  We analyze the two-body spectrum within the Hofstadter-Hubbard model on a
square lattice through an exact variational ansatz and study the topological
properties of its low-lying two-body bound-state branches. In particular we
discuss how the Hofstadter-Hubbard butterfly of the two-body branches evolves
as a function of onsite interactions and how to efficiently calculate their
Chern numbers using the Fukui-Hatsugai-Suzuki approach. Our numerical results
are fully consistent with the simple picture that appears in the
strong-coupling limit, where the attraction between fermions forms a composite
boson characterized by an effective hopping parameter and an effective
magnetic-flux ratio.
