Obstructions to Gapped Phases from Non-Invertible Symmetries

TL;DR

The paper identifies precise arithmetical constraints on (3+1)d theories with non-invertible duality symmetries that must be satisfied to realize a symmetry-preserving gapped phase. By analyzing gauging operations $S$ and $T$ and the resulting duality defects, it derives when duality-invariant SPTs and TQFTs exist and when they cannot, linking the obstruction to the number $N$ via $N=k^{2}\ell$ with $-1$ a quadratic residue mod $\ell$. It constructs explicit duality-invariant TQFTs and SPTs, extends the analysis to triality and more general non-invertible symmetries, and corroborates the results with lattice gauge theory examples, including the $\mathbb{Z}_{N}$ Villain model and Cardy-Rabinovici model. A hidden time-reversal structure on duality defects and Gauss-sum computations underpin the anomaly and central charge analyses, clarifying when gapless phases or symmetry breaking are inevitable. Overall, the work provides a cohesive framework linking higher-form non-invertible symmetries, topological phases, and lattice realizations to map out the landscape of allowed gapped, duality-preserving phases.

Abstract

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.

Figures (6)

• Figure 1: The definition of the duality defect $\mathcal{D}$ via gauging in half of spacetime. The left region couples to a background two-form gauge field $B_L$ associated to the $\mathbb{Z}_{N}^{(1)}$ global symmetry. In the right region this symmetry is gauged with dynamical field $b$. The right region recovers the $\mathbb{Z}_{N}^{(1)}$ global symmetry through the Wilson surface operators of $b$ which couple to the background field $B_R$.
• Figure 2: $\mathbb{Z}_{N}^{(1)} \times \mathbb{Z}_{N}^{(1)}$ symmetry of $\mathcal{D}$ arises from ending of one-form symmetry defects from $\mathcal{Q}$ (purple) and $S\mathcal{Q}$ (green).
• Figure 3: The duality defect $\mathcal{D}$ in a bulk SPT phase. The defect $\mathcal{D}$ is a well-defined (2+1)d TQFT which is indentified with a minimal abelian TQFT.
• Figure 4: $\mathsf{T}$ symmetry of the duality defect separating phases $\mathcal{Q}$ and $S\mathcal{Q}$. $R_{\pi}$ combined with gauging of one-form symmetry in all of spacetime maps the boundary conditions to themselves, but reverses the orientation of $\mathcal{D}$. Hence this composite operation defines a time-reversal symmetry $\mathsf{T}$ of the duality defect worldvolume.
• Figure 5: The $\mathsf{T}$ symmetry of the duality defect $\mathcal{D}$ is realized by intersecting duality defects (shown as black dots). At each such intersection the duality defect orientations reverse. Colliding two such junctions leads to the bulk fusion algebra of $\mathcal{D}$ with its orientation reversal $\overline{\mathcal{D}}$ resulting in a condensation of one-form symmetry surfaces shown in magenta. Within the duality defect worldvolume, this restricts to a condensation of abelian anyons that produces the charge conjugation symmetry $C$.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem