High-Temperature Expansion in Thermodynamics of a System with Electron-Boson Interaction

In the last decades, quantum systems with strong interaction have attracted special attention of the researchers. This is primarily connected with important discoveries of high-temperature superconductivity and Hall's and other quantum effects in solids. It is well known that electron-phonon interaction (EPI) is the main mechanism of superconductivity. Apart from EPI, relaxation in metals occurs in the process of electron interactions with collective excitations (spin or charge fluctuations) or in any more complex processes. However, EPI dominates at high temperatures, because its reciprocal relaxation time exceeds the corresponding times of other mechanisms by several orders of magnitude. The present paper is aimed at studying the quantum properties of a systems with strong EPI at high temperatures within the framework of perturbation theory including changes in the energy spectrum of the Bloch electrons caused by scattering on bosons with S = 0 (on the example of phonons of the crystal lattice). To solve this problem, the Matsubara scattering operator at a high temperature \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{S} = e^{\beta \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{H} _0 } e^{ - \beta \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{H} } \) is expanded in a series in a small parameter (in the present work, the ratio of the EPI coupling constant to the reciprocal temperature β = 1/kT is used as a small parameter). Calculations are performed by the method of generating functional.