Finite-temperature phase transitions of third and higher order in gauge theories at large $N$

We study phase transitions in $SU(\infty )$ gauge theories at nonzero temperature using matrix models. Our basic assumption is that the effective potential is dominated by double trace terms for the Polyakov loops. As a function of the various parameters, related to terms linear, quadratic, and quartic in the Polyakov loop, the phase diagram exhibits a universal structure. In a large region of this parameter space, there is a continuous phase transition whose order is larger than second. This is a generalization of the phase transition of Gross, Witten, and Wadia. Depending upon the detailed form of the matrix model, the eigenvalue density and the behavior of the specific heat near the transition differ drastically. We speculate that in the pure gauge theory, although the deconfining transition is thermodynamically of first order, it can be nevertheless conformally symmetric at infinite $N$.

Figure 1

Phase diagrams for the matrix model in Eq. (23), this figure and Fig. 2. This figure shows zero external field, $h=0$, varying the mass term, $a_1$, and the quartic coupling, $b_1$, for the first Polyakov loop, ${\rho }_1$. The red dashed line is a first-order transition; the blue dash-dotted line is a second-order transition; the green solid line is for the generalized Gross-Witten-Wadia (GWW) transition. The critical first order is located at the origin where all three phase transition lines meet. For illustration we use the model with a Vandermonde determinant for the red dashed line, $s=1$ in Sec. 4. The confined phase is the region to the right of the red dashed and blue dash-dotted lines, the deconfined to the left of the lines. The green shaded and red hatched regions are the projections of the surfaces of the GWW and first-order phase transitions onto the $h=0$ plane, respectively.

Figure 2

As in Fig. 1, versus the background field $h$ and the quadratic coupling $a_1$. We also illustrate the (small) difference between $s=1$ and $s=4$ in the figure on the right-hand side.

Figure 3

The eigenvalue density as a function of $\omega$. Below and at the GWW point the density is independent of the model and driven by the first Polyakov loop, ${\rho }_1=\omega$. Above the GWW point, the density depends on the coefficients of the double trace terms.

Figure 4

Numerical computations of the eigenvalues as a function of $\delta \omega =\omega -1/2$ in the model given in Eq. (56) with $s=2$, 4 and $N=55$. This demonstrates a pileup of eigenvalues when $s=4$, which motivated the ansatz of Eq. (82).

Figure 5

The derivatives of the free energy near the GWW point. The plot on the top left-hand side shows that the transition for $s=1$ is of third order. The plot on the top right-hand side shows that $s=2$ and 3 are transitions of fourth order. The plot on the bottom shows that $s=4$ are transition of fifth order.

Figure 6

The first Polyakov loop ${\rho }_1$ as a function of the coefficient of its quadratic term, $-a_1$. From bottom to top, $s=1$, 2, 3, 4.

Figure 7

The dependence of the first Polyakov loop ${\rho }_1$ on the quartic coupling $b_1$ in zero external field: from bottom to top, $s=1$, 2, 3, 4.

Figure 8

The dependence of the first-order line in zero field, $h=0$: from left to right, $s=1$, 2, 3, and 4. The blue solid line for $s=1$ corresponds to the red dashed line in Fig. 1.

Figure 9

Phase diagram as a function of heavy quark mass $m$. For an infinite number of colors and flavors, the phase diagram depends on the value of quartic coupling $b_1$ for the first Polyakov loop ${\rho }_1$.