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.

The Aubry-André 1D lattice model describes a particle hopping in a pseudo-random potential. Depending on its strength λ, all eigenstates are either localized (λ > 1) or delocalized (λ < 1). Near the transition, the localization length diverges like ξ ∼ (λ−1) −ν with ν = 1. We show that when the particle is initially prepared in a localized ground state and the potential strength is slowly ramped down across the transition, then -in analogy with the Kibble-Zurek mechanism -it enters the delocalized phase having finite localization lengthξ ∼ τ ν/(1+zν) Q . Here τQ is ramp/quench time and z is a dynamical exponent. At λ = 1 we determine z 2.37 from the power law scaling of energy gap with lattice size L. Even though for infinite L the model is gapless, we show that the gap relevant for excitation during the ramp remains finite. Close to the critical point it scales like ξ −z with the value of z determined by the finite size scaling. It is the gap between the ground state and the lowest of those excited states that overlap with the ground state enough to be accessible for excitation. We propose an experiment with a non-interacting BEC to test our prediction. Our hypothesis is further supported by considering a generalized version of Aubry-André model possessing an energydependent mobility edge. arXiv:1811.05496v2 [cond-mat.stat-mech]

A short quasi-monochromatic wave packet incident on a semi-infinite disordered medium gives rise to a reflected wave. The intensity of the latter decays as a power law 1/t α in the long-time limit. Using the one-dimensional Aubry-André model, we show that in the vicinity of the critical point of Anderson localization transition, the decay slows down and the power-law exponent α becomes smaller than both α = 2 found in the Anderson localization regime and α = 3/2 expected for a one-dimensional random walk of classical particles.