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Recently, two‐dimensional band insulators with a topologically nontrivial (almost) flat band in which integer and fractional quantum Hall effect can be realized without an orbital magnetic field have been studied extensively. 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. In this letter, 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\vertC_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  σ  model on the Brillouin zone torus, which are elliptic functions up to a normalization factor. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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