Universality class of replica symmetry breaking, scaling behavior, and the low-temperature fixed-point order function of the Sherrington-Kirkpatrick model

TL;DR

This paper develops a scaling theory for replica symmetry breaking in the SK model, revealing critical behavior, divergent correlation lengths, and rational exponents through high-order RSB analysis and a new fixed point extrapolation method.

Contribution

It introduces a novel scaling hypothesis for the free energy in the SK model and provides detailed numerical analysis of RSB criticality and exponents.

Findings

Identification of separated temperature and magnetic field scaling regimes

High-precision rational-valued critical exponents

Divergent correlation length indicating long-range RSB criticality

Abstract

A scaling theory of replica symmetry breaking (RSB) in the SK-model is presented in the framework of critical phenomena for the scaling regime of small inverse RSB-orders, small temperatures, small magnetic fields, and near opposite pseudo-dynamic limits. We employ the pseudo-dynamical picture (PRL98, 127201 (2007)) with two critical points, where separated temperature- and magnetic field-scaling is obtained. An unconventional scaling hypothesis for the free energy is given, modeling this separated scaling in accordance with detailed numerical self-consistent solutions for up to 200 orders of RSB. A divergent correlation-length describes the RSB-criticality as a long-range correlation effect occurring on the pseudo-lattice of RSB-orders. Rational-valued exponents are concluded with high precision from high-order RSB scaling and by using a new fixed point extrapolation method. Power laws, scaling relations, and scaling functions are analyzed.