Topological magnetoelectric pump in three dimensions

TL;DR

This paper investigates a three-dimensional topological pump in lattice fermion models, deriving the U(1) current density, demonstrating quantized particle pumping linked to topological invariants, and exploring the relationship between different Chern numbers through numerical analysis.

Contribution

It introduces a lattice fermion model in continuous time to study topological pumping in 3D, deriving the conserved current, and connecting topological invariants with quantized particle transport.

Findings

The U(1) current density includes a lattice effect characterized by the Chern number.

The particle number pumped is quantized due to topological reasons.

Numerical results relate the second Chern number to spectral asymmetry and bulk-edge correspondence.

Abstract

We study the topological pump for a lattice fermion model mainly in three spatial dimensions. We first calculate the U(1) current density for the Dirac model defined in continuous space-time to review the known results as well as to introduce some technical details convenient for the calculations of the lattice model. We next investigate the U(1) current density for a lattice fermion model, a variant of the Wilson-Dirac model. The model we introduce is defined on a lattice in space but in continuous time, which is suited for the study of the topological pump. For such a model, we derive the conserved U(1) current density and calculate it directly for the $1+1$ dimensional system as well as $3+1$ dimensional system in the limit of the small lattice constant. We find that the current includes a nontrivial lattice effect characterized by the Chern number, and therefore, the pumped particle number is quantized by the topological reason. Finally we study the topological temporal pump in $3+1$ dimensions by numerical calculations. We discuss the relationship between the second Chern number and the first Chern number, the bulk-edge correspondence, and the generalized Streda formula which enables us to compute the second Chern number using the spectral asymmetry.