Two-dimensional topological order of kinetically constrained quantum particles

0.52 MB·English

by  Stefanos Kourtis

Two-dimensional topologicalorder ofkineticallyconstrained quantum particles
Stefanos Kourtis and Claudio Castelnovo
TCMgroup, CavendishLaboratory, UniversityofCambridge, CambridgeCB30HE,UnitedKingdom
(Dated:April21,2015)
Weinvestigatehowimposingkineticrestrictionsonquantumparticlesthatwouldotherwisehopfreelyona
two-dimensionallatticecanleadtotopologicallyorderedstates.Thekineticallyconstrainedmodelsintroduced
here are derived as a generalization of strongly interacting particle systems in which hoppings are given by
flux-latticeHamiltoniansandmayberelevanttoopticallydrivencold-atomsystems. Afterintroducingabroad
classofmodels, wefocusonparticularrealizationsandshownumericallythattheyexhibittopologicalorder,
5
aswitnessedbytopologicalground-statedegeneraciesandthequantizationofcorrespondinginvariants. These
1
resultsdemonstratethatthecorrelationsresponsibleforfractional quantumHallstatesinlatticescanarisein
0
2 modelsinvolvingtermsotherthandensity-densityinteractions.

r
p
I. INTRODUCTION teractionsnow mix bandswith opposite Chern numbers12,26.
A
Strongly correlated materials without sharply defined bands

0 maythereforebeconsideredascandidatesfortherealization
The pursuitof topologicalstates ofmatter hasfueledsub-
2 ofFCIorsimilarstates.
] sFtraonmtiatlheexpinetreimgeerntqaulaanntdumtheHoraeltliceaflfedcetv1eltooptmopeonltosginicpahlyinsiscus-. Whenparticlesrepeltheirneighborsverystrongly,itisrea-
el lators2,3 and beyond, topological states possess remarkable sonabletoapproximatetheinteractionasahardcoreconstraint
which does not allow particles to occupy neighboring sites.
- propertiesofbothfundamentalandtechnologicalinterest.For
r On two-dimensionallattices of corner- or side-sharingtrian-
t instance,certaincorrelatedtopologicallyorderedstates,such
s gles,theconfigurationsallowedbythisconstraintareidentical
as fractional quantum Hall (FQH) states4, have the poten-
at. tial to be used for fault-tolerantquantumcomputation5. The tloattthicoesegaosfetsh2e7,s28o.-cInaltlhedehqauradn-thuemxavgeorsni(oHnHof)mthoesdeelHoHfcmlaosdsieclasl,
m functionalityrequiredfornovelhigh-endtechnologicalappli-
transitions from one configuration to another are caused by
cations,however,steersmaterialdesigntowardsmicroscopic
- hopping terms. When the hoppings considered give rise to
d structureswithincreasinglycomplexfeatures.
Chernbands,imposingahardcoreconstraintinpartiallyfilled
n
o Several steps have been taken in order to bridge the gap latticescanleadtoFQH-likegroundstates26.
c betweentheoreticaldescriptionsofFQH-typetopologicalor- Here we explore the possibility of obtaining the same
[ derandexperimentallyaccessiblesystems.Followingthefirst physics by replacing the hardcore constraint with a kinetic

2 theoreticalproposalsuggestingthepresenceofFQHstatesin one. Insteadofenergeticallypenalizingor—in theinfinite-
v a lattice model6, FQH physics was predicted to arise in the interaction limit — removing configurations from the Fock
7 contextof optically trapped cold atoms7–9. More recently, a spacealtogether,westartwithallconfigurationsbeingapriori
1 class of lattice models called Chern insulators10 have been equivalent. Constraints are then introducedonly in the tran-
9 shown to develop FQH-like ground states, termed fractional sitionsbetweenconfigurations. Theresultingsystemscanbe
6 Chern insulators (FCI)11–13, upon introduction of a short- thoughtofasquantumversionsofcooperativelatticegases29.
0
range repulsion between particles occupying a fractionally Similarlyconstrainedquantumparticleshavebeenstudiedin
.
1 filledband,evenintheabsenceofanexternalmagneticfield severalcontextsinthepast,withthekineticconstraintsoften
0 (see Refs. 14,15 for comprehensivereviews). Initially, these givingrisetonontrivialcorrelationeffects30–33. Inthispaper,
5 modelswerefinetunedsothatthelowestbandoftheirenergy weshowthatwhentheconstrainedkinetictermsareendowed
1
dispersionimitatesthelowestLandaulevel: Itisalmostper- with appropriate Berry phases that break time-reversal sym-
:
v fectlyflatandtopological,i.e.,characterizedbyaChernnum- metry, the obtainedgroundstates show definitivefeaturesof
Xi berC = 1andhencenamedChernband. Proposalsforre- FQHtopologicalorder.
alizations±ofFCIstatesbasedonoxideheterostructures16,lay-
This result is far from being purely academic: Cur-
r
a ered multiorbitalsystems17,18, optical lattices19, and strained rent experiments on optical lattices can generate a partially
orirradiatedgraphene20,21followedsoonthereafter.
filled Chern band in the laboratory, with tunable interaction
As bands in solids are generally not flat and interaction strengths34,35. Some of the terms that arise in the model-
strengthsvary,anobviouswaytobringFQH-likestatescloser ing of the experimental setups exhibit precisely the type of
torealityistorelaxtheenergeticconditionsimposedonrele- constraintsstudiedhere34,36. Thesamekineticconstraintsare
vant systems in order to emulate Landau levels. It was rec- alsoanapproximatelimitofstrongshort-rangerepulsionand
ognized early on that interaction strengths may be allowed thereforerelevanttotherealizationofFQHphysicsinstrongly
toincreasebeyondbandgaps12,17,22,23,andsubsequentresults correlatedmaterials,aswillbeshownbelow.
showedthatafinitedispersionmayactuallyfavorcertainFCI Therestofthispaperisstructuredasfollows:InSec.IIwe
states18,24. One can then venture into the strong-correlation present a general formalism of tightly-boundparticles inter-
regimeby allowingforarbitrarilystrongrepulsionstrengths. acting via strong nearest-neighborrepulsion. In the infinite-
Surprisingly, one still finds robust FCI states — as well as interaction limit particles obey a hardcore nearest-neighbor
moreexotic,topologicallyorderedstates25—eventhoughin- exclusion principle, motivating us to re-interpret the prob-
2
lem as a quantum hard-hexagon model. We then relax the wheret aregenerallycomplex-valuedhoppingamplitudes.
ij
static exclusionconstrainttoakineticone,derivetwoexam- The strong-repulsion limit can therefore be recast into a
plesofconstrainedfermionicmodels,andposethequestion: HamiltonianoftheformofEq.(1),withFˆ beingaproduct
ij
Can topological order arise from the kinetic constraints? In ofhole-densityoperatorsin the vicinityof i andj. On two-
Sec. III we investigate the topological nature of the ground dimensionallattices of corner- or edge-sharingtriangles, the
states of theconstrainedmodelsusingexactdiagonalization. allowed states are exactly the configurationsof classical HH
Theresultsweobtainpromptustoanswerthequestionaffir- models, widely studied in the context of glassiness. Due to
matively.OurconclusionsaresummarizedinSec.IV. thisresemblance,weshallcalltheaboveHamiltoniansquan-
tumhard-hexagon(QHH)models.
We nowwish to reducethe abovehardcoreconstrainttoa
II. MODELS:FROMSTRONGINTERACTIONSTO new one that does not correspond to a density-density inter-
KINETICCONSTRAINTS action. One motivation for doing so is that density-hopping
terms of the form cˆ†(1 nˆ)cˆ have been shown to arise in
−
A. Generalformalism the description of optical lattice experiments, where trapped
atomsareperiodicallydrivenusingcircularlypolarizedlight
Toestablishageneralformalism,weconsiderN quantum or equivalentsettings34,35. The theoretical treatment of rele-
particles of a single species that hop on a two-dimensional vant models describes how the driving, apart from affecting
lattice Λ. The discussion below will be limited to spin- thepreexistinghoppinganddensity-densityterms,introduces
less fermions, but there is no fundamental obstacle to treat- new frequency-dependentdensity-hoppingterms36. The rel-
ing bosons in the same fashion. Distances between nearest- evanceof these termsin the experimentalsetupsdependson
neighboring sites of Λ are set to unity for convenience, and theinteractionstrengthbetweenparticles,whichisinprinci-
periodicboundaryconditionsare assumedin bothspatialdi- ple tunable. Nevertheless, their precise effect in that setting
rections. The generalform of the Hamiltonianswe consider requires careful study, as it may very well be scrambled by
belowreads otherprocesses.Asafirststep,however,itisofintrinsicinter-
esttoisolate thekineticallyconstrainedtermsanddetermine
ˆ = cˆ†Fˆ cˆ +H.c. , (1) theirpropertiesinasimplercontext.Asweshallshowbelow,
H X(cid:16) i ij j (cid:17) suchtermscangeneratenontrivialbehaviorevenontheirown.
i6=j
NoticethatFˆHC isjustaproductofhole-densityoperators
where cˆ(cˆ†) is a regular particle annihilation (creation) op- at different sites. In any partially filled system, products of
i i
erator acting at position i of Λ. An appropriate choice of (1 nˆ) operators that act on different sites are more likely
−
the operator-valued function Fˆ can result in any possible tovanishthansingle(1 nˆ)operators. Wecanthereforeat-
ij temptto restricttheprod−uctinEq.(3) toonlyafew(1 nˆ)
n-body intersite term. For example, setting Fˆ = 1 yields −
ij operators and see whether this captures the same physics as
all possible hopping terms with equal amplitude, whereas
thefullproduct. Todecidewhichtermstotruncate,wedraw
Fˆij = cˆicˆ†j yields all possible density-density interactions inspirationfromclassicalmodelsofdiffusion. Asmentioned
withequalmagnitude. above, FˆHC is the quantumcounterpartof the HH modelon
In the following, we restrict the Fˆij to products of hole- the triangularlattice. In the HH model, each particle can be
densityoperatorsoftheform(1 nˆ)actingintheneighbor- visualizedasaharddiskofradius1/2 < r < 1,sothatcon-
−
hoodof i and j. Such terms emergeeffectivelyfrom strong figurationswithparticlesonnearest-neighboringsitesarefor-
interactions37. Toseethis,considerparticlesthathoparound bidden.Ifthediskradiusisreducedto√3/4<r<1/2,then
the lattice and at the same time interact with each other via all particle configurations are a priori allowed. The particle
a nearest-neighbor repulsion of strength V. In the infinite- motion, however, is still constrained if we assume that hard
V limit, particlescannotoccupynearest-neighborsites. This discs move from site to site along straight lines: On lattices
suggests that states containing nearest-neighboring particles of edge- or corner-sharing equilateral triangles, they cannot
can be removed from the Fock space. Accordingly, for any movepast one another (see Fig. 1). This is an example of a
sitei Λwedefinetheprojectedoperatorc˜† bydemanding cooperativelattice-gasmodel38.
∈ i
thatitsactiononanystateistocreateaparticleatiifandonly Aquantumanalogofthekineticallyconstrainedmodelde-
ifthissiteandallofitsnearestneighborsareempty.Formally, scribed above can be straightforwardly constructed by dis-
carding all factors in Eq. (3) apart from those which pertain
c˜† :=cˆ† 1 nˆ , (2) tocommonneighborsofiandj. Withthischoice,
i i Y (cid:0) − j(cid:1)
{j:|j−i|=1} FˆKC =t (1 nˆ ) . (4)
ij ij Y − l
where nˆ := cˆ†cˆ . If we allow particlesto hop while obey- {l:|l−i|=|l−j|=1}
j j j
ingtheabovehard-corecondition,thenthesystemcanbede- Examplesoftheresultingtermsarepictoriallyrepresentedin
scribedbyEq.(1)with Fig.1fortwolattices.Followingtheclassicalterminology,we
shallcalltheprocessesrepresentedby FˆKC vacancy-assisted
ij
FˆHC =t (1 nˆ ) , (3) hoppings(VAH).Note thatthistypeofconstraintisnotspe-
ij ij Y − l
{l:|l−i|=1∨|l−j|=1} cifictolatticescontainingtriangularplaquettes: Next-nearest
3
(cid:5)(cid:6)(cid:7) (cid:1)(cid:2)(cid:7)(cid:4)(cid:9) (cid:5)(cid:8)(cid:7) (cid:1)(cid:2)(cid:9)
(cid:11)(cid:12)(cid:13)(cid:6)(cid:14)(cid:15)(cid:16)(cid:17)(cid:6)(cid:12)(cid:18)(cid:19)(cid:3)(cid:19)(cid:20)(cid:19)(cid:21)(cid:18)(cid:19)(cid:4)(cid:19)(cid:20)(cid:19)(cid:22) (cid:11)(cid:12)(cid:13)(cid:6)(cid:14)(cid:15)(cid:16)(cid:17)(cid:6)(cid:12)(cid:18)(cid:19)(cid:9)(cid:23)(cid:14)(cid:24)(cid:11)(cid:12)(cid:6)(cid:13)(cid:14)(cid:25)(cid:10)(cid:18)(cid:19)(cid:4)(cid:19)(cid:20)(cid:19)(cid:22)
(cid:1)(cid:1)(cid:2)(cid:1)(cid:2)(cid:7)(cid:4)(cid:8) (cid:1)(cid:2)(cid:9)(cid:4)(cid:2)
(cid:1)(cid:2)(cid:3)
(cid:1)(cid:2)(cid:9)(cid:4)(cid:7)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)
(cid:1)(cid:2)(cid:9)(cid:4)(cid:3)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:7)
(cid:1)(cid:2)(cid:9)(cid:4)(cid:6)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:3)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:6) (cid:1)(cid:2)(cid:9)(cid:4)(cid:5)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5) (cid:1)(cid:2)(cid:9)(cid:4)(cid:11)
(cid:10) (cid:10)(cid:4)(cid:5) (cid:2) (cid:10) (cid:10)(cid:4)(cid:5) (cid:2)
(cid:2)(cid:1)(cid:3)(cid:4) (cid:2)(cid:1)(cid:3)(cid:4)
(cid:3) (cid:3)
FIG.1.(Coloronline)Illustrationofvacancy-assistedhoppingmod- (cid:5)(cid:9)(cid:7) (cid:5)(cid:10)(cid:7)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:7)(cid:5) (cid:1)(cid:2)(cid:3)(cid:4)(cid:9)(cid:11)
elson(a)thetriangularlattice,withnearestandthird-nearestneigh- (cid:26)(cid:6)(cid:15)(cid:23)(cid:27)(cid:25)(cid:18)(cid:19)(cid:3)(cid:19)(cid:20)(cid:19)(cid:21)(cid:18)(cid:19)(cid:4)(cid:19)(cid:20)(cid:19)(cid:28) (cid:26)(cid:6)(cid:15)(cid:23)(cid:27)(cid:25)(cid:18)(cid:19)(cid:9)(cid:23)(cid:14)(cid:24)(cid:11)(cid:12)(cid:6)(cid:13)(cid:14)(cid:25)(cid:10)(cid:18)(cid:19)(cid:4)(cid:19)(cid:20)(cid:19)(cid:28)
borhoppingsasdefinedinRefs.17and18,and(b)thekagomelat- (cid:1)(cid:1)(cid:2)(cid:1)(cid:2)(cid:3)(cid:4)(cid:3) (cid:1)(cid:2)(cid:3)(cid:4)(cid:9)(cid:9)
tice, withnearest andsecond-nearest neighbor hoppings asdefined (cid:1)(cid:2)(cid:3)(cid:4)(cid:3)(cid:5) (cid:1)(cid:2)(cid:3)(cid:4)(cid:8)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:6)
inRef.39. Solidanddashedcirclesdenoteparticlesandvacancies, (cid:1)(cid:2)(cid:3)(cid:4)(cid:8)(cid:7)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:6)(cid:5)
respectively. Thehoppings(arrows)areonlyallowedifavacancyis (cid:1)(cid:2)(cid:3)(cid:4)(cid:8)(cid:6)
located at the position shown. Note that some of the hoppings are (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:8)(cid:11)
imaginary;seeEqs.(5)and(6). (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:5)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:11) (cid:1)(cid:2)(cid:3)(cid:4)(cid:8)(cid:9)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:11)(cid:5) (cid:1)(cid:2)(cid:6)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:12) (cid:1)(cid:2)(cid:6)(cid:4)(cid:10)(cid:7)
neighbor hoppings on checkerboard or honeycomb lattices (cid:10) (cid:10)(cid:4)(cid:5) (cid:2) (cid:10) (cid:10)(cid:4)(cid:5) (cid:2)
(cid:2)(cid:1)(cid:3)(cid:4) (cid:2)(cid:1)(cid:3)(cid:4)
canbesimilarlyconstrained. Herewefocusonthetriangular (cid:3) (cid:3)
andkagomelatticesbecauseallhoppingspresentinthemod-
els introducedbelow are constrainedin the mannerenforced
FIG.2. (Coloronline)Flowofeigenvaluesunderinsertionofmag-
byEq.(4),andalsobecausethesemodelsareknowntoyield neticfluxcorrespondingtoaphaseϕ2throughoneofthehandlesof
themostrobustfermionicFCIstates. thetoroidalsystemfor(a),(b)thetriangularand(c),(d)thekagome
lattice. (a),(c)correspondtoQHHmodels(i.e.,Fˆ = FˆHC)and(b),
(d)toVAHmodels(i.e.,Fˆ =FˆKC).
B. Examplemodels
We now ask whether the correlations induced by the d˜i,k = −2δcos(k·ai)+2δ2cos(k·bi), i=1,2,3,
vacancy-assistedhoppingsdescribedabovecangeneratetopo- (6c)
logical order. To answer this, we shall impose this kinetic
constrainttotight-bindingmodelswithtopologicallynontriv- where a = (1,0)T, a = (1/2,√3/2)T, a = a a ,
1 2 3 2 1
ial bands. We choose tij that correspondto two thoroughly b1 = a2 + a3, b2 = a3 a1, b3 = a1 + a2 and−λ
studiedFCImodels,diagonalizetheconstrainedHamiltonian (λ ,λ ,λ ), λ˜ (λ˜ ,λ˜ ,λ˜−) are vectors of the Gell-Man≡n
1 2 3 1 2 3
exactly on finite periodic clusters and look for signatures of matrices ≡
topologicallynontrivialstates.
Tight-binding Hamiltonians are represented as := 0 1 0 0 0 1 0 0 0
Pkψk†Hkψk,whereψk ≡(c1,k,c2,k,...,cn,k)TistheHspinor λ1 =1 0 0,λ2 =0 0 0,λ3 =0 0 1,
of annihilation operators on each of the n sublattices of the 0 0 0 1 0 0 0 1 0
lattice model at momentum k. The triangular-lattice model (7a)
introducedinRefs.17and18isgivenby
0 i 0 0 0 i 0 0 0
Hk△ :=gk·τ, (5a) λ˜1 =−0i 00 00,λ˜2 =0i 00 −00,λ˜3 =00 0i 0i.
3 −
(7b)
g =2t cos(2k a ), (5b)
0,k 3X · i
j=1
We set t /t = 0.19 for the triangular lattice and t /t =
3 2
g =2tcos(k a ), i=1,2,3, (5c)
i,k · i 0.3,δ/t=0.28,δ2/t=0.2forthekagomelattice. Theseval-
uesofthehoppingtermsarechosentoyieldrelativelyflatlow-
where a1 = (1/2, √3/2)T, a2 = (1/2,√3/2)T, a3 = estbandswithChernnumberC = 1. Wehaveverifiedthat
(a1 + a2) and τ − (τ0,τ1,τ2,τ3) is the vector of Pauli allourresultsremainvalidinafinit−erangearoundthevalues
− ≡
matricesincludingthe2 2unitmatrixasτ0. Thekagome- chosenhere. Wefocusonaverageparticledensitiesρ = 1/6
×
latticemodelintroducedinRef.39isdefinedas forthetriangularlatticeandρ=1/9forthekagomelattice,at
C whichthelowestbandofthepurehoppingcounterpartsofthe
H :=d λ+d˜ λ˜, (6a)
k k· k· modelswe study herewouldbe atfilling ν = 1/3. We then
di,k = −2tcos(k·ai)+2t2cos(k·bi), i=1,2,3, impose the constraintencapsulated in Eq. (4). The resulting
(6b) vacancy-assistedhoppingmodelsaresketchedinFig.1.
4
(cid:9)(cid:15)(cid:13)(cid:6) (cid:1)(cid:2)(cid:3)(cid:8)(cid:5) (cid:9)(cid:16)(cid:13) (cid:8) (cid:4)(cid:5)(cid:6)(cid:2)(cid:3)(cid:8) (cid:4)(cid:7)(cid:6)(cid:2)(cid:3)(cid:6)(cid:5)
(cid:8)(cid:9)(cid:9) (cid:8)(cid:9)(cid:9)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:3)(cid:2)(cid:8)(cid:7) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:4)(cid:6)(cid:7)(cid:4)(cid:6)(cid:7)(cid:5)(cid:5)(cid:5) (cid:3)(cid:5)(cid:6)(cid:7)(cid:5)(cid:2)(cid:9)(cid:10)(cid:2)(cid:11)(cid:12)(cid:13)(cid:8)(cid:2)(cid:2)(cid:3)(cid:2)(cid:2)(cid:3)(cid:2)(cid:3)(cid:2)(cid:8)(cid:8)(cid:8) (cid:1)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:12)(cid:15)(cid:16)(cid:17)(cid:18)(cid:13)(cid:15)(cid:19)(cid:12)(cid:20)(cid:13)(cid:10)(cid:13)(cid:15)(cid:20)(cid:12)(cid:21)(cid:18)(cid:22)(cid:12)(cid:23)(cid:22)(cid:18)(cid:13)(cid:20)(cid:12)(cid:14)(cid:24)(cid:12)(cid:25) (cid:1)(cid:1)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:6)(cid:5)(cid:4)(cid:7)(cid:5)(cid:6)(cid:2) (cid:10)(cid:11)(cid:9) (cid:2)(cid:2)(cid:2)(cid:2)(cid:3)(cid:3)(cid:3)(cid:3)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:3)(cid:5)(cid:7)(cid:26)(cid:27)(cid:6)(cid:2) (cid:10)(cid:11)(cid:9)
(cid:2) (cid:8) (cid:7) (cid:6) (cid:8) (cid:8)(cid:2) (cid:8)(cid:2)(cid:2)
(cid:1)(cid:1)(cid:2)(cid:3)(cid:4) (cid:2)(cid:14) (cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:3)(cid:2)(cid:5) (cid:2)(cid:3)(cid:2)(cid:5)(cid:8) (cid:2)(cid:3)(cid:2)(cid:4) (cid:2)(cid:3)(cid:2)(cid:4)(cid:8) (cid:2)(cid:3)(cid:2)(cid:7) (cid:2)(cid:3)(cid:2)(cid:7)(cid:8) (cid:1)(cid:2)(cid:3)(cid:2)(cid:2)(cid:5)(cid:3)(cid:2)(cid:6)(cid:5) (cid:2)(cid:3)(cid:2)(cid:6)(cid:7) (cid:2)(cid:3)(cid:2)(cid:6)(cid:26) (cid:2)(cid:3)(cid:2)(cid:6)(cid:27) (cid:2)(cid:3)(cid:2)(cid:5) (cid:2)(cid:3)(cid:2)(cid:5)(cid:5) (cid:2)(cid:3)(cid:2)(cid:5)(cid:7) (cid:2)(cid:3)(cid:2)(cid:5)(cid:26) (cid:2)(cid:3)(cid:2)(cid:5)(cid:27)
(cid:1)(cid:2)(cid:2) (cid:1)(cid:2)(cid:2)
(cid:3) (cid:3)
FIG. 3. (Color online) (a) Berry curvature of the triangular-lattice
model on a 42-site cluster with N = 7 as a function of the mag- FIG.4. (Coloronline) Gaptoexcitedstatesatzerofluxasafunc-
neticfluxesϕ1andϕ2threadingthetwohandlesofthetoroidalsys- tionofinversenumberoflatticesitesNs forQHH(solidtriangles)
tem;(b)errorinthequantizationofσH asafunctionofthenumber andVAH(emptytriangles)on(a)thetriangular-latticemodeland(b)
ofsubintervals inthepartitionoftheBrillouinzoneoffluxes. The thekagome-latticemodel. Thevalueofthegapistheenergydiffer-
unidirectionalityofthestripesalongϕ1 ismerelyanartifactofthe ence between the highest in energy quasidegenerate FCI level and
choice of the alignment of the two-site unit cell of the model [see thelowestlevelthatdoesnotbelongtotheFCIstatemanifold.
Fig.1(a)].
allowforadefinitiveanswerastowhetherthereistopological
III. RESULTS orderinthegroundstateoftheVAHmodelsinthethermody-
namiclimit.Wenote,however,thatthegapsoftheVAHmod-
els do seem to follow the same trend as those of their QHH
FQH-like topological order at ν = 1/3 manifests itself
counterparts,whichremainwellgappedupto thelargestac-
in the many-body energy spectrum as spectral flow, which
cessiblesystemsizes(seealsoRef.26). Thefinite-sizeeffects
is the exchange of the threefold quasidegeneratemany-body
observedinourcalculationsaresimilartotheonesobservedin
ground-statelevelsuponinsertionofafluxquantumthrough
previous,moredetailedstudiesofthedependenceofFCIstate
oneofthehandlesofthetoroidal(i.e.,periodic)system. This
energiesonclustersizeandaspectratio,whichconcludedthat
spectralpropertyisshowninFig.2. Toestablishconsistency
with previous results26, Fig. 2(a) shows the spectral flow of theseeffectsareindeedirrelevanttotheunderlyingphysics41.
Furthermore,wefindnosignaturesofachargeinstability,so
theQHHversionofthetriangular-latticemodel,wheresimul-
theonlyevidentcompetitorforthegroundstateisacompress-
taneousoccupationofnearest-neighboringsitesisprohibited
ible metallic state. The latter is clearly disfavoredaccording
but hoppings are unconstrained. In Fig. 2(c) we show that
topreviousdetailedstudiesofFCIsinthemodelsdefinedby
the same physicsoccursin the kagome-latticemodel. Apart
Eqs.(5)and(6)intheparameterregimeschosenhere18,42.
fromthecharacteristicFQHfeaturesintheenergyspectra,the
ground states obtained have a very precisely quantized Hall
conductivityσ =e2/(3h).
H
We nowreducetheHHconstraintofEq.(3)tothekinetic IV. CONCLUSION
constraint of Eq. (4), as outlined above. We notice that the
energyspectra,shownontherightofFig.2,arequalitatively Regardlessofthedetailedenergeticsofthemodelsderived
equivalent. Thesymmetrysectorsinwhichthequasidegener- above, the key conclusion of this work is that kinetic con-
ate ground states reside are the same as in the case of hard- straints in partially filled Chern bandscan generate topolog-
coreinteractionsandcanbepredictedbymethodsalreadyin ically ordered states. This can be intuitively understood by
use for finite FQH and FCI systems13,40. More importantly, notingthatkineticconstraintscanbeseenasanapproximate
the ground states of the VAH models have nontrivial topo- generalization of hardcore nearest-neighbor repulsion. The
logicalcharacteristics. Theirmany-bodyBerrycurvatureisa ground states of the latter can be adiabatically connected to
smooth functionthat integratesto a precisely quantizedHall FCI states, which in turn means that correlations crucial for
conductivity,evenforthefinitesystemsconsideredhere. An FQH-likestatescanbegeneratedbykineticconstraintsalone.
example of this quantization is presented in Fig. 3 for the Thismaybeasteppingstoneforfurtherintuitionintothemi-
triangular-lattice model. Equivalent results can be obtained croscopicsoftheFQH effectitself. Morehintsin thisdirec-
forthekagome-latticemodelaswell. tioncanbefoundinrecentwork43,wheretheintimaterelation
TheFQH-likegroundstatesoftheVAHmodelsaregeneri- between FQH and FCI states gives rise to similar density-
callylessgappedthanthoseoftheirQHHcounterparts,ascan hopping terms. Finally, we briefly comment on choosing to
beseeninFig.4. Nevertheless,thereisafinitevolumeinpa- introducefermionicmodelsinthiswork. Thereasonforour
rameterspaceinwhichoneobtainsgappedFQH-likeground choice is that the ν = 1/3 fermionic FCI state is the most
states evenin the VAH modelsformost clusters. Finite-size robuststate — and thereforethe easiest one to study — that
effects — such as the noticeable energy separation between requiresshort-rangeinteractions. Ifoneisinterestedinmod-
quasidegenerateground-statelevels—aresizableanddonot eling currentexperimentsprecisely, it would be more mean-
5
ingfultopursuebosonicFCIstates. However,themostrobust deredcold-atomsystems34–36.
bosonic FCI state at ν = 1/2 requiresonly an on-site inter-
action, whereas the ν = 1/4 state requires longer-range re-
pulsion, which wouldconsequentlygive rise to less intuitive ACKNOWLEDGMENTS
kineticconstraints.Sincehereweareinterestedinwhetherki-
neticconstraintscaninprincipleinducetopologicalorder,we ThisworkwassupportedinpartbyEngineeringandPhysi-
have focused on the simpler case of the ν = 1/3 fermionic calSciencesResearchCouncilGrantNo. EP/G049394/1,the
FCI.Nevertheless,similartendenciestowardstopologicalor- HelmholtzVirtual Institute “New States of Matter and Their
deringaretobeexpectedwhensuitablekineticconstraintsare Excitations”andtheEPSRCNetworkPluson“Emergenceand
introducedinbosonicversionsofHaldane-likemodels44,with Physics far from Equilibrium”. S.K. acknowledgesfinancial
potentialrepercussionsontherealizationoftopologicallyor- supportbytheICAMBranchContributions. Theauthorsare
gratefultoM.Bukov,C.Chamon,N.R.Cooper,M.Daghofer,
A.G.Grushin,C.Mudry,T.Neupert,andJ.K.Pachosforstim-
ulatingdiscussions.
1 K. V. Klitzing, G. Dorda, and M. Pepper, 22 M. Hafezi, A. S. Sø rensen, E. Demler, and M. D. Lukin,
Phys.Rev.Lett.45,494(1980). Phys.Rev.A76,023613(2007).
2 M.Z.HasanandC.L.Kane,Rev.Mod.Phys.82,3045(2010). 23 A. Sterdyniak, N. Regnault, and G. Mo¨ller,
3 X.-L. Qi, T. L. Hughes, and S. C. Zhang, Phys.Rev.B86,165314(2012).
Phys.Rev.B78,195424(2008). 24 A. G. Grushin, T. Neupert, C. Chamon, and C. Mudry,
4 D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys.Rev.B86,205125(2012).
Phys.Rev.Lett.48,1559(1982). 25 S.KourtisandM.Daghofer,Phys.Rev.Lett.113,216404(2014).
5 C. Nayak, A. Stern, M. Freedman, and S. Das Sarma, 26 S. Kourtis, T. Neupert, C. Chamon, and C. Mudry,
Rev.Mod.Phys.80,1083(2008). Phys.Rev.Lett.112,126806(2014).
6 G. S. Kliros and N. D’Ambrumenil, 27 L.K.RunnelsandL.L.Combs,J.Chem.Phys.45,2482(1966).
J.Phys.CSolidStatePhys.3,4241(1991). 28 R.J.Baxter,J.Phys.A.Math.Gen.13,L61(1980).
7 N. R. Cooper, N. K. Wilkin, and J. M. F. Gunn, 29 F.RitortandP.Sollich,Adv.Phys.52,219(2003).
Phys.Rev.Lett.87,120405(2001). 30 G.I.JaparidzeandA.P.Kampf,Phys.Rev.B59,12822(1999).
8 M. A. Cazalilla, N. Barbera´n, and N. R. Cooper, 31 L.A.Pe´rezandC.Wang,SolidStateCommun.118,589(2001).
Phys.Rev.B71,121303(R)(2005). 32 G.Go´rskiandJ.Mizia,Phys.Rev.B83,064410(2011).
9 N. R. Cooper and J. Dalibard, 33 G.Mo¨llerandN.R.Cooper,Phys.Rev.Lett.108,045306(2012).
Phys.Rev.Lett.110,185301(2013). 34 M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T. Bar-
10 F.D.M.Haldane,Phys.Rev.Lett.61,2015(1988). reiro,S.Nascimbe`ne, N.R.Cooper,I.Bloch, andN.Goldman,
11 T. Neupert, L. Santos, C. Chamon, and C. Mudry, Nat.Phys. 11,162(2015).
Phys.Rev.Lett.106,236804(2011). 35 G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger,
12 D. N. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, D.Greif, andT.Esslinger,Nature515,237(2014).
Nat.Commun.2,389(2011). 36 M. Bukov, L. D’Alessio, and A. Polkovnikov,
13 N.RegnaultandB.A.Bernevig,Phys.Rev.X1,021014(2011). arXiv:1407.4803 (2014).
14 S. A. Parameswaran, R. Roy, and S. L. Sondhi, 37 P. Fendley, K. Schoutens, and J. de Boer,
ComptesRendusPhys.14,816(2013). Phys.Rev.Lett.90,120402(2003).
15 E. J. Bergholtz and Z. Liu, 38 J.Ja¨ckle,J.Phys.Condens.Matter14,1423(2002).
Int.J.Mod.Phys.B27,1330017(2013). 39 E. Tang, J.-W. Mei, and X.-G. Wen,
16 D. Xiao, W. Zhu, Y. Ran, N. Nagaosa, and S. Okamoto, Phys.Rev.Lett.106,236802(2011).
Nat.Commun.2,596(2011). 40 B.A.BernevigandN.Regnault,Phys.Rev.B85,075128(2012).
17 J.W.F.Venderbos,S.Kourtis,J.vandenBrink, andM.Daghofer, 41 A. M. La¨uchli, Z. Liu, E. J. Bergholtz, and R. Moessner,
Phys.Rev.Lett.108,126405(2012). Phys.Rev.Lett.111,126802(2013).
18 S. Kourtis, J. W. F. Venderbos, and M. Daghofer, 42 Y.-L. Wu, B. A. Bernevig, and N. Regnault,
Phys.Rev.B86,235118(2012). Phys.Rev.B85,075116(2012).
19 N.Y.Yao,A.V.Gorshkov,C.R.Laumann,A.M.La¨uchli,J.Ye, 43 C.H.LeeandX.-L.Qi,Phys.Rev.B90,085103(2014).
andM.D.Lukin,Phys.Rev.Lett.110,185302(2013). 44 Y.-F. Wang, Z.-C. Gu, C.-D. Gong, and D. N. Sheng,
20 P. Ghaemi, J. Cayssol, D. N. Sheng, and A. Vishwanath, Phys.Rev.Lett.107,146803(2011).
Phys.Rev.Lett.108,266801(2012).
21 A. G. Grushin, A. Go´mez-Leo´n, and T. Neupert,
Phys.Rev.Lett.112,156801(2014).