An Extended Variational Principle for the SK Spin-Glass Model

TL;DR

This paper introduces a broader variational principle for the SK spin-glass model that extends beyond Parisi's ultrametric ansatz, providing a new framework for evaluating the model's free energy and questioning the ansatz's validity.

Contribution

It develops a generalized variational principle that encompasses Parisi's approach as a special case, allowing for a more comprehensive analysis of the SK model.

Findings

The new principle provides a lower bound and actual free energy through optimization.

Ultrametric structures are shown to be a subset of the broader variational class.

The validity of Parisi's ultrametric ansatz remains uncertain within this framework.

Abstract

The recent proof by F. Guerra that the Parisi ansatz provides a lower bound on the free energy of the SK spin-glass model could have been taken as offering some support to the validity of the purported solution. In this work we present a broader variational principle, in which the lower bound, as well as the actual value, are obtained through an optimization procedure for which ultrametic/hierarchal structures form only a subset of the variational class. The validity of Parisi's ansatz for the SK model is still in question. The new variational principle may be of help in critical review of the issue.