Spin

Spin is an intrinsic form of angular momentum carried by elementary particles, with no classical counterpart: it is not a rotation of anything, and it cannot be spun up or slowed down. It is quantised, taking values in half-integer multiples of ħ, and an electron's spin of 1/2 means a measurement along any axis returns one of exactly two outcomes, conventionally called up and down. Because a charged particle with angular momentum carries a magnetic moment, spin is the origin of most magnetism in matter.

In thermodynamics and statistical physics, 'spin' is most often used in a narrower and cruder sense: the two-valued variable σ = ±1 sitting on each site of a lattice model. The Ising model's spins are these — bare labels retaining nothing of the quantum object except that they have two states and that neighbours have an energetic preference about whether they match. The interaction between real spins that drives ferromagnetism is the exchange interaction, a consequence of the Pauli exclusion principle rather than of magnetic forces, which are far too weak to hold order together at hundreds of kelvin.

The crudeness is deliberate and, near a critical point, harmless. Universality guarantees that critical exponents depend only on dimensionality and the symmetry of the order parameter, so a lattice of abstract ±1 labels reproduces the exponents of a real ferromagnet — and of a fluid — exactly, despite discarding all the quantum mechanics. The full treatment of spin as angular momentum, with its half-integer quantisation and its role in the exclusion principle, belongs to quantum mechanics.

Proposed by Goudsmit and Uhlenbeck in 1925 to explain atomic spectra, and grounded by Dirac's relativistic electron equation of 1928. The Stern–Gerlach experiment of 1922 had already demonstrated the two-valued measurement outcome before anyone knew what it meant. Lenz and Ising's use of a two-state variable on a lattice dates from 1920, predating the discovery of spin itself.