Majorana bound states and subgap states in three-terminal topological superconducting nanowire-quantum dot hybrid devices

Three-terminal topological superconducting nanowire (TSNW)-quantum dot (QD) hybrid junction devices are studied. The energy spectra and the wave functions of the subgap states are calculated as a function of the superconducting phase differences between TSNWs and as a function of the QD level energy based on the Bogoliubov-de Gennes tight-binding Hamiltonians. It is shown that when the QD level is located near or inside the superconducting gap, there can exist eight subgap states. Among them, four low energy (two positive and two negative) subgap states are essentially formed by linear combinations of the six Majorana bound states (MBSs) located at the ends of the three TSNWs. The remaining four high energy subgap states are mainly built from linear combinations of the QD state and the three MBSs of the TSNWs adjacent to the QD. When there is no QD level near or inside the superconducting gap, only six subgap states built from linear combinations of the six MBSs of the three TSNWs can be present in the system. It is also shown that there exists a unique point in the parameter space of the superconducting phase differences between TSNWs, at which the energies of the six low energy subgap states move close to each other towards nearly zero energies. Simple but general effective model Hamiltonians for the three-terminal TSNW-QD hybrid devices have also been developed. Based on the effective model Hamiltonians, the subgap states of the three-terminal TSNW-QD hybrid devices in the limit of the three infinitely long TSNWs are studied. The results of the calculations and the effective model Hamiltonians could be used as a starting point to construct and investigate the braiding schemes of MBSs in TSNW junction devices.