Exceptional Andreev spectrum and supercurrent in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -wave non-Hermitian Josephson junctions

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 quasibound states. The two exceptional points with zero energy locate symmetrically with respect to the Josephson phase difference $\ensuremath{\phi}=\ensuremath{\pi}$, 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 the transport properties in NHJJs. The supercurrent varies continuously as a function of $\ensuremath{\phi}$ across the exceptional points. No enhancement of critical current is observed. We also generalize our analysis to a mixed $s\text{\ensuremath{-}}p$ wave NHJJ. Our results provide rich insights on the transport properties of Josephson junctions in the presence of Majorana zero modes, exceptional points, and non-Hermiticity.