Momentum-space instantons and maximally localized flat-band topological Hamiltonians

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.