Superconductivity and the BCS-Bogoliubov Theory

TL;DR

This paper reformulates the BCS-Bogoliubov theory of superconductivity using linear algebra, introduces a new superconducting state with lower energy, and derives results consistent with the traditional theory.

Contribution

It provides a linear algebraic reformulation of the BCS-Bogoliubov theory and introduces a new superconducting state with lower energy than the BCS state.

Findings

Existence of an energy gap in the new formulation

Introduction of a new superconducting state with lower energy

Results consistent with traditional BCS-Bogoliubov theory

Abstract

First, we reformulate the BCS-Bogoliubov theory of superconductivity from the viewpoint of linear algebra. We define the BCS Hamiltonian on $\mathbb{C}^{2^{2M}}$, where $M$ is a positive integer. We discuss selfadjointness and symmetry of the BCS Hamiltonian as well as spontaneous symmetry breaking. Beginning with the gap equation, we give the well-known expression for the BCS state and find the existence of an energy gap. We also show that the BCS state has a lower energy than the normal state. Second, we introduce a new superconducting state explicitly and show from the viewpoint of linear algebra that this new state has a lower energy than the BCS state. Third, beginning with our new gap equation, we show from the viewpoint of linear algebra that we arrive at the results similar to those in the BCS-Bogoliubov theory.