# Superconductivity near a quantum critical point in the extreme retardation
regime

Emil A. Yuzbashyan  Department of Physics and Astronomy, Rutgers University,
Piscataway, New Jersey 08854, USA  Michael K.-H. Kiessling  Department of
Mathematics, Rutgers University, Piscataway, New Jersey 08854, USA  Boris L.
Altshuler  Physics Department, Columbia University, 538 West 120th Street, New
York, New York 10027, USA

###### Abstract

We study fermions at quantum criticality with extremely retarded interactions
of the form $V(\omega_{l})=(g/|\omega_{l}|)^{\gamma}$, where $\omega_{l}$ is
the transferred Matsubara frequency. This system undergoes a normal-
superconductor phase transition at a critical temperature $T=T_{c}$. The order
parameter is the frequency-dependent gap function $\Delta(\omega_{n})$ as in
the Eliashberg theory. In general, the interaction is extremely retarded for
$\gamma\gg 1$, except at low temperatures $\gamma>3$ is sufficient. We
evaluate the normal state specific heat, $T_{c}$, the jump in the specific
heat, $\Delta(\omega_{n})$ near $T_{c}$, and the Landau free energy. Our
answers are asymptotically exact in the limit $\gamma\to\infty$. At low
temperatures, we prove that the global minimum of the free energy is
nondegenerate and determine the order parameter, the free energy, and the
specific heat. These answers are exact for $T\to 0$ and $\gamma>3$. We also
uncover and investigate an instability of the $\gamma$ model: negative
specific heat at $T\to 0$ and just above $T_{c}$.

##  I Introduction

Critical fluctuations of the order parameter field mediate strong electron-
electron interactions near metallic quantum critical points. Coupling to this
bosonic order parameter field induces attraction between electrons, leading to
the enhancement of superconductivity near such points in many quantum
materials, such as the cuprates, iron pnictides, and certain heavy-fermion
materials [1, 2, 3, 4]. On the other hand, same interactions destroy Fermi
liquid quasiparticles thus undermining superconductivity which builds on
quasiparticle coherence. As a result, the enhancement does not occur in other
quantum critical materials, such as heavy-fermion metals CeCu6-xAux and
YbRh2Si2 [5].

It has been argued [6] that many quantum critical system are described by the
$\gamma$ model – effective electron-electron interactions of the form
$V(\omega_{l})=(g/|\omega_{l}|)^{\gamma}$, where $\omega_{l}=2\pi Tl$ is the
transferred Matsubara frequency. Examples include a 2D nematic ($\gamma=1/3$)
and magnetic ($\gamma=1/2$) quantum critical points, a spin-liquid model for
the cuprates ($\gamma=0.7$), 2D pairing mediated by an undamped propagating
boson ($\gamma=1$), and the strong coupling limit of phonon mediated
superconductivity ($\gamma=2$). Interactions of this form obtain by
integrating out the bosonic fields and averaging over the Fermi surface,
taking into account that the order parameter field is massless at criticality.

Here we study the thermodynamic properties of the $\gamma$ model in the
extreme retardation regime, when $V(\omega_{l})$ is effectively local in the
Matsubara frequency domain. This corresponds to $\gamma>3$ at low temperatures
and to $\gamma\gg 1$ at arbitrary $T$. Even though in all the above examples
$0\leq\gamma\leq 2$, there is no fundamental reason why larger $\gamma$ are
impossible. Moreover, properties of the system at finite temperature are
continuous in $\gamma$. For example, the relative difference between the exact
superconducting transition temperature $T_{c}$ for $\gamma=2$ and the large
$\gamma$ asymptote for $T_{c}$ is only 6% (see below).

The attractive feature of the $\gamma$ model is that there is a single
intrinsic dimensionless parameter in the model – the exponent $\gamma$ itself.
The coupling constant $g$ sets the energy units. The momentum-dependence of
the fermion-boson coupling and the spectrum of mediating bosons become
irrelevant at criticality. The Fermi energy is the largest energy in the
problem, i.e., we are in the limit $\varepsilon_{F}/g\to\infty$. This suggests
that the $\gamma$ model is the minimal description of the strong coupling
fixed point (quantum critical point) in certain materials.

The interaction as we wrote it diverges at $\omega_{l}=0$. Moving slightly
away from criticality regularizes this divergence. The particular form of the
regularization is unimportant, because it does not affect our results. A
convenient regularization is
$V(\omega_{l})=\frac{g^{\gamma}}{|\omega_{l}|^{\gamma}+\Omega^{\gamma}}$,
where $\Omega$ has the meaning of the critical boson mass. Now it is
straightforward to analyze the interaction in the time domain. In the
$\gamma\to\infty$ limit,
$V(0)=\frac{g^{\gamma}}{\Omega^{\gamma}}\equiv\lambda$ and $V(\omega_{l})=0$
for $\omega_{l}\neq 0$. Its Fourier transform is
$V(\tau^{\prime}-\tau)=\lambda T$. This interaction is maximally retarded –
independent of the time separation between the interacting particles both in
real and imaginary time domains. By contrast, instantaneous interactions,
e.g., the pairing interaction in the Bardeen-Cooper-Schrieffer (BCS) theory,
are proportional to $\delta(\tau^{\prime}-\tau)$. When $\gamma$ is large but
finite, the interaction is extremely retarded.

The $\gamma$ model is a generalization of the strong coupling limit of the
Eliashberg theory [7, 8, 9], which corresponds to $\gamma=2$. It presents two
phases: normal and superconducting. The order parameter is the frequency
dependent gap function $\Delta_{n}\equiv\Delta(\omega_{n})$ as in the
Eliashberg theory. Here $\omega_{n}=\pi T(2n+1)$ is the fermionic Matsubara
frequency. We focus on the thermodynamic properties near the superconducting
transition temperature $T_{c}$ and at low $T$. Our results near $T_{c}$ are
asymptotically exact in the limit $\gamma\to\infty$ and the low temperature
answers are similarly exact for $T\to 0$ at any $\gamma>3$.

First, we determine the $T_{c}$ itself,

$T_{c}(\gamma)=\frac{ga^{\frac{1}{\gamma}}}{2\pi},\quad a\approx 1.1843,$ (1)

and then the order parameter near $T_{c}$

$\Delta_{n}=\eta\psi
a^{-\frac{1}{\gamma}}|\omega_{n}|J_{n+\frac{1}{2}+a^{-1}}(a^{-1}),$ (2)

where $\eta\approx 19.20$, $J_{\alpha}(x)$ is the Bessel function of the first
kind, and the complex number $\psi$ is the Landau order parameter. Note that
Eq. (1) appeared in an earlier paper [10].

The free energy expanded to the fourth power of $|\psi|$ near the transition
(the Landau free energy) is

$f_{L}=R\nu_{0}g^{2}\left[\frac{2\pi\gamma}{g}(T-T_{c})|\psi|^{2}+\frac{|\psi|^{4}}{2}\right],$
(3)

where $R\approx 0.62$ and $\nu_{0}$ is the density of states at the Fermi
level. This allows us to evaluate $|\psi|$ near $T_{c}$,

$|\psi|=\left[\frac{2\pi\gamma}{g}\right]^{1/2}(T_{c}-T)^{1/2},$ (4)

the jump in the specific heat at $T_{c}$,

$\delta c=c_{\mathrm{s}}-c_{\mathrm{n}}=2\pi
R\nu_{0}g\gamma^{2}a^{\frac{1}{\gamma}},$ (5)

and the thermodynamic critical field,

$H_{c}=4\pi^{3/2}\gamma\sqrt{R\nu_{0}}(T_{c}-T).$ (6)

As expected $|\psi|$ and $H_{c}$ have the usual mean-field critical exponents,
but note also their $\gamma$-dependence.

We prove that the global minimum of the free energy is unique for $T\to 0$ up
to an overall phase $e^{i\phi}$ of the gap function $\Delta(\omega_{n})$ and
determine the leading small $T$ asymptotic behavior of $\Delta(\omega_{n})$.
Specifically,

$\frac{\Delta(\omega_{n})}{\omega_{*}}=Y\\!\left(\frac{\omega_{n}}{\omega_{*}}\right),\quad
Y(x)=x\tan\theta_{0}(x),$ (7)

where

$\omega_{*}=g\frac{[\zeta(\gamma-2)]^{\frac{1}{3}}}{2^{\frac{1}{3}}}\left(\frac{g}{2\pi
T}\right)^{\\!\\!\frac{\gamma}{3}-1}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!,$ (8)

and $\theta_{0}(x)$ is the solution of the universal gap equation
$\theta^{\prime\prime}=x\sin\theta$. Equation (7) says that plots of the gap
functions $\Delta(\omega_{n})$ vs $\omega_{n}$ collapse onto the same curve
(Fig. 4 below) for all $\gamma>3$, $g$, and $T\to 0$, when both
$\Delta(\omega_{n})$ and $\omega_{n}$ are measured in units of $\omega_{*}$.
Further, we evaluate $2\Delta(0)/T_{c}$ and find that it is finite for all
$\gamma>3$ and $T>0$, but diverges as $|\gamma-3|^{-1/3}$ for $\gamma\to
3^{+}$ and as $T^{1-\gamma/3}$ for $T\to 0$.

We also determine the normal state specific heat $c_{\mathrm{n}}$ at all
temperatures and the specific heat $c_{\mathrm{s}}$ in the superconducting
state just below $T_{c}$ and for $T\to 0$. The analysis of the specific heat
reveals a pathology in the $\gamma$ model: $c_{\mathrm{n}}$ is negative just
above $T_{c}$ and $c_{\mathrm{s}}$ is negative at $T\to 0$. This indicates
that the $\gamma$ model is thermodynamically unstable [11] at these
temperatures. The resolution of this problem depends on the microscopic
Hamiltonian underlying the $\gamma$ model, which is not given. We discuss
several scenarios. In particular, there is a scenario that removes this
pathology without affecting any of the above answers.

The content of this paper is as follows. In Sec. II, we derive the free energy
functional of the $\gamma$ model and map it to a classical Heisenberg spin
chain using the approach we proposed in an earlier paper. In Sec. III, we
obtain the equation for the stationary points of the effective action for
arbitrary $\gamma$ and in Sec. IV we specialize the free energy to the extreme
retardation regime. In Sec. V, we first device a numerical approach for
determining $T_{c}$ for arbitrary $\gamma$ and then obtain the solution of the
linearized gap equation for $\gamma\to\infty$. We derive the large $\gamma$
asymptote of $T_{c}$ in Sec. VI. Section VII contains our results for the
Landau free energy, jump in the specific heat at $T_{c}$, Landau order
parameter, and thermodynamic critical field. We work out the specific heat in
the normal state and in the superconducting state just below $T_{c}$ in Sec.
VIII. In Sec. IX, we solve the gap equation at low temperatures, and evaluate
the specific heat in the superconducting state at $T\to 0$. In the final
section, we summarize and pose an important open problem. The Appendix
provides a proof of the uniqueness of the global minimum of the free energy at
small $T$.

##  II Mapping to the spin chain

In previous paper [9] we mapped the effective action for electrons interacting
via phonons to a classical spin chain. In the strong coupling limit where
phonon frequencies go to zero, this is the $\gamma$ model with $\gamma=2$. The
same approach works for arbitrary $\gamma$. We define the $\gamma$ model by
its Euclidean action

$\begin{split}S=T\sum_{\mathsf{p}\sigma}&\psi^{*}_{\mathsf{p}\sigma}G_{\mathsf{p}}^{-1}\psi_{\mathsf{p}\sigma}-\\\
&\frac{T^{3}\delta}{2}\\!\\!\sum_{\mathsf{p}_{i}\sigma\sigma^{\prime}}\frac{g^{\gamma}}{|\omega_{l}|^{\gamma}+\Omega^{\gamma}}\psi^{*}_{\mathsf{p}_{1}\sigma}\psi_{\mathsf{p}_{3}\sigma}\psi^{*}_{\mathsf{p}_{2}\sigma^{\prime}}\psi_{\mathsf{p}_{4}\sigma^{\prime}},\end{split}$
(9)

where $G_{\mathsf{p}}^{-1}=-i\omega_{n}+\xi_{\bf p}$, $\omega_{n}$ are
fermionic Matsubara frequencies, $\xi_{\bf p}$ are single-particle levels
counted from the chemical potential, $\psi_{\mathsf{p}\sigma}$ is the
fermionic field, and $\mathsf{p}_{i}$ are frequency-momentum 4-vectors
constrained by
$\mathsf{p}_{1}-\mathsf{p}_{3}=\mathsf{p}_{2}-\mathsf{p}_{4}=(\omega_{l},\bm{q})$.
The single-particle level spacing $\delta$ is a combination of the density of
states at the Fermi energy $\nu_{0}$ and the system volume $V$,
$\delta=(\nu_{0}V)^{-1}$. We added a constant $\Omega\ll T$ to avoid
divergence at zero frequency. In the end we take the limit $\Omega\to 0$.

This action is exact for the electron-phonon system ($\gamma=2$) in the limit
when phonon frequencies go to zero, the Fermi energy
$\varepsilon_{F}\to\infty$, and assuming the electron-phonon coupling depends
only on the magnitude of the transferred momentum. Its status is unclear in
most other examples listed above, when the bosonic field is a collective mode
containing fermions themselves, such as, e.g., the nematic or magnetic order
parameter. In this paper, we simply postulate Eq. (9) and study its properties
regardless of its origin.

The mapping to the spin chain involves several steps [9]. First, we decouple
the interaction with three Hubbard-Stratonovich fields $\Sigma_{\uparrow}$,
$\Sigma_{\downarrow}$, and $\Phi$ and integrate out the fermions [12].
Stationary point equations for these fields are the Eliashberg equations
generalized to arbitrary $\gamma$ from $\gamma=2$. Fluctuations around the
stationary point are negligible as long as the Fermi energy is the largest
energy in the problem [13]. To unveil the spin chain, we expand the action in
$\Omega$ and then send $\Omega\to 0$ [14]. A key ingredient of the mapping is
the observation that at the stationary points the normal and anomalous Green’s
functions integrated over $\xi_{\bf p}$ satisfy the constraint
$G_{n}^{2}+|F_{n}|^{2}=1$. The normal average $G_{n}$ is real and the
anomalous average $F_{n}$ is complex. Moreover, the mass of fluctuations
violating this constraint is proportional to $\Omega^{-\gamma}\to\infty$ [13].
The constraint is therefore rigidly enforced even away from the stationary
point.

Figure 1: Transition from (a) the normal state to (b) a superconducting state
in terms of classical spins. Positions of the spins $\bm{S}_{n}$ are the
fermionic Matsubara frequencies $\omega_{n}$. Spin-spin interactions are
purely ferromagnetic and the spins are subject to a Zeeman magnetic field
$2\pi\omega_{n}$ along the $z$ axis. In the superconducting state, spins
acquire $x$-components, which implies nonzero anomalous Green’s function. The
normal-superconductor transition is seen as the softening of the sharp domain
wall at the origin of the spin chain.

It is now natural to introduce classical spins $\bm{S}_{n}$ of unit length as

$S_{n}^{z}=G_{n},\quad S_{n}^{x}=\mathrm{Re}(F_{n}),\quad
S_{n}^{y}=\mathrm{Im}(F_{n}).$ (10)

In terms of the spins, the action expanded in $\Omega$ becomes
$S_{\mathrm{eff}}=\nu_{0}VH_{s}$, where $H_{s}$ is the spin chain Hamiltonian

$H_{s}=-2\pi\sum_{n}\omega_{n}S_{n}^{z}-\pi^{2}Tg^{\gamma}\sum_{nm}\frac{{\bm{S}_{n}}\cdot{\bm{S}_{m}}-1}{|\omega_{n}-\omega_{m}|^{\gamma}}.$
(11)

In addition to the expansion in $\Omega$, we regularized the interaction term
[9] by replacing
${\bm{S}_{n}}\cdot{\bm{S}_{m}}\to{\bm{S}_{n}}\cdot{\bm{S}_{m}}-1$ and then
took the limit $\Omega\to 0$ which is now harmless. The free energy density of
a given field configuration is [15]

$f=\frac{TS_{\mathrm{eff}}}{V}=\nu_{0}TH_{s}.$ (12)

The Boltzmann weight of a given field configuration is
$e^{-fV/T}=e^{-H_{s}/\delta}$, i.e., the spin chain is at an effective
temperature $T_{s}=\delta$. We work in the thermodynamic limit where
$\delta\to 0$ and therefore the spin chain is at zero temperature.

We introduce the frequency-dependent superconducting gap function
$\Delta_{n}\equiv\Delta(\omega_{n})$ through [9, 16]

$S_{n}^{z}=\frac{\omega_{n}}{\sqrt{\omega_{n}^{2}+|\Delta_{n}|^{2}}},\quad
S_{n}^{+}=\frac{\Delta_{n}}{\sqrt{\omega_{n}^{2}+|\Delta_{n}|^{2}}},$ (13)

where $S_{n}^{+}=S_{n}^{x}+iS_{n}^{y}$.

In the normal state, $\Delta_{n}=0$ and as a consequence
$\bm{S}_{n}=\mbox{sgn}(\omega_{n})\hat{z}$, where $\hat{z}$ is the unit vector
along the $z$ axis. A characteristic feature of this state is a sharp domain
wall between $\omega_{-1}$ and $\omega_{0}$. In the superconducting state,
$\Delta_{n}\neq 0$ and the domain wall softens, see Fig. 1.

##  III Stationary points of the free energy

In Ref. [9], we discussed stationary points of the free energy functional $f$
for $\gamma=2$. These results in fact apply to any $\gamma>1$. In particular,
we know that with a proper choice of the arbitrary overall phase $e^{i\phi}$,
the gap function at the minimum of the free energy has the following
properties: $\Delta(\omega_{n})>0$ for all $\omega_{n}$ or
$\Delta(\omega_{n})=0$ for all $\omega_{n}$, $\Delta(\omega_{n})$ is an even
function, and $\Delta(\omega_{n})\to 0$ as $\omega_{n}\to\pm\infty$. These
properties hold for $T=0$ as well, in which case $\omega_{n}$ becomes a
continuous variable $\omega$ and $\Delta(\omega_{n})>0$ implies $\Delta(0)>0$.
We also know that the stationary points of $f$ and of the spin chain $H_{s}$
are identical, because we are in the strong coupling limit
$\lambda=\frac{g^{\gamma}}{\Omega^{\gamma}}\to\infty$ (equivalent to
$\Omega\to 0$).

Given that $\Delta_{n}\geq 0$, Eq. (10) implies $S_{n}^{y}=0$ and
$S_{n}^{x}\geq 0$. Let $\theta_{n}\equiv\theta(\omega_{n})$ be the angle
$\bm{S}_{n}$ makes with the $z$ axis. Then

$S_{n}^{z}=\cos\theta_{n},\quad S_{n}^{x}=\sin\theta_{n},\quad
0\leq\theta_{n}\leq\pi.$ (14)

The angle $\theta_{n}$ cannot exceed $\pi$ due to the condition $S_{n}^{x}\geq
0$. This also prevents $\theta_{n}$ from winding. The remaining properties of
the gap function

$\Delta_{n}=\omega_{n}\tan\theta_{n}$ (15)

translate into

$\theta(-\omega_{n})=\pi-\theta(\omega_{n}),\quad\lim_{\omega_{n}\to\pm\infty}\theta_{n}=\frac{\pi}{2}\mp\frac{\pi}{2}.$
(16)

The spin chain Hamiltonian reads in terms of $\theta_{n}$

$H_{s}=-2\pi\sum_{n}\omega_{n}\cos\theta_{n}-\pi^{2}Tg^{\gamma}\sum_{nm}\frac{\cos(\theta_{n}-\theta_{m})-1}{|\omega_{n}-\omega_{m}|^{\gamma}}.$

Differentiating this with respect to $\theta_{n}$, we obtain the equation for
the stationary points of the free energy of the $\gamma$ model

$\omega_{n}\sin\theta_{n}=g^{\gamma}\pi T\sum_{m\neq
n}\frac{\sin(\theta_{m}-\theta_{n})}{|\omega_{m}-\omega_{n}|^{\gamma}}.$ (17)

This is the superconducting gap equation. It is a version of the Eliashberg
gap equation as it follows from the generalized Eliashberg equations mentioned
in the previous section. It takes the form of the standard gap equation for
the strong coupling limit of the Eliashberg theory [8, 7] when we rewrite it
in terms of $\Delta_{n}$ and set $\gamma=2$.

The normal state, $\Delta_{n}=0$, is always a stationary point. The
corresponding spin configuration $\bm{S}_{n}=\mbox{sgn}(\omega_{n})\hat{z}$ is
shown in Fig. 1(a). In this state $\theta_{n}=0$ for $\omega_{n}>0$ and
$\theta_{n}=\pi$ for $\omega_{n}<0$, which indeed satisfies the stationary
point equation (17). It is also not difficult to show that since $S_{n}^{z}$
is the normal Green’s function integrated over $\xi_{\bf p}$, in the normal
state all states below the chemical potential are occupied and above – empty.
At $T=0$, the Matsubara frequency $\omega_{n}\to\omega$ takes values on the
entire real axis. The angle $\theta(\omega)$ is then discontinuous at
$\omega=0$ in the normal state. In contrast, in the ground state (free energy
minimum at $T=0$) we have

$\theta(0)=\frac{\pi}{2}.$ (18)

This follows from Eqs. (13) and (14) together with the condition
$\Delta(0)>0$.

The symmetry (16) enables us to express the free energy density (12) in terms
of $\theta_{n\geq 0}$ only. It is also convenient to introduce along the way
the non-dimensionalized free energy density $\bar{f}$ as follows:

$\begin{split}&\bar{f}\equiv\frac{f}{\nu_{0}g^{2}}=-2\tau\sum_{n=0}^{\infty}\bar{\omega}_{n}\cos\theta_{n}-\\\
&\frac{\tau^{2-\gamma}}{2}\\!\\!\\!\\!\sum_{n,m=0}^{\infty}\\!\left[\frac{\cos(\theta_{n}-\theta_{m})-1}{(n-m)^{\gamma}}-\frac{\cos(\theta_{n}+\theta_{m})+1}{(n+m+1)^{\gamma}}\right]\\!,\end{split}$
(19)

where the dimensionless temperature $\tau$ and fermionic Matsubara frequency
$\bar{\omega}_{n}$ are

$\tau=\frac{2\pi
T}{g},\quad\bar{\omega}_{n}=\frac{\omega_{n}}{g}=\tau\left(n+\frac{1}{2}\right)\\!\\!.$
(20)

##  IV Extreme retardation regime

In this paper we are primarily interested in the extreme retardation regime of
the $\gamma$ model. In this regime, the range of the interaction is extremely
short in the frequency domain and therefore extremely long (retarded) in the
time domain. In terms of the spin chain (11), this means that a given spin
$\bm{S}_{n}$ interacts only with its nearest neighbors or, in the continuum
$T\to 0$ limit, only with spins in its immediate vicinity. There are two
overlapping cases when this happens. One is $\gamma\gg 1$ at arbitrary $T$.
The other is low $T$ for any $\gamma>3$. Let us consider these two cases.

###  IV.1 Large $\gamma$, arbitrary $T$

At large $\gamma$, the $|n-m|\geq 2$ interaction terms in Eq. (11) are
suppressed by a factor of $2^{-\gamma}$ or smaller compared to the $|n-m|=1$
terms. Discarding exponentially small terms, we obtain a spin chain with
nearest neighbor interactions,

$H_{s}=-2\pi\sum_{n}\omega_{n}S_{n}^{z}-\frac{\pi g^{\gamma}}{(2\pi
T)^{\gamma-1}}\sum_{n}({\bm{S}_{n}}\cdot{\bm{S}_{n+1}}-1).$ (21)

The corresponding non-dimensionalized free energy density for the coplanar
spin distribution (14) is

$\begin{split}\bar{f}\equiv\frac{f}{\nu_{0}g^{2}}=&-\tau\sum_{n}\bar{\omega}_{n}\cos\theta_{n}-\\\
&\frac{\tau^{2-\gamma}}{2}\sum_{n}\left[\cos(\theta_{n+1}-\theta_{n})-1\right],\end{split}$
(22)

and the expression (19) in terms of $\theta_{n\geq 0}$ becomes

$\begin{split}\bar{f}=-2\tau\sum_{n\geq
0}&\bar{\omega}_{n}\cos\theta_{n}+\tau^{2-\gamma}\frac{\cos(2\theta_{0})+1}{2}\\\
&-\tau^{2-\gamma}\sum_{n\geq
0}\left[\cos(\theta_{n+1}-\theta_{n})-1\right].\end{split}$ (23)

We use this expression for the free energy in subsequent sections to determine
the critical temperature $T_{c}$, the jump in the specific heat at the
transition, etc.

###  IV.2 Any $\gamma>3$, low $T$

We will see below that at $T\ll g$ the $\gamma$ model is local for any
$\gamma>3$. The gap equation becomes a nonlinear ordinary differential
equation (ODE), and the interaction part of the free energy corresponds to the
continuum limit of the classical ferromagnetic Heisenberg spin chain with
nearest-neighbor interactions. Again, the interaction is as local in the
frequency domain as it can be.

Consider the solution $\theta_{n}=\theta(\omega_{n})$ of the gap equation (17)
under the conditions (16). One can always extend $\theta(\omega_{n})$ to an
infinitely differentiable function $\theta(\omega)$ for all real $\omega$ that
coincides with $\theta_{n}$ when $\omega=\omega_{n}$, and such that conditions
(16) hold with $\omega$ in place of $\omega_{n}$. This also extends
$\Delta(\omega_{n})$ to $\Delta(\omega)$ with the help of Eq. (15) and fixes
$\theta(0)=\frac{\pi}{2}$ [17] by continuity of $\theta(\omega)$. There are
uncountably many such functions but we will determine a distinguished
$\theta(\omega)$ that is real analytic on the entire real $\omega$ axis [18]
and captures the small $T$ asymptotic behavior of $\theta(\omega_{n})$. We
will see in Sec. IX that $\theta(\omega)$ varies on a scale $\omega_{*}\propto
T^{1-\frac{\gamma}{3}}$. The difference $|\omega_{m}-\omega_{n}|$ in Eq. (17)
is of the order of $\omega_{*}$ when $|m-n|\sim(T/g)^{-\frac{\gamma}{3}}\gg
1$. Terms with $|m-n|\gg 1$ are negligible due to the rapid convergence of the
sum over $m$. Therefore, $\theta(\omega_{m})$ is close to $\theta(\omega_{n})$
for $m$ whose contribution to Eq. (17) is significant and

$\begin{split}\sin(\theta_{m}-\theta_{n})\approx\theta_{m}-\theta_{n}\phantom{+\frac{d\theta_{n}}{d\omega_{n}}(\omega_{m}-\omega_{n})+}\\\
\approx\frac{d\theta_{n}}{d\omega_{n}}(\omega_{m}-\omega_{n})+\frac{1}{2}\frac{d^{2}\theta_{n}}{d\omega_{n}^{2}}(\omega_{m}-\omega_{n})^{2}.\end{split}$
(24)

Substituting Eq. (24) into the gap equation (17) and dropping the discrete
index $n$ [$\omega_{n}\to\omega$ and $\theta_{n}\to\theta(\omega)$], we find
the nonlinear ODE

$\bar{\omega}\sin\theta=\frac{\tau^{3-\gamma}}{2}\frac{d^{2}\theta}{d\bar{\omega}^{2}}\sum_{k=1}^{\infty}\frac{1}{k^{\gamma-2}},$
(25)

where $\bar{\omega}=\omega/g$. We see that the approximations made require
$\gamma>3$ and low $T$ so that the summation over $k$ converges and the
replacement (24) is accurate. More precisely, in Sec. IX we show that the
local approximation (25) for the gap equation and the corresponding
approximation for the free energy are accurate when
$\tau^{\frac{r\gamma}{3}}\ll 1$, where $r=\gamma-3$ for $3<\gamma\leq 5$ and
$r=2$ for $\gamma\geq 5$. We use Eq. (25) in Sec. IX to determine the scaling
laws at low $T$.

The non-dimensionalized free energy density in this approximation becomes

$\begin{split}\bar{f}=-\tau\sum_{n}\bar{\omega}_{n}\cos\theta_{n}+\zeta(\gamma-2)\frac{\tau^{4-\gamma}}{4}\sum_{n}\left(\frac{d\theta_{n}}{d\bar{\omega}}\right)^{\\!\\!2}\\!\\!,\end{split}$
(26)

where $\zeta(x)$ is the Riemann zeta function. The variation of this
expression with respect to $\theta_{n}$ gives Eq. (25). It is also instructive
to compare formulas (26) and (22) for the free energy. The low temperature
limit of Eq. (22) and the large $\gamma$ limit of Eq. (26) should coincide.
This is indeed the case, because
$1-\cos(\theta_{n+1}-\theta_{n})\approx\frac{1}{2}\tau^{2}(d\theta_{n}/d\bar{\omega}_{n})^{2}$
at low $T$ and $\zeta(\gamma-2)\to 1$ as $\gamma\to\infty$.

In terms of spins the free energy (26) takes a particularly simple form,

$\bar{f}=\kappa^{-2}\int_{-\infty}^{\infty}dx\left\\{-xS^{z}+\frac{1}{2}\left(\frac{d\bm{S}}{dx}\right)^{\\!\\!2}\right\\}\\!,$
(27)

where

$\kappa^{3}=\frac{2\tau^{\gamma-3}}{\zeta(\gamma-2)},\quad
x=\kappa\bar{\omega}.$ (28)

This is the continuum limit of the classical ferromagnetic Heisenberg spin
chain with nearest neighbor interactions and a Zeeman field. Note that the
free energy diverges as $T^{2-\frac{2\gamma}{3}}$ in the $T\to 0$ limit. This
divergence is cutoff by the mass $\Omega$ of the critical boson. In other
words, the $\gamma$ model requires an infrared cutoff for $\gamma>3$.
Physically, we are working in the regime $T\gg\Omega>0$ and by low $T$ we mean
$g\gg T\gg\Omega$.

##  V Superconducting transition

Here we first investigate the superconducting transition in the $\gamma$ model
for arbitrary $\gamma$ and then turn to $\gamma\to\infty$ limit. As mentioned
in Sec. III, the normal state is a stationary point of the free energy
functional at any temperature. It is the global minimum above a certain
(critical) temperature $T_{c}$. Below $T_{c}$, which is nonzero for all
$\gamma$, it is a saddle point implying the emergence of a new
(superconducting) global minimum at $T=T_{c}$.

Interestingly, there is a single unstable direction (normal mode) at any
$T<T_{c}$ for $\gamma\gtrsim 2.7$ and for $\gamma\lesssim 0.2$. In other
words, there is an orthonormal basis in the configuration space of the free
energy, such that it increases along all axes except one, along which it
decreases. Therefore, a single mode is responsible for the Cooper instability
at all $T$ in the $\gamma$ model with these values of $\gamma$. The same is
true in the BCS theory [19, 20], which corresponds to $\gamma=0$. It is an
amplitude (Higgs) mode, because the phase of $\Delta_{n}$ is zero for it.

Coherent dynamics of the BCS condensate in response to sudden perturbations
can be understood as undamped, underdamped, and overdamped oscillations of the
Higgs mode [20, 21, 22]. At the first glance, this suggests that the $\gamma$
model for $\gamma\gtrsim 2.7$ and $\gamma\lesssim 0.2$ has the same three
nonequilibrium phases as the BCS condensate. In Phase I the order parameter
decays to zero at long times, in Phase II it goes to a nonzero constant, and
in Phase III it oscillates persistently and periodically [23]. However, we
have to keep in mind that the normal modes we obtain for the $\gamma$ model
are the modes of the free energy functional and not of the Hamiltonian
dynamics as in the BCS case. Because of this it is far from certain that the
far from equilibrium dynamics of the two systems are indeed similar.

###  V.1 Arbitrary $\gamma$

Figure 2: Three lowest eigenvalues of the second derivative matrix $X$ vs.
reduced temperature $\tau$ for three different values of $\gamma$ [(a), (b),
(c)] and the second lowest eigenvalue $\chi_{1}$ as a function of $\gamma$ at
$\tau=0.01$ (d). The inset in (d) magnifies the small $\gamma$ region of the
graph to show that $\chi_{1}(\gamma)$ becomes positive again at small
$\gamma$. Eigenvalues of $X$ determine the stability of the normal state – it
is stable when all of them are positive and unstable otherwise. For any
$\gamma$ there is a critical temperature $\tau_{c}$ below which the normal
state is unstable. When $0.2\lesssim\gamma\lesssim 2.7$ two eigenvalues are
negative at $\tau\to 0$. Otherwise, there is a single negative eigenvalue at
all $\tau<\tau_{c}$.

As usual, to determine the type of the stationary point, we need to expand the
free energy to quadratic order around it. It is convenient to work with the
free energy (19) formulated on the positive Matsubara axis. Expanding Eq. (19)
around the normal state, we obtain a quadratic form

$\delta\bar{f}=\tau\\!\\!\\!\sum_{n,m=0}^{\infty}\theta_{n}X_{nm}\theta_{m}=\tau\bm{\theta}^{T}X\bm{\theta},$
(29)

where $\delta\bar{f}=\bar{f}-\bar{f}_{\mathrm{n}}$, $\bar{f}_{\mathrm{n}}$ is
the non-dimensionalized normal state free energy, $\bm{\theta}$ is a column
vector with components $\theta_{n}$, and the second derivative matrix $X$ (the
Hessian) reads

$\begin{split}\tau^{\gamma-1}X_{nn}&=\tau^{\gamma}\left(n+\frac{1}{2}\right)+\sum_{k=1}^{n}\frac{1}{k^{\gamma}}-\frac{1}{2(2n+1)^{\gamma}},\\\
\tau^{\gamma-1}X_{n\neq
m}&=-\frac{1}{2(n+m+1)^{\gamma}}-\frac{1}{2|n-m|^{\gamma}}.\end{split}$ (30)

The normal state is a minimum when all eigenvalues of $X$ are positive. It
becomes a saddle point and a transition to the superconducting state takes
place when one of the eigenvalues of $X$ vanishes. In a mechanical
interpretation, where $\delta\bar{f}$ is the potential energy of a system of
1D point particles with coordinates $\theta_{n}$ and equal mass, eigenvalues
and eigenvectors of $X$ have the meaning of squares of the frequencies and
normal modes of small oscillations around the normal state, respectively. The
normal state is unstable when one of the frequencies becomes imaginary.

To analyze the eigenvalues of $X$ numerically, we truncate it to a finite
$L\times L$ matrix. The size $L$ should be such that the Zeeman field
overwhelms the interaction near the ends of the chain as it does at large
Matsubara frequencies in an infinite chain. The first line in Eq. (30) shows
that this requires $L\gg\tau^{-\gamma}$ for $\gamma>1$ and
$(L\tau)^{\gamma}\gg 1$ for $\gamma<1$. For comparison, the critical
dimensionless temperature is $\tau_{c}\approx 1$ for $\gamma\geq 2$ (see
below). It increases with decreasing $\gamma$ and diverges,
$\tau_{c}\sim\gamma^{-\frac{1}{\gamma}}$ as $\gamma\to 0^{+}$ [10].

Figure 2 shows three lowest eigenvalues $\chi_{0}<\chi_{1}<\chi_{2}$ of $X$ as
functions of the reduced temperature $\tau$ for several $\gamma$ and the
second eigenvalue $\chi_{2}$ as a function of $\gamma$ at a fixed temperature
$\tau=0.01<\tau_{c}$. We observe that: (a) $\chi_{0}$ is negative below a
certain value of $\tau$ and positive above it for all $\gamma$, i.e., there is
a superconducting transition for any $\gamma$; (b) $\chi_{2}$ is positive at
all $\gamma$ and temperatures; and (c) $\chi_{1}$ changes sign twice as a
function of $\gamma$ at low temperatures.

Point (c) comes as a surprise – the second eigenvalue $\chi_{1}$ also becomes
negative at very low temperatures for $0.2\lesssim\gamma\lesssim 2.7$. Outside
of this interval of $\gamma$ only one eigenvalue is negative for all
$T<T_{c}$. It is interesting to see whether this means that the dynamics of
the Cooper instability for $\gamma\gtrsim 2.7$ and $\gamma\lesssim 0.2$ are as
in the BCS model, where there is a single negative eigenvalue as well. Suppose
we prepare the system in the normal state and then suddenly turn on the
interaction (interaction quench). The BCS order parameter first grows
exponentially with the growth exponent set by the negative eigenvalue [19,
24]. Nonlinear effects stop the growth at some point after which the order
parameter oscillates persistently and periodically. The fact that oscillations
occur with a single basic frequency is a consequence of having only one
unstable direction.

###  V.2 Large $\gamma$

The problem of finding the large $\gamma$ asymptotic behavior of $T_{c}$ as
well as of the eigenvalues and eigenvectors of $X$ is exactly solvable. First,
it is not difficult to see that at $\gamma=\infty$ the critical temperature
$\tau_{c}=1$ ($T_{c}=\frac{g}{2\pi}$). Indeed, the interaction term in Eq.
(22) is proportional to $\tau^{2-\gamma}$. At $\gamma=\infty$ the interaction
vanishes for $\tau>1$. Then, the system simply minimizes the Zeeman term as
shown in Fig. 1(a), i.e., we are in the normal state, where
$\theta_{n}=\frac{\pi}{2}-\frac{\pi}{2}\mbox{sgn}(\omega_{n})$. For $\tau<1$
the interaction diverges. The sharp domain wall at the origin (the jump from
$\theta_{-1}=\pi$ to $\theta_{0}=0$) now costs infinite energy as the
interaction is ferromagnetic and favors parallel spin alignment. It is more
advantageous to spread the jump in $\theta_{n}$ over a large energy interval,
i.e., the domain wall softens [25]. In fact, in this limit all spins at finite
$\omega_{n}$ are along the $x$-axis and $\bar{f}=0$ for $\tau<1$ [26].

Now let us evaluate the leading order asymptotic behavior of $T_{c}$ and other
observables for large $\gamma$. Near the transition the system is close to the
normal state. We therefore expand the free energy (23) around the normal state
to the second order,

$\delta\bar{f}=\tau\sum_{n=0}^{\infty}\bar{\omega}_{n}\theta_{n}^{2}+\frac{\tau^{2-\gamma}}{2}\sum_{n=0}^{\infty}(\theta_{n+1}-\theta_{n})^{2}-\tau^{2-\gamma}\theta_{0}^{2}.$
(31)

The notations here the same as in Eq. (29).

Finding eigenvalues of the Hessian $X$ is equivalent to finding the stationary
points of $\bar{f}_{\chi}=\bar{f}-\chi\tau\sum_{n}\theta_{n}^{2}$. Setting the
derivative of $f_{\chi}$ with respect to $\theta_{n}$ to zero, we obtain one
equation for $n\geq 1$,

$\theta_{n+1}=2\left[\tau^{\gamma}\left(n+\frac{1}{2}\right)-\tau^{\gamma-1}\chi+1\right]\theta_{n}-\theta_{n-1},$
(32)

and another one for $n=0$,

$\theta_{1}=[\tau^{\gamma}-1-2\tau^{\gamma-1}\chi]\theta_{0}.$ (33)

Similar to a second order linear differential equation, the recurrence
relation (32) has two linearly independent solutions. Equation (33) provides
one boundary condition. The second boundary condition is $\theta_{n}\to 0$ for
$n\to\infty$, see Eq. (16).

Observe that Eq. (32) is the recurrence relation for Bessel functions,

$Z_{\alpha+1}(x)=\frac{2\alpha}{x}Z_{\alpha}-Z_{\alpha-1},$ (34)

with $x=\tau^{-\gamma}$ and
$\alpha=n+\frac{1}{2}-\tau^{-1}\chi+\tau^{-\gamma}$. The Bessel function
$Z_{\alpha}(x)$ that goes to zero as $\alpha\to\infty$ is $J_{\alpha}(x)$ –
the Bessel function of the first kind. Therefore, the normal mode with
eigenvalue $\chi$ is

$\theta_{n\chi}=J_{n+\frac{1}{2}+\tau^{-\gamma}-\chi\tau^{-1}}(\tau^{-\gamma}),$
(35)

up to a normalization constant. The boundary condition (33) now determines the
eigenvalues $\chi$ at temperature $\tau$,

$\frac{J_{\frac{3}{2}+\tau^{-\gamma}-\chi\tau^{-1}}(\tau^{-\gamma})}{J_{\frac{1}{2}+\tau^{-\gamma}-\chi\tau^{-1}}(\tau^{-\gamma})}=\tau^{\gamma}-1-2\tau^{\gamma-1}\chi.$
(36)

We use this formula to determine the critical temperature and derive the
Landau free energy near the critical point.

##  VI Critical temperature

As discussed above, the superconducting transition temperature
$\tau_{c}=\frac{2\pi T_{c}}{g}$ corresponds to one of the eigenvalues $\chi$
of the matrix $X$ crossing zero. Setting $\chi=0$ in Eq. (36), we obtain an
equation for $a=\tau_{c}^{\gamma}$,

$\frac{J_{\frac{3}{2}+a^{-1}}(a^{-1})}{J_{\frac{1}{2}+a^{-1}}(a^{-1})}=a-1.$
(37)

Numerically, we find that this equation has a unique solution

$a\approx 1.1843,$ (38)

and therefore

$T_{c}(\gamma\to\infty)=\left[a+O(2^{-\gamma})\right]^{\frac{1}{\gamma}}\frac{g}{2\pi}.$
(39)

The correction $O(2^{-\gamma})$ to $a$ is due to interactions with non-
nearest-neighbor spins which we neglected. This formula gives the leading term
in the asymptotic expansion of $T_{c}(\gamma)$ around $\gamma=\infty$. Note
that $T_{c}(\infty)=\frac{g}{2\pi}$ in agreement with our reasoning at the
beginning of Sec. V.2.

$\gamma$ |  $\frac{T_{c}(\gamma)}{g}$ |  $\frac{T_{c}(\gamma\to\infty)}{g}$ |  $\frac{T_{c}(\gamma)-T_{c}(\gamma\to\infty)}{T_{c}(\gamma\to\infty)}$
---|---|---|---
2 |  0.183 |  0.173 |  $6\%$
3 |  0.171 |  0.168 |  $2\%$
4 |  0.1669 |  0.1660 |  $0.5\%$
8 |  0.16258 |  0.16256 |  $0.01\%$
20 |  0.160506897 |  0.160506899 |  $-10^{-6}\%$
Table 1: Values of the superconducting transition temperature $T_{c}(\gamma)$
for various $\gamma$ versus the large $\gamma$ asymptote (39). The agreement
is reasonable already for $\gamma=2$. The relative error – the last column of
the table – is roughly $\frac{2^{-\gamma}}{2\gamma}$ consistent with
$O(2^{-\gamma})$ term in Eq. (39).

In Table 1, we compare the large $\gamma$ asymptote (39) with numerically
exact values of $T_{c}$ for several $2\leq\gamma\leq 20$. The agreement is
reasonable already at $\gamma=2$. To determine $T_{c}$ numerically, we
diagonalize the Hessian $X$ in Eq. (30) as a function of the reduced
temperature $\tau=\frac{2\pi T}{g}$ and compute the value of $\tau$ at which
the lowest eigenvalue of $X$ crosses zero.

##  VII Thermodynamics near the transition

Near $T_{c}$ we need to keep only the unstable mode $\theta_{n\chi_{0}}$ whose
eigenvalue $\chi_{0}$ changes sign at the transition. Then,
$\theta_{n}=\epsilon\theta_{n\chi_{0}}$, where the amplitude of the unstable
mode $\epsilon$ is our order parameter. The free energy $\bar{f}$ expanded
near the transition to order $\epsilon^{4}$ is the Landau free energy from
which various thermodynamic properties, such as the jump in the specific heat,
thermodynamic critical field $H_{c}$ etc., follow.

At the transition, the unstable mode [Eq. (35) with $\chi=0$ and
$\tau=\tau_{c}$] is

$\theta_{n0}=J_{n+\frac{1}{2}+a^{-1}}(a^{-1}),$ (40)

where we used $\tau_{c}^{\gamma}=a$. This function is positive for all $n\geq
0$. It decays with $n$ monotonically and very quickly, approximately as
$n^{-n}$. Its values at the first ($n=0$) and second ($n=1$) Matsubara
frequencies contribute $96.66\%$ and $3.28\%$ to
$\sum_{n=0}^{\infty}\theta_{n0}^{2}$. The superconductivity is therefore
confined to few small Matsubara frequencies, mostly to $\pm\pi T$, consistent
with the short ranginess of the interaction, cf. [10]. In terms of spins, the
domain wall that develops below $T_{c}$ (see Fig. 1) measures only a couple of
sites long.

To obtain the Landau free energy, we expand Eq. (22) to quartic order in
$\theta_{n}$ and substitute $\theta_{n}=\epsilon\theta_{n\chi_{0}}$. The
quadratic part simplifies since $\theta_{n\chi_{0}}$ is the eigenstate of
matrix $X$ with eigenvalue $\chi_{0}$ and we find

$\delta\bar{f}=\chi_{0}\epsilon^{2}\tau_{c}b_{2}+\epsilon^{4}\tau_{c}^{2}b_{4}.$
(41)

As $\chi_{0}\sim\epsilon^{2}$ at the minimum with respect to $\epsilon$, it is
sufficient to evaluate the coefficients at $\chi_{0}\epsilon^{2}$ and
$\epsilon^{4}$ at $\tau=\tau_{c}$ – corrections to them in
$\delta\tau=\tau-\tau_{c}$ contribute terms of order $\epsilon^{6}$ to the
free energy. Then, $b_{2}=\sum_{n}\theta_{n0}^{2}\approx 6.074\times 10^{-2}$
and similarly we determine $b_{4}\approx 8.445\times 10^{-4}$.

We also need $\chi_{0}$ as a function of $\delta\tau$. Expanding Eq. (36) in
$\chi$ and $\delta\tau$ around $\chi=0$ and $\tau=\tau_{c}$, we find
$\chi_{0}=h\gamma\delta\tau$, where $h\approx 0.5339$. It is convenient to
redefine the order parameter as $\tau_{c}\epsilon^{2}=\eta^{2}|\psi|^{2}$,
where $\eta^{2}=\frac{hb_{2}}{2b_{4}}$. We have

$\delta\bar{f}=R\left[\gamma\delta\tau|\psi|^{2}+\frac{|\psi|^{4}}{2}\right],\quad
R\approx 0.6226.$ (42)

We made $\psi=|\psi|e^{i\phi}$ complex to restore the arbitrary overall phase
$e^{i\phi}$ of $\Delta_{n}$ which we set to one until now. Since
$\theta_{n\geq 0}$ is small, Eqs. (13) and (14) imply
$\Delta_{n}=\omega_{n}\theta_{n}$ for $\omega_{n}>0$ and therefore

$\begin{split}\Delta(\omega_{n})=\eta|\psi|a^{-\frac{1}{\gamma}}\omega_{n}J_{n+\frac{1}{2}+a^{-1}}(a^{-1})e^{i\phi},\quad\mbox{$\omega_{n}>0$},\\\
\Delta(-\omega_{n})=\Delta(\omega_{n}),\quad\eta\approx 19.20.\end{split}$
(43)

Note that $\Delta(\omega_{n})e^{-i\phi}>0$ in agreement with the theorem
mentioned earlier that $\Delta(\omega_{n})$ must be nonnegative at the minimum
of the free energy up to an overall phase.

Let us also restore the units in the Landau free energy (42),

$f_{L}\equiv
f-f_{\mathrm{n}}=R\nu_{0}g^{2}\left[\frac{2\pi\gamma}{g}(T-T_{c})|\psi|^{2}+\frac{|\psi|^{4}}{2}\right],$
(44)

where $f_{\mathrm{n}}$ is the normal state free energy density.

Minimizing the condensation energy (44) with respect to $|\psi|^{2}$, we find

$\displaystyle|\psi|=\left[\frac{2\pi\gamma}{g}\right]^{1/2}(T_{c}-T)^{1/2},$
(45a) $\displaystyle
f-f_{\mathrm{n}}=-2\pi^{2}R\nu_{0}\gamma^{2}(T-T_{c})^{2}.$ (45b)

From here the jump in the specific heat $c=-T\frac{\partial^{2}f}{\partial
T^{2}}$, i.e., the difference between superconducting and normal state
specific heats at $T=T_{c}$ is

$\delta c=c_{\mathrm{s}}-c_{\mathrm{n}}=2\pi
R\nu_{0}g\gamma^{2}a^{\frac{1}{\gamma}},$ (46)

where we used Eq. (39). For $\gamma=2$ this formula gives $\delta c\approx
24\nu_{0}g$, which is not too far from the exact answer (rounded to two
significant digits) $\delta c=17\nu_{0}g$ [7] considering that $\gamma=2$ is
well outside of $\gamma>3$ range where our large $\gamma$ theory is supposed
to be accurate.

The thermodynamic critical field $H_{c}$ is the magnetic field above which the
energy cost $\frac{H^{2}}{8\pi}$ of expelling the magnetic field (Meissner
effect) exceeds the energy gain (45b) due to superconductivity [27]. We find

$H_{c}=4\pi^{3/2}\gamma\sqrt{R\nu_{0}}(T_{c}-T).$ (47)

Both $|\psi|$ and $H_{c}$ have the usual mean-field scaling with $(T_{c}-T)$
for a scalar theory.

##  VIII Heat capacity above and below $T_{c}$

Here we evaluate the normal state specific heat $c_{\mathrm{n}}$ for
$\gamma>2$ and the specific heat $c_{\mathrm{s}}$ in the superconducting state
just below $T_{c}$. It turns out that $c_{\mathrm{n}}$ is negative in a
temperature interval $(T_{c},T_{\mathrm{n}})$ above $T_{c}$ for any
$\gamma\geq 2$. We will discuss the significance of this later in this
section.

We saw above that in the normal state $S_{n}^{z}=\mbox{sgn}(\omega_{n})$. The
free energy density in terms of the spin chain is $f=\nu_{0}TH_{s}$. The
Zeeman term in Eq. (21) contributes

$f^{Z}_{\mathrm{n}}=-2\pi\nu_{0}T\\!\\!\\!\sum_{n=-\infty}^{\infty}|\omega_{n}|=-8\pi^{2}\nu_{0}T^{2}\sum_{n=0}^{\infty}\left(n+\frac{1}{2}\right)$
(48)

to the normal state free energy. Looking back at the derivation of the spin
chain [9], we see that up to a $T$-independent constant, $f^{Z}_{\mathrm{n}}$
must be the $\varepsilon_{F}\to\infty$ limit of the free energy of the
noninteracting Fermi gas [11],

$f_{\mathrm{n}}^{(0)}=u_{0}\varepsilon_{F}-\frac{1}{3}\pi^{2}\nu_{0}T^{2},$
(49)

where $u_{0}$ is a constant that depends only on the number of spatial
dimensions, e.g., $u_{0}=3/5$ in 3D.

Summations in Eq. (48) diverge. The reason is that we took the limit
$\varepsilon_{F}\to\infty$ when deriving the spin chain and the Fermi gas free
energy (49) diverges in this limit. However, this affects only the
$T$-independent part, which is of no interest to us here. The standard way to
deal with this divergence is to apply the Poisson summation formula to Eq.
(48) discarding the $T$-independent part. We obtain

$f^{Z}_{\mathrm{n}}=-\frac{1}{3}\pi^{2}\nu_{0}T^{2}.$ (50)

It is instructive to also derive this answer using the zeta function
regularization technique [28]. Recall the definition of the Hurwitz zeta
function

$\zeta(s,p)=\sum_{n=0}^{\infty}\frac{1}{(n+p)^{s}}.$ (51)

We interpret the second summation in Eq. (50) as $\zeta(-1,\frac{1}{2})$ and
since $\zeta(-1,\frac{1}{2})=\frac{1}{24}$, we obtain Eq. (50) with this
approach too.

Now note that
${\bm{S}_{n}}\cdot{\bm{S}_{m}}-1=\mbox{sgn}(\omega_{n}\omega_{m})-1$ vanishes
when $\omega_{n}$ and $\omega_{m}$ have the same sign and is equal to $-2$
otherwise. This observation allows us to rewrite the interaction part of the
free energy as

$\begin{split}f_{\mathrm{int}}=&\nu_{0}g^{\gamma}(2\pi
T)^{2-\gamma}\sum_{l=1}^{\infty}\frac{l}{l^{\gamma}}=\\\
&=4\pi^{2}\nu_{0}\zeta(\gamma-1)\left(\frac{g}{2\pi}\right)^{\gamma}T^{2-\gamma}.\end{split}$
(52)

Here we reduced the summation over $n$ and $m$ to a single sum over $l=n+m+1$
using $\omega_{n}-(-\omega_{m})\propto(n+m+1)$ and the fact that there are $l$
ways to choose $n$ and $m$ for a given $l$.

Adding Eq. (52) to Eq. (50), we obtain the normal state free energy up to a
$T$-independent constant

$f_{\mathrm{n}}=-\frac{1}{3}\pi^{2}\nu_{0}T^{2}+4\pi^{2}\nu_{0}\zeta(\gamma-1)\left(\frac{g}{2\pi}\right)^{\gamma}T^{2-\gamma}$
(53)

and

$c_{\mathrm{n}}(T)=\frac{2\pi^{2}\nu_{0}}{3}T\left[1-\left(\frac{T_{\mathrm{n}}}{T}\right)^{\\!\\!\gamma}\right],$
(54)

where

$T_{\mathrm{n}}=[3(\gamma-1)(\gamma-2)\zeta(\gamma-1)]^{\frac{1}{\gamma}}\frac{g}{2\pi}>T_{c}.$
(55)

The inequality $T_{\mathrm{n}}(\gamma)>T_{c}(\gamma)$ holds for all $\gamma>2$
because the difference $T_{\mathrm{n}}(\gamma)-T_{c}(\gamma\to\infty)$ grows
with $\gamma$ and exceeds $0.25T_{c}$ already at $\gamma\to 2^{+}$ [29], while
$T_{c}(\gamma\to\infty)$ given by Eq. (39) underestimates $T_{c}$ by 6% or
less as seen from Table 1.

Figure 3: Specific heat $c_{\mathrm{s}}$ in the superconducting state just
below $T_{c}$ as a function of $\gamma$, normalized by $\nu_{0}g\gamma^{2}$.
We see that $c_{\mathrm{s}}$ is positive at $T=T_{c}^{-}$ for all $\gamma>3$.
Since $c_{\mathrm{s}}(T)$ is continuous, it will remain positive for a finite
temperature range below $T_{c}$. Note also that
$c_{\mathrm{s}}(T_{c}^{-})\approx 1.3\nu_{0}g\gamma^{2}$ at large $\gamma$.

Therefore, the heat capacity is negative for $T_{c}<T<T_{\mathrm{n}}$. Note
that $T_{\mathrm{n}}\to T_{c}$ as $\gamma\to\infty$. Nevertheless, there is a
sliver of temperature where the heat capacity is negative. The interpretation
of this depends on the origin of the $\gamma$ model. Any subsystem that does
not interact with other subsystems and has negative heat capacity is
thermodynamically unstable [11]. One scenario is that the quasiparticles are
ill defined, they do not form a Fermi liquid, and therefore the stationary
point we started with (the solution of the gap equation) is not the global
minimum of the total free energy. Another scenario is that the effective
fermion-fermion interaction $V(\omega_{l})$ changes with temperature, so that
$V(\omega_{l})=\frac{g^{\gamma}}{|\omega_{l}|^{\gamma}}$ below $T_{c}$ and
something else above $T_{c}$. In other words, the $\gamma$ model kicks in only
below $T_{c}$. This scenario is in principle possible when the bosons that
mediate the interaction are collective excitations of the fermions themselves
as is the case for many $\gamma<1$. Then, the superconducting transition
modifies the bosonic propagator and therefore $V(\omega_{l})$. However, if the
interaction is mediated by phonons or other true bosons, the $\gamma$ model is
unphysical for $\gamma\geq 2$ at least in a certain range of temperatures, see
also Ref. [30] where we addressed this issue for phonon mediated electron-
electron interactions ($\gamma=2$).

Due to the jump at $T_{c}$, the specific heat becomes positive in the
superconducting state for a range of temperatures $T_{-}<T\leq T_{c}$. We
encountered this situation before in the $\gamma=2$ case, where the
superconducting state was free of the pathologies of the normal state – the
opening of the gap stabilized the system. To evaluate the specific heat
$c_{\mathrm{s}}$ just below $T_{c}$, i.e., at $T=T_{c}^{-}$, we use Eqs. (46)
and (54) and

$c_{\mathrm{s}}(T_{c}^{-})=c_{\mathrm{n}}(T_{c}^{-})+\delta c.$ (56)

We plot $c_{\mathrm{s}}(T_{c}^{-})$ normalized by $\nu_{0}g\gamma^{2}$ as a
function of $\gamma$ in Fig. 3. We see that it is positive at $T=T_{c}^{-}$
for all $\gamma>3$. By continuity it must also remain positive in a certain
finite temperature range $(T_{-},T_{c}^{-}]$. Note also the large $\gamma$
asymptote $c_{\mathrm{s}}(T_{c}^{-})\approx 1.3\nu_{0}g\gamma^{2}.$ We will
see in Sec. IX.2 that $T_{-}>0$, i.e., $c_{\mathrm{s}}$ becomes negative at
low temperatures. This is unlike the $\gamma=2$ case where the specific heat
is always positive in the superconducting state and vanishes when $T\to 0$
[30] as it should.

##  IX Low temperature properties – universal gap function and specific heat

We saw that the $\gamma$ model is a superconductor in thermal equilibrium at
$T<T_{c}$ – the anomalous averages are nonzero. Having addressed its
properties near $T_{c}$ and in the normal state, we now turn to the
superconducting state at $T\ll T_{c}$.

For all $\gamma>3$, the problem of determining the gap function
$\Delta(\omega)$ and thermodynamic properties in the low temperature regime
reduces to a single parameterless second order ODE up to corrections of
relative order $\left(T/g\right)^{r\gamma/3},$ where $r>0$ is a function of
$\gamma$. In this section, we first solve this ODE to evaluate the gap
function on the Matsubara axis, the free energy and the specific heat, and
then discuss the corrections. Similar to the normal state, the absolute
specific heat is negative, but the difference between the superconducting and
normal state specific heats is positive.

###  IX.1 Gap function

We showed in Sec. IV.2 that at low temperatures and $\gamma>3$ the gap
equation reduces to Eq. (25). Rescaling the variable $\bar{\omega}$ as in Eq.
(28), we obtain a universal low-temperature gap equation

$\frac{d^{2}\theta}{dx^{2}}=x\sin\theta(x),$ (57)

where

$x=\frac{\omega}{\omega_{*}},\quad\omega_{*}=g\frac{[\zeta(\gamma-2)]^{\frac{1}{3}}}{2^{\frac{1}{3}}}\left(\frac{g}{2\pi
T}\right)^{\\!\\!\frac{\gamma}{3}-1}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!.$ (58)

The energy constant $\omega_{*}$ is the coupling $g$ rescaled by a factor that
depends on $g/T$ and $\gamma$. Eq. (57) is universal in that it is
parameterless and therefore independent of $\gamma$, $g,$ and $T$ – all
dependence on these parameters is in the energy scale $\omega_{*}$.

Figure 4: Universal gap function $\Delta(\omega)$ and the corresponding angle
$\theta_{0}(x)$ spins make with the $z$ axis (inset). Gap functions
$\Delta(\omega_{n})$ for all $\gamma>3$ and all values of the coupling $g$
collapse onto this plot in the limit $T\to 0$ as long as both $\Delta$ and
$\omega$ are measured in units of the energy constant $\omega_{*}$ given by
Eq. (58).

Recall from Sec. IV.2 that $\theta(0)=\frac{\pi}{2}$. Further, the requirement
$\Delta_{n}\geq 0$ discussed in the beginning of Sec. III together with Eqs.
(14) and (15) imply $0\leq\theta_{n}\leq\frac{\pi}{2}$ for $\omega_{n}>0.$
Therefore, for $x\geq 0$ we have

$\theta(0)=\frac{\pi}{2},\quad\theta(+\infty)=0,\quad
0\leq\theta(x)\leq\frac{\pi}{2}.$ (59)

In the Appendix, we show that under the conditions (59) there exists a unique
solution $\theta_{0}(x)$ of the nonlinear differential equation (57) and note
that $\theta_{0}(x)$ is real analytic on the entire $x$-axis by standard
theorems of the theory of ODEs. This is a parameterless function which we plot
in the inset to Fig. 4 for $x\geq 0$. Observe also that since
$\theta(x)\to\pi-\theta(-x)$ leaves Eq. (57) invariant and by uniqueness,
$\theta_{0}(x)$ must map into itself under this transformation, i.e.,
$\theta_{0}(x)=\pi-\theta_{0}(-x)$.

Having determined $\theta_{0}(x)$, we know the gap function for all low
temperatures, $g$, and $\gamma>3$. Indeed, Eq. (15) implies

$\frac{\Delta(\omega)}{\omega_{*}}=Y\\!\left(\frac{\omega}{\omega_{*}}\right),\quad
Y(x)=x\tan\theta_{0}(x).$ (60)

[Recall that we extended $\Delta(\omega_{n})$ to $\Delta(\omega)$ defined on
the entire real $\omega$ axis in Sec. IV.2.] This equation provides the
leading small $T$ asymptotic behavior of $\Delta(\omega)$ for any $g$ and
$\gamma$. Graphically, plots of the gap function $\Delta(\omega_{n})$ vs.
Matsubara frequency $\omega_{n}$ for any $\gamma$ and $g$ tend to the same
universal curve shown in Fig. 4 as $T\to 0$, when both $\Delta(\omega_{n})$
and $\omega_{n}$ are measured in units of $\omega_{*}$.

It is straightforward to work out the expansion of $\Delta(\omega)$ at small
$\omega$ and its large $\omega$ asymptote. Both depend on a single constant
that needs to be determined numerically. At large $\omega/\omega_{*}$, the
angle $\theta_{0}$ is small and $Y(x)\approx x\theta_{0}(x)$. Equation (57)
becomes the Airy equation. Its solution that goes to zero at infinity is the
Airy function of the first kind, $\theta_{0}(x)\propto\mathrm{Ai}(x)$.
Therefore at large $\omega/\omega_{*}\equiv x$

$\frac{\Delta(\omega)}{\omega_{*}}\approx 4.58\mathrm{Ai}(x)\approx
1.29x^{\frac{3}{4}}\exp\biggl{(}-\frac{2x^{\frac{3}{2}}}{3}\biggr{)},$ (61)

where we determined the constant of proportionality 4.58 numerically and used
the known asymptotic expansion for $\mathrm{Ai}(x)$ [31].

Similarly, at small $x=\omega/\omega_{*}$

$\frac{\Delta(\omega)}{\omega_{*}}\approx 0.91-0.10x^{2}+0.05x^{4}+O(x^{6}).$
(62)

In particular, recalling the definition (58) of $\omega_{*}$, we find

$\Delta(0)=0.72g[\zeta(\gamma-2)]^{\frac{1}{3}}\left(\frac{g}{2\pi
T}\right)^{\\!\\!\frac{\gamma}{3}-1}.$ (63)

Since this result is for $\gamma>3$, $\Delta(0)$ diverges as
$T^{1-\frac{\gamma}{3}}$ for ${T\to 0}$ [32]. For fixed $T$ and $\gamma\to
3^{+}$ we have

$\frac{2\Delta(0)}{T_{c}}\approx 8.47|\gamma-3|^{-\frac{1}{3}},$ (64)

where we took the value of $T_{c}$ for $\gamma=3$ from Table 1. Reference [32]
found the same answer, but for $\gamma\to 3^{-}$ and with a prefactor $4\pi$
instead of 8.47. Note that unlike in the BCS theory, $2\Delta(0)$ here is not
the gap in the spectrum [33].

###  IX.2 Low $T$ free energy and specific heat

To evaluate the free energy $\bar{f}$, we use Eq. (27). As discussed in Sec.
VIII, the free energy contains a diverging $T$-independent constant because we
took the limit $\varepsilon_{F}\to\infty$. This divergence is present for
superconducting states as well, since in these states
$S_{n}^{z}\to\mathrm{sgn}(\omega_{n})$ when $|\omega_{n}|\to\infty$, same as
in the normal state. To isolate it, we subtract and add the noninteracting
part of the normal state free energy $f_{\mathrm{n}}^{Z}$ given by Eq. (48),
i.e., $f=f-f_{\mathrm{n}}^{Z}+f_{\mathrm{n}}^{Z}$. The difference
$f-f_{\mathrm{n}}^{Z}$ is finite and we have already evaluated the $T$
dependence of $f_{\mathrm{n}}^{Z}$ in Sec. VIII.

Taking the difference $\bar{f}-\bar{f}_{\mathrm{n}}^{Z}$, where
$\bar{f}_{\mathrm{n}}^{Z}=f_{\mathrm{n}}^{Z}/(\nu_{0}g^{2})$, corresponds to
the replacement $S^{z}\to S^{z}-\mbox{sgn}(x)$ in the Zeeman term in Eq. (27),
since $S^{z}(x)=\mbox{sgn}(x)$ in the normal state. Using also
$S^{z}=\cos\theta$, $d\bm{S}/dx=d\theta/dx\equiv\theta^{\prime}$, and the
symmetry property (16) of $\theta(x)$, we obtain the non-dimensionalized free
energy in the superconducting state at $T\ll T_{c}$ as

$\begin{split}\bar{f}_{\mathrm{s}}=&\frac{[\zeta(\gamma-2)\tau^{3-\gamma}]^{\frac{2}{3}}}{2^{\frac{2}{3}}}\\!\\!\\!\int\limits_{0}^{\infty}\\!\\!dx\\!\left\\{2x(1-\cos\theta_{0})+(\theta^{\prime}_{0})^{2}\right\\}\\\
&-\frac{\tau^{2}}{12},\quad\mbox{where}\quad\tau=\frac{2\pi
T}{g},\quad\bar{f}=\frac{f}{\nu_{0}g^{2}}.\end{split}$ (65)

Substituting the numerical solution for $\theta_{0}(x)$ discussed in the
previous subsection into the integral, we find

$\bar{f}_{\mathrm{s}}=1.11[\zeta(\gamma-2)]^{\frac{2}{3}}\tau^{2-\frac{2\gamma}{3}}-\frac{\tau^{2}}{12}.$
(66)

The specific heat $c_{\mathrm{s}}=-T\frac{\partial^{2}f}{\partial T^{2}}$ in
the superconducting state is

$c_{\mathrm{s}}=\frac{2\pi^{2}\nu_{0}}{3}T\left[1-\left(\frac{T_{s}}{T}\right)^{\\!\\!\frac{2\gamma}{3}}\right],$
(67)

where

$T_{s}=\left[1.48(\gamma-3)(2\gamma-3)\right]^{\frac{3}{2\gamma}}\left[\zeta(\gamma-2)\right]^{\frac{1}{\gamma}}\frac{g}{2\pi}.$
(68)

It is evident that $c_{\mathrm{s}}$ becomes negative as $T\to 0$ for any
$\gamma>3$.

This indicates that the $\gamma$ model is pathological for $T\to 0$ just as it
is above $T_{c}$, see the discussion at the end of Sec. VIII. Again, there
cannot be an unambiguous explanation of this pathology without the knowledge
of the system Hamiltonian. To fix it, we need to modify the effective
fermionic action (9), i.e., the $\gamma$ model itself. One scenario is as
follows. Consider the difference between $c_{\mathrm{s}}$ and the normal state
specific heat (54) $c_{\mathrm{n}}$,

$c_{\mathrm{s}}-c_{\mathrm{n}}=\frac{2\pi^{2}\nu_{0}}{3}T\left[\left(\frac{T_{\mathrm{n}}}{T}\right)^{\\!\\!\gamma}-\left(\frac{T_{s}}{T}\right)^{\\!\\!\frac{2\gamma}{3}}\right].$
(69)

This is positive for $T<T_{\mathrm{n}}^{3}/T_{s}^{2}$. A straightforward
numerical analysis shows that $T_{\mathrm{n}}^{3}/T_{s}^{2}>T_{c}$. Therefore,
this difference is always positive when $T<T_{c}$. Suppose the modification of
the fermionic action is such that it introduces a new order parameter
independent of $\bm{S}_{n}$. Further, suppose this adds terms to $H_{s}$ that
depend on the new order parameter and not on $\bm{S}_{n}$ [see Eqs. (11) and
(12)]. Then, it does not affect the superconducting transition and the gap
equation, but adds $T$-dependent terms to the free energy and to the specific
heat. The change in the specific heat must be the same for $c_{\mathrm{s}}$
and $c_{\mathrm{n}}$. Provided the new $c_{\mathrm{n}}$ is positive, so is the
new $c_{\mathrm{s}}$, because Eq. (69) remains valid. This scenario is
favorable for the results of this section in the sense that it fixes the
pathology while leaving them intact with the exception of the answers for
$\bar{f}_{\mathrm{s}}$ and $c_{\mathrm{s}}$. However, note that
$\bar{f}_{\mathrm{s}}-\bar{f}_{\mathrm{n}}$ and
$c_{\mathrm{s}}-c_{\mathrm{n}}$ do not change.

###  IX.3 Accuracy of the local approximation

In this section, we solved the gap equation on the Matsubara axis and
evaluated the specific heat at low temperatures for $\gamma>3$, by replacing
the gap equation (17), which we copy here for convenience,

$\omega_{n}\sin\theta_{n}=g^{\gamma}\pi T\sum_{m\neq
n}\frac{\sin(\theta_{m}-\theta_{n})}{|\omega_{m}-\omega_{n}|^{\gamma}},$ (70)

with the differential equation (57). Now let us investigate the accuracy of
this local approximation.

To derive Eq. (57), we expanded $\sin(\theta_{m}-\theta_{n})$ in
$(\omega_{m}-\omega_{n})$ to second order, see Eq. (24). The contribution from
odd powers of $(\omega_{m}-\omega_{n})$ to the right hand side of Eq. (70)
cancels after summation over $m$. Therefore, the error comes from terms of
order 4 and higher. The Lagrange error bound $M(\omega_{m}-\omega_{n})^{4}/4!$
then gives an upper bound on the difference between
$\sin(\theta_{m}-\theta_{n})$ and our approximation to it. Here $M$ is the
maximum value of $d^{4}\theta_{n}/d\omega_{n}^{4}$.

Let us split the summation in Eq. (70) into two parts: $|m-n|<L$ and
$|m-n|\geq L$, where $L$ is to be determined. For $|m-n|\leq L$ we use the
Taylor series expansion of $\sin(\theta_{m}-\theta_{n})$ to third order plus
the Lagrange error bound. For $|m-n|>L$, we replace
$\sin(\theta_{m}-\theta_{n})\to 1$, which provides an upper bound for the
error from neglecting these terms. Pulling out an overall factor of $2(2\pi
T)^{-\gamma}$, we obtain

$\begin{split}\frac{(2\pi
T)^{2}}{2}\frac{d^{2}\theta_{n}}{d\omega_{n}^{2}}\zeta(\gamma-2)-\frac{(2\pi
T)^{2}}{2}\frac{d^{2}\theta_{n}}{d\omega_{n}^{2}}\sum_{k=L}^{\infty}k^{2-\gamma}\\\
+\frac{(2\pi
T)^{4}}{4!}M\sum_{k=1}^{L}k^{4-\gamma}+\sum_{k=L}^{\infty}k^{-\gamma}.\end{split}$
(71)

The first term leads to Eq. (57) after the change of variables (58). The
remaining three terms are the error. To estimate the second derivative of
$\theta(\omega_{n})$ and $M$, we use the numerical solution of Eq. (57)
plotted in Fig. 4. For example,
$d^{2}\theta/d\omega^{2}=\omega_{*}^{-2}d^{2}\theta/dx^{2}$ with $0\leq
d^{2}\theta/dx^{2}<0.6$ and $M\approx 1.4\omega_{*}^{-4}$. We replace the
summations over $k$ in Eq. (71) with integrals and then minimize the error
with respect to $L$. As $L$ turns out to be large, corrections due to the
replacement of the summations with integrals are negligible. In this way, we
obtain that the relative error [magnitude of the ratio of the sum of the last
three terms in Eq. (71) to the first term] is

$\mathrm{RE}\sim\left(\frac{2\pi
T}{g}\right)^{\\!\\!\frac{r\gamma}{3}}\\!\\!\\!\\!\\!\\!,\quad
r=\left\\{\begin{array}[]{ll}\gamma-3,&\mbox{for $3<\gamma\leq 5,$}\\\
2,&\mbox{ for $\gamma>5$}.\\\ \end{array}\right.$ (72)

##  X Conclusions

In this paper, we studied the thermodynamics of a system of fermions near a
quantum critical point with extremely retarded interactions of the form
$V(\omega_{l})=(g/|\omega_{l}|)^{\gamma}$, where $\omega_{l}$ is a bosonic
Matsubara frequency. The case $\gamma=2$ of this $\gamma$ model corresponds to
the strong coupling limit of the Eliashberg theory, which is intermediately
retarded. Extreme retardation at generic $T$ kicks in for $\gamma\gg 1$ and
for $\gamma>3$ at $T\to 0$. Note that for $\gamma>2$, the $\gamma$ model is a
model without a Hamiltonian and is defined through a nonlocal effective
Euclidian action only.

The $\gamma$ model shows two phases: normal and superconducting. Similarly to
the Eliashberg theory, the order parameter is the frequency dependent gap
function $\Delta(\omega_{n})$, where $\omega_{n}$ is the fermionic Matsubara
frequency. We determined the superconducting transition temperature $T_{c}$
and the order parameter $\Delta(\omega_{n})$ near $T_{c}$. The amplitude
$\psi$ of $\Delta(\omega_{n})$ serves as the Landau order parameter. We
expanded the free energy functional to order $|\psi|^{4}$ near the transition
to obtain the Landau free energy, from which we derived the jump in the
specific heat and the thermodynamic critical field. These answers are
asymptotically exact in the limit $\gamma\to\infty$. We also evaluated the
normal state specific heat $c_{\mathrm{n}}$ for arbitrary $T$ and $\gamma>2$.

Next, we turned our attention to the properties of the $\gamma$ model at low
temperatures. We proved that the global minimum of the free energy is unique
(nondegenerate). We derived the universal gap equation, which is a
parameterless second order ODE, and determined the scaling of $\Delta(\omega)$
with $T$, $g$, and $\gamma$ for all $g$, $\gamma>3$, and $T\to 0$. Building on
this, we obtained explicit expressions for the free energy and specific heat
in the superconducting state for this range of parameters. We also evaluated
$2\Delta(0)/T_{c}$ and found that it is finite for $T>0$ and $\gamma>3$, but
diverges as $|\gamma-3|^{-1/3}$ for $\gamma\to 3^{+}$ and as $T^{1-\gamma/3}$
for $T\to 0$. These results are exact for any $\gamma>3$ at $T\to 0$.

Note that “exact” and “asymptotically exact” here and elsewhere in this paper
mean exact for the $\gamma$ model defined by the effective action (9) in the
thermodynamic and $\varepsilon_{F}\to\infty$ limits. In this regime,
fluctuational corrections to the spin chain are negligible and it is at zero
effective temperature, i.e., its ground state [which is determined by the gap
equation (17)] captures all thermodynamical properties that we evaluated.

We found that the $\gamma$ model is thermodynamically unstable for $\gamma\geq
2$. Its specific heat is negative above $T_{c}$ for $\gamma\geq 2$ and also at
$T\to 0$ for $\gamma>3$. We saw in an earlier paper [30] that when this model
is understood as an effective description of the phonon mediated electron-
electron interaction ($\gamma=2$), this instability implies the emergence of a
new order above $T_{c}$. The new phase breaks the lattice translational
symmetry and invalidates the $\gamma$ model at least in a certain temperature
range. For other $\gamma$, a microscopic Hamiltonian is similarly necessary to
resolve this issue. An interesting open problem is therefore to construct
classes of physical many-body Hamiltonians that correspond to the $\gamma$
model with arbitrary $\gamma>2$.

###### Acknowledgements.

We thank Ar. Abanov, A. V. Chubukov, G. Kotliar, and T.-H. Lee for helpful
discussions.

*

##  Appendix A Existence and uniqueness of solutions to the universal gap
equation

In this appendix we show that under the conditions (59) the universal gap
equation (57) has a unique solution. Uniqueness is easier to show than
existence, so we start with uniqueness, then discuss existence of a solution.
We also offer a section about the properties of solutions of Eq. (57) when
conditions (59) are dropped.

###  A.1 Uniqueness of a solution satisfying conditions (59)

Suppose that both $\theta_{0}(x)$ and $\tilde{\theta}_{0}(x)$ are twice
continuously differentiable solutions of the ODE (57) that satisfy the
conditions (59). Then $\Theta(x):=\theta_{0}(x)-\tilde{\theta}_{0}(x)$
vanishes when $x\to 0$ from the right, and when $x\to\infty$. Moreover,
$\Theta(x)$ is also twice continuously differentiable, and its second
derivative is given by

$\frac{d^{2}}{dx^{2}}\Theta(x)=x\left(\sin\big{[}\theta_{0}(x)\big{]}-\sin\big{[}\tilde{\theta}_{0}(x)\big{]}\right),\quad
x\in(0,\infty).$ (73)

We multiply Eq. (73) by $\Theta(x)$, integrate from $x=0$ to $x=\infty$,
integrate by parts on the left-hand side, and obtain

$0\geq-\int_{0}^{\infty}|\Theta^{\prime}(x)|^{2}dx=\int_{0}^{\infty}x|\Theta(x)|^{2}\,\frac{\sin\big{[}\theta_{0}(x)\big{]}-\sin\big{[}\tilde{\theta}_{0}(x)\big{]}}{\theta_{0}(x)-\tilde{\theta}_{0}(x)}dx\geq
0,$ (74)

where the prime denotes derivative with respect to $x$. The second inequality
in (74) follows from the fact that $\sin\theta$ is an increasing function for
$\theta\in[-\pi/2,\pi/2]$. Since both inequalities in (74) are strict if
$\theta_{0}(x_{0})\neq\tilde{\theta}_{0}(x_{0})$ for some $x_{0}>0$ (and
therefore in some open neighborhood of $x_{0}$), the only option compatible
with (74) is: $\theta_{0}(x)=\tilde{\theta}_{0}(x)$ for all $x\geq 0$.

Thus uniqueness holds.

###  A.2 Existence of a solution satisfying conditions (59)

The uniqueness proof only shows that there cannot exist more than one twice
continuously differentiable solution to Eq. (57) that satisfies the conditions
(59), while leaving open whether a solution exists at all. In this subsection
we show that such a solution does exist, indeed.

We now consider the initial value problem for Eq. (57), with initial data

$\theta(0)=\frac{\pi}{2};\qquad\theta^{\prime}(0)=\alpha,$ (75)

and we treat $\alpha$ as a real parameter that we exhibit explicitly in the
solution to the initial value problem, written as $\theta(x;\alpha)$. Our goal
is to show that there exists at least one particular value $\alpha_{0}$
(possibly not unique, in this section) such that the pertinent solution
$\theta(x;{\alpha_{0}})$ of the initial value problem satisfies the remaining
conditions in (59); i.e., $\theta(x;{\alpha_{0}})$ vanishes as $x\to\infty$,
and $\theta(x;{\alpha_{0}})$ takes values only in $[0,\pi/2]$ for $x\geq 0$.
In this case we may identify $\theta(x;{\alpha_{0}})$ with a sought-after
solution $\theta_{0}(x)$ of Eq. (57) under the conditions (59).

We begin by noting that the Picard-Lindelöf theorem [34] guarantees that for
each $\alpha$ the initial value problem for Eq. (57) with initial data (75)
has a unique twice continuously differentiable solution $\theta(x;\alpha)$,
which exists for all $x\in\mathbb{R}$. The regularity follows from the facts
that the function $x\sin\theta$ is continuous in both $x$ and $\theta$. The
uniqueness and global character of such a solution follow from the additional
feature that the derivative of $\sin\theta$ is uniformly bounded in absolute
value. In fact, $\theta(x;\alpha)$ is analytic in $x$; this follows from the
fact that the function $x\sin\theta$ is analytic in both $x$ and $\theta$; see
Ref. [35]. Thus it remains to show that there is a solution $\theta_{0}(x)$
that converges to 0 as $x\to\infty$, and that this solution does not take
values outside of the interval $[0,\pi/2]$ when $x\geq 0$.

Since we demand that the solution $\theta_{0}(x)\leq\pi/2$, with
$\theta_{0}(0)=\pi/2$, it now follows from Eq. (57) that _a necessary
condition for the existence of such a solution is that the initial slope_
$\alpha<0$. Indeed, for $\theta\in(0,\pi)$ and $x>0$, the right-hand side of
Eq. (57) is $>0$, so that the unique solution $\theta(x;\alpha)$ of Eq. (57)
that satisfies the initial data (75) is convex as long as
$\theta(x;\alpha)\in(0,\pi)$. But this means that if there is any $x_{0}\geq
0$ for which $\theta(x_{0};\alpha)\in(0,\pi)$ and
$\theta^{\prime}(x_{0};\alpha)=0$, then $x_{0}$ is a local minimum point of
$\theta(x;\alpha)$, and this solution will inevitably increase to values
$>\pi/2$.

On the other hand, since we also demand that the solution $\theta_{0}(x)\geq
0$, its negative initial slope $\alpha$ cannot be too large in magnitude, for
it is straightforward to show that there is some $\alpha_{0}<0$ such that for
$\alpha<\alpha_{0}$ there is an $x_{0}>0$ such that the pertinent solution
$\theta(x;\alpha)$ satisfies $\theta(x_{0};\alpha)=0$ and
$\theta^{\prime}(x_{0};\alpha)<0$. (We have recycled the symbol $x_{0}$ with a
new meaning.) In this case it follows right away that
$\theta(x_{0}+\epsilon;\alpha)<0$ for some $\epsilon>0$. To see that there is
such an $\alpha_{0}<0$, recall that $\sin\theta\leq 1$, so that we obtain the
estimate

$\theta^{\prime\prime}(x;\alpha)\leq x,$ (76)

and integrating this estimate twice for the stipulated initial data we find
that

$\theta(x;\alpha)\leq\tfrac{\pi}{2}+\alpha x+\tfrac{1}{6}x^{3}.$ (77)

The cubic polynomial at the right-hand side may have no, or one, or two
positive roots, depending on $\alpha$. If there is at least one positive root,
let $x_{*}$ denote either the unique positive root or the smaller of the two
positive roots. Such a root $x_{*}$ exists if and only if
$\alpha+\frac{1}{2}x_{*}^{2}\leq 0$. Setting
$\alpha_{*}=-\frac{1}{2}x_{*}^{2}$, our cubic problem becomes
$\tfrac{\pi}{2}-\frac{1}{2}x_{*}^{3}+\frac{1}{6}x_{*}^{3}=0$, viz.
$x_{*}=\left(\frac{3\pi}{2}\right)^{\frac{1}{3}},$ which returns

$\alpha_{*}=-\tfrac{1}{2}\left(\tfrac{3\pi}{2}\right)^{\frac{2}{3}}.$ (78)

And so, when $\alpha\leq\alpha_{*}$ the cubic upper bound to the solution
$\theta_{\alpha}(x)$ of our initial value problem, i.e., Eq. (57) with initial
conditions (75), vanishes at $x_{*}$ with a slope $\leq 0$. Hence,
$\theta(x;\alpha)$ itself must have a zero at some $x_{0}<x_{*}$ when
$\alpha\leq\alpha_{*}$. Moreover, $\theta^{\prime}(x_{0};\alpha)<0$. For
suppose $\theta^{\prime}(x_{0};\alpha)=0$; then both $\theta(x;\alpha)$ and
its first $x$ derivative would vanish at $x_{0}$, and by the uniqueness of the
solution of the second-order initial value problem formulated with these data
at $x_{0}$, the function $\theta(x;\alpha)$ would have to vanish identically,
which is a contradiction to the fact that $\theta(0;\alpha)=\frac{\pi}{2}$. So
$\theta^{\prime}(x_{0};\alpha)<0$, and since $\theta(x_{0};\alpha)=0$, it
follows that $\theta(x+\epsilon;\alpha)<0$, which violates the required lower
bound 0 for $\theta_{0}$. It follows that there is some $\alpha_{0}$ with
$\alpha_{*}<\alpha_{0}<0$ such that _a further necessary condition for the
existence of the desired solution_ $\theta_{0}(x)$ _is that the initial slope_
$\alpha\geq\alpha_{0}$.

Now consider what happens to $\theta(x;\alpha)$ if we start with
$\alpha=\alpha_{*}$ and continuously increase $\alpha$ from there. As just
discussed, the solution $\theta(x;{\alpha_{*}})$ to the initial value problem
(57), (75) has a smallest positive zero at $x_{0}(\alpha_{*})<x_{*}$, and
$\theta^{\prime}\big{(}x_{0}(\alpha_{*});\alpha_{*}\big{)}<0$. Moreover,
$\theta^{\prime}(x;\alpha_{*})<0$ for all $x\in[0,x_{0}(\alpha_{*})].$ Indeed,
if there was some $x_{\star}<x_{0}$ with
$\theta^{\prime}(x_{\star};\alpha_{*})=0$, then
$\theta(x_{\star};\alpha_{*})\in(0,\pi/2)$, and as discussed above,
$x_{\star}$ would be a local minimum point of $\theta(x;{\alpha_{*}})$. This
solution would increase for $x>x_{\star}$ to values $>\pi/2$ – in
contradiction to the cubic upper bound we derived. Thus
$\theta(x;{\alpha_{*}})$ decreases monotonically from the value
$\frac{\pi}{2}$ at $x=0$ to the value 0 at $x=x_{0}(\alpha_{*})$. Now increase
$\alpha$ continuously above $\alpha_{*}$. It is easy to see [just formally
integrate Eq. (57) twice, using Eq. (75)] that for all $0<x\leq x_{0}(\alpha)$
one has $\frac{\partial}{\partial\alpha}\theta^{\prime}(x;\alpha)>0$ and
$\frac{\partial}{\partial\alpha}\theta(x;\alpha)>0$. Thus the zero
$x_{0}(\alpha)$ moves continuously to the right as $\alpha$ increases.
Moreover, as long as $x_{0}(\alpha)<\infty$, the function $\theta(x;\alpha)$
reaches its first zero at $x_{0}(\alpha)$ with a nonzero negative slope,
$\theta^{\prime}\big{(}x_{0}(\alpha);\alpha\big{)}<0$. This follows from the
already made observation that $\theta(x;\alpha)$ must be identically zero if
it vanishes at a finite location $x_{0}(\alpha)$ with vanishing slope, in
contradiction to the initial data $\theta(0;\alpha)=\frac{\pi}{2}$. Thus we
can increase $\alpha$ until a value $\alpha_{0}$ is reached at which
$x_{0}(\alpha_{0})=\infty$, with
$\lim_{\alpha\to\alpha_{0}}\theta^{\prime}\big{(}x_{0}(\alpha);\alpha\big{)}=0$
(limit from the left). The solution $\theta(x;{\alpha_{0}})$ is a sought-after
solution $\theta_{0}(x)$.

This demonstrates the existence of a solution to Eq. (57) that satisfies (59).

###  A.3 The types of solutions for general initial data

We briefly discuss the general initial value problem for Eq. (57) with initial
data at $x=0$. By the $2\pi$-periodicity of the sine function, it suffices to
restrict the discussion to data

$\theta(0)=\vartheta\in[-\pi,\pi];\qquad\mbox{and}\qquad\theta^{\prime}(0)=\alpha\in\mathbb{R}.$
(79)

For any pair of such initial data $(\vartheta,\alpha)$ there exists a unique
analytical solution $\theta(x;{\vartheta,\alpha})$ of Eq. (57) for all $x\geq
0$; cf. [35]. The purpose of this subsection is to present a mostly
qualitative and partly quantitative overview of the behavior of these
solutions.

To get a more intuitive grasp of the possible solutions
$\theta(x;{\vartheta,\alpha})$ it is helpful to note that for $x>0$ the
variable transformation

$\theta(x)=\phi(t),\qquad\mbox{with}\qquad t={\textstyle{2\over 3}}x^{3/2},$
(80)

changes Eq. (57) into

$\ddot{\phi}(t)+{\textstyle{1\over 3t}}\dot{\phi}(t)=\sin\phi(t)$ (81)

for $t>0.$ Here we have introduced Newton’s dot notation to denote derivatives
with respect to $t$. Equation (81) describes a damped rigid pendulum, or,
equivalently, a point mass moving on an upright circle subject to uniform
gravity and Newtonian friction, with a friction coefficient inversely
proportional to time $t$ [36]. We count the angle $\phi$ from the “up”
position, i.e., $\phi=0$ corresponds to the unstable inverted pendulum
equilibrium and $\phi=\pi$ to the stable equilibrium. Of course, a priori any
integer multiple of $2\pi$ may be added to either $0$ or $\pi$ to obtain yet
another unstable, respectively stable equilibrium solution for Eq. (81),
because Eq. (81) in itself does not restrict $\phi(t)$ to lie in any
particular interval such as $[0,\pi/2]$.

The damped pendulum Eq. (81) intuitively suggests that the damping will force
the evolution of $\phi(t)$ to converge at $t\to\infty$ to one of the
infinitely many copies of the two possible equilibrium states. Most initial
data would lead to a (copy of the) stable equilibrium state $\phi_{s}=\pi$.
However, for each $\vartheta$ there should be a discrete set of slopes
$\alpha$ which launch a solution that asymptotically approaches (a copy of)
the unstable equilibrium state $\phi_{u}=0$. The approach to any stable
equilibrium is damped oscillatory, while the approach to an unstable
equilibrium is monotonic. Copies of the stable pendulum equilibrium have
Newtonian energy $=-1$ and the copies of the unstable one $=+1$.

There is one caveat to what we just wrote. Since the friction coefficient
vanishes as $t\to\infty$, it is in principle conceivable that there also exist
solutions that asymptotically approach a dynamical solution of the undamped
pendulum equation, instead of converging to an equilibrium solution. We can
rigorously rule out an asymptotic approach to a so-called librating solution
of the undamped pendulum equation (see below), but we have not rigorously
ruled out an asymptotic approach to an oscillating undamped pendulum solution;
we expect that it does not occur, though.

In addition to these intuitive insights into the possible types of solution,
we offer some exact results.

First, one can show the convergence of the Newtonian energy of the pendulum
evolution

$E[\phi,\dot{\phi}](t):={\textstyle{1\over
2}}\big{|}\dot{\phi}(t)\big{|}^{2}+\cos\phi(t).$ (82)

The energy is monotonically decreasing with $t$ due to friction. Indeed,

${\textstyle{d\over
dt}}E[\phi,\dot{\phi}](t)=\dot{\phi}(t)\big{[}\ddot{\phi}(t)-\sin\phi(t)\big{]}=-{\textstyle{1\over
3t}}|\dot{\phi}(t)|^{2}\leq 0,$ (83)

and for any $0<t<\infty$ the right-hand side in Eq. (83) is “$<0$” if
$\dot{\phi}\neq 0$. Since $E$ is bounded below, it must converge as
$t\to\infty$. However, it may or may not converge to its minimum value
$E_{\min}=-1$.

We remark that convergence of $E[\phi,\dot{\phi}](t)$ to a constant as
$t\to\infty$ does not in itself imply convergence of $\phi(t)$ as
$t\to\infty$. After all, in the absence of friction the energy is conserved,
hence trivially converges to a constant. At the same time, $\phi(t)$ keeps
oscillating or librating forever in this case, unless $\phi(t)$ is one of the
two equilibrium states. With the help of our energy dissipation identity (83)
we can rule out an asymptotic approach to a librating solution, though.
Namely, since $-1\leq\cos\phi\leq 1$, we can extract from Eq. (83) the two-
sided bounds

$-1+C_{1}t^{-2/3}\leq E[\phi,\dot{\phi}](t)\leq 1+C_{2}t^{-2/3},$ (84)

with $C_{1}$ and $C_{2}$ some positive constants. Since any asymptotically
librating solution has energy $E\geq 1+\epsilon$ for some $\epsilon>0$, it
follows from the upper bound in Eq. (84) that the only possibilities are the
asymptotic approach to the stable equilibrium ($E=-1$), to the unstable one
($E=1)$, or to an undamped oscillating pendulum solution ($-1<E<1$).

Second, whenever $\phi(t)$ does converge to one of the equilibrium points of
the damped rigid pendulum, i.e., to a zero of $\sin\phi$, linearization about
that equilibrium yields an accurate approximation at large $t$. Letting
$\xi(t)$ denote the deviation of $\phi(t)$ from the equilibrium point, and
dropping terms nonlinear in $\xi$, we find that $\xi(t)$ is (asymptotically) a
solution of one of the following two ODEs, viz.,

$\ddot{\xi}(t)+\tfrac{1}{3t}\dot{\xi}(t)=\pm\xi(t).$ (85)

These are Bessel differential equations. Their solutions for positive $t$ map
into the Airy function $\mathrm{Ai}$ in the original variables. More
precisely, for large enough $x$ the function $\theta(x;{\vartheta,\alpha})$
approaches the asymptotic form $C_{1}+C_{2}\;$Ai$(\pm x+C_{3})$. The constant
$C_{1}$ is an integer multiple of $\pi$, and the “$+$” sign is to be chosen
when an unstable pendulum equilibrium is approached monotonically, while the
“$-$” sign pertains to the oscillatory approach to a stable pendulum
equilibrium. The constants $C_{1}$, $C_{2}$, and $C_{3}$ depend on the initial
data.

## References

  * [1] D. Broun, What lies beneath the dome?, Nat. Phys. 4, 170 (2008).
  * [2] T. Shibauchi, A. Carrington, and Y. Matsuda, A Quantum Critical Point Lying Beneath the Superconducting Dome in Iron Pnictides, Annu. Rev. Condens. Matter Phys. 5, 113 (2014).
  * [3] G. Stewart, Heavy-fermion systems, Rev. Mod. Phys. 56, 755 (1984).
  * [4] H. Wang, S. Raghu, and G. Torroba, Non-Fermi-liquid superconductivity: Eliashberg approach versus the renormalization group, Phys. Rev. B 95, 165137 (2017).
  * [5] H. v. Löhneysen, A. Rosch, M. Vojta, and P. Wölfle, Fermi-liquid instabilities at magnetic quantum phase transitions, Rev. Mod. Phys. 79, 1015 (2007).
  * [6] A. V. Chubukov, A. Abanov, Y. Wang, Y.-M. Wu, The interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal, Ann. Phys. 417 168142 (2020).
  * [7] J.P. Carbotte, Properties of boson-exchange superconductors, Rev. Mod. Phys. 62, 1027 (1990).
  * [8] R. Combescot, Strong-coupling limit of Eliashberg theory, Phys. Rev. B 51, 11625 (1995).
  * [9] E. A. Yuzbashyan and B. L. Altshuler, Migdal-Eliashberg theory as a classical spin chain, Phys. Rev. B 106, 014512 (2022).
  * [10] Y. Wang, A. Abanov, B. L. Altshuler, E. A. Yuzbashyan, and A. V. Chubukov, Superconductivity near a Quantum-Critical Point: The Special Role of the First Matsubara Frequency, Phys. Rev. Lett. 117, 157001 (2016).
  * [11] L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed. (Butterworth-Heinemann, Oxford, 1980).
  * [12] $\Sigma_{\sigma}\equiv\Sigma_{\sigma\mathsf{p}_{1}\mathsf{p}_{2}}$ and $\Phi\equiv\Phi_{\mathsf{p}_{1}\mathsf{p}_{2}}$, where $\sigma=\uparrow$ or $\downarrow$, couple to $-i\psi^{*}_{\mathsf{p}_{1}\sigma}\psi_{\mathsf{p}_{2}\sigma}$ and $\psi^{*}_{\mathsf{p}_{1}\uparrow}\psi^{*}_{-\mathsf{p}_{2}\downarrow}$, respectively, see Ref. [9] for more detail.
  * [13] E. A. Yuzbashyan and B. L. Altshuler, Fluctuations in boson-mediated superconductors and the meaning of the Migdal theorem (tentative title), unpublished.
  * [14] This derivation of the spin chain holds everywhere in the configuration space (not only at the stationary points) when $\Omega\to 0$. A different derivation is available for general $\Omega$. See Secs. II and VII in Ref. [9].
  * [15] To be more precise, taken at its minimum $f$ is the grand potential per volume as we work at fixed chemical potential. Nevertheless, we colloquially refer to it as the free energy or free energy density.
  * [16] Here we do not consider the spin-flip saddle points [9] where the sign of the right hand sides of Eqs. (13) flips for certain $n$.
  * [17] Cf. Eq. (18) and observe that an infinitely differentiable $\theta(\omega)$ exists for the normal state at any finite $T$ as well. However, in this case $\theta(\omega)$ changes abruptly from $\frac{\pi}{2}$ to 0 near $\omega=0$ for small $T$ and dropping higher order derivatives in Eq. (24) is not justified. In this way, the boundary condition $\theta(0)=\frac{\pi}{2}$ eliminates the normal state as a solution of Eq. (25).
  * [18] Not to be confused with the real frequency axis for the original complex frequency $z$. In terms of $z$, Matsubara frequencies $\omega_{n}$ correspond to discrete points $z_{n}=i\omega_{n}$ on the imaginary axis and the “real $\omega$ axis” – to imaginary $z$ axis, $z=i\omega$.
  * [19] R. A. Barankov, L. S. Levitov, and B. Z. Spivak, Collective Rabi Oscillations and Solitons in a Time-Dependent BCS Pairing Problem, Phys. Rev. Lett. 93, 160401 (2004).
  * [20] E. A. Yuzbashyan, O. Tsyplyatyev and B. L. Altshuler, Relaxation and Persistent Oscillations of the Order Parameter in Fermionic Condensates, Phys. Rev. Lett. 96, 097005 (2006).
  * [21] R. A. Barankov and L. S. Levitov, Synchronization in the BCS Pairing Dynamics as a Critical Phenomenon Phys. Rev. Lett. 96, 230403.
  * [22] R. A. Barankov and L. S. Levitov, Excitation of the dissipationless Higgs mode in a fermionic condensate, arXiv:0704.1292 (2007).
  * [23] E. A. Yuzbashyan, M. Dzero, V. Gurarie, and M. S. Foster, Quantum quench phase diagrams of an $s$-wave BCS-BEC condensate, Phys. Rev. A 91, 033628 (2015).
  * [24] E. A. Yuzbashyan and O. Tsyplyatyev, Dynamics of emergent Cooper pairing at finite temperatures, Phys. Rev. B 79, 132504 (2009).
  * [25] At any finite $\gamma$, spins approach their normal state positions when $\omega_{n}\to\pm\infty$, because $\Delta_{n}\to 0$ in these limits, see Eq. (13). For $\gamma=\infty$ we impose this as boundary conditions at infinity.
  * [26] Consider $\theta_{n}$ that changes slowly with $n$. The interaction part of the free energy (22) is then proportional to $\int_{-\infty}^{\infty}dn(\theta^{\prime})^{2}$, where $\theta^{\prime}=d\theta/dn$. The Zeeman term is negligible at any finite $n$. The interaction is to be minimized subject to the constraint $\int_{-\infty}^{\infty}dn\theta^{\prime}=-\pi$. Take $\theta^{\prime}=-\pi/L$ for $|n|<L/2$ and $\theta^{\prime}=0$ otherwise. This satisfies the constraint and it is also clear that the interaction attains its absolute minimum when $L\to\infty$.
  * [27] G. Grosso, G. P. Parravicini, Solid State Physics, 2nd ed. (Elsevier, Amsterdam, 2013).
  * [28] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (World Scientific, Singapore, 2009).
  * [29] As $\gamma\to 2^{+}$, we have $(\gamma-2)\zeta(\gamma-1)\to 1$ and $T_{\mathrm{n}}/T_{c}\to(3/a)^{\frac{1}{2}}\approx 1.28$.
  * [30] E. A. Yuzbashyan and B. L. Altshuler, Breakdown of the Migdal-Eliashberg theory and a theory of lattice-fermionic superfluidity, Phys. Rev. B 106, 054518 (2022).
  * [31] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (U.S. Government Printing Office, Washington, D.C., 10th Printing, 1972), p. 448, Eqs. 10.4.59 and 10.4.61.
  * [32] Y.-M. Wu, A. Abanov, A. V. Chubukov, Pairing in quantum-critical systems: Transition temperature, pairing gap, and their ratio, Phys. Rev. B 99, 014502 (2019).
  * [33] The spectral gap is twice the solution of the equation $\Delta(\omega)=\omega$ on the real axis.
  * [34] G. Teschl, Ordinary Differential Equations and Dynamical Systems (American Mathematical Society, Providence, Rhode Island, 2012).
  * [35] H.T. Davis, Introduction to Nonlinear Differential and Integral Equations, 1st ed. (Dover Publications, New York, 1962).
  * [36] There is no complication at $t=0$ due to this $t$-dependent damping term because we already know that the initial value problem is well-defined in the $x$ variable at $x=0$. It is therefore sufficient to consider the problem in the $t$ variable with data at some $t_{0}>0.$


# Meta-Learning and representation learner: A short theoretical note

Mouad El Bouchattaoui
E-mail<EMAIL_ADDRESS>Personal e-mail:
<EMAIL_ADDRESS>
(October 2020)

## Acronyms

Meta-learning, or "learning to learn," is a subfield of machine learning where
the goal is to develop models and algorithms that can learn from various tasks
and improve their learning process over time. Unlike traditional machine
learning methods focusing on learning a specific task, meta-learning aims to
leverage experience from previous tasks to enhance future learning.

This approach is particularly beneficial in scenarios where the available data
for a new task is limited, but there exists abundant data from related tasks.
By extracting and utilizing the underlying structure and patterns across these
tasks, meta-learning algorithms can achieve faster convergence and better
performance with fewer data. The following notes are mainly inspired from
[Vanschoren, 2018], [Baxter, 2019], and [Maurer, 2005].

##  1 Intuition and a Loose Formalism

In conventional supervised machine learning, we have a dataset
$D=\\{(x_{1},y_{1}),\dots,(x_{n},y_{n})\\}$ on which we train a predictive
model $\hat{y}=f_{\theta}(x)$. The parameters $\theta$ are chosen by
minimizing the loss $\mathcal{L}(D,\theta,\omega)$, where $\omega$ encodes
what we call the meta-knowledge: choice of the optimizers, the class of
functions, estimate of initial parameters, etc. Meta-knowledge can thus be any
element of the "how-to-learn" strategy.

In classical training, the optimization is performed for a given dataset of
points $D$ and the meta-knowledge is pre-specified. However, meta-learning
consists of learning the "how-to-learn" strategy for a given set of tasks.
Loosely speaking, a task can be defined by the result of a dataset and a loss
function: $T=\\{D,\mathcal{L}\\}$. Meta-learning can thus be formulated as a
minimization problem: the search for the best meta-knowledge $\omega$ that
minimizes on average the loss of a model trained on $D$ for a certain
$\omega$:

$\underset{\omega}{\min}\\{\mathbb{E}_{T\sim p(T)}[\mathcal{L}(D,\omega)]\\}$

The average is performed here on a set of tasks on which we define a
probability distribution $p(T)$. In practice, we assume to have access to a
set of tasks sampled from $p(T)$. For example, we can suppose to have $M$
source tasks to be used in the meta-training:
$D_{src}=\\{(D_{src}^{train},D_{src}^{val})^{(i)}\mid i=1,\dots,M\\}$.

In the meta-training step, we find the best $\omega$ such that:

$\omega^{*}=\underset{\omega}{\arg\max}\\{\log(p(\omega\mid D_{src}))\\}$

Very often, the optimization described in the meta-training step can be
described by a bi-level optimization problem:

$\begin{cases}\omega^{*}=\underset{\omega}{\arg\min}\\{\sum_{i=1}^{M}\mathcal{L}^{meta}(\theta^{*(i)}(\omega),\omega,D_{src}^{val(i)})\\}\\\
\theta^{*(i)}(\omega)=\underset{\theta}{\arg\min}\\{\mathcal{L}^{task}(\theta,\omega,D_{src}^{train(i)})\\}\end{cases}$

$\mathcal{L}^{meta}$ denotes the "outer objective"; the one used in meta-
learning. $\mathcal{L}^{task}$ denotes the inner objective; the one used for
the base model learning.

In the meta-testing, we suppose to have $Q$ target tasks:
$D_{target}=\\{(D_{target}^{train},D_{target}^{val})^{(i)}\mid
i=1,\dots,Q\\}$. During the meta-testing step, each target task will be
trained using the meta-knowledge $\omega^{*}$. For $i=1,\dots,Q$:

$\theta^{*(i)}=\underset{\theta}{\arg\max}\\{\log(p(\theta\mid\omega^{*},D_{target}^{train(i)}))\\}$

One should mention that transfer learning is not a meta-learning strategy. In
transfer learning, the prior can be seen as the result of learning on a source
task but without a meta-objective that optimizes the "how-to-learn" strategy.
The lack of meta-learning loss is also encountered in domain adaptation and
data generalization. On the contrary, we can formulate hyperparameter
optimization as a meta-learning problem. This gave birth, for example, to the
field of neural architecture search in deep learning. A real challenge in
meta-learning is the choice of $\omega$. In practice, this means defining
aspects of the learning strategy to be learned and those to be fixed.

##  2 Formal Definition

Let $z$ be a point from the task dataset sampled according to a distribution
$D$ on the data: $z\sim D$. The distribution thus defines the learning task. A
dataset point is generated such that $z=(x,y)\in Z=X\times Y$ with $X$ as the
input space and $Y$ as the output space. We define the hypothesis map as
$h:X\longrightarrow W$ (where $W$ is the action space), whose performance on
the data distribution $D$ will be theoretically assessed by the following
risk:

$R(h,D)=\underset{z\sim D}{\mathbb{E}}[l(h,z)]$ $R(h,D)=\int_{X\times
Y}l(y,h(x))\,dD(x,y)$ $R(h,D)=\int_{Z}l_{h}(z)\,dD(z)$

where $l_{H}=\\{l_{h}\mid h\in\mathcal{H}\\}$, $l:Y\times
W\longrightarrow[0,M]$ is the function that defines the loss function,
measuring the similarity/dissimilarity between $y$ and $h(x)$. $\mathcal{H}$
denotes the hypothesis space.

Since a hypothesis depends on the sample, we define a learning algorithm $A$
that maps a data sample (m-sample) to a hypothesis such that
$A:S=(z_{1},\dots,z_{m})\sim D^{\otimes m}\longrightarrow A(S)$. The
hypothesis $A(S)$ should perform well on the learning task $D$ by ensuring a
small risk $R(A(S),D)$. We can use the risk on a sample to theoretically
assess the performance of a learning algorithm $A$ on the task $D$ by
$\underset{S\sim D^{\otimes m}}{\mathbb{E}}[R(A(S),D)]$. In general, the
distribution of $D$ is unknown; therefore, we prefer to measure the
performance of the learning algorithm $A$ conditioned on a data sample $S$ by
ensuring a statistical guarantee of the type $\forall
D\,\mathbb{P}(R(A(S),D)\leq B(\delta,S))\geq 1-\delta$.

In the meta-learning setting, we have a sequence of samples
$\textbf{S}=(S_{1},\dots,S_{n})$, where each sample originates from a certain
task $S_{i}\sim D_{i}^{\otimes m}$. In Baxter’s formalism, an environment of
tasks is given by the probability distribution $E$ defined on the set of all
possible tasks $\\{D_{1},D_{2},\dots\\}$. A task is now a random realization
following the distribution probability $E$. The probability of an m-sample
(task sample, not a meta-sample) being generated is
$\mathbb{D}_{E}(S)=\underset{D\sim E}{\mathbb{E}}[D^{\otimes m}(S)]$.

A formal definition of a meta-sample is a sample whose elements are generated
independently according to
$\textbf{S}=(S_{1},\dots,S_{n})\sim\mathbb{D}_{E}^{\otimes n}$. A meta-
algorithm A is thus a mapping from a meta-sample S to a learning algorithm
$\textbf{A}(\textbf{S})$. The performance of the learning algorithm $A$ on the
environment $E$ is measured by the transfer risk:

$\textbf{R}(A,E)=\underset{D\sim E}{\mathbb{E}}\left\\{\underset{S\sim
D^{\otimes m}}{\mathbb{E}}[R(A(S),D)]\right\\}$

Very often, the generalization power in the case of the meta-algorithm is
given by a control over the generalization error in the following sense:

$\forall E\,\text{Proba}(\textbf{R}(\textbf{A}(\textbf{S}),E)\leq
B(\delta,\textbf{S}))\geq 1-\delta$

One should note that the fact that the tasks are not sampled uniformly — $E$
is a non-uniform distribution — gives a chance to minimize $R(A,E)$ as the No-
Free-Lunch theorem is not applicable.

A meta-sample is also called an n-m sample and can be written in the form of a
matrix of data points:

$\textbf{z}=\begin{bmatrix}z_{11}&\dots&z_{1m}\\\
z_{n1}&\dots&z_{nm}\end{bmatrix}$

We denote $Z^{(n,m)}$ as the set of such matrices.

##  3 Concept of Representation and Generalization Guarantees

We now split the hypothesis space $\mathcal{H}=\\{X\longrightarrow W\\}$ into
$\\{X\xlongrightarrow{F}V\xlongrightarrow{G}W\\}=G\circ F$. Thus,
$\mathcal{H}$ can be written as $\mathcal{H}=\\{g\circ f\mid g\in G,f\in
F\\}$. $F$ will be called the representation space, and $f\in F$ will be
called the representation. A representation learner is defined as a mapping
$\mathcal{A}:\cup_{n,m\geq 1}Z^{(n,m)}\longrightarrow F$. The idea is to find
a good representation $f$ from information given on the environment $E$ in the
n-m sample z. Therefore, we aim to have a small empirical loss (where
$\langle\cdot\rangle$ refers to the empirical mean and $z_{i}$ to the i-th row
of z):

$E^{*}_{G}(F,\textbf{z})=\frac{1}{n}\sum_{i=1}^{n}\underset{g\in
G}{\inf}\langle l_{g\circ f}\rangle_{z_{i}}$

$E^{*}_{G}(F,\textbf{z})$ is an estimate of
$E^{*}_{G}(F,E)=\int_{\mathcal{D}}\inf\langle l_{g\circ f}\rangle\,dE$. If $n$
tasks are sufficient, $\textbf{D}=(D_{1},\dots,D_{n})$, we will have $n$
hypotheses $\mathbf{g}\circ\bar{f}=(g_{1}\circ f,\dots,g_{n}\circ f)$ such
that $\mathbf{g}\circ\bar{f}\in G^{n}\circ\bar{F}$. We thus minimize the
empirical loss:

$\frac{1}{n}\sum_{i=1}^{n}\inf\langle l_{g_{i}\circ f}\rangle_{z_{i}}$

For the sake of good generalization, one should have a bound over
$\mathbb{P}\\{Z\in Z^{(n,m)}\mid
d_{v}(E(\textbf{A}(z),Z),E(\textbf{A}(Z),\textbf{D}))>\alpha\\}$ with
$\mathbb{P}=P_{1}^{m}\otimes\dots\otimes P_{n}^{m}$ or over $\mu\\{Z\in
Z^{(n,m)}\mid
d_{v}(E^{*}_{G}(\textbf{A}(Z),Z),E^{*}_{G}(\textbf{A}(Z),E))>\alpha\\}$ with
$\mu(S)=\int P_{1}^{m}\dots P_{n}^{m}\,dE(P_{1},\dots,P_{n})$. Here,
$P_{1},\dots,P_{n}$ model the $n$ tasks.

Let’s define $l_{G}$ the same way as $l_{H}$. Let $P$ be the probability
measure on $V\times Y$. We define the pseudo-metric:

$d_{P}:=\int_{V\times Y}|l_{g}(v,y)-l_{g^{\prime}}(v,y)|\,dP(v,y)$

Let $\mathcal{N}(\epsilon,l_{G},d_{P})$ be the size of the smallest
$\epsilon$-cover of the pseudo-metric space $(l_{g},d_{P})$. We define the
$\epsilon$-capacity as
$C(\epsilon,l_{G})=\underset{P}{\mathcal{N}(\epsilon,l_{G},d_{P})}$.
Similarly, we define a pseudo-metric on $F$ as
$d^{*}_{[P,l_{G}]}(f,f^{\prime})=\int_{Z}\sup|l_{g\circ f}-l_{g\circ
f^{\prime}}|$. In this pseudo-metric space, the $\epsilon$-capacity will be
denoted $C_{l_{G}}^{*}=(\epsilon,F)$. We present the following two theorems
[Baxter, 2019]. The first theorem bounds the number $m$ so that we can achieve
good generalization given a representation learner averaging over all tasks.
The second theorem bounds the number of tasks $n$ and the number of examples
$m$ per task to achieve good generalization power for a given representation
learner and under the same environment. We first introduce some necessary
definitions.

######  Definition 3.0.1 (Polish space)

A topological space $X$ is completely metrizable if there exists a metric $d$
such that $(X,d)$ is complete. A separable, completely metrizable space is
called a Polish space.

######  Definition 3.0.2 (Analytic subset)

An analytic subset of a Polish space can be defined as the continuous image of
a Borel set in the Polish space.

######  Definition 3.0.3 (Indexed map)

A map $\mathcal{H}:Z\longrightarrow[0,M]$ is said to be indexed by the set $T$
if there exists a function $f:Z\times T\longrightarrow[0,M]$ such that
$\mathcal{H}=\\{f(.,t)\mid t\in T\\}$.

######  Definition 3.0.4 (Permissibility)

$\mathcal{H}$ is permissible if it can be indexed by a set $T$ which is an
analytic subset of a Polish space $\bar{T}$, and the indexing function $f$ is
measurable with respect to the product $\sigma$-algebra
$\sigma(H)\otimes\sigma(T)$. $\sigma(T)$ is the Borel $\sigma$-algebra induced
by the topology on $T$.

We now extend the concept of permissibility to cover families of hypothesis
spaces $\textbf{H}=\\{\mathcal{H}\\}$.

######  Definition 3.0.5 (f-Permissible, Extension)

H is $f$-permissible if there exist sets $T$ and $S$ that are analytic subsets
of the Polish spaces $\bar{T}$ and $\bar{S}$, respectively, and a function
$f:Z\times T\times S\longrightarrow[0,M]$ measurable with respect to
$\sigma_{H_{\sigma}}\otimes\sigma(H)\otimes\sigma(T)$, where
$H_{\sigma}=\\{h\mid h\in\mathcal{H}\text{ and }\mathcal{H}\in\textbf{H}\\}$,
such that:

$\textbf{H}=\left\\{\\{f(.,t,s)\mid t\in T\\},s\in S\right\\}$

###### Theorem 3.1

Suppose $F,G$, and $l$ are such that the family of hypothesis spaces
$\\{l_{G\circ f|f\in F}\\}$ is $f$-permissible. For all $0<\alpha<1$,
$0<\delta<1$, $v>0$, for any representation learner $\mathcal{A}$ with values
in $G^{n}\circ\bar{F}$, if

$m\geq\frac{8M}{\alpha^{2}v}\left(\ln(C(\epsilon_{1},l_{G}))+\frac{1}{n}\ln\left(\frac{4C^{*}_{l_{G}}(\epsilon_{2},F)}{\delta}\right)\right)$

where $\epsilon_{1}+\epsilon_{2}=\frac{\alpha v}{8}$, then

$\mathbb{P}\\{Z\in Z^{(n,m)}\mid
d_{v}(E(\mathcal{A}(z),Z),E(\mathcal{A}(Z),\textbf{D}))>\alpha\\}\leq\delta$

###### Theorem 3.2

Let $F,G$, and $l$ be as in the previous theorem. Let $Z$ be an $n$-$m$ sample
drawn according to the environmental measure $Q$. For
$0<\alpha,\delta,\epsilon_{1},\epsilon_{2}<1$, $v>0$, and
$\epsilon_{1}+\epsilon_{2}=\frac{\alpha v}{16}$, if

$n\geq\frac{32M}{\alpha^{2}}\ln\left(\frac{8C_{l_{G}}^{*}\left(\frac{\alpha
v}{16},F\right)}{\delta}\right)$

and

$m\geq\frac{32M}{\alpha^{2}v}\left[\ln(C(\epsilon,l_{G}))+\frac{1}{n}\ln\left(\frac{8C_{l_{G}}^{*}(\epsilon_{2},F)}{\delta}\right)\right]$

for any representation learner $\mathcal{A}:\cup_{n,m\geq
1}Z^{(n,m)}\longrightarrow F$, we have:

$\mathbb{P}\\{Z\in Z^{(n,m)}\mid
d_{v}(E_{G}^{*}(\mathcal{A}(Z),Z),E_{G}^{*}(\mathcal{A}(Z),Q))>\alpha\\}\leq\delta$

## References

  * [Baxter, 2019] Baxter, J. (2019).  Learning internal representations (colt 1995).  arXiv, pages arXiv–1911.
  * [Maurer, 2005] Maurer, A. (2005).  Algorithmic stability and meta-learning.  Journal of Machine Learning Research, 6(Jun):967–994.
  * [Vanschoren, 2018] Vanschoren, J. (2018).  Meta-learning: A survey.  arXiv preprint arXiv:1810.03548.