Exceptional Andreev spectrum and supercurrent in p-wave non-Hermitian Josephson junctions

TL;DR

The paper addresses transport and spectral properties of a one-dimensional $p$-wave non-Hermitian Josephson junction by solving the non-Hermitian BdG equation with a dissipative barrier, revealing a pair of zero-energy exceptional points that descend from Majorana zero modes and are topologically protected. It derives the complex Andreev spectrum, analyzes inelastic scattering, and computes the supercurrent from inelastic Andreev reflections, finding a continuous current-phase relation and a linear decrease of the critical current with the dissipation strength $Z$, with no enhancement at the EPs. The study further extends the analysis to a mixed $s$-$p$ wave NHJJ, where additional EPs appear but without Majorana protection, illustrating how non-Hermiticity and topology shape transport in superconducting junctions. Overall, the work provides new insights into Majorana physics, exceptional points, and transport in non-Hermitian Josephson devices, suggesting experimental signatures in spectroscopic measurements.

Abstract

We investigate the spectrum of Andreev bound states and supercurrent in a $p$-wave non-Hermitian Josephson junction (NHJJ) in one dimension. The studied NHJJ is composed of two topological $p$-wave superconductors connected by a non-Hermitian dissipative junction. Starting from the effective non-Hermitian Bogoliubov-de Gennes bulk Hamiltonian, we find that a pair of exceptional points emerge in the complex spectrum of Andreev quasi-bound states. The two exceptional points with zero energy locate symmetrically with respect to Josephson phase difference $φ=π$, at which a Majorana zero mode persists. Notably, the exceptional points descend from a pair of Majorana zero modes after turning on the non-Hermiticity and are topologically protected. By analyzing the non-Hermitian scattering process at the junction, we explicitly demonstrate the loss of quasiparticles through the decay of scattering amplitude probabilities. Furthermore, we obtain the supercurrent directly by the inelastic Andreev reflection amplitudes, which provides a more intuitive interpretation of transport properties in NHJJs. The supercurrent varies continuously as a function of $φ$ across the exceptional points. No enhancement of critical current is observed. We also generalize our analysis to a mixed $s$-$p$ wave NHJJ. Our results provide new insights on transport properties of Josephson junctions in presence of Majorana zero modes, exceptional points, and non-Hermiticity.

Figures (4)

• Figure 1: (a) Sketch of a 1D $p$-wave NHJJ: two triplet $p$-wave superconductors are connected by a short junction, which is coupled to the environment. The two superconductors possess a phase difference $\phi=\phi_{2}-\phi_{1}$. (b) Andreev spectrum as a function of $\phi$ corresponding to the setup in (a). A pair of exceptional points appear, implying unique non-Hermitian spectral features. We choose $Z=0.2$ for the non-Hermitian barrier strength.
• Figure 2: (a) Scattering probabilities $|A_{\eta s}|^{2}$ as a function of $Z$ with $\eta=e/h$ and $s=\pm$. $R_{A}(R)$ stands for Andreev (normal) reflection, and $T_{A}(T)$ stands for Andreev (normal) tunneling. We choose $\phi=\pi/2$. (b) Scattering probabilities as a function of $\phi$ for fixed $Z=0.2$. Other parameters are chosen as $\Delta=0.2$ and $E=0.25$.
• Figure 3: (a) Current-phase relation for different $Z$ with temperature $T\rightarrow0$ ($\beta=1/k_{B}T\rightarrow\infty$). The solid lines are from Eq. \ref{['eq:currentBefore']} away from EPs, and dashed lines are from Eq. \ref{['eq:currentAfter']} between EPs. (b) Current-phase relation of the Hermitian limit with $Z=0$ at different temperatures. (c) Current-phase relation with $Z=0.2$ at different temperatures. (d) The critical supercurrent $I_{c}$ as a function of parameter $Z$ at different temperatures.
• Figure 4: (a) Sketch of a 1D mixed $s$-$p$ wave NHJJ: the left superconductor is of $s$-wave paring and the right superconductor is of $p$-wave paring. The two superconductors possess a phase difference $\phi=\phi_{2}-\phi_{1}$. (b) Andreev spectrum as a function of $\phi$ corresponding to setup in (a). Four pairs of exceptional points appear, implying unique non-Hermitian spectral features. Solid lines are for the real energy and dashed lines are for the corresponding imaginary parts. Red and blue colors are for different pairs of Andreev spectra. We choose $Z=0.2$ in this plot.