Methods for constructing parameter-dependent flat band lattices

TL;DR

This paper introduces two novel methods for constructing flat band systems in tight-binding models, enabling the design of lattices with flat energy dispersions through graph-based conditions and site modification techniques.

Contribution

It provides new systematic approaches for creating flat band lattices, including parameter tuning and site addition/removal rules, expanding the possibilities beyond existing graph-based methods.

Findings

Derived conditions for flat band appearance in graphs

Constructed multiple flat band systems from a single lattice

Demonstrated lattice construction starting from kagome lattice

Abstract

We present two methods for constructing a flat band (FB) system having a flat energy dispersion over the entire Brillouin zone within tight-binding model, where the resulting Hamiltonian may not be easily obtained by existing methods based on a bipartite graph and line graph techniques. In the first method, we derive a set of conditions equivalent to the appearance of FBs for a given graph structure. This method allows parameter to be tuned so that systems with a small number of sites per unit cell has a FB. In the second method, we show that FB systems can be obtained by removing or adding sites to an existing FB system under specific rules. In particular, the site addition method enables us to construct multiple FB systems stemming from a single FB system. The FB system obtained by the second method has the characteristics that the component ratios in the FB eigenstate are partially common to the original system. We illustrate how lattices having a FB can be constructed by applying the latter method starting from an existing lattice such as a kagome lattice, demonstrating that a wide variety of lattices can possess a FB in the band structure.