Valence bond states on quantum spin chains as ground states with spectral gap

The authors 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 they show the existence of a nonzero spectral gap above the ground state energy. A special example of this structure is a state recently studied by Affleck et al. (1987). For the Hamiltonian studied by these authors they estimate the gap, and prove that it lies between 1/3 and 10/27.