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arXiv:1912.02815 · Physical review. B./Physical review. B

Sonic horizons and causality in phase transition dynamics

Debasis Sadhukhan, Aritra Sinha, Anna Francuz, Justyna Stefaniak, Marek M. Rams, Jacek Dziarmaga, Wojciech Hubert Zurek

Published Apr 21, 2020• 44 citations
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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 nonadiabatic near the critical point. In the simplest approximation, that stage can be regarded as an ``impulse'' where the state of the system remains unchanged. It leads to the correct KZM scaling laws. However, such a ``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 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 the 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 nonadiabatic stage it lags behind the adiabatic evolution.

DOI
10.1103/physrevb.101.144429

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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 nonadiabatic near the critical point. In the simplest approximation, that stage can be regarded as an impulse'' where the state of the system remains unchanged. It leads to the correct KZM scaling laws. However, such a 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=1z=1z=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>1z>1z>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 increasing transition time. In the extreme case, the dynamical exponent zzz can diverge such as in the Griffiths region of the random Ising chain where the 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<1z<1z<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 nonadiabatic stage it lags behind the adiabatic evolution.
When a system is driven across a quantum critical point at a constant rate, its evolution must become nonadiabatic as the relaxation time $\ensuremath{\tau}$ diverges at the critical point. According to the Kibble-Zurek mechanism (KZM), the emerging post-transition excited state is characterized by a finite correlation length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$ set at the time $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{t}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\tau}}$ when the critical slowing down makes it impossible for the system to relax to the equilibrium defined by changing parameters. This observation naturally suggests a dynamical scaling similar to renormalization familiar from the equilibrium critical phenomena. We provide evidence for such KZM-inspired spatiotemporal scaling by investigating an exact solution of the transverse field quantum Ising chain in the thermodynamic limit.
In the nonadiabatic dynamics across a quantum phase transition, the Kibble-Zurek mechanism predicts that the formation of topological defects is suppressed as a universal power law with the quench time. In inhomogeneous systems, the critical point is reached locally and causality reduces the effective system size for defect formation to regions where the velocity of the critical front is slower than the sound velocity, favoring adiabatic dynamics. The reduced density of excitations exhibits a much steeper dependence on the quench rate and is also described by a universal power law that we demonstrated in a quantum Ising chain.
The Kibble-Zurek mechanism (KZM) successfully predicts the density of topological defects deposited by the phase transitions, but it is not clear why. Its key conjecture is that, near the critical point of the second-order phase transition, critical slowing down will result in a period when the system is too sluggish to follow the potential that is changing faster than its reaction time. The correlation length at the freeze-out instant t^\hat tt^ when the order parameter catches up with the post-transition broken symmetry configuration is then decisive, determining when the mosaic of broken symmetry domains locks in topological defects. To understand why the KZM works so well we analyze Landau-Ginzburg model and show why temporal evolution of the order parameter plays such a key role. The analytical solutions we obtain suggest novel, hitherto unexplored, experimentally accessible observables that can shed light on symmetry breaking dynamics while testing the conjecture on which the KZM is based.
We study the effects of symmetry-breaking perturbations in the out-of-equilibrium quantum dynamics of many-body systems slowly driven by a time-dependent symmetry-preserving parameter, across the quantum critical regime associated with a continuous quantum transition (CQT). For this purpose, we analyze the out-of-equilibrium dynamics arising from generalized Kibble-Zurek (KZ) protocols, within a dynamic renormalization-group framework allowing for finite-size systems. We show that the time dependence of generic observables develops an out-of-equilibrium finite-size scaling (FSS) behavior, arising from the interplay between the timescale ts{t}_{s}ts​ of the parameter variations in the KZ protocol, the size LLL of the system, and the strength hhh of the symmetry-breaking perturbation, in the limit of large ts{t}_{s}ts​ and LLL. Moreover, scaling arguments based on the first-order adiabatic approximation of slow variations in quantum systems allow us to characterize the approach to the adiabatic regimes for some limits of the model parameters (for example, when we take ${t}_{s}\ensuremath{\rightarrow}\ensuremath{\infty}$ before $L\ensuremath{\rightarrow}\ensuremath{\infty}$), predicting asymptotic power-law suppressions of the nonadiabatic behaviors in the adiabatic limits. This out-of-equilibrium FSS is supported by numerical analyses for the paradigmatic quantum Ising chain along generalized KZ protocols, with a time-dependent transverse field crossing its CQT, in the presence of a static longitudinal field breaking the Z2{\mathbb{Z}}_{2}Z2​ symmetry.
The Kibble-Zurek mechanism demands an initial adiabatic stage before an impulse stage to have a frozen correlation length that generates topological defects in a cooling phase transition. Here we study such a driven critical dynamics but with an initial condition that is near the critical point and that is far away from equilibrium. In this case, there is no initial adiabatic stage at all and thus adiabaticity is broken. However, we show that there again exists a finite length scale arising from the driving that divides the evolution into three stages. A relaxation--finite-time-scaling--adiabatic scenario is then proposed in place of the adiabatic-impulse-adiabatic scenario of the original Kibble-Zurek mechanism. A unified scaling theory, which combines finite-time scaling with critical initial slip, is developed to describe the universal behavior and is confirmed with numerical simulations of a two-dimensional classical Ising model.
Driving a homogeneous system across a quantum phase transition in a quench time ${\ensuremath{\tau}}_{Q}$ generates excitations on wavelengths longer than the Kibble-Zurek (KZ) length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}\ensuremath{\propto}{\ensuremath{\tau}}_{Q}^{\ensuremath{\nu}/(1+z\ensuremath{\nu})}$ within the KZ time window $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{t}\ensuremath{\propto}{\ensuremath{\tau}}_{Q}^{z\ensuremath{\nu}/(1+z\ensuremath{\nu})}$, where zzz and $\ensuremath{\nu}$ are the critical exponents. Quenches designed with local time-dependent inhomogeneity can introduce a gap in the spectrum. They can be parametrized by a time- and space-dependent distance $\ensuremath{\epsilon}$ from the critical point in the parameter space of the Hamiltonian: $\ensuremath{\epsilon}(t,x)=\frac{t\ensuremath{-}x/v}{{\ensuremath{\tau}}_{Q}}\ensuremath{\equiv}\ensuremath{\theta}(vt\ensuremath{-}x).$ For a variety of setups with short-range interactions, they have been shown to suppress excitations if the spatial velocity vvv of the inhomogeneous front is below the characteristic KZ velocity $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{v}\ensuremath{\propto}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}/\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{t}$. Ising-like models with long-range interactions can have no sonic horizon, spreading information instantaneously across the system. Usually, this should imply that inhomogeneous transitions will render the dynamics adiabatic regardless of the velocity of the front. However, we show that we get an adiabatic transition with no defects only when the inhomogeneous front moves slower than the characteristic crossover velocity $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}\ensuremath{\propto}{\ensuremath{\theta}}^{(z\ensuremath{-}1)\ensuremath{\nu}/(1+\ensuremath{\nu})}$, where $\ensuremath{\theta}$ is the slope of the inhomogeneous front at the critical point. The existence of this crossover velocity and adiabaticity of the model results from the energy gap in the quasiparticle spectrum that is opened by the inhomogeneity. This effect can be employed for efficient adiabatic quantum state preparation in systems with long-range interactions.