Learning topological defects formation with neural networks in a quantum phase transition

TL;DR

This paper uses neural networks to analyze the formation and statistics of topological defects during a quantum phase transition in a one-dimensional quantum Ising model, revealing universal power-law behaviors.

Contribution

It introduces a neural network-based approach to study non-equilibrium dynamics and defect formation in quantum phase transitions, highlighting universal statistical relationships.

Findings

Excitation energies follow a power-law with quench rate.

Kink number cumulants scale universally with quench rate.

Kink-kink correlations match analytical predictions.

Abstract

Neural networks possess formidable representational power, rendering them invaluable in solving complex quantum many-body systems. While they excel at analyzing static solutions, nonequilibrium processes, including critical dynamics during a quantum phase transition, pose a greater challenge for neural networks. To address this, we utilize neural networks and machine learning algorithms to investigate the time evolutions, universal statistics, and correlations of topological defects in a one-dimensional transverse-field quantum Ising model. Specifically, our analysis involves computing the energy of the system during a quantum phase transition following a linear quench of the transverse magnetic field strength. The excitation energies satisfy a power-law relation to the quench rate, indicating a proportional relationship between the excitation energy and the kink numbers. Moreover, we establish a universal power-law relationship between the first three cumulants of the kink numbers and the quench rate, indicating a binomial distribution of the kinks. Finally, the normalized kink-kink correlations are also investigated and it is found that the numerical values are consistent with the analytic formula.