Monte Carlo study of duality and the Berezinskii-Kosterlitz-Thouless phase transitions of the two-dimensional $q$-state clock model in flow representations

The two-dimensional $q$-state clock model for $q\ge 5$ undergoes two Berezinskii-Kosterlitz-Thouless (BKT) phase transitions as temperature decreases. Here we report an extensive worm-type simulation of the square-lattice clock model for $q=5$–9 in a pair of flow representations, from high- and low-temperature expansions, respectively. By finite-size scaling analysis of susceptibilitylike quantities, we determine the critical points with a precision improving over the existing results. Due to the dual flow representations, each point in the critical region is observed to simultaneously exhibit a pair of anomalous dimensions, which are ${\eta }_1=1/4$ and ${\eta }_2=4/q^2$ at the two BKT transitions. Further, the approximate self-dual points ${\beta }_{\mathrm{sd}}\left(L\right)$, defined by the stringent condition that the susceptibilitylike quantities in both flow representations are identical, are found to be nearly independent of system size $L$ and behave as ${\beta }_{\mathrm{sd}}\simeq q/2\pi$ asymptotically at the large-$q$ limit. The exponent $\eta$ at ${\beta }_{\mathrm{sd}}$ is consistent with $1/q$ within statistical error as long as $q\ge 5$. Based on this, we further conjecture that $\eta \left({\beta }_{\mathrm{sd}}\right)=1/q$ holds exactly and is universal for systems in the $q$-state clock universality class. Our work provides a vivid demonstration of rich phenomena associated with the duality and self-duality of the clock model in two dimensions.

Figure 1

Unit vectors of the $q$-state clock model and the Potts model. For $q=2$ and 3, the possible directions taken by spins are identical in both models. They become different for $q\ge 4$, and (a) and (b) correspond to the four-state clock model and the four-state Potts model, respectively.

Figure 2

(a) Plot of ${\beta }_{c1}, {\beta }_{c2}$, and ${\beta }_{\mathrm{sd}}$ versus $q$. The blue dashed line represents ${\beta }_{c2}\left(q\right)=0.041q^2-0.11q+0.65$ and the black dashed line represents ${\beta }_{\mathrm{sd}}\left(q\right)=q/2\pi +0.31$. (b) Anomalous dimension $\eta \left({\beta }_{\mathrm{sd}}\right)$ at the self-dual point as a function of $q$ on a log-log scale. The data points can be fitted accurately by the function $1/q$.

Figure 3

(a) Semilogarithmic plot of the correlation length $\xi$ versus $b/\sqrt{t}$ for the 2D $XY, q=5,6$ clock model, where $t=({\beta }_{c1}-\beta )/{\beta }_{c1}$, illustrating the exponential growth of $\xi$. Different models are distinguished by different colors, as defined in the legend. The corresponding nonuniversal constant $b=1$ ($XY$), 0.94 ($q=5$), and 0.994 ($q=6$). (b) Semilogarithmic plot of the ratio $\xi /L$ as a function of $b/\sqrt{t_L}$, where $t_L=t{[ln(L/L_0)]}^2$ with the nonuniversal length scale simply set by $L_0=1$. The data points collapse onto a single curve, indicating $\xi /L$ is a universal function of the scaling field $b/\sqrt{t_L}$. These plots clearly support that the $q=5$ and 6 clock models are in the universality class of the $XY$ model. In particular, the resemblance of the three systems is clearly illustrated by the approximately identical values of $b$; actually, the $b$ value for $q=6$ is hardly different from that for the 2D $XY$ model, which is a rather surprising observation.

Figure 4

Illustration of two types of flow configurations in the high-$T$ expansion for $q=3$: (a) closed configuration with $(\nabla ·\mathbf{N})\text{mod}q=0$ and (b) open configuration with defects $\mathcal{I}$ and $\mathcal{M}$ (gray circles) violating the $q$-modular flow conservation. The direction of horizontal (vertical) edges of $G$ is specified by the positive direction of the $x$ axis ($y$ axis).

Figure 5

Illustration of a closed flow configuration in the low-$T$ expansion for $q=3$ with periodic boundary conditions. The closed circles and dashed gray lines constitute the original lattice $G$. The dual vertices are at the faces of $G$, with dual edges connecting them. The direction of the horizontal (vertical) edges of $G^{*}$ is specified by the positive direction of the $x$ axis ($y$ axis). The value of the flow variable on each edge is determined by the difference between its right and left spins.

Figure 6

Scaled susceptibilities (a) ${\chi }_H(\beta ,L)/L^{7/4}{(lnL+C_1)}^{1/8}$ and (b) ${\chi }_L(\beta ,L)/L^{7/4}{(lnL+C_1)}^{1/8}$ versus the inverse temperature $\beta$ for $q=5$. The constant $C_1$ is set to (a) 3.6 and (b) 4.0. The vertical red dashed line presents the central value of our estimate and the shadow shows the error bar.

Figure 7

Scaled susceptibility ${\chi }_L(\beta ,L)/L^{7/4}{(lnL+C_1)}^{1/8}$ versus the inverse temperature $\beta$ for (a) $q=6$, (b) $q=7$, (c) $q=8$, and (d) $q=9$. The constants $C_1$ are set to 4.9, 4.9, 5.0, and 5.3, respectively. The vertical red line represents the central value of ${\beta }_{c2}$ and the shadow shows the error bar.

Figure 8

Scaled susceptibilities ${\chi }_L({\beta }_{c1},L)/L^2$ and ${\chi }_H({\beta }_{c2},L)/L^2$ versus $L$ on a log-log scale. The dashed line represents $\chi \left(L\right)/L^2=a_0{L}^{-\eta }$ with $\eta =4/q^2$ for $q=5,6,9$. For each $q$, the data points of ${\chi }_H({\beta }_{c2},L)$ and ${\chi }_L({\beta }_{c1},L)$ are very close to each other, which indicates that they not only share the same scaling behavior but also have the same amplitude.

Figure 9

Linear plot of ${\chi }_H$ and ${\chi }_L$ versus $\beta$ for $q=5$. The red dots and blue dots correspond to ${\chi }_H$ and ${\chi }_L$, respectively, with system size $L=128,256,512,1024$ (from bottom to top). The black vertical dashed line shows the exact self-dual point ${\beta }_{\mathrm{sd}}=1.076318...$.

Figure 10

Plots of ${\chi }_{\mathrm{diff}}$ versus the inverse temperature $\beta$ for (a) $q=6$, (b) $q=7$, (c) $q=8$, and (d) $q=9$. The vertical red line represents the central value of ${\beta }_{\mathrm{sd}}$ and the shadow shows the error bar.

Figure 11

Linear plot of the scaled magnetic susceptibility ${\chi }_H({\beta }_{c1},L)L^{-7/4}{(lnL+C_1)}^{-1/8}$ versus the inverse temperature $\beta$ for (a) $q=6$ and (b) $q=9$. The constant $C_1$ is set to the corresponding estimated value for each case. The vertical red line represents the central value of ${\beta }_{c1}$ and the shadow shows the error bar.