Edge states and the integer quantum Hall effect of spin-chiral ferromagnetic kagome lattice with a general spin coupling

0.42 MB·English

by  Zhigang Wang

Edge states and the integer quantum Hall effect of spin-chiral ferromagnetic kagom´e
lattice with a general spin coupling
Zhigang Wang and Ping Zhang
Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, P.R. China
ThechiraledgestatesandthequantizedHallconductance(QHC)inthetwo-dimensionalkagom´e
latticewithspinanisotropiesincludedinageneralHund’scouplingregionarestudied. Thiskagom´e
lattice system is periodic in the xdirection but hastwo edges in they direction. Numerical results
8 show that the strength of the Hund’s coupling, as well as the spin chirality, affects the edge states
0 andthecorrespondingQHC.Withinthetopologicaledgetheory,wegivetheexpressionoftheQHC
0 with thewinding numberof thechiral edge states on theRiemman surface. This expression is also
2 compaired with that within thetopological bulk theoryand theyare found tokeep consistent with
n each other.
a
J PACSnumbers: 73.43.-f,73.43.Cd,71.27.+a

0
3
I. INTRODUCTION coplanar and the spin chirality appears. Ohgushi et al.

[6]havefirstpointedoutthatthechiralspinstatecanbe
]
l realized by the introduction of spin anisotropy in an or-
l ThequantumHalleffect(QHE),inwhichtheHallcon-
a dered spinsystemonthe 2Dkagom´elattice,whichisthe
h ductance (HC) σxy is quantized with extremely high ac- cross section of the pyrochlore lattice perpendicular to
- curacy, is a typical realization of topological effects in
s the(1,1,1)direction[16]. Inthiscase,ithasbeenshown
e condensed matter physics [1]. The topological aspect of in the topological bulk theory [6, 17] that the presence
m the QHE with a periodic potential was first discussed
of chiral spin state may induce gauge-invariant nonzero
by Thouless, Kohmoto,Nightingale, and den Nijs [2]. In
. Chern number, thus resulting in a QHE in insulating
at theirwork,theHCisgivenbytheChernnumber[3]over state. However, as stressed by Halperin [18], this gauge
m the magnetic Brillouin zone, which is a topological ex-
invariance has a relation with the edge states which are
pression by the bulk state. Later Hatsugai [4] suggested
- localized near the sample boundaries. Recently we rein-
d that the HC can be given by another topological quan-
vestigated the kagom´e lattice with boundaries. Using
n tity, i.e., the “windingnumber” ofthe edgestates onthe
the topological edge theory established by Hatsugai in
o
complex-energy surface which is generally a high-genus
c the last decade [4], we interpreted the quantized Hall
Riemann surface. To distinguish them, we call the for-
[ conductance (QHC) in insulating state with the winding
mer “the bulk theory” and the latter “the edge theory”.
numberofthe edgestatesonthe complex-energysurface
1 The two topological expressions for the HC, which look
[19].
v
quite different, actually give the same integer number.
8
8 A recent established recognition points out that the Intheabove-mentionedtheoreticalworks[6,17,19]on
5 conventional QHE originates from the non-local effect the 2D kagom´e lattice, a very important limit has been
4 provided by the external magnetic field, more exactly used. Thatisthehoppingelectronspinsarealignedwith
1. speaking,bythevectorpotentialthatdescribesthemag- localized spins at each site of the lattices, which is also
0 netic field. That means a non-zero vector potential, for calledthe“infinite(strong)Hund’scouplinglimit”. How-
8 example,whichcanbeprovidedbyspin-orbitinteraction ever, the latest detailed experiments on the pyrochlores
0 or by spin chirality, will induce the QHE even if the ex- [20]showedthatthespin-chiralmechanismalone cannot
:
v ternal magnetic field vanishes. This type of QHE can explaintheanomaloustransportphenomenainthesesys-
i be called “the anomalous QHE” [5, 6, 7]. Since Haldane tems. So in this paper we study the 2D kagom´e lattice
X
performed the famous “Haldane model” in his pioneer- with boundaries in a generalHund’s coupling region, es-
ar ing work [5] in 1988, the anomalous QHE in spin-orbit pecially in the weak coupling region [21]. We find that
coupled [8, 9, 10] or spin-chiral ferromagnetic systems theedgestatesandQHCarenotonlyaffectedbythespin
[11, 12, 13, 14] has been a hot topic in condensed mat- chirality, but also affected by the strength of the Hund’s
ter physics. Based on the tight-binding two-dimenisonal coupling. Varying the spin chirality and the strength of
(2D) graphite model [15], Haldane model includes the the Hund’s coupling, two different types of phenomena
next-nearestneighboringinteractionanda periodiclocal areobtained. InonecasethatthestrengthoftheHund’s
magnetic-flux density, which breaks time-reversalinvari- coupling is larger than its critical value (see [22] or Sec.
ance and createsa chirality. However,because the intro- III), there are only two edge-state energies in each bulk
duction of local flux is technically difficult, it is not so energy gap. With the help of the topological edge the-
easytorealizeHaldanemodelinrealmaterials. Another ory,weobtainthatthecorrespondingHCisquantizedas
typical spin-chiral system is the ferromagnetic system, σ = e2 in insulating state. However, in another case
xy ±h
whichisrepresentedbypyrochlorecompoundsR Mo O thattheHund’scouplingstrengthissmallerthanitscrit-
2 2 7
(R=Nd,Sm,Gd),inwhichthespinconfigurationisnon- ical value, there are possibly four edge-state energies in
2
some one bulk gap. In this case, within the topologi-
cal edge theory, we obtain that the corresponding HC
is quantized as σ = 2e2 in this insulating state, which
xy ± h
can not occur in the strong Hund’s coupling case. We
also make a comparison between these results and those
inthebulktheory[22],andfindthattheykeepconsistent
with each other.
This paper is organized as the following. In sec. II,
we introduce the tight-binding model of the 2D kagom´e
lattice with boundaries along the y direction and obtain
the eigenvalue equations for sites. We also write out the
Hamiltonian of the 2D kagom´e lattice without bound-
aries in the reciprocal space in this section. Because the
analytical derivation is very difficult, then in sec. III,
FIG. 1: (Color online) (a) 2D spin-chiral ferromagnetic
wenumericallycalculatetheenergyspectrum. Usingthe kagom´e lattice. The dashed line represents the Wigner-Seitz
charactersofthe edge-stateenergiesin the spectrum, we unit cell, which contains three independent sites (A, B, C).
study the QHE within the topological edge theory. As (b) The umbrella structure on the triangular cell of the 2D
comparison, we also recalculate the HC in the infinite kagom´e lattice. (c) The 2D kagom´e lattice system with
system in this section. Finally we make a conclusion in boundaries along the y direction. The bold lines are corre-
the last section. spond to thedown and up edges, respectively.
II. MODEL 1(c)). Since the system is periodic in the x direction,
we can use a momentum representation of the electron
operator
We consider the double-exchangeferromagnetkagom´e
lattice schematically shown in Fig. 1(a). The triangle
1
is one face of the tetrahedron. Here we consider a pure c(lmsσ) = √L eikxX(lms)γmsσ(kx), (2)
spinmodelwithanisotropicDzyaloshinskii-Moriyainter- x Xkx
actions on a kagom´e lattice. It consists of an umbrella
of three spins per unit cell of the kagom´e lattice. Each where R(lms)= X(lms),Y(lms) are the coordinate of the
umbrella can be described by the spherical coordinates sitesintheunitcell(lm)andkx isthemomentumalong
(cid:0) (cid:1)
of the three spins (π/6,θ), (5π/6,θ), and ( π/2,θ), as the x direction. Let us consider the one-particle state
shown in Fig. 1(b). The angle θ ranges from−0 to π. |Ψ(kx)i= msσ Ψms(kx)γm†s(kx)|0i. Inserting it into the
Thetight-binding modelofthe 2Dkagom´esystemcan Schr¨odinger equation H Ψ =E Ψ , we can easily obtain
P | i | i
be written as the following [22] the following three eigenvalue equations for sites A, B,
and C,
H = t c† c +H.c. J c† (σ n )c ,
hiX,ji,σ ij(cid:16) iσ jσ (cid:17)− 0iX,α,β iα αβ · i iβ EΨmAσ =e−ik4xΨmBσ+eik4xΨm+1Bσ
(1) +eik4xΨmCσ+e−ik4xΨm+1Cσ
wheret isthehoppingintegralbetweentwoneighboring
ij
σJ cosθΨ +iσ¯J sinθΨ ,
sitesiandj;c†iσ andciσ arethecreationandannihilation − 0 mAσ 0 mAσ¯
operatorsofanelectronwithspinσonthesitei. J0isthe EΨmBσ =eik4xΨmAσ+e−ik4xΨm−1Aσ
effective coupling constantto eachlocalmoment S , and
i +2cos(k /2)Ψ
x mCσ
these moments are treated below as classical variables.
ni is a unit vector collinear with the local moment Si. −σJ0cosθΨmBσ−J0eiσ¯π6 sinθΨmBσ¯,
σαβ are the Pauli matrices. In the following we change EΨmCσ =e−ik4xΨmAσ+eik4xΨm−1Aσ
notation i (lms), where (lm) labels the kagom´e unit
cell and s d→enote the sites A, B and C in this cell. The +2cos(kx/2)ΨmBσ
sizeoftheunitcellissettobeunitthroughoutthispaper. σJ0cosθΨmCσ J0eiσ¯56π sinθΨmCσ¯. (3)
− −
This Hamiltonian has already been discussed in the infi-
nite Hund’s coupling limit J in Refs. [6, 17, 19]. Here we set t =1 as the energy unit, σ= 1 denotes the
0 ij
→ ∞ ±
Inthislimitthetwoσ= , bandsareinfinitely splitand spin up and down, respectively. σ¯= σ. Eliminating the
↑ ↓ −
the model describes a fully polarized electron subject to B- and C-sublattice sites, one can obtain a difference
a modulation of a fictitious magnetic field, correspond- equation for Ψ , which is the famous Harper equation
Aσ
ing to the molecular field associated with the magnetic [23]. BecausetheexpressionoftheHarperequationistoo
texture. sophisticatedtoperformahelpinanalyticalresolvingthe
Now we suppose that the system is periodic in the x edgestates,weturntomakeanumericalcalculationand
direction but has two edges in the y direction (see Fig. analysis from Eq. (3).
3
If the system is also periodic in the y direction, the one-particle state Ψ(k) = Ψ (k)γ† (k)0 into the
| i sσ sσ sσ | i
Hamiltoniancanberewritteninthereciprocalspace. We Schr¨odinger equation H Ψ =E Ψ , we can easily obtain
use the momentum representation of the electron oper- the Hamiltonian in the |recPiipro|cail space H(k), which is
ator c(lmsσ) = √L1xLy keik·R(lms)γsσ(k). Inserting the given by
P
J cosθ p1 p3 iJ sinθ 0 0
0 k k 0
− p1k J0cosθ p2k 0 J0e−iπ6 sinθ 0
H(k)= p3k − p2k −J0cosθ 0 − 0 −J0e−i56π sinθ , (4)
 iJ0sinθ 0 0 J0cosθ p1k p3k 
 − 0 J0eiπ6 sinθ 0 p1k J0cosθ p2k 
 − 
 0 0 −J0ei56π sinθ p3k p2k J0cosθ 
 
where p1k=2cos(kx/4 + √3ky/4), p2k=2cos(kx/2) and 4 4
p3=2cos( k /4+√3k /4). (a) (b)
k x y
−
III. EDGE STATES AND THE HALL Energy2 Energy2
CONDUCTANCE 0 0
Beforeinvestigatingthe chiraledgestatesandQHCof -2 -2
the 2D kagom´e lattice with boundaries, we simply dis- 0.0 0.5 1.0 K M
cuss the energy structures of this system with different kx(2 )
exchange interaction J . Firstly in the absence of the
0
exchange interaction, i.e., J0=0, Eq. (2) becomes to be FIG. 2: (Color online) The energy spectrums of the 2D
independent of both the spin index σ and the chiral pa- kagom´e lattice (a) with boundaries along they direction and
rameter θ. In this case the two σ= , bands are com- (b)withoutboundaries. Inbothfigures,theexchangeinterac-
pletely degenerate and there are onl↑y t↓hree bulk energy tion J0=0. There are only three energy bands and there are
no bulk gaps between them. The lower band is flat, which
bands, between which there are no bulk energy gaps, as
reflectsthatthe2Dkagom´elatticeisalinegraphofthehon-
showninFig. 2. FromFig. 2,one cansee thatthe lower
eycomb structure. In Fig. 2(a), the blue lines denote the
energy band becomes dispersionless (E= 2), which re-
− up-edge-stateenergies.
flects the fact that the 2D kagom´e lattice is a line graph
of the honeycomb structure [24]. This flat band touches
at k =0 with the middle band, while the middle band
x model. In the following, we investigate the HC in both
touches at k =2π, 4π with the upper band.
x 3 3 cases.
In the existence of exchange interaction, i.e., J = 0,
0
6
theenergyspectrumsplitsintotwopartsduetothespin-
dependent potential. For very large values of J , the
0
spectrumisdividedintotwogroupsofthreebands,which A. Case I: J0 <Jc
is the infinite Hund’s coupling case studied in Ref. [19].
FornonzerobutnottoolargevaluesofJ ,wenumerically In the following we study the QHE of 2D kagom´e sys-
0
calculate the energy spectrum and find that there are tem with boundaries in both cases. First, we investigate
main two different phenomena for QHE happening. The the case J0 < Jc. As an example, we choose the fixed
critical value of the exchange interaction J depends on chiralparameterasθ=π/3,andthe exchangeinteraction
c
the chiral parameter θ. In the topological bulk theory asJ =1(J =4/√7 1.51). The numberofsite A(orB,
0 c
≈
[22], this critical value is analytically obtained, which C) in the y direction is chosento be L =31. By numeri-
y
reads as cal calculating Eq. (3), we draw in Fig. 3(a) the energy
spectrum in this case. From this figure, one can clearly
J (θ)= 2/ 1+3cos2θ. (5) seethatthe twogroupsofthreebulk bandsarenotcom-
c
±
pletely divided (i.e., there is no Mott gap) and there are
p
Althoughinthepresentmodelwithboundaries,thecrit- two bulk gaps appearing. The range of the lower energy
icalvalueJ (θ)cannotbeanalyticallyobtained,thenu- gap is between 2.68 and 2.6, and that of the higher
c
− −
merical calculations tell us that the above expression of energy gapis between 1.1 and 1.4, which are enlargedin
J (θ)[Eq. (5)]isalsoapproximatelyvalidforthepresent Figs. 3(b) and 3(c), respectively.
c
4
In the topologicaledge theory [4], when the Fermi en-
ergyliesinoneenergygap,theHCofthesystemisgiven (a) -2.0
4 (b)
bythewindingnumberoftheedgestatesI,σxedyge=−eh2I. -2.5
Thewindingnumberisgivenbythenumberoftheinter- A LG
section between the canonical loop α on the Riemann 2 f
i y
g -3.0
surface (the complex energy surface) and the trace of ner HG 0.0 0. 5 1.0
the edge-state energy µ . Although in present model, E
i 0
the Riemann surface is more sophisticated than that in 1.5
B
tahnealiynsfiisniitnetHheunpdre’svicoouuspwlinogrkc[a1s9e][t1o9]d,iswceuscsanthfeolwloiwndtinhge -2 1.0 (c) HG C f
LG
numberofthe edgestate. Becausethe analyticalprocess
is same as that in the strong Hund’s coupling case, here 0.0 0.5 1.0 0.0 0.5 1.0
we only present the results as the following: kx(2 ) kx(2 )
The winding number of the edge states is givenby the
sumoftheintersectionnumberbetweentheFermienergy FIG.3: (Coloronline)(a)Theenergyspectrumof2Dkagom´e
ǫ and the energies of the down (or up) edge state in lattice with edges along the y direction. The parameters are
f
one period. The down (or up) edge states refer to the chosen as J0=1, θ=π/3. The shaded areas are energy bands
and the lines are the spectrum of the edge states. In this
eigenstates which are localized near the down (or) up
case, there is no Mott gap and there are two energy gaps,
boundaries of the sample. The intersection number is
whichareenlargedin(b)and(c). In(b)and(c),theredand
obtainedas: Intheleftneighboringoftheintersection,if
bluelinescorrespondtothedownandupedge-stateenergies,
the down-edge-state energy is connected with the lower
respectively.
energy band, or the up-edge-state energy is connected
with the upper energy band, the intersection number is
given by “+1”. If not, it is given by “ 1”.
−
Now with the above results, we investigate the HC of
the system. From Fig. 3(b), one can see that when the
Fermi energy ǫ lies in the lower gap (LG), there is only
f
one intersection between ǫ and the down edge-state en-
f
ergy,whichislabeledasAinFig. 3(b). Intheleftneigh-
boring of the intersection A, the down edge-state energy
isconnectedwiththelowerband. Sothewindingnumber
of the edge state is I =+1. With the same method, one
1
can find in Fig. 3(c) there are four edge-state energies
lie in the higher gap (HG). When the Fermi energy ǫ
f
lies in the HG, there are two intersections (labeled as B FIG. 4: (a) The energy spectrum of the 2D kagom´e lattice
and C in Fig. 3(b)) between ǫ and the down edge-state without edges. The parameters are same as that in Fig. 3.
f
energies. Because in the left neighboring of the intersec- (b) TheHall conductance σxy of thissystem as a function of
theFermienergyǫ atthetemperatureT=0. Inbothfigures,
tions, the down edge-state energies are connected with f
theshaded areas are thebulk energy gaps.
the lower band, we can obtain that the winding number
oftheedgestatesisI =+2. Accordingtothetopological
2
edge theory, we can get the QHC as
numbers are 1, 3, 2, 2, 3, 1 from the lowest to
− − − −
e2I = e2, ǫ the lower gap the topmost band. So, when the Fermi energy lies in
σxedyge =(−−eh2hI21=−−2heh2, ǫff ∈∈the higher gap . (6) tthheeLHGG,,σσxbybuulklk==eh2e2C1=4−eh2C;w=hen2et2h.eFCeormmipeanreerwgyitlhiesEiqn.
xy h n=1 n − h
To check the above results, we recalculate the energy (6), one obtain that σedge=σbulk. In Fig. 4(b), we plot
P xy xy
spectrum and the HC of the 2D kagom´e lattice without the HC as a function of the Fermi energy ǫ using the
f
bteomunhdaasribeseewnitshtuinditehdebtoypToaloilgleicfaulmbiuerlketthaelo.ry[2.2T].hIisnstyhse- finite-temperature formula σxy=eh2 n 21π fnkΩznkd2k,
where fnk is the Fermi distribution function. From this
bulk theory, when the Fermi energyǫf lies in the energy figure, one can see that when the syPstem iRs in the insu-
gap, the HC of the system in units of e2/h is given by
lating state, the HC is quantized. In the LG, the con-
the sum of the Chern numbers of all the occupied en- ductance is σ = e2, and in the HG, the conductance
ergy bands [3]: σbulk=e2 occuC . The Chern number xy −h
xy h n=1 n is σ = 2e2. Fig. 4(a) plots the energy spectrum of the
Ωofznkth=e∇nk-t×hAbnakn,dainsddAefinnkeP=d−aishuCnnk=|∇2u1πnRkiΩiznsktdh2ekg,ewomheerte- sysNtxeoymte−wthitahhtoiunttbheousntrdoanrgieHs.und’s couplinglimit[19], the
ric vector potential. With the Hamiltonian in the mo-
mentum representation H(k) [Eq. (4)], after a straight- non-zero QHC only takes two values, ±eh2. So this is a
forwardnumericalcalculation,one canobtain the Chern novel result that the HC can take the value 2e2 in the
± h
5

weak Hund’s coupling.
-2.5
(a)
(b)
4 -3.0
B. Case II: J0 >Jc -3.5 A G-I f
0.0 0. 5 1.0
2 G-IV -1.2 (c)
Thenweconsiderthe othercase,inwhichJ >J . As
0 c y
aanndexthame epxlec,hwanegcehionotseeratchteiofinxJed0=ch2i.raTlhpeanraummebteerroθf=sπit/e3s Energ 0 G-III(Mott gap) --21..48 G-II B f
A(orB,C)inthey directionisalsochosentobeLy=31. 0.0 0. 5 1.0
WithEq. (3),weplotinFig. 5(a)theenergyspectrumof -2 G-II 2.4
the 2D kagom´e lattice with boundaries. From Fig. 5(a), (d)
1.8 G-IV C
onecanfindthattherearefourgapsinthiscase. Wecall G-I f
themG-I,G-II,G-III (whichisthe Mottgap),andG-IV -4 1.2
correspondingtothatbetween 3.54and 3.36,between 0.0 0.5 1.0 0.0 0.5 1.0
2.0and 1.65,between 1an−d0,andbe−tween1.65and kx(2 ) kx(2 )
− − −
2.0, respectively. To clearly see these gaps and the edge
states, we enlargethe G-I, -II, and -IV in Figs. 5(b)-(d), FIG.5: (Coloronline)(a)Theenergyspectrumof2Dkagom´e
respectively. lattice with edges along the y direction. The parameters are
From Figs. 5(b)-(d), one can see that when the Fermi chosen as J0=2, θ=π/3. The shaded areas are energy bands
andthelinesarethespectrumoftheedgestates. Inthiscase,
energy ǫ lies in the G-I (or -II, -IV), there is only
f
thereisaMottgap. Besidesthat,therearethreeotherenergy
one intersection between ǫ and the down edge-state en-
f
gaps, which are enlarged in (b), (c) and (d). In (b)-(d), the
ergy, which is labeled as point A (B, C) in Fig. 5(b)
red and bluelines correspond to thedown and up edge-state
(5(c), 5(d)). In the left neighboring of the intersection
energies, respectively.
A (B), the down edge-stateenergy is connected with the
lower band. So the winding number of the edge state
is I =+1. But in the left neighboring of the intersec-
I(II)
tionC,thedownedge-stateenergyisconnectedwiththe
upper band. The winding number of the edge state is
I = 1. By the way, since there is no edge-state energy
IV
−
lies in the Mott gap, there is no intersection between ǫ
f
and the left edge-state energy when ǫ lies in the Mott
f
gap, and correspondingly, the winding number is I =0.
III
So, with the topological edge theory, one can easily ob-
tainthatwhenthe Fermienergyǫ lies inthe bulk gaps,
f
the QHC is
FIG. 6: (a) The energy spectrum of the 2D kagom´e lattice
σxedyge =−−−−eheeh2he22h2IIIIIIIIIV=I=I==−−eeh0he2h22,,,, ǫf ∈ǫǫfǫftfh∈e∈∈GMGGo---IItIVIt gap . (7) ttw(hhbiee)thFTsohehuraetmdHeeiddeagnlaleersrce.goaynsTdǫahufreceatpttahantrhecaeebmtuσeelxmtkyepreosnefreatarrhtgeuiysrsegsayamTspt=ese.0ma.satIshnaabtofuitnhncFfitgiigou.nreo5sf.,

Similar to the discussion in case-I, to check the above Hund’s coupling limit. The strength J0 is more larger,
results, we also recalculate the HC of the correspond- the results more tend to those in the strong Hund’s cou-
ing 2D kagom´e lattice without boundaries. After a pling limit. Fig. 6(a) plots the energy spectrum of the
straightforward numerical calculation, one obtains that system without boundaries.
theChernnumbersare 1,0,1,1,0, 1fromthelowest
− −
to the topmost band. When the Fermi energy ǫ lies in
f
G-I, only the lowest band is fully filled. So the HC is IV. SUMMARY
σI,bulk=e2C = e2; When ǫ lies in G-II, the lowest two
xy h 1 −h f
bands are fully occupied, σII,bulk=e2 (C +C )= e2; In summary, in this paper we have studied the chi-
xy h 1 2 −h ral edge states and the QHC of the 2D kagom´e lattice
When ǫ lies in the Mott gap, σIII,bulk=e2 3 C =0;
f xy h n=1 n with spin anisotropies included in a general Hund’s cou-
When ǫ lies in G-IV,the lowestfour bands are fully oc-
f pling region. This system is periodic in the x direction
cupied, σxIVy,bulk=eh2 4n=1Cn=eh2. ComparPing with Eq. but has two edges in the y direction. By numerical cal-
(7),onecanalsoobtainthatσedge=σbulk. InFig. 6(b)we culation, we find that both the strength of the Hund’s
P xy xy
plot the HC as a function of the Fermi energy ǫ . From coupling and the spin chirality affect the edges states
f
thisfigure,onecanseeσ issimilartothatinthestrong and the corresponding QHC. Upon varying the chirality
xy
6
and the strength of the Hund’s coupling, two types of boundaries within the topologicalbulk theory. Bothtwo
phenomena occur, which are distinguished by the criti- topological expressions give the same QHC.
cal relation between these two parameters (Eq. (5)). A
remarkable difference with the infinite Hund’s coupling
limitisthattherearepossiblyfouredge-stateenergiesin Acknowledgments
onebulkgapinthecaseJ <J . IftheFermienergylies
0 c
in this gap, the HC is quantized as σ = 2e2. We also This workwas supportedby NSFC under Grants Nos.
xy ± h
give the corresponding expression of the QHC without 10604010and 60776063.
[1] K. von Klitzing, G. Dorda and M Pepper, Phys. Rev. (2000).
Lett.45, 494 (1980). [13] J.Ye,Y.B.Kim,A.J.Millis,B.I.Shraiman,P.Majum-
[2] D.J. Thouless, M. Kohmoto, P. Nightingale and M. den dar,andZ.Teˇsanovi´c,Phys.Rev.Lett. 83,3737(1999).
Nijs, Phys. Rev.Lett. 49, 405 (1982). [14] Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagaosa, and
[3] D.J.Thouless,Topological Quantum Numbers inNonrel- Y. Tokura, Science 291, 2573 (2001).
ativistic Physics (World Scientific, Singapore, 1998). [15] G. Semenoff,Phys. Rev.Lett. 53, 2449 (1984).
[4] Y.Hatsugai, Phys.Rev.B 48, 11851 (1993); Phys.Rev. [16] A. P. Ramirez, Annu.Rev.Mater. Sci. 24, 453 (1994).
Lett.71, 3697 (1993). [17] Z. Wang and P Zhang, Phys. Rev.B 76, 064406 (2007).
[5] F.D.M. Haldane, Phys.Rev.Lett. 61, 2015 (1988). [18] B. I.Halperin, Phys.Rev.B 25, 2185 (1982).
[6] K. Ohgushi, S. Murakami, and N. Nagaosa, Phys. Rev. [19] Z.WangandP.Zhang,cond-mat/0711.2725v1.toappear
B 62, R6065 (2000). in Phys. Rev.B.
[7] R.ShindouandN.Nagaosa,Phys.Rev.Lett.87,116801 [20] Y. Taguchi and Y. Tokura, Europhys. Lett. 54, 401
(2001); Z.Wang,P.Zhang,andJ.Shi,Phys.Rev.B76, (2001); Y. Taguchi, T. Sasaki, S. Awaji, Y. Iwasa, T.
094406 (2007). Tayama,T.Sakakibara,S.Iguchi,T.Ito,andY.Tokura,
[8] T.Jungwirth,Q.Niu,andA.H.MacDonald,Phys.Rev. Phys. Rev.Lett. 90, 257202 (2003); Y.Yasui, S.Iikubo,
Lett.88, 207208 (2002). H. Harashina, T. Kageyama, M. Ito, M. Sato, and K.
[9] Z. Fang, N. Nagaosa, K.S. Takahashi, A. Asamitsu, R. Kakurai, J. Phys.Soc. Jpn.72, 865 (2003).
Mathieu, T. Ogasawara, H. Yamada, M. Kawasaki, Y. [21] G.TataraandH.Kawamura,J.Phys.Soc.Jpn.71,2613
Tokura, and K. Terakura, Science 302, 92 (2003). (2002).
[10] Y. Yao, L. Kleinman, A.H. MacDonald, J. Sinova, T. [22] M. Taillefumier, B. Canals, C. Lacroix, V. K. Dugaev,
Jungwirth,D.-S.Wang,E.Wang,andQ.Niu,Phys.Rev. and P. Bruno, Phys.Rev.B 74, 085105 (2006).
Lett.92, 037204 (2004). [23] P. G. Harper, Proc. Phys. Soc. London Sect. A 68, 874
[11] P.Matl, N.P. Ong,Y.F. Yan,Y.Q.Li, D.Studebaker, (1955).
T. Baum, and G. Doubinina, Phys. Rev. B 57, 10248 [24] A. Mielke, J. Phys. A 24, L73(1991); 24, 3311 (1991);
(1998). 25, 4335 (1992).
[12] S. H. Chun, M. B. Salamon, Y. Lyanda-Geller, P. M.
Goldbart, and P. D. Han, Phys. Rev. Lett. 84, 757