Band flatness optimization through complex analysis

Abstract

Narrow band electron systems are particularly likely to exhibit correlated many-body phases driven by interaction effects. Examples include magnetic materials, heavy fermion systems, and topological phases such as fractional quantum Hall states and their lattice-based cousins, the fractional Chern insulators (FCIs). Here we discuss the problem of designing models with optimal band flatness, subject to constraints on the range of electron hopping. In particular, we show how the imaginary gap, which serves as a proxy for band flatness, can be optimized by appealing to Rouche's theorem, a familiar result from complex analysis. This leads to an explicit construction which we illustrate through its application to two-band FCI models with nontrivial topology (i.e. nonzero Chern numbers). We show how the imaginary gap perspective leads to an elegant geometric picture of how topological properties can obstruct band flatness in systems with finite ranged hopping.

8 pages, 4 figures

References in corpus (19)

• Topological Field Theory of Time-Reversal Invariant Insulators
• High temperature fractional quantum Hall states
• Fractional quantum Hall states at zero magnetic field
• Nearly-flat bands with nontrivial topology
• Flat bands and Wigner crystallization in the honeycomb optical lattice
• Bosonic Condensation and Disorder-Induced Localization in a Flat Band
• Strongly correlated flat-band systems: The route from Heisenberg spins to Hubbard electrons
• Exotic electronic states in the world of flat bands: from theory to material
• The insulator/Chern-insulator transition in the Haldane model
• Spin Hamiltonian for which the Chiral Spin Liquid is the Exact Ground State
• Position-Momentum Duality and Fractional Quantum Hall Effect in Chern Insulators
• Flat-Band Ferromagnetism as a Pauli-Correlated Percolation Problem
• Scattering matrix invariants of Floquet topological insulators
• Free-fermion Entanglement Spectrum through Wannier Interpolation
• Route to ferromagnetism in organic polymers
• Fractional Chern Insulators in Bands with Zero Berry Curvature
• A simple model for multiple-choice collective decision making
• Search for localized Wannier functions of topological band structures via compressed sensing
• Variational Mote Carlo Study of Flat Band Ferromagnetism -- Application to CeRh_3 B_2

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