Simplicity of State and Overlap Structure in Finite-Volume Realistic Spin Glasses

TL;DR

The paper argues that realistic spin glasses have a simple pure state and overlap structure, with at most a pair of states, challenging more complex mean field pictures and supporting simpler two-state or chaotic pairs models.

Contribution

It provides heuristic and rigorous evidence that realistic spin glasses exhibit a simple state and overlap structure, ruling out complex mean field scenarios.

Findings

Pure state structure is simple with at most a pair of states.

Overlap distribution shows at most a pair of delta-functions.

Supports the two-state and chaotic pairs models over mean field pictures.

Abstract

We present a combination of heuristic and rigorous arguments indicating that both the pure state structure and the overlap structure of realistic spin glasses should be relatively simple: in a large finite volume with coupling-independent boundary conditions, such as periodic, at most a pair of flip-related (or the appropriate number of symmetry-related in the non-Ising case) states appear, and the Parisi overlap distribution correspondingly exhibits at most a pair of delta-functions at plus/minus the self-overlap. This rules out the nonstandard SK picture introduced by us earlier, and when combined with our previous elimination of more standard versions of the mean field picture, argues against the possibility of even limited versions of mean field ordering in realistic spin glasses. If broken spin flip symmetry should occur, this leaves open two main possibilities for ordering in the spin glass phase: the droplet/scaling two-state picture, and the chaotic pairs many-state picture introduced by us earlier. We present scaling arguments which provide a possible physical basis for the latter picture, and discuss possible reasons behind numerical observations of more complicated overlap structures in finite volumes.