# Multi-fractality of the entanglement Hamiltonian eigen-modes

## TL;DR

This paper investigates the fractal characteristics of entanglement Hamiltonian eigen-modes in free fermion models, revealing their potential to characterize quantum phase transitions through fractal analysis.

## Contribution

It demonstrates that the fractal properties of maximally entangled modes mirror those of Hamiltonian eigen-modes, offering a new way to identify quantum phase transitions.

## Key findings

- MEM and Hamiltonian eigen-modes share the same fractal dimension in delocalized phase.
- Both modes exhibit multi-fractality at the phase transition point.
- Fractal behavior of MEM can serve as a quantum phase transition indicator.

## Abstract

We study the fractal properties of single-particle eigen-modes of entanglement Hamiltonian in free fermion models. One of these modes that has the highest entanglement information and thus called maximally entangled mode (MEM) is specially considered. In free Fermion models with Anderson localization, fractality of MEM is obtained numerically and compared with the fractality of Hamiltonian eigen-mode at Fermi level. We show that both eigen-modes have similar fractal properties: both have same single fractal dimension in delocalized phase which equals the dimension of the system, and both show multi-fractality at phase transitio point. Therefore, we conclude that, fractal behavior of MEM -- in addition to the fractal behavior of Hamiltonian eigen-mode -- can be used as a quantum phase transition characterization.