Tunable $\varphi$-Josephson junction with a quantum anomalous Hall insulator

We theoretically study the Josephson current in a superconductor/quantum anomalous Hall insulator/superconductor junction by using the lattice Green function technique. When an in-plane external Zeeman field is applied to the quantum anomalous Hall insulator, the Josephson current $J$ flows without a phase difference across the junction $\theta$. The phase shift $\varphi$ appealing in the current-phase relationship $J\propto \sin(\theta-\varphi$) is proportional to the amplitude of Zeeman fields and depends on the direction of Zeeman fields. A phenomenological analysis of the Andreev reflection processes explains the physical origin of $\varphi$. A quantum anomalous Hall insulator breaks time-reversal symmetry and mirror reflection symmetry simultaneously. However it preserves magnetic mirror reflection symmetry. Such characteristic symmetry property enable us to have a tunable $\varphi$-junction with a quantum Hall insulator.