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Europhysics Letters (EPL)

Novel Method for Calculating the Path Integral for the Partition Function of a Many-Fermion System

Alexander V. Gorokhov, Joseph L. Birman

Published Jul 15, 1991• 0 citations
Publisher

Abstract

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.

DOI
10.1209/0295-5075/15/6/009

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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.
Abstract A many‐fermion interacting system is investigated within the scenario of the Feynman path integral representation of quantum mechanics. Short‐time propagator algorithms and a basis set, closely related to the coherent states, are used to obtain the many‐body analytic propagator. A second‐quantized Hamiltonian involving a restricted set of two‐body interactions and the whole set of Coulomb interactions are separately and shown to lead to an exact and an approximate propagator, respectively. In the latter case, use of a grand canonical ensemble allows the grand partition function and the density operator matrix to be readily obtained. No further approximations are required in the calculation of the trace of the evolution operator involved in the evaluation of statistical expectation values. © 1994 John Wiley & Sons, Inc.
We exploit the grassmannian nature of the variables involved in the path integral expression of the grand canonical partition function for self--interacting fermionic models to show, in one-space dimension, a general relation among the terms of it expansion in the high temperature limit and a combination of co-factors of a suitable matrix with commuting entries. As an application, we apply this framework to calculate the exact coefficients, up to order β^3, of the expansion of the grand canonical partition function for the Hubbard model in d=(1+1) in the high temperature limit. The results are valid for any set of parameters that characterize the model.
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Charret et al. applied the properties of Grassmann generators to develop a new method to calculate the coefficientes of the high temperature expansion of the grand canonical partition function of self-interacting fermionic models on d-dimensions (d > or = 1). The method explores the anti-commuting nature of fermionic fields and avoids the calculation of the fermionic path integral. We apply this new method to the relativistic free Dirac fermions and recover the known results in the literature without the beta-independent and mu-independent infinities that plague the continuum path integral formulation.
Quantum statistical mechanics has developed primarily through two approaches, pioneered by Gibbs and Feynman, respectively. In Gibbs' method one calculates partition functions from phase-space integrations or sums over stationary states. Alternatively, in Feynman's approach, the focus is on the path-integral formulation. The Hubbard-Stratonovich transformation leads to a functional-integral formulation for calculating partition functions. We outline here the functional integral approach to quantum statistical mechanics, including generalizations and improvements to Hubbard's formulation. We show how the dimensionality of the integrals is reduced exactly, how the problem of assuming an unknown canonical transformation is avoided, how the reality of the partition function in the complex representation is guaranteed, and how the extremum conditions are simplified. This formulation can be applied to general systems, including superconductors.