Fragile ergodic phases in logarithmically-normal Rosenzweig-Porter model
In this talk I will consider an extension of the Gaussian Rosenzweig-Porter (RP) model, the Log-normal-RP (LN-RP) model, with a logarithmically-normal distribution of off-diagonal matrix elements.
Unlike its Gaussian counterpart, the LN-RP shows true multifractal behavior in the non-ergodic extended phase both in the coordinate and energy spaces. The tails of the log-normal distribution are governed by the parameter p and interpolate the model between the Gaussian RP model (p=0) and Levy matrices (p -> infinity) with a special point p=1 associated with Anderson localization model on the random regular graph. We show that for all p>=1 the multifractal phase (present in the RP model) collapses and gives place to a new weakly ergodic one. This phase is characterized by the broken basis-rotation symmetry, which the fully-ergodic phase relevant for RMT respects, and it is fragile with respect to the truncation of the log-normal tails. Thus, in the LN-RP model, in addition to the localization and ergodic transitions there exists also the transition between the above two weakly and fully ergodic phases. In the talk, I will formulate the criteria of all 3 phase transitions, and provide some physical intuition of the first two ones in terms of the hybridization of fractal wave functions. As a by-product of the latter description I will demonstrate that in this case, unlike the RP model, the Anderson transition is discontinuous, and confirm my statements by the numerical simulations.
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