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

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.