Partition function zeros of the $p$-state clock model in the complex temperature plane

We investigate the partition function zeros of the two-dimensional $p$-state clock model in the complex temperature plane by using the Wang-Landau method. For $p=5$, 6, 8, and 10, we propose a modified energy representation to enumerate exact irregular energy levels for the density of states without any binning artifacts. Comparing the leading zeros between different $p$'s, we provide strong evidence that the upper transition at $p=6$ is indeed of the Berezinskii-Kosterlitz-Thouless (BKT) type in contrast to the claim of the previous Fisher zero study [Phys. Rev. E 80, 042103 (2009)]. We find that the leading zeros of $p=6$ at the upper transition collapse onto the zero trajectories of the larger $p$'s including the $XY$ limit while the finite-size behavior of $p=5$ differs from the converged behavior of $p\ge 6$ within the system sizes examined. In addition, we argue that the nondivergent specific heat in the BKT transition is responsible for the small partition function magnitude that decreases exponentially with increasing system size near the leading zero, fundamentally limiting access to large systems in search for zeros with an estimator under finite statistical fluctuations.

Figure 1

Leading Fisher zeros at the upper transition in the $p$-state clock model with $p=5$, 6, 8, and 10. The uncertainty shown by the error bar is given as a 95% confidence interval estimated from the bootstrap resampling with the WL DOS samples. The error bar is omitted if it is smaller than the symbol size. The data points for the $XY$ limit are from the previous higher-order tensor renormalization-group (HOTRG) calculations [35].

Figure 2

Scaling relation between the real and imaginary parts of the leading Fisher zero. The power-law relation $\mathrm{Im}\left[{\beta }_1\right]\propto {({\beta }_c-\mathrm{Re}\left[{\beta }_1\right])}^{1+\nu }$ is examined with the previous estimates of critical points ${\beta }_c$ [12, 18]. The arbitrary factor $b_p$ is adjusted for a graphical comparison between the data points of different $p$'s.

Figure 3

Leading Fisher zeros at the lower transition. The real and imaginary parts of the leading zeros are rescaled with the factor $(1-cos\frac{2\pi }{p})$. The system sizes are limited to $L\le 16$ for reliable identification of the leading zeros.

Figure 4

Reliability test of the leading-zero identification for the upper transition in the six-state clock model. (a) The normalized partition function $\tilde{\mathcal{Z}}\left(\beta \right)$ is evaluated for $L=28$ as a function of ${\beta }_I$ at ${\beta }_R=\mathrm{Re}\left[{\beta }_1\right]$. The leading zero is marked by the square symbol. The error bar displayed at the points along the oscillations presents the statistical fluctuation measured by bootstrap resampling, indicating that its magnitude is typical in the range of ${\beta }_I$ for both $\mathrm{Re}\left[\tilde{\mathcal{Z}}\right]$ and $\mathrm{Im}\left[\tilde{\mathcal{Z}}\right]$. The envelope function $f\left({\beta }_I\right)$ is obtained from the Gaussian approximation [48]. (b) The maximum oscillation amplitude of $\mathrm{Re}\left[\tilde{\mathcal{Z}}\right]$ for ${\beta }_I>\mathrm{Im}\left[{\beta }_1\right]$ is shown for comparison with its fluctuation as a function of system size $L$.