7 - BCS Theory

Abstract

This chapter discusses the Bardeen, Cooper, and Schrieffer (BCS) theory. BCS proposed a microscopic theory of superconductivity that predicts quantitatively many of the properties of elemental superconductors. If the order parameter is to be different from zero, the statistical operator cannot be invariant under a global gauge transformation. In the case of a ferromagnet, the normal state is rotationally invariant and the thermal average of the magnetization, which is a vector, vanishes. Below the Curie temperature, however, there is a spontaneous magnetization that clearly breaks the rotational symmetry of the high temperature phase. The phenomenon of superconductivity is characterized by the breaking of global gauge symmetry. It is observed that the BCS Hamiltonian is bilinear in the electron operators, and it can be diagonalized by a unitary transformation, called the Bogoliubov transformation, that mixes electron creation and annihilation operators. Treating order parameter in the mean-field approximation the BCS Hamiltonian breaks global gauge symmetry. The response of a superconductor to a magnetic field is explained in the chapter.