How a magnet makes up its mind

A magnet with no instructions still picks a direction

Take a lump of iron above a few hundred degrees and it is not a magnet. Its atoms each carry a tiny magnetic moment, a little arrow, but the arrows point every which way and average to nothing. Now let it cool. At one specific temperature, with no magnet nearby, no field, nothing outside telling the arrows which way to face, a billion of them suddenly agree. The lump becomes a magnet, and it had to choose a direction, north or south, out of thin air.

That is the strange bit. The laws governing the arrows are perfectly even-handed: flip every arrow at once and the energy doesn't change, so "up" and "down" are equally good. Yet below a critical temperature the material commits to one of them. The symmetry of the rules is not the symmetry of the outcome. Physicists call this spontaneous symmetry breaking, and the cleanest place to watch it happen is a toy model a graduate student wrote down in 1925.

The only two rules

Each spin feels two competing pressures. The first is its neighbours: aligning with them lowers the energy. The second is heat, which jostles the spins at random and cares nothing for alignment. Write the total energy of a configuration as a sum over neighbouring pairs, plus a term for any external field $h$ leaning on the spins:

$$E = -J \sum_{\langle i,j \rangle} s_i s_j - h \sum_i s_i$$

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With $J > 0$, a pair pointing the same way ($s_i s_j = +1$) contributes $-J$ and lowers the energy; a mismatched pair costs $+J$. So the energy is happiest when everyone agrees. Set the field $h$ to zero and the tie between all-up and all-down is exact, which is the even-handedness we need for the magnet to choose.

Heat enters through statistics, not force. At temperature $T$ the probability of finding the system in a given configuration follows the Boltzmann weight: low-energy configurations are exponentially more likely, but the bias softens as $T$ climbs.

$$P(\text{config}) = \frac{1}{Z}\, e^{-E / k_B T}$$

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Here $k_B$ is Boltzmann's constant and $Z$, the partition function, is just the sum of $e^{-E/k_B T}$ over every configuration so the probabilities add to one. Two knobs, $J$ and $T$, and their ratio decides everything.

Simulating it: Metropolis Monte Carlo

You can't sum over every configuration by hand. A modest 32×32 grid has $2^{1024}$ of them, more than there are atoms in the observable universe. So the lab does what every working physicist does and samples instead. The recipe is Metropolis Monte Carlo (Metropolis and co-workers, 1953), and it is almost suspiciously simple:

1. Pick a spin at random.
2. Work out $\Delta E$, the energy change if you flip it.
3. If the flip lowers the energy ($\Delta E \le 0$), accept it.
4. If it raises the energy, accept it anyway with probability $e^{-\Delta E / k_B T}$.

Repeat a few million times. That fourth step is the whole trick: the system is sometimes allowed an "uphill" move, and the temperature sets how often. Cold, and uphill moves almost never happen, so the lattice freezes into agreement. Hot, and they happen constantly, so it never settles. Run it long enough and the configurations you see appear with exactly the Boltzmann probability of equation (2), no derivation required.

The transition, and the number Onsager found

Turn the temperature in the lab slowly downward and there is a moment where the fizz stops being fizz and starts being domains. For the 2D square lattice that moment sits at a precise value:

$$k_B T_c = \frac{2J}{\ln(1 + \sqrt{2})} \approx 2.269\,J$$

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Above $T_c$ the net magnetisation is zero: as many spins up as down, no overall direction. Below it, a nonzero fraction of the spins agree, and the material carries a real magnetisation even with the field switched off. That fraction is the order parameter $M$, the single number that tells you which phase you're in. It is zero in the disordered phase and climbs towards one as you cool.

Lars Onsager then did something people had thought impossible. In 1944 he solved the 2D model exactly, with no external field, and wrote the magnetisation down in closed form.

$$M(T) = \left[\, 1 - \sinh^{-4}\!\left(\frac{2J}{k_B T}\right) \right]^{1/8}$$

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A magnet is a room full of arrows agreeing on a direction that nothing outside the room ever specified.

Look at how $M$ switches on. Just below $T_c$ it doesn't ramp up gently. It rises with that $1/8$ power, a curve that leaves zero with an infinite slope. Near the critical point we write $M \sim (T_c - T)^\beta$, and Onsager's formula hands you $\beta = 1/8$ exactly. That $\beta$ is our first critical exponent, and it is where the story stops being about magnets in particular.

Critical temperature

2.269 J

kT_c = 2J / ln(1+√2) for the 2D square lattice

Order-parameter exponent

β = 1/8

M ~ (T_c − T)^β as you cool through the transition

Exact 2D solution

1944

Onsager solved it in closed form, with no external field

Why the same numbers keep turning up

Approach $T_c$ and a second quantity misbehaves: the correlation length $\xi$, the typical size of an aligned domain. Far from $T_c$ it is a few lattice spacings across. As you close in on $T_c$ it grows without bound, and at $T_c$ it diverges. That divergence is exactly what you're watching when domains appear at every size at once. With no characteristic length left, the lattice looks statistically the same whether you squint at ten cells or a thousand. It has gone scale-free.

Kenneth Wilson explained why in the early 1970s, and it won him a Nobel Prize. His renormalisation group takes the scale-free lattice at $T_c$ and repeatedly blurs it: average each block of spins into one, zoom out, repeat. Because the critical system looks the same at every scale, the blurring flows towards a fixed point, and everything that once told one system apart from another (the lattice, the exact $J$) gets averaged away in the first few steps. What survives the blurring is only the dimension and the symmetry, and those alone fix the exponents. Universality is not a coincidence. It is what is left once you throw away everything that isn't scale-invariant.

This is why the Ising model earns your time even if you never touch a magnet. The liquid-gas critical point (the temperature and pressure where the distinction between water and steam vanishes) sits in the same universality class as the 3D Ising magnet. Swap "spin up or down" for "dense or sparse region" and the order parameter, the exponents, the whole critical behaviour carry straight across. Boiling water and a cooling magnet are, near their critical points, the same problem wearing different clothes.

The same trick, elsewhere

Local rules producing global order is not unique to spins. The reaction-diffusion system below has a completely different cast, two chemicals that react and spread, and yet it plays the same game: nothing in the equations specifies a stripe or a spot, and the pattern that emerges still spans the whole dish. Watch a global structure organise itself out of purely neighbour-to-neighbour rules.

The connection reaches into computing, too. A Hopfield network, one of the founding models of neural memory, is an Ising model in disguise: neurons standing in for spins, learned synaptic weights standing in for the uniform coupling $J$. Storing a memory means carving an energy minimum for the network to roll into, exactly as a magnet rolls into its all-up or all-down ground state. Recalling a memory is the network relaxing to the nearest minimum, the same settling that let our lattice pick a direction. The maths that decides which way a magnet points also decides how a network remembers a face.

So the next time your compass swings north, remember there was a moment, deep in the iron's history, when north and south were equally on offer and the atoms simply had to pick. The lab up top lets you rerun that decision as often as you like: cool it through 2.269 and watch a billion arrows make up their collective mind, with nobody in charge. Next time we can let the field $h$ back in and watch hysteresis appear, the magnet's memory of which way it was last pushed. Have a play with the temperature knob first.

Reading further

• Ernst Ising, "Beitrag zur Theorie des Ferromagnetismus" (Zeitschrift für Physik, 1925). The original thesis paper that solved the 1D chain and found no transition. Short, and a lesson in how a "negative" result can still name a field.
• Lars Onsager, "Crystal Statistics. I." (Physical Review 65, 117, 1944). The exact 2D solution, one of the hardest and most celebrated calculations in statistical mechanics. Equations (3) and (4) are its headline results.
• Kenneth G. Wilson, "The renormalization group: Critical phenomena and the Kondo problem" (Reviews of Modern Physics 47, 773, 1975). The Nobel-winning account of why universality holds: blur the scale-free critical system and watch the microscopic details flow away. Builds on Leo Kadanoff's block-spin picture of 1966.
• James P. Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity. The modern textbook that puts order parameters and universality at the centre. Free online, and the single best source for everything above.