Modular Hamiltonian and modular flow of massless fermions on a cylinder

TL;DR

This paper explicitly computes the modular flow and Hamiltonian for massless free fermions on a cylinder, revealing new non-local behaviors and the effects of state purity on modular data.

Contribution

It provides the first explicit analysis of modular flow and Hamiltonian for generic periodic ground states of massless fermions on a cylinder, including non-local and chirality-mixing effects.

Findings

Modular flow and Hamiltonian are generally non-local.

In the pure state limit, modular data becomes local.

Modular flow can mix chiralities even in local cases.

Abstract

We determine explicitly the modular flow and the modular Hamiltonian for massless free fermions in diamonds on a cylinder in 1+1 dimensions. We consider both periodic and antiperiodic boundary conditions, the ground state in the antiperiodic case and the most general family of quasi-free zero-energy ground states in the periodic case, which depend on four parameters and are generally mixed. While for the antiperiodic ground state and one periodic ground state (the maximally mixed zero-temperature state) the modular data is known, our results for the generic ground state in the periodic case are completely new. We find that generically both the modular flow and the modular Hamiltonian are non-local, and we show that in the parametric limit where the state becomes pure the modular data becomes local. Moreover, even in the local case the modular flow generically mixes the two chiralities. This kind of behavior has not been observed previously.