  Topological phase transitions in free fermion systems can be characterized by
closing of single-particle gap and change in topological invariants. However,
in the presence of electronic interactions, topological phase transitions are
more complicated. In paper I of this series (arXiv:1510.07816), we have
developed an efficient scheme to evaluate the topological invariants based on
Green's function formalism. Here, in paper II, we demonstrate four
interaction-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 invariants
presented in paper I. Across these transitions, the defining symmetries of the
TIs have been neither explicitly nor spontaneously broken. In the first two
models, the topological invariants calculated from Green's function formalism
succeed in characterizing interaction-driven TPTs. However, in the second two
models, we find single-particle gap does not close and the topological
invariants constructed from single-particle Green's function acquire no change
across the TPTs. Unexpected breakdown of the Green's function formalism in
constructing topological invariants is thus discovered. We thence classify the
TPTs in interacting TIs into two categories: those have noninteracting
correspondence can be characterized successfully by the topological invariants
constructed from Green's functions, while for the others that do not have
noninteracting correspondence, the Green's function formalism experiences a
breakdown but more interesting and exciting phenomena, such as emergent
collective critical modes at the transition, arise. Discussion on the success
and breakdown of topological invariants constructed from the Green's function
formalism in the context of symmetry protected topological (SPT) states is
presented.
