Spin-triplet superconductor$-$quantum anomalous Hall insulator$-$spin-triplet superconductor Josephson junctions: $0$-$\pi$ transition, $\phi_{0}$ phase and switch effects

We study the Josephson effect in spin-triplet superconductor$-$quantum anomalous Hall insulator$-$spin-triplet superconductor junctions using the nonequilibrium Green function method. The current-phase difference relations show strong dependence on the orientations of the $\bf{d}$-vectors in superconductors. We focus on two $\bf{d}$-vector configurations, the parallel one with the left and right ${\bf{d}}$-vectors being in the same direction, and the nonparallel one with the left ${\bf{d}}$-vector fixed at the $z$-axis. For the parallel configuration, the $0$-$\pi$ transition can be realized when one rotates the ${\bf{d}}$-vectors from the parallel to the junction plane to the perpendicular direction. The $\phi_{0}$ phase with nonzero Josephson current at zero phase difference can be obtained as long as ${d_{x}}{d_{z}}\ne0$. For the nonparallel configuration, the $0$-$\pi$ transition and the $\phi_{0}$ phase still exist. The condition for the formation of the $\phi_{0}$ phase becomes $d_{Rx}\ne0$. The switch effects of the Josephson current are found in both configurations when the ${\bf{d}}$-vectors are rotated in the $xy$ plane. Furthermore, the symmetries satisfied by the current-phase difference relations are analysed in details by the operations of the time-reversal, mirror-reflections, the spin-rotation and the gauge transformation, which can well explain the above selection rules for the $\phi_{0}$ phase. Our results reveal the peculiar Josephson effect between spin-triplet superconductors and the quantum anomalous Hall insulator, which provide helpful phases and effects for the device designs. The distinct current-phase difference relations for different orientations may be used to determine the direction of the ${\bf{d}}$-vector in the spin-triplet superconductor.