Machine Learning Topological Defect Formation: When Are the Defects Made?

Topological defects that form in a nonequilibrium second-order phase transition are presumably seeded by fluctuations of the order parameter in the vicinity of the critical point. Motivated by this conjecture that underlies the Kibble-Zurek mechanism (KZM), we investigate whether machine learning can anticipate their locations from the fluctuations in the "impulse regime," the time interval when the evolution of the order parameter cannot keep up with the conditions imposed by the quench. Going beyond the conventional KZM focus on defect density, we show that a recurrent neural network can predict locations of topological defects from short-time dynamical data deep within the impulse regime. We thus demonstrate that defects are sown in the immediate vicinity of the critical point. The seeds of defects-fluctuations imprinted on the evolving order parameter-are exponentially small near the critical point, but become amplified by the evolution and play a dominant role in breaking symmetry. This occurs before the freeze-out time t[over ^] that concludes the impulse regime, and well before the order parameter assumes its final symmetry-broken configuration. Furthermore, we find that the predictive power of the machine learning also exhibits power-law scaling consistent with KZM.