Perturbation theory in statistical mechanics and the theory of superconductivity

Abstract

The connection between formal perturbation theory and the modern theory of superconductivity is investigated. It is found that the condition for ladder diagrams to give a convergent sum is identical with the condition for the temperature to be above the critical temperature. The effect of the residual terms of the Hamiltonian is investigated and found to be small. They give rise to a correlation between electrons in the normal state, and to a |T − T c| − 1 2 singularity in the specific heat, but with a very small coefficient, in both the normal and superconducting states. These effects are caused by the existence of a collective mode whose spectrum becomes imaginary at the critical temperature. It is found that, below the critical temperature, most of the divergence is removed by using the Bardeen et al. (referred to throughout this paper as BCS) Hamiltonian as the unperturbed Hamiltonian, but that ladder diagrams with momentum exactly zero still diverge. These results are not affected by the Coulomb interaction, and it is suggested that the phonon-like collective mode continues to exist at nonzero temperatures, although it has been shown not to exist at zero temperature. The convergence of the ladder diagrams is suggested as a criterion which the BCS solution must satisfy, and it is shown that this is equivalent to requiring the BCS solution to give a local minimum of the thermodynamic potential. This criterion is used to investigate some more complicated interactions. It is found that there is an interaction for which pairing of particles with opposite spin or with the same spin is not possible, and a more complicated trial wave function must be used. A predominantly P-state force is found to give a solution of the equations which appears to represent a state with ferromagnetic properties.