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The 2D Ising Phase Transition

A square lattice of spins si=±1s_i=\pm1 with energy H=−J∑⟨ij⟩sisjH=-J\sum_{\langle ij\rangle}s_is_j, evolved by single-spin Metropolis updates in checkerboard order. Above the Onsager temperature Tc=2J/ln⁡(1+2)≈2.269T_c=2J/\ln(1+\sqrt2)\approx2.269 the lattice is a disordered fizz with no net magnetization. Cool through TcT_c and the Z2\mathbb{Z}_2 symmetry breaks: domains coarsen, a net magnetization appears and grows as m=[1−sinh⁡−4(2J/T)]1/8m=[1-\sinh^{-4}(2J/T)]^{1/8}, so the order-parameter exponent is β=1/8\beta=1/8. Right at TcT_c the susceptibility diverges and the dynamics critically slow down (clusters of every size, sluggish to relax).

Figure 1. Live 2D Ising lattice and the magnetization curve across the order-disorder transition. Method: checkerboard single-spin Metropolis (shared lattice-MC engine), susceptibility from fluctuations.
temperature T2.00
sweeps / frame3
start

WHAT TO TRY

  • Sweep the temperature through the Onsager point Tc = 2J/ln(1+sqrt 2): below it the lattice locks into one magnetised domain, above it the spins disorder, and right at Tc you see scale-free critical clusters.
  • Compare the measured magnetisation against the Onsager exact curve: the live |M| point tracks the analytic line and drops sharply to zero at Tc, the second-order phase transition.
  • Start hot versus cold: a cold aligned start melts as you heat it, a hot random start orders as you cool it, and near Tc the susceptibility chi spikes as fluctuations diverge.