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

Abstract

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.

3 pages, 1 figure

References in corpus (12)

• High temperature fractional quantum Hall states
• Fractional quantum Hall states at zero magnetic field
• Nearly-flat bands with nontrivial topology
• Fractional quantum Hall effect in the absence of Landau levels
• Fractional Chern Insulator
• Fractional Quantum Hall Effect of Hard-Core Bosons in Topological Flat Bands
• Generic Wavefunction Description of Fractional Quantum Anomalous Hall States and Fractional Topological Insulators
• Emergent Many-Body Translational Symmetries of Abelian and Non-Abelian Fractionally Filled Topological Insulators
• Fractional Chern Insulators and the W-Infinity Algebra
• Non-Abelian Quantum Hall Effect in Topological Flat Bands
• Topological Hubbard model and its high-temperature quantum Hall effect
• Quantum phases of disordered flatband lattice fractional quantum Hall systems

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• Symmetries as the guiding principle for flattening bands of Dirac fermions
• Origin of band flatness and constraints of higher Chern numbers
• Theory of topological exciton insulators and condensates in flat Chern bands
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