Entanglement spectrum and Wannier center flow of the Hofstadter problem

TL;DR

This paper investigates the entanglement spectrum and Wannier functions of the Hofstadter model, revealing spectral flow characteristics and their relation to topological invariants like Chern numbers, with implications for understanding topological band structures.

Contribution

It demonstrates the connection between entanglement spectra, Wannier functions, and topological invariants in the Hofstadter model, extending previous work on topological band structures.

Findings

Entanglement levels exhibit spectral flow similar to energy spectra.

Entanglement spectra show discontinuities linked to edge states.

Connections established between entanglement properties and topological invariants.

Abstract

We examine the quantum entanglement spectra and Wannier functions of the square lattice Hofstadter model. Consistent with previous work on entanglement spectra of topological band structures, we find that the entanglement levels exhibit a spectral flow similar to that of the full system's energy spectrum. While the energy spectra are continuous, with open boundary conditions the entanglement spectra exhibit discontinuities associated with the passage of an energy edge state through the Fermi level. We show how the entanglement spectrum can be understood by examining the band projectors of the full system and their behavior under adiabatic pumping. In so doing we make connections with the original TKNN work on topological two-dimensional band structures and their Chern numbers. Finally we consider Wannier states and their adiabatic flows, and draw connections to the entanglement properties.