Domain-size heterogeneity in the Ising model: Geometrical and thermal transitions

A measure of cluster size heterogeneity $\left(H\right)$, introduced by Lee et al. [Phys. Rev. E 84, 020101 (2011)] in the context of explosive percolation, was recently applied to random percolation and to domains of parallel spins in the Ising and Potts models. It is defined as the average number of different domain sizes in a given configuration and a new exponent was introduced to explain its scaling with the size of the system. In thermal spin models, however, physical clusters take into account the temperature-dependent correlation between neighboring spins and encode the critical properties of the phase transition. We here extend the measure of $H$ to these clusters and, moreover, present new results for the geometric domains for both $d=2$ and 3. We show that the heterogeneity associated with geometric domains has a previously unnoticed double peak, thus being able to detect both the thermal and percolative transitions. An alternative interpretation for the scaling of $H$ that does not introduce a new exponent is also proposed.

Figure 1

Size heterogeneity $H$ for the $d=2$ Ising model in equilibrium as a function of the temperature for several system sizes (top row) and the corresponding collapses (bottom row) using the reduced temperature $t\equiv T/T_{\mathrm{c}}-1$. The vertical dotted line locates the critical temperature $T_{\mathrm{c}}=2/ln(1+\sqrt{2})$. First (small) peak for the geometric domains (a), very close to $T_{\mathrm{c}}$. The best collapse (d), keeping $\nu =1$, was obtained with $\tau \simeq 2.016\left(4\right)$. Second, large and broad peak for the geometric domains (b). The height of the peak grows with an effective $\tau \simeq 2.048$ that is definitively different from the first peak. A good collapse (e) is obtained with $\nu \simeq 1.25$. The horizontal lines correspond to an independent measure of $H$ at $T\to \infty$ using a random configuration. Measure of $H_{\mathrm{p}}$ for the physical domains (c) whose exponents are ${\tau }_{\mathrm{p}}=31/15$ and ${\nu }_{\mathrm{H}}=31/16$ and give, again, an excellent collapse (f). In this case, there is a single peak close to $T_{\mathrm{c}}$. Notice the different horizontal scales in all graphs and the logarithmic one in (b).

Figure 2

Collapsed $H_{\mathrm{p}}$ for $d=2$, analogous to Fig. 1, but using the alternative (but equivalent) scaling with $\nu =1$.

Figure 3

Size of the largest and the largest non-percolating clusters, normalized by $N=L^d$, as a function of temperature for $L=640$. A cluster is considered percolating when it touches all sides of the square lattice. Below $T_{\mathrm{c}}$, indicated by a vertical dashed line, there is almost always a large, percolating cluster and the available space for the second largest cluster is small. For large temperatures, it is almost certain that there is no percolating cluster and both curves coincide. The insets show snapshots for $L=320$ with the largest nonpercolating geometric cluster in light gray at two different temperatures (that roughly correspond to the local maxima of the bottom curve). At the critical temperature, left snapshot, there is a percolating cluster (not shown) and the second largest is compact. In red (dark gray) we show, among the several physical subclusters in which it divides, the largest one. The right snapshot shows that, at higher temperature (second maximum), although the largest nonpercolating geometric cluster has almost the same average size as at $T_{\mathrm{c}}$, the physical subclusters are much smaller.

Figure 4

(a) Heterogeneity $H$ for both geometric (left peak, filled symbols) and physical clusters (right peak, empty symbols) for the 3d Ising model. The vertical lines are located at $T_{\mathrm{perc}}\simeq 0.95T_{\mathrm{c}}$ (pure geometric percolation) and $T_{\mathrm{c}}\simeq 4.511$ (Ising critical temperature). Inset: the behavior of $H_{\mathrm{g}}$ at the thermal critical point, crossing over to a different declivity. Collapse of both $H_{\mathrm{g}}$ (b) and $H_{\mathrm{p}}$ (c). For $H_{\mathrm{g}}$, $t\equiv (T-T_{\mathrm{perc}})/T_{\mathrm{perc}}$, and the best collapse was obtained with $T_{\mathrm{perc}}\simeq 0.95T_{\mathrm{c}}$, $\tau \simeq 2.08$, and $\nu \simeq 0.75$. For $H_{\mathrm{p}}$, on the other hand, $t\equiv (T-T_{\mathrm{c}})/T_{\mathrm{c}}$, and an excellent collapse is obtained with 3d Ising model exponents: $\nu \simeq 0.63$, $\tau \simeq 2.209$.