Sonic horizons and causality in phase transition dynamics

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 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.I.

Topological defects as relics of spontaneous symmetry breaking from black hole physics

“…The same scaling laws (although, again, with somewhat different prefactors) follow from the arguments based on the "sonic horizon" paradigm [5,28,39,40]. Equations (2.3), (2.4) and (2.5) can be used to test the validity of KZM in laboratory experiments and in numerical simulations.…”

“…The new, post-transition order parameter will randomly select how to break symmetry in domains that are this far apart, as they have no time to communicate with one another. This sonic horizon argument [5] leads to domains having the size ∼ξ -same scaling (although different pre-factors [39,40]) as these given by the "freeze-out".…”

Report on scipost_202007_00061v1

“…Still, nontrivial predictions can be made concerning the non-adiabatic or impulse region −τ KZ < t < τ KZ [29,31,32] due to the fact that the KZ time and correlation length, τ KZ and ξ KZ , are the only relevant scales for a slow enough ramp protocol. Consequently, timedependent correlation functions are described in terms of scaling functions of the rescaled variables t/τ KZ and x/ξ KZ in the KZ scaling limit τ KZ → ∞.…”

“…The simplest approximation which leads to the right scaling exponents assumes that when adiabaticity is lost, the system becomes completely frozen and reenters the dynamics only some time after crossing the critical point. This freeze-out scenario or impulse approximation has been refined recently by taking into account the actual evolution of the system in the non-adiabatic time window [15,[27][28][29][30][31][32][33]. Since the Kibble-Zurek length and time scales are the only relevant scales, the non-adiabatic evolution features dynamical scaling, i.e.…”

“…The various facets of the quantum KZM was demonstrated and analysed on the quantum Ising chain [6-8, 10, 13, 28, 31, 33, 34, 37, 38, 47-50], the XY spin chain [11,12,51] or other exactly solvable systems [15,29,48,52,[52][53][54] (see however e.g. [9,18,32,55,56]).…”