  We consider the reduced density matrix for a disc in the ground state of the
interacting fixed points of the large-N O(N) vector model in 2+1 dimensions.
Using the map to the free energy on $\mathbb{H}_2 \times S^1$, we show that
there is an instability in the R\'enyi entropies at $n = 1$, in the $N
\rightarrow \infty$ limit, indicating a phase transition at or near this
R\'enyi parameter. We close with a discussion of the finite-N case.
