Entanglement entropy of two disjoint intervals separated by one spin in a chain of free fermion

TL;DR

This paper computes the entanglement entropy of two disjoint fermion chain segments separated by one site, using advanced mathematical techniques to analyze the asymptotic behavior of related matrices.

Contribution

It provides a rigorous calculation of entanglement entropy for a specific non-contiguous subsystem in a free fermion chain, extending previous formulas to this configuration.

Findings

Calculated mutual information between two blocks separated by one site.

Analyzed asymptotic behavior of inverse Toeplitz matrices with Fisher-Hartwig symbols.

Applied Riemann-Hilbert method for asymptotic analysis.

Abstract

We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, \texttt{arXiv:1104.1004}, for the reduced density of states of two disjoint intervals with lattice sites $P=\{1,2,\dots,m\}\cup\{2m+1,2m+2,\dots, 3m\}$, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher-Hartwig symbol using the the Riemann--Hilbert method.