| Summary: | We study the phase transitions in the metal/superconductor system using topological invariants of the Ryu-Takayanagi (RT) surface and the volume enclosed by the RT surface in the Lifshitz black hole background. It is shown that these topological invariant quantities identify not only the phase transition but also its order. According to these findings a discontinuity slope is observed at the critical points for these invariant quantities that correspond to the second order of phase transition. These topological invariants provide a clearer illustration of the superconductor phase transition than do the holographic entanglement entropy and the holographic complexity. Also, the backreaction parameter, k, is found to have an important role in distinguishing the critical points. The reducing values of the parameter k means that the backreaction of the matter fields are negligible. A continuous slope is observed around the critical points which is characteristic of the probe limit. In addition, exploring the nonlinear electrodynamic, the effects of the nonlinear parameter, β, is investigated. Finally the properties of conductivity are numerically explored in our model.
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