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

2016/07/31 by Davide Facoetti, Pierpaolo Vivo, Giulio Biroli
Physics and Astronomy · Mathematics · #cond-mat.dis-nn #cond-mat.stat-mech #math-ph #math.MP #quant-ph

paper · pdf · doi:10.1209/0295-5075/115/47003

published as EPL 115 (2016) 47003 · 7 pages, 2 figures. Final version EPL (2016)

arxiv created 2016/09/27 · arxiv updated 2016/09/29

Abstract

We study the statistics of the local resolvent and non-ergodic properties of eigenvectors for a generalised Rosenzweig-Porter N× 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-γ sites close in energy |a_j-a_i|≤ N^1-γ in agreement with Kravtsov 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.

Citations

• A random matrix model with localization and ergodic transitions