Abstract
We present an anomalous Josephson effect in a TI-based Josephson junction with a ferromagnetic insulator (FI) trilayer which has noncoplanar magnetizations. It is shown that there exist equal spin-singlet and -triplet correlations due to the magnetism-tuning chiral Dirac energy band structure combined with the spin precession and selective equal spin Andreev reflection by chiral Majorana modes (CMMs). The consequent anomalous Josephson supercurrent is exhibited, in which a 0-π or similar 0 - π state transition through phase shift is induced only by exchange field strengths of the first FI region, while the ϕ0 supercurrent and the maximum one gradually drop with the increase of exchange field strengths of the second and third FI regions without relative state transitions. The much different features are found by varying the lengths of trilayer. The Andreev bound states without hybridization for the CMM administrate these features, which could be used to probe and confirm the zero energy CMM. In addition, the corresponding free energies are presented and discussed.