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

الأصل · EN

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 C₁ and total number of bands N has a universal lower bound of √4|C₁|/πN. 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.