The Ising Model Explained: How Tiny Arrows Explain Magnets, Boiling and Phase Transitions

Heat a permanent magnet past a certain critical temperature and it loses its magnetisation. Water does something similar when it boils: below a certain temperature it is one thing, above it another. These sudden changes are called phase transitions, and the simplest model that captures one is the Ising model. It is the physics behind our Ising Model hoodie, slogan “Every Scale at Once”, and this post connects to our explainer What Happens at Absolute Zero?

What is the Ising model?

Picture a grid. On every square sits a tiny arrow, a “spin”, which can point only up (s = +1) or down (s = −1). Each spin interacts only with its nearest neighbours. Neighbours that match lower the energy; neighbours that disagree raise it. That is the whole model:

E = −J Σ sᵢsⱼ

E is the total energy in joules, the sum runs over every pair of nearest-neighbour sites i and j, sᵢ and sⱼ are the spins (each +1 or −1), and J is the coupling energy in joules. When J is positive, aligned neighbours (product +1) lower the energy, so the spins “want” to line up, as they do in a ferromagnet such as iron.

Temperature fights that wish. At temperature T, thermal energy of order k_BT (k_B = 1.380 649 × 10⁻²³ J/K is the Boltzmann constant, an exact value (NIST CODATA^[1])) keeps flipping spins at random. The contest between order (J) and disorder (k_BT) is the entire story. If you have read our post on what entropy is, you will recognise the same tug of war between energy and the number of possible arrangements.

Who invented the Ising model?

Not only Ising. Historian Stephen Brush’s review in Reviews of Modern Physics describes it as the “Ising model,” “discussed by Ernst Ising in 1925 but suggested earlier (1920) by Wilhelm Lenz” (Brush 1967^[2]). Brush himself titled his review “History of the Lenz-Ising Model”. Ising worked out the one-dimensional version, a single row of spins, in his 1925 paper in Zeitschrift für Physik (volume 31, pages 253–258). It is one more case for our post on who really discovered it: the model carries one name, but Brush credits the suggestion to Lenz five years earlier.

What did Onsager prove in 1944?

The two-dimensional version, a flat grid of spins, is much harder. In 1944 Lars Onsager solved it exactly. His abstract states that the infinite two-dimensional crystal “has an order-disorder transition at a temperature T = T_c” given by the condition sinh(2J/kT_c) sinh(2J′/kT_c) = 1, where J and J′ are the couplings along the two directions of the grid (Onsager 1944^[3]). On a square grid with equal couplings, J = J′, the condition becomes sinh(2J/k_BT_c) = 1, which gives

k_BT_c/J = 2/ln(1+√2) ≈ 2.269

Here T_c is the critical temperature in kelvin and ln is the natural logarithm. As a worked number, if J were 1 meV (1.602 × 10⁻²² J), then T_c = 2.269 × (1.602 × 10⁻²² J)/(1.381 × 10⁻²³ J/K) ≈ 26.3 K. Onsager also found that at T_c “the specific heat becomes infinite as −log|T − T_c|” (Onsager 1944^[3]): the material’s appetite for heat spikes without limit right at the transition.

This was a landmark because it showed, with no approximations, that a simple model of local interactions really does produce a sharp phase transition.

How fast does the magnetisation appear below T_c?

In 1952 C. N. Yang calculated the spontaneous magnetisation of the two-dimensional Ising model exactly; his abstract notes that “the result also gives the long-range order in the lattice” (Yang 1952^[4]). In modern notation, for T below T_c, the average spin per site is

M = [1 − sinh⁻⁴(2J/k_BT)]^(1/8)

M runs from 1 (every spin aligned) at absolute zero down to 0 at T_c, and stays 0 above it. The striking part is the power 1/8. Close to T_c the magnetisation does not fade gently; it collapses steeply. At 99% of T_c, the formula gives M ≈ 0.69, still most of the full value. Our chart below plots the curve.

What does a phase transition look like up close?

We ran a small computer simulation of a 2D Ising grid using a standard Monte Carlo method, in which spins flip at random with the probabilities that thermal physics prescribes. Below T_c, large single-colour domains take over. Far above T_c, the grid looks like television static. Right at T_c, something remarkable appears: islands inside islands inside islands, patches of every size from single squares up to the whole grid. There is no typical size. That is “every scale at once”.

The Nobel committee described this behaviour when Kenneth Wilson won the 1982 Nobel Prize in Physics “for his theory for critical phenomena in connection with phase transitions” (NobelPrize.org^[5]). Its press release explains that critical phenomena “have to do with the very large fluctuations that occur in the system when we come close to the critical point”, and that one must deal with fluctuations “over widely different scales of length” (Nobel press release 1982^[6]).

Why do magnets and boiling liquids behave alike?

This is the deepest lesson of the Ising model. The 1982 press release lists everyday transitions such as “a permanent magnet loses its magnetization above a certain critical temperature”, then states that “liquids, mixtures of liquids, ferromagnets, and binary alloys show the same critical behaviour” (Nobel press release 1982^[6]). In the committee’s words, “experimental and theoretical work from the sixties suggested this form of universality, but Wilson’s theory gave a convincing proof from fundamental principles” (Nobel press release 1982^[6]).

The Ising model also translates directly into other systems. Brush notes the “mathematically equivalent ‘lattice gas’ model” used to study gas–liquid and liquid–solid transitions (Brush 1967^[2]): read “spin up” as “site occupied by a molecule” and “spin down” as “site empty”, and the magnet becomes a fluid.

Is the 3D Ising model solved?

No. Onsager’s exact result is for the two-dimensional crystal (Onsager 1944^[3]). A 2022 review calls an exact solution of the Ising model on the simple cubic lattice “one of the long-standing open problems in rigorous statistical mechanics,” and notes that that all attempts so far to find an exact partition function have failed (Viswanathan et al. 2022^[7]). It remains open: its critical temperature is known only numerically, from simulations.

FAQ

Are the spins in the Ising model real electrons?

They are a simplification. Real electron spins are quantum objects that can point in any direction. The Ising model keeps only two states per site, which is enough to capture the ordering transition.

Why is T_c higher when J is larger?

Because k_BT_c/J is a fixed number (≈ 2.269 on the square grid), a stronger coupling J needs more thermal energy to scramble.

Where can I see more thermal physics?

Browse Waves, Chaos & Matter for collective behaviour, and read our post on negative absolute temperature, where spin systems again do strange things.

References

1. NIST CODATA, Boltzmann constant. https://physics.nist.gov/cgi-bin/cuu/Value?k
2. S. G. Brush (1967), “History of the Lenz-Ising Model,” Rev. Mod. Phys. 39, 883–893 (DOI 10.1103/RevModPhys.39.883). https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.39.883
3. L. Onsager (1944), “Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition,” Phys. Rev. 65, 117–149 (DOI 10.1103/PhysRev.65.117). https://journals.aps.org/pr/abstract/10.1103/PhysRev.65.117
4. C. N. Yang (1952), “The Spontaneous Magnetization of a Two-Dimensional Ising Model,” Phys. Rev. 85, 808–816 (DOI 10.1103/PhysRev.85.808). https://journals.aps.org/pr/abstract/10.1103/PhysRev.85.808
5. NobelPrize.org, The Nobel Prize in Physics 1982 (Kenneth G. Wilson). https://www.nobelprize.org/prizes/physics/1982/summary/
6. NobelPrize.org, Press release: The 1982 Nobel Prize in Physics. https://www.nobelprize.org/prizes/physics/1982/press-release/
7. G. M. Viswanathan, M. A. G. Portillo, E. P. Raposo & M. G. E. da Luz (2022), “What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution,” Entropy 24, 1665 (DOI 10.3390/e24111665). https://www.mdpi.com/1099-4300/24/11/1665

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.