When and why do zero-modes cause a divergence in the entanglement entropy?

TL;DR

This paper investigates the relationship between zero-modes and divergences in ground state entanglement entropy across Bosonic and Fermionic chains, identifying conditions and signatures for when divergences occur.

Contribution

It provides a comprehensive analysis of zero-modes' role in entanglement entropy divergences, including necessary and sufficient conditions and extensions to Fermionic systems.

Findings

Zero modes are necessary but not sufficient for entropy divergences.

Zero modes of the entanglement Hamiltonian serve as signatures for divergences.

Methods for Bosonic chains have Fermionic analogs and are supported by numerical studies.

Abstract

We examine the correlations between divergences in ground state entanglement entropy and emergent zero-modes of the underlying Hamiltonian in the context of one-dimensional Bosonic and Fermionic chains. Starting with a pair of coupled Bosonic degrees of freedom, we show that zero modes are necessary, but not sufficient for entanglement entropy divergences. We then list sufficient conditions that identify divergences. Next, we extend our analysis to Bosonic chains, where we demonstrate that zero modes of the entanglement Hamiltonian provide a signature for divergences independent of the entanglement Hamiltonian. We then generalize our results to one-dimensional Fermionic lattices for a chain of staggered Fermions which is a discretized version of the Dirac field. We find that the methods detailed for Bosonic chains have Fermionic analogs and follow this up with a numerical study of the entanglement in the Fermionic chain. Finally, we discuss our results in light of the factorization algebra theorem.