A simple method to construct Flat Band lattices

Abstract

We develop a simple and general method to construct arbitrary flat-band lattices. We identify the basic ingredients behind zero-dispersion bands and develop a method to construct extended lattices based on a consecutive repetition of a given miniarray. The number of degenerated localized states is defined by the number of connected miniarrays times the number of modes preserving the symmetry at a given connector site. In this way, we create one or more (depending on the lattice geometry) complete degenerated flat bands for quasi-one-and two-dimensional systems. We probe our method by studying several examples and discuss the effect of additional interactions such as anisotropy or nonlinearity. We test our method by studying numerically a ribbon lattice using a continuous description