Replica Symmetry Breaking — Parisi Order Function

SK spin glass: q(x) encodes the ultrametric hierarchy of metastable states
PARISI ORDER PARAMETER q(x)
PARISI MATRIX (replica overlap structure)
ULTRAMETRIC TREE OF STATES
Parisi ansatz: the replica overlap matrix Q_{ab} has a hierarchical block structure parameterized by q(x), x ∈ [0,1]. At T_c: q(x)≡0 (replica symmetric, paramagnetic). Below T_c: q(x) is a monotone non-decreasing step function. Full RSB (T→0): q(x) → continuous function from 0 to 1. Parisi 1979 (Fields Medal 2021 to Parisi).
Replica symmetry breaking (Parisi 1979): In the SK spin glass, the free energy calculation requires the "replica trick" — computing [log Z]_J = lim_{n→0} (Z^n - 1)/n. The replica overlap matrix Q_{ab} = (1/N)Σ_i ⟨s_i⟩_a⟨s_i⟩_b must be determined self-consistently. Replica symmetry (Q_{ab} = q for all a≠b) gives a solution unstable below T_c — the AT line (de Almeida-Thouless 1978). Parisi's hierarchical ansatz replaces q with a function q(x) — k-step RSB approximates it as a step function with k steps. The full RSB (continuous q(x)) gives the exact free energy, confirmed rigorously by Guerra (2003) and Talagrand (2006). The ultrametric tree visualizes state organization: all pairs of states satisfy the ultrametric inequality d(A,C) ≤ max(d(A,B), d(B,C)).