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arXiv:2102.03134 · Physical review. B./Physical review. B

Driven quantum spin chain in the presence of noise: Anti-Kibble-Zurek behavior

Manvendra Singh, Suhas Gangadharaiah

Published Aug 31, 2021• 22 citations
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Abstract

We study defect generation in a quantum XY-spin chain arising due to the linear drive of the many-body Hamiltonian in the presence of a time-dependent fast Gaussian noise. The main objective of this work is to quantify analytically the effects of noise on the defect density production. In the absence of noise, it is well known that in the slow sweep regime, the defect density follows the Kibble-Zurek (KZ) scaling behavior with respect to the sweep speed. We consider time-dependent fast Gaussian noise in the anisotropy of the spin-coupling term $[{\ensuremath{\gamma}}_{0}=({J}_{1}\ensuremath{-}{J}_{2})/({J}_{1}+{J}_{2})]$ and show via analytical calculations that the defect density exhibits anti-Kibble-Zurek (AKZ) scaling behavior in the slow sweep regime. In the limit of large chain length and long time, we calculate the entropy and magnetization density of the final decohered state and show that their scaling behavior is consistent with the AKZ picture in the slow sweep regime. We have also numerically calculated the sublattice spin correlators for finite separation by evaluating the Toeplitz determinants and find results consistent with the KZ picture in the absence of noise, while in the presence of noise and slow sweep speeds the correlators exhibit the AKZ behavior. Furthermore, by considering the large $n$-separation asymptotes of the Toeplitz determinants, we further quantify the effect of the noise on the spin-spin correlators in the final decohered state. We show that while the correlation length of the sublattice correlator scales according to the AKZ behavior, we obtain different scaling for the magnetization correlators.

DOI
10.1103/physrevb.104.064313

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We study defect generation in a quantum XY-spin chain arising due to the linear drive of the many-body Hamiltonian in the presence of a time-dependent fast Gaussian noise. The main objective of this work is to quantify analytically the effects of noise on the defect density production. In the absence of noise, it is well known that in the slow sweep regime, the defect density follows the Kibble-Zurek (KZ) scaling behavior with respect to the sweep speed. We consider time-dependent fast Gaussian noise in the anisotropy of the spin-coupling term [\ensuremathγ0=(J1\ensuremath−J2)/(J1+J2)][{\ensuremath{\gamma}}_{0}=({J}_{1}\ensuremath{-}{J}_{2})/({J}_{1}+{J}_{2})][\ensuremathγ0​=(J1​\ensuremath−J2​)/(J1​+J2​)] and show via analytical calculations that the defect density exhibits anti-Kibble-Zurek (AKZ) scaling behavior in the slow sweep regime. In the limit of large chain length and long time, we calculate the entropy and magnetization density of the final decohered state and show that their scaling behavior is consistent with the AKZ picture in the slow sweep regime. We have also numerically calculated the sublattice spin correlators for finite separation by evaluating the Toeplitz determinants and find results consistent with the KZ picture in the absence of noise, while in the presence of noise and slow sweep speeds the correlators exhibit the AKZ behavior. Furthermore, by considering the large nnn-separation asymptotes of the Toeplitz determinants, we further quantify the effect of the noise on the spin-spin correlators in the final decohered state. We show that while the correlation length of the sublattice correlator scales according to the AKZ behavior, we obtain different scaling for the magnetization correlators.
We study a one-dimensional Kitaev chain driven by the anisotropy parameter ${J}_{\ensuremath{-}}=t/{\ensuremath{\tau}}_{Q}$, in the presence of weak Gaussian noise in the parameter J+{J}_{+}J+​. The system is prepared in the ground state of the Hamiltonian at the initial time $t\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$. The defect density and the residual energy at the end of the drive protocol reveals anti-Kibble Zurek (AKZ) behavior which deviates from the earlier reported linear in quench time dependence for the defect density. The entropy density at the end of the protocol is found to exhibit the signature of the AKZ behavior. In the context of the Kitaev chain, the two-point spin correlators are short ranged. The hidden topological order across the quantum phase transition point is probed via the nonlocal string order parameters. The two-point Majorana correlator and the hidden string correlator calculated in the final decohered state at the end of the noisy drive protocol is consistent with the AKZ picture. We have analyzed the kink statistics in the dual spin space, the first three cumulants at the end of the noisy drive also exhibit the AKZ scaling behavior for slower sweeps similar to the defect density. The kink distribution function is well approximated with the normal distribution with nonuniversal noise dependent mean and variance.
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.
We present an exact solution for a quantum spin chain driven through its critical points. Our approach is based on a many-body generalization of the Landau-Zener transition theory, applied to a fermionized spin Hamiltonian. The resulting nonequilibrium state of the system, while being a pure quantum state, has local properties of a mixed state characterized by finite entropy density associated with Kibble-Zurek defects. The entropy and the finite spin correlation length are functions of the rate of sweep through the critical point. We analyze the anisotropic XYXYXY spin-1∕21∕21∕2 model evolved with a full many-body evolution operator. With the help of Toeplitz determinant calculus, we obtain an exact form of correlation functions. The properties of the evolved system undergo an abrupt change at a certain critical sweep rate, signaling the formation of ordered domains. We link this phenomenon to the behavior of complex singularities of the Toeplitz generating function.
We study the dynamics of a transverse-field XY\mathrm{XY}XY chain driven across quantum critical points by noisy control fields. We characterize the defect density as a function of the quench time and the noise strength and demonstrate that the defect productions for three quench protocols with different scaling exponents exhibit the anti-Kibble-Zurek behavior, whereby slower driving results in more defects. The protocols are quenching through the boundary line between paramagnetic and ferromagnetic phases, quenching across the isolated multicritical point, and quenching along the gapless line, respectively. We also show that the optimal quench time to minimize defects scales as a universal power law of the noise strength in all three cases. Furthermore, by using quantum simulation of the quench dynamics in the spin system with well-designed Landau-Zener crossings in pseudomomentum space, we propose an experimentally feasible scheme to test the predicted anti-Kibble-Zurek behavior of this noisy transverse-field XY\mathrm{XY}XY chain with two-level systems under controllable fluctuations.
The Kibble-Zurek (KZ) mechanism is a universal framework that can in principle describe nonequilibrium phase transition phenomena in any system with the required symmetry properties. However, a conflicting phenomenon termed anti-KZ behavior has been reported in the study of ferroelectric phase transitions, in which slower driving results in more topological defects [S. M. Griffin et al., Phys. Rev. X 2, 041022 (2012)]. Although this finding is significant, related experimental work remains scarce. In this work, we experimentally demonstrate anti-KZ behavior under a noisy control field in three different quantum phase transition protocols using a single trapped 171Yb+{}^{171}{\mathrm{Yb}}^{+}171Yb+ ion. The density of defects is studied as a function of the quench time and the noise intensity. We experimentally verify that the optimal quench time to minimize excitations scales as a universal power law of the noise intensity. Our work provides an experimental verification of anti-KZ behavior in a two-level system.
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.