  Recently, two-dimensional band insulators with a topologically nontrivial
(almost) flat band has been studied extensively, which can realize integer and
fractional quantum Hall effect in a system without an orbital magnetic field.
Realizing a topological flat band generally requires longer range hoppings in a
lattice Hamiltonian. It is natural to ask what is the minimal hopping range
required.% for a topological flat-band Hamiltonian. In this paper, we prove
that the mean hopping range of the flat-band Hamiltonian with Chern number
$C_1$ and total number of bands $N$ has a universal lower bound of
$\sqrt{4|C_1|/\pi N}$. Furthermore, for the Hamiltonians that reach this lower
bound, the Bloch wavefunctions of the topological flat band are instanton
solutions of a $CP(N-1)$ non-linear $\sigma$ model on the Brillouin zone torus,
which are elliptic functions up to a normalization factor.
