Equation of state of two-dimensional Yang-Mills theory

We study the pressure, $P$, of $\mathrm{SU}(N)$ gauge theory on a two-dimensional torus as a function of area, $A=l/t$. We find a crossover scale that separates the system on a large circle from a system on a small circle at any finite temperature. The crossover scale approaches zero with increasing $N$ and the crossover becomes a first-order transition as $N\to \infty$ and $l\to 0$ with the limiting value of $\frac{2Pl}{(N-1)t}$ depending on the fixed value of $Nl$.

Figure 1

The equation of state for SU(2) gauge theory on a two-dimensional torus is shown as the solid curve. The asymptotic values of $Pl/t$ at small area is $-0.5$. At very large area, $Pl/t$ behaves as $0.75\exp (-0.75l/t)l/t$, which is shown as the dotted curve. There is a crossover between the two limits.

Figure 2

Behavior of Casimir as a function of $\lambda$. The upper solid curve is the maximum value of Casimir given a value of $\lambda$, as a function of $\lambda$. Similarly, the lower solid curve is the minimum value of Casimir given a value of $\lambda$, as a function of $\lambda$. The dotted line is where $C_r^{(2)}=N^2$.

Figure 3

$Q$ as a function of the scaled area $\alpha =NA$ is shown in the top panel. It is seen that $Q$ as a function of $\alpha$ has a large-$N$ limit. For very small values of $\alpha$, $Q$ approaches 1. In the bottom panel, the large area behavior of $Q$ in the large-$N$ limit is shown. In this case, $QN$ behaves as $\exp (-\sigma \alpha /N)$.

Figure 4

Susceptibility $\chi$ as a function of the scaled area $\alpha =NA$. The crossover coupling ${\alpha }_c$ is shown by the vertical line.

Figure 5

Scatter plot of $C_r^{(2)}/N^2$ versus $\lambda /N$ at various area $A$. The top panel is for $N=47$ and the bottom for $N=101$. Each point corresponds to a $C_r^{(2)}$ and $\lambda$ measured in the course of a Monte Carlo simulation at a particular $\alpha$ specified by the color. The upper and lower solid curves are the maximum and minimum values of $C_r^{(2)}$ at a given $\lambda$, respectively.

Figure 6

The large-$N$ limit of $Q$ as a function of $\alpha$.

Figure 7

Phase diagram. Various approaches to vanishing area at large $N$ are indicated by lines with arrows. The critical value of the slope $AN={\alpha }_c$ is shown as the dot-dashed line. For values of $AN\gg {\alpha }_c$ (the dotted line), $Q$ decays exponentially with area. For values of $AN\ll {\alpha }_c$ (the dashed line), $Q\approx 1$. In particular, when $A$ is reduced to 0 after taking the large-$N$ limit (i.e., along the $y$ axis), $Q$ vanishes. When the two limits are interchanged (i.e., along the $x$ axis), $Q$ becomes 1.