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

Abstract

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 $e^{i\varphi }$, 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 $\varphi$. 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 $\varphi$ 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 $\varphi$ 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 $\varphi$-half-vortex. This is similar to the way a p-wave superconducting vortex core binds a Majorana fermion.