Absence of Mobility Edge in Short-range Uncorrelated Disordered Model: Coexistence of Localized and Extended States

Abstract

Unlike the well-known Mott's argument that extended and localized states should not coexist at the same energy in a generic random potential, we provide an example of a nearest-neighbor tight-binding disordered model which carries both localized and extended states without forming the mobility edge (ME). Unexpectedly, this example appears to be given by a well-studied $\beta$-ensemble with independently distributed random diagonal potential and inhomogeneous kinetic hopping terms. In order to analytically tackle the problem, we locally map the above model to the 1D Anderson model with matrix-size- and position-dependent hopping and confirm the coexistence of localized and extended states, which is shown to be robust to the perturbations of both potential and kinetic terms due to the separation of the above states in space. In addition, the mapping shows that the extended states are non-ergodic and allows to analytically estimate their fractal dimensions.

arXiv comment4.5 pages, 4 figures, 60 references + 3.5 pages, 5 figures in Appendices

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• Proposal for many-body quantum chaos detection
• Multifractal phase in the weighted adjacency matrices of random Erdös-Rényi graphs
• Robust non-ergodicity of ground state in the $\beta$ ensemble
• Dynamical Signatures of Chaos to Integrability Crossover in $2\times 2$ Generalized Random Matrix Ensembles
• Reducing dynamical fluctuations and enforcing self-averaging by opening many-body quantum systems
• Long-range spectral statistics of the Rosenzweig-Porter model
• Emergent multifractality in power-law decaying eigenstates
• Spectral form factor and energy correlations in banded random matrices
• Re-entrant localization induced by short-range hopping in the fractal Rosenzweig-Porter Model
• Soft modes in vector spin glass models on sparse random graphs
• Superdiffusion in random two dimensional system with time-reversal symmetry and long-range hopping
• Anomalous energy correlations and spectral form factor in the nonergodic phase of the $\beta$-ensemble
• Power-law banded random matrix ensemble as a model for quantum many-body Hamiltonians
• Quasiperiodic Skin Criticality in an Exactly Solvable Non-Hermitian Quasicrystal
• Ergodicity-breaking phase diagram and fractal dimensions in long-range models with generically correlated disorder
• Identifying mobility edge from finite temperature spectral form factor