Obstructions to gapped phases from noninvertible symmetries

Quantum systems in 3+1 dimensions that are invariant under gauging a one-form symmetry enjoy novel noninvertible duality symmetries encoded by topological defects. These symmetries are renormalization group invariants which constrain dynamics. We show that such noninvertible symmetries often forbid a symmetrypreserving vacuum state with a gapped spectrum. In particular, we prove that a self-dual theory with Z(1) N oneform symmetry is gapless or spontaneously breaks the self-duality symmetry unless N = k2t where -1 is a quadratic residue modulo t. We also extend these results to noninvertible 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 noninvertible symmetries are realized in lattice gauge theories, which serve to illustrate our results.

Automated summary

In this paper, we study the novel noninvertible duality symmetries encoded by topological defects in 3+1-dimensional quantum systems that are invariant under gauging a one-form symmetry. These symmetries, which are renormalization group invariants, impose constraints on the dynamics. We find that these noninvertible symmetries often prevent the existence of a symmetry-preserving vacuum state with a gapped spectrum. Furthermore, we extend our results to noninvertible symmetries arising from more general gauging operations, including triality symmetries. Along the way, we discover the hidden time-reversal symmetry of duality defects in symmetry-protected topological phases, and these noninvertible symmetries are realized in lattice gauge theories.