Kibble-Zurek Dynamics in Two-dimensional Frustrated Systems with a Neural Foundation-state Subspace Method

Abstract

Universal scaling generated when a strongly interacting quantum many-body system is driven across a continuous phase transition provides a dynamical probe of equilibrium criticality. Accessing this regime numerically in two dimensions is challenging because it requires accurate real-time evolution of correlated many-body states over many system sizes and driving rates. We introduce a Neural Foundation-state Subspace (NFS) method for near-adiabatic dynamics. A foundation neural-network quantum state represents the ground-state manifold along the driving path, and a small fidelity-selected subset defines a fixed variational subspace. The many-body Schr\"odinger equation then reduces to the evolution of a few linear coefficients, with projected operators reusable across ramp times. We validate the method on the two-dimensional transverse-field Ising model, recovering the expected Kibble-Zurek scaling and critical exponents in quantitative agreement with ground-state quantum Monte Carlo estimates. Applied to the frustrated square-lattice $J_1$-$J_2$ Heisenberg model up to $16 \times 16$ clusters, our approach provides strong numerical evidence of Kibble-Zurek mechanism across the N\'eel-to-spin-liquid transition at $J_2/J_1=0.49$, yielding ${\nu=1.23(15)}$ and ${\eta=0.409(19)}$ at fixed $z=1$, consistent with static estimates and supporting the proposed continuous critical behavior.