Table Of ContentBosonic integer quantum Hall effect as topological pumping
Masaya Nakagawa1,∗ and Shunsuke Furukawa2
1Department of Physics, Kyoto University, Kyoto 606-8502, Japan
2Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
(Dated: January 6, 2017)
Based on a quasi-one-dimensional limit of quantum Hall states on a thin torus, we construct a
model of interaction-induced topological pumping which mimics the Hall response of the bosonic
integer quantum Hall (BIQH) state. The quasi-one-dimensional counterpart of the BIQH state is
identified as the Haldane phase composed of two-component bosons which form effective spin-1
degrees of freedom. An adiabatic change between the Haldane phase and trivial Mott insulators
constituteoff-diagonal topological pumpingin which thetranslation ofthe latticepotential forone
component induces a current in the other. The mechanism of this pumping is interpreted in terms
7
of changes in polarizations between symmetry-protected quantized values.
1
0
2 PACSnumbers: 05.30.Jp,03.75.Mn,73.43.Cd
n
a I. INTRODUCTION pumping have also been discussed.11–14
J
The integer quantum Hall effect and its 1D coun-
4 Adiabatic change of parameters of a Hamiltonian terpart, the Thouless pumping, are composed of non-
] sometimes causes nontrivial effects which cannot be interacting free fermions. It is well known that in high
l found in its instantaneous ground state. Topological magnetic fields, interactions between particles can gen-
e
- pumping, originally proposed by Thouless,1 provides a erate highly entangled ground states and result in vari-
tr prototypical example of such phenomena. Thouless con- ous quantum Hall states with fascinating features. For
s sidered a one-dimensional (1D) band insulator in a peri- example, the fractional quantum Hall (FQH) states15,16
t. odic lattice potential V(x) = V(x+a). Let us consider anditsnon-Abeliangeneralizations17–19 exhibitfraction-
a
m an adiabatic shift of the lattice potential V(x − at/T) ally quantized Hall conductivity and host fractionalized
parametrized by t. Since the lattice potential is peri- excitations with exotic quantum statistics which differs
-
d odic,soistheHamiltonianwithrespecttoitsparameter: from ordinary bosons and fermions. The FQH states are
n H(t+T)=H(t). The one-particlequantum states inan examples of topologically ordered states,20 and exhibit
o energy band are then specified by the wave number k ground-state degeneracy if the system is put on a topo-
x
c and the parameter t. When the lattice potential is adia- logicallynontrivialsurfacewithnonzerogenus, suchasa
[ baticallyshiftedbyvaryingt,theparticlesmovewiththe torus.21
1 lattice, and the Hamiltonian returns to its original form OtherthantheFQHstates,strongcorrelationscancre-
v afterashiftbyonelatticespacing. Thegroundstatealso ate yet another interesting integer quantum Hall state.
7 returns to the initial one as long as the particles stay ThebosonicintegerquantumHall(BIQH)state,22 which
2 in the same band. However, the change in the ground is formed by two-component bosons with the total fill-
1
state during the cycle can cause a nonvanishing parti- ing factor ν = 1 + 1, is such an example. Since non-
1
0 cle current. The total current over one cycle is given by interacting bosons form a Bose-Einstein condensate in
. theChernnumber,whichtakesanintegerandcharacter- the ground state, quantum Hall states of bosons in-
1
izestopologicallydistinctclassesofthesetofone-particle evitably require interactions. Although the BIQH state
0
7 statesdefinedinthe(kx,t)plane. Hencethetotalcurrent doesnotexhibitanyfractionalizedexcitationortopolog-
1 is quantized, and this phenomenon is called the topolog- ical ground-state degeneracy, it is strictly distinguished
: ical Thouless pumping. If we identify the (k ,t) plane from a trivial phase as long as the U(1) symmetry asso-
v x
with the two-dimensional (2D) reciprocal space, we find ciated with the conservation of the total particle num-
i
X thattheThoulesspumpingsharesthesameoriginasthe ber is preserved.23 In this sense, the BIQH state is an
r integer quantum Hall effect,2,3 in which the quantized example of a 2D symmetry-protected topological (SPT)
a pumpingcorrespondstothequantizedHallconductivity. phase24–26 of bosons, and there have been studies on
Owing to its topological nature, the topological pump- physical models that realize this state (particularly in
ing is robust against small deformation of the pumping cold-atom setups)27–34 and exotic phase transitions out
protocol, and can be realized by various types of cycles of this phase.34,35 The Hall response of the BIQH state
usinglatticemodels.4–7Afteralmost30yearssinceThou- is described by the effective Chern-Simons theory
less’s prediction, the topological pumping was finally re-
1
alizedexperimentallybyusingultracoldatomsinoptical L=−4πεµνλ(A1µ∂νA2λ+A2µ∂νA1λ), (1)
superlattices,8–10 inwhichtheperiodicchangeofthelat-
tice potential was created by the change of the phase of whereA (A )istheU(1)gaugefieldwhichcouplesto
1µ 2µ
the standing wave potential. Other schemes of manipu- the first (second) component of bosons. Here we assume
latinganopticalsuperlatticeforrealizingthetopological that the number of particles is separately conserved in
2
each component. From the effective theory (1), we can
read off that a probe electric field for one component
inducesthequantizedHallresponseintheother. Suchan
“off-diagonal” response is a unique feature of the BIQH
state, which we highlight in this paper.
In this paper, we aim to construct nontrivial classes of
topological pumping which correspond to quantum Hall
states created by strong interactions. We first consider
topological pumping which mimics the Hall response FIG.1. (coloronline)QuantumHallsystemonathintorus
of the FQH states, and then focus on the case of the (left) and topological pumping (right). A flux φ (in units
of (cid:126)/Q) is inserted through a thin torus on which the LLL
BIQH state. Based on a quasi-1D limit of quantum Hall
orbitals (2) are formed with a constant spacing a. In a thin-
states,36–39 we can systematically construct strongly in-
torus limit L /(cid:96) (cid:28) 1, the flux insertion argument for the
teracting models of topological pumping and thus nat- y
Laughlin FQH states can be restated as the change between
urally extend the connection between the topological
degenerateCDWgroundstatesbyatranslation,whichresults
pumping and the quantum Hall effect to interacting sys-
in fractional Thouless pumping. The figure is the case of
tems. We show that the quasi-1D limit of the BIQH ν =1/3.
state is given by the Haldane phase,40–43 which is a cele-
brated example of a SPT phase in one dimension.24,44,45
The mechanism of the resulting topological pumping E = (cid:126)Ω(n + 1) (n = 0,1,...), where (cid:126)Ω = (cid:126)2 is
n 2 2M(cid:96)2
is interpreted in terms of changes in polarizations be- the cyclotron energy. The states in each level are N -
φ
tween quantized values that correspond to two distinct fold degenerate, and labeled in the present gauge by the
gappedphasesinthepresenceoftheinversionsymmetry. wave number k = 2πm (m=0,1,··· ,N −1) in the y
Theobtainedoff-diagonaltopologicalpumpingintertwin- direction. In pmarticuLlayr, the lowest-Landφau-level (LLL)
ing two-component bosons provides novel interaction- wavefunctions are given by46,47
induced topological pumping, and suggests an intriguing
connection between the 2D topological phases and 1D ψ (r)= 1 (cid:88)exp(cid:104)− 1 (x−k (cid:96)2−nL )2
gapped phases. m (cid:112)π1/2(cid:96)L 2(cid:96)2 m x
y n∈Z
The rest of this paper is organized as follows. In Sec.
(cid:16) nL (cid:17) (cid:105)
II, we describe our idea for obtaining models of topolog- +i k + x y . (2)
m (cid:96)2
ical pumping systematically from quantum Hall states
on a thin torus. We take the case of FQH states as ex- This wavefunction is localized around x = k (cid:96)2 = ma
m
amples, and explain that their thin-torus counterparts with a width (cid:96) in the x direction, and delocalized in
naturally give rise to fractional charge pumping. In Sec. the y direction; see the left panel of Fig. 1. Here,
III, we construct a 1D lattice model that corresponds to a = 2π(cid:96)2/L = L /N is the spacing between neigh-
y x φ
the thin-torus limit of two-component bosons in a mag- boring wavefunctions, and used as an effective “lattice
netic field. We then identify the Haldane state as the constant” later.
thin-torus counterpart of the BIQH state. In Sec. IV, In Laughlin’s flux insertion argument,48 the quantized
the topological pumping which mimics the BIQH effect Hallconductivitycanbederivedasaresponsetoanadi-
isdescribedandinterpretedintermsofchangesinpolar- abatic insertion of a flux quantum through the torus.
izations between symmetry-protected quantized values. The effect of flux insertion is expressed by a twisted
Finally, we summarize our results in Sec. V. boundary condition, which results in the replacement
k →k + φ with0≤φ≤2π. Afterinsertingoneflux
m m Ly
quantum (φ = 2π), the Hamiltonian of the system goes
II. TOPOLOGICAL PUMPING AS FLUX back to its original form. However, each Landau-level
INSERTION THROUGH A THIN TORUS orbital shifts its position during this process. Hence, if
some of the Landau levels are completely occupied and
others are empty, the system exhibits the integer quan-
In thissection, we explainour idea forobtaining mod-
tum Hall effect. The FQH effect is also understood in
els of 1D topological pumping systematically from a so-
a similar manner by taking into account the topologi-
called thin-torus limit of 2D quantum Hall states.36–39
cal ground-state degeneracy on the torus.21,47,49 In this
Let us consider a bosonic or fermionic system com-
case, the insertion of one flux quantum transfers the ini-
posed of N particles of charge Q and mass M in a uni-
tialgroundstatetoanotherdegenerategroundstate,and
form magnetic field B on a 2D torus of size L × L .
x y some integer multiple of flux quanta are required to go
We take the Landau gauge A = (0,Bx) and assume
back to the initial ground state.
QB > 0. The total number of flux quanta piercing the
(cid:113) Keepingthispictureinmind,letusgraduallydecrease
system is N = LxLy, where (cid:96) = (cid:126) is the magnetic
φ 2π(cid:96)2 QB the length Ly in the y direction while keeping the total
length. The filling factor is defined as ν = N/N . The area L L fixed. By identifying the locations x = ma
φ x y
single-particle spectrum is given by the Landau levels of the LLL orbitals (2) as a lattice coordinate, the orig-
3
inal 2D system can be viewed as an effective 1D lattice CDW ground states.56,57,59,61,62 Although the topologi-
model.36–39Theeffectivemodelcontainssomelong-range cal pumping constructed here is just a translational op-
interactions, whose coefficients are given by interaction eration of the entire system and seems to be somewhat
matrixelementswithrespecttotheLLLorbitalsoncon- trivial, we will see that the case of the BIQH state pro-
cerned sites. However, as we decrease L , the lattice vides more nontrivial topological pumping.
y
constant a increases, and the LLL orbitals are mutually
separated further. Interactions for longer distances are
thus suppressed more rapidly in this process, and the
physics for L /(cid:96) (cid:28) 1 is expected to be dominated by a
y
few interaction terms at short distances. III. THIN-TORUS LIMIT OF THE BOSONIC
If the ground state of a given quantum Hall state is INTEGER QUANTUM HALL STATE
smoothlychangedwithoutclosinganexcitationgapwith
decreasing Ly, the ground state is expected to gradually Based on the correspondence between the quantum
acquiresa1Dcharacterowingtothesuppressionofinter-
Hall state and the topological pumping described in
actions for long distances. We can then view the ground
the previous section, we here construct the thin-torus
state on a thin torus as the 1D lattice counterpart of the
counterpart of the BIQH state. Let us start with two-
quantumHallstate. Itisthenexpectedthattheinsertion
componentbosonsinauniformmagneticfieldonatorus
of a flux quantum through the thin torus induces quan-
described by the Hamiltonian
tizedcurrentinthexdirection. Thisphenomenoncanbe
viewed as the topological pumping in the 1D model.50,51
LetusillustratetheaboveideausingthesimplestFQH H = (cid:88) (cid:90) d2rΨ(α)†(r)[p−QA(r)]2Ψ(α)(r)
state, the Laughlin state of fermions at ν = 1/3. It was 2M
α=1,2
shown that with decreasing L , the ν = 1/3 Laughlin
state is smoothly deformed inyto a charge-density-wave +(cid:88)g(αβ) (cid:90) d2rΨ(α)†(r)Ψ(β)†(r)Ψ(β)(r)Ψ(α)(r),
(CDW) state in which every third site is occupied by a 2
α,β
particle.37 Here, density-density interactions for nearest-
(3)
neighbor and next-nearest-neighbor pairs of sites play a
dominant role and stabilize the CDW state. If a flux
quantum is adiabatically inserted through a thin torus, where Ψ(α)(r) (α = 1,2) denotes the bosonic field oper-
theLLLorbitalsshifttheirpositionsbyonelatticespac- ator for the α-th component. We assume repulsive con-
ing a, and the CDW ground state changes into another tact interactions g(αβ) > 0 between particles, and set
degenerate ground state51 as shown in Fig. 1. The to- g(11) = g(22) ≡ g for simplicity. The filling factor for
tal current during this shift of the ground state is equal each component is set to unity so that the total filling
to 1/3 if averaged in space; this corresponds to the frac- factor is given by ν = 1+1. The system possesses the
tionalHallconductivityσ = 1Q2 oftheLaughlinstate. U(1)×U(1) symmetry associated with the particle num-
xy 3 h
Hence, as expected, the flux insertion for the 1D coun- ber conservation in each component. Through exact di-
terpart of the FQH state results in fractional Thouless agonalization analyses,27–29 it was shown that the BIQH
pumping. Similarly to Laughlin’s argument for the FQH state described by the effective Chern-Simons theory (1)
effect,49 here the degeneracy of the ground states is es- appears when the ratio of the intercomponent to intra-
sential in obtaining the fractional pumping. component interactions, δ ≡g(12)/g, is close to unity.
For a filling fraction with a denominator larger than WithintheLLLapproximation,thefieldoperatorsare
3, it is not clear whether the FQH state is adiabatically expanded as
connected to a CDW state with decreasing L . This is
y
because in a thin torus, the density-density interactions
that stabilize a CDW state can severely compete with N(cid:88)φ−1
Ψ(α)(r)= b(α)ψ (r,φ ), (4)
other interaction terms in which two particles hop in m m α
a center-of-mass-conserving manner.37–39,52,53 However, m=0
if we keep only the density-density interactions and ne-
glectotherinteractionterms,thesystemexhibitsaCDW where b(α) annihilates a particle in the m-th LLL orbital
m
ground state with q-fold degeneracy at every rational fil-
and satisfies the commutation relations [b(α),b(β)†] =
ingfractionν =p/q(withpandqbeingcoprime).37–39,54 m n
The adiabatic shift of such a CDW state by one lat- δαβδmn and[bm(α),b(nβ)]=0. Tofacilitatelaterdiscussions
tice spacing clearly results in fractional Thouless pump- on topological pumping, we have introduced a magnetic
ing. Similar schemes for realizing fractional pumping by fluxφα whichcouplestotheα-thcomponentsothatthe
CDW states have been discussed in literature,55–59 es- LLL wave function ψm(r,φα) is given by the right-hand
pecially in connection with the “synthetic dimension” side of Eq. (2) with km replaced by km + φLαy. Corre-
technique60 in cold atoms. In this technique, infinite- spondingly, the positions of the LLL orbitals are shifted
range interactions in the synthetic dimension stabilize to x = (m+ φα)a in the α-th component. Substituting
2π
4
the expansion (4) into the Hamiltonian (3), we obtain Forδ >1,wheretheintercomponentinteractionisdomi-
nant(2V <V(12)),thegroundstatesareferromagnetic
H = 1(cid:126)Ω(cid:88)N(α) states 00 00
2
α (cid:12)··· n(1) ···(cid:69) (cid:12)··· 2 2 2 2 ···(cid:69) (cid:12)··· 0 0 0 0 ···(cid:69)
+(cid:88)(cid:88) (cid:88) Vmnb(jα+)n†b(jα+)m†b(jα+)m+nb(jα) (cid:12)(cid:12)··· nj(2) ··· =(cid:12)(cid:12)··· 0 0 0 0 ··· , (cid:12)(cid:12)··· 2 2 2 2 ··· ,
j
α j |n|≤m≤Nφ (10)
2
+(cid:88) (cid:88) V(12)(φ −φ )b(1)†b(2)† b(2) b(1). if we fix only the total number of particles. If we fix the
mn 1 2 j+n j+m j+m+n j number of particles in each component, a phase separa-
j −N2φ<m,n≤N2φ (5) tion occurs. The point δ = 1, at which 2V00 = V0(012), is
special—the on-site energy is the same for |n(1),n(2)(cid:105) =
j j
We note that the interactions in this Hamiltonian pre- |2,0(cid:105),|1,1(cid:105),|0,2(cid:105), leading to 3Nφ-fold degeneracy of the
ground state. This macroscopic degeneracy in the thin-
servesthecenter-of-masspositioninthexdirection. The
interaction matrix elements are calculated by using the torus limit is lifted by fluctuations as we increase Ly.
LLL wavefunctions as39,46 To obtain a unique ground state at δ =1, we proceed
awayfromthethin-toruslimitbyincreasingL andcon-
y
Vmn = LzmxLngy (cid:88)q (cid:104)δn(cid:48),nye−21q2(cid:96)2cos(qxkm(cid:96)2)+(m↔n)(cid:105), sairdeerthleeadnienagreflsut-cntueiagthiobnosr. oTnhees,nwexhti-clheadairnegVin10te,rVa1(c01t2i)o(n=s
(6a) V(12)),andV(12)terms. TheV(12)terminvolveshopping
−1,0 01 01
Vm(1n2)(φ)= Lg(1L2) (cid:88)δn(cid:48),nye−12q2(cid:96)2cos(cid:20)qx(cid:18)km− Lφ (cid:19)(cid:96)2(cid:21), otyfppea.rtTicoledsiswcuhsilsetthheecootmhperetoitnioesnaorfethofesaenteerlmecst,rowstearteic-
x y q y strictourselvestothelow-energymanifoldoftheHilbert
(6b) space spanned by the 3Nφ-fold degenerate ground states
of Eq. (8) at δ = 1. The thin-torus ground states (9)
where the sum is over the wave vector q = (2πnx,2πny) and (10) for δ (cid:54)= 1 also reside in this manifold. In this
(nx,ny ∈ Z), δn(cid:48),ny is the modulo-Nφ KronecLkxer dLelyta, risesstartiicstfieeddsuatbsepvaecrey,siintew,hthicehotpheeracotonrsstraint (cid:80)αn(jα) =2
and zmn = 2−δm,|n|(1+δm,0)2−δm,Nφ/2(1+δ|n|,Nφ/2) is a fac-
tor for fixing the double counting of some terms. If we S = 1(cid:88)b(α)†σ b(β) (11)
take the limit Lx/(cid:96) → ∞ (and thus Nφ → ∞) while j 2 j αβ j
keepingL /(cid:96)fixed,theseelementsaregivenmoresimply α,β
y
by
satisfythecommutationrelationsoftheSU(2)generators
andhavethefixedmagnitudeS2 =1(1+1)asisknownin
2z g j
Vmn = √ mn e−21(km2 +kn2)(cid:96)2, (7a) the Schwinger boson formalism.63 Here, σ =(σx,σy,σz)
2πL (cid:96)
y is a set of Pauli matrices. The Hamiltonian can thus be
Vm(1n2)(φ)= √g(12) e−12[(km−φ/Ly)2+kn2](cid:96)2. (7b) written in terms of the spin-1 operators as
2πLy(cid:96) H =(cid:88)(cid:2)J (SxSx +SySy )+J SzSz +D(Sz)2(cid:3),
xy j j+1 j j+1 z j j+1 j
Let us first consider the case when no flux is inserted j
(12)
throughthetorus: φ =φ =0. Whenwetakethethin-
1 2
torus limit L → 0, the only remaining interactions are
on-siteones,wyhicharethem=n=0componentsofEq. whereJxy =2V0(112),Jz =2(V10−V1(012)),andD =2V00−
(5). We thus obtain V(12). ThishastheformoftheXXZchainwithasingle-
00
ion anisotropy.64–66 At δ = 1, in particular, since V =
10
H = (cid:88) (cid:88)V n(α)(n(α)−1)+(cid:88)V(12)n(1)n(2), 2V(12) = 2V(12), the effective Hamiltonian is given by
00 j j 00 j j 10 01
α=1,2 j j the spin-1 antiferromagnetic Heisenberg chain
(8)
(cid:88)
H =J S ·S , (13)
j j+1
where we ignore the constant kinetic energy of the LLL. j
The ground state of the thin-torus Hamiltonian (8) is
where we set V ≡J >0. This Hamiltonian has a non-
easily obtained. For δ < 1, where the intracomponent 10
degenerategroundstateknownastheHaldanestate.40–43
interaction is dominant (2V >V(12)), the ground state
00 00 The macroscopic degeneracy of the ground state of Eq.
is the product state of Bose Mott insulators
(8) is thus lifted by the leading fluctuations.
At this stage, it is interesting to compare the phases
(cid:12)(cid:12)··· n(j1) ···(cid:69)=(cid:12)(cid:12)··· 1 1 1 1 ···(cid:69). (9) of the spin-1 chain (12)64–66 with the phase diagram of
(cid:12)··· n(2) ··· (cid:12)··· 1 1 1 1 ··· the original 2D system. The two-component bosons (3)
j
5
state and high entanglement between the two compo-
nents, it is natural to speculate that this phase evolves
into the BIQH phase in the 2D case.
Next, let us consider the thin-torus limit in the case
when some fluxes φ are inserted through the torus.
α
Since the Hamiltonian (5) depends on the fluxes only
through φ −φ , we set φ (cid:54)= 2πn (n ∈ Z) and φ = 0
1 2 1 2
without loss of generality. In this case, since the flux
causeslatticemismatchbetweenthetwocomponents,the
only remaining interaction in the limit L /(cid:96) → 0 is the
y
on-site intracomponent one V , and the intercomponent
00
interactions completely disappear. The ground state is
FIG. 2. (color online) Schematic phase diagram of two- thus the doubled Mott insulators (9) at any δ. We note
component bosons (3) at ν = 1 + 1 in the space of the
that this fact does not contradict the above identifica-
ratio of the intercomponent to intracomponent interactions,
tion of the BIQH state with the Haldane phase, since
δ=g(12)/g,andthelengthL inthey direction. Aquasi-1D
y the flux insertion breaks the inversion symmetry of the
limitL /(cid:96)(cid:28)1isdescribedbythespin-1chain(12)whilethe
y system (except at φ =π) and thus there is no clear dis-
2D phase diagram has been studied in Ref. 29. The product 1
ofBoseMottinsulators[(Mott)2]inthequasi-1Dlimitisex- tinction between the Haldane phase and the Bose Mott
pected to evolve into the the product of Moore-Read states insulator.7,69,70 In the next section, we show that the
[(MR)2] for δ (cid:46)0.2 in the 2D case. A phase separation (PS) change in the ground state during the adiabatic flux in-
occurs for δ >1 in the quasi-1D limit and for δ (cid:38)2.5 in the sertionisrelatedtotheHallresponseoftheBIQHstate,
2D case. A Haldane phase that intervenes between the re- leading to the off-diagonal topological pumping in the
gions of (Mott)2 and PS in the quasi-1D limit is expected to quasi-1D system.
evolve into the BIQH phase in the 2D system.
IV. OFF-DIAGONAL TOPOLOGICAL
in two dimensions show a couple of phases when vary- PUMPING IN THE THIN-TORUS LIMIT
ing the interaction ratio δ (Fig. 2).29 When δ is small
(numerically δ (cid:46) 0.2), the two components are nearly Inthissection,wedescribethe1Dtopologicalpumping
decoupled and form the Moore-Read states67 indepen- which mimics the BIQH effect, and thereby reinterpret
dently. In the opposite limit, when δ is large (numer- thefactthatthethin-toruscounterpartoftheBIQHstate
ically δ (cid:38) 2.5), the 2D ground state exhibits a phase is the Haldane phase. We start from the two-component
separation. The BIQH state appears around δ = 1 in- Bose system (3) on a 2D torus, and consider its Hall re-
tervening the two limiting cases.27–29 Our mapping to a sponse. AsdiscussedinSec.III,weintroduceamagnetic
spin chain (12) qualitatively reproduces these phases as flux φ (α = 1,2) through the torus, which results in
α
summarized in Fig. 2. When Ly/(cid:96) is sufficiently small, the twisted boundary condition for the α-th component
the single-ion anisotropy D is the most dominant term in the y direction. We also introduce a magnetic flux
in Eq. (12). For δ < 1, we have D > 0, and the ground θ (α = 1,2) through the other direction of the torus,
α
state is a large-D state, which is equivalent to the dou- which results in the analogous twisted boundary condi-
bled Mott insulators in Eq. (9). For a single-component tion in the x direction. Then the Hall response of the
Bosegasatthefillingfactorν =1,itwasshownthatthe α-th component to the flux insertion for the β-th com-
ground state in the thin-torus limit is given by a Bose ponent (α,β =1,2) can be expressed by the many-body
Mott insulator |··· n(α) ···(cid:105) = |··· 1 1 1 1 ···(cid:105), and Chern number51
j
then two CDW states |··· n(α) ···(cid:105) = |··· 2 0 2 0 ···(cid:105) 1 (cid:90) 2π (cid:90) 2π
j C = dθ dφ ((cid:104)∂ ψ|∂ ψ(cid:105)−(cid:104)∂ ψ|∂ ψ(cid:105)),
and |··· 0 2 0 2 ···(cid:105) become nearly degenerate with the αβ 2πi α β θα φβ φβ θα
0 0
ground state as we increase L /(cid:96); these three states nat- (14)
y
urallyevolveintothethree-folddegenerategroundstates where |ψ(θ ,φ )(cid:105) is the many-body ground state for the
α β
of the bosonic Moore-Read state on a 2D torus.68 Simi- twists θ and φ for the α- and β-th components in the
α β
larly, the doubled Mott insulators in Eq. (9) is expected x and y directions, respectively, while the other twist-
to evolve into the doubled Moore-Read states found in ing angles are set to zero. The Chern-Simons theory
the2Dsystem. Forδ >1,wehaveD <0andJ <0,and (1) of the BIQH effect corresponds to C = C = 0
z 11 22
the spin chain (12) exhibits ferromagnetic ground states and C = C = 1. These responses result in the off-
12 21
asinEq.(10); ifthetotalmagnetizationofthesystemis diagonal topological pumping in the thin-torus limit as
fixed at zero, a phase separation occurs as found in the we see below.
2D system. As we increase L /(cid:96), the Haldane phase ap- Hereafter we focus on the responses corresponding to
y
pears between the large-D and ferromagnetic phases in C and C . These can be analyzed by setting φ = 0
11 21 2
thespin-chainmodel,anditsrangealongtheδaxisgrad- and adiabatically changing the pumping parameter t ≡
ually increases. Owing to the uniqueness of the ground φ1T from 0 to T. The Hall current in the x direction is
2π
6
identified as the pumped charge. For fixed t, it is useful The BIQH effect should thus correspond to the topolog-
to introduce the Berry phase ical pumping with ∆P =0 and ∆P =1.
1 2
Let us now discuss in detail the topological pumping
(cid:90) 2π
in the thin-torus limit of the BIQH state. The pumping
γ (t)=− dθ (cid:104)ψ(θ ,t)|∂ |ψ(θ ,t)(cid:105) (mod 2π),
α α α θα α protocol in the present case is not just the translation
0
(15) but involves the change of intercomponent interactions,
which is associated with the change in the ground state insharpcontrastwiththeFQHcasesdiscussedinSec. II.
|ψ(θ ,t)(cid:105) when θ is adiabatically changed from 0 to 2π. Since the Hamiltonian (5) in the LLL basis is invariant
α α
The Chern number (14) can then be rewritten as under the combined operations of the spatial inversion
j →N −jandtheinterchangeoftwocomponents1↔2,
φ
1 (cid:90) T z (t) in Eq. (18) satisfies
C =− dt ∂ γ (t). (16) α
α,β=1 2π t α
0 z (t)=e2πit/Tz∗(t), (22)
1 2
In this expression, the quantized Hall response or equiv-
alentlythequantizedchargepumpingcanbeunderstood which indicates P1(t)+P2(t)=t/T. Thus, ∆P1 =0 im-
as 2πn (n∈Z) change in the Berry phase γα(t) over the plies∆P2 =1andviceversa. Furthermore,byexploiting
pumping process.71 the invariance of the pumping protocol under the com-
Amoreintuitiveunderstandingofthequantizedpump- binedoperationsofthespatialinversionandtimereversal
ing can be gained by introducing the polarization. To t → −t. we find Pα(−t) = −Pα(t), from which we ob-
introduce it, we define tain∆P =2(cid:82)T/2dt∂ P (t). Thus,halfoftheexpected
α 0 t α
changes ∆P = δ in the polarizations over one cycle
z (t)=(cid:104)ψ(t)|exp(cid:20)2πi(cid:90) dr xΨ(α)†(r)Ψ(α)(r)(cid:21)|ψ(t)(cid:105), must occur dαuringα,t2∈[0,T/2]:
α
L
x
1
(17)
P (T/2)−P (0)= δ . (23)
α α 2 α,2
where |ψ(t)(cid:105) = |ψ(θ =0,t)(cid:105). This is convenient in de-
α To discuss the variation of the polarizations P (t) as
scribingthecenter-of-masspositionoftheparticlesinthe α
functions of t, it is important to notice that the sys-
α-th component in the x direction since the position x
tem possesses the spatial inversion symmetry at t = 0
is defined modulo L under the periodic boundary con-
x and t = T/2. Therefore, the polarization P must
dition. Within the LLL approximation, we can exploit α
be quantized to 0 or 1/2 at these values of t. It is
the fact that j-th LLL orbital for the α-th component is
known that in the presence of the inversion symmetry, a
localized around x (j)=(j+ tδ )a in the x direction
α T α,1 change in the polarization between these quantized val-
(withlargerspacingaforsmallerL /(cid:96)),andapproximate
y ues in general signals a phase transition.75,76 The polar-
z (t) as
α ization can therefore be used as an order parameter for
zα(t)≈(cid:104)ψ(t)|exp(cid:104)N2πia(cid:88)xα(j)n(jα)(cid:105)|ψ(t)(cid:105) dsieotnecstyinmgm1eDtryt.o7p5,o7l6o,8g3i,c8a4lSpihmaisleasrprerosuteltcstecdanbyaltshoebienvoebr--
φ j tained through the quantization of the Berry phase.85–89
=(cid:104)ψ(t)|exp(cid:104)2πi(cid:88)jn(α)+ 2πitδ (cid:105)|ψ(t)(cid:105). At t=0, the thin-torus limit of the BIQH state is given
N j T α,1 bytheHaldanestateasdiscussedinSec.III;theHaldane
φ
j phaseisknowntobeatopologicalphaseprotectedbythe
(18)
inversionsymmetry,24,44,45 andhasthepolarizations75,76
ThisresemblestheexpectationvalueoftheLieb-Schultz-
P (0)=P (0)=N /2. (24)
Mattis twist operator.72–77 The phase of z (t) gives the 1 2 φ
α
polarization78,79 At t = T/2, the ground state in the thin-torus limit is
givenbythedoubledMottinsulators;adirectcalculation
1
P (t)= Imlnz (t) (mod 1). (19) using Eq. (9) yields
α 2π α
P (T/2)=N /2, P (T/2)=(N +1)/2. (25)
Importantly, the polarization is directly related to the 1 φ 2 φ
Berry phase as78,80–82
TheHaldanephaseandthedoubledMottinsulatorshave
the different polarizations (24) and (25), and thus are
1
Pα(t)=−2πγα(t). (20) distinctphasesaslongasthesystempossessestheinver-
sion symmetry. The topological pumping can be inter-
The Chern number can therefore be rewritten as the pretedasaprocessofconnectingbetweenthetwophases
change in the polarization over the pumping cycle: smoothly by breaking the inversion symmetry. The idea
of utilizing symmetry-breaking perturbations to connect
(cid:90) T between otherwise distinct gapped phases has also been
C =∆P = dt ∂ P (t). (21)
α,β=1 α t α used in other examples of topological pumping such as
0
7
00..1155
((aa)) NN == 66
gg φφ
22//ℓℓ 00..11
y) y)
gg
nerner 00..0055
EE
((
FIG.3. (coloronline)Schematicpictureofoftheoff-diagonal
00
topologicalpumpinginthethin-toruslimitoftheBIQHstate.
Theupperandlowerchainsdescribethefirstandsecondcom- 00..1155
ponents, respectively. ((bb)) NN == 77
gg φφ
22//ℓℓ 00..11
y) y)
gg
those based on the Su-Schrieffer-Heeger model,4,5,90 the nerner 00..0055
spin-Peierls phases6 (equivalent to an interacting Su- EE
((
Schrieffer-Heeger model via the Jordan-Wigner transfor-
00
mation),andtheHaldaneinsulatorphaseofanextended
Bose-Hubbardmodel.7 Wenotethatthepolarizationsin 22 33 44 LL // ℓℓ 55 66
yy
Eqs. (24) and (25) are consistent with the relations (23);
conversely, the relations (23) require the appearance of FIG. 4. Energy spectrum of the Hamiltonian (5) as a func-
topologically distinct phases (in an inversion-symmetry- tion of L /(cid:96) for (a) N = 6 and (b) N = 7. The ground-
y φ φ
protected sense) at t = 0 and T/2. This gives an expla- stateenergyissubtractedfromthespectrum. Circlesindicate
nationonwhytheHaldanephase, atypicalexampleofa eigenenergies in the equal-population case N1 = N2 = Nφ.
1D SPT phase, should emerge in the thin-torus limit of Crosses indicates eigenenergies in the minimally imbalanced
the BIQH state. case(N1,N2)=(Nφ+1,Nφ−1). Thetwolowestenergiesfor
each pseudomomentum are shown. The data for the largest
Figure 3 summarizes the process of the off-diagonal
L /(cid:96)correspondtothecaseofL =L . Dashedlinesindicate
topologicalpumpinginthethin-toruslimit. TheHaldane y x y
the energy gaps 0.721J and 0.857J of the spin-1 Heisenberg
phaseatt=0canintuitivelybedescribedaslocalization
chain (13) with N =6 and 7 spins, respectively, where J is
φ
ofbosonsatbondsofthelatticeinawaysimilartothatof givenbyV =2V(12) inEq.(7)(wenotethatthegapinthe
one-component bosons.70 By shifting the lattice for the thermodyn1a0mic lim10it91,92 is given by 0.410J).
first component, the inversion symmetry is broken and
the ground state is smoothly deformed into the doubled
Mott insulators at t = T/2, and finally returns to the
original ground state at t = T. During this process, theHamiltonian(5)withthenumberoffluxquantaupto
the bosons in the second component are pumped by one N =7. Weconsidercontactinteractionswithg(12) =g.
φ
lattice spacing while the bosons in the first component Figure 4 presents the energy spectrum as a function of
stay around the same positions. L /(cid:96). The ground state is found to remain in the sector
y
Tworemarksareinorder. First,althoughtheHaldane with zero pseudomomentum, indicating that the BIQH
phaseinthespin-1chaincanbeprotectednotonlybythe state in two dimension is smoothly deformed into the
inversion symmetry but also by the time-reversal or the Haldane state in the thin-torus limit. For L /(cid:96)(cid:46)5, the
y
spinrotationsymmetry,24,44,45 thelattertwosymmetries energy gap above the ground state agrees well with the
do not protect the Haldane phase in the case of soft-core finite-sizeenergy gapof the spin-1Heisenberg chain(13)
bosons as in the present case.7,69,70 This is because the (dashed lines) calculated by KOBEPACK.93,94 Reflect-
spin-1degreesoffreedomrequiredforthesymmetrypro- ing the large ground-state degeneracy in the thin-torus
tectionargumentarenotperfectlyformedinthepresence limit, a large number of eigenenergies collapse onto the
offluctuationsinon-siteparticlenumbers. Theinversion ground-stateenergywithdecreasingL /(cid:96). ForL /(cid:96)(cid:38)5,
y y
symmetry is thus the only symmetry that protects the the energy gap stay around a constant value, indicating
Haldane phase at t = 0 in the present case. The sec- the convergence to a 2D system. Figure 5 presents the
ond remark is on the mapping to a spin-1 chain done in polarizations P , the amplitudes |z | = |z | of the twist
α 1 2
Sec.III.Inthemappingfromthethin-torusHamiltonian operators, and the energy spectrum as functions of the
toaspinchainintheFQHcases,theinversionsymmetry pumping parameter t. The ground state is found to re-
is sometimes broken since the mapping process involves maininthezero-pseudomomentumsectorinthisprocess
groupingofseveralneighboringsitesstartingfromCDW also. The calculated polarizations smoothly connect be-
groundstates.52,53 InthecaseoftheBIQHstate, incon- tween Eqs. (24) and (25) for the Haldane state (t = 0)
trast, the mapping to a spin chain retains the inversion and the doubled Mott insulators (t=±T/2). While the
symmetryoftheoriginalsystem,sinceeachspin-1degree polarization of the first component stays around P =0,
1
of freedom is composed of bosons at the same site. the second component shows ∆P = 1 over the cycle,
2
To support the above picture of topological pumping, clearly signaling the off-diagonal topological pumping.
we haveperformed exactdiagonalizationcalculations for As we decrease L /(cid:96), long-range interactions are sup-
y
8
(a) Ly / ℓ = 2.5 1.25 regime—while |zα| converges to unity in the thermody-
0.5 namic limit in 1D gapped systems, its value can be sup-
n
zatio 0 P1 2/g 1.2 pbrleestsoedthwehloecnatlihzeatsiyosntelemngstizhe.7i5s,7s6m,7a9lTlehritshbaenhoarvicoormcapnarbae-
Polari-0.5 P2 ergy) ℓ 1.15 eflxupcltauianteidonbsywthheensudpecprreeasssiinong Lofy/o(cid:96)n:-sIitfesupcahrtiflculectnuuamtiobnesr
z|2 1 (En 1.1 arecompletelyabsent,thepumpingcycleisdescribedby
| = |1 0.5 1.05 tahnedspthine-1docuhbailnedmModoetlt(1in2s)u,liantowrshicchantnhoetHbaeldcaonnenpehctaesde
|z 0 without closing a gap.
-0.5 0 0.5 0 0.1 0.2 0.3 0.4 0.5
t / T t / T
(b) L / ℓ = 3.5
y 1.35 V. SUMMARY
0.5
n
o
zati 0 P1 2/g 1.3 In summary, we have constructed strongly interacting
Polari-0.5 P2 ergy) ℓ 1.25 mlimoditelosfo2fDtoqpuoalongtuicmal Hpuamllpsitnagtesb.y Tushiengthtihne-ttohruins-tloimruist
= |z|2 0.5 (En 1 .11.52 obfattihcealFlyQcHonstnaetcetsinigs gbivetewnebeynCdDegWenegrraotuendCDstWatesg;roaudniad-
| 1 states gives the fractional Thouless pumping. As a
|z 0 more nontrivial example, we have constructed topolog-
-0.5 0 0.5 0 0.1 0.2 0.3 0.4 0.5
ical pumping which corresponds to the BIQH effect of
t / T t / T
two-componentbosons. Thequasi-1Dcounterpartofthe
(c) L / ℓ = 5.0
y 1.35 BIQH state is identified as the Haldane phase, and adi-
n 0.5 abatically connecting between the topological Haldane
olarizatio 0 PP12 2gy) /gℓ 1.3 ptcuihdtaaestetehdaentohdffe-tdnhiaeatgutorrnievailoatflotpdhooeluobtgoliecpdaolloMpguoicmtatpliipnnugsu.mlWaptioenrghsavvcioeanestlhtuie--
P-0.5 er changeinthepolarizationsbetweeninversion-symmetry-
= |z|2 00..45 (En pinrgotbeecttwedeeqnutahnetiHzeadldvaanleueasn.dStriinvciealtphheaisdeesabyofincvoenrnsieocnt--
| 1 0.3 1.25 symmetry-breaking perturbations does not depend on
|z 0.2
-0.5 0 0.5 0 0.1 0.2 0.3 0.4 0.5 thedetailsofthesystem, theobtainedoff-diagonaltopo-
t / T t / T logical pumping should not be limited to the thin-torus
model considered in this paper. While the time-reversal
FIG.5. (coloronline)ThepolarizationsP (lefttop),theam- symmetry is broken in quantum Hall states and related
α
plitudes |z | = |z | of the twist operators (left bottom), and topological pumping, it is an intriguing direction to con-
1 2
the energy spectrum (right) as functions of the pumping pa- struct a bosonic version of time-reversal-symmetric Z
2
rameter t∈[−T/2,T/2), for Nφ =6 and (a) Ly/(cid:96)=2.5, (b) pumping95 which may correspond to 2D bosonic topo-
3.5,and(c)5.0. Circlesandcrossesindicateseigenenergiesin logical insulators.96,97
theequal-populationandminimallyimbalancedcases,respec-
tively, as in Fig. 4. Since the energy spectrum is symmetric
around t=0, it is shown only for t∈[0,T/2].
ACKNOWLEDGMENTS
pressed and the system gradually acquires a 1D char- WearegratefultoMasaakiNakamura,IppeiDanshita,
acter. Correspondingly, the change in the polarization Masahito Ueda, and Norio Kawakami for helpful discus-
becomesshaperinthequasi-1Dlimit. Arapidchangein sions. M. N. acknowledges a short-stay research pro-
the center-of-mass position (and thus the polarization) gramsupportedbyaGrant-in-AidforScientificResearch
hasalsobeenobservedin1Dtopologicalpumping.8,9,12,14 on Innovative Areas “Topological Materials Science”, in
When the rapid crossover from the Haldane state to the which this work was initiated. M. N. was supported by
doubled Mott insulators occurs, the energy gap becomes JSPSKAKENHIGrantNo.JP14J01328andaJSPSRe-
small and the amplitudes |z | = |z | of the twist op- search Fellowship for Young Scientists. S. F. was sup-
1 2
erators decrease. This indicates the increase in the lo- ported by JSPS KAKENHI Grant No. JP25800225 and
calization length of the many-body wavefunction in this Matsuo Foundation.
9
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