Monte Carlo study of a $U\left(1\right)\times U\left(1\right)$ loop model with modular invariance

We study a $U\left(1\right)\times U\left(1\right)$ system in (2+1) dimensions with long-range interactions and mutual statistics. The model has the same form after the application of operations from the modular group, a property which we call modular invariance. Using the modular invariance of the model, we propose a possible phase diagram. We obtain a sign-free reformulation of the model and study it in Monte Carlo. This study confirms our proposed phase diagram. We use the modular invariance to analytically determine the current-current correlation functions and conductivities in all the phases in the diagram, as well as at special “fixed” points which are unchanged by an operation from the modular group. We numerically determine the order of the phase transitions, and find segments of second-order transitions. For the statistical interaction parameter $\theta =\pi$, these second-order transitions are evidence of a critical loop phase obtained when both loops are trying to condense simultaneously. We also measure the critical exponents of the second-order transitions.

Figure 1

Phase diagram for the model in Eq. (1), with fixed $\theta =2\pi /3$. The dashed line is the “symmetric” line where the potentials are equal, $v_1=v_2$, which is also assumed everywhere in the phase diagram in Fig. 2. In phase “$\infty$” both $J$ variables are gapped, while they condense in phase “0”. In phases I and II only one species is gapped. In the lower left corner different composite variables are gapped; here the structure can be significantly different for different values of $\theta$.

Figure 2

The phase diagram for the “symmetric” model with $g_1=g_2=g$, for one period in $\eta$. The $J$ variables are gapped in phase $\infty$, and the (dual vortex) $Q$ variables are gapped in phase 0. In other phases, the gapped particles are linear combinations of $J$ and $Q$. Every phase can be mapped to phase $\infty$ by an operation in the modular group. Solid symbols show where the locations of phase transitions have been confirmed by our numerical study. There are infinitely many phase transitions in the model, so at small $g$ our diagram does not show every transition. The labels on the phases are explained in the text.

Figure 3

A section of Fig. 2, blown up to show the labels on the various phases. Not all transitions are shown as there are infinitely many of them at small $g$. The dashed line is at $\eta =2/5$, where the data in Figs. 4, 5, 6 were taken.

Figure 4

$C_J^{11}\left(k_{\min }\right)L$ as a function of $g$ for $\eta =\frac{2}{5}$ (i.e., along the dashed line in Fig. 3), for system sizes $L=6$ and $L=8$. The dashed curves correspond to the theoretical predictions in the different phases. The dotted vertical lines denote the phase transitions, which occur at $g=\frac{1}{5}$ and $g=\frac{1}{5\sqrt{21}}$.

Figure 5

$C_J^{12}\left(k_{\min }\right)L$ as a function of $g$ for the same system as in Fig. 4. The dashed curves correspond to the theoretical predictions. The dotted vertical lines denote the phase transitions.

Figure 6

$2\pi {\sigma }_{xy}^{12}\left(k_{\min }\right)$ as a function of $g$ for the same system as in Fig. 4. We observe that the conductivity approaches $\frac{1}{2\pi }2$ in phase 1/2 and $\frac{1}{2\pi }\frac{5}{2}$ in phase 2/5, as expected. The dotted vertical lines denote the phase transitions. In phase 0, ${\sigma }_{xy}^{12}$ approaches nonuniversal values.

Figure 7

Histograms of $\epsilon$ at various points on the phase transition between phase $\infty$ and phase 0, using sizes $L=10,12,14,16$. A normally distributed histogram implies that the transition is second order, while a transition with multiple peaks implies first order. The first panel corresponds to $\eta =\frac{1}{5}$; the next two panels correspond to $\eta =\frac{1}{4}$, $\frac{1}{3}$. We see no evidence of first-order behavior at these sizes.

Figure 8

Histograms of $\epsilon$ at various points on line of phase transitions at $\eta =\frac{1}{2}$, using sizes $L=10,12,14,16$. Each point is related to a point on the $\infty$-0 phase transition by the operation $T^{1}S{T}^{1}S$. The first panel maps to $\eta =\frac{1}{5}$; the next two panels map to $\eta =\frac{1}{4}$, $\frac{1}{3}$. We see no evidence of first-order behavior. The histograms are also identical to those in Fig. 7, which provides a check on our Monte Carlo.

Figure 9

Same as Fig. 8, but the operation to map these points to $\infty$-0 phase transition is $S{T}^{-1}S$, as explained in the text.

Figure 10

Histograms of $\epsilon$ at some special “fixed” points in Fig. 2 using sizes $L=10,12,14,16$. The first two panels show histograms at “triple points” where three phase transitions meet. Panel (a) shows the triple point at the upper end of the line of phase transitions at $\eta =\frac{1}{2}$, which has $g=\sqrt{3}/2$. Panel (b) shows the point at the lower end with $g=\sqrt{3}/6$. Both histograms have a double-peaked structure which indicates that the transitions at these points are first order. Panel (c) shows histograms at the fixed point $\eta =\frac{1}{2},g=\frac{1}{2}$. These histograms show no sign of first-order behavior, which we expect since this point should have the same properties as the point $\eta =0,g=1$ which is known to be continuous (Ref. 21).

Figure 11

(a) Schematic blowup of the phase diagram in the $v_1$, $v_2$ (Fig. 1) variables, near a tetracritical point. At such a point we expect two scaling directions with distinct critical exponents ${\nu }_s$ and ${\nu }_a$. Due to the symmetry on interchange of $J_1$ and $J_2$, we expect the scaling directions to be in the symmetric and antisymmetric directions, shown by the dotted lines. The shape of the phase transition lines implies that ${\nu }_s>{\nu }_a$. (b) Phase diagram for the special case of $\theta =\pi$ $(\eta =1/2)$. We can see that along the symmetric direction the system goes along the phase boundary, and we can only drive a phase transition across the antisymmetric direction.

Figure 12

Plots of the derivatives $\partial \left(C_J^{11}\left(k_{\min }\right)L\right)/\partial t_s$ and $\partial \left(C_J^{11}\left(k_{\min }\right)L\right)/\partial t_a$ at various points on the $\infty$-0 phase transition. Error bars were obtained by comparing the results of independent simulations. We expect such plots to scale as $L^{1/{\nu }_{s,a}}$. The values shown in Fig. 13 were extracted by fitting these curves to the function $a+b{L}^{1/\nu }$.

Figure 13

Critical exponents along the (a) symmetric and (b) antisymmetric scaling direction, extracted from the data in Fig. 12. Error bars come from the fits, and are further increased to account for finite-size effects as discussed in the text. The blue circles represent the value of $\nu$ obtained in Ref. 21. ${\nu }_s$ cannot be measured when $\theta =\pi$, since the system does not cross a phase transition in this direction [see Fig. 11b]. On the $\theta =\pi$ line there are two points which map to each point on the $\infty$-0 semicircle.