Spherical Spin-Glass–Coulomb-Gas Duality: Solution beyond Mean-Field Theory

Abstract

We present an alternate solution of a Gaussian spin-glass model with infinite ranged interactions and a global spherical constraint at zero magnetic field. The replicated spin-glass Hamiltonian is mapped onto a Coulomb gas of logarithmically interacting particles confined by a logarithmic single particle potential. The precise free energy is obtained by analyzing the Painlevé τ{IV}[n] function in the n→0 limit. The large-N thermodynamics exactly recovers that of Kosterlitz, Thouless, and Jones [Phys. Rev. Lett. 36, 1217 (1976)10.1103/PhysRevLett.36.1217]. It is hoped that the approach here can be extended to apply to systems beyond the spherical model, particularly those in which destabilizing terms lead to replica symmetry breaking.

Simplicity of the spherical spin-glass model

“…In this mapping, there is one charge for each system replica, and correlations between the charges carry information regarding the overlap between the different replicas. Our analysis here differs from that in the previous works [8,9], in that we apply a saddle point approximation that allows us to demonstrate the irrelevance of the interactions in the thermodynamic limit. This leads to a rigorous, ansatz-free evaluation of the replicated partition sum that reveals for the first time the full simplicity of the spherical spin glass thermodynamics.…”

“…We briefly review the charge mapping approach to the replicated spherical spin glass partition function in this section. For background material we refer the reader to [1,8,9] and to the text by De Dominicis and Giardina [10]. Here, we shall take as our starting point the replicated hamiltonian for the mean-spherical spin glass model,…”

“…It is the purpose of this paper to demonstrate this result. We employ an approach, previously discussed in [8] and [9], that takes advantage of a mapping of the problem onto the finite-temperature partition function of a set of logarithmically interacting charges. In this mapping, there is one charge for each system replica, and correlations between the charges carry information regarding the overlap between the different replicas.…”

Edge effects in some perturbations of the Gaussian unitary ensemble

“…For γ (1) p this is to be solved subject to the initial condition γ 0 , γ 1 = 0, while for γ (2) p it is subject to the initial condition γ 0 = 0, γ 1 = 1. Solving the recurrence gives the stated result.…”

“…By orthogonal invariance of the distribution of X the corresponding ensemble of matrices has identical spectral properties to the shifted mean GOE ensemble in which X 0 has all entries equal to ε. This ensemble appeared in an analysis of a spherical spin glass due to Kosterlitz, Thouless and Jones [14] (see also [1]), and it has also been used as a model Hamiltonian in the study of mesoscopic quantum structures [19].…”