Notes on analytical study of holographic superconductors with Lifshitz scaling in external magnetic field

We employ the matching method to analytically investigate the holographic superconductors with Lifshitz scaling in an external magnetic field. We discuss systematically the restricted conditions for the matching method and find that this analytic method is not always powerful to explore the effect of external magnetic field on the holographic superconductors unless the matching point is chosen in an appropriate range and the dynamical exponent $z$ satisfies the relation $z=d-1$ or $z=d-2$. From the analytic treatment, we observe that Lifshitz scaling can hinder the condensation to be formed, which can be used to back up the numerical results. Moreover, we study the effect of Lifshitz scaling on the upper critical magnetic field and reproduce the well-known relation obtained from Ginzburg-Landau theory.

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