  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, $\beta$, is
investigated. Finally the properties of conductivity are numerically explored
in our model.
