Exact Haldane mapping for all $S$ and super universality in spin chains

TL;DR

This paper presents an exact mapping of antiferromagnetic Heisenberg spin chains of any spin magnitude onto the O(3) nonlinear sigma model, revealing super universal features and clarifying Haldane's original semiclassical approach.

Contribution

It introduces a novel exact mapping that bypasses large S approximations, highlighting super universal properties and topological aspects of spin chains.

Findings

Exact mapping for all spin values S

Super universal features of the theta angle

Explanation of Haldane's original results

Abstract

The low energy dynamics of the anti-ferromagnetic Heisenberg spin $S$ chain in the semiclassical limit $S\to\infty$ is known to map onto the O(3) nonlinear $\sigma$ model with a $\theta$ term in 1+1 dimension. Guided by the underlying dual symmetry of the spin chain, as well as the recently established topological significance of "dangling edge spins," we report an {\em exact} mapping onto the O(3) model that avoids the conventional large $S$ approximation altogether. Our new methodology demonstrates all the super universal features of the $\theta$ angle concept that previously arose in the theory of the quantum Hall effect. It explains why Haldane's original ideas remarkably yield the correct answer in spite of the fundamental complications that generally exist in the idea of semiclassical expansions.