Spontaneous $\pi$ flux trapping in granular rings of unconventional superconductors

Abstract

We study Josephson couplings in unconventional superconductors and generalize the Sigrist-Rice formula by incorporating symmetry constraints and interface orientation disorder. Applying this framework to granular superconducting rings, we establish a no-go result that single-band chiral superconductors cannot spontaneously trap a magnetic flux. This rules out chiral $p$-wave pairing in $\beta$-Bi$_2$Pd, in light of the half-quantum flux observed in recent Little-Parks experiments. Incorporating the full crystalline and time-reversal symmetries, we show that a fully gapped helical equal-spin pairing state, naturally stabilized by spin-orbit coupling arising from local inversion-symmetry breaking, is instead favored. We further find that granular rings of such superconductors can trap a spontaneous $\pi$ flux in a manner robust against interface disorder.