Quenches in the Sherrington-Kirkpatrick model

TL;DR

This paper investigates the energy convergence of different quench dynamics in the SK model, revealing that reluctant algorithms can reach ground state energies and introducing a novel backtracking DMFT approach.

Contribution

It provides the first application of backtracking DMFT to fully connected disordered models and compares the effectiveness of greedy and reluctant algorithms in reaching ground states.

Findings

Reluctant algorithms can converge to ground state energy density.

Greedy algorithms generally do not reach ground state energies.

Synchronous greedy algorithms may also achieve ground state energies and exhibit a dynamical phase transition.

Abstract

The Sherrington-Kirkpatrick (SK) model is a prototype of a complex non-convex energy landscape. Dynamical processes evolving on such landscapes and locally aiming to reach minima are generally poorly understood. Here, we study quenches, i.e. dynamics that locally aim to decrease energy. We analyse the energy at convergence for two distinct algorithmic classes, single-spin flip and synchronous dynamics, focusing on greedy and reluctant strategies. We provide precise numerical analysis of the finite size effects and conclude that, perhaps counter-intuitively, the reluctant algorithm is compatible with converging to the ground state energy density, while the greedy strategy is not. Inspired by the single-spin reluctant and greedy algorithms, we investigate two synchronous time algorithms, the sync-greedy and sync-reluctant algorithms. These synchronous processes can be analysed using dynamical mean field theory (DMFT), and a new backtracking version of DMFT. Notably, this is the first time the backtracking DMFT is applied to study dynamical convergence properties in fully connected disordered models. The analysis suggests that the sync-greedy algorithm can also achieve energies compatible with the ground state, and that it undergoes a dynamical phase transition.