Interacting CFTs for all couplings: Thermal versus Entanglement Entropy at Large $N$

In this paper, I calculate the large $N$ limit of marginal $O(N)$ models with non-polynomial potentials in arbitrary odd dimensions $d$. This results in a new class of interacting pure conformal field theories (CFTs) in $d=3+4n$ for any $n \in \mathbb{Z}_+$. Similarly, in $d=3+4n$ I calculate the thermal entropy for all couplings on $R^{2+4n} \times S^1$ for $n=0,1,2,3$. In 2+1 dimensions I find the strong-to-weak coupling ratio of the thermal entropy to be 4/5, matching recent results, and further extend this analysis to higher odd dimensions. Next, I calculated the vacuum entanglement entropy $s^d_{\text{EE}}$ on $S^{d-2}$ for all couplings in arbitrary odd $d$ in the large N limit. I find the vacuum entanglement entropy on $S^{d-2}$ to be not only solvable but also constant for all couplings $\lambda$. Thus, in the large $N$ limit, the vacuum entanglement entropy on $S^{d-2}$ for odd $d$ is constant for all $\lambda$, in contrast to the thermal entropy which is shown to also be monotonically decreasing with $\lambda$ in $d=3+4n$.