Kibble-Zurek mechanism is widely known to appear in the transverse-field quantum Ising chain in the thermodynamic limit at zero temperature, having notorious characteristics, like the divergence of its relaxation time. In this work, I present the same effect in a simple system, the driven underdamped Brownian motion. Using linear response theory, I show the appropriate limits where the Kibble-Zurek mechanism happens. The divergence of the relaxation time, the high-temperature condition in the initial thermal equilibrium state, a new Kibble-Zurek scaling at sudden processes, and the pausing effect in the optimal protocol are presented as consequences.
Abstract For systems performing a weakly isothermal process, the decorrelation time dictates how fast the relaxation function decorrelates. However, like many other thermally isolated systems, the transverse-field quantum Ising chain presents an ill-defined decorrelation time. On the other hand, the Kibble–Zurek mechanism uses a heuristic relaxation time to achieve its famous scaling. The problem however of having a well-defined decorrelation time, derived from first principles, agreeing with the Kibble–Zurek mechanism is still open. Such a solution is proposed here by measuring the work using the time-averaged relaxation function of the system, which offers a new and well-defined decorrelation time for thermally isolated systems. I recover with this the Kibble–Zurek mechanism in the finite-time and weak driving regime, and new features in the slowly-varying one. The gain in control over the system in such distinction is desirable for potential applications.
While quantum phase transitions share many characteristics with thermodynamic phase transitions, they are also markedly different as they occur at zero temperature. Hence, it is not immediately clear whether tools and frameworks that capture the properties of thermodynamic phase transitions also apply in the quantum case. Concerning the crossing of thermodynamic critical points and describing its non-equilibrium dynamics, the Kibble-Zurek mechanism and linear response theory have been demonstrated to be among the very successful approaches. In the present work, we show that these two approaches are also consistent in the description of quantum phase transitions, and that linear response theory can even inform arguments of the Kibble-Zurek mechanism. In particular, we show that the relaxation time provided by linear response theory gives a rigorous argument for why to identify the "gap" as a relaxation rate, and we verify that the excess work computed from linear response theory exhibits Kibble-Zurek scaling.
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.
We study the out-of-equilibrium Kibble-Zurek (KZ) dynamics in quantum Ising chains in a transverse field, driven by a time-dependent longitudinal field
h(t)=t/ts (
ts is the time scale of the protocol), across their first-order quantum transitions (FOQTs) at
h=0. The KZ protocol starts at time
ti<0 from the negatively magnetized ground state for
hi=ti/ts<0. Then, the system evolves unitarily up to a time
tf>0, such that the magnetization of the state at time
tf is positive. In finite-size systems, the KZ dynamics develops out-of-equilibrium finite-size scaling (OFSS) behaviors. Their scaling variables depend either exponentially or with a power law on the size, depending on the boundary conditions (BC). The OFSS functions can be computed in effective models restricted to appropriate low-energy (magnetized and/or kink) states. The KZ scaling behavior drastically changes in the thermodynamic limit (TL), defined as the infinite-size limit keeping
t and
ts fixed, which appears substantially unrelated with the OFSS regime, because it involves higher-energy multi-kink states, which are irrelevant in the OFSS limit. The numerical analyses of the KZ dynamics in the TL show the emergence of a quantum spinodal-like scaling behavior at the FOQTs for all considered BC, which is independent of the BC. The longitudinal magnetization changes sign at
h(t)=h∗>0, where
h∗ decreases with increasing
ts, as
h∗∼1/lnts. Moreover, in the large-
ts limit, the time-dependence of the magnetization is described by a universal function of
Ω=t/τs, with
τs=ts/lnts.
The quantum Ising chain has shortcuts to adiabaticity when operated with weak processes. However, when exactly do the non-equilibrium effects of the Kibble-Zurek mechanism, inherent to the system, appear in the optimal protocols in such a context? I propose here that such contrasting difference occurs due to the manner by which one measures the excitation spent energy of the system. Therefore, in this work, I made a qualitative analysis of a quantum annealing procedure of the time-averaged excess work, where the system acquires as a diverging decorrelation time the heuristic Kibble-Zurek mechanism relaxation time. Four important effects are then observed: the absence of shortcuts to adiabaticity, the pausing effect around the critical point in the optimal protocol when the Kibble-Zurek mechanism holds, the persistence of the time-averaged work to avoid slowly-varying regime even for large switching times, and diverging fluctuations of the time-averaged work. In the end, by comparing the excess and the time-averaged excess works, I conclude that this last one is not useful to measure the excitation spent energy in weak processes, although brings an intuition to what happens in the strong driving case.
We consider the time-dependent transverse field Ising chain with time-periodic perturbations. Without perturbations, this model is one of the famous models that obeys the scaling in the adiabatic limit predicted by the quantum Kibble-Zurek mechanism (QKZM). However, it is known that when oscillations are added to the system, the nonperturbative contribution becomes larger and the scaling may break down even if the perturbation is small. Therefore, we analytically analyze the density of defects in the model and discuss how much the oscillations affect the scaling. As a result, although the nonperturbative contribution does not become zero in the adiabatic limit, the scaling does not change from the prediction of the QKZM. This indicates that the QKZM is robust to the perturbations.