# Finite-size effects in the two-dimensional BCS-BEC crossover

M. Lanaro1,2, G. Bighin1, L. Dell’Anna1,2,3 and L. Salasnich1,2,3,4
1Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università di Padova,
Via Marzolo 8, I-35131 Padova, Italy
2Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Via Marzolo 8,
I-35131 Padova, Italy
3Padua QTech Center, University of Padova, Via Gradenigo 6, I-35131 Padova,
Italy
4Istituto Nazione di Ottica del Consiglio Nazionale delle Ricerche, Unita di
Sesto Fiorentino, Via Carrara 2, I-50019 Sesto Fiorentino, Italy

###### Abstract

We study the finite-size effects on the BCS-BEC crossover in two dimensions,
occurring in confined fermionic superfluids. We analyze several thermodynamic
properties, such as the chemical potential, the energy gap and the superfluid
density, taking into account unavoidable quantum fluctuations, and, by means
of renormalization group procedure, we detect the putative Berezinskii-
Kosterlitz-Thouless phase transition at finite-size.

Bose and Fermi superfluids in lower spatial dimension constitute an extremely
active research topic, both theoretically and experimentally. Contrary to the
three-dimensional case, mean-field approaches are unreliable in two- and one-
dimensional systems and the inclusion of quantum fluctuations is crucial to
compare correctly the theoretical predictions with the experimental data and
Monte Carlo simulations Stoof2009Ultracold ; book-stringa . In two-dimensional
(2D) systems the Berezinskii-Kosterlitz-Thouless (BKT) phase transition marks
the morphing from a superfluid phase characterized by quasi-condensation to a
normal phase, in which vortex proliferation destroys progressively
superfluidity jose40 . The BKT phase transition has been recently observed
experimentally jochim ; enss ; moritz  in Fermi gases made of alkali-metal
atoms, and widely investigated theoretically by several groups sademelo ;
bighin ; bighin2017vortices . However an accurate study which goes beyond the
mean-field level for finite-size systems is still laking. The study of the
role of the confinement for Fermi gasses is an experimentally relevant but
theoretically challenging problem which requires a significant theoretical
effort in order to include finite-size effects and the quantum fluctuations.
In this Letter, inspired by the recent experimental achievement of box
potentials box1 ; box2 , we investigate this puzzling problem studying a 2D
Fermi gas in the crossover from the Bardeen-Cooper-Schrieffer (BCS) phase
bardeen1957microscopic  to the Bose-Einstein Condensation (BEC) phase
bose1924plancks ; einstein1924quantum . First we will see how, at zero
temperature, the energy $\Delta$ of the Cooper pairs and the chemical
potential $\mu$ of the fermionic system change along the BCS-BEC crossover due
to the presence of a finite-size confining box potential. We also determine
the BKT critical temperature by calculating the superfluid fraction and, then,
implementing the renormalization group Kosterlitz-Thouless equations
bighin2017vortices , taking into account the finite size of the system
benfatto2009broadening ; tononi2022topological ; sala-bkt .

Theoretical framework. We will use the binding energy $\epsilon_{B}$ of Cooper
pairs as tunable interaction parameter along the BCS-BEC crossover.
Remarkably, $\epsilon_{B}$ depends on the ratio $\mu/\Delta$ by the following
simple formula marini1998evolution ,
$\frac{\epsilon_{B}}{\epsilon_{F}}=2\frac{\sqrt{1+\left({\mu\over\Delta}\right)^{2}}-{\mu\over\Delta}}{\sqrt{1+\left({\mu\over\Delta}\right)^{2}}+{\mu\over\Delta}}$,
where $\epsilon_{F}=\pi\hbar n/m$ is the 2D Fermi energy of non-interacting
two-spin-component fermions with number density $n$ and individual mass $m$.
This expression is a mean-field result which, nevertheless, works also beyond
the mean-field level, including Gaussian fluctuations bighin ;
bighin2017vortices . As shown in Fig. 1, the ratio $\mu/\Delta$ ranges from
$-\infty$, when $\epsilon_{B}\to+\infty$ (BEC regime), to $+\infty$, when
$\epsilon_{B}\to 0$ (BCS regime). Thus, contrary to the 3D case, in the 2D
problem the Cooper pairs always form a bound state. In Fig. 1 we show how the
binding energy depends on $\mu/\Delta$. The crossover regime (green region) is
approximately chosen for $\mu/\Delta$ in the range $(-1,1)$ since at $\mu=0$
the Fermi surface is lost.

Figure 1: Binding energy $\epsilon_{B}/\epsilon_{F}$ versus parameter
$x_{0}=\mu/\Delta$. We show with two different boxes the regions of BCS
($x_{0}\xrightarrow{}\infty$) and BEC ($x_{0}\xrightarrow{}-\infty$). The
green region points the relevant values for the BCS-BEC crossover.

The finite-size effects will be included through an infrared cutoff in
momentum space ($k_{min}$), which corresponds to a maximum wavelength
($\Lambda_{max}$), equal to the length of the side of the box confinement $L$,
such that $L=\Lambda_{max}=2\pi/k_{min}$, in order to mimic achievable results
in real experiments. We will analyze the BCS-BEC crossover in two dimensions
going beyond the mean-field analysis (both mean-field calculations and details
of the beyond-mean-field approach are reported in the supplemental material
supplemental_material ). The putative Berezinskii-Kosterlitz-Thouless phase
transition for finite systems will be obtained by means of the renormalization
group (RG) procedure.

Figure 2: In the higher panel we show $\Delta/\epsilon_{F}$ versus
$\epsilon_{B}/\epsilon_{F}$, while in the lower $\mu/\epsilon_{F}$ versus
$\epsilon_{B}/\epsilon_{F}$ for different values of the cutoff, respectively
$L\sqrt{n}$=5 (blue squared points), 10 (red triangled points), 100 (green
rounded points) and $\infty$ (black plus points). Here L is the total length,
while $\sqrt{n}=1/d$, with $d$ inter-particle distance. In this way,
$L\sqrt{n}$ is an adimensional quantity.

Beyond Mean Field analysis. The role of fluctuations is crucial in low
dimensions. We aim at investigating how some quantities, such as the chemical
potential and the energy gap, change in the presence of Gaussian fluctuations.
From the gap equation, in presence of a cutoff, we get supplemental_material

$\Delta=\sqrt{\epsilon_{B}^{2}+2(\epsilon_{min}+\mu)\epsilon_{B}}$ (1)

where $\epsilon_{min}=\hbar^{2}k_{min}^{2}/(2m)$, $\mu$ is obtained by solving
the number equation at the mean-field level supplemented by the Gaussian
contribution, $-\frac{d\Omega_{\rm GF}(\mu)}{d\mu}$. $\Omega_{\rm GF}$ is the
beyond-mean-field gran potential which takes the form

$\Omega_{\rm GF}=\frac{1}{2\beta}\sum_{\textbf{q},m}\ln{(\text{det
}\mathbb{M}(\textbf{q},\text{i}\Omega_{m}))}$ (2)

where the matrix $\mathbb{M}(\textbf{q},\text{i}\Omega_{m})$ is the inverse
pair fluctuation propagator, reported in the supplemental material
supplemental_material . Following the approach in Ref. he2015quantum , we,
then, calculate $\mu$ looking at the maximum of the energy density
$E(\mu)=\Omega(\mu)+\mu n$, at a given density $n$, where
$\Omega(\mu)=\Omega_{MF}+\Omega_{GF}$, namely it is composed by the mean-field
term and the contribution coming from the Gaussian fluctuations. This
procedure for determining the chemical potential beyond the mean-field
approach at finite size is computationally demanding and requires a big
computational effort cloud  and polynomial fit to smooth noisy data.

Finally, let us consider the superfluid density. In order to go beyond the
mean-field analysis we have to consider also the collective excitations, so
that the superfluid density reads

$n_{s}=n-n_{sp}-n_{col}$ (3)

where $n$ is the total density, $n_{sp}$ is the mean-field fermionic single
particle contribution to the normal density (see Res. supplemental_material )
while $n_{col}$ is the beyond-mean-field contribution coming from the bosonic
collective modes

$n_{col}=\frac{\beta}{2}\int_{k_{m}}^{\infty}\frac{d^{2}\bf{k}}{(2\pi)^{2}}\,{\hbar^{2}k^{2}}{m}\frac{\exp[\beta{\cal
E}_{col}({\bf k})]}{\left(\exp[\beta{\cal E}_{col}({\bf k})]-1\right)^{2}}$
(4)

If we make the following ansatz

${\cal E}_{col}({\bf k})\simeq\hbar c_{s}k$ (5)

where $c_{s}$ is the sound velocity, which in its turn, depends on
$\epsilon_{B}$, we can calculate Eq. (4) and, defining $x_{m}=\beta\hbar
c_{s}k_{m}$, we get

$\displaystyle
n_{col}=\frac{m}{4\pi\hbar^{2}c_{s}^{4}}\frac{1}{\beta^{3}}\left[6\,\textrm{Li}_{3}(e^{x_{m}})-6x_{m}\,\textrm{Li}_{2}(e^{x_{m}})\right.$
$\displaystyle\left.-x_{m}^{2}\left(\frac{x\,e^{x_{m}}}{1-e^{x_{m}}}+3\ln[1-e^{x_{m}}]\right)\right]$
(6)

In the limit $x_{m}\rightarrow 0$, namely for $k_{m}\rightarrow 0$, for an
infinite system, only the first term survives and we get

$n_{col}=\frac{3m}{2\pi\hbar^{2}c_{s}^{4}}\frac{\zeta(3)}{\beta^{3}}$ (7)

which is the maximum value for $n_{col}$. We see, therefore, that the
collective excitations further contribute to suppressing the superfluid
density. However such a suppression is less effective for finite size systems
and, therefore, the superfluid density is expected to be enhanced by finite-
size effects.

Berezinskii-Kosterlitz-Thouless phase transition. From the Mermin-Wagner-
Hohenberg theorem hohenberg1967existence ; mermin1966absence  we know that in
two dimensions the critical temperature at which we have the superconductive
phase transition is $T_{c}=0$ Altland2006CondensedMF . However, it is possible
to recognize a quasi-condensate phase and the temperature under which we can
see this type of behaviour is called Berezinskii-Kosterlitz-Thouless
temperature ($T_{BKT}$).
Another important feature characterizing this phase transition is the
universal jump of the superfluid density

$n_{s}(T^{-}_{BKT})\neq n_{s}(T^{+}_{BKT})=0$ (8)

that is due to the abrupt unbinding of two vortices that typically describes
this transition. Kosterlitz derived a set of renormalized group (RG) equations
able to describe the flow of vortices in the two dimensional Berezinskii-
Kosterlitz-Thouless phase transition kosterlitz1974critical . The RG equations
are the following

$\left\\{\begin{array}[]{@{}l@{}}\frac{d}{dl}K^{-1}_{l}(T)=4\pi^{3}y^{2}_{l}(T)\\\
\\\ \frac{d}{dl}y_{l}(T)=\left[2-\pi
K_{l}(T)\right]\;y_{l}(T)\end{array}\right.\ $ (9)

where $l=\ln{\left(r/d\right)}$, $r$ is the size of the superfluid system,
characterized by the presence of vortices, and $d$ can be identified with the
inter-particle distance.

Figure 3: (Main plot) The superfluid density $n_{s}/n$ versus $T/T_{F}$ at
fixed value of $\ln{\left(\epsilon_{B}/\epsilon_{F}\right)}=0.92$, for
$L\sqrt{n}$=5 (blue dashed line), 10 (red solid line), 100 (green dashed-
dotted line) and $\infty$ (black dotted line). (Inset) Zoom of the main plot.
The inflection points of the superfluid density are depicted by round points
while the triangular points are placed at the intersections of the slope at
the inflection points with the $T$-axis. Figure 4: The critical temperature
obtained through the intersection between the tangent to the superfluid
density at the inflection point and the horizontal line $n_{s}/n=0$ (the
triangular points in Fig. 3) i.e. $(T_{BKT}/T_{F})_{c_{1}}$ against
$\ln{(\epsilon_{B}/\epsilon_{F})}$ (see the main text).

An important quantity describing the system is the phase stiffness. A
fermionic superfluid can be described by the effective Hamiltonian of two
dimensional XY model, which reads

$H=\frac{J}{2}\int d^{2}\textbf{r}(\nabla\theta(\textbf{r}))^{2},$ (10)

where the phase stiffness $J$ is a function of the fermion-fermion attractive
strength and of the temperature bighin2017vortices . This quantity is related
to the superfluid density via fisher1973helicity

$J=\frac{\hbar^{2}}{4m}n_{s}$ (11)

and the superfluid density $n_{s}$ found through the Beyond Mean Field
analysis must be computed with the gap energies $\Delta$ in presence of the
cutoff. From the phase stiffness, we define, then, the following quantity

$K=\frac{J}{k_{B}T}$ (12)

The quantity $y$, instead, is related to the the core vortex energy $\mu_{c}$
such that

$y=\exp{\left[-\frac{\mu_{c}}{k_{B}T}\right]}$ (13)

Higher order terms amit1984field , slightly modify Eqs. (9) therefore can be
neglected. In the framework of the two-dimensional XY model the vortex core
energy is commonly expressed as $\mu_{c}=\frac{\pi^{2}}{4}J$, within the
Ginzburg-Landau theory of superconducting films minnhagen1985charge .
The starting point for the RG flow is determined by the bare superfluid
density, calculated at a fixed temperature $T$, which defines the initial
superfluid stifness $J_{0}(T)$. The initial conditions for the RG equations
are, therefore,

$\left\\{\begin{array}[]{@{}l@{}}K_{0}(T)=\frac{J_{0}(T)}{k_{B}T}\\\ \\\
y_{0}(T)=\exp{\left[-\frac{\pi^{2}}{4}\frac{J_{0}(T)}{k_{B}T}\right]}\end{array}\right.\
$ (14)

Superfluid density and critical temperature. We must notice that also the
adimensional scaling parameter $l$ of the RG equations will be influenced by
the cutoff benfatto2009broadening . The inter-particle distance $d$ is taken
as the square-root of the total density of particles $n$, i.e. $d=1/\sqrt{n}$,
while $L$ is the size our system imposed by a confining potential, therefore,
the maximum value of $l$ is

$l_{max}=\ln{\left(L/d\right)}=\ln{\left(L\sqrt{n}\right)}$ (15)

In Fig. 3 we show how the inclusion of the cutoff modifies the behavior of
$n_{s}/n$ as a function of $T/T_{F}$. The main feature is the disappearance of
the universal jump at the critical temperature, replaced by a curve that
becomes smoother and smoother upon decreasing the size of the system.
Our aim is, now, to obtain the critical temperature $T_{BKT}/T_{F}$ as a
function of the normalized binding energy $\epsilon_{B}/\epsilon_{F}$. In the
limit of infinite size, the slope of the tangent of the superfluid density
right at the critical temperature goes to infinity. For finite-size systems,
instead, the superfluid density is a smooth function. The criterion for
identifying the critical temperature, however, can be still based of the
maximum value of the slope of the tangent of the superfluid density. The
criterion has a certain degree of arbitrariness therefore we can decide to
follow two different procedures. In the first one, we focus on the
intersection between the tangent of $n_{s}/n$ at the inflection point, where
the absolute value of the slope is maximum, and the temperature-axis,
identifying this intersection as the critical temperature for a specific value
of $\epsilon_{B}/\epsilon_{F}$. As shown in Fig.3, the critical temperatures
obtained by this procedure are represented by the triangular points. The
slopes are the straight lines and we call these critical temperatures
$(T_{BKT}/T_{F})_{c_{1}}$. In this way we obtained $T_{BKT}/T_{F}$ as a
function of $\epsilon_{B}/\epsilon_{F}$ for different values of $L\sqrt{n}$,
shown in Fig.4. In the second procedure, also adopted in Ref.
tononi2022topological , we identify the critical temperatures as the
x-coordinates of the inflection points, where $n_{s}/n$ has the maximum
absolute value of the slope. In Fig.3 these points are the round marked ones.
The plots of the critical temperature obtained by this latter procedure,
called $(T_{BKT}/T_{F})_{c_{2}}$, as a function of the binding energy, for
different values of $L\sqrt{n}$, are showed in the supplemental material.

Conclusions. The main objective of this work has been to understand how the
size of the system, finite in feasible experiments, affects the BCS-BEC
crossover in two dimensions. The finite-size has been introduced through an
infrared cutoff in momentum space. Setting a minimum value to the wave-vector
corresponds, indeed, to setting a maximum value to the wave-length, which is
the size of the system. We have analyzed the effects of the finite size in
several thermodynamic properties, such as the chemical potential, the energy
gap and the superfluid density, going beyond the mean-field level by including
Gaussian quantum fluctuations. Finally, by means of a renormalization group
analysis and defining a couple of criteria useful to identify an effective
critical temperature tailored for finite sizes, we have characterized the
Berezinskii-Kosterlitz-Thouless phase transition in such systems. We believe
that in the near future our theoretical predictions could be compared with the
available experimental setups of strongly confined gases of fermionic atoms in
quasi-two dimensional geometry. In particular, the behavior of the measured
superfluid density and critical temperature will highlight the reliability and
accuracy of the beyond-mean-field theoretical method we have proposed.

Acknowledgements. CloudVeneto is acknowledged for the use of computing and
storage facilities. LD and LS are partially supported by “Iniziativa Specifica
Quantum” of INFN and by the Project ”Frontiere Quantistiche” (Dipartimenti di
Eccellenza) of the Italian Ministry for Universities and Research. ML and LS
are respectively fully and partially supported by the European Quantum
Flagship Project ”PASQuanS2”. LD and LS are partially supported by the
European Union-NextGenerationEU within the National Center for HPC, Big Data
and Quantum Computing [Spoke 10: Quantum Computing]. LS is partially supported
by the BIRD Project ”Ultracold atoms in curved geometries” of the University
of Padova and by PRIN Project ”Quantum Atomic Mixtures: Droplets, Topological
Structures, and Vortices”. LD acknowledges support from the Project n.
BIRD211534 granted by the University of Padova.

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