Generating-function approach to the resonating-bond state on the triangular and square ladders

We calculate the energy of a resonating-valence-bond (RVB) state for the Heisenberg antiferromagnet Hamiltonian on the triangular and square ladders as N→∞. We take the RVB state to be a linear combination of all states in which all spins are bonded pairwise into nearest-neighbor singlets. The amplitude of each such state in the RVB wave function is proportional to ${\gamma }^{n/2\mathrm{}}$, where γ is a variational parameter, and n is the number of horizontal bonds in the state. The optimal γ is very close to 1, when all states have equal amplitude. We compare our results to Anderson’s finite-size calculation for the triangular ladder and to spin-wave theory for the two-dimensional lattices.