Flatband generators

TL;DR

This paper develops systematic methods to identify and construct flatband Hamiltonians in 1D, 2D, and non-Hermitian systems, enabling better design and understanding of flatband phenomena in various physical contexts.

Contribution

It introduces a comprehensive flatband generator scheme for different lattice dimensions and types, expanding the toolkit for flatband lattice design and analysis.

Findings

Classified flatband lattices by their compact localized states properties

Generated all possible 1D flatband Hamiltonians with arbitrary bands and CLS sizes

Proposed a 2D flatband generator with CLSs in a 2x2 plaquette

Abstract

Flatbands (FBs) are dispersionless energy bands in the single-particle spectrum of a translational invariant tight-binding network. The FBs occur due to destructive interference, resulting in macroscopically degenerate eigenstates living in a finite number of unit cells, which are called compact localized states (CLSs). Such macroscopic degeneracy is in general highly sensitive to perturbations, such that even slight perturbation lifts the degeneracy and leads to various interesting physical phenomena. In this thesis, we develop an approach to identify and construct FB Hamiltonians in 1D, 2D Hermitian, and 1D non-Hermitian systems. First, we introduce a systematic classification of FB lattices by their CLS properties, and propose a scheme to generate tight-binding Hamiltonians having FBs with given CLS properties---a FB generator. Applying this FB generator to a 1D system, we identify all possible FB Hamiltonians of 1D lattices with arbitrary numbers of bands and CLS sizes. Extending the 1D approach, we establish a FB generator for 2D FB Hamiltonians that have CLSs occupying a maximum of four unit cells in a $2\times2$ plaquette. Employing this approach in the non-Hermitiaon regime, we realize a FB generator for a 1D non-Hermitian lattice with two bands. Ultimately, we apply our methods to propose a tight-binding model that explains the spectral properties of a microwave photonic crystal. Our results and methods in this thesis further our understanding of the properties of FB lattices and their CLSs, provide more flexibility to design FB lattices in experiments, and open new avenues for future research.