# Analytical study of quasi-one dimensional flat band networks and slow light analogue

## TL;DR

This paper presents an analytical approach to understanding flat band states in quasi-one-dimensional structures, revealing their localization properties and potential for slow light applications, supported by numerical and experimental discussions.

## Contribution

It introduces an exact analytical method within the tight-binding framework to identify and analyze flat, non-dispersive eigenstates and their localization in quasi-one-dimensional networks.

## Key findings

- Identification of localized flat band states in specific lattice structures
- Analytical scheme for characterizing dispersionless modes using real space renormalization
- Discussion of slow light phenomena and mode switching in photonic analogues

## Abstract

Exact method of analytical solution of flat, non-dispersive eigenstates in a class of quasi-one dimensional structures is reported within the tight-binding framework. The states are localized over certain sublattice sites. One such finite size cluster of atomic sites is decoupled from the rest of the system by the special non-permissible vertex having zero amplitude. This immediately leads to the self-trapping of the incoming excitation. We work out an analytical scheme to discern the localizing character of the diffraction free dispersionless modes using real space renormalization group technique. Supportive numerical calculations of spectral profile and transport are demonstrated to substantiate the essence of compact localized states. Possible experimental scope regarding the photonic analogue of the tight-binding electronic case is also discussed elaborately. This eventually unfolds the concepts of slow light and the related re-entrant mode switching from the study of optical dispersion.