A Dirac fermion model associated with second order topological insulator

TL;DR

This paper links the topological properties of a Dirac fermion model to the quadrupole phase in second order topological insulators, revealing the topological origin of corner states and quantized charge pumping.

Contribution

It demonstrates that the index of the Hamiltonian equals the Higgs field's winding number, connecting lattice corner states to vortex-like states and analyzing topological charge pumping.

Findings

Corner states correspond to Jackiw-Rossi vortex states.

The Hamiltonian index equals the Higgs field's winding number.

Quantized charge pumping is topologically protected.

Abstract

We study topological aspects of a Dirac fermion coupled with a Higgs field associated with the lattice model introduced by Benalcazar et al. which shows the topological quadrupole phase. Using the index theorem, we show that the index of the Hamiltonian is just given by the winding number of the Higgs field, implying that a corner state of the lattice model belongs to the same class of the Jackiw-Rossi states localized at a vortex. We also calculate the current density of the Dirac fermion with a symmetry breaking term dependent on time, which is associated with the dipole pump proposed by Benalcazar et al. We argue that it is indeed a topological current, and the total pumped charge is given by an integer related with the index.