Spontaneous $π$ flux trapping in granular rings of unconventional superconductors

We study Josephson couplings in unconventional superconductors from a symmetry-based perspective, extending the Sigrist-Rice formula to broader classes of pairing symmetries and in the presence of junction interface disorder effects. Applying this formalism to granular superconducting rings, we show that single-band chiral superconductors cannot spontaneously trap magnetic flux, which rules out chiral p-wave pairing as a candidate for $β$-Bi₂Pd superconductor, given the observation of half quantum flux in recent Little-Parks experiments. Taking into account the full crystalline symmetry and time-reversal symmetry of the superconducting state of $β$-Bi₂Pd, we propose that the effective pairing is a triplet helical equal-spin state, enforced by spin-orbit couplings arising from local inversion symmetry breaking. We further demonstrate that granular rings of such helical superconductors can support spontaneous flux trapping, even in the presence of strong junction interface orientation disorder. Our proposal reconciles apparent inconsistencies among different experimental observations and highlights the crucial role of local inversion symmetry breaking in understanding the pairing symmetry in $β$-Bi₂Pd.