Local and non-local properties of the entanglement Hamiltonian for two disjoint intervals

TL;DR

This paper investigates the entanglement Hamiltonian of two disjoint intervals in free-fermion chains, revealing long-range hopping effects and continuum limits, and identifies a commuting operator for eigenstate determination.

Contribution

It provides a detailed analysis of the entanglement Hamiltonian for disjoint intervals, including continuum expressions and the existence of a commuting operator, advancing understanding of non-local entanglement properties.

Findings

Long-range hopping terms are present in the entanglement Hamiltonian.

Continuum expressions can be derived from lattice models.

A commuting operator exists for the double interval case.

Abstract

We consider free-fermion chains in the ground state and the entanglement Hamiltonian for a subsystem consisting of two separated intervals. In this case, one has a peculiar long-range hopping between the intervals in addition to the well-known and dominant short-range hopping. We show how the continuum expressions can be recovered from the lattice results for general filling and arbitrary intervals. We also discuss the closely related case of a single interval located at a certain distance from the end of a semi-infinite chain and the continuum limit for this problem. Finally, we show that for the double interval in the continuum a commuting operator exists which can be used to find the eigenstates.