Effective field theory with a $\theta$-vacua structure for 2d spin systems

TL;DR

This paper develops a 2+1 dimensional nonlinear sigma model with a tunable topological theta-term to describe 2d spin systems, revealing a rich vacuum structure influenced by interchain interactions and bond alternation.

Contribution

It introduces a novel 2+1d $O$(4) nonlinear sigma model with a variable theta-term, connecting 2d spin systems to topological field theory concepts and extending previous 1+1d insights.

Findings

The theta angle varies continuously with bond alternation strength.

The model captures competition between magnetic and valence-bond-solid orders.

The vacuum structure can be tuned by adjusting interchain coupling.

Abstract

We construct a nonlinear sigma (NL$\sigma$) model description of 2+1d spin systems, by coupling together antiferromagnetic spin chains via interchain exchange terms. Our mapping incorporates methods developed recently by ourselves and by Senthil and Fisher, which aim at describing competition between antiferromagnetic and valence-bond-solid orders in quantum magnets. The resulting 2+1d $O$(4) NL$\sigma$ model contains a topological $\theta$-term whose vacuum angle $\theta$ varies continuously with $\delta$, the bond-alternation strength of the interchain interaction. This implies that the $\theta$-vacua structure for this NL$\sigma$ model can be explored by tuning $\delta$ in a suitable 2d spin system, which is strongly reminiscent of the situation for 1+1 AF spin chains with bond-alternation.