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Register a new userQuantum ergodicity in the many-body localization problem
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We generalize Pages result on the entanglement entropy of random pure states to the many-body eigenstates of realistic disordered many-body systems subject to long range interactions. This extension leads to two principal conclusions: first, for increasing disorder the shells of constant energy supporting a systems eigenstates fill only a fraction of its full Fock space and are subject to intrinsic correlations absent in synthetic high-dimensional random lattice systems. Second, in all regimes preceding the many-body localization transition individual eigenstates are thermally distributed over these shells. These results, corroborated by comparison to exact diagonalization for an SYK model, are at variance with the concept of non-ergodic extended states in many-body systems discussed in the recent literature.
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We discuss the problem of localization in two dimensional electron systems in the quantum Hall (single Landau level) regime. After briefly summarizing the well-studied problem of Anderson localization in the non-interacting case, we concentrate on the problem of disorder induced many-body localization (MBL) in the presence of electron-electron interactions using numerical exact diagonalization and eigenvalue spacing statistics as a function of system size. We provide evidence showing that MBL is not attainable in a single Landau level with short range (white noise) disorder in the thermodynamic limit. We then study the interplay of topology and localization, by contrasting the behavior of topological and nontopological subbands arising from a single Landau level in two models - (i) a pair of extremely flat Hofstadter bands with an optimally chosen periodic potential, and (ii) a Landau level with a split-off nontopological impurity band. Both models provide convincing evidence for the strong effect of topology on the feasibility of many-body localization as well as slow dynamics starting from a nonequilibrium state with charge imbalance.
Ergodicity in quantum many-body systems is - despite its fundamental importance - still an open problem. Many-body localization provides a general framework for quantum ergodicity, and may therefore offer important insights. However, the characterization of many-body localization through simple observables is a difficult task. In this article, we introduce a measure for distances in Hilbert space for spin-1/2 systems that can be interpreted as a generalization of the Anderson localization length to the many-body Hilbert space. We show that this many-body localization length is equivalent to a simple local observable in real space, which can be measured in experiments of superconducting qubits, polar molecules, Rydberg atoms, and trapped ions. Using the many-body localization length and a necessary criterion for ergodicity that it provides, we study many-body localization and quantum ergodicity in power-law-interacting Ising models subject to disorder in the transverse field. Based on the nonequilibrium dynamical renormalization group, numerically exact diagonalization, and an analysis of the statistics of resonances we find a many-body localized phase at infinite temperature for small power-law exponents. Within the applicability of these methods, we find no indications of a delocalization transition.
We study the eigenstates of a paradigmatic model of many-body localization in the Fock basis constructed out of the natural orbitals. By numerically studying the participation ratio, we identify a sharp crossover between different phases at a disorder strength close to the disorder strength at which subdiffusive behaviour sets in, significantly below the many-body localization transition. We repeat the analysis in the conventionally used computational basis, and show that many-body localized eigenstates are much stronger localized in the Fock basis constructed out of the natural orbitals than in the computational basis.
We investigate a many-body localization transition based on a Boltzmann transport theory. Introducing weak localization corrections into a Boltzmann equation, Hershfield and Ambegaokar re-derived the Wolfle-Vollhardt self-consistent equation for the diffusion coefficient [Phys. Rev. B {bf 34}, 2147 (1986)]. We generalize this Boltzmann equation framework, introducing electron-electron interactions into the Hershfield-Ambegaokar Boltzmann transport theory based on the study of Zala-Narozhny-Aleiner [Phys. Rev. B {bf 64}, 214204 (2001)]. Here, not only Altshuler-Aronov corrections but also dephasing effects are taken into account. As a result, we obtain a self-consistent equation for the diffusion coefficient in terms of the disorder strength and temperature, which extends the Wolfle-Vollhardt self-consistent equation in the presence of electron correlations. Solving our self-consistent equation numerically, we find a many-body localization insulator-metal transition, where a metallic phase appears from dephasing effects dominantly instead of renormalization effects at high temperatures. Although this mechanism is consistent with that of recent seminal papers [Ann. Phys. (N. Y). {bf 321}, 1126 (2006); Phys. Rev. Lett. {bf 95}, 206603 (2005)], we find that our three-dimensional metal-insulator transition belongs to the first order transition, which differs from the Anderson metal-insulator transition described by the Wolfle-Vollhardt self-consistent theory. We speculate that a bimodal distribution function for the diffusion coefficient is responsible for this first order phase transition.
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