Gross-Witten-Wadia transition in a matrix model of deconfinement

We study the deconfining phase transition at nonzero temperature in a $SU(N)$ gauge theory, using a matrix model which was analyzed previously at small $N$. We show that the model is soluble at infinite $N$, and exhibits a Gross-Witten-Wadia transition. In some ways, the deconfining phase transition is of first order: at a temperature $T_d$, the Polyakov loop jumps discontinuously from 0 to $\frac{1}{2}$, and there is a nonzero latent heat $\sim N^2$. In other ways, the transition is of second order: e.g., the specific heat diverges as $C\sim 1/(T-T_d{)}^{3/5}$ when $T\to T_d^{+}$. Other critical exponents satisfy the usual scaling relations of a second order phase transition. In the presence of a nonzero background field $h$ for the Polyakov loop, there is a phase transition at the temperature $T_h$ where the value of the $\mathrm{\text{loop}}=\frac{1}{2}$, with $T_h<T_d$. Since $\partial C/\partial T\sim 1/(T-T_h{)}^{1/2}$ as $T\to T_h^{+}$, this transition is of third order. These properties, closely analogous to those on a femtosphere at zero coupling, suggest that in infinite volume, the Gross-Witten-Wadia transition may be an infrared stable fixed point of a $SU(\infty )$ gauge theory.

Figure 1

Plot of the specific heat, divided by $(N^2-1)T^3$, for different values of $N$.