Critical Binder cumulant and universality: Fortuin-Kasteleyn clusters and order-parameter fluctuations

We investigate the dependence of the critical Binder cumulant of the magnetization and the largest Fortuin-Kasteleyn cluster on the boundary conditions and aspect ratio of the underlying square Ising lattices. By means of the Swendsen-Wang algorithm, we generate numerical data for large system sizes and we perform a detailed finite-size scaling analysis for several values of the aspect ratio $r$, for both periodic and free boundary conditions. We estimate the universal probability density functions of the largest Fortuin-Kasteleyn cluster and we compare it to those of the magnetization at criticality. It is shown that these probability density functions follow similar scaling laws, and it is found that the values of the critical Binder cumulant of the largest Fortuin-Kasteleyn cluster are upper bounds to the values of the respective order-parameter's cumulant, with a splitting behavior for large values of the aspect ratio. We also investigate the dependence of the amplitudes of the magnetization and the largest Fortuin-Kasteleyn cluster on the aspect ratio and boundary conditions. We find that the associated exponents, describing the aspect-ratio dependencies, are different for the magnetization and the largest Fortuin-Kasteleyn cluster, but in each case are independent of boundary conditions.

Figure 1

Illustration of the finite-size behavior of critical Binder cumulants for the magnetization $U_M^{*}$ and the LFKC $U_{{\mathrm{l}}_{\infty }}^{*}$. Cases of PBC with $r=1$ and $r=4$ and FBC with $r=1$ are illustrated. The dashed lines are the extrapolated limits.

Figure 2

Dependence of critical Binder cumulants $U_M^{*}$ and $U_{{\mathrm{l}}_{\infty }}^{*}$ on the logarithm of the aspect ratio $ln\left(r\right)$ for PBC and FBC.

Figure 3

Scaled pdfs of the magnetization, $x=\left|M\right|/\sqrt{\langle M^2\rangle }$, for (i) PBC with $r=1$ and $r=4$ and (ii) FBC with $r=1$. Sizes $L^{*}=60$ and $L^{*}=120$ are illustrated.

Figure 4

Scaled probability pdfs of the LFKC, $x=l_{\infty }/\sqrt{\langle l_{\infty }^2\rangle }$, corresponding to the cases of Fig. 3.

Figure 5

Percolation probabilities $p_{\mathrm{both}}$ and $p_{\mathrm{short}}$ with respect to $ln\left(r\right)$. Dashed lines indicate the $r=1$ values in the thermodynamic limit (see discussion in the text), and the expected large-$r$ behavior. The crossover behavior for the PBC is associated with the appearance of the double-peak structure in Fig. 4. In the inset, we compare the $r=1$ shoulderlike behavior in Fig. 4 with a restricted pdf describing only the LFKC that percolates simultaneously in both directions.

Figure 6

Main panel: Scaled pdfs of the LFKC, $x=l_{\infty }/\sqrt{\langle l_{\infty }^2\rangle }$, for PBC and FBC. Illustration of striking differences for $r=4$ and approach to the same universal pdf as $r$ increases ($r=36$). Inset: Illustration of the similarity of the magnetization's pdfs, $x=\left|M\right|/\sqrt{\langle M^2\rangle }$, for large $r$ ($r=36$).

Figure 7

Scaled pdfs of the magnetization, $x=\left|M\right|/\sqrt{\langle M^2\rangle }$, and the LFKC, $x=l_{\infty }/\sqrt{\langle l_{\infty }^2\rangle }$, for PBC as we vary the aspect ratio in the window $r=1$–16. This figure elucidates the similarity among the two functions for $r=1$, but also highlights their striking differences with increasing $r$.

Figure 8

Amplitude dependence on $r$ and illustration of the power law $A_Q\left(r\right)=a{r}^{-z}$ on a double logarithmic scale. The lines through the estimates in the range $r\ge 3$ represent only the asymptotic behaviors, as also discussed in the main text.