Posed by Lenz in 1920 · solved in one dimension by Ising in 1924 · solved in two by Onsager in 1944 — the standard test bench for phase transitions
Linked from: Topic 16 · Phase Transitions and Universality
01The question it poses
A lump of iron is a magnet below 770 °C and not one above it. That looks like something the chemistry of iron ought to answer for — after all, the temperature differs wildly between metals.
But in 1920 Wilhelm Lenz asked a different question: what is the smallest set of rules that still makes "the whole thing suddenly acquires a direction" happen? Could you throw out the chemistry entirely and keep only two sentences — each unit has two possible states, and adjacent units prefer to agree?
He handed the problem to his student Ernst Ising. The answer turned out to matter far more than the question: this almost empty model not only has a phase transition, its critical behaviour is identical to that of real magnets, of the liquid–gas critical point, and of ordering in alloys. It became the prototype of the whole idea of a universality class.
02The rules themselves
Take a square grid. Each cell holds a variable with two possible values: +1 or −1 (think "up/down", or "for/against").
Any two adjacent cells contribute energy −J if they agree and +J if they disagree, where J is some positive number. In other words: agreeing with a neighbour is cheaper.
The system's total energy is the sum over all adjacent pairs.
Fix a temperature T. Pick a cell at random and work out how much the energy would change if you flipped it; call that ΔE.
If ΔE ≤ 0 (flipping is cheaper), flip it. If ΔE > 0 (flipping costs), flip it anyway with probability e−ΔE/kT, where k is Boltzmann’s constant, a fixed factor converting temperature into energy — the hotter it is, the more readily the system does something that costs.
Repeat steps 4 and 5 until the statistics stop changing.
That is the whole model. No electrons, no quantum mechanics, no difference between iron and nickel. Two numbers remain: J (how sticky neighbours are) and T (how strong the random disturbance is) — and really only their ratio decides anything.
Rule 5 is the source of all the drama. If only energy-lowering moves were allowed, the system would freeze solid in whatever configuration it stumbled into. Allowing costly moves with some probability keeps it probing, and lets the two tendencies — agree with the neighbours, or flip at random — actually compete. Temperature is the balance on which that competition is weighed.
What a flip costs depends only on how many of the four neighbours currently agree with the cell. The entire physics of the model is in this one picture.
03What you see when it runs
Sweep T from low to high and you get three quite different pictures.
All three are 64×64 grids under identical rules; only the temperature differs. In the middle panel the blobs run from a few cells across to spanning the whole grid — there is no typical size.
Cold: almost the entire field is one colour, with a scattering of isolated opposite cells. The system has picked a direction and can hold it.
Hot: speckle everywhere, with blobs no more than a few cells across. No agreement survives more than a few cells of distance.
At one particular temperature in between: blobs of every size. Zoom in or out and what you see is statistically the same. This is what a diverging correlation length looks like — cells arbitrarily far apart are still pulling on each other.
Plot the net alignment (average the cell values, then take the absolute value) against temperature and you get the curve below. On a two-dimensional square grid it is known exactly.
Onsager's exact solution of 1944. The critical temperature is kTc/J = 2/ln(1+√2) ≈ 2.269, and the curve is continuous there but with infinite slope.
Approaching Tc, the net alignment falls to zero as (Tc − T) to the power 1/8. That 1/8 is exact, not fitted. Three dimensions has no exact solution, but numerical methods and the conformal bootstrap give an exponent of about 0.3264 — which is precisely the number measured in real magnets and at the liquid–gas critical point.
04What it explains in the real world
The Curie point of a ferromagnet. This is its home ground: why magnetism does not fade away gradually but hits zero at a definite temperature.
The liquid–gas critical point. Read +1/−1 as "this cell contains a molecule / does not" and the Ising model becomes the lattice gas — a correspondence written down by Lee and Yang in 1952. Mathematically, water's critical point and a magnet's Curie point are then the same thing.
Order–disorder transitions in alloys. In brass, copper and zinc atoms alternate neatly at low temperature and sit at random at high temperature. Read "copper or zinc on this site" as ±1 and it is the same model again.
Two liquids separating. Oil and water mixing above some temperature and separating below it also lands in this class.
What these four have in common is not that they resemble each other but that their critical exponents agree to within experimental precision. That is the entire content of the name "three-dimensional Ising universality class". It is also one of the very few places in complexity science where "the same thing happening in different fields" can be pinned down to three decimal places — most cross-domain analogies cannot do that.
What it cannot explain
It does not explain where real magnetism comes from. Why electron spins prefer to align is a quantum-mechanical matter (the exchange interaction), and the Ising model simply stuffs it into the constant J as an input. Real spins can also point in any direction in three-dimensional space, not just up or down; that case belongs to a different universality class (the Heisenberg class) with different exponents.
It cannot tell you what Tc is. The model computes the value of kTc/J, but J is entirely determined by the material and the model has nothing to say about it. So it can never tell you iron's Curie point is 770 °C. It gives shape, not position.
It has no time in it. The model defines an equilibrium statistical distribution, not a dynamical process. The flipping procedure in steps 4 and 5 (the Metropolis algorithm) is an invented sampling device, not how matter actually moves. Asking it how long a transition takes, or by what route it got there, is asking for something the model never defined.
In one dimension nothing happens at all. This is exactly what Ising computed in 1924: on a line there is no transition at any finite temperature. He concluded from it that there was none in any dimension — famously wrong, but the one-dimensional result itself stands. (Which makes "my system is like a one-dimensional chain" a real boundary condition that cancels the conclusion.)
It does not transfer to people as it stands. Reading ±1 as "for/against" and T as "social noise" is fluent, but the premises behind the derivation essentially all fail for human systems: equilibrium (societies are not), homogeneous units (people are not), symmetric coupling (I influence you exactly as much as you influence me — not in a society), and a scale large enough to ignore finite-size effects (an organization has tens or hundreds of members, nowhere near). Worse, social networks are full of long-range links, and range changes the universality class outright. As a metaphor for how local agreement grows into global agreement it is excellent; as a model you compute with, check those four first.
Further Reading
Ising model (entry, with what is known in each dimension)