Quantum Inductance as a Phase-Sensitive Probe of Fermion Parity Switching in Majorana Nanowires

Abstract: We study the flux-dependent quantum inductance of a one-dimensional (1D) semiconductor-superconductor (SM-SC) Majorana nanowire coupled to a quantum dot in an interferometric setup. Although quantum capacitance in this setup enables fast fermion parity readout, as has been demonstrated experimentally, it cannot by itself reliably confirm a protected fermion parity switch, a key signature of non-trivial topology and the existence of Majorana zero modes (MZMs). In realistic devices, disorder can produce avoided crossings or narrow double crossings between the two parity sectors that can mimic the behavior of a protected single parity switching, leading to false positives for non-trivial topological behavior. We show that quantum inductance provides a complementary probe that is directly sensitive to the phase structure of the low energy spectrum, allowing us to distinguish genuine fermion-parity crossings from avoided crossings or narrow double crossings. Using a general Lehmann framework applied to both effective models and full microscopic simulations with disorder, we demonstrate that only a true fermion-parity switch produces the characteristic inductive response of a protected crossing. In contrast, topologically trivial avoided crossings or narrow double crossings yield quantum inductance signatures that are markedly different from those of topologically nontrivial fermion parity crossings. Therefore, our results show that combined measurements of quantum capacitance and quantum inductance provide a robust and experimentally accessible means to identify true fermion-parity switches, corresponding to a nontrivial Pfaffian invariant.