Streda formula for the Hofstadter-Wilson-Dirac model in two and four dimensions

TL;DR

This paper rederives the spectral asymmetry of the Wilson-Dirac model in external fields, linking it to the Streda formula and Chern numbers, and demonstrates its validity through numerical calculations in various dimensions.

Contribution

It provides a novel interpretation of spectral asymmetry as the Streda formula in higher-dimensional Wilson-Dirac models, extending its applicability.

Findings

Streda formula reproduces known Chern numbers in weak magnetic fields

Numerical evidence supports the formula's validity in higher dimensions

Conjecture that the formula applies to stronger fields and more general systems

Abstract

We rederive the spectral asymmetry of the Wilson-Dirac model in external fields, paying attention to the Chern number due to the Berry connection. We interpret the smooth part of the spectral asymmetry as the Streda formula that is originally derived for the two-dimensional quantum Hall effect (QHE). We show by numerical calculations that the Streda formula reproduces the known first and second Chern numbers in a weak magnetic field limit. We conjecture that the Streda formula is valid even for stronger fields and for more generic systems in higher dimensions.