Dynamical phases in a "multifractal" Rosenzweig-Porter model

Abstract

We consider the static and dynamic phases in a Rosenzweig-Porter (RP) random matrix ensemble with the tailed distribution of off-diagonal matrix elements of the form of the large-deviation ansatz. We present a general theory of survival probability in such a random-matrix model and show that the {\it averaged} survival probability may decay with time as the simple exponent, as the stretch-exponent and as a power-law or slower. Correspondingly, we identify the exponential, the stretch-exponential and the frozen-dynamics phases. As an example, we consider the mapping of the Anderson model on Random Regular Graph (RRG) onto the "multifractal" RP model and find exact values of the stretch-exponent $\kappa$ depending on box-distributed disorder in the thermodynamic limit. As another example we consider the logarithmically-normal RP (LN-RP) random matrix ensemble and find analytically its phase diagram and the exponent $\kappa$. In addition, our theory allows to compute the shift of apparent phase transition lines at a finite system size and show that in the case of RP associated with RRG and LN-RP with the same symmetry of distribution function of hopping, a finite-size multifractal "phase" emerges near the tricritical point which is also the point of localization transition.

31 pages, 8 figures, 73 references + 10 pages, 5 figures in Appendices and references; Conclusions and Appendix C added, File style changed

References in corpus (16)

• Many body localization and thermalization in quantum statistical mechanics
• Anomalous diffusion and Griffiths effects near the many-body localization transition
• Absence of diffusion in an interacting system of spinless fermions on a one-dimensional disordered lattice
• Anomalous thermalization in ergodic systems
• Non-ergodic phases in strongly disordered random regular graphs
• Fractality of wave functions on a Cayley tree: Difference between a tree and a locally tree-like graph without boundary
• From non-ergodic eigenvectors to local resolvent statistics and back: a random matrix perspective
• 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
• Robustness of delocalization to the inclusion of soft constraints in long-range random models
• Percolation in Fock space as a proxy for many-body localisation
• Absolutely continuous spectrum implies ballistic transport for quantum particles in a random potential on tree graphs
• Exact solution of a percolation analogue for the many-body localisation transition
• Statistical properties of the Green function in finite size for Anderson Localization models with multifractal eigenvectors
• Multifractality of random eigenfunctions and generalization of Jarzynski equality

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