Davide Venturelli — LTPMC (Sorbonne Université) #
Multifractality in disordered graphs: insights from two random matrix models #
In recent years, there has been a notable resurgence of interest in the spectral properties of random graphs. In particular, the existence of eigenstates exhibiting properties intermediate between full localization and full ergodicity has acquired significance in various physical contexts, including Anderson localization and the Many-Body localization. In the many-body setting, multifractal eigenstates that do not cover the entire accessible Hilbert space may violate the eigenstate thermalization hypothesis, and thus they are often termed "non-ergodic".  Random matrix models have been an invaluable tool to describe and help understanding complex physical systems, in particular those with quenched randomness. The physical mechanism at the origin of multiftactal eigenstates is one such problem, and various matrix ensembles (either sparse or dense) have been introduced as proxies to capture their peculiar spectral properties.  In this talk, I will first address the generalized Rosenzweig-Porter model, which has recently been shown to host non-ergodic states [1]. Using replica methods, we have derived novel exact results about its spectral correlations [2], and here I will discuss their physical significance.  Next, I will consider the Erdös-Rényi random graph with randomly distributed weights. Despite this model being well-known, we have recently revealed a previously unnoticed multifractal phase in a broad region of its parameter space, using a combination of analytical (cavity-based) and numerical approaches [3]. I will elucidate the physical mechanisms underlying the emergence of these states, rooted in the pronounced heterogeneity in the graph's topology.  REFERENCES:  [1] V. E. Kravtsov, I. M. Khaymovich, E. Cuevas, M. Amini, "A random matrix model with localization and ergodic transitions", New J. Phys. 17 122002 (2015). [2] D. Venturelli, L. F. Cugliandolo, G. Schehr, M. Tarzia, "Replica approach to the generalized Rosenzweig-Porter model", SciPost Phys. 14, 110 (2023). [3] L. F. Cugliandolo, G. Schehr, M. Tarzia, D. Venturelli, "Multifractal phase in the weighted adjacency matrices of random Erdös-Rényi graphs", arXiv:2404.06931 (2024).