Entanglement Spectrum in General Free Fermionic Systems

TL;DR

This paper develops a Riemann-Hilbert mathematical framework to analyze the entanglement spectrum in free fermionic systems with two disjoint intervals, providing insights into quantum correlations in such systems.

Contribution

It introduces a novel Riemann-Hilbert approach to compute entanglement measures in free fermionic systems with disjoint intervals in the thermodynamic limit.

Findings

Computed entanglement and negativity spectra for large interval separations.

Provided a method to expand entanglement measures in the ratio of interval distance to size.

Offered a mathematical tool for analyzing quantum correlations in fermionic systems.

Abstract

The statistical mechanics characterization of a finite subsystem embedded in an infinite system is a fundamental question of quantum physics. Nevertheless, a full closed form { for all required entropic measures} does not exist in the general case even for free systems when the finite system in question is composed of several disjoint intervals. Here we develop a mathematical framework based on the Riemann-Hilbert approach to treat this problem in the one-dimensional case where the finite system is composed of two disjoint intervals and in the thermodynamic limit (both intervals and the space between them contains an infinite number of lattice sites and the result is given as a thermodynamic expansion). To demonstrate the usefulness of our method, we compute the change in the entanglement and negativity namely the spectrum of eigenvalues of the reduced density matrix with our without time reversal of one of the intervals. We do this in the case that the distance between the intervals is much larger than their size. The method we use can be easily applied to compute any power in an expansion in the ratio of the distance between the intervals to their size. {We expect these results to provide the necessary mathematical apparatus to address relevant questions in concrete physical scenarios, namely the structure and extent of quantum correlations in fermionic systems subject to local environment.