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. 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 X ⊂ V the ground state projector can be approximated by the product of two projections, one supported on X and one supported on X , and a bounded observable supported on a boundary region in such a way that as the boundary region increases, the approximation becomes better. Such an approximation was useful in proving an area law in one dimension, and this result corresponds to a multi-dimensional analogue.