Semi-classical description of the frustrated antiferromagnetic chain

The antiferromagnetic Heisenberg model on a chain with nearest and next nearest neighbor couplings is mapped onto the $SO(3)$ nonlinear sigma model in the continuum limit. In one spatial dimension this model is always in its disordered phase and a gap opens to excited states. The latter form a doubly degenerate spin-1 branch at all orders in $1/N$. We argue that this feature should be present in the spin-1 Heisenberg model itself. Exact diagonalizations are used to support this claim. The inapplicability of this model to half-integer spin chains is discussed.