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

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