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arXiv:cond-mat/9807059 · Physical review. B, Condensed matter

Nonzero-temperature path-integral method for fermions and bosons: A grand canonical approach

Maksim A. Skorobogatiy, John D. Joannopoulos

Published Jul 15, 1999• 4 citations
arXivHTMLPDF

Abstract

The calculation of the density matrix for fermions and bosons in the grand canonical ensemble allows an efficient way for the inclusion of fermionic and bosonic statistics at all temperatures. It is shown that in a path- integral formulation the one-particle density matrix can be expressed via an integration over a novel representation of the universal temperature-dependent functional. In this paper we discuss a representation for the universal functional in terms of Hankel functions which is convenient for computational applications. Temperature scaling for the universal functional and its derivatives is also introduced thus allowing an efficient rescaling rather then recalculation of the functional at different temperatures. We expect that our method will give rise to a numerically efficient path-integral approach for calculation of a density matrix in the grand canonical ensemble.

DOI
10.1103/physrevb.60.1433

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A new method ( PI-DFT ) which combines path integrals and density functional theory is proposed as a pathway to many fields of physics. Within path integral theory it is possible to construct particle densities without explicitly calculating individual wave functions. These densities can directly be used as an input to energy density functionals. Thus our method makes full use of the theorem of Hohenberg, Kohn and Sham which shows, that the energy of a many electron system only depends on the particle density. At glance we present a recursion formula for the calculation of many fermion and boson particle densities from one-particle densities at a set of different temperatures. For both statistics the numerical effort of our method increases only with the square of the particle number.
The simple relationship between the wavefunctions of a system of impenetrable one-dimensional bosons and impenetrable one-dimensional fermions is exploited to derive an expansion of the boson reduced density matrices in terms of the fermion reduced density matrices and vice versa. This expansion is independent of the statistical ensemble used, and of the interparticle potential (subject to the impenetrability condition). The special case of zero impenetrability radius with no other forces is treated in detail, using the grand canonical ensemble. This leads to a natural generalization of a formula previously found for zero temperature only, for the one-particle reduced density matrix.
This chapter investigates a quantum gas in the grand-canonical ensemble with the grand canonical Hamiltonian ensemble. The path integral formalism pioneered by Feynman has proven to be an extremely useful tool for understanding and handling quantum mechanics, quantum field theory, and statistical mechanics. The path integral approach is a powerful tool that considers nonperturbative calculations for the investigation of many-particle systems. It is worth noting that while the path integral approach seems to be quite cumbersome in quantum mechanics, it does provide a powerful tool in quantum field theory. The entire aforementioned derivation is calculated in order to justify the evaluation of the path integral at equal times. The chapter examines the example of an atomic Fermi gas by considering an electron gas. The Fermi liquid concept is the deviation of the electron gas from the ideal gas behavior. It utilizes the so-called diagram technique to calculate some quantities.
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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