Spin transport in Heisenberg antiferromagnets in ... - Institute of Physics

PHYSICAL REVIEW B 75, 214403 共2007兲

Spin transport in Heisenberg antiferromagnets in two and three dimensions M. Sentef, M. Kollar, and A. P. Kampf Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, 86135 Augsburg, Germany 共Received 21 December 2006; revised manuscript received 7 February 2007; published 1 June 2007兲 We analyze spin transport in insulating antiferromagnets described by the XXZ Heisenberg model in two and three dimensions. Spin currents can be generated by a magnetic-field gradient or, in systems with spin-orbit coupling, perpendicular to a time-dependent electric field. The Kubo formula for the longitudinal spin conductivity is derived analogously to the Kubo formula for the optical conductivity of electronic systems. The spin conductivity is calculated within interacting spin-wave theory. In the Ising regime, the XXZ magnet is a spin insulator. For the isotropic Heisenberg model, the dimensionality of the system plays a crucial role: In d = 3 the regular part of the spin conductivity vanishes linearly in the zero frequency limit, whereas in d = 2 it approaches a finite zero frequency value. DOI: 10.1103/PhysRevB.75.214403

PACS number共s兲: 75.40.Gb, 75.10.Jm, 75.30.Ds

I. INTRODUCTION

The challenge of spintronics research is to exploit the spin degree of freedom as an additional tool in electronic devices.1–5 This task demands to explore the basic physical principles underlying the generation and decay of spinpolarized charge currents. In this context, metallic and semiconducting devices have been examined both theoretically and experimentally. Simultaneously, the synthesis of quasione-dimensional correlated insulators such as Sr2CuO3 共Ref. 7兲 and CuGeO3 has renewed the general interest in the spin and thermal transport properties of low-dimensional quantum spin systems. Also fundamental questions have been raised, e.g., how the integrability of a system influences its spin8–12 and thermal12–16 conductivities. In two-dimensional high-mobility electron systems with Rashba spin-orbit coupling charge currents are necessarily accompanied by spin currents.17 These spin currents flow perpendicular to the charge current direction and therefore lead to an intrinsic spin Hall effect.18,19 The spin conductivity of Heisenberg chains has previously been computed within linear-response theory by adopting an analogy to the Kubo formula for charge transport.20 The low-frequency behavior of the optical conductivity ␴共␻兲 = ␴⬘共␻兲 + i␴⬙共␻兲 provides a transparent scheme to distinguish the charge transport properties of ideal conductors, insulators, and nonideal conductors.20,21 Decomposing the real part of the longitudinal conductivity as ␴⬘共␻兲 = D␦共␻兲 + ␴reg共␻兲, a finite Drude weight or charge stiffness20,22 D ⬎ 0 is the characteristic of ideal conductors. A similar classification scheme can be carried over to spin transport and the spin conductivity in order to distinguish between spin conductors, spin insulators or even spin superfluids.23,24 The optical conductivity is conveniently derived as the current response to a time-dependent electromagnetic vector potential. Similarly, for the spin current response the concept of a fictitious spin vector potential can be introduced, which is related to a twist in the direction of the spin quantization axis.25 However, the physical realization of this perturbation and its relation to externally applied magnetic or electric fields is not obvious. Indeed, spin currents flow in response 1098-0121/2007/75共21兲/214403共9兲

to a magnetic-field gradient. In this case, the analogy to the generation of electric currents by a potential gradient is straightforwardly established for one-dimensional systems by a Jordan-Wigner transformation.26–28 This transformation maps spin- 21 operators to creation and annihilation operators of spinless fermions; a magnetic-field gradient for the spins thereby translates into a potential gradient for the fermions. In this paper, we do not appeal to the analogy to the charge current response. Rather, we derive the Kubo formula for the spin conductivity of XXZ Heisenberg magnets directly 共Sec. II兲. In particular, a magnetic-field gradient or, in the presence of spin-orbit coupling,29 a time-dependent electric field can be used to drive a spin current 共Sec. III兲. The longitudinal spin conductivity of the Heisenberg antiferromagnet in two and three dimensions is computed using spinwave theory 共Sec. IV兲 for both the noninteracting-magnon approximation 共Sec. V B兲 and the ladder approximation for repeated two-magnon scattering processes 共Sec. V C兲. In particular, the low-frequency behavior of the spin conductivity will be analyzed in detail revealing distinct differences between the spin transport properties of two-dimensional 共2D兲 and three-dimensional 共3D兲 antiferromagnets. We find that only for an isotropic 2D antiferromagnet the spin conductivity remains finite in the dc limit. II. KUBO FORMULA

Specifically, consider the antiferromagnetic XXZ Heisenberg model 共HAFM兲

冉

1 H = J 兺 ⌬Szi Szj + 共S+i S−j + S−i S+j 兲 2 具i,j典

冊

共1兲

with nearest-neighbor exchange coupling J ⬎ 0, an anisotropy parameter ⌬, and local spin operators Si of length S. The sum over 具i , j典 extends over the nearest-neighbor bonds on a d-dimensional hypercubic lattice with lattice constant a = 1 and N sites. An external space- and time-dependent magnetic field Bz共l , t兲 couples to the spin system via the Zeeman energy. The time-dependent Hamiltonian thus reads

214403-1

©2007 The American Physical Society


SENTEF, KOLLAR, AND KAMPF

H共t兲 = H − 兺 Sz共l兲hz共l,t兲,

共2兲

l

where h共l , t兲 = hz共l , t兲ez = g␮BBz共l , t兲ez. The spin current density operator ji→j for the magnetization transport from site i to site j is defined via the continuity equation,

⳵tSzi + 兺 ji→j = 0.

FIG. 1. 共Color online兲 Setup for a spin current generated by a magnetic-field Bz gradient along the x direction.

共3兲

␹ jS共q, ␻兲 =

j

Here, 兺 j ji→j is the lattice divergence of the local spin current density at site i. The operator jx共l兲 = jl→l+x thus follows from Heisenberg’s equation of motion S˙zi = i关H , Szi 兴 and Eq. 共3兲 as jx共l兲 =

Ji + − + 共S S − S−l Sl+x 兲, 2 l l+x

具jx共q, ␻兲典 = ␹ jS共q, ␻兲h 共q, ␻兲, z

共5兲

where the Fourier transforms in time and space are defined as A共q, ␻兲 = 兺 l

冕

⬁

dtei共␻t−q·l兲A共l,t兲.

冕

⬁

dte

具关jx共q,t兲,S 共− q,0兲兴典. z

+

冊

dtei共␻+i0 兲t具关jx共q,t兲,iqx jx共− q,0兲兴典 . 共9兲

0

The response formula therefore becomes 具jx共q, ␻兲典 = −

具− Kx典 − ⌳xx共q, ␻兲 iqxhz共q, ␻兲, i共␻ + i0+兲

共10兲

where we have introduced the spin-flip part of the exchange interaction along the x direction, 具Kx典 =

1 − + + S−l Sl+x 典, 兺 J具S+l Sl+x 2បN l

共11兲

and the longitudinal retarded current-current correlation function, ⌳xx共q, ␻兲 =

i បN

冕

⬁

+

dtei共␻+i0 兲t具关jx共q,t兲, jx共− q,0兲兴典. 共12兲

0

For a closer analogy to charge transport, we consider the transport of magnetization ml = g␮BSzl instead of 共dimensionless兲 spin and define the magnetization current operator jm,x = g␮B jx. The longitudinal spin conductivity ␴xx共␻兲 is then defined as the linear magnetization current response to a long wavelength 共q → 0兲, frequency dependent magnetic-field gradient 共Fig. 1兲. Setting hz = g␮BBz, the relation jm,x共q, ␻兲 = ␴xx共q, ␻兲iqxBz共q, ␻兲

共13兲

thus yields the Kubo formula for the spin conductivity in the long-wavelength limit,

共6兲

具− Kx典 − ⌳xx共q = 0, ␻兲 . i共␻ + i0+兲

共14兲

This result indeed establishes the perfect analogy to the formula for the optical conductivity of interacting lattice electrons.20 The real part of the spin conductivity is given by

−⬁

i共␻+i0+兲t

冕

⬁

␴xx共␻兲 = − 共g␮B兲2

In Eq. 共5兲 we have introduced the dynamic susceptibility i ␹ jS共q, ␻兲 = បN

+

共4兲

where l + x is the nearest-neighbor site of site l in the positive x direction. In our subsequent calculation we assume longrange antiferromagnetic order oriented along the z direction and thus restrict the analysis to a scalar spin current operator for the magnetization transport. The definition of a proper spin current vector operator Ii→j is rather subtle, as discussed in detail in Ref. 30. In fact, if 具Si典 and 具S j典 are not collinear, only the projection of Ii→j onto the plane spanned by the local order parameters 具Si典 and 具S j典 may be interpreted as a physical transport current. If 具Si典 and 具S j典 are collinear, however, the magnetization transport is indeed correctly described by the scalar current density operator defined in Eq. 共4兲. We note that the proper definition of a spin current operator is also a controversial issue in the context of spin transport in semiconductors with spin-orbit coupling,31 but these issues are of no concern for the purpose of our analysis. The linear spin current response to an external magneticfield gradient is

冉

1 i − 具关jx共q兲,Sz共− q兲兴典 បN i共␻ + i0+兲

共7兲

0

reg ⬘ 共␻兲 = DS␦共␻兲 + ␴xx ␴xx 共␻兲

共15兲

with the spin stiffness23,24 or “spin Drude weight” DS, DS ⬘ 共q = 0, ␻ → 0兲兴, = 共g␮B兲2关具− Kx典 − ⌳xx ␲

Using the spatial Fourier transform of the continuity equation 共3兲,

共16兲

and the regular part ˙z

S 共q,t兲 + iq · j共q,t兲 = 0,

共8兲

and assuming a magnetic-field gradient of Bz only along the x direction, ␹ jS is transformed by partial integration as

reg ␴xx 共␻兲 =

⬙ 共q = 0, ␻兲 ⌳xx . ␻

The spin conductivity also fulfills the “f-sum rule”32,33

214403-2

共17兲


SPIN TRANSPORT IN HEISENBERG ANTIFERROMAGNETS…

2 ␲

冕

⬁

⬘ 共␻兲 = 具− Kx典, d␻␴xx

共18兲

0

which can be derived by using the Kubo formula and the Kramers-Kronig relations for ⌳xx. Note that the integral in Eq. 共18兲 contains one-half of the possible spin Drude weight peak at zero frequency. The structure of the spin conductivity formula emerges from the straightforward calculation for the linear current response to an external magnetic-field gradient without introducing Peierls-like phase factors with a fictitious spin vector potential.9,34 However, in an external electric field a moving magnetic dipole does acquire an Aharonov-Casher phase.35 Katsura et al.29 pointed out that Aharonov-Casher phase factors and a corresponding “vector potential” can be introduced on the basis of the Dzyaloshinskii-Moriya interaction,36,37 which may be induced in an external electric field due to spin-orbit coupling. As we show in the next section, this alternative approach leads to the same result for the spin conductivity, Eq. 共14兲. III. RESPONSE TO AN ELECTRIC FIELD

Spin currents are also generated by applying a timedependent electric field in the presence of spin-orbit coupling. In low-symmetry crystals with localized spin moments spin-orbit coupling, parametrized by a coupling constant ␭so, leads to the Dzyaloshinskii-Moriya 共DzM兲 antisymmetric exchange interaction36,37 HDzM = 兺 Dij · 共Si ⫻ S j兲.

共19兲

具i,j典

In high-symmetry crystals an inversion symmetry breaking external electric field E induces a DzM vector Dij ⬀ E ⫻ eij,38 where eij denotes the unit vector connecting the neighboring sites i and j. Specifically, for a field E = 共0 , Ey , 0兲, the DzM vector on the bonds in the x-y plane takes the form Di,i+x = ␣E ⫻ ex = − ␣Eyez,

the DzM

HDzM = − 兺 Ax共l,t兲jx共l兲,

共21兲

= 21 共S+i S−j − S−i S+j 兲,

l

where jx共l兲 is the spin current density operator defined in Eq. 共4兲 and Ax共l,t兲 =

jm,x共l;A兲 = g␮B jx共l;A = 0兲 + 共g␮B兲2Kx共l兲Ax共l兲.

共24兲

As for the charge current response to an electromagnetic vector potential,20 the linear spin current response to the spin vector potential in the long-wavelength limit becomes 具jm,x共0, ␻兲典 = 共g␮B兲2关具Kx典 + ⌳xx共0, ␻兲兴Ax共0, ␻兲.

共25兲

The spin conductivity is therefore obtained by identifying the time derivative of the spin vector potential, i共␻ + i0+兲Ax共0 , ␻兲, as the driving force for the spin current. As a consequence, in the presence of spin-orbit coupling, a spin current can also be driven perpendicular to a time-dependent electric field 共see Fig. 2兲. Physically, the origin of this phenomenon is contained in the Maxwell equation rot B =

1 ⳵E 4␲ , jc + c c ⳵t

共26兲

in the absence of charge currents, jc = 0. Assuming a magnetic field in the z direction, the y component of Eq. 共26兲 reduces to

⳵ xB z =

−1 −J ⳵ tE y = ⳵ tA x . c 2␣c

共27兲

Equation 共27兲 identifies the time-dependent electric field and the magnetic field gradient as the same driving forces for the spin current.

共20兲

Di,i+y = 0,

where ␣ ⬀ ␭so. With 共Si ⫻ S j兲 Hamiltonian can therefore be rewritten as z

FIG. 2. 共Color online兲 Setup for a transverse spin current driven by a time-dependent electric field in the presence of spin-orbit coupling.

2␣Ey共l,t兲 . J

共22兲

IV. SPIN-WAVE THEORY FOR THE ANTIFERROMAGNETIC XXZ MODEL

In our subsequent analysis of the dynamic spin current correlation function ⌳xx共0 , ␻兲 we employ interacting spinwave theory. Even in d = 2 and for S = 1 / 2, spin-wave theory has been shown to provide quantitatively accurate results.40 The Dyson-Maleev 共DM兲 transformation41–43 for the antiferromagnetic Heisenberg model on bipartite lattices is applied. In the DM representation the spin operators are replaced by bosonic operators according to

The electric field therefore acts as a “spin vector potential” in formal analogy to the electromagnetic vector potential in charge transport. If the DzM exchange interaction is added to the XXZ spin Hamiltonian, the Heisenberg equation of motion yields an additional contribution to the spin current operator, 1 + − − + jDzM i→j = 共D ji − Dij兲 2 共Si S j + Si S j 兲,

共23兲

Szi = S − a†i ai ,

冉

S+i = 冑2S 1 −

冊

a†i ai a i, 2S

S−i = 冑2Sa†i

for the up-spin sublattice A and by Szj = − S + b†j b j ,

and the total magnetization current operator is given by 214403-3

共28兲



冉

S+j = 冑2Sb†j 1 −

冊

b†j b j , 2S

S−j = 冑2Sb j

VDM = − ⌬J

共29兲

+ V共2兲␣†1␤2␣3␣4 + V共3兲␣†1␣†2␤†3␣4 + V共4兲␣†1␣3␤†4␤2

for the down-spin sublattice B. Due to the bosonic commutation relations for the a and b operators, the spin algebra is preserved. The spatial Fourier transformation of the bosonic operators is conveniently written as44 ai =

冑

2 兺 e−ik·Riak, N k

bj =

冑

2 兺 e+ik·R jbk , N k

共30兲

where the momentum k is restricted to the magnetic Brillouin zone 共MBZ兲. For the 2D square lattice MBZ= 兵k : 兩kx兩 + 兩ky兩艋 ␲其. Inserting the DM representation into the XXZ Heisenberg model, the quadratic part of the resulting bosonic Hamiltonian is diagonalized by the Bogoliubov transformation ak = uk␣k + vk␤†k,

bk = uk␤k + vk␣†k .

The coefficients uk and vk are given by uk =

冑

1 + ␧k , 2␧k

vk = − sgn共␥k兲

冑

1 − ␧k . 2␧k

共31兲

共32兲

The DM transformed Hamiltonian reads HDM = E0 + H0 + VDM

共33兲

with the diagonalized quadratic part H0 = 兺 ប⍀k共␣†k␣k + ␤†k␤k兲.

共34兲

k

+ V共5兲␤†4␣3␤2␤1 + V共6兲␤†4␤†3␣†2␤1 + V共7兲␣†1␣†2␤†3␤†4 + V共8兲␤1␤2␣3␣4 + V共9兲␤†4␤†3␤2␤1兲.

␧k = 冑1 − ␥2k/⌬2

cos共k␣兲. The magnon-vacuum energy is E0 with = −N⌬JS ␣ 共S兲d. The Oguchi correction factor

␣共S兲 = 1 +

r , 2S

2 兺 ␧k , N k

V. EVALUATION OF THE SPIN CONDUCTIVITY

共35兲

␥k = d1 兺␣d =1 2 2

共39兲

The interaction vertices V共i兲, i = 1 , . . . , 9 depend on the wave vectors k1 , . . . , k4, abbreviated by 1,…,4, and are explicitly given in Ref. 39. The Kronecker delta, ␦G共1 + 2 − 3 − 4兲, ensures momentum conservation to within a reciprocal lattice vector of the MBZ. We briefly comment on two difficulties with the DysonMaleev transformation. First, two constraints, a†i ai 艋 2S and b†j b j 艋 2S, are required for the Fock space of bosonic excitations in order to avoid unphysical spin excitations. The spinwave analysis without constraints can nevertheless be quantitatively justified noting that 具a†i ai典 ⬇ 0.197 in d = 2 at T = 0 for the S = 21 isotropic Heisenberg model, calculated in linear spin-wave theory 共LSW兲, where VDM is neglected. This result implies for the sublattice magnetization 具mA典 = 21 − 具a†i ai典 ⬇ 0.303 共⌬ = 1兲, supporting a posteriori the validity of spin-wave theory even for S = 21 . Second, the Dyson-Maleev transformation is not Hermitian, since S+ ⫽ 共S−兲† when the spin operators are expressed in terms of bosonic operators. As a consequence, the quartic part of the transformed Hamiltonian is non-Hermitian, too. However, its non-Hermiticity does not affect our calculations since we will take into account the only quantitatively relevant Hermitian V共4兲 term 共see Sec. V C兲.

For a d-dimensional hypercubic lattice the spin-wave dispersion is ប⍀k = 2d⌬JS␣共S兲␧k,

d 兺 ␦G共1 + 2 − 3 − 4兲 ⫻ 共V共1兲␣†1␣†2␣3␣4 N 共1234兲

A. Basic propagators and correlation functions

We proceed by defining the time-ordered magnon propagators 共see Refs. 39 and 44 for more details兲,

共36兲

G␣␣共k,t兲 ⬅ − i具0兩T␣k共t兲␣†k共0兲兩0典,

共40兲

arises from the normal-ordering of quartic terms45 in the interaction part VDM. In d = 2,

G␤␤共k,t兲 ⬅ − i具0兩T␤†k共t兲␤k共0兲兩0典.

共41兲

1 1 1 1 r = 1−3F2 − , , ;1,1; 2 , 2 2 2 ⌬

The bare Fourier-transformed propagators in the absence of interactions are

r=1−

冉

冊

共37兲

with the generalized hypergeometric function 3F2.46 Specifically, for d = 2, ⌬ = 1, and S = 21 , the Oguchi correction factor is given by

␣

冉冊

1 =2− 2

8␲ 1 ⌫ 4

冉冊

冉冊

4

1 ⌫ 4 − 8␲

共0兲共k, ␻兲 = G␣␣

1 , ␻ − ⍀k + i0+

共0兲 G␤␤ 共k, ␻兲 =

−1 . ␻ + ⍀k − i0+ 共42兲

4

⬇ 1.157 947,

共38兲

The regular part of the longitudinal spin conductivity reg ␴xx 共␻兲 = 共g␮B兲2

where ⌫ denotes the gamma function. Finally, the normalordered quartic interaction part of the Dyson-Maleev Hamiltonian 共33兲 reads

⬙ 共q = 0, ␻兲 ⌳xx 共g␮B兲2 =− G⬙j 共␻兲 ␻ ␻

共43兲

is determined by the imaginary part of the Fourier transformed time-ordered spin current correlation function

214403-4



G j共t兲 ⬅ −

i 具0兩Tjx共t兲jx共0兲兩0典 បN

1 G j共␻兲 = បN

共44兲

in the ground state 兩0典 with the spin current operator jx = 兺l jx共l兲关see Eq. 共4兲兴. jx is transformed by the same steps which are used for the Dyson-Maleev transformation of the HAFM Hamiltonian. The result is jx = j0 + j1, where j0 = 2JS␣共S兲兺 sin共kx兲 ⫻

冉

k

1 ␥k † 共␣k␣k + ␤†k␤k兲 − 共␣†k␤†k + ␣k␤k兲 ⌬␧k ␧k

冊

共45兲

and j1 = J

2 兺 ␦G共1 + 2 − 3 − 4兲sin共k1,x兲 ⫻ 共j共1兲␣†1␣†2␣3␣4 N 共1234兲

+ j共2兲␣†1␤2␣3␣4 + j共3兲␣†1␣†2␤†3␣4 + j共4兲␣†1␣3␤†4␤2 + j共5兲␤†4␣3␤2␤1 + j共6兲␤†4␤†3␣†2␤1 + j共7兲␣†1␣†2␤†3␤†4 + j共8兲␤1␤2␣3␣4 + j共9兲␤†4␤†3␤2␤1兲,

共46兲

where, for brevity, we have omitted here the explicit expressions of the spin current vertices j共i兲; for completeness, they are listed in the Appendix. The quadratic part j0 is of order S1, but it also contains Oguchi corrections of order S0 arising from the normal ordering of quartic terms in j1. The calculation of G j共t兲 involves the following four correlation functions: 共i兲 A two-magnon correlation function, G00 j 共t兲 = − i具0兩Tj 0共t兲j 0共0兲兩0典;

共47兲

共ii兲 a four-magnon correlation function, G11 j 共t兲 = − i具0兩Tj 1共t兲j 1共0兲兩0典;

共48兲

共iii兲 and two cross-correlation functions, G01 j 共t兲 = − i具0兩Tj 0共t兲j 1共0兲兩0典,

共49兲

G10 j 共t兲 = − i具0兩Tj 1共t兲j 0共0兲兩0典.

共50兲

⫻

Before we continue with the calculation of G j共␻兲, we discuss the selection rules for the matrix elements of the spin current operator, which provide insight into the relevant physical processes for spin transport in antiferromagnets. The Lehmann representation

冉

兺n

兩具0兩jx兩n典兩2

1 1 + − ␻ − 共En − E0兲 + i0 ␻ + 共En − E0兲 − i0+

冊共52兲

shows that the spin current correlation function is determined by the matrix elements 具0兩jx兩n典, where 兩n典 denotes the excited states with energy En. The ground state and the relevant excited states must therefore have 共i兲 the same total momentum and 共ii兲 the same spin quantum numbers, but 共iii兲 opposite parity in order to have nonvanishing matrix elements 具0兩jx兩n典. These conditions imply that only multimagnon excitations with vanishing total momentum contribute to the spin conductivity; one-magnon excitations are forbidden by both 共i兲 and 共ii兲. Condition 共iii兲 is reflected in the sin共kx兲 vertex in Eq. 共46兲, which selects the x direction for the spin current. Below we will focus on the two-magnon contribution to the spin conductivity. Apart from the sin共kx兲 vertex function, our calculation proceeds analogously to the twomagnon analysis of Raman scattering in antiferromagnets.39 In fact, the selection rules for spin transport are the same as for the two-magnon Raman scattering intensity in B1g scattering geometry. In the analysis of the spin current correlation function, Eq. 2 共51兲, we focus on G00 j because it provides the leading 共S 兲 1 order contribution and also the dominant S corrections for the spin conductivity. The product j0共t兲j0共0兲 contains 16 terms. However, following the arguments of Canali and Girvin in Ref. 39, the contributions of 14 of these terms are negligibly small due to two arguments: First, the 12 terms containing prefactors ␥k / ⌬␧k are small because the absolute value of this factor is small near the MBZ boundary, where the free magnon density of states has a van-Hove singularity. Specifically, the four regions of the MBZ boundary where 兩sin共kx兲兩 ⬃ 1 provide the quantitatively relevant contributions. Second, only two of the four remaining terms are nonzero if magnon interactions are neglected. The two remaining dominant terms lead to G j共␻兲 ⬇ G+j 共␻兲 + G+j 共− ␻兲,

共53兲

where

The regular part of the spin conductivity 共43兲 is then obtained with 1 01 10 共␻兲 + G11 G j共␻兲 = 关G00 j 共␻兲 + G j 共␻兲 + G j 共␻兲兴. 共51兲 ប j

En⫽E0

G+j 共␻兲 =

sin共kx兲sin共kx⬘兲关2JS␣共S兲兴2 ⌸kk⬘共␻兲. 兺 បN ␧ k␧ k⬘ k,k⬘

共54兲

The two terms which are involved in Eq. 共53兲 are thus calculated from the same two-magnon Green function, †

†

⌸kk⬘共t兲 = − i具0兩T␣k共t兲␤k共t兲␣k⬘共0兲␤k⬘共0兲兩0典.

共55兲

Its Fourier transform is expressed in terms of magnon propagators and a three-point vertex function47

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⌸kk⬘共␻兲 = i

冕

⬁

−⬁

d␻⬘ G␣␣共k, ␻ + ␻⬘兲G␤␤共k, ␻⬘兲⌫kk⬘共␻, ␻⬘兲. 2␲ 共56兲

The vertex function ⌫kk⬘共␻ , ␻⬘兲 satisfies the Bethe-Salpeter equation ⌫kk⬘共␻, ␻⬘兲 = ␦kk⬘ − ⫻

i ⌬Jd 兺 ប N k1

冕

⬁

−⬁

d␻1 2␲

␣␤ Vkk 共␻⬘, ␻1兲G␣␣共k1, ␻ 1k1k

+ ␻ 1兲

⫻ G␤␤共k1, ␻1兲⌫k1k⬘共␻, ␻1兲.

共57兲

␣␤ 共␻⬘ , ␻1兲 is the sum of all Here, the four-point vertex Vkk 1k1k 47 the irreducible interaction parts. In terms of magnon propagators and the vertex function, G+j 共␻兲 is therefore given by

G+j 共␻兲 =

i关2JS␣共S兲兴2 បN

冕

⬁

−⬁

sin共kx兲 d␻⬘ 兺 2␲ k ␧k

⫻ G␣␣共k, ␻ + ␻⬘兲G␤␤共k, ␻⬘兲⌫k共␻, ␻⬘兲,

FIG. 3. 共Color online兲 Regular part of the longitudinal spin conductivity for noninteracting magnons in two dimensions for different values of the anisotropy parameter ⌬. The gap in ␴reg xx 共␻兲 increases with increasing ⌬ and vanishes for ⌬ = 1. The dotted lines show the expansion around ␻gap, Eq. 共64兲. reg ␴xx 共␻兲 = −

共58兲

where the reduced vertex function ⌫k is defined by ⌫ k共 ␻ , ␻ ⬘兲 = 兺 k⬘

sin共kx⬘兲 ⌫kk⬘共␻, ␻⬘兲. ␧ k⬘

= 共59兲

Equations 共57兲 and 共58兲 define the integral equations which must be solved in the calculation of the correlation function G+j 共␻兲. We will proceed in two steps: In a first approximation, the interactions between magnons are omitted 共Sec. V B兲. Subsequently, interactions are treated within a ladder approximation for repeated magnon-magnon scattering processes 共Sec. V C兲. B. Noninteracting magnons

We start in a first step by neglecting magnon-magnon interactions, which amounts to the replacements G → G共0兲 and ⌫k → sin共kx兲 / ␧k, i.e., V␣␤ = 0. The required complex contour integral in Eq. 共57兲 then straightforwardly leads to the result G+j 共␻兲 =

sin2共kx兲 1 关JS␣共S兲兴2 . 兺 2 បN ␧k ␻ − 2⍀k + i0+ k

˜兲 = ᐉ共m兲共␻

sin2共kx兲 1 2 . 兺 m ˜ − 2␧k + i0+ N k ␧k ␻

再

reg 2 d/2 ˜2 − ␻ ˜ gap ␲ , d = 2, ␴xx 共␻兲兲 ⌬dd共d/2兲−1 共␻ ⯝ ⫻ 2 3d/2 2 1, d = 3, 共g␮B兲 /h 2 ˜ ␻

冎

共63兲 ˜ 艌␻ ˜ gap, with corrections of order 共␻ ˜ −␻ ˜ gap兲共d/2兲+1. Spefor ␻ cifically in d = 2, the result of the expansion is

冉

共61兲

˜ ⬎ 0兲 of the spin conductivity within LSW The regular part 共␻ is then given by

sin2共kx兲共g␮B兲2 ␲2 2 ˜ − 2␧k兲. 共62兲 ␦共␻ 2˜ N 兺 h 共d⌬兲 ␻ k ␧2k

Equation 共62兲 directly reflects the spin current selection rules ˜ − 2␧k兲 accounts for two magnon exciexplained above, ␦共␻ tations at energy ␧k. Momentum conservation and spin conservation are fulfilled by a combination of ␣ and ␤ magnons, which carry opposite spin 共Sz = + 1 or −1兲 and momentum 共k and −k兲. In a fully polarized Heisenberg ferromagnet, there is only one magnon species and hence the spin conductivity vanishes.6 The spin Drude weight 关Eq. 共16兲兴 DS = 0 for any ⌬ 艌 1 within LSW. Furthermore, the regular part of the spin con˜ ductivity diverges at the maximum two-magnon energy 共␻ = 2兲 in dimensions d = 2 and d = 3 due to the neglect of magnon interactions. The LSW result for the spin conductivity in d = 2 is shown in Fig. 3. A special feature of the regular part of ␴xx ⬘ 共␻兲 is its finite zero frequency limit at the isotropic ˜ gap point ⌬ = 1. An expansion near the magnon energy gap ␻ ⬅ 2冑1 − 1 / ⌬2 yields the leading contribution as

共60兲

For convenience we introduce the dimensionless frequency ˜ = ␻ / ⍀max, where ⍀max = 2d⌬JS␣共S兲 / ប is the maximum ␻ one-magnon energy, and for m 苸兵0 , 1 , 2其 we define the functions

共g␮B兲2 ␲ ˜兲 Im ᐉ共2兲共␻ ˜ h 共d⌬兲2␻

reg * ␴xx 共␻兲 ⯝ ␴LSW ⌬2 1 −

2 ␻gap ␻2

冊

共64兲

˜ gap. The finite zero frequency value of for ␻ 艌 ␻gap ⬅ ⍀max␻ the spin conductivity for ⌬ = 1 is thus obtained as

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reg * ␴xx 共␻ → 0兲 = ␴LSW =

共g␮B兲2 ␲ . h 8

共65兲

The expansion 共63兲 for the spin conductivity within LSW in d = 2 is also shown in Fig. 3. This expansion reveals that in d = 3, as opposed to the two-dimensional case, the spin conductivity for ⌬ = 1 vanishes linearly in the zero frequency limit within LSW 共see Fig. 5兲. The unphysical divergence at the upper edge of the LSW spectrum will be cured by including magnon-magnon interactions as discussed in the next section. C. Ladder approximation: two-magnon scattering

For the calculation of G+j 共␻兲 magnon-magnon interactions are taken into account to lowest order39 by approximating the four-point vertex V␣␤ by its first-order irreducible interaction part, 共4兲 ␣␤ Vkk k k共␻⬘, ␻1兲 = Vkk k k , 1 1

1 1

共66兲

which is explicitly given by 共4兲 Vkk k

1 1k

4

=

␥k−k1 2

冉

冊

␥ k␥ k1 1 +1 − , ␧ k␧ k1 2⌬2␧k␧k1

共67兲

i.e., we neglect all the contributions to V␣␤ where two or more of the bare interactions V共i兲 are involved. Then the magnon propagators can again be replaced by the bare ex共0兲共0兲 pressions G␣␣ and G␤␤ in Eq. 共57兲 since all the first-order diagrams for the magnon self-energy vanish at T = 0.39 The algebraic solution of the coupled integral equations 共57兲 and 共58兲 is based on the decoupling of the sums over k and k1 by means of the identities

兺k sin共kx兲␥kgk = 0, 兺k sin共kx兲␥k−k gk = 1

sin共k1,x兲兺k sin2共kx兲gk , d

共68兲

FIG. 4. 共Color online兲 Regular part of the longitudinal spin conductivity within the ladder approximation for magnon-magnon in1 teractions for S = 2 in d = 2. The gap in ␴reg xx 共␻兲 increases with increasing ⌬ and vanishes for ⌬ = 1. The dotted lines represent the linear expansions around ␻gap, Eq. 共71兲.

Figures 4 and 5 show the S = 21 spin conductivity within the ladder approximation for d = 2 and d = 3, respectively. Scattering between magnons removes the divergence of the LSW result at ␻ = 2⍀max. The frequencies for the maximal spin conductivity increase with increasing anisotropy parameter ⌬. In contrast to the spin conductivity of the 3D Heisenberg antiferromagnet, which vanishes at the isotropic point ⌬ = 1 for ␻ → 0, the most notable feature of the regular part of the spin conductivity of the isotropic 2D Heisenberg antiferromagnet remains its finite value in the zero frequency limit. For ␻ 艌 ␻gap we find the following leading-order linear expansion in d = 2,

共69兲

which hold for any function gk which has the symmetry of the hypercubic lattice. We proceed by repeatedly using Eqs. 共68兲 and 共69兲 to decouple Eqs. 共57兲 and 共58兲 leading to a ladder approximation of the Bethe-Salpeter equation for the vertex part ⌫k共␻ , ␻⬘兲. This has been demonstrated in Ref. 39 for the vertex function 关cos共kx兲 − cos共ky兲兴 / 2 of the Raman B1g mode instead of the spin current vertex sin共kx兲. By virtue of the identities Eqs. 共68兲 and 共69兲 the analogous analytical steps can be performed, and we obtain for the spin conductivity reg ␴xx 共␻兲 = −

共g␮B兲2 ␲ ˜ h 共d⌬兲2␻

⫻ Im

ᐉ共2兲 − ␬共ᐉ共1兲ᐉ共1兲 − ᐉ共0兲ᐉ共2兲兲 1 + ␬共ᐉ共0兲 + ᐉ共2兲兲 − ␬2共ᐉ共1兲ᐉ共1兲 − ᐉ共0兲ᐉ共2兲兲

,

共70兲 ˜ 兲 and ␬−1 = 2dS␣共S兲. where ᐉ共i兲 ⬅ ᐉ共i兲共␻

FIG. 5. 共Color online兲 Regular part of the longitudinal spin con1 ductivity within the ladder approximation for S = 2 for the isotropic 关green 共light gray兲 line兴 and one anisotropic case 共black line兲 in d = 3. The dashed lines show the LSW results.

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冉

reg * ␴xx 共␻兲 ⯝ ␴ladder A共⌬兲 1 −

冊

␻gap , ␻

共71兲

which is also included in Fig. 4. A共⌬兲 is a numerical prefactor with A共1兲 = 1, and the zero frequency limit of the spin conductivity for ⌬ = 1 is given by reg * ␴xx 共␻ → 0兲 = ␴ladder = Y␴

Y ␴ ⬇ 1.856 851

共g␮B兲2 ␲ , h 8

共72兲

1 for S = . 2

共73兲

The renormalization factor Y ␴ can be expressed in terms of the gamma function, 1 1 − b0␬ + b20␬2 4 Y␴ = 1 1 1 − 共b0 + b2兲␬ − 共b21 − b0b2兲␬2 2 4

冋

b0 =

32␲ , 1 4 ⌫ 4

冉冊

b1 = 2 −

4 , ␲

册

冉冊

2,

共74兲

ACKNOWLEDGMENT

This work was supported by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 484.

1 4 16␲ 4 b2 = − . 1 4 4␲3 ⌫ 4 ⌫

sionality of the model is important for the low-frequency behavior of the spin conductivity, especially upon approaching the isotropic point 共⌬ = 1兲 from the gapped Ising regime 共⌬ ⬎ 1兲. In d = 3, the regular part of the spin conductivity vanishes for ⌬ = 1 in the dc limit. In d = 2, however, the regular part of the spin conductivity remains finite for ␻ → 0 at the isotropic point, which separates the Ising regime from the XY regime.48 Experimentally the spin conductivity can be determined by measurements of magnetization currents. This issue was discussed by Meier and Loss in Ref. 6, where possible experimental setups were proposed. In these setups the magnetization current is detected via the electric field which it generates. Given the estimates in Ref. 6 for the expected voltages in moderate magnetic field gradients, measurements of the spin conductivity indeed appear experimentally feasible.

APPENDIX: SPIN CURRENT VERTICES

冉冊

The spin current vertices involved in Eq. 共47兲 are given by j共1兲 = u1v2u3u4 + v1u2v3v4 ,

The inclusion of magnon-magnon interactions thus renormalizes ␴* from its value for noninteracting magnons. The reg 共␻兲 is finite in the limit ␻ → 0, however, turns fact that ␴xx out to be a robust feature of the isotropic 2D Heisenberg antiferromagnet. On the other hand, for 3D or anisotropic 2D antiferromagnets the regular part of the spin conductivity is suppressed at low frequencies.

j共2兲 = u1u2u3u4 + v1v2v3v4 + v1v2u3u4 + u1u2v3v4 , j共3兲 = 2共u1v2v3u4 + v1u2u3v4兲, j共4兲 = 2共u1u3v4u2 + u1v3u4u2 + v1u3v4v2 + v1v3u4v2兲, j共5兲 = 2共v4u3u2v1 + u4v3v2u1兲,

VI. CONCLUSIONS AND OUTLOOK

In Heisenberg magnets, spin currents flow along a magnetic-field gradient or, in the presence of spin-orbit coupling, perpendicular to a time-dependent electric field. We presented an explicit derivation of the Kubo formula for the spin conductivity for the antiferromagnetic XXZ Heisenberg model and showed that the magnetization transport arises from two-magnon processes, which provide the dominant contribution to the spin conductivity. In close analogy to the calculation of the B1g two-magnon Raman light scattering intensity, the spin conductivity was evaluated using interacting spin-wave theory. The dimen-

1

S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488 共2001兲. 2 Semiconductor Spintronics and Quantum Computation, edited by D. D. Awschalom, D. Loss, and N. Samarth 共Springer-Verlag, Berlin, 2002兲, and references therein.

j共6兲 = v4v3v2v1 + u4u3u2u1 + v4v3u2u1 + u4u3v2v1 , j共7兲 = u1v2v3v4 + v1u2u3u4 , j共8兲 = v1u2u3u4 + u1v2v3v4 , j共9兲 = v4v3u2v1 + u4u3v2u1 . The coefficients ui = uki and vi = vki with i = 1 , . . . , 4 are defined by Eq. 共32兲.

J. C. Slonczewski, Phys. Rev. B 39, 6995 共1989兲. topic of spintronics was recently reviewed by I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 共2004兲. 5 J. König, M. C. Bonsager, and A. H. MacDonald, Phys. Rev. Lett. 87, 187202 共2001兲. 6 F. Meier and D. Loss, Phys. Rev. Lett. 90, 167204 共2003兲. 3

4 The

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SPIN TRANSPORT IN HEISENBERG ANTIFERROMAGNETS… 7 M.

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