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.

Figures (22)

• Figure 1: Schematic representation of the semiconductor nanowire in proximity to an $s$-wave superconductor, with both ends of the wire coupled to a quantum dot (QD). The effective coupling to the left and right ends of the nanowire is controlled by the constants $\lambda_L$ and $\lambda_R$, respectively. A back-gate electrode tunes the dot potential $V_{QD}$, while a magnetic flux $\Phi$ threads the loop formed between the nanowire and the QD. In the topological regime, the nanowire hosts a pair of Majorana modes, $\gamma_L$ and $\gamma_R$, that are localized near its opposite ends.
• Figure 2: Flux dependence of quantum capacitance and quantum inductance in low energy effective models.(a) $h/e$-periodic quantum capacitance $\mathcal{C}(\Phi)$ for a pair of topological MZMs (model 1 in Sec. \ref{['Model1']}), where the ground state switches fermion parity at $\Phi=h/2e$ (even and odd parity branches shown in blue and red, respectively). (b) $h/2e$-periodic $\mathcal{C}(\Phi)$ in the trivial regime for a pair of Andreev bound states localized near the nanowire ends (model 2 in Sec. \ref{['Model2']}), illustrating an avoided crossing near $\Phi=h/2e$. (c) Same as (b), but for a trivial narrow double crossing scenario near $\Phi=h/2e$. (d)-(f) Corresponding quantum inductance curves $\mathcal{L}^{-1}(\Phi)$ for the cases shown in (a)-(c), respectively, highlighting the response of the phase derivative operators to the flux driven evolution of the low energy spectrum. In (a)-(c), shaded regions indicate the sign of the local curvature of $\mathcal{C}(\Phi)$ (green: positive; yellow: negative). The phase window $\Delta\phi$ marks the avoided and narrow double crossing region in (b) and (c), which produces a extrema in $\mathcal{L}^{-1}(\Phi)$ in (e) and (f), respectively.
• Figure 3: (a) Normalized quantum capacitance $C/C_{0}$ versus magnetic flux $\Phi$ threading the topological superconducting ring, expressed in units of the flux quantum $\Phi_{0}=h/2e$. (b) Normalized quantum inductance $\mathcal{L}^{-1}/\mathcal{L}_{0}^{-1}$ as a function of $\Phi/\Phi_{0}$. Even and odd parity responses are indicated by the blue and red curves, respectively. Both quantum capacitance and quantum inductance exhibit an $h/e$ periodicity, and the even and odd branches cross at $\Phi=\Phi_{0}$, signaling a fermion parity switch.
• Figure 4: (a) Normalized quantum capacitance $C/C_{0}$ versus magnetic flux $\Phi$, expressed in units of the flux quantum $\Phi_{0}=h/2e$. (b) Normalized quantum inductance $\mathcal{L}^{-1}/\mathcal{L}_{0}^{-1}$ as a function of $\Phi/\Phi_{0}$. Even and odd parity responses are indicated by the blue and red curves, respectively. Both capacitance and quantum inductance are $h/2e$ periodic, and the absence of a true crossing (presence of an avoided crossing) in the capacitance is reflected in the quantum inductance curves by the pair of extrema at the corresponding flux values $\Phi \approx 0.325\,\Phi_{0}$ and $1.325\,\Phi_{0}$.
• Figure 5: (a) BdG low energy spectrum of the clean nanowire ($V_{0}=0$ meV) in the open boundary configuration, i.e., with the ends of the wire not coupled to the quantum dot (QD). For Zeeman fields exceeding the topological quantum phase transition (TQPT) point $\Gamma_{c}=\sqrt{\mu^{2}+\Delta^{2}}=0.54$ meV, a pair of Majorana zero modes (highlighted in red) emerges and persists until the superconducting gap collapses near $\Gamma\approx1.25$ meV. The light blue shaded region denotes the parameter range where the Pfaffian invariant (refer to Eq. \ref{['eq:pfaff']}) is negative, corresponding to the operationally topological regime. The black circle marks the point that is further examined once the wire is coupled to the QD. (b) Low energy modes (fixed fermion parity) as a function of the QD potential $V_{QD}$ for $\Phi=0$ (blue solid line) and $\Phi=h/2e$ (green dashed line), evaluated at the Zeeman field indicated in panel (a). The tunnel couplings are $\lambda_{L}=\lambda_{R}=0.025$. (c) Difference in quantum capacitance between the even and odd parity sectors, $\Delta \mathcal{C}_\Phi = \mathcal{C}_{\mathrm{even}} - \mathcal{C}_{\mathrm{odd}}$, plotted as a function of $V_{QD}$ for the same flux values, $\Phi=0$ (blue solid line) and $\Phi=h/2e$ (green dashed line). (d) Difference in quantum inductance, $\Delta \mathcal{L}_\Phi^{-1}=\mathcal{L}^{-1}_{\mathrm{even}} - \mathcal{L}^{-1}_{\mathrm{odd}}$, for the same parameters. Panels (b)-(d) show the detailed behavior of the system in the vicinity of the point highlighted in panel (a).