The 2D spin glass

Spin glasses were first seen in alloys such as gold with a little iron, or copper with a little manganese. The magnetic atoms sit at random places, and the interaction between two of them, carried by the conduction electrons, is ferromagnetic or antiferromagnetic depending on their distance. In 1972 Cannella and Mydosh found a sharp cusp in the magnetic susceptibility of such an alloy: below a freezing temperature, the spins freeze in random-looking directions. There is no magnetization, as in a paramagnet, but no motion either, as in a magnet: disorder frozen in place, like the atoms of a glass.

The Edwards–Anderson model

In 1975 Sam Edwards and Philip Anderson proposed a minimal model: Ising spins s_i = ±1 on a lattice, with random couplings J_ij between neighbors and energy

E = −∑_⟨ij⟩ J_ij s_i s_j.

Here the lattice is a square grid covering the screen, and each coupling is +1 (the two spins prefer to agree) or −1 (they prefer to disagree) with equal probability. The couplings are drawn once and then stay fixed: the disorder is quenched.

Frustration

Take four spins around a square, with three couplings +1 and one −1. Satisfying the three positive bonds forces the four spins to be equal, and then the negative bond is unsatisfied. Whatever you try, an odd number of the four bonds stays unsatisfied. Gérard Toulouse called such a square, or plaquette, frustrated (1977): this happens exactly when the product of its four couplings is −1, so about half of the plaquettes of the grid are frustrated.

Strings

The background draws the frustration rather than the spins. Orange dots mark the frustrated plaquettes, and each unsatisfied bond is drawn in blue as a short segment that crosses it, joining the centers of the two plaquettes on either side. By the parity rule, an odd number of segments meet at each frustrated plaquette and an even number at every other one: the segments form strings that start and end on orange dots, plus closed loops. Flipping a spin toggles its four bonds and moves the strings, but never their ends, which are fixed by the couplings.

An unsatisfied bond costs 2 more than a satisfied one, so up to a constant the energy is twice the total length of the strings. A ground state thus pairs up the frustrated plaquettes with strings of minimum total length. On a planar lattice this is a minimum-weight perfect matching problem, solved by efficient (polynomial-time) algorithms, as shown by Bieche, Uhry, Maynard and Rammal in 1980; in three dimensions, finding a ground state is NP-hard (Barahona, 1982). Even in the best configuration about 15% of the bonds stay unsatisfied, since the ground-state energy is about −1.40 per spin, and a huge number of ground states share that energy.

On this site

At low temperature the strings are short, mostly joining neighboring dots, and they keep rearranging: a spin with two satisfied and two unsatisfied bonds can flip at no cost, which moves strings from one optimal pairing to another. When the temperature rises (slider of the animation menu, in units of J, from 0.2 to 2.5, by default 0.8), thermal excitations lengthen the strings into a tangle. There is no sharp transition along the way: in two dimensions the spin-glass phase only exists at zero temperature, although relaxation becomes extremely slow at low temperature. In three dimensions the spin-glass phase survives up to a finite temperature.

One spin every 26 pixels. The dynamics is Metropolis at random sites, about 1.5 attempts per spin per second; to keep the motion calm, zero-cost flips are accepted only 12% of the time, which does not change the equilibrium distribution. Before it appears, the system is slowly cooled from T = 2.5. The couplings are drawn once per visit and kept from page to page.

Why it matters

The theory of spin glasses, in particular Giorgio Parisi’s solution of the mean-field Sherrington–Kirkpatrick model with replica symmetry breaking (Nobel Prize in Physics 2021), became a toolbox used far beyond magnetism: for hard optimization problems, error-correcting codes, neural networks, and the high-dimensional inference and learning problems our group works on.

Spin glass and Geometrical frustration, on Wikipedia.