Modular conjugation for the chiral fermion in multicomponent regions on the torus

TL;DR

This paper derives an explicit expression for the modular conjugation of a massless Dirac field in a thermal state on a circle, revealing non-local behavior and a new contribution from the purification process in multicomponent regions.

Contribution

It provides the first explicit computation of modular conjugation for a thermal state on a torus in 1+1 dimensions, highlighting non-local effects and the role of the second world algebra.

Findings

Explicit modular conjugation expression on the torus

Non-local behavior in the modular conjugation for thermal states

Identification of a new contribution from the second world algebra

Abstract

We continue the study of the Tomita-Takesaki modular conjugation for a massless Dirac field in a generic multicomponent region in $1+1$ spacetime dimensions. In this paper we focus on the computations for a thermal state on a circle, namely on the euclidean torus. By analytic continuation from the modular flow we arrive at an explicit expression for the modular conjugation in this scenario and derive its relevant limits. In contrast to the case of the vacuum on the line, this new result has a non-local behaviour even for connected regions. It also presents a novel contribution coming from the purification one has to introduce in order to deal with a mixed state: a term that maps the algebra of operators of the region to a copy of the global one, the so called 'second world' algebra.