The homogeneous plaquette Ising model in two and three dimensions is investigated by means of Monte Carlo simulations. By introducing a suitable order parameter for the two-dimensional lattice, and the finite-size scaling of the corresponding fourth-order cumulant, it is found that, consistent with the previous theoretical indications, the model in two dimensions is disordered at finite temperature and exhibits a zero-temperature phase transition characteristic of the one-dimensional Ising model with an essential (exponential) singularity of the order-parameter susceptibility as opposed to a Curie-law (power-law) divergence. In three dimensions, however, the model is believed to have a first-order phase transition at $T_c\simeq 3.6$ screened by strong metastability leading to a so-called “glassy transition” at $T\simeq 3.4$ when subjected to slow cooling. By computing the configurational entropy $S_c\equiv S\left(\text{liquid}\right)-S\left(\text{crystal}\right)$ in the supercooled temperature range via thermodynamic integration of the internal energy results, the Kauzmann temperature defined as that temperature where the extrapolated configurational entropy $S_c\left(T\right)$ vanishes, is estimated to be $T_K\simeq 3.18$. By finding ways to estimate the equilibration time of the supercooled liquid and the nucleation time of the stable crystal droplets, it is shown that $T\simeq 3.4$ is indeed the limit of stability or the effective spinodal temperature $T_{\text{sp}}$, at which the two time-scales associated with the quasiequilibration of the supercooled liquid, ${\tau }_{\text{eq}}$, and the nucleation of the stable crystal droplets, ${\tau }_{\text{nuc}}$, cross one another, with the former rising above the latter such that the supercooled liquid state becomes physically irrelevant below $T_{\text{sp}}\simeq 3.4$ and the impending entropy crisis at $T_K\simeq 3.18$ $\left(<T_{\text{sp}}\right)$ is thus avoided. Hence, what is sometimes called “glassy temperature,” is really a kinetic spinodal temperature that may be regarded as the remnant of the mean-field spinodal.

• Received 21 October 2009

DOI:https://doi.org/10.1103/PhysRevE.81.031501