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arXiv:2403.16868

Electron localization in disordered quantum systems at finite temperatures

Chong Sun

Published Mar 25, 2024• 0 citations
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Abstract

We study electron localization in disordered quantum systems, focusing on both individual eigenstates and thermal states. We employ complex polarization as a numerical indicator to characterize the system's localization length. Furthermore, we assess the efficacy of mean-field approximation in providing a quantitative analysis of such systems. Through this study, we seek to provide insight into the following aspects: the behavior of electron localization as a function of interaction, disorder, and temperature, whether thermal states and highly excited states exhibit similar properties in many-body localized systems, and the reliability of the mean-field approximation in weak-interaction scenarios.

DOI
10.48550/arxiv.2403.16868

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We study electron localization in disordered quantum systems, focusing on both individual eigenstates and thermal states. We employ complex polarization as a numerical indicator to characterize the system's localization length. Furthermore, we assess the efficacy of mean-field approximation in providing a quantitative analysis of such systems. Through this study, we seek to provide insight into the following aspects: the behavior of electron localization as a function of interaction, disorder, and temperature, whether thermal states and highly excited states exhibit similar properties in many-body localized systems, and the reliability of the mean-field approximation in weak-interaction scenarios.
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Abstract Statistical mechanics provides a framework for describing the physics of large, complex many-body systems using only a few macroscopic parameters to determine the state of the system. For isolated quantum many-body systems, such a description is achieved via the eigenstate thermalization hypothesis (ETH), which links thermalization, ergodicity and quantum chaotic behavior. However, tendency towards thermalization is not observed at finite system sizes and evolution times in a robust many-body localization (MBL) regime found numerically and experimentally in the dynamics of interacting many-body systems at strong disorder. Although the phenomenology of the MBL regime is well-established, the central question remains unanswered: under what conditions does the MBL regime give rise to an MBL phase , in which the thermalization does not occur even in the asymptotic limit of infinite system size and evolution time? This review focuses on recent numerical investigations aiming to clarify the status of the MBL phase, and it establishes the critical open questions about the dynamics of disordered many-body systems. The last decades of research have brought an unprecedented new variety of tools and indicators to study the breakdown of ergodicity, ranging from spectral and wave function measures, matrix elements of observables, through quantities probing unitary quantum dynamics, to transport and quantum information measures. We give a comprehensive overview of these approaches and attempt to provide a unified understanding of their main features. We emphasize general trends towards ergodicity with increasing length and time scales, which exclude naive single-parameter scaling hypothesis, necessitate the use of more refined scaling procedures, and prevent unambiguous extrapolations of numerical results to the asymptotic limit. Providing a concise description of numerical methods for studying ETH and MBL, we explore various approaches to tackle the question of the MBL phase. Persistent finite size drifts towards ergodicity consistently emerge in quantities derived from eigenvalues and eigenvectors of disordered many-body systems. The drifts are related to continuous inching towards ergodicity and non-vanishing transport observed in the dynamics of many-body systems, even at strong disorder. These phenomena impede the understanding of microscopic processes at the ETH-MBL crossover. Nevertheless, the abrupt slowdown of dynamics with increasing disorder strength provides premises suggesting the proximity of the MBL phase. This review concludes that the questions about thermalization and its failure in disordered many-body systems remain a captivating area open for further explorations.
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We develop a numerical technique to study Anderson localization in interacting electronic systems. The ground state of the disordered system is calculated with quantum Monte Carlo simulations while the localization properties are extracted from the "Thouless conductance" g, i.e., the curvature of the energy with respect to an Aharonov-Bohm flux. We apply our method to polarized electrons in a two-dimensional system of size L. We recover the well-known universal beta(g)=dlogg/dlogL one parameter scaling function without interaction. Upon switching on the interaction, we find that beta(g) is unchanged while the system flows toward the insulating limit. We conclude that polarized electrons in two dimensions stay in an insulating state in the presence of weak to moderate electron-electron correlations.
Strong long-range interaction leads to localization in a closed quantum system without disorders. Employing the exact diagonalization method, I numerically investigate thermalization and many-body localization in translational invariant quantum chains with finite Coulomb interactions. In the computational basis, excluding all trivial degeneracies, the interaction-induced localization is well demonstrated in aspects of level statistics, eigenstate expectation values, and Anderson localization on graphs constructed of many-body bases. The nature of localization for generic eigenstates is attributed to the quasidisorder from the power-law interactions, and the full localization in the Hilbert space is similar to that in the disorder case. However, due to real-space symmetries, the long-time dynamics is dominated by the degenerated eigenstates and eventually reaches homogeneity in real space. On the other hand, the entanglement entropy exhibits size dependence beyond the area law for the same reason, even deep in the localized state, indicating an incomplete localization in real space.
Quantum coherence quantifies the amount of superposition a quantum state can have in a given basis. Since there is a difference in the structure of eigenstates of the ergodic and many-body localized systems, we expect them also to differ in terms of their coherences in a given basis. Here, we numerically calculate different measures of quantum coherence in the excited eigenstates of an interacting disordered Hamiltonian as a function of the disorder. We show that quantum coherence can be used as an order parameter to detect the well-studied ergodic to many-body localized phase transition. We also perform quantum quench studies to distinguish the behavior of coherence in thermalized and localized phases. We then present a protocol to calculate measurement-based localizable coherence to investigate the thermal and many-body localized phases. The protocol allows one to investigate quantum correlations experimentally in a nondestructive way, in contrast to measures that require tracing out a subsystem, which always destroys coherence and correlation.