Obstructions to Gapped Phases from Non-Invertible Symmetries

Quantum systems in 3+1-dimensions that are invariant under gauging a one-form symmetry enjoy novel non-invertible duality symmetries encoded by topological defects. These symmetries are renormalization group invariants which constrain dynamics. We show that such non-invertible symmetries often forbid a symmetry-preserving vacuum state with a gapped spectrum. In particular, we prove that a self-dual theory with $\mathbb{Z}_{N}^{(1)}$ one-form symmetry is gapless or spontaneously breaks the self-duality symmetry unless $N=k^{2}\ell$ where $-1$ is a quadratic residue modulo $\ell$. We also extend these results to non-invertible symmetries arising from invariance under more general gauging operations including e.g. triality symmetries. Along the way, we discover how duality defects in symmetry protected topological phases have a hidden time-reversal symmetry that organizes their basic properties. These non-invertible symmetries are realized in lattice gauge theories, which serve to illustrate our results.