Half-vortices, confinement transition and chiral Josephson effect in superconducting Weyl/Dirac semimetals

We start by showing that the most generic spin-singlet pairing in a superconducting Weyl/Dirac semimetal is specified by a $U(1)$ phase $e^{i\phi}$ and $two~real~numbers$ $(\Delta_s,\Delta_5)$ that form a representation of complex algebra. Such a "complex" superconducting state realizes a $Z_2\times U(1)$ symmetry breaking in the matter sector where $Z_2$ is associated with the chirality. The resulting effective XY theory of the fluctuations of the $U(1)$ phase $\phi$ will be now augmented by coupling to another dynamical variable, the $chiral~angle$ $\chi$ that defines the polar angle of the complex number $(\Delta_s,\Delta_5)$. We compute this coupling by considering a Josephson set up. Our energy functional of two phase variables $\phi$ and $\chi$ allows for the realization of a half-vortex (or double Cooper pair) state and its BKT transition. The half-vortex state is sharply characterized by a flux quantum which is half of the ordinary superconductors. Such a $\pi$-periodic Josephson effect can be easily detected as doubled ac Josephson frequency. We further show that the Josephson current $I$ is always accompanied by a $chiral~Josephson~current$ $I_5$. Strain pseudo gauge fields that couple to the $\chi$, destabilize the half-vortex state. We argue that our "complex superconductor" realizes an extension of XY model that supports confinement transition from half-vortex to full vortex excitations.