Hofstadter Butterfly Diagram in Noncommutative Space

TL;DR

This paper investigates the Hofstadter butterfly energy spectrum of electrons in a two-dimensional noncommutative space under a magnetic field, revealing a fractal structure with notable differences from the commutative case.

Contribution

It introduces a gauge-invariant method to compute the Hofstadter butterfly in noncommutative space using lattice models derived via Bopp's shift and Peierls substitution.

Findings

Fractal structure observed in the noncommutative Hofstadter diagram.

Global features similar to the commutative case, but with distinct detailed structures.

Method developed for calculating energy spectra in noncommutative quantum systems.

Abstract

We study an energy spectrum of electron moving under the constant magnetic field in two dimensional noncommutative space. It take place with the gauge invariant way. The Hofstadter butterfly diagram of the noncommutative space is calculated in terms of the lattice model which is derived by the Bopp's shift for space and by the Peierls substitution for external magnetic field. We also find the fractal structure in new diagram. Although the global features of the new diagram are similar to the diagram of the commutative space, the detail structure is different from it.