Ising model

The Ising model places a variable on each site of a lattice that takes only the values +1 or −1, and lets each interact solely with its nearest neighbours through the energy H = −J Σ σ_i σ_j − h Σ σ_i. With J > 0 neighbours prefer to agree, and the model is ferromagnetic; h is an external field. That is the entire specification: no molecules, no positions, no dynamics. The behaviour comes from the competition in the free energy F = E − TS between an energy that wants order and an entropy that wants disorder, with temperature setting the exchange rate.

Its dimensional dependence is the model's most famous feature. In one dimension Ising solved it exactly in 1924 and found no ordered phase at any temperature above zero: a domain wall costs a fixed energy 2J but gains entropy k_B ln N, so order never pays. In two dimensions Onsager solved it exactly in 1944 and found a transition at k_BT_c = 2J/ln(1 + √2) ≈ 2.269J, with a magnetisation vanishing as (T_c − T)^(1/8) — the first exact critical exponent ever obtained. In three dimensions no exact solution is known, but Monte Carlo and the conformal bootstrap agree that β = 0.326419.

The model matters far beyond magnetism, and universality is why: its 3D exponents are exactly those measured in the liquid–vapour transition of real fluids and in binary-mixture demixing, because it shares their dimension and order-parameter symmetry while differing in every microscopic respect. Reinterpreting the two states as occupied/empty, copper/zinc, firing/quiet or infected/susceptible yields the lattice gas, the binary alloy, the Hopfield network and epidemic models on networks.

Posed by Wilhelm Lenz to his student Ernst Ising in 1920 and solved by Ising in one dimension in his 1924 Hamburg thesis; he concluded the model failed to explain ferromagnetism and left research. Lars Onsager's exact two-dimensional solution in 1944 found the transition Ising had ruled out, and the model has been central to statistical physics ever since.