We report a theoretical study on the spin and charge transport in hybrid
triplet Josephson junctions, of which the triplet pair potentials could have both
different orbital symmetries and spin states. Based on a lattice model and a
Hamiltonian method, we find that the spin/charge supercurrent is proportional to
sin2φ with
φ
being the macroscopic superconducting phase, coming from the second-order Josephson
effect, when the orbital symmetries of pair potentials in the two triplet superconductors are
orthogonal to each other. A dissipationless transverse spin current is also found flowing at
the interface of the junction and its polarization points along the cross-product of two
d
vectors, which arises from the combined effect of the orthogonal orbital symmetries and misalignment
of d
vectors. In a special hybrid junction, where the zero-energy states are absent at the interface of
the junction, there is no net spin supercurrent flowing through the junction although two
d
vectors can be perpendicular to each other, whereas a mode-resolved spin supercurrent is
flowing in the system instead.