Entanglement entropy for a Maxwell field: Numerical calculation on a two dimensional lattice

TL;DR

This paper numerically investigates entanglement entropy for a Maxwell field in 2+1 dimensions on a lattice, revealing universal terms, a novel divergent component, and proposing a generalized strong subadditivity principle.

Contribution

It extends entropy calculation methods to gauge fields with complex algebraic structures and uncovers new phenomena in entanglement properties of Maxwell fields.

Findings

Mutual information has a well-defined continuum limit.

Identified a logarithmic divergence with topological coefficient.

Discovered a new form of strong subadditivity for non-tensor product algebras.

Abstract

We study entanglement entropy (EE) for a Maxwell field in 2+1 dimensions. We do numerical calculations in two dimensional lattices. This gives a concrete example of the general results of our recent work on entropy for lattice gauge fields using an algebraic approach. To evaluate the entropies we extend the standard calculation methods for the entropy of Gaussian states in canonical commutation algebras to the more general case of algebras with center and arbitrary numerical commutators. We find that while the entropy depends on the details of the algebra choice, mutual information has a well defined continuum limit. We study several universal terms for the entropy of the Maxwell field and compare with the case of a massless scalar field. We find some interesting new phenomena: An "evanescent" logarithmically divergent term in the entropy with topological coefficient which does not have any correspondence with ultraviolet entanglement in the universal quantities, and a non standard way in which strong subadditivity is realized. Based on the results of our calculations we propose a generalization of strong subadditivity for the entropy on some algebras that are not in tensor product.