Line of continuous phase transitions in a three-dimensional U(1) loop model with $1/r^2$ current-current interactions

We study a lattice model of interacting loops in three dimensions with a $1/r^2$ interaction. Using Monte Carlo methods, we have found that the phase diagram contains a line of second-order phase transitions between a phase where the loops are gapped and a phase where they proliferate. The correlation length exponent and critical conductivity vary continuously along this line. Our model is exactly self-dual at a special point on the critical line, which allows us to calculate the critical conductivity exactly at this point.

Figure 1

An example of the $\sigma \left(k_{\min }\right)$ data used to determine the location of the phase transition for the model at fixed $g=0.3125$, varying $t$. From these data, we determine the phase transition to take place at $t_{\mathrm{crit}}=2.05$, and estimate the crossing value ${\sigma }^{\mathrm{crossing}}\left(k_{\min }\right)=0.067\pm 0.005$.

Figure 2

The phase diagram of our model, Eq. (10). The model contains a phase with small loops at large $t$ and a proliferated phase at small $t$, separated by a critical line of continuous phase transitions. The phase boundary is accurately determined from crossings like those in Fig. 1 (see also Table ). For large negative $t$, the model becomes ill defined.

Figure 3

$L·d\rho /dt$ as a function of $t$ for the same system as Fig. 1. Upon increasing $L$, the peak evolves slowly toward the critical $t_{\mathrm{crit}}=2.05$. For each $L$, we extract the maximum value and show it in Fig. 4a. We fit this to Eq. (13) to extract $\nu \left(g\right)$.

Figure 4

Peak values as a function of $L$ for (a) $L·d\rho /dt$ and (b) $C_3$, plotted on a log-log scale. We can see that the slopes of the lines decrease with increasing $g$ for $0\le g\le 1$, which corresponds to an increasing $\nu$. The $C_3$ data at $g=2$ were not shown because the line overlaps with the other lines and makes the figure hard to read.

Figure 5

Values of $\nu$ determined from scaling of the ${|L·d\rho /dt|}_{\max }$ and ${\left[C_3\right]}_{\max }$ (as shown in Fig. 4) with system size.

Figure 6

The value of ${\sigma }^{\mathrm{crossing}}$ vs the coupling $g$ of the marginally long-ranged interaction. For each $g$, ${\sigma }^{\mathrm{crossing}}$ was determined from plots like those shown in Fig. 1. We can see that ${\sigma }^{\mathrm{crossing}}$ varies as we move along the critical line. The crossings are narrow for $g\le 1$, while for $g>1$, we notice a weak drift in the crossings as $L$ is increased, and the reported values are from the $L=12$ and $L=14$ data.