Characterizing local Majorana properties using Andreev states

TL;DR

The paper introduces a local spectroscopic probe for Majorana physics in quantum-dot based Kitaev chains by using an Andreev bound state (ABS) as a high-resolution subgap spectrometer. In the weak-coupling limit, the differential conductance exhibits four subgap peaks at voltages $e|V|=\epsilon_A\pm\epsilon_B$, whose heights encode the BdG coherence factors of both the probing ABS and the target subgap state; two ratios, $\xi_1$ and $\xi_2$, directly yield $|u_A|^2/|v_A|^2$ and $|u_B|^2/|v_B|^2$, enabling access to the local SC charge and Majorana polarization. The authors develop a Green's function formalism to treat the ABS–KC–lead system, derive expressions for the current and conductance in the linear regime, and provide analytical relations linking peak heights to coherence factors and relaxation rates, including explicit formulas for the conductance at the four thresholds. The method is shown to scale to longer Kitaev chains, robust to finite broadening and thermal effects, and capable of identifying sweet spots with vanishing local charge and zero-energy splitting, thus offering a practical, in situ diagnostic of local Majorana character in scalable architectures.

Abstract

We propose using Andreev bound states (ABS) as spectroscopic probes to characterize Majorana zero modes (MZMs) in quantum-dot based minimal Kitaev chains. Specifically, we show that tunneling conductance measurements with a superconducting probe hosting an ABS reveal four subgap peaks whose voltage positions and relative heights enable extraction of the MZM energy splitting and Bogoliubov-de Gennes coherence factors. This provides direct access to zero-splitting regimes and to the local Majorana polarization - a measure of the Majorana character. The method is compatible with existing experimental architectures and remains robust in extended chains.

Figures (7)

• Figure 1: (a) ABS behavior in a superconductor-quantum dot (S-QD) junction and the corresponding SC charge as a function of the QD chemical potential $\mu_A$. (b) Schematic representation of a generic ABS-ABS tunneling scheme. (c) Diagrams of the on-resonance dominant transport mechanisms. (d) SC differential conductance between generic ABSs as a function of the target subgap state detuning $\mu_B$, showing both the dispersion of conductance peaks and the emergence of a peak height asymmetry. We set $\mu_A=0$ to avoid extra asymmetries coming from the probe. (e) Symmetric conductance cut along the yellow line at $\mu_B = 0$ showing the distinctive four peaks associated to different tunneling processes, and with negative differential conductance values characteristic of the weak-coupling regime.
• Figure 2: (a) Illustration of the MZM localization in a minimal KC at (left) and away (right) from the Majorana sweet spot. (b, c) Evolution of $\log\,\xi_1$ and $\log\,\xi_2$ in Eq.\ref{['coeff_charge']} as a function of the ABS detuning $\mu_A$, for several values of the outer QDs chemical potential $\mu = 2.45$, $2.5$, $2.55$, $2.62$, with $\mu_c = 1.2$, all in $\Delta$ units. Red dashed line marks the dispersion of the probing ABS. Arrows in panel (c) indicate the $|u_B|^2/|v_B|^2$ values obtained from Hamiltonian diagonalization. (d, e) Phase diagram of the minimal chain as a function of $\mu$ and $\mu_c$. Panel (d) shows the calculated local MP at QD1 ($|{\cal P}_1|$), while (e) shows the corresponding $\log , \xi_2$ extracted from transport. Dark blue line denotes the analytical contour for $\epsilon_B = 0$, cf. the SM SM.
• Figure 3: Phase diagram for chains with $n=5$ (top panels) and $n=10$ (bottom panels) sites, as a function of the chemical potentials $\mu$ and $\mu_c$, all set to the same value. Left column, Local MP at the boundary site, $|{\cal P}_1|$. Right column, $\log \, \xi_2$ extracted from transport conductance. Dark blue line denotes the contour for $\epsilon_B = 0$.
• Figure S1: (a) Schematics of resonant Andreev reflection processes occurring at voltages $[\alpha_-]$ and $[\alpha_+]$. (b) SC differential conductance for various coupling strengths between the KC and the ABS, $\lambda/\Delta = 0.01$, $0.05$, and $0.1$, normalized to the main peak value, with $G_0 = 2e^2/h$. We set , and $\mu_A=0$ to avoid extra asymmetries coming from the probe. (c) Conductance at $[\alpha_+]$ as a function of the ABS detuning $\mu_A$, and normalized to the resonance value at $\mu_A/\Delta \approx -0.7$. Red dashed line indicates the dispersion of the probing ABS. We use $\mu/\Delta = 2$, $\mu_c/\Delta = -1$, and the broadening $\Gamma_t = \Lambda = 5 \times 10^{-3} \cdot \Delta$, chosen for visualization clarity.
• Figure S2: Phase diagram of a 5‑site Kitaev. (a) Global Majorana polarization $|{\cal P}|$, reflecting the non-local topological character of the ground state. (b) Map of the coefficient $\log , \xi_2$ extracted from transport measurements. Dark blue line denotes the contour for $\epsilon_B = 0$.