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

Cluster or domain size distributions are commonly used in statistical mechanics
to unveil geometric properties and characterize both the equilibrium critical
behavior and the out of equilibrium dynamics of several models. If the system is
finite, not all possible domain sizes are present on a single configuration.
These sample-to-sample fluctuations disappear and the distributions become dense
in the infinite size limit, or after ensemble averages are taken. In order to
characterize these fluctuations, a quantity associated with how heterogeneously
sized the equilibrium domains are, $H$, was recently proposed in the context of
explosive percolation and then applied to ordinary random percolation and to
domains of parallel spins in the Ising and Potts models. Differently from the
cluster size distribution, the heterogeneity $H$ only takes into account whether
a given size is present in each configuration and gives the number of such
distinct sizes. In addition to geometric domains of parallel spins, for thermal
spin models one can consider physical clusters, which take into account the
temperature-dependent correlation between neighboring spins and encode the
critical properties of phase transitions. In this work we extend the measure of
$H$ to these clusters and, moreover, present new results for the geometric
domains in the Ising model with $d=2$ and $3$. We show through extensive Monte
Carlo simulations that the heterogeneity associated with physical domains for
$d=2$ has a single peak that diverges at the critical temperature, while for the
geometric domains it has a previously unnoticed double peak, presenting
signatures for both the percolative and thermal transitions [1].

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[1] Phys. Rev. E 91, 042113 (2015)