Band rearrangement through the 2D-Dirac equation: Comparing the APS and the chiral bag boundary conditions

The Dirac equation on a two-disk is studied under the chiral bag boundary condition, where the mass is treated as a parameter ranging over all real numbers. The eigenvalues as functions of the parameter are compared with those obtained under the APS boundary condition studied in a previous paper of authors (Iwai and Zhilinskii, 2015). Discrete symmetry (or pseudo-symmetry) of the boundary condition as well as the Hamiltonian is studied to explain the difference between the patterns of eigenvalues under the chiral bag and the APS boundary conditions. The spectral flow for a one-parameter family of operators is the net number of eigenvalues passing through zero in the positive direction as the parameter runs. It was demonstrated in the previous paper that the spectral flow is useful to understand the characteristic of eigenvalue pattern of the Dirac equation with the APS boundary condition. However, to capture the feature of eigenvalue pattern under the chiral bag boundary condition, one needs to introduce an extended notion of spectral flow. The eigenvalue patterns under the both boundary conditions are compared with a semi-quantum description of energy-band rearrangement.