  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.
