Anomaly Constraints on Gapped Phases with Discrete Chiral Symmetry

TL;DR

The paper investigates how discrete $\\mathbb{Z}_N$ anomalies in $(3+1)d$ QFTs constrain long-distance phases, showing they can forbid symmetry-preserving gapped vacua. It leverages anomaly inflow from a $(4+1)d$ $\\mathbb{Z}_N$ SPT, denoted $\\mathcal{T}^k_N$, whose boundary hosts $k$ Weyl fermions; a boundary-obstruction analysis reveals that if $N$ does not divide $2k$, no symmetry-preserving gapped boundary exists. This obstruction is then applied to gauge theories with discrete chiral symmetry $\\mathbb{Z}_{N_f I(R)}$, forcing symmetry breaking or nontrivial IR vacua when $I(R) mid 2\\dim(R)$, with concrete examples including adjoint QCD. The results also extend to condensed matter contexts, such as Weyl semimetals, where certain lattice and symmetry conditions protect gapless modes via $\\mathbb{Z}_N$-gravity anomalies, linking high-energy anomaly physics to crystallographic topological order and long-distance constraints in materials.

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.

Figures (1)

• Figure 1: The schematic picture of the band structure \ref{['eq:Bloch']}. (a) When $-1\le\xi\le$, the band has two crossings at $k_x = \pm k_0$ with energy $E=0$ which leads to the effective relativistic Weyl fermion description. (b) When $|\xi|>1$ the system is trivially gapped when half-filled.