Spectral flow for Aharonov--Bohm rings generated by zero-mass lines

We found that the spectrum of the linear dispersion mode localized along the zero-mass line for variable mass Dirac fermions shows the nonzero spectral flow as a function of the external magnetic field B. This happens due to the Aharonov–Bohm effect, leading to the linear dependence of every eigenvalue on B. The nonzero spectral flow allows to use the magnetic field slowly varying in time to control the energy of the linear dispersion mode.