Phases and phase transitions in a $U\left(1\right)\times U\left(1\right)$ system with $\theta =2\pi /3$ mutual statistics

We study a $U\left(1\right)\times U\left(1\right)$ system with short-range interactions and mutual $\theta =2\pi /3$ statistics in (2+1) dimensions. We are able to reformulate the model to eliminate the sign problem and perform a Monte Carlo study. We find a phase diagram containing a phase with only small loops and two phases with one species of proliferated loop. We also find a phase where both species of loop condense, but without any gapless modes. Lastly, when the energy cost of loops becomes small, we find a phase that is a condensate of bound states, each made up of three particles of one species and a vortex of the other. We define several exact reformulations of the model that allow us to precisely describe each phase in terms of gapped excitations. We propose field-theoretic descriptions of the phases and phase transitions, which are particularly interesting on the “self-dual” line where both species have identical interactions. We also define irreducible responses useful for describing the phases.

Figure 1

The contribution to the partition function is multiplied by a phase $e^{i\theta }$ for each cross linking of the two currents shown in the figure on the left. If we change the direction of one of the currents and get the figure on the right, the phase is $e^{-i\theta }$. When considering symmetries of our model for $\theta \ne \pi$, we must only consider operations that leave the relative orientation of the current loops unchanged.

Figure 2

The phase diagram for the model in Eq. (1) with $\theta =2\pi /3$. 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. In phase (IV), both species of loops are condensed in single-strength loops. Phase (III) is a condensate of bound states comprised of three charges from one species of loop and a vortex from the other species. The precise description of these condensates is given in the text.

Figure 3

A sample of the $C^{22}\left(k_{\min }\right)L$ crossings from which the location of the phase transition was determined. Error bars come from comparing runs with different random seeds. Here, the transition is from phase (IV) to phase (II). The data are taken for $t_1=1.0$, and we determine the phase transition to be at $t_2=1.80\pm 0.01$. There is no sign of drift in the location of the crossings, suggesting a second-order transition. Note that the ${\vec{J}}_2$ variables are condensed in both phases (IV) and (II). Nevertheless, $C^{22}\left(k_{\min }\right)L\sim 1/L$ in phase (IV) and $C^{22}\left(k_{\min }\right)L\sim L$ in phase (II), so the crossing analysis on the plot detects the transition. This subtlety is discussed in the text.

Figure 4

$C^{22}\left(k_{\min }\right)L$ along the self-dual line. We can see that $C^{22}\left(k_{\min }\right)L$ vanishes in phases (0) and (IV), despite its unusual behavior at the transition between these phases. $C^{22}\left(k_{\min }\right)L$ diverges in phase (III) due to the proliferation of triple-strength loops in the $\vec{J}$ variables. The vertical lines mark the locations of the phase transitions. More detailed data at the phase transitions is shown in Figs. 7 and 10.

Figure 5

$C^{12}\left(k_{\min }\right)L$ along the self-dual line. $C^{12}\left(k_{\min }\right)L$ vanishes in phase (0), but it approaches a universal value in phase (IV). We predict this value to be at $-2\pi /\theta =-3$, which is shown by a horizontal line. In phase (IV), $C^{12}\left(k_{\min }\right)L$ approaches a nonuniversal value. The vertical lines mark the locations of the phase transitions.

Figure 6

Different scenarios for how the phases can meet on the self-dual line. In scenario (a), all four phases meet at a point. In (b), phases (I) and (II) meet on a line segment on the self-dual line and in (c), phases (0) and (IV) meet on a line segment perpendicular to the self-dual line. We believe that scenario (c) is unlikely, but we cannot distinguish between (a) and (b) at finite size. We can only state that if such a segment exists in scenario (b), it is in the narrow range $t=0.335–0.35$.

Figure 7

$C^{22}\left(k_{\min }\right)L$ along the self-dual line at the bottom multicritical point. $C^{22}\left(k_{\min }\right)L$ vanishes as $1/L$ for sufficiently large $L$ in both phase (0) and phase (IV), but is nonzero at the transition.

Figure 8

Heat capacity along the self dual line at the bottom multicritical point. The sharpness of the peaks suggests either a small $\nu$ or a first-order phase transition between phases (0) and (IV). The inset shows histograms of the energy per site $\epsilon$ at $L=28,32$, $t=0.345$. The irregular shape of the histogram suggests that this is a first-order transition.

Figure 9

Heat capacity along the self-dual line at the upper multicritical point. The behavior of the peak with size suggests a second-order transition. The inset shows a histogram of the energy per site $\epsilon$, at $L=32\text{and}t=2.66$, which is where $C$ has a maximum for this size. The single peak further suggests a second-order transition. We also studied histograms at all of the other points where heat-capacity was measured, and found only single-mode distributions everywhere.

Figure 10

$C^{22}\left(k_{\min }\right)L$ along the self-dual line at the upper multicritical point. The crossings do not drift with increasing size, implying a second-order transition at $t_{\mathrm{crit}}=2.62$.

Figure 11

Illustration of the “molecules” that are condensed in phase (III). Each molecule contains three charges (stars) and one antivortex (circles). Black objects on white backgrounds are of species 1 and white objects on black backgrounds are of species 2.

Figure 12

The conductivity ${\sigma }^{22}$, along the self-dual line, obtained from the raw data in Figs. 4 and 5. Vertical lines mark the phase boundaries. We can see that ${\sigma }^{22}$ diverges in phase (IV), but is zero in phase (III).

Figure 13

The transverse conductivity ${\sigma }_{xy}^{12}$, along the self-dual line, at the boundary between phases (IV) and (III). A vertical line marks the phase boundary. In phase (IV), ${\sigma }_{xy}^{12}$ approaches a nonuniversal value, while in phase (III), it approaches the universal value $3/2\pi$, which is indicated by the horizontal line in the figure. Calculations of ${\sigma }_{xy}^{12}$ in phase (IV) involve cancellations of similar quantities, which greatly increases the noise in this region. The origin of the cancellation is in the quantization of the $C^{12}$ values in this phase, as seen in Fig. 5. Therefore we have chosen not to display our data in phase (IV). The reader may note that the data is irregular in the region of the phase transition; based on estimates of the error bars we believe that this is due to noise and is not of physical significance.