A spin glass is a disordered magnetic system with random, competing ferromagnetic and antiferromagnetic interactions. The system gets "frustrated" — no spin configuration satisfies all interactions simultaneously. At low temperatures, it freezes into one of exponentially many metastable states. The replica method computes the free energy by analytically continuing the replica count n → 0.
0.0
Energy E/N
0.0
Replica overlap q
0.0
|Magnetization|
0%
Frustration
Spin configuration (blue=+1, red=−1)
Energy history & replica overlap
Edwards-Anderson order parameter q(T)
1.00
12
1.0
Edwards-Anderson Hamiltonian: H = −Σ_{⟨ij⟩} J_{ij} σ_i σ_j
where J_{ij} ~ N(0,1) (Gaussian random couplings)
Replica overlap:q_{ab} = (1/N) Σ_i ⟨σ_i^a σ_i^b⟩
For T > T_c (paramagnet): q = 0. For T < T_c (glass): q > 0 — replicas remember each other's configuration, signaling broken ergodicity.
Parisi solution (1979–1983): The true solution requires Replica Symmetry Breaking (RSB) with an ultrametric organization of states. The order parameter becomes a function q(x) on [0,1], not just a number — an infinite hierarchy of broken symmetries. This was one of the deepest results in statistical mechanics of the 20th century (Nobel Prize 2021 to Parisi).