Analytical coexistence of s, p, s + p phases of a holographic superconductor

We analytically study the critical phase of a mixed system of the U(2) gauge fields and global symmetry on the boundary using gauge/gravity. A variational minimization problem has been formulated. The numerical results pertinently show that there exists a minimum chemical potential in which both scalar (s-wave) and vector (p-wave) condensates exist in a mixture, as well as in the distinct s- and p-phases. This result is obtained by breaking the symmetry into U(1) symmetry and rotational symmetry. While the analytical solutions of condensates and charge densities are achieved in both cases: the balanced and unbalanced holographic superconductors. This is the first an- alytical study of the coexistence of two modes of the superconductivity with different order parameters. The realistic model consists of two different phases of the superfluidity in Helium.