Tunable anyonic permeability across ${\mathbb{Z}_2}$ spin liquid junctions

We introduce two classes of junctions in a toric code, a prototypical model of a $\mathbb{Z}_2$ quantum spin liquid, and study the nature of anyonic transport across them mediated by Zeeman fields. In the first class of junctions, termed potential barrier junctions, the charges sense effective static potentials and a change in the band mass. In a particular realization, while the junction is completely transparent to the electric charge, magnetic charge transmission is allowed only after a critical field strength. In the second class of junctions we stitch two toric codes with operators which do not commute at the junction. We show that the anyonic transmission gets tuned by effective pseudospin fluctuations at the junction. Using exact analytical mappings and numerical simulations, we compute charge-specific transmission probabilities. Our work, apart from uncovering the rich physical mechanisms at play in such junctions, can motivate experimental work to engineer defect structures in topologically ordered systems for tunable transport of anyonic particles.