Controllable phase transitions and novel selection rules in Josephson junctions with inherent orthogonality

We propose a new type of Josephson junction consisting of topologically nontrivial superconductors with inherent orthogonality and a ferromagnetic interface. It is found this type of junction can host rich ground states: 0 phase, π phase, 0+π phase, φ0 phase and φ0±φ phase. Phase transitions can be controlled by changing the direction of the interfacial magnetization. Phase diagrams are presented in the orientation space. Novel selection rules for the lowest order current, sin⁡ϕ or cos⁡ϕ, of this kind of junction are derived. General conditions for the formation of various ground states are established, which possess guiding significance to the experimental design of required ground states for practical applications. We construct the succinct form of a Ginzburg-Landau type of free energy from the viewpoint of the interplay between topological superconductivity and ferromagnetism, which can immediately lead to the selection rules. The constructed terms are universally available to the topological Josephson junctions with or without inherent orthogonality reported recently. The spin supercurrent, its selection rules and their relations to the constructed energy are also investigated.