Dirac fermion model associated with a second-order topological insulator

We study topological aspects of a Dirac fermion coupled with a Higgs field associated with the lattice model introduced by Benalcazar et al. [Science 357, 61 (2017)] which has 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 the total pumped charge is given by an integer related with the index.