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We consider the reduced density matrix for a disc in the ground state of theinteracting 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 thatthere is an instability in the R\'enyi entropies at $n = 1$, in the $N\rightarrow \infty$ limit, indicating a phase transition at or near thisR\'enyi parameter. We close with a discussion of the finite-N case.

On phase transitions in the R\'enyi entropies of 2+1-d large-N interacting vector models