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arXiv:2404.17247 · Physical review. B./Physical review. B

Analytical derivation and extension of the anti-Kibble-Zurek scaling in the transverse field Ising model

Kaito Iwamura, Takayuki Suzuki

Published Oct 4, 2024• 8 citations
arXivHTMLPDF

Abstract

A defect density which quantifies the deviation from the spin ground state characterizes non-equilibrium dynamics during phase transitions. The widely recognized Kibble-Zurek scaling predicts how the defect density evolves during phase transitions. However, it can be perturbed by a noise, leading to the anti-Kibble-Zurek scaling. In this research, we analytically investigate the effect of Gaussian white noise on the transition probabilities of the Landau-Zener model. We apply this analysis to the one-dimensional transverse field Ising model and obtain an analytical approximate solution of the defect density. Our analysis reveals that when the introduced noise is small, the model follows the previously known anti-Kibble-Zurek scaling. Conversely, when the noise increases, the scaling can be obtained by using the adiabatic approximation. This result indicates that deriving the anti-Kibble-Zurek scaling does not require solving differential equations, instead, it can be achieved simply by applying the adiabatic approximation. Furthermore, we identify the parameter that minimizes the defect density based on the new scaling, which allows us to verify how effective the already known scaling of the optimized parameter is.

DOI
10.1103/physrevb.110.144102

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A defect density which quantifies the deviation from the spin ground state characterizes non-equilibrium dynamics during phase transitions. The widely recognized Kibble-Zurek scaling predicts how the defect density evolves during phase transitions. However, it can be perturbed by a noise, leading to the anti-Kibble-Zurek scaling. In this research, we analytically investigate the effect of Gaussian white noise on the transition probabilities of the Landau-Zener model. We apply this analysis to the one-dimensional transverse field Ising model and obtain an analytical approximate solution of the defect density. Our analysis reveals that when the introduced noise is small, the model follows the previously known anti-Kibble-Zurek scaling. Conversely, when the noise increases, the scaling can be obtained by using the adiabatic approximation. This result indicates that deriving the anti-Kibble-Zurek scaling does not require solving differential equations, instead, it can be achieved simply by applying the adiabatic approximation. Furthermore, we identify the parameter that minimizes the defect density based on the new scaling, which allows us to verify how effective the already known scaling of the optimized parameter is.
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.
We revisit the quantum annealing problem in the non-Hermitian transverse-field Ising model. We determine, both analytically and numerically, the intrinsic transition probabilities and the resulting defect density. Our results reveal that, unlike the Hermitian case where defect production is dominated by modes near the gap-closing point, the non-Hermitian dynamics involve significant contributions from broad momentum sectors. We find that, depending on the dissipation strength, the defect density exhibits standard Kibble-Zurek scaling, anti-Kibble-Zurek behavior, and a suppression faster than the Kibble-Zurek prediction. We demonstrate that these deviations from the standard Kibble-Zurek scaling can be understood in terms of the underlying excitation probabilities. Specifically, the fast decay of the defect density originates from a vanishing excitation probability spanning a range of annealing times across all allowed modes, even at the gap-closing points. In contrast, the anti-Kibble-Zurek behavior arises from supplementary excitations facilitated by dissipation over a broad range of allowed modes, particularly those situated away from the gap-closing region.
We propose an optimized adiabatic-impulse (OAI) protocol that substantially reduces the evolution time for crossing a quantum phase transition while preserving Kibble-Zurek (KZ) scaling. Near criticality, the control parameter is ramped linearly across the critical point at a rate characterized by a quench time τQτ_QτQ​. Away from criticality, the evolution remains adiabatic and is tuned close to the threshold of adiabatic breakdown, as quantified by an adiabatic coefficient ζζζ that scales as τQατ_Q^ατQα​. As a consequence, the total evolution time exhibits a sublinear power-law dependence on τQτ_QτQ​, and the conventional linear quench is recovered in the limit α→∞α\rightarrow\inftyα→∞. We apply the OAI protocol to the transverse Ising chain and numerically determine the minimal ζζζ required for KZ scaling. We further investigate the nonequilibrium dynamics in the presence of a noisy field that can induce anti-Kibble-Zurek (AKZ) behavior. Within the OAI protocol, noise-induced defects is significantly attenuated due to the shorter evolution time. The optimal quench time at which the defect density is minimized obeys an altered universal power-law scaling with the noise strength. Finally, we generalize the OAI protocol to the nonlinear quenches and numerically demonstrate a marked reduction in noise-induced defects.
In the nonadiabatic dynamics across a quantum phase transition, the Kibble-Zurek paradigm describes that the average number of topological defects is suppressed as a universal power law with the quench time scale. A conflicting observation, termed anti-Kibble-Zurek dynamics, has been reported in several studies, specifically in the driven systems with an uncorrelated stochastic (white) noise. Here, we study the defect generation in the driven transverse field/anisotropy quantum XY model in the presence of a correlated (colored) Gaussian noise. We propose a generic conjecture that properly captures the noise-induced excitation features, which shows good agreement with the numerical simulations. We show that the dynamical features of defect density are modified by varying the noise correlation time. Our numerical simulations confirm that, for fast noises, the dynamics of the defect density is the same as that of the uncorrelated (white) noise, as is expected. However, the larger ratio of noise correlation time to the annealing time results in larger defect density formation and reforms the universal dynamical features. Our finding reveals that the noise-induced defects scale linearly with the annealing time for fast noises, while in the presence of slow noises the noise-induced defects scale linearly with the square of the annealing time. The numerical simulations confirm that the optimal annealing time, at which the defect density is minimum, scales linearly in logarithmic scale with the total noise power having different exponents for fast and slow noises.
The Kibble-Zurek mechanism (KZM) predicts that the average number of topological defects generated upon crossing a continuous or quantum phase transition obeys a universal scaling law with the quench time. Fluctuations in the defect number near equilibrium are approximately of Gaussian form, in agreement with the central limit theorem. Using large deviations theory, we characterize the universality of fluctuations beyond the KZM and report the exact form of the rate function in the transverse-field quantum Ising model. In addition, we characterize the scaling of large deviations in an arbitrary continuous phase transition, building on recent evidence establishing the universality of the defect number distribution.
We analyze mechanisms for universal out-of-equilibrium dynamics near criticality by exploring the effect of randomized quantum resetting (QR) under a finite-time quench across a quantum phase transition. Using the transverse-field Ising chain as a generic model and exploiting its exact solution, QR is found to cause a crossover of the scaling of the topological defect density with the time scale τττ of the quench, from Kibble-Zurek to anti-Kibble-Zurek scaling as τττ increases. This reflects a competition between non-adiabatic quench-driven excitations and QR, giving rise to local minima of the defect densities at optimal annealing times. These times and the corresponding local minima are shown to scale as universal power laws with the rate of QR. Additional results for the scaling of the mean excess energy suggest that a system driven across a quantum critical point exhibits the same scaling behavior under a linear quench with QR as with uncorrelated noise.