Chirality Josephson current in inversion-asymmetric Weyl semimetals

In this report, I will talk about Josephson junctions based on inversion-asymmetric but time-reversal symmetric Weyl semimetals under the influence of Zeeman fields. Due to distinct spin textures, the Weyl nodes of opposite chirality respond differently to an external magnetic field. Remarkably, a Zeeman field perpendicular to the junction direction results in a phase shift of opposite sign in the current-phase relations of opposite chirality. This leads to a finite chirality Josephson current even in the absence of a phase difference across the junction. In the long junction and zero temperature limit, the chirality Josephson current embodies a novel quantum anomaly at $\pi$ phase difference which is associated with a $\mathbb{Z}_2$ symmetry at low energies. It can be detected experimentally via an anomalous Fraunhofer pattern.