Electrically modulated anomalous phase shift in Andreev bound states mediated by chiral Majorana modes

Chiral Majorana modes (CMMs) have attracted considerable interest due to non-Abelian statistics and potential applications in topological quantum computations, but still lack conclusive experimental evidence. Here, we propose that the nonlocal nature of CMMs leads to another transport signature, an anomalous Josephson effect with a tunable ground-state phase difference modulated by a transverse electric field. The CMMs can mediate nonlocal Andreev reflections from the top edge state to the bottom edge state in the quantum anomalous Hall insulator and vice versa. This nonlocal Andreev reflection leads to the spatial separation of electrons and holes. We refer to such Andreev bound states (ABSs) as nonlocal ABSs. It is shown that such nonlocal ABSs and the corresponding Josephson current can be shifted in the dependence on the superconducting phase difference by a transverse electric field. As a result, an electrically modulated anomalous Josephson effect should be observable in experiments. Moreover, when the Josephson junction is sandwiched between two quantum anomalous Hall insulator leads, the electrically tunable phase shift in ABSs results in a conductance oscillation which is applicable in transistors. These findings provide different proposals to experimentally verify the existence of CMMs, and as well as promise potential applications in phase-controllable Josephson devices and topological transistors.