From fractional Chern insulators to Abelian and non-Abelian fractional quantum Hall states: adiabatic continuity and orbital entanglement spectrum

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by  Zhao Liu

FromfractionalCherninsulatorstoAbelianandnon-AbelianfractionalquantumHallstates:
adiabaticcontinuityandorbitalentanglementspectrum
Zhao Liu1,2 and Emil J. Bergholtz3
1Beijing Computational Science Research Center, Beijing, 100084, China
2Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
3Dahlem Center for Complex Quantum Systems and Institut für Theoretische Physik,
Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany
(Dated:September25,2012)
The possibility of realizing lattice analogues of fractional quantum Hall (FQH) states, so-called fractional
Cherninsulators(FCIs),innearlyflattopological(Chern)bandshasattractedalotofrecentinterest. Here,we
maketheconnectionbetweenAbelianaswellasnon-AbelianfractionalquantumHall(FQH)statesandfrac-
2
tionalCherninsulators(FCIs)moreprecise. Usingagauge-fixedversionofQi’sWannierbasisrepresentation
1
ofaChernband,wedemonstratethattheinterpolationbetweenseveralFCIstates,obtainedbyshort-rangelat-
0
ticeinteractionsinaspin-orbitcoupledkagomelatticemodel,andthecorrespondingcontinuumFQHstatesis
2
smooth: thegapremainsapproximatelyconstantandextrapolatestoafinitevalueinthethermodynamiclimit,

p whilethelowlyingpartoftheorbitalentanglementspectrumremainsqualitativelyunaltered. Corroborating
e theseresults,wefindthatthesquaredoverlapsbetweentheFCIandFQHgroundstatesareaslargeas98.7%
S fortheeightelectronLaughlinstateatν = 1/3and97.8%forthetenelectronMoore-Readstateatν = 1/2.
Forthebosonicanaloguesofthesestates,theadiabaticcontinuityisalsoshowntohold,albeitwithsomewhat
4
smallerassociatedoverlapsetc. WefurtherprovideanexamplewheretheinterpolationbetweenHamiltonians
2
ofthetwosystemsresultsinaphasetransitionbyconsideringfermionsatfillingfractionν =4/5.

]
l PACSnumbers:73.43.Cd,71.10.Fd,73.21.Ac
e
-
r
t
s I. INTRODUCTION stateonthehoneycomblatticeisindeedsmoothlyconnected
. to the Laughlin state describing the continuum FQH state at
t
a the same filling fraction22. However, a crude implementa-
m AfterHaldane’sseminalworkmodelinganintegerquantum
Hall(IQH)effectinasimplelatticemodel1ittookovertwenty tionofQi’sWanniermappingisnotalwayssuccessful—e.g.,
- wavefunctionoftenoverlapswithFQHmodelstatesturnout
d yearsuntilitwasrecentlyrealizedthatsimilarideascouldbe
to be minuscule even when there is a well established FCI
n see to emulate lattice analogues of fractional quantum Hall
o (FQH)states2–6. Thesestates,termedfractionalCherninsula- phase—duetothefinitesizeproperties(non-orthogonality)of
c tors(FCIs),haveanumberofappealingtraits: mostsaliently theWannierfunctionsand,inprinciple,alsobecausethetwo
[ systemscarryindependentgaugedegreesoffreedom.Thisis-
theydonotrequireanexternalmagneticfieldandtheymight,
suewasconsideredindetailbyWuet. al. whoalsocameup
1 inprinciple,persistatelevatedtemperatures.
v While the basic ingredient needed for the IQH effect is a with an involved, yet elegant, prescription that remedy these
0 problems and showed that it lead to impressive overlaps be-
band with non-zero Chern number (and a finite band gap),
1 tween the fermionic FCI at ν = 1/3 and the corresponding
an additional perquisite for the FCIs is that these bands are
3 LaughlinFQHstate23.
5 only weakly dispersive, thus enhancing the effect of interac-
. tions within the band. Following the initial suggestions2–4, Inthiswork,weapplytheWannierstatemapping21adopted
9
therearebynowmanyknownmodelswithnearlyflatbands to finite size systems23 to both Abelian and non-Abelian
0
2 carrying non-zero Chern number, including intriguing solid- FCI phases. We demonstrate the adiabatic continuity be-
1 stateproposals2,7–12andpossiblecoldatomrealizations13.Al- tween these states and their corresponding FQH analogues
: though there is plenty of numerical evidence for FCI ana- (Laughlin24 and Moore-Read25 states) by showing that the
v
logues of Laughlin states4–6, hierarchy/composite fermion gapremainsessentiallyunalteredwheninterpolatingbetween
i
X states14–16 as well as non-abelian states17–19, the physics in theFCIandFQHHamiltoniansaswellasstudyingtheover-
r Chern bands is only identical to that of a Landau level in a laps with the model FQH states which turnout to remain
a veryidealizedlimit17,20.Inactuallatticemodels,however,the high throughout the interpolation. Moreover, we report on
distinctionsareratherstrikingsuchasparticlehole-symmetry the first studies of orbital entanglement spectra26 (OES) of
breaking15,16 andtheemergenceofqualitativelynewcompet- the FCI states. In contrast to the earlier particle entangle-
ing compressible states15, underscoring the need for a better ment spectrum27 (PES) studies6 which probe the quasi-hole
understandingofthesessystemsataquantitativelevel. physics,ourOESstudies,basedonacutin(Wannier)orbital
In an insightful paper, Qi introduced a Wannier basis rep- space28,providesafirsttestoftheedgephysicsinFCIphases.
resentationofaChernbandmincingtheLandaugaugewave The upshot of these studies is that the FCI states considered
functions in the continuum, and thereby paved the way to- hereare,inawell-definedsense,closertotheidealizedmodel
wards a more direct comparison between FCI and FQH FQHwavefunctionsthanFQHstatesobtainedformorereal-
states21. Indeed, Scaffidi and Möller recently used this map- istic(Coulomb)interactionsincontinuumLandaulevel. Un-
ping to convincingly show that the ν = 1/2 bosonic FCI derscoring that these results are indeed non-trivial, we also
2
provideanexamplewheretheinterpolationbetweentheLan-
dau level physics and the interacting Chern band problem is
notsmoothbyconsideringfermionsatν =4/5. t ,�
1 1
Theremainderofthisworkisorganizedasfollows. InSec-
tion II we give relatively detailed description of the Wannier
mappingprovidingabridgebetweenthedescriptionofChern
bands and continuum Landau levels on a torus. Section III
contains our main results on the adiabatic continuity and the
OES studies focusing on electronic (fermonic) states [corre-
sponding results for bosons are contained in the Appendix].
Finally,wediscussourfindingsinSectionIV.
Figure1. (Coloronline)Schematicpictureofthemappingofaflat
ChernbandinthelatticemodeltoacontinuumLandaulevelinterms
ofWannierstates.
II. MODELANDMETHODS
Inthissection,weputthedescriptionoffractionalquantTuuemsday, Augusνt 28=, 12p/q,becauseT2q commuteswithT1,wecandiagonalize
Hall(FQH)systemsinthecontinuumandfractionalChernin- certainmany-bodyHamiltonianHFQH intheLLLorbitalba-
sulators (FCIs) in the lattice on the same footing. First, we sisandobtainthemany-bodygroundstates|ΨFQH(K1,K2)(cid:105)
discussthelowestLandaulevel(LLL)onatorus29,andthen as the common eigenstates of T1 and T2q with eigenvalues
wegoontodiscussasuitablyadaptedversionoftheWannier e2πiK1/Ns and e2πiK2/N0, where K1 can be regarded as the
function mapping of Chern bands in a finite-size systems23. total momentum in the ex-direction. It directly follows that
This provides the necessary framework for a direct quantita- the degeneracy of |ΨFQH(K1,K2)(cid:105) is at least q-fold, among
tive comparison between FCIs with FQH states despite the whichwecanalwayspickupq-foldcenter-of-massdegener-
fact that the two systems have different symmetries. Finally, atestateswithdifferentK1thatareconnectedbytheoperator
wegiveaspecifickagomelatticemodelthatweusethrough- Tk(k =0,1,...,q−1).
2
outthisworktostudytheFQH-FCIcorrespondence. Forlaterconvenience,wealsointroduceanalternativede-
scriptionofthetranslationalsymmetryonthetorus. Suppose
N hastwofactorsN andN ,namelyN = N ×N . Af-
s 1 2 s 1 2
A. QuantumHallstates ter defining N ≡ GCD(N,N ) and q ≡ N /N , we
0,1 1 1 1 0,1
can introduce two translation operators S1 = (T2)q/q1 and
We consider N particles projected to the lowest Landau R2 = T1q1. Because S1 commutes with R2, we can make
level on a twisted torus spanned by two basic vectors L = the many-body ground states as their common eigenstates.
1
L v (α)andL = L v ,wherev (α) = sinαe +cosαe , Within this description of the translational symmetry, the q-
1 1 2 2 2 1 x y
v =e ,whereαisthetwistedangleofthetorus,andL is foldcenter-of-massdegeneratestatesare
2 y 1(2)
thelengthofthebasicvector(inunitsofthemagneticlength).
Athsesutomruinsgisthaenninutmegbeerr,othfeflumxagqnueatnicta,trNanss,lathtiroonuginhvsaurirafnacceeoinf |ΨFQH(s,r)(cid:105)= √1q q(cid:88)1−1e2πim(sNN0−qκ12)S1mT2r|ΨFQH(K1,K2)(cid:105),
1 m=0
v and v direction leads to L L sinα = 2πN . There are
1 2 1 2 s (2)
precisely N single particle states, |ψ (cid:105), in the lowest Lan-
s j
dau level that we choose as maximally to be localized in the where s = 0,1,...,q − 1, r = 0,1,...,q/q − 1. If we
1 1
ex-direction(butdelocalizedintheey-direction)as chooseN1 andN2 appropriately,wecanmakeq1 = 1. Then
|Ψ (s,r)(cid:105)and|Ψ (K ,K )(cid:105)reducetothesamedescrip-
(cid:104)x,y|ψ (cid:105)=(cid:16)√1 (cid:17)12 (cid:88)+∞ exp(cid:110)i(cid:16)2πj +nL sinα(cid:17)(cid:16)y tioFnQ.H FQH 1 2
j 1
πL L
2 n=−∞ 2
2πj (cid:17) 1(cid:16) 2πj (cid:17)2(cid:111)
− cotα−nL cosα − x− −nL sinα , B. Cherninsulators
L 1 2 L 1
2 2
(1)
Now we move our attention from FQH states in the con-
wherej =0,1,2,...,N −1isthesingle-particlemomentum tinuum to the FCIs in the lattice. We consider a two-
s
inunitsof2π/L . Notethatψ isquasi-periodicandcentered dimensional (2D) lattice on the torus with two lattice vec-
2 j
along the line x = 2πj/N2. We define N0 is the greatest tors v1(β) = sinβex + cosβey and v2 = ey. The num-
common divisor of N and Ns, namely N0 ≡ GCD(N,Ns). ber of unit cells is N1 and N2 in respective direction and
Then p ≡ N/N and q ≡ N /N are coprime. There are there are s sites in each unit cell. The states in the first
0 s 0
two translation operators, T (α = 1,2), that commute with Brillouin zone (1BZ) can be labelled by a 2D momentum
α
themany-bodyHamiltonian(aswithanytranslationalinvari- k = (k1,k2) where ki = 0,1,...Ni − 1. In momen-
antoperator)andobeyT T = e2πip/qT T . T corresponds tum space, the single-particle Hamiltonian can be written as
1 2 2 1 1
to a e -translation and T translates a many-body state one H = (cid:80) (c† ,...,c† )h(k)(c ,...,c )T and a band
y 2 k∈BZ k,1 k,s k,1 k,s
lattice constant 2π/L in the e -direction. At filling factor structure is formed. We focus on a single, isolated band
2 x
3
|k(cid:105)=(cid:80)s u (k)c† |vac(cid:105),whereu (k)isthecorrespond- vastly different origins. Moreover, as discussed below, the
α=1 α k,α α
ing eigenfunction of h(k), and suppose N interacting parti- symmetriesofthetwomodelsaredifferent.
cles fractionally fill in this band. If the interaction Hamilto-
nian H is chosen appropriately, the ground states of this
FCI
interactingmany-bodysystemareFCIstates|Ψ (cid:105)atcertain C. Symmetries
FCI
fillingfactorsν =N/(N N ).
1 2
TocomparetheFCIstateswiththetheoreticallymuchbet- The FQH Hamiltonian in the LLL on the torus conserves
terunderstoodFQHstates,weneedtoexpand|ΨFCI(cid:105)inabasis center on mass position corresponding to momentum K1 =
withsingle-particlestatesthatmimictheLLLstates[Eq. (1)]. (cid:80)N j (mod N ). However, the corresponding quantity is
An appropriate choice is the Wannier basis, whose single- i=1 i s
notconservedfortheFCIproblemdespitethefactthatthere
particle state |X,k2(cid:105) is localized in the v1 direction but de- isaaone-to-onecorrespondencejX,k2 = N2X +Ck2 +δ2
localizedinthev2direction,whereX isthepositioninthev1 between |X,k (cid:105) and |ψ (cid:105) which allow us to calculate a to-
2 j
dviredcirtieocntioann2d1,k232.isthemomentum(inunitsof2π/N2)inthe tal 1D momentum (cid:80)Ni=1(jX,k2)i (mod N1N2). Instead, the
2 translationalsymmetry(inreal-space)inthedirectionsofthe
InaN ×N finite-sizelattice,whenfocusingononefrac-
1 2 twolatticevectorsintheChernbandimpliesaconservedthe
tionally filled band with Chern number C, the (lattice ver-
two-dimensional momentum, which leads to a reduced sym-
sion of the) Berry connection in the v direction can be de-
1 metry for the FCI Hamiltonian in the Wannier basis: only
1fi,nked),aswAhe1r(ek1(cid:15),1k2i)s t=he (cid:80)v α-deir−eic2tπio(cid:15)1αn/Nre1lua∗αti(vke1,dki2sp)ulaαc(ekm1e+nt J1 = (cid:80)Ni=1(jX,k2)i (mod N2) is conserved. [Another man-
2 α 1 ifestation of the lower symmetry in the FCI problem is re-
of site α in an unit cell. Similarly we can define the
flectedinthelackofparticle-holesymmetry15.]
Berry connection in the v direction as A (k ,k ) =
2 2 1 2 The symmetry difference is indeed a generic effect due to
(cid:80)αe−i2π(cid:15)2α/N2u∗α(k1,k2)uα(k1,k2 +1). To restore the or- the underlying lattice where the Berry curvature necessarily
thogonality between different Wannier functions, we need
variesinreciprocalspaceaslongasthenumberofbandsisfi-
to introduce unitary Berry connections23 A (k ,k ) =
1(2) 1 2 nite.Inanideallimit,however,theFCIHamiltonianwillhave
A1(2)(k1,k2)/|A1(2)(k1,k2)|. Then the unitary Wilson thesameemergentsymmetriesastheFQHHamiltonian20.
loops are W (k ) = (cid:81)N1−1A (k ,k ) and W (k ) =
1 2 k1=0 1 1 2 2 1
(cid:81)N2−1A (k ,k ), whose argument angles are picked in
(−k22π=,00].2Af1ter2defining a shift δ as the cardinality of the D. Kagomelatticemodel
2
set {k = 0,1,...,N |arg[W (k )] > arg[W (0)]}, we can
2 2 1 2 1
introduceaprincipleBrillouinzone(pBZ)asthesetofk sat- In the following, we focus on a special lattice model,
2
isfyingCk +δ ∈[0,N )andmovek from1BZtopBZ. namely the kagome lattice model proposed in Ref. 2, to in-
2 2 2 2
Afterintroducing[λ1(k2)]N1 ≡W1(k2)and[λ2(k1)]N2 ≡ vestigate the FCI-FQH correspondence. The single-particle
W (k ),wherewechoosetheargumentanglearg[λ (k )] ∈ Hamiltonian of the kagome lattice model [cf. Fig. 1] in the
2 1 1 2
(−2π/N ,0]andarg[λ (k )] ∈ (−2π/N ,0],wecandefine realspaceis
1 2 1 2
theWannierfunctionlocalizedinthev directionas
1 H =t (cid:88) c† c +iλ (cid:88) (Eˆ ×Rˆ )·σ c† c ,
|X,k2(cid:105)= e√iΦN(k12) Nk(cid:88)11=−01e−i2Nπk11X(cid:40)(cid:81)kκ1[=λ−011(Ak21)(]κk1,k2)(cid:41)|k1,k2(cid:105), whereE1ˆ(cid:104)i,ji(cid:105)s,σtheiσnojσrmaliz1ed(cid:104)i,,j|E(cid:105)ˆ,α,β| =i1j, elecitjricfieαlβdaiαrisjinβg
ij ij
wherek isinpBZandΦ(k )isindependentofX andneeds fromanionatthecenterofeachhexagonasexperiencedbya
2 2
tobefixedbyaspecialprescription(seeAppendixCforde- particlehoppingalongtheunit-vectorRˆ fromsiteitositej.
ij
tails). Letting jX,k2 = N2X + Ck2 + δ2, we can build a Inthisworkweconsiderelectronsarespin-polarized(allspin
one-to-onemapbetween|X,k (cid:105)and|ψ (cid:105). up)particlesandsett =−1whileusingbandstructurescor-
2 j 1
Considering the one-to-one map between the Wannier or- respondingtovariousλ asinputtoourstudiesofinteractions
1
bital and the LLL orbital as well as their similar localizing projected to non-trivial bands. In the momentum space we
properties,theFCIstates|Ψ (cid:105)intheWannierbasiswillbe have three energy bands, the lowest one of which has Chern
FCI
very well approximated by the lattice version of FQH states numberC =1. Ascustomary6 wetaketheflatbandlimitand
constructedas30 projecttheinteractionHamiltonianH tothisC =1band.
FCI
(cid:88)
|Ψlat (s,r)(cid:105)= |{X,k }(cid:105)(cid:104){jX,k2}|Ψ (s,r)(cid:105),(3)
FQH 2 FQH
{X,k2} III. CONTINUITYBETWEENFQHANDFCI
where|{•}(cid:105)isthemany-bodyoccupationconfigurationover
thesingle-particlestate|•(cid:105)(onecanfindthat|Ψ (s,r)(cid:105)in As discussed above, the FQH Hamiltonian and FCI
FQH
the LLL orbital basis and its lattice version |Ψlat (s,r)(cid:105) in Hamiltonian have different symmetries. However, they
FQH
the Wannier basis have a common description). However, it have similar expression written in second-quantized form.
is important to note that |Ψlat (s,r)(cid:105) will in general differ Taking two-body interactions as an example, we have
FQH
from|Ψ (cid:105),sincetheHamiltoniansofthetwosystemshave H = (cid:80)Ns−1 δmodNs VFQH c† c† c c , for
FCI FQH j1j2j3j4=0 j1+j2,j3+j4 j1j2j3j4 j1 j2 j3 j4
4
the FQH case, where c† (c ) creates (annihilates) a
j j
particle in the state |ψ (cid:105), while we have H =
j FCI
(cid:80)N1N2−1 δmodN2 VFCI c† c† c c , where c† (c )
j1j2j3j4=0 j1+j2,j3+j4 j1j2j3j4 j1 j2 j3 j4 j j
creates (annihilates) a particle in the state |X,k (cid:105) with j =
2
XN +Ck +δ fortheFCIcase. Therefore,thestructureof
2 2 2
the Hilbert space of FCIs in the Wannier basis is the same
as that of FQH systems in the LLL orbital basis if we set
N = N × N . This makes it meaningful to consider an
s 1 2
interpolatingHamiltonianasfollows,
H(λ)=λw H +(1−λ)w H , (4)
FCI FCI FQH FQH
where λ ∈ [0,1] is the interpolation parameter and w and
FCI
w are the energy rescaling factors that can make the en-
FQH
ergy gap at λ = 0 and λ = 1 equal to 1 (in the cases where
we find adiabatic continuity below it is well established that
the gap survives in the thermodynamic limit at λ = 0 and
λ = 1 respectively). We then diagonalize Eq. (4) in each
J =0,1,...,N −1sectorandanalyzetheenergygapandthe
1 2
groundstatesasafunctionofλ,inordertoexaminewhether
Figure 2. (Color online) Results of the interpolation Eq. (4) for
the FCI states are adiabatically connected to the correspond- fermions at ν = 1/3 with N = 4 (red), N = 6 (green), and
e e
ingFQHmodelstates. N =8(blue).IntheFCIpart,thelatticesizeisN ×N =4×3,
e 1 2
N ×N =6×3andN ×N =4×6,respectively;andλ =1.
1 2 1 2 1
(a)Theenergygap∆doesnotcloseforanyintermediateλ.(b)The
total overlap O (circle) and the average weight W (dot) are still
A. Fermionsatν =1/3 tot
close to 1 at λ = 1. All of those demonstrate that the continuity
holdsforfermionsatν =1/3.
Westartourdiscussionbyfocusingonthefermionsatfill-
ingfactorν =1/3.OntheFQHside,wechoosetheHamilto-
nianasHFQH =(cid:80)i<j∇2iδ2(ri−rj). Thenthegroundstates FCIstatesareindeedverywellcapturedbythelatticeversion
areexactthree-folddegeneratefermionicLaughlinstateswith ofFQHstatesconstructedbyEq. (3).
zero energy. On the FCI side, the Hamiltonian is set as the The entanglement spectrum26 (ES) can usually provide us
(cid:80)
nearest-neighborinteractionHFCI = (cid:104)ij(cid:105)ninj. Theground moreinsightsthantheoverlap,whichisonlyasinglenumber
states are three nearly-degenerate states separated by a gap and will necessarily vanish in the thermodynamic limit. For
from the excited states. Since β = π/3 for the kagome lat- any bipartite pure state |Ψ(cid:105) , it can be decomposed using
AB
tice,wesetα = π/3alsoforthetwistedLandauleveltorus. theSchmidtdecomposition,
This choice is further justified by the large overlap between
theFCIstatesatλ = 1andtheLaughlinstatesatλ = 0(see |Ψ(cid:105) =(cid:88)e−ξi/2|φA(cid:105)⊗|φB(cid:105),
AB i i
thefollowingdiscussion).
i
We find that for each λ ∈ [0,1], there are three nearly-
degenerate states separated by a sizable gap ∆ from excited where the states |φA(cid:105) (|φB(cid:105)) form an orthonormal basis for
i i
levels in the energy spectrum. We report the evolution of ∆ the subsystem A (B). {ξ ≥ 0} is defined as entanglement
i
with λ for various system sizes in Fig. 2(a). It can be seen spectrumandarerelatedtotheeigenvalues,η ,ofthereduced
i
that the gap never closes for any intermediate λ—in fact it densitymatrix,ρA =trB(|Ψ(cid:105)ABAB(cid:104)Ψ|),ofAasηi =e−ξi.
is always greater than one and has a maximal value for λ ≈ Insomepreviousworks,theESforparticlecuthavebeenin-
0.4 − 0.6. This provides strong evidence for the adiabatic vestigatedextensivelytoprobethequasiholeexcitationprop-
continuitybetweenFQHstatesandFCIstates. ertiesofFCIstates. HerewefocusonanotherkindofES,the
To further confirm that there is no phase transition be- OESforacutinorbitalspace,totesttheedgephysicsofFCI
tween λ = 0 and λ = 1, we study the properties of states.
the ground manifold. We can define the total overlap as We first briefly recall the OES of FQH states on the torus
O = 1(cid:80)d (cid:80)d |(cid:104)Ψi |Ψj(λ)(cid:105)|2,wheredisthenum- that has been studied in Refs. 31 and 32. The three-fold
tot d i=1 j=1 FQH
berof(nearly-)degeneratestates(hered=3),|Ψj(λ)(cid:105)isthe degenerate fermionic ν = 1/3 Laughlin states have follow-
(nearly-) degenerate state of H(λ), and |Ψj (cid:105) = |Ψj(λ = ingsimplerepresentationsintheorbitalbasisinthethin-torus
FQH limit33L =0(forN =8,N =24),
0)(cid:105)istheFQHstate(heretheexactν = 1/3Laughlinstate). 2 e s
WefindthatO decreasesfrom1atλ=0smoothlytoabout
tot 100100|100100100100|100100
0.987atλ=1forourlargestsystemsizeN =8[Fig. 2(b)].
e
010010|010010010010|010010
Therefore, thegroundstatesdonotchangequalitativelydur-
ingtheinterpolationfromλ=0toλ=1,supportingthatthe 001001|001001001001|001001. (5)
5
Figure3. (Coloronline)Theorbitalentanglementspectra(OES)ofexactfermionicLaughlinstates(bluediamond)andtheprojectednearly-
degeneratestates|ΨJ1(λ)(cid:105)(redcross)atν = 1/3,N = 8. ThelatticesizeisN ×N = 4×6andλ = 1fortheFCIpart. Intheleft
prj e 1 2 1
column,J =0,correspondingtotheLaughlinstateinK =12sector. Inthemiddlecolumn,J =2,correspondingtotheLaughlinstate
1 1 1
inK =20sector. Intherightcolumn,J =4,correspondingtotheLaughlinstateinK =4sector. In(a),(b)and(c),λ=0.5. (a): The
1 1 1
unprojected|ΨJ1=0(λ)(cid:105)hasweightW ≈0.99730ontheK1 =12sector.TheoverlapwiththeLaughlinstateO=|(cid:104)ΨKLa1u=12|ΨJ1=0(λ)(cid:105)|2
is 0.99654. (b): The unprojected |ΨJ1=2(λ)(cid:105) has weight W ≈ 0.99732 on the K1 = 20 sector. The overlap with the Laughlin state
O = |(cid:104)ΨKLa1u=20|ΨJ1=2(λ)(cid:105)|2 is0.99638. (c): Theunprojected|ΨJ1=4(λ)(cid:105)hasweightW ≈ 0.99732ontheK1 = 4sector. Theoverlap
O = |(cid:104)ΨK1=4|ΨJ1=4(λ)(cid:105)|2 is 0.99638. In (d), (e) and (f), λ = 1. (d): The unprojected |ΨJ1=0(λ)(cid:105) has weight W ≈ 0.99062 on the
Lau
K1 =12sector. TheoverlapwiththeLaughlinstateO =|(cid:104)ΨLKa1u=12|ΨJ1=0(λ)(cid:105)|2is0.98733. (e): Theunprojected|ΨJ1=2(λ)(cid:105)hasweight
W ≈ 0.99034ontheK1 = 20sector. TheoverlapwiththeLaughlinstateO = |(cid:104)ΨKLa1u=20|ΨJ1=2(λ)(cid:105)|2 is0.98667. (f): Theunprojected
|ΨJ1=4(λ)(cid:105)hasweightW ≈ 0.99034ontheK1 = 4sector. TheoverlapwiththeLaughlinstateO = |(cid:104)ΨKLa1u=4|ΨJ1=4(λ)(cid:105)|2 is0.98667.
TheweightandoverlapinthemiddlecolumnandrightcolumnarealwaysthesamebecauseoftheinversionsymmetryoftheWannierbasis.
TheorangeshadowsindicatethegenericlevelsintheOESof|ΨJ1(λ)(cid:105)whichdeviatefromthelevelsoftheexactLaughlinstate.
prj
Then, we bipartite the system into blocks A and B, which Considering the structure of the Hilbert space does not
consistofl consecutiveorbitsandtheremainingN −l or- changeduringtheinterpolation,wecanmakeacutintheba-
A s A
bits,respectively[TheboldblockinEq. (5)isoursubsystem sis and extract the OES of |Ψi(λ)(cid:105) by the same method as
A]. After extracting the ES from the ground states, we label thatforFQHstates. ForpureFCIstates,thiscorrespondstoa
(cid:80)
every ES level by the particle number N = n and cutinthelocalizedWannierorbitals. However,thetotalmo-
A j∈A j
thetotalmomentumKA = (cid:80)j∈Ajnj (modNs)inblockA, mentumK1in|Ψi(λ)(cid:105)isnotagoodquantumnumber(except
wheren istheparticlenumberinthestate|ψ (cid:105). Inthiswork, at λ = 0). This means |Ψi(λ)(cid:105) may have weight on some
j j
weconcentrateonthecaselA =Ns/2. K1 that |ΨFQH(cid:105) does not have weight on. We can calculate
the weight Wi of each |Ψi(λ)(cid:105) on the K sectors of FQH
1
In Refs. 31 and 32, it was shown that the resulting OES statesandobtainanaverageweightW = 1(cid:80)d Wi. From
d i=1
spectrafortheFQHstateformtowersthatcanbedecomposed Fig. 2(b), we can see that the |Ψi(λ > 0)(cid:105) indeed has some
intotheedgemodesoftheunderlyingconformalfieldtheory ’momentum leakage’ leading to W < 1. However, even for
(CFT).Thiscombinationcomesaboutasthenaturalpartition, the pure FCI states at λ = 1, K is also almost conserved
1
Eq. (5), gives a subsystem A with the geometry of a cylin- (W ≈0.990forN =8). Therefore,weproject|Ψi(λ)(cid:105)into
e
der which has two edges on which gapless edge states with the K sector of the corresponding FQH states and consider
1
oppositechiralityreside. Anilluminatingrecentdiscussionof theOESof|Ψi (λ)(cid:105).
prj
theconnectionbetweentheOES,theCFTdescribingtheedge
andmatrixproductstateswasgiveninRef. 35. InFig. 3,wedisplayourOESresultsforN = 8(thelat-
e
6
ticesizeisN ×N =4×6fortheFCIpart). Forthissystem
1 2
size, |Ψi(λ)(cid:105) are located in the J = 0, J = 2 and J = 4
1 1 1
sectors. TheycorrespondtotheLaughlinstatewithK = 12
1
(100sector),K = 20(010sector)andK = 4(001sector),
1 1
respectively.WefindthattheOESalmostperfectlymatchthat
ofthecorrespondingLaughlinstatesuptoξ = ξ forallof
max
the three |Ψi(λ)(cid:105): ξ ≈ 13.3 at λ = 0.5, while it reduces
max
slightlytoabout12.3atλ = 1. Thisimpressivematchofthe
OESwiththemodelstatestronglysuggeststhattheedgeex-
citationpropertiesoftheLaughlinstatesarepreservedduring
the interpolation and furthermore corroborates the adiabatic
continuity between FCI states and FQH states at ν = 1/3.
In Ref. 31, the OES of the ν = 1/3 Coulomb ground states
was investigated and compared to the model states. In this
casetherewasamatchoftheOESlevelswithexactν = 1/3
Laughlin state up to ξ ≈ 8 (the number of levels below
max
this value increase with system size). That we find higher
ξ here indicates that the FCI states in the lattice are actu-
max
allyclosertotheLaughlinmodelstatesthanisthecaseforthe
CoulombFQHgroundstates.
Figure 4. (Color online) Results of the interpolation Eq. (4) for
fermionsatν = 1/2forN = 6(red),N = 8(green),andN =
e e e
10 (blue). In the FCI part, the lattice size is N ×N = 3×4,
1 2
N ×N =4×4andN ×N =5×4,respectively;andλ =0.8.
1 2 1 2 1
(a)Theenergygap∆doesnotcloseforanyintermediateλ.(b)The
B. Fermionsatν =1/2
total overlap O (circle) and the average weight W (dot) are still
tot
close to 1 at λ = 1. All of those demonstrate that the continuity
It is also interesting to investigate whether the adiabatic holds for fermions at ν = 1/2. (One may note that for N = 8,
e
continuity holds also for some non-Abelian states. In fact, N1 ×N2 = 4×4, W = 1 for all λ. This is accidental for this
none of the two previous Wannier basis studies22,23 consid- particularlatticesize.)
eredstatesinthisclass. Tothisend, weturnourattentionto
theν =1/2fermionicMoore-Readphase.Toobtaintheexact
statesare34(forN =10,N =20)
fermionic Moore-Read states in the continuum on the torus, e s
wechooseHFQH =(cid:80)i<j<kSijk∇4i∇2jδ2(ri−rj)δ2(rj−rk), 010101|0101010101|0101
where S is the symmetrizing operator. The ground states
ijk 101010|1010101010|1010
are exact six-fold degenerate fermionic Moore-Read states
with zero energy. On the FCI side, we construct the Hamil- 01100|1100110011|00110±10011|0011001100|11001
(cid:80)
tonianasathree-bodyinteractionH = n n n be- 11001|1001100110|01100±00110|0110011001|10011.
FCI (cid:104)ijk(cid:105) i j k
tweenthreenearest-neighborsites. Thegroundstatesaresix
Their total momentum is K = 0, K = 10, K = 15
nearly-degenerate states separated by a gap from the excited 1 1 1
(two-fold) and K = 5 (two-fold), respectively. Among the
states.Herewechooseadifferenttwistedangleα=2π/3for 1
six |Ψi(λ)(cid:105), one locates in J = 0 sector [corresponding to
thetorusandthisisjustifiedbythelargeoverlapbetweenthe 1
the K = 0 Moore-Read state (0101 sector)], one locates in
FCIstatesatλ = 1andtheMoore-Readstatesatλ = 0(for 1
J = 2 sector [corresponding to the K = 10 Moore-Read
α=π/3thisoverlapisrelativelysmall). 1 1
state(1010sector)],twolocateinJ =1sector[correspond-
1
Wefindthattherearesixnearly-degeneratestates|Ψi(λ)(cid:105) ing to the two K = 5 Moore-Read states (1100±0011 sec-
1
for each λ ∈ [0,1]. The evolution of the energy gap [Fig. tors)],andtwolocateinK = 3sector[correspondingtothe
2
4(a)],totaloverlapandaverageweight[Fig.4(b)]behavesim- twoK = 15Moore-Readstates(0110±1001sectors)]. The
1
ilarly with those for the ν = 1/3 fermionic Laughlin phase. two |Ψi(λ)(cid:105) with the same J = 1 (J = 3) will mix with
1 1
Whilebothofthetotaloverlapandaverageweightareslightly eachother, sotheydo nothavea one-to-onecorrespondence
smaller(Otot ≈0.977andW ≈0.984atλ=1forourlargest tothe1100+0011stateand1100-0011state(0110+1001state
systemsizeNe =10),thesenumbersarewayabovetheover- and 0110-1001 state). Therefore, we only consider the OES
laps found between the Moore-Read state and the Coulomb forthe|Ψi(λ)(cid:105)inJ =0andJ =2sectorsherebyproject-
1 1
ground state in the second Landau level39. [Of course, the ingthemintotheK sectorofthecorrespondingMoore-Read
1
three-bodylatticeHamiltonianusedherefortheFCIissome- states.InFig.5,wecanseethatthelow-lyingpartoftheOES
what artificial to begin with making a direct comparison of of |Ψi (λ)(cid:105) also match that of the exact Moore-Read state
prj
overlapsabitbiased.] very well. As expected, ξ here (ξ ≈ 12.5 at λ = 0.5
max max
We also consider the OES for N = 10 (the lattice size and ξ ≈ 11 at λ = 1) is lower slightly than that for the
e max
is N × N = 5 × 4 for the FCI part). In the continuum, ν = 1/3 fermionic Laughlin case reflecting the lower over-
1 2
thethin-torusconfigurationofthesixfermionicMoore-Read lap. We also find that the OES of |Ψi (λ)(cid:105) lacks inversion
prj
7
Figure 5. (Color online) The orbital entanglement spectra (OES) of exact fermionic Moore-Read states (blue diamond) and the projected
nearly-degeneratestates|ΨJ1(λ)(cid:105)(redcross)atν =1/2,N =10.ThelatticesizeisN ×N =5×4andλ =0.8fortheFCIpart.Inthe
prj e 1 2 1
leftcolumn,J =0,correspondingtotheMoore-ReadstateinK =0sector.Intherightcolumn,J =2,correspondingtotheMoore-Read
1 1 1
stateinK1 = 10sector. In(a)and(b),λ = 0.5. (a): Theunprojected|ΨJ1=0(λ)(cid:105)hasweightW ≈ 0.99403ontheK1 = 0sector. The
overlapwiththeMoore-ReadstateO = |(cid:104)ΨK1=0|ΨJ1=0(λ)(cid:105)|2 is0.99119. (b): Theunprojected|ΨJ1=2(λ)(cid:105)hasweightW ≈ 0.99436
MR
ontheK1 = 10sector. TheoverlapwiththeMoore-ReadstateO = |(cid:104)ΨKMR1=10|ΨJ1=2(λ)(cid:105)|2 is0.99094. In(c)and(d),λ = 1. (c): The
unprojected|ΨJ1=0(λ)(cid:105)hasweightW ≈0.98488ontheK1 =0sector.TheoverlapwiththeMoore-ReadstateO=|(cid:104)ΨKMR1=0|ΨJ1=0(λ)(cid:105)|2
is0.97792.(d):Theunprojected|ΨJ1=2(λ)(cid:105)hasweightW ≈0.98583ontheK1 =10Moore-Readstate.TheoverlapwiththeMoore-Read
stateO = |(cid:104)ΨK1=10|ΨJ1=2(λ)(cid:105)|2 is0.97750. TheorangeshadowsindicatethegenericlevelsintheOESof|ΨJ1(λ)(cid:105)whichdeviatefrom
MR prj
thelevelsoftheexactMoore-Readstate.
(left-right)symmetry,whichisnotsoobviousintheν =1/3 the five-fold nearly-degeneracy15. Here we set β = π/3,
state. However,takentogetherthereisnodoubtthattheFCI w = w = 1 and choose |Ψi(λ)(cid:105) as the ground states
FCI FQH
phaseisexcellentlydescribedbytheMoore-Readwavefunc- inJ sectorswheretheFQHstateslocateinatλ=0. Weuse
1
tion and the low energy physics of the FCI problem should thetotaloverlaptoprobethephasetransitionbetweenλ = 0
thusbewithinthesameuniversalityclass. and λ = 1. In Fig. 6, it is clear that the total overlap drops
downtoaverysmallnumberatintermediateλ.
Wealsofindasimilarphasetransitionforfermionsatν =
C. Fermionsatν =4/5andν =2/3 2/3. However, ν = 2/3 is probably on the border between
competingcompressiblestatesandFCIstates(seeRef. 15for
Finally, we consider a case where there is a lack of adia- suchstudyincheckerboardlattice),thusitisquitelikelythat
batic continuity between the low energy sector of the FQH the appearence of this phase transition may depends on the
and FCI Hamiltonians. To this end we focus on fermions systemsizeandtheshapeofthesamples. Onthecontrary,we
at ν = 4/5. On the FQH side, we choose the Hamilto- expecttheν =4/5resultsshowingaclearphasetransitionto
nian as H = (cid:80) ∇6δ2(r − r ). Then the ground berobusttosuchdetails.
FQH i<j i i j
statesarefive-folddegeneratestatesthataretheparticle-hole
conjugate (phc) of ν = 1/5 Laughlin states. On the FCI
side, the Hamiltonian is set as the nearest-neighbor interac- IV. DISCUSSION
(cid:80)
tion H = n n . Due to the particle-hole symme-
FCI (cid:104)ij(cid:105) i j
try breaking, the ground states are no longer FCI states but In this paper, we have investigated the interpolation be-
competing compressible (Fermion liquid like) states without tween the FCI states and FQH states with the help of appro-
8
priatelygauge-fixedWannierwavefunctions21,23. Bydemon-
strating an almost constant gap and a large overlap during
the interpolation, we provide the strong evidence that both
Abelian and non-Abelian FCI states are adiabatically con-
nected to their corresponding FQH (Laughlin and Moore-
Read respectively) states for fermions as well as for bosons.
The method used here may be seen as an improved version
of the first study of adiabatic continuity22, which studied
ν = 1/2bosons,byutilizingtherecentgauge-fixinginsights
of Ref. 23. It also provides a more direct and quantitative
comparisonbetweenFCIandFQHthanthelike-wiseelegant
connection recently established via relating Chern bands to
Figure 6. (Color online) Results of the interpolation Eq. (4) for
theLandaubandsoftheHofstadterprobleminRef. 40.
fermions at ν = 4/5 for N = 16 (red), N = 20 (green), and
e e
N =24(blue).IntheFCIpart,thelatticesizeisN ×N =4×5, Tounderscorethenon-trivialityofourresults,wehavealso
e 1 2
N ×N =5×5andN ×N =6×5,respectively;andλ =1. consideredfermionsatfillingfactorν =4/5,forwhichthere
1 2 1 2 1
The total overlap O shows a clear drop at intermediate λ, which isnoFCIstateduetotheparticle-holesymmetrybreaking15in
tot
demonstratesthatthecontinuitydoesnotholdforfermionsatν = theChernband(whichisabsentinaLandaulevel). Theover-
4/5. lap and gap indeed drops drastically during the interpolation
reflectingaphasetransition.
Moreover, we have given a first report of the orbital en-
tanglement spectrum (OES) of FCI states based on the or-
bital cut in the localized Wannier basis. The low-lying parts
oftheOESofFCIstatesmatchthoseofcorrespondingideal
model FQH states very well. By comparing the OES of FCI
states with those of FQH states obtained from realistic inter-
action in Landau levels, we find FCI states are closer to the
idealFQHstates. TheanalysisoftheOESgeneralizesearlier
works on FQH states on the torus, and thereby provides an
appealing picture of the FCI OES as composed of two CFT
spectra with opposite chirality31,32. Invoking the bulk-edge
correspondence26,36–38,ourresultsalsoprovideafirstglimpse
ofthegaplessedgephysicsoftheFCIphases.
Ourworkalsoinvitesanumberofinterestingfuturedirec-
tions. Perhaps most interestingly, it suggests a natural gen-
eralization studying FCI states in Chern bands with higher
Chern number. Indeed, novel series of FCI states with arbi-
trary Chern number, |C| = N, have recently been observed
innumerics41,42. Whilethesenewstatesmightcorrespondto
appropriatelysymmetrizedversionsofmulti-componentFQH
states, such ideas need to be substantiated by further inves-
tigations and more direct comparisons as would be possible
withintheframeworkusedhere.
ACKNOWLEDGMENTS
We acknowledge Andreas Läuchli for several related col-
Figure 7. (Color online) The orbital entanglement spectra (OES)
laborations. E.J.B. is supported by the Alexander von Hum-
ofexactfermionicphcstate(bluediamond)andtheprojectedstate
|ΨJ1(λ)(cid:105) (red cross) at ν = 4/5, N = 24. |ΨJ1(λ)(cid:105) is in boldtfoundation.
prj e prj
J = 2 sector and corresponds to the phc state in K = 12 sec-
1 1
tor. The lattice size is N × N = 6 × 5 and λ = 1 for the
1 2 1
FCI part. (a): λ = 0.1. The unprojected |ΨJ1(λ)(cid:105) has weight AppendixA:Bosonsatν =1/2
W ≈ 0.98645 on the K = 12 sector. The overlap with the phc
1
stateO = |(cid:104)ΨK1=12|ΨJ1=2(λ)(cid:105)|2 is0.97213. (b): λ = 0.5. The
phc Inthissectionwefocusonthecontinuityproblemofbosons
unprojected|ΨJ1(λ)(cid:105)hasweightW ≈ 0.06792ontheK1 = 12 at filling factor ν = 1/2. On the FQH side, we choose the
sector. TheoverlapwiththephcstateO =|(cid:104)ΨKphc1=12|ΨJ1=2(λ)(cid:105)|2 HamiltonianasHFQH = (cid:80)i<jδ2(ri−rj). Thentheground
isalmost0.
states are exact two-fold degenerate bosonic Laughlin states
with zero energy. On the FCI side, the Hamiltonian is set as
9
lower overlap. However, all of our results strongly support
that theFQH states areadiabatically connected toFCI states
forbosonsatν =1/2.
AppendixB:Bosonsatν =1
Inthissectionwefocusonthecontinuityproblemofbosons
atfillingfactorν =1.OntheFQHside,wechoosetheHamil-
tonianasH =(cid:80) δ2(r −r )δ2(r −r ). Thenthe
FQH i<j<k i j j k
groundstatesareexactthree-folddegeneratebosonicLaugh-
lin states with zero energy. On the FCI side, the Hamilto-
nianistacticallychosenastheon-sitethree-bodyinteraction
(cid:80)
H = n (n −1)(n −2). Thegroundstatesarethree
FCI i i i i
nearly-degenerate states separated by a gap from the excited
states. Wesetα=2π/3forthetwistedtorusaswasdonefor
thefermioncaseatν =1/2.
We find that there are three nearly-degenerate states
|Ψi(λ)(cid:105)foreachλ ∈ [0,1]. Theevolutionoftheenergygap
Figure 8. (Color online) Results of the interpolation Eq. (4) for [Fig. 10(a)], total overlap and average weight [Fig. 10(b)]
bosonsatν = 1/2forNb = 4(red), Nb = 6(green), andNb = is similar with that for the ν = 1/2 fermionic Moore-Read
8 (blue). In the FCI part, the lattice size is N1 ×N2 = 4×2, phase. However,bothofthetotaloverlapandaverageweight
N ×N =3×4andN ×N =4×4,respectively;andλ =1.
1 2 1 2 1 aresmaller(O ≈ 0.869andW ≈ 0.9073atλ = 1forour
(a)Theenergygap∆doesnotcloseforanyintermediateλ.(b)The tot
largestsystemsizeN =10).
total overlap O (circle) and the average weight W (dot) are still b
tot WeconsidertheOESforN =12(thelatticesizeisN ×
close to 1 at λ = 1. All of those demonstrate that the continuity b 1
holdsforbosonsatν =1/2. N2 = 3×4 for the FCI part). In the continuum, the thin-
torusconfigurationofthetwobosonicLaughlinstatesare(for
N =12,N =12)
b s
(cid:80)
the on-site interaction H = n (n −1). The ground
FCI i i i 111|111111|111
states are two nearly-degenerate states separated by a gap
fromtheexcitedstates. Wesetα = π/3alsoforthetwisted 020|202020|202±202|020202|020.
torus.
Wefindthattherearetwonearly-degeneratestates|Ψi(λ)(cid:105) Their total momentum is K1 = 6 and K1 = 0 (two-fold).
for each λ ∈ [0,1]. The evolution of the energy gap [Fig. The three |Ψi(λ)(cid:105) are in J1 = 0 sector (two-fold) and J1 =
8(a)], total overlap and average weight [Fig. 8(b)] is similar 2 sector. Because the two states in J1 = 0 sector will mix
with that for the ν = 1/3 fermionic Laughlin phase. How- with each other, we only concentrate on the single state in
ever,bothofthetotaloverlapandaverageweightaresmaller J1 = 2 sector, which corresponds to the K1 = 6 Moore-
(O ≈ 0.950 and W ≈ 0.9663 at λ = 1 for our largest Read state. Compared with the fermionic Moore-Read case,
tot
systemsizeN =8). theasymmetryproblemintheOESismoreserious. Although
b
WeconsidertheOESforN = 8(thelatticesizeisN × theOESof|ΨJ1=2(λ)(cid:105)doesnotpreciselymatchthatofexact
b 1
N = 4×4 for the FCI part). In the continuum, the thin- bosonicMoore-Readstate,wecanstillfindarelativelygood
2
torusconfigurationofthetwobosonicLaughlinstatesare(for correspondence between their OES levels up to ξmax ≈ 13.8
Nb =8,Ns =16) atλ=0.5andξmax ≈11.5atλ=1.
0101|01010101|0101
1010|10101010|1010. AppendixC:FurtherdetailsontheWannierbasisconstruction
Theirtotalmomentumis K = 0 and K = 8, respectively.
1 1
Thetwo|Ψi(λ)(cid:105)arebothinJ = 0sector,sotheymixwith To make this paper self-contained we give the prescrip-
1 tion of how to fix the phases Φ(k ) in the Wannier func-
eachotheranddonothaveagoodone-to-onecorrespondence 2
with the two Laughlin states. This means that each |Ψi(λ)(cid:105) tion |X,k2(cid:105) in this Appendix. The results were first found
in Ref. 23. The essential point in fixing the phase is
hasweightontheK =0andK =8sectorssimultaneously.
Hanodwpervoejre,cwtethceanothste1irllipnroKje1ct=on18es|Ψecit(oλr.)(cid:105)IinnFKig1.=9,0wseecctaonr tW(cid:80)oamnnaikeer tshteatecsoninnedcetpioennd(cid:104)eXnt,ko2f|YˆX|X(cid:48)a,nkd2(cid:48)(cid:105)kb2e.tweHeenreadYjˆacen=t
seethatthelow-lyingpartoftheOESof|Ψiprj(λ)(cid:105)alsomatch k1,k2|k1,k2(cid:105)A2(k1,k2)(cid:104)k1,k2+1|istheunitaryprojected
thatoftheexactLaughlinstateverywell. Ofcourseξmaxhere position operator in the ey-direction, and adjacent Wannier
(ξmax ≈ 13.2atλ = 0.5andξmax ≈ 11.2atλ = 1)islower states are defined by jX(cid:48),k2(cid:48) = jX,k2 +C. Because jX,k2 =
than that in the ν = 1/3 fermionic Laughlin case due to the N X +Ck +δ , k(cid:48) = k +1 (mod N ) for two adjacent
2 2 2 2 2 2
10
Figure9. (Coloronline)Theorbitalentanglementspectra(OES)ofexactbosonicLaughlinstates(bluediamond)andtheprojectednearly-
degeneratestates|ΨJ1(λ)(cid:105)(redcross)atν = 1/2,N = 8. ThelatticesizeisN ×N = 4×4andλ = 1fortheFCIpart. Bothofthe
prj b 1 2 1
twonearly-degeneratestatesareinJ =0sector. Intheleftcolumn,weconsidertheonewithlowerenergyandprojectitinK =0sector.
1 1
Intherightcolumn,weconsidertheonewithhigherenergyandprojectitinK = 8sector. In(a)and(b),λ = 0.5. (a): Theunprojected
1
|ΨJ1(λ)(cid:105)hasweightW ≈ 0.79721ontheK1 = 0sectorandW ≈ 0.19096ontheK1 = 8sector. TheoverlapwiththeLaughlinstate
O= 12(|(cid:104)ΨKLa1u=0|ΨJ1=0(λ)(cid:105)|2+|(cid:104)ΨKLa1u=8|ΨJ1=0(λ)(cid:105)|2)is0.98264.(b):Theunprojected|ΨJ1(λ)(cid:105)hasweightW ≈0.19276ontheK1 =0
sectorandW ≈0.79902ontheK1 =8sector.TheoverlapwiththeLaughlinstateO= 21(|(cid:104)ΨKLa1u=0|ΨJ1=0(λ)(cid:105)|2+|(cid:104)ΨKLa1u=8|ΨJ1=0(λ)(cid:105)|2)
is0.98825. In(c)and(d),λ = 1. (c): Theunprojected|ΨJ1(λ)(cid:105)hasweightW ≈ 0.75966ontheK1 = 0sectorandW ≈ 0.19968on
the K1 = 8 sector. The overlap with the Laughlin state O = 12(|(cid:104)ΨKLa1u=0|ΨJ1=0(λ)(cid:105)|2 +|(cid:104)ΨLKa1u=8|ΨJ1=0(λ)(cid:105)|2) is 0.93967. (d): The
unprojected |ΨJ1(λ)(cid:105) has weight W ≈ 0.20603 on the K1 = 0 sector and W ≈ 0.76720 on the K1 = 8 sector. The overlap with the
LaughlinstateO = 1(|(cid:104)ΨK1=0|ΨJ1=0(λ)(cid:105)|2+|(cid:104)ΨK1=8|ΨJ1=0(λ)(cid:105)|2)is0.96111. Theorangeshadowsindicatethegenericlevelsinthe
2 Lau Lau
OESof|ΨJ1(λ)(cid:105)whichdeviatefromthelevelsoftheexactLaughlinstate.
prj
Wannierstates. Ifincreasingfromk2tok2+1doesnotcross W(cid:48)(k1,k2)=[µ1(k2)]k1,and
theboundaryofpBZ,X(cid:48) =X. Otherwise,X(cid:48) =X+C.
FromthedefinitionoftheWannierstate,onecanobtainthat (cid:40) λ1(k2) ,k +1∈pBZ
µ (k )= λ1(k2+1) 2 .
1 2 e2πiC/N1 λ1(k2) ,otherwise
(cid:104)X,k2|Yˆ|X(cid:48),k2(cid:48)(cid:105)=ei[Φ(k2(cid:48))−Φ(k2)]A2(0,k2)U2(k2), λ1(k2+1)
(C1) Theproductof(cid:104)X,k |Yˆ|X(cid:48),k(cid:48)(cid:105)hasaverysimpleform,
2 2
where
N(cid:89)1−1N(cid:89)2−1(cid:104)X,k |Yˆ|X(cid:48),k(cid:48)(cid:105)=(cid:34)W (0)N(cid:89)2−1U (k )(cid:35)N1.
U2(k2)= N11 N(cid:88)1−1WW(cid:48)((kk11,,kk22)), X=0 k2=0 2 2 2 k2=0 2 2 (C2)
k1=0
with Defining U2(k2) = U2(k2)/|U2(k2)| and introducing a
W(k1,k2)= (cid:81)(cid:81)kκ1=−kκ01=1−0A11A(κ1(,κk,2k+2)1)AA22((k01,,kk22)), pgλhl2ea(0sien)ω((2−ω|U2π2)/N(Nk222),|=π,/wNh(cid:81)2ic)kNh,22w=s−a0e1tiUscfia2ne(skc2Eh)qo.ows(eiCth(cid:104)2X).th,Feki2n|aaYˆrlgl|yuX,mb(cid:48)e,ynkt2(cid:48)c(cid:105)oamn=--