On Frustration-Free Quantum Spin Models

The goal of our work is to characterize the landscape of the frustration-free quantum spin models over the Cayley graph of a finitely generated group $G$. This is achieved by establishing $G$-equivariant morphisms from the partially ordered space of frustration-free models to the partially ordered spaces 1) of hereditary $C^\ast$-algebras of the underlying UHF quasi-local algebra of observables, 2) of open projections in its double dual, and 3) of subsets of pure state space. Our main result consists of an intrinsic characterization of the images of these morphisms, which captures the essence of frustration-freeness and enables us to extend the concept to generic AF-algebras. Additionally, using well established facts about AF-algebras, we prove density theorems, provide intrinsic characterizations of frustration-free ground states, and propose a definition of a boundary algebra for models constrained to half-lattices, under the sole assumption of frustration-freeness.