Effective Actions for Spin Ladders

Abstract

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