Valence bonds and the Lieb-Schultz-Mattis theorem

The decoupling of short-range valence-bond states into topologically distinct sectors is discussed. On a square-lattice strip with odd width, this decoupling implies that short-range resonating-valence-bond (RVB) states are doubly degenerate. This result is precisely equivalent to the Lieb-Schultz-Mattis (LSM) theorem. Consequently short-range RVB states are consistent with the LSM theorem whether or not they possess gapless excitations. In the limit of infinite strip width, the decoupling becomes fourfold, in agreement with a recent conjecture of Haldane.