# Generalized Hofstadter model on a cubic optical lattice: From nodal bands to the three-dimensional quantum Hall effect

## TL;DR

This paper proposes a tunable 3D Hofstadter model in optical lattices, enabling exploration of Weyl semimetals, nodal loops, and 3D quantum Hall effects with ultracold atoms, advancing topological phase research.

## Contribution

It introduces a method to realize a generalized 3D Hofstadter Hamiltonian in optical lattices, enabling study of complex topological phases including Weyl points, nodal loops, and 3D quantum Hall states.

## Key findings

- Realization of Weyl points and nodal loops in the bulk bands.
- Demonstration of 3D quantum Hall effect with nonzero Chern numbers.
- Proposal of a platform for exploring exotic 3D topological phases.

## Abstract

We propose that a tunable generalized three-dimensional Hofstadter Hamiltonian can be realized by engineering the Raman-assisted hopping of ultracold atoms in a cubic optical lattice. The Hamiltonian describes a periodic lattice system under artificial magnetic fluxes in three dimensions. For certain hopping configurations, the bulk bands can have Weyl points and nodal loops, respectively, allowing the study of both the two nodal semimetal states within this system. Furthermore, we illustrate that with proper rational fluxes and hopping parameters, the system can exhibit the three-dimensional quantum Hall effect when the Fermi level lies in the band gaps, which is topologically characterized by one or two nonzero Chern numbers. Our proposed optical-lattice system provides a promising platform for exploring various exotic topological phases in three dimensions.