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True to their unconventional nature, multi-band alkaline Fe-selenides and,more recently, the heavy-fermion CeCu$_{2}$Si$_{2}$ have shown signatures offully-gapped but sign-changing superconductivity (SC). A two-orbital pairingstate, called $s\tau_{3}$, with non-trivial matrix structure, was proposed as acandidate able to reconcile the seemingly contradictory properties of theseSC's. Motivated by the non-trivial orbital structure of the proposed$s\tau_{3}$ state, which has orbital-selective pairing structure, we studyprototypical Josephson junctions where at least one of the leads is in a SCstate of this kind. An analysis of these junctions in the limit of twodegenerate orbitals (bands) and with a simple form of junction hybridizationreveals several remarkable properties. One is the emergence of gapless, purelyelectron- and hole-like bound states for $s\tau_{3}-N-s\tau_{3}$ junctions witharbitrary global phase difference between the leads, and likewise for$s\tau_{3}-N-I$ junctions. The other is the absence of static Josephsoncurrents when both leads are SC's. In both of these signatures, $s\tau_{3}$junctions are dramatically different from conventional Josephson junctions. Wealso find that the gapless bound states are protected by an orbital-exchangesymmetry, although the protection is not topological. Junctions which breakthis symmetry, such as $s\tau_{3}-N-s$, have gapped Andreev bound states. Ingeneral, the Josephson effect also re-emerges once the degeneracy of the twoorbitals is lifted. We support these conclusions via analytical and numericalresults for the bound states, together with microscopic calculations of theJosephson current. Our results indicate that junctions involving $s\tau_{3}$pairing in alkaline Fe-selenides's will generically have bound states with asmall gap together with a greatly suppressed Josephson current.

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