Three-Band Model for Quantum Hall and Spin Hall Effects

TL;DR

This paper explores the topological properties of three-band Hamiltonians, demonstrating their relation to Skyrmion numbers and analyzing a Kagome lattice model to reveal quantum and spin Hall effects with non-Abelian gauge connections.

Contribution

It introduces a topological framework for three-band Hamiltonians, extending the understanding of quantum Hall effects beyond two-band models and analyzing their symmetry classes.

Findings

Topological properties are characterized by Skyrmion numbers.

Kagome lattice Hamiltonian exhibits non-Abelian gauge connections.

Pseudo-spin Hall conductance can be computed from conserved pseudo-spin currents.

Abstract

Topological properties of a certain class of spinless three-band Hamiltonians are shown to be summed up by the Skyrmion number in momentum space, analogous to the case of two-band Hamiltonian. Topological tight-binding Hamiltonian on a Kagome lattice is analyzed with this view. When such a Hamiltonian is "folded", the two bands with opposite Chern numbers merge into a degenerate band exhibiting non-Abelian gauge connection. Conserved pseudo-spin current operator can be constructed in this case and used to compute the pseudo-spin Hall conductance. Our model Hamiltonians belong to the symmetry class D and AI according to the ten-fold classification scheme.