# Finite temperature CFT results for all couplings: O(N) model in 2+1 dimensions

## TL;DR

This paper analytically studies the finite-temperature behavior of the massless O(N) model in 2+1 dimensions across all couplings, revealing universal entropy ratios and complex frequency plane features in the large N limit.

## Contribution

It provides an exact large N solution for the O(N) model at finite temperature for all couplings, including entropy behavior and analytic structure of correlators.

## Key findings

- Entropy density decreases from Stefan-Boltzmann limit to 4/5 at infinite coupling.
- Retarded correlator exhibits two logarithmic branch cuts with no singularities.
- The ratio 4/5 and branch point locations are universal in a class of bosonic CFTs.

## Abstract

A famous example of gauge/gravity duality is the result that the entropy density of strongly coupled ${\cal N}=4$ SYM in four dimensions for large N is exactly 3/4 of the Stefan-Boltzmann limit. In this work, I revisit the massless O(N) model in 2+1 dimensions, which is analytically solvable at finite temperature $T$ for all couplings $\lambda$ in the large N limit. I find that the entropy density monotonically decreases from the Stefan-Boltzmann limit at $\lambda=0$ to exactly 4/5 of the Stefan-Boltzmann limit at $\lambda=\infty$. Calculating the retarded energy-momentum tensor correlator in the scalar channel at $\lambda=\infty$, I find that it has two logarithmic branch cuts originating at $\omega=\pm 4 T \ln \frac{1+\sqrt{5}}{2}$, but no singularities in the whole complex frequency plane. I show that the ratio 4/5 and the location of the branch points both are universal within a large class of bosonic CFTs in 2+1 dimensions.