Fermion parity and quantum capacitance oscillation with partially separated Majorana and quasi-Majorana modes

In a recent experiment, flux-dependent oscillations of the quantum capacitance were observed in a one-dimensional spin-orbit-coupled semiconductor–superconductor heterostructure connected end-to-end via a quantum dot and threaded by a magnetic flux. In the topological superconducting phase hosted by this type of heterostructure, the oscillations corresponding to different fermion parity sectors are shifted by half a period and can serve as a mechanism for fermion parity readout or fusion operations involving a pair of localized, well-separated Majorana modes. In this work, we demonstrate that flux-induced fermion parity-dependent oscillations of the quantum capacitance in a disordered semiconductor–superconductor-quantum dot system can originate not only from topologically protected, spatially well-separated Majorana zero modes (MZMs) localized at the wire ends, but also, generically, from partially-separated Majorana modes with significant overlap, as well as from quasi-Majorana modes in the topologically trivial phase, which can be viewed as Andreev bound states whose constituent Majorana wave functions are slightly shifted relative to each other and have non-zero amplitude at opposite ends of the wire. Therefore, while the detection of flux-dependent oscillations of quantum capacitance marks an important experimental advance, such observations alone do not constitute conclusive evidence of the presence of topological Majorana zero modes. Except Fig4Left and Fig4Right, all data sets have the quantity corresponding to the x-axis (e.g., position, QD potential, magnetic flux, etc.) in the first column and the values of the functions represented in the figures in columns 2, 3, .... For Fig4Left, the first column contains the Zeeman field in meV x 100; second column - chemical potential in meV x 100; third column - topological invariant. For Fig4Right: first column - Zeeman field in meV x 100; second column - chemical potential in meV x 100; third column - energy in micro eV; fourth column - localization length (in lattice sites).