Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics

TL;DR

This paper explores the complex fractal energy spectra of a biased quantum particle in a non-Hermitian Hofstadter model, revealing intricate, space-filling energy levels with fractal geometries and a unique band structure.

Contribution

It introduces the concept of fractal energy carpets in non-Hermitian Hofstadter quantum mechanics, highlighting the fractal nature of the energy spectrum and its differences from the traditional Hofstadter butterfly.

Findings

Energy spectrum forms infinitely many fractals in quasi-momentum space.

Energy levels are space-filling curves with Hausdorff dimension 2.

The band structure resembles a fractal spider net, unlike the Hofstadter butterfly.

Abstract

We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasi-momentum space the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider net in contrast to the Hofstadter butterfly for unbiased motion.