Asymmetric and symmetric exchange in a generalized 2D Rashba ferromagnet
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Asymmetric and symmetric exchange in a generalized 2D Rashba ferromagnet
I. A. Ado, A. Qaiumzadeh,
2, 3
R. A. Duine,
2, 4, 5
A. Brataas, and M. Titov
1, 6 Radboud University, Institute for Molecules and Materials, NL-6525 AJ Nijmegen, The Netherlands Center for Quantum Spintronics, Department of Physics,Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran Institute for Theoretical Physics and Centre for Extreme Matter and Emergent Phenomena,Utrecht University, 3584 CE Utrecht, The Netherlands Department of Applied Physics, Eindhoven University of Technology,P.O. Box 513, 5600 MB Eindhoven, The Netherlands ITMO University, Saint Petersburg 197101, Russia
Dzyaloshinskii-Moriya interaction (DMI) is investigated in a 2D ferromagnet (FM) with spin-orbitinteraction of Rashba type at finite temperatures. The FM is described in the continuum limit byan effective s - d model with arbitrary dependence of spin-orbit coupling (SOC) and kinetic energyof itinerant electrons on the absolute value of momentum. In the limit of weak SOC, we derivea general expression for the DMI constant D from a microscopic analysis of the electronic grandpotential. We compare D with the exchange stiffness A and show that, to the leading order in smallSOC strength α R , the conventional relation D = (4 mα R / ~ ) A , in general, does not hold beyondthe Bychkov-Rashba model. Moreover, in this model, both A and D vanish at zero temperature inthe metal regime (i. e., when two spin sub-bands are partly occupied). For nonparabolic bands ornonlinear Rashba coupling, these coefficients are finite and acquire a nontrivial dependence on thechemical potential that demonstrates the possibility to control the size and chirality of magnetictextures by adjusting a gate voltage. Chiral magnetic structures have attracted a great dealof interest in recent years with the observation of novelexotic magnetic phases such as skyrmion lattices [1],single skyrmions [2–4], chiral domain walls [5–7], chi-ral magnons [7–9], and helimagnets [10]. The source ofchiral symmetry breaking, required for the formation ofsuch structures, is the asymmetric exchange interactionthat is referred to as Dzyaloshinskii-Moriya interaction(DMI) [2, 11–17]. DMI originates from spin-orbit cou-pling (SOC) in magnetic systems with broken inversionsymmetry, e. g., in noncentrosymmetric crystals or at sur-faces and interfaces of thin magnetic films. The latter,effectively low-dimensional systems, which are of particu-lar interest for applications, are in the focus of our study.Recently, both bulk and interfacial DMI have beenmeasured by employing Brillouin light scattering and,indirectly, using spin-polarized electron energy-loss spec-troscopy [18–29]. On the other hand, for calculation ofDMI in realistic materials, there exist effective compu-tational techniques that provide decent agreement withcertain experimental data [30–33]. A comprehensive un-derstanding of the asymmetric exchange in generic sys-tems requires model studies as well.A widely used strategy for addressing DMI in systemswith magnetic order is to utilize an s - d type model ap-proach with noninteracting itinerant electrons mediatingmagnetic interactions. Within this ideology, the authorsof Ref. [34] derived formulas for the asymmetric exchangebetween two single magnetic ions embedded in a 1D- or2DEG with Rashba SOC. A decade later, their resultwas generalized by allowing for finite uniform magneti-zation [35]. As far as smooth noncollinear magnetic structures areconcerned (e. g., domain walls or skyrmions), it is moreconvenient to describe a magnet in the continuum limitby sending the lattice spacing to zero in the first place.In this paradigm, Berry phase type expressions for theasymmetric exchange have been recently derived [36] andthe relation between DMI and ground-state spin currentshas been pointed out [37, 38]. Surprisingly, though, theonly 2D ferromagnet (FM) model for which DMI has,so far, been calculated in the continuum limit refers tothe system of a FM deposited on top of a topologicalinsulator [39–41].In this Letter, we focus on a less exotic model thatcaptures the effects of both Rashba SOC and the s - d type exchange interaction between localized FM spinsand 2DEG. The following Hamiltonian of one conductionelectron is considered: H = ξ ( p ) + α R ζ ( p ) [ p × σ ] z + J sd S n ( r , t ) · σ , (1)where ξ ( p ) and ζ ( p ) are arbitrary functions of the abso-lute value of momentum that parametrize free electrondispersion (kinetic energy) and momentum dependentRashba SOC, respectively. The last term stands for theeffective s - d exchange interaction with strength J sd . Weassume that the system is deep in the FM phase and thetemperature is far below the corresponding Curie tem-perature; hence, the localized spins of the absolute value S can be described by the continuous vector field n ( r , t )with the constraint | n | ≡
1. We also assume the dynam-ics of itinerant electrons to be much faster than that ofFM spins and treat the field n as time independent. Thenotation σ refers to a vector of Pauli matrices.The model of Eq. (1) describes a generic FM layercoupled to 2DEG with spin-orbit interaction of Rashbatype. One possible realization of such a system is aLaAlO /SrTiO interface [42]. The model might also beused to describe a SrRuO /SrIrO interface, which hasrecently gained considerable attention in the context ofthe so-called topological Hall effect – the phenomenonintrinsically linked to DMI [43, 44].In the continuum limit, DMI (or the antisymmetricexchange) is recognized as a contribution Ω D [ n ] to themicromagnetic free energy density that is linear with re-spect to the first spatial derivatives of the vector field n .The symmetric exchange, on the other hand, is associ-ated with a contribution Ω A [ n ] that is quadratic withrespect to the first spatial derivatives of n . The ratiobetween the two contributions plays a key role in forma-tion of chiral magnetic structures, affecting their stabil-ity and size. Relation between Ω D [ n ] and Ω A [ n ] for themodel of Eq. (1) is interesting for one more, historical,reason. Standard symmetry analysis shows [17] that inan isotropic 2D FM system, one hasΩ A [ n ] = A (cid:2) ( ∇ x n ) + ( ∇ y n ) (cid:3) , (2)where A is the exchange stiffness. For a particular choice, ξ ( p ) = p / m and ζ ( p ) ≡ D [ n ] = D n · [[ e z × ∇ ] × n ] (3)and, moreover, to D = (4 mα R / ~ ) A . Unfortunately, theactual calculation of Ω D [ n ] has been performed neitherin Ref. [46] nor, to the best of our knowledge, anywhereelse even for the particular case of the Bychkov-Rashbamodel.Below, we undertake an accurate microscopic treat-ment of the model of Eq. (1) in the leading order with re-spect to small α R and, under rather general assumptionson ξ ( p ) and ζ ( p ) [47], directly derive Eqs. (2) and (3).Furthermore, we report that the exchange stiffness A andthe DMI constant D are given by remarkably concise ex-pressions, namely, A = ∆ sd π ∂∂ ∆ sd (cid:20)Z ∞ dp p [ ξ ′ ( p )] ∆ sd ( f − − f + ) (cid:21) , (4) D = α R ∆ sd π ~ ∂∂ ∆ sd (cid:20)Z ∞ dp p ζ ( p ) ξ ′ ( p )∆ sd ( f − − f + ) (cid:21) , (5)where ∆ sd = | J sd | S is half of the exchange splitting, ξ ′ ( p ) = ∂ξ/∂p , and f ± = f ( ξ ( p ) ± ∆ sd ) are expressedvia the Fermi-Dirac distribution f ( ε ) = (1 + exp [( ε − µ ) /T ]) − (6)with the chemical potential µ and temperature T . We would like to draw the reader’s attention to thefact that the result of Eq. (4) is well-known, though, ina different form (see, e. g., Eq. (70) in Ref. [48]). It is,however, useful to cast A in the form of Eq. (4) in orderto compare the symmetric and asymmetric exchange forseveral particular choices of ξ ( p ) and ζ ( p ) as we do laterin the text.We have checked that the DMI constant of Eq. (5) canalso be obtained either by evaluation of ground-state spincurrents [37, 38] or by using the formalism of Ref. [36].We have also checked that one may restore both Eqs. (4)and (5) by calculation of spin density of conduction elec-trons s [49] followed by an integration of the relation n × ( δ Ω /δ n ) = J sd S n × s , as it was done in Ref. [39]for DMI in the Dirac model. It must also be possible tocompute A and D from an effective action [41, 50].Nevertheless, we believe that the most natural andstraightforward way to derive Eqs. (4) and (5) is to ex-tract Ω A [ n ] and Ω D [ n ] from the electronic grand poten-tial density Ω. In this approach, there is no need toassume a priori the symmetry form of the final resultas it is often done in the literature. Using the standardformulation of statistical physics, we express the grandpotential density at r = r asΩ = − T πi Tr Z dε g ( ε ) (cid:2) G A ( r , r ) − G R ( r , r ) (cid:3) , (7)where G A ( R ) = ( ε ∓ i − H ) − is the advanced (retarded)Green’s function for the model of Eq. (1), Tr stands forthe matrix trace, and the notation g ( ε ) = ln (1 + exp [( µ − ε ) /T ]) (8)is employed.Now, let us show how Eq. (7) can be used to obtain theDMI contribution to micromagnetic free energy density.First, one should Taylor expand n ( r ) around n ( r ) anduse the result to generate the Dyson series G ( r , r ) = G ( r − r ) + J sd S Z d r ′ G ( r − r ′ ) × X βγ ( r ′ − r ) β ∇ β n γ ( r ) σ γ G ( r ′ − r ) , (9)where G is the Green’s function of a homogeneous sys-tem with fixed n ( r ) ≡ n ( r ). In Eq. (9), we have disre-garded all the gradients of n but the first, which is onlyaccounted for in the linear order. The second term inEq. (9) is precisely the one that determines the asym-metric exchange. Substituting it into Eq. (7), we switchto momentum representation and symmetrize the resultto obtain the general formulaΩ D [ n ] = X βγ Ω DMI βγ ∇ β n γ , (10)with the DMI tensor defined asΩ DMI βγ = T J sd S π ~ Re Z dε g ( ε ) Z d p (2 π ) × Tr (cid:16) G R σ γ G R v β G R − G R v β G R σ γ G R (cid:17) , (11)where v = ∂ H /∂ p is the velocity operator. Note thatwe have dropped the argument of n ( r ) in Eq. (10) andfurther below.Evaluation of Eq. (11) for the present model isperformed with the help of the momentum-dependentGreen’s function G R ( A ) = ε − ξ ( p ) + α R ζ ( p ) [ p × σ ] z + J sd S n · σ ( ε − ε + ( p ) ± i ε − ε − ( p ) ± i , (12)where we introduce the spectral branches ε ± ( p ) = ξ ( p ) ± p ( J sd S ) + [ α R p ζ ( p )] − α R J sd S p ζ ( p ) sin θ sin φ , theangle θ stands for the polar angle of n with respect tothe z axis, while φ is the angle between the momentum p and the in-plane projection of the vector n . We sub-stitute Eq. (12) into Eq. (11), calculate the matrix trace,expand the integrands to the linear order in α R , andstraightforwardly integrate over φ . This results in thefollowing form of the DMI tensor:Ω DMI βγ = D X ij n i ǫ ijγ ǫ jzβ , (13)where ǫ q q q denotes the three-dimensional Levi-Civitasymbol, while D = α R ∆ π ~ T Z ∞ p dp Z ∞−∞ dε g ( ε ) × Im (cid:18) ζ ( p ) + p ζ ′ ( p )[ ε − ε +0 ( p ) + i [ ε − ε − ( p ) + i (cid:19) , (14)where ζ ′ ( p ) = ∂ζ/∂p and ε ± ( p ) = ξ ( p ) ± ∆ sd . FromEqs. (13) and (14), it is already evident that, up to thelinear order in α R , the asymmetric exchange does, indeed,have the form of Eq. (3) with the DMI constant D whichis totally independent of the direction of magnetization.Integration over ε in Eq. (14) leads to D = α R ∆ sd π ~ T Z ∞ dp g ′− − g ′ + ∆ sd ∂ (cid:2) p ζ ( p ) (cid:3) ∂p − α R ∆ sd π ~ T Z ∞ dp g − − g + ∆ ∂ (cid:2) p ζ ( p ) (cid:3) ∂p , (15)where g ′± = ∂g ± /∂ ∆ sd and g ± = g ( ξ ( p ) ± ∆ sd ) [51].Eventually, the above two integrals are combined to forma full derivative with respect to ∆ sd . Partial integrationconcludes the derivation of the DMI constant D of Eq. (5)once the identity ∂g ( ε ) /∂ε = − f ( ε ) /T is used.The symmetric exchange can be treated similarly.In order to derive Eqs. (2) and (4), one should take FIG. 1: Dzyaloshinskii-Moriya interaction constant D in theBychkov-Rashba model as a function of the chemical potential µ at different temperatures T . Both µ and T are normalizedby half of the exchange splitting ∆ sd = | J sd | S . α R = 0 and extract all terms proportional to ∇ β n γ ∇ β ′ n γ ′ and ∇ β ∇ β ′ n γ in Eq. (7). We relegate the details of thecalculation to the Supplementary Material [47].In the rest of the Letter, we apply the general expres-sions of Eqs. (4) and (5) to three particular cases. Allfurther analytical results are presented in Table I, andthe corresponding plots are given in Figs. 1, 2, and 3.To begin with, we return to the Bychkov-Rashba (BR)model characterized by ξ ( p ) = p / m and ζ ( p ) ≡
1. Ascan be immediately seen from Eqs. (4) and (5), the rela-tion D BR = (4 mα R / ~ ) A BR , indeed, holds, and the predic-tion of Ref. [46] is validated. Furthermore, in the limitof zero temperature, one finds from Eq. (5) that D BR = ∆ sd mα R π ~ ( − ( µ/ ∆ sd ) , | µ | < ∆ sd , µ > ∆ sd . (16)Thus, if SOC is weak, both A and D are finite in theBychkov-Rashba model at T = 0 only in the half-metalregime | µ | < ∆ sd .In fact, DMI in this model vanishes identically in themetal regime µ > ∆ sd irrespective of the SOC strength.At larger α R , the asymmetric exchange ceases to havethe simple symmetry of Eq. (3) in the form of Lifshitzinvariants. However, contributions from the two Fermisurfaces still cancel each other within each component ofthe DMI tensor Ω DMI , no matter what the SOC strengthis [52–54]. A nonperturbative in SOC study of DMI inthe model of Eq. (1) will be presented elsewhere.Next, it is instructive to see how the deviations fromparabolic dispersion, a common property of, e. g., narrowgap semiconductors and quantum wells [55–57], affect A and D and the relation between them. To model non-parabolicity (NP) we use ξ ( p ) = ( p / m )(1 + Υ p / m )and ζ ( p ) ≡ ξ ( p )is an increasing function even for negative values of Υ; FIG. 2: Dzyaloshinskii-Moriya interaction constant D and“normalized” exchange stiffness (4 mα R / ~ ) A as functions ofthe chemical potential µ at zero temperature for different val-ues of nonparabolicity coefficient Υ. Both µ and 1 / Υ arenormalized by half of the exchange splitting ∆ sd = | J sd | S . i. e., our choice of ξ ( p ) is understood as an approximationat small values of p . Temperature is set to zero.We find, in this case, that the DMI constant and theexchange stiffness remain finite for all values of µ . More-over, the NP corrections to D and (4 mα R / ~ ) A are ofdifferent signs, but have equal magnitudes, D NP − D BR = − (4 mα R / ~ )( A NP − A BR ) , (17)independently of the sign of Υ (see Fig. 2 and Table I).This leads, in the metal regime, to a particularly unex-pected relation D NP = − (4 mα R / ~ ) A NP , µ > ∆ sd (18)(cf. the relation D BR = (4 mα R / ~ ) A BR for the Bychkov-Rashba model).For Υ <
0, the exchange stiffness becomes negative inthe metal regime, which may eventually make the FMphase unstable. Of course, within our study, we do notconsider direct contributions to magnetic exchange thatmay remain sufficiently large to be overcome by nega-tive A NP . Nevertheless, the reduction of the direct ex-change in nonparabolic FM layers may have a seriousimpact on the size of noncollinear magnetic textures. In a particular case of a single skyrmion, a simple estimate ofits size is ∝ A/D [58]. We note that, for Υ <
0, the DMIconstant is enhanced; hence, the deviations from parabol-icity may reduce the size of magnetic skyrmions leadingto miniaturization of skyrmion-based technology. In gen-eral, nontrivial dependence of A and D on the chemicalpotential shown in Fig. 2 clearly demonstrates the possi-bility to control the size of skyrmions by means of a gatevoltage.Finally, motivated by theoretical [59], computa-tional [60], and experimental [61] demonstrations of gen-erally nonlinear (NL) dependence of Rashba SOC on mo-mentum, we model the effect of the latter on the asym-metric exchange. Since Rashba spin splitting is usuallyreported [59–61] to either saturate or decrease with in-creasing p , we use ζ ( p ) = 1 / (cid:0) λ p / m (cid:1) with positiveparameter λ and ξ ( p ) = p / m . At zero temperature,we then find a finite DMI constant D NL for any value ofthe chemical potential (see Fig. 3 and Table I). Moreover, D NL exhibits a sign change around µ = ∆ sd . This demon-strates that a gate voltage can also be used to manipulatechirality of magnetic order in 2D FM.Tuning of DMI has, so far, been realized by differentapproaches to interface engineering [62, 63]. The am-bition to manipulate the stability parameter, size, anddensity of skyrmions was very recently achieved as well,by means of similar methods [64]. Based on our find-ings, we argue that a gate voltage variation may add yetanother important and flexible tool for controlling chiralmagnetic domains, paving the way towards novel mate-rial design.To conclude, we considered the asymmetric exchangein generalized 2D Rashba FM. In the weak SOC limit, weestablished the full form of the corresponding contribu-tion to micromagnetic free energy density and deriveda general formula for the DMI constant. We showed FIG. 3: Dzyaloshinskii-Moriya interaction constant D as afunction of the chemical potential µ at zero temperature fordifferent values of nonlinearity coefficient λ . Both µ and 1 /λ are normalized by half of the exchange splitting ∆ sd = | J sd | S . Cases D and (4 mα R / ~ ) A in units mα R ∆ sd / π ~ | µ | < ∆ sd µ > ∆ sd ξ ( p ) = p / m , ζ ( p ) ≡ D − ( µ/ ∆ sd ) − ( π /
3) (
T / ∆ sd ) + S ( µ/ ∆ sd , T / ∆ sd )(4 mα R / ~ ) Aξ ( p ) = p / m , ζ ( p ) ≡ T = 0 D − ( µ/ ∆ sd ) mα R / ~ ) Aξ ( p ) = ( p / m ) (cid:0) p / m (cid:1) , ζ ( p ) ≡ D − ( µ/ ∆ sd ) − u ( µ/ ∆ sd , Υ∆ sd ) − U ( µ/ ∆ sd , Υ∆ sd )(4 mα R / ~ ) A − ( µ/ ∆ sd ) + u ( µ/ ∆ sd , Υ∆ sd ) U ( µ/ ∆ sd , Υ∆ sd ) ξ ( p ) = p / m , ζ ( p ) = 1 / (cid:0) λ p / m (cid:1) D − ( µ/ ∆ sd ) − w ( µ/ ∆ sd , λ ∆ sd ) − W ( µ/ ∆ sd , λ ∆ sd )Notations s ( a, b ) = 2 b { ln (1 + exp [ − ( a + 1) /b ]) − b Li ( − exp [ − ( a + 1) /b ]) } , S ( a, b ) = s ( a, b ) + s ( − a, b ) u ( a, b ) = ( q − { b − ( q − q + 1) } / b , q = p a + 1) b , U ( a, b ) = u ( a, b ) − u ( − a, − b ) w ( a, b ) = { [2 b ( r − − r ( r − r − − r ln r } /rb , r = 1 + ( a + 1) b , W ( a, b ) = w ( a, b ) − w ( − a, − b )TABLE I: Analytical results for D and (4 mα R / ~ ) A for particular choices of ξ ( p ) and ζ ( p ). Results that correspond to theBychkov-Rashba model are shown both with full temperature dependence (upper row) and, for clarity, at zero temperature(second row). Notation Li ( · ) stands for the dilogarithm, which is the polylogarithm of the order 2. Sign of Υ can be takenarbitrary, whereas λ is assumed positive. For Υ < µ < (4 | Υ | ) − − ∆ sd . Heresigns of Υ, u , and U coincide, while w and W are positive. that, to the leading order in small α R in the Bychkov-Rashba model, a linear relation between the exchangestiffness A and the DMI constant D , indeed, holds, whileat zero temperature, both vanish once the two spin sub-bands are partly occupied. At the same time, deviationsfrom the Bychkov-Rashba model prevent this cancella-tion. There is no general linear dependence between A and D . In particular, the relation D = (4 mα R / ~ ) A forthe Bychkov-Rashba model is replaced at zero temper-ature by the relation D = − (4 mα R / ~ ) A in the metalregime of the same model if nonparabolicity of the kineticterm is taken into account. For nonparabolic bands ornonlinear Rashba coupling, both A and D acquire a non-trivial dependence on the chemical potential that demon-strates the possibility of controlling the size and chiralityof magnetic textures by adjusting a gate voltage.We are grateful to M. I. Katsnelson, P. M. Ostrovsky,S. Brener, O. Gomonay, and K.-W. Kim for helpful dis-cussions. This research was supported by the EuropeanResearch Council via Advanced Grant No. 669442 “In-sulatronics”, the Research Council of Norway throughits Centres of Excellence funding scheme, Project No.262633, “QuSpin”, and by the JTC-FLAGERA ProjectGRANSPORT. M.T. acknowledges support from theRussian Science Foundation under Project No. 17-12-01359. 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ONLINE SUPPLEMENTARY MATERIALAsymmetric and symmetric exchange in a generalized 2D Rashba ferromagnet
I. A. Ado, A. Qaiumzadeh, R. A. Duine, A. Brataas, and M. TitovIn this Supplementary Material, we formulate the assumptions on ξ ( p ) and ζ ( p ) and also derive Eqs. (2)and (4) of the main text of the Letter. A. Assumptions on ξ ( p ) and ζ ( p ) The result of Eq. (5) assumes that the derivative ∂D/∂α R at α R = 0 does exist. The latter is not the case, e. g.,for the model of Dirac fermions, where D ∝ /α R [39–41]. Thus, the necessary condition for the validity of Eq. (5) is ξ ( p )
0. In order to establish the sufficient conditions, one should investigate the convergence of the integrals thatdefine ∂D/∂α R . Given ξ ( p ) and ζ ( p ) have no singularities at finite values of p , it would be a study of convergenceof the corresponding integrals at p = ∞ . Uniform convergence is guaranteed, for instance, if distribution functions f ( ε ± ( p )) decay at infinity well enough. This will be the case if at large p function ξ ( p ) is positive, unbounded, andgrows faster than | p ζ ( p ) | .The result of Eq. (4) provides the value of the exchange stiffness in the absence of SOC, hence it depends on ξ ( · )only. If ξ ( p ) has no singularities at finite values of p , and it is positive and unbounded at large p , Eq. (4) is valid. B. Derivation of Eqs. (2) and (4) of the main text of the Letter
In order to compute the symmetric exchange contribution to micromagnetic free energy density, one has to extractall terms proportional to ∇ β n γ ∇ β ′ n γ ′ and ∇ β ∇ β ′ n γ in the electronic grand potential, Eq. (7). To do that, we extendthe Dyson series of Eq. (9) as G ( r , r ) = G ( r − r ) + J sd S Z d r ′ G ( r − r ′ ) X βγ ( r ′ − r ) β ∇ β n γ ( r ) σ γ G ( r ′ − r )+ ( J sd S ) Z d r ′ d r ′′ G ( r − r ′ ) X βγ ( r ′ − r ) β ∇ β n γ ( r ) σ γ G ( r ′ − r ′′ ) X β ′ γ ′ ( r ′′ − r ) β ′ ∇ β ′ n γ ′ ( r ) σ γ ′ G ( r ′′ − r )+ J sd S Z d r ′ G ( r − r ′ ) X ββ ′ γ ( r ′ − r ) β ( r ′ − r ) β ′ ∇ β ∇ β ′ n γ ( r ) G ( r ′ − r ) , (s1)where the first line has been already analysed in the main text, the second line is a second order correction to theGreen’s function due to the first spatial derivatives of n , while the third line is a first order correction due to thesecond spatial derivatives of n . We substitute the latter two into Eq. (7), switch to momentum representation, andsymmetrize the outcome, arriving atΩ A [ n ] = X ββ ′ γγ ′ Ω exc-I ββ ′ γγ ′ ∇ β n γ ∇ β ′ n γ ′ + X ββ ′ γ Ω exc-II ββ ′ γ ∇ β ∇ β ′ n γ , (s2)where the tensors are defined asΩ exc-I ββ ′ γγ ′ = T ( J sd S ) π Im Z dε g ( ε ) Z d p (2 π ) Tr (cid:16) G R v β G R σ γ G R σ γ ′ G R v β ′ G R + G R v β ′ G R σ γ ′ G R σ γ G R v β G R (cid:17) (s3)and Ω exc-II ββ ′ γ = − T J sd S π Im Z dε g ( ε ) Z d p (2 π ) Tr (cid:18) ∂ G R ∂p β ∂p β ′ σ γ G R + G R σ γ ∂ G R ∂p β ∂p β ′ (cid:19) . (s4)The notation of the argument of n ( r ) is dropped in Eq. (s2) and further below.2The Green’s functions entering Eqs. (s3) and (s4) are taken in the momentum representation of Eq. (12) of the maintext, but with α R = 0. Taking a matrix trace calculation and performing an integration over the angle, we obtainΩ exc-I ββ ′ γγ ′ = A δ ββ ′ δ γγ ′ + W δ ββ ′ n γ n γ ′ , (s5)Ω exc-II ββ ′ γ = A δ ββ ′ n γ , (s6)where δ q q is Kronecker delta, while A = ∆ π T ∞ Z p dp ∞ Z −∞ dε g ( ε ) Im [ ξ ′ ( p )] (cid:2) + ( ε − ξ ( p )) (cid:3) [ ε − ξ ( p )][ ε + i − ε +0 ( p )] [ ε + i − ε − ( p )] ! , (s7) A = − ∆ π T ∞ Z p dp ∞ Z −∞ dε g ( ε ) Im [ ξ ′ ( p ) + p ξ ′′ ( p )] (cid:2) ∆ + 3( ε − ξ ( p )) (cid:3) p [ ε + i − ε +0 ( p )] [ ε + i − ε − ( p )] + 2 [ ξ ′ ( p )] (cid:2) ∆ + ( ε − ξ ( p )) (cid:3) [ ε − ξ ( p )][ ε + i − ε +0 ( p )] [ ε + i − ε − ( p )] ! , (s8)and the actual value of W is not relevant for the final result. Combining Eqs. (s2), (s5), and (s6) we findΩ A [ n ] = A (cid:2) ( ∇ x n ) + ( ∇ y n ) (cid:3) + A (cid:2) n ∇ x n + n ∇ y n (cid:3) + W ( n ∇ x n ) + W ( n ∇ y n ) . (s9)Before we proceed, it is important to notice two consequences of the constraint n ≡
1, namely,12 ∇ β n = n ∇ β n = 0 and 12 ∇ β n = ∇ β ( n ∇ β n ) = ( ∇ β n ) + n ∇ β n = 0 . (s10)With the help of Eq. (s10) we are able to bring Eq. (s9) to the formΩ A [ n ] = ( A − A ) (cid:2) ( ∇ x n ) + ( ∇ y n ) (cid:3) , (s11)proving Eq. (2) of the main text with A = A − A .To complete the calculation of the exchange stiffness A , one should perform a partial fraction decomposition of theintegrands in Eqs. (s7), (s8) and make use of the formulaIm (cid:0) [ ε − ε ± ( p ) + i − n − (cid:1) = ( − n +1 n ! π δ ( n ) ( ε − ε ± ( p )) (s12)to integrate over ε with the result A = ∆ sd π T Z ∞ dp p [ ξ ′ ( p )] ∆ ( g ′− − g ′ + ) + ∆ sd π T Z ∞ dp p [ ξ ′ ( p )] ∆ sd ( g ′′− + g ′′ + )+ ∆ sd π T Z ∞ dp [ ξ ′ ( p ) + p ξ ′′ ( p )]( g ′′− − g ′′ + ) + ∆ sd π T Z ∞ dp p [ ξ ′ ( p )] ( g ′′′− − g ′′′ + ) , (s13)where ξ ′ ( p ) = ∂ξ/∂p and the derivatives of g ± = g ( ε ± ( p )) = g ( ξ ( p ) ± ∆ sd ) are taken with respect to the argument.The latter can also be assumed to be the derivatives with respect to ξ , g ( n ) ± = ∂ n g ± ∂ξ n . (s14)The third term cancels out the fourth term in Eq. (s13) after integration by parts with the help of ξ ′ ( p ) + p ξ ′′ ( p ) = ∂ [ p ξ ′ ( p )] /∂p. (s15)In the remaining terms, one replaces the derivatives of g ± = g ( ξ ( p ) ± ∆ sd ) with respect to ξ by the derivatives withrespect to ∆ sd , reduces the resulting expression to a form of a full derivative with respect to ∆ sd , and uses the relation ∂g ( ε ) /∂ε = − f ( ε ) /T/T