Compact localized states and flatband generators in one dimension

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. We classify FB networks through the properties of compact localized states (CLS) which are exact FB eigenstates and occupy $U$ unit cells. We obtain necessary and sufficient conditions for a network to be of FB class $U$. These conditions are turned into a simple local FB testing routine which avoids Bloch based band structure calculations. The tester in turn is used to introduce a novel FB generator based on local algebraic network properties. We obtain the complete FB family of two-band networks in one dimension with nearest unit cell interaction, for which $U \leq 2$. We find that the CLS set is generically linearly independent and spans the complete FB Hilbert space. With the CLS structure we obtain the Bloch polarization vectors of the FB.