Matrix-pairing states in the alkaline Fe-selenide superconductors: exotic Josephson junctions

True to their unconventional nature, multi-band alkaline Fe-selenides and, more recently, the heavy-fermion CeCu$_{2}$Si$_{2}$ have shown signatures of fully-gapped but sign-changing superconductivity (SC). A two-orbital pairing state, called $stau_{3}$, with non-trivial matrix structure, was proposed as a candidate able to reconcile the seemingly contradictory properties of these SCs. Motivated by the non-trivial orbital structure of the proposed $stau_{3}$ state, which has orbital-selective pairing structure, we study prototypical Josephson junctions where at least one of the leads is in a SC state of this kind. An analysis of these junctions in the limit of two degenerate orbitals (bands) and with a simple form of junction hybridization reveals several remarkable properties. One is the emergence of gapless, purely electron- and hole-like bound states for $stau_{3}-N-stau_{3}$ junctions with arbitrary global phase difference between the leads, and likewise for $stau_{3}-N-I$ junctions. The other is the absence of static Josephson currents when both leads are SCs. In both of these signatures, $stau_{3}$ junctions are dramatically different from conventional Josephson junctions. We also find that the gapless bound states are protected by an orbital-exchange symmetry, although the protection is not topological. Junctions which break this symmetry, such as $stau_{3}-N-s$, have gapped Andreev bound states. In general, the Josephson effect also re-emerges once the degeneracy of the two orbitals is lifted. We support these conclusions via analytical and numerical results for the bound states, together with microscopic calculations of the Josephson current. Our results indicate that junctions involving $stau_{3}$ pairing in alkaline Fe-selenidess will generically have bound states with a small gap together with a greatly suppressed Josephson current.