Dynamics and Correlations among Soft Excitations in Marginally Stable Glasses

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) pseudogap in the distribution of soft excitations. We examine how such pseudogaps appear dynamically by studying 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 soft spins to be frustrated among each other, with a correlation diverging as $C(\lambda )\sim 1/\lambda$, where $\lambda$ is the stability of the more stable spin. 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 analogous frustrations among soft excitations in short range glasses and how to detect them experimentally. We also show how these findings apply to hard sphere packings.

Figure 1

(a) Hysteresis loop: Magnetization $M$ under a periodic quasistatic driving of the external field $h$. Inset: magnified segment of the hysteresis loop of a finite size system. (b) Distribution of local stabilities, $\rho (\lambda )$, in locally stable states along the hysteresis loops for different system sizes $N$. (c) Finite size scaling of the avalanche size distribution $D(n)$ confirms $\tau =\sigma =1$ up to logarithmic corrections. (d) Correlation $C(\lambda )$ between the least stable spin and spins of stability $\lambda$ in locally stable states along the hysteresis loop. The data for different system sizes collapse, implying $C(\lambda \ll 1)\sim 1/\lambda$ in the thermodynamic limit.

Figure 2

(a) The average dissipated energy $\mathrm{\Delta }\mathcal{H}$ in avalanches of size $n$ scales as $\mathrm{\Delta }\mathcal{H}\sim n\ln n/\sqrt{N}$. $-\mathrm{\Delta }\mathcal{H}/n$ is a measure of the typical value of the stability of most unstable spins, ${\lambda }_0(n)$. Thus, in the thermodynamic limit, ${\lambda }_0\sim \ln n/\sqrt{N}\ll 1$ even for very large avalanches. (b) The average number of times, $F(n)$, spins active in avalanches of size $n$ reflip later on in the avalanche.

Figure 3

Illustration of a step in the dynamics, in the SK model and the random walker model. Circles on the $\lambda$ axis represent the spins or walkers. At each step, the most unstable spin (in red) is reflected to the stable side, while all others (in green or blue) receive a kick and move. The dashed and solid line outlines the density profile $\rho (\lambda )\sim \lambda$ for $\lambda >1/\sqrt{N}$. The blue spins were initially frustrated with the flipping spin 0. They are stabilized and are now unfrustrated with 0. In contrast, green spins become frustrated with spin 0 and are softer now. Because the motion of spins depends on their frustration with spin 0, a correlation builds up at small $\lambda$, leading to an overall frustration of “soft” spins among each other.