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We consider an annular superconductor-insulator-superconductorJosephson-junction, with the insulator being a double layer of electron andholes at Abelian fractional quantum Hall states of identical fillings. When thetwo superconductors gap out the edge modes, the system has a topological groundstate degeneracy in the thermodynamic limit akin to the fractional quantum Halldegeneracy on a torus. In the quasi-one-dimensional limit, where the width ofthe insulator becomes small, the ground state energies are split. We discussseveral implications of the topological degeneracy that survive the crossoverto the quasi-one-dimensional limit. In particular, the Josephson effect shows a$2\pi d$-periodicity, where $d$ is the ground state degeneracy in the 2dimensional limit. We find that at special values of the relative phase betweenthe two superconductors there are protected crossing points in which thedegeneracy is not completely lifted. These features occur also if the insulatoris a time-reversal-invariant fractional topological insulator. We describe thelatter using a construction based on coupled wires. Furthermore, when thesuperconductors are replaced by systems with an appropriate magnetic order thatgap the edges via a spin-flipping backscattering, the Josephson effect isreplaced by a spin Josephson effect.

Imprint of topological degeneracy in quasi-one-dimensional fractional quantum Hall states