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Zeroth Order Phase Transition in a Holographic Superconductor with Single Impurity

We investigate the single normal impurity effect in a superconductor by the holographic method. When the size of impurity is much smaller than the host superconductor, we can reproduce the Anderson theorem, which states that a conventional s-wave superconductor is robust to a normal (non-magnetic) impurity with small impurity strength. However, by increasing the size of the impurity in a fixed-size host superconductor, we find a decreasing critical temperature $T_c$ of the host superconductor, which agrees with the results in condensed matter literatures. More importantly, the phase transition at the critical impurity strength (or the critical temperature) is of zeroth order.