Fermion-boson transmutation and comparison of statistical ensembles in one dimension

TL;DR

This paper explains fermion-boson transmutation in one-dimensional systems without quantum field theory, applying it to derive exact thermodynamic results and compare statistical ensembles, highlighting differences between microcanonical, canonical, and grand canonical approaches.

Contribution

It provides a non-field-theoretic presentation of fermion-boson transmutation and applies it to obtain exact thermodynamic results in different statistical ensembles.

Findings

Exact low-temperature thermodynamics derived for 1D fermions

Comparison shows differences between microcanonical, canonical, and grand canonical ensembles

Fermion-boson transmutation simplifies understanding of 1D interacting fermions

Abstract

The theoretical description of interacting fermions in one spatial dimension is simplified by the fact that the low energy excitations can be described in terms of bosonic degrees of freedom. This fermion-boson transmutation (FBT) which lies at the heart of the Luttinger liquid concept is presented in a way which does not require a knowledge of quantum field theoretical methods. As the basic facts can already be introduced for noninteracting fermions they are mainly discussed. As an application we use the FBT to present exact results for the low temperature thermodynamics and the occupation numbers in the microcanonical and the canonical ensemble. They are compared with the standard grand canonical results.