On the Pure States of the Replica Symmetry Breaking ansatz

TL;DR

This paper redefines pure states in spin systems without disorder averaging or replicas, offering a new perspective on the RSB scheme and deriving the Parisi formula for the SK model.

Contribution

It introduces a novel interpretation of pure states in spin systems and redefines the RSB scheme without using replicas or disorder averaging.

Findings

Reinterpreted pure states as disjoint subsets with product-like measures

Defined an approximate probability measure for RSB scheme

Derived the Parisi formula for the SK model

Abstract

We discuss the concept of Pure State of the Replica Symmetry Breaking ansatz in finite and infinite spin systems without averaging on the disorder, nor using replicas. Consider a system of n spins $\sigma\in\Omega^{n}$ with the usual set $\Omega=\left\{ -1,1\right\}$ of inner states and let $G:\,\Omega^{n}\rightarrow\left[0,1\right]$ a Gibbs measure on it of Hamiltonian $\mathcal{H}$ (also non random). We interpret the pure states of a model $\left(\Omega^{n},\mu\right)$ as disjoint subsets $\Omega^{n}$ such that the conditional measures behaves like product measures as in usual mean field approximations. Starting from such definition we try to reinterpret the RSB scheme and define an approximated probability measure. We then apply our results to the Sherrington-Kirkpatrick model to obtain the Parisi formula.