# Anisotropic resistivity and superconducting instability in ferroelectric-
like metals

Vladimir A. Zyuzin  L. D. Landau Institute for Theoretical Physics, Semenova
1-a, 142432, Chernogolovka, Russia  Alexander A. Zyuzin  QTF Centre of
Excellence, Department of Applied Physics, Aalto University, FI-00076 AALTO,
Finland  A. F. Ioffe Physical–Technical Institute, 194021 St. Petersburg,
Russia

###### Abstract

We propose a theoretical model of a ferroelectric-like metal where spontaneous
electric polarization coexists with the conducting electrons. In our model we
adopt a scenario when conducting electrons interact with two soft transverse
optical phonons, generalize it to the case when there is a spontaneous
ferroelectric polarization in the system, and show that a linear coupling to
the phonons emerges as a result. We find that this coupling results in
anisotropic electric transport which has a transverse to the current voltage
drop. Importantly, the obtained transverse component of the resistivity has
distinct linear dependence with temperature. Moreover, we show that the
coupling enhances superconducting transition temperature of the ferroelectric-
like metal. We argue that our results help to explain recent experiments on
ferroelectric-like strontium titanate, as well as provide new experimental
signatures to look for.

The strontium titanate based (STO) compound has a rich phase diagram upon
chemical doping and temperature variation, which includes seemingly self-
exclusive ferroelectric and superconducting states, see for a review [1, 2,
3]. To begin with, we note that the pristine $\mathrm{SrTiO}_{3}$ is a wide-
band gap quantum paraelectric insulator. It can be tuned into a ferroelectric
state by partial substitution of Sr ions with Ca, Ba, or Pb [4, 5, 6], isotope
substitution of oxygen [7], and applying stress [8]. The structural formation
is associated with the soft transverse optical (TO) lattice vibration. The gap
in the dispersion of this phonon mode vanishes at the transition [1, 2].

On the other hand, STO becomes semiconducting (or metallic) with partial
substitution of Sr with Nb, La or with oxygen reduction, see for details [1,
2, 3]. The charge transport measurements show square temperature dependence of
resistivity within unusually wide region of material parameters, [2, 9].
Despite the study of superconductivity in this material has a very long
history, dating back to work [10], the mechanism of Cooper pairing is far from
being understood as well. One of the reasons for that is the proximity to a
ferroelectric quantum critical point. Another is that superconductivity
emerges in the situation of Fermi energy an order of magnitude smaller than
the Debye frequency. Moreover, the superconducting transition temperature
mediated by acoustic phonons in this system was estimated to be negligibly
small [11].

A possible solution to the problem of superconductivity in STO was proposed a
long time ago by Ngai, who introduced a model of superconducting instability
based on electron coupling tuned by a pair of TO phonons [12]. In paraelectric
STO the electron scattering by transverse phonons is proportional to second
power in displacement amplitude. It is distinct from the electron scattering
by acoustic phonons, which is described by the gradient of lattice
deformation. Later Epifanov, Levanyuk, and Levanyuk studied the T-squared
dependence of conductivity due to interaction of fermions with TO phonons near
ferroelectric phase transition [13, 14]. Below we will be referring to this
mechanism as the two-phonon.

Recent observation of the enhancement of superconducting transition
temperature upon oxygen isotope substitution, which brings a system closer to
the ferroelectric transition, has reinvigorated this subject, [15]. Proximity
to a ferroelectric instability suggests soft TO phonons playing an influential
role. And the model of two-phonon scattering was brought forward to revisit
the temperature dependence of charge transport [16, 17] and study the effect
of oxygen isotope substitution on superconducting transition temperature [18,
19].

The story is far from being complete. Recently, the signatures of
ferroelectric-like instability has been observed in n-doped
$\mathrm{Sr}_{1-x}\mathrm{Ca}_{x}\mathrm{TiO}_{3-\delta}$, [20, 21, 22]. It
was found that superconductivity and ferroelectricity may coexist in this
material [20, 22]. The resistivity showed anomalous temperature dependence,
suggesting the emergence of additional scattering channel, [21, 22]. The
coexistence of metallic and ferroelectric phases might be understood within
the dipole model. The ferroelectric instability is based on the interplay
between the long-range dipole-dipole interaction and the short-range repulsion
between ions. The former favours ferroelectric structure formation with the
emergence of electric dipole moment per unit cell and the latter supports the
paraelectric phase, please see [23] and for a review [24]. In metals,
itinerant electrons screen dipole-dipole interaction and eliminate
ferroelectricity. Although, it may survive in weakly doped semiconductors due
to long Thomas-Fermi screening length.

Motivated by recent experiments, [20, 21, 22], we propose a model of electron
scattering by one-phonon in the presence of spontaneous ferroelectric
polarization. This scattering mechanism adds up with two-phonon contribution,
[12, 13, 14]. We show that below the structural transition, the decrease of
temperature enhances electric resistivity due to onset of ferroelectric
polarization. At lower temperatures, we predict linear in temperature
suppression of resistivity. We also calculate the contribution of one-phonon
scattering processes to the superconducting transition temperature. We argue
that our mechanism might dominate over the two-phonon mediated pairing deep in
the ferroelectric-like phase.

Model. We propose to study a model of a ferroelectric-like metal (polar
metal), which consists of electrons interacting with TO phonons. The TO
phonons are responsible for the ferroelectricity in the system, while the
electrons are responsible for the electric conduction. The electrons are
described by the Hamiltonian

$\displaystyle H_{\mathrm{e}}=\int_{\bf
r}\psi^{{\dagger}}(x)\left(-\frac{\partial^{2}_{\bf
r}}{2m}-\mu\right)\psi(x),$ (1)

where $\psi^{\dagger}(x),\psi(x)$ are the electron creation and annihilation
operators, $\mu$ is the chemical potential, $m$ is the mass of electrons,
$x=({\bf r},t)$ is a general coordinate, and $\int_{\bf r}(..)\equiv\int d{\bf
r}(..)$. The Hamiltonian of two degenerate denoted by $a=1,2$ branches of TO
phonons is

$\displaystyle H^{(a)}_{\mathrm{ph}}=\sum_{\bf q}\omega_{\bf q}\left(b_{a;{\bf
q}}^{{\dagger}}b_{a;{\bf q}}+\frac{1}{2}\right),$ (2)

where $\omega_{\mathbf{q}}=\sqrt{\omega_{\mathrm{TO}}^{2}+s^{2}q^{2}}$ is the
spectrum with $s$ being the speed of sound and $\omega_{\mathrm{TO}}$ being
the TO phonon gap. Polarization vector $\mathbf{P}(x)$ describing the TO
phonons is

$\displaystyle\mathbf{P}(x)=\sum_{a;\mathbf{q}}\frac{\mathbf{e}_{a;\mathbf{q}}}{\sqrt{V}}A_{\mathbf{q}}\left[b_{a;\mathbf{q}}(t)e^{i\mathbf{qr}}+b^{{\dagger}}_{a;\mathbf{q}}(t)e^{-i\mathbf{qr}}\right],~{}~{}~{}$
(3)

where $V$ is the volume of the material, $\mathbf{e}_{a;\mathbf{q}}$ is the
unit vector in the direction of polarization of $a=1,2$ branches of TO phonons
with wave-vector $\mathbf{q}$, and
$b^{{\dagger}}_{a;\mathbf{q}},~{}b_{a;\mathbf{q}}$ are the bosonic creation
and annihilation operators. We consider
$A_{\mathbf{q}}^{2}=\Omega_{0}^{2}/(4\pi\omega_{\bf q}),$ where $\Omega_{0}$
is material dependent coefficient determined via Lyddane-Sachs-Teller relation
for the static dielectric function
$\epsilon_{0}(\mathbf{q})=\Omega_{0}^{2}/\omega_{\mathbf{q}}^{2}$, [16]. The
identity unit vectors satisfy is

$\displaystyle\sum_{a=1,2}e_{a;{\bf q},\alpha}e_{a;{\bf
q},\beta}=\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}},$ (4)

where $\alpha,\beta=x,y,z$. We assume that $\omega_{\mathrm{TO}}$ vanishes at
temperature $T_{\mathrm{FE}}$ (ferroelectric transition temperature), and the
system becomes ferroelectric below this temperature forming spatially
homogeneous spontaneous electric polarization $\langle{\bf P}(x)\rangle={\bf
P}_{0}$. For temperatures $T<T_{\mathrm{FE}}$ the phonon gap is
$\omega_{\mathrm{TO}}\propto|{\bf P}_{0}|$.

We assume that electrons interact with two TO phonons, [12, 13, 14]. Namely,
it is impossible for electrons to interact with one TO phonon via
$\propto\left[\mathrm{div}\cdot{\bf P}(x)\right]\psi^{\dagger}(x)\psi(x)$ term
(of the Frohlich type), however, it is possible for electrons to interact with
two TO phonons via $\propto{\bf P}(x)\cdot{\bf P}(x)\psi^{\dagger}(x)\psi(x)$
term. Below we will be referring to this mechanism as the two-phonon
mechanism. In what follows, we generalize the two-phonon mechanism to the
ferroelectric case, when $\langle{\bf P}(x)\rangle={\bf P}_{0}$ in the system.
We then write for the interaction between electrons and the TO phonons,

$H_{\mathrm{e-ph}}=g\int_{\bf
r}[\mathbf{P}_{0}+\mathbf{P}(x)]^{2}\psi^{\dagger}(x)\psi(x),$ (5)

where $g$ is the electron-phonon interaction constant. We will be using
$\hbar=k_{\mathrm{B}}=1$ units throughout the paper.

Figure 1:  Feynman diagrams for (a,b) the effective electron interaction and
(c,d) self-energy. Here the wavy line is the phonon Green function. Object
${\bf P}_{0}$ in (a,c) denotes the one-phonon contribution to the interaction
in case of ferroelectric order. Figures (b,d) describe two-phonon interaction.

We note in passing that we do not consider Coulomb repulsion between the
electrons and focus on the electron-phonon interaction only. The reason for
that is the large dielectric constant. Moreover, we assume that the electrons
don’t screen finite electric polarization ${\bf P}_{0}$.

Effective electron interaction. Here we discuss corrections to electron
quantum life-time and conductivity due to the electron-phonon interaction. We
choose to work in Keldysh formalism as it conveniently describes systems in
equilibrium at finite temperatures, please see Supplemental Material [25].

In order to obtain effective electron interaction, as usual, we integrate out
the phonons. To second order in electron-phonon interaction we obtain two
processes shown in Fig. (1a) and (1b). Processes Fig. (1b) describe
interaction of electrons with two TO phonons [due to $\propto{\bf P}^{2}(x)$
in Eq. (5)], while those in Fig. (1a) describe interaction of electrons with
one TO phonon [due to $\propto{\bf P}_{0}\cdot{\bf P}(x)$ in Eq. (5)] given
that there is a spontaneous electric polarization ${\bf P}_{0}$ in the system.
The processes in Fig. (1b) were studied in Refs. [12, 13, 14, 16, 17, 18, 19],
and we refer to these works for details, the processes in Fig. (1a) are new
and are subject of the following analysis. The effective electron interaction
due to these processes [wavy line in Fig. (1a) and (1c)] is

$\displaystyle V_{1}^{\mathrm{R}/\mathrm{A}}({\bf q};\omega)=(2gP_{0}A_{\bf
q})^{2}\sin^{2}(\phi_{{\bf q}{\bf P}_{0}})D^{\mathrm{R}/\mathrm{A}}({\bf
q};\omega),$ (6)

where $D^{\mathrm{R}/\mathrm{A}}({\bf q};\omega)=2\omega_{{\bf q}}/[(\omega\pm
i0)^{2}-\omega_{{\bf q}}^{2}]$ is the TO phonon (retarded/advanced) Green
function. We have defined an angle between ${\bf q}$ and ${\bf P}_{0}$ as
$\cos(\phi_{{\bf q}{\bf P}_{0}})\equiv({\bf q}\cdot{\bf P}_{0})/(qP_{0})$.

With all the details given in the SM [25] we here present essential results
and experimental predictions of our model.

Self-energy and resistivity. The experiments, see for a review Refs. [1, 2]
show that the resistivity of metallic STO is proportional to $T^{2}$ at low
temperatures. In typical Fermi liquid such temperature dependence originates
from the electron-electron Coulomb interaction. However, in metallic STO, due
to a large dielectric constant, Coulomb interaction is expected to be weak. In
Refs. [13, 14, 16, 17, 18] it was theoretically suggested that the $T^{2}$
contribution to the resistivity due to the two-phonon mechanism can be
significant. Namely, the self-energy Fig. (1d) due to the electron-phonon
processes which are depicted in Fig. (1b) results in the elastic inverse
electron life-time proportional to the $T^{2}$ at low temperatures.

In another set of experiments Refs. [20, 21, 22] it was observed that
substitution of Sr atoms with Ca, i.e. by creating Sr1-xCaxTiO3 compound,
results in the ferroelectric transition of the material. Moreover, it was
shown that the ferroelectric order survives when the Sr1-xCaxTiO3 is made
metallic by doping it. Furthermore, the experiments clearly observe a new
boson mode the conducting electrons scatter by in the ferroelectric phase,
which results in non-monotoneous temperature dependence of the resistivity in
the vicinity of the ferroelectric transition [20, 21, 22].

Figure 2:  (Color online). (a) Temperature dependence of the decay rate due to
electron scattering by one TO phonon. Assuming temperature dependence in
$P_{0}^{2}(T)\propto\omega_{\mathrm{TO}}^{2}(T)=\omega_{\mathrm{TO}}^{2}(0)(1-T/T_{\mathrm{FE}})$,
the curves are plotted for
$T_{\mathrm{BG}}/\omega_{\mathrm{TO}}(0)=\\{0.2,0.5,1.5\\}$. The amplitude
decreases with the increase of $T_{\mathrm{BG}}/\omega_{\mathrm{TO}}(0)$. (b)
Schematics of the Hall bar to measure the electric transport anisotropy given
in Eq. (8). With the direction of the electric polarization being in $x-y$
plane, the current is passed at some angle $\chi$ with respect to the
polarization, for example, in $y-$direction. The voltage is measured across
the system in opposite gates (gold) or along the diagonals to specifically
study the angle dependence of Eq. (8). (c) Schematics of the temperature
dependence of the longitudinal resistivity given in Eq. (8). For
$T>T_{\mathrm{FE}}$ the resistivity is expected to be $\propto T^{2}$ due to
two-phonon mechanism, Refs. [13, 14, 16, 17]. Below $T_{\mathrm{FE}}$ the one-
phonon mechanism Eq. (Anisotropic resistivity and superconducting instability
in ferroelectric-like metals) sets in, in addition to two-phonon $\propto
T^{2}$, resulting in an increase of the resistivity.

Let us analyze how the electron scattering on one-phonon interaction derived
in Eq. (6) affects the resistivity of the system. Taking the imaginary part of
the electron-self energy Fig. (1c) and setting frequency to zero, we find the
electron decay rate due to scattering via one-phonon mechanism, $\tau_{1{\bf
k}}^{-1}$. Setting momentum of the electron
$k=k_{\mathrm{F}}\equiv\sqrt{2m\mu}$, we obtain

$\displaystyle\frac{1}{\tau_{1{\bf k}}}$ $\displaystyle=$ $\displaystyle
2\left[1+\cos^{2}(\phi_{{\bf k}{\bf P}_{0}})\right]\left(2\pi
g\frac{\Omega_{0}P_{0}}{T_{\mathrm{BG}}}\right)^{2}\nu$ $\displaystyle\times$
$\displaystyle
T\ln\left|\mathrm{tanh}\left(\frac{\sqrt{T^{2}_{\mathrm{BG}}+\omega_{\mathrm{TO}}^{2}}}{2T}\right)\mathrm{coth}\left(\frac{\omega_{\mathrm{TO}}}{2T}\right)\right|,$

where $\nu=mk_{\mathrm{F}}^{2}/2\pi^{2}$ is the density of electron states per
spin and $T_{\mathrm{BG}}=2sk_{\mathrm{F}}$ is the Bloch-Gruneisen
temperature. Let us separate isotropic part from the angle-dependent one by a
redefinition of $\tau_{1{\bf
k}}^{-1}\equiv\tau_{1}^{-1}\left[1+\cos^{2}(\phi_{{\bf k}{\bf
P}_{0}})\right]$.

The temperature dependence of the isotropic part of the decay rate is shown in
Fig. (1a). At the ferroelectric transition $P_{0}^{2}\rightarrow 0$ in the
prefactor of the Eq. (Anisotropic resistivity and superconducting instability
in ferroelectric-like metals), which takes care of the logarithm which
formally diverges due to $\omega_{\mathrm{TO}}\rightarrow 0$ at the
transition. Hence, right at the transition the decay rate vanishes as shown in
Fig. (1a). Slightly below the ferroelectric transition,
$T_{\mathrm{FE}}\gtrsim T$, the decay rate increases with the decrease of
temperature as shown in Fig. (2). We may assume $P_{0}$ and
$\omega_{\mathrm{TO}}$ to be temperature independent far from the transition.
In this case, at $T_{\mathrm{FE}}>T\gtrsim
T_{\mathrm{BG}},\omega_{\mathrm{TO}}$, the decay rate scales linearly with
temperature, and for the region
$T_{\mathrm{FE}},T_{\mathrm{BG}}>T>\omega_{\mathrm{TO}}$, we find
$\tau_{1}^{-1}\propto T\ln|2T/\omega_{\mathrm{TO}}|$.

It is instructive to compare the decay rate due to electron scattering on
acoustic longitudinal phonons is temperature independent, $\propto\nu T_{BG}$,
at $T\gg T_{\mathrm{BG}}$ and $\propto\nu T^{3}/T_{\mathrm{BG}}^{2}$ at
$T\rightarrow 0$.

Let us calculate the electric current in the system. In the SM [25], we
construct the kinetic equation with the collision integral defined by the
impurity, one- and two-phonon scattering processes. The kinetic equation is
solved to give the electric current

$\displaystyle{\bf
j}=\sigma_{\Sigma}\left(1-\frac{\tau_{\Sigma}}{5\tau_{1}}\right){\bf
E}-\sigma_{\Sigma}\frac{2\tau_{\Sigma}}{5\tau_{1}}({\bf E}\cdot{\bf
P}_{0}){\bf P}_{0}\frac{1}{P_{0}^{2}},$ (8)

where $\sigma_{\Sigma}=e^{2}\nu D_{\Sigma}$ is the conductivity, where
$D_{\Sigma}=v_{\mathrm{F}}^{2}\tau_{\Sigma}/3$ is the diffusion coefficient,
and $\tau_{\Sigma}^{-1}=\tau_{\mathrm{imp}}^{-1}+\tau_{1}^{-1}+\tau_{2}^{-1}$
is the total decay rate due to the isotropic impurity, one- and two-phonon
scattering processes (in general decay rate due to the electron-electron
Coulomb interaction should also be included).

Obtained anisotropy of the electric current [second term in Eq. (8)] results
in the transverse responses. For example, in Fig. (2b) we schematically show
Hall-bar setup, where electric current is passed in $y-$ direction at some
angle $\chi$ to the electric polarization, which is in $x-y$ direction. The
voltage drop in the direction perpendicular to the current is measured either
in opposite or along diagonal gates to study the angle dependence,
$\propto\cos(\chi)\sin(\chi)$, of the transverse part of Eq. (8). Most
importantly, as can be deduced from Eq. (8), only the anisotropic part of the
resistivity depends on temperature as shown in Fig. (2a).

Longitudinal part of the resistivity in the ferroelectric phase is more
complicated as the two-phonon mechanism (also impurity scattering and others)
contributes to it as well. In Fig. (2c) we schematically plot the temperature
dependence of the longitudinal part of the resistivity Eq. (8). There, for
$T>T_{\mathrm{FE}}$ the resistivity is expected to be $\propto T^{2}$ due to
two-phonon mechanism as predicted in Refs. [13, 14, 16, 17]. Below
$T_{\mathrm{FE}}$ the one-phonon mechanism Eq. (Anisotropic resistivity and
superconducting instability in ferroelectric-like metals) sets in resulting in
an increase of the resistivity with a characteristic dip at $T_{\mathrm{FE}}$.
The two-phonon mechanism exists in the ferroelectric phase as well because
$\omega_{\mathrm{TO}}\rightarrow 0$ at the transition, allowing for the
$\propto T^{2}$ results of Refs. [13, 14, 16, 17] to be applicable. We argue
that this picture explains experimental results of [20, 21, 22].

We conclude that by subtracting from the longitudinal resistivity its
temperature-squared part of the fit in the ferroelectric phase, one should
explicitly obtain a one-phonon contribution with a characteristic temperature
dependence shown in Fig. (2a). In addition, the same temperature dependence,
shown in Fig. (2a), is expected in the measured transverse voltage.

Superconducting transition temperature. Let us now proceed with the
calculation of correction of one-phonon process to the Cooper pairing in
ferroelectric-like metal. To qualitatively estimate the superconducting
transition temperature, we adopt approach of Refs. [26, 27]. The pole in the
fermion scattering amplitude in the Cooper channel determines the
superconducting transition temperature. Equation for the respective vertex
part is given by

$\displaystyle\Gamma({\bf q})=V({\bf q})-\int_{\bf p}V({\bf q}-{\bf
p})\frac{\mathrm{tanh}\left(\frac{\xi_{{\bf p}}}{2T}\right)}{2\xi_{{\bf
p}}}\Gamma({\bf p}).$ (9)

where $\xi_{{\bf p}}=\mathbf{p}^{2}/2m-\mu$, $V({\bf q})=V_{1}({\bf
q})+V_{2}({\bf q})$ is the electron-electron attraction potential to be
specified for one and two TO phonon coupling mechanisms, $V_{1}({\bf q})$ and
$V_{2}({\bf q})$ respectively, in what follows.

Let us first revisit two-phonon mechanism of superconductivity in paraelectric
metal, [12, 19]. At $T_{\mathrm{BG}},\omega_{\mathrm{TO}}>T$, estimating
$\mathrm{coth}(\omega_{\mathbf{k}}/2T)\approx 1$, we obtain

$\displaystyle V_{2}(\mathbf{q})=$
$\displaystyle-\left(\frac{g\Omega_{0}^{2}}{2\pi}\right)^{2}\int_{\mathbf{k}}\left\\{1+\frac{[\mathbf{k}\cdot(\mathbf{k}+\mathbf{q})]^{2}}{\mathbf{k}^{2}|\mathbf{k}+\mathbf{q}|^{2}}\right\\}$
$\displaystyle\times\frac{1}{\omega_{\mathbf{k}}\omega_{\mathbf{k}+\mathbf{q}}}\frac{1}{\omega_{\mathbf{k}}+\omega_{\mathbf{k}+\mathbf{q}}}.$
(10)

The first factor under the integral originates from the transverse
polarization of phonons, while the terms on the second line originate from the
dispersion dependence of the phonon Green function. To estimate the transition
temperature, we consider (Anisotropic resistivity and superconducting
instability in ferroelectric-like metals) in the long-wave limit. Setting
$V_{2}(0)$, we reproduce previous result [19]:

$\displaystyle T_{c}\propto\mu e^{-1/\lambda_{2}\nu},$
$\displaystyle\lambda_{2}=\left(\frac{g\Omega_{0}^{2}}{2\pi}\right)^{2}\frac{1}{2s^{3}}\ln\left[\frac{sq_{0}}{\max(T_{\mathrm{BG}},\omega_{\mathrm{TO}})}\right],$
(11)

where $q_{0}$ is the large momentum cut-off which is determined by the lattice
spacing. It was noted that as the paraelectric system is tuned closer to the
ferroelectric instability, the softening of TO phonon gap might enhance the
superconducting transition temperature, [19].

We argue that in the ferroelectric-like metal one-phonon coupling processes
shall be taken into account as well. The respective interaction term is given
by the static part of Eq. (6), $V_{1}(\mathbf{q})=-\frac{2}{\pi\omega_{{\bf
q}}}(g\Omega_{0}P_{0})^{2}\sin^{2}\phi_{\mathbf{q}\mathbf{P}_{0}}$.

We seek for the transition temperature to s-wave superconducting state.
Substituting $V_{1}(\mathbf{q})$ into Eq. 9 and integrating the resulting
equation over the directions of momentum $\mathbf{q}$, in the long wave limit
we obtain

$\displaystyle
T_{c}\propto\mu\exp\left(-\frac{1}{\lambda_{1}\nu}-\frac{1}{\lambda_{2}\nu}\right),$
$\displaystyle\lambda_{1}=\frac{4}{3\pi}\left(\frac{g\Omega_{0}P_{0}}{T_{\mathrm{BG}}}\right)^{2}\ln\left(\frac{\omega_{\mathrm{TO}}^{2}+T_{\mathrm{BG}}^{2}}{\omega_{\mathrm{TO}}^{2}+\frac{s^{2}}{v_{F}^{2}}T^{2}}\right).$
(12)

As the system is tuned deep into the ferroelectric state, the phonon gap
$\omega_{\mathrm{TO}}$ increases and, hence, the two-phonon contribution to
the interaction potential decreases logarithmically, in accordance with
(Anisotropic resistivity and superconducting instability in ferroelectric-like
metals). On the other hand, the one-phonon interaction term is proportional to
the ferroelectric polarization squared, which increases as the system enters
into the ferroelectric state. Hence, one-phonon processes might determine the
superconducting transition temperature at $\omega_{\mathrm{TO}}\gg
T_{\mathrm{BG}}$ provided
$P_{0}/\omega_{\mathrm{TO}}\gtrsim(\Omega_{0}/4s)\sqrt{\ln|sq_{0}/\omega_{\mathrm{TO}}|}$.

It is instructive to comment on the density of states dependence of the one-
phonon contribution to the transition temperature. At
$\omega_{\mathrm{TO}}>T_{\mathrm{BG}}$, the exponent in (Anisotropic
resistivity and superconducting instability in ferroelectric-like metals)
follows the usual BCS dependence. However, at
$T_{\mathrm{BG}}>\omega_{\mathrm{TO}}$, the interaction constant is inversely
proportional to the Bloch-Gruneisen temperature squared,
$\lambda_{1}\nu\propto\nu/T_{\mathrm{BG}}^{2}\propto m/k_{F}$. In this case
the increase of the density of states might suppress one-phonon contribution
and diminish $T_{c}$.

Conclusions. We proposed a mechanism of electron interaction with transverse
optical phonons in the presence of ferroelectric polarization. We calculated
the temperature dependence of resistivity and predicted anisotropic electric
current response as one of the smoking gun signatures of the ferroelectric
polarization onset. We also analyzed the superconducting transition
temperature in the ferroelectric phase. We think that our results for the
temperature dependence of the resistivity in the vicinity of the ferroelectric
transition given in Eqs. (Anisotropic resistivity and superconducting
instability in ferroelectric-like metals) and (8), as well as for the increase
of the superconducting transition temperature given in Eq. (Anisotropic
resistivity and superconducting instability in ferroelectric-like metals)
qualitatively agree with the experimentally observed ones in Sr1-xCaxTiO3
systems [20, 21, 22]. In particular, the one-phonon scattering mechanism
explains the experimentally observed dip in the resistivity at the
ferroelectric transition temperature.

Acknowledgments. We would like to thank the Pirinem School of Theoretical
Physics, where all work was initiated for the warm hospitality. We thank A.T.
Burkov and A.Yu. Zyuzin for helpful discussions. VAZ thanks Abhishek Kumar and
D.L. Maslov for hospitality during his stay at the UF. VAZ is supported by the
Russian Foundation for Basic Research (grant No. 20-52-12 013), Deutsche
Forschungsgemeinschaft (grant No. EV 30/14-1) cooperation, and by the
Foundation for the Advancement of Theoretical Physics and Mathematics BASIS.
AAZ is supported by the Academy of Finland (project 308339) and in parts by
the Academy of Finland Centre of Excellence program (project 336810).

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## Supplemental Material to
”Anisotropic resistivity and superconducting instability in ferroelectric-like
metals”

###  .1 Model of a ferroelectric metal

The Hamiltonian includes electrons, phonons and interaction between the two.

$\displaystyle H_{\mathrm{F}}=\int_{\bf
r}\psi^{{\dagger}}(x)\left(-\frac{\partial^{2}_{\bf r}}{2m}-\mu\right)\psi(x)$
(1)

where $x=({\bf r},t)$ is a general coordinate, $\mu$ is the chemical
potential, and $m$ is the mass of electrons. Interaction of electrons with two
phonons is

$\displaystyle H_{\mathrm{e}-2\mathrm{ph}}=g\int_{{\bf
r}}\psi^{{\dagger}}(x)\psi(x)\left[{\bf P}_{0}+{\bf
P}(x)\right]\cdot\left[{\bf P}_{0}+{\bf P}(x)\right],$ (2)

where $g$ is a constant. Here ${\bf P}(x)$ is the phonon displacement field,

$\displaystyle{\bf P}(x)=\sum_{a=1,2}\sum_{{\bf q}}\frac{{\bf e}^{a}_{{\bf
q}}}{\sqrt{V}}A_{\bf q}\left[b_{a;{\bf q}}(t)e^{i{\bf q}{\bf r}}+b_{a;{\bf
q}}^{{\dagger}}(t)e^{-i{\bf q}{\bf r}}\right],$ (3)

where $A_{\bf q}^{2}=\left[\epsilon_{0}({\bf
q})-\epsilon_{\infty}\right]\frac{\omega_{\bf q}}{4\pi}$ with
$\epsilon_{0}({\bf q})=\frac{\Omega_{0}^{2}}{\omega_{\bf q}^{2}}$ and
$\epsilon_{0}({\bf q})\gg\epsilon_{\infty}$, and $b_{a;{\bf q}}$ and
$b_{a;{\bf q}}^{{\dagger}}$ are boson fields. Polarization vectors ${\bf
e}^{a}_{{\bf q}}$ of the two branches $a=1,2$ of transverse optical phonons
satisfy

$\displaystyle\sum_{a=1,2}e^{a}_{\alpha;{\bf p}}e^{a}_{\beta;{\bf
p}}=\delta_{\alpha\beta}-\frac{p_{\alpha}p_{\beta}}{p^{2}},$ (4)

where $\alpha,~{}\beta=x,y,z$. Hamiltonian of phonons is

$\displaystyle H^{(a)}_{\mathrm{ph}}=\int_{\bf q}\omega_{\bf q}\left(b_{a;{\bf
q}}^{{\dagger}}b_{a;{\bf q}}+\frac{1}{2}\right),$ (5)

where $\omega_{\bf q}=\sqrt{\omega_{\mathrm{TO}}^{2}+(sq)^{2}}$ is the
dispersion of phonons with $\omega_{\mathrm{TO}}$ being the mass of phonons
and $s$ being the speed of sound. Constant ${\bf P}_{0}$ in Eq. (2) is
electric polarization occuring in a ferroelectric, and which can understood as
a condensation of soft optical phonons when $\omega_{\mathrm{TO}}\rightarrow
0$ as the temperature is reduced to ferroelectric transition temperature,
$T\rightarrow T_{\mathrm{C}}^{\mathrm{FE}}$. The condensation not only results
in non-zero ${\bf P}_{0}$ but also in a finite gap $\omega_{\mathrm{TO}}$
below ferroelectric transition temperature, i.e.
$T<T_{\mathrm{C}}^{\mathrm{FE}}$. The gap there is propotional to the absolute
value of ${\bf P}_{0}$, i.e. $\omega_{\mathrm{TO}}\propto|{\bf P}_{0}|$. We
assume that the situation is such that the density of conducting electrons is
low and they don’t screen the finite polarization ${\bf P}_{0}$.

Although, the Matsubara technique is certainly the most conventional and
historic choice to study superconducting instability of the system, here we
wish to work in the Keldysh technique. The Keldysh technique has a power to
describe not only equilibrium but also non-equilibrium systems, allows for a
convenient derivation of the kinetic equation, and is probably the best in
describing disordered electron systems (by avoiding replicas). Therefore, we
will make an effort to exercise the Keldysh technique. We will follow the book
[1].

As far as notations are concerned, electron fields are promoted to Grassmann
fields, $\psi^{{\dagger}}(x)\rightarrow\bar{\psi}(x)$ and
$\psi(x)\rightarrow\psi(x)$, in accord with construction of electronic path
integral. General Hamiltonian bilinear in electron operators reads as

$\displaystyle\int_{{\cal
C}}dt~{}\bar{\psi}(t)\theta(t)\psi(t)=\int_{-\infty}^{+\infty}dt~{}\hat{\bar{\psi}}(t)\left[\begin{array}[]{cc}\theta_{+}(t)&0\\\
0&-\theta_{-}(t)\end{array}\right]\hat{\psi}(t)=\int_{-\infty}^{+\infty}dt~{}\hat{\bar{\Psi}}(t)\hat{L}\left[\begin{array}[]{cc}\theta_{+}(t)&0\\\
0&\theta_{-}(t)\end{array}\right]\hat{L}^{-1}\hat{\Psi}(t)$ (10)

where

$\displaystyle\hat{\bar{\psi}}=[\bar{\psi}_{+},~{}\bar{\psi}_{-}],~{}~{}\hat{\psi}=\left[\begin{array}[]{c}\psi_{+}\\\
\psi_{-}\end{array}\right]$ (13)

were introduced in the first step, and where we performed the Larkin-
Ovchinnikov rotation with the help of a matrix

$\displaystyle\hat{L}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{cc}1&-1\\\
1&1\end{array}\right],~{}~{}\hat{L}^{-1}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{cc}1&1\\\
-1&1\end{array}\right],$ (18)

in the next step, and as a consequence introduced rotated electron fields

$\displaystyle\hat{\bar{\Psi}}=\hat{\bar{\psi}}\hat{L}^{-1}=[\bar{\Psi}^{\mathrm{q}},~{}\bar{\Psi}^{\mathrm{cl}}],~{}~{}\hat{\Psi}=\hat{L}\hat{\sigma}_{3}\hat{\psi}=\left[\begin{array}[]{c}\Psi^{\mathrm{cl}}\\\
\Psi^{\mathrm{q}}\end{array}\right],$ (21)

where
$\bar{\Psi}^{\mathrm{q}}=\frac{1}{\sqrt{2}}(\bar{\Psi}^{+}-\bar{\Psi}^{-})$
and
$\bar{\Psi}^{\mathrm{cl}}=\frac{1}{\sqrt{2}}(\bar{\Psi}^{+}+\bar{\Psi}^{-})$,
and the same for the fields without the bars. The Hamiltonian gets rotated as

$\displaystyle\hat{L}\left[\begin{array}[]{cc}\theta_{+}&0\\\
0&\theta_{-}\end{array}\right]\hat{L}^{-1}=\frac{1}{2}\left[\begin{array}[]{cc}\theta_{+}+\theta_{-}&\theta_{+}-\theta_{-}\\\
\theta_{+}-\theta_{-}&\theta_{+}+\theta_{-}\end{array}\right]\equiv\left[\begin{array}[]{cc}\theta^{\mathrm{cl}}&\theta^{\mathrm{q}}\\\
\theta^{\mathrm{q}}&\theta^{\mathrm{cl}}\end{array}\right].$ (28)

For example, if $\theta(x)={\bf P}_{0}\cdot{\bf P}(x)$, we have

$\displaystyle\left[\begin{array}[]{cc}\theta^{\mathrm{cl}}&\theta^{\mathrm{q}}\\\
\theta^{\mathrm{q}}&\theta^{\mathrm{cl}}\end{array}\right]=\left[\begin{array}[]{cc}{\bf
P}_{0}\cdot{\bf P}^{\mathrm{cl}}&{\bf P}_{0}\cdot{\bf P}^{\mathrm{q}}\\\ {\bf
P}_{0}\cdot{\bf P}^{\mathrm{q}}&{\bf P}_{0}\cdot{\bf
P}^{\mathrm{cl}}\end{array}\right].$ (33)

If now the field $\theta$ is a composite field, such as $\theta={\bf
P}\cdot{\bf P}$, then we get for the Hamiltonian

$\displaystyle\left[\begin{array}[]{cc}\theta^{\mathrm{cl}}&\theta^{\mathrm{q}}\\\
\theta^{\mathrm{q}}&\theta^{\mathrm{cl}}\end{array}\right]=\frac{1}{4}\left[\begin{array}[]{cc}{\bf
P}^{\mathrm{cl}}\cdot{\bf P}^{\mathrm{cl}}+{\bf P}^{\mathrm{q}}\cdot{\bf
P}^{\mathrm{q}}&{\bf P}^{\mathrm{cl}}\cdot{\bf P}^{\mathrm{q}}+{\bf
P}^{\mathrm{q}}\cdot{\bf P}^{\mathrm{cl}}\\\ {\bf P}^{\mathrm{cl}}\cdot{\bf
P}^{\mathrm{q}}+{\bf P}^{\mathrm{q}}\cdot{\bf P}^{\mathrm{cl}}&{\bf
P}^{\mathrm{cl}}\cdot{\bf P}^{\mathrm{cl}}+{\bf P}^{\mathrm{q}}\cdot{\bf
P}^{\mathrm{q}}\end{array}\right].$ (38)

We will be needing correlators of ${\bf P}$ fields when deriving the effective
interaction between electrons. They are

$\displaystyle\langle
P^{\mathrm{cl}}_{\alpha}(x_{1})P^{\mathrm{cl}}_{\beta}(x_{2})\rangle$
$\displaystyle=\frac{i}{V}\sum_{\epsilon,{\bf
q}}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)A_{{\bf
q}}^{2}\left[e^{i{\bf q}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon(t_{1}-t_{2})}+e^{-i{\bf q}({\bf r}_{1}-{\bf
r}_{2})+i\epsilon(t_{1}-t_{2})}\right]d^{\mathrm{K}}(\epsilon,{\bf q}),$ (39)
$\displaystyle\langle
P^{\mathrm{cl}}_{\alpha}(x_{1})P^{\mathrm{q}}_{\beta}(x_{2})\rangle$
$\displaystyle=\frac{i}{V}\sum_{\epsilon,{\bf
q}}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)A_{{\bf
q}}^{2}\left[e^{i{\bf q}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon(t_{1}-t_{2})}d^{\mathrm{R}}(\epsilon,{\bf q})+e^{-i{\bf
q}({\bf r}_{1}-{\bf
r}_{2})+i\epsilon(t_{1}-t_{2})}d^{\mathrm{A}}(\epsilon,{\bf q})\right],$ (40)
$\displaystyle\langle
P^{\mathrm{q}}_{\alpha}(x_{1})P^{\mathrm{q}}_{\beta}(x_{2})\rangle$
$\displaystyle=0.$ (41)

Here the phonon Green functions read

$\displaystyle d^{\mathrm{R}/\mathrm{A}}(\epsilon,{\bf
q})=\frac{1}{\epsilon\pm i0-\omega_{{\bf q}}},$ (42) $\displaystyle
d^{\mathrm{K}}(\epsilon,{\bf q})={\cal
F}^{\mathrm{B}}_{\epsilon}\left[d^{\mathrm{R}}(\epsilon,{\bf
q})-d^{\mathrm{A}}(\epsilon,{\bf q})\right],$ (43)

where ${\cal F}^{\mathrm{B}}_{\epsilon}=\coth\left(\frac{\epsilon}{2T}\right)$
is the boson distribution function. We can simplify the correlators by
$\epsilon\rightarrow-\epsilon$ and ${\bf q}\rightarrow-{\bf q}$ under the sum
in corresponding terms,

$\displaystyle\langle
P^{\mathrm{cl}}_{\alpha}(x_{1})P^{\mathrm{cl}}_{\beta}(x_{2})\rangle$
$\displaystyle=\frac{i}{V}\sum_{\epsilon,{\bf
q}}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)A_{{\bf
q}}^{2}e^{i{\bf q}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon(t_{1}-t_{2})}D^{\mathrm{K}}(\epsilon,{\bf q}),$ (44)
$\displaystyle\langle
P^{\mathrm{cl}}_{\alpha}(x_{1})P^{\mathrm{q}}_{\beta}(x_{2})\rangle$
$\displaystyle=\frac{i}{V}\sum_{\epsilon,{\bf
q}}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)A_{{\bf
q}}^{2}e^{i{\bf q}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon(t_{1}-t_{2})}D^{\mathrm{R}}(\epsilon,{\bf q}),$ (45)
$\displaystyle\langle
P^{\mathrm{q}}_{\alpha}(x_{1})P^{\mathrm{cl}}_{\beta}(x_{2})\rangle$
$\displaystyle=\frac{i}{V}\sum_{\epsilon,{\bf
q}}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)A_{{\bf
q}}^{2}e^{i{\bf q}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon(t_{1}-t_{2})}D^{\mathrm{A}}(\epsilon,{\bf q}),$ (46)

where now updated phonon Green functions read as

$\displaystyle D^{\mathrm{R}/\mathrm{A}}(\epsilon,{\bf q})=\frac{2\omega_{{\bf
q}}}{(\epsilon\pm i0)^{2}-\omega_{{\bf q}}^{2}},$ (47) $\displaystyle
D^{\mathrm{K}}(\epsilon,{\bf q})={\cal
F}^{\mathrm{B}}_{\epsilon}\left[D^{\mathrm{R}}(\epsilon,{\bf
q})-D^{\mathrm{A}}(\epsilon,{\bf q})\right].$ (48)

It is also worth noticing that the displacement fields commute

$\displaystyle
P^{\mathrm{cl}}_{\alpha}(x_{1})P^{\mathrm{cl}}_{\beta}(x_{2})=P^{\mathrm{cl}}_{\beta}(x_{2})P^{\mathrm{cl}}_{\alpha}(x_{1}).$
(49)

###  .2 Effective interaction between electrons: one-phonon interaction

Interaction between electrons with two phonons is

$\displaystyle H_{\mathrm{e}-2\mathrm{ph}}=g\int_{{\bf
r}}\psi^{{\dagger}}(x)\psi(x)[{\bf P}_{0}+{\bf P}(x)]\cdot[{\bf P}_{0}+{\bf
P}(x)]=g\int_{{\bf r}}\psi^{{\dagger}}(x)\psi(x)[{\bf P}_{0}^{2}+{\bf
P}^{2}(x)+2{\bf P}_{0}\cdot{\bf P}(x)].$ (50)

First term above, i.e. with ${\bf P}_{0}^{2}$, is just the chemical potential
of electrons. Second term was studied in previous works and the results are
understood. Third term is new, and is a subject of the present study.

$\displaystyle H_{\mathrm{e}-2\mathrm{ph}}\rightarrow 2g\int_{{\bf
r}}\psi^{{\dagger}}(x)\psi(x){\bf P}_{0}\cdot{\bf P}(x).$ (51)

Effective interaction between electrons is obtained by integrating the phonons
out. We assume that ${\bf P}_{0}=P_{0}{\bf e}_{\alpha}$, i.e. the
ferroelectric spontaneous polarization is pointing in $\alpha$ direction. To
the second order in electron-phonon interaction we get

$\displaystyle
iS^{(2)}=\frac{1}{2}\langle\left(iS_{\mathrm{e}-2\mathrm{ph}}\right)\left(iS_{\mathrm{e}-2\mathrm{ph}}\right)\rangle_{\mathrm{phonons}}$
(52) $\displaystyle=$
$\displaystyle-i\frac{2P_{0}^{2}g^{2}}{V}\int_{\epsilon_{1},\epsilon_{2},\omega}\int_{{\bf
k}_{1},{\bf k}_{2},{\bf q}}\left[\hat{\bar{\Psi}}(\epsilon_{1}+\omega;{\bf
k}_{1}+{\bf q})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{1};{\bf
k}_{1})\right]\left[\hat{\bar{\Psi}}(\epsilon_{2}-\omega;{\bf k}_{2}-{\bf
q})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{2};{\bf
k}_{2})\right]\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf
q}}^{2}D^{\mathrm{K}}(\omega;{\bf q})$ (53)
$\displaystyle-i\frac{2P_{0}^{2}g^{2}}{V}\int_{\epsilon_{1},\epsilon_{2},\omega}\int_{{\bf
k}_{1},{\bf k}_{2},{\bf q}}\left[\hat{\bar{\Psi}}(\epsilon_{1}+\omega;{\bf
k}_{1}+{\bf q})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{1};{\bf
k}_{1})\right]\left[\hat{\bar{\Psi}}(\epsilon_{2}-\omega;{\bf k}_{2}-{\bf
q})\hat{\sigma}_{1}\hat{\Psi}(\epsilon_{2};{\bf
k}_{2})\right]\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf
q}}^{2}D^{\mathrm{R}}(\omega;{\bf q})$ (54)
$\displaystyle-i\frac{2P_{0}^{2}g^{2}}{V}\int_{\epsilon_{1},\epsilon_{2},\omega}\int_{{\bf
k}_{1},{\bf k}_{2},{\bf q}}\left[\hat{\bar{\Psi}}(\epsilon_{1}+\omega;{\bf
k}_{1}+{\bf q})\hat{\sigma}_{1}\hat{\Psi}(\epsilon_{1};{\bf
k}_{1})\right]\left[\hat{\bar{\Psi}}(\epsilon_{2}-\omega;{\bf k}_{2}-{\bf
q})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{2};{\bf
k}_{2})\right]\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf
q}}^{2}D^{\mathrm{A}}(\omega;{\bf q}).$ (55)

###  .3 Cooper ladder

Here we derive a prescription how to deal with the Cooper channel. Although,
it is a standard procedure and the resulting equations are standard textbook,
we wish to derive it from scratch. General static four electron interaction
relevant for the problem at hand after regrouping of the Grassmann fields for
the Cooper channel appears to be

$\displaystyle iS_{\mathrm{int}}=$
$\displaystyle-i\int_{\\{\epsilon\\};\\{{\bf k}\\}}V({\bf k}_{1}-{\bf
k}_{2})\left[(\hat{\bar{\Psi}}_{1;\alpha}\hat{\sigma}_{0}\hat{\Psi}_{2;\alpha})(\hat{\bar{\Psi}}_{3;\beta}\hat{\sigma}_{1}\hat{\Psi}_{4;\beta})+(\hat{\bar{\Psi}}_{1;\alpha}\hat{\sigma}_{1}\hat{\Psi}_{2;\alpha})(\hat{\bar{\Psi}}_{3;\beta}\hat{\sigma}_{0}\hat{\Psi}_{4;\beta})\right]\delta_{q_{1}-q_{2},q_{4}-q_{3}}$
(56) $\displaystyle=$ $\displaystyle-i\int_{\\{\epsilon\\};\\{{\bf
k}\\}}V({\bf k}_{1}-{\bf
k}_{2})\left(\bar{\Psi}^{\mathrm{q}}_{1;\alpha}\bar{\Psi}^{\mathrm{cl}}_{3;\beta}+\bar{\Psi}^{\mathrm{cl}}_{1;\alpha}\bar{\Psi}^{\mathrm{q}}_{3;\beta}\right)\left(\Psi^{\mathrm{cl}}_{4;\beta}\Psi^{\mathrm{cl}}_{2;\alpha}+\Psi^{\mathrm{q}}_{4;\beta}\Psi^{\mathrm{q}}_{2;\alpha}\right)\delta_{q_{1}-q_{2},q_{4}-q_{3}}$
(57) $\displaystyle-i\int_{\\{\epsilon\\};\\{{\bf k}\\}}V({\bf k}_{1}-{\bf
k}_{2})\left(\bar{\Psi}^{\mathrm{q}}_{1;\alpha}\bar{\Psi}^{\mathrm{q}}_{3;\beta}+\bar{\Psi}^{\mathrm{cl}}_{1;\alpha}\bar{\Psi}^{\mathrm{cl}}_{3;\beta}\right)\left(\Psi^{\mathrm{q}}_{4;\beta}\Psi^{\mathrm{cl}}_{2;\alpha}+\Psi^{\mathrm{cl}}_{4;\beta}\Psi^{\mathrm{q}}_{2;\alpha}\right)\delta_{q_{1}-q_{2},q_{4}-q_{3}},$
(58)

where $\alpha$ and $\beta$ are spins, and where $q_{i}=(\epsilon_{i},{\bf
k}_{i})$. Under the sign of integral we replace ${\bf k}_{3/4}\rightarrow-{\bf
k}_{3/4}$ and $\epsilon_{3/4}\rightarrow-\epsilon_{3/4}$ so as to meet the
structure of the Cooper channel, and rewrite the interaction

$\displaystyle iS_{\mathrm{int}}=$
$\displaystyle-i\int_{\\{\epsilon\\};\\{{\bf k}\\}}V({\bf k}_{1}-{\bf
k}_{2})\left(\bar{\Psi}^{\mathrm{q}}_{1;\alpha}\bar{\Psi}^{\mathrm{cl}}_{-3;\beta}+\bar{\Psi}^{\mathrm{cl}}_{1;\alpha}\bar{\Psi}^{\mathrm{q}}_{-3;\beta}\right)\left(\Psi^{\mathrm{cl}}_{-4;\beta}\Psi^{\mathrm{cl}}_{2;\alpha}+\Psi^{\mathrm{q}}_{-4;\beta}\Psi^{\mathrm{q}}_{2;\alpha}\right)\delta_{q_{1}-q_{3},q_{2}-q_{4}}$
(59) $\displaystyle-i\int_{\\{\epsilon\\};\\{{\bf k}\\}}V({\bf k}_{1}-{\bf
k}_{2})\left(\bar{\Psi}^{\mathrm{q}}_{1;\alpha}\bar{\Psi}^{\mathrm{q}}_{-3;\beta}+\bar{\Psi}^{\mathrm{cl}}_{1;\alpha}\bar{\Psi}^{\mathrm{cl}}_{-3;\beta}\right)\left(\Psi^{\mathrm{q}}_{-4;\beta}\Psi^{\mathrm{cl}}_{2;\alpha}+\Psi^{\mathrm{cl}}_{-4;\beta}\Psi^{\mathrm{q}}_{2;\alpha}\right)\delta_{q_{1}-q_{3},q_{2}-q_{4}},$
(60)

where attraction corresponds to $V({\bf q})<0$.

Let us now iterate the interaction and construct a Cooper ladder. We first
relabel the indeces

$\displaystyle iS_{\mathrm{int}}=$
$\displaystyle-i\int_{\\{\epsilon\\};\\{{\bf k}\\}}V({\bf k}_{1}-{\bf
k}_{4})\left(\bar{\Psi}^{\mathrm{q}}_{1;\alpha}\bar{\Psi}^{\mathrm{cl}}_{-2;\beta}+\bar{\Psi}^{\mathrm{cl}}_{1;\alpha}\bar{\Psi}^{\mathrm{q}}_{-2;\beta}\right)\left(\Psi^{\mathrm{cl}}_{-3;\beta}\Psi^{\mathrm{cl}}_{4;\alpha}+\Psi^{\mathrm{q}}_{-3;\beta}\Psi^{\mathrm{q}}_{4;\alpha}\right)\delta_{q_{1}-q_{2},q_{4}-q_{3}}$
(61) $\displaystyle-i\int_{\\{\epsilon\\};\\{{\bf k}\\}}V({\bf k}_{1}-{\bf
k}_{4})\left(\bar{\Psi}^{\mathrm{q}}_{1;\alpha}\bar{\Psi}^{\mathrm{q}}_{-2;\beta}+\bar{\Psi}^{\mathrm{cl}}_{1;\alpha}\bar{\Psi}^{\mathrm{cl}}_{-2;\beta}\right)\left(\Psi^{\mathrm{q}}_{-3;\beta}\Psi^{\mathrm{cl}}_{4;\alpha}+\Psi^{\mathrm{cl}}_{-3;\beta}\Psi^{\mathrm{q}}_{4;\alpha}\right)\delta_{q_{1}-q_{2},q_{4}-q_{3}}.$
(62)

This is done for our convenience and for further construction of the ladder.
To second order in interaction,

$\displaystyle\frac{1}{2}\langle\left(iS_{\mathrm{int}}\right)\left(iS_{\mathrm{int}}\right)\rangle$
(63) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\\{\epsilon\\}}\int_{\\{{\bf k}\\}}V({\bf
k}_{1}-{\bf k}_{5})V({\bf k}_{5}-{\bf k}_{4})\delta_{q_{1}-q_{2},q_{4}-q_{3}}$
(64)
$\displaystyle\times\left[G^{\mathrm{R}}(-\epsilon_{5}+\epsilon_{4}-\epsilon_{3};-{\bf
k}_{5}+{\bf k}_{4}-{\bf k}_{3})G^{\mathrm{K}}(\epsilon_{5};{\bf
k}_{5})+G^{\mathrm{K}}(-\epsilon_{5}+\epsilon_{4}-\epsilon_{3};-{\bf
k}_{5}+{\bf k}_{4}-{\bf k}_{3})G^{\mathrm{R}}(\epsilon_{5};{\bf
k}_{5})\right]$ (65)
$\displaystyle\times\left[\left(\bar{\Psi}^{\mathrm{q}}_{1;\alpha}\bar{\Psi}^{\mathrm{cl}}_{-2;\beta}+\bar{\Psi}^{\mathrm{cl}}_{1;\alpha}\bar{\Psi}^{\mathrm{q}}_{-2;\beta}\right)\left(\Psi^{\mathrm{cl}}_{-3;\beta}\Psi^{\mathrm{cl}}_{4;\alpha}+\Psi^{\mathrm{q}}_{-3;\beta}\Psi^{\mathrm{q}}_{4;\alpha}\right)+\left(\bar{\Psi}^{\mathrm{q}}_{1;\alpha}\bar{\Psi}^{\mathrm{q}}_{-2;\beta}+\bar{\Psi}^{\mathrm{cl}}_{1;\alpha}\bar{\Psi}^{\mathrm{cl}}_{-2;\beta}\right)\left(\Psi^{\mathrm{q}}_{-3;\beta}\Psi^{\mathrm{cl}}_{4;\alpha}+\Psi^{\mathrm{cl}}_{-3;\beta}\Psi^{\mathrm{q}}_{4;\alpha}\right)\right]$
(66)

where we have relabelled the indeces for our convenience. Recall that
$G^{\mathrm{K}}(\epsilon;{\bf k})={\cal
F}^{\mathrm{F}}_{\epsilon}\left(G^{\mathrm{R}}(\epsilon;{\bf
k})-G^{\mathrm{A}}(\epsilon;{\bf k})\right)=-2\pi i{\cal
F}^{\mathrm{F}}_{\epsilon}\delta(\epsilon-\xi_{\bf k})$ where ${\cal
F}_{\epsilon}^{\mathrm{F}}=\tanh\left(\frac{\epsilon}{2T}\right)$ is the
electron distribution function in equilibrium. Integral over the
$\epsilon_{5}$ frequency is

$\displaystyle\int_{\epsilon_{5}}\left[G^{\mathrm{R}}(-\epsilon_{5}+\epsilon_{4}-\epsilon_{3};-{\bf
k}_{5}+{\bf k}_{4}-{\bf k}_{3})G^{\mathrm{K}}(\epsilon_{5};{\bf
k}_{5})+G^{\mathrm{K}}(-\epsilon_{5}+\epsilon_{4}-\epsilon_{3};-{\bf
k}_{5}+{\bf k}_{4}-{\bf k}_{3})G^{\mathrm{R}}(\epsilon_{5};{\bf
k}_{5})\right]$ (67) $\displaystyle=-i\frac{{\cal F}^{\mathrm{F}}_{\xi_{{\bf
k}_{5}}}+{\cal F}^{\mathrm{F}}_{\xi_{-{\bf k}_{5}+{\bf k}_{4}-{\bf
k}_{3}}}}{\epsilon_{4}-\epsilon_{3}-\xi_{{\bf k}_{5}}-\xi_{-{\bf k}_{5}+{\bf
k}_{4}-{\bf k}_{3}}+i0}\rightarrow i\frac{{\cal F}^{\mathrm{F}}_{\xi_{{\bf
k}_{5}}}+{\cal F}^{\mathrm{F}}_{\xi_{-{\bf k}_{5}+{\bf k}_{4}-{\bf
k}_{3}}}}{\xi_{{\bf k}_{5}}+\xi_{-{\bf k}_{5}+{\bf k}_{4}-{\bf k}_{3}}-i0},$
(68)

where static limit was taken in the transformation under the right arrow. In
our case the static part of the interaction is

$\displaystyle V({\bf
q})=\frac{2g^{2}P_{0}^{2}}{V}\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf
q}}^{2}D^{\mathrm{R}}(\omega=0;{\bf q})=-\frac{g^{2}P_{0}^{2}}{\pi
V}\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)\frac{\Omega_{0}^{2}}{\omega_{\bf
q}^{2}},$ (69)

recall that $A_{{\bf
q}}^{2}\approx\frac{1}{4\pi}\frac{\Omega_{0}^{2}}{\omega_{{\bf q}}}$. It is
clear that the interaction between electrons is attractive. The Cooper ladder
then reads

$\displaystyle\Gamma({\bf k}_{1}-{\bf k}_{4})=V({\bf k}_{1}-{\bf
k}_{4})-\frac{1}{2}\int_{\bf p}V({\bf k}_{1}-{\bf p})\frac{{\cal
F}^{\mathrm{F}}_{\xi_{{\bf p}}}+{\cal F}^{\mathrm{F}}_{\xi_{-{\bf p}+{\bf
k}_{4}-{\bf k}_{3}}}}{\xi_{{\bf p}}+\xi_{-{\bf p}+{\bf k}_{4}-{\bf
k}_{3}}}\Gamma({\bf p}-{\bf k}_{4}),$ (70)

assuming that ${\bf k}_{4}={\bf k}_{3}$, we get

$\displaystyle\Gamma({\bf k}_{1}-{\bf k}_{4})=V({\bf k}_{1}-{\bf
k}_{4})-\frac{1}{2}\int_{\bf p}V({\bf k}_{1}-{\bf p})\frac{{\cal
F}^{\mathrm{F}}_{\xi_{{\bf p}}}}{\xi_{{\bf p}}}\Gamma({\bf p}-{\bf k}_{4}).$
(71)

It is convenient to apply a change of variables from ${\bf p}$ to $Q=|{\bf
p}-{\bf k}_{1}|$ and $p$ as

$\displaystyle\int\frac{d^{3}p}{(2\pi)^{3}}(..)=\frac{1}{k_{1}}\int_{0}^{\infty}\frac{pdp}{2\pi^{2}}\int_{|p-k_{1}|}^{{|p+k_{1}|}}\frac{QdQ}{2}(..).$
(72)

For example, such a trick is given in Ref. [2]. We put
$\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)\rightarrow\frac{1}{2}$ in the
interaction, so it becomes $V({\bf q})=-\frac{g^{2}P_{0}^{2}}{2\pi
V}\frac{\Omega_{0}^{2}}{\omega_{\bf q}^{2}}$. We assume that $\Gamma({\bf
p}-{\bf k}_{4})$ is momentum independent, estimated at the Fermi momentum.
Then, after the transform Eq. (72), we get for the integral over $Q$

$\displaystyle\int_{|p-k_{1}|}^{|p+k_{1}|}\frac{QdQ}{2}V(Q)=-\frac{g^{2}P_{0}^{2}\Omega_{0}^{2}}{2\pi
V}\int_{|p-k_{1}|}^{|p+k_{1}|}\frac{QdQ}{2}\frac{1}{\omega_{Q}^{2}}=-\frac{g^{2}P_{0}^{2}\Omega_{0}^{2}}{8\pi
Vs^{2}}\ln\left[\frac{\omega_{\mathrm{TO}}^{2}+s^{2}|p+k_{1}|^{2}}{\omega_{\mathrm{TO}}^{2}+s^{2}|p-k_{1}|^{2}}\right]\approx-\frac{g^{2}P_{0}^{2}\Omega_{0}^{2}}{8\pi
Vs^{2}}\ln\left[\frac{\omega_{\mathrm{TO}}^{2}+\omega_{\mathrm{BG}}^{2}}{\omega_{\mathrm{TO}}^{2}+\frac{s^{2}}{v_{\mathrm{F}}^{2}}\xi_{\bf
p}^{2}}\right],$ (73)

where $\omega_{\mathrm{BG}}=s2p_{\mathrm{F}}$ is the Bloch-Gruneisen
frequency. If $v_{\mathrm{F}}\gg s$, then the
$\frac{s^{2}}{v_{\mathrm{F}}^{2}}\xi_{\bf p}^{2}$ term can be dropped as
compared with $\omega_{\mathrm{TO}}^{2}$. Remaining integral is

$\displaystyle\int_{0}^{\infty}\frac{pdp}{2\pi^{2}}\frac{{\cal
F}^{\mathrm{F}}_{\xi_{{\bf p}}}}{\xi_{{\bf
p}}}=\frac{m}{4\pi^{2}}\int_{-\mu}^{\omega_{\mathrm{D}}}d\xi_{\bf
p}\frac{{\cal F}^{\mathrm{F}}_{\xi_{{\bf p}}}}{\xi_{{\bf
p}}}=\frac{m}{4\pi^{2}}\left(\int_{0}^{\mu}+\int_{0}^{\mu}\right)d\xi_{\bf
p}\frac{{\cal F}^{\mathrm{F}}_{\xi_{{\bf p}}}}{\xi_{{\bf
p}}}=\frac{m}{2\pi^{2}}\ln\left(\frac{\mu}{T}\right),$ (74)

valid when $\omega_{\mathrm{D}}\gg\mu$, which is the case in STO. It is
important that both upper limits are the same. Overall, we have

$\displaystyle\int_{\bf p}V({\bf k}_{1}-{\bf p})\frac{{\cal
F}^{\mathrm{F}}_{\xi_{{\bf p}}}}{\xi_{{\bf
p}}}=-\frac{g^{2}mP_{0}^{2}\Omega_{0}^{2}}{16\pi^{3}Vs^{2}p_{\mathrm{F}}}\ln\left[\frac{\omega_{\mathrm{TO}}^{2}+\omega_{\mathrm{BG}}^{2}}{\omega_{\mathrm{TO}}^{2}}\right]\ln\left(\frac{\mu}{T}\right).$
(75)

Superconducting transition temperature is

$\displaystyle T_{\mathrm{c}}^{\mathrm{SC}}=\mu e^{-\frac{\nu}{G}},$ (76)

where $\nu=\frac{mp_{F}}{2\pi^{2}}$ is the density of states, and

$\displaystyle
G=\frac{g^{2}m^{2}P_{0}^{2}\Omega_{0}^{2}}{32\pi^{5}Vs^{2}}\ln\left[\frac{\omega_{\mathrm{TO}}^{2}+\omega_{\mathrm{BG}}^{2}}{\omega_{\mathrm{TO}}^{2}}\right]$
(77)

is the effective interaction. There is a maximum in
$T_{\mathrm{c}}^{\mathrm{SC}}$ at some value of $\mu$.

###  .4 Self-energy due to one-phonon interaction: imaginary part

To derive kinetic equation, we need to calculate self-energy due to electron-
phonon interaction. We contract the interaction

$\displaystyle-i\frac{2P_{0}^{2}g^{2}}{V}\int_{\epsilon_{1},\epsilon_{2},\omega}\int_{{\bf
k}_{1},{\bf k}_{2},{\bf
q}}\left\langle\left[\hat{\bar{\Psi}}(\epsilon_{1}+\omega;{\bf k}_{1}+{\bf
q})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{1};{\bf
k}_{1})\right]\left[\hat{\bar{\Psi}}(\epsilon_{2}-\omega;{\bf k}_{2}-{\bf
q})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{2};{\bf
k}_{2})\right]\right\rangle\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf
q}}^{2}D^{\mathrm{K}}(\omega;{\bf q})$ (78)
$\displaystyle-i\frac{2P_{0}^{2}g^{2}}{V}\int_{\epsilon_{1},\epsilon_{2},\omega}\int_{{\bf
k}_{1},{\bf k}_{2},{\bf
q}}\left\langle\left[\hat{\bar{\Psi}}(\epsilon_{1}+\omega;{\bf k}_{1}+{\bf
q})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{1};{\bf
k}_{1})\right]\left[\hat{\bar{\Psi}}(\epsilon_{2}-\omega;{\bf k}_{2}-{\bf
q})\hat{\sigma}_{1}\hat{\Psi}(\epsilon_{2};{\bf
k}_{2})\right]\right\rangle\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf
q}}^{2}D^{\mathrm{R}}(\omega;{\bf q})$ (79)
$\displaystyle-i\frac{2P_{0}^{2}g^{2}}{V}\int_{\epsilon_{1},\epsilon_{2},\omega}\int_{{\bf
k}_{1},{\bf k}_{2},{\bf
q}}\left\langle\left[\hat{\bar{\Psi}}(\epsilon_{1}+\omega;{\bf k}_{1}+{\bf
q})\hat{\sigma}_{1}\hat{\Psi}(\epsilon_{1};{\bf
k}_{1})\right]\left[\hat{\bar{\Psi}}(\epsilon_{2}-\omega;{\bf k}_{2}-{\bf
q})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{2};{\bf
k}_{2})\right]\right\rangle\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf
q}}^{2}D^{\mathrm{A}}(\omega;{\bf q}),$ (80)

where the contractions read as

$\displaystyle\left\langle\left[\hat{\bar{\Psi}}(\epsilon_{1}+\omega;{\bf
k}_{1}+{\bf q})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{1};{\bf
k}_{1})\right]\left[\hat{\bar{\Psi}}(\epsilon_{2}-\omega;{\bf k}_{2}-{\bf
q})\hat{\sigma}_{1}\hat{\Psi}(\epsilon_{2};{\bf k}_{2})\right]\right\rangle$
(81) $\displaystyle=$ $\displaystyle\bar{\Psi}^{\mathrm{q}}(\epsilon_{1};{\bf
k}_{1})\Psi^{\mathrm{cl}}(\epsilon_{1};{\bf
k}_{1})iG^{\mathrm{K}}(\epsilon_{1}-\omega;{\bf k}_{1}-{\bf q})$ (82)
$\displaystyle+$ $\displaystyle\bar{\Psi}^{\mathrm{cl}}(\epsilon_{1};{\bf
k}_{1})\Psi^{\mathrm{q}}(\epsilon_{1};{\bf
k}_{1})iG^{\mathrm{K}}(\epsilon_{1}+\omega;{\bf k}_{1}+{\bf q})$ (83)
$\displaystyle+$ $\displaystyle\bar{\Psi}^{\mathrm{cl}}(\epsilon_{1};{\bf
k}_{1})\Psi^{\mathrm{cl}}(\epsilon_{1};{\bf
k}_{1})\left[iG^{\mathrm{R}}(\epsilon_{1}+\omega;{\bf k}_{1}+{\bf
q})+iG^{\mathrm{A}}(\epsilon_{1}-\omega;{\bf k}_{1}-{\bf q})\right]$ (84)
$\displaystyle+$ $\displaystyle\bar{\Psi}^{\mathrm{q}}(\epsilon_{1};{\bf
k}_{1})\Psi^{\mathrm{q}}(\epsilon_{1};{\bf
k}_{1})\left[iG^{\mathrm{R}}(\epsilon_{1}-\omega;{\bf k}_{1}-{\bf
q})+iG^{\mathrm{A}}(\epsilon_{1}+\omega;{\bf k}_{1}+{\bf q})\right],$ (85)

where we have relabelled the indexes where needed. Similarly

$\displaystyle\left\langle\left[\hat{\bar{\Psi}}(\epsilon_{1}+\omega;{\bf
k}_{1}+{\bf q})\hat{\sigma}_{1}\hat{\Psi}(\epsilon_{1};{\bf
k}_{1})\right]\left[\hat{\bar{\Psi}}(\epsilon_{2}-\omega;{\bf k}_{2}-{\bf
q})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{2};{\bf k}_{2})\right]\right\rangle$
(86) $\displaystyle=$ $\displaystyle\bar{\Psi}^{\mathrm{q}}(\epsilon_{1};{\bf
k}_{1})\Psi^{\mathrm{cl}}(\epsilon_{1};{\bf
k}_{1})iG^{\mathrm{K}}(\epsilon_{1}+\omega;{\bf k}_{1}+{\bf q})$ (87)
$\displaystyle+$ $\displaystyle\bar{\Psi}^{\mathrm{cl}}(\epsilon_{1};{\bf
k}_{1})\Psi^{\mathrm{q}}(\epsilon_{1};{\bf
k}_{1})iG^{\mathrm{K}}(\epsilon_{1}-\omega;{\bf k}_{1}-{\bf q})$ (88)
$\displaystyle+$ $\displaystyle\bar{\Psi}^{\mathrm{cl}}(\epsilon_{1};{\bf
k}_{1})\Psi^{\mathrm{cl}}(\epsilon_{1};{\bf
k}_{1})\left[iG^{\mathrm{R}}(\epsilon_{1}-\omega;{\bf k}_{1}-{\bf
q})+iG^{\mathrm{A}}(\epsilon_{1}+\omega;{\bf k}_{1}+{\bf q})\right]$ (89)
$\displaystyle+$ $\displaystyle\bar{\Psi}^{\mathrm{q}}(\epsilon_{1};{\bf
k}_{1})\Psi^{\mathrm{q}}(\epsilon_{1};{\bf
k}_{1})\left[iG^{\mathrm{R}}(\epsilon_{1}+\omega;{\bf k}_{1}+{\bf
q})+iG^{\mathrm{A}}(\epsilon_{1}-\omega;{\bf k}_{1}-{\bf q})\right].$ (90)

Last contraction reads as

$\displaystyle\left\langle\left[\hat{\bar{\Psi}}(\epsilon_{1}+\omega;{\bf
k}_{1}+{\bf q})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{1};{\bf
k}_{1})\right]\left[\hat{\bar{\Psi}}(\epsilon_{2}-\omega;{\bf k}_{2}-{\bf
q})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{2};{\bf k}_{2})\right]\right\rangle$
(91) $\displaystyle=$ $\displaystyle\bar{\Psi}^{\mathrm{q}}(\epsilon_{1};{\bf
k}_{1})\Psi^{\mathrm{cl}}(\epsilon_{1};{\bf
k}_{1})\left[iG^{\mathrm{R}}(\epsilon_{1}+\omega;{\bf k}_{1}+{\bf
q})+iG^{\mathrm{R}}(\epsilon_{1}-\omega;{\bf k}_{1}-{\bf q})\right]$ (92)
$\displaystyle+$ $\displaystyle\bar{\Psi}^{\mathrm{cl}}(\epsilon_{1};{\bf
k}_{1})\Psi^{\mathrm{q}}(\epsilon_{1};{\bf
k}_{1})\left[iG^{\mathrm{A}}(\epsilon_{1}+\omega;{\bf k}_{1}+{\bf
q})+iG^{\mathrm{A}}(\epsilon_{1}-\omega;{\bf k}_{1}-{\bf q})\right]$ (93)
$\displaystyle+$ $\displaystyle\bar{\Psi}^{\mathrm{q}}(\epsilon_{1};{\bf
k}_{1})\Psi^{\mathrm{q}}(\epsilon_{1};{\bf
k}_{1})\left[iG^{\mathrm{K}}(\epsilon_{1}+\omega;{\bf k}_{1}+{\bf
q})+iG^{\mathrm{K}}(\epsilon_{1}-\omega;{\bf k}_{1}-{\bf q})\right].$ (94)

Let us first write down a general expression for the self-energy.

$\displaystyle\langle S_{\mathrm{int}}\rangle=$
$\displaystyle-i\int_{\epsilon;{\bf k}}\bar{\Psi}^{\mathrm{cl}}(\epsilon;{\bf
k})\Psi^{\mathrm{cl}}(\epsilon;{\bf k})\Sigma^{\mathrm{cl-cl}}(\epsilon;{\bf
k})-i\int_{\epsilon;{\bf k}}\bar{\Psi}^{\mathrm{q}}(\epsilon;{\bf
k})\Psi^{\mathrm{q}}(\epsilon;{\bf k})\Sigma^{\mathrm{K}}(\epsilon;{\bf k})$
(95) $\displaystyle-i\int_{\epsilon;{\bf
k}}\bar{\Psi}^{\mathrm{q}}(\epsilon;{\bf k})\Psi^{\mathrm{cl}}(\epsilon;{\bf
k})\Sigma^{\mathrm{R}}(\epsilon;{\bf k})-i\int_{\epsilon;{\bf
k}}\bar{\Psi}^{\mathrm{cl}}(\epsilon;{\bf k})\Psi^{\mathrm{q}}(\epsilon;{\bf
k})\Sigma^{\mathrm{A}}(\epsilon;{\bf k})$ (96) $\displaystyle=$
$\displaystyle-i\int_{\epsilon;{\bf k}}\hat{\bar{\Psi}}(\epsilon;{\bf
k})\left[\begin{array}[]{cc}\Sigma^{\mathrm{R}}(\epsilon;{\bf
k})&\Sigma^{\mathrm{K}}(\epsilon;{\bf k})\\\ \Sigma^{\mathrm{cl-
cl}}(\epsilon;{\bf k})&\Sigma^{\mathrm{A}}(\epsilon;{\bf
k})\end{array}\right]\hat{\Psi}(\epsilon;{\bf k}),$ (99)

where, recall, the fields with a hat are the spinors in the Keldysh space,

$\displaystyle\hat{\bar{\Psi}}=[\bar{\Psi}^{\mathrm{q}},~{}\bar{\Psi}^{\mathrm{cl}}],~{}~{}\hat{\Psi}=\left[\begin{array}[]{c}\Psi^{\mathrm{cl}}\\\
\Psi^{\mathrm{q}}\end{array}\right].$ (102)

It is expected that $\Sigma^{\mathrm{cl-cl}}(\epsilon;{\bf k})=0$, which we
will show below for our model. The expressions for the components of the self-
energy are

$\displaystyle\Sigma^{\mathrm{cl-cl}}(\epsilon;{\bf k})=$
$\displaystyle\int_{\omega;{\bf q}}\left[iG^{\mathrm{R}}(\epsilon+\omega;{\bf
k}+{\bf q})+iG^{\mathrm{A}}(\epsilon-\omega;{\bf k}-{\bf
q})\right]U^{\mathrm{R}}(\omega;{\bf q})$ (103) $\displaystyle+$
$\displaystyle\int_{\omega;{\bf q}}\left[iG^{\mathrm{R}}(\epsilon-\omega;{\bf
k}-{\bf q})+iG^{\mathrm{A}}(\epsilon+\omega;{\bf k}+{\bf
q})\right]U^{\mathrm{A}}(\omega;{\bf q}),$ (104)
$\displaystyle\Sigma^{\mathrm{K}}(\epsilon;{\bf k})=$
$\displaystyle\int_{\omega;{\bf q}}\left[iG^{\mathrm{R}}(\epsilon-\omega;{\bf
k}-{\bf q})+iG^{\mathrm{A}}(\epsilon+\omega;{\bf k}+{\bf
q})\right]U^{\mathrm{R}}(\omega;{\bf q})$ (105) $\displaystyle+$
$\displaystyle\int_{\omega;{\bf q}}\left[iG^{\mathrm{R}}(\epsilon+\omega;{\bf
k}+{\bf q})+iG^{\mathrm{A}}(\epsilon-\omega;{\bf k}-{\bf
q})\right]U^{\mathrm{A}}(\omega;{\bf q})$ (106) $\displaystyle+$
$\displaystyle\int_{\omega;{\bf q}}\left[iG^{\mathrm{K}}(\epsilon+\omega;{\bf
k}+{\bf q})+iG^{\mathrm{K}}(\epsilon-\omega;{\bf k}-{\bf
q})\right]U^{\mathrm{K}}(\omega;{\bf q}),$ (107)

$\displaystyle\Sigma^{\mathrm{R}}(\epsilon;{\bf k})=$
$\displaystyle\int_{\omega;{\bf q}}\left[iG^{\mathrm{K}}(\epsilon+\omega;{\bf
k}+{\bf q})U^{\mathrm{A}}(\omega;{\bf q})+iG^{\mathrm{K}}(\epsilon-\omega;{\bf
k}-{\bf q})U^{\mathrm{R}}(\omega;{\bf q})\right]$ (108) $\displaystyle+$
$\displaystyle\int_{\omega;{\bf q}}\left[iG^{\mathrm{R}}(\epsilon+\omega;{\bf
k}+{\bf q})+iG^{\mathrm{R}}(\epsilon-\omega;{\bf k}-{\bf
q})\right]U^{\mathrm{K}}(\omega;{\bf q}),$ (109)

$\displaystyle\Sigma^{\mathrm{A}}(\epsilon;{\bf k})=$
$\displaystyle\int_{\omega;{\bf q}}\left[iG^{\mathrm{K}}(\epsilon+\omega;{\bf
k}+{\bf q})U^{\mathrm{R}}(\omega;{\bf q})+iG^{\mathrm{K}}(\epsilon-\omega;{\bf
k}-{\bf q})U^{\mathrm{A}}(\omega;{\bf q})\right]$ (110) $\displaystyle+$
$\displaystyle\int_{\omega;{\bf q}}\left[iG^{\mathrm{A}}(\epsilon+\omega;{\bf
k}+{\bf q})+iG^{\mathrm{A}}(\epsilon-\omega;{\bf k}-{\bf
q})\right]U^{\mathrm{K}}(\omega;{\bf q}).$ (111)

Let us demonstrate that $\Sigma^{\mathrm{cl-cl}}(\epsilon;{\bf k})=0$. We use
$D^{\mathrm{A}}(-\omega;{\bf q})=D^{\mathrm{R}}(\omega;{\bf q})$, which allows
to double certain terms. Furthermore, we notice that in the expression for the
$\Sigma^{\mathrm{cl-cl}}(\epsilon;{\bf k})$ all the residues are on the same
half of the complex space plane. Thus, indeed, due to integration over the
frequency $\omega$

$\displaystyle-i\frac{4P_{0}^{2}g^{2}}{V}\int_{\epsilon;{\bf
k}}\bar{\Psi}^{\mathrm{cl}}(\epsilon;{\bf k})\Psi^{\mathrm{cl}}(\epsilon;{\bf
k})\int_{\omega;{\bf q}}\left[iG^{\mathrm{R}}(\epsilon+\omega;{\bf k}+{\bf
q})+iG^{\mathrm{A}}(\epsilon-\omega;{\bf k}-{\bf
q})\right]\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf
q}}^{2}D^{\mathrm{R}}(\omega;{\bf q})=0.$ (112)

The Keldysh part of the self-energy reads

$\displaystyle-i\frac{4P_{0}^{2}g^{2}}{V}\int_{\epsilon;{\bf
k}}\bar{\Psi}^{\mathrm{q}}(\epsilon;{\bf k})\Psi^{\mathrm{q}}(\epsilon;{\bf
k})$ (113) $\displaystyle\times\int_{\omega;{\bf
q}}\left\\{\left[iG^{\mathrm{R}}(\epsilon-\omega;{\bf k}-{\bf
q})+iG^{\mathrm{A}}(\epsilon+\omega;{\bf k}+{\bf
q})\right]D^{\mathrm{R}}(\omega;{\bf q})+iG^{\mathrm{K}}(\epsilon+\omega;{\bf
k}+{\bf q})D^{\mathrm{K}}(\omega;{\bf
q})\right\\}\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf q}}^{2}$ (114)
$\displaystyle=$ $\displaystyle-i\frac{4P_{0}^{2}g^{2}}{V}(2\pi
i)\int_{\epsilon;{\bf k}}\bar{\Psi}^{\mathrm{q}}(\epsilon;{\bf
k})\Psi^{\mathrm{q}}(\epsilon;{\bf k})\int_{\bf q}\left[{\cal
F}^{\mathrm{F}}_{\xi_{{\bf k}+{\bf q}}}{\cal F}^{\mathrm{B}}_{\xi_{{\bf
k}+{\bf q}}-\epsilon}-1\right]\left[\delta\left(\epsilon-\xi_{{\bf k}+{\bf
q}}-\omega_{\bf q}\right)-\delta\left(\epsilon-\xi_{{\bf k}+{\bf
q}}+\omega_{\bf
q}\right)\right]\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf q}}^{2},$
(115)

where

$\displaystyle\int_{\omega;{\bf q}}\left[iG^{\mathrm{R}}(\epsilon-\omega;{\bf
k}-{\bf q})+iG^{\mathrm{A}}(\epsilon+\omega;{\bf k}+{\bf
q})\right]\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf
q}}^{2}D^{\mathrm{R}}(\omega;{\bf q})$ (116) $\displaystyle=$
$\displaystyle-2\pi i\int_{\bf q}\left[\delta\left(\epsilon-\xi_{{\bf k}+{\bf
q}}-\omega_{\bf q}\right)-\delta\left(\epsilon-\xi_{{\bf k}+{\bf
q}}+\omega_{\bf
q}\right)\right]\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf q}}^{2},$
(117)

and

$\displaystyle\int_{\omega;{\bf q}}iG^{\mathrm{K}}(\epsilon+\omega;{\bf
k}+{\bf q})D^{\mathrm{K}}(\omega;{\bf
q})\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf q}}^{2}$ (118)
$\displaystyle=$ $\displaystyle 2\pi i\int_{\bf q}{\cal
F}^{\mathrm{F}}_{\xi_{{\bf k}+{\bf q}}}{\cal F}^{\mathrm{B}}_{\xi_{{\bf
k}+{\bf q}}-\epsilon}\left[\delta\left(\epsilon-\xi_{{\bf k}+{\bf
q}}-\omega_{\bf q}\right)-\delta\left(\epsilon-\xi_{{\bf k}+{\bf
q}}+\omega_{\bf
q}\right)\right]\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf q}}^{2}.$
(119)

Another term after relabelling ceratin indexes is

$\displaystyle-i\frac{4P_{0}^{2}g^{2}}{V}\int_{\epsilon;{\bf
k}}\bar{\Psi}^{\mathrm{cl}}(\epsilon;{\bf k})\Psi^{\mathrm{q}}(\epsilon;{\bf
k})\int_{\omega;{\bf q}}\left[iG^{\mathrm{K}}(\epsilon+\omega;{\bf k}+{\bf
q})D^{\mathrm{R}}(\omega;{\bf q})+iG^{\mathrm{A}}(\epsilon+\omega;{\bf k}+{\bf
q})D^{\mathrm{K}}(\omega;{\bf
q})\right]\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf q}}^{2}$ (120)
$\displaystyle=$ $\displaystyle-i\frac{4P_{0}^{2}g^{2}}{V}\int_{\epsilon;{\bf
k}}\bar{\Psi}^{\mathrm{cl}}(\epsilon;{\bf k})\Psi^{\mathrm{q}}(\epsilon;{\bf
k})\int_{\bf q}\left[{\cal F}^{\mathrm{F}}_{\xi_{{\bf k}+{\bf
q}}}\frac{2\omega_{\bf q}}{\left(\epsilon-\xi_{{\bf k}+{\bf
q}}+i0\right)^{2}-\omega_{\bf q}^{2}}+{\cal F}^{\mathrm{B}}_{\omega_{\bf
q}}\frac{2\left(\epsilon-\xi_{{\bf k}+{\bf
q}}\right)}{\left(\epsilon-\xi_{{\bf k}+{\bf q}}+i0\right)^{2}-\omega_{\bf
q}^{2}}\right]\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf q}}^{2},$
(121)

where the integral is

$\displaystyle\int_{\omega;{\bf q}}\left[iG^{\mathrm{K}}(\epsilon+\omega;{\bf
k}+{\bf q})D^{\mathrm{R}}(\omega;{\bf q})+iG^{\mathrm{A}}(\epsilon+\omega;{\bf
k}+{\bf q})D^{\mathrm{K}}(\omega;{\bf
q})\right]\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf q}}^{2}$ (122)
$\displaystyle=$ $\displaystyle\int_{\bf q}\left[{\cal
F}^{\mathrm{F}}_{\xi_{{\bf k}+{\bf q}}}\frac{2\omega_{\bf
q}}{\left(\epsilon-\xi_{{\bf k}+{\bf q}}+i0\right)^{2}-\omega_{\bf
q}^{2}}+{\cal F}^{\mathrm{B}}_{\omega_{\bf q}}\frac{2\left(\epsilon-\xi_{{\bf
k}+{\bf q}}\right)}{\left(\epsilon-\xi_{{\bf k}+{\bf
q}}+i0\right)^{2}-\omega_{\bf
q}^{2}}\right]\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf q}}^{2}.$
(123)

Finally,

$\displaystyle-i\frac{4P_{0}^{2}g^{2}}{V}\int_{\epsilon;{\bf
k}}\bar{\Psi}^{\mathrm{q}}(\epsilon;{\bf k})\Psi^{\mathrm{cl}}(\epsilon;{\bf
k})\int_{\omega;{\bf q}}\left[iG^{\mathrm{K}}(\epsilon+\omega;{\bf k}+{\bf
q})D^{\mathrm{A}}(\omega;{\bf q})+iG^{\mathrm{R}}(\epsilon+\omega;{\bf k}+{\bf
q})D^{\mathrm{K}}(\omega;{\bf
q})\right]\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf q}}^{2}$ (124)
$\displaystyle=$ $\displaystyle-i\frac{4P_{0}^{2}g^{2}}{V}\int_{\epsilon;{\bf
k}}\bar{\Psi}^{\mathrm{q}}(\epsilon;{\bf k})\Psi^{\mathrm{cl}}(\epsilon;{\bf
k})\int_{\bf q}\left[{\cal F}^{\mathrm{F}}_{\xi_{{\bf k}+{\bf
q}}}\frac{2\omega_{\bf q}}{\left(\epsilon-\xi_{{\bf k}+{\bf
q}}-i0\right)^{2}-\omega_{\bf q}^{2}}+{\cal F}^{\mathrm{B}}_{\omega_{\bf
q}}\frac{2\left(\epsilon-\xi_{{\bf k}+{\bf
q}}\right)}{\left(\epsilon-\xi_{{\bf k}+{\bf q}}-i0\right)^{2}-\omega_{\bf
q}^{2}}\right]\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf q}}^{2}.$
(125)

Therefore, we conclude with the expressions for the self-energy,

$\displaystyle\Sigma^{\mathrm{cl-cl}}(\epsilon;{\bf k})=0$ (126)
$\displaystyle\Sigma^{\mathrm{K}}(\epsilon;{\bf
k})=\frac{4P_{0}^{2}g^{2}}{V}(2\pi i)\int_{\bf q}\left[{\cal
F}^{\mathrm{F}}_{\xi_{{\bf k}+{\bf q}}}{\cal F}^{\mathrm{B}}_{\xi_{{\bf
k}+{\bf q}}-\epsilon}-1\right]\left[\delta\left(\epsilon-\xi_{{\bf k}+{\bf
q}}-\omega_{\bf q}\right)-\delta\left(\epsilon-\xi_{{\bf k}+{\bf
q}}+\omega_{\bf
q}\right)\right]\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf q}}^{2},$
(127) $\displaystyle\Sigma^{\mathrm{R}/\mathrm{A}}(\epsilon;{\bf
k})=\frac{4P_{0}^{2}g^{2}}{V}\int_{\bf q}\left[{\cal
F}^{\mathrm{F}}_{\xi_{{\bf k}+{\bf q}}}\frac{2\omega_{\bf
q}}{\left(\epsilon-\xi_{{\bf k}+{\bf q}}\pm i0\right)^{2}-\omega_{\bf
q}^{2}}+{\cal F}^{\mathrm{B}}_{\omega_{\bf q}}\frac{2\left(\epsilon-\xi_{{\bf
k}+{\bf q}}\right)}{\left(\epsilon-\xi_{{\bf k}+{\bf q}}\pm
i0\right)^{2}-\omega_{\bf
q}^{2}}\right]\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf q}}^{2}.$
(128)

Imaginary part of the self-energy reads as

$\displaystyle\mathrm{Im}\Sigma^{\mathrm{R}/\mathrm{A}}(\epsilon;{\bf k})=$
$\displaystyle\mp\pi\frac{4P_{0}^{2}g^{2}}{V}\int_{\bf
q}\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf q}}^{2}$ (129)
$\displaystyle\times\left\\{{\cal F}^{\mathrm{F}}_{\xi_{{\bf k}+{\bf
q}}}\left[\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}-\omega_{\bf
q}\right)-\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}+\omega_{\bf
q}\right)\right]+{\cal F}^{\mathrm{B}}_{\omega_{\bf
q}}\left[\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}-\omega_{\bf
q}\right)+\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}+\omega_{\bf
q}\right)\right]\right\\}.$ (130)

Let us now estimate the life-time of electrons. We use ${\bf e}_{k}=\frac{{\bf
k}}{k}$ and ${\bf e}_{0}=\frac{{\bf P}_{0}}{|P_{0}|}$ unit vectors to write
components of the ${\bf q}$ vector. We have

$\displaystyle\frac{q_{\alpha}^{2}}{q^{2}}=\cos^{2}(\theta)({\bf
e}_{k}\cdot{\bf
e}_{0})^{2}+\sin^{2}(\theta)\cos^{2}(\phi)\left(\left[\left[{\bf
e}_{k}\times{\bf e}_{0}\right]\times{\bf e}_{k}\right]\cdot{\bf
e}_{0}\right)^{2}+2\cos(\theta)\sin(\theta)\cos(\phi)({\bf e}_{k}\cdot{\bf
e}_{0})\left(\left[\left[{\bf e}_{k}\times{\bf e}_{0}\right]\times{\bf
e}_{k}\right]\cdot{\bf e}_{0}\right),$ (131)

where the third term in the right hand side will not survive the angle
integration.

$\displaystyle\cos^{2}(\theta)=\frac{1}{2kq}\left(Q^{2}-k^{2}-q^{2}\right),~{}\sin^{2}(\theta)=1-\cos^{2}(\theta).$
(132)
$\displaystyle\int_{0}^{2\pi}\frac{d\phi}{2\pi}\cos^{2}(\phi)=\frac{1}{2}.$
(133)

Then

$\displaystyle\int\frac{d\phi}{2\pi}\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)=1-\frac{m}{kq}\left(z-\xi_{k}-\frac{q^{2}}{2m}\right)({\bf
e}_{k}\cdot{\bf
e}_{0})^{2}-\frac{1}{2}\left[1-\frac{m}{kq}\left(z-\xi_{k}-\frac{q^{2}}{2m}\right)\right]\left(\left[\left[{\bf
e}_{k}\times{\bf e}_{0}\right]\times{\bf e}_{k}\right]\cdot{\bf
e}_{0}\right)^{2},$ (134)

where $z=\xi_{Q}$ as before. We pick only the momentum ($z$ and $q$)
independent terms from the expression for
$\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)$, and get for the imaginary part
of the self-energy

$\displaystyle\mathrm{Im}\Sigma^{\mathrm{R}/\mathrm{A}}(\epsilon;{\bf
k}_{\mathrm{F}})\approx$
$\displaystyle\mp\pi\frac{4P_{0}^{2}g^{2}}{V}\frac{m}{2k}\int\frac{qdq}{2\pi^{2}}A_{{\bf
q}}^{2}\left[1-\frac{1}{2}\left(\left[\left[{\bf e}_{k}\times{\bf
e}_{0}\right]\times{\bf e}_{k}\right]\cdot{\bf e}_{0}\right)^{2}\right]$ (135)
$\displaystyle\times\int_{\xi_{k-q}}^{\xi_{k+q}}dz\left\\{{\cal
F}^{\mathrm{F}}_{z}\left[\delta\left(\epsilon-z-\omega_{\bf
q}\right)-\delta\left(\epsilon-z+\omega_{\bf q}\right)\right]+{\cal
F}^{\mathrm{B}}_{\omega_{\bf q}}\left[\delta\left(\epsilon-z-\omega_{\bf
q}\right)+\delta\left(\epsilon-z+\omega_{\bf q}\right)\right]\right\\}.$ (136)

The imaginary part of the self-energy is then taken at the mass-shell, i.e.
$\epsilon=\xi_{\bf k}$,

$\displaystyle\mathrm{Im}\Sigma^{\mathrm{R}/\mathrm{A}}(\epsilon;{\bf
k}_{\mathrm{F}})\approx$
$\displaystyle\mp\pi\frac{4P_{0}^{2}g^{2}}{V}\frac{m}{2k_{\mathrm{F}}}\left[1-\frac{1}{2}\left(\left[\left[{\bf
e}_{k}\times{\bf e}_{0}\right]\times{\bf e}_{k}\right]\cdot{\bf
e}_{0}\right)^{2}\right]\int_{0}^{2k_{\mathrm{F}}}\frac{qdq}{2\pi^{2}}A_{{\bf
q}}^{2}\left({\cal F}^{\mathrm{F}}_{\epsilon-\omega_{\bf q}}-{\cal
F}^{\mathrm{F}}_{\epsilon+\omega_{\bf q}}+2{\cal F}^{\mathrm{B}}_{\omega_{\bf
q}}\right).$ (137)

The upper limit is due to the restriction imposed by the delta functions.
Compare this expression with the corresponding one in paragraph 21.3 in Ref.
[2]. The two expressions are analogous to each other. We conclude that the
advantage of the Keldysh technique is to skip possible complications occuring
in the course of analytic continuation. We simplify

$\displaystyle 1-\frac{1}{2}\left(\left[\left[{\bf e}_{k}\times{\bf
e}_{0}\right]\times{\bf e}_{k}\right]\cdot{\bf
e}_{0}\right)^{2}=1-\frac{1}{2}\sin^{2}(\phi_{{\bf k}{\bf P}_{0}}),$ (138)

where $\phi_{{\bf k}0}$ is an angle between ${\bf k}$ and ${\bf P}_{0}$.

Recalling that $\omega_{\bf q}=\sqrt{\omega_{\mathrm{TO}}^{2}+(sq)^{2}}$ and
assuming equilibrium distribution functions, we proceed in estimating the
integral. For $|\epsilon|<\omega_{\mathrm{TO}}$ we get

$\displaystyle\mathrm{Im}\Sigma^{\mathrm{R}/\mathrm{A}}(\epsilon;{\bf
k}_{\mathrm{F}})\approx$
$\displaystyle\pm\pi\frac{8P_{0}^{2}g^{2}}{2\pi^{2}V}\frac{m}{2k_{\mathrm{F}}}\frac{\Omega_{0}^{2}T}{2\pi
s^{2}}\left[1-\frac{1}{2}\sin^{2}(\phi_{{\bf k}{\bf P}_{0}})\right]$ (139)
$\displaystyle\times$
$\displaystyle\left\\{\ln\left[\frac{\cosh\left(\frac{\omega_{\mathrm{TO}}-\epsilon}{2T}\right)\cosh\left(\frac{\omega_{\mathrm{TO}}+\epsilon}{2T}\right)}{\sinh^{2}\left(\frac{\omega_{\mathrm{TO}}}{2T}\right)}\right]-\ln\left[\frac{\cosh\left(\frac{\sqrt{\omega^{2}_{\mathrm{TO}}+\omega^{2}_{\mathrm{BG}}}-\epsilon}{2T}\right)\cosh\left(\frac{\sqrt{\omega^{2}_{\mathrm{TO}}+\omega^{2}_{\mathrm{BG}}}+\epsilon}{2T}\right)}{\sinh^{2}\left(\frac{\sqrt{\omega^{2}_{\mathrm{TO}}+\omega^{2}_{\mathrm{BG}}}}{2T}\right)}\right]\right\\}$
(140)
$\displaystyle\equiv\pm\left(\frac{P_{0}^{2}g^{2}m\Omega_{0}^{2}\omega_{\mathrm{TO}}}{4\pi^{2}Vs^{2}k_{\mathrm{F}}}\right)\left[1-\frac{1}{2}\sin^{2}(\phi_{{\bf
k}{\bf P}_{0}})\right]\frac{2T}{\omega_{\mathrm{TO}}}f(\epsilon,T),$ (141)

where we have made a

$\displaystyle\int\frac{qdq}{\omega_{\bf q}}(..)=\frac{1}{s^{2}}\int
d\omega_{\bf q}(..)$ (142)

change of variable. We assume that
$\omega_{\mathrm{BG}}\gg\omega_{\mathrm{TO}}$. Here are three limiting
behavior of the defined above function for $\epsilon=\xi_{\bf k}=0$,

$\displaystyle f(0,T)\approx
4e^{-\frac{\omega_{\mathrm{TO}}}{T}},~{}~{}~{}~{}T<\omega_{\mathrm{TO}},$
(143) $\displaystyle f(0,T)\approx
2\ln\left(\frac{2T}{\omega_{\mathrm{TO}}}\right),~{}~{}~{}~{}\omega_{\mathrm{BG}}>T>\omega_{\mathrm{TO}},$
(144) $\displaystyle f(0,T)\approx
2\ln\left(\frac{\omega_{\mathrm{BG}}}{\omega_{\mathrm{TO}}}\right),~{}~{}~{}~{}T>\omega_{\mathrm{BG}},$
(145)

###  .5 Collision integral. Electric transport.

We anticipate that the resistivity will also have the same temperature
dependence as the inverse life-time. To explicitly check that, we need to
construct kinetic equation and solve for the distribution function when the
electric field is applied. Including only the imaginary part of the self-
energy, we write for the collision integral

$\displaystyle I_{\mathrm{CI}}(\epsilon;{\bf k})=$ $\displaystyle
i\Sigma^{\mathrm{K}}(\epsilon;{\bf k})-i\Sigma^{\mathrm{R}}(\epsilon;{\bf
k}){\cal F}_{\epsilon}^{\mathrm{F}}+i{\cal
F}_{\epsilon}^{\mathrm{F}}\Sigma^{\mathrm{A}}(\epsilon;{\bf
k})=-\mathrm{Im}\Sigma^{\mathrm{K}}(\epsilon;{\bf k})+2{\cal
F}_{\epsilon}^{\mathrm{F}}\mathrm{Im}\Sigma^{\mathrm{R}}(\epsilon;{\bf k})$
(146) $\displaystyle=-$ $\displaystyle 2\pi\frac{4P_{0}^{2}g^{2}}{V}\int_{\bf
q}\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf
q}}^{2}\left[\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}-\omega_{\bf
q}\right)-\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}+\omega_{\bf
q}\right)\right]\left({\cal F}^{\mathrm{F}}_{\xi_{{\bf k}+{\bf q}}}{\cal
F}^{\mathrm{F}}_{\epsilon}-1\right)$ (147) $\displaystyle-$ $\displaystyle
2\pi\frac{4P_{0}^{2}g^{2}}{V}\int_{\bf
q}\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf
q}}^{2}\left[\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}-\omega_{\bf
q}\right)+\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}+\omega_{\bf
q}\right)\right]{\cal F}_{\omega_{\bf q}}^{\mathrm{B}}\left({\cal
F}^{\mathrm{F}}_{\epsilon}-{\cal F}^{\mathrm{F}}_{\xi_{{\bf k}+{\bf
q}}}\right).$ (148)

It can be checked that the collision integral is zero when equilibrium
distribution functions with the same temperatures are plugged in. This can be
seen by applying the $\left({\cal F}^{\mathrm{F}}_{\xi_{{\bf k}+{\bf q}}}{\cal
F}^{\mathrm{F}}_{\epsilon}-1\right)=-{\cal F}_{\epsilon-\xi_{{\bf k}+{\bf
q}}}^{\mathrm{B}}\left({\cal F}^{\mathrm{F}}_{\epsilon}-{\cal
F}^{\mathrm{F}}_{\xi_{{\bf k}+{\bf q}}}\right)$ identity to the first term,
and then substituting $\epsilon-\xi_{{\bf k}+{\bf q}}=\pm\omega_{\bf q}$ there
to ${\cal F}_{\epsilon-\xi_{{\bf k}+{\bf q}}}^{\mathrm{B}}$ in accord with
corresponding delta-function.

The collision integral is no longer zero when, for example, the electric field
is applied to the system. In this case we search for the solution in the
following form

$\displaystyle{\cal F}^{\mathrm{F}}_{\epsilon}\approx{\cal
F}^{\mathrm{F}0}_{\epsilon}+{\cal F}^{\mathrm{F}1}_{\epsilon},$ (149)
$\displaystyle{\cal F}_{\omega_{\bf
q}}^{\mathrm{B}}=\coth\left(\frac{\omega_{\bf q}}{2T}\right),$ (150)

where ${\cal
F}^{\mathrm{F}0}_{\epsilon}=\tanh\left(\frac{\epsilon}{2T}\right)$ is electron
equilibrium distribution function, and to be found ${\cal
F}^{\mathrm{F}1}_{\epsilon}$ is proportional to the electric field. This
approximation assumes that both electrons and phonons are at the same
temperature.

$\displaystyle I_{\mathrm{CI}}(\epsilon;{\bf
k})=-2\pi\frac{4P_{0}^{2}g^{2}}{V}{\cal F}^{\mathrm{F}1}_{\xi_{\bf k}}$ (151)
$\displaystyle\times\int_{\bf
q}\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf q}}^{2}\left\\{{\cal
F}^{\mathrm{F}0}_{\xi_{{\bf k}+{\bf q}}}\left[\delta\left(\epsilon-\xi_{{\bf
k}+{\bf q}}-\omega_{\bf q}\right)-\delta\left(\epsilon-\xi_{{\bf k}+{\bf
q}}+\omega_{\bf q}\right)\right]+{\cal F}^{\mathrm{B}}_{\omega_{\bf
q}}\left[\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}-\omega_{\bf
q}\right)+\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}+\omega_{\bf
q}\right)\right]\right\\}$ (152) $\displaystyle-$ $\displaystyle
2\pi\frac{4P_{0}^{2}g^{2}}{V}\int_{\bf
q}\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf
q}}^{2}\left[\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}-\omega_{\bf
q}\right)-\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}+\omega_{\bf
q}\right)\right]{\cal F}^{\mathrm{F}0}_{\xi_{{\bf k}}}{\cal
F}^{\mathrm{F}1}_{\xi_{{\bf k}+{\bf q}}}$ (153) $\displaystyle+$
$\displaystyle 2\pi\frac{4P_{0}^{2}g^{2}}{V}\int_{\bf
q}\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf
q}}^{2}\left[\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}-\omega_{\bf
q}\right)+\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}+\omega_{\bf
q}\right)\right]{\cal F}^{\mathrm{B}}_{\omega_{\bf q}}{\cal
F}^{\mathrm{F}1}_{\xi_{{\bf k}+{\bf q}}}.$ (154)

If first integral is proportional to the already calculated imaginary self-
energy, the other two require some analysis. We approximate the collision
integral at the mass-shell $\epsilon=\xi_{\bf k}$ and set
$|k|=k_{\mathrm{F}}$, which gives $\xi_{\bf k}=0$. This approximation
essentially leaves the dependence of the collision integral only on the angle
between the momentum at the Fermi surface and the electric field ${\bf E}$.
Then in the first term out of the remaining two, ${\cal
F}^{\mathrm{F}0}_{\xi_{{\bf k}}}=0$, and the whole term drops out. In the
remaining we assume that we already know the solution, which is

$\displaystyle{\cal F}^{\mathrm{F}1}_{\xi_{{\bf k}+{\bf q}}}\propto\left[({\bf
k}+{\bf q})\cdot{\bf E}\right]\delta(\xi_{{\bf k}+{\bf q}}),$ (155)

then, setting $\epsilon=\xi_{\bf k}=0$, we get for the expression under the
integral

$\displaystyle\delta(\xi_{{\bf k}+{\bf
q}})\left[\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}-\omega_{\bf
q}\right)+\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}+\omega_{\bf
q}\right)\right]\rightarrow 2\delta(\xi_{{\bf k}+{\bf q}})\delta(\omega_{\bf
q}),$ (156)

which equals zero because $\omega_{\bf q}\neq 0$ away from the transition.
Finally,

$\displaystyle I_{\mathrm{CI}}(\phi_{{\bf
kE}})=-2\pi\frac{4P_{0}^{2}g^{2}}{V}{\cal F}^{\mathrm{F}1}_{\xi_{\bf k}}$
(157) $\displaystyle\times\int_{\bf
q}\left(1-\frac{q_{\alpha}^{2}}{q^{2}}\right)A_{{\bf q}}^{2}\left\\{{\cal
F}^{\mathrm{F}0}_{\xi_{{\bf k}+{\bf q}}}\left[\delta\left(\epsilon-\xi_{{\bf
k}+{\bf q}}-\omega_{\bf q}\right)-\delta\left(\epsilon-\xi_{{\bf k}+{\bf
q}}+\omega_{\bf q}\right)\right]+{\cal F}^{\mathrm{B}}_{\omega_{\bf
q}}\left[\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}-\omega_{\bf
q}\right)+\delta\left(\epsilon-\xi_{{\bf k}+{\bf q}}+\omega_{\bf
q}\right)\right]\right\\},$ (158)

where $\phi_{{\bf kE}}$ is an angle between ${\bf k}$ and ${\bf E}$. Let us
rewrite the collision integral in a more convenient form,

$\displaystyle I_{\mathrm{CI}}(\phi_{{\bf kE}})=-\frac{1}{\tau_{1{\bf
k}}}{\cal F}^{\mathrm{F}1}_{\xi_{\bf k}}.$ (159)

where we have defined

$\displaystyle\frac{1}{\tau_{1{\bf
k}}}=\frac{1}{\tau_{1}}\left[1+\cos^{2}(\phi_{{\bf k}{\bf P}_{0}})\right]$
(160)

Expression defining anisotropy of the electric current reads

$\displaystyle\frac{1}{k^{2}P_{0}^{2}}\int_{0}^{2\pi}\frac{d\phi}{2\pi}\int_{0}^{\pi}\sin(\theta)d\theta{\bf
k}\left({\bf k}\cdot{\bf E}\right)\left({\bf k}\cdot{\bf
P}_{0}\right)^{2}=\frac{2}{15}k^{2}\left[{\bf E}+2({\bf E}\cdot{\bf
P}_{0}){\bf P}_{0}\frac{1}{P_{0}^{2}}\right].$ (161)

Approximating the kinetic equation in the usual way, we obtain for the
electric current

$\displaystyle{\bf j}=e\int_{\bf k}{\bf v}_{\bf k}{\cal F}^{\mathrm{F}}_{\bf
k}$ $\displaystyle=\frac{2\mu e^{2}\tau_{1}\nu_{3\mathrm{D}}}{3m}{\bf
E}-\frac{2\mu e^{2}\tau_{1}\nu_{3\mathrm{D}}}{15m}\left[{\bf E}+2({\bf
E}\cdot{\bf P}_{0}){\bf P}_{0}\frac{1}{P_{0}^{2}}\right]$ (162)
$\displaystyle=\sigma_{1}{\bf E}-\frac{1}{5}\sigma_{1}\left[{\bf E}+2({\bf
E}\cdot{\bf P}_{0}){\bf P}_{0}\frac{1}{P_{0}^{2}}\right],$ (163)

where $\nu_{3\mathrm{D}}=\frac{mk_{\mathrm{F}}}{2\pi^{2}}$ is the density of
states, and $\sigma_{1}=e^{2}\nu_{3\mathrm{D}}D_{1}$ with
$D_{1}=\frac{1}{3}v_{\mathrm{F}}^{2}\tau_{1}$ is the conductivity due to the
one-phonon scattering processes. In deriving the expression for the current,
we have expanded in $\cos^{2}(\phi_{{\bf k}{\bf P}_{0}})$ as

$\displaystyle\frac{1}{1+\cos^{2}(\phi_{{\bf k}{\bf P}_{0}})}\approx
1-\cos^{2}(\phi_{{\bf k}{\bf P}_{0}})$ (164)

in order to estimate the integrals and extract the main anisotropy of the
electric current. To complete the analysis, we recall that there are other
scattering processes, for example impurity, two-phonon and other scattering
processes, in the system which change the collision integral to

$\displaystyle I_{\mathrm{CI}}(\phi_{{\bf
kE}})=-\left[\frac{1}{\tau_{\mathrm{imp}}}+\frac{1}{\tau_{1{\bf
k}}}+\frac{1}{\tau_{2}}\right]{\cal F}^{\mathrm{F}1}_{\xi_{\bf k}},$ (165)

where $\frac{1}{\tau_{\mathrm{imp}}}$ and $\frac{1}{\tau_{2}}$ are the inverse
scattering times due to the impurities and two-phonons. Both are isotropic in
momentum direction. The current will then be

$\displaystyle{\bf j}=\sigma_{\Sigma}{\bf
E}-\frac{1}{5}\sigma_{\Sigma}\frac{\tau_{\Sigma}}{\tau_{1}}\left[{\bf
E}+2({\bf E}\cdot{\bf P}_{0}){\bf P}_{0}\frac{1}{P_{0}^{2}}\right],$ (166)

where $\sigma_{\Sigma}=e^{2}\nu_{3\mathrm{D}}D_{\Sigma}$ with
$D_{1}=\frac{1}{3}v_{\mathrm{F}}^{2}\tau_{\Sigma}$ and

$\displaystyle\frac{1}{\tau_{\Sigma}}=\frac{1}{\tau_{\mathrm{imp}}}+\frac{1}{\tau_{1}}+\frac{1}{\tau_{2}}.$
(167)

For a general mutual in-plane alignmenent of the ${\bf E}$ and ${\bf P}_{0}$,
with an angle $\chi$ between them, the conductivity tensor is

$\displaystyle\hat{\sigma}_{\Sigma}=\left[\begin{array}[]{cc}\sigma_{xx}&\sigma_{xy}\\\
\sigma_{yx}&\sigma_{yy}\end{array}\right],$ (170)

where

$\displaystyle\sigma_{xx}=\sigma_{yy}=\sigma_{\Sigma}-\frac{1}{5}\sigma_{\Sigma}\frac{\tau_{\Sigma}}{\tau_{1}}[1+2\cos^{2}(\chi)],~{}~{}\sigma_{xy}=\sigma_{yx}=-\frac{2}{5}\sigma_{\Sigma}\frac{\tau_{\Sigma}}{\tau_{1}}\cos(\chi)\sin(\chi).$
(171)

Then

$\displaystyle\rho_{xx}=\rho_{yy}=\frac{\sigma_{xx}}{\sigma_{xx}^{2}-\sigma_{xy}^{2}},~{}~{}\rho_{xy}=\rho_{yx}=-\frac{\sigma_{xy}}{\sigma_{xx}^{2}-\sigma_{xy}^{2}}.$
(172)

We can claim that at small temperatures $\rho_{xy}\propto T$ given that
$\frac{1}{\tau_{\mathrm{imp}}}>\frac{1}{\tau_{1/2}}$ happening in a range of
temperatures $T>\omega_{\mathrm{TO}}$. This is the experimental signature of
the ferroelectric metal phase, and is one of the main results of the present
paper.

###  .6 Effective interaction between electrons: two-phonon interaction

If now the field $\theta$ is the composite field, such as $\theta={\bf P}{\bf
P}$, then we get for the Hamiltonian

$\displaystyle\left[\begin{array}[]{cc}\theta^{\mathrm{cl}}&\theta^{\mathrm{q}}\\\
\theta^{\mathrm{q}}&\theta^{\mathrm{cl}}\end{array}\right]=\frac{1}{4}\left[\begin{array}[]{cc}{\bf
P}^{\mathrm{cl}}{\bf P}^{\mathrm{cl}}+{\bf P}^{\mathrm{q}}{\bf
P}^{\mathrm{q}}&{\bf P}^{\mathrm{cl}}{\bf P}^{\mathrm{q}}+{\bf
P}^{\mathrm{q}}{\bf P}^{\mathrm{cl}}\\\ {\bf P}^{\mathrm{cl}}{\bf
P}^{\mathrm{q}}+{\bf P}^{\mathrm{q}}{\bf P}^{\mathrm{cl}}&{\bf
P}^{\mathrm{cl}}{\bf P}^{\mathrm{cl}}+{\bf P}^{\mathrm{q}}{\bf
P}^{\mathrm{q}}\end{array}\right].$ (177)

The interaction between electrons now read as

$\displaystyle iS_{\mathrm{int}}=$ $\displaystyle-\frac{g^{2}}{2}\int_{{\bf
r}_{1},{\bf
r}_{2}}\int_{t_{1},t_{2}}\left[\hat{\bar{\Psi}}(x_{1})\hat{\sigma}_{0}\hat{\Psi}(x_{1})\right]\left[\hat{\bar{\Psi}}(x_{2})\hat{\sigma}_{0}\hat{\Psi}(x_{2})\right]\langle\theta^{\mathrm{cl}}(x_{1})\theta^{\mathrm{cl}}(x_{2})\rangle$
(178) $\displaystyle-\frac{g^{2}}{2}\int_{{\bf r}_{1},{\bf
r}_{2}}\int_{t_{1},t_{2}}\left[\hat{\bar{\Psi}}(x_{1})\hat{\sigma}_{0}\hat{\Psi}(x_{1})\right]\left[\hat{\bar{\Psi}}(x_{2})\hat{\sigma}_{1}\hat{\Psi}(x_{2})\right]\langle\theta^{\mathrm{cl}}(x_{1})\theta^{\mathrm{q}}(x_{2})\rangle$
(179) $\displaystyle-\frac{g^{2}}{2}\int_{{\bf r}_{1},{\bf
r}_{2}}\int_{t_{1},t_{2}}\left[\hat{\bar{\Psi}}(x_{1})\hat{\sigma}_{1}\hat{\Psi}(x_{1})\right]\left[\hat{\bar{\Psi}}(x_{2})\hat{\sigma}_{0}\hat{\Psi}(x_{2})\right]\langle\theta^{\mathrm{q}}(x_{1})\theta^{\mathrm{cl}}(x_{2})\rangle$
(180) $\displaystyle-\frac{g^{2}}{2}\int_{{\bf r}_{1},{\bf
r}_{2}}\int_{t_{1},t_{2}}\left[\hat{\bar{\Psi}}(x_{1})\hat{\sigma}_{1}\hat{\Psi}(x_{1})\right]\left[\hat{\bar{\Psi}}(x_{2})\hat{\sigma}_{1}\hat{\Psi}(x_{2})\right]\langle\theta^{\mathrm{q}}(x_{1})\theta^{\mathrm{q}}(x_{2})\rangle,$
(181)

where we need to integrate over the phonon modes. Using the intermediate
results of the previous subsections, we calculate the first correlator to be

$\displaystyle\langle\theta^{\mathrm{cl}}(x_{1})\theta^{\mathrm{cl}}(x_{2})\rangle$
(182) $\displaystyle=-\frac{1}{8}\left(\frac{1}{V}\right)^{2}$
$\displaystyle\sum_{\epsilon,{\bf
q}}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)A_{{\bf
q}}^{2}e^{i{\bf q}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon(t_{1}-t_{2})}D^{\mathrm{K}}(\epsilon,{\bf
q})\sum_{\epsilon^{\prime},{\bf
q}^{\prime}}\left(\delta_{\alpha\beta}-\frac{q^{\prime}_{\alpha}q^{\prime}_{\beta}}{(q^{\prime})^{2}}\right)A_{{\bf
q}^{\prime}}^{2}e^{i{\bf q}^{\prime}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon^{\prime}(t_{1}-t_{2})}D^{\mathrm{K}}(\epsilon^{\prime},{\bf
q}^{\prime})$ (183) $\displaystyle-\frac{1}{8}\left(\frac{1}{V}\right)^{2}$
$\displaystyle\sum_{\epsilon,{\bf
q}}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)A_{{\bf
q}}^{2}e^{i{\bf q}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon(t_{1}-t_{2})}D^{\mathrm{R}}(\epsilon,{\bf
q})\sum_{\epsilon^{\prime},{\bf
q}^{\prime}}\left(\delta_{\alpha\beta}-\frac{q^{\prime}_{\alpha}q^{\prime}_{\beta}}{(q^{\prime})^{2}}\right)A_{{\bf
q}^{\prime}}^{2}e^{i{\bf q}^{\prime}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon^{\prime}(t_{1}-t_{2})}D^{\mathrm{R}}(\epsilon^{\prime},{\bf
q}^{\prime})$ (184) $\displaystyle-\frac{1}{8}\left(\frac{1}{V}\right)^{2}$
$\displaystyle\sum_{\epsilon,{\bf
q}}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)A_{{\bf
q}}^{2}e^{i{\bf q}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon(t_{1}-t_{2})}D^{\mathrm{A}}(\epsilon,{\bf
q})\sum_{\epsilon^{\prime},{\bf
q}^{\prime}}\left(\delta_{\alpha\beta}-\frac{q^{\prime}_{\alpha}q^{\prime}_{\beta}}{(q^{\prime})^{2}}\right)A_{{\bf
q}^{\prime}}^{2}e^{i{\bf q}^{\prime}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon^{\prime}(t_{1}-t_{2})}D^{\mathrm{A}}(\epsilon^{\prime},{\bf
q}^{\prime}).$ (185)

The other two read, which are related to each other by a $x_{1}\leftrightarrow
x_{2}$ replacement, as

$\displaystyle\langle\theta^{\mathrm{cl}}(x_{1})\theta^{\mathrm{q}}(x_{2})\rangle=-\frac{1}{4}\left(\frac{1}{V}\right)^{2}$
$\displaystyle\sum_{\epsilon,{\bf
q}}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)A_{{\bf
q}}^{2}e^{i{\bf q}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon(t_{1}-t_{2})}D^{\mathrm{K}}(\epsilon,{\bf q})$ (186)
$\displaystyle\times$ $\displaystyle\sum_{\epsilon^{\prime},{\bf
q}^{\prime}}\left(\delta_{\alpha\beta}-\frac{q^{\prime}_{\alpha}q^{\prime}_{\beta}}{(q^{\prime})^{2}}\right)A_{{\bf
q}^{\prime}}^{2}e^{i{\bf q}^{\prime}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon^{\prime}(t_{1}-t_{2})}D^{\mathrm{R}}(\epsilon^{\prime},{\bf
q}^{\prime}),$ (187)

and finally the third one is

$\displaystyle\langle\theta^{\mathrm{q}}(x_{1})\theta^{\mathrm{q}}(x_{2})\rangle=-\frac{1}{4}\left(\frac{1}{V}\right)^{2}$
$\displaystyle\sum_{\epsilon,{\bf
q}}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)A_{{\bf
q}}^{2}e^{i{\bf q}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon(t_{1}-t_{2})}D^{\mathrm{R}}(\epsilon,{\bf q})$ (188)
$\displaystyle\times$ $\displaystyle\sum_{\epsilon,{\bf
q}}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)A_{{\bf
q}^{\prime}}^{2}e^{i{\bf q}^{\prime}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon^{\prime}(t_{1}-t_{2})}D^{\mathrm{A}}(\epsilon^{\prime},{\bf
q}^{\prime}).$ (189)

We are now ready to outline the effective interaction between the electrons.
We do it term by term.

####  .6.1 First term

$\displaystyle iS_{\mathrm{int};1}=$ $\displaystyle-\frac{g^{2}}{2}\int_{{\bf
r}_{1},{\bf
r}_{2}}\int_{t_{1},t_{2}}\left[\hat{\bar{\Psi}}(x_{1})\hat{\sigma}_{0}\hat{\Psi}(x_{1})\right]\left[\hat{\bar{\Psi}}(x_{2})\hat{\sigma}_{0}\hat{\Psi}(x_{2})\right]\langle\theta^{\mathrm{cl}}(x_{1})\theta^{\mathrm{cl}}(x_{2})\rangle$
(190) $\displaystyle=$ $\displaystyle\frac{g^{2}}{8V^{2}}\int_{\\{{\bf
r}\\},\\{t\\}}\int_{\\{{\bf
k}\\},\\{\epsilon\\}}\left[\hat{\bar{\Psi}}(\epsilon_{1};{\bf
k}_{1})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{2};{\bf
k}_{2})\right]\left[\hat{\bar{\Psi}}(\epsilon_{3};{\bf
k}_{3})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{4};{\bf
k}_{4})\right]e^{i\epsilon_{1}t_{1}-i{\bf k}_{1}{\bf
r}_{1}}e^{-i\epsilon_{2}t_{1}+i{\bf k}_{2}{\bf
r}_{1}}e^{i\epsilon_{3}t_{2}-i{\bf k}_{3}{\bf
r}_{2}}e^{-i\epsilon_{4}t_{2}+i{\bf k}_{4}{\bf r}_{2}}$ (191)
$\displaystyle\times\bigg{\\{}\sum_{\alpha,\beta}\sum_{\epsilon,{\bf
q}}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)A_{{\bf
q}}^{2}e^{i{\bf q}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon(t_{1}-t_{2})}D^{\mathrm{K}}(\epsilon,{\bf
q})\sum_{\epsilon^{\prime},{\bf
q}^{\prime}}\left(\delta_{\alpha\beta}-\frac{q^{\prime}_{\alpha}q^{\prime}_{\beta}}{(q^{\prime})^{2}}\right)A_{{\bf
q}^{\prime}}^{2}e^{i{\bf q}^{\prime}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon^{\prime}(t_{1}-t_{2})}D^{\mathrm{K}}(\epsilon^{\prime},{\bf
q}^{\prime})$ (192) $\displaystyle+\sum_{\alpha,\beta}\sum_{\epsilon,{\bf
q}}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)A_{{\bf
q}}^{2}e^{i{\bf q}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon(t_{1}-t_{2})}D^{\mathrm{R}}(\epsilon,{\bf
q})\sum_{\epsilon^{\prime},{\bf
q}^{\prime}}\left(\delta_{\alpha\beta}-\frac{q^{\prime}_{\alpha}q^{\prime}_{\beta}}{(q^{\prime})^{2}}\right)A_{{\bf
q}^{\prime}}^{2}e^{i{\bf q}^{\prime}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon^{\prime}(t_{1}-t_{2})}D^{\mathrm{R}}(\epsilon^{\prime},{\bf
q}^{\prime})$ (193) $\displaystyle+\sum_{\alpha,\beta}\sum_{\epsilon,{\bf
q}}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)A_{{\bf
q}}^{2}e^{i{\bf q}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon(t_{1}-t_{2})}D^{\mathrm{A}}(\epsilon,{\bf
q})\sum_{\epsilon^{\prime},{\bf
q}^{\prime}}\left(\delta_{\alpha\beta}-\frac{q^{\prime}_{\alpha}q^{\prime}_{\beta}}{(q^{\prime})^{2}}\right)A_{{\bf
q}^{\prime}}^{2}e^{i{\bf q}^{\prime}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon^{\prime}(t_{1}-t_{2})}D^{\mathrm{A}}(\epsilon^{\prime},{\bf
q}^{\prime})\bigg{\\}}$ (194) $\displaystyle=$ $\displaystyle-i\int_{\\{{\bf
k}\\},\\{\epsilon\\}}\left[\hat{\bar{\Psi}}(\epsilon_{1};{\bf
k}_{1})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{2};{\bf
k}_{2})\right]\left[\hat{\bar{\Psi}}(\epsilon_{3};{\bf
k}_{3})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{4};{\bf
k}_{4})\right]\delta_{\epsilon_{1}-\epsilon_{2},\epsilon_{4}-\epsilon_{3}}\delta_{{\bf
k}_{1}-{\bf k}_{2},{\bf k}_{4}-{\bf k}_{3}}$ (195) $\displaystyle\times$
$\displaystyle i\frac{g^{2}}{8V^{2}}\int_{{\bf
q};\epsilon}\sum_{\alpha\beta}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)\left(\delta_{\alpha\beta}-\frac{({\bf
q}-{\bf Q})_{\alpha}({\bf q}-{\bf Q})_{\beta}}{({\bf q}-{\bf
Q})^{2}}\right)A_{{\bf q}}^{2}A_{{\bf q}-{\bf Q}}^{2}$ (196)
$\displaystyle\times$ $\displaystyle\left[D^{\mathrm{K}}(\epsilon,{\bf
q})D^{\mathrm{K}}(-\epsilon+\Omega,{\bf q}-{\bf
Q})+D^{\mathrm{R}}(\epsilon,{\bf q})D^{\mathrm{R}}(-\epsilon+\Omega,{\bf
q}-{\bf Q})+D^{\mathrm{A}}(\epsilon,{\bf
q})D^{\mathrm{A}}(-\epsilon+\Omega,{\bf q}-{\bf Q})\right]$ (197)
$\displaystyle\equiv-i\int_{\\{{\bf
k}\\},\\{\epsilon\\}}\left[\hat{\bar{\Psi}}(\epsilon_{1};{\bf
k}_{1})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{2};{\bf
k}_{2})\right]\left[\hat{\bar{\Psi}}(\epsilon_{3};{\bf
k}_{3})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{4};{\bf
k}_{4})\right]\delta_{\epsilon_{1}-\epsilon_{2},\epsilon_{4}-\epsilon_{3}}\delta_{{\bf
k}_{1}-{\bf k}_{2},{\bf k}_{4}-{\bf k}_{3}}U^{\mathrm{K}}(\Omega;{\bf Q}),$
(198)

where $\Omega=\epsilon_{1}-\epsilon_{2}$ and ${\bf Q}={\bf k}_{1}-{\bf k}_{2}$
are the frequency and momentum transfer. The integral over the frequency reads
as

$\displaystyle\int_{\epsilon}\left[D^{\mathrm{K}}(\epsilon,{\bf
q})D^{\mathrm{K}}(-\epsilon+\Omega,{\bf q}-{\bf
Q})+D^{\mathrm{R}}(\epsilon,{\bf q})D^{\mathrm{R}}(-\epsilon+\Omega,{\bf
q}-{\bf Q})+D^{\mathrm{A}}(\epsilon,{\bf
q})D^{\mathrm{A}}(-\epsilon+\Omega,{\bf q}-{\bf Q})\right]$ (199)
$\displaystyle=$ $\displaystyle-2\pi{\cal F}_{\mathrm{B}}(\omega_{{\bf
q}}){\cal F}_{\mathrm{B}}(\omega_{{\bf q}-{\bf Q}})\left[\delta(\omega_{{\bf
q}}-\omega_{{\bf q}-{\bf Q}}-\Omega)+\delta(\omega_{{\bf q}}-\omega_{{\bf
q}-{\bf Q}}+\Omega)\right]$ (200) $\displaystyle-2\pi{\cal
F}_{\mathrm{B}}(\omega_{{\bf q}}){\cal F}_{\mathrm{B}}(\omega_{{\bf q}-{\bf
Q}})\left[\delta(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}}-\Omega)+\delta(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf Q}}+\Omega)\right]$
(201) $\displaystyle+i\frac{2(\omega_{{\bf q}}-\omega_{{\bf q}-{\bf
Q}})}{(\Omega+i0)^{2}-(\omega_{{\bf q}}-\omega_{{\bf q}-{\bf
Q}})^{2}}+i\frac{2(\omega_{{\bf q}}-\omega_{{\bf q}-{\bf
Q}})}{(\Omega-i0)^{2}-(\omega_{{\bf q}}-\omega_{{\bf q}-{\bf Q}})^{2}}$ (202)
$\displaystyle+i\frac{2(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})}{(\Omega+i0)^{2}-(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})^{2}}+i\frac{2(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})}{(\Omega-i0)^{2}-(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf Q}})^{2}}.$ (203)

Therefore, we write for the interaction

$\displaystyle U^{\mathrm{K}}(\Omega;{\bf Q})=-\frac{g^{2}}{4V^{2}}$
$\displaystyle\int_{{\bf q}}\left(1+\frac{[{\bf q}({\bf q}-{\bf Q})]^{2}}{{\bf
q}^{2}({\bf q}-{\bf Q})^{2}}\right)A_{{\bf q}}^{2}A_{{\bf q}-{\bf Q}}^{2}$
(204) $\displaystyle\times\bigg{[}$ $\displaystyle-2\pi{\cal
F}_{\mathrm{B}}(\omega_{{\bf q}}){\cal F}_{\mathrm{B}}(\omega_{{\bf q}-{\bf
Q}})\left[\delta(\omega_{{\bf q}}-\omega_{{\bf q}-{\bf
Q}}-\Omega)+\delta(\omega_{{\bf q}}-\omega_{{\bf q}-{\bf Q}}+\Omega)\right]$
(205) $\displaystyle-2\pi{\cal F}_{\mathrm{B}}(\omega_{{\bf q}}){\cal
F}_{\mathrm{B}}(\omega_{{\bf q}-{\bf Q}})\left[\delta(\omega_{{\bf
q}}+\omega_{{\bf q}-{\bf Q}}-\Omega)+\delta(\omega_{{\bf q}}+\omega_{{\bf
q}-{\bf Q}}+\Omega)\right]$ (206) $\displaystyle+i\frac{2(\omega_{{\bf
q}}-\omega_{{\bf q}-{\bf Q}})}{(\Omega+i0)^{2}-(\omega_{{\bf q}}-\omega_{{\bf
q}-{\bf Q}})^{2}}+i\frac{2(\omega_{{\bf q}}-\omega_{{\bf q}-{\bf
Q}})}{(\Omega-i0)^{2}-(\omega_{{\bf q}}-\omega_{{\bf q}-{\bf Q}})^{2}}$ (207)
$\displaystyle+i\frac{2(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})}{(\Omega+i0)^{2}-(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})^{2}}+i\frac{2(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})}{(\Omega-i0)^{2}-(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})^{2}}\bigg{]}$ (208)

####  .6.2 Second term

$\displaystyle iS_{\mathrm{int};2}=$ $\displaystyle-\frac{g^{2}}{2}\int_{{\bf
r}_{1},{\bf
r}_{2}}\int_{t_{1},t_{2}}\left[\hat{\bar{\Psi}}(x_{1})\hat{\sigma}_{0}\hat{\Psi}(x_{1})\right]\left[\hat{\bar{\Psi}}(x_{2})\hat{\sigma}_{1}\hat{\Psi}(x_{2})\right]\langle\theta^{\mathrm{cl}}(x_{1})\theta^{\mathrm{q}}(x_{2})\rangle$
(209) $\displaystyle=$ $\displaystyle\frac{g^{2}}{8V^{2}}\int_{\\{{\bf
r}\\},\\{t\\}}\int_{\\{{\bf
k}\\},\\{\epsilon\\}}\left[\hat{\bar{\Psi}}(\epsilon_{1};{\bf
k}_{1})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{2};{\bf
k}_{2})\right]\left[\hat{\bar{\Psi}}(\epsilon_{3};{\bf
k}_{3})\hat{\sigma}_{1}\hat{\Psi}(\epsilon_{4};{\bf
k}_{4})\right]e^{i\epsilon_{1}t_{1}-i{\bf k}_{1}{\bf
r}_{1}}e^{-i\epsilon_{2}t_{1}+i{\bf k}_{2}{\bf
r}_{1}}e^{i\epsilon_{3}t_{2}-i{\bf k}_{3}{\bf
r}_{2}}e^{-i\epsilon_{4}t_{2}+i{\bf k}_{4}{\bf r}_{2}}$ (210)
$\displaystyle\times\sum_{\alpha,\beta}\sum_{\epsilon,{\bf
q}}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)A_{{\bf
q}}^{2}e^{i{\bf q}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon(t_{1}-t_{2})}D^{\mathrm{K}}(\epsilon,{\bf
q})\sum_{\epsilon^{\prime},{\bf
q}^{\prime}}\left(\delta_{\alpha\beta}-\frac{q^{\prime}_{\alpha}q^{\prime}_{\beta}}{(q^{\prime})^{2}}\right)A_{{\bf
q}^{\prime}}^{2}e^{i{\bf q}^{\prime}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon^{\prime}(t_{1}-t_{2})}D^{\mathrm{R}}(\epsilon^{\prime},{\bf
q}^{\prime})$ (211) $\displaystyle=$ $\displaystyle-i\int_{\\{{\bf
k}\\},\\{\epsilon\\}}\left[\hat{\bar{\Psi}}(\epsilon_{1};{\bf
k}_{1})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{2};{\bf
k}_{2})\right]\left[\hat{\bar{\Psi}}(\epsilon_{3};{\bf
k}_{3})\hat{\sigma}_{1}\hat{\Psi}(\epsilon_{4};{\bf
k}_{4})\right]\delta_{\epsilon_{1}-\epsilon_{2},\epsilon_{4}-\epsilon_{3}}\delta_{{\bf
k}_{1}-{\bf k}_{2},{\bf k}_{4}-{\bf k}_{3}}$ (212) $\displaystyle\times$
$\displaystyle i\frac{g^{2}}{8V^{2}}\int_{{\bf
q};\epsilon}\sum_{\alpha\beta}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)\left(\delta_{\alpha\beta}-\frac{({\bf
q}-{\bf Q})_{\alpha}({\bf q}-{\bf Q})_{\beta}}{({\bf q}-{\bf
Q})^{2}}\right)A_{{\bf q}}^{2}A_{{\bf q}-{\bf
Q}}^{2}D^{\mathrm{K}}(\epsilon,{\bf q})D^{\mathrm{R}}(-\epsilon+\Omega,{\bf
q}-{\bf Q})$ (213)

where ${\bf Q}={\bf k}_{1}-{\bf k}_{2}$ and $\Omega=\epsilon_{1}-\epsilon_{2}$
are the transferred momentum and frequency. Integration over frequency gives

$\displaystyle\int_{\epsilon}D^{\mathrm{K}}(\epsilon,q)D^{\mathrm{R}}(-\epsilon+\Omega,-{\bf
q}+{\bf Q})$ $\displaystyle=-i{\cal F}_{\mathrm{B}}(\omega_{{\bf
q}})\left[\frac{2\omega_{{\bf q}-{\bf Q}}}{(\Omega-\omega_{{\bf
q}}+i0)^{2}-\omega_{{\bf q}-{\bf Q}}^{2}}+\frac{2\omega_{{\bf q}-{\bf
Q}}}{(\Omega+\omega_{{\bf q}}+i0)^{2}-\omega_{{\bf q}-{\bf Q}}^{2}}\right]$
(214) $\displaystyle=i2{\cal F}_{\mathrm{B}}(\omega_{{\bf
q}})\left[\frac{(\omega_{\bf q}-\omega_{{\bf q}-{\bf
Q}})}{(\Omega+i0)^{2}-(\omega_{{\bf q}}-\omega_{{\bf q}-{\bf
Q}})^{2}}-\frac{(\omega_{\bf q}+\omega_{{\bf q}-{\bf
Q}})}{(\Omega+i0)^{2}-(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})^{2}}\right].$ (215)

We then get for the effective interaction

$\displaystyle iS_{\mathrm{int};2}=-i$ $\displaystyle\int_{\\{{\bf
k}\\},\\{\epsilon\\}}\left[\hat{\bar{\Psi}}(\epsilon_{1};{\bf
k}_{1})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{2};{\bf
k}_{2})\right]\left[\hat{\bar{\Psi}}(\epsilon_{3};{\bf
k}_{3})\hat{\sigma}_{1}\hat{\Psi}(\epsilon_{4};{\bf
k}_{4})\right]\delta_{\epsilon_{1}-\epsilon_{2},\epsilon_{4}-\epsilon_{3}}\delta_{{\bf
k}_{1}-{\bf k}_{2},{\bf k}_{4}-{\bf k}_{3}}$ (216) $\displaystyle\times$
$\displaystyle\left(-\frac{g^{2}}{4V^{2}}\right)\int_{{\bf
q}}\left(1+\frac{[{\bf q}({\bf q}-{\bf Q})]^{2}}{{\bf q}^{2}({\bf q}-{\bf
Q})^{2}}\right)A_{{\bf q}}^{2}A_{{\bf q}-{\bf Q}}^{2}{\cal
F}_{\mathrm{B}}(\omega_{{\bf q}})\left[\frac{(\omega_{{\bf q}}-\omega_{{\bf
q}-{\bf Q}})}{(\Omega+i0)^{2}-(\omega_{{\bf q}}-\omega_{{\bf q}-{\bf
Q}})^{2}}-\frac{(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})}{(\Omega+i0)^{2}-(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})^{2}}\right].$ (217)

From where we deduce

$\displaystyle U^{\mathrm{R}}(\Omega;{\bf Q})=-\frac{g^{2}}{4V^{2}}\int_{{\bf
q}}\left(1+\frac{[{\bf q}({\bf q}-{\bf Q})]^{2}}{{\bf q}^{2}({\bf q}-{\bf
Q})^{2}}\right)A_{{\bf q}}^{2}A_{{\bf q}-{\bf Q}}^{2}{\cal
F}_{\mathrm{B}}(\omega_{{\bf q}})\left[\frac{(\omega_{{\bf q}}-\omega_{{\bf
q}-{\bf Q}})}{(\Omega+i0)^{2}-(\omega_{{\bf q}}-\omega_{{\bf q}-{\bf
Q}})^{2}}-\frac{(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})}{(\Omega+i0)^{2}-(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})^{2}}\right].$ (218)

When $\Omega=0$ the sign of the interaction is determined from

$\displaystyle\frac{(\omega_{{\bf q}}-\omega_{{\bf q}-{\bf
Q}})}{(\Omega+i0)^{2}-(\omega_{{\bf q}}-\omega_{{\bf q}-{\bf
Q}})^{2}}-\frac{(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})}{(\Omega+i0)^{2}-(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})^{2}}=-\frac{1}{\omega_{{\bf q}}-\omega_{{\bf q}-{\bf
Q}}}+\frac{1}{\omega_{{\bf q}}+\omega_{{\bf q}-{\bf Q}}}.$ (219)

Both terms are attractive. If it is clear with the second term, the first is
simplified with the following transformations,

$\displaystyle\frac{{\cal F}_{\mathrm{B}}(\omega_{{\bf q}})}{\omega_{{\bf
q}}-\omega_{{\bf q}-{\bf Q}}}\rightarrow\frac{{\cal
F}_{\mathrm{B}}(\omega_{{\bf q}+{\bf Q}})}{\omega_{{\bf q}+{\bf
Q}}-\omega_{{\bf q}}}\rightarrow\frac{{\cal F}_{\mathrm{B}}(\omega_{{\bf
q}-{\bf Q}})}{\omega_{{\bf q}-{\bf Q}}-\omega_{{\bf q}}},$ (220)

which should be understood as transformations performed under the integral
over ${\bf q}$. Therefore, the first term can be rewritten as

$\displaystyle\frac{{\cal F}_{\mathrm{B}}(\omega_{{\bf q}})}{\omega_{{\bf
q}}-\omega_{{\bf q}-{\bf Q}}}\rightarrow\frac{1}{2}\frac{{\cal
F}_{\mathrm{B}}(\omega_{{\bf q}})-{\cal F}_{\mathrm{B}}(\omega_{{\bf q}-{\bf
Q}})}{\omega_{{\bf q}}-\omega_{{\bf q}-{\bf Q}}},$ (221)

which is also attractive.

####  .6.3 Third term

Third term is just an advanced version of already calculated second term,

$\displaystyle iS_{\mathrm{int};3}=-i$ $\displaystyle\int_{\\{{\bf
k}\\},\\{\epsilon\\}}\left[\hat{\bar{\Psi}}(\epsilon_{1};{\bf
k}_{1})\hat{\sigma}_{1}\hat{\Psi}(\epsilon_{2};{\bf
k}_{2})\right]\left[\hat{\bar{\Psi}}(\epsilon_{3};{\bf
k}_{3})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{4};{\bf
k}_{4})\right]\delta_{\epsilon_{1}-\epsilon_{2},\epsilon_{4}-\epsilon_{3}}\delta_{{\bf
k}_{1}-{\bf k}_{2},{\bf k}_{4}-{\bf k}_{3}}$ (222) $\displaystyle\times$
$\displaystyle\left(-\frac{g^{2}}{4V^{2}}\right)\int_{{\bf
q}}\left(1+\frac{[{\bf q}({\bf q}-{\bf Q})]^{2}}{{\bf q}^{2}({\bf q}-{\bf
Q})^{2}}\right)A_{{\bf q}}^{2}A_{{\bf q}-{\bf Q}}^{2}{\cal
F}^{\mathrm{B}}_{\omega_{{\bf q}}}\left[\frac{(\omega_{{\bf q}}-\omega_{{\bf
q}-{\bf Q}})}{(\Omega-i0)^{2}-(\omega_{{\bf q}}-\omega_{{\bf q}-{\bf
Q}})^{2}}-\frac{(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})}{(\Omega-i0)^{2}-(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})^{2}}\right].$ (223)

From where we deduce

$\displaystyle U^{\mathrm{A}}(\Omega;{\bf Q})=-\frac{g^{2}}{4V^{2}}\int_{{\bf
q}}\left(1+\frac{[{\bf q}({\bf q}-{\bf Q})]^{2}}{{\bf q}^{2}({\bf q}-{\bf
Q})^{2}}\right)A_{{\bf q}}^{2}A_{{\bf q}-{\bf Q}}^{2}{\cal
F}^{\mathrm{B}}_{\omega_{{\bf q}}}\left[\frac{(\omega_{{\bf q}}-\omega_{{\bf
q}-{\bf Q}})}{(\Omega-i0)^{2}-(\omega_{{\bf q}}-\omega_{{\bf q}-{\bf
Q}})^{2}}-\frac{(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})}{(\Omega-i0)^{2}-(\omega_{{\bf q}}+\omega_{{\bf q}-{\bf
Q}})^{2}}\right].$ (224)

####  .6.4 Fourth term

$\displaystyle iS_{\mathrm{int};4}=$ $\displaystyle-\frac{g^{2}}{2}\int_{{\bf
r}_{1},{\bf
r}_{2}}\int_{t_{1},t_{2}}\left[\hat{\bar{\Psi}}(x_{1})\hat{\sigma}_{1}\hat{\Psi}(x_{1})\right]\left[\hat{\bar{\Psi}}(x_{2})\hat{\sigma}_{1}\hat{\Psi}(x_{2})\right]\langle\theta^{\mathrm{cl}}(x_{1})\theta^{\mathrm{q}}(x_{2})\rangle$
(225) $\displaystyle=$ $\displaystyle\frac{g^{2}}{8V^{2}}\int_{\\{{\bf
r}\\},\\{t\\}}\int_{\\{{\bf
k}\\},\\{\epsilon\\}}\left[\hat{\bar{\Psi}}(\epsilon_{1};{\bf
k}_{1})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{2};{\bf
k}_{2})\right]\left[\hat{\bar{\Psi}}(\epsilon_{3};{\bf
k}_{3})\hat{\sigma}_{1}\hat{\Psi}(\epsilon_{4};{\bf
k}_{4})\right]e^{i\epsilon_{1}t_{1}-i{\bf k}_{1}{\bf
r}_{1}}e^{-i\epsilon_{2}t_{1}+i{\bf k}_{2}{\bf
r}_{1}}e^{i\epsilon_{3}t_{2}-i{\bf k}_{3}{\bf
r}_{2}}e^{-i\epsilon_{4}t_{2}+i{\bf k}_{4}{\bf r}_{2}}$ (226)
$\displaystyle\times$ $\displaystyle\sum_{\alpha,\beta}\sum_{\epsilon,{\bf
q}}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)A_{{\bf
q}}^{2}e^{i{\bf q}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon(t_{1}-t_{2})}D^{\mathrm{R}}(\epsilon,{\bf
q})\sum_{\epsilon^{\prime},{\bf
q}^{\prime}}\left(\delta_{\alpha\beta}-\frac{q^{\prime}_{\alpha}q^{\prime}_{\beta}}{(q^{\prime})^{2}}\right)A_{{\bf
q}^{\prime}}^{2}e^{i{\bf q}^{\prime}({\bf r}_{1}-{\bf
r}_{2})-i\epsilon^{\prime}(t_{1}-t_{2})}D^{\mathrm{A}}(\epsilon^{\prime},{\bf
q}^{\prime})$ (227) $\displaystyle=$ $\displaystyle-i\int_{\\{{\bf
k}\\},\\{\epsilon\\}}\left[\hat{\bar{\Psi}}(\epsilon_{1};{\bf
k}_{1})\hat{\sigma}_{0}\hat{\Psi}(\epsilon_{2};{\bf
k}_{2})\right]\left[\hat{\bar{\Psi}}(\epsilon_{3};{\bf
k}_{3})\hat{\sigma}_{1}\hat{\Psi}(\epsilon_{4};{\bf
k}_{4})\right]\delta_{\epsilon_{1}-\epsilon_{2},\epsilon_{4}-\epsilon_{3}}\delta_{{\bf
k}_{1}-{\bf k}_{2},{\bf k}_{4}-{\bf k}_{3}}$ (228) $\displaystyle\times$
$\displaystyle i\frac{g^{2}}{8V^{2}}\int_{{\bf
q};\epsilon}\sum_{\alpha\beta}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)\left(\delta_{\alpha\beta}-\frac{({\bf
q}-{\bf Q})_{\alpha}({\bf q}-{\bf Q})_{\beta}}{({\bf q}-{\bf
Q})^{2}}\right)A_{{\bf q}}^{2}A_{{\bf q}-{\bf
Q}}^{2}D^{\mathrm{R}}(\epsilon,{\bf q})D^{\mathrm{A}}(-\epsilon+\Omega,{\bf
q}-{\bf Q})$ (229)

The integral over the frequency vanishes,

$\displaystyle\int_{\epsilon}D^{\mathrm{R}}(\epsilon,{\bf
q})D^{\mathrm{A}}(-\epsilon+\Omega,{\bf q}-{\bf
Q})=\int_{\epsilon}D^{\mathrm{R}}(\epsilon,{\bf
q})D^{\mathrm{R}}(\epsilon-\Omega,{\bf q}-{\bf Q})=0.$ (230)

Therefore, this interaction vanishes as is expected.

###  .7 References in Supplemental Material

  1. 1.

A. Kamenev, Field theory of non-equilibrium systems (Cambridge, University
Press, 2012).

  2. 2.

A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinskii, Methods of Quantum Field
Theory in Statistical Physics (Dover Publications, New York, 1975)