Bulk-edge correspondence for the Dirac oscillator on the two-torus as a magnetic unit cell

Abstract

The 2D Dirac oscillator can be viewed as the Dirac operator for a charged particle in a constant magnetic field. This system admits a non-abelian magnetic translation symmetry. Under a cocycle condition with respect to this symmetry, the 2D Dirac oscillator Hamiltonian is made into a Dirac Hamiltonian defined on the two-torus as a magnetic unit cell. A remarkable feature of this system is the existence of an edge-state (or extremal) eigenvalue which is attributed to different bands, according to the range to which the value of the control parameter belongs. The other eigenvalues, which are called bulk-state eigenvalues, are attributed to either of two bands for all values of the control parameter. For a sufficiently weak magnetic flux density, the Dirac oscillator Hamiltonian is brought into a semi-quantum Hamiltonian after the Landau–Peierls substitution. Then, a bulk-edge correspondence is shown to hold in terms of spectral flow for the quantum Hamiltonian and change in (formal) Chern number for the semi-quantum Hamiltonian.