Revisiting the replica trick: Competition between spin glass and conventional order

TL;DR

This paper clarifies the correct application of the replica trick in spin glass models with multiple order parameters, resolving ambiguities and elucidating the relationship between quenched and annealed free energies.

Contribution

It establishes that the proper procedure is to maximize over replica order parameters first, then minimize over conventional order parameters, clarifying the replica trick's application.

Findings

Correct procedure for replica trick application identified

Relationship between quenched and annealed free energies clarified

Quenched and annealed free energies can differ with zero replica order parameters

Abstract

There is an ambiguity in how to apply the replica trick to spin glass models which have additional order parameters unrelated to spin glass order -- with respect to which quantities does one minimize vs maximize the action, and in what sequence? Here we show that the correct procedure is to first maximize with respect to "replica" order parameters, and then minimize with respect to "conventional" order parameters. With this result, we further elucidate the relationship between quenched free energies, annealed free energies, and replica order -- it is possible for the quenched and annealed free energies to differ even while all replica order parameters remain zero.