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We study the two-species Product Vacua and Boundary States (PVBS) models on the integer lattice $${\mathbb {Z}}^d$$Zd and prove the existence and non-existence of a spectral gap for all choices of parameters. The PVBS models are spin-1 quantum spin systems which are translation-invariant, frustration-free, and composed of nearest-neighbor non-commuting interactions with both an exclusion property and an interchange interaction between particle species. These models serve as possible representatives of families of automorphically equivalent gapped quantum spin-1 systems on $${\mathbb {Z}}^d$$Zd. The main result is that the two-species PVBS Hamiltonians have a positive spectral gap when gapped on both of the single-species subspaces and are gapless if gapless on either single-species subspace. The addition of a new particle species does not create any new gapless phases.

Spectral Gaps for the Two-Species Product Vacua and Boundary States Models on the d-Dimensional Lattice