Analytic Study of First-Order Phase Transition in Holographic Superconductor and Superfluid

We use the matching method to investigate the first-order phase transition in holographic superconductor and superfluid. We first use the simple holographic superconductor model to describe the matching method and mention how to see the first-order phase transition. Next, we study the holographic superconductor with St\"uckelberg term and see that the analytic results indicate the existence of first-order phase transition. Finally, we study the holographic superfluid and show that the first-order phase transition can be found for some values of parameters. We determine the critical value analytically and compare it with the previous numerical result.