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Topological phase transitions in free fermion systems can be characterized byclosing of single-particle gap and change in topological invariants. However,in the presence of electronic interactions, topological phase transitions aremore complicated. In paper I of this series (arXiv:1510.07816), we havedeveloped an efficient scheme to evaluate the topological invariants based onGreen's function formalism. Here, in paper II, we demonstrate fourinteraction-drive topological phase transitions (TPTs) in two-dimensional (2D)interacting topological insulators (TIs) via large-scale quantum Monte Carlo(QMC) simulations, based on the scheme of evaluating topological invariantspresented in paper I. Across these transitions, the defining symmetries of theTIs have been neither explicitly nor spontaneously broken. In the first twomodels, the topological invariants calculated from Green's function formalismsucceed in characterizing interaction-driven TPTs. However, in the second twomodels, we find single-particle gap does not close and the topologicalinvariants constructed from single-particle Green's function acquire no changeacross the TPTs. Unexpected breakdown of the Green's function formalism inconstructing topological invariants is thus discovered. We thence classify theTPTs in interacting TIs into two categories: those have noninteractingcorrespondence can be characterized successfully by the topological invariantsconstructed from Green's functions, while for the others that do not havenoninteracting correspondence, the Green's function formalism experiences abreakdown but more interesting and exciting phenomena, such as emergentcollective critical modes at the transition, arise. Discussion on the successand breakdown of topological invariants constructed from the Green's functionformalism in the context of symmetry protected topological (SPT) states ispresented.

Topological invariants for interacting topological insulators: II. Breakdown of the Green's function formalism