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

TL;DR

This work shows that 1+1D SU(2)-symmetric critical phases are partitioned into two symmetry-protected classes by a discrete $\mathbb{Z}_2$ symmetry related to lattice translations. Using modular invariance and the Gepner–Witten global anomaly, the authors derive a selection rule: RG flows between WZW theories $\text{WZW}_k\to\text{WZW}_{k'}$ are allowed only if $k'\equiv k \pmod{2}$, with $k'<k$ as enforced by the $c$-theorem. Consequently, spin chains with integer spin realize even-$k$ WZW fixed points, while half-odd-integer spins realize odd-$k$ fixed points, establishing a parity-protected distinction in gapless critical behavior. The analysis connects field-theoretic anomalies to condensed-matter physics via $\mathbb{Z}_2$ orbifolds, parity symmetry, and a robust RG-flow framework, and suggests Raman spectroscopy as a probe of the level $k$. This symmetry-protected view broadens the classification of critical phases and links anomalies to observable condensed-matter phenomena.

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.