Gapped PVBS models for all species numbers and dimensions

Product vacua with boundary states (PVBS) are cousins of the Heisenberg XXZ spin model and feature $n$ particle species on ${\mathbb{Z}}^d$. The PVBS models were originally introduced as toy models for the classification of ground state phases. A crucial ingredient for this classification is the existence of a spectral gap above the ground state sector. In this work, we derive a spectral gap for PVBS models at arbitrary species number $n$ and in arbitrary dimension $d$ in the perturbative regime of small anisotropy parameters. Instead of using the more common martingale method, the proof verifies a finite-size criterion in the spirit of Knabe.