Renyi entropy of interacting thermal bosons in the large- N approximation

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 $\ensuremath{\rho}$. For a $\mathcal{U}(N)$-symmetric model, we show that the Renyi entropy of the system in the large-$N$ limit can be understood in terms of an effective noninteracting 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) the 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 subsystem $A$ and a different value outside it. The potential varies in space near the boundary of 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 noninteracting answer.