2020/02/07 by V. E. Kravtsov, Kravtsov, V. E., Ivan M. Khaymovich +5 · 1 citation
Physics and Astronomy · Mathematics · #Quantum many-body systems #Random Matrices and Applications #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2002.02979
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→ 0 and Levy ensembles at p→∞. 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.