Fermions on a torus knot

TL;DR

This paper explores the thermodynamics of noninteracting fermions constrained to a torus knot, analyzing key functions like free energy and magnetization, and discusses potential extensions to interacting fermions.

Contribution

It provides an analytical thermodynamic analysis of fermions on a torus knot, considering noninteracting particles and outlining approaches for interacting cases.

Findings

Analytical expressions for Helmholtz free energy, mean energy, magnetization, and susceptibility.

Insights into the behavior of Fermi energy in this topology.

Framework for extending analysis to interacting fermions.

Abstract

In this work, we investigate the effects of a nontrivial topology (and geometry) of a system considering \textit{interacting} and \textit{noninteracting} particle modes, which are restricted to follow a closed path over the torus surface. In order to present a prominent thermodynamical investigation of this system configuration, we carry out a detailed analysis using statistical mechanics within the grand canonical ensemble approach to deal with \textit{noninteracting} fermions. In an analytical manner, we study the following thermodynamic functions in such context: the Helmholtz free energy, the mean energy, the magnetization and the susceptibility. Further, we take into account the behavior of Fermi energy of the thermodynamic system. Finally, we briefly outline how to proceed in case of \textit{interacting} fermions.