Phase Transitions

Other transitions followed the same pattern. Iron is magnetic below 770 °C, its Curie point, and not above it. In 1937 Lev Landau found what they share. The low-temperature phase is less symmetric. A magnet picks a direction, although the laws of physics favour none. An order parameter, such as the magnetisation, measures how much symmetry is broken, and near the transition it grows from zero.

Could statistical mechanics produce such sharpness at all? Every sum over states is a smooth function of temperature. Ernst Ising had found in 1925 that a chain of tiny magnets has no transition. In 1944 Lars Onsager solved the two-dimensional version exactly and showed a sharp transition emerging when the number of magnets becomes infinite. His exact answers also disagreed with Landau's theory near the critical point.

Experiments showed that near critical points, quantities such as the magnetisation or the density difference change as powers of the distance from the transition, and that the powers are the same for carbon dioxide, xenon and certain magnets. Why should a fluid and a magnet agree? Leo Kadanoff suggested in 1966 that at a critical point, fluctuations exist at every scale, so the system looks the same when viewed from further away. Kenneth Wilson turned this into the renormalisation group in 1971: zoom out step by step, averaging over small details, and follow what survives. Only a few features do, such as the dimension of space and the symmetry of the order parameter. Everything else is washed out, which is why different materials share their critical behaviour. With Michael Fisher he computed the exponents, and they matched experiment.

The same ideas transformed quantum field theory, where renormalisation had been a mathematical trick. In the 1970s Michael Kosterlitz and David Thouless, and independently Vadim Berezinskii, found a new kind of transition driven by topology, the start of the study of topological matter.

Take a square grid of tiny magnets, each pointing up or down, with energy lowered by an amount $J$ for each pair of aligned neighbours. The simplest approximation, mean-field theory, replaces each magnet's four neighbours by their average. It predicts a transition at

and near it, the magnetisation should grow as $(T_c - T)^{\beta}$ with $\beta = \tfrac12$. Onsager's exact solution, completed for the magnetisation by C. N. Yang in 1952, gives

Mean-field theory overestimates the critical temperature by more than 75%, and gets the exponent badly wrong. The reason is fluctuations. Averaging ignores the large, correlated patches of up and down magnets that form near the transition, and those patches are what destroy order at a lower temperature than the average predicts.

In three dimensions, fluctuations matter less but still matter. The exponent is $\beta \approx 0.326$, known from the renormalisation group and computer calculations. It is the same number measured at the liquid–gas critical point of carbon dioxide, xenon and water, and in uniaxial magnets. None of those materials resemble a grid of magnets in detail. At the critical point, the details stop mattering. Above four dimensions, fluctuations become unimportant and mean-field theory's $\beta = \tfrac12$ becomes exact. The dimension of space is one of the few features that survives the zooming out.

Phase transitions now appear throughout physics and beyond: superconductors, liquid crystals, the early universe, and the droplets that organise the inside of living cells. The three-dimensional Ising model, the simplest description of the critical point of every fluid on Earth, has still not been solved exactly. And many systems never settle into any phase at all. They are driven, flowing and alive, and belong to non-equilibrium physics.