Dynamical detection of extended nonergodic states in many-body quantum systems

TL;DR

This paper introduces a dynamical method to directly measure the correlation fractal dimension in many-body quantum systems, enabling analysis of larger systems without finite-size scaling.

Contribution

The authors demonstrate that the correlation fractal dimension can be obtained from long-time dynamics, bypassing finite-size scaling, and validate the approach on various random and many-body models.

Findings

The dynamical method accurately measures fractal dimensions.

The approach extends analysis to larger systems than traditional methods.

Analytical results are derived for full random matrices.

Abstract

Fractal dimensions are tools for probing the structure of quantum states and identifying whether they are localized or delocalized in a given basis. These quantities are commonly extracted through finite-size scaling, which limits the analysis to relatively small system sizes. In this work, we demonstrate that the correlation fractal dimension $D_2$ can be directly obtained from the long-time dynamics of interacting many-body quantum systems. Specifically, we show that it coincides with the exponent of the power-law decay of the time-averaged survival probability, defined as the fidelity between an initial state and its time-evolved counterpart. This dynamical approach avoids the need for scaling procedures and enables access to larger systems than those typically reachable via exact diagonalization. We test the method on various random matrix ensembles, including full random matrices, the Rosenzweig-Porter model, and power-law banded random matrices, and extend the analysis to interacting many-body systems described by the one-dimensional Aubry-Andr\'e model and the disordered spin-1/2 Heisenberg chain. In the case of full random matrices, we also derive an analytical expression for the entire evolution of the time-averaged survival probability.