# PHASE TRANSITION IN A PHONON GAS WITH PAIR CORRELATIONS

Yu.M. Poluektov<EMAIL_ADDRESS><EMAIL_ADDRESS>National Science Center “Kharkov Institute of Physics and Technology”, 61108
Kharkov, Ukraine

###### Abstract

The phase transition to the state of a phonon gas with pairwise correlations
of interacting phonons with opposite momenta is studied. A method for
describing such phonon systems within the framework of the self-consistent
field model is developed and their thermodynamic characteristics are
calculated. It is shown that a phonon gas with pair correlations can exist in
a state of unstable thermodynamic equilibrium. The possibility of experimental
observation of a solid in such a phase is discussed.
Key words: phonon, phase transition, phonon-phonon interaction, Debye model,
entropy, heat capacity

###### pacs:

05.30.Jp, 05.70.Fh, 63.20.-e, 63.20.K, 63.20.Ry, 65.40.-b, 67.80.-s

##  I Introduction

A quantum mechanical description of a system of interacting bosons by the
second quantization method was first realized by Bogolyubov R01 . He
established the important role of correlations of particles with opposite
momenta. Accounting for such pair correlations in Fermi systems later became
the basis for constructing the theory of superconductivity R02 ; R03 . The
role of pair correlations in Bose systems was studied in works R04 ; R05 ; R06
; R07 ; R08 ; R09 . Of undoubted interest is the theoretical study of the role
of similar correlations in a system of massless particles (phonons). This
problem is also relevant in connection with experimental studies of the
properties of quantum crystals R10 .

The purpose of this work is a theoretical study of the possibility of a phase
transition of a gas of phonons in a solid into a state with correlations of
pairs of phonons with opposite momenta. A general method for describing such
states within the framework of the self-consistent field model is developed.
It is shown that in the case of the existence of attraction between phonons,
at a certain critical temperature there becomes possible a second-order phase
transition to a state with correlations of phonon pairs, in which the symmetry
with respect to the phase transformation is broken. The thermodynamic
functions characterizing such a state are calculated: entropy, pressure,
energy, heat capacities, thermodynamic coefficients. It is shown that, in
contrast to the standard theory of second-order phase transitions, the
transition of a phonon gas to the asymmetric state is accompanied not by a
decrease, but by an increase in the free energy. As a consequence, the state
with pair correlations of phonons is a state of unstable thermodynamic
equilibrium. The possibility of experimental detection of such a state is
discussed.

##  II Hamiltonian of interacting phonons

We choose the Hamiltonian that takes into account the interaction of phonons
with opposite momenta in the form analogous to the Bardeen-Cooper-Schrieffer
(BCS) Hamiltonian in the theory of superconductivity R02

$\begin{array}[]{l}\displaystyle{H=\frac{1}{2}\sum_{k}\hbar\omega_{\bf
k}\Big{(}a_{\bf k}^{+}a_{\bf
k}+a_{\bf-k}a_{\bf-k}^{+}\Big{)}+\sum_{k,k^{\prime}}U_{{\bf k}{\bf
k}^{\prime}}a_{\bf k}^{+}a_{\bf-k}^{+}a_{\bf-k^{\prime}}a_{\bf
k^{\prime}},}\end{array}$ (1)

where $\omega_{\bf k}=\omega_{-\bf k}\equiv\omega_{k}=ck$, and the operators
$a_{\bf k}^{+},a_{\bf k}$ are subject to the usual commutation conditions for
Bose particles. For simplicity, we will use the scalar model, disregarding the
polarization of phonons, and assume that the phonon velocity $c$ is
independent of temperature. Accounting for polarization presents no
fundamental difficulties. The difference from the BCS theory is that the
Hamiltonian (1) describes a system of Bose particles, the number of which is
not fixed and is determined by temperature. We will study the phonon gas in
the model of a self-consistent field. The technique for studying systems of
interacting phonons in this approach with the fulfillment of all thermodynamic
relations was developed in works R11 ; R12 . To formulate the model of a self-
consistent field, we break the total Hamiltonian (1) into the sum of two terms

$\begin{array}[]{l}\displaystyle{H=H_{S}+H_{C},}\end{array}$ (2)

where

$\begin{array}[]{l}\displaystyle{H_{S}=\frac{1}{2}\sum_{k}\Big{[}\hbar\omega_{\bf
k}\Big{(}a_{\bf k}^{+}a_{\bf k}+a_{\bf-k}a_{\bf-k}^{+}\Big{)}+\Delta_{\bf
k}a_{\bf k}^{+}a_{-\bf k}^{+}+\Delta_{-\bf k}^{*}a_{-\bf k}a_{\bf
k}\Big{]}+E_{0},}\end{array}$ (3)
$\begin{array}[]{l}\displaystyle{H_{C}=\frac{1}{2}\sum_{k,k^{\prime}}U_{{\bf
k}{\bf k}^{\prime}}a_{\bf k}^{+}a_{-\bf k}^{+}a_{-\bf k^{\prime}}a_{\bf
k^{\prime}}-\frac{1}{2}\sum_{k}\Big{[}\Delta_{\bf k}a_{\bf k}^{+}a_{-\bf
k}^{+}+\Delta_{-\bf k}^{*}a_{-\bf k}a_{\bf k}\Big{]}-E_{0}.}\end{array}$ (4)

Let us note that, along with the phase-invariant operators $a_{\bf
k}^{+}a_{\bf k},\,a_{\bf-k}a_{\bf-k}^{+}$, the Hamiltonian (3) also contains
operators $a_{\bf k}^{+}a_{-\bf k}^{+},\,a_{-\bf k}a_{\bf k}$, leading to a
violation of the symmetry of the state with respect to the phase
transformation $a_{\bf k}\rightarrow e^{i\theta}a_{\bf k}$, $a_{\bf
k}^{+}\rightarrow e^{-i\theta}a_{\bf k}^{+}$, where $\theta$ is a real number.
Within the framework of the self-consistent field approximation, we will
describe the phonon system with the help of the approximating Hamiltonian
$H_{S}$, which includes so far unknown parameters $\Delta_{\bf k}=\Delta_{-\bf
k}$ and $E_{0}$ which will be determined later. The effects associated with
the existence of the correlation Hamiltonian (4) can be taken into account
using the perturbation theory R13 ; R14 , but we restrict ourselves to the
main approximation. The operators will be averaged using the statistical
operator

$\begin{array}[]{l}\displaystyle{\rho=\exp\beta\big{(}F-H_{S}\big{)},}\end{array}$
(5)

where $\beta=1/T$ is the inverse temperature and $F$ is the normalization
constant, the meaning of which will be clarified below. The average of an
arbitrary operator is $\big{\langle}A\big{\rangle}={\rm Sp}\rho A$, where the
trace is taken over all possible states in the self-consistent field
approximation.

The Hamiltonian (3) is quadratic in the phonon creation and annihilation
operators and, as a consequence, can be reduced to a diagonal form using the
canonical Bogolyubov transformation R01

$\begin{array}[]{l}\displaystyle{a_{\bf k}=u_{\bf k}\gamma_{\bf
k}+\upsilon^{*}_{\bf k}\gamma^{+}_{-\bf k},\qquad a_{\bf k}^{+}=u_{\bf
k}^{*}\gamma_{\bf k}^{+}+\upsilon_{\bf k}\gamma_{-\bf k},}\end{array}$ (6)

while the condition

$\begin{array}[]{l}\displaystyle{|u_{\bf k}|^{2}-|\upsilon_{\bf
k}|^{2}=1}\end{array}$ (7)

ensures the fulfillment of the correct commutation relationships for new
operators $\gamma_{\bf k}^{+},\gamma_{\bf k}$. In order for off-diagonal terms
to drop out in the Hamiltonian $H_{S}$, it is necessary the fulfillment of the
condition

$\begin{array}[]{l}\displaystyle{2\hbar\omega_{k}u_{\bf k}\upsilon_{\bf
k}+\Delta_{\bf k}^{*}u_{\bf k}^{2}+\Delta_{\bf k}\upsilon_{\bf
k}^{2}=0.}\end{array}$ (8)

This leads to the system of linear homogeneous equations

$\begin{array}[]{ccc}\displaystyle{\big{(}\hbar\omega_{k}-\varepsilon_{\bf
k}\big{)}\upsilon_{\bf k}+\Delta_{\bf k}^{*}u_{\bf k}=0,\vspace{3mm}}\\\
\displaystyle{\Delta_{\bf k}\upsilon_{\bf
k}+\big{(}\hbar\omega_{k}+\varepsilon_{\bf k}\big{)}u_{\bf k}=0,}\end{array}$
(9)

and therefore

$\begin{array}[]{ccc}\displaystyle{\varepsilon_{\bf
k}=\sqrt{\big{(}\hbar\omega_{k}\big{)}^{2}-\big{|}\Delta_{\bf
k}\big{|}^{2}}.}\end{array}$ (10)

Taking into account condition (7), we find

$\begin{array}[]{ccc}\displaystyle{\big{|}u_{\bf
k}\big{|}^{2}=\frac{1}{2}\left(\frac{\hbar\omega_{k}}{\varepsilon_{\bf
k}}+1\right),\qquad\big{|}\upsilon_{\bf
k}\big{|}^{2}=\frac{1}{2}\left(\frac{\hbar\omega_{k}}{\varepsilon_{\bf
k}}-1\right),\vspace{3mm}}\\\ \displaystyle{u_{\bf k}\upsilon_{\bf
k}^{*}=-\frac{\Delta_{\bf k}}{2\varepsilon_{\bf k}}.}\end{array}$ (11)

As a result, the Hamiltonian (3) takes the form of the Hamiltonian of free
phonons with energy $\varepsilon_{\bf k}$ (10):

$\begin{array}[]{ccc}\displaystyle{H_{S}=\sum_{k}\varepsilon_{\bf
k}\gamma_{\bf k}^{+}\gamma_{\bf k}+\frac{1}{2}\sum_{k}\varepsilon_{\bf
k}+E_{0}.}\end{array}$ (12)

Here, the second term describes zero oscillations, and the quantity $E_{0}$
characterizes the shift in the energy of the ground state upon transition to a
description in terms of new operators. The phonons described by the operators
$a_{\bf k},a_{\bf k}^{+}$ will be called “bare” phonons. The phonons described
by the operators $\gamma_{\bf k},\gamma_{\bf k}^{+}$ will be called “dressed”
phonons. The distribution function of the “dressed” phonons has the form of
the Planck function

$\begin{array}[]{ccc}\displaystyle{f_{\bf k}\equiv\big{\langle}\gamma_{\bf
k}^{+}\gamma_{\bf k}\big{\rangle}=\frac{1}{e^{\beta\varepsilon_{\bf
k}}-1},}\end{array}$ (13)

and their anomalous averages are equal to zero: $\big{\langle}\gamma_{\bf
k}^{+}\gamma_{-\bf k}^{+}\big{\rangle}=\big{\langle}\gamma_{-\bf k}\gamma_{\bf
k}\big{\rangle}=0$. The total number of the “dressed” phonons is

$\begin{array}[]{ccc}\displaystyle{N_{ph}=\sum_{k}\frac{1}{e^{\beta\varepsilon_{\bf
k}}-1}.}\end{array}$ (14)

The pair averages for the “bare” phonons are given by the relations:

$\begin{array}[]{ccc}\displaystyle{\big{\langle}a_{\bf k}^{+}a_{-\bf
k}^{+}\big{\rangle}=-\frac{\Delta_{\bf k}^{*}}{2\varepsilon_{\bf
k}}\big{(}1+2f_{\bf k}\big{)},\qquad\big{\langle}a_{-\bf k}a_{\bf
k}\big{\rangle}=-\frac{\Delta_{\bf k}}{2\varepsilon_{\bf k}}\big{(}1+2f_{\bf
k}\big{)},\vspace{3mm}}\\\ \displaystyle{\big{\langle}a_{\bf k}^{+}a_{\bf
k}\big{\rangle}=\frac{\hbar\omega_{k}}{2\varepsilon_{\bf k}}\big{(}1+2f_{\bf
k}\big{)}-\frac{1}{2}.}\end{array}$ (15)

The total number of the “bare” phonons is determined by the formula

$\begin{array}[]{ccc}\displaystyle{N_{ph}^{(0)}=\sum_{{\bf
k}}\big{\langle}a_{\bf k}^{+}a_{\bf k}\big{\rangle}=\frac{1}{2}\sum_{{\bf
k}}\bigg{[}\frac{\hbar\omega_{k}}{\varepsilon_{\bf k}}\big{(}1+2f_{\bf
k}\big{)}-1\bigg{]}.}\end{array}$ (16)

At $T\rightarrow 0$, the total number of the “dressed” phonons $N_{ph}$ tends
to zero, while the total number of the “bare” phonons $N_{ph}^{(0)}$ remains
finite. The temperature dependences of the phonon numbers are shown in Fig. 4.

As we can see, along with the phase-invariant “normal” average
$\big{\langle}a_{\bf k}^{+}a_{\bf k}\big{\rangle}$, there are also phase-
noninvariant “anomalous” averages $\big{\langle}a_{\bf k}^{+}a_{-\bf
k}^{+}\big{\rangle}$, $\big{\langle}a_{-\bf k}a_{\bf k}\big{\rangle}$, which
under phase transformations of operators are transformed as follows:
$\big{\langle}a_{-\bf k}a_{\bf k}\big{\rangle}\rightarrow
e^{2i\theta}\big{\langle}a_{-\bf k}a_{\bf k}\big{\rangle}$,
$\big{\langle}a_{\bf k}^{+}a_{-\bf k}^{+}\big{\rangle}\rightarrow
e^{-2i\theta}\big{\langle}a_{\bf k}^{+}a_{-\bf k}^{+}\big{\rangle}$. With this
in mind, it is obvious that the operator of the observed energy quantity (3)
does not depend on the choice of phase and on the whole remains invariant with
respect to phase transformations. The next task is to study the state of the
phonon gas, in which the “anomalous” averages that violate the phase symmetry
are nonzero.

##  III Parameters of the self-consistent Hamiltonian

Let us move on to finding the parameters $\Delta_{\bf k}$ and $E_{0}$, which
enter into the approximating Hamiltonian (3). The normalization condition of
the statistical operator (5) ${\rm Sp}\rho=1$ determines the normalization
constant

$\begin{array}[]{ccc}\displaystyle{F=-T\ln{\rm Sp}e^{-\beta
H_{S}}.}\end{array}$ (17)

Taking into account the definition of entropy $S=-{\rm Sp}\rho\ln\rho$, we
find

$\begin{array}[]{ccc}\displaystyle{F=\big{\langle}H_{S}\rangle-
TS,}\end{array}$ (18)

whence it follows that the quantity $F$ has the meaning of free energy in the
self-consistent field model. The parameter $E_{0}$ is determined from the
requirement that the approximating Hamiltonian (3) be as close as possible to
the initial Hamiltonian (1), which can be formulated as
$\big{\langle}(H-H_{S})\big{\rangle}=0$ or $\big{\langle}H_{C}\rangle=0$,
which gives

$\begin{array}[]{ccc}\displaystyle{E_{0}=\sum_{k,k^{\prime}}U_{{\bf k}{\bf
k}^{\prime}}\big{\langle}a_{\bf k}^{+}a_{-\bf
k}^{+}a_{-\mathbf{k}^{\prime}}a_{\mathbf{k}^{\prime}}\big{\rangle}-\frac{1}{2}\sum_{k}\Big{[}\Delta_{\bf
k}\big{\langle}a_{\bf k}^{+}a_{-\bf k}^{+}\big{\rangle}+\Delta_{-\bf
k}^{*}\big{\langle}a_{-\bf k}a_{\bf k}\big{\rangle}\Big{]}.}\end{array}$ (19)

The average of four operators is expressed in terms of the products of the
averages of pairs of operators

$\begin{array}[]{ccc}\displaystyle{\big{\langle}a_{\bf k}^{+}a_{-\bf
k}^{+}a_{-\mathbf{k}^{\prime}}a_{\mathbf{k}^{\prime}}\big{\rangle}=\big{\langle}a_{\bf
k}^{+}a_{-\bf
k}^{+}\big{\rangle}\big{\langle}a_{-\mathbf{k}^{\prime}}a_{\mathbf{k}^{\prime}}\big{\rangle}+\big{\langle}a_{\bf
k}^{+}a_{-\mathbf{k}^{\prime}}\big{\rangle}\big{\langle}a_{\bf
k}^{+}a_{-\mathbf{k}^{\prime}}\big{\rangle}+\big{\langle}a_{\bf
k}^{+}a_{\mathbf{k}^{\prime}}\big{\rangle}\big{\langle}a_{-\bf
k}^{+}a_{-\mathbf{k}^{\prime}}\big{\rangle}.}\end{array}$ (20)

Since we are interested in the effects associated with the existence of
anomalous averages in the chosen model, we will take into account in expansion
(20) the contribution of only the first term. Then, taking into account (15),
we obtain

$\begin{array}[]{ccc}\displaystyle{E_{0}=\frac{1}{4}\sum_{k,k^{\prime}}U_{{\bf
k}{\bf k}^{\prime}}\frac{\Delta_{\bf k}^{*}\Delta_{\bf
k^{\prime}}}{\varepsilon_{\bf k}\varepsilon_{\bf k^{\prime}}}\big{(}1+2f_{\bf
k}\big{)}\big{(}1+2f_{\bf
k^{\prime}}\big{)}+\frac{1}{2}\sum_{k}\frac{\big{|}\Delta_{\bf
k}\big{|}^{2}}{\varepsilon_{\bf k}}\big{(}1+2f_{\bf k}\big{)}.}\end{array}$
(21)

Taking into account the form of the Hamiltonian (12), we find the formula for
the free energy

$\begin{array}[]{ccc}\displaystyle{F=E_{0}+\frac{1}{2}\sum_{k}\varepsilon_{\bf
k}-T\ln{\rm Sp}\,e^{-\beta\sum_{k}\varepsilon_{\bf k}\gamma_{\bf
k}^{+}\gamma_{\bf k}}.}\end{array}$ (22)

Here, the last term describes the free energy of an ideal gas of “dressed”
phonons:

$\begin{array}[]{ccc}\displaystyle{F_{0}=-T\ln{\rm
Sp}\,e^{-\beta\sum_{k}\varepsilon_{\bf k}\gamma_{\bf k}^{+}\gamma_{\bf
k}}=T\sum_{k}\ln\big{(}1-e^{-\beta\varepsilon_{\bf k}}\big{)}.}\end{array}$
(23)

Comparing formulas (12), (13), (18) and taking into account the definition of
entropy, we find for entropy a natural expression in terms of the distribution
function of “dressed” phonons:

$\begin{array}[]{ccc}\displaystyle{S=\sum_{k}\Big{[}\big{(}1+f_{\bf
k}\big{)}\ln\big{(}1+f_{\bf k}\big{)}-f_{\bf k}\ln f_{\bf
k}\Big{]}.}\end{array}$ (24)

Since in the limit $T\rightarrow 0$, the distribution function (13) tends to
zero, then the entropy, as required, also tends to zero.

Since the dispersion law for “bare” phonons $\omega_{k}=ck$ is linear in the
wave number $k$, then it is natural to assume, and this is justified by the
result, that the same dependence is also preserved for the parameter
$\Delta_{\bf k}$, so we set

$\begin{array}[]{ccc}\displaystyle{\Delta_{\bf k}=\sigma\hbar
ck=\sigma\hbar\omega_{k},}\end{array}$ (25)

where $\sigma$ has the meaning of the order parameter in the asymmetric state,
which does not depend on the wave number of the photon. Then it turns out that
the energy of the “dressed” photon (10)

$\begin{array}[]{ccc}\displaystyle{\varepsilon_{\bf
k}=\xi\\!\cdot\\!\hbar\omega_{k}=\hbar\omega_{k}\sqrt{1-|\sigma|^{2}},\qquad\xi\equiv\sqrt{1-|\sigma|^{2}}}\end{array}$
(26)

decreases with increasing $|\sigma|$. In this case, formulas (11) will take
the form

$\begin{array}[]{ccc}\displaystyle{\big{|}u_{\bf
k}\big{|}^{2}\equiv\big{|}u\big{|}^{2}=\frac{1}{2}\left(\frac{1}{\xi}+1\right),\qquad\big{|}\upsilon_{\bf
k}\big{|}^{2}\equiv\big{|}\upsilon\big{|}^{2}=\frac{1}{2}\left(\frac{1}{\xi}-1\right),\vspace{3mm}}\\\
\displaystyle{u_{\bf k}\upsilon_{\bf k}^{*}\equiv
u\upsilon^{*}=-\frac{\sigma}{2\xi}.}\end{array}$ (27)

We will describe the interaction between phonons with the help of one constant
$g_{0}$ , assuming

$\begin{array}[]{ccc}\displaystyle{U_{{\bf k}{\bf
k}^{\prime}}=\frac{g_{0}}{V},}\end{array}$ (28)

where $V$ is the volume. We will not specify the nature of the interaction
between phonons, assuming that the effective interaction between them can be
caused by nonlinear effects. Then the energy of the ground state (21)

$\begin{array}[]{ccc}\displaystyle{E_{0}=\frac{g_{0}}{4V}\frac{|\sigma|^{2}}{\xi^{2}}\bigg{(}\sum_{k}\big{(}1+2f_{\bf
k}\big{)}\bigg{)}^{\\!2}+\frac{\hbar
c}{2}\frac{|\sigma|^{2}}{\xi}\sum_{k}k\big{(}1+2f_{\bf k}\big{)},}\end{array}$
(29)

and free energy (22) can be written in the form

$\begin{array}[]{ccc}\displaystyle{F=\frac{g_{0}}{4V}\frac{|\sigma|^{2}}{\xi^{2}}I_{2}^{2}+\frac{\hbar
c}{2}\frac{|\sigma|^{2}}{\xi}I_{3}+\frac{\hbar
c}{2}\,\xi\\!\cdot\\!I_{1}-T\\!\cdot\\!I_{4}.}\end{array}$ (30)

Here $T$ is temperature, and

$\begin{array}[]{ccc}\displaystyle{I_{0}\equiv\sum_{k}1,\quad
I_{1}\equiv\sum_{k}k,\quad I_{2}\equiv\sum_{k}\big{(}1+2f_{\bf k}\big{)},\quad
I_{3}\equiv\sum_{k}k\big{(}1+2f_{\bf k}\big{)},\quad
I_{4}\equiv\sum_{k}\ln\\!\big{(}1+f_{\bf k}\big{)}.}\end{array}$ (31)

These quantities can be calculated using the transition in (31) from summation
to integration. Integration is carried out from zero to the Debye wave number
$k_{D}$, which is determined by the density of number of particles in a solid

$\begin{array}[]{ccc}\displaystyle{n\equiv\frac{N}{V}=\frac{k_{D}^{3}}{6\pi^{2}}.}\end{array}$
(32)

The calculation results are expressed in terms of the Debye functions

$\begin{array}[]{ccc}\displaystyle{D_{n}(x)=\frac{n}{x^{n}}\int_{0}^{x}\frac{z^{n}dz}{e^{z}-1},\qquad(n\geq
1),}\end{array}$ (33)

and related to them functions

$\begin{array}[]{ccc}\displaystyle{\Phi_{n}(x)=1+\frac{2(1+n)}{n}\frac{D_{n}(x)}{x}.}\end{array}$
(34)

Some properties of these functions are given in Appendix A. The calculation
gives:

$\begin{array}[]{ccc}\displaystyle{I_{0}=\frac{Vk_{D}^{3}}{6\pi^{2}}\equiv
N,\quad I_{1}=\frac{3}{4}Nk_{D},\quad
I_{2}=N\Phi_{2}\bigg{(}\frac{\xi}{\tau}\bigg{)},\quad
I_{3}=\frac{3}{4}Nk_{D}\Phi_{3}\bigg{(}\frac{\xi}{\tau}\bigg{)},\vspace{3mm}}\\\
\displaystyle{I_{4}=\frac{N}{3}\bigg{[}D_{3}\bigg{(}\frac{\xi}{\tau}\bigg{)}-3\ln\\!\Big{(}1-e^{-\frac{\xi}{\tau}}\Big{)}\bigg{]}.}\end{array}$
(35)

Here, the reduced dimensionless temperature is defined

$\begin{array}[]{ccc}\displaystyle{\tau\equiv\frac{T}{\Theta_{D}},}\end{array}$
(36)

where $\Theta_{D}\equiv\hbar ck_{D}$ is Debye energy or Debye temperature in
energy units. In what follows, we will also use the notation
$\overline{\tau}\equiv 1/\tau$ for the reciprocal reduced temperature. Taking
into account relations (34), (35), the free energy for the state with pair
correlations takes the form

$\begin{array}[]{ccc}\displaystyle{\frac{F}{\Theta_{D}N}\equiv
f(\xi,\tau)=\frac{\xi}{2}+\frac{3}{16}g\big{(}\xi^{-2}-1\big{)}\Phi_{2}^{2}\big{(}\xi\overline{\tau}\big{)}+\frac{3}{8}\Big{(}\xi^{-1}-\frac{4}{3}\xi\Big{)}\Phi_{3}\big{(}\xi\overline{\tau}\big{)}+\tau\ln\\!\big{(}1-e^{-\xi\overline{\tau}}\big{)},}\end{array}$
(37)

where the notation for the dimensionless interaction constant of phonons is
used

$\begin{array}[]{ccc}\displaystyle{g=\frac{4}{3}\frac{g_{0}n}{\Theta_{D}}.}\end{array}$
(38)

Thermodynamic potentials, which, along with equilibrium values, additionally
include arbitrary parameters, in our case it is $\sigma$ or
$\xi\equiv\sqrt{1-|\sigma|^{2}}$, are potentials with incomplete thermodynamic
equilibrium. The equilibrium value of the parameter is found from the
condition of the extremum of such a thermodynamic potential

$\begin{array}[]{ccc}\displaystyle{\frac{\partial
F}{\partial|\sigma|}=-\frac{\partial
F}{\partial\xi}\frac{|\sigma|}{\xi}=0.}\end{array}$ (39)

This equation has two solutions. The first solution $\sigma=0$ corresponds to
the normal phase, for which the Debye theory is valid and the free energy is
given by

$\begin{array}[]{ccc}\displaystyle{\frac{F_{D}}{\Theta_{D}N}\equiv
f_{D}(\tau)=\frac{1}{2}-\frac{1}{8}\Phi_{3}\big{(}\overline{\tau}\big{)}+\tau\ln\\!\big{(}1-e^{-\overline{\tau}}\big{)}.}\end{array}$
(40)

We will also refer to this normal state as the Debye phase. The second
solution, where $\partial F/\partial\xi=0$ and $\sigma\neq 0$, corresponds to
the phase with broken phase symmetry, in which the correlations of phonons
with opposite momenta are nonzero. The condition $\partial
F/\partial\xi=\partial f/\partial\xi=0$ leads to the equation

$\begin{array}[]{ccc}\displaystyle{\frac{3}{4}\,g\bigg{[}\big{(}\xi^{-2}-1\big{)}\,{\rm
cth}\frac{\xi}{2\tau}-\\!\Big{(}\,\frac{4}{3}\,\xi^{-2}-1\Big{)}\Phi_{2}\big{(}\xi\overline{\tau}\big{)}\bigg{]}\Phi_{2}\big{(}\xi\overline{\tau}\big{)}+\xi\bigg{[}\big{(}\xi^{-2}-1\big{)}\,{\rm
cth}\frac{\xi}{2\tau}-\\!\Big{(}\,\frac{5}{4}\,\xi^{-2}-1\Big{)}\Phi_{3}\big{(}\xi\overline{\tau}\big{)}\bigg{]}=0.}\end{array}$
(41)

This equation together with formula (37) in parametric form determines the
temperature dependence of the equilibrium free energy in the asymmetric phase.
The order parameter can vary within limits $\xi_{0}\leq\xi\leq 1$. Equation
(41) has a solution only for a negative interaction constant $g=-|g|$, such
that $|g|<1$. At zero temperature, as follows from (41), we have
$\xi_{0}=|g|$. The transition from the asymmetric phase, where $\sigma\neq 0$,
to the symmetric phase occurs when the order parameter vanishes, $\sigma=0$ or
$\xi=1$. Thus, the phase transition temperature $\tau_{c}=T_{c}/\Theta_{D}$ is
determined by the equation

$\begin{array}[]{ccc}\displaystyle{|g|\Phi_{2}^{2}(\overline{\tau}_{c}\big{)}-\Phi_{3}(\overline{\tau}_{c}\big{)}=0,}\end{array}$
(42)

where $\overline{\tau}_{c}=1/\tau_{c}$. The dependence of the critical
temperature $\tau_{c}$ on the modulus of the interaction constant $|g|$ is
shown in Figure 1. The unusual character of this dependence is noteworthy. As
we can see, the critical temperature increases with decreasing the modulus of
the interaction constant, and when $|g|=1$ the critical temperature turns to
zero. It would seem that, by decreasing the interaction we can thereby
increase the phase transition temperature, but, as will be shown below, with
decreasing $|g|$ the stability region of the asymmetric phase narrows. It
should be emphasized that the states of free phonons and systems of phonons
with pair correlations are qualitatively different due to their different
symmetries. Since the dependence of thermodynamic quantities on the
interaction constant in the asymmetric phase is non-analytical, it is
impossible to pass to the limit $|g|\rightarrow 0$. Therefore, the asymmetric
phase does not pass into the symmetric phase when the interaction is weakened.
This situation is typical for all systems with broken phase symmetry R15 .

Figure 1:  Dependence of the critical temperature $\tau_{c}=T_{c}/\Theta_{D}$
on the interaction constant $|g|$.

Formulas (37) and (41) make it possible to construct the temperature
dependence of the equilibrium free energy, which is shown in Fig. 2. This
figure also shows the temperature dependence (40) of the free energy of the
Debye phase.

Figure 2:  The free energies in the asymmetric (1) and symmetric (2) states at
$|g|=0.6$.

For the asymmetric phase to be thermodynamically stable, the extremum of the
free energy must be a minimum. In present case, this condition is not
satisfied at all temperatures, since $\partial^{2}f\big{/}\partial\xi^{2}<0$
and the free energy of the asymmetric phase is always greater than the free
energy of the symmetric phase. Thus, the state with pair correlations of
phonons is a state with unstable thermodynamic equilibrium. The possibility of
experimental observation of such a state will be discussed below.

##  IV Thermodynamic quantities

Although the equilibrium state of phonons with pair correlations corresponds
not to the minimum, but to the maximum of the free energy, and therefore is a
state with an unstable equilibrium, nevertheless, let us analyze the
thermodynamic properties of such a state. Having expressions (37), (40) for
the equilibrium free energy, we can calculate all thermodynamic quantities in
the asymmetric phase. We first write down the free energy differential

$\begin{array}[]{ccc}\displaystyle{dF=\bigg{(}\frac{\partial
F}{\partial\xi}\bigg{)}_{\\!T,V}d\xi+\bigg{(}\frac{\partial F}{\partial
T}\bigg{)}_{\\!V,\xi}dT+\bigg{(}\frac{\partial F}{\partial
V}\bigg{)}_{\\!T,\xi}dV.}\end{array}$ (43)

When condition (41) is satisfied, the first term in (43) is equal to zero.
Taking into account the general thermodynamic identities $dF=-SdT-pdV$ and
$E=F+TS$, we find that the entropy, pressure and energy in the self-consistent
field model are determined by the usual formulas

$\begin{array}[]{ccc}\displaystyle{S=-\bigg{(}\frac{\partial F}{\partial
T}\bigg{)}_{\\!V,\xi},\qquad p=-\bigg{(}\frac{\partial F}{\partial
V}\bigg{)}_{\\!T,\xi},\qquad E=F-T\bigg{(}\frac{\partial F}{\partial
T}\bigg{)}_{\\!V,\xi}.}\end{array}$ (44)

It should be emphasized that, due to (39), here it is not necessary to
differentiate with respect to the parameter $\xi$. Since we assumed that the
speed of “bare” phonons does not depend on temperature, then in this
approximation the Debye energy is a function of only density or, at a constant
number of particles in a solid $N$, a function of only volume
$\Theta_{D}\equiv\Theta_{D}(V)$. To characterize this dependence, the
Grüneisen parameter is introduced

$\begin{array}[]{ccc}\displaystyle{\Gamma\equiv-\frac{V}{\Theta_{D}}\frac{\partial\Theta_{D}}{\partial
V}=\frac{n}{\Theta_{D}}\frac{\partial\Theta_{D}}{\partial
n}=\frac{1}{3}+\frac{n}{c}\frac{dc}{dn}.}\end{array}$ (45)

If we neglect the dependence of the phonon velocity on density, then
$\Gamma=1/3$. If the phonon velocity depends on density in a power-law manner
$c\sim n^{k}$, then the Grüneisen parameter in this case is also just a number
$\Gamma=1/3+k$. In the following, we assume the Grüneisen parameter to be a
positive number independent of temperature and density. This assumption is
satisfied with good accuracy in a wide temperature range R16 . All
thermodynamic characteristics can be expressed in terms of the function
$f(\xi,\tau)$ (37) and its derivatives. Let us introduce the definitions of
the following functions of variables $\xi,\tau$:

$\begin{array}[]{ccc}\displaystyle{\psi(\xi,\tau)\equiv\frac{\partial
f}{\partial\xi}=\frac{1}{2}\,{\rm
cth}\bigg{(}\frac{\xi}{2\tau}\bigg{)}-\frac{3g}{8}\xi^{-3}\Phi_{2}^{2}\big{(}\xi\overline{\tau}\big{)}+\frac{3g}{8}\big{(}\xi^{-2}-1\big{)}\Phi_{2}\big{(}\xi\overline{\tau}\big{)}\frac{1}{\tau}\,\dot{\Phi}_{2}\big{(}\xi\overline{\tau}\big{)}-\vspace{3mm}}\\\
\displaystyle{-\frac{3}{8}\Big{(}\xi^{-2}+\frac{4}{3}\,\Big{)}\Phi_{3}\big{(}\xi\overline{\tau}\big{)}+\frac{3}{8}\Big{(}\xi^{-1}-\frac{4}{3}\,\xi\,\Big{)}\frac{1}{\tau}\,\dot{\Phi}_{3}\big{(}\xi\overline{\tau}\big{)},}\end{array}$
(46) $\begin{array}[]{ccc}\displaystyle{s(\xi,\tau)\equiv-\frac{\partial
f}{\partial\tau}=\frac{3g}{8}\frac{\xi}{\tau^{2}}\big{(}\xi^{-2}-1\big{)}\Phi_{2}\big{(}\xi\overline{\tau}\big{)}\dot{\Phi}_{2}\big{(}\xi\overline{\tau}\big{)}++\frac{3}{8}\frac{\xi}{\tau^{2}}\Big{(}\xi^{-1}-\frac{4}{3}\,\xi\,\Big{)}\dot{\Phi}_{3}\big{(}\xi\overline{\tau}\big{)}-\vspace{3mm}}\\\
\displaystyle{-\ln\\!\big{(}1-e^{-\xi\overline{\tau}}\big{)}+\frac{\xi}{\tau}\frac{1}{e^{\xi\overline{\tau}}-1},}\end{array}$
(47) $\begin{array}[]{ccc}\displaystyle{\varphi(\xi,\tau)\equiv
f-\tau\frac{\partial
f}{\partial\tau}=\frac{3}{2}\big{(}\xi^{-1}-\xi\big{)}\,{\rm
cth}\bigg{(}\frac{\xi}{2\tau}\bigg{)}+\vspace{3mm}}\\\
\displaystyle{+\frac{3g}{16}\big{(}\xi^{-2}-1\big{)}\Phi_{2}\big{(}\xi\overline{\tau}\big{)}\bigg{[}6\,{\rm
cth}\bigg{(}\frac{\xi}{2\tau}\bigg{)}-5\,\Phi_{2}\big{(}\xi\overline{\tau}\big{)}\bigg{]}-\frac{9}{8}\Big{(}\xi^{-1}-\frac{4}{3}\,\xi\,\Big{)}\Phi_{3}\big{(}\xi\overline{\tau}\big{)}.}\end{array}$
(48)

Taking into account the condition $\psi(\xi,\tau)=0$ equivalent to (41), the
corresponding equilibrium functions $s(\tau)$ and $\varphi(\tau)$ can be
calculated, which always take positive values. The equilibrium values of
entropy, pressure and energy are expressed through these functions:

$\begin{array}[]{ccc}\displaystyle{\frac{S}{N}=s(\tau),\qquad p=\Gamma
n\Theta_{D}\varphi(\tau),\qquad E=N\Theta_{D}\varphi(\tau).}\end{array}$ (49)

From (49), in particular, it follows that pressure and energy are related as

$\begin{array}[]{ccc}\displaystyle{p=\Gamma\frac{E}{V}.}\end{array}$ (50)

At $\Gamma=1/3$, this formula coincides with the equation of state for a
photon gas R17 ; R18 . The temperature dependences of entropy, pressure and
energy are shown in Fig. 3.

Figure 3:  Entropy $s=S/N$ (a), pressure and energy $\varphi=p\big{/}\Gamma
n\Theta_{D}=E\big{/}N\Theta_{D}$ (b) in the asymmetric (1) and symmetric (2)
states at $|g|=0.6$.

The numbers of “dressed” $N_{ph}$ and “bare” $N_{ph}^{(0)}$ phonons, according
to (14) and (16), are expressed by the formulas

$\begin{array}[]{ccc}\displaystyle{N_{ph}=\frac{3}{2}N\frac{\tau}{\xi}D_{2}\bigg{(}\frac{\xi}{\tau}\bigg{)},\qquad
N_{ph}^{(0)}=\frac{N}{2}\bigg{(}\frac{1}{\xi}-1\bigg{)}+\frac{3}{2}N\frac{\tau}{\xi^{2}}D_{2}\bigg{(}\frac{\xi}{\tau}\bigg{)}.}\end{array}$
(51)

The temperature dependences of the number of phonons are shown in Fig. 4. As
was noted, the number of “dressed” phonons, which determine the temperature
dependences of thermodynamic quantities, also tends to zero as the temperature
approaches zero (curve 1 in Fig. 4). In this case, the number of “bare”
phonons, which do not contribute to entropy, remains finite and even increases
(curve 2 in Fig. 4).

Figure 4:  Temperature dependencies of the number of phonons: $N_{ph}$ –
“dressed” (1); $N_{ph}^{(0)}$ – “bare” (2); $N_{ph}^{(D)}$ – “Debye” (3)
phonons at $|g|=0.6$.

In order to calculate the heat capacities and thermodynamic coefficients, one
should find the entropy and pressure differentials. Here it is necessary to
take into account that, due to condition $d\psi(\xi,\tau)=0$, the
differentials of $\xi$ and $\tau$ are related as follows

$\begin{array}[]{ccc}\displaystyle{d\xi=-\frac{\big{(}\partial\psi\big{/}\partial\tau\big{)}_{\xi}}{\big{(}\partial\psi\big{/}\partial\xi\big{)}_{\tau}}\,d\tau,}\end{array}$
(52)

where

$\begin{array}[]{ccc}\displaystyle{d\tau=\frac{dT}{\Theta_{D}}+\Gamma\frac{T}{\Theta_{D}}\frac{dV}{V}.}\end{array}$
(53)

As a result, we get:

$\begin{array}[]{ccc}\displaystyle{\frac{dS}{N}=\tau\Omega\bigg{(}\frac{dT}{T}+\Gamma\frac{dV}{V}\bigg{)},}\end{array}$
(54) $\begin{array}[]{ccc}\displaystyle{dp=\Gamma n\Omega
T\frac{dT}{\Theta_{D}}+\Gamma\big{(}1+\Gamma\big{)}n\Theta_{D}\varphi\bigg{[}\frac{\Gamma}{\big{(}1+\Gamma\big{)}}\frac{\Omega\tau^{2}}{\varphi}-1\bigg{]}\frac{dV}{V}.}\end{array}$
(55)

Here the quantity

$\begin{array}[]{ccc}\displaystyle{\Omega=\Omega(\xi,\tau)\equiv\frac{\big{(}\partial^{2}f\big{/}\partial\xi\partial\tau\big{)}^{2}-\big{(}\partial^{2}f\big{/}\partial\xi^{2}\big{)}\big{(}\partial^{2}f\big{/}\partial\tau^{2}\big{)}}{\big{(}\partial^{2}f\big{/}\partial\xi^{2}\big{)}}}\end{array}$
(56)

is expressed in terms of the second derivatives of the function $f(\xi,\tau)$
(37), the explicit form of which is given in Appendix B. Taking into account
the condition $\psi(\xi,\tau)=0$, the equilibrium function $\Omega(\tau)$ is
calculated, through which are expressed the heat capacities and thermodynamic
coefficients.

Heat capacities. Assuming in formulas (54), (55) first $dV=0$, and then
$dp=0$, we obtain expressions for the isochoric and isobaric heat capacities:

$\begin{array}[]{ccc}\displaystyle{C_{V}=T\bigg{(}\frac{\partial S}{\partial
T}\bigg{)}_{\\!V}=\frac{NT\Omega}{\Theta_{D}},}\end{array}$ (57)
$\begin{array}[]{ccc}\displaystyle{C_{p}=T\bigg{(}\frac{\partial S}{\partial
T}\bigg{)}_{\\!p}=\frac{NT\Omega}{\Theta_{D}Z}.}\end{array}$ (58)

In (58), it is the notation $\displaystyle{Z\equiv
1-\frac{\Gamma\tau^{2}}{\big{(}1+\Gamma\big{)}}\frac{\Omega}{\varphi}}$.

Thermodynamic coefficients. Similarly, using formulas (54), (55), formulas can
be obtained for the isobaric volume expansion coefficient $\alpha_{p}$,
isochoric pressure coefficient $\beta_{V}$ and isothermal compressibility
coefficient $\gamma_{T}$:

$\begin{array}[]{ccc}\displaystyle{\beta_{V}=\frac{1}{p}\bigg{(}\frac{\partial
p}{\partial
T}\bigg{)}_{\\!V}=\frac{T}{\Theta_{D}^{2}}\frac{\Omega}{\varphi},}\end{array}$
(59)
$\begin{array}[]{ccc}\displaystyle{\alpha_{p}=\frac{1}{V}\bigg{(}\frac{\partial
V}{\partial
T}\bigg{)}_{\\!p}=\frac{T\Omega}{\Theta_{D}^{2}\big{(}1+\Gamma\big{)}\varphi
Z},}\end{array}$ (60)
$\begin{array}[]{ccc}\displaystyle{\gamma_{T}=-\frac{1}{V}\bigg{(}\frac{\partial
V}{\partial
p}\bigg{)}_{\\!T}=\frac{1}{\Gamma\big{(}1+\Gamma\big{)}n\Theta_{D}\varphi
Z}.}\end{array}$ (61)

The remaining thermodynamic coefficients, which can be constructed from the
quantities $T,\,S,\,p$ and $V$, are expressed in terms of the coefficients and
heat capacities given here R18 . The difference in heat capacities, as
required, is expressed through the given above thermodynamic coefficients R17
; R18 :

$\begin{array}[]{ccc}\displaystyle{C_{p}-C_{V}=\frac{VT}{\gamma_{T}}\,\alpha_{p}^{2}=N\frac{\Gamma}{\big{(}1+\Gamma\big{)}}\frac{T^{3}}{\Theta_{D}^{3}}\frac{\Omega^{2}}{\varphi
Z}=\frac{NT\Omega}{\Theta_{D}}\frac{\big{(}1-Z\big{)}}{Z}.}\end{array}$ (62)

Similar formulas for thermodynamic quantities of the normal state according to
the Debye theory are given for reference in Appendix C.

Figure 5:  The isochoric heat capacity in the asymmetric phase (1) and the
Debye heat capacity (2) at $|g|=0.7$.

##  V A jump of heat capacity

Above the phase transition temperature $\tau>\tau_{c}$, in the self-consistent
field model under consideration the phonon system is described by the Debye
model with the free energy $f_{D}(\tau)$ (40). The transition to the phase-
asymmetric state with pair correlations is accompanied by an appearance of an
order parameter $\sigma$. This occurs at the temperature
$\tau_{c}=T_{c}/\Theta_{D}$ determined by formula (42). Let us consider the
behavior of the heat capacity near the transition temperature, assuming
$|\sigma|\ll 1$ and $\xi\approx
1-\frac{1}{2}|\sigma|^{2}-\frac{1}{8}|\sigma|^{4}$. Then the expansion of the
free energy in the state with broken symmetry at $\tau<\tau_{c}$ in powers of
the order parameter takes the form

$\begin{array}[]{ccc}\displaystyle{f(\tau)\approx
f_{D}(\tau)+\alpha_{0}(\tau_{c})\frac{(\tau-\tau_{c})}{\tau_{c}}|\sigma|^{2}+C(\tau_{c})|\sigma|^{4},}\end{array}$
(63)

where

$\begin{array}[]{ccc}\displaystyle{\alpha_{0}(\tau_{c})=\frac{3}{8}\bigg{[}3g\Phi_{2}\big{(}\overline{\tau}_{c}\big{)}{\rm
cth}\bigg{(}\frac{1}{2\tau_{c}}\bigg{)}+\Phi_{3}\big{(}\overline{\tau}_{c}\big{)}+2{\rm
cth}\bigg{(}\frac{1}{2\tau_{c}}\bigg{)}\bigg{]},}\end{array}$ (64)
$\begin{array}[]{ccc}\displaystyle{C(\tau_{c})=\frac{9}{16}\Big{[}g\Phi_{2}\big{(}\overline{\tau}_{c}\big{)}+1\Big{]}{\rm
cth}\bigg{(}\frac{1}{2\tau_{c}}\bigg{)}+\frac{3}{64}\Phi_{3}\big{(}\overline{\tau}_{c}\big{)}.}\end{array}$
(65)

Here, just like before, $\overline{\tau}_{c}=1/\tau_{c}$. From condition
$\partial f\big{/}\partial|\sigma|=0$ there follows the expression for the
equilibrium value of the order parameter

$\begin{array}[]{ccc}\displaystyle{|\sigma|^{2}=-\frac{\alpha_{0}(\tau_{c})}{2C(\tau_{c})}\frac{(\tau-\tau_{c})}{\tau_{c}}.}\end{array}$
(66)

A substitution of (66) into (63) gives the expression for the equilibrium free
energy

$\begin{array}[]{ccc}\displaystyle{f(\tau)\approx
f_{D}(\tau)-\frac{\alpha_{0}^{2}}{4C}\frac{(\tau-\tau_{c})^{2}}{\tau_{c}^{2}}.}\end{array}$
(67)

It is usually assumed in the theory of phase transitions R17  that the
coefficients $\alpha_{0}(\tau_{c})$ and $C(\tau_{c})$ are positive, so that
the transition to the asymmetric phase is accompanied by a decrease in the
free energy. In the case under consideration both coefficients
$\alpha_{0}(\tau_{c})$ and $C(\tau_{c})$ are negative, so that, as noted
above, the transition to the asymmetric phase is accompanied by an increase in
the free energy. Formula (67) makes it easy to obtain a formula for the jump
in the isochoric heat capacity during the transition from the symmetric to the
asymmetric phase:

$\begin{array}[]{ccc}\displaystyle{\Delta
C_{V}=C_{V}(\tau_{c}+0)-C_{V}(\tau_{c}-0)=N\Theta_{D}\frac{\alpha_{0}^{2}}{2CT_{c}}.}\end{array}$
(68)

Jumps of other quantities can be found using well-known formulas R17 ; R18 .
The change in the heat capacity with temperature in the asymmetric phase and
the temperature dependence of the Debye heat capacity are shown in Fig. 5.

In the conventional theory of second-order phase transitions R17 , the heat
capacity increases by a jump upon transition from the symmetric to the
asymmetric phase. In this case, due to the fact that $C(\tau_{c})<0$, the heat
capacity jump is negative. Thus, upon transition from the normal Debye phase
to the phase where there exists a correlation of phonons with opposite
momenta, the heat capacity decreases by a jump.

##  VI Region of low temperatures

Let us consider the thermodynamic quantities in the asymmetric phase at low
temperatures $T\ll\Theta_{D}$. It follows from formula (41) that at $T=0$ the
parameter $\xi$ takes the finite value $\xi_{0}=|g|$. Therefore, at $\tau\ll
1$ we have $\frac{\xi}{\tau}\gg 1$. The temperature dependence of $\xi$ at
$\tau\ll 1$, up to exponentially small terms, has the form

$\begin{array}[]{ccc}\displaystyle{\xi=\xi_{0}\big{(}1+a_{3}\tau^{3}-a_{4}\tau^{4}\big{)},}\end{array}$
(69)

where

$\begin{array}[]{ccc}\displaystyle{a_{3}=\frac{12\zeta(3)}{\xi_{0}}\bigg{(}\frac{5}{\xi_{0}^{2}}-3\bigg{)},\qquad
a_{4}=\frac{8\pi^{4}}{15}\frac{1}{\xi_{0}^{2}}\bigg{(}\frac{5}{\xi_{0}^{2}}-4\bigg{)},}\end{array}$
(70)

and $\zeta(3)=1.202$. The temperature dependence of the free energy in the
low-temperature limit is given by the formula

$\begin{array}[]{ccc}\displaystyle{f(\tau)=f_{0}-b_{3}\tau^{3}+b_{4}\tau^{4},}\end{array}$
(71)

where

$\begin{array}[]{ccc}\displaystyle{f_{0}\equiv\frac{3}{16}\bigg{(}\xi_{0}+\frac{1}{\xi_{0}}\bigg{)},\qquad
b_{3}\equiv\frac{9\zeta(3)}{2}\frac{1}{\xi_{0}^{2}}\bigg{(}\frac{1}{\xi_{0}^{2}}-1\bigg{)},\qquad
b_{4}\equiv\frac{\pi^{4}}{5}\frac{1}{\xi_{0}^{3}}\bigg{(}\frac{1}{\xi_{0}^{2}}-\frac{4}{3}\bigg{)}.}\end{array}$
(72)

As we can see, in contrast to the corresponding dependence in the Debye theory

$\begin{array}[]{ccc}\displaystyle{f_{D}(\tau)=\frac{3}{8}-\frac{\pi^{4}}{15}\,\tau^{4},}\end{array}$
(73)

in the case under consideration there is the main contribution to the free
energy of the cubic term with respect to temperature, and the term of the
fourth degree enters with the opposite sign. It follows from (71) that at
$\tau\ll b_{3}\big{/}b_{4}$ both entropy and heat capacity tend to zero at
$T\rightarrow 0$ according to a quadratic law with a cubic correction:

$\begin{array}[]{ccc}\displaystyle{\frac{S}{N}\approx
3b_{3}\tau^{2}-4b_{4}\tau^{3},\qquad\frac{C_{V}}{N}\approx
6b_{3}\tau^{2}-12b_{4}\tau^{3}.}\end{array}$ (74)

The decrease of these quantities in the low-temperature region is slower than
their cubic dependence according to the Debye theory. The pressure and energy
in this region are determined by the formula

$\begin{array}[]{ccc}\displaystyle{\frac{p}{\Gamma
n\Theta_{D}}=\frac{E}{N\Theta_{D}}=f_{0}+2b_{3}\tau^{3}-3b_{4}\tau^{4}.}\end{array}$
(75)

The first term on the right side of (75) describes the contribution to the
pressure and energy of zero-point lattice vibrations.

##  VII Region of existence of the asymmetric phase

The state of phonons with pair correlations considered in the article can
exist in a state of unstable thermodynamic equilibrium only in those regions
of temperatures and densities where the known thermodynamic inequalities are
satisfied R17

$\begin{array}[]{ccc}\displaystyle{C_{V}>0,\qquad\big{(}\partial
p\big{/}\partial V\big{)}_{T}<0.}\end{array}$ (76)

The satisfiability of inequalities (76) in the asymmetric phase depends on the
value of the interaction constant $|g|$, so that they are satisfied under the
condition

$\begin{array}[]{ccc}\displaystyle{0<\Omega<\frac{\big{(}1+\Gamma\big{)}}{\Gamma}\frac{\varphi}{\tau^{2}}.}\end{array}$
(77)

Condition (77) also ensures the fulfillment of the inequality $C_{p}-C_{V}>0$,
since in this case $0<Z<1$. If inequalities (76) are violated, then the system
cannot exist even in a metastable state.

Figure 6 shows the change in the isochoric heat capacity with temperature for
various values of the interaction constant. There are two characteristic
values $|g|_{e}=0.66$ and $|g|_{n}=0.425$, at which the behavior of the heat
capacity changes qualitatively. In the interval $|g|_{e}<|g|<1$, the heat
capacity increases monotonically with increasing temperature (curve 1 in Fig.
6). At $|g|=|g|_{e}$, an inflection point appears on the temperature
dependence of the heat capacity at $\tau_{e}/\tau_{c}=0.235$ (curve 2, Fig.
6). In the range of values $|g|_{n}<|g|<|g|_{e}$, the heat capacity remains
positive, so that the stability conditions (76) are still satisfied, but it
has a minimum (curve 3, Fig. 6). At $|g|=|g|_{n}$ and temperature
$\tau_{n}/\tau_{c}=0.151$, the heat capacity turns to zero, and therefore the
system loses stability (curve 4, Fig. 6). In the range of values
$0<|g|<|g|_{n}$, a temperature region arises where conditions (76) are not
satisfied (curve 5, Fig. 6). As the interaction constant decreases, this
region of instability expands.

Figure 6:  The isochoric heat capacity in the asymmetric phase for different
values of the interaction constant: 1) $|g|=0.8$; 2) $|g|=0.66$; 3) $|g|=0.5$;
4) $|g|=0.425$; 5) $|g|=0.35$.

As noted above, the critical temperature increases as the magnitude of the
interaction decreases (Fig. 1), but, as we see, there narrows the temperature
range where the metastable existence of a new phase is possible. Thus, the
transition temperature is not likely to be high in reality.

##  VIII Discussion. Conclusions

It is shown in the paper that a gas of phonons, between which there is an
attraction, can be in the unstable equilibrium state with broken phase
symmetry in which there exist pair correlations between phonons with opposite
momenta. The transition to such a state from the normal state, which is
described by the Debye theory, can occur at a certain critical temperature as
a result of the second-order phase transition. The critical temperature is
determined by the value of the interaction constant, and it turns out that the
critical temperature increases with the weakening of the interaction (Fig. 1).
However, with a decrease in the modulus of the interaction constant and an
increase in the critical temperature, there expands the temperature range in
which the thermodynamic stability of the asymmetric phase is completely lost.
In contrast to conventional equilibrium phase transitions, in this case it
turns out that the free energy of the asymmetric state is higher than the
energy of the normal state. This leads to the fact that such a state turns out
to be metastable, and upon transition from the normal state to the asymmetric
phase the heat capacity decreases by a jump.

Qualitatively, the situation considered in the paper can be explained by a
simple example of a physical pendulum, which can be in two extreme states. In
the lower position its energy is minimal and this state is stable. In the
upper position the energy of the pendulum is maximum, its position is unstable
and it can be withdrawn from it at the slightest fluctuation. If, however, the
friction in the suspension of the pendulum is large, then it can remain in the
upper metastable state for quite a long time.

Similarly, if the relaxation time of a state of a solid with pairwise phonon
correlations turns out to be long, then such a state may be accessible for
observation. The question of how the system can be transferred to the
considered unstable state requires a separate study. There are considerations
that this can be done with the help of external high-frequency forces. It is
also possible that the new phase can be identified by its influence on the
propagation and absorption of sound R19 . The considered phase may turn out to
be more stable under inhomogeneous conditions, for example, near boundaries.
Theoretically, such states can also be studied within the framework of the
generalized Debye model R20 .

In this paper, the problem was considered in the scalar approximation without
taking into account phonon polarizations, under the assumption that the speed
of all phonons is the same. However, already in an isotropic medium there are
two elastic moduli and, consequently, two types of phonons – longitudinal and
transverse. This circumstance must be taken into account in a more realistic
model R21 . In addition, nonlinear effects in the considered model were taken
into account in a minimal way by introducing the interaction of phonons with
opposite momenta into the Hamiltonian (1). In real crystals, a significant
role can be played by nonlinear effects described by elastic moduli of the
third and fourth order R12 ; R22 . Nonlinear effects play a particularly
important role in crystals in which quantum regularities are manifested R23 .
In our calculations we neglected the contribution of the correlation
Hamiltonian (4) to the total energy. Accounting for the correlation energy can
also lead to the stabilization of the considered state. Thus, the model of a
crystal with correlations of phonons with opposite momenta needs further
development and generalization. Phase symmetry breaking in multiparticle
systems is accompanied by the manifestation of superfluid properties R24 ; R25
; R26 ; R27 . In the considered model, the phase transition is also
accompanied by the breaking of the phase symmetry. In this connection, as well
as in connection with experimental studies of quantum crystals R10 , the
question of the manifestation of superfluidity effects in phonon systems is of
considerable interest.

The author is grateful to A.A. Soroka for help with numerical calculations.

##  Appendix A Debye functions with integer index $n\geq 1$ and related
functions

Definitions:

$\begin{array}[]{ccc}\displaystyle{D_{n}(x)=\frac{n}{x^{n}}\int_{0}^{x}\frac{z^{n}dz}{e^{z}-1},\qquad(n\geq
1),}\end{array}$ (78)
$\begin{array}[]{ccc}\displaystyle{\Phi_{n}(x)=1+\frac{2(1+n)}{n}\frac{D_{n}(x)}{x}.}\end{array}$
(79)

Functions (78) can be represented as

$\begin{array}[]{ccc}\displaystyle{D_{n}(x)=\frac{n}{x^{n}}\bigg{[}n!\zeta(n+1)-\sum_{m=0}^{\infty}\int_{x}^{\infty}e^{-(m+1)z}z^{n}dz\bigg{]},}\end{array}$
(80)

$\zeta(n+1)$ is the Riemann zeta function. The functions with $n=2,3$ are used
in this paper, so that

$\begin{array}[]{ccc}\displaystyle{\Phi_{2}(x)=1+3\frac{D_{2}(x)}{x},\qquad\Phi_{3}(x)=1+\frac{8}{3}\frac{D_{3}(x)}{x}.}\end{array}$
(81)

Large arguments $x\gg 1$: Up to exponentially small terms:

$\begin{array}[]{ccc}\displaystyle{D_{n}(x)\approx\frac{n\\!\cdot\\!n!}{x^{n}}\zeta(n+1),\qquad
D_{2}(x)\approx\frac{4}{x^{2}}\zeta(3),\qquad
D_{3}(x)\approx\frac{\pi^{4}}{5x^{3}},\vspace{3mm}}\\\
\displaystyle{\Phi_{n}(x)\approx
1+\frac{2(1+n)\\!\cdot\\!n!}{x^{n+1}}\zeta(n+1),\qquad\Phi_{2}(x)\approx
1+\frac{12}{x^{3}}\zeta(3),\qquad\Phi_{3}(x)\approx
1+\frac{8\pi^{4}}{15x^{4}}.}\end{array}$ (82)

Small arguments $x\ll 1$:

$\begin{array}[]{ccc}\displaystyle{D_{n}(x)\approx
1-\frac{n}{2(n+1)}\,x+\frac{n}{12(n+2)}\,x^{2},\quad D_{2}(x)\approx
1-\frac{x}{3}+\frac{x^{2}}{24},\quad D_{3}(x)\approx
1-\frac{3}{8}\,x+\frac{x^{2}}{20},\vspace{3mm}}\\\
\displaystyle{\Phi_{n}(x)\approx\frac{2(n+1)}{nx}+\frac{(n+1)}{6(n+2)}\,x,\quad\Phi_{2}(x)\approx\frac{3}{x}+\frac{x}{8},\quad\Phi_{3}(x)\approx\frac{8}{3x}+\frac{2}{15}\,x.}\end{array}$
(83)

Derivatives (used designations are $dD_{n}(x)\big{/}dx\equiv\dot{D}_{n}(x)$,
$d^{2}D_{n}(x)\big{/}dx^{2}\equiv\ddot{D}_{n}(x)$ and so on):

$\begin{array}[]{ccc}\displaystyle{\dot{D}_{n}(x)=\frac{n}{e^{x}-1}-\frac{n}{x}D_{n}(x),\vspace{3mm}}\\\
\displaystyle{\dot{D}_{2}(x)=\frac{2}{e^{x}-1}-\frac{2}{x}D_{2}(x),\qquad\dot{D}_{3}(x)=\frac{3}{e^{x}-1}-\frac{3}{x}D_{3}(x);}\end{array}$
(84)
$\begin{array}[]{ccc}\displaystyle{\dot{\Phi}_{n}(x)=\frac{2(n+1)}{x(e^{x}-1)}+\frac{(n+1)}{x}-\frac{(n+1)}{x}\,\Phi_{n}(x)=\frac{(n+1)}{x}\bigg{[}{\rm
cth}\frac{x}{2}-\Phi_{n}(x)\bigg{]},\vspace{3mm}}\\\
\displaystyle{\dot{\Phi}_{2}(x)=\frac{6}{x(e^{x}-1)}+\frac{3}{x}-\frac{3}{x}\,\Phi_{2}(x)=\frac{3}{x}\bigg{[}{\rm
cth}\frac{x}{2}-\Phi_{2}(x)\bigg{]},\vspace{3mm}}\\\
\displaystyle{\dot{\Phi}_{3}(x)=\frac{8}{x(e^{x}-1)}+\frac{4}{x}-\frac{4}{x}\,\Phi_{3}(x)=\frac{4}{x}\bigg{[}{\rm
cth}\frac{x}{2}-\Phi_{3}(x)\bigg{]}.}\end{array}$ (85)
$\begin{array}[]{ccc}\displaystyle{\ddot{\Phi}_{n}(x)=-\frac{2(n+1)}{x(e^{x}-1)}\bigg{(}1+\frac{(n+2)}{x}\bigg{)}-\frac{2(n+1)}{x(e^{x}-1)^{2}}-\frac{(n+1)(n+2)}{x^{2}}+\frac{(n+1)(n+2)}{x^{2}}\,\Phi_{n}(x),\vspace{3mm}}\\\
\displaystyle{\ddot{\Phi}_{2}(x)=-\frac{6}{x(e^{x}-1)}\bigg{(}1+\frac{4}{x}\bigg{)}-\frac{6}{x(e^{x}-1)^{2}}-\frac{12}{x^{2}}+\frac{12}{x^{2}}\,\Phi_{2}(x),\vspace{3mm}}\\\
\displaystyle{\ddot{\Phi}_{3}(x)=-\frac{8}{x(e^{x}-1)}\bigg{(}1+\frac{5}{x}\bigg{)}-\frac{8}{x(e^{x}-1)^{2}}-\frac{20}{x^{2}}+\frac{20}{x^{2}}\,\Phi_{3}(x).}\end{array}$
(86)

##  Appendix B The function $f(\xi,\tau)$ (37) and its derivatives:

$\begin{array}[]{ccc}\displaystyle{f(\xi,\tau)=\frac{\xi}{2}+\frac{3}{16}\,g\big{(}\xi^{-2}-1\big{)}\Phi_{2}^{2}(\xi\overline{\tau})+\frac{3}{8}\Big{(}\xi^{-1}-\frac{4}{3}\xi\Big{)}\Phi_{3}(\xi\overline{\tau})+\tau\ln\\!\big{(}1-e^{-\xi\overline{\tau}}\big{)},}\end{array}$
(87) $\begin{array}[]{ccc}\displaystyle{\frac{\partial
f}{\partial\xi}\equiv\psi(\xi,\tau)=\frac{1}{2}\,{\rm
cth}\frac{\xi}{2\tau}-\frac{3g}{8}\xi^{-3}\Phi_{2}^{2}(\xi\overline{\tau})+\frac{3g}{8\tau}\big{(}\xi^{-2}-1\big{)}\Phi_{2}(\xi\overline{\tau})\dot{\Phi}_{2}(\xi\overline{\tau})\,-\vspace{3mm}}\\\
\displaystyle{-\frac{3}{8}\Big{(}\xi^{-2}+\frac{4}{3}\Big{)}\Phi_{3}(\xi\overline{\tau})+\frac{3}{8\tau}\Big{(}\xi^{-1}-\frac{4}{3}\xi\Big{)}\dot{\Phi}_{3}(\xi\overline{\tau}),}\end{array}$
(88) $\begin{array}[]{ccc}\displaystyle{\frac{\partial
f}{\partial\tau}\equiv-s(\xi,\tau)=-\frac{3g}{8}\frac{\xi}{\tau^{2}}\big{(}\xi^{-2}-1\big{)}\Phi_{2}(\xi\overline{\tau})\dot{\Phi}_{2}(\xi\overline{\tau})-\frac{3}{8}\frac{\xi}{\tau^{2}}\Big{(}\xi^{-1}-\frac{4}{3}\xi\Big{)}\dot{\Phi}_{3}(\xi\overline{\tau})\,+\vspace{3mm}}\\\
\displaystyle{+\ln\\!\big{(}1-e^{-\xi\overline{\tau}}\big{)}-\frac{\xi}{\tau}\frac{1}{e^{\xi\overline{\tau}}-1},}\end{array}$
(89) $\begin{array}[]{ccc}\displaystyle{\varphi(\xi,\tau)\equiv
f-\tau\frac{\partial f}{\partial\tau}=\frac{\xi}{2}\,{\rm
cth}\frac{\xi}{2\tau}+\frac{3g}{16}\big{(}\xi^{-2}-1\big{)}\Phi_{2}(\xi\overline{\tau})\Big{[}\Phi_{2}(\xi\overline{\tau})+2\frac{\xi}{\tau}\dot{\Phi}_{2}(\xi\overline{\tau})\Big{]}+\vspace{3mm}}\\\
\displaystyle{+\frac{3}{8}\Big{(}\xi^{-1}-\frac{4}{3}\xi\Big{)}\Big{[}\Phi_{3}(\xi\overline{\tau})+\frac{\xi}{\tau}\dot{\Phi}_{3}(\xi\overline{\tau})\Big{]},}\end{array}$
(90)
$\begin{array}[]{ccc}\displaystyle{\frac{\partial^{2}f}{\partial\xi^{2}}=\frac{1}{4\tau}\bigg{[}1-{\rm
cth}^{2}\frac{\xi}{2\tau}\bigg{]}+\vspace{3mm}}\\\
\displaystyle{+\frac{3g}{8}\bigg{[}\frac{3}{\xi^{4}}\Phi_{2}^{2}(\xi\overline{\tau})-\frac{4}{\xi^{3}\tau}\Phi_{2}(\xi\overline{\tau})\dot{\Phi}_{2}(\xi\overline{\tau})+\frac{1}{\tau^{2}}\big{(}\xi^{-2}-1\big{)}\dot{\Phi}_{2}^{2}(\xi\overline{\tau})+\frac{1}{\tau^{2}}\big{(}\xi^{-2}-1\big{)}\Phi_{2}(\xi\overline{\tau})\ddot{\Phi}_{2}(\xi\overline{\tau})\bigg{]}+\vspace{3mm}}\\\
\displaystyle{+\frac{3}{4\xi^{3}}\Phi_{3}(\xi\overline{\tau})-\frac{3}{4\tau}\Big{(}\xi^{-2}+\frac{4}{3}\Big{)}\dot{\Phi}_{3}(\xi\overline{\tau})+\frac{3}{8\tau^{2}}\Big{(}\xi^{-1}-\frac{4}{3}\xi\Big{)}\ddot{\Phi}_{3}(\xi\overline{\tau}),}\end{array}$
(91)
$\begin{array}[]{ccc}\displaystyle{\frac{\partial^{2}f}{\partial\tau^{2}}=\frac{3g}{4}\frac{\xi}{\tau^{3}}\big{(}\xi^{-2}-1\big{)}\Phi_{2}(\xi\overline{\tau})\dot{\Phi}_{2}(\xi\overline{\tau})+\frac{3g}{8}\frac{\xi^{2}}{\tau^{4}}\big{(}\xi^{-2}-1\big{)}\Big{[}\dot{\Phi}_{2}^{2}(\xi\overline{\tau})+\Phi_{2}(\xi\overline{\tau})\ddot{\Phi}_{2}(\xi\overline{\tau})\Big{]}+\vspace{3mm}}\\\
\displaystyle{+\frac{3}{4}\frac{\xi}{\tau^{3}}\Big{(}\xi^{-1}-\frac{4}{3}\xi\Big{)}\dot{\Phi}_{3}(\xi\overline{\tau})+\frac{3}{8}\frac{\xi^{2}}{\tau^{4}}\Big{(}\xi^{-1}-\frac{4}{3}\xi\Big{)}\ddot{\Phi}_{3}(\xi\overline{\tau})-\frac{\xi^{2}}{\tau^{3}}\frac{e^{\xi\overline{\tau}}}{\big{(}e^{\xi\overline{\tau}}-1\big{)}^{2}},}\end{array}$
(92)
$\begin{array}[]{ccc}\displaystyle{\frac{\partial^{2}f}{\partial\xi\partial\tau}=\frac{3g}{8}\frac{1}{\tau^{2}}\big{(}\xi^{-2}+1\big{)}\Phi_{2}(\xi\overline{\tau})\dot{\Phi}_{2}(\xi\overline{\tau})-\frac{3g}{8}\frac{\xi}{\tau^{3}}\big{(}\xi^{-2}-1\big{)}\Big{[}\dot{\Phi}_{2}^{2}(\xi\overline{\tau})+\Phi_{2}(\xi\overline{\tau})\ddot{\Phi}_{2}(\xi\overline{\tau})\Big{]}+\vspace{3mm}}\\\
\displaystyle{+\frac{\xi}{\tau^{2}}\dot{\Phi}_{3}(\xi\overline{\tau})-\frac{3}{8}\frac{\xi}{\tau^{3}}\Big{(}\xi^{-1}-\frac{4}{3}\xi\Big{)}\ddot{\Phi}_{3}(\xi\overline{\tau})+\frac{\xi}{\tau^{2}}\frac{e^{\xi\overline{\tau}}}{\big{(}e^{\xi\overline{\tau}}-1\big{)}^{2}}.}\end{array}$
(93)

##  Appendix C Thermodynamic functions in the Debye free phonon model

The free energy:

$\begin{array}[]{ccc}\displaystyle{\frac{F_{D}}{\Theta_{D}N}\equiv
f_{D}(\tau)=\frac{1}{2}-\frac{1}{8}\Phi_{3}(\overline{\tau})+\tau\ln\\!\big{(}1-e^{-\overline{\tau}}\big{)},}\end{array}$
(94)

the entropy

$\begin{array}[]{ccc}\displaystyle{\frac{S}{N}=-\frac{\partial
f_{D}}{\partial\tau}=\frac{4}{3}D_{3}(\overline{\tau})-\ln\\!\big{(}1-e^{-\overline{\tau}}\big{)},}\end{array}$
(95)

the pressure

$\begin{array}[]{ccc}\displaystyle{p=\Gamma
n\Theta_{D}\bigg{(}f_{D}-\tau\frac{\partial
f_{D}}{\partial\tau}\bigg{)}=\frac{3}{8}\Gamma
n\Theta_{D}\Phi_{3}(\overline{\tau}),}\end{array}$ (96)

the energy

$\begin{array}[]{ccc}\displaystyle{E_{D}=N\Theta_{D}\bigg{(}f_{D}-\tau\frac{\partial
f_{D}}{\partial\tau}\bigg{)}=\frac{3}{8}N\Theta_{D}\Phi_{3}(\overline{\tau})=\frac{pV}{\Gamma},}\end{array}$
(97)

the phonon number

$\begin{array}[]{ccc}\displaystyle{N_{ph}^{(D)}=\frac{3}{2}N\tau
D_{2}(\overline{\tau}).}\end{array}$ (98)

The entropy and pressure differentials:

$\begin{array}[]{ccc}\displaystyle{\frac{dS}{N}=\tau\Omega_{0}\bigg{(}\frac{dT}{T}+\Gamma\frac{dV}{V}\bigg{)},}\end{array}$
(99) $\begin{array}[]{ccc}\displaystyle{dp=\Gamma
n\Omega_{0}T\frac{dT}{\Theta_{D}}+\Gamma\big{(}1+\Gamma\big{)}n\Theta_{D}\varphi_{0}\bigg{[}\frac{\Gamma}{\big{(}1+\Gamma\big{)}}\frac{\Omega_{0}\tau^{2}}{\varphi_{0}}-1\bigg{]}\frac{dV}{V},}\end{array}$
(100)

where

$\begin{array}[]{ccc}\displaystyle{\Omega_{0}\equiv\frac{1}{\tau^{2}}\bigg{[}4\tau
D_{3}(\overline{\tau})-\frac{3}{e^{\overline{\tau}}-1}\bigg{]}=-\frac{3}{8}\frac{\dot{\Phi}_{3}(\overline{\tau})}{\tau^{3}},\qquad\varphi_{0}\equiv\frac{3}{8}\Phi_{3}(\overline{\tau}).}\end{array}$
(101)

The formulas for the heat capacities and thermodynamic coefficients coincide
with formulas (56) – (61) for the asymmetric phase, if we make the following
substitutions in them

$\begin{array}[]{ccc}\displaystyle{\Omega\rightarrow\Omega_{0},\quad\varphi\rightarrow\varphi_{0},\quad
Z\rightarrow Z_{0}\equiv
1-\frac{\Gamma}{\big{(}1+\Gamma\big{)}}\frac{\Omega_{0}\tau^{2}}{\varphi_{0}},}\end{array}$

so that we obtain

$\begin{array}[]{ccc}\displaystyle{C_{V}=\frac{NT\Omega_{0}}{\Theta_{D}},\qquad
C_{p}=\frac{NT\Omega_{0}}{\Theta_{D}}\bigg{(}1+\frac{\Gamma}{1+\Gamma}\frac{\Omega_{0}\tau^{2}}{\varphi_{0}Z_{0}}\bigg{)},\vspace{3mm}}\\\
\displaystyle{C_{p}-C_{V}=\frac{VT}{\gamma_{T}}\alpha_{p}^{2}=N\frac{\Gamma}{\big{(}1+\Gamma\big{)}}\frac{T^{3}}{\Theta_{D}^{3}}\frac{\Omega_{0}^{2}}{\varphi_{0}Z_{0}},}\end{array}$
(102)

and also

$\begin{array}[]{ccc}\displaystyle{\beta_{V}=\frac{T}{\Theta_{D}^{2}}\frac{\Omega_{0}}{\varphi_{0}},\qquad\alpha_{p}=\frac{T\Omega_{0}}{\Theta_{D}^{2}\big{(}1+\Gamma\big{)}\varphi_{0}Z_{0}},\qquad\gamma_{T}=\frac{1}{\Gamma\big{(}1+\Gamma\big{)}n\Theta_{D}\varphi_{0}Z_{0}}.}\end{array}$
(103)

The stability condition (77) for the Debye state is always satisfied.

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