Fractality of wave functions on a Cayley tree: Difference between a tree and a locally tree-like graph without boundary

Abstract

We investigate analytically and numerically eigenfunction statistics in a disordered system on a finite Bethe lattice (Cayley tree). We show that the wave function amplitude at the root of a tree is distributed fractally in a large part of the delocalized phase. The fractal exponents are expressed in terms of the decay rate and the velocity in a problem of propagation of a front between unstable and stable phases. We demonstrate a crucial difference between a loopless Cayley tree and a locally tree-like structure without a boundary (random regular graph) where extended wavefunctions are ergodic.

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Cited by in corpus (16)

• From non-ergodic eigenvectors to local resolvent statistics and back: a random matrix perspective
• Delocalized Glassy Dynamics and Many Body Localization
• Critical behavior at the localization transition on random regular graphs
• Dynamical phases in a "multifractal" Rosenzweig-Porter model
• Critical properties of the Anderson localization transition and the high dimensional limit
• Multifractality of wave functions on a Cayley tree: From root to leaves
• The Lévy-Rosenzweig-Porter random matrix ensemble
• Eigenvectors under a generic perturbation: non-perturbative results from the random matrix approach
• Multifractal finite-size scaling at the Anderson transition in the unitary symmetry class
• Statistical properties of the Green function in finite size for Anderson Localization models with multifractal eigenvectors
• Fully localized and partially delocalized states in the tails of Erdös-Rényi graphs in the critical regime
• Multifractality in random networks with power-law decaying bond strengths
• Localization in random bipartite graphs: numerical and empirical study
• Random Transverse Field Spin-Glass Model on the Cayley tree : phase transition between the two Many-Body-Localized Phases
• Spectral correlations in Anderson insulating wires
• Localization properties of the sparse Barrat-Mézard trap model