Constructing vortex functions and basis states of Chern insulators: ideal condition, inequality from index theorem, and coherent-like states on von Neumann lattice

Abstract

In the field of fractional Chern insulators, a great deal of effort has been devoted to characterizing Chern bands that exhibit properties similar to the Landau levels. Among them, the concept of the vortex function, which generalizes the complex coordinate used for the symmetric-gauge Landau-level basis, allows for a concise description. In this paper, we develop a theory of constructing the vortex function and basis states of Chern insulators in the tight-binding formalism. In the first half, we consider the optimization process of the vortex function, which minimizes an indicator that measures the difference from the ideal Chern insulators. In particular, we focus on the sublattice position dependence of the vortex function or the quantum geometric tensor. This degree of freedom serves as a discrete analog of the non-uniformity in the spatial metric and magnetic field in a continuous model. In the second half, we construct two types of basis sets for a given vortex function: radially localized basis set and coherent-like basis set. The former basis set is defined as the eigenstates of an analogy of the angular momentum operator. Remarkably, one can always find exact zero mode(s) for this operator, which is explained by the celebrated Atiyah-Singer index theorem. As a byproduct, we propose an inequality rooted in the band topology. We also discuss the subtle differences between our formalism and the previous works about the momentum-space Landau level. The latter basis set generalizes the concept of coherent states on von Neumann lattice. While this basis set is not orthogonal, it is useful to compare the LLL and the given Chern insulator directly in the Brillouin zone. These basis sets are expected to be useful for many-body calculations of fractional Chern insulators.

arXiv comment15 pages, 4 figures, corrected several mistakes (mainly sign errors) after publication in PRB. (see red text)

References in corpus (38)

• Topological Insulators
• Topological insulators and superconductors
• Chern Numbers in Discretized Brillouin Zone: Efficient Method of Computing (Spin) Hall Conductances
• Symmetry and Topology in Non-Hermitian Physics
• Fractional quantum Hall states at zero magnetic field
• Nearly-flat bands with nontrivial topology
• Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors
• Signatures of Fractional Quantum Anomalous Hall States in Twisted MoTe2 Bilayer
• Observation of Fractionally Quantized Anomalous Hall Effect
• Fractional Chern Insulator
• Integer and fractional Chern insulators in twisted bilayer MoTe2
• Topological Flat Band Models and Fractional Chern Insulators
• Fractional Quantum Hall Physics in Topological Flat Bands
• Observation of integer and fractional quantum anomalous Hall effects in twisted bilayer MoTe2
• Non-Hermitian topological phenomena: A review
• Band geometry of fractional topological insulators
• Generic Wavefunction Description of Fractional Quantum Anomalous Hall States and Fractional Topological Insulators
• Zoology of Fractional Chern Insulators
• Spontaneous fractional Chern insulators in transition metal dichalcogenides Moire superlattices
• Position-Momentum Duality and Fractional Quantum Hall Effect in Chern Insulators
• Fractional Chern Insulators and the W-Infinity Algebra
• Vortexability: A Unifying Criterion for Ideal Fractional Chern Insulators
• Topological insulators and fractional quantum Hall effect on the ruby lattice
• Geometric stability of topological lattice phases
• Thirty Years of Composite Fermions and Beyond
• Wavefunctionology: The Special Structure of Certain Fractional Quantum Hall Wavefunctions
• Smooth gauge for topological insulators
• Band structure engineering of ideal fractional Chern insulators
• Pseudopotential Formalism for Fractional Chern Insulators
• Engineering geometrically flat Chern bands with Fubini-Study Kähler structure
• Fractional Topological Insulators -- a Pedagogical Review
• Ideal Chern bands are Landau levels in curved space
• Topological Lattice Models with Constant Berry Curvature
• Lattice construction of pseudopotential Hamiltonians for Fractional Chern Insulators
• Recent Developments in Fractional Chern Insulators
• Field Theory on the von Neumann Lattice and the Quantized Hall Conductance of Bloch Electrons
• Crystal-symmetry preserving Wannier states for fractional chern insulators
• A Z${}_2$ index of Dirac operator with time reversal symmetry

Cited by in corpus (4)

• Nonlocal Moments in the Chern Bands of Twisted Bilayer Graphene
• Stability of quasi-particle creation and multiband geometry in fractional Chern insulators under magnetic fields
• Refining Heuristic Predictors of Fractional Chern Insulators using Machine Learning
• Topological two-body interaction obstructing trivial ground states: an indicator of fractional Chern insulators