  Extensive investigations on topological phase transition (TPT) in
three-dimensional compounds have been done. whereas, rare in two-dimensional
systems, let alone noncentrosymmetric materials. In this work, based on
first-principles calculations, we explore an inversion symmetry broken
structural ReC2H monolayer. We reveal that it undergoes two TPTs, namely from
normal insulator to Z2 topological insulator and back to normal insulator, at
the critical biaxial strain of 2.3% and 7.8%, respectively. The band inversion
occurs at the generic momentum in the first TPT, while at high symmetric K
point in the second one. Usually, band inversion is identified by the exchange
in the components or irreducible representations of the wavefunctions. These
quantities can be easily obtained in theoretical calculation but hard to be
detected in experimental techniques like Angle-resolved photoemission
spectroscopy. It is well known that nonlinear optical (NLO) response is very
sensitive to the components and symmetries of the engaged bands, which also
incorporates information of band topology. Therefore, we study the shift
current, one of the widely explored NLO responses in noncentrosymmetric
systems, during the two TPTs. We find that in both cases band inversion leads
to the sign change of shift vectors around the momenta where the bandgap closes
and reopens. Whereas the shift current, as the overall contribution of shift
vectors weighted by the absorption rate in the whole Brillouin zone, may keep
its direction. This work offers insight that a scrutinized examination is
highly demanded in utilizing shift current to detect TPT.
