Conductance of a superconducting Coulomb blockaded Majorana nanowire

In the presence of an applied magnetic field introducing Zeeman spin splitting, a superconducting (SC) proximitized one-dimensional (1D) nanowire with spin-orbit coupling can pass through a topological quantum phase transition developing zero-energy topological Majorana bound states (MBSs) on the wire ends. One of the promising experimental platforms in this context is a Coulomb blockaded island, where by measuring the two-terminal conductance one can in principle investigate the MBS properties. We theoretically study the tunneling transport of a single electron across the superconducting Coulomb blockaded nanowire at finite temperature to obtain the generic conductance equation. By considering all possible scenarios where only MBSs are present at the ends of the nanowire, we compute the nanowire conductance as a function of the magnetic field, the temperature, and the gate voltage. In the simplest 1D topological SC model, the oscillations of the conductance peak spacings (OCPSs) arising from the MBSs overlap from the two wire ends manifest an increasing oscillation amplitude with increasing magnetic field (in agreement with theories without Coulomb blockade and in disagreement with a recent experimental observation). We develop a generalized finite temperature master equation theory including not only multiple subbands in the nanowire, but also the possibility of ordinary Andreev bound states in the non-topological regime. Inclusion of all four effects (temperature, multiple subbands, Andreev bound states, and MBSs) provides a complete picture of the tunneling transport properties. Based on this complete theory, we indeed obtain OCPSs whose amplitudes decrease with increasing magnetic field in qualitative agreement with recent experimental results, but this happens only for rather high temperatures with multisubband occupancy and the presence of both Andreev bound states and MBSs.