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

Abstract

AbstractWe 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 $$\Delta _s+i\gamma ^5\Delta _5$$ Δ s + i γ 5 Δ 5 where $$\Delta _s$$ Δ s is the usual order parameter and $$\Delta _5$$ Δ 5 is the pseudo-scalar order parameter. After factoring out the U(1) phase $$e^{i\phi }$$ e i ϕ , this form of superconductivity admits yet additional complex structure in the plane of $$(\Delta _s,\Delta _5)$$ ( Δ s , Δ 5 ) . The polar angle $$\chi $$ χ 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 $$\pm \Phi $$ ± Φ leads to $$\Delta _5=\Delta _0 \sin (\chi )$$ Δ 5 = Δ 0 sin ( χ ) , where $$\Delta _0$$ Δ 0 is the superconducting order parameter of the SC layers, and the chiral angle $$\chi ={2\pi }\Phi /\Phi _0$$ χ = 2 π Φ / Φ 0 is directly given by the flux $$\Phi $$ Φ in units of the flux quantum $$\Phi _0=h/(2e)$$ Φ 0 = h / ( 2 e ) . This can be used as a building block to construct a two-dimensional Josephson array. In this setup $$\chi $$ χ will be a background field defining a pseudoscalar $$\Delta _5$$ Δ 5 that can be tuned to desired configuration. While in a uniform background field $$\Delta _5$$ Δ 5 the dynamics of $$\phi $$ ϕ is given by standard XY model and its associated vortices, a staggered background $$\pm \Delta _5$$ ± Δ 5 (or equivalently $$\chi $$ χ and $$\chi +\pi $$ χ + π in alternating lattice sites) creates a new set of minima for the $$\phi $$ ϕ field that will support half-vortex excitations. An isolated single engineered “half-vortex” in the $$\chi $$ χ field in an otherwise uniform background will bind a $$\phi $$ ϕ -half-vortex. This is similar to the way a p-wave superconducting vortex core binds a Majorana fermion.