Topological antiferromagnetic spin-density-wave phase in an extended Kondo lattice model

0.32 MB·English

by  Yin Zhong

A novel topological antiferromagnetic spin-density-wave phase in an extended Kondo
lattice model
Yin Zhong,1, Yu-Feng Wang,1 Yong-Qiang Wang,2 and Hong-Gang Luo1,3,
∗ †
1Center for Interdisciplinary Studies & Key Laboratory for Magnetism and
2
Magnetic Materials of the MoE, Lanzhou University, Lanzhou 730000, China
1
0 2Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China
2 3Beijing Computational Science Research Center, Beijing 100084, China
(Dated: October30, 2012)
t
c
Byusinganextendedmean-fieldtheorywestudythephasediagramofthetopologicalKondolat-
O
ticemodel on thehoneycomblattice at half-filling, in which theconduction electrons are described
8 bytheHaldanemodel. Besidesthewell-definedKondoinsulatorandnormalantiferromagneticspin-
2 density-wave (N-SDW) state, it is found that a novel and nontrivial topological antiferromagnetic
SDWstate(T-SDW)withaquantizedHallconductanceispossible ifthequasiparticlegap isdom-

] inatedbythenext-nearestneighborhoppingratherthantheantiferromagnetic order. Byanalyzing
el the low-energy effective Chern-Simon action and the corresponding chiral edge-state, the T-SDW
- couldbeconsideredasaquantumanomalousHallinsulatorwithantiferromagneticlong-rangeorder.
r This novel state is apparently beyond Landau-Ginzburg paradigm, which can be attributed to the
t
s interplay of quantum anomalous Hall effect and the subtle antiferromagnetic order in the Kondo
.
t lattice-like model. While the transition between the SDW states and the Kondo insulator is found
a
tobeconventional(afirstordertransition),thetransitionbetweentheN-andT-SDWsis,however,
m
a topological quantum phase transition. Interestingly, such topological quantum phase transition
- canbedescribedbyDiracfermionscoupledtoaU(1)Chern-Simongauge-field,whichresemblesthe
d
criticaltheorybetweenbosonicintegerquantumHallphasesandsuperfluidphaseandalsoindicates
n
such topological quantum phase transition may fall into the 3D-XY universal class. It is expected
o
thatthepresentworkmayshedlightontheinterplaybetweenconductionelectronsandthedensely
c
localized spins on thehoneycomb lattice.
[

3
v I. INTRODUCTION the correspondinglocalorderparametersintroduced,in-
1 stead He et al. used the quantized Hall conductance or
1 the topological matrices to identify these novel topo-
4 Understanding the emergent novel quantum phases K
logical states.38 The success in identifying novel phases
6 and corresponding quantum criticality in heavy fermion
byusing the quantizedHallconductance motivatesus to
. compoundsisstillachallengeinmoderncondensedmat-
9 further explore whether such a novel phase exists or not
0 ter physics.1–13 To capture the essential physics in these
in other strongly correlated models.
2 phenomena, the Kondo lattice model is introduced to
1 describe the interplay between the Kondo screening and In this work we consider a modified Kondo lattice
v: the magnetic interaction, namely, the Ruderman-Kittel- model, where conduction electrons are described by the
i Kasuya-Yosida (RKKY) exchange interaction, which Haldane model on the honeycomb lattice at half-filling.
X is mediated by conduction electrons among localized According to Ref. [38], this model can be called ”topo-
r spins.14 While the former favorsa nonmagnetic spin sin- logical Kondo lattice”. We use the extended mean-field
a
glet state in strong coupling limit, the latter one tends decoupling16 to explore its phase diagram. Besides the
to stabilize usual magnetic ordered states in weak cou- well-defined Kondo insulator and the N-SDW phase, a
pling limit. There seems to exist a quantum phase tran- nontrivial T-SDW phase with quantum anomalous Hall
sition or even a coexistence regime between these two effect is possible if the quasiparticle gap is dominated
kinds of well-defined states,15–20 however, a more rad- by the next-nearest neighbor hopping rather than the
ical critical quantum phase perhaps has been observed antiferromagnetic order. Furthermore, such novel state
inthecompoundofYbRh (Si Ge ) ,5,21 whichfur- can be fully encoded by its low energy effective Chern-
2 0.95 0.05 2
ther motivates people to study more rich phases beyond Simon action, which underlies the nontrivial quantized
those mentioned above. Hallresponsetotheexternalelectromagneticfield. Then,
by examining the stability of the gapless chiral edge-
In recent years much progress in fact has been made
state derived from the Chern-Simon action, the T-SDW
in exploring the novel quantum phases,22–50 which is
state could be considered as an example of the quantum
obviously beyond the conventional Landau-Ginzburg
anomalous Hall insulator with antiferromagnetic long-
paradigm,wherethestatesofmatterareclassifiedbased
ranged order.
on the conventional symmetry-breaking picture. For ex-
ample, the so-called topological spin-wave-density (T- Moreover,the transitionbetweenthe mentioned SDW
SDW) state found by He et al.38 in the extended Hub- states and the Kondo insulator is found to be conven-
bard model can not distinguish from the normal SDW tional first order transition in the sense of the Landau-
(N-SDW) state according to the broken-symmetry and Ginzburg broken-symmetry picture. However, the tran-
2
sitionbetweentheT-andN-SDWsisatopologicalquan- at half-filling,
tum phase transition. Interestingly, such topological
H =H +H +H ,
quantumphasetransitioncanbe describedbytwo-flavor H
k ⊥
fiDeilrda,cwfhericmhiorenssemcobulepsletdhetocriaticUa(l1t)heCohryerbne-tSwimeeonnbgoasuongeic- HH =−t X c†iσcjσ −t′ X eiϕijc†iσcjσ,
ij σ ij σ
integerquantumHallphasesandsuperfluidphaseinRef. h i hh ii
J
[t4h9e]3aDnd-XinYdiucantievsertshael cclraitsisc.alInbeohuarvivoiresws,htohueldsifmalillairnittyo Hk = 4k Xi (c†i↑ci↑−c†i↓ci↓)(d†i↑di↑−d†i↓di↓),
between our case and the bosonic integer quantum Hall J
tdreasnirsaitbiloentoissienedmeeodreaenxianmteprleesstiwnhgenreewsofimnedfienrgmainnidcittheis- H⊥ = 2⊥ Xi (c†i↑ci↓d†i↓di↑+c†i↓ci↑d†i↑di↓), (1)
ories can dual to certain kinds of bosonic ones. where H is the spinful Haldane model,53 which sup-
H
Additionally, to our knowledge, since no realistic ma- ports the quantum anomalous Hall (QAH) effect with
terials could be modeled by the proposed topological two chiral edge states (one for spin-up and the other for
Kondo lattice model, we have to expect such model and spin-down), t and t′ are the nearest-neighbor and the
the novel T-SDW state may be realized in experiments next-nearest-neighborhopping,respectively.30Thephase
of ultra-cold atoms on the honeycomb optical lattices in ϕij =±21πisintroducedtogiverisetoaquantumHallef-
nearfuture. Thepresentworkhasanattempttouncover fectwithoutexternalmagneticfields(theso-calledQAH)
novel quantum states beyond the conventional Landau- and the positive phase is gained with anticlockwise hop-
Ginzburg paradigm for the Kondo lattice-like models ping. Besides, the pseudofermion representation for lo-
on the honeycomb lattice, thus sheds light on the rich cal spins has been utilized as Siα = 12 σσ′d†iστσασ′diσ′
physics involved in the heavy fermion systems. The fine with τα being usual Pauli matrix and aPlocal constraint
butmoresophisticatednumericalapproachescanbeused d†i di +d†i di =1 enforced in each site. H denotes the
for further study on the system. m↑agn↑etic i↓nst↓ability due to the polarizatiokn of conduc-
tionelectronsby localspinswhile H describesthe local
The remainder of this paper is organized as follows. Kondoscreeningeffectresultingfrom⊥spin-flipscattering
In Sec. II, we first introduce the topological Kondo lat-
process of conduction electrons by local moments.
tice model on the honeycomb lattice and provide a brief
The interplay of the mentioned Kondo screening and
discussion on the Haldane model and its quantized Hall
the magnetic instability on the honeycomb lattice with-
conductivity. Then Sec. III is devoted to the mean-field
out the nontrivial next-nearest-neighbor hopping term
treatment of the topological Kondo lattice. Three dis-
(t) has been studied by the present authors in the pre-
tinctstatesarefoundandoneofthemisidentifiedasthe ′
vious work.51 There either a direct first-order transition
T-SDWstate,whichshowsthe quantumanomalousHall
or a possible coexistence of the Kondo insulator and the
effect in spite of the established antiferromagnetic long-
N-SDW state was obtained by the extended mean-field
ranged order. In Sec. IV, the global ground-state phase
decoupling.
diagram is proposed based on the mean-field decoupling
Here, we would like to see whether a novel quantum
and the corresponding quantum phase transitions are
state could be found in the topological Kondo lattice
also discussed. In Sec. V, the critical theory for the
(Eq. [1]). This is motivated by the recent work in the
topological quantum phase transition is studied and we
so-called topological Hubbard model, where the usual
find that its critical behaviors should fall into the usual
Haldane model is complemented with the Hubbard on-
3D-XY universal class though the critical theory is for-
site repulsion interaction U.38 In such a model, the T-
mulatedbyDiracfermionscoupledtoU(1)Chern-Simon
SDWstateswithnontrivialedgeexcitationswerediscov-
gauge-field. The Kane-Mele-Kondo lattice model36, the
ered and classified by the effective Chern-Simon theory
extended Bernevig-Hughes-Zhang model37 and the spin
with different matrices. Most importantly, allof these
fluctuation effect in SDW states beyond the mean-field K
T-SDW states have the same physical symmetries with
treatment of Sec. III are also briefly discussed in this
antiferromagnetic long-range order. Thus, in the sense
section. Finally,Sec. VI is devotedto a briefconclusion.
of Landau-Ginzburg paradigm they are the same states.
However,these stateshavedifferentedge statesandHall
conductance fromthe N-SDW states,which display that
the T-SDW states are indeed distinguished with their
normal counterparts, namely, the N-SDW states.
ItshouldbeemphasizedthattheT-SDWstatesshould
II. THE TOPOLOGICAL KONDO LATTICE
MODEL not be identified as symmetry-protected-topological
(SPT)states42,43sincebreakingthesymmetryofthecon-
servation of total electron number (this symmetry pro-
ThemodelweconsideredistheanisotropicKondolat- tects the edge-state) will lead to superconducting states
tice model, where the conductionelectronsare described ratherthana usualantiferromagneticphase. By original
by the spinful Haldane model on the honeycomb lattice definition in Ref.[42], the SPT states should smoothly
3
evolve into their corresponding conventional state, e.g. It is easy to check that both the quasiparticle energy
thetopologicalinsulatorevolvesintousualbandinsulator E and the excitation gap∆ can be reproducedby
qσ gap
when breaking the time-reversal symmetry.29,30 There- the±following effective massive Dirac action
fore, based on the above definition, we will not consider
theT-SDWstatesascertainkindsofSPTphases. Mean- S = d2xdτ = d2xdτ [ψ¯ (γ ∂ +m)ψ ],
H 0 aσ µ µ aσ
while,italsocannotbeconsideredastheCherninsulator Z L Z Xaσ
because as a Chern insulator, it should both exhibit a
nonzeroHall conductance andpreservethe lattice trans- where γµ = (τz,τx,τy) and ∂µ = (∂τ,∂x,∂y) with
lationalsymmetry,52 while the T-SDWstatesdouble the τz,τx,τy the usual Pauli matrices. Here we introduce
effectivelatticeconstant,whichobviouslybreaksoriginal the effective mass m = 3√3t′ of Dirac fermions and
lattice translational symmetry, due to the antiferromag- set the effective Fermi ve−locity vF = 23t to unit. The
netic long-range order. Dirac fields are defined as ψ = (c ,c )T, ψ =
1σ 1Aσ 1Bσ 2σ
Before moving to the discussion of the topological (c , c )T and ψ¯ = ψ γ with a = 1,2 denot-
2Aσ − 2Bσ aσ a†σ 0
Kondo lattice model in the next section, it is helpful to ing the states near the two nonequivalent Dirac points
giveabriefargumentonthelowenergyeffectivetheoryof K~ = (0, 4π ) and T implying the transposition ma-
± ± 3√3
HaldanemodelanditscorrespondingChern-Simontreat-
nipulation.
ment since these issues may not appear in literature of
heavy fermions and the same technique will be used in
the next section.
B. The Chern-Simon action for the massive Dirac
fermions in 2+1D
A. The massive Dirac fermions from Haldane
Having obtained the effective massive Dirac action in
model
2+1D,it is interesting to see its physicalresponse to the
external electromagnetic field A = (iφ,A ,A ) with φ
µ x y
In this subsection, we will derive an effective action and A~ =(A ,A ) representing the usual scalar and vec-
x y
of the Haldane model, which can be described by free
torpotential, respectively. The electromagneticfield can
massive Dirac fermions in 2+1D.
be readily introduced into the Dirac action by the con-
Our starting point is the spinful Haldane model53
ventional minimal coupling, namely, ∂ ∂ ieA .
µ µ µ
→ −
Thus, the resulting effective Dirac action coupled with
HH =−t c†iσcjσ −t′ eiϕijc†iσcjσ. (2) the external electromagnetic field reads
X X
ij σ ij σ
h i hh ii
It is useful to rewrite this single-particle Hamiltonian in S = d2xdτ [ψ¯aσ(γµ(∂µ ieAµ)+m)ψaσ]. (5)
Z −
the momentum space as Xaσ
By integrating out the Dirac fields, we get an effec-
HH= −t[f(k)c†kAσckBσ +f⋆(k)c†kBσckAσ] tive Chern-Simon action, which represents the electro-
Xkσ magnetic response of the massive Dirac fermions to the
+2t′γ(k)[c†kAσckAσ −c†kBσckBσ], (3) external electromagnetic field Aµ,38(For details, see Ap-
pendix A.)
where we have defined f(k)=e−ikx+2eikx/2cos(√23ky), im
γ(k)=sin(√3ky)−2cos(32kx)sin(√23ky)andA,B repre- SCS =Z d2xdτ[Ne28−π m ǫµνλAµ∂νAλ], (6)
senting two nonequivalent sublattices of the honeycomb | |
lattice, respectively. Then, by diagonalizing the above
where N = 4 (two from spins and the other two from
Hamiltonian, one obtains the quasiparticle energy band the nonequivalent Dirac points), ǫµνλ is the usual all-
as
antisymmetrictensorandwehavedroppedouttheregu-
larMaxwellterm( F2 )sincethe lowenergyphysicsis
Ekσ± =± t2|f(k)|2+4t′2γ(k)2, (4) dominated by the ∼Cheµrνn-Simon term alone. We should
p
which preserves the particle-hole symmetry and also the emphasize that although the effective Chern-Simon ac-
spin degeneracy. It is well-known that for 3√3t < t, tion is used here, it does not imply any fractionalization
′
the excitation gap mainly opens near six Dirac points ornontrivialtopologicalorder(Acharacteristicsignature
(Only two of them are nonequivalent in fact).31 Then, of the topological order is the ground-state degeneracy
expanding both f(k) and γ(k) near two nonequivalent depending on the topology of the system.) because no
Dirac points K~ = (0, 4π ), respectively, the gap can emergentgauge fields or fractionalizedquasiparticles ex-
± ± 3√3 ist in the present case.24
be found as ∆ = 6√3t and the quasiparticle energy
gap ′ Now,one cansee that the quantizedHall conductance
(rkeaxd,ksyE∓qσ3±4√π3≃).±q(32tq)2+(3√3t′)2 with q = (qx,qy) ≡ ∂σ0HAy=) =2he222eπ2frEomy wJhxer=e h∂∂S=ACxS2|πA~~→i0s =reinNtreo2d4−uπic|mmed|(∂aynAd0w−e
4
havealsousedm/m = 1andN =4.24Sincethequan- screening existing (V = 0), we can easily derive ground-
| | −
tized Hall conductance is realized without any external stateenergyofthe antiferromagneticSDWstateper site
magnetic field (A~ 0), the Haldane model (or its low as
→
energy theory) just provides an example of the quantum
EAFM=J m (2m 1)
anomalousHall effect, which is defined as a band insula- g k c d−
tor with quantized Hallconductance but without orbital 1 σJ md
magnetic field.30 −N r( k2 +γ(k))2+t2|f(k)|2
s Xkσ
andtwoself-consistentequationsfromminimizingEAFM
III. MEAN-FIELD TREATMENT OF g
with respect to magnetization m and m , respectively.
TOPOLOGICAL KONDO LATTICE d c
J m (2m 1)=0,
c d
In this section, we will use mean-field decoupling to k −
1 J md/2+σγ(k)
study the ground-state phase diagram of the topological mc = k .
Kondolatticemodel(Eq. [1]). Fluctuationeffectbeyond 4Ns Xkσ (σJkmd +γ(k))2+t2 f(k)2
q 2 | |
thepresentmean-fieldtreatmentwillbeanalyzedinSec.
V. For further analytical treatment, we may use a sim-
By utilizing the mean-field decoupling introduced by plified linear density of state (DOS) ρ(ε) = ε/Λ2 when
| |
Zhang and Yu16 for the longitudinal and transverse in- transformingthesummationovermomentumkintointe-
teractiontermH ,H ,respectively,itisstraightforward gralonenergyεwithΛ 2.33tbeinghigh-energycutoff.
≃
to obtain a meank-field⊥Hamiltonian Thus, tf(k) can be replaced by ε to simplify corre-
| | | |
sponding calculations and γ(k) is replaced by 3√3t
HMF =HH +HMF +HMF +E0, near two nonequivalent Dirac points K~ = (0,±4π ). ′
k ⊥ ± ± 3√3
J
HkMF = 2k Xkσ[σ(−mcd†kAσdkAσ +mdc†kAσckAσ) From1these two equations, one obtains md =1/2 and
−(A→B)], mc= 2Λ2 Xσ (σ3√3t′+Jk/4)[qΛ2+(Jkσ3√3t′+Jk/4)2
J V
H⊥MF = ⊥2 Xkσ(c†kAσdkAσ +c†kBσdkBσ +h.c.), −q(Jkσ3√3t′+Jk/4)2]
E0 =Ns(2J mdmc+J V2), (7) while the ground-state energy per site for the antiferro-
k ⊥
magnetic SDW state reads
where we have defined several mean-field parameters as
−hd2†iAm↑dd,iA↑hc−†iA↑dc†iAiA↓↑di−A↓ic†i=A↓c2imA↓di, hd=†iB↑−di2Bm↑c−, dhc†iB†iB↓↑dciBiB↓↑i =− EgAFM=−3Λ22 Xσ [(Λ2+(Jkσ3√3t′+Jk/4)2)3/2
c†iB↓ciB↓i = 2mc and −V = hc†i↑di↑+d†i↓ci↓i = hc†i↓di↓+ −((Jkσ3√3t′+Jk/4)2)3/2]. (8)
dti†iz↑acti↑ioin. Iotfclaoncablesspeiennstahnadtmcodn,dmucctcioonrreeslepcotnrodntso,mreasgpnece-- Apparently, the local spins are fully polarized (md =
1/2) while the conduction electrons have small mag-
tively,whilenon-vanishingV denotestheonsetofKondo
netization as shown in Fig. 1. It is also noted that
screeningeffect. Besides,since we areconsideringa half-
when t = 0 (no next-nearest-neighbor hopping), the
filled lattice, the local constraint of the pseudofermion ′
above m correctly recovers the value in our previ-
has been safely neglected at the present mean-field level c
ous work.51 Meanwhile, the low-lying quasiparticle ex-
with chemical potential setting to zero.16
citations in the antiferromagnetic SDW state has the
Firstly, we proceed to discuss two simple but physi-
cwahlliychinctoerrreesstpinogndlimtoitsthfeoranKtoifnedrroomcoaugpnleintigc JSkDWandstJa⊥te, eEn3e,4rg(ky)E=±1,σ2(kJ)m=/±2.qI(t3tskh/o2u)l2d+b(eσJnko/te4d±t3h√at3tt′h)2e agnadp
(J J ) and Kondo insulating state (J J ), ar±oσund the±Dirkaccpoints only closes when the condition
reskpe≫ctive⊥ly.14 k ≪ ⊥ J /4 = 3√3t is fully satisfied, otherwise, any low-lying
′
qukasiparticle excitations in the antiferromagnetic SDW
stateareclearlygapped. Thus,wemayconcludethatthe
A. The antiferromagnetic spin-density-wave state antiferromagnetic SDW state we obtained is mainly an
insulating state (except for the case with J /4=3√3t)
′
For the case with J J , in general, one expects withfullypolarizedlocalspins(m =1/2)wkhileconduc-
d
that the antiferromagnket≫ic SD⊥W state to be the stable tion electrons only partially polarize (m < 1/2). This
c
ground-state of Kondo lattice model on the honeycomb feature is similarto the previousstudy onsquarelattice,
lattice due to its bipartite feature.14 To study the pos- thus confirms the validity of our current treatment.16
sible antiferromagnetic ordered state, diagonalizing the Additionally, as a matter of fact, with the help of a
mean-fieldHamiltonianEq. [7]with assumingnoKondo low energy effective theory similar to the one in Sec.II,
5
J°(cid:144)L
4
TQPT
N-SDW
3
m
c
2
T-SDW
1
J°(cid:144)L t'(cid:144)L
0.05 0.10 0.15 0.20
t¢(cid:144)L
FIG.2. Thephasediagramofthetopologicalquantumphase
transition (TQPT) in the antiferromagnetic SDW states.
N-SDW denotes the normal antiferromagnetic SDW state
FIG. 1. Magnetization of conduction electrons (mc) versus whiletheT-SDWrepresentsthetopologicalantiferromagnetic
next-nearest-neighborhopping t′ and theKondo coupling Jk SDWstatewithquantumanomalousHalleffect. Thebound-
in the antiferromagnetic SDW state with Λ 2.33t being aryofthesetwokindsofstatesisdeterminedby3√3t′=J /4
≃ k
high-energy cutoff. with Λ 2.33t being high-energy cutoff.
≃
the mentioned case for vanished gap (J /4=3√3t) can can be found (W = 0) and the corresponding effective
′
be identified as a topological quantumkphase transition Chern-Simon term vanishes.
betweena N-SDWstate andaT-SDWone, whichshows Therefore, it seems that even in the antiferromagnetic
aquantumanomalousHalleffectinspiteoftheantiferro- SDW state, there exists a quantized Hall conductance
magneticlong-rangedorder. Moredetailswillbepursued without externalmagnetic fields, if the quasiparticlegap
in the next subsection. is still dominated by the next-nearest neighbor hopping
(3√3t) rather than the antiferromagnetic order (J /4).
′
Thus,wehaveuncoveredatopologicalantiferromagknetic
B. Topological quantum phase transition and the SDW state, namely, the T-SDW state with a quantum
T-SDW state anomalous Hall effect for 3√3t > J /4 (Recalling that
′
the quantum anomalous Hall effect isk defined as a band
After obtained the condition for the vanished gap in insulator with quantized Hall conductance but without
the last subsection, it is interesting to see what new orbital/externalmagnetic field.).
physics this will lead to and whether a novel SDW state It is also interesting to see that in the T-SDW state,
may be uncovered. one may use the following effective Chern-Simon action
Performing the same treatment as for the Haldane to reproduce the quantized Hall conductance
model in Sec.II on the mean-field Hamiltonian Eq.[7],
i ie
we obtains the following effective action for the antifer- S = d2xdτ[K − ǫµνλa ∂ a + q ǫµνλA ∂ a ],
IJ Iµ ν Jλ I µ ν Iλ
Z 4π 2π
romagnetic SDW state
where we have defined the so-called K matrix as K =
IJ
S = d2xdτ [ψ¯ (γ (∂ ieA )+m )ψ ],(9) δ for I,J = 1,2 and the corresponding charge vec-
aσ µ µ µ aσ aσ IJ
Z Xaσ − tor reads q = (1,1)T. Physically, the effective gauge
field a and a are introduced to give rise to the con-
where the effective mass is defined as m =m =m 1µ 2µ
1↑ 2↓ − served current for spin-up and spin-down electrons, re-
J /4 and m = m = m+ J /4 with m = 3√3t.
1 2 ′ spectively. (So the system has U(1) U((1) symmetry
Tkhen, it is str↓aightfor↑ward to dekrive an effective−Chern- ×
fortwokindsofspinsandaglobalU(1)symmetryforthe
Simon action by integrating out the Dirac fermions
conservation of total electron number (charge) as well.)
Forthephysicalobservable,thefillingfactorν,whichde-
i
SCS = d2xdτ[We2− ǫµνλAµ∂νAλ] termines the quantized Hall conductance as σ = νe2,
Z 8π H h
can be calculated by ν = qTK 1q = 2, thus the effec-
−
and we have also defined W = maσ . It is easy tive action correctly reproduces the quantized Hall con-
to see that a quantized Hall conduPctaaσn|cmeawσ|ith the value ductance obtainedin the previous paragraph. Moreover,
σ = 2e2/h (W = 4) is obtained if 3√3t > J /4 (See since Det[K] = 1, no states with topological order and
H ′
the second subsectio−n of Sec.II for the calculatiokn of the fractionalexcitationsareinvolved47 (Generally,fraction-
quantizedHallconductance.). Incontrast,when3√3t is alized excitations require a K-matrix with Det[K] > 1
′
| |
smaller than J /4, no such quantized Hall conductance since the ground-state degeneracy on a torus is equal to
k
6
Det[K].48). In fact, the elementary quasiparticle (the Moreover, when the gap closes, a topological quan-
electron)canalsobeobtainedinthepresentformulismby tum phase transition similar to the one in Ref. [38] ap-
checking the so-call exchange statistical angle θ. With- pearsbetweenthesetwodistinctantiferromagneticSDW
outanysurprise,wefindsuchangleisπ whichmeansex- statesandgaplessDiracfermionsreappearneartheDirac
change two identical quasiparticle give rise to a π phase points. However, in contrast to the case in Ref. [38], no
in their wavefunction. Therefore, the elementary quasi- discontinuity of the third derivative of the ground-state
particle is the usual electron as expected. energyisfoundnearthetopologicalquantumphasetran-
Todeepenthediscussion,onemayutilizethebulk-edge sition. This is due to the fact that in our case the effec-
correspondence for the above effective abelian Chern- tive magnetic field, which the conduction electron expe-
Simontheorytoderivetwodecoupledgaplesschiraledge riences, is fixed (m = 1/2) throughout the whole an-
d
states as follows tiferromagnetic region while the effective magnetic field
varies in Ref. [38]. More importantly, when considering
1
Sedge = dxdτ [ i∂τφI∂xφI +cI∂xφI∂xφI] fluctuation effect, in Sec. V we will find that the men-
Z 4π −
IX=1,2 tioned topological quantum phase transition should fall
into the 3D-XY universal class though the critical the-
withc denotingthenon-universalvelocityofedgestates
I
ory is fermionic. We note that such new feature is not
and φ being the bosonic representation for the two
I
reported in Ref. [38]. The main findings in this subsec-
edge-state modes (I = 1 for the spin-up mode while
tion are systematically summarized in Fig. 2.
I =2 for the spin-down mode). Because each edge-state
mode contributes e2/h to the Hall conductance, the to-
tal quantized Hall conductance is obviously 2e2/h as ex-
C. The Kondo insulating state
pected. It is noted that the above quantized value for
Hallconductanceisprotectedbythechiralfeatureofthe
edge-state even when the conservation of electron num- Another interesting case appears when J J . It
ber with different spins is broken (Such breaking could is natural to expect that a Kondo insulatingkst≪ate ⊥arises
happenwhenfluctuationsbeyondthepresentmean-field- in this situation for half-filling.14,16 Following the same
like treatment is consideredand the stability of the gap- methology of treating antiferromagnetic SDW state, we
less chiral edge-stat is discussed in Appendix B.). In cangetthe ground-stateenergyper siteforthe expected
contrast, if one breaks the U(1) symmetry for conser- Kondo insulating state with V = 0 but no magnetic or-
6
vation of total electron number (e.g. by some super- ders md =mc =0
conducting paring terms), the T-SDW state cannot be
4
stable anymore since its quantized Hall conductance (or EKondo=J V2 [(Λ2+(3√3t)2+J2V2)3/2
the gapless chiral edge-state) will be destroyed generi- g ⊥ − 3Λ2 ′ ⊥
cally. (Our case is similar to the bosonic example pro- ((3√3t′)2+J2V2)3/2]. (10)
vided by Senthil and Levin very recently,48 where the − ⊥
Minimizing EKondo with respect to Kondo hybridiza-
conservation of total boson number protects the quan- g
tized (charge) Hall conductance.) However, the T-SDW tion parameter V, we obtain V2 = 161J⊥4 [Λ4−8Λ2J⊥2 −
state may not be considered as a kind of the symmetry- 16J2(3√3t)2 +16J4] which implies a critical coupling
′
protected-topological states42,43 since breaking the con- ⊥ ⊥
Jc = 2 [ Λ2+(3√3t)2 3√3t]correspondingtovan-
sseurpveartcioonndoufcttinotgasltealteecstrbount nnuotmtbheereoxnpleyctleedadussutoalsaonmtie- is⊥hingΛV2.qIt is noted ′tha−t V ′(J Jc) in contrast
ferromagnetic SDW states. to usual mean-field result β =∝ 1/⊥2,−and⊥ this can be
However, one finds that there is only a usual antifer- attributed to the low-energy linear DOS of conduction
romagnetic SDW state, namely, the N-SDW when the electrons on the honeycomb lattice at half-filling. Sim-
gapisdominatedbytheantiferromagneticorder(J /4> ilar critical behavior for onset of Kondo screening on
3√3t). Obviously, these two SDW states have theksame the honeycomb lattice has been obtained in the study
′ of Kondo breakdown mechanism as well.23 As a mat-
physicalsymmetry andat the same time break the spin-
ter of fact, the existence of the critical coupling Jc re-
rotation invariance, however, they are rather different
sultsfromthecompetitionbetweentheKondoinsul⊥ating
states of matter due to the response to the electromag-
state andthe nontrivialdecoupledstate (This decoupled
netic field (The former has quantized Hall conductance
state is in fact the quantum anomalous Hall state de-
while the latter does not.). Therefore, this provides a
scribed by the spinful Haldane model in Eq.[2].) where
simple example beyond the Landau-Ginzburg paradigm,
V = m = m = 0, its ground-state energy E0 =
particularly the symmetry-breaking based classification d c g
ofstatesofmatter.24 We note thatalthoughthe T-SDW −3Λ42[(Λ2 +(3√3t′)2)3/2 −(3√3t′)3] comes solely from
state and the N-SDW states cannot be fully identified free conduction electrons. Comparing E0 and EKondo,
g g
by symmetry-breaking paradigm, one may use the their oneclearlyrecoversthe criticalcoupling Jc,whichjusti-
distinct quantized Hall conductance (or their low energy fies the above simple picture. ⊥
effective Chern-Simon action and the robust chiral edge However, since the Kondo insulating state is unsta-
states) to identify and classify them completely. ble to the decoupled state when J < Jc, one may
⊥ ⊥
7
J(cid:144)L which has been studied in previous section. However, in
4
contrast to one’s expectation, one can check that these
KI
putative continuous second-order quantum phase transi-
3 tion between the Kondo insulating state and the antifer-
romagneticSDWstates areinfactfirst-orderwhencom-
paring the first-order derivative of EKondo and EAFM
2 N-SDW with respect to the Kondo coupling Jgfor differentgfixed
t. Thus,wedonotexpectradicalcriticalbehaviorsnear
T-SDW ′
1 such first-order quantum phase transition points in the
spirit of Landau-Ginzburg paradigm.2
Additionally,itisnotedthatthe mentionedfirst-order
3 3t'(cid:144)L
0.2 0.4 0.6 0.8 1.0 quantum phase transition has also been obtained on the
squarelattice54 (with t =0)andwe suspectthis feature
′
FIG. 3. The ground-state phase diagram of the topological may be generic for conventional mean-field treatment
Kondolatticemodel. KIistheKondoinsulatorandthequan-
of Kondo lattice models according to standard Landau-
tum phase transition between the KI and the antiferromag- Ginzburg phase transition theory.16 However, we point
neticSDWstatesisfirstorder. IntheantiferromagneticSDW
out that it is a subtle issue to compare the results of
region, N-SDW denotes the normal antiferromagnetic SDW
a first-order quantum transition with numerical simu-
state while the T-SDW represents the topological antiferro-
lations, particularly when the first-order transition is a
magnetic SDW state with quantum anomalous Hall effect.
TheboundaryofthesetwokindsofSDWstatesisdetermined weakone.55 Moreover,generically,apossible coexistence
by 3√3t′ =J /4 with Λ 2.33t being high-energy cutoff. region of the Kondo insulating state and the antiferro-
k ≃
magnetic ordered state cannot be excluded but we will
leave this interesting issue for future work.16
wonderwhetheranontrivialdecoupledstateappearsbe-
tween the Kondo insulating state and the antiferromag-
netic SDW states. It is easy to see thatthe ground-state V. EXTENSION AND DISCUSSIONS
energy of the antiferromagnetic SDW state EAFM is al-
g
ways lower than E0 for any positive coupling J and the A. Spin fluctuation effect in antiferromagnetic
g
next-nearest-neighborhoppingt. Therefore,ankontrivial spin-density wave states
′
quantum anomalous Hall state in intermediate coupling
seemsunfavorablebasedonourcurrentmean-fieldtreat- Here,wediscussthe spinfluctuationeffectinthe anti-
ment. However,onenotesthattheT-SDWstatefoundin ferromagneticspin-density wave states, which is omitted
the previous subsectionin the antiferromagneticordered in the previous mean-field treatment in Sec. III.
region shows the quantum anomalous Hall effect. First, let us recall the effective Dirac action (Eq. 9)
S = d2xdτ [ψ¯ (γ (∂ ieA )+m )ψ ],
aσ µ µ µ aσ aσ
Z −
IV. THE GLOBAL PHASE DIAGRAM OF Xaσ
TOPOLOGICAL KONDO LATTICE
wherewehavedefinedtheeffectivemassasm =m =
1 2
↑ ↓
m J /4andm =m =m+J /4withm= 3√3t.
Havingfoundthe antiferromagneticSDWstates(both F−ollkowingRef.1↓38,wh2e↑ntheantikferromagnetic−orderi′s
theN-andT-SDWstates)andthekondoinsulatorinthe
well established, the effect of spin fluctuations is equiv-
previous section by mean-field decoupling, here we pro-
alent to introduce an effective U(1) dynamic gauge-field
ceed to discuss the possible ground-statephase diagram.
b into the above action.
Sinceanontrivialdecouplingstate(thequantumamo- µ
malous Hall state) is excluded in the last subsection, in-
S = d2xdτ [ψ¯ (γ (∂ ieA ieσb )+m )ψ ].
stead, we may consider possible second-order quantum ′ aσ µ µ µ µ aσ aσ
Z − −
phasetransitionbetweenKondoinsulating stateandthe Xaσ
antiferromagnetic SDW states. The boundary of the It is notedthat different spinflavorofelectrons havethe
transitions canbe determinedby comparingthe ground- oppositegaugechargeofb ,whichjustindicatesthatthe
µ
state energies of EgAFM (Eq.[8]) and EgKondo (Eq.[10]). gauge-field bµ describes the spin fluctuations of ordered
For physically interesting case with J = J = J, the magnetic background.
phase diagramis shownin Fig.3, from⊥whichkwe can see After integrating out Dirac fermions, one can obtain
that there exist three distinct states, one is the Kondo
iannsdultahteorT(-KSDI)WansdtattheewoitthheqrutawnotuamreatnhoemNal-oSuDsWHasltlaetfe- SC′ S = Z d2xdτ[−8πiem2maσǫµνλ(Aµ+σbµ)∂ν(Aλ+σbµ)].
Xaσ | aσ|
fect, which is stable for large next-nearest-neighborhop-
ping t. The phase transition between the two kinds of For normal antiferromagnetic spin-density wave(N-
′
SDW states is the topological quantum phase transition SDW) states, we have m = m < 0 and m =
1 2 1
↑ ↓ ↓
8
m >0, thus the resulting action vanishes. In contrast, Forourcase,theT-SDWstatehasquantizedHallcon-
2
whe↑nweconsiderthetopologicalantiferromagneticspin- ductance σ = 2e2 while the N-SDW state has van-
H h
density wave(T-SDW) states, the effective action reads ished quantized Hall conductance. In some sense, one
2ie2 may identify N-SDW and T-SDW states as the super-
SC′ S =Z d2xdτ[−4π ǫµνλ(Aµ∂νAλ+bµ∂νbλ)].(11) fluidphase andthe bosonicinteger quantumHallphases
in Ref. 49, respectively, according to distinct features
AsarguedinRef. 38,theChern-Simontermofb gives
µ on the quantized Hall conductance. Therefore, we may
rise to two gapless chiral modes, which only carry spin
conclude that the critical behaviors of topological quan-
degrees of freedom of physical electrons. However, since
tumphasetransitionbetweenN-SDWandT-SDWstates
these two modes do not carrychargedegreesof freedom,
couldfallintothe3D-XY universalclassthoughwecan-
they are not protected by the conservation of particle
notfindaunambiguousbosoniclocalorderparameterfor
number. Therefore, we suspect two chiral modes from
ourcase. Inourview,thesimilaritybetweenourcaseand
thespinfluctuation(b )willbegappedandnonoticeable
µ the bosonic integer quantum Hall transitionis indeed an
spin current could exist in T-SDW states.
interestingnewfindinganditisdesirabletoseemoreex-
amples where some ferminic theories can dual to certain
kinds of bosonic ones.
B. Spin fluctuation effect at the topological
quantum phase transition point and relation to 3D
XY universal class
C. Kane-Mele-Kondo lattice model
In this subsection, we devote to discuss the Spin fluc-
tuation effect at the topological quantum phase tran- A careful reader may wonder whether there exists a
sition point between N-SDW and T-SDW states(m = similar T-SDW state in the Kane-Mele-Kondo lattice
3√3t′ = J /4) in Sec. III. Here, the effective mass model studied in Ref.[36]. We have studied this ques-
−of Dirac a−ctiokn is m1 = m2 = J /2 < 0 and tion by using the same treatment in the present paper.
m1 = m2 = 0. Thu↑s, ψ1 ,ψ2↓ are−makssive and can Itisfoundthattheedgestatesresemblethecaseinusual
be↓safely in↑tegrated out whil↑e on↓e should not integrate 2+1D topological insulator, where the gapless helical
out ψ1 ,ψ2 due to gaplessnessof them. Then, following edge state exists and the spin Hall conductance is quan-
the tre↓atme↑nt of last subsection, one obtains tized due to the gaplessedge state. However,inour case
S =S +S +S , theantiferromagneticorderbreaksthetime-reversalsym-
′′ cs 1 2
↓ ↑ metry, thus the gapless helical edge state will be gapped
ie2
S = d2xdτ[− ǫµνλ(A ∂ A +b ∂ b )]. since it cannot be stable due to impurities and weak in-
cs µ ν λ µ ν λ
Z 4π teraction without the protection from the time-reversal
S = d2xdτ[ψ¯ γ (∂ ieA +ieb )ψ ], symmetry.29,30 Physically, the quantized spin Hall con-
1 1 µ µ µ µ 1
↓ Z ↓ − ↓ ductancecontributedbysuchgaplessedgestatewillalso
bedestroyedgenericallyandweconcludethatnosuchT-
S = d2xdτ[ψ¯ γ (∂ ieA ieb )ψ ], (12)
2 2 µ µ µ µ 2 SDW-like states could appear in the Kane-Mele-Kondo
↑ Z ↑ − − ↑
lattice model.
where we have integrated out two massive modes, which
leadstoaChern-SimontermforA andb ,respectively.
µ µ
To our surprise, one may note that the above action re-
semblestotheoneinRef. 49,wheresuchactiondescribes D. Bernevig-Hughes-Zhang model with Hubbard-U
the phase transition between the bosonic integer quan- term
tum Hall phases48 and superfluid phase. (Please note
that in our case, we have two flavor fermions with op- Recently,itisnotedthatinRef.[37],Bernevig-Hughes-
posite gauge charge while there exists only one flavor in Zhang model with an extra Hubbard interaction has
Ref. 49.) InRef. 49,thosebosonicintegerquantumHall been studied by applying the dynamical mean field the-
phaseshavequantizedHallconductanceσ =2ne2 with ory. Those authors find that the Hartree-Fock mean-
H h
n = 0,1,2,3,.... (The one with n = 0 just corresponds field theory cannot capture a topological antiferromag-
to the trivial Bose-Mott insulator.) Obviously, one ex- netic phase, which is verified to exist by their dynami-
pects there will be a usual 3D-XY transition between cal mean-field theory (DMFT) calculation. In contrast
the bosonic integer quantum Hall phases and the super- to the case of the just mentioned extended Bernevig-
fluidphasesincewehaveanaturallocalorderparameter Hughes-Zhang model, in the present paper, the T-SDW
ifapproachingthecriticalpointfromthewell-definedsu- state is well-captured by our mean-field decoupling and
perfluid phase. Interestingly, it is known that the above is further inspected by effective Chern-Simon actionand
fermionic action in fact provides an alternative descrip- chiraledge-statein Sec. III. We suspect thatsuchdiffer-
tion for the usual 3D-XY transition,49,56 thus the cor- ence may result from the different models one used. It
rectness of the fermionic effective action is justified and seems that the mean-field decoupling works more effec-
does not violate our basic physical intuition. tively in the Kondo lattice-like model than in extended
9
Bernevig-Hughes-Zhang model and it will be interesting Then, integrating out Dirac fermions one obtains
to clarify this point in future work.
S =NlnDet[γ (∂ ieA )+m]
eff µ µ µ
−
=NTrln[γ (∂ ieA )+m]
µ µ µ
−
VI. CONCLUSION =NTr[ln[γµ∂µ+m]+ln[1 ie(γµ∂µ+m)−1γµAµ]]
−
d3q
N A Π (q)A (A2)
Insummary,wehaveobtainedthe globalground-state ≃ Z (2π)3 µ µν ν
phase diagram of the topological Kondo lattice model
on the honeycomb lattice at half-filling by using an ex- whereΠµν(q)= −2e2 (2dπ3k)3Tr[ikmµ2γ+µ−k2mγνi(kmµ2++q(µk)+γµq)−2mγµ]=
tended mean-field decoupling. It is found that besides e2 d3k [ 1 R 1 ][ imq Tr(γ γ γ )]+...
the well-defined Kondo insulator and the normal anti- −2 (2π)3 m2+k2m2+(k+q)2 − λ µ ν λ ≃
e2mRǫµνλq is calculated at one-loop level. We
ferromagnetic SDW state, there can exist a novel topo- −8πm λ
logical SDW state with quantum anomalous Hall effect. ha|ve| also used the identity Tr(γµγνγλ) = 2iǫµνλ
ThetopologicalSDWstatecannotbe distinguishedwith with γ0 = τz, γ1 = τx and γ2 = τy while
the normalantiferromagnetic SDW state in terms of the arcsin( |q| )
d3k [ 1 1 ] = √q2+4m2 1
Landau-Ginzburg paradigm, specifically, the symmetry- (2π)3 m2+k2m2+(k+q)2 4πq ≃ 8πm
breaking based classification of states. However, such a fRor q m. Therefore, the effec|ti|ve Chern-Sim|on|
| | ≪ | |
novel SDW state can be indeed fully encoded by its low action Eq.[6] is obtained as
energy effective Chern-Simon action with gapless chiral
d3q Ne2m
edge-state. S = A − ǫµνλq A
eff Z (2π)3 µ 8π m λ ν
Moreover, the phase transition between the two kinds | |
imN
of SDW states is a topologicalquantumphase transition = d2xdτ[e2− ǫµνλA ∂ A ]. (A3)
µ ν λ
while afirst-orderquantumphasetransitionis foundbe- Z 8π m
| |
tween the Kondo insulating state and the antiferromag-
netic SDW states. Interestingly, the mentioned topolog-
Appendix B: Stability of the gapless chiral
icalquantum phase transitioncanbe described by Dirac
Edge-state
fermions coupled Chern-Simon gauge-field, which indi-
catessuchtopologicalquantumphasetransitionmayfall
into the 3D-XY universal class. It is expected both the In Sec.III, the decoupled gapless chiraledge states are
global ground-state phase diagram and the novel topo- described as follows
logical SDW state could be realized by experiments of 1
ultra-cold atoms on the honeycomb optical lattice.57,58 Sedge = dxdτ [ i∂τφI∂xφI +cI∂xφI∂xφI].
Z 4π −
We hope the present work may be helpful for further IX=1,2
studies on the interplay between conduction electrons To discuss its stability to weak impurities or interaction
and the densely localized spins for the honeycomb lat- effect, it is helpful to write down the corresponding fer-
tice. monic formalism59
He0dge = Z dx[ψI†(−icI∂x)ψI]. (B1)
ACKNOWLEDGMENTS IX=1,2
If the total particle number is conserved, in general,
The authors would like to thank Su-Peng Kou for il- the interacting term will be the following form59
luminating andhelpful communicationonrelatedissues.
H =H +H +H
The work was supported partly by NSFC, the Program int 1 2 3
ftorarlNUCnEivTe,rtshiteieFsuannddamtheentnaaltRioenseaalrpcrhoFguranmdsffoorrbthaesiCcerne-- H1 =g1Z dx[ψ1†ψ1ψ1†ψ1+ψ2†ψ2ψ2†ψ2]
search of China.
H2 =g2Z dx[ψ1†ψ1ψ2†ψ2]
Appendix A: Derivation of Chern-Simon action H3 =g3Z dx[ψ1†ψ1†ψ2ψ2+ψ2†ψ2†ψ1ψ1]. (B2)
Obviously, the dangerous H vanishes due to ψ ψ =
3 I I
Here, we would like to give a brief derivation of the ψI†ψI† =0. H1,H2 donotvanishbuttheycannotgapout
effectiveChern-SimonactionEq.[6]fromtheDiracaction the edge states since they only correspond to the usual
Eq.[5]. First, the Dirac action is written as forward scattering.
Besides these interacting terms, the mass term H =
m
S = d2xdτ [ψ¯aσ(γµ(∂µ ieAµ)+m)ψaσ].(A1) M dx[ψ1†ψ2+ψ2†ψ1], which canresultfromweakimpu-
Z Xaσ − rityRscattering, may be important when considering the
10
stability of the gapless chiral edge-state.59 However,due be easily compensated by adjusting the chemical poten-
to the chiral feature of the edge-state, such mass term tial. Therefore, we may conclude that the gapless chiral
does not lead to a gap but a shift of energy (the quasi- edge-state is at least stable under weak impurities or in-
particle energy spectrum is E = M+ck), which can teraction effect as one expects.
k
± ±
∗ [email protected] 30 X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057
† [email protected] (2011).
1 S. Doniach, Physica B and C 91, 231 (1977). 31 S.RachelandK.LeHur,Phys.Rev.B82,075106(2010).
2 S. Sachdev, Quantum Phase Transition, 2nd ed. (Cam- 32 M. Hohenadler, T.-C. Lang and F. F. Assaad, Phys. Rev.
bridge University Press, Cambridge, England, 2011). Lett. 106, 100403 (2011).
3 H. V.Lo¨hneysen, A.Rosch, M. Vojta and P. Wo¨lfle, Rev. 33 A. Ru¨egg and G. A. Fiete, Phys. Rev. Lett. 108, 046401
Mod. Phys79, 1015 (2007). (2012).
4 M. Vojta, J Low Temp Phys161, 203 (2010). 34 R. S. K. Mong, A. M. Essin and J. E. Moore, Phys. Rev.
5 J. Custers, P. Gegenwart, H. Wilhelm, K. Neumaier, Y. B 82, 075106 (2010).
Tokiwa, O.Trovarelli, C.Geibel, F.Steglich, C.P´epin and 35 A. M. Essin and V. Gurarie, Phys. Rev. B 85, 195116
P. Coleman, Nature (London) 424, 524 (2003). (2012).
6 J. Custers, P. Gegenwart, C. Geibel, F. Steglich, P. Cole- 36 X.-Y. Feng, J. Dai, C.-H. Chung and Qimiao Si,
manandS.Paschen,Phys.Rev.Lett.104,186402(2010). arXiv:cond-mat/1206.0979v1 (2012).
7 Y.Matsumoto,S.Nakatsuji,K.Kuga,Y.Karaki,N.Horie, 37 T. Yoshida, R. Peters, S. Fujimoto and N. Kawakami,
Y. Shimura, T. Sakakibara, A. H. Nevidomskyy and P. arXiv:cond-mat/1207.4547v2 (2012).
Coleman, Science331, 316 (2011). 38 J.He,Y.-H.Zong,S.-P.Kou,Y.LiangandS.Feng,Phys.
8 T.Senthil,S.SachdevandM. Vojta,Phys.Rev.Lett.90, Rev.B 84, 035127 (2011).
216403 (2003). 39 J. He, Y. Liang and S.-P. Kou, Phys. Rev. B 85, 205107
9 T. Senthil, M. Vojta and S. Sachdev, Phys. Rev. B 69, (2012).
035111 (2004). 40 A. Kitaev, the Proceedings of the L.D.Landau Memorial
10 C. P´epin, Phys.Rev.Lett. 94, 066402 (2005). Conference“AdvancesinTheoretical Physics”(2008) [See
11 K. S. Kim and C. L. Jia, Phys. Rev. Lett. 104, 156403 also arXiv:cond-mat/0901.2686v2 (2009)].
(2010). 41 A.P.Schnyder,S.Ryu,A.FurusakiandA.W.W.Ludwig,
12 T.GroverandT.Senthil,Phys.Rev.B81,205102(2010). Phys. Rev.B 78, 195125 (2008).
13 Y.Zhong,K.Liu, Y.Q.WangandH.-G.Luo,Phys.Rev. 42 X.Chen,Z.-G.Gu,Z.-X.LiuandX.-G.Wen,arXiv:cond-
B 86, 115113 (2012). mat/1106.4772v4 (2011).
14 H. Tsunetsugu, M. Sigrist and K. Ueda, Rev. Mod. Phys. 43 X. Chen, Z.-X. Liu and X.-G. Wen, Phys. Rev. B 84,
69, 809 (1997). 235141 (2011).
15 C. Lacroix and M. Cyrot, Phys.Rev. B 20, 1969 (1979). 44 X.-G. Wen,Phys. Rev.B 85, 085103 (2012).
16 G. M. Zhangand L. Yu,Phys. Rev.B 62, 76 (2000). 45 Z.-G. Gu and X.-G. Wen, arXiv:cond-mat/1201.2648v1
17 S. Capponi and F. F. Assaad, Phys. Rev. B 63, 155114 (2012).
(2001). 46 M. Levin and Z.-C. Gu, Phys. Rev.B 86, 115109 (2012).
18 H. Watanabe and M. Ogata, Phys. Rev. Lett. 99, 136401 47 Y.-M. Lu and A. Vishwanath, Phys. Rev. B 86, 125119
(2007). (2012).
19 G.-B. Li and G.-M. Zhang, Phys. Rev. B 81, 094420 48 T. Senthil and M. Levin, arXiv:cond-mat/1206.1604v1
(2010). (2012).
20 G.-M. Zhang, Y.-H. Su, and Lu Yu, Phys. Rev. B 83, 49 T. Grover and A. Vishwanath, arXiv:cond-
033102 (2011) mat/1210.0907v1 (2012).
21 J. Custers, P. Gegenwart, C. Geibel, F. Steglich, P. Cole- 50 Y.-M. Lu and D.-H. Lee arXiv:cond-mat/1210.0909v1
manandS.Paschen,Phys.Rev.Lett.104,186402(2010). (2012).
22 P. A. Lee, N. Nagaosa and X. G. Wen, Rev. Mod. Phys 51 Y.Zhong,K.Liu,Y.F.Wang,Y.Q.WangandH.-G.Luo,
78, 17 (2006). arXiv:cond-mat/1208.1332v2 (2012).
23 S.Saremi and P.A.Lee,Phys.Rev.B75, 165110 (2007). 52 N.Regnault, and B. A.Bernevig, Phys.Rev.X 1, 021014
24 Xiao-Gang Wen, Quantum Field Theory of Many-Body (2011).
Systems (Oxford Graduate Texts, NewYork, 2004). 53 F. D. M. Haldane, Phys. Rev.Lett. 61, 2015 (1988).
25 Z. Y. Meng, T. C. Lang, S. Wessel, F. F. Assaad, and A. 54 Using the ground-state energy of antiferromagnetic state
Muramatsu, Nature(London) 464, 847 (2010). (Eq.7)andKondoinsulator(Eq.8)inRef.[16],itiseasyto
26 B. K. Clark, D. A. Abanin and S. L. Sondhi, Phys. Rev. check that at the putative QCP Jc = 0.58D, there exists
Lett. 107, 087204 (2011). a jump about 0.1 for the first-order derivative of ground-
27 F. Mezzacapo and M. Boninsegni, Phys. Rev. B 85, state energy of antiferromagnetic state and Kondo insula-
060402(R) (2012). torwithrespecttotheKondocoupling.Thus,afirst-order
28 Y.Zhong,K.Liu, Y.Q.WangandH.-G.Luo,Phys.Rev. transition is clearly observed in the mean-field treatment
B 86, 165134 (2012). of Ref. [16].
29 M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 55 M. A.Continentino,Quantum Scaling in Many-Body Sys-
(2010). tems (World ScientificPress, Singapore, 2001).