Axionic Antiferromagnetic Insulator Phase in a Correlated and Spin-Orbit Coupled System

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by  Akihiko Sekine

JournalofthePhysicalSocietyofJapan FULLPAPERS
AxionicAntiferromagneticInsulatorPhaseinaCorrelatedandSpin-OrbitCoupled
System
AkihikoSekine∗andKentaroNomura
InstituteforMaterialsResearch,TohokuUniversity,Sendai980-8577,Japan
We study theoretically a three-dimensional correlated and spin-orbit coupled system, the half-filled extended Fu-
Kane-Mele-Hubbardmodelonadiamondlattice,focusingonthetopologicalmagnetoelectricresponseoftheantifer-
romagneticinsulatorphase.Intheantiferromagneticinsulatorphase,theDirac-likelow-energyeffectiveHamiltonianis
obtained.Thenthethetaterm,whichresultsinthemagnetoelectricresponse,isderivedasaconsequenceofthechiral
4 anomaly.Therealizationofthedynamicalaxionfieldinourmodelisdiscussed.Therelationwithasymmetrybroken
1 phaseinducedbyinteractionsinlatticequantumchromodynamicsisalsodiscussed.
0
2

p
e
S 1. Introduction accordingtotheformula2
4 A great number of studies on topological insulators have 1 (cid:90) (cid:34) 2 (cid:35)
2 beendonesincethepioneeringworks1appeared,inthesearch θ= 4π d3k(cid:15)ijkTr Ai∂jAk−i3AiAjAk , (2)
BZ
for novel phenomena due to the topological properties of
l] the system. The most prominent feature common to two- where Aµjν = i(cid:104)uµ|∂/∂kj|uν(cid:105), and |uν(cid:105) is the periodic Bloch
e dimensional (2D) and 3D topological insulators is the exis- functionwithνtheoccupiedbands.Wecancalculateθfrom
-
r tenceoftheedge(surface)stateswhichareprotectedbytime- otherequivalentexpressions.10–12 However,sometechniques
t
s reversalsymmetry.Theseedgeorsurfacestatesareknownto (such as choosing a gauge for A) are required to calculate
t. be robust against perturbations. On the one hand, one of the numerically.Insystemswherethesingle-particleHamiltonian
ma noteworthycharacterspeculiarto3Dtopologicalinsulatorsis can be described as H(k) = (cid:80)5µ=1Rµ(k)αµ with matrices αµ
the topological magnetoelectric effect which is described by satisfyingtheCliffordalgebra{αµ,αν}=2δµν,thereexistsan
d- thethetaterm.2Thethetatermiswrittenas explicitexpressionforθ:8,13
con Sθ =(cid:90) dtd3x4πθe2(cid:126)2cE·B, (1) θ= 41π(cid:90)BZd3k(|R2||+R|R+4)R2|4R|3(cid:15)ijklRi∂∂Rkxj∂∂Rkyk ∂∂Rkzl, (3)
[ (cid:113)
where E and B are an electric field and magnetic field, re- wherei, j,k,l = 1,2,3,5and|R| = (cid:80)5 R2.Herenotethat
3 spectively. From this action, we obtain the cross-correlated µ=1 µ
v responses expressed by P = θe2/(4π2(cid:126)c)B and M = only the matrix α4 is even under time-reversal. In this work,
3 θe2/(4π2(cid:126)c)E, with P the electric polarization and M the we derive an analytical expression for θ in a time-reversal
2 symmetry broken phase with the use of a field-theoretical
magnetization.
5
method.
4 In the field theory literature, the action (1) is termed the
. axion electrodynamics.3 The axion is an elementary particle The spin-orbit interaction has been revealed to be impor-
1
tant to realize topologically nontrivial phases. On the other
40 pCrPopporosebdleamboinutqfuoarntytuymeacrhsroamgoodtoynsaomlviecsth(eQCsoD-c)a.4l–l6edBsytrsounbg- hand,theeffectsoftheelectron-electroninteractionhasbeen
a central subject in modern condensed matter physics. Now,
1 sequent studies, the axion is now considered to be essential
: the interplay of spin-orbit coupling and electron correlation
v to explain experimental results in particle physics and as-
i trophysics.7 The axion is also considered as a candidate for isahottopic.Oneofthetriggersisthediscoveryofanovel
X dark matter.7 However, regardless of intensive experimental Mottinsulatingstateinacorrelated5delectronsystem,which
r revealed that the Mott insulating state is induced by strong
searches,theaxionhasnotyetbeenfound.Theaxionsinter-
a
spin-orbitcoupling.14,15Recentstudieshaveshownthatnovel
actwithphotons,andtheaxion-photoncouplingisdescribed
phases,suchasthequantumspinHallinsulator,16thetopolog-
byEq.(1)withθ beingtheaxionfield.Therefore,observing
ical Mott insulator,17 the topological magnetic insulator,8,13
themagnetoelectricresponsesoriginatingfromEq.(1)incon-
the Weyl semimetal,18 and the phase which is a condensed-
densedmatterisequivalenttorealizingthe(dynamical)axion
matteranalogofaphaseinlatticeQCD,19 emergebythein-
field.Therehavebeensometheoreticalstudieswhichpropose
terplayofspin-orbitcouplingandelectroncorrelation.
ways to observe experimentally the dynamical axion field in
condensedmatter.8,9 Electron correlation effects in topological insulators have
also been investigated intensively.20 The Kane-Mele model
When the system is time-reversal invariant, the condition
thatθ=π(mod2π)isimposedfor3Dtopologicalinsulators, on the honeycomb lattice is well known as a model which
and θ = 0 for normal insulators. On the other hand, when describesa2Dtopologicalinsulator.1 TheKane-Melemodel
with on-site interaction, the Kane-Mele-Hubbard model, has
time-reversalsymmetryofthesystemisbroken,thevalueof
been one of the most investigated system so far. In this sys-
θcanbearbitrary.Ingeneral,thevalueofθcanbecalculated
tem,theantiferromagneticinsulatorphasedevelopsinthere-
gion where the on-site interaction strength U is strong.21–26
∗E-mail:[email protected]
1
J.Phys.Soc.Jpn. FULLPAPERS
When the strength U is intermediate, it has been shown that
the spin liquid phase emerges22–26 and pointed out the pos-
sibility of the fractional topological insulator phase.21 In an-
othermodelofa2Dtopologicalinsulatorwithon-siteinterac-
tion, the Bernevig-Hughes-Zhang-Hubbard model, the exis-
tenceofthetopologicalantiferromagneticinsulatorphasehas
been pointed out.27 On the other hand, in the case of three-
dimensions,theFu-Kane-Melemodelonthediamondlattice,
the3DanalogoftheKane-Melemodel,isknownasamodel
for a 3D topological insulator.28,29 What is the properties
of an interacting Fu-Kane-Mele model, the Fu-Kane-Mele-
Fig. 1. (Coloronline)(a)Adiamondlattice,whichconsistsoftwosub-
Hubbardmodel?Sofartherehasbeennostudyonthismodel, lattices(redandblue),andeachsublatticeformsafcclattice.(b)Thefirst
althoughinterestingphenomenaareexpectedtoemerge. Brillouinzoneofafcclattice.GreencirclesrepresenttheXpoints.
Inthispaper,wefocusonthetopologicalmagnetoelectric
response of the antiferromagnetic insulator phase in the ex-
tended Fu-Kane-Mele-Hubbard model on a diamond lattice
athalf-filling,withinthemean-fieldapproximation.Thispa- ces. The diamond lattice consists of two sublattices (A and
per is organized as follows. In Sec. 2, the model we adopt B),witheachsublatticeformingafcclattice.Insuchacase,
is explained. We take into account the on-site and nearest- we can define the basis c ≡ [c ,c ,c ,c ]T where
k kA↑ kA↓ kB↑ kB↓
neighbor repulsive electron-electron interactions. In Sec. 3, thewavevectork isgivenbythepointsinthefirstBrillouin
themean-fieldphasediagramispresented.InSec.4,weob- zoneofthefcclattice[seeFig.1(b)].Thenthesingle-particle
tainanalyticallythevalueofθintheantiferromagneticinsula- HamiltonianH (k)[H ≡(cid:80) c†H (k)c ]iswrittenas28,29
0 0 k k 0 k
torphase.FirstweshowthatwecanderivetheDiracHamil-
tonian in the antiferromagnetic insulator phase. Then based (cid:88)5
H (k)= R (k)α , (6)
on the Fujikawa’s method,30,31 we obtain the theta term as 0 µ µ
µ=1
a consequence of the chiral anomaly. In Sec. 5, we discuss
therealizationofthedynamicalaxionfieldinourmodel.we wherethecoefficientsRµ(k)aregivenby
alsodiscusstherelationbetweenourantiferromagneticinsu- R (k)=λ[sinu −sinu −sin(u −u )+sin(u −u )],
1 2 3 2 1 3 1
latorphaseandtheso-called“Aokiphase”,asymmetrybro-
kenphaseinducedbyinteractionsinlatticeQCD.32 R2(k)=λ[sinu3−sinu1−sin(u3−u2)+sin(u1−u2)],
R (k)=λ[sinu −sinu −sin(u −u )+sin(u −u )], (7)
2. Model 3 1 2 1 3 2 3
Letusconsidera3Dlatticemodelwithelectroncorrelation R4(k)=t+δt1+t(cosu1+cosu2+cosu3),
andspin-orbitcoupling.Themodelweadoptistheextended R (k)=t(sinu +sinu +sinu ).
5 1 2 3
Fu-Kane-Mele-Hubbard model on a diamond lattice at half-
filling, in which the Hamiltonian is given by H = H0 +Hint Here u1 = k · a1, u2 = k · a2, and u3 = k · a3 with
withthenon-interactingpart a1 = 2a(0,1,1),a2 = 2a(1,0,1)anda3 = 2a(1,1,0)beingthe
primitive translation vectors. In the following, we set a = 1.
(cid:88) 4λ (cid:88)
H0 = tijc†iσcjσ+ia2 c†iσ·(d1ij×d2ij)cj, (4) Thealphamatricesαµaregivenbythechiralrepresentation:
(cid:104)i,j(cid:105),σ (cid:104)(cid:104)i,j(cid:105)(cid:105) (cid:34)σ 0 (cid:35) (cid:34)0 1(cid:35) (cid:34)0 −i(cid:35)
α = j , α = , α = , (8)
andtheinteractionpart j 0 −σ 4 1 0 5 i 0
j
(cid:88) (cid:88)
H =U n n + V nn , where j = 1,2,3. In the present basis, the time-reversal op-
int i↑ i↓ ij i j (5)
i (cid:104)i,j(cid:105) erator and spatial inversion (parity) operator are given by
T = 1⊗(−iσ )K (K is the complex conjugation operator)
where c†iσ is an electron creation operator at a site i with andP=τ ⊗1,2respectively.Wehaveintroducedthehopping
spin σ(=↑,↓), n = c† c , n = n +n , and a is the lat- 1
iσ iσ iσ i i↑ i↓ strength anisotropy δt1 due to the lattice distortion along the
tice constant of the fcc lattice. The first and second terms [111] direction. Namely, we have set such that t = t +δt
ij 1
of H0 represent the nearest-neighbor hopping and the next- for the [111] direction, and t = t for the other three di-
ij
nearest-neighborspin-orbitcoupling,respectively.d1ijandd2ij rections. When δt1 = 0, the system is a semimetal, i.e., the
are the two vectors which connect two sites i and j of the energy bands touch at the three points Xr = 2π(δ ,δ ,δ )
rx ry rz
same sublattice. They are given by two of the four nearest- (r= x,y,z).Finiteδt opensagapof2|δt |attheXr points.
1 1
neighbor vectors, a(1,1,1), a(−1,−1,1), a(1,−1,−1), and
4 4 4 TheZ2invariantofthesystemisgivenby
a(−1,1,−1), with proper signs (directions of the vectors).
oσ4f f=ree(σdo1,mσ.2T,σhe3)fiarrset athnedPsaeucolinmdatetrrimcessofforHthedsepsicnridbeegtrheee (−1)ν0 =(cid:89)8 sgnt+δt1+t(cid:88)3 cos(cid:16)Γi·ap(cid:17), (9)
int i=1 p=1
on-siteandnearest-neighborrepulsiveelectron-electroninter-
actions,respectively.Thelatticestructureofadiamondlattice where Γi are the eight time-reversal invariant momenta:
isshowninFig.1(a). (0,0,0), (2π,0,0), (0,2π,0), (0,0,2π), (π,π,π), (π,π,−π),
It is convenient to express the non-interacting part H of (π,−π,π),and(−π,π,π).Weseethatthesystemisatopologi-
0
the Hamiltonian in terms of the 4×4 alpha (gamma) matri- calinsulator(normalinsulator)when0<δt1 <2t(δt1 <0or
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δt >2t).Notethatinthispaperwedonotdistinguishaweak Mele-Hubbardmodelonahoneycomblattice.21Herewehave
1
topologicalinsulatorfromanormalinsulator. omittedirrelevantconstanttermsinEqs.(14)and(15).
LetuslookatcloselyH (k)aroundthe Xr points.Setting Secondly,weapproximatethenearest-neighborinteraction
0
k = Xr +q and retaining the terms up to the order of q, we H =(cid:80) V nn as
V (cid:104)i,j(cid:105) ij i j
obtain the low-energy effective Hamiltonian near the Fermi (cid:88)(cid:88) (cid:104)
levelaroundeachXpoint:28,29 HV ≈− Vij (cid:104)c†iσcjσ(cid:48)(cid:105)c†jσ(cid:48)ciσ+(cid:104)c†jσ(cid:48)ciσ(cid:105)c†iσcjσ(cid:48)
(cid:104)i,j(cid:105)σ,σ(cid:48) (16)
H0(Xx+q)=tqxα5+2λqyα2−2λqzα3+δt1α4, (cid:105)
−(cid:104)c† c (cid:105)(cid:104)c† c (cid:105) .
H (Xy+q)=tq α +2λq α −2λq α +δt α , (10) iσ jσ(cid:48) jσ(cid:48) iσ
0 y 5 z 3 x 1 1 4
H (Xz+q)=tq α +2λq α −2λq α +δt α . Weassumethatthevaluesof(cid:104)c†iσcjσ(cid:48)(cid:105)dependonthehopping
0 z 5 x 1 y 2 1 4 strength,namelyweset(cid:104)c† c (cid:105) = −∆δ t /t.Ontheother
Theseareso-calledtheDiracHamiltonian.Forexample,the iσ jσ(cid:48) σσ(cid:48) ij
hand,weneglecttheinteractionstrengthanisotropyduetothe
energyspectrumaroundtheXx pointisreadilyobtainedas latticedistortionforsimplicity,i.e.,wesetV =V.Thisdoes
ij
(cid:113) not change the resultingphase diagram qualitatively. After a
E(Xx+q)=± (tq )2+(2λq )2+(2λq )2+(δt )2. (11)
x y z 1 calculation,weobtain
Asmentionedabove,weseethatthesystemisgaplesswhen HMF =2NV(cid:104)3+(1+δt /t)2(cid:105)∆2+V∆/t (cid:88) t c† c .
δt = 0andnonzeroδt isregardedasthemassoftheDirac V 1 ij iσ jσ
1 1 (cid:104)i,j(cid:105),σ
quasiparticles.AteachXr point,oneofthethreecomponents
(17)
which originate from spin-orbit coupling R (k) disappears
r
and instead R (k) compensates for the q -dependence of the FinallycombiningEqs.(6),(14),(15),and(17),themean-
5 r
effectiveHamiltonian. fieldHamiltonianofthesystemisgivenby
(cid:104) (cid:105)
HMF =2NUm2+2NV 3+(1+δt /t)2 ∆2
3. Mean-fieldphasediagram SDW 1
 
fielSdpianp-dperonxsiitmyawtiaovnetionstthaebiliintyte.—racLtieotnutserpmerfaonrmd dtheerivmeeathne- +(cid:88)c†k(cid:88)5 R˜µ(k)αµck, (18)
mean-field Hamiltonian of the system. First we consider the k µ=1
spin-densitywave(SDW)instability.Todothis,wefirstlyap- whereR˜ (k)=R (k)−Um ,R˜ (k)=R (k)−Um ,R˜ (k)=
proximatetheon-siteinteractionHU =U(cid:80)ini↑ni↓as R3(k)−1Um3, R˜41(k) = (1+1 V∆2/t)R4(k2), and R˜5(k2) =3 (1+
HU ≈U(cid:88)(cid:2)(cid:104)ni↓(cid:105)ni↑+(cid:104)ni↑(cid:105)ni↓−(cid:104)ni↑(cid:105)(cid:104)ni↓(cid:105) V∆/t)R5(k).Notethatm21+m22+m23 =m2.Thefreeenergyat
zero temperature for the SDW instability is readily obtained
i (12)
(cid:105) as
−(cid:104)c†c (cid:105)c†c −(cid:104)c†c (cid:105)c†c +(cid:104)c†c (cid:105)(cid:104)c†c (cid:105) . (cid:104) (cid:105)
i↑ i↓ i↓ i↑ i↓ i↑ i↑ i↓ i↑ i↓ i↓ i↑ F (m,θ,ϕ,∆)=2NUm2+2NV 3+(1+δt /t)2 ∆2
SDW 1
Due to the spin-orbit coupling, the spin SU(2) symmetry is
(cid:114)
brokenandtheorientationsofthespinsarecoupledtothelat- −2(cid:88) (cid:88)5 (cid:104)R˜ (k)(cid:105)2.
ticestructure.Weassumetheantiferromagneticorderingbe- µ=1 µ
k
tweenthetwosublatticesintermsofthesphericalcoordinate (19)
(m,θ,ϕ):
Charge-density wave instability.— Next we consider the
(cid:104)Si(cid:48)A(cid:105)=−(cid:104)Si(cid:48)B(cid:105)=(msinθcosϕ,msinθsinϕ,mcosθ) charge-density wave (CDW) instability. To do this, we ap-
(13)
≡m e +m e +m e , proximatetheinteractiontermsHU andHV as
1 x 2 y 3 z
(cid:88)
where(cid:104)S (cid:105)= 1(cid:104)c† σ c (cid:105)(µ= A,B)withi(cid:48)denotingthe HU ≈U (cid:2)(cid:104)ni↓(cid:105)ni↑+(cid:104)ni↑(cid:105)ni↓−(cid:104)ni↑(cid:105)(cid:104)ni↓(cid:105)(cid:3), (20)
i(cid:48)µ 2 i(cid:48)µα αβ i(cid:48)µβ
i(cid:48)-thunitcell.Thenafteracalculation,weobtain i
(cid:88)(cid:104) (cid:105) (cid:88) 
U −(cid:104)ni↑(cid:105)(cid:104)ni↓(cid:105)+(cid:104)c†i↑ci↓(cid:105)(cid:104)c†i↓ci↑(cid:105) =2NU m2f, (14) H ≈(cid:88)V (cid:104)n(cid:105)n +(cid:104)n (cid:105)n −(cid:104)n(cid:105)(cid:104)n (cid:105)−(cid:88)(cid:104)(cid:104)c† c (cid:105)c† c
i f V ij i j j i i j iσ jσ(cid:48) jσ(cid:48) iσ
(cid:104)i,j(cid:105) σ,σ(cid:48)
U(cid:88)(cid:104)(cid:104)ni↓(cid:105)ni↑+(cid:104)ni↑(cid:105)ni↓−(cid:104)c†i↑ci↓(cid:105)c†i↓ci↑−(cid:104)c†i↓ci↑(cid:105)c†i↑ci↓(cid:105) +(cid:104)c†jσ(cid:48)ciσ(cid:105)c†iσcjσ(cid:48) −(cid:104)c†iσcjσ(cid:48)(cid:105)(cid:104)c†jσ(cid:48)ciσ(cid:105)(cid:105)(cid:111).
i (21)
(cid:88) (cid:88) (cid:88) (cid:88)
=−U c† [m σ ]c +U c† [m σ ]c
i(cid:48)A f f i(cid:48)A i(cid:48)B f f i(cid:48)B We assume a charge imbalance between the two sublattices
i(cid:48) f=1,2,3 i(cid:48) f=1,2,3 such that (cid:104)n (cid:105) = (1+ρ)/2 and(cid:104)n (cid:105) = (1−ρ)/2. As for
i(cid:48)Aσ i(cid:48)Bσ
=−U(cid:88)c†k[m1α1+m2α2+m3α3]ck, cHaVse,wofeSaDssWumiens(cid:104)tca†ibσiclijσty(cid:48)(cid:105).T=he−n∆tδhσeσm(cid:48)tiej/atna-finedldViHj a=mViltaosniiannthoef
k
(15) theinteractiontermisobtainedas
(cid:104) (cid:105)
whereNisthenumberoftheunitcellsandthewavevectorsk HiMntF = NCρ2+2NV 3+(1+δt1/t)2 ∆2
takeNpointsinthefirstBrillouinzoneofthefcclattice.This (cid:88) (cid:88) (22)
−Cρ c†(τ ⊗1)c +V∆/t t c† c ,
equationmeansthattheon-siteinteractiontermhasthesame k 3 k ij iσ jσ
matrix form as the spin-orbit interaction term in the mean- k (cid:104)i,j(cid:105),σ
field level. A similar result has been obtained in the Kane- whereC =4V−U/2,Nisthenumberoftheunitcells,andthe
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J.Phys.Soc.Jpn. FULLPAPERS
studiesbeyondthemean-fieldapproximation,forexample,in
thehalf-filledone-dimensionalextendedHubbardmodel.34,35
Asisshownlater,whatisdifferentfromusualsystemsinour
modelisthatthetopologicalmagnetoelectricresponsedueto
theexistenceofthethetatermcanariseintheSDWphase.In
thissense,wecalltheSDWphaseinourmodelthe“axionic
SDW”(orthe“axionicantiferromagneticinsulator”).Forthe
purposeofthisstudy,thatwederivethethetaterminatime-
reversalsymmetrybrokenphase,wefocusontheSDWphase
inthefollowing.
4. Magnetoelectricresponseoftheantiferromagneticin-
sulatorphase
Low-energyeffectiveHamiltonian.—Letusinvestigatethe
Fig. 2. (Coloronline)Mean-fieldphasediagramoftheextendedFu-Kane-
Mele-Hubbardmodelathalf-filling.Thestrengthsofthespin-orbitinterac- properties of the SDW phase, namely the antiferromagnetic
tionandlatticedistortionareλ/t = 0.4andδt1/t = −0.4,respectively.The insulator phase. We consider the general case characterized
antiferromagneticorderingissettobethe[111]directionasanexample.The bytheorderparameter(13).WhenUm (cid:28) 2λ(f = 1,2,3),
f
spin-densitywave(SDW)andcharge-densitywave(CDW)phasesaregiven we can derive the Dirac Hamiltonian around the X˜r points
bynonzeromandρ,respectively.Whenm=ρ=0,thesystemisanormal
insulator(NI).Inallthephases,∆takenonzeropositivevalues.IntheSDW whichareslightlydeviatedfromtheXr points:
phase,thetopologicalmagnetoelectricresponsedescribedbythethetaterm H(X˜x+q)=t(cid:48)q α +2λq α −2λq α +δt(cid:48)α −Um α ,
[Eq.(1)]arises. x 5 y 2 z 3 1 4 1 1
H(X˜y+q)=t(cid:48)q α +2λq α −2λq α +δt(cid:48)α −Um α ,
y 5 z 3 x 1 1 4 2 2
H(X˜z+q)=t(cid:48)q α +2λq α −2λq α +δt(cid:48)α −Um α ,
z 5 x 1 y 2 1 4 3 3
(24)
wavevectorsktakeN pointsinthefirstBrillouinzoneofthe
fcclattice.CombiningEqs.(6)and(22),weobtainthemean- where t(cid:48) = t(1 + V∆/t), δt(cid:48) = δt (1 + V∆/t),
(cid:16) (cid:17) (cid:16) 1 1(cid:17)
fieldHamiltonianofthesystem.Thematrixτ3⊗1isdifferent X˜x = 2π,Um2,−Um3 , X˜y = −Um1,2π,Um3 , and X˜z =
from the alpha matrices αµ, and thus the free energy for the (cid:16)Um1,−Um2,22πλ(cid:17). Fo2rλexample, the e2λnergy sp2λectrum around
CDW instability is a little complicated but can be obtained 2λ 2λ
theX˜x pointisreadilyobtainedas
analyticallyas
(cid:104) (cid:105) E(X˜x+q)
F (ρ,∆)= NCρ2+2NV 3+(1+δt /t)2 ∆2
CDW 1 (cid:113) (25)
(cid:88)(cid:88) (cid:114) (cid:113) (23) =± (t(cid:48)qx)2+(2λqy)2+(2λqz)2+(δt1(cid:48))2+(Um1)2.
− R˜2+C2ρ2+2Cρ(cid:15) γ2,
We see from Eq. (24) that the antiferromagnetic ordering
k (cid:15)=±1
opensagapattheX˜rpoints,i.e.,lowerstheenergyofthesys-
where R˜2 = (cid:80)5µ=1[R˜µ(k)]2 and γ2 = (cid:80)3j=1[R˜j(k)]2 with tem.WhenUmf isnotsmallcomparedto2λ,itisnotapparent
R˜1(k) = R1(k), R˜2(k) = R2(k), R˜3(k) = R3(k), R˜4(k) = thattheDiracHamiltoniancanbederived.Thusinthefollow-
(1+V∆/t)R4(k),andR˜5(k)=(1+V∆/t)R5(k). ing,weassumethatUmf issmall,althoughitisexpectedthat
Mean-field phase diagram.— To obtain the mean-field the momentum points around which the Dirac Hamiltonians
ground-state phase diagram, we have to minimize the free canbederivedexistevenwhenUm isnotsmall.
f
energies (19) and (23) by the conditions ∂FSDW/∂m = LetusanalyzeEq.(24).Theimportantpointisthatallthe
∂FSDW/∂θ = ∂FSDW/∂ϕ = ∂FSDW/∂∆ = 0and∂FCDW/∂ρ = five alpha matrices which anticommute with each other are
∂FCDW/∂∆ = 0, and then we have to compare them. The used.Tobespecific,letusfirstconsiderH(X˜x +q).Wecan
phase diagram with the antiferromagnetic ordering set to be redefinethealphamatricesbecausetherepresentationofthe
the [111] direction is shown in Fig. 2 as an example. The matrices is arbitrary. Redefining such that α → α , α →
5 1 3
phase diagrams for the other directions and for the positive −α ,andα → −α (α = α α α α )forthealphamatrices,
3 1 5 5 1 2 3 4
δt1 are qualitatively the same as Fig. 2. It was found that and t(cid:48)qx → qx, 2λqy → qy, and 2λqz → qz for the wave
thetransitionfromthenormalinsulator(ortopologicalinsu- vector,36weobtain
lator) phase to the SDW phase is of the second-order, and
H(X˜x+q)=q α +q α +q α +δt(cid:48)α +Um α . (26)
that the transition from the topological insulator (or normal x 1 y 2 z 3 1 4 1 5
insulator) phase to the CDW phase is of the first-order. The Inthesamemanner,H(X˜y+q)andH(X˜z+q)canberewritten
values of ∆ are always nonzero and positive when V (cid:44) 0. as
The obtained phase diagram looks similar to those of con- H(X˜y+q)=q α +q α +q α +δt(cid:48)α +Um α ,
ventionalcorrelatedelectronsystems(i.e.,theHubbardmod- x 1 y 2 z 3 1 4 2 5 (27)
els).33 Namely, strong on-site electron-electron interaction H(X˜z+q)=qxα1+qyα2+qzα3+δt1(cid:48)α4+Um3α5.
induces SDW phase and strong nearest-neighbor electron- We see that all the three effective Hamiltonians above are
electroninteractioninducesCDWphase.However,notethat
equivalent. Hence we can regard the Dirac quasiparticles
other phases might be found when our model is studied be- aroundtheX˜rpointsasthequasiparticlesofthreeflavorschar-
yond the mean-field approximation. Actually, another phase
acterized by their masses Um . Note that the mass of Dirac
f
has been reported between the SDW and CDW phases by quasiparticles δt is renormalized to be δt(cid:48) due to the near-
1 1
4
J.Phys.Soc.Jpn. FULLPAPERS
estneighborelectron-electroninteraction,andthatthesecond This is because the system with negative mass of the Dirac
massUm isinducedbytheon-siteinteraction. quasiparticles is identified from the Z invariant as a normal
f 2
Thethetaterm.—Herewederivethethetatermintheanti- insulator.S isthethetatermintheEuclideanspacetime:
θ
ferromagneticinsulatorphase,inthesamewayasthatof3D (cid:90) ((cid:80) κ )e2
topologicalinsulatorsisderived.Fromthediscussionabove, S =i d4x f f (cid:15)µνρλF F . (37)
we can write down the low-energy effective (Euclidean) ac- θ 32π2(cid:126)c µν ρλ
tionofthesystem,i.e.,theactionoftheDiracquasiparticles After dropping the irrelevant surface term [the second term
interactingwithanexternalelectromagneticfieldA as oftheexponentinEq.(35)],weobtainthetotalactionofthe
µ
(cid:90) systemas
(cid:88) (cid:104) (cid:105)
SAFI = d4x ψ¯f(x) γµDµ−Mfeiκfγ5 ψf(x), (28) S =S +S . (38)
f=1,2,3 AFI NI θ
whereψ (x)isafour-componentspinor,D =∂ +ieA ,M = Actually Sθ is also a surface term, since we can rewrite as
(cid:113) f µ µ µ f (cid:15)µνρλF F = 2(cid:15)µνρλ∂ (A F ). However, we are now inter-
(δt(cid:48))2+(Um )2,cosκ = |δt(cid:48)|/M ,sinκ = Um /M ,and µν ρλ µ ν ρλ
1 f f 1 f f f f estedinthemagnetoelectricresponseofthesystem.Thuswe
wehaveusedthefactthatα4 =γ0,α5 =−iγ0γ5andαj =γ0γj denotethetotalactionasabove.Rewritingthethetaterm(37)
(j=1,2,3).Thesubscript f denotestheflavor.Herewehave intherealtime(t=−iτ),weobtainEq.(1).
consideredthecaseofδt(cid:48) <0,namelythesystemisanormal The value of θ in the antiferromagnetic insulator phase is
1
insulatorwhentheinteractionsareweak. given as θ = (cid:80) κ = (cid:80) tan−1(Um /|δt(cid:48)|). It is known that
f f f f 1
We follow the Fujikawa’s method30,31 and write down a θ = π (mod 2π) in 3D topological insulators and is θ = 0 in
calculationbrieflyinwhatfollows.Letusconsiderainfinites- normalinsulators.However,θcanbearbitrarybetween0and
imalchiraltransformationforeachflavor: πiftime-reversalsymmetryofthesystemisbroken.Wecan
ψf →ψ(cid:48)f =e−iκfdφγ5/2ψf, ψ¯f →ψ¯(cid:48)f =ψ¯fe−iκfdφγ5/2, (29) obtainthevalueofθinthecaseofδt1 >0inthesamemanner
asabove.Combiningbothcases,θiswrittenas
whereφ∈[0,1].Thethetatermisgeneratedasaconsequence (cid:34) (cid:35)
of the chiral anomaly after the transformation. The partition θ= π2(cid:2)1+sgn(δt1)(cid:3)− (cid:88) tan−1 δt (1U+mVf∆/t) , (39)
functionistransformedas f=1,2,3 1
(cid:90) (cid:90)
Z = D[ψ,ψ¯]e−SAFI[ψ,ψ¯] →Z(cid:48) = D[ψ(cid:48),ψ¯(cid:48)]e−SAFI[ψ(cid:48),ψ¯(cid:48)]. wheretheconditionthatUmf (cid:28)2λisrequired,andwehave
written δt(cid:48) = δt (1+V∆/t) explicitly. Note that (1+V∆/t)
1 1
(30) is always positive, and thus the value of θ when m = 0 is
f
TheintegrandsinEq.(28)istransformedas determinedbythesignofδt1.Theregionwherethevalueof
θbecomesnonzeroisshowninFig.2asthe“axionicSDW”.
ψ¯fMfeiκfγ5ψf →ψ¯fMfeiκf(1−dφ)γ5ψf, Here we compare our analytical result for the value of θ,
ψ¯ γ D ψ →ψ¯ γ D ψ +(i/2)κ dφ∂ (ψ¯ γ γ ψ ). Eq. (39), with an exact numerical value calculated by Eq.
f µ µ f f µ µ f f µ f µ 5 f
(2). A numerical study on the value of θ in the Fu-Kane-
(31)
Mele model on a diamond lattice10 indicates the relation
ThendefiningtheJacobian Jf whichisinducedbythechiral θ ∝(cid:80) tan−1(Um /|δt(cid:48)|)when(Um /|δt(cid:48)|)issmall.Thusour
transformationforeachflavorD[ψ ,ψ¯ ]→ J D[ψ ,ψ¯ ],the f f 1 f 1
f f f f f result is in qualitative agreement with the exact numerical
partitionfunctionbecomes result. From our analytical result θ = (cid:80) tan−1(Um /|δt(cid:48)|),
Z(cid:48) =(cid:90) D[ψ,ψ¯]e−S(cid:48)+2i(cid:80)fκf(cid:82) d4xdφ∂µ(ψ¯fγµγ5ψf)+(cid:80)flnJf, (32) ∞it.isTfhoeunndumthearticaθl →study3πs/h2owins ththeatlimθiftdo(Unmotf/h|δatfv1(cid:48)|e) t1→he
(cid:80) tan−1(Um /|δt(cid:48)|) dependence when (Um /|δt(cid:48)|) is large,
where f f 1 f 1
and that θ takes some value between 0 and π in the limit
(cid:90)
S(cid:48) = d4x(cid:88)ψ¯f(x)(cid:104)γµDµ−Mfeiκf(1−dφ)γ5(cid:105)ψf(x), (33) (Umf/|δt1(cid:48)|) → ∞.10 This suggests that our analytical result
is considered not to be valid when (Um /|δt(cid:48)|) is large, i.e.,
f f 1
Um is large. The difference between our result and the ex-
andtheJacobianJ iscalculatedtobe30,31 f
f actnumericalresultintheregionwhereUm isnotsmallcan
f
(cid:34) (cid:90) κ e2 (cid:35) resultfromthat(i)theothercontributionstoθbecomesimpor-
Jf =exp −i d4xdφ32πf2(cid:126)c(cid:15)µνρλFµνFρλ . (34) tantandunignorableasUmf becomeslarger,and(ii)itmight
become impossible to derive the Dirac Hamiltonians around
HereFµν = ∂µAν−∂νAµ andwehavewritten(cid:126)andcexplic- thepointsneartheoriginalXpointsasUmf becomeslarger.
itly.Werepeatthisprocedureinfinitetimes,i.e.,integratethe
exponentofEq.(32)overthevariableφfrom0to1.Thenwe 5. Discussions
obtain Itshouldbenotedthatthethetatermisderivedonlyinodd
(cid:90)
Z(cid:48) = D[ψ,ψ¯]e−SNI+2i(cid:80)fκf(cid:82) d4x∂µ(ψ¯fγµγ5ψf)−Sθ, (35) spatialdimensions.IntheKane-Mele-Hubbardmodelonthe
honeycomblatticeathalf-filling,whichisatwo-dimensional
analog of the Fu-Kane-Mele-Hubbard model, the antiferro-
whereS istheactionwhichrepresentsthenormalinsulator
NI magnetic insulator phase is also realized.21–26 However, the
phaseinthepresentcase:
magnetoelectric response which results from the theta term
(cid:90)
(cid:88) (cid:104) (cid:105)
SNI = d4x ψ¯f(x) γµDµ−Mf ψf(x). (36) doesnotappearinthatmodel.
Theoriginthatgeneratessmalldeviationsofthevalueofθ
f
5
J.Phys.Soc.Jpn. FULLPAPERS
from0orπintheantiferromagneticinsulatorphaseoftheFu- thefluctuationofθuptothelinearorderinδm as
f
Kane-Mele-Hubbard model is the existence of the γ5 (or α5 δθ(r,t)≈ (cid:88) tan−1(cid:110)U(cid:104)m +δm (r,t)(cid:105)/|δt(cid:48)|(cid:111)
in our notation) term, which breaks time-reversal symmetry, f f 1
in the low-energy effective action (28). What we would like f=1,2,3
(cid:88) (cid:16) (cid:17)
tostresshereisthatwefoundtheappearanceoftheγ5 term − tan−1 Um /|δt(cid:48)|
f 1 (40)
intheantiferromagneticinsulatorphase.Thisisnotapparent
f=1,2,3
at first sight of the mean-field Hamiltonian (18). Expanding (cid:88)
the mean-field Hamiltonian around the X˜ points (which are ≈U/|δt1(cid:48)| δmf(r,t).
slightly deviated from the original X points) and relabeling f
thealphamatricesareessential. Thisequationsuggeststhatthedynamicalaxionfieldcanbe
Here we mention the relation between the antiferromag- realized by the fluctuations of the spins, i.e., the spin-wave
netic insulator phase in our model and the “Aoki phase”, a excitations,asinthecaseofRef.8.Anadvantageofourana-
phasewithbrokentime-reversalandparitysymmetriesinlat- lyticalderivationoftheexpressionofθisthatwecanseethe
tice QCD. The Aoki phase is characterized as the phase in- realization of the dynamical axion field immediately, as Eq.
duced by interactions with the γ5 term, i.e., (cid:104)ψ¯iγ5ψ(cid:105) (cid:44) 0 in (40). In the case where we use expressions (2) or (3), it will
addition to the usual mass renormalization (cid:104)ψ¯ψ(cid:105).32 It can be noteasytonoticetherealizationinourmodel.
seenfromtheeffectiveHamiltonian[Eqs.(26)and(27)]that
the situation in our model is analogous. Thus it can be said 6. Summary
thattheantiferromagneticinsulatorphaseoftheextendedFu- Insummary,wehavestudiedthegroundstateandthetopo-
Kane-Mele-Hubbard model is a condensed matter analog of logicalmagnetoelectricresponsedescribedbythethetaterm,
theAokiphaseinlatticeQCD.Inotherwords,theAokiphase intheextendedFu-Kane-Mele-Hubbardmodelonadiamond
incondensedmatterinthreespatialdimensionscanbechar- lattice at half-filling, within the mean-field approximation.
acterizedbythemagnetoelectricresponsewhichresultsfrom The mean-field phase diagram was presented. It was found
thethetatermwithnon-quantizedvalueofθ,i.e.,bytheaxion that the transition from the normal insulator (or topological
electrodynamics.Theexistenceofasimilarcondensedmatter insulator)phasetotheantiferromagneticinsulatorphaseisof
analogoftheAokiphaseinthreespatialdimensionshasbeen the second-order. We obtained the Dirac-like low-energy ef-
pointedoutina3Dtopologicalinsulatorwithon-siteinterac- fectiveHamiltonianintheantiferromagneticinsulatorphase.
tions.19 Wefoundthatthereexiststheγ termintheeffectiveHamilto-
5
TheAokiphasehasbeenfoundinlatticemodelsforQCD nian.Thisantiferromagneticinsulatorphaseisdifferentfrom
such as the Wilson fermions,32,37 the Nambu-Jona-Lasinio conventional one, and can be regarded as a condensed mat-
modelonalattice,38,39 andtheGross-Neveumodelonalat- ter analog of a symmetry broken phase in lattice QCD. We
tice.32 In the latter two models, the interactions are local. derived the theta term by following the Fujikawa’s method
Namely, from the viewpoint of the form of interactions, it and obtained the analytical value of θ. We have proposed a
can be said that Hubbard-like models in condensed matter concrete model to describe the axion electrodynamics in the
aresimilartotheNambu-Jona-LasiniomodelandtheGross- antiferromagneticinsulatorphase.Thedynamicalaxionfield
Neveumodel.ItisknownthattheexistenceoftheAokiphase can be induced by the fluctuation of the order parameter. In
can solve the U(1) problem in QCD. However, although the our model, the interplay of spin-orbit coupling and electron
importance of the Aoki phase has been confirmed theoreti- correlationresultsintheemergenceofthetopologicalmagne-
cally, the phase is not a realistic phase. This is because the toelectricresponseandtherealizationofthedynamicalaxion
phaseisanartifactduetothenonzerolatticespacingoflattice field.
QCD,37 andinaddition,theappearanceofthephasedepends
onthevalueofbarequarkmass.32 Ontheotherhand,incon- Acknowledgments
densed matter, electron systems can be naturally defined on A.S. is supported by the JSPS Research Fellowship for
lattices, and the value of the bare mass of Dirac fermions is Young Scientists. This work was supported by Grant-in-Aid
tunable. In our model, as mentioned in Sec. 2, the value is for Scientific Research (No. 25103703, No. 26107505 and
determinedbythestrengthoflatticedistortion.Moreover,ex- No. 26400308) from the Ministry of Education, Culture,
perimentalsearchesincondensedmatterarepossibleinprin- Sports,ScienceandTechnology(MEXT),Japan.
ciple. Further investigations of the Aoki phase in condensed
mattermightenableustosuggestsomeperceptiontothefield
oflatticeQCD.Thisisaninterestingfuturesubject.
Finallyletusconsiderbrieflythedynamicalbehaviorofθ 1) C.L.KaneandE.J.Mele,Phys.Rev.Lett.95,146802(2005);C.L.
inourmodel,thedynamicalaxionfield,discussedinRef.8. KaneandE.J.Mele,Phys.Rev.Lett.95,226801(2005).
When the spins fluctuate, i.e., when (cid:104)S (cid:105) = −(cid:104)S (cid:105) = [m + 2) X.-L.Qi, T.L.Hughes, and S.-C.Zhang, Phys.Rev.B 78, 195424
A B 1 (2008).
δm (r,t)]e +[m +δm (r,t)]e +[m +δm (r,t)]e ,weobtain
1 x 2 2 y 3 3 z 3) F.Wilczek,Phys.Rev.Lett.58,1799(1987).
4) R.D.PecceiandH.R.Quinn,Phys.Rev.Lett.38,1440(1977);R.D.
PecceiandH.R.Quinn,Phys.Rev.D16,1791(1977).
5) S.Weinberg,Phys.Rev.Lett.40,223(1978).
6) F.Wilczek,Phys.Rev.Lett.40,279(1978).
7) Markus Kuster, Georg Raffelt, and Berta Beltra´n (Eds.), Axions
(SpringerBerlinHeidelberg,2008).
8) R.Li,J.Wang,X.-L.Qi,andS.-C.Zhang,Nat.Phys.6,284(2010).
9) H.OoguriandM.Oshikawa,Phys.Rev.Lett.108,161803(2012).
6
J.Phys.Soc.Jpn. FULLPAPERS
10) A.M.Essin, J.E.Moore, and D.Vanderbilt, Phys.Rev.Lett.102, 25) A.Vaezi, M.Mashkoori, and M.Hosseini, Phys.Rev.B 85, 195126
146805(2009). (2012).
11) A.M.Essin,A.M.Turner,J.E.Moore,andD.Vanderbilt,Phys.Rev. 26) W.Wu,S.Rachel,W.-M.Liu,andK.LeHur,Phys.Rev.B85,205102
B81,205104(2010). (2012).
12) S.Coh,D.Vanderbilt,A.Malashevich,andI.Souza,Phys.Rev.B83, 27) T.Yoshida,R.Peters,S.Fujimoto,andN.Kawakami,Phys.Rev.B87,
085108(2011). 085134(2013).
13) J.Wang,R.Li,S.-C.Zhang,andX.-L.Qi,Phys.Rev.Lett.106,126403 28) L.Fu,C.L.Kane,andE.J.Mele,Phys.Rev.Lett.98,106803(2007).
(2011). 29) L.FuandC.L.Kane,Phys.Rev.B76,045302(2007).
14) B.J.Kim,H.Jin,S.J.Moon,J.-Y.Kim,B.-G.Park,C.S.Leem,J.Yu, 30) K.Fujikawa,Phys.Rev.Lett.42,1195(1979).
T.W.Noh,C.Kim,S.-J.Oh,J.-H.Park,V.Durairaj,G.Cao,andE. 31) K.Fujikawa,Phys.Rev.D21,2848(1980).
Rotenberg,Phys.Rev.Lett.101,076402(2008). 32) S.Aoki,Phys.Rev.D30,2653(1984);S.Aoki,Phys.Rev.Lett.57,
15) B.J.Kim,H.Ohsumi,T.Komesu,S.Sakai,T.Morita,H.Takagi,and 3136(1986).
T.Arima,Science323,1329(2009). 33) See,forexample,Y.Zhang,Y.Ran,andA.Vishwanath,Phys.Rev.B
16) A.Shitade,H.Katsura,J.Kunesˇ,X.-L.Qi,S.-C.Zhang,andN.Nagaosa, 79,245331(2009);M.Kurita,Y.Yamaji,andM.Imada,J.Phys.Soc.
Phys.Rev.Lett.102,256403(2009). Jpn.80,044708(2011).
17) D.PesinandL.Balents,Nat.Phys.6,376(2010). 34) M.Nakamura,J.Phys.Soc.Jpn.68,3123(1999);M.Nakamura,Phys.
18) X.Wan,A.M.Turner,A.Vishwanath,andS.Y.Savrasov,Phys.Rev.B Rev.B61,16377(2000).
83,205101(2011). 35) M.TsuchiizuandA.Furusaki,Phys.Rev.Lett.88,056402(2002).
19) A.SekineandK.Nomura,J.Phys.Soc.Jpn.82,033702(2013). 36) Wheninteractingwithanelectromagneticfield,thevectorpotentialA
20) M.HohenadlerandF.F.Assaad,J.Phys.Condens.Matter25,143201 isrescaledaswellasthespatialcoordinatex.Thenweseethatthetheta
(2013). term(37)isinvariantunderrescaling.Namely,thevalueofθdoesnot
21) S.RachelandK.LeHur,Phys.Rev.B82,075106(2010). dependonthevaluesofλandtinthelow-energylimit.
22) M.Hohenadler, T.C.Lang, and F.F.Assaad, Phys.Rev.Lett.106, 37) S.SharpeandR.Singleton,Phys.Rev.D58,074501(1998).
100403(2011). 38) S.Aoki,S.Boettcher,andA.Gocksch,Phys.Lett.B331,157(1994).
23) Y.YamajiandM.Imada,Phys.Rev.B83,205122(2011). 39) V.Azcoiti,G.DiCarlo,E.Follana,M.Giordano,andA.Vaquero,Nucl.
24) M.Hohenadler,Z.Y.Meng,T.C.Lang,S.Wessel,A.Muramatsu,and Phys.B875,45(2013).
F.F.Assaad,Phys.Rev.B85,115132(2012).
7