Band rearrangement through the 3D-Dirac equation with boundary conditions, and the corresponding topological change

Rearrangement of energy bands against a parameter is studied through the 3D-Dirac equation on a ball in $\mathbb{R}^3$ under the APS and the chiral bag boundary conditions on the boundary two-sphere, where APS is an abbreviation of Atiyah-Patodi-Singer. The notion of spectral flow and its extension is introduced to characterize the energy eigenvalue redistribution against the parameter, which reflects an analytical property of edge eigenstates. It is shown that though band rearrangement takes place the net spectral flow is zero ($1+(-1)=0$) for both boundary conditions. The corresponding semi-quantum Hamiltonian defined on the 3D momentum space is studied in parallel and it is shown that a change against the parameter is observed in the mapping degree defined for the semi-quantum model Hamiltonian, which is viewed as reflecting a topological property of the bulk eigenstates of the full quantum system. Specifically, there are two mappings assigned, for which changes in mapping degree take the values $\pm1$. The present correspondence is viewed as a bulk-edge correspondence.