Synthetic complex Weyl superconductors, chiral Josephson effect and synthetic half-vortices

We show that the most generic form of spin-singlet superconducting order parameter for chiral fermions in systems with broken time reversal symmetry and inversion symmetry is of the [Formula: see text] where [Formula: see text] is the usual order parameter and [Formula: see text] is the pseudo-scalar order parameter. After factoring out the U(1) phase [Formula: see text], this form of superconductivity admits yet additional complex structure in the plane of [Formula: see text]. The polar angle [Formula: see text] in this plane, which we call the chiral angle, can be controlled by the external flux bias. We present a synthetic setup based on stacking of topological insulators (TIs) and superconductors (SCs). Alternatively flux biasing the superconductors with a fluxes [Formula: see text] leads to [Formula: see text], where [Formula: see text] is the superconducting order parameter of the SC layers, and the chiral angle [Formula: see text] is directly given by the flux [Formula: see text] in units of the flux quantum [Formula: see text]. This can be used as a building block to construct a two-dimensional Josephson array. In this setup [Formula: see text] will be a background field defining a pseudoscalar [Formula: see text] that can be tuned to desired configuration. While in a uniform background field [Formula: see text] the dynamics of [Formula: see text] is given by standard XY model and its associated vortices, a staggered background [Formula: see text] (or equivalently [Formula: see text] and [Formula: see text] in alternating lattice sites) creates a new set of minima for the [Formula: see text] field that will support half-vortex excitations. An isolated single engineered "half-vortex" in the [Formula: see text] field in an otherwise uniform background will bind a [Formula: see text]-half-vortex. This is similar to the way a p-wave superconducting vortex core binds a Majorana fermion.