A holographic model of d-wave superconductor vortices with Lifshitz scaling

Abstract: We study analytically the $d$-wave holographic superconductors with Lifshitz scaling in the presence of external magnetic field. The vortex lattice solutions of the model have also been obtained with different Lifshitz scaling. Our results imply that holographic $d$-wave superconductor is indeed a type II one even for different Lifshitz scaling. This is the same as the conventional $d$-wave superconductors in the Ginzburg-Landau theory. Our results also indicate that the dynamical exponent $z$ has no effect to the shape of the vortex lattice even after higher order corrections (away from the phase transition point $B_c$) are included. However, it has effects on the upper critical magnetic field $B_{c_2}$ through the fact that a larger $z$ results in a smaller $B_{c_2}$ and therefore influences the size (characterized by $r_0\equiv 1/\sqrt{B_{c_2}}$) of the vortex lattices. Furthermore, close comparisons between our results and those of the Ginzburg-Landau theory reveal the fact that the upper critical magnetic field $B_{c_2}$ is inversely proportional to the square of the superconducting coherence length $\xi$, regardless of the anisotropy between space and time.