Bose-Hubbard triangular ladder in an articial gauge eld Catalin-Mihai Halati1and Thierry Giamarchi1 1Department of Quantum Matter Physics University of Geneva Quai Ernest-Ansermet 24 1211 Geneva Switzerland

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Bose-Hubbard triangular ladder in an artiﬁcial gauge ﬁeld

Catalin-Mihai Halati1and Thierry Giamarchi1

1Department of Quantum Matter Physics, University of Geneva, Quai Ernest-Ansermet 24, 1211 Geneva, Switzerland

(Dated: February 21, 2023)

We consider interacting bosonic particles on a two-leg triangular ladder in the presence of an

artiﬁcial gauge ﬁeld. We employ density matrix renormalization group numerical simulations and

analytical bosonization calculations to study the rich phase diagram of this system. We show that

the interplay between the frustration induced by the triangular lattice geometry and the interactions

gives rise to multiple chiral quantum phases. Phase transition between superﬂuid to Mott-insulating

states occur, which can have Meissner or vortex character. Furthermore, a state that explicitly

breaks the symmetry between the two legs of the ladder, the biased chiral superﬂuid, is found for

values of the ﬂux close to π. In the regime of hardcore bosons, we show that the extension of the

bond order insulator beyond the case of the fully frustrated ladder exhibits Meissner-type chiral

currents. We discuss the consequences of our ﬁndings for experiments in cold atomic systems.

I. INTRODUCTION

The interplay between kinetic energy and interactions

leads, for quantum systems, to a very rich set of many-

body phases with remarkable properties, such as super-

conductivity, or Mott insulators. This is particularly true

in reduced dimensionality, where the eﬀects of interac-

tions are at their maximum. This leads in one dimension

to a set of properties, known as Tomonaga-Luttinger liq-

uids [1]. These are quite diﬀerent from the typical physics

that exists in higher dimensions, characterized by ordered

states with single particle type excitations, such as Bo-

goliubov excitations for bosons, or Landau quasiparticles

for fermions.

An intermediate situation is provided by ladders, i.e. a

small number of one-dimensional (1D) chains coupled by

tunneling. Such systems possess some unique properties,

diﬀerent from both the one-and the high-dimensional

ones. For example fermionic ladders exhibit supercon-

ductivity with purely repulsive interactions, at variance

with isolated 1D chains that are dominated by antiferro-

magnetic correlations [2].

Ladders are also the minimal systems in which the or-

bital eﬀects of a magnetic ﬁeld can be explored. For

bosonic ladders this has allowed to predict [3] the exis-

tence of quantum phase transitions as a function of the

ﬂux between a low ﬁeld phase with current along the

legs (Meissner phase) and a high ﬁeld phase with cur-

rents across the rungs and the presence of vortices (vortex

phase), akin to the transition occurring in type II super-

conductors. Ultracold atomic systems oﬀer the possibil-

ity of studying such systems coupled to artiﬁcial gauge

ﬁelds [4,5], and the Meissner to vortex phase transition

has been observed experimentally [6]. These works have

paved the way for a ﬂurry of studies for other situations

both for bosonic and fermionic ladders [3,7–30]. Further-

more, properties beyond the phase diagram, such as the

Hall eﬀect, were also studied [31,32] and even measured

[33–35].

These extensive studies of ladders have however con-

centrated mostly on square ladders, for which the eﬀect

of hopping is unfrustrated, leaving the case of triangular

ladders under ﬂux relatively unexplored, despite some

previous studies focusing on particular setups, or corners

of the phase diagram [36–45]. The triangular structure is

not bipartite and, thus, prevents the particle-hole sym-

metry that occurs naturally in square lattices. This has

drastic consequences since it leads to frustration of the

kinetic energy and, thus, to quite diﬀerent properties, as

was largely explored for two-dimensional systems [46–50].

In this paper, we explore the phase diagram of a tri-

angular two leg bosonic ladder under an artiﬁcial mag-

netic ﬁeld. We consider bosons with a contact repulsive

interaction. We study, using a combination of analyti-

cal bosonization and numerical density matrix renormal-

ization group (DMRG) techniques the phase diagram of

such a system as a function of the magnetic ﬁeld, ﬁlling

and repulsion between the bosons. We discuss in partic-

ular our ﬁndings in comparison with the phases found for

the square ladders.

The plan of the paper is as follows, in Sec. II we de-

scribe the model considered, its non-interacting limit and

the observables of interest. In Sec. III we brieﬂy dis-

cuss the methods employed in this work. We present

the results regarding the phase diagram at half ﬁlling in

Sec. IV. In this regime, we identify the following quan-

tum phases, the Meissner superﬂuid (M-SF), the vortex

superﬂuid (V-SF) and the biased chiral superﬂuid (BC-

SF), which breaks the Z2symmetry of the ladder. For

the fully frustrated π-ﬂux ladder, Sec. IV C, we obtain a

transition between superﬂuid and chiral superﬂuid states.

In the limit of hardcore bosons, Sec. V, at πﬂux we have

successive phase transitions between superﬂuid, bond or-

der insulator and chiral superﬂuid states. The bond order

extends in the phase diagram for lower values of the ﬂux

to the chiral bond order insulator (C-BOI). At unity ﬁll-

ing for interacting bosons, Sec. VI, also a Meissner Mott

insulator (M-MI) can be found in the phase diagram. We

discuss our results in Sec. VII and conclude in Sec. VIII.

arXiv:2210.14594v2 [cond-mat.quant-gas] 20 Feb 2023

FIG. 1: Sketch of the setup. The bosonic atoms are conﬁned

in a quasi-one-dimensional triangular ladder. The legs are

numbered by m= 1,2 and the sites on each leg by j. The

atoms tunnel along the legs with the amplitude Jk, along the

rungs with the amplitude Jand have an on-site interaction

of strength U. Each triangular plaquette is pierced by a ﬂux

χ.

II. MODEL

A. Setup

We consider interacting bosonic atoms conﬁned to a

triangular ladder in an artiﬁcial gauge ﬁeld, as sketched

in Fig. 1. The Bose-Hubbard Hamiltonian of the system

is given by

H=Hk+H⊥+Hint,(1)

Hk=−Jk

L−1

j=1 e−iχb†

j,1bj+1,1+eiχb†

j,2bj+1,2+ H.c.,

H⊥=−J

j=1 b†

j,1bj,2+ H.c.

−J

L−1

j=1 b†

j+1,1bj,2+ H.c.,

Hint =U

j=1

m=1

nj,m(nj,m −1).

The bosonic operator bj,m and b†

j,m are the annihila-

tion and creation operators of the particles at position

jand leg m= 1,2. We consider a total number of

N=PL

j=1 P2

m=1 nj,m atoms and that the ladder has

Lsites on each leg. The atomic density is given by

ρ=N/(2L). Hkdescribes the tunneling along the two

legs of the ladder, indexed by j, with amplitude Jk. The

complex factor in the hopping stems from the artiﬁcial

magnetic ﬁeld, with ﬂux χ[4,5]. The tunneling along the

rungs of the ladder is given by H⊥and has amplitude J.

The atoms interact repulsively with with an on-site in-

teraction strength U > 0.

(a) (b)

FIG. 2: Single particle dispersion of model (1), E±(k) (2), as

a function of momentum k, for (a) Jk/J = 0.2, (b) Jk/J = 1,

χ∈ {π/4, π/2,3π/4, π}. Note the presence of the two minima

away from k= 0 for some values of the ﬂux and transverse

hopping.

B. Non-interacting limit

In the non-interacting, U= 0, limit we can exactly

diagonalize the Hamiltonian (1) (see Appendix A) and

obtain the following dispersion relation

E±(k) = −2Jkcos(k) cos(χ) (2)

±q2J2[1 + cos(k)] + 4J2

ksin2(k) sin2(χ).

The non-interacting bands, E±(k), are represented in

Fig. 2for several values of Jk/J and χ. We can observe

that the lower band can have either a single minimum at

k= 0, e.g. for Fig. 2(a) for Jk/J = 0.5 and χ= 0.25π,

or two minima at ﬁnite values of k, e.g. for Fig. 2(c)

for Jk/J = 2 and χ= 0.5π. The position of the double

minima depends on Jk/J and χ.

The topology of the lower band can already provide

some hints regarding the nature of the ground state in

the case of weakly interacting bosons. Similarly with the

analysis performed in the case of the square ladder with

ﬂux [12,18] we expect phases of the following natures:

Meissner states in the case in which the bosons condense

in the k= 0 minimum; vortex phases, in the case of

two condensates in the two minima of the lower band;

and states which break the Z2symmetry of the ladder,

corresponding to a condensate in just one of the double

minima. In Sec. IV and Sec. VI we show how these states

are realized on the triangular ladder in the interacting

regime.

C. Observables of interest

In the rest of this section, we describe some of the ob-

servables which are suitable for the investigation of the

chiral phases we obtain in this system. We deﬁne the lo-

cal currents on the leg jk

j,m and the rung j⊥

j, respectively,

j,m =−iJkeiχ(−1)mb†

j,mbj+1,m −H.c.,(3)

j⊥

2j−1=−iJ(b†

j,1bj,2−H.c.),

j⊥

2j=−iJ(b†

j+1,1bj,2−H.c.).

In addition to the local currents, the chiral current Jcand

the average rung current Jrare of interest and deﬁned as

Jc=1

2(L−1) X

jDjk

j,1−jk

j,2E,(4)

Jr=1

2L−1X

jj⊥

j.

In order to identify biased phases, in which the Z2

symmetry between the two legs of the ladder is broken,

we compute the density imbalance

∆n=1

2LX

(nj,1−nj,2).(5)

Furthermore, we compute the central charge c, which

can be interpreted as the number of gapless modes. We

extract the central charge from the scaling of the von

Neumann entanglement entropy SvN (l) of an embedded

subsystem of length lin a chain of length L. For open

boundary conditions the entanglement entropy for the

ground state of gapless phases is given by [51–53]

SvN =c

6log L

πsin πl

L+s1,(6)

where s1is a non-universal constant, and we neglect log-

arithmic corrections [54] and oscillatory terms [55] due

to the ﬁnite size of the system.

III. METHODS

A. Bosonization

The low-energy physics of one dimensional interact-

ing quantum systems, corresponding to the Tomonaga-

Luttinger liquids universality class, can be described in

terms of two bosonic ﬁelds φand θ[1]. These bosonic

ﬁelds are related to the collective excitations of density

and currents and fulﬁll the canonical commutation rela-

tion, [φ(x),∇θ(x0)] = iπδ(x−x0). In the bosonized rep-

resentation, the single particle operator of the bosonic

atoms can be written as [1]:

b†

j= (aρ)1

2





e−iθ(aj)+X

p6=0

ei2p[πρaj−φ(aj)]e−iθ(aj)





(7)

with ρthe density and athe lattice spacing. In the fol-

lowing, we take a= 1.

B. MPS ground state simulations

The numerical results were obtained using a ﬁnite-

size density matrix renormalization group (DMRG) algo-

rithm in the matrix product state (MPS) representation

[56–60], implemented using the ITensor Library [61]. We

compute the ground state of the model (1) for ladders

with a number of rungs between L= 60 and L= 180,

and with a maximal bond dimension up to 1800. This en-

sures that the truncation error is at most 10−9. Since we

are considering a bosonic model with ﬁnite interactions

the local Hilbert space is very large, thus, a cutoﬀ for

its dimension is needed. We use a maximal local dimen-

sion of at least four or ﬁve bosons per site. We checked

that the local states with a higher number of bosons per

site do not have an occupation larger than 10−5for the

parameters considered. We make use of good quantum

numbers in our implementation as the number of atoms

is conserved in the considered model.

IV. PHASE DIAGRAM AT HALF FILLING,

ρ= 0.5

M-SF

V-SF

BC-SF

FIG. 3: Sketch of the phase diagram for half-ﬁlling, ρ= 0.5,

for U/J = 2.5. The identiﬁed phases (see text) are the Meiss-

ner superﬂuid (M-SF), the vortex superﬂuid (V-SF) and the

biased chiral superﬂuid (BC-SF). At χ=π(marked by a thick

vertical line) we have a phase transition between a superﬂuid

(green) and a chiral superﬂuid (dark blue).

(a)

(b)

(c)

V-SF

M-SF, M-MI

CBO-I

BC-SF

C-SF

(d)

(e)

FIG. 4: The pattern of currents, depicted with arrows, and

local densities, depicted with red disks, obtained in the nu-

merical ground state results for the (a) Meissner states (M-SF,

or M-MI) (b) vortex superﬂuid phases (V-SF), (c) biased chi-

ral superﬂuid (BC-SF), (d) chiral superﬂuid (C-SF), and (e)

chiral bond order insulator (CBO-I), where we also marked

the bond ordering on every second rung. We note that the

local currents are not normalized to the same value for the

diﬀerent phases represented in the four sketches.

In this section, we focus for the case in which we have

one bosonic atom every two sites, ρ= 0.5. In Fig. 3we

sketch the phase diagram we obtain from our numerical

and analytical results which we detail in the following. In

particular, we focus on several regions on the phase dia-

gram. We investigate the limit of small Jk(see Sec. IV A),

where we obtain a Meissner superﬂuid (M-SF). At large

Jkwe observe a phase transition between the Meissner

superﬂuid (M-SF) and a vortex superﬂuid (V-SF) state

(see Sec. IV B). At χ=πa transition between a super-

ﬂuid and a chiral superﬂuid state is present (Sec. IV C),

and for χ.πthe chiral superﬂuid extends to a biased

chiral superﬂuid phase (BC-SF). Throughout this section

the value of the on-site interaction is U/J = 2.5.

A. Small Jklimit - single chain limit

In the regime of small Jk/J it is useful to rewrite the

Hamiltonian given in (1) as a single chain with long range

FIG. 5: Numerical ground state results for the average rung

current, Jr, and the chiral current, Jc, as a function of the ﬂux

χfor Jk/J = 0.2, U/J = 2.5, ρ= 0.5, L∈ {60,120,180}. The

inset contains the dependence of the central charge, c, on the

ﬂux. We can identify the Meissner superﬂuid phase. We note

that the markers corresponding to the smaller system sizes are

below the ones for L= 180. The maximal bond dimension

used was m= 500 for L= 60, m= 900 for L= 120 and

m= 1500 for L= 180.

complex hopping

Hchain =−JkX

jhei(−1)jχb†

jbj+2 + H.c.i(8)

−JX

jb†

jbj+1 + H.c.

nj(nj−1).

In this case the bosonized Hamiltonian is

Hchain =Zdx

2πnuK + 16πρJkcos(χ)∂xθ(x)2(9)

K∂xφ(x)2o

+ρ2UZdx cos [2pφ(x)] ,

with the velocity u, Luttinger parameter K, and p= 1

for ρ= 1 and p= 2 for ρ= 0.5. For the atomic density

considered in this section we expect that the interaction

term does not dominate and we obtain a Luttinger liq-

uid for which the Luttinger parameter depends on the

ﬂux χ. As we see in the following this is in agreement

with our numerical results. However, we note slight devi-

ations from the analytical expectation of the dependence

of eﬀective Luttinger parameter on χand the numerical

results (see Appendix B).

In the single chain limit the local current observables

(4) can be rewritten as

j⊥

j=−iJ b†

jbj+1 −H.c.(10)

j=−iJkhe(−1)jiχb†

jbj+2 −H.c.i.

(a)

(b)

(c)

FIG. 6: Numerical ground state results for (a) the average

rung current, Jr, and the chiral current, Jc, (b) the central

charge, c, and (c) the absolute value of the density imbalance,

|∆n|, as a function of the ﬂux χfor Jk/J = 0.5, U/J = 2.5,

ρ= 0.5, L∈ {60,120,180}. We observe a transition from

the Meissner superﬂuid to the biased chiral superﬂuid at χ≈

0.75π. The maximal bond dimension used was m= 600 for

L= 60, m= 1200 for L= 120 and m= 1800 for L= 180.

In terms of the bosonic ﬁeld the currents read

j⊥

j= 2ρJ sin (∂xθ),(11)

j= 2ρJksin h2∂xθ+ (−1)jχ

2i,

j+1 −jk

j= 2ρJkcos (2∂xθ) sin(χ).

In the obtained gapless phase the expectation value of

the rung currents will average to zero, and the chiral

current has a ﬁnite value Jc∝PjDjk

j,1−jk

j,2E∝sin(χ).

These results are consistent with the currents expected

in the Meissner superﬂuid phase, which are depicted in

Fig. 4(a), and their values are shown in Fig. 5.

In Fig. 5at small values of the leg tunneling ampli-

tude, Jk/J = 0.2, we observe that the currents on the

rungs are close to zero and the chiral current, Jc, has a

ﬁnite value stable with increasing the system size. The

central charge is c≈1 for all values of the ﬂux, implying

the existence of one gapless mode. Furthermore, in this

phase, the single particle correlations decay algebraically

with the distance (see Appendix B). Based on these con-

siderations we can identify the Meissner superﬂuid state.

However, we identify some small deviations from the ex-

pected sin(χ) dependence of the chiral current (Fig. 5).

We can observe in Eq. (9) for large values of Jkand χ≈

πthe coeﬃcient of the ﬁrst term in the Hamiltonian will

vanish and eventually become negative. This instability

in our bosonized model could signal a phase transition.

In the numerical results for Jk/J = 0.5, presented in

Fig. 6, we see above χ&0.75 a phase with strong currents

and central charge c≈1. Furthermore, a ﬁnite density

imbalance between the two legs of the ladder is present.

We associate this regime with the biased chiral superﬂuid

phase. We describe in more details the nature of this

phase in Sec. IV D.

We observe in the numerical results that the value of

the Luttinger parameter extracted from the algebraic de-

cay of the single particle correlations decreases and it is

close to zero as we increase χtowards the phase tran-

sition between the Meissner superﬂuid and the biased

chiral superﬂuid (see Appendix B).

B. Large Jklimit - two coupled chains limit

We bosonize the Hamiltonian of the two coupled chains

(1) in the limit where tunneling Jkalong the two chains

dominate. In this regime we have a pair of bosonic ﬁelds

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Bose-HubbardtriangularladderinanarticialgaugeeldCatalin-MihaiHalati1andThierryGiamarchi11DepartmentofQuantumMatterPhysics,UniversityofGeneva,QuaiErnest-Ansermet24,1211Geneva,Switzerland(Dated:February21,2023)Weconsiderinteractingbosonicparticlesonatwo-legtriangularladderinthepresenceofanarticialg...

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Bose-Hubbard triangular ladder in an articial gauge eld Catalin-Mihai Halati1and Thierry Giamarchi1 1Department of Quantum Matter Physics University of Geneva Quai Ernest-Ansermet 24 1211 Geneva Switzerland

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