Anomaly Constraints on Gapped Phases with Discrete Chiral Symmetry

Understand

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.