  Lattice generalizations of fractional quantum Hall (FQH) systems, called
fractional Chern insulators (FCIs), have been extensively investigated in
strongly correlated systems. Despite many efforts, previous studies have not
revealed all of the guiding principles for the FCI search. In this paper, we
investigate a relationship between the topological band structure in the
two-particle problem and the FCI ground states in the many-body problem. We
first formulate the two-particle problem of a bosonic on-site interaction
projected onto the lowest band of a given tight-binding Hamiltonian. We
introduce a reduced Hamiltonian whose eigenvalues correspond to the
two-particle bound-state energies. By using the reduced Hamiltonian, we define
the two-particle Chern number and numerically check the bulk-boundary
correspondence that is predicted by the two-particle Chern number. We then
propose that a nontrivial two-particle Chern number of dominant bands roughly
indicates the presence of bosonic FCI ground states at filling factor
$\nu=1/2$. We numerically investigate this relationship in several
tight-binding models with Chern bands and find that it holds well in most of
the cases, albeit two-band models being exceptions. Although the two-particle
topology is neither a necessary nor a sufficient condition for the FCI state as
other indicators in previous studies, our numerical results indicate that the
two-particle topology characterizes the degree of similarity to the FQH
systems.
