Ground states with cluster structures in a frustrated Heisenberg chain

We examine the ground state of a Heisenberg model with arbitrary spin S on a one-dimensional lattice composed of diamond-shaped units. A unit includes two types of antiferromagnetic exchange interactions which frustrate each other. The system undergoes phase changes when the ratio $\lambda$ between the exchange parameters varies. In some phases, strong frustration leads to larger local structures or clusters of spins than a dimer. We prove for arbitrary S that there exists a phase with four-spin cluster states, which was previously found numerically for a special value of $\lambda$ in the S=1/2 case. For S=1/2 we show that there are three ground state phases and determine their boundaries.