Valence Bond States on Quantum Spin Chains as Ground States with Spectral Gap

We show that every translation invariant valence bond state on a one-dimensional quantum spin chain arises as the unique ground state of a certain family of finite range interactions. For each interaction in this family we show the existence of a non-zero spectral gap above the ground state energy. A special example of this structure is a state recently studied by Affleck, Lieb, Kennedy, and Tasaki. For the Hamiltonian studied by these authors we can estimate the gap, and prove that it lies between 1/3 and 10/27