Dynamics and Correlations among Soft Excitations in Marginally Stable Glasses

TL;DR

This paper investigates how marginal stability in glasses leads to a pseudo-gap in soft excitations, revealing diverging correlations among soft spins in the SK model and drawing parallels with random walks, with implications for sphere packings.

Contribution

It corrects the multi-spin-flip stability criterion and uncovers the diverging correlations among soft spins in the SK model, linking spin dynamics to random walk behavior.

Findings

Soft spins are frustrated among each other with diverging correlations.

The correlation function scales as C(λ) ~ 1/λ.

Analogy established between spin dynamics and 2D random walks.

Abstract

Marginal stability is the notion that stability is achieved, but only barely so. This property constrains the ensemble of configurations explored at low temperature in a variety of systems, including spin, electron and structural glasses. A key feature of marginal states is a (saturated) pseudo-gap in the distribution of soft excitations. We study how such a pseudo-gap appears dynamically in the case of the Sherrington-Kirkpatrick (SK) spin glass. After revisiting and correcting the multi-spin-flip criterion for local stability, we show that stationarity along the hysteresis loop requires that soft spins are frustrated among each other, with a correlation that diverges as $C(\lambda)\sim 1/\lambda$, where $\lambda$ is the larger of two considered local fields. We explain how this arises spontaneously in a marginal system and develop an analogy between the spin dynamics in the SK model and random walks in two dimensions. We discuss the applicability of these findings to hard sphere packings.