Replica-symmetry breaking: discrete and continuous schemes in the Sherrington–Kirkpatrick model

Abstract

Abstract. We study hierarchies of replica-symmetry-breaking solutions of the Sherrington-Kirkpatrick model. Stationarity equations for order parameters of solutions with an arbitrary number of hierarchies are set and the limit to infinite number of hierarchical levels is discussed. In particular, we demonstrate how the continuous replica-symmetry breaking scheme of Parisi emerges and how the limit to infinite-many hierarchies leads to equations for the order-parameter function of the continuous solution. The general analysis is accompanied by an explicit asymptotic solution near the de Almeida-Thouless instability line in the nonzero magnetic field.

Free-energy functional for the Sherrington-Kirkpatrick model: The Parisi formula completed

“…Such an approach becomes asymptotically exact near the de Almeida-Thouless instability line. 10 We hence can analyze the critical behavior without the necessity to come over to a truncated model. Further on, we can use higher orders of the power expansion of the order-parameter function m͑͒ to systematically improve the asymptotic solution and extend it from the critical region below the instability line to a rather accurate approximation in the entire spin-glass phase.…”

“…We demonstrated it explicitly in the asymptotic region below the critical temperature of the spin-glass phase in zero magnetic field 9 and recently also in the nonzero magnetic field. 10 On the other hand, there are models, such as the Potts spin glass, 11 where a discrete onestep RSB appears to be stable on a finite temperature interval. 12 We demonstrate in this paper that independently of where the absolute maximum of the RSB free energy may lie, we can always construct a solution with a continuous distribution of differences ⌬ l and a single order-parameter function m͑͒ on the defining interval ͓0, 1͔ determined from explicit equations for a local maximum of the free energy.…”

Mean-field solution of the Potts glass near the transition temperature to the ordered phase

“…In the asymptotic solution near the critical temperature of the Ising spin glass the new solution with m K+1 < m K leads to a higher free energy f K+1 > f K . 21,22 There is also another stationary solution for m K+1 > m K that, in the Ising model, lowers the free energy and worsens thermodynamic inhomogeneity. Hence, it is unacceptable.…”

Continuous replica-symmetry breaking in mean-field spin-glass models: Perturbation expansion without the replica trick

“…Alternatively, one can expand the full solution near the critical transition point to the spin-glass phase. [6][7][8] Presently, the most advanced construction of the solution with continuous RSB is a high-order perturbation expansion of a solution of the Parisi nonlinear differential equation resolved numerically by means of a pseudospectral code and Padé approximants. 9 These expansions are applicable only to continuous transitions and to temperatures not too far below the critical point.…”