Replica Trick & Spin Glass

Replica trick: Computing log Z (quenched average) is hard, so use log Z = lim_n→0 (Zⁿ−1)/n. We replicate the system n times, average over disorder (coupling constants J_ij drawn Gaussian), then analytically continue n→0. The saddle-point is characterized by the overlap matrix q_ab = (1/N)Σ⟨σ^a_iσ^b_i⟩. Replica symmetry (RS): all q_ab=q (Sherrington-Kirkpatrick ansatz). Parisi RSB: hierarchical block structure — physical below T_c=J. The canvas shows sampled overlap histograms and the Parisi matrix pattern.