On the Discontinuous Breaking of Replica Symmetry and Shattering in Mean-Field Spin Glasses

TL;DR

This paper investigates the nature of phase transitions in mean-field spin glasses, revealing that a discontinuous replica symmetry breaking leads to a shattered phase, while providing examples of continuous transitions with shattering.

Contribution

It demonstrates the link between discontinuous replica symmetry breaking and shattering, and provides an example of shattering with a continuous transition in spherical spin glasses.

Findings

Discontinuous RSB implies an intermediate shattered phase.

Shattering can occur with a continuous phase transition.

Examples of spherical spin glasses with shattering and continuous transitions.

Abstract

We show that in mean-field spin glasses, a discontinuous breaking of replica symmetry at the critical inverse temperature $\beta_c$ implies the existence of an intermediate shattered phase. This confirms a prediction from physics regarding the nature of random first order phase transitions. On the other hand, we give an example of a spherical spin glass which exhibits shattering, yet the transition is continuous at $\beta_c$.