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

Abstract

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 iφ and two real numbers (∆s, ∆5) that form a representation of complex algebra. Such a "complex" superconducting state realizes a Z2 × U (1) symmetry breaking in the matter sector where Z2 is associated with the chirality. The resulting effective XY theory of the fluctuations of the U (1) phase φ will be now augmented by coupling to another dynamical variable, the chiral angle χ that defines the polar angle of the complex number (∆s, ∆5). We compute this coupling by considering a Josephson set up. Our energy functional of two phase variables φ and χ 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 π-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 I5. Strain pseudo gauge fields that couple to the χ, 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.