Ground-state energies of the nonlinear σ model and the Heisenberg spin chains

We prove a theorem on the O(3) nonlinear σ model with the topological θ term which states that the grround-state energy at θ=π is always higher than the ground-state energy at θ=0, for the same value of the coupling constant g. Provided that the nonlinear σ model gives the correct description for the Heisenberg spin chains in the large-s limit, this theorem makes a definite prediction relating the ground-state energies of the half-integer- and the integer-spin chains. The ground-state energies obtained from the exact Bethe Ansatz solution for the spin-(1/2 chain and the numerical diagonalizaton on the spin-1,- (3/2, and -2 chains support this prediction.