The Non Linear Sigma Model and Spin Ladders

TL;DR

This paper extends the Haldane map to spin ladders, explaining the difference in behavior between even and odd ladders through topological terms, and predicts gap formation or gaplessness based on ladder parity and spin.

Contribution

It generalizes the Haldane map to spin ladders, clarifying the role of topological terms in their quantum behavior and deriving consequences for their coupling constants.

Findings

Even ladders have a dynamically generated spin gap.

Odd ladders with half-integer spin remain gapless.

The topological term's presence or absence explains qualitative differences.

Abstract

The well known Haldane map from spin chains into the $O(3)$ non linear sigma model is generalized to the case of spin ladders. This map allows us to explain the different qualitative behaviour between even and odd ladders, exactly in the same way it explains the difference between integer and half-integer spin chains. Namely, for even ladders the topological term in the sigma model action is absent, while for odd ladders the $\theta$ parameter, which multiplies the topological term, is equal to $2 \pi S$, where $S$ is the spin of the ladder. Hence even ladders should have a dynamically generated spin gap, while odd ladders with half-integer spin should stay gapless, and physically equivalent to a perturbed $SU(2)_1$ Wess-Zumino -Witten model in the infrared regime. We also derive some consequences from the dependence of the sigma model coupling constant on the ladder Heisenberg couplings constants.