Topological magnetoelectric pump in three dimensions

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.Comment: 16 pages, 8 figures; final versio

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“…Second, we interpret the vector current J i as the vacuum expectation value (VEV) of an axial vector current J i 5 involving the matrix operator γ 5 generating chirality. In other words, we have J i = J i 5 reading in terms of the Hamiltonian H (2.1)-(2.6) as follows [34,[38][39][40][41][42] ,…”

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