Finite-Volume Fractional-Moment Criteria for Anderson Localization

Abstract

A technically convenient signature of localization, exhibited by discrete operators with random potentials, is exponential decay of the fractional moments of the Green function within the appropriate energy ranges. Known implications include: spectral localization, absence of level repulsion, strong form of dynamical localization, and a related condition which plays a significant role in the quantization of the Hall conductance in two-dimensional Fermi gases. We present a family of finite-volume criteria which, under some mild restrictions on the distribution of the potential, cover the regime where the fractional moment decay condition holds. The constructive criteria permit to establish this condition at spectral band edges, provided there are sufficient `Lifshitz tail estimates' on the density of states. They are also used here to conclude that the fractional moment condition, and thus the other manifestations of localization, are valid throughout the regime covered by the ``multiscale analysis''. In the converse direction, the analysis rules out fast power-law decay of the Green functions at mobility edges.

arXiv commentLaTeX, 46 pages, one figure (epsf). To appear in Comm. Math. Phys

Cited by in corpus (44)

• Moment Analysis for Localization in Random Schroedinger Operators
• Localization Bounds for Multiparticle Systems
• Eigenvector localization for random band matrices with power law band width
• Fractional moment bounds and disorder relevance for pinning models
• Equality of the bulk and edge Hall conductances in a mobility gap
• New characterizations of the region of complete localization for random Schrödinger operators
• Multi-Particle Anderson Localisation: Induction on the Number of Particles
• Correlations Estimates in the Lattice Anderson Model
• Dynamical Localization for Unitary Anderson Models
• Generalized eigenvalue-counting estimates for the Anderson model
• Anderson localization for a class of models with a sign-indefinite single-site potential via fractional moment method
• The bootstrap multiscale analysis for the multi-particle Anderson model
• Eigenfunctions in a two-particle Anderson tight binding model
• Absence of mobility edge for the Anderson random potential on tree graphs at weak disorder
• Quantum harmonic oscillator systems with disorder
• Multiparticle localization for disordered systems on continuous space via the fractional moment method
• Multi-Scale Jacobi Method for Anderson Localization
• Resonances for large one-dimensional "ergodic" systems
• Large Block Properties of the Entanglement Entropy of Disordered Fermions
• An eigensystem approach to Anderson localization
• From Extreme Values of I.I.D. Random Fields to Extreme Eigenvalues of Finite-volume Anderson Hamiltonian
• Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails
• On the Wegner orbital model
• Exponential dynamical localization for the almost Mathieu operator
• Localization criteria for Anderson models on locally finite graphs
• The exponential decay of eigenfunctions for tight binding Hamiltonians via landscape and dual landscape functions
• Decorrelation estimates for the eigenlevels of the discrete Anderson model in the localized regime
• Fractional Moment Estimates for Random Unitary Operators
• Localization via fractional moments for models on $\mathbb{Z}$ with single-site potentials of finite support
• Transport of a quantum particle in a time-dependent white-noise potential
• Localization on quantum graphs with random vertex couplings
• How large is large? Estimating the critical disorder for the Anderson model
• Localization in the Disordered Holstein model
• Exponential scaling limit of the single-particle Anderson model via adaptive feedback scaling
• Lyapunov exponents of random walks in small random potential: the lower bound
• Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials
• Localization and IDS Regularity in the Disordered Hubbard Model within Hartree-Fock Theory
• Diffusion in the Mean for an Ergodic Schrödinger Equation Perturbed by a Fluctuating Potential
• Eigensystem multiscale analysis for the Anderson model via the Wegner estimate
• A bound on the averaged spectral shift function and a lower bound on the density of states for random Schrödinger operators on ${\mathbb{R}}^d$
• Dynamical Evolution of Entanglement in Disordered Oscillator Systems
• Resonant tunneling in a system with correlated pure point spectrum
• Wegner estimate for discrete Schrödinger operators with Gaussian random potentials
• Dynamical Localization and Transport properties of Quantum Walks on the hexagonal lattice