Instantons in Schwinger Model

TL;DR

This paper studies instanton solutions in the Schwinger model, analyzing their properties at different temperatures and their role in the fermion condensate and correlators, revealing their static nature and impact on the high-temperature phase.

Contribution

It provides a detailed interpretation of instantons in the Schwinger model, especially their behavior at high temperature and their contribution to the partition function.

Findings

Instantons have a size of order the photon Compton wavelength.

At high temperature, instantons become quasistatic and resemble walls.

The instanton gas density at small fermion mass is proportional to m times an exponential factor.

Abstract

The known calculations of the fermion condensate $<\bar{\psi}\psi>$ and the correlator $<\bar{\psi}\psi(x) ~\bar{\psi}\psi(0)>$ have been interpreted in terms of {\em localized} instanton solutions minimizing the {\em effective} action. Their size is of order of massive photon Compton wavelength $\mu^{-1}$. At high temperature, these instantons become quasistatic and present the 2-dimensional analog of the `walls' found recently in 4-dimensional gauge theories. In spite of the static nature of these solutions, they should not be interpreted as `thermal solitons' living in Minkowski space: the mass of these would-be solitons does not display itself in the physical correlators. At small but nonzero fermion mass, the high-T partition function of $QED_2$ is saturated by the rarefied gas of instantons and antiinstantons with density $\propto m~\exp\{-S^{inst.}\}~=~m~\exp\{-\pi T/\mu\}$ to be confronted with the dense strongly correlated instanton-antiinstanton liquid saturating the partition function at $T=0$.