Resolving modular flow: a toolkit for free fermions

TL;DR

This paper introduces a new resolvent-based formula for modular flow in free chiral fermions in 1+1 dimensions, enabling analysis beyond symmetric cases and revealing diverse local and non-local behaviors.

Contribution

It provides the first explicit resolvent-based formula for modular flow in free fermions, extending understanding beyond conformal symmetry and across various geometries and boundary conditions.

Findings

Derived modular two-point functions consistent with KMS condition

Revealed local and non-local behavior depending on temperature and boundary conditions

Presented novel results for disjoint regions on different geometries

Abstract

Modular flow is a symmetry of the algebra of observables associated to spacetime regions. Being closely related to entanglement, it has played a key role in recent connections between information theory, QFT and gravity. However, little is known about its action beyond highly symmetric cases. The key idea of this work is to introduce a new formula for modular flows for free chiral fermions in $1+1$ dimensions, working directly from the \textit{resolvent}, a standard technique in complex analysis. We present novel results -- not fixed by conformal symmetry -- for disjoint regions on the plane, cylinder and torus. Depending on temperature and boundary conditions, these display different behaviour ranging from purely local to non-local in relation to the mixing of operators at spacelike separation. We find the modular two-point function, whose analytic structure is in precise agreement with the KMS condition that governs modular evolution. Our ready-to-use formulae may provide new ingredients to explore the connection between spacetime and entanglement.