Product vacua with boundary states

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 Affleck-Kennedy-Lieb-Tasaki (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 $J\ge 1$, that are related to a recently introduced class of $\text{SO}(2J+1)$-invariant quantum spin chains.