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Introduction to Mean-Field Theory of Spin Glass Models

We discuss the mean-field theory of spin-glass models with frustrated long-range random spin exchange. We analyze the reasons for breakdown of the simple mean-field theory of Sherrington and Kirkpatrick. We relate the replica-symmetry breaking to ergodicity breaking and use the concept of real replicas to restore thermodynamic homogeneity of the equilibrium free energy in a replicated phase space. Embedded replications of the spin variables result in a set of hierarchical free energies and overlap susceptibilities between replica hierarchies as order parameters. The limit to infinite number of replica hierarchies leads to the Parisi solution with a continuous replica-symmetry breaking. We present a closed-form representation of the Parisi mean-field theory that is independent from stability of solutions with finite-many replica hierarchies. Hence, solutions with continuous and discrete replica-symmetry breaking can coexist. We demonstrate the construction of the spin-glass mean-field solutions via real replicas on three spin-glass models: Ising, Potts and p-spin. An asymptotic expansion in each model is used to demonstrate various types of the transition from the paramagnetic to the glassy phase.