Fractonic Luttinger Liquids and Supersolids in a Constrained Bose-Hubbard Model

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Philip Zechmann,1, 2 Ehud Altman,3Michael Knap,1, 2 and Johannes Feldmeier1, 2, 4

1Technical University of Munich, TUM School of Natural Sciences, Physics Department, 85748 Garching, Germany

2Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 M¨unchen, Germany

3Department of Physics, University of California, Berkeley, CA 94720, USA

4Department of Physics, Harvard University, Cambridge, MA 02138, USA

(Dated: July 6, 2023)

Quantum many-body systems with fracton constraints are widely conjectured to exhibit uncon-

ventional low-energy phases of matter. In this paper, we demonstrate the existence of a variety

of such exotic quantum phases in the ground states of a dipole-moment conserving Bose-Hubbard

model in one dimension. For integer boson ﬁllings, we perform a mapping of the system to a model

of microscopic local dipoles, which are composites of fractons. We apply a combination of low-

energy ﬁeld theory and large-scale tensor network simulations to demonstrate the emergence of a

dipole Luttinger liquid phase. At non-integer ﬁllings our numerical approach shows an intriguing

compressible state described by a quantum Lifshitz model in which charge density-wave order co-

exists with dipole long-range order and superﬂuidity – a “dipole supersolid”. While this supersolid

state may eventually be unstable against lattice eﬀects in the thermodynamic limit, its numerical

robustness is remarkable. We discuss potential experimental implications of our results.

I. INTRODUCTION

The current advent of quantum simulation technology

is marked by rapid progress in controlling strongly in-

teracting many-body systems. In particular, the abil-

ity to engineer highly speciﬁc quantum Hamiltonians has

raised immense interest in the physics of quantum sys-

tems subjected to dynamical constraints. A particularly

exciting class of systems that has caught much atten-

tion in this regard are so-called fracton models [1–11].

These are characterized by elementary excitations with

restricted mobility (the fractons), whereas non-trivial dy-

namics can be carried by multi-fracton composites. Re-

cently, fractonic systems conserving both a global U(1)

charge as well as its associated dipole moment have suc-

cessfully been implemented in cold atomic quantum sim-

ulation platforms via the application of strong linear po-

tentials [12–15]. In this context, much eﬀort – both in

theory and experiment – has been devoted to uncovering

the highly exotic nonequilibrium properties of fractonic

systems with dipole conservation. These range from dy-

namical localization [13,14,16–20] over novel hydrody-

namic universality classes [12,21–31] and glassy dynam-

ics [3,32] to unconventionally slow spreading of quantum

information [33,34].

Less attention has been devoted to understand the

ground states of fractonic systems. Nonetheless, a gap-

less Luttinger liquid has been identiﬁed as ground state

in certain strongly fragmented dipole-conserving spin

chains [18]. Furthermore, a recent duality mapping be-

tween fracton gauge theories and elasticity theory [35–

40] suggests the possible existence of new phases with

highly unconventional properties, such as dipole super-

ﬂuids or fracton condensates [35,41–45]. Similar phases

have recently also been predicted in a mean-ﬁeld study of

a Bose-Hubbard lattice model subject to dipole conser-

vation [46]. However, in one spatial dimension, where

generically quantum ﬂuctuations are expected to be

strong, an understanding of the phases and phase tran-

sitions has been lacking so far.

In this paper, we address this challenge by studying

the Bose-Hubbard model with dipole conservation in one

spatial dimension. The one-dimensional character of the

system enables us to employ an established toolbox of

eﬃcient theoretical techniques. On the one hand, we re-

solve the question of a consistently-deﬁned local dipole

density, which subsequently allows us to use bosoniza-

tion [47] for constructing eﬀective low-energy ﬁeld the-

ories of the fracton model. On the other hand, we ap-

ply tensor network techniques as eﬃcient numerical tools

for the computation of ground-state properties of one-

dimensional systems [48,49].

The microscopic model we focus on throughout this

paper consists of interacting lattice bosons on a chain

subject to the conservation of both charge (i.e., the boson

particle number) and dipole moment (i.e., the boson cen-

ter of mass). In such a constrained Bose-Hubbard model

the single particle hopping term is absent and is instead

replaced by symmetric correlated hopping processes of

two bosons. Our microscopic model is described by the

Hamiltonian

H=−tX

(ˆ

b†

jˆ

bj+1ˆ

b†

j+2 + H.c.)

ˆnj(ˆnj−1) −µX

ˆnj.

(1)

Here, tdenotes the dipole hopping amplitude, Uthe

strength of on-site interactions, µthe chemical potential,

and ˆnj=b†

jbjthe local boson number operator. Both

the total charge ˆ

Q(or particle number ˆ

N) and its asso-

ciated dipole moment ˆ

Pare conserved quantities, which

arXiv:2210.11072v2 [cond-mat.quant-gas] 4 Jul 2023

we deﬁne as

j=1

ˆqj=X

(ˆnj−n) = 0

j=1

(L−j) ˆqj=X

(L−j) (ˆnj−n) = const.,

(2)

where ˆqjdenotes the local deviation from the average bo-

son density n=⟨ˆn⟩. Selecting the reference position of

the dipole moment as in Eq. (2) will turn out convenient

in the following. We introduce the notation of a dipole

operator d†

j=b†

jbj+1, such that the kinetic term d†

jdj+1

may be viewed as regular nearest-neighbor hopping for

a particle-hole dipole-like degree of freedom. We empha-

size, however, that the ˆ

d(†)

jdo not satisfy the commuta-

tion relations of creation/annihilation operators. Accord-

ingly, ˆ

d†

jˆ

djis in general not the local dipole density. How-

ever, under certain circumstances it can be, such as in

the low-energy subspace considered in Ref. [50]. Longer

range correlated kinetic terms may in principle be in-

cluded and should not qualitatively aﬀect the low-energy

physics. In our numerical computations we restrict our-

selves to the simplest case of Eq. (1).

Our analysis of the zero-temperature phases of Eq. (1)

yields several key results, which we present as follows. In

Sec. II, we ﬁrst establish the presence of area-law cumu-

lative charge ﬂuctuations as a general criterion for the ex-

istence of a consistently deﬁned local dipole density; see

Fig. 1(a) for an illustration. Using an explicit mapping

to microscopic dipole degrees of freedom, we determine

the ground-state phases of the model Eq. (1) at integer

boson ﬁlling as a function of correlated hopping strength

t/U in Sec. III. We predict that the system undergoes

a BKT (Berezinskii-Kosterlitz-Thouless) transition be-

tween a dipole Mott insulator (d-Mott) and a dipole Lut-

tinger liquid (d-Luttinger). In the dipole Mott insula-

tor both charges and dipoles are gapped, whereas in the

dipole Luttinger liquid dipoles are gapless but charge ex-

citations retain a ﬁnite energy gap. The dipole Luttinger

liquid persists when increasing t/U up until an instabil-

ity towards boson bunching occurs. We conﬁrm these

analytical predictions numerically using large-scale den-

sity matrix renormalization group (DMRG) calculations.

As a next step, we consider the model away from integer

ﬁlling in Sec. IV. Our numerical analysis in this regime

is consistent with an exotic ground state with vanish-

ing charge gap and thus ﬁnite compressibility, described

by a quantum Lifshitz model (see e.g. [51]). This state

spontaneously breaks the continuous dipole symmetry,

which, as has recently been shown, is allowed in princi-

ple even in one dimension, due to a modiﬁed Mermin-

Wagner theorem in systems with multipole conservation

laws [52,53]. In Ref. [46], the quantum Lifshitz model

was proposed as low-energy eﬀective theory for the con-

strained Bose-Hubbard model in a phase termed “Bose

Einstein insulator”. In our one-dimensional scenario, we

demonstrate that this state is characterized by a coexis-

FIG. 1. Fractonic phases of matter in one dimen-

sion. (a) At low energies, area law ﬂuctuations of the charge

q(x) permit the deﬁnition of a local dipole density qd(x)

as ∂xqd(x) = q(x). This allows us to apply bosonization

to construct a low-energy eﬀective ﬁeld theory for micro-

scopic dipoles, which are composites of fractons. (b) The

grand-canonical phase diagram of the dipole-conserving Bose-

Hubbard model features three distinct phases: an incompress-

ible dipole Mott insulator (d-Mott), shown in blue, within

lobes of integer ﬁlling; an incompressible dipole condensate

in form of a Luttinger liquid of dipoles (d-Luttinger), located

in the red region at the tips of lobes which extends to the

bunching instability (grey region); and a compressible su-

persolid of dipoles (d-Supersolid) at non-integer ﬁlling in the

green region. Solid black lines correspond to estimated phase

boundaries from grand-canonical iDMRG computations. The

dashed black lines indicate the energies for adding or remov-

ing a single particle (see text below). The regions between

the Mott lobes at small dipole hopping t/U (hatched region)

additionally host a Mott insulating phase at non-integer ﬁll-

ing, which for instance at n= 3/2 is stable up to t/U ≈0.14.

tence of density-wave order and dipole superﬂuidity. We

thus refer to this situation as a “dipole supersolid” (d-

Supersolid). Generic theoretical arguments suggest that

the dipole supersolid will eventually become unstable in

the thermodynamic limit due to lattice eﬀects. Nonethe-

less, the full consistency of our results with a dipole su-

persolid phase within all numerically accessible system

sizes demonstrates that the phenomenology of the dipole

supersolid is remarkably robust. Our results can be sum-

marized in the phase diagram of Fig. 1(b). We conclude

in Sec. Vwith a discussion of the implications of our

results for potential future experimental and theoretical

investigations.

II. CONSTRUCTING A LOCAL DIPOLE

DENSITY

The ground-state phases studied in this paper require

the existence of a bounded local density of microscopic

dipoles. This property will be instrumental for us in

devising an appropriate low-energy description for the

model Eq. (1). Such a local dipole density can be seen as

an emergent property whose deﬁnition is consistent only

at low energies and does not extend to high energy states

of such dipole-conserving systems. In the following, we

express the conserved global dipole moment in terms of a

local density that will remain bounded if charge ﬂuctua-

tions can be shown to be bounded. The most natural way

to satisfy this criterion is the presence of a ﬁnite charge

gap, corresponding to an incompressible state. In such a

scenario, the low-energy theory of the system is naturally

given in terms of eﬀective dipole degrees of freedom as

described in [52]. Here, we show how this applies even to

a microscopic description of the system.

A. In the continuum

Let us ﬁrst consider the scenario of a continuum charge

density q(x) in a closed system of length L. We require

both the total charge and the associated dipole moment

to be conserved,

Q=ZL

dx q(x) = 0,

P=ZL

dx (L−x)q(x) = const.

(3)

Here, q(x) = n(x)−ndenotes again the deviation of

the local particle density n(x) from the average den-

sity n. Our goal is to express the dipole moment as

P=RL

0dx qd(x) in terms of a local and bounded dipole

charge density qd(x). We emphasize that the naive choice

qd(x) = xq(x) suggested by Eq. (3) is not suitable since

x q(x) is manifestly unbounded. Instead, we can use

Cauchy’s formula for repeated integration to rewrite the

dipole moment as

P=ZL

dx (L−x)q(x) = ZL

dx Zx

dx′q(x′),(4)

Based on Eq. (4) we deﬁne the local dipole charge density

qd(x) = Zx

dx′q(x′),(5)

or alternatively, in diﬀerential form,

∂xqd(x) = q(x).(6)

The ﬁeld qd(x) is thus related to a “height ﬁeld” repre-

sentation of the dipole constraint [54]. We now see that

while xq(x) is unbounded, qd(x) deﬁned in Eq. (5) re-

mains bounded if the charge ﬂuctuations within a region

of size xremain of order O(1) as x→ ∞. As the ﬂuc-

tuations do not scale with the “volume” xof the region

but originate solely from its boundaries, we will refer to

these ﬂuctuations as “area-law” in the following. Such

area-law-type charge ﬂuctuations are guaranteed for the

ground state in the presence of a ﬁnite charge gap, which

induces a ﬁnite correlation length for charged degrees of

freedom. We therefore obtain a consistently deﬁned local

dipole density upon which we can construct an eﬀective

model of the low-energy behavior.

B. On the lattice

The description of the system in terms of a ﬁnite den-

sity of microscopic dipole charges introduced in Eq. (6)

can also be realized on a lattice. For this purpose, we

substitute the continuum derivative with a discrete lat-

tice derivative, ∆xqd:= qd,x+1/2−qd,x−1/2. We have thus

deﬁned the local dipole charge as a local bond degree of

freedom.

For simplicity, we focus on integer ﬁlling n∈N, where

any occupation number basis state |n⟩=|n1, ..., nL⟩

gives rise to a charge density |q⟩=|n1−n, ..., nL−n⟩

in terms of the local deviation from average ﬁlling. The

corresponding local dipole charge density state |qd⟩=

|qd,3/2, ..., qd,L−1/2⟩can thus be obtained by sweeping

through the system from left to right and applying the

relation

qd,x+1/2=qd,x−1/2+qx,(7)

where we for now set qd,1/2= 0. The so-deﬁned local

dipole charge can assume both positive and negative val-

ues. Much like the conventional charge density, we would

like to rewrite the local dipole charge in terms of a non-

negative local occupation number nd,x+1/2of microscopic

dipoles. This can be achieved simply by adding a suitable

integer constant m∈Nto the local dipole charge

nd,x+1/2=qd,x+1/2+m=nd,x−1/2+qx,(8)

where now nd,1/2=m. Note that the addition of such a

constant leaves the diﬀerential relation Eq. (6) invariant.

The constant mcan be chosen arbitrarily, and we obtain

non-negative local dipole occupation numbers nd,x+1/2≥

0 for all xwhen

m≥mmin =−min0,min

xqd,x+1/2.(9)

An illustration of the mapping between nxand nd,x is

provided in Fig. 2. We emphasize that in the presence

−1

nd,x+1/2

qd,x+1/2

−1

FIG. 2. Microscopic dipole density. Mapping between

product states in the boson occupation number basis (up-

per panel) and microscopic dipole occupation numbers on the

bonds of the lattice (lower panel). Here, nxand nd,x+1/2are

non-negative, whereas qxand qd,x+1/2are deﬁned with re-

spect to the average densities (dashed line). For a state at

integer boson ﬁlling within the same dipole moment sector

as the uniform state |n⟩, the resulting dipole model exhibits

integer ﬁlling as well. See main text for a detailed description

of the mapping.

of a ﬁnite charge gap the local dipole charge is always of

order O(1), and thus the required mmin in Eq. (9) remains

bounded as well.

The mapping between boson occupation numbers and

bounded dipole occupation numbers can in principle also

be performed for states at non-integer boson ﬁllings, pro-

vided the charge ﬂuctuations are bounded. In such a

case, however, the dipole density Eq. (5) is deﬁned with

respect to a nontranslationally invariant reference state

n0(x), such that q(x) = n(x)−n0(x). The resulting

model for microscopic dipoles is then not translation-

ally invariant. It is an interesting open question how

an analysis of such a model can prove useful. Formally,

the mapping could even be performed for arbitrary states

|n⟩in the Hilbert space. However, for most states this

will lead to an unbounded local dipole density that di-

verges with system size. The presence of a ﬁnite charge

gap then ensures that only such occupation number ba-

sis states that yield a bounded local dipole density con-

tribute signiﬁcantly to the ground-state wave function.

The contribution of states requiring high local dipole

density decays exponentially with mmin and can thus be

safely discarded. Furthermore, while the presence of a

ﬁnite charge gap is a suﬃcient condition to ensure area-

law charge ﬂuctuations, it is not a necessary one. We will

encounter such a situation in Sec. IV in which the charge

gap vanishes but cumulative charge ﬂuctuations obey an

area law. We further emphasize that the resulting de-

scription in terms of microscopic dipole bond degrees of

freedom remains valid for dipole-conserving systems with

longer-range terms than in the present microscopic model

Eq. (1).

III. INTEGER FILLING: LOW-ENERGY

DIPOLE THEORY

We start our analysis of the constrained Bose-Hubbard

model of Eq. (1) by considering the system at a ﬁxed in-

teger ﬁlling n∈Nas a function of the relative strength

t/U of the correlated hopping. For t/U being suﬃciently

small, we expect a Mott insulating state with gapped

charge (i.e., single particle) excitations. We then per-

form the mapping to a system of microscopic dipoles and

construct a low-energy eﬀective theory by bosonization

of these lattice dipoles.

A. Eﬀective action of dipoles

In order to determine the proper low-energy model

in the dipole language, we extract the resulting aver-

age dipole density ndthat results at integer boson ﬁlling

n∈N. In particular, in the following we ﬁx the sector

of the total dipole moment P= 0 that is associated with

the homogeneous boson state n=|n, ..., n⟩. For this

state, the local deviation from the average boson ﬁlling

is qx= 0 for all x, and therefore the local deviation from

the average dipole ﬁlling is qd,x+1/2= 0 for all xas well

according to Eq. (7). As a result of Eq. (8), the average

dipole density is thus given by

nd=m∈N,(10)

i.e., microscopic dipoles are at integer ﬁlling as well. This

feature will become relevant upon constructing an appro-

priate low-energy theory. We emphasize that the states

in the sector connected to the homogeneous root state

|n⟩=|n, ..., n⟩are obtained by simple hopping processes

of the microscopic dipoles, and thus feature the same in-

teger dipole ﬁlling.

The presence of a charge gap allows us to rewrite the

constrained Bose-Hubbard model at integer boson ﬁlling

in terms of microscopic bond dipoles at integer ﬁlling

nd∈N. The Hamiltonian may then be expressed in this

basis, leading to a hopping of bond dipoles as well as

dipole density interactions. In order to understand the

low-energy properties of this system we may then proceed

by standard bosonization [47] of the newly found dipole

objects. In particular, we introduce a counting ﬁeld ϕd

for the bond dipoles, in terms of which the local dipole

density reads

nd(x) = hnd−1

π∇ϕd(x)iX

e2ip(πndx−ϕd(x)).(11)

We further introduce a conjugate dipole phase ﬁeld θd,

which satisﬁes the relation

1

π∇ϕd(x), θd(x′)=−iδ(x−x′).(12)

The low-energy eﬀective Hamiltonian for the system is

generically given by the kinetic energy (∇θd)2as well as

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摘要：

FractonicLuttingerLiquidsandSupersolidsinaConstrainedBose-HubbardModelPhilipZechmann,1,2EhudAltman,3MichaelKnap,1,2andJohannesFeldmeier1,2,41TechnicalUniversityofMunich,TUMSchoolofNaturalSciences,PhysicsDepartment,85748Garching,Germany2MunichCenterforQuantumScienceandTechnology(MCQST),Schellingstr.4...

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