Localization transition on the Random Regular Graph as an unstable tricritical point in a log-normal Rosenzweig-Porter random matrix ensemble

Abstract

Gaussian Rosenzweig-Porter (GRP) random matrix ensemble is the only one in which the robust multifractal phase and ergodic transition have a status of a mathematical theorem. Yet, this phase in GRP model is oversimplified: the spectrum of fractal dimensions is degenerate and the mini-band in the local spectrum is not multifractal. In this paper we suggest an extension of the GRP model by adopting a logarithmically-normal (LN) distribution of off-diagonal matrix elements. A family of such LN-RP models is parametrized by a symmetry parameter $p$ and it interpolates between the GRP at $p\to 0$ and Levy ensembles at $p\to \infty$. A special point $p=1$ is shown to be the simplest approximation to the Anderson localization model on a random regular graph.We study in detail the phase diagram of LN-RP model and show that $p=1$ is a tricritical point where the multifractal phase first collapses. This collapse is shown to be unstable with respect to the truncation of the log-normal distribution. We suggest a new criteria of stability of the non-ergodic phases and prove that the Anderson transition in LN-RP model is discontinuous at all $p>0$.

arXiv comment11 pages, 9 figures, 2 tables, 48 references + 8 pages, 6 figures in Appendices

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• Rare thermal bubbles at the many-body localization transition from the Fock space point of view
• The Many-Body localization transition in the Hilbert space
• Dynamical phases in a "multifractal" Rosenzweig-Porter model
• The Lévy-Rosenzweig-Porter random matrix ensemble
• Emergent fractal phase in energy stratified random models
• Non-ergodic delocalized phase with Poisson level statistics
• Universality in Anderson localization on random graphs with varying connectivity
• Out of equilibrium Phase Diagram of the Quantum Random Energy Model
• Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization
• Interacting Thermofield Doubles and Critical Behavior in Random Regular Graphs
• Spectral statistics of a minimal quantum glass model
• Anatomy of the eigenstates distribution: a quest for a genuine multifractality
• Statistics of Green's functions on a disordered Cayley tree and the validity of forward scattering approximation
• Dirty bosons on the Cayley tree: Bose-Einstein condensation versus ergodicity breaking
• Investigating finite-size effects in random matrices by counting resonances
• Robust extended states in Anderson model on partially disordered random regular graphs
• Circular Rosenzweig-Porter random matrix ensemble
• Replica approach to the generalized Rosenzweig-Porter model