Asymptotic Expansion of the Bardeen-Cooper-Schrieffer Partition Function by Means of the Functional Method

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Abstract
The canonical operator exp [−β(H − μN)] associated with the Bardeen‐Cooper‐Schrieffer (BCS) model Hamiltonian of superconductivity is represented as a functional integral by the use of Feynman's ordering parameter. General properties of the partition function in this representation are discussed. Taking the inverse volume of the system as an expansion parameter, it is possible to calculate the thermodynamic potential including terms independent of the volume. This yields a new proof that the BCS variational value is asymptotically exact. The behavior of the canonical operator for large volume is described and related to the state of free quasiparticles. A study of the terms of the thermodynamic potential which are of smaller order in the volume in the low‐temperature limit, shows that the ground state energy is nondegenerate and belongs to a number eigenstate.