Strong violation of critical phenomena universality: Wang-Landau study of the two-dimensional Blume-Capel model under bond randomness

We study the pure and random-bond versions of the square lattice ferromagnetic Blume-Capel model, in both the first-order and second-order phase transition regimes of the pure model. Phase transition temperatures, thermal and magnetic critical exponents are determined for lattice sizes in the range $L=20–100$ via a sophisticated two-stage numerical strategy of entropic sampling in dominant energy subspaces, using mainly the Wang-Landau algorithm. The second-order phase transition, emerging under random bonds from the second-order regime of the pure model, has the same values of critical exponents as the two-dimensional Ising universality class, with the effect of the bond disorder on the specific heat being well described by double-logarithmic corrections, our findings thus supporting the marginal irrelevance of quenched bond randomness. On the other hand, the second-order transition, emerging under bond randomness from the first-order regime of the pure model, has a distinctive universality class with $\nu =1.30\left(6\right)$ and $\beta ∕\nu =0.128\left(5\right)$. These results amount to a strong violation of universality principle of critical phenomena, since these two second-order transitions, with different sets of critical exponents, are between the same ferromagnetic and paramagnetic phases. Furthermore, the latter of these two sets of results supports an extensive but weak universality, since it has the same magnetic critical exponent (but a different thermal critical exponent) as a wide variety of two-dimensional systems with and without quenched disorder. In the conversion by bond randomness of the first-order transition of the pure system to second order, we detect, by introducing and evaluating connectivity spin densities, a microsegregation that also explains the increase we find in the phase transition temperature under bond randomness.

Figure 1

Behavior of the pure 2d BC model at $\Delta =1.975$: (a) FSS behavior of the specific heat peaks giving a clear $L^2$ divergence. (b) The same for the susceptibility maxima. (c) Simultaneous fitting of the maxima of the averaged logarithmic derivatives of the order parameter defined in Eq. (4). (d) Power-law behavior of the averaged absolute order-parameter derivative in Eq. (5). (e) The DP structure of the energy PDF at $T=T_h$ via the two different implementations of the WL scheme. The multi-R implementation is displaced very slightly to the right. (f) Limiting behavior for the surface tension $\Sigma \left(L\right)$ defined in the text and the latent heat $\Delta e\left(L\right)$ shown in panel (e). The linear fits shown include only the five points corresponding to the larger lattice sizes.

Figure 2

Behavior of the pure 2d BC model at $\Delta =1$: (a) Simultaneous fitting of six pseudocritical temperatures defined in the text for $L⩾50$. (b) FSS of the specific heat peaks. The inset shows a linear fit of the specific heat data on a log scale for $L⩾50$. (c) FSS of the susceptibility peaks on a log-log scale. (d) FSS of the order parameter at the estimated critical temperature also on a log-log scale. Linear fits are applied in panels (c) and (d).

Figure 3

(Color online) Behavior of the random-bond 2d BC model at $\Delta =1.975$: (a)–(d) Softening effects on the first-order transition features of the 2d BC model, induced by bond randomness of various strengths. In (a) and (b), the very rough $P\left(e\right)$ is for the pure model, whereas the smoothed curves are for $r=0.90$ (deeper) and $r=0.82$ (shallower). The curves in (c) and (d) are for different seeds. Panels (e)–(h) illustrate the $⟨e⟩\text{−}T$ behavior and $⟨M⟩\text{−}T$ behavior, for the size $L=30$ and various disorder strengths $r$ for different disorder realizations. The curves in (e) and (f) are for $r=0.90,0.67,0.54,0.43,0.33$, top to bottom on the right of (e), bottom to top on the right of (f), and top to bottom on the left of (f). The curves in (g) are for $r=0.74,0.60,0.54,0.48,0.43,0.33,0.14$ top to bottom on the left and bottom to top on the right. The curves in (h) are for different seeds.

Figure 4

Behavior of the random-bond 2d BC model at $\Delta =1.975$: (a) Illustration of the clear saturation of the specific heat $\left({\left[C\right]}_{\mathit{av}}^{*}\right)$ for the random-bond (open symbols) 2d BC model. (b) FSS behavior of five pseudocritical temperatures defined in the text for $r=3∕5=0.6$. The lines show a simultaneous fit for the larger lattice sizes $L⩾50$. (c)–(f) Estimation of critical exponents $\nu$, $\gamma ∕\nu$, and $\beta ∕\nu$ for the case of $r=3∕5=0.6$. In all panels the fits have been performed for the larger lattice sizes shown $(L⩾50)$.

Figure 5

Behavior of the random-bond $(r=0.6)$ 2d BC model at $\Delta =1$: (a) Simultaneous fitting of six pseudocritical temperatures defined in the text for $L⩾50$. (b) FSS of the averaged specific heat peaks. A double-logarithmic fit is applied for $L⩾50$. The inset shows a linear fit on a double-logarithmic scale. (c) FSS behavior of the averaged susceptibility peaks on a log-log scale. (d) FSS of the averaged order parameter at the estimated critical temperature, also on a log-log scale. Linear fits are applied for $L⩾50$.

Figure 6

(Color online) Temperature behavior of the connectivity spin densities $Q_n$ defined in the text for $\Delta =1$ (upper curves) and $\Delta =1.975$ (lower curves) for disorder strength $r=0.6$ and lattice size $L=60$. In each group, the curves are for $n=0,1,2,3,4$ from bottom to top.