Product vacua with boundary states

Abstract

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