Effects of short-ranged interactions on the Kane-Mele model without discrete particle-hole symmetry

0.32 MB·

by  Hsin-Hua Lai

Effects ofshort-ranged interactions onthe Kane-Melemodel withoutdiscrete particle-hole
symmetry
Hsin-Hua Lai
National HighMagneticFieldLaboratory, FloridaStateUniversity, Tallahassee, Florida32310, USA
Hsiang-Hsuan Hung
Department of Physics, The University of Texas at Austin, Austin, Texas78712, USA
(Dated:September16,2014)
4 Westudytheeffectsof short-ranged interactionsontheZ2 topological insulator phase, alsoknown asthe
quantum spin Hall phase, in the Kane-Mele model at half-filling with staggered potentials which explicitly
1
breaks the discrete particle-hole symmetry. Within Hartree-Fock mean-field analysis, we conclude that the
0
on-siterepulsiveinteractionshelpstabilizethetopologicalphase(quantumspinHall)againstthestaggeredpo-
2
tentialsbyenlargingtheregimeofthetopologicalphasealongtheaxisoftheratioofthestaggeredpotential

p strengthandthespin-orbitcoupling. Insharpcontrast, theon-siteattractiveinteractionsdestabilizethetopo-
e logicalphase. Wealsoexaminetheattractiveinteractioncasebymeansoftheunbiaseddeterminantprojector
S
quantumMonteCarloandtheresultsarequalitativelyconsistentwiththeHartree-Fockpicture.

3
1
I. INTRODUCTION j A,B ,whereAandB arethesublatticelabeling. The
∈ { }
] natural question is if the stabilization of QSH against other
l
e effectivegapclosingperturbationsdueto short-rangedinter-
Topologicalinsulators(TI)1–5areperhapssomeofthemost
- actions can exist in a KMH model with lower symmetries.
r intriguingstatesofmatterfoundinrecentyears.Muchinterest
t Toanswerthisquestion,atleastinsomesimplemodel,inthis
s intheseisinitiallymotivatedbythefirstexperimentalrealiza-
. workwefocusontheKMHmodelwithoutPHSathalf-filling.
t tion in HgTe/(Hg,Cd)Te quantum wells which shows quan-
a WeexplicitlybreakthePHSbyincludingthestaggeredpoten-
tized quantum spin Hall conductance protected by the time-
m tials in the originalKM model.9 It is well known that in the
reversalsymmetry(TRS).6–8Themostimportantofallisthat
- noninteractinglimit,themodelcontainsbothQSHandtrivial
d the existence of the topological state and many of its prop- phasedependingontheratioofthestrengthofthestaggered
n erties can be well understood in some noninteracting mod-
potentials(m )andthatofspin-orbitalcouplings(λ ). The
o els. Amongthem, theZ topologicalinsulator(TI)orquan- a so
2 phasetransitionbetweenthesetwophaseshappenswhenthe
c tum spin Hall (QSH) insulator can be realized in the non-
[ interacting Kane-Mele (KM) model9. The KM model can bacatniodngsa,pitsicsloesxep.ecItnedthtehaptrethseenbceanodfgthapesshgoetrt-rreannogremdaliinzteedr-,
4 be described as two copies of the original Haldane model
whichcausestheshiftofthephaseboundary.WithinHartree-
v on the two-dimensional (2D) honeycomb lattice10 with the
Fockmean-fieldapproach,23wefindthattheshort-rangedre-
7 usual real nearest-neighbor hopping and imaginary second-
6 neighborhoppingwhicharisesfromthe spin-orbitcouplings pulsive interactions increase the threshold value of ma/λso
1 towidentheQSHregime,suggestinginthismodeltheQSH
(SOC).DuetothesimplicityoftheKMmodel,ithasrecently
2 stateismorestabilizedagainstthestaggeredpotentialsbythe
served as the theoretical framework to study the phase dia-
. short-rangedrepulsiveinteractions.Onthecontrary,theshort-
1 gram of the interacting Z TIs.11–21 Based on the numerical
0 2 rangedattractive interactionsdestabilize the QSH, making it
studiesontheKane-Mele-Hubbard(KMH)model,severalex-
4 morefragiletothestaggeredpotentials.
oticquantumphaseshavebeenproposed,includingthequan-
1
: tum spin liquid (QSL).17,22 Even though several phase dia- InordertogobeyondtheHartree-Fockmean-fieldpicture,
v
grams have been proposed in the KMH model, so far there
i numericaltoolsarehighlydemanding. Thedeterminantpro-
X havejusta fewstudiesfocusingonthe interactingeffectson
jector QMC can be used to detect the topologicalphase but,
the stabilizationofthe QSH. Specifically, thequestionabout
r however,inthecasewithrepulsiveinteractionsitsuffersfrom
a if the short-ranged interactions enlarge (stabilize) the QSH
the well-known sign problem. On the other hand, the QMC
regime found in the noninteracting KM model has not been
canstillbeperformedontheattractiveinteractionside.Within
paidmuchattentioninmostofthepreviousstudies.
the Hartree-Fock picture, opposite to the result in the repul-
Recently, Ref. 19 studies the KMH model with third- siveinteractioncase,theon-siteattractiveinteractionstendto
neighborhoppingsathalf-fillingusingdeterminantprojector shrinktheQSHregime.ThispictureisconfirmedbytheQMC
quantum Monte Carlo (QMC) and conclude that the short- analysis in Sec. III, which may imply that our Hartree-Fock
rangedon-siterepulsiveHubbardinteractionstendtopushthe picturein the repulsiveinteractionside ispossibly plausible.
QSH phase boundaryto larger threshold values of the third- In appendix A we also perform renormalizationgroup (RG)
neighborhoppingsresultinginthestabilizationoftheQSHin analysis at the tree-level in the gapless critical phase at the
a larger parameter regime due to the interactions. The gen- phaseboundarybetweenthetopologicallytrivialandnontriv-
eralized KM model in Ref. 19 as well as the original KM ialphaseswherethebandgapsclosetoformDiracpoints.We
modelathalf-fillingpossessesthediscreteparticle-holesym- concludethatsuchacoarse-grainedpicturecannotcorrectly
metry (PHS),13 c c† ,c† c with for predictashiftofthephaseboundary.
{ jα → ± jα jα → ± jα} ±
2
symmetry,whiletheTRSandPHSstillremain.Thepresence
ofthestaggeredpotentialsfurtherbreakthePHS.
For the case of the U > 0, the previous numerical
studies12–17 on the KMH model shows that the magnetic
transition to the topologically trivial insulator phase such as
antiferro-magnetic Mott insulator (AFM) happens at a quite
largeU/t value. Thecriticalvalueofthe Hubbardrepulsion
forλ /t = 0.1isroughlyU /t 5,17 andthecriticalU /t
so c c
≃
isevenlargerforlargerλ /t. Ontheotherhand,forU < 0,
so
FIG.1. Illustrationofthehoneycomblattice. Therearetwosublat- the theoretical mean-field studies by Yuan et. al24 suggests
ticesperunitcelllabeledasAandB.~e1/2 =(±1/2,√3/2)arethe thatthe criticalvalue ofthe attractiveHubbardinteractionis
two vectors connecting the same sublattices in different unit cells. U /t< 2forfinitespin-orbitcouplings.IfU issmallerthan
c
Wesetthelatticeconstanttobe1. the critic−al value, there would be a possible phase transition
tothesuperconductingstatewhosebulkisstillinsulatingbut
superconductivityappearsneartheedges.24Suggestedbythe
This paper is organized as follows. In Sec. II we de-
previousnumericalstudies in both U-axis (negativeor posi-
fine explicitly the model Hamiltonian that we will study. In
tive), in this work we restrict our analysis within U/t < 2
Sec.IIAweintroducetheHartree-Fockapproachtodecouple | |
to ignore possibly magnetic phase transitions and within the
theshort-rangedinteractionsandinSec.IIBwenumerically
regimeitisappropriatetoignorethemagneticphases.
solve for the transition point between the QSH and the triv-
For clarification, from now on we replace the site label-
ialphasebyfindingthegapclosingpointself-consistently.In
ing j with j = r,a , where r runsover the Bravais lattice
Sec.IIIwepresenttheQMCresultsintheKMHmodelwith { }
of unit cells of the honeycombnetwork and a runs over the
attractiveinteractions.InSec.IVweconcludewithsomedis-
two sites (A and B) in the unit cell. We define the poten-
cussions. In Appendix A we show, at the long-wavelength
tial on sublattice A(B) as m (m ). In the noninteracting
description, the tree-level RG analysis on the critical phase A B
limit, the Hamiltonian in the momentum space is H =
at band gap closing point. In Appendix B we provide the KMs
Hartree-Fockmean-fieldstudiesonthecasewiththemoreex- k∈B.Z.Ψ†khKMs(k)Ψk,with
tended interactions, includingboth on-site interactionU and P m tf(k)
thenearest-neighborV. hKMs(k)= tf∗(Ak) mB ⊗12×2+
(cid:18) − (cid:19)
2λ g(k) 0
+ so σ .(3)
II. KANE-MELE-HUBBARDMODELWITHSTAGGERED 0 2λsog(k) ⊗ z
(cid:18) − (cid:19)
ON-SITEPOTENTIALS
σ is the Pauli matrix and ΨT
z k
≡
c (k,A) c (k,B) c (k,A) c (k,B) . f(k)
↑ ↑ ↓ ↓
Themodelwearefocusinginthisworkisthehoneycomb 1 + eik·~e1 + eik·~e2, and g(k) sin(k ~e ) sin(k≡
Kane-Mele-Hubbardmodelin the presence of staggered po- (cid:0) ≡ (cid:1) · 1 − ·
~e ) sin(k (~e ~e )).Withoutlackofgenerality,wechoose
tentials at half-filling9. The honeycomb lattice is shown in 2 − · 1− 2
m ,m ,andλ tobepositive.
Fig.1andthemodelHamiltonianisH =H +H ,where A B so
KMs U
H =t c† c +iλ σc† ν c
KMs jσ kσ so jσ jk kσ A. Hartree-Fockapproach
hXjkiXσ hhXjkiiXσ
+ ǫ m c† c , (1) Inorderto take theon-siteinteractioninto considerations,
j j jσ jσ
we use Hartree-Fock mean-field decoupling approach to de-
j σ
XX
coupleitas
H =U n n , (2)
U j↑ j↓
j H =U n (r,a)n (r,a)
X U ↑ ↓
withǫ = 1forsublatticej A,B . Theν = +1( 1) (Xr,a)
j jk
for (counte±r-)clockwise secon∈d-n{eighb}or hopping, and U−is U
n(r,a) n(r,a)+ s (r,a) s (r,a) ,(4)
the strength of the interaction that can be positive and nega- ≃ 2 h i h z i z
tive.ThepositiveU correspondstotheon-siteHubbardrepul- (Xr,a)(cid:20) (cid:21)
sion. We theoreticallyconsiderthe negativeU side notonly with n n +n , and s n n . We have explicitly
↑ ↓ z ↑ ↓
forthe completenessofthisworkbutalso forcomparingthe neglecte≡d the constant n 2≡appea−ring in the Hartree-Fock
j
h i
results obtained in the mean-field analysis in Sec. IIB with decouplingsince they only shift the total energy. The terms
those obtained in the unbiased determinant projector QMC c†c alsovanishsincetheydonotconserveSz. Sincethere
h σ σ¯i
studiesinSec.III.TheQMCworksinthenegativeU sidebut is no local magnetic field at each site, the local magnetiza-
suffersfromthesignprobleminthepositiveU side. Without tioniszero,whichmeansthesecondtermintheEq.(4)van-
the staggeredpotentials, thefinite spin-orbitcouplingsbreak ishes. The on-site interaction within the Hartree-Fock pic-
the spin SU(2) down to U(1) and also break the sublattice tureaddsdensitymodulationstothediagonalelementsinthe
3
KMs matrix (3). In addition, due to the translational sym- 7 λso=0.4t 5 λso=0.4t
mtheattrythehnd(ren,asi)tiies=athnsi(tae)iA≡andhnBaiaarneddtihffeercernutcidaulepotointthies —λλm6.65 Trivial —λm4.5 Trivial
presenceofthestaggeredpotential. Inthemomentumspace, s5o.5 λso4
5
h(k)=h (k)+h (k),withh definedinEq.(3)and Topological (QSH) 3.5 Topological (QSH)
KMs u KMs 4.5
4 3
hu(k)= U2 hn0Ai nB ⊗12×2. (5) 0 0.5 (aU)1/t 1.5 2 0 0.5 (b-U)1/t 1.5 2
(cid:18) h i(cid:19)
Since the phase boundary between the QSH and the triv-
FIG. 2. The phase boundary between the topologically nontrivial
ialphaseintheKMmodelislocatedattheparameterregime phase (QSH)and thetrivialphase withintheHartree-Fockpicture.
in which the band gaps close, the key point is to find when (a)Thenumerical dataintherepulsiveinteractioncase, during the
the band gaps close in the presence of the short-ranged in- numerics, wesett = 1, λ = 0.4andU > 0. TheHubbardre-
so
teractions. Within Hartree-Fock picture, since no symmetry pulsiontendstoscreenthestaggerpotentialstopushtheboundaryof
isspontaneousbroken,thebandgapsassociatedwiththefull thenontrivialphasetowardthetrivialside,whichwidenstheregime
Hamiltonianmatrix still close at K = K′ = 4π/3,0 in ofthenontrivialphase.(b)Thenumericaldateintheattractiveinter-
theBrillouinzone(B.Z.).NearKandK−′,theoff{-diagona}lel- actioncase, U = −|U|. Contrarytotheresultin(a),theattractive
interactionsenhancesthestaggeredpotentialandshrinktheregime
ementsinEq.(3)becomelinearinkandvanishexactlyatK
ofthetopologicalphase.
andK′. SincethetwopointsarerelatedbytheTRS,thegaps
mustsimultaneouslycloseatbothKandK′. Itisthensuffi-
cienttoexaminethegapnearKonly.FocusingonK,wefind
TheattractiveU ontheotherhandenhancesthestaggeredpo-
thatthetwoofthefourbandswitheigenvaluesE = m
2λ g(K)+ U n andE = m +2λ g(K)1+ U nA − tentials,whichleadstotheoppositeresultshowninFig.2(b).
so 2h Ai 2 − B so 2h Bi Inordertohaveamorecompleteandunbiasedanalysis,next
would get inverted by tuning the mass m and λ . We
A/B so section we perform QMC to compare qualitatively with the
thendefinethegapfunction∆(K)as
resultsobtainedintheHartree-Fockpicturehere. Forthere-
pulsiveinteractioncase,thereisasignproblemandcannotbe
U
∆(K)=m +m 4λ g(K)+ n n , accessedbytheQMC.Wethusfocusontheattractiveinterac-
A B so A B
− 2 h i−h i
(cid:18) (cid:19) tioncase inwhichtheQMCanalysisis freeofsign-problem
(6)
inSec.III
withg( K)= 3√3/2.When∆(K)>0,thebandsarenot
± ±
invertedand we are in the topologicallytrivialphase. When
III. SIGN-FREEDETERMINANTPROJECTORQMC
∆(K) < 0,thebandsareinvertedandweareinthetopolog-
STUDIESINTHEKANE-MELEMODELWITHON-SITE
ical phase. The transition happens at ∆(K) = 0 when the
ATTRACTIVEINTERACTION
bandgapsclosetoformDiracpoints.
Inthissection,wewillstudytheKMmodelwithstaggered
potentialsusingtheunbiasedprojectorQMCmethod. Dueto
B. Self-consistentnumericalcalculations
asignproblemintheQMCinthecaseoftherepulsiveHub-
bardinteraction,Eq.(2)withU >0,wewillfocusonthecase
Forsimplicity,intheself-consistentnumericalcalculations,
withtheattractiveinteractionandcomparetheresultqualita-
we set m = m λ . Before jumping to the nu-
A B ≡ m tively with that in the Hartree-Fockpicture. We remark that
merical calculations, we first give a simple physical picture
the presence of the staggered potentials breaks the PHS, so
here.Sincethepresenceofthestaggeredpotentialslowersthe
even at half-filling, the QMC result in the attractive interac-
chemicalpotentialatsublatticeB,thetotalnumberdensityat
tioncaseisdifferentfromthatintherepulsivecase. We will
sublatticeBisexpectedtobehigherthanthatinsublatticeA.
seethatthebehaviorgivenbytheQMCisconsistentwiththat
ThecontributionfromthelastterminEq.(6)isthennegative
from the Hartree-Fockanalysis, where the attractive interac-
(positive)forrepulsive(attractive)interaction. Forfixedλ ,
so tionsdestabilizetheQSHphase.
the gap function reaches zero at larger (smaller) mass com-
In the QMC method, the expectationvalue of an arbitrary
pared with the case without the on-site repulsive (attractive)
observableOˆ isevaluatedas
interaction. Therefore,theregimeofthetopologicalphaseis
ternalcatrivgee)di(nstherraucntkio)nin. thepresenceoftheon-siterepulsive(at- Oˆ = lim hψT|e−Θ2HOˆe−Θ2H|ψTi, (7)
h i Θ→∞ ψT e−ΘH ψT
In the numerical calculations, the honeycomblattice con- h | | i
sists of200 200unitcellsandwe choosetosett = 1and whereΘistheprojectiveparameter.Thetrivialwavefunction
×
λ = 0.4. The result is shown in Fig. 2. We can see that ψ isrequiredtohavenonvanishingoverlapwiththeground
so T
| i
the on-site repulsive interaction widens the regime of QSH, statewavefunction ψ ,i.e. ψ ψ = 0. Numerically,the
0 T 0
Fig. 2(a). The physical picture is that the on-site Hubbard projectionoperatore|−ΘiH is dhiscr|etizied6 intoe−∆τH, written
repulsion screens the staggered potentials resulting in stabi- ase−ΘH =(e−∆τH)M,whereΘ=∆τM;M isthenumber
lizing the topologicalphase against the staggered potentials. of time slices and ∆τ is chosen as a small number. In the
4

first-orderSuzuki-Trotterdecomposition,wecanhave
1.0
e−∆τH e−∆τHKMse−∆τHU, (8) SO=0.4t U= 0, 6x6
≃ 0.8 U= -2t, 6x6
whereH istheHamiltonianoftheKMmodelwithstag- U= -4t, 6x6
KMs U= -4t, 12x12
geredpotentials.HU istheattractiveHubbardon-siteinterac- 0.6
tion C
0.4
H = U n n , (9)
U j↑ j↓
−| |
j
X 0.2
wheren = c† c ,n = c† c . Athalf-filling,we
j,σ j,σ j,σ j σ j,σ j,σ
canrecastH as 0.0
U 0.0 0.5 1.0 1.5 2.0 2.5
P
H = |U| (n 1)2, (10) m/t
U j
− 2 −
Xj FIG.3. ThespinChernnumberCσ vsthemagnitudesofstaggered
potentialsλ /tusingQMCatλ = 0.4t. Thedotlineindicates
m SO
Fortheattractiveinteractioncase,wecanimplementtheexact thenoninteractingphaseboundaryλc =3√3λ =2.0784t. The
Hubbard-Stratonovichtransformation25 solidsymbolsdenotethe6 6QMCmresultsatUS=O 0, 2tand 4t.
× − −
e−∆τHU =e∆τ|U2|(nj−1)2 = 1 esα(nj−1), (11) Thehollowsymbolsdenotethe12×12QMCatU =−4t.
2
s=±1
X
a propertopologicalmeasurementto probe the phase transi-
|U|∆τ
where α = arcCosh(e 2 ). By the implementation, the tion. The spin Chern number formalism has been proposed
denominator of Eq. (7), also named the projector partition
usingtheQMCmethodtoacquirethezero-frequencysingle-
function,26canbenumericallyevaluatedas13,17,27 particleGreen’sfunctionsG(iω = 0,~k)20,28–31 andthencon-
ψ e−ΘH ψ structprojectoroperatorsintermsoftheR-zeroeigenstatesof
T T
h | | i theGreen’sfunctionstoevaluatethespinChernnumber32
M
∼=hψT| e−∆τHKMse−∆τHU,τ|ψTi C = i d2k ǫµνTr[P (k)∂ P (k)∂ P (k)], (14)
τY=1 σ 2π σ µ σ ν σ
M Z
∼ Tr e−∆τPi,jc†i,σ[HσKMs]ijcj,σeαsi,τ(ni,σ−12) wherePσ(k) = n|vnkσihvnkσ|isthesingleparticlespec-
{Xsi,τ}Yσ (cid:16)τY=1 (cid:17) tral projector onPto R-zero modes; i.e., G(0,~k)|vnkσi =
λ v and λ > 0. Although the spin Chern num-
nkσ nkσ nkσ
| i
det O [s ] det O [s ] berhasbeenshowntosufferstrongfinite-sizeeffect,itisstill
↑ i,τ ↓ i,τ
∼ ( ) usefultodeterminethetopologicalphaseboundaryinthein-
{Xsi,τ} (cid:16) (cid:17) (cid:16) (cid:17)
teractioncase by observingthe dramaticchangein the topo-
= w[si,τ], (12) logicalnumber.20,28–31
{Xsi,τ} TheQMCresultsareshowninFig.3.Thespin-orbitalcou-
plingisused atλ = 0.4t. Inthe noninteractinglimit, the
SO
where det Oσ[si,τ] is the matrix determinant of topologicalphaseboundaryisidentifiedatλc =3√3λ =
m SO
e−∆τHKMse(cid:16)−∆τHU w(cid:17)ith fermion trace ”Tr” and at a 2.0784t,10,33 depictedasthedotlineinFig. 3. Asλm < λcm,
given auxiliary configuration s . In the QSH with the the system is a QSH (left hand side of the dot line); other-
i,τ
{ } ∗ wise it is a trivial insulator. It is obviousto see that C has
attractive interaction, we have O [s ] = O [s ] . σ
↑ i,τ ↓ i,τ strongfinite-sizeeffect,sinceitispoorlyquantized.However,
Therefore, the weight w[si,τ] > 0 and the sam(cid:16)pling in E(cid:17)q. evenonthe noninteracting6 6 clusterat the criticalpoint,
(12) is sign-free. In the current literature, ∆τt = 0.05 and asignificantdropisclearlyse×enandthelocationisconsistent
Θt = 40areusedthroughthecontent. Theeigenstatesofthe with the analytical prediction. Upon introducing the attrac-
noninteracting Hamiltonian HKMs is used as the trial wave tive interaction, the topological phase boundary shifts away
function. from λc . One can observe that at U = U = 2t and
To characterize the QSH state and a trivial insulator, we 4t, thme critical points can be roughlyide−nt|ifie|d at−1.4t and
usuallyneedtoevaluatetheZ2 topologicalinvariant.1 Inthe −0.9tbyQMCsimulation. Uponincreasingthestrengthofthe
KM model, however, the existence of the staggered poten- Hubbard interaction, the critical point moves towards to the
tialbreakstheinversionsymmetry;thustheconventionalap- QSHphase,i.e.theattractiveinteractiondestabilizestheQSH
proachtoevaluatetheZ2index,∆, phase. ForthecaseofU/t = 2,wecanseethatthebound-
ary is roughly at λc /λ −1.4/0.4 = 3.5, which is quite
(−1)∆ = η˜i (13) closetotheHartreem-Focskoan≃alysisfocusingontheKpointin
ki∈YTRIM B.Z.inSec.IIA,λc /λ 3.4.
m so ≃
is not applicable. However, the model still does not break On the other hand, the locationof the boundaryis subject
the S conservation, and thus the spin Chern number C is toweakerfinite-sizeeffect.ForU = 4t,thelocationswhere
z σ
−
5
the spinChernnumbersdropfor6 6 and12 12clusters aU-V model(includingbothon-siteU andnearest-neighbor
× ×
are fairly close. As a consequence, one can merely perform V).InAppendixBwecheckthecasewithbothon-siterepul-
theQMC simulationona small-size clusterto determinethe sive (attractive)HubbardU anda nearestneighborrepulsive
topological phase boundary. Moreover, in comparison with (attractive)V. Theinclusionofthenearest-neighborV com-
the6 6and12 12clusters,thecomputedspinChernnum- plicatestheanalysisandwefindthatwhetherornotthetopo-
× ×
bersshowclosertothequantizednumberwithincreasingsys- logicalphaseisstabilizedduetotheshort-rangedinteractions
tem sizes, so the poor quantization is expected to vanish in dependsonthedetailsofthecompetitionbetweentheon-site
the thermodynamic limit. Therefore, by the unbiased QMC U and the nearest-neighborV. From the low energy analy-
simulation,wecanarriveatasummarythattheattractivein- sisfocusingonKpointintheB.Z., weconcludethatwithin
teractionbringsaminuseffecttothemass,whichisconsistent theHartree-FockpictureifU isdominantoverV (U > 6V )
withtheHartree-Fockapproach. thequalitativeresultobtainedfromthecasewithonlyon-site
U,repulsive(attractive)interactionsstabilize(destabilize)the
topological phase, is still correct in the U-V model. How-
IV. DISCUSSION ever, from the studies of the Kane-Mele-U-V model, it may
suggestalong-rangedrepulsiveinteractionsuchasCoulomb
interactionthatdecaysveryslowmaycompletelydestabilize
In this work, we study the effects of the short-ranged in-
theQSHphase.
teraction on the Kane-Mele model with staggered potentials
As a final remark, there is a recent paper addressing the
which breaks the discrete particle-hole symmetry. Within
Hubbardinteractioneffectsonthetopologicalinsulatorsprop-
the Hartree-Fock mean-field approach, we conclude that the
ertiesusingtheslave-rotorformalism.38Inthatapproach,both
short-rangedrepulsiveinteractionshelpstabilizethetopolog-
thespinorbitalcoupling,λ ,andthestaggeredpotential,λ
icalphase(QSH)againstthestaggeredpotentialsbyenlarging so m
arerenormalizedandthesituationofwhetherornottheHub-
the regime of the topologicalphase (QSH) along the axis of
bardinteractionsstabilizetheQSHphaseisnotclearyet,de-
theratioofthestaggeredpotentialstrengthandthespin-orbit
pendingonthedetailsoftherenormalizedratiosofλ∗ /λ∗ .
coupling. In sharp contrast, the short-ranged attractive in- so m
Though, it is very likely that the results obtained in that ap-
teractionsdestabilizethetopologicalphase(QSH),makingit
proach are consistent with what we find in this work. Be-
morefragiletothestaggeredpotentials. BeyondtheHartree-
sides, the slave-rotor formalism39 can also be applied to the
Fock picture, we study the interaction effects using the pro-
caseinthegeneralizedKane-Melemodelwiththird-neighbor
jector QMC. Due to the sign problem, the KM model with
hoppings20 in which the Hartree-Fock mean-field approach
repulsive interaction can not be accessed by the QMC, and
failstopredictanyboundaryshiftduetothepresenceofHub-
thus we focus on the case of the attractive interaction. The
bardinteractionfoundintheQMCstudies.
QMC results show that the attractive interactions shrink the
QSH regime, which is consistent with that in the Hartree-
Fock approach. The qualitative consistency between these
twoapproachesinthecaseofattractiveinteractionmayimply ACKNOWLEDGMENTS
thattheHartree-Fockanalysisin therepulsiveside isplausi-
ble. Because QMC suffers from sign problem in this case,
One of the authors, H.-H. Lai, would like to thank Kun
other numerical tool such as Dynamical Mean Field Theory
Yang for helpful discussions. H.-H. Lai acknowledges the
(DMFT)16,34–37 maybeusedtoconfirmourconjectureinthe
support by the National Science Foundation through grant
futurework.
No. DMR-1004545. H.-H. Hung acknowledges compu-
The boundaryshifts due to the presence of the on-site in-
tational support from the Center for Scientific Computing
teraction in this model can not be explained by the contin-
at the CNSI and MRL: an NSF MRSEC (DMR-1121053)
uumlow-energytheoryatthecriticalphaseinwhichthegaps
and NSF CNS-0960316, as well as the support from ARO
closetoformDiracpoints. Explicitly,inAppendixAwefo-
GrantNo. W911NF-09-1-0527,and Grant No.W911NF-12-
cus on the low-energy descriptions at the critical phase and
1-0573fromtheArmyResearchOfficewithfundingfromthe
performtree-levelRGanalysis. Thestraightforwardthoughts
DARPAOLEProgram.
isthateventhoughthelocalfour-fermioninteractionsare ir-
relevant in the critical phase, before they flow to negligible
valuesunderRGtheycanstillgenerateafinitebilinearmass
term,whichcanpossiblyshiftthephaseboundary. However, AppendixA:RGanalysisofthecriticalphaseintheKane-Mele
we find that the tree-level RG correctionscompletely cancel modelwithweakU inthepresenceofstaggeredpotentials
eachother,whichgivesnogenerationofabilinearmassterm.
Hence we concludethat the boundaryshiftcan only be cap- In this model, since Sz is still conserved, the spin-up and
turedbythephysicsofthelatticemodelandcannotbecap- spin-down Hamiltonian can be treated separately. For each
turedby the coarse-grainedcontinuumtheoryaroundK and spin species, we can diagonalize the Hamiltonian matrix for
K′. spectra. Thereare 2 bandsforeach spin species. Thebands
Inthiswork,weonlyconsidertheon-siteinteractioneffects can be characterized by the eigenvector-eigenenergy pairs
ontheKane-Melemodelwithstaggeredpotentials.Itisinter- ~vα(k),ǫα(k) ,whereb=1,2arebandindices.TheHamil-
{ b b }
esting to consider a more extended interaction case such as toniancan be diagonalizedby rewritingthe originalfermion
6
fieldsintermsofthecomplexfermionfieldsdα(k)inthedi- sionaroundK′withsmallmomentumshiftδkgives
b
agonalbasis,
HK′ ≃ vF|δk| ψ1†L↑(δk)ψ1L↑(δk)−
c (r,a)= 1 vα(k,a)dα(k)eik·r,(A1) δXk<Λ (cid:20)
α rNuc bX=1,2k∈XB.Z. b b −ψ2†L↑(δk)ψ2L↑(δk) , (A6)
(cid:21)
fwehrmerieonNfiuecldisftshaetisnfiuemsbtheer uosfuaulniatntciecllosmamndutatthieoncroemlaptiloenx cwahnedreefisnimeialasrilmyiwlaertdraenfisnfeodrmda↑bt(iKon′t+otδhke)re≡alsψpbaLc↑e(aδska).boWvee,
{dαb†(k),dαb′′(k′)} = δαα′δbb′δkk′. Intermsofthenewcom- and therefore the spin-down fermion field can be effectively
plexfermionfields,theHamiltonianbecomes expressedas
c (r,a) v (a)ψ (r)eiK′·r, (A7)
H = ǫα(k)dα†(k)dα(k). (A2) ↑ ≃ bL↑ bL↑
KMs b b b b=1,2
b=1,2α=↑,↓k∈B.Z. X
X X X
withv↑(K′+δk,a) v (a),LlabelingthevalleyatK′,
Atthe criticalphase(λc = 3√3λ ), the gapscloseat mo- andremb emberK′ = ≡Kb.LT↑heactionforthelow-energyde-
m so −
mentums K and K′ = K. Around these points, only the scriptionis
−
spin-down fermions are gapless at K and spin-up fermions
d2qdω
aregaplessatK′. Asfarasthelong-wavelength(low-energy) S = ψ† (q)( iω)ψ (q)+H ,(A8)
descriptionisconcerned,wecanfocusontheKandK′points 0,P Z (2π)3 h bPαP − bPαP Pi
andperformexpansionaroundthesespointsbyintroducinga withP =R/L=K/K′andα = / andweuse2+1
R/L
smallmomentumshiftδk. dimensionalvectorq representingfreq↓uen↑cyandmomentum
For the low-energy description at momentum K, we find (ω, q). WecanalsodefinetheGreen’sfunctionsas
thatonlythespin-downfermionsaregaplessandcanexpan-
sion around K by introducing k = K + δk, with |δk| < hψb†L↑(q)ψbL↑(q′)i=hψb†R↓(q)ψbR↓(q′)i
Λ, Λ≪|K|,gives = iω−(−1)bvF|q|δ(3), (A9)
(iω)2 (v q)2 qq′
F
− | |
HK ≃ vF|δk| ψ1†R↓(δk)ψ1R↓(δk)− and we introduce the abbreviation δ(3) = (2π)3δ(ω
|δk|<Λ (cid:20) qq′ −
X ω′)δ(2)(q q′).
−
ψ† (δk)ψ (δk) , (A3) In order to write down the generalexpressionof the four-
− 2R↓ 2R↓ fermion interactions, we need first to obtain the symmetry
(cid:21)
transformation of the fields defined above. There are Sz-
whereweintroducedd↓(K+δk) ψ (δk)withRlabeling conservation,U(1)-charge,TRS, andC3 inthissystem. Ex-
b ≡ bR↓ ceptTRS, the other else symmetrytransformationsare quite
thevalleyatKandv √3t/2istheFermivelocityofeach
F ≡ transparent. Let’sfocusonthe symmetrytransformationun-
bandatK. Itismoreconvenienttotransformthecontinuum
derTRS( ),andwefind
fieldsdefinedabovetorealspace,definingfields T
v↑∗(a) ψ↑ −1 = v↓ (a)ψ↓ , (A10)
bL T bLT − bR bR
ψbR↓(r)= 1 eiδk·rψbR↓(δk). (A4) vb↓R∗(a)Tψb↓RT−1 =vb↑L(a)ψb↑L. (A11)
N
r uc |δXk|<Λ With TRS, the eigenvector-eigenvalue pairs have the prop-
erty,~v↑(k) = [~v↓( k)]∗ and ǫ↑(k) = ǫ↓( k), which gives
Therefore, in the low-energy description, we can re-express v↑∗(a)b=v↓ (a)b. W−ecanusethbepropertibes−abovetosimplify
thespin-downfermionfieldas bL bR
theTRStransformationin(A10)and(A11),butasfarasthe
RG analysispresentedbelow is concerned,we don’tneed to
c (r,a) v (a)ψ (r)eiK·r, (A5) dothat.
↓ bR↓ b↓
≃ The general expressions of the local four-fermion inter-
b=1,2
X actions in terms of the continuum fields defined above are
shownbelow. Forsimplicityintheexpression,wedefinebe-
wherewedefinedvb↓(K+δk,a)≡vbR↓(a). lowfbPα(a)≡vbαP(a)ψbαP(r),andthelocalfour-fermionac-
Similarly,wecanalsoobtainthelow-energydescriptionat tioncanbewrittenas(repeatedameanssummationoverthe
K′. AtK′,onlythespin-upfermionsaregaplessandexpan- eigenvectorelements)
7
S = ωa f† (a)f (a)f† (a)f (a)+ωa f† (a)f (a)f† (a)f (a)+
int 11 1L↑ 1L↑ 1R↓ 1R↓ 22 2L↑ 2L↑ 2R↓ 2R↓
+ωa f† (a)f (a)f† (a)f (a)+f† (a)f (a)f† (a)f (a) +
12 1L↑ 1L↑ 2R↓ 2R↓ 1R↓ 1R↓ 2L↑ 2L↑
h i
+λa (f† (a)f (a)f† (a)f (a)+f† (a)f (a)f† (a)f (a))+H.c. +
12 1L↑ 1L↑ 1R↓ 2R↓ 1R↓ 1R↓ 1L↑ 2L↑
h i
+λa (f† (a)f (a)f† (a)f (a)+f† f (a)f† (a)f (a))+H.c. +
21 2L↑ 2L↑ 1R↓ 2R↓ 2R↓ 2R↓ 1L↑ 2L↑
h i
+ua f† (a)f (a)f† (a)f (a)+H.c. +ua f† (a)f (a)f† (a)f (a)+H.c. , (A12)
12 1L↑ 2L↑ 1R↓ 2R↓ 21 1L↑ 2L↑ 2R↓ 1R↓
h i h i
and we remarkthatin the presenceof the on-site interaction wherethesubscript>meansmomentumshellintegralofthe
U,Eq.(2),allthebarecouplingsabovearesimplyequaltoU. fast-momentummodes.
The RG analysis in a nutshell is to integral out the fast-
momentummodesdefinedwithinamomentumshellbetween
[Λ/b,Λ],withb edℓ 1+dℓslightlybiggerthanone.The
≡ ≃
mathematicalformatthetree-levelis
S = S , (A13) Atthetree-level,wefindthecorrectionsare
eff,< int >
h i
Λ2 Λ2
S = dℓ ωa v (a) 2 ωa v (a) 2 f† (a)f (a)+ dℓ ωa v (a) 2 ωa v (a) 2 f† (a)f (a)
h inti> −4π 11 1R↓ − 12 2R↓ 1L↑ 1L↑ 4π 22 2R↓ − 12 1R↓ 2L↑ 2L↑
(cid:20) (cid:21) (cid:20) (cid:21)
Λ2dℓ ωa (cid:12)(cid:12)v (a)(cid:12)(cid:12)2 ωa (cid:12)(cid:12)v (a)(cid:12)(cid:12)2 f† (a)f (a)+ Λ2dℓ ωa (cid:12)(cid:12)v (a)(cid:12)(cid:12)2 ωa (cid:12)(cid:12)v (a)(cid:12)(cid:12)2 f† (a)f (a)
−4π 11 1L↑ − 12 2L↑ 1R↓ 1R↓ 4π 22 2L↑ − 12 1L↑ 2R↓ 2R↓
(cid:20) (cid:21) (cid:20) (cid:21)
Λ2dℓ λa (cid:12)(cid:12)v (a)(cid:12)(cid:12)2 λa (cid:12)(cid:12)v (a)(cid:12)(cid:12)2 f† (a)f (a)+H.c. (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
−4π 12 1L↑ − 21 2L↑ 1R↓ 2R↓
Λ2dℓ(cid:20)λa (cid:12)(cid:12)v (a)(cid:12)(cid:12)2 λa (cid:12)(cid:12)v (a)(cid:12)(cid:12)2(cid:21)(cid:16)f† (a)f (a)+H.c.(cid:17), (A14)
−4π 12 1R↓ − 21 2R↓ 1L↑ 2L↑
(cid:20) (cid:12) (cid:12) (cid:12) (cid:12) (cid:21)(cid:16) (cid:17)
(cid:12) (cid:12) (cid:12) (cid:12)
andallthefour-fermioncouplingsareirrelevant, AppendixB:Kane-Mele-U-Vmodelinthepresenceofstaggered
potentials
dg
= g, (A15) In this appendix, we will consider a more extended inter-
dℓ − action which includes both the Hubbard U and the nearest-
neighborV. The presence of the nearest-neighborV within
Hartree-Fockcontributeboththe diagonaltermsandthe off-
withg =ωa-s,λa-s,ua-sintroducedaboveandℓistheloga-
diagonal terms to the original Hamiltonian. The diagonal
rithmofthelengthscaleinRGanalysis..
terms obviously renormalize the mass terms and the off-
Thebarecouplingsofωa (ℓ = 0) = ωa (0) = ωa (0) = diagonaltermsrenormalizethenearest-neighborhoppingam-
11 22 12
2 2 plitudet, resultinginrenormalizingthevelocityoftheDirac
U, andwenumericallycheckthat v (a) = v (a) ,
1R↓ 2R↓ fermions in the critical phase. Within the Hartree-Fock pic-
2 2
v1L↑(a) = v2L↑(a) . At the t(cid:12)ree-level(cid:12)RG a(cid:12)nalysis, (cid:12)all ture,besidestheexpectationvaluesofthedensitiesdefinedin
the couplings decays at the same (cid:12)rate unde(cid:12)r RG(cid:12)flow. B(cid:12)e-
theon-siteHubbardcase,wealsoneedtointroduce
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)foreallth(cid:12)e fou(cid:12)r-fermion(cid:12) couplingsflow tonegligiblevalues,
at some small ℓ , we have ωa (ℓ ) = ωa (ℓ ) = ωa (ℓ ) =
c 11 c 22 c 12 c
λa12(ℓc) = λa21(ℓc), andhencethebilinearcorrectionsgener- c†(r,B)c (r,A) (χ )∗, (B1)
ated bythese irrelevantfour-fermioninteractionscompletely σ σ ≡ σ
(cid:28) (cid:29)
canceleachother,whichleavesnocorrectionsatthetree-level
RG analysis. Therefore, we conclude such long-wavelength c†σ(r+~ea,B)cσ(r,A) ≡(χσ(~ea))∗, (B2)
analysis can not capture the shift of the boundary between (cid:28) (cid:29)
thetopologicallytrivialandnontrivialphases. Theboundary
shiftcanonlybecaptured,atleastinthismodel,bythelattice where~e definedinFig.1. Thenearest-neighborV con-
a=1,2
Hamiltonianwhichisnotcoarse-grained. tributesadditionaltermstothefullHamiltonian.Inthematrix
8
8 UUU,, o VVn==lyUU//120 Trivial 89 UUU,, o VVn==lyUU//120 Trivial wherefσ(k) ≡ (χσ)∗ +eik·~e1(χσ(~e1))∗ +eik·~e2(χσ(~e2))∗.
By C symmetry, we can simplify the result by identifying
7 3
6
—λλm4 —λλsmo56 tχioσna=ltoχσf((~ek1))d=efinχeσd(i~en2E).q.W3eancdanthseereeftohraetvfaσn(iskh)aitsmporompeonr--
so
2 Topological (QSH) 34 Topological (QSH) tumsKandK′. WealsosetλAm = λBm = λm forsimplicity.
Focusing on momentum K, we can see that the two of the
0 2
0 0.2 0.4 0.6 U0/.t8 1 1.2 1.4 0 0.2 0.4 0.6 -U0./8t 1 1.2 1.4 fourbandswiththeeigenvaluesE1/2 = λm 2λsog(K)+
± ∓
(a) (b) U2hnA/Bi+3VhnB/Aicanbeinvertedduetothetuningofthe
ratioofλ andλ . Therefore,wedefinethegapfunctionas
m so
FIG.4. Illustrationofthephaseboundarybetweentrivialandnon-
∆(K)=E E
trivial phase in the present of both U and V repulsion. For illus- 1 2
−
tration, we take t = 1, λ = 0.4, and choose V = U/2 and U
so
=2λ 4λ g(K)+ 3V n n .(B4)
V = U/10. The blue open dot squares represent the boundary in m so A B
− 2 − h i−h i
thecaseofV =U/10andtheopenreddiamondslinerepresentsthe (cid:18) (cid:19)
(cid:2) (cid:3)
boundary inthecaseofV = U/10. Theblacklinerepresentsthe Duetothepresenceofthestaggeredpotentials,thedensityat
boundaryinthepresentofon-siteHubbardU. (a)Therepulsivein- B is largerthan thatat A, n > n , andthe sign of the
B A
teractioncase,U,V >0.Wecanseethatinthiscase.Ifthenearest- correctionofthelasttermihnthiegaphfunictiondependsonthe
neighbor V is much smaller than U, V = U/10, the interactions
competition between U and V. For repulsive U,V > 0, if
stilltendtostabilizethetopologicalphase. Butduetothecompeti-
U > 6V, the last term is negativeand the repulsive interac-
tionbetweenU andV thetopologicalwidowislesswidenedbythe
tionsstabilizes the topologicalphase. However,if U < 6V,
interactions. IfV ismorecomparabletoU,V = U/2inthiscase,
the last term is positive and then the interactions destabi-
theeffectsofV willbedominantoverU andwilltendtodestabilize
thetopologicalphase. Accordingtothegapfunctiondefinedaround lizesthe topologicalphase. On theotherhand, the attractive
momentumKpoint,Eq.(B4),thetransitionpointisatU =6V.(b) U,V <0wouldgivetheoppositeresultstotherepulsivecase.
Theattractiveinteractioncase,U,V <0.Theresultsinthiscaseare Forillustration,wenumericallycheckthecaseswitht=1,
qualitativelyoppositetothoseintherepulsivecase.WhenV ismore λ = 0.4, and chooseV = U/10 andV = U/2 ona hon-
so
comparabletotheU,theinteractionstendtostabilizethetopological eycomblatticeconsistingof200 200unitcells. Theresults
phase. onshownin Fig. 4. The blueop×endotsquaresrepresentthe
boundaryinthecaseofV =U/10andtheopenreddiamonds
line representsthe boundaryin the case of V = U/10. The
black dashed line represents the boundary in the present of
form,theadditionaltermscanbeexpressedas
only on-site Hubbard U shown in Fig. 2. Qualitatively, the
resultsbetweentherepulsiveandtheattractivecaseareoppo-
3 n f (k) 0 0 site. In the repulsiveinteractioncase, the more short-ranged
B ↑
h i −
(f (k))∗ 3 n 0 0 therepulsiveinteractionsare,themorestablethetopological
hv(k)=V − ↑0 h0Ai 3 n f (k), is. On the other hand, in the attractive interaction case, the
B ↓
h i −
0 0 (f (k))∗ 3 n moreextendedtheattractiveinteractionsare,themorestable
 − ↓ h Ai
 (B3) thetopologicalphaseis.
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