Orbital magnetization and its effect in antiferromagnets on the distorted fcc lattice

Zhigang Wang and Ping Zhang*Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, People’s Republic of China

Junren ShiInstitute of Physics, CAS, P.O. Box 603, Beijing 100080, People’s Republic of China

�Received 7 June 2007; published 12 September 2007�

We study the intrinsic orbital magnetization �OM� in antiferromagnets on the distorted face-centered-cubiclattice. The combined lattice distortion and spin frustration induce nontrivial k-space Chern invariant, whichturns to result in profound effects on the OM properties. We derive a specific relation between the OM and theHall conductivity, according to which it is found that the intrinsic OM vanishes when the electron chemicalpotential lies in the Mott gap. The distinct behavior of the intrinsic OM in the metallic and insulating regionsis shown. The Berry phase effects on the thermoelectric transport is also discussed.

DOI: 10.1103/PhysRevB.76.094406 PACS number�s�: 75.20.�g, 72.15.Jf, 75.47.�m

I. INTRODUCTION

The orbital magnetism of Bloch electrons has been anoutstanding problem in solid state physics, and attracted re-newed interest due to the recent recognition1–3 that the Berryphase effect plays a very important role in the orbital mag-netism. The issue was carried out in the powerful semiclas-sical formalism,4,5 in which the Bloch electron for nth bandis treated as a wave packet �wn�rc ,kc�� with its center �rc ,kc�in the phase space. The orbital magnetic moment character-izes the rotation of the wave packet around its centroid and is

given by mn�kc�=�−e�

2 �wn��r−rc�� v�wn�, where �−e� is thecharge of the electron and v is the velocity operator. Bywriting the wave packet in terms of the Bloch state, oneobtains �kc is abbreviated as k�

mn�k� = − i�e/2����kunk��Hk − �nk�0����kunk� , �1�

where �unk� is the periodic part of the Bloch state with band

energy �nk�0�, and Hk is the crystal Hamiltonian acting on �unk�.

Equation �1� can be alternatively derived by taking the dif-ferential of the electron energy, which within first order inthe perturbative magnetic field B turns to be �nk=�nk

�0�

−mn�k� ·B, with respect to B. It was further found1 that thepresence of a weak magnetic field B will result in a modifi-cation of the density of states in the semiclassical phasespace, d3k→d3k�1+eB ·�n /��, where �n�k�= i��kunk�� ��kunk� is the Berry curvature in k space. Due to thisweak-field modification, a quantum-state summation �kO�k�of some physical quantity O�k� should be converted to anintegral according to d3k�1+eB ·�n /��O�k�. Additionalthermodynamic average over Bloch bands should be in-cluded at finite temperature. Therefore, the total free energyfor an equilibrium ensemble of electrons in the weak fieldmay be written as1

F = −1

��

n d3k�1 +

e

�B · �n�k��ln�1 + e���−�nk�� .

�2�

where � is the electron chemical potential and �=1/kBT.The equilibrium orbital magnetization �OM� density is given

by the field derivative at fixed temperature and chemical po-

tential, M� =−��F /�B��,T, with the result

M� = �n d3kmn�k�fn

+1

��

n d3k

e

��n�k�ln�1 + e���−�nk��

Mc + M�, �3�

where fn is the local equilibrium Fermi function for nth band.In addition to the conventional term Mc in terms of the or-bital magnetic moment mn�k�, the extra term M� in Eq. �3�is a Berry phase effect and exposes a new topological ingre-dient to the orbital magnetism. Interestingly, it is this Berryphase correction that eventually enters the thermal transportcurrent.3 At zero temperature and magnetic field, the generalexpression �3� is reduced to1

M� = �n�0

d3k�mn�k� +e

��n�k���0 − �nk�� , �4�

where the upper limit means that the integral is over stateswith energies below the zero-temperature chemical potential�Fermi energy� �0.

The Berry phase effect on orbital magnetism was untilnow partially presented by very few studies. Recent observa-tion of the anomalous Nernst effect �ANE� in CuCr2Se4−xBrxcompound6 was attributed3 to the manifestation of the Berryphase effect in the OM. Also, the orbital magnetism wasrecently studied by use of two-dimensional �2D� Haldanemodel and ferromagnetic kagomé lattice with spinchirality.7,8 These two models are rare examples to show thezero-field quantum Hall effect �QHE�.9,10 From Ref. 8, onelearns that the Berry phase effect causes the OM to displaydifferent behavior in metallic and insulating regions. Thisdifference may be explained in parallel with Haldane’s recentfinding11 of the Berry phase effect in the intrinsic Hall con-ductivity �including QHE and AHE�.

The objective of the present paper is dual. First, we re-mark that the Berry phase effect on the orbital magnetism

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has been included in the well-known Kubo-Streda formula.12

Therefore, a full quantum-mechanical linear response theoryof the OM can be developed as a useful complement of thesemiclassical formalism, although the latter looks more el-egant and practical for calculation on clean samples. Thenwe examine the three-dimensional �3D� problem by studyingthe orbital magnetism in antiferromagnets on the distortedface-centered-cubic �fcc� lattice. The results reveal that ageneral topological orbital magnetism theory that takes intoaccount Berry phase effect must now be developed.

This paper is organized as follows. In the next section, weaddress that the intrinsic OM given in the semiclassical for-malism is consistent with the well-known quantum-mechanical Kubo-Streda formula in the clean-sample limit.Section III describes the physical model that is used in thiswork. The topological property and the consequent intrinsicHall effect associated with the model are also given in thissection. In Sec. IV, we present a detailed study of the prop-erties of the OM and its effects on transport response inantiferromagnets on the distorted fcc lattice. Finally, in Sec.V we present our conclusions.

II. KUBO-STREDA FORMULA OF THE ORBITALMAGNETIZATION

Due to the above mentioned modification of the density ofstates, the particle number in the weak magnetic field �say,along z axis� is given by1

N�B,�� = �n d3k�1 +

e

�B�n

z�k�� fn. �5�

It is easy to see that a link between Eq. �3� and Eq. �5� is� �Mz

���B= � �N

�B��, which is nothing but the usual thermodynamic

Maxwell relation and therefore should be free from theweak-field limit used by the semiclassical approach. Thus,the zero-field OM is given by

Mz = limB→0

� � �N�B,����B

���

d��. �6�

On the other side, the integrand in Eq. �6� can be written interms of Kubo-Streda12 formula for electrons as follows:

��xy�� = ��xyI �� − e

�N�B,���B

, �7�

where ��xy�� is the Hall conductivity and

��xyI �� = i

e2�

2 d

�f�,���

Tr�vxG+��vy� − H�

− vx� − H�vyG−��� . �8�

Here, G±���=lim�→0+��− H± i��−1 is the operator Greenfunction and v� is the velocity operator. In Bloch-state rep-resentation, the trace in Eq. �6� is equivalent to�nk�unk��¯��unk� with the Hamiltonian transformed to Hk

=eik·rHe−ik·r and the velocity to v��k�= 1�

�Hk

�k�. Replacing

�f�,��

� in Eq. �8� by −�f�,��

�� and using the completeness rela-tion of the Bloch states, �nk�unk��unk�=1, one has

��

�xyI ���d�� = − i

e2�

2lim

�→0+ df�,�� �

n,k

n�,k�

� � − �nk� − �n�k + i�

�unk�vx�k��un�k���un�k��vy�k��unk� −� − �nk�

− �n�k� − i��unk�vy�k��un�k��

��un�k��vx�k��unk�� =e2

��n,k

n�,k�

f��nk,��Im��unk�

�Hk

�kx�un�k���un�k��

�Hk

�ky�unk��

�nk − �n�k�. �9�

By use of the identity

�Hk

�k�

�unk� =��nk

�k�

�unk� + ��nk − Hk�� �unk

�k�� �10�

and after a transformation of k sum to an integral, Eq. �9� isready to be simplified as

�

��xyI ���d�� =

e2

��

n d3kfn

�Im�� �unk

�kx�Hk − �nk�

�unk

�ky�� . �11�

A comparison of Eq. �11� with Eq. �1� immediately gives thefollowing relation:

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�

��xyI ���d�� = e�

n d3kmn

z�k�fn = eMc�z�, �12�

where the second line is obtained by using the definition ofMc in Eq. �3�. Thus, one finds that the semiclassical expres-sion for Mc is equivalent to the quantum-mechanical expres-sion for �1/e����xy

I ���d��. Note that although �xyI is a

Fermi-surface term, the quantity ���xyI ���d�� is a Fermi-sea

term and all the Bloch states below � should be accountedwhen calculating the OM. On the other side, in the cleanlimit, the Kubo formula for the Hall conductivity ��xy�� canbe written in terms of the Berry curvatures �n

z ,13

��xy�� = −e2

��

n d3kfn�n

z . �13�

From Eq. �13�, one has

�

��xy���d�� = −1

�

e2

��

n d3k�n�k�ln�1 + e���−�nk�� ,

�14�

which is exactly the semiclassical expression for the Berryphase term eM�

�z� in Eq. �3�. Thus, the semiclassical OM inEq. �3� can actually be written as a Kubo-Streda formula:

Mz =1

e�

��xyI ���d�� −

1

e�

��xy���d��. �15�

The other two components Mx and My are given in a simi-lar manner. This equivalence between the semiclassical andquantum-mechanical description for the OM is only valid inthe intrinsic region and will break down when the impurityscattering effect is included. Thus, while the semiclassicalformula of the OM is more suitably employed to study theintrinsic property of the OM, the Kubo-Streda formula mustbe used when one takes into account the impurity scattering.Another aspect is that the Kubo-Streda formula is valid inarbitrary strength of the external magnetic field, while thesemiclassical formula only works in the weak-field limit. Insome special cases, for example, when one wants to knowthe edge state effect on the OM in a finite-size sample in astrong magnetic field B,14,15 a more apparent approach can beemployed by directly calculating the total free energy andthen a finite-field OM is obtained by a B derivative of thefree energy.

III. SPECIFIC MODEL AND TWO-DIMENSIONAL CHERNNUMBER

Now we focus our attention to the properties of the OM ina specific spin-frustrated system. As in Ref. 16, the model weused describes the chiral spin state in the ordered antiferro-magnet �AF� on the three-dimensional fcc lattice. The AF onthe fcc lattice is a typical frustrated system, and nontrivialtriple-Q spin structure with finite spin chirality has been re-vealed by band structure calculation17 and observed inexperiments.18,19 The anomalous behaviors in the fcc AFwere also observed. For example, there occurs mysterious

weak ferromagnetism in NiS2 below the second AF transitiontemperature.20 The Hall conductivity in this material is alsolarge and strongly temperature dependent.21 In Co�SxSe1−x�2,the AHE is enhanced in the intermediate x region, where thenontrivial magnetism is realized.22

The triple-Q spin structure on fcc lattice is shown in Fig.1.16 Here, the lattice points are divided into four sublattices

with different local spins S�a �a=1,2 ,3 ,4� on them. The AF

nature requires �aS�a=0. The minimization of the two-spinexchange interaction energy cannot uniquely determine thesublattice spin orientation. The inclusion of higher order

�four-spin exchange� interaction H4=J4�a�b�S�a ·S�b�2 withpositive J4 gives the ground-state spin configuration23,24 as

S�1= � 1�3

, 1�3

, 1�3

�, S�2= � 1�3

,− 1�3

,− 1�3

�, S�3= �− 1�3

, 1�3

,− 1�3

�, and S�4

= �− 1�3

,− 1�3

, 1�3

�, where each direction corresponds to the fourcorners from the center of a tetrahedron �Fig. 1�b��. In theinfinite strong-coupling limit, the effective Hamiltonian forthe hopping electrons strongly coupled to the mean-field ef-fective magnetic field caused by these local spins isgiven by H=�NNtij

ef f i† j with tij

ef f = t��i �� j�= teiaij cos�ij

2 .Here, the spin wave function ��i� is explicitly givenby ��i�=�cos

�i

2 ,ei�i sin�i

2�T

, where the polar coordinates are

pinned by the local spins, i.e., ��i�S� i��i�= 1

2 �sin �i cos �i , sin �i sin �i , cos �i�. �ij is the angle be-

tween the two spins S� i and S� j. The phase factor aij can beregarded as the gauge vector potential a��r�, and the corre-sponding gauge flux is related to scalar spin chirality �ijk

=S� i · �S� j �S�k�.25 In periodic crystal lattices, the nonvanishingof the gauge flux relies on the multiband structure with eachband being characterized by a Chern number. The Chernnumber appears as a result of the spin-orbit interactionand/or spin chirality in ferromagnets. In ferromagnets, thetime-reversal broken symmetry is manifest, while in AF, thetime-reversal operation combined with the translation opera-tion often constitutes the unbroken symmetry. In the lattercase, the nonzero Hall conductivity �xy is forbidden. How-ever, when there are more than two sublattices and the spinstructure is noncollinear, this combined symmetry would beabsent and finite �xy is not forbidden.16

The net spin chirality for the ideal fcc AF lattice in Fig. 1is zero, because the spin chiralities are the vector quantitiesand the sum of these four vectors is zero. However, when thelattice is distorted along the �1,1,1� direction, then the non-

FIG. 1. �a� Triple-Q spin structure on fcc lattice �Ref. 16�. �b�Relation of the four-spin moments S�a �a=1,2 ,3 ,4�.

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zero net spin chirality occurs. Following Ref. 16, we expressthe distortion along the �1,1,1� direction by putting the trans-fer integral within the �1,1,1� plane as tintra=1, while thatbetween the planes as tinter=1−d. As the unit cell is cubicshown in Fig. 1, the first Brillouin zone �BZ� is cubic: �− �

a , �a�.3 From now on, we set a=1. Then the Hamiltonian

matrix Hk for each k is given by16

Hk =�0 e−i�/6f2 ei�/6f1 f3

ei�/6f2 0 e−i�/6f3 ei2�/3f1

e−i�/6f1 ei�/6f3 0 e−i2�/3f2

f3 e−i2�/3f1 ei2�/3f2 0� , �16�

where f1=2�1−d�cos� kz

2 +kx

2�+2 cos�− kz

2 +kx

2�, f2=2�1

−d�cos� kx

2 +ky

2�+2 cos�− kx

2 +ky

2�, f3=2�1−d�cos� ky

2 +kz

2�+2 cos�

−ky

2 +kz

2�. In this Hamiltonian, the two lower bands are fully

degenerate, �1k=�2k�k�=−�f12+ f2

2+ f32, while the two upper

bands are also degenerate with �3,4k=�f12+ f2

2+ f32. The band

structure along high-symmetry lines in the first BZ is shownin Fig. 2. At d=0 �Fig. 2�a��, the upper and lower dispersionstouch along the edge of the BZ and comprises an assemblyof the massless Dirac fermions �Weyl fermions� in �2+1�D.For d�0 �elongation along �1,1,1� direction�, all the Weylfermions along the edge open a gap and turn into the massiveDirac fermions �Fig. 2�b��. Therefore, the gap fully opens inthe density of states centered at zero energy. For d�0 �sup-pression along �1,1,1� direction�, all the �2+1�D Weyl fermi-ons along the edges open the gap as in the case of d�0.However, there occurs two additional �3+1�D Weyl fermi-ons �Fig. 2�c�� at �kx ,ky ,kz�= ±D�1,1 ,1�, where D arccos� 1

d−1� and � correspond to the right- and the left-

handed chirality.26 Thus unlike d�0, the gap does not fullyopen in the case of d�0 due to the presence of a new singlecontact between the upper and lower bands in the BZ. Thenormalized eigenvectors are given by

�u1k� =1�2�−

f1�1k + if2f3

�1k�f1

2 + f22

, e−i�/3 f2�1k − if1f3

�1k�f1

2 + f22

, 0,

−�f1

2 + f22

�1k�T

,

�u2k� =1�2

�ei�/6 f2

�2k, e−i�/6 f1

�2k, 1, iei�/6 f3

�2k�T

,

�u3k� =1�2� f1�3k + if2f3

�3k�f1

2 + f22

,

− e−i�/3 f2�3k − if3f1

�3k�f1

2 + f22

, 0,�f1

2 + f22

�3k�T

,

�u4k� =1�2

�ei�/6 f2

�4k, e−i�/6 f1

�4k, 1, iei�/6 f3

�4k�T

.

�17�

The Berry curvatures for these four Bloch bands are derivedto have the form

�n��k� =

F��

2�nk3 �18�

with

F�� = f1�f2

�k�

�f3

�k�

+ f3�f1

�k�

�f2

�k�

− f2�f1

�k�

�f3

�k�

, �19�

where �� ,� ,�� represent a cyclic permutation of �x ,y ,z�.Let us see the Hall conductivity of this system16 with d

�0. In the integer filling case, the zero-temperature Hallconductivity is a sum of Chern invariant27 over occupiedBloch bands,

�xy = �e2/h��n

occu �−�:��

dkz

2�Cn�kz� , �20�

where the 2D Chern number28 Cn�kz� is given by

Cn�kz� = −1

2�

�− �:��2dkxdky�n

z�k�

= −1

2�

�− �:��2dkxdkyz · ��k � An�k�� . �21�

Here, An�k�= i�unk ��kunk� is the Berry phase connection�vector potential� for nth band. To proceed one may firsttransform the integral of �k�An over the first BZ to the lineintegral of An along the BZ boundary by use of Stokes’theorem, and then apply the complex contour integrationtechnique and residue theorem to sinusoidal functions. Aftera straightforward derivation, one obtains the nonzero Chernnumber, C1�kz�=−sgn�g�kz��, C3=sgn�g�kz��, where

FIG. 2. �Color online� Energy spectrum along high-symmetry lines in the first BZ for �a� d=0, �b� d=0.2, and �c� d=−0.2.

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g�kz� = 2 + 2�1 − d�cos�kz + 2kP� �22�

and kP=arctan� �d−1�cos kz−1

�d−1�sin kz�. It is easy to verify that for d

�0, the value of g�kz� is always positive, independent of kz.For d�0, the sign of g�kz� depends on kz in such a way thatg�kz��0 for kz� �−D ,D� and g�kz��0 for kz� �−� ,−D�� �D ,��. Note that the present choice of the other twoBloch states �u2k� and �u4k� makes them to have no contribu-tion to the Chern number.

However, the above purely mathematical calculation of2D Chern number is not favored by theoretical physicists,who would like to resort to the physical connotation that thevector potential An and gauge flux �n are endowed with.Correspondingly, here we present this gauge-field analysis ofthe lower band �1k as an example. The value of 2D Chernnumber C1�kz�, which is confined to the �kx ,ky� subspace atfixed kz, is invariant under gauge transformation �u1k� �=ei�1�k��u1k�, A1��k�=A1�k�−�k�1�k�, where �1�k� is an ar-bitrary smooth function of k. If the gauge choice for �u1k� iswell defined everywhere in the whole �kx ,ky� subspace in thefirst BZ, then its Chern number C1�kz� will obviously bezero. However, at point k0= �kz+2kP ,kz ,kz�, one can find thatthe wave function �u1k� in Eq. �17� is ill defined since both itsdenominator and numerator are zero at this point. This meansthat the used gauge cannot apply to the whole BZ and oneneeds to render a gauge transformation to avoid the singular-ity at k0. For this, one transforms the wave function to

�u1k� � =1�2�−

f2�1k + if1f3

�1k�f2

2 + f32

, e−i�/3 f3�1k − if1f2

�1k�f2

2 + f32

, 0,

−�f2

2 + f32

�1k�T

. �23�

The new eigenvector �u1k� � recovers the well-defined behav-ior at k0; the new singularity brought about is at k0�= �kz ,kz

+2kP ,kz�. Thus according to the two different gauge choices,the BZ cross section at fixed kz is now divided into tworegions V and V�, as shown in Fig. 3 �kz=0�. The wavefunctions �u1k� are used onto the region V, while �u1k� � applyto V�. Note that there remains some freedom in the divisionof the BZ. Because �u1k� and �u1k� � are ill defined only at k0

and k0�, respectively, we are free to deform this division aslong as k0�k0���V�V��. This corresponds to the gauge degreeof freedom.29,30 At k�V�V�, the two choices of wave func-tions are different by a phase factor �u1k� �=ei�1�k��u1k�, i.e.,A1��k�=A1�k�−�k�1�k�, where

ei�1�k� =� f22 + f1

2

f32 + f2

2

f2�1k + if3f1

f1�1k + if2f3. �24�

Thus, one obtains the value of nonzero 2D Chern number forlower band �1�k� as follows:

C1�kz� = −1

2��

�V

�A1�k� − A1��k�� · dk

= −1

2��

�V

d�1�k� = − sgn�g�kz�� , �25�

which is consistent with the explicit calculation based on thecomplex-contour integration technique.

Consider the �=0 case, i.e., the two lower degeneratebands are fully filled while the two upper degenerate bandsare empty. Then a further kz integral of C1�kz� gives the Hallconductivity �xy =− e2

h for d�0 and �xy = e2

h� 2D

� −1� for d�0.The asymmetry of �xy between d�0 and d�0 will be ex-plained below together with the behavior of the OM. When

the local spins �S� i� are inverted �which means that the spinchirality is also inverted�, then the Hall conductivity changesits sign.16

IV. ORBITAL MAGNETIZATION AND ITS EFFECTS

Now we turn to study the OM and its various effects.Without loss of generality, the present attention is only on thez component of the OM which is connected with Hall con-ductivity �xy. First, after a straightforward derivation, oneobtains the k-space orbital magnetic moment as follows:

mnz�k� =

Fxy

2�n2�k�

. �26�

See Eq. �19� for Fxy. One can see that the orbital magneticmoment is identical for upper and lower bands, while theBerry curvatures �Eq. �18�� for upper and lower bands differby a sign. A comparison between Eqs. �26� and �18� gives aninteresting relation for the present model

mnz�k� = �n

z�k��nk. �27�

Given the expressions for mnz�k� and �n

z�k�, the OM Mz cannow be systematically studied. At zero temperature, in par-ticular, by substituting Eq. �27� into Eq. �4� one finds animportant relation

Mz =e

��0�

n�0

d3k�nz�k� = −

�0

e�xy , �28�

which indicates that the OM is proportional to the Hall con-ductivity with coefficient �−�0 /e�. Equation �28� also holdsat low temperature. Although this remarkable relation be-

FIG. 3. �Color online� Division of cross section �kz=0� of thefirst BZ into two regions V �red area� and V� �blue area�. Note thatthe projection of the singularities k0 and k0� onto �kx ,ky� subspacevary with kz.

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tween the OM and the Hall conductivity is specific to thepresent Hamiltonian model, it definitely tells one that thetopological ingredient in the OM may be faithfully mappedout through the Hall conductivity. If the band is partiallyfilled, then after integrating by parts one finds that Eq. �28�can be written as a pure Fermi-surface integral. In the integerfilling case, on the other hand, the OM and Hall conductivitydisplay a Fermi-sea feature. Figure 4 shows Mz and �xy as afunction of the distortion d ��xy has been previously plottedin Ref. 16�. The d�0 and d�0 cases are asymmetric by theobservation that �xy is quantized for d�0 while nonquan-tized for d�0. To understand this asymmetry, one may treatthe distortion d as a control parameter of the 3D band struc-ture and start from d=0, at which the upper and lower bandsare degenerate along the BZ edge, i.e., �kx= ±� ,ky = ±� ,kz�,�kx= ,ky = ±� ,kz= ±��, �kx= ±� ,ky ,kz= ±��. When d is var-ied from d=0 to d�0, then these initial degenerate pointscompletely split into two groups of “Dirac point” singulari-ties, which play the role of positive and negative monopolesources, respectively �see, as an example, k0 and k0� points inFig. 2�. The upper and lower bands are now tightly coupledby a series of “Berry flux loops.” Along each loop Berrycurvature flux 2� passes from the lower bands to the upperbands through one Dirac point, then returns through the othercorresponding one. The positive �negative� monopoles of thelower bands and the negative �positive� monopoles of theupper bands may recombine by a relative displacement of aprimitive reciprocal lattice vector G. In the present cubic

lattice, the G is along one selective k� axis ��=x ,y ,z� withamplitude 2� �the lattice constant a has been scaled to beunity�. This means that during the Dirac point splitting pro-cess, the individual Chern invariant �the 3D generalization of2D Chern number� for the lower and upper bands changes by�G, respectively, while their sum is conserved. As a result,the Hall conductivity �xy for lower/upper bands is quantizedto be a product of e2

h and the following Chern invariant:

1

2�

�− �:��3d3k�n

z�k� = −1

2�CnGz. �29�

When d is varied from d=0 to d�0, Eq. �29� breaks down,since the 2D Chern number Cn is now k dependent by thepresence of the additional two �3+1�D Weyl fermions at�kx ,ky ,kz�= ±D�1,1 ,1�. In this case, although the gap is

opened at Dirac points along the BZ edge, the emergence ofnew Weyl fermions contributes nonzero density of states inthe gap. It is then straightforward to repeat the integration inEq. �29� by parts to expose the nonquantized part of the 3Dintrinsic Hall conductivity as a Fermi-surface property.

In the half filled case, i.e., when the chemical potential isin the Mott gap, the Hall conductivity vanishes due to thecancellation of the Chern invariants of the upper and lowerbands. As a consequence, the OM in Eq. �29� also vanishes.This result is prominently different from that in Ref. 16, inwhich a finite OM with amplitude smoothly varied with dis-tortion d was given via a tight-binding calculation. Reference16 employed the Kubo-Streda formula in calculating theOM. As we have shown above, however, the Kubo-Stredaformula and the semiclassical formula give the same expres-sion for the intrinsic OM. So no discrepancy is expected tooccur between the two approaches. The most possible expla-nation is the different treatments of local Hund’s exchangeinteraction �between the conduction electrons and the localspins� in our present derivation and in the numerical calcu-lations in Ref. 16. Here, the exchange interaction has beenassumed infinite, which completely polarize the spins of theconduction electrons and results in the description of theconduction electrons in terms of the spinless fermions andthe effective four-band Hamiltonian, Eq. �16�, while in Ref.16 the exchange interaction was taken to be finite in calcu-lating the OM. Thus the spin degrees of freedom have beenaccounted for in Ref. 16 and an eight-band model has beennumerically solved. Due to the presence of the finite mean-field exchange field, the lower four Hubbard bands below theMott gap may no longer be featured by the two degeneratedoublets, which is essential for the present derivation of Eq.�28�. From this aspect, by tuning the exchange field morestronger �with respect to the hopping integral tij� to approachinfinity, the finite OM found in Ref. 16 is expected to begreatly suppressed and tend to vanish according to Eq. �28�in the Mott spin-gap region.

Figure 5�a� shows the Mz as a function of the electronchemical potential � for the distortion d=0.2. One can seethat initially the OM rapidly decreases as the filling of thelower bands increases, arriving at a minimum at �=−0.4, avalue corresponding to the top of the lower band. Then, asthe chemical potential continues to vary in the gap �shadedregion in Fig. 5�a�� between the lower and upper bands, the

FIG. 4. �a� Zero-temperature orbital magnetization Mz and �b�Hall conductivity �xy �Ref. 16� as a function of distortion d. TheFermi energy in �a� is chosen to be �0=0.1, while in �b� the Fermienergy is chosen to be �0=0.0 which corresponds to case that onlythe lower bands are fully occupied.

FIG. 5. �Color online� �a� Orbital magnetization Mz and �b� itstwo components Mc

z �red curve� and M�z �blue curve� as a function

of the electron chemical potential � for distortion d=0.2. Theshaded area is the gap between the lower and upper bands. Tosuppress the divergence at band and/or gap contact, we have usedthe temperature of kBT=0.05.

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OM goes up and increases as a linear function of �. Thislinear relationship in the insulating region is explicit fromEq. �28�. When the chemical potential touches the bottom ofthe upper band, then the linear increase in Mz suddenlystops and the OM rapidly decreases again by the chemicalpotential going through the upper bands. The turning behav-ior at the band and/or gap contacts becomes numerically di-vergent at kBT=0. This discontinuity is due to the singularbehavior of �n�k� at the BZ edge point k=k0, which willplay its role when the k integral is over the entire BZ.

The distinct behavior of the OM in the metallic and insu-lating regions, as shown in Fig. 5�a�, reflects the differentroles that its two components Mc

z and M�z play in these two

regions. To see this, we show in Fig. 5�b� Mcz �blue curve�

and M�z �red curve� as a function of �. One can see that Mc

z

and M�z oppose each other, which implies that they are car-

ried by opposite-circulating currents. Also one can see that inthe band insulating regime, the conventional term Mc

z keeps aconstant during variation of �. This behavior is due to thefact that the upper limit of the k integral of mn�k� is invariantas the chemical potential varies in the band gap. In the me-tallic region, however, since the occupied states vary with thechemical potential �, thus Mc also varies with �, resulting ina decreasing slope shown in Fig. 5�b�. The Berry phase termM� also displays different features in the insulating and me-tallic regions. In the insulating region, M� linearly increaseswith �, as is expected from Eq. �28�. In the metallic region,however, this term keeps a constant with the amplitude sen-sitively depending on the topological property of the band inwhich the chemical potential is located. On the whole, itreveals in Fig. 3 that the metallic behavior of the OM isdominated by its conventional term Mc, while in the insulat-ing regime the Berry phase term M� comes to play a mainrole in determining the behavior of the OM.

The above separate discussion of Mc and M� can betransferred to study the ANE. The relation between the OMand ANE has been recently found.3 To discuss the transportmeasurement, it is important to discount the contributionfrom the magnetization current, a point which has attractedmuch discussion in the past. Cooper et al.31 have argued thatthe magnetization current cannot be measured by conven-tional transport experiments. Xiao et al.3 have adopted thispoint and built up a remarkable picture that the conventionalorbital magnetic moment Mc does not contribute to the trans-port current, while the Berry phase term in Eq. �3� directlyenters and therefore modifies the intrinsic Hall current asfollows:3

jH = −e2

�E � �

n d3k

�2��3 fn�r,k��n�k� − � � M��r� .

�30�

In the case of uniform temperature and chemical potential,obviously, the second term is zero and the Hall effect of thedistorted fcc lattice is featured by the first term in Eq. �30�. Inthe following, however, we turn to study another situation,where the driving force is not provided by the electric field.Instead, it is provided by a statistical force, i.e., the gradientof temperature T. In this case, Eqs. �30� and �3� give the

expression of intrinsic thermoelectric Hall current as jx=�xy�−�yT�, where the anomalous Nernst conductivity �xy isgiven by3

�xy =1

T

e

��

n d3k

�2��3�n

���nk − ��fn + kBT ln�1 + e−��nk−���� . �31�

Figure 6 shows �xy of the distorted fcc lattice as a func-tion of the chemical potential for d=0.2 and kBT=0.05. Onecan see that the ANE disappears in the insulating regions,and when scanning � through the contacts between the bandsand gaps, there will appear peaks and valleys. Remarkably, asimilar peak-valley structure was also found by the recentfirst-principles calculations in CuCr2Se4 compound.3 TheANE of this compound was recently measured by Lee et al.6

as a function of Br doping which substitutes Se in the com-pound and changes the chemical potential �. Due to thescarce data available, until now the peak-valley structure of�xy revealed in Fig. 6 and in Ref. 3 has not been found inexperiment, and more direct experimental results are neededfor quantitative comparison with the theoretical results. In-terestingly, the expression for �xy can be simplified at lowtemperature as the Mott relation,3

�xy = −�2

3

kB2T

e

��xy��0���0

. �32�

Thus, one can see that the low-temperature nonzero ANE is aFermi-surface Berry phase effect. Another unique feature of�xy is its linear dependence of temperature.

V. CONCLUSION

In summary, after pointing out the equivalence of thesemiclassical approach and the quantum Kubo-Streda for-mula in description of the intrinsic orbital magnetization, wehave theoretically studied the properties of the OM in anti-ferromagnets on the distorted 3D fcc lattice. The distortionparameter d in the fcc lattice produces nonzero 2D Chernnumber and results in profound effects on the OM properties.An explicit relation between the OM and the Hall conductiv-ity in this system has been derived. According to this relationwe have found that the OM vanishes when the electronchemical potential lies in the Mott gap, which is in contrast

FIG. 6. The intrinsic anomalous Nernst conductivity �xy as afunction of the electron chemical potential � for d=0.2 and kBT=0.05. The shaded area is the gap between the lower and upperbands.

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with the results in Ref. 16. We have shown that the two partsMc and M� in the OM oppose each other and yield theparamagnetic and diamagnetic responses, respectively. Inparticular, due to its Fermi-sea topological property, the mag-netic susceptibility of M� remains to be a nonzero constantwhen the Fermi energy is located in the energy gap. It hasbeen further shown that the OM displays distinct behavior inthe metallic and Chern-insulating regions, because of differ-ent roles Mc and M� play in these two regions. The anoma-

lous Nernst conductivity has been studied, which displays apeak-valley structure as a function of the electron chemicalpotential. We expect that these results will be experimentallyverified in the spin-frustrated systems.

ACKNOWLEDGMENT

This work was supported by NSFC under Grants Nos.10604010 and 10544004.

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