Effective actions for spin ladders

We derive a path-integral expression for the effective action in the continuum limit of an AFM Heisenberg spin ladder with an arbitrary number of legs. The map is onto an $O(3)$ nonlinear $\sigma$-model (NL$\sigma$M) with the addition of a topological term that is effective only for odd-legged ladders and half-odd integer spins. We derive the parameters of the effective NL$\sigma$M and the behaviour of the spin gap for the case of even-legged ladders.