Aspects of quantum information in finite density field theory

TL;DR

This paper explores quantum information measures in finite density quantum field theories, revealing unique entanglement features, oscillations, and correlations specific to Fermi surfaces in 1+1 dimensions.

Contribution

It introduces a finite, non-monotonic entropic c-function, analyzes Friedel oscillations in Renyi entropies, and demonstrates how mutual information and relative entropy reveal Fermi surface and superselection sector effects.

Findings

The entropic c-function violates monotonicity in finite density theories.

Friedel oscillations are observed in Renyi entropies and explained via defect OPE.

Mutual information detects Fermi surface correlations at leading order.

Abstract

We study different aspects of quantum field theory at finite density using methods from quantum information theory. For simplicity we focus on massive Dirac fermions with nonzero chemical potential, and work in $1+1$ space-time dimensions. Using the entanglement entropy on an interval, we construct an entropic $c$-function that is finite. Unlike what happens in Lorentz-invariant theories, this $c$-function exhibits a strong violation of monotonicity; it also encodes the creation of long-range entanglement from the Fermi surface. Motivated by previous works on lattice models, we next calculate numerically the Renyi entropies and find Friedel-type oscillations; these are understood in terms of a defect operator product expansion. Furthermore, we consider the mutual information as a measure of correlation functions between different regions. Using a long-distance expansion previously developed by Cardy, we argue that the mutual information detects Fermi surface correlations already at leading order in the expansion. We also analyze the relative entropy and its Renyi generalizations in order to distinguish states with different charge and/or mass. In particular, we show that states in different superselection sectors give rise to a super-extensive behavior in the relative entropy. Finally, we discuss possible extensions to interacting theories, and argue for the relevance of some of these measures for probing non-Fermi liquids.