Spin Glass
When magnetic couplings are random — some ferromagnetic, some antiferromagnetic — spins become frustrated. No configuration fully satisfies all bonds, producing an exponentially complex energy landscape and glassy dynamics.
The Edwards-Anderson Model
In a spin glass, each pair of neighboring spins i,j has a coupling Jij drawn from a random distribution (here Gaussian). The Hamiltonian is:
Bonds can be ferromagnetic (J>0, wants spins aligned) or antiferromagnetic (J<0, wants spins opposed). When competing, no spin can satisfy all its neighbors simultaneously — this is frustration.
Frustration & Glassy Dynamics
The highlighted bonds are frustrated — they cannot simultaneously be in their preferred state. The fraction of frustrated bonds never reaches zero in a true spin glass.
Below the glass transition temperature Tg, the system gets trapped in one of exponentially many metastable states. Relaxation becomes logarithmically slow — the defining signature of glass, and a model for optimization hardness.
Phase Diagram
High T: paramagnetic phase — spins fluctuate randomly, no order.
Low T, weak disorder: ferromagnetic order emerges, spins align collectively.
Low T, strong disorder: spin glass phase — spins freeze in random orientations, but the pattern is stable. The Edwards-Anderson order parameter q = ⟨σᵢ⟩² > 0 despite zero magnetization.
The Anneal Button
Simulated annealing slowly lowers the temperature from a high value, allowing the system to explore the energy landscape and find lower-energy states than direct cooling would.
This mirrors the physical annealing process used in metallurgy — and gave its name to the optimization algorithm. Try pressing Anneal and watch the energy decrease toward a local minimum of the complex landscape.