Quantum spin torque driven transmutation of antiferromagnetic Mott insulator
Marko D. Petrovic, Priyanka Mondal, Adrian E. Feiguin, Branislav K. Nikolic
QQuantum spin torque driven transmutation of antiferromagnetic Mott insulator
Marko D. Petrovi´c, Priyanka Mondal, Adrian E. Feiguin, and Branislav K. Nikoli´c ∗ Department of Physics and Astronomy, University of Delaware, Newark DE 19716, USA Department of Physics, Northeastern University, Boston, MA 02115, USA
The standard model of spin-transfer torque (STT) in antiferromagnetic spintronics considers ex-change of angular momentum between quantum spins of flowing electrons and noncollinear-to-themlocalized spins treated as classical vectors. These vectors are assumed to realize N´eel order inequilibrium, ↑↓ . . . ↑↓ , and their STT-driven dynamics is described by the Landau-Lifshitz-Gilbert(LLG) equation. However, many experimentally employed materials (such as archetypical NiO) arestrongly electron-correlated antiferromagnetic Mott insulators (AFMI) where localized spins forma ground state quite different from the unentangled N´eel state |↑↓ . . . ↑↓(cid:105) . The true ground stateis entangled by quantum spin fluctuations, leading to expectation value of all localized spins being zero , so that LLG dynamics of classical vectors of fixed length rotating due to STT cannot evenbe initiated. Instead, a fully quantum treatment of both conduction electrons and localized spinsis necessary to capture exchange of spin angular momentum between them, denoted as quantumSTT . We use a recently developed time-dependent density matrix renormalization group approachto quantum STT to predict how injection of a spin-polarized current pulse into a normal metallayer coupled to AFMI overlayer via exchange interaction and possibly small interlayer hopping—which mimics, e.g., topological-insulator/NiO bilayer employed experimentally—will induce nonzeroexpectation value of AFMI localized spins. This new nonequilibrium phase is a spatially inhomoge-neous ferromagnet with zigzag profile of localized spins. The total spin absorbed by AFMI increaseswith electron-electron repulsion in AFMI, as well as when the two layers do not exchange any charge. Introduction .—The emergence of antiferromagneticspintronics [1–4] has elevated antiferromagnetic (AF) in-sulators (AFIs) and metals into active elements of spin-tronic devices. They exhibit dynamics of their localizedspins at a much higher frequencies, reaching THz [4],when compared to ferromagnetic spintronics. Further-more, the absence of net magnetization forbids any straymagnetic fields, making them largely insensitive to per-turbations by external fields. They also exhibit mag-netoresistance effects [5, 6] enabling electric readout ofchanges in the orientations of their localized spins.Basic spintronic phenomena like spin-transfer torque(STT) [7–10]—where spin angular momentum is ex-changed between flowing conduction electrons andnoncollinear-to-them [11] localized AF—and spin pump-ing [12]—where precessing localized AF spins pump purespin current in the absence of any bias voltage—havebeen demonstrated recently using different AF materi-als. The theoretical description [13–22] of these phenom-ena invariably assumes that localized magnetic momentson two sublattices of the AF material, M Ai and M Bi ,are classical vectors with net zero total magnetization inequilibrium due to assumed N´eel classical ground state(GS), ↑↓ . . . ↑↓ . Out of equilibrium, the dynamics ofsuch classical vectors of fixed length is described by theLandau-Lifshitz-Gilbert (LLG) equation [23]. The STTis typically introduced into the LLG equation either asa phenomenological term [17–20], or it is calculated mi-croscopically by using steady-state single-particle quan-tum transport formalism applied to model [13, 14, 21] orfirst-principles [15, 16, 22] Hamiltonians of AF materials.Recently STT [24] from time-dependent single-particlequantum transport formalism [25] has been coupled [26] to the LLG equation, capturing additional quantum ef-fects like electronic spin pumping by moving M Ai ( t )and M Bi ( t ) and the corresponding enhanced damping onthem, but this remains conventional [11] quantum -for-electrons– classical -for-localized-spins theory of STT.However, AFIs employed in spintronics experimentsare typically strongly electron-correlated transition metaloxides due to narrow d bands. For example, widelyused [6–10] NiO shares features of both Mott and charge-transfer insulators [27, 28]. Due to quantum (or zero-point) spin fluctuations [29–31], the AF GS is highly entangled [30, 32–34], which results in zero expectationvalue of all localized spins, S i = 0 ( M i ∝ S i = 0).Thus, conventional [11] STT ∝ s i × S i = 0 due to in-jected nonequilibrium electronic spin density s i cannot be initiated because S i ( t = 0) ≡
0. Even if | S i ( t =0) | (cid:54) = 0 is provoked by spin-rotation-symmetry-breakinganisotropies [35] or impurities (see Supplemental Mate-rial [36] for illustrations), the LLG equation is inappli-cable [38, 39] because the length | S i ( t ) | < | S N´eel i | willbe changing in time, with smaller value signifying higherentanglement (unobserved quantum systems exhibit uni-tary evolution toward states of higher entanglement [40]).Thus, both situations necessitate to describe localizedspins fully quantum mechanically where S i ( t ) is calcu-lated only at the end.The entanglement in the AF GS leading to S i = 0can be illustrated using an example [41, 42] of a one-dimensional (1D) AF quantum spin- Heisenberg chainˆ H AFI = J N AFI − (cid:88) i =1 ˆ S i · ˆ S i +1 , (1) a r X i v : . [ c ond - m a t . s t r- e l ] O c t t < t > N conf x y z U v V v ' ' N AFMI N R N L N AFI
2i i U (a) Mott Insulator (b) B e Normal Metal J v (c) J v AFI J FIG. 1. (a) Schematic view of a “bilayer” [10] for tDMRGcalculations where 1D TB chain (blue dots) of N = N L + N AFMI + N R = 92 sites, with intrachain hopping γ mod-els the NM surface (such as that of Bi Se in experimentof Ref. [10]) through which spin-polarized current pulse is in-jected. The pulse exerts quantum STT on a Hubbard chain of N AFMI = 12 sites with the on-site Coulomb repulsion U , mod-eling the surface of strongly electron-correlated AFMI (suchas that of NiO in Ref. [7–10]). The electronic spins in twochains interact via interchain exchange interaction J v , and weconsider both γ v = 0 and γ v (cid:54) = 0 interchain hopping wherethe latter mimics possible hybridization of NM and AFMIvia evanescent wavefunctions [22]. For times t < N e = 12noninteracting electrons are confined by potential V within N conf = 25 sites of the L lead (composed of N L = 40 sites), aswell as spin-polarized by an external magnetic field B e point-ing along the z -axis. Concurrently, N AFMIe = 12 electronshalf-fill the AFMI chain. For times t ≥ V and B e are re-moved, so that electrons propagate as spin-polarized currentpulse from the L to the R lead, as illustrated in (b). For com-parison, in panel (c) we replace the AFMI from (a) and (b)with AF quantum Heisenberg chain where spin- operatorsreside on each (orange) site and interact via J = 4 γ /U inEq. (1) with no charges allowed within this chain. on N AFI sites. Here ˆ S αi = ˆ I ⊗ . . . ⊗ ˆ σ α ⊗ . . . ⊗ ˆ I N AFI acts nontrivially, as the Pauli matrix ˆ σ α , only on theHilbert space of site i ; ˆ I i is the unit operator; and J > N AFI , such as for N AFI = 4 we find | GS (cid:105) = 1 √ (cid:0) |↑↓↑↓(cid:105) + 2 |↓↑↓↑(cid:105) − |↑↑↓↓(cid:105) − |↑↓↓↑(cid:105)− |↓↓↑↑(cid:105) − |↓↑↑↓(cid:105) (cid:1) . (2)Its energy, (cid:104) GS | ˆ H AFI | GS (cid:105) = − J , is lower than the en-ergy of the unentangeled (i.e., direct-product) N´eel state, (cid:104)↑↓↑↓| ˆ H AFI |↑↓↑↓(cid:105) = − J . This is in sharp contrast to fer- romagnets where quantum spin fluctuations are absent,and both classical ↑↑ . . . ↑↑ and its unentangled quantumcounterpart |↑↑ . . . ↑↑(cid:105) are GS of the respective classicaland quantum Hamiltonian [such as Eq. (1) with J < M i in spintronics [11] and micromagnetics [23], even as thesize of the localized spin is reduced to that of a singleelectron spin. Conversely, in the case of many-body en-tangled [30, 32–34] AF GS, the quantum state of eachlocalized spin subsystem must be described by the re-duced density matrix, ˆ ρ i = Tr other | GS (cid:105)(cid:104) GS | , where par-tial trace is performed in the Hilbert subspace of all otherlocalized spins j (cid:54) = i . The expectation value S i ≡ (cid:104) ˆ S i (cid:105) = Tr [ˆ ρ i ˆ S i ] , (3)is then identically zero vector, S i = 0, on all sites (seethe SM [36]). The GS in the limit N AFI → ∞ is com-putable by Bethe ansatz [42], and its entanglement en-sures S i = 0. The entanglement in the GS of crystallinerealization of a two-dimensional (2D) quantum Heisen-berg antiferromagnet or antiferromagnetic Mott insulator(AFMI) realized with cold atoms on a square lattice hasbeen detected by neutron scattering [34] or optically [43],respectively, at ultralow temperatures.In this Letter, we employ the emerging conceptof quantum STT [44–47] where both conduction elec-trons and localized spins are treated fully quantum-mechanically to describe the exchange of spin angularmomentum between them. This allows us to predict nonequilibrium phase transition of AFMI driven by ab-sorption of spin angular momentum from spin-polarizedcurrent pulse injected into an adjacent normal metal(NM). To model such genuine quantum many-body prob-lem, we evolve in time a nonequilibrium quantum stateof NM/AFMI system via very recently developed [46]time-dependent density matrix renormalization group(tDMRG) approach [48–51] to quantum STT.Our system geometry in Fig. 1 consists of a NM mod-eled as 1D tight-binding (TB) chain, which is split intothe left (L) and the right (R) leads sandwiching a cen-tral region. The conduction electron spins in the centralregion are exchange coupled to an AFMI chain modeledby Hubbard model with the on-site Coulomb repulsion U . The current pulse, carrying electrons initially spin-polarized in the direction perpendicular to the interface(i.e., along the z -axis in Fig. 1), is injected from theL lead into the central region of NM in order to initi-ate the AFMI dynamics via quantum STT. Our geom-etry mimics recent experiment [10] on injection of cur-rent pulses into metallic surface of topological insulatorBi Se , which then exert spin torque on the surface ofNiO overlayer covering Bi Se , except that in the exper-iment spin-orbit coupling polarizes injected electrons inthe plane of the interface (i.e., along the y -axis in Fig. 1).Nevertheless, since singlet with s (cid:48) i ( t = 0) ≡ S i t e I nd e x NM/AFMIU = 8 γ (a) NM/NMU = 0(c)0 10 20 30 40Time (¯h /γ )120406080 S i t e I nd e x (b) 0 10 20 30 40 50Time (¯h /γ )(d)0 .
00 0 .
05 0 .
10 0 .
15 0 . s zi , s zi FIG. 2. Spatio-temporal profiles of the z -component of theelectronic spin within: (a) AFMI chain with the Coulombrepulsion U = 8 γ ; and (c) the same chain with U = 0 actingas the second NM chain half-filled with electrons. In bothpanels s (cid:48) zi ( t = 0) ≡
0, so that only s (cid:48) zi ( t ) (cid:54) = 0 component isinduced by current pulse spin-polarized along the z -axis inFig. 1(b) and flowing within NM chain whose s zi profiles inpanels (b) and (d) are driving the profiles in panels (a) and (c),respectively, via quantum STT. The dotted horizontal linesin (b) and (d) mark the boundaries between the leads andthe central region of the NM chain in Fig. 1. The interchainexchange is J v = 0 . γ and hopping γ v = 0 in Eq. (6). AFMI driven by quantum STT will be the same for ar-bitrary spin-polarization of injected electrons.Our main results in Figs. 3–5 demonstrate how quan-tum STT deposits spin angular momentum [Figs. 4 and5] into the AFMI by driving its on-site electronic spinexpectation value from s (cid:48) i ( t = 0) ≡ s (cid:48) zi ( t ) (cid:54) = 0[ s (cid:48) xi ( t ) = 0 = s (cid:48) yi ( t )], with zigzag pattern s (cid:48) z j − ( t )
2. The total spin angu-lar momentum absorbed by AFMI increases with theon-site Coulomb repulsion [Fig. 5(a)], but it is reduced[Figs. 4(c)] when the interchain hopping allows for hy-bridization of NM and AFMI and electron leakage fromAFMI [Fig. 4(a)] into NM [Fig. 4(b)]. Prior to delvinginto these results, we introduce rigorous notation anduseful concepts.
Hamiltonian models and tDMRG method .—Thesecond-quantized many-electron Hamiltonian describingthe NM/AFMI system in Fig. 1(a) consists of four termsˆ H = ˆ H NM + ˆ H AFMI + ˆ H NM − AFMI + ˆ H V , B ( t < . (4)The first term is 1D TB Hamiltonian of noninteractingelectrons within NM chain ˆ H NM = − γ (cid:80) Ni =1 (ˆ c † i ↑ ˆ c i +1 ↑ +ˆ c † i ↓ ˆ c i +1 ↓ + h . c . ) where ˆ c † iσ (ˆ c iσ ) creates (annihilates) anelectron with spin σ = ↑ , ↓ at site i , and γ is the intrachainhopping. These operators act on four possible states ateach site i —vacuum | (cid:105) , spin-up |↑(cid:105) , spin-down |↓(cid:105) , and U = Time = FIG. 3. Spatial profile of the z -component s (cid:48) zi of electronicspin within AFMI chain in Fig. 1(b) driven by quantum STTfrom NM chain: (a) at different times using U = 8 γ in Eq. (5);and (b) for different U values at time t = 25 (cid:126) /γ . The inter-chain exchange is J v = 0 . γ and hopping γ v = 0 in Eq. (6). doubly occupied state |↑↓(cid:105) , so that total Hilbert space ofNM/AFMI system has dimension 4 × . The inter-acting electrons within the AFMI chain are described bythe Hubbard Hamiltonian [41, 42]ˆ H AFMI = − γ N AFMI − (cid:88) i =1 (cid:16) ˆ d † i ↑ ˆ d i +1 ↑ + ˆ d † i ↓ ˆ d i +1 ↓ + h . c . (cid:17) + U N AFMI (cid:88) i =1 ˆ n (cid:48) i ↑ ˆ n (cid:48) i ↓ . (5)Here, ˆ n (cid:48) iσ = ˆ d † iσ ˆ d iσ are local particle number operators forspin σ at site i of AFMI. The on-site Coulomb repulsion,such as U = 0–10 γ in Fig. 3(b), is expressed in the unitsof hopping γ (typically γ = 1 eV) which we use as a unitof energy. The operators for the total number of elec-trons, ˆ N AFMI e = (cid:80) i ˆ n (cid:48) i , and total electronic spin alongthe α -axis, ˆ s (cid:48) α = (cid:80) i ˆ s (cid:48) αi , are given by sums of local (per-site) charge and spin operators, ˆ n (cid:48) i = (cid:80) σ = {↑ , ↓} ˆ d † iσ ˆ d iσ and ˆ s α (cid:48) i = (cid:80) σ = {↑ , ↓} ˆ d † iσ ˆ σ ασσ (cid:48) ˆ d iσ (cid:48) , respectively. The in-terchain exchange interaction J v between electronic spinswithin NM and AFMI is described byˆ H NM − AFMI = − J v N AFMI (cid:88) i =1 ˆ s i + N L · ˆ s (cid:48) i − γ v N AFMI (cid:88) i =1 (cid:16) ˆ c † i + N L ↑ ˆ d i ↑ + ˆ c † i + N L ↓ ˆ d i ↓ + h . c . (cid:17) , (6)where ˆ s i and ˆ s (cid:48) i are local electron spin operators in NMand AFMI chains, respectively. Here we also add a termwith possible γ v (cid:54) = 0 hopping between N AFMI sites of thecentral region of the NM chain and N AFMI sites of AFMIin Fig. 1(a), which can arise in realistic devices used inspintronics [7–10] due to evanescent wavefunctions pene-trating [22] from the NM surface into the region of AFMI N A F M I e (a) γ v = 0 γ v = 0 . γ /γ )1213 N e (b) 0123 A F M I ( U = γ ) (c) Su m o f Sp i n E x p e c t a t i o n V a l u e s /γ )9101112 N M (d) FIG. 4. Time dependence of the total number of electronswithin (a) AFMI and (b) NM chains in the setup of Fig. 1(b)for two different interchain hoppings γ v = 0 (blue lines) and γ v = 0 . γ (red lines). Panels (c) and (d) show the correspond-ing time dependence of the sum of the z -component of elec-tronic spin expectation values, (cid:80) i s (cid:48) zi and (cid:80) i s zi , respectively.The on-site Coulomb repulsion is U = 8 γ [Eq. (5)] within theAFMI and interchain exchange interaction is J v = 0 . γ . near the interface, thereby leading to charge transfer inequilibrium or current leakage [22] between the two ma-terials. Such normal-metal proximity effect on finite-sizeMott insulators can also create exotic many-body statesin equilibrium [52]. To prepare the initial state of theconduction electrons in the NM chain, we confine themwithin N conf sites of the L lead in Fig. 1(a) and polar-ize their spins along the + z -axis by means of an addi-tional term ˆ H V , B ( t <
0) = − V (cid:80) N conf i =1 (cid:16) ˆ c † i ↑ ˆ c i ↑ + ˆ c † i ↓ ˆ c i ↓ (cid:17) − (cid:80) N conf i =1 gµ B ˆ s zi B z e . Here V = 2 γ is the confining potential; B z e is the external magnetic field; and gµ B B z e = 10 γ ,where g is the electron gyromagnetic ratio and µ B is theBohr magneton. After the initial state is prepared for t <
0, ˆ H V , B ( t ≥
0) is set to zero, so that spin-polarizedelectrons from the L lead propagate toward the R lead,as illustrated in Fig. 1(b) and computed in Fig. 2.In the limit U (cid:29) γ , the half-filled ( n i = 1) 1D Hub-bard model describes electrons localized one per site, soit can be mapped [41, 42] to isotropic AF quantum spin- Heisenberg chain with the effective Hamiltonian givenin Eq. (1). Therefore, for comparison we also analyze theNM/AFI setup in Fig. 1(c) where AFI sites hosts local-ized spin- operators ˆ S i , as described by the Hamiltonianˆ H = ˆ H NM + ˆ H AFI + ˆ H NM − AFI + ˆ H V , B ( t < H NM is the same as in Eq. (4); ˆ H AFI is the same as in Eq. (1)where we use J = 4 γ /U as the exchange interactionin the limit U (cid:29) γ [41, 42]; the interchain interactionis described by ˆ H NM − AFI = − J v (cid:80) N AFI i =1 ˆ s i + N L · ˆ S i where J v = 0 . γ ; and ˆ H V , B ( t <
0) is the same as in Eq. (4).The tDMRG simulations [48–51] evolve thenonequilibrium state of the whole system in
FIG. 5. (a) Time evolution of the sum of electronic spin ex-pectation values within NM chain (cid:80) i s zi (dashed lines) andAFMI chain (cid:80) i s (cid:48) zi (solid lines) in the setup of Fig. 1(b) fordifferent value of the on-site Coulomb repulsion U within theAFMI chain. For comparison, panel (b) plots the same infor-mation for the setup in Fig. 1(c) where AFMI is replaced byAF quantum spin- Heisenberg chain with no electrons, sothat solid lines are (cid:80) i S zi defined in Eq. (3). For each U in(a), we set the corresponding intrachain exchange interaction J [Eq. (1)] within AFI in (b) as J = 4 γ /U . Fig. 1, | Ψ( t + δt ) (cid:105) = e − i ˆ Hδt/ (cid:126) | Ψ( t ) (cid:105) , using time step δt = 0 . (cid:126) /γ . We start the propagation with m = 100states and limit the truncation error to 10 − , whilethe maximal number of states allowed during theevolution is set to m max = 400. Any single-particleexpectation value at site i can be obtained fromˆ ρ i ( t ) = Tr other | Ψ( t ) (cid:105)(cid:104) Ψ( t ) | , as exemplified by Eq. (3).Since fermionic leads are not semi-infinite as in usualquantum transport calculations [26], the system in Fig. 1can be evolved only for a limited time [53–56] beforeelectrons are backscattered by the right boundary whichbreaks L → R current flow. For example, in Fig. 2 suchbackscattering occurs at t (cid:39) (cid:126) /γ . Nevertheless, thequantum dynamics of the conduction electrons in theNM chain and charge and spin confined within theAFMI chain can be safely assumed to be effectivelyequivalent to that in an infinite open quantum systembefore the boundary reflection takes place. Results and discussion .—The Hubbard 1D chain mod-eling the AFMI possesses a sizable energy gap ∆ c forcharge excitations at U (cid:38) γ , whose value is exactlyknown [42] in the limit N AFMI → ∞ (∆ c = 0 . γ at U = 2 γ ; or ∆ c = 0 . γ at U = 3 γ ). In chains of fi-nite length, such as ours with N AFMI = 12 sites, DMRGpredicts slightly larger ∆ c values [57]. However, the spinsector of the half-filled Hubbard chain is gapless in thethermodynamic limit. This means that injecting a chargein the AFI is energetically costly, but creating a spin ex-citation is not. Figures 2(a) and 3 demonstrate thatAFMI with U (cid:38) γ will be driven out of its GS with s (cid:48) i = 0 on all sites toward a nonequilibrium phase with s (cid:48) zi ( t ) (cid:54) = 0 and s (cid:48) xi = 0 = s (cid:48) yi due to quantum STT exertedby injected current pulse in the NM chain that is spin-polarized along the z -axis. The spatial profile of s (cid:48) zi ( t ) isinhomogeneous with a zigzag pattern deep in the Mottinsulator phase, which distinguishes it from the weak re-sponse of the borderline case with U = 2 γ [Fig. 3(b)] ornoninteracting chain with U = 0 [Figs. 2(c) and 3(b)].Even after the current pulse in the NM chain hasended, the spin angular momentum remains depositedwithin AFMI, with its total value increasing with U [Fig. 5(a)]. Such Mott insulator transmuted into a phasewith nonzero total magnetization remains magnetizedalso when intrachain hopping is switched on, γ v = 0 . γ ,in Fig. 4(c). However, γ v = 0 . γ allows electrons to leakfrom AFMI [Fig. 4(a)] to NM [Fig. 4(b)] chain, so thattotal spin deposited into AFMI is reduced in Fig. 4(c)when compared to isolated AFMI.Figure 5 explains, pedagogically and at the micro-scopic level, quantum STT [44–47] as the transfer of totalspin angular momentum from NM conduction electrons(dashed lines in Fig. 5) to confined electrons within theAFMI [solid lines in Fig. 5(a)] or to localized spins withinthe AFI [solid lines in Fig. 5(b)]. The NM/AFMI casewith U = 10 γ shows that (cid:80) i s (cid:48) zi ( t ) within AFMI is nearlyidentical to (cid:80) i S zi ( t ) within AFI with J = 4 γ /U , asanticipated from mapping [41, 42] of AFMI to AFI inthe limit U (cid:29) γ . However, this correspondence failsfor U < γ . The absorbed spin by AFMI or AFI canbe viewed as multiple excitations of any two-spinon orhigher-order spinon states [58], as long as they are com-patible with total angular momentum conservation [46]. Conclusions.—
In conclusion, we demonstrate howthe very recently developed tDMRG [46] approach to quantum STT [44–47] makes it possible to study spintorque on strongly electron-correlated antiferromagnets.In contrast, quantum-classical theory of conventionalSTT [11, 13–22] would conclude that entangled AF trueGS does not undergo any current-driven dynamics whenits localized spins have zero expectation value at t = 0 asthe initial state used in this study. Although tDMRG hasbeen previously applied to study charge current throughAFMI [53–55] or spin-charge separation [56] in geome-tries where electrons are injected into AFMI by finitebias voltage, spin-dependent transport phenomena in ge-ometries like Fig. 1 of relevance to spintronics [7–10] re-main unexplored. Realistic spintronic devices would re-quire to consider two- or three-dimensional geometries.But Keldysh Green functions [25, 59], as the only avail-able nonequilibrium quantum many-body formalism forhigher dimensions and longer times, cannot at presentaccess large U with perturbative self-energies [57, 59],or its nonperturbative implementation can handle [60]only a very few sites. Therefore, this study represents apivotal test case that provides intuition about quantumSTT phenomena in strongly correlated and/or entangledquantum materials, as well as a benchmark [59] for any future developments via the Keldysh Green functions.M. D. P., P. M. and B. K. N. were supported bythe U.S. National Science Foundation (NSF) Grant No.ECCS 1922689. A. E. F. was supported by the U.S. De-partment of Energy (DOE) Grant No. DE-SC0019275. ∗ [email protected][1] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono,and Y. Tserkovnyak, Antiferromagnetic spintronics, Rev.Mod. Phys. , 015005 (2018).[2] T. 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