On the relationship between sigma models and spin chains

TL;DR

This paper explores the connection between the two-dimensional O(3) sigma model with a topological term and one-dimensional spin-$s$ Heisenberg models, using lattice regularization to analyze their low-energy behaviors.

Contribution

It establishes a detailed relationship between the sigma model with topological term and spin chains, extending the analysis to models with different symmetries.

Findings

Quantization of the topological term coefficient as $ heta=2\pi s$

Correspondence between low-energy behaviors of sigma models and spin chains

Generalization to sigma models with other symmetries

Abstract

We consider the two-dimensional $\rm O(3)$ non-linear sigma model with topological term using a lattice regularization introduced by Shankar and Read [Nucl.Phys. B336 (1990), 457], that is suitable for studying the strong coupling regime. When this lattice model is quantized, the coefficient $\theta$ of the topological term is quantized as $\theta=2\pi s$, with $s$ integer or half-integer. We study in detail the relationship between the low energy behaviour of this theory and the one-dimensional spin-$s$ Heisenberg model. We generalize the analysis to sigma models with other symmetries.