# Quantum Fluids in Thermodynamic Geometry

L. F. Escamilla-Herrera<EMAIL_ADDRESS>División de Ciencias e
Ingenierías Campus León, Universidad de Guanajuato, A.P. E-143, C.P. 37150,
León, Guanajuato, México.  J. L. López-Picón<EMAIL_ADDRESS>División
de Ciencias e Ingenierías Campus León, Universidad de Guanajuato, A.P. E-143,
C.P. 37150, León, Guanajuato, México.  José Torres-Arenas
<EMAIL_ADDRESS>División de Ciencias e Ingenierías Campus León,
Universidad de Guanajuato, A.P. E-143, C.P. 37150, León, Guanajuato, México.
Alejandro Gil-Villegas<EMAIL_ADDRESS>División de Ciencias e Ingenierías
Campus León, Universidad de Guanajuato, A.P. E-143, C.P. 37150, León,
Guanajuato, México.

(August 28, 2024)

###### Abstract

In this work, the Thermodynamic Geometry (TG) of quantum fluids (QF) is
analyzed. We present results for two models. The first one is a quantum hard-
sphere fluid (QHS) whose Helmholtz free energy is obtained from Path Integrals
Monte Carlo simulations (PIMC). It is found that due to quantum contributions
in the thermodynamic potential, the anomaly found in TG for the classical
hard-sphere fluid related to the sign of the scalar curvature, is now avoided
in a considerable region of the thermodynamic space. The second model is a
semi-classical square-well fluid (QSW), described by a quantum hard-sphere
repulsive interaction coupled with a classical attractive square-well
contribution. Behavior of the semi-classical curvature scalar as a function of
the thermal de Broglie wavelength $\lambda_{B}$ is analyzed for several
attractive-potential ranges, and description of the semi-classical R-Widom
lines defined by the maxima of the curvature scalar, are also obtained and
compared with classical results for different square-well ranges.

Statistical Mechanics, Quantum Hard-Spheres, Thermodynamic Geometry, Square-
well fluid

###### pacs:

05.20.Jj, 95.30.Sf, 95.30.Tg

##  I Introduction

Quantum fluids are an important subject within the molecular liquids research
community. Unlike classical fluids, in the quantum regime the dual wave-
particle nature of the individual constituents becomes dominant and manifests
in their collective properties. These systems defy our classical understanding
of matter and offer exciting prospects for technological advancements,
considering substances like hydrogen and helium, and their transport and
storage. Additionally, the quantum nature of the hydrogen bond has been of
increasing interest recentlyLi et al. (2011) as well as its consequence in the
description of the phase diagram of associating fluids Contreras et al.
(2020a). On the other hand, research on QF gives insight into the fundamental
principles of quantum mechanics, shedding light on the nature of matter and
its interactions under extreme conditions. In this paper, we explore some
aspects of the thermodynamic properties of QF using the Thermodynamic Geometry
formalism, highlighting the main aspects such as the Riemann scalar curvature,
and the R-Widom line.

Quantum hard spheres fluid is a natural system to study, since it is possible
to compare quantum effects with respect to the very well-known classical hard
spheres system (HS). Analytical results are known for the Slater sum of QHS at
low densities, obtaining interesting insights of quantum effects in semi-
classical computer simulation studies Yoon and Scheraga (1988); Thermodynamic,
structural and dynamic properties of QHS have been studied for a wide range of
densities, temperatures and thermal de Broglie wavelength values using the
more robust Path-Integral Monte Carlo approach Runge and Chester (1988); Sesé
(1991); Markland et al. (2011a). Besides, in order to consider attractive
interactions for a more realistic description of real fluid systems, different
hard-core potential equations of state have been developed, such as for
quantum square-well Singh and Sinha (1978a); Serna and Gil-Villegas (2016) and
quantum Yukawa Singh and Sinha (1978b) fluids.

In the past few decades, several approaches have been developed in order to
describe thermodynamic properties of systems using differential geometry. Most
of these proposals start at the well known fluctuation theory Rao (1945);
Mrugała (1984); Mrugala et al. (1990), assuming the existence of the notion of
distance between a pair of points, each one representing a thermodynamic
state, located on the space composed by thermodynamic parameters; in the
context of Riemannian geometry, such a distance plays the role of the metric.
In particular, Hessian metrics were postulated by Weinhold Weinhold (1975) and
Ruppeiner Ruppeiner (1979, 1995); both proposed a metric defined by the
Hessian of a given thermodynamic potential in the thermodynamic equilibrium
space. The Weinhold case is constructed using the Hessian of the internal
energy, while Ruppeiner constructed its metric with the negative of the
Hessian of the entropy; it was later proved that both metrics are related by a
conformal factor Salamon et al. (1984). The theoretical framework of such
Hessian metrics is usually referred in the literature as Thermodynamic
Geometry. This formalism will be considered in this work to study the physical
properties of the QHS fluid.

In this work we are also interested in the study of the supercritical region
of quantum systems using the tools of TG, an aspect which is of great interest
for both theoretical and experimental research, and particularly the
estimation of the existing boundary which separates the gas-like and liquid-
like phases, which has been observed in the supercritical region Fisher and
Widom (1969); Xu et al. (2005); Bolmatov et al. (2015). One of these proposed
boundaries is the so-called Widom line Xu et al. (2005); Corradini et al.
(2015); Jaramillo-Gutiérrez et al. (2022), which is defined as the locus of
points that maximize the correlation length. It is known that the correlation
length diverges at the critical point, but in the vicinity above it, changes
with a defined power law Fisher and Widom (1969). This behavior makes possible
to characterize this curve through extreme values of the correlation length,
and near to the critical point, via the extreme values of different response
functions. It is well known that such curves exhibit a lineal behavior in the
region close to the critical point, to later separate from each other as
temperature increases. Therefore, coincidence of such lines in the $(P,T)$
plane has been used previously in the literature as a definition for a unique
Widom line Xu et al. (2005); Brazhkin et al. (2011); Zerón et al. (2019).

A more recent proposal to define the Widom line comes from the TG framework,
that defines this curve as the locus of maxima of the isotherms of the scalar
curvature obtained for the thermodynamic metric constructed using this
formalism Ruppeiner (2012); May and Mausbach (2012). In this work, this
definition for the Widom line will be used to explore the supercritical
behavior of the QHS fluid.

This work is organized as follows. In section II, we briefly present how the
metric and curvature are calculated in terms of the dimensionless Helmholtz
free energy representation. Subsequently, equations of state related to the
classical and quantum HS systems are presented together with the geometrical
analysis of their curvatures, as well as a discussion about its consequences.
Section III contains the same geometrical analysis for the semi-classical
square-well fluid and includes for this case the Widom lines, for different
values of the SW range $\lambda$. In section IV, we present the general
results and discussions about the comparison with classical and quantum
models. Finally, in section conclusions and future perspectives of this
research are drawn and discussed.

##  II Thermodynamic Geometry: Classical and quantum hard sphere fluid

In this section we will consider the TG formalism Ruppeiner (1979, 1995) for a
QHS fluid. We use the Helmholtz free energy representation for which the
thermodynamic metric components are obtained as second derivatives of the free
energy per volume $f=A/V$ with respect to absolute temperature $T$ and number
density $\rho=N/V$. This metric is given in its matrix form by,

$[g_{ij}]=\frac{1}{kT}\
\left(\begin{array}[]{cc}-\frac{\partial^{2}f}{\partial T^{2}}&0\\\
0&\frac{\partial^{2}f}{\partial\rho}\end{array}\right)\,.$ (1)

In the SAFT-VR approach Gil-Villegas et al. (1997), which will be followed in
subsequent sections, the dimensionless free energy is defined by

$\displaystyle\frac{A}{NkT}=a(T,\rho)\,.$ (2)

The TG metric components of the system are determined by

$\displaystyle g_{{}_{TT}}=$
$\displaystyle\frac{6}{\pi\sigma^{3}}\left(\frac{2\eta}{T}\frac{\partial
a}{\partial T}-\frac{\partial^{2}a}{\partial T^{2}}\right)\,,$ (3)
$\displaystyle g_{\eta\eta}=$
$\displaystyle\frac{\pi\sigma^{3}}{6}\left(\eta\frac{\partial^{2}a}{\partial\eta^{2}}+2\frac{\partial
a}{\partial\eta}\right)\,;$

where $\eta$ is the packing fraction for hard-spheres particles with diameter
$\sigma$, obtained from the density number as $\eta=\pi\rho\sigma^{3}/6$.

The corresponding reduced curvature scalar, $R^{*}=R/\sigma^{3}$, is given by

$\displaystyle
R^{*}=\frac{\pi^{2}\sigma^{3}}{36\sqrt{g}}\frac{\partial}{\partial\eta}\left(\frac{1}{\sqrt{g}}\frac{\partial
g_{{}_{TT}}}{\partial\eta}\right)-\frac{1}{\sigma^{3}\sqrt{g}}\frac{\partial}{\partial
T}\left(\frac{1}{\sqrt{g}}\frac{\partial g_{\eta\eta}}{\partial T}\right)\,.$
(4)

This form of $R^{*}$ is particularly simple because the metric is diagonal in
the Helmholtz free energy representation Ruppeiner (1979, 2012). According to
the TG approach, negative (positive) values for $R$ indicates that attractive
(repulsive) interactions are predominant on the system under study. However,
it is well known that the TG description for a classical HS fluid presents a
fundamental inconsistency related to the interpretation of the sign of $R$
(see for instance Ref. Ruppeiner (2012)), since the corresponding scalar
curvature, calculated for different equations of state, is always negative,
which contradicts the repulsive nature of the hard sphere potential.

In the fluid region of the phase diagram of a classical hard spheres fluid,
the Helmholtz free energy can be accurately described by the Carnahan and
Starling equationCarnahan and Starling (1970),

$\frac{A_{HS}}{NkT}=\ln{(\rho\lambda_{B}^{3})}-1+\frac{4\eta-3\eta^{2}}{(1-\eta)^{2}}\,$
(5)

where $\lambda_{B}$ is the thermal de Broglie wavelength, given by

$\lambda_{B}=h/\sqrt{2\pi mkT}\,,$ (6)

The TG applied to the classical HS fluid was partially analyzed in Ref.
Jaramillo-Gutiérrez et al. (2020) where the curvature $R$ of the free energy
(5) was computed and compared with $R$ values determined with another HS model
Ruppeiner (2012). It was observed for both models that the interaction
hypothesis related to the thermodynamic curvature of the HS system seemed to
be invalidated, since the HS curvature remains always negative. Given the fact
that there is not an attractive contribution to the HS interaction, then the
sign of its curvature contradicts the aforementioned hypothesis. This is known
in the literature as the BPH anomaly Brańka et al. (2018). In what follows, a
model of a QHS will be analyzed in order to gain insight into a possible
quantum formulation of TG and with the premise that the BPH anomaly could be
solved when quantum contributions are taken into account.

Quantum corrections to classical equations of state (EOS) can be introduced
according to semi-classical thermodynamic perturbation methods Serna and Gil-
Villegas (2016); Contreras et al. (2020b). These EOS are given as functions of
the thermal de Broglie wavelength $\lambda_{B}$. In Serna and Gil-Villegas
(2016), the functional expression of the Carnahan-Starling excess Helmholtz
free energy was used to construct a parametrization for a QHS fluid, fitting
the expression in order to reproduce Path Integral Monte Carlo (PIMC)
simulation values for this system.

The corresponding quantum expression for the QHS fluid can be expressed in
terms of an effective temperature dependent packing fraction $\eta_{e}$,

$\frac{A^{QHS}}{NkT}=\frac{4\eta_{e}-3\eta_{e}^{2}}{(1-\eta_{e}^{2})}\,;$ (7)

where $\eta_{e}$ is a function of the packing fraction and the reduced de
Broglie thermal wavelength $\lambda_{B}^{*}=\lambda_{B}/\sigma$,

$\eta_{e}=\left(1+d_{1}\lambda_{B}^{*}\right)\eta+\left(d_{2}\lambda_{B}^{*}+d_{3}\lambda_{B}^{*2}\right)\eta^{2}\,,$
(8)

with $d_{1}=1.6593854484$, $d_{2}=-1.0927115150$, $d_{3}=-1.1188233921$.

For the purpose to explore geometric properties of Eq. (7) it is convenient to
construct the TG formulation in terms of $\lambda_{B}$ instead of the usual
variable temperature $T$. In order to do this, it is useful to introduce the
reduced temperature $T^{*}=kT/\epsilon$, where the energy parameter $\epsilon$
is related to the potential depth of a particular attractive interaction.
Although for the HS system there is not such additional interaction, it is
possible to define a suitable energy parameter, that will be denoted as
$\epsilon_{0}$, given by

$\epsilon_{0}=\frac{h^{2}}{2\pi m\sigma^{2}}\,.$ (9)

and then $T^{*}_{{}_{HS}}=kT/\epsilon_{0}$.

The reduced thermal wavelength $\lambda_{B}^{*}$ can also be written in terms
of the de Boer parameter $\Lambda=h/\sigma\sqrt{m\epsilon}$ in the following
way,

$\lambda_{B}^{*}=\frac{\Lambda}{\sqrt{2\pi}\sqrt{T^{*}}},\quad
T^{*}=\frac{kT}{\epsilon}\,;$ (10)

for the QHS system the corresponding expression is,

$\lambda_{B}^{*}=\frac{\Lambda_{0}}{\sqrt{2\pi}\sqrt{T^{*}_{{}_{HS}}}}=\frac{1}{\sqrt{T^{*}_{{}_{HS}}}}\,;$
(11)

where $\Lambda_{0}=\sqrt{2\pi}$. This particular choice for the de Boer
parameter is close to the corresponding value for molecular Hydrogen. This
procedure for $\Lambda$ will be also used to obtain the geometric properties
of the semi-classical SW systems in the next section; since this
simplification allows to write geometric properties of the HS system in terms
of the reduced thermal wavelength $\lambda_{B}^{*}$ by using the variable
change $\lambda^{*}_{B}=1/\sqrt{T^{*}_{{}_{{HS}}}}$.

The geometric properties of the QHS system can be rewritten in terms of
derivatives with respect to $\lambda^{*}_{B}$ using Eq.(11,

$\displaystyle g_{\lambda_{B}^{*}\lambda_{B}^{*}}$
$\displaystyle=-\frac{3\eta}{2\pi\sigma^{3}}\left(\lambda^{*2}_{B}\frac{\partial^{2}a}{\partial\lambda_{B}^{*2}}-\lambda_{B}^{*5}\frac{\partial
a}{\partial\lambda_{B}^{*}}\right),$ (12a) $\displaystyle g_{\eta\eta}$
$\displaystyle=\frac{\pi\sigma^{3}}{6}\left(\eta\frac{\partial
a^{2}}{\partial\eta^{2}}+2\frac{\partial a}{\partial\eta}\right)\,;$ (12b)

and hence the curvature scalar is given by :

$R^{*}=\frac{\pi^{2}\sigma^{3}}{36\sqrt{g}}\frac{\partial}{\partial\eta}\left(\frac{1}{\sqrt{g}}\frac{\partial
g_{\lambda_{B}^{*}\lambda_{B}^{*}}}{\partial\eta}\right)-\frac{\lambda^{*3}_{B}}{4\sigma^{3}\sqrt{g}}\frac{\partial}{\partial\lambda_{B}^{*}}\left(\frac{\lambda^{*3}_{B}}{\sqrt{g}}\frac{\partial
g_{\eta\eta}}{\partial\lambda_{B}^{*}}\right)\,.$ (13)

As long as the free energy representation is used, the curvature $R^{*}$ given
in (13) remains the same, regardless the particular model, therefore it will
be used to calculate $R^{*}$ for the classical and semi-classical SW fluid
described in the next section.

Figure 1: Reduced curvature for the classical and quantum hard sphere systems
for different isotherms in the ($\lambda_{B}^{*}$,$\eta$) representation. The
solid line for $\lambda_{B}^{*}\to 0$ represents the behavior for a classical
HS fluid; as $\lambda_{B}^{*}$ increases (temperature decreases), the quantum
contribution becomes more important in the region where the curvature becomes
negative, crossing the $R=0$ line, reversing the known anomaly for the
classical HS curvature.

Results for $R^{*}$ for the QHS system are presented in Fig. 1. Several
isotherms in the $(\lambda_{B}^{*},\eta)$ representation are plotted. The
solid line represent the classical case (since this line correspond to the
isotherm where $\lambda_{B}^{*}\to 0$), which in the range of validity for the
considered EOS given by (7), $R^{*}$ is always positive (i.e. negative in the
$T-\eta$ thermodynamic space). As $\lambda_{B}^{*}$ increases, quantum effects
become more noticeable, since a change in the sign of the curvature appears at
small densities. In such region, the usual interpretation given in literature
of the sign of $R^{*}$ is then fulfilled. However, for even greater values of
$\lambda_{B}^{*}$, another change of sign appears, returning to the BPH
anomaly for $R^{*}$ in the region of higher values of $\eta$, therefore
constraining the region where quantum effects are able to revert this anomaly.
For instance, in the isotherm $\lambda_{B}^{*}=0.9$ (dash-dot-dot line) a bump
underneath the line $R^{*}=0$ is observed, this is the region for which the
usual TG interpretation of a repulsive interaction holds.

Figure 2: Reduced curvature for the QHS system in the
($\lambda_{B}^{*}$,$\eta$) representation for different values of $\eta$. The
crossing through zero of the quantum curvature is also favored by increasing
density. This behavior for both $\lambda_{B}^{*}$ and $\eta$ is better
depicted in the three dimensional representation of $R^{*}$.

This interpretation is reinforced in Fig. 2, where $R^{*}$ is given in the
($\lambda_{B}$,$\eta$) representation for several constant values of the
packing fraction. A change of sign appears in $R^{*}$ starting from the curve
$\eta=0.1$; the bump region where the BPH anomaly is reverted can also be
noticed in this Figure, appearing in at higher densities, and it is clearly
visible from $\eta=0.2$, and around the interval $0.1\leq\lambda_{B}^{*}\leq
0.6$.

Figure 3: 3D reduced curvature scalar for the QHS system (orange) as a
function of $\lambda^{*}_{B}$ and $\eta$. The crossing with $R^{*}=0$ plane
(blue) is also presented. The main feature of this plot is the existence of a
region of $(\lambda^{*}_{B},\eta)$ for which the interpretation of the sign of
the curvature scalar holds, located in the strip around the middle section of
$\eta$.

In order to better depict the behavior of the curvature of the semi-classical
QHS system, a 3D plot in the ($\lambda_{B},\eta$) representation is presented
in Fig. 3. This plot also presents the plane of flat curvature $R^{*}=0$. The
region below this plane with negative curvature, as observed in Fig. 1,
becomes a strip that cross along the middle section of the plot. This is the
region where quantum effects are able to revert the BPH anomaly in the
curvature of the QHS system, since in the usual $(T,\eta)$ representation,
curvature becomes positive in this strip due to quantum effects, consistent
with the interpretation given in the TG literature of a repulsive potential,
as is the case of the HS fluid.

Figure 4: Contour plot for the curvature scalar for the QHS system in the
$(\lambda_{B}^{*},\eta)$ space. The region where $R^{*}<0$ (i.e., where the
BPH anomaly is reverted) is located in the strip in the middle of the plot
limited by the lower and upper contour-lines (purple section), whereas for
high values of $\lambda_{B}^{*}$ and $\eta$, the value of $R^{*}$ becomes
positive once again. A region of minimum values for the curvature also appears
in the upper-middle section of the plot (violet section), which can be
interpreted as the region of maximal repulsion of the QHS potential.

Finally, a contour plot of $\lambda_{B}^{*}$ vs $\eta$ for the QHS curvature
$R^{*}$ is presented in Fig. 4. This plot describes the values that curvature
$R^{*}$ takes in the ($\lambda_{B}^{*}$, $\eta$) representation. The region
where the BPH anomaly is reverted due to quantum effects is also present in
the middle section, divided by the lines where $R^{*}=0$. This region is the
same one observed in Fig. 3. Additionally, the existence of a small U-shaped
region around $\lambda_{B}^{*}=0.8$ and $\eta=0.2$ can be noticed, that is the
region where the lowest value (or highest value in the ($T$,$\eta$)
representation) of the curvature is reached, where the QHS fluid becomes more
repulsive.

##  III Thermodynamic Geometry of the semi-classical square-well fluid

In this section, the TG of a semi-classical SW fluid is explored. This is
achieved by coupling the QHS equation of state (7) with a classical SW
potential, a first approach to explore thermodynamic geometric properties of a
quantum SW fluid. In this regard, the SW fluid is a more general and
interesting system whose classical thermodynamics has been widely studied
Alder et al. (1972); Smith et al. (1977); Gil-Villegas et al. (1996);
Benavides and Gil-Villegas (1999); Khanpour (2011); Vega et al. (1992). In the
case of QSW systems, several models have been developed previously, see for
instance Refs. Singh and Sinha (1978c); Serna and Gil-Villegas (2016);
Contreras et al. (2020b).

In this section a semi-classical analytic model is used in order to analyze
its thermodynamic geometric properties via the TG framework. We consider a
thermodynamic perturbation approach with the QHS contribution described in
section II as the reference system, given by the semi-classical Helmholtz free
energy (7), coupled with the classical SW contribution presented in López-
Picón et al. (2022) as a perturbation,

$\frac{A}{Nk_{{}_{B}}T}=\frac{A^{\text{ideal}}}{Nk_{{}_{B}}T}+\frac{A^{\text{QHS}}}{Nk_{{}_{B}}T}+\frac{A_{1}}{Nk_{{}_{B}}T}+\frac{A_{2}}{Nk_{{}_{B}}T}\,;$
(14)

where $A_{1}$ and $A_{2}$ are the first SW perturbation terms used in the
SAFT-VR approach Patel et al. (2005a). The first contribution $A_{1}/NkT$ is
given by

$\frac{A_{1}}{Nk_{{}_{B}}T}=-4\left(\frac{\epsilon}{kT}\right)(\lambda^{*3}-1)\eta
G_{\text{HS}}(1;\eta_{\text{eff}})\,.$ (15)

where $G^{\text{HS}}(1;\eta_{\text{eff}})$ is the radial distribution function
contact value evaluated at an effective packing fraction $\eta_{\text{eff}}$ ,

$G_{\text{HS}}(1;\eta_{\text{eff}})=\frac{1-\eta_{\text{eff}}/2}{(1-\eta_{\text{eff}})^{3}}\,.$
(16)

The effective packing fraction is parameterized using a Padé approximation,

$\eta_{\text{eff}}=\frac{c_{1}\eta+c_{2}\eta^{2}}{(1+c_{3}\eta)^{3}}\,,$ (17)

where the indexed constants $c_{i}$ $(i=1,2,3)$ are given in terms of the
reduced SW range, $\lambda^{*}=\lambda/\sigma$, which can be written as,

$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle-\frac{3.1649}{\lambda^{*}}+\frac{13.3501}{\lambda^{*2}}-\frac{14.8057}{\lambda^{*3}}+\frac{5.7029}{\lambda^{*4}}\,,$
(18) $\displaystyle c_{2}$ $\displaystyle=$
$\displaystyle\frac{43.0042}{\lambda^{*}}-\frac{191.6623}{\lambda^{*2}}+\frac{273.8968}{\lambda^{*3}}-\frac{128.9334}{\lambda^{*4}}\,,$
$\displaystyle c_{3}$ $\displaystyle=$
$\displaystyle\frac{65.0419}{\lambda^{*}}-\frac{266.4627}{\lambda^{*2}}+\frac{361.0431}{\lambda^{*3}}-\frac{162.6996}{\lambda^{*4}}\,.$

These inverse-power expansions in $\lambda^{*}$ ensure that
$\eta_{\text{eff}}\to 0$ for $\lambda^{*}\to\infty$, from which, the desired
behavior in the mean-field limit, $G_{\text{HS}}(1;\eta_{\text{eff}})\to 1$,
is recovered. The second-order fluctuation term $A_{2}/Nk_{{}_{B}}T$ is given
by through the local-compressibility approximation Gil-Villegas et al. (1997);
Patel et al. (2005b),

$\frac{A_{2}}{Nk_{{}_{B}}T}=\frac{1}{2}\left(\frac{\epsilon}{kT}\right)K^{\text{HS}}\eta\frac{\partial}{\partial\eta}\left(\frac{A_{1}}{Nk_{{}_{B}}T}\right).$
(19)

where $K^{\text{HS}}$ is the HS isothermal compressibility Lafitte et al.
(2013),

$K^{\text{HS}}=\frac{(1-\eta)^{4}}{1+4\eta+4\eta^{2}-4\eta^{3}+\eta^{4}}.$
(20)

Before proceeding to obtain the curvature of the free energy given in Eq.
(14), it is important to remark the following point, as mentioned before when
the geometric properties of the QHS system were derived, it is convenient to
express the free energy of the semi-classical and classical SW fluid in terms
of the thermal de Broglie wavelength $\lambda^{*}_{B}$, including the explicit
terms depending on temperature, according to the relation
$\lambda_{B}^{*}=\Lambda/\sqrt{2\pi T^{*}}$, with $T^{*}=kT/\epsilon$. In the
case of the QHS system the energy parameter $\epsilon_{0}$ is used, defined in
Eq. (9). However, the SW potential already possess an energy parameter for
temperature reduction the potential depth $\epsilon$. Therefore, we have two
different reduced temperatures $T^{*}_{SW}=kT/\epsilon$ and
$T^{*}_{{HS}}=kT/\epsilon_{0}$ for the SW and HS potentials, respectively.

From the relation between the thermal de Broglie wavelength and the de Boer
parameter, presented in Eq. (10) , it follows that when this parameter is
fixed to $\Lambda=\sqrt{2\pi}$, the energy parameters $\epsilon$ and
$\epsilon_{0}$, are indeed equal, and the same is also true for the reduced
temperatures. This allows us to write the complete free energy in terms of
only one reduced temperature. Besides, the relation between $\lambda^{*}_{B}$
and $T^{*}$ is simply $\lambda^{*}_{B}=1/\sqrt{T^{*}}$ and the change of
variable applied to calculate the curvature of the QHS system can also be
applied without any further modifications, and the corresponding expressions
for the metric components and curvature are the same than the ones given by
Eq. (12) and Eq. (13), respectively.

$\lambda^{*}$ | $\eta_{\text{qcrit}}$ | $T^{*}_{\text{qcrit}}$ | $P^{*}_{\text{qcrit}}$ | $\eta_{\text{ccrit}}$ | $T^{*}_{\text{ccrit}}$ | $P^{*}_{\text{ccrit}}$
---|---|---|---|---|---|---
1.25 | 0.0314 | 0.2378 | 0.0047 | 0.2239 | 0.83627 | 0.1649
1.50 | 0.0404 | 0.4932 | 0.0126 | 0.1499 | 1.32907 | 0.1434
1.75 | 0.0475 | 0.8487 | 0.0255 | 0.1232 | 1.9073 | 0.1583
2.00 | 0.0566 | 1.3782 | 0.0493 | 0.1247 | 2.8248 | 0.2325
2.50 | 0.0783 | 3.3019 | 0.1673 | 0.1330 | 5.8015 | 0.5226
3.00 | 0.0931 | 6.7089 | 0.4201 | 0.1336 | 10.3288 | 2.3393
Table 1: Theoretical critical values in the ($T^{*},\eta$) representation of
the semi-classical and classical SW fluid are presented for different
potential ranges between 1.25 an 3.0, denoted by qcrit and ccrit,
respectively.

The TG metric of the semi-classical SW fluid is obtained through second-order
derivatives of the Helmholtz free energy given in Eq. (14) and applying Eq.
(1). In order to compare how this geometric property changes by introducing
the QHS potential given in Eq. (7), the classical SW fluid model presented in
López-Picón et al. (2022) is used as a reference. In Table 1 the critical
values for different potential ranges for the classical and semi-classical SW
fluid are presented in the $(T^{*},\eta)$ representation. It can be noticed
that the critical values of the semi-classical system are considerably lower
when compared to the classical one.

Figure 5: Isotherms of the curvature $R$ of the semi-classical and classical
SW fluid for a potential range of $\lambda^{*}=1.5$ in the supercritical
region are presented, given by solid and dashed lines, respectively. To have a
better insight on the behavior of both systems, $T^{*}$ were chosen to be at
$1.25\ T_{\text{crit}}^{*}$ (bottom), $2\ T_{\text{crit}}^{*}$ (middle) and
$3\ T_{\text{crit}}^{*}$ (upper), namely at the same percentage away from its
corresponding critical value.
Figure 6: 3D plot of the reduced classical and semi-classical SW curvatures,
represented by blue and orange surfaces, respectively. Both plots are
determined for the same SW range, $\lambda^{*}=1.5$. The corresponding thermal
de Broglie wavelength values at the critical temperatures are
$\lambda_{B}^{SC}=1.4239$ and $\lambda_{B}^{C}=0.8674$ for the semi-classical
and classical curvatures, respectively. In order to avoid singularities due to
the gas-liquid transition, the plot is constructed considering only
supercritical values of $\lambda_{B}^{*}$ for each system between
$0.3\leq\lambda_{B}^{*}/\lambda_{B\text{crit}}^{*}\leq 0.8$.

Once both metric elements and curvatures are calculated for the semi-classical
and classical SW fluids using Eqs. (3) and (13) in the
$(\lambda_{B}^{*},\eta)$ representation, an analysis of their behavior was
performed in the supercritical region of $\lambda_{B}^{*}$, in order to avoid
any singularity due to the gas-liquid phase transition. First, a 2D comparison
of the curvature scalar for different isotherms is presented in Fig. 5 for a
SW range $\lambda^{*}=1.5$ for both semi-classical and classical systems,
solid and dashed lines, respectively. Three different temperatures have been
considered for this comparison, namely, $T^{*}=1.25\ T_{\text{crit}}^{*}$,
$T^{*}=2.0\ T_{\text{crit}}^{*}$ and $T=3.0\ T_{\text{crit}}^{*}$; these
particular values were chosen to fairly compare the corresponding curvatures
since the critical points for each system are not close to each other, as
noticed in Table 1. The main remark about Fig. 5 is that the maximum of
$R^{*}$ reached for the semi-classical SW is higher when compared to the
classical one at the same “distance” to their corresponding critical point;
with a steeper maximum for the semi-classical system and located at smaller
packing fractions compared to the classical one. This feature is more evident
as reaching the critical point of each system.

Fig. 6 presents 3D plots of the semi-classical (orange surface) and classical
(blue surface) SW fluid for the curvature scalars in the thermodynamic space
$(\eta,\lambda_{B}^{*})$ for a SW $\lambda^{*}=1.5$. As in the case of the
two-dimensional plots presented in Fig. 5, these surfaces were obtained for
the supercritical region to avoid any singularity in $R^{*}$ close to the
critical point, starting from $0.3\ \lambda_{B\text{crit}}^{*}$ up to $0.8\
\lambda_{B\text{crit}}^{*}$, i.e. from 30 to 80 percent of the critical value
in $\lambda_{B}^{*}$. From this figure it can be noticed that the curvature of
the semi-classical and classical SW fluids greatly differs in the region
$\eta\to 0$, i.e., for small densities, where quantum effects are more
relevant. This behavior is also captured in the 2D plots.

Figure 7: Differences between classical and semi-classical SW curvatures for
$\lambda^{*}=1.5$. These differences were computed approaching the critical
point in four different directions: (1) Both parameters in the sub-critical
region (upper left); (2) approaching critical density from the sub-critical
region and critical temperature from the supercritical region (upper right);
(3) density in the supercritical region ant temperature approaching the
critical point from the sub-critical region (bottom left);(4) both parameters
in the supercritical region approaching the critical point (bottom right).

In order to display the differences in the behavior between the semi-classical
and classical curvature scalars of the SW fluid, the difference
$R^{*}_{Q}-R^{*}_{C}$ of the reduced semi-classical and classical curvatures
for $\lambda^{*}=1.5$, $R^{*}_{Q}$ and $R^{*}_{C}$, respectively, is presented
in Fig. 7. As can be seen in Fig 7, each curve is always positive for the
considered interval of $\eta/\eta_{\textbf{crit}}$, indicating that
$R^{*}_{Q}$ is always greater than $R^{*}_{C}$.

To remark these differences, four different temperature regions are shown
separately: (1) $\eta<\eta_{B\text{crit}}$ and $T^{*}<T_{B\text{crit}}^{*}$ in
the upper left plot; (2) $\eta<\eta_{B\text{crit}}$ and
$T^{*}>T_{B\text{crit}}^{*}$ in the upper right plot; (3)
$\eta>\eta_{B\text{crit}}$ and $T^{*}<T_{B\text{crit}}^{*}$ in the bottom
left; (4) $\eta>\eta_{B\text{crit}}$, and $T^{*}>T_{B\text{crit}}^{*}$ in the
bottom right side. The behavior of the differences in the supercritical region
of $\eta$ is the same with $\left(R^{*}_{Q}-R^{*}_{C}\right)\to 0$ as
$\eta^{*}$ goes far away from its critical value. This is a hint that at
higher densities, geometric properties of the semi-classical system resembles
those of the classical SW fluid. In the opposite way, in the subcritical and
supercritical region of temperature, the difference
$\left(R^{*}_{Q}-R^{*}_{C}\right)\to\infty$ as $\eta^{*}\to 0.9$, which
represents the divergence that appears at $R$ as the critical point is
reached.

Figure 8: Comparison of $R$-Widom lines obtained for the semi-classical and
classical SW fluids, given by dots and diamonds, respectively, for the
following values of the SW range $\lambda^{*}$: 1.5 (bottom), 1.75 (middle
bottom), 2 (middle upper), and 2.5 (upper). Results are presented as a
function of the reduced de Broglie thermal wavelength
$\lambda^{+}_{B}=\lambda_{B}/{\lambda_{B}}_{\text{crit}}$ and the reduced
pressure $P^{+}=P/P_{\text{crit}}$. In this representation, higher
supercritical states are reached as $\lambda_{B}^{*}\to 0$. Semi-classical and
classical lines exhibit the same linear behavior in the region near the
critical point. The supercritical line can be extended further for the semi-
classical SW fluid for smaller potential ranges. Interestingly, this behavior
is reversed as $\lambda^{*}$ increases.

An additional comparison of the differences in the geometric properties of the
semi-classical and classical SW fluid is also performed, via an analysis on
the behavior of the $R$-Widom lines for both systems. As mentioned in the
introduction section, these lines are defined in the supercritical region as
the locus of the maxima for the isotherms of the scalar scalar curvature $R$;
this line separates a gas-like from a liquid like phase in the supercritical
region for the semi-classical and classical SW fluid. This comparison is
presented in Fig. 8 in the $(\lambda^{+}_{B},P^{+})$ representation, instead
of the usual temperature-pressure one, therefore, separation of phases is
inverted with respect to the usual case. The most relevant feature of these
plots is that when semi-classical and classical Widom lines are compared,
these lines are range dependent. For smaller ranges ($\lambda^{*}<2.0$), a
shorter Widom line is obtained for the semi-classical system. For instance,
when $\lambda^{*}=1.5$ the semi-classical line cuts around $P^{+}\approx 3$,
while its classical counterpart have a linear behavior up to $P^{+}\approx 4$.
This situation is reversed, however, as $\lambda^{*}$ is increased, as can be
seen for $\lambda^{*}=2.0$; in this case the longest Widom line is the semi-
classical. Therefore, for greater values of the SW range, the line which
separates the aforementioned liquid-like phase from the gas-like is more
persistent for the semi-classical system.

##  IV Conclusions

In this work, a short glance of the Thermodynamic Geometry of quantum fluids
was explored. Two models different were analyzed, a quantum hard-sphere fluid
whose Helmholtz free energy is obtained from Path Integrals Monte Carlo
simulations and a semi-classical square-well fluid, described by a quantum
hard-sphere repulsive interaction coupled with a classical attractive square-
well contribution. The curvature obtained from the QHS potential presents
important differences compared to its classical counterpart.

It is found that when quantum contributions are taken into account, the BPH
anomaly is partially reverted, in a region in the middle of the plane
$(\lambda_{B}^{*},\eta)$ where the considered QHS equation of state is valid.
For the regions of low and high densities, $R_{\text{QHS}}$ exhibits the same
anomalous behavior than its classical counterpart. It is not surprising that
at lower densities the classical anomalous behavior remains, however, it is
interesting that at the regime of high densities the QHS curvature tends to
return to the anomalous classical behavior since it resembles the behavior of
other quantum systems Markland et al. (2011b).

Regarding the geometric properties of the semi-classical SW fluid, when
compared to the classical one, interesting differences were found between both
systems. For instance, the critical point in temperature and density of the
semi-classical system is significantly lower that the classical one, which can
be noticed in the direct comparison of the supercritical curvatures depicted
in Fig. 6. This comparison is further explored at the difference
$R^{*}_{Q}-R^{*}_{C}$, where we have found, for each of the four cases
explored, that the semi-classical curvature is always greater than the
classical one. Finally, the supercritical $R$-Widom lines for both systems
were also determined and compared, showing that near the critical point both
lines practically overlap.Interestingly, it was found that the length of these
lines are also potential-range dependent, and the length behavior for ranges
below $\lambda^{*}=2.0$ is larger for the classical SW fluid, which is
reversed for ranges above this value. Then we can distinguish the liquid-like
phase from the gas-like one deeper into the supercritical region when compared
to the classical one.

It is clear that this research represents only a first step in the study of
quantum fluids within the framework of Thermodynamic Geometry, and more work
is needed in order to clarify the influence and consequences of quantum
effects in these systems.

###### Acknowledgements.

J. Torres-Arenas acknowledge support by University of Guanajuato through grant
174/2023 of Convocatoria Institucional de Investigación Científica 2023. L.F.
Escamilla-Herrera acknowledge support from CONAHCYT through posdoctoral grant:
Estancias Posdoctorales por México para la Formación y Consolidación de las y
los Investigadores por México.

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