  We consider the two-body problem in a periodic potential, and study the
bound-state dispersion of a spin-$\uparrow$ fermion that is interacting with a
spin-$\downarrow$ fermion through a short-range attractive interaction. Based
on a variational approach, we obtain the exact solution of the dispersion in
the form of a set of self-consistency equations, and apply it to tight-binding
Hamiltonians with onsite interactions. We pay special attention to the
bipartite lattices with a two-point basis that exhibit time-reversal symmetry,
and show that the lowest-energy bound states disperse quadratically with
momentum, whose effective-mass tensor is partially controlled by the quantum
metric tensor of the underlying Bloch states. In particular, we apply our
theory to the Mielke checkerboard lattice, and study the special role played by
the interband processes in producing a finite effective mass for the bound
states in a non-isolated flat band.
