The large $N$ vector model with angular velocity

Abstract

We study the free energy of a critical vector model at large $N$ on $S^1\times S^2$ with an angular velocity $\hat{\mu }$ without the singlet constraint. We study the model for which the large $N$ dynamics is controlled by the uniform saddle point of the auxiliary field arising in the Hubbard-Stratanovich transformation. The leading high-temperature behaviour is determined analytically both as an expansion about $\hat{\mu }r=0$ and ${\hat{\mu }}^2{r}^2=1$ where $r$ is the radius of the sphere. We supplement the analytic results with a numerical analysis that agrees with both the analytical expansions in their respective regimes and smoothly interpolates between them. The leading high-temperature contribution to the free energy develops a pole at ${\hat{\mu }}^2{r}^2=1$, in agreement with expectations from the thermal effective field theory. Its residue coincides with that of the massless free theory. Sub-leading terms, however, exhibit non-analytic dependence on the angular velocity and distinguish the critical fixed-point result from the free theory answer. The residue at the pole can also be obtained by placing the model on the pp-wave geometry. We show that the residue agrees with that obtained from the direct computation. The free energy of the model connects the non-trivial fixed point of the $O(N)$ model at $\hat{\mu }r=0$ to its free fixed point at ${\hat{\mu }}^2{r}^2=1$.

46 pages, 4 figures, clarification about the model added, appendix D about the non-uniformity of O(N) saddle added, acknowledgements added