Zonal Estimators for Quasiperiodic Bosonic Many-Body Phases

Abstract

In this work, we explore the relevant methodology for the investigation of interacting systems with contact interactions, and we introduce a class of zonal estimators for path-integral Monte Carlo methods, designed to provide physical information about limited regions of inhomogeneous systems. We demonstrate the usefulness of zonal estimators by their application to a system of trapped bosons in a quasiperiodic potential in two dimensions, focusing on finite temperature properties across a wide range of values of the potential. Finally, we comment on the generalization of such estimators to local fluctuations of the particle numbers and to magnetic ordering in multi-component systems, spin systems, and systems with nonlocal interactions.

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Citation: Ciardi, M.; Macrì, T.; Cinti,
F. Zonal Estimators for Quasiperiodic
Bosonic Many-Body Phases. Entropy
2022,24, 265. https://doi.org/
10.3390/e24020265
Academic Editor: Antonio M.
Scarfone
Received: 17 January 2022
Accepted: 9 February 2022
Published: 12 February 2022
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entropy
Article
Zonal Estimators for Quasiperiodic Bosonic Many-Body Phases
Matteo Ciardi 1,2,* , Tommaso Macrì 3,4 and Fabio Cinti 1,2,5
1
Dipartimento di Fisica e Astronomia, Università di Firenze, I-50019 Sesto Fiorentino, Italy; fabio.cinti@unifi.it
2INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Italy
3Departamento de Física Teórica e Experimental and International Institute of Physics, Universidade Federal
do Rio Grande do Norte, Natal 59078-970, RN, Brazil; macri@fisica.ufrn.br
4Harvard-Smithsonian Center for Astrophysics, Institute for Theoretical Atomic, Molecular and Optical
Physics (ITAMP), Cambridge, MA 02138, USA
5Department of Physics, University of Johannesburg, P.O. Box 524, Auckland Park 2006, South Africa
*Correspondence: matteo.ciardi@unifi.it
Abstract:
In this work, we explore the relevant methodology for the investigation of interacting
systems with contact interactions, and we introduce a class of zonal estimators for path-integral Monte
Carlo methods, designed to provide physical information about limited regions of inhomogeneous
systems. We demonstrate the usefulness of zonal estimators by their application to a system of trapped
bosons in a quasiperiodic potential in two dimensions, focusing on finite temperature properties
across a wide range of values of the potential. Finally, we comment on the generalization of such
estimators to local fluctuations of the particle numbers and to magnetic ordering in multi-component
systems, spin systems, and systems with nonlocal interactions.
Keywords:
quantum phases; quasicrystals; trapped bosons; superfluidity; path-integral Monte Carlo
1. Introduction
Path-integral Monte Carlo (PIMC) methods [
1
] are of great importance for the sim-
ulation of strongly correlated systems where other techniques fail, especially in two and
three spatial dimensions. Over the last thirty years, this has been amply demonstrated
on quantum fluids [
2
–
4
] and, more recently, in ultra-cold gases like, for instance, dipolar
systems [
5
–
9
] and Rydberg atoms [
10
–
13
]. For strongly-interacting quantum fluids, there
is at present considerable interest towards the exploration of patterns owing to peculiar
symmetries such as quantum-cluster crystals [
14
,
15
], stripe phases [
16
], or cluster qua-
sicrystals [
17
–
19
], with the aim of understanding fundamental physical phenomena. In this
regard, also thanks to the increase in computational capabilities, advancements in PIMC
methods continue to play a key role.
In this work, we detail the numerical techniques required to investigate the quantum
properties of interacting trapped bosons in an external quasiperiodic potential at finite
temperature, with specific attention to superfluidity. Quasicrystals are a fascinating state of
solid-state matter exhibiting behaviors halfway between a periodically ordered structure
and a fully disordered system. They were first synthesized in 1982 (and their discovery
later announced in 1984) by Shechtman et al. [
20
]. Later, Bindi et al. demonstrated that
quasicrystals can also originate naturally, in the presence of extreme conditions such as
collisions between asteroids [
21
]. Their properties have already been the subject of extensive
theoretical investigation [
22
], motivated in part by the discovery of aperiodic tilings that
can cover the plane without being bounded by the symmetries of classical crystallography,
such as the Penrose tiling [
23
]. At finite temperature, the thermodynamic features of
quasicrystals can be established in terms of the interplay between different length and
energy scales pertaining to the inter-particle potentials [
24
]. These classical systems were
found to remain stable even at zero temperature [
25
]. Quasicrystalline properties have
been observed in a variety of physical systems, for instance, in nonlinear optics [
26
–
28
],
on twisted bilayer graphene [
29
] and in ultra-cold trapped atoms [
30
,
31
]. In the latter
Entropy 2022,24, 265. https://doi.org/10.3390/e24020265 https://www.mdpi.com/journal/entropy

Entropy 2022,24, 265 2 of 18
case, quasicrystalline structures generated by means of optical lattices are employed to
experimentally investigate remarkable effects such as many-body localization in one and
two-dimensions [
32
], and have been suggested as a candidate to probe the existence of two-
dimensional Bose glasses [
33
]. In this regard, recent PIMC simulations support the existence
of a Bose glass phase, fully stable and robust at finite temperature, in a region of parameters
suitable for experimental setups [
34
,
35
]. Other works have delineated zero-temperature
phase diagrams, in the mean-field approximation as well as for a strong interactions using
ab-initio techniques [36–39].
Here, we summarize the derivation of the pair-product approximation for particles
interacting through hard-core interactions in two or three dimensions, and we present the
details of our implementation. Then, we explore a new zonal estimator, which gives access
to local information about the superfluidity in finite regions of trapped systems, and is
therefore well-suited to the study of spatially inhomogeneous potentials. Zonal estimators
can be relevant to the detection of correlated phases, such as the Bose glass phase, which is
characterized by rare regions where superfluidity and finite compressibility coexist [34].
This paper is organized as follows. In Section 2, we present and discuss the PIMC
methodology for ensembles of interacting bosons through the pair-product approximation.
In Section 3, we introduce a model Hamiltonian describing interacting trapped bosons
subjected to a quasiperiodic potential. Structural properties such as density profiles and
diffraction patterns are shown in Section 4, whereas we examine global quantum features in
Section 5. The zonal estimator of the superfluid fraction is explored in Section 6. To conclude,
Section 7is devoted to the discussion of our findings, drawing some conclusions.
2. Methodology
In this section, we review the implementation of the PIMC to the study of an inter-
acting Bose gas in an external potential at finite temperature. This methodology aims
to sample the partition function of a quantum system at finite temperature. In line with
Feynman’s path integral theory [
40
,
41
], thermodynamic properties are addressed by con-
sidering an equivalent classical system, in which each quantum particle is represented
by a classical polymer. As a result, quantum quantities, like for instance superfluidity or
Bose–Einstein condensation, can be mapped across the equivalence as properties of the
polymers themselves [
42
]. The evaluation of those quantities takes place via a standard
classical Monte Carlo procedure such as the Metropolis algorithm [
43
], allowing us to sam-
ple thermodynamic properties within a precision limited only by numerical and statistical
errors. At present, one of the most efficient ways of sampling configurations of connected
polymers is operated through the so-called worm algorithm [
2
,
44
]; originally developed
for the grand-canonical ensemble, we routinely use the worm algorithm in its canonical
version to sample superfluid fraction, condensate fraction, or ground state energy.
In the following, we recall the formalism and derivations at the core of PIMC. The par-
tition function,
Z
, is defined as the trace of the equilibrium density matrix operator,
ρ
,
at temperature TRef. [45]:
ρ=e−βˆ
H,Z=Tr e−βˆ
H, (1)
βbeing the inverse temperature parameter, β=1/kBT.
For
N
distinguishable particles, denoting with
ri
the position of the
i
-th particle,
and introducing
R
=
(r1
,
r2
,
. . .
,
rN)
, we can project the density matrix operator on the basis
of spatial coordinates |Ri, obtaining
ρ(R,R,β) = hR|e−βˆ
H|Ri,Z=ZdRρ(R,R,β). (2)
For an ensemble of bosons, taking into account permutations, we arrive at
Z=1
N!∑
PZdRρ(R,PR,β), (3)

Entropy 2022,24, 265 3 of 18
where PR=(rP(1),rP(2), . . . , rP(N))denotes a permutation of the particle coordinates.
Introducing a decomposition of the density matrix operator into a convolution of
density matrices at a higher temperature, Equation (4) yields
Z=1
N!∑
PZdR0dR1. . . dRM−1ρ(R0,R1,τ)ρ(R1,R2,τ)··· ρ(RM−1,PR0,τ), (4)
with
β
breaking up into
M
smaller intervals
τ=β/M
, and
Rm= (rm
1
,
rm
2
,
. . .
,
rm
N)
the coor-
dinates of particles on a given time slice. To each particle
ri
corresponds, then, a classical
polymer made of
m=
1, 2,
. . .
,
M
beads, connected with each other through harmonic
springs [
45
]. Errors introduced by the equivalence are reduced as
M
increases. Like-
wise, the same decomposition can be applied to the evaluation of observables by Monte
Carlo sampling.
For a generic diagonal observable,
ˆ
A
such that
hR|ˆ
A|R0i=A(R)◦(R−R0)
, it fol-
lows that
hAi =1
ZN!∑
PZdR0dR1. . . dRM−1A(R0)ρ(R0,R1,τ)ρ(R1,R2,τ)···ρ(RM−1,PR0,τ).(5)
Equation
(5)
is evaluated through a stochastic process, consisting of the generation of
random configurations {R0,R1, . . . RM−1}from the probability distribution
π(R0,R1, . . . RM−1) = 1
ZN!∑
P
ρ(R0,R1,τ)ρ(R1,R2,τ)··· ρ(RM−1,PR0,τ). (6)
The thermodynamic average then is measured as an average of
{A(R0)}
over the
sampled configurations [46].
Having to employ a finite number of time slices,
M
, the most sensitive step of the
procedure lies in finding a good approximation of the high-temperature density-matrix
elements
ρ(Rm
,
Rm+1
,
τ)
in
(4)
[
46
–
48
]. Due to the nature of the two-body interaction po-
tential between the bosons in Hamiltonian
Vint (|ˆ
ri−ˆ
rj|)
(see
(37)
for an application), and to
the density regime of interest, in the present work we apply a pair-product approximation
(PPA) ansatz [
45
]. The rest of this section is devoted to the treatment and implementation
of contact interactions in this context; similar derivations and other details can be found,
e.g., in [48,49] and references therein, and the supplemental material of [37].
We express the density-matrix terms as
ρ(R,R0;τ) = ρf ree (R,R0;τ)∏
i<j
ρrel
int (rij,r0i j;τ)
ρrel
f ree (rij ,r0
ij ;τ)+O(τ2), (7)
(to keep Equation (7) simple, we omit the indices m). Here,
ρf ree (R,R0;τ) = hR|exp{−τˆ
Hf ree }|R0i=
N
∏
i=1
1
(4πλiτ)d/2 exp−(r−r0)2
4λiτ(8)
with
λi=¯h2/
2
mi
, is the density matrix of the non-interacting Hamiltonian of
N
particles,
ˆ
Hf ree =∑N
i=1ˆ
p2
i/2mi.
For two particles, labeled
i
and
j
, we can decompose the Hamiltonian into a center-
of-mass term and a relative term, with the relative term being
ˆ
Hrel
int =ˆ
p2
ij
2mr+Vint (ˆ
rij )
; for
free particles, the relative Hamiltonian is only
ˆ
Hrel
f ree =ˆ
p2
ij
2mr
. Here we have introduced

Entropy 2022,24, 265 4 of 18
the relative coordinates
rij =rj−ri
,
pij = (mipi−mjpj/(mi+mj)
, and the reduced mass
mr=mimj/(mi+mj). We can then write the propagators
ρrel
f ree (rij ,r0ij ;τ) = hR|exp{−τˆ
Hrel
f ree }|R0i=1
(4πλrτ)d/2 exp(−|rij −r0
ij |2
4λrτ), (9)
ρrel
int (rij,r0i j;τ) = hR|exp{−τˆ
Hrel
int }|R0i, (10)
where λr=¯h2/2mr=λi+λj.
We use a standard Metropolis procedure, which consists of generating new config-
urations according to the free particle distribution, and then accepting or rejecting them
according to a statistical weight, which takes external potentials and interactions into ac-
count. The form
(7)
is best suited for this procedure, as long as we can efficiently determine
the terms under the product symbol. For ease of notation, we now take r=rij and define
ρint (r,r0,τ) = ρrel
int (r,r0,τ)
ρrel
f ree (r,r0,τ). (11)
In order to estimate
ρrel
int
for the model proposed in Equation
(37)
, we can expand it on
the eigenfunctions of the relative Schrödinger equation
ˆ
Hrel
int ψ(r) = h−λr∇2
r+Vint (r)iψ(r) = Eψ(r). (12)
For central potentials, which only depend on
r=|r|
, like the one considered in this
study, the equation splits into an angular part, giving rise to spherical harmonics in
d
dimensions, and a radial part:
−λr∂2
∂r2+1
r
∂
∂r−m2
r2ukm(r) + Vint (r)ukm (r) = λrk2ukm(r),d=2 (13)
−λr1
r2
∂
∂rr2∂
∂r−l(l+1)
r2ukl (r) + Vint(r)ukl(r) = λrk2ukl (r).d=3 (14)
In terms of these wavefunctions, we can expand the relative density matrix into
ρrel
int (r,r0,τ) =
∞
∑
m=0
cmcos(mθ)Z∞
0dk e−λrτk2ukm (r)∗ukm(r0) + ∑
n
e−τEnφn(r)∗φn(r0),d=2 (15)
ρrel
int (r,r0,τ) = 1
2π2
∞
∑
l=0
(2l+1)Pl(cos θ)Z∞
0dk k2e−λrτk2ukl (r)∗uk l (r0) + ∑
n
e−τEnφn(r)∗φn(r0).d=3 (16)
The coefficients
cm
are defined as
c0=
1,
cm=
2 for
m>
0. The functions
Pl(cos θ)
are
the Legendre polynomials of degree
l
[
50
]. Finally, the
φn(r)
are the bound states of the
potential, if any, and
En
their energies; they play no role in the study of repulsive potentials,
such as the one we are considering.
The free problem has straightforward solutions that, in the two-dimensional case, yield
ukm(r) = rk
2πJm(kr)(17)
ρrel
f ree (r,r0,τ) =
∞
∑
m=0
cmcos(mθ)Z∞
0dk e−λrτk2k
2πJm(kr)Jm(kr0). (18)

Entropy 2022,24, 265 5 of 18
In three dimensions, we get
ukl (r) = jl(kr)(19)
ρrel
f ree (r,r0,τ) = 1
2π2
∞
∑
l=0
(2l+1)Pl(cos θ)Z∞
0dk k2e−λrτk2jl(kr)jl(kr0). (20)
Jm(x)
are the Bessel functions of the first kind, and
jl(x) = √π/2x Jl+1/2 (x)
are the
spherical Bessel functions of the first kind; the factor appearing in the two-dimensional
case is due to the normalization and orthogonality relations. In both cases, the sum can be
computed analytically by employing tabulated integrals, leading back to the simple form
of (9).
More generally, it is necessary to solve Equations
(13)
or
(14)
numerically to find the
eigenfunctions. If the interaction is a short-range potential, so that
Vint (r) =
0 when
r>r0
,
for some value of
r0
, the eigenfunctions in the region
r>r0
are a generalization of the
free case:
ukm(r) = rk
2π[cos(δm(k))Jm(kr)−sin(δm(k))Ym(kr)],d=2 (21)
ukl (r) = cos(δl(k))jl(kr)−sin(δl(k))yl(kr).d=3 (22)
Ym(x)
and
yl(x) = √π/2xYl+1/2(x)
are, respectively, the Bessel and spherical Bessel
functions of the second kind. For
r<r0
, it is still necessary to solve the Schrödinger
equation. The phase shifts
δl
are determined by imposing smoothness conditions on
the wavefunction at
r=r0
. In the particular case of a hard-core potential of radius
r0
,
the requirement is
tan δm(k) = Jm(kr0)
Ym(kr0),d=2 (23)
tan δl(k) = jl(kr0)
yl(kr0).d=3 (24)
In order to implement the above formalism efficiently in our simulations, we recast
Equation (11) as follows:
ρint (r,r0,τ) = ρrel
f ree (r,r0,τ) + ρrel
int (r,r0,τ)−ρrel
f ree (r,r0,τ)
ρrel
f ree (r,r0,τ)
=1+ρrel
int (r,r0,τ)−ρrel
f ree (r,r0,τ)
ρrel
f ree (r,r0,τ)
=








1+1
ρrel
f ree (r,r0,τ)∑∞
m=0cmcos(mθ)Im(r,r0),d=2
1+1
ρrel
f ree (r,r0,τ)∑∞
l=0(2l+1)Pl(cos θ)Il(r,r0),d=3
(25)
where
Im(r,r0) = 1
2π2Zdk e−τλrk2ukm (r)∗ukm (r0)−k
2πJm(kr)Jm(kr0),d=2 (26)
Il(r,r0) = 1
2π2Zdk e−τλrk2k2ukl (r)∗ukl (r0)−jl(kr)jl(kr0).d=3 (27)

Entropy 2022,24, 265 6 of 18
The interacting propagator cannot be calculated analytically; the integrals must be
computed numerically and tabulated before running the simulations. While, in principle,
we could tabulate the entire propagator as a function of
r
,
r0
, and
θ
, the decomposition
of
(25)
has several advantages. First of all, it cleanly separates the contributions from the
free propagator, which are relevant at any
m
at large enough distances, so that we only
need to write tables for those values of
m
that actually present a variation with respect to
the free case. Moreover, since the angular variable
θ
is explicitly considered in the sum,
the integrals only need to be tabulated as a function of
r
and
r0
, reducing computational
time and memory usage considerably.
In our two-dimensional simulations, at all temperatures, we have found the contribu-
tions from the harmonics m≥1 to be negligible, so that we only use
ρint (r,r0,τ) = 1+I0(r,r0)
ρrel
f ree (r,r0,τ). (28)
Having established the form of the propagator, we can now use it to compute values of
thermodynamic observables. Some special care must be devoted to the thermal estimator
of the total or kinetic energy, for which the effective potential leads to a contribution of
the form ∂
∂τ u(r,r0,τ), (29)
with
u(r,r0,τ) = −ln ρint (r,r0,τ). (30)
For a generic ρint of the form
ρint (r,r0,τ) = 1+I(r,r0)
ρrel
f ree (r,r0,τ), (31)
with
I=Zdk e−τλrk2F(r,r0,k), (32)
we can introduce
J=λrZdk k2e−τλrk2F(r,r0,k); (33)
it is then possible to show that
∂
∂τ u(r,r0,τ) = 1
ρrel
f ree (r,r0,τ) + I(r,r0)J(r,r0) + 1
τ(r−r0)2
4λrτ−d
2I(r,r0), (34)
d
being the dimensionality of the system. In particular, this applies to the propagator
(28)
used in the present work.
As a final technical consideration, we note that the above treatment of the interaction
has been carried out in the absence of external potentials, which represent, instead, a crucial
component of our physical problem. The Trotter decomposition of
(4)
allows us to treat
different potential and interaction terms independently, or to group them together as
needed, as long as we pick a fittingly small value of
τ
. For a problem in the harmonic trap,
it is most efficient to sample configurations from the harmonic propagator
ρosc (r,r0,τ) = hr|exp(−τˆ
p2
2m+mω2ˆ
r2
2)|r0i, (35)

Entropy 2022,24, 265 7 of 18
which can be computed analytically [
42
,
51
]. We obtained satisfying results by employing
this distribution together with the pair-product propagator in the absence of external
potentials (11), as displayed in Figure 1, so that the complete form of our propagator is
ρ(R,R0;τ) =
N
∏
i=1
ρosc (ri,ri0;τ)
N
∏
i=1
e−τVext(ri)∏
i<j
ρint (rij,r0i j;τ)(36)
6420246
x
/
losc
0
20
40
60
80
nl
2
osc
g
0=0.0217
g
0=0.0664
g
0=0.217
g
0=0.75
g
0=2.17
Figure 1.
Comparison between exact density profiles from PIMC simulations (solid lines) and those
obtained from the 2D Thomas–Fermi approximation (dashed lines). The shades of purple correspond
to different values of the interaction, from small (light) to strong (dark), as indicated in the legend.
3. Application: Trapped Bosons in a Quasicrystal Potential
As motivated in Section 1, we aim to discuss the utilization of PIMC methods, imple-
menting zonal estimators for the quantum properties, in systems displaying a non-periodic
patterning. We introduce a model of
N
identical bosons in continuous two-dimensional
space described by the Hamiltonian
H=
N
∑
i=1 ˆ
p2
i
2m+mω2
2ˆ
r2
i+Vqc (ˆ
ri)!+∑
i<j
Vint (|ˆ
ri−ˆ
rj|), (37)
where
m
is the particle mass,
ˆ
pi
and
ˆ
ri
are the momentum and position operators of the
i
-th particle,
ω
is the frequency of the two-dimensional harmonic trap confining the bosons.
Vqc is an external potential defined by
Vqc (r) = V0
4
∑
i=1
cos2(ki·r), (38)
k1=klat 1
0,k2=klat /√21
1,k3=klat 0
1,k4=klat /√21
−1, (39)
with
klat
setting the spatial modulation, and
V0
the strength of the external potential.
The structure of maxima and minima of this potential displays the geometry of the aperiodic,
eightfold-symmetrical Ammann–Beenker tiling [
52
], therefore underlying a quasicrystalline
structure.
Vint
is a contact potential with scattering length
a
. Figure 2depicts the
sna pshot
configurations of interacting bosons described by Hamiltonian
(37)
with only the harmonic
trap (a) and in the presence of the quasicrystalline potential (b).

Entropy 2022,24, 265 8 of 18
Figure 2.
Snapshots of particle configurations at
˜
g0=
2.1704, for two values of
V0
. Red lines represent
the links between successive beads in the equivalent polymer system, as described in Section 2.
The shaded background represents the potential, with darker to brighter areas corresponding to
lower to higher values. On the
left
, we show a configuration in the absence of the quasiperiodic
potential (
V0/Er=
0.0), where the only external potential is the harmonic trap. On the
right
,
the presence of the quasiperiodic potential is reflected in the distribution of the polymers, which tend
to localize at the minima.
In Figure 3, we show the convergence of the thermodynamic average of the total
energy as we increase the number of slices
M
, and reduce the time step
τ
. Varying the
strength
V0
at
T=
0.25
Tc
, convergence is already established around
τ≈
0.1
βc
. Other
observables, such as the superfluid fraction, are found to converge even faster.
0.0 0.1 0.2 0.3 0.4 0.5
/
c
0
50
100
150
200
E
/
Er
V
0/
Er
= 0.5
V
0/
Er
= 1.0
V
0/
Er
= 1.5
Figure 3.
Trotter limit. The circles correspond to the measured values of the average energy
hEi
,
at decreasing values of the imaginary time step
τ
. Different colors represent different values of the
potential: V0/Er=0.5 (dark blue), V0/Er=1.5 (grey blue), V0/Er=2.5 (light blue).
We now briefly review the ground-state mean-field approach, which we use to plot
density profiles and display the validity of the pair-product approximation. The system is
described by a two-dimensional time-independent Gross–Pitaevskii equation [53],
µΨ(r) = −¯h2∇2
2m+mω2r2
2+Vqc (r) + g|Ψ(r)|2!Ψ(r). (40)

Entropy 2022,24, 265 9 of 18
The two-dimensional mean-field parameter gsatisfies, up to logarithmic corrections,
g=¯h2
m
4π
ln(4.376 /a2n(0)), (41)
with
n(
0
)
the particle density at the center of the two-dimensional trap, and
a
the scattering
length of the repulsive interaction [
54
–
56
]. Contrary to the three-dimensional case, where
the relationship between mean-field constant and scattering length is linear, the logarithm
in (41) implies that amust be vanishingly small for small but finite values of g.
We introduce the harmonic oscillator length losc =√¯h/mω, and rewrite (41) as
g=¯h2
m˜
g=¯h2
m
4π
ln(l2
osc /a2)+ln(4.376 /n(0)l2
osc )≈¯h2
m
2π
ln(losc /a), (42)
where the last approximate equality applies in the limit
a/losc 
1. We use this approximate
form to describe the interaction [37], defining
g0=¯h2
m˜
g0=2πln losc
a−1
. (43)
In Figure 1, we plot the density profiles obtained for interacting bosons in a two-
dimensional harmonic trap, sliced across the center. The measured profiles (solid lines) are
compared with those predicted by a two-dimensional Thomas–Fermi approximation
n(r) = ((µ−V(r))/g0,V(r)<µ
0 , V(r)>µ(44)
In practice, since in our simulations we work at fixed particle number
N=
500, we first
derive
µ
as
µ=¯hωpN˜
g0/π
. We find good agreement between the numerical profiles and
analytical ones, even for large values of
g0
, supporting the validity of the approach used to
treat the propagator
(36)
. In the following, we express lengths in units of
losc
, and energies
in units of
Er=¯h2k2
lat /
2
m
; we employ a weak harmonic trapping with
¯hω≈
0.02
Er
.
The connection between the values relative to the harmonic trap and those pertaining to
the lattice is the ratio losc /λl at ≈1.6.
4. Density Profiles and Diffraction Patterns
In the present section, we introduce the physical estimators obtained via PIMC, and we
discuss the results achieved for trapped bosons in a quasiperiodic potential in two dimensions.
There are several complementary ways to display spatial configuration of the quantum
system and its classical-polymer equivalent system. One is to select a system’s configuration
at a given simulation step, plotting the position of the beads
ri,m
: the resulting snapshots
provide a first graphical estimate of the particle’s probability distribution. In Figure 2we
show snapshots of particle configurations at
˜
g0=
2.1704, for two values of
V0
. Red lines
represent the links between successive beads in the equivalent polymer system, as described
in Section 2. The shaded background represents the potential, with darker to brighter areas
corresponding to lower to higher values. On the left, we show a configuration in the
absence of the quasiperiodic potential (
V0/Er=
0.0), where the only external potential is
the harmonic trap. On the right, the presence of the quasiperiodic potential is reflected in
the distribution of the polymers, which tend to localize at the minima.

Entropy 2022,24, 265 10 of 18
Our approach give access to density profiles, which are obtained as averages over
simulation steps, as well as over the positions of all different beads associated to each
particle. In continuous space, the average is performed by separating the simulation area
into bins, and counting the number of beads in each at every simulation step. In Figure 4,
we show two-dimensional density patterns for the system at
˜
g0=
0.0217 and
T/Tc=
0.25.
Figure 4a is the density profile in the harmonic trap, the same shown in Figure 3. As the
value of
V0
increases, the quasicrystalline structure appears initially as a modulation (b)
and then as localization at the deepest minima (c–d).
42024
x
/
losc
4
3
2
1
0
1
2
3
4
y
/
losc
V
0/
Er
=0.0
(a)
42024
x
/
losc
V
0/
Er
=0.5
(b)
42024
x
/
losc
V
0/
Er
=1.5
(c)
42024
x
/
losc
V
0/
Er
=2.5
(d)
Figure 4.
Density patterns at
˜
g0=
0.0217 and
T/Tc=
0.25. In each picture, the normalized two-
dimensional density takes on values going from 0 (black) to 1 (white). The density patterns are shown
at four values of V0:V0/Er=0.0 (a), V0/Er=0.5 (b), V0/Er=1.5 (c), and V0/Er=2.5 (d).
Diffraction patterns can be investigated employing the structure factor that, for exam-
ple, is observed experimentally in scattering experiments. For a particle density distribution
n(r) = ∑
i
δ(ri), (45)
the structure factor reads
I(q) = hn(q)n(−q)i, (46)
where n(q)is defined as following
n(q) = Zd2re−iq·rn(r) = ∑
j
e−iq·rj(47)
which is the Fourier transform of the particle distribution (45) [57].
Figure 5displays some examples of diffraction patterns. As we should expect, the struc-
ture factor evolves from a single peak in the fluid phase to a typical quasicrystalline pattern
as
V0
increases. The three rows in the figure correspond to three different temperatures; we
can see that, aside for some smearing of the peaks due to thermal fluctuations, the structure
remains essentially unchanged. This is expected since, even above
Tc
, strong intensities of
the lattice lead to localization, and the formation of a classical insulator.

Entropy 2022,24, 265 11 of 18
4
2
0
2
4
ky
/
klat
V
0/
Er
=0.0
(a)
V
0/
Er
=0.5
(b)
V
0/
Er
=1.5
(c)
V
0/
Er
=2.5
(d)
V
0/
Er
=3.5
(e)
4
2
0
2
4
ky
/
klat
(f) (g) (h) (i) (j)
42024
kx
/
klat
4
2
0
2
4
ky
/
klat
(k)
42024
kx
/
klat
(l)
42024
kx
/
klat
(m)
42024
kx
/
klat
(n)
42024
kx
/
klat
(o)
Figure 5.
Diffraction patterns at
˜
g0=
0.0217. In each picture, the density in
k
-space is shown, taking
on values from a lower cutoff (black) to the maximum (white). Each column displays a value of
V0
:
V0/Er=
0.0 (
a
,
f
,
k
),
V0/Er=
0.5 (
b
,
g
,
l
),
V0/Er=
1.5 (
c
,
h
,
m
),
V0/Er=
2.5 (
d
,
i
,
n
),
V0/Er=
2.5 (
e
,
j
,
o
).
Each row displays a value of T:T/Tc=0.25 (a–e), T/Tc=0.4 (f–j), T/Tc=0.7 (k–o).
5. Global Quantum Estimators
A first impression of the importance of quantum effects in the simulation can be
obtained by looking at particle permutations [
58
]. As mentioned above, due to the bosonic
nature of the particles, the polymers of the equivalent classical system connect to each
other, forming long cycles containing varying number of particles. We call the number of
particles in a cycle
Nperm
. At each simulation step, we can construct a histogram, counting
how many cycles contain exactly
Nperm
particles for each value of 0
<Nperm ≤N
. We can
then average the histogram over simulation steps, and normalize it, so that for each value
of
Nperm
we obtain the probability
pperm
of finding a cycle with exactly
Nperm
particles.
The histogram of
pperm
is shown in Figure 6for different values of
V0
. The distribution of
pperm in the superfluid phase shows that permutations entail cycles comprising almost all
particles in the trap. Intermediate values of
V0
lead to the disappearance of permutation
cycles comprising of all particles in the system, but still allow for cycles of a few hundred
particles. Even larger values lead to a sharp drop in
pperm
as a function of
Nperm
, with only
cycles of the order of ten particles remaining relevant.
We further characterize the quantum regime by considering the superfluid fraction.
We recall that, in the context of the two-fluid model [
57
,
59
,
60
], the density of a quantum
system displaying superfluidity at low temperature can be described by decomposing its
density
ρ
into a sum of two fields:
ρ=ρn+ρs
, where the first component describes the
normal density, whereas the second is the superfluid one. In this context, the superfluid
fraction is defined as the ratio of the superfluid density to ρ:
ns=ρs
ρ. (48)
The two components exhibit contrasting behavior in terms of, for example, flow
transport and entropy [
61
]. The superfluid component displays zero viscosity, and is
therefore unresponsive to the application of external velocity fields. In particular, when
subjected to an angular velocity field, the superfluid exhibits a reduction of the total

Entropy 2022,24, 265 12 of 18
moment of inertia compared to a classical fluid in the same conditions. This phenomenon
is encoded in a fundamental relation, which links nsto the system’s moments of inertia:
ns=1−I
Icl
. (49)
0 100 200 300 400
Nperm
17.5
15.0
12.5
10.0
7.5
5.0
pperm
V
0= 0
V
0= 1.0
V
0= 1.5
V
0= 2.0
V
0= 2.5
V
0= 3.0
Figure 6.
Probabilities of permutation cycles
Pperm
as a function of cycle length, at
˜
g0=
0.0217
and
T/Tc=
0.25. Each color corresponds to a different value of
V0
, as indicated in the legend.
Although histograms are generally more appropriate, we use solid lines for ease of comparison
between different cases. All plots are in logarithmic scale.
Here, we indicated
I
as the moment of inertia related to
ρn
,
Icl
representing the
classical
moment of inertia, which is the one the same mass of fluid would have if it behaved classically.
By applying linear response theory on Equation
(49)
, one can extract
ns
through PIMC.
In fact, it is possible to show that the expectation value of the angular momentum in the
quantum system is given in terms of the area encircled by tangled paths in the classical
system of polymers [
3
,
62
]. This way to evaluate Equation
(49)
results as particularly
appropriate for all trapped and finite-size bosonic systems. Since we are dealing with a
pure two-dimensional system, we are interested in studying infinitesimal rotations around
an axis perpendicular to the
xy
plane. Following the formulation given in Ref. [
62
],
ns
in its
complete form reads
ns=4m2
¯h2βIcl hA2
zi− hAzi2, (50)
where
Az
is the component of the total area enclosed by particle paths on the
xy
plane,
defined as
Az=1
2
N
∑
i=1
M−1
∑
m=0rm
i×rm+1
iz, (51)
with
rm
i
, the position of the bead corresponding to the
i
-th particle on the
m
-th time slice,
the same as in Section 2, whereas the classical moment of inertia in Equation
(50)
is
computed as
Icl =mN
∑
i=1
M−1
∑
m=0
rm
i·rm+1
i. (52)

Entropy 2022,24, 265 13 of 18
Usually, the
hAzi
term is set to zero, or ignored altogether [
45
]. The most convincing
argument is that, from an energetic point-of view, and therefore as far as equilibrium
probabilities are concerned, any configuration is equivalent to a symmetric one where the
directions of all links between nearest-neighbor beads have been reversed; if both configu-
rations can be accessed, they should be visited with equal probability. Since reversing all
links also changes the sign in the area corresponding to the configuration, the immediate
consequence is that the average value of the area will be zero, leading to
hAzi=
0. However,
in configurations where bosons are localized into clusters (such as the deepest minima of
the quasiperiodic optical lattices, e.g., see Figure 2), this term does not necessarily average
to zero, and it must be kept into account. This observation is justified by the fact that the
system might spend a long period of time (compared to simulation times) in a region of
configuration space where hAzi 6=0, leading to a manifestation of ergodicity breaking.
In Figure 7, we show plots of
ns
as a function of
V0
, at different values of temperature
and interaction strength. In all cases, deeper quasiperiodic lattices bring about a reduction
of the global superfluidity, up to a critical value at which it is completely depleted, and the
system transitions to a localized phase.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
V
0/
Er
0.0
0.2
0.4
0.6
0.8
1.0
ns
T
/
Tc
= 0.1
T
/
Tc
= 0.25
T
/
Tc
= 0.4
T
/
Tc
= 0.7
g
0= 0
g
0= 0.0217
g
0= 2.1704
Figure 7.
Measured values of the global superfluid fraction at different values of
V0
. Colored
markers correspond to different values of the interaction:
˜
g0=
0 (teal circles),
˜
g=
0.0217 (black
triangles),
˜
g0=
2.1704 (green diamonds). Lines are a guide for the eye, and also serve to distinguish
different temperatures:
T/Tc=
0.1 (solid lines),
T/Tc=
0.25 (dashes),
T/Tc=
0.4 (dashes and dots),
T/Tc=0.7 (dots).
6. Zonal Superfluid Estimators
In dealing with inhomogeneous systems, it also worthwhile to extrapolate superfluid
features that may be spatially dependent, introducing density fields
ρ(r)
,
ρs(r)
, and
ρn(r)
,
which extend the uniform quantities introduced above. We can then combine Equation
(48)
and Equation
(49)
, and the definition of the classical moment of inertia
Icl =Rdrρ(r)r2
to write
ns=1
Icl Zdrρs(r)r2, (53)
with
r
representing the distance from the center of the coordinate system. The definition
of a local superfluid fraction follows from
(48)
, as
ns(r) = ρs(r)/ρ(r)
; with this, we can
rewrite (53) as

Entropy 2022,24, 265 14 of 18
1
Icl Zdrns(r)ρ(r)r2, (54)
meaning that the global superfluid fraction is an average of the local superfluid fraction,
weighted by the local moment of inertia. While the introduction of inhomogeneous density
fields is natural, their expression in terms of of the classical polymers requires some
elaboration; following Kwon et al. [
63
], who have introduced a physically motivated and
consistent definition of ρs(r), we write
ρs(r) = 4m2
¯h2βhAzAz(r)i− hAzihAz(r)i
r2, (55)
where
Az(r) = 1
2
N
∑
i=1
M−1
∑
m=0
r×rm+1
iδ(r−rm
i). (56)
Equations
(53)
,
(55)
and
(56)
allow us to investigate the superfluid behavior locally,
usually by sampling
(55)
on a grid. However, and especially for strongly inhomogeneous
systems such as
(37)
, the amount of detail proves to be excessive, and the estimators too
noisy. In order to overcome this issue, and to extract some degree of spatial information
about the system, we act on a middle level by introducing a zonal estimator.
We divide the system into
K
regions, labeled by
k=
1, 2
. . . K
, and introduce a zonal
superfluid fraction by specializing (53) to
ns,k=1
Icl,kZkdrρs(r)r2, (57)
where
Icl,k
is the fraction of the total classical moment of inertia corresponding to region
k
, so that
Icl =∑K
k=1Icl,k
. The zonal superfluid fractions can be recombined, using
(54)
,
to give
ns=
K
∑
k=1
Icl,k
Icl
ns,k. (58)
In the present case, we exploit the circular symmetry provided by the trap to divide
the system into three concentric belts. It is important to stress that, for instance, the decom-
position
(58)
may display sectors with a finite superfluid fraction, but still give a negligible
contribution to the global
ns
, if the associated moment of inertia is small. This is the case
for regions close to the trap center, as we show in Figure 8, which displays the superfluid
fraction ns,kin different regions against temperature.
Most familiar is the behavior in the case of
V0
, in Figure 8a, where we can see the
global superfluidity drop from nearly
ns=
1 at low temperature to
ns=
0 at the critical
temperature
Tc
. The depletion of superfluidity proceeds at different rates across the trap:
the inner regions display a value of
ns,k
, which is still close to 1, even at high values of
T
,
such as
T/Tc=
0.7. The global superfluidity, however, is dominated by the contribution of
bosons in the outer region, due to their larger classical moment of inertia.
As
V0
increases, the effects of the quasiperiodic lattice become prominent; the global
superfluidity is depleted by localization, as already shown in Figure 7, but, for some values
of
V0
, the zonal superfluidity remains finite in the central region. In [
34
], we used this
information, coupled with the fluctuations in particle number, to characterize a Bose Glass
phase induced by the quasiperiodic potential.

Entropy 2022,24, 265 15 of 18
Figure 8.
Temperature behavior of the superfluid fraction in different regions, at
˜
g0=
0.0217.
The regions are depicted on the right, where they are superimposed to the density profiles of Figure 4,
for
V0/Er=
0.0 (
e
) and
V0/Er=
1.5 (
f
). The inner radius is
ra
(purple line) and the outer radius is
rb
(red line). Colored markers in the plots (
a
–
d
) correspond to values of
ns,k
measured in different
regions:
r<ra
(blue circles),
ra<r<rb
(purple squares),
r>rb
(red diamonds). The same
colors are used to shade the regions in (
e
,
f
). We also report the global superfluid fraction, which
was already displayed in Figure 7(black triangles). The four plots each correspond to a different
value of
V0
:
V0/Er=
0.0 (
top left
),
V0/Er=
1.5 (
top right
),
V0/Er=
2.5 (
bottom left
),
V0/Er=
3.5
(bottom right).
7. Discussion and Conclusions
The present work we detailed PIMC methods to explore both local and global quan-
tum properties of interacting bosons confined in external quasiperiodic potential. Those
properties have been probed in a finite temperature regime with specific attention to super-
fluidity. In detail, we summarized the derivation of the pair-product approximation for
particles interacting through hard-core interactions in two or three dimensions, and we
presented the details of our implementation. Then, we explored a new zonal estimator,
which gives access to local information about the superfluidity in finite regions of trapped
systems, and it is therefore well-suited to the study of spatially inhomogeneous potentials.
For the example presented in this work, zonal estimators are relevant to the detection of
correlated phases, such as the Bose glass phase, which is characterized by rare regions
where superfluidity and finite compressibility coexist [
34
]. Similar zonal estimators can
be applied to other quantities, such as regional fluctuations of particle number, associated
with density compressibility, and magnetic ordering in multi-component systems or spin
systems related to the spin compressibility.
Moreover, one might apply such estimators to the characterization of local properties
of self-assembled quasicrystalline phases in free space generated by two-body nonlocal
interactions [
64
]. A prime example of a two-body model potential leading to quasiperi-
odic patterns is in the paradigmatic hard-soft corona potential, which is largely used to
investigate purely classical systems [
65
]. The same model has also been applied to bosonic
systems, where the effects of zero point motion, as well as quantum exchanges, disclose
rich phase diagrams including quantum quasicrystal with 12-fold rotational symmetry [
17
].
Additionally, quantum properties of self-assembled cluster quasicrystals revealed that,
in some cases, quantum fluctuations do not jeopardize dodecagonal structures, showing a
small but finite local superfluidity [
19
]. Cluster quasicrystals display peculiar features, not
exhibited by simpler quasiperiodic structures. By increasing quantum fluctuations, in fact,
a structural transition from quasicrystal to cluster triangular crystal featuring the properties

Entropy 2022,24, 265 16 of 18
of a supersolid is observed [
19
,
66
,
67
]. We point out that the discussed methodology is also
useful to analyze the superfluid character of further peculiar inhomogeneous systems such
as, for instance, bosons enclosed within spherical traps or subject to a polyhedral-symmetric
substrate potential [68–70].
Author Contributions:
Conceptualization, M.C., T.M. and F.C.; Data curation, M.C., T.M. and F.C.;
Formal analysis, M.C., T.M. and F.C.; Investigation, M.C., T.M. and F.C.; Methodology, M.C., T.M. and
F.C.; Writing—original draft, M.C., T.M. and F.C. All authors have read and agreed to the published
version of the manuscript.
Funding:
T.M. acknowledges CNPq for support through Bolsa de produtividade em Pesquisa
n.311079/2015-6 and support from CAPES. This work was supported by the Serrapilheira Institute
(grant number Serra-1812-27802).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Acknowledgments:
We acknowledge V. Zampronio for discussions and for a careful reading of the
manuscript. T.M. acknowledges the hospitality of ITAMP-Harvard where part of this work was
done. We thank the High Performance Computing Center (NPAD) at UFRN and the Center for High
Performance Computing (CHPC) in Cape Town for providing computational resources.
Conflicts of Interest: The authors declare no conflict of interest.
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