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 is of the $\Delta_s+i\gamma^5\Delta_5$ where $\Delta_s$ is the usual order parameter and $\Delta_5$ is the pseudo-scalar order parameter. After factoring out the $U(1)$ phase $e^{i\phi}$, this form of superconductivity admits yet additional complex structure in the plane of $(\Delta_s,\Delta_5)$. The polar angle $\chi$ in this plane dubbed chiral angle will be locked to the $U(1)$ phase $\phi$. We propose a synthetic setup based on stacking of topological insulators (TIs) and superconductors (SCs). Alternatively flux biasing the superconductors with a fluxes $\pm\Phi$ leads to $\Delta_5=\Delta_0 \sin(\chi)$, where $\Delta_0$ is the superconducting order parameter of the SC layers, and the chiral angle $\chi=\Phi/\Phi_0$ is directly given by the flux $\Phi$ in units of the flux quantum $\Phi_0=h/(2e)$. This can be used as a building block to construct a two-dimensional Josephson array. In this setup $\chi$ will be a background field defining a pseudoscalar $\Delta_5$ that can be tuned to desired configuration. While in a uniform background field $\Delta_5$ the dynamics of $\phi$ is given by standard XY model and its associated vortices, a {\em staggered} background $\pm\Delta_5$ (or equivalently $\chi$ and $\chi+\pi$ in alternating lattice sites) creates a new set of minima for the $\phi$ field that will support half-vortex excitations. An isolated single synthetic "half-vortex" in the $\chi$ field in an otherwise uniform background will bind a $\phi$-half-vortex. This is similar to the way a p-wave superconducting vortex core binds a Majorana fermion.