Finite Temperature Schwinger Model with Chirality Breaking Boundary Conditions

TL;DR

This paper solves the finite temperature N_f-flavor Schwinger Model with boundary conditions that break chirality, providing analytical results on thermal correlators, effective action, and chiral condensate dependence on temperature and system size.

Contribution

It introduces a boundary-condition approach to study chiral symmetry breaking in the Schwinger Model at finite temperature, avoiding small mass approximations and enabling analytical calculations.

Findings

Boundary conditions induce a CP-odd theta-term.

Chiral condensate depends on the order of limits for temperature and system size.

Analytical expressions for thermal correlators and effective action.

Abstract

The $N_f$-flavour Schwinger Model on a finite space $0\leq x^1\leq L$ and subject to bag-type boundary-conditions at $x^1=0$ und $x^1=L$ is solved at finite temperature $T=1/\beta$. The boundary conditions depend on a real parameter $\theta$ and break the axial flavour symmetry. We argue that this approach is more appropriate to study the broken phases than introducing small quark masses, since all calculations can be performed analytically. In the imaginary time formalism we determine the thermal correlators for the fermion-fields and the determinant of the Dirac-operator in arbitrary background gauge-fields. We show that the boundary conditions induce a CP-odd $\theta$-term in the effective action. The chiral condensate, and in particular its T- and L- dependence, is calculated for $N_f$ fermions. It is seen to depend on the order in which the two lengths $\beta=1/T$ and $L$ are sent to infinity.