We introduce a family of quantum spin chains with nearest-neighbor
interactions that can serve to clarify and refine the classification of gapped
quantum phases of such systems. The gapped ground states of these models can be
described as a product vacuum with a finite number of particles bound to the
edges. The numbers of particles, n_L and n_R, that can bind to the left and
right edges of the finite chains serve as indices of the particular phase a
model belongs to. All these ground states, which we call Product Vacua with
Boundary States (PVBS) can be described as Matrix Product States (MPS). We
present a curve of gapped Hamiltonians connecting the AKLT model to its
representative PVBS model, which has indices n_L=n_R=1. We also present
examples with n_L=n_R=J, for any integer J\geq 1, that are related to a
recently introduced class of SO(2J+1)-invariant quantum spin chains.Comment: New section on SO(2J+1)-invariant model
