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arXiv:2410.08086

Localization and Dynamics in One-dimensional and Two-dimensional Gaussian Disordered Quantum Spin Systems

Dongyan Guo, Taotao Hu, Jiameng Hong

Published Oct 10, 2024• 0 citations
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Abstract

This paper introduces Gaussian disorder, characterized by two parameters:the expected value and the standard deviation.Studying this type of disorder enhances our understanding of how many-body localization (MBL) transition is influenced by the central value and breadth of disorder.We apply Gaussian disorder to a one-dimensional spin-1/2 chain to reveal its features.Our findings indicate that the crucial parameter driving localization transitions is the standard deviation; as it increases, the system transitions from the ergodic phase to the localized phase, while the expected value has no effect,so the critical factor influencing MBL transition is the breadth of disorder rather than its central value.We then explore Gaussian disorder in two system-environment models (hereafter referred to as the sye-env models, consisting of ladder configuration and staggered configuration),analyzing the mutual influence when they are coupled.We extensively explore the impacts brought about by system disorder,environment disorder,initial state,and the strength of the sys-env interaction.Finally,for theses two different models,we consider the system and environment as a whole,studying how the localization properties of the entire system are influenced by the disorder strength of each chain and the interaction strength between the two chains.In the ladder configuration, we observe a strong dependence on the initial state,which can lead to either the loss or retention of information,whereas no such dependence on the initial state is found in the staggered configuration.We emphasize that when using dynamic indicators to analyze system characteristics, it is essential to consider the influence of different initial states,as this may lead to significant differences under certain structural conditions,and not all initial states can accurately reflect the localization properties of the many-body system.

DOI
10.48550/arxiv.2410.08086

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This paper introduces Gaussian disorder, characterized by two parameters:the expected value and the standard deviation.Studying this type of disorder enhances our understanding of how many-body localization (MBL) transition is influenced by the central value and breadth of disorder.We apply Gaussian disorder to a one-dimensional spin-1/2 chain to reveal its features.Our findings indicate that the crucial parameter driving localization transitions is the standard deviation; as it increases, the system transitions from the ergodic phase to the localized phase, while the expected value has no effect,so the critical factor influencing MBL transition is the breadth of disorder rather than its central value.We then explore Gaussian disorder in two system-environment models (hereafter referred to as the sye-env models, consisting of ladder configuration and staggered configuration),analyzing the mutual influence when they are coupled.We extensively explore the impacts brought about by system disorder,environment disorder,initial state,and the strength of the sys-env interaction.Finally,for theses two different models,we consider the system and environment as a whole,studying how the localization properties of the entire system are influenced by the disorder strength of each chain and the interaction strength between the two chains.In the ladder configuration, we observe a strong dependence on the initial state,which can lead to either the loss or retention of information,whereas no such dependence on the initial state is found in the staggered configuration.We emphasize that when using dynamic indicators to analyze system characteristics, it is essential to consider the influence of different initial states,as this may lead to significant differences under certain structural conditions,and not all initial states can accurately reflect the localization properties of the many-body system.
We propose a Gaussian disorder to study many-body localization (MBL) in one-dimensional quantum system and coupled system-environment models. Unlike uniform disorder, Gaussian disorder is controlled by the standard deviation σ (driving MBL transitions) and the mean μ (causing global energy shifts). By calculating the von Neumann entropy, mean gap ratio, and dynamical indicators, we elucidate the ergodic-to-MBL transition as σ , with Gaussian disorder exhibiting more pronounced finite-size critical point drift effects compared to conventional random disorder. Through finite-size scaling analysis, it shows that there are no significant differences between the two disorder types at our assumed system sizes. We construct two system-environment coupling configurations (ladder and staggered), showing that under weak coupling, the system and environment evolve independently, whereas strong coupling induces cooperative localization. The ladder configuration shows a strong dependence on the initial state, which may lead to either the loss or retention of information, while no such dependence is observed in the staggered configuration. This can be explained by the changes in energy density of the initial state caused by variations in different model structures. When treating the system-environment as a unified two-dimensional ladder, asymmetric disorder on the two chains can drive a global MBL phase transition, with slightly different behaviors compared to symmetric disorder. Additionally, we emphasize the necessity of correctly selecting the initial state when characterizing the phase transition through dynamic indicators.
It is known that strong disorder in closed quantum systems leads to many-body localization (MBL), and that this quantum phase can be destroyed by coupling to an infinitely large Markovian environment. However, the stability of the MBL phase is less clear when the system and environment are of finite and comparable size. Here, we study the stability and eventual localization properties of a disordered Heisenberg spin chain coupled to a finite environment, and extensively explore the effects of environment disorder, geometry, initial state, and system-bath coupling strength, using the steady-state value of magnetization as a probe. Focusing on nonequilibrium dynamics and steady-state properties, our results indicate that within system sizes amenable to exact diagonalization, a strongly localized system interacting in a junction configuration retains remnant information on its initial state at long times despite coupling to a finite ergodic environment. In contrast, in a ladder configuration, strong dependencies on the initial state and coupling strength are observed, which can lead to either the loss or retention of information. Finally, we highlight and discuss discrepancies that can arise when similar methodologies are employed to infer localization or thermalization, revealing the need for careful interpretation.
It is known that strong disorder in closed quantum systems leads to many-body localization (MBL), and that this quantum phase can be destroyed by coupling to an infinitely large Markovian environment. However, the stability of the MBL phase is less clear when the system and environment are of finite and comparable size. Here, we study the stability and eventual localization properties of a disordered Heisenberg spin chain coupled to a finite environment, and extensively explore the effects of environment disorder, geometry, initial state and system-bath coupling strength. Our numerical results indicate that in most cases, the system retains its localization properties despite the coupling to the finite environment, albeit to a reduced extent. However, in cases where the system and environment is strongly coupled in the ladder configuration, the eventual localization properties are highly dependent on the initial state, and could lead to either thermalization or localization.
In this paper, we theoretically investigate the many-body localization properties of one-dimensional Ising spin-1 chains by using the methods of exact matrix diagonalization. We compare it with the MBL properties of the Ising spin-1/2 chains. The results indicate that the one-dimensional Ising spin-1 chains can also undergo MBL phase transition. There are various forms of disorder, and we compare the effects of different forms of quasi-disorder and random disorder on many-body localization in this paper. First, we calculate the exctied-state fidelity to study the MBL phase transtion. By changing the form of the quasi-disorder, we study the MBL transition of the system with different forms of quasi-disorder and compare them with those of the random disordered system. The results show that both random disorder and quasi-disorder can cause the MBL phase transition in the one-dimensional Ising spin-1 chains. In order to study the effect of spin interactions, we compare Ising spin-1 chains and spin-1/2 chains with the next-nearest-neighbour(N-N) two-body interactions and the next-next-nearest-neighbour (N-N-N)interactions. The results show that the critical point increases with the addition of the interaction. Then we study the dynamical properties of the model by the dynamical behavior of diagonal entropy (DE), local magnetization and the time evolution of fidelity to further prove the occurrence of MBL phase transition in the disordered Ising spin-1 chains with the (N-N) coupling term and distinguish the ergodic phase (thermal phase) and the many-body localized phase. Lastly, we delve into the impact of periodic driving on one-dimensional Ising spin-1 chains. And we compare it with the results obtained from the Ising spin-1/2 chains. It shows that periodic driving can cause Ising spin-1 chains and Ising spin-1/2 chains to occur the MBL transition.
We study the transitions between ergodic and many-body localized phases in spin systems, subject to quenched disorder, including the Heisenberg chain and the central spin model. In both cases systems with common spin lengths 1/21/21/2 and 1 are investigated via exact numerical diagonalization and random matrix techniques. Particular attention is paid to the sample-to-sample variance ${({\mathrm{\ensuremath{\Delta}}}_{s}r)}^{2}$ of the averaged consecutive-gap ratio $\ensuremath{\langle}r\ensuremath{\rangle}$ for different disorder realizations. For both types of systems and spin lengths we find a maximum in ${\mathrm{\ensuremath{\Delta}}}_{s}r$ as a function of disorder strength, accompanied by an inflection point of $\ensuremath{\langle}r\ensuremath{\rangle}$, signaling the transition from ergodicity to many-body localization. The critical disorder strength is found to be somewhat smaller than the values reported in the recent literature. Further information about the transitions can be gained from the probability distribution of expectation values within a given disorder realization.
We analyze a disordered central spin model, where a central spin interacts equally with each spin in a periodic one-dimensional (1D) random-field Heisenberg chain. If the Heisenberg chain is initially in the many-body localized (MBL) phase, we find that the coupling to the central spin suffices to delocalize the chain for a substantial range of coupling strengths. We calculate the phase diagram of the model and identify the phase boundary between the MBL and ergodic phase. Within the localized phase, the central spin significantly enhances the rate of the logarithmic entanglement growth and its saturation value. We attribute the increase in entanglement entropy to a nonextensive enhancement of magnetization fluctuations induced by the central spin. Finally, we demonstrate that correlation functions of the central spin can be utilized to distinguish between MBL and ergodic phases of the 1D chain. Hence, we propose the use of a central spin as a possible experimental probe to identify the MBL phase.