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Journal of Mathematical Physics

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

Bernhard Mühlschlegel

Published May 1, 1962• 81 citations
Publisher

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.

DOI
10.1063/1.1724250

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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.
The ground-state wave function and some of the excited states of the BCS reduced Hamiltonian are found. In the limit of large volume, the boundary and continuity conditions on the exact wave function lead directly to the equations which Bardeen, Cooper, and Schrieffer found by a variational technique. It is also shown in what sense the BCS trial wave function may be considered asymptotically exact in this limit. Finite-volume corrections are included in an appendix, and explicit calculations are carried out for a one-step model of the kinetic energy which has possible applications to the problem of the finite nucleus.
The spectrum and eigenstates of Bardeen-Cooper-Schrieffer's reduced Hamiltonian in the theory of superconductivity are investigated in a mathematically rigorous manner. The Hamiltonian is defined first for a finite number of electrons contained in a finite volume. The interaction Hamiltonian is assumed to take a simple form. The Hamiltonian after the Bogoliubov transformation (transformed Hamiltonian) has a limit in the weak topology of Hilbert space when the volume tends to infinity with the number density of electrons kept constant. In the limit the interaction part vanishes and the free part gives a continuous excitation spectrum of Bogolons (transformed electrons). It should be noticed that the operator norm of the former does not tend to zero and has no strong limit as well as the total Hamilton. An equivalent boson which consists of two Bogolons with opposite momenta and spins is introduced. In the Fock space for the bosons the Hamiltonian has a strong limit for infinite volume. The limit operator has the same excitation spectrum of Bogolons as the free Hamiltonian and the existence of the lower limit of the spectrum is shown for some special class of the interactions. For large volume the limit operator is a good approximation to be transformed Hamiltonian. The ground state of the free part of the transformed Hamiltonian (i.e. the ground state of Bardeen-Cooper-Schrieffer's calculation) is not the exact eigenstate of the transformed total Hamiltonian and the difference does not vanish when the volume tends to infinity.
Superconducting pairing of electrons in nanoscale metallic particles with discrete energy levels and a fixed number of electrons is described by the reduced Bardeen, Cooper, and Schrieffer model Hamiltonian. We show that this model is integrable by the algebraic Bethe ansatz. The eigenstates, spectrum, conserved operators, integrals of motion, and norms of wave functions are obtained. Furthermore, the quantum inverse problem is solved, meaning that form factors and correlation functions can be explicitly evaluated. Closed form expressions are given for the form factors and correlation functions that describe superconducting pairing.
A method for projecting the Bardeen-Cooper-Schrieffer states with respect to the particle number is suggested. This method does not involve the Fowler-Darwin integrals which are usually applied for this purpose. Energy matrix elements of an arbitrary single-particle operator and overlap integrals are calculation in the quasi- particle representation and are expressed analytically in terms of the u and v parameters. The volume of calculations according to the method suggested increases linearly with the size of the system in the case of factorizable pairing matrix elements (the weaker the particle number fluctuations, the smaller the quantity of calculations). The method is illustrated by some model systems. In the paper the limiting case of weak correlations is considered and comparison with perturbation theory is carried out.