Nonrelativistic Dirac fermions on the torus

TL;DR

This paper analyzes a deformed Dirac fermion on the torus, exploring its spectrum and partition function, and derives Cardy-like formulas for high energy states, bridging gaps in understanding non-critical, nonrelativistic fermionic theories.

Contribution

It introduces a novel analysis of a massive, chemically potential-deformed Dirac fermion on the torus, extending modular form techniques beyond conformal theories.

Findings

Derived a Fourier-transform based representation of the torus free energy.

Obtained Cardy-like formulas for the high energy density of states.

Explored limits including massless relativistic and nonrelativistic fermions.

Abstract

Two dimensional conformal feld theories have been extensively studied in the past. When considered on the torus, they are strongly constrained by modular invariance. However, introducing relevant deformations or chemical potentials pushes these theories away from criticality, where many of their aspects are still poorly understood. In this note we make a step towards filling this gap, by analyzing the theory of a Dirac fermion on the torus, deformed by a mass term and a chemical potential for the particle number symmetry. The theory breaks conformal and Lorentz invariance, and we study its spectrum and partition function. We also focus on two limits that are interesting on their own right: a massless relativistic fermion with nonzero chemical potential (a simple model for CFTs at finite density), and nonrelativistic Schrodinger fermions (of relevance in condensed matter systems). Taking inspiration from recent developments in massive modular forms, we obtain a representation of the torus free energy based on Fourier-transforming over a twisted boundary condition. This dual representation fullfills many properties analogous to modular invariance in CFTs. In particular, we use this result to derive Cardy-like formulas for the high energy density of states of these theories.