Sonic horizons and causality in the phase transition dynamics

TL;DR

This paper investigates how the sonic horizon influences causality and correlation growth during phase transitions, especially in systems with different dynamical exponents, extending the understanding beyond the quantum Ising chain.

Contribution

It analyzes the role of the sonic horizon in systems with various dynamical exponents, including cases with diverging and vanishing z, to understand causality in phase transition dynamics.

Findings

In systems with z>1, the sonic horizon speed decreases with transition time.

In the Griffiths region with diverging z, quasiparticle localization freezes correlation growth.

In the long-range Ising chain with z<1, correlations grow adiabatically initially, then lag behind.

Abstract

A system gradually driven through a symmetry-breaking phase transition is subject to the Kibble-Zurek mechanism (KZM). As a consequence of the critical slowing down, its state cannot follow local equilibrium, and its evolution becomes non-adiabatic near the critical point. In the simplest approximation, that stage can be regarded as "impulse" where the state of the system remains unchanged. It leads to the correct KZM scaling laws. However, such "freeze-out" might suggest that the coherence length of the nascent order parameter remains unchanged as the critical region is traversed. By contrast, the original causality-based discussion emphasized the role of the {\it sonic horizon}: domains of the broken symmetry phase can expand with a velocity limited by the speed of the relevant sound. This effect was demonstrated in the quantum Ising chain where the dynamical exponent $z=1$ and quasiparticles excited by the transition have a fixed speed of sound. To elucidate the role of the sonic horizon, in this paper we study two systems with $z>1$ where the speed of sound is no longer fixed, and the fastest excited quasiparticles set the size of the sonic horizon. Their effective speed decays with the increasing transition time. In the extreme case, the dynamical exponent $z$ can diverge such as in the Griffiths region of the random Ising chain where localization of excited quasiparticles freezes the growth of the correlation range when the critical region is traversed. Of particular interest is an example with $z<1$ -- the long-range extended Ising chain, where there is no upper limit to the velocity of excited quasiparticles with small momenta. Initially, the power-law tail of the correlation function grows adiabatically, but in the non-adiabatic stage it lags behind the adiabatic evolution -- in accord with a generalized Lieb-Robinson bound.