$1\mathrm{/}3$-Flux Bound States in Multicomponent Superconductor Exhibiting Non-Abelian Statistics of ${\mathbb{Z}}_3$ Parafermions

Multicomponent superconductors (MSCs) are predicted to host magnetic flux quanta carrying arbitrary fractions of the superconducting flux quantum. Recent advances have raised the expectation that $1\mathrm{/}3$ flux quanta may emerge in MSCs with $C_3$ rotational symmetry. Electrons bound to such fractional flux are long regarded to form anyons, yet their explicit braiding statistics remain unexplored. We propose that these 1/3-flux bound states (1/3-FBSs) exhibit the intriguing non-Abelian statistics of ${\mathbb{Z}}_3$ parafermions. Under no-double-occupancy constraint, two successive braiding operations of 1/3-FBSs are equivalent to a single ${\mathbb{Z}}_3$ parafermion braiding operation. The parafermion parity encoding the braiding outcome can be read out via the fermionic occupation number of the 1/3-FBSs. Combined with a crossed-Andreev-reflection-induced Hadamard gate, one can realize the complete set of ${\mathbb{Z}}_3$ parafermion braiding operations.