Majorana fermions and multiple topological phase transition in Kitaev ladder topological superconductors

Motivated by the InSb nanowire superconductor system, we investigate a system where one-dimensional topological superconductors are placed in parallel. It would be simulated well by a ladder of the Kitaev chains. The system undergoes multiple topological phase transitions, where the number of Majorana fermions changes as a function of the interchain superconducting pairings. We analytically determine the topological phase diagram by explicitly calculating the topological number and the band structure. They show even-odd effects with respect to the number of legs of the ladder. When the relative phase between the inter- and intrachain superconducting pairings is 0 or pi, the system belongs to the class BDI characterized by the Z index, and otherwise it belongs to the class D characterized by the Z(2) index. This topological class change would be caused by applying the Josephson current or an external magnetic field, and could be observed by measuring the zero-bias differential conductance.