Quench

This is the two-dimensional Ising model, the simplest thing in physics that orders. Every site is a spin, up or down, light or dark, and each one feels nothing but its four neighbours. No spin can see the lattice. No spin decides anything. And yet below one exact temperature the whole field takes a side, because agreeing with your neighbours, repeated a hundred thousand times, is indistinguishable from a decision nobody made.

The temperature where that happens is not approximate. Onsager solved this model exactly in 1944: Tc equals 2 divided by ln(1 + sqrt(2)), which is 2.269185. It is one of the few places in physics where a collective surprise has a closed-form address, and it is marked on the slider below the lattice.

The model on this page was made to find that address on its own before being allowed to claim it. Measured blind by the standard method, a Binder-cumulant crossing at three lattice sizes, it put the critical point at 2.2784, within 0.405 percent of the exact answer, and away from the transition it follows Yang's exact 1952 magnetization curve to three decimals. The probe that checks this first proves itself against a small lattice solved by brute force, all 65,536 states summed directly.

Near the critical point the piece runs the Wolff cluster algorithm, and the reason is measured rather than aesthetic: close to Tc, flipping one spin at a time stops working, because order lives in correlated clusters at every scale and local moves cannot turn a cluster over. Decorrelating the field there cost 338 times more work by single flips than by cluster moves. When you write into the lattice the piece switches to local dynamics, because the cluster algorithm is only valid with no field applied.

The writing is the interaction worth sitting with. Drag across the lattice and a region aligns under your hand. Above the critical temperature the field forgets your mark in seconds, because thermal noise outvotes any local agreement. Quench below it and the mark keeps itself, because now the neighbours hold each other. Same lattice, same rule, and whether it can remember you is decided by one number.

The honest limits. This is the equilibrium Ising model with invented local dynamics for the transitions between states, not a simulation of any real magnet. The lattice is finite, so the transition is rounded into a region rather than a point, and the measured critical temperature carries the finite-size shift its uncertainty states. Nothing about a real material's atoms, impurities, or long-range forces is solved here.