Emergent higher-symmetry protected topological orders in the confined phase of $U(1)$ gauge theory

Abstract

We consider compact $U^{\kappa }(1)$ gauge theory in 3+1D with the $2\pi$-quantized topological term ${\sum }_{I,J=1}^{\kappa }\frac{K_{IJ}}{4\pi }{\int }_{M^4}{F}^I\wedge F^J$. At energies below the gauge charges' gaps but above the monopoles' gaps, this field theory has an emergent ${\mathbb{Z}}_{k_1}^{(1)}\times {\mathbb{Z}}_{k_2}^{(1)}\times \cdots$ 1-symmetry, where $k_i$ are the diagonal elements of the Smith normal form of $K$ and ${\mathbb{Z}}_0^{(1)}$ is regarded as $U(1{)}^{(1)}$. In the $U^{\kappa }(1)$ confined phase, the boundary's IR properties are described by Chern-Simons field theory and has a ${\mathbb{Z}}_{k_1}^{(1)}\times {\mathbb{Z}}_{k_2}^{(1)}\times \cdots$ 1-symmetry that can be anomalous. To show these results, we develop a bosonic lattice model whose IR properties are described by this field theory, thus acting as its UV completion. The lattice model in the aforementioned limit has an exact ${\mathbb{Z}}_{k_1}^{(1)}\times {\mathbb{Z}}_{k_2}^{(1)}\times \cdots$ 1-symmetry. We find that a gapped phase of the lattice model, corresponding to the confined phase of the $U^{\kappa }(1)$ gauge theory, is a symmetry protected topological (SPT) phase for the ${\mathbb{Z}}_{k_1}^{(1)}\times {\mathbb{Z}}_{k_2}^{(1)}\times \cdots$ 1-symmetry, whose SPT invariant is $e^{i\pi {\sum }_{I,J}{K}_{IJ}\int B_I⌣B_J+B_I\underset{1}{⌣}d{B}_J}{e}^{i\pi {\sum }_{I<J}{K}_{IJ}\int d{B}_I\underset{2}{⌣}d{B}_J}$. Here, the background 2-cochains $B_I$ satisfy $d{B}_I={\sum }_I{B}_I{K}_{IJ}=0$ mod $1$ and describe the symmetry twist of the ${\mathbb{Z}}_{k_1}^{(1)}\times {\mathbb{Z}}_{k_2}^{(1)}\times \cdots$ 1-symmetry. We apply this general result to a few examples with simple $K$ matrices. We find the non-trivial SPT order in the confined phases of these models and discuss its classifications using the fourth cohomology group of the corresponding 2-group.

Published version

References in corpus (11)

• Generalized Global Symmetries
• Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d
• Exotic Symmetries, Duality, and Fractons in 2+1-Dimensional Quantum Field Theory
• Intrinsically Gapless Topological Phases
• Generalized Symmetry Breaking Scales and Weak Gravity Conjectures
• Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes
• Mean string field theory: Landau-Ginzburg theory for 1-form symmetries
• Lattice models that realize ${\mathbb{Z}}_n$-1-symmetry protected topological states for even $n$
• Physics of Symmetry Protected Topological phases involving Higher Symmetries and their Applications
• "Light from chaos" in two dimensions
• Lattice realization of compact $U(1)$ Chern-Simons theory with exact 1-symmetries

Cited by in corpus (12)

• Exact emergent higher-form symmetries in bosonic lattice models
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Aspects of ${\mathbb{Z}}_N$ rank-2 gauge theory in $(2+1)$ dimensions: construction schemes, holonomies, and sublattice one-form symmetries
• Theory of oblique topological insulators
• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
• Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases
• Dynamical Axions in $U(1)$ Quantum Spin Liquids
• Hybrid Dyons, inverted Lorentz force and magnetic Nernst effect in quantum spin ice
• A stabilizer code model with non-invertible symmetries: Strange fractons, confinement, and non-commutative and non-Abelian fusion rules
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Theta electromagnetism in quantum spin ice: Microscopic analysis of improper symmetries
• Generalized symmetry enriched criticality in (3+1)d