Monte Carlo study of an improved clock model in three dimensions

We study a generalized clock model on the simple cubic lattice. The parameter of the model can be tuned such that the amplitude of the leading correction to scaling vanishes. In the main part of the study, we simulate the model with $Z_8$ symmetry. At the transition, with increasing length scale, $O\left(2\right)$ symmetry emerges. We perform Monte Carlo simulations using a hybrid of local Metropolis and cluster algorithms of lattices with a linear size up to $L=512$. The field variable requires less memory and the updates are faster than for a model with $O\left(2\right)$ symmetry at the microscopic level. Our finite-size scaling analysis yields accurate estimates for the critical exponents of the three-dimensional $XY$-universality class. In particular, we get $\eta =0.03810\left(8\right),\nu =0.67169\left(7\right)$, and $\omega =0.789\left(4\right)$. Furthermore, we obtain estimates for fixed point values of phenomenological couplings and critical temperatures.

Figure 1

We plot the number of measurements times the volume $L^3$ as a function of the linear lattice size $L$ for $N=8$ at $D=1.05$ and $D=1.07$.

Figure 2

We give the results for ${(Z_a/Z_p)}^{*}$ fitting with the ansätze (49), (51), and (52) with ${\epsilon }_3=4$, corresponding to fits 1, 3, and 4 in the legend of the figure, as a function of the minimal lattice size $L_{\min }$ that is included in the fit. Data for $D=1.02$, 1.05, and 1.07 are jointly fitted. The solid line gives our final estimate and the dashed ones the corresponding error.

Figure 3

We plot $U_4$ at $Z_a/Z_p=0.32037$ for $N=8$ at $D=0.45$, 0.9, 1.05, 1.24, and $\infty$ as a function of the linear lattice size $L$.

Figure 4

We plot $U_4$ at $Z_a/Z_p=0.32037$ for $N=8$ at $D=0.45$, 0.0, $-0.5$, and $-0.7$ as a function of the linear lattice size $L$.

Figure 5

We plot the estimates of the correction exponent $\omega$ obtained by fitting $U_4$ at $Z_a/Z_p=0.32037$ using the ansatz (57), where all linear lattice sizes with $L_{\min }\le L$ are included. Data for $N=8$ at $D=-0.7,-0.5$, 0.0, 0.45, 0.9, 1.02, 1.05, 1.07, 1.24, and $\infty$ are taken into account. The lines connecting the data points should only guide the eye. The $L_{\min }$ are slightly shifted for different fits to make the figure readable.

Figure 6

We plot $U_4(Z_a/Z_p=0.32037)+0.121(D-1.06)L^{-0.789}$ for $N=8$ at $D=1.02$, 1.05, and 1.07. Note that we have shifted the values of $L$ for $D=1.02$ and 1.07 to make the figure readable.

Figure 7

We plot estimates of $D^{*}$ obtained from fits of $U_4$ at $Z_a/Z_p=0.32037$ for $N=8$ at $D=1.02$, 1.05, and 1.07 as a function of the minimal lattice size $L_{\min }$ taken into account. The ansätze (59, 60, 61) are used. The corresponding correction exponents are given in the legend. Our preliminary estimate $D^{*}=1.055\left(10\right)$ is indicated by the straight solid line. The dashed lines give the error bar.

Figure 8

We plot the Binder cumulant $U_4$ at $Z_a/Z_p=0.32037$ for $N=8$ at $D=-0.85,-0.86$, and $-0.87$ for linear lattice sizes $8\le L\le 48$. The lines connecting the data points should only guide the eye.

Figure 9

Estimates of the RG exponent $y_t$ obtained from fitting the improved slopes of $U_4,Z_a/Z_p$, and ${\xi }_{2\mathrm{nd}}/L$ at ${\xi }_{2\mathrm{nd}}/L=0.59238$ for $N=8$ at $D=1.05$ and 1.07 as a function of the mininal linear lattice size $L_{\min }$ that is taken into account. The ansatz (69) is used. To make the figure readable, we shifted the values of $L_{\min }$ by $-0.3$ and 0.3, for two of the fits. The straight solid line gives our preliminary estimate obtained from the improved slopes at ${\xi }_{2\mathrm{nd}}/L=0.59238$. The dashed lines indicate our preliminary error estimate.

Figure 10

Estimates of the RG exponent $y_t$ obtained from fitting the improved slopes of $U_4,Z_a/Z_p$, and ${\xi }_{2\mathrm{nd}}/L$ at $Z_a/Z_p=0.32037$ for $N=8$ at $D=1.05$ and 1.07 as a function of the minimal linear lattice size $L_{\min }$ that is taken into account. The ansatz (69) is used. To make the figure readable, we shifted the values of $L_{\min }$ by $-0.3$ and 0.3, for two of the fits. The straight lines indicate our preliminary result and its error estimate.

Figure 11

Estimates of the RG exponent $y_t$ obtained from fitting the improved slopes of $U_4,Z_a/Z_p$, and ${\xi }_{2\mathrm{nd}}/L$ at ${\xi }_{2\mathrm{nd}}/L=0.59238$ for $N=8$ at $D=1.05$ and 1.07 as a function of the minimal linear lattice size $L_{\min }$ that is taken into account. The ansatz (68) is used. To make the figure readable, we shifted the values of $L_{\min }$ by $-0.3$ and 0.3, for two of the slopes.

Figure 12

Estimates for $y_t$ obtained from analyzing the energy density. We fitted the data by using the ansätze (71, 72, 73). The corresponding correction exponents are given in the legend. $L_{\min }$ is the minimal linear lattice size that is included in the fits. Data for $N=8$ at $D=1.05$ and 1.07 are taken into account. For comparison, we give the estimate of $y_t$ obtained in the previous section by a straight solid line. The dashed lines give the error bar.

Figure 13

Estimates of the critical exponent $\eta$ obtained from fitting the improved magnetic susceptibility ${\chi }_{imp}$ at $Z_a/Z_p=0.32037$ for $D=1.05$ and 1.07 as a function of the minimal linear lattice size $L_{\min }$ that is taken into account. The ansätze (76) and (77) are used. To make the figure readable, we shifted the values of $L_{\min }$ by $-0.3$ and 0.3, for two of the fits.

Figure 14

Estimates of the critical exponent $\eta$ obtained from fitting the improved magnetic susceptibility ${\chi }_{imp}$ at ${\xi }_{2\mathrm{nd}}/L=0.59238$ for $D=1.05$ and 1.07 as a function of the minimal linear lattice size $L_{\min }$ that is taken into account. The ansätze (76) and (77) are used. To make the figure readable, we shifted the values of $L_{\min }$ by $-0.3$ and 0.3, for two of the fits.