Compact localized states and flatband generators in one dimension

TL;DR

This paper introduces a new method to generate flat band Hamiltonians in one-dimensional networks using local properties, classifies these networks via compact localized states, and discovers a novel high-symmetry sawtooth chain.

Contribution

It provides a comprehensive flat band generator based on local network properties and classifies flat band networks through compact localized states, including a new high-symmetry chain.

Findings

Complete two-parameter flat band family for two-band 1D networks with U=2.

Discovery of a high-symmetry sawtooth chain with identical hoppings.

Framework for extending flat band analysis to higher dimensions.

Abstract

Flat bands (FB) are strictly dispersionless bands in the Bloch spectrum of a periodic lattice Hamiltonian, recently observed in a variety of photonic and dissipative condensate networks. FB Hamiltonians are finetuned networks, still lacking a comprehensive generating principle. We introduce a FB generator based on local network properties. We classify FB networks through the properties of compact localized states (CLS) which are exact FB eigenstates and occupy $U$ unit cells. We obtain the complete two-parameter FB family of two-band $d=1$ networks with nearest unit cell interaction and $U=2$. We discover a novel high symmetry sawtooth chain with identical hoppings in a transverse dc field, easily accessible in experiments. Our results pave the way towards a complete description of FBs in networks with more bands and in higher dimensions.