Anomaly Constraints on Gapped Phases with Discrete Chiral Symmetry

TL;DR

This paper demonstrates that in certain 3+1 dimensional quantum field theories with discrete symmetries, anomalies prevent the existence of a gapped, symmetry-preserving vacuum, impacting both high-energy and condensed matter physics.

Contribution

It establishes a new class of anomaly constraints on gapped phases with discrete chiral symmetry in continuum quantum field theories.

Findings

Anomalies forbid gapped, symmetric vacua in specific 3+1d theories.

Results apply to discrete chiral symmetries in gauge theories.

Implications for condensed matter systems like Weyl semimetals.

Abstract

We prove that in $(3+1)d$ quantum field theories with $\mathbb{Z}_N$ symmetry, certain anomalies forbid a symmetry-preserving vacuum state with a gapped spectrum. In particular, this applies to discrete chiral symmetries which are frequently present in gauge theories as we illustrate in examples. Our results also constrain the long-distance behavior of certain condensed matter systems such as Weyl-semimetals and may have applications to crystallographic phases with symmetry protected topological order. These results may be viewed as analogs of the Lieb-Schultz-Mattis theorem for continuum field theories.