Overdamped Landau-Lifshitz-Gilbert dynamics of the Sherrington-Kirkpatrick model

We study the dynamics of a continuous variable version of the Sherrington-Kirkpatrick (SK) model, where the system is composed of all-to-all randomly coupled 3-component spins. The spins interact with Heisenberg couplings and have an on-site uniaxial magnetic anisotropy to bias the system to end up with spins up or down along the anisotropy axis. When subject to stochastic Landau-Lifshitz-Gilbert dynamics (LLG), a continuous time Markov process, the system can be shown to equilibrate more efficiently than the discrete-time Glauber dynamics of the associated SK model. Here, we study the overdamped limit of the LLG dynamics and find that it does not significantly affect the quality of the ground-state solutions. We also look at how the components of the connected correlation matrix decay in time and find a phase transition as we vary the temperature.