Approximating the ground state of gapped quantum spin systems

We consider quantum spin systems defined on finite sets V equipped with a metric. In typical examples, V is a large, but finite subset of Z_d. For finite range Hamiltonians with uniformly bounded interaction terms and a unique, gapped ground state, we demonstrate a locality property of the corresponding ground state projector. In such systems, this ground state projector can be approximated by the product of observables with quantifiable supports. In fact, given any subset XV the ground state projector can be approximated by the product of two projections, one supported on X and one supported on X_c, and a bounded observable supported on a boundary region in such a way that as the boundary region increases, the approximation becomes better. This result generalizes to multidimensional models, a result of Hastings that was an important part of his proof of an area law in one dimension ["An area law for one dimensional quantum systems," J. Stat. Mech.: Theory Exp. 2007, 08024].