Dipole condensates in tilted Bose-Hubbard chains Ethan Lake1Hyun-Yong Lee2 3 4Jung Hoon Han5and T. Senthil1 1Department of Physics Massachusetts Institute of Technology Cambridge MA 02139

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Dipole condensates in tilted Bose-Hubbard chains

Ethan Lake,1Hyun-Yong Lee,2, 3, 4 Jung Hoon Han,5and T. Senthil1

1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, 02139

2Division of Display and Semiconductor Physics, Korea University, Sejong 30019, Korea

3Department of Applied Physics, Graduate School, Korea University, Sejong 30019, Korea

4Interdisciplinary Program in E·ICT-Culture-Sports Convergence, Korea University, Sejong 30019, Korea

5Department of Physics, Sungkyunkwan University, Suwon 16419, Korea

We study the quantum phase diagram of a Bose-Hubbard chain whose dynamics conserves both

boson number and boson dipole moment, a situation which can arise in strongly tilted optical

lattices. The conservation of dipole moment has a dramatic eﬀect on the phase diagram, which

we analyze by combining a ﬁeld theory analysis with DMRG simulations. In the thermodynamic

limit, the phase diagram is dominated by various types of incompressible dipolar condensates. In

ﬁnite-sized systems however, it may be possible to stabilize a ‘Bose-Einstein insulator’: an exotic

compressible phase which is insulating, despite the absence of a charge gap. We suggest several

ways by which these exotic phases can be identiﬁed in near-term cold atom experiments.

I. INTRODUCTION AND SUMMARY

Many of the most fascinating phenomena in quantum

condensed matter physics arise from the competition be-

tween kinetic energy and interactions, and it is there-

fore interesting to examine situations in which the roles

played by either kinetic energy or interactions can be

altered. One way of doing this is by ﬁnding a way to

quench the system’s kinetic energy. This can be done

with strong magnetic ﬁelds—which allows one to explore

the rich landscape of quantum Hall phenomenology—or

by engineering the system to have anomalously ﬂat en-

ergy bands, as has been brought to the forefront of con-

densed matter physics with the emergence of Moire ma-

terials [1].

A comparatively less well understood way to quench

kinetic energy occurs when exotic conservation laws in-

hibit particle motion. One large class of models in which

this mechanism is operative are systems whose dynam-

ics conserves the dipole moment (i.e. center of mass) of

the system’s constituent particles, in addition to total

particle number [2,3]. This conservation law can be

easily engineered as an emergent symmetry in strongly-

tilted optical lattices, where energy conservation facil-

itates dipole-conserving dynamics over arbitrarily long

pre-thermal timescales [4–6] (other physical realizations

are discussed below).

Dipole conservation prevents individual particles from

moving independently on their own [Fig. 1(a)]. In-

stead, motion is possible in only one of two ways: ﬁrst,

two nearby particles can ‘push’ oﬀ of each other, and

move in opposite directions. This type of motion allows

particles to hop over short distances, since this process

freezes out as the particles get far apart. Second, a par-

ticle and a hole (where ‘hole’ is deﬁned with respect to a

background density of particles) can team up to form a

dipolar bound state, which — by virtue of the fact that

it is charge neutral — may actually move freely, without

constraints. Dipole conservation thus forces the system’s

‘kinetic energy’ to be intrinsically nonlinear, as the way

constrained hopping tilted optical lattice

✓

(a)(b)

FIG. 1: (a) The restricted kinematics of dipole

conservation. An isolated boson cannot move (top),

while two nearby bosons can move only by coordinated

hopping in opposite directions (middle). A boson and a

hole (blue circle) can move freely in both directions

(bottom). (b) Approximate dipole conservation can be

engineered in tilted optical lattices with large tilt

strength V. Energy conservation then forbids single

bosons from hopping (top), while dipole-conserving

hopping processes are allowed (bottom).

in which any given particle moves is always conditioned

on the charge distribution in its immediate vicinity. This

leads to a blurring of the lines between kinetic energy

and interactions, producing a wide range of interesting

physical phenomena.

The eﬀects of dipole conservation on quantum dynam-

ics have been explored quite intensely in recent years,

with the attendant kinetic constraints often leading to

Hilbert space fragmentation, slow thermalization, and

anomalous diﬀusion (e.g. [8–21]). On the other hand,

there has been comparatively little focus on understand-

ing the quantum ground states of dipole conserving mod-

els [22–25]. Addressing this problem requires developing

intuition for how interactions compete with an intrinsi-

cally nonlinear form of kinetic energy, and understanding

the types of states favored when the dipolar kinetic en-

ergy dominates the physics.

arXiv:2210.02470v2 [cond-mat.quant-gas] 20 Oct 2022

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FIG. 2: (a) thermodynamic phase diagram of the 1d DBHM as a function of boson ﬁlling ρand hopping strength

t/U (with t4=t3≡t). The blue region denotes a dipole condensate (DC) which for non-integer ρhas CDW order.

In ﬁnite-sized systems the DC may give way to a Bose-Einstein insulator, although where exactly this is most likely

to occur is non-universal. Red lines denote a diﬀerent type of DC marked as ‘bDC’ (here ‘b’ stands for

bond-centered CDW; see text for details), whose existence relies on having a nonzero t4. Black lines at integer ρ

denote Mott insulators, and in the shaded pink region phase separation between the MI and bDC phases occurs.

The green region at largest t/U denotes the fractured Bose droplet (FBD) phase. The small white circles signify the

location of the phase boundary as obtained in DMRG. (b) Plots of the real-space average density hρiifor the four

points marked on the phase diagram. (c) Entanglement entropy in the DC phase (top) and the bDC (bottom). The

red lines are c= 1 ﬁts to the Calabrese-Cardy formula for the entanglement entropy Si= (c/6) log[(2L/π) sin(πi/L)]

for a ﬁnite chain of length L[7].

One concrete step towards addressing these questions

was given in Ref. [22], which put forward the dipolar

Bose-Hubbard model (DBHM) as a representative model

that succinctly captures the eﬀects of dipole conserva-

tion. The DBHM is simply a dipole-conserving version

of the well-loved Bose Hubbard model, and displays a va-

riety of interesting phases with rather perplexing proper-

ties, all driven by the physics of the dipolar bound states

mentioned above. Of particular interest is the Bose-

Einstein insulator (BEI) phase identiﬁed in Ref. [22],

which is realized in the regime where the nonlinear kinetic

energy dominates. The BEI is compressible, and contains

a Bose-Einstein condensate, but remarkably is neverthe-

less insulating, and has vanishing superﬂuid weight.

The aim of the present work is to perform a detailed

investigation of the DBHM in one dimension, with the

aim of fully understanding the phase diagram and mak-

ing concrete predictions for near-term cold atom experi-

ments. We are able to understand the entire phase dia-

gram within a concise ﬁeld theory framework, whose pre-

dictions we conﬁrm with extensive DMRG simulations.

We will see that the physics of the DBHM in 1d is slightly

diﬀerent from the 2d and 3d versions, in that the BEI

phase is absent in the thermodynamic limit, being ren-

dered unstable by a particular type of relevant perturba-

tion. However, if the bare strength of these destabilizing

perturbations is very small it may be possible to stabilize

a BEI regime up to (potentially very large) length scales.

In the following we will see various pieces of numerical

evidence that this indeed can occur at fractional ﬁllings.

The concrete model we will study in this paper is a

modiﬁed version of the standard Bose-Hubbard model,

with the hopping terms modiﬁed to take into account

dipole conservation:

HDBHM =−X

it3b†

i−1b2

ib†

i+1 +t4b†

i−1bibi+1b†

i+2 +h.c.

(1)

where ni=b†

ibiis the boson number operator on site

i, and t3,4≥0 determine the strength of the dipole-

conserving hopping processes (see Sec. II for an expla-

nation of how HDBHM arises in the tilted optical lattice

context). In this paper we will always work in the canon-

ical ensemble, with the boson density ﬁxed at

ρ=n

m,gcd(n, m)=1,(2)

where we are working in units where the lattice spacing

is equal to unity. Our goal is to study the behavior of

the ground states of HDBHM as a function of t3,4/U and

ρ.1

Let us now give a brief overview of our results. The

phase diagram we obtain is shown in Fig. 2(a), and

contains a lot of information. At the present juncture

we will only mention its most salient features, leaving a

more detailed treatment to the following sections.

Our ﬁrst result is that—at least in the thermodynamic

limit—there is a nonzero charge gap to exciting single

bosons throughout the phase diagram. Regardless of t/U

and ρ, the boson correlation functions always decay as

hb†

ibji ∼ e−|i−j|/ξ, with the ground state being incom-

pressible across the entire phase diagram. This is rather

remarkable, as the system thus remains incompressible

over a continuous range of ﬁlling fractions, and moreover

does so without disorder, and with only short-ranged in-

teractions. It manages to do this by having vortices in the

boson phase condense at all points of the phase diagram,

a situation which is made possible by the way in which

dipole conservation modiﬁes vortex energetics. This is of

course in marked contrast to the regular Bose-Hubbard

model, which has a compressible superﬂuid phase which

is broken by incompressible Mott insulators only at inte-

ger ﬁlling fractions and weak hopping strengths [26].

While the statements in the above paragraph are cor-

rect in the thermodynamic limit, the ubiquitous vortex

condensation just discussed may be suppressed in ﬁnite

systems. This will happen if the operators which cre-

ate vortices can take a long time to be generated under

RG, becoming appreciable only at extremely large length

scales. In our model this is particularly relevant at frac-

tional ﬁllings, where a BEI seems to be realized in our

DMRG numerics. In Sec. Vwe study this phenomenon

from a diﬀerent point of view within the context of a

dipolar rotor model.

The broad-strokes picture of our phase diagram is dic-

tated by the physics of neutral ‘excitonic’ dipolar bound

states of particles and holes annihilated by the dipole

operators

di≡b†

ibi+1.(3)

As discussed above, the motion of these dipolar particle-

hole bound states is not constrained by dipole conserva-

tion. This can be seen mathematically by noting that

the hopping terms in HDBHM can be written using the

dioperators as

Hhop =−X

(t3d†

idi+1 +t4d†

idi+2 +h.c.),(4)

and thus constitute conventional hopping terms for the

dipolar bound states. Since the dipolar bound states are

allowed to move freely, it is natural to expect that the

1See also Ref. [11] for a discussion of the ground state physics of

a dipolar spin-1 model, which in some aspects behaves similarly

to our model at half-odd-integer ﬁlling and small t3,4/U.

most eﬃcient way for the system to lower its energy is

for them to Bose condense. Indeed, this expectation is

born out in both our ﬁeld theory analysis and in our

numerical simulations. The result of this condensation is

an exotic gapless phase we refer to as a dipole condensate

(DC). Unlike the superﬂuid that occurs in the standard

Bose-Hubbard model, the dipole condensate realized here

is formed by charge neutral objects, and in fact actually

has vanishing DC conductivity.

A second interesting feature of the DBHM is an insta-

bility to an exotic glassy phase we dub the fractured Bose

droplet (FBD) phase. This instability occurs in the green

region drawn in the phase diagram of Fig. 2(a), and is in

fact common to all dipole-conserving boson models of the

form (1). It arises simply due to the √nfactors appearing

in b|ni=√n|n−1i, b†|n−1i=√n|ni. These factors

mean that when acting on a state of average density ρ,

the dipolar hopping terms in the Hamiltonian scale as

−2(t3+t4)ρ2at large ρ, precisely in the same way as the

Hubbard repulsion term, which goes as + 1

2Uρ2. Thus

when

t3+t4≥U

4,(5)

it is always energetically favorable to locally make ρas

large as possible (in our phase diagram, DMRG ﬁnds an

instability exactly when this condition is satisﬁed). Once

this occurs, the lowest-energy state of the system will

be one in which all of the bosons agglomerate into one

macroscopic droplet [Fig. 2(b), panel 4].

The physics of the FBD phase is actually much more

interesting than the above discussion might suggest:

rather than simply forming a giant droplet containing an

extensively large number of bosons, dipole conservation

means that the system instead fractures into an interest-

ing type of metastable glassy state (the physics of which

may underlie the observation of spontaneous formation

in the dipole-conserving 2d system of Ref. [27]). Un-

derstanding the physics of this phase requires a set of

theoretical tools better adapted to addressing dynamical

questions, and will be addressed in a separate upcoming

work [28] (see also the fractonic microemulsions of Ref.

[23]).

The remainder of this paper is structured as follows. In

the next section (Sec. II), we brieﬂy discuss various possi-

ble routes to realizing the DBHM in experiment, focusing

in particular on the setup of tilted optical lattices. In Sec.

III we develop a general ﬁeld theory approach that we use

to understand the phase diagram in broad strokes. This

approach in particular allows us to understand the role

that vortices in the boson phase play in determining the

nature of the phase diagram. In Sec. IV we discuss a few

characteristic features possessed by the dipole conden-

sate, as well as how to detect its existence in experiment.

In Sec. V, we give a more detailed discussion of how the

exotic physics of the Bose-Einstein insulator may appear

in small systems due to ﬁnite-size eﬀects. The following

sections VI,VII, and VIII discuss in detail the physics

at integer, half-odd-integer, and generic ﬁlling fractions,

respectively. We conclude with a summary and outlook

in Sec. IX.

II. EXPERIMENTAL REALIZATIONS

Before discussing the physics of the DBHM Hamilto-

nian (1) in detail, we ﬁrst brieﬂy discuss pathways for

realizing dipole conserving dynamics in experiment.

The simplest and best-explored way of engineering a

dipole conserving model is to realize HDBHM as an ef-

fective model that describes the prethermal dynamics of

bosons in a strongly tilted optical lattice [4,5,8,9]. In

this context, the microscopic Hamiltonian one starts with

Htilted =−tsp X

(b†

ibi+1 +b†

i+1bi) + X

V ini+HU,

(6)

where HUdenotes the Hubbard repulsion term, and V

is the strength of the tilt potential (which in practice is

created with a magnetic ﬁeld gradient). In the strong

tilt limit where tsp/V, U/V 1,2energy conservation

prevents bosons from hopping freely, but does not forbid

coordinated hopping processes that leave the total boson

dipole moment invariant [Fig. 1(b)]. Perturbation the-

ory to third order [5,12] then produces the dipolar model

(1) with t3=t2

spU/V 2, t4= 0, and an additional nearest-

neighbor interaction (2t2/V 2)Pinini+1—see App. Bfor

the details. A nonzero t4will eventually be generated at

sixth order in perturbation theory (or at third order, if

one adds an additional nearest-neighbor Hubbard repul-

sion), but in the optical lattice context we generically

expect t4/t31. We note however that the DMRG

simulations that we discuss below are performed with a

nonzero t4(in fact for simplicity, we simply set t4=t3).

This is done both because one can imagine other physical

contexts in which an appreciable t4coupling is present,

and because the t4term moderately helps DMRG conver-

gence. In any case, the qualitative features of the t4= 0

and t3=t4models are largely the same, with the only

diﬀerences arising near certain phase transitions, and at

certain ﬁlling fractions (as will be discussed in Sec. VII).

An interesting aspect of the eﬀective dipolar Hamilto-

nian that arises in this setup is that t3/U always scales

as (tsp/V )21 [33]. One may worry that this could

lead to problems when trying to explore the full phase

diagram, since we will be unable to access regimes in

which t3/U &1. This however does not appear to be

a deal-breaker, since all of the action in the DBHM will

turn out to occur at t3/U .0.1 (and at large ﬁllings, the

dipole condensate in particular turns out to be realizable

at arbitrarily small t3/U).

2See e.g. [29–32] for discussions of tilted Bose-Hubbard models in

regimes with weaker V.

Another possible realization of the 1d DBHM is in

bosonic quantum processors based on superconducting

resonators [34–36], where the dipolar hopping terms can

be engineered directly, and there are no fundamental con-

straints on t3,4/U. In this setup there is no way to for-

bid single particle hopping terms on symmetry grounds

alone, and generically the Hamiltonian will contain a

term of the form

Hsp =−t0X

(b†

i+1bi+b†

ibi+1).(7)

The presence of such dipole-violating terms is actually

not a deal-breaker, as long as t0is suﬃciently small com-

pared to t3,4. Indeed, the fact that single bosons are

gapped throughout the entire phase diagram means that

a suﬃciently small Hsp will always be unimportant, a

conclusion that we verify in DMRG.

III. MASTER FIELD THEORY

In the remainder of the main text, we will ﬁx

t3=t4≡t(8)

for concreteness, which matches the choice made in the

numerics discussed below. Those places where setting

t4= 0 qualitatively changes the physics will be men-

tioned explicitly.

In this section we discuss a continuum ﬁeld theory ap-

proach that we will use in later sections as a guide to

understand the phase diagram. Our ﬁeld theory involves

two ﬁelds θand φ, which capture the long-wavelength

ﬂuctuations of the density and phase, respectively. In

terms of these ﬁelds, the boson operator is

b=√ρeiφ, ρ =ρ+1

2π∂2

xθ, (9)

Note that density ﬂuctuations are expressed as the double

derivative of θ(in the standard treatment [37] there is

only a single derivative). This gives the commutation

relations

[φ(x), ∂2

yθ(y)] = 2πiδ(x−y).(10)

The reason for writing the ﬂuctuations in the density in

this way will become clear shortly.

Before discussing how to construct our ﬁeld theory,

let us discuss how φ, θ transform under the relevant

symmetries at play. Dipole symmetry leaves θalone,

but acts as a coordinate-dependent shift of φ, mapping

U(1)D:φ(x)7→ φ(x) + λx for constant λ. Thus ei∂xφis

an order parameter for the dipole symmetry, since

U(1)D:ei∂xφ7→ eiλei∂xφ.(11)

The operators ei∂xθ,eiθ create vortices3in the phase

3Since we are in 1d it is more correct to use the word ‘instanton’,

but we will stick to ‘vortex’ throughout.

φand its gradient ∂xφrespectively, which can be shown

using the commutation relation (B10). Vortices in ∂xφ

are not necessarily objects that we are used to dealing

with, but indeed they are well-deﬁned on the lattice [38],

and are the natural textures to consider in a continuum

limit where ∂xφbecomes smooth but φdoes not (a limit

that dipole symmetry forces us to consider, as this turns

out to be relevant for describing the dipole condensate).

In a background of charge density ρvortices carry

momentum 2πρ, and so a translation through a dis-

tance δacts as Tδ:ei∂xθ7→ ei2πδn/mei∂xθ(recall that

ρ=n/m). To understand this, consider moving a

vortex created by ei∂xθ(x)through a distance δto the

right. Doing so passes the vortex over an amount of

charge equal to ρδ, which in our continuum notation is

created by an operator proportional to eiδρφ(x). Since

ei∂xθ(x)eiδρφ(x)=ei2πδρeiδρφ(x)ei∂xθ(x), a phase of ei2πδρ

is accumulated during this process. Consistent with this,

a more careful analysis in App. Ashows that

Tδ:θ(x)7→ θ(x+δ)+2πρxδ. (12)

For our discussion of the phases that occur at fractional

ﬁllings, we will also need to discuss how θtransforms

under both site- and bond-centered reﬂections Rsand

Rb=T1/2RsT1/2. Using (12) and the fact that Rs:

ρ(x)7→ ρ(−x), we see that

Rs:θ(x)7→ θ(−x),

Rb:θ(x)7→ θ(−x)−π

2ρ. (13)

We now need to understand how to write down a ﬁeld

theory in terms of φand θwhich faithfully captures

the physics of HDBHM . The most naive approach is to

rewrite HDBHM as

HDBHM =tX

(|bi+1bi−1−b2

i|2+|bi+2bi−1−bibi+1|2)

i(U/2−t)n2

i−t(nini+1 +nini+2 +nini+3),

(14)

and to then perform a gradient expansion. Using the

representation (9) and keeping the lowest order deriva-

tives of θand φ, this produces a continuum theory with

Hamiltonian density

H=KD

2(∂2

xφ)2+u

2(∂2

xθ)2,(15)

where we have deﬁned the dipolar phase stiﬀness KDand

charge stiﬀness uas

KD≡4ρ2t, u ≡U−8t

(2π)2.(16)

Taking the above Hamiltonian density Has a starting

point and integrating out θproduces the Lagrangian of

the quantum Lifshitz model studied in Refs. [22,39],

which describes the BEI phase:

LBEI =Kτ

2(∂τφ)2+KD

2(∂2

xφ)2,(17)

where Kτ≡1/(8π2u).

The steps leading to (17) miss an essential part of the

physics, since they neglect vortices in the phase φ(as

well as vortices in the dipole phase ∂xφ). In the regu-

lar Bose-Hubbard model, vortices can be accounted for

using the hydrodyanmic prescription introduced by Hal-

dane in Ref. [37]. Using our representation of the den-

sity ﬂuctuations as ∂2

xθ/2π, a naive application of this

approach would lead to a Lagrangian containing cosines

of the form cos(l∂xθ), l∈N. This however turns out to

not fully account for the eﬀects of vortices in the DBHM,

which require that the terms cos(lθ) be added as well.

The exact perscription for including vortices is worked

out carefully in App. Ausing lattice duality, wherein we

derive the eﬀective Lagrangian

LDBHM =i

2π∂τφ(2πρ +∂2

xθ) + KD

2(∂2

xφ)2+u

2(∂2

xθ)2

−yD,4mcos(4mθ)−ymcos(m∂xθ),

(18)

where the coupling constants yl, yD,l are given by the

l-fold vortex and dipole vortex fugacities

yl∼e−l2c√KD/u, yD,l ∼e−l2cD√KD/u,(19)

where c, cDare non-universal O(1) constants (App. A

contains the derivation). The appearance of min the

term ymcos(m∂xθ) is due to (12), which ensures that

the leading translation-invariant interactions are those

which create m-fold vortices (recall ρ=n/m). The fac-

tor of 4 in yD,4mcos(4mθ) is due to the bond-centered

reﬂection symmetry Rbwhich shifts θaccording to (13)

(with cos(mθ) being the most relevant cosine of θin the

absence of Rbsymmetry).

From the above expression (16) for u, we see that an

instability occurs when

t>tF BD ≡U

8,(20)

which is precisely the condition given earlier in (5). When

t>tF BD,ubecomes negative, and the system is unsta-

ble against large density ﬂuctuations—this leads to the

glassy phase discussed in the introduction. In the rest of

this paper, we will restrict our attention to values of tfor

which u > 0, where the above ﬁeld theory description is

valid.

To understand the physics contained in the Lagrangian

LDBHM , the ﬁrst order of business is to evaluate the im-

portance of the cosines appearing therein. It is easy to

check that at the free ﬁxed point given by the quadratic

terms in LDBHM (the ﬁrst line of (18)), cos(lθ) has ultra

short-ranged correlations in both space and time, for any

l∈Z. This is however not true for cos(l∂xθ), whose corre-

lation functions are constant at long distances, regardless

of l. This means that cos(m∂xθ) is always relevant, im-

plying that ∂xθwill always pick up an expectation value

in the thermodynamic limit, and that vortices will con-

dense at all rational ﬁllings. This is physically quite rea-

sonable due to dipole symmetry forbidding a (∂xφ)2term

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DipolecondensatesintiltedBose-HubbardchainsEthanLake,1Hyun-YongLee,2,3,4JungHoonHan,5andT.Senthil11DepartmentofPhysics,MassachusettsInstituteofTechnology,Cambridge,MA,021392DivisionofDisplayandSemiconductorPhysics,KoreaUniversity,Sejong30019,Korea3DepartmentofAppliedPhysics,GraduateSchool,KoreaUnive...

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Dipole condensates in tilted Bose-Hubbard chains Ethan Lake1Hyun-Yong Lee2 3 4Jung Hoon Han5and T. Senthil1 1Department of Physics Massachusetts Institute of Technology Cambridge MA 02139

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