Thermodynamic Phase Diagram of Two-Dimensional Bosons in a Quasicrystal Potential Zhaoxuan Zhu1Hepeng Yao2and Laurent Sanchez-Palencia1

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Thermodynamic Phase Diagram of Two-Dimensional Bosons in a Quasicrystal

Potential

Zhaoxuan Zhu,1Hepeng Yao,2and Laurent Sanchez-Palencia1

1CPHT, CNRS, Ecole Polytechnique, IP Paris, F-91128 Palaiseau, France

2Department of Quantum Matter Physics, University of Geneva,

24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland

(Dated: July 13, 2023)

Quantum simulation of quasicrystals in synthetic bosonic matter now paves the way to the explo-

ration of these intriguing systems in wide parameter ranges. Yet thermal ﬂuctuations in such systems

compete with quantum coherence, and signiﬁcantly aﬀect the zero-temperature quantum phases.

Here we determine the thermodynamic phase diagram of interacting bosons in a two-dimensional,

homogeneous quasicrystal potential. Our results are found using quantum Monte Carlo simulations.

Finite-size eﬀects are carefully taken into account and the quantum phases are systematically dis-

tinguished from thermal phases. In particular, we demonstrate stabilization of a genuine Bose glass

phase against the normal ﬂuid in sizable parameter ranges. Our results for strong interactions are

interpreted using a fermionization picture and experimental relevance is discussed.

The discovery of quasiperiodic structures in plane

tilings [1] and material science [2,3] has profoundly al-

tered our dichotomous perception of order and disorder.

Lying at the interface of the two realms, quasicrystals

display a number of intriguing properties, including un-

usual localization and fractal properties, anomalous crit-

ical scalings, and phasonic degrees of freedom [4–9]. So

far, quasicrystals have been observed in their natural

state in meteorites [10,11] and nuclear blast residues [12]

or in the laboratory after fast solidiﬁcation of certain al-

loys [2,13], and have been extensively studied in solid-

state physics [2,5,6,14–16]. Moreover, artiﬁcial qua-

sicrystals can now be engineered in synthetic quantum

matter with unique control knobs, using photonic crys-

tals [8,17–19], quantum ﬂuids of light [20–22], and ultra-

cold quantum gases [23–25]. In the latter, defectless and

phononfree quasicrystal potentials can be emulated in

a variety of conﬁgurations using appropriately-arranged

sets of laser beams [26–31]. Furthermore, two-body in-

teractions can be tuned using magnetic control [32–35],

hence paving the way to the exploration of quantum

phase diagrams in wide parameter ranges.

In past years, one-dimensional (1D) quasiperiodic

models of ultracold atoms have been discussed quite ex-

haustively [36–54] but exploration of their 2D counter-

parts has only recently gained momentum, mostly in

tight-binding models [55–57]. So far, theoretical and

experimental work has demonstrated the emergence of

quasicrystalline order through matterwave interferome-

try [28,30], Anderson-like localization [28,31,58], and

Bose glass (BG) physics [31,55,56,59]. The BG is an

emblematic compressible insulator, characteristic of dis-

ordered or quasi-disordered systems and distinct from the

superﬂuid (SF) and Mott insulator (MI) phases, which

also appear in periodic systems [60–62]. In bosonic mod-

els, however, thermal ﬂuctuations compete with (quasi-

)disorder, which has so far hindered the observation of

the BG phase [43,44]. It has been recently proposed that

this issue may be overcome by scaling up characteristic

energies using shallow quasiperiodic potentials [47]. Up

to now, this has been investigated only in 1D [48] and 2D

harmonically trapped [63] systems. In contrast, the case

of a 2D Bose gas with genuine long-range quasicrystal

order remains unexplored. Moreover, the central issue of

discriminating the BG phase from trivial thermal phases

has been hardly addressed. As argued below, this cannot

be achieved as in 1D and requires speciﬁc analysis in 2D.

In this Letter, we determine the ﬁrst thermodynamic

phase diagrams of weakly to strongly interacting 2D Bose

gases in a shallow quasicrystal potential at ﬁnite temper-

atures. Quantum Monte Carlo simulations are performed

in quasicrystal, homogeneous potentials and ﬁnite-size ef-

fects are carefully taken into account. The SF, MI, and

BG quantum phases, induced by the competition of in-

teractions and quasicrystal potential, are systematically

discriminated from the normal ﬂuid (NF), which is in-

stead dominated by thermal ﬂuctuations. Most impor-

tantly, we ﬁnd that the BG phase survives up to signif-

icantly high temperatures. Our results in the strongly-

interacting regime are interpreted using a fermionization

picture and implications to experiments in ultracold atom

systems are discussed.

Model.— The dynamics of the 2D Bose gas is gov-

erned by the Hamiltonian

H=ˆdrΨ(r)†−ℏ2∇2

2m+V(r)Ψ(r),(1)

2ˆdrdr′Ψ(r)†Ψ(r′)†U(r−r′)Ψ(r′)Ψ(r),

where Ψ(r)is the bosonic ﬁeld operator at position rand

mis the particle mass. The quasicrystal potential,

V(r) = V0

k=1

cos2(Gk·r),(2)

arXiv:2210.15526v2 [cond-mat.quant-gas] 11 Jul 2023

Figure 1. Thermodynamic phase diagrams of 2D bosons in the eightfold quasicrystal potential of Eq. (2) with amplitude

V0= 2.5Erand diﬀerent interaction strengths, (a) ˜g0= 0.05, (b) ˜g0= 0.86, and (c) ˜g0= 5. The quantum phases, SF (blue),

BG (yellow), and MI (red), are distinguished from the NF regime (green). Note the small MI lobes in panel (c) at µ≃4.1Er

and µ≃5.1Er, which survive only at very low temperatures. QMC results are shown as data points with errorbars, while color

boundaries are guides to the eye.

is the sum of four standing waves with amplitude V0

and lattice period a=π/|Gk|, and successively rotated

by an angle of 45◦. This potential is characterized by

an eightfold discrete rotational symmetry, incompatible

with periodic order, hence forming a quasiperiodic pat-

tern. The bosons interact via the two-body scattering

potential U(r−r′). At low energy, the collisions are

dominated by s-wave scattering and hence fully charac-

terized by the sole 2D scattering length a2D . Due to the

logarithmic scaling of the interaction strength versus the

scattering length in 2D [64–66], it is convenient to use

the interaction parameter

˜g0=2π

ln(a/a2D ).(3)

The model considered here is similar to that recently em-

ulated in ultracold-atom quantum simulators in Refs. [30,

31]. The typical potential amplitude V0ranges from zero

to a few tens of recoil energies, Er=π2ℏ2/2ma2. In

the eightfold quasicrystal potential (2), the critical am-

plitude for single-particle localization is V0≃1.76Er[59].

So far, ultracold bosons in such 2D quasicrystal potential

have been studied for vanishing or weak interactions, up

to ˜g0≃0.86 [31]. However, signiﬁcantly higher values

can be realized using transverse conﬁnement or Feshbach

resonances, up to the strongly-interacting regime, where

˜g0∼1−5[67]. The typical temperature in ultracold

atom experiments is kBT/Er∼0.01 −0.5with kBthe

Boltzmann constant.

Finite-temperature phase diagrams.— Figure 1shows

the thermodynamic phase diagrams of the interacting

Bose gas in a quasicrystal potential of amplitude V0=

2.5Er(above the critical localization potential) for three

values of the interaction parameter ˜g0, ranging from

weak to strong interactions. The numerical calculations

are performed using path-integral quantum Monte Carlo

(QMC) simulations within the grand-canonical ensem-

ble at temperature Tand chemical potential µ. Details

about the analysis of the numerical results, in particular

as regards ﬁnite-size eﬀects, appear below. In brief, we

compute the compressibility κ=L−2∂N/∂µ, where N

is the average particle number and Lthe system’s lin-

ear size, as well as the superﬂuid fraction fs, found using

the winding number estimator with periodic boundary

conditions [68]. These two quantities are suﬃcient to

identify the expected zero-temperature quantum phases:

SF (κ̸= 0 and fs̸= 0), BG (κ̸= 0 and fs= 0), and MI

(κ= 0 and fs= 0). For high enough temperatures, how-

ever, one expects a NF regime, dominated by thermal

ﬂuctuations. It is characterized by a ﬁnite compressibil-

ity and absence of superﬂuidity (κ̸= 0 and fs= 0), just

as the BG phase.

To discriminate a genuine BG against a trivial NF,

we use the criterion that phase coherence and superﬂu-

idity must be destroyed by quasi-disorder and not ther-

mal ﬂuctuations [60,61]. In 1D, any ﬁnite temperature

destroys superﬂuidity so that the BG phase is strictly

well deﬁned only at zero temperature. In practice, it is

thus suﬃcient to identify a NF by the onset of a sizable

temperature dependence of characteristic quantities, as

done in Refs. [43,44,48]. In dimensions higher than one,

however, quantum phases can survive at ﬁnite tempera-

ture while showing a signiﬁcant temperature dependence

of the characteristic quantities, and the above criterion

breaks down. To discriminate the BG from the NF in the

2D Bose gas, we thus proceed diﬀerently and systemati-

cally compare the obtained phases in the presence of the

quasicrystal potential with those of the homogeneous gas

for the same temperature and the same average number

of particles: If the gas is a SF in the absence of the qua-

sicrystal potential, we identify a BG phase as soon as the

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ThermodynamicPhaseDiagramofTwo-DimensionalBosonsinaQuasicrystalPotentialZhaoxuanZhu,1HepengYao,2andLaurentSanchez-Palencia11CPHT,CNRS,EcolePolytechnique,IPParis,F-91128Palaiseau,France2DepartmentofQuantumMatterPhysics,UniversityofGeneva,24QuaiErnest-Ansermet,CH-1211Geneva,Switzerland(Dated:July13,20...

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Thermodynamic Phase Diagram of Two-Dimensional Bosons in a Quasicrystal Potential Zhaoxuan Zhu1Hepeng Yao2and Laurent Sanchez-Palencia1

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