Dynamics of Spin Glasses

This project contributes new equations for understanding the dynamics of mean-field spin glass systems. It focusses on the Sherrington-Kirkpatrick spin glass system: with both discrete `hard' spins, and a soft-spin spherical model. This system involves pairwise all-to-all interaction, where the strength of the interaction scales as N^{-1/2} multiplied by a centered Gaussian variable of unit variance. It is widely considered a paradigm for glassy dynamics, in both condensed matter physics and data science. The dynamics is initiated from a deep quench (random initial conditions).

To date, the established means of studying spin glass dynamics is through correlation / response functions (Crisanti-Horner-Sommers 1993 and Cugliandolo-Kurchan 1994), which constitute a set of coupled delay differential equations. Recent numerical results (Bernaschi et al 2020) have cast doubt on some of the assumptions (i.e. `weak ergodicity breaking') used to establish these equations. New autonomous (i.e. non-delayed) equations are established for the emergent dynamics of spin glass systems in the high temperature regime (MacLaurin 2020). They are shown to be highly accurate through comparison with numerical simulations. They are shown to accurately characterize the exponential convergence to equilibrium in the high temperature regime.

At the point of bifurcation (from high temperature fast dynamics to glassy slow dynamics) the dynamics is studied by performing a perturbation expansion about the autonomous solution. Preliminary numerical results suggest that the convergence to equilibrium becomes algebraic. Various new equation ansatzes are suggested to describe the dynamics.