Percolation in the two-dimensional Ising model

The study of the Ising model from a percolation perspective has played a significant role in the modern theory of critical phenomena. We consider the celebrated square-lattice Ising model and construct percolation clusters by placing bonds, with probability $p$, between any pair of parallel spins within an extended range beyond nearest neighbors. At the Ising criticality, we observe two percolation transitions as $p$ increases: starting from a disordered phase with only small clusters, the percolation system enters into a stable critical phase that persists over a wide range $p_{c_1} < p < p_{c_2}$, and then develops a long-ranged percolation order with giant clusters for both up and down spins. At $p_{c1}$ and for the stable critical phase, the critical behaviors agree well with those for the Fortuin-Kasteleyn random clusters and the spin domains of the Ising model, respectively. At $p_{c2}$, the fractal dimension of clusters and the scaling exponent along $p$ direction are estimated as $y_{h2} = 1.958\,0(6)$ and $y_{p2} = 0.552(9)$, of which the exact values remain unknown. These findings reveal interesting geometric properties of the two-dimensional Ising model that has been studied for more than 100 years.