Renyi Entropy of Interacting Thermal Bosons in Large $N$ Approximation

TL;DR

This paper develops a large N approximation framework using a Wigner function approach to analyze the Renyi entropy of interacting thermal bosons, revealing how interactions modify entanglement entropy in different temperature regimes.

Contribution

It introduces a self-consistent mean field method to compute Renyi entropy for interacting bosons, extending the understanding of entanglement in many-body thermal systems.

Findings

Interaction effects peak at intermediate temperatures.

Potential profiles near subsystem boundaries influence entropy.

High and low temperature behaviors resemble non-interacting systems.

Abstract

Using a Wigner function based approach, we study the Renyi entropy of a subsystem $A$ of a system of Bosons interacting with a local repulsive potential. The full system is assumed to be in thermal equilibrium at a temperature $T$ and density $\rho$. For a ${\cal U}(N)$ symmetric model, we show that the Renyi entropy of the system in the large $N$ limit can be understood in terms of an effective non-interacting system with a spatially varying mean field potential, which has to be determined self consistently. The Renyi entropy is the sum of two terms: (a) Renyi entropy of this effective system and (b) the difference in thermal free energy between the effective system and the original translation invariant system, scaled by $T$. We determine the self consistent equation for this effective potential within a saddle point approximation. We use this formalism to look at one and two dimensional Bose gases on a lattice. In both cases, the potential profile is that of a square well, taking one value in the subsystem $A$ and a different value outside it. The potential varies in space near the boundary of the subsystem $A$ on the scale of density-density correlation length. The effect of interaction on the entanglement entropy density is determined by the ratio of the potential barrier to the temperature and peaks at an intermediate temperature, while the high and low temperature regimes are dominated by the non-interacting answer.