Non-Abelian Landau-Ginzburg Theory of Ferromagnetic Superconductivity and Photon-Spinon Mixing

We propose an effective theory of non-Abelian superconductivity, an SU(2)xU(1) extension of the Abelian Landau-Ginzburg theory, which could be viewed as an effective theory of ferromagnetic superconductivity made of spin-up and spin-down doublet Cooper pair. Just like the Abelian Landau-Ginzburg theory it has the U(1) electromagnetic interaction, but the new ingredient is the non-Abelian SU(2) gauge interaction between the spin doublet Cooper pair. A remarkable feature of the theory is the mixing between the photon and the diagonal part of the SU(2) gauge boson. After the mixing it has massless gauge boson (the massless non-Abelian spinon) and massive gauge boson (the massive photon), in addition to the massive off-diagonal gauge bosons (the massive non-Abelian spinons) which induces the spin-flip interaction between the spin up and down components of the Cooper pair. So, unlike the ordinary Landau-Ginzburg theory it has a long range interaction mediated by the massless non-Abelian spinon, which could be responsible for the long range magnetic order and spin waves observed in ferromagnetic superconductors. The theory is characterized by three scales. In addition to the correlation length fixed by the mass of the Higgs field it has two different penetration lengths, the one fixed by the mass of the photon (which generates the well known Meissner effect) and the other fixed by the mass of the off-diagonal spinons (which determines the scale of the spin flip interaction). The non-Abelian structure of the theory naturally accommodates new topological objects, the non-Abrikosov quantized spin vortex (as well as the well known Abrikosov vortex) and non-Abelian spin monopole. We discuss the physical implications of the non-Abelian Landau-Ginzburg theory.