Dynamics of the order parameter in symmetry breaking phase transitions

TL;DR

This paper shows that simplified ordinary differential equations can effectively describe the dynamics of symmetry breaking phase transitions and the formation of topological defects, providing new analytical and computational insights into the Kibble-Zurek mechanism.

Contribution

It demonstrates that ODEs derived from the Langevin PDEs can capture key aspects of phase transition dynamics and defect formation, simplifying analysis and expanding the range of quench timescales studied.

Findings

ODEs predict the adiabatic-impulse scenario and freeze-out scaling.

Analytical solutions for overdamped cases are obtained.

Exploration of Kibble-Zurek scaling over broad quench timescales.

Abstract

The formation of topological defects in second-order phase transitions can be investigated by solving partial differential equations for the evolution of the order parameter in space and time, such as the Langevin equation. We demonstrate that the ordinary differential equations governing either the temporal or spatial dependence in the Langevin equation provide surprisingly substantial insights into the dynamics of the phase transition. The temporal evolution of the order parameter predicts the essence of the adiabatic-impulse scenario, including the scaling of the freeze-out time, which is crucial to the Kibble-Zurek mechanism (KZM). In particular, Bernoulli differential equations that arise in the overdamped case can be solved analytically. The spatial part of the evolution, in turn, leads to the characteristic size of domains that choose the same broken symmetry. Apart from the fundamental insights into the KZM, this finding enables the exploration of Kibble-Zurek scaling using ordinary differential equations over a large range of quench timescales, which would otherwise be difficult to achieve with numerical simulations of the full partial differential equations.