Zeeman Field Effect on Zero-energy Vortex and Corner Modes in a Two Dimensional Second-order Topological Superconductor

https://doi.org/10.1007/s10948-025-07110-7

We study the effect of out-of-plane Zeeman field on zero-energy vortex- and corner-localized excitations in a two-dimensional second-order topological superconductor. It is shown analytically and numerically that zero-energy vortex modes are robust against the Zeeman-field effect as long as superconductivity is not destroyed by the field and a superconducting bulk gap is open. On the other hand, Majorana Kramers pairs existing at each corner of a square lattice at zero field and protected by time-reversal symmetry are gapped, in general, by the field. Despite that, we also found the specific model parameters supporting pairs of Majorana corner modes with zero energy in the presence of a magnetic field. In this parametric regime, the corner modes can coexist with zero-energy vortex modes in the presence of a vortex in the model. The precursor for coexisting zero-energy vortex and corner modes under the Zeeman field is the gapless bulk spectrum with Dirac cones in the normal (nonsuperconducting) state.