Non-Hermitian Quantum Fractals

TL;DR

This paper introduces a new class of non-Hermitian quantum fractals arising in coupled Hatano-Nelson models on a tree lattice, revealing self-similar structures and Mandelbrot set-like spectra without external magnetic fields.

Contribution

It is the first to identify and analytically characterize non-Hermitian quantum fractals in lattice models, linking their self-similarity to non-Hermiticity and iterative lattice configurations.

Findings

Revealed self-similar recursive structures in energy spectra and wave functions.

Proved the spectrum's resemblance to the Mandelbrot set.

Extended the concept to generalized non-Hermitian systems with iterative lattices.

Abstract

The first quantum fractal discovered in physics is the Hofstadter butterfly. It stems from large external magnetic fields. We discover instead a new class of non-Hermitian quantum fractals (NHQFs) emerging in coupled Hatano-Nelson models on a tree lattice in absence of any fields. Based on analytic solutions, we are able to rigorously identify the self-similar recursive structures in energy spectrum and wave functions. We prove that the complex spectrum of NHQFs bears a resemblance to the Mandelbrot set in fractal theory. The self-similarity of NHQFs is rooted in the interplay between the iterative lattice configuration and non-Hermiticity. Moreover, we show that NHQFs exist in generalized non-Hermitian systems with iterative lattice structures. Our findings open a new avenue for investigating quantum fractals in non-Hermitian systems.