Topological term in the non-linear $\sigma$ model of the SO(5) spin chains

TL;DR

This paper identifies a topological Berry phase term in the non-linear sigma model of SO(5) spin chains, revealing its role in distinguishing representations and affecting the chain's spectral properties.

Contribution

It demonstrates the existence of a topological term in the SO(5) sigma model, analogous to the theta-term in SO(3), and explores its implications for the spin chain's spectrum and ground state.

Findings

Topological term distinguishes linear and projective representations.

Presence of the topological term is linked to the non-trivial second homotopy group.

Connection to a solvable SO(5) spin model with valence bond solid ground state.

Abstract

We show that there is a topological (Berry phase) term in the non-linear $\sigma$ model description of the SO(5) spin chain. It distinguishes the linear and projective representations of the SO(5) symmetry group, in exact analogy to the well-known $\theta$-term of the SO(3) spin chain. The presence of the topological term is due to the fact that $\pi_2(\frac{SO(5)}{SO(3)\times SO(2)})= \mathbb{Z}$. We discuss the implication of our results on the spectra of the SO(5) spin chain, and connect it with a recent solvable SO(5) spin model which exhibits valence bond solid ground state and edge degeneracy.