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We study the phase transitions in the metal/superconductor system usingtopological invariants of the Ryu-Takayanagi ($RT$) surface and the volumeenclosed by the $RT$ surface in the Lifshitz black hole background. It is shownthat these topological invariant quantities identify not only the phasetransition but also its order. According to these findings a discontinuityslope is observed at the critical points for these invariant quantities thatcorrespond to the second order of phase transition. These topologicalinvariants provide a clearer illustration of the superconductor phasetransition than do the holographic entanglement entropy and the holographiccomplexity. Also, the backreaction parameter, $k$, is found to have animportant role in distinguishing the critical points. The reducing values ofthe parameter $k$ means that the backreaction of the matter fields arenegligible. A continuous slope is observed around the critical points which ischaracteristic of the probe limit. In addition, exploring the nonlinearelectrodynamic, the effects of the nonlinear parameter, $\beta$, isinvestigated. Finally the properties of conductivity are numerically exploredin our model.

Topological invariants of the Ryu-Takayanagi ($RT$) surface used to observe holographic superconductor phase transition