Thermal Fluctuations of Induced Fermion Number

TL;DR

This paper investigates how finite temperature affects the induced fermion number, revealing that it becomes non-topological and fluctuates, contrasting with the sharp, topological zero-temperature case, through analysis of specific 1+1D models.

Contribution

It provides a detailed analysis of finite temperature effects on induced fermion number, highlighting the transition from topological to non-topological behavior in different models.

Findings

Finite temperature induces fluctuations in fermion number.

In certain backgrounds, fermion number remains topological at finite temperature.

In general, finite temperature causes the fermion number to become non-topological and non-sharp.

Abstract

We analyze the phemomenon of induced fermion number at finite temperature. At finite temperature, the induced fermion number $<N>$ is a thermal expectation value, and we compute the finite temperature fluctuations, $(\Delta N)^2=<N^2>-<N>^2$. While the zero temperature induced fermion number is topological and is a sharp observable, the finite temperature induced fermion number is generically nontopological, and is not a sharp observable. The fluctuations are due to the mixing of states inherent in any finite temperature expectation value. We analyze in detail two different cases in 1+1 dimensional field theory: fermions in a kink background, and fermions in a chiral sigma model background. At zero temperature the induced fermion numbers for these two cases are very similar, but at finite temperature they are very different. The sigma model case is generic and the induced fermion number is nontopological, but the kink case is special and the fermion number is topological, even at finite temperature. There is a simple physical interpretation of all these results in terms of the spectrum of the fermions in the relevant background, and many of the results generalize to higher dimensional models.