Linear independence of nearest-neighbor valence bond states in several two-dimensional lattices

We show for several two-dimensional lattices that the nearest neighbor valence bond states are linearly independent. To do so, we utilize and generalize a method that was recently introduced and applied to the kagome lattice by one of the authors. This method relies on the choice of an appropriate cell for the respective lattice, for which a certain local linear independence property can be demonstrated. Whenever this is achieved, linear independence follows for arbitrarily large lattices that can be covered by such cells, for {\em both} open and periodic boundary conditions. We report that this method is applicable to the kagome, honeycomb, square, squagome, two types of pentagonal, square-octagon, the star lattice, two types of archimedean lattices, three types of "martini" lattices, and to fullerene-type lattices, e.g., the well known "Buckyball". Applications of the linear independence property, such as the derivation of effective quantum dimer models, or the constructions of new solvable spin-1/2 models, are discussed.