Geometric Entropy of Nonrelativistic Fermions and Two Dimensional Strings

TL;DR

This paper investigates the geometric entropy of nonrelativistic fermions and its relation to two-dimensional string theory, revealing nonperturbative effects, divergences, and potential stability issues at high temperatures.

Contribution

It demonstrates the ultraviolet finiteness and infrared divergence of geometric entropy for 2D fermions and links these effects to nonperturbative phenomena in string theory.

Findings

Geometric entropy is UV finite for finite Fermi energies.

Infrared divergence occurs in the geometric entropy.

High-temperature thermodynamic entropy shows nonperturbative effects and potential instability.

Abstract

We consider the geometric entropy of free nonrelativistic fermions in two dimensions and show that it is ultraviolet finite for finite fermi energies, but divergent in the infrared. In terms of the corresponding collective field theory this is a {\em nonperturbative} effect and is related to the soft behaviour of the usual thermodynamic entropy at high temperatures. We then show that thermodynamic entropy of the singlet sector of the one dimensional matrix model at high temperatures is governed by nonperturbative effects of the underlying string theory. In the high temperature limit the ``exact'' expression for the entropy is regular but leads to a negative specific heat, thus implying an instability. We speculate that in a properly defined two dimensional string theory, the thermodynamic entropy could approach a constant at high temperatures and lead to a geometric entropy which is finite in the ultraviolet.