Presets
Playback
Monte Carlo
Temperature, annealing and update speed.
Bonds and lattice
The bond disorder p and lattice size rebuild the system.
Layers
Model checks
The science
The model
Each site carries a spin s = ±1 and neighbours interact through bonds Jij = ±1: H = - Σ Jij si sj. A bond is antiferromagnetic (J = -1, red) with probability p. With p = 0 this is the ordinary Ising ferromagnet, solved exactly by Onsager with Tc = 2/ln(1+√2) = 2.269. Metropolis dynamics flips a spin with probability min(1, e-ΔE/T).
Frustration
A plaquette is frustrated when the product of its four bonds is negative: no assignment of spins can satisfy all four bonds. For independent random bonds a plaquette is frustrated with probability (1 - (1 - 2p)4)/2. Because every bond belongs to two plaquettes, at least F/2 bonds are unsatisfied, giving the rigorous bound E/N ≥ -2 + F/N. In a spin glass these competing constraints leave many nearly degenerate states, so cooling gets trapped in metastable configurations.
Replica overlap
Two replicas with the same bonds but independent random starts and updates are compared through q = (1/N) Σ siA siB. A ferromagnet at low T has |q| ≈ 1 (both fall into the same state or its mirror), while a glass shows small |q| at accessible times because the replicas freeze into different valleys.
Honest simplifications
The 2D ±J spin glass has no finite-temperature glass transition (Tc = 0), unlike the 3D model; here the slowing of dynamics at low T is shown, not a true phase transition. Lattices are small (up to 64 x 64), averages are short running means over one disorder sample, and screen time is arbitrary Monte Carlo time.
What to try
- Choose the ferromagnet and watch |m| jump near T = 2.27, following the Onsager curve.
- Set p = 0.5 and quench to T = 0.2: domains freeze and |q| stays small.
- Use "Anneal the glass" for a slow cooling run and compare the final energy with the bound line.
- At p = 1 the antiferromagnet is unfrustrated on this even lattice: yellow dots disappear and |m| is zero.