  A generalization of the Lieb-Schultz-Mattis theorem to higher dimensional
spin systems is shown. The physical motivation for the result is that such spin
systems typically either have long-range order, in which case there are gapless
modes, or have only short-range correlations, in which case there are
topological excitations. The result uses a set of loop operators, analogous to
those used in gauge theories, defined in terms of the spin operators of the
theory. We also obtain various cluster bounds on expectation values for gapped
systems. These bounds are used, under the assumption of a gap, to rule out the
first case of long-range order, after which we show the existence of a
topological excitation. Compared to the ground state, the topologically excited
state has, up to a small error, the same expectation values for all operators
acting within any local region, but it has a different momentum.
