Anatomy of the eigenstates distribution: a quest for a genuine multifractality

TL;DR

This paper develops an analytical approach to understand the conditions for genuine multifractality in random matrix models, concluding that only the distribution of on-site energies can induce such phases, challenging previous claims.

Contribution

It introduces a graphical method for calculating fractal dimensions in generalized Rosenzweig-Porter models and clarifies the role of on-site energy distribution in multifractality.

Findings

Only on-site energy distribution leads to genuine multifractality.

Homogeneous diagonal disorder models do not host multifractal phases.

Analytical framework for fractal dimension calculation in RP-like models.

Abstract

Motivated by a series of recent works, an interest in multifractal phases has risen as they are believed to be present in the Many-Body Localized (MBL) phase and are of high demand in quantum annealing and machine learning. Inspired by the success of the RosenzweigPorter (RP) model with Gaussian-distributed hopping elements, several RP-like ensembles with the fat-tailed distributed hopping terms have been proposed, with claims that they host the desired multifractal phase. In the present work, we develop a general (graphical) approach allowing a self-consistent analytical calculation of fractal dimensions for a generic RP model and investigate what features of the RP Hamiltonians can be responsible for the multifractal phase emergence. We conclude that the only feature contributing to a genuine multifractality is the on-site energies' distribution, meaning that no random matrix model with a statistically homogeneous distribution of diagonal disorder and uncorrelated off-diagonal terms can host a multifractal phase.