# The entanglement spectrum of chiral fermions on the torus

## TL;DR

This paper computes the entanglement spectrum of chiral fermions on a torus, revealing complex non-local structures and the effects of zero modes on the modular Hamiltonian across various interval configurations.

## Contribution

It provides a detailed derivation of the reduced density matrix and modular Hamiltonian for chiral fermions on a torus, including the impact of zero modes and bi-local couplings.

## Key findings

- Infinite series of bi-local couplings near endpoints
- Condensation of points in the presence of zero modes at low temperatures
- Non-locality becomes continuous due to zero mode effects

## Abstract

We determine the reduced density matrix of chiral fermions on the torus, for an arbitrary set of disjoint intervals and generic torus modulus. We find the resolvent, which yields the modular Hamiltonian in each spin sector. Together with a local term, it involves an infinite series of bi-local couplings, even for a single interval. These accumulate near the endpoints, where they become increasingly redshifted. Remarkably, in the presence of a zero mode, this set of points 'condenses' within the interval at low temperatures, yielding continuous non-locality.