Symmetry protection of critical phases and global anomaly in $1+1$ dimensions

TL;DR

This paper establishes a classification of critical phases in 1+1D SU(2) Wess-Zumino-Witten theories based on a global anomaly, revealing symmetry-protected distinctions between even and odd levels affecting quantum spin chain criticality.

Contribution

It introduces a novel selection rule linking level changes in Wess-Zumino-Witten theories to a global anomaly, classifying critical behaviors into symmetry-protected categories.

Findings

Critical behavior is classified into two categories based on parity of the level.

A selection rule constrains RG flows between different levels.

The results restrict possible gapless phases in translation-invariant quantum spin chains.

Abstract

We derive a selection rule among the $(1+1)$-dimensional SU(2) Wess-Zumino-Witten theories, based on the global anomaly of the discrete $\mathbb{Z}_2$ symmetry found by Gepner and Witten. In the presence of both the SU(2) and $\mathbb{Z}_2$ symmetries, a renormalization-group flow is possible between level-$k$ and level-$k'$ Wess-Zumino-Witten theories only if $k\equiv k' \mod{2}$. This classifies the Lorentz-invariant, SU(2)-symmetric critical behavior into two "symmetry-protected" categories corresponding to even and odd levels,restricting possible gapless critical behavior of translation-invariant quantum spin chains.