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Journal of Mathematical Physics

Functional Integrals for Many-Boson Systems

A. Casher, D. Lurié, M. Revzen

Published Aug 1, 1968• 45 citations
Publisher

Abstract

We express the grand canonical partition function (GCPF) for a system of interacting bosons as a functional integral over one complex function. Our derivation is based on the so-called coherent-state representation [R. J. Glauber, Phys. Rev. 136, 2766 (1963)]. We show how to extract the perturbative expansion of the GCPF and the various Green's functions from our functional-integral representation and we indicate the relevance of our formalism to the theory of superfluidity.

DOI
10.1063/1.1664716

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We express the grand canonical partition function (GCPF) for a system of interacting bosons as a functional integral over one complex function. Our derivation is based on the so-called coherent-state representation [R. J. Glauber, Phys. Rev. 136, 2766 (1963)]. We show how to extract the perturbative expansion of the GCPF and the various Green's functions from our functional-integral representation and we indicate the relevance of our formalism to the theory of superfluidity.
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Formal functional integrals are commonly used as theoretical tools and as sources of intuition for predicting phase transitions of many-body systems in Condensed Matter Physics. In this thesis, we derive rigorous versions of these functional integrals for two types of quantum many-particle systems. We begin with a brief review of quantum statistical mechanics in Chapter 2 and the formalism of coherent states in Chapter 3, which form the basis for our analysis in Chapters 4 and 5. In Chapter 4, we study a mixed gas of bosons and/or fermions interacting on a finite lattice, with a general Hamiltonian that preserves the total number of particles in each species. We rigorously derive a functional integral representation for the partition function, which employs a large-field cutoff for the boson fields. We then expand the resulting “action” in powers of the fields and find a recursion relation for the coefficients. In the case of a two-body interaction (such as the Coulomb interaction), we also find bounds on the coefficients, which give a domain of analyticity for the action. This domain is large enough for use of the action in the functional integral, provided that the large-field cutoffs are taken to grow not too quickly. In Chapter 5, we study a system of electrons and phonons interacting in a finite lattice, using the Holstein Hamiltonian. Again, we rigorously derive a coherent-state functional integral representation for the partition function of this system and then prove that the “action” in the functional integral is an entire-analytic function of the fields. However, since the Holstein Hamiltonian does not preserve the total number of bosons, the approach from Chapter 4 requires some modification. In particular, we repeatedly use Duhamel expansions in powers of the interaction, rather than sums over particle numbers.
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We introduce a Grassmannian version of the Stratonovich-Hubbard transformation and derive a new functional-integral representation for the grand partition function F (β) of many-fermion N -particle systems. A new separable form for Z (β) is found using both a disentangling method for "the evolution operator" and the dynamical super-Weyl group SW ( N ). Employing fermionic coherent states the relevant trace is calculated exactly. The case of a separable potential is analyzed in some detail.
This book describes the theory and selected applications of one of the most important mathematical tools used in the theoretical investigation of collective excitations in statistical physics, such as occur in superfluidity, superconductivity, plasma dynamics, superradiation, and in phase transitions. The author, who is a distinguished physicist and leading researcher in this area, begins with an introduction to functional integral techniques in equilibrium statistical thermodynamics, and discusses the expression of partition functions and Green functions in terms of functional integrals. Subsequent sections deal with the application of functional integrals in superfluid Bose systems, systems with Coulomb interaction, and superfluid Fermi systems. The final section considers the application of the concept of Bose-condensation of auxiliary fields to the theory of crystals, heavy atoms and also to the theory of model Hamiltonians (BCS and Dicke models).