Finite Temperature Path Integral Method for Fermions and Bosons: a Grand Canonical Approach

TL;DR

This paper introduces a new computationally efficient path integral method for calculating fermionic and bosonic density matrices at all temperatures within the Grand Canonical Ensemble, using a Hankel function representation.

Contribution

It presents a novel Hankel function-based representation of the universal temperature-dependent functional, improving computational efficiency for density matrix calculations.

Findings

Hankel function representation simplifies calculations

Temperature rescaling is more efficient

Method applied to noninteracting electrons in a confining potential

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 fermionic density matrix can be expressed via an integration over a novel representation of the universal temperature dependent functional. While several representations for the universal functional have already been developed, they are usually presented in a form inconvenient for computer calculations. In this work we discuss a new representation for the universal functional in terms of Hankel functions which is advantageous for computational applications. Temperature scaling for the universal functional and its derivatives are also introduced thus allowing an efficient rescaling rather then recalculation of the functional at different temperatures. A simple illustration of the method of calculation of density profiles in Grand Canonical ensemble is presented using a system of noninteracting electrons in a finite confining potential.