A critical Dyson hierarchical model for the Anderson localization transition

Abstract

A Dyson hierarchical model for Anderson localization, containing non-random hierarchical hoppings and random on-site energies, has been studied in the mathematical literature since its introduction by Bovier [J. Stat. Phys. 59, 745 (1990)], with the conclusion that this model is always in the localized phase. Here we show that if one introduces alternating signs in the hoppings along the hierarchy (instead of choosing all hoppings of the same sign), it is possible to reach an Anderson localization critical point presenting multifractal eigenfunctions and intermediate spectral statistics. The advantage of this model is that one can write exact renormalization equations for some observables. In particular, we obtain that the renormalized on-site energies have the Cauchy distributions for exact fixed points. Another output of this renormalization analysis is that the typical exponent of critical eigenfunctions is always ${\alpha }_{typ}=2$, independently of the disorder strength. We present numerical results concerning the whole multifractal spectrum $f(\alpha )$ and the compressibility $\chi$ of the level statistics, both for the box and the Cauchy distributions of the random on-site energies. We discuss the similarities and differences with the ensemble of ultrametric random matrices introduced recently by Fyodorov, Ossipov and Rodriguez [J. Stat. Mech. L12001 (2009)].

21 pages, 11 figures; v2=final version

References in corpus (47)

• Anderson Transitions
• Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices
• Fluctuations of the inverse participation ratio at the Anderson transition
• Multifractality and critical fluctuations at the Anderson transition
• Exact relations between multifractal exponents at the Anderson transition
• Many-body localization transition in a lattice model of interacting fermions: statistics of renormalized hoppings in configuration space
• Asymptotics of Universal Probability of Neighboring Level Spacings at the Anderson Transition
• Renormalization Group and Probability Theory
• Multifractal analysis with the probability density function at the three-dimensional Anderson transition
• Multifractal analysis of the metal-insulator transition in the 3D Anderson model I: Symmetry relation under typical averaging
• Eigenfunction entropy and spectral compressibility for critical random matrix ensembles
• Anderson localization transition and eigenfunction multifractality in ensemble of ultrametric random matrices
• Point-Contact Conductances at the Quantum Hall Transition
• Weak Chaos in a Quantum Kepler Problem
• Fluctuations of the correlation dimension at metal-insulator transitions
• Multifractal analysis of the metal-insulator transition in the 3D Anderson model II: Symmetry relation under ensemble averaging
• Critical statistics in a power-law random banded matrix ensemble
• Critical Hamiltonians with long range hopping
• Fluctuation of the Correlation Dimension and the Inverse Participation Number at the Anderson Transition
• Dynamical scaling for critical states: is Chalker's ansatz valid for strong fractality?
• The Hierarchical Random Energy Model
• Anderson localization on the Cayley tree : multifractal statistics of the transmission at criticality and off criticality
• Probing the eigenfunction fractality with a stop watch
• $f(\alpha )$ Multifractal spectrum at strong and weak disorder
• Anomalously large critical regions in power-law random matrix ensembles
• Virial expansion for almost diagonal random matrices
• Supersymmetry Approach to Almost Diagonal Random Matrices
• Poisson statistics of eigenvalues in the hierarchical Anderson model
• Boundary multifractality in critical 1D systems with long-range hopping
• Statistics of renormalized on-site energies and renormalized hoppings for Anderson localization models in dimensions d=2 and d=3
• Overlap Interfaces in Hierarchical Spin-Glass models
• Interaction-dependent enhancement of the localisation length for two interacting particles in a one-dimensional random potential
• Multifractality of Hamiltonians with power-law transfer terms
• Symmetry relation for multifractal spectra at random critical points
• Level compressibility in a critical random matrix ensemble: The second virial coefficient
• Scattering at the Anderson transition: Power--law banded random matrix model
• Metal-insulator transition in two-dimensional disordered systems with power-law transfer terms
• Universal and non-universal features of the multifractality exponents of critical wave-functions
• Power spectrum for critical statistics: A novel spectral characterization of the Anderson transition
• Anderson localization transition with long-ranged hoppings : analysis of the strong multifractality regime in terms of weighted Levy sums
• Supersymmetric Virial Expansion for Time-Reversal Invariant Disordered Systems
• Statistics of the two-point transmission at Anderson localization transitions
• Random elastic networks : strong disorder renormalization approach
• Critical properties in long-range hopping Hamiltonians
• Random cascade models of multifractality : real-space renormalization and travelling-waves
• Critical level spacing distribution in long-range hopping Hamiltonians
• Anderson transitions : multifractal or non-multifractal statistics of the transmission as a function of the scattering geometry

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• Transition between localized and extended states in the hierarchical Anderson model
• Quantum percolation transition in 3d: density of states, finite size scaling and multifractality
• Dyson Hierarchical Long-Ranged Quantum Spin-Glass via real-space renormalization
• Random field Ising model : statistical properties of low-energy excitations and of equilibrium avalanches
• One-dimensional Ising spin-glass with power-law interaction : real-space renormalization at zero temperature
• Renormalization Group Analysis of the Hierarchical Anderson Model
• Critical Exponents of the Random Field Hierarchical Model
• Towards rigorous analysis of the Levitov-Mirlin-Evers recursion
• The renormalization flow of the hierarchical Anderson model at weak disorder
• Spreading of a local excitation in a Quantum Hierarchical Model
• Singular Spectrum and Recent Results on Hierarchical Operators