The 2D Ising Phase Transition

A square lattice of spins $s_i=\pm1$ with energy $H=-J\sum_{\langle ij\rangle}s_is_j$, evolved by single-spin Metropolis updates in checkerboard order. Above the Onsager temperature $T_c=2J/\ln(1+\sqrt2)\approx2.269$ the lattice is a disordered fizz with no net magnetization. Cool through $T_c$ and the $\mathbb{Z}_2$ symmetry breaks: domains coarsen, a net magnetization appears and grows as $m=[1-\sinh^{-4}(2J/T)]^{1/8}$, so the order-parameter exponent is $\beta=1/8$. Right at $T_c$ the susceptibility diverges and the dynamics critically slow down (clusters of every size, sluggish to relax).