n−/summationdisplay\nj/planckover2pi12Ω\nmA±\ns,j(q)f′(ω)kj/bracketleftig/parenleftig\nk+q\n2/parenrightig\nigr\nk−q\n2,∓,ωgr\nk+q\n2,±,ωga\nk+q\n2,±,ω+/parenleftig\nk−q\n2/parenrightig\nigr\nk−q\n2,∓,ωga\nk−q\n2,∓,ωga\nk+q\n2,±,ω/bracketrightig/bracketrightbigg\n+O(Ω2). (203)\nSimilarly, we have\n˜s±,(1b)(q) =−ie\nmVlim\nΩ→0/integraldisplaydω\n2π/summationdisplay\nkAi(Ω)A±\ns,i(q)/bracketleftbigg\n−f/parenleftbigg\nω+Ω\n2/parenrightbigg\ngr\nk−q\n2,∓,ω−Ω\n2gr\nk+q\n2,±,ω+Ω\n2+f/parenleftbigg\nω−Ω\n2/parenrightbigg\nga\nk−q\n2,∓,ω−Ω\n2ga\nk+q\n2,±,ω+Ω\n2\n+Ωf′(ω)gr\nk−q\n2,∓,ω−Ω\n2ga\nk+q\n2,±,ω+Ω\n2/bracketrightig\n+O(Ω2). (204)\nThe sum of the two contributions thus is (using −iΩAi=Ei)\n˜s±,(1)(q)≡˜s±,(1a)(q)+ ˜s±,(1b)(q)\n=e\nmV/summationdisplay\niEi/integraldisplaydω\n2π/summationdisplay\nk\n×/bracketleftbigg/summationdisplay\nj/planckover2pi12\nmA±\ns,j(q)f′(ω)kj/bracketleftig/parenleftig\nk+q\n2/parenrightig\nigr\nk−q\n2,∓,ωgr\nk+q\n2,±,ωga\nk+q\n2,±,ω+/parenleftig\nk−q\n2/parenrightig\nigr\nk−q\n2,∓,ωga\nk−q\n2,∓,ωga\nk+q\n2,±,ω/bracketrightig\n+A±\ns,i(q)f′(ω)gr\nk−q\n2,∓,ωga\nk+q\n2,±,ω\n+A±\ns,i(q)f(ω)/bracketleftig\ngr\nk−q\n2,∓,ω(gr\nk+q\n2,±,ω)2−(gr\nk−q\n2,∓,ω)2gr\nk+q\n2,±,ω−(c.c.)/bracketrightig\n+/summationdisplay\njA±\ns,j(q)f(ω)kjqi[(gr\nk−q\n2,∓,ω)2(gr\nk+q\n2,±,ω)2−(ga\nk−q\n2,∓,ω)2(ga\nk+q\n2,±,ω)2]/bracketrightbigg\n. (205)\n.\nConsidering the adiabatic limit, electron Green’s functions are now ev aluated atq= 0. Using f′(ω) =\n−δ(ω) valid at low temperatures, we obtain\n˜s±,(1)(q) =−e\n2πmV/summationdisplay\nijEiA±\ns,j(q)/summationdisplay\nk\n×/bracketleftbigg/planckover2pi12\nmkikj/bracketleftbig\ngr\nk,∓gr\nk,±ga\nk,±+gr\nk,∓ga\nk,∓ga\nk,±/bracketrightbig\n+δijgr\nk,∓ga\nk,±\n−δij/integraldisplay\ndωf(ω)/bracketleftbig\ngr\nk,∓,ω(gr\nk,±,ω)2−(gr\nk,∓,ω)2gr\nk,±,ω−(c.c.)/bracketrightbig/bracketrightbigg\n, (206)\nwherega\nk,σ≡ga\nk,σ,ω=0and c.c. denotes complex conjugate. The summation over kin the first term on the\nright-hand side is carried out as follows. Using gr\nk,σga\nk,σ=iτe(gr\nk,σ−ga\nk,σ) (we assume that elastic lifetime\nτefor the two spins are equal), the first term of the right-hand side o f Eq. (206) is evaluated as\n/summationdisplay\nkkikj[gr\nk,∓gr\nk,±ga\nk,±+gr\nk,∓ga\nk,∓ga\nk,±] =iτeδij\n3/summationdisplay\nkk2[gr\nk,∓gr\nk,±−ga\nk,∓ga\nk,±]. (207)\n42Retarded contribution is evaluated by use of contour integral as\n1\nV/summationdisplay\nkk2gr\nk,∓gr\nk,±=/integraldisplay∞\n−ǫFdǫν(ǫ)(k(ǫ))21\nǫ±M−iηe1\nǫ∓M−iηe\n=iπ\n2M[ν+(kF+)2−ν−(kF−)2], (208)\nwhereν(ǫ)≡mk(ǫ)\n2π2,k(ǫ)≡/radicalbig\n2m(ǫ+ǫF)//planckover2pi1andηe≡/planckover2pi1\n2τe. Other terms of Eq. (206) are smaller by order of\n/planckover2pi1\nǫFτeand are neglected. The result of induced spin density is therefore\n˜s±,(1)(q) =−1\n2Mjs·A±\ns,, (209)\nwhere\njs≡1\ne(σ+−σ−)E≡P\nej, (210)\nandP≡σ+−σ−\nσ++σ−is the spin polarization of the current, σ±≡e2n±τe\nmbeing spin-resolved conductivity ( n±is\nspin-resolved electron density).\nThe non-adiabatic contribution (finite qcontribution) has a form like ˜ s±,na(q) =χ±\nij(q)EiA±\ns,j(q), where\nχ±\nijdenotes a correlation function, and has a non-local form in the rea l space as (Tatara et al., 2007, 2008)\n˜s±,na(r) =Ei/integraldisplay\nd3r′χ±\nij(r−r′)A±\ns,j(r′). (211)\nThis nonlocal spin polarization represents the force due to electro n reflection by localized spin structure like\nin the case of domain wall (Tatara and Kohno, 2004), as will be discus sed later.\nFrom Eqs.(198)(209)(211), the spin polarization density is obtaine d as\n˜s=−1\n2M/bracketleftbigg\nsAs,t+js,iAs,i/bracketrightbigg⊥\n+˜sna. (212)\nThe laboratory frame spin density is finally obtained as\ns≡R˜s=1\n2M/bracketleftbigg\ns(n×˙n)+[n×(js·∇)n]/bracketrightbigg\n+sna, (213)\nwheresna≡R˜sna.\nWhen spin relaxation is included in the electron Hamiltonian He, spin polarization perpendicular to the\nadiabatic case is induced (Zhang and Li, 2004). The effect of spin rela xation (sr) leads to spin polarization\nof (Kohno et al., 2006; Kohno and Shibata, 2007; Tatara and Ente l, 2008)\nssr=−1\nJsda3/bracketleftbigg\nαsr˙n+a3\n2Sβsr(js·∇)n/bracketrightbigg\n, (214)\nwhereαsrandβsrare dimensionless parameters proportional to spin relaxation rate .\nThe LLG equation including the effects of conduction electron, Eq. ( 182), is explicitly given by\n(1+δ)˙n=−γB×n−αsr(n×˙n)−a3\n2S(js·∇)n−βsra3\n2S[n×(js·∇)n]+τna, (215)\nwhereδ≡sa3\n2Sis the renormalization of localized spin due to the conduction electron spin polarization.\n4312. Current-driven domain wall motion\nWe discuss here dynamics of a domain wall based on the LLG equation in cluding the current-induced\ntorques, Eq. (215). Applied magnetic field and pinning, represente d by including a local magnetic field, are\nnot considered. Similarly to the field-driven case discussed in Sec. 3.4 , the equation of motion for current-\ndriven wall is obtained by putting the wall profile (51) in Eq. (215) and integrating over spatial coordinate\nas\n˙φ+α˙X\nλ=βw\nλ˜j\n˙X−αλ˙φ=−vcsin2φ+˜j, (216)\nwhere both vc≡K⊥λS\n2/planckover2pi1and˜j≡a3P\n2eSjhave dimension of velocity, P≡ejs/jbeing spin polarization of the\ncurrent. Here\nβw≡βsr+βna, (217)\nrepresents the total current-induced force as a result of spin r elaxation and electron reflection ( βna). The\nreflection force term βnaarises from the nonadiabatic (nonlocal) torque τna, and is written in terms of\nelectric registance due to the wall, Rw(Tatara and Kohno, 2004; Tatara et al., 2008) as\nβna≡e2\nhNRw, (218)\nwhereN=nAλis the total number of spins in wall with thickness λ, wherenandAare electron density\nand cross sectional area of the system, respectively.\nWhenβw= 0 and without magnetic field, the wall velocity when a constant ˜jis applied is easilyobtained\nas (Tatara and Kohno, 2004)\n˙X=/braceleftigg\n0 ( ˜j <˜ji\nc)\n1\n1+α2/radicalig\n˜j2−(˜jic)2 (˜j≥˜ji\nc)(219)\nand˜ji\nc≡vcis the intrinsic threshold current density (Tatara and Kohno, 2004 ). Namely, the wall cannot\nmove if the applied current is lower than the threshold value as shown in black line in Fig. 22. This is\nbecause the torque supplied by the current is totally absorbed by t he wall by tilting the out of plane angle\nto be sin2φ0=˜j/vcwhen the current is weak ( |˜j/vc|≤1) and thus the wall cannot move. This effect is\ncalled the intrinsic pinning effect (Tatara et al., 2008). For larger cur rent density, the torque carried by the\ncurrent induces an oscillation of the angle similar to the Walker’s break down in an applied magnetic field,\nand the wall speed also becomes an oscillating function of time.\nWhen nonadiabaticity parameter βwis finite, the behavior changes greatly and intrinsic pinning effect is\nremoved and the wall can move with infinitesimal applied current if ext rinsic pinning is neglected. In fact,\nwhen the applied current density is |˜j|>|˜ja|, where\n˜ja≡vc\n1−βw/α, (220)\nthe solution of Eq. (216) is an oscillating function given by (Tatara et al., 2008)\n˙X=βw\nα˜j+vc\n1+α2/parenleftbig˜j/˜ja/parenrightbig2−1\n˜j/˜ja−sin(2ωt−ϑ), (221)\nwhere\nω≡vc\nλα\n1+α2/radicaligg/parenleftbigg˜j\n˜ja/parenrightbigg2\n−1, sinϑ≡vc\n(βw/α−1)˜j. (222)\n44Figure 22: Time averaged wall velocity vwas function of applied spin-polarized current jforα= 0.01. Intrinsic pinning\nthreshold ji\ncexists only for βw= 0. The current density where derivative of vwis discontinuous corresponds to ˜ja.\nThe time-average of the wall speed is\n˙X=βw\nα˜j+vc\n1+α21\n˜ja/radicalig\n˜j2−˜j2a. (223)\nFor current density satisfying ˜j <˜ja,ωbecomes imaginary and the oscillation in Eq. (221) is replaced by\nan exponential decay in time. The wall velocity then reaches a termin al value of\n˙X→βw\nα˜j. (224)\nThe angle of the wall also reaches a terminal value determined by\nsin2φ0→/parenleftbiggβw\nα−1/parenrightbigg˜j\nvc. (225)\nThe averaged wall speed (Eq. (223)) is plotted in Fig. 22.\n12.1. Threshold current for domain wall motion\nThe intrinsic pinning is a unique feature of current-driven domain wall, as the wall cannot move even in\nthe absence of pinning center. In the unit of A/m2, the intrinsic pinning threshold is\nji\nc=eS2\nPa3/planckover2pi1K⊥λ. (226)\nFor device applications, this threshold needs to be lowered by reduc ing the hard-axis anisotropy and wall\nwidth (Fukami et al., 2008). The intrinsic pinning regime is promising fo r stable device operations, be-\ncause the threshold current and dynamics is insensitive to extrinsic pinning and external magnetic field\n(Tatara and Kohno, 2004), as was confirmed experimentally (Koya ma et al., 2011). This is due to the fact\nthat the wall dynamics in the intrinsic pinning regime is governed by a to rque, which governs the wall ve-\nlocity˙X, while extrinsic pinning and magnetic field induce force, which governs ˙φ; The forces due to sample\nirregularity therefore does not modify the motion induced by a torq ue in the intrinsic pinning regime. The\ninsensitivity is therefore a consequence of the fact that the wall c arries both linear and angular momenta,\nand thus has two different mechanisms for driving.\nExperimentally, intrinsic pinning is observed in perpendicularly magnet ized materials (Koyama et al.,\n2011), while materials with in-plane magnetization mostly are in the ext rinsic pinning regime governed by\n45the nonadiabatic parameter βwand extrinsic pinning. In this regime, the threshold current of the w all\nmotion is given by (Tatara et al., 2006)\nje\nc∝Ve\nβw, (227)\nwhereVerepresents strength of extrinsic pinning potential like those gene rated by geometrical notches and\ndefects. Control of nonadiabaticity parameter is therefore exp ected to be useful for driving domain walls at\nlow current density.\nClosetothethresholdcurrentdensity,thermalassist(Tatara e t al.,2005)andcreepmotion(Lemerle et al.,\n1998) becomes important.\nOf recent interest from the viewpoint of low current operation is to use multilayer structures. For\ninstance, heavy metal layers turned out to lower the threshold cu rrent by exerting a torque as a result\nof spin Hall effect (Emori et al., 2013), and synthetic antiferroma gnets turned out to be suitable for fast\ndomain wall motion at low current (Saarikoski et al., 2014; Yang et al. , 2015; Lepadatu et al., 2017).\nIt was recently shown theoretically that strong Rashba-induced m agnetic field works as a strong pinning\ncenter when introduced locally, and that this Rashba pinning effect is useful for highly reliable control of\ndomain walls in racetrack memories (Tatara et al., 2016).\n13. Magnon gauge field\nLet us discuss that effective gauge field exist for magnons. Magnon (spin wave) is an excitation repre-\nsenting fluctuation of localized spin. For localized spin configuration p olarized uniformly along zdirection,\nmagnon is introduced by use of the Holstein-Primakov boson, repre sented by field operators bandb†. For\ntreating localized spins with spatial and temporal variations, we intr oduce unitary transformation, in the\nsame manner as the case of electron. The localized spin is represent ed as\nS=U3(r,t)/tildewideS, (228)\nwhere/tildewideSis spin vector with average along the zdirection and U3is a 3×3 unitary matrix describing a\nrotation of a vector ˆzto the direction ( θ,φ). The spin vector /tildewideSis repressented by magnon field as\n/tildewideS≡Sˆz+δs\nδs=\nγ(b†+b)\niγ(b†−b)\n−b†b\n, (229)\nwhereγ≡/radicalig\nS\n2, and the terms third- and higher-order in boson operators are ne glected. The unitary matrix\nis chosen as Thiele (1973)\nU=\ncosθcosφ−sinφsinθcosφ\ncosθsinφcosφsinθsinφ\n−sinθ0 cos θ\n. (230)\nThe unitary transformation modifies derivatives of spin as\n∇iS=U3/parenleftbigg\n∇i−i\n/planckover2pi1AU,i/parenrightbigg\n/tildewideS, (231)\nwhere\nAU,i≡iU−1\n3∇iU3, (232)\n46is a spin gauge field represented by a 3 ×3 matrix. Explicitly, the spin gauge field reads\nAU,i=i/planckover2pi1\n∇µθ\n0 0 1\n0 0 0\n−1 0 0\n+∇µφ\n0−cosθ0\ncosθ0 sinθ\n0−sinθ0\n\n. (233)\nHere we consider the simple system with ferromagnetic exchange ine teraction, represented by the La-\ngrangian\nLF=/integraldisplayd3r\na3/bracketleftbigg\n−/planckover2pi1S(1−cosθ)˙φ−J\n2(∇S)2/bracketrightbigg\n. (234)\nThe exchange interaction is written in the rotated frame as\n(∇S)2= (∇/tildewideS)2+i\n/planckover2pi1/tildewideS†(AU·↔\n∇)/tildewideS+O((AU)2). (235)\nThe second term contains terms linear in boson operators. They de scribe the interaction of localized spin\nstructure and magnons, and are neglected. We thus obtain to the linear order in the derivative of localized\nspin structure\n(∇S)2= 2S/bracketleftbigg\n|∇b|2−i\n/planckover2pi1As(b†↔\n∇b)/bracketrightbigg\n, (236)\nwhereAs=/planckover2pi1\n2(∇φ)cosθagrees with the spin gauge field for conduction electron. (Although magnon spin is\n−1, we keep the prefactor of1\n2for electron spin in Asto avoid confusion.) The time-derivative term of Eq.\n(234),LB, is written in terms of magnon opearators as\nLB=−i/planckover2pi1\n2/bracketleftbigg\nb†∂tb+i\n/planckover2pi1As,tb†b/bracketrightbigg\n, (237)\nwhereAs,tis the time component of spin gauge field. The Lagrangian (234) is the refore written as a\nLagrangian for a boson interacting with an effective U(1) gauge field (Tatara, 2015a),\nLF=/integraldisplayd3r\na3/bracketleftbigg\n−i/planckover2pi1\n2b†∂tb+As,tb†b−JS|∇b|2−As·jm/bracketrightbigg\n, (238)\nwhere\njm≡−iJS\n/planckover2pi1b†↔\n∇b, (239)\nis magnon current. As magnon carry negative spin, the sign of the g auge coupling term, As·jm, of Eq. (238)\nis negative. Equation (238) indicates an interesting fact that magn on feels an effective U(1) gauge field that\nis the same as the one As,µfor conduction electron spin. This fact might be natural from symm etry point\nof view, but is not obvious. At equilibrium, magnon chemical potential is zero, and expectation value of\nmagnon number is determined by temperature. Equation (238) indic ates that an effective chemical potential\nformagnon, As,t, emergesfromdynamicsoflocalizedspin. Experimentalobservatio nofdynamically-induced\nmagnon chemical potential was reported recently (Du et al., 2017) . Spin structure with finite effective\nmagnetic field Bslike magnetic skyrmion induces magnon Hall effect (Dugaev et al., 200 5).\n14. Interface Rashba spin-orbit effects\nAlthough theoretical physics has been focusing on infinite systems for exploring beautiful general law\nsupported by symmetries, studying such ’beautiful’ systems seem s to becoming insufficient in condensed\nmatter physics. This is because demands to understand interface s and surfaces has been increasing rapidly\n47as devices are becoming smaller and smaller to meet the needs for fas t processing of huge data. Systems\nwith lower symmetry are therefore important subjects of materia l science today.\nSurfaces and interfaces have no inversion symmetry, and this lead s to emergence of an antisymmetric\nexchange interaction (Dzyaloshinskii-Moriyainteraction) (Dzyalos hinsky, 1958; Moriya, 1960) in magnetism\n. As for electrons, broken inversion symmetry leads to a peculiar sp in-orbit interaction, called the Rashba\ninteraction (Rashba, 1960), whose quantum mechanical Hamiltonia n is\nHR=iαR·(∇×σ), (240)\nwhereσis the vector of Pauli matrices and αRis a vector representing the strength and direction of the\ninteraction. The form of the interaction is the one derived directly f rom the Dirac equation as a relativistic\ninteraction, but the magnitude can be strongly enhanced in solids ha ving heavy elements compared to the\nvacuum case.\nAs isobviousfromtheformoftheHamiltonian, the Rashbainteractio ninduces electromagneticcrosscor-\nrelation effects where a magnetization and an electric current are in duced by external electric and magnetic\nfield,EandB, respectively, like represented at finite frequency ωas\nM=κME(αR×E), j=i/planckover2pi1ωγκME(αR×B), (241)\nwhereκMEis a coefficient depending on frequency (Shibata et al., 2016). The em ergence of spin accumu-\nlation from the applied electric field, mentioned in Refs. (Rashba, 196 0; Dyakonov and Perel, 1971), was\nstudied by Edelstein (Edelstein, 1990) in detail, and the effect is some times called Edelstein effect. The gen-\neration of electric current by magnetic field or magnetization, called the inverse Edelstein effect (Shen et al.,\n2014), was recently observed in multilayer of Ag, Bi and a ferromag net (Sanchez et al., 2013).\n14.1. Effective magnetic field\nWhen a current density jis applied, the conduction electron has average momentum of p=m\nenj(nis\nelectron density), and Eq. (240) indicates that an effective magne tic field of Be=ma3\ne/planckover2pi12γαR×j,acts on the\nconduction electron spin ( γ(=e\nm) is the gyromagnetic ratio). When the sdexchange interaction between\nthe conduction electron and localized spin is strong, this field multiplied by the the spin polarization, P, is\nthe field acting on the localized spin. Namely, the localized spin feels a cu rrent-induced effective magnetic\nfield of\nBR=Pma3\ne/planckover2pi12γαR×j. (242)\nThe strength of the Rashba-induced magnetic field is estimated (ch oosinga= 2˚A) asBR= 2×1016×\nαR(Jm)js(A/m2); For a strong Rashba interaction αR= 1 eV˚A like at surfaces (Ast et al., 2007), BR=\n4×10−2Tatjs= 1011A/m2. Thisfieldappearsnotverystrong,butissufficientatmodifythema gnetization\ndynamics. Infact, forthedomainwallmotion, whentheRashba-ind ucedmagneticfieldisalongthemagnetic\neasy axis, the field is equivalent to that of an effective βparameter of\nβR=2mλ\n/planckover2pi12αR, (243)\nwhereλis the wall thickness. If αR= 1 eV˚A,βRbecomes extremely large like βR≃250 forλ= 50 nm.\nNote thatβarising from spin relaxation is the same order as Gilbert damping const ant, namely of the order\nof 10−2. Such a large effective βRfrom Rashba effect is expected to leads to an extremely fast domain wall\nmotion under current (Obata and Tatara, 2008; Manchon and Zha ng, 2009).\nExperimentally, it was argued that fast domain wall motion observed in Pt/Co/AlO was due to the\nRashba interaction (Miron et al., 2010), but the result is later asso ciated with the torque generated by spin\nHall effect in Pt layer (Emori et al., 2013).\n48Figure 23: Domain wall (DW) transported with current jin a magnetic wire with a local Rashba spin-orbit pinning pot ential\nαRof finite width used to capture and pin a moving wall.\nDomain walls in ferromagnetic nano wires are potential building-blocks of future technologies such as\nracetrackmemories(Yang et al.,2015). Forsuchmemories,efficien tmechanismstoinitiateandstopdomain-\nwall motion are necessary. It was pointed out theoretically that a lo cally embedded spin-orbit interaction\nof Rashba type (Fig. 23) acts as a strong pinning center for curre nt-driven domain walls and that efficient\ncapturing and depinning of the wall is realized even using a weak spin-o rbit interaction (Tatara et al., 2016).\n14.2. Rashba-induced spin gauge field\nSince the interaction (240) is the one coupling to the spin current, t he Rashba interaction is regarded\nas a gauge field acting on electron spin as far as the linear order conc erns. In the field representation, the\ninteraction is\nHR=i/integraldisplay\nd3rc†αR·(↔\n∇×σ)c. (244)\nConsidering the case of strong sdexchange interaction, the interaction is expressed in terms of the rotated\nframe electron, a≡U−1c, whereUis defined in Eq. (65), as\nHR=−e\n2m/integraldisplay\nd3ra†(−i↔\n∇)·ARa, (245)\nwhere\nAR,α≡−m\ne/planckover2pi1ǫαβγαR,βRγδσδ≡Aδ\nR,ασδ, (246)\nandRis defined in (125). We neglect contributions including derivatives of lo calized spin structure, namely,\nspin gauge fieldAs. In the strong sdexchange interaction case, a gauge field ARis projected to the diagonal\ncomponent,AR,α→Az\nR,α≡AR,α, giving rise to a U(1) effective gauge field of\nAR≡−m\ne/planckover2pi1(αR×n). (247)\nExistence of a gauge field naturally leads to an effective electric and m agnetic field (Kim et al., 2012;\nNakabayashi and Tatara, 2014)\nER=−˙AR=m\ne/planckover2pi1(αR×˙n)\nBR=∇×AR=−m\ne/planckover2pi1∇×(αR×n). (248)\nIn the presence of electron spin relaxation, the electric field has a p erpendicular component (Tatara et al.,\n2013a)\nE′\nR=m\ne/planckover2pi1βR[αR×(n×˙n)], (249)\nwhereβRis a coefficient representing the strength of spin relaxation. For th e case of strong Rashba inter-\naction ofαR= 3 eV˚A, as realized in Bi/Ag, the magnitude of the electric field is |ER|=m\ne/planckover2pi1αRω= 26kV/m\n49E' , jRBRαRnjs\nBiNF\nFigure 24: Schematic figure depicting (spin relaxation cont ribution of) Rashba-induced spin electric field E′\nRgenerated by\nmagnetization precession in a junction of a ferromagnet (F) , a nonmagnetic spacer (N) and a heavy atom layer (Bi), where t he\nRashba interaction is induced. Electric current jis induced as a result of motive force E′\nRin the direction perpendicular to\nbothn×˙nand Rashba field αR. The magnetic field component BRlies in-plane and is expected to induce ’giant’ spin Hall\neffect when an electric field is applied perpendicular to the p lane.\nif the angular frequency ωof magnetization dynamics is 10 GHz. The magnitude of relaxation con tribution\nis|E′\nR|∼260V/m ifβR= 0.01. The effective magnetic field in the case of spatial length scale of 1 0 nm is\nhigh as well; BR∼260T.\nThe Rashba-induced electric fields, ERandE′\nR, are important from the viewpoint of spin-charge con-\nversion. In fact, results (248)(249) indicates that a voltage is ge nerated by a dynamics magnetization if the\nRashba interaction is present. Importantly, this effect emerges e ven from a spatially uniform magnetization\nprecession, in sharp contrast to the conventional adiabatic effec tive electric field (spin motive force) (Eq.\n(77)). In the case of a think film with Rashba interaction perpendicu lar to the plane and with a precessing\nmagnetization, the component ER∝˙nhas no DC component, while the relaxation contribution E′\nRhas\na DC component perpendicular to n×˙n∝ba∇dbln, wheren×˙nandndenote time-averages. The geometry of\nthis current pumping effect, j∝E′\nR∝αR×n(Fig. 24), is therefore the same as the one expected in the\ncase of inverse Edelstein effect, conventionally discussed in terms o f spin current Shen et al. (2014).\nIn the present form, Eqs. (248)(249), the Rashba-induced elec tric field is a local quantity; a voltage is\ngenerated by a direct contact between the Rashba interaction an d magnetization. It is expected, however,\nto become long-ranged if electron diffusion is taken into account. Th e inverse Edelstein effect reported in\nRef. (Sanchez et al., 2013) in a system with a Ag spacer, would ther efore be explained by the long-ranged\nRashba-induced voltage. For this scenario to be justified, it is cruc ial to confirm the existence of magnetic\ncomponent, BR, which can be of the order of 100T. In the setup of Fig. 24, BRis alongn. The field can\ntherefore be detected by measuring “giant” in-plane spin Hall effec t when a current is injected perpendicular\nto the plane.\nElectric current generation by spin dynamics with Rashba spin-orbit interaction was theoretically dis-\ncussedinthecaseofadot(Levitov and Rashba,2003)andtwo-dim ensionalelectrongas(Tokatly and Sherman,\n2010)\n15. Anomalous optical properties of Rashba conductor\nThe idea of effective gauge field is useful for extending the discussio n to include other degrees of freedom,\nlike optical properties. In fact, the fact that the Rashba interac tion coupled with magnetization leads to\nan effective vector potential AR(Eq. (247)) for electron spin indicates that the existence of intrin sic spin\nflow. Such intrinsic flow affects the optical properties, as incident e lectromagnetic waves get Doppler shift\nwhen interacting with flowing electrons, resulting in a transmission de pending on the direction (directional\ndichroism), as was theoretically demonstrated in Refs. (Shibata et al., 2016; Kawaguchi and Tatara, 2016).\nThe magnitude of the directional dichroism for the case of wave vec torqis determined by q·(αR×n).\nThe vector ( αR×n), which breaks both time-reversal and spatial inversion symmetr ies, is called in the\ncontext of multiferroics the toroidal moment, and it was argued to acts as an effective vector potential for\nlight (Sawada and Nagaosa, 2005).\n5015.1. Electromagnetic metamaterial property\nIt was shown that Rashba conductor itself, without magnetization , shows peculiar optical properties\nsuch as negative refraction as a result of spin-charge mixing effect s (Shibata et al., 2016, 2018). In fact,\nspin-charge mixing effects of Eq. (241) leads to a current generat ed by applied electric field, E, given by\n(schematically shown in Fig. 25)\njIE·E=−i/planckover2pi1ωγ(κME)2[αR×(αR×E)]. (250)\nAs it is opposite to the applied field, the mixing effect results in a soften ing of the plasma frequency as\nfor theEhaving components perpendicular to αR. The electric permittivity of the system is therefore\nanisotropic; Choosing αRalong thezaxis, we have\nεz= 1−ω2\np\nω(ω+iη), ε x=εy= 1−ω2\nR\nω(ω+iη), (251)\nwhereωp=/radicalbig\ne2ne/ε0mis the bare plasma frequency ( neis the electron density), and\nωR≡ωp/radicalbig\n1+ReC(ωR)<ωp, (252)\nis the plasma frequency reduced by the spin mixing effect. Here\nC(ω)≡−αR2kF2\nǫFn/integraldisplayd3k\n(2π)3γksk\n(/planckover2pi1ω+iη)2−4γ2\nk, (253)\nwithsk≡/summationtext\nσ=±σfkσis the electron spin polarizationand γk≡|k×αR|, representsthe correlationfunction\nrepresentingtheRashba-Edelsteineffect(Shibata et al.,2016). Therealpartof C(ω)isnegativenear ω∼ωp\nand thusωR< ωp. The frequency region ωR< ω < ω pis of interest, as the system is insulating ( εz>0)\nin the direction of the Rashba field but metallic in the perpendicular dire ction (εx<0). The dispersion\nin this case becomes hyperbolic, and the group velocity and phase ve locity along qcan have opposite\ndirection, resulting in negative refraction. Rashba system is, ther efore a natural hyperbolic metamaterial\n(Narimanov and Kildishev, 2015). A great advantage of Rashba con ductors are that the metamaterial\nbehavior arises in the infrared or visible light region, which is not easily a ccessible in artificially fabricated\nsystems. In the case of BiTeI with Rashba splitting of α= 3.85 eV˚A(Ishizaka et al., 2011), the plasma\nfrequency is ωp= 2.5×1014Hz (corresponding to a wavelength of 7 .5µm) forne= 8×1025m−3and\nǫF= 0.2 eV (Demk´ o et al., 2012). We then have ωR/ωp= 0.77 (ωR= 1.9×1014Hz, corresponding to the\nwavelength of 9 .8µm), and hyperbolic behavior arises in the infrared regime.\nSpintronics devices have potential applications for electromagnet ic metamaterials as was argued for spin\ntorque oscillators (Tatara et al., 2013b).\n15.2. Directional dichroism of magnetic Rashba conductor\nRashba interaction breaking spatial inversion symmetry leads to va rious interesting spin transport and\noptical responses as we have seen. When the time reversal symme try is broken in addition, other anomalous\nproperties are expected. Here we discuss such a optical propert y. The system we consider is a Rashba\nconductor with magnetization M(or in a magnetic field), which breaks the time-reversal symmetry. Unique\nfeature in this system is that we have a vector\nu≡αR×M, (254)\nthat has the same symmetry as a velocity; Namely, urepresents a sort of intrinsic flow in magnetic Rashba\nconductor. Such a vector can induce intriguing cross correlation e ffect, because it allows a coupling between\nthe electric and the magnetic fields like (Kawaguchi and Tatara, 201 6)\nHu=gu·(E×B), (255)\n51Figure 25: Schematic figure show-\ning the cross-correlation effects in the\nplane perpendicular to the Rashba\nfieldαR. Edelstein effect (E) gener-\nates spin density, sE, from the applied\nelectric field, and inverse Edelstein ef-\nfect (IE) generates current jIE·Efrom\nmagnetization ME.\nFigure 26: Electric permittivity of\nRashba conductor with ˜ α≡mα\n/planckover2pi12kF=\n1.0 andη/(/planckover2pi1ωp) = 0.01. The shaded\nregion between ωRandωpis the hy-\nperbolic region showing metamaterial\nbehavior.\nFigure 27: Schematic picture show-\ning negative refraction of Rashba con-\nductor. The physical flow of light (the\nPoyinting vector S) incident with an\nangleθinwith respect to αRget re-\nfracted in the negative direction. The\nwave vector qof the refracted wave are\ninthe normaldirection, butitdoesnot\ndescribe physical flow.\nwheregis a coefficient. The vector1\nµE×B(whereµis the magnetic permeability of solids) is the Poynting\nvector representing the momentum of the electromagnetic wave. The vector coupling of Eq. (255) is thus\nessentially u·k(kis the wave vector of the electromagnetic wave), and represents the Doppler shift with\nrespecttotheintrinsic’velocity’, u. Inotherwords, uactsasaneffectivevectorpotentialforelectromagnetic\nvector potential A. Quantity uhas the same symmetry as toroidal moment in multiferroics, and thu s it\nis reasonable that it causes directional dichroism. As seen from Eq. (247),uis also an effective vector\npotential for conduction electron spin. It is interesting that the s ame quantity works as a vector potential\nfor both electron spin and photon. Note that the intrinsic flow repr esented by uis not charge current nor\nspin current; those currents are driven by effective electric field o f Eq. (248) and are proportional to time\nderivative of u.\nThe Maxwell’s equations taking account of Eq. (255) read\n∇·E=ρ\nǫ0−1\nǫ0∇·(u×B),\n∇×B=µ0j+ǫ0µ0∂E\n∂t+µ0∂\n∂t(u×B)−µ0∇×(u×E). (256)\nTherefore the total electric and magnetic fields in the present sys tem read\nEtot=E+1\nǫ0(u×B),\nBtot=B+µ0(u×E). (257)\nThese relations are the same as what we obtain in a moving frame and t hey clearly represent a cross-\ncorrelation effect as a result of the Doppler shift. The interaction ( 255) modifies the electric permittivity\nas\nǫij=ǫ(0)\nij−1\nǫ0ω(uiδlj+δiluj)kl+2\nǫ0ωδij(u·k). (258)\nThe last diagonal term linear in kleads to transmission and reflection depending on the direction kwith\nrespect to a vector u, namely, an asymmetric light propagation (directional dichroism).\n16. Summary\nWe have discussed spintronics effects from a unified viewpoint of effe ctive gauge field (spin gauge field)\ncoupling to electron spin current. Effective gauge field is a general c oncept to describe low energy behavior\n52of smooth backgroundstructure, which is localized spin (magnetiza tion) structure in the present case. As we\nhave seen, gauge field description has an advantage that driving fie ld is clearly identified and provides solid\nphysical picture of the effects. In fact, spin motive force, a volta ge generated by dynamic magnetization,\nis induced by the adiabatic component of spin gauge field, while spin pum ping effect is driven by the\nnonadiabaticcomponent. Spin-charge conversionand anomalouso ptical properties due to Rashba spin-orbit\ninteraction were also discussed. We have seen that localized spin str uctures generates the same effective\nadiabatic gauge field for both magnon and conduction electron spin. Moreover, a ’troidal’ moment arising\nfrom the Rashba interaction and localized spin acts as an effective ga uge field for electron spin and photons.\nEffective gauge field therefore describes transport and respons es of various degrees of freedom in a unified\nviewpoint.\nAcknowledgements\nThe author thanks a Grant-in-Aid for Scientific Research (B) (No. 17H02929) from the Japan Society\nfor the Promotion of Science and a Grant-in-Aid for Scientific Resea rch on Innovative Areas (No.26103006)\nfrom The Ministry of Education, Culture, Sports, Science and Tech nology (MEXT), Japan for financial\nsupport.\nAppendix A. Summary of path-ordered Green’s function\nHere we present a practival introduction for non-equilibrium Green ’s function.\nAppendix A.1. Obseervable and path-ordering\nAn expectation value of observable ˆOat timetis\nO(t) =1\nZtr/bracketleftig\ne−βˆH0ˆOH(t)/bracketrightig\n, (A.1)\nwhereZ≡tr[e−βˆH0] is the partition function,\nˆOH(t)≡ˆU(t−t0)†ˆOˆU(t−t0), (A.2)\nis the Heisenberg representation with unitary operator represen ting the time-development\nˆU(t−t0) =Te−i\n/planckover2pi1/integraltextt\nt0dtˆH(t), (A.3)\nTbeing the time-ordering operator. Writing explicitly the time-depend ence, we have\nO(t) =1\nZtr/bracketleftig\ne−βˆH0[Tei\n/planckover2pi1/integraltextt\nt0dtˆH(t)]ˆO[Te−i\n/planckover2pi1/integraltextt\nt0dtˆH(t)]/bracketrightig\n, (A.4)\nwhereTis the anti time-ordering operator. Physical observable is theref ore written as a path-ordered\nexpectation value along a path C≡C→+C←+Cβ, whereC→is a path from t0tot∞,C←is fromt∞to\nt0and imaginary direction, Cβ(Fig. A.28), as\nO(t) =1\nZtr/bracketleftig\nTCe−i\n/planckover2pi1/integraltext\nCdtˆH(t)ˆO/bracketrightig\n, (A.5)\nwhereTCis the ordering operator along C.\n53tC→\nC← Cβ\n−i¯hβ\nFigure A.28: The contour C≡C→+C←+Cβwhich arises to describe physical observables. Path-order ed Green’s function\nis defined on this contour.\nAppendix A.2. Interaction representation\nTime-development operator ˆUis separated into a free part ˆU0described by ˆH0and an interaction part\nas\nˆU(t−t0) =ˆU0(t−t0)T[e−i\n/planckover2pi1/integraltextt\nt0dt/tildewideV(t)] (A.6)\nObservable of Eq. (A.4) then reads\nO(t) =1\nZtr/bracketleftig\ne−βˆH0T[ei\n/planckover2pi1/integraltextt\nt0dt/tildewideV(t)]/tildewideO(t)T[e−i\n/planckover2pi1/integraltextt\nt0dt/tildewideV(t)]/bracketrightig\n, (A.7)\nwhere\n/tildewideO(t)≡[ˆU0(t−t0)]−1ˆOˆU0(t−t0). (A.8)\nIf the observable is quadratic in the field operators as ˆO=/summationtext\nαβˆcαOαβˆcβ(αandβare index specifying\nthe field, like spin indices), physical ovservable is written as\nO(t) = tr[OG(tC,tC+0)], (A.9)\nwhere\nGc(tC,t′\nC)≡−i\nZtr/bracketleftig\nTCe−i\n/planckover2pi1/integraltext\nCdtˆH(t)ˆc(tC)ˆc†(t′\nC)/bracketrightig\n, (A.10)\nis path-ordered Green’s function defined for complex time tC,t′\nC,TCbeing the path-ordering operator. To\ncalculate physical observable, we need to know how to calculate the path-ordered Green’s function, and to\nevaluate real-time value of path-ordered Green’s function, define d on a complex time contour. By definition,\nthe path-ordered Green’s function agrees with the time-ordered Green’s function if the two time are on the\nupper contour, C→, and it is anti-time-ordered if two times are on C←;\nGc(tC∈C→,t′\nC∈C→) =−i/angbracketleftig\nTcH(t)c†\nH(t′)/angbracketrightig\n≡Gt(t,t′)\nGc(tC∈C←,t′\nC∈C←) =−i/angbracketleftig\nTcH(t)c†\nH(t′)/angbracketrightig\n≡Gt(t,t′). (A.11)\nWhen one of the two times is on C→, the ordering of the operator is fixed irrespective of tCandt′\nC. When\ntC∈C→andt′\nC∈C←, we have the lesser Green’s function,\nGc(tC∈C→,t′\nC∈C←) =i/angbracketleftig\nc†\nH(t′)cH(t)/angbracketrightig\n≡G<(t,t′), (A.12)\nwhile we have the greater Green’s function for the opposite case,\nGc(tC∈C←,t′\nC∈C→) =−i/angbracketleftig\ncH(t)c†\nH(t′)/angbracketrightig\n≡G>(t,t′). (A.13)\n54Those four Green’s functions are not independent, because\nGt(t,t′) =θ(t−t′)G>(t,t′)+θ(t′−t)G<(t,t′). (A.14)\nIn fact,\nGt(t<t′,t′) =i/angbracketleftig\nc†\nH(t′)cH(t)/angbracketrightig\n=G<(t,t′)\nGt(t>t′,t′) =−i/angbracketleftig\ncH(t)c†\nH(t′)/angbracketrightig\n=G>(t,t′). (A.15)\nAppendix A.3. Free electron Green’s functions\nLet us derive explicit form of Green’s functions for free electron de scribed by the Hamiltonian\nˆH=/integraldisplay\nd3rˆc†/parenleftbigg−/planckover2pi12∇2\n2m−ǫF/parenrightbigg\nˆc. (A.16)\nThe field operator in the momentum representation is\n˜ck(t) =e−i\n/planckover2pi1ǫktˆck, (A.17)\nwhereǫk≡/planckover2pi12k2\n2m−ǫF. The time-ordered Green’s function of free electron is therefore\ngt\nk(t−t′) =−ie−i\n/planckover2pi1ǫkt(θ(t−t′)/angbracketleftig\nˆckˆc†\nk/angbracketrightig\n−θ(t′−t)/angbracketleftig\nˆc†\nkˆck/angbracketrightig\n)\n=−ie−i\n/planckover2pi1ǫkt(θ(t−t′)(1−fk)−θ(t′−t)fk), (A.18)\nwhere\nfk≡1\neβǫk+1, (A.19)\nis the Fermi distribution function. In the frequency representat ion, we have\ngt\nk,ω=1−fk\n/planckover2pi1ω−ǫk+i0−fk\n/planckover2pi1ω−ǫk−i0, (A.20)\nwhere±i0 denote infinitesimal positive imaginary part. Lessor and greater G reen’s function are\ng<\nk(t−t′) =ie−i\n/planckover2pi1ǫktfk\ng>\nk(t−t′) =−ie−i\n/planckover2pi1ǫkt(1−fk), (A.21)\nand the Fourier representations are\ng<\nk,ω= 2πifkδ(/planckover2pi1ω−ǫk)\ng>\nk,ω=−2πi(1−fk)δ(/planckover2pi1ω−ǫk). (A.22)\nAppendix A.4. Calculation of lesser component\nWhen physical quantities are calculated perturbatively, we need to calculate the lesser component of\nproducts of path-ordered Green’s functions. We consider here t he case of a scattering by a potential V. The\nDyson equation for the path-ordered Green’s function reads\nGc\nα,β(tC,t′\nC) =∞/summationdisplay\nn=0/parenleftbigg\n−i\n/planckover2pi1/parenrightbiggn/bracketleftiggn/productdisplay\ni=1/integraldisplay\nCdtCi/bracketrightigg\n[Gc(tC,tC1)V(t1)Gc(tC1,tC2)···V(tn)Gc(tCn,t′\nC)]αβ.(A.23)\n55The lesser component of the first order term is\nG<(1)(t,t′) =Gc(1)(tC∈C→,t′\nC∈C←)\n=/integraldisplay\nCdtC1Gc(tC,tC1)V(t1)Gc(tC1,t′\nC). (A.24)\nWriting explicitly the contributions from the contour C→andC←, we have\nG<(1)(t,t′) =/integraldisplay\nC→dtC1Gt(tC,tC1)V(t1)G<(tC1,t′\nC)+/integraldisplay\nC←dtC1G<(tC,tC1)V(t1)Gt(tC1,t′\nC)\n=/integraldisplay∞\n−∞dt1/bracketleftig\nGt(t,t1)V(t1)G<(t1,t′)−G<(t,t1)V(t1)Gt(t1,t′)/bracketrightig\n. (A.25)\nUsing identities\nGt(t,t′) =Gr(t,t′)+G<(t,t′) (A.26)\nGt(t,t′) =−Ga(t,t′)+G<(t,t′), (A.27)\nwe have a formula for evaluating the lesser component,\n/bracketleftbigg/integraldisplay\nCdtC1Gc(tC,tC1)V(t1)Gc(tC1,t′\nC)/bracketrightbigg<\n=/integraldisplay∞\n−∞dt1/bracketleftbig\nGr(t,t1)V(t1)G<(t1,t′)+G<(t,t1)V(t1)Ga(t1,t′)/bracketrightbig\n.\n(A.28)\nLet us denote Eq. A.28 simply as\n[GcGc]<=GrG<+G<Ga. (A.29)\nFor retarded and advanced components,\n[GcGc]a=GaGa\n[GcGc]r=GrGr. (A.30)\nThis formula is called the Langreth theorem. Terms with more Green’s functions are evaluated by applying\nthe formula multiply. For the three Green’s function case, we have\n[GcGcGc]<=Gr[GcGc]<+G<[GcGc]a=GrGrG<+GrG<Ga+G<GaGa. (A.31)\nReferences\nReferences\nAidelsburger M, Nascimbene S, Goldman N. Artificial gauge fie lds in materials and engineered systems. arXiv:171000851 2 017;.\nAst CR, Henk J, Ernst A, Moreschini L, Falub MC, Pacil´ e D, Bru no P, Kern K, Grioni M. 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URL:\nhttps://link.aps.org/doi/10.1103/PhysRevB.71.064420 . doi:10.1103/PhysRevB.71.064420 .\n61This figure \"hedgehog16_n2.png\" is available in \"png\"\n format from:\nhttp://arxiv.org/ps/1712.03489v2"    },    {        "title": "1202.5352v3.Dynamical_Magnetic_and_Nuclear_Polarization_in_Complex_Spin_Systems__Semi_magnetic_II_VI_Quantum_Dots.pdf",        "content": "Dynamical Magnetic and Nuclear Polarization in Complex Spin Systems:\nSemi-magnetic II-VI Quantum Dots\nRamin M. Abolfath1;2;3;y, Anna Trojnar2;3, Bahman Roostaei4, Thomas Brabec2, Pawel Hawrylak3\n1School of Natural Sciences and Mathematics, University of Texas at Dallas, Richardson, TX 75080\n2University of Ottawa, Physics Department 150 Louis Pasteur, Ottawa, ON, K1N 6N5, Canada\n3Institute for Microstructural Sciences, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6\n4Institut f ur Theoretische Physik, Universit at zu K oln, K oln, Germany\n(Dated: June 13, 2021)\nDynamical magnetic and nuclear polarization in complex spin systems is discussed on the example\nof transfer of spin from exciton to the central spin of magnetic impurity in a quantum dot in the\npresence of a \fnite number of nuclear spins. The exciton is described in terms of the electron\nand heavy hole spins interacting via exchange interaction with magnetic impurity, via hyper\fne\ninteraction with a \fnite number of nuclear spins and via dipole interaction with photons. The\ntime-evolution of the exciton, magnetic impurity and nuclear spins is calculated exactly between\nquantum jumps corresponding to exciton radiative recombination. The collapse of the wavefunction\nand the re\flling of the quantum dot with new spin polarized exciton is shown to lead to build up\nof magnetization of the magnetic impurity as well as nuclear spin polarization. The competition\nbetween electron spin transfer to magnetic impurity and to nuclear spins simultaneous with the\ncreation of dark excitons is elucidated. The technique presented here opens up the possibility of\nstudying optically induced Dynamical Magnetic and Nuclear Polarization in Complex Spin Systems.\nPACS numbers: 75.50.-y,75.50.Pp,85.75.-d\nI. INTRODUCTION\nThere is currently interest in developing means of lo-\ncalizing and controlling complex spin systems in solid\nstate devices1. This includes electron and/or hole spins in\ngated2,3, self-assembled4, nanocrystal5and carbon nan-\notube quantum dots6, nitrogen vacancies in diamond7\nand magnetic impurities in II-VI8{13and III-V14,15semi-\nconductors. The complex spin systems involved include\nheavy valence holes with spin J= 3=2, nitrogen vacan-\ncies with spin M= 1, half-\flled shell electrons of mangan\nMn2+impurity atom with spin M= 5=2 orMn3+atom\nwithM= 3=2 in II-VI semiconductor quantum dots,\nor strongly coupled valence hole-Mn atom in InAs/GaAs\nquantum dots. Extensive theoretical studies have been\ncarried out, predicting rich phase diagram for these sys-\ntems17{21. For NV centers in diamond, carbon nanotube\nbased quantum dots and magnetic impurities in II-VI\nsemiconductor quantum dots the complex spin system\ninteracts with only a \fnite number of nuclear spins. The\ncontrolling of magnetization of complex spin systems is\noften carried out optically and involves transfer of pho-\nton angular momentum into exciton spin, and exciton\nspin into the spin of the complex spin system22,23. This\ndynamical magnetic polarization (DMP) process is de-\ncohered by photon and nuclear spin baths. Recently,\n\frst optical experiments on single magnetic impurities\nin II-VI quantum dots measured the dynamic evolution\nof the magnetization process8{10with theoretical models\nof DMP based on rate equations24,25.\nIn this work, we develop a microscopic theory of opti-\ncally driven dynamical magnetic polarization of complex\nspin systems. The theory describes the transfer of spin\nfrom exciton to the central spin of magnetic impurity in aquantum dot in the presence of a \fnite number of nuclear\nspins using quantum jump approach26{28. The exciton is\ndescribed in terms of the electron and heavy hole spins\ninteracting via exchange interaction with magnetic impu-\nrity, via hyper\fne interaction with a \fnite number of nu-\nclear spins and via dipole interaction with photons. The\ntime-evolution of the exciton, magnetic impurity and nu-\nclear spins is calculated exactly between quantum jumps\ncorresponding to exciton radiative recombination. The\ncollapse of the wavefunction and the re\flling of the quan-\ntum dot with new spin polarized exciton as in recent ex-\nperiment by Goryca et al.8is shown to lead to build up\nof magnetization of the magnetic impurity as well as nu-\nclear spin polarization. The competition between elec-\ntron spin transfer to magnetic impurity and to nuclear\nspins simultaneous with the creation of dark excitons is\nelucidated.\nThe paper is organized as follows. In Section II we\ndescribe our model. Section III describes quantum jump\napproach to time evolution of a single MI and a single ex-\nciton in the absence of nuclear spins. Section IV contains\nquantum jump approach and the dynamical evolution of\nMI interacting with a train of excitons in the presence of\nnuclear spins. In sections V and VI we present numerical\nresults, discussions, conclusion and the summary.\nII. THE MODEL\nWe consider a semiconductor QD containing a complex\nspin system M, e.g., magnetic ion (MI), coupled with\nfew nuclear spins of the host material, as in, e.g., CdTe\nquantum dots. The quantum dot with MI is attached to\na smaller quantum dot with no MI where the electronsarXiv:1202.5352v3  [cond-mat.mes-hall]  17 May 20122\n(a) \n(b) \n(c) \n+\n(d) \nFIG. 1: Schematic representation of the DMP process.\nand valence holes with de\fnite spin polarization are gen-\nerated optically by circularly polarized light. This is il-\nlustrated in Fig.1a where circles describe quantum dots,\nblue arrow corresponds to electron spin Sz= +1=2 and\nwhite arrow corresponds to heavy hole spin Jz=\u00003=2\nin the smaller dot. The larger dot contains a randomly\noriented complex spin M, represented by a magenta ar-\nrow and a number of randomly oriented nuclear spins\nrepresented by small arrows. The DMP process starts\nwith transfer of spin polarized exciton from the smaller\nQD to the larger QD, as illustrated in Fig.1b. As a re-\nsult of interactions, the spin of electron, MI and nuclear\nspins undergo a \rip-\rop process as the wavefunction of\nthe larger dot evolves and forms an entangled state, a\nlinear combination of bright and dark excitons, as shown\nin Fig.1c. During this process the smaller dot is re\flled\nwith spin polarized exciton. Simultaneously, the bright\nexciton decays due to interaction with the photon \feld\nwith a random recombination time, resulting in a photon\nemission and a quantum jump takes place. As a result of\nthis process, the states of the magnetic ion and nuclear\nspins are modi\fed, the polarization is increased and the\nlarger dot is re\flled with spin polarized exciton and the\nDMP process continues.We now quantitatively describe the DMP process. We\nstart with the Hamiltonian describing the quantum dot\ncoupled with the photon bath H=HQD+HphQD +\nHph. HereHphis the photon Hamiltonian, HphQD is\nthe Hamiltonian describing coupling of photons with the\nexciton in a QD and HQDis the QD hamiltonian.\nThe QD Hamiltonian describes the coupling between\nexcitonXand the magnetic moment of the complex\nspin system, consisting of MI and nuclear spins I. It\nis given by: HQD=Hm+Hx+Hxm+Hn+Hxn+Hmn.\nHereHxdescribes the exciton internal energy, Hmde-\nscribes the MI internal energy and the remaining terms\ninHQDrepresent X-MI, X-I and MI-I exchange cou-\nplings. The exciton Hamiltonian describes the low en-\nergy quadruplet jS;Jicharacterized by quantum spin\nnumbers of an electron, S=\u00061=2, and a heavy hole,\nJ=\u00063=2 in the QD. The complex spin system is de-\nscribed by a total spin ~M=PN\ni=1~ ui, whereNis the\nnumber of spins u= 1=2 building up the MI system, and\nHm=P\ni<jJij~ ui\u0001~ uj+DM2\nzin whichJijare exchange\nmatrix elements building the total spin M. In quantum\ndots one often includes strain \feld Dleading to splitting\nof the di\u000berent Mzlevels16. Similarly ~I=PNb\ni=1~Iiwhere\nNbis total number of nuclear spins.\nWe assume that exchange coupling constants of MI\nspins with the environment are identical and the full QD\nHamiltonian can be written as21:\nHQD=Hm+Hx+JhmJzMz\u0000Jem~S\u0001~M\n+NbX\nn[Jne~In\u0001~S+JnhIz;nJz] +NbX\nnNbX\nn06=nJnn0~In\u0001~In0\n+NbX\nnAn~In\u0001~M: (1)\nThe exciton Hamiltonian Hx= \u0001 0SzJz+ \u0001 1(S+J\u0000+\nS\u0000J+) describes splitting \u0001 0between the low energy\ndark exciton doublet j\";*i=j+ 1=2;+3=2i;j#;+i=\nj\u00001=2;\u00003=2iwith total angular momentum jz=\u00062\nalong quantization-axis, ^ z, and higher energy bright ex-\nciton doubletj #;*i=j\u00001=2;+3=2i;j \";+i=j\u0000\n3=2;+1=2iwithjz=\u00061. Here\"=#and*=+\nrepresent spin of electron and hole30. The bright exci-\nton doublet is split by the anisotropic electron-hole ex-\nchange interaction characterized by parameter \u0001 1which\nmeasures the splitting of the two bright exciton states\nj+ 1=2;\u00003=2i;j\u00001=2;+3=2i. \u0001 1is zero for cylindri-\ncal quantum dots and the two bright exciton states cor-\nrespond to circular photon polarization. The exciton-\nMI coupling in Eq.1 is given as a sum of the ferromag-\nnetic Heisenberg electron-MI exchange Hem=\u0000Jem~S\u0001~M\nand anti-ferromagnetic Ising exchange interaction HhM=\n+JhMJzMz.21Only electron-MI interaction is responsi-\nble for the e-MI spin \rip-\rop process. The interaction\nof complex spin MI with nuclear spin associated with\nthe spin complex is denoted here by HMI=A~IM\u0001~M.\nThis interaction might, for example, describe coupling3\nof manganese d-shell electron spins with manganese ion\nnuclear spin16. With hole spin strongly aligned along\nthe growth zdirection the coupling of electron and hole\nspins to surrounding nuclear spins of isotopes of the\nQD and barrier material with \fnite nuclear spin readsPNb\nn[Jne~In\u0001~S+JnhIz;nJz] +P\nn;n0Jnn0~In\u0001~In0whereNb\nis the number of nuclear spins in the QD and the last\nterm describes nuclear spin interaction. We note that for\nisotropic QD the long range e-h exchange \u0001 1is zero and\nthe heavy hole spin Jz=\u00063=2 is preserved.\nIII. SINGLE MI, SINGLE EXCITON AND NO\nNUCLEAR SPIN\nWe start our discussion of DMP by discussing time\nevolution of magnetization of X-MI complex interacting\nwith harmonic \felds of photons in the absence of nuclear\nspins. To focus on quantum dynamics in the simplest\nspin system, we consider MI with M= 1=2 and just two\nstates,j \"i =jMz= 1=2iandj #i =jMz=\u00001=2i,\nand Hamiltonian HQD=Hx+HxmwhereHxm=\n\u0000Jem~Se\u0001~M+JhmSz;hMz. We also consider a CW laser\n\feld with one type of circular polarization, e.g., \u001b= +1,\nthat generates excitons with one type of polarization,\njz= +1, corresponding to jXbi=j#;*i. Because we\nneglect the hole spin-\rip in the spin \rip-\rop process of\nX-MI complex, as discussed in Sec. II, the dark exciton\njXdi=j\";*iwithjz= +2 is the only state generated\nthroughout the electron-MI spin-\rip. Hence the space of\na single X-MI complex can be spanned by j1i=jXb;#i,\nj2i=jXb;\"i,j3i=jXd;#i,j4i=jXd;\"i,j5i=j0;#i,\nj6i=j0;\"i. In this basis the exciton Hamiltonian is di-\nagonalHx= diag(Eb;Eb;Ed;Ed;0;0). HereEb, andEd\nare the energy of bright and dark excitons measured rel-\native to the vacuum. In the basis of fj1i;j4ig,fj2i;j3ig,\nandfj5i;j6ig, the X-MI Hamiltonian is block-diagonal\nHxm=Hxm;1\bHxm;2\bHxm;3whereHxm;1= (\u0000Jem+\nJhm)=41,Hxm;2=\u0012\n(Jem\u0000Jhm)=4\u0000Jem=2\n\u0000Jem=2 (Jem\u0000Jhm)=4\u0013\n,\nandHxm;3= 0 respectively. Here 1is a 2\u00022 unit ma-\ntrix.\nThe o\u000b-diagonal elements of Hxm;2describe mix-\ning ofXbandXdvia spin-1/2 MI. Hence j (t)i=\nCb;\"(t) exp(\u0000iEbt=\u0016h)jXb;\"i+Cd;#(t) exp(\u0000iEdt=\u0016h)jXd;#\niwith initial condition j (t= 0)i=jXb;\"iis one of\nthe solutions of the time-dependent Schr odinger equa-\ntion\u0000i\u0016h@\n@tj (t)i=HQDj (t)i. This state describes a\ncoherent Rabi-oscillations between bright and dark exci-\ntons due to MI spin \rip-\rop.\nNote that inj (t)ithere is no mixing with the vac-\nuum,j0;Mz=\u00061=2i, unless we take into account the\ncoupling of bright-exciton with radiation \feld. In the in-\nteraction and rotating wave approximation the electron-\nphoton coupling is described by the Hamiltonian that\n(a) \n (b) \n(c) \n (d) FIG. 2: Time evolution of density matrix with the initial\ncondition\u001a11(t= 0) =\u001a22(t= 0) = 0:5 and\u001aij(t= 0) = 0\nfor otheriandj's, corresponding to initial random state of\nMI, are shown in (a), (c), and (d) for \u000e= 0;1;5 meV. In\n(b) the expectation value of MI spin for \u000e= 0 is shown.\nHerej1i=jXb;#i,j2i=jXb;\"i,j3i=jXd;#i,j4i=jXd;\"i,\nj5i=j0;#i,j6i=j0;\"i. The population of vacuum can be\ncalculated by \u001avacuum =\u001a55+\u001a66. The elements of density\nmatrix, not plotted in this \fgure, are all identical to zero.\ndoes not directly change the state of MI\nHphQD (t) = \u0016hX\n~k;Mzg~k[by\n~kj0;MzihXb;Mzje\u0000i(!b\u0000!~k)t\n+b~kjXb;Mzih0;Mzje+i(!b\u0000!~k)t]; (2)\nwhereby\n~kandb~kare creation and annihilation operators\nof photon with speci\fc circular polarization \u001b= +1.g~k,\nand!~kare the photon-X coupling constant and photon\nfrequency, respectively, and !b=Eb=\u0016h. The equation of\nmotion of the QD density matrix, \u001a, coupled with ther-\nmal bath of photons can be calculated after tracing over\nphoton degrees of freedom. Here \u001arepresents the density\nmatrix of a single exciton interacting with a single MI.\nAssuming that photons are in thermal equilibrium and\nare weakly coupled with excitons in QDs, the equation of\nmotion for exciton density matrix, \u001a, can be calculated\nperturbatively. Up to the second order of perturbation,\nit is straightforward to show that28\nd\u001a\ndt=\u0000i\n\u0016h[HQD;\u001a]\u0000\u0000\n2nBX\nMz(j0;Mzih0;Mzj\u001a\n\u00002jXb;Mzih0;Mzj\u001aj0;MzihXb;Mzj\n+\u001aj0;Mzih0;Mzj)\n\u0000\u0000\n2(nB+ 1)X\nMz(jXb;MzihXb;Mzj\u001a\n\u00002j0;MzihXb;Mzj\u001ajXb;Mzih0;Mzj4\n+\u001ajXb;MzihXb;Mzj); (3)\nwherenB= 1=(e\u0016h!=kBT\u00001) is Bose-Einstein distribu-\ntion function and \u0000 =4!3\nb\u00182\n3\u0016hc3is the transition rate for\nthe spontaneous emission of photons. \u0018is the dipole mo-\nment matrix element. Note that in Eq.(3) vacuum can\nbe considered as a shelving-state.\nThe numerical solutions of Eq.(3) at zero-temperature\n(nB= 0) are shown in Fig. 2 for a QD with Ed= 2 eV\nand\u000e=Eb\u0000Ed= 0;1;5 meV. Here we used Jem= 1\nmeV andJhm= 4 meV. The initial state of MI is com-\npletely uncorrelated with half of the spins populated in\nup-direction. As it is shown, because of the coupling\nwith the bath of photons, bright-exciton decays to vac-\nuum without \ripping the MI spin and mixing with Xd,\ne.g.,jXb;Mzi!j 0;Mzi. However, a coherent Rabi os-\ncillation between XbandXdvia exchange with MI is\nresponsible for spin-transfer to MI. In Fig. 2(b) the time\nevolution of the components of the ensemble-averaged\nmagnetization of MI, hM\u000bi= Tr(\u001aM\u000b) with\u000b=x;y;z\nare depicted for \u000e= 0. As it is shown, h~M(t)iexhibits\nunder-damped oscillations around a positive \feld that\ndecays to zero as a function of time. In Fig. 2, we \fnd\nthat\u001a11=e\u0000\u0000t=2 and\u001a55= (1\u0000e\u0000\u0000t)=2 \ft perfectly the\nnumerical solution of \u001a11and\u001a55for all\u000es. The decay\nchannel of dark-exciton is through a transition to bright-\nexciton and spin-\rip of MI. This process is schematically\ndepicted in the inset of Fig. 3. A strong dependence of\ndark-exciton population on \u000eis seen in Fig. 2.\nConsistent with the time-evolution of the density ma-\ntrix, we propose an exciton wave-function that \fts the\ndensity matrix via \u001a(t) =j (t)ih (t)j:\nj (t)i=Cb#e\u0000\u0000t=2jXb;#i+Cb#p\n1\u0000e\u0000\u0000tj0;#i\n+Cb\"cos(Jemt=\u0016h)e\u0000\u0000t=2jXb;\"i+C0\"p\n1\u0000e\u0000\u0000tj0;\"i\n+Cd#sin(Jemt=\u0016h)jXd;#i+Cd\"jXd;\"i; (4)\nwithCb#= 1=p\n2. Note thatjXb;\"iandjXd;#icoher-\nently oscillate because Jemin o\u000b-diagonal elements of\nHxmmix these two states. Also from \u001a66(\u0000t >> 1)!\n1=2 we deducejC0\"(\u0000t >> 1)j ! 1=p\n2, and \fnally\nCd\"(t) = 0 because \u001a44= 0. The rest of coe\u000ecients\ninj (t)ican be determined numerically by \ftting to the\nsolutions of density matrix that also ful\flls the normal-\nization of the wavefunction h j i= 1.\nThe results obtained in this section illustrate that a\nbright-exciton transfers the angular-momentum of the\nCW laser \feld to the magnetization of the MI during\nthe transient time, t < tr\u001910\u0000\u00001. However, it gradu-\nally losses the magnetization to the environment via an-\nnihilation of the exciton within the exciton annihilation\ntimetr. Two vacuum states j0;\"iandj0;#iare equally\npopulated within tr, hence the \fnal MI magnetization\nis randomized and its ensemble average vanishes. Note\nthat the lack of DMP and build up of MI magnetization\nis the consequence of the ground state with random and\nuncorrelated states of MI. If the annihilation of the exci-\nton is selectively blocked for one type of spin of MI, e.g.,\nXbJem \nMI \nvacuum XdFIG. 3: Time evolution of Szof a train of injected photo-\nelectrons inside QD (circles) and a single MI (stars) with spin\nS= 1=2 and no nuclear spin. The inset shows the \u0003-shape\nthree level optical resonance of bright- and dark-exciton ( Xb\nandXd). The Rabbi oscillation between XbandXdoccurs\nbecause of the exchange interaction between the exciton and\nthe system of MI and nuclear spins. The optical selection rule\nallows decay of Xbto vacuum. However, the population of Xd\ndecreases indirectly through the conversion of XdtoXb.\nby interruption of the decay process by quantum jumps\nwithint<tr, a dramatic change in the dynamics of the\nsystem occurs due to interaction between MI and other\nexcitons in the environment and a \fnal state with non-\nvanishing MI magnetization appears. As seen in Fig. 1,\nthe time evolution of the density matrix predicts that the\nstate of MI after \frst quantum jump ( t<tr) is partially\nspin-polarized. Thus the MI with partial spin polariza-\ntion interacts with second exciton tunneling in from the\nsmall quantum dot and as a result a net spin polarization\nbuilds up . The rest of this paper is devoted to discussion\nof the DMP by quantum jumps.\nA. Quantum jump algorithm\nAs noted in Ref. 27, detection of photons from a\nsingle quantum system requires spontaneous emission\ndue to vacuum \ructuations, i.e., the photon emission\nis a stochastic process, described by quantum jump ap-\nproach26,27. The time evolution of the density matrix of a\nQD interacting with photons is given by Eq.(3). At zero\ntemperature nB= 0 and the time evolution of the den-\nsity matrix is described by a standard Lindblad master5\nequation (ME)26\nd\u001a\ndt=\u0000i\n\u0016h[HQD;\u001a]\u0000\u0000\n2X\nMz(fPy\nMzPMz;\u001ag\u00002PMz\u001aPy\nMz);\n(5)\nwheref:::gis the anti-commutator. Comparing Eq.(5)\nwith Eq.(3) we identify quantum jump operator ^PMz=\nj0;MzihXb;Mzjand its Hermitian conjugate ^Py\nMz=\njXb;Mzih0;Mzjthat project the excitonic state onto vac-\nuum and vice versa without \ripping spin of MI, henceP\nMz^Py\nMz^PMz=jXb;\"ihXb;\"j+jXb;#ihXb;#j. At each\ninstance of time, t, the density matrix can be divided\ninto series of density matrices, each representing a spe-\nci\fc quantum trajectory associated with a sequence of\nrandomly generated quantum jumps in the interval of\ntime [0;t]. Hence the QD density matrix can be calcu-\nlated by ensemble average of density matrices over all\nquantum jump trajectories.\nThe formulation of quantum jump starts from Eq.(5).\nIn the absence of MI, a recipe for quantum jump al-\ngorithm can be found in Ref. 26. For completeness of\nour presentation we \frst review this algorithm and then\ngeneralize it to exciton in the presence of MI and nu-\nclear spins. We consider optical transition in a two-\nlevel system consisting of bright-exciton and vacuum\nwithout considering an intermediate transition to dark-\nexciton. This condition is ful\flled if we disregard pres-\nence of any MI and nuclear spin. Here the quantum\njump operators are ^P=j0ihXbj,^Py=jXbih0j, hence\n^Py^P=jXbihXbj. Starting at t= 0 with the initial con-\nditionj (t= 0)i=jXbi, we calculate the time evolution\nof the system in discrete time-steps \u000et. In each time-step\nwe evaluate the quantum jump probability by calculat-\ning\u000eq0= \u0000(\u000et)h j^Py^Pj i= \u0000\u000etand drawing a random\nnumberr. Ifr<\u000eq 0a quantum jump occurs and j icol-\nlapses toj0i, otherwisej (t+\u000et)i=e\u0000\u0000\u000et(^Py^P)=2j (t)i.\nHere the generator for the time-evolution operator is a\nnon-Hermitian Hamiltonian He\u000b=\u0000i\u0016h\u0000(^Py^P)=2. So at\nt= 0 +\u000etwe havej (0 +\u000et)i=e\u0000\u0000\u000et(jXbihXbj)=2jXbi=\ne\u0000\u0000\u000et=2jXbi+p\n1\u0000e\u0000\u0000\u000etj0i. The last term keeps the\nnorm ofj iconstant (if we use the norm of wave-function\nas a constraint in our calculation). At this time \u000eq1=\n\u0000(\u000et)e\u0000\u0000\u000et. We draw rand ifr<\u000eq 0+\u000eq1thenjXbi!\nj0iand a photon is detected and calculation is termi-\nnated. Otherwise, j (\u000et+\u000et)i=e\u0000\u0000\u000et(jXbihXbj)=2j (0 +\n\u000et)i=e\u0000\u0000(2\u000et)=2jXbi+p\n1\u0000e\u0000\u0000(2\u000et)j0i. Innth-step\n\u000eqn= \u0000(\u000et)e\u0000n\u0000\u000etthus we calculate a cumulative quan-\ntum jump probability:\n\u000epn=nX\nk=0\u0000(\u000et)e\u0000k\u0000\u000et= \u0000(\u000et)1\u0000e\u0000(n+1)\u0000\u000et\n1\u0000e\u0000\u0000\u000et;(6)\nand ifr < \u000ep nquantum jump occurs. As the\ntime advances, the chance for a quantum jump be-\ncomes more likely, however, the probability amplitude\nforXbinj idecreases with the same rate simul-\ntaneously. In nth-step if there is still no quantumjump, thenj (n\u000et)i=e\u0000\u0000\u000et(jXbihXbj)=2j ([n\u00001]\u000et)i=\ne\u0000\u0000(n\u000et)=2jXbi+p\n1\u0000e\u0000\u0000(n\u000et)j0i.\nThe quantum jump algorithm in the presence of MI is\nsimilar to the one in the absence of MI, with a di\u000berence\nthat the time evolution of the wavefunction is generated\nby an e\u000bective Hamiltonian He\u000b=HQD\u0000i\u0016h\u0000\u000et(^Py^P)=2\nthat allows an intermediate transition to the dark-exciton\ndue to spin-exchange with MI. Therefore the description\nof quantum jump process in the presence of MI is based\non a three level system depicted in the inset of Fig. 3 and\nconsist ofj0i,jXbi,jXdiand MI.\nIV. DYNAMICAL EVOLUTION OF MI BY\nTRAIN OF EXCITONS IN THE PRESENCE OF\nNUCLEAR SPINS\nAs illustrated in Fig.1 a small quantum dot is con-\ntinuously re\flled by a non-resonant circularly polarized\nCW laser. The spin polarized excitons transfer into the\nQD containing the complex spin system MI. We assume\ntherefore a train of incoming bright excitons jXbi\u0011j#\n;*iinteracting with MI in the quantum dot. Each elec-\ntron in the exciton transfers spin to MI and creates a\nsuperposition of dark and bright excitons entangled with\nMI and nuclear spins. At the bright exciton recombina-\ntion time,tr, photon is detected, quantum jump takes\nplace, dark exciton wavefunction is erased and MI and\nnuclear spin complex is left in a modi\fed state. The ex-\nciton removal is performed by using the quantum jump\nprojector method26,27described below which yields the\nmodi\fed wavefunction of the MI and nuclear spins. New\nspin polarized exciton tunnels into the quantum dot and\nbegins interaction with the MI and nuclear spins modi\fed\nby electron spin of previous exciton.\nThe basis for combined exciton-spin sys-\ntem is composed of three groups of basis\nstates: vacuum j0;Mz;Iz1;:::;IzNbi, bright ex-\ncitonjXb;Mz;Iz1;:::;IzNbiand dark exciton\njXd;Mz;Iz1;:::;IzNbi. Only the vacuum and bright\nexciton group of states are coupled to the pho-\nton \feld via projectors P\u0015=j0;\u0015ihXb;\u0015j. Here\nstatesj\u0015i=jMz;Iz1;:::;IzNbidescribe a total of\nNS= (2Mz+ 1)(2Iz+ 1)Nbcomplex spin MI and\nnuclear spin states. In the following symbols j\u0015iandj\u0016i\nrepresentjMz;Iz1;:::;IzNbi.\nThe time evolution of the density matrix \u001a=j\tih\tj\nin ME, Eq.(5) can be generalized by Mz!\u0015. As de-\nscribed in section III A, the wave-function j\tisubjected\nto stochastic \\birth-death\" process29of recombination\nand photo-excitation can be used to describe the time\npropagation of the system coupled with radiation-\feld\nand undergoing quantum jump process. At t= 0 we start\nwith the initial state j\tn=0(t= 0)i=j0ij\u001f0iof MI and\nthe nuclear spin-bath. The index ncounts the number of\nquantum jump events. The state j\u001f0i=P\n\u0015Cn=0\n\u0015j\u0015i\nis a random linear combination of all possible con\fg-\nurations with the coe\u000ecients C(0)\n\u0015being uniformly dis-6\ntributed random complex numbers. We note that if we\nwere to compute expectation value hMzifor this random\nstate we would obtain a \fnite value. However, averaging\nover many sets of C(0)\n\u0015yields no initial magnetization.\nAtt= 0+a bright exciton created in neighboring QD\nenters the central QD. The creation of jXbiand annihi-\nlation ofj0iare described by operator jXbih0j, hence the\nwave-function of the system with one exciton is given by\nj\tn=1(t= 0+)i=jXbih0j\tn=0(t= 0)i=jXbij\u001f0i\n=jXbiX\n\u0015Cn=0\n\u0015(t= 0)j\u0015i\n=X\n\u0015Cn=1\n\u0015(t= 0+)jXb;\u0015i (7)\nwhereCn=1\n\u0015(t= 0+) =Cn=0\n\u0015(t= 0). The initial wave-\nfunction of the injected bright exciton, MI and nuclear-\nspins is an uncorrelated state. However, the Hamiltonian\nHQDthat accounts for the exchange coupling, creates\nquantum correlation in the exciton-MI-nuclei complex\nandj\tievolves into a linear combination of all con\fgu-\nrations, including an entangled state between bright and\ndark excitons. As a function of time, the bright-exciton\ndecays into vacuum because of coupling with quantized\nelectromagnetic-\feld.\nTo be consistent with the quantum jump algorithm\nwe discretize the time tinto small steps \u000et. Note that\nbecause of the small eh- and MI-nuclear-spin couplings\n(Jne,JnhandJnh) the excitons and MI evolve in a frozen-\n\ructuating \feld of nuclear-spins32{34. The eh recombi-\nnation time, tris the smallest time-scale in our model,\nhence\u000et<<tr.\nThe time evolution of the wave-function of eh-MI-\nnuclei complex is calculated in the Schr odinger picture33\nby using the relation j\t(t+\u000et)i= exp(\u0000iHe\u000b\u000et=\u0016h)j\t(t)i.\nHere,j\t(t)iis the wave-function of the entire system,\nHe\u000b=HQD\u0000i\u0016h(\u0000=2)P\n\u0015Py\n\u0015P\u0015where the last term de-\nscribes the decay of bright exciton due to coupling with\nphoton-\feld, hence\nj\tn=1(\u000et)i= exp(\u0000iHe\u000b\u000et=\u0016h)j\tn=1(t= 0+)i:(8)\nIn Eq. 8 we use exp( \u0000iHe\u000b\u000et=\u0016h)\u0019\nexp(\u0000iHQD\u000et=\u0016h) exp(\u0000\u0000\u000et=2P\n\u0015Py\n\u0015P\u0015) +O((\u000et)2)\nwithPy\n\u0015P\u0015 =jXb;\u0015ihXb;\u0015jand the follow-\ning identities exp( \u0000\u0000\u000etP\n\u0015Py\n\u0015P\u0015)jXb;\u0016i =Q\n\u0015exp(\u0000\u0000\u000etPy\n\u0015P\u0015)jXb;\u0016i= exp(\u0000\u0000\u000et)jXb;\u0016i, as\nexp(\u0000\u0000\u000etPy\n\u0015P\u0015)jXb;\u0015i= exp(\u0000\u0000\u000et)jXb;\u0015i, and\nexp(\u0000\u0000\u000etPy\n\u0015P\u0015)jXb;\u0016i=jXb;\u0016iwhere\u00166=\u0015,\nas well as exp( \u0000\u0000\u000etPy\n\u0015P\u0015)jXd;\u0015i=jXd;\u0015i, and\nexp(\u0000\u0000\u000etPy\n\u0015P\u0015)j0;\u0015i=j0;\u0015i.\nBecauseHe\u000bis time-independent, we employ the\nmethod based on Bessel-Chebyshev polynomial expan-\nsion33to calculate the time evolution of the wave-\nfunction\nj\tn=1(\u000et)i\u0019e\u0000\u0000\n2\u000etP\n\u0015Py\n\u0015P\u0015e\u0000i\n\u0016hHQD\u000etj\tn=1(t= 0+)i=e\u0000\u0000\n2\u000etP\n\u0015Py\n\u0015P\u0015X\n\u0016[~Cn=1\nXb;\u0016(\u000et)jXb;\u0016i+\n+~Cn=1\nXd;\u0016(\u000et)jXd;\u0016i]: (9)\nNote thatHQDis Hermitian and thus exp( \u0000iHQD\u000et=\u0016h)\nis a unitary operator that conserves the norm of wave-\nfunction, hencej~Cn=1\nXb;\u0016(\u000et)j2+j~Cn=1\nXd;\u0016(\u000et)j2= 1. This\nis in contrast with the operator exp( \u0000\u0000\u000et=2P\n\u0015Py\n\u0015P\u0015)\nthat is non-unitary and does not preserve norm of\nwave-function, however, because it describes the de-\ncay ofXbinto vacuum, we build a norm-conserving\nwave-function by adding vacuum. After applying\nexp(\u0000\u0000\u000et=2P\n\u0015Py\n\u0015P\u0015) in Eq. 9 we \fnd\nj\tn=1(\u000et)i=X\n\u0015j\u0015i[exp(\u0000\u0000\u000et=2)Cn=1\nXb;\u0015(\u000et)jXbi\n+p\n1\u0000exp(\u0000\u0000\u000et)Cn=1\n0;\u0015(\u000et)j0i\n+Cn=1\nXd;\u0015(\u000et)jXdi]: (10)\nIn the limit of \u0000 = 0 the coe\u000ecients with and with-\nout tilde used in Eqs. 9-10 are identical. Matching the\ninitial conditions between Eq.(7) and Eq.(10) implies\nCn=1\nXb;\u0015(\u000et= 0) =Cn=1\n\u0015(t= 0+) andCn=1\nXd;\u0015(\u000et= 0) = 0.\nUsing an iterative procedure to propagate the wavefunc-\ntion in time we \fnd\nj\tn=1(t)i=X\n\u0015j\u0015i[Cn=1\nXb;\u0015(t)jXbi+Cn=1\n0;\u0015(t)j0i\n+Cn=1\nXd;\u0015(t)jXdi]; (11)\nwhere the coe\u000ecients Cn=1\nXb;\u0015(t),Cn=1\n0;\u0015(t), andCn=1\nXd;\u0015(t) are\ndetermined numerically. This wave-function describes a\ncorrelated state of bright and dark excitons as well as\nvacuum. Because of the spin \rip-\rop process of electron\nwith MI and nuclear-spins the initially formed bright ex-\ncitonjXbimixes with the dark-exciton jXdi.\nFrom the Lindblad ME, the quantum jump transi-\ntion rate is given by \u0000 jump = \u0000\u001ab.\u001abrepresents the\npopulation of the bright exciton obtained from the full\nQD density matrix after tracing over MI and nuclear\nspin degrees of freedom. The quantum jump probability\n\u000epjump = \u0000 jump\u000etis then calculated and compared with\na random number rgenerated between zero and one. If\nr <Rtr\n0dt\u0000jump a quantum jump takes place, photon is\nrecorded and the quantum dot is in the ground state.\nThe elapsed time trrecorded for this quantum jump is\nthe eh-recombination time. At t=trwe allow exciton\nto annihilate by spontaneous emission of a photon. The\noperator that allows annihilation of bright exciton and\ncreation of vacuum is j0ihXbj. Hence\nj\tn=2(t=tr)i=j0ihXbj\tn=1(t=tr)i\n=j0iX\n\u0015Cn=1\nXb;\u0015(t=tr)j\u0015i\n=X\n\u0015Cn=2\n\u0015(t=tr)j0;\u0015i(12)7\nwhereCn=2\n\u0015(t=tr) =Cn=1\nXb;\u0015(t=tr).\nImmediately after annihilation of exciton, a new spin\npolarized exciton tunnels into the quantum dot from the\nneighboring dot. The spin polarized exciton interacts\nwith the MI spin Mand nuclear spins Iin a state mod-\ni\fed by the previous exciton. At t=tr+ 0+, second\nbright exciton Xbcreated in the neighboring QD tunnels\ninto the central QD\nj\tn=3(t=tr+ 0+)i=jXbih0j\tn=2(t=tr)i\n=jXbiX\n\u0015Cn=2\n\u0015(t=tr)j\u0015i\n=X\n\u0015Cn=3\n\u0015(t=tr+ 0+)jXb;\u0015i (13)\nwith matching the initial conditions that requires\nCn=3\n\u0015(t=tr+ 0+) =Cn=2\n\u0015(t=tr). Note that this\nstate is not correlated. The quantum correlation appears\nfrom the time evolution of wave-function generated by\nexchange couplings in He\u000bright aftert=tr\nj\tn=3(t)i= exp(\u0000iHe\u000bt=\u0016h)j\tn=3(t=tr+ 0+)i\n=X\n\u0015j\u0015i[Cn=3\nXb;\u0015(t)jXbi+Cn=3\n0;\u0015(t)j0i\n+Cn=3\nXd;\u0015(t)jXdi]; (14)\nwhereCn=3\nXb;\u0015(tr+ 0+) =Cn=3\n\u0015(tr+ 0+) andCn=3\nXd;\u0015(tr+\n0+) = 0 are matching conditions. As we see there is no\ntype of linear combination between j0iandfjXbi;jXdig\nbecause there is no Rabi-oscillations between vacuum and\nexcitons.\nTo summarize the above procedure and make connec-\ntion between tunneling of exciton and photo-emission we\nformally introduce a projector Qn!n+1=jXn+1\nbihXn\nbj\ninnth step of quantum jump. The superscripts re-\nfer to annihilated nth and created n+ 1th exciton.\nNote thatQn!n+1jXn\nbi=jXn+1\nbiandQn!n+1jXdi=\nQn!n+1j0i= 0, hence the quantum jump operator\nprojects out any correlated state composed of superpo-\nsition of bright and dark exciton to a new born bright\nexciton. The new wavefunction in the QD then can be\nconstructed asj\t(t=t+\nr)i=Qn!n+1j\t(t=t\u0000\nr)iwhere\nt\u0006\nr=tr\u0006\u0011and\u0011!0. In this state,jXn+1\nbiis initially\nuncorrelated from MI and nuclear-spins. At t=t+\nrit can\nbe expressed as\nj\tn+1i=jXn+1\nbiX\n\u0015ANCn\nXb;\u0015(tr)j\u0015i: (15)\nWe observe that after quantum jump the new injected\nexcitonXn+1\nbstarts with the normalized (factor AN)\nstate of MI and nuclear spinsP\n\u0015Cn\nXb;\u0015(tr)j\u0015iwhich was\nleft over by the previous bright exciton Xn\nbat the time\nof radiative recombination. Detecting a photon erased\nthe dark exciton wave-function and modi\fed the state of\nboth MI and nuclear spins. This is the DMP mechanism\ndiscussed here. With initial condition established, the\ntime evolution of the entangled state of photo-carriers\n0 5 10 15\nΓt00.10.20.30.40.5<Sz>eh\nMI\nnucleiFIG. 4: Time evolution of Szof a train of injected photo-\nelectrons inside QD (circles), a single MI (stars) and the av-\nerage ofNb= 15 nuclear-spin polarization (triangles).\nwith MI and spin-bath then can be calculated after up-\ndating the coe\u000ecients C's. At the end one needs to aver-\nage over initial conditions. Although the procedure dis-\ncussed here describes the immediate re\flling of central\nQD after annihilation of the exciton, we can always im-\nplement a waiting time between the recombination and\nre\flling process.\nV. NUMERICAL RESULTS AND DISCUSSION\nOur approach to DMP is illustrated using parameters\nbased on (Cd,Mn)Te QDs with ~Jem= 15 meV nm3,\n~Jhm= 60 meV nm3corresponding to the exchange cou-\npling in the bulk materials, hence Jem=~Jemj\u001ee(Rm)j2\nandJhm=~Jhmj\u001eh(Rm)j2. The circular symmetry of\nquantum dots is implemented by assuming \u0001 1= 0. Here\n\u001ee=h(~Rm) is the e/h envelope-wavefunction in the cen-\ntral dot at ~Rm, the position of MI. We assume Jeh= 0:6\nmeV30and initialize Jne,Jnh,JnmandJnn0as random\nnumbers with a mean value of the order of 1 \u0016eV. How-\never we note that the realistic value for nuclear hyper-\n\fne interaction is reported within 1 neV35three orders\nof magnitude smaller than the energy scales used in our\n\fnite size calculation.\nHere we discuss numerical results with nuclear spins,\nimmediately after re\flling of QD by bright exciton.\nFigs. 4-7 illustrate DMP/DNP and the quantum jump\ntrajectories for exciton, MI and nuclear-spins. In Fig. 4\na single quantum jump trajectory for spin-1/2 MI is plot-\nted. In Figs. 5-7 the ensemble average of twenty quantum\njump trajectories for spin-1/2 (Fig. 5), spin-1 (Fig. 6)8\n0 5 10 15\nΓt00.10.20.30.40.5<Sz>eh\nMI\nnuclei\nFIG. 5: Time evolution of ensemble average of 20 trajectories\nforSzof a train of injected photo-electrons inside QD (circles),\na single MI (stars) and the average of Nb= 15 nuclear-spin\npolarization (triangles).\nand spin-5/2 (Fig. 7) MI are plotted. The trajectories are\ntime-evolution of the initial spin wavefunctions which are\ngenerated in random linear combination of spin con\fgu-\nrations. Each curve consists of thousands of time-steps\nand points. For clarity of the legends, after every hundred\npoints symbols like circle, star and triangle are superim-\nposed on each curve. As shown the MI and the average\npolarization of Nb= 15 nuclear spins gradually builds-up\nby a train of injected bright excitons. At t=tr, one pair\nof eh collapses into vacuum with hSe;z(t)i<1=2 as part\nof the e-spin is transferred to MI. An empty dot instanta-\nneously absorbs the second photo-generated eh pair with\ntotal angular momentum jz=\u00001 which transfers to\nspin of MI and nuclei before its removal. We repeat this\nprocedure until the spin polarization of MI and nuclear-\nspins is built-up. The method presented here is limited\nto a \fnite number of nuclear spins because of exponen-\ntially increasing computational e\u000bort with the number of\nspins. However, a systematic study of the convergence of\nthe numerical results by increasing Nbshows satisfactory\noutcomes around Nb= 15.\nWe now discuss DMP for MIs with more than one local-\nized electron. We consider two cases of MI with two and\n\fve electrons localized in open-shell p/d-orbitals. The\nspin Hund's rule implies that the total spin of the elec-\ntronic ground state of MIs is maximum. For two electrons\nthe spin triplet manifold ( M= 1) is separated from the\nhigher energy spin-singlet state ( M= 0) with the singlet-\ntriplet energy gap EM=0\u0000EM=1=jJmj. HereJmis\nthe ferromagnetic exchange coupling between two elec-\ntrons localized in MI. Similarly the lowest energy state\nof MI (e.g., Mn) with \fve d-electrons corresponds to\n0 5 10 15\nΓt-1-0.500.51<Mz>\nMz = -1\nMz = 0\nMz = +1\naverage(a)\n0 5 10 15\nΓt00.10.20.30.40.50.6<Sz>\neh\nMI\nnuclei(b)FIG. 6: (a) Time evolution of magnetic moment of ensemble\nof MIs with two localized p/d-electrons in ferromagnetically\nordered spin-triplet M= 1 andMz=\u00061;0 interacting with\na train of injected photo-electrons inside QD, and Nb= 15\nnuclear-spins. Unlike Mz=\u00061 that interact strongly with\nexcitons,Mz= 0 exhibits weak interaction. The average of\nthree states show saturation close to hMzi=1\n3(2\u00021 + 0) =\n0:67 as the ensemble is populated equally among all possible\nspin-triplet states. (b) Time evolution of ensemble averaged\nof spin of MI (stars), photo-electrons (circles), and Nb= 15\nnuclear-spins (triangles).\ntotal spinM= 5=2 with six-fold degeneracy. These\nstates are separated from higher energy spin-manifold\nwith energy gap proportional to Jm. Considering Jmfew\ntimes larger than other exchange couplings avoids mixing\nground state with the higher energy excited states of MI.\nFor a system containing spin-1 MI, we consider an en-\nsemble with equally populated states Mz=\u00061;0 (1/3\nfor eachMz). Similarly the ensemble of Mn's contains\nMz=\u00065=2;\u00063=2;\u00061=2 with equal population (1/6 for9\n0 10 20 30 40 50 60\nΓt-2.5-2-1.5-1-0.500.511.522.5<Mz>Mz = -5/2\nMz = -3/2\nMz = -1/2\nMz = +1/2\nMz = +3/2\nMz = +5/2(a)\n0 10 20 30 40 50\nΓt00.511.5<Sz>eh\nMn\nnuclei(b)\nFIG. 7: Time evolution of magnetic moment of individual\nstates of Mn (a) and an ensemble of Mn's (b) with \fve lo-\ncalized d-electrons (stars) interacting with a train of injected\nphoto-electrons inside QD (circles), and Nb= 15 nuclear-\nspins (triangles). The ensemble is constructed with equally\npopulatedMz=\u00065=2;\u00063=2;\u00061=2 among \fnite number of\nMn's. The average of six states show saturation close to\nhMzi=1\n6(2\u00025\n2+ 2\u00023\n2+ 2\u00021\n2) =9\n6= 1:5 as the ensemble\nis populated equally among all possible spin-triplet states.\neachMz). Thus ensemble average over all possible QJ\ntrajectories includes summation over all Sz. In Fig. 6(a),\nwe show the time evolution of spin-triplet states initially\nstarted from Mz=\u00061;0. We observe that MI with Mz=\n\u00001 switches polarization to hMzi\u0019+1 because of strong\ninteraction with excitons that allows DMP mechanism to\nproceed e\u000eciently. It is therefore expected that the po-\nlarization of Mz(t= 0) = +1 does not alter dramatically,\nalthough it is initially decohered by nuclear spins but we\n\fnd that it stays polarized with hMzi\u0019+1 because of\nstrong interaction with train of excitons. On the otherhand, MI with initial polarization Mz(t= 0) = 0 \ructu-\nates around Mz= 0. Analogue to its spin-singlet counter\npart, the spin-triplet Mz= 0 weakly interacts with exci-\ntons and nuclear spins36. Hence the the polarization ob-\ntained for this ensemble indicates that the \fnal state is a\nmixture of all spin-triplet con\fgurations with maximum\nachievable polarization hMzi=1\n3(2\u00021 + 0) =2\n3= 0:67\nasMz=\u00061 equally contribute to ensemble average of\nhMzi. Similarly we can predict the maximum spin polar-\nization achievable for ensemble of Mn can be calculated\nbyhMzi=1\n6(2\u00025\n2+ 2\u00023\n2+ 2\u00021\n2) =9\n6= 1:5 as shown\nin Fig. 7. It is straightforward to show that mixing with\nhigher energy excited states suppress the magnetic satu-\nration down tohMzi= 1:1 if the \fnal state is a mixture\nof equally populated all spin multiplicities of \fve spin-\n1/2 electrons. The results shown in Fig. 7 suggest that\nthe saturation ofhMzioccurred between 1.1 and 1.5 that\nmight be interpreted as an indication of leakage of the op-\ntically pumped ground state of Mn to its excited states.\nVI. SUMMARY\nIn conclusion, dynamical magnetic and/or nuclear po-\nlarization in single quantum complex spin systems is dis-\ncussed for the case of spin transfer from exciton to the\ncentral spin of magnetic impurity in a quantum dot in the\npresence of a \fnite number of nuclear spins. The exciton\nis described in terms of the electron and heavy hole spins\ninteracting with magnetic impurity via exchange interac-\ntion, with a \fnite number of nuclear spins via hyper\fne\ninteraction and with photons via dipole interaction. The\ntime-evolution of the exciton, magnetic impurity and nu-\nclear spins is calculated exactly between quantum jumps\ncorresponding to exciton radiative recombination. 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B 79, 245314 (2009).\n36Under particular conditions, beyond the present model,\nstrong coupling between spin singlet states and MI leads to\nformation of spin texture and molecular states of magnetic11\npolarons. Such possibilities were investigated recently in:\nR. M. Abolfath, M. Korkusinski, T. Brabec, P. Hawrylak,Phys. Rev. Lett. (in press)."    },    {        "title": "1502.01343v1.Effective_one_body_Hamiltonian_with_next_to_leading_order_spin_spin_coupling.pdf",        "content": "E\u000bective-one-body Hamiltonian with next-to-leading order spin-spin coupling\nSimone Balmelli\u0003and Philippe Jetzer\nPhysik-Institut, Universit at Z urich, Winterthurerstrasse 190, 8057 Z urich, Switzerland\nWe propose a way of including the next-to-leading (NLO) order spin-spin coupling into an e\u000bective-\none-body (EOB) Hamiltonian. This work extends [S. Balmelli and P. Jetzer, Phys. Rev. D 87,\n124036 (2013)], which is restricted to the case of equatorial orbits and aligned spins, to general\norbits with arbitrary spin orientations. This is done applying appropriate canonical phase-space\ntransformations to the NLO spin-spin Hamiltonian in Arnowitt-Deser-Misner (ADM) coordinates,\nand systematically adding \\e\u000bective\" quantities at NLO to all spin-squared terms appearing in the\nEOB Hamiltonian. As required by consistency, the introduced quantities reduce to zero in the test-\nmass limit. We expose the result both in a general gauge and in a gauge-\fxed form. The last is\nchosen such as to minimize the number of new coe\u000ecients that have to be inserted into the e\u000bective\nspin squared. As a result, the 25 parameters that describe the ADM NLO spin-spin dynamics get\ncondensed into only 12 EOB terms.\nPACS numbers: 04.25.-g, 04.25.dg\nI. INTRODUCTION\nThanks to the LIGO/Virgo network of second-\ngeneration ground based interferometers, a \frst direct\ndetection of gravitational waves (GW) is expected to oc-\ncur in few years [1]. Furthermore, in the next decades,\nspace-born GW detectors [2] (such as the planned eLISA)\nwill open an entire new window to astrophysics, and allow\ntests of General Relativity at an unprecedented level [3].\nBoth types of detectors rely on coalescing (and possible\nspinning) black hole binaries as a primary GW source.\nIn order to extract the GW signal from the noise, very\nprecise waveform templates need to be constructed. Cur-\nrently, the most accurate description of coalescing black\nholes is provided by numerical relativity (NR) (see e.g.\n[4{7] for some recent advances). However, the full param-\neter space is too large (especially in the case of nonzero\nspins) for being densely covered by NR simulations .\nThis is the main reason why semi-analytical methods,\nby far less computationally expensive, can turn out to be\nvery useful. At the present time, the e\u000bective-one-body\n(EOB) approach (we refer to [8] for a general review) is\nthe only semi-analytical method that has been able to ac-\ncurately describe the complete waveform of a coalescing\nprocess. After years of constant development [9{24] the\nEOB has now reached an excellent agreement with NR\nwaveforms in the case of nonspinning binaries [7, 25]. A\nlot of e\u000bort has also been put into the modeling of spins\n[26{40], leading to a good overlap with NR waveforms in\nthe case of nonprecessing (aligned or anti-aligned) spins\n[7, 38, 40]. By contrast, the description of precessing\nspins still needs some work before reaching a comparable\nperformance [39], and is currently one of the most urgent\ntasks. In view of this, it may be crucial to incorporate\nmore analytical information from the spin-orbit and spin-\nspin Hamiltonian computed within the post-newtonian\n(PN) theory.\n\u0003balmelli@physik.uzh.chIn Ref. [36], a possible way of including the next-to-\nleading (NLO) spin-spin coupling [41, 42] (see also [43{\n48]) into the spinning EOB model of Refs. [26, 27, 34] has\nbeen exposed for the special case of equatorial orbits and\nnonprecessing spins. A general inclusion of NLO spin-\nspin e\u000bects for fully precessing orbits would be a neces-\nsary step for extending the reliabilty of EOB waveforms\nto a signi\fcantly larger portion of the parameter space.\nThe present paper aims at \flling this gap, providing a\npossible implementation of the missing terms.\nRecently, an improved (and calibrated) EOB model\nfor spinning binaries has been proposed [40], where the\nNLO spin-spin coupling is only incorporated for the case\nof circular orbits. The present paper could be a \frst step\nfor developing, at a next stage, a more general version of\nthat improved model.\nThe paper has the following structure: in Sec. (II)\nwe summarize the main concepts and the formalism of\nRef. [36]. In Sec. (III), which is the central part of the\npaper, we propose a way of including the NLO spin-spin\nterms and show the explicit result, both in a general form\n(with a given number of free gauge parameters) and in\na gauge-\fxed formulation. Finally, Sec. (IV) discusses a\nsecond, slightly di\u000berent way of incorporating the wished\nspin-spin terms. Both approaches are compared plotting\nthe (gauge invariant) angular frequency and binding en-\nergy at the last stable orbit (LSO). The plot also shows\nthe prediction of the calibrated models of Refs. [35, 38].\nThroughout the paper, we use geometric units with\nG\u0011c\u00111. When writing formulae in a PN ex-\npanded form, however, we will reintroduce the usage of\nc, prepending a factor (1 =c)2nwith the mere purpose of\nlabeling the PN order n.\nII. SUMMARY OF THE PREVIOUS WORK\nIn this section, we outline the method followed in\nRef. [36], which forms the basis of the current paper. We\nwork in EOB coordinates, with Rbeing the radial coor-arXiv:1502.01343v1  [gr-qc]  4 Feb 20152\ndinate,nthe unit radial vector and Pthe momentum\nvector. We will often use the rescaled variables r\u0011R=M\nandp\u0011P=\u0016. Here,M\u0011m1+m1is the central EOB\nmass (m1andm2being the individual masses of the two\nblack holes), and \u0016\u0011m1m2=Mis the reduced mass.\nMoreover, we denote by \u0017\u0011\u0016=M the symmetric mass\nratio.\nThe starting point is the EOB model of Ref. [26] (which\nincludes both spin-spin and spin-orbit coupling at lead-\ning order (LO)), together with its extensions to the NLO\n[27] and to the next-to-next-to leading order (NNLO) [34]\nspin-orbit coupling. As already mentioned, in Ref. [36]\nthe NLO spin-spin Hamiltonian in ADM coordinates has\nbeen reformulated and inserted into this EOB model for\nthe special case of two black holes whose spins are aligned\n(or anti-aligned) with the orbital angular momentum. In\nparticular, it has been shown that it is su\u000ecient to re-\nplace the spin parameter a0of the e\u000bective metric (see\nfor instance Eqs. (4.7) and (4.8) of Ref. [36]), when-\never it appears as a second power, by a new, e\u000bective\nsquared spin parameter. Using the dimensionless nota-\ntion\u001f0=a0=M, this prescription takes the form\n\u001f2\n0!(\u001f2)e\u000b=\u001f2\n0+ \u0001\u001f2\ne\u000b: (2.1)\nWe recall that \u001f0is a combination of the dimensionless\nspin parameters \u001f1and\u001f2(\u001fa=Sa=m2\na) of the two\nbodies:\n\u001f0=m1\nM\u001f1+m2\nM\u001f2: (2.2)\nThe additional term \u0001 \u001f2\ne\u000bis of fractional 1PN order with\nrespect to\u001f2\n0and carries the information for reproducing\nthe correct NLO spin-spin coupling. It reads as\n\u0001\u001f2\ne\u000b=1\nc2\u0014\u0010\na11p2+c11\nr\u0011\n\u001f2\n1\n+\u0010\na22p2+c22\nr\u0011\n\u001f2\n2\n+\u0010\na12p2+c12\nr\u0011\n\u001f1\u001f2\u0015\n: (2.3)\nThe calculation of the coe\u000ecients aabandcabis the\nmain result of Ref. [36], given by Eq. (5.12) there. All\nof them vanish in the test mass limit \u0017!0, consistently\nwith the requirement that the EOB metric must reduce\nto the Kerr one.\nPN results in ADM coordinates can be included into\nan EOB model after suitable canonical transformations.\nWe denote here by GPN\noandGPN\nssthe generating func-\ntions of the corresponding purely orbital and spin-spin\ntransformation, respectively. The procedure for trans-\nforming the NLO spin-spin Hamiltonian from ADM into\nEOB coordinates is made of three steps:1\n1Here, we treat the PN expansion under the assumption of rapidly1) A purely orbital transformation\nHNLO0\nss =HNLO(ADM)\nss +\b\nG1PN\no;HLO(ADM)\nss\t\n:(2.4)\n2) A LO spin-spin transformation\nHNLO00\nss =HNLO0\nss +\b\nGLO\nss;H1PN0\no\t\n; (2.5)\nwhere\nH1PN0\no =H1PN(ADM)\no +\b\nG1PN\no;HN(ADM)\no\t\n: (2.6)\n3) A NLO spin-spin transformation\nHNLO000\nss =HNLO00\nss +\b\nGNLO\nss;HN(ADM)\no\t\n: (2.7)\nNotice that, for consistency, the transformations must\nbe performed in a well-de\fned order. As indicated above,\nwe make \frst use of the orbital transformation Go, and\nthen of the spin-spin transformation Gss2. Notice that,\nin Ref. [34], the spin-orbit generating function Gsois also\napplied after Go. By contrast, an evaluation order pre-\nscription between spin-orbit and spin-spin transforma-\ntion would \frst be necessary when taking into account\ncontributions that are cubic in the spins.\nThe \fnal Hamiltonian must be equal to the corre-\nsponding term arising from the PN expansion of the EOB\nHamiltonian. Since, for the moment, this is only true in\nthe spin-aligned case, we are just allowed to write:\nHNLO000\nss,al =HNLO(EOB)\nss,al: (2.8)\nThe NLO spin-spin transformation ^GNLO\nss,al\u0011GNLO\nss,al=\u0016\nis given by\n^GNLO\nss,al=(n\u0001p)\nc6r\u001a\u0014\n\u000b11p2+\u0012\n\r11\u00001\n2\u00131\nr\u0015\n\u001f2\n1\n+\u0014\n\u000b22p2+\u0012\n\r22\u00001\n2\u00131\nr\u0015\n\u001f2\n2(2.9)\n+\u0014\n\u000b12p2+\r121\nr\u0015\n(\u001f1\u0001\u001f2)\u001b\n:\nrotating black holes ( Sa=m2\na\nc\u001fa, withj\u001faj.1), which as-\nsigns a well-de\fned PN order to the spin-dependent terms. As\na consequence, throughout this paper, GssandHssare of 2PN\norder when labeled with \\LO\", of 3PN order when labeled with\n\\NLO\", and so on.\n2Since the set of canonical transformations carries a group struc-\nture, the successive evaluation of Go(r;p0) and ofGss(r0;p00)\nis a canonical transformation itself (with generating function\nGss,o(r;p00) =Go(r;p0) +Gss(r0;p00)\u0000r0\u0001p0). Despite tak-\ning a unique generating function would avoid the necessity of\n\fxing an evaluation order prescription, we prefer here to use two\nseparated transformations, so as to mantain the continuity with\nrespect to Ref. [36]. As a second reason, the transformation of\nthe Hamiltonian is more easily calculated here than for a sin-\ngle generating function, since in that last case e\u000bects quadratic\nin the transformation must be considered (see e.g. Eq. (6.9) of\nRef. [9]).3\nNotice that, in view of the generalization to precessing\norbits, we have written the individual dimensionless spins\n\u001fa=Sa=m2\naas vectors. The coe\u000ecients \u000baband\rabcan\nbe found in Eq. (5.13) of Ref. [36]. It is also provided\nan expression for ^GNLO\nssin the test mass limit (assuming\nm1>m 2):\nlim\n\u0017!0^GNLO\nss=1\nc6r2\u0014\n\u00001\n2\u0000\n\u001f2\n1+ (n\u0001\u001f1)2\u0001\n(n\u0001p)\n+ (p\u0001\u001f1)(n\u0001\u001f1)\u0015\n:(2.10)\nThe purpose of this paper is that of generalizing the\nprescription (2.1) to the case of general orbits, i.e., when\nthe scalar products ( n\u0001\u001fa) and (p\u0001\u001fa) (a= 1;2) can-\nnot be set to zero.\nIII. INCLUDING NLO SPIN-SPIN TERMS\nINTO THE EOB FOR GENERAL ORBITS\nA. The prescription\nIn Ref [36], the EOB metric is written in Boyer-\nLindquist-like coordinates. When spin precessions must\nbe taken into account, however, it is necessary to switch\nto another system of coordinates. Following Ref. [26]\n(and reformulating the angular variable \u0012according to\na0cos(\u0012)\u0011(n\u0001a0)), we can write the e\u000bective metric inCartesian-like coordinates,\ng00\ne\u000b=1\n\u001a2 \na2\n0\u0000(n\u0001a0)2\u0000\u0000\nR2+a2\n0\u00012\n\u0001t!\n(3.1)\ng0i\ne\u000b=R\n\u001a2\u0012\n1\u0000R2+a2\n0\n\u0001t\u0013\n(a0\u0002n)i(3.2)\ngij\ne\u000b=1\n\u001a2\u0012\n\u0001Rninj+R2\u0000\n\u000eij\u0000ninj\u0001\n(3.3)\n\u0000R2(a0\u0002n)i(a0\u0002n)j\n\u0001t\u0013\n; (3.4)\nwhere\u001a2=R2+ (n\u0001a0)2. \u0001tand \u0001Rcan be found\ne.g. in Eq. (4.9) of Ref. [36] (notice that they both depend\non the spin through a term \u0018a2\n0). The e\u000bective Hamil-\ntonian (that we denote here as \\old\", in order to avoid\nconfusion with the modi\fed version that is presented in\nthis paper) takes the form\nHold\ne\u000b= \u0001Hso+\fiPi+\u000bq\n\u00162+\rijPiPj+Q4(Pi);(3.5)\nwith a quartic-in-momenta term Q4(Pi) [11, 26] and with\n\u000b=1p\n\u0000g00\ne\u000b(3.6a)\n\fi=g0i\ne\u000b\ng00\ne\u000b(3.6b)\n\rij=gij\ne\u000b+\fi\fj\n\u000b2: (3.6c)\n\u0001Hsohas been introduced to describe higher-order spin-\norbit couplings. It is de\fned in terms of the gyro-\ngravitomagnetic ratios ge\u000b\nSandge\u000b\nS\u0003, see Eqs. (4.15) and\n(4.16) of Ref. [27].\nWith the reduced quantities \u001f0=a0=M= (m1\u001f1+\nm2\u001f2)=M,^H=H=\u0016,^\u0001r= \u0001R=Mand ^\u0001t= \u0001t=M, we\ncan write, more explicitly:\n\u0001^Hso+\fipi=r\u0017\n2~r4(r2+\u001f2\n0\u0000^\u0001t)\u0012\u0012m1\nm2ge\u000b\nS+ge\u000b\nS\u0003\u0013\n(n\u0002p)\u0001\u001f1+\u0012m2\nm1ge\u000b\nS+ge\u000b\nS\u0003\u0013\n(n\u0002p)\u0001\u001f2\u0013\n(3.7a)\n\u000b= ^\u0001t\u0000\nr2+ (n\u0001\u001f0)2\u0001\n~r4!1=2\n(3.7b)\n\rijpipj=r2\nr2+ (n\u0001\u001f0)2\"\np2+ ^\u0001r\nr2\u00001!\n(n\u0001p)2\u00001\n~r4\u0010\n2r2\u0000^\u0001t+\u001f2\n0+ (n\u0001\u001f0)2\u0011\n((n\u0002p)\u0001\u001f0)2#\n;(3.7c)\nwhere\n~r4=\u0000\nr2+\u001f2\n0\u00012\u0000^\u0001t\u0000\n\u001f2\n0\u0000(n\u0001\u001f0)2\u0001\n: (3.8)The spin-squared term (( n\u0002p)\u0001\u001f0)2, generated by the\ncontraction of \rijpipj, can be expressed through the sim-\nple scalarsp2, (n\u0001p),\u001f2\n0, (n\u0001\u001f0) and (p\u0001\u001f0) according4\nto\n((n\u0002p)\u001f0)2=\u0000\np2\u0000(n\u0001p)2\u0001\u0000\n\u001f2\n0\u0000(n\u0001\u001f0)2\u0001\n\u0000((p\u0001\u001f0)\u0000(n\u0001p)(n\u0001\u001f0))2:(3.9)\nLet us now do some considerations:\ni) The prescription (2.1) acts selectively - leaving all\nnon-squared spins untouched - and cannot be truly\nconsidered as a rede\fnition of the e\u000bective spin of\nthe EOB metric. It is, rather, a direct modi\fcation\nof the e\u000bective Hamiltonian.\nOne can say the same for the inclusions of spin-orbit\nterms done in Refs. [26, 27, 34]. Adding the spin-\norbit coupling requires a modi\fcation of all terms\nin the metric that are linear in the spin - or, equiv-\nalently, a direct modi\fcation of the Hamiltonian\nthrough an additional quantity \u0001 Hso(Eq. (4.16)\nof Ref. [27]).ii) Changing the Hamiltonian itself rather than the\nmetric is of course not unreasonable. Since the mo-\ntion of a spinning particle is non-geodesic, there is\nno reason to believe that the dynamics of spinning\nbodies can be accurately described by geodesics in\nan e\u000bective metric. One should in principle not\nworry about intervening on the structure of the ef-\nfective Hamiltonian itself.\nIn view of these remarks, and noting, in addition, that\nthe e\u000bective Hamiltonian depends on the spin squared\nonly through the scalars \u001f2\n0, (n\u0001\u001f0)2, (p\u0001\u001f0)2and\n(n\u0001\u001f0)(p\u0001\u001f0), we see a natural way to generalize (2.1).\nWe treat the spin di\u000berently whether it is contracted with\nitself,norp, and propose the following type of replace-\nments in Eq. (3.7):\n\u001f2\n0!\u0000\n\u001f2\u0001\ne\u000b=\u001f2\n0+1\nc2\u0014\nz(\u001f)\n11\u001f2\n1+z(\u001f)\n22\u001f2\n2+z(\u001f)\n12\u001f1\u0001\u001f2\u0015\n(3.10a)\n(n\u0001\u001f0)2! (n\u0001\u001f)2\ne\u000b= (n\u0001\u001f0)2+1\nc2\"\nz(n)\n11(n\u0001\u001f1)2+z(n)\n22(n\u0001\u001f2)2+z(n)\n12(n\u0001\u001f1)(n\u0001\u001f2)#\n(3.10b)\n(p\u0001\u001f0)2! (p\u0001\u001f)2\ne\u000b= (p\u0001\u001f0)2+1\nc2\"\nz(p)\n11(p\u0001\u001f1)2+z(p)\n22(p\u0001\u001f2)2+z(p)\n12(p\u0001\u001f1)(p\u0001\u001f2)#\n(3.10c)\n(n\u0001\u001f0)(p\u0001\u001f0)!((n\u0001\u001f)(p\u0001\u001f))e\u000b= (n\u0001\u001f0)(p\u0001\u001f0) +1\nc2\"\nz(np)\n11(n\u0001\u001f1)(p\u0001\u001f1) +z(np)\n22(n\u0001\u001f2)(p\u0001\u001f2)\n+1\n2\u0010\nz(np)\n12(n\u0001\u001f1)(p\u0001\u001f2) +z(np)\n21(p\u0001\u001f1)(n\u0001\u001f2)\u0011#\n; (3.10d)\nwhere\nz(x)\nab\u0011 \na(x)\nabp2+b(x)\nab(n\u0001p)2+c(x)\nab\nr!\n; (3.11)\nthe symbol (x) corresponding to ( \u001f), (n), (p) or (np).\nRecall that the functions ^\u0001rand ^\u0001tdepend on\u001f2\n0, and\nthus need to be transformed according to (3.10a). The\ne\u000bective Hamiltonian that results applying (3.10) onto\n^Hold\ne\u000bwill be simply denoted as ^He\u000b.\nThe coe\u000ecients a(\u001f)\nab,b(\u001f)\nabandc(\u001f)\nabare already known,their explicit expression being given by Eq. (5.12) of\nRef. [36] (where the label ( \u001f) had not been used). In\nparticular, since the b(\u001f)\nab's vanish, (3.10a) is consistent\nwith Eq. (2.3).\nDetermining the z(x)\nab's from the Hamiltonian in ADM\ncoordinates cannot be done without \fnding, simultane-\nously, the generating function ^GNLO\nssof the corresponding\ncanonical transformation (see Eq. (2.7)). We are looking\nfor a su\u000eciently general ansatz that implements its al-\nready known test-mass limit (2.10), and that mantains,\nin addition, the symmetry under exchange of the labels 1\nand 2. The searched canonical tranformation may have\nthe following form:5\n^GNLO\nss=1\nc6r(\n(n\u0001p)\u0014\u0012\n\u0010(\u001f)\n11\u00001\n2r\u0013\n\u001f2\n1+\u0012\n\u0010(n)\n11\u00001\n2r\u0013\n(n\u0001\u001f1)2+\u000e(p)\n11(p\u0001\u001f1)2\u0015\n+\u0012\n\u0010(np)\n11+1\nr\u0013\n(n\u0001\u001f1)(p\u0001\u001f1)\n+ (n\u0001p)\u0014\u0012\n\u0010(\u001f)\n22\u00001\n2r\u0013\n\u001f2\n2+\u0012\n\u0010(n)\n22\u00001\n2r\u0013\n(n\u0001\u001f2)2+\u000e(p)\n22(p\u0001\u001f2)2\u0015\n+\u0012\n\u0010(np)\n22+1\nr\u0013\n(n\u0001\u001f2)(p\u0001\u001f2)\n+ (n\u0001p)h\n\u0010(\u001f)\n12(\u001f1\u0001\u001f2) +\u0010(n)\n12(n\u0001\u001f1)(n\u0001\u001f2) +\u000e(p)\n12(p\u0001\u001f1)(p\u0001\u001f2)i\n+1\n2\u0010\n\u0010(np)\n12(n\u0001\u001f1)(p\u0001\u001f2) +\u0010(np)\n21(p\u0001\u001f1)(n\u0001\u001f2)\u0011)\n; (3.12)\nwhere\n\u0010(x)\nab\u0011 \n\u000b(x)\nabp2+\f(x)\nab(n\u0001p)2+\r(x)\nab\nr!\n(3.13)\nfor (x) = (\u001f);(n) and (np). The\u000e(p)\nab's are, instead, con-\nstant coe\u000ecients. Notice that a canonical transformation\nof this type applies an in\fnitesimal rotation on the spins\naccording to\n\u001f0\na=\u001fa+\u0012@G\n@\u001fa\u0002\u001fa\u0013\n:\nIn our speci\fc case, the spins are left invariant under\nthe constraint of aligned spins and equatorial orbits:\n\u0012@GNLO\nss\n@\u001fa\u0002\u001fa\u0013\f\f\f\f\nal= 0:\nThus, nonprecessing orbits are preserved under the trans-formation given by (3.12), which is a consistency require-\nment for the approach we are following.\nB. The general solution\nIn order to determine all coe\u000ecients, we \frst explic-\nitly calculate the transformations given by Eqs. (2.4)-\n(2.7). All needed expressions are already collected in\nRef. [36], and speci\fcally: ^HN(ADM)\no can be found in\nEq. (2.3) there, ^H1PN(ADM)\no in Eq. (2.4); ^HLO(ADM)\nss in\nEq. (2.7), ^HNLO(ADM)\nss in Eq. (2.9); ^G1PN\noin Eq. (3.3),\nand ^GLO\nssin Eq. (3.5). We need to rearrange two of\nthese formulae, namely ^HNLO(ADM)\nss and ^GLO\nss, that had\nbeen originally written in a form that is not suitable for\nour purpose. In Eq. (2.9b) of Ref. [36], it appears the\nscalar ((p\u0002\u001f1)\u0001n) ((p\u0002\u001f2)\u0001n). Using the identity\n(3.9) (but with the vector \u001f1+\u001f2instead of\u001f0) it is\neasy to show that it can be decomposed as\n\u0000\n(p\u0002\u001f1)\u0001n\u0001\u0000\n(p\u0002\u001f2)\u0001n\u0001\n=\u0000\np2\u0000(n\u0001p)2\u0001\n(\u001f1\u0001\u001f2)\u0000p2(n\u0001\u001f1)(n\u0001\u001f2)\u0000(p\u0001\u001f1)(p\u0001\u001f2)\n+ (n\u0001p)\u0000\n(p\u0001\u001f1)(n\u0001\u001f2) + (n\u0001\u001f1)(p\u0001\u001f2)\u0001\n: (3.14)\nEq. (2.9b) of Ref. [36] then becomes\n^HNLO\nS1S2=3\u0017\nr3\u0014\n\u0000\u00121\n2+\u0017\n3\u0013\np2(\u001f1\u0001\u001f2) +\u0010\n1\u0000\u0017\n4\u0011\n(n\u0001p)2(\u001f1\u0001\u001f2)\n+\u00121\n2+3\n4\u0017\u0013\np2(n\u0001\u001f1)(n\u0001\u001f2) +5\n2\u0017(n\u0001p)2(n\u0001\u001f1)(n\u0001\u001f2)\n+\u00121\n2+\u0017\n6\u0013\n(p\u0001\u001f1)(p\u0001\u001f2)\u0000\u0012\n1 +\u0017\n4\u0000\u0017\n2m1\nm2\u0013\n(n\u0001p)(n\u0001\u001f1)(p\u0001\u001f2)\n\u0000\u0012\n1 +\u0017\n4\u0000\u0017\n2m2\nm1\u0013\n(n\u0001p)(p\u0001\u001f1)(n\u0001\u001f2)\u0015\n+\u0017\nr4\u0014\n6(\u001f1\u0001\u001f2)\u000012(n\u0001\u001f1)(n\u0001\u001f2)\u0015\n: (3.15)\nFurthermore, using basic vector identities, Eq. (3.5) of\nRef. [36] is simplifed as follows:6\n^GLO\nss=\u00001\nc41\n2r2n\u0002\n\u001f2\n0\u0000(\u001f0\u0001n)2\u0003\n(r\u0001p)\n+\u0000\n\u001f0\u0001n\u0001\n(r\u0002p)\u0001\u0000\n\u001f0\u0002n\u0001o\n=\u00001\nc41\n2rh\n(n\u0001p)\u001f2\n0\u0000(n\u0001\u001f0)(p\u0001\u001f0)i\n:(3.16)\nAfter all transformations (2.4)-(2.7), the resulting\nHNLO000\nss must be equated to the corresponding term\nHNLO(EOB)\nss obtained by a PN expansion of the EOB\nHamiltonian\n^HEOB=1\n\u0017vuut1 + 2\u0017 ^He\u000b\n\u0016\u00001!\n: (3.17)\nThe expansion can be done simply replacing r!r=\"2,\np!\"pand performing a Taylor series in the small num-\nber\".HNLO(EOB)\nss is then de\fned as the part propor-\ntional to\"8which is quadratic in the spins. Finding a\nsolution for the equation\nHNLO000\nss (r;p) =HNLO(EOB)\nss (r;p) (3.18)\nis equivalent to solving an inhomogeneous system of 57\nlinear equations (18 for the spin(1)-spin(1) combination,\n18 for the spin(2)-spin(2) and 21 for the spin(1)-spin(2)one), with 72 variables. This means that, if the system\nadmits a solution, there will be at least 15 undetermined\nvariables, that, as we shall see, will play the role of gauge\ncoe\u000ecients. Notice that, because of the symmetry under\nexchange of the particle label 1 and 2, the system can be\nreduced to 39 equations and 50 variables. The general\nset of equations is solved by:\n\u000b(\u001f)\n11=11\u00172\n32+3\u00172\n4m1\nm2\n\f(\u001f)\n11= 0\n\r(\u001f)\n11=5\u0017\n4+\u0012\u0017\n2\u0000\u00172\n4\u0013m2\nm1\n\u000b(n)\n11= 0\n\f(n)\n11= 0\n\u000b(\u001f)\n12=11\u00172\n16+\u0017\n2\n\f(\u001f)\n12= 0\n\r(\u001f)\n12=\u0000\u00172\n2\n\u000b(n)\n12= 0\n\f(n)\n12= 0: (3.19)\na(\u001f)\n11=\u000011\u00172\n16\u00003\u00172\n2m1\nm2\nb(\u001f)\n11= 0\nc(\u001f)\n11=\u000029\u00172\n16\u00003\u00172\n2m1\nm2\na(\u001f)\n12=\u0000\u0017\u000011\u00172\n8\nb(\u001f)\n12= 0\nc(\u001f)\n12=\u0000\u0017+19\u00172\n8\na(n)\n11=\u000019\u0017\n4+39\u00172\n8+\u0012\n\u0000\u0017\n2+15\u00172\n4\u0013m2\nm1+\r(n)\n11\u0000\u000b(np)\n11\nb(n)\n11=5\u0017\n4+15\u00172\n2+\u00125\u0017\n2+15\u00172\n4\u0013m2\nm1\u00005\r(n)\n11\u0000\f(np)\n11\nc(n)\n11=\u000011\u0017\n4+\u00172+\u0012\n\u0000\u0017\n2+7\u00172\n4\u0013m2\nm1\u0000\r(n)\n11\u0000\r(np)\n11\na(n)\n12=\u00002\u0017\u00003\u00172\n4+\r(n)\n12\u00001\n2\u0010\n\u000b(np)\n21+\u000b(np)\n12\u0011\nb(n)\n12=\u00005\u0017\u000015\u00172\n2\u00005\r(n)\n12\u00001\n2\u0010\n\f(np)\n21+\f(np)\n12\u0011\nc(n)\n12=\u00002\u0017+3\u00172\n2\u0000\r(n)\n12\u00001\n2\u0010\n\r(np)\n21+\r(np)\n12\u00117\na(p)\n11= 2(\u000b(np)\n11+\u000e(p)\n11)\nb(p)\n11= 2(\f(np)\n11\u00002\u000e(p)\n11)\nc(p)\n11= 4\u0017+ 2m2\nm1\u0017+\u00172\n2+ 2(\r(np)\n11\u0000\u000e(p)\n11)\na(p)\n12=\u000b(np)\n21+\u000b(np)\n12+ 2\u000e(p)\n12\nb(p)\n12=\f(np)\n21+\f(np)\n12\u00004\u000e(p)\n12\nc(p)\n12=\u0000\u0017+\u00172+\r(np)\n21+\r(np)\n12\u00002\u000e(p)\n12\na(np)\n11= 2(\u000b(np)\n11\u0000\f(np)\n11)\nb(np)\n11= 4\f(np)\n11\nc(np)\n11=13\u0017\n2+15\u00172\n4+\u0012\n4\u0017+3\u00172\n2\u0013m2\nm1+1\n4(\u00008\r(n)\n11+ 8\u000b(np)\n11+ 8\f(np)\n11+ 12\r(np)\n11+ 8\u000e(p)\n11)\na(np)\n12= 2(\u000b(np)\n12\u0000\f(np)\n12)\nb(np)\n12= 4\f(np)\n12\nc(np)\n12=\u00005\u00172\n2\u000012\u00173\u0000\u0000\n5\u00172+ 6\u00173\u0001m1\nm2\u0000\u0000\n2\u00172+ 6\u00173\u0001m2\nm1\u00002\r(n)\n12+ 2\u000b(np)\n12+ 2\f(np)\n12+ 3\r(np)\n12+ 2\u000e(p)\n12\na(np)\n21= 2(\u000b(np)\n21\u0000\f(np)\n21)\nb(np)\n21= 4\f(np)\n21\nc(np)\n21=\u00005\u00172\n2\u000012\u00173\u0000\u0000\n5\u00172+ 6\u00173\u0001m2\nm1\u0000\u0000\n2\u00172+ 6\u00173\u0001m1\nm2\u00002\r(n)\n12+ 2\u000b(np)\n21+ 2\f(np)\n21+ 3\r(np)\n21+ 2\u000e(p)\n12: (3.20)\nAs already mentioned, the coe\u000ecients \u000b22,\f22,\r22,\na22,b22andc22directly follow from the solution above\nexchanging the particle labels 1 and 2. In regard to this\npoint, it is worth discussing the Kerr limit \u0017!0. In this\ncase, indeed, one has to choose which one of the two spins\nvanishes - thus somehow breaking the symmetry between\nthem. For\u0017!0 (andm2=m1!0) all coe\u000ecents \u000b,\f\nand\rmust vanish, with the only exception of \r(\u001f)\n22,\r(n)\n22\nand\r(np)\n22. The reason of that lies in the form of the\ncanonical transformation (3.12) and its limit (2.10). The\nprice of having enforced the formal symmetry between\nspin(1) and spin(2), by adding \u0017-independent terms also\nto the spin(2)-spin(2) contribution of ^GNLO\nss, is that of\ngenerating coe\u000ecients \r22that do not tend to zero in\nthe Kerr limit (and speci\fcally: \r(\u001f)\n22!1=2,\r(n)\n22!1=2\nand\r(np)\n22!\u00001, which can be easily veri\fed taking into\naccount that \u0017m1=m2!1).\nBy contrast, all coe\u000ecients a,bandcvanish for\u0017!\n0, as required by consistency. In particular, the non-\nzero limit of \r(\u001f)\n22,\r(n)\n22and\r(np)\n22is responsible for the\nconvergence towards zero of a(n)\n22,b(n)\n22,c(n)\n22,c(p)\n22andc(np)\n22,\nwhich might not have seemed immediately obvious from\nEq. (3.20). Alternatively, to make this convergence more\nexplicit, one could rede\fne~\r(\u001f)\n22\u0011\r(\u001f)\n22\u00001\n2=\u00172\n2+m2\nm1\u0012\n\u0000\u0017\n2+\u00172\n4\u0013\n~\r(n)\n22\u0011\r(n)\n22\u00001\n2\n~\r(np)\n22\u0011\r(np)\n22+ 1; (3.21)\nwhich absorb the \u0017-independent terms in the spin(2)-\nspin(2) part of (3.12), and satisfy ~ \r(\u001f)\n22, ~\r(n)\n22,~\r(np)\n22!0.\nNow we can do the following reformulation:\na(n)\n22=\u000021\u00172\n8+\u0012\u0017\n2\u000015\u00172\n4\u0013m2\nm1+ ~\r(n)\n22\u0000\u000b(np)\n22\nb(n)\n22=\u0000\u00125\u0017\n2+15\u00172\n4\u0013m2\nm1\u00005~\r(n)\n22\u0000\f(np)\n22\nc(n)\n22=\u00005\u00172\n2+\u0012\u0017\n2\u00007\u00172\n4\u0013m2\nm1\u0000~\r(n)\n22\u0000~\r(np)\n22\nc(p)\n22=\u00172\n2\u00002\u0017m2\nm1+ 2~\r(np)\n22\u00002\u000e(p)\n22\nc(np)\n22=3\u00172\n4\u0000\u0012\n4\u0017+3\u00172\n2\u0013m2\nm1\u00002~\r(n)\n22\n+ 2\u000b(np)\n22+ 2\f(np)\n22+ 3~\r(np)\n22+ 2\u000e(p)\n22: (3.22)\nWe stress that, in this case, the formal symmetry with\nrespect to the corresponding a11,b11andc11is not di-\nrectly visible.8\nC. Gauge \fxing\nThe general solution (3.20) contains 18 gauge param-\neters, namely \r(n)\n11,\u000e(p)\n11,\u000b(np)\n11,\f(np)\n11,\r(np)\n11;\r(n)\n22,\u000e(p)\n22,\n\u000b(np)\n22,\f(np)\n22,\r(np)\n22; and\r(n)\n12,\u000e(p)\n12,\u000b(np)\n12,\u000b(np)\n21,\f(np)\n12,\n\f(np)\n21,\r(np)\n12,\r(np)\n21. An appropriate choice of them can\nbe used to simplify the 30 coe\u000ecients a(x)\nab,b(x)\nabandc(x)\nab\n(for (x) = ( n);(p);(np)), that are not uniquely deter-\nmined. One could try to impose the so-called Damour-\nJaranowski-Sch afer (DJS) gauge (see e.g. Refs. [27, 34])\nwhich would consist in eliminating those new terms pro-\nportional top2, i.e., all coe\u000ecients a(x)\nab. We cannot, how-\never, make all of them vanish: there is no way, within the\nmethod we have followed, to impose the DJS gauge. No-\ntice that this does not mean that the DJS gauge is a bad\nchoice in general: in the spin-orbit sector, it is a very\nuseful gauge, independently of how the spin-spin sector\nlooks like. Moreover, the impossibility of imposing it in\nthis paper is strictly related with the type of prescription\nwe have followed. It would be interesting to investigate,\nin a future work, if a di\u000berent method for including NLO\nspin-spin terms would allow to impose a DJS gauge in\nthe spin-spin sector too.\nHere, we nevertheless wish to give an example of gauge\n\fxing, and choose an alternative approach. We can, for\nexample, search a gauge for which the maximal number of\ncoe\u000ecients can be set to zero. One \fnds that at most 24\nof them can vanish, the remarkable fact being that this\nhappens for a unique gauge choice, and namely when\nthe non-zero coe\u000ecients are a(n)\n11,c(n)\n11,a(n)\n22,c(n)\n22,a(n)\n12\nandc(n)\n12. In this case, the gauge coe\u000ecients are \fxed as\nfollows:\n\r(n)\n11=\u0017\n4+3\n2\u00172+\u0012\u0017\n2+3\n4\u00172\u0013m2\nm1\n\u000b(np)\n11= 0\n\f(np)\n11= 0\n\r(np)\n11=\u00002\u0017\u0000\u00172\n4\u0000\u0017m2\nm1\n\u000e(p)\n11= 0\n\r(n)\n12=\u0000\u0017\u00003\u00172\n2\u000b(np)\n21= 0\n\f(np)\n21= 0\n\r(np)\n21=\u00172\n2+\u00172m2\nm1\n\u000b(np)\n12= 0\n\f(np)\n12= 0\n\r(np)\n12=\u00172\n2+\u00172m1\nm2\n\u000e(p)\n12= 0: (3.23)\nWe may also write\n~\r(n)\n22=\u0000\u0012\u0017\n2+3\n4\u00172\u0013m2\nm1\n~\r(np)\n22=\u0000\u00172\n4+m2\nm1\u0017: (3.24)\nThe coe\u000ecients of the e\u000bective spin squared take the fol-\nlowing, remarkably simple form:\na(\u001f)\n11=\u0000\u001211\n16+3\n2m1\nm2\u0013\n\u00172\nc(\u001f)\n11=\u0000\u001229\n16+3\n2m1\nm2\u0013\n\u00172\na(\u001f)\n12=\u0000\u0017\u000011\u00172\n8\nc(\u001f)\n12=\u0000\u0017+19\u00172\n8\na(n)\n11=\u0000\u001221\n8+9\n2m1\nm2\u0013\n\u00172\nc(n)\n11=\u0000\u00129\n4+m1\nm2\u0013\n\u00172\na(n)\n12=\u00003\u0017\u00009\u00172\n4\nc(n)\n12=\u00003\u0017\n2+7\u00172\n2; (3.25)\nwhich is the main result of this paper. For clarity pur-\nposes, we summarize the whole, new e\u000bective Hamilto-\nnian:\n^He\u000b=\u0017r\n2~r4\ne\u000b\u0010\nr2+\u0000\n\u001f2\u0001\ne\u000b\u0000^\u0001e\u000b\nt\u0011\u0014\u0012m1\nm2ge\u000b\nS+ge\u000b\nS\u0003\u0013\n(n\u0002p)\u0001\u001f1+\u0012m2\nm1ge\u000b\nS+ge\u000b\nS\u0003\u0013\n(n\u0002p)\u0001\u001f2\u0015\n+ ^\u0001e\u000b\nt\n~r4\ne\u000b!1=2\u0000\nr2+ (n\u0001\u001f)2\ne\u000b\u00011=2\"\n1 +1\u0010\n1 +(n\u0001\u001f)2\neff\nr2\u0011 \np2+ ^\u0001e\u000b\nr\nr2\u00001!\n(n\u0001p)2\n\u00001\n~r4\ne\u000b\u0012\n2r2\u0000^\u0001e\u000b\nt+\u0000\n\u001f2\u0001\ne\u000b+ (n\u0001\u001f)2\ne\u000b\u0013\n((n\u0002p)\u0001\u001f)2\ne\u000b!\n+Q4(p)#1=2\n; (3.26)9\nwith\n~r4\ne\u000b=\u0000\nr2+\u0000\n\u001f2\u0001\ne\u000b\u00012\u0000^\u0001e\u000b\nt\u0000\u0000\n\u001f2\u0001\ne\u000b\u0000(n\u0001\u001f)2\ne\u000b\u0001\n;\n(3.27)\nand with\n((n\u0002p)\u0001\u001f)2\ne\u000b= 2(n\u0001p)(n\u0001\u001f0)(p\u0001\u001f0)\u0000p2(n\u0001\u001f)2\ne\u000b\n\u0000(p\u0001\u001f0) +\u0000\np2\u0000(n\u0001p)2\u0001\u0000\n\u001f2\u0001\ne\u000b:\n(3.28)\nThe \u0001-potentials are^\u0001e\u000b\nt=r2P1\n3h\n1\u00002u+ 2\u0017u3+\u001294\n3\u000041\n32\u00192\u0013\n\u0017u4+\u0000\n\u001f2\u0001\ne\u000bu2i\n^\u0001e\u000b\nr=^\u0001e\u000b\nt\u0000\n1 + 6\u0017u2+ 2(26\u00003\u0017)\u0017u3\u0001\n;\nwhereP1\n3denotes the (1,3)-Pad\u0013 e approximant taken with\nrespect to the variable u\u0011r\u00001. Notice that, as in\nRef. [36], we do nottake the Pad\u0013 e with respect to the\nvariablercontained in\u0000\n\u001f2\u0001\ne\u000b.\nWe recall that the LO e\u000bective spin is de\fned as \u001f0=\nm1\nM\u001f1+m2\nM\u001f2, while the implementation of NLO spin-\nspin e\u000bects reads as follows:\n\u0000\n\u001f2\u0001\ne\u000b=\u001f2\n0\u00001\nc2\u001a\u0014\u001211\n16+3\n2m1\nm2\u0013\n\u00172p2+\u001229\n16+3\n2m1\nm2\u0013\u00172\nr\u0015\n\u001f2\n1\n+\u0014\u001211\n16+3\n2m2\nm1\u0013\n\u00172p2+\u001229\n16+3\n2m2\nm1\u0013\u00172\nr\u0015\n\u001f2\n2\n+\u0014\u0012\n\u0017+11\u00172\n8\u0013\np2+\u0012\n\u0017\u000019\u00172\n8\u00131\nr\u0015\n\u001f1\u001f2\u001b\n(3.29a)\n(n\u0001\u001f)2\ne\u000b= (n\u0001\u001f0)2\u00001\nc2\u001a\u0014\u001221\n8+9\n2m1\nm2\u0013\n\u00172p2+\u00129\n4+m1\nm2\u0013\u00172\nr\u0015\n(n\u0001\u001f1)2\n+\u0014\u001221\n8+9\n2m2\nm1\u0013\n\u00172p2+\u00129\n4+m2\nm1\u0013\u00172\nr\u0015\n(n\u0001\u001f2)2\n+\u0014\u0012\n3\u0017+9\u00172\n4\u0013\np2+\u00123\u0017\n2\u00007\u00172\n2\u00131\nr\u0015\n(n\u0001\u001f1)(n\u0001\u001f2)\u001b\n: (3.29b)\nThe 25 coe\u000ecients that build the rather complex NLO\nspin-spin Hamiltonian in ADM coordinates get con-\ndensed into 12 new contributions in the EOB. This re-\nsult is maybe not as striking as in the nonspinning sector\n(where, at 3PN order, the 11 ADM coe\u000ecients are re-\nduced to 3 EOB terms only), still it con\frms the notable\nability of the EOB in reproducing higher-order e\u000bects.\nIV. DISCUSSION: \\FULL\" AND \\PARTIAL\"\nINCLUSION\nOne of the basic ideas behind the EOB is that of de\fn-\ning a map between real and e\u000bective quantities. The\ncomplicated PN dynamics of spinning bodies, however,\nhas forced a splitting of the concept of \\e\u000bective spin\"\nin the EOB. Instead of having one single map between\n(S1;S2) andSe\u000b, one has to distinguish between linear\nand squared spin, and even, as shown in this paper, be-\ntween di\u000berent contractions of the spin vector with the\ndynamical variables, and map each component in a dif-\nferent way. Having thus lost, so to say, the concept of an\nunique e\u000bective spin, one might ask up to which pointare we allowed to split the handling of it. The inclu-\nsion of NLO spin-spin e\u000bects we have presented in this\npaper obeys a simple rule: the spin mapping is only\ndiversi\fed when strictly necessary, in other words, the\nspin terms are as much as possible equally treated. We\nmay call this the \\full\" approach, because of its formal\nand intuitive simplicity. One could also have followed\nanother line of thought, pursuing the computational ef-\n\fciency rather than the conceptual unity. Such an al-\nternative approach would consist in leaving untouched\nsome spin terms of the e\u000bective Hamiltonian that actu-\nally don't contribute to the NLO spin-spin part of the PN\nexpanded EOB. For example, the spin-orbit contribution\nHso= \u0001Hso+\fipiof the e\u000bective Hamiltonian only\ngenerates terms that are odd in the spins, and is thus\nunrelated to the spin-spin coupling. We may therefore\napply the prescription (3.10) only to the orbital (even in\nthe spins) part Horb=\u000bp\n1 +\rijpipj, leavingHsoas\ngiven by the \\old\" formulation (3.7a). We denote this\nkind of inclusion as \\partial\". It is clear that this way of\nproceeding leads to an explicit expression for He\u000bwhich\nis shorter and simpler than in the \\full\" case, but it im-\nplies, at the same time, an additional and unnecessary10\nfragmentation of the e\u000bective spin. In addition, a prob-\nlem related to this approach is the fact that the \\partial\"\nHamiltonian contains two di\u000berent Delta potentials, \u0001 t\nand \u0001e\u000b\nt. Since these potentials are de\fned through a\nPad\u0013 e approximant, which may contain poles, the \\par-\ntial\" Hamiltonian will in principle show twice as many\npoles than the \\full\" one, if the spin is nonzero.\nLet us \fnally remark that the \\full\" and \\partial\"\nHamiltonians di\u000ber by terms that are odd (and at least\ncubic) in the spins, and therefore comparing them may\nhelp to distinguish the e\u000bects intrinsically related to the\npure NLO spin-spin coupling from higher-order terms\nthat get automatically resummed into the EOB after the\nNLO spin-spin inclusion.\n0.060.080.100.120.140.160.18ˆωLSOtar12\ntar14\n4PN NLOssF\n4PN NLOssP\n3PN NLOssF\n3PN NLOssP\n−1.0−0.5 0.0 0.5 1.0\nχ0−0.04−0.03−0.02−0.01ˆeLSO\n−0.05 0.00 0.05−1.8−1.7−1.6×10−2\nFigure 1. Reduced angular frequency (top) and binding en-\nergy (bottom) at the LSO as a function of the e\u000bective spin\nparameter \u001f0, in the case of equal masses and equal spins, for\ndi\u000berent EOB models.\nFig. 1 shows the angular freguency ^ !LSOand the bind-\ning energy ^ eLSOat the LSO (see also Refs. [36, 37]) as\na function of the spin parameter \u001f0for di\u000berent EOB\nmodels, in the case of equal masses ( \u0017= 1=4). The\ncurves \\3PN NLOssF\",\\3PN NLOssP\", \\4PN NLOssF\"\nand \\4PN NLOssP\" denote the purely analytical EOB\nmodel discussed in this paper and in Ref. [36], with\nNLO spin-spin coupling (\\F\" and \\P\"indicating the\n\\full\" or \\partial\" inclusion) and with next-to-next-to-\nleading order (NNLO) spin-orbit coupling (that is repro-duced by the gyro-gravitomagnetic factors calculated in\nRefs. [27, 34]). The label \\3PN\" or \\4PN\" refers to the\npurely orbital order, resummed with Pad\u0013 e P1\n3orP1\n4, re-\nspectively. The 4PN orbital order needs an additional\nterm\u0017\u0000\nac\n5(\u0017) +aln\n5(\u0017) lnu\u0001\nu5(Eq. (5) of Ref. [49]) in-\nside of ^\u0001t. The \fgure clearly shows that the e\u000bect of the\n4PN terms at the LSO is repulsive, since the frequency\nand the binding energy get increased.\nIn Ref. [36, 37], it has already been pointed out that a\nsmaller e\u000bective spin squared leads to more bounded or-\nbits. It is thus not surprising that the \\partial\" inclusion\nis less bounded than the \\full\" one, as it can be seen in\nthe \fgure.\nFor completeness, Fig. 1 also shows two curves gen-\nerated by a di\u000berent EOB model. Here, \\tar12\" and\n\\tar14\" denote the calibrated model of Refs. [35, 38].\nThey both contain analytical information up to LO in the\nspin squared sector, with a calibration at the NLO level,\nand to NNLO in the spin-orbit sector, with calibration\nat next-to-next-to-next-to-leading order (NNNLO). The\nmain di\u000berence between the two models is that \\tar14\"\nreproduces the 4PN purely orbital order and is calibrated\nfor a varying mass-ratio, while \\tar12\" only contains the\n3PN orbital order, and is calibrated for equal masses.\nSince the calibration is done at the waveform level,\ntogether with the tuning of some parameters related\nto the dissipative part, it is di\u000ecult to have a precise\nguess about the real accuracy of the curves \\tar12\" and\n\\tar14\". Hoping that the deviation from reality is not\nso large as to compromise a qualitative discussion of the\nplot, we can observe that the \\partial\" approach does\nnot seem to show any particular advantage with respect\nto the \\full\" one. On the contrary, the \\full\" curves\nare generally closer to the corresponding calibrated ones.\nThis might be interpreted as a further argument in favour\nof the \\full\" approach. Sistematically replacing, every-\nwhere in the EOB Hamiltonian, terms by their \\e\u000bec-\ntive\" equivalent is thus not only conceptually robust, but\nseems even to behave well numerically.\nA second interesting point is that the di\u000berence be-\ntween our EOB model and the calibrated \\tar\" mod-\nels is signi\fcantly smaller at the 4PN level than at the\n3PN. In particular, the frequency ^ !LSO predicted by\n\\tar14\" roughly follows the curves \\4PN LSOssF\"and\n\\4PN LSOssP\", and lies in the gap between the two for\n\u001f0\u00150:5. The maximal deviation between \\4PN\" and\n\\tar14\" is 6 :2% (\\P\") and 8 :7% (\\F\"), while between\n\\3PN\" and \\tar12\" is 24 :6% (\\F\") and 26 :7% (\\P\"). The\nsame is true in the nonspinning regime. For \u001f0= 0, the\n\\4PN\" value of ^ !LSOdeviates from \\tar14\" by only 2 :6%,\nwhereas the di\u000berence between \\tar12\" and \\3PN\" is of\n12:7%.\nIn the case of the binding energy, there is a good corre-\nspondence for spin equal to zero (the deviations are 1 :4%\nat 4PN and 5 :0% at 3PN). However, for large spins, the\ndi\u000berence is quite large (up to 39 :2% (\\F\") and 43 :5%\n(\\P\") at the 3PN level, and up to 20 :9% (\\F\") and 27 :1%\n(\\P\") at 4PN). Nevertheless, from 3PN to 4PN there is11\nstill an improvement up to a factor \u00182. This fact may\nstrengthen the hope that the need for a calibration be-\ncomes less urgent when higher-order analytical terms are\nincluded.\nV. CONCLUSION\nWe have proposed a prescription for modifying the\nEOB Hamiltonian of Ref. [27], so that, when expanded\nin PN terms, it reproduces the correct NLO spin-spin\ncoupling for general precessing orbits. This is a general-\nization of the result of Ref. [36], where only equatorial or-\nbits with aligned spins had been taken into account. The\nimplementation of the correct spin-spin terms is possible\nafter a suitable canonical phase-space transformation of\nthe ADM Hamiltonian. We have \frst shown the result\nin a rather general gauge, with 18 free gauge parameters\n(that actually reduce to 10 if we require symmetry un-\nder exchange of the particles). Then, a speci\fc gauge is\nchosen. Being impossible, under the type of prescription\nwe have considered, to impose a DJS-type gauge (i.e., to\nremove all new inclusions of type p2\u001f2), we have done\nthe simple choice of looking for a gauge for which the\nmaximal number of the coe\u000ecients entering the e\u000bective\nspin squared can be set to zero. It turned out that thereis a unique gauge satisfying this criterion.\nIn the end, a slightly di\u000berent approach is taken into\naccount, where the e\u000bective spin squared is left at LO in\nthe spin-orbit sector of the e\u000bective Hamiltonian. This\npartial inclusion is less repulsive than the full one, and\nhas the unpleasant feature of increasing the number of\npoles of the Pad\u0013 e approximant by a factor 2. A com-\nparison of the (gauge invariant) angular frequency and\nbinding energy, taken at the LSO, with the calibrated\nEOB models of Refs. [35, 38], leads to the conclusion\nthat the \\partial\" approach do not show any particular\nadvantage with respect to the \\full\" one. By an Occam's\nrazor-like argument, the \\full\" approach, which is con-\nceptually more consistent, may thus be preferable. As\na last thing, the plot also brings the encouraging evi-\ndence that, as higher order terms are included, the purely\nanalytical EOB approaches more and more a calibrated\nmodel even in the strong \feld.\nACKNOWLEDGMENTS\nWe are thankful to Thibault Damour for a very useful\ndiscussion. Moreover, we would also like to thank Andrea\nTaracchini for having provided us some information. S.B.\nis supported by the Swiss National Science Foundation.\n[1] J. Aasi et al. (2013), arXiv:1304.0670 [gr-qc].\n[2] S. Vitale, Gen. Relat. Gravit. 46(2014).\n[3] P. Amaro-Seoane et al. (2013), arXiv:1305.5720 [astro-\nph.CO].\n[4] G. Lovelace, M. Boyle, M. A. Scheel, and B. Szilagyi,\nClass. Quant. Grav. 29, 045003 (2012).\n[5] L. T. Buchman, H. P. Pfei\u000ber, M. A. Scheel, and\nB. Szil\u0013 agyi, Phys. Rev. D 86, 084033 (2012).\n[6] A. Mrou\u0013 e, M. Scheel, B. Szil\u0013 agyi, H. Pfei\u000ber, M. Boyle,\nD. Hemberger, L. Kidder, G. Lovelace, S. Ossokine,\nN. Taylor, et al., Phys. Rev. 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D 87, 121501 (2013)."    },    {        "title": "1002.2699v1.Electronic_Dynamics_Due_to_Exchange_Interaction_with_Holes_in_Bulk_GaAs.pdf",        "content": "Electronic Dynamics Due to Exchange Interaction with Holes\nin Bulk GaAs\nHans Christian Schneider and Michael Krau\u0019\nPhysics Department and Research Center OPTIMAS, University of Kaiserslautern, 67663\nKaiserslautern, Germany\nABSTRACT\nWe present an investigation of electron-spin dynamics in p-doped bulk GaAs due to the electron-hole exchange\ninteraction, aka the Bir-Aronov-Pikus mechanism. We discuss under which conditions a spin relaxation times for\nthis mechanism is, in principle, accessible to experimental techniques, in particular to 2-photon photoemission,\nbut also Faraday/Kerr e\u000bect measurements. We give numerical results for the spin relaxation time for a range\nof p-doping densities and temperatures. We then go beyond the relaxation time approximation and calculate\nnumerically the spin-dependent electron dynamics by including the spin-\rip electron-hole exchange scattering\nand spin-conserving carrier Coulomb scattering at the level of Boltzmann scattering integrals. We show that\nthe electronic dynamics deviates from the simple spin-relaxation dynamics for electrons excited at high energies\nwhere the thermalization does not take place faster than the spin relaxation time. We also present a derivation\nof the in\ruence of screening on the electron-hole exchange scattering and conclude that it can be neglected for\nthe case of GaAs, but may become important for narrow-gap semiconductors.\n1. INTRODUCTION\nMuch experimental and theoretical work in the area of spintronics has been focused on the control and manipula-\ntion of the electrons' spin degree of freedom independently charge. Oftentimes the ultimate goal is the realization\nof spintronics devices, in which electronic spins are the carriers of the information.1,2A particularly appealing\nproperty of the manipulation of electronic spins in semiconductors that sets it apart from existing electronics\nin ferromagnetic materials is that the carrier densities can be varied over several orders of magnitude, and the\nexistence of band gaps makes a controlled excitation and read-out with optical \felds comparatively easy. The\nlimiting factor for the usefulness of the information encoded in a spin polarization in a non-ferromagnetic semi-\nconductor is the spin-polarization decay, which is caused by a variety of interaction mechanisms.3,4A famous\nresult on the physical limit of spin manipulation by relaxation phenomena is the measurement of a spin lifetime\nof about 1 nanosecond in n-doped GaAs at very low temperatures.3,4However, for actual devices, one might be\nforced to use materials and excitation conditions that do not yield these extremely long spin relaxation-times\nfor several reasons. First, one would likely prefer to work with undoped semiconductors, or even with dilute\nmagnetic semiconductors, in which holes are generally the majority carriers. Second, carrier injection through\nmetal-semiconductor interfaces or by optical \felds may create nonequilibrium electronic distributions at energies\nfar from the bottom of the bands.5Third, a knowledge of the spin-dependent dynamics at moderate to high\ndensities at room temperature is needed for most devices that interact with optical \felds and may be used for\nswitching of optical signals, such as ampli\fers and lasers.\nMuch work has been done to characterize the spin-\rip interaction mechanisms in semiconductors in terms of\nspin relaxation times. State-of-the-art ultrafast optical techniques such as time-resolved Faraday rotation6and\nspin-resolved 2-photon photoemission7make it possible to probe the spin and energy resolved carrier dynamics\non short timescales. In this regime, one has to reexamine whether spin relaxation-times constitute a complete\ndescription of spin-dependent carrier dynamics, or whether one needs to resort to a microscopic description by\nincluding the relevant scattering processes that change the carrier momentum and their spin.8{10\nThis paper is devoted to a theoretical study of the spin-dependent scattering processes due to the electron-hole\nexchange interaction in bulk GaAs. The spin-\rip mechanism due to this interaction, which is the most important\nH.C.S.:Email: hcsch@physik.uni-kl.dearXiv:1002.2699v1  [cond-mat.mtrl-sci]  13 Feb 2010one in semiconductors with high p-doping concentration, is the Bir, Aronov, and Pikus (BAP)11mechanism.\nWe \frst address the question about the conditions under which spin relaxation-times are appropriate to de-\nscribe the spin-dependent dynamics of electrons in p-doped bulk GaAs. We therefore start our presentation in\nSection 2 with an analysis of how one can de\fne a spin-relaxation time for the BAP process starting from the\nBoltzmann equation for electron-hole exchange scattering. We then show how this quantity is related to experi-\nmental results, and present numerical results for the BAP spin-relaxation time that are suitable for comparison\nwith experiments over most of the experimentally interesting density and temperature range. In Section 3, we\nderive the dynamical Boltzmann equations for the electronic distributions including both exchange and direct\nscattering. This Boltzmann equation was already used in Section 2, but here we go back to the basic Hamilto-\nnian that describes carriers in semiconductors interacting via the direct Coulomb and the electron-hole exchange\ninteraction. Although these equations have been derived before, we include a discussion of screening e\u000bects in a\nfashion analogous with the usual random-phase approximation (RPA) treatment of screening and we pay special\nattention to the long-range and short-range contributions to the electron-hole exchange interaction. We then\npresent numerical solutions of the Boltzmann equation including both the electron-hole exchange interaction and\ndirect electron-hole Coulomb interaction for the case of the ultrafast optical excitation of a nonequilibrium spin-\npolarized distribution of electrons and hole. We compare the the full energy-resolved spin polarization dynamics\nwith the BAP spin relaxation-time.\n2. SPIN RELAXATION-TIME\n2.1 Derivation from Boltzmann equation\nIn this Section, we discuss a derivation for the spin relaxation time due to the electron-hole exchange interaction in\np-doped semiconductors (GaAs), i.e., the BAP spin relaxation mechanism. We start from the general Boltzmann\nequation for the electronic distribution functions. We assume in the following spherically symmetric energy\ndispersions so that the distribution functions depend only on the carrier energy instead of the vector momentum.\nWe therefore neglect in this article other spin-relaxing mechanisms, such as the Dyakonov-Perel mechanism,\nwhich depends on anisotropies of the electronic band structure. We also assume that anisotropic electronic\ndistributions, which are created by optical excitations are washed out in a few hundred femtoseconds by carrier-\ncarrier scattering, so that for the timescales of interest for the electron-hole exchange interaction, we always have\nspherically symmetric electronic distributions. With these assumptions we can write the Boltzmann equation\nfor the spin-dependent electronic distributions ns(E) in the form\n@\n@tns(E;t) =Z\nf\u0000wxc(E;E0) [1\u0000n\u0000s(E0)]ns(E) +wxc(E0;E)n\u0000s(E0) [1\u0000ns(E)]gD(E0)dE0(1)\n+Z\nf\u0000wdir(E;E0) [1\u0000ns(E0)]ns(E) +wdir(E0;E)ns(E0) [1\u0000ns(E)]gD(E0)dE0\nHere,s=\";#denotes the electron spin projection quantum number along the quantization axis, which is chosen\nto be thezaxis. The details of the electron-hole exchange and direct scattering, such as direct carrier Coulomb\nscattering and/or carrier-phonon scattering, are contained in the scattering kernels which will be discussed in\nthe next section. The exact form of the density of states is not important in the following, but for de\fniteness\nwe consider parabolic electron bands with e\u000bective mass me, so thatD(E) = (2me=\u0016h2)3=2=(2\u0019)2\u0002p\nE. This\nequation is derived and solved numerically in the following Section 3. Instead of a full numerical solution, we\nanalyze \frst how a simpli\fed description in terms of a spin-relaxation time is possible,\nFor the derivation of the spin-relaxation time, we focus on a simple pump-probe scenario. We assume that\nsuch an experiment is performed with a weak picosecond or sub-picosecond pump pulse that excites electrons close\nto the bottom of the band. Then a detailed microscopic description of the coherent carrier-creation process is not\nimportant for the electron spin dynamics because the time scale for the spin-dependent dynamics is much longer,\nnamely on the order of several hundred picoseconds. The equilibration of the photoexcited electrons occurs via\nthe direct scattering processes, i.e., via Coulomb scattering with equilibrated holes, which are present in the\nsystem due to the p-doping, or via electron-phonon scattering. For the present discussion, it is not important\nwhich mechanism is dominant, only that the electrons are thermalized after shortly after the pump pulse. If thepump is weak enough, the thermalized electrons will be non-degenerate and described by a Maxwell-Boltzmann\ndistribution\nns(E;t) =ns(t)f\f(E) (2)\nf\f(E) =N\u00122\u0019\u0016h2\nme\f\u00133=2\nexp(\u0000\fE): (3)\nHere,\f= 1=(kBT),kBis Boltzmann's constant, and Nis the total density of electrons. With this normalization,\nns(E) is in the range of 0 to 1. For this low density of nondegenerate electrons, the e\u000bect of the direct scattering\nis to keep the energy dependence of the electronic distributions constant. This is already built into the electronic\ndistribution function, Eq. (2), and its time development can therefore be calculated by inserting Eq. (2) in\nEq. (1), and keeping only the exchange contribution, i.e., the \frst line, in Eq. (1). In addition, for low density\ndistributions as assumed here, 1 \u0000ns(E;t)\u00191. Integrating the resulting equation overR1\n0D(E)dEand using\nthe normalization Z1\n0dED(E)f\f(E) =N (4)\none obtains from Eq. (1) a dynamical equation for the density of electrons with spin s,\n@\n@tns(t) =\u0000h\rxci\f\u0002\nns(t)\u0000n\u0000s(t)\u0003\n(5)\nThe energy-dependent out-scattering rate \rxc, de\fned by\n\rxc(E) =Z1\n0w(E;E0)D(E0)dE0; (6)\nenters Eq. (5) only in an energy-averaged sense. [See Appendix B for the numerical solution of Eq. (6)]. The\naverage over the electronic Maxwell-Boltzmann distribution is de\fned by\nh\rxci\f=1\nNZ1\n0\rxc(E)f\f(E)D(E)dE: (7)\nFrom Eq. (5) and Eq. (2), the spin relaxation time, i.e., the decay time of the spin polarization, de\fned by\nP(E;t) =n\"(E;t)\u0000n#(E;t)\nn\"(E;t) +n#(E;t); (8)\ncan be deduced. The energy resolved spin polarization is experimentally accessible, e.g., by spin resolved 2-\nphoton photoemission techniques.12However, for low electronic excitation conditions, the energy dependence of\nthe non-degenerate electronic distributions of the form (2) cancels in Eq. (8), so that the polarization P(E) is\nenergy independent and equal to the total spin polarization P(E;t) =P(t)\u0011[N\"(t)\u0000N#(t)]=[N\"(t) +N#(t)].\nFrom (5) it then follows that\nd\ndtP(t) =\u00001\n\u001cspinP(t) (9)\nwhere the spin relaxation-time is de\fned by\n1\n\u001cspin= 2h\rxci\f: (10)\nFor a low density of electrons in an energy range, in which the direct scattering leads to an e\u000bective equilibration\non timescales shorter than the spin relaxation-time, one therefore obtains an energy independent spin relaxation\ntime. Only this quantity is experimentally accessible, even with experimental methods that possess an energy\nresolution that is better than the spread in energy of the Maxwell-Boltzmann distribution f(E), which is used tocalculate the average in Eq. (7). Moreover, it is impossible to measure an energy resolved spin-relaxation time,\nwhich is sometimes de\fned by\n1\n\u001cspin(E)= 2\rxc(E) (11)\nThus the only meaningful spin relaxation time is the one originally introduced by Bir, Aronov, and Pikus,11but\nwithout the energy dependence, which has been discussed in the literature.13,14For the theoretical description\nof experimental results that do not meet the conditions for the derivation of a single, energy-independent spin\nrelaxation-time, one needs to solve the full Boltzmann equation numerically to obtain the energy or momentum-\nresolved electron spin dynamics. If the form of the electronic distribution functions changes appreciably over\nthe time of the spin relaxation, one should not expect a single electron spin-relaxation time, but an energy-\ndependent one. However, from the energy-dependent dynamics one can determine the time evolution accessible\nby the experiment and try to \ft the spin-polarization dynamics by an exponential decay, if possible. Another\nexperimental consequence of this analysis is that a measurement of the electronic spin dynamics by di\u000berent\ntechniques, such as Faraday rotation, 2-photon photoemission or di\u000berential transmission, may yield di\u000berent\nresults, and one may extract di\u000berent spin relaxation-times from these measurements, as has been shown for the\ncase of hole spin-relaxation.10\n2.2 Numerical results for spin relaxation-times\nAs discussed above, a spin relaxation-time for the BAP process, Eq. (10), can be rigorously de\fned for experi-\nments probing a low density of thermalized (non-degenerate) electrons. For this case, we numerically evaluate\nthe expression (10) using also Eqs. (7) and (6); see Appendix B and the set of parameters shown in Table 1\nin that Appendix. The results obtained in this Section can therefore be compared to the spin relaxation-times\nmeasured by optical pump-probe techniques, such as 2-photon photoemission or Faraday-rotation measurements,\nas long as the experiments only probes equilibrated electrons. In this case, as stressed above, the spin relaxation\ntime is energy independent.\nAn important characteristic of a spin dependent scattering process is its temperature dependence because\ndi\u000berent mechanisms di\u000ber strongly in their respective temperature dependences and a measurement of this\n101001000\n050100150200250300spin relaxation time τspin (ps)\ntemperature T (K)1018 cm-3\n1019 cm-35 x 1018 cm-3\nFigure 1. Temperature dependence of \u001cspinfor di\u000berent p-doping concentrations.101001000\n110100spin relaxation time τpol (ps)\nhole concentration N (1018 cm-3) 50 K\n300 K150 K75 K25 KFigure 2. Dependence of \u001cspinon the p-doping concentration for di\u000berent temperatures.\nquantity can sometimes be used to identify the dominant scattering process for the material and doping concen-\ntration under study. Fig. 1 shows the computed spin relaxation-time \u001cspinfrom 25 K up to room temperature.\nIt con\frms the well-known general trend that the BAP mechanism becomes important for \\high\" temperatures,\ni.e., around room temperature, where the spin relaxation-time can reach values below 100 ps for typical doping\ndensities of N= 5\u00021018cm\u00003and beyond.\nAnother important \\\fngerprint\" of the BAP process is its dependence on the density of holes introduced by\nthe p-doping. The evaluation of Eq. (7) allows one to obtain the BAP spin relaxation-time for arbitrary hole\nconcentrations, as shown in Fig. (2). The doping-density dependence is obtained for the whole density range\nwithout having to patch together analytical results, which can be obtained for the limiting cases of low and\nhigh doping densities. Since both the analytical results are not valid around the densities, for which the holes\nbecome degenerate, a discontinuity results if one plots the standard analytical results in one \fgure over the\nwhole doping density range.15One can comare the results of Fig. 2 with recent evaluations of the contribution\nof the Dyakonov-Perel e\u000bect16,17or with measurements18for undoped GaAs, i.e., hole concentrations of well\nbelow 1017cm\u00003. For these low densities, even around room temperature, the BAP process is not the dominant\nrelaxation process. This result shows that there is no \\rule of thumb\" that (for intrinsic GaAs) the Dyakonov-\nPerel e\u000bect is dominant at low temperatures and the BAP becomes important for \\higher\" temperatures. The\nBAP vs. Dyakonov-Perel problem, including the e\u000bects of carrier heating, has been explored in more detail\nrecently.16,19\nIn Fig. 3 we examine the importance of di\u000berent contributions to the spin-relaxation time due to the electron-\nhole exchange interaction. The long-range contribution to the electron-hole exchange interaction is clearly the\ndominant one, as can be seen from comparing the top and bottom solid lines in Fig. 3. If one leaves out the\nshort-range contribution one gets a good approximation to the true spin-relaxation time especially for higher\np-doping concentrations approaching or exceeding N= 1020cm\u00003. It is even a good approximation to only\ninclude the contribution of the heavy-hole to heavy-hole transitions to the electron spin relaxation-time. Since\nthe results of Fig. 3 again cover the p-doping density range around hole degeneracy where the approximate\nresults are not valid,15using just the long-range contribution due to heavy-hole to heavy-hole transitions will\ngive a much better estimate of the true spin relaxation-time than using any one of the approximate expressions.101001000\n110100spin relaxation time τpol (ps)\nhole concentration N (1018 cm-3) Figure 3. Dependence of \u001cspinon the p-doping concentration for T= 300 K. (a) The bottom solid line is the result including\nshort and long range contributions to the electron-hole exchange scattering and contributions from hh-hh, hh-lh, and lh-lh\ntransitions. (b) The long-dashed line is the result including only the long range contribution and all hole bands. (c) The\nshort-dashed is the same as (b), but includes only the hh-hh scattering transitions. (d) The top solid line includes only\nthe short-range contribution to the electron-hole exchange interaction and all relevant transitions between hole bands.\n3. MICROSCOPIC ELECTRON DYNAMICS DUE TO EXCHANGE SCATTERING\nWe now turn towards a derivation of the microscopic spin-dependent carrier dynamics that underlies the concept\nof the spin relaxation time derived in the previous section. The purpose of this derivation is to make the\npresentation self-contained and pay attention to the screening properties of the exchange interation. In particular,\nall the parameters needed to describe the exchange interaction quantitatively are included in this section and\nthe appendices, and the steps for a numerical solution of the Boltzmann equation are presented in some detail\nin Appendix B.\n3.1 Hamiltonian and dynamical equations\nBecause we have the application to p-doped GaAs in mind, we treat the case of electronic dynamics in the\npresence of a high density of thermalized holes. We do not include electron-phonon interactions, and do not\ntreat the hole dynamics and hole-hole Coulomb explicitly. Rather, we assume that the hole system is always in\nequilibrium with the lattice, i.e., we have equilibrium hole distributions at the lattice temperature. Furthermore,\nelectron-electron scattering is neglected because electron-hole scattering is the dominant scattering mechanism\ndue to the low electronic density. Under these assumptions, the starting point is the basic Hamiltonian\nH=Hkin+HCoul+Hexc; (12)\nwhich includes the electron kinetic energy and the electron-hole Coulomb interaction as well as the electron-hole\nexchange interaction. The kinetic and the direct electron-hole Coulomb term have the familiar forms20,21\nHkin=X\nsk\u000fskcy\nskcsk (13)\nand\nHCoul=1\n2X\n~k~k0~ qX\nsj;s0j02hj0;s0jVCoul(~ q)jj;sicy\nj0;~k+~ qcy\ns0~k0\u0000~ qcs~k0cj~k: (14)In Eqs. (13) and (14), cs,cy\nsandcj,cy\njare conduction and valence-band destruction and creation operators,\nrespectively. Conduction-electron states are labeled by the electron spin projection quantum number s=\u00061=2,\nwhereas valence-band electrons are labeled by the projection quantum numbers j=\u00061=2,\u00063=2 corresponding\nto the total angular momentum J= 3=2. In this paper, we will keep the conduction-valence (cv) picture for\ncarrier states21,22and the index structure of direct and exchange terms in the Hamiltonian and throughout the\ncalculations. Only in the \fnal expressions for the scattering kernels a change to electron and hole distributions\nis made by replacing nj;k\u0000!1\u0000nj;kand\u000fj;k\u0000!\u0000\u000fj;k. However, the linear and angular momentum labels, j\nandk, as well as the interaction matrix elements from the conduction-valence band picture are retained. For a\ncomplete transformation to the electron-hole picture, one would have to change the angular and linear momenta\nto\u0000jand\u0000k, de\fne hole operators,23and calculate the interaction matrix elements using the time-reversed\nvalence-band wave functions, which leads to a switch of the jandj0labels in the matrix elements.13,24The\nmatrix element for the direct Coulomb interaction is\nhj0;s0jVCoul(~ q)jj;si\u0011\u000ej0;j\u000es0;svq (15)\nwith the Coulomb potential energy\nvq=1\nL3e2\n\"0\"bgq2(16)\nin SI units, where \"bgis the dimensionless background dielectric constant. The electron-hole exchange interaction\nis described by\nHexc=X\nk;k0;qX\nj;s;j0;s0hj0s0jVexc(~ q)jsjicy\nj0;~k+~ qcy\ns0;~k0\u0000~ qcj;~k0cs;~k: (17)\nNote in particular that the Hamiltonian of the exchange interaction (17) di\u000bers from its direct interaction\ncounterpart (14) in the last two creation operators and the order of the band indices in the matrix element. The\nmatrix element of the exchange interaction consists of a \\long range\" and a \\short range\" part23\nhj0;s0jVexc(~ q)js;ji=hj0;s0jVlr\nexc(~ q)js;ji+hj0;s0jVsr\nexc(~ q)js;ji (18)\nwhere the long range part is given by23\nhj0;s0jVlr\nexc(~ q)js;ji=vq\u0016h2\n2m0e2(~ q\u0001~ p\u0003\ns;j0)(~ q\u0001~ ps0;j)\n(\u000f0s\u0000\u000f0\nj0)(\u000f0\ns0\u0000\u000f0\nj)=vq\u0000\n~ q\u0001~ x\u0003\ns;j0\u0001\n(~ q\u0001~ xs0;j) (19)\nIn Eq. (19), ~ psjis the momentum and ~ xsjthe dipole operator matrix element between the conduction electron\nstates,k= 0 and the valence-band state j,k= 0. The energies \u000f0are the electronic energies at the \u0000 point,\nk= 0, where the conduction and valence electron bands are respectively degenerate. Using the standard basis\nfor the coordinate representation of conduction ( S= 1=2) and valence ( J= 3=2) electrons,25the short range\npart can be brought into the form23\nhj0s0jVsr\nexc(q)jsji=3\n2\u0001sr\nexchj0;s0j\u00143\n4+ (~J\u0001~S)\u0015\njj;si (20)\nIn Eq. (20), \u0001sr\nexcdenotes the excitonic exchange splitting, and ~Jand~Sare the hole and electron angular\nmomentum operators, respectively. Explicit matrix expression for Eqs. (19) and (20) are compiled in Appendix A.\nIn order to describe the scattering dynamics on a microscopic footing, exchange and direct Coulomb interac-\ntion need to be included in the equations of motion,\n@\n@tns(k) =@\n@tns(k)\f\f\f\nCoul+@\n@tns(k)\f\f\f\nexc(21)\nWe therefore present a derivation of the exchange scattering in a fashion analogous to the derivation of the\nBoltzmann equation for direct electron-hole Coulomb scattering in the random-phase approximation (RPA).Employing a Green's function approach21,22and using a quasiparticle ansatz one obtains in the electron-hole\npicture\n@\n@tns(k)\f\f\f\nexc=X\n~k0\u0002\n\u0000Wexc\nk;k0(1\u0000n\u0000s;k0)ns;k+Wexc\nk0;kn\u0000s;k0(1\u0000ns;k)\u0003\n(22)\n@\n@tns(k)\f\f\f\ndir=X\n~k0\u0002\n\u0000Wdir\nk;k0(1\u0000ns;k0)ns;k+Wdir\nk0;kns;k0(1\u0000ns;k)\u0003\n(23)\nwhere\nWexc\nk;k0=2\u0019\n\u0016hX\nqX\nj;j0jhj;\u0000sj~Vexc(~ q;EG+\u000f\u0000s;k0+\u000fj0;k0+q)jsj0ij2nj;k+q(1\u0000nj0;k0+q)\n\u0002\u000e(\u000fj;k+q+\u000fs;k\u0000\u000f\u0000s;k0\u0000\u000fj0;k0+q) (24)\nWdir\nk;k0=2\u0019\n\u0016hX\nqX\nj;j0jhj;\u0000sjVscr(~ q;\u000fs\u0000\u000fs;k0)jsj0ij2nj;k+q(1\u0000nj0;k0+q)\u000e(\u000fj;k+q+\u000fs;k\u0000\u000fs;k0\u0000\u000fj0;k0+q) (25)\nNote that, instead of writing separate expressions for the in and out-scattering kernels, we have displayed the\nsymmetry of the in and out-scattering processes by introducing only one kernel Wk;k0. This symmetric form\nwith one kernel is especially convenient for a numerical solution because it avoids the separate calculation of\ntwo kernels, which may introduce numerical problems if one violates the balance between in and out scattering,\nwhich is responsible for the conservation of the carrier density.\nThe occurrence of an interband energy in the dynamical exchange interaction is analogous to the dynamically\nscreened of the direct Coulomb interaction Vscr, and can be derived in exactly the same fashion.21,22The result\nfor the dynamical exchange interaction is\nhjs0j~Vexc(~ q;!)jsj0i=hjs0jVexc(~ q)jsj0i+X\ns1j1hjs1jVexc(~ q)jsj1i\u0005s1j1(~ q;!)hj1s0j~Vexc(~ q;!)js1j0i (26)\nwith the retarded RPA electron-hole polarization function\n\u0005sj(~ q;!) =\u0000i\u0016hX\nkGs;k+q(!)Gj;k(!) =X\nkns;k+q+nj;k\n\u0016h!\u0000(\u000fs;k+q+\u000fj;k+EG) +i\u0016h\r(27)\nHere, a zero-density contribution that occurs on transforming into the electron-hole picture has been subtracted.\nIn Eq. (27) an interband energy enters the denominator. On evaluating Eq. (27) for the case of GaAs, the\nproduct of the exchange interaction and the polarization function in Eq. (26) turns out to so small that the e\u000bect\nof the screening term on the exchange interaction is negligible for the numerical results in this paper. Therefore\nonly the unscreened exchange interaction will be used in the following, i.e., ~Vexc(~ q;!)!Vexc(~ q). Even though\nthe e\u000bect of screening on the exchange interaction is comparatively weak for the case of GaAs considered here,\nit might become more important in narrow band-gap systems. In the following numerical results we will also\nuse a statically screened Coulomb interaction Vscr(q;!)!\"\u00001(q)vq. A discussion of the static approximation is\ngiven by Collet.26Although dynamical screening e\u000bects can become important for quantitative results on direct\nCoulomb scattering, for the purposes of this paper it is mainly important that the direct electron-hole Coulomb\nscattering occurs much faster than the electron-hole exchange scattering.\nThe connection to Boltzmann equation (1) used previously is made in the following way. In the general\nBoltzmann equation (22), which contains distribution functions and energies that may be anisotropic, one as-\nsumes spherical symmetry and employs an e\u000bective mass description for electrons and holes. Then the carrier\ndistributions depend on the energy, i.e.,\nn\u0017;k!n\u0017(E) whereE=\u0016h2\n2m`k2(28)for\u0017= electrons (e), heavy holes (HH), and light holes (LH). As mentioned above we assume further a high\ndensity of equilibrated ,unpolarized holes. Then the distribution functions can be expressed as Fermi functions\nnj;k=f(\u000fHH\nk\u0000\u0016) forj=\u00063=2 andnj;k=f(\u000fLH\nk\u0000\u0016) forj=\u00061=2 and the~ qsummation can be converted into\nan integral that can be evaluated mostly analytically as shown in Appendix B. The chemical potential \u0016is \fxed\nby the temperature and density of the equilibrated holes. Due to the assumption of unpolarized holes and the\nsymmetry properties of the interaction matrix element, which are discussed in the Appendix A, the kernel 24\nis a function W(E;E0) independent of the electron spin. Writing the wave-vector summations in Eq. (22) as\nenergy integrals including the density of states, one \fnally obtains Eq. (1), on which the analysis of the BAP\nspin-relaxation time is based.\n3.2 Numerical Results for Spin-Polarization Dynamics\nIn this Section, the dynamics of spin polarized electrons under the in\ruence of exchange (spin-\rip) and direct\n(non spin-\rip) electron-hole scattering processes are numerically evaluated using Eqs. (22), (24) and the static\n(non-screened) exchange interaction matrix element (18). Here, we analyze the energy and spin-resolved electron\ndynamics for excitation conditions that would be used in experiments designed to measure the energy-resolved\npolarization dynamics. One of our goals here is to show, which energy-resolved quantities can be extracted\nfrom typical experiments. For experiments with sub-picosecond excitation pulses, the a detailed microscopic\ndescription of the coherent carrier-creation process is again neglected because it occurs on a shorter time scale\nthan for the the spin-dependent dynamics. As a model for such an experiment that creates a nonequilibrium\nelectron distribution by a short laser pulse, we consider as an initial condition a Lorentzian electron distribution\npeaked around an electronic energy \u000fkL, or modulus of the electron wave vector kL. In an experiment, the\nwave vector kLand the energy would be determined by the laser wavelength and the band structure.12,27We\nrestrict ourselves to excited electron densities that are small enough so that the neglect of the electron-electron\ninteractions is justi\fed. The initial low-density (nondegenerate) electron distribution is chosen with initial spin\npolarization of 0.5 and with the same temperature as the holes.\nFigure 4 shows the dynamical results for the spin polarization P(E;t) de\fned in Eq. (8). The curves were\ncalculated numerically as described in Appendix B with both the short-range and the long-range contribution\nto the electron-hole exchange interaction matrix element taken into account. The parameters are again those\ngiven in Table 1. We assume unpolarized holes with density NA= 5\u00021018cm\u00003and temperature Th= 75 K.\nThe spin polarization is monitored at the energy E, at which the carriers are initially created in the form of a\nnonequilibrium electron distribution peaked at E. At a carrier energy E= 5 meV, i.e., close to the bottom of the\nelectron band, the decay of the spin polarization after the excitation of the electrons is almost exponential. For\nhigher peak energies of the excited electrons, already starting below an excess energy of 70 meV, the dynamics\nis not described by an exponential drop at all. Instead, the P(E;t) rises after an initial steep drop. Only for\n0.250.5\n020406080100120Spin PolarizationTime (ps)E = 460 meVE = 70 meVE = 200 meVE = 5 meV\nE = 140 meV\nFigure 4. Spin polarization dynamics for a p-doping concentration of 5 \u00021018cm\u00003and temperature T= 75 K for\nexcitation of electrons at di\u000berent energies Eabove the bottom of the band.00.250.5\n020406080100120Spin PolarizationTime (ps)E = 460 meVE = 820 meVE = 1 eVE = 200 meVFigure 5. Spin polarization dynamics for a p-doping concentration of 5 \u00021018cm\u00003and temperature T= 300 K for\nexcitation of electrons at di\u000berent energies Eabove the bottom of the band.\nlonger times, the spin polarization dynamics resembles an exponential decay with time constants of about 305 ps,\nwhich is identical to the time constant of the decay of the total spin polarization and almost identical to the\nBAP spin relaxation-time as shown in Figs. 1 and 2. The reason for this behavior is the interplay between the\nspin-\rip exchange scattering and the direct scattering. If the polarization is determined at an energy where\nelectrons undergo in and out-scattering processes, a quasi-equilibrium is reached due to direct electron-hole\nscattering events on a timescale faster than the spin-\rip scattering, so that the spin-\rip scattering occurs for\ne\u000bectively equilibrated electrons. For these, the concept of a spin relaxation time exists, as shown in Section 2.\nFor electrons with higher energies out-scattering processes, dominate for several ten picoseconds up to more\nthan 100 ps. On this timescale, the initial peaked electronic distribution has been already been washed out,\nbut no quasi-equilibrium has been established for electrons at these energies. The polarization dynamics is\ntherefore due to a combination of spin-conserving and spin-\rip scattering processes, and the dynamics depend\non the form of the actual nonequilibrium electron distribution. This non-exponential dynamics occurs until a\nquasi-equilibrium is established due to direct scattering processes. Then the spin polarization decay becomes\nexponential. Thus experiments exciting and probing the electronic spin polarization at higher energies will\nnot yield energy-resolved information about the the energy resolved spin-relaxation time due to the exchange\ninteraction as de\fned in Eq. (11).\nFigure 5 shows the same scenario for a hole temperature of T= 300 K. In this case, again, the spin polarization\ndecays exponentially only for electrons close to the bottom of the band, with a spin-relaxation time of about\n70 ps, which is again almost identical to the spin-relaxation time of the total polarization and the BAP spin\nrelaxation time. At higher energies, the initial polarization dynamics is non-exponential. However, the onset of\nthe nonexponential dynamics occurs only for electrons with much higher excess energy, here for energies well above\n200 meV because at higher energies the equilibrated \\tail\" of the distributions stretches out to higher energies.\nThis indicates that the nonexponential dynamics is not as important and will not be as easily observed as it is\nthe case for T= 75 K. The dynamics generally occur on a shorter time scale than in the T= 75 K case (note the\ndi\u000berentyaxes in the two \fgures), but the \\deviation\" from the exponential decay is qualitatively di\u000berent. Now\nthe nonequilibrium dynamics leads to a slower initial decay of the polarization, before the dynamics becomes\nexponential when a quasi-equilibrium is established by the direct scattering processes. Since the initial dynamics\nis so much di\u000berent for the di\u000berent temperatures there is no simple rule of thumb which qualitative behavior of\nthe spin dynamics occurs for nonequilibrium conditions. Certainly, the spin relaxation-time cannot serve even\nas guideline in this case, and one has to model the excitation conditions and energy-dependent measurement\naccurately to obtain reliable results that may be compared to a given experiment.4. CONCLUSION\nWe have investigated how spin relaxation-times can be used to describe the spin polarization dynamics of electrons\ndue to the electron-hole exchange interaction, i.e., the Bir-Aronov-Pikus mechanism. It is found that there is\nonly one ensemble averaged spin relaxation time that can be accessed by optical experiments, such as Faraday\nrotation, di\u000berential transmission, or 2-photon photoemission, even if these experiments have an energy resolution\nthat is smaller than the spread of excited electronic energies. In order to derive a meaningful spin-relaxation\ntime, one has to assume that a nondegenerate electronic distribution is kept a quasi-equilibrium by the direct\nscattering processes while the exchange scattering e\u000bects the spin polarization. Thus the energy-dependence of\nthe spin relaxation dynamics is washed out by the thermal average. If one tries to get around measuring the\nensemble average and intends to probe the electronic spin polarization at energies, at which there is initially no\nquasi-equilibrium established, one still does not get a unique energy dependent spin relaxation-time, but rather\na complicated spin polarization dynamics that is determined by the interplay of direct and spin-\rip scattering\nprocesses. It is therefore not possible to extract an energy-dependent spin relaxation-time because the of the\ndirect scattering mechanisms, although an energy-dependent energy relaxation time can be de\fned, as is usually\ndone for electronic lifetimes.\nACKNOWLEDGMENTS\nWe thank M. Aeschlimann and W. H ubner for helpful discussions and acknowledge \fnancial support by the DFG\nthrough the Graduiertenkolleg 792 \\Nonlinear Optics and Physics on Ultrashort Time Scales.\"\nAPPENDIX A. MATRIX ELEMENTS\nThe exchange interaction between electrons and holes consists of a short-range and long-range contribution In\nthe following, electron-hole states are labeled by conduction-valence band angular momentum labels ( js) =\n(\u00003\n2\");(\u00003\n2#);(\u00001\n2\");(\u00001\n2#);(+1\n2\");(+1\n2#);(+3\n2\");(+3\n2#)\nThelong-range part to the exchange interaction matrix results from the evaluation of Eq. (19) in the standard\nangular-momentum basis and reads:\nVlr\nexc(~ q) =vqjrcvj20\nBBBBBBBBBBBB@0 0 01\n2p\n3q2\n? 01p\n3q+qz 0\u00001\n2q2\n+\n01\n2q2\n? 01p\n3qzq+ 0\u00001\n2p\n3q2\n+ 0 0\n0 01\n6q2\n?1\n3qzq\u00001\n3q+qz2\n3q2\nz\u00001\n2p\n3q2\n+\u00001p\n3qzq+\n1\n2p\n3q2\n?1p\n3qzq+1\n3qzq+2\n3q2\nz\u00001\n6q2\n+\u0000qz\n3q+ 0 0\n0 01\n3q\u0000qz\u00001\n6q2\n\u00002\n3q2\nz\u00001\n3q\u0000qz\u00001p\n3q2\n+qz1\n2p\n3q2\n?\n1p\n3q\u0000qz\u00001\n2p\n3q2\n\u00002\n3q2\nz\u00001\n3q\u0000qz\u00001\n3q+qz1\n6q2\n? 0 0\n0 0 \u00001\n2p\n3q2\n\u0000 0\u00001p\n3q\u0000qz 01\n2q2\n? 0\n\u00001\n2q2\n\u0000 0\u00001p\n3q\u0000qz 01\n2p\n3q2\n? 0 0 01\nCCCCCCCCCCCCA\n(29)\nHere,q2\n?=q2\nx+q2\nyis modulus of the transverse component of ~ q;andqzis thezcomponent. Then\nq\u0006=qx\u0006iqy=q?e\u0006i'(30)\nwhere'is the azimuthal angle of ~ q:\nPikus and Bir23also show how the short-range contribution arises directly from the Coulomb interaction.\nHere, one evaluates the following exchange Coulomb matrix elements\nhj0~k1;s0~k2jvjs~k3;j~k4i=X\n\u001b;\u001b0Z\n'j0~k1(~ x\u001b)\u0003's0~k2(~ x0\u001b0)\u0003v(~ x\u0000~ x0)'s~k3(~ x\u001b)'j~k4(~ x0\u001b0)d3xd3x0(31)\nThe carrier states are of the usual form\n'\u000b~k(~ x;\u001b) =1p\nL3exp(\u0000i~k\u0001~ x)u\u000b(~ x\u001b); (32)whereu\u000b(~ x\u001b) are the Bloch functions at k= 0. For electrons, \u000b=s, and for holes, \u000b=j, so that the angular\nmomentum dependence of the uis determined by their angular momentum quantum number.\nThe contribution due of the exponential factors can be treated as in the spin-diagonal Coulomb matrix ele-\nment by dividing the integral over the crystal into integrals over unit cells and pulling the exponential factors\nout of the unit cell integrals. The remaining integrals over the unit cell are evaluated by inserting the ex-\nplicit expressions for the us, whose angular momentum dependence is given by their total angular momentum\nprojection quantum number j, i.e., by linear combinations of spherical harmonics. Expanding the interaction\nv(~ x\u0000~ x0) =P\n`V`(r;r0)P`\nm=\u0000`Y`\nm(^x1)Y`\nm(^x2)\u0003, the angular momentum integrals can be calculated and the\nremaining integrals over randr0can be absorbed into a quantity that is accessible experimentally. One \fnds in\nthe same matrix representation as Eq. (29)\nVsr\nexc=1\nL30\nBBBBBBBBBB@0 0 01\n2p\n3 0 0 0 0\n03\n20 0 0 0 0 0\n0 01\n20 0 1 0 0\n1\n2p\n3 0 0 1 0 0 0 0\n0 0 0 0 1 0 0p\n3\n2\n0 0 1 0 01\n20 0\n0 0 0 0 0 03\n20\n0 0 0 01\n2p\n3 0 0 01\nCCCCCCCCCCA\u00013\n2Vuc\u0001SR (33)\nwhere\n\u0001SR=X\n\u001b;\u001b0Z\nunit cellu\u000b0(~ x\u001b)\u0003u\f0(~ x0\u001b0)\u0003[v(~ x\u0000~ x0)]periodicu\u000b(~ x\u001b)u\f(~ x0\u001b0)d3xd3x0(34)\nis the exchange splitting energy, Vucis the volume of the unit cell and L3the crystal volume. The Coulomb\npotentialvappearing in Eq. (34) is lattice periodic because it results from a summation over reciprocal lattice\nvectors. The energy \u0001 SRis related to the 1s-exciton splitting (into one triplet and one quintuplet) \u0001exciton\nSR due\nto the exchange interaction by\nVuc\u0001SR=1\n2\u0019a3\nB\u0001exciton\nSR; (35)\nwhereaBis the excitonic Bohr radius.\nUsing the matrix expressions (29) and (33) in (22) one calculates the squared matrix elements for electronic\nspin-\rip scattering due to transitions between the di\u000berent hole bands. For instance, for heavy-hole to heavy-hole\nscattering one has\njh3=2;#jVxc(~ q)j\";3=2ij2=\f\f\f\fvqjrcvj2(\u00001\n2q2\n+)\f\f\f\f2\n=\u0012e2\n\"0\"q2jrcvj2q2\n?\u00132\n=\u000b2\u0000\n1\u0000z2\u00012(36)\nwhere\u000b=e2\n\u000f0\u000fjrcvj2, and we have de\fned z=qz=qso thatq2\n?=q2\u0000q2\nz\u0011q2(1\u0000z2). In a similar fashion one\nobtains for heavy-hole to light-hole scattering (two contributions in the scattering kernel)\n1\n3\u0012\u000b\n2(1\u0000z2) +3\n2\u0001SR\u00132\n(37)\nand for light-hole to light-hole scattering\n\u00122\n3z2\u000b+ \u0001 SR\u00132\n+ (1\u0000z2)\u000b2\n36(38)\nThe full squared interaction matrix element in Eq. (22) is a sum of Eqs. (36){(38), but the heavy-hole to heavy-\nhole contribution (36) is the dominant one for the conditions investigated in this paper. Note that there is\nno dependence of the in-plane angle '(betweenqxandqy) for matrix elements that also have a short-range\ncontribution. Therefore there is no explicit 'dependence in the terms that have combined short-range and\nlong-range contributions.Physical quantity Symbol Value\nDipole matrix element dcv=ercv0:6enm\nExcitonic exchange splitting \u0001exciton\nSR 0:05 meV\nElectron e\u000bective mass me 0.067\nHeavy hole e\u000bective mass mHH 0.457\nLight hole e\u000bective mass mLH 0.087\nBackground dielectric constant \"bg 12.5\nTable 1. Values of constants used in the numerical calculations.\nAPPENDIX B. NUMERICAL EVALUATION OF BAP SPIN RELAXATION-TIME\nTo obtain the BAP spin relaxation-time one evaluates the expression, with momentum indices chosen for nu-\nmerical convenience and as a function of kinstead ofE, but cf. Eq.(28),\n\rxc(k) =2\u0019\n\u0016hX\n~ q;~ pX\nj;j0jhj0;\u0000sjVxc(~ q)js;jij2(1\u0000nj;~ p+~ q)nj0;~k+~ q\u000e(\u000fe\nk+\u000fj\n~k+~ q\u0000\u000fe\n~ p\u0000\u000fj0\n~ p+~ q): (39)\nFirst, the momentum sums over ~ qand~ pare replaced by two three-dimensional integrals using spherical co-\nordinates with the pintegration as the inner integral. One measures pagainstq;andqagainstk;denoting\nz0= cos6(~ p;~ q) andz= cos6(~ q;~k) together with their respective azimuthal angles '0and'. The'0and'\nintegrations yield only 2 \u0019each because there is no explicit dependence on these angles in the momentum labels\n~k+~ qor~k0\u0000~ q, and the exchange matrix element does not depend on '0, only onqandz0. Then one analyti-\ncally evaluates the integral over z0making use of the properties of \u000edistribution. For these manipulations, it is\nconvenient to introduce a unit of length, denoted by `, and make the integrations over qandk0dimensionless by\nde\fning ~q=q`and~k0=~k`. It is also convenient to introduce the energy\nEkin=\u0016h2\n2m0`2(40)\nto make the kinetic energies dimensionless, \u000fj;e\np=Ekin= ~p2=mj;e. The masses are expressed in terms of the vacuum\nelectron mass m0. The result is (expressed in our dimensionless units and leaving the tildes o\u000b)\n\rxc(k) =1\n32\u001931\n\u0016h\fE2\nkinZ1\n0dqqZ1\n\u00001dzX\njj0f\u0012\n\u000fjp\nk2+q2+2kqz\u0000\u0016\u0013\njhj0;#jVxc(z)j\";jij2memj0log\u00121 +e\u0010max\n1\u0000e\u0010min\u0013\n(41)\nIn this equation, the squared interaction matrix elements Eqs. (36){(38) are used, and we have used the following\nde\fnitions. First, the Fermi-Dirac funtion f(\u0018) = [1 + exp( \f\u0018)] and the hole chemical potential \u0016, further\n\u0010min;max=\f\u00121\nmj0p2\nmin;max\u0000\u0016j0\u0013\n(42)\nwhere\u0016j0is the reduced mass of an electron and a hole in band j0. Further,\npmin;max=\f\f\f\f\fs\n\u0016\u00141\nmj(k2+q2+ 2kqz) +1\nmek2\u0000Mj0\nmeq2\u0015\n\u0000\u0016j0\nmeq\f\f\f\f\f(43)\nwithMj0=me+mj0.\nREFERENCES\n[1] Kikkawa, J. M. and Awschalom, D. D., \\Electron spin and optical coherence in semiconductors,\" Physics\nToday 32, 33 (1999).\n[2]\u0014Zuti\u0013 c, I., Fabian, J., and Das Sarma, S., \\Spintronics: Fundamentals and applications,\" Reviews of Modern\nPhysics 76, 323 (2004).[3] Kikkawa, J. M. and Awschalom, D. D., \\Resonant spin ampli\fcation in n-type GaAs,\" Physical Review\nLetters 80, 4313 (1998).\n[4] Sandhu, J. S., Heberle, A. P., Baumberg, J. J., and Cleaver, J. R. A., \\Gateable suppression of spin\nrelaxation in semiconductors,\" Physical Review Letters 86, 2150 (2001).\n[5] Furis, M., Smith, D. L., Crooker, S. A., and Reno, J. L., \\Bias-dependent electron spin lifetimes in n-gaas\nand the role of donor impact ionization,\" Applied Physics Letters 89, 102102 (2006).\n[6] Crooker, S. A., Awschalom, D. 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(2004).\n[21] Sch afer, W. and Wegener, M., Semiconductor Optics and Transport Phenomena , Springer (2002).\n[22] Binder, R. and Koch, S. W., \\Nonequilibrium semiconductor dynamics,\" Prog. Quantum Electron-\nics19(4/5), 307 (1995).\n[23] Pikus, G. E. and Bir, G. L., \\Exchange interaction in excitons in semiconductors,\" Sov. Physics JETP 33,\n208 (1971).\n[24] Maialle, M. Z., Silva, E. A. D. E., and Sham, L. J., \\Exciton spin dynamics in quantum-wells,\" Physical\nReview B 47, 15776 (1993).\n[25] Cohen-Tannoudji, C., Diu, B., and Lalo e, F., Quantum Mechanics , Wiley (1977).\n[26] Collet, J. H., \\Screening and exchange in the theory of the femtosecond kinetics of the electron-hole plasma,\"\nPhysical Review B 47, 10279 (1993).\n[27] Hilton, D. J. and Tang, C. L., \\Optical orientation and femtosecond relaxation of spin-polarized holes in\nGaAs,\" Physical Review Letters 89, 146601 (2002)."    },    {        "title": "1410.1597v1.Y3Fe5O12_Spin_Pumping_for_Quantitative_Understanding_of_Pure_Spin_Transport_and_Spin_Hall_Effect_in_a_Broad_Range_of_Materials.pdf",        "content": "1\nY3Fe5O12 Spin Pumping for Quantitative Understanding of Pure Spin \nTransport and Spin Hall Effect i n a Broad Range of Materials (I nvited) \nChunhui Du, Hailong Wang, and P. Chr is Hammel, Fengyuan Yang \nDepartment of Physics, The Ohio S tate University, Columbus, OH,  43210, USA \n \n \nAbstract \nUsing Y 3Fe5O12 (YIG) thin films grown by our sputtering technique, we study d ynamic \nspin transport in nonmagnetic (NM), ferromagnetic (FM) and anti ferromagnetic (AF) materials by \nferromagnetic resonance (FMR) spin pumping.  From both inverse spin Hall effect (ISHE) and \ndamping enhancement, we determine the spin mixing conductance a nd spin Hall angle in many \nmetals.  Surprisingly, we observe robust spin conduction in AF insulators excited by an adjacent \nYIG at resonance.  This demonstrates that YIG spin pumping is a  powerful and versatile tool for \nunderstanding spin Hall physics, s pin-orbit coupling (SOC), and  magnetization dynamics in a \nbroad range of materials.    \n  2\nI. Introduction \nFMR spin pumping is an emerging technique for dynamic injection  of a pure spin current \nfrom a FM into a NM without an accompanying charge current [1-1 6], which offers the potential \nto enable low energy cost, high efficiency spintronics.  The pe rformance of these future spin-based \napplications relies on the efficiency of spin transfer across t he FM/NM interfaces [3-16].  YIG has \nbeen widely used in microwave applications and is particularly desirable for dynamic spin \ntransport due to its exceptiona lly low damping [17].  Its insul ating nature also allows clean \ndetection of the pure spin curre nt by ISHE in the NM.  Here we present our recent results on FMR \nspin pumping using YIG thin films grown by a sputtering techniq ue that we developed for \ndeposition of high quality epitaxi al films of complex materials  [12, 18-22].  In particular, we focus \non the characterization of structural and magnetic quality of t he YIG films, spin pumping from \nYIG into a number of metals and ISHE detection of the spin curr ents, determination of interfacial \nspin mixing conductance ( ↓↑݃ )and spin Hall angle ( SH), and spin transport i n an AF insulator. \nII. Crystal and Magnetic Structures of Y 3Fe5O12  \nY3Fe5O12 has a cubic crystal structure with space group Ia-3d  as shown in Fig. 1.  The \ncubic unit cell has a lattice constant of a = 12.376 Å and contains 8 formula units (f.u.) with 160 \natoms, of which only the Fe3+ ions carry magnetic moment (5 B each).  Of the 40 Fe3+ ions in a \nunit cell, 16 are on octahedral sites and 24 are on tetrahedral  sites.  Each octahedral Fe is connected \nto 6 tetrahedral Fe and each tetrahedral Fe is connected to 4 o ctahedral Fe through corning sharing \nan oxygen, resulting in an intert wining octahedron-tetrahedron network.  The magnetic moments \nof all the octahedral Fe are aligned in parallel.  The same is true for all the tetrahedral Fe, but in \nopposite direction to that of the octahedral Fe, resulting in a  ferrimagnetic order.  The octahedron-\ntetrahedron Fe network leads to very low magnetic anisotropy, w hich in turn leads to exceptionally 3\nlow damping in YIG.  We will discuss later the critical importa nce of preserving the stoichiometry \nand ordering of YIG, both in the bulk of the films and at the i nterfaces, for achieving high-\nefficiency spin pumping.  \nIII. Experimental Details \nThe attractive properties of YIG such as low damping and magnet ic softness require \nstoichiometric samples with a high degree of crystalline perfec tion and magnetic ordering.  Liquid-\nphase epitaxy (LPE) has been the dominant technique for growing  YIG epitaxial films and single \ncrystals in the past few decades [23].  Pulsed laser deposition  (PLD) has recently been used to \ngrow epitaxial YIG thin films [24-26].  Using a new off-axis sp uttering approach, we deposit YIG \nthin films on Gd 3Ga5O12 (GGG) substrates in a custom ultrahigh vacuum sputtering syste m [12, \n27].  Our sputtering technique is different from conventional o n-axis sputtering [28] and high \npressure off-axis sputtering [ 29].  For on-axis sputtering geom etry, the energetic bombardment of \nthe sputtered atoms limits the  crystalline quality and ordering  of the films.  For h igh pressure off-\naxis sputtering at ~200 mTorr [ 29], the frequent scattering by the sputter gas typically results in \noff-stoichiometry in the films, degrading the film quality of c omplex materials [22].  Our low-\npressure (~10 mTorr) off-axis sputtering technique simultaneous  minimizes the bombardment \ndamage and maintains the desired stoichiometry in the films.   \nW e  d e t e r m i n e  t h e  c r y s t a l l i n e  q u a l i t y  o f  t h e  Y I G  f i l m s  b y  a  B r u k er triple-axis x-ray \ndiffractometer (XRD) and measure the surface smoothness by a Br uker atomic force microscope \n(AFM).  FMR absorption and spin pumping measurements are perfor med using a Bruker electron \nparamagnetic resonance (EPR) spectrometer in a cavity at a radi o-frequency (rf) f = 9.65 GHz.  We \nmeasure the frequency dependencies of the FMR linewidth using a  microstrip transmission line at \na frequency range bet ween 10 and 20 GHz.  Magnetic hysteresis l oops of our YIG films are taken 4\nby a LakeShore vibrating sample magnetometer (VSM). \nIV. Results and Discussion \nIV(a). Structural and magnetic quality of the YIG films  \nFigure 2(a) shows a 2 θ- XRD scan of a 50-nm YIG films deposited on GGG (111), which \nexhibits clear Laue oscillations , reflecting the high crystalli ne quality of the YIG film.  The out-\nof-plane lattice constant of the YIG film is determined to be c = 12.383 Å which is identical to the \nlattice constant of GGG ( a = 12.383 Å) and only 0.06% large r than the bulk value ( a = 12.376 Å) \nof YIG.  The full-width-at-half-maximum (FWHM) of a XRD rocking  curve is a widely used \nmeasure of crystalline quality for epitaxial films.  The left i nset to Fig. 2(a) give a FWHM of \n0.0072 for the first Laue oscillation peak to the left of YIG (444), demonstrating the state-of-the-\nart crystalline uniformity of the YIG film.  The right inset to  Fig. 2(a) shows an x-ray reflectometry \n(XRR) scan of a YIG/Pt bilayer with two periods of oscillations , corresponding to the 34-nm YIG \nand 4.1-nm Pt layers.  A fit to the XRR scan gives a YIG/Pt int erfacial roughness of 0.22 nm, \nindicating the sharpness of the interface.  The smooth surface of the YIG films is confirmed by the \nAFM image in Fig. 2(b) from which we obtain a root-mean-square (rms) roughness of 0.10 nm \nover an area of 10 m  10 m.  Figure 2(c) shows a room temperature in-plane magnetic \nhysteresis loop for a 20-nm YIG f ilm, which exhibits a very sma ll coercivity ( Hc) of 0.35 Oe and \nexceptionally sharp reversal: the magnetic switching is complet ed within 0.1 Oe.  This indicates \nthe high magnetic uniformity of the YIG film.  The high crystal line and magnetic uniformity of \nour YIG films provide the material platform for spin pumping st udy of a broad range of materials. \nIV(b). FMR spin pumping of YIG/metal bilayers  \nFigure 3(a) shows the derivative of a FMR absorption spectrum o f a YIG(20 nm) film at \nan rf power Prf = 0.2 mW in an in-plane field, wh ich gives a peak-to-peak line width H = 7.4 Oe.  5\nFigure 3(b) illustrates the geomet ry for ISHE detection of spin  pumping in a FMR cavity where a \nDC magnetic field H is applied in the xz plane at an angle H with respect to the film surface and \nan rf field hrf is applied along the y-axis.  All samples are approximately 5 mm long and 1 mm \nwide.  An ISHE voltage ( VISHE) vs. H plot for a YIG/Pt(5 nm) bilayer at Prf = 200 mW is shown in \nFig. 3(c) which exhibits VISHE = 1.74 mV and antisymmetric dependence on H as expected from \nthe ISHE.  VISHE h a s  a  l i n e a r  Prf dependence [left inset to Fig. 3(c)], indicating that the spin  \npumping is in the linear regime up to 200 mW.  The right inset to Fig. 3(c) shows a sinusoidal \nbehavior of the normalized VISHE as a function of H, which is expected from spin pumping. \nWe measure the ISHE in a broad range of metals under the same F MR conditions.  Figure \n4 shows our results for Cr(5 nm), Fe(10 nm), Co(10 nm), Ni 80Fe20 [Py(5 nm)], Ni(10 nm), Cu(10 \nnm), Nb(10 nm), Ag(5 nm), Ta(5 nm), W(5 nm), Pt(5 nm), and Au(5  nm) on YIG [12, 14, 16, 30], \nwhich give VISHE = -5.01 mV , -13.5 V , -23.5 V , +70.5 V , +42.1 V , +1.06 V , -666 V , +1.60 \nV , -5.10 mV , -5.26 mV , +3.04 mV , and +72.6 V , respectively.  The sign and magnitude of VISHE \nreflects the variation in SH, which can be calculated from [5, 8, 10, 11, 14],   \n୍ܸୗୌൌିఏౄ\nఙొ௧ొߣୗୈtanhቀ௧ొ\nଶఒీቁܲܮ݂↓↑݃ቀఊ౨\nସగఈቁଶ\n,     ( 1 )  \nwhere e is the electron charge, ߪ, ݐ, ߣୗୈ, and L are the conductivity, thickness, spin diffusion \nlength, and sample length of the NM layer, respectively, hrf = 0.25 Oe [14] in our FMR cavity at \nPrf = 200 mW,  is the gyromagnetic ratio, and  is the Gilbert damping constant of YIG.  The \nfactor P = 1.21 [14] arises from the el lipticity of the magnetization p recession.   \nEq. (1) indicates that calculation of SH relies on accurate measurement of ↓↑݃ ,which can \nvary from sample to sample.  In literatures, it is uncommon tha t VISHE and ↓↑݃ are measured \nindependently.  Our thin YIG film s with narrow FMR linewidth an d low damping allows us to \nindependently determine ↓↑݃ from the damping enhancement [2, 5, 6, 8, 11, 14], 6\n↓↑݃ൌସగெ౩௧ౕృ\nఓాሺଢ଼୍ୋ/െଢ଼୍ୋሻ      ( 2 )  \nwhere ݃ ,ߤ, ܯୱ and ݐଢ଼୍ୋ are the Landé ݃ factor, Bohr magneton, satu ration magnetization and \nthickness of the YIG films, respectively.  We obtain the dampin g constants from the frequency \ndependencies of the FMR linewidth: Δܪ ൌ Δܪ ୧୬୦ସగఈ\n√ଷఊ [31], where Hinh is the inhomogeneous \nbroadening as shown in Fig. 5 for several YIG-based structures.   Using least-squares fits to the \ndata in Fig. 5, we obtain the damping constants [ ଢ଼୍ୋൌ (8.7 േ 0.6)  10-4 for a bare YIG film] for \neach structure.  Table I lists the values of ↓↑݃ for 13 metals on YIG that we have studied [12, 14, \n16, 30], among which the YIG/Pt exhibits the highest ↓↑݃ of (6.9 േ 0.6)  1018 m-2.   \nRegarding SD, since the term ߣୗୈtanh\tሺ௧ొ\nଶఒీሻ in Eq. (1) is virtually insensitive to the value \nof SD when SD  tNM [14], we only need to measure ߣୗୈ for materials with short ߣୗୈ.  For those \nwith long SD, such as Cu, we use values repor ted in literature.  Figure 6(a ) plots Pt thickness ( tPt) \ndependence of the ISHE-induced charge current Ic = VISHE/Rw for YIG/Pt bilayers, where R and w \nare the resistance and width of the YIG/Pt samples.  Given that  Ic is proportional to the pure spin \ncurrent pumped into Pt, we obtain ߣୗୈ = 7.3  0.8 nm for Pt by fitting to ౄు\nோ௪∝ߣୗୈtanhቀ௧ౌ౪\nଶఒీቁ \n[32].  Similarly, we calculate ߣୗୈ = 1.9 േ 0.2, 2.1 േ 0.2, and 13.3 േ 2.1 nm for Ta, W, and Cr, as \nshown in Figs. 6(b), 6(c), and 6(d), respectively.  Using the o btained values of ↓↑݃ and ߣୗୈ, we \ncalculate SH for the 13 metals (Table I) [14, 16, 30], which reveal the imp ortant role of d-electron \nconfiguration of trans ition metals in spin Hall effect (SHE).  This is consistent with the prediction \nof Tanaka et al. [33] who calculated the spin  Hall conductivitie s (SHC) of the  4d (5d) transition \nmetals by considering the role of the total number of 4 d (5d) and 5 s (6s) electrons.  We list in Table \nI the total number of d and s electrons, nd+s, in the conduction bands and plot SH vs. nd+s for four \n3d and four 5 d metals in Fig. 7.  The calculated SHC of 5 d metals [33] are also shown for 7\ncomparison.  Both 3 d and 5 d metals show the same systematic behavior in SH w h i c h  v a r i e s  \nsignificantly both in sign and magnitude as a function of nd+s.  While the behavior of SH in 4d and \n5d transition metals are underst ood theoretically [33, 34], theor etical calculations for spin Hall \neffect in 3 d metals are needed to explain the observed ISHE in 3 d metals.  \nIV(c). Spin transport in antiferromagnetic insulators  \nFollowing the spin pumping study from YIG into metals, we inves tigate spin transport in \nYIG/insulator/Pt trilayer system s, where the insulator spacer i s either a diamagnet, SrTiO 3 (STO), \nor an AF, NiO.  Figure 8(a) shows a semi-log plot of VISHE as a function of the SrTiO 3 thickness \n(tSTO) in YIG/STO/Pt(5 nm ) trilayers, which ex hibit a clear exponent ial decay with a decay length \nof  = 0.19 nm [13].  The short decay length is comparable to the i nter-atomic spacing, which \nindicates that the Pt conducti on electrons tunnel across the ST O barrier and exchange couple to \nthe precessing YIG magnetization to acquire spin polarization.  This result demonstrates that \nexchange coupling is the dominant mechanism responsible for spi n pumping [2, 13].  More \ninterestingly, when an AF insulator NiO is used, spin currents can propagate over a much longer \ndistance, as shown in Fig. 8(b) f or YIG/NiO/Pt trilayers which exhibits a decay length of  = 9.4 \nnm [35].  Strikingly, we observe a significant enhancement in I SHE voltages at small NiO \nthicknesses tNiO: VISHE increases from 0.604 mV for the YIG/Pt bilayer to 1.30 mV for the \nYIG/NiO(2 nm)/Pt trilayer.  This is in drastic contrast to the suppression of VISHE by more than \ntwo orders of magnitude when a 1-nm SrTiO 3 is inserted between YIG and Pt.  The surprising \nenhancement of VISHE, long spin decay length in YIG/N iO/Pt structures point toward the AF nature \nof NiO as the underlying mechanism for the observed robust spin  transport in NiO [35].  Since the \nrange of tNiO covers NiO layers with orderi ng temperatures [36] both above ( large tNiO) and below \n(small tNiO) room temperature, this indicates that both AF ordered and AF fluctuating [37] spins in 8\nNiO can be excited by exchange coupling to the precessing YIG m agnetization and efficiently \ntransporting spin current  over a long distance.  \nIV(d). Key factors contributi ng to large ISHE signals  \nLastly, we comment below on seve ral important factors that are critical for achieving high \nspin current and large ISHE signa ls.  (1) The spin current gene rated in YIG/NM bilayers depends \non how out of equilibrium that the YIG magnetization can be exc ited, which can be described by \nthe precession cone angle [2, 5, 8].  This demands YIG films wi th low damping, a widely accepted \ncriterion in spin pumping.  (2) However, the measured value of  is a “bulk” property of the YIG \nfilm, while spin pumping is determined predominantly by the YIG /NM interface because of the \nexchange mechanism with an effective distance of ~0.2 nm [2, 13 ].  As an example, the insertion \nof a 1-nm SrTiO 3 layer between YIG and Pt reduces VISHE by a factor of 256 [Fig. 8(a)].  This \nsuggests that a large precession cone angle persisting close to  the interface with Pt is critical for \nhigh-efficiency spin transfer.  Thus, the YIG film needs to mai ntain correct stoichiometry, high \ncrystalline quality, and uniform  magnetic ordering from inside to the top atomic layer of the YIG \nfilm.  Post-deposition treatments, such as polishing, etching, or sometimes annealing, could \njeopardize the YIG surface.  (3) Spin mixing conductance is a p henomenological parameter that \ndescribes the quality of the YIG/NM interface in conducting spi n currents.  The values of ↓↑݃ vary \nsignificantly for the same YIG/Pt  structure made by different t echniques and research groups due \nto variation of the interface quality [6, 10, 12].  Since chara cterizing the chemical, structural, and \nmagnetic uniformity of the YIG surface is rather challenging, I SHE voltage is a good quantity for \ncomparing the spin pumping effi ciency and YIG/Pt interface qual ity.  After all, VISHE is a direct \nmeasure of the pure spin currents pumped into Pt.  (4) We find that oxygen content during the YIG \ngrowth is critical for its spin pumping performance.  Our best YIG thin films for spin pumping can 9\nonly be grown within a narrow window of the oxygen partial pres sure, outside which, the ISHE \nsignals degrade dramatically.  Som e of these points are support ed by experimental evidence and \nsome are our speculations.  More thorough characterizations of the YIG/NM interfaces are needed \nto understand the nature of spin t ransfer from dynamically exci ted YIG to metals. \nV. Conclusion \nEpitaxial YIG thin films provide a material platform for high-s ensitivity investigation of \ndynamic spin transport in a broad range of nonmagnetic, ferroma gnetic, and antiferromagnetic \nmaterials, be they conducting or insulating.  The phenomena unc overed in these materials and \nstructures provide guidelines fo r potential spintronics applica tions and enable further \nunderstanding of fundamental inter actions such as spin-orbit co upling and magnetic excitations.  \nVI. Acknowledgements \nThis work was primarily supported by the U.S. Department of Ene rgy (DOE), Office of \nScience, Basic Energy Sciences, under Award No. DE-FG02-03ER460 54 (FMR and spin pumping \ncharacterization) and No. DE-SC 0001304 (sample synthesis and ma gnetic characterization).  This \nwork was supported in part by the Center for Emergent Materials , an NSF-funded MRSEC under \naward No. DMR-1420451 (structural characterization).  Partial s upport was provided by Lake \nShore Cryogenics Inc. and the Na noSystems Laboratory at the Ohi o State University. \n  10\nReference:  \n1. I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys . 76, 323 (2004). \n2. Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin,  Rev. Mod. Phys.  77, 1375 \n(2005). \n3. A. Azevedo, L. H. Vilela Leão, R. L. Rodriguez-Suarez, A. B. Ol iveira, and S. M. Rezende, J. \nAppl. Phys. 97, 10C715 (2005). \n4. Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizu guchi, H. Umezawa, H. \nKawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature  464, 262 (2010). \n5. O. Mosendz, J. Pearson, F. Fradin, S. Bader, and A. Hoffmann, Appl. Phys. Lett.  96, 022502 \n(2010). \n6. B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y.-Y . Song, Y. Y. Sun, and M. 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Interfacial spin mixing conductance ↓↑݃ ,spin diffusion length ߣୗୈ, spin Hall angle SH, \nand total number of d and s electrons in conduction bands, nd+s, for metals and alloys studied by \nour YIG spin pumping. \nStructure ↓↑݃(m-2) ߣୗୈ (nm)  SH nd+s \nYIG/ Ti (3.5 േ 0.3)  1018 ~13.3 -(3.6 േ 0.4)  10-4 4 \nYIG/ V (3.1 േ 0.3)  1018 ~13.3 -(1.0 േ 0.1)  10-2 5 \nYIG/ Cr (8.3 േ 0.7)  1017 13.3 -(5.1 േ 0.5)  10-2 6 \nYIG/ Mn (4.5 േ 0.4)  1018 ~13.3 -(1.9 േ 0.1)  10-3 7 \nYIG/ FeMn  (4.9 േ 0.4)  1018 3.8 [38] -(7.4 േ 0.8)  10-5 7.5 \nYIG/Cu/ Py (6.3 േ 0.5)  1018 1.7 [30] (2.0 േ 0.5)  10-2 9.6 \nYIG/Cu/ Ni (2.0 േ 0.2)  1018 3.2 [16] (4.9 േ 0.5)  10-2 10 \nYIG/ Cu (1.6 േ 0.1)  1018 500 [39] (3.2 േ 0.3)  10-3 11 \nYIG/ Ag (5.2 േ 0.5)  1017 700 [40] (6.8 േ 0.7)  10-3 11 \nYIG/ Ta (5.4 േ 0.5)  1018 1.9 -(6.9 േ 0.6)  10-2 5 \nYIG/ W (4.5 േ 0.4)  1018 2.1 -(1.4 േ 0.1)  10-1 6 \nYIG/ Pt (6.9 േ 0.6)  1018 7.3 (1.0 േ 0.1)  10-1 10 \nYIG/ Au (2.7 േ 0.2)  1018 60 [39] (8.4 േ 0.7)  10-2 11 \n \n  15\nFigure captions: \nFigure 1.   Schematic of the cubic unit ce ll of YIG garnet structure.  \nFigure 2.   (a) Semi-log 2 θ- XRD scan of a 50-nm YIG film gr own on a GGG (111) substrate, i n \nwhich the clear satellite peaks are Laue oscillations.  Left in set: XRD rocking curve of the first \nsatellite peak on the left of YIG (444) peak showing a FWHM of 0.0072.  Right inset: X-ray \nreflectometry spectrum (red) of a YIG(34 nm)/Pt(4.1) bilayer on  GGG, where the blue curve is a \nfit by Bruker Leptos.  (b) AFM image of a YIG film with a rough ness of 0.10 nm over an area of \n10 m  10 m.  (c) A room temperature in-plane magnetic hysteresis loop of  a 20-nm YIG film \nwhich gives a small coercivit y of 0.35 Oe and very sharp magnet ic reversal (switching completed \nwithin 0.1 Oe). Figure 3.   (a) Room-temperature derivativ e of a FMR absorption spectrum of a YIG film with an \nin-plane field at P\nrf = 0.2 mW, which gives a peak-to-peak linewidth of 7.4 Oe.  (b)  Schematic of \nexperimental setup for ISHE measurements.  (c) VISHE vs. H spectra at θH = 90 for a YIG(20 \nnm)/Pt(5 nm) bilayer.  Left inset: rf power dependence of VISHE for the YIG/Pt bilayer, where the \nblue line is a least-squares fit to the data.  Right inset: ang ular dependence of the normalized VISHE \nfor the YIG/Pt bilayer, where the red curve is a sin θH fit. \nFigure 4.  VISHE vs. H – Hres spectra of (a) YIG/Cr(5 nm), (b) YIG/Fe(10 nm), (c) YIG/Co(10 nm), \n(d) YIG/Py(5 nm), (e) YIG/Ni(10 nm), (f) YIG/Cu(10 nm), (g) YIG /Nb(10 nm), (h)YIG/Ag(5 nm), \n(i) YIG/Ta(5 nm), (j) YIG/W(5 nm), (k) YIG/Pt(5 nm), and (l) YI G/Au (5 nm) bilayers at θH = \n90(red) using Prf = 200mW.   \nFigure 5.   (a) Frequency dependencies of F MR linewidth of a bare YIG fil m, YIG/Cr, YIG/Cu, \nYIG/Au, YIG/W, YIG/Ta, YIG/Pt  bilayers, and a YIG/Cu/Py trilaye r.  \nFigure 6.   ISHE-induced charge current ( VISHE/R) normalized by sample width w of (a) YIG/Pt, 16\n(b) YIG/Ta, (c) YIG/W, and (d) YIG/Cr bilayers as a function of  the metal layer thicknesses, from \nwhich the spin diffusion length of SD = 7.3 േ 0.8, 1.9 േ 0.2, 2.1 േ 0.2, and 13.3 േ 2.1 nm, \nrespectively, are obtained. \nFigure 7.   The obtained spin Hall angles ( SH) as a function of the total number of d and s electrons, \nnd+s, for selected 3d and 5d metals.  The theoretical calculations of spin Hall conductivity (SHC) \nfor 5d metals by Tanaka et al . [ref] are also shown (open green  circles). \nFigure 8.   (a) Semi-log plot of VISHE as a function of the SrTiO 3 barrier thickness for \nYIG/SrTiO 3/Pt(5 nm) trilayers.  (b) Semi- log plot of the NiO thickness de pendencies of the ISHE \nvoltages for YIG(20 nm)/NiO( tNiO)/Pt(5 nm) trilayers. Inset: VISHE as a function of NiO thickness \nfrom 0 to 10 nm.  \n  17\n \n \nFigure 1 . \n  \n18\n \n \n \nFigure 2.  \n  100102104106108\n50 51 52 53Intensity (c/s)\n2 (deg)(a) YIG(50 nm)/GGG\nGGG(444)\nYIG(444)\n-101\n-2 -1 0 1 2M/Ms\nH (Oe)(d)YIG\n(b)25 25.05\n (deg)101103105\n2468 1 0\n2 (deg)experiment\nfit19\n \n \nFigure 3  \n \n  -2-1012\n2500 2600 2700dIFMR/dH (a.u.)\nH (Oe)H = 7.4 OeYIG(20 nm) (a)\nH = 90o\n-2-1012\n-4000 -2000 0 2000 4000VISHE (mV)\nH (Oe)H = 90oYIG/Pt\n(c)012\n0 100 200VISHE (mV)\nPrf (mW)\n-101\n0 90 180 270 360Norm.  VISHE\nH (deg)\n(b)20\n \n \nFigure 4.    -6000-4000-20000\nYIG/Cr(5 nm)(a)\n-15-10-50\nYIG/Fe(10 nm)(b)\n-30-20-100\n(c)\nYIG/Co(10 nm)020406080 YIG/Py(5 nm) (d)\n01020304050YIG/Ni(10 nm)\n(e)VISHE (V)\n00.51\n(f)YIG/Cu(10 nm)\n-800-600-400-2000\nYIG/Nb(10 nm)(g)\n012(h)YIG/Ag(5 nm)\n-6000-4000-20000\n-100 0 100(i)\nYIG/Ta(5 nm)\nH - Hres (Oe)-6000-4000-20000\n-100 0 100\nH - Hres (Oe)(j)\nYIG/W(5 nm)0100020003000\n-100 0 100YIG/Pt(5 nm) (k)\nH - Hres (Oe)020406080\n-100 0 100(l)YIG/Au(5 nm)\nH - Hres (Oe)21\n \n \nFigure 5.  \n  01020304050\n0 5 10 15 20H (Oe)\nf (GHz)YIG/CuYIG/Ta\nYIG/Cu/Py\nYIG/Au\nYIG/CrYIG/W\nYIGYIG/Pt22\n \n \nFigure 6.  \n  012345\n0 1 02 03 04 05 0|VISHE|/Rw (mA/m)\ntPt (nm)SD = 7.3 nm(a)Pt\n00.20.40.6\n0 5 10 15|VISHE|/Rw (mA/m)\ntTa (nm)SD = 1.9 nm(b)Ta\n00.20.40.6\n0 5 10 15|VISHE|/Rw (mA/m)\ntW (nm)SD = 2.1 nm(c)W\n00.511.5\n0 20 40 60 80 100|VISHE|/(Rw) (mA/m)\ntCr (nm)SD = 13.3 nm(d)Cr23\n \n \nFigure 7  \n  -0.15-0.1-0.0500.050.1\n-0.1-0.0500.05\n56789 1 0 1 1nd+sSHSHC (103 -1cm-1)\nTheoryExperiment, 5 dExperiment, 3 dTaWPt Au\nV\nCrNi\nCu24\n \n \nFigure 8  \n 101102103\n0 0.2 0.4 0.6 0.8 1VISHE (V)\n tSTO (nm)YIG/ STO (tSTO)/Pt\n = 0.19 nm(a)\n101102103\n0 1 02 03 04 05 06 07 0VISHE (V)\n9.4 nm(b)\nYIG/ NiO(tNiO)/Pt\n tNiO (nm)050010001500\n02468 1 0tNiO (nm)"    },    {        "title": "1702.04918v1.Microscopic_theory_of_electrically_induced_spin_torques_in_magnetic_Weyl_semimetals.pdf",        "content": "Microscopic theory of electrically induced spin torques in magnetic Weyl semimetals\nDaichi Kurebayashi1,\u0003and Kentaro Nomura1\n1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n(Dated: February 17, 2017)\nWe theoretically study electrical responses of magnetization in Weyl semimetals. The Weyl semimetal is a\nnew class of topological semimetals, possessing hedgehog type spin textures in momentum space. Because\nof this peculiar spin texture, an interplay of electron transport and spin dynamics might provide new method\nto electrical control of magnetization. In this paper, we consider the magnetically doped Weyl semimetals,\nand systematically study current- and charge-induced spin torque exerted on the local magnetization in three-\ndimensional Dirac-Weyl metals. We determine all current-induced spin torques including spin-orbit torque,\nspin-transfer torque, and the so-called \f-term, up to first order with respect to spatial and temporal derivation and\nelectrical currents. We find that spin-transfer torque and \f-term are absent while spin-orbit torque is proportional\nto the axial current density. We also calculate the charge-induced spin torque microscopically. We find the\ncharge-induced spin torque originates from the chiral anomaly due to the correspondence between spin operators\nand axial current operators in our model.\nI. INTRODUCTION\nElectrical manipulation of magnetization is of major inter-\nest in the field of spintronics for achieving low-energy con-\nsumption electronic devices1. In ferromagnetic metals, spin-\npolarized currents are widely used to control magnetization\ndynamics such as spin-transfer torque and spin pumping2–5.\nUtilizing charge currents, however, su \u000bers from the Joule\nheating, limiting the energy e \u000eciency. Recently, the applica-\ntion of the topological properties of materials has drawn much\ninterest to achieve a more e \u000ecient manipulation of the mag-\nnetization. For instance, in magnetically doped topological\ninsulators and ferromagnetic insulators deposited on the sur-\nface of a topological insulator6,7, electrical manipulation of\nmagnetic textures8,9, electric field- or current-induced magne-\ntization switching10–13, and spin-charge conversion14–16have\nbeen considered theoretically and experimentally. Topologi-\ncal materials are often realized by strong spin-orbit coupling,\nthus the spin degrees of freedom couple to the momenta,\nknown as the spin-momentum locking. Spin-momentum lock-\ning may provide a new means to achieve a reliable and less\ndissipative control of magnetization.\nAs a new class of topological materials, Weyl semimet-\nals are intensively researched. Weyl semimetals are charac-\nterized by pairs of bulk gapless points each distinguished by\ntheir chirality.17,18Close to the gapless points, excitations are\ndescribed by a three-dimensional linear dispersion which is\nanalogous to the Weyl fermion in high-energy physics. To\nrealize Weyl semimetals, at least time-reversal or inversion\nsymmetry must be broken.19,20Weyl semimetals whose time-\nreversal symmetry is broken by the magnetic ordering of lo-\ncal moments are especially promising for spintronics appli-\ncations, because they possess both magnetic and topological\nproperties. To realize magnetic Weyl semimetals, there are a\nlot of theoretical proposals such as pyrochlore iridates21, mul-\ntilayers of topological insulators and normal insulators22, and\nmagnetically doped topological insulators23–25Recently, the\nnoncollinear antiferromagnets Mn 3X (X =Sn;Ge) were theo-\nretically and experimentally proposed as time-reversal broken\nWeyl semimetals.In our previous papers26,27, we found that spin torque can be\ngenerated by an applied gate voltage in a magnetic field, so-\ncalled charge-induced spin torque, and proposed a spintronic\ndevice incorporating magnetic Weyl semimetals. Our deriva-\ntion was phenomenological, based on the peculiar coupling\nbetween magnetization and charge density emerging from the\ntopological nature of Weyl semimetals. The microscopic ori-\ngin of the e \u000bect still has to be clarified. One of the main fo-\ncuses of this paper is to give the physical understanding of the\ncharge-induced spin torque. In addition to the charge-induced\nspin torque, the magnetization responses to the applied cur-\nrents are also not well known. In this paper, we study mag-\nnetization dynamics in magnetic Weyl semimetals by con-\nsidering spin torques. We present a microscopic calculation\nof spin torques, specifically spin-orbit torque, spin-transfer\ntorque,\f-term, and the charge-induced torque in our contin-\nuum model for magnetic Weyl semimetals. Consequently,\nwe obtain the analytical form of the spin-orbit torque and\nthe charge-induced spin torque in magnetic Weyl semimetals.\nThe results concerning the charge-induced spin torque is con-\nsistent with that proposed in Ref. 26, and is understood by the\nchiral anomaly. We also find that the spin-transfer torque and\n\f-term are absent in magnetic Weyl semimetals. The absence\nof spin-transfer torque and \f-term is understood by the corre-\nspondence of transport and spin phenomena which our model\npossesses.\nII. FORMALISM\nMagnetization dynamics is described by the Landau-\nLifshitz-Gilbert equation as\ndˆM\ndt=\r0H\u0002ˆM+\u000b0ˆM\u0002dˆM\ndt+Te; (1)\nwhere\r0is the gyromagnetic ratio, His an external magnetic\nfield, and\u000b0is the damping constant. The e \u000bect of the back-\nground of conduction electrons is described in terms of spin\ntorques as\nTe(r)=JSˆM\u0002hˆ\u001b(r)i (2)arXiv:1702.04918v1  [cond-mat.mes-hall]  16 Feb 20172\nwherehˆ\u001b(r)iis the spin density of conduction electrons.28The\nspin torques can be obtained by calculating hˆ\u001b(r)i. In a weak\nspin-orbit coupled system, the spin density can be expanded\nas\nhˆ\u001b(r)i=a0dˆM\ndt+b0ˆM\u0002dˆM\ndt+(a\u0001r)ˆM+ˆM\u0002(b\u0001r)ˆM\n(3)\nin the first order of time derivative and spatial gradient. The\nfirst two terms are a correction to the Gilbert damping and\nthe spin Berry phase term renormalizing the magnitude of\nspin, respectively. The last two terms describe the current-\ninduced spin torques, the spin-transfer torque and so-called \f-\nterm. With conventional Schr ¨odinger like electrons, the spin-\ntransfer torque and the \f-term are proportional to a spin cur-\nrentjS, described by TSTT/(jS\u0001r)ˆMandT\f/ˆM\u0002(jS\u0001r)ˆM,\nrespectively.29In strong spin-orbit coupled systems, other\ntypes of current-induced spin torques are expected. For ex-\nample, in two-dimensional Rashba systems or surface of the\nthree-dimensional topological insulators, spin-orbit torque is\nexpected due to spin-momentum locking, TSO/ˆz\u0002jwhere\nˆzis the normal vector perpendicular to the surface and jis an\nelectric current.11,12\nIn addition to current-induced torques, the spin torque gen-\nerated by a local charge density in the presence of an external\nmagnetic field is expected in anomalous Hall ferromagnets.26\nThere is a relationship, in anomalous Hall ferromagnets,\nbetween the charge density and the magnetization, \u001aind/\n\u001bAHEˆM\u0001B26,30,31where\u001aindis the induced charge-density\nand\u001bAHEis the magnitude of the anomalous Hall conductiv-\nity. The relation states that the local charge density increases\nwhen the magnetization is parallel to the external magnetic\nfield and decreases when it’s antiparallel. As an inverse e \u000bect\nof this relation, a spin torque is generated by modifying the lo-\ncal charge density to flip the magnetization. This spin torque\nis called charge-induced spin torque. Time-reversal symme-\ntry broken Weyl semimetals are known to host the anomalous\nHall e \u000bect32, so charge-induced spin torque is also to be ex-\npected.\nIn this paper, we calculate the non-equilibrium electronic\nspin polarization in magnetic Weyl semimetals generated by\nelectric voltage and currents. As a simple model, we con-\nsider a Weyl semimetal realized in a Dirac semimetal coupled\nto the local moments of magnetic dopants.24,25This might be\nrealized in magnetically doped topological insulator systems\nsuch as chromium-doped Bi 2Se3and (Bi;Sb) 2Te3, where in-\ncreasing chromium concentration reduces the strength of spin-\norbit coupling and leads to a topological phase transition into\na normal insulator phase with band gap closing.33,34For clar-\nify, we note that the Dirac semimetal considered here is not\na symmetry-protected Dirac semimetal such as Cd 2As3and\nNa3Bi.35–37The low-energy e \u000bective Hamiltonian of the mag-\nnetic Weyl semimetals22,31is\nHWSM=Z\ndr y(r)\u0014\nfvF\u001cz\u001b\u0001(\u0000ir+eA(r;t))\u0000EFg\u0000\u001cz\u0001\n+JSˆM(r;t)\u0001\u001b\u0015\n (r)+ˆVimp:(4)The first term describes a Dirac semimetal consisting of de-\ngenerate three-dimensional massless electrons where vFis the\nFermi velocity, \u001b=(\u001bx;\u001by;\u001bz) are Pauli matrices repre-\nsenting real spin operators, and \u001cz=\u00061 labels the chiral-\nity of two Weyl nodes. In the second and third terms, \u0001\nparameterizes the inversion-symmetry breaking, being finite\nin noncentrosymmetric materials, and Jis the exchange cou-\npling constant between conduction electron spins and local\nmagnetic moments, Sis spin of localized magnetic moments.\nˆM=(Mx;My;Mz) is a normalized directional vector of the\nmagnetic moments. The last term is the coupling to random\nimpurities given as ˆVimp=R\ndr y(r)Vimp(r) (r). We con-\nsider non-magnetic impurities Vimp(r)=P\niu(r\u0000Ri) with\nshort-range potential u(r)!u0\u000e(r). A Gaussian average is\ntaken for impurity positions as Vimp(r)Vimp(r0)=niu2\n0\u000e(r\u0000r0)\nwhere niis the concentration of impurities. Without impuri-\nties ( ˆVimp=0), the band structure of this Hamiltonian consists\nof two Weyl nodes with separation 2 JSˆM=vFin momentum\nspace and 2 \u0001in energy. For clarification, \u0001is di\u000berent from\nthe chiral chemical potential describing Fermi energy di \u000ber-\nence between two Weyl nodes.\nThe Green’s function of the Hamiltonian Eq. (4) is given as\nG\u0015(i!n;k)=[i!n+EF+\u0015\u0001\u0000\u0015vFk\u0001\u001b\u0000\u0006\u0015(i!n;k)]\u00001(5)\nwhere \u0006\u0015is the self energy induced by impurity scattering,\nand\u0015=\u00061 are the eigenvalues of \u001czlabeling the chiral-\nity. In the first Born approximation, the imaginary part of\nthe self energy is given as Im \u0006\u0015=niu2\n0P\nqImG\u0015(i!n;q)=\n\u0019niu2\n0D\u0015(EF)sgn(!n)\u0011\u0011\u0015sgn(!n) where D\u0015(EF) is the den-\nsity of states at the Fermi energy for a node with the chirality\n\u0015, and\u0011\u0015is the electron damping rate by impurity scattering.\nWe neglect the real part of the self energy because they can be\nabsorbed into the Fermi energy. In the following, we assume\nthat the damping rate is the same in both nodes, \u0011\u0011\u0011+=\u0011\u0000,\nand\u0011 << EFcorresponding to a weak impurity scattering\nregime. We calculate spin torques in the lowest order of \u0011.\nIII. CURRENT-INDUCED SPIN TORQUES\nIn this section, we discuss the current-induced magne-\ntization dynamics. For current-induced spin torques, we\nconsider small transverse magnetization fluctuations, \u000eM=\n(\u000eMx;\u000eMy;0);k\u000eMk<<1, in the presence of the static elec-\ntronic fields. Coupling to an electric field is introduced by\nH0=R\ndr j(r)\u0001A(r) where j(r)= y(r)ˆj (r) is a current\noperator. Here the saturation magnetization axis is taken in\nz-axis. We calculate the conduction electron’s spin density in\nresponse to an electric field within linear response theory as\nh\u001b\u000b(q)i=\u001fi\n\u000b(q)Ei;\n\u001fi\n\u000b(q)=lim\n\n!0Ki\n\u000b(q;\n +i0)\u0000Ki\n\u000b(q;0)\ni\n(6)\nwhere Ki\n\u000b(q;\n)=iR1\n0dtei\nth[\u001b\u000b(q;t);ji]iis a dynamical\ncorrelation function between the electron spin and the elec-\ntric currents. Here we consider a uniform electric field, thus3\n\tB\n\tC\nFIG. 1. Feynman diagrams for the current-induced spin torques, (a)\nthe spin-orbit torque and (b) the spin-transfer torque and the \f-term.\nthe wave vector qcomes only from the magnetization fluc-\ntuations. Furthermore, we assume that the spatial variation of\nthe magnetization fluctuation is small and expand the response\nfunction up to linear order in qand\u000eMas\nKi\n\u000b(q;\n)\u0019Ki\n\u000b(\n)+2Ki j\n\u000b\f(\n)qj\u000eM\f; (7)\nwhere indices \u000b;\f=1;2 denote direction in spin space and\ni;j=1\u00183 denote direction in momentum space. The first\nterm corresponds to the spin-orbit torque while the second\nterm corresponds to torque appearing when the magnetization\nis not uniform such as the spin-transfer torque and the \f-term.\nThe response functions are obtained by\nKi\n\u000b(i\nm)=kBTeX\nn;k;\u0015Tr\u0002\u001b\u000bG\u0015vi;\u0015G+\n\u0015\u0003(8)\nKi j\n\u000b\f(i\nm)=kBTeJSX\nn;k;\u0015Trh\n\u001b\u000bG\u0015v\u0015;jG\u0015\u001b\fG\u0015v\u0015;iG+\n\u0015\n\u0000\u001b\u000bG\u0000\n\u0015v\u0015;iG\u0015\u001b\fG\u0015v\u0015;jG\u0015i\n(9)\nwhere v\u0015is velocity operators of electrons with the chirality \u0015,\nandG\u0015;G\u0006\n\u0015denote G\u0015(i!n;k);G\u0015(i!n\u0006i\nm;k), respectively,\nand are expressed as Feynman diagrams in Fig.1.\nBefore moving on to the detail calculation of the response\nfunctions, let us examine Ki j\n\u000b\fwith symmetry arguments. Af-\nter some calculation shown in Appendix A, the response func-\ntion can be written as\nKi j\n\u000b\f=A\u000e\u000b\f\u000ei j+B\u000e\u000bi\u000e\fj+C\u000e\u000bj\u000e\fi: (10)\nBy substituting this into Eq.(6) and (7), the spin density asso-\nciated with the electric field and magnetic textures is\nh\u001bi=A(E\u0001r)\u000eM+BE(r\u0001\u000eM)+Cr(E\u0001\u000eM):(11)\nIn the linearized Hamiltonian, Eq.(4), there is a correspon-\ndence between the axial velocity operators and the spin oper-\nators, i.e.\nv5=\u001cz@HWSM\n@k=vF y\u001b : (12)This suggests that the spin operators and the axial current op-\nerators are identical except for a constant factor38\nj5=\u0000e yv5 =\u0000evF y\u001b : (13)\nThe electron spins couple to the magnetization by the ex-\nchange interaction as \u0000M\u0001\u001b, while the axial currents couple\nto the axial vector potential as \u0000A5\u0001j5. Due to the corre-\nspondence between the spin operators and the axial current\noperators, magnetization and the axial vector potentials cou-\nple to electrons in the same way, namely the electrons cannot\ndistinguish the perturbations. By using this correspondence,\nEq.(11) can be interpreted in the axial transport picture as\nhj5i=A(E\u0001r)A5+BE(r\u0001A5)+Cr(E\u0001A5) (14)\nin the expression explicitly depending on the axial vector po-\ntentials. This expression is not gauge invariant under the lo-\ncal chiral gauge transformation. Namely, these terms are pro-\nhibited when local chiral gauge symmetry is present. Conse-\nquently, coe \u000ecients of these terms including the spin-transfer\ntorque and the \f-term are going to be zero.39On the other\nhand, the spin density associated with Ki\n\u000bis gauge invariant\nsince it only includes electric field. Therefore only the spin\nsusceptibility of spin-orbit torque is finite while the others\nmust be zero. A similar discussion exists for a ferromagnetic\ninsulator on the surface of a topological insulator12, however\nthe responsible symmetries are di \u000berent, i.e. the U(1) gauge\nsymmetry for the TI surface but the chiral gauge symmetry\nfor the Weyl semimetals. Note that the correspondence be-\ntween the spin operators and the axial velocity operators re-\ntains when the band dispersion close to the Fermi energy is\ndescribed by k-linear term. If the dispersion deviates from lin-\near, including k2-term, there is no restriction from local chiral\ngauge symmetry, and the spin-transfer torque and the \f-term\nare also expected.\nThe leading order of the spin response function associated\nwith the spin-orbit torque in damping rate, \u0011, is calculated in\nAppendix A, giving\nKi\n\u000b(\n)=i\n\u000e\u000bie\n6\u00192vFX\n\u0015(EF+\u0015\u0001)2\n\u0011\n=i\n\u000e\u000bieEF\u0001\n3\u00192v2\nF\u0011; (15)\nand the spin density induced by the applied electric fields is\ngiven by\nh\u001bi=eEF\u0001\n3\u00192v2\nF\u0011E: (16)\nThe associated spin torque is therefore\nTe(r)=eJS E F\u0001\n3\u00192v2\nF\u0011ˆM\u0002E=\u0000JS\nevFˆM\u0002hj5i (17)\nwherehj5i=\u0000(e2EF\u0001=3\u00192vF\u0011)Eis the axial current density.\nThe second equality is obtained from the correspondence be-\ntween the spin and the axial transport phenomena. Equation\n(16) indicates that the spin density is induced in the direction\nof the applied electric field. The e \u000bect can be understood by4\nthe Rashba-Edelstein e \u000bect. In the Weyl Hamiltonian, mo-\nmentum and spin are fixed to be parallel or antiparallel de-\npending on the chirality, the spin-momentum locking. Sup-\npose that the electric field is applied in the +xdirection, the\nFermi surface is shifted along \u0000kxdirection, and the number\nof the electrons with +kxdecreases whereas the number with\n\u0000kxincreases. Because of spin-momentum locking, the im-\nbalance of the momentum induces positive xcomponent of\nthe net spin density for the right-handed and negative for the\nleft-handed. When \u0001 = 0, meaning two Weyl nodes are lo-\ncated at the same energy, the induced spin density of each\nnode has same magnitude but opposite direction, thus the con-\ntributions cancel out. Once \u0001becomes non-zero, the cancella-\ntion is incomplete and the net spin density becomes finite. In\nthe presence of inversion-symmetry, \u0001has to vanish. There-\nfore, to observe the spin-orbit torque, both the inversion and\ntime-reversal symmetries must be broken.\nIV . CHARGE-INDUCED SPIN TORQUE\nIn this section, we microscopically derive the charge-\ninduced spin torque, and discuss the relation to the chiral\nanomaly. The charge-induced spin torque is described as\na spin response to the external electric voltage and mag-\nnetic fields. Coupling to external fields is introduced by\nH0=R\ndr j(r)\u0001A(r) where j(r)= y(r)ˆj (r) and H00=\neR\ndr y(r)\u001e(r) (r) where\u001e(r) is an electric potential. We\ncalculate the electron’s spin density h\u001biup to linear order in\nboth the electric scalar potential, \u001e, and the vector potential\nA. The electron’s spin density can be obtained as\nh\u001b\u000b(q)i=2\u001fi\n\u000b(q)\u001eAi(q) (18)\nwhere\u000bandidenote spin and velocity direction, respectively,\nand\u001fi\n\u000b(q) is the response function of the spin polarization\ngiven by\n\u001fi\n\u000b(q)=\u0000e2vFX\n!;k;\u0015n\nTrh\n\u001b\u000b˜G\u0000\n\u0015G\u0015vi;\u0015G\u0015i\n+Trh\n\u001b\u000bG\u0015vi;\u0015G\u0015˜G+\n\u0015io\n; (19)\nwhere ˜G\u0006\n\u0015\u0011G\u0015(!;k\u0006q) and G\u0015\u0011G\u0015(!;k). The response\nfunction is represented diagrammatically in Fig. 2. This is cal-\nculated in Appendix B, and the result up to O(\u00110), the leading\norder in the scattering rate for the charge-induced spin torque,\nis given by\n\u001fi\n\u000b(q)=\u0000ie2\n4\u00192vF\u000f\u000bi jqj; (20)\nwhere\u000fi jkis antisymmetric under any permutation of indexes.\nFrom Eq.(18), the spin density induced by the external fields\nis given as\nh\u001b\u000b(q)i=e2\n2\u00192vF\u001e\u000f\u000bi jiqiAj(q)=e2\n2\u00192vF\u001eB\u000b(q);(21)\n\fFIG. 2. Feynman diagrams for the charge-induced spin torque.\nwhere B\u000b(q)=\u000f\u000bi jiqiAj(q) is the external magnetic field.\nTherefore, the spin torque associated with this process is ex-\npressed by\nTe(r)=e2JS\n2\u00192vF\u001eˆM\u0002B(r)=\u001bAHE\u001eˆM\u0002B(r);(22)\nwhere\u001bAHE=e2S J\n2\u00192vFis the anomalous Hall conductivity of\nmagnetic Weyl semimetals. By replacing the electric poten-\ntial with the shift in the chemical potential, \u001e! \u0000\u000e\u0016F=e,\nEq. (22) is consistent with our previous result obtained\nphenomenologically.26\nThe physical origin of the spin polarization, Eq.(21), can\nbe understood by considering the Landu levels of chiral\nfermions. Here we consider Weyl fermions in a uniform mag-\nnetic field applied along the zdirection. Defining canonical\nmomenta as \u0019i=\u0000iri+eAi+\u001cz(JS=vF)Miwhere i=x;y, the\nHamiltonian given in Eq.(4) can be written as31\nHWSM=\u001czvF \nkz+\u001czJS\nvFMz!\n\u001bz+\u001czvF\u0010\n\u0019x\u001bx+\u0019y\u001by\u0011\n:(23)\nThe spin operators are described by\n\u001bi=\u001cz\u000eHWSM\nvF\u000e\u0019i; \u001bz=\u001cz\u000eHWSM\nvF\u000ekz: (24)\nThe correspondence between the spin operators and the ax-\nial velocity operators still holds in the presence of magnetic\nfields. The Hamiltonian, Eq.(23) can be solved obtaining the\nenergy eigenvalues\nE\u0015\n0(kz)=\u0000\u0015vFkz\u0000JS M z; (25)\nE\u0015\nn(kz)=\u0000vFsgn(n)s \nkz+\u0015JS M z\nvF!2\n+2eBzjnj(26)\nwhere n=\u00061;\u00062;\u0001\u0001\u0001are the Landau indices, \u0015=\u00061 labels\nthe chirality. The zcomponent of the total spin density is given\nby the sum of contributions from all states below the Fermi\nenergy\nh\u001bzitot=1\nvFX\nkzEF>En(kz)X\nnh\nv+\nn;kz\u0000v\u0000\nn;kzi\n; (27)\nwhere v\u0015\nn;kz=dE\u0015\nn(kz)\ndkzis the group velocity, while xandycom-\nponents become zero because the Landau levels disperse only\nalong the kzaxis.\nLet us first consider contributions from the zeroth Landau\nlevels. The group velocity of the zeroth Landau levels is \u0000vF5\nfor the right-handed electrons ( \u0015= +1), and +vFfor the left-\nhanded electrons ( \u0015=\u00001). Therefore, in the zeroth Landu\nlevels, both right and left-handed electrons have down spins\nwith respect to the magnetic field. By contrast, the non-zero\nLandau levels do not contribute to the total spin density. Sup-\npose\"is a some energy which cuts the non-zero Landau lev-\nels below Fermi energy, the momenta associated with \"are\nobtained as\nk\u0015\n\u0006=\u0000\u0015JS M z\nvF\u0006q\n(\"=vF)2\u00002eBzjnj: (28)\nThe group velocity of the non-zero Landau levels at energy \"\nis\nv\u0015\nn;k\u0015\n\u0006=1\nvFdE\u0015\nn(kz)\ndkz\f\f\f\f\f\fkz=k\u0015\n\u0006=\u0007vFsgn(n)q\n(\"=vF)2\u00002eBzjnj\n\"=vF\nwhenj\"j>p\n2eBzjnjvF. So there always exist two states pos-\nsessing opposite sign of the velocity within a Landau level\nwith the chirality \u0015. According to the above arguments, con-\ntributions to the spin density from the non-zero Landau levels\nexactly cancel out.\nThe total spin density, therefore, is given only by the ze-\nroth Landau levels and estimated as h\u001bzitot=\u0000n0(EF) where\nn0(EF) is electron density in the zeroth Landau levels below\nthe Fermi energy. When the Fermi energy is shifted by the\napplied gate voltage, \u001e, the induced spin density is given by\nh\u001bzi=\u0000gLLZEF\u0000e\u001e\nEFdED (E)\n=e2\n4\u00192vF\u001eBz (29)\nwhere gLL=\f\f\f\feBz\n2\u0019\f\f\f\fis the zeroth Landau level degeneracy\nandD(E)=2=2\u0019vFis the density of the states for the one-\ndimensional linear dispersion. This result is identical to Eq.\n(21).\nAbove results are derived from the continuum model where\nthe energy dispersion is linearized and each chirality is sepa-\nrated. It is, however, not obvious that analysis with the con-\ntinuum model always gives the same results as a lattice model\nwhere the energy-momentum cuto \u000bis naturally introduced\nand the chirality is no longer a good quantum number. For ex-\nample, the chiral magnetic e \u000bect in equilibrium, predicted us-\ning a continuum model, is actually absent in lattice systems.40\nIn the following, we numerically examine the spin density of\nWeyl semimetals on a lattice to check whether the predicted\nphenomenon survives.\nWe describe a Dirac-Weyl semimetal in a magnetic field\nusing the lattice Hamiltonian41\nHLattice =X\nj=x;y;z\u0010\nTj+Ty\nj\u0011\n+HW+Hexc (30)\nwhere Tj=P\nRcy\nR+ej\u0010\n\u0000it\u000bj+r\u000b4\u0011\neiAj(R)cRare the the trans-\nlation operators in j=x;y;zdirections, \u000bj=\u001bj\u001czand\n\u000b4=\u0000\u001b0\u001cxbeing 4\u00024 Dirac matrices satisfying the anticom-\nmutation relation, HW=Pcy\nR(3r\u000b4)cRis the Wilson term in\n    \n-0.002-0.001 0 0.001 0.002\n-0.3 -0.2 -0.1  0  0.1  0.2  0.3\n 0\n-0.3 -0.2 -0.1  0  0.1  0.2  0.3Fermi energy        [eV]\n[eV](a)\n(b)FIG. 3. (Color online) (a) Electron spin density calculated as a func-\ntion of the Fermi energy. (b) Landau level structure of the lattice\nHamiltonian of the Weyl semimetal. The zeroth landau levels are\nshown as a solid line, while non-zero Landu levels are dashed lines.\nThe parameters are chosen as t=\u00000:248 eV , r=0:414 eV , J=0:2\neV , ˆM=ˆz,ˆB=B0ˆz, and B0=\u001e0=128 where\u001e0is the magnetic flux\nquantum.\nthe massless case, and Hexcis the exchange Hamiltonian. The\ne\u000bect of the magnetic field is introduced by the Peierls phase.\nWe use typical materials parameters for Bi 2Se3determined to\nfitab initio calculations.42,43The external magnetic field is\ntaken to be uniform and pointing in the +zdirection with the\nLandau gauge, A(R)=(0;B0x;0).\nWe calculate the electron spin density below the Fermi en-\nergy,h\u001bzi(EF)=hcyˆ\u001bzci, by numerical diagonalization of the\nHamiltonian. To measure the response to external fields, we\nevaluateh\u0001\u001bzi(EF)\u0011 h\u001bzi(EF)jBz=B0\u0000h\u001bzi(EF)jBz=0which\nis shown in Fig. 3 (a). When the Fermi energy is \u00000:2 [eV]\n<\u0018EF<\u00180.2 [eV], the proportion relation between zcompo-\nnent of the spin density and the Fermi energy is clearly seen\nas we expected from our analysis with the continuum model,\nEq.(29), whereas the perpendicular component to the mag-\nnetic field is always zero, independent of EF. Within this en-\nergy scale, the Landau levels shown in Fig.3 (b) behave as the\nEq.(26), namely the linearized continuum model where the\nchirality is a good quantum number describes the spectrum\nwell. On the other hand, when jEFj>\u00180:2 [eV], the spin den-\nsity deviates from EF-linear dependence. In this regime, two\nLandau level branches with opposite chirality hybridize while\nthey are approaching kz=0 point. In above argument based\non the continuum model, we have neglected the hybridization\nof the chirality, therefore our theory, Eq.(21), is unapplicable\nin this energy regime. According to the numerical results, we\nconclude that the spin density response to the applied electric\nvoltage shown in Eq. (21) can be expected even in the lattice\nsystem when the Fermi energy is located in the linear disper-6\nsion regime.\nWe now discuss the relationship between the charge-\ninduced spin torque and the chiral anomaly. As we already\nmentioned in the previous section, the spin operators can be\nexpressed in terms of the axial velocity operators in Weyl\nsemimetals. The correspondence suggests that we can replace\nthe spin vertex with the axial current vertex in Fig. 2, and Eq.\n(21) can be interpreted as\nhj5i=e3\n2\u00192\u001eB: (31)\nAccording to the discussion in the previous section, Eq.(31)\nis disallowed if U(1) gauge symmetries are present because\nit explicitly depends on a gauge potential, \u001e. However, the\npresence of the chiral anomaly makes this an exceptional case.\nThe conservation law of axial currents can be violated even in\nthe presence of the chiral gauge symmetry, which is known\nas the chiral anomaly.44–46The violation of the axial current\nconservation is governed by the anomaly equation. By taking\nthe divergence, we obtain\nr\u0001hj5i=\u0000e3\n2\u00192r\u001e\u0001B=e3\n2\u00192E\u0001B:\nThis equation coincides with the anomaly equation in a static\ncase, suggesting that Eq.(31) is allowed as a part of the\nanomaly equation. This indicates that the charge-induced spin\ntorque is a consequence of the chiral anomaly.\nRecently many attempts have been made to observe evi-\ndence of the chiral anomaly47, for example through the ob-\nservation of a negative magnetoresistance48–53. Observing\na charge-induced torque is another direct piece of evidence\nof the chiral anomaly. Experimentally, this phenomenon is\ncaptured as a frequency shift in the ferromagnetic resonance\n(FMR) peak. In ferromagnetic metals, the interaction energy\nof the magnetization is given by the sum of the Zeeman en-\nergy and the exchange energy with the background conduction\nelectron spins, U=(\u0000\r0\u001aSS B0+JSh\u001bzi)ˆMzwhere\u001aSis the\ndensity of local moments. By substituting the contribution\nfrom the charge-induced spin torque, Eq.(21), the interaction\nenergy is obtained as\nU=\u0000\r0\u001aSS \n1\u0000e2J\n2\u00192\r0\u001aSvF\u001e!\nB0ˆMz\n\u0011\u0000\r0\u001aSS \n1\u0000\u0001B\nB0!\nB0ˆMz (32)\nwhere \u0001B=B0=e2J\n2\u00192\r0\u001aSvF\u001e=eJ\n2\u00192\r0\u001aSvF\u000e\u0016Fcharacterizes the\nshift in the FMR peak. With typical material parameters for\nmagnetic doped topological insulators such as Cr xBi2\u0000xTe3\n(J=10 meV , ~vF=2:2 eVÅ\u00001, and\u001aS=1:1\u000210\u00004Å\u00003\nwith chromium concentration x=0:1), we qualitatively esti-\nmate the shift in the FMR peak as\n\u0001B\nB0\u00190:3\u0002 \u000e\u0016F\n[eV]!\n: (33)\nThis indicates that when a FMR measurement is performed\nwhile changing the gate voltage, the FMR peak shifts linearly\nin the applied gate voltage. If the chemical potential is shifted\n0.1 [eV] by the gate voltage, the shift of the FMR peak can be\nup to 3%.V . DISCUSSION AND CONCLUSION\nIn this paper, we have studied electrically-induced spin\ntorques, charge-induced spin torque, spin-orbit torque, spin-\ntransfer torque, and \f-term. They are summarized as\nTe=\u001bAHE\u001eˆM\u0002B+\u001fSOTˆM\u0002E (34)\nwhere\u001fSOT=(eJS E F\u0001)=(3\u00192vF\u0011), and the spin-transfer\ntorque and the \f-term are absent because of the chiral gauge\nsymmetry. The first term denotes the charge-induced torque\ngenerated by an applied voltage, while the second term is the\nspin-orbit torque driven by an electric field due to the spin-\nmomentum locking. These two terms are both electrically-\ndriven spin torques, however, the origins are completely dif-\nferent. The charge-induced spin torque is an adiabatic phe-\nnomenon in which all states below Fermi energy contribute.\nOn the other hand, the spin-orbit torque is a non-equilibrium\nspin torque, associated with electrons near the Fermi surface.\nIn our analysis, we have taken account only of intra-nodal\nscattering. This assumption is valid when the Fermi energy\nis located close to the Weyl nodes and the two nodes are well\nseparated. The absence of the spin-transfer torque and the \f-\nterms is guaranteed as long as this assumption is valid. When\nthe Fermi energy is far from the Weyl nodes or the distance be-\ntween nodes is small, states with di \u000berent chirality hybridize\nand chiral gauge symmetry is no longer present. In this case,\nthere is no symmetry restriction, and the spin-transfer torque\nand the\f-terms are also to be expected.\nIt is straight forward to introduce vertex corrections. Sev-\neral theoretical papers have concluded that, in Weyl semimet-\nals, ladder type corrections only modify the Fermi velocity of\nthe current vertex.54–56Because our model includes the cor-\nrespondence between the velocity and the spin operators, the\nvertex is also only modified in the coe \u000ecient. Therefore our\nresults are not cancelled by the vertex correction, and valid\nafter a modification in the Fermi velocity vF!\u0003F, being the\nrenormalized velocity.\nFinally we discuss other spin torques within the chiral trans-\nport picture. As we introduced in Eq.(3), the amplitude of the\nspin and the Gilbert damping are modified by the conduct-\ning electrons. They are expressed as h\u001bi=(\u000e\u000b=JS)\u000e˙M+\n(\u000eS=JS)ˆz\u0002\u000e˙Mwhere\u000e\u000band\u000eSare the modification to the\nGilbert damping constant and the amplitude of spins, respec-\ntively. Here we take the saturated magnetization axis in z\naxis. In the chiral transport picture, the coe \u000ecients are noth-\ning but the longitude and Hall conductivity for the axial cur-\nrents, expressed as hj5i=(evF=JS)a0e+(evF=JS)b0ˆz\u0002e\nwhere e=\u000e˙Mis the chiral electric field. If two Weyl nodes\nare well separated, the coe \u000ecients are equivalent to the con-\nductivities of the vector currents. Namely, the correction to\nthe Gilbert damping constant and the amplitude of spin are ex-\nplicitly written as \u000e\u000b=(JS=evF)\u001bxxand\u000eS=(JS=evF)\u001bxy.\nThese relations coincide with those of a ferromagnetic insula-\ntor deposited on the surface of a topological insulator.12\nIn conclusion, we microscopically derived the electrically\ninduced spin torques in magnetically doped Weyl semimetals.\nFirstly, we examine current-induced spin torques. Because of\nthe spin-transport correspondence and the local chiral gauge7\nsymmetry, we found that the spin-transfer torque and the \f-\nterms are absent as long as the two Weyl nodes with opposite\nchirality is well separated in momentum space. Consequently,\nin the magnetic Weyl semimetals, only the spin-orbit torque\nis expected as current-induced spin torques up to first order\nwith respect to spatial and temporal derivation and electrical\ncurrents. We derived the analytical expression and found the\nnon-equilibrium spin density is induced in the applied elec-\ntric field direction when the inversion symmetry is broken, in\nother words, \u0001,0. The phenomenon is understood by the\nRashba-Edelstein e \u000bect because of the spin-momentum lock-\ning.\nSecondly, by using a linear response theory, we derived the\nanalytical expression of the charge-induced spin torque in-\nduced by the gate voltage in the presence an external mag-\nnetic field, and succeeded to reproduce our previous resultsobtained phenomenologically. We also discussed that the\ncharge-induced spin torque is understood in terms of the chiral\nanomaly and the correspondence between the spin phenomena\nand the chiral transport phenomena. This indicates that the\nobservation of the charge-induced spin torque might be direct\nevidence of the chiral anomaly. Experimentally, it is expected\nto be observed as a frequency shift of the ferromagnetic reso-\nnance peak which is estimated as \u00183%.\nACKNOWLEDGMENTS\nWe thank Hiroshi Kohno, Yasufumi Araki, Joseph Barker,\nYuya Ominato, and Takahiro Chiba for useful discussions and\ncomments. D.K. is supported by a JSPS Research Fellow-\nship for Young Scientists. This work was supported by JSPS\nKAKENHI Grants No. JP15H05854 and No. JP26400308.\nAppendix A: Current-induced spin torques\n1. Spin-orbit torque\nThe detail calculations for the spin-orbit torque is presented in this section. In this section, \u0016;\u0017;\u001a and\rare dummy indices,\nrunning over 1\u00183 and summed in all the values of the indices. Let us start with the Eq.(8), and proceed our calculation as\nKi\n\u000b(i\nm)=kBTev FX\nn;k;\u0015\u0015Trh\n\u001b\u000bG\u0015\u001biGy\n\u0015i\n=kBTev FX\nn;k;\u0015\u0015D\u0015(i!n+i\nm;k)D\u0015(i!n;k)Trh\n\u001b\u000bg0\u0015(i!n+i\nm;k)\u001big0\u0015(i!n;k)+v2\nFk\u0016k\u0017\u001b\u000b\u001b\u0016\u001bi\u001b\u0017i\n=kBTev FX\nn;k;\u0015\u0015D\u0015(i!n+i\nm;k)D\u0015(i!n;k)(\ng0\u0015(i!n+i\nm;k)g0\u0015(i!n;k)Tr[\u001b\u000b\u001bi]+v2\nFk2\n3\u000e\u0016\u0017Trh\n\u001b\u000b\u001b\u0016\u001bi\u001b\u0017i)\n(A1)\nafter dropping terms which are odd in momentum, where D\u0015(i!n;k);g0\u0015(i!n;k) are defined by\nD\u0015(i!n;k)\u0011h\n(i!n+EF+\u0015\u0001\u0000\u0006\u0015(i!n;k))2\u0000v2\nFk2i\u00001(A2)\ng0\u0015(i!n;k)\u0011i!n+EF+\u0015\u0001\u0000\u0006\u0015(i!n;k) (A3)\nsatisfying G\u0015(i!n;k)=D\u0015(i!n;k)(g0\u0015+\u0015vFk\u0001\u001b);and we have used the relation k\u0016k\u0017=k2\n3\u000e\u0016\u0017under symmetrical integration.\nThe trace calculations are performed as\nTr[\u001b\u000b\u001bi]=2\u000e\u000bi; \u000e\u0016\u0017Trh\n\u001b\u000b\u001b\u0016\u001bi\u001b\u0017i\n=\u00002\u000e\u000bi: (A4)\nBy defining the Green’s function projected onto conduction and valence band as\ng\u0015;\u0006(i!n)\u0011\u0002i!n+EF+\u0015\u0001\u0007vFk+i\u0011sgn(!n)\u0003\u00001(A5)\nand using the following relations,\nD\u0015(i!n;k)g0\u0015(i!n;k)=1\n2\u0002g\u0015;+(i!n)+g\u0015;\u0000(i!n)\u0003(A6)\nD\u0015(i!n;k)vFk=1\n2\u0002g\u0015;+(i!n)\u0000g\u0015;\u0000(i!n)\u0003; (A7)\nthe response function is expressed by\nKi\n\u000b(i\nm)=kBTev F\u000e\u000biX\nn;k;\u0015\u0015\"1\n3\bg\u0015;+(i!n)g\u0015;+(i!n+i\nm)+g\u0015;\u0000(i!n)g\u0015;\u0000(i!n+i\nm)\t\n+2\n3\bg\u0015;+(i!n)g\u0015;\u0000(i!n+i\nm)+g\u0015;\u0000(i!n)g\u0015;+(i!n+i\nm)\t#\n(A8)8\nNext we are going to perform the Matsubara summation,\nkBTX\nng\u0015;\u0006(i!n)g\u0015;\u0006(i!n+i\nm)=i\n2\u0019Z\nd\u000ff(\u000f)hn\ngR\n\u0015;\u0006(\u000f)\u0000gA\n\u0015;\u0006(\u000f)o\ngR\n\u0015;\u0006(\u000f+ \n)+gA\n\u0015;\u0006(\u000f\u0000\n)n\ngR\n\u0015;\u0006(\u000f)\u0000gA\n\u0015;\u0006(\u000f)oi\n=i\n2\u0019Z\nd\u000fh\nff(\u000f+ \n)\u0000f(\u000f)ggR\n\u0015;\u0006(\u000f+ \n)gA\n\u0015;\u0006(\u000f)+f(\u000f)n\ngR\n\u0015;\u0006(\u000f)gR\n\u0015;\u0006(\u000f+ \n)\u0000gA\n\u0015;\u0006(\u000f\u0000\n)gA\n\u0015;\u0006(\u000f)oi\n\u0019i\n2\u0019Z\nd\u000ff(\u000f)h\n(gR\n\u0015;\u0006)2\u0000(gA\n\u0015;\u0006)2i\n+i\n2\u0019Z\nd\u000fh\nf0(\u000f)gR\n\u0015;\u0006gA\n\u0015;\u0006\u0000f(\u000f)n\n(gR\n\u0015;\u0006)3+gR\n\u0015;\u0006)3oi\n+O(\n2)\n=i\n2\u0019Z\nd\u000ff0(\u000f)\u0010\ngR\n\u0015;\u0006\u0000gA\n\u0015;\u0006\u0011\n+i\n2\u0019Z\nd\u000ff0(\u000f)\"\ngR\n\u0015;\u0006gA\n\u0015;\u0006\u00001\n2n\n(gR\n\u0015;\u0006)2+(gA\n\u0015;\u0006)2o#\n=\u0000i\n2\u0019\u0010\ngR\n\u0015;\u0006\u0000gA\n\u0015;\u0006\u0011\n\u0000i\n4\u0019\u0010\ngR\n\u0015;\u0006\u0000gA\n\u0015;\u0006\u00112(A9)\nkBTX\nng\u0015;\u0006(i!n)g\u0015;\u0007(i!n+i\nm)=i\n2\u0019Z\nd\u000ff(\u000f)hn\ngR\n\u0015;\u0006(\u000f)\u0000gA\n\u0015;\u0006(\u000f)o\ngR\n\u0015;\u0007(\u000f+ \n)+g\u0015;\u0006(\u000f\u0000\n)n\ngR\n\u0015;\u0007(\u000f)\u0000gA\n\u0015;\u0007(\u000f)oi\n\u0019i\n2\u0019Z\nd\u000ff(\u000f)\u0010\ngR\n\u0015;\u0006gR\n\u0015;\u0007\u0000gA\n\u0015;\u0006gA\n\u0015;\u0007\u0011\n+i\n2\u0019Z\nd\u000fh\nf0(\u000f)gR\n\u0015;\u0007gA\n\u0015;\u0006\u0000f(\u000f)n\ngR\n\u0015;\u0006(gR\n\u0015;\u0007)2+(gA\n\u0015;\u0006)2gA\n\u0015;\u0007oi\n+O(\n2) (A10)\nkBTX\nn\u0002g\u0015;+(i!n)g\u0015;\u0000(i!n+i\nm)+g\u0015;\u0000(i!n)g\u0015;+(i!n+i\nm)\u0003=i\n\u0019Z\nd\u000ff(\u000f)\u0010\ngR\n\u0015;+gR\n\u0015;\u0000\u0000gA\n\u0015;+gA\n\u0015;\u0000\u0011\n+i\n2\u0019Z\nd\u000fh\nf0(\u000f)gR\n\u0015;+gA\n\u0015;\u0000\u0000f(\u000f)gR\n\u0015;+gR\n\u0015;\u0000\u0010\ngR\n\u0015;++gR\n\u0015;\u0000\u0011\n+c:ci\n=O(\n0)+i\n2\u0019Z\nd\u000ff0(\u000f)h\ngR\n\u0015;+gA\n\u0015;\u0000\u0000gR\n\u0015;+gR\n\u0015;\u0000+c:ci\n=O(\n0)\u0000i\n2\u0019Z\nd\u000ff0(\u000f)\u0010\ngR\n\u0015;+\u0000gA\n\u0015;+\u0011\u0010\ngR\n\u0015;\u0000\u0000gA\n\u0015;\u0000\u0011\n=O(\n0)+i\n2\u0019\u0010\ngR\n\u0015;+\u0000gA\n\u0015;+\u0011\u0010\ngR\n\u0015;\u0000\u0000gA\n\u0015;\u0000\u0011\n(A11)\nwhere gR;(A)\n\u0015;\u0006\u0011gR;(A)\n\u0015;\u0006(\u000f),f(\u000f) is the Fermi-Dirac distribution function, and we have substituted i\nm!\n. We have also used\nthe relationd\nd\u000fgR;(A)\n\u0015;\u0006=\u0000(gR;(A)\n\u0015;\u0006)2, and assumed zero temperature, namely f(\u000f)=\u0000\u000e(\u000f). In the weak scattering regime where\n\u0011<< EF, the Green’s functions can be expanded by gR\n\u0015;\u0006\u0000gA\n\u0015;\u0006=\u00002\u0019i\u000e(EF+\u0015\u0001\u0007vFk), and momentum integrals are evaluated9\nas\nX\nk;\u0015(A9) =X\n\u0015\u0015Zdk3\n(2\u0019)3i\n4\u0019\u0014\u0010\ngR\n\u0015;+\u0000gA\n\u0015;+\u00112+\u0010\ngR\n\u0015;\u0000\u0000gA\n\u0015;\u0000\u00112\u0015\n\u0019\u0000\n4\u00192v3\nFX\n\u0015\u0015Z\nd\u0018\u00182\"\n\u000e(EF+\u0015\u0001\u0000\u0018) 1\nEF+\u0015\u0001\u0000\u0018+i\u0011\u00001\nEF+\u0015\u0001\u0000\u0018\u0000i\u0011!\n+\u000e(EF+\u0015\u0001 +\u0018) 1\nEF+\u0015\u0001 +\u0018+i\u0011\u00001\nEF+\u0015\u0001 +\u0018\u0000i\u0011!#\n=i\n2\u00192v3\nFX\n\u0015\u0015(EF+\u0015\u0001)\n\u0011=i\nEF\u0001\n\u00192v3\nF\u0011; (A12)\nX\nk;\u0015(A11) =X\n\u0015\u0015Zdk3\n(2\u0019)3i\n2\u0019\u0010\ngR\n\u0015;+\u0000gA\n\u0015;+\u0011\u0010\ngR\n\u0015;\u0000\u0000gA\n\u0015;\u0000\u0011\n\u0019\n2\u00192v3\nFX\n\u0015\u0015Z\nd\u0018\u00182i\n2\u0019 \n\u00002\u0019i\u000e(EF+\u0015\u0001\u0000\u0018)\u0000i\u0011\n(EF+\u0015\u0001\u0000\u0018)2+\u00112!\n\u0002 \n\u00002\u0019i\u000e(EF+\u0015\u0001 +\u0018)\u0000i\u0011\n(EF+\u0015\u0001 +\u0018)2+\u00112!\n=\u0000i\n2\u00192v3\nFX\n\u0015\u0015Z\nd\u0018\u00182\"\n\u000e(EF+\u0015\u0001\u0000\u0018)\u0011\n(EF+\u0015\u0001 +\u0018)2+\u000e(EF+\u0015\u0001 +\u0018)\u0011\n(EF+\u0015\u0001\u0000\u0018)2#\n=\u0000i\n2\u00192v3\nFX\n\u0015\u0015\u0011\n4=0 (A13)\nwhere we have neglected O(\n0) terms because they are cancelled by the definition of the spin susceptibility, Eq.(6). By substi-\ntuting these result for Eq.(A8), the response function is obtained by\nKi\n\u000b(\n)=i\neEF\u0001\n3\u00192v2\nF\u0011(A14)\nwhich is presented in Eq.(15).\n2. Spin-transfer torque and \f-term\nIn this section, we present the detail calculations for obtaining Eq.(10). Let us start with the first term of Eq.(9),\nKi j\n\u000b\f(i\nm)=kBTeJS v2\nFX\nn;k;\u0015Trh\n\u001b\u000bG\u0015\u001bjG\u0015\u001b\fG\u0015\u001biG+\n\u0015i\n=kBTeJS v2\nFX\nn;k;\u0015D\u0015(i!n+i\nm;k)D\u0015(i!n;k)3n\ng0\u0015(i!n;k)3g0\u0015(i!n+ \n m;k)Trh\n\u001b\u000b\u001bj\u001b\f\u001bii\n+\u000e\u0016\u0017(vFk)2\n3g0\u0015(i!n;k)2\u0010\nTrh\n\u001b\u000b\u001bj\u001b\f\u001b\u0016\u001bi\u001b\u0017i\n+Trh\n\u001b\u000b\u001bj\u001b\u0016\u001b\f\u001bi\u001b\u0017i\n+Trh\n\u001b\u000b\u001b\u0016\u001bj\u001b\f\u001bi\u001b\u0017i\u0011\n+\u000e\u0016\u0017(vFk)2\n3g0\u0015(i!n;k)g0\u0015(i!n+i\nm;k)\u0010\nTrh\n\u001b\u000b\u001bj\u001b\u0016\u001b\f\u001b\u0017\u001bii\n+Trh\n\u001b\u000b\u001b\u0016\u001bj\u001b\f\u001b\u0017\u001bii\n+Trh\n\u001b\u000b\u001b\u0016\u001bj\u001b\u0017\u001b\f\u001bii\u0011\n+v4\nFk2k2\n15\u0010\n\u000e\u0016\u0017\u000e\u001a\r+\u000e\u0016\u001a\u000e\u0017\r+\u000e\u0016\r\u000e\u0017\u001a\u0011\nTrh\n\u001b\u000b\u001b\u0016\u001bj\u001b\u0017\u001b\f\u001b\u001a\u001bi\u001b\ri)\n:\nThe second term can be estimated by just exchanging i$jand replacing \nm!\u0000\nm. The equation looks cumbersome,\nhowever the Pauli matrices always appear with even number in trace. This constrains the response function to being proportional\nto pairs of Kronecker deltas with all possible combination of indices,\nKi j\n\u000b\f(i\nm)=A\u000e\u000b\f\u000ei j+B\u000e\u000bi\u000e\fj+C\u000e\u000bj\u000e\fi (A15)\nwhere A;B;Care constants of proportionality.10\nAppendix B: Charge-induced torque\nThe detail calculations for the charge-induced spin torque is presented in here. Because the charge-induced spin torque is an\nadiabatic spin torque, namely the leading order of the spin torque is give in O(\u00110). We focus on the non dissipation case, \u0011=0.\nWe also assume that the Fermi energy is located on the Weyl nodes, EF=0. For calculating the charge-induced spin torque,\nit is easier to introduce the Dirac’s Gamma matrices and express the Green’s function as G(k)=[vFki\ri]\u00001where i=1\u00184,\nk=(k;!=vF) is a four-momentum and the gamma matrices are defined as\n\ri\u0011i\u001cy\u001bi(i=1;2;3); \r4\u0011\u0000i\u001cx (B1)\nsatisfyingf\ri;\rjg=2\u0011i jwith the Euclidean metric \u0011i j=diag(\u00001;\u00001;\u00001;\u00001).31The interaction with the electromagnetic fields\nand the magnetization are given as\nHEM=\u0000Z\nd3r ev FAi i\ri (B2)\nHexc=\u0000Z\nd3r JS ˆM\u0001 i\r\r5 (B3)\nwhere Ai=(A;\u001e=vF) for i=1\u00184, and \u0011 y(r)\u001cx;  \u0011c(r), and\r5\u0011\r1\r2\r3\r4=\u001cz, thus the current and spin operators are\ngiven by\nji=\u000eHEM\n\u000eAi=\u0000ievF i\ri ; \u001b i=\u000eHexc\n\u000e(JS M i)=\u0000i \ri\r5 :\nWith this notation, Eq.(19) is rewritten as\n\u001fi\n\u000b(q)=ie2\nvFZd4k\n(2\u00194)Tr\"\n\r\u000b\r56k\nk2\r46k\nk2\ri(6k+6q)\n(k+q)2+\r\u000b\r5(6k\u00006q)\n(k\u0000q)2\ri6k\nk2\r46k\nk2#\n(B4)\nwhere four-momenta are expressed in the Feynman’s slash notation, 6k=ki\ri. Here the\u000b=1\u00183 and i=1\u00184 denote the spin\nand velocity direction respectively. Note that, because the integral in Eq.(B4) is divergent, it should be evaluated with a proper\nregularization scheme. In this paper, we introduce dimensional regularization. The basic idea of the dimensional regularization\nis to perform an integral of an analytic function in the space-time dimension dwhere the integral converges, then taking limit\nd!4 after obtaining the analytical expression in d-dimension. By expanding Eq.(B4) in terms of the incident momentum q,\none can obtain \u001fi\n\u000b(q)=i(e2=vF)\u0010\n\u0005\u000bi\u0000\u0005\u000bi jqj\u0011\n+O(q2) where\n\u0005\u000bi=Zddk\n(2\u0019)dk\u0016k\u0017k\u001a\nk6Trh\n\r\u000b\r5\r\u0016\r4\r\u0017\ri\r\u001ai\n+(i$4) (B5)\n\u0005\u000bi j=Zddk\n(2\u0019)dk\u0016k\u0017k\u001ak\u0015\nk8Trh\n\r\u000b\r5\r\u0016\r4\r\u0017\ri\r\u001a\rj\r\u0015i\n\u0000(j$4)\n(B6)\nwith dummy indices \u0016;\u0017;\u001a;\u0015 =1\u0018d. Since the integral is odd in four-momentum, O(q0) term is zero, \u0005\u000bi=0.O(q1) term,\ntherefore, becomes the leading order in incident momentum, which is evaluated as\n\u0005\u000bi j=1\nd(d+2)Zddk\n(2\u0019)d1\nk4\u0010\nT\u000b4i j\u0000T\u000bji4\u0011\n(B7)\nTi jkl\u0011\u0000\u0010\n\u000e\u0016\u0017\u000e\u001a\u0015+\u000e\u0016\u001a\u000e\u0017\u0015+\u000e\u0017\u001a\u000e\u0016\u0015\u0011\nTrh\n\r5\ri\r\u0016\rj\r\u0017\rk\r\u001a\rl\r\u0015i\nwhere we have used\nk\u0016k\u0017k\u001ak\u0015=k4\nd(d+2)\u0010\n\u000e\u0016\u0017\u000e\u001a\u0015+\u000e\u0016\u001a\u000e\u0017\u0015+\u000e\u0017\u001a\u000e\u0016\u0015\u0011\n(B8)\nunder the symmetrical integration. To evaluate the trace, we have to take care of the commutation relation between \r5and\r\u0016\nbecause\r5is essentially defined in four-dimension. If we impose all gamma matrices to satisfy the anticommutation relations,\n\r5commutes with \r\u0016for\u0016=1\u00184 while commutes with \r\u0016for other\u0016. Keeping this in mind, trace parts with Kronecker delta\nare evaluated as\nT\u000b4i j=(d\u00004)(d+2)Trh\n\r5\r4\r\u000b\ri\rji\nd!4=\u00004(d\u00004)(d+2)\u000f\u000bi j (B9)11\nThe momentum integral in d-dimension is also performed as\nZddk\n(2\u0019)d1\nk4=\u00001\n(4\u0019)d=2\u0000(2\u0000d\n2)\n\u0000(2)\nd!4=1\n(4\u0019)2 1\nd=2\u00002\u0000\r!\f\f\f\f\f\fd!4(B10)\nwhere we have used the expansion of the Gamma function near the pole, \u0000(x)\u00191=x\u0000\r, and\r\u00190:5772 is the Euler-Mascheroni\nconstant. By substituting Eq.(B9) and (B10) for Eq. (B7), we finally obtain\n\u0005\u000bi j=\u000016\n(4\u00192)\u000f\u000bi j1\nd(d+2)(d\u00004)(d+2)1\nd\u00004\f\f\f\f\fd!4\n=\u00001\n4\u00192\u000f\u000bi j; (B11)\nand, as a result, the susceptibility, Eq. 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Ronning, Sci.\nRep. 6, 27294 (2016), arXiv:1601.05524.\n54R. R. Biswas and S. Ryu, Phys. Rev. B 89, 014205 (2014),\narXiv:1309.3278v1.\n55A. A. Burkov, J. Phys. Condens. Matter 27, 113201 (2015),\narXiv:1502.07609.\n56Y . Ominato and M. Koshino, Phys. Rev. B 89, 054202 (2014)."    },    {        "title": "1712.08505v1.Spinning_and_tumbling_of_micron_sized_triangles_in_a_micro_channel_shear_flow.pdf",        "content": "Spinning and tumbling of micron-sized triangles in a micro-channel shear flow\nJ. Fries,1,∗M. Vijay Kumar,1,∗B. Mekonnen Mihiretie,1D. Hanstorp,1and B. Mehlig1\n1Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden\nWe report on measurements of the angular dynamics of micron-sized equilaterally triangular\nplatelets suspended in a micro-channel shear flow. Our measurements confirm that such particles\nspin and tumble like a spheroid in a simple shear. Since the triangle has corners we can observe the\nspinning directly. In general the spinning frequency is different from the tumbling frequency, and\nthe spinning is affected by tumbling. This gives rise to doubly-periodic angular dynamics.\nPACS numbers: 83.10.Pp,47.15.G-,47.55.Kf,47.10.-g\nI. INTRODUCTION\nThe motion of very small particles in viscous flow is\ndetermined by two principles. First, the hydrodynamic\nforce and torque on the particle are linearly related to\nits translational and rotational slip velocities, and to the\nstrain-rate matrix of the flow. The constants of propor-\ntionality are the elements of the ‘resistance tensors’ of\nthe particle [1, 2]. These elements depend on the parti-\ncle shape and on its orientation with respect to the flow.\nSecond, the particle moves in such a way that the force\nand the torque on the particle vanish at every instant.\nSo to compute the particle motion, one must determine\nthe resistance tensors by solving the Stokes problem with\nno-slip boundary conditions on the particle surface and\nthe given flow velocities far from the particle.\nForsmallneutrallybuoyantspheroidsthisproblemwas\nsolved by Jeffery [3]. He showed that translation and an-\ngular dynamics decouple. The centre-of-mass is advected\nby the flow. In a simple shear flow the particle tumbles\nperiodically. The vector naligned with the axis of rota-\ntional symmetry of the spheroid follows one of infinitely\nmany marginally stable periodic orbits, the ‘Jeffery or-\nbits’. More generally, symmetric particles with continu-\nous rotation symmetry tumble in this way [4].\nOften we encounter rigid particles that do not possess\ncontinuous rotation symmetry. Examples are hexagonal\nice crystals [5], and many plankton species [6]. The al-\ngaeTriceratium , for example may assume the shape of\nflat triangles, squares, or pentagons [7]. In a recent pa-\nper [8] we used symmetry considerations to analyse the\nangular dynamics of such particles with discrete k-fold\nrotational symmetry ( k>2) and certain mirror symme-\ntries. The theory predicts that the symmetry vector n\nfollows a Jeffery orbit if the particle possesses a mirror\nplane that contains the axis of rotation. An example of\nsuch a particle is the equilateral triangle shown schemat-\nically in Fig. 1. Since this particle has corners, one can\nobserve how the particle spins around n. This is de-\nscribed by the dynamics of the vector pthat points to\none of the corners of the triangle (Fig. 1).\n∗These authors contributed equally to this work.\nqpn\nFIG. 1: Triangular particle (schematic). The unit\nvector npoints along the axis of 3-fold rotational\nsymmetry, the unit vector pis orthogonal to nand\npoints to a corner of the triangle, and q≡p∧n.\nTo understand how small particles spin and tumble in\nturbulence is a question of current research [9–15]. In\nthe past most experimental and numerical studies of the\nangular dynamics of small particles in viscous flow con-\ncentrated on non-spherical particles with a continuous\nrotation symmetry. For such particles the tumbling of\nthe symmetry vector nis readily observed. The particle\nspin is however hard to measure experimentally, and it\nis often not analysed. Exceptions are, first, some papers\nthat analyse the angular dynamics of triaxial particles in\na simple shear [16–18], using Jeffery’s theory which also\napplies to ellipsoids in its general form. Second, Voth et\nal.[19] measured the dynamics of small crosses in turbu-\nlence and showed that they tumble like spheroids. Also\nin this case the spin can be determined. Finally, Variano\nand co-workers studied the angular dynamics of agarose-\nhydrogel cylinders in turbulence, and tracked particle\nspin by following micron-sized glass spheres contained\nwithin the cylinders using optical velocimetry [13, 20].\nIn this paper we report on measurements of the an-\ngular dynamics of neutrally buoyant micron-sized disk-\nlike equilateral triangles that are suspended in a micro-\nchannel shear flow. These particles possess a 3-fold ro-\ntation axis and a mirror plane that contains this axis.\nOur measurements support the recent theoretical pre-\ndiction [8] that the angular dynamics of such particles\nfollows Jeffery orbits. Figures 2 and 3 show the dynam-\nics of the vectors pandncorresponding to two different\nJeffery orbits. Also shown in these Figures are fits to\nJeffery orbits (gray lines), in reasonable agreement with\nthe experimental data. We see that the dynamics of p\nis doubly periodic. This is explained by the fact thatarXiv:1712.08505v1  [physics.flu-dyn]  22 Dec 20172\npx\npz\nnx\nny\nnz\n∆x\nFIG. 2: Angular dynamics of a triangular particle in a\nshear flow. In the first two panels, the x- and\nz-components of p(Fig. 1) are shown as functions of\nthe distance ∆xcovered by the centre-of-mass of the\nparticle in the channel direction. Thin blue line:\nexperimental data obtained before flow reversal. Thin\nred line: experimental data obtained after flow reversal.\nThick gray line: fit to Jeffery orbit as described in the\ntext. Dashed red and blue lines indicate data not part\nof a reversible trajectory and therefore not fitted.\nPanels 3 to 5 show the components of ncomputed from\npxandpz. Experimental parameters are shown in\nTable I (trajectory ID: A I).\nthe dynamics of p(spinning) is coupled to the dynamics\nofn(tumbling), and that the two have different peri-\nods. It is important that the angular dynamics is doubly\nperiodic, because this implies that small shape changes\nthat break the symmetry can result in chaotic motion.\nFor nearly spheroidal particles this was shown theoret-\nically [16, 17]. Ref. [18] verified the sensitivity of the\nangular dynamics to small shape changes experimentally\nusing glass rods with almost circular cross section in a\nmicro-channel flow. For particles with discrete rotation\nsymmetry this effect remains to be analysed. Here we\nconcentrate on the doubly periodic angular dynamics.\nThe remainder of this paper is organised as follows.\nIn Section II we briefly summarise the theoretical ex-\npectations. In Section III we describe the experimental\nmethod. Section IV contains our experimental results,\ntheir theoretical analysis, and a discussion of the exper-\nimental factors that we found difficult to control. We\npx\npz\nnx\nny\nnz\n∆x\nFIG. 3: Same as Fig. 2, but for a different Jeffery orbit.\nTrajectory ID: B I(Table I).\nconclude with Section V. The Supplemental Material [21]\ncontains additional plots of experimental data (similar to\nFigs. 2 and 3) as well as video recordings of the angular\ndynamics displayed in Figs. 2 and 3.\nII. BACKGROUND AND THEORY\nFigure 1 shows a platelet in the shape of an equilateral\ntriangle (schematic). Its orientation is determined by the\nunit vectors nandp, where npoints along the axis of\ndiscrete rotation symmetry. The vector pis orthogonal\ntonand points to one of the corners of the triangle. The\nbasis is completed by q≡p∧n. This coordinate system\nrotates with the particle.\nConsider a particle suspended in a simple shear. De-\nfine the lab-fixed coordinate system by the flow direction\nˆex, the shear direction ˆey, and ˆez≡ˆex∧ˆey, and denote\nthe components of a vector vin this basis by vx,vyand\nvz. Our experiments are conducted in a micro-channel\nflow. Neutrally buoyant particles are placed in the chan-\nnel cross section in such a way that the undisturbed fluid\nvelocity near the particle takes the form U∞=±syˆex.\nHeres>0denotes the undisturbed shear rate.\nWe denote the symmetric part of the matrix of fluid-\nvelocity gradients by S∞, the strain-rate matrix. The\nasymmetric part, O∞, describes the rotation of the flow.\nIts elements are related to those of the vector Ω∞≡3\n1\n2∇∧U∞byO∞x=Ω∞∧xfor an arbitrary vector\nx. The vector Ω∞points in the∓ˆez-direction (‘vorticity\ndirection’).\nWhen diffusion and inertial effects are negligible then\nthe centre-of-mass of triangle simply follows the flow.\nSince the particle has a discrete threefold rotational sym-\nmetry, and since it has a mirror plane containing the axis\nof rotation, its angular dynamics is determined by Jef-\nfery’s angular velocity [8]:\nω=Ω∞−Λ (S∞n)∧n. (1)\nThe corresponding equations of motion for nandpread\nin vector notation [18]:\n˙n=O∞n+ Λ [S∞n−(n·S∞n)n],(2a)\n˙p=O∞p−Λ (n·S∞p)n. (2b)\nDots denote time derivatives, and the ‘Bretherton con-\nstant’[4] Λparametrisestheshapeoftheparticle. Oblate\nspheroidshave−1<Λ<0, andweexpectthattheshape\nparameter for our triangular platelets lies in this range.\nBut we do not know the precise numerical value of Λfor\nour particles. We remark that Eqs. (2) ensure that nand\npremain orthonormal.\nThe dynamics of nis independent of that of p. The\nvector nmoves in one of infinitely many marginally sta-\nble Jeffery orbits, distinguished by the ‘orbit constant’ C.\nOur definition of the orbit constant is slightly different\nfrom that used in Ref. [16], but equivalent. We use\nC≡nzwhenny= 0. (3)\nWhenC= 0, the vector ntumbles in the flow-shear\nplane. It aligns for some time with the shear direction\nˆeyand then rapidly flips by 180 degrees. By contrast,\nC=±1correspondstothelog-rollingorbitwhere nstays\naligned with the vorticity direction. All Jeffery orbits\nhave the same tumbling period TJand frequency:\nωT≡2π\nTJ=s\n2/radicalbig\n1−Λ2. (4)\nNow consider the dynamics of p. Eq. (1) shows that it\ndepends on n. The spinning frequency is given by\nωS≡ω·n=Ω∞·n (5)\nfrom Eq. (1). The spinning frequency vanishes for tum-\nbling in the flow-shear plane, and tends to ±s/2for log\nrolling. In general ωSvaries as ntumbles, and it is in\ngeneral different from ωT. Therefore the dynamics of p\nis doubly periodic. If we follow the p-dynamics we can\nmeasure both sandΛ. Usually this is not possible be-\ncause most experiments are conducted for particles with\na continuous rotation symmetry.\nEqs. (2) require that rotational diffusion [22] is negli-\ngible, which in turn requires that the Péclet number is\nlarge. A lower bound for the Péclet number is given by\nPe≡s/Dmax, where Dmaxdenotes the largest eigenvalue\n     \nˆeyˆexU∞\nx=symicroscopefluid PDMS moving stage glass plate\nmicro-channelpumpillumination\nFIG. 4: Experimental setup (schematic). The ˆez-axis\npoints into the plane. Details are given in the text.\noftherotationdiffusiontensor. Foraflatdiskwithradius\naone estimates [2] Dmax∼kBT/(8πµa3), independently\nof its thickness. Here Tis the temperature, µis the dy-\nnamic viscosity of the fluid, and kBis the Boltzmann\nconstant. Also, Eqs. (2) are derived using the Stokes ap-\nproximation, assuming that particle inertia [23] and fluid\ninertia [24–29] are negligible. For neutrally buoyant par-\nticles this is ensured by requiring that the shear Reynolds\nnumber Res=sa2/νis small. Here νis the kinematic\nviscosity of the fluid.\nStokesflowisreversible[30]. ThisensuresthatEqs.(2)\ntoo are reversible: the particle orientation retraces its\nangular dynamics upon reversal of the flow direction [4].\nDiffusion and inertial effects break reversibility. Testing\nfor reversibility allows us to rule out that diffusion and\ninertia have substantial effects, at least for the duration\nof the experiment.\nIII. METHOD\nThe experimental setup is schematically shown in\nFig. 4. It is similar to the setup used in Ref. [18]. The\nmain differences are, firstly, that the particles are differ-\nent, and secondly that we used a different fluid in this\npaper, namely PEG-10K (aq.).\nThe density of the fluid was measured using an Anton\nPaar DMA 5000 instrument yielding a value of ρ=1.05\ng/cm3. Fluid-particle density matching was achieved by\ntitration while observing the particle with the micro-\nscope. The dynamic viscosity of the fluid was measured\nto 80.2 mPa·s using an Ubbelohde viscometer at a tem-\nperature of 20◦C. During the viscosity measurement, the\nshear rate in tube of the viscometer was in the range 50\nto100s−1. Thedensityandviscositymeasurementsyield\na kinematic viscosity of 76.3 mm2/s. González-Tello et\nal.[31] have shown that similar polyethylen glycoles are4\n10µm\nFIG. 5: Triangular particles obtained from LIQUIDIA\nTechnologies [32]. Reproduced with permission of\nLIQUIDIA Technologies.\nNewtonianuptoshearratesmuchhigherthanthoseused\nin this experiment.\nMicro-channels were made out of polydimethylsilox-\nane (PDMS). The channel was covered by a glass plate\nand fixed to a translation stage. The channel dimen-\nsions were: length 40(1) mm, width 2.5(1) mm, and\ndepth 200(10)µm. In the experiment, the channel was\nmounted on a stage, illuminated from above, and imaged\nfrom below through the glass plate, using a Nikon X60\nmicroscope objective. The objective was fixed in the lab\nframe, while the stage was moved to keep the particle in\nthe view of the objective.\nFor our measurements we considered only particles\nlocated near the horizontal channel centre, at approxi-\nmately equal distance between the channel walls. This\nwas ensured by visual inspection of the particle position\nin the camera plane, and it means that the shear points\nin the vertical direction (Fig. 4). Our setup does not al-\nlow us to directly measure the depth (distance from the\ntop or bottom plates) at which the particles are located.\nIt is in principle possible to estimate the particle depth\nby focusing on the particle with the microscope [18], but\nthis method is not very precise. Therefore we decided to\nobtain the depth by fitting the angular particle dynamics\nto Jeffery orbits in an assumed parabolic velocity profile.\nWe obtained 10µm ‘pollen particles’ from LIQUIDIA\nTechnologies [32]. A representative image of the par-\nticles is shown in Fig. 5. According to the data sheet\nprovided by the company [32], the particles are triangu-\nlar synthetic polymer micro-platelets, 0.5to1µm thick,\nand with side length 10µm. The particle corners form\nan equilateral triangle with a center-to-corner distance\na= 5.8µm. Their refractive index is between 1.5and\n1.6, and their mass density is ρp= 1.05g/cm3. Our\nparticle suspension was highly diluted in order to avoid\ninteraction between particles.\nThe flow rate in the experiments varied. The mea-\nsurements reported in this paper are consistent with a\nflow rate of the order of several µl min−1. Assuming aparabolic profile and a flow rate of 8 µl/min (see below),\nthe flow velocity at the channel centre is U∞\nx= 0.42\nmm/s. A rough estimate (see below) says that the par-\nticle was located between 20and50µm from the top or\nbottom plates. In a parabolic flow profile this means that\nthe shear rate varied between 0 and ∼8 s−1. Together\nwith the numbers quoted above this allows us to estimate\nthe shear Reynolds number to Res≡sa2/ν∼10−6, and\nthe Péclet number to Pe≡s/Dmax∼105. This means\nthat inertial effects are negligible on the time scales of\nthe experiment, and that generally rotational diffusion\nis negligible. When the particle aligns with the flow di-\nrection however, rotational diffusion may kick it out of\nalignment [22].\nTo confirm that neither rotational diffusion nor inertial\neffects significantly affect the particle orientation during\nthe experiment, we reverted the pressure and checked\nthat the particle orientation retraces its trajectory. In\norder to avoid large particle or fluid accelerations, the\nreversals were not instantaneous. The flow velocity was\nreduced slowly, reversed, and increased in a symmetric\nfashion. Reversals lasted for approximately 10 s. Dur-\ning this time rotational diffusion and external noise may\nturn the particle and thus break reversibility. As men-\ntioned above, in this study we discarded data that did\nnot exhibit reversal symmetry.\nSince the flow speed U∞\nxvaried during reversals, we\nplot the angular dynamics not as a function of time tbut\nas a function of the distance ∆x(t)that the centre-of-\nmass position of the particle covers:\n∆x(t) =/integraldisplayt\n0dt/primeU∞\nx(t/prime). (6)\nIn the creeping-flow limit, changes in U∞\nxcorrespond to\nlinear changes of the time scale [33]. In this way we can\ndisregard changes in U∞\nx, provided that the Stokes ap-\nproximation holds. Reversibility implies that the angular\ndynamics collapses onto its reversed counterpart when\nplotted as a function of centre-of-mass position [4].\nThe position of the translation stage was recorded at\na rate of 33 Hz, and the video frames were recorded at\na rate of 20 Hz. We synchronised the two time series by\nvisually matching two video frames with their respective\nstage positions. The position of a video frame relative to\nthe stage was then obtained by linear interpolation. In\nany given video frame, the centre-of-mass position and\nthe orientation of the triangular particle were determined\nmanually. Extracting the time series of pandnis com-\nplicated by two factors. First, the n-dynamics is sensi-\ntive to noise. Second, the projection of a triangle onto\nthe camera plane is ambiguous in that two differently ori-\nented triangles have the same projection onto the camera\nplane. The two possible reconstructions are mirror im-\nages in the camera plane. They have px,pzandnyin\ncommon, but py, andnxandnzare of opposite signs.\nUsing that ωz=nx˙ny−ny˙nxis parallel to the vortic-\nity direction of the undisturbed flow, it follows that their\nmotion corresponds to motion in opposite shears. To re-5\nsolvetheambiguitywechosethereconstructionforwhich\nωz≥0whenU∞\nx>0.\nJeffery orbits were fitted to the particle trajectories\nmeasured before the reversal of the flow. The fit mini-\nmizes the sum of squared distances between the experi-\nmental and theoretical corner locations. The fitting pa-\nrameters are ∆x(TJ),Λ,a, and the three Euler angles\ndescribing the initial orientation of the particle.\nIV. RESULTS AND DISCUSSION\nThe first two panels of Fig. 2 show how the x- andz-\ncomponents of the vector pchange as the particle travels\nalong the channel. We show two trajectories, measured\nbefore and after reverting the channel flow (thin blue\nand red lines). We see that the dynamics reverses fairly\ny[µm]s[s−1]\ny[µm]U∞\nx[mm/s]\nFIG. 6: (Top) shear rate versus particle depth (distance\nfrom bottom plate), as inferred from fits to Jeffery\norbits (see text). The dashed line is the shear rate at a\nflow rate of 8 µl/min. Data are from the experiments\nsummarised in Table I: A I(/triangleleft), A II(/triangle), B I(/triangleinv), B II(⊿),\nBIII(/oblong), and B IV(◦). (Bottom) particle velocity vs.\nparticle depth as inferred from fits to Jeffery orbits (see\ntext). The dashed line is the flow velocity at a flow rate\nof 8µl/min, for a parabolic velocity profile.well, so that inertial effects and rotational diffusion are\nnegligible. In steady flow this is expected, since Resis\nsmall and Peis large. But it is important to note that\nwe estimated the Péclet number for steady flow. During\nreversals the shear rate stends to zero. Reversals last\nfor approximately 10 s, and during this time rotational\ndiffusion and external noise can turn the particle and\nthus break reversibility. In this study we discarded data\nthat does not exhibit reversal symmetry.\nThe gray lines in Fig. 2 show fits of Eqs. (2) to the\nexperimental p-dynamics. From the fits we can infer s,\nΛ, andC. The values obtained are shown in Table I. The\nfit results in Λ≈−0.95. For the orbit constant we find\nC≈0.96, quite close to unity. This means that the an-\ngular dynamics is close to the log-rolling orbit where the\ntriangle lies mostly flat in the flow-shear plane. The cor-\nresponding video-microscopy recording is included in the\nSupplemental Material [21]. The n-dynamics is shown in\nthe lower panels of Fig. 2. The data is noisier than the\np-dynamics because it is reconstructed from a projection\nof the p-dynamics. But it is clear that nzis always quite\nclose to unity, as it must for a log-rolling orbit.\nWe measured the centre-of-mass velocity of the parti-\ncle, ameasureoftheundisturbedfluidvelocity U∞\nxatthe\nparticle position. We obtained U∞\nx∼0.31mm/s. The\nfluctuations in U∞\nxwere negligible for large parts of the\nparticle trajectory. Using ∆x(TJ) =U∞\nxTJwe obtained\nthe Jeffery period from the measured value of ∆x(TJ).\nUsing Eq. (4) and the fitted value of Λwe determined\nthe shear rate to s≈4.3s−1. These values of U∞\nxands\nare consistent with a flow rate 8µl min−1if one assumes\na parabolic flow profile, and that the particle was located\nat a distance of 50µfrom the bottom or top plates. This\nflow rate and particle depth yield U∞\nx= 0.31 mm s−1and\ns= 4.2 s−1(dashed lines in Fig. 6).\nAs pointed out in the Introduction, we can determine\nboth tumbling and spinning frequencies by fitting the\nexperimental data to Jeffery orbits. For Fig. 2, the tum-\nbling frequency (4) evaluates to ωT= 0.65s−1. The spin-\nning frequency, ωS, changes between 1.0s−1and2.1s−1\nfor the smallest and largest values of nz. Since the p-\ndynamics couples to that of n, and since the spinning\nand tumbling frequencies are different, the p-dynamics\nis doubly periodic (first two panels of Fig. 2). The n-\ndynamics, by contrast, is approximately periodic - disre-\ngarding the noise in the experimental data.\nFig. 3 shows data for a different Jeffery orbit. The\ncorresponding values of Λ,C, andsare given in Ta-\nble I. A slight dephasing of an otherwise reversible mo-\ntion is seen in this experiment. This may be due to a\nsmall density mismatch between particle and fluid that\ncauses the particle to rise or sink. For this experiment,\nwe findωT= 1.35s−1, andωSchanges between 0.18s−1\nand 0.94s−1. The orbit constant for this particle is\nC=−0.25, so the n-vector tumbles near the flow-shear\nplane.\nMore experimental results are given in the Supplemen-\ntal Material [21], Figs. S1-S4. All data described in the6\nTABLE I: Summary of experimental data. Steady flow speed U∞\nxestimated from centre-of-mass speed of particle.\nCentre-of-mass distance ∆x(TJ)covered in one Jeffery period TJ, Bretherton constant Λ, orbit constant Cinferred\nfrom the fit to Jeffery orbit, shear rate sinferred from U∞\nx,∆x(TJ), and Λas described in the text. Particle height\nabove bottom inferred from U∞\nxands. Figures S1-S4 are in the Supplemental Material [21].\nParticle ID Trajectory ID Figure U∞\nx ∆x(TJ) ΛC s height above\n[mm/s] [µm] [ s−1] bottom [ µm]\nA I (Fig. 2) 0.31 2962 -.95 .964.30 49\nB I (Fig. 3) -0.19 -904 -.94 .257.60 22\nA II (Fig. S1) 0.32 3097 -.96 -.93 4.47 49\nB II (Fig. S2) 0.17 811 -.95 -.72 8.50 18\nB III (Fig. S3) 0.18 904 -.95 .71 8.23 19\nB IV (Fig. S4) 0.14 859 -.95 -.80 6.83 18\nmaintextandintheSupplementalMaterialwasobtained\nfor two particles, labeled ’A’ and ’B’ in Table I.\nThe inferred shear rates, particle velocities, and parti-\ncle depths for all experiments are also given in this Table.\nFig. 6 shows how the shear rate sand velocity U∞\nxde-\npend on the particle depth, as obtained from the fits to\nJefferyorbits(symbols). Alsoshownaretheexpectedde-\npendencies assuming a parabolic flow profile with a flow\nrate of 8µl/min (dashed lines). Most data points display\nreasonable agreement with this flow rate. The fits imply\nthat particle A was located approximately ∼50µm from\nthe top or bottom plates, and particle B was located at\n∼20µm. In the latter case, the theory based on a flow\nrate of 8µl/min underestimates the measured shear rates\nsomewhat.\nWe remark that Fig. S2 shows that particle B moves\nreversibly, but deviates somewhat from the Jeffery or-\nbit fitted. We see that the n-dynamics agrees well with\nthe fit, but that there are deviations in the p-dynamics.\nSince the wall distance for this particle was only about\n20µm, these deviations could be caused by particle-wall\ninteractions. Numerical simulations [34] show that the\nn-dynamics of an oblate spheroid tumbling close to a\nflat wall is slightly modified by the presence of the wall.\nThe trajectory in Fig. S2 may indicate that the particle\nspin is more sensitive to the presence of the wall than the\ntumbling.\nFinally, Fig. 7 shows a summary of the experimentally\ndetermined periodic n-dynamics. The Figure shows sub-\nsequent values of nzatny= 0, for the orbits shown in\nFigs. 2, 3, and S1-S4. Since the ndynamics is periodic,\nthese values of nzshould remain constant, equal to the\norbit constant C[18], as shown by the dashed line in\nFig. 7. We see that the experimental data is roughly\nconsistent with this expectation. In the experiment the\nvalues fluctuate (Fig. 7). There are several sources of\nerror that could explain the scatter. First, inaccuracies\nin the manual determination of the particle corners in\nthe experimental images could be caused by systematic\nchanges in the diffraction patterns upon changing orien-\ntation. Second, effects of particle-wall interactions can-\nnot be excluded, especially since some particles were lo-\nn(i)\nzn(i+1)\nz\nFIG. 7: Periodic dynamics of nz. We measured nz\nwhenny= 0to obtain a discrete time series n(i)\nzfor\ni= 1,2,.... The graph shows n(i+1)\nzversusn(i)\nz.\nSymbols as in Fig. 6. Also shown is the theoretical\nexpectation (dashed line) derived from the periodicity\nof the n-dynamics: n(i+1)\nz =n(i)\nz.\ncated near the wall. Wall irregularities may enhance the\neffect. Third, we cannot rule out that small particle-\nshape irregularities may cause deviations from the dy-\nnamics predicted for symmetric particles [18]. We note\nthat Figs. 2, 3, and S1-S4 show reversible angular dy-\nnamics. This rules out that the fluctuations are due to\nrotational diffusion.\nV. CONCLUSIONS\nWe measured the angular dynamics of micron-sized\nplatelets with 3-fold rotation symmetry in a micro-\nchannel shear flow. The angular dynamics is well de-\nscribed by Jeffery’s equation of motion. This supports\nthe conclusion [8] that a particle with a discrete k-fold\nrotation symmetry ( k >2) and a mirror plane contain-\ning the rotation axis moves according to Jeffery’s equa-7\ntion. The resulting angular trajectories are doubly peri-\nodic (Figs. 2 and 3) because tumbling and spinning fre-\nquencies are not commensurate.\nOur measurements are not accurate enough to system-\natically study the effect of minute shape asymmetries.\nBut since we can track the corners of the particle, we\ncould in principle describe the particle orientation by its\nEuler angles [35] (θ,φ,ψ )and represent the particle dy-\nnamics by a Poincaré surface-of-section [17, 18]. This\nwould make it possible to study how breaking of the dis-\ncrete rotation symmetry may give rise to chaotic tum-\nbling. It would also be interesting to measure the angu-\nlar dynamics of particles with k-fold rotation symmetry\nthat lacks a mirror plane containing the rotation axis.\nWe predicted [8] that there are such particles that tum-\nble like spheroids, but spins differently: ω·ndoes notvanish when ntumbles in the flow-shear plane.\nACKNOWLEDGMENTS\nWe thank E. Variano and G. Voth for discussions. We\nthank LIQUIDIA for permission to reproduce Fig. 5. We\nacknowledge support by Vetenskapsrådet [grant numbers\n2013-3992 and 2017-03865], Formas [grant number 2014-\n585], by the grant ‘Bottlenecks for particle growth in\nturbulent aerosols’ from the Knut and Alice Wallenberg\nFoundation, Dnr. KAW 2014.0048, by the Carl Trygger\nFoundation for Scientific Research, and by the MPNS\nCOST Action MP1305 ‘Flowing matter’.\n[1] J. Happel and H. Brenner, Low Reynolds number hydro-\ndynamics: with special applications to particulate media ,\nVol. 1 (Springer Science & Business Media, 1983).\n[2] S. Kim and S.J. Karrila, Microhydrodynamics: Princi-\nples and Selected Applications. 2005 (DoverPublications,\n2005).\n[3] G.B. Jeffery, “The motion of ellipsoidal particles im-\nmersed in a viscous fluid,” Proceedings of the Royal Soci-\nety of London A: Mathematical, Physical and Engineer-\ning Sciences 102, 161–179 (1922).\n[4] F.P. Bretherton, “The motion of rigid particles in a shear\nflow at low Reynolds number,” J. Fluid Mech. 14, 284–\n304 (1962).\n[5] U. Nakaya, Snow Crystals: Natural and Artificial (Har-\nvard University Press, 1954).\n[6] J. S. Guasto, R. Rusconi, and R. Stocker, “Fluid me-\nchanics of planktonic microorganisms,” Ann. Rev. Fluid\nMech.44, 373 (2012).\n[7] M. D. Guiry and G. M. Guiry, AlgaeBase,\nwww.algaebase.org , accessed 19 July (2016).\n[8] Johan Fries, Jonas Einarsson, and Bernhard Mehlig,\n“Angular dynamics of small crystals in viscous flow,”\nPhysical Review Fluids 2, 014302 (2017).\n[9] S. Parsa, E. Calzavarini, F. Toschi, and G. A. Voth,\n“Rotationrate ofrodsin turbulent fluid flow,” Phys.Rev.\nLett.109, 134501 (2012).\n[10] K. Gustavsson, J. Einarsson, and B. Mehlig, “Tumbling\nof small axisymmetric particles in random and turbulent\nflows,” Phys. Rev. Lett. 112, 014501 (2014).\n[11] R. Ni, N. T. Ouelette, and G. A. Voth, “Alignment of\nvorticityandrodswithLagrangianfluidstretchingintur-\nbulence,” J. Fluid Mech. 743, R3 (2014).\n[12] L. Chevillard and C. Meneveau, “Orientation dynamics\nof small, triaxial-ellipsoidal particles in isotropic turbu-\nlence,” J. Fluid Mech. 737, 571 (2013).\n[13] M. Byron, J. Einarsson, K. Gustavsson, G. Voth,\nB. Mehlig, and E. Variano, “Shape-dependence of par-\nticle rotation in isotropic turbulence,” Phys. Fluids 27,\n035101 (2015).\n[14] L. Zhao, N. R. Challabotla, H. I. Andersson, and E. A.\nVariano, “Rotation of nonspherical particles in turbulent\nchannel flow,” Phys. Rev. Lett. 115, 244501 (2015).[15] G. Voth and A. Soldati, Annu. Rev. Fluid Mech. (2016).\n[16] E. J. Hinch and L. G. Leal, “Rotation of small non-\naxisymmetric particles in a simple shear flow,” J. Fluid\nMech.92, 591–608 (1979).\n[17] A.L.Yarin, O.Gottlieb, andI.V.Roisman,“Chaoticro-\ntation of triaxial ellipsoids in simple shear flow,” J. Fluid\nMech.340, 83–100 (1997).\n[18] J. Einarsson, B. M. Mihiretie, A. Laas, S. Ankardal,\nJ. R. Angilella, D. Hanstorp, and B. Mehlig, “Tumbling\nof asymmetric microrods in a microchannel flow,” Phys.\nFluids28, 013302 (2016).\n[19] G. G. Marcus, S. Parsa, S. Kramel, Ni. R., and\nG. A. Voth, “Measurements of the solid-body rotation\nof anisotropic particles in 3d turbulence,” New J. Phys.\n16, 102001 (2014).\n[20] A. Bordoloi and E Variano, “Rotational kinematics of\nlarge cylindrical particles in turbulence,” J. Fluid Mech.\n815, 199 (2017).\n[21] Supplemental Material. Figs. S1–S4 and corresponding\nvideo-microscopy recordings.\n[22] E. J. Hinch and L. G. Leal, “The effect of Brownian mo-\ntion on the rheological properties of a suspension of non-\nspherical particles,” J. Fluid Mech. 52, 683–712 (1972).\n[23] J. Einarsson, J. R. Angilella, and B. Mehlig, “Orienta-\ntional dynamics of weakly inertial axisymmetric particles\nin steady viscous flows,” Physica D: Nonlinear Phenom-\nena278–279 , 79–85 (2014).\n[24] G. Subramanian and D. L. Koch, “Inertial effects on fibre\nmotion in simple shear flow,” J. Fluid Mech. 535, 383–\n414 (2005).\n[25] J. Einarsson, F. Candelier, F. Lundell, J.R. Angilella,\nand B. Mehlig, “Rotation of a spheroid in a simple shear\nat small Reynolds number,” Phys. Fluids 27, 063301\n(2015).\n[26] J. Einarsson, F. Candelier, F. Lundell, J.R. Angilella,\nand B. Mehlig, “Effect of weak fluid inertia upon Jeffery\norbits,” Phys. Rev. E 91, 041002(R) (2015).\n[27] F. Candelier, J. Einarsson, F. Lundell, B. Mehlig, and\nJ.R. Angilella, “The role of inertia for the rotation of a\nnearly spherical particle in a general linear flow,” Phys.\nRev. E91, 053023(erratum 059901) (2015).\n[28] T. Rosén, J. Einarsson, A. Nordmark, C. K. Aidun,8\nF. Lundell, and B. Mehlig, “Numerical analysis of the\nangular motion of a neutrally buoyant spheroid in shear\nflowatsmallReynoldsnumbers,” Phys.Rev.E 92(2015),\n063022.\n[29] J. Meibohm, F. Candelier, T. Rosen, J. Einarsson,\nF. Lundell, and B. Mehlig, “Angular velocity of a\nspheroid log rolling in a simple shear at small Reynolds\nnumber,” Phys. Rev. Fluids 1, 084203 (2016).\n[30] G. I. Taylor, Low Reynolods number Flow, 16mm film\n(Newton, Massachusetts: Educational Services Incorpo-\nrated, 1960).\n[31] Pedro Gonzalez-Tello, Fernando Camacho, and Gabriel\nBlazquez, “Density and viscosity of concentrated aque-\noussolutionsofpolyethyleneglycol,” JournalofChemicaland Engineering Data 39, 611–614 (1994).\n[32]LIQUIDIA Technologies, www.liquidia.com (P.O. Box\n110085, Research Triangle Park, NC 27709).\n[33] J. Einarsson, A. Johansson, S. K. Mahato, Y. N. Mishra,\nJ.R. Angilella, D. Hanstorp, and B. Mehlig, “Peri-\nodic and aperiodic tumbling of microrods advected in\na microchannel flow,” Acta Mechanica 224, 2281–2289\n(2013).\n[34] Nipa A Mody and Michael R King, “Three-dimensional\nsimulations of a platelet-shaped spheroid near a wall in\nshear flow,” Physics of Fluids 17, 113302 (2005).\n[35] H. Goldstein, Classical Mechanics (Addison-Wesley,\nReading, Massachusetts, 1980).Supplementaray online material\nSpinning and tumbling of micron-sized triangles in a micro-channel shear flow\nJ. Fries,1,a)M. Vijay Kumar,1,a)B. Mekonnen Mihiretie,1D. Hanstorp,1and B. Mehlig1\nDepartment of Physics, University of Gothenburg, SE-41296 Gothenburg,\nSweden\n(Dated: 9 November 2018)\na)These authors contributed equally to this work.\n1arXiv:1712.08505v1  [physics.flu-dyn]  22 Dec 2017px\npz\nnx\nny\nnz\n∆x\nFIG. S1. Same as Fig. 1 in the main text, but for a different Jeffery orbit. Trajectory ID: A II.\nSee Table I in the main text.\n2px\npz\nnx\nny\nnz\n∆x\nFIG. S2. Same as Fig. 1 in the main text, but for a different Jeffery orbit. Trajectory ID: B II.\nSee Table I in the main text.\n3px\npz\nnx\nny\nnz\n∆x\nFIG. S3. Same as Fig. 1 in the main text, but for a different Jeffery orbit. Trajectory ID: B III.\nSee Table I in the main text.\n4px\npz\nnx\nny\nnz\n∆x\nFIG. S4. Same as Fig. 1 in the main text, but for a different Jeffery orbit. Trajectory ID: B IV.\nSee Table I in the main text.\n5"    },    {        "title": "2403.00524v1.Chaos_assisted_Turbulence_in_Spinor_Bose_Einstein_Condensates.pdf",        "content": "Chaos-assisted Turbulence in Spinor Bose-Einstein Condensates\nJongmin Kim,1,∗Jongheum Jung,1,∗Junghoon Lee,1Deokhwa Hong,1and Y . Shin1, 2,†\n1Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea\n2Institute of Applied Physics, Seoul National University, Seoul 08826, Korea\nWe present a turbulence-sustaining mechanism in a spinor Bose-Einstein condensate, which is based on the\nchaotic nature of internal spin dynamics. Magnetic driving induces a complete chaotic evolution of the lo-\ncal spin state, thereby continuously randomizing the spin texture of the condensate to maintain the turbulent\nstate. We experimentally demonstrate the onset of turbulence in the driven condensate as the driving frequency\nchanges and show that it is consistent with the regular-to-chaotic transition of the local spin dynamics. This\nchaos-assisted turbulence establishes the spin-driven spinor condensate as an intriguing platform for exploring\nquantum chaos and related superfluid turbulence phenomena.\nTurbulence, a ubiquitous phenomenon in fluids, poses a ma-\njor challenge in physics owing to its complexity [1]. To effec-\ntively investigate this phenomenon, it is essential to generate\nturbulence in a controlled manner. Various existing methods,\nsuch as moving grids [2–4], impinging jets [5, 6], and rotating\ndrums [7–9], have provided valuable insights into the differ-\nent aspects of turbulent states, highlighting the strong connec-\ntion between the generation method and flow dynamics. How-\never, these approaches are primarily based on external forces\nand inertial energy cascades [10–12], which might limit the\npossibilities of exploring novel and unconventional turbulent\nregimes. In this study, we propose an alternative approach to\ngenerate turbulence by harnessing the intrinsic chaos within\nthe fluid itself. By utilizing the unpredictable and sensitive\nnature of chaotic systems, this method introduces energy di-\nrectly into the flow through self-amplifying mechanisms such\nas the stretching and folding of fluid elements [13]. This\nchaos-based method holds the potential for improved mixing\nand the creation of a more energetic and sustained turbulent\nstate, thereby offering the opportunity to explore previously\nuncharted turbulent regimes.\nIn this Letter, we present a case study of chaos-assisted tur-\nbulence in a driven spin-1 atomic Bose-Einstein condensate\n(BEC). Previous studies have observed a stationary turbulent\nstate in the BEC when it is subjected to continuous radio fre-\nquency (rf) magnetic fields [14, 15]. Under specific driving\nconditions, the turbulence attains its maximum intensity, ac-\ncompanied by an isotropic spin composition. Although the\noscillating magnetic field, aided by noise, has been identified\nas the driving force, the mechanism that sustains the gener-\nated turbulence while maintaining the isotropic spin nature\nhas remained unclear. Here, through numerical simulations\nand experimental validation, we demonstrate that the applied\nmagnetic field induces chaotic motion within local spin states,\nthus sustaining the turbulent state. This finding concretizes the\nnotion of chaos-assisted turbulence generation, particularly in\na superfluid system, paving the way for a deeper understand-\ning and control of this complex and fascinating phenomenon.\nA superfluid with internal spin degrees of freedom presents\na departure from conventional superfluid dynamics. The order\nparameter of the superfluid can be expressed as Ψ =√neiφζ,\nwhere nis the superfluid density, φis the superfluid phase,andζis the spinor for the spin state [16]. Due to the intricate\nrelation between the superfluid phase and the spin rotation,\nmass superflow is associated not only with the spatial varia-\ntions of φbut also with those of ζ, i.e., with the spin texture.\nThe superfluid velocity and its vorticity are given by\nvs=ℏ\nm(∇φ−iζ†∇ζ),\n∇ ×vs=−iℏ\nm∇ζ†× ∇ζ, (1)\nwhere mis the particle mass and ℏis the reduced Planck con-\nstant, h/2π. Therefore, a mass superflow can be generated by\nmanipulating the spin texture in space.\nIn this work, we consider a situation in which the local dy-\nnamics of ζis in a classically chaotic regime. Then, because\nof the sensitivity on initial conditions, small spatial fluctua-\ntions in a superfluid, even starting with a uniform spin tex-\nture, would develop into complex spatial variations of the spin\nstates over time evolution, leading to an irregular spin texture.\nIf the spin texture is kept irregular because of the chaotic na-\nture of the local spin dynamics, the resulting turbulent flow in\nthe superfluid would be sustained. This is a scenario of the\naforementioned chaos-assisted turbulence generation.\nOur superfluid system is a BEC of23Na atoms in the F=1\nhyperfine state with antiferromagnetic interactions [17, 18].\nA uniform external magnetic field of Bzis applied along ˆz,\nand for spin driving, an rf magnetic field with oscillating fre-\nquency of ωis applied along the transverse direction. In a\nmean-field description, neglecting the spatial modes of the\nBEC and taking the rotating wave approximation, the local\ndynamics of the spin state ζ= (ζ+1, ζ0, ζ−1)Tis described by\nthe following Hamiltonian per particle,\nHs=ℏδfz−ℏΩfx+qζ†f2\nzζ+εs|f|2, (2)\nwhere f= (fx,fy,fz)are the spin operators of the spin-1 sys-\ntem and f=ζ†fζis the spin vector with fx,y,z represent-\ning the magnetizations in x, y, andzdirections, respectively.\nδ=ω−ω0is the frequency detuning of the rf magnetic field\nfrom the Larmor frequency ω0=1\n2µBBz/ℏwithµBbeing the\nBohr magneton, Ωis the Rabi frequency of the rf field, and q\ndenotes the quadratic Zeeman energy. The last term represents\nthe energy of spin interaction, which introduces nonlinearity\nto the system.arXiv:2403.00524v1  [cond-mat.quant-gas]  1 Mar 20242\n(a)\n(c) (d)(b)\nI\nIIIIIIV \nI IIIIIIV fxfyfz\nI II\nIII IV \nx yz\n-į0+D \n-į0-D -į0\n\nfxfyfzfxfyfz\nfxfyfz\nFIG. 1. Chaotic spin dynamics in a spinor Bose-Einstein condensate\nunder magnetic field modulations. (a) The linear Zeeman field in\nEq. (2) oscillates as Ωˆx−δ(t)ˆzwithδ(t) =δ0+Dsin(2πνt+ϕ)\nandν= 60 Hz. (b) Magnetization, f= (fx, fy, fz), distributions\nof a spin state ensemble after 1-s spin driving for four different sets\nof driving parameters: (I) {δ0, D}/2π={250,0}, (II){250,700},\n(III){750,700}, and (IV) {1000,700}Hz. The ensemble consists\nof1282spin states, which initially have f≈0(see text for details).\n(c) Time evolution of the second-momentum trace distances ∆(2)\nbetween the spin state ensemble and the Haar random ensemble for\n(I)-(IV). The dashed line represents the expectation value for a set of\n1282spin states sampled from the Haar ensemble. (d) ∆(2)after 1-s\nspin driving as a function of δ0andD.\nWhen the energy scales of εs,q, and ℏΩare comparable and\nδ= 0, it is known that the spin dynamics of the Hamiltonian\nHsbecomes chaotic [19, 20]. Here, we investigate a magnetic\ndriving scheme where the external magnetic field Bzis modu-\nlated such that δ(t) =δ0+Dsin (2 πνt+ϕ)[Fig. 1(a)]. This\nfield modulation breaks the energy conservation constraint,\npossibly enhancing the chaoticity of the system and facilitat-\ning complete randomization of the spin state [21, 22].\nTo demonstrate the chaotic behavior of the spin system,\nwe numerically investigate the time evolution of an ensemble\nof spin states, E, and characterize it in the coordinate space\nof magnetization f. The ensemble Econsists of 1282spin\nstates, whose initial states are constructed from the mF= 0\nstate ( ζ= (0 ,1,0)T) with small Gaussian random noise\nadded [23], so f≈0. The system parameters are set to\nεs/h= 45 Hz,q/h = 47 Hz,Ω/2π= 200 Hz and\nν= 60 Hz. Figure 1(b) shows the fdistributions of the\nensemble after the evolution of 1 s for various sets of driving\nparameters {δ0, D}. Without field modulation ( D= 0) and\nfor small δ0(case I), the magnetization is dispersed on the\nplane perpendicular to the linear Zeeman field, but far fromfully randomized. Meanwhile, when subjected to the field\nmodulation, the ensemble spreads over a broader region of the\nmagnetization space (cases II and III), implying the enhanced\nspin mixing of the system. [23]\nAs a proxy of the randomness of the spin states in E, we\nestimate a trace distance,\n∆(2)≡1\n2||ρ(2)\nE−ρ(2)\nHaar||1, (3)\nwhere ρ(2)\nE=P\nζ∼E\u0000\nζζ†\u0001⊗2is the second momentum of the\nensemble and ρ(2)\nHaaris that of the Haar random ensemble EHaar,\nwhich is a unitarily invariant, maximally randomized ensem-\nble of the spin-1 system [22, 24, 25]. Here, || · || 1denotes the\ntrace norm, and ∆(2)represents the difference from the fully\nrandom ensemble in second momentum. Figure 1(c) shows\nthe evolution curves of ∆(2)for the different driving parame-\nters. In a specific driving condition, Ebecomes fully random-\nized as ∆(2)≃0. In Fig. 1(d), the value of ∆(2)after 1 s of\nspin driving is displayed in the plane of δ0andD. Interest-\ningly, the dynamic behavior of the system exhibits a sudden\ntransition from chaotic to regular as δ0exceeds a threshold\nvalue δth. The value of δthis found to be linearly proportional\nto the magnitude of field modulation as δth≃D, highlighting\nthe role of field modulation in the randomization of the spin\nstate.\nWe extend our discussion to the dynamics of a BEC with\nspatial extent, e.g., in two dimensions. Here, the spinor con-\ndensate can be viewed as the spatial array of the local non-\nlinear spin subsystems that are coupled to their neighbors.\nWe numerically calculate the system’s evolution using spin-\n1 Gross-Pitaevskii equations (GPEs) [23, 26]. We prepare the\ninitial condensate in the mF= 0 state with small random\nnoises [15, 23], and observe that an irregular spin texture de-\nvelops under spin driving, e.g., with δ0/2π= 500 Hz and\nD/2π= 800 Hz [Fig. 2(a) case B]. The irregular spin texture\nimplies turbulent superflow in the spinor BEC, as verified by\nthe vorticity distribution of the superfluid velocity in Fig. 2(b).\nThe range of driving parameters for the generation of turbu-\nlence is well aligned with that for the chaotic spin dynamics.\nIn Figs. 2(c) and 2(d), we plot the trace distance ∆(2)and the\nkinetic energy Kof the system in the δ0-Dplane, respectively.\nHere, ∆(2)is measured from the spin state ensemble Eob-\ntained by position projection of the condensate [15] and the ki-\nnetic energy is calculated as K=R\nd2rΨ†(−ℏ2\n2m∇2)Ψ. The\nrelationship of δth≃Dis still observed as in Fig. 1(d), which\nclearly demonstrates that the turbulence generation originates\nfrom chaotic spin dynamics. Meanwhile, the value of ∆(2)\nfor turbulent BEC appears to be relatively higher than the cor-\nresponding value in Fig. 1(d). It is attributed to the energy\ndissipation process through spin-wave relaxation [14], which\nwould homogenize the spin texture. Further characterization\nof the velocity fields of the turbulent BEC is provided in [23],\nwhere a −5/3power-law scaling was observed in the incom-\npressible part of the kinetic energy [34].\nNext, we conduct an experimental verification of the turbu-\nlence generation mechanism assisted by chaotic spin dynam-3\nFIG. 2. Turbulence generation in a spinor BEC by spin driving. (a)\nMagnetization Fzand (b) vorticity ∇ ×vsdistributions in a two-\ndimensional BEC after 1-s spin driving for three different driving\nconditions: (A) {δ0, D}/2π={0,0}, (B){500,800}, and (C)\n{1250,500}Hz.Fz=nfzis the density of magnetization along\nthezdirection and vsis the superflow velocity. n0is the mean parti-\ncle density, εsdenotes the spin interaction energy scale, and ξsis the\nspin healing length. (c) ∆(2)and (d) kinetic energy K, as functions\nofδ0andD.Nis the total particle number and µdenotes the chem-\nical potential of the BEC.\nics. We prepare a BEC consisting of about 3.5×106 23Na\natoms in the |F=1, mF=0⟩state. The BEC is trapped in an\noptical dipole trap of highly oblate geometry and its Thomas-\nFermi (TF) radii are (Rx′, Ry′, Rz′)≈(160,75,1.6)µm,\nwhere x′, y′andz′denote spatial coordinates. For the peak\natomic density at the trap center, the spin interaction en-\nergy is εs≈h×53Hz [27] and the spin healing length\nξs=ℏ/√2mεs≈2.0µm, comparable to the thickness of the\nsample. A uniform external magnetic field of Bz≈0.41G is\napplied along ˆz= (−ˆx′+ˆy′)/√\n2, giving ω0≈2π×291kHz\nandq≈h×47Hz. An rf magnetic field is applied along ˆy′\nwithΩ = 2 π×150Hz. The external magnetic field Bzis si-\nnusoidally modulated at ν= 60 Hz with a variable magnitude\nof less than a few mG.\nFigure 3(a) shows an optical density (OD) image of a\nBEC, taken along the ˆz′direction after a 18-ms time-of-\nflight [28]. The BEC was spin-driven for 2 s with δ0= 0and\nD/2π= 950 Hz. A stationary turbulent state appears in the\ndriven BEC, manifested by an irregular density distribution\nthat arises after free expansion from the chaotic velocity field\nof the driven condensate. Furthermore, the turbulent BEC\nexhibits equal populations in all three spin states. The spin\nisotropy of the turbulent BEC was demonstrated in Ref. [14]\nFIG. 3. Observation of the turbulence onset in a driven spinor BEC.\n(a)-(c) Time-of-flight images of BECs after spin driving for 2 s with\nδ0= 0,D/2π= 950 Hz,ν= 60 Hz, and ϕ= 0. Turbulence\nappears as irregular density patterns in the freely expanding BECs.\nA magnetic field gradient B′was applied along the x′direction dur-\ning spin driving. (d) Image after Stern-Gerlach spin separation for\nB′= 0.17G/cm. The deformation of the clouds due to separation is\ncompensated by perspective transformation [23]. (e) Integrated den-\nsity profiles of the three spin components. The width wof the spin\nturbulence region was determined from the density distribution of the\nmF= 0 component. (f) 1/was a function of B′. Each data point\nwas obtained from the twenty to thirty measurements and its error\nbar indicates their standard deviation. The solid line is a linear line\nofw−1=αB′withα= 3.7×104/G, fit to the data. The inset shows\nthe values of δth=µB\n4ℏB′w. The horizontal line and the shaded re-\ngion indicate their mean and standard deviation, respectively.\nby showing that the populations of the three spin states are\nequal regardless of the quantization axis.\nWe investigate how the frequency detuning δ0of the rf driv-\ning field affects turbulence generation by conducting a similar\nexperiment with an additional magnetic field gradient B′ap-\nplied along the long axis of the condensate. This field gradi-\nent renders the frequency detuning of the rf driving spatially\nvaring across the condensate as δ0(x′) = (1\n2µBB′x′)/ℏ. We\nobserve that spin-isotropic turbulence occurs within a central\nregion of the condensate [Fig. 3(b) and 3(c)], whose spatial\nextent decreases as B′increases, implying the existence of a\nrange of δ0effective for the generation of turbulence.\nThe image obtained after Stern-Gerlach spin separation re-\nveals that the turbulent region is positioned between two fer-\nromagnetic domains with opposite magnetizations [Fig. 3(d)].\nFor high |δ|, the ground state of the driven BEC is ferromag-\nnetic owing to the large Zeeman energy. As the BEC is driven,4\nthe ferromagnetic domains gradually appear at both ends of\nthe condensate and reach saturation in the steady state, con-\ntaining vortices [23]. The interface between the turbulent re-\ngion and the ferromagnetic domains is clearly defined, show-\ning sudden changes in density for the different spin compo-\nnents [Fig. 3(d) and 3(e)]. This allows us to reliably deter-\nmine the turbulent region from the density distribution of the\nmF= 0component.\nWe measure the spatial extent wof the spin-isotropic tur-\nbulent region along the direction of the field gradient, and\nfind that its inverse is proportional to the magnitude of the\nfield gradient [Fig. 3(f)]. This means that the threshold value\nofδ0for turbulence generation is consistently determined as\nδth=δ0(w\n2) =µBB′w/4ℏ. The presence of such a threshold\ndetuning is in excellent agreement with the numerical results\nshown in Fig. 2, providing clear evidence of the regular-to-\nchaotic transition in the driven spinor BEC system.\nThe relationship of δthand the magnitude Dof the field\nmodulations is investigated (Fig. 4). As Dincreases, in gen-\neral,δthincreases, which is consistent with the expected be-\nhavior of the spin dynamics. However, we observe that a tur-\nbulent region is formed even in the absence of field modula-\ntions, indicating a threshold detuning of δth,0= 0.74(9) kHz\nforD= 0. We find that it is caused by ambient magnetic field\nfluctuations, which amounts to about 1 mG [14]. Magnetic\nfield noises are likely generated by current ripples in the mag-\nnetic coils that leak from the 60-Hz AC power line. In fact, for\nν= 60 Hz,δthshows a sinusoidal dependence on the phase ϕ\nof the external field modulation [Fig. 4(c)], and it arises from\ninterference with the background field component oscillating\nat the same frequency.\nTaking into account such ambient field fluctuation effects,\nwe present a model of the threshold detuning as\nδ2\nth=β2\nν\u0000\nD2+ 2DDνcos(ϕ−ϕν)\u0001\n+δ2\nth,0, (4)\nwhere Dνandϕνrepresent the amplitude and phase of the\nbackground field that oscillates at frequency ν, respectively,\nandβνdenotes the proportionality of δthto the field mod-\nulation [23]. Our measurement results of δthfor various D\nandϕare well described by the model [Fig. 4(a)], suggesting\nDν/2π= 0.39(5) kHz and βν= 1.05(6) forν= 60 Hz.\nIn Fig. 4(b), we show the measurement data of δthforν=\n200Hz and 1 kHz, where the model curve fitting to the data\nprovides Dν≈0andβν= 1.15(4) and 0.59(4), respectively.\nIt is evident that when the modulation frequency surpasses the\nenergy scales relevant to the system, the efficiency of the mix-\ning caused by the external field modulation is reduced.\nIn conclusion, we have presented a turbulence generation\nmechanism based on chaos and demonstrated it with the\nspinor BEC system under magnetic field driving. The pres-\nence of the threshold frequency detuning δthand its linear re-\nlationship with Dwere demonstrated, in excellent agreement\nwith the predicted regular-to-chaotic transition. The station-\nary turbulence state of the spinor BEC provides interesting\nopportunities to explore nonequilibrium superfluid dynamics.\nFIG. 4. Threshold detuning δthof the spin driving for turbulence gen-\neration. Dependence of δthon the driving amplitude Dfor (a) ν=\n60Hz , (b) 200 Hz and 1 kHz. The gray horizontal lines indicate the\nthreshold detuning value of δth,0/2π= 0.74(9) kHz due to ambient\nfield noise. Corresponding δthdata for different modulation phases ϕ\nwithD/2π= 1kHz for (c) ν= 60 Hz , (d) 200 Hz and 1 kHz. Each\ndata point indicates the mean value of ten to twenty measurements,\nand its error bar is the standard deviation of the measurements. The\nsolid lines in (a) and (c) are the model curves of Eq. (4) with the\nparameter values of {βν, Dν/2π, ϕν}={1.05,0.39kHz,0.86π},\nwhich were determined from a model fit to the data in (c). The solid\nlines in (b) are model curves fit to the data with Dν= 0. The dashed\nlines indicate the corresponding model curves with δth,0= 0.\nAn important next step is to compare this chaos-assisted tur-\nbulence with conventional hydrodynamic turbulence, where\nenergy is injected on a certain length scale and an inertial en-\nergy cascade follows [10–12, 23]. Since our driven system\nexhibits spatio-temporally chaotic flow induced by its internal\nspin dynamics, it might be discussed in the context of active\nturbulence [29, 30], which is self-driven turbulence by sponta-\nneous flow instability. Further research should focus on quan-\ntifying the spin texture [31, 32] and the mechanisms of energy\ntransfer and dissipation [33, 34] of the turbulent BEC.\nThis work was supported by the National Research Foun-\ndation of Korea (Grants No. NRF-2023R1A2C3006565 and\nNo. NRF-2023M3K5A1094811).\n∗These authors contributed equally to this work\n†yishin@snu.ac.kr\n[1] R. P. Feynmann, R. B. Leighton, M. Sands, The Feynman lec-\ntures on physics , V ol. 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Gardiner,\nDynamics and statistical mechanics of ultra-cold Bose gases us-\ning c-field techniques, Advances in Physics 57, 363 (2008).6\nSupplemental Material:\nChaos-assisted Turbulence in Spinor Bose-Einstein Condensates\nA. Numerical simulation of the spin dynamics\nFrom the Hamiltonian of Eq. (2), the equation of motion for the spin state of the BEC is given by\niℏ∂tζ=h\nℏδ(t)fz+qf2\nz−ℏΩfx+εsX\ni=x,y,zfifii\nζ. (S1)\nWe numerically solved it using a relaxation scheme [35], with q/h= 47 Hz,Ω/2π= 200 Hz,εs/h= 45 Hz for various\ndriving conditions of δ(t) =δ0+Dsin(2πνt+ϕ). The initial state was the easy-axis polar (EAP) state along the zaxis, that\nis,ζ0= (0,1,0)T. To consider the effect of quantum noise, we added a Gaussian noise of κη=κ(η1, η2, η3)Tto the initial\nstate, where ηj=ηj,R+iηj,I(j= 1,0,−1) are complex random variables with ηj,Randηj,Ibeing real random numbers of the\nstandard normal distribution. We set κ= 0.005[15], which represents the ratio of the amount of noise to the particle density.\nIn the analysis of the time evolution of a spin state ensemble, we consider a set of 1282spin states whose noise components are\nchosen independently.\nFIG. S1. Magnetization f={fx, fy, fz}distributions of the spin state ensemble at various rf driving times τdfor (a) {δ0, D}/2π=\n{250,0}Hz (case I in Fig. 1) and (b) {250,700}Hz (case II in Fig. 1), and ν= 60 Hz. (c) Trajectory of a spin state in the magnetization\nspace for 0< τd<20ms, where the initial state is ζ0= (0,1,0)Tand the driving parameters are the same in (b). The color of the trajectory\ndenotes the evolution time τd.\nB. Gross-Pitaevskii equations of spin-1 Bose-Einstein condensates\nThe dynamics of a two-dimensional BEC under spin driving was numerically investigated using spin-1 Gross-Pitaevskii\nequations (GPEs),\niℏ∂tΨ =h\n−ℏ2\n2m∇2+ℏδ(t)fz+qf2\nz−ℏΩfx+gnn+gsX\ni=x,y,zFifi−µi\nΨ, (S2)\nwhere gnandgsdenote the particle and spin interaction coupling constants, respectively, µ=gnn0=γ2εsis the chemical\npotential of the condensate, and F= (Fx, Fy, Fz) =n(fx, fy, fz)is the magnetization density. For F= 123Na BECs,\n∗These authors contributed equally to this work\n†yishin@snu.ac.kr7\nγ=p\ngn/gs= 5.3. The initial state was the EAP state, Ψ0=√n0(0,1,0)Twithn0= 104/ξ2\nsand noise was added. To\nincorporate quantum fluctuations, we used the truncated Wigner approximation [36]. The noise distribution corresponds to the\nsummation of a half quantum for each Bogoliubov excitation mode in the EAP state for q→ ∞ [37]. The GPEs in Eq. (S2)\nwere solved numerically using a relaxation pseudospectral scheme [35]. In Fig. 2, the size of the system was 160ξs×160ξs,\ncovered by a 256×256grid of equally spaced points, where ξs≃2.3µm.\nC. Power spectrum of incompressible kinetic energy\nTsubota et al. [31, 32, 34] investigated spin-superflow turbulence in ferromagnetic spin-1 BECs using analytical and numerical\nmethods, and showed that the turbulent states arising from counterflow instability exhibit −7/3and−5/3power-laws in the\nspectra of the spin interaction energy and the incompressible part of the kinetic energy, respectively. In our numerical study\nof BEC dynamics under spin driving, we have observed that the same power-law behavior appears in the driven BEC with\nantiferromagnetic interactions, not only in the spin interaction energy, as described in our earlier work [15], but also in the\nincompressible part of the kinetic energy.\nFor the analysis of the kinetic energy spectrum, we followed the method described in [34]. The energy of superfluid velocity\nis given by Ev=m\n2R\nd2r|Av|2, where Av=√nvs. According to the Helmholtz theorem, the weighted velocity field Avcan\nbe decomposed as Av=Aiv+Acvsuch that ∇ ·Aiv= 0and∇ ×Acv=0. Then, the kinetic energy is expressed as the sum\nof two parts as Ev=Eiv+EcvwithEαv=m\n2R\nd2r|Aαv|2(α∈ {i, c}).Eiv(Ecv) is called the incompressible (compressible)\npart of the superfluid velocity energy. The power spectrum of Eivis obtained as follows:\nAiv(r) =X\nk˜Aiv(k)eik·r\nEiv(k) =Nm\n2n0X\nk<|k|<k+∆k|˜Aiv(k)|2. (S3)\nSimilarly, the power spectrum of the spin interaction energy Es=gs\n2R\nd2r|F(r)|2is obtained as follows:\nF(r) =X\nk˜F(k)eik·r\nEs(k) =Nεs\n2n2\n0X\nk<|k|<k+∆k|˜F(k)|2(S4)\nIn Fig. S2, we present the energy spectra Eiv(k)andEs(k)of the spin-driven BEC for different driving times, obtained\nfrom our numerical results using Eq. (S2) for {δ0, D}/2π={0,1}kHz and ν= 60 Hz. In the calculation, we used a grid of\n1024×1024 configuration to expand the range of the wave number k[15]. When the driven BEC reaches a stationary turbulent\nstate for τd>0.3s,Eiv(k)andEs(k)approximately follow power laws with exponents of −5/3and−7/3, respectively, within\nthe range of kb< k < k s, where ks= 2π/ξsandkb=ks/√\n2π[15, 32]. It is important to note that the observation of the\n−5/3power-law scaling does not necessarily imply a direct energy cascade [34].\nD. Magnetic domain formation in a driven BEC\nIn Fig. S3, we present experimental images of the BECs for different spin-driving times, τd, to demonstrate the process\nof magnetic domain formation under a field gradient. The initial BEC was prepared in the mF= 0 state, consisting of ap-\nproximately 2.5×106 23Na atoms. We applied a magnetic field gradient of B′= 0.14G/cm along the x′direction and a rf\ndriving field with Ω = 2 π×150Hz. For short spin-driving times, τd≤10 ms, a stripe density pattern emerges in the central\nregion of the BEC, indicating the development of a helical spin structure owing to spatial variations in frequency detuning as\nδ0(x′) = (1\n2µBB′x′)/ℏ. Subsequently, for 10 ms < τd<200 ms, the stripe pattern breaks into smaller segments and the spin\ntexture becomes irregular, starting from the center region and then spreading throughout the BEC. Meanwhile, the mF=±1\nspin components move in opposite directions owing to the field gradient B′, resulting in the buildup of net magnetization at both\nends. After τd>200ms, the ferromagnetic domains, although still turbulent, become more distinct, and their boundary with the\nspin-isotropic turbulent region becomes more pronounced. Remarkably, the boundary exhibits a sudden change in the density of\nthe corresponding spin component [Fig. 3(e)]. For τd>0.5s, the driven BEC reaches a state with a stationary magnetic domain\nstructure, where the spin-turbulent region is sandwiched between two ferromagnetic domains.8\nFIG. S2. Energy spectrum and scaling behavior. Power spectra of (a) the incompressible kinetic energy Eiv(k)and (b) the spin interaction\nenergy Es(k)of a spin-driven BEC for various driving times. The xandyaxes are on logarithmic scale. The black dashed lines represent\nk−5/3in (a) and k−7/3in (b), respectively. The power-law exponents were determined from the fitting of Eiv(s)(k) =E′ka(b)to the data\nwithin the range of kb=√\n2/ξs< k < k s= 2π/ξs, indicated by the vertical gray dashed-dotted lines. In (c) and (d), the resulting exponent\nvalues of aandbare displayed as functions of the driving time, respectively. For (b,d), we extracted the data from the study reported in\nRef. [15].\nFIG. S3. Turbulence generation and magnetic domain formation in a spin-driven BEC. Time-of-flight images of BECs after Stern-Gerlach\nspin separation for various rf driving times, τd. A magnetic field gradient of B′= 0.14G/cm was applied along ˆx′before rf field driving.\nE. Image reconstruction for compensating Stern-Gerlach spin separation\nIn the Stern-Gerlach (SG) spin separation imaging method, we applied a magnetic field gradient pulse right after releasing\nthe trapping potential and before imaging. This procedure enabled the spatial separation of spin components during the time\nof flight, allowing for examination of the spin texture of the BEC. However, the non-uniformity of the field gradient hinders a\nprecise quantitative analysis of the spin texture because it causes the clouds of the three spin components to deform differently\nwhile being spatially separated. In our experiment, we observed that the mF= 1(−1)component was slightly elongated\n(compressed) along the x′axis.\nTo mitigate the influence of the deformation effect on our examination of the spin texture, we introduce an image reconstruc-\ntion method based on a model that accounts for the stretching and compression caused by the field gradient. In the model, we\nposit that the magnetic force acting on the mF=±1components is directed radially from a distant zero-field center. Con-\nsequently, the mF=−1component, which seeks low fields, contracts as it is drawn toward the center, while the mF= 1\ncomponent expands as it moves away. Assuming a constant field gradient magnitude, the classical equation of motion indicates\nthat the mF=±1component initially positioned at r′±1is moved to r′(r′\n±1;r′\nc, l) =r′\n±1±r′\n±1−r′\nc\n|r′\n±1−r′c|l, where r′\ncis the position\nof the zero-field center and l=|r′−r′\n±1|is the travel distance during spin separation. Using the relationship between r′±1and\nr′, the initial density distributions of the mF=±1components can be reconstructed from the measured OD image.9\nFIG. S4. Stern-Gerlach image reconstruction. (a) Original experimental image of a BEC with helical spin texture, taken after spin separa-\ntion, and (b) its reconstructed image. (c) Experimental image of a turbulent BEC and (d) its reconstructed image (same image in Fig. 3(d)).\n(e)-(h) Integrated 1D density profiles of the three spin components in (a)-(d). (i) Determination of the width wof the spin turbulent re-\ngion from the density profile of the mF=0 spin component. The density profile of the condensate was obtained by removing the ther-\nmal contribution using a Gaussian function fitted to the outer region of the entire profile (dashed line in the upper panel). The width\nwwas determined as the distance between the boundary positions L1andL2by fitting the condensate profile with an empirical function\nn(x′) =A\u0014\nmax\u0012\n1−(x′−x′\nc)2\nR2\nx′,0\u0013\u00152\u0012\n1 +eL1−x′\nσ\u0013−1\u0012\n1 +ex′−L2\nσ\u0013−1\n, where Rx′= 210 µm and σ= 1.6µm.\nWe determine the model parameters r′\ncandl, using an image of a BEC with a helical spin texture as a reference, where each\nspin component shows density stripes, as shown in Fig. S4(a). The BEC was initially prepared in the |F= 1, mF= 0⟩state\nand its spin state was rotated to1√\n2(|mF= +1⟩+|mF=−1⟩)by applying aπ\n2rf pulse. Subsequently, after an time interval\nofτ= 100 ms under a field gradient of B′= 0.028 G/cm along ˆx′, anotherπ\n2rf pulse was applied to the BEC. Assuming a\nThomas-Fermi (TF) density profile for the BEC trapped in a harmonic potential, the column density profiles of the three spin\ncomponents are expected to be\nnmF=0(r′) = n0cos2(kx′x′+θ0)\"\nmax \n1−x′2\nR2\nx′−y′2\nR2\ny′,0!#3/2\n,\nnmF=±1(r′) =n0\n2sin2(kx′x′+θ0)\"\nmax \n1−x′2\nR2\nx′−y′2\nR2\ny′,0!#3/2\n, (S5)\nincluding the Ramsey interference pattern, where n0is the peak column density of the BEC, Rx′(y′)is the TF radus along the\nx′(y′)direction, kx′=µBB′τ\n2ℏis the wave number of the fringe pattern, and θ0is the fringe phase at the center of the BEC. We\nobtain the values of r′\ncandlby fitting the following function to the reference OD image:\nn(r′;n0, Rx′, Ry′, kx′, θ0,r′\nc, l) =nmF=0(r′) +X\ni=±1χinmF=i(r′\ni(r′;r′\nc, l)). (S6)\nHere χ±1is the OD conversion factor due to changes in cloud size as well as absorption coefficient.\nFigure S4(b) shows the density distributions of the three spin components reconstructed from the reference image of Fig. S4(a).\nAs shown in the integrated 1D density profiles in Figs. S4(e) and S4(f), the relative shift and deformation between different spin\ncomponents are well compensated for in the reconstructed image. The same image reconstruction method is applied to the\nexperimental image data taken after the SG spin separation [see Figs. S4(c) and S4(d)]10\nF. Background field noise contribution\nFluctuations in the ambient magnetic field can modify the effect of spin driving. If there is a background field oscillating at\nthe modulation frequency ν, e.g., δBz(t) =Bνsin(2πνt+ϕν), it interferes with the controlled modulation field, and the driving\nfrequency detuning δis modified accordingly as δ(t) =δ0+Dtotalsin(2πνt+ϕtotal)with\nDtotal =p\nD2+D2ν+ 2DDνcos(ϕ−ϕν),\ntanϕtotal=Dsinϕ+Dνsinϕν\nDcosϕ+Dνcosϕν, (S7)\nwhere Dν=µB\n2ℏBν. Taking into account the contribution of background field fluctuations with frequencies other than ν, we\nmodel the threshold detuning as\nδth=q\nβ2ν\u0000\nD2+ 2DDνcos(ϕ−ϕν)\u0001\n+δ2\nth,0, (S8)\nwhere δth,0denotes the threshold detuning at D= 0, purely originating from ambient, uncontrolled field fluctuations. The\nproportionality factor βνrepresents the efficiency of field modulation to generate turbulence and can be dependent on ν, as\nobserved in the experiment [Fig. 4(b)]."    },    {        "title": "1710.09582v1.Spin_dynamics_in_helical_molecules_with_non_linear_interactions.pdf",        "content": "Spin dynamics in helical molecules with non-linear\ninteractions\nE. D\u0013 \u0010az1, P.Albares2, P. G. Est\u0013 evez2, J. M. Cerver\u0013 o2, C. Gaul1;3,\nE. Diez2and F. Dom\u0013 \u0010nguez-Adame1\n1GISC, Departamento de F\u0013 \u0010sica de Materiales, Universidad Complutense, E{28040\nMadrid, Spain\n2NANOLAB, Departamento de F\u0013 \u0010sica Fundamental, Universidad de Salamanca,\nE{37008 Salamanca, Spain\n3Cognitec Systems GmbH, Gro\u0019enhainer Str. 101, 01127 Dresden, Germany\nE-mail: elenadg@ucm.es\nAbstract.\nIt is widely admitted that the helical conformation of certain chiral molecules may\ninduce a sizable spin selectivity observed in experiments. Spin selectivity arises as a\nresult of the interplay between a helicity-induced spin-orbit coupling and electric dipole\n\felds in the molecule. From the theoretical point of view, di\u000berent phenomena might\na\u000bect the spin dynamics in helical molecules, such as quantum dephasing, dissipation\nand the role of metallic contacts. Previous studies neglected the local deformation\nof the molecule about the carrier thus far, but this assumption seems unrealistic to\ndescribe charge transport in molecular systems. We introduce an e\u000bective model\ndescribing the electron spin dynamics in a deformable helical molecule with weak\nspin-orbit coupling. We \fnd that the electron-lattice interaction allows the formation\nof stable solitons such as bright solitons with well de\fned spin projection onto the\nmolecule axis. We present a thorough study of these bright solitons and analyze their\npossible impact on the spin dynamics in deformable helical molecules.\nPACS numbers: 05.45.Yv, 72.25.-b, 73.63.-b\nKeywords : Spin dynamics, chiral molecules, solitons.arXiv:1710.09582v1  [cond-mat.mes-hall]  26 Oct 2017Spin dynamics in helical molecules with non-linear interactions 2\n1. Introduction\nManipulation and control of the electron spin degree of freedom in nanoscale materials\nlies at the very core of spintronics. Among the large variety of materials with\ntechnological interest in this \feld, organic systems are gaining signi\fcance as active\ncomponents in spintronics nanodevices. Although large spin-orbit coupling (SOC) is\nuncommon in carbon-based materials, recent experiments on electron transport have\nbrought with them a considerable e\u000bort to uncover the origin of the observed high spin\nselectivity in DNA [1,2] and bacteriorhodopsin on non-magnetic metallic substrates [3].\nAs a working hypothesis, it has been suggested that spin selectivity may be related to\nthe speci\fc geometric structure of the involved molecular systems, namely their helical\nconformation [2]. A number of theoretical models have been put forward to explain the\nobserved spin selectivity in helical molecules. Usually they rely on large SOC [4{11], the\nneed for dephasing when SOC is weak [12,13], the leakage of electrons from the molecule\nto the environment [14], the role of the bonding of the molecule to the metallic leads that\nenhance the e\u000bect [15,16], or the interplay between a helicity-induced SOC and a strong\ndipole electric \feld, which is characteristic of these molecules [17] (see Refs. [18, 19]\nfor a recent review). Theoretical models usually assume rigid lattices and neglect the\nlocal deformation of the molecule about the carrier. However, this assumption seems\nunrealistic to describe charge transport in molecular systems like DNA [20].\nDepending on the various energy scales involved (electron bandwidth, zero-\npoint energy of molecular vibrations, thermal energy), lattice deformation can play\na signi\fcant role on transport properties. This is particularly relevant when charge\ncarriers interact with intramolecular modes that occur at high frequency due to the\nstretching of sti\u000b covalent bonds. Coupling to those modes may strongly alter charge\ntransport [21] and even lead to self-trapping of carriers, provided that the relaxation\nenergy (the energy gained upon the deformation of the lattice about the carrier) exceeds\nthe band width [22]. Self-trapping has been commonly formulated within the framework\nof the small polaron theory based on a local Holstein-type coupling [23] between the\ncarrier and the intramolecular mode. This model was later extended by Peyrard and\nBishop to study the ac response of a DNA molecule, where the charge in the \u0019-stack\ninteracts with the base-pair opening dynamics of the double strand [24{27].\nDavydov's soliton theory of charge and energy transfer in \u000b-helix and acetanilide\nprovides another paradigmatic example on how the interaction of carriers and vibrational\ndegrees of freedom can induce self-trapping phenomena [28]. Starting from a Fr olich-\nlike Hamiltonian [29] and assuming the adiabatic approximation, Davydov put forward a\nsoliton theory of long-range energy transfer of excitations interacting with intramolecular\nvibrational modes in a quasi-one-dimensional lattice. In the adiabatic approximation,\nthe continuous limit of the Davydov's equations reduce to the non-linear Schr odinger\n(NLS) equation for the elementary excitations.\nInspired by the success of the Peyrard-Bishop-Holstein and Davydov's approaches,\nin this work we introduce an e\u000bective self-focusing non-linear model describing theSpin dynamics in helical molecules with non-linear interactions 3\ndynamics of a single charge carrier in the electrostatic potential due to a helical\narrangement of dipoles. The proposal generalizes the linear model formerly introduced\nby Guti\u0013 errez et al. considering spin-selective transport of electrons through a helically\nshaped electrostatic potential [6]. This model has been recently revisited and extended\nto study the coherent spin dynamics in helical molecules [11]. The strong interaction\nwith the lattice vibrations will be addressed by adding a non-linear term to the\nSchr odinger equation within the adiabatic approximation [30]. The resulting equation\nturns out to be integrable, thus allowing us to obtain a family of bright solitons\ndescribing the coherent spin dynamics in deformable helical molecules.\n2. Electron spin dynamics in a rigid helical molecule\nFollowing Ref. [11], we start out by revisiting and amending the model introduced in\nRef. [6]. Two main factors determine the high spin selectivity found: an unconventional\nRashba-like SOC, re\recting the helical symmetry of molecules, and a weakly dispersive\nelectronic band. \u000b-helix proteins and other macromolecules present a net dipole moment\nalong the helix axis due to the helical arrangement of peptide dipoles [31]. Thus we\nconsider the electron motion through a very helical arrangement of peptide dipoles\ndirected along the Zaxis. The dipoles are located at rj=j\u0001zbez+ab\u001ajand their\ndipole moments are dj=db'j. Here, we have used cylindrical coordinates with\nb\u001aj= (cos'j;sin'j;0),b'j= (\u0000sin'j;cos'j;0), and'j= 2\u0019j=Nd+\u0019. The orientation\nof the individual dipoles does not a\u000bect much the results, provided they are arranged\nhelically [11]. Typical values are Nd= 10 dipoles per turn in DNA, a= 0:7 nm, and\nb=Nd\u0001z= 3:2 nm. The total electric \feld on the molecule axis due to the dipoles is\nthen found to be [11]\nE(z) =1\n4\u0019\u000f0X\njdj\u0002\na2+ (z\u0000j\u0001z)2\u00033=2: (1)\nTo estimate the SOC, we need E(z) =\u0000i[Ex(z)\u0000iEy(z)] = exp(\u0000i2\u0019z=b)D(z), where\nD(z)\u0019 E 0\u0011d(\u000f0ab\u0001z)\u00001K1(2\u0019a=b),K1being the modi\fed Bessel function of the\nsecond kind.\nThe SOC Hamiltonian stems from the classical formula \u001b\u0001(bp\u0002E), symmetrized\nsuch that the Hamiltonian is Hermitian. Here \u001bis a vector whose components are the\nPauli matrices \u001bx,\u001by, and\u001bz. Forbp=bpzbezthe SOC Hamiltonian simpli\fes to\nbHSO=\u0015\n2\"\nbpz \n0E(z)\nE\u0003(z) 0!\n+ \n0E(z)\nE\u0003(z) 0!\nbpz#\n; (2)\nwhere\u0015=e~=(2mc)2. The electron Hamiltonian bH=bp2\nz=2m+bHSOcan be cast in the\nformbH=EbbHwhere the dimensionless Hamiltonian bHreads\nbH=\u0000@2\n\u0018\u00002\u0019\rcM : (3a)Spin dynamics in helical molecules with non-linear interactions 4\nHere we have de\fned Eb=~2=2mb2,\u0018=z=b,@\u0018=@=@\u0018 and the dimensionless spin-\norbit parameter \r=~\u0015E0=(2\u0019bEb). The matrix operator cMis given by\ncM= \n0e\u0000i2\u0019\u0018\nei2\u0019\u00180! \ni@\u0018\u0000\u0019 0\n0i@\u0018+\u0019!\n: (3b)\nThe dimensionless Hamiltonian (3a) is readily diagonalized and the eigenenergies\nare found to be\n\"qs=q2+\u00192\u00002\u0019sp\n1 +\r2q ; s =\u00061: (4a)\nThe corresponding normalized eigenfunctions are\n\u001fqs(\u0018) = \n\f\"(s)ei(q\u0000\u0019)\u0018\n\f#(s)ei(q+\u0019)\u0018!\n; (4b)\nwhere\n\f\"(s) =1\n2\u0002\n(1 +s) cos\u001e+ (1\u0000s) sin\u001e\u0003\n;\n\f#(s) =1\n2\u0002\n(1\u0000s) cos\u001e\u0000(1 +s) sin\u001e\u0003\n; (4c)\nsatisfying\f2\n\"(s) +\f2\n#(s) = 1, and\ntan\u001e=\r\n1 +p\n1 +\r2: (4d)\nOnce the eigenvectors of the Hamiltonian (3a) have been obtained, we focus on the\ndynamics of an electron wave packet of the form\n\u001f(\u0018;t) =X\nsZ1\n\u00001dq\n2\u0019Cqs\u001fqs(\u0018)e\u0000i\"qst; (5a)\nwhere time is expressed in units of ~=Eband\nCqs=Z1\n\u00001d\u0018\u001fy\nqs(\u0018)\u0001\u001f(\u0018;0): (5b)\nOur magnitude of interest will be the time-dependent spin projection onto the molecule\naxis, also referred as helicity, which is calculated as follows\nSP(t) =Z1\n\u00001d\u0018\u001fy(\u0018;t)\u001bz\u001f(\u0018;t) =Z1\n\u00001dq\n2\u0019h\u0000\njCq;+1j2\u0000jCq;\u00001j2\u0001\ncos(2\u001e)\n+ 2 sin(2\u001e)Re\u0000\nC\u0003\nq;+1Cq;\u00001ei(\"q;+1\u0000\"q;\u00001)t\u0001i\n: (6)\nConsider an initial wave packet of dimensionless width Wwith an arbitrary state of\nspin polarization \u001f(\u0018;0) =f(\u0018)\u0002\ncos(\u0012)u\"+ei'sin(\u0012)u#\u0003\n. Hereu\u001bwith\u001b=\";#denotes\nan eigenvector of \u001bzand the polarization state is de\fned by the angle \u0012. For the sake\nof concreteness we set '= 0 hereafter. After a straightforward calculation one can\nobtain a closed expression for SP( t) that has a transient contribution which vanishes\nat large times t\u001dW=\u0000\n4\u0019p\n1 +\r2\u0001\n. A transient time of t\u001840 fs is roughly estimatedSpin dynamics in helical molecules with non-linear interactions 5\nfor a highly localized initial state with W\u00181 passing through a DNA molecule with\na SOC parameter of the order of \r\u00180:1. Thus, after a quick transient state, the spin\nprojection reaches the asymptotic value given as SP 1= SP(t!1 ) where\nSP1=1\n1 +\r2h\ncos(2\u0012)\u0000\rsin(2\u0012)Z1\n\u00001d\u0018jf(\u0018)j2cos(2\u0019\u0018)i\n: (7)\nNotice that if we consider an initial fully polarized state with\u0012= 0 or\u0012=\u0019=2, namely\nwith spin parallel (antiparallel) to the molecule axis, the larger the SOC parameter, the\nsmaller the asymptotic spin polarization, as expected.\nIn experiments, however, an initially unpolarized current becomes spin polarized\nafter being transmitted through the helical molecule. Therefore, our case of interest is\nan initial fully unpolarized wave packet with spin projection out of the molecule axis,\ni.e. along the Xaxis such as \u0012=\u0019=4 or\u0012= 3\u0019=4 and\u001f(\u0018;0) =f(\u0018) (\u0000u\"+u#)=p\n2.\nIn such a case, the asymptotic polarization has a non-monotonous dependence with the\nmagnitude of the SOC parameter according to Eq (8). The integral in equation (7)\napproaches unity for a narrow wave packet, and the asymptotic spin projection along\nthe molecule axis becomes\njSP1j=\r\n1 +\r2: (8)\nTherefore, the SOC \rips the electron spin after a quick transient and the spin projection\nalong the molecule axis becomes nonzero.\n3. Electron spin dynamics in a deformable helical molecule\nIn order to describe a deformable helical molecule where the electron dynamics is\na\u000bected by the lattice vibrations, we will add a non-linear term to the Hamiltonian (3a).\nThis additional term can be justi\fed within the adiabatic approximation, according\nto Davydov's theory [28]. In such a scenario, the dimensionless NLS describing the\ndynamics of the spinor state \u001f(\u0018;t) reads\ni@t\u001f(\u0018;t) =bH\u001f(\u0018;t)\u00004g\u0002\n\u001fy(\u0018;t)\u0001\u001f(\u0018;t)\u0003\n\u001f(\u0018;t); (9)\nwherebHis given in equation (3a).\nThe integrability of this equation can be analyzed by using the Painlev\u0013 e test [32].\nThis test proves the integrability of equation (9) and yields its three component Lax\npair. The Painlev\u0013 e property can be also used to derive Darboux transformations and an\niterative procedure for obtaining solutions [33]. It can be also proved that equation (9) is\nthe only integrable case of a model very recently put forward by Kartashov and Konotov\nto study the dynamics of Bose-Einstein condensates with helical SOC [34]. Furthermore,\nit reduces to the Manakov non-linear system when the SOC vanishes [35,36].\nIn this regard, for self-focusing non-linear interaction ( g > 0), it can be\ndemonstrated that the following bright solitons (similar to the case of Davydov's soliton)Spin dynamics in helical molecules with non-linear interactions 6\nare a solution to equation (9)\n\u001fs(\u0018;t) =rg\n2sech\u0002\ng(\u0018+ct)\u0003\ne\u0000i's(\u0018;t) \n\f\"(s)e\u0000i\u0019\u0018\n\f#(s)ei\u0019\u0018!\n; s =\u00061; (10)\nwhere\f\"(s) and\f#(s) are given by (4c). Here cis a free parameter representing the\nvelocity of the soliton. The phase is de\fned as 's(\u0018;t) = [c=2\u0000s\u0019cos\u00001(2\u001e)]\u0018+\n(c2=4\u0000g2\u0000\u00192\r)t. The existence of two di\u000berent bright solitons due to the arbitrary\nchoice of the constant s=\u00061 is known as the Kramer doublet and it is directly related\nto the preservation of the time-reversal symmetry in the model. Notice that these\nsolitons have a well de\fned state of spin polarization that depends on the SOC due to\nequation (4d). Most importantly, this polarization is preserved along its propagation\nand it is found to be\njSPsolj=j\f2\n\"(s)\u0000\f2\n#(s)j=1p\n1 +\r2: (11)\n4. Connection with experiments\nHaving presented the salient features of solitons in deformable helical molecules, we\nnow turn to discuss their relevance in experiments. In recent experiments on electron\ntransport in organic helical molecules, it has been clearly demonstrated that an\ninitially unpolarized current turns out to be highly polarized when passing through\nthe molecule [1{3, 37{42]. Since the intrinsic SOC e\u000bects are rather weak in these\nmolecules, theoretical models proposed to describe the experiments rely on SOC related\nto the peculiarities of the helical geometry. It is worth mentioning that all these\nmodels strongly depend on a phenomenological SOC which was roughly estimated to\nbe\u000b= 4\u000012 meV nm [6]. It seems that this coupling, even in the best scenario, is not\nlarge enough to support the high degree of spin polarization observed in the experiments.\nAll these approaches, however, neglected lattice deformations that strongly a\u000bect the\nelectron dynamics in organic molecules.\nIn order to show the relevance of the non-linear interaction between the lattice and\nthe spin degree of freedom, \fgure 1 compares the spin projection achieved with the linear,\nSP1, and the non-linear model, SP sol, as given by equations (8) and (11) respectively. To\nunderstand the experimental situation when a spin unpolarized current is injected into\nthe molecule, we will consider an initial localized wave packet with a polarization state\nsuch that\u0012= 3\u0019=4 (fully unpolarized in the sense discussed above). The wave packet\nevolves in time and, after a short transient time, it reaches a steady polarization given by\nequation. (8). This polarization has to be compared with that obtained for the stable\nsoliton (11). Figure 1 clearly shows an outstanding result, namely, the non-linearity\nstrongly enhances the resulting spin projection of a coherent electron passing through a\ndeformable helical molecule. Thus, lattice vibrations lead to a larger e\u000bective SOC in the\nmolecule. We de\fne the enhancement factor as\u0011=\u0000\njSPsolj\u0000jSP1j\u0001\n=\u0000\njSPsolj+jSP1j\u0001\n,\nas depicted in the inset of \fgure 1. This parameter assess the e\u000bects of the lattice onSpin dynamics in helical molecules with non-linear interactions 7\nFigure 1. Asymptotic spin projection as a function of the dimensionless SOC\nparameter \rfor a rigid helical molecule (dashed line) and a non-linear deformable\nhelical molecule (solid line). The inset shows the enhancement factor \u0011=\u0000\njSPsolj\u0000\njSP1j\u0001\n=\u0000\njSPsolj+jSP1j\u0001\n. The gray area highlights the region with realistic values of\n\raccording to our estimations.\nthe spin polarization capability of the deformable helical molecule. According to the\ncurve shown in the inset of \fgure 1, the self-focusing non-linear interaction has a major\ne\u000bect when the SOC is weak. And this is precisely the case of interest in experiments,\nas discussed below.\nThe gray area of \fgure 1 highlights the region where the dimensionless SOC of\nour model takes values consistent with previous estimations \r=\u000b=(2\u0019bEb)\u00140:2\nin DNA. Our approach explains the physical scenario in experiments as follows. The\ninitial unpolarized electron is injected in a deformable helical molecule whose vibrations\ninteract continuously with the electronic dynamics inside the system. Such non-\nlinear interaction transforms the initial arbitrary state till it reaches the most stable\ncon\fguration, namely, a solitonic solution. Once the soliton state is conformed, it can\npropagate with no dispersion along the molecule with a well de\fned high degree of\nspin polarization. The formation of solitons after a short transient time could be an\nimportant e\u000bect for the resulting highly polarized currents found in experiments.\n5. Conclusions\nIn conclusion, we have presented a non-linear model to study the spin dynamics of\nelectrons in a deformable helical molecule. Such dynamics is subjected to: i) the electric\n\feld created by the helical arrangement of molecular dipoles and ii) the interaction\nbetween the electron and the lattice vibrations. On the one hand, the dipole electric\n\feld induces a Rashba-like SOC for electrons moving along the helical axis. On the\nother hand, the electron-lattice interaction allows the formation of stable solitons.Spin dynamics in helical molecules with non-linear interactions 8\nOnce the model was presented, we were able to prove that the system supports the\nformation of stable bright solitons. Remarkably, the spin polarization in such solitons\nis preserved during the propagation across the helical molecule. We also calculated the\nspin projection onto the molecule axis as a \fgure of merit to assess the spin dynamics.\nFor completeness, we also compared this result with those obtained for the linear model\ncorresponding to the rigid molecule. In particular, we focused on the most relevant\nsituation for experiments, namely the partial spin polarization achieved by an initial\nunpolarized electron when passing through the helical molecule. In such scenario,\nthe spin polarization capability of a deformable helical molecule is largely increased\ncompared to the rigid one, even in the case of weak SOC.\nAcknowledgments\nThe authors thank R. Guti\u0013 errez and V. Mujica for helpful discussions. This research\nhas been supported by MINECO (Grants MAT2013-46308 and MAT2016-75955) and\nJunta de Castilla y Le\u0013 on (Grant SA045U16).\nReferences\n[1] G ohler B, Hamelbeck V, Markus T Z, Kettner M, Hanne G F, Vager Z, Naaman R and Zacharias\nH 2011 Science 331894\n[2] Xie Z, Markus T Z, Cohen S R, Vager Z, Guti\u0013 errez R and Naaman R 2011 Nano Lett. 114652\n[3] Mishra D, Markus T Z, Naaman R Kettner M, Gohler B, Zacharias H, Friedman N, Sheves M and\nFontanesi C 2013 Proc. Nat. Acad. Sci. USA 11014872\n[4] Yeganeh S, Ratner M A, Medina E and Mujica V 2009 J. Chem. Phys. 131014707\n[5] Medina E, L\u0013 opez F, Ratner M A and Mujica V 2012 Europhys. 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C 11914542\n[39] Mondal P C, Fontanesi C, Waldeck D H and Naaman R 2015 ACS Nano 93377\n[40] Einati H, Mishra D, Friedman N, Sheves M and Naaman R 2015 Nano Lett. 151052\n[41] K V, Mathew S P, Cohen S R, Hern\u0013 andez Delgado I, Lacour J and Naaman R 2016 Adv. Mat. 28\n1957\n[42] Aragon\u0013 es A C, Medina E, Ferrer-Huerta M, Gimeno N, Teixid\u0013 o M, Palma J L, Tao N, Ugalde J M,\nGiralt E, D\u0013 \u0010ez-P\u0013 erez I and Mujica V 2017 Small 131613"    },    {        "title": "2303.04742v1.Spin_selective_charge_recombination_in_chiral_donor_bridge_acceptor_triads.pdf",        "content": "Spin selective charge recombination in chiral donor-bridge-acceptor triads\nThomas P. Fay1,a)and David T. Limmer1, 2, 3, 4,b)\n1)Department of Chemistry, University of California, Berkeley, CA 94720, USA\n2)Kavli Energy Nanoscience Institute at Berkeley, Berkeley, CA 94720, USA\n3)Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA\n4)Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA\nInthispaperweoutlineaphysicallymotivatedframeworkfordescribingspin-selectiverecombinationprocessesinchiral\nsystems, from which we derive spin-selective reaction operators for recombination reactions of donor-bridge-acceptor\nmolecules, where the electron transfer is mediated by chirality and spin-orbit coupling. In general the recombination\nprocess is selective only for spin-coherence between singlet and triplet states, and it is not in general selective for spin\npolarisation. Wefindthatspinpolarisationselectivityonlyarisesinhoppingmediatedelectrontransfer. Wedescribehow\nthiseffectivespin-polarisationselectivityisaconsequenceofspin-polarisationgeneratedtransientlyintheintermediate\nstate. Therecombinationprocessalsoaugmentsthecoherentspindynamicsofthechargeseparatedstate,whichisfound\nto have a significant effect on recombination dynamics and to destroy any long-lived spin polarisation. Although we\nonlyconsiderasimpledonor-bridge-acceptorsystem,theframeworkwepresentherecanbestraightforwardlyextended\nto describe spin-selective recombination processes in more complex systems.\nI. INTRODUCTION\nTherehasrecentlybeengrowinginterestinspin-selectivity\nof processes in molecular chiral donor-acceptor systems.1–4\nThese systems may provide a versatile platform for exploring\nthe molecular origins of the chirality induced spin selectiv-\nity (CISS) effect,5–8without the complications of interactions\nwithelectrodes,3,9–12andithasalsobeensuggestedthatchiral\ndonor-acceptor systems could be exploited in various quan-\ntum information science applications.1,13,14Several theories\nhave been proposed for the molecular CISS effect and how\nspin polarisation is generated in the formation of charge sep-\narated states,7,8,15–17and a handful of experimental protocols\nhavebeenproposedtotestthesetheories.2,5,8,15,18Whilstmost\nstudies to date have focussed on the CISS effect in the for-\nmation of charge seperated states in donor-acceptor systems,\nit has also been suggested that CISS could play a role in the\nchargerecombinationofdonor-acceptorsystems.8,15However\nthe proposed theories of CISS in charge recombination are\neither limited to simple one-step electron transfer15or purely\nphenomenological,8and thus far the role of CISS in experi-\nmentallyrealiseddonor-bridge-acceptorsystems2,19–21hasnot\nbeen explored. Motivated by this, in this paper we aim to an-\nswer the question: does chirality lead to spin-selective charge\nrecombination in donor-bridge-acceptor molecules?\nIn systems where chirality and spin-orbit coupling have\nno effect on charge recombination of charge separated\n(CS) states, the treatment of spin-selective recombination is\nwell understood.22–25We describe the system with a time-\ndependent spin density operator for the CS state, ˆ𝜎CS, which\nspans the set of near-degenerate singlet and triplet spin states\nof the CS state. This spin density operator obeys the well-\nestablished Haberkorn quantum master equation\nd\nd𝑡ˆ𝜎CS¹𝑡º=\u0000𝑖»ˆ𝐻ˆ𝜎CS¹𝑡º¼¸Kˆ𝜎CS¹𝑡º (1)\na)Electronic mail: tom.patrick.fay@gmail.com\nb)Electronic mail: dlimmer@berkeley.eduwhere ˆ𝐻isthe spinHamiltonian fortheCS stateand Kisthe\nreaction superoperator, which is given by\nKˆ𝜎CS=\u0000\bˆ𝐾ˆ𝜎CS\t\n=\u0000\u001a𝑘CRS\n2ˆ𝑃S¸𝑘CRT\n2ˆ𝑃Tˆ𝜎CS\u001b\n(2)\nin which and 𝑘CRSand𝑘CRTare the singlet and triplet spin\nselective reaction rate constants and ˆ𝑃S=jSihSjand ˆ𝑃T=Í\n𝛼=𝑥𝑦𝑧jT𝛼ihT𝛼jare projection operators onto singlet and\ntriplet spin states of the CS state.22,23,25Kdescribes the full\neffectofthereactionprocessonthedynamicsoftheCSstate,\nand the reaction operator ˆ𝐾encodes how population is lost\nfrom the CS state. In what follows we will refer to Kas\nthe reaction superoperator and ˆ𝐾as the reaction operator.\nIt should be noted that this equation does not conserve the\ntrace of ˆ𝜎CSbecause population is lost from the CS state by\nrecombination, and it further assumes that the recombination\nof the CS state is irreversible.\nIt has been postulated by Luo & Hore that in chiral donor-\nacceptorsystems,thereactionoperator ˆ𝐾foraCISSmediated\nrecombination process should be given by8\nˆ𝐾=𝑘CR\n2\f\f𝜙𝜒\u000b\n𝜙𝜒\f\f (3)\nin which\f\f𝜙𝜒\u000b\n=cos¹𝜒2ºjSi¸sin¹𝜒2ºjT𝑧i,jSi=\n1p\n2¹j\"D#Ai\u0000j#D\"AiºandjT𝑧i=1p\n2¹j\"D#Ai¸j#D\"Aiº,where\nthe quantisation axis, 𝑧, is the spin-polarisation axis, and\nthe mixing angle 𝜒is a phenomenological parameter that\nparametrises the extent of spin polarisation selectivity. This\nform of the reaction operator is based on the assumption that\nin a chiral molecule the recombination process is partially\nspin selective, and it is straightforward to verify that the total\ndecay rate of the CS state with this model is dependent on\nthe spin-polarisation hΔ𝑆𝑧i=h𝑆D𝑧\u0000𝑆A𝑧iof the CS state.\nThe limit of full spin-selectivity is recovered for 𝜒=𝜋2\nand\f\f𝜙𝜒\u000b\n=j\"D#Ai, where the recombination only occurs if\nthe donor and acceptor electrons have specific opposite spin\norientations in the molecular frame. It should be noted that\nthis reaction operator has not been derived from any micro-\nscopicmodelsofchiralityinducedspinselectivity,andthereisarXiv:2303.04742v1  [physics.chem-ph]  8 Mar 20232\nno direct experimental evidence that it provides a reasonable\nmodelofspinselectiverecombinationinchiraldonor-acceptor\nsystems.\nInpreviousworkwehavederivedadifferentreactionsuper-\noperatortothatgivenbyEq.(3)forasimpleone-stepelectron\ntransfer between a donor and an acceptor D\u000f¸–A\u000f\u0000!D–A,\ninvoking the modest approximations.25Specifically it is as-\nsumed that the coupling between the charge transfer states\nis weak, that the nuclear degrees of freedom are initially at\nlocalthermalequilibriumononeofthechargetransferpoten-\ntialenergysurfaces,andthattheCondonapproximationholds\nfor the direct charge transfer coupled, 𝑉DA, and the spin-orbit\nmediated charge transfer coupling ΛDA(essentially the same\napproximations as Marcus theory26,27,30). The model elec-\ntronic Hamiltonian for the singlet and triplet D\u000f¸–A\u000f\u0000(CS)\nand D–A ( S0) states in this theory is given by,\nˆ𝐻DA=𝐸S0jS0ihS0j¸∑︁\nΘ=ST𝑥T𝑦T𝑧𝐸CSΘjCSΘihCSΘj\n¸𝑉DA¹jCSSihS0j¸jS0ihCSSjº\n¸𝑖ΛDA\n2¹jCST𝑧ihS0j\u0000jS0ihCST𝑧jº(4)\nand from this the reaction superoperator can be derived to be\nKˆ𝜎CS=\u0000\u001a𝑘CR\n2j𝜓𝜃ih𝜓𝜃jˆ𝜎CS\u001b\n\u0000𝑖»𝛿𝜖j𝜓𝜃ih𝜓𝜃jˆ𝜎CS¼(5)\nwherej𝜓𝜃i=cos𝜃jSi¸𝑖sin𝜃jT𝑧i,with tan𝜃=Λ DA¹2𝑉DAº,\nwherethequantizationaxis 𝑧isdefinedbythespin-orbitcou-\npling vector. This state is not spin polarised, i.e. h𝑆D𝑧i=\nh𝑆A𝑧i=0, so in this case the recombination process is not\nselective for spin polarisation. However, the recombination\nprocess is selective for the imaginary part of the coherence\nbetweenjSiandjT𝑧istates, ˆΠST𝑧=\u0000𝑖¹jSihT𝑧j\u0000jT𝑧ihSjº2.\nThe shift term 𝛿𝜖\u0019¹𝑉2\nDA¸¹ΛDA2º2º¹𝐸S0\u0000𝐸CSºappear-\ninginthisreactionsuperoperatorisasuperexchangemediated\nspin-orbit coupling interaction in the CS state that emerges as\na result of the spin-orbit interaction between the CS state and\ntheS0ground-state, which cannot in general be neglected.7,15\nIn subsequent work it was found that an interplay of spin-\norbit coupling and exchange interactions in a two-step charge\nseparation following photo-excitation can produce spin po-\nlarisation, where a single-step electron transfer cannot.7This\nnaturally raises the question: what role does chirality induced\nspin selectivity play in the charge recombination of a donor-\nbridge-acceptorsystem? Toaddressthisquestionwewillcon-\nsider charge recombination in a chiral D–B–A molecule and\nderive a reaction operator which describes the superexchange\nand incoherent hopping limits of the charge recombination\nprocess.\nII. CISS IN DONOR-BRIDGE-ACCEPTOR CHARGE\nRECOMBINATION\nAs a minimal model for a spin-orbit coupling mediated\ncharge recombination process in a chiral donor-acceptor sys-\ntem, we consider recombination of a donor-bridge-acceptor\nD–B–A\n∙+\n∙−\nD–B–AD–B–A\n∙+\n∙−\nkCRVBA VDB\nEnergy\nD–B–A\n∙+\n∙−\nD–B–AD–B–A\n∙+\n∙−\nkCR\nkfkb\nSuperexchange HoppingS0CS\nS0CSCS\n⋆\nCS\n⋆\nΛDBFIG. 1. Diagram summarising the charge transfer states involved\nin the donor-bridge-acceptor system, indicating the incoherent rate\nprocesseswitharrowsandsuperexchangecouplingwithdashedlines,\nfor the superexchange limit (left) and hopping limit (right).\nsystemfromachargeseparated(CS)state,D\u000f¸–B–A\u000f\u0000,back\nto a closed shell ground state ( S0), D–B–A. We will also\nassumethatdirectA\u000f\u0000!D\u000f¸tunnellingoftheelectrondoes\nnotoccur,sotheelectroneitherhopsviaanintermediateCS★\nstate, D\u000f¸–B\u000f\u0000–A, in which the bridge B is charged, or the\nelectron tunnels via indirect superexchange coupling through\nvirtual transitions to intermediate charge transfer states. Both\ntheCSand CS★statescanexistineithersingletortripletelec-\ntronspinstates,whereasthe S0groundstateisassumedtoonly\nexistinapuresingletstate. Forsimplicitywewillassumethat\nonly the B\u000f\u0000!D\u000f+electron transfer is mediated by spin orbit\ncoupling,andtheinitialhoppingstepisnotspinselective. The\ntransitionfromsuperexchangemediatedtunnellingtohopping\nis controlled by the energy of the CS★state (at fixed elec-\ntron transfer reorganisation energy). When the intermediate\nCS★state is very high in energy (compared to thermal energy\n𝑘B𝑇) hopping is unfeasible, thus superexchange dominates,\nbut when the intermediate CS★is thermally accessible, hop-\npingbecomesthedominantmechanism.28–30Thisisillustrated\nschematically in Fig. 1.20,28\nA. The superexchange limit\nFirstly, in order to understand the superexchange mediated\nelectrontransferlimit,weconsiderthefollowingmodelforthe\nelectronic Hamiltonian for the S0, CS and CS★states,7,28,30\nˆ𝐻DBA=𝐸S0jS0ihS0j¸∑︁\nΘ=ST𝑥T𝑦T𝑧𝐸CS★Θ\f\fCS★Θ\u000b\nCS★Θ\f\f\n¸∑︁\nΘ=ST𝑥T𝑦T𝑧𝐸CSΘjCSΘihCSΘj\n¸∑︁\nΘ=ST𝑥T𝑦T𝑧𝑉BA\u0000\f\fCS★Θ\u000b\nCSΘ\f\f¸\f\fCSΘ\u000b\nCS★Θ\f\f\u0001\n¸𝑉DB\u0000\f\fCS★S\u000b\nS0\f\f¸\f\fS0\u000b\nCS★S\f\f\u0001\n¸𝑖ΛDB\n2\u0000\f\fCS★T𝑧\u000b\nS0\f\f\u0000\f\fS0\u000b\nCS★T𝑧\f\f\u0001\n(6)\nHereΘ = ST𝑥T𝑦orT𝑧, denotes the total electron spin\nstateforthechargeseparatedstates, 𝑉DBand𝑉BAdenotespin-\nconserving diabatic (i.e. direct tunnelling) couplings between3\nelectronic states, and ΛDBdenotes the spin-orbit coupling be-\ntween the CS★andS0ground state. For simplicty we assume\nthat SOC only mediates the D$Bcoupling, but the gen-\neral result we obtain holds when SOC mediates the B$A\nelectron transfer process as well.\nInthesuperexchangemediatedelectrontransferlimit,where\n𝐸CS★Θ\u001d𝐸CSΘ𝐸S0,wedonotneedtoexplicitlyincludethe\nintermediate charged bridge state, CS★, and we can treat the\ncharge recombination of the CS state to the S 0state as a two\nelectron transfer state problem, with an effective Hamiltonian\ngiven by\nˆ𝐻DBAeff=\u0000𝐸S0¸𝛿𝐸S0\u0001jS0ihS0j\n¸∑︁\nΘ=ST𝑥T𝑦T𝑧\u0000𝐸CSΘ¸𝛿𝐸CSΘ\u0001jCSΘihCSΘj\n¸𝑉DAeff¹jCSSihS0j¸jS0ihCSSjº\n¸𝑖ΛDAeff\n2¹jCST𝑧ihS0j\u0000jS0ihCST𝑧jº(7)\nAderivationofthiseffectiveHamiltonianisgiveninappendix\nA. We see from this that in the superexchange limit there is\naneffectivespin-conservingdiabaticcouplingbetweenthe S0\nandCSstatesgivenby 𝑉DAeff=𝑉DB𝑉BA¹¯𝐸\u0000𝐸CS★Sºaswell\nas an effective spin-orbit interaction with coupling strength\nΛDAeff= Λ DB𝑉BA¹¯𝐸\u0000𝐸CS★T𝑧º, where ¯𝐸is the average\nenergy of the S0andCSstates.\nBecause the electronic Hamiltonian reduces to an effective\ntwo-statemodelinthesuperexchangelimit,theelectrontrans-\nfer can be regarded as occurring in a single step,\nD\u000f¸\u0000B\u0000A\u000f\u0000𝑘CR\u0000\u0000\u0000! D\u0000B\u0000A\nThe electron transfer rate 𝑘CRis mediated by the spin-\nconservingsuperexchangecouplingterm 𝑉DAeff,andthespin-\norbit coupling mediated superexchange coupling ΛDAeff, and\n(when the Condon approximation is applied to the interstate\nsuperexchange couplings30) the theory reduces to that pre-\nsentedinRef.15andtherecombinationsuperoperatorisgiven\nexactly by Eq. (5), where ΛDA= Λ DAeffand𝑉DA=𝑉DAeff.\nWe see that in the superexchange limit there is no selectivity\nfor spin-polarisation and only selectivity for spin-coherence.\nB. The hopping limit\nIn the limit where charge recombination is controlled by\nhopping via an intermediate charge seperated state in which\nthebridgeischarged,28D\u000f¸–B\u000f\u0000–A,whichwedenoteCS★.\nThe kinetic scheme for this mechanism is simply\nD\u000f¸\u0000B\u0000A\u000f\u0000𝑘f\u0000\u0000\u0000⇀↽\u0000\u0000\u0000𝑘bD\u000f¸\u0000B\u000f\u0000\u0000A𝑘CR\u0000\u0000\u0000! D\u0000B\u0000A\nThe forward and backward hopping rates are given by 𝑘fand\n𝑘brespectively, and the recombination from the CS★state\nis assumed to occur irreversibly at a rate 𝑘CR. We start by\nwritingdowncoupledequationsofmotionforthespindensityoperators of the CS★[D\u000f¸–B\u000f\u0000–A] and CS [D\u000f¸–B–A\u000f\u0000]\nstates, as derived in Refs. 7, 15, and 25.\nd\nd𝑡ˆ𝜎CS★¹𝑡º=\u0000\u001a𝑘CR\n2j𝜓𝜃ih𝜓𝜃jˆ𝜎CS★¹𝑡º\u001b\n\u0000𝑘bˆ𝜎CS★¹𝑡º\n\u0000𝑖»𝛿𝜖j𝜓𝜃ih𝜓𝜃j¸2𝐽ˆ𝑃Sˆ𝜎CS★¹𝑡º¼¸𝑘fˆ𝜎CS¹𝑡º\n=LCS★ˆ𝜎CS★¹𝑡º¸𝑘fˆ𝜎CS★¹𝑡º(8)\nd\nd𝑡ˆ𝜎CS¹𝑡º=\u0000𝑖»ˆ𝐻ˆ𝜎CS¹𝑡º¼¸𝑘bˆ𝜎CS★¹𝑡º\u0000𝑘fˆ𝜎CS¹𝑡º(9)\nwhereLCS★=\u0000f𝑘CR2j𝜓𝜃ih𝜓𝜃j\u0001g\u0000𝑘b\u0000𝑖»𝛿𝜖j𝜓𝜃ih𝜓𝜃j¸\n2𝐽ˆ𝑃S\u0001¼. Here we have assumed that the intermediate state\nis sufficiently short lived that we need only include the ex-\nchange interaction 𝐽and the spin-orbit coupling shift term 𝛿𝜖\nin the spin Hamiltonian for the CS★state, and we can safely\nneglecthyperfine,dipolarandZeemaninteractionterms. This\nequation assumes there is no long-lived coherence between\ndifferent charge transfer states, and thus individual hopping\nsteps can be treated as incoherent processes. We also assume\ntheCS★spin states are near-degenerate (relative to thermal\nenergy)sowedonotneedtoaccountforspin-selectivityinthe\nforward/backward hopping processes.7\nBefore proceeding further we will outline a qualitatively\nhow spin polarisation selectivity can emerge in hopping me-\ndiated charge recombination. When spin-orbit coupling me-\ndiates the charge recombination from the intermediate CS★\nstate, it will selectively remove the j𝜓𝜃istate. Molecules in\nelectronspinstatesorthogonalto j𝜓𝜃i,whichhaveanon-zero\nspin-coherence, e.g. j𝜓𝜃?i=sin𝜃jSi\u0000𝑖cos𝜃jT𝑧i, will not\nreact and subsequently they will evolve coherently under the\nexchange interaction 𝐽in the intermediate state to generate a\nspinpolarisedstate.7Thisspinpolarisationisthentransferred\nbacktotheCSstate. Inthissenseweseethattheeffectiveloss\nofspinpolarisationfromtheCSstateisaresultoftheopposite\nspin polarisation being generated in the intermediate state in\nmolecules which do not recombine, which is transferred back\nto the CS state.\nIn order to derive the effective reaction superoperator, we\napply the steady-state approximation to the CS★spin density\noperator,d\nd𝑡ˆ𝜎CS★¹𝑡º\u00190, from which we obtain the following\nequation for CS★density operator in terms of the CS density\noperator,\nˆ𝜎CS★¹𝑡º\u0019\u0000𝑘fL\u00001\nCS★ˆ𝜎CS★¹𝑡º (10)\nandfromthiswecanobtaintheeffectivereactionsuperoperator\nas\nKˆ𝜎CS¹𝑡º=\u0000𝑘f¹1¸𝑘bL\u00001\nCS★ºˆ𝜎CS¹𝑡º(11)\nAlternativelywecanderivethehoppingreactionsuperoperator\nwithoutinvokingthesteadystateapproximationbyfirstsolving\nEq. (8) for ˆ𝜎CS★¹𝑡ºto give\nˆ𝜎CS★¹𝑡º=𝑘f∫𝑡\n0d𝜏𝑒LCS★𝜏ˆ𝜎CS¹𝑡\u0000𝜏º(12)\nAssuming𝑒LCS★𝜏decaystozeroonatime-scalefasterthanthe\ndynamics of ˆ𝜎CS¹𝑡\u0000𝜏º, we can invoke a Markovian approxi-\nmationwherewereplace ˆ𝜎CS¹𝑡\u0000𝜏º! ˆ𝜎CS¹𝑡ºintheintegral4\nand we set the upper limit of the integral to 𝜏=1. With this\nwe arrive at the same result as the steady-state approximation\nforˆ𝜎CS★¹𝑡ºgivenbyEq.(10),butthisapproachshowsthatthe\nSSA approximation can be expected to be accurate provided\nthe spin dynamics of the CS★occur on a much faster time\nscale than those of the CSstate. Unfortunately this reaction\nsuperoperator does not reduce to a simple form in the general\ncase,althoughananalyticalexpressioncanbeobtained. How-\neverwecanexamineparticularlimitsandextracttheeffective\nreaction operator ˆ𝐾.\nFirstlyitisinstructivetoconsiderthecasewhere 𝜃=0and\n𝛿𝜖=0, i.e. when there is no spin-orbit coupling involved in\nthe charge recombination process and there is no CISS effect.\nIn this case the reaction operator reduces to\nKˆ𝜎CS=\u0000\u001a˜𝑘CR\n2ˆ𝑃Sˆ𝜎CS\u001b\n\u0000𝑖\u0002\n2˜𝐽ˆ𝑃Sˆ𝜎CS\u0003\n¸˜𝑘D\u0012\nˆ𝑃Sˆ𝜎CSˆ𝑃S\u00001\n2\bˆ𝑃Sˆ𝜎CS\t\u0013 (13a)\nwhere ˜𝑘CR=𝑘f𝑘b¹𝑘b¸𝑘CRºis the effective reaction rate,\n2˜𝐽=8𝐽𝑘b𝑘f»¹4𝐽º2¸¹𝑘CR¸2𝑘bº2¼is an effective exchange\ncoupling, and ˜𝑘D=𝑘f¹¹4𝐽º2¸𝑘CR¹𝑘CR¸2𝑘bºº»¹ 4𝐽º2¸\n¹𝑘CR¸2𝑘bº2¼\u0000˜𝑘CR2,isaneffectivesinglet-tripletdephasing\nrate. Inthelimitwhere 𝐽=0,wecansimilarlyobtainasimple\nexpression for the reaction operator\nKˆ𝜎CS=\u0000\u001a˜𝑘CR\n2j𝜓𝜃ih𝜓𝜃jˆ𝜎CS\u001b\n\u0000𝑖»𝛿˜𝜖j𝜓𝜃ih𝜓𝜃jˆ𝜎CS¼\n¸˜𝑘D\u0012\nj𝜓𝜃ih𝜓𝜃jˆ𝜎CSj𝜓𝜃ih𝜓𝜃j\u00001\n2fj𝜓𝜃ih𝜓𝜃jˆ𝜎CSg\u0013\n(14)\nwhere the effective charge recombination rate ˜𝑘CR, decoher-\nence rate ˜𝑘D, and spin-orbit interaction 𝛿˜𝜖, are given by the\nabove expressions with the simple replacement 2𝐽!𝛿𝜖. In\neachcaseweseethattheeffectiverecombinationoperatorcan\nbedecomposedintoashiftinthespinHamiltonian 𝛿ˆ𝐻,aLind-\nbladian decoherence term, with Lindblad rates and operators\n𝛾𝑗andˆ𝐿𝑗, and a reaction term with the reaction operator ˆ𝐾,\nKˆ𝜎CS=\u0000\bˆ𝐾ˆ𝜎CS\t\n\u0000𝑖\u0002\n𝛿ˆ𝐻ˆ𝜎CS\u0003\n¸∑︁\n𝑗𝛾𝑗\u0012\nˆ𝐿𝑗ˆ𝜎CSˆ𝐿y\n𝑗\u00001\n2n\nˆ𝐿y\n𝑗ˆ𝐿𝑗ˆ𝜎CSo\u0013\n(15)\nFirstly we will consider the reaction operator ˆ𝐾, although the\ndecoherenceandHamiltonianshifttermscannotingeneralbe\nneglected and we will later evaluate their importance. In gen-\neral ˆ𝐾can be decomposed into the following set of operators\nˆ𝐾=˜𝑘S\n2ˆ𝑃S¸˜𝑘T𝑧\n2ˆ𝑃T𝑧¸˜𝑘𝑧\n2Δˆ𝑆𝑧¸˜𝑘ST𝑧\n2ˆΠST𝑧(16)\nwhere ˆ𝑃T𝑧=jT𝑧ihT𝑧jisaprojectionoperatorontothe T𝑧state,\nΔˆ𝑆𝑧=¹jSihT𝑧j¸jT𝑧ihSjº2=j\"D#Aih\"D#Aj\u0000j#D\"Aih#D\"Aj\nis the spin polarisation operator and ˆΠST𝑧=\u0000𝑖¹jSihT𝑧j\u0000\njT𝑧ihSjº2is the spin coherence operator. In general the ratesmust satisfy ˜𝑘S˜𝑘T𝑧\u00150and˜𝑘S˜𝑘T𝑧\u0015¹˜𝑘2\nST𝑧¸˜𝑘2\n𝑧º4in order\nto preserve positivity of the density operator. We note that in\nthe case of Luo & Hore’s phenomenological theory, we have\n˜𝑘S=𝑘CRcos2¹𝜒2º,˜𝑘T𝑧=𝑘CRsin2¹𝜒2º,˜𝑘𝑧=𝑘CRsin¹𝜒º\nand˜𝑘ST𝑧=0.8\nWefindingeneralthatboth ˜𝑘𝑧and˜𝑘ST𝑧areproportionalto\nsin¹2𝜃º, meaning they only emerge in chiral systems. This is\nbecause the spin coherence that is generated by the spin-orbit\nmediated recombination is proportional to sin¹2𝜃º. Further-\nmore we find that the ratio of ˜𝑘𝑧to˜𝑘ST𝑧is given by\n˜𝑘𝑧\n˜𝑘ST𝑧=4𝐽¹2𝑘b¸𝑘CRº\n¹2𝑘b¸𝑘CRº2¸4𝛿𝜖2¸8𝐽𝛿𝜖cos 2𝜃(17)\nwhichshowsthatspinpolarisationselectivitycanonlyariseif 𝐽\nisnon-zero,andifthebackreactionandchargerecombination\nfromtheintermediatestatearesufficientlyslowtoallowsome\ndegree of coherent spin evolution in the intermediate state to\ngenerate spin polarisation.\nThefullexpressionsforthespin-selectiverateconstantsare\nsomewhat cumbersome, although straightforward to evaluate\nnumerically,soherewewillanalysesomeoftheirpropertiesin\nspecificlimits. InappendixBweshowhowtoobtain Kinthe\nweak spin-orbit coupling limit, which we expect to be appli-\ncable to many organic donor-bridge-acceptor systems, where\nthereactionsuperoperatorparametersarerelativelysimple. In\nthis limit we find there are two dominant decoherence pro-\ncesses, with Lindblad operators ˆ𝐿𝑗=ˆ𝑃Sand ˆ𝑃T𝑧, and the\nrecombinationprocessisselectiveforboththe SandT𝑧states,\nas well as the spin polarisation and S-T𝑧coherence ˆΠST𝑧. We\nalso find that the spin-coherence and spin-polarisation selec-\ntivityonlyemergeatfirst-orderinthespinorbitcoupling,and\nspin-polarisation selectivity is only non-zero if 𝐽is non-zero.\nStarting from the weak spin-orbit coupling limit results in\nappendix B, it is instructive to consider the limit where co-\nherent dynamics of the CS★spins are much slower than the\nincoherent recombination process, i.e. 𝑘b𝑘CR\u001d𝐽𝛿𝜖. In\nthis limit we find\n˜𝑘S\u0019cos2𝜃𝑘f𝑘CR\n𝑘CR¸𝑘b˜𝑘T𝑧\u0019sin2𝜃𝑘f𝑘CR\n𝑘b(18a)\n˜𝑘ST𝑧\u0019sin¹2𝜃º𝑘f𝑘CR\n𝑘CR¸𝑘b˜𝑘𝑧\u0019𝑘f𝑘CR\n𝑘CR¸𝑘b4𝐽sin 2𝜃\n2𝑘b¸𝑘CR(18b)\nWeseethatthespinpolarisationremovedfromtheCSstateis\nproportional to 𝐽in the intermediate state. Conversely, in the\nlimit where𝐽\u001d𝑘b𝑘CR𝛿𝜖, it can be found that ˜𝑘𝑧\u00190and\n˜𝑘ST𝑧\u00190. Thisisbecauseinthelarge 𝐽limitspinpolarisation\ngenerated by the exchange interaction oscillates many times\nin the intermediate state prior to transfer back to the CS state,\nwhichaveragesthespinpolarisationthatistransferredbackto\nzero.\nItisalsointerestingtoconsiderthelimitwhere 𝑘CRissmall\nand𝛿𝜖=0. In this case we find\n˜𝑘S\u0019cos2𝜃𝑘f𝑘CR\n𝑘b˜𝑘T𝑧\u0019sin2𝜃𝑘f𝑘CR\n𝑘b(19a)\n˜𝑘ST𝑧\u0019𝑘f𝑘CR𝑘bsin 2𝜃\n𝑘2\nb¸¹2𝐽º2˜𝑘𝑧\u0019¹2𝐽º𝑘f𝑘CRsin 2𝜃\n𝑘2\nb¸¹2𝐽º2(19b)5\nIn this case, if the initial charge separation goes via the same\nintermediate CS★state and if the initial charge separation fol-\nlowing photo-excitation forms the CS★state in thej𝜓𝜃istate,\ni.e. assuming ΛD★B𝑉D★B= Λ DB𝑉DB(where D★denotes\nthe excited precursor donor orbital), then the spin polarisa-\ntion in the initial CS state is \u0000¹2𝐽º𝑘bsin 2𝜃¹𝑘2\nb¸¹2𝐽º2ºand\nthe initial spin coherence is 𝑘2\nbsin 2𝜃¹𝑘2\nb¸¹2𝐽º2º. So in\nthis limit the charge recombination is selective for the same\nspin-coherence as is initially generated but the oppositespin\npolarisation. It is straightforward to show that this result also\nholdsinthecasewhere 𝛿𝜖≠0. Itshouldbenotedthatthesitu-\nationismorecomplicatedifthephase ΛD★B𝑉D★Bisdifferent\ntoΛDB𝑉DB, which could be the case in a real system.\nWe now turn to the shift term, 𝛿ˆ𝐻, in the full effective\nreaction superoperator, which arises from coherent dynamics\nwhich occur transiently in the intermediate state. We can\nexpand this shift term in terms of the operators ˆ𝑃Sˆ𝑃T𝑧,Δˆ𝑆𝑧\nand ˆΠST𝑧, as we did for ˆ𝐾,\n𝛿ˆ𝐻=𝛿˜𝜖Sˆ𝑃S¸𝛿˜𝜖T𝑧ˆ𝑃T𝑧¸𝛿˜𝜖𝑧Δˆ𝑆𝑧¸𝛿˜𝜖ST𝑧ˆΠST𝑧(20)\nEvaluating the reaction superoperator, we find in general that\n𝛿˜𝜖𝑧=0, whilst the other terms are non-zero. The non-zero\nterms account for the shift in energy of the singlet and triplet\nstates due to the exchange coupling in the intermediate state,\nand an effective spin-orbit coupling that arises due to the su-\nperexchange spin-orbit interaction in the CS★state (the term\nproportional to 𝛿𝜖in Eq. (8)). As with ˆ𝐾it is possible to\nobtain a exact expression for 𝛿ˆ𝐻, but it is very complicated,\nhowever in the weak spin-orbit coupling limit (presented in\nappendix B) it is relatively straightforward to evaluate. The\npresence of this shift term (and the decoherence terms) cou-\nples and subsequently mixes spin polarised states of the CS\nstate, which means there exists no “protected” spin polarised\nCS state that can be generated from an non-polarised initial\nspinstate. Importantlythisimpliestherecanbenolong-lived\nspin polarisation in the CS state. In numerical tests presented\ninthenextsection,wewillshowthatthisshifttermisessential\nin accurately calculating the spin polarisation and coherence\ndynamics of the CS state.\nIII. NUMERICAL TESTS FOR THE HOPPING LIMIT\nInthissectionweaimtoevaluatetheaccuracyofthesteady\nstate approximation used in the previous section to obtain the\nreaction superoperator, as well as the relative importance of\nspinselectivityinthereactionoperator ˆ𝐾andaugmentedspin\ndynamics generated by 𝛿ˆ𝐻. As a first test, in Fig. 2 we show\nthe dynamics of CS state population and the spin polarization\nin the CS state as a function of time for the CS state, for a\nmodel chiral D–B–A system undergoing hopping mediated\ncharge recombination, where the intermediate CS★state has\na lifetime of approximately 10 ns (the complete set of model\nparameters are given in the figure caption). For this example\nwe can calculate the spin dynamics with the CS state initial-\nized in the singlet spin state ( ˆ𝜎CS¹𝑡=0º=jSihSj) for the full\nmodel, Eqs. (8) and (9), for the full steady state approxima-\ntion, Eq. (11), and for the weak spin-orbit coupling reaction\nSteady state approximation\nWeak SOC approximation\nWeak SOC approximation (no shift)FIG. 2. Dynamics of the hopping model calculated with the full\nset of density operator equations (solid lines) and various levels of\napproximation for the reaction superoperator (dashed lines). In this\nexample𝐽=1000mT,𝛿𝜖=\u0000098𝐽,𝑘f=100𝜇s\u00001,𝑘b=100𝑘f,\nand𝑘CR=10𝑘f.\nsuperoperatorobtainedinappendixB,andneglectingtheweak\nspin-orbit coupling contribution to the 𝛿ˆ𝐻(i.e. setting 𝛿𝜖¹1º\nST𝑧\nin Eq. (B20) to zero).\nIn the top panel of Fig. 2 we show the CS survival proba-\nbility (blue), 𝑝¹𝑡º=Tr»ˆ𝜎CS¹𝑡º¼, and spin polarization (red),\nhΔ𝑆𝑧i, and spin coherence (gold),\nΠST𝑧\u000b\n, for the full model,\nEqs.(8)and(9)],(solidlines)andforthefulleffectivereaction\nsuperoperator, Eq. (11), (dashed lines). In this example both\nspin coherenceand spinpolarisation aregenerated transiently\nby the hopping process, and oscillations in these quantities\ninduce oscillations in the CS state decay rate. We also see\nthat the net spin-polarisation generated in the CS state even-\ntually decays to zero. This is because all spin-polarised states\nare coupled by effective interactions in 𝛿ˆ𝐻, which means the\nCS state fully decays, leaving zero net spin-polarisation. We\nsee the full model result and effective reaction superoperator\nobtained with the steady-state approximation agree to graph-\nical accuracy, even in this model with physically reasonable\nparameters where the frequency associated with 2𝐽is signif-\nicantly faster than the decay rate of the intermediate state. In\nthe middle panel we compare the full model (solid lines) and\nthe weak spin-orbit coupling reaction superoperator, Eq. (B7)\n(dashed lines), which recovers most of the population and\nspin-polarisation dynamics, but is not quantitatively accurate6\n.\nFIG. 3. Dynamics of the hopping model calculated with the full set\nof density operator equations (solid lines) and the full steady state\nreactionsuperoperatorwith 𝛿ˆ𝐻=0(dashedlines). Inthisexample 𝐽\nis varied and in each case 𝛿𝜖=\u0000098𝐽,𝑘f=100𝜇s\u00001,𝑘b=100𝑘f,\nand𝑘CR=10𝑘f.\nforthisexample. InthebottompanelofFig.2weshowresults\nfor the same calculation neglecting O¹sin𝜃ºcontributions to\n𝛿ˆ𝐻in the weak spin-orbit coupling reaction superoperator,\nEq. B7. This approximation fails to capture the spin polarisa-\ntiondynamicsaccuratelyinthisexample,whichdemonstrates\nthat the emergent spin polarisation in the CS state is primar-\nily generated by augmentation of the coherent spin dynamics,\nrather than spin-selective recombination.\nTo further test the importance of 𝛿ˆ𝐻we have calculated\nthe decay rate, 𝑘¹𝑡º=\u0000¤𝑝¹𝑡º𝑝¹𝑡º, spin polarization and spin\ncoherence for the same model for a range exchange couplings\n𝐽and𝛿𝜖values in the intermediate state, as shown in Fig. 3.\nIn each case we calculate the full spin dynamics of the CS\nstate (solid lines), which agree to graphical accuracy with the\nfull steady-state approximation results, and compare this to\nthedynamicswiththesteady-stateapproximationreactionsu-\nperoperator with 𝛿ˆ𝐻=0(dashed lines). We see that across a\nrangeof𝐽values(with 𝛿𝜖=\u0000098𝐽ineachcase)that 𝛿ˆ𝐻can-\nnotbeneglected,evenwhenthefullstead-stateapproximation\nforthereactionoperatoranddecoherencetermareused. This\napproximation can even be qualitatively wrong, for example\nwhen𝛿ˆ𝐻is neglected in the 𝐽=100mT and𝐽=1000mT\nexamples the sign of the predicted spin polarization is wrong.\nWe also see that even when 𝛿ˆ𝐻is set to zero, decoherence\ncontributionsto Kdestroylong-livedspinpolarisation. Over-\nall these tests show that both the reaction operator and the\nHamiltonian shift play a significant role in determining the\nspin polarisation dynamics and survival probability of the CSstate in this example, and the Hamiltonian shift 𝛿ˆ𝐻cannot\nsimply be ignored, so the full reaction superoperator should\nalways be used in calculations.\nIV. CONCLUSIONS\nIn this work we have derived a description of spin-\nselectiveelectrontransferinchiraldonor-bridge-accpetorsys-\ntems through the reaction superoperator formalism. We have\nobtained expressions for this superoperator applicable in both\nthesuperexchangeandhoppingregimesfortherecombination\nprocess. Theformofthereactionsuperoperatorisverysimple\ninthesuperexchangelimit,whereitisnotselectiveforspinpo-\nlarisation. Howeverinthehoppingmediatedlimitthereaction\nsuperoperator becomes more complicated and we have found\nthe recombination is selective for both spin-polarisation and\nspin-coherence in this case. The spin polarisation selectivity\nshould be understood as arising from spin-polarisation being\ngenerated in molecules initially in non-reactive spin states,\nwhich is transferred back to the CS state in reverse-hopping.\nWe also find that spin-polarisation selectivity emerges only\nwhen the intermediate charge separated state is sufficiently\nlong lived, and when an exchange coupling in this state is\nlarge enough to generate spin polarisation in molecules in\nnon-reactive spin states. Spin coherence selectivity and spin-\npolarisation selectivity can differ from the chirality induced\nspincoherenceandpolarisationgeneratedbyphoto-excitation,\nand the selectivity depends on the phases couplings between\nbridge orbitals and the orbitals in the ground state and the ex-\ncited precursor. This inverse spin polarisation selectivity of\nformationrecombinationcouldhavesomebiologicalfunction,\nforexampleinmagnetoreception8,31,32orinhinderingreverse\nelectron transfer in photosynthetic reaction centres.18,33\nNumericaltestshaveshownthatinhoppingmediatedcharge\nrecombination chirality dependent shifts in the spin Hamilto-\nnian, which are induced by transient dynamics in the inter-\nmediate state, also play an essential role in determining the\nsurvival probability and spin polarisation in the charge sep-\narated state. This shift Hamiltonian mixes all spin-polarised\nstates, thereby destroying long-lived spin polarisation in the\nCS state. In order to accurately describe dynamics of charge\nrecombination in chiral molecules it is clearly necessary to\naccount for both spin-selectivity in the reaction and augmen-\ntationofthecoherentdynamicsbytherecombinationprocess.\nIt is important to note that the theory proposed here is not\nfullyconsistentwiththephenomenologicaltreatmentsofCISS\nwhichhavebeenproposedrecently,8andwhichhavebeenused\ninvestigate the role of CISS in avian magnetoreception.8,31,32\nAlthoughthetheorywehavepresenteddoesnothavethesim-\nplicityofthisphenomenologicaltreatment,itisderivedfroma\nphysically reasonable model, and all parameters appearing in\nthe theory (such as forward and backward hopping rates) can\nin principle be measured experimentally or calculated using\ncomputational approaches. Experimental and computational\nstudies suggest hopping mediated charge recombination may\nplay a role in avian magnetoreception,34,35and with the the-\nory presented here it would be possible to rigorously study7\nchirality-mediated spin effects in magnetoreception. The spin\ndensityoperatorframeworkusedherecanbeextendedstraight-\nforwardlytoincludeotherimportantphysicsofspin-correlated\ncharge separated states necessary to understand real systems,\nsuch as hyperfine coupling effects and spin relaxation.5,36–40\nWe anticipate that this framework for understanding spin se-\nlectivechargerecombinationinchiralsystemswillbeusefulin\nnumerouscontextsinvolvingmolecularCISS,suchasindevis-\ning systems exploiting CISS for quantum information science\nand in studies of CISS effects in biological electron transfer.\nACKNOWLEDGEMENTS\nT. P. F and D.T.L. were supported by the U.S. Department\nof Energy, Office of Science, Basic Energy Sciences, CPIMS\nProgram Early Career Research Program under Award DE-\nFOA0002019.\nAppendix A: Effective Hamiltonian theory for superexchange\nelectron transfer\nHere we derive the effective Hamiltonian Eq. (7) starting\nfrom the full model electronic state Hamiltonian given by\nEq. (6). We assume that the couplings ΛDB,𝑉DBand𝑉BA\nare small and that 𝐸CS★\u001d𝐸S0𝐸CSΘ. In this case the eigen-\nvectorsof ˆ𝐻=ˆ𝐻DBAcanbepartitionedintotwoapproximate\nsubspaces, one spanned by the high energy CS★states and\none spanned by the lower energy CSandS0states. With\nthis observation, we can derive an approximate form for the\nelectronic Hamiltonian within lower energy subspace. We\nfirst define a projection operator onto the low energy sub-\nspace ˆ𝑃=jS0ihS0j¸Í\nΘjCSΘihCSΘj, and its complement\nˆ𝑄=1\u0000ˆ𝑃. We project the electronic energy eigenstate equa-\ntion ˆ𝐻jΨi=𝐸jΨitoobtainequationsfor jΨiinthe ˆ𝑃and ˆ𝑄\nprojected spaces,41\n𝐸ˆ𝑃jΨi=ˆ𝑃ˆ𝐻ˆ𝑃jΨi¸ˆ𝑃ˆ𝐻ˆ𝑄jΨi (A1)\n𝐸ˆ𝑄jΨi=ˆ𝑄ˆ𝐻ˆ𝑃jΨi¸ˆ𝑄ˆ𝐻ˆ𝑄jΨi (A2)\nSolving the equation for ˆ𝑄jΨi, and substituting this into the\nequation for ˆ𝑃jΨiyields an effective equation for ˆ𝑃jΨi\n𝐸ˆ𝑃jΨi=»ˆ𝑃ˆ𝐻ˆ𝑃¸ˆ𝑃ˆ𝐻ˆ𝑄¹𝐸\u0000ˆ𝑄ˆ𝐻ˆ𝑄º\u00001ˆ𝑄ˆ𝐻ˆ𝑃¼ˆ𝑃jΨi(A3)\nTheeffectiveHamiltonianappearingontheright-handsideof\nthis equation is dependent on E, but by exploiting the large\nseparation between the low and high energy states, we can\nreplace¹𝐸\u0000ˆ𝑄ˆ𝐻ˆ𝑄º\u00001!¹ ¯𝐸\u0000ˆ𝑄ˆ𝐻ˆ𝑄º\u00001where ¯𝐸isthemean\nenergy of the CS and S0states. With this approximation and\nbynotingthat ˆ𝑃ˆ𝐻ˆ𝑃,ˆ𝑄ˆ𝐻ˆ𝑄and ˆ𝑃ˆ𝐻ˆ𝑄,with ˆ𝐻=ˆ𝐻DBA,canbewritten as\nˆ𝑃ˆ𝐻ˆ𝑃=𝐸S0jS0ihS0j¸∑︁\nΘ𝐸CSΘjCSΘihCSΘj(A4)\nˆ𝑄ˆ𝐻ˆ𝑄=∑︁\nΘ𝐸CS★Θ\f\fCS★Θ\u000b\nCS★Θ\f\f (A5)\nˆ𝑃ˆ𝐻ˆ𝑄=∑︁\nΘ𝑉BA\f\fCSΘ\u000b\nCS★Θ\f\f\n¸𝑉DB\f\fS0\u000b\nCS★S\f\f\u0000𝑖ΛDB\n2\f\fS0\u000b\nCS★T𝑧\f\f(A6)\nˆ𝑄ˆ𝐻ˆ𝑃=¹ˆ𝑃ˆ𝐻ˆ𝑄ºy (A7)\nwe straightforwardly obtain Eq. (7) for ˆ𝐻DBAeff, where the\nenergy level shifts in Eq. (7) are given by,\n𝛿𝐸S0=𝑉2\nDB\n¯𝐸\u0000𝐸CS★S¸Λ2\nDB\n4¹¯𝐸\u0000𝐸CS★T𝑧º(A8)\n𝛿𝐸CSΘ=𝑉2\nBA\n¯𝐸\u0000𝐸CS★Θ (A9)\nAppendix B: The weak SOC reaction operator\nIn order to derive the weak spin-orbit coupling reaction\nsuperoperator, we partition LCS★into a sum of a reference\nterm,Ldwhich is diagonal in the singlet-triplet basis, and a\ntermwhichcouplessingletandtripletstates, Lc. Thediagonal\nterm is given by\nLd=\u0000\u001a𝑘S\n2ˆ𝑃S¸𝑘T𝑧\n2ˆ𝑃T𝑧¸𝑘b\n2\u0001\u001b\n\u0000𝑖»𝜖Sˆ𝑃S¸𝜖T𝑧ˆ𝑃T𝑧\u0001¼\n(B1)\nwhere the effective rate constants and energies are given by\n𝑘S=𝑘CRcos2𝜃 (B2)\n𝑘T𝑧=𝑘CRsin2𝜃 (B3)\n𝜖S=2𝐽¸𝛿𝜖cos2𝜃 (B4)\n𝜖T𝑧=𝛿𝜖sin2𝜃 (B5)\nThe coupling term is given by\nLc=\u0000\u001a𝑘CR\n2sin¹2𝜃ºˆΠST𝑧\u0001\u001b\n\u0000𝑖»𝛿𝜖sin¹2𝜃ºˆΠST𝑧\u0001¼\n(B6)\nWe can expand the reaction superoperator to first-order in the\ncoupling term, which gives the following expression for the\nreaction superoperator\nK\u0019K¹0º¸K¹1º(B7)\nK¹0º=\u0000𝑘f¹1¸𝑘bL\u00001\ndº (B8)\nK¹1º=𝑘f𝑘bL\u00001\ndLcL\u00001\nd (B9)8\nTheK¹0ºis given by\nK¹0º=\u00008>> <\n>>:˜𝑘¹0º\nS\n2ˆ𝑃S¸˜𝑘¹0º\nT𝑧\n2ˆ𝑃T𝑧\u00019>> =\n>>;\n\u0000𝑖h\n˜𝜖¹0º\nSˆ𝑃S¸˜𝜖¹0º\nT𝑧ˆ𝑃T𝑧\u0001i\n¸𝛾¹0º\nS\u0012\nˆ𝑃S\u0001ˆ𝑃S\u00001\n2\bˆ𝑃S\u0001\t\u0013\n¸𝛾¹0º\nT𝑧\u0012\nˆ𝑃T𝑧\u0001ˆ𝑃T𝑧\u00001\n2\bˆ𝑃T𝑧\u0001\t\u0013(B10)\nwhere the effective rate constants are given by\n˜𝑘¹0º\nS=𝑘f𝑘S\n𝑘S¸𝑘b(B11)\n˜𝑘¹0º\nT𝑧=𝑘f𝑘T𝑧\n𝑘T𝑧¸𝑘b(B12)\nthe effective energy shifts are given by\n˜𝜖¹0º\nS=2𝑘f𝑘b\u0012𝜖S\u0000𝜖T𝑧\u00002𝑘b¸𝑘S¸𝑘T𝑧\u00012¸4\u0000𝜖S\u0000𝜖T𝑧\u00012\n¸𝜖S\n¹2𝑘b¸𝑘Sº2¸4𝜖2\nS\u0013 (B13)\n˜𝜖¹0º\nT𝑧=2𝑘f𝑘b\u0012𝜖T𝑧\u0000𝜖S\u00002𝑘b¸𝑘S¸𝑘T𝑧\u00012¸4\u0000𝜖S\u0000𝜖T𝑧\u00012\n¸𝜖T𝑧\u00002𝑘b¸𝑘T𝑧\u00012¸4𝜖2\nT𝑧\u0013 (B14)and the decoherence rates are given by\n𝛾¹0º\nS=𝑘f\u0000𝑘f𝑘b\u00002𝑘b¸𝑘T𝑧¸𝑘S\u0001\n\u00002𝑘b¸𝑘T𝑧¸𝑘S\u00012¸4\u0000𝜖T𝑧\u0000𝜖S\u00012\n\u0000𝑘f𝑘b¹2𝑘b¸𝑘Sº\n¹2𝑘b¸𝑘Sº2¸4𝜖2\nS(B15)\n𝛾¹0º\nT𝑧=𝑘f\u0000𝑘f𝑘b\u00002𝑘b¸𝑘S¸𝑘T𝑧\u0001\n\u00002𝑘b¸𝑘S¸𝑘T𝑧\u00012¸4\u0000𝜖S\u0000𝜖T𝑧\u00012\n\u0000𝑘f𝑘b\u00002𝑘b¸𝑘T𝑧\u0001\n\u00002𝑘b¸𝑘T𝑧\u00012¸4𝜖2\nT𝑧(B16)\nIgnoring any decoherence corrections to K¹1º,K¹1ºcan be\nwritten as\nK¹1º\u0019\u00008>> <\n>>:˜𝑘¹1º\n𝑧\n2Δˆ𝑆𝑧¸˜𝑘¹1º\nST𝑧\n2ˆΠST𝑧\u00019>> =\n>>;\u0000𝑖h\n𝛿˜𝜖¹1º\nST𝑧ˆΠST𝑧\u0001i\n(B17)\nwhere the spin polarisation selective rate is given by\n˜𝑘¹1º\n𝑧=\u00002 sin¹2𝜃º𝑘f𝑘b\u00002𝑘b¸𝑘S¸𝑘T𝑧\u0001 \u0000𝛿𝜖¹𝑘S\u0000𝑘T𝑧º¸𝑘CR\u0000𝜖T𝑧\u0000𝜖S\u0001\u0001\n¹𝑘b¸𝑘Sº\u0000𝑘b¸𝑘T𝑧\u0001 \u0000\u00002𝑘b¸𝑘S¸𝑘T𝑧\u00012¸4\u0000𝜖S\u0000𝜖T𝑧\u00012\u0001 (B18)\nthe spin coherence selective rate is given by\n˜𝑘¹1º\nST𝑧=sin¹2𝜃º𝑘f𝑘b¹4𝑘CR𝑘b¹𝑘S¸𝑘T𝑧¸𝑘bº¸𝑘CR¹𝑘S¸𝑘T𝑧º2¸4𝛿𝜖¹𝑘S\u0000𝑘T𝑧º¹𝜖S\u0000𝜖T𝑧ºº\n¹𝑘b¸𝑘Sº\u0000𝑘b¸𝑘T𝑧\u0001 \u0000\u00002𝑘b¸𝑘S¸𝑘T𝑧\u00012¸4\u0000𝜖S\u0000𝜖T𝑧\u00012\u0001 (B19)\nand the Hamiltonian correction term is given by\n𝛿˜𝜖¹1º\nST𝑧=2\n3sin¹2𝜃º𝑘f𝑘b\u00122\u0000\u00002𝑘b¸𝑘T𝑧\u0001¹2𝛿𝜖𝑘 b¸𝛿𝜖𝑘 S\u0000𝑘CR𝜖Sº\u0000𝜖T𝑧¹2𝑘CR𝑘b¸𝑘CR𝑘S¸4𝛿𝜖𝜖 Sº\u0001\n\u0010\n¹2𝑘b¸𝑘Sº2¸4𝜖2\nS\u0011 \u0010\u00002𝑘b¸𝑘T𝑧\u00012¸4𝜖2\nT𝑧\u0011\n¸4𝛿𝜖𝑘2\nb¸4𝛿𝜖𝑘 b\u0000𝑘S¸𝑘T𝑧\u0001¸𝛿𝜖𝑘2\nS¸𝑘S\u0000\u0000𝑘CR𝜖S¸2𝛿𝜖𝑘 T𝑧¸𝑘CR𝜖T𝑧\u0001¸𝑘T𝑧\u0000𝑘CR𝜖S¸𝛿𝜖𝑘 T𝑧\u0000𝑘CR𝜖T𝑧\u0001\n¹𝑘b¸𝑘Sº\u0000𝑘b¸𝑘T𝑧\u0001 \u0000\u00002𝑘b¸𝑘S¸𝑘T𝑧\u00012¸4\u0000𝜖S\u0000𝜖T𝑧\u00012\u0001\u0013\n(B20)\n1C. 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Inpa rticular,thehelicalspintexturesofWeylsemimetals\nmanifest the spin-momentum locking of Weyl fermions in a vis ible manner. The spin-wave fluctuations of the\nhelical order carry electric charge density; therefore, th e spin textures can be electrically controlled in a simple\nandpredictable manner.\nPACS numbers: 73.43.-f,71.70.Ej,75.70.Tj\nRelativistic electrons governed by the Dirac equation had\nbeen thought to be remote from condensed matter physics.\nThe developments in the last decade, especially the dis-\ncovery of graphene[1] and topological insulators (TIs)[2–\n4], however, have established the ubiquitousness of Dirac\nfermions in condensed matter. The topologically protected\nsurface states of topologicalinsulatorsare generallymas sless\nDirac fermions, meanwhile, the bulk states of many topo-\nlogical insulators can be described by massive Dirac equa-\ntions. More recently, massless Dirac[5–8] (and Weyl[9–26] )\nfermionshavealsobeendiscoveredinbulkmaterials(arece nt\nexperimentallydiscoveredmaterialistheTaAsclass[27–3 5]).\nThe interactions among the nominal massless Dirac\nfermions, if sufficiently strong, can dynamically generate\na Dirac mass and fundamentally change the properties of\nfermions. Thisphenomenonwasfirststudiedinthecontextof\nparticle physics[36]. In this paperwe investigatethe phys ical\nconsequencesofdynamicalmassgenerationinseveralclass es\nof topological Dirac metals. We find that the dynamically\ngenerated masses manifest themselves as helical spin order s.\nThese types of spin order have attracted considerableinter est\ninothermaterialsandmodels[37–42]. Asweshallshow,thei r\nemergence in topological Dirac semimetals is a quite robust\nphenomenon,which is independentof material details. Phys -\nically, the helical spin orders result from the spin-orbit c ou-\npling. Their descriptions as Dirac fermions permit a unified\ntreatment.\nWe present the results for three classes of materials. The\nfirst example is the edge state of quantum spin Hall (QSH)\ninsulators[43–45]. Dynamically generated Dirac mass man-\nifests itself as helical order at the edge [see Fig.1]. Quan-\ntum fluctuations, however, can destroy this order. Proxim-\nity to other materials (e.g. proximity to certain superlatt ice\nstructures) can stabilize the helical order. The second ex-\nample is the surface states of weak topological insulators,\nwhich are closely related to the first example, while avoidin g\nthe strong quantumfluctuationsbecause of the higherdimen-\nsionality. The third example is Weyl semimetal in the mag-\nnetic field. Here, the helical order depends on the direction\nof the magnetic field in a specific manner, which providesa sharp experimental signature for the identification of Wey l\nsemimetals[74]. The second and the third examples will be\nourfocuses.\nHelical order at QSH edge. The edge of QSH accommo-\ndates two counterpropagating modes, whose spin is locked\nwiththepropagatingdirection[46,47]. TheHamiltonianre ads\nH(k)=/summationdisplay\nkvF(k−kF)c†\n↑kc↑k−/summationdisplay\nkvF(k+kF)c†\n↓kc↓k(1)\nwhere the chemical potential has been absorbed into the def-\ninition of kF. This form of H(k) is dictated by time reversal\nsymmetry. We can introduce cR/Lbyc↑(x)=eikFxcR(x) and\nc↓(x)=e−ikFxcL(x),thentheHamiltonianbecomes\nH(p)=vF/summationdisplay\nppc†\npσzcp, (2)\nwherecp≡(cRp,cLp)Tis the Fourier transformation of\ncR/L(x), andσzis the spin operator. This Hamiltonian de-\nscribes a one-dimensional massless Dirac metal (more pre-\ncisely,aWeyl metal),with σzplayingtheroleofchirality.\nIn this paper we focus on the possibility of dynamical\nfermionmassgenerationandsymmetrybreaking[48–50]. For\nsimplicity let us take the interaction to be short-ranged,\nnamely,HI=−g(c†\nLcR)(c†\nRcL) (withg>0, which means re-\npulsive interaction among electrons with opposite chirali ty).\nIn the mean field theory, we define m=g/angbracketleftc†\nLcR/angbracketright, and obtain\nits mean field value as |m|=m0≡vFΛexp(−2πvF\ng), where\nΛis a momentum cuto ff. In this mean field calculation, only\n|m|can be obtained, while the phase of mis arbitrary. It is\nconvenientto write m(x)=m0exp(iθ(x)), neglectingthe fluc-\ntuation of|m|. The ground state without Goldstone modes is\nθ(x)=θ0,whereθ0isanarbitraryrealconstant.\nTheexpectationofthe x-componentofspinisgivenby\n/angbracketleftσx(x)/angbracketright=e−iQx/angbracketleftc†\nL(x)cR(x)/angbracketright+eiQx/angbracketleftc†\nR(x)cL(x)/angbracketright\n=2m0\ngcos(Qx−θ(x)) (3)\nwhereQ≡2kF. Similarly,we have\n/angbracketleftσy(x)/angbracketright=−ie−iQx/angbracketleftc†\nL(x)cR(x)/angbracketright+ieiQx/angbracketleftc†\nR(x)cL(x)/angbracketright2\nFIG.1: Helicalorder atthe edge of QSH.\n=−2m0\ngsin(Qx−θ(x)) (4)\nand/angbracketleftσz(x)/angbracketright=/angbracketleftc†(x)σzc(x)/angbracketright=0. The helical order is illus-\ntratedinFig.(1). Sofaraphysicallyintuitivemean-fielda nal-\nysis is presented. A more rigorousapproach is the bosoniza-\ntion. The helical spin ordering is favored in the presence\nof sufficiently strong interaction and certain perturbation en-\nabling the umklapp processes (see the online Supplemental\nMaterial).\nHelical order at the surface of weak TI. A natural recipe\ntoavoidstrongquantumfluctuationsinone-dimensional(1D )\nis to couple many 1D systems together to form a 2D system.\nThispicturebringsustothestudyofthissection.\nWeak TIs are characterizedby the so-called weak topolog-\nical indices[51, 52]. The simplest models of weak topologi-\ncal insulators consist of layered QSH [see Fig.2]. Therefor e,\nthe surface states can be obtained by coupling the QSH edge\nstates. SupposethatthesurfaceoftheweakTIcoincideswit h\nthexz-plane. To simplify the problem, we include hopping\nterms among only adjacent layers. The low energy Hamilto-\nnianforthesurfacereads\nH(k)=/summationdisplay\nkvF(kx−kF)c†\n↑kc↑k−/summationdisplay\nkvF(kx+kF)c†\n↓kc↓k\n−2t/bardbl/summationdisplay\nkcoskz(c†\n↑kc↑k+c†\n↓kc↓k), (5)\nwherek≡(kx,kz), andt/bardblis theinterlayerhopping.\nA notable feature of this Hamiltonian is as follows. The\nenergy of electrons with spin up and spin down is E↑(k)=\nvF(kx−kF)−2t/bardblcoskzandE↓(k)=−vF(kx+kF)−2t/bardblcoskz,\nrespectively,therefore,we have\nE↑(k+Q)=vF(kx+kF)+2t/bardblcoskz=−E↓(k) (6)\nwhereQ≡(2kF,π). Therefore, there is perfect Fermi sur-\nfacenestingat wavevector Q(Thissalient featureis absentin\nthe surface states of the strong topologicalinsulator[53] ). An\ninfinitesimal interaction HI=−g(c†\nLcR)(c†\nRcL) can generate a\ngap, analogous to the case of the QSH edge. The order pa-\nrameter of this symmetrybreakingis m(x)=g/angbracketleftc†\nL(x)cR(x)/angbracketright≡\nm0exp(iθ(x)). Thespindensitiesaregivenby\n/angbracketleftσx(x)/angbracketright=e−iQ·x/angbracketleftc†\nL(x)cR(x)/angbracketright+eiQ·x/angbracketleftc†\nR(x)cL(x)/angbracketright\n=2m0\ngcos(Q·x−θ(x)) (7)\nwhereQ≡(2kF,π),and\n/angbracketleftσy(x)/angbracketright=−ie−iQ·x/angbracketleftc†\nL(x)cR(x)/angbracketright+ieiQ·x/angbracketleftc†\nR(x)cL(x)/angbracketright\n=−2m0\ngsin(Q·x−θ(x)) (8)FIG. 2: (a) Weak TI as layered QSH. (b) Fermi surface nesting a t\nthe surface (inthe xzplane) of weak TI.\nFIG. 3: Helical order at the surface of weak TIs. This spin ord er is\nhelical inthe xdirection andstaggered inthe zdirection.\nThe spin texture at the weak TI surface is illustrated in Fig. 3\n(Wehavetaken θ(x)tobeaconstant). Thespinorderishelical\nin thexdirection, while staggered in the zdirection. Finally,\nweremarkthatinrealmaterialstheperfectnestingisrepla ced\nbyapproximatenesting. Tobemoreconclusive,wehavestud-\nied a realistic lattice model in the random phase approxima-\ntion(RPA),andfoundthattheapproximatenestingfavorshe -\nlical spin order even if the interaction is quite weak(see Su p-\nplemental Material), therefore, we expect it to occur in rea l\nmaterialsoftheweaktopologicalinsulator.\nHelicalspinorderinWeylsemimetals. Thedynamicalmass\ngeneration induced by interaction and the resultant charge\ndensity wave state has been studied before[49, 50, 54–57].\nWith an external magnetic field, the Fermi surface instabil-\nity becomesinfinitesimal, i.e. an infinitesimal interactio ncan\nopenupagapattheFermisurface[18,58]. Thechargedensity\nwavepatternis,however,toocrudetoidentifytheuniquefe a-\ntureofWeylsemimetals,namely,thespin-momentumlocking\ndescribed by the Weyl equation. Here we show the existence3\nof helical spin orders, which providesa finer signature of th e\nWeyl-typespectrum.\nFor simplicity let us consider a single pair of Weyl points\nlocated at K1andK2respectively. We define the shorthand\nnotationQ=K1−K2. The low-energy Weyl Hamiltonian\nreadshs(p)=vF/summationtext\ni,j=x,y,zpies\nijσj, in which s=1,2,pi≡\nki−Ks,i, andesis a 3×3 matrix. For each i=x,y,zwe\ncan define a vector es\ni=(es\nix,es\niy,es\niz); thushsbecomes more\ncompact:\nhs(p)=vF/summationdisplay\ni=x,y,zpi(es\ni·σ), (9)\nwherepiis small compared to |Q|. Hereafter we shall focus\non the cases with es\ni·es\nj=δij, for which compact analytical\ntreatmentispossible.\nNow we add a magnetic field B=Bˆz(We can always\nrotate the coordinate system such that Bpoints in the ˆ z\ndirection)[75]. The energy eigenvalues for nonzero Landau\nlevelsare\nEl(p)=±vF/radicalBig\np2z+2eBl(l=1,2,3,...),(10)\nwhich are all gapped. To obtain the zeroth Landaulevel, first\nwecansolvethe2Dprobleminthe xy-planebyletting pz=0.\nIntheLandaugaugethe 2DHamiltonianis\nh2D(qx,qy)=vF[(px+eBy)es\nx·σ+pyes\ny·σ)].(11)\nWe can find that the zeroth Landau level wavefunction\nψpx(x,y)=1√\n2πeBexp[−1\n2eB(eBy+px)2]exp(ipxx)|es\nx×es\ny/angbracketright,\nwhere we have introducedthe notation |n/angbracketrightto denote the two-\ncomponent spinor satisfying /angbracketleftn|σi|n/angbracketright=nifor an vector n.\nAddingthe pztermisnowstraightforwardbecause |es\nx×es\ny/angbracketrightis\naneigenvectorof pzes\nz·σ(Thisisthe simplificationoftaking\nes\ni·es\nj=δij). Thesingle-particlewavefuncitonis\nψs\npx,pz(x)=1√\n2πeBexp[−1\n2eB(eBy+px)2]\n×exp(ipxx+ipzz)|χses\nz/angbracketright,(12)\ninwhichwe haveintroducedthechirality\nχs=(es\nx×es\ny)·es\nz=±1, (13)\nandthesingle-particleenergyis\nEs\npx,pz=χsvFpz, (14)\nAccording to the above wavefunction structure, the fermion\noperatorscanbeexpandedas\nc(x)=/summationdisplay\nseiKs·x|χses\nz/angbracketrightcs(x)+..., (15)\nwherecs(x) are low-energy fermion operators analogous to\ncR/L(x) in the QSH section, and “ ...” denotes high energy\nmodes. Suppose that χ1=−χ2=1, then the indexidentification 1(2) ↔R(L) is valid, and the analysis of\ndynamical symmetry breaking in the QSH section applies,\nnamely, a four-fermion interaction induces a chiral conden -\nsation/angbracketleftc†\nL(x)cR(x)/angbracketright=m(x)\ng≡m0\ngexp(iθ(x)). Thex-component\nofspindensitybecomes\n/angbracketleftσx(x)/angbracketright=/angbracketleftc†(x)σxc(x)/angbracketright\n=e−iQ·xm(x)\ng/angbracketleftχ2e2\nz|σx|χ1e1\nz/angbracketright+H.c.(16)\nIf we write|χ1e1\nz/angbracketright=|e1\nz/angbracketright=[cos(φ1/2),sin(φ1/2)eiϕ1]T\nand|χ2e2\nz/angbracketright=| −e2\nz/angbracketright=[sin(φ2/2),−cos(φ2/2)eiϕ2]T,\nthen we have/angbracketleftχ2e2\nz|σx|χ1e1\nz/angbracketright=sin(φ1/2)sin(φ2/2)eiϕ1−\ncos(φ1/2)cos(φ2/2)e−iϕ2.Similarly,we have\n/angbracketleftσy(x)/angbracketright=/angbracketleftc†(x)σyc(x)/angbracketright\n=e−iQ·xm(x)\ng/angbracketleftχ2e2\nz|σy|χ1e1\nz/angbracketright+H.c.,(17)\nwith/angbracketleftχ2e2\nz|σy|χ1e1\nz/angbracketright=−i[cos(φ1/2)cos(φ2/2)e−iϕ2+\nsin(φ1/2)sin(φ2/2)eiϕ1],and\n/angbracketleftσz(x)/angbracketright=/angbracketleftc†(x)σzc(x)/angbracketright\n=e−iQ·xm(x)\ng/angbracketleftχ2e2\nz|σz|χ1e1\nz/angbracketright+H.c.,(18)\nwith/angbracketleftχ2e2\nz|σz|χ1e1\nz/angbracketright=cos(φ1/2)sin(φ2/2)+\nsin(φ1/2)cos(φ2/2)ei(ϕ1−ϕ2). Finally, the charge density\nis\n/angbracketleftσ0(x)/angbracketright ≡ /angbracketleftc†(x)c(x)/angbracketright\n=e−iQ·xm(x)\ng/angbracketleftχ2e2\nz|χ1e1\nz/angbracketright+H.c.,(19)\nwith/angbracketleftχ2e2\nz|χ1e1\nz/angbracketright=cos(φ1/2)sin(φ2/2)−\nsin(φ1/2)cos(φ2/2)ei(ϕ1−ϕ2).\nStudying some special cases helps us to understand these\nresults. Forinstance,we consider\nh1(p)=vF(pxσx−pyσz+pzσy),\nh2(p)=vF(pxσx+pyσz+pzσy). (20)\nIt is readily seen that e1\nz=e2\nz=(0,1,0), and the previous\ngeneralresultstell usthat\n/angbracketleftσx(x)/angbracketright=2m0\ngsin(Q·x−θ(x)),\n/angbracketleftσy(x)/angbracketright=0,\n/angbracketleftσz(x)/angbracketright=2m0\ngcos(Q·x−θ(x)). (21)\nThis case is shown in Fig.(4a). If we take a di fferent Weyl\nHamiltonian\nh1(p)=vF(pxσx+pyσy+pzσz),\nh2(p)=vF(pxσx−pyσy+pzσz), (22)4\nFIG. 4: Helical spin orders in Weyl semimetals (with dynamic al\nmassgeneration)for(a)HamiltonianinEq.(20)and(b)Hami ltonian\ninEq.(22). Here Qistaken tobe inthe ˆ zdirection.\nthea simplecalculationyields\n/angbracketleftσx(x)/angbracketright=−2m0\ngcos(Q·x−θ(x)),\n/angbracketleftσy(x)/angbracketright=−2m0\ngsin(Q·x−θ(x)),\n/angbracketleftσz(x)/angbracketright=0. (23)\nThiscaseisshowninFig.(4b). Thechargedensity /angbracketleftσ0(x)/angbracketright=0\nfor these two cases. Eq.(20) and Eq.(22) qualitatively re-\nsemble the Weyl semimetal materials and models. For in-\nstance,thesimplemodel[18] h(k)=2txsinkxσx+[2ty(cosky−\ncosk0)+m(2−coskx−coskz)]σy+2tzsinkzσzhas a pair of\nWeyl nodes at (0 ,±k0,0), withhs=1,2=vxσxpx±vyσypy+\nvzσzpzas its low-energyapproximation,whichis the same as\nEq.(22) except for the possible velocity anisotropy. A more\nquantitative study of these materials shall be presented el se-\nwhere.\nBy changing the direction of the magnetic field, |es\nz/angbracketrightis\nchangedaccordingly,andthehelicalspintexturechangesi na\nprescribed way. We also remark that if the Pauli matrices in\nEq.(9) are not associated with spin but some other degreesof\nfreedom, the helical order is a straightforward generaliza tion\noftheaboveresults.\nThe spin helix predicted here can be observed by the spin-\nresolved scanning tunneling microscope (STM). It is unclea r\nwhether the magnitude of electron-electron interaction (a nd\nthe samplequality)of the recentlydiscoveredWeyl semimet -\nalsfavorsthegenerationofthespinhelix,butweareprobab ly\njustifiedinbeingoptimisticaboutitspossiblerealizatio n,con-\nsideringtheongoingrapidprogressinthisfield.\nElectric manipulation of spin texture. So far we have\nnot investigated the e ffects of fluctuations of θ. Such\nphase fluctuationsare termed“axions”[59–61], and have als o\nbeen studied in the context of topological insulators andsuperconductors[62–67]. In our present study, the axions a re\nmuchmorevisiblebecauseoftheirsimplegeometricalmean-\ning: theyarethephaseangleofspinpolarization(rotatedf rom\nQ·x).\nNowwe shall show that spintexturescarryelectriccharge;\nmoreover,the chargedensity dependson the spin texturein a\nprecisemanner. Forconcreteness,letustaketheQSHedgeas\nan example. In the presenceof θ(x,t), the spin polarizationis\npointing to angle Q·x−θ(x,t). Let us consider the simplest\ncase,θ=θ0−Acos(qx),whereA<<1istheamplitudeofthe\nspin modulation on the background of the helical order. For\nsuch a slow modulation of the phase θof the Dirac mass, the\nGoldstone-Wilczekformula[68]relatesthegradientof θtothe\nchargedensity:\nρ(x)=1\n2π∂xθ=A\n2πqsin(qx). (24)\nOn the other hand, if the charge density is given as ρ(x)=\nρ0cos(qx),thenwe have\nθ(x)=2πρ0\nqsin(qx). (25)\nTherefore,gatingthesystemperiodically,suchthatthech arge\ndensity modulates periodically, can control the spin modul a-\ntionina predicablemanner.\nWe can also consider gating the system to induce a con-\nstant charge density ρ0(x)=C. According to the Goldstone-\nWilczek formula, we have θ(x)=2πCxfor this case. Now\nthephase Qx−θ(x)=(Q−2πC)x,namely,thewavevectorof\nhelical spin order becomes Q−2πC. This is consistent with\nthe relation Q=2kF: a constant charge density amounts to\nshiftingkFin the underlyingFermi surface “before”dynami-\ncalmassgeneration.\nFinally,we remarkthat taking kF=0 bringsusback to the\nresult of Ref.[69]; namely, a magnetic domain wall between\n+xand−xmagnetization generates a fractional charge ±e/2.\nFor a general kF/nequal0, the±xmagnetization is replaced by a\nspin helix with a wavevector Q=2kF, i.e. both sides of the\ndomainwallare spinhelices,witha phasedi fferenceπ.\nConclusions. A most prominent feature of topological\nDirac and Weyl semimetals is the spin-momentum locking,\nwhich is a dramatic consequence of spin-orbit coupling. We\nhaveshownthatthisspin-momentumlockingcanbe“frozen”\nas helical spin ordering in the presence of dynamical insta-\nbility (“helicalsolids” fromhelical liquids). These real -space\n(notthe reciprocal-space)helicalspin texturesshouldbe visi-\nbleinthespin-resolvedSTM.Apartfromitsintrinsicinter est,\nit has potential applications due to the electric tunabilit y of\nhelicalspintexture.\nAcknowledgements. Z.W. is supported by NSFC under\nGrant No. 11304175 and the Tsinghua University Initiative\nScientific Research Program. S.C.Z. is supportedby the NSF\nunderGrant numbersDMR-1305677andthe US Department\nof Energy, Office of Basic Energy Sciences under Contract\nNo. DE-AC02-76SF00515.5\n∗Corresponding author. wangzhongemail@tsinghua.edu.cn\n[1] A. K. Geim and K. S. 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Pisarski, Nucl ear\nPhysics A 843, 37 (2010).\n[74] A closely related phenomenon called “chiral magnetic s pi-\nral” has been investigated in the context of quark matters, s ee\nRef.[70–73].\n[75] We focus on the orbital e ffects of magnetic field, omitting the\nZeeman energyat this stage."    },    {        "title": "1805.05833v1.Spin_orbit_effects_on_the_spin_and_pseudospin_polarization_in_ac_driven_silicene.pdf",        "content": "arXiv:1805.05833v1  [cond-mat.mes-hall]  15 May 2018Spin-orbit effects on the spin and pseudospin polarization i n\nac-driven silicene\nAlexander L´ opez\nEscuela Superior Polit´ ecnica del Litoral, ESPOL, Departa mento de F´ ısica,\nCampus Gustavo Galindo Km. 30.5 V´ ıa Perimetral,\nP. O. Box 09-01-5863, Guayaquil, Ecuador∗\nFrancisco Mireles\nDepartamento de F´ ısica Te´ orica, Centro de Nanociencias y Nanotecnolog´ ıa,\nUniversidad Nacional Aut´ onoma de M´ exico,\nApdo. Postal 14, 22800 Ensenada B.C., M´ exico\nJohn Schliemann\nInstitute of Theoretical Physics, University of Regensbur g,\nUniversit¨ at Str. 31, 93053, Regensburg, Germany\nBenjamin Santos\nINRS -´Energie Mat´ eriaux T´ el´ ecommunications, Varennes, QC J3 X 1S2, Canada\n(Dated: April 29, 2022)\n1Abstract\nWestudythepseudospinandspindynamicaleffects insingle-l ayer siliceneduetoaperpendicular\nelectricfieldperiodicallydrivenanditsinterplay withth eintrinsicandextrinsic(Rashba)spin-orbit\ninteraction. We find that the spin nonconserving processes o f the real spin of the quasiparticles\nin silicene, – induced by the rather weak spin-orbit mechani sms –, manifest themselves as shifts\nof the resonances of its quasienergy spectrum in the low coup ling regime to the driving field. We\nshow that there is an interesting cooperative effect among the , in principle, competing Rashba\nand intrinsic spin-orbit contributions. This is explicitl y illustrated by exact and approximated\nanalytical solutionsofthedynamicalequations. Inadditi on, weshowthatafiniteRashbaspin-orbit\ninteraction is indeed necessary in order to achieve a nonvan ishing spin polarization. As additional\nfeature, trivial and nontrivial topological phases might b e distinguished from each other as fast\nor slow dynamical fluctuations of the spin polarization. We m ention the possible experimental\ndetection schemes of our theoretical results and their rele vance in new practical implementation of\nperiodically driven interactions in silicene physics and r elated two-dimensional systems.\n2I. INTRODUCTION\nSilicene1–5is an interesting material belonging to the family of two-dimensional s ystems\nthat support massive Dirac fermions within their low energy regime. A s well as graphene6–8,\nits carbon counterpart, silicene has a honeycomb lattice structur e with two interpenetrated\ntriangular sublattices. Yet, in silicene, its lattice structure is natur ally buckled. This in\nturn implies that fermions at each sublattice site will respond differen t to perpendicular\napplied electric fields. Within the static scenario, previous works hav e shown that ap-\nplying a constant and perpendicular electric field, a topological phas e transition can be\nrealized9. In addition, recent proposals that employ laser irradiation have sh own to protect\ntopological phases in Weyl semimetals10. Studies of silicene in the nonequilibrium regime\nhave shown the realization of dynamical topological phases within th e so-called off resonant\napproximation11,12. Yet, these works neglected the Rashba spin-orbit interaction sin ce it is\ntypically much smaller than the other energy scales in the problem. We show in this work\nthat nevertheless, even a rather weak Rashba interaction in silicen e can induce a nontrivial\nspin polarization effect when a periodically varying electric field is perpe ndicularly applied\nto the sample and as long as the system is in the resonant regime.\nThe study is focused within the Kane and Mele formulation for Dirac fe rmions in the\nlow energy approximation and investigate the spin-pseudospin effec ts due to a periodic time\nmodulated perpendicular electric field Ez(t)13. We show that a necessary requirement for\ninducing a nontrivial dynamical evolution of physical observables, s uch as the spin polar-\nization, does require the nonvanishing of the Rashba spin-orbit inte raction. In this manner,\nour work differs from those previous approaches that have overlo oked the role of this spin\nnonconserving interaction in silicene. In order to tackle the problem of silicene with peri-\nodically driven electric field, we make use of the standard Floquet’s th eorem approach14–16\nand compare the exact numerical results with the rotating wave ap proximation (RWA) for\nexperimentally achievable parameter regimes. We focus on the desc ription of the physics\nclose to resonance and near the critical value of the static field tha t induces an energy gap\nclosing of its spectrum. As it is already well established in the literatur e, the behavior of the\nenergy gap is important in order to determine the topological phase of the driven system\nboth in the static regime17–19as well as in the driven scenario20–26. We show that our results\nof the nontrivial spin dynamical effects are qualitatively dependent on the zero momentum\n3bandgap.\nThe structure of the paper is as follows. In the next section we pre sent the single particle\nDirac-like model for silicene under the presence of a time-varying ele ctric field and spin-orbit\ninteraction andgive numerical results forthe quasienergy spectr um by full diagonalizationof\nthe Floquet Hamiltonian. These exact numerical results are compar ed to the approximate\nquasienergy spectra obtained via the RWA scheme for experimenta lly suitable parameter\nregimes. Then, in section III we evaluate the spin and pseudospin po larizations induced by\nthe driving field using the analytical RWA solution which allows us in turn t o gain a better\nphysicalinsightontherelevanceoftheresonantlyinducedproces sesaswelloftheimportance\nof the Rashba spin-orbit interaction. In section IV we present con cluding remarks and argue\non the experimental feasibility of the realization of our findings.\nII. MODEL\nWe consider a monolayer silicene sample subject to a perpendicular st atic electric field\nEs. In addition, we consider a homogeneous time-dependent electric fi eld component Ez(t)\nthat drives the system out of equilibrium. Then, following reference s9,11the long wavelength\n(low energy) approximation leads to the 8 ×8 Dirac-like Hamiltonian ( /planckover2pi1= 1),\nHη(/vectork,t) =vF(ηkxσx+kyσy)+ηszσzλI (1)\n+ηh11σz+h22−ℓ[Es+Ez(t)]σz,\nwherevF=√\n3atb\n2≈105m/s is the Fermi velocity for charge carriers in silicene with\na= 3.86˚A the lattice constant and tb= 1.6eV is the hopping parameter. Here ℓ= 0.23˚A\nmeasures half the separation among the two sublattice planes. In a ddition,η=±1 describes\ntheK,K′Dirac points, σiandsiare Pauli matrices describing pseudo and real spin degrees\nof freedom respectively, whereas /vectork= (kx,ky) is the in plane momentum measured from the\nDirac point. The parameter λI= 3.9meV represents the strength of the effective intrinsic\nspin-orbit contribution within a tight binding approximation. Further more, the two contri-\nbutions given by the terms h11=aλR2(kysx−kxsy) andh22=λR1(ηsyσx−sxσy)/2 describe\nthe Rashba spin-orbit coupling associated to the nearest neighbor hopping and next-nearest\nneighbors, respectively. It is found that λR2= 0.7meV and typically h22is much smaller\nthan the other energy scales in the problem9. Thus, we keep h11and neglect h22in the\n4following since our results will show that this Rashba spin-orbit contr ibution is a necessary\ningredient for nontrivial manipulation of the spin polarizaton of Dirac fermions. Using the\nbasis{ψA↑,ψB↑,ψA↓,ψB↓}, at theKDirac point ( η= +1) we have the reduced Hamiltonian\nH(/vectork,t) =\nE−(t)vFk−iv2k−0\nvFk+−E−(t) 0 −iv2k−\n−iv2k+0−E+(t)vFk−\n0iv2k+vFk+E+(t)\n, (2)\nwherek±=kx±iky,E±(t) =λI±ℓ[Es+Ez(t)] andv2=aλR2. If we now define\ntanφ=ky/kxand perform a unitary transformation with H1(k,t) =R†\nφH(/vectork,t)Rφwith\nRφ= Diag(e−iφ,1,1,eiφ) we get\nH1(k,t) =\nE−(t)vFk iv 2k0\nvFk−E−(t) 0 −iv2k\n−iv2k0−E+(t)vFk\n0iv2k v Fk E +(t)\n. (3)\nTo simplify the notationwe will denote λR2=λRwhenever we refer to the Rashba spin-orbit\nstrength. Introducing the 4 ×4 matrix Λ = sx⊗1, which explicitly reads\nΛ =\n01\n10\n, (4)\nwith1being the 2 ×2 identity matrix, the Hamiltonian takes a block diagonal form\nH2(t) =T†\nαH(t)Tα=\nH2−(t) 0\n0H2+(t)\n, (5)\nwhere theunitarytransformationhastheexplicit form Tα= exp(−iαΛ/2) andαischosen to\nget rid of the off-diagonal terms. After some straightforward alg ebra one gets the condition\nforblockdiagonalizationtofixtheangleoftransformationbythepa rameter relationtan α=\nv2k/λI. Then, we find that\nH2±(t) =∓∆±σz+vFkσx−ℓEz(t)σz, (6)\nwhere we have introduced the momentum-dependent static bandg ap ∆±=/radicalig\nλ2\nI+(v2k)2±\nVz, withVz=ℓEs. In absence of the Rashba term ( v2= 0), the static electric field has a\n5critical value Es=Ec≡λI/ℓfor which the static bandgap closes (∆ −= 0) at one of the\nDirac cones. This is the so-called single-valley Dirac cone phase. Thus , in the vicinity of\nthe band inversion, one would expect that the typically small Rashba term proportional\ntov2might have a significant role in the dynamics, at least at large momenta . This max-\nimum value of the particle’s momentum is, for consistency, determine d by the validity of\nthe linear dispersion approximation. For instance, in a typical scatt ering problem this max-\nimum value corresponds to the Fermi momentum kF. We are interested in determining\nthe interplay among the Rashba nonconserving spin processes and the modulation effects\ndue to the driving field ℓEz(t), focusing in the near resonant parameter regime where the\nRWA holds. Before proceeding further we perform an additional un itary transformation\nD= Diag{D−,D+}, giving\nH3(t) =D†H2(t)D=\nH3−(t) 0\n0H3+(t)\n, (7)\nwhereD±=σzcosγ±\n2∓σxsinγ±\n2and we have defined tan γ±=vFk/∆±. We find then\nH3±(t) =∓ω±σz−ℓEz(t)(cosγ±σz∓sinγ±σx), (8)\nwhere we have introduced the effective energy ω±=/radicalig\n(vFk)2+∆2\n±. In order to explore\nthe single-valley Dirac cone scenario beyond the static regime we con sider a periodic time-\ndependent electric field Ez(t+T) =Ez(t), whereTis the period. Then, both of the\nHamiltonians H3±(t) inherit its periodicity and we can resort to Floquet theorem14–16which\nstates that the general solution to the dynamics\ni∂tU3(t) =H3(t)U3(t), (9)\ncan be written as\nU3(t) =P(t)e−iHFt, (10)\nwithP(t) periodic and HFa constant matrix, respectively. The eigenvalues of HFgive\nthe quasienergy spectrum of the periodically driven problem. In ord er to determine these\nquasienergies one standard approach consists of performing an e xpansion in the (infinite)\neigenbasis of time periodic functions Fn(t) =einΩt, where the frequency is defined as\nΩ = 2π/T. The resulting infinite eigenvalue problem is solved by a truncation pro ce-\ndure to determine the Floquet exponents in such a way that adding m ore modes dot not\n6FIG. 1. (Color online) Exact (blue continuous line) and appr oximate (yellow dashed line)\nquasienergy spectrum for one of the effective spin components gapless (left panel) and the other\ngapped (right panel) at the Dirac cone kx= 0. This is the so-called single-valley Dirac cone\nconfiguration. The parameters have been chosen as ξ= 0.25Ω,λI=Vz= 0.2Ω andλR= 0.05Ω.\nqualitatively modify the obtained quasienergies. In order to find the quasienergy spectrum\none needs to evaluate the evolution operator at t=Tand diagonalize the resulting matrix\nU3(T). In the following section we find an approximate semi-analytical solu tion to the en-\nergy spectrum and compare to the exact result obtained via numer ical diagonalization of the\nFloquet Hamiltonian in Fourier representation. We also evaluate the d ynamics of the spin\npolarization and show that a finite Rashba spin-orbit interaction con tribution is necessary\nin order to obtain nontrivial dynamical effects on the spin polarizatio n.\nIII. ANALYTICAL RESULTS WITHIN THE RWA\nLet us now discuss an approximate solution leading to analytical expr essions for the\ndynamical evolution of the physical quantities. This in turn allows us t o highlight the main\nphysical mechanisms underlying the spin and pseudospin control of the Dirac fermions.\nWe begin by rewriting the upper and lower components of the block dia gonal Hamiltonian\nH3±(t) =∓ω±σz−ℓEz(t)(σzcosγ±∓σxsinγ±), (11)\nwhere we recall that ω±=/radicalig\n(vFk)2+∆2\n±. Changing to the interaction representation the\ndynamics is dictated by VI±(t) =e∓iω±tσzV±(t)e±iω±tσz, whereV±(t) is the time-dependent\n7contribution shown in equation (11). Choosing the time dependence of the driving field as\nℓE(t) =ξsinΩtand using the algebra of the Pauli matrices we get,\nVI±(t) =−ξsinΩt[σzcosγ±∓sinγ±\n(cos2ω−tσx±sin2ω−tσy)]. (12)\nIn order to implement the rotating wave approximation (RWA) we wou ld like to remind\nthat it is a standard approximation scheme that renders the Floque t Hamiltonian time-\nindependent by restricting to small values of the effective coupling s trength to the driving\nfield and for frequencies with values near those of the energies of t he static Hamiltonian,\ni.e., near resonances. Hence, if we invoke the RWA, we get the appro ximate interaction\ncontribution\nVI±(t)≈ξsinγ±\n2i(e∓iδ±tσ+−e±iδ±tσ−), (13)\nwhere the frequency detuning that characterizes near resonan ce contributions is defined as\nδ±= 2ω±−Ω. In addition, we have used the standard definitions σ±= (σx±iσy)/2. To\nget (13) we have assumed Ω −2ω±∼0 and thus have disregarded the quickly oscillating\ntermsproportionalto ±Ωaswell asthosecorrespondingtothesecular contributionsΩ+2 ω±.\nSwitching back to the Schr¨ odinger representation we get the effe ctive RWA Hamiltonian\nh±(t) =∓σzω±−ig±/parenleftbig\ne±iΩtσ+−e∓iΩtσ−/parenrightbig\n, (14)\nwhere we have introduced the effective momentum-dependent cou pling constant g±, that\nreads explicitly\ng±=ξsinγ±\n2=ξvFk\n2ω±. (15)\nThe Hamiltonian (14) is taking to a time-independent form, hF±, by means of periodic\nunitary transformations in each subblock matrix P±(t) =e±i(1+σz)Ωt/2, which relies on the\nstandard transformation rule hF±=P†\n±(t)h±(t)P±(t)−iP†\n±(t)∂tP±(t). Explicitly, we have\nhF±=±Ω\n21∓2ω±−Ω\n2σz+g±σy, (16)\nfor which the approximate quasienergies are found to be given as\nεν±=1\n2/parenleftbigg\nν/radicalig\n4g2\n±+(2ω±−Ω)2±Ω/parenrightbigg\n,modΩ\n(17)\n8whereν=±1 is the energy sub-band index. Since the Hamiltonian preserves its in -plane\nrotational invariance, we set from now on ky= 0, without loss of generality and denote\nkx=k. In FIG. 1 we show the quasienergy spectrum for the configuratio n when one spin\ncomponent is gapped and the other gapless at k= 0. The continuous line is the exact\nresult obtained from a numerical diagonalization of both H3±(t), whereas the dashed line\ncorrespondstotheRWAscheme foraset ofparameterswhichhas beensetas λI=Vz= 0.2Ω\nandλR= 0.05Ω. We find an excellent agreement between the RWA and exact res ults for an\neffective value of the coupling strength ξ= 0.25Ω. Indeed, we have checked that the RWA\nproperly describes the quasienergies even for values of the effect ive coupling strengh as large\nasξ= 0.5Ω where they begin to differ from each other. If we choose the set of parameters\nλI= 0.2Ω andVz= 0.1Ω withλR= 0.05Ω same coupling strength ξ= 0.25Ω, we find that\nboth pseudospin sectors are gapped at k= 0, as it is shown in FIG. 2.\nHaving checked the quasienergy spectrum, we turn our attention to the interplay of the\nRashba spin-orbit coupling and driving field. Previous works have neg lected the Rashba\nspin-orbit in silicene9,11,12because it possesses a value much smaller than the other energy\nscales in the problem. Our purpose now is to show that it does have an important role\nin determining the time evolution of the spin polarization Sz(t) and its interplay with the\nac-driven component of the electric field renders nontrivial the tim e evolution of Sz(t) and\nalthough small, we show that λRis indeed a necessary ingredient for getting a finite Sz(t)\nfor a properly chosen initially prepared state. Our purpose is to sho w how the dynamics of\nthe Dirac fermions can be modified via this parameter. Let us discuss these issues in detail.\nWe first note that the block-diagonal structure of the Hamiltonian allows us to define an\neffective spin corresponding to these subblocks. Thus, we could ex pect that this block\ndiagonal condition could imply that effective spin is an approximate con served quantity.\nYet, in the following we show this is not the case due to the competing e ffects of the Rashba\nspin-orbit interaction and the driving field. In order to quantify the robustness of spin\npolarization under the driving field and the spin nonconserving effect s due to the Rashba\nspin-orbitcoupling, weevaluatenowthetimeevolutionofthespinpola rizationSz(t), defined\nas\nSz(t) =∝an}bracketle{tΨ(0)|U†(t)SzU(t)|Ψ(0)∝an}bracketri}ht, (18)\nwhere|Ψ(0)∝an}bracketri}htis the initially prepared electronic state and Sz=sz⊗1. The explicit form\nof the evolution operator U(t) would be given below. In order to highlight the role of the\n9FIG. 2. (Color online) Exact (blue continuous line) and appr oximate (yellow dashed line)\nquasienergy spectrum for the two effective spin sectors being gapless at the Dirac cone k= 0.\nThe parameters have been chosen as ξ= 0.25Ω,λI= 0.2Ω,Vz= 0.1Ω andλR= 0.05Ω.\ndriving field and Rashba term we will choose the initial state |Ψ(0)∝an}bracketri}htas\n|Ψ(0)∝an}bracketri}ht=1√\n2\n|ψ−(0)∝an}bracketri}ht\n|ψ+(0)∝an}bracketri}ht\n, (19)\nwith vanishing initial spin polarization Sz(0) = (∝an}bracketle{tψ−(0)|ψ−(0)∝an}bracketri}ht−∝an}bracketle{tψ+(0)|ψ+(0)∝an}bracketri}ht)/2 = 0.\nThe simplest choice is to set sz|ψ±(0)∝an}bracketri}ht=±|ψ±(0)∝an}bracketri}ht. Using the transformations that render\nthe evolution operator block diagonal, it reads explicitly\nU3(t) =\nU3−(t) 0\n0U3+(t)\n (20)\nwith\nU3±(t) =\na±(t)b±\n−b∗\n±a∗\n±(t)\n (21)\nwhere the dimensionless functions a±(t) andb±(t) read for the RWA result as\na±(t) =e±iΩt/2/bracketleftbigg\ncosΓ±t±iδ±t\n2sincΓ±t/bracketrightbigg\n(22)\nb±(t) =g±te±iΩt/2sincΓ±t, (23)\nand satisfy the restriction\n|a±(t)|2+|b±(t)|2= 1. (24)\n10FIG. 3. (Color online) Effective spin polarization Sz(t) as a function of time and momentum for\nλR= 0.05Ω. The upper panels correspond to vanishing ξ= 0 with left (right) panel having the\nother parameters as given in FIG. 1 (FIG. 2), respectively. T he lower left (right) panel shows the\nSz(t) for finite ξ= 0.25Ω as in FIG. 1 (FIG. 2). We notice that near the Dirac point ( k= 0)\nadditional spin inversion flips signal the competing effects a mong the Rashba and driving field.\nFor large momenta the Rashba contribution dominates and we o bserve no dynamical effects due\nto the driving field.\nIn addition, we have defined Γ ±= 1/2/radicalbig\n4g2\n±+δ2\n±, withδ±= 2ω±−Ω and we remind\nthat tanα=v2k/λImeasures the ratio of the two spin-orbit contributions that rende r the\nHamiltonian block diagonal as was described previously. Thus we get\nSz(t) =sinα\n2{Re[W−−(t)−W++(t)]sinα\n+ Im[W−+(t)−W+−(t)]cosα+Im[W+−(t)+W+−(t)]}, (25)\nwhereW−−(t) =∝an}bracketle{tψ−(0)|W(t)|ψ−(0)∝an}bracketri}ht, and so on, are the matrix elements of the operator\nW(t) =D−U†\n3−(t)D−D+U3+(t)D+. (26)\n11After some algebraic operations we get the result,\nSz(t) =sinα×\n[(Ima−Reb++Rea+Imb−)cosγ−−\n(Imb+Rea−+Ima+Reb−)cosγ++\n(Reb+Imb−−Rea+Ima−)sinγ−+\n(Imb+Reb−−Ima+Rea−)sinγ+], (27)\nSince the parameterization of the evolution operator given in equat ion (21) is valid in gen-\neral, we must remark that the result given in equation (27) is exact a nd thus allows us to\nverify both, the numerical calculation and the RWA result. Indeed, the functional form of\nthe dimensionless functions (22) for the exact result would differ fr om the relations corre-\nsponding to the RWA, but the polarization would still be calculated fro m expression (27),\nwith the corresponding dimensionless quantities for the exact resu lt. Since we have verified\nthat RWA is valid for values of the coupling strength close to ξ= 0.5Ω we rely on the\nphysics within the RWA which is encoded in the explicit form of the funct ions defined in\nequations (22). Moreover, the result given in equation (27) shows that the vanishing of the\nfirst termintheparenthesis ofequation(25)means thatallpolariz ationeffects arefirst order\nin the Rashba spin-orbit strength since sin α∝λRwhereas the vanishing of the second term\ncan be explained via the fact that it is proportional to the intrinsic sp in-orbit interaction\ncosα∝λIwhich should not induce any spin-flipping processes.\nLet us now discuss the results for the spin polarization Sz(t) within the RWA as given in\nequation (27). In FIG. 3 we show a density plot of Sz(t) as a function of momentum and\ntime for different parameter configurations. The upper panels sho w the dynamical evolution\nof the polarization in absence of the driving field ( ξ= 0) for the single-valley Dirac cone\nconfiguration FIG. 1 and gapless (FIG. 2) scenario. Yet, as seen in the two lower panels\nof FIG. 3, once ξ= 0.25Ω is finite a richer physical scenario is observed. In general, it is\nobserved that spin-flipping processeses only require a finite value o f the Rashba spin-orbit\ninteraction and thus it should be considered as a valuable parameter in experimental setups.\nWe would like to remark that previous works9,11,12have disregarded the role of the Rashba\nspin-orbit coupling in the physics of monolayer silicene based on its rat her small value as\ncompared to the other energy scales in the problem. However, our results for the dynamical\nevolution of the spin polarization show that for vanishing λR, there are not pseudospin-spin\n12oscillations and Sz(t) vanishes exactly for all values of the time and momentum variables f or\nthe initially chosen state. As discussed before, mathematically spea king, this can be seen\nby the sinαprefactor in equation (27), but the physical reason for this resu lt relies essen-\ntially in the nonconserving nature of the Rashba coupling. In presen ce of the driving field\n(ξ= 0.25Ω), the dynamics of Sz(t) is determined by the effective spin sectors being in the\nconfiguration of FIG. 1 (lower left panel) or FIG. 2 (lower right pane l) of the quasienergies\nshown with Vz=λI(Vz∝ne}ationslash=λI). In the first situation, the dynamics could be said to be fast,\nmeaning that, within the same time window, more spin-flipping occurs f or those quasiener-\ngies corresponding to the so-called single-valley Dirac cone configur ation of FIG. 111. In\nthis manner, the analysis of the dynamics of Sz(t) could afford an experimental verification\nfor the topological features of Dirac fermions in silicene and related materials. Moreover,\nfor momenta in the vicinity of the Dirac point k= 0, the role of the driving field is en-\nhanced since it qualitatively modifies the time evolution of Sz(t) which can in principle be\nunderstood as the resonant processes that lead to additional sp in-flips and steems from the\nnoncommuting nature of the Rashba spin-orbit interaction and the driving electric field and\namounts to a direct manipulation of the pseudospin degree of freed om as we show below.\nTherefore, we find that even though the single-valley Dirac cone co nfiguration Ez=Ec\nmight be expected as the only requirement for a nontrivial spin-pse udospin behavior of the\nDirac fermions on a single layer of silicene, this is not the only paramete r condition that\nrenders nontrivial behavior on other quantities of interest, such as the spin polarization.\nThus, our results show that indeed the, in principle, competing effec ts of both the Rashba\nterm and the oscillating electric field are necessary in order to get a n ontrivial evolution of\nthe spin polarization. Of course, the Rashba term encodes the pse udospin-spin coupling and\naccounts for the angular momentum exchange among these two de grees of freedom whereas\nthe resonant nature of the driven field is encoded in the Ω condition a s well and the ξcou-\npling strength that render the RWA valid.\nIn order to further explore the dynamical features of the model, we now focus on the pseu-\ndospin polarization dynamics, given as\nPz(t) =∝an}bracketle{tΨ(0)|U†(t)PzU(t)|Ψ(0)∝an}bracketri}ht, (28)\n13FIG. 4. (Color online) Effective pseudospin polarization Pz(t) as a function of time and momen-\ntumfortheparameters chosen as inFIG. 3. Near theDiracpoin t (k= 0) thedrivingfieldeffectively\ninduces pseudospin inversion at finite driving.\nwithPz=1⊗σz. After some algebraic manipulations we get the result\nPz(t) =cosα×\n/bracketleftbig\n1/2/parenleftbig\ncos2γ−−sin2γ−/parenrightbig/parenleftbig\nIma2\n−−Imb2\n−/parenrightbig\n+1/2/parenleftbig\nRea2\n−−Reb2\n−/parenrightbig\n+cosγ−sinγ−Ima−Imb−\n−1/2/parenleftbig\ncos2γ+−sin2γ+/parenrightbig/parenleftbig\nIma2\n+−Imb2\n+/parenrightbig\n−1/2/parenleftbig\nRea2\n+−Reb2\n+/parenrightbig\n−cosγ+sinγ+Ima+Imb+/bracketrightbig\n−sinα×\n[cosγ−(Ima−Reb−−Imb−Rea−)+sinγ−(Ima−Rea−+Imb−Reb−)\n−cosγ+(Ima+Reb+−Imb+Rea+)−sinγ+(Ima+Rea+−Imb+Reb+)],(29)\nThe results are shown in FIG. 4 for the same parameters as in fugur e FIG.3. We note that\ncontrary to what happens for Sz(t), for vanishing value of the driving field strength ξ= 0\n(upperpanels)showaqualitativebehaviourwhichisnotonlydominate dbytheRashbaspin-\norbit interaction λR= 0.05Ω. First we notice that the dynamics of Pz(t) is much faster than\nthat ofSz(t) since more spin-flipping processes occur within a shorter time windo w. These\nmight stem from the fact that even without Rashba spin-orbit inter action, the pseudospin\n14is a nonconserved quantity since the static Hamiltonian is nondiagona l (even ifλR= 0)\nfor finite momentum. In addition, in the lower panels (corresponding to finite driving) we\nnotice that there is a more effective modulation of the pseudospin po larization. This is a\nnatural consequence of the effective role of perpendicular electr ic fields acting differently at\neach sublattice in silicene. Moreover, from the figures in the lower pa nels of FIG. 4 we can\ninfer that driving the system in the single-valley Dirac cone configura tion of FIG. 1 (lower\nleft panel) and the gapless scenario of FIG. 2 (lower right panel) migh t be distinguished from\neachother byselecting thoseresonant procesesses happening in thevicinity oftheresonances\nvFk/Ω =±0.5 where pseudospin-flipping might be accounted for the avoiding cro ssings than\ncan be observed around these values of momenta that qualitatively can be understood as an\nadmixture of both pseudospin components with equal weights. We a lso notice in the lower\nleft (right) panel that for small momenta, the long term dynamics ( Ωt/π≥4) the trivial\n(nontrivial) topological scenario is dominated by vanishing (finite) va lue of the pseudospin\nclose to zero momentum.\nIV. CONCLUSIONS\nWe have shown that the realization of effective spin-pseudospin exc hange requires the in-\nterplayofboththespinnonconserving termsassociatedtothera therweakRashbaspin-orbit\nterm as well as the time-dependent field strength. In fact, for th e configuration where both\nof theeffective spin sectors have anenergy spectrum thatis gapp edatk= 0, the cooperative\neffects of the Rashba term and the driving interaction are enhance d as the spin polarization\nshows additional flipping processes which require a shorter phase d ifference among the two\nspin components. Since this phase difference shifts the maxima and m inima ofSz(t), addi-\ntional minima appear in its dynamical evolution. Indeed we could expec t that the closer in\nvalue the quasienergy spectra of the two effective spin states are , the probability for spin-\nflipping processes would in turn be reduced and the additional minima s hould disappear.\nWe think that these findings could be interesting for a better under standing of the role of\nRashba spin-orbit coupling in silicene and our results show that this pr eviously disregarded\nspin interaction in silicene could indeed be accounted for in experiment al setups where fine\ncontrol of the spin degree of freedom would be required. For an ac tual experimental real-\nization, we could expect that standard spectroscopic techniques could allow the validation\n15of our results, for instance via time-resolved spin-dependent pho toemission measurements.\nWe also have found that resonant processes play an important role in pseudospin dynamics\nand the evolution of these two quantities could afford a probe for th e dynamical evolution\nand possible modulation of the topological properties of these new m aterials27for instance,\nby time-resolved ultrafast optical spectroscopy studies of topo logical insulators28or by us-\ning time, spin and angle-resolved photoemission of the ultrafast dyn amics of topological\ninsulators29that can be extended to systems where large values of the Rashba spin-orbit\ninteraction could be achieved in group IV films30.\nACKNOWLEDGMENTS\nThis work has been supported by Deutsche Forschungsgemeinsch aft via GRK 1570 “Elec-\ntronic Properties of Carbon Based Nanostructures”. F. M. and A . 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Wang, Sci entific Reports 7, 45923 (2017).\n17"    },    {        "title": "1411.3249v1.Spin_Hall_effect.pdf",        "content": "Spin Hall e\u000bect\nJairo Sinova\nInstitut f ur Physik, Johannes Gutenberg Universit at Mainz, 55128 Mainz, Germany and\nInstitute of Physics ASCR, v.v.i., Cukrovarnick\u0013 a 10, 162 53 Praha 6, Czech Republic\nSergio O. Valenzuela\nICN2 - Institut Catala de Nanociencia i Nanotecnologia, Campus UAB, Bellaterra, 08193 Barcelona, Spain and\nICREA - Instituci\u0013 o Catalana de Recerca i Estudis Avan\u0018 cats, 08010 Barcelona, Spain\nJ. Wunderlich\nInstitute of Physics ASCR, v.v.i., Cukrovarnick\u0013 a 10, 162 53 Praha 6, Czech Republic and\nHitachi Cambridge Laboratory, Cambridge CB3 0HE, United Kingdom\nC. H. Back\nUniversit at Regensburg, Universit atstra\u0019e 31, 93040 Regensburg, Germany\nT. Jungwirth\nInstitute of Physics ASCR, v.v.i., Cukrovarnick\u0013 a 10, 162 53 Praha 6, Czech Republic and\nSchool of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom\n(Dated: November 13, 2014)\nSpin Hall e\u000bects are a collection of relativistic spin-orbit coupling phenomena in which electrical\ncurrents can generate transverse spin currents and vice versa. Although \frst observed only a\ndecade ago, these e\u000bects are already ubiquitous within spintronics as standard spin-current gener-\nators and detectors. Here we review the experimental and theoretical results that have established\nthis sub-\feld of spintronics. We focus on the results that have converged to give us a clear under-\nstanding of the phenomena and how they have evolved from a qualitative to a more quantitative\nmeasurement of spin-currents and their associated spin-accumulation. Within the experimental\nframework, we review optical, transport, and magnetization-dynamics based measurements and\nlink them to both phenomenological and microscopic theories of the e\u000bect. Within the theoreti-\ncal framework, we review the basic mechanisms in both the extrinsic and intrinsic regime which\nare linked to the mechanisms present in their closely related phenomenon in ferromagnets, the\nanomalous Hall e\u000bect. We also review the connection to the phenomenological treatment based\non spin-di\u000busion equations applicable to certain regimes, as well as the spin-pumping theory of\nspin-generation which has proven important in the measurements of the spin Hall angle. We\nfurther connect the spin-current generating spin Hall e\u000bect to the inverse spin galvanic e\u000bect,\nwhich often accompanies the SHE, in which an electrical current induces a non-equilibirum spin\npolarization. These e\u000bects share common microscopic origins and can exhibit similar symmetries\nwhen present in ferromagnetic/non-magnetic structures through their induced current-driven spin\ntorques. Although we give a short chronological overview of the evolution of this \feld, the main\nbody of this review is structured from a pedagogical point of view, focusing on well-established\nand accepted physics. In such a young \feld, there remains much to be understood and explored,\nhence we outline from our own perspective some of the future challenges and opportunities of this\nrapidly evolving area of spintronics.\nContents\nI. Introduction 2\nII. Overview 3\nA. Spin Hall, anomalous Hall, and Mott polarimetry 3\nB. Intrinsic spin Hall and quantum Hall e\u000bects 5\nC. Spin Hall e\u000bect and non-magnetic/ferromagnetic\nhybrid structures 5\nD. Spin Hall e\u000bect, spin galvanics, and spin torques 6\nIII. Theory of spin Hall e\u000bect 8\nA. Mechanisms of the spin Hall e\u000bect 9\n1. Intrinsic mechanism 10\n2. Skew scattering mechanism 11\n3. Side-jump mechanism 13\nB. Phenomenological Drift-Di\u000busion Theory 14C. The Cr\u0013 epieux-Bruno model: extrinsic side-jump and\nskew scattering model 14\nD. Theory of inverse spin Hall e\u000bect induced by\nspin-pumping 16\nE. Kubo formalism 18\nIV. Experimental studies of spin Hall e\u000bect 20\nA. Early experiments of anomalous Hall e\u000bect in\nparamagnets 20\nB. Optical tools in spin Hall experiments 21\n1. Optical detection of the spin Hall e\u000bect 21\n2. Optical generation of the inverse spin Hall e\u000bect 23\n3. All-optical generation and detection 24\n4. Electrical manipulation 25\nC. Transport experiments 26\n1. Concepts of nonlocal spin transport. Electrical\ninjection and detection 26arXiv:1411.3249v1  [cond-mat.mtrl-sci]  12 Nov 20142\n2. Nonlocal detection of inverse spin Hall e\u000bect with\nlateral spin-current 27\n3. Nonlocal detection of spin Hall e\u000bects with vertical\nspin-current 28\n4. Direct detection of the spin Hall induced spin\naccumulation 31\n5. Spin Hall injection and detection without\nferromagnets 31\n6. Spin Hall magnetoresistance 33\nD. Spin Hall e\u000bect coupled to magnetization dynamics 34\n1. Ferromagnetic resonance spin pumping 34\n2. Spin Hall e\u000bect modulation of magnetization\ndamping 37\n3. Spin Hall e\u000bect - spin transfer torque 38\n4. Spin Hall e\u000bect induced switching of the\nmagnetization 39\nE. Spin Hall angles 41\nV. Future directions and remaining challenges 41\nList of symbols and abbreviations 44\nAcknowledgements 44\nReferences 44\nI. INTRODUCTION\nSpintronics is a \feld that jointly utilizes the spin and\ncharge degrees of freedom to control equilibrium and non-\nequilibrium properties of materials and devices (Bader\nand Parkin, 2010; Wolf et al. , 2001; Zutic et al. , 2004).\nThe generation, manipulation, and detection of spin-\ncurrents is one of the key aspects of the \feld of spintron-\nics. Among the several possibilities to create and con-\ntrol spin-currents the spin Hall e\u000bect (SHE) has gained\nits distinct place since its \frst observation a decade ago\n(Day, 2005; Kato et al. , 2004b; Wunderlich et al. , 2004,\n2005). In the direct SHE, an electrical current passing\nthrough a material with relativistic spin-orbit coupling\ncan generate a transverse pure spin-current polarized per-\npendicular to the plane de\fned by the charge and spin-\ncurrent. Its reciprocal e\u000bect, the so called inverse SHE\n(ISHE), is the phenomenon in which a pure spin-current\nthrough the material generates a transverse charge cur-\nrent.\nThe SHE borrows its concept from the well established\nanomalous Hall e\u000bect (AHE) where relativistic spin-orbit\ncoupling generates an asymmetric de\rection of the charge\ncarriers depending on their spin direction (Nagaosa et al. ,\n2010). The AHE can be detected electrically in a ferro-\nmagnet (FM) via a transverse voltage because of the dif-\nference in population of majority and minority carriers.\nThe generalization of this e\u000bect to a pure spin-current\ngenerated by the SHE in a non-magnetic material (NM)\nwas proposed over four decades ago (Dyakonov and Perel,\n1971b) based on the idea of asymmetric Mott scattering\n(Mott, 1929). This so called extrinsic SHE remained un-\nexplored until recent proposals that put forward a similar\nprediction (Hirsch, 1999; Zhang, 2000) as well as the pos-\nsibility of a strong intrinsic e\u000bect (Murakami et al. , 2003;\nSinova et al. , 2004).The initial challenge for SHE detection was primarily\nthe lack of direct electrical signals; therefore initial exper-\niments detected it by optical means, both in the extrin-\nsic regime (Kato et al. , 2004b) and the intrinsic regime\n(Wunderlich et al. , 2004, 2005). The ISHE was detected\nsoon thereafter (Saitoh et al. , 2006; Valenzuela and Tin-\nkham, 2006; Zhao et al. , 2006). Early measurements were\nmostly qualitative. However, more accurate quantitative\nmeasurements of spin Hall angles have been established\nin later experiments through the aid of FM detectors in\nstatic or dynamic magnetization regimes, and a much\n\frmer situation has arisen in the \feld.\nAdding to this \rurry of activity and increased under-\nstanding, recent experiments in magnetic tunnel junc-\ntions have aimed to use spin-currents injected from an\nadjacent spin Hall NM for spin-transfer torque (STT)\nswitching of a FM (Liu et al. , 2012; Miron et al. , 2011a).\nIn addition to this SHE induced torque there is also\na spin-orbit torque (SOT) (Bernevig and Vafek, 2005;\nChernyshov et al. , 2009), which is generated via the in-\nverse spin galvanic e\u000bect (ISGE) (Belkov and Ganichev,\n2008). In the ISGE, a charge current can generate a\nnon-equilibrium homogeneous spin-polarization via rela-\ntivistic spin-orbit coupling and it is often a companion\ne\u000bect to the spin-current generating SHE (Kato et al. ,\n2004b,c; Wunderlich et al. , 2004, 2005). These results\nunderscore the relevance of the SHE for applications.\nAs already mentioned, the SHE borrows directly from\nthe physics and mechanisms of the AHE and correspond-\ningly much of their descriptions are parallel. The family\nof these three key spin-dependent Hall e\u000bects is illus-\ntrated in Fig. 1. The important caveat is that, unlike\nthe AHE which correlates charge degrees of freedom via\nrelativistic spin-orbit interaction, the SHE and ISHE cor-\nrelate the charge degree of freedom, a conserved quantity,\nand the spin degree of freedom, a non-conserved quantity\nsubject to decay and dephasing.\nmagnetic Mz≠0\noptical detectionAHE\nSHE\nnon-magnetic Mz=0 non-magnetic Mz=0\nI=0SHE-1\nelectrical detection\nelectrical detection\nFIG. 1 An illustration of the connected family of the spin-\ndependent Hall e\u000bects. In the AHE, a charge current gen-\nerates a polarized transverse charge current. In the SHE an\nunpolarized charge current generates a transverse pure spin-\ncurrent. In the ISHE a pure spin-current generates a trans-\nverse charge current.3\nThe aim of this review is to survey the rapid develop-\nments of the SHE \feld, to give an overview of its current\nexperimental understanding, the basic theoretical tools\nthat are being applied to describe it and their current\nlevel of success and limitations, the connection to impor-\ntant related phenomena, as well as the potential of the\nSHE for applications, particularly in the area of magne-\ntization dynamics.\nGiven the enormous work that has been done in just a\ndecade, we can only highlight what we have deemed the\nreports that have contributed signi\fcantly to the \feld.\nAs with any review we are shackled by our own views and\nthe reader interested in this \feld should complement this\nreading with other recent reviews of the subject, such\nas (Culcer, 2009; Hankiewicz and Vignale, 2009; Ho\u000b-\nmann, 2013; Jungwirth et al. , 2012; Maekawa and Taka-\nhashi, 2012; Raimondi et al. , 2012; Sinova and MacDon-\nald, 2008; Valenzuela and Kimura, 2012; Vignale, 2010).\nII. OVERVIEW\nIn this section we provide an overview that starts from\nthe original seeds of the SHE \feld and connects after-\nwards to the broader context of the phenomenon within\nspintronics. The overview is organized as follows: First\nwe look back to how the Mott scattering of electron\nbeams in vacuum and the skew scattering of electrons in\nFMs germinated into the prediction of the extrinsic SHE\nin NMs. Second we discuss that in a solid-state system\nthere is in addition an intrinsic spin-de\rection, arising\nfrom the internal spin-orbit coupling forces in a perfect\ncrystal. This key distinction from electrons in a vacuum\nmakes the spin-dependent Hall physics in condensed mat-\nter systems much richer. We also note here the connec-\ntion of this intrinsic mechanism to the quantum Hall ef-\nfects. Third, we summarize studies of spin injection and\ndetection in hybrid FM/NM structures, which were par-\nticularly impactful on the research of SHE. Here we high-\nlight DC transport as well as AC ferromagnetic resonance\n(FMR) experiments. Finally, we connect the physics of\nthe SHE, which considers pure spin-currents and non-\nuniform spin accumulations, to the physics of the spin\ngalvanic e\u000bects. The latter e\u000bects represents a seemingly\ndistinct family of relativistic phenomena relating to the\ngeneration or detection of uniform non-equilibrium spin-\npolarizations. However, as we point out, the spin Hall\nand spin galvanic e\u000bects can have common features in\ntheir microscopic physical origins and both can generate\ncurrent-induced torques in magnets. These two relativis-\ntic e\u000bects are now at the forefront of current-induced\nmagnetization dynamics research aimed at future spin-\ntronic technologies.A. Spin Hall, anomalous Hall, and Mott polarimetry\nIn their original work, Dyakonov and Perel, 1971b re-\nferred to the phenomena of Mott scattering (Mott, 1929)\nand of the AHE (Hall, 1881) to theoretically predict the\nextrinsic SHE. In particular, they pointed out the fol-\nlowing: (i) Spin-dependent asymmetric de\rection is ob-\nserved in electron beams in vacuum due to Mott scatter-\ning (Gay and Dunning, 1992; Mott, 1929, 1932; Shull\net al. , 1943). (ii) Mott's skew scattering is regarded\namong the origins of the AHE of electron carriers in\nFMs (Berger, 1979; Karplus and Luttinger, 1954; Na-\ngaosa et al. , 2010; Smit, 1955, 1958). The two points\nimply that under an applied electrical current, asym-\nmetric spin-dependent de\rection should occur in NMs.\nUnlike in FMs, NMs in equilibrium have the same num-\nber of spin-up and spin-down electrons and no transverse\ncharge imbalance will occur. Instead, the SHE generates\nan edge spin-accumulation that has opposite polarization\nat opposite edges.\nLet us now explore the Mott scattering seed of the\nSHE in more detail. Mott proposed its scattering ex-\nperiment (Mott, 1929, 1932) to provide a direct evidence\nthat spin, inferred four years earlier from atomic spectra\n(Uhlenbeck and Goudsmit, 1925), is an intrinsic prop-\nerty of a free electron. Mott anchored his proposal in the\nthen recently derived Dirac equation (Dirac, 1928). The\nensuing quest for the experimental veri\fcation of Mott\nscattering (Shull et al. , 1943) was among the founding\npillars not only for verifying the electron spin but over\nthe entire relativistic quantum mechanics concept. Since\nMott scattering of electron beams from heavy nuclei in a\nvacuum chamber can be regarded as the SHE in a non-\nsolid-state environment, the seeds of the SHE date back\nto the very foundations of the electron spin and relativis-\ntic quantum mechanics.\nFigure 2(a) shows Mott, 1929 double-scattering exper-\niment proposal. First, an unpolarized beam of electrons\nis scattered from heavy nuclei in a target. Because of the\nrelativistic spin-orbit coupling, large angle ( \u001890\u000e) scat-\ntering from the \frst target produces a polarized beam\nwith the spin-polarization transverse to the scattering\nplane. Scattering of these polarized electrons from the\nsecond target results, again due to the spin-orbit cou-\npling, in a left-right scattering asymmetry that is propor-\ntional to the polarization induced by the \frst scattering.\nIn a complete analogy to the Mott double-scattering ef-\nfect, but instead of vacuum now considering a solid state\nsystem, Hankiewicz et al. , 2004 proposed an H-bar mi-\ncrodevice schematically shown in Fig. 2(b). An unpolar-\nized electrical current driven through the \frst leg of the\ndevice generates a transverse spin-current due to the rel-\nativistic SHE. The spin-current injected into the second\nleg generates, via the ISHE, an electrical current, or in\nan open circuit geometry a voltage across the second leg.\nThis H-bar SHE/ISHE experiment has been realized in\na NM semiconductor by Br une et al. , 2010.\nIn Fig. 2(c) we show an earlier variant of the double-4\n(a)\n(b)\n(c)\nSHE\nISHEFSOFSO\nFSOFSOFSOFSO\nFSOFSO\nFIG. 2 (a) Schematics of Mott, 1929 original double-\nscattering proposal, (b) SHE/ISHE analogue of Mott double-\nscattering in Hankiewicz et al. , 2004 H-bar device, (c) SHE\n(left) and ISHE (right) wired as proposed by Hirsch, 1999. In-\nstead of directly injecting a spin-current generated in the SHE\npart of the experiment, as suggested by Mott and by Han-\nkiewicz et al. , Hirsch considered that the pure spin-current is\ngenerated from the opposite spin-accumulations at the edges\nof the SHE part of the \"double-scattering\" device. FSOrep-\nresents an e\u000bective spin-orbit force that de\rects the spins in\nthe SHE/ISHE. Panel (a) from Ref. Gay and Dunning, 1992.\nscattering experiment proposed by Hirsch, 1999 for ob-\nserving the SHE/ISHE in a solid state device. Instead\nof considering the spin-current produced directly by the\ncharge current via the SHE, Hirsch focused on the edges\nof the SHE sample. Here the transverse spin-current ac-\ncumulates, forming a non-equilibrium spin polarization\nof opposite sign at the two opposite edges. In NMs, the\nnon-equilibrium spin polarization corresponds to a split-\nting of the spin-up and spin-down chemical potentials.\nWhen connecting the two edges, the gradient of the spin-\ndependent chemical potentials will generate a circulating\nspin-current which is then detected by the ISHE spin-\ncurrent meter inserted into the closed spin-current cir-\ncuit. The idea for the experiment was borrowed from the\nordinary Hall e\u000bect (HE) in which opposite charge accu-\nmulates at opposite edges due to the Lorentz force, and\nthe resulting electro-chemical potential gradient gener-\nates a circulating charge current when the two edges areconnected in the closed circuit geometry.\nRealizing Hirsch, 1999 SHE/ISHE device remains a\nchallenge. Similarly to the Hankiewicz et al. , 2004 de-\nsign directly copying the Mott double-scattering experi-\nment, the wires connecting the SHE and ISHE parts of\nHirsch's device have to be shorter than the characteris-\ntic spin-conserving length-scale. The spin-orbit coupling\nrequired for the SHE/ISHE in the \frst place, however,\ntends to make the spin life-time short. The additional\ncomplication is that the spin-orbit coupling also limits,\nagain via the \fnite spin life-time, the width of the sam-\nple edge with non-zero spin accumulation from which the\nspin-current is extracted in Hirsch's device proposal.\nWhile di\u000ecult to realize experimentally, Hirsch's con-\ncept is stimulating for comprehending the general key\ndistinctions between charge and spin-current. Electron\ncharge is a conserved quantity but its spin direction is\nnot conserved. In the charge HE, the di\u000berence be-\ntween electro-chemical potentials at the edges determines\nthe uniform charge current which in steady-state \rows\nthrough the closed circuit. In the SHE, on the other\nhand, the spin-current in the connecting wire of Hirsch's\ndevice is not uniform and is not determined by the dif-\nference between the spin-dependent chemical potentials\nat the left and right edges. It is determined by the lo-\ncal gradient of the spin-dependent chemical potentials\nwhich vanishes, i.e. also the spin-current vanishes, on\nthe length-scale given by the spin life-time. As long as\nthe connecting wire is longer than the characteristic spin-\nconserving length-scale, there is no di\u000berence between a\nclosed and an open spin-current circuit.\nHirsch's concept also points to the general applicability\nof the ISHE as an electrical spin detector. Even in elec-\ntrically open circuits, the non-conserving, non-uniform\nspin-current can still \row. It is then readily separated\nfrom the charge current and can be detected by the\nISHE. The Mott polarimetry of electron beams in vac-\nuum chambers and AHE polarimetry of charge currents\nin itinerant magnets can, therefore, be complemented by\nthe ISHE polarimetry of pure spin-currents.\nA spin-current in a NM of any origin (not only of the\nSHE origin) can be detected by the ISHE. Indeed, ISHE\ndetectors of pure spin-currents became a standard mea-\nsurement tool. They led to, e.g., the discovery of the spin\nSeebeck e\u000bect (Jaworski et al. , 2010; Uchida et al. , 2008,\n2010) and helped establishing the emerging \feld of spin\ncaloritronics (Bauer et al. , 2012).\nGiven the inherent challenges in realizing Hirsch's de-\nvice it is not surprising that experimentalists initially\navoided attempts to perform the SHE/ISHE \"double-\nscattering\" experiments and that the \frst observations\nof the SHE (Kato et al. , 2004b; Wunderlich et al. , 2004,\n2005) and ISHE (Saitoh et al. , 2006; Valenzuela and Tin-\nkham, 2006; Zhao et al. , 2006) were made separately.\nWhen the Hankiewicz et al. , 2004 H-bar microdevice was\neventually realized in experiment by Br une et al. , 2010,\nboth the SHE and ISHE had been already independently\nestablished.5\nB. Intrinsic spin Hall and quantum Hall e\u000bects\nRemarkably, the H-bar experiment (Br une et al. , 2010)\ndiscussed in the previous subsection was performed in a\nballistic transport regime where the picture of Mott scat-\ntering, single or double, did not apply. A fundamental\nphysics principle makes the SHE in solid-state systems\nricher than in the Mott electron beams scattered from\nspin-orbit coupled targets in vacuum chambers. For elec-\ntrons moving in a crystal, a transverse spin-dependent\nvelocity can be generated by the relativistic spin-orbit\n\feld of a perfect crystal even in the absence of scatter-\ning. The roots of this intrinsic SHE are clearly distinct\nfrom the Mott (skew) scattering AHE and from the Mott\nscattering of free electron beams.\nThe reactive term responsible for the intrinsic SHE is\nakin to the ordinary HE in which the transverse de\rec-\ntion of electrons is a reaction to the Lorentz force of the\napplied magnetic \feld acting on the moving carriers (see\nFig. 3(a)). In strong magnetic \felds, the quantum Hall\ne\u000bect (QHE) becomes a precise, disorder-independent\nmeasure of the quantum conductance e2=h, and the inte-\nger multiples of e2=hobserved in the QHE correspond to\nthe number of occupied dissipationless chiral edge states\nin the conductor (see Fig. 3(b)).\nB (M )HE(int.-AHE) int.-SHE\nBQHE QSHE(a) (c)\n(b) (d)I I\nFIG. 3 Schematics of the HE and the AHE (a), the QHE (b),\nSHE (c), and the QSHE (d). In the HE and QHE, the carrier\nde\rection is a reaction to the Lorentz force. In the cases of\nthe intrinsic AHE, SHE and QSHE, the carriers experience\nan internal spin-orbit force.\nBesides the externally applied Lorentz force, electrons\nmoving in a crystal can experience an internal spin-orbit\nforce. The e\u000bect was \frst recognized in FMs where it\ngenerates the intrinsic AHE (see Fig. 3(a)) (Jungwirth\net al. , 2002; Karplus and Luttinger, 1954; Onoda and\nNagaosa, 2002). Murakami et al. , 2003 and Sinova et al. ,\n2004 predicted that the same spin-orbit force derived di-\nrectly from the relativistic band structure of a NM can\ninduce the SHE without involving Mott scattering (seeFig. 3(c)). The \frst experimental observations con\frmed\nthat the SHE can indeed have the two distinct origins.\nWhile Wunderlich et al. , 2004, 2005 ascribed the circu-\nlarly polarized luminescence signal from the edge of the\np-GaAs sample to the intrinsic SHE, Kato et al. , 2004b\ndetected an edge Kerr rotation signal in n-GaAs due to\nthe extrinsic, skew-scattering SHE.\nFollowing the discovery of the phenomenon, the SHE\nexperiments in semiconductors using optical spin detec-\ntion have explored the basic phenomenologies of the ex-\ntrinsic and intrinsic SHEs (Chang et al. , 2007; Kato et al. ,\n2004b; Matsuzaka et al. , 2009; Nomura et al. , 2005; Sih\net al. , 2006, 2005; Stern et al. , 2006, 2007, 2008; Wunder-\nlichet al. , 2004, 2005). They also experimentally demon-\nstrated the potential of the SHE as a spin-current source\n(Sih et al. , 2006). A two-color optical excitation tech-\nnique with perpendicular linear polarizations of the inci-\ndent laser beams was used to detect the ISHE in a semi-\nconductor (Zhao et al. , 2006). The spin-current produced\nby the laser excitation is transferred due to the ISHE into\na transverse electrical current, resulting in a spatially de-\npendent charge accumulation which was detected by the\noptical transmission signal of a probe laser beam. These\nall-optical measurements in an intrinsic semiconductor\nwere eventually performed on timescales shorter than the\nscattering time and have provided a direct demonstration\nof the intrinsic SHE signal (Werake et al. , 2011).\nThe intrinsic SHE proposal triggered an intense the-\noretical debate which is summarized in several review\narticles (Culcer, 2009; Engel et al. , 2006; Hankiewicz and\nVignale, 2009; Murakami, 2005; Raimondi et al. , 2012;\nSchliemann, 2006; Sinova and MacDonald, 2008; Sinova\net al. , 2006; Vignale, 2010). Combined with the estab-\nlished physics of the dissipationless QHE, it led to the\nprediction and subsequent experimental veri\fcation of\nthe quantum spin Hall e\u000bect (QSHE) (Bernevig et al. ,\n2006; Hasan and Kane, 2010; Kane and Mele, 2005;\nK onig et al. , 2007; Murakami et al. , 2004). In the time-\nreversal symmetric QSHE, the chiral edge states of the\nQHE are replaced by pairs of helical spin-edge states (see\nFig. 3(d)). This leads to a 2 e2=hquantization of the ob-\nserved transport signal (K onig et al. , 2007) and resistance\nvalues in nonlocal experiments that can be expressed\nas speci\fc integer fractions of the inverse conductance\nquanta (B uttiker, 2009; Roth et al. , 2009). The QSHE\ninitiated the new research \feld of topological insulators\n(Hasan and Kane, 2010; Moore, 2010).\nC. Spin Hall e\u000bect and non-magnetic/ferromagnetic hybrid\nstructures\nAmong the early SHE device proposals, Zhang, 2000\nsuggested to electrically detect the edge spin accumula-\ntion produced by the SHE using an attached FM probe\n(Johnson and Silsbee, 1985; Silsbee, 1980). In a broader\ncontext, the idea of connecting the SHE with the more\nmature \feld which utilized FMs for injection and detec-6\ntion of spins in NMs fueled numerous studies of funda-\nmental importance for the SHE \feld. Electrical spin in-\njection from a FM contact and electrical observation of\nthe ISHE on a Hall cross patterned in the NM was demon-\nstrated by Valenzuela and Tinkham, 2006.\nMetal spin Hall devices provided the demonstration of\nthe electrical measurement of the SHE by an attached\nFM contact (Kimura et al. , 2007), as proposed originally\nby Zhang, 2000. They showed that the same NM elec-\ntrode can generate the SHE or the ISHE, i.e., can be used\nas an electrical spin injector or detector (Kimura et al. ,\n2007; Mihajlovic et al. , 2009; Seki et al. , 2008; Valenzuela\nand Tinkham, 2006; Vila et al. , 2007).\nCompared to metals, semiconductor spin transport de-\nvices with FM metal electrodes can su\u000ber from the prob-\nlem of the resistance mismatch which hinders e\u000ecient\nspin transport across the interface (Schmidt et al. , 2000).\nThe introduction of a highly resistive tunnel barrier be-\ntween the FM metal electrode and the semiconductor\nchannel solves this problem (Lou et al. , 2007; Rasbha,\n2000) and FM tunnel contacts were successfully used to\ndetect the SHE-induced spin accumulation in a semicon-\nductor (Garlid et al. , 2010). Similarly, an electrical spin-\ninjection from a FM/semiconductor tunnel contact was\nused to demonstrate, side by side, the electrical spin de-\ntection by the ISHE and by the FM detection electrode\n(Olejn\u0013 \u0010k et al. , 2012).\nUsing FMs contributed signi\fcantly to the basic un-\nderstanding of the SHE. Apart from the transport mea-\nsurements, NM/FM hybrid structures also allow to com-\nbine the SHE physics with the \feld of magnetization dy-\nnamics. The ISHE and SHE can be investigated using\nspin-pumping (SP) and other related dynamic methods\nin structures comprising FMs and NMs, as illustrated\nin Fig. 4 (Ando et al. , 2009, 2008; Bai et al. , 2013a;\nCzeschka et al. , 2011; Liu et al. , 2011, 2012; Miron et al. ,\n2011a; Mosendz et al. , 2010a,b; Saitoh and Ando, 2012;\nSaitoh et al. , 2006; Wei et al. , 2014). In return, the SHE\nwas found to provide e\u000ecient means for injecting spin-\ncurrents into the FM, generating the STT (Ralph and\nStiles, 2008), and by this electrically controlling magne-\ntization in FMs with potential applications in spintronic\ninformation technologies (Emori et al. , 2013; Liu et al. ,\n2012; Miron et al. , 2011a,b; Ryu et al. , 2013). More-\nover, the ISHE detection of pure spin-currents did not\nremain limited to NMs but is now used also in FMs\n(Azevedo et al. , 2014; Miao et al. , 2013) and antiferro-\nmagnets (Freimuth et al. , 2010; Mendes et al. , 2014).\nIn general, when SHE induced torques in the adja-\ncent FM are considered in the description of the dy-\nnamic magnetization (the Landau-Lifshitz-Gilbert equa-\ntion), two types of torques can occur. An (anti)damping-\nlike torque which has the same functional shape as the\nGilbert damping term (and thus can manifest itself in an\nincreased or decreased Gilbert damping) and a \feld-like\nterm which alters the magnetic energy landscape and can\nbe observed as a shift of the resonance line in a FMR ex-\nperiment. FMR allows the determination of the total in-ternal magnetic \feld in a sample as well as investigation\nof dissipation. Thus, in principle FMR like techniques\nenable determination of \feld-like and (anti)damping-like\ncontributions of SHE induced torques.\nLETTERS\nPUBLISHED ONLINE: 26 JUNE 2011 | DOI: 10.1038/NMAT3052\nElectrically tunable spin injector free from the\nimpedance mismatch problem\nK. Ando1*, S.\nTakahashi1,2, J. Ieda2,3, H. Kurebayashi4, T. Trypiniotis4, C. H. W. Barnes4, S. Maekawa2,3\nand E. Saitoh1,2,3\nInjection of spin currents into solids is crucial for exploring spin\nphysics and spintronics1,2. There has been significant progress\nin recent years in spin injection into high-resistivity materials,\nfor example, semiconductors and organic materials, which\nuses tunnel barriers to circumvent the impedance mismatch\nproblem3–14; the impedance mismatch between ferromagnetic\nmetals and high-resistivity materials drastically limits the\nspin-injection efficiency15. However, because of this problem,\nthere is no route for spin injection into these materials\nthrough low-resistivity interfaces, that is, Ohmic contacts,\neven though this promises an easy and versatile pathway\nfor spin injection without the need for growing high-quality\ntunnel barriers. Here we show experimental evidence that spin\npumping enables spin injection free from this condition; room-\ntemperature spin injection into GaAs from Ni 81Fe19through\nan Ohmic contact is demonstrated through dynamical spin\nexchange. Furthermore, we demonstrate that this exchange can\nbe controlled electrically by applying a bias voltage across a\nNi81Fe19=GaAs interface, enabling electric tuning of the spin-\npumping efficiency.\nElectron transport across a ferromagnetic metal/non-magnetic\nmaterial gives rise to non-equilibrium spin currents in the non-\nmagnetic layer. However, the spin polarization of the current across\nthe interface is strongly reduced, especially when the electrical\nresistance of these two layers is considerably different, for example\na ferromagnetic metal/semiconductor (FM/SC) contact. This is\nknown as the impedance-mismatch problem3,15.\nThe above problem arises from the fact that spins are injected\nby carrier transport across a FM/SC interface (see Fig. 1a). It\nseems natural to consider that this problem disappears when\nspins are injected directly into the SC layer without using\ncharge transport across the interface (see Fig. 1b); the driving\nforce for the spin flow (not the charge-carrier flow) is expected\nto offer a method for versatile spin injection free from the\nimpedance mismatch problem. In this work, we experimentally\ndemonstrate that spin pumping, generation of pure spin currents\nfrom magnetization precession16\u001518, provides a powerful method\nfor direct spin injection. The spin-angular momentum of the\nprecessing magnetization in the FM layer is transferred to the\ncarriers in the SC layer through dynamical exchange interaction at\nthe FM/SC interface, inducing a pure spin voltage, the potential\nacting on spins not on carriers, in the SC layer. This enables spin\ninjection into both p- and n-doped GaAs from Ni 81Fe19through\nboth Ohmic and Schottky contacts in a Ni 81Fe19=GaAs interface\neven at room temperature. The Ohmic contact case is the one\nshowing the higher spin-current injection in our experiments, in\n1Institute for Materials Research, T ohoku University, Sendai 980-8577, Japan,2CREST, Japan Science and T echnology Agency, Sanbancho, T okyo\n102-0075, Japan,3The Advanced Science Research Center, Japan Atomic Energy Agency, T okai 319-1195, Japan,4Cavendish Laboratory, University of\nCambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, UK. *e-mail: ando@imr.tohoku.ac.jp.\nab\nM\nSpin current Spin currentNon-magneticmaterial\nFerromagneticmetalNon-magneticmaterial\nFerromagneticmetalM(t)Figure 1jSpin injection through carrier transport and dynamical exchange\ninteraction. a, A schematic of the conventional electrical spin injection in a\nnon-magnetic material/ferromagnetic metal junction. Mdenotes the static\nmagnetization. The spin current is driven by electron transport across the\ninterface, which is drastically limited by the impedance mismatch.\nb, A schematic of spin injection through dynamical exchange interaction in\na non-magnetic material/ferromagnetic metal junction. M(t) shows the\nprecessing magnetization. The dynamical exchange interaction at the\ninterface transfers the spin-angular momentum from the magnetization in\nthe ferromagnetic layer to the electrons in the non-magnetic layer, inducing\na pure spin voltage in the non-magnetic layer.\ncontrast to the expectation in the case of spin injection by charge\ntransport where the impedance mismatch problem is present.\nFurthermore, the spin-pumping efficiency is demonstrated to be\ncontrolled electrically through electric modulation of the dynamical\nexchange interaction at the Ni 81Fe19=GaAs interface, providing a\nclear picture of the dynamical spin injection.\nThe room-temperature spin injection through Ohmic and\nSchottky contacts is observed using the inverse spin-Hall effect\n(ISHE; refs 19\u001523). The ISHE converts a spin current into an\nelectric voltage, enabling electric detection of the spin current.\nFigure 2a shows a schematic illustration of the samples used\nin this study. The samples are a Ni 81Fe19/Zn-doped GaAs (a\nNi81Fe19=p-GaAs film) with a doping concentration of NDD1:4\u0002\n1019cm\u00003and a Ni 81Fe19/Si-doped GaAs (a Ni 81Fe19=n-GaAs film)\nwith NDD1:2\u00021018cm\u00003(for details, see Methods). Here, note that\nthe current\u0015voltage characteristics shown in Fig. 2b,c indicate the\nformation of Ohmic and Schottky contacts in the Ni 81Fe19=p-GaAs\nand Ni 81Fe19=n-GaAs interfaces, respectively.\nWe measured the ferromagnetic resonance (FMR) signal and\nthe electric-potential difference Vbetween the electrodes attached\nto the GaAs layer to detect spin injection; as shown in Fig. 2a, in\nNATURE MATERIALS jVOL 10jSEPTEMBER 2011jwww.nature.com/naturematerials 655\n© 2011 Macmillan Publishers Limited.  All rights reserved.   \nFIG. 4 Illustration of the SP spin-current generation by mag-\nnetization dynamics from a FM into a NM. From Ref. Ando\net al. , 2011.\nIn ISHE experiments using FMR techniques a detailed\nanalysis of \feld-symmetric and \feld-antisymmetric con-\ntributions of the detected DC output voltage at FMR\nallows in principle a quantitative determination of the\nstrength and symmetry of the SHE induced torques, as\nwell as the spin Hall angle.\nNote that the torque can induce not only the small-\nangle FMR precession but the lateral current along a\nNM/FM interface can drive domain walls at high veloci-\nties (Emori et al. , 2013; Miron et al. , 2011b; Ryu et al. ,\n2013) or switch the magnetization in the FM (Liu et al. ,\n2012; Miron et al. , 2011a). This may have practical im-\nplications for designing domain-wall based memories or\nfor three-terminal magnetic tunnel junction bits with the\nlateral writing current decoupled from the perpendicular\nread-out current.\nD. Spin Hall e\u000bect, spin galvanics, and spin torques\nFrom the early experiments with the relativistic\ntorques it was realized that the SHE is not the only pos-\nsible mechanism responsible for torques induced by the\nlateral current in the NM/FM bilayers (Manchon and\nZhang, 2008). The interface breaks the structural inver-\nsion symmetry which implies that the SHE-STT can be\naccompanied by another microscopic mechanism. Its ori-\ngin is in the so called spin galvanic phenomena that were\nexplored earlier in inversion-asymmetric NMs (Ivchenko\nand Ganichev, 2008). In the picture discussed in the pre-\nvious section, the spin-current generated in the NM via\nthe relativistic SHE is absorbed in the FM and induces\nthe STT. In the competing scenario, a non-equilibrium\nspin-density of carriers is generated in inversion asym-\nmetric systems via the relativistic ISGE (Belkov and7\nby the dashed arrows in Fig. 1, which result in a current flow. This\nmeans that a current is driven by a homogeneous spin polarization.\nBelow we describe the observation of such a current.\nPhenomenologically, an electric current can be linked by general\nsymmetry arguments to the electron’s averaged spin Sby\nja\ngX\nQagSg 1\nwhere jis an electric current density, Qis a second-rank pseudo-\ntensor, and a,g1, 2 indicate coordinates. Non-vanishing tensor\ncomponents Qagcan only exist in non-centrosymmetric systems\nbelonging to one of the gyrotropic classes6. In zinc-blende-based\nheterojunctions with a 2DEG, non-zero components of Qagexist in\ncontrast to the corresponding bulk crystals7. In our (001)-grown\nheterojunctions with the C2nsymmetry only two linearly indepen-\ndent components, Qxyand Qyx, are different from zero ( xk[11¯0]\nand yk[110]) where xand yare cartesian coordinates. Hence, to\nobserve a spin-polarization-driven current a spin component lying\nin the plane of the heterojunctions is required (for example, Syin\nFig. 2).\nT o achieve an in-plane spin orientation in experiment one could\neither use spin selective contacts8(see Fig. 2a) or optical orientation9\nby using circularly polarized light. Although significant progressconcerning electrical spin injecting has been made recently\n10–12,\nreliable spin-injection into lateral low-dimensional electron sys-\ntems —at room temperature —is still a challenge. Furthermore,\nelectrical spin injection causes, apart from the driving current, a\nlaterally inhomogeneous spin polarization and hence additional\ndriving forces for current flow which would hamper the unam-\nbiguous demonstration of the effect described here. Instead, we use\noptical spin orientation, which ensures a homogeneous non-equili-\nbrium spin polarization and directly proves the spin-galvanic effect.\nIn this case of exciting electrons from the valence band to the\nconduction band the conduction band gets selectively spin-popu-\nlated owing to selection rules which allow transitions by circularly\npolarized light only if the spin of the electron is changed by ^1. In\naddition to this method we have also used circularly polarized\nterahertz radiation causing intraband instead of interband exci-\ntation. One interesting aspect of employing terahertz radiation isthat only one type of carriers, electrons or holes, is involved. In this\nrespect, the effect of intraband spin orientation13is indeed very close\nto electrical spin injection.\nIt has been shown before that irradiation of quantum wells with\ncircularly polarized light can result in a photocurrent caused by\nnon-uniformly distributing photoexcited carriers in k-space\naccording to optical selection rules and energy and momentum\nconservation. This is the circular photogalvanic effect observed\nrecently in a 2DEG14. In order to achieve a uniform distribution in\nspin sub-bands and to exclude this circular photogalvanic effect we\nuse the geometry depicted in Fig. 2b, where the photogalvanic\ncurrent is identical to zero for normal incidence of the light14. In this\ngeometry, optical excitation yields a steady-state spin orientation\nS0zin the zdirection with the generation rate S˙zproportional to the\nintensity of the radiation. T o obtain an in-plane component of the\nspins, necessary for the effect we describe here, a magnetic field, B,\nwas applied (Fig. 2b). The field perpendicular to both the light-\npropagation direction eˆzand the optically oriented spins rotates the\nspins into the plane of the 2DEG owing to Larmor precession. Withthe magnetic field oriented along the xaxis we obtain a non-\nequilibrium spin polarization S\nywhich reads, after time averaging7:\nSy2qLts’\n1qLts2S0z 2\nwhere tstskts’pandtsk,ts’are the longitudinal and transverse\nelectron-spin-relaxation times, qLgmBBx=\u0016his the Larmor fre-\nquency, gis the in-plane effective electron g-factor, mBis the Bohr\nmagneton, and S0ztsk_Szis the steady-state electron spin polar-\nization in the absence of the magnetic field. Using the Larmor\nprecession we prepared the situation sketched in Fig. 1 where the\nspin polarization Sylies in the plane. The denominator in equation\n(2) yielding the decay of SyforqLexceeding the inverse spin-\nrelaxation time is well known from the Hanle effect15.\nThe experiments were carried out at room temperature\n(T293 K) and at liquid helium temperature on n-GaAs/AlGaAs\nsingle quantum wells of 15-nm width and on GaAs single hetero-\njunctions. The (001)-oriented samples grown by molecular beam\nepitaxy contain 2DEG systems with electron densities ns.2£\n1011cm22and mobilities mabove 106cm2V21s21atT4.2 K.\nTwo pairs of contacts were centred on opposite sample edges\nalong the directions xk[11¯0] and yk[110] (see inset in Fig. 3).\nComplementary measurements were also carried out on p-GaAs\nmultiple quantum-well structures containing 20 wells of 15-nm\nwidth with hole densities in each well ps2£1011cm22andm\n5£105cm2V21s21:\nAt room temperature a magnetic field of up to 1 T was generated\nby an electromagnet. For the low-temperature measurements the\nsamples were placed in a cryostat with a split-coil superconducting\nmagnet yielding a field Bof up to 3 T. For optical interband\nexcitation a continuous-wave Ti:sapphire laser was used at a\nwavelength of l0.777mm. In order to extract the spin-galvanic\ncurrent the linearly polarized laser beam was transmitted through a\nFigure 1 Microscopic origin of the spin-galvanic current in the presence of k-linear terms\nin the electron hamiltonian. The jykxterm in the hamiltonian splits the conduction band\ninto two parabolas with the spin ^1=2in the ydirection. If one spin sub-band is\npreferentially occupied, for example, by spin injection (the j21=2ly-states shown in the\nfigure) asymmetric spin-flip scattering results in a current in the xdirection. The rate of\nspin-flip scattering depends on the value of the initial and final k-vectors. There are four\ndistinct spin-flip scattering events possible, indicated by the arrows. The transitions\nsketched by dashed arrows yield an asymmetric occupation of both sub-bands and hence\na current flow. If, instead of the spin-down sub-band, the spin-up sub-band is\npreferentially occupied the current direction is reversed.Figure 2 Two ways of generating an in-plane spin-polarization. a, The spin injection from\nferromagnetic contacts of magnetization Minto the two-dimensional electron gas (2DEG).\nb, The optical orientation in combination with an in-plane magnetic field Bx. Spins, initially\naligned along the zdirection are rotated into the ydirection by Bx.letters to nature\nNATURE | VOL 417 | 9 MAY 2002 | www.nature.com 154 © 2002 Macmillan Magazines Ltd\nFIG. 5 Microscopic origin of the spin galvanic current in the\npresence of k-linear terms in the electron Hamiltonian. The\n\u001bykxterm in the Hamiltonian splits the conduction band into\ntwo parabolas with the spin \u00061=2 in they-direction. If one\nspin sub-band is preferentially occupied, for example, by spin\ninjection (thej\u00001=2iy-states shown in the \fgure) asymmet-\nric spin-\rip scattering results in a current in the x-direction.\nThe rate of spin-\rip scattering depends on the value of the\ninitial and \fnal k-vectors. There are four distinct spin-\rip\nscattering events possible, indicated by the arrows. The tran-\nsitions sketched by dashed arrows yield an asymmetric oc-\ncupation of both sub-bands and hence a current \row. If,\ninstead of the spin-down sub-band, the spin-up sub-band is\npreferentially occupied the current direction is reversed. From\nRef. Ganichev et al. , 2002.\nGanichev, 2008; Ganichev et al. , 2004; Ivchenko and\nGanichev, 2008; Kato et al. , 2004c; Silov et al. , 2004;\nWunderlich et al. , 2004, 2005). A SOT is then directly\ninduced if the carrier spins are exchange coupled to mag-\nnetic moments (Bernevig and Vafek, 2005; Chernyshov\net al. , 2009; Manchon and Zhang, 2008; Miron et al. ,\n2010).\nFrom the early observations in NM semiconductors,\nSHE and ISGE are known as companion phenomena,\nboth allowing for electrical alignment of spins in the same\nstructure (Kato et al. , 2004b,c; Wunderlich et al. , 2004,\n2005). Hand in hand, SHE and ISGE evolved from subtle\nacademic phenomena to e\u000ecient means for electrically re-\norienting magnets. Understanding the relation between\nthe spin Hall and spin galvanic phenomena is, therefore,\nimportant not only from the basic physics perspective\nbut has also practical implications for spintronic devices.\nThe term spin galvanic e\u000bect (SGE) is derived from\nthe analogy to the galvanic (voltaic) cell. Instead of a\nchemical reaction, however, it is the spin polarization\nthat generates an electrical current (voltage) in the SGE.\nInversely, an electrical current generates the spin polar-\nization in the ISGE.\nFollowing theoretical predictions of the phenom-\nena (Aronov and Lyanda-Geller, 1989; Edelstein, 1990;\nIvchenko et al. , 1989; Ivchenko and Pikus, 1978; Mal-shukov and Chao, 2002), it was the SGE that was initially\nobserved in an asymmetrically con\fned 2D electron gas\nin a GaAs quantum well (Ganichev et al. , 2002). The key\nsignature of the SGE is the electrical current induced by\na non-equilibrium, but uniform polarization of electron\nspins. The microscopic origin of the e\u000bect is illustrated\nin Fig. 5. In the non-equilibrium steady-state, the spin-\nup and spin-down sub-bands have di\u000berent populations,\ninduced in the Ganichev et al. , 2002 experiment by a\ncircularly polarized light excitation. Simultaneously, the\ntwo sub-bands for spin-up and spin-down electrons are\nshifted in momentum space due to the inversion asym-\nmetry of the semiconductor structure which leads to an\ninherent asymmetry in the spin-\rip scattering events be-\ntween the two sub-bands. This results in the \row of the\nelectrical current.\nky!kx!\nJ!\nkx!ky!\nFIG. 6 Left panel: Rashba spin-texture in equilibrium with\nzero net spin-density. Right panel: Non-equilibrium redistri-\nbution of eigenstates in an applied electric \feld resulting in\na non-zero spin-density due to broken inversion symmetry of\nthe spin-texture.\nA microscopic picture of the ISGE is illustrated in\nFig. 6. The uniform non-equilibrium spin-density oc-\ncurs as a consequence of an electric-\feld and scattering\ninduced redistribution of carriers on the Fermi surface\nwhose texture of spin expectation values has a broken\ninversion symmetry. For the Rashba spin-orbit coupling,\nillustrated in Fig. 6, the uniform in-plane spin polariza-\ntion is perpendicular to the applied electrical current.\nInitial observations of the ISGE were made in par-\nallel with the initial SHE experiments, in both cases\nemploying the Kerr/Faraday magneto-optical detection\nmethods or circularly polarized luminescence (Belkov and\nGanichev, 2008; Ganichev et al. , 2004; Ivchenko and\nGanichev, 2008; Kato et al. , 2004b,c; Silov et al. , 2004;\nWunderlich et al. , 2004, 2005). Kato et al. , 2004b,c ob-\nserved the SHE and ISGE in the same strained bulk n-\nInGaAs sample and Wunderlich et al. , 2004, 2005 de-\ntected the two e\u000bects in the same asymmetrically con-\n\fned 2D hole gas (2DHG) in a AlGaAs/GaAs het-\nerostructure.\nSubsequently, it was predicted (Bernevig and Vafek,\n2005) and experimentally veri\fed (Chernyshov et al. ,\n2009) that the ISGE can generate relativistic SOTs in\na FM semiconductor (Ga,Mn)As with broken inversion8\nsymmetry in the strained crystal structure of a thin \flm\nsample. Both the ISGE and SHE based mechanisms have\nbeen found to contribute to the relativistic spin torques\nin the NM/FM bilayers with broken structural inversion\nsymmetry (Garello et al. , 2013; Kim et al. , 2013; Man-\nchon and Zhang, 2008; Miron et al. , 2011a, 2010; Pai\net al. , 2014; Pi et al. , 2010; Suzuki et al. , 2011).\nAs mentioned earlier, the SHE and the Mott scattering\nof free electron beams can have the same skew scatter-\ning origin which is captured by the second-order Born\napproximation (third order in the scattering potential).\nMoreover, in condensed matter systems, the SHE can\narise from the spin-dependent transverse de\rection in-\nduced by the intrinsic spin-orbit coupling in a perfect\ncrystal with no impurities. We have also already men-\ntioned that this intrinsic SHE has its direct counterpart\nin systems with broken time reversal symmetry in the\nintrinsic AHE.\nThe spin galvanic phenomena, on the other hand, are\ntraditionally considered to originate in NMs only from\nextrinsic origins (seen already in the \frst-order Born ap-\nproximation scattering). Nevertheless, the physics of the\nSHE, AHE, spin galvanics, and relativistic spin torques\ncan be entangled even when considering the intrinsic ef-\nfects. In Fig. 7 we illustrate that the same current-\ninduced reactive mechanism that generates the trans-\nverse spin-current in the intrinsic SHE can induce a uni-\nform spin polarization, i.e. a signature characteristic of\nthe ISGE, in systems with broken space and time rever-\nsal symmetry. Relativistic SOTs generated by the non-\nequilibrium uniform spin polarization of this intrinsic ori-\ngin were identi\fed in the FM semiconductor (Ga,Mn)As\n(Kurebayashi et al. , 2014).\nIII. THEORY OF SPIN HALL EFFECT\nThe SHE is a prime example of a \feld germinated di-\nrectly from several key theoretical predictions and one\nwhich needed the correct timing to come to its full life.\nIt all began with the seminal prediction of the extrin-\nsic SHE by Dyakonov and Perel, 1971b based on a phe-\nnomenological theory that considered the consequences\nof chiral Mott scattering in a solid-state system. This\nprediction laid dormant for almost three decades until\nHirsch, 1999 and Zhang, 2000 made a similar prediction,\nbut did it at a time that the nascent \feld of spintronics\ncould fully exploit the notion of the SHE.\nShortly after this, Murakami et al. , 2003 and Sinova\net al. , 2004 predicted the intrinsic SHE based on linear\nresponse microscopic theories of strong spin-orbit cou-\npled materials. It is perhaps at this point that the \feld\nof SHE surged forward in a \rurry of enormous theoreti-\ncal activity culminating later on the parallel discoveries\nof the extrinsic (Kato et al. , 2004a) and intrinsic (Wun-\nderlich et al. , 2004, 2005) SHE.\nThe theories of the SHE have naturally emerged from\nthe theory of the AHE. However, the latter, although re-\nIntrinsic spin-Hall effect: the Rashba SOC example\nHR=~2k2\n2m\u0000µB~\u0000·~Beff(~k)\n=~2k2\n2m+↵R(\u0000xky\u0000\u0000ykx)\nJ!\nIntrinsic (Berry phase) spin-orbit torque from Bloch eq. \nLarge exchange limit and Rashba SOC\nJ!ky! ky!\nky! ky!kx!kx!\nkx!kx!(a)\t\r  (b)\t\r \n(c)\t\r  (d)\t\r FIG. 7 (a) A model equilibrium spin texture in a 2D Rashba\nspin-orbit coupled system with spins (red arrows) pointing\nperpendicular to the momentum. (b) In the presence of an\nelectrical current along the x-direction the Fermi surface (cir-\ncle) is displaced along the same direction. When moving in\nmomentum space, electrons experience an additional spin-\norbit \feld (purple arrows). In reaction to this non-equilibrium\ncurrent induced \feld, spins tilt up for ky>0 and down for\nky<0, creating a spin-current in the y-direction. (c) A model\nequilibrium spin texture in a 2D Rashba spin-orbit coupled\nsystem with an additional time-reversal symmetry breaking\nexchange \feld of a strength much larger than the spin-orbit\n\feld. In equilibrium, all spins in this case align approximately\nwith the direction of the exchange \feld. (d) The same reac-\ntive mechanism as in (b) generates a uniform, non-equilibrium\nout-of-plane spin-polarization. From Refs. Kurebayashi et al. ,\n2014; Sinova et al. , 2004.\nmaining complex and for many decades quite controver-\nsial, relies on the very important pillar of charge conser-\nvation for its development. The ever-present key di\u000ber-\nence between the SHE and the AHE is that spin, unlike\ncharge, is not a conserved quantity in most cases, which\nmakes the examination of experiments and predictions\nmore involved in the SHE.\nIn the initial predictions of the extrinsic SHE this was\ndealt with by writing down phenomenological theories\nbased on coupled spin-charge drift-di\u000busion equations\nderived from symmetry considerations. This approach\nis well justi\fed in the weak spin-orbit coupling regime\n(Dyakonov and Khaetskii, 2008; Dyakonov and Perel,\n1971b; Hirsch, 1999; Zhang, 2000). However, within the\nstrong spin-orbit couple regime of, e.g., heavy transition\nmetals, the dominant coherent e\u000bects of the intrinsic SHE\nare more di\u000ecult to couple to such phenomenological the-\nories.\nIt is within this strong spin-orbit coupling regime that\nthe AHE has made its furthest progress within the last\ndecade by systematic approaches that aimed at reaching\nagreement in non-trivial models among linear response\ntheories based on di\u000berent formalisms that, ultimately,\nshould provide equivalent predictions. This has led to a\nbetter established microscopic theory of the AHE (Ko-\nvalev et al. , 2010; Nagaosa et al. , 2010; Sinitsyn et al. ,9\n2007) and a full understanding of its mechanisms that\nhave now been linked to the earlier ad-hock introduction\nof the semiclassically de\fned mechanisms: intrinsic, skew\nscattering, and side-jump scattering.\nWe spend the \frst part of this section, Sec. III.A, de\fn-\ning and explaining each of those contribution and their\norigins in the more modern parsing of the spin-dependent\nHall transport theory. We will try to clarify in particular\nthe typical misconceptions that sometimes linger in the\nliterature regarding which aspect of the spin-orbit cou-\npling - within the crystal itself or within the disorder po-\ntential - contribute to each mechanism. We will borrow\nin this part extensively from Nagaosa et al. , 2010 and we\nwill direct the reader to this previous review for detailed\nexplanations of the di\u000berent linear response theories and\nthe resolution of some of the historical controversies.\nWe follow in Sec. III.B by a description of the phe-\nnomenological spin-charge drift-di\u000busion equations that\nare often used to \ft experiments. Because of the chal-\nlenge of merging the strong spin-orbit coupled micro-\nscopic theories and the phenomenological weak spin-orbit\ncoupling theories, one of the more popular models that\nis used to describe the SHE is based on a simple Hamil-\ntonian in which the spin-orbit coupling is only present\nin the disorder potential. We discuss such a model in\nSec. III.C. This theory has the bene\ft of having a sin-\ngle parameter - the strength of the spin-orbit coupling\nparameter for the disorder - which can be \ftted to the\nspin-di\u000busion length and from this value estimate the\nSHE angle (Cr\u0013 epieux and Bruno, 2001; Engel et al. , 2005;\nMaekawa and Takahashi, 2012; Zhang, 2000). However,\nas seen by comparing to experiment, it gives sensible re-\nsults in the weak spin-orbit coupling regime but misses\nthe coherent e\u000bects of the band structure in strongly spin-\norbit coupled materials.\nA smaller number of microscopic ab-initio theory stud-\nies on impurities indicate also a signi\fcant contribution\ndue to skew scattering (Long et al. , 2014; Zimmermann\net al. , 2014).\nIn Sec. III.D we discuss in detail the theory of SP and\nhow it is utilized to measure the ISHE and the spin Hall\nangle. It introduces the concept of the spin-mixing con-\nductance (Tserkovnyak et al. , 2002a), another parame-\nter borrowed from the weak spin-orbit coupled systems,\nwhich is at present often used in analyzing magnetization\ndynamics experiments in connection to the physics of the\nSHE (Saitoh et al. , 2006). Besides introducing the basic\nconcepts of SP and its connection to the measurements\nof the ISHE, we discuss the range of assumptions and\nlimits which are often used in experiments.\nLastly, in Sec. III.E, we present the formalism primar-\nily used in the strong spin-orbit coupled systems. This\nformalism is based in the Kubo formula and exploited\nsuccessfully in transition metals (Freimuth et al. , 2010;\nTanaka et al. , 2008). Calculations seem to indicate that\nthe principal contribution - as in the AHE - arises from\nthe intrinsic de\rection mechanism.A. Mechanisms of the spin Hall e\u000bect\nThe family of spin-dependent Hall e\u000bects (AHE, SHE,\nand ISHE) originate from three distinct microscopic\nmechanism that they all share. These mechanisms have\nbeen \frst identi\fed in the AHE (Nagaosa et al. , 2010).\nThe mechanisms originate from coherent band mixing\ne\u000bects due to the external electric \feld and the disor-\nder potential. It makes them more complex than the\nsimpler single-band diagonal transport. As with other\ncoherent interference transport phenomena, they cannot\nbe satisfactorily explained using traditional semiclassi-\ncal Boltzmann theory. It is then not surprising that the\noriginal proposals of the intrinsic, skew scattering, and\nside-jump mechanisms which were based on semiclassi-\ncal theory considerations brought in both insightful new\nconcepts but also seeds for ensuing controversies in the\ndebate over the quantum-mechanical microscopic origins\nof the AHE. While initially based in the semiclassical the-\nory, there exists now a more modern strincter de\fnition\nof the mechanisms within microscopic theories. However,\nto keep continuity and not create further confusion, this\nmore modern parsing of the di\u000berent contributions has\ninherited the already established lexicon (see Ref. Na-\ngaosa et al. , 2010, Sec. IV).\nThe modern parsing of the microscopic mechanisms is\nbased on both experimental and microscopic transport\ntheory considerations, rather than on the identi\fcation\nof one particular e\u000bect from within semiclassical theory.\nThe justi\fcation here is of course primarily on the AHE,\nnot the SHE, for which the spin-Hall conductivity and its\nconsequences have to be ultimately coupled to the spin-\naccumulation that it induces and can therefore depend on\nthe method of measurement. In other words, depending\non the measurement the spin accumulation may vary,\ne.g. in non-local transport measurements vs. FMR based\nmeasurements.\nThe link to semiclassically de\fned processes, as they\nhave been historically attributed, is established through\nthe works on AHE after developing a fully generalized\nBoltzmann transport theory which takes inter-band co-\nherence e\u000bects into account and is fully equivalent to\nmulti-band microscopic theories. The key recent devel-\nopments have been in understanding the link between\nsemiclassical and microscopic theory of spin-dependent\nHall transport.\nA very natural classi\fcation of contributions based on\nthe AHE, which is guided by experiment and by micro-\nscopic theory of metals, is to separate them according to\ntheir dependence on the Bloch state transport lifetime \u001c.\nWithin the metallic regime, disorder is treated perturba-\ntively and higher order terms vary with a higher power\nof the quasiparticle scattering rate \u001c\u00001. As we will dis-\ncuss, it is relatively easy to identify contributions to the\nanomalous or spin Hall conductivity, \u001bH\nxy, which vary as\n\u001c1and as\u001c0. In experiments of the AHE a similar sepa-\nration can sometimes be achieved by plotting \u001bxyvs. the\nlongitudinal conductivity \u001bxx/\u001c, when\u001cis varied by10\naltering disorder or varying temperature.\nWe emphasize that several microscopically distinct\nterms can share the same \u001c-dependence. We also note\nthat in this parsing of the AHE and SHE contributions\nit is the dependence on \u001c(or\u001bxx) which de\fnes it, not\na particular mechanism linked to a microscopic or semi-\nclassical theory. The contribution proportional to \u001c1we\nde\fne as the skew-scattering contribution, \u001bH\u0000skew\nxy . The\nsecond contribution proportional to \u001c0(or independent\nof\u001bxx) we further separate into two terms: intrinsic and\nside-jump .\nThe \frst term arises from the evolution of spin-orbit\ncoupled quasiparticles as they are accelerated by an ex-\nternal electric \feld, whereas the second term arises from\nscattering events from impurities that do not include the\nskew scattering contribution. Although the intrinsic and\nside-jump terms cannot be separated simply experimen-\ntally by DC measurements, they in principle canbe sep-\narated experimentally (as well as theoretically) by de\fn-\ning the intrinsic term, \u001bH\u0000int\nxy , as the extrapolation of\nthe ac-Hall conductivity to zero frequency in the limit\nof\u001c! 1 , with 1=\u001c!0 faster than !!0. This\nthen leaves a unique de\fnition for the side-jump term, as\n\u001bH\u0000sj\nxy\u0011\u001bH\nxy\u0000\u001bH\u0000skew\nxy\u0000\u001bH\u0000int\nxy .\nWe further describe these contributions below. We\nnote that the above de\fnitions have not relied on link-\ning the terms to semiclassical processes such as side-jump\nscattering (Berger, 1979) or skew-scattering from asym-\nmetric contributions to the semiclassical scattering rates\n(Smit, 1958) identi\fed in earlier theories.\nThe ideas explained brie\ry in this section are substan-\ntiated in the recent review Nagaosa et al. , 2010, which\nanalyses the tendencies in the AHE data of several ma-\nterial classes, and extensively discusses the AHE the-\nory. The extensions to the other spin-dependent Hall\ne\u000bects, such as SHE and ISHE, require the coupling of\nthis spin-current generating mechanisms to spin-charge\ndrift-di\u000busion transport equations that are appropriate\nto describe the particular experimental measurement, be\nit optical or electrical.\n1. Intrinsic mechanism\nAmong the three contributions, the easiest to evaluate\naccurately and the one that has dominated most theoret-\nical studies is the intrinsic contribution. We have de\fned\nthe intrinsic contribution microscopically as the dclimit\nof the interband spin Hall conductivity with 1 =\u001c!0\nfaster than the frequency. There is however a direct link\nto the semiclassical theory in which the induced inter-\nband coherence is captured by a momentum-space Berry-\nphase related contribution to the anomalous velocity.\nIn the context of the AHE, this contribution was \frst\nderived by Karplus and Luttinger, 1954 but its topologi-\ncal nature was not fully appreciated until recently (Jung-\nwirth et al. , 2002; Onoda and Nagaosa, 2002). The work\nof Jungwirth et al. , 2002 was motivated by the exper-imental importance of the AHE in FM semiconductors\nand also by the analysis of the relationship between mo-\nmentum space Berry phases and anomalous transverse\nvelocities in semiclassical transport theory (Sundaram\nand Niu, 1999; Xiao et al. , 2010). Its connection to the\nSHE was described by Murakami et al. , 2003 and Sinova\net al. , 2004.\nThe intrinsic contribution to the spin Hall conductivity\nis dependent only on the band structure of the perfect\ncrystal, hence its name. Pictorially it can be seen to\narise from the non-equilibrium electron dynamics of the\nBloch electrons as they are accelerated in an electric \feld\nand undergo spin-precession due to the induced electric\n\feld, as illustrated in Sec. II.D Fig. 7 for a Rashba spin-\norbit coupled Hamiltonian. Here the 2D Rashba system\nis described by the Hamiltonian:\nH=p2\n2m\u0000\u0015\n~~ \u001b\u0001(^z\u0002~ p); (3.1)\nwherep=~kis the 2D electron momentum, \u0015is the\nRashba coupling constant, ~ \u001bthe Pauli matrices, mthe\nelectron e\u000bective mass, and ^ zthe unit vector perpendic-\nular to the 2D electron gas (2DEG) plane.\nFor this example the dynamics of an electron spin in\nthe presence of time-dependent spin-orbit coupling is de-\nscribed by the Bloch equation (Sinova et al. , 2004):\n~d^n\ndt= ^n\u0002~\u0001(t) +\u000b~d^n\ndt\u0002^n; (3.2)\nwhere ^nis the direction of the spin and \u000bis a damping\nparameter that we assume is small. For the application\nof Eq. (3.2) we have in mind the ~ pdependent spin-orbit\ncoupling term in the spin-Hamiltonian, \u0000~ s\u0001~\u0001=~, where\n~\u0001 = 2\u0015=~(^z\u0002~ p). For a Rashba e\u000bective magnetic \feld\nwith magnitude \u0001 1that initially points in the ^ x1di-\nrection then tilts (arbitrarily slowly) slightly toward ^ x2,\nwhere ^x1and ^x2are orthogonal in-plane directions, it\nfollows from the linear response limit of Eq. (3.2) that\n~dn2\ndt=nz\u00011+\u000bdnz=dt\n~dnz\ndt=\u0000\u00011n2\u0000\u000bdn 2=dt+ \u0001 2; (3.3)\nwhere \u0001 2=~\u0001\u0001^x2. By solving these inhomogeneous\ncoupled equations, it follows that to leading order in the\nslow-time dependences n2(t) = \u0001 2(t)=\u00011,i.e., the ^x2-\ncomponent of the spin rotates to follow the direction of\nthe spin-orbit \feld, and that\nnz(t) =1\n\u00012\n1~d\u00012\ndt: (3.4)\nThese dynamics give rise to the spin-current in the ^ y\ndirection:\njs;y=Z\nannulusd2~ p\n(2\u0019~)2~nz;~ p\n2py\nm\n=\u0000eEx\n16\u0019\u0015m(pF+\u0000pF\u0000); (3.5)11\nwherepF+andpF\u0000are the Fermi momenta of the major-\nity and minority spin Rashba bands (Sinova et al. , 2004).\nWe choose the example based on the Rashba sys-\ntem because it is simple to see pictorially the intrin-\nsic contribution. However, for this particular simple\nexample, in a large range of Fermi energies the result\nfor the intrinsic spin Hall conductivity turns out to be\n\u001bH\u0000int\nxy =\u0000(e=~)js;y=Ex=e2=8\u0019~. This contribution\nis eventually cancelled by short-range disorder scattering\nbecause the induced spin-current is proportional to the\nspin-dynamics, which should vanish in the steady state\n(Sinova et al. , 2006).\nFor other spin-orbit coupled Hamiltonians, corre-\nsponding to realistic materials system, this cancellation\nfrom the vertex corrections arising from disorder does\nnot exist. The above result, illustrated in a simple semi-\nclassical form, is usually best evaluated directly from the\nKubo formula for the spin Hall conductivity for an ideal\nlattice (Sinova et al. , 2004):\n\u001bH\u0000int\nxy =e2\nVX\nk;n6=n0(fn0;k\u0000fn;k)\n\u0002Im[hn0kj^jz\nspin xjnkihnkjvyjn0ki]\n(Enk\u0000En0k)(Enk\u0000En0k\u0000i\u0011)(3.6)\nwheren;n0are band indices, ~jz\nspin=~\n4f\u001bz;~ vgis the spin-\ncurrent operator, !and\u0011are set to zero in the DC clean\nlimit, and the velocity operators at each ~ pare given by\n~vi=~@H(~ p)=@pi.\nWhat makes this contribution quite unique, particu-\nlarly in the AHE, is that it is directly linked to the topo-\nlogical properties of the Bloch states. Speci\fcally it is\nproportional to the integration over the Fermi sea of the\nBerry's curvature of each occupied band, or equivalently\n(Haldane, 2004) to the integral of Berry phases over cuts\nof Fermi surface segments. This same linear response\ncontribution to the AHE and SHE conductivity can be\nobtained from the semiclassical theory of wave-packets\ndynamics (Jungwirth et al. , 2002; Sundaram and Niu,\n1999; Xiao et al. , 2010).\nOne of the motivations for identifying the intrinsic\ncontribution \u001bH\u0000int\nxy is that it can be evaluated accu-\nrately even for materials with relatively complex bands\nusing microscopic electronic structure theory techniques.\nIn many materials which have strongly spin-orbit cou-\npled bands, the intrinsic contribution seems to dominate\nthe SHE and AHE. Particularly in metals the calcula-\ntions have given semi-quantitative predictions of the ex-\npected spin Hall angles. This is illustrated in the density-\nfunctional calculation for Pt (Guo et al. , 2008), shown in\nFig. 8, and in the microscopic tight-binding calculations\nfor other 4d and 5d metals (Tanaka et al. , 2008), shown in\nFig. 9. As it is clear from Fig. 8 the largest contributions\nto the spin Hall conductivity arise, similar to AHE, when-\never bands connected via spin-orbit coupling are near\neach other at the Fermi energy. The calculated spin Hall\nconductivities are predicted to be large in these transi-\ntion metals, and, in particular, a sign change is predicted\nin the Brillouin zone (BZ) for the fcc lattice. They give a\nprominent enhancement of SHC in Pt. We determine aneffective Hamiltonian near XandLpoints, and we dem-\nonstrate robustness of the SHE against impurities.\nThe band structure of Pt is calculated using a fully\nrelativistic extension [ 25] of the all-electron linear\nmuffin-tin orbital method [ 26] based on the density func-\ntional theory with local density approximation [ 27]. The\nlattice constants for Pt and Al used are 3.92 and 4.05 A ˚,\nrespectively. The basis functions used are s,p,d, andf\nmuffin-tin orbitals for Pt but s,p, anddmuffin-tin orbitals\nfor Al [ 26]. In the self-consistent band structure calcula-\ntions, 89 kpoints in the fcc irreducible wedge (IW) of the\nBZ were used in the BZ integration. The SHC is evaluated\nby the Kubo formula [ 28]. A fine mesh of 60 288 kpoints\non a larger IW (3 times the fcc IW) is used. These corre-\nspond to the division of the /.0255Xline into 60 segments.\nComparison with test calculations with 102 315 kpoints\n(72 divisions of the /.0255Xline) for Pt indicate that the\ncalculated SHC converges within 1%.\nFigure 1shows the relativistic band structure of Pt, and\nalso the SHC ( /.0027\nxy) as a function of EF. Remarkably, the\nSHC peaks at the true Fermi level (0 eV), with a large value\nof2200/.0133@=e/.0134/.0133/.0010cm/.0134/.02551. This gigantic value of the SHC is\norders of magnitude larger than the corresponding value in\np-type semiconductors Si, Ge, GaAs, and AlAs [ 28,29].\nFurthermore, the calculated SHC in simple metal Al is only/.025517/.0133@=e/.0134/.0133/.0010cm/.0134\n/.02551, being 2 orders of magnitude smaller\nthan that of Pt. Interestingly, the SHC in Pt decreases\nmonotonically as the EFis artificially raised and becomes\nrather small above 3.0 eV . When the EFis artificially\nlowered, the SHC also decreases considerably, and changes\nits sign at /.02551:1e V . As theEFis further lowered, the SHC\nincreases in magnitude again and becomes peaked at/.02554:2e V with a large value of /.02551970/.0133@=e/.0134/.0133/.0010cm/.0134\n/.02551.\nThe SHC decreases again when the EFis further lowered,\nand finally becomes very small below /.02556:0e V . Note thatthe bands below /.02558:0e V and also above 2.0 eV are\npredominantly of 5scharacter and the effect of the spin-\norbit coupling is negligible.\nWe notice that a peak in the SHC appears at the double\ndegeneracies on the LandXpoints near EF(0 eV) in the\nscalar-relativistic band structure (i.e., without the spin-orbit coupling) while the other peak at /.02554:2e V occurs\nnear the double degeneracies at the Land/.0255points (see\nFig. 1). The double degeneracy (bands 5 and 6) at Lis\nmade mostly (93%) of d\nx0z0anddy0z0(z0is the threefold\naxis), being consistent with the point group D3datL. The\ndouble degeneracy (bands 4 and 5) at Xconsists mainly of\ndx0z0anddy0z0(z0: fourfold axis), being consistent with the\npoint group D4h. These double degeneracies are lifted by\nthe spin-orbit coupling, with large spin-orbit splittings(/.00240:66, 0.93 eV , respectively).\nOne may attribute the large SHC in Pt to these double\ndegeneracies. To see this, let us consider the k-resolved\ncontribution to the SHC, i.e., Berry curvature /.0010\nzn/.0133k/.0134;\n \n/.0027zxy/.0136e\n@X\nk/.0010z/.0133k/.0134/.0136e\n@X\nkX\nnfkn/.0010zn/.0133k/.0134;\n/.0010zn/.0133k/.0134/.0136X\nn0/.0222n2Im/.0137hknjjzxjkn0ihkn0jvyjkni/.0138\n/.0133/.0015kn/.0255/.0015kn0/.01342;(1)\nwhere the spin current operator jzx/.01361\n2fsz;vg, with spin sz\ngiven by sz/.0136@\n2/.0012/.0006z(/.0012,/.0006z:4/.00024Dirac matrices) [ 28].\nfknis the Fermi distribution function for the nth band at k.\n/.0010znis an analogue of the Berry curvature for the nth band,\nand it is enhanced when other bands come close in energy(i.e., near degeneracy). Figure 2(a) shows clearly that\n/.0010\nz/.0133k/.0134is large only near the LandXpoints. Inter-\nestingly, Berry curvature /.0010zn/.0133k/.0134for the doublet bands 4\nand 5 near the Xpoint are large but have opposite signs\n[Fig. 2(b)]. However, because band 5 near the Xpoint is\nunoccupied, only /.0010zn/.0133k/.0134for band 4 contributes to the SHC,\nresulting in the large positive peak in /.0010z/.0133k/.0134near theX\npoint [Fig. 2(a)]. Figure 2(c)shows that the SHC decreases\nmonotonically as the temperature ( T) is raised. This rather\nstrong temperature dependence is also due to the near\ndegeneracies since the small energy scale is relevant tothe SHC there. Nevertheless, the SHC /.0027\nxy/.0136240/.0133@=e/.0134/.0002\n/.0133/.0010cm/.0134/.02551atT/.0136300 K is still large and is close to the\nmeasured value (240) [ 4]. The SHC for Al at 4 and 300 K is\n/.025517and/.02556/.0133@=e/.0134/.0133/.0010cm/.0134/.02551, respectively. The former\nvalue is similar to the experimental values ( /.025527,/.025534)a t\n4.2 K [ 5].\nIn order to study the role of near degeneracies in more\ndetail, we construct two effective Hamiltonians H/.0133k/.0134for\nthe two doubly degenerate bands at XandLpoints, re-\nspectively. At the Xpoint, by imposing the D4hsymmetry\nand the time-reversal symmetry, the effective Hamiltonianwith basis j/.0133x\n0/.0007iy0/.0134z0\"iandj/.0133x0/.0006iy0/.0134z0#i(z0: fourfold\naxis) can be written in terms of 4/.00024Clifford/.0255matrices\n(/.02551/.0136/.0028x,/.02552/.0136/.0027z/.0028y,/.02553/.0136/.0027x/.0028y,/.02554/.0136/.0027y/.0028y,/.02555/.0136/.0028z)a s\nH/.0133k/.0134/.0136/.0015/.0133k/.0134/.0135P5\na/.01361da/.0133k/.0134/.0255a. By expanding the coeffi--11-10-9-8-7-6-5-4-3-2-101234Energy (eV)SOC\nnoSOC\n-20 0 20\nσxyz (102Ω-1cm-1)-11-10-9-8-7-6-5-4-3-2-101234\nW L X W Γ Γ(a) (b)fcc Pt\nFIG. 1 (color online). (a) Relativistic band structure and\n(b) spin Hall conductivity of fcc Pt. The zero energy and thedotted line is the Fermi level. The dashed curves in (a) are the\nscalar-relativistic band structure.PRL 100, 096401 (2008)PHYSICAL REVIEW LETTERSweek ending\n7 MARCH 2008\n096401-2\ncientsdawith respect to the wave numbers k0measured\nfromXandLpoints ( k0/.0136k/.0255ki,i/.0136L,X), we have\nconstructed the effective Hamiltonian. Fitting with the\ncalculated energy bands and wave functions, we deter-\nmined the expansion coefficients to k04order. This effec-\ntive model is an even function of k0and is similar to the\nLuttinger model, representing the valence bands of cu-\nbic semiconductors [ 30], or the valence and conduction\nbands of zero-gap cubic semiconductors [ 31] near the /.0255\npoint. The previous analysis for the p-type semiconduc-\ntors [ 30] are equally applied. The effective Hamiltonian\nhas the eigenvalues El/.0133k/.0134/.0136/.0015/.0133k/.0134/.0255d/.0133k/.0134, andEu/.0133k/.0134/.0136\n/.0015/.0133k/.0134/.0135d/.0133k/.0134for the lower and upper bands, respectively,\nwhered/.0136/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129P5\ni/.01361d2\niq\n, and these bands correspond to the\nheavy-hole and light-hole bands, respectively. From\nEq. (35) of Ref. [ 30], the response of a generalized spin\ncurrent (corresponding to /.0255ab) is given by /.0027ab\nij/.01364Zdk\n/.01332/.0025/.01343/.0133fkl/.0255fku/.0134Gab\nij; (2)\nwherefkuandfklare the Fermi functions of the upper and\nthe lower bands, and Gab\nij/.01361\n4d3/.0015abcdedc@dd\n@ki@de\n@kj, where\n/.0015abcde is the totally antisymmetric tensor with /.001512345/.01361.\nWe flipped the sign of /.0027ab\nijbecause the sign of the charge of\nthe carriers is opposite from Ref. [ 30].Gab\nijdescribes the\nmapping of an area form from the three-dimensional k\nspace to the five-dimensional dspace. It can be regarded as\na ‘‘solid angle’’ enclosed by the dvector when the wave\nnumber kruns over the domain between the two Fermi\nsurfaces. Hence, it becomes larger for smaller d/.0133k/.0134/.01361\n2/.0002\n/.0133Eu/.0255El/.0134. The spin operators are given by sx/.0136/.025535=2,\nsy/.0136/.025545=2, andsz/.0136/.025534=2, where /.0255ab/.01361\n2i/.0137/.0255a;/.0255b/.0138.\nUsing these relations, one can calculate the SHC /.0027zxy\nfrom Eq. ( 2) by summing over the three Xpoints and\nfourLpoints.\nThe next issue is whether the contributions from various\nbands cancel or not. From Eq. ( 2), the SHC from the X\npoints and that from the Lpoints are calculated as a\nfunction of the EF, as shown in Fig. 3. Here we put a cutoff\nfor the kintegral as /.0025=/.01335a/.0134. The integrand is dominated by\nthe contribution near the Lor theXpoints, and cancellation\ndoes not occur when the Fermi energy is in the gap. It is\nanalogous to the zero-gap semiconductors rather than\nGaAs [ 31,32]. Thus we can identify the peaks at EF/.00240\nwith the peak of the SHC in Fig. 1, and the enhancement of\nSHC in Pt is attributed to the near degeneracies at the Land\nXpoints.\nAs is similar to the p-type semiconductors [ 18], this\nintrinsic SHE is robust against impurity scattering [ 33]. To\nsee this, we consider dilutely distributed short-ranged im-\npuritiesV/.0133r/.0134/.0136P\niV/.0014/.0133r/.0255ri/.0134. It is justified in Pt, because\nscreening is prominent compared with semiconductors.Then the vertex corrections from the impurity scatteringfor the SHC vanishes in the clean limit from the follow-ing reason. Because the effective Hamiltonian satisfies\nσ σ(a ()b )\no\n‘‘\nFIG. 3 (color online). Spin Hall con-\nductivity of platinum calculated from theeffective Hamiltonian for (a) the Lpoints\nand (b) the Xpoints, as a function of E\nF\n[32].0100200300Ωz(k) (atomic units)\n0 150 300\nT (K)012σ (103Ω-cm-1)\n-500050010001500200025003000Ωz\nn(k) (atomic units)W L Γ XW Γ\nband 1\nW L Γ XW Γ(a)\n(b)(c)\nband 2band 3band 4band 5band 6\nFIG. 2 (color online). (a) Berry curvature /.0010z/.0133k/.0134at zero tem-\nperature, and (b) band ( n)-decomposed Berry curvature /.0010zn/.0133k/.0134\nalong the symmetry lines in the fcc Brillouin zone. In (b), /.0010zn/.0133k/.0134\nfor thenth band has been shifted upwards by /.0133n/.02551/.0134/.0002500for\nclarity. The inset (c) shows the temperature dependence of the\nspin Hall conductivity /.0027zxy.PRL 100, 096401 (2008)PHYSICAL REVIEW LETTERSweek ending\n7 MARCH 2008\n096401-3FIG. 8 Band structure for Pt calculated with (solid lines)\nand without (dotted lines) spin-orbit coupling. The spin Hall\nconductivity (b) is shown calculated at each energy. In the\nlower \fgure the Berry curvature is calculated (total) as well\nas the one corresponding for each sub band. From Ref. Guo\net al. , 2008.\ngoing from Pt to Ta which has been directly observed\nin experiments. More recent density-functional calcula-\ntions on a range of hcp metals and antiferromagnetic Cr,\nshow a strong anisotropy of the spin Hall conductivity\n(Freimuth et al. , 2010), as illustrated in Fig. 10.\n2. Skew scattering mechanism\nThe skew scattering contribution to the SHE and the\nAHE is the term which is proportional to the Bloch state\ntransport lifetime \u001c. It will therefore tend to dominate in\nnearly perfect crystals. It is the only contribution to the\nSHE and AHE which appears in traditional Boltzmann\ntransport theory where interband coherence e\u000bects are\nusually neglected. Skew scattering is due to chiral fea-12\nusing the band structure of Pt, which is represented by theopen symbols in Fig.5/H20849a/H20850. We see that the magnitude of SHCin Pt withn=9 does not reproduce that in Ir. The same is truein Ta and W/H20849bcc structure/H20850. Therefore, we need to calculateSHC using a correct band structure for each metal.Next, we examine the/H9261dependence of the SHC. We veri-fied that the SHC in each metal increases approximately pro-portional to/H9261as shown in Fig.4. To elucidate the origin ofSHC, we calculated the SHC when SOI is anisotropic:/H92611/H20858/H20849lˆzsˆz/H20850+/H92612/H20858/H20849lˆxsˆx+lˆysˆy/H20850. We find that the SHC forHSO=/H9261/H20858/H20849lˆzsˆz/H20850/H20849/H92611=/H9261,/H92612=0/H20850is as large as the SHC in the isotro-pic case/H20849/H92611=/H92612=/H9261/H20850. In contrast, the SHC forHSO=/H9261/H20858/H20849lˆxsˆx+lˆysˆy/H20850/H20849/H92611=0,/H92612=/H9261/H20850is 1 order of magnitudesmaller than the isotropic case. Therefore, thezcomponentof the SOI gives the decisive contribution to the SHC. Thematrix element oflˆzis finite only for/H20855yz/H20841lz/H20841zx/H20856=−/H20855zx/H20841lz/H20841yz/H20856=iand/H20855xy/H20841lz/H20841x2−y2/H20856=−/H20855x2−y2/H20841lz/H20841xy/H20856=2i. Note thatdxyanddx2−y2orbitals/H20849dyzanddzxorbitals/H20850are given by the linearcombination oflz=/H110062/H20849lz=/H110061/H20850. Here, we examine whichorbitals cause a significant contribution to the SHC. Thezcomponent of SOI is rewritten as/H92613/H20858i/H20853P/H20849lz2=1/H20850/H20849lˆzsˆz/H20850/H20854i+/H92614/H20858i/H20853P/H20849lz2=4/H20850/H20849lˆzsˆz/H20850/H20854i, whereP/H20849lz2=n/H20850represents the projec-tion operator. SHC caused bydxyanddx2−y2orbitals is givenby setting/H92613=0 and/H92614=/H9261which are represented aslz=/H110062 in TableII. Similarly, SHC causeddyzanddzxorbitalsis given by setting/H92613=/H9261and/H92614=0, which are represented aslz=/H110061 in TableII. We see that the interorbital transitionbetweendxyanddx2−y2orbitals causes a significant contribu-tion to the SHC in many metals. Only in the case of Mo, W,and Ir, the contribution ofdzxanddyzorbitals is comparableto that ofdxyanddx2−y2orbitals. In other metals,dxyanddx2−y2orbitals give the dominant contribution to the SHC.Here, we show the OHCs for/H9253=0.02 in various transitionmetals in Fig.6. We see that all the 4dand 5dtransitionmetals show huge and positive OHCs, which are almost 1order of magnitude larger than the SHCs. In Au/H20849Ag/H20850, theOHC takes a small value since thed-electron DOS is small atthe Fermi level. Therefore, a huge and positive OHC is auniversal nature of transition metals. As in the case of theSHE, the intrinsic OHE shows the crossover behavior: theOHC is independent of/H9267in the low resistive regime,whereas it decreases in proportion to/H9267−2in the high resistiveregime.7,8In a later publication, we will present an intuitive/H20849semiclassical/H20850explanation of the origin of the OHC.34Now, we discuss the/H9253dependences of SHC and OHC.The/H9253dependence of intrinsic SHCs in Ta and W are shownSHC ( )−0.50.00.00.51.0γ = 0.024d5dSHC ( )−0.50.00.00.51.0γ = 0.024d5dSHC ( )−0.1−0.050.00.00.055d4d4d5dγ = 0.2n79 11AuAg658 10Tc RuOsRenSHC ( )−0.1−0.050.00.00.055d4d4d5dγ = 0.2n79 11AuAg658 10Tc RuOsRen579 11SHC ( )−1.00.00.01.04d5d\nMoNbRhPdTaWIr Ptγ = 0.002(a)(b)8 10610 cm3−1Ω−110 cm3−1Ω−110 cm3−1Ω−1579 11SHC ( )−1.00.00.01.04d5d\nMoNbRhPdTaWIr Ptγ = 0.002(a)(b)8 10610 cm3−1Ω−110 cm3−1Ω−110 cm3−1Ω−1\nFIG. 5./H20849Color online/H20850ndependence of SHC for/H9253=0.002, 0.02,and 0.2. In/H20849a/H20850, we see that Pt shows the largest SHC for/H9253=0.002.The open symbols represents the SHC in Pt forn=5–9. In/H20849b/H20850, theSHCs obtained in the present model forn=7 and 8/H20849hcp structure/H20850are also shown. SHC in W takes the largest value for/H9253=0.2.TABLE II. SHC that originates from thedzx,dyz,dxy, anddx2−y2orbitals. Here, we set/H9253=0.02.lz=/H110062/H20849lz=/H110061/H20850represents the SHCcaused bydxyanddx2−y2orbitals/H20849dyzanddzxorbitals/H20850. The ratiorepresents/H20849SHC fromlz=/H110062/H20850//H20849SHC fromlz=/H110061/H20850. We see thatdxyanddx2−y2orbitals cause a significant contribution to the SHC inmany metals.Metalslz=/H110061lz=/H110062 RatioNb/H208494d45s1/H20850−0.0332 −0.0770 2.32Mo/H208494d55s1/H20850−0.0474 −0.0587 1.24Rh/H208494d85s1/H208500.0847 0.269 3.18Pd/H208494d105s0/H208500.0847 0.455 5.37Ag/H208494d105s1/H208500.00224 0.0181 8.08Ta/H208495d36s2/H20850−0.0222 −0.254 11.4W/H208495d46s2/H20850−0.174 −0.205 1.18Ir/H208495d96s0/H208500.0123 0.0231 1.89Pt/H208495d96s1/H208500.136 0.678 4.98Au/H208495d106s1/H208500.0177 0.0987 5.59\n6 8 10OHC ( )0.00.01.02.03.04.05.0γ = 0.024d5dn579 114d5dRhPdIrPtAgAuNbTaMoWRuTcReOs10 cm3−1Ω−16 8 10OHC ( )0.00.01.02.03.04.05.0γ = 0.024d5dn579 114d5dRhPdIrPtAgAuNbTaMoWRuTcReOs10 cm3−1Ω−1\nFIG. 6./H20849Color online/H20850ndependence of OHC for/H9253=0.02. Theobtained OHCs are positive for all metals, and they are about tentimes larger than the SHCs except for Pt and Pd.TANAKAet al.PHYSICAL REVIEW B77, 165117/H208492008/H20850\n165117-8\nFIG. 9 Intrinsic spin Hall conductivity for 4d and 5d transi-\ntion metals. A key feature is the change in sign from Pt to\nTa. From Ref. Tanaka et al. , 2008.\nconsidered in the integration in order to reproduce the SHC\nquantitatively correctly. This makes it hardly possible tointerpret the spin Hall conductivity in terms of a smallnumber of virtual interband transitions. Even the sign andorder of magnitude of the SHC are difficult to predict based\non simple arguments. Recently, the variation of the sign of\nthe Fermi-surface contribution to the SHC along the 4d\nand5dtransition metal series has been attributed to the\nvariation of the sign of the spin-orbit polarization on theFermi-surface [ 22]. However, we find that the spin-orbit\npolarization does not change sign for Sc, Ti and Ru whilethe SHC changes sign as the spin polarization is rotatedfrom the zaxis to the yaxis.\nIn conclusion, we have investigated the dependence of\nthe SHE on the directions of electric field and spin polar-ization. For the special cases of hexagonal and tetragonalmetals we derived the general expression for the SHCvector. We predict that in hcp metals and antiferromagneticCr the SHE is strongly anisotropic. For Sr, Ti, and Ru theanisotropy is particularly strong since the sign of the SHCdepends on the orientation of spin polarization. In this case\ncollinearity of spin polarization and electric field (or spin\npolarization and spin current) can be achieved for specialdirections of the electric field (or of the spin current).\nWe gratefully acknowledge computing time on the\nsupercomputers JUGENE and JUROPA at JSC and fund-ing under the HGF-YIG programme VH-NG-513.[1] N. Nagaosa et al. ,Rev. Mod. 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Gradhand et al. ,Phys. Rev. Lett., 104, 186403 (2010) ;\nPhys. Rev. B 81, 245109 (2010)\n[17] See http://www.flapw.de .\n[18] X. Wang et al. ,Phys. Rev. B 74, 195118 (2006) .\n[19] J. R. Yates et al. ,Phys. Rev. B 75, 195121 (2007) .\n[20] A. A. Mostofi et al. ,Comput. Phys. Commun. 178, 685\n(2008) .\n[21] F. Freimuth et al. ,Phys. Rev. B 78, 035120 (2008) .\n[22] H. Kontani et al. ,Phys. Rev. Lett. 102, 016601 (2009) .\nFIG. 2 (color online). (a) For the hcp metals Sc, Ti, Zn, Y, Zr, Tc, Ru, Cd, La, Hf, Re, and Os and for antiferromagnetic Cr the spin\nHall conductivities /C27xyzand/C27zxyare shown as light (red) and dark (blue) bars, respectively. (b) Decomposition of the SHC of Sc into\nperpendicular and parallel components following Eq. ( 6). The angle /C11enclosed by the direction of the spin current and the direction of\nthe spin polarization is also shown. At the angle /C180¼62:2/C14the component of the spin polarization perpendicular to the spin current\nvanishes and /C11¼180/C14.PRL 105, 246602 (2010) PHYSICAL REVIEW LETTERSweek ending\n10 DECEMBER 2010\n246602-4\nFIG. 10 Intrinsic spin Hall conductivity for hcp metals Sc,\nTi, Zn, Y, Zr, Tc, Ru, Cd, La, Hf, Re, and Os and for anti-\nferromangetic Cr. From Ref. Freimuth et al. , 2010.\ntures which appear in the disorder scattering in the pres-\nence of spin-orbit coupling. This mechanism was \frst\nidenti\fed in FMs by Smit (Smit, 1958) and has its ori-\ngins in the Mott scattering in relativistic physics (Mott,\n1929, 1932).\nTypical treatments of semi-classical Boltzmann trans-\nport theory found in textbooks often appeal to the prin-\nciple of detailed balance which states that the transi-\ntion probability Wn!mfrom state ntomis identical\nto the transition probability in the opposite direction\n(Wm!n). Although these two transition probabilities are\nidentical in the Fermi's golden-rule approximation, since\nWn!n0= (2\u0019=~)jhnjVjn0ij2\u000e(En\u0000En0);whereVis the\nperturbation inducing the transition, detailed balance in\nthis microscopic sense is not generic. In the presenceof spin-orbit coupling, either in the Hamiltonian of the\nperfect crystal or in the disorder Hamiltonian, a tran-\nsition which is right-handed with respect to the mag-\nnetization direction has a di\u000berent probability than the\ncorresponding left-handed transition. When the transi-\ntion rates are evaluated perturbatively, asymmetric chiral\ncontributions appear at third order. In simple models the\nasymmetric chiral contribution to the transition proba-\nbility is often assumed to have the form:\nWA\nkk0=\u0000\u001c\u00001\nA(k\u0002k0)\u0001Ms: (3.7)\nInserting this asymmetry into the Boltzmann equation\nleads to a current proportional to the longitudinal current\ndriven byEand perpendicular to both EandMs. This\ncontribution to the Hall conductivity \u001bH\u0000skew\nxy and the\nconductivity \u001bxxare approximately proportional to the\ntransport lifetime \u001cand the Hall resistivity, \u001aH\u0000skew\nxy =\n\u001bH\u0000skew\nxy\u001a2\nxx, is therefore proportional to the longitudinal\nresistivity\u001axxwhenever this contribution dominates.\nThere are several speci\fc mechanisms for skew scat-\ntering (see Sec. IV-B and Sec. V-A. in Nagaosa et al. ,\n2010). Evaluation of the skew scattering contribution\nto the Hall conductivity or resistivity requires that the\nconventional linearized Boltzmann equation be solved us-\ning a collision term with accurate transition probabilities,\nsince these will generically include a chiral contribution.\nIn practice our ability to accurately estimate the skew\nscattering contribution to the SHE and AHE of a real\nmaterial is limited only by the accuracy of the character-\nization of its disorder.\nIn simple models, the skew scattering contributions to\nthe SHE or AHE are considered to arise only from the\nspin-orbit coupling in the disorder potential. This can\nonly be considered valid when the spin-orbit coupling in\nthe bands is not strong enough to split the degenerate\nspin subbands when compared to the typical disordered\nbroadening. In systems with strong spin-orbit coupling\nin the bands, such as heavy transition metals, consider-\ning the spin-orbit coupling only in the disorder poten-\ntial would be incorrect. The reason is because a strong\ncontribution to the skew scattering also arises from the\nscattering of the spin-orbit coupled quasiparticle from\nthe scalar potential. In fact, the spin-orbit coupling of\nthe disordered potential is typically strongly renormal-\nized by the other nearby subbands as well, and therefore\nthe e\u000bect of the multi-band character cannot be ignored\nin these materials, even when discussing skew scattering\nalone.\nRecent studies of the skew scattering based on ab-initio\nstudies and Boltzmann equation in systems with impu-\nrities of Cr, Mn, Fe, Co, and Ni in Pt, Au and Pd hosts\nhave yielded contributions to the spin Hall angle of a frac-\ntion of a per-cent (Long et al. , 2014; Zimmermann et al. ,\n2014). The result related to 1% doping of impurities to\na Pt host is shown in Fig. 11.\nWe end this subsection with a note directed to the\nreader who is more versed in the latest development of13\n3\n−2.0−1.5−1.0−0.50.00.51.01.5σyx [(mΩ cm)−1]\nSc Ti VCrMn Co NiCu−0.10.00.10.20.3αAHE [%]Meth. A\nMeth. B\nMeth. C\nFIG. 1: Skew-scattering contribution to the anomalous Hall\nconductivity (upper panel) and the anomalous Hall angle\n(lower panel) for the ferromagnetic Fe host with 3 dimpurities\nwith concentration of 1 at. %.\n−0.50.00.5αAHE [%](A) (B) (C)\n−1.0−0.50.00.5αSHE [%]\nCr Mn Fe Co Ni−3036σyx [(mΩ cm)−1]\nσ↑\nyx σ↓\nyx\nFIG. 2: Computed with three different methods skew-\nscattering contribution to the AHA (upper panel), the SHA\n(middle panel), together with spin-resolved conductivities\n(lower panel, method A only) in five dilute alloys based on\na Pt host with an impurity concentration of 1 at. %.\nfor the AHA and the SHA are shown in Fig. 2. One imme-\ndiately notices the large magnitude of the obtained Hall\nangles, which are almost an order of magnitude larger\nthan in the respective Fe dilute alloys. Remarkably, the\nV Cr Mn Fe Co Ni−0.50.00.51.01.5αAHE, αSHE [%]Au, αAHE\nAu, αSHE\nPd, αAHE\nPd, αSHEFIG. 3: Skew-scattering contribution to the AHA and SHA\nin alloys based on the non-magnetic Au and Pd hosts with all\nmagnetic 3dimpurities (concentration of 1 at. %) as computed\nwith method A.\nmagnitude of the SHA is comparable to that of the AHA\nin these systems. Moreover, with the only exception of\nNi impurities within all three approaches and Cr impu-\nrities as computed with method C, the sign of the AHA\nand SHA is in one-to-one correspondence. As shown in\nFig. 3 we also observe a similar trend for the Au and Pd\nhosts with the magnetic 3 dimpurities from V (Cr) to Co.\nTo understand the obvious correlation between the\nSHA and AHA, we analyse the spin-resolved Hall con-\nductivities defined as σ↑\nyx= (1/2)(σyx+σs\nyx) andσ↓\nyx=\n(1/2)(σyx−σs\nyx), which, within the two-current model,\nwould correspond to the conductivities of the spin-up and\nspin-down electrons, respectively. The values of the spin-\nresolved conductivities computed with method A for Pt\nand presented in Fig. 2, point at consistent suppression of\nthe skew-scattering for spin-down electrons in Pt doped\nwith Cr, Mn, Fe and Co. The situation in Au and Pd\n(not shown) is exactly analogous to that in Pt. Thus,\nin the majority of considered systems the transverse cur-\nrent which is responsible for both AHE and SHE is al-\nmost purely spin-up polarized (in Pt(Ni) the situation is\nreversed).\nThe reason behind this can be explained based on the\nlocal densities of states (LDOS) of the host and of the\nimpurity atoms. Taking the Pt host, for which the DOS\nis dominated by the d-electrons at EF, and Mn impurity\nas a representative defect, we can understand the weak\nspin-up scattering with enhanced σ↑\nyxfrom the fact of\nthe similar behaviour and orbital character of the host\nand impurity LDOS at EF: the spin-up Mn LDOS at\nEFis also predominantly of d-character. For the spin-\ndown channel the character of the Mn LDOS at the EF\niss-like due to the exchange splitting −hence the host\nand the impurity LDOS are different, and the scattering\nFIG. 11 Skew scattering spin Hall angle in a Pt host with 1%\nlevel of impurities. From Ref. Zimmermann et al. , 2014.\nthe links between the full semiclassical and the micro-\nscopic theory of the SHE and AHE. We have been careful\nabove not to de\fne the skew-scattering contribution as\nthe sum of allthe contributions arising from the asym-\nmetric scattering rate present in the collision term of the\nBoltzmann transport equation. We know from micro-\nscopic theory that this asymmetry also makes an AHE\ncontribution of order \u001c0(Sinitsyn et al. , 2007). There ex-\nists a contribution from this asymmetry which is present\nin the microscopic theory treatment associated with the\nso called ladder diagram corrections to the conductivity,\nand therefore of order \u001c0. In our parsing of the contri-\nbutions to the SHE and AHE we do not associate this\ncontribution with skew-scattering but place it under the\numbrella of side-jump scattering even though it does not\nphysically originate from any side-step type of scattering.\n3. Side-jump mechanism\nGiven the sharp de\fnition we have provided for the\nintrinsic and skew scattering contributions to the SHE\nand AHE conductivity, the equation\n\u001bH\nxy=\u001bH\u0000int\nxy +\u001bH\u0000skew\nxy +\u001bH\u0000sj\nxy (3.8)\nde\fnes unambiguously the side-jump contribution as the\ndi\u000berence between the full SHE/AHE conductivity and\nthe skew and intrinsic contributions. In using the term\nside-jump for the remaining contribution, we are appeal-\ning to the historically established taxonomy outlined in\nthe previous section. Establishing this connection rigor-\nously has been the most controversial aspect of the AHE\ntheory and, not surprisingly, some confusion has spilled\nover to the discussion of the SHE.\nThe basic semiclassical argument for a side-jump con-\ntribution can be stated straight-forwardly: when consid-\nering the scattering of a Gaussian wave-packet from a\nspherical impurity with spin-orbit interaction ( HSO=\n(1=2m2c2)(r\u00001@V=@r )SzLz), a wave-packet with inci-\ndent wave-vector kwill undego a displacement transverse\ntokequal to1\n6k~2=m2c2. This type of contribution was\n\frst noticed, but discarded, by Smit (Smit, 1955, 1958)\nand reintroduced by Berger (Berger, 1964) who argued\nthat it was the key contribution to the AHE. Most of the\nearlier developments were based on physical arguments\nof how to incorporate the physics in a semiclassical Boltz-mann formalism, although it cannot be done systemati-\ncally and errors have ensued (Nagaosa et al. , 2010).\nA very common misconception is that the side-jump\ncan be generally computed by taking only into account\nthe spin-orbit coupling interaction of the disorder scat-\ntering potential. This can only be justi\fed in a weak\nspin-orbit coupled system, e.g. n-doped GaAs, where\nindeed this is likely to be the case. This is the considera-\ntion in the Cr\u0013 epieux-Bruno model (Cr\u0013 epieux and Bruno,\n2001) where the spin-orbit coupling is only present in\nthe disorder potential and which has been subsequently\nused by others (Engel et al. , 2005; Maekawa and Taka-\nhashi, 2012). However, when addressing materials with\nstrong spin-orbit coupling it is important to remember\nthat there is always twosources of side-jump scattering:\n1. Extrinsic-side-jump: The contribution arising from\nthe non-spin-orbit coupled part of the wave-packet\nscatting o\u000b the spin-orbit coupled disordered.\n2. Intrinsic-side-jump: The contribution arising from\nthe spin-orbit coupled part of the wave-packet\nformed by the Bloch electrons scattering o\u000b the\nscalar potential alone without spin-orbit coupling.\nBoth can be important and independent of each other\ndepending on the crystalline environment and the type\nof scattering impurity. In heavy-element materials, such\nas Pt and Ta, that have become quite important for pos-\nsible future technological applications, the dominant con-\ntribution is likely to be the second type of contribution.\nIn FMs it has been demonstrated that the second type\nof contribution, termed here intrinsic-side-jump to dis-\ntinguish them clearly, can be very large. Both of these\nside-jump contributions add to the scattering indepen-\ndent mechanisms, i.e., they are independent of \u001c, which\nincorporate both side-jump types and the intrinsic mech-\nanism (Weischenberg et al. , 2011).\nBoth types of side-jump and the intrinsic contributions\nhave quite di\u000berent dependences on more speci\fc system\nparameters, particularly in systems with complex band\nstructures (for a detailed review on these delicate issues\nwe recommend Sinitsyn, 2008). Most of the prior mis-\ntakes surrounding the theory of side-jump can be traced\nback to the physical meaning ascribed to quantities which\nwere gauge dependent, like the Berry's connection that\nis typically identi\fed as the de\fnition of the side-step\nupon scattering. Studies of simple models, e.g. semicon-\nductor conduction bands, also gave results in which the\nside-jump contribution seemed to be the same magnitude\nbut opposite in sign compared to the intrinsic contribu-\ntion (Nozieres and Lewiner, 1973). It is well understood\nnow that these cancellations are unlikely in more com-\nplex models (Sinitsyn et al. , 2007; Weischenberg et al. ,\n2011). The prior cancellations can be traced back to the\nfact that in these very simple band structures the Berry's\ncurvature of the Bloch electrons is a constant indepen-\ndent of momentum. One is reminded in this case of the\nfamous quote by Albert Einstein which states that things\nshould be made as simple as possible but not simpler.14\nIt is only through comparison between di\u000berent fully\nmicroscopic linear response theory calculations, based\non equivalently valid microscopic formalisms such as\nKeldysh (non-equilibrium Green's function) or Kubo for-\nmalisms, and the systematically developed semi-classical\ntheory that the speci\fc contribution due to the side-\njump mechanism can be separately identi\fed with con-\n\fdence. Nonsystematic treatments can lead to miscon-\nceptions that linger for a long time in the community\n(Maekawa and Takahashi, 2012).\nVery recently, there has been strong steps forward in\nthe theory of the AHE in developing full theories with\npredictive power to calculate all these contribution in FM\nmaterials with a complex band structure (Czaja et al. ,\n2013; Freimuth et al. , 2010; Kovalev et al. , 2010; Low-\nitzer et al. , 2010; Weischenberg et al. , 2011). In the the-\nory of the SHE on the other hand, perhaps because of\nthe complexity of the measurements, the fact that spin\ndecays in a non-trivial way, and the lack of practical gen-\neral theories that can bring one from a weak to a strong\nspin-orbit coupled regime, such progress still remains to\nbe undertaken fully.\nB. Phenomenological Drift-Di\u000busion Theory\nDyakonov and Perel, 1971a,b considered the phe-\nnomenological theory of the SHE by coupling the usual\ndrift di\u000busion equation for charge transport to the spin-\ncurrent drift-di\u000busion equations. Hence, the spin-charge\ndrift-di\u000busion equations applicable to electrical transport\nmeasurements can be written from symmetry considera-\ntions as (Dyakonov and Khaetskii, 2008):\njc=e\u0016nE+Drn+e\u000bSH\u0016(E\u0002P) +e\u000bSHD(r\u0002P);\n(3.9)\njs\nij=\u0000~\u0016nEiPj+D@Pj\n@xi\u0000~\u000bSH\u000fijk\u0012\n\u0016nEk+D@n\n@xk\u0013\n;\n(3.10)\nwhere the \frst two terms of Eqs. (3.9) and (3.10) cor-\nrespond to the de\fnition of the uncoupled charge and\nspin-currents. Here Pcorresponds to the spin polariza-\ntion,Dis the electron di\u000busion constant, \u0016is the spin-\nindependent electron mobility, nis the electron density,\nEis the electric \feld, and \u000bSHis the spin Hall angle\nde\fned by the ratio of the spin Hall conductivity to the\ndiagonal charge conductivity.\nIn Eq. (3.9) the third term corresponds to the AHE.\nThe fourth term describes the ISHE if a charged di\u000busive\ncurrent is absent, i.e., in the case of the pure spin-current\nin the system. We distinguish this from the situation in\nwhich a polarized charge di\u000busive current, e.g. generated\nby optical excitation (Bakun, 1984), leads to a charge\ntransverse current which we associate here with a regime\ncloser to the AHE. This distinction is made more clear by\nthe fact that the SHE has a precise de\fnition of a pure\nspin-current being generated by a charge current, and\ntherefore its inverse is associated with a pure spin-currentgenerating a transverse charge current. In Eq. (3.10) the\nthird term represents the SHE from an electric \feld, while\nthe fourth term represent its di\u000busion driven counterpart.\nThe equations are written to \frst order in the spin Hall\nangle.\nThese equations can be directly extended to include\nthermal gradients and junctions (Johnson and Silsbee,\n1987). Recently, there has been an extension of these to\nincorporate thermal SHEs within the emerging \feld of\nspin caloritronics (Bauer et al. , 2010, 2012). The treat-\nment, for the most part, remains phenomenological with\nconnections in particular to the Onsager relations be-\ntween the thermodynamic forces and their corresponding\nentropy \ruxes. Within this emerging sub\feld of spintron-\nics many theoretical challenges remain, not least a better\ntreatment of scattering coherent e\u000bects driven by statis-\ntical forces and the ability of going beyond the simple\nadiabatic frozen phonon approximations.\nC. The Cr\u0013 epieux-Bruno model: extrinsic side-jump and\nskew scattering model\nWe discuss here the Cr\u0013 epieux and Bruno, 2001 model\nthat incorporates spin-orbit coupling only through the\ndisorder potential, i.e., there is no spin-orbit coupling\npresent directly in the Bloch electron bands at the Fermi\nsurface. This model has been applied to weak spin-orbit\ncoupled materials, such as n-GaAs, to explain the ex-\ntrinsic origin of their SHE (Engel et al. , 2005) and has\nalso been applied to weakly spin-orbit coupled metals\n(Maekawa and Takahashi, 2012). The model builds on\nthe in\ruential work of Nozieres and Lewiner (Nozieres\nand Lewiner, 1973), where they studied the AHE in semi-\nconductors with simple band structure and how to ac-\ncount for the e\u000bects of the spin-orbit coupling when pro-\njecting to an e\u000bective two-band model. There are many\nsubtle issues in such treatment already at this simple-\nmodel level, but extrapolating some of its results to gen-\neralities, e.g. speci\fc cancellations, is dangerous since\n\"the side-jump is no longer given by the simple expres-\nsion\" derived in these works, as the authors themselves\nstate (Nozieres and Lewiner, 1973).\nThe model two-band Hamiltonian is given by\nH=~2k2\n2m\u0003+V(r)+\u0015e\u0000so\u001b\u0001(k\u0002rV) =H0+W:(3.11)\nHerem\u0003is the e\u000bective mass of the Bloch electron, V(r)\nis the disorder potential, \u001bare the Pauli metrices, and\n\u0015e\u0000sois the e\u000bective spin-orbit coupling parameter. For\na free electron, \u0015e\u0000so=~2=2m2c2is an extremely small\nparameter, but in a solid state environment it is strongly\nrenormalized by nearby bands. For the e\u000bective two-\nband model of conduction electrons, obtained from the\n8\u00028 Kane description of the semiconductor band struc-\nture,\u0015e\u0000so= (P2=3)[1=E2\ng\u00001=(Eg+ \u0001so)], withEg\nbeing the gap, Pthe s-p dipole matrix element, and \u0001 so\nthe spin-orbit splitting of the valence band. For n-GaAs,15\nk (a)\nk2/c100 (b)\nFIG. 12 Schematic picture of the skew-scattering (a) and the\nextrinsic side-jump (b) mechanisms from a quantum point of\nview (\fcorresponds to spin up and \nto spin down). The\nbold curves represent the anisotropic enhancement of the am-\nplitude of the wave-packet due to spin-orbit coupling. Here\nthe electrons themselves contain no spin-orbit coupling. From\nRef. Cr\u0013 epieux and Bruno, 2001.\nfor example, this value is 5 :3\u0017A2. The total scattering\npotential is W.\nIn this model the velocity operator is modi\fed by the\nspin-orbit coupled term to read\n^v=^p\nm\u0003+\u0015e\u0000so\n~[\u001b\u0002rV]; (3.12)\nand the scattering amplitude due to the disorder poten-\ntial from statejk;sitojk0;s0iis given by\nhk0;s0jWjk;si=~Vkk0(\u000ess0+i\u0015e\u0000so(\u001bs0s\u0002k0)\u0001k);\n(3.13)\nwhere ~Vkk0is the Fourier transform of V. The disor-\nder potential is considered to be short ranged for sim-\nplicity (for ionic scattering see Engel et al. , 2005) such\nthatV(r) =uiP\nj\u000e(r\u0000rj) and ~Vkk0=ui\u000ek;k0. One can\nthen connect this procedure with the Boltzman equation,\nwhich will also yield the spin-di\u000busion equation (Zhang,\n2000). Microscopically the scattering from this disorder\npotential induces a collision integral in the Boltzmann\nformalism of the form\n\u0012@fk;s\n@t\u0013\ncoll=X\nk0;s0[Pks;k0s0fk0s0\u0000Pk0s0;ksfks];(3.14)\nwith the transition scattering probabilities available from\nthe T-matrix which yields a symmetric and antisymmet-\nric contribution:\nPsym\nk0s0;ks=2\u0019\n~ni\nVu2\ni\u0000\n\u000es;s0+\u00152\ne\u0000soj(k0\u0002k)\u0001\u001bs;s0\u0001\n\u0002\u000e(\u000fk0\u0000\u000fk); (3.15)\nPant\nk0s0;ks=\u0000(2\u0019)2\n~\u0015e\u0000soni\nVu3\niN(0)\u000es;s0(k0\u0002k)\u0001\u001bs;s0)\n\u0002\u000e(\u000fk0\u0000\u000fk): (3.16)Hereniis the density of impurities and N(0) is the den-\nsity of states at the Fermi level. Within the framework\nof the semiclassical Bolzmann equation one then writes\n(Maekawa and Takahashi, 2012; Zhang, 2000):\nvk\u0001rfks+eE\n~\u0001rkfks=\u0000\u000efks\n\u001ctr\u0000f0\nks\u0000f0\nk0s\n\u001csf(\u0012);(3.17)\nwhere\n\u001c\u00001\ntr=X\nk0;s0Psym\nks;k0s0=1\n\u001c0\ntr(1 + 2k4\nF\u00152\ne\u0000so=3);(3.18)\nand\n\u001c\u00001\nsf=X\nk0Psym\nk1;k0\u00001=k4\nF\u00152\ne\u0000so\n3\u001c0\ntr(1 + cos2(\u0012)):(3.19)\nHere\u001ctris the transport lifetime, \u001c0\ntris the transport life-\ntime when neglecting the spin-orbit coupling part of the\ndisorder potential, and \u001csfis the spin-\rip time. Further\nexpanding the equilibrium and non-equilibrium distribu-\ntion function yields the spin di\u000busion equation\nr2(\u00161\u0000\u0016\u00001) =1\n\u00152\nsd(\u00161\u0000\u0016\u00001); (3.20)\nwith\u00152\nsd=D\u001csf=2,D= (1=3)\u001ctrv2\nF, and\u0016srepresent-\ning the spin-dependent chemical potentials ( s=\u00061 is\nthe spin index). Averaging over the scattering angle one\nobtains that the ratio of spin-\rip time and transport time\nfor this particular model is approximately\n\u001ctr\n\u001csf\u00191\n2k4\nF\u00152\ne\u0000so: (3.21)\nThis is one of the key aspects that has made this model\nappealing. The model provides a means to obtain its ef-\nfective spin-orbit parameter \u0015e\u0000soby measuring the spin-\ndi\u000busion length, independently of the spin Hall angle. At\nthis point, it should to be emphasized that the model is\napplicable to the weak spin-orbit coupling regime, i.e.,\nwhen\u001ctr=\u001csf\u001c1.\nFrom either a microscopic or Boltzmann like analysis\nthe result for this model for the side-jump contribution to\nthe spin Hall angle is (Cr\u0013 epieux and Bruno, 2001; Engel\net al. , 2005):\n\u000bsj\nSH\u0011\u001bH\u0000sj\nxy\n\u001bxx=\u00002\u0015e\u0000som\u0003\n\u001ctr=\u00002k2\nF\u0015e\u0000so\nkFl:(3.22)\nHerel=\u001ctrkF=m\u0003is the mean free path. The skew\nscatting contribution is given by\n\u000bsk\nSH=4\u0019\n3k2\nF\u0015e\u0000soN(0)ui: (3.23)\nIn the SHE experiments in n-GaAs, the spin Hall\nangle is dominated by the skew scattering contribu-\ntion vs. the side-jump contribution by a factor of 2,\n\u001bH\u0000skew\nxy=\u001bH\u0000sj\nxy\u0018\u00001:7=0:8 (Engel et al. , 2005).16\n\u0015sdkFl k2\nF\u0015e\u0000so\u000bSH\f\f\f\f\u000bsj\nSH\n\u000bSH\f\f\f\fRef.\n(nm) (%)\nAl (4.2K) 455(15) 73 0.0079 0 :032\u00060:006 0.67 1 NL\nAl (4.2K) 705\u000630 118 0.0083 0 :016\u00060:004 0.88 1NL\nAu (295K) 86\u000610 371 0.3 11.3 0.014 2 NL\nAu (295K) 35\u00063\u0003253 0.52 0 :35\u00060:03 1.17 3 SP\nCuIr (10K) 5\u000030 2 :1\u00060:6 4 NL\nMo (10K) 10 36.8 0.32 -0.20 8.7 5 NL\nMo (10K) 10 8.11 0.07 -0.075 23 5 NL\nMo (10K) 8 :6\u00061:3 34.1 0.34 \u0000(0:8\u00060:18) 2.5 6 NL\nMo (295K) 35\u00063\u000356.7 0.14\u0000(0:05\u00060:01) 9.9 3 SP\nNb (10K) 5 :9\u00060:3 11.3 0.14 \u0000(0:87\u00060:20) 2.9 6 NL\nPd (295K) 9\u000324.0 0.23 1 :0 1.9 7 SP\nPd (10K) 13\u00062 26.8 0.18 1 :2\u00060:4 1.1 6 NL\nPd (295K) 15\u00064\u000348.6 0.28 0 :64\u00060:10 1.8 3 SP\nPt (295K) 77.9 0.74 0.37 5.1 8 NL\nPt (5K) 14 97.3 0.61 0.44 2.9 9 NL\nPt (295K) 10 67.6 0.58 0.9 1.9 9 NL\nPt (10K) 11\u00062 98.5 0.77 2 :1\u00060:5 0.74 5 NL\nPt (295K) 7\u000377.8 0.97 8 :0 0.31 10 SP\nPt (295K) 3\u00006 60.8 0.88-1.75 7 :6+8:4\n\u00002:00.57 11 SP\nPt (295K) 10\u00062\u000329.2 0.25 1 :3\u00060:2 1.31 3 SP\nTa (10K) 2 :7\u00060:4 3.90 0.17 \u0000(0:37\u00060:11) 24 6 NL\nTABLE I Experimental spin Hall angles and e\u000bective spin-\norbit coupling parameters, k2\nF\u0015e\u0000so. The values marked by\n(\u0003) are not measured but taken from the literature. The\nFermi momenta are taken to be kF= 1:75\u0002108cm\u00001(Al),\n1:21\u0002108cm\u00001(Au), 1:18\u0002108cm\u00001(Nb), and 1 :0\u0002108\ncm\u00001(Mo,Pd,Ta,Pt). Here kFl= (3\u0019=2)\u001b=kF(h=e2). Ref-\nerences: 1) (Valenzuela and Tinkham, 2006, 2007), 2) (Seki\net al. , 2008), 3) (Mosendz et al. , 2010b), 4) (Niimi et al. ,\n2011), 5)(Morota et al. , 2009), 6) citeMorota2011, 7) (Ando\net al. , 2010a), 8) (Kimura et al. , 2007), 9) (Vila et al. , 2007),\n10) (Ando and Saitoh, 2010), 11) (Liu et al. , 2011)\nWhen this simpli\fed model is used for metals it yields\na mixture of results and inconsistencies. This can be\nseen in Table I where we show for a series of metals the\nexperimental SHE angles and the independently exper-\nimentally inferred e\u000bective spin-orbit coupling parame-\ntersk2\nF\u0015e\u0000so. Skew scattering is not possible to estimate\nfrom the model expression (3.23) without knowing the\nspeci\fc value for ui. However, we know from the AHE\nthat skew scattering has only been seen to dominate for\nextremely conductive metals so it is neglected in the dis-\ncussion of Table I. When comparing the side-jump con-\ntribution estimates to the measured values of \u000bSHthe\nresults vary extensively. In some cases, like Ta, the the-\noretical side-jump contribution is 24 \u0002larger than the\nmeasured\u000bSH. The failure of Eq. (3.22), derived assum-\ning the weak spin-orbit coupling regime, is not surprising\nhere since Ta is a 5d heavy transition metal. In others,\nlike in Pt, Eq. (3.22) gives a value that is smaller than\nthe measured \u000bSH, in some cases approaching the ex-\nperimental value rather closely. However, even in this\ncase ascribing the measured spin Hall angle to the side-\njump contribution of Eq. (3.22) is questionable because\nthe independently inferred parameter k2\nF\u0015e\u0000sois close to\n1. This is inconsistent with \u001ctr=\u001csf\u001c1, i.e., with the\nweak spin-orbit coupling regime assumed in the model.\nIn the heavy element materials the intrinsic SHE esti-\nmates have had much more success. Hence, the simple\nmodel expression Eq. (3.22), although illustrative and ap-pealing, should perhaps only be consider as such, not as\na quantitative predictive theory of the SHE.\nD. Theory of inverse spin Hall e\u000bect induced by\nspin-pumping\nWhen measuring the ISHE it is necessary to generate a\nspin-current that \rows into the NM whose spin Hall an-\ngle is being measured. In the non-local transport schemes\nthis is achieved indirectly by spin-di\u000busion into the NM.\nThis can restrict both the types of materials and reliabil-\nity of the measurements since there are several interfaces\ninvolved and the distances of the devices have to be kept\nshorter than the spin-di\u000busion length in the area where\nthe spin-current is di\u000busing.\nA key alternative to generating spin-currents is to ex-\nploit SP in a FM/NM bilayer system. This phenomenon\nwas observed experimentally in early 2000's (Mizukami\net al. , 2001, 2002; Urban et al. , 2001). The experiments\nshowed an enhanced Gilbert damping in FMR associated\nwith the loss of angular momentum by a spin-current\n\rowing from the FM to the NM which served as a spin\nsink.\nThe associated SP theory based on the scatter-\ning formalism was developed by Tserkovnyak et al.\n(Tserkovnyak et al. , 2002a,b, 2005). It extends the theory\nof adiabatic quantum pumping (Brouwer, 1998; B uttiker\net al. , 1993) by incorporating the spin degrees of freedom.\nIt can be shown that the precessing magnetization in FM\ngenerates a time-dependent spin-current at the FM/NM\ninterface that \rows into the NM given by,\njs;pump~ \u001b(t) =~\n4\u0019Ar^~ m\u0002d^~ m\ndt: (3.24)\nHere ^~ m(t) is the unit vector of the magnetization, ~ \u001bis\nthe unit vector of the spin-current polarization, js;pump\nits magnitude, and Aris de\fned as the SP conductance\nof the particular sample. The spin-current generated at\nthe interface which propagates into the NM decays on\na length scale connected to the e\u000bective spin di\u000busion\nlength\u0015sdof NM. A sketch of the physics is shown in\nFig. 13. Note that in systems with strong spin-orbit cou-\npling this length scale can be di\u000ecult to de\fne since it\ncan be as short as several atomic layers. Also, proximity\ne\u000bects as well as roughness at the interface with the NM\ncan blur the sharpness of such an interface.\nThe scattering-matrix theory introduces the concept\nof a complex spin-mixing conductance at the interface\nbased on spin-conserving channels and no spin-loss at\nthe interface. Theoretical ab-initio calculations, and the\nphase randomization at the scattering interface, seem to\nindicate that only the real part of the mixing conduc-\ntance dominates the physics and that in the di\u000busive\nregime this will be approximately the Sharvin conduc-\ntance given by the number of conducting channels. In17\njs,pumpjs,back-flowxyzΘHM\nFIG. 13 Schematic of SP comprising of a spin-pump current\n\rowing from the FM to the NM and a back \row current that\ndepends on the thickness of the NM.\nthis approximation (Tserkovnyak et al. , 2002b):\nAr\u0019Re[g\"#] =k2\nF\n4\u0019\u00191\n4\u0019(3\u00192n)2=3; (3.25)\nwherekFandnare the Fermi wavevector and electron\ndensity in the NM, respectively. Direct ab-initio calcula-\ntions of the mixing conductance (Carva and Turek, 2007;\nXia et al. , 2002; Zwierzycki et al. , 2005) have veri\fed\nthat for a FM/NM interface with moderate spin-orbit\ncoupling the spin-mixing conductance is of this order of\nmagnitude.\nAs the magnetization rotates, the spin-current injected\nfrom the FM to the NM is time dependent but the AC\nspin-current when averaged over time has a non-zero DC\ncomponent, hence the notion of pumping, which is given\nby:\njs;dc=~!\n4\u0019Arsin2\u0002: (3.26)\nHere!is the driving radio frequency (RF) and \u0002 is the\ncone angle of precession (see Fig. 13). Under the as-\nsumption of the NM being a perfect spin-sink the SP\nconductance will be the spin-mixing conductance. How-\never, in the systems where the NM has a \fnite thickness\nof the order of the spin-di\u000busion length, the induced spin\naccumulation in the NM due to the pumped spin-current\nfrom the FM will create a spin accumulation which in\nturn will generate a spin-current back \row. The spin ac-\ncumulation in the NM within the spin-di\u000busive regime,\n\u0016s\u0011\u0016\"\u0000\u0016#, is governed by the equation:\nd\u0016s\ndt=D@2\nz\u0016s\u0000\u0016s\n\u001csf; (3.27)\nwith the boundary conditions:\nz= 0 :@z\u0016s=\u00004e2\u001a\n~js;0\nz=tNM :@z\u0016s= 0; (3.28)\nwhere\u001ais the NM resistance and tNMis the thickness of\nthe NM. In the NM, the spin-current decays away fromthe FM/NM interface due to the combination of spin\ndi\u000busion and spin \rip scattering. The z-dependent spin-\ncurrent density js(z) in the NM (Azevedo et al. , 2011;\nMosendz et al. , 2010a) with the above boundary condi-\ntions reads:\njs(z) =\u0000~\n4e2\u001a@z\u0016s(z) =js;0sinh\u0010\ntNM\u0000z\n\u0015sd\u0011\nsinh\u0010\ntNM\n\u0015sd\u0011:(3.29)\nThe back-\row current density js;back at the interface\ncan be taken into account by the relation js;back(0)\u0019\n2Re[g\"#]\u0016s(0). This allows the following expression to be\nsolved for the total spin-current crossing the interface:\njs;0~ \u001b(t) = (js;pump\u0000js;back)~ \u001b(t) =~\n4\u0019~Ar^~ m\u0002d^~ m\ndt:\n(3.30)\nThe result is that the e\u000bective spin-mixing conduc-\ntance gets reduced due to a back \row factor given by\n(Tserkovnyak et al. , 2002b):\n\f\u0011\u001csf\u000esd=h\ntanh (tNM=\u0015sd); (3.31)\nwhere\u000esdis an e\u000bective spin-\rip scattering energy ob-\ntained by the inverse of the product of the volume de-\n\fned by the scattering cross section and spin-di\u000busion\nlength and the density of states. This then gives the\nresult (Tserkovnyak et al. , 2002b):\n~Ar\u0019g\"# 1\n1 +\fg\"#(3.32)\n~Ar\u0019g\"# 1\n1 +1\n4p\u000f\n3tanh\u0010tNM\n\u0015sd\u0011\u0019g\"#\ne\u000b; (3.33)\nwhereg\"#is now the real part of the spin mixing conduc-\ntance. The last approximation assumes a weak spin-orbit\ncoupling limit. More explicitly it assumes \u000f=\u001ctr=\u001csf\u001c\n1. Hence, the larger \u000f, the more e\u000ecient the injected\nspin-current is relaxed in the NM and the smaller is the\namount of back\row (Tserkovnyak et al. , 2002b). How-\never, one has to be aware of the limitations of the approx-\nimation since in the strongly spin-orbit coupled systems\nmany of these assumptions fail.\nThe detection of the net spin-current \rowing into the\nNM can be done electrically via the ISHE as was demon-\nstrated by Saitoh et al. , 2006. By measuring the Hall\nvoltage induced by the spin-current one can infer the spin\nHall angle of the material:\n~jc=\u000bSH2e\n~~js\u0002~ \u001b(t): (3.34)\nHere, the vector of the spin-current density ~jspoints\nperpendicular to the NM/FM interface. Note that the\nvector of the spin-current polarization ~ \u001b(t) is a time\nvarying quantity. In the geometry sketched in Fig. 13\nthe propagation direction of the spin-current is along z18\nand its polarization is along x-axis. For the detection of\na DC voltage along the y-direction one has to consider\nthe charge current jc^y=\u000bSH2e\n~(1=tNM)RtNM\n0js(z)^z\u0002^x\nwith magnitude (Azevedo et al. , 2011):\njc=\u000bSH2e\n~js,0\u0015sd\ntNMtanh\u0012tNM\n2\u0015sd\u0013\n: (3.35)\nTo convert this charge current density into the actual\nmeasured voltage one has to consider the details of the\nmeasurement geometry and the resistivity of the bilayer\nwhich will be discussed in Sec. IV.D. In addition, as de-\nscribed as well in Sec. IV.D, the AC-component can be\ndirectly measured. An extension of the above theory to\nincorporate the AC-component has been done by Jiao\nand Bauer, 2013, with the result that back-\row is im-\nportant to distinguish between the measured voltages for\nboth the AC and DC con\fgurations.\nWe conclude this section with a discussion regarding\nthe assumptions of the SP theory. In the above deriva-\ntions, whenever \u000f\u00181, the approximations do not hold\nanymore since for the given boundary conditions and\nfor the use of the spin di\u000busion equation (and the spin-\nresolved spin-mixing conductance) one assumes \u000f\u001c1\n(Tserkovnyak et al. , 2002b). However, for \u000f >0:1 most\nof the spin scattering occurs right at the interface and\nconsequently the \flms are almost perfect sinks. Hence,\nin this case there is no dependence on the thickness of\nthe \flm. Since in such \flms the interface plays the\nprominent role and scattering occurs at or near the inter-\nface, many issues regarding proximity e\u000bects, the induced\nspin-accumulation, and the spin Hall angle inferred from\nthe measurements should be taken as phenomenological\nparameters rather than direct connections to a quantita-\ntive value of the bulk spin Hall angles.\nE. Kubo formalism\nIn this section we review the Kubo formalism employed\nin the calculations of the intrinsic SHE which incorpo-\nrate the e\u000bects of disorder at its simplest level through\n\fnite quasiparticle lifetime. It provides a fully quantum\nmechanical formally exact expression for the spin and\nanomalous conductivity in linear response theory (Ma-\nhan, 2000). Here we emphasize the key issues in study-\ning the SHE within this formalism and how it relates\nto the semiclassical formalism described in the previous\nsections.\nFor the purpose of studying the SHE and AHE it is\nbest to reformulate the current-current Kubo formula\nfor the conductivity in the form of the Bastin formula\n(see appendix A in Cr\u0013 epieux and Bruno, 2001) which\ncan be manipulated into the more familiar form for the\nconductivity of the Kubo-Streda formula for the zero-\ntemperature Hall conductivity, \u001bH\nxy=\u001bI(a)\nxy+\u001bI(b)\nxy+\u001bII\nxy,where\n\u001bI(a)\nxy=e2\n2\u0019VTrhf^sz;^vgxGR(\u000fF)^vyGA(\u000fF)ic;(3.36)\n\u001bI(b)\nxy=\u0000e2\n4\u0019VTrhf^sz;^vgxGR(\u000fF)^vyGR(\u000fF) +c:c:ic;\n(3.37)\n\u001bII\nxy=e2\n4\u0019VZ+1\n\u00001d\u000ff(\u000f)Tr[f^sz;^vgxGR(\u000f)vyGR(\u000f)\nd\u000f\n\u0000f^sz;^vgxGR(\u000f)\nd\u000fvyGR(\u000f) +c:c:]: (3.38)\nHere the subscript cindicates a disorder con\fguration av-\nerage. The last contribution, \u001bII\nxy, was originally derived\nby Streda in the context of the QHE (Streda, 1982). In\nthese equations GR=A(\u000fF) = (\u000fF\u0000H\u0006i\u000e)\u00001are the re-\ntarded and advanced Green's functions evaluated at the\nFermi energy of the total Hamiltonian.\nLooking more closely at \u001bII\nxywe notice that every term\ndepends on products of retarded Green's functions only,\nor on products of advanced Green's functions only. It\ncan be shown that only the disorder free part of \u001bII\nxyis\nimportant in the weak disorder limit, i.e., this contribu-\ntion is zeroth order in the parameter 1 =kFl. The only\ne\u000bect of disorder on this contribution (for metals) is to\nbroaden the Green's functions (see below) through the\nintroduction of a \fnite lifetime (Sinitsyn et al. , 2007).\nBy a similar argument, \u001bIb\nxyis of order 1 =kFland can\nbe neglected in the weak scattering limit (Mahan, 2000).\nThus, important disorder e\u000bects beyond simple quasipar-\nticle lifetime broadening are contained only in \u001bIa\nxy. For\nthese reasons, it is standard within the Kubo formalism\nto neglect\u001bIb\nxyand evaluate the \u001bII\nxycontribution with a\nsimple lifetime broadening approximation to the Green's\nfunction.\nIn this formalism the e\u000bect of disorder on the disorder-\ncon\fguration averaged Green's function is captured by\nthe use of the T-matrix, de\fned by the integral equation\nT=W+WG 0T, whereW=P\niV0\u000e(r\u0000ri) is a delta-\nscatterers potential and G0are the Green's function of\nthe pure lattice. From this one obtains\n\u0016G=G0+G0TG0=G0+G0\u0006\u0016G: (3.39)\nUpon disorder averaging we obtain\n\u0006 =hWic+hWG 0Wic+hWG 0WG 0Wic+:::(3.40)\nTo linear order in the impurity concentration, ni, this\ntranslates to\n\u0006(z;k) =niVk;k+ni\nVX\nkVk;k0G0(k0;z)Vk0;k+\u0001\u0001\u0001;\n(3.41)\nwithVk;k0=V(k\u0000k0) being the Fourier transform of\nthe single impurity potential, which in the case of delta\nscatterers is simply V0. Note that \u0016GandG0are diagonal19\nin momentum but, due to the presence of spin-orbit cou-\npling, non-diagonal in spin-index in the Pauli spin-basis.\nOne e\u000bect of disorder on the spin and anomalous Hall\nconductivity is taken into account by inserting the dis-\norder averaged Green's function, \u0016GR=A, directly into the\nexpressions (3.36) and (3.38) for \u001bIa\nxyand\u001bII\nxy, respec-\ntively. This step captures the intrinsic contribution to\nthe SHE and AHE and the e\u000bect of disorder on it, which\nis generally weak in metallic systems.\nThe so-called ladder diagram vertex corrections con-\ntribute to the AHE and SHE at the same order in\n1=kFlas the intrinsic contribution. It is useful to de-\n\fne a ladder-diagram corrected velocity vertex ~ v\u000b(\u000fF)\u0011\nv\u000b+\u000e~v\u000b(\u000fF), where\n\u000e~v\u000b(\u000fF) =niV2\n0\nVX\nk\u0016GR(\u000fF)(v\u000b+\u000e~v\u000b(\u000fF))\u0016GA(\u000fF):\n(3.42)\nNote again that ~ v\u000b(\u000fF) andv\u000b=@^H0=@~k\u000bare matrices\nin the spin-orbit coupled band basis. The skew scattering\ncontributions are obtained by evaluating, without doing\nan in\fnite partial sum as in the case of the ladder dia-\ngrams, third order processes in the disorder scattering.\nAs may seem obvious from the above machinery, calcu-\nlating the intrinsic contribution is not very di\u000ecult, while\ncalculating the full e\u000bects of the disorder in a systematic\nway (beyond calculating a few diagrams) is challenging\nfor any disorder model beyond the simple delta-scattering\nmodel.\nAn important recent development has taken place\nwithin the theory of the AHE which we hope will have\na direct analogy to the spin Hall conductivity. Assum-\ning uncorrelated Gaussian noise disorder, i.e., ignoring\nany skew scattering contribution, all the scattering inde-\npendent contributions - side-jump and intrinsic - can be\nformulated in terms of the band structure of the crystal\nalone (Kovalev et al. , 2010; Weischenberg et al. , 2011).\nFor a short-range scattering disorder model, e.g. scalar\ndelta-correlated Gaussian disorder, the starting point for\nthis theory of the scattering-independent side-jump is\nthe retarded Green's function in equilibrium and the\nHamiltionian Hof a general multiband noninteracting\nsystem. The \frst step is to expand the self-energy of\nthe system \u0006 eqin powers of potential V(r), which de-\nscribes scattering o\u000b impurities. Inserting the expression\nfor the self-energy within these simple disorder models\ninto the appropriate equations for the current densities\nderived following the Kubo-St\u0014 reda formalism mentioned\nabove, rotating into eigenstate representation and keep-\ning only the leading order terms in the limit of vanishing\ndisorder parameter V0, the scattering-independent part\nof the AHE conductivity may be written as \u001bH\u0000(0)\nxy =\n\u001bH\u0000int\nxy +\u001bH\u0000sj\nxy , where\n\u001bH\u0000int\nij =2e2\n~Zd3k\n(2\u0019)3ImX\nn6=m(fn\u0000fm)vnm;i(k)vmn;j(k)\n(!n\u0000!m)2\n(3.43)TABLE II AHE conductivities for bcc Fe and hcp Co in\nS/cm for selected high-symmetry orientations of the magneti-\nzation.\u001bH\u0000int\nxy ,\u001bH\u0000sj\nxy and\u001bH\u0000int+sj\nxy stand for intrinsic con-\ntribution, side-jump contribution and their sum, respectively.\nThe experimental values are for the scattering-independent\nconductivity.\nFe [001] [111] [110] Co [001] [100]\n\u001bH\u0000int\nxy 767 842 810 \u001bH\u0000int\nxy 477 100\n\u001bH\u0000sj\nxy 111 178 141 \u001bH\u0000sj\nxy 217\u000045\n\u001bH\u0000int+sj\nxy 878 1020 951 \u001bH\u0000int+sj\nxy 694 55\nExp. 1032 Exp. 813 150\nis the intrinsic contribution. In this expression the band\nindicesnandmrun from 1 to N,vnm;iare the matrix el-\nements of the velocity operator ^ vi=@~ki^Hand!n(k) =\n\"n(k)=~:The scattering-independent side-jump contribu-\ntion to AHE conductivity for inversion-symmetric sys-\ntems reads:\n\u001bH\u0000sj\nij =e2\n~NX\nn=1Zd3k\n(2\u0019)3Re Tr\u001a\n\u000e(\"F\u0000\"n)\rc\n[\rc]nn\u0002\n\u0002\u0014\nSnAki(1\u0000Sn)@\"n\n@kj\u0000S\u0011Akj(1\u0000Sn)@\"n\n@ki\u0015\u001b\n:\n(3.44)\nHere the imaginary part of the self-energy Im\u0006 eq=\n\u0000~V0\ris taken to be in the eigenstate representation,\ni.e.\rc=Uy\rU, with\n\r=1\n2NX\nn=1Zd3k\n(2\u0019)2USnUy\u000e(!F\u0000!n); (3.45)\nUas the unitary matrix that diagonalizes the Hamilto-\nnian at point k,\n[UyH(k)U]nm=\"n(k)\u000enm; (3.46)\nSnis aN\u0002Nmatrix that is diagonal in the band in-\ndices, [Sn]ij=\u000eij\u000ein, and the so-called Berry connec-\ntion matrix is given by Ak=iUy@kU. Not included in\nEq. (3.44) are the vertex corrections, which vanish for\nan inversion-symmetric system in the Gaussian disorder\nmodel. Because this side-jump contribution in the short-\nrange disorder model is solely determined by the elec-\ntronic structure of the pristine crystal, it is thus directly\naccessible by ab initio methods.\nTable II shows a comparison of the improvement in\npredictive power of the AHE theory when including the\nside jump term. Fig. 14 shows the non-trivial angu-\nlar dependence, within the Fermi surface, of the side\njump and intrinsic contributions. This is reminiscent\nof the spin-hot spots observed previously in the the-\nory of spin-dephasing, and emphasizes the importance\nof anisotropies induced by the band structure itself.20\n(a)Ni [001] \u001bsj\n(b)Ni [110] \u001bsj\n(c)Ni [001] \u001bint\n(d)Fe [001] \u001bsj\nFIG. 14 (Color online) Angular distribution of the a) Side-\njump contribution for Ni [001], b) Side-jump contribution for\nNi [110], c) Intrinsic contribution for Ni [001], and d) Side-\njump contributionfor Fe [001] on a sphere in the Brillouin\nzone. The color code of each surface point corresponds to the\nsum of all contributions along the path from the origin to the\nparticular surface point. From Ref. Weischenberg et al. , 2011.\nIV. EXPERIMENTAL STUDIES OF SPIN HALL EFFECT\nSeveral experimental schemes to detect the SHE were\noutlined already by Dyakonov and Perel, 1971b in their\nseminal theory work. They proposed to use paramag-\nnetic resonance for detecting the edge spin polarization,\nto measure the nuclear magnetization resulting from the\nOverhauser e\u000bect, to exploit the gyrotropy, i.e., the dif-\nference in the propagation of electro-magnetic waves with\nopposite helicities through the spin polarized edges, or,\nin semiconductors, to detect circular polarization of the\nluminescence excited by an unpolarised light.\nVariants of the two latter schemes, namely the Kerr\nmagneto-optical microscopy and circularly polarized elec-\ntroluminescence from the sample edges, were indeed em-\nployed in the pioneering SHE experiments (Kato et al. ,\n2004a; Wunderlich et al. , 2004, 2005). These were,\nhowever, performed more than thirty years after the\nDyakonov and Perel, 1971b original proposal. Within\nthese three decades, the interest in the phenomenon was\nscarce. The experimental SHE research only picked up\nmomentum after the theoretical work by Hirsch, 1999\nwho rediscovered the phenomenology of the extrinsic\nSHE, and after the prediction of the intrinsic SHE (Mu-\nrakami et al. , 2003; Sinova et al. , 2004).\nThe renewed theoretical interest occured in the midstof an extraordinary growth of the nascent \feld of spin-\ntronics (Zutic et al. , 2004), which had already found im-\nportant applications in the hard disk drive industry and\npromised revolutionary concepts for memory and logic\ndevices. In this setting, the theoretical SHE proposals\nnot only ignited an extensive theoretical debate for their\ninherent fundamental interest but also attracted signif-\nicant attention due to the potential of spin Hall phe-\nnomena as new spin injection and detection tools. The\nproposals started to materialize shortly thereafter with\nthe observations of the SHE in n-doped semiconductors\n(Kato et al. , 2004a) and in the 2DHG (Wunderlich et al. ,\n2004, 2005), and of the ISHE in metallic systems (Saitoh\net al. , 2006; Valenzuela and Tinkham, 2006, 2007).\nIn this section we review the experimental studies of\nthe spin Hall phenomena. In Sec. IV.A we summarize\nAHE experiments in non-ferromagnetic materials that\nwere performed within the three decades separating the\n\frst theoretical proposal and the experimental observa-\ntions of the SHE. The rest of the section is devoted to\nmodern experiments divided according to the techniques\nused to generate, detect, and manipulate the SHE and\nISHE in experimental samples (Sec. IV.B-IV.D). The\noverall understanding of the experiments is still incom-\nplete regarding some materials and structures, in par-\nticular when trying to quantify the magnitude of the\nSHE. Therefore, in those cases, we attempt to provide an\noverview of the current status of the \feld while stressing\nthe strengths and weaknesses of the di\u000berent techniques\nand methods employed.\nApart from the basic research interest in this relativis-\ntic quantum-mechanical phenomenon, Sec. IV.D provides\nan illustration of the application potential of the SHE in\nspintronic devices. This prompted detailed studies of the\nSHE e\u000eciency for the charge-spin conversion in a variety\nof materials. Measurements of the corresponding spin-\nHall angles are summarized in the Sec. IV.E.\nA. Early experiments of anomalous Hall e\u000bect in\nparamagnets\nChazalviel and Solomon, 1972 reported a pioneering\nwork on a spin dependent Hall e\u000bect in non-magnetic\nsemiconductors. They detected the AHE in InSb and n-\ndoped Ge at low temperatures ( <25 K), where the spin\npolarization was created by the application of a mag-\nnetic \feld and the spin-dependent Hall e\u000bect was sepa-\nrated from the larger ordinary HE by magnetic resonance\nof the conduction electrons (Chazalviel, 1975; Chazalviel\nand Solomon, 1972). The magnitude of the measured\nanomalous Hall angles was of the order of 10\u00004for InSb,\nand of 10\u00005for Ge, while its sign was observed to change\ndepending on the degree of carrier compensation (InSb)\nand temperature (Ge). The change in sign was associ-\nated with competing contributions from the side-jump\nand skew scattering mechanisms. The former was ex-\npected to be favored in low mobility samples, which was21\ncon\frmed in the experiment.\nIn early 1980's, Fert and collaborators studied diluted\nmagnetic alloys based on non-magnetic hosts, such as\nAu and Cu, and magnetic impurities such as Mn, Fe, or\nCr (Fert et al. , 1981). They found that CuMn showed\nnegligible skew scattering e\u000bects, but that the exchange\nscattering by polarized Mn impurities created a spin po-\nlarized current. They also noted that the addition of\nnon-magnetic impurities to CuMn gave rise to skew scat-\ntering of the polarized current by the unpolarised impuri-\nties. By analyzing variations of the Hall coe\u000ecient, they\nwere able to extract the Hall angle for the non-magnetic\nimpurities. They found that they varied from \u00001:4\u000210\u00002\nfor Lu to\u00002:6\u000210\u00002for Ir.\nIn another type of AHE measurements, a circularly\npolarized beam at the normal incidence to the surface of\na bulk semiconductor was used to excite spin-polarized\nphoto-electrons (Bakun, 1984; Miah, 2007). These elec-\ntrons di\u000bused in the vertical direction from the surface\nand after aligning their spins along an axis parallel to the\nsurface by an applied magnetic \feld (via Hanle preces-\nsion), an electrical voltage was detected in the transverse\nin-plane direction (Bakun, 1984). Alternatively, the ver-\ntically spin-polarized electrons can be accelerated in the\nin-plane direction by an applied electrical bias yielding\nalso a transverse in-plane voltage (Miah, 2007). Since\nin these experiments the source spin-current is accompa-\nnied by a di\u000busive or drift charge current the geometry\ncorresponds to the AHE.\nB. Optical tools in spin Hall experiments\n1. Optical detection of the spin Hall e\u000bect\nFollowing Dyakonov and Perel, 1971b prediction of the\nextrinsic SHE, the proposed concepts by Hirsch, 1999\nand Zhang, 2000 for its experimental detection relied on\nelectrical measurements. The concepts considered SHE\nchannels in NMs, in the latter case attached to a FM\ndetection electrode.\nHowever, the experimental discovery of the SHE was\nprompted also by theory works which approached the\nSHE origin and detection from a di\u000berent angle. In-\nspired by studies of the intrinsic nature of the closely\nrelated AHE in FMs, (Jungwirth et al. , 2002; Onoda and\nNagaosa, 2002), Murakami et al. , 2003 and Sinova et al. ,\n2004 predicted that a spin-dependent transverse de\rec-\ntion of electrons in non-magnetic systems can originate\ndirectly from the relativistic band structure of the con-\nductor without involving the Mott scattering. Unlike\nHirsch, 1999 and Zhang, 2000 who considered the ex-\ntrinsic, scattering induced SHE and electrical detection\nschemes designed for metals, the intrinsic SHE proposals\nfocused on semiconductors and suggested to utilize the\noptical activity of these materials for detecting the SHE.\nAs in the original work by Dyakonov and Perel, 1971b,\ncircularly polarized electro-luminescence was suggestedin Ref. (Murakami et al. , 2003), while spatially resolved\nmagneto-optical Faraday or Kerr e\u000bects were discussed\nin Refs. (Murakami et al. , 2003; Sinova et al. , 2004).\nThese methods were indeed used in the \frst measure-\nments of the phenomenon. Kato et al. , 2004b employed a\nmagneto-optical Kerr microscope to scan the spin polar-\nization across the channel while Wunderlich et al. , 2004,\n2005 used co-planar p\u0000ndiodes to detect circularly po-\nlarized electro-luminescence at opposite edges of the spin\nHall channel. Wunderlich et al. , 2004, 2005 ascribed the\nobserved signal to the intrinsic SHE while Kato et al. ,\n2004b to the extrinsic SHE.\nKato et al. , 2004b performed the experiments in n-\nGaAs and n-In0:07Ga0:93As \flms grown by molecular\nbeam epitaxy on (001) semi-insulating GaAs substrates.\nThe \flms were doped with Si with n= 3\u00021016cm\u00003in\norder to obtain long spin relaxation lifetimes of \u001cs\u001810\nns, which result in spin di\u000busion lengths \u0015sd=pD\u001cs\u0018\n10\u0016m. The unstrained GaAs sample consisted of 2 \u0016m\nofn-GaAs grown on 2 \u0016m of undoped Al 0:4Ga0:6As,\nwhereas the strained InGaAs sample had 0.5 \u0016m ofn-\nIn0:07Ga0:93As and 0.1\u0016m of undoped GaAs. Static Kerr\nrotation measurements were performed at 30 K with a\npulsed Ti:sapphire laser tuned to the absorption edge of\nthe semiconductor with normal incidence to the sample.\nIn this technique, the laser beam is linearly polarized and\nthe polarization axis of the re\rected beam is determined.\nThe rotation angle is proportional to the net magnetiza-\ntion along the beam direction.\nFigure 15(a) shows a schematic of the experimental ge-\nometry. The epilayers were patterned into 300 \u000277\u0016m2\n(GaAs) and n-InGaAs 300\u000233\u0016m2(InGaAs) channels.\nAn electric \feld was applied along the channel while a\nmagnetic \feld Bcould be applied perpendicular to it in\nthe \flm plane. Figure 15(b) shows a two dimensional\nscan of the GaAs sample, which demonstrates the exis-\ntence of spin accumulation close to the edges. The am-\nplitude of the measured edge spin polarizations reaches\n\u00180:1%. The polarization has opposite sign at the two\nedges and decreases rapidly with the distance from the\nedge as expected for the SHE. This is clearly seen in\nthe one dimensional pro\fle in Fig. 15(c). Further experi-\nments demonstrated the e\u000bect of spin (Hanle) precession,\nand associated the suppression of the observed signal to\nthe applied magnetic \feld, as predicted by Dyakonov and\nPerel, 1971a and Hirsch, 1999.\nZhang, 2000 showed by solving the spin-dependent\ndrift-di\u000busion equations for a \fnite width channel, that\nthe spin di\u000busion length \u0015sdde\fnes the length scale of\nthe edge spin accumulation. By \ftting to the spin-drift\ndi\u000busion equation, Kato et al. , 2004b extracted the trans-\nverse spin-current and the spin Hall resistivity \u001aH. This\nanalysis, which assumes well-resolved spin-up and spin-\ndown transport channels (Hirsch, 1999; Zhang, 2000), is\nvalid in the weak spin-orbit limit, which is veri\fed by\nnoting that \u0001 so\u001c=~\u001810\u00003\u001c1, where \u0001 sois the spin-\norbit coupling energy and \u001cthe momentum scattering\ntime. The measured value of \u001aH\u00182 \nm is consistent22\nwith that obtained from modeling based on scattering by\nscreened and short-range impurities (Engel et al. , 2006;\nTse and Das Sarma, 2006). Noting that the charge re-\nsistivity\u001a\u00184\u000210\u00006\nm, this corresponds to a spin\nHall angle of\u001810\u00004. In the weak spin-orbit coupling\nregime the spin-orbit splitting of the quasiparticle bands\nis smeared out by disorder which favors the extrinsic SHE\ninterpretation of the measured signal. The absence of\nthe intrinsic SHE was con\frmed by measurements in the\nstrained InGaAs sample which showed no dependence of\nthe SHE signal on the strain induced anisotropies of the\nspin-orbit coupled band structure.\nEBext\n(d)(c)(a)\n (b)\nFIG. 15 Observation of the SHE by the magneto-optical\nKerr microscope. (a) Schematics of the GaAs sample. (b)\nTwo-dimensional images of the spin density ns(left) and\nre\rectivity R(right) for an unstrained GaAs sample mea-\nsured at temperature 30 K and applied driving electric \feld\nE=10 mV\u0016m\u00001. (c) Kerr rotation as a function of xand\nexternal magnetic \feld BextforE=10 mV\u0016m\u00001. (d) Spatial\ndependence of the peak Kerr rotation A0across the GaAs\nchannel. From Ref. Kato et al. , 2004b.\nExperiments in 2DHG devices (Fig. 17) were carried\nout in the strong spin-orbit coupling limit, \u0001 so\u001c=~\u00184,\nwhich favors the intrinsic mechanism (Nomura et al. ,\n2005; Wunderlich et al. , 2005). The device comprised\ncoplanarp\u0000njunction light emitting diodes (LEDs)\nthat were patterned in (Al,Ga)As/GaAs heterostructures\ngrown by molecular beam epitaxy and using modulation\ndonor (Si) and acceptor (Be) doping in (Al,Ga)As barrier\nmaterials. The heterostructure consisted of an n-doped\nAlGaAs/GaAs heterojunction, followed by the growth of\n90 nm of intrinsic GaAs and a p-doped AlGaAs/GaAs\nheterojunction. The coplanar p\u0000njunctions were cre-\nated by removing the p-doped layer of the wafer and thus\ncreating a hole channel, with a carrier density 2 \u00021012\ncm\u00002. The 2DEG at the bottom heterojunction was al-most depleted. The removal of p-doped surface layer pop-\nulated the 2DEG, forming the n-side of the coplanar p\u0000n\njunction.\nA currentIpwas applied to drive the electrolumines-\ncence at the edge of the channel due to recombination\nnearp\u0000njunctions. The detection of spin-polarization\nin the 2DHG was done by measuring the circular po-\nlarization of the emitted light, shown in Figs. 17(d),(e).\nThe magnitude of the signal reached \u00181% at 4 K. Con-\nsistent with the SHE phenomenology, the experiments\ndemonstrated that the spin accumulation was opposite\nat opposite sides of the channel and that it reversed sign\nfollowing current reversal.\nCalculations of the SHE conductivity showed that the\nSHE originating from the spin-orbit coupled quasiparti-\ncle bands of the 2DHG is only weakly a\u000bected by dis-\norder for the parameters of the studied system (Wun-\nderlich et al. , 2005). A quantitative microscopic descrip-\ntion of the measured edge spin accumulation signal was\ndeveloped and further experimentally tested by Nomura\net al. , 2005. The theory analysis pointed out that the\nlength scale of the edge spin accumulation is de\fned in\nthe strong spin-orbit coupling regime by the spin-orbit\nprecession length Lso=vF\u001cso, where\u001cso=~\u0019=\u0001sois the\nprecession time of the spin in the internal spin-orbit \feld\nandvFis the Fermi velocity. With increasing strength\nof the spin-orbit coupling, the edge spin accumulation\nregion narrows down and, simultaneously, the amplitude\nof the spin polarization increases. For the experimental\nparameters of the studied 2DHG, Lso\u001810 nm and the\ncalculated amplitude of the edge spin polarization was\n8%, in good agreement with the 1% polarization of the\nmeasured electro-luminescence signal which was averaged\nover a\u0018100 nm sensitivity range of the co-planar light\nemitting diode. A comparison between measurements in\ndevices with 1.5 and 10 \u0016m wide channels con\frmed the\nexpectation that the SHE signal is independent of the\nchannel width.\nSubsequent magneto-optical measurements of the SHE\nin then-GaAs 3D epilayers have experimentally demon-\nstrated that the SHE-induced spin accumulation is due\nto a transverse spin-current which can drive spin polar-\nization tens of microns into a region in which there is\nminimal electric \feld (Sih et al. , 2006). The work proved\nexperimentally that the SHE can be used as a source of\nspin-current generated in a NM.\nA systematic doping dependence of the SHE angle was\nstudied in n-GaAs 3D epilayers with electron densities\nn= 1:8\u00021016\u00003:3\u00021017cm\u00003and the results were\nfound consistent with theory predictions for the extrinsic\nSHE (Matsuzaka et al. , 2009). The measured SHE angles\nof\u00185\u000210\u00004\u00005\u000210\u00003increase with increasing doping\nwith a tendency to saturate at the high doping end of the\nstudied set of samples at a value corresponding to \u00181%\nedge spin polarization. It was concluded from this sys-\ntematic analysis that the spin accumulation is reduced by\nan enhanced spin relaxation due to the Dyakonov-Perel\nmechanism, while the spin-current induced by the SHE23\n1.500 1.505 1.510 1.515 1.520-2-1012\n  Polarization [%]\nEnergy [eV]\n-2-1012\n  Polarization [%]\nn-channel\nILED2DHG\nILED ILED\nIP  downIP  up\nnp 10 µm(a) \n(b) \n(c) \n(d) \n(e) \nFIG. 16 Observation of the SHE by the circularly polarized\nelectro-luminescence of co-planar p\u0000ndiodes. (a) Schematic\ncon\fguration of the lateral p\u0000njunction to detect spin accu-\nmulation. (b) Light emission from the p\u0000njunction recorded\nby a charged-couple device camera. (c) Electron microscope\nimage of the microdevice with symmetrically placed p\u0000n\ndiodes at both edges of the 2DHG channel. (d),(e) Emitted\nlight polarization of recombined light in the p\u0000njunction for\nthe current \row indicated in (c) at 4 K. From Ref. Nomura\net al. , 2005.\nis enhanced with increasing n(Matsuzaka et al. , 2009).\nThe SHE was observed also in other semiconductor sys-\ntems including n-ZnSe 3D epilayers (Stern et al. , 2006),\nand InGaN/GaN superlattices (Chang et al. , 2007).\n2. Optical generation of the inverse spin Hall e\u000bect\nA traditional way of generating spin-polarized photo-\ncarriers in semiconductors is by absorption of circularly\npolarized light (Meier and Zakharchenya, 1984). Because\nof the optical selection rules, the out-of-plane spin polar-\nization of photo-carriers is determined in this technique\nby the sense and degree of the circular polarization of ver-\ntically incident light. This technique was used to observe\nthe AHE in semiconductors (Bakun, 1984; Miah, 2007),\nwhich we have already discussed in Sec. IV.A, and even-\ntually led also to the detection of the ISHE generated bythe pure spin current.\nAndo et al. , 2010b reported an experiment in a\nNM/semiconductor hybrid structure in which they\ndemonstrated the conversion of circularly polarized light\nabsorbed in a semiconductor to an electrical signal in the\nattached NM ISHE sensor. The photo-induced ISHE was\nobserved in a Pt/GaAs hybrid structure. In the GaAs\nlayer, circularly polarized light generates spin-polarized\ncarriers, inducing a pure spin-current into the Pt layer\nthrough the interface. This pure spin-current is con-\nverted into an electrical voltage due to the ISHE in Pt.\nSystematic changes of the ISHE signal were observed\nupon changing the direction and ellipticity of the cir-\ncularly polarized light, consistent with the expected phe-\nnomenology of the photo-induced ISHE. The observed\nphenomenon allows the direct conversion of circular-\npolarization information into the electrical voltage and\ncan be used as a spin photodetector.\nUsing a similar detector con\fguration, Kampfrath\net al. , 2013 demonstrated the control of the transmis-\nsion of terahertz spin-current pulses. The samples con-\nsisted of Fe/Au and Fe/Ru heterostructures. The ab-\nsorption of a femtosecond laser pulse in the Fe layer gen-\nerates a non-equilibrium electron distribution and asso-\nciated spin-current, dominated by the majority-spin sp-\nlike electrons, that \rows into the Au(Ru) non-magnetic\nlayer. The transport dynamics is di\u000berent in the Fe/Au\nand Fe/Ru heterostructures because of the much larger\nelectron mobility of Au; the \row of the non-equilibrium\nelectrons occur much more slowly in Ru than in Au, and\nare accompanied by signi\fcantly more spin accumula-\ntion. The non-magnetic layer can thus be used to either\ntrap or transmit electrons, and thus engineer ultrafast\nspin pulses, which change in temporal shape and de-\nlay. The detection of the spin-current pulses used by\nKampfrath et al. , 2013 relied on the ISHE.\nWhile in the static experiments by Ando et al. , 2010b\nthe resulting charge current is measured as a voltage,\nKampfrath et al. , 2013 detected the electromagnetic\npulse emitted by the charge current burst by electro-\noptical sampling using a GaP crystal. The feasibility of\nthe experiment demonstrated the operation of the ISHE\nas a spin-current detector up to frequencies as high as 20\nTHz.\nWunderlich et al. , 2009, 2010, using the same type of\nlateralp\u0000ndiodes as in Refs. (Nomura et al. , 2005;\nWunderlich et al. , 2005), exploited optical spin injec-\ntion by a circularly polarized laser beam to observe the\nISHE and to fabricate experimental opto-spintronic and\nspin-transistor devices. In the SHE measurements in\nRefs. (Nomura et al. , 2005; Wunderlich et al. , 2005), the\np\u0000njunctions were fabricated along the edges of the\n2DHG channel and under forward bias could sense the\nspin state of recombining electrons and holes through po-\nlarized electro-luminescence. In Refs. (Wunderlich et al. ,\n2009, 2010), on the other hand, the spin Hall channel\nwas fabricated in the etched part of the epilayer with the\n2DEG, the channel was oriented perpendicular to the24\nPhotoinduced inverse spin-Hall effect: Conversion of light-polarizationinformation into electric voltageK. Ando,1,2,a/H20850M. Morikawa,2T. Trypiniotis,3Y. Fujikawa,1C. H. W. Barnes,3andE. Saitoh1,2,41Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan2Department of Applied Physics and Physico-Informatics, Keio University, Yokohama 223-8522, Japan3Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE,United Kingdom4PRESTO, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075, Japan/H20849Received 26 November 2009; accepted 30 January 2010; published online 23 February 2010/H20850The photoinduced inverse spin-Hall effect was observed in a Pt/GaAs hybrid structure. In the GaAslayer, circularly polarized light generates spin-polarized carriers, inducing a pure spin current intothe Pt layer through the interface. This pure spin current is, by the inverse spin-Hall effect in the Ptlayer, converted into electric voltage. By changing the direction and ellipticity of the circularlypolarized light, the electromotive force varies systematically, consistent with the prediction of thephotoinduced inverse spin-Hall effect. The observed phenomenon allows the direct conversion ofcircular-polarization information into electric voltage; this phenomenon can be used as a spinphotodetector. ©2010 American Institute of Physics./H20851doi:10.1063/1.3327809/H20852Recent developments in optical and material sciencehave led to remarkable industrial applications, including op-tical data recording and optical communication.1In order toextend the scope of these conventional technologies, inten-sive experimental and theoretical interests have been focusedon simple and effective methods for detecting light circularpolarization,2–4since this polarization carries single-photoninformation and it is essential in optical technology, includ-ing quantum cryptography and quantum communication.5Light circular polarization is coupled with electrons’spins in semiconductors.6,7In a semiconductor crystal, theoptical selection rules for interband transitions induce spin-polarized electrons in the conduction band via the absorptionof circularly polarized light/H20851see Fig.1/H20849a/H20850/H20852. This process con-verts light circular polarization into electron-spin polariza-tion, enabling the integration of light-polarization informa-tion into spintronic technologies.8Recently, in the field of spintronics, the inverse spin-Halleffect/H20849ISHE/H20850was discovered.9–18The ISHE converts a spincurrent, a flow of electron spins in a solid, into an electro-motive force using the spin-orbit interaction, which enablesthe transcription of electron-spin information into an electricsignal. This suggests that light-polarization information canbe converted into an electric signal by combining the opticalselection rules and the ISHE. In this letter, we demonstratethis conversion of light-circular-polarization information intoelectric voltage in a Pt/GaAs structure: the photoinduced in-verse spin-Hall effect.Figure1/H20849b/H20850shows a schematic illustration of the sampleused in this study. The sample is a Pt/GaAs hybrid structure.The 5-nm-thick Pt layer was sputtered on the Si-doped GaAssubstrate. The donor impurity Si concentration isND=4.7/H110031018cm−3. Immediately before the sputtering, the surfaceof the GaAs layer was cleaned by chemical etching.19Twoelectrodes are attached to the ends of the Pt layer. During themeasurement, circularly polarized light with a wavelength of/H9261=670 nm/H20849h/H9263=1.85 eV/H20850and a power ofIi=10 mW wasirradiated to the sample at an in-plane angle of/H9258to thedirection across the two electrodes, as shown in Fig.1/H20849b/H20850.\na/H20850Electronic mail: ando@imr.tohoku.ac.jp.\nV\nPtGaAs\n(a) (b) (c)HHLHCB\nSOkS1/2P1/2P3/2E\n0\nσ−Eg∆\nθ0\nV\nVθ\ntopviewside viewxyz\nEISHEσ\n/CID14807/CID14809GaAsPt\nJs\nFIG. 1./H20849Color online/H20850/H20849a/H20850A schematic illustration of the band structure of GaAs and spin-polarized electrons generated by the absorption of circularlypolarized light./H20849b/H20850A schematic illustration of the Pt/GaAs hybrid structure used in this study./H9258is the in-plane angle between the incident direction of theillumination and the direction across the two electrodes attached to the Pt layer./H92580=65° is the angle of the light illumination to the normal axis of the filmplane./H20849c/H20850A schematic illustration of the inverse spin-Hall effect induced by photoexcited pure spin currents in the Pt/GaAs system.APPLIED PHYSICS LETTERS96, 082502/H208492010/H20850\n0003-6951/2010/96/H208498/H20850/082502/3/$30.00 © 2010 American Institute of Physics96,0 8 2 5 0 2 - 1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:147.231.126.31 On: Fri, 12 Sep 2014 08:39:29\nThe angle of the light illumination to the normal axis of thefilm plane/H92580is set at 65°/H20851see Fig.1/H20849b/H20850/H20852. In the GaAs layer,electrons with a spin polarization/H9268along the light propaga-tion direction are excited to the conduction band by the cir-cularly polarized light due to the optical selection rules.7This spin polarization then diffuses as a pure spin currentinto the spin-sink Pt layer across the Schottky barrier at thePt/GaAs interface via the electron tunneling and/or thermalprocesses/H20851see Fig.1/H20849c/H20850/H20852. Here, note that the measurementswere performed at zero applied bias across the junction, orthe open-circuit condition. The injected pure spin current isconverted into an electromotive forceEISHEby the ISHE inthe Pt layer due to the strong spin-orbit interaction in Pt20asEISHE=DISHEJs/H11003/H9268,/H208491/H20850whereJsand/H9268are the spatial direction and the spin polar-ization vector of the spin current/H20851see Fig.1/H20849c/H20850/H20852.DISHEis theISHE efficiency. We measured the difference in the gener-ated voltage between illumination with right circularly polar-ized/H20849RCP/H20850and left circularly polarized/H20849LCP/H20850light,VR−VL, by a polarization-lock-in technique using a photoelasticmodulator/H20849PEM/H20850operated atp=50 kHz. The difference inthe intensities of RCP and LCP light incident on the samplewas confirmed to be vanishingly small. Here, note that holespin polarization plays a minor role in this setup, since itrelaxes in/H11011100 fs,21which is much faster than the relax-ation time of/H1101135 ps for electron spin polarization.22All themeasurements were performed at room temperature.Figure2shows the in-plane illumination angle/H9258depen-dence ofVR−VLmeasured for the Pt/GaAs system. Withincreasing illumination angle/H9258from/H9258=0,VR−VLincreasesmonotonically below/H9258=90°. Above/H9258=90°,VR−VLde-creases with/H9258and changes its sign at/H9258=180°. This variationis well reproduced by a function proportional to sin/H9258asshown in Fig.2, being consistent with the model of thephotoinduced ISHE described in Eq./H208491/H20850; since/H9268andJsaredirected along the light propagation direction and the normalvector of the film plane, respectively, Eq./H208491/H20850predictsVR−VL/H11008/H20841Js/H11003/H9268/H20841x/H11008sin/H9258, consistent with the experimental re-sult. Here,/H20841Js/H11003/H9268/H20841xdenotes thexcomponent ofJs/H11003/H9268/H20851seeFig.1/H20849c/H20850/H20852. We found that this electromotive force is not de-tectable in a Cu/GaAs system, where the Pt layer is replacedby Cu with the very weak ISHE, supporting that the ISHE isresponsible for the observedVR−VLsignal.TheVR−VLsignal depends strongly on the ellipticity ofthe irradiated light polarization. The ellipticityAis definedas the ratio of the minor to major radiuses of the ellipticallypolarized light. Here, the angular momentum component of aphoton along the light propagation direction is zero/H20849maxi-mized/H20850whenA=0/H208491/H20850. As shown in Fig.3/H20849a/H20850, theVR−VLsignal increases withA, supporting that the electromotiveforce is due to the ISHE induced by photoexcited spin cur-rents.In order to investigate in detail theAdependence ofVR−VLshown in Fig.3/H20849a/H20850, we define the degree of circularpolarizationPcircas the difference in the numbers betweenRCP and LCP photons,Pcirc/H11013I+−I−I++I−=2A1+A2,/H208492/H20850whereI+andI−are the intensities of the RCP and LCP light,respectively. Using Eq./H208492/H20850and theAdependence ofVR−VLshown in Fig.3/H20849a/H20850,VR−VLis replotted as a function ofthe degree of circular polarizationPcircof the illuminatedlight as shown in Fig.3/H20849b/H20850. Figure3/H20849b/H20850shows thatVR−VLisproportional toPcirc, which is consistent with the predictionof the photoinduced ISHE as follows. Since the electron spinpolarization generated by circularly polarized light is propor-tional to the difference in the numbers between RCP andLCP photons absorbed in GaAs,6,7the electromotive forcedue to the photoinduced ISHE is proportional toItGaAsPcircGaAs:VR−VL/H11013QItGaAsPcircGaAs,/H208493/H20850whereItGaAsandPcircGaAsare the light intensity and thedegree of circular polarization of the light injected into theGaAs layer, respectively. The proportionality constantQisproportional to sin/H92582because of Eq./H208491/H20850:Q/H11013Q/H11032sin/H92582=Q/H11032/H20849n0/n2/H20850sin/H92580. Here, note that the light intensity and thePtGaAs\nV\nθ0 90 180 270 360-3-2-10123\nθ(deg)\nV+−= 90°θ\nV−+\n= 180°θVR−VL(µV)\nFIG. 2./H20849Color online/H20850The illumination angle/H9258dependence ofVR−VLmeasured for the Pt/GaAs hybrid structure./H9258is the in-plane angle betweenthe incident direction of the illumination and the direction across the elec-trodes attached to the edges of the Pt layer as shown in the inset.VR−VListhe difference in the electromotive force for illumination with right and leftcircularly polarized light. The filled circles are the experimental data. Thesolid curve shows a fitting result using a function proportional to sin/H9258.When/H92580=0, theVR−VLsignal is found to be negligibly small compared toVR−VL/H110151.9/H9262V measured when/H92580=65°, consistent with the prediction ofEq./H208496/H20850. The Hanle measurement of theVR−VLsignal shows negligiblysmall variation, consistent with the prediction of the photoinduced ISHE;pure-spin currents travel only near the Pt/GaAs interface.(a)\n0.5 1.000.21.0\nA(VR−VL)/(VR−VL)max0.40.60.8\n0.5 1.000.20.40.60.81.0\nPcirc(VR−VL)/(VR−VL)max(b)n0n1n20θd1PtGaAs0θ1θ2θEs(p)iEs(p)tFIG. 3./H20849a/H20850The irradiation light ellipticityAdependence ofVR−VLmea-sured for the Pt/GaAs system, whereAis defined as the ratio of the minor tomajor radiuses of the elliptically polarized light. The inset shows the defi-nitions of/H9258r/H20849r=0,1,2/H20850./H20849b/H20850The degree of circular polarizationPcircof theilluminated light dependence ofVR−VLmeasured for the Pt/GaAs system.The filled circles are the experimental data. The solid line shows a linear fitto the data.Pcircis obtained from the irradiation light ellipticityAandEq./H208492/H20850.082502-2 Andoet al.Appl. Phys. Lett.96,0 8 2 5 0 2/H208492010/H20850\n This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:147.231.126.31 On: Fri, 12 Sep 2014 08:39:29Photoinduced inverse spin-Hall effect: Conversion of light-polarizationinformation into electric voltageK. Ando,1,2,a/H20850M. Morikawa,2T. Trypiniotis,3Y. Fujikawa,1C. H. W. Barnes,3andE. Saitoh1,2,41Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan2Department of Applied Physics and Physico-Informatics, Keio University, Yokohama 223-8522, Japan3Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE,United Kingdom4PRESTO, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075, Japan/H20849Received 26 November 2009; accepted 30 January 2010; published online 23 February 2010/H20850The photoinduced inverse spin-Hall effect was observed in a Pt/GaAs hybrid structure. In the GaAslayer, circularly polarized light generates spin-polarized carriers, inducing a pure spin current intothe Pt layer through the interface. This pure spin current is, by the inverse spin-Hall effect in the Ptlayer, converted into electric voltage. By changing the direction and ellipticity of the circularlypolarized light, the electromotive force varies systematically, consistent with the prediction of thephotoinduced inverse spin-Hall effect. The observed phenomenon allows the direct conversion ofcircular-polarization information into electric voltage; this phenomenon can be used as a spinphotodetector. ©2010 American Institute of Physics./H20851doi:10.1063/1.3327809/H20852Recent developments in optical and material sciencehave led to remarkable industrial applications, including op-tical data recording and optical communication.1In order toextend the scope of these conventional technologies, inten-sive experimental and theoretical interests have been focusedon simple and effective methods for detecting light circularpolarization,2–4since this polarization carries single-photoninformation and it is essential in optical technology, includ-ing quantum cryptography and quantum communication.5Light circular polarization is coupled with electrons’spins in semiconductors.6,7In a semiconductor crystal, theoptical selection rules for interband transitions induce spin-polarized electrons in the conduction band via the absorptionof circularly polarized light/H20851see Fig.1/H20849a/H20850/H20852. This process con-verts light circular polarization into electron-spin polariza-tion, enabling the integration of light-polarization informa-tion into spintronic technologies.8Recently, in the field of spintronics, the inverse spin-Halleffect/H20849ISHE/H20850was discovered.9–18The ISHE converts a spincurrent, a flow of electron spins in a solid, into an electro-motive force using the spin-orbit interaction, which enablesthe transcription of electron-spin information into an electricsignal. This suggests that light-polarization information canbe converted into an electric signal by combining the opticalselection rules and the ISHE. In this letter, we demonstratethis conversion of light-circular-polarization information intoelectric voltage in a Pt/GaAs structure: the photoinduced in-verse spin-Hall effect.Figure1/H20849b/H20850shows a schematic illustration of the sampleused in this study. The sample is a Pt/GaAs hybrid structure.The 5-nm-thick Pt layer was sputtered on the Si-doped GaAssubstrate. The donor impurity Si concentration isND=4.7/H110031018cm−3. Immediately before the sputtering, the surfaceof the GaAs layer was cleaned by chemical etching.19Twoelectrodes are attached to the ends of the Pt layer. During themeasurement, circularly polarized light with a wavelength of/H9261=670 nm/H20849h/H9263=1.85 eV/H20850and a power ofIi=10 mW wasirradiated to the sample at an in-plane angle of/H9258to thedirection across the two electrodes, as shown in Fig.1/H20849b/H20850.\na/H20850Electronic mail: ando@imr.tohoku.ac.jp.\nV\nPtGaAs\n(a) (b) (c)HHLHCB\nSOkS1/2P1/2P3/2E\n0\nσ−Eg∆\nθ0\nV\nVθ\ntopviewside viewxyz\nEISHEσ\n/CID14807/CID14809GaAsPt\nJs\nFIG. 1./H20849Color online/H20850/H20849a/H20850A schematic illustration of the band structure of GaAs and spin-polarized electrons generated by the absorption of circularlypolarized light./H20849b/H20850A schematic illustration of the Pt/GaAs hybrid structure used in this study./H9258is the in-plane angle between the incident direction of theillumination and the direction across the two electrodes attached to the Pt layer./H92580=65° is the angle of the light illumination to the normal axis of the filmplane./H20849c/H20850A schematic illustration of the inverse spin-Hall effect induced by photoexcited pure spin currents in the Pt/GaAs system.APPLIED PHYSICS LETTERS96, 082502/H208492010/H20850\n0003-6951/2010/96/H208498/H20850/082502/3/$30.00 © 2010 American Institute of Physics96,0 8 2 5 0 2 - 1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:147.231.126.31 On: Fri, 12 Sep 2014 08:39:29(a)\t\r \n(c)\t\r (b)\t\r \nFIG. 17 (a) A schematic illustration of the band structure of\nGaAs and spin-polarized electrons generated by the absorp-\ntion of circularly polarized light. (b) A schematic illustration\nof the ISHE induced by photoexcited pure spin-currents in the\nPt/GaAs system. (c) The illumination angle \u0012dependence of\nVR\u0000VLmeasured for the Pt/GaAs hybrid structure. \u0012is\nthe in-plane angle between the incident direction of the illu-\nmination and the direction across the electrodes attached to\nthe edges of the Pt layer as shown in the inset. VR\u0000VLis\nthe di\u000berence in the electromotive force for illumination with\nright and left circularly polarized light. The \flled circles are\nthe experimental data. The solid curve shows a \ftting result\nusing a function proportional to sin \u0012. From Ref Ando et al. ,\n2010b.\np\u0000njunction, and the diode was under zero or reverse\nbias, operating as a photocell as shown in Fig. 18. The\noptical activity of the lateral diode con\fned to a submi-\ncron depletion region, combined with a focused ( \u00181\u0016m)\nlaser beam, allowed for a well localized injection of spin-\npolarized photo-electrons into the planar 2DEG channel.\nThe Hall signals were detected electrically on multiple\nHall-crosses patterned along the channel. Two regimes\nof operation of the device are distinguished: One cor-\nresponds to an AHE regime, in which the reverse-bias\ncharge current is drained behind the Hall crosses at the\nopposite end of the channel from the p\u0000njunction in-\njection point (Fig. 18(a)). The other regime corresponds\nto the ISHE measurement since in this case the charge\ncurrent is drained before the Hall crosses, allowing only\nthe pure spin-current to di\u000buse further in the channel\n(Fig. 18(b)). In both cases the measured transverse elec-\ntrical signals were consistent with the phenomenology of\nthe spin-dependent Hall phenomena (Wunderlich et al. ,2009, 2010). The sign of the voltage was opposite for op-\nposite helicities of the incident light, i.e., opposite spin-\npolarizations of injected photo-electrons. Moreover, the\namplitude of the electrical signals was found to depend\nlinearly on the degree of circular polarization of the light,\nrendering the device an electrical polarimeter (Wunder-\nlichet al. , 2009). The electrical signals were observable\nover a wide temperature range with spin Hall angles of\n10\u00003\u000010\u00002. The measured 2DEG was in the weak spin-\norbit coupling regime, \u0001 so\u001c=~\u001810\u00001, and the measured\ndata were consistent with the extrinsic mechanism (Wun-\nderlich et al. , 2009).\nRH2RH1VB xIPHa \nb FIG. 4: iSHE based transistor. (a) Schematics of the spin injection Hall e↵ect measurementsetup with optically injected spin-polarized electrical current propagating through the Hall barand corresponding experimental Hall e↵ect signals at crosses H1 and H2. The Hall resistances,RH=VH/IPH, for the two opposite helicities of the incident light are plotted as a function ofthe focused light spot position, i.e., of the position of the injection point. The optical currentIPHis independent of the helicity of the incident light and varies only weakly with the light spotposition. (b) Same as (a) for the iSHE measurement geometry in which electrical current is closedbefore the first detecting Hall cross H1. (c) Schematics of the setup of the spin Hall transistor andexperimental Hall signals as a function of the gate voltage at a Hall cross placed behind the gateelectrode for two light spot positions with a relative shift of 1µm. Figure form Ref. 38.23\n 0.8RHVGVBxIPHc \nFIG. 4: iSHE based transistor. (a) Schematics of the spin injection Hall e↵ect measurementsetup with optically injected spin-polarized electrical current propagating through the Hall barand corresponding experimental Hall e↵ect signals at crosses H1 and H2. The Hall resistances,RH=VH/IPH, for the two opposite helicities of the incident light are plotted as a function ofthe focused light spot position, i.e., of the position of the injection point. The optical currentIPHis independent of the helicity of the incident light and varies only weakly with the light spotposition. (b) Same as (a) for the iSHE measurement geometry in which electrical current is closedbefore the first detecting Hall cross H1. (c) Schematics of the setup of the spin Hall transistor andexperimental Hall signals as a function of the gate voltage at a Hall cross placed behind the gateelectrode for two light spot positions with a relative shift of 1µm. Figure form Ref. 38.23\nb RH2RH1VbxIPHB \nc FIG. 4: iSHE based transistor. (a) Schematics of the spin injection Hall e↵ect measurementsetup with optically injected spin-polarized electrical current propagating through the Hall barand corresponding experimental Hall e↵ect signals at crosses H1 and H2. The Hall resistances,RH=VH/IPH, for the two opposite helicities of the incident light are plotted as a function ofthe focused light spot position, i.e., of the position of the injection point. The optical currentIPHis independent of the helicity of the incident light and varies only weakly with the light spotposition. (b) Same as (a) for the iSHE measurement geometry in which electrical current is closedbefore the first detecting Hall cross H1. (c) Schematics of the setup of the spin Hall transistor andexperimental Hall signals as a function of the gate voltage at a Hall cross placed behind the gateelectrode for two light spot positions with a relative shift of 1µm. Figure form Ref. 38.23\n-10-50510050100150 RH [ :\u0003]2.52.01.51.00.50.0IPHRH2 x [ Pm ]     RH1IPH [ nA ]\n   V+ V- V+ V- V+ V- \nR [\u0003:\u0003]FIG. 4: iSHE based transistor. (a) Schematics of the spin injection Hall e↵ect measurementsetup with optically injected spin-polarized electrical current propagating through the Hall barand corresponding experimental Hall e↵ect signals at crosses H1 and H2. The Hall resistances,RH=VH/IPH, for the two opposite helicities of the incident light are plotted as a function ofthe focused light spot position, i.e., of the position of the injection point. The optical currentIPHis independent of the helicity of the incident light and varies only weakly with the light spotposition. (b) Same as (a) for the iSHE measurement geometry in which electrical current is closedbefore the first detecting Hall cross H1. (c) Schematics of the setup of the spin Hall transistor andexperimental Hall signals as a function of the gate voltage at a Hall cross placed behind the gateelectrode for two light spot positions with a relative shift of 1µm. Figure form Ref. 38.23\n-20-1001020050100150IPH [ nA ] \n x [ Pm ] RH [\u0003:\u0003]2.52.01.51.00.50.0 \nV [ PV ]FIG. 4: iSHE based transistor. (a) Schematics of the spin injection Hall e↵ect measurementsetup with optically injected spin-polarized electrical current propagating through the Hall barand corresponding experimental Hall e↵ect signals at crosses H1 and H2. The Hall resistances,RH=VH/IPH, for the two opposite helicities of the incident light are plotted as a function ofthe focused light spot position, i.e., of the position of the injection point. The optical currentIPHis independent of the helicity of the incident light and varies only weakly with the light spotposition. (b) Same as (a) for the iSHE measurement geometry in which electrical current is closedbefore the first detecting Hall cross H1. (c) Schematics of the setup of the spin Hall transistor andexperimental Hall signals as a function of the gate voltage at a Hall cross placed behind the gateelectrode for two light spot positions with a relative shift of 1µm. Figure form Ref. 38.23\n-0.50.00.51.0-3-2-101\nVG [ V ]VH [ PV ] \n0.00.20.40.60.8IPH [ PA ] 'x=1Pm V+ FIG. 4: iSHE based transistor. (a) Schematics of the spin injection Hall e↵ect measurementsetup with optically injected spin-polarized electrical current propagating through the Hall barand corresponding experimental Hall e↵ect signals at crosses H1 and H2. The Hall resistances,RH=VH/IPH, for the two opposite helicities of the incident light are plotted as a function ofthe focused light spot position, i.e., of the position of the injection point. The optical currentIPHis independent of the helicity of the incident light and varies only weakly with the light spotposition. (b) Same as (a) for the iSHE measurement geometry in which electrical current is closedbefore the first detecting Hall cross H1. (c) Schematics of the setup of the spin Hall transistor andexperimental Hall signals as a function of the gate voltage at a Hall cross placed behind the gateelectrode for two light spot positions with a relative shift of 1µm. Figure form Ref. 38.23\nFIG. 18 ISHE based transistor. (a) Schematics of the spin in-\njection Hall e\u000bect measurement setup with optically injected\nspin-polarized electrical current propagating through the Hall\nbar and corresponding experimental Hall e\u000bect signals at\ncrosses H1 and H2. The Hall resistances, RH=VH=IPH,\nfor the two opposite helicities of the incident light are plot-\nted as a function of the focused light spot position, i.e., of\nthe position of the injection point. The optical current IPH\nis independent of the helicity of the incident light and varies\nonly weakly with the light spot position. (b) Same as (a) for\nthe ISHE measurement geometry in which electrical current is\nclosed before the \frst detecting Hall cross H1. (c) Schematics\nof the setup of the spin Hall transistor and experimental Hall\nsignals as a function of the gate voltage at a Hall cross placed\nbehind the gate electrode for two light spot positions with a\nrelative shift of 1 \u0016m. From Ref. Wunderlich et al. , 2010.\n3. All-optical generation and detection\nThe SHE and the ISHE were also observed using two-\ncolor optical coherence control techniques in intrinsic\nGaAs at 80 K with polarized 70 fs, 715 and 1430 nm\npulses (Zhao et al. , 2006). When the pulses were orthog-25\nonally polarized, a pure spin source current was gener-\nated that yielded a transverse Hall pure charge current\nvia the ISHE. When the pulses were parallel polarized,\na pure charge source current was generated that yielded\na pure spin-current via the SHE. By varying the relative\nphase or polarization of the incident pulses, the type,\nmagnitude, and direction of both the source and trans-\nverse currents were tuned without applying electric or\nmagnetic \felds. In contrast to the previous steady-state\nexperiments, where drift currents are generated by elec-\ntric \felds, the injected currents are ballistic currents with\nelectrons traveling initially at \u00181000 km/s.\nThe generation of spin and charge currents results from\nthe quantum interference between absorption pathways\nfor one- and two-photon absorption connecting the same\ninitial and \fnal states as illustrated in Fig. 19(a). For\na spin-current, a coherent pulse centered at frequency !\nwith phase '!is normally incident along ^ zand linearly\npolarized along ^ x-direction that can be arbitrary with\nsespect to crystal axes since the e\u000bects are not strongly\nsensitive to crystal orientation. A co-propagating 2 !\npulse with phase '2!is linearly polarized along the or-\nthogonal ^y-direction. Excited spin-up electrons are po-\nlarized along ^ zand move preferentially in one direction\nalong ^x, while spin-down electrons move in the opposite\ndirection. Together they generate a spin-current propor-\ntional to cos(\u0001 ') where \u0001'= 2'!\u0000'2!. The spin-\ncurrent is dominated by electrons as holes lose their spin\nin<100 fs. Due to the ISHE, a charge current is gener-\nated (Fig. 19(a)) that has the same cosine dependence as\nthe spin-current source. Consistent with the ISHE phe-\nnomenology, the excess charge on one side and the de\fcit\non the other side of the sample, shown in Fig. 19(b), was\nobserved along the direction perpendicular to the driving\nspin-current.\nThe ballistic nature of transport in these experiments\nwas fully exploited in fs time-resolved measurements\n(Werake et al. , 2011). They allowed to infer the mo-\nmentum scattering time \u001c\u00190:45 ps and with a much\nshorter time delay of the probe pulses to observe in real\ntime the transverse charge current. The measurements\nshowed that the charge current was generated well before\nthe \frst scattering event, providing a direct demonstra-\ntion of the intrinsic ISHE.\n4. Electrical manipulation\nA distinct feature of the ISHE experiments in the\n2DEG is the observed spin precession due to internal\nRashba and Dresselhaus spin-orbit \felds (Wunderlich\net al. , 2009, 2010). Since the spin di\u000busion length scales\napproximately (Wunderlich et al. , 2010) as\u0018L2\nso=w, it\nwas possible to observe a few spin precessions in chan-\nnels of a width w= 1\u0016m forLso\u00181\u0016m of the studied\n2DEG. The corresponding oscillations of the spin Hall\nvoltages were consistently observed by measuring at dif-\nferent Hall crosses along the channel or by shifting the\n/.0001/.0030/.01362/.0030!/.0255/.00302!. We define Kx>0if\"electrons pref-\nerentially move along /.0135^xwith#electrons preferentially\nmoving along /.0255^xwhen/.0001/.0030/.01360. Optically injected holes\nmove in opposition to electrons but lose their spin on a time\nscale<100 fs [16]; therefore, hole spin is ignored in sub-\nsequent discussions. Spin-dependent skew, side-jump[17,18], and perhaps other types of asymmetric scattering\nsend\"and#in the same direction along ^y, yielding a Hall\ncharge current, as shown by the curved arrows in Fig. 1(a).Since the spin current is the source for this Hall current,\nthese two currents have the same /.0001/.0030dependence with the\ntransverse current taking the form J\ny\nHall/.0136/.0255AHallcos/.0133/.0001/.0030/.0134.\nAlong with the Hall current, a transverse current produced\nby the QUIC process [straight transverse arrow in Fig. 1(a)]\nhas been predicted [9], but not yet observed, with Jy\nQUIC/.0136\nAQUICsin/.0133/.0001/.0030/.0134, whereAQUIC andAHallare positive coeffi-\ncients. The spin and charge currents decay via momentum\nrelaxation, although the latter is also influenced by space\ncharge effects. Note that, in comparison with other obser-\nvations of the spin Hall effect, our sample is intrinsic, with\nthe holes playing the role of spin-scattering ‘‘impurities’’;\nthe currents are ballistic and can be produced anywhere onthe sample. Given that the spin-dependent scattering time\nis much longer than a typical momentum relaxation time,\none can expect that /.0024a single scattering event will lead to\na particular electron contributing to the Hall current.\nFor the experiments, a sample consisting of 10 periods\nof 14 nm wide GaAs wells ( E\ng/.01361:542 eV at 80 K) alter-\nnating with 14 nm thick Al0:3Ga0:7Asbarriers was grownon a (100)-oriented GaAs substrate (removed for the ex-\nperiments). All effects reported here have also been ob-served in bulk, intrinsic GaAs. An optical parametricoscillator produces 70 fs (full width at half maximum) !\npulses centered at 1430 nm, with 2!pulses centered at\n715 nm obtained by frequency doubling. The /.0001/.0030is\ncontrolled using a scanning interferometer and pulse polar-izations are selected independently. The pulses are cofo-cused to a w/.00243/.0022mdiameter spot on the sample. For\n2@!/.0255E\ng/.0024193 meV , the electrons are injected with a\nspeed/.0024103km=s.F o ra2!pulse fluence of 1:5/.0022J=cm2,\nthe injected electron density is /.00241017cm/.02553; the!pulse\nfluence is set to generate approximately the same density.For this density the momentum relaxation time is /.0024100 fs ,\ngoverned by phonon and carrier-carrier scattering [19].\nThe source spin current is detected via spin accumula-\ntion that follows spin motion [12]; such accumulation is\nproportional to the source current. Upon injection, \"and#\nelectrons have the same /.0024Gaussian spatial profile with the\nsame peak density, i.e., N\n#/.0133x;y;t/.01360/.0134/.0136N\"/.0133x;y;t/.01360/.0134.\nThe two profiles move in opposite directions and separate\nby an effective distance LSby the time momentum and\nskew scattering are complete. This spin separation persists\nlong after the currents have vanished, decaying eventually\nby spin relaxation, carrier diffusion, and recombination aswe have independently verified. The magnitude of L\nSis\ndetermined by: (i) the fraction of the total injected densitythat is spin polarized and that travels along ^x; (ii) the initial\ninjection velocity [20]; and (iii) the momentum relaxationtime. For our experimental conditions, L\nSis expected to be\n<100 nm , and therefore, we use a derivative technique to\nachieve sub-optical-wavelength resolution [12]. WhenL\nS/.0028w, the net spin accumulation /.0001S/.0136N\"/.0133x;y/.0134/.0255\nN#/.0133x;y/.0134/.0024LSdN\"=dx, with the sign giving the spin flow\ndirection. The /.0001S, centered on /.0133x;y/.0134, is measured using a\n100 fs, 808 nm, linearly polarized probe pulse that isfocused onto the sample with a /.00243/.0022m(FWHM) diameter.\nThe probe arrives /.002410 ps after the pump pulses by which\ntime carrier thermalization and momentum relaxation arecomplete, but not electron recombination or spin relaxa-tion. The probe spectrum is centered on the heavy-holeexciton for efficient sampling of N\n#andN\". The efficiency\nof this sampling is the reason for choosing to present\nresults for the quantum well sample. Circular dichroism,\ni.e., absorption difference for right and left circularly po-\nlarized probe beam components in the presence of \"and#\nelectrons, is used to deduce [12] /.0001S.\nThe transverse charge current is similarly measured via\nelectron (and hole) accumulation [11]. For the chargecurrent,\"and#electrons move in the same direction and\nso the entire charge profile moves by L\nC, where\n/.0001N/.0133x;y/.0134/.0136LCdN=dx andN/.0133x;y/.0134denotes the total elec-\ntron density. In contrast to LS, the magnitude of LCis\ninfluenced by space charge fields produced by electron-hole separation; thus, L\nCandLScannot be directly com-\nz [100]^x^\ny^2 2\nk2(a) (b)\nFIG. 1 (color online). (a) Illustration of orthogonally polarized\n!and2!pulses producing a pure spin current (double headed\nstraight arrow) along !beam polarization direction ( ^x). A Hall\ncharge current (curved arrows) and a transverse QUIC chargecurrent along ^ylead to electron accumulation near one edge of\nthe illuminated region. (b) Same as (a) except for collinearly\npolarized pulses producing a QUIC charge current along ^xand\nleading to a Hall pure spin current (curved arrows) and a\ntransverse QUIC pure spin current (double headed arrow) result-\ning in up and down spin accumulations along /.0006^y.\nInset: schematic showing how !and2!pulses connect valence\nand conduction band states for current generation.PRL 96,246601 (2006)PHYSICAL REVIEW LETTERSweek ending\n23 JUNE 2006\n246601-2\npared without additional information. The N/.0133x;y/.0134is mea-\nsured through differential transmission /.0001T/.0133N/.0134(i.e., the\ntransmission with and without pump beams) of a linearly\npolarized probe pulse, for which /.0001T/.0133N/.0134=T/Nwhen\n/.0001T/.0133N/.0134/.00281, whereTis the linear transmission. Finally,\n/.0001Nis measured by monitoring the phase-dependent com-\nponent of the differential transmission /.0001T/.0133/.0001/.0030/.0134=T//.0001N\nwith a lock-in amplifier slaved to the interferometer scan-\nning frequency. Using these procedures, we simulta-\nneously measure /.0001Sand/.0001Nat each/.0133x;y/.0134in the spot as\na function of /.0001/.0030.\nFigures 2(a) and 2(b) show the measured /.0001/.0030depen-\ndence of /.0001Sand/.0001Nas the probe is scanned first along ^x\n(fory/.01360), then along ^y(forx/.01360) with the pump focal\nspot center defining x/.0136y/.01360. The scans along ^x\n[Fig. 2(a)] show \"accumulation for one sign of x,#accu-\nmulation for the other sign, and periodic reversal with /.0001/.0030.\nNo charge accumulation is measured anywhere along ^x.\nThese observations confirm a pure spin source current\nalong^x. The scans along ^y[Fig. 2(b)], which show regions\nof charge accumulation that oscillate with /.0001/.0030, confirm a\ntransverse charge current along ^y. The charge current has a\ndifferent/.0001/.0030dependence from the spin current, suggesting\nthe presence of in-phase (Hall) and quadrature (QUIC)components.\nThe Hall and QUIC contributions to the transverse cur-\nrent can be unambiguously distinguished by measuring the\ncharge and spin accumulations as a function of /.0133x;y/.0134for\n/.0001/.0030/.01360and/.0001/.0030/.0136/.0025=2, as shown in Fig. 3. For the data in\nFigs. 3(a) and 3(b) we adjusted /.0001/.0030so that the source\ncurrent is maximal ( /.0001/.0030/.01360;/.0025;2/.0025...), with the trans-\nverse QUIC current therefore zero. Under these conditions,the/.0001Nshown in Fig. 3(b) results from the Hall current\ngenerated by the source spin current displayed in Fig. 3(a).For Figs. 3(c) and 3(d) /.0001/.0030is chosen so that the spin source\ncurrent, and therefore, the Hall current is zero. Figure 3(d)\nis thus a contour plot of the /.0001Nproduced by the transverse\nQUIC current predicted theoretically [9]. Through theappropriate choice of /.0001/.0030one can ‘‘tune’’ the source\ncurrent magnitude and direction as well as that of thetwo transverse currents.\nBy polarizing both beams along the same ( ^x) direction as\ndepicted in Fig. 1(b) one can produce a QUIC source\ncharge current of the form [8] J\nx/sin/.0133/.0001/.0030/.0134that is capable\nof producing the ballistic analogue to the previously ob-served spin Hall current [4,5]. In this case, spin-dependentscattering deflects \"and#electrons in opposite directions\nalong^yto produce a pure spin current given by K\ny\nHall/.0136\n/.0255BHallsin/.0133/.0001/.0030/.0134. Also along ^y, a current of the form\nKy\nQUIC/.0136/.0255BQUICcos/.0133/.0001/.0030/.0134is generated by the QUIC pro-\ncess, as earlier predicted [9], but not yet observed. The\nprocedures used to measure the charge and spin currentsvia accumulation effects are analogous to those used to\nFIG. 2 (color). (a) Contour map showing dependence of /.0001S\ninduced by a source spin current on /.0001/.0030and position along ^x\n(y/.01360). (b) Corresponding contour map of combined Hall and\nQUIC/.0001Nmeasured along ^y(x/.01360).\nFIG. 3 (color). Spatial contour maps showing: (a) the spin /.0001S\nand (b) the charge /.0001Naccumulations for /.0001/.0030/.01360associated\nwith the pure spin source and Hall charge currents, respectively;(c) and (d) show /.0001Sand/.0001Nfor/.0001/.0030/.0136/.0025=2, when the pure spin\nsource current is zero and only a transverse QUIC charge current\nis present. Approximate spin and charge densities can be ob-tained by multiplying /.0001Sand/.0001Nby10\n15cm/.02553.PRL 96,246601 (2006)PHYSICAL REVIEW LETTERSweek ending\n23 JUNE 2006\n246601-3\npared without additional information. The N/.0133x;y/.0134is mea-\nsured through differential transmission /.0001T/.0133N/.0134(i.e., the\ntransmission with and without pump beams) of a linearly\npolarized probe pulse, for which /.0001T/.0133N/.0134=T/Nwhen\n/.0001T/.0133N/.0134/.00281, whereTis the linear transmission. Finally,\n/.0001Nis measured by monitoring the phase-dependent com-\nponent of the differential transmission /.0001T/.0133/.0001/.0030/.0134=T//.0001N\nwith a lock-in amplifier slaved to the interferometer scan-\nning frequency. Using these procedures, we simulta-\nneously measure /.0001Sand/.0001Nat each/.0133x;y/.0134in the spot as\na function of /.0001/.0030.\nFigures 2(a) and 2(b) show the measured /.0001/.0030depen-\ndence of /.0001Sand/.0001Nas the probe is scanned first along ^x\n(fory/.01360), then along ^y(forx/.01360) with the pump focal\nspot center defining x/.0136y/.01360. The scans along ^x\n[Fig. 2(a)] show \"accumulation for one sign of x,#accu-\nmulation for the other sign, and periodic reversal with /.0001/.0030.\nNo charge accumulation is measured anywhere along ^x.\nThese observations confirm a pure spin source current\nalong^x. The scans along ^y[Fig. 2(b)], which show regions\nof charge accumulation that oscillate with /.0001/.0030, confirm a\ntransverse charge current along ^y. The charge current has a\ndifferent/.0001/.0030dependence from the spin current, suggesting\nthe presence of in-phase (Hall) and quadrature (QUIC)components.\nThe Hall and QUIC contributions to the transverse cur-\nrent can be unambiguously distinguished by measuring the\ncharge and spin accumulations as a function of /.0133x;y/.0134for\n/.0001/.0030/.01360and/.0001/.0030/.0136/.0025=2, as shown in Fig. 3. For the data in\nFigs. 3(a) and 3(b) we adjusted /.0001/.0030so that the source\ncurrent is maximal ( /.0001/.0030/.01360;/.0025;2/.0025...), with the trans-\nverse QUIC current therefore zero. Under these conditions,the/.0001Nshown in Fig. 3(b) results from the Hall current\ngenerated by the source spin current displayed in Fig. 3(a).For Figs. 3(c) and 3(d) /.0001/.0030is chosen so that the spin source\ncurrent, and therefore, the Hall current is zero. Figure 3(d)\nis thus a contour plot of the /.0001Nproduced by the transverse\nQUIC current predicted theoretically [9]. Through theappropriate choice of /.0001/.0030one can ‘‘tune’’ the source\ncurrent magnitude and direction as well as that of thetwo transverse currents.\nBy polarizing both beams along the same ( ^x) direction as\ndepicted in Fig. 1(b) one can produce a QUIC source\ncharge current of the form [8] J\nx/sin/.0133/.0001/.0030/.0134that is capable\nof producing the ballistic analogue to the previously ob-served spin Hall current [4,5]. In this case, spin-dependentscattering deflects \"and#electrons in opposite directions\nalong^yto produce a pure spin current given by K\ny\nHall/.0136\n/.0255BHallsin/.0133/.0001/.0030/.0134. Also along ^y, a current of the form\nKy\nQUIC/.0136/.0255BQUICcos/.0133/.0001/.0030/.0134is generated by the QUIC pro-\ncess, as earlier predicted [9], but not yet observed. The\nprocedures used to measure the charge and spin currentsvia accumulation effects are analogous to those used to\nFIG. 2 (color). (a) Contour map showing dependence of /.0001S\ninduced by a source spin current on /.0001/.0030and position along ^x\n(y/.01360). (b) Corresponding contour map of combined Hall and\nQUIC/.0001Nmeasured along ^y(x/.01360).\nFIG. 3 (color). Spatial contour maps showing: (a) the spin /.0001S\nand (b) the charge /.0001Naccumulations for /.0001/.0030/.01360associated\nwith the pure spin source and Hall charge currents, respectively;(c) and (d) show /.0001Sand/.0001Nfor/.0001/.0030/.0136/.0025=2, when the pure spin\nsource current is zero and only a transverse QUIC charge current\nis present. Approximate spin and charge densities can be ob-tained by multiplying /.0001Sand/.0001Nby10\n15cm/.02553.PRL 96,246601 (2006)PHYSICAL REVIEW LETTERSweek ending\n23 JUNE 2006\n246601-3FIG. 3: Observation of the iSHE using the two-color optical pump-and-probe technique. (a)\nIllustration of orthogonally polarized ωand 2ωpulses producing a pure spin current (double headed\nstraight arrow) along ωbeam polarization direction (ˆ x). The charge current due to the iSHE\n(curved arrows) along ˆ yleads to electron accumulation near one edge of the illuminated region.\n(b) Measured charge accumulation due to the iSHE. Figure from Ref. 31.\nthe opposite end of the channel from the p-n junction injection point (Fig. 4(a)). The other\n11FIG. 19 Observation of the ISHE using the two-color optical\npump-and-probe technique. (a) Illustration of orthogonally\npolarized!and 2!pulses producing a pure spin-current (dou-\nble headed straight arrow) along !beam polarization direc-\ntion (^x). The charge current due to the ISHE (curved arrows)\nalong ^yleads to electron accumulation near one edge of the\nilluminated region. (b) Measured charge accumulation due to\nthe ISHE. From Ref. Zhao et al. , 2006.\nlaser spot, i.e., the spin injection point (Fig. 18). The lat-\neral ISHE channels also allow to place top gate electrodes\nin between the Hall crosses as shown in Fig. 18(c). (The\ngates are formed by unetched regions of the wafer). The\nstrength of the Rashba and Dresselhaus spin-orbit \felds\nand, therefore, also the spin precession can be manipu-\nlated electrically in the device shown in Fig. 18(c). To\ndemonstrate an AND logic functionality, two gates were\nfabricated on top of the channel and the Hall electrical\nsignal was measured at a cross placed behind both gates.\nIntermediate gate voltages on both gates represented the\ninput value 1 and gave the largest electrical ISHE signal,\nrepresenting the output value 1. When a large reverse\ngate voltage was applied to any of the two gates, repre-\nsenting input 0, the electrical ISHE signal disappeared,\ni.e., the output was 0.\nA di\u000berent approach to achieve the control of spin-\ncurrents is by directly modifying the spin-orbit coupling\nstrength on a given material, which in turn determines\nthe spin Hall angle. The electronic band structure and\nimpurity states are weakly dependent on an external elec-\ntric \feld and therefore cannot be used to change the spin-\norbit strength. However, Okamoto et al. , 2014 noted that\nthe electric \feld can induce a carrier redistribution within\na band or multiple bands. Therefore, if the electrons gen-\nerating the SHE can be controlled by populating di\u000berent\nareas (valleys) of the electronic structure, the spin-orbit\ninteraction (and the spin Hall angle) can be tuned di-\nrectly within a single sample. Okamoto et al. , 2014 re-\nported such a tuning in bulk GaAs at room temperature\nby means of an electrical intervalley transition induced\nin the conduction band. The spin Hall angle was de-26\ntermined by measuring an electromotive force driven by\nphotoexcited spin-polarized electrons drifting through n-\nGaAs Hall bars. By controlling electron populations in\nthe \u0000 and L valleys with an applied electric \feld (part of\nthep-character in the L valley provides a larger e\u000bective\nspin-orbit interaction), the angle was changed by a fac-\ntor of 40, from 0.0005 to 0.02 for moderate electric \felds\nbeyond 100 kVm\u00001. Thus the highest spin Hall angle\nachieved is comparable to that of Pt.\nC. Transport experiments\nHirsch, 1999 and Zhang, 2000, discussed speci\fc con-\ncepts for the experimental detection of the SHE and ISHE\nusing DC transport techniques. Hirsch, 1999 proposed a\ndevice that consists of a metallic slab in which spin ac-\ncumulation is generated by an electrical current via the\nSHE, as described in Sec. II (see Fig. 2). A transverse\nstrip connects the edges of the slab, allowing the spin-\ncurrent to \row through it. Due to the ISHE, a voltage\nis generated that can be measured with a voltmeter. In\nan alternative approach, Zhang, 2000 proposed to detect\nthe spin accumulation electrically using a FM probe. The\nconcept borrows from techniques for spin injection and\ndetection in NM implemented in nonlocal spin devices\n(Johnson and Silsbee, 1985; Silsbee, 1980).\nShortly after the optical SHE detection in semiconduc-\ntors (Kato et al. , 2004a; Wunderlich et al. , 2004, 2005),\nValenzuela and Tinkham, 2006 reported an observation\nof the voltage generated by the ISHE. Instead of gener-\nating the spin-current by the SHE, which would render\na second order voltage in the spin Hall angle, they used\nelectrical spin injection from a FM in combination with\na Hall cross patterned in the ISHE paramagnet. Simul-\ntaneously, Saitoh et al. , 2006 observed the voltage gen-\nerated by the ISHE in a set-up where the spin-injection\nfrom the FM to the NM was achieved using the SP tech-\nniques. Kimura et al. , 2007 combined the concept of the\nspin Hall cross and the proposal by Zhang, 2000 to de-\ntect both the SHE and the ISHE in the same device. It\ntook a few more years to demonstrate the idea of Hirsch\nof simultaneously exploiting both the SHE and the ISHE\nin an electrical device. Eventually, Br une et al. , 2010\nperformed the experiment in a ballistic H-bar semicon-\nductor device (Hankiewicz et al. , 2004). The transport\nSHE and ISHE experiments are described in detail below\nin Sec. IV.C.1-IV.C.5.\nMore recently, the SHE was also detected via the ma-\nnipulation of magnetization in FMs (Liu et al. , 2011;\nMiron et al. , 2011b). Spin-currents generated by the\nSHE were shown to be su\u000eciently large to induce mag-\nnetization dynamics, drive domain walls, or switch mag-\nnetization in the FM, demonstrating the potential of the\nSHE for applications (Emori et al. , 2013; Liu et al. , 2012;\nMiron et al. , 2011a,b; Ryu et al. , 2013). These SHE ex-\nperiments together with the ISHE measurements via SP\nare discussed in Sec. IV.D.\nSource F1\nIDetector F2(a) (b)2dm\nxzyN(c)N\nVNL+VNL-FIG. 20 (a) Nonlocal spin detection and spin accumulation\n(a) Schematic illustrations of the device layout. An injected\ncurrentIon the source (F1) generates spin accumulation in\nthe NM (N) which is quanti\fed by the detector (F2) voltage\nVNL. (b) Schematic representation of the spin splitting in\nthe electrochemical potential induced by spin injection. The\nsplitting decays over characteristic lengths \u0015sfover the N side.\n(c) Detector behavior for an idealized Stoner FM with a full\nspin subband for the parallel magnetization orientation (top)\nand for the antiparallel magnetization orientation (bottom).\n1. Concepts of nonlocal spin transport. Electrical injection and\ndetection\nJohnson and Silsbee, 1985 reported the injection and\ndetection of nonequilibrium spins using a device that con-\nsisted of a NM, N, with two attached FM electrodes (F1,\nF2), illusrated in Fig. 20. In this device, spin-polarized\nelectrons are injected from F1 into N by applying a cur-\nrentIfrom F1 that results in spin accumulation in N.\nThe population of, say, spin-up electrons in N increases\nby shifting the electrochemical potential by \u000e\u0016N, while\nthe population of spin-down electrons decreases by a sim-\nilar shift of\u0000\u000e\u0016N. Overall, this corresponds to a spin-\naccumulation splitting of 2 \u000e\u0016N. The spin accumulation\ndi\u000buses away from the injection point and reaches the F2\ndetector, which measures its local magnitude.\nAs \frst suggested by Silsbee, 1980, the spin accumu-\nlation in N can be probed by the voltage VNL, which is\ninduced at F2. Silsbee noted that the density polariza-\ntion in N, or equivalently the nonequilibrium magnetiza-\ntion, acts as the source of spin electromotive force that\nproducesVNL. The magnitude of VNLis associated with\n\u000e\u0016N, while its sign is determined by the relative magne-\ntization orientation of F1 and F2.\nBecause the current is applied to the left on N, there\nis no charge current towards the right, where the de-\ntector F2 lies (Fig. 20(a)). For this reason, the spin\ndetection is said to be implemented nonlocally, where\nno charge current circulates by the detection point, and\nthusVNLis sensitive to the spin degree-of-freedom only.\nAccordingly, nonlocal measurements eliminate the pres-\nence of spurious e\u000bects associated to charge transport,\nsuch as anisotropic magnetoresistance (AMR) or the or-\ndinary HE that could mask subtle signals related to spin\ninjection. Typically, nonlocal devices exhibit a small\noutput background allowing sensitive spin-detection ex-\nperiments. This approach has been widely used in re-\ncent years to characterize the spin transport in met-27\nals, semimetals, semiconductors, superconductors, car-\nbon nanotubes and graphene. It has also been used to\nstudy the spin transfer properties of FM/NM material\ninterfaces.\n2. Nonlocal detection of inverse spin Hall e\u000bect with lateral\nspin-current\nValenzuela and Tinkham, 2006, 2007 adapted the non-\nlocal detection techniques to study the ISHE. Their de-\nvice is schematically shown in Fig. 21(a). By using a FM\nelectrode F, a spin-polarized current is injected in a non-\nmagnetic strip. It propagates to both sides away from the\ninjection point and decays with the spin di\u000busion length,\n\u0015sd. A laterally induced voltage VSH, which results from\nthe conversion of the injected spin-current into charge im-\nbalance owing to the ISHE, is then measured using a Hall-\ncross structure. The magnitude of VSHis determined by\nthe anomalous Hall operator, \u001bSH^\u001b\u0002E\u001b, where\u001bSHde-\nnotes the spin Hall conductivity, \u001bis the spin index, and\nE\u001bis an e\u000bective spin-dependent \\electric\" \feld, which\nfollows from the spin-dependent electrochemical poten-\ntial\u0016\u001balong the NM al strip, i.e. E\u001b(r) =\u0000r\u0016\u001b(r).\nxzy\nF1F2\nAl\n500 nmB\nqM(a) (b)\nIB (M )\nVV\nNL+NL-F\nN\nFIG. 21 (a) Spin-current induced Hall e\u000bect or inverse spin\nHall e\u000bect (ISHE). Schematic representation of an actual de-\nvice where the pure spin-current is generated by spin injec-\ntion through a FM (F) with out-of-plane magnetization. (a)\nDevice fabricated with CoFe electrodes (light color) and an\nAl channel (dark color). Adapted from Ref. Valenzuela and\nTinkham, 2006.\nIn the device of Fig. 21(b), the injector electrode F1 is\nmade of CoFe, while the strip material is Al with thick-\nnesstAl. The full device is fabricated without break-\ning vacuum using electron beam evaporation and shadow\nevaporation techniques. An Al 2O3tunnel barrier is used\nfor spin-current injection. The purpose of the barrier is\ntwo-fold. First, it enhances the polarization of the in-\njected electrons and, second, it assures a uniform current\ninjection. The latter is essential because it suppresses\nthe \row of charge current towards the Hall cross, pre-\nserving the nonlocal character of the measurements and\neliminating the previously mentioned spurious e\u000bects.\nThe FM electrode is magnetized in-plane at zero mag-\nnetic \feld due to shape anisotropy and thus an out-of-\nplane magnetic \feld B?is used to generate a perpendicu-\nlarly polarized spin-current at the Hall-cross (Fig. 21(b)).\nSpin imbalance in the Al \flm occurs with a de\fned spindirection given by the magnetization orientation of the\nF1 electrode. Consequently, VSHis expected to vary\nwhenB?is applied and the magnetization Mof the elec-\ntrode is tilted out of the substrate plane. De\fning \u0012as\nthe angle between Mand the electrode axis, it follows\nfrom the cross product in the anomalous Hall operator\nthatVSHis proportional to sin \u0012, correlating with the\ncomponent of Mnormal to the substrate (Fig. 21(b)).\nI+I-\nV+V-F1F2\n3 2 1 0-202\n0.4 0.2 0.0 -0.2 -0.4\n-202V/I(mW) \nB^(T)LFM = 2 mm\n V/I(mW) B^(T)\nI+I-\nV+V-F1F2\n-101\n-4 -2 0 2 4-0.10.00.1\nB^(T)RSH(mW)\nsinqDRSH\nFIG. 22 Observation of the ISHE (right) in a metal device\nwith an electrical spin injection from a FM, compared with\nthe spin detection by the non-local spin valve e\u000bect (left).\nThe light color FM electrodes in the micrographs are made of\na CoFe alloy. The dark color Hall cross is made of Al. From\nRef. Valenzuela and Tinkham, 2006.\nThe device layout in Fig. 21(b) is more sophisticated\nthan the schematics in Fig. 21(a), where only F1 is re-\nquired. The second electrode (F2) together with F1\nand the N strip form a spin injection/detection device\n(Fig. 22(a)) for the purpose of calibration. Calibra-\ntion procedures are necessary to demonstrate consistency\nwith standard nonlocal methods. Explicitly, this de-\nvice can be utilized to measure the spin accumulation\nin the NM and then determine its associated spin dif-\nfusion length \u0015sd, the spin polarization of the injected\nelectronsP, and the magnetization orientation of the\nFM electrodes \u0012in the presence of an external magnetic\n\feld (perpendicular to the substrate). For this purpose,\nbatches of samples are commonly used where the distance\nbetween the two FMs, LF, is modi\fed and the spin pre-\ncession signal acquired (Fig. 22(b)). The distance of F1\nrelative to the Hall cross, LSH, is also modi\fed in order\nto test the consistency of the spin di\u000busion results. Sub-\nsequent measurements in the con\fguration of Fig. 22(c),\nperformed in Al of di\u000berent tAl, and thus di\u000berent \u0015sd,\nyielded\u001bSH\u001820\u000040 (\ncm)\u00001and\u000bSH\u00181\u00003\u000210\u00004,\nwhich compares well with theoretical estimates based on\nextrinsic mechanisms (Shchelushkin and Brataas, 2005).\nOlejn\u0013 \u0010k et al. , 2012 used the same geometry to de-\ntect the ISHE in n-GaAs using epitaxial ultrathin-\nFe/GaAs injection contacts with strong in-plane mag-28\nnetic anisotropy. Hybrid semiconductor/metal-FM\nstructures su\u000bered for a long time from the resistance\nmismatch problem (Schmidt et al. , 2000). Since the spin\ntransport relies on di\u000berent conductivities for spin-up\nand spin-down electrons and is governed by the least\nconductive part of the device, the e\u000bects are weak in\ndevices in which the non-magnetic semiconductor with\nequal spin-up and spin-down conductivities dominates\nthe resistance of the device (Rasbha, 2000). The in-\ntroduction of a highly resistive tunnel barrier between\nthe FM metal electrode and the semiconductor channel\nsolved the problem (Lou et al. , 2007; Rasbha, 2000).\nThe device of Olejn\u0013 \u0010k et al. , 2012, shown in Fig. 23(a),\ncomprised the n-GaAs channel, a Hall-cross, and two\nFe electrodes as in Fig. 21(a) with Fe Schottky injec-\ntion contact. The Fe/ n-GaAs heterostructure was grown\nepitaxially in a single molecular beam epitaxy chamber\nwithout breaking ultrahigh vacuum. The heterostructure\ncontained 250 nm of low Si-doped GaAs (5 \u00021016cm\u00003),\n15 nm of GaAs with graded doping, and 15 nm of highly\nSi-doped GaAs (5 \u00021018cm\u00003). The purpose of the dop-\ning pro\fle was to create a narrow tunnel Schottky barrier\nbetween GaAs and Fe favorable for spin injection or de-\ntection. It was then possible to simultaneously detect the\nspin-current in n-GaAs generated by nonlocal injection\nfrom a Fe contact by using the ISHE and the spin accu-\nmulation by using the additional Fe contact (Figs. 23(b)\nand (c)). The spins were manipulated by spin precession\nwith an external magnetic \feld combined with drift us-\ning an external bias (Huang et al. , 2007). In this case,\nthe magnetic \feld was applied in-plane ( x-direction) to\nprecess the spin accumulation into the out-of-plane di-\nrection, so that it could be detected by the ISHE. The\nsignal \frst increases at low \felds but then is suppressed\ndue to spin dephasing (Fig. 23(c)).\nDevices described above required the application of a\nmagnetic \feld for observing the ISHE. Seki et al. , 2008\nused a FM (FePt) with an out-of-plane anisotropy, which\nenabled them to measure the ISHE in Au without mag-\nnetic \felds. The device was fabricated with the geometry\nin Fig. 21 using ohmic contacts. The measurements pre-\nsented a rather large background voltage, which is likely\ndue to the \row of charge current at the position of the\nHall cross (Mihajlovic et al. , 2009). The use of ohmic\ncontacts, as opposed to tunnel barriers, results in inho-\nmogeneous in-plane current injection. Because the width\nof the Au wire and the distance of the Hall cross were\ncomparable, some current reached the Hall cross con-\ntributing to the background. By considering that the\nvoltage was independent of the magnetization of the in-\njector electrodes, Seki et al. deduced\u000bSH= 0:113 for Au\nat 295 K, which was weakly dependent on temperature.\nThis large\u000bSHwas \frst attributed to resonant scattering\nin the orbital-dependent Kondo e\u000bect of Fe impurities in\nthe Au host metal (Guo, 2009). In the follow-up work,\nSugai et al. , 2010 found that \u000bSH\u00180:07 was approxi-\nmately independent of the Fe concentration. Seki et al. ,\n2010 further observed a reduction of \u000bSHfrom 0.1 to\naFIG. 23 (a) Schematic of the device used to detect the ISHE\ninn-GaAs. Current is injected on the Fe-electrode on the\nright, the voltage generated by ISHE and by spin accumula-\ntion are detected simultaneously with the Hall cross and the\nFe-electrode on the left, respectively. The spin transport can\nbe further modi\fed by a drift current applied between the\noutermost Au electrodes. (b) and (c) show the experimen-\ntal symmetrized nonlocal spin injection/detection signal and\nthe antisymmetrized ISHE signal in the in-plane hard-axis\n\feld for constant spin-injection bias current (300 \u0016A) and for\nthree di\u000berent drift currents. From Ref. Olejn\u0013 \u0010k et al. , 2012\n0.03 when the thickness was decreased from 10 nm to 20\nnm in Au. Additionally, Gu et al. , 2010 obtained sim-\nilar results in Pt-doped Au by co-deposition of Pt and\nAu with magnetron sputtering (1.4% Pt). These results\nin combination with ab initio and quantum Monte Carlo\ncalculations for the skew scattering due to a Pt impurity\nled to the proposal of a much larger \u000bSHin the surface\nof Au than in the bulk (Gu et al. , 2010).\n3. Nonlocal detection of spin Hall e\u000bects with vertical\nspin-current\nThe approach described in the previous section en-\nables proper quanti\fcation of the spin Hall angle because\nof the direct measurements of the spin di\u000busion length.\nHowever, it is suitable for materials that have spin dif-\nfusion lengths beyond tens of nanometers. For smaller\nspin di\u000busion lengths, Kimura et al. , 2007 modi\fed this\napproach using the device structure shown in Fig. 24(a).\nThe structure comprises a Hall cross where the material\nof the transverse arm is the large spin-orbit coupling NM\nN2 with short \u0015sd, which acts as a spin-current absorber\nthat induces VSHvia the ISHE. The longitudinal arm,\non the other hand, is made of a NM N1 with long spin29\nIB (M )\nV+V-\nIISIS\nISHE SHEN1\nN2N1\nN2N1\nN2F(a) (b)\nFIG. 24 (a) Schematic illustration of a nonlocal device to\nmeasure the direct and inverse spin Hall e\u000bect in materials\n(N2) with short spin relaxation length \u0015N2\nsd. (b) Schematic\nillustration of the charge accumulation process in N2 (left)\ndue to the ISHE when a spin-current is injected from F as\nin (a). Schematic illustration of the charge to spin-current\nconversion due to the SHE when a current is applied to N2.\nThis process generates spin accumulation that is detected by\nmeasuring the voltage at which F \roats. See (Kimura et al. ,\n2007). Adapted from Ref. Valenzuela and Kimura, 2012.\ndi\u000busion length that ful\fls the purpose of transporting\nspin information between the FM electrode (F) and N2.\nThe way the measurements are performed is sketched\nin Fig. 24(b) (left). A charge current is injected from\nF into N1 that induces a spin-current towards N2 po-\nlarized in-plane in the direction parallel to the N1 arm.\nWhen the distance between F and the cross is smaller\nthan the spin di\u000busion length in N1, the spin-current is\npreferably absorbed into the transverse arm N2 because\nof the strong spin relaxation in N2. The injected vertical\nspin-current into N2 vanishes in a short distance from\nthe N1/N2 interface because of the short spin di\u000busion\nlength of N2 and generates a transverse voltage via the\nISHE.\nThis device can be also used to measure the SHE.\nThe bias con\fguration is modi\fed as shown in Fig. 24(b)\n(right). Here, N2 acts as a spin-current source, which in-\nduces a spin accumulation in N1 that is detected with the\nFM electrode F, as originally proposed by Zhang, 2000.\nKimura et al. , 2007 used permalloy (Py) as the FM\nsource, and Cu and Pt as N1 and N2, respectively (see\nFig. 25). The materials were deposited by electron-beam\nevaporation. The devices were fabricated with transpar-\nent interfaces between Py and Cu and between Pt and\nCu. Ar ion beam etching was done prior to depositing\nCu in order to clean the surfaces of Py and Pt, a method\nthat has been repeated in the other studies described be-\nlow. The long spin di\u000busion length of Cu (about 500\nnm) assured that the spin-current reached Pt, which was\n4 nm thick. The measurements were interpreted with\na one-dimensional model by assuming that the induced\nspin-current at the Cu/Pt interface was completely ab-\nsorbed by the Pt. The spin relaxation length for Pt was\nassumed (not measured) to be \u0015sd= 3 nm. Kimura et al. ,\n2007 then obtained that \u001bSH\u00182:4\u0002102(\ncm)\u00001and\n\u000bSH= 3:7\u000210\u00003.\nOver the last few years some of the initial simpli\f-\ncations that are mentioned above have been removed,\nb cFIG. 25 (a) Scanning electron microscope (SEM) image of\nthe fabricated spin Hall device to measure the SHE in Pt to-\ngether with a schematic illustration of the fabricated device.\n(b) Signal due to de ISHE at 77 K. The black and grey curves\nshow measurements for the two opposite sweeps of the mag-\nnetic \feld. Spin-accumulation signal generated by SHE at 77\nK. Insets: measurement set-up. NiFe, Cu and Pt are in grey,\npink and yellow, respectively. From Ref. Kimura et al. , 2007.\nleading to more reliable quantitative interpretations of\nthe experimental results. Vila et al. , 2007 noted that the\nabsorption e\u000eciency of the spin-current may depend on\nthe device geometry and temperature. They modi\fed\nthe design of Fig. 25 to a conventional nonlocal spin in-\njection/detection structure where a Pt electrode was in-\nserted between the FM Py electrodes (see Fig. 26). This\nchange enabled them to determine explicitly the magni-\ntude of the absorbed spin-current. By comparing with\nreference devices without the Pt insertion, they observed\nthat the ratio between the spin signal with and without\nPt varied from 0.35 at 5 K to 0.2 at room temperature,\nirrespective of the Pt thickness. They then performed\nsystematic spin absorption studies as a function of the\nPt thickness, obtaining that \u0015sdfor Pt was 10 nm and\n14 nm at room temperature and at 5 K, respectively.\nThe Pt thickness dependence of the ISHE signal re-\nsulted in somewhat lower \u0015sdfor Pt of 7 nm and 8 nm\nat room temperature and at 5 K, respectively, still more\nthan a factor of 2 larger than previously assumed. The\nobtained value of \u001bSH\u00183:5\u0002102\ncm\u00001was larger\nthan that in the Kimura et al. , 2007 experiment; this is\nbecause the assumption of the complete spin-current ab-\nsorption into the Pt wire led to underestimating the spin\nHall conductivity.\nAdditionally, Vila et al. , 2007 found that the spin Hall\nconductivity was nearly constant as a function of tem-\nperature indicating that the spin Hall resistivity likely\nevolves in a quadratic form with the Pt resistivity in the\nanalyzed temperature range, which was initially associ-\nated to a side jump origin of the SHE. However, this\nresistivity dependence can also be associated with the30\niSHESHE\nFIG. 26 (a) SEM image of the typical device for SHE mea-\nsurements and an illustration of the device. (b) Direct and\ninverse SHE (SHE and ISHE) recorded at T= 10 K using a\ndevice with a Pt-thicness of 20 nm, altogether with the AMR\nfrom the Py wire measured on the same condition. SHE mea-\nsurement corresponds to VBC=IAE, and ISHE to VEA=IBC;\nwithVthe voltage, Ithe applied current; A, B, C and E\nare the contact leads as denoted in the SEM image. From\nRef. Vila et al. , 2007.\nintrinsic mechanism (Kontani et al. , 2009; Tanaka et al. ,\n2008).\nNiimi et al. , 2011 further included a correction factor\n0<x< 1 that accounted for the fact that the transverse\ncharge current induced by the ISHE is partially shunted\nby the wire N1 above the N1/N2 interface or, conversely,\nthat the charge current that induces the spin-current via\nthe SHE does not only \row through N2 but also leaks\ninto N1 (see also (Liu et al. , 2011)). In order to deter-\nminexexperimentally, they measured the voltage drop\nof two identical N2 nanowires with and without shunting\nN1 bridges. Within a one-dimensional circuit model, the\ncurrent \rowing into the N2 wire I0was assumed to divide\ninto two components at the N1/N2 interface: xI0for the\nN2 wire and (1\u0000x)I0for the N1 bridge. With this, x\nwas estimated to be 0 :36\u00060:08 for Cu (N1), when using\na number of transition metals and alloys as N2 (Morota\net al. , 2011; Niimi et al. , 2011), therefore appearing to\nbe rather insensitive to the resistivity of N2. Because of\nthis correction, former reports underestimated \u001bSHby a\nfactorx\u00001\u00182:8. Such large correction is to be expected\ngiven that the N1 wire (usually Cu or Ag) is highly con-\nductive (conductivity \u00183\u00005\u0002107(\nm)\u00001) and thick\n(\u0018100 nm), when comparing with N2 ( \u0018105\u0000107\n(\nm)\u00001and\u001810 nm).\nIn addition, Niimi et al. , 2011 and Morota et al. , 2011\npointed out that the spin-currents injected in N2 should\ndilute when its thickness tN2is larger than the spin di\u000bu-\nsion length in N2 leading to smaller spin Hall signals. To\ncorrect for this e\u000bect, they obtained an aggregate spin-\ncurrent in N2 by integrating over tN2, which was then\ndivided by tN2; they also forced the spin-current to bezero at the bottom surface of N2.\nNiimi et al. , 2011 reported \u000bSH= 0:021\u00060:006 for\nthe skew scattering o\u000b Ir in a Cu matrix, which is con-\nsistent with experimental work relying on spin polarized\ncurrents generated by dilute Mn impurities, for which\n\u000bSH= 0:026 (Fert et al. , 1981; Fert and Levy, 2011). The\nspin Hall angle was extracted with CuIr wires that were\nprepared with di\u000berent Ir concentrations (0%, 1%, 3%,\n6%, 9%, and 12%) using magnetron sputtering. They\nmeasured\u001aHof CuIr as a function of the resistivity in-\nduced by the Ir impurities, de\fned as \u001aimp=\u001aCuIr\u0000\u001aCu,\n\fnding a simple linear dependence up to Ir concentration\nof 12%. This is presented as a proof that the dominant\nmechanism of the extrinsic SHE induced by the Ir impu-\nrities is the skew scattering, with \u000bSH=\u001aH=\u001aimp.\nMorota et al. , 2011 investigated the ISHE and SHE\nin 4d and 5d transition metals, Nb, Ta, Mo, Pd, and\nPt. Nb, Ta, and Mo wires were deposited by magnetron\nsputtering while Pd and Pt wires were grown by electron-\nbeam evaporation. In particular, for Pt, they obtained a\nspin Hall angle \u001bSH= 0:021\u00060:005 that was roughly 6 \u0002\nlarger than that in Kimura et al. , 2007. Such a di\u000berence\ncan be explained with the above corrections. They also\nfound that the sign of the spin Hall conductivity changes\nsystematically depending on the number of delectrons,\na tendency that is in good agreement with theoretical\ncalculations based on the intrinsic SHE (Kontani et al. ,\n2009).\nMore recently, Niimi et al. , 2012 studied the ISHE\nand SHE by introducing a small amount of Bi impuri-\nties in Cu. The alloy Cu 1\u0000xBixwere deposited by mag-\nnetron sputtering from Bi-sintered Cu targets with dif-\nferent Bi concentrations (0%, 0.3%, 0.5%, 1%, 3%, and\n6%). The spin Hall resistivity was derived by 1D and\n3D calculations as a function of the resistivity induced\nby the Bi impurities. As for the case with Ir impuri-\nties, the experimental results follow the linear variation\nof the spin Hall resistivity characteristic of skew scatter-\ning by dilute impurities but only at the lowest concen-\ntrations (<1%). At larger concentrations, inhomoge-\nneous distribution on Bi results in the departure from\nthe dilute impurity regime. From the slope \u001aH=\u001aimp\nin the linear regime, \u000bSHwas estimated with the stan-\ndard 1D analysis above, and with more accurate 3D cal-\nculations, resulting in \u000bSH(1D) =\u0000(0:12\u00060:04) and\n\u000bSH(3D) =\u0000(0:24\u00060:09) at 10 K.\nThe 3D calculations yield a larger \u000bSHbecause spin\naccumulation is observed to spread at the side edges of\nthe CuBi/Cu junction, which is not taken into account in\nthe 1D model. For the calculations with the 1D model,\nthe spin-current is considered to \row vertically into the\nCuBi wire, therefore, they cannot take into account the\nspin escape by lateral spreading. In general, the cor-\nrection is observed to become important when the spin\ndi\u000busion length in N2 is longer than tN2. For the cases of\nCuIr or Pt, it produces a small additional error because\nthe spin di\u000busion length in N2 is usually shorter than\ntN2. For Pt,\u000bSHwas estimated to increase from 0.02131\n(1D model) to 0.024 (3D model).\nNonlocal methods have been used to estimate spin Hall\nangles in a number of other materials, including IrO 2(Fu-\njiwara et al. , 2013), and Bi (Fan and Eom, 2008); it was\nalso applied to determine the sign of the spin injection\npolarization of FMs by using materials with a well estab-\nlished spin Hall angle, which is not possible with stan-\ndard nonlocal spin injection and detection methods using\nthe same FM material for the two electrodes. This pro-\ncedure was demonstrated for the Heusler alloy Co 2FeSi\n(Oki et al. , 2012). The ISHE in nonlocal geometries was\nalso used as a probe of spin \ructuations in weak FM\nNiPd alloys (Wei et al. , 2012). An anomaly near the\nCurie temperature was explained by the \ructuation con-\ntributions to skew scattering via spin-orbit interactions;\nthe total magnetic moment involved in the experiment\nwas extremely small (less than 10\u000014emu), highlighting\nthe very high sensitivity of the technique.\n4. Direct detection of the spin Hall induced spin accumulation\nAs discussed in Sec. II.C, Zhang, 2000 proposed to de-\ntect the spin accumulation induced by the SHE via a\nFM probe directly attached in the side of a thin conduc-\ntor. The magnetization of the FM points to the direction\nperpendicular to the plane of the \flm. The method is\nbased on measuring the voltage at which the FM \roats\ndepending on the direction of its magnetization, which\ngives direct information of the spin accumulation at the\nedge of the conductor (see section IV.C.1). The imple-\nmentation of the method took several years because of\nthe local currents that circulate nearby the FM, which\nresult in spurious signals that are avoided by the nonlo-\ncal methods, as described above.\nGarlid et al. , 2010 implemented a similar device based\non epitaxial Fe/In xGa1\u0000xAs heterostructures (Fig. 27).\nThe active layers consisted of a 2.5 \u0016m thick Si-doped\n(3\u00005\u00021016cm\u00003) channel, a highly doped Schottky\ntunnel barrier (5\u00021018cm\u00003), and a 5 nm thick Fe layer.\nHeterostructures with In concentrations 0, 0.03, 0.05,\nand 0.06 were processed using lithographic and etching\ntechniques into devices with 30 \u0016m-wide channels ori-\nented along the [110] direction, which is the xdirection\nin Fig. 27.\nIt is technically di\u000ecult to fabricate a thin \flm with\na FM attached at its edge with the magnetization ori-\nentation proposed by Zhang, 2000. To circumvent this\nobstacle, Garlid et al. , 2010 patterned pairs of Fe elec-\ntrodes so that the centers of the contacts in each pair are\n2, 6, or 10 \u0016m from the edges of the channel. However,\nsince the contacts are magnetized along x, and the spin\npolarization generated by the SHE is oriented along z,\na magnetic \feld along ywas applied to precess the spin\naccumulation into the xdirection so that it could be de-\ntected. The spin accumulation is identi\fed through the\nobservation of the Hanle e\u000bect in the voltage measured\nbetween the pairs of FM contacts. The voltage \frst in-creases at low \felds but then is suppressed due to spin\ndephasing in large \felds.\nThe local character of the measurement causes a large\nbackground signal due to imperfect cancelation of the\nbackground HE voltage induced by the applied magnetic\n\feld, of local HEs due to fringe \felds generated by the\nFM contacts, and voltages due to the small fraction of the\nchannel current that is shunted through the Fe contacts.\nThe HE voltages were eliminated by using the expected\nsymmetries of the signal, while the shunting e\u000bect was\nreduced by subtracting the voltages for the two current\ndirections.\nThe results showed that the magnitude of the spin Hall\nconductivity was in agreement with models of the ex-\ntrinsic SHE due to ionized impurity scattering. The bias\nand temperature dependences of the SHE indicated that\nboth skew and side-jump scattering contribute to the to-\ntal spin Hall conductivity. By analyzing the dependence\nof the SHE on channel conductivity, which was modi\fed\nwith the In content, Garlid et al. , 2010 determined the\nrelative magnitudes of the skew and side-jump contribu-\ntions to the total spin Hall conductivity.\nEhlert et al. , 2014, 2012 reported measurements of\nthe SHE using a similar structure based on n-GaAs lay-\ners with relatively low carrier concentration (5 \u00021016\ncm\u00003) and corresponding low conductivity. The FM\nvoltage probes were implemented with (Ga,Mn)As/GaAs\nEsaki diode structures. The heterostructures were grown\nby molecular-beam epitaxy and consisted of a 1 \u0016m-\nthickn-type transport channel, a 15-nm thick n!\nn+GaAs transition layer (5 \u00021018cm\u00003), a 2.2-nm\nAl0:36Ga0:64As di\u000busion barrier, and a 15-nm-thick layer\nof Ga 0:95Mn0:05As. The highly doped (Ga,Mn)As/GaAs\np\u0000njunction forms an Esaki diode. This structure was\ncovered on the top by 2 nm of Fe and 4 nm of Au. The\npurpose of Fe was to make the contacts harder magnet-\nically, which helped to keep the magnetization aligned\nalong their long axes during Hanle measurements. The\nvalues of spin Hall conductivities that were extracted are\nconsistent with those calculated by Engel et al. , 2005\nbut smaller than those observed by Garlid et al. , 2010.\nEhlert et al. , 2014, 2012 observe that the combined re-\nsults of these two experiments show that both the skew\nand side-jump contributions to the spin Hall conductivity\ncannot be treated as fully independent of the conductiv-\nity of the channel.\n5. Spin Hall injection and detection without ferromagnets\nSpin injection by the SHE combined with spin detec-\ntion by the ISHE in one device (Hirsch, 1999) was im-\nplemented by Br une et al. , 2010 using a device geometry\nproposed by Hankiewicz et al. , 2004. The original Hirsch,\n1999 proposal required a transverse strip connecting the\nedges of a slab on which spin accumulation was generated\ndue to the SHE. A spin-current would circulate in the\ntransverse strip which would then generate a measurable32\nfrom the same wafer, but with only a series of contacts\nacross the channel) are shown in Fig. 1(b). These data\nestablish that the ferromagnetic (FM) contacts are sensitive\nto the spin polarization generated by spin injection into thechannel as well as its dephasing by precession in a mag-netic field applied in the zdirection. The Fe contacts,\nwhich have a strong easy axis along [110], show sharpand reproducible switching behavior as well as nearly\nperfect remanence.\nSince the contacts are magnetized along [110] ( ^x), and\nthe spin polarization generated by the spin Hall effect isoriented along [001] ( ^z), a field B\nyis applied to precess the\nspin accumulation into the [110] direction [ 8,17]. We\ntherefore expect to observe an increase in the current-\ninduced spin accumulation at low fields followed by asuppression due to spin dephasing in large fields. Thesignal should reverse sign when B\nyis reversed or when\nthe contact magnetizations are reversed. The voltage Vab/C0\nVcd, shown in Fig. 1(c)for the two different magnetization\ndirections on the GaAs sample with a channel current jx¼5:7/C2103A=cm2atT¼30 K , shows that the expected\nbehavior is superimposed on a background that remains inspite of the differential measurement. The background is\ndue to (1) imperfect cancelation of the background Hall\nvoltage due to the applied field, (2) imperfect cancelationof local Hall effects due to the fringe fields generated bythe FM contacts, and (3) voltages due to the small fraction(0.1%) of the channel current that is shunted through the Fecontacts. We can eliminate the first two effects based on theexpected symmetries of the signal. For example, reversingthe magnetizations of both Hall contacts from the þxto\n/C0xdirections reverses the sign of the spin-dependent\nvoltage but not an ordinary Hall voltage, which can there-fore be removed by subtracting the data obtained in the twodifferent parallel states. At low applied fields, local Halleffects are due predominantly to the xcomponents of the\ncontact magnetization. The corresponding fringe fields,which are in the /C6zdirection are even with respect to\nB\ny, while the spin-dependent signal is odd in By. We can\ntherefore eliminate local Hall effects by retaining only the\ncomponents of the signal that are odd with respect to By.\nThe data from Fig. 1(c) are shown in Fig. 1(d) after\nremoving the first two backgrounds. By construction, thesedata are odd with respect to B\ny, and they show extrema at\nintermediate fields (approximately 250 Oe) as expected.\nThe magnitude of the voltage at these maxima corresponds\nto a spin polarization P¼ðn\"/C0n#Þ=ðn\"þn#Þ/C251:3%at\nthe sample edges, where\nP¼e/C1V\n/C17PFe3m/C3\n@2ð3/C252nÞ2=3: (1)\nIn this expression, which follows from the usual relation-\nship between the spin accumulation and the density ofstates [ 16],P\nFe¼0:42is the spin polarization of Fe at\nthe Fermi level and /C17/C240:5is the interfacial transparency.\nThere are, however, additional features in the field sweeps\nnear 1 kOe that do not reverse sign when the current is\nreversed, and hence cannot be due to a Hall effect. Thesederive from the features (at the same fields) in the data ofFig.1(c)and result from the current that is shunted through\nthe Fe contacts (and hence has a component perpendicularto the plane) in combination with tunneling anisotropicmagnetoresistance (TAMR) at the Schottky contact [ 18].\nThis final background contribution can be minimized by\nsubtracting the voltages for the two current directions, as\nwill be done for all subsequent data shown in this Letter.\nWe have also performed the same measurements with\nthe FM contacts on opposite sides of the channel initializedin either of the antiparallel states \"#or#\". The data in this\ncase are shown as the solid line in Fig. 1(d)after removal of\nall three backgrounds. This curve shows no signal, indicat-ing that the spin accumulations at opposite edges of the\nsample are opposite in sign.\nData taken at different contact separations for the x¼0,\n0.03, 0.05 and 0.06 devices at T¼30 K andj\nx¼/C62:9/C2\n103A=cm2are shown in Fig. 2. The SHE-induced spin\nFIG. 1 (color online). (a) Micrograph of a spin Hall device\nwith Fe contacts located 10/C22mfrom the edges of the GaAs\nchannel. The devices with contacts closer to the edges have athird contact in the center of the channel that is not used in this\nexperiment. The contact pairs abandcdare used to measure the\nspin accumulation. (b) Nonlocal spin valve (—) and Hanle effect(d) data obtained on a GaAs device at T¼60 K forj\nInj¼\n8:2/C2102A=cm2. Hanle data are shown for both parallel and\nantiparallel states of injector and detector. (c) The measuredvoltage V\nab/C0Vcdfor a GaAs device with Fe contacts 2/C22m\nfrom the edges at T¼30 K forjx¼5:7/C2103A=cm2in the\ntwo different parallel states ( \"\",d) and ( ##,/C13). An offset\nvoltage of 13.2 mV has been subtracted from both sets of data.(d) The same data after extraction of the spin Hall signal for both\npositive ( d) and negative ( /C13) currents. The spin Hall signal in\nthe antiparallel state is shown as the solid line.PRL 105, 156602 (2010) PHYSICAL REVIEW LETTERSweek ending\n8 OCTOBER 2010\n156602-2\nFIG. 27 (a) Micrograph of a spin Hall device with Fe con-\ntacts located 10 \u0016m from the edges of the GaAs channel. The\ncontact pairs aband cdare used to measure the spin accu-\nmulation. (b) Nonlocal spin valve (red lines) and Hanle e\u000bect\n(black dots) data obtained on a GaAs device at T = 60 K\nfor injection current 8 :2\u0002102A/cm2. (c) Measured voltage\nVab\u0000Vcdfor a GaAs device with Fe contacts 2 \u0016m from the\nedges at T = 30 K for a channel current 5 :7\u0002103A/cm2. An\no\u000bset voltage of 13.2 mV has been subtracted from the data.\nIn (b) and (c) data is shown for both parallel and antipar-\nallel states of injector and detector. (d) spin Hall signal for\nboth positive (full circle) and negative (open circle) currents,\nafter removing background and extracting antisymmetric sig-\nnal. The spin Hall signal in the antiparallel state is shown as\nthe solid red line. From Ref. Garlid et al. , 2010.\nvoltage transverse to it (see also Sec. II and Fig. 2(c)).\nThe fabrication of such structure is challenging, albeit\nnot impossible. Hankiewicz et al. , 2004 considered the\nsame concept but on a planar structure shaped as an H,\nwhich is much simpler to fabricate. The device and mea-\nsurement principle is shown in Fig. 28 (see also Sec. II\nand Fig. 2(b)). An electric current is applied in one of\nthe legs of the H-shaped structure and generates a trans-\nverse spin-current owing to the SHE. The spin-current\npropagates towards the other leg through the connecting\npart and produces a measurable voltage via the ISHE.\nThis non-local voltage in the second leg dominates local\ncontributions if the separation between the legs is large\nenough.\nBr une et al. , 2010 used devices based on high-mobility\nHgTe/(Hg, Cd)Te quantum wells with a top gate elec-\ntrode. The H-structures consisted of legs 1 \u0016m long and\n200 nm wide, with the connecting part being 200 nm wide\nand 200 nm long. The estimated mean free path in the\nsystem was\u00152:5\u0016m, i.e., the samples are well within the\nV+\nV-I-I+\na\nbFIG. 28 (a) Scanning electron micrograph of a H-shape de-\nvice and probe con\fguration for spin injection via SHE and\nspin detection via ISHE. (b) The inset indicates the measure-\nment con\fguration for current injection (arrows) and volt-\nage probes. The black curve in the main panel shows the\nnon-local ISHE resistance signal. The blue solid curve indi-\ncates the residual voltage owing to current spreading. From\nRef. Br une et al. , 2010.\nquasi-ballistic regime. Sweeping the gate voltage in the\nsample allowed to vary the strength of the Rashba spin-\norbit coupling by a variation of both the electrical \feld\nacross the quantum well and the Fermi level in the quan-\ntum well. In the sample it was possible to electrically\ntune the carrier density from strongly n-type, through\ninsulating, down to a p-type regime. This resulted in\na strong modulation of the ISHE voltage, as shown in\nFig. 28. In the p-regime, where the spin-orbit coupling is\nstrong, the signal is at least one order of magnitude larger\nthan in the weakly spin-orbit coupled n-regime. Detailed\nnumerical calculations con\frmed that the observed spin\nHall signals had the ballistic intrinsic origin (Br une et al. ,\n2010).\nAn H-shaped structure was also used in graphene de-\nvices (Abanin et al. , 2011a). Here, a large Hall response\nwas observed near the graphene neutrality point in the33\npresence of an external magnetic \feld. The results were\nascribed to spin-currents that resulted from the imbal-\nance of the Hall resistivity for the spin-up and spin-down\ncarriers induced by the Zeeman interaction; a process\nthat does not involve a spin-orbit interaction, i.e. is not\nof the SHE origin, and that is largest in the cleanest\ngraphene samples (Abanin et al. , 2011b). More recently,\nthe controlled addition of small amounts of covalently\nbonded hydrogen atoms has been reported to induce an\nenhancement of the spin-orbit interaction by three or-\nders of magnitude in graphene (Balakrishnan et al. , 2013;\nCastro Neto and Guinea, 2009). Such large enhancement\nwas estimated from nonlocal signals of up to 100 \n, which\nare observed at zero external magnetic \felds and at room\ntemperature. From the magnetic \feld and the length de-\npendence of the non-local signal, a spin orbit strength\nof 2.5 meV was extracted for samples with 0.05% hydro-\ngenation.\n6. Spin Hall magnetoresistance\nIn bilayer FM/NM systems a new type of magnetore-\nsistance has been recently discovered which is directly as-\nsociated with the SHE (Huang et al. , 2012; Weiler et al. ,\n2012). The observed magnetoresistance is given by\n\u001a=\u001a0+\u001a1( ^m\u0001(^j\u0002^z))2; (4.1)\nwhere\u001a0is the normal resistance, \u001a1is the anisotropic\nresistance amplitude, and ^j, ^m, and ^zare the directional\nvectors of the current, the magnetization, and the normal\nto the interface. This means that the magnetoresistance\ndepends on the in-plane component of the magnetization\nperpendicular to the current. In contrast, the conven-\ntional non-crystalline AMR (McGuire and Potter, 1975)\nhas the form of\n\u001a=\u001a0+\u001a1(^j\u0001^m)2; (4.2)\nwith ^j\u0001^m= cos(\u0012j\u0000m), where\u0012j\u0000mis the angle between\nthe current and the magnetization.\nThis phenomenon has been termed the spin Hall mag-\nnetoreristance (SHMR) (Chen et al. , 2013; Hahn et al. ,\n2013; Nakayama et al. , 2013; Vlietstra et al. , 2013; Weiler\net al. , 2012). Its origin is illustrated in Fig. 29. When\na current \rows parallel to the FM/NM interface a SHE\nspin-current is generated in the NM directed to the in-\nterface. If the magnetization is parallel to the polariza-\ntion of the spin-current generated by the SHE, it gets re-\n\rected at the interface and a spin-current back \rows, as\nsketched in Fig. 29(a). This back \row spin-current then\ngets transformed into a charge current via the ISHE in\nthe direction of the longitudinal current. If the magneti-\nzation is instead perpendicular to the polarization of the\nspin-current generated by the SHE, it can enter the FM\nand dephase, as shown in Fig. 29(b). In this case there\nis no spin-current back \row and no contribution via the\nISHE to the longitudinal current in the NM.\nSpin Hall Magnetoresistance Induced by a Nonequilibrium Proximity EffectH. Nakayama,1,2M. Althammer,3,4Y .-T. Chen,5K. Uchida,1,6Y. Kajiwara,1D. Kikuchi,1,7T. Ohtani,1S. Gepra¨gs,3M. Opel,3S. Takahashi,1R. Gross,3,8G. E. W. Bauer,1,5,7,*S. T. B. Goennenwein,3,†and E. Saitoh1,7,9,10,‡1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan2Laboratory for Nanoelectronics and Spintronics, Research Institute of Electrical Communication, Tohoku University,Sendai 980-8577, Japan3Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany4Center for Materials Information Technology MINT and Department of Chemistry, University of Alabama,Tuscaloosa, Alabama 35487, USA5Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands6PRESTO, Japan Science and Technology Agency, Saitama 332-0012, Japan7WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan8Physik-Department, Technische Universita¨t Mu¨nchen, 85748 Garching, Germany9CREST, Japan Science and Technology Agency, Tokyo 102-0076, Japan10The Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan(Received 31 October 2012; published 13 May 2013)We report anisotropic magnetoresistance inPtjY3Fe5O12bilayers. In spite ofY3Fe5O12being a verygood electrical insulator, the resistance of the Pt layer reflects its magnetization direction. The effectpersists even when a Cu layer is inserted between Pt andY3Fe5O12, excluding the contribution of inducedequilibrium magnetization at the interface. Instead, we show that the effect originates from concertedactions of the direct and inverse spin Hall effects and therefore call it ‘‘spin Hall magnetoresistance.’’DOI:10.1103/PhysRevLett.110.206601PACS numbers: 72.25.Ba, 72.25.Mk, 75.47./C0m, 75.76.+jThe resistance of a metallic magnet depends on itsmagnetization direction, a phenomenon called magnetore-sistance (MR). Several types of MR, i.e., anisotropic mag-netoresistance (AMR) [1], giant magnetoresistance [2–4],and tunnel magnetoresistance [5–9] are presently indis-pensable in data storage technology. For these MRs tooccur, conduction electrons must pass through the magnet.Here we report the discovery of a fundamentally differentMR that is caused bynonequilibriumproximity magneti-zation of a metallic Pt film attached to an electricallyinsulating magnetY3Fe5O12(YIG). Although the conduc-tion electrons in the Pt film cannot enter the magneticinsulator, thePtjYIGbilayer resistance reflects the magne-tization direction of insulating YIG.Spin transport and charge transport phenomena areinterconnected. For example, the spin Hall effect (SHE)refers to conversion of an electric current into a transversespin current, i.e., a net flow of electron magnetic moments,due to the spin-orbit interaction (SOI). The conversionefficiency of the SHE is enhanced in heavy metals suchas Pt in which the SOI is very strong [Fig.1(a)]. Thereciprocal of the SHE is the inverse spin Hall effect(ISHE), i.e., the conversion of an injected spin currentinto a transverse electric current or voltage [Fig.1(c)].Here the directions of electric-current flowJe, spin-currentflowJs, and spin-current polarization/C27are at right anglesto one another [10–17].The SHE generates spin currents and spin accumula-tions. On the other hand, the ISHE has become useful fordetecting spin currents and spin-based electric powergeneration [10–17]. Here a question arises: Is it possiblethat SHE and ISHE operate simultaneously? Based onour recent understanding of interfacial spin mixing at theinterface between a magnetic insulator and a metal[18–22], we can now answer this question affirmatively.\n(a) (c)(b)\n(d)Jsback(e) (f)JeσJsparamagnet (N)Je´\nM || σJeMJsback´ M ⊥σJeMJsbackJsabs´JePt = NYIG = FPt = NJsabsJsJsFIG. 1 (color). (a), (b), (c) Illustrations of the magnetic controlof the conductivity due to the direct and inverse spin Hall effects(SHE and ISHE) in a paramagnetic thin film metal (N) withstrong spin-orbit interaction attached to a ferromagnetic insula-tor (F). (d), (e) Illustrations of the geometric relation betweenthe flow of electrons and accumulated spins inN¼Ptand themagnetization in the magnetic insulatorF¼YIG. (f) Schematicillustration of the spin accumulation generated by nonequilib-rium proximity due to the SHE inN. At the interfaces ofN, thespin accumulation is formed depending on its spin polarizationdirection. Dashed curves inNshow the electron motions withdifferent spin polarization directions; the blue (red) arrows moveto the upper (lower) side.PRL110,206601 (2013)\nSelected for aViewpointinPhysicsPHYSICAL REVIEW LETTERSweek ending17 MAY 2013\n0031-9007=13=110(20)=206601(5) 206601-1/C2112013 American Physical Society(a)\t\r (b)\t\r FIG. 29 Illustration of the SHE magnetoresistance. (a) When\nthe magnetization aligns with the polarization of the SHE\nspin-current, its back \row re\rection generates an ISHE cur-\nrent that contributes to the longitudinal current. (b) When\nthe magnetization is perpendicular to polarization of the SHE\nspin-current, the spin-current is absorbed and no ISHE cur-\nrent a\u000bects the longitudinal current. From Ref. Nakayama\net al. , 2013.\nThe typical experimental results are illustrated in\nFig. 30, where the bilayer system was YIG/Pt. The mag-\nnetoresistance traces are measured as a function of the\nmagnetization angle in the x-y plane parallel to the inter-\nface, and in the z-y and z-x planes that are perpendicu-\nlar to the interface. The measured angular dependencies\nare consistent with the SHMR phenomenology described\nby Eq. (4.1) and are inconsistent with the AMR expres-\nsion (4.2). The theory of the e\u000bect was derived by Chen\net al. , 2013 based on the scattering formalism and the\nspin-charge drift-di\u000busion equations.\nat the interface and dissipated as a spin-transfer torque to\nthe magnetization, thereby modulating the spin-current\ndistribution in Pt. Because the current-induced spin accu-\nmulation is polarized along the ydirection, the polarization\ndirection of the modulated spin current flowing along z\nvaries as /m/C2ðm/C2^yÞ. This in turn modulates the lon-\ngitudinal (applied) electric current as /^y/C1½m/C2ðm/C2^yÞ/C138 ¼\nm2y/C01and induces a transverse electric current\n/^x/C1½m/C2ðm/C2^yÞ/C138 ¼ mxmyin the ydirection due to\nthe ISHE, where mxandmyare the Cartesian components\nofm. The prefactors of these dependencies can be com-\nputed by spin diffusion theory [ 15] and quantum mechani-\ncal boundary conditions in terms of the spin-mixing\nconductance [ 32], thereby fully explaining the observed\nSMR in PtjYIG [26]. The SMR resistivity change can\nhence be formulated as\n/C26xx¼/C260/C0/C1/C26Sm2y;/C26 xy¼/C1/C26Smxmy: (1)\nThis is very different from the AMR phenomenology of\npolycrystalline conductive ferromagnets [ 1]\n/C26xx¼/C26?þ/C1/C26Am2x;/C26 xy¼/C1/C26Amxmy:(2)\nIn both expressions the resistivity /C26xxis measured along\nthe direction of the electric-current flow Je(along the x\ndirection, cf. Fig. 4), while /C26xyis the resistivity componentrecorded in the sample plane perpendicular to Je(along\ntheydirection), which typically appears in the magneto-\nresistive properties of ferromagnets [ 1]./C260is a constant\nresistivity offset, /C1/C26Sand/C1/C26A(¼/C26k/C0/C26?) are the mag-\nnitude of the resistivity change as a function of the mag-\nnetization orientation, /C26kand/C26?are the resistivities for\nmagnetizations aligned along and perpendicular to Je,\nrespectively. In Figs. 4(a)and4(b), we show the evolution\nof the MR xxandMR xy¼/C26xyðHÞ=/C26xxðH¼0Þin sample 1\nas a function of H, applied at different angles /C11.T o\nquantitatively evaluate this dependence, we show the\nevolution of the MR xxandMR xyas a function of /C11in\nFigs. 4(c) and4(d), respectively (symbols). The MR xxfor\nrotations of the magnetization in the plane perpendicular to\ntheydirection (angle /C13) and perpendicular to the xdirec-\ntion (angle /C12) are summarized in Figs. 4(e)and4(f), while\nFigs. 4(i)and4(j)show corresponding transport data for\nthe sample 2. The behavior of the electric resistanceexpected from the AMR according to Eq. ( 2) is shown as\nblue curves in these panels, while the SMR predicted by\nEq. ( 1) is depicted by red curves. The out-of-plane rotation\ndata are consistently described in terms of the SMR; theangle-dependent MR data thus show that the MR observed\nin experiment indeed is due to the SMR effect. For a\n12-nm-thick Pt film with the resistivity 8:6/C210\n/C07/C10m\nthe theory sketched above agrees with the experimental-100 0 100\nH (Oe)0.01 %α= 0\n75°15°\n30°\n45°\n60°\n90°\n105°\n120°\n135°\n150°\n165°\n180°\n-100 0 100\nH (Oe)0.01 %α= 0\n75°15°\n30°\n45°\n60°\n90°\n105°\n120°\n135°\n150°\n165°\n180°MR xy (%)(k) α:\nyyz\nx, J eαH\n β:\nz\nHβ\ny\nx, J e γ:\nzHγ\ny\nx, J e\n α, β, γ-90° 0° 90° 180° 270°\nMR xy-11-11\n-101\n-101MR xx(g)\nexp. SMR calc.\nAMR calc.(h)\n(j)(i)MRxx (α) [sample 2]\nMRxy (α)\nMRxx (γ)\nMRxx (β)-90° 0° 90° 180° 270°\n α, β, γ-11\n-101\n-101-11\nMR xy(d)\n(e)(c)\n(f)exp. SMR calc.\nAMR calc.MRxx (α) [sample 1]\nMRxy (α)\nMRxx (γ)\nMRxx (β)(a) diagonal (b)  off-diagonal\nPt/YIGVxx VxyMR xx (%)Pt/YIG\nMRxx\nnorm normnorm norm\nFIG. 4 (color). (a), (b) Diagonal and off-diagonal components of the MR in PtjYIG films as a function of in-plane angle /C11. (c), (d),\n(e), (f), (g), (h), (i), (j) /C11,/C12, and/C13dependence of the normalized MR xx[MRnormxx¼MR xx=MR xxð/C11¼0Þ] and MR xy[MRnormxy¼\nMR xy=MR xyð/C11¼/C045/C14Þ] in two different samples, where the angles /C11,/C12, and/C13are defined in (k). The red and blue curves show MR\nexpected according to the SMR model and the AMR model, respectively. (c) and (g) show /C11dependence of MR xx, (d) and (h) show /C11\ndependence of MR xy, (e) and (i) show /C13dependence of MR xx, and (f) and (j) show /C12dependence of MR xx.PRL 110, 206601 (2013) PHYSICAL REVIEW LETTERSweek ending\n17 MAY 2013\n206601-4at the interface and dissipated as a spin-transfer torque to\nthe magnetization, thereby modulating the spin-current\ndistribution in Pt. Because the current-induced spin accu-\nmulation is polarized along the ydirection, the polarization\ndirection of the modulated spin current flowing along z\nvaries as /m/C2ðm/C2^yÞ. This in turn modulates the lon-\ngitudinal (applied) electric current as /^y/C1½m/C2ðm/C2^yÞ/C138 ¼\nm2y/C01and induces a transverse electric current\n/^x/C1½m/C2ðm/C2^yÞ/C138 ¼ mxmyin the ydirection due to\nthe ISHE, where mxandmyare the Cartesian components\nofm. The prefactors of these dependencies can be com-\nputed by spin diffusion theory [ 15] and quantum mechani-\ncal boundary conditions in terms of the spin-mixing\nconductance [ 32], thereby fully explaining the observed\nSMR in PtjYIG [26]. The SMR resistivity change can\nhence be formulated as\n/C26xx¼/C260/C0/C1/C26Sm2y;/C26 xy¼/C1/C26Smxmy: (1)\nThis is very different from the AMR phenomenology of\npolycrystalline conductive ferromagnets [ 1]\n/C26xx¼/C26?þ/C1/C26Am2x;/C26 xy¼/C1/C26Amxmy:(2)\nIn both expressions the resistivity /C26xxis measured along\nthe direction of the electric-current flow Je(along the x\ndirection, cf. Fig. 4), while /C26xyis the resistivity componentrecorded in the sample plane perpendicular to Je(along\ntheydirection), which typically appears in the magneto-\nresistive properties of ferromagnets [ 1]./C260is a constant\nresistivity offset, /C1/C26Sand/C1/C26A(¼/C26k/C0/C26?) are the mag-\nnitude of the resistivity change as a function of the mag-\nnetization orientation, /C26kand/C26?are the resistivities for\nmagnetizations aligned along and perpendicular to Je,\nrespectively. In Figs. 4(a)and4(b), we show the evolution\nof the MR xxandMR xy¼/C26xyðHÞ=/C26xxðH¼0Þin sample 1\nas a function of H, applied at different angles /C11.T o\nquantitatively evaluate this dependence, we show the\nevolution of the MR xxandMR xyas a function of /C11in\nFigs. 4(c) and4(d), respectively (symbols). The MR xxfor\nrotations of the magnetization in the plane perpendicular to\ntheydirection (angle /C13) and perpendicular to the xdirec-\ntion (angle /C12) are summarized in Figs. 4(e)and4(f), while\nFigs. 4(i)and4(j)show corresponding transport data for\nthe sample 2. The behavior of the electric resistanceexpected from the AMR according to Eq. ( 2) is shown as\nblue curves in these panels, while the SMR predicted by\nEq. ( 1) is depicted by red curves. The out-of-plane rotation\ndata are consistently described in terms of the SMR; theangle-dependent MR data thus show that the MR observed\nin experiment indeed is due to the SMR effect. For a\n12-nm-thick Pt film with the resistivity 8:6/C210\n/C07/C10m\nthe theory sketched above agrees with the experimental-100 0 100\nH (Oe)0.01 %α= 0\n75°15°\n30°\n45°\n60°\n90°\n105°\n120°\n135°\n150°\n165°\n180°\n-100 0 100\nH (Oe)0.01 %α= 0\n75°15°\n30°\n45°\n60°\n90°\n105°\n120°\n135°\n150°\n165°\n180°MR xy (%)(k) α:\nyyz\nx, J eαH\n β:\nz\nHβ\ny\nx, J e γ:\nzHγ\ny\nx, J e\n α, β, γ-90° 0° 90° 180° 270°\nMR xy-11-11\n-101\n-101MR xx(g)\nexp. SMR calc.\nAMR calc.(h)\n(j)(i)MRxx (α) [sample 2]\nMRxy (α)\nMRxx (γ)\nMRxx (β)-90° 0° 90° 180° 270°\n α, β, γ-11\n-101\n-101-11\nMR xy(d)\n(e)(c)\n(f)exp. SMR calc.\nAMR calc.MRxx (α) [sample 1]\nMRxy (α)\nMRxx (γ)\nMRxx (β)(a) diagonal (b)  off-diagonal\nPt/YIGVxx VxyMR xx (%)Pt/YIG\nMRxx\nnorm normnorm norm\nFIG. 4 (color). (a), (b) Diagonal and off-diagonal components of the MR in PtjYIG films as a function of in-plane angle /C11. (c), (d),\n(e), (f), (g), (h), (i), (j) /C11,/C12, and/C13dependence of the normalized MR xx[MRnormxx¼MR xx=MR xxð/C11¼0Þ] and MR xy[MRnormxy¼\nMR xy=MR xyð/C11¼/C045/C14Þ] in two different samples, where the angles /C11,/C12, and/C13are defined in (k). The red and blue curves show MR\nexpected according to the SMR model and the AMR model, respectively. (c) and (g) show /C11dependence of MR xx, (d) and (h) show /C11\ndependence of MR xy, (e) and (i) show /C13dependence of MR xx, and (f) and (j) show /C12dependence of MR xx.PRL 110, 206601 (2013) PHYSICAL REVIEW LETTERSweek ending\n17 MAY 2013\n206601-4at the interface and dissipated as a spin-transfer torque to\nthe magnetization, thereby modulating the spin-current\ndistribution in Pt. Because the current-induced spin accu-\nmulation is polarized along the ydirection, the polarization\ndirection of the modulated spin current flowing along z\nvaries as /m/C2ðm/C2^yÞ. This in turn modulates the lon-\ngitudinal (applied) electric current as /^y/C1½m/C2ðm/C2^yÞ/C138 ¼\nm2y/C01and induces a transverse electric current\n/^x/C1½m/C2ðm/C2^yÞ/C138 ¼ mxmyin the ydirection due to\nthe ISHE, where mxandmyare the Cartesian components\nofm. The prefactors of these dependencies can be com-\nputed by spin diffusion theory [ 15] and quantum mechani-\ncal boundary conditions in terms of the spin-mixing\nconductance [ 32], thereby fully explaining the observed\nSMR in PtjYIG [26]. The SMR resistivity change can\nhence be formulated as\n/C26xx¼/C260/C0/C1/C26Sm2y;/C26 xy¼/C1/C26Smxmy: (1)\nThis is very different from the AMR phenomenology of\npolycrystalline conductive ferromagnets [ 1]\n/C26xx¼/C26?þ/C1/C26Am2x;/C26 xy¼/C1/C26Amxmy:(2)\nIn both expressions the resistivity /C26xxis measured along\nthe direction of the electric-current flow Je(along the x\ndirection, cf. Fig. 4), while /C26xyis the resistivity componentrecorded in the sample plane perpendicular to Je(along\ntheydirection), which typically appears in the magneto-\nresistive properties of ferromagnets [ 1]./C260is a constant\nresistivity offset, /C1/C26Sand/C1/C26A(¼/C26k/C0/C26?) are the mag-\nnitude of the resistivity change as a function of the mag-\nnetization orientation, /C26kand/C26?are the resistivities for\nmagnetizations aligned along and perpendicular to Je,\nrespectively. In Figs. 4(a)and4(b), we show the evolution\nof the MR xxandMR xy¼/C26xyðHÞ=/C26xxðH¼0Þin sample 1\nas a function of H, applied at different angles /C11.T o\nquantitatively evaluate this dependence, we show the\nevolution of the MR xxandMR xyas a function of /C11in\nFigs. 4(c) and4(d), respectively (symbols). The MR xxfor\nrotations of the magnetization in the plane perpendicular to\ntheydirection (angle /C13) and perpendicular to the xdirec-\ntion (angle /C12) are summarized in Figs. 4(e)and4(f), while\nFigs. 4(i)and4(j)show corresponding transport data for\nthe sample 2. The behavior of the electric resistanceexpected from the AMR according to Eq. ( 2) is shown as\nblue curves in these panels, while the SMR predicted by\nEq. ( 1) is depicted by red curves. The out-of-plane rotation\ndata are consistently described in terms of the SMR; theangle-dependent MR data thus show that the MR observed\nin experiment indeed is due to the SMR effect. For a\n12-nm-thick Pt film with the resistivity 8:6/C210\n/C07/C10m\nthe theory sketched above agrees with the experimental-100 0 100\nH (Oe)0.01 %α= 0\n75°15°\n30°\n45°\n60°\n90°\n105°\n120°\n135°\n150°\n165°\n180°\n-100 0 100\nH (Oe)0.01 %α= 0\n75°15°\n30°\n45°\n60°\n90°\n105°\n120°\n135°\n150°\n165°\n180°MR xy (%)(k) α:\nyyz\nx, J eαH\n β:\nz\nHβ\ny\nx, J e γ:\nzHγ\ny\nx, J e\n α, β, γ-90° 0° 90° 180° 270°\nMR xy-11-11\n-101\n-101MR xx(g)\nexp. SMR calc.\nAMR calc.(h)\n(j)(i)MRxx (α) [sample 2]\nMRxy (α)\nMRxx (γ)\nMRxx (β)-90° 0° 90° 180° 270°\n α, β, γ-11\n-101\n-101-11\nMR xy(d)\n(e)(c)\n(f)exp. SMR calc.\nAMR calc.MRxx (α) [sample 1]\nMRxy (α)\nMRxx (γ)\nMRxx (β)(a) diagonal (b)  off-diagonal\nPt/YIGVxx VxyMR xx (%)Pt/YIG\nMRxx\nnorm normnorm norm\nFIG. 4 (color). (a), (b) Diagonal and off-diagonal components of the MR in PtjYIG films as a function of in-plane angle /C11. (c), (d),\n(e), (f), (g), (h), (i), (j) /C11,/C12, and/C13dependence of the normalized MR xx[MRnormxx¼MR xx=MR xxð/C11¼0Þ] and MR xy[MRnormxy¼\nMR xy=MR xyð/C11¼/C045/C14Þ] in two different samples, where the angles /C11,/C12, and/C13are defined in (k). The red and blue curves show MR\nexpected according to the SMR model and the AMR model, respectively. (c) and (g) show /C11dependence of MR xx, (d) and (h) show /C11\ndependence of MR xy, (e) and (i) show /C13dependence of MR xx, and (f) and (j) show /C12dependence of MR xx.PRL 110, 206601 (2013) PHYSICAL REVIEW LETTERSweek ending\n17 MAY 2013\n206601-4\nMRxx\t\r norm\t\r (a)\t\r \n(b)\t\r \n(c)\t\r \nFIG. 30 Magnetoresistance curves as a function of the angles\n(a)\u000b, (b)\r, and (c)\f, illustrated in the right panel. The key\ncontrast to conventional AMR is the trace in (b), where no\ndependence is observed, while conventional AMR would give\nthe sinusoidal form illustrated in the dashed-blue line. From\nRef. Nakayama et al. , 2013.34\nD. Spin Hall e\u000bect coupled to magnetization dynamics\nWhen the SHE is studied by coupling to magnetiza-\ntion dynamics three di\u000berent FMR-based techniques can\nbe found: (i) Ferromagnetic resonance { spin pumping\n(FMR-SP), (ii) modulation of damping (MOD) experi-\nments, and (iii) spin Hall e\u000bect { spin transfer torque\n(SHE-STT). The general underlying principle for the\nthree methods is similar. In a bilayer NM/FM structure,\nthe FM is used to inject or absorb a dynamic spin-current\ninto or from the NM. (Note that these studies have been\nalso extended to replacing the SHE/ISHE generating NM\nwith another FM or antiferromagnet (Azevedo et al. ,\n2014; Freimuth et al. , 2010; Mendes et al. , 2014; Miao\net al. , 2013).)\nIn FMR-SP, a spin-current is injected from the FM\ninto the NM. The injected spin-current is a pure AC spin-\ncurrent which is not accompanied by a charge current but\nwhich nevertheless can be detected electrically since it is\nconverted into a charge current by means of the ISHE in\nthe NM (Saitoh et al. , 2006). The e\u000eciency of the con-\nversion process can be quanti\fed by the spin Hall angle.\nSince in the process of spin injection angular momentum\nis lost in the FM, the FMR-SP leads to a broadening\nof the FMR line (Heinrich et al. , 2003; Mizukami et al. ,\n2001; Urban et al. , 2001).\nIn MOD experiments, the direct SHE induced in the\nNM by a DC electrical current is used to modify the\ndamping in the FM which is concomitantly driven into\nFMR by the application of an RF magnetic \feld. In this\napproach, the DC spin-current generated by the SHE and\ninjected across the NM/FM interface leads to a damp-\ning or antidamping-like torque acting on the precessing\nmagnetization of the FM. Modulation of the damping is\nobserved as a function of the applied DC charge current\nand a detailed line-width analysis allows extraction of the\nspin Hall angle (Saitoh and Ando, 2012). Note that the\npure DC spin-current is generated in the bulk of the NM\nand that in order to quantitatively determine the spin\nHall angle it is important to know the transmissibility of\nthe NM/FM interface for the pure spin-current.\nIn the SHE-STT, a spin-current is used to transfer spin\nangular momentum and thus to exert a torque on the\nmagnetic moments. In these experiments an AC current\nsent along the NM/FM interface can create a RF excita-\ntion of the magnetization of the FM via the SHE-STT.\nIn conventional STT junctions, an electrical current is\nsent perpendicular to a stack with two FM electrodes to\ntransfer angular momentum from one FM to the other\nFM (Ralph and Stiles, 2008). SHE-STT experiments, on\nthe other hand, exploit the use of a perpendicular pure\nspin-current generated by an in-plane electrical current\nin the attached NM via the SHE.\nIn both the MOD experiments and the SHE-STT, the\ntorques in the FM that are generated by the SHE in\nthe NM would be in addition to the ISGE-related SOTs\npresent at the inversion asymmetric FM/NM interface\n(Freimuth et al. , 2013; Garello et al. , 2013; Kurebayashi\nFIG. 31 A spin-current is generated by SP at the FM/NM\ninterface (grey arrows). The time dependent spin polariza-\ntion of this current (indicated as a dark grey arrow) rotates\nalmost entirely in the y\u0000zplane. The small time averaged\nDC component (yellow arrow) appears along the xaxis. Both\ncomponents lead to charge currents in NM and can be con-\nverted into AC and DC voltages by placing probes along the\nxandydirection, respectively. From Ref. Wei et al. , 2014.\net al. , 2014). Hence, in these experiments the spin Hall\nangle is in reality a parametrization of the total torques\ngenerated by the currents and therefore it should be con-\nsidered instead as the e\u000bective spin Hall angle for the\nspeci\fc bilayer system.\nIn the rest of the section we expand on the details and\nrecent results of each of these FMR-based techniques.\nFMR-SP is the more widely used technique to measure\nthe e\u000bective spin Hall angle thus we detail this technique\nmore extensively.\n1. Ferromagnetic resonance spin pumping\nAs described in the theory section (Sec. III.D),\nTserkovnyak et al. , 2002a, 2005 have shown that the pre-\ncessing magnetization in a FM generates a spin-current\nstrictly at the FM/NM interface, as sketched in Fig. 31.\nThe spin-current generated at the interface propagates\ninto the NM and consequently decays on a length scale\nconnected to the e\u000bective spin di\u000busion length \u0015sdof the\nNM. As mentioned in the theory section, we note that\nthe term e\u000bective is used here, since the determination\nof the spin di\u000busion length for a NM interfaced with a\nFM may also be connected to spin memory loss and prox-\nimity polarization at the interface. In the case of Pt and\nPd in contact with a FM metal, proximity e\u000bects are\nwell known from x-ray magnetic circular dichroism ex-\nperiments.\nThe direction of the injected pure spin-current points\nfrom the FM to the NM and its polarization is time-\ndependent. Its projection onto the static magnetization\ndirection of the FM leads to a small DC component of\nthe injected spin-current into the NM. Performing time\naveraging one obtains a net DC spin-current given by35\nEq. (4.3) from Sec. III.D:\njs;dc=~!\n4\u0019~Arsin2\u0002;\nwhere!is the driving RF and \u0002 is the cone angle of pre-\ncession. Here ~Aris the e\u000bective SP conductance. If the\nthickness of the NM is smaller than the spin-di\u000busion\nlength, the build-up of spin accumulation will yield a\nback-\row spin-current which will reduce the total spin-\ncurrent into the NM. The SP conductance, ~Ar, is propor-\ntional to the real part of the mixing conductance, dis-\ncussed in Sec. III.D, and is reduced by this back \row.\nThe reduction depends on the ratio \u001ctr=\u001csf, the reduc-\ntion being strongest as this ratio increases. Hence, the\ne\u000bective spin mixing conductance may become small even\nthough a pure spin-current is e\u000eciently transferred across\nthe FM/NM interface. Recently, spin \rip scattering near\nthe FM/NM interface has been divided up into a spin\nmemory loss occuring directly at the interface (interface\nscattering) and the decay of the spin polarization as de-\nscribed above (Rojas-S\u0013 anchez et al. , 2014).\nThe ISHE is used to electrically detect pure spin-\ncurrents generated by the SP (Saitoh et al. , 2006), as\nshown Fig 32. In spin-orbit coupled NMs like Pt or Pd,\nthe ISHE converts the pure spin-current into a detectable\ncharge current given by Eq. (3.34) Sec. III.D:\n~jc=\u000bSH2e\n~~js\u0002~ \u001b(t):\nHere, the vector of the spin-current density ~jspoints per-\npendicular to the NM/FM interface into the NM. Note\nthat the vector of the spin-current polarization ~ \u001b(t) is\na time varying quantity, which we do not average here,\nsince it has now been demonstrated that the AC compo-\nnent is also measurable (Wei et al. , 2014; Weiler et al. ,\n2014). In Fig. 32, only the DC component of the spin-\ncurrent polarization is depicted.\nTo measure the e\u000bect of the injected spin-current via\nISHE, i.e. to measure the generated charge current, con-\ntact electrodes have to be attached to the sample. If the\ncoordinate system of Fig. 31 is considered, placing elec-\ntrodes along the y-direction allows detecting the small\nDC component of the SP-induced ISHE. In contrast, if\nthe contact electrodes are attached along the x-direction,\nthe much larger AC component in the GHz frequency\nrange can be detected when high frequency lines are used.\nIn case of DC detection the time averaged DC com-\nponent of the injected spin-current pointing along the\nx-direction (yellow arrow in Fig. 31) leads to a charge\ncurrent which is converted to a potential drop across the\nresistance of the NM and can be measured as a voltage\nsignal. When performing FMR-SP experiments not only\nvoltages due to ISHE are generated, but also due to, e.g.,\nthe AMR or the AHE. Thus, great care has to be taken\nto disentangle these contributions.\nIn the geometry sketched in Fig. 31 the propagation\ndirection of the spin-current is along zand its polariza-\ntion is along x-direction. Equation (3.35) from Sec. III.Dis then used to convert between this spin-current and the\nmeasured voltage.\nIn the original experiments by Saitoh et al. , 2006, the\nbilayer is placed in a FMR cavity in which the magnetic-\n\feld component of the microwave mode with frequency\n9.45 GHz is maximized while the electric-\feld compo-\nnent is minimized. The voltage probes are placed on the\nsides of the millimeter-sized sample (see Fig. 32). A sim-\nilar set-up was used by Azevedo et al. , 2011. Here, the\nsample is rotatable in the cavity and the cavity (i.e. the\ndirection of the RF excitation \feld) is kept \fxed with\nrespect to the DC external magnetic \feld. This experi-\nmental geometry has advantages and disadvantages. The\nmain advantage is that it is possible to \fnd an in-plane\nangle between excitation RF \feld and angular position of\nthe voltage probes where the AMR contribution to the\nsignal vanishes exactly while ISHE is detectable. Sec-\nond, in the in-plane excitation geometry typically used,\nthe sensitivity is large due to the large in-plane suscepti-\nbility at FMR. A major disadvantage is that it is not easy\nto perform frequency dependent measurements and that\ndue to the use of a cavity the exact amplitude of the exci-\ntation \feld, and thus the cone angle of precession which\nenters Eq. (4.3) in Sec. III.D, is usually not well known.\nFinally, since typically large, millimeter sized samples are\nused in the experiments, spurious RF electric \felds may\nlead to additional contributions due to the AHE. It is\ntherefore not straightforward to obtain an exact quanti-\ntative value of the spin Hall angle from cavity FMR-type\nmeasurements.\nIn experiments shown in Fig. 32, the measured FMR\nspectrum of the NiFe/Pt sample is compared to a refer-\nence NiFe sample (see Fig. 32(b)). The FMR line width\nof the NiFe/Pt sample is larger than that of the reference\nNiFe \flm which demonstrates the presence of the SP ef-\nfect in the NiFe/Pt. The induced voltage signal measured\nsimultaneously across the sample along an axis parallel\nto the NiFe/Pt interface is shown in Fig. 32(c). Saitoh\net al. , 2006 and Ando et al. , 2008 demonstrated that the\nsignal is present only when the spin polarization vector of\nthe injected spin-current has a component perpendicular\nto the measured electric \feld across the sample, consis-\ntent with the ISHE.\nAndo et al. , 2009 reported electrical detection of a\nspin wave resonance in nanostructured NiFe/Pt samples.\nElectrical tuning of the spin signal in a semiconductor has\nbeen recently demonstrated also by Ando et al. , 2011. In\nthe experiment, spins were injected from NiFe into GaAs\nthrough a Schottky contact using the FMR-SP. Tuning of\nthe SP e\u000eciency was achieved by applying a bias voltage\nacross the NiFe/GaAs Schottky barrier and interpreted\nas a consequence of a suppressed or enhanced spin cou-\npling across the interface. The FM/semiconductor SP\nexperiments in Ref. Ando et al. , 2011 were performed\nalso on samples with an ohmic contact between NiFe\nand GaAs. The measurements indicate that the resis-\ntance mismatch problem in ohmic metal/semiconductor\nspin-injection devices can be circumvented by using the36\nVHPtNi81Fe19\nNi81Fe19θMicrowave\nMagnetizationMagnetizationSpin pumpingJsJsJcJsPt+–σσ\n100150H (mT)100150H (mT)FMRNi81Fe19/Pt\nNi81Fe19/PtNi81Fe19θ = 90°θ = 90°dI(H) (arbitrary units)\ndV(H) /dH0010–4 V/THabc\nVHPtNi81Fe19\nNi81Fe19θMicrowave\nMagnetizationMagnetizationSpin pumpingJsJsJcJsPt+–σσ\n100150H (mT)100150H (mT)FMRNi81Fe19/Pt\nNi81Fe19/PtNi81Fe19θ = 90°θ = 90°dI(H) (arbitrary units)\ndV(H) /dH0010–4 V/THabc\nVHPtNi81Fe19\nNi81Fe19θMicrowave\nMagnetizationMagnetizationSpin pumpingJsJsJcJsPt+–σσ\n100150H (mT)100150H (mT)FMRNi81Fe19/Pt\nNi81Fe19/PtNi81Fe19θ = 90°θ = 90°dI(H) (arbitrary units)\ndV(H) /dH0010–4 V/THabc\nFIG. 32 Observation of the ISHE in a metal device with spin\ninjection from a FM by FMR-SP. (a) Schematic illustration\nof the NiFe/Pt sample system used in the study and of the\nSP e\u000bect and the ISHE. (b) Magnetic \feld dependence of the\nFMR signal for the NiFe/Pt bilayer \flm and a bare NiFe \flm.\nIdenotes the microwave absorption intensity. (c) Magnetic\n\feld dependence of dV(H)=dH for the NiFe/Pt sample. Vde-\nnotes the electric-potential di\u000berence between the electrodes\non the Pt layer. From Ref. Saitoh et al. , 2006.\nFMR-SP technique. Similar experiments have recently\nbeen performed also for spin injection into Si (Ando et al. ,\n2010b), Ge (Jain et al. , 2012) and organic semiconductors\n(Watanabe et al. , 2014).\nA second possibility to quantify the spin Hall angle\nhas been pioneered by Mosendz et al. , 2010a. They use a\nmicrostructured co-planar wave guide (CPW) with inte-\ngrated bilayer structure on top of the center wave guide.\nThis geometry allows excitation of FMR in the FM layer\nover a wide frequency range while the driving RF \feld\nis in the plane of the bilayer at 90\u000eto the long axis of\nthe several hundred micrometer long device (see Fig. 34).\nThe use of a wave guide structure allows precise knowl-\nedge of the amplitude of the RF \felds and thus the cone\nangle of the precessing magnetization. Voltage pick-up\nat the ends of the wire are used, perpendicular to the\ndirection of the RF driving \feld. Mosendz et al. , 2010a\napplied the external magnetic bias \feld at an angle of 45\u000e\nto the long axis of the wave guide. In this experimental\ngeometry both ISHE and AMR signals are detected at\nthe voltage probes as can be seen directly in the recorded\nvoltage traces (see Fig. 33).\nAMR leads to a parasitic DC voltage signal at FMR\ndue to the mixing of the time dependent resistivity (AMR\nand precessing magnetization) with a capacitively or in-\nductively coupled microwave current I(t) in the bilayer.\nThe AMR of the bilayer can be taken into account by con-\nsidering the orientation of the magnetization with respect\nto the current direction: RA=Rk\u0000R?. The general\n \nFIG. 33 (a),(b) Derivative of FMR spectra for Py/Pt (blue\nopen circles) and Py (black triangles). The solid lines are \fts\nto a Lorentzian FMR absorption function. (c) Voltage along\nthe samples vs. \feld DC magnetic \feld (Py/Pt: blue open\ncircles; Py: black triangles). Dotted and dashed lines show\nthe decomposition of the spectrum into a symmetric (ISHE)\nand antisymmetric (AMR) contribution. The solid line shows\nthe combined \ft for the Py/Pt sample. From Ref. Mosendz\net al. , 2010a.\nformula describing the parasitic voltage pick-up due to\nthe AMR is given by hV(t)i=hI(t)RA\u000bip(t) sin(2'H)i,\n(Mecking et al. , 2007; Obstbaum et al. , 2014) and it fol-\nlows that this time-averaged DC voltage is to \frst order\nproportional to the in-plane dynamic cone angle of the\nmagnetization \u000bip(t). The cone angle of precession can\neasily be calculated from the simultaneously measured\nsusceptibility at FMR in the exactly known geometry of\nthe CPW structure. The angle 'His de\fned in Fig. 34.\nNote that according to Bai et al. , 2013b, spurious e\u000bects\ndue to the AMR can be excluded by carefully analyz-\ning the high frequency characteristics of the CPWs used\nin the experiments with in-plane excitation, leading to a\nquantitative determination of the spin Hall angles.\nAnother possibility is to place the bilayer in the gap\nof the CPW (see Fig. 34). Now the in-plane dynamic\ncone angle relevant for the AMR is given by \u000bip(t) =37\n\u001fy'y'hx(t) sin('H)+\u001fy'zhz(t). The formula contains both\nin-plane and out-of-plane magnetic \felds, together with\nthe corresponding tensor elements of the susceptibility\n(\u001fij). Since the out-of-plane \feld produced by the CPW\nis about three orders of magnitude larger than its in-plane\ncomponent, one is tempted to simply neglect the terms\narising form the in-plane \feld. This approach is justi-\n\fed as long as only a single layer is studied. However, as\nsoon as a FM/NM bilayer with a highly conductive NM is\nused, the inductively or capacitively coupled microwave\ncurrent largely \rows in the NM and therefore generates\nan in-plane Oersted \feld of the same frequency and phase\nand with an amplitude comparable to the RF \feld gener-\nated by the CPW. Hence the RF current distribution in\nthe bilayer has a signi\fcant e\u000bect on the magnetization\ndynamics in the FM layer and can even be the dominat-\ning source of DC voltage generation by the AMR (Ob-\nstbaum et al. , 2014). Using standard electro-magnetic\nwave simulation codes, the RF magnetic \feld contribu-\ntion can be calculated rather accurately.\nWhen performing angular dependent measurements,\nthe symmetric and antisymmetric contributions due to\nthe ISHE and the AMR can be traced (see Fig. 34(a) and\n(b)). While for in-plane excitation the signal shows the\nsame angular dependence, for the out-of-plane excitation\ncase the antisymmetric contribution can be suppressed\ncompletely at an angle of 'H= 0 (see Fig. 34(d)). The\nvoltage contribution at this angle is thought to arise from\nISHE exclusively and allows quantitative determination\nof the spin Hall angle. Note that in these measurements\nboth symmetric and antisymmetric contributions can be\nobserved in a bare FM layer when the angle is set to\n'H= 45\u000e(see Fig. 34(c)).\n2. Spin Hall e\u000bect modulation of magnetization damping\nA MOD experiment that is the inverse of the FMR-\nSP was proposed by Ando et al. , 2008. In the MOD de-\nscribed in Fig. 36, a FM/NM bilayer (in this case Py/Pt)\nis placed in a microwave cavity (frequency 9.4 GHz) and\nsubjected to an RF driving \feld. By adjusting the exter-\nnal \feld, the bilayer can be brought into FMR. A typical\nFMR trace dI(H)=dH is shown in Fig. 36(b). The di-\nrection of the external magnetic \feld encloses an angle\n\u0012with the direction of current \row. Since the mm-sized\nsample consists of 10 nm NiFe and 10 nm Pt, the e\u000bect of\nSP which contributes to the relaxation of the precessing\nmagnetization can be observed as a line width broad-\nening when comparing to the data obtained for a plain\nNiFe \flm. Fig. 36(c) illustrates the e\u000bect of a DC cur-\nrent sent through the bilayer sample due to the combined\naction of the SHE and STT. Due to the SHE a spin-\ncurrent is generated in the Pt layer and enters the NiFe\n\flm. Its \row direction is perpendicular to the interface\nand its polarization direction ~ \u001bdepends on the direc-\ntion of current \row. The spin-current exerts a torque\non the precessing magnetization which either adds to\n \nFIG. 34 Symmetric (red dots) and antisymmetric (blue open\nsquares) voltage signals amplitudes at FMR (at 12 GHz) for a\nPy/Pt bilayer as a function of angle 'H. In (a) the magnetic\nexcitation \feld is in-plane placing a Py/NM bilayer on top of\nthe signal line of a CPW. Both symmetric and antisymmetric\namplitudes obey a sin( 'H) sin(2'H) behavior. (b) The mag-\nnetic excitation \feld generated by the CPW is out-of-plane\nwith respect to the Py/Pt layers. The amplitudes of the an-\ntisymmetric part follow a ( asin('H) +b) sin(2'H) behavior.\nThe symmetric part obeys ( csin('H)+d) sin(2'H)+ecos('H),\nwhich re\rects the fact, that the symmetric part is due to AMR\nand ISHE. (c) Voltage at FMR for 'H= 45\u000e, and (d)'H= 0\u000e\nfor a single Py layer and a PyPt bilayer. From Ref. Obstbaum\net al. , 2014.\nthe damping torque or opposes it. The e\u000bect is maxi-\nmized when the external magnetic \feld points perpen-\ndicular to the direction of current \row. For the situation\nsketched here, the spin-current density can be written as\n~js=\u000bSH~\n2e^n\u0002~jc=\u000bSH~\n2e\f\f\f~jc\f\f\f^\u001b. The e\u000bect of the in-\njected spin-current on the precessing magnetization can\nbe modelled in terms of an additional STT contribution\nto the Landau-Lifshitz Gilbert equation (Ando et al. ,\n2008; Liu et al. , 2011) that has to be added on top of\nthe SP contribution:\n~ \u001cSTT=\u0000\u00160\r\u000bSH\u0011~\n2ejc\n\u00160M2sdPy~M\u0002\u0010\n~M\u0002^\u001b\u0011\n(4.3)\nHere,dPyis the thickness of the Py layer. For the sake\nof simplicity, the factor38\n\u0014=\u000bSH\u0011~\n2ejc\n\u00160M2sdPy(4.4)\nis introduced. Note that this factor is dimensionless and\n\u0014 < 0 forjc>0 due to the negative electron charge.\nThe parameter \u0011de\fnes the so called injection e\u000eciency\nand contains the e\u000bects of spin-current losses near the\ninterface. There is no consensus on the exact ingredients\nfor this parameter, so it could be useful to use \u0011\u0001\u000bSHas\nan e\u000bective quantity parametrizing the STT e\u000eciency.\n \nFIG. 35 (a) A schematic illustration of the MOD experi-\nment to determine the spin Hall angle. His the external\nmagnetic \feld, and Jcrepresents the applied electric current\ndensity. (b) Magnetic \feld dependence of the FMR signal\nfor a NiFe/Pt bilayer \flm (red) and a pure NiFe \flm (black).\nNote the linewidth broadening for NiFe/Pt due to SP. (c)\nSchematic illustration of the spin Hall and the spin-torque\ne\u000bects.~M,~Js, and~ \u001bdenote the magnetization, the \row di-\nrection of the spin-current density, and the spin-polarization\nvector of the spin-current, respectively. From Ref. Ando et al. ,\n2008.\nFigure 36 shows the MOD experimental \fndings.\nWhen a current \rows through the FM/NM bilayer,\nthe STT generated by the spin-current traversing the\nNM/FM interface due to the SHE alters the FMR line\nwidth when the current \row direction and the external\nmagnetic \feld direction enclose an angle of 90\u000ewhile no\ne\u000bect is observed for collinear orientation, consistent with\nthe theoretical expectation.\nSimilar experiments have been performed by Demidov\net al. , 2011a,b using Brillouin Light scattering methods.\nThe key \fnding in these experiments is the control of the\nFMR line width of the FM \flm by employing the SHE\nwhich generates a pure spin-current in the adjacent NM.\n \nFIG. 36 FMR spectra for the NiFe/Pt bilayer measured at\nvarious electric current density values Jcwhen the magnetic\n\feld direction is (a) 90\u000eand (b) 0\u000e. The inset shows magni\fed\nviews around the peaks of the spectra, where the solid and\ndashed curves are the FMR spectra measured with electric\ncurrent densities Jcand\u0000Jc, respectively. From Ref. (Ando\net al. , 2008).\nUltimately, in suitable nanostructured materials, the ap-\nplication of a large enough charge current density should\nlead to the generation of coherent auto-oscillations in the\nFM nano object due to a DC charge current (Demidov\net al. , 2012).\n3. Spin Hall e\u000bect - spin transfer torque\nFinally, a third FMR technique has been employed that\nallows accessing the spin Hall angle experimentally. Liu\net al. , 2011 applied a microwave frequency charge current\nin the plane of a NiFe/Pt sample and observed the FMR\nin NiFe. Due to the action of the SHE a transverse spin-\ncurrent is generated in the NM, in this case Pt, which is\ninjected into the FM layer. Consequently, an oscillatory\nSTT acts on the magnetic moments in the FM, induc-\ning precession of the magnetization (see Fig. 37). The\noscillatory magnetiziation in the FM leads to an oscilla-39\ntory AMR which in turn leads to an oscillatory resistance.\nThis high frequency resistance mixes with the RF current\nand leads to a detectable DC voltage across the device\nwhich can be picked up using a bias tee (Fig. 37(c)).\nIn these experiments the external magnetic \feld is typ-\nically \fxed at an angle of 45\u000eand swept in the plane\nof the \flms to achieve the FMR condition. In the set-\nup, di\u000berent torques act on the magnetization of the FM\nwhich is aligned along the magnetic \feld direction as de-\npicted in Fig. 37(a). The torques include all the STTs\ndue to the SHE in the NM, the torque induced by the\nOersted \feld due to the RF current through the device,\nand the torque already modi\fed by SP. We also empha-\nsize that the torques generated by the SHE in the NM\nwould be in addition to the ISGE-related SOTs present in\nthe FM near the interface (Freimuth et al. , 2013; Garello\net al. , 2013; Kurebayashi et al. , 2014).\nLandau-Lifshitz-Gilbert equations including all rele-\nvant torques can be used to model the DC voltage re-\nsponse of the bilayer device and the result shows that the\nmixing voltage contains the contributions of symmetric\nand anti-symmetric Lorentzian lines (Liu et al. , 2011).\nAccording to Liu et al. , 2011, the detailed analysis of the\nresonance properties of this voltage enables a quantita-\ntive measure of the spin-current absorbed by the FM and\nof the spin Hall angle. Liu et al. , 2011 shows that the ra-\ntio of the symmetric and antisymmetric components of\nthe resonance curve, when scaled properly by material\nparameters like the saturation magnetization, thickness\nand width of the FM, and the external magnetic \feld, is\nlinked to the ratio of spin and charge currents and thus\nto the spin Hall angle. The authors emphasize that the\nmeasurement method is (in a reasonable thickness regime\nof the FM and the NM) self calibrating since the strength\nof the torque from the spin current is measured relative\nto the torque from the RF magnetic \feld, which can be\ncalculated from the geometry of the sample. The same\nmethod has been applied to various combinations of FMs\nand NMs (Liu et al. , 2011, 2012; Pai et al. , 2012).\nAlso in these types of experiments the tunability of\nthe e\u000bective damping parameter has been demonstrated\nby Kasai et al. , 2014; Liu et al. , 2011. An example is\nillustrated in Fig. 38 for the case of Py/Pt where the\ne\u000bective damping parameter is shown to be tunable as\na function of the current direction and amplitude (Liu\net al. , 2011).\n4. Spin Hall e\u000bect induced switching of the magnetization\nFor su\u000eciently large current densities pushed through\nthe NM and large spin Hall angles, it is possible to even\nreverse the magnetization in a FM nanoelement placed on\ntop of the NM current carrying line, as has been demon-\nstrated by Liu et al. , 2012 (see Fig 39). In these ex-\nperiments it is important to use a NM/FM combination\nwhere, when placing the NM in contact with the FM, the\ninduced damping due to SP remains negligible. This is\nSpin-Torque Ferromagnetic Resonance Induced by the Spin Hall Effect\nLuqiao Liu, Takahiro Moriyama, D. C. Ralph, and R. A. Buhrman\nCornell University, Ithaca, New York, 14853\n(Received 12 October 2010; published 20 January 2011)\nWe demonstrate that the spin Hall effect in a thin film with strong spin-orbit scattering can excite\nmagnetic precession in an adjacent ferromagnetic film. The flow of alternating current through a Pt=NiFe\nbilayer generates an oscillating transverse spin current in the Pt, and the resultant transfer of spin angularmomentum to the NiFe induces ferromagnetic resonance dynamics. The Oersted field from the currentalso generates a ferromagnetic resonance signal but with a different symmetry. The ratio of these twosignals allows a quantitative determination of the spin current and the spin Hall angle.\nDOI: 10.1103/PhysRevLett.106.036601 PACS numbers: 72.25.Ba, 72.25.Mk, 72.25.Rb, 76.50.+g\nThe spin Hall effect (SHE), the conversion of a longitu-\ndinal charge current density JCinto a transverse spin\ncurrent density JS@=2e, originates from spin-orbit scatter-\ning [1–4], whereby conduction electrons with opposite spin\norientations in a nonmagnetic metal [ 5] or semiconductor\n[6] are deflected in opposite directions. Several techniques\n[5,7,8] have been developed to determine the magnitude of\nthe SHE, which is generally characterized by the spin Hallangle, /C18\nSH¼JS=JC. For thin-film Pt, estimates of /C18SH\nobtained using different approaches differ by more than\nan order of magnitude [ 8–10], but already there have been\nefforts to utilize the spin current that arises from the SHE,first to tune the damping coefficient in a ferromagneticmetal [ 8], and, most recently, to induce a spin wave oscil-\nlation in a ferrimagnetic insulator having small damping\n[11]. Here we show that the SHE can be used to excite\ndynamics in an ordinary metallic ferromagnet. Our experi-ment also allows a quantitative determination of the SHEstrength that is self-calibrated as explained below.\nWe study Pt=Permalloy bilayer films with a microwave-\nfrequency (rf) charge current applied in the film plane(Permalloy ¼Py¼Ni\n81Fe19). An oscillating transverse\nspin current is generated in the Pt by the SHE and injected\ninto the adjacent Py [Fig. 1(a)], thereby exerting an oscil-\nlating spin torque (ST) on the Py that induces magnetiza-tion precession. This leads to an oscillation of the bilayerresistance due to the anisotropic magnetoresistance of Py.A dc voltage signal is generated across the sample from themixing of the rf current and the oscillating resistance,similar to the signal that arises from ST induced ferromag-\nnetic resonance (FMR) in spin valves and magnetic tunnel\njunctions [ 12–15]. The resonance properties enable a quan-\ntitative measure of the spin current absorbed by the Py.\nOur measurement setup is shown in Fig. 1(c).Pt=Py\nbilayers were grown by dc magnetron sputter deposition.The starting material for the Pt was 99.95% pure. Highlyresistive Ta (1 nm) was employed as the capping layer toprevent oxidation of the Py. The bilayers were subse-\nquently patterned into microstrips of 1 to 20/C22mwide\nand 3 to 250/C22mlong. By using a bias tee, we were ableto apply a microwave current and at the same time measure\nthe dc voltage. A sweeping magnetic field H\nextwas applied\nin the film plane, with the angle /C18between Hextand micro-\nstrip kept at 45/C14unless otherwise indicated. The output\npower of the microwave signal generator was varied from 0to 20 dBm and the measured dc voltage was proportional to\nthe applied power, indicating that the induced precession\nwas in the small angle regime. All the measurements wepresent were performed at room temperature with a powerof 10 dBm.\nWe model the motion of the Py magnetic moment ^mby\nthe Landau-Lifshitz-Gilbert equation containing the STterm [ 16]:\nd^m\ndt¼/C0/C13^m/C2~Heffþ/C11^m/C2d^m\ndtþ/C13@\n2e/C220MSt\n/C2JS;rfð^m/C2^/C27/C2^mÞ/C0/C13^m/C2~Hrf: (1)\nFIG. 1 (color online). (a) Schematic of a Pt=Pybilayer thin\nfilm illustrating the spin transfer torque /C28STT, the torque /C28H\ninduced by the Oersted field Hrf, and the direction of the\ndamping torque /C28/C11./C18denotes the angle between the magneti-\nzationMand the microstrip. Hextis the applied external field.\nThe spin Hall effect causes spins in the Pt pointing out of the\npage to be deflected towards the top surface, generating a spin\ncurrent incident on the Py. (b) Left-side view of the Pt=Py\nsystem, with the solid line showing the Oersted field generated\nby the current flowing just in the Py layer, which should produce\nno net effect on the Py anisotropic magnetoresistance.(c) Schematic circuit for the ST-FMR measurement.PRL 106, 036601 (2011) PHYSICAL REVIEW LETTERSweek ending\n21 JANUARY 2011\n0031-9007 =11=106(3) =036601(4) 036601-1 /C2112011 American Physical SocietyFIG. 37 a) Schematic of Pt/Py bilayer thin \flm illustrat-\ning the STT induced by the SHE rising form the RF current\nthrough NM as well as the damping torque and the torque\ndue to the Oersted \feld when the magnetization of FM is\naligned in an external magnetic \feld. b) Shows the dimen-\nsions of the sample and the Oersted \feld due to a current\n\rowing through FM. c) Depicts the electrical measurement\nscheme. Figure from Ref. Liu et al. , 2011.\n \nFIG. 38 E\u000bective damping as a function of current density\nthrough the Pt layer in a Py(4 nm)/Pt(6 nm) bilayer. From\nRef. Liu et al. , 2011.\nthe case for CoFeB/Ta. On one hand, \f-Ta shows a gi-\nant spin Hall angle (Liu et al. , 2012)), on the other hand,\nenhancement of damping due to SP is not observed in\nthe CoFeB layer. Furthermore, due to the large resis-\ntivity of the CoFeB layer a large portion of the applied\ncurrent is pushed through the Ta layer where it produces\nthe pure spin-current due to the SHE. Another impor-\ntant feature is that the bilayer is capped with MgO to\ninduce a large perpendicular anisotropy in CoFeB. The\nthin layer of MgO (1.6 nm) is used as a tunnel barrier be-\ntween the thin CoFeB free layer (1.6 nm) and the thicker\nCoFeB reference layer (3.8 nm) so that the tunnelling\nmagnetoresistance (TMR) e\u000bect can be used to deter-\nmine the relative orientation of their magnetization. The\nresults of these experiments are summarized in Fig 39\nand may be viewed as a paradigm change in the mecha-40\n|Ic0.P-AP |≠|Ic0.AP-P | due to spin accumulation in\nthe spin valve and the MTJ magnetoresistance\nbehavior, respectively. The equivalence of the two\ncritical currents for a SHE-ST switching device\ncould be a major technical advantage. From our\nmeasured values of | Ic0| and using Eq. 1 with\nm0Meff=0 . 7 6T( 32), we determine JS/Jefor\nthis device to be 0.12 T0.04 ( 32), in accord with\nour two other spin Hall angle measurements. Wenote that our three determinations of J\nS/Jeare\nconsistent for FM layer thicknesses ranging from1 to 4 nm and are not sensitive to whether the FM\nlayer is magnetized in plane or out of plane.\nTechnology applications. Improvements to\nthis initial three-terminal SHE device can be very\nreasonably expected to result in substantial\nreductions in the switching currents for thermally\nstable nanomagnets. By reducing the width of the\nTa microstrip to be equal to the dimension of the\nlong axis of the nanopillar, we can easily decrease\nI\nc0by a factor of 3 without affecting thermal sta-\nbility. A further reduction in Ic0could be achieved\nby reducing the demagnetization field of the FM\nfree layer from 700 mT to ≤100 mT (37, 38).\nWith such improvements, Ic0could be reduced to\n<100 mA, at which point the three-terminal SHE\ndevices would be competitive with the efficiencyof conventional ST switching in optimized MTJs(31,33,39) while providing the added advantage\nof a separation between the low-impedance switch-ing (write) process and high-impedance sensing\n(read) process. This separation solves the reliability\nchallenges that presently limit applications based\non conventional two-terminal MTJs while also giving\nimproved output signals. Oth er three-terminal spin-\ntorque devices based on conv entional spin-filtering\nhave been demonstrated previously ( 40–43), but\nthe SHE-ST design can provide better spin-torqueefficiency and is much easier to fabricate. More-\nover, the discovery of materials with even larger\nvalues of the spin Hall angle than in b-Ta could\nalso add to the competitiveness of the SHE-ST.\nReferences and Notes\n1. J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n2. L. Berger, Phys. Rev. B 54, 9353 (1996).\n3. J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nD. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n4. S. I. Kiselev et al .,Nature 425, 380 (2003).\n5. D. C. Ralph, M. D. Stiles, J. Magn. Magn. Mater. 320,\n1190 (2008).\n6. M. I. Dyakonov, V. I. Perel, Phys. Lett. A 35, 459\n(1971).\n7. J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n8. S. F. Zhang, Phys. Rev. Lett. 85, 393 (2000).\n9. J. Sinova et al .,Phys. Rev. Lett. 92, 126603 (2004).10. S. Murakami, N. Nagaosa, S.-C. Zhang, Science 301,\n1348 (2003).\n11. Y. K. Kato, R. C. Myers, A. C. Gossard, D. D. Awschalom,\nScience 306, 1910 (2004).\n12. S. O. Valenzuela, M. Tinkham, Nature 442, 176\n(2006).\n13. Y. Kajiwara et al .,Nature 464, 262 (2010).\n14. L. Q. Liu, O. J. Lee, T. D. Gudmundsen, D. C. Ralph,\nR. A. Buhrman, http://arXiv.org/abs/1110.6846.\n15. M. H. Read, C. Altman, Appl. Phys. Lett. 7, 51 (1965).\n16. R. Hoogeveen, M. Moske, H. Geisler, K. Samwer, Thin\nSolid Films 275, 203 (1996).\n17. T. Tanaka et al .,Phys. Rev. B 77, 165117 (2008).\n18. M. Morota et al .,Phys. Rev. B 83, 174405 (2011).\n19. L. Q. Liu, T. Moriyama, D. C. Ralph, R. A. Buhrman,\nPhys. Rev. Lett. 106, 036601 (2011).\n20. L. Q. Liu, R. A. Buhrman, D. C. Ralph, http://arXiv.org/abs/\n1111.3702.\n21. T. Kimura, Y. Otani, T. Sato, S. Takahashi, S. Maekawa,\nPhys. Rev. Lett. 98, 156601 (2007).\n22. K. Ando et al .,Phys. Rev. Lett. 101, 036601 (2008).\n23. O. Mosendz et al .,Phys. Rev. Lett. 104, 046601\n(2010).\n24. O. Mosendz et al .,Phys. Rev. B 82, 214403 (2010).\n25. B. Gu et al .,Phys. Rev. Lett. 105, 216401 (2010).\n26. The antisymmetric peaks shown here for these two\nsamples are opposite to what was illustrated in ( 20)f o ra\nsubstrate/Pt/Permalloy sample, because the relative order\nof the FM/nonmagnetic layers was reversed in that case.\n27. S. Mizukami, Y. Ando, T. Miyazaki, J. Magn. Magn. Mater.\n226–230, 1640 (2001).\n28. Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, Phys. Rev.\nLett. 88, 117601 (2002).\n29. H. Lee et al .,J. Phys. D 41, 215001 (2008).\n30. I. M. Miron et al .,Nature 476, 189 (2011).\n31. S. Ikeda et al .,Nat. Mater. 9, 721 (2010).\n32. Supplementary materials are available on Science Online.\n33. J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n34. I. M. Miron et al .,Nat. Mater. 9, 230 (2010).\n35. E. B. Myers et al .,Phys. Rev. Lett. 89, 196801\n(2002).\n36. T. Suzuki et al .,Appl. Phys. Lett. 98, 142505 (2011).\n37. L. Q. Liu, T. Moriyama, D. C. Ralph, R. A. Buhrman, Appl.\nPhys. Lett. 94, 122508 (2009).\n38. T. Moriyama et al .,Appl. Phys. Lett. 97, 072513\n(2010).\n39. T. Kishi et al ., in Proceedings of the IEEE International\nElectron Devices Meeting 2008 , San Francisco, 15 to\n17 December 2008 (IEEE, New York, 2008); 10.1109/\nIEDM.2008.4796680.\n40. T. Kimura, Y. Otani, J. Hamrle, Phys. Rev. Lett. 96,\n037201 (2006).\n41. T. Yang, T. Kimura, Y. Otani, Nat. Phys. 4, 851\n(2008).\n42. J. Z. Sun et al .,Appl. Phys. Lett. 95, 109901 (2009).\n43. P. M. Braganca et al .,IEEE Trans. NanoTechnol. 8, 190\n(2009).\nAcknowledgments: We acknowledge support from the Army\nResearch Office, Defense Advanced Research ProjectsAgency, Office of Naval Research, and NSF/Materials\nResearch Science and Engineering Center (DMR-1120296)\nthrough the Cornell Center for Materials Research (CCMR),\nas well as the NSF/Nanoscale Science and Engineering CenterProgram through the Cornell Center for Nanoscale Systems.\nWe also acknowledge NSF support through use of the Cornell\nNanofabrication Facility/National Nanofabrication InfrastructureNetwork and the CCMR facilities. Patent disclosures have\nbeen filed on behalf of the authors regarding the use of the\nspin Hall effect in Ta for magnetic memory and logic\napplications.\nSupplementary Materials\nwww.sciencemag.org/cgi/content/full/336/6081/555/DC1Materials and Methods\nSupplementary Text\nFigs. S1 to S5\nReference ( 44)\n20 December 2011; accepted 29 March 201210.1126/science.1218197\nFig. 3. Spin Hall effect– induced switching for an in-plane magnetized nanomagnet at room\ntemperature. ( A) Schematic of the three-terminal SHE devices and the circuit for measurements. The\ndirection of the spin Hall spin transfer torque is no t the same as in Fig. 1A because the CoFeB layer now\nlies above the Ta rather than below. ( B) TMR minor loop of the MTJ as a function of the external applied\nfield Bextapplied in-plane along the long axis of the sample. (Inset) TMR major loop of the device. ( C)\nTMR of the device as a function of applied dc current IDC. An in-plane external field of –3.5 mT is\napplied to set the device at the center of the minor loop. ( D) Switching currents as a function of the\nramp rate for sweeping current. Red squares indicate switching from AP to P; blue triangles indicate\nswitching from P to AP. Solid lines represent linear fi ts of switching current versus log(ramp rate). Error\nbars are smaller than the symbol size.\n4 MAY 2012 VOL 336 SCIENCE www.sciencemag.org 558RESEARCH ARTICLE\n on March 15, 2013 www.sciencemag.org Downloaded from \nFIG. 39 SHE induced switching for an in-plane magnetized\nnanomagnet at room temperature. (A) Schematic of the\nthree-terminal SHE devices and the circuit for measurements.\n(B) TMR minor loop of the magnetic tunnel junction as a\nfunction of the external applied \feld Bextapplied in-plane\nalong the long axis of the sample. (Inset) TMR major loop\nof the device. (C) TMR of the device as a function of ap-\nplied dc current IDC. An in-plane external \feld of -3.5 mT\nis applied to set the device at the center of the minor loop.\n(D) Switching currents as a function of the ramp rate for\nsweeping current. Red squares indicate switching from an-\ntiparallel (AP) to parallel (P) magnetizations; blue triangles\nindicate switching from P to AP. Solid lines represent linear\n\fts of switching current versus log(ramp rate). Error bars are\nsmaller than the symbol size. From Ref. Liu et al. , 2012.\nnism for switching magnetic nanoelements in spintronic\ndevices since here switching is driven be a purely in-plane\nelectrical current and not via a current perpendicular to\nthe layer stack. Similar results have been obtained for\nW/CoFeB layers (Pai et al. , 2012). One should note that\nwhile the exact value for the spin Hall angle extracted\nfrom these experiments is still under debate, the fact that\nswitching can be achieved for these devices underpins not\nonly the technological relevance, but also that a sizeable\nSHE (possibly in combination with other ISGE-related\nSOTs) must be generated in these structures.\nHaazen et al. , 2013; Miron et al. , 2010, 2011b have\ndemonstrated similar results in earlier experiment us-\ning ultrathin FM layers. Their devices are based on\nthe Pt/Co/AlOx system with ultrathin Co layers of a\nthickness of only 0.6 nm sandwiched between Pt (3 nm)\nand AlOx (1.6 nm). The use of ultra thin Co in con-\ntact with Pt leads to a strong perpendicular anisotropy.\nWhen a current is driven through the Pt layer, switching\nof the Co magnetization can be observed by monitor-\ning the AHE of the device (see Fig. 40). While in the\noriginal interpretation the driving force for the observed\nswitching was thought to arise mostly from the Rashba\nsymmetry ISGE due to the broken inversion symmetry\nalong the growth direction of the layer stack, detailed\nanalysis in later three-dimensional vector measurement\n \nFIG. 40 Top left: Device schematic and current-induced\nswitching. Hall cross geometry. Black and white arrows in-\ndicate the up and down equilibrium magnetization states of\nthe Co layer, respectively. Bottom left: Scanning electron\nmicrograph of the sample and electric circuitry used in the\nmeasurements. Shown are the terminals for the Hall voltage\nmeasurements as well as the current line where a pulsed cur-\nrent is applied for the switching experiments. Middle: The\nstate of the perpendicular magnetization is measured via the\nanomalous Hall resistance as a function of applied \feld, B.\nAfter injection of positive (black squares) and negative (red\ncircles) current pulses of amplitude Ip= 52:58 mA the Hall\nresistance is measured. The data are reported during a single\nsweep ofB. Right: The measurement schematics and pulse\nsequence. From Ref. Miron et al. , 2011b.\n(Garello et al. , 2013) point towards signi\fcant contri-\nbutions from the SHE. Similarly, the interpretation of\nthe results of current driven domain wall motion exper-\niments in the same type of layer stacks has to be revis-\nited (Miron et al. , 2010). Current and even \feld induced\ndomain wall motion experiments in layer stacks where\nISGE, SHE, and proximity polarization of the NM can\ncontribute are complicated for interpretation, and disen-\ntangling the relative strength of these contributions is not\nstraightforward. Experimentally, however, it has been\nobserved that the inclusion of relativistic torques, of ei-\nther the SHE or ISGE origin, leads to a large increase\nof domain wall velocities for optimally tuned materials\nwhich is potentially of great technological interest Emori\net al. , 2013; Ryu et al. , 2013.\nSpin-orbit coupling together with broken inversion\nsymmetry introduces yet another important aspect into\nthe physics of these systems. To fully understand the un-\nderlying mechanisms in these experiments one needs to\ntake into account also the fact that these domain walls\nare chiral due to the Dzyaloshinski-Moriya interaction\npresent at the FM/NM interface. This opens a new \feld\nconnecting spintronics with the skyrmion physics.\nWe conclude by discussing in more detail that in the\nNM/FM bilayer systems the relativistic torques induc-\ning magnetization dynamics are, in general, not only\ndue to the SHE but the ISGE-induced SOTs may also\ncontribute (Chernyshov et al. , 2009; Fang et al. , 2011;\nManchon and Zhang, 2009; Miron et al. , 2010). The\nISGE originate from spin-orbit coupling which, combined\nwith broken inversion symmetry in the crystal, can pro-\nduce spin-polarization when electrical current is driven\nthrough a NM. In combination with FMs, the ISGE and41\nthe SHE can drive magnetization dynamics in devices\nwith similar geometries. Disentangling these contribu-\ntions in NM/FM bilayer systems and engineering them\nfor maximal e\u000bect is at present a highly active \feld in\nspintronics.\nHowever, the discrimination of the SHE and ISGE\nbased microscopic mechanisms between the \feld-like and\nthe antidamping-like torque components is di\u000ecult to\nachieve for several conceptual reasons. The original theo-\nretical proposals (Aronov and Lyanda-Geller, 1989; Edel-\nstein, 1990; Malshukov and Chao, 2002) and experimen-\ntal observations (Ganichev et al. , 2004; Kato et al. , 2004c;\nSilov et al. , 2004; Wunderlich et al. , 2004, 2005) of the\nISGE were made in NMs with no FM component in\nthe structure. The corresponding non-equilibrium spin-\ndensity, generated in the ISGE by inversion-asymmetry\nterms in the relativistic Hamiltonian, has naturally no\ndependence on magnetization. Hence, in the context\nof magnetic semiconductors (Bernevig and Vafek, 2005;\nChernyshov et al. , 2009; Endo et al. , 2010; Fang et al. ,\n2011) or FM/NM structures (Manchon and Zhang, 2008;\nMiron et al. , 2011a, 2010; Pi et al. , 2010; Suzuki et al. ,\n2011), the ISGE may be expected to yield only the \feld-\nlike component of the torque \u0018^ m\u0002^\u0010, where the vector ^\u0010\nis independent of the magnetization vector ^ m. However,\nwhen carriers experience both the spin-orbit coupling\nand magnetic exchange coupling, the inversion asymme-\ntry can generate a non-equilibrium spin-density compo-\nnent of extrinsic, scattering-related (Pesin and MacDon-\nald, 2012; Wang and Manchon, 2012) or intrinsic, Berry-\ncurvature (Freimuth et al. , 2013; Garate and MacDonald,\n2009; Kurebayashi et al. , 2014) origin which is magneti-\nzation dependent and yields an antidamping-like torque\n\u0018^ m\u0002(^ m\u0002^\u0010). Experiments in (Ga,Mn)As con\frmed\nthe presence of the ISGE-based mechanism (Chernyshov\net al. , 2009; Endo et al. , 2010; Fang et al. , 2011) and\ndemonstrated that the \feld-like and the Berry-curvature\nantidamping-like SOT components can have comparable\nmagnitudes (Kurebayashi et al. , 2014).\nThe STT is dominated by the antidamping-like com-\nponent (Ralph and Stiles, 2008) in weakly spin-orbit cou-\npled FMs with \u001cex\u001c\u001cs, where\u001cexis the precession time\nof the carrier spins in the exchange \feld of the FM and\n\u001csis the spin life-time in the FM. This, in principle, ap-\nplies also to the case when the spin-current is injected to\nthe FM from a NM via the SHE. However, at \fnite \u001cs,\nthe STT also acquires a \feld-like component (Ralph and\nStiles, 2008). Experiments in W/Hf/CoFeB structures\ncon\frmed the presence of the SHE-based mechanism in\nthe observed torques and showed that the SHE-STT can\nhave both antidamping-like and \feld-like components of\ncomparable magnitudes (Pai et al. , 2014).\nIn the commonly studied polycrystalline transition-\nmetal FM/NM samples, the dependence of the torques\non the angle of the driving in-plane current also does\nnot provide the direct means to disentangle the two mi-\ncroscopic origins. The lowest order inversion-asymmetry\nspin-orbit terms in the Hamiltonian have the Rashbaform for which the vector ^\u0010is in the plane parallel to\nthe interface and perpendicular to the current, inde-\npendent of the current direction. The same applies to\nthe spin-polarization of the SHE spin-current propagat-\ning from the NM to the FM. The ^ mand ^\u0010functional\nform of the \feld-like and antidamping-like SHE-STTs is\nthe same as of the corresponding SOT components. In\nthe observed lowest order torque terms in Pt/Co and\nTa/CoFeB structures (Garello et al. , 2013) the ISGE-\nbased and the SHE-based mechanism remained, there-\nfore, indistinguishable. The simultaneous observation of\nhigher order torque terms in these samples pointed to\nSOTs due to structural inversion-asymmetry terms be-\nyond the basic Rashba model. From the Ta thickness\ndependence measurements in the Ta/CoFeB structure it\nwas concluded that in these samples both the ISGE-based\nand the SHE-based mechanisms contributed to both the\n\feld-like and the antidamping-like torques (Kim et al. ,\n2013).\nE. Spin Hall angles\nIn this subsection we collect within Table III experi-\nmental measurements of the SHE in di\u000berent materials.\nThe list, in such an active and evolving \feld, is by no\nmeans exhaustive. As discussed in this experimental sec-\ntion, more things are learned about the techniques and\nsystematic errors are better understood and corrected,\nthe measurements begin to converge for several materi-\nals, particularly for the transition metals.\nIn this table we show the material, the temperature\nthe measurement was taken at, the spin di\u000busion length\neither measured or used in the analysis, the conductivity,\nthe spin Hall angle, the reference of the work and the type\nof technique as well as relevant comments.\nV. FUTURE DIRECTIONS AND REMAINING\nCHALLENGES\nWe conclude this review with our own personal view of\npossible interesting directions and remaining challenges.\nAs such, this is not entirely scienti\fc and it re\rects\nmerely our own preference and intuition and should only\nbe taken as such. We apologize for any omissions of the\nmany interesting possibilities that others may consider.\nWe only know for certain that such future outlook is\nbound to always fail in a \feld that continues to bring\nunexpected surprises.\nTransition metals have traditionally played a dominant\nrole in spintronics both in basic research and, in partic-\nular, in applications. It is therefore not surprising that\nthe SHE \feld has gained new momentum when bringing\nnon-magnetic transition metals in the game. And they\nhave played their role particularly well. When brought\nout of equilibrium by an applied electric \feld, the SHE\nin some non-magnetic transition metals can generate suf-42\nT[K] \u0015sd\u001bNM[106S/m] \u000bSH(%) Comment Ref.\nAl 4.2 455 \u000615 10.5 0 :032\u00060:006 NL (Valenzuela et al 2006,2007)\n4.2 705\u000630 17 0 :016\u00060:004 NL (Valenzuela et al 2006,2007)\nAu 295 86 \u000610 37 11.3 NL (10-nm thick \flms) (Seki et al 2008,2010)\n295 83 37 3 NL (20-nm thick \flms) (Seki et al. , 2010)\n4.5 65\u000348.3 <2:3 NL (SHE/iSHE) (Mihajlovic et al. , 2009)\n295 36\u000325.7 <2:7 NL (SHE/iSHE) (Mihajlovic et al. , 2009)\n295 35\u00064 28 7 :0\u00060:1 NL (Sugai et al. , 2010)\n295 27\u00063 14 7 :0\u00060:3 NL (0.95 at.% Fe) (Sugai et al. , 2010)\n295 25\u00063 14.5 12 \u00064 NL (1.4 at.% Pt, 10-nm thick \flms) (Gu et al. , 2010)\n295 50\u00068 16.7 0 :8\u00060:2 NL (1.4 at.% Pt, 20-nm thick \flms) (Gu et al. , 2010)\n<10 40\u000616 25 1 :4\u00060:4 NL (Niimi et al. , 2014)\n295 35\u00063\u000325.2 0 :35\u00060:03 SP (Mosendz et al. , 2010a)\n295 35 20 0 :25\u00060:1 SP (Vlaminck et al. , 2013)\n295 35\u00063\u00035.25 1 :6\u00060:1 SP (Hung et al. , 2013)\n295 35\u00063\u00037 0:335\u00060:006 SP (Hung et al. , 2013)\n295 35\u00031:1\u00060:3 SP (Obstbaum et al. , 2014)\n295 60 20.4 8 :4\u00060:7 SP (Wang et al. , 2014)\nAg 295 700 15 0 :7\u00060:1 SP (Wang et al. , 2014)\nBi 3 0 :3\u00060:1 - >0:3 local, signal decreases with \u001aN (Fan and Eom, 2008)\n295 - 2 :4\u00060:3(I)\u0000(7:1\u00060:8)(I) SP as a function of Bi thickness (Hou et al. , 2012)\n50\u000612(V) 1 :9\u00060:2(V) volume (V) and interfacial (I) param.\nCu 295 500 16 0 :32\u00060:03 SP (Wang et al. , 2014)\nCuIr 10 5 \u000030 2 :1\u00060:6 NL (Ir concentrations from 0 to 12 %) (Niimi et al. , 2011)\nCuMnxTy - 0.7(Ta);2.6(Ir) T = Lu, Ta, Ir, Au, Sb ( y\u00181\u000020\u000210\u00004) (Fert et al. , 1981)\n1.35(Au);1.15(Sb) Mn ( x\u00181\u00002\u000210\u00004) createsIs\n-1.2(Lu) Note factor 2 in the de\fnition of \u000bskew\nSH Ref. 15 in (Fert and Levy, 2011)\nCuBi 10 \u0018100;\u001830 \u000011 NL (Bi = 0.3%; 0.5%), \u000bskew\nSH =\u0000(24\u00069) on Bi (Niimi et al. , 2012)\n\u001810;\u00187 similar in AgBi (Niimi et al. , 2014)\nn-GaAs 4.2 2200 0.0056 0.15 NL, n\u00191017cm\u00003(Olejn\u0013 \u0010k et al. , 2012)\n4.2 8500 0.00137 0.08 LSP, n\u00191016cm\u00003(Ehlert et al. , 2012)\n30 0.0036 0.08 LSP, n\u00193\u00005\u00021016cm\u00003(Garlid et al. , 2010)\nGe 2 \u0019\u00000:001 MR, \u000bSHT-dependent, sign change at \u001910 K (Chazalviel, 1975)\n295 0.027 0 :001; 0:00044 SP, not annealed and annealed values (Rojas-S\u0013 anchez et al. , 2013)\nn-InGaAs 30 \u00183000\u00180:002 \u00190:02 KRM, x= 0:07,n\u00193\u00021016cm\u00003(Kato et al. , 2004a)\n(Si-doped) 30 0.003-0.005 \u00190:1;\u00190:25;\u00190:38 LSP,x= 0:03;0:05;0:06,n\u00193\u00005\u00021016cm\u00003(Garlid et al. , 2010)\nInSb 1.3 \u00000:026\u00060:005 MR, n\u00191014cm\u00003,\u0016\u00192:2\u0002104cm2/Vs (Chazalviel et al 1972)\n1.3 0 :003 MR n\u00191014cm\u00003,\u0016\u00194\u0002104cm2/Vs (Chazalviel et al 1972)\nIrO2 300 3.8(P) 0.5(P);0.18(A) 4(P);6.5(A) NL, polycryst. (P), amorphous (A) (Fujiwara et al. , 2013)\nMo 10 10 3.03 -0.20 NL (Morota et al. , 2009)\n10 10 0.667 -0.075 NL (Morota et al. , 2009)\n10 8:6\u00061:3 2.8 \u0000(0:8\u00060:18) NL (Morota et al. , 2011)\n295 35\u00063\u00034.66\u0000(0:05\u00060:01) SP (Mosendz et al. , 2010a)\nNb 10 5 :9\u00060:3 1.1 \u0000(0:87\u00060:20) NL (Morota et al. , 2011)\nPd 10 13 \u00062 2.2 1 :2\u00060:4 NL (Morota et al. , 2011)\n295 9\u00031.97 1 :0 SP (Ando et al. , 2010b)\n295 15\u00064\u00034.0 0 :64\u00060:10 SP (Mosendz et al. , 2010a)\n295 5:5\u00060:5 5 1 :2\u00060:3 SP (Vlaminck et al. , 2013)\n295 2:0\u00060:1 3.7 0 :8\u00060:20 STT+SHE (Kondou et al. , 2012)\nPt 295 3\u00036.41 0.37 NL (Kimura et al. , 2007)\n5 8 8.0 0.44 NL ( \u0015N= 14 nm from spin-absorption) (Vila et al. , 2007)\n295 7 5.56 0.9 NL ( \u0015N= 10 nm from spin-absorption) (Vila et al. , 2007)\n10 11\u00062 8.1 2 :1\u00060:5 NL (Morota et al. , 2011)\n10\u001810 8.1 2 :4 NL (3D corrected (Morota et al. , 2011)) (Niimi et al. , 2012)\n295 7\u00036.4 8 :0 SP (Ando and Saitoh, 2010)\n295 10\u00062\u00032.4 1 :3\u00060:2 SP (Mosendz et al. , 2010a)\n295 10\u00032 4 :0 SP (Ando et al. , 2011)\n295 3:7\u00060:2 2.42 8 \u00061 SP (Azevedo et al. , 2011)\n295 8:3\u00060:9 4:3\u00060:2 1 :2\u00060:2 SP (Feng et al. , 2012)\n295 7:7\u00060:7 1:3\u00060:1 1 :3\u00060:1 SP (Nakayama et al. , 2012)\n295 1:5\u000010\u00032:45\u00060:1 3+4\n\u00001:5SP, spin Hall MR (Hahn et al. , 2013)\n295 4 4 2 :7\u00060:5 SP (Vlaminck et al. , 2013)\n295 8\u00061\u00031.02 2 :012\u00060:003 SP (Hung et al. , 2013)\n295 1 :3\u00032.4 2 :1\u00061:5 SP (Bai et al. , 2013b)\n295 1 :4\u000312\u00064 SP (Obstbaum et al. , 2014)\n295 7 :3 2.1 10 pm1 SP (Wang et al. , 2014)\n295 1:2\u00060:1 3.6 2 :2\u00060:4 STT+SHE (Kondou et al. , 2012)\n295 3( <6) 5.0 7 :6+8:4\n\u00002:0STT+SHE (Liu et al. , 2011)\n295 2:1\u00060:2 3.6 2 :2\u00060:8 STT+SHE (Ganguly et al. , 2014)\n295 2:1\u00060:2 3.6 8 :5\u00060:9 STT+SHE, modulation of damping (Ganguly et al. , 2014)\n295 2 :4\u00031:2 \u00184 spin Hall MR (Nakayama et al. , 2013)\n295 1:5\u00060:5 0.5-3 11 \u00068 spin Hall MR (variable Pt thickness) (Althammer et al. , 2013)\np-Si 295 \u00190:01 SP,\u001cs\u001810 psn\u00192\u00021019cm\u00003(Ando and Saitoh, 2012)\nTa 10 2 :7\u00060:4 0.3 \u0000(0:37\u00060:11) NL (Morota et al. , 2011)\n295 1 :9 0.34 \u00007:1\u00060:6 SP (Wang et al. , 2014)\n295 1:8\u00060:7 0:08\u00000:75\u0000(2+0:8\n\u00001:5) SP, spin Hall MR (variable Ta thickness) (Hahn et al. , 2013)\n295 0.53 \u0000(12\u00064) STT+SHE ( \f-Ta) (Liu et al. , 2012)\nW 295 2 :1 0.55 \u000014pm1 SP (Wang et al. , 2014)\n295 0 :38\u00060:06\u0000(33\u00066) STT+SHE ( \f-W, lower in \u000b-W\u000bSH) (Pai et al. , 2012)\nTABLE III Experimental spin Hall angles and related parameters. SP=spin pumping, NL=nonlocal, STT+SHE=spin transfer\ntorque combined with spin Hall e\u000bect, MR=magnetoresistance, LSA=local spin accumulation, MR=Magnetic Resonance,\nKRM=Kerr rotation microscopy. Values marked with \u0003are taken not measured but taken from the literature.43\n\fcient \rux of spin angular momentum to reorient mag-\nnetization in an adjacent transition metal FM. Entirely\nnew concepts for writing information in magnetic tunnel\njunctions or domain-wall based spintronic devices have\nemerged from this discovered surprising strength of the\nSHE in the common and technologically relevant family\nof materials.\nTa, W, Ir, or Pt are examples among the non-magnetic\ntransition metals with large SHE. The strength of the ef-\nfect is derived from the large spin-orbit coupling in these\nheavy elements. Apart from the new opportunities for\napplications, this brings also new challenges for the ba-\nsic research of the SHE in transition mentals. We have\nmentioned in the review the pitfalls in attempting to mi-\ncroscopically describe the SHE in structures comprising\nheavy transition metals from theories of spin transport in\nweakly spin-orbit coupled systems. The proper descrip-\ntion and microscopic understanding of the SHE struc-\ntures in the strong spin-orbit coupling regime is among\nthe key remaining challenges in the SHE \feld.\nThe \rurry of recent SHE studies in transition met-\nals may give an impression that the \feld is forgetting\nits semiconductor roots. Robust FMs are typically dense\nmoment systems and their switching requires comparably\nlarge electrical current densities generating the SHE spin-\ncurrent. Highly conductive transition metals are clearly\nfavorable from this perspective when compared to semi-\nconductors. Moreover, the reported spin Hall angles in\nsemiconductors do not reach the record values in transi-\ntion metals.\nWe nevertheless foresee semiconductors playing vital\nrole in future SHE research; in particular when consider-\ning spintronics concepts without FMs. In the transition\nmetal context the SHE is used as an e\u000ecient spin-current\ngenerator or detector but these studies rarely consider\nspin manipulation in the non-magnetic SHE system. Es-\npecially in the strongly spin-orbit couple heavy metals,\nthe spin di\u000busion length is of the order of nanometers,\ntoo short for implementing any spin manipulation tools\nalong the non-magnetic transport channel. For semicon-\nductors, on the other hand, we have mentioned in the\nreview several examples of electrical manipulation of the\noutput SHE signal. A gate electrode can be used to\ncontrol coherent spin precession along the channel, addi-\ntional drift current was shown to modify the spin-current\npro\fle along the channel, or non-linear inter-valley trans-\nport can strongly enhance the spin Hall angles bringing\nthe values close to their heavy metal counterparts.\nThe physics is, however, no di\u000berent in principle be-\ntween metals and semiconductors. Large SHE requires\nlarge spin-orbit coupling which, on the other hand, tends\nto suppress spin coherence/di\u000busion length. Semicon-\nductors with their simpler electronic structure and model\nspin-orbit \felds o\u000ber unique ways how to get around this\nproblem. As has been already demonstrated, a proper\ntuning of the Rashba and Dresssehaus spin-orbit \felds\ncan signi\fcantly enhance spin coherence in the presence\nof strong spin-orbit coupling. Experiments outside theSHE \feld have recently made major progress in control-\nling these two canonical spin-orbit \felds in common semi-\nconductor structures and we envisage new developments\nin semiconductor SHE devices utilizing the coherent spin-\nmanipulation techniques.\nCombining optical selection rules with SHE makes\nsemiconductors also favorable materials for exploring new\nconcepts in opto-spintronics. These may include opti-\ncal spin-torque structures, electrical polarimeters, spin-\nphotovoltaic cells, switches, invertors and interconnects.\nThe opto-spintronic sub\feld of the SHE research is still\nat its infancy and we expect growing activity in this di-\nrection in the future.\nThe fascinating feature of the SHE is that it can gen-\nerate a large spin-current, and a resulting large spin-\naccumulation by bringing weakly out of equilibrium a\nnon-magnetic system. It is, therefore, natural that non-\nmagnetic materials have been traditionally in the cen-\nter of interest of the SHE research. However, limit-\ning ourselves to paramagnetic or diamagnetic materi-\nals, whether metallic or semiconducting, is not necessary\nwhen considering the spin Hall phenomena. Recently,\nseveral transition metal FMs and antiferromagnets were\ndemonstrated to act as e\u000ecient ISHE spin-current detec-\ntors which opens a new broad area of future materials\nresearch in the SHE.\nIt also brings us back to the opening paragraphs of this\nsection where we mentioned SHE-induced spin torques\nin NM/FM heterostructures. Since in strongly spin-orbit\ncoupled systems these torques are limited to a few atomic\nlayers around the NM/FM interface, and considering\nthe likely material intermixing at the interface, it is not\nmeaningful to speak strictly about a non-magnetic layer\nSHE in these structures. The di\u000berence than becomes\nblurred whether including magnetism via intermixing or\nproximity polarization at hetero-interfaces, or directly\nconsidering the SHE in bulk FM or antiferromagnetic\nmaterials. Within this notion, an important challenge\narises not only for the normal-metal/magnet interfaces\nbut also for monolayer magnets to identify the micro-\nscopic origin of the observed spin torques. It remains an\nopen question whether the current induced torques in the\nmagnet are better linked to a SHE-induced spin-current\norigin or to one of the variants of the ISGE-induced non-\nequilibrium spin-polarization. Resolving these contribu-\ntions is an important academic exercise with potentially\nlarge implications for the utility of these spin-orbit cou-\npling phenomena in spintronic information technologies.\nWe conclude by emphasizing that the \feld of SHE does\nnot live in a vacuum. Its interconnects to other emerging\n\felds, e.g. topological insulators and spin-caloritronics,\nmakes its growth and possibilities very di\u000ecult to predict\nsince many things that we have discussed here and that\nhave emerged from its link to these \felds were not known\nor expected a few years ago. It is a rapid evolving \feld\nthat produces discoveries at a neck breaking speed and\nwe all look forward to its exciting future.44\nList of symbols and abbreviations\n2DEG : Two-dimensional electron gas\n2DHG : Two-dimensional hole gas\nAC: Alternating current\nAHE : Anomalous Hall e\u000bect\nAMR : Anistropic magnetoresistance\nCPW : Coplanar wave guide\nDC: Direct current\nFM: Ferromagnet\nFMR : Ferromagnetic resonance\nHE: Hall e\u000bect\nISGE : Inverse spin galvanic e\u000bect\nMOD : Modulation of damping\nMRAM Magnetic random access memory\nNM: Non-magnetic material\nQHE : Quantum Hall e\u000bect\nQSHE : Quantum spin Hall e\u000bect\nRF: Radio frquency\nSGE : Spin galvanic e\u000bect\nSHE : Spin Hall e\u000bect\nSOT : Spin orbit torque\nSP: Spin pumping\nSTT : Spin transfer torque\nTMR : Tunnelling magnetoresistance\nAcknowledgements\nWe thank all colleagues who have given us the permis-\nsion to show their results in this review. We also thank\nall our colleagues within the spintronics community that\nengaged us in many fruitful interactions and spirited\ndiscussions. J. S. acknowledges partial support by the\nAlexander von Humboldt Foundation and the European\nResearch Council (ERC) Synergy Grant No. 610115.\nS. O. V. acknowledges partial support from the Euro-\npean Research Council under grant agreement 308023\nSPINBOUND and from the Spanish Ministry of Econ-\nomy and Competitiveness (MINECO) under contracts\nMAT2013-46785-P and SEV-2013-0295. J. W. acknowl-\nedges partial support from EMRP JRP IND08 MetMags\nand the ERC Synergy Grant No. 610115. C. B. acknowl-\nedges partial support from the German research foun-\ndation (DFG) through programs SFB 689 and SPP 1538\nand from the European Research Council (ERC) through\nstarting grant no. 280048 ECOMAGICS. T. 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Institut f¨ ur Theoretische Physik, Universit¨ at Hambur g, Jungiusstraße 9, 20355 Hamburg, Germany\nWe provide a general procedure to calculate the current-ind uced spin-transfer torque which acts\non a general steep magnetic texture due to the exchange inter action with an applied spin-polarized\ncurrent. As an example, we consider a one-dimensional ferro magnetic quantumwire and also include\na Rashba spin-orbit interaction. The spin-transfer torque becomes generally spatially non-local.\nLikewise, the Rashba spin-orbit interaction induces a spat ially nonlocal field-like nonequilibrium\nspin-transfer torque. We also find a spatially varying nonad iabaticity parameter and markedly\ndifferent domain wall dynamics for very steep textures as com pared to wide domain walls.\nPACS numbers: 75.78.Fg, 75.70.Tj, 75.25.-b\nThe exchange interaction of a spin-polarized electron\ncurrent with localized magnetic moments in a ferromag-\nnetic wire typically induces a spin transfer torque (STT).\nA pronounced consequence is the coordinated switching\nof the localized magnetic moments of a ferromagneticdo-\nmain wall (DW) in the wire generating a net DW motion\n[1–3]. Other magnetic textures also rose to recent promi-\nnence, such as magnetic vortices and skyrmions [4–6], or\none-dimensionalspinchains[7]. There, themagnetization\nchangesonmuchshorterlengthscalesascomparedtothe\nconventional broad mesoscopic Bloch domain walls. In\naddition, these textures are in general strongly affected\nbysymmetrybreakinginteractions,suchasthespin-orbit\n[8] or the Dzyaloshinskii-Moriya interaction [9, 10].\nClearly, in small structures, the backaction of the local\nmagnetic moments on the polarized itinerant electron\nspinscanbecomerelevant. Inparticular,theythemselves\nexperience a back-actingSTT as well. For wide magnetic\ntextures, theimpact ofbackactionis generallysmallsince\nthe itinerant spins relax much faster than they have time\nto interact with the local moments. Hence, for wide tex-\ntures, it is reasonable to assume a stationary spin po-\nlarized current which generates the (non-)adiabatic STT\n[11]. This assumption, however, becomes increasingly\ninvalid in more narrow or steep magnetic textures. In\nthis context, questions have been raised why the nona-\ndiabaticity parameter βis much larger in small vortex\nstructures [12, 13] than compared to spin-waves in ex-\ntended structures [14]. This effect has been traced back\nto a non-standard description of the STT [15].\nFor a unified description of the STT for arbitrary mag-\nnetic textures, several approaches have been developed\n[15–22]. Nevertheless, a complete picture is still miss-\ning. Spin relaxation has either been neglected [16, 17] or\nincluded [18–20], quantum corrections are considered[23]\nor a semiclassical approach on the basis of spin diffusion\nhas been formulated [15, 24]. Spin-orbit interaction has\nbeen considered for broad textures [21, 22] only.\nIn this work, we provide a general and conceptually sim-\nple scheme to calculate the STT for arbitrary magnetic\ntextures. To show the flexibility of the approach, we also\ninclude the Rashba spin-orbit interaction in the itiner-ant electrons, whose contribution to the STT in steep\nmagnetic structures is of high interest [25, 26]. The STT\nand the resulting nonadiabatic Rashba field-like STT are\nshown to become non-local in space for steep textures.\nThe known results for broad Bloch DWs [27–29] are re-\ncovered as a limiting case. We calculate DW velocities\nfor a broad range of widths of a Bloch DW and find a\nnon-local nonadiabaticity parameter for steep textures.\nInterestingly, the overall direction of the DW movement\ncan be determined by the DW width.\nWe consider a one-dimensional (1D) ferromagnetic quan-\ntum wire with a magnetic texture formed by localized\nmagnetic moments Msn(x,t) with a saturation magneti-\nzationMsand unit vectors n(x,t). Their dynamics fol-\nlows from the Landau-Lifshitz-Gilbert (LLG) equation\n∂tn=−γ0n×Heff+αn×∂tn+T,(1)\nwith the effective magnetic field Heff, the gyro-magnetic\nratioγ0, and the Gilbert damping constant α.Tde-\nnotes the spin torque which is induced by the exchange\ninteraction of the localized moments with the polarized\nspins of the current carrying electrons. The latter and\nthe interaction are described by the Hamiltonian\nH=Hkin+HRashba+Hsd+Hrelax.(2)\nIt contains the kinetic energy, the Rashba spin-orbit in-\nteraction, the sdinteraction with the magnetic moments\nand the relaxation of the electron spins. It is convenient\nto use the low-energy description of this Hamiltonian,\nas it is accurate for 1D quantum wires [30] and yields\na simple derivation of the STT [27, 28]. Then, all rele-\nvant electron operators cσ,c†\nσwith spin index σcan be\nincluded in spin-like operators ˆJr=1\n2:c†\nrσσσσ′crσ′:.\nThe index r=R/L= +/−refers to right or left mov-\ning electrons in the wire. The Rashba Hamiltonian as-\nsumes the simple form HRashba= ∆R/summationtext\nrp/integraltext\ndxˆJr·ey,\nwith the Rashba splitting ∆ R= 2˜αRkF[29]. All terms\nof Eq. (2) can be expressed in terms of ˆJr, for which the\nHeisenberg equations of motion ∂tˆJr=−i\n/planckover2pi1/bracketleftBig\nˆJr,H/bracketrightBig\n−can\nreadily be evaluated [27–29]. We find for the expectation2\nvalueJr=∝angbracketleftˆJr∝angbracketright\n∂(r)Jr=−∆sd\n/planckover2pi1[Jr×mr+β(Jr−Jrel\nr)],(3)\nwith the derivative ∂(r)=∂t+vr∂x,mr=n+rαRey, the\nFermi velocity v, the exchange interaction strength ∆ sd,\nthe Rashba parameter αR= ∆R/∆sd, the nonadiabatic-\nity parameter β=/planckover2pi1/(∆sdτ), relaxation time τand the\nrelaxed spin density Jrel\nr. In fact, this is the continuity\nequation of the spin current[31–34] when we introduce\nthe spin density s=JR+JLand the spin current den-\nsityJ=v(JR−JL) and sum over r.\nTosolvethe equationsofmotion (3), weset upthe ansatz\nJr=armr+br∂(r)mr+crmr×∂(r)mr,(4)\nwith the unit vectors mr=mr/|mr|and∂(r)mr=\n∂(r)mr/|∂(r)mr|. It is the central observation of this\nwork to use space- and time-dependent coefficients\nar(x,t),br(x,t),cr(x,t). All three vectorsin Eq. (4) form\na complete orthonormal basis in spin space. With the\nprefactors determined below, the STT T=−∆sd\n/planckover2pi1n×s=\n−∆sd\n/planckover2pi1n×/summationtext\nrJrcan be calculated. Straightforward alge-\nbra yields\nT=/summationdisplay\nν=x,t/bracketleftbig\nTnonad\nν(br)∂νn+Tad\nν(cr)n×∂νn/bracketrightbig\n−HR(ar,br)n×ey. (5)\nIn particular, the ansatz yields the adiabatic STT Tad\nx,t,\nthe nonadiabatic STT Tnonad\nx,t, as well as the field-like\nRashba term with the nonadiabatic Rashba field HR. By\nconstruction, the antidamping field Hanti\nR=−βn×HRis\nincluded, although it does not appear explicitly. It eas-\nily can be recovered by choosing a suitable overcomplete\nbasis in the ansatz (4). Notice that we have defined the\nSTT in terms of the normalized vectors of the derivative\n∂xn=∂xn/|∂xn|to avoid a divergence of the prefactors.\nIn general, also terms ∝∂tn,n×∂tnappear. They act\nas additional damping terms in Eq. (1) and are conve-\nniently absorbed in a rescaled Gilbert parameter.\nThe remaining step is to determine the coefficients ar,br\nandcr. For this, we insert the Ansatz Eq. (4) into Eq.\n(3) and order it according to linearly independent parts.\nThis yields to an ordinary differential equation\n∂(r)\nar\nbr\ncr\n=Ar\nar\nbr\ncr\n+∆sdβ\n/planckover2pi1\narel\nr\nbrel\nr\ncrel\nr\n,(6)\nwherearel\nr,brel\nrandcrel\nrare the corresponding coefficients\nof the relaxed spin density Jrel\nr. The coefficient matrix\nAr=\n−∆sdβ//planckover2pi1|∂(r)mr| 0\n−|∂(r)mr| −∆sdβ//planckover2pi1−∆sd|mr|//planckover2pi1−g\n0 ∆ sd|mr|//planckover2pi1+g−∆sdβ//planckover2pi1\n\n(7)\nFIG. 1. (Color online) Sketch of the configurations of the lo-\ncalized DWspins (grayarrows) andtheitinerant spinsflowin g\nfrom negative to positive x-values, for a broad (a) and a steep\n(b) domain wall (distances are normalized to the DW width).\nRed (blue) arrows show the direction of the z-component of\nthe itinerant spins in positive (negative) direction. The l ines\nare guides for the eyes. While for broad DWs ( λ≫x∞\nosc),\ntheitinerantspinsfollow theDWmagnetization adiabatica lly,\nthis is not possible for steep DWs ( λ≤x∞\nosc).\nis space- and time-dependent due to |∂(r)mr|(x,t) and\ng= [∂(r)mr·(mr×∂2\n(r)mr)]/(∂(r)mr)2. Hence, a gen-\neral analytical solution of Eq. (6) cannot be found. Even\nthough these equations are derived within a 1D model,\nthey readily can be used for higher dimensional mag-\nnetic textures, if the electronic motion perpendicular to\nthe current is less important.\nIn the following, we solve Eq. (6) numerically for the ex-\nample of a Bloch DW. Other textures can be treated in\nthe similar way. As both Eqs. (6) and (1) depend on n, it\nis necessary to solve them self-consistently for each time\nstep until convergence is achieved. This is still demand-\ningandwecanusephysicalargumentstofurther simplify\nthe equation. First, we may assume that all coefficients\ndepend only on the distance to the DW center xDW(t),\nsince we do not consider explicit time-dependent mod-\nifications of the shape of the initially prepared domain\nwall [27]. Hence, ar(x,t) =ar[∆x=x−xDW(t)]. Then,\nit follows that ∂(r)ar(∆x) = [pv−∂txDW(t)]∂∆xar(∆x).\nAs the DW velocity ∂txDW(t) is generally much smaller\nthan the Fermi velocity v, we can neglect the terms in-\nvolving∂txDW(t). Effectively, every time derivative in\nEq. (6) is neglected, and thereby also the damping parts\nin Eq. (5). We do not simplify the LLG equation (1)\nby this assumption, so that a precessional motion of the\nDW is still possible.\nBefore we discuss the numerical results, it is instruc-\ntive to analyze the generic qualitative behavior which\ncan be extracted from Eq. (6). The eigenvalues of the\nmatrix in Eq. (7) are of the form ξ1=−β∆sd/(/planckover2pi1v) and\nξ±= [−β∆sd±i/radicalbig\n(∆sd|mr|+g)2+(/planckover2pi1v∂x|mr|)2]/(/planckover2pi1v).\nFor constant coefficients, the solutions of Eq. (6) would\nbe combinations of exponentials ∝exp(ξix). Thus, the\nreal part Re ξidetermines a damped, and the imaginary\npart Imξian oscillating behavior. As the real part is3\nFIG. 2. (Color online) Spatial dependence of the three com-\nponents to the STT: the adiabatic STT (top), the nonadia-\nbatic STT (middle), and, the Rashba field-like term (bottom)\nfor varying DW widths λ. Blue (red) colors indicate positive\n(negative) deviation from their relaxed values. Two cases f or\nλ/x∞\nosc= 1.7,5.7 are highlighted (shifted thick black curves).\nThe parameters are αR= 0.4,β= 0.2.\nthe same for all ξi, we find a unique damping length\nxdamp=/planckover2pi1v/(β∆sd). The oscillation length 2 πxosccan\nonly be estimated. Far away from the DW center, when\n|∂xmr| ∝ |∂xn|= 0, it is determined by the inverse\nsdcoupling strength such that xosc≈x∞\nosc≡/planckover2pi1v/∆sd.\nClose to the DW center, xoscis modified by the term\n|∂xmr| ∝1/λ, whereλis the DW width. Thus, in par-\nticular for steep DWs, the oscillation period is reduced.\nThe qualitative observations are supported by ex-\nplicit numerical calculations. We consider a Bloch-\nz DW with boundary conditions nz(x→ ±∞) =\n±1. The DW is formed by the effective field Heff=\n(2Aex/Ms)∂2\nxn+K/bardbln/bardble/bardbl−K⊥n⊥e⊥, with an easy(hard)\naxis along e/bardbl(⊥). Further, we choose the physical pa-\nrameters of Pt/Co/AlO xand setAex= 10−11J/m and\nMs= 1090kA /m. In addition, we fix α= 0.1 and\nset ∆sd= 0.25eV,vF= 106m/s and the current’s po-\nlarization P= 1. The Rashba interaction is chosen\naround ˜αR= 10−10eVm corresponding to ∆ R= 0.1eVandαR= ∆R/∆sd= 0.4. We vary the DW width\nλ=/radicalbig\nJ/K/bardbl, withJ= 2Aex/Ms, by setting J=λJ(0)\nandK/bardbl=K(0)\n/bardbl/λfor a fixed ratio/radicalBig\nJ(0)/K(0)\n/bardbl= 1. The\nperpendicular anisotropy is set to K⊥= 0.3K/bardbl. We in-\nject polarized electrons from the right in their relaxed\nstate. Thus, JR(x→ −∞) =Jrel\nR, withJrel\nR= (Is/v)n\nwiththesaturationspincurrent Is≡PIc/(2eMs). More-\nover, we use the oscillation period x∞\nosc=/planckover2pi1v/∆sd= 2.6\nnm as a length scale. This uniquely determines the ini-\ntial values of Eq. (6).\nTwo typical situations emerge and are illustrated in Fig.\n1. For broad DWs, the itinerant spins follow the DW\nadiabatically. In contrast, for steep DWs, the spins ex-\nperience a mismatch of nandJRparticularly at and\nbeyond the DW center. This induces a significant back-\naction on the STT. In Fig. 2, the three contributions\ntoTare shown for varying DW widths. For broad\nDWs (λ≫x∞\nosc), the conventional contributions Tad=\nIs|∂xn|,Tnonad=−βIs|∂xn|andH∞\nR= ∆sdαRIs/(v/planckover2pi1γ0)\nare recovered (cf. Appendix). In fact, Eq. (6) contains\nthese solutions for λ→ ∞. Differences occur for the\nRashba field-like torque due to corrections in first or-\nder in the derivative of the magnetization [28, 29]. In\ncontrast, for steep DWs ( λ < x∞\nosc), every STT compo-\nnent is altered. In particular, both the adiabatic and\nthe nonadiabatic STT show spatial oscillations which\nare damped after crossing the DW with the damping\nlengthxdamp=/planckover2pi1v/(β∆sd) =β−1x∞\nosc. For sharp DWs\n(λ≪x∞\nosc), the adiabatic STT is suppressed while the\nnonadiabatic STT is strongly increased. This behavior\nis even more pronounced when we exclude relaxation\n(β= 0). Then, the oscillations remain undamped be-\nyondtheDW(cf. Appendix). Besidesthe(non)adiabatic\nSTT, the Rashba field-like torque is also enhanced and\nmay even change its sign at the DW center. As we solved\nEq. (6) exactly no divergent terms appear for HRwhich\ncould be the case in approximate calculations[29].\nAn interesting quantity for technological applications is\nthe velocity of the DW center for a given applied current\ndensityIc. We show the results for the combinations of\nβ= 0,β= 0.2,αR= 0 and αR= 0.4 in Fig. 3. For\nbroad DWs in absence of the Rashba coupling ( λ≫x∞\nosc\nandαR= 0), we recover the well-known results. With-\nout relaxation, β= 0, and for small currents, there is\nno DW movement. A finite velocity ∝angbracketleftvDW∝angbracketrightarises when\nthe current density exceeds a critical value. For a finite\nβ= 0.2 = 2α, the Walker breakdown is observed, i.e.,\na strong increase of the velocity for small current and a\ndecrease beyond a critical Ic.\nHowever,the picturechangessignificantlyforsteepDWs.\nHere, even for β= 0, a finite DW velocity arises for\n0< λ < x∞\nosc. It actually resembles the Walker break-\ndown for β > αin the case of broad DWs. For λ→0,\nthe DW velocity decreases again.\nA finite Rashba field-like STT for αR= 0.4 mainly af-4\nβ= 0\n β= 0.2\nFIG. 3. (Color online) DW velocity vs DW width λand\napplied current density Icfor the cases without relaxation\n(β= 0) in (a,c,e), for a finite relaxation ( β= 0.2) in (b,d,f)\nfor either vanishing Rashba coupling ( αR= 0) (a,b), finite\nαR= 0.4 (c,d), or, for a width dependent αR= 0.42\nλ/x∞osc\n(e,f). Blue (red) colors emphasize a positive (negative) ve loc-\nity.\nfects broad DWs, since the nonadiabatic STT only can\nbe dominant for narrow DWs. So far, we have as-\nsumed a constant Rashba coupling independent of λ.\nThus, the Rashba field-like torque is comparable for all\nλand every magnetic field, such as an externally applied\nglobal field, moves broader DWs faster [35]. To remove\nthis trivial width dependence, we introduce a rescaled\nαR= 0.42\nλ/x∞osc. Indeed, the DW velocity then becomes\nindependent of λfor broad DWs. However, for small λ,\na strong influence of HRon∝angbracketleftvDW∝angbracketrightarises. In particular,\nfor a finite β, even a substantial backward motion of the\nDWcanbegenerated. ThemotionoftheDWagainstthe\ncurrent flow is induced by the antidamping field [29, 36]\nHanti\nRwhich is implicitly included in the STT.\nIt is illuminating to discuss the case of small λin terms\nof the nonadiabaticity parameter β∗=Tnonad/Tad[15].\nFor a nonlocal STT, β∗becomes x-dependent with sin-\ngularities at the roots of Tad. Hence, an averaged nona-\ndiabaticity parameter\n∝angbracketleftβ∝angbracketright=/integraltext\nβ∗(x)(∂xn)2dx/integraltext\n(∂xn)2dx(8)\nhasbeenintroduced [15]. In Fig. 4, weshow ∝angbracketleftβ∝angbracketrightforsmall\ncurrent densities. For broad DWs and αR= 0, the ex-\npected result ∝angbracketleftβ∝angbracketright=βis recovered. In contrast, for steep\nDWs,λ/lessorsimilarx∞\nosc, the nonadiabaticity strongly increases.\nThis clearly shows that the standard description of the\nSTT fails. What is more, for finite αR, even for broad\nDWs, it holds that ∝angbracketleftβ∝angbracketright ∝negationslash=β, because the (non)adiabatic\nSTT also implicitly includes the antidamping field and\nFIG. 4. (Color online) Averaged nonadiabaticity parameter\n/angbracketleftβ/angbracketrightvs DW width for β= 0 (red, no symbols) and β= 0.2\n(black, circles). Solid lines correspond to αR= 0.4 while\ndashedlinesreferto αR= 0. Theinsetshows βdv/dI cobtained\nfrom the derivative dv/dI cat small Ic→0. The values in the\ninset are normalized such that βdv/dI c(λ→ ∞,αR= 0) =β.\nthe standard solutions for these STTs no longer hold.\nAn alternativecharacterizationofthe nonadiabaticityre-\nsults from observing that, for a constant STT, the DW\nvelocity increases with d ∝angbracketleftvDW∝angbracketright/dIc∝βfor small cur-\nrent densities [37]. Thus, we define βdv/dIcby the deriva-\ntive of the DW velocity with respect to Icand normalize\nit to the conventional βvalid for large λ. The inset in\nFig. 4 shows that also βdv/dIcbecomes maximal around\nλ≈x∞\nosc, but approaches zero for λ→0. This further\nunderpins the non-trivial behavior for steep DWs which\nis not captured by the average ∝angbracketleftβ∝angbracketright.\nFinally, our findings apply also to magnetic textures\nother than Bloch DWs. For example, the increase of the\nnonadiabaticity in steep textures is responsible for the\ndifferences between steep vortices and broad spin waves\n[13, 14], with an increased βvortex≈5βspin-waves.\nIn summary, we have introduced a general procedure to\ncalculate the complete spin-transfer torque in arbitrary\n1D magnetic textures. It applies to the entire range\nof steep to broad domain walls and also can include\nother symmetry breaking interactions. Here, we have\ndiscussed the Rashba spin-orbit interaction. For abrupt\nchanges of the magnetic texture, the STT, the DW ve-\nlocity and the nonadiabaticity are qualitatively modified\nincludingbackwardmotionofsteep DWs againstthe cur-\nrent. This shows that steep magnetic textures require\na fully nonadiabatic description. An extension to two-\ndimensional structures such as vortices or skyrmions is\nstraight-forward for the case of vanishing Rashba cou-\npling, while current flow perpendicular to the direct cur-\nrent has to be considered for finite couplings.\nWe acknowledge support from the DFG SFB 668\n(project B16).5\nAppendix\nIn this Appendix, we present the full explicit equa-\ntions for the spin-transfer torque (STT) calculated from\nthe differential equation (6). In addition, we solve this\ndifferential equation in the limiting case of infinitely wide\ndomain walls. Finally, we show an additional plot of the\ncomponents of the STT with vanishing relaxation β= 0.\nExplicit expressions of the spin-transfer torque\nStarting from the Ansatz\nJr=armr+br∂(r)mr+crmr×∂(r)mr,(9)\nand with ∂(r)mr=∂(r)n\n|mr|−mr·∂(r)n\n|mr|mr, the STT T=\n−∆sdn×/summationtext\nrJrcan be calculated. Its components in\nEq. (5) are obtained as\nTad\nν=∆sd\n/planckover2pi1/summationdisplay\nrcr(mr·n)|∂(r)n|\nm2r|∂(r)mr|×/braceleftBigg\nvr ν=x\n1ν=t\nTnon−ad\nν=−∆sd\n/planckover2pi1/summationdisplay\nrbr|∂(r)n|\n|mr||∂(r)mr|×/braceleftBigg\nvr ν=x\n1ν=t\nHR=∆sdαR\n/planckover2pi1γ0/summationdisplay\nrp/parenleftbiggar\n|mr|−brmr·∂(r)n\nm2r|∂(r)mr|/parenrightbigg\n.(10)\nSolution of the differential equation for wide domain\nwalls\nWe show how the standard solutions for the STT are\nrecovered in the form of [11]\nT=˜Tad∂xn+˜Tnonadn×∂xn−HRn×ey(11)\nas the limiting case of an infinitely wide domain wall\n(DW) from the differential equation. Here, the standard\nexpressions [11] ˜Tad=Is,˜Tnonad=−βIsandHR=\n∆sdαRIs/(v/planckover2pi1) are known. To derive this result, we use a\nslightly modified Ansatz\nJr=armr+rv˜bb∂xmr+rv˜crmr×∂xmr.(12)\nHere, we have not normalized the derivatives and have\nexcluded the temporal derivatives from the beginning.\nWiththisAnsatz,aslightlymodifieddifferentialequation\nresults in the form\nvp∂x\nar\n˜br\n˜cr\n=Ar\nar\n˜br\n˜cr\n+∆sdβ\n/planckover2pi1\nmr·Jrel\nr\n∂xmr·Jrel\nr\n(mr×∂xmr)·Jrel\nr,\n\n(13)withJrel\nr=a(0)\nrnand/summationtext\nrva(0)\nr=Is. The coefficient\nmatrixAr(x,t) now reads\nAr=\n−∆sdβ//planckover2pi1v2|∂xmr|20\n−1−∆sdβ//planckover2pi1−f−∆sd|mr|//planckover2pi1−g\n0 ∆ sd|mr|//planckover2pi1+g−∆sdβ//planckover2pi1−f\n\n(14)\nwithf=rv(∂xmr·∂2\nxmr)/(∂xmr)2andg=rv[∂xmr·\n(mr×∂2\nxmr)]/(∂xmr)2. Since∂xmr∝∂xn∝1/λ, we\ncan neglect all derivatives in Eq. (13) in the limit of a\nbroad DW with λ→ ∞. When we further assume con-\nstant coefficients ar,˜brand ˜crand set|αR| ≪1, Eq. (13)\nreduces to a purely algebraic system of equations\n0 =ar−a(0)\nr,\n0 =−ar−∆sdβ\n/planckover2pi1˜br−∆sd\n/planckover2pi1˜cr,\n0 =∆sd\n/planckover2pi1˜br−∆sdβ\n/planckover2pi1˜cr. (15)\nThe solutions are ar=a(0)\nr,˜br=−a(0)\nrβ/planckover2pi1/[∆sd(1+β2)]\nand ˜cr=−a(0)\nr/planckover2pi1/[∆sd(1 +β2)]. Thus, we get the STT\nT=−∆sdn×/summationtext\nrJrin the final form of\nT|α2\nR|,β2≪1=Is∂xn−βIsn×∂xn−∆sdαRIs\nv/planckover2pi1n×ey.\n(16)\nIn fact, these are the standard solutions for the\n(non)adiabatic STT and the Rashba field-like torque.\nIt can be easily recognized how much the STT at small\nDW widths differs from the “broad DW limit” in the\ndefinition of Eq. (11) from numerical calculations. In\nFig. 5 we show the STT for several DW widths – for\nsimplicity for a vanishing αR. While for λ= 100∆ osc\nthe results of Eq. (16) are recovered, the components\nfor smaller DW widths actually differ at and away from\nthe DW center. This happens when the DWs are too\nsteep to allow the spin current density to remain aligned\nmainly parallel to the magnetization n, as it can be\nseen in Fig. (5) from the component in ndirection.\nSince the electron spin can change its orientation due\nto thesdinteraction over a length scale of the order\nx∞\nosc=/planckover2pi1v/∆sdit may be almost unchanged for λ≪x∞\nosc.\nAs the magneticdomains changein this lengthscale from\nn(x < xDW)→ −n(x > xDW) it means an anti-parallel\naligned electron spin. In fact, for a perfect anti-parallel\nalignment and β= 0 the electron spin swould never\nchange back to a parallel alignment as it would provide\nno finite torque T=−∆sdn×s. With a finite βthe elec-\ntron spin eventually relaxes back to the parallel align-\nment, but for typical β≪1 on an even larger length\nscalexdamp=x∞\nosc/β.\nFrom the results above we find that a STT description\ncontaining a constant spin current density exceeds is re-\nliability for steep magnetic textures.6\n-1-0,500,51(aR-aL) / IS\n010T~non-ad / ISλ = ∆osc\nλ = 10∆osc\nλ = 100∆osc\n-4 -2 0 2 4 6\nx / λ-10-50T~ad / IS\nFIG. 5. (Color online) Components of the STT T=\n˜Tnon−ad∂xn+˜Tadn×∂xnvs. reduced distance from the DW\ncenter for three different DW widths. In addition, the mag-\nnitude of the spin current density in n-direction, aR−aL, is\nshown. Note that the components are defined according to\nthe non-normalized derivatives of n. Parameters are αR= 0\nandβ= 0.2.\nComponents of the spin-transfer torque without\nrelaxation\nIntheabsenceofanyrelaxation,i.e., β= 0, neitherthe\nadiabaticnorthe nonadiabaticSTT approachesthe stan-\ndardstationarysolutionsafterthecurrenthascrossedthe\ndomain wall center xDW. Deviations from a non-parallel\nalignment arenot damped and mayexist overa verylong\ndistance. This can be seen in Fig. 6, where we plotted\nthe STT components according to the definition in Eq.\n(5) as it allows for more covenient physical interpreta-\ntion. Actually, oscillations with a period of 2 πv/planckover2pi1/∆sd\nare sustained for all x > xDWfor both the adiabatic and\nFIG. 6. (Color online) Spatial dependence of the\n(non)adiabatic STT and the Rashba field vs the reduced DW\nwidthλ/x∞\noscforβ= 0. Blue (red) values indicate posi-\ntive (negative) deviations from the standard solutions. 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One consequence of this is the possibility\nof the Cooper-pair spin current analogous to the magnon spin current in magnetic insulators, the\nanalogy also extending to the existence of the Gilbert damping of the collective spin-triplet dynamics.\nThe recently fabricated heterostructure of the thin \flm of the itinerant ferromagnet SrRuO 3on\nthe bulk Sr 2RuO 4, the best-known candidate material for the spin-triplet superconductor, o\u000bers a\npromising platform for generating such spin current. We will show how such heterostructure allows\nus to not only realize the long-range spin valve but also electrically drive the collective spin mode\nof the spin-triplet order parameter. Our proposal represents both a new realization of the spin\nsuper\ruidity and a transport signature of the spin-triplet superconductivity.\n^x^y φ\nJzsp(a) ^d\nφ\nJzsp^n\nJ ↑↑\nJ ↓↓(b) (c)\nFIG. 1. Schematic illustration of the analogy between the\nmagnetic insulator and the spin-triplet superconductor. (a)\nThe planar spiraling of the magnetic order parameter ^ nleads\nto spin current. (b) The same phenomena occurs for that of\nthe spin component ^dof the spin-triplet superconductor order\nparameter, (c) the dual picture of which is the counter\row of\nthe spin up-up and down-down pairs.\nIntroduction : Harnessing spin rather than charge in\nelectronic devices has been a major topic in solid state\nphysics, which not only has been utilized for various\nmemory devices but is also expected to play a key role\nin processing quantum information [1]. In order for vari-\nous spin devices to function robustly, the long-range spin\ntransport needs to be achieved. Metallic wires, however,\ntypically do not transport spins beyond the spin-di\u000busion\nlength due to the single electron spin relaxation [2].\nIn recent years, it has been shown that the exponential\ndamping can be circumvented in the spin transport via\ncollective magnetic excitations. For example, easy-plane\n(ferro- and antiferro-)magnetic insulators, as the U(1)\norder parameter can characterize them, may be consid-\nered analogous to the conventional super\ruid [3{5]. As\nFig. 1 (a) illustrates schematically, the planar spiraling\nof the magnetic order parameter in such magnetic insu-\nlators can give rise to the spin supercurrent, just as the\nphase gradient of the conventional super\ruid gives rise\nto the mass supercurrent; in this sense these magnetic\ninsulators can be regarded as spin super\ruids [6].\nInterestingly, there exists a class of super\ruids and su-\nperconductors which can support both mass and spin su-\npercurrent. Such super\ruids and superconductors wouldneed to involve both spin ordering and gauge symme-\ntry breaking. This occurs in the condensate of both\nthe spin-1 bosons [7] and the spin-triplet Cooper pairs\nof3He atoms [8, 9] or electrons [10, 11]; in the latter\ncase, the dissipationless spin current would be carried by\nthe Cooper pairs. While the vortices with spin supercur-\nrent circulation have been observed in all theses systems\n[12, 13], the bulk spin supercurrent has not been detected\nin the superconductor.\nIn this Letter, we will show how this existence of spin\nsuper\ruidity in the spin-triplet superconductor allows\nnot only the long-range spin current but also electrically\nexciting the spin wave in the bulk. For realizing these\nphenomena, we propose a two-terminal setup with volt-\nage bias between ferromagnetic metal leads in contact\nwith the spin-triplet superconductor. While the static\norder-parameter case [14] can be essentially reduced to\nthe Blonder-Tinkham-Klapwijk type formalism [15] for\nthe interfacial transport, here we need to complement\nit with the appropriate equations of motion for the col-\nlective spin dynamics in the superconductor. Recently,\na thin \flm of the itinerant ferromagnet SrRuO 3has\nbeen epitaxially deposited on the bulk Sr 2RuO 4, the best\nknown candidate material for the spin-triplet supercon-\nductor [16], yielding, due to their structural compatibil-\nity, an atomically smooth and highly conductive interface\n[17] with a strong Andreev conductance [18]. This makes\nSr2RuO 4and SrRuO 3the most suitable candidate mate-\nrials for the bulk and the leads, respectively, of our setup\n[19]. For the remainder of this paper, we will \frst show\nhow the simplest e\u000bective spin Hamiltonian for the spin-\ntriplet superconductor and the resulting spin dynamics\nare analogous to those of the antiferromagnetic insula-\ntor; then, we will discuss the magnetoresistance for the\nDC bias voltage and the coupling between the AC bias\nvoltage and the spin wave.\nGeneral considerations : We \frst point out the close\nanalogy between the spin order parameter of the antifer-arXiv:1802.01599v1  [cond-mat.supr-con]  5 Feb 20182\nromagnet and the spin-triplet superconductor. De\fned\ni(d\u0001\u001b)\u001by=\u0014\u0000dx+idydz\ndzdx+idy\u0015\n\u0011\u0014\u0001\"\"\u0001\"#\n\u0001#\"\u0001##\u0015\n;(1)\nthe d-vector of the spin-triplet pairing, which\nparametrizes the Cooper-pair spin state, be-\nhaves similarly under spin rotations to the N\u0013 eel\norder parameter of an antiferromagnet, i.e.,\n[Si(r);dj(r0)] =i\u0016h\u000fijk\u000e(r\u0000r0)dk(r) and [di;dj] = 0\nfor the condensate spin S(unlike the magnetization,\nneither the N\u0013 eel order parameter nor the d-vector\ngenerate the spin rotation in themselves) [8, 9, 11].\nGiven that the commutation relations establish S\u0002^d\nas the conjugate momentum to din both cases, it is\nnatural that the simplest e\u000bective Hamiltonian for the\nspin-triplet superconductor ^d-vector,\nH=1\n2Z\ndr[A(r^d)2+K^d2\nz+\r2\neS2=\u001f]; (2)\nwhere\reis the electron gyromagnetic ratio, Athe^d-\nvector sti\u000bness, and \u001fthe magnetic susceptibility, should\nbe equivalent to that of the antiferromagnet N\u0013 eel order\nparameter, once we identify the ^d-vector with the N\u0013 eel\norder parameter [4]. In the latter, antiferromagnetic case,\na (xy) planar texture of the orientational order param-\neter^ n!(cos\u001e;sin\u001e;0) is associated with a collective\n(z-polarized) spin current Jz/z\u0001^ n\u0002@i^ n!@i\u001e\row-\ning in theith direction. While this extends directly to\nour spin-triplet case, Eq. (1) gives the intuitive dual pic-\nture of Fig. 1 (c) for the planar spiraling of the d-vector,\ni.e.,^d= (cos\u000b;sin\u000b;0). Namely, as the phase of \u0001 \"\"\n(\u0001##) is given by \u001ec\u0007\u000b(where\u001ecis the overall phase\nof the superconductor), the spiraling of the d-vector on\nthexyplane as shown in Fig. 1 (b), or the gradient of\n\u000b, would imply the counter\row of the spin up-up and\ndown-down pairs. The resultant ( z-polarized) spin cur-\nrent is/\u0000r\u000b. Given the same commutation relation\nand the same e\u000bective Hamiltonian, it is natural that, in\nabsence of dissipation, the equations of motion for these\ntwo cases, the Leggett equations the ^d-vector [8, 9, 20]\nand the Landau-Lifshitz type equation for the N\u0013 eel order\nparameter, are identical.\nWe further argue that both cases have the same phe-\nnomenological form of dissipation as well. For the case of\nthe N\u0013 eel order parameter ^ n, such dissipation, /\u000b(@t^ n)2,\nknown generally as Gilbert damping for collective mag-\nnetic dynamics, has been understood phenomenologically\n[4, 21, 22]. That such dissipation has not been featured in\nthe 3He super\ruid literature can be attributed not to the\nintrinsic nature of the spin-triplet pairing but rather to\nthe very weak relativistic spin-orbit coupling of the 3He\natoms originating solely from the nuclear dipole-dipole\ninteraction [8]. In contrast, electrons in Sr 2RuO 4are\nsubject to the Ru atomic spin-orbit coupling [23] esti-\nmated to be of order 0.1 eV [24]. In this work, we willconsider the decay rate of \u000bn\u0016h\r2\ne=\u001ffor the condensate\nspin, the addition of which makes the Leggett equations\nof motion for spin [25] equivalent to the Landau-Lifshitz-\nGilbert type equations for antiferromagnets:\n@t^d=\u0000^d\u0002\r2\ne\n\u001fS;\n@tS=^d\u0002(Ar2^d\u0000K^dz^ z\u0000\u000bn\u0016h@t^d); (3)\nwhere\u000bis the dimensionless Gilbert damping parame-\nter andnthe Cooper-pair density. This set of equations\nshows how the e\u000bective Hamiltonian of Eq. (2) provides\nthe simplest method for considering the local ^d-vector\ndynamics, including the spin-wave excitation and the col-\nlective dissipation.\nFor the boundary conditions, at the interface between\nthe ferromagnetic lead and the spin-triplet superconduc-\ntor, we consider a two-channel interface conductance due\nto the spins aligned or anti-aligned to the lead magne-\ntization We note, in this regard, that the SrRuO 3thin\n\flm has a very high transport spin polarization, with\na 3-to-1 ratio between the majority and minority spin\nchannels [26{28], while the magnetization gets enhanced\nin the heterostructure [17]. In this Letter, for the sake of\nsimplicity, we shall only consider the case where the lead\nmagnetizations are collinear. Furthermore, the d-vector\nof the bulk spin-triplet superconductor will be taken to be\nperpendicular to the lead magnetization, i.e., the Cooper\npairs are equal-spin paired along the quantization axis\nparallel to the magnetization; it has been claimed for\nthe Sr 2RuO 4superconductor, based on the c-axis NMR\nmeasurement, that its d-vector can be rotated into the\nab-plane by applying magnetic \feld larger than 200 G\n[29], well below the upper critical \feld.\nLong-range spin valve : The simplest physics that can\narise in our two-terminal setup is the spin-valve magne-\ntoresistance due to the relative alignment of the leads.\nWe consider the case where the spin-triplet supercon-\nductor has the easy-plane anisotropy, that is, K > 0\nin Eq. (2), while the lead magnetization is perpendic-\nular to this plane; as already mentioned, the former\ncan be realized for the SrRuO 3/Sr2RuO 4heterostruc-\nture by applying a \u0015200 G \feld along the c-axis. In\nthis case, we can take ^dzto be a small parameter in ^d=\n(q\n1\u0000^d2zcos\u001ez;q\n1\u0000^d2zsin\u001ez;^dz) andjSx;yj\u001cjSzj. In\nsuch a case, [ \u001ez(r);Sz(r0)] =i\u0016h\u000e(r\u0000r0) gives us the con-\njugate pair, leading to the equations of motion\n@t\u001ez=\r2\ne\n\u001fSz; @tSz=Ar2\u001ez\u0000\u000bn\u0016h@t\u001ez;(4)\nwhere the \frst equation is a spin analogue of the Joseph-\nson relation and the second is the spin continuity equa-\ntion with the relaxation term. Note that we measure\nSzwith respect to its equilibrium value. One con\frms\nthe condensate spin imbalance relaxation time to be3\nHa\nVLIVR\n^\n^yz\nx^\nI\nVLVR\nz^\n^y\n^x\nFIG. 2. The setup for the DC voltage bias for the spin valve\n(upper) and the AC bias voltage for the spin-wave detection\n(lower), where ^ x;^ y;^ zcoincide with the crystalline a; b; c -axes,\nrespectively. For the upper \fgure, the lead magnetization is\nalong the c-axis, with the applied magnetic \feld Ha\u0015200 G\nalong the c-axis giving us the easy plane d-vector con\fgura-\ntion on the ab-plane, hence the spiraling in the ab-plane. For\nthe lower \fgure, the lead magnetization is along the a-axis; as\nthe easy-axis d-vector anisotropy favors the alignment along\nthec-axis, in the absence of an applied \fled, the AC bias volt-\nage gives us the low-frequency standing wave of the d-vector\noscillating around the c-axis in the bc-plane.\n\u001f=\u000bn \u0016h\r2\nefrom Eq. (4) through deriving @tSz+r\u0001Jsp\nz=\n\u0000\u000bn\u0016h\r2\neSz=\u001f, where Jsp\nz=\u0000Ar\u001ez. It is also impor-\ntant to note here that the magnitude of the d-vector\nanisotropy Khas no e\u000bect on the in-plane d-vector pre-\ncession, which allows us to ignore the fact that our ap-\nplied \feld gives us the Abrisokov vortices in the spin-\ntriplet superconductor and hence a non-uniform K.\nWe consider the spin-up current and the spin-down\ncurrent to be independent at the interface:\nI\u001b\nL;R=\u0006g\u001b\u001b\nL;R(VL;R\u0000\u0016h@t'\u001b=2e); (5)\nwhereg\u001b\u001b\nL;R's are the conductances for the \u001b-spin,IL;R\nthe\u001b-spin current into (out of) the left (right) lead, and\nVL;Rthe bias voltage of the left (right) lead; this is due\nto the spin-triplet superconductor having the equal spin\npairing axis collinear with the lead magnetization and\ntakingg\"#= 0. From Eq. (1), we see that the overall\n(or charge) phase of the superconductor is given by the\naverage of the spin up-up and the spin down-down con-\ndensate phase, \u001ec=P\n\u001b'\u001b=2, while\u001ezof Eq. (4) is given\nby\u001ez=P\n\u001b\u001b'\u001b=2. We are interested here in the steady-\nstate solution, i.e., @t'\u001b= const, for which we de\fne the\nconstant precession rate of !c\u0011P\n\u001b@t'\u001b=2 for the over-\nall phase\u001ecand \ns\u0011P\n\u001b\u001b@t'\u001b=2 for\u001ez. For such\nsolution, the following continuity conditions can be ap-plied to the charge and spin supercurrents, respectively:\nX\n\u001b(I\u001b\nL\u0000I\u001b\nR)=0;X\n\u001b\u001b(I\u001b\nL\u0000I\u001b\nR)=2\u000bne\nsSL (6)\n(Sis the bulk cross section area and Lthe spacing be-\ntween the two leads), the former from the charge con-\nservation and the latter from applying the steady-state\ncondition on Eq. (4), along with the spin current loss\n/\u000bLin the superconductior.\nThe current through the Sr 2RuO 4bulk can be ob-\ntained from the interface boundary conditions and the\ncontinuity conditions above, with the larger magni-\ntude for the parallel magnetization than the antipar-\nallel magnetization. We de\fne the total conductance\ngL;R\u0011P\n\u001bg\u001b\u001b\nL;R and the conductance polarization\npL;R\u0011P\n\u001b\u001bg\u001b\u001b\nL;R=gL;R, which de\fnes the relevant trans-\nport spin polarization. Applying the continuity condi-\ntions Eq. (6) on the interface boundary conditions Eq. (5)\nand setting VL=\u0000VR=V=2, we obtain\n\u0012\ngL+gRpLgL+pRgR\npLgL+pRgRgL+gR+g\u000b\u0013\u0012\n!c\n\ns\u0013\n=eV\n\u0016h\u0012gL\u0000gR\npLgL\u0000pRgR\u0013\n;\n(7)\nwhereg\u000b\u00114\u000bne2SL\n\u0016h. We can now obtain the dependence\nof the charge current on the conductance polarization:\nIc=X\n\u001bI\u001b=I0\u0014\n1\u0000gLgR(pL\u0000pR)2\n(gL+gR)(gL+gR+g\u000b)\u0000(pLgL+pRgR)2\u0015\n;\n(8)\nwhereI0\u0011gLgRV=(gL+gR). Note that Icis max-\nimized atpL=pR, when the steady-state angle \u001ezre-\nmains static. Di\u000berent spin polarizations at the two ends,\non the other hand, would trigger spin dynamics and re-\nsult in a nonzero dissipation rate of R=1\n2\u000bn\u0016h\n2\ns=\nR0(1\u0000Ic=I0)2=(pL\u0000pR)2per volume of the supercon-\nducting bulk, where R0= 8\u000bn(eV)2=\u0016h. Given that pL;R\nchange sign on the magnetization reversal, the above re-\nsults e\u000bectively give us the spin-valve magnetoresistance\nof our heterostructure, i.e., a larger conductance for the\nparallel magnetizations than for the antiparallel. Any\ne\u000bect that the spin-triplet pairing may have on the mag-\nnetization, hence the conductance polarization, can be\nignored when the Curie temperature of SrRuO 3(\u0018160K)\n[30] is two orders of magnitude higher than the supercon-\nducting critical temperature ( \u00181.5K) Sr 2RuO 4.\nWe emphasize that the above magnetoresistance re-\nsult is obtain solely for the current carried by Cooper\npairs. At a \fnite-temperature, quasiparticle contribu-\ntion would generally result in an exponentially-decaying\nmagnetoresistance, negligible for the lead spacing be-\nyond the spin-di\u000busion length. By contrast, the cur-\nrent of Eq. (8), which is carried by the Cooper pairs,\ngives us the\u00181=Lbehavior for the large spacing limit.\nTherefore, any magnetoresistance beyond the quasipar-\nticle spin-di\u000busion length should arise only below the su-\nperconducting transition at Tc, upon the emergence of4\n0.80.91.|Ic|/I0\n12340.50.60.70.80.91.\nω/ω0|Ic|/I0\nFIG. 3. Charge current versus frequency plotted for ~ g= 0:5,\n~L= 2, \u0000 =!0= 0:1 and ~A= 0:2, with the orange curve\nrepresenting pL=pR=pand the blue pL=\u0000pR=p. Note\nthatp= 0:8 for the top plot and p= 0:2 for the bottom plot.\na Cooper-pair condensate. For our Sr 2RuO 4/ SrRuO 3\nheterostructure, detection of magnetoresistance in the su-\nperconducting state for the lead spacing larger than the\nSr2RuO 4spin-di\u000busion length can be taken as a trans-\nport evidence for the spin-triplet superconductivity. The\nvalue of the spin-di\u000busion length itself can be extracted\nby measuring the exponential decay of the (normal) mag-\nnetoresistance, both above and below the transition.\nElectrically driven spin collective mode : For the case of\nthe easy-axis anisotropy of the d-vector, hence K < 0 in\nEq. (2),the spin collective excitation of the Cooper pairs\n[8, 9, 31, 32] will modify the supercurrent transport under\nthe AC bias voltage. We shall still continue to consider\nthe case where Eq. (5) would be valid, i.e., the equal spin\npairing axis of the spin-triplet superconductor collinear\nto the lead magnetizations. One way to satisfy this con-\ndition would be to have the lead magnetizations collinear\nto thea-axis, with no applied magnetic \feld; that would\nleave thea-axis as the equal spin pairing axis, with the\nd-vector moving on the the bc-plane. The equations of\nmotion, corresponding to spin injection polarized along\nthex-direction, are then modi\fed to\n@t\u001ex=\r2\ne\n\u001fSx; @tSx=Ar2\u001ex\u0000!2\n0\u001f\n\r2ecos\u001exsin\u001ex\u0000\u000b\u0016h@t\u001ex;\n(9)\nwhere\u001exis conjugate to Sxand!2\n0\u0011 jKj\r2\ne=\u001fis\nthe spin-wave energy gap. For the AC voltage bias\nV=V0exp(\u0000i!t), the steady-state solution for thespin phase \u001ex(x;t) =f(x) exp(\u0000i!t) and the charge\nphase\u001ec(x;t) =g(x) exp(\u0000i!t) behave di\u000berently, fo-\ncusing on the frequencies far below the plasma fre-\nquency. Hence the spin equations of motion Eq. (9)\ngives usf(x) =C+cosh\u0014x+C\u0000sinh\u0014x, wherev2\u00142=\n!2\u0000!2\n0\u0000i!\u0000, withv\u0011\rep\nA=\u001f (the^d-vector sti\u000b-\nnessAde\fned in Eq. (2)) being the spin-wave veloc-\nity and \u0000\u0011\u000bn\u0016h\r2\ne=\u001fthe damping rate. By contrast,\nthe charge current Jc(x;t) =\u0000\u001a@x\u001ec, where\u001ais the\n\u001ecsti\u000bness, should be uniform, which means we can set\n\u001ec(x;t) = const:\u0000x(Jc\n0=\u001a) exp(\u0000i!t), with a constant\nJc\n0. By imposing consistency between the current ob-\ntained from the boundary conditions of Eq. (5) and the\ndynamics of Eq. (9), we can solve for Jc\n0andC\u0006; Fig. 3\nshows the numerical results for Ic=Jc\n0Sfor the case of\nbothpL=pRandpL=\u0000pR.\nOur numerical results show that magnetoresistance be-\ncomes signi\fcant at !>\u0018!0, where the collective spin\nmode of the Cooper pairs is activated. For simplicity\nwe have set gL=gR=gand used the dimensionless\nparameters ~ g\u0011g\u0016hv=2eA,~L\u0011!0L=2v, and ~A=A=\u001a.\nFor! <! 0, in addition to barely noticeable magnetore-\nsistance, the charge current amplitude does not oscillate\nwith frequency; it remains close to the DC value I0, which\ncontrasts with the complete transport suppression ob-\ntained for the magnetic insulator [3]. In contrast, for\n! > ! 0, we see an oscillation with the !=! 0period of\nabout\u0019=~L, where the current amplitude maxima for the\nantiparallel lead magnetization occur at the current am-\nplitude minima for the parallel lead magnetization and\nvice versa. As in the ferromagnetic insulator [3], we ex-\npect that for ~L\u001c1 (whileLis still larger than the quasi-\nparticle spin-di\u000busion length), the magnetoresistance of\nEq. (8) is recovered for the static bias, i.e.,!!0.\nWe point out that the detection of the oscillation\nshown in Fig. 3 would determine the yet-unknown energy\nparameters for the spin-triplet pairing of Sr 2RuO 4. From\nthe e\u000bective Hamiltonian of Eq. (2), if we had known\naccurately the \feld Hcalong thec-axis that would ex-\nactly restore the d-vector isotropy, the gap frequency !0\nshould be just the electron Larmor frequency of this \feld\nfrom the spin equations of motion of Eq. (9). However,\nwe know no more than the upper bound Hc<200 G,\nhence only !0< \re\u0002200 G = 3:5 GHz, while the AC\nbias experiment, as shown in in Fig. 3, would allow us to\nde\fnitely identify the spin collective mode gap.\nConclusion and discussion : We have studied the DC\nand AC current transport between the itinerant ferro-\nmagnetic lead with collinear magnetization through the\nspin-triplet superconductor. We showed here that mag-\nnetoresistance can arise for both cases due to the Cooper-\npair spin transport. For the DC bias, the persistence\nof magnetoresistance for the lead spacing larger than\nthe quasiparticle spin-di\u000busion length can be taken as\na transport evidence for the spin-triplet pairing. For\nthe AC bias, the activation of magnetoresistance and5\nfrequency dependent oscillation above the threshold fre-\nquency will allow us to determine the spin anisotropy\nenergy scale. All together, our work shows both a new\nrealization of the spin super\ruidity and a transport sig-\nnature of the spin-triplet superconductivity. The recently\nfabricated SrRuO 3/Sr2RuO 4heterostructure provides a\npromising experimental setup.\nAcknowledgement : We would like to thank Young Jun\nChang, Bongju Kim, Han Gyeol Lee, Seung Ran Lee,\nYoshiteru Maeno, Tae Won Noh, S. Raghu, Manfred\nSigrist and So Takei for sharing their insights. The hos-\npitalities of Natal International Institute for Physics dur-\ning the workshop \\Collective Spin Transport in Electrical\nInsulators\" (S.B.C. and Y.T.) and of Kyoto University\nduring the workshop \\Oxide Superspin 2017\" (S.B.C.),\nwhere parts of this work has been completed, are grate-\nfully acknowledged. This research was supported by IBS-\nR009-Y1 (S.B.C. and K.H.L.), and United States Army\nResearch O\u000ece under Contract No. W911NF-14-1-0016\n(S.K.K. and Y.T.).\n\u0003sbchung0@uos.ac.kr\nyevol@physics.ucla.edu\n[1] S. A. 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B 94, 064508 (2016)."    },    {        "title": "2306.15227v1.Quantum_spin_fluctuations_in_dynamical_quantum_phase_transitions.pdf",        "content": "Quantum spin fluctuations in dynamical quantum phase transitions\nCheuk Yiu Wong,1Hadi Cheraghi,2and Wing Chi Yu1,∗\n1Department of Physics, City University of Hong Kong, Kowloon, Hong Kong\n2Institute of Physics, Maria Curie-Sk lodowska University, 20-031 Lublin, Poland\n(Dated: June 28, 2023)\nQuantum phase transitions have long been studied in their relation to quantum fluctuations.\nThese fluctuations can be quantified as the degree of spin squeezing in spin models, where one\nof the two non-commutative observables breaks the standard quantum limit of measurement by\nminimizing its uncertainty. However, the understanding of their role in dynamical quantum phase\ntransitions (DQPTs) is still incomplete. In this work, we combine the Loschmidt amplitude, which\ndetects DQPTs, and the spin-squeezing parameter (SSP), the quantification of spin squeezing, to\nstudy the spin dynamics in a quenched interacting spin model around DQPT. We show that the\nextremal, mostly maximal, of SSP occurs near DQPTs when the system is quenched between dif-\nferent phases. These phenomena further unveil the spin correlations during DQPTs, for which the\nhighest contribution aligns with the preferred direction of spin interactions in the post-quenched\nphase. We also demonstrate the time evolution of SSP differs for various quench scenarios. These\nfindings provide us with physical insights into the dynamics of quantum fluctuations around DQPTs\nand their relation to the equilibrium phase diagrams.\nI. INTRODUCTION\nDynamical quantum phase transitions (DQPTs) have\nbeen serving as a theoretical framework on far-from-\nequilibrium physics of quantum many-body systems.\nThe theory originates as an analogy to the equilibrium\nstatistical mechanics about the phase transitions [1–3].\nDQPTs can be triggered through a sudden quench pro-\ntocol from the initially prepared Hamiltonian Hiin one\nphase or at phase boundary to the final Hamiltonian Hf\nin another phase. DQPTs occur at times when the dy-\nnamic quantity Loschmidt amplitude (LA) defined as\nG(t) =⟨ψi\n0|ψ(t)⟩=⟨ψi\n0|e−iHft/ℏ|ψi\n0⟩, (1)\nthe overlap of the time-evolved state |ψ(t)⟩onto the ini-\ntial ground state |ψi\n0⟩, vanishes, or equivalently the dy-\nnamic free energy or Loschmidt rate (LR)\nλ(t) =−lim\nN→∞1\nNln[L(t)], (2)\nwhere L(t) =|G(t)|2is termed Loschmidt echo (LE),\nbehaves nonanalytically [1–3]. This is the definition of\ntype-II DQPTs. On the other hand, the type-I DQPTs is\ndefined under the long-term dynamics of a system’s order\nparameter on a sudden quench [4–8]. Supported by the\nrealization of DQPTs in experiments for various quan-\ntum systems via quantum simulators [9–18], the estab-\nlishment of the whole DQPT theories thrive. There has\nbeen research on the fundamental properties of DQPTs\nand the theoretical studies of DQPTs in various quantum\nmodels [1–8, 19–72]. DQPTs are also substantially shown\nto have deep connections to entanglement spectrum and\ncorrelation matrix [2, 26–31, 46, 68].\n∗wingcyu@cityu.edu.hkThe wealth of linkage to the entanglement motivates\nthe investigation of a more experimentally accessible\nquantity regarding spin dynamics, the spin-squeezing pa-\nrameter (SSP). SSP quantifies the quantum fluctuation\nwhen one measures one of two non-commutative spin ob-\nservables and it has various definitions fitting the pur-\npose of studies [73–75]. The idea of using SSP as a tool\nto study quantum phase transitions was introduced in\nthe past twenty years [76–78]. A recent paper showed\nthat a certain definition of SSP can be a probe to detect\nthe type-I DQPT, where the SSP behaves nonanalytically\nagainst the tunable parameter of a Bose-Einstein conden-\nsate after a sudden quench across certain phase bound-\nary [79]. Similar effect is also observed in experiment for\nanother model between different dynamical phases [16].\nA recent work on SSP in the one-dimensional (1D) XY\nmodel, where the authors showed the possibility to gen-\nerate spin-squeezed state from originally unsqueezed sys-\ntem by quenching, also stated the nonanalyticities found\nin long-time-averaged SSP against post-quenched Hamil-\ntonian for an arbitrary pre-quenched Hamiltonian, and\nsuggested a potential linkage to equilibrium phase transi-\ntions [80]. These works seem to suggest potential connec-\ntion between type-I DQPTs and quantum fluctuations,\nwhile the short-term physical nature behind the type-II\nDQPTs is not well-addressed.\nInspired by the extensive use of spin squeezing in the\nmentioned research, we investigate the spin dynamics by\nobserving the spin squeezing of a spin-1\n2model around\nDQPTs under different quench protocols in this work.\nWe adopt the definition of SSP by Ueda and Kitagawa\n[81]\nξ2\nS=4(∆Jˆn⊥)2\nN, (3)\nwhere (∆ Jˆn⊥)2denotes the variance of the spin opera-\ntors orthogonal to the mean-spin direction (MSD) of thearXiv:2306.15227v1  [cond-mat.str-el]  27 Jun 20232\nsystem, with the spin operators defined as\nJa=1\n2NX\nj=1σa\nj a=x, y, z. (4)\nThe state is considered to be squeezed if ξ2\nS<1, and the\nlarger the SSP is, the less the state is squeezed.\nIn this work, we combine the conventional LA analysis\nunder a sudden quench and the study of time evolution of\nSSP defined by Eq. (3) in the XY model. Along with ex-\ntracting explicitly the two-point correlation functions in\nthe analytical expression of SSP, we demonstrate how the\nevolution of quantum fluctuation unveils the spin corre-\nlations during DQPT. Namely, our results show the SSP\nattains local maximum near DQPT in most of the cases.\nFurthermore, we show the cause of this increase aligns\nwith the physical properties of the underlying equilibrium\nphase the system is quenched to. We also distinguish the\nevolution pattern of SSP for different types of quench,\nnamely quenches across and from the equilibrium phase\nboundary, and within one phase, in the regime around\nthe first critical time. We believe our study provides in-\nsights on filling the knowledge gap of spin fluctuation in\nDQPTs as well as the physical essence of DQPTs.\nThe manuscript is organized as follows: Sec. II covers\nthe analytical details necessary for the study, including\nthe expression of LA and SSP. Sec. III demonstrates the\nfindings in different quench scenarios, with subsections\ndivided as follows: Sec. III A contains the analysis on\nquenches between Ising ferromagnetic (FM) and para-\nmagnetic (PM) phases; Sec. III B analyzes the results\nfor quenches between anisotropic phases, and Sec. III C\nanalysis of quenches from the multicritical point of the\nXY model. Finally, a conclusion and some future per-\nspectives regarding this work are given in the last sec-\ntion.\nII. FORMULATION AND 1D XY MODEL\nWe demonstrate our analysis on the 1D XY model\nwhich is an integrable model and was proven the exis-\ntence of DQPTs [38, 69]. The Hamiltonian is the follow-\ning\nH(δ, g) =−ℏ\n2NX\nj=1\u0014\nJ\u00121 +δ\n2σx\njσx\nj+1+1−δ\n2σy\njσy\nj+1\u0013\n+gσz\nj\u0015\n,\n(5)\nwhere Nspecifies the system size, Jis the coupling\nstrength, δgoverns the anisotropic coupling between\nspins along the xandydirections, gis the external mag-\nnetic field strength along the zdirection, and periodic\nboundary condition is adopted. Below we set ℏ=J= 1\nfor convenience. The model exhibits competitions be-\ntween anisotropic and magnetic coupling which results\nin the existence of multiple phases. In particular, g <1\ngives rise to the FM phase, where it is further dividedinto x-spin dominant FM xphase and y-spin dominant\nFMyphase depending on the sign of δ. When g >1, the\nmodel transits to the PM phase with the ground-state\nMSD aligning parallel to the external magnetic field. Fig-\nure 1 shows the equilibrium phase diagram of the 1D XY\nmodel. We will perform quenches across and from differ-\nent phase boundaries and critical points.\nWe study the quench dynamics of the model by first\nJordan-Wigner transformation σ+\nj= exp[ iπP\nl<jc†\nlcl]cj\nandσz\nj= 1−2c†\njcj, followed by a Fourier transformation\ncj= (1/√\nN)P\nke−ijkck[82–84] and turns the Hamilto-\nnian into\nH(δ, g) =X\nk>0η†\nkH(k)ηk, (6)\nwhere ηk= (ck, c†\n−k)Tand H(k) = ⃗d(k)·⃗ σis\nthe Bloch Hamiltonian with the Bloch vector ⃗d(k) =\n(0,−δsink, g−cosk)Tand⃗ σ= (σx, σy, σz) are the Pauli\nmatrices [35]. For the sake of analysis, we concern the\nunit Bloch vectors ˆd(k) =⃗d(k)/|⃗d(k)|. We restrict our\ncalculations in the even-parity subspace and Nbeing\neven with the k’s of the form k= (2m−1)π/N, where\nm= 1,2, . . . , N/ 2. Performing the Bogoliubov transfor-\nmation ck= cos[ θk(δ, g)]βk+isin[θk(δ, g)]β†\n−k, the diag-\nonalized Hamiltonian reads\nH(δ, g) =X\nk>0εk(δ, g)(β†\nkβk−β−kβ†\n−k), (7)\nwhere εk(δ, g) =|⃗d(k)|=p\n(cosk−g)2+ (δsink)2is\nthe quasiparticle eigenenergy, and θk(δ, g)∈[0, π/2] is\nthe Bogoliubov angle defined such that tan[2 θk(δ, g)]≡\nδsink/(cosk−g).\nFMx\nFMyPM\nFIG. 1. Schematic drawing of the ground-state phase diagram\nof the equilibrium 1D XY model.3\nConsider the system initially prepared in Hi=\nH(δi, gi) and quenched to Hf=H(δf, gf). Using the\nvacuum state as the initial state, LA in Eq.(1) can be\ndecomposed into momentum-wise form, namely G(t) =Q\nk>0Gk(t), where\nGk(t) = cos2ϕkeiεk(δf,gf)t+ sin2ϕke−iεk(δf,gf)t(8)\nwith ϕk=θi\nk−θf\nkbeing the difference between the initial\nand the final Bogoliubov angles. The nonanalyticities in\nLA occurs when ϕk=±π/4 at critical times\ntp\nc=π\nεf\nk∗\u0012\np−1\n2\u0013\np∈N, (9)\nwhere k∗is the critical momentum when the correspond-\ning initial and final unit Bloch vectors ˆdi(k∗) and ˆdf(k∗)\nare orthogonal to each other, i.e. ˆdi(k∗)·ˆdf(k∗) = 0.\nThe behavior of the momentum-wise LA can be visu-\nalized by the vector introduced by Ding [52]:\n⃗ rk(t) = (xk, yk) =|Gk(t)|eiϕG\nk(t), (10)\nwhere ϕG\nk(t) =ϕG\nk(t)−ϕdyn\nk(t) is the Pancharactnam ge-\nometric phase (PGP) [85, 86], defined by the difference\nbetween the phase of LA ϕG\nk(t) = Arg[ Gk(t)] and the dy-\nnamical phase ϕdyn\nk(t) =−Rt\n0dt′⟨ψk(t′)|Hf(k)|ψk(t′)⟩=\nεf\nkcos(2 ϕk)t. The PGP characterises the dynamical topo-\nlogical order parameter (DTOP) [35]\nνD=1\n2πIπ\n0∂ϕG\nk(t)\n∂k, (11)\nwhich changes its value at DQPTs.\nOn the other hand, we adopt the formulation in Ref.\n[80] regarding the spin dynamics analysis. The MSD of\nXY model is along zdirection as a result of the Z2sym-\nmetry which guarantees ⟨Jx⟩=⟨Jy⟩= 0. The factor\n(∆Jˆn⊥)2in Eq. (3) is then taken to be [(cos α)Jx+\n(sinα)Jy]2. Setting ⟨O⟩ ≡ ⟨ ψ(t)|O|ψ(t)⟩for any oper-\natorOfor simplicity, the SSP reads\nξ2\nS(t) =2\nNmin{⟨J2\nx+J2\ny⟩+J(α, t)}, (12)\nwhere\nJ(α, t) = cos(2 α)⟨J2\nx−J2\ny⟩+ sin(2 α)⟨JxJy+JyJx⟩.\n(13)\nThe angle which yields the minimum of Eq. (12) is\ntermed the squeezing angle and is denoted as αsin the\nfollowing. This quantity has been studied in the con-\ntext of qubit systems relating its time evolution to the\nenergy spectrum of the systems [87]. In this paper, we\nwill demonstrate how it embeds the competition between\nthe net parallel-spin contribution and the cross-spin con-\ntribution over time. With transitional symmetry of thechain, the time evolution of SSP can be written as\nξ2\nS(t) = 1 + 2N−1X\nn=1(Gxx\nn(t) +Gyy\nn(t))\n−2(\u0014N−1X\nn=1(Gxx\nn(t)−Gyy\nn(t))\u00152\n+\u0014N−1X\nn=1(Gxy\nn(t) +Gyx\nn(t))\u00152)1\n2\n,(14)\nwhere Gab\nn(t) =⟨σa\n1σb\n1+n⟩/4 is the correlation function\nalong direction aandbandGaa\n0(t) =⟨σa\n1σa\n1⟩/4 = 1 /4.\nThe calculation of the correlation functions involves\ncomputing strings of operators of the forms [88]\nGxx\nn(t) =1\n4⟨B1A2B2···AnBnAn+1⟩\nGyy\nn(t) =(−1)n\n4⟨A1B2A2···BnAnBn+1⟩\nGxy\nn(t) =i\n4⟨B1A2B2···AnBnBn+1⟩\nGyx\nn(t) =i(−1)n−1\n4⟨A1B2A2···BnAnAn+1⟩,(15)\nwhere A=c†\nj+cjandB=c†\nj−cjin Jordan-Wigner ba-\nsis. By Wick’s theorem, the expectation values turn into\nsums of all possible permutations of products of opera-\ntor pairs, which turns out to be a problem of calculating\npfaffians of 2 n×2nskew-symmetric matrices\nGab\nn(t)∼pf\n⟨O1O2⟩ ⟨O 1O3⟩ ⟨O 1O4⟩ ··· ⟨O 1O2n⟩\n⟨O2O3⟩ ⟨O 2O4⟩ ··· ⟨O 2O2n⟩\n⟨O3O4⟩ ··· ⟨O 3O2n⟩\n......\n⟨O2n−1O2n⟩\n.\n(16)\nNote that at t= 0 for parallel-spin correlation a=b=\nx, y, the pfaffians reduce to Toeplitz determinants known\nfor the analytical works of Lieb et al. in correlation func-\ntion calculations in XY model [88]. The elements of the\npfaffians are four types of two-point functions as follows:\n⟨AjAj+n⟩=⟨BjBj+n⟩\n=−i2\nNX\nk>0sin(2ϕk) sin( nk) sin(2 εf\nkt)\n⟨AjBj+n⟩=2\nNX\nk>0[cos(2 ϕk) cos(2 θf\nk+nk)\n−sin(2ϕk) sin(2 θf\nk+nk) cos(2 εf\nkt)]\n⟨BjAj+n⟩=−2\nNX\nk>0[cos(2 ϕk) cos(2 θf\nk−nk)\n−sin(2ϕk) sin(2 θf\nk−nk) cos(2 εf\nkt)].(17)\nIn the next section, we study the spin dynamics in\nterms of the SSP and the correlation functions in relation\nto DQPT under three scenarios:4\n(b) (a)\n(c) (d)\nFIG. 2. Scaled time plots of LR, DTOP (top panels) and SSP\n(bottom panels) at δ= 0.8 for (a) gi= 0.4→gf= 2, (b)\ngi= 2→gf= 0.1, (c) gi= 1→gf= 2 and (d) gi= 1→\ngf= 0.5. Red and blue dashed lines indicate the first peak\nof LRs and SSPs respectively. Black dashed horizontal lines\nindicate ξ2\nS(t) = 1.\n1. quenches between Ising phases, where we fixed the\nδand alter along g;\n2. quenches between anisotropic phases, where we\nfixed gand quench along δin the region bounded\nby−∞< δ < ∞and 0 ≤g <1;\n3. quenches from the multicritical point ( δc, gc) =\n(0,1).\nAlternatively we use the shorthanded notation ( δi, gi)→\n(δf, gf) for any quench when necessary. All quenches\nwere performed on an N= 100 system, and we concen-\ntrate our discussions around the first critical time.\nIII. DYNAMICS OF SPIN SQUEEZING\nA. Quenches between Ising phases\nWe performed quench cases involving the Ising bound-\narygc= 1 while fixing δ= 0.8. The quench dynamics\nof forward quench ( gi< gf), backward quench ( gi> gf)\nand quench from critical point ( gi=gc= 1) are shown\nin FIG. 2. The DTOPs show the discontinuous jumps\nat critical times as expected. Note that all the extra\nspikes on the DTOPs in this paper are finite-size ef-\nfect as confirmed numerically and they would vanish to\ngive true DTOP evolution pattern shown in Ref. [35]\nfor larger systems. The vectors ⃗ rk’s plotted in FIG. 3a\nand 3b swipe a full circle passing through the origin for\n(a)\n (b)\n(c)\n (d)FIG. 3. Trajectories of ⃗ rk’s at the corresponding critical times\nfor the same quench cases in FIG. 2. Black dots indicate\nthe origin the vectors are located. The color scale shows the\nprogression of ⃗ rk’s from k= 0 to π.\nquenches across the equilibrium phase boundary. The\nmomentum at which the vector is zero is the critical mo-\nmentum, where Gk∗(tc) = 0. ⃗ rk’s only swipe half a circle\nfor quenches from the critical point, which is consistent\nwith the observation in Ref. [52] and this leads to the\nhalf-integer jump in DTOP.\nFrom FIG. 2, we observe that the peaks of SSPs in\nwhichever case occurs slightly sooner than the DQPTs.\nNote that the critical times indicated in the figures are\ncalculated by Eq. (9) in the thermodynamic limit. For\nsystem sizes N≥100, the peak times extracted from SSP\ndo not change significantly, suggesting this time discrep-\nancy remains in larger systems. Nevertheless, the SSP\nis a local maximum, which implies the state is the least\nspin-squeezed, in the vicinity of DQPT. The difference\nbetween forward and backward quenches is the SSP keeps\nincreasing for forward quenches (left column of FIG. 2)\nwhile it stays near and below 1 for backward quenches\n(right column of FIG. 2) , but they both have some re-\nvival effects during DQPT where the spin-squeezing pa-\nrameter falls following its peak, decreasing in parallel to\nthe Loschmidt rate. This suggests that the system is\naround its most uncertain spin state along the direction\nperpendicular to the MSD around the critical time.\nThe evolution pattern of SSPs can be qualitatively un-\nderstood via the summed time-evolved correlation func-\ntions and squeezing angle, which are plotted in FIG.\n4 and 5, respectively. For forward quenches including\nquenches from Ising boundary (FIG. 4a and 4c), the\nx-direction correlation dominates initially as expected.\nAs one approaches the first critical time, the y-direction\ncorrelation functions surpass the x-direction correlations.\nAt the same time, the effect of cross-direction correlations5\n(b)\n(d) (c)(a)\nFIG. 4. Scaled time plots of sums of correlation functions\nGxx\nn(t) (top panels), Gyy\nn(t) (middle panels) and Gcross\nn(t)\n(bottom panels) for the same quench cases in FIG. 2. Red\nand blue dashed lines indicate the first peak of LRs and SSPs\nrespectively.\nGcross\nn(t) =Gxy\nn(t) +Gyx\nn(t) grows rapidly. In the vicin-\nity of the first DQPT, this surpassing effect is among the\nhighest while the contribution from the cross-direction\ncorrelations sinks to near zero. After the DQPT, Gxx\nn(t)’s\nrevive partly, and both Gxx\nn(t)’s and Gyy\nn(t)’s have small\nbut non-zero contributions while the cross-spin corre-\nlations oscillate around zero in the later time. This\nphenomenon resonates with the fact that the quench is\nfrom FM phase, which is xx-correlation dominant, to PM\nphase, in which none of x- and y-correlations dominate.\nAround DQPT, the system not only stays temporarily\nleast spin-squeezed, but also the furthest away from the\noriginal phase where yy-correlations thrive and surpass\nxx-correlations.\nIn terms of the squeezing angle, we see from Eq. 13\nthat J(αs, t) exhibits dynamics resembling the rotation\nof a vector during quenching. In particular, we define a\nsqueezing vector with xcomponent ⟨J2\nx−J2\ny⟩andycom-\nponent ⟨JxJy+JyJx⟩and an angle 2 αsbetween them.\nBy observing the evolution of the squeezing angle αsas\nindicated by the grey dots in the plots, the vector has\n”rotated” in the clockwise direction for about half a cir-\nFIG. 5. Colormap of J(α, t) for the same quench in FIG.\n2. Grey dots emphasize the squeezing angle αs. Red and\nblue dashed lines refers to the first peak of LRs and SSPs\nrespectively.\ncle before reaching the first critical time. The change\nofαsseems to be the most rapid across DQPT. After\none revival of the LE where the LR attains a local mini-\nmum (at t/tc≈2), the vector has completed about a full\nrevolution.\nOn the other hand, from FIG. 4b and 4d, backward\nquenches seem to align with our observation to the spin\nnature of the system established before: The correlation\nfunctions along xdirection are the strongest alongside\nwith a local minimum of y-direction correlations in the\nvicinity of DQPT, marking the temporary transition to\na state with FM xcharacter. Apart from that, the contri-\nbution from the cross-spin correlations briefly vanishes.\nIn fact, in all quenches except those from Ising bound-\nary the Gcross\nn(t)’s change their sign when passing the\ncritical time. The sign change is opposite to that for\nforward quenches. Another qualitative difference from\nforward quenches is the rotation direction of the squeez-\ning vector, where it rotates anticlockwise initially and\nthe motion oscillate around α=π/2 (see FIG. 5b and\n5d). Like the forward quenches, the rate of change of αs\naround DQPT is the strongest and the features are more\napparent for “longer” (∆ g=gf−giis large) quench.\nB. Quenches between anisotropic phases\nThe dynamics of the system in quenches between the\nanisotropic phases behaves differently from that between\nIsing phases. One major difference is that there will be\ntwo critical momenta k∗\n1andk∗\n2whenever one quenches\nacross the anisotropic boundary, which in turn give two\ncritical times tc,1andtc,2. Budich and Heyl showed that6\n(a)\n(b)\n(c)\n(d)\nFIG. 6. Scaled time plots of LR, DTOP (top panels) and\nSSP (bottom panels) at g= 0.5 for (a) δi=−1.2→δf= 0.8,\n(b)δi= 0.8→δf=−0.8, (c) δi= 0→δf= 0.8 and (d)\nδi= 0→δf=−1.2. Red and blue dashed lines indicate\nthe first peak of LRs and SSPs respectively. The second red\ndashed lines in (a) and (b) represents the second critical time\ntc,2for each quench. Black dashed horizontal lines indicate\nξ2\nS(t) = 1. The inset plots in (c) and (d) displays the zoomed\nSSP around the first critical time.\nat the two critical momenta, the Bloch vectors rotate in\nopposite direction when passing through the respective\ncritical time [35]. Here we demonstrate this idea through\nthe vector defined in Eq. (10) and further explore the\nsystem’s dynamics through the spin-squeezing parame-\nter.\nThe anisotropic boundary is at δc= 0, separating two\nphases – the x-spin dominant FM xphase ( δ > 0) and\nthey-spin dominant FM yphase ( δ <0). We fix gto 0.5\nand quench from FM yto FM xphase (upward quench),\nFMxto FM yphase (downward quench) and also from the\nequilibrium phase boundary. Their dynamics are shown\nin FIG. 6. Clearly from the DTOPs in FIG. 6a and 6b,\nwe can observe the negative and positive jumps at tc,1\nandtc,2respectively. For each critical time, there is an\nSSP peak slightly before it, which is the same as those\nquenches in the case involving the Ising phases in Sec.\nIII A. Notice that the upward and downward quenches\nare similar for a fixed g, unlike the forward and backward\nquenches in Ising scenarios.\nThe existence of two critical momenta can be realized\nin the trajectory of ⃗ rk. Namely, there are two “loops” in\nthe trajectory plot. At whichever critical time, the cor-\nresponding vector ⃗ rk∗vanishes. FIG. 7a and 7b demon-\nstrate the phenomena at tc,1, the critical time for k∗\n1.\nWhen the system keeps evolving, the blue loop would\nexpand further and it touches the origin at tc,2, the crit-\n(a)\n(b)\n(c)\n(d)\n(e)\n(f)FIG. 7. Trajectories of ⃗ rk’s at the critical times: (a,c) up-\nward quench (FIG. 6a), (b,d) downward quench (FIG. 6b),\nquenches from anisotropic boundary to (e) FM x(FIG. 6c)\nand (f) FM y(FIG. 6d) phase. Black dots indicate the origin\nthe vectors are located. The color scale shows the progression\nof⃗ rk’s from k= 0 to π.\nical time for k∗\n2(as shown in FIG. 7c and FIG. 7d). The\nopposite circular motion of the trajectories provides a vi-\nsualization of the integer jumps in DTOP when passing\nthrough critical times.\nOn the other hand, the dynamics of quench from the\nanisotropic boundary is qualitatively very different from\nother quenches. The DTOP behaves not as expected (see\nthe top panel of FIG. 6c and 6d), it has nonanalyticities\nat critical times, yet it does not jump between adjacent\nDQPTs. This can be seen by the ⃗ rkin FIG. 7e and 7f,\nwhere the trajectory does not change smoothly at DQPT\nwhen crossing the critical momentum, but the angles do.\nThe evolution of SSP goes from a very high value to a\nminimum around DQPT. Unlike other quench scenarios,\nthe system is among the most squeezed state near DQPT.\nThe cause of this might be the equilibrium SSP in the\ndomain δ= 0, g∈[0,1) diverges. When quenched out of\nthe phase boundary, the SSP falls rapidly and reaches its\nminimum near DQPT.\nSince anisotropic phases involves competition between\nx- and y-spin interactions, a more direct interpretation7\n(b)\n(d)(a)\n(c)\nFIG. 8. Scaled time plots of sums of correlation functions\nGxx\nn(t) (top panels), Gyy\nn(t) (middle panels) and Gcross\nn(t)\n(bottom panels) for the same quench cases in FIG. 6. Red\nand blue dashed lines indicate the first peak of LRs and SSPs\nrespectively. The second red dashed lines in (a) and (b) rep-\nresents the second critical time tc,2for each quench.\nabout what is happening during DQPT via the time evo-\nlution of the summed correlation functions can be ob-\nserved in FIG. 8. In particular, the upward quenches in\nFIG. 8a do show an increase in Gxx\nn(t) and a decrease\ninGyy\nn(t), and they reach maximum and minimum re-\nspectively in the vicinity of first DQPT. The downward\nquenches in FIG. 8b shows the opposite phenomenon.\nIn both cases, the contribution of cross-spin correlations\nvanish at the first critical times. One observes the same\neffect for quenches from anisotropic boundary, where the\nxxcorrelations reach a higher value whilst yycorrelations\ndrop to near zero (see FIG. 8c). The feature appearing\nonly in quenches from anisotropic boundary is that at\nDQPT, the effect of cross-direction correlation functions\nreach their extrema (FIG. 8c and 8d), which differs from\nall other quenches studied.\nThe squeezing angle dynamics shown in FIG. 9a and 9b\nare similar to the quenches between Ising phases, where\nthe squeezing vector rotates in the clockwise direction at\nthe beginning and the decrease of the squeezing angles αs\nis the most rapid at DQPT. In between the first and the\nFIG. 9. Colormap of J(α, t) for the same quench cases in\nFIG. 6. Grey dots emphasize the squeezing angle αs. Red\nand blue dashed lines refers to the first peak of LRs and SSPs\nrespectively. The second red dashed lines in (a) and (b) rep-\nresents the second critical time tc,2for each quench.\n(b) (a)\nFIG. 10. Scaled time plots of LR, DTOP (top panels) and SSP\n(bottom panels) for (a) ( δi, gi) = (0 ,1)→(δf, gf) = (1 ,0)\nand (b) ( δi, gi) = (0 ,1)→(δf, gf) = (0 .5,0.5). Red and\nblue dashed lines indicate the first peak of LRs and SSPs\nrespectively and the two lines overlap with each other in (a).\nBlack dashed horizontal lines indicate ξ2\nS(t) = 1.\nsecond critical times, the squeezing vector turns its rota-\ntion direction and the increase in αsis the fastest at the\nsecond critical time. On the other hand, the squeezing\nangle evolution for quenches from anisotropic boundary\n(FIG. 9c and 9d) reaches its minimum at DQPT, mark-\ning the distinction from the cross-boundary quench cases.\nThen the whole evolution pattern follows what backward\nquenches have in Ising phases, where it oscillates around\nπ/2 orπand never turns a full round.\nC. Quenches from multicritical point\nWe study two cases when quenched from the multicrit-\nical point ( δi, gi) = (0 ,1). Figure 10 shows the Loschmidt8\n(a)\n (b)\nFIG. 11. Trajectories of ⃗ rk’s at the corresponding critical\ntimes for the same quench cases in FIG. 10. Black dots indi-\ncate the origin the vectors are located. The color scale shows\nthe progression of ⃗ rk’s from k= 0 to π.\n(b) (a)\nFIG. 12. Scaled time plots of sums of correlation functions\nGxx\nn(t) (top panels), Gyy\nn(t) (middle panels) and Gcross\nn(t)\n(bottom panels) for the same quench cases in FIG. 10. Red\nand blue dashed lines indicate the first peak of LRs and SSPs\nrespectively and the two lines overlap with each other in (a).\nFIG. 13. Colormap of J(α, t) for the same quench cases in\nFIG. 10. Grey dots emphasize the spin squeezing angle αs.\nRed and blue dashed lines refers to the first peak of LRs and\nSSPs respectively.\nrate and spin-squeezing parameter evolutions. Notice the\nperfect oscillation of both LR and SSP for quench to\n(δf, gf) = (1 ,0), the case of a classical Ising model, in\nFIG. 10a. The periodicity of SSP is half the periodicity\nof LR. This can be understood from the exact expressionsof the LR and SSP, namely the LR is of the form\nλ(t) =1\n2ln 8−1\n4ln[38 + 24 cos(2 t) + 2 cos(4 t)] (18)\nand the SSP\nξ2\nS(t) =1\n8\u0014\n9−cos(4 t)−1√\n2p\ncos(8 t)−68 cos(4 t) + 67\u0015\n.\n(19)\nThe half periodicity of SSP comes from the dominat-\ning cos(4 t) term in SSP as compared to the dominating\ncos(2 t) term in LR in Eq. (18) and (19) respectively.\nFrom equations (18), we see that the LR is independent\nof the system size N. The size independence also holds for\nSSP due to the only few non-zero correlation functions\nas follows:\nGyy\n2(t) =Gyy\nN−2(t) =1\n16sin2(2t)\nGxy\n1(t) =Gxy\nN−1(t) =Gyx\n1(t) =Gyx\nN−1(t) =1\n8sin(2t),\n(20)\nand all other correlation functions vanish. In other\nwords, in this quench case, all parallel-spin correlations\nalong xare zero, whereas the next-nearest neighbor cor-\nrelations along yare finite, resulting in the summed cor-\nrelation functions in FIG. 12a. The observation here is\ncounterintuitive to the fact that the system is quenched\nto the point where the y-direction interaction terms are\nless involved. Notice however that at DQPT, the Gyy\nn(t)’s\ndrop to zero as expected.\nThe aforementioned perfect pattern in the LR and\nSSP evolutions are destroyed when ( δf, gf) differs from\nthe classical Ising model. As shown in FIG. 10b, the\nSSP evolution is distorted and resembles the pattern for\na backward quench between Ising phases described in\nSec. III A. It has twice as many peaks as the LR, and\nthe major peaks occur near the respective critical times.\nNevertheless, for both quench scenarios here, although\nthey start from a critical point, the DTOPs show inte-\nger jumps, aligning with the case of quenches across the\nphase boundary instead of that from a phase boundaries\npresented in the previous sections. The trajectories of\nthe vectors ⃗ rk(t) are also similar to the quenches across\nthe phase boundary (FIG. 11).\nThe sums of the time evolution of the correlation func-\ntions for the quench to ( δf, gf) that differs from the clas-\nsical Ising model are shown in FIG. 12b. The distortion\nemerges and the x-direction correlations rise a brief mo-\nment and then slowly decrease around DQPT. The per-\nfect oscillation pattern of yy-correlation functions is de-\nformed and interestingly they surpass the xx-correlation\nfunctions at the first critical time. On the other hand,\nthe contribution of cross-spin correlations mostly vanish\nat DQPT, while before and after DQPT they are of dif-\nferent signs, which resembles the quenches across critical\nboundary to the FM xphase.\nIn terms of the squeezing angle, the quench (0 ,1)→\n(1,0) acts like the limiting case of backward quenches9\n(b) (a)\nFIG. 14. Correlation functions for quenches across anisotropic\nboundary: (a) ( −1.2,0.5)→(0.8,0.5) and (b) (0 .8,0.5)→\n(−0.8,0.5). We show here n≤10 from light green to dark\ngreen with increasing n. Correlations for larger n’s converge\nto the curve in yellow. The first and the second red dashed\nlines indicate tc,1andtc,2respectively. Blue dashed lines rep-\nresents the maximum of SSPs.\nbetween Ising phases shown in FIG. 5, where in FIG.\n13a the slope of the squeezing angle approaches −∞ at\nthe first DQPT. The squeezing vector never completes\na full rotation, rather it zigzags between the fourth and\nthe first quadrants, i.e. αs∈[3π/4, π) in 0 < t < t cand\nαs∈(0, π/4] in tc< t < 2tc. Note that in the evolution\nofαsfor the case shown in FIG.13b, the squeezing vector\nrotates anticlockwise and complete a full circle for one\nrevival time. Again we observe the same phenomenon\nthat the change of αsis the maximum at DQPT.\nIV. DISCUSSIONS\nIn the previous section, we showed that the time evo-\nlution of the correlation functions matches the physical\nproperties of the phase one quenches to in most of the\ncases. Here we provide an insight for the whole quench\ndynamics of an interacting spin model: Once the sys-\ntem is being quenched, the cross-direction correlations\neither begin growing or shrinking for around half the crit-\nical time for all kinds of quench. At the same time for\nquenches between FM xand FM yphases, the parallel-\nspin correlations evolve according to what phase the sys-\ntem is quenched to – when the final phase flavors x-\nspin, Gxx\nn(t)’s grow, whereas when the final phase fla-\nvors y-spin, Gyy\nn(t)’s grow. In the vicinity before the\nfirst DQPT, the flavoring parallel-spin correlation func-\ntions reach their maximal values while the contribution\nof cross-spin correlations diminish. In this sense, the sys-\ntem does appear to “transit” to the final phase at DQPT\nfor a brief amount of time. For the case of quenching\nto PM phase, interestingly, the supposedly low yy-spin\ncorrelations rise and peak just before DQPT. However,\nthe originally flavored xx-spin correlations shrink to near\n(b) (a)\n(d) (c)FIG. 15. LR and SSP plots of (a),(b) quenches across Ising\nboundary for fixed gfwith δ= 0.8 and (c),(d) quenches\nacross anisotropic boundary with fixed δfwith g= 0.5. Black\ndashed lines indicate the line where ξ2\nS(t) = 1.\nminimum. This serves as an evidence of “leaving” the\noriginal phase. On the other hand, the cross-spin corre-\nlation functions rarely play a role during DQPT caused\nby quenching across the critical boundary, but they do\ncontribute for quenches from critical boundaries, espe-\ncially for the quench from anisotropic boundary, where\nthe cross-spin correlations contribute a big part in mini-\nmizing SSP around DQPT.\nFollowing this interpretation, perhaps we can pro-\nvide crucial evidence suggesting the second DQPT for\nquenches across the anisotropic boundary is more of a\n“partially return transition” to the initial state. Namely,\nFIG. 7(a-d) show the trajectory of the loops including k∗\n1\n(red loop) and k∗\n2(blue loop) progresses in opposite di-\nrection. The strongest evidence comes from the sums of\ncorrelation functions in FIG. 8a and 8b. In the summed\nGxx\nn(t) and summed Gyy\nn(t) plots, the first DQPTs oc-\ncur when the corresponding flavoring correlations of the\npostquenched Hamiltonian peak, while at the second crit-\nical time tc,2this correlations become the lowest. On the\ncontrary, the originally flavored correlations do not re-\nstore their domination, yet the nearest-neighbor as well\nas the first few n’s correlations do rise and peak be-\nfore the second DQPT, as shown in FIG. 14, making\nthe summed quantities rise a little between the two crit-\nical times. The cross-spin correlation functions vanish\nas expected for DQPTs. The evolution of the squeezing\nangles also strengthen the argument, where the squeez-\ning vectors change their rotation directions after tc,1and\nthe squeezing angles pass through tc,2with the greatest\nslope. After tc,2, the squeezing vectors return back to10\nthe original rotation direction to complete a full turn. In\nshort, the above evidences suggest the system may par-\ntially and temporarily returned to a state having the spin\ncharacter of the initial state in the vicinity of the second\nDQPT.\nIn studying the dynamics around DQPT, often the\ntime-evolved SSP peaks around the critical time. This\nimplies the system is the least squeezed around DQPT.\nHere we exploit some possible reasons behind regard-\ning the first critical time. From the analytical expres-\nsion of the SSP in Eq. (14), peaked SSP occurs when\nGxx\nn(t) +Gyy\nn(t) is high, Gxx\nn(t)−Gyy\nn(t) and Gcross\nn(t)\nare minimal. We can immediately see that for all quench\ncases we studied, except quenches from phase boundary\nto the ordered phase, the cross-spin correlation functions\nare near zero, and the difference of xcorrelations and y\ncorrelations is minimized. This gives a small square-root\nterm in Eq. (14) around DQPT and thus a relatively\nhigh SSP. In some cases like the backward quenches in\nIsing phases and quenches from multicritical point, some\nGyy\nn(t)’s are negative in the transient regime, plus the\ngrowing contribution from Gcross\nn(t), the SSP in these\ncases drops at the beginning. Shortly after, the quanti-\nties get lowered to minimal and thus SSPs grow to their\nmaxima near critical time. To put it more generally,\nwhen quenched from some phase to the FM xphase, the\nsums of Gxx\nn(t)’s are around their maxima and sums of\nGxx\nn(t)’s would behave such that Gxx\nn(t)−Gyy\nn(t) is low at\nDQPT. The opposite is true for quenches to FM yphase.\nIn other words, the xandyspins are of the most uncer-\ntain around DQPT. The cross-spin correlations behave\nin such a way that vanish at DQPT while before and af-\nter DQPT they evolve in opposite sign, making the SSP\naway from critical time drops.\nBesides the different evolutions of spin-related quan-\ntities, the overall spin dynamics relating to the types\nof quenches within one quench scenario is also studied\nand we found a general relationship between cases with\nand without successful DQPTs as well as the critical spin\ndynamics around the first critical time. In FIG. 15, we\nplot the rate functions and spin-squeezing parameters for\nquenches starting from various points in the equilibrium\nphase diagram. The case involving Ising phases show an\napparent distinction from quenches across critical point\nand within one phase. Namely, SSPs are general high and\ngrowing for forward quenches, while low and oscillating\nbelow 1 mostly for backward quenches. Quenches within\none phases, on the other hand, evolve below 1 for forward\nquenches and above 1 for backward quenches. The case\nwhen the initial point is at the critical point behaves be-\ntween the two, where the SSP oscillates around 1. Special\nfeatures can also be observed during successful DQPTs\nfor quenches across anisotropic boundary. In FIG. 15c\nand 15d, quenches across critical point results a sharp\npeak in SSP at the two critical times, while those peaks\nare absent in quenches within one phase where no DQPT\noccurs. Also for the quenches within the phase, the SSPs\nevolve around and below 1. The critical quench, on theother hand, differs drastically from other cases, where the\nSSP falls from infinity to a minimum at DQPT, as stated\nin Sec. III B. In either case, we further confirm the phe-\nnomenon that SSP reaches its local extremum (usually\nmaximum) when approaching DQPT.\nV. CONCLUSION\nWe analyzed three distinct quench scenarios in the XY\nmodel, for which we successfully detected some physical\nproperties an integrable spin-1\n2model might have when\nquenched between different phases. Firstly, the collec-\ntive spin of the system tends to fluctuate more when ap-\nproaching critical times. This can be observed by the\ntime evolution of the spin-squeezing parameter of the\nsystem, namely the SSP peaks, or the system is the\nleast spin-squeezed, just before DQPT occurs. This phe-\nnomenon seems to persist for longer critical times and\nfor most types of quenches - whether quenched across\nor from critical boundaries, except when the system is\ninitially in the most unsqueezed state where the SSP di-\nverges. In either case, the system exhibits extreme spin-\nsqueezing dynamics relative to the initial state, being the\nleast or the most spin-squeezed situation, in the vicinity\nof DQPTs.\nBy observing the exact analytical expressions for the\ntime evolution of spin-squeezing parameter, we also re-\nveal the spin dynamics by studying the competition be-\ntween parallel- and cross-spin correlations. We find that\nduring DQPTs, the system does appear to evolve in such\na way to match the spin properties of the phase one\nquenches to. For instance, when quenched to xx-spin-\nflavored phase, the x-direction correlations grows to its\nmaximum at DQPTs. This maximum reflects the system\nis in the least unsqueezed state during DQPTs. Further-\nmore, we also showed that there is possibly some kind\nof “return transition” occurring for models having two\ncritical momenta during the second DQPTs.\nIn short, the time evolution of SSP provides us non-\ntrivial physical insights into the spin correlations around\nDQPTs, though we would like to remark that the SSP is\nnot a precise order parameter of DQPTs and this work\nis not intended to address if the spin correlations are\nthe driver of DQPTs. Nevertheless, the above-mentioned\nobservations welcomes further confirmation in experi-\nments, and in other models , for example, the long-range\ntransverse-field Ising model and the Lipkin-Meshhov-\nGlick model which have been extensively studied for their\nintriguing dynamical behaviors around various types of\nphase transitions [5, 51, 71, 72]. 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