Entanglement Hamiltonian and orthogonal polynomials

TL;DR

This paper explores the entanglement Hamiltonian in inhomogeneous free-fermion chains, revealing a connection to orthogonal polynomials and a local inverse temperature interpretation that approximates the entanglement spectrum.

Contribution

It introduces a novel approach linking entanglement Hamiltonians to orthogonal polynomials, enabling an exact commuting operator and a local temperature interpretation.

Findings

The commuting operator closely approximates the entanglement spectrum.

The local inverse temperature can be derived from conformal field theory.

Eigenvalues of the commuting operator match the entanglement entropy predictions.

Abstract

We study the entanglement Hamiltonian for free-fermion chains with a particular form of inhomogeneity. The hopping amplitudes and chemical potentials are chosen such that the single-particle eigenstates are related to discrete orthogonal polynomials of the Askey scheme. Due to the bispectral properties of these functions, one can construct an operator which commutes exactly with the entanglement Hamiltonian and corresponds to a linear or parabolic deformation of the physical one. We show that this deformation is interpreted as a local inverse temperature and can be obtained in the continuum limit via methods of conformal field theory. Using this prediction, the properly rescaled eigenvalues of the commuting operator are found to provide a very good approximation of the entanglement spectrum and entropy.