Two-component quantum Hall effects in topological flat bands

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by  Tian-Sheng Zeng

Two-component quantum Hall effects in topological flat bands
Tian-Sheng Zeng,1 W. Zhu,2 and D. N. Sheng1
1Department of Physics and Astronomy, California State University, Northridge, California 91330, USA
2Theoretical Division, T-4 and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
(Dated: March 29, 2017)
WestudyquantumHallstatesfortwo-componentparticles(hardcorebosonsandfermions)loading
intopologicallatticemodels. Bytuningtheinterplayofinterspeciesandintraspeciesinteractions,we
demonstratethattwo-componentfractional quantumHallstatesemergeat certain fractional filling
7 factors ν = 1/2 for fermions (ν = 2/3 for bosons) in the lowest Chern band, classified by features
1 fromgroundstatesincludingtheuniqueChernnumbermatrix(inverseofK-matrix),thefractional
0 charge and spin pumpings, and two parallel propagating edge modes. Moreover, we also apply our
2
strategytotwo-componentfermionsatintegerfillingfactorν =2,whereapossibletopologicalNeel

r antiferromagnetic phase is under intense debate very recently. For the typical π-flux checkerboard
a lattice, by tuning the onsite Hubbard repulsion, we establish a first-order phase transition directly
M
from a two-component fermionic ν = 2 quantum Hall state at weak interaction to a topologically
trivial antiferromagnetic insulator at strong interaction, and therefore excludethe possibility of an

8 intermediate topological phase for our system.
2

] I. INTRODUCTION interesting to study the effect of interspecies interac-
l
e tions on two-component quantum Hall states in topo-
-
r Fractionalized topological ordered phases in topolog- logicallattice models. Experimentally,the Haldane hon-
t eycombinsulatorhasbeenachievedfromtwo-component
s ical flat bands have attracted intense attention in the
t. past few years1. They emerge as the ground state of fermionic 40K atoms with a tunable Hubbard repulsion
a in a periodically modulated honeycombopticallattice37.
interacting many-body systems at the fractional fillings
m For two-component bosonic atoms 87Rb in different hy-
of topological bands in the absence of a magnetic field,
- perfine spin channels, the two-dimensional Hofstadter-
in analogy to the fractional quantum Hall (FQH) states
d
n in two-dimensional Landau levels2–5. Topological flat Harper Hamiltonian is also engineered in the optical
lattice with time-reversal symmetry38. These advances
o bands with higher Chern number C have been revealed
c to host a series of Abelian single-component FQH states would open up new possibilities of studying the multi-
[ component quantum Hall effect for bosons and fermions
emerge at fillings ν = 1/(C +1) (for hardcore bosons)

2 and ν = 1/(2C + 1) (for spinless fermions)6–13, which in topological lattice models.
v arebelievedtobecolor-entangledlatticeversionsofmul- The aimof this paper is to providecompelling numer-
1 ticomponent Halperin (mmn) FQH states14 in a single ical evidence of K matrix classifications for both frac-
4 Chern band15–17. Much stronger evidence comes from a tional Halperin and integer quantum Hall states in sev-
4 hidden symmetry of particle entanglement spectrum, as eralmicroscopictopologicallatticemodelsthroughexact
2
discussed in Ref.9. However,a closely related problem is diagonalization(ED)anddensity-matrixrenormalization
0
. that no direct evidence for the integer valued symmetric group(DMRG)methods. Bytuningtheinterspeciesand
1 K-matrixhasbeenrevealed,whichis believedtoclassify intraspecies interactions, we show that for a given frac-
0
the topologicalorderatdifferent fillings for multicompo- tional filling factor, the many-body ground states in the
7
nent systems18–21. decoupledlimitevolveto asetofdegeneratestatessepa-
1
: Here we numerically address the possibility of multi- ratedfromthe higher-energyspectrum by a finite gapin
v component quantum Hall states in topological flat band thestronglyinteractingregime,whosetopologicalnature
Xi modelsfilledwithinteractingtwo-component(orbilayer) isdescribedbytheK-matrix20. Inaddition,thetopolog-
r particles, where “two-component” serves as a generic la- icalpropertiesofthesestatesarealsocharacterizedby(i)
a bel for spin or pseudo-spin(bilayer,etc.) quantum num- fractionalquantizedtopologicalinvariantsrelatedtoHall
ber. Different from the single-component case, there conductance, and (ii) degenerate ground states manifold
aremoretunable parametersintwo-componentsystems, under the adiabatic insertion of flux quanta. For inte-
like interspecies interaction whose magnitude can be ger quantum Hall states, we mainly focus on their phase
tuned through Feshbach resonance in cold atom set- transition nature driven by onsite interspecies interac-
ting22. When the spin degrees of freedom are included, tions, and demonstrate their first-order characteristics
one would expect many more exotic phases to occur, from discontinuous behaviors of related physical quan-
such as quantum Hall ferromagnetism23–25, and a rich tities at the transition point.
classoffractionalquantumHallstatesincludingHalperin Thispaperisorganizedasfollows. InSec.II,wegivea
(331) states for two-component fermions26–31 and spin- description of the Hamiltonian of two-component quan-
singlet incompressible states including Halperin (221) tum particles in two types of topological lattice mod-
states for two-component bosons32–36 previously stud- els,suchasπ-fluxcheckerboardandHaldane-honeycomb
ied for the lowest Landau level systems. Thus it is lattices. In Sec. III, we study the ground states of these
2
two-componentparticlesinthestronginteractionregime, (a) 0.5 (b)
presentnumericalresults ofthe K matrix by exactdiag-
0.25
onalizationatfillingsν =2/3forspinfulhardcorebosons 0.4
csaeunrsdtsioνtnh=eofp1r/flo2upxeforrqtiuesaspniontfafu.tlhefIesnremSgierocon.usnIIdinICsSt,aetcwe.esIIucInaBldc,euralantthdeedtihinse-- E−E) /t000..23 CB,N=6,Ns=2×3×3 E−E) /t000.1.25
adiabatic charge and spin pumping from DMRG, and ( CB,N=8,Ns=2×3×4 ( 0.1 CB,N=6,N=2×3×4
demonstrate the quantized drag Hall conductance. In 0.1 HC,N=6,Ns=2×3×3 0.05 HC,N=6,Nss=2×4×3
Sec. IIID, we discuss the momentum-resolved entangle-
ment spectrum of these two-component FQH states. In 0 0 5 10 0 0 5 10
Sec. IV, we discuss the role of interspecies onsite inter- K +N K K +N K
y y x y y x
action on the C = 2 integer quantum Hall state for
q
two-componentfermions at ν =2. Finally, in Sec. V, we
FIG. 1. (Color online) Numerical results for the low energy
summarize our results and discuss the prospect of inves- spectrum of (a) bosonic systems ν =2/3,Ns =3×N in dif-
tigating nontrivial topological states in multicomponent ferenttopologicallatticesatU =10t,V =0and(b)fermionic
quantum gases. systems ν =1/2,Ns =N ×4 in different topological lattices
at U =10t,V =20t.
II. THE MODEL HAMILTONIAN
two-component FQHE system in Landau levels30. For
Our starting point is the following noninteract- strong Hubbard repulsion, two-component fermions in
ing Hamiltonian of two-component particles (hardcore theHaldane-honeycombmodelmayexhibitvariouschiral
bosons and fermions) in topological lattice models, such magneticorderingsortopologicalMottinsulatorphaseat
as the π-flux checkerboard(CB) lattice, half-filling42,43.
In the ED study, we explore the many-body ground
HCB =−t c†r′,σcr,σexp(iφr′r)+H.c. state of H in a finite system of Nx×Ny unit cells (the
Xσ hXr,r′i(cid:2) (cid:3) total number of sites is Ns = 2 ×Nx ×Ny). The to-
tal filling of the lowestChern band is ν =2N/N , where
±t′ c†r′,σcr,σ−t′′ c†r′,σcr,σ+H.c., (1) N =N↑+N↓isthetotalparticlenumberwithglosbalcon-
XσhhXr,r′ii XσhhXr,r′ii servation U(1)-symmetry. With the translational sym-
metry,theenergystatesarelabeledbythe totalmomen-
and Haldane-honeycomb (HC) lattice
tum K = (K ,K ) in units of (2π/N ,2π/N ) in the
x y x y
Brillouinzone. For largersystems,we exploitDMRG on
HHC =−t′ [c†r′,σcr,σexp(iφr′r)+H.c.] cylinder geometry, and keep the number of states 1200–
Xσ hhXr,r′ii 2400 to obtain accurate results.
−t c† c −t′′ c† c +H.c., (2)
r′,σ r,σ r′,σ r,σ
XσhXr,r′i XσhhXr,r′ii
III. HALPERIN (221) AND (331) STATES
wherec† istheparticlecreationoperatorofspinσ =↑,↓
r,σ
at site r, h...i,hh...ii and hhh...iii denote the nearest-
In this section, we present the numerical evidences of
neighbor, the next-nearest-neighbor, and the next-next-
two-component FQH states at filling factors ν = 2/3
nearest-neighborpairs of sites, respectively. We take the
and ν = 1/2 for bosons and fermions, respectively. Im-
parameterst′ =0.3t,t′′ =−0.2tforcheckerboardlattice,
portantly, the Chern number matrix (the inverse of K-
as in Ref.39, while t′ = 0.6t,t′′ = −0.58t for honeycomb
matrix) uniquely identifies these two-component FQH
lattice, as in Refs.40,41. These parameters enhance the
states. The further information from ground state de-
flatness of the topological band and make quantum Hall
generaciesandcharge(spin)pumpingsiscomplementary
states more robust.
to and consistent with the Chern number matrix. The
Considertheon-siteinterspeciesandnearestneighbor-
Chern number matrix also provides an accurate predic-
ing intraspecies interactions
tionforthetransportmeasurementforexperimentalsys-
tems. Thus it is very important to numerically extract
Vint =U nr,↑nr,↓+V nr′,σnr,σ (3) this topologicalinformation for two-componentparticles
Xr Xσ hXr,r′i in topological flat band models.
where n is the particle number operator of spin-σ at
r,σ
siter. ThemodelHamiltonianbecomesH =H +V
CB int
(H = H + V ). Here, U is the strength of the A. Ground-state degeneracy
HC int
interspecies interaction while V is the strength of in-
traspecies correlations in topological flat bands, play- First, we demonstrate the groundstate degeneracyon
ing the analogous role of Haldane pseudopotentials for torus geometry, which serves as a primary signature of
3
0.4 (a) ν=1/2,θ↑=θ, θ↓=0 (b) ν=1/2,θ↑=θ↓=θ
(a) CB,N=6,N=2x3x3 (b) CB,N=6,N=2x3x4
s s
0.25
0.25 0.25
0.3
0.2
(E−E) /t000..12 K=(0,0) 0.01.51 KKK===(((000,,,012))) (E−E) /t00.001..512 00..0023 CCCBBB,,,KKK===(((000,,,012))) 0.001..51200..0023 CCCBBB,,,KKK===(((000,,,012)))
Others 0.05 K=(0,3) CB,K=(0,3) 0.01 CB,K=(0,3)
Others 0.05 0.010 2 4 0.05 0 1 2
0 0
0 5 10 0 5 10 0 0
U /t U /t 0 1 2 3 4 0 0.5 1 1.5 2
θ/2π θ/2π
FIG.2. (Coloronline)Robustnessofdegenerategroundstates
FIG. 3. (Color online) Numerical results for the y-direction
on the π-flux checkerboard lattice vs the variations of in-
2te×rsp3ec×ie3s raetpuνls=ion2/fo3r,:V(a=) b0;os(obn)icfesrymstioenmics Nsyst=em6s,NNs == aspteνctr=al 1fl/o2w,Uof=fer1m0ito,nVic=sys2t0etm:s(aN) θ=↑α6=,Nθs,θ=↓α2=×03;×(b4)
6,Ns = 2×3×4 at ν = 1/2,V = 20t. The energies are θ↑α=θ↓α =θ.
measured relative to the lowest ground energy. The momen-
tumsectorsinwhichthesedegenerateFQHstatesemergeare
labeledexplicitlywithdifferentcolorswhiletheothermomen- valuesformanonuniformdistributionswithstandardde-
tum sectors are labeled bythecyan circle. viationsoftheorder0.2,whicharepossiblycompressible
liquid states. Thus, we remark that the emergence of
two-component FQH states is induced by suitable inter-
an incompressible FQH state. We consider finite-size species and intraspecies repulsions.
systems up to maximum particle number N = 8. In
Figs.1(a)and1(b), weshowthe energyspectrumofsev-
eraltypicalsystemsinstronginteractingregime. Thekey B. Chern number matrix and K-matrix
featureisthat,thereexistsawell-definedanddegenerate
groundstate manifold separatedfrom higher-energylev- To uncover the topological nature of the ground-state
elsbyarobustgap. Fortwo-componenthardcorebosons manifoldwe extractthe Chernnumber matrix. Here, we
at ν = 2/3, the ground states show three-fold quasi- utilize the scheme proposed by one of the current au-
degeneracies; For two-component fermions at ν = 1/2, thors in Refs.45,46. With twisted boundary conditions
we find that the ground state manifold hosts eight-fold ψ(r + N ) = ψ(r )exp(iθα) where θα is the twisted
σ α σ σ σ
quasi-degeneracies. We also calculate the density and angle for spin-σ particles in the direction-α, we plot the
spin structure factors for the ground states, and exclude low energy spectra flux under the variation of θα. As
σ
any possible charge or spin density wave orders as the shown in Figs. 3(a) and 3(b), these ground states evolve
competinggroundstates,duetotheabsenceoftheBragg into each other without mixing with the higher levels.
peaksintheresults(WecheckthemuptoNs =2×4×4 For two-species hardcore bosons at ν = 2/3, the energy
sites using DMRG with periodic boundary conditions). recoversitself after the insertion of three flux quanta for
Next, we consider the effects of interspecies repulsion θα = θα = θ or θα = θ,θα = 0, indicating its 1/3 frac-
↑ ↓ ↑ ↓
on these topological phases. We first use ED to study tional quantization of quasiparticles. Interestingly, for
the evolution of the energy spectrum with U for the pe- two-speciesfermionsatν =1/2,theenergyrecoversitself
riodic system. As shown in Figs. 2(a) and 2(b), with afterthe insertionoftwo flux quantafor θα =θα =θ, or
↑ ↓
the decrease of U, the degeneracy is lifted and finally after the insertionof four flux quanta for θα =θ,θα =0,
↑ ↓
disappears at U = 0, where the ground state becomes indicating its 1/4 fractional quantization of quasiparti-
a possibly metallic phase with vanishingly small exci- cles. Meanwhile, the many-body Chern number of the
tation energy gap. In usual FQH states, the occupa- ground state wavefunction ψ is defined as45,46
i
tion of each single-particle orbital is constant and equal
to the filling factor44. By diagonalizing the N × N - 1 ∂ψi ∂ψi ∂ψi ∂ψi
matrixρσ =hc†r′,σcr,σi,weobtainreducedsingle-sparticsle Cσi,σ′ = 2π Z dθσxdθσy′Im(cid:18)h∂θσx|∂θσy′i−h∂θσy′|∂θσxi(cid:19)
eigenstates ρσ|φαi = ρασ|φαi where |φαi (α = 1,...,Ns) (4)
are the effective orbitals as eigenvectors for ρ and ρα
σ σ
(ρ1 ≥ ... ≥ ρNs) are interpreted as occupations. We ForthethreegroundstateswithK =(0,0)oftwo-species
σ σ
find that the occupations are close to the uniform filling hardcore bosons at N = 6,N = 2×3×3, by numeri-
s
ρα ≃ ν/2 for α ≤ N /2 with standard deviations of the callycalculatingthe Berrycurvaturesusing m×mmesh
σ s
order 0.01, while ρ ≪ 1 for α > N /2 in the strongly squares in the boundary phase space with m ≥ 9 we
α s
interacting regime, indicating an incompressible liquid obtain 3 Ci = 2, and 3 Ci = −1. Due to
i=1 ↑,↑ i=1 ↑,↓
with particles uniformly occupying the Ns/2-orbitals of the symPmetry c↑ ↔ c↓, one hPas the symmetric proper-
thelowestChernband. However,forU ≪t,theseeigen- ties Ci = Ci . All of the above imply a symmetric
σ,σ′ σ′,σ
4
C-matrix in the spanned Hilbert space, namely, 1
(a) (b)
C= C↑,↑ C↑,↓ = 1 2 −1 (5) 0.8 ∆ S,θ↑=θ,θ↓=0 0.5 ∆ S,θ↑=θ,θ↓=0
(cid:18)C↓,↑ C↓,↓(cid:19) 3(cid:18)−1 2 (cid:19) ∆ Q,θ↑=θ,θ↓=0 0.4 ∆ Q,θ↑=θ,θ↓=0
0.6
where the off-diagonal part C is related to the drag 0.3
↑,↓
0.4
Hall conductance. Thus we can obtain the K-matrix 0.2
whichistheinverseoftheC-matrix,namelyK=C−1 =
0.2
0.1
2 1
. Therefore, we establish that three-fold ground
(cid:18)1 2(cid:19) 0 0
0 0.5 1 0 0.5 1
states for two-species hardcore bosons at ν = 1/3 are θ/2π θ/2π
indeed Halperin (221) states in the lattice version, and
thethree-folddegeneracycoincideswiththedeterminant FIG. 4. (Color online) The charge and spin transfer with
detK, as predicted in Ref.47. inserting flux θ↑ = θ,θ↓ = 0 in the cylinder: (a) bosonic
Similarly, for the eight ground states with K = systems on the Ny = 3 cylinder at ν = 2/3,U = 10t,V = 0;
(0,i),(i = 0,1,2,3) of two-species fermions at N = (b)fermionicsystemsontheNy =4cylinderatν =1/2,U =
6,N = 2×3×4, by numerically calculating the Berry 10t,V =20t. Herethecalculation is performed usinginfinite
s
DMRGwith keepingup to 1200 states.
curvatures(usingm×mmeshsquaresinboundaryphase
space with m ≥ 9), we obtain 8 Ci = 3, and
i=1 ↑,↑
8 Ci = −1. The above resulPts imply a symmetric C. Fractional charge and spin pumpings
i=1 ↑,↓
PC-matrix, namely,
To uncover the topological nature of two-component
C C 1 3 −1
C= ↑,↑ ↑,↓ = (6) FQHstates,wefurthercalculatethechargepumpingun-
(cid:18)C↓,↑ C↓,↓(cid:19) 8(cid:18)−1 3 (cid:19) dertheinsertionoffluxquantaoncylindersystemsbased
onthenewlydevelopedadiabaticDMRG48 inconnection
Thus we canobtainthe K-matrixwhich is the inverseof
to the quantized Hall conductance. It is expected that a
3 1
the C-matrix, namely K = C−1 = . Therefore, quantized charge will be pumped from the right side to
(cid:18)1 3(cid:19)
the left side by inserting a U(1) charge flux θ =0→2π.
weestablishthateight-foldgroundstates fortwo-species
Thenettransferofthetotalchargefromtherightsideto
fermion at ν = 1/2 are indeed Halperin (331) states in
theleftsideisencodedbyQ(θ)=NL+NL =tr[ρ (θ)Q]
lattice version, and the eight-fold degeneracy coincides ↑ ↓ L
with the determinant detK, as predicted in Ref.47. (ρL the reduceddensity matrix ofthe left part)49b. Inobr-
With θx = θx = θx and θy =θy =θy, the many-body dertoquantifythedragHallconductance,wealsodefine
charge Ch↑ern n↓umber of the↑grou↓nd state wavefunction, thbe spin transfer ∆S by S(θ) = N↑L−N↓L = tr[ρL(θ)S]
related to charge Hall conductance, reads in analogy to the charge transfer.
AsshowninFigs.4(a)and4(b),forbosonsatνb=2/b3,
dθxdθy ∂ψ ∂ψ ∂ψ ∂ψ afractionalcharge∆Q=Q(2π)−Q(0)≃0.33ispumped
C = Im h | i−h | i
q Z 2π (cid:18) ∂θx ∂θy ∂θy ∂θx (cid:19) by threading one flux quanta with θ↑ = θ,θ↓ = 0 in one
species of two-component gases, and a fractional charge
=q·C·qT = Cσ,σ′ =ν, (7) ∆Q = Q(2π)−Q(0) ≃ 0.66 ≃ ν would be pumped by
Xσ,σ′ threading one flux quanta with θ = θ = θ in both of
↑ ↓
two-component gases. For fermion at ν = 1/2, a frac-
where q = (1,1) is the charge eigenvector of K-matrix.
Similarly, with θx = −θx = θx and θy = −θy = θy, tional charge ∆Q = Q(2π)−Q(0) ≃ 0.25 is pumped by
↑ ↓ ↑ ↓ threading one flux quanta with θ = θ,θ = 0 in one
we can also define the many-body spin Chern number ↑ ↓
species of two-component gases, and a fractional charge
of the ground state wavefunction, related to spin Hall
∆Q = Q(2π)−Q(0) ≃ 0.50 ≃ ν would be pumped by
conductance, as
threading one flux quanta with θ = θ = θ in both of
↑ ↓
dθxdθy ∂ψ ∂ψ ∂ψ ∂ψ two-component gases.
Cs =Z 2π Im(cid:18)h∂θx|∂θyi−h∂θy|∂θxi(cid:19) The dynamical pumping process reveals the fractional
statistics of the pumped quasiparticle. Based on these
=s·C·sT =C↑,↑+C↓,↓−C↑,↓−C↓,↑, (8) observations,weclaimthatthenumberofphysicallydis-
tinct stable quasiparticles is equal to the rank of the K-
where s = (1,−1) is the spin eigenvector of K-matrix.
matrix, and establish the pumping relationship that (i)
From Eqs. 7 and 8, we conclude that to identify the na-
bythreadingthefluxθ =θ,θ =0fromθ =0toθ =2π
↑ ↓
tureofthesedegeneratestates,onecancalculatetheadi-
abatic charge and spin pumping by performing different ∆Q=C +C , (9)
↑,↑ ↓,↑
flux insertion simulations, which identify the total frac-
∆S =C −C , (10)
↑,↑ ↓,↑
tionally quantized Hall and drag Hall conductances in
experiments, as will be shown below in Sec. IIIC. and (ii) by threading the flux θ =θ =θ from θ =0 to
↑ ↓
5
(a)12 (b)12 (c)12
(a) (b)
10 10
10 10 10
8 8 8 8 8
6 6 6
6 6
5 5
4 4 5 5 4
2 2 2 2 4 5 5 4
02 1 ∆ N= −1 02 1 1 ∆ N=0 02 1 ∆ N=1 2 1 2 1 2 2 1 4 1 3
0 2∆ K 4 0 2∆ K 4 0 2∆ K 4 ∆ N=−1 1 ∆ N=0
0 0
0 2 4 6 0 2 4 6
∆ K ∆ K
FIG.5. (Coloronline)Themomentum-resolvedentanglement
spectrumforbosonicsystemsontheNy =6cylinderofsingle
layer square lattice with Chern number two at ν =1/3,U = FIG. 6. (Color online) The momentum-resolved entangle-
10t,V = 0. In each tower, the horizontal axis shows the ment spectrum for fermionic systems on the Ny = 8 cylin-
relative momentum ∆K = Ky −Ky0 (in units of 2π/Ny) in der of single layer square lattice with Chern number two at
the transverse direction of the corresponding eigenvectors of ν = 1/4,U = 10t,V = 20t. In each tower, the horizontal
density matrix ρL. The numbers below the red dashed line axis shows the relative momentum ∆K =Ky−Ky0 (in units
label the nearly degenerating pattern for the low-lying ES of 2π/Ny) in the transverse direction of the corresponding
with different momenta: 1,2,5,··· eigenvectors of density matrix ρL. The numbers below the
red dashed line label the nearly degenerating pattern for the
low-lying ES with different momenta.
θ =2π
As shown in Figs. 6(a) and 6(b), for odd charge sector
∆Q= Cσ,σ′, (11)
∆N = −1, two symmetric branches of ES appear with
Xσ,σ′
the same level counting 1,2,5,···, differing by a phase
∆S =0. (12)
π in the momentum. However, for even charge sector
∆N = 0, two asymmetric branches of ES appear with
different level countings 1,1,4,··· and 0,1,3,···, differ-
D. Chiral edge spectrum ing by a phase π in the momentum. Thus, without the
π-phase shift, the total counting of two branches of ES
Another “fingerprint”ofchiraltopologicalorderis the wouldbe (a) 2,4,10,··· for odd chargesector∆N =−1
characteristicedgestateusuallydescribedbythespecific and (b) 1,2,7,··· for even charge sector ∆N =0, which
Luttinger liquid theory, which can be revealed through are consistent with the analysis of the counting of the
the low-lying entanglement spectrum (ES)50. Here we rootconfigurationsofthebilayerHalperin(331)edgeex-
examine the ES of these FQH states based on DMRG citations in the lowest Landau level at half filling54.
method. Due to its difficulty of DMRG convergence for
two-component gases on the N > 6 cylinder, in order
y
to calculate the momentum-resolved entanglement spec- IV. INTEGER QUANTUM HALL STATE
trum, we stack the two-component topological checker-
board lattices into an equivalent single-layer square lat- In this section, we consider the ground states at inte-
tice with Chern number two, as engineered in Ref.8, ger filling ν = 2N/N = 2 for two-component fermions
s
wherehardcorebosonsatone-thirdfillinghostthree-fold (N =N +N ), namely, one fermion per site. In nonin-
↑ ↓
degenerate ground states reminiscent of Halperin (221) teracting cases U =0,V =0, the system is a topological
states. In such a construction, the original two degener- Chern insulator with quantized Hall conductivity two.
ate edge modes in bilayer checkerboard lattices are dif- In the large repulsive case U ≫ t, it would be a trivial
fered by a momentum phase π in the Brillouin zone of antiferromagnetic Mott insulator as expected. Several
the single-layer square lattice. meanfieldstudiesofHaldane-honeycombmodelindicate
For hardcore bosons, as shown in Figs. 5(a-c), the a topologically nontrivial N´eel antiferromagnetic insu-
two branches of low-lying bulk ES appear with the level lating phase in the intermediate interaction regime55–57,
counting 1,2,5,···, implying the gapless edge modes. supported by quantum cluster methods43. However re-
Since K has two positive eigenvalues, the edge excita- cent dynamical cluster approximation have revealed a
tions would have two forward-moving branches in the first order transition from C =2 Chern insulator into a
q
same direction in the charge sector ∆N = 0 as shown trivialMottinsulator,withnoevidenceoftopologicalan-
in Fig. 5(b), consistent with theoretical analysis of spin- tiferromagnetic insulator as an intermediate phase58,59.
singlet quantum Hall state in Refs.51–53. In the following discussion, we take the typical example
Similarly for fermions, the edge excitations would also ofπ-fluxcheckerboardlattice,andinvestigatethe nature
havetwoforward-movingbranchesinthesamedirection. of the interaction-driven transition using both ED and
6
(a) (b)3
(a) L =6
CB,Ns=16 3 y
K=(0,0)
E) /t02 2 OPSu(v mQerp)la ∆p QF(U)
− Entropy S
E 1 L
( 1

0
0
2 4 6 8 10 12 14 2 4 6 8 10
U/t U/t
(c)1.5 5
(b) L =8
y
4
1
C
↑,↑ 3 Overlap F(U)
S( Q)
S( Q)
0.5 ∆s/∆0 2 Entropy SL
1
0
2 4 6 8 10 12 14 0
U/t 2 4 6 8 10
U/t
FIG. 7. (Color online) Numerical results for the y-direction
FIG. 8. (Color online) Numerical DMRG results on a cylin-
spectralflowoffermionicsystems: (a)Twodifferentchecker-
boardlatticegeometriesconsideredinourcalculation;(b)The der with finite width Ly =2×Ny and fixed length Lx =18
lowenergyspectraasafunctionofonsiterepulsionU forsys- for N↑ = N↓ = Lx ×Ny,ν = 2. The evolutions of physi-
cal quantities vs onsite repulsion U, including the absolute
tem Ns = 16,N↑ = N↓ = 8 at ν = 2; (c) The manybody
wavefunction overlap F(U),charge pumping∆Q,spin struc-
ChernnumberC↑,↑ andantiferromagneticspinstructurefac-
tor S(Q) of the lowest ground state, and the spin excitation ture factor S(Q), and entanglement entropy SL are shown
gap ∆s as a function of onsite repulsion U. Here the param- for (a) Ly =6,Ny =3 and (b) Ly =8,Ny =4, respectively.
eters V =0,t′=0.3t,t′′ =0. The first-order transition is characterized by the discontin-
uous behavior of these physical quantities at the transition
point. Here the parameters V = 0,t′ = 0.3t,t′′ = 0, and the
DMRG calculations. maximally kept numberof states 2020.
For small system sizes, we present a ED diagnosis of
quantumphase transitionofFermi-Hubbardmodel from
weakinteractionstostronginteractionsinπ-fluxchecker- the critical threshold U , demonstrating the discontinu-
c
board lattice. Our current system size limit for ED cal- ous first-order transition. In the Mott regime, the anti-
culation is Ns = 16,N↑ = N↓ = 8. For simplicity, we ferromagneticorderdominates andwe obtainthe charge
take t′ = 0.3t,t′′ = 0. The lattice geometry is indicated Hall conductance C = C + C = 0, as expected.
q ↑,↑ ↓,↓
in Fig. 7(a), with two different lattice sizes Ns = 16,12. Meanwhile, the spin gap ∆s =E0(Sz =1)−E0(Sz =0)
In Fig. 7(b), we plot its low energy evolution as onsite tends to collapse (here a finite small value is limited by
repulsion U increases. For weak interactions, there al- finite size effects).
waysexistsastableuniquegroundstatewithalargegap
Forlargersystemsizes,weexploitanunbiasedDMRG
separated from higher levels. When one flux quanta is
approach to study this phase transition from four dif-
inserted, it would evolves back to itself. With twisted
ferent aspects, using a cylinder geometry up to a max-
boundaryconditionsasdescribedinSec.IIIB,weobtain
imum width L = 8 (N = 4). As shown in Figs. 8(a)
the K-matrix as a unit matrix y y
and 8(b), first, we calculate the absolute wavefunction
1 0 overlapF(U)=|hψ(U)|ψ(U +δU)i| where δU is a small
K=C= . (13)
(cid:18)0 1(cid:19) finite value (here we take δU as small as 0.05t near the
transition region). For U < U (or U > U ), F(U) has
When U increases further, this ground state undergoes c c
a large value close to 1, indicating the same structure
anenergylevelcrossingwithotherlevelsaroundU ≃9t.
c of wavefunctions. When U approaches a critical value
In order to clarify the nature of the phase transition, we
U 60, F(U) suddenly drops down to a very small value
calculateitsChernnumbersC ,C andantiferromag- c
↑,↑ ↓,↓ close to zero, separating two different phases. Second,
netic spin structure factor
we calculate the charge pumping by inserting one flux
S(Q)= 1 eiQ·(r−r′)hSrSr′i, (14) θ↑ = θ,θ↓ = 0 from θ = 0 to θ = 2π, and obtain : (i)
Ns Xr,r′ z z ∆Q=1,∆S =1 in weak interacting regime U <Uc; (ii)
∆Q = 0,∆S = 0 in strongly interacting regime U > U .
c
where Q = (π,π),S = (n − n )/2. As indicated in Third, the spin structure factor S(Q) is also calculated,
z ↑ ↓
Fig. 7(c), both C = C and S(Q) experience a sud- and it exhibits a discontinuous jump near the transition
↑,↑ ↓,↓
den dramatic jump as the interaction U increases across point, similar to our ED analysis.
7
Finally, we uncover its entanglement signatures of diagram and phase transition.
quantum Hall transitions61,62. We partition the system
into two halves at the cylinder center and trace out the
right part to obtain the entanglement spectrum ξ (the V. SUMMARY AND DISCUSSIONS
i
eigenvalues of reduced density matrix of the left part)
and entanglement entropy SL = − iξilnξi. As shown In summary, we show that two-component hardcore
inFigs.8(a)and8(b),forU <Uc,SLPisalmostaconstant bosons and fermions in topological lattice models could
for a given system, indicating that the same topological hostHalperinFQHstatesatapartialfillingofthelowest
properties as integer quantum Hall state. SL starts to Chern band, with fractional topological properties char-
drop at U ≃ Uc, with a discontinuous first-order deriva- acterized by K-matrix, including the degeneracy, frac-
tive ∂SL/∂U due to the change of topology63. By com- tionalquantizedHallconductanceandchiraledgemodes.
paring the complete spectra of reduced density matrices The roleofonsite interspeciesrepulsiononintegerquan-
between two groundstates, the first-ordertransition can tumHallstateontheπ-fluxcheckerboardtopologicallat-
be extracted from the discontinuous jump of majoriza- ticeisexamined,andshowntoleadtoafirst-orderphase
tion62. Recently, a diagnostic of phase transition driven transition from a C = 2 integer quantum Hall state to
q
by disorderfromquantumHallstates toaninsulatorvia a trivial Mott insulator for two-component fermions at
entanglement entropy is also examined64. integer filling factor ν =2. Another interesting issue for
The above pictures from both ED and DMRG cal- two-componenthardcore bosons at ν =2 filling, but not
culations indicate a first-order phase transition directly discussed here, is related to the possibility for a bosonic
from a C = 2 integer quantum Hall state to a Mott integerquantumHallstate34,65–71,whichisleftforfuture
q
insulator driven by onsite repulsion. We did not ob- study.
serve any evidence of topological Neel antiferromagnetic
insulating phase in the intermediate interaction regime,
which is consistent with recent studies on the Haldane- ACKNOWLEDGMENTS
honeycomb lattice from different methods58,59. Further,
withtheinclusionofnext-next-nearest-neighborhopping ThisworkissupportedbytheU.S.DepartmentofEn-
t′′ = −0.2t, we numerically arrive at the same phase ergy, Office of Basic Energy Sciences under Grants No.
transitionnatureasabove. Forthefuturework,itwould DE-FG02-06ER46305 (T.S.Z, and D.N.S.), and and by
be interesting to consider the effects of the band flatness U.S. Department of Energy’s National Nuclear Security
modulatedbynext-nearest-neighborhoppingt′andnext- Administration through Los Alamos National Labora-
next-nearest-neighborhopping t′′ on the quantum phase tory LDRD Program(W.Z.).
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