Finite-size scaling analysis of pseudocritical region in two-dimensional continuous-spin systems

At low temperatures, the two-dimensional continuous-spin systems exhibit large correlation lengths. Some of them show the Berezinskii–Kosterlitz–Thouless-like transitions, and some others show pseudocritical behaviors for which correlation lengths are extremely large but finite. To distinguish pseudo and genuine critical behaviors, it is important to understand the nature of spin-spin correlations and topological defects at low temperatures in continuous-spin systems. In this paper, I develop a finite-size scaling analysis which is suitable for distinguishing the critical behavior and its applications to the two-dimensional $XY$, Heisenberg, and ${\mathrm{RP}}^2$ models.

Figure 1

(a) FSF-FSS plot of correlation ratio for 2D $XY$ model. Error bars are smaller than size of marks. (b) A semilogarithmic plot of an FSF-FSS function $U_{\mathcal{C}}$ for the 2D $XY$ model. Since $ln{U}_{\mathcal{C}}$ is a concave function, $U_{\mathcal{C}}$ reaches zero at the critical temperature.

Figure 2

(a) FSF-FSS plot of correlation ratio for 2D Heisenberg model. Error bars are smaller than size of marks. (b) A semilogarithmic plot of an FSF-FSS function $U_{\mathcal{C}}$ for the 2D Heisenberg model. Since $ln{U}_{\mathcal{C}}$ is a convex function in a sufficiently large $\xi \left(L\right)/L$ region, $U_{\mathcal{C}}$ never reaches zero at a finite temperature.

Figure 3

(a) FSF-FSS plot of correlation ratio for 2D ${\mathrm{RP}}^2$ model. (b) A semilogarithmic plot of a FSF-FSS function $U_{\mathcal{C}}$ for the 2D ${\mathrm{RP}}^2$ model. Since $ln{U}_{\mathcal{C}}$ is a convex function in a sufficiently large $\xi \left(L\right)/L$ region, $U_{\mathcal{C}}$ never reaches zero at a finite temperature. The slope abruptly changes at the inflection point ${\xi }_{\times }/L(\sim 0.6)$. The solid line is a guide to the eye.

Figure 4

FSS plot of correlation ratio $\mathcal{C}$ for 2D ${\mathrm{RP}}^2$ model.

Figure 5

FSS plot of distribution functions of the nematic order parameter $m$. (a) FSS plot at a high temperature ($T=1.0$). (b) FSS plot at the crossover temperature ($T=0.3431$). (c) FSS plot at a low temperature ($T=0.3$).