Monte Carlo study of a $U\left(1\right)\times U\left(1\right)$ system with $\pi$-statistical interaction

We study a $U\left(1\right)\times U\left(1\right)$ system with two species of loops with mutual $\pi$ statistics in (2 + 1) dimensions. We are able to reformulate the model in a way that can be studied by Monte Carlo and we determine the phase diagram. In addition to a phase with no loops, we find two phases with only one species of loop proliferated. The model has a self-dual line, a segment of which separates these two phases. Everywhere on the segment, we find the transition to be first-order, signifying that the two loop systems behave as immiscible fluids when they are both trying to condense. Moving further along the self-dual line, we find a phase where both loops proliferate, but they are only of even strength and, therefore, avoid the statistical interactions. We study another model, which does not have this phase, and also find first-order behavior on the self-dual segment.

Figure 1

The phase diagram for the unrestricted model. Phase (0) contains no loops. Phase (I) contains proliferated loops in $J_1$ and no loops in $J_2$, while in phase (II) the variables are interchanged. Phase (III) contains proliferated double-strength loops in both variables.

Figure 2

Plot of the energy difference between the peaks of the histograms, as a function of $t_1=t_2=t$ on the self-dual line. Higher values denote a stronger first-order nature. The inset shows the energy histogram for $t_1=t_2=0.6$. The dual-peaked shape of the histogram implies a first-order transition. The histograms were performed at $L=24$ with ${10}^6$ Monte Carlo sweeps. The heat capacity intercepts were found using data from $L=8,10,12,14$, and 16.

Figure 3

${\rho }_2\left(q_{\min }\right)·L$ as a function of $t_2$ at $t_1=1.2$. Based on this data, we conclude that the phase transition occurs at approximately $t_2=1.168$, and is second-order in nature. Scans like this were used to determine the phase diagram described in the legend of Fig. 1. Here the transition is between (I) and (III). Each data point is the result of $5\times {10}^6$ Monte Carlo sweeps. Error bars were determined by looking at the difference between runs with different initial conditions.

Figure 4

The phase diagram of the “restricted” model. The region with +45${}^{\circ }$ lines is inaccessible in the formulation Eq. (5), and the cross-hatched region is inaccessible in this work. The transition is first-order everywhere on the accessible portion on the self-dual line, but the strength decreases as $t$ is increased.

Figure 5

Same as described in the legend of Fig. 2 but for the restricted model. The estimates from the heat capacity come from the $y$ intercepts in Fig. 6.

Figure 6

$C/L^3$ vs. $1/L^3$ for the restricted model. The data shows nonzero $y$ intercepts for all $t$, confirming the first-order nature of the transition on the self-dual line. The black lines are least-squares fits to the data, and uncertainties come from comparing runs with different initial conditions