Surprises in the phase diagram of the Anderson model on the Bethe lattice

  title={Surprises in the phase diagram of the Anderson model on the Bethe lattice},
  author={Simone Warzel},
  journal={arXiv: Mathematical Physics},
  • S. Warzel
  • Published 18 December 2012
  • Physics
  • arXiv: Mathematical Physics
rst for which an energy regime of extended states and a separate regime of localized states could be established. In this paper, we review recently discovered surprises in the phase diagram. Among them is that even at weak disorder, the regime of diusive transport extends well beyond energies of the unperturbed model into the Lifshitz tails. As will be explained, the mechanism for the appearance of extended states in this non-perturbative regime are disorder-induced resonances. We also present… 

Figures from this paper

Random Schr\"odinger Operators on discrete structures
The Anderson model serves to study the absence of wave propagation in a medium in the presence of impurities, and is one of the most studied examples in the theory of quantum disordered systems. In
Anderson localization on the Bethe lattice using cages and the Wegner flow
Anderson localization on treelike graphs such as the Bethe lattice, Cayley tree, or random regular graphs has attracted attention due to its apparent mathematical tractability, hypothesized
The large connectivity limit of the Anderson model on tree graphs
  • V. Bapst
  • Computer Science, Mathematics
  • 2014
A rigorous lower bound on the free energy function recently introduced by Aizenman and Warzel is derived and an upper bound is derived on the critical disorder such that all states at a given energy become localized.
Poisson Eigenvalue Statistics for Random Schrödinger Operators on Regular Graphs
For random operators it is conjectured that spectral properties of an infinite-volume operator are related to the distribution of spectral gaps of finite-volume approximations. In particular,
Lower bounds on the localisation length of balanced random quantum walks
We consider the dynamical properties of Quantum Walks defined on the d-dimensional cubic lattice, or the homogeneous tree of coordination number 2d, with site-dependent random phases, further
Convergence of the density of states and delocalization of eigenvectors on random regular graphs
Consider a random regular graph of fixed degree $d$ with $n$ vertices. We study spectral properties of the adjacency matrix and of random Schr\"odinger operators on such a graph as $n$ tends to
Uniqueness for solutions of the Schrödinger equation on trees
We prove that if a solution of the time-dependent Schrödinger equation on an homogeneous tree with bounded potential decays fast at two distinct times then the solution is trivial. For the free
Dependence of the Density of States on the Probability Distribution for Discrete Random Schrödinger Operators
We prove that the density of states measure (DOSm) for random Schrödinger operators on $\mathbb{Z}^d$ is weak-$^{\ast }$ Hölder-continuous in the probability measure. The framework we develop is
The Anderson model on the Bethe lattice: Lifshitz Tails
This paper is devoted to the study of the (discrete) Anderson Hamiltonian on the Bethe lattice, which is an infinite tree with constant vertex degree. The Hamiltonian we study corresponds to the sum
Delocalization for the 3D discrete random Schrödinger operator at weak disorder
We apply a recently developed approach (Liaw 2013 J. Stat. Phys. 153 1022–38) to study the existence of extended states for the three-dimensional discrete random Schrödinger operator at small


Extended states in a Lifshitz tail regime for random Schrödinger operators on trees.
It is found that extended states appear through disorder enabled resonances well beyond the energy band of the operator's hopping term for weak disorder this includes a Lifshitz tail regime of very low density of states.
Difference between level statistics, ergodicity and localization transitions on the Bethe lattice
We show that non-interacting disordered electrons on a Bethe lattice display a new intermediate phase which is delocalized but non-ergodic, i.e. it is characterized by Poisson instead of GOE
Localization bounds for an electron gas
Mathematical analysis of the Anderson localization has been facilitated by the use of suitable fractional moments of the Green function. Related methods permit now a readily accessible derivation of
Analyticity of the density of states in the anderson model on the Bethe lattice
Let H=1/2Δ+V on l2(B), whereB is the Bethe lattice andV(x),xεB, are i.i.d.r.v.'s with common probability distributionμ. It is shown that for distributions sufficiently close to the Cauchy
A selfconsistent theory of localization
A new basis has been found for the theory of localization of electrons in disordered systems. The method is based on a selfconsistent solution of the equation for the self energy in second order
Bounds on the density of states in disordered systems
It is proven that the averaged density of states does neither vanish nor diverge inside the band for a class of tight-binding models governed by short-range one-particle Hamiltonians with site-diagonal and/or off- diagonal disorder and continuous distribution of the matrix elements.
Self-consistent theory of localization. II. Localization near the band edges
For pt. I see abstr. A43970 of 1973. The solution of the integral equation which arises in the self-consistent theory of localization has been explored for a Cauchy distribution of site energies, for
Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model
We study a lattice field model which qualitatively reflects the phenomenon of Anderson localization and delocalization for real symmetric band matrices. In this statistical mechanics model, the field
Absence of diffusion in the Anderson tight binding model for large disorder or low energy
We prove that the Green's function of the Anderson tight binding Hamiltonian decays exponentially fast at long distances on ℤv, with probability 1. We must assume that either the disorder is large or
An elementary proof is given of localization for linear operators H = Ho + λV, with Ho translation invariant, or periodic, and V (·) a random potential, in energy regimes which for weak disorder (λ →