From non-ergodic eigenvectors to local resolvent statistics and back: a random matrix perspective

Abstract

We study the statistics of the local resolvent and non-ergodic properties of eigenvectors for a generalised Rosenzweig-Porter $N\times N$ random matrix model, undergoing two transitions separated by a delocalised non-ergodic phase. Interpreting the model as the combination of on-site random energies $\{a_i\}$ and a structurally disordered hopping, we found that each eigenstate is delocalised over $N^{2-\gamma }$ sites close in energy $|a_j-a_i|\le N^{1-\gamma }$ in agreement with Kravtsov \emph{et al}, arXiv:1508.01714. Our other main result, obtained combining a recurrence relation for the resolvent matrix with insights from Dyson's Brownian motion, is to show that the properties of the non-ergodic delocalised phase can be probed studying the statistics of the local resolvent in a non-standard scaling limit.

7 pages, 2 figures. Final version EPL (2016)

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Cited by in corpus (5)

• Delocalized Glassy Dynamics and Many Body Localization
• Multifractality of wave functions on a Cayley tree: From root to leaves
• Eigenvectors under a generic perturbation: non-perturbative results from the random matrix approach
• Statistical properties of the Green function in finite size for Anderson Localization models with multifractal eigenvectors
• Random Transverse Field Spin-Glass Model on the Cayley tree : phase transition between the two Many-Body-Localized Phases