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

RESUMO

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 $1/3$ 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.