Topological two-body interaction obstructing trivial ground states: An indicator of fractional Chern insulators

The search for candidate materials for fractional Chern insulators (FCIs) has mainly focused on the topological and geometrical structures of single-particle Chern bands. However, there are inherent limitations in approaches that neglect interaction effects, highlighting the need for complementary methods. In this work, we discuss how the Chern number defined for the effective interaction projected onto a Chern band is related to the stabilization of FCIs. Specifically, by formulating both the effective interaction and the two-particle problem using a common matrix, we establish a connection between the two-particle band structure and the effective interaction. This formulation allows us to characterize the effective interaction through the topology of the two-particle band. To investigate the relationship between topological effective interactions and FCIs, we perform numerical calculations primarily based on exact diagonalization. We find a notable correlation between the fact that the dominant two-particle bands carry a unit Chern number and the realization of a robust FCI at the filling fraction $\ensuremath{\nu}=1/3$. This result is consistent with the presumed correspondence between pseudopotentials in the fractional quantum Hall effect and the two-particle band structure. From another perspective, our findings suggest that the topology inherent in the interaction itself can obstruct trivial ground states. We also discuss this in the context of scattering channels. Extending such topological two-body interactions could pave the way for realizing exotic states beyond FCIs.