Finite-Temperature Conformal Field Theory Results for All Couplings: $\mathrm{O}(N)$ Model in $2+1$ Dimensions

A famous example of gauge-gravity duality is the result that the entropy density of the strongly coupled $\mathcal{N}=4$ supersymmetric Yang-Mills theory in four dimensions for large $N$ is exactly $3/4$ of the Stefan-Boltzmann limit. In this work, I revisit the massless $\mathrm{O}(N)$ model in $2+1$ dimensions, which is analytically solvable at a 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 $\omega =\pm 4T\ln (1+\sqrt{5}/2)$ 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 conformal field theories in $2+1$ dimensions.

Figure 1

The ratio $s/s_{\text{free}}$ for the $\mathrm{O}(N)$ model in $2+1$ dimensions in the large $N$ limit for all temperatures and couplings. Results are shown using a compactified interval $(T/\lambda +T)\in [0,1]$ in order to show all coupling values. Arrows indicate the free theory result (19) and the strong-coupling result (20). See the text for details.