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arXiv:1412.6008

New theory of superconductivity. Method of equilibrium density matrix

Boris Volodumurovich Bondarev

Published Sep 22, 2013• 5 citations
arXivHTMLPDF

Abstract

A new variational method for studying the equilibrium states of an interacting particles system has been proposed. The statistical description of the system is realized by means of a density matrix. This method is used for description of conduction electrons in metals. An integral equation for the electron distribution function over wave vectors has been obtained. The solutions of this equation have been found for those cases where the single-particle Hamiltonian and the electron interaction Hamiltonian can be approximated by a quite simple expression. It is shown that the distribution function at temperatures below the critical value possesses previously unknown features which allow to explain the superconductivity of metals and presence of a gap in the energy spectrum of superconducting electrons.

DOI
10.48550/arxiv.1412.6008

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A new variational method for studying the equilibrium states of an interacting particles system has been proposed. The statistical description of the system is realized by means of a density matrix. This method is used for description of conduction electrons in metals. An integral equation for the electron distribution function over wave vectors has been obtained. The solutions of this equation have been found for those cases where the single-particle Hamiltonian and the electron interaction Hamiltonian can be approximated by a quite simple expression. It is shown that the distribution function at temperatures below the critical value possesses previously unknown features which allow to explain the superconductivity of metals and presence of a gap in the energy spectrum of superconducting electrons.
This book contains detailed insights on the calculus of variations that studies the equilibrium density matrix for many-particle Fermi systems. There are two approximations taken into account in the book. The simplest one is the mean field approximation. The second approximation applies not only to the particle distribution pattern but also to the correlation function. The variational principle for electron distribution function among wave vectors has been denoted in the work. The method that makes using the density matrix for finding the valence electron energy in metal has been proposed. It has been proved that the Coulomb interaction of electrons in a crystal lattice results in production of the model Hamiltonian consisting of two components. One component describes attraction of electrons with the equal wave vectors inducing formation of specific electron pairs. The other component describes repulsion of electrons with the wave vectors being equal by size and opposite by direction. This component denotes the anisotropy regarding distribution of electrons among wave vectors thus indicating the superconducting substance ability. The Fermi-Dirac distribution function, as considered together with the model Hamiltonian, explains all the properties exhibited by superconductors. Thus, the effect of a magnetic field on the superconducting states is studied in the book. The book can be interesting for senior schoolchildren, students of higher educational institutions, postgraduates and teachers.
We present the current-density functional theory for the superconductor immersed in the magnetic field. The order parameter of the superconducting state, transverse component of the paramagnetic current-density, and electron density are chosen as basic variables that uniquely determine the equilibrium properties of the system. In order to construct this theory, the development of the approximate form of the exchange-correlation (xc) energy functional is indispensable as well as the derivation of the effective single-particle equation which makes it possible to reproduce the equilibrium densities mentioned above. The rigorous expression of the xc-energy functional is derived using the technique of the coupling-constant integration. Furthermore, the approximate form of the xc energy functional is proposed such that the energy gap resulting from the effective single-particle equation is consistent with the attractive interaction energy of the system.
We present the current–density functional theory for the superconductor immersed in the magnetic field. The order parameter of the superconducting state, transverse component of the paramagnetic current–density, and electron density are chosen as basic variables that uniquely determine the equilibrium properties of the system. In order to construct this theory, the development of the approximate form of the exchange–correlation (xc) energy functional is indispensable as well as the derivation of the effective single-particle equation which makes it possible to reproduce the equilibrium densities mentioned above. The rigorous expression of the xc-energy functional is derived using the technique of the coupling-constant integration. Furthermore, the approximate form of the xc energy functional is proposed such that the energy gap resulting from the effective single-particle equation is consistent with the attractive interaction energy of the system.
The variational theory of equilibrium boson system state to have been previously developed by the author under the density matrix formalism is applicable for researching equilibrium states and thermodynamic properties of the quantum Bose gas which consists of zero-spin particles. Particle pulse distribution function is obtained and duly employed for calculation of chemical potential, internal energy and gas capacity temperature dependences. It is found that specific phase transition, which is similar to transition of liquid helium to its superfluid state, occurs at the temperature exceeding that of the Bose condensation.
The evaluation of one electron thermodynamic Green’s functions using the equation of motion technique of quantum dynamics via newly formulated Hamiltonian and using Dyson’s equation approach. This involves approximation free approach and different cooper pairs are emerged automatically in the system and stands as an ab-initio approach. The detailed description of enhancement in the electron density of states (EDOS) for high temperature superconductors has been investigated. The investigated expressions of EDOS in the new framework are found responsible to describe a large number of dynamical properties of high temperature superconductors. The temperature dependence of EDOS has been found as a unique feature of the theory, which certainly becomes the outcome of the anharmonic interactions. The presence of electron-phonon interaction parameter in each term is an additional and new feature of the theory.
A fermion system with a simple attractive interaction is discussed with the aid of time-dependent correlation functions. Although perturbation theory is inapplicable, a sequence of correlation approximations described in the first paper of this series can be employed. The lowest approximation in the sequence expresses the two-particle correlation function in terms of single-particle functions and leads to the Hartree approximation; the second expresses three-particle correlation function in terms of one- and two-particle correlation functions and leads to the time-dependent correlation functions that characterize the super-conducting model of Bardeen, Cooper, and Schrieffer. In the second section of this paper these correlation functions are determined and the thermodynamic properties of the superconductor are calculated from them.In the third section of the paper, the electromagnetic effects of the superconductor predicted by the Bardeen-Cooper-Schrieffer time-dependent correlation functions are considered. Their unsatisfactory description of current conservation is indicated and overcome in the fourth section by a more accurate solution valid at nonvanishing temperature. This solution predicts different diffusive properties but the same Meissner effect and superconductive behavior, since the longitudinal current correlation function is modified while the transverse current correlation functions is not.The fifth section of the paper is devoted to the properties of a pure superconductor which depend on the lifetimes of the single-particle excitations. The effect of these lifetimes on the static electrical conductivity is determined, and it is shown that they do not destroy supercurrents although they eliminate a gap in the single-particle excitation spectrum. Their effect on the thermal conductivity is also calculated using heat current correlation functions. It is shown that a model which treats the lifetime of the single-particle excitation due to lattice interactions as constant yields results in agreement with observed thermal conductivities.