The Ising spin glass on random graphs at zero temperature: not all spins are glassy in the glassy phase

TL;DR

This paper analyzes the zero-temperature RSB phase of spin glasses on Bethe lattices, revealing that not all spins are glassy and that RSB manifests only on a subset of spins, with implications for critical behavior and avalanches.

Contribution

It introduces a closed equation for cavity field extremes, showing RSB is not homogeneous and identifying fixed versus fluctuating spins in the glassy phase.

Findings

Spontaneous RSB affects only a vanishing fraction of spins at criticality.

The local field statistics relate to correlation functions in the paramagnetic phase.

Estimated upper critical dimension for spin glasses is at least 8.

Abstract

We investigate the replica symmetry broken (RSB) phase of spin glass (SG) models in a random field defined on Bethe lattices at zero temperature. From the properties of the RSB solution we deduce a closed equation for the extreme values of the cavity fields. This equation turns out not to depend on the parameters defining the RSB, and it predicts that the spontaneous RSB does not take place homogeneously on the whole system. Indeed, there exist spins having the same effective local field in all local ground states, exactly as in the replica symmetric (RS) phase, while the spontaneous RSB manifests only on the remaining spins, whose fraction vanishes at criticality. The characterization in terms of spins having fixed or fluctuating local fields can be extended also to the random field Ising model (RFIM), in which case the fluctuating spins are the only responsible for the spontaneous magnetization in the ferromagnetic phase. Close to criticality we are able to connect the statistics of the local fields acting on the spins in the RSB phase with the correlation functions measured in the paramagnetic phase. Identifying the two types of spins on given instances of SG and RFIM, we show that they participate very differently to avalanches produced by flipping a single spin. From the scaling of the number of spins inducing RSB effects close to the critical point and using the $M$-layer expansion we estimate the upper critical dimension $D_U \geq 8$ for SG.