Classical and Quantum Gases on a Semiregular Mesh

Abstract

The main objective of a statistical mechanical calculation is drawing the phase diagram of a many-body system. In this respect, discrete systems offer the clear advantage over continuum systems of an easier enumeration of microstates, though at the cost of added abstraction. With this in mind, we examine a system of particles living on the vertices of the (biscribed) pentakis dodecahedron, using different couplings for first and second neighbor particles to induce a competition between icosahedral and dodecahedral orders. After working out the phases of the model at zero temperature, we carry out Metropolis Monte Carlo simulations at finite temperature, highlighting the existence of smooth transitions between distinct “phases”. The sharpest of these crossovers are characterized by hysteretic behavior near zero temperature, which reveals a bottleneck issue for Metropolis dynamics in state space. Next, we introduce the quantum (Bose-Hubbard) counterpart of the previous model and calculate its phase diagram at zero and finite temperatures using the decoupling approximation. We thus uncover, in addition to Mott insulating “solids”, also the existence of supersolid “phases” which progressively shrink as the system is heated up. We argue that a quantum system of the kind described here can be realized with programmable holographic optical tweezers.

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applied
sciences
Article
Classical and Quantum Gases on a Semiregular Mesh
Davide De Gregorio †and Santi Prestipino *,†


Citation: De Gregorio, D.; Prestipino,
S. Classical and Quantum Gases on a
Semiregular Mesh. Appl. Sci. 2021,11,
10053. https://doi.org/ 10.3390/
app112110053
Academic Editor: Alberto Milani
Received: 8 September 2021
Accepted: 25 October 2021
Published: 27 October 2021
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Copyright: © 2021 by the authors.
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Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Dipartimento di Scienze Matematiche ed Informatiche, Scienze Fisiche e Scienze della Terra, Università degli
Studi di Messina, Viale F. Stagno d’Alcontres 31, 98166 Messina, Italy; davide.degregorio@studenti.unime.it
*Correspondence: sprestipino@unime.it
† These authors contributed equally to this work.
Abstract:
The main objective of a statistical mechanical calculation is drawing the phase diagram of a
many-body system. In this respect, discrete systems offer the clear advantage over continuum systems
of an easier enumeration of microstates, though at the cost of added abstraction. With this in mind,
we examine a system of particles living on the vertices of the (biscribed) pentakis dodecahedron,
using different couplings for first and second neighbor particles to induce a competition between
icosahedral and dodecahedral orders. After working out the phases of the model at zero temperature,
we carry out Metropolis Monte Carlo simulations at finite temperature, highlighting the existence of
smooth transitions between distinct “phases”. The sharpest of these crossovers are characterized by
hysteretic behavior near zero temperature, which reveals a bottleneck issue for Metropolis dynamics
in state space. Next, we introduce the quantum (Bose-Hubbard) counterpart of the previous model
and calculate its phase diagram at zero and finite temperatures using the decoupling approximation.
We thus uncover, in addition to Mott insulating “solids”, also the existence of supersolid “phases”
which progressively shrink as the system is heated up. We argue that a quantum system of the kind
described here can be realized with programmable holographic optical tweezers.
Keywords:
lattice-gas models; spherical boundary conditions; ultracold quantum gases; decoupling
approximation; supersolid phases
1. Introduction
Investigating the behavior of a many-particle system has an undeniable charm: despite
microscopic interactions being undirected, various forms of self-organization (“order”) can
develop at the macroscale. In the last century, countless examples of emergent order have
been described, each with its own practical realization, and many more can be devised by
exploring through theory physical situations that are somehow atypical. These indications
can stimulate new experimental work or simply be aimed to clarify and expand the scope
of the theory itself.
A way to produce novel, unconventional phase behaviors is to consider many-body
systems under geometric constraints, since local interactions are frustrated and unusual
ground states then appear. A classic example is a (finite) system of hard particles confined
in the surface of a sphere [
1
]. The sphere topology forces an excess of five-fold coordinated
particles over seven-fold ones, leading to high-density packings with defects [
2
–
8
]. We note
that bosonic atoms confined in thin spherical shells [
9
] have already been realized [
10
,
11
]
and are currently studied in microgravity [
12
,
13
]. In other cases, frustration is directly
embodied in the interaction law—like in spin glasses or in antiferromagnets on a triangular
lattice [14,15].
In this paper, we consider a discrete system of particles (“lattice gas”) on a spherical
mesh of points, which is chosen such that a rich interplay arises between distinct “phases”
having the symmetries of a Platonic solid. Clearly, on a finite mesh well-definite phases
only exist at zero temperature (
T=
0), since for
T>
0 any phase transition will be smeared
Appl. Sci. 2021,11, 10053. https://doi.org/10.3390/app112110053 https://www.mdpi.com/journal/applsci

Appl. Sci. 2021,11, 10053 2 of 21
out, i.e., replaced by a smooth crossover region. Using a finite mesh, we greatly reduce the
computational effort without however making the phase behavior trivial.
If a toy model of classical particles on a finite mesh may look somewhat artificial and
hardly corresponds to a real-world system, its quantum counterpart might be different.
The last decades have witnessed a considerable progress in the manipulation of quantum
atoms at low temperature, opening the way to a systematic study of correlation effects in
many-body systems [
16
–
19
]. While optical lattices [
20
] are routinely employed in numerous
laboratories worldwide as a tool for confinement of quantum atoms, in the last few years
a laser technology has been invented, based on the use of optical tweezers [
21
,
22
], which
allows virtually any type of structure (not necessarily a lattice) to be realized with cold
atoms. We are thus encouraged to consider the quantum (Bose-Hubbard) counterpart of
the lattice gas on a spherical mesh, with the explicit purpose to compare their thermal
behaviors. In particular, we devise a quantum variational theory that predicts supersolid
phases and returns the results of a classical mean-field theory when quantum tunneling
is precluded.
The rest of the paper is organized as follows. In Section2we introduce our model
and the methods used to investigate its phase behavior. Next, we present our results, first
at zero temperature (Section 2.1) and then at finite temperatures (Section 2.2). In
Section 3
we deal with the quantum extension of the model in Section 2. Using the decoupling
approximation, we not only work out the ground-state diagram (Section 3.1) but also a
few finite-temperature properties (Section 3.2). Lastly, we give our concluding remarks in
Section 4.
2. Lattice-Gas Models on a Spherical Mesh
As anticipated in the Introduction, we hereafter explore the possibility of unusual
orderings in a system of particles occupying the nodes of a spherical mesh, chosen to be
sufficiently regular so that some polyhedral (Platonic) arrangement can occur.
We focus our attention on the pentakis dodecahedron [
23
] (PD, see Figure 1), a Catalan
solid with 32 vertices obtained by augmenting the dodecahedron with 12 right pyramids
on its pentagonal faces, in such a way that the resulting polyhedron is dual to the truncated
icosahedron (clearly, the pyramidal apices form the vertices of an icosahedron). Even
though the PD is not inscribable, implying that no spherical mesh can be drawn from its
vertices, it is straightforward to obtain a biscribed solid by a small distortion of the PD that
preserves its connectivity properties and its full icosahedral symmetry. A biscribed solid is
any convex polyhedron that has concentric circumscribed and inscribed spheres, where
the sphere center is also the centroid of the vertices. The five Platonic solids are biscribed
solids, but none of the Archimedean or Catalan solids are. As for the PD, it suffices to
adjust the height of the pentagonal pyramids only slightly to force all the vertices to be on
the same sphere. The outcome of this construction is the biscribed form of the PD. It is this
variant of the PD that is considered hereafter.
The PD has 60 faces (isosceles triangles) and 90 edges (60 short and 30 long). We
call PD mesh the skeleton of the PD, i.e., the mesh formed by its edges. The PD mesh is
the finite analog of a lattice; the nodes of the mesh (i.e., the PD vertices) are its “sites”.
While five edges depart from an icosahedral vertex/site, the number of edges departing
from a dodecahedral vertex/site is six (in this sense, icosahedral and dodecahedral sites
are “inequivalent”). Setting the circumscribed radius equal to 1, the short-edge length
(i.e., the shortest distance in the mesh) is
`1=r3015 −q15(5+2√5)/
15
'
0.64085
. . .
,
whereas the long-edge length (the second shortest distance in the mesh) is
`2=√15 −
√3/3 '0.71364 . . .

Appl. Sci. 2021,11, 10053 3 of 21
Figure 1.
The pentakis dodecahedron has 12 icosahedral vertices (red dots) and 20 dodecahedral
vertices (yellow dots). There are five distinct ways to choose eight yellow dots forming a cube—then,
the other 12 dodecahedral vertices are said to form a “co-cube”. The short edges of the PD mesh
are colored in grey and the long edges in blue. The couplings entering the model Hamiltonian (1)
are indicated.
The phase behavior of a lattice-gas model on the PD mesh is better studied in the grand-
canonical ensemble. Upon increasing the chemical potential
µ
at fixed
T
, the mesh becomes
increasingly populated, with the possibility of “transitions” between qualitatively distinct
arrangements. Denoting
ci=
0, 1 the occupation number of site
i
, we call first (second)
neighbors any two sites/particles that are linked by a short (long) edge. According to
our nomenclature, an icosahedral site has five first-neighbor sites and no second-neighbor
site, whereas a dodecahedral site has three first neighbors and three second neighbors (see
Figure 1). With these specifications, the grand Hamiltonian of our model reads:
H=V1∑
hi,ji
cicj+V2∑
hhk,lii
ckcl−µ∑
i
ci, (1)
where
V1>
0 (
V2
) is the coupling between two first (second) neighbor particles. This model
can mimic a system of particles adsorbed on a “substrate” sculpted like a PD mesh (think
of, e.g., the interstices between atoms in a C
60
molecule), and interacting via a spherically-
symmetric potential with a hard core followed, at larger distances, by a soft short-range
repulsion. By suitably tuning the ratio
γ=V2/V1
between the couplings, we anticipate
the existence of a competition between icosahedral order and dodecahedral order at low
temperature, with the further possibility of arrangements with intermediate order.
In the following, we use
V1
as the unit of energy; in turn, this defines a reduced
temperature,
T∗=kBT/V1
(
kB
being the Boltzmann constant), and a reduced chemical
potential, µ∗=µ/V1.
2.1. Zero-Temperature Phases
At zero temperature, the stable phase at fixed
µ
is the one minimizing the grand
potential
Ω
. We expect the absolute minimum
Ω
to be reached in one of a few mi-
crostates/configurations, chosen among those exhibiting a homogeneous occupancy of
equivalent sites. To be clear, there are five ways to select—out of 20 dodecahedral sites—8
sites forming a cube [
24
] (similarly, each cube is the union of 2 tetrahedra). Then, the PD
vertices are naturally grouped in three sets of equivalent nodes: we call A the set of

Appl. Sci. 2021,11, 10053 4 of 21
icosahedral sites, B any set of cubic sites, and C the set comprising the remaining do-
decahedral sites (“co-cubic” sites). For example, in the icosahedral phase (ICO) only A
sites are occupied at
T=
0; in the dodecahedral phase (DOD) only B and C sites are
occupied. In our setting, ICO and DOD play a role analogous to two distinct crystalline
phases. By a straightforward count of neighbors, the grand potential of the most rele-
vant phases is readily calculated: in an obvious notation,
Ωempty =
0,
ΩICO =−
12
µ
,
ΩCOC =
6
V2−
12
µ
,
ΩDOD =
30
V2−
20
µ
, and
Ωfull =
60
V1+
30
V2−
32
µ
. The 10-fold-
degenerate tetrahedral phase (TET,
Ω=−
4
µ
) and the 5-fold-degenerate cubic phase
(CUB,
Ω=−
8
µ
) are never stable, since they are less stable than “empty” or ICO. Further
possibilities are configurations where, in addition to icosahedral sites, also a selection
of dodecahedral sites are occupied:
ΩICO+TET =
12
V1−
16
µ
,
ΩICO+CUB =
24
V1−
20
µ
,
and ΩICO+COC =36V1+6V2−24µ.
By a lengthy (though elementary) calculation, we arrive at the
T=
0 phase diagram
depicted in Figure 2. Here, PQ, PR, and RS are straight lines with equations
µ∗=
3
+
3
γ
,
µ∗=
9
−
6
γ
, and
µ∗=
3
+ (
3
/
2
)γ
, respectively; the two half-lines departing from the
origin, namely OT and OU, have equations
µ∗= (
15
/
4
)γ
and
µ∗= (
3
/
2
)γ
, respectively.
All the phase boundaries are first-order (since the number of particles changes discontinu-
ously from one phase to the other), except for RT (since both the particle number
N
and
the energy
E
change continuously through it). Looking at Figure 2, we note that: (i) the
ICO phase is only stable for
γ>
0, in a region delimited by the
µ∗=
0 and
µ∗=
3 lines;
(ii) the DOD phase is stable in a wide region of the
γ
-
µ
plane, bounded from the right
by
γ=
4
/
5 and from the above by
µ∗=
5; (iii) ICO+CUB and ICO+COC are each stable
in an unbounded set of positive
γ
values, and coexist along the RS line; (iv) for each
γ
,
“empty” and “full” are respectively stable for all values of
µ
that are sufficiently small or
sufficiently large.
µ
/V1
γ
-2
0
2
4
6
-1 0 1
e m p t y
f u l l
DOD
ICO
ICO+CUB
ICO+COC
P
Q
R
S
T
O
U
Figure 2.
Zero-temperature phase diagram of the lattice gas on a PD mesh. The phase boundaries
are reported in the main text.

Appl. Sci. 2021,11, 10053 5 of 21
2.2. Finite-Temperature Behavior
For
T>
0 the study of model (1) cannot be fully analytical since the number of
microstates, 2
32
, is huge. Thus, we have resorted to grand-canonical Monte Carlo (MC)
simulation for the computation of thermal averages. We employ the standard Metropolis
algorithm with single-site moves: at each step of the Markov chain, a move is attempted by
changing the state of a randomly selected site
j
from
cj
to 1
−cj
. Calling
∆H
the virtual
change in
H
, the move is accepted according to a probability given by
min{
1,
exp(−β∆H)}
with
β= (kBT)−1
, as usual. Once equilibrium has been reached, statistical averages are
computed over no less than five million MC cycles (one cycle consisting of 32 trial moves).
We monitor a number of equilibrium properties as a function of
µ
: the number of
particles (
N=h∑icii
) and the energy (
E=hHi+µN
), together with their self- and
cross-correlations; the reduced isothermal compressibility,
ρkBTKT=hδN2i
Nwith ρ=N/32 and δN=∑
i
ci−N; (2)
and two specific heats, namely
Cµ=T
N
∂S
∂TV,µ
and CN=T
N
∂S
∂TV,N
(3)
(
S
being the entropy), expressed in terms of grand-canonical averages through
Equations (A6)
and (A15) of Appendix A. In addition, we also determine a number of order parameters
(OPs, see Appendix Bfor a definition), in order to establish the nature of the system “phase”
and the crossover behavior at its boundaries.
For the sake of illustration, take γ=1/2. For T=0 the succession of phases is
empty 0
−→ ICO 1.875
−→ DOD 5
−→ full . (4)
We explore the phase behavior of this model at five temperatures,
T∗=
0.1, 0.2,
. . .
0.5,
and in the
µ∗
range from
−
2 to 7. To account for the possibility of hysteresis, we carry
out our simulation runs in sequence: for a given value of
µ∗
, the run is started from the
last configuration generated in the previous run at a slightly larger or smaller
µ∗
. Our
results are collected in Figures3and 4. In the former figure, we plot
N
,
E
,
OICO
, and
ODOD
as a function of
µ∗
. As
T
progressively grows, icosahedral and dodecahedral orders
become increasingly weakened, and the crossover region between distinct “phases” gets
wider and wider. An interesting behavior occurs for
T∗=
0.1, where we observe a large
hysteresis loop near the ICO-DOD “transition”. In other words, despite the absence of
sharp transitions for
T>
0, at sufficiently low temperature the order can be so robust that
we observe hysteresis—the most characteristic feature of a first-order transition. This is the
clue to insufficient sampling of the equilibrium distribution, which can be cured by either
substantially increasing the length of the MC trajectory or changing the MC algorithm (see
more below).
Curiously enough, hysteresis is only found for one of the three transitions present at
T=
0. To clarify this point, it is instructive to compute the acceptance probability of the
Metropolis move(s) that initially drive the system out of one phase into another. The crucial
quantity to look at is the ensuing variation in
H
, i.e.,
∆H=∆E−µ∆N
, with
µ
given by the
transition-point value.

Appl. Sci. 2021,11, 10053 6 of 21
Figure 3.
Lattice gas on a PD mesh, for
γ=
1
/
2 and
T∗=
0.1, 0.2,
. . .
, 0.5.
Top left
: number of
particles.
Top right
: reduced energy.
Bottom left
: icosahedral OP.
Bottom right
: dodecahedral OP.
For the lowest temperatures, MC data refer to two distinct sequences of runs where
µ∗
is respectively
increased or decreased in steps of 0.1 (
T∗=
0.1, blue and cyan dots;
T∗=
0.2, emerald and green;
T∗=
0.3, red and pink;
T∗=
0.4, brown; and
T∗=
0.5, black). For all temperatures but the lowest
one, MC data are reported as lines. Hysteresis is evident for
T∗=
0.1 and barely visible already for
T∗=0.2.
Starting from the empty mesh at
µ=
0, the repeated addition of particles in icosahedral
sites occurs with probability 1, since at each step
∆E=
0 and
∆N=
1. If we start from ICO,
the repeated removal of particles again occurs with probability 1, since at each step
∆E=
0
and
∆N=−
1. This means that for
µ=
0 the system has equal probability to be in either
of the two phases. Hence, no hysteresis would be observed at the empty-ICO transition,
as indeed found.
Now consider the ICO-DOD transition at
µ∗=
15
/
8. Starting from ICO, the cost to
annihilate a particle is
∆H∗=
15
/
8 (since
∆E=
0); the next step of creating a particle in a
dodecahedral site has a minimum cost of
∆H∗=
31
/
8 (since
∆E∗≥
2). Thus, for
µ∗=
15
/
8
there is a free-energy barrier for the transition to DOD, and the system then remains for
long in ICO notwithstanding DOD is more stable. If we instead start from DOD, we should
first annihilate a particle (
∆H∗=
3
/
8) and then create another particle in an icosahedral
site (
∆H∗≥
17
/
8). Again, the transition from DOD to ICO is discouraged at low
T
(though
apparently less so than the opposite transition), meaning that for µ∗=15/8 the system is
more likely to be found in DOD than in ICO. As a result, hysteresis will be observed at the
ICO-DOD transition.

Appl. Sci. 2021,11, 10053 7 of 21
Figure 4.
Lattice gas on a PD mesh, for
γ=
1
/
2 and
T∗=
0.1, 0.2,
. . .
, 0.5.
Top
: reduced compress-
ibility.
Bottom left
: Constant-
µ
specific heat.
Bottom right
: Constant-
N
specific heat. Symbols and
notation as in Figure 3.
Lastly, we consider the DOD-full transition at
µ∗=
5. If we start from DOD, the first
step towards “full” is creating a particle in an icosahedral site (
∆E∗=
5) and the probability
for this move is one. If we instead start from “full”, the cost for annihilating a particle in
an icosahedral site is zero again, since
∆E∗=−
5. Indeed, no hysteresis is detected at the
DOD-full transition.
In Figure 4we plot a few response functions for
γ=
1
/
2. At the lowest temperatures
all these quantities exhibit a distinct peak near each
T=
0 transition point, which is where
the energy and particle number are subject to the sharpest variations. In the “empty” phase,
the reduced isothermal compressibility is close to 1 (the ideal-gas value); upon heating,
every asperity in its profile becomes smoothened until
ρKT
becomes a monotonously
decreasing function of
µ
. As
T
grows, both specific heats develop a broad maximum inside
the ICO and DOD regions. Admittedly, these are the locations where, in the moderately
hot system, the fluctuations of energy and particle number are stronger. Eventually, both
maxima gradually deflate, the DOD bump being the last to disappear. Finally notice
that inside the “empty” region the two specific heats have different behaviors at low
temperature: while the covariances involving energy are practically zero, the mean square
fluctuation of
N
is of order
N
(see Equation (2)). Looking at Equations (A6) and (A15), we
thus conclude that Cµ/kB≈(βµ)21 and CN≈0.
As a second example, consider
γ=
1 and
T∗=
0.05 close to
µ∗=
9
/
2—which
is where, at
T=
0, ICO+CUB transforms into ICO+COC. By the same argument put
forward before, we would conclude that this transition is accompanied by hysteresis at
low temperature. However, when plotting
N
as a function of
µ
, in a small neighborhood
of
µ∗=
9
/
2 we see a narrow plateau in the middle between the
N
levels in the two
phases (top-left panel of Figure 5), and a similar occurrence is found for
E
(not shown).
This unexpected outcome would suggest the existence of a yet undetected phase near
µ∗=
9
/
2, characterized by
N=
22 and
E∗=
33. In fact, when the length of the MC
trajectory is increased by a factor of 10 the plateau at
N∗=
22 changes to a more or less
smooth crossover between the phases (top-right panel of Figure 5). To clarify things better,

Appl. Sci. 2021,11, 10053 8 of 21
we have enumerated the microstates with all icosahedral sites occupied, thus confirming
that ICO+CUB (ICO+COC) is the stable phase for
µ∗<
9
/
2 (
µ∗>
9
/
2); instead, exactly
for
µ∗=
9
/
2 we have counted as many as 240 distinct microstates, all with
N=
22 and
E∗=
33, having the same grand potential as ICO+CUB and ICO+COC. It is the existence of
such configurations that makes the transition between ICO+CUB and ICO+COC smoother
than expected. We care to stress that this occurrence is rather exceptional; usually, phases
are well separated in free energy and compete with each other only in pairs.
Figure 5.
Lattice gas on a PD mesh, for
γ=
1 and
T∗=
0.05:
N
vs.
µ
across the transition between
ICO+CUB and ICO+COC. We plot data from two sequences of runs, ascending (blue) and descending
(cyan). The
top left
and
top right
panels differ for the number of equilibrium cycles performed
in each run, which is 5
×
10
6
and 5
×
10
7
, respectively.
Bottom
panels: OPs for ICO+CUB and
ICO+COC, for the simulation with 5
×
10
7
equilibrium cycles per run. We see a narrow interval of
µ∗
values around 9/2 where the order is neither ICO+CUB nor ICO+COC.
As anticipated, hysteresis is an annoying problem due to the inadequacy of Metropolis
dynamics to overcome free-energy barriers. To solve this issue we have employed the
Wang-Landau algorithm [
25
], which directly computes the density of states and is thus
particularly suited for a free-energy landscape with multiple minima. Compared to the
original algorithm, the refinement parameter ln fwas reduced at a slower rate (at regular
intervals,
ln f
is divided by 1.1 rather than 2), which greatly reduces the (already small)
saturation error [
26
]. We report results for
γ=
1
/
2 in Figures 6and 7. In the former
figure,
N
and
E∗
are plotted as a function of
µ
across the ICO-DOD transition; we see
that Metropolis sampling is indeed adequate for
T∗=
0.3 (notice, in particular, how the
Wang-Landau data carefully interpolate the Metropolis data in the low-
µ
region, where
a small bump is present in the energy). At lower temperatures, only the Wang-Landau
simulation is unaffected by hysteresis. For completeness, for
T∗=
0.2 we plot in Figure 7
the density of states
g
as a function of
N
and
U
(i.e., the sum of the first two terms on the
r.h.s. of (1)) and the probability density of the particle number,
P(N)
. In the ICO region,
well before the transition at
µ∗=
15
/
8, a second peak builds up in
P(N)
, which, as
µ
is

Appl. Sci. 2021,11, 10053 9 of 21
increased, is gradually shifted to larger and larger
N
values until becoming centered at
N=20.
Figure 6.
MC results for
γ=
1
/
2 and
T∗=
0.1, 0.2, and 0.3. The colored points and lines are the
same MC data shown in Figure 3(
T∗=
0.1, blue and cyan dots;
T∗=
0.2, emerald and green squares;
T∗=
0.3, red and pink). The superimposed black lines are the outcome of a Wang-Landau simulation.
(Left): average number of particles. (Right): average energy.
Figure 7.
Wang-Landau simulation for
γ=
1
/
2 and
T∗=
0.2. (
Left
): logarithm of the density of
states
g
, defined in terms of
N
and
U
(see text). The four points where
g=
1 correspond to the
T=
0
phases: “empty”, perfect ICO, perfect DOD, and “full”. (
Right
): probability function of the particle
number N, plotted as a function of the reduced chemical potential.
Finally, it is useful to compare MC results with the outcome of a mean-field (MF)
theory. The simplest approach is to estimate the grand potential of (1) using the Gibbs-
Bogoliubov (GB) inequality with a trial probability density
π[c]
given as an uncorrelated
product of one-site terms:
π[c] =
32
∏
i=1
π(1)
i(ci), (5)
with
π(1)
i(c) = 


π(c;ρA),i∈A
π(c;ρB),i∈B
π(c;ρC),i∈C .
(6)
In the previous equation,
π(c;ρA) = ρAδc,1 + (1−ρA)δc,0, (7)
with 0
≤ρA≤
1, and similarly for B and C. The rationale behind Equation (6) is that the
average occupancy takes a possibly different value in each set of equivalent nodes, being
ρAfor the icosahedral set, ρBfor the cubic set, and ρCfor the co-cubic set.
An upper bound to the exact grand potential Ωis the GB grand potential Ω∗,
Ω∗=hHi+kBThln πi, (8)

Appl. Sci. 2021,11, 10053 10 of 21
where hO[c]i=∑{c}π[c]O[c]. Then, it is a simple matter to show that
Ω∗=24V1ρAρB+36V1ρAρC+24γV1ρBρC+6γV1ρ2
C−12µρA−8µρB−12µρC
+12kBT[ρAln ρA+ (1−ρA)ln(1−ρA)]+8kBT[ρBln ρB+ (1−ρB)ln(1−ρB)]
+12kBT[ρCln ρC+ (1−ρC)ln(1−ρC)].(9)
Observe that the value of
Ω∗
in the putative ground states listed in Section 2.1 exactly
reproduces their respective grand potentials. The stationary values of (9) fulfill the cou-
pled equations
ρA=1
eβ(2V1ρB+3V1ρC−µ)+1,ρB=1
eβ(3V1ρA+3γV1ρC−µ)+1, and
ρC=1
eβ(3V1ρA+2γV1ρB+γV1ρC−µ)+1. (10)
These equations are solved numerically, seeking the solution that provides the absolute
minimum Ω∗for the given Tand µ.
To have a flavor of how MF theory works, we draw in Figure 8the theoretical phase
diagram on the
µ
-
T
plane for
γ=
2
/
3, corresponding to a lattice gas where the icosahedral
phase covers a
µ
range as wide as that of the dodecahedral phase, see Figure 2. Owing to
a symmetry property of Equation (10), the phase diagram is symmetric around
µ=
5
/
2
(see also Section 3.2). At
T=
0 the phase boundaries in Figure 8are exact. As
T
grows,
the theory predicts a gradual weakening of ICO and DOD orders, as witnessed by the
decrease of
|ρA−ρB|
on heating, eventually resulting in an abrupt (first-order) transition
to either “empty” (
ρA=ρB=ρC<
0.5) or “full” (
ρA=ρB=ρC>
0.5), according to
whether
µ<
5
/
2 or
µ>
5
/
2. Clearly, this singularity is an artifact of MF theory, since
no sharp transition is present in our system for
T>
0. Another unphysical prediction
of the theory concerns the behavior of the model in a narrow strip of temperatures and
chemical potentials near
µ∗=
0 and
µ∗=
5. In Figure 8we have denoted ICO
0
a phase
where 0.5
>ρA>ρB=ρC
and DOD
0
a phase where 0.5
<ρA<ρB=ρC
. These two
phases have no counterpart in the simulation, hence they are just an unwanted outcome of
MF theory.
kBT/V1
µ
/V1
0
0.5
1
-1 0 1 2 3 4 5 6
e m p t y f u l l
DODICO
ICO’ DOD’
Figure 8.
Phase diagram of the lattice gas on a PD mesh as predicted by MF theory, for
γ=
2
/
3. All
phase boundaries are first-order lines, see more in the text.

Appl. Sci. 2021,11, 10053 11 of 21
3. Hard-Core Bosons on a Spherical Mesh: Extended Bose-Hubbard Model
The PD mesh is regular enough that we can study the quantum analog of the lattice-gas
model in relatively simple terms. The obvious bosonic counterpart of (1) is the hard-core
limit of the extended Bose-Hubbard (BH) model
H=−t∑
hi,jia†
iaj+a†
jai−t∑
hhk,liia†
kal+a†
lak
+U
2∑
i
ni(ni−1) + V1∑
hi,ji
ninj+V2∑
hhk,lii
nknl−µ∑
i
ni, (11)
where
ai
,
a†
i
are bosonic field operators and
ni=a†
iai
is a number operator. Moreover,
t≥
0 is the hopping amplitude, taken for simplicity to be the same for first- and second-
neighbor pairs, whereas
U>
0 is the on-site repulsion. In the hard-core limit
U→+∞
,
the site occupancies are effectively restricted to zero or one and the
U
term can thus be
discarded. For hard-core bosons, creation and annihilation operators at different sites
commute, whereas
ai
and
a†
i
are anticommuting operators as a result of the dynamical
suppression of Fock states with two or more particles in the same site [27].
In the original BH model [
28
–
30
], where the second, fourth, and fifth terms in
H
are
absent, the tunneling term (kinetic energy) is minimized by a condensed state spread over
the entire volume of the system, whereas the potential energy favors particle localization.
As a result, the
T=
0 system exists as either a superfluid (large
t/U
) or a Mott insulating
ground state (small
t/U
), separated by a quantum transition. The Bose-Hubbard Hamilto-
nian can be derived starting from the second-quantized Hamiltonian describing a gas of
ultracold bosonic atoms subject to an optical-lattice potential [18].
The phase scenario becomes richer when the range of interaction is increased: depend-
ing on the lattice, other Mott insulating ground states (density waves) may appear; more-
over, crystalline order may coexist with superfluidity (supersolids) [
31
–
36
]. Supersolidity
is a fascinating property of quantum matter, which has only recently been experimentally
detected in a gas of dipolar atoms [
37
–
39
]. In a supersolid, atoms can simultaneously
support frictionless flow and form a crystal. As suggested by Leggett, a rotating supersolid
should have a moment of inertia that is reduced with respect to its classical value [
40
]. This
phenomenon is called “nonclassical rotational inertia” and its first observation is reported
in a paper published this year [41].
Due to the semiregular character of the PD mesh, in our system the superfluid phase
would be discouraged in favor of less-symmetric condensed phases, and a supersolid
region will then occur at low temperature. To check this expectation, we employ a mean-
field theory, an approach known to give accurate results in the continuum [42–44].
3.1. Decoupling Approximation
As done in Refs. [
24
,
45
], we analyze the phase diagram of the extended BH model in
the hard-core limit using the decoupling approximation (DA) [
46
,
47
]. The latter approach
consists in linearizing the hopping and interaction terms in (11) as
a†
iaj≈a†
iaj+a†
iaj−a†
iajand ninj≈ninj+hniinj−hniinj, (12)
where the thermal averages
haii≡φi
and
hnii≡ρi
are determined self-consistently;
φi
and
ρi
represent the superfluid OP and the average occupancy in the
i
-th site, respectively
(the condensed fraction is
|φi|2
, see e.g., [
45
]). The DA Hamiltonian is a sum of one-site
terms, given by
HDA =−t∑
iFia†
i+F∗
iai−Fiφ∗
i
+V1
2∑
i
(2Rini−Riρi)+V2
2∑
i2R0
ini−R0
iρi−µ∑
i
ni(13)

Appl. Sci. 2021,11, 10053 12 of 21
with
Fi=∑j∈NNi,NNNiφj
,
Ri=∑j∈NNiρj
, and
R0
i=∑j∈NNNiρj
(denoting NN
i
and NNN
i
the first and second neighbors of
i
, respectively). While referring to [
45
] for a full justifi-
cation of DA, it is worth to underline that the self-consistency equations for
φi
and
ρi
are
also the conditions under which the grand potential of (13) is stationary. If more stationary
solutions are found, we must select the one providing the minimum grand potential.
As discussed before, the 32 nodes of the PD mesh are naturally classified as icosahedral
(A), cubic (B), or co-cubic (C), implying that the number of variational parameters in
(13)
is reduced to six. A phase with
φA=φB=φC=
0 is a Mott insulator, whereas a
homogeneous occupancy together with
φA=φB=φC6=
0 defines a superfluid. Any
unbalance between φA,φB, and φCcorresponds to a supersolid.
Looking at Figure 1we soon realize that
FA=2φB+3φC,FB=3φA+3φC,FC=3φA+2φB+φC;
RA=2ρB+3ρC,RB=3ρA,RC=3ρA;
R0
A=0 , R0
B=3ρC,R0
C=2ρB+ρC, (14)
in such a way that the DA Hamiltonian becomes
HDA =12h(A)+8h(B)+12h(C)(15)
with
h(A)=E(A)
0−th(2φB+3φC)a†
A+ (2φ∗
B+3φ∗
C)aAi+ (2V1ρB+3V1ρC−µ)nA;
h(B)=E(B)
0−3th(φA+φC)a†
B+ (φ∗
A+φ∗
C)aBi+ (3V1ρA+3V2ρC−µ)nB;
h(C)=E(C)
0−th(3φA+2φB+φC)a†
C+ (3φ∗
A+2φ∗
B+φ∗
C)aCi+ (3V1ρA+2V2ρB+V2ρC−µ)nC(16)
and
E(A)
0=tφ∗
A(2φB+3φC)−V1ρAρB−3
2V1ρAρC;
E(B)
0=3tφ∗
B(φA+φC)−3
2V1ρAρB−3
2V2ρBρC;
E(C)
0=tφ∗
C(3φA+2φB+φC)−3
2V1ρAρC−V2ρBρC−1
2V2ρ2
C. (17)
In the hard-core limit, the eigenvalues of each partial Hamiltonian in (16) follow from
the diagonalization of a 2 ×2 matrix. We easily obtain:
λ(A)
±=E(A)
0+2V1ρB+3V1ρC−µ
2±s2V1ρB+3V1ρC−µ
22
+t2|2φB+3φC|2;
λ(B)
±=E(B)
0+3V1ρA+3V2ρC−µ
2±s3V1ρA+3V2ρC−µ
22
+9t2|φA+φC|2;
λ(C)
±=E(C)
0+3V1ρA+2V2ρB+V2ρC−µ
2
±s3V1ρA+2V2ρB+V2ρC−µ
22
+t2|3φA+2φB+φC|2. (18)
Therefore, for T=0 the grand potential of (13) reads
Ω=12λ(A)
−+8λ(B)
−+12λ(C)
−
=E0+6(2V1ρB+3V1ρC−µ) + 4(3V1ρA+3V2ρC−µ)
+6(3V1ρA+2V2ρB+V2ρC−µ)−6qA−4qB−6qC (19)

Appl. Sci. 2021,11, 10053 13 of 21
with E0=12E(A)
0+8E(B)
0+12E(C)
0and
A= (2V1ρB+3V1ρC−µ)2+4t2|2φB+3φC|2;
B= (3V1ρA+3V2ρC−µ)2+36t2|φA+φC|2;
C= (3V1ρA+2V2ρB+V2ρC−µ)2+4t2|3φA+2φB+φC|2. (20)
For T>0, the partition function of (13) reads:
Ξ= ∑
±
e−βλ(A)
±!12 ∑
±
e−βλ(B)
±!8 ∑
±
e−βλ(C)
±!12
, (21)
yielding the grand potential
Ω=−kBTln Ξ=E0+30V1ρA+12(V1+V2)ρB+18(V1+V2)ρC−16µ
−12kBTln2 cosh1
2βqA−8kBTln2 cosh1
2βqB
−12kBTln2 cosh1
2βqC. (22)
In seeking the stationary solutions of (22), it can be assumed—without loss of
generality—that
φA
,
φB
,
φC
are real and positive [
24
,
45
]. Putting the derivative of (22)
with respect to each free parameter equal to zero, and suitably rearranging the formulae,
we arrive at the coupled equations:
ρA=1
2−tanh1
2βqA2V1ρB+3V1ρC−µ
2qA
;
ρB=1
2−tanh1
2βqB3V1ρA+3V2ρC−µ
2pB;
ρC=1
2−tanh1
2βqC3V1ρA+2V2ρB+V2ρC−µ
2qC
;
φA=tanh1
2βqAt(2φB+3φC)
qA
;
φB=tanh1
2βqB3t(φA+φC)
pB;
φC=tanh1
2βqCt(3φA+2φB+φC)
qC
. (23)
The above equations can be solved numerically by, e.g, the method described in [
24
].
Upon combining the six Equation (23) together, the following identities are easily derived:
ρA−1
22
+φ2
A=1
4tanh21
2βqA;
ρB−1
22
+φ2
B=1
4tanh21
2βqB;
ρC−1
22
+φ2
C=1
4tanh21
2βqC, (24)
indicating that the value of each superfluid OP is comprised between 0 and 1/2.

Appl. Sci. 2021,11, 10053 14 of 21
Before moving to numerical results, we show that for
t=
0 the DA theory is equivalent
to the MF theory described in Section 2.2. Indeed, for
t=
0 the grand potential (22) becomes
Ω=E0−12kBTlnh1+e−β(2V1ρB+3V1ρC−µ)i−8kBTlnh1+e−β(3V1ρA+3V2ρC−µ)i
−12kBTlnh1+e−β(3V1ρA+2V2ρB+V2ρC−µ)i(25)
with
E0=−24V1ρAρB−36V1ρAρC−24V2ρBρC−6V2ρ2
C. (26)
Upon differentiating (25) with respect to each density parameter and putting the result
equal to zero, the same results as in Equation (10) are eventually obtained. If these equations
are substituted back into (25), then the grand potential (9) is obtained, indicating that the
DA phase diagram of the
t=
0 quantum system is exactly identical to the phase diagram
of the lattice-gas system in the MF approximation.
3.2. Numerical Results
Until now, the value of
V2
was arbitrary. For the sake of example, we henceforth take
γ=V2/V1=2/3. We note that the case γ=1 was considered in [45].
We have first solved Equation (23) numerically for
T=
0 and various
µ
values, being
careful that the minimum
Ω
solution is picked out in each case. The resulting phase
diagram is shown in Figure 9. In addition to the “empty” phase (
ρA=ρB=ρC=
0) and
the “full” phase (
ρA=ρB=ρC=
1), we observe an icosahedral phase (
ρA=
1,
ρB=
ρC=
0) and a dodecahedral phase (
ρA=
0,
ρB=ρC=
1). All these phases are insulating
(
φA=φB=φC=
0) and incompressible (i.e., the density is constant throughout the phase).
Upon increasing
t
at fixed
µ
a supersolid phase invariably appears, characterized by
ρA6=ρB=ρC
. We also note that our phase diagram is symmetric around
µ=
5
/
2. Indeed,
we see from Equation (23) that
ρA,B(µ) =
1
−ρA,B(
5
−µ)
and
φA,B(µ) = φA,B (
5
−µ)
,
and an analogous symmetry property holds for (22). It is worth stressing the similarities
and differences between Figure9and the phase diagram of hard-core bosons on a triangular
lattice [
33
,
47
,
48
]: The overall structure is the same, but the nature of the condensed phase at
large
t
is different, being herein supersolid rather than superfluid—owing to the frustration
effect associated with the existence of inequivalent nodes.
We actually distinguish four different supersolid phases (see Figure10, where the OPs
are plotted for two representative values of
t
). While
φB
is always slightly larger than
φA
,
in the region between the ICO and DOD lobes (where, in particular,
t<
0.266) we find
ρA>ρB
for
µ<
5
/
2 (supersolid 1b) and
ρA<ρB
for
µ>
5
/
2 (supersolid 2b). Outside
the lobes, we instead find
ρA<ρB
for
µ<
5
/
2 (supersolid 1) and
ρA>ρB
for
µ>
5
/
2
(supersolid 2), namely the densities are in reverse order with respect to the reference “solid”
phase. For
t.
0.278, the values of
ρA
and
ρB
jump discontinuously at
µ=
5
/
2, signaling
that the phase transitions along this line are first-order. A further first-order line runs
vertically near
t=
0.266, separating the supersolid phases 1b and 2b from the supersolid
phases 1 and 2, respectively. Finally, there are four second-order transition lines: the two
lines separating “empty” and “full” from the adjacent supersolid, the descending part of
the boundary between ICO and supersolid 1b, and the ascending part of the boundary
between supersolid 2b and DOD.

Appl. Sci. 2021,11, 10053 15 of 21
µ
/V1
t/V1
-3
0
3
6
0 0.1 0.2 0.3 0.4
empty
full
DOD
ICO
supersolid 1
supersolid 2
1b
2b
Figure 9.
DA phase diagram of the extended BH model at
T=
0, for
γ=
2
/
3. The open dots
mark transition points. The dashed red curves are the continuous-transition loci derived in the text
(cf. Equation (27)). The black dashed line marks the passing from supersolid 1 to 2 for
t&
0.278.
Here the system is superfluid (see right panel of Figure10). The remaining black lines represent
first-order transitions.
Figure 10.
DA results for
γ=
2
/
3. The OPs are plotted as a function of
µ
for fixed
t
(
left
,
t/V1=
0.25;
right,t/V1=0.4).

Appl. Sci. 2021,11, 10053 16 of 21
Imposing B-C symmetry, we may simplify Equation (23) and then determine the
equations for the continuous-transition loci, following the same procedure as illustrated in
Ref. [45]. We eventually find:
µ=−3+√69
2t(“empty”-supersolid boundary);
µ=5V1+3+√69
2t(“full”-supersolid boundary);
µ=3V1−3t±q9V2
1−18V1t−51t2
2(ICO-supersolid boundary);
µ=7V1+3t±q9V2
1−18V1t−51t2
2(DOD-supersolid boundary). (27)
In fact, looking at Figure 9we see that the lower branch of the ICO-supersolid locus is
only virtual, since this transition is preempted by a first-order phase transition. A similar
comment applies for the upper branch of the DOD-supersolid locus.
For
T>
0 the phase diagram evolves in the way illustrated in Figure 11. Clearly,
the indications of DA for non-zero temperatures are less accurate; moreover, the prediction
of sharp phase boundaries is an artifact of the approximation, the transitions actually being
smooth crossovers. Already for
T∗=
0.5 we observe a retreat of every supersolid phase
with respect to
T=
0. The smallest
t
value for which the system can be supersolid is now
slightly larger than 0.20. ICO and DOD also lose ground in favor of “empty” and “full”,
respectively, a trend that will become more marked on increasing
T
further. An effect
of finite temperature is that the occupation unbalance between A and B/C is no longer
sharp in ICO and DOD, and is moreover
µ
-dependent; but, similarly to
T=
0, the site
occupancies are independent of
t
. Notice that the ICO and DOD densities as well as
the ranges of stability are exactly the same as predicted by the MF theory of Section2.2
(see Figure 8). For any
T>
0 the densities are
µ
-dependent and independent of
t
also in
the former “empty” and “full” phases, though still homogeneous throughout the mesh.
For
T∗=
0.7 the supersolid 1b and 2b phases nearly disappeared; furthermore, the
µ
extent
of ICO and DOD is more than halved with respect to
T=
0. Finally, in the phase diagram
for
T∗=
1 not only 1b and 2b but also ICO and DOD are no longer present. Moreover,
the supersolid sector lies to the right of
t=
0.35; for
t<
0.35, the occupancies evolve
continuously through µ∗=5/2.
In spite of the elementary character of the DA, the main traits of the thermal evolu-
tion outlined above are correct, being in line with other theoretical studies [
49
–
51
] and
simulations [
52
–
54
]. To recapitulate, the quantum phases are weakened by the thermal
fluctuations associated with finite temperatures. Thus, for
T>
0 a normal fluid appears in
the system. Here, the superfluid OP is zero and the density at each site of the mesh becomes
non-integer. This is to be contrasted with the incompressibility of insulating quantum
phases, which have integer occupancy at each site. Thermal fluctuations also undermine
the supersolid phases, which are shifted towards higher and higher hopping amplitudes
as Tis progressively increased.

Appl. Sci. 2021,11, 10053 17 of 21
µ
/V1
t/V1
-3
0
3
6
0 0.1 0.2 0.3 0.4
empty
full
DOD
ICO
supersolid 1
supersolid 2
1b
2b
Figure 11.
DA phase diagram of the extended BH model for
γ=
2
/
3, plotted for three reduced
temperatures
T∗
(0.5, blue dots; 0.7, brown dots; 1, red dots). Lines through the points are drawn as a
guide to the eye. For the sake of comparison, in the figure we have also reported the transition lines
and phases for T=0.
4. Conclusions
Using a combination of analytic calculations and numerical experiments we have
worked out the phase behavior of a lattice-gas model on the skeleton of the PD. Only
particles connected by a PD edge are allowed to mutually interact, with different couplings
for short and long edges. Depending on the ratio
γ
between these couplings, various
ordered phases are observed at
T=
0 (in addition to “empty” and “full”): icosahedral,
dodecahedral, icosahedral+cubic, and icosahedral+co-cubic. For
T>
0 we study the
phase behavior of the lattice gas by MC simulation. The total number of sites (32) is
sufficiently small that the system is quickly equilibrated at any temperature, with negligible
uncertainties on the thermal properties. Despite the absence of sharp phase transitions in a
finite system, at low temperature we observe strong hysteresis at some of these boundaries.
We have shown that hysteresis, which would occur with any MC algorithm with local
updates, can be cured by making an entropic sampling. In this respect, it would be
intriguing to examine whether anything similar to the concept of nucleation barrier [
55
,
56
]
applies for this model (however, we will leave this for future work).
A variation on the theme of the present model is one where the occupancy of sites is
unrestricted. In this case, for high chemical potentials we expect to observe the formation
of cluster phases, as in [
7
]. A mean-field theory similar to that formulated in [
57
] would
most likely suffice to obtain accurate predictions for the phase behavior of this system, thus
making the simulation unnecessary.
Finally, we have considered the quantum analog of the lattice-gas model and solved it
using the decoupling approximation. This theory predicts various Mott insulating ground
states, each being the counterpart of a phase of the classical model, as well as a number of
supersolids for higher hopping amplitudes. Admittedly, it is the frustration effect due to
the semiregular character of the PD mesh that causes the superfluid phase to be superseded
by a supersolid phase. The take-home message is that confining bosonic particles in a
semiregular mesh is an easy way to stabilize a supersolid in an ultracold quantum gas.

Appl. Sci. 2021,11, 10053 18 of 21
Upon heating, the extent of all the
T=
0 phases gets progressively reduced, leaving room
to normal-fluid behavior.
Author Contributions:
Conceptualization, S.P.; methodology, D.D.G. and S.P.; software, S.P.; vali-
dation, D.D.G. and S.P.; formal analysis, S.P.; investigation, D.D.G. and S.P.; resources, D.D.G. and
S.P.; data curation, D.D.G.; writing—original draft preparation, D.D.G. and S.P.; writing—review and
editing, D.D.G. and S.P.; visualization, D.D.G. and S.P.; supervision, S.P.; project administration, S.P.
All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data are available from the authors upon request.
Acknowledgments:
We express our gratitude to one of the referees for suggesting entropic sampling
as a cure to the hysteresis apparent in Figures3and 4.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A. Calculation of Specific Heats
In this Appendix, we derive the statistical-mechanical formulae for the specific heats
in Equation (3).
Working in the grand-canonical ensemble, the partition function (a sum over mi-
crostates) reads
Ξ=∑
σ
eβµNσe−βEσ, (A1)
with
T
,
V
, and
µ
as control parameters (
V
is the system volume). The grand potential
Ω
, that
is the thermodynamic potential in the
T
,
V
,
µ
representation, is given by
Ω=−kBTln Ξ
.
The constant-µspecific heat is
Cµ=T
N
∂S
∂TV,µ
=−T
N
∂2Ω
∂T2V,µ
, (A2)
where the number of particles Nis
N=−∂Ω
∂µ T,V
=∂ln Ξ
∂βµ β,V
=∑σNσeβµNσe−βEσ
∑σeβµNσe−βEσ≡ hNi. (A3)
From ∂
∂T=−kBβ2∂
∂β (A4)
it soon follows that
Cµ=1
hNikBβ2∂2ln Ξ
∂β2V,µ
. (A5)
Upon inserting Equation (A1) into (A5) we eventually obtain:
Cµ
kB
=β2
hNih(E−hEi)2i − 2µ(hE Ni−hEihNi) + µ2h(N− hNi)2i
≡β2
hNihδE2i−2µhδEδNi+µ2hδN2i. (A6)
The averages in (A6) can be computed in a grand-canonical MC simulation.

Appl. Sci. 2021,11, 10053 19 of 21
Next, we focus on a different specific heat, calculated by keeping N(and V) fixed:
CN=T
N
∂S
∂TV,N
. (A7)
To enforce N=const., the chemical potential must be in a suitable relation with Tand V:
N(T,V,µ) = const. =⇒µ=µ(T,V). (A8)
Hence
∂µ
∂TV
=−∂N
∂TV,µ
∂N
∂µ T,V
(A9)
and ∂S
∂TV,N
=∂
∂TS(T,V,µ(T,V))V
=∂S
∂TV,µ
+∂S
∂µ T,V
∂µ
∂TV
. (A10)
In turn, the µderivative of Sis given by a Maxwell relation:
∂S
∂µ T,V
=∂N
∂TV,µ
=−∂N
∂µ T,V
∂µ
∂TV
. (A11)
Putting Equations (A7)–(A11) together, we end up with:
CN=Cµ−T
hNi∂N
∂T2
V,µ
∂N
∂µ T,V
. (A12)
Now, using (A3) we obtain:
∂N
∂µ T,V
=βhδN2i(A13)
and ∂N
∂TV,µ
=kBβ2hδEδNi− µhδN2i. (A14)
Finally, substituting Equations (A6), (A13), and (A14) into (A12), we arrive at
CN
kB
=β2
hNihδE2i− hδEδNi2
hδN2i. (A15)
Appendix B. Order Parameters
We hereby introduce a few quantities, to be computed within the simulation of the
lattice gas on a PD mesh, allowing us to identify the order present in the system at fixed
T
and
µ
. Clearly, there is no unique way to define these OPs—our proposal below is just
one possibility.
For our definition we need the current number of particles,
N=∑ici
, and of occupied
dodecahedral sites,
ND=∑i∈B∪Cci
. A quantity sensitive to dodecahedral order is then
obtained as follows. Let
OD
be
ND/
20 if 19
≤ N ≤
21 and
ND≥
19, and 0 otherwise.
Then,
ODOD =hODi
. Similarly, to measure the amount of icosahedral order we compute a
quantity
OI
, defined as
(N −ND)/
12 if 11
≤ N ≤
13 and
ND≤
1, and 0 otherwise. Then,
OICO =hOIi.
Measuring the degree of ICO+CUB order is more subtle, since in the phase region
where this “phase” is stable the simulated system circulates, even at the lowest temper-
atures, between five different basins of microstates. As a result, in a long simulation
the average occupancy will be the same for all dodecahedral sites. Here, the crucial ob-

Appl. Sci. 2021,11, 10053 20 of 21
servation is that the cosine of the angle
θij
formed by the vector radii relative to two
distinct cubic sites
i
and
j
is either
−
1 or
±
1
/
3. In view of this, let
OCUB
be defined as
1
−(
1
/
3
)∑0
i,j∈B∪Ccicj(cos θij +
1
)(cos2θij −
1
/
9
)2
if 19
≤ N ≤
21 and 7
≤ ND≤
9, and 0
otherwise (the sum is over all distinct pairs of dodecahedral sites; the sum prefactor is just
a reasonable choice). Then,
OICO+CUB =hOCUB i
. The amount of ICO+COC order is simi-
larly defined. Let
OCOC
be 1
−(
1
/
3
)∑0
i,j∈B∪C(
1
−ci)(
1
−cj)(cos θij +
1
)(cos2θij −
1
/
9
)2
if 23 ≤ N ≤ 25 and 11 ≤ ND≤13, and 0 otherwise. Then, OICO+COC =hOCOC i.
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