Effective actions for spin ladders

TL;DR

This paper derives an effective field theory for antiferromagnetic Heisenberg spin ladders, revealing topological effects for odd-legged ladders and analyzing spin gaps for even-legged cases.

Contribution

It provides a path-integral formulation mapping spin ladders to an $O(3)$ nonlinear sigma model with topological terms, extending understanding of their quantum properties.

Findings

Effective action expressed as an $O(3)$ NL$\sigma$M with topological terms for odd-legged ladders.

Parameters of the NL$\sigma$M derived for arbitrary-leg ladders.

Behavior of the spin gap analyzed for even-legged ladders.

Abstract

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.