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We consider a superconductor-barrier-superconductor (S-B-S) sandwichconfiguration built with Rarita-Schwinger-Weyl semimetal featuring four bandcrossings at a single nodal point. Assuming a homogenous s-wave pairing in eachsuperconducting region, and the barrier region created by applying a voltage ofmagnitude $V_0 $ across a piece of normal state semimetal, we apply the BdGformalism to compute the discreet energy spectrum $\varepsilon $ of the subgapAndreev bound states in the short-barrier regime. In contrast with the two-bandsemimetals studied earlier, we find upto four pairs of localized states (ratherthan one pair for two-band semimetals) in the thin-barrier limit, and eachvalue of $\varepsilon $ has a complicated dependence on the phase difference$\varphi_{12} $ via cosine and sine functions, which cannot be determinedanalytically. These are artifacts of multiple band crossings hostingquasiparticles of pseudospin value greater than $1/2$. Using the bound stateenergies, we compute the Josephson current across the junction configuration.

Andreev bound states in superconductor-barrier-superconductor junctions of Rarita-Schwinger-Weyl semimetals