# Superconductivity from incoherent Cooper pairs in strong-coupling regime

Alexander A. Zyuzin  QTF Centre of Excellence, Department of Applied Physics,
Aalto University, FI-00076 Aalto, Finland  A. Yu. Zyuzin  Ioffe
Physical–Technical Institute, 194021 St. Petersburg, Russia

###### Abstract

We propose a scenario for superconductivity at strong electron-electron
attractive interaction, in the situation when the increase of interaction
strength promotes the nucleation of the local Cooper pairs and forms a state
with a spatially phase incoherent Cooper pair order parameter. We show that
this state can be characterized by a pseudogap which is determined by the
electron scattering by phase fluctuations. At low temperatures, however, long-
range correlations between the regions with different phases become important
and establish global phase coherence hence superconductivity in the system. We
develop a mean-field theory to describe a phase transition between the
preformed Cooper pair and superconducting states. The superconducting
transition temperature and the upper critical magnetic field are shown to be
enhanced in the strong coupling case. The mean-field approach is justified by
the small value of the Ginzburg-Levanyuk parameter. This scenario of
superconductivity applies not only to conductors with parabolic bands but also
to the flat-band systems in which flat and dispersive bands coexist and
responsible for the Cooper pairs formation as well as their phase-
synchronization.

##  I Introduction

The study of superconductivity in systems with strong electron-phonon Migdal
(1958); Eliashberg (1960) or non-retarded pairing Nozières and Schmitt-Rink
(1985) interactions encounters the problem of ultraviolet divergence, commonly
solved by introducing a regularized electron-electron scattering length
Gor’kov (2016); Gor’kov and Melik-Barkhudarov (1961). The regularized theory
Engelbrecht  _et al._ (1997) is applied to discuss the crossover between weak
Bardeen-Cooper-Schrieffer (BCS) and strong Bose-Einstein condensation (BEC)
regimes, for instance, in ultra cold atoms and nuclear systems, as reviewed in
Ohashi  _et al._ (2020); Giorgini  _et al._ (2008); Strinati  _et al._ (2018).
However, strictly speaking, this theory can be justified either in the
situations when the scattering length of pairing interaction potential is
smaller than the Fermi wave length Nozières and Schmitt-Rink (1985), or when
the Fermi energy exceeds the Debye energy, as in the conventional BCS model.
In this paper, we propose a solution for the problem of ultraviolet divergence
in superconductors.

We investigate a scenario of high-temperature superconductivity with strong
attractive interaction between electrons in the intermediate BCS to BEC
crossover regime. Specifically, we consider a situation where the normal metal
state becomes unstable towards the formation of local Cooper pairs at
temperatures much higher than the superconducting transition temperature.

As demonstrated below, the ultraviolet divergence describes these local
correlations, representing a preformed Cooper pair state. This state manifests
as a system of small domains, each with a size on the order of the Fermi
wavelength, exhibiting spatially uncorrelated random phase of the pairing
order parameter. The coupling of these domains through Andreev reflections
plays a role in the emergence of phase synchronization and superconductivity
at elevated temperatures Zyuzin (2022).

Our model differs from the one involving Cooper pairs preformed on spatial
fluctuations due to disorder Sacépé  _et al._ (2020). Instead, our approach
aligns with the scenario of flat-band superconductivity, recently proposed for
systems with an exotic dispersion relation featuring a flat band in the
electron spectrum Zyuzin and Zyuzin (2022); Islam  _et al._ (2023).

The study of flat-band materials is considered to be of significant importance
for understanding the mechanisms of high-temperature superconductivity Volovik
(2018). One of the reasons for that is the singular density of states at the
flat-band, which might support elevated superconducting transition
temperatures Khodel’ and Shaginyan (1990); E. (1994); Imada and Kohno (2000);
Miyahara  _et al._ (2007); Kopnin  _et al._ (2011); Peotta and Törmä (2015).

On the other hand, it has recently been recognized that the flatness of the
band dispersion guarantees physics similar to that in the strong interaction
case of Cooper instability Zyuzin and Zyuzin (2022). In particular, the cut-
off for the ultraviolet divergent superconducting condensation energy can be
naturally determined by the width of the flat-band in momentum space Kopnin
_et al._ (2011), which, in extreme limit, extends up to edges of the first
Brillouin zone Peotta and Törmä (2015). Thus, in addition to the elevated
superconducting transition temperature, a non-BCS type of phase transition in
flat-band systems may be expected. Indeed, more recently, it has been noted
that the spatial locality of flat-band electrons can be a precursor of a
pseudogap state with an uncorrelated random phase of the Cooper pair order
parameter Zyuzin and Zyuzin (2022); Islam  _et al._ (2023). In the flat-band
scenario, what renders the system of preformed pairs superconducting is the
coupling between the pairs, ensured by electrons from the dispersive bands,
which inevitably correlate with the flat-bands in realistic materials Zyuzin
and Zyuzin (2022); Islam  _et al._ (2023).

Here in the case of conductors with parabolic bands at strong attractive
interaction, we find that the pairing instability is governed by two
physically distinct short and long-range scale electron-electron correlations,
the former being overlooked previously, supporting local pair binding at high
temperatures and long-range phase-synchronization between such preformed pairs
at low temperatures.

The rest of the paper is organized as follows: In section (II), we introduce a
model for superconductivity in the strong interaction limit. We emphasize the
significance of short and long-range scale correlations in the paring
instability and discuss solutions to the self-consistency equation for the
order parameter in both weak and strong interaction regimes. In section (III),
we present the mean-field theory of superconductivity in the strong coupling
case. This involves the computation of the mean-field transition temperature,
upper critical magnetic field, and analysis of the applicability of the mean-
field approach within the Ginzburg-Levanyuk framework. In section (IV),
explicit calculations of the electron self-energy, due to scattering by random
phase fluctuations, are provided. Our findings are summarized in section (V).

##  II Model

Here we consider three dimensional electron gas system in the presence of
attractive interaction between electrons. The conduction band electrons with
parabolic dispersion are described by the Hamiltonian $H=\int
d\mathbf{r}\psi_{\sigma}^{\dagger}(\mathbf{r})[-\bm{\nabla}^{2}_{r}/(2m)-\mu]\psi_{\sigma}(\mathbf{r}),$
where $\mu>0$ and $m$ are the chemical potential and effective mass of
electrons, $\psi^{{\dagger}}_{\sigma}(\mathbf{r}),\psi_{\sigma}(\mathbf{r})$
are the electron operators with spin projection $\sigma=\uparrow,\downarrow$
($\hbar=k_{\mathrm{B}}=1$ units are used henceforth). The interaction
Hamiltonian given by

$\displaystyle H_{\mathrm{int}}=\int
d\mathbf{r}d\mathbf{r}^{\prime}\psi_{\uparrow}^{\dagger}(\mathbf{r})\psi_{\downarrow}^{\dagger}(\mathbf{r}^{\prime})U(\mathbf{r}-\mathbf{r}^{\prime})\psi_{\downarrow}(\mathbf{r}^{\prime})\psi_{\uparrow}(\mathbf{r}),~{}~{}$
(1)

is modelled by a potential well of size $a$ and depth $U>0$

$U(\mathbf{r})=-U\theta(a-r).$ (2)

We consider a short-range potential with a radius smaller than the Fermi
wavelength of electrons, $a<\lambda_{\mathrm{F}}\equiv 2\pi/\sqrt{2m\mu}$. In
the BCS theory, conventionally, the pairing potential is taken to be local.
However, in the strong coupling limit, it suffices to incorporate finite-size
effects to accurately account for the ultraviolet divergence.

As derived in Appendix (A), the potential (2) creates a two-particle bound
state provided:

$U>\frac{\pi^{2}}{4ma^{2}}\equiv
U_{c}=\frac{\mu}{2}\left(\frac{\lambda_{\mathrm{F}}}{2a}\right)^{2}\geq\mu.$
(3)

This means that the system becomes unstable towards the local pair binding at
the critical strength of attraction $U=U_{c}$. At $U>U_{c}$, the binding
energy increases as

$E_{0}=U_{c}\left[\frac{\pi}{4}\left(1-\frac{U}{U_{c}}\right)\right]^{2}.$ (4)

Interestingly, the increase in the mass of quasiparticles suppresses the
interaction threshold. Thus, the critical interaction strength must be smaller
in heavy-fermion materials or artificial flat-band system.

Mathematically, the emergence of pair instability can be uncovered by
investigating the divergent poles in the particle-particle scattering ladder
or equivalently by demonstrating the existence of a nontrivial solution for
the order parameter $\Delta(\mathbf{R},\mathbf{r})$ in the equation (Appendix
(A))

$\Delta(\mathbf{R},\mathbf{r})=U\int d\mathbf{R}^{\prime}\int_{r^{\prime}\leq
a}d\mathbf{r}^{\prime}\Pi\left(\mathbf{R}-\mathbf{R}^{\prime},\frac{\mathbf{r}-\mathbf{r}^{\prime}}{2}\right)\Delta(\mathbf{R}^{\prime},\mathbf{r}^{\prime}),$
(5)

where $\mathbf{R}$ and $\mathbf{r}$ are the center of mass and relative
coordinates, respectively. The length of lowercase vectors
$(\mathbf{r},\mathbf{r}^{\prime})$ is bounded by the radius of the potential
$r,r^{\prime}\leq a$. The kernel of integral Eq. (5) is defined as

$\Pi(\mathbf{R},\mathbf{r})=T\sum_{\omega_{n}}G_{\omega_{n}}(\mathbf{R}+\mathbf{r})G_{-\omega_{n}}(\mathbf{R}-\mathbf{r}),$
(6)

where $T$ is the temperature and the Green function in the Matsubara frequency
$\omega_{n}=(2n+1)\pi T$ and spatial coordinate representation
$G_{\omega_{n}}(\mathbf{R})=\int\frac{d^{3}p}{(2\pi)^{3}}\exp(i\mathbf{p}\cdot\mathbf{R})(i\omega_{n}-\xi_{\mathbf{p}})^{-1}$,
in which $\xi_{\mathbf{p}}=\mathbf{p}^{2}/2m-\mu$, is given by

$\displaystyle G_{\omega_{n}}(\mathbf{R})=-\frac{m}{2\pi
R}\exp\left\\{i\mathrm{sgn}(\omega_{n})\sqrt{1+\frac{i\omega_{n}}{\mu}}\frac{2\pi
R}{\lambda_{\mathrm{F}}}\right\\}.$ (7)

###  II.1 The separation of scales

The calculation of the kernel (6) uncovers a separation of scales in the
superconductivity problem at $a<\lambda_{F}$, as we shall demonstrate below.
This separation is based on the asymptotic coordinate dependence of
$\Pi(\mathbf{R},\mathbf{r})$ on small and large length scales with respect to
$\lambda_{F}$.

At $\lambda_{F}>R>r\approx a$, in Appendix (A) it is shown that the kernel (6)
diverges as

$\Pi(\mathbf{R},\mathbf{r})=\frac{m}{(2\pi)^{3}}\frac{1}{(R^{2}+r^{2}/4)^{2}}-\frac{m^{3}\lambda_{F}T^{2}}{24\pi^{2}R}.$
(8)

This is the short-range part of the kernel, which gives the ultraviolet-
diverging contribution to (5) at small distances. The first term in expression
(8) can be evaluated by setting the chemical potential in the Green functions
(7) to zero, as in the problem of two-particle scattering. The second term
describes a temperature-dependent correction. We note that expression (8) is
cut at short distances $r\sim a$ by the radius of the pairing potential.

In the BCS weak coupling limit, the local contribution does not lead to Cooper
instability and is conventionally incorporated into the renormalized
interaction constant, see for example Gor’kov (2016); Gor’kov and Melik-
Barkhudarov (1961). On the other hand, local correlations are detrimental to
the pairing instability in the strong interaction limit, with a flat-band
superconductor being a primary example Volovik (2018).

At $R>\lambda_{\mathrm{F}}$, the long-range part of the kernel is given by

$\Pi(\mathbf{R},\mathbf{r})=\frac{m}{2\pi\lambda_{\mathrm{F}}R^{2}\ell_{\mathrm{T}}}\frac{1}{\sinh(2\pi
R/\ell_{\mathrm{T}})}.$ (9)

Physically, it is responsible for the long-range correlations between the
Cooper pairs as it exponentially decays on thermal length
$\ell_{\mathrm{T}}=v_{\mathrm{F}}/T$. Mathematically, at large distances
($R>\lambda_{\mathrm{F}}$), the dependence on $r$ in (6) might be neglected,
which allows us using notation $\Pi(\mathbf{R})$ for
$\Pi(\mathbf{R},\mathbf{r})$. Taking this into account, at lengths smaller
than the thermal length $\ell_{\mathrm{T}}>R$, expression in (9) may be
brought to the form

$\Pi(\mathbf{R})=\frac{m}{4\pi^{2}\lambda_{\mathrm{F}}R^{3}}.$ (10)

To sum up, we find that the kernel (6) contains two qualitatively distinct
assymptotics (8) and (10), which describe short and long-range correlations
between the particles in the Cooper channel. We want to highlight that the
obtained separation of scales is strikingly comparable to the one found in the
exotic flat energy band scenario, thus making the latter resemble the problem
with a simple parabolic electron energy dispersion.

###  II.2 Weak and strong interactions

Let us now investigate the manifestation of scale separation in the weak and
strong interaction regimes. In Appendix (A), we show that solution of (5)
might be represented as

$\Delta(\mathbf{R},\mathbf{r})=\frac{\Delta(\mathbf{R})}{r}\sin\left(\frac{\pi
r}{2a}\sqrt{\frac{U}{U_{c}}}\right),$ (11)

where $\Delta(\mathbf{R})$ changes slowly on the scale of the wave-length
$\lambda_{\mathrm{F}}$, while the $r$-term describes the position dependence
inside the potential well as $r<a$. We will investigate the self-consistency
equation for $\Delta(\mathbf{R})$ in the weak and strong interaction regimes
by considering local and nonlocal parts of the kernel as perturbations,
respectively.

In Appendix (A), from Eq. (5), we derive an equation that governs the
superconducting transition temperature at $\mu\gg T$:

$\Delta(\mathbf{R})=\frac{U}{U_{\mathrm{c}}}\left\\{1+\frac{32a}{\pi^{2}\lambda_{\mathrm{F}}}\ln\left|\frac{\ell_{\mathrm{T}}}{\pi\lambda_{\mathrm{F}}}\right|\right\\}\Delta(\mathbf{R}).$
(12)

The first term on the right hand side of Eq. (12) comes from the local
contribution (8) after integration over the lengths smaller than
$\lambda_{\mathrm{F}}$ (temperature-dependent corrections were neglected
here). The second term represents the BCS logarithm, derived under the
condition that integration over the coordinate is bounded by
$\ell_{\mathrm{T}}$ and $\lambda_{\mathrm{F}}$ from above and below,
respectively.

The solution to Eq. (12) exists provided $U/U_{c}<1$. It is due to the
logarithm contribution at low temperatures,
$\ell_{\mathrm{T}}>\lambda_{\mathrm{F}}$. This is the weak interaction regime.
Thus, the metal to superconductor transition temperature is given by the BCS
expression

$T_{c}\simeq\frac{\mu}{\pi^{2}}\exp\left\\{-\frac{1-U/U_{c}}{\nu
U\Omega_{a}}\right\\},$ (13)

where $\Omega_{a}=4\pi a^{3}/3$ is the volume of the potential well (2),
$\nu=mp_{\mathrm{F}}/2\pi^{2}$ is the electron density of states per spin with
$p_{\mathrm{F}}=\sqrt{2m\mu}$ being the Fermi momentum. The local contribution
can be incorporated into the renormalized interaction constant,
$U/(1-U/U_{c})$, making the exponent proportional to the inverse electron-
electron scattering length Gor’kov (2016); Gor’kov and Melik-Barkhudarov
(1961).

However, as the electron-electron attraction increases ($U/U_{c}\rightarrow
1$), it becomes apparent that $\ell_{\mathrm{T}}$ must be on the order of
$\lambda_{\mathrm{F}}$ to fulfil Eq. (12). In this case, the correction
stemming from the local contribution in Eq. (12) becomes significant,
suggesting a reevaluation of Eq. (5).

In our model, under the strong coupling conditions, the system partitions into
domains with volumes proportional to $\lambda_{F}^{3}$, as illustrated in Fig.
1. Within each domain, the order parameter can be independently identified,
subject to a random phase. At $U/U_{c}\geq 1$, we find that equation for the
order parameter takes the form, Appendix (A)

$\Delta(\mathbf{R})=\frac{U}{U_{\mathrm{c}}}\left\\{1-\frac{(4\pi)^{2}aT^{2}}{3\lambda_{\mathrm{F}}\mu^{2}}\right\\}\Delta(\mathbf{R}).$
(14)

As a result, at $U\gtrsim U_{c}$, we obtain the crossover temperature for the
formation of the local pairing instability

$T_{\mathrm{PG}}=\frac{\mu}{4\pi}\left(\frac{3\lambda_{\mathrm{F}}}{a}\right)^{1/2}\left(1-\frac{U_{c}}{U}\right)^{1/2}.$
(15)

In derivation of Eq. (14) we neglect the long-range nonlocal correlations
(10), as they contribute on lengths greater than $\lambda_{\mathrm{F}}$ and
average out due to random phase of the order parameter at such scales. We will
later investigate the role of nonlocality in establishing long-range
correlations among the domains.

It is also interesting to note that $T_{\mathrm{PG}}$ is larger than two
particle binding energy (4)

$\frac{E_{0}}{T_{\mathrm{PG}}}\sim\left(\frac{U}{U_{c}}\right)^{2}\left[\left(1-\frac{U_{c}}{U}\right)\frac{\lambda_{\mathrm{F}}}{a}\right]^{3/2}<1.$
(16)

Therefore, there might be only a small fraction of bound particles at
temperatures near the crossover $T_{\mathrm{PG}}$.

We emphasize that the local nature of strong coupling instability implies that
the system is not in a phase-coherent superconducting state, but rather
contains preformed Cooper pairs with spatially uncorrelated random phase. As
the temperature decreases, the correlations establish global phase coherence.
In the upcoming section, we will analyze the superconducting transition
temperature within the framework of the preformed Cooper pair model.

##  III  Mean field theory

The Hubbard-Stratonovich transformation allows to reduce the four-fermion
interaction action in the stationary approximation to the functional over the
complex bosonic field $\Delta(\mathbf{r}_{1};\mathbf{r}_{2})$ (here we
consider static approximation with time-independent $\Delta$)

$\displaystyle
S=\int_{\mathbf{r},\tau}\overline{\Psi}_{\sigma}(\mathbf{r},\tau)\left[\partial_{\tau}-\frac{\bm{\nabla}^{2}}{2m}-\mu\right]\Psi_{\sigma}(\mathbf{r},\tau)~{}~{}$
$\displaystyle+\int_{\mathbf{r}_{1},\mathbf{r}_{2},\tau}\bigg{\\{}\frac{|\Delta(\mathbf{r}_{1};\mathbf{r}_{2})|^{2}}{U}-\Delta^{*}(\mathbf{r}_{1};\mathbf{r}_{2})\Psi_{\downarrow}(\mathbf{r}_{2},\tau)\Psi_{\uparrow}(\mathbf{r}_{1},\tau)$
$\displaystyle-\Delta(\mathbf{r}_{1};\mathbf{r}_{2})\overline{\Psi}_{\uparrow}(\mathbf{r}_{2},\tau)\overline{\Psi}_{\downarrow}(\mathbf{r}_{1},\tau)\bigg{\\}}\theta(a-|\mathbf{r}_{1}-\mathbf{r}_{2}|).$
(17)

It suffices to separate the spatial variables
$(\mathbf{r}_{1}+\mathbf{r}_{2})/2=\mathbf{R}$ and
$\mathbf{r}_{1}-\mathbf{r}_{2}=\mathbf{r}$. After integrating out the fermions
$\overline{\Psi}_{\sigma}(\mathbf{r},\tau),\Psi_{\sigma}(\mathbf{r},\tau)$ in
(III), we arrive at the bosonic action for $\Delta(\mathbf{R},\mathbf{r})$.

In the strong interaction case due to local contribution the typical size of
the Cooper pair is of the order of Fermi wave-length. Hence, we divide the
space volume into equal domains with volume
$\lambda_{\mathrm{F}}\times\lambda_{\mathrm{F}}\times\lambda_{\mathrm{F}}\equiv\Omega_{\mathrm{F}}$.
We choose to centre these domains at points $\mathbf{R}_{N}$ and numerate as
$N=1,2,3...$ Using (11), we discretize the field $\Delta(\mathbf{R})$ by
introducing $\Delta(N)\equiv\Delta(\mathbf{R}_{N})$ at $N$-th domain and
integrate over the coordinate $\mathbf{r}$ around points $\mathbf{R}_{N}$ in
the bosonic action. In this model, $\Delta(N)$ has a random uncorrelated phase
on each domain. Including the normalization term $\Omega_{a}$ into the
definition of $\Delta(N)$, one obtains the following expression

$\displaystyle TS$ $\displaystyle=$
$\displaystyle\sum_{N}\bigg{\\{}\frac{\Omega_{\mathrm{F}}}{\Omega_{a}}\left(\frac{1}{U}-\frac{1}{U_{c}(T)}\right)|\Delta(N)|^{2}+b|\Delta(N)|^{4}$
(18) $\displaystyle-$
$\displaystyle\Omega^{2}_{\mathrm{F}}\sum_{N^{\prime}\neq
N}\Delta(N)\Pi(N,N^{\prime})\Delta^{*}(N^{\prime})\bigg{\\}},$

where
$U_{c}(T)=U_{c}\left\\{1-\frac{(4\pi)^{2}aT^{2}}{3\lambda_{\mathrm{F}}\mu^{2}}\right\\}^{-1}$.
The derivation of the quadratic term is given in Appendix (A). We also take
into account quartic term $b\propto 1/\mu^{3}$ as the $1/U-1/U_{c}(T)$ term
can change sign. The local contribution to the quartic term is given by

$b=T\sum_{\omega_{n}}\int_{r_{i}<\lambda_{\mathrm{F}}}G_{\omega_{n}}(\mathbf{r}_{12})G_{-\omega_{n}}(\mathbf{r}_{23})G_{\omega_{n}}(\mathbf{r}_{24})G_{-\omega_{n}}(\mathbf{r}_{41}),$
(19)

where $\mathbf{r}_{ij}=\mathbf{r}_{i}-\mathbf{r}_{j}$ is introduced for
brevity.

The last term on the right hand side of (18) describes the interaction between
the regions $N$ and $N^{\prime}$. At
$\lambda_{\mathrm{F}}<|\mathbf{R}_{N}-\mathbf{R}_{N^{\prime}}|$, the kernel is
given by

$\Pi(N,N^{\prime})=\nu\frac{\mathrm{csch}(2\pi|\mathbf{R}_{N}-\mathbf{R}_{N^{\prime}}|/\ell_{\mathrm{T}})}{2\ell_{\mathrm{T}}|\mathbf{R}_{N}-\mathbf{R}_{N^{\prime}}|^{2}}.$
(20)

Expression (20) describes the long-range interaction between the domains,
being exponentially suppressed at lengths larger than the thermal length
$\ell_{\mathrm{T}}$.

Figure 1:  (a) Schematics of the system divided into equal domains with volume
$\lambda_{\mathrm{F}}\times\lambda_{\mathrm{F}}\times\lambda_{\mathrm{F}}$,
where $\lambda_{\mathrm{F}}$ is the Fermi wave-length. At strong interaction
$U>U_{c}$, one can define the local order parameter $\Delta_{i}$ on each
domain separately. The phases of the order parameter on different domains are
uncorrelated. This is the high temperature pseudogap state. The correlations
between the domains on larger length scales
$\propto\ell_{\mathrm{T}}=v_{\mathrm{F}}/T$ establish the phase coherence in
the system at low temperatures. (b) Schematics of the phase diagram in the
plane of normalized temperature $T/\mu$ and interaction constant $U/U_{c}$.
One distinguishes between the normal metal, superconductor, and phase-
incoherent gapless and pseudogap states. The crossover between the normal
metal and pseudogap states (dashed curve) is defined by temperature
$T_{\mathrm{PG}}$. The superconducting transition is defined by temperature
$T_{c}$ (solid curve). The BCS regime establishes in the weak interaction
limit while it changes to a phase-synchronization regime at strong
interaction.

At the vicinity of temperature (15), using (18), we find the amplitude of the
local order parameter $\Delta(N)=|\Delta_{\mathrm{PG}}|\exp[i\phi(N)]$ in the
form

$|\Delta_{\mathrm{PG}}|^{2}=\frac{1}{2b}\frac{\Omega_{\mathrm{F}}}{\Omega_{a}}\left(\frac{1}{U_{c}(T)}-\frac{1}{U}\right).$
(21)

Restoring the temperature dependence, we obtain

$|\Delta_{\mathrm{PG}}|\propto
T_{\mathrm{PG}}\sqrt{1-T^{2}/T^{2}_{\mathrm{PG}}}.$ (22)

The energy of interaction between two nearest neighbour domains with random
phases $\phi_{1}$ and $\phi_{2}$ is given by $\propto
E_{\mathrm{int}}\cos(\phi_{2}-\phi_{1})$, where
$E_{\mathrm{int}}\propto|\Delta_{\mathrm{PG}}|^{2}\Pi(\lambda_{\mathrm{F}})\propto\frac{\lambda_{\mathrm{F}}}{a}\mu(1-U_{c}/U)$.
We note that $E_{\mathrm{int}}$ is smaller than the crossover temperature
$T_{\mathrm{PG}}$. Comparing the two, we obtain
$E_{\mathrm{int}}/T_{\mathrm{PG}}\propto[\frac{\lambda_{\mathrm{F}}}{a}(1-U_{c}/U)]^{1/2}<1$.
Under this condition, two neighbouring domains might be considered non-
interacting, thus possessing uncorrelated phases.

Having identified the amplitude of the local order parameter and the crossover
temperature for local instability, let us now investigate the transition
between preformed Cooper pair state and the superconducting state tuned by
long-range correlations (20).

At the superconducting phase transition, we have a nonzero average for the
order parameter defined as

$\displaystyle\langle\Delta(N)\rangle=\frac{\int
D[\Delta,\Delta^{*}]\Delta(N)\exp\\{-S[\Delta,\Delta^{*}]\\}}{\int
D[\Delta,\Delta^{*}]\exp\\{-S[\Delta,\Delta^{*}]\\}}.$ (23)

To evaluate the superconducting transition temperature in Eq. (23), we develop
the mean-field theory of superconductivity mapping it to the Langevin theory
of magnetism Mal’shukov (1991); Zyuzin (2022). Namely, substituting
$\Delta(N)$ with its mean value $\langle\Delta(N)\rangle$ at each domain
except $N=0$, allows one to obtain the mean-field approximation for the action
$S[\Delta,\Delta^{*}]$ as

$\displaystyle
TS_{\mathrm{MF}}=\left(\frac{1}{U}-\frac{1}{U_{c}(T)}\right)\frac{\Omega_{\mathrm{F}}}{\Omega_{a}}|\Delta(0)|^{2}+b|\Delta(0)|^{4}$
(24)
$\displaystyle-\Omega_{\mathrm{F}}^{2}\sum_{N}\Pi(0,N)[\Delta(0)\langle\Delta^{*}(N)\rangle+\langle\Delta(N)\rangle\Delta^{*}(0)].$

Subsequently, we simplify Eq. (23) utilizing the action given in (24) under
the condition of $\langle\Delta(N)\rangle\rightarrow 0$. As a result, Eq. (23)
yields

$\displaystyle
1=\frac{\langle|\Delta(0)|^{2}\rangle}{T}\Omega_{\mathrm{F}}^{2}\sum_{N}\Pi(0,N),$
(25)

where

$\displaystyle\langle|\Delta(0)|^{2}\rangle=\frac{\int_{0}^{\infty}dxxe^{-\frac{1}{T}[(\frac{1}{U}-\frac{1}{U_{c}})\frac{\Omega_{\mathrm{F}}}{\Omega_{a}}x+bx^{2}]}}{\int_{0}^{\infty}dxe^{-\frac{1}{T}[(\frac{1}{U}-\frac{1}{U_{c}})\frac{\Omega_{\mathrm{F}}}{\Omega_{a}}x+bx^{2}]}}.$
(26)

The solution of Eq. (25) gives us a superconducting transition temperature
$T_{c}$. To investigate the solution of Eq. (25) in the regimes of weak and
strong interactions, we shall further consider the continuum limit by
substituting $\Omega_{\mathrm{F}}\sum_{N}\rightarrow\int d\mathbf{R}$.

###  III.1 Weak coupling regime

In the weak coupling limit $U\ll U_{c}$, we can set $U_{c}(T)\equiv U_{c}$.
Thus, at
$1/U-1/U_{c}\gg\sqrt{bT\Omega_{a}/\Omega_{\mathrm{F}}}\propto\sqrt{T/U_{c}^{3}}$,
which allows to neglect the $b$-term in (26), we reproduce the standard BCS
result (13) for the superconductor-metal transition temperature.

Figure 2:  Schematics of the phase transition between a zero temperature
superconducting state and high-temperature normal metal state via a preformed-
pair state.

###  III.2 Strong coupling regime

Analyzing Eq. (25) in the strong coupling regime, at
$1/U_{c}(T)-1/U>\sqrt{bT\Omega_{a}/\Omega_{\mathrm{F}}}$, the transition
temperature between the preformed Cooper pair state and the superconducting
state yields
$T_{c}=2|\Delta_{\mathrm{PG}}|^{2}\Omega_{\mathrm{F}}\nu\ln|\mu/\pi^{2}T_{c}|$.
Here, substituting the expression for $|\Delta_{\mathrm{PG}}|$ from Eq. (21),
we obtain

$T_{c}=\left(\frac{1}{U_{c}}-\frac{1}{U}\right)\frac{\Omega^{2}_{\mathrm{F}}}{b\Omega_{a}}\nu\ln|\mu/\pi^{2}T_{c}|.$
(27)

In (27), we set $U_{c}(T_{c})\equiv U_{c}$, because the temperature of
superconducting phase coherence, $T_{c}$, is smaller than the temperature of
the Cooper pair formation, $T_{\mathrm{PG}}$, given by (15). One notices the
non-exponential dependence of $T_{c}$ on the density of states, $\nu$, which
is in contrast to the BCS result. The superconducting transition temperature
is proportional to the energy of interaction between two nearest neighbour
domains
$T_{c}\propto\mu(\lambda_{\mathrm{F}}/a)\left(1-U_{c}/U\right)\ln|a/\lambda_{\mathrm{F}}(1-U_{c}/U)|\propto
E_{\mathrm{int}}$, where $(\lambda_{\mathrm{F}}/a)\left(1-U_{c}/U\right)<1$.
The schematics of the phase diagram is shown in Fig. (1).

###  III.3 Low temperature limit at strong coupling

The theory of superconductivity at strong coupling is developed for
temperatures much larger than the superconducting gap. However, we note that
at large distances, $\Pi(\mathbf{R})$, which describes Andreev coupling among
the preformed pairs, is cut by the thermal length $R\sim\ell_{\mathrm{T}}$ and
diverges at $T\rightarrow 0$. Within the textbook BCS model, at zero
temperature, the cutoff changes as $T\rightarrow|\Delta|$. Similarly, under
the substitution $T\rightarrow\sqrt{T^{2}+|\Delta_{\mathrm{PG}}|^{2}}$, our
approach might be continued to zero temperatures as well.

In this low temperatures regime, the superconducting energy gap is given by
the expression (26), leading to $\langle\Delta\rangle=\Delta_{\mathrm{PG}}$.
Noting different temperature dependencies of $\langle\Delta\rangle$ and
$\Delta_{\mathrm{PG}}$, we find an interesting phase diagram, as shown in Fig.
(2). Namely, the averaged order parameter vanishes at the transition to the
preformed pair state $T\rightarrow T_{c}$ as

$\langle\Delta\rangle\sim\sqrt{1-T/T_{c}},$ (28)

while $\Delta_{\mathrm{PG}}$ decreases only at $T\sim T_{\mathrm{PG}}>T_{c}$.
This is a two stage process: with increasing temperature, the phase disorder
terminates superconducting state. As the temperature increases further, the
number of pre-bound electron states decreases terminating local
$\Delta(\textbf{R})$. Let us also emphasize, that we keep quartic term $b$ not
because of the large temperatures, but because of small parameter
$|U-U_{c}|\ll U_{c}$.

We also note that the investigation of quantum fluctuations at strong
interaction is beyond the present work.

###  III.4 Upper critical magnetic field

It is interesting to consider the orbital pair-braking effect of the magnetic
field on $T_{c}$ in the quasiclassical approximation. In this case, the mean-
field order parameter $\langle\Delta(N)\rangle$ satisfies the following
equation

$\displaystyle\frac{T_{c}-T}{T_{c}}\Omega_{\mathrm{F}}\sum_{N^{\prime}}\Pi(0,N^{\prime})\langle\Delta(N)\rangle$
$\displaystyle+D(\bm{\nabla}_{R_{N}}-2ie\mathbf{A}/c)^{2}\langle\Delta(N)\rangle=0,$
(29)

where $e<0$ is the electron charge, $\mathbf{A}$ is the vector potential, and
$D=\Omega_{\mathrm{F}}\sum_{N}R_{N}^{2}\Pi(0,N)=(7\zeta(3)/12\pi)\ell^{2}_{\mathrm{T}}\approx
0.2\ell^{2}_{\mathrm{T}}$ is the superconducting phase stiffness.

At $U\ll U_{c}$, one reproduces the standard result for the upper critical
magnetic field $H_{c2}=(\Phi_{0}/2\pi\xi_{0}^{2})(1-T/T_{c})$, where
$\Phi_{0}=\pi\hbar c/|e|$ is the magnetic superconducting flux quantum (here
we restored $\hbar$) and
$\xi_{0}=\sqrt{7\zeta(3)/(48\pi^{2})}(v_{\mathrm{F}}/T_{c})\approx
0.1v_{\mathrm{F}}/T_{c}$ is the superconducting zero temperature coherence
length. However, we find that $H_{c2}$ logarithmically increases in the strong
interaction limit

$H_{c2}=\frac{\Phi_{0}}{2\pi\xi_{0}^{2}}\frac{T_{c}-T}{T_{c}}\ln\left|\frac{\mu}{\pi^{2}T_{c}}\right|.$
(30)

One observes that the average Fermi velocity defined as
$v_{\mathrm{F}}^{*}\equiv T_{c}\sqrt{\Phi_{0}/2\pi H_{c2}}\propto
0.1v_{\mathrm{F}}/\sqrt{\ln|\mu/\pi^{2}T_{c}|}$ is suppressed compared to the
BCS result.

###  III.5  Ginzburg-Levanyuk criterion

The Landau mean-field theory becomes invalid at the vicinity of the second-
order phase transition due to strong fluctuations of the order parameter. The
Ginzburg - Levanyuk criterion determines the region of temperatures (and other
parameters) in which such fluctuations can be neglected. This region can be
defined through the condition that the fluctuations of the order parameter are
small within the correlation volume, which is given here by
$\Omega_{c}\propto\ell_{\mathrm{T}}^{3}/\tau^{3/2}$, where $\tau=1-T/T_{c}$ at
$T<T_{c}$

$\frac{\int_{\Omega_{c}}d\mathbf{r}_{0}d\mathbf{r}_{1}\langle\Delta(\mathbf{r}_{0})\Delta(\mathbf{r}_{1})\rangle}{\Omega_{c}^{2}\langle\Delta(\mathbf{r})\rangle^{2}}<1,$
(31)

In Appendix (B), with the help of the Ornstein-Zernike theory Stanley (1987),
we demonstrate that for transition from the preformed pair state to the
superconducting state, condition (31) yields

$\tau>\left(\frac{\lambda_{\mathrm{F}}}{\ell_{\mathrm{T}}}\right)^{6}\sim\left(\frac{T_{c}}{\mu}\right)^{6}.$
(32)

To compare, for the clean superconductors in the weak coupling BCS regime, the
condition (31) imposes more stringent requirement
$\tau>\left(T_{c}/\mu\right)^{4}$, yet the ratio $T_{c}/\mu$ in weak coupling
is much smaller, Larkin and Varlamov (2005).

The Ginzburg-Levanyuk parameter has been estimated utilizing the Ornstein-
Zernike theory to justify the validity of the mean-field approach. It is
crucial to emphasize that the mean-field theory is applied to describe the
phase transition to superconducting state at $T_{c}$.

In the regime of strong interactions, the temperature $T_{\mathrm{PG}}$ does
not signify a phase transition but rather manifests a crossover from a normal
metal to the preformed pair region. The mean-field approach can not be
employed in this scenario.

Instead, we demonstrate that the attractive potential harbors a two-particle
bound state. At $T\propto T_{\mathrm{PG}}$, the temperature significantly
exceeds the energy of interaction between the preformed pairs situated in two
neighbouring domains. Consequently, at $T_{\mathrm{PG}}$, the domains can be
treated as non-interacting, thereby possessing uncorrelated phases.

##  IV Pseudogap phase

Having identified the phase-incoherent preformed pair state within the
temperature range $T_{\mathrm{PG}}>T>T_{c}$ under the strong coupling
conditions ($U>U_{c}$), we shall now investigate the fundamental spectral
properties of quasiparticles in this intriguing state.

Figure 3:  The self-energy due to electron scattering by random phase disorder
within Born approximation in Nambu representation and in the static limit (no
frequency transfer via bosonic correlator). The vertex represents a matrix in
Nambu space
$\hat{\Delta}=\mathrm{Re}[\Delta]\tau_{x}-\mathrm{Im}[\Delta]\tau_{y}$, in
which $\tau_{x,y,z}$ are the Pauli matrices in Nambu space and
$\Delta=\mathrm{Re}[\Delta]+i\mathrm{Im}[\Delta]$, while the solid line stands
for the Green function in Nambu space
$\hat{G}_{\omega_{n}}(\mathbf{p})=G_{\omega_{n}}(\mathbf{p})(1+\tau_{z})/2-G_{-\omega_{n}}(\mathbf{p})(1-\tau_{z})/2$.

As we discussed above, the preformed Cooper pair state can be seen as a
cluster of domains of with spatially random phase or the order parameter
$\Delta(\mathbf{R}_{N})$. The back action of the random phase on electrons can
be studied in analogy to the problem of electron scattering by random disorder
in metals. In our case the scattering of quasiparticles on
$\Delta(\mathbf{R}_{N})$ with zero phase average $\langle\Delta\rangle=0$ is
described by the Hamiltonian
$H_{\mathrm{int}}=\Omega_{\mathrm{F}}\sum_{N}[\Delta(\mathbf{R}_{N})\psi^{{\dagger}}_{\uparrow}(\mathbf{R}_{N})\psi^{{\dagger}}_{\downarrow}(\mathbf{R}_{N})+\mathrm{h.c.}]$.

At $T\rightarrow T_{c}$, to evaluate the correlation function of the
scattering potential, we express
$\Delta(\mathbf{R}_{N})=|\Delta_{\mathrm{PG}}|e^{i\phi(\mathbf{R}_{N})}$ and,
similarly to Eq. (23), obtain

$\displaystyle\frac{\langle\Delta(0)\Delta^{*}(\mathbf{R})\rangle}{|\Delta_{\mathrm{PG}}|^{2}}=\frac{\int_{0}^{2\pi}d\phi\cos(\phi)\exp\\{E_{\mathrm{J}}(R)\cos(\phi)/T\\}}{\int_{0}^{2\pi}d\phi\exp\\{E_{\mathrm{J}}(R)\cos(\phi)/T\\}},~{}~{}$
(33)

where $E_{\mathrm{J}}(R)$ is the energy of Josephson coupling between the two
domains

$E_{\mathrm{J}}(R)=T\frac{2\pi\mathcal{L}_{\mathrm{T}}^{3}}{\ell_{\mathrm{T}}R^{2}}\frac{1}{\sinh(2\pi
R/\ell_{\mathrm{T}})}.$ (34)

The length
$\mathcal{L}_{\mathrm{T}}=\lambda_{\mathrm{F}}(2|\Delta_{\mathrm{PG}}|^{2}/T\mu)^{1/3}$
is much smaller than the thermal correlation length $\ell_{\mathrm{T}}$,
$\mathcal{L}_{\mathrm{T}}/\ell_{\mathrm{T}}\propto(|\Delta_{\mathrm{PG}}|T/\mu^{2})^{2/3}\ll
1$.

The correlation function (33) can be written as the ratio of two modified
Bessel functions of first kind:

$\langle\Delta(0)\Delta^{*}(\mathbf{R})\rangle=|\Delta_{\mathrm{PG}}|^{2}\frac{I_{1}(E_{\mathrm{J}}(R)/T)}{I_{0}(E_{\mathrm{J}}(R)/T)}.$
(35)

Using asymptotic of the Bessel functions, one observes that the phase exhibits
slow changes at small length scales $R<\mathcal{L}_{\mathrm{T}}$, namely
$\langle e^{i(\phi_{0}-\phi_{N}}\rangle\simeq 1$. However, in the case
$\ell_{\mathrm{T}}>R>\mathcal{L}_{\mathrm{T}}$, the correlation function
decays as power law $\propto\mathcal{L}_{\mathrm{T}}^{3}/R^{3}$, while at
$R>\ell_{\mathrm{T}}$ the decay is exponential,
$\propto\mathcal{L}_{\mathrm{T}}^{3}\exp(-2\pi
R/\ell_{\mathrm{T}})/(\ell_{\mathrm{T}}R^{2})$, indicating a phase loosening
at larger distances.

We are now in the position to evaluate the correction to the density of states
of quasiparticles due to scattering by phase disorder, given by the diagram in
Fig. (3). Performing Fourier transformation of the correlation function
$\langle\Delta\Delta^{*}\rangle_{\mathbf{q}}=\int
d\mathbf{R}\langle\Delta(\mathbf{R})\Delta^{*}(0)\rangle
e^{-i\mathbf{q}\cdot\mathbf{R}}$ in (35), we obtain equation for the Green
function in the Born approximation
$[i\omega_{n}-\xi_{\mathbf{p}}-\Sigma_{\omega_{n}}(\mathbf{p})]^{-1}G_{\omega_{n}}(\mathbf{p})=1$,
where the self-energy is given by

$\displaystyle\Sigma_{\omega_{n}}(\mathbf{p})=\int\frac{d^{3}q}{(2\pi)^{3}}\frac{\langle\Delta\Delta^{*}\rangle_{\mathbf{p}-\mathbf{q}}}{i\omega_{n}+\xi_{\mathbf{q}}}.$
(36)

Noting that $\mathcal{L}_{\mathrm{T}}/\lambda_{\mathrm{F}}<1$, or
equivalently, $|\Delta_{\mathrm{PG}}|<\sqrt{T\mu}$, we can take the asymptote
of the correlation function $\langle\Delta(0)\Delta^{*}(\mathbf{R})\rangle\sim
1/R^{3}$. Correction to the electron density of states per spin due to
electron scattering by phase disorder is given by

$\delta\nu(\omega)=-\frac{1}{\pi}\mathrm{Im}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{\Sigma_{\omega+i\delta}(\mathbf{p})}{(\omega-\xi_{\mathbf{p}}+i\delta)^{2}},$
(37)

which can be brought to the form

$\frac{\delta\nu(\omega)}{\nu}=-\frac{4|\Delta_{\mathrm{PG}}|^{2}}{v^{2}_{\mathrm{F}}\sqrt{1+\omega/\mu}}\int_{0}^{\infty}\frac{I_{1}(\frac{E_{\mathrm{J}}(r)}{T})}{I_{0}(\frac{E_{\mathrm{J}}(r)}{T})}\cos\left(\frac{2\omega}{v_{\mathrm{F}}}r\right)rdr,$
(38)

At frequencies $\mu>|\omega|$, subtracting the part,
$\delta\nu(0)/\nu=-4\pi^{2}|\Delta_{\mathrm{PG}}|^{4}/(T\mu^{3})$, we obtain

$\frac{1}{\nu}[\delta\nu(\omega)-\delta\nu(0)]=2\pi^{4}\frac{|\Delta_{\mathrm{PG}}|^{4}}{\mu^{4}}\frac{|\omega|}{T}.$
(39)

The frequency-independent correction has a negative sign, which together with
(39) gives a dip in the density of states, mimicking the pseudogap behaviour.

The linear frequency dependence,
$\delta\nu(\omega)-\delta\nu(0)\propto|\omega|/T$, holds in two-dimensional
superconductor case as well.

At temperatures $T\rightarrow T_{\mathrm{PG}}$, one cannot use the correlator
(33). Instead, the disorder averaging technique within the Gaussian
approximation gives the scattering time.

In conclusion to this section, we find the pseudogap state at temperatures
above the superconducting transition temperature $T\rightarrow T_{c}$.

##  V Conclusion

To summarize, we developed a model of superconductivity in conductors with a
parabolic band spectrum and strong attractive interactions between electrons.
We noted the significance of the separation of spatial scales in analyzing the
Cooper pair instability.

We demonstrated that the strong interaction potential, which might bind
electrons into pairs, results in a state characterized by a spatially random
phase of the order parameter. This inhomogeneous state lacks global phase
coherence. However, it can be characterized by the pseudogap arising from
electron scattering by phase disorder. At low temperatures, the long-range
correlation between regions possessing different phases plays an important
role. This correlation is responsible for phase-synchronization and,
therefore, superconductivity in the system. We emphasized that the preformed
Cooper pair model in conductors with parabolic bands and strong interaction
can be mapped to the model of superconductivity in flat-band systems.

Finally, we note that at the transition $\mu\rightarrow 0$, the size of the
Cooper pair becomes macroscopic, causing the contribution of nonlocal terms to
vanish. Therefore, we expect a different superconducting state compared to the
one considered here.

##  VI Acknowledgements

We would like to thank Mikhail Feigel’man for raising the question of the
Ginzburg-Levanyuk criterion for the validity of the mean-field theory. This
work was performed as part of the Academy of Finland Centre of Excellence
program (project 352925). A.A.Z. acknowledges support from the QuantERA II
Programme that has received funding from the European Union’s Horizon 2020
research and innovation programme under Grant Agreement No 101017733 and
funding from the Finnish-India consortia for research and education (FICORE).
We thank the Pirinem School of Theoretical Physics where our research was
initiated for warm hospitality.

##  Appendix A  Self-consistency equation

The linearized self-consistency equation is given by

$\displaystyle\Delta(\mathbf{r},\mathbf{r}^{\prime})\Theta(a-|\mathbf{r}-\mathbf{r}^{\prime}|)=UT\sum_{n}\int
d\mathbf{r}_{1}d\mathbf{r}_{2}\Theta(a-|\mathbf{r}_{1}-\mathbf{r}_{2}|)$
$\displaystyle\times\Theta(a-|\mathbf{r}-\mathbf{r}^{\prime}|)G_{\omega_{n}}(\mathbf{r}-\mathbf{r}_{1})\Delta(\mathbf{r}_{1},\mathbf{r}_{2})G_{-\omega_{n}}(\mathbf{r}^{\prime}-\mathbf{r}_{2}),$
(40)

where

$\displaystyle G_{\omega_{n}}(\mathbf{R})=-\frac{m}{2\pi
R}\exp\left\\{i\mathrm{sgn}(\omega_{n})\sqrt{1+\frac{i\omega_{n}}{\mu}}\frac{2\pi
R}{\lambda_{\mathrm{F}}}\right\\}$ (41)

is the Green function in Matsubara representation and
$\lambda_{\mathrm{F}}=2\pi/\sqrt{2m\mu}$ is the Fermi wavelength. Transforming
to the centre of mass $\mathbf{R}$ and relative $\mathbf{r}$ coordinates, we
obtain

$\displaystyle\Delta(\mathbf{R},\mathbf{r})=UT\sum_{n}\int
d\mathbf{R}^{\prime}\int_{r^{\prime}<a}d\mathbf{r}^{\prime}\Theta(a-r)\Delta\left(\mathbf{R}^{\prime},\mathbf{r}^{\prime}\right)$
$\displaystyle\times
G_{\omega_{n}}\left(\mathbf{R}-\mathbf{R}^{\prime}+\frac{\mathbf{r}-\mathbf{r}^{\prime}}{2}\right)G_{-\omega_{n}}\left(\mathbf{R}-\mathbf{R}^{\prime}-\frac{\mathbf{r}-\mathbf{r}^{\prime}}{2}\right).$
(42)

As in the main text, we introduce

$\displaystyle\Pi(\mathbf{R},\mathbf{r})=T\sum_{n}G_{\omega_{n}}\left(\mathbf{R}+\mathbf{r}\right)G_{-\omega_{n}}\left(\mathbf{R}-\mathbf{r}\right).$
(43)

Formal rewriting gives

$\displaystyle\Delta(\mathbf{R},\mathbf{r})=U\int
d\mathbf{R}^{\prime}\int_{r^{\prime}<a}d\mathbf{r}^{\prime}\Theta(a-r)$
$\displaystyle\times\Pi\left(\mathbf{R}-\mathbf{R}^{\prime},\frac{\mathbf{r}-\mathbf{r}^{\prime}}{2}\right)\Delta\left(\mathbf{R}^{\prime},\mathbf{r}^{\prime}\right).$
(44)

We separate the local and nonlocal contributions. We start with the local
term. At zero temperature, we find

$\displaystyle\Pi(\mathbf{R},\mathbf{r})=T\sum_{n}\frac{m^{2}}{4\pi^{2}}\frac{1}{|\mathbf{R}+\mathbf{r}||\mathbf{R}-\mathbf{r}|}$
$\displaystyle\times\exp\left\\{i\mathrm{sign}(\omega_{n})[\sqrt{2im\omega_{n}}|\mathbf{R}+\mathbf{r}|-\sqrt{-2im\omega_{n}}|\mathbf{R}-\mathbf{r}|]\right\\}$
$\displaystyle=\frac{m}{2\pi^{3}}\frac{1}{(|\mathbf{R}+\mathbf{r}|^{2}+|\mathbf{R}-\mathbf{r}|^{2})^{2}}.$
(45)

At $R<\lambda_{\mathrm{F}}$, the temperature-dependent correction can be
evaluated as

$\displaystyle\delta\Pi(\mathbf{R},\mathbf{r})$ $\displaystyle=$
$\displaystyle\frac{m^{2}}{4\pi^{2}R^{2}}\int_{-\mu}^{\mu}\frac{d\epsilon}{2\pi}\left[\tanh\left(\frac{\epsilon}{2T}\right)-\mathrm{sign}(\epsilon)\right]$
(46) $\displaystyle\times$
$\displaystyle\sin\\{p_{\mathrm{F}}R[\sqrt{1+\epsilon/\mu}-\sqrt{1-\epsilon/\mu}]\\}$
$\displaystyle\sim$ $\displaystyle-\frac{m^{3}\lambda_{F}T^{2}}{24\pi^{2}R}.$

The nonlocal contribution can be evaluated as

$\displaystyle\Pi(\mathbf{R},\mathbf{r})=T\sum_{n>0}\frac{m^{2}}{2\pi^{2}|\mathbf{R}+\mathbf{r}||\mathbf{R}-\mathbf{r}|}$
(47)
$\displaystyle\times\cos\left\\{\frac{2\pi}{\lambda_{\mathrm{F}}}(|\mathbf{R}+\mathbf{r}|-|\mathbf{R}-\mathbf{r}|)\right\\}$
$\displaystyle\times\exp\left\\{-\frac{\omega_{n}}{v_{\mathrm{F}}}(|\mathbf{R}+\mathbf{r}|+|\mathbf{R}-\mathbf{r}|)\right\\}$
$\displaystyle=\frac{m^{2}T}{4\pi^{2}|\mathbf{R}+\mathbf{r}||\mathbf{R}-\mathbf{r}|}\frac{\cos\left\\{\frac{2\pi}{\lambda_{\mathrm{F}}}(|\mathbf{R}+\mathbf{r}|-|\mathbf{R}-\mathbf{r}|)\right\\}}{\mathrm{sinh}(\pi(|\mathbf{R}+\mathbf{r}|+|\mathbf{R}-\mathbf{r}|)T/v_{\mathrm{F}})}.$

At small distances $|\mathbf{R}\pm\mathbf{r}|\rightarrow\lambda_{\mathrm{F}}$,
the spatial dependence in
$\mathrm{sinh}(\pi(|\mathbf{R}+\mathbf{r}|+|\mathbf{R}-\mathbf{r}|)T/v_{\mathrm{F}})\rightarrow\mathrm{sinh}(2\pi\lambda_{\mathrm{F}}T/v_{\mathrm{F}})$
is cut by the Fermi wave-length. Hence, we can substitute
$|\mathbf{R}\pm\mathbf{r}|=R$ in this term at what follows.

###  A.1 Summary

At $\lambda_{F}>R$, the kernel is determined by the local term

$\Pi(\mathbf{R},\mathbf{r})=\frac{m}{2\pi^{3}}\frac{1}{(|\mathbf{R}+\mathbf{r}|^{2}+|\mathbf{R}-\mathbf{r}|^{2})^{2}}-\frac{m^{3}\lambda_{F}T^{2}}{24\pi^{2}R}$
(48)

At larger distances $R>\lambda_{F}$, the kernel is determined by the nonlocal
term

$\Pi(\mathbf{R},\mathbf{r})=\frac{m^{2}T}{4\pi^{2}R^{2}}\frac{1}{\mathrm{sinh}(2\pi
RT/v_{\mathrm{F}})}.$ (49)

There are two different regimes which will be justified a posteriori, namely
the weak coupling $U\ll\pi^{2}/4ma^{2}$ and strong coupling
$U>\pi^{2}/4ma^{2}$. At $U\ll\pi^{2}/4ma^{2}$, we have standard BCS regime.
Nonlocal term contributes to the self-consistency equation and Cooper
instability. The nonlocal term can be considered as a perturbation. On the
other hand, at $U>\pi^{2}/4ma^{2}$, as we will see below, the solution of the
self-consistency equation is determined by the local contribution. The
nonlocal term can be treated as a perturbation. In the situation when local
contribution determines the Cooper instability, it suffices to discretize the
system into the small regions with volume
$\lambda_{\mathrm{F}}\times\lambda_{\mathrm{F}}\times\lambda_{\mathrm{F}}$
each. Neglecting nonlocal contribution means that the self-consistency
equation for the Cooper instability can be considered independently in each
region. Since the order parameter is determined up to a phase it has
uncorrelated phase in different regions.

###  A.2 Strong coupling

At strong coupling the nonlocal contribution can be considered as a
perturbation. We can focus on a single region only. We seek for the s-wave
solution of Eq. (5) $\psi(\mathbf{r})$ as

$\Delta(\mathbf{R},\mathbf{r})=\Delta(\mathbf{R})\psi(\mathbf{r}).$ (50)

Considering that $\Delta(\mathbf{R})$ changes slowly on the Fermi wave length,
we set $\Delta(\mathbf{R})\approx\Delta(\mathbf{R}^{\prime})$. We note that
$|\mathbf{R}-\mathbf{R}^{\prime}|$ is bounded by the typical size of the
preformed Cooper pair region given by the Fermi wave-length. Hence we
integrate the local term as

$\int_{R<\lambda_{\mathrm{F}}}\frac{d\mathbf{R}}{(R^{2}+|\mathbf{r}-\mathbf{r}^{\prime}|^{2}/4)^{2}}=\frac{2\pi^{2}}{|\mathbf{r}-\mathbf{r}^{\prime}|}.$
(51)

Here we neglect term $\propto\lambda^{-1}_{\mathrm{F}}$, which might be summed
up into the long distance part. As a result, the self-consistency equation can
be brought to the form

$\displaystyle\psi(\mathbf{r})$ $\displaystyle=$
$\displaystyle\frac{\pi\Theta(a-r)}{16}\frac{U}{U_{c}}$ (52)
$\displaystyle\times$
$\displaystyle\int_{r^{\prime}<a}d\mathbf{r}^{\prime}\left(\frac{1}{|\mathbf{r}-\mathbf{r}^{\prime}|}-\frac{1}{6}m^{2}\lambda^{3}_{\mathrm{F}}T^{2}\right)\psi(\mathbf{r}^{\prime}),~{}~{}~{}~{}$

where

$U_{c}=\frac{\pi^{2}}{4ma^{2}}$ (53)

is the critical interaction strength. After integrating over the angle
$\psi(r)=\int\frac{d\vec{\theta}}{4\pi}\psi(\mathbf{r})$, we obtain

$\displaystyle\psi(r)$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{4a^{2}}\frac{U}{U_{c}}\int_{0}^{a}dr^{\prime}r^{\prime
2}$ (54) $\displaystyle\times$
$\displaystyle\left(\frac{r+r^{\prime}-|r-r^{\prime}|}{2rr^{\prime}}-\frac{1}{6}m^{2}\lambda^{3}_{\mathrm{F}}T^{2}\right)\psi(r^{\prime}).~{}~{}~{}~{}~{}$

From (54), for $r\psi(r)$, we have differential equation

$\frac{d^{2}}{dr^{2}}(r\psi(r))=-\frac{\pi^{2}U}{4a^{2}U_{c}}(r\psi(r))$ (55)

with boundary condition for the first derivative at $r=a$:

$\displaystyle\frac{d}{dr}(r\psi)|_{r=a}=$ $\displaystyle-$
$\displaystyle\frac{\pi
U}{16U_{c}a^{2}}\left(\frac{2}{\pi\lambda_{\mathrm{F}}}+\frac{1}{6}m^{2}\lambda^{3}_{\mathrm{F}}T^{2}\right)$
(56) $\displaystyle\times$ $\displaystyle\int^{a}_{0}dr^{\prime}r^{\prime
2}\psi(r^{\prime}).$

A nonzero solution is

$\psi(\mathbf{r})=\sin\left(\frac{\pi r}{2a}\sqrt{\frac{U}{U_{c}}}\right)$
(57)

at

$\frac{U_{c}}{U}=1+\frac{(4\pi)^{2}a}{3\lambda_{\mathrm{F}}}\left(\frac{T}{\mu}\right)^{2}.$
(58)

As a result, at $U\gtrsim U_{c}$, we obtain the crossover temperature for the
formation of the local Cooper pairing

$T_{\mathrm{PG}}=\frac{\mu}{4\pi}\left(\frac{3\lambda_{\mathrm{F}}}{a}\right)^{1/2}\left(1-\frac{U_{c}}{U}\right)^{1/2}.$
(59)

###  A.3 Weak coupling limit

In the limit $U\ll U_{c}$, we obtain

$\displaystyle\int_{r<a}d^{3}r\Delta(\mathbf{R},\mathbf{r})=$ (60)
$\displaystyle=U\int_{r,r^{\prime}<a}d^{3}rd^{3}r^{\prime}\left\\{\frac{m}{4\pi|\mathbf{r}-\mathbf{r}^{\prime}|}-\nu\ln\left|\frac{\pi\lambda_{\mathrm{F}}}{\ell_{\mathrm{T}}}\right|\right\\}\Delta(\mathbf{R},\mathbf{r}^{\prime}).$

Using

$\Delta(\mathbf{R},\mathbf{r})=\frac{\Delta(\mathbf{R})}{r}\psi(r),$ (61)

we obtain an equation

$1-\frac{U}{U_{c}}=-U\nu\Omega_{a}\ln\left|\frac{\pi\lambda_{\mathrm{F}}}{\ell_{\mathrm{T}}}\right|,$
(62)

where $\Omega_{a}=4\pi a^{3}/3$. The above equation gives the superconducting
transition temperature

$T_{c}=\frac{\mu}{\pi^{2}}\exp\left\\{-\frac{1-U/U_{c}}{U\nu\Omega_{a}}\right\\},$
(63)

which is the expression (13) of the main text. To compare Eq. 63 with Gor’kov
and Melik-Barkhudarov renormalization Gor’kov and Melik-Barkhudarov (1961), we
note that

$\left(\frac{1}{U}-\frac{1}{U_{c}}\right)\frac{1}{\Omega_{a}^{3}}\equiv\frac{1}{|f_{0}|},$
(64)

where $f_{0}$ is the scattering amplitude, which is calculated in the second
order scattering approximation as it was done in Ref. Gor’kov and Melik-
Barkhudarov (1961) and reviewed in Ref. Ohashi  _et al._ (2020).

##  Appendix B  Ginzburg - Levanyuk criterion

The Landau mean field theory becomes invalid at the vicinity of the second-
order phase transition due to strong fluctuations of the order parameter. The
Ginzburg - Levanyuk criterion determines the region of temperatures (and other
parameters) in which such fluctuations can be neglected. The region is defined
through the condition that the fluctuations are small within the correlation
volume $\Omega_{c}$:

$\frac{\int_{\Omega_{c}}d\mathbf{r}_{0}d\mathbf{r}_{1}\langle\Delta(\mathbf{r}_{0})\Delta(\mathbf{r}_{1})\rangle}{\Omega_{c}^{2}\langle\Delta(\mathbf{r})\rangle^{2}}<1,$
(65)

where $\langle\Delta(\mathbf{r}_{0})\Delta(\mathbf{r}_{1})\rangle$ is the
correlation function of the order parameter and

$\langle\Delta(\mathbf{r})\rangle\equiv\langle\Delta\rangle$

is the (position independent) mean-field value. One can seek for the
correlation function both above and below the transition temperature. However,
the mean-field value only exists below the transition temperature, the region
we shall focus on in what follows.

###  B.1 Ornstein-Zernike theory for the correlation function

To derive the equation for the correlation function in the strong coupling
limit $U>U_{c}c$, we shall use the Ornstein-Zernike theory Stanley (1987). For
simplicity, let us consider a real valued order parameter
$\Delta(\mathbf{r})=\Delta^{*}(\mathbf{r})$. In our model, we split the
superconductor into a system of regions with volume
$\Omega_{\mathrm{F}}\equiv\lambda_{\mathrm{F}}\times\lambda_{\mathrm{F}}\times\lambda_{\mathrm{F}}$.
Let us by hand determine the order parameter $\Delta(\mathbf{r}_{0})$ in the
region $\Omega_{\mathrm{F}}$ around point $\mathbf{r}_{0}$. As a result of our
two-point Andreev interaction model, the field that acts on the order
parameter $\Delta(\mathbf{r}_{1})$ at some other point $\mathbf{r}_{1}$ is
given by the Andreev coupling terms in the Ginzburg-Landau functional:

$\displaystyle W(\mathbf{r}_{1},\mathbf{r}_{0})=$ $\displaystyle-$
$\displaystyle\Omega^{2}_{\mathrm{F}}\Pi(\mathbf{r}_{1}-\mathbf{r}_{0})\Delta(\mathbf{r}_{0})$
(66) $\displaystyle-$ $\displaystyle\Omega^{2}_{\mathrm{F}}\sum_{j\neq
0,1}\Pi(\mathbf{r}_{1}-\mathbf{r}_{j})\Delta(\mathbf{r}_{j};\Delta(\mathbf{r}_{0})),~{}~{}~{}~{}$

where

$\Pi(\mathbf{r}_{1}-\mathbf{r}_{0})=\nu\frac{\mathrm{csch}(2\pi|\mathbf{r}_{1}-\mathbf{r}_{0}|/\ell_{\mathrm{T}})}{2\ell_{\mathrm{T}}|\mathbf{r}_{1}-\mathbf{r}_{0}|^{2}}.$
(67)

In expression (66), the coupling between $\Delta(\mathbf{r}_{0})$ and
$\Delta(\mathbf{r}_{1})$ is split into the direct and indirect terms following
the Ornstein-Zernike theory. The indirect term
$\Delta(\mathbf{r}_{j};\Delta(\mathbf{r}_{0}))$ defines the averaged order
parameter which is calculated under the condition of the fixed value
$\Delta(\mathbf{r}_{0})$. It accounts for the contribution in which the order
parameter $\Delta(\mathbf{r}_{1})$ is being correlated with
$\Delta(\mathbf{r}_{j})$ where $j\neq 0,1$, while $\Delta(\mathbf{r}_{j})$
correlates directly with the order parameter $\Delta(\mathbf{r}_{0})$.

The conditionally averaged value
$\Delta(\mathbf{r}_{j};\Delta(\mathbf{r}_{0}))$ is defined as

$\displaystyle\Delta(\mathbf{r}_{j};\Delta(\mathbf{r}_{0}))=\frac{1}{\int[D\Delta(\mathbf{r})]e^{-S[\Delta(\mathbf{r})]/T}}$
$\displaystyle\times\int[D\Delta(\mathbf{r})]\delta(\Delta(\mathbf{r})-\Delta(\mathbf{r}_{0}))\Delta(\mathbf{r}_{j})e^{-S[\Delta(\mathbf{r})]/T}.$
(68)

Using $\Delta(\mathbf{r}_{j};\Delta(\mathbf{r}_{0}))$, the two-point
correlation function reads

$\langle\Delta(\mathbf{r}_{0})\Delta(\mathbf{r}_{j})\rangle=\int\Delta(\mathbf{r}_{0})\Delta(\mathbf{r}_{j};\Delta(\mathbf{r}_{0}))[D\Delta(\mathbf{r}_{0})].$
(69)

For a given field $W(\mathbf{r}_{1})$ at point $\mathbf{r}_{1}$, one can
calculate the conditionally averaged value at fixed $\Delta(\mathbf{r}_{0})$:

$\displaystyle\Delta(\mathbf{r}_{1};\Delta(\mathbf{r}_{0}))=\frac{1}{\int
e^{-W(\mathbf{r}_{1},\mathbf{r}_{0})\Delta(\mathbf{r}_{1})/T}\rho(\Delta(\mathbf{r}_{1}))[D\Delta(\mathbf{r}_{1})]}$
$\displaystyle\times\int\Delta(\mathbf{r}_{1})e^{-W(\mathbf{r}_{1},\mathbf{r}_{0})\Delta(\mathbf{r}_{1})/T}\rho(\Delta(\mathbf{r}_{1}))[D\Delta(\mathbf{r}_{1})],$
(70)

where

$\displaystyle\rho(\Delta(\mathbf{r}_{1}))=\exp\bigg{\\{}$ $\displaystyle-$
$\displaystyle\frac{1}{T}\bigg{[}\left(\frac{1}{U}-\frac{1}{U_{c}(T)}\right)\frac{\Omega_{\mathrm{F}}}{\Omega_{a}}|\Delta(\mathbf{r}_{1})|^{2}$
(71) $\displaystyle+$ $\displaystyle
b|\Delta(\mathbf{r}_{1})|^{4}\bigg{]}\bigg{\\}}$

is the onsite distribution function of $|\Delta(\mathbf{r}_{1})|$.

Taking into account the smallness of the averaged order parameter at the
vicinity of the transition temperature, we obtain

$\displaystyle\Delta(\mathbf{r}_{1};\Delta(\mathbf{r}_{0}))$ $\displaystyle=$
$\displaystyle\frac{1}{T}\Omega^{2}_{\mathrm{F}}\bigg{\\{}\langle\Delta^{2}(\mathbf{r}_{1})\rangle\Pi(\mathbf{r}_{1}-\mathbf{r}_{0})\Delta(\mathbf{r}_{0})$
(72) $\displaystyle+$ $\displaystyle\sum_{j\neq
0,1}\Pi(\mathbf{r}_{1}-\mathbf{r}_{j})\Delta(\mathbf{r}_{j};\Delta(\mathbf{r}_{0}))\bigg{\\}}.~{}~{}~{}~{}$

We then multiply Eq. (72) with $\Delta(\mathbf{r}_{0})$ and average the
resulting equation. Using definition (69), one obtains

$\displaystyle\langle\Delta(\mathbf{r}_{0})\Delta(\mathbf{r}_{1})\rangle$
$\displaystyle=$
$\displaystyle\frac{\langle\Delta^{2}\rangle^{2}}{T}\Omega^{2}_{\mathrm{F}}\Pi(\mathbf{r}_{1}-\mathbf{r}_{0})$
$\displaystyle+$
$\displaystyle\frac{\langle\Delta^{2}\rangle}{T}\Omega^{2}_{\mathrm{F}}\sum_{j\neq
0,1}\Pi(\mathbf{r}_{1}-\mathbf{r}_{j})\langle\Delta(\mathbf{r}_{0})\Delta(\mathbf{r}_{j})\rangle,$

where $\langle\Delta^{2}(\mathbf{r})\rangle\equiv\langle\Delta^{2}\rangle$
notation is applied. Eq. (B.1) can be rewritten in the continuum limit as

$\displaystyle\langle\Delta(\mathbf{r})\Delta(0)\rangle$ $\displaystyle=$
$\displaystyle\frac{\langle\Delta^{2}\rangle^{2}}{T}\Omega_{\mathrm{F}}^{2}\Pi(\mathbf{r})$
$\displaystyle+$
$\displaystyle\frac{\langle\Delta^{2}\rangle}{T}\Omega_{\mathrm{F}}\int
d\mathbf{r}^{\prime}\Pi(\mathbf{r}-\mathbf{r}^{\prime})\langle\Delta(\mathbf{r}^{\prime})\Delta(0)\rangle.$

Using equation for the superconducting transition temperature (Eq. (25) of the
main text)

$T_{c}=\langle\Delta^{2}\rangle\Omega_{\mathrm{F}}\int
d\mathbf{r}\Pi(\mathbf{r}),$ (75)

one can rewrite equation for the correlation function in the form

$\displaystyle\langle\Delta(\mathbf{r})\Delta(0)\rangle$ $\displaystyle=$
$\displaystyle\frac{T_{c}}{T}\langle\Delta^{2}\rangle\Omega_{\mathrm{F}}K(\mathbf{r})$
$\displaystyle+$ $\displaystyle\frac{T_{c}}{T}\int
d\mathbf{r}^{\prime}K(\mathbf{r}-\mathbf{r}^{\prime})\langle\Delta(\mathbf{r}^{\prime})\Delta(0)\rangle,$

where

$K(\mathbf{r})=\frac{\Pi(\mathbf{r})}{\int d\mathbf{r}\Pi(\mathbf{r})}.$ (77)

###  B.2 Solution of the correlation function

Eq. (B.1) can be further analyzed at the vicinity of the phase transition in
the limit of large correlation radius.

$\displaystyle\left(1-\frac{T_{c}}{T}\right)\langle\Delta(\mathbf{r})\Delta(0)\rangle$
$\displaystyle=$
$\displaystyle\frac{T_{c}}{T}\langle\Delta^{2}\rangle\Omega_{\mathrm{F}}\delta(\mathbf{r})$
$\displaystyle+$
$\displaystyle\alpha\ell_{\mathrm{T}}^{2}\bm{\nabla}^{2}\langle\Delta(\mathbf{r})\Delta(0)\rangle,~{}~{}~{}~{}$

where $\alpha\approx 1$ is the numerical constant and
$\ell_{\mathrm{T}}=v_{\mathrm{F}}/T$ is the thermal correlation length. The
solution of Eq. (B.2) is given by

$\displaystyle\langle\Delta(\mathbf{r})\Delta(0)\rangle$ $\displaystyle=$
$\displaystyle\langle\Delta^{2}\rangle\Omega_{\mathrm{F}}\int\frac{d\mathbf{q}}{(2\pi)^{3}}\frac{e^{i\mathbf{q}\cdot\mathbf{r}}}{\tau+\alpha\ell_{\mathrm{T}}^{2}q^{2}}$
(79) $\displaystyle=$
$\displaystyle\frac{\Omega_{\mathrm{F}}\langle\Delta^{2}\rangle}{4\pi\alpha\ell_{\mathrm{T}}^{2}}\frac{e^{-r/r_{c}}}{r},$

where $r_{c}$ is the correlation radius

$r_{c}=\ell_{\mathrm{T}}\sqrt{\alpha/\tau},~{}~{}~{}\tau=1-T/T_{c},$ (80)

and

$\Omega_{c}=r_{c}^{3}$ (81)

is the correlation volume. Using (65), we obtain

$\frac{\int_{\Omega_{c}}d\mathbf{r}_{0}d\mathbf{r}_{1}\langle\Delta(\mathbf{r}_{0})\Delta(\mathbf{r}_{1})\rangle}{\Omega_{c}^{2}\langle\Delta\rangle^{2}}=\frac{\Omega_{\mathrm{F}}}{\Omega_{c}}\frac{1}{\tau}\frac{\langle\Delta^{2}\rangle}{\langle\Delta\rangle^{2}}.$
(82)

To find the square of the averaged $\Delta$, one can use the definition

$\displaystyle\langle\Delta\rangle=\frac{\int
D[\Delta,\Delta^{*}]\Delta(0)\exp\\{-S_{\mathrm{MF}}[\Delta,\Delta^{*}]\\}}{\int
D[\Delta,\Delta^{*}]\exp\\{-S_{\mathrm{MF}}[\Delta,\Delta^{*}]\\}}.$ (83)

Here we can rewrite as follows

$\displaystyle
TS_{\mathrm{MF}}=\left(\frac{1}{U}-\frac{1}{U_{c}(T)}\right)\frac{\Omega_{\mathrm{F}}}{\Omega_{a}}|\Delta(0)|^{2}+b|\Delta(0)|^{4}$
(84) $\displaystyle-\Omega_{\mathrm{F}}\int
d\mathbf{r}\Pi(\mathbf{r})\Delta(0)\langle\Delta(\mathbf{r})\rangle=TS_{\mathrm{MF}}[\Delta(0)]-T_{c}\frac{\langle\Delta\rangle\Delta(0)}{\langle\Delta^{2}\rangle},$

where we used the definition
$T_{c}=\langle\Delta^{2}\rangle\Omega_{\mathrm{F}}\int
d\mathbf{r}\Pi(\mathbf{r})$. We obtain at $\tau\ll 1$

$\displaystyle\langle\Delta\rangle^{2}=s\tau\langle\Delta^{2}\rangle,$ (85)

where $s\approx 1$.

###  B.3 Result for Ginzburg-Levanyuk criterion

Noting the ratio

$\frac{\Omega_{\mathrm{F}}}{\Omega_{c}}\frac{1}{\tau}\frac{\langle\Delta^{2}\rangle}{\langle\Delta\rangle^{2}}\sim\frac{\Omega_{\mathrm{F}}}{\Omega_{c}}\frac{1}{\tau^{2}}\sim\frac{\lambda_{\mathrm{F}}^{3}}{\ell^{3}_{\mathrm{T}}}\frac{1}{\sqrt{\tau}}<1,$
(86)

we obtain the condition for small fluctuations in the strong coupling case

$\tau>\left(\frac{\lambda_{\mathrm{F}}}{\ell_{\mathrm{T}}}\right)^{6}\sim\left(\frac{T_{c}}{\mu}\right)^{6}.$
(87)

We shall compare this condition with the one, which is obtained in the BCS
model in the clean limit

$\tau>\left(\frac{T_{c}}{\mu}\right)^{4}.$ (88)

It gives more stringent condition on the validity of the mean field approach.

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